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PhysRevB.74.085301.pdf	Inﬂuence of the contacts on the conductance of interacting quantum wires K. Janzen, V . Meden, and K. Schönhammer Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany /H20849Received 19 April 2006; published 1 August 2006 /H20850 We investigate how the conductance Gthrough a clean interacting quantum wire is affected by the presence of contacts and noninteracting leads. The contacts are deﬁned by a vanishing two-particle interaction to the leftand a ﬁnite repulsive interaction to the right or vice versa. No additional single-particle scattering terms/H20849impurities /H20850are added. We ﬁrst use Bosonization and the local Luttinger liquid picture and show that within this approach Gis determined by the properties of the leads regardless of the details of the spatial variation of the Luttinger liquid parameters. This generalizes earlier results obtained for steplike variations. In particular, nosingle-particle backscattering is generated at the contacts. We then study a microscopic model applying thefunctional renormalization group and show that the spatial variation of the interaction produces single-particlebackscattering, which in turn leads to a reduced conductance. We investigate how the smoothness of thecontacts affects Gand show that for decreasing energy scale its deviation from the unitary limit follows a power law with the same exponent as obtained for a system with a weak single-particle impurity placed in thecontact region of the interacting wire and the leads. DOI: 10.1103/PhysRevB.74.085301 PACS number /H20849s/H20850: 71.10.Pm, 72.10. /H11002d, 73.21.Hb I. INTRODUCTION The low-energy physics of one-dimensional /H208491D/H20850metals is not described by the Fermi-liquid theory if two-particleinteractions are taken into account. Such systems fall into theLuttinger liquid /H20849LL/H20850universality class 1that is characterized by power-law scaling of a variety of correlation functionsand a vanishing quasiparticle weight. For spin-rotational in-variant interactions and spinless models, on which we focushere, the exponents of the different correlation functions canbe parametrized by a single number, the interaction depen-dent LL parameter K/H110211/H20849for repulsive interactions; K=1 in the noninteracting case /H20850. As a second independent LL param- eter one can take the velocity vcof charge excitations /H20849for details see below /H20850. Instead of being quasiparticles the low- lying excitations of LLs are collective density excitations.This implies that impurities or more generally inhomogene-ities have a dramatic effect on the physical properties ofLLs. 2–5 In the presence of only a single impurity on asymptoti- cally small energy scales observables behave as if the 1Dsystem was cut in two halves at the position of the impurity,with open boundary conditions at the end points /H20849open chain ﬁxed point /H20850. 6–8In particular, for a weak impurity and de- creasing energy scale sthe deviation of the linear conduc- tance Gfrom the impurity-free value scales as /H20849s/s0/H208502/H20849K−1/H20850, with Kbeing the scaling dimension of the perfect chain ﬁxed point and s0a characteristic energy scale /H20849e.g., the band- width /H20850. This holds as long as /H20841Vback/s0/H208412/H20849s/s0/H208502/H20849K−1/H20850/H112701, with Vbackbeing a measure for the strength of the 2 kFbackscatter- ing of the impurity and kFthe Fermi momentum. For smaller energy scales or larger bare impurity backscattering this be-havior crosses over to another power-law scaling G/H20849s/H20850 /H11011/H20849s/s 0/H208502/H208491/K−1/H20850, with the scaling dimension of the open chain ﬁxed point 1/ K. This scenario was veriﬁed for inﬁnite LLs,6–8as well as ﬁnite LLs connected to Fermi-liquid leads,9,10a setup that is closer to systems that can be realized in experiments. In the latter case the scaling holds as long asthe contacts are modeled to be “perfect,” that is free of any bare and effective single-particle backscattering, and the im- purity is placed in the bulk of the interacting quantum wire.For an impurity placed close to perfect contacts the expo-nents change to 2 /H20849K−1/H20850//H20849K+1/H20850/H20849close to the perfect chain ﬁxed point /H20850and 1/ K−1 /H20849close to the open chain ﬁxed point /H20850. 9,10 The role of an inhomogeneous two-particle interaction, that is an interaction that depends not only on the relativedistance of the two particles, but also the center of mass isless well understood. In the present publication we will ﬁllthis gap. Such an inhomogeneity will generically appearclose to the interface of the interacting quantum wire and theleads and a detailed understanding is thus essential for theinterpretation of transport experiments on quasi-1D quantumwires. 11We here use two models to study the effect of two- particle inhomogeneities on the linear conductance. We ﬁrstinvestigate the so-called local Luttinger liquid /H20849LLL /H20850 model, 12–14that is characterized by a spatial dependence of the LL parameters Kandvc, with K=1 and vc=vFin the leads /H20849vFis the noninteracting Fermi velocity /H20850. We show that regardless of the details of the spatial variation the conduc-tance always takes the perfect value 1/ /H208492 /H9266/H20850/H20849in units such that/H6036=1 and the electron charge e=1/H20850. Thus the LLL de- scription cannot produce any effective single-particle back-scattering from the contact region generated by an inhomo-geneous two-particle interaction. Our results generalizeearlier ﬁndings obtained for steplike variations of the LLparameters. 12–14 We then study a microscopic lattice model with a spatially dependent nearest-neighbor interaction. Across the contactbetween the left lead and the wire the interaction is turned onfrom zero to a bulk value Uand correspondingly turned off close to the right contact. We show that this two-particleinhomogeneity generically leads to an effective single-particle backscattering and a reduced conductance. To com-pute the latter we use an approximation scheme that is basedon the functional renormalization group /H20849fRG /H20850. 10We numeri-PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 1098-0121/2006/74 /H208498/H20850/085301 /H208499/H20850 ©2006 The American Physical Society 085301-1cally and analytically investigate the dependence of the con- ductance on U, the length of the interacting wire N, and the “smoothness” with which the interaction is turned on and off.For weak effective inhomogeneities we analytically showthat 1/ /H208492 /H9266/H20850−Gdisplays scaling with the energy scale /H9254N =/H9266vF/Nset by the length of the wire. The exponent we ﬁnd is consistent with the one found for a weak single-particleimpurity close to a perfect contact. 9,10We give the energy scale up to which this scaling holds. It depends on the bulkinteraction Uand the 2 k FFourier component of the function with which the interaction is varied close to the contacts. Thelatter provides a quantitative measure of the smoothness ofthe contacts. This ﬁnding suggests a similarity between theuniversal low-energy properties of a LL with a single-particle inhomogeneity and a two-particle inhomogeneity.We discuss this similarity but also point out differences. This paper is organized as follows. In Sec. II we study the two-particle inhomogeneity within the LLL picture. To mo-tivate the LLL model in Sec. II A we present a brief intro-duction into Bosonization and the Tomonaga-Luttingermodel. The transport properties of the LLL model are inves-tigated in Sec. II B. In Sec. III we introduce our microscopicmodel and give the basic equations of the fRG approach. Wethen discuss our numerical results in Sec. III A and the ana-lytical ﬁndings in Sec. III B. We close in Sec. IV with asummary. II. LOCAL LUTTINGER LIQUID DESCRIPTION In this section we discuss how sufﬁciently smooth con- tacts /H20849perfect contacts /H20850can be properly described in a LLL picture. This approach was successfully used to show that forperfect contacts the properties of the leads and not the ﬁnite-size quantum wire determine the conductance. 12–14Up to now the LL parameters were always assumed to vary step-like. Here we present a simple derivation of this result for anarbitrary variation of the LL parameters. Also the role ofimpurities in the interacting wire was investigated within theLLL picture. 9,15It was shown that the exponent of the tem- perature dependence of the conductance is different for im-purities placed in the bulk and close to the contacts. 9Using the fRG this was later conﬁrmed in a microscopic model inwhich the contacts were modeled to be arbitrarily smooth. 10 A. Model and generalized wave equations In order to also elucidate the limitations of the LLL pic- ture we ﬁrst shortly recall the basic ideas necessary to justifythe Tomonaga-Luttinger model 1,16,17for interacting fermions in one dimension in the homogeneous case. We consider asystem of length Lwith periodic boundary conditions. In second quantization the two-body interaction reads V=1 2/H20885/H20885 v/H20849x−x/H11032/H20850/H9254/H9267/H20849x/H20850/H9254/H9267/H20849x/H11032/H20850dxdx /H11032 +1 2LN2v˜/H208490/H20850−1 2v/H208490/H20850N, /H208491/H20850 where /H9254/H9267/H20849x/H20850=/H9274†/H20849x/H20850/H9274/H20849x/H20850−N/Lis the operator of the particledensity relative to its homogeneous value, with Nthe par- ticle number operator. The Fourier transform of the two-body interaction is denoted as v˜/H20849k/H20850. The last term is usually dropped as it only modiﬁes the chemical potential. If the range of the two-body interaction is much larger than themean particle distance only particle-hole pairs in the vicinityof the two Fermi points are present in the ground state andthe eigenstates with low excitation energy. 16This allows us to linearize the dispersion around the two Fermi points andto introduce two independent types of fermions, the right andleft movers with particle density operators /H9254/H9267±/H20849x/H20850=/H9267±/H20849x/H20850 −N± /H9254/H9267/H9251/H20849x/H20850=1 L/H20858 n/HS110050eiknx/H9267n,/H9251=/H11509 /H11509x/H20875−i 2/H9266/H20858 n/HS110050eiknx n/H9267n,/H9251/H20876/H11013/H11509/H9021/H9251 /H11509x. /H208492/H20850 As shown below the ﬁeld operators /H9021/H9251are convenient ob- jects for the solution of the problem.18In the subspace of low-energy states the Fourier components of the density /H9267n,/H9251 obey the commutation relations16 /H20851/H9267m,/H9251,/H9267n,/H9252/H20852=/H9251m/H9254/H9251/H9252/H9254m,−n. /H208493/H20850 After proper normalization they take the form of boson com- mutation relations. One can write the operator of the kinetic energy as a quadratic form of the /H9267±.16The Tomonaga model then results from replacing /H9254/H9267/H20849x/H20850by/H9254/H9267+/H20849x/H20850+/H9254/H9267−/H20849x/H20850andNbyN++N−in Eq. /H208491/H20850. With the well known “g-ology” generalization1the boson part of the interaction reads VTL,b=1 2/H20885/H20885 /H208532g2/H20849x−x/H11032/H20850/H9254/H9267+/H20849x/H20850/H9254/H9267−/H20849x/H11032/H20850+g4/H20849x−x/H11032/H20850 /H11003/H20851/H9254/H9267+/H20849x/H20850/H9254/H9267+/H20849x/H11032/H20850+/H9254/H9267−/H20849x/H20850/H9254/H9267−/H20849x/H11032/H20850/H20852/H20854dxdx /H11032. /H208494/H20850 This reduces to Tomonaga’s original model for g2/H11013g4/H11013v, while Luttinger17later independently studied the model with g4/H110130. He also discussed the special case g2/H20849x/H20850/H11011/H9254/H20849x/H20850, which corresponds to the interaction term in the massless Thirring model.19This simpliﬁes various aspects but brings in inﬁni- ties which have to be removed, e.g., by normal ordering. Inthe boson part of the kinetic energy T b=/H9266vF/H20885/H20851/H9254/H9267+2/H20849x/H20850+/H9254/H9267−2/H20849x/H20850/H20852dx /H208495/H20850 the integrand has to be normal ordered. This, as above, is usually suppressed. Inhomogeneous Luttinger liquids can be realized by add- ing an external one-particle potential or by allowing the two-body interaction to depend on both spatial variables, i.e., v/H20849x−x/H11032/H20850in Eq. /H208491/H20850is replaced by v/H20849x,x/H11032/H20850. The operator of a one-particle potential can only be expressed in terms of the /H9254/H9267/H9251if it is sufﬁciently smooth in real space, i.e., the external potential has a vanishing 2 kFFourier component. Similarly only for sufﬁciently smooth variations—in Sec. III B we willspecify the meaning of “sufﬁciently smooth”—of vwith /H20849x +x/H11032/H20850/2 the analogous steps from Eq. /H208491/H20850to Eq. /H208494/H20850, i.e.,JANZEN, MEDEN, AND SCHÖNHAMMER PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-2gi/H20849x−x/H11032/H20850→gi/H20849x,x/H11032/H20850are allowed without changing the low- energy physics. The standard LLL model is obtained by as- suming gi/H20849x,x/H11032/H20850=gi,x/H9254/H20849x−x/H11032/H20850 HLLL, b=/H9266/H20885/H20877/H20873vF+g4,x 2/H9266/H20874/H20851/H9254/H9267+2/H20849x/H20850+/H9254/H9267−2/H20849x/H20850/H20852 +g2,x /H9266/H9254/H9267+/H20849x/H20850/H9254/H9267−/H20849x/H20850/H20878dx /H208496/H20850 =/H9266 2/H20885/H20877/H20873vF+g4,x+g2,x 2/H9266/H20874/H20875/H11509/H9021/H20849+/H20850/H20849x/H20850 /H11509x/H208762 +/H20873vF+g4,x−g2,x 2/H9266/H20874/H20875/H11509/H9021/H20849−/H20850/H20849x/H20850 /H11509x/H208762/H20878dx /H11013/H9266 2/H20885/H20877vN/H20849x/H20850/H20875/H11509/H9021/H20849+/H20850/H20849x/H20850 /H11509x/H208762 +vJ/H20849x/H20850/H20875/H11509/H9021/H20849−/H20850/H20849x/H20850 /H11509x/H208762/H20878dx, /H208497/H20850 where we have deﬁned /H9021/H20849±/H20850/H20849x/H20850/H11013/H9021+/H20849x/H20850±/H9021−/H20849x/H20850, as well as the velocities vN/H20849x/H20850andvJ/H20849x/H20850. As in the homogeneous model the right and left movers are coupled by the g2interaction and using the /H9021/H20849±/H20850/H20849x/H20850as the basic ﬁelds simpliﬁes the solu- tion as /H9021/H20849/H9263/H20850/H20849x/H20850and/H9021/H20849/H9263/H20850/H20849x/H11032/H20850commute. Apart from a term which vanishes in the thermodynamic limit /H9021/H20849−/H9263/H20850/H20849x/H11032/H20850and /H11509/H9021/H20849/H9263/H20850/H20849x/H20850//H11509xobey canonical commutation relations after proper normalization, /H20875/H11509/H9021/H20849/H9263/H20850/H20849x/H20850 /H11509x,/H9021/H20849−/H9263/H20850/H20849x/H11032/H20850/H20876=i /H9266/H20875/H9254L/H20849x−x/H11032/H20850−1 L/H20876. /H208498/H20850 This follows from Eqs. /H208492/H20850and /H208493/H20850. Here /H9254Ldenotes the L-periodic delta function. Neglecting the correction term yields the following Heisenberg equations of motion for the/H9021 /H20849/H9263/H20850/H20849x,t/H20850: /H11509 /H11509t/H9021/H20849+/H20850/H20849x,t/H20850=−vJ/H20849x/H20850/H11509/H9021/H20849−/H20850/H20849x,t/H20850 /H11509x, /H11509 /H11509t/H9021/H20849−/H20850/H20849x,t/H20850=−vN/H20849x/H20850/H11509/H9021/H20849+/H20850/H20849x,t/H20850 /H11509x. /H208499/H20850 Therefore the /H9021/H20849/H9263/H20850/H20849x,t/H20850obey generalized wave equations, e.g., for the ﬁeld related to the change of the total density, /H115092 /H11509t2/H9021/H20849+/H20850/H20849x,t/H20850−vJ/H20849x/H20850/H11509 /H11509xvN/H20849x/H20850/H11509/H9021/H20849+/H20850/H20849x/H20850 /H11509x=0 . /H2084910/H20850 The spatial derivative of the ﬁrst equation in Eq. /H208499/H20850yields the continuity equation for the total charge density /H9254/H9267=/H9254/H9267+ +/H9254/H9267−, /H11509 /H11509t/H9254/H9267/H20849x,t/H20850+/H11509 /H11509x/H20875vJ/H20849x/H20850/H11509/H9021/H20849−/H20850/H20849x,t/H20850 /H11509x/H20876=0 /H2084911/H20850 which implies the conservation of the total charge Q =/H20848/H9254/H9267dx. As Eqs. /H2084910/H20850and /H2084911/H20850are linear the expectation values discussed in the following obey the same equations.In a homogeneous system the velocities vJand vNare constant and the corresponding “sound velocity” called the charge velocity vcis given by vc=/H20881vJvN. It constitutes the ﬁrst LL parameter characterizing the system. The second LL parameter Kis deﬁned as K/H11013/H20881vJ/vN=vc/vN. The general solution of the wave equation for constant vcandKreads f/H20849x−vct/H20850+g/H20849x+vct/H20850, where fandgare arbitrary functions. B. Computing the transmitted charge For general dependencies of vJ/H20849x/H20850andvN/H20849x/H20850on the posi- tion the generalized wave equation /H2084910/H20850cannot be solved analytically. In the following we discuss properties of thesolution in the thermodynamic limit when the velocities vJ/H20849x/H20850andvN/H20849x/H20850are constant for x/H11021aandx/H11022b/H11022abut have arbitrary /H20849bounded /H20850variation in the interval /H20851a,b/H20852. Figure 1 shows two qualitatively different types of behavior of vN/H20849x/H20850. The dashed curve presents a monotonous transition between the asymptotic values while the full curve corresponds to an“interacting wire” region in /H20851a,b/H20852with a higher /H20849constant /H20850 value of vN,wcorresponding to a stronger repulsive interac- tion. In order to describe the charge transport through thisregion we consider an incoming density with compact sup-port which at t=0 is completely to the left of aand moves towards it with constant velocity vc,L/H20849see Fig. 1/H20850. Before the disturbance reaches point athe time evolution is /H9021/H20849+/H20850/H20849x,t/H20850=fin/H20849x−vc,Lt/H20850,/H9254/H9267/H20849x,t/H20850=fin/H11032/H20849x−vc,Lt/H20850./H2084912/H20850 The total incoming charge Qinis given by Qin=fin/H20849a/H20850 −fin/H20849−/H11009/H20850. This short-time solution for /H9021/H20849+/H20850/H20849x,t/H20850determines /H9021/H20849−/H20850/H20849x,t/H20850up to a constant. In the long-time limit there will again be no charge in the region /H20851a,b/H20852with the total trans- mitted charge Qtransmoving to the right with constant veloc- ityvc,Rand the total reﬂected charge Qrefmoving to the left with velocity vc,L, where charge conservation implies Qtrans +Qref=Qin. In the following we show that it is possible to determine Qref/Qinwithout the explicit solution for /H9254/H9267/H20849x,t/H20850 in this long-time limit and without any additional assump- tions on the spatial variation of vN/H20849x/H20850andvJ/H20849x/H20850. The trick is to introduce the quantity S/H20849t/H20850/H11013/H20885 −/H11009/H110091 vJ/H20849x/H20850/H11509/H9021/H20849+/H20850/H20849x,t/H20850 /H11509tdx=/H9021/H20849−/H20850/H20849−/H11009,t/H20850−/H9021/H20849−/H20850/H20849/H11009,t/H20850 /H2084913/H20850 and to show that Sis time independent. This can be seen by discussing the expression in the second line which follows FIG. 1. /H20849Color online /H20850Spatial dependence of velocity vN/H20849x/H20850used for the discussion of the generalized wave equation /H2084910/H20850. Two dif- ferent realizations /H20849full and dashed curves /H20850of the transition region /H20851a,b/H20852are shown. The shaded area shows the incoming density /H9254/H9267/H20849x,0/H20850.INFLUENCE OF THE CONTACTS ON THE ¼ PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-3using Eq. /H208499/H20850or if one wants to argue with the density only by taking the time derivative and using the generalized waveequation S˙/H20849t/H20850=/H20885 −/H11009/H110091 vJ/H20849x/H20850/H115092/H9021/H20849+/H20850/H20849x,t/H20850 /H11509t2dx=/H20885 −/H11009/H11009/H11509 /H11509x/H20875vN/H20849x/H20850/H11509 /H11509x/H9021/H20849+/H20850/H20849x/H20850/H20876dx =vN,R/H9254/H9267/H20849/H11009,t/H20850−vN,L/H9254/H9267/H20849−/H11009,t/H20850=0 , /H2084914/H20850 where we have used that the assumption about the initial state implies that /H9267/H20849±/H11009,t/H20850vanishes for all ﬁnite times. The constant value of Sfollows from Eq. /H2084912/H20850, S=−/H20885 −/H11009avc,L vJ,Lfin/H11032/H20849x−vc,Lt/H20850dx=−vc,L vJ,LQin=−Qin KL./H2084915/H20850 When the incoming density hits the transition region /H20851a,b/H20852it is partially reﬂected and transmitted, where the detailed be- havior is very different for the two realizations of vN/H20849x/H20850 shown in Fig. 1. For sufﬁciently large times the total charge in the transition region goes to zero and /H9021/H20849+/H20850/H20849x,t/H20850takes the form /H9021/H20849+/H20850/H20849x,t/H20850=/H20902fref/H20849x+vc,Lt/H20850, for x/H11021a const, for a/H11021x/H11021b. ftrans/H20849x−vc,Rt/H20850, for x/H11022b/H2084916/H20850 With the deﬁnition of Sin Eq. /H2084913/H20850this yields S=vc,L vJ,L/H20885 −/H11009a fref/H11032/H20849x+vc,Lt/H20850dx−vc,R vJ,R/H20885 b/H11009 ftrans/H11032/H20849x−vc,rt/H20850dx =Qref KL−Qtrans KR. /H2084917/H20850 The comparison of Eqs. /H2084915/H20850and /H2084917/H20850as well as charge conservation leads to the result derived earlier assuming astepwise variation of the LL parameters, 12–14 Qref=KL−KR KL+KRQin,Qtrans=2KR KL+KRQin. /H2084918/H20850 Our result shows that the properties of the transition region play no role at all. The ratio Qtrans/Qinis positive, while Qref/Qinhas no deﬁnite sign. From this result one can easily obtain the linear conduc- tance Gthrough the system. We ﬁrst discuss a homogeneous system with constant LL parameters KLandvc,L/H20849the use of the index Lbecomes clear later /H20850. We consider a current free initial state in which the density is increased by /H9254/H92670in the left half of the inﬁnite system. Then half of the additionaldensity moves to the right with velocity vc,Lwhich corre- sponds to the current, j=1 2vc,L/H9254/H92670=1 2vc,L/H11509/H9267 /H11509/H9262/H9254/H92620/H11013G/H9254/H92620, /H2084919/H20850 in the central region extending linearly with time. Here /H9262 denotes the chemical potential. With /H11509/H9267//H11509/H9262=1/ /H20849/H9266vN/H20850this yields for the homogeneous systemGhom=KL 2/H9266. /H2084920/H20850 We now switch to an inhomogeneous system as in Fig. 1.I n analogy to the initial condition in the homogeneous case weraise the density for x/H11021aby /H9254/H92670. The stationary current is then obtained by multiplying the result for the homogeneouscase by the fraction of transmitted charge through /H20851a,b/H20852, computed in Eq. /H2084918/H20850. For the conductance this leads to G inhom =2KRKL KL+KR1 2/H9266, /H2084921/H20850 which again is independent of the details of the transition region. For an interacting quantum wire in region /H20851a,b/H20852attached to noninteracting leads /H20849KL=KR=1/H20850the conductance is 1//H208492/H9266/H20850independently of how quickly the LL parameters vary spatially near the contact points aandb. This shows that the LLL description cannot produce one-particle back-scattering from the contact region generated by an inhomo-geneous two-particle interaction. As we will discuss next us-ing a microscopic model and the fRG approach contactsdeﬁned by a vanishing interaction to the left and a positiveﬁnite interaction to the right /H20849or vice versa /H20850generically pro- duce an effective single-particle backscattering. This clearlyshows the limitations of the LLL picture. With the approxi-mation to describe the two-body interaction as a quadraticform in the Bose ﬁelds the dependence of the conductanceon the sharpness of the transition is lost. In contrast, ourmicroscopic model directly allows to study the transitionfrom smooth to abrupt contacts with the concomitant changeof the conductance. III. A MICROSCOPIC MODEL As our microscopic model we consider the spinless tight- binding Hamiltonian with nearest-neighbor hopping /H9270/H110220 and a spatially dependent nearest-neighbor interactionU j,j+1=Uj+1,j, H=−/H9270/H20858 j=−/H11009/H11009 /H20849cj+1†cj+cj†cj+1/H20850+/H20858 j=1N−1 Uj,j+1/H20849nj− 1/2 /H20850/H20849nj+1− 1/2 /H20850, /H2084922/H20850 where we used standard second-quantized notation with cj† and cj†being creation and annihilation operators on site j, respectively, and nj=cj†cjthe local-density operator. The in- teraction acts only between the bonds of the sites 1 to N, which deﬁne the interacting wire. We later take Uj,j+1=Uh/H20849j/H20850,j=1,2, ..., N−1 , /H2084923/H20850 with a function hthat is different from 1 only in regions close to the contacts at sites 1 and N. We here mainly con- sider the half ﬁlled band case n=1/2. In the interacting re- gion the fermions have higher energy compared to the leads.To avoid that the interacting wire is depleted /H20849implying a vanishing conductance /H20850we added a compensating single- particle term. It can be included as a shift of the local-densityJANZEN, MEDEN, AND SCHÖNHAMMER PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-4operator by 1/2. This ensures that the average ﬁlling in the leads and the wire is n=1/2. It is important to have a deﬁnite ﬁlling in the interacting wire as the LL parameters depend onn. For general ﬁllings the required shift of the density de- pends on nand the interaction. 10 The model equation /H2084922/H20850with nearest-neighbor interac- tionUacross all bonds /H20849not only the once within 1 and N/H20850is a LL for all Uand ﬁllings, except at half ﬁlling for /H20841U/H20841 /H110222/H9270.20Atn=1/2 the LL parameter Kis given by20 K=/H208752 /H9266arccos/H20873−U 2/H9270/H20874/H20876−1 , /H2084924/H20850 for /H20841U/H20841/H113492/H9270. To leading order in U//H9270this gives K=1−U /H9266/H9270+O/H20849/H20851U//H9270/H208522/H20850. /H2084925/H20850 At temperature T=0 the linear conductance of the system described by Eq. /H2084922/H20850can be written as21,22 G/H20849N/H20850=1 2/H9266/H20841t/H208490,N/H20850/H208412/H2084926/H20850 with the effective transmission /H20841t/H20849/H9255,N/H20850/H208412=/H208494/H92702−/H20851/H9255 +/H9262/H208522/H20850/H20841G1,N/H20849/H9255,N/H20850/H208412. The one-particle Green function Ghas to be computed in the presence of interaction and in contrast to the noninteracting case acquires an Ndependence. To com- puteGand thus Gwe use a recently developed fRG scheme.23The starting point is an exact hierarchy of differ- ential ﬂow equations for the self-energy /H9018/H9011and higher-order vertex functions, where /H9011/H33528/H20849/H11009,0/H20852denotes an infrared en- ergy cutoff which is the ﬂow parameter. We here introduce /H9011 as a cutoff for the Matsubara frequency. A detailed accountof the method is given in Ref. 10. We truncate the hierarchy by neglecting the ﬂow of the two-particle vertex only con-sidering /H9018 /H9011, which is then energy independent. The self- energy /H9018/H9011=0at the end of the fRG ﬂow provides an approxi- mation for the exact /H9018. This approximation scheme and variants of it were successfully used to study a variety oftransport problems through 1D wires of correlated electrons.In particular, in all cases of inhomogeneous LLs studied theexponents of power-law scaling were reproduced correctly toleading order in U. 10,24–26 On the present level of approximation the fRG ﬂow equa- tion for the self-energy reads /H11509 /H11509/H9011/H90181/H11032,1/H9011=−1 2/H9266/H20858 /H9275=±/H9011/H20858 2,2/H11032ei/H92750+G2,2/H11032/H9011/H20849i/H9275/H20850/H90031/H11032,2/H11032;1,2,/H2084927/H20850 where the lower indices 1, 2, etc., label single-particle states, /H90031/H11032,2/H11032;1,2is the bare antisymmetrized two-particle interaction and G/H9011/H20849i/H9275/H20850=/H20851G0−1/H20849i/H9275/H20850−/H9018/H9011/H20852−1, /H2084928/H20850 with the noninteracting propagator G0. For the initial condi- tion of the self-energy ﬂow see below. As our single-particlebasis we later use Wannier states /H20851as in the Hamiltonian /H2084922/H20850/H20852 as well as momentum states.A. Numerical results In the real-space Wannier basis the self-energy matrix is tridiagonal and the set of coupled differential equations /H2084927/H20850 reads /H20849with j/H33528/H208511,N/H20852/H20850 d d/H9011/H9018j,j/H9011=−1 2/H9266/H20858 l=±1/H20858 /H9275=±/H9011Uj,j+lGj+l,j+l/H9011/H20849i/H9275/H20850ei/H92750+, /H2084929/H20850 d d/H9011/H9018j,j±1/H9011=Uj,j±1 2/H9266/H20858 /H9275=±/H9011Gj,j±1/H9011/H20849i/H9275/H20850ei/H92750+. /H2084930/H20850 To derive these equations we have used that /H9003j1/H11032,j2/H11032;j1,j2=U¯j1,j2/H20849/H9254j1,j1/H11032/H9254j2,j2/H11032−/H9254j1,j2/H11032/H9254j2,j1/H11032/H20850/H20849 31/H20850 with U¯j1,j2=Uj1,j1+1/H20849/H9254j1,j2−1+/H9254j1,j2+1/H20850. The initial condition of /H9018/H9011at/H9011=/H11009is10,23 /H9018j,j/H11009=− /H20849Uj−1,j+Uj,j+1/H20850/2, /H9018j,j±1/H11009=0 . /H2084932/H20850 Even for very large N/H20849earlier results were obtained for up to N=107/H20850,10,24the system Eqs. /H2084929/H20850and /H2084930/H20850can easily be integrated numerically starting at a large but ﬁnite /H90110. Due to the slow decay of the right-hand side /H20849rhs/H20850of the ﬂow equa- tion the integration from /H11009to/H90110gives a nonvanishing con- tribution on the diagonal of the self-energy matrix that doesnot vanish even for /H9011 0→/H11009and that exactly cancels the term in Eq. /H2084932/H20850such that /H9018j,j/H90110=0 , /H9018j,j±1/H90110=0 . /H2084933/H20850 For half ﬁlling the Hamiltonian /H2084922/H20850is particle-hole sym- metric. For Nodd and symmetric h/H20849j/H20850, that is h/H20849j/H20850=h/H20849N−1 −j/H20850, on which we focus here, it is furthermore invariant un- der inversion at site j=/H20849N+1/H20850/2. Together these symmetries lead to a vanishing conductance as was discussed earlier.27,28 We thus only consider even N. The two extreme cases of /H20849a/H20850 an abrupt turning on and off of the interaction h/H20849j/H20850=1 for j =1,2,..., N−1, and zero otherwise, and /H20849b/H20850a very smooth variation of h/H20849j/H20850were considered earlier. In case /H20849a/H20850 1//H208492/H9266/H20850−Gincreases for ﬁxed Nand increasing24,28Uas well as for ﬁxed Uand increasing N.24Using our fRG scheme it was shown that for asymptotically large N,Gvan- ishes as G/H11011/H20849/H9254N//H9270/H208502/H208491/K−1/H20850, /H2084934/H20850 with the energy scale /H9254N=/H9266vF/N. The scale on which the asymptotic low-energy scaling sets in strongly depends on U and even for up to N=106it was only reached for fairly large U, e.g., U=1.5/H9270. As discussed in the Introduction scaling of Gwith the same exponent as in Eq. /H2084934/H20850is found for a single impurity in an otherwise perfect chain. This suggests that thelow-energy physics of a correlated quantum wire with twocontact inhomogeneities due to the two-particle interaction issimilar to that of a wire with a single impurity /H20849single- particle term /H20850. Here we will further investigate this relation. Already at this stage it is important to note that the simi- larity is not complete. As shown elsewhere 29G/H20849T/H20850for twoINFLUENCE OF THE CONTACTS ON THE ¼ PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-5contacts, ﬁxed U, ﬁxed large N, and decreasing temperature Tvanishes as /H20849forT/greatermuch/H9254N/H20850 G/H11011/H20849T//H9270/H208501/K−1, /H2084935/H20850 that is with an exponent that is only one-half of the one found in the /H9254Nscaling equation /H2084934/H20850. This is in clear contrast to the temperature dependence of Gfor a single impurity in an otherwise perfect chain. In this case the exponent 2 /H208491/K −1/H20850is found in the scaling with any infrared energy scale, e.g., T,/H9254N, and/H9011. For a given NandUit is always possible to ﬁnd a sufﬁ- ciently smooth function h/H20849j/H20850such that 1/ /H208492/H9266/H20850−Gis smaller then an arbitrarily small upper bound /H20851case /H20849b/H20850/H20852.10,24In that case the contacts are perfect and the effect of impurities inthe interacting wire connected to Fermi liquid leads can bestudied. Further down we will give a quantitative deﬁnitionof the meaning of “sufﬁciently smooth.” Here we study the dependence of the conductance on N, U, and the function h/H20849j/H20850by numerically solving the ﬂow equations /H2084929/H20850and /H2084930/H20850with the initial condition Eq. /H2084933/H20850.I n Fig. 2we show 1−2 /H9266Gas a function of /H9254N//H9270/H110111/Non a log-log scale /H20849upper panel /H20850forU//H9270=0.2 and three different h/H20849j/H20850of increasing smoothness. In our numerical calculations for simplicity we focus on contact functions h/H20849j/H20850that are symmetric around the middle of the interacting wire /H20849equal contacts /H20850and therefore only give their deﬁnition for the ﬁrst half of the wire /H20849with j/H33528/H208511,N−1/H20852/H20850 h1/H20849j/H20850=1 , j=1, ..., N/2, h2/H20849j/H20850=/H20877j/m, j=1, ..., m−1 1, j=m, ..., N/2, andh3/H20849j/H20850=/H209022/H20849j/m/H208502, j=1, ..., /H20849m−1/H20850/2 1−2 /H20849j/m−1/H208502, j=/H20849m+1/H20850/2, ... , m−1 1, j=m, ..., N/2, where mis odd and measures the length of the contact re- gion. In Fig. 2we chose m=7. The lower panel of Fig. 2 shows the logarithmic derivative of the data. A plateau in thisﬁgure corresponds to power-law scaling of the original datawith an exponent given by the plateau value. For ﬁxed /H9254N, 1−2/H9266Gdecreases quickly with increasing smoothness of h/H20849j/H20850. For sufﬁciently small /H9254N//H9270all curves follow power-law scaling. The smoother the contact the smaller /H9254N//H9270has to be before the scaling sets in. The numerical exponent is close to2/H20849K−1/H20850//H20849K+1/H20850/H20849shown as the thin solid line /H20850, with Kfrom Eq. /H2084924/H20850. The latter is the exponent found for a system with perfect contacts and a weak single-particle impurity placed inthe contact region. We also studied other values for thelength of contact mand found similar results. The deviation 1−2 /H9266Gfrom the unitary limit decreases if mis increased while all other parameters are ﬁxed. Within our approximation scheme we map the many-body problem on an effective single-particle problem with the en-ergy independent /H9018 /H9011=0as an impurity potential.10,23During the fRG ﬂow the interplay of the two-particle inhomogeneityand the bulk interaction generates an oscillating self-energywith an amplitude that decays slowly away from the twocontacts. This is in close analogy to the case of a singleimpurity in an otherwise perfect chain. 10,23,30Scattering off this effective potential leads to the reduced conductance. TheUdependence of 1−2 /H9266Gand the effective exponent is shown in Fig. 3forh2/H20849j/H20850. At ﬁxed /H9254Nthe deviation of the conductance from the unitary limit increases with U. For all U//H9270shown we ﬁnd power-law behavior with exponents that are close to 2 /H20849K−1/H20850//H20849K+1/H20850/H20849again shown as the thin solid lines /H20850. The larger U//H9270the larger is the deviation of the nu- merical exponent and 2 /H20849K−1/H20850//H20849K+1/H20850. This is not surprising FIG. 2. /H20849Color online /H20850Upper panel: The conductance 1−2 /H9266Gas a function of /H9254N//H9270forU//H9270=0.2, contacts of m=7 lattice sites, and three different contact functions hi/H20849j/H20850of increasing smoothness. Lower panel: The effective exponent /H20849logarithmic derivative /H20850of the data. The thin solid line shows the exponent 2 /H20849K−1/H20850//H20849K+1/H20850with K/H20849U//H9270=0.2 /H20850=0.9401. FIG. 3. /H20849Color online /H20850Upper panel: The conductance 1−2 /H9266Gas a function of /H9254N//H9270for the contact function h2/H20849j/H20850, contacts of m=7 lattice sites, and U//H9270=0.2,0.4,0.6. Lower panel: The effective ex- ponent /H20849logarithmic derivative /H20850of the data. The thin solid lines show the exponents 2 /H20849K−1/H20850//H20849K+1/H20850with K/H20849U//H9270=0.2 /H20850=0.9401, K/H20849U//H9270=0.4 /H20850=0.8864, and K/H20849U//H9270=0.6 /H20850=0.8375.JANZEN, MEDEN, AND SCHÖNHAMMER PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-6as our approximate scheme can only be expected to capture exponents to leading order in U//H9270.10 For later reference we note that power-law behavior is not only found in the /H9254N/H20849that is 1/ N/H20850dependence of 1−2 /H9266Gbut also in the dependence on the fRG ﬂow parameter /H9011at a large ﬁxed N. In this case Gis computed using Eq. /H2084926/H20850with the Green function at scale /H9011. This is shown in Fig. 4where we compare the dependence on /H9254N//H9270and/H9011//H9270forU=0.2 and h2/H20849j/H20850. Scaling with /H9011//H9270holds roughly down to the scale /H9254N//H9270=/H9266vF//H20849/H9270N/H20850set by the length of the interacting wire. Beyond this scale Gsaturates. We now analytically conﬁrm the power-law scaling of 1//H208492/H9266/H20850−Gfor weak effective inhomogeneities and compute the exponent to leading order in the bulk interaction. B. Analytical results Numerically we found that 1/ /H208492/H9266/H20850−Gshows power-law scaling in both effective low-energy cutoffs /H9254Nand/H9011,a s long as the scaling variable considered is sufﬁciently largerthan the other ﬁxed energy scale. We note that the sameholds for the temperature Tas the variable. 31In all cases the same exponent is found. Analytically the easiest access toscaling is by considering N→/H11009,T=0, and taking /H9011as the variable. We will here follow this route. For the case of asingle impurity with bare backscattering amplitude V backin an inﬁnite LL /H20849no Fermi-liquid leads /H20850a similar calculation was performed within the fRG,23to show that for small /H9011 /H9018kF,−kF//H9270/H11011/H20849Vback//H9270/H20850/H20849/H9011//H9270/H20850−U//H20849/H9266/H9270/H20850. /H2084936/H20850 Within the Born approximation this leads to K 2/H9266−G/H11011/H20841Vback//H9270/H208412/H20849/H9011//H9270/H20850−2U//H20849/H9266/H9270/H20850, /H2084937/H20850 where −2 U//H20849/H9266/H9270/H20850is the leading-order approximation of the weak impurity exponent 2 /H20849K−1/H20850/H20851see Eq. /H2084925/H20850/H20852. This scaling holds as long as the rhs stays small. For analytical calcula-tions it is advantageous to switch from the real-space basis to the momentum states. In the momentum state basis the ﬂow equation /H2084927/H20850is given by /H11509 /H11509/H9011/H9018k/H11032,k/H9011=−1 2/H9266/H20858 /H9275=±/H9011/H20885 −/H9266/H9266 dqdq /H11032ei/H92750+Gq,q/H11032/H9011/H20849i/H9275/H20850/H9003k/H11032,q/H11032;k,q, /H2084938/H20850 with /H9003k/H11032,q/H11032;k,q=1 2/H9266/H20851ei/H20849q−q/H11032/H20850−ei/H20849q−k/H11032/H20850+ei/H20849k−k/H11032/H20850−ei/H20849k−q/H11032/H20850/H20852 /H11003U˜/H20849k+q−k/H11032−q/H11032/H20850/H20849 39/H20850 and the Fourier transform of the interaction U˜/H20849k/H20850=1 2/H9266/H20858 j=1N−1 Uj+1,jeijk=Uh˜/H20849k/H20850. /H2084940/H20850 Here h˜/H20849k/H20850denotes the Fourier transform of the function h/H20849j/H20850 that contains the shape of the turning on and off of the inter- action. In the present section we do not assume special sym-metry properties of h/H20849j/H20850. Thus the two contacts might be different, as it is generically the case in experiments. At /H9011 =/H9011 0all matrix elements of /H9018/H90110are zero. We now expand the rhs of Eq. /H2084938/H20850to ﬁrst order in /H9018/H9011. This expansion is justiﬁed as long as /H9018/H9011remains small /H20849it is certainly small at the beginning of the fRG ﬂow /H20850. The ﬂow equation then becomes an inhomogeneous, linear differential equation /H11509 /H11509/H9011/H9018k/H11032,k/H9011=Fk/H11032,k/H208491/H20850/H20849/H9011/H20850+Fk/H11032,k/H208492/H20850/H20849/H9011,/H9018/H9011/H20850, /H2084941/H20850 with Fk/H11032,k/H208491/H20850/H20849/H9011/H20850=−1 2/H9266/H20858 /H9275=±/H9011/H20885 −/H9266/H9266 dq/H9003k/H11032,q;k,q i/H9275−/H9264q =−U 2/H9266/H9270h˜/H20849k−k/H11032/H20850/H20849eik+e−ik/H11032/H20850/H208731−/H9011 /H20881/H90112+4/H92702/H20874, /H2084942/H20850 where we used /H9264k=−2/H9270cos/H20849k/H20850−/H9262and Eq. /H2084939/H20850. The factor ei/H92750+was dropped as the integration starts at /H90110/H11021/H11009. Later we will primarily be interested in the kF,−kFmatrix element /H20849backscattering /H20850of/H9018. For these momenta the rhs simpliﬁes to FkF,−kF/H208491/H20850/H20849/H9011/H20850=−U /H9266/H9270h˜/H20849−2kF/H20850e−ikF/H208731−/H9011 /H20881/H90112+4/H92702/H20874./H2084943/H20850 The term on the rhs of the linearized ﬂow equation that con- tains/H9018/H9011is FIG. 4. /H20849Color online /H20850Upper panel: The conductance 1−2 /H9266Gas a function of /H9254N//H9270and/H9011//H9270/H20849with N=216sites /H20850for the contact function h2/H20849j/H20850, contacts of m=7 lattice sites, and U//H9270=0.2. Lower panel: The effective exponent /H20849logarithmic derivative /H20850of the data. The thin solid line shows the exponents 2 /H20849K−1/H20850//H20849K+1/H20850with K/H20849U//H9270=0.2 /H20850=0.9401.INFLUENCE OF THE CONTACTS ON THE ¼ PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-7Fk/H11032,k/H208492/H20850/H20849/H9011,/H9018/H9011/H20850=−1 2/H9266/H20858 /H9275=±/H9011/H20885 −/H9266/H9266 dqdq /H11032/H9003k/H11032,q/H11032;k,q/H20849G0/H9018/H9011G0/H20850q,q/H11032. /H2084944/H20850 Before further evaluating this term we take N→/H11009and com- pute/H9003k/H11032,q;k,qin this limit. As we will see in this case the information about the smoothness of h/H20849j/H20850is lost on the rhs of Eq. /H2084944/H20850. It nevertheless enters the solution of the differential equation via the inhomogeneity Fk/H11032,k/H208491/H20850in Eq. /H2084941/H20850.A sw ea r e only interested in the leading divergent behavior of the linear part on the rhs of Eq. /H2084941/H20850this procedure is justiﬁed. To be speciﬁc we ﬁrst assume that the interaction is turned on andoff abruptly, that is h/H20849j/H20850=1 for j=1,2,..., N−1 and zero otherwise. To obtain the Fourier transform of the two-particle interaction we have to perform U˜/H20849k/H20850=Uh˜/H20849k/H20850=U 2/H9266/H20858 j=1N−1 eijk=U 2/H9266eik−eiNk 1−eik→N→/H11009 U1 2/H92542/H9266/H20849k/H20850. /H2084945/H20850 With this result the two-particle vertex simpliﬁes to /H9003k/H11032,q;k,q→N→/H11009U /H9266/H20851cos/H20849k−k/H11032/H20850− cos /H20849q−k/H11032/H20850/H20852 /H11003/H92542/H9266/H20849k+q−k/H11032−q/H11032/H20850. /H2084946/H20850 Up to a factor 1/2 the vertex is then equivalent to the one obtained for a homogeneous nearest-neighbor interaction. With this vertex Fk/H11032,k/H208492/H20850/H20849/H9011,/H9018/H9011/H20850Eq. /H2084944/H20850reads Fk/H11032,k/H208492/H20850/H20849/H9011,/H9018/H9011/H20850=−U 2/H92662/H20858 /H9275=±/H9011/H20885 −/H9266/H9266 dq/H20851cos/H20849k−k/H11032/H20850− cos /H20849q−k/H11032/H20850/H20852 /H11003/H9018q+k−k/H11032,q /H20849i/H9275−/H9264q+k−k/H11032/H20850/H20849i/H9275−/H9264q/H20850. /H2084947/H20850 The differential equation /H2084941/H20850can be solved by the varia- tion of constant method. To apply this we ﬁrst determine thesolution of the homogeneous equation. The scaling can beextracted if only the leading singular contribution /H20849for/H9011 →0/H20850ofF k/H11032,k/H208492/H20850/H20849/H9011,/H9018/H9011/H20850Eq. /H2084947/H20850atk/H11032=kFandk=−kFis kept. Following the same steps as in Ref. 23we ﬁnd /H11509 /H11509/H9011/H20851/H9018kF,−kF/H9011/H20852hom/H11015−U 2/H9266/H92701 /H9011/H20851/H9018kF,−kF/H9011/H20852hom, /H2084948/H20850 with the solution /H20851/H9018kF,−kF/H9011/H20852hom//H9270/H11015c0/H20849/H9011//H9270/H20850−U//H208492/H9266/H9270/H20850, /H2084949/H20850 where c0is a dimensionless constant. The singular part of the solution of the inhomogeneous linear differential equation isthen given by /H9018 kF,−kF/H9011//H9270/H11015c/H20849/H9011/H20850/H20849/H9011//H9270/H20850−U//H208492/H9266/H9270/H20850, /H2084950/H20850 withc/H20849/H9011/H20850=/H20885 /H90110/H9011 d/H9011/H110321 /H20851/H9018kF,−kF/H9011/H11032/H20852homFkF,−kF/H208491/H20850/H20849/H9011/H11032/H20850. /H2084951/H20850 For small /H9011the function c/H20849/H9011/H20850is nonsingular and given by lim /H9011→0c/H20849/H9011/H20850=−U /H9270h˜/H20849−2kF/H20850e−ikFc¯ /H2084952/H20850 with c¯being a constant of order 1. This gives /H9018kF,−kF/H9011//H9270/H11015−U /H9270h˜/H20849−2kF/H20850e−ikFc¯/H20849/H9011//H9270/H20850−U//H208492/H9266/H9270/H20850/H2084953/H20850 and with the Born approximation the ﬁnal result 1 2/H9266−G/H11011/H20841U//H9270/H208412/H20841h˜/H20849−2kF/H20850/H208412/H20849/H9011//H9270/H20850−U//H20849/H9266/H9270/H20850. /H2084954/H20850 To obtain the singular part of the solution of the homoge- neous differential equation /H20851/H9018kF,−kF/H9011/H20852homwe assumed that the interaction is turned on and off abruptly. One can show that the parts of h/H20849j/H20850with a smooth variation of ﬁnite length do not contribute to the singular part of Eq. /H2084944/H20850. Thus the same singular part is found independently of how the interaction isvaried and Eq. /H2084954/H20850is valid for general h/H20849j/H20850. The power-law scaling Eq. /H2084954/H20850holds for /H9011 c/lessmuch/H9011/lessmuch/H9270 with a scale /H9011cset by /H20841U//H9270/H208412/H20841h˜/H20849−2kF/H20850/H208412/H20849/H9011c//H9270/H20850−U//H20849/H9266/H9270/H20850/H110111//H208494/H9266/H20850. /H2084955/H20850 To leading order in U//H9270the exponent U//H20849/H9266/H9270/H20850agrees with the exponent 2 /H20849K−1/H20850//H20849K+1/H20850/H20851see Eq. /H2084925/H20850/H20852found for a single weak impurity placed close to a perfect /H20849that is arbitrarily smooth /H20850contact.9,10We thus have analytically shown, that with respect to the scaling exponent weak single-particle andweak two-particle inhomogeneities are indeed equivalent.The leading-order exponent is furthermore consistent withthe numerical results of the last section. For the single impurity case the prefactor of the power law is given by /H20841V back//H9270/H208412. In case of the two-particle inho- mogeneity this is replaced by the square of the bulk interac-tion and the square of the 2 k FFourier component /H20849back- scattering /H20850of the function h/H20849j/H20850with which the interaction is turned on and off. The smaller this component the smaller is the perturbation due to the two contacts. Therefore thesmoothness of the turning on and off is directly measured by h˜/H20849−2kF/H20850. The presence of the factor /H20841U//H9270/H208412explains why in the numerical study the weak inhomogeneity exponent for larger U//H9270is only observable for fairly smooth contacts, that ish/H20849j/H20850’s with small 2 kFcomponent. For larger U//H9270and fairly abrupt contacts the rhs of Eq. /H2084954/H20850becomes too large already on intermediate energy scales and no energy window forscaling is left. For /H9011/lessmuch/H9011 cdeﬁned in Eq. /H2084955/H20850the inhomo- geneity is effectively large and Eq. /H2084934/H20850holds. We here con- sidered the half ﬁlled band case, but also other ﬁllings can bestudied following the same steps. 23 The analysis Eqs. /H2084938/H20850–/H2084954/H20850also holds if in addition weak single-particle impurities are placed close to the contacts, with the only difference that /H9018kF,−kF/H90110now has a nonvanishing initial condition set by the backscattering of the bare impu-JANZEN, MEDEN, AND SCHÖNHAMMER PHYSICAL REVIEW B 74, 085301 /H208492006 /H20850 085301-8rity. The prefactor in Eq. /H2084954/H20850is determined by either the square of the single impurity backscattering amplitude or/H20841U/ /H9270/H208412/H20841h/H20849−2kF/H20850/H208412depending on the relative size. IV . SUMMARY In the present paper we have investigated the role of con- tacts, deﬁned by an inhomogeneous two-particle interaction,on the linear conductance through an interacting 1D quantumwire. The wire and contacts were assumed to be free of anybare single-particle impurities. We ﬁrst showed that withinthe LLL picture the contacts are always perfect, that is theconductance is 1/ /H208492 /H9266/H20850independent of the strength of the interaction, the length of the wire, and the spatial variation of the LL parameters. Earlier only stepwise changes of the LLparameters were considered. We then studied the problemwithin a microscopic lattice model. Similar to the case of asingle impurity the interplay of the two-particle inhomoge-neity and the bulk interaction generates an oscillating self-energy with an amplitude that decays slowly away from thecontacts. Scattering off this effective potential leads to thediscussed effects. We showed that within the microscopicmodeling 1/ /H208492 /H9266/H20850−Gincreases with increasing bulk interac- tionU, increasing wire length N, and decreasing smoothness. The measure for smoothness is given by h˜/H20849−2kF/H20850, that is thebackscattering Fourier component of the spatial variation h/H20849j/H20850of the two-particle interaction. As long as the inhomo- geneity stays effectively small 1/ /H208492/H9266/H20850−Gshows power-law scaling with an exponent that is consistent with 2 /H20849K−1/H20850/ /H20849K+1/H20850, the scaling exponent known for the case of a wire with a single impurity placed close to one of the two smooth contacts. In experiments on quantum wires the leads are electroni- cally two or three dimensional. Depending on the systemsstudied the contacts are either regions in which the systemgradually crosses over from higher dimensions to quasi-1Dor the contact regions extend over a ﬁnite part of the wire/H20849see the experiments on carbon nanotubes /H20850. 11This shows that our simpliﬁed description—1D leads, spatially dependent in-teractions close to end contacts, no explicit single-particlescattering at the contacts—provides only an additional steptowards a detailed understanding of the role of contacts intransport through interacting quantum wires, such as carbonnanotubes and cleaved edge overgrowth samples. 11 ACKNOWLEDGMENTS We thank S. Jakobs and H. Schoeller for very fruitful discussions. The authors are grateful to the DeutscheForschungsgemeinschaft /H20849SFB 602 /H20850for support. 1For a review, see K. Schönhammer, in Interacting Electrons in Low Dimensions , edited by D. Baeriswyl /H20849Kluwer Academic Publishers, Dordrecht, 2005 /H20850. 2A. Luther and I. Peschel, Phys. Rev. B 9, 2911 /H208491974 /H20850. 3D. C. Mattis, J. Math. Phys. 15, 609 /H208491974 /H20850. 4W. Apel and T. M. Rice, Phys. Rev. B 26, 7063 /H208491982 /H20850. 5T. Giamarchi and H. J. Schulz, Phys. Rev. B 37, 325 /H208491988 /H20850. 6C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 /H208491992 /H20850; Phys. Rev. B 46, 15233 /H208491992 /H20850. 7A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 /H208491993 /H20850. 8P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. Lett. 74, 3005 /H208491995 /H20850. 9A. Furusaki and N. Nagaosa, Phys. Rev. B 54, R5239 /H208491996 /H20850. 10T. Enss, V . Meden, S. Andergassen, X. Barnabé-Thériault, W. Metzner, and K. Schönhammer, Phys. Rev. B 71, 155401 /H208492005 /H20850. 11M. Bockrath, D. Cobden, J. Lu, A. Rinzler, R. Smalley, L. Balents, and P. McEuen, Nature /H20849London /H20850397, 598 /H208491999 /H20850;Z . Yao, H. Postma, L. Balents, and C. Dekker, ibid. 402, 273 /H208491999 /H20850; O. M. Auslaender, A. Yacoby, R. de Picciotto, K. W. Baldwin, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 84, 1764 /H208492000 /H20850; R. de Picciotto, H. Stormer, L. Pfeiffer, K. Bald- win, and K. West, Nature /H20849London /H20850411,5 1 /H208492001 /H20850. 12I. Saﬁ and H. J. Schulz, Phys. Rev. B 52, R17040 /H208491995 /H20850. 13D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539 /H208491995 /H20850. 14V . V . Ponomarenko, Phys. Rev. B 52, R8666 /H208491995 /H20850. 15D. L. Maslov, Phys. Rev. B 52, R14368 /H208491995 /H20850.16S. Tomonaga, Prog. Theor. Phys. 5, 544 /H208491950 /H20850. 17J. M. Luttinger, J. Math. Phys. 4, 1154 /H208491963 /H20850. 18There is no generally accepted use of prefactors in the deﬁnition of the /H9021/H9251in the literature. We therefore did not include any. 19W. Thirring, Ann. Phys. /H20849N.Y . /H208503,9 1 /H208491958 /H20850. 20F. D. M. Haldane, Phys. Rev. Lett. 45, 1358 /H208491980 /H20850. 21Under quite weak assumptions on the frequency dependence of the self-energy, that are fulﬁlled for the present situation, thereare no current vertex corrections at T=0 and Eq. /H2084926/H20850is exact /H20849Ref. 22/H20850. 22A. Oguri, J. Phys. Soc. Jpn. 70, 2666 /H208492001 /H20850. 23V . Meden, W. Metzner, U. Schollwöck, and K. Schönhammer, J. Low Temp. Phys. 126, 1147 /H208492002 /H20850. 24V . Meden, S. Andergassen, W. Metzner, U. Schollwöck, and K. Schönhammer, Europhys. Lett. 64, 769 /H208492003 /H20850. 25X. Barnabé-Thériault, A. Sedeki, V . Meden, and K. Schönham- mer, Phys. Rev. Lett. 94, 136405 /H208492005 /H20850. 26X. Barnabé-Thériault, A. Sedeki, V . Meden, and K. Schönham- mer, Phys. Rev. B 71, 205327 /H208492005 /H20850. 27A. Oguri, Phys. Rev. B 59, 12240 /H208491999 /H20850; Phys. Rev. B 63, 115305 /H208492001 /H20850. 28V . Meden and U. Schollwöck, Phys. Rev. B 67, 193303 /H208492003 /H20850. 29S. Jakobs, V . Meden, H. 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PhysRevB.101.235160.pdf	PHYSICAL REVIEW B 101, 235160 (2020) Temperature dependence of the Kondo resonance in the photoemission spectra of the heavy-fermion compounds Yb XCu4(X=Mg,Cd,and Sn) Hiroaki Anzai ,1,Kohei Morikawa,1Hiroto Shiono,1Hitoshi Sato,2Shin-ichiro Ideta,3Kiyohisa Tanaka,3Tao Zhuang,4 Keisuke T. Matsumoto,4and Koichi Hiraoka4,† 1Graduate School of Engineering, Osaka Prefecture University, Sakai 599-8531, Japan 2Hiroshima Synchrotron Radiation Center, Hiroshima University, Higashi-Hiroshima 739-0046, Japan 3UVSOR Synchrotron Facility, Institute for Molecular Science, Okazaki 444-8585, Japan 4Graduate School of Science and Engineering, Ehime University, Matsuyama, Ehime 790-8577, Japan (Received 16 March 2020; revised manuscript received 9 June 2020; accepted 12 June 2020; published 25 June 2020) We report a temperature-dependent study of Kondo resonance in the heavy-fermion compounds Yb XCu4with X=Mg, Cd, and Sn. A sharp peak of the Yb2+4f7/2state has been observed in photoemission spectra, and its energy position in the limit of zero temperature is in agreement with the energy scale of the Kondo temperatureT K. The peak develops in the form of a dispersionless peak with large spectral weight at Tcohwell below TK.T h e onset temperature of the robust Kondo state is comparable to the temperature at a local maximum in magneticsusceptibility. This nontrivial development of changes in the Kondo resonance at T cohdemonstrates the formation of the coherent heavy-fermion state. DOI: 10.1103/PhysRevB.101.235160 I. INTRODUCTION In rare-earth compounds, the interaction between conduc- tion electrons and localized-4 felectrons ( c–fhybridization) causes a loss in strength of the local magnetic moments andforms a correlated Fermi-liquid state with large effective elec-tron mass at low temperatures. This heavy-fermion behavioris characterized by a resonant state near the Fermi energy(E F). According to the single-impurity Anderson model, a sharp resonance peak in the electronic excitation spectrumappears at the energy scale of Kondo temperature T K, which is a criterion for the magnetic response to a single magneticimpurity [ 1]. For ytterbium-based compounds, the spin-orbit split Yb 2+4f7/2state is interpreted as the Kondo resonance state [ 2,3]. Its intensity is enhanced when the temperature varies from well above to well below TK[4,5]. Therefore, studying the temperature dependence of the Kondo resonanceover a wide temperature range will provide important insightsfor understanding the heavy-fermion state. The periodic Anderson lattice model suggests a distinct temperature scale T coh, which is associated with the onset of coherent screening at the lower temperatures and a cri-terion for the collective magnetic response to a lattice oflocal moments [ 6–8]. At temperatures above T coh, the screen- ing of a single local moment by conduction electrons isdominant. At temperatures T<T coh, the collective screening results for coherent excitations of the heavy quasiparticlesforms hybridized bands near E F. Transport investigations have long provided evidence for the existence of Tcohin the Yb-based heavy-fermion systems. For example, the magnetic anzai@pe.osakafu-u.ac.jp †hiraoka.koichi.mk@ehime-u.ac.jpsusceptibility changes from Curie-Weiss behavior to beingessentially independent of temperature at approximately T coh [9,10]. It is generally accompanied by a drop in electrical resistivity, indicating the onset of coherence and the emer-gence of a Fermi-liquid behavior far below T K[11,12]. Such a crossover towards the coherent states in the heavy-fermionsystems has been rarely studied from microscopic measure-ments. Photoemission spectroscopy is an excellent tool for study- ing the structure of single-particle excitations and allowsdirect observation of the characteristic resonant states inthe heavy-fermion materials. We focus on the spectroscopicsignature of Yb XCu 4(X=Mg,Cd,and Sn) with the same AuBe 5-type ( C15b) crystal structure [ 10–13]. The compounds with X=Mg and Cd have large linear coefﬁcients of spe- ciﬁc heat ∼62 and ∼175 mJ K−2mol−1, respectively [ 11,12]. The fascinating property of Yb XCu4is a variety of Kondo temperatures: TKforX=Mg is approximately four times larger than TK=287 K for X=Cd [11,12]. In our previous report, the Kondo temperature for X=Sn was determined to be TK=503 K, indicating a strong hybridization of the Yb 4 fstates with the conduction bands [ 10]. The values of TKare summarized in Table I. Coherent transport generally develops at well below TKof the material [ 14]. Thus, the spectroscopic information at the experimentally accessibletemperature of T cohcontributes to the understanding of the incoherent-coherent crossover in the heavy-fermion systems. In this paper, we report the temperature dependence of the Kondo resonance in photoemission spectra of Yb XCu4with X=Mg, Cd, and Sn. A sharp 4 f7/2peak is observed clearly near EFand exhibits a clear material dependence. Quanti- fying the energy position and spectral weight of the 4 f7/2 peak, we demonstrate the development of the coherent Kondo state in the form of a sharp and dispersionless peak at low 2469-9950/2020/101(23)/235160(7) 235160-1 ©2020 American Physical SocietyHIROAKI ANZAI et al. PHYSICAL REVIEW B 101, 235160 (2020) TABLE I. Characteristic temperatures in Yb XCu4(X=Mg, Cd, and Sn). The Kondo temperature TKis calculated by using the Wilson’s formula TK=1.29T0,w h e r e T0is the temperature scale determined from the susceptibility at zero temperature [ 15,16]. The values of T0andχ(0) are reported in Refs. [ 10–12]. The temperatures at the local maximum in the magnetic susceptibility Tmax χforX=Cd and Sn are determined from χ(T) in Fig. 1(a). The value of Tmax χfor X=Mg is given in Refs. [ 11,12]. The onset temperatures Tcohare estimated from the temperature dependence of the Yb2+4f7/2peak in Fig. 3(a). XT K(K) Tmax χ(K) Tcoh(K) Mg 1109 130 122 Cd 287 35 60 Sn 503 40 59 temperatures much lower than TK. This observation enabled us to obtain the consistent and reasonable spectroscopic dataof the Yb XCu 4compounds. High-quality single crystals of Yb XCu4with X=Mg, Cd, and Sn were synthesized by the ﬂux method [ 13]. The mea- surements of photoemission spectroscopy were performedon BL7U of the UVSOR Synchrotron Facility. The energyresolution was 16 meV . The data were collected with photonenergy of hν=24 eV and in a wide temperature range below T Kof the samples. The samples were cleaved in situ and maintained under ultrahigh vacuum (8 ×10−9Pa) during the measurements. Energies were calibrated with the Fermi edgeof polycrystalline gold. II. RESULTS AND DISCUSSION Figure 1(a) shows the temperature dependences of the magnetic susceptibility χ(T)f o r X=Cd and Sn on a FIG. 1. (a) Temperature dependence of magnetic susceptibility χ(T)f o rY b XCu4with X=Cd and Sn. The original data are reported in Refs. [ 10,13]. Arrows represent temperatures at the local maximum in χ(T). (b) The photoemission spectra of Yb XCu4 measured at T=8 K. The vertical bar indicates the characteristic energy state.logarithmic scale. The original data are reported in our pre- vious publications [ 10,13]. The susceptibility deviates from the Curie-Weiss law with decreasing temperature and thenexhibits a broad peak at low temperatures. A small upturnbelow 20 K may be due to the presence of impurity phasesof Yb 2O3[13]. The ratio χ(T)/χ(0) is a universal scaling with respect toT/T0, where T0is the temperature scale deduced from the magnetic susceptibility at zero temperature within theCoqblin-Schrieffer model [ 15,16]. For the data in Fig. 1(a),i t is difﬁcult to determine χ(0) due to the upturn at T<20 K. Here, we estimated the temperature at the local maximumT max χas a characteristic temperature. The temperature is in- dicated by arrows in Fig. 1(a)and summarized in Table I.T h e value for X=Mg has been reported in the literature [ 11,12]. We note that the weak peak in χ(T)f o r X=Cd causes large error bars in Tmax χ. It is revealed that TKis proportional to Tmax χ by a factor of ∼9.5. In Fig. 1(b), the photoemission spectra of Yb XCu4mea- sured at T=8 K are presented. The spectra consist of a spin- orbit doublet of the Yb2+4fstates: the 4 f5/2and 4 f7/2states are peaked at \|ω\|/similarequal1.35 eV and near EF, respectively. These assignments are in agreement with the previous photoemis-sion studies [ 17–20]. The full energy width at half maximum 0.107 eV of the 4 f 7/2peak for X=Mg is much narrower than the results obtained by using a He discharge lamp [ 17], indicating the high-quality data in this paper. A shoulder structure is seen at the high-energy side of the 4f7/2peak as indicated by the vertical bar in the spectra of X=Sn. The relative intensity and energy difference coincide well with the those of the spin-orbit partner of the 4 f5/2 state, suggesting the 4 f-derived electronic states. A similar structure has been observed in the spin-orbit split states ofYbInCu 4[18,19]. For the case of X=Mg and Cd, the 4 f5/2 and 4 f7/2peaks exhibit an asymmetric line shape. We consider that a shoulder of the 4 f7/2peak also exists in the spectra of X=Mg and Cd, but its small spectral weight may be smeared due to the high-energy tail of the main peak and the limitationof the experimental energy resolution. A candidate for the origin of the shoulder structure is the 4fstates of the subsurface region that intervenes between the surface and the bulk regions of the crystal [ 18–21]. The elec- tronic state of the subsurface region exhibits characteristicsintermediate between those at the surface and those in thebulk [ 21]. The observed shoulder is, indeed, located between the surface-derived state at \|ω\|/similarequal1.1 eV and the bulk-derived 4f 7/2state near EF[20]. It is, therefore, reasonable to assign the shoulder of the 4 f7/2peak to the 4 f-derived states in the Yb subsurface atomic layers. The temperature dependences of the 4 f7/2peak for X= Mg, Cd, and Sn, respectively, are shown in Figs. 2(a)–2(c). The peaks at T=8 K are located away from the energy width of the thermal broadening 4 kBTin the neighborhood of EF, where kBis Boltzmann’s constant. Therefore, we can rule out the possibility that the intense peak near EF, such as the Kondo resonance in the Ce compounds, is observed due to thecutoff of the Fermi-Dirac function [ 22]. The enhancement of the peak intensity with decreasing temperature is essentiallyconsistent with the predictions of the Anderson impuritymodel [ 1]. 235160-2TEMPERATURE DEPENDENCE OF THE KONDO RESONANCE … PHYSICAL REVIEW B 101, 235160 (2020) FIG. 2. (a) Temperature dependence of the Yb2+4f7/2peak in photoemission spectra for X=Mg. The spectra are normalized to the intensity at \|ω\|/similarequal0.3 eV. (b) and (c) The same spectra as in panel (a) but for (b) X=Cd and (c) X=Sn. (d) Temperature dependence of the symmetrized spectra for X=Mg. The spectra at six characteristic temperatures are selected from panel (a). An offset is used for clarity. (e) and (f) The same spectra as in panel (d) but for (e) X=Cd and (f) X=Sn. The vertical bars denote the energy positions of the 4 f7/2peak. The colored area indicates the low-energy spectral weight WLE. The area in the neighborhood of EFis also shaded in gray. A key ﬁnding from the raw spectra is a distinct temperature evolution of the 4 f7/2peak. The peak intensity of X=Cd and Sn decreases rapidly with increasing temperature. Theintensity of X=Mg, however, is maintained even at the high temperature of T/similarequal190 K. Furthermore, the peak positions ofX=Mg are less sensitive to temperature compared with those of X=Cd and Sn as shown in Figs. 2(a)–2(c).T h i si s consistent with the robustness of the Kondo singlet state inX=Mg, where T Kis larger than that of X=Cd and Sn. To quantify the energy and intensity of the 4 f7/2peak precisely, we have symmetrized the photoemission spectrawith respect to E F. This method can remove the effects of the Fermi-Dirac cutoff on spectra near EFand provides a clear view of temperature dependence of the photoemissionspectra [ 23]. The symmetrized spectra are shown with a constant offset in Figs. 2(d)–2(f). We have determined the energy positions of the peak and plotted them as a functionof temperature in Fig. 3(a), which reveals a universal trend in Yb XCu 4. The peak energy of X=Mg decreases with decreasing temperature from T=187 to 122 K and then saturates at low temperatures. The similar saturation is seenin the results for X=Cd and Sn. It is noteworthy that the onset temperatures, which start to approach saturation, are comparable to T max χ shown in Fig. 1(a) and Table I. The consistency of the spectroscopic data with the magnetic susceptibility data indicates that thedispersionless feature of the 4 f 7/2peak at low temperatures is relevant to the coherent excitations of the heavy quasiparticlesbelow T coh. We have estimated the onset temperatures of the peak saturation as Tcohand summarized them in Table I.The appearance of the Kondo resonance at kBTKin the limit of zero temperature represents the formation of the Kondosinglet state [ 4]. In this context, we have applied the average of the peak energy over the region of the saturation to the energyscale of the Kondo temperature. As shown by the dashed linein Fig. 3(a), the average energies for X=Mg, Cd, and Sn are determined to be approximately 69, 30, and 42 meV , respec-tively. These energy scales are in agreement with the sample’sT Kshown in Table I. Therefore, the observed 4 f7/2state is responsible for the heavy-fermion behavior of Yb XCu4. The intensity of the Kondo resonance indicates the strength of the c–fhybridization [ 4,5]. We have checked the temper- ature dependence of the spectral weight at low energies. Asshown in Figs. 2(a)–2(c), the line shape near \|ω\|/similarequal0.3e V is identical within the noise level. Thus, we have subtractedthe constant intensity at ∼0.3 eV from each spectrum and displayed them as colored areas in Figs. 2(d)–2(f).T h e spectral weights for X=Cd and Sn are rapidly suppressed with increasing temperature. In contrast, the spectral weightforX=Mg remains largely unchanged even at T/similarequal190 K. These results indicate the robustness of the Kondo state inYbMgCu 4. In the research of high- Tcsuperconductors, a decrease in the dip minimum intensity at EFis interpreted as an increase in the energy gap in the neighborhood of EF[23,24]. For the spectra of Yb XCu4in Figs. 2(d)–2(f),t h e4 f7/2peak shifts to higher energy with keeping the same dip minimum,demonstrating no gap opening at E F. Previous Hall effect mea- surements, indeed, reveal metallic behaviors of the Yb XCu4 compounds with a healthy density of states [ 25]. Therefore, 235160-3HIROAKI ANZAI et al. PHYSICAL REVIEW B 101, 235160 (2020) FIG. 3. (a) Temperature dependence of energy positions of the Yb2+4f7/2peak. The arrows represent the onset temperature for the saturation of peak energy. The dashed lines indicate the average of the peak energy over the region of the saturation. (b) Temperature dependence of low-energy spectral weight WLE, which is determined by the integration of peak intensity over the colored area of the symmetrized spectra in Figs. 2(d)–2(f). The weight is plotted as a percentage of the total weight at T=8 K. The shaded area indicates the crossover region in WLE. The error bars derived from statistical and reﬂected by noise in the data. the characteristic dip in the symmetrized spectra is indicative of the shift in energy of the Kondo resonance. The low-energy spectral weight WLEis determined by integrating the peak intensity over the energy window of thecolored areas in Figs. 2(d)–2(f). This method is simple and reasonable for extracting W LEfrom different samples with different types of spectral shapes [ 26]. Another method for the background subtraction and its consistency are discussed inAppendix A. Figure 3(b) shows the temperature dependence ofW LEas a percentage of the total weight at T=8K .W e note that WLEforX=Sn reﬂects mainly the development of the 4 f7/2peak because the background-subtracted intensity of the shoulder structure is insensitive to temperature. As tem-perature increases, W LEforX=Cd and Sn rapidly decreases and is less than 30% at T/similarequal150 K. For X=Mg, in contrast, the weight of 65% persists at the same temperature. Theseresults suggest strong hybridization of the 4 felectrons with the conduction electrons in YbMgCu 4[10,11]. The rate of decrease in WLEchanges at ∼78% as indicated by the gray-shaded area in Fig. 3(b). The ratio is quantitatively similar to the temperature region where the saturation of thepeak energy occurs in Fig. 3(a). Therefore, these remarkable changes in the Kondo resonance demonstrate the emergenceof a crossover between the two regimes of Kondo screeningat the higher scale and coherent screening at the lower scale.The crossover temperature is considered to be T cohshown in Fig. 3(a) and Table I. The two relevant energy scales of Tcoh andTKin the heavy-fermion compounds are consistent with the protracted screening behavior predicted by the periodicAnderson lattice model [ 6,7].We next discuss the electronic band structure near E F. The crystal-ﬁeld splitting of 4 fstates in YbRh 2Si2has been observed by angle-resolved photoemission spectroscopy [ 27]. The hybridization of the 4 fstates with the conduction band locally deforms the band structure at their crossing point inmomentum space [ 27–31]. Thus far, the energy dispersion of the crystal-ﬁeld-split 4 fstates as well as that of the conduction bands has not been observed in Yb XCu 4.T h e eightfold degenerate Hunds rule ground state of Yb3+with J=7/2 splits into two Kramers doublets and one quartet in the cubic site symmetry of the C15bstructure [ 32]. The overall splitting energy is estimated to be ∼8m e V[ 33,34]. This value is within the error bars of the data in Fig. 3(a). Even though the low-energy and high-energy tails of the 4 f7/2 peak in Fig. 2may be related to the energy dispersion of the conduction bands, the observed 4 f7/2peak in the momentum- integrated spectra reﬂects the essential features of the Kondoresonance. Under these circumstances, the Kondo resonance develops in the form of a sharp and dispersionless peak below T coh as demonstrated by the peak position and spectral weight in Fig. 3. Moreover, we have conﬁrmed the consistency of the peak width of X=Cd in Appendix B. Such an evolution of the Kondo resonance indicates the formation of coherentheavy quasiparticle bands, which are accompanied by theopening of a hybridization gap at the crossing point of theconduction and 4 fbands [ 27–31]. We assume a reconstruc- tion of the Fermi surface at T cohof Yb XCu4, such as the change from small to large Fermi surfaces with increasing thestrength of c–fhybridization [ 28–30]. The Fermi surfaces and their orbital characters are key ingredients for understandingthe low-energy physics. Further investigations, such asangle-resolved photoemission spectroscopy measurementsare required to fully describe the electronic band structure ofYbXCu 4. III. CONCLUSION We have revealed the temperature dependence of the Kondo resonance in the heavy-fermion compound Yb XCu4 by synchrotron radiation photoemission spectroscopy. The intense peak of the Yb2+4f7/2state is observed near EFin all spectra. The 4 f7/2peak shifts closer to EFand becomes sharper with decreasing temperature. Below Tcoh, the peak with a large spectral weight does not show temperature de-pendence. The peak positions in the limit of zero temperatureare quantitatively consistent with the energy scales of Kondotemperature, demonstrating that the observed 4 f 7/2peak is certainly the Kondo resonance in Yb XCu4. It is interesting to note that Tcohis comparable with the crossover temperature Tmax χto the ground-state Fermi liquid with coherent transport. Our results suggest the development of the coherent heavy-fermion state below T coh. ACKNOWLEDGMENTS We thank Y . Taguchi and K. Mimura for valuable discus- sions. The experiments were performed under the approval ofUVSOR (Proposals No. 28-838, No. 29-556, and No. 30-579). 235160-4TEMPERATURE DEPENDENCE OF THE KONDO RESONANCE … PHYSICAL REVIEW B 101, 235160 (2020) FIG. 4. (a) The Yb2+4f7/2peak in the photoemission spectra of X=Cd is taken from the original data of Fig. 2(b). The black curve represents the temperature-independent integral-type background.(b) The difference spectra between the raw data and the background in panel (a). (c) Comparison of the low-energy spectral weight obtained by two different methods for background subtraction. Theopen diamonds represent the results extracted from the subtraction of the integral-type background W IBG LE. The same data of WLEfor X=Cd in Fig. 3(b) are shown as ﬁlled diamonds. APPENDIX A: BACKGROUND SUBTRACTION In Appendix A, we compare the low-energy spectral weight WLEwith that obtained by another method for the background subtraction. Figure 4(a) shows again the 4 f7/2 peak in the photoemission spectra of X=Cd at T=8K . Here, we adopted the temperature-independent integral-typebackground (IBG) for extracting W IBG LEand plotted it as the black curve [ 35]. This method is widely used for the evalua- tion of the photoemission spectra [ 17,18,20,21]. The differ- ence between the raw data and the background is depictedby the open circles in Fig. 4(b). The high-energy tail of the difference spectra may be derived from the energy dispersionof the conduction band as discussed in the main text. We integrated the intensity of the difference spectra over an energy window of 0 /lessorequalslant\|ω\|/lessorequalslant0.3 eV and then normalized the spectral weight as a percentage of the total weight at T=8K . The temperature dependence of W IBG LEis shown by the open di- amonds in Fig. 4(c)together with WLEextracted from the sub- traction of the constant background. One clearly sees that therate of decrease in W IBG LEandWLEchanges at the same temper- ature. This consistency demonstrates that the choice of back-ground subtraction does not affect the results of our analysis. APPENDIX B: PEAK WIDTH We now examine the peak width of the 4 f7/2state as a function of temperature. Figure 5(a) shows theFIG. 5. (a) Temperature evolution of peak width for X=Cd plotted on a relative energy scale ε. The intensity of difference spectra is normalized to the highest intensity of each spectrum. The energy position of spectral midpoint, deﬁned as energies where the spectral peaks become half in intensity, is shown by the ar-row. (b) Temperature dependence of peak width 2 ×HWHM. The HWHM is determined by the half width at half maximum in the low- energy region ( ε/greaterorequalslant0 eV). The arrow indicates the onset temperature for the saturation of the peak width. The dashed line represents the average of the peak width over the region of the saturation. difference spectra of X=Cd plotted on a relative energy scale εwith respect to the 4 f7/2peak. The intensity is normalized to the highest intensity of each spectrum. Thespectra in the high-energy region ( ε<0 eV) have an al- most identical shape. On the other hand, the spectra inthe low-energy region ( ε/greaterorequalslant0 eV) exhibit a clear tempera- ture dependence: The energy position of spectral midpointrapidly shifts above T/similarequal60 K as shown by an arrow in Fig.5(a). We have determined the peak width by using the half width at half maximum (HWHM) in the low-energy regionbecause the full width at half maximum does not yield thecorrect width due to the asymmetric line shape. The peakwidth 2 ×HWHM is plotted in Fig. 5(b) as a function of temperature. The width decreases with decreasing temper-ature from T=242 to 60 K and then saturates at low temperatures. This trend is in good agreement with the tem-perature dependence of the energy position of the 4 f 7/2peak in Fig. 3(a). The average of the peak width over the region of the saturation is estimated to be 40 meV , which is com-parable to or larger than the energy scale of T K=287 K for X=Cd. The experimental and thermal broadenings have a non- negligible effect on the width of the Kondo resonance [ 36]. Moreover, the effect of a repulsive term describing d–f Coulomb interaction and the scattering processes of pho-toelectrons to phonons or electron-hole excitations makea broadening of spectral features [ 36,37]. We emphasize here that the development of the Kondo resonance at T coh is not only evident from the peak position and spectral weight in Fig. 3, but also evident from the peak width in Fig.5. 235160-5HIROAKI ANZAI et al. PHYSICAL REVIEW B 101, 235160 (2020) [ 1 ] R .I .R .B l y t h ,J .J .J o y c e ,A .J .A r k o ,P .C .C a n ﬁ e l d , A. B. Andrews, Z. Fisk, J. D. Thompson, R. J. Bartlett,P. Riseborough, J. Tang, and J. M. 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PhysRevB.96.054509.pdf	PHYSICAL REVIEW B 96, 054509 (2017) Magnetic ﬁeld induced emergent inhomogeneity in a superconducting ﬁlm with weak and homogeneous disorder Rini Ganguly,1Indranil Roy,1,Anurag Banerjee,2Harkirat Singh,1,†Amit Ghosal,2,‡and Pratap Raychaudhuri1,§ 1Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India 2Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Nadia 741252, India (Received 13 April 2017; revised manuscript received 20 July 2017; published 11 August 2017) When a magnetic ﬁeld is applied on a conventional type-II superconductor, the superconducting state gets destroyed at the upper critical ﬁeld, Hc2, where the normal vortex cores overlap with each other. Here, we show that in the presence of weak and homogeneous disorder the destruction of superconductivity with the magneticﬁeld follows a different route. Starting with a weakly disordered NbN thin ﬁlm ( T c∼9 K), we show that under the application of a magnetic ﬁeld the superconducting state becomes increasingly granular, where regions ﬁlledwith chains of vortices separate the superconducting islands. Consequently, phase ﬂuctuations between theseislands give rise to a ﬁeld induced pseudogap state, which has a gap in the electronic density of states, but wherethe global zero resistance state is destroyed. DOI: 10.1103/PhysRevB.96.054509 I. INTRODUCTION Over the past decade, the notion of emergent granularity has evolved from a theoretical proposition [ 1–3]t oa n alternative paradigm to understand the superconducting statein the presence of strong homogeneous disorder [ 4–10]. Clean s-wave superconductors which are well described by Bardeen- Cooper-Schrieffer (BCS) theory [ 11] are characterized by an energy gap ( /Delta1) in the electronic density of states (DOS) centered around the Fermi level, and two sharp coherencepeaks at the gap edge. Theoretically, /Delta1is the Cooper pairing energy scale which determines the superconducting transitiontemperature, T c, whereas the coherence peaks signify the establishment of long-range phase coherence. Since the late1950s [ 12,13], it was known that /Delta1remains ﬁnite even at strong (nonmagnetic) disorder, leading to the belief thatthe superconducting transition would also remain robustagainst disorder. However, this conclusion is invalidated bythe emergence of granularity. It is now understood that inthe presence of strong disorder, the superconducting statecan segregate into superconducting and insulating regions,where zero resistance is achieved through Josephson tunnelingbetween superconducting islands. Consequently, the super-conducting state can get destroyed through phase ﬂuctuationsbetween the islands, even when the pairing amplitude remainsﬁnite [ 1,14]. Experimentally, this manifests as a pseudogap [15,5,6,16], which persists well above T c, where the zero resistance state is destroyed. Experimental and theoreticalevidence also suggests that at a critical disorder Cooper pairscan eventually get localized, giving rise to an insulator madeout of Cooper pairs [ 17,18]. Another aspect of strongly disordered superconductors, namely, the magnetic ﬁeld induced superconductor-insulatortransition (SIT) [ 19,20], has also attracted considerable atten- tion. Under the application of a magnetic ﬁeld, the ground state indranil.roy@tifr.res.in †Present address: IK Gujral Punjab Technical University Batala Campus, Kahnuwan Road, Batala, Punjab 143505, India. ‡ghosal@iiserkol.ac.in §pratap@tifr.res.inof several strongly disordered superconductors transform into an insulator, with sheet resistance exceeding 109/Omega1. While no consensus has so far emerged on the origin of this insulatorstate, it is now widely accepted that it is related in some way tothe superconducting correlations [ 21–23]. Several theoretical scenarios, such as Coulomb blockade in the emergent granularsuperconducting state [ 24] and boson localization [ 25], as well as theories invoking charge vortex duality [ 26,27], have been proposed to explain this phenomenon. From an experimentalstandpoint it is therefore important to obtain microscopicinformation on the evolution of the superconducting statewith magnetic ﬁeld in order to discriminate between variouspossibilities. Here, using low-temperature scanning tunneling spec- troscopy (STS) we investigate the magnetic ﬁeld evolutionof the superconducting state in a weakly disordered NbN thinﬁlm [ 28]. The sample under investigation is an NbN ﬁlm with T c∼9 K, corresponding to kFl∼4 (where kFis the Fermi wave vector and lis the electronic mean free path) [ 5]. For this kFl, the coherence length [ 29]ξ∼5−10 nm, and the magnetic penetration depth [ 28]λ∼800 nm. It has been shown earlier [28] that by controlling deposition parameters the disorder in NbN ﬁlms can be tuned over a large range, from kFl∼10 to kFl∼0.42. As the disorder is increased the superconductor progressively passes through three regimes: Regime I (10 /greaterorsimilar kFl/greaterorsimilar3) where the superconducting energy gap in zero ﬁeld vanishes at the same temperature where resistance appears; Regime II (3 /greaterorsimilarkFl/greaterorsimilar1), where a pronounced pseudogap state appears above Tc; and Regime III ( kFl/lessorsimilar1), where superconductivity is completely suppressed down to 300 mK.In the present context, “weak disorder” refers to Regime Iwhere the zero-ﬁeld state follows the BCS paradigm. However,it is important to note that in the BCS sense, NbN ﬁlms in general (in all three regimes) are in the strong disorder limit [30], where ξ BCS=¯hvf π/Delta1/greatermuchl(vFis the Fermi velocity). The normal state exhibits a weak negative temperature coefﬁcient of resistance withR(300 K) R(15 K)∼0.78, consistent with earlier reports at this level of disorder [ 31]. The central result of this paper is that when a magnetic ﬁeld ( H) is applied on the sample, the superconducting state becomes inhomogeneous,in a manner similar to what disorder alone would have done 2469-9950/2017/96(5)/054509(14) 054509-1 ©2017 American Physical SocietyRINI GANGULY et al. PHYSICAL REVIEW B 96, 054509 (2017) FIG. 1. Superconducting state in zero ﬁeld. (a) Representative normalized tunneling conductance spectra [ GN(V)v sV] in zero ﬁeld at 450 mK along a 200-nm line. The spectra show a uniform energy gap and a ﬁnite coherence peak at the gap edge over the entire line. (b) Coherence peak height ( GNp) map over a 200-nm ×200-nm area, forming an inhomogeneous structure. (c) Temperature variation of the average superconducting energy gap /Delta1(green squares), and broadening parameter /Gamma1(red circles) obtained by ﬁtting the average GN(V)v sV spectra. The ﬁts to the tunneling spectra are shown in the inset. The light green line is the expected variation of /Delta1(T) obtained from BCS theory. The blue line shows temperature variation of resistance measured on the same sample. The resistance appears exactly at the same temperature, Tc, where the BCS gap vanishes. at a much larger strength. This experimental observation is backed by numerical simulations which show that the ﬂuxtubes enter a disordered superconductor at locations wheredisorder partially suppresses the superconducting correlations.Thus disorder creates a network of weak links where vortices enter, and additionally suppress the superconducting order pa- rameter. The resulting superconducting state contains regions,a few tens of nanometers in size, where the superconductingorder parameter is ﬁnite, separated by chains of vortices wherethe superconducting order is suppressed. Consequently, thesystem exhibits a ﬁeld induced pseudogap that progressivelywidens as the magnetic ﬁeld is increased. II. SAMPLE DETAILS AND EXPERIMENTAL METHODS Sample. The epitaxial NbN thin ﬁlm with thickness ∼25 nm was deposited on single-crystalline MgO (100) substrate using reactive pulsed laser deposition using a KrF excimer laser (248 nm). A pure Nb target was ablated inan ambient N 2atmosphere of 40 mTorr, while the substrate was kept at 600◦C. For the ablation process, laser pulses with energy density 200 mJ /mm2were used with a repetition rate of 10 Hz. To maintain a pristine surface while trans-porting the sample for STS measurements, the sample wastransferred in situ in an ultrahigh-vacuum suitcase with base pressure ∼10 −10Torr and transferred to the low-temperature scanning tunneling microscope (STM) without exposure toair. Transport (and magnetic) measurements were performedon the sample after all STS measurements were completed.The resistance measurements as a function of temperature andmagnetic ﬁeld were performed using conventional four-probetechnique, using a current of 0.5 mA. Scanning tunneling spectroscopy measurements. All STS measurements were performed in a home-built STM [ 32] operating down to 350 mK and ﬁtted with a superconductingsolenoid with maximum ﬁeld of 90 kOe. The STM tip wasmade out of a mechanically cut Pt-Ir tip, which was sharpenedin situ by ﬁeld emission on an Ag single crystal. The tunneling conductance [ G(V)= dI dV\|V] was measured by adding a150-μV , 2-kHz ac voltage to the dc bias voltage ( V) and recording the ac response in the tunneling current ( I)u s i n g standard lock-in technique. For the conductance maps at ﬁxedbias voltage, the ac response in the tunneling current was mea-sured while the tip was rastered over the sample surface. Thefull area spectroscopy was performed by stabilizing the tip atevery point, then momentarily switching off the feedback loopand sweeping Vfrom+5m Vt o −5 mV and recording the ac response in the tunneling current as a function of bias voltage. III. RESULTS A. Zero-ﬁeld superconducting state The tunneling conductance between a normal tip and a superconductor, measured using an STM, provides the mostdirect access to the local density of states in a superconductor.We characterize the zero-ﬁeld state by measuring the tunnelingconductance spectra [ G(V)v sV] over a 200-nm ×200-nm area at 450 mK. The tunneling conductance reveals a uniformenergy gap and the presence of a coherence peak at thegap edge over the entire area. The representative spectraalong a 200-nm line are shown in Fig. 1(a). To look at the spatial variation of the zero bias conductance (ZBC) and thecoherence peak height we normalize the tunneling spectrausing the tunneling conductance well above /Delta1, i.e.,G N(V)= G(V) G(V=3.5 meV).GN(0) shows a very narrow distribution sharply peaked at 0.05 consistent with a fully gapped superconducting state, as described below. On the other hand the coherencepeak height ( G Np), extracted from the average of the maxima ofGN(V) at positive and negative bias voltages, shows a large spatial variation. It has been demonstrated from quantumMonte Carlo simulations that in a disordered superconductorG Npprovides a measure of the local superconducting order parameter [ 33]. We observe that GNpvaries smoothly from 1 . 1t o1 . 8[ F i g . 1(b)] forming an inhomogeneous structure varying tens of nanometers on the length scale. However, thecoherence peak is present at all locations showing that thesuperconducting order parameter is ﬁnite at all points. 054509-2MAGNETIC FIELD INDUCED EMERGENT INHOMOGENEITY . . . PHYSICAL REVIEW B 96, 054509 (2017) FIG. 2. Superconducting state in magnetic ﬁeld. (a)–(c) Conductance maps at 40, 60, and 75 kOe respectively, taken at 450 mK at ﬁxed bias 2.2 mV over an area of 200 nm ×200 nm. The red dots show the position of the vortices obtained from the local minima in the conductance. (d)–(f) Normalized tunneling spectra along the green lines shown in the conductance maps (a)–(c), respectively, which pass through the center of the vortices. The vertical black dashed line denotes the center of the vortex. (g)–(i) Representative spectra for three different ﬁelds, respective ly, at the center of the vortex core (black) and away from the core (violet) corresponding to the black and violet dashed lines shown in panels (d)-(f). In contrast to a conventional superconductor, we observe a soft gap at the core of the vortices. We now turn our focus to the temperature variation of /Delta1. To obtain /Delta1,w eﬁ tt h e GN(V)v sVspectra averaged over the entire area [Fig. 1(c)] with the tunneling equation, G(V)∝∫∞ −∞Ns(E)[−∂f(E−eV) ∂E]dE.H e r e eis the electronic charge, f(E) is the Fermi Dirac distribution function, and Ns(E)=\|E\|+i/Gamma1√ (\|E\|+i/Gamma1)2−/Delta12is the BCS quasiparticle DOS, where a phenomenological broadening parameter /Gamma1is incorporated to take into account nonthermal sources of broadening inthe DOS [ 34]. Comparing with the temperature variation of resistance, we observe that /Delta1follows the usual BCS temperature dependence and vanishes at T c./Gamma1on the other hand is nearly temperature independent and varies between0.23 and 0.33. The dominant role of /Gamma1is to broaden the coherence peak and suppress its height. The relativelylarge value of /Gamma1obtained by ﬁtting the average spectra reﬂects the presence of regions where the coherence peak issuppressed.B. Emergence of granular superconducting state in magnetic ﬁeld When a magnetic ﬁeld is applied, the ﬁeld enters a type-II superconductor in the form of vortices comprising a circulatingsupercurrent and enclosing a magnetic ﬂux quantum, /Phi1 0= h/2e. At the center of each vortex is the vortex core, where the superconducting order is destroyed and the circulatingsupercurrent is zero. In an STS experiment, these cores canbe identiﬁed by recording the conductance map with biasvoltage close to the coherence peak, where the vortices appearas low-conductance points owing to the suppression of thecoherence peak [ 35]. To identify the vortex cores in our sample, we record the conductance maps at 450 mK in differentmagnetic ﬁelds ( H/22a5ﬁlm surface), at a ﬁxed bias voltage of 2.2 mV [Figs. 2(a)–2(c)]. The vortices, corresponding to the local minima in the conductance, are shown as red dots.In this context, we would like to note that with increasing 054509-3RINI GANGULY et al. PHYSICAL REVIEW B 96, 054509 (2017) ﬁeld, we also observe a decrease in the maximum value of conductance even far from the vortex core. This is a directresult of the orbital supercurrent, which in an extreme type-IIsuperconductor ( λ/greatermuchξ), extends well beyond the core of the vortex up to a length of the order of λ, and causes partial suppression of the coherence peaks. Therefore the upper limitof the color scale in Figs. 2(a)–2(c)is adjusted to the maximum value of the conductance within the area and the depth of thecolor is kept the same in all three panels, such that each imageis effectively normalized to the maximum conductance valuewithin the area. The area was kept the same for differentﬁelds and small drifts ( <10 nm) were corrected by looking at topographic features of the surface. To avoid large driftsduring large temperature sweeps, the sample was ﬁrst cooledin zero ﬁeld and the data were taken by gradually rampingup the ﬁeld and stabilizing at speciﬁc values. In principle,this zero-ﬁeld-cooled (ZFC) protocol could suffer from thedrawback that in a disordered superconductor the ﬂux densityat the center of the sample might be lower than the appliedﬁeld owing to strong pinning. To verify whether the magneticﬁeld is uniformly entering at the ﬁelds where we performedthe STS measurements, careful magnetization measurementswere performed on the ﬁeld-cooled (FC) and ZFC state as afunction of temperature. We observed that the magnetizationof the two states becomes indistinguishable above 3 kOe (seeAppendix A), conﬁrming that the ﬂux completely penetrates the sample above this ﬁeld. At 40 kOe we observe that chainsof vortices form a laminar structure, separating regions whereG(V=2.2 mV) is high. As the ﬁeld is increased, further entry of vortices progressively widen the regions with suppressedcoherence peak, and the laminar structure becomes denser,thereby shrinking the puddles where the coherence peak ishigh. We would like to note that at 60 and 75 kOe, whencounting the number of vortices we observe that multiplyingthe number of vortices with /Phi1 0does not account for the entire ﬂux expected to pass through the area for the appliedmagnetic ﬁeld. This discrepancy arises because of our inabilityto account for two very closely placed vortices which appearas a single patch where the coherence peak is suppressed. Thisis consistent with numerical simulations that we present later. We now investigate the nature of the vortex core, by recording the tunneling spectra along a line passing throughthe center of a vortex [Figs. 2(d)–2(f)]. Due to the uncertainty in the number of vortices in every patch at 60 and 75 kOe adirect comparison of these line scans is difﬁcult. Nevertheless,all three line scans display a surprising common feature. In aconventional superconductor, the core of an Abrikosov vortexbehaves like a normal metal, where the tunneling spectrum iseither ﬂat [ G N(V)∼1] or displays a small peak at zero bias owing to the formation of Caroli–De Gennes–Matricon [ 36] (CDM) bound states in very clean samples (see Appendix B). In contrast here, we observe that a soft gap continues to surviveeven at the center of the vortex core [Figs. 2(g)–2(i)] although the coherence peak gets suppressed [ 37]. The suppression of the coherence peak suggests that the superconducting orderparameter is suppressed in the core of the vortex even thoughthe pairing amplitude remains ﬁnite. Since the proliferation of vortices locally suppresses the superconducting order, the inhomogeneous distribution ofvortices produces boundaries of suppressed superconductivitywhich separate the superconducting patches. To visualize this superconducting state more clearly, we measured the tunnelingconductance spectra over the same area on a 32 ×32 grid in different magnetic ﬁelds. Figures 3(a)–3(d) show the G N(0) maps at different ﬁelds up to 75 kOe corresponding to the samearea as in Figs. 2(a)–2(c). We observe that with increase in ﬁeld the superconducting state develops large inhomogeneity,forming regions where G N(0) is large and regions where GN(0) is small. This is also reﬂected in the distribution ofGN(0), which with increase in ﬁeld develops large tails [Figs. 3(e)–3(h)]. Figures 3(i)–3(k) show the coherence peak height maps corresponding to the same ﬁelds. We observean inverse correlation of the G Npmaps with the GN(0) maps ,implying that in regions where GN(0) is large, the coherence peak is suppressed. The anticorrelation is alsoapparent from the two-dimensional histogram of G N(0) and GNpwhich shows a negative slope over a large scatter, which suggests that the anticorrelation is not perfect. We quantify theanticorrelation using the cross correlator, I=1 n/summationdisplay i,j/parenleftbig Gi,j N(0)−/angbracketleftGN(0)/angbracketright/parenrightbig/parenleftbig Gi,j Np−/angbracketleftGNp/angbracketright/parenrightbig σ0σp, where σ0andσpare the standard deviations in the values of GN(0) and GNp, respectively; i,jrefer to the pixel index of the image; and nis the total number of pixels. We obtain I∼−0.15 to −0.2 where I=−1 implies perfect anticor- relation. This is qualitatively similar to earlier observationin strongly disordered NbN samples in zero ﬁeld [ 7]. The weak anticorrelation suggests that G N(0) is probably not governed by the local superconducting order parameter alone.As expected, the vortices (shown in red rods) are preferentiallylocated in the regions where G N(0) is high. C. Field induced pseudogap state We now investigate the temperature evolution of the super- conducting state in a magnetic ﬁeld. Figures 4(a)–4(e)show the temperature variation of the average GN(V)-Vspectra over a 200-nm ×200-nm area at different magnetic ﬁelds along with the temperature variation of resistance. As the magnetic ﬁeld isincreased we observe that a soft gap in the tunneling spectrumcontinues to persist up to a temperature T ∗, well above Tc(H). (For consistency, we deﬁne T∗as the temperature where GN(0) is 95% of the normal state value.) Here, Tc(H) is deﬁned as the temperature where the resistance is 0 .05% of its normal state value (for more details, see Appendix C). This is analogous to the pseudogap state observed earlier in zero ﬁeld in stronglydisordered superconductors [ 4,5,7,15]. Plotting T c(H) andT∗ in the H-Tparameter space [Fig. 4(f)], we observe that the pseudogap state becomes progressively wider as the magneticﬁeld is increased. To rule out the possibility that the observedpseudogap is caused from a local distribution in temperature atwhich /Delta1→0, we have also separately tracked the temperature dependence of the tunneling spectra at locations where thecoherence peak at low temperature is ﬁnite and locationswhere the coherence peak is suppressed. Figures 4(h)–4(i) show the temperature evolution corresponding to two suchlocations [Fig. 4(g)] at 40 kOe. We observe that at both 054509-4MAGNETIC FIELD INDUCED EMERGENT INHOMOGENEITY . . . PHYSICAL REVIEW B 96, 054509 (2017) FIG. 3. Magnetic ﬁeld induced granularity. (a)–(d) ZBC [ GN(0)] maps for ﬁelds 0, 40, 60, and 75 kOe, respectively, over the same 200-nm ×200-nm area at 450 mK, obtained from area spectroscopy over a 32 ×32 pixels grid. The red dots show the positions of the vortices. (e)–(h) Distribution of GN(0) for 0, 40, 60, and 75 kOe, respectively. With increasing ﬁeld the distributions develop large tails, signifying emerging inhomogeneity with ﬁeld. (i)–(k) Coherence peak height ( GNp) maps for 40, 60, and 75 kOe. The upper limit of the color scale is set to the maximum value of GNpat that ﬁeld. (l)–(n) Cross-correlation histograms between GN(0) and GNpfor corresponding ﬁelds, showing inverse correlation between the two quantities; in all these three histograms the bin size is adjusted to segment GN(0) and GNpinto 15 and 18 bins, respectively, over their plotted range. locations GN(0)→1, at the same temperature, conﬁrming that the pairing amplitude uniformly disappears at the sametemperature [Fig. 4(j)]. IV . COMPARISON WITH NUMERICAL SIMULATIONS We next carry out numerical simulation in order to develop further insight into our experimental ﬁndings. We describeour system through an attractive Hubbard Hamiltonian, whichin the presence of disorder and applied magnetic ﬁeld has theform H=−t/summationdisplay /angbracketlefti,j/angbracketright,σeiφijc+ iσcjσ−\|U\|/summationdisplay iˆni↑ˆni↓+/summationdisplay i,σ(Vi−μ)ˆniσ, (1) where ciσ(c+ iσ) annihilates (creates) an electron with spin σ at site iof a two-dimensional square lattice, ˆniσ=c+ iσciσ is the occupation number of site iwith spin σ, and the phases φij=π /Phi10/integraltextj i¯A·dlare the Peierls factor of an applied orbital magnetic ﬁeld. We use the Landau gauge ¯A=Bxˆy for all our calculations. The attraction Uinduces s-wave 054509-5RINI GANGULY et al. PHYSICAL REVIEW B 96, 054509 (2017) FIG. 4. Field induced pseudogapped state. (a)–(e) Temperature variation of the average GN(V)-Vspectra at 0, 20, 40, 60, and 75 kOe, respectively, along with the temperature variation of resistance. The vertical dashed lines correspond to Tc, where resistance appears and T∗, where the pseudogap in the density of states disappears. At H=0 these two happen at the same temperature. The range of the temperature axes in all plots has been kept the same for visual comparison. (f) TcandT∗are plotted on the H-Tspace, which shows that the pseudogap state widens as the ﬁeld is increased. (g) Conductance map at ﬁxed bias; V=2.2 mV at 40 kOe at 450 mK. The blue and green boxes show two representative areas where the conductance is high and low, respectively. (h),(i) Average GN(V)-Vfor different temperatures inside the blue and the green boxes, respectively. (j) Temperature dependence of GN(0) for the average spectra inside the blue and green box, respectively; we observe that in both cases GN(0) goes to 1 at the same temperature. superconductivity in the system. The disorder at site iis given byVi, which is chosen as an independent random variable from a uniform distribution between −VandVwhich quantiﬁes the disorder strength V. The chemical potential μﬁxes the average density ( ρ=(1/N)/summationtext iˆni, where ˆniis the occupancy of the i-th site and Nis the total number of sites) which we ﬁx atρ=0.875 for all our calculations. We carry out a fully self-consistent mean ﬁeld analysis of ( 1)u s i n gt h e Bogoliubov–de Gennes (BdG) technique following Refs. [ 38– 40]o na3 6 ×36 two-dimensional grid. Considering the lattice spacing of NbN, which is ∼4.4˚A, the size of our simulation would translate to an area of 16 nm ×16 nm. The small size of the simulation is necessitated due to available computational resources and results in two caveats. First, we need to use a large value of \|U\|, beyond the weak-coupling BCS value, to keep the coherence length well within the system size. We use\|U\|=1.2t,which in the clean limit gives a coherence length, ξ c∼10−12 lattice spacing [ 38]. This is further reduced in the presence of disorder yielding an operational coherencelength, ξ∼5−6 lattice spacing. This is also consistent with the dirty limit relation, ξ∼(ξ cl)0.5. Secondly, due to the small simulation area the effective magnetic ﬁeld for a given number of vortices ( n) is much larger than the experimental value.While this drawback prevents quantitative comparison with the experimental data, it has been shown through several studies that the ﬁnite size simulations capture the broad qualitative features of disordered superconductors [ 1,2,24,33,38]. Since these simulations are restricted to T=0, we compare our simulations with experimental data taken at the lowest tem- perature, 450 mK. (For further details see, Appendix D.) We ﬁrst investigate the zero-ﬁeld state obtained from the simulations. Figure 5(a)shows the single-particle DOS, D(E), normalized to its value at E=−0.2t,averaged over the entire lattice for V=0.5t. The average D(E) shows a fully formed gap and sharp coherence peaks consistent with the zero-ﬁeldtunneling spectra at 450 mK. Figure 5(b) shows the spatial variation of D(E) at the coherence peak, D p. We observe that Dpshows large spatial variation forming an inhomogeneous structure similar to that in Fig. 1(b). To conﬁrm that this disorder strength Vis indeed appropriate for our experiments, we compare the width of the normalized distribution of GNp at 450 mK in zero ﬁeld, deﬁned as ˜GNp=GNp−GMin Np GMax Np−GMin Npwith the corresponding normalized distribution of Dp, namely, ˜Dp=Dp−Dmin p Dmaxp−Dminp[where Gmin Np(Dmin p) and Gmax Np(Dmax p)a r e 054509-6MAGNETIC FIELD INDUCED EMERGENT INHOMOGENEITY . . . PHYSICAL REVIEW B 96, 054509 (2017) FIG. 5. Simulation of the superconducting state in zero ﬁeld. (a) Density of states, D(E), in the absence of vortices averaged over a 36×36 lattice showing a fully formed gap and sharp coherence peaks. (b) Spatial variation of Dpcalculated for 36 ×36 lattice. (c) Normalized distribution of ˜GNpobtained from experiments at 450 mK in zero ﬁeld, and ˜Dp. The distribution of ˜GNphas a standard deviation, σ˜GNp=0.1358, whereas the distribution of ˜Dphas a standard deviation, σ˜Dp=0.141. the minimum and maximum of GNp(Dp)]. The distributions [Fig. 5(c)] have similar width as measured from standard deviations from the mean value showing that we are workingat comparable disorder strength. We now track the evolution of the superconducting state with magnetic ﬁeld. Since the vortex simulations are carriedout by repeating the simulation box periodically, ncan only be even [ 39]. Figures 6(a)–6(c) show the spatial variation of the phase of the superconducting order parameter φfor n=2, 4, and 6; the color scale corresponds to the local pairing amplitude deﬁned as \|/Psi1 i\|=\|U\|/angbracketleftci↓ci↑/angbracketright. The positions of the vortices can be identiﬁed from the locations where φ twists around a point and \|/Psi1\|∼0. The spatial variation of D(0) and Dpcorresponding to these ﬂux ﬁllings shown in Figs. 6(d)–6(i) qualitatively captures all the broad features observed in our experiment. The presence of the vortex resultsin a local increase in D(0). From the map of D p,f o rn=2 we clearly see that the spatial variation of Dpis anticorrelated withD(0); i.e., Dpis small at locations where D(0) is large. This anticorrelation is weaker for n=4, and not discernible forn=6. This is in contrast with the experimental data where the anticorrelation is nearly independent of magnetic ﬁeld.This disagreement with experiment is possibly due to the veryhigh effective magnetic ﬁeld for n=4 and n=6 due to the small size of our simulation. Furthermore, we observe fromtheD(0) maps that the regions with large D(0) around two closely located vortices coalesce to form one continuous largerpatch. (The length scale of these patches is of the order of thecoherence length ξin our simulation.) Therefore, it is likely that at high ﬁelds we are unable to resolve all the individualvortices from the conductance images, which accounts forthe apparent nonconservation of magnetic ﬂux in our sample.We also observe that the distribution of D(0) progressively increases with increasing n[Figs. 6(j)–6(l)] and forms long tails consistent with experiments. Finally, we dwell on the issue of the soft gap observed inside the vortex core in our experiments. In Fig. 6(k) (inset) we compare the average D(E) close to the center of the vortices and at regions far from it, for n=4. Close to the center of the vortices the coherence peak is completely suppressed but asoft gap continues to survive. For a consistency check we have performed the same calculations without any disorder ( V=0) (see Appendix D). In that case, we realize an Abrikosov lattice commensurate with the lattice geometry, and D(E)s h o w sa large zero energy peak inside the vortex core consistent withthe CDM bound state [ 36]. To understand physically the origin of this behavior we note the circulating supercurrent densityaround a vortex, J∝(1/r), where ris the radial distance from the center of the vortex. In a clean superconductor,the normal vortex core appears below a limiting r, where the increase in kinetic energy of the Cooper pair exceeds thepairing energy, 2 /Delta1, and destroys the superconducting pairing. In the presence of disorder a completely different scenariocan emerge. Here, disorder scattering reduces the superﬂuidstiffness, J s, making the superconductor susceptible to phase ﬂuctuations [ 28]. The survival of the soft gap in the vortex core, in our opinion, is strongly tied to the phase ﬂuctuationsof the order parameter due to the inhomogeneous backgroundthat depletes the superﬂuid stiffness in the core regions, butkeeps the pairing amplitude ﬁnite. V . SUMMARY AND OUTLOOK The emerging physical picture from the experiments and the simulations is as follows: The random disorder potential makesthe superconducting order parameter spatially inhomogeneouseven in the absence of magnetic ﬁeld. As a result, the ﬂux tubesfrom the applied ﬁeld thread the system through locationswhere the local amplitude of the order parameter, \|/Psi1\|,i s low. Such spatial organization lessens the energy cost byaccumulating the phase-twist in regions of low \|/Psi1\|.T h i s naturally makes the ﬁeld induced vortex lattice aperiodic, andﬂux tubes wipe out remnants of pairing amplitude in a regionof size ∼ξaround vortex centers. Such local annihilation of superconducting correlations with magnetic ﬁeld introducesgranularity in the superconducting state even with low disorderstrengths, in a manner similar to what disorder alone wouldhave done for much larger strengths. Consequently, thepseudogap is observed in the region of H-Tparameter space where Cooper pairs continue to survive even when the zero 054509-7RINI GANGULY et al. PHYSICAL REVIEW B 96, 054509 (2017) FIG. 6. Simulation in the presence of vortices. (a)–(c) The spatial variation of φshown as arrows for n=2, 4, and 6 vortices on the 36 ×36 lattice. The colors of the arrows stand for the strength of superconducting order parameter \|/Psi1\|on a scale of 0 to 1 on the lattice. The positions of the vortices can be identiﬁed from the locations where φtwists around a point and \|/Psi1\|has a value close to zero. (d)–(f) The spatial variation ofD(0) corresponding to n=2, 4, and 6 vortices on the 36 ×36 lattice. (g)–(i) The spatial variation of Dpcorresponding to n=2, 4, and 6 vortices on the 36 ×36 lattice. (j)–(l) Distribution of D(0) for n=2, 4, and 6 vortices, respectively. With increasing ﬁeld the distributions develop large tails, signifying emerging inhomogeneity with ﬁeld. The inset of (k) shows the D(E) as a function of E/t forn=4a v e r a g e d over regions close to the center of the vortices (black) and for regions far from the vortices (blue). 054509-8MAGNETIC FIELD INDUCED EMERGENT INHOMOGENEITY . . . PHYSICAL REVIEW B 96, 054509 (2017) resistance state is destroyed due to phase ﬂuctuation between superconducting puddles. This is consistent with earlier planartunneling measurements on Pb-Bi ﬁlms [ 41]. Our results provide valuable clues to understanding the magnetic ﬁeld induced superconductor-insulator transition inmuch more strongly disordered samples, which are very closeto, but on the superconducting side of, disorder driven SIT.There, even the zero-ﬁeld state consists of regions where \|/Psi1\| is completely suppressed such that the superconducting stateis composed of superconducting puddles that are Josephsoncoupled through insulating regions. When a magnetic ﬁeldis applied, the superconducting puddles will further fragmentthrough vortex proliferation, until they reach a critical sizewhere Coulomb blockade makes it energetically unfavorablefor the current to pass through the superconducting islands[24]. At this point we would expect to see a transition from a su- perconductor to an insulatorlike behavior in transport measure-ments. Therefore, we propose that the disorder and magneticﬁeld driven SITs are both manifestations of the same micro-scopic phenomenon: the granularity that emerges naturally inthe superconducting state. Microscopic validation of this sce-nario could be obtained through STS measurements on morestrongly disordered superconductors at very low temperatures. ACKNOWLEDGMENTS We would like to thank Jim Valles, Nandini Trivedi, Lara Benfatto, and Vikram Tripathi for useful discussions. The workwas supported by the Department of Atomic Energy, Govern-ment of India and the Department of Science and Technology,Government of India (Grant No: EMR/2015/000083). R.G. and I.R. performed the measurements. I.R. and R.G. analyzed the data. H.S. prepared the sample. A.B. carriedout the simulations under the supervision of A.G. P.R.conceived the problem and supervised the experiments. P.R.and A.G. wrote the paper. All authors discussed the resultsand commented on the manuscript. R.G. and I.R. contributedequally as joint ﬁrst authors of this paper. APPENDIX A: FLUX PENETRATION IN THE SUPERCONDUCTOR IN THE ZERO-FIELD-COOLED STATE In a strongly pinned superconductor, when a magnetic ﬁeld is applied after the superconductor is cooled to lowtemperatures in zero magnetic ﬁeld [i.e., the zero-ﬁeld-cooled(ZFC) state], the entry of magnetic ﬂux is hindered by thepinning potential. Consequently, the ﬂux density graduallydecays from the edge towards the center of the superconductor,and the resulting ﬂux density gradient is determined by thelocal critical current density [ 42]. In contrast, when the sample is cooled from above T cin the presence of a magnetic ﬁeld [i.e., the ﬁeld-cooled (FC) state] the presence of strong pinningtraps the magnetic ﬂux threading the sample in the normal statein the form of vortices, producing a nearly uniform ﬂux densityproﬁle, and a magnetization ( M) that is higher than in the ZFC state. As the magnetic ﬁeld is gradually increased towardslarger values, the critical current density of the superconductordecreases and the difference between the FC and ZFC statebecomes smaller. FIG. 7. Flux penetration in the NbN ﬁlm. Temperature depen- dence of magnetization of the NbN ﬁlm measured using the FC and ZFC protocol. The magnetization in the FC state is close tozero showing a nearly complete ﬂux penetration. Above 3 kOe the magnetization values of the FC and ZFC state are identical within experimental resolution. The substrate contribution in themagnetization has been subtracted from all curves. To assess the ﬂux penetration in our sample we performed careful M-Tmeasurements in the ZFC and FC state for dif- ferent magnetic ﬁelds using a SQUID magnetometer (Fig. 7). The ZFC state is created by applying the magnetic ﬁeld aftercooling the sample to the base temperature (1.8 K) in zero ﬁeld.The FC state is created by applying the ﬁeld at 15 K and coolingthe sample to the base temperature in the magnetic ﬁeld. TheM-Tmeasurements are carried while warming up the sample from this initial ZFC or FC state. At 10 Oe the ZFC curve showspronounced diamagnetic response whereas the FC curve is ﬂatand very close to zero as expected for a strongly pinned type-IIsuperconductor. However, as the magnetic ﬁeld is increasedthe difference between the FC and ZFC curves progressivelydecreases, and at 3 kOe the two become indistinguishablewithin experimental resolution. Thus beyond this ﬁeld the ﬂuxcompletely enters the ZFC state and the role of ﬂux pinningon the entry of ﬂux in the ZFC state is negligible. APPENDIX B: COMPARISON BETWEEN ABRIKOSOV VORTEX CORES AND THE VORTEX CORE IN DISORDERED NbN Here we compare the vortex core in disordered NbN with the Abrikosov vortex core in a clean single crystal ofNbSe 2(Tc∼7.2 K). Figures 8(a) and8(d) show the images of the vortices acquired at 450 mK in NbN and in a pureNbSe 2crystal, respectively. Figures 8(b) and8(e) show the normalized tunneling spectra along a line passing throughthe center of the vortex for the two samples. For NbSe 2 we observe that for the normalized tunneling spectrum atthe center of the vortex G(V)/greaterorequalslant1 at all biases and the spectrum shows a zero bias conductance peak associated withCaroli–de Gennes–Matricon (CDM) state [ 36,43] [Fig. 8(f)]. In contrast, at the center, the normalized tunneling spectrumfor NbN shows a pseudogap [Fig. 8(c)], characterized by a suppression of the coherence peak and a soft gap characteristicof superconducting pairing. 054509-9RINI GANGULY et al. PHYSICAL REVIEW B 96, 054509 (2017) FIG. 8. Comparison of vortex core in NbSe 2and NbN. (a) Conductance map over 200-nm ×200-nm area of NbN at 40 kOe at ﬁxed bias voltage [ G(V=2.2 mV)]. The vortices are shown as red dots. (b) Normalized tunneling spectra [ GN(V)-V] along the green line on conductance map in (a) going through the center of one vortex core; the black line corresponds to the spectrum at the vortex core. (c) Representative spectra for NbN at the center of the vortex core (black) and a point away from the core (violet). (d) Conductance map over a 250-nm ×85-nm area of pure NbSe 2at 7 kOe at ﬁxed bias voltage [ G(V=1.3 mV)]. The dark regions are the vortices with lower values of conductance forming a hexagonal Abrikosov lattice. (e) Tunneling spectra [( GN(V)-V)] along the green line on conductance map in (d) going through one vortex core of NbSe 2; dark black line corresponds to the spectrum at the vortex core. (f) Representative spectra for NbSe 2at the center of the vortex core (black) and a point away from the core (violet). APPENDIX C: CRITERION FOR THE UPPER CRITICAL FIELD FROM TRANSPORT MEASUREMENTS While determining Hc2(T) [or equivalently, Tc(H)] from transport measurements different criteria are used in theliterature. While sometimes it is deﬁned as the locus of pointswhere the resistance drops to 90% of its normal state value,in other instances it is deﬁned as the locus of points wherethe resistance falls essentially below the measurable limit.Other criteria, such as 50% of the normal state value, arealso occasionally used. To determine the most appropriate criterion in our context, we measured the diamagnetic shielding response of thesuperconducting ﬁlm using a two-coil mutual inductancetechnique [ 44,45]. In this technique the superconducting ﬁlm is sandwiched between a quadrupolar primary coil and a dipolarsecondary coil and the mutual inductance ( m) is measured between the two [Fig. 9(a)]. Below the superconducting tran- sition, the superconducting ﬁlm partially shields the magneticﬁeld produced by the primary coil from the secondary coil andthe real part of the mutual inductance ( m /prime) decreases, signaling the onset of the diamagnetic response.Figure 9(b) shows the m/prime-H(upper panel) and R-H(lower panel) measured at different temperatures. To measure m/primewe use an ac excitation ﬁeld with an amplitude of 3.5 mOe and afrequency of 31 kHz. We observe that the onset of ac shieldingresponse coincides with the ﬁeld where the resistance dropsto 0.05% of its normal state value, below which the resistance drops below our lower measurable limit. Consequently, we de-ﬁne this ﬁeld as our upper critical ﬁeld [ 46]. We conclude that the broad transition region above this ﬁeld observed in R-H measurements consists of phase ﬂuctuating superconductingpuddles where the global superconducting order is destroyed.The same locus of points in the H-Tparameter space can also be obtained by measuring T c(H)f r o m R-Tmeasurements in constant magnetic ﬁelds, where Tc(H) is deﬁned as the temperature where the resistance drops to 0 .05% of its normal state value [Fig. 9(c)]. APPENDIX D: DETAILS OF NUMERICAL SIMULATIONS General features of simulations. We consider our two- dimensional s-wave superconductor to lie on the xy-plane and represent it as an attractive Hubbard model, and add to it 054509-10MAGNETIC FIELD INDUCED EMERGENT INHOMOGENEITY . . . PHYSICAL REVIEW B 96, 054509 (2017) (a) (c) (b) FIG. 9. Comparison of magnetic ﬁeld variation of resistance and diamagnetic shielding response. (a) Schematic diagram of the two- coil mutual inductance setup; the superconducting ﬁlm is sandwiched between a quadrupolar primary coil and a dipolar secondary coil. (b)Magnetic ﬁeld variation of m /prime(upper panel) and resistance (lower panel) at different magnetic ﬁelds; the vertical dashed lines show the onset of the diamagnetic shielding response which coincide with theﬁeld where the resistance goes below our measurable limit (at 0 .05% of its normal state value). The resistance is plotted in log scale for clarity. (c) The loci of H c2(T)[ o rTc(H)] in the H-Tparameter space fromm/prime-H,R-H,a n dR-Tmeasurements. nonmagnetic random potential on lattice sites as disorder. This is then subjected to an orbital magnetic ﬁeld, Bˆz, which is set up by using vector potential in the Landau gauge ¯A=Bxˆy. We carry out our numerical simulation of this system on asimulation box of size L x×Ly, (linear dimension is expressed in terms of the number of the lattice spacing). While themicroscopic BdG calculation is performed on a simulationbox mentioned above, we repeat such simulation box m x(my) times along the ˆx(ˆy) direction, so that the BdG calculations can be performed on each of them, which are then combinedthrough Bloch translation to produce results on a much biggersystem of size m xLx×myLy[39]. Note that the idea of repe- tition of a simulation box for a periodic vortex lattice in a cleansystem has an exact parallel of such method of repetition of aunit cell in a solid to describe energy band structure. In order torepresent a bulk system, we get rid of boundary effects imple-menting periodic boundary conditions in both xandydirec- tions and allow each simulation box to contain an even numberof ﬂux quanta. This is because a superconducting ﬂux quantumhas the half strength of a “regular” ﬂux quantum. It is thus theﬂux that is directly set in our simulation, and not the magneticﬁeld strength. The strength, thus, can be tuned either by chang-ing discretely the (even) number of ﬂux quanta through thesimulation box, or by changing the size of the simulation box. Chosen parameters. Using presently available computa- tional resources, BdG simulations for superconductors arerestricted to relatively small system sizes. Therefore, the parameters for such simulations have to be carefully chosenso that they represent the actual experimental system and yetthe resulting physical quantities such as ξremain within the size of the simulations. Here we explain the rationale of thechosen parameters in our simulation. (1)Choice of \|U\|.A small value ( \|U\|/t < 1) is desired for representing a truly weakly coupled s-wave superconductor. However, a small \|U\|makes the coherence length ξlarge. Because our numerical resources in the presence of disorder(as discussed below) limit the simulation box to 36 ×36 (in the unit of lattice spacing in both xandydirections), we choose \|U\|=1.2twhich leads to a clean limit coherence length, ξ c∼10–12 lattice spacing [ 38]. In order to track the spatial reorganization, it is important that the linear dimensionof our simulation box is at least a few times the coherencelength. But we emphasize that a small tuning of \|U\|does not change any of our qualitative claim. We have checked this bymaking some test runs for \|U\|=1.5, 2.0. (2)Choice of ρ.The choice of ρ=0.875 is also is decided by a trade-off. The strength of superconductivity(say, the magnitude of pairing amplitude) in the clean systemis maximum at half ﬁlling (i.e., ρ=1.0) for the attractive Hubbard model we study. However, this model, in the absenceof disorder, produces a doubly degenerate ground state withordering in the charge density wave (CDW) channel competingwiths-wave superconductivity. While disorder wipes out the global CDW, a local and short-ranged CDW orderingpersists. In our calculations, we wanted to stay away fromsuch competing orders, as we focused only on the orbital ﬁeldeffects on disordered superconductors. Thus we had to useρ< 1. Pairing amplitude would be very weak even without disorder and magnetic ﬁeld, if we choose too small a value forρ. The chosen value ρ=0.875 is a reasonable trade-off, and for this same reason disordered superconductivity is widelystudied using this value [ 38,47,48]. (3)Choice of V . The disorder strength, V, in our numerical simulation is chosen by matching the width of the distributionof˜D pfrom theory with the distribution of ˜GNpfrom our experiment. Note that both these distributions will reduce toδfunctions at the corresponding BCS values in the absence of disorder. The width of these (normalized) distributionsincreases with disorder strength, and thus the width is taken tobe the signature for the match of the extent of disorder. (4)Normalization of the density of states. Because we carry out our numerics on a lattice, the superconducting DOSdoes not become ﬂat at energy scales larger than /Delta1.O nt h e other hand, for a tight binding model the DOS on lattice evenwithout Hubbard attraction is not ﬂat. It is understood thatsuperconductivity arises by opening up a gap in the DOSin a tight binding model at the chemical potential. Whilethis reorganizes the structure of DOS at low \|E\|, it remains unchanged for large E. We found that for our model parameters the threshold \|E\|beyond which the DOS remains unaffected by superconductivity is roughly 0 .2t, and hence it is used for normalization. We avoided normalization by the value of DOSat positive E, because of the presence of a close-by Van Hove singularity for our parameters. The singularity falls within thegap for V=0 [see Fig. 5(a)], and does not create trouble, but the situation is more complex in the presence of ﬁeld with Van 054509-11RINI GANGULY et al. PHYSICAL REVIEW B 96, 054509 (2017) FIG. 10. V ortex simulation in a clean superconductor. (a) Surface plot of the pairing amplitude \|/Psi1\|on a scale of 0 to 1 in a square simulation box illustrating the vortex. (b) Phase of the superconducting order parameter in the simulation box. (c) Local density of states for the vortex core region showing the CDM peak (black) and at regions away from the vortex core region (blue). Hove singularity and partial gap ﬁlling. Thus we renormalize the DOS by its value at E=−0.2t. Simulation of clean systems ( V=0).Focusing on a clean system, we choose to work with a rectangular simulation boxof size L×2Lthat contains two superconducting ﬂux quanta (leading to two vortices through it). In this case, the magneticﬁeld is tuned by changing L. Our construction thus forces a square vortex lattice. This commensurability of the vortexlattice with the underlying lattice on which the electrons liveis unavoidable for a simulation box that is not too large,like in the present case. Two vortex centers can appear any-where in the simulation box due to the translation symmetry,provided the relative distance remains L, and we choose them to lie at the center of each square-shaped half simulation box.We start the BdG calculation with guess values of parameters,on which the ﬁnal self-consistency would be achieved. Forexample, the guess for the order parameter proﬁle /Psi1(¯r)i st a k e n by applying the analytical solution of Abrikosov close to H c2 [49]. This allows a rather accurate starting point for the phase of the order parameter, aiding signiﬁcantly the convergence of theBdG self-consistency. The procedure outlined here reproducesstandard results, e.g., the depletion of pairing amplitude at thevortex core of the diameter ∼ξ, expected curling of the phase of the order parameter, and the zero-bias peak at E/t=0i n the local DOS at the vortex core due to the formation of theCDM bound state [ 36]. To describe our clean system, we use L=40, and m x=20,my=10, which leads to an effective system size of 800 ×800. The vortex lattice becomes of size 20×20, in terms of intervortex spacing. For our clean system(V=0), we only show the results on the square area through which only one ﬂux quantum passes. These results are shownin Figs. 10(a) –10(c) . Simulation of disordered systems. N o wm o v i n go nt ot h e disordered situation, the lack of translation symmetry in thepresence of disorder does not offer a good guess for the localorder parameter causing the convergence to self-consistencyto be very slow. We used combinations of Anderson, Broyden,and modiﬁed Broyden mixing methods to accelerate theconvergence to self-consistency [ 40]. Even then, the number of iterations for self-consistency for n=2 on a given realization of disorder at V=0.5 grows by two orders of magnitude compared to the clean case. As a consequence we cannotsimulate a large simulation box unlike the clean case. Thus weuse a simulation box of size 36 ×36 and increase the magnetic ﬁeld by changing the number of ﬂux through the simulationbox in the presence of disorder. We, however, continue to usethe repeated zone scheme [ 39] with m x,my=20. 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PhysRevB.89.075432.pdf	PHYSICAL REVIEW B 89, 075432 (2014) Gate-deﬁned coupled quantum dots in topological insulators Christian Ertler, Martin Raith, and Jaroslav Fabian Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany (Received 8 October 2013; published 25 February 2014) We consider electrostatically coupled quantum dots in topological insulators, otherwise conﬁned and gapped by a magnetic texture. By numerically solving the (2 +1) Dirac equation for the wave packet dynamics, we extract the energy spectrum of the coupled dots as a function of bias-controlled coupling and an external perpendicularmagnetic ﬁeld. We show that the tunneling energy can be controlled to a large extent by the electrostatic barrierpotential. Particularly interesting is the coupling via Klein tunneling through a resonant valence state of the barrier.The effective three-level system nicely maps to a model Hamiltonian, from which we extract the Klein couplingbetween the conﬁned conduction and valence dots levels. For large enough magnetic ﬁelds Klein tunneling canbe completely blocked due to the enhanced localization of the degenerate Landau levels formed in the quantumdots. DOI: 10.1103/PhysRevB.89.075432 PACS number(s): 73 .63.Kv,75.75.−c,73.20.At In topological insulators (TIs), according to the bulk- boundary correspondence principle [ 1,2], topologically pro- tected surface states are formed, which are robust against time-reversal (TR) elastic perturbations. In the long-wavelengthlimit the two-dimensional (2D) electron states at the surfaces ofthree-dimensional (3D) TIs can be described as massless Diracelectrons with the peculiar property that the spin is locked tothe momentum, thereby forming a helical electron gas. Chargeand spin properties become strongly intertwined, opening newopportunities for spintronic [ 3,4] applications [ 5–10]. To build functional nanostructures, such as quantum dots (QDs) or quantum point contacts, additional conﬁnementof the Dirac electrons is needed. However, conventionalelectrostatic conﬁnement in a massless Dirac system isineffective due to Klein (interband) tunneling. In graphenethis problem could be overcome by either mechanicallycutting or etching QD islands out of graphene ﬂakes [ 11–13] or by inducing a gap by an underlying substrate, which breaksthe pseudospin symmetry [ 14,15]. Another promising idea to overcome the restrictions given by Klein tunneling is to usegraphene strips or nanoribbons. An electrostatic conﬁnementin such a system has been proposed in Ref. [ 16] by employing the transversal electron motion. Moreover, an effective spinexchange coupling of two gate-deﬁned quantum dots becomespossible in a graphene nanoribbon by indirectly coupling thedots via the tunneling to a common continuum of delocalizedstates [ 17]. In TIs a mass gap can be created by breaking the TR symmetry at the surface by applying a magnetic ﬁeld. Thiscould be achieved by proximity to a magnetic material[18,19], or by coating the surface randomly with magnetic impurities [ 20–22]. By modifying the magnetic texture of the deposited magnetic ﬁlm, a spatially inhomogeneous mass termis induced, opening the possibility to deﬁne quantum dot (QD)regions [ 23], or waveguides formed along the magnetic domain wall regions [ 24]. Another interesting, possibly more feasible way of deﬁning conﬁnement regions, is to induce a uniformmass gap and to deﬁne the QDs by electrostatic gates, whichare energetically shifting the band gap [ 25,26]. In this paper we will focus on such gate-deﬁned topological insulator quantumdots.Single QDs conﬁning Dirac electrons have been thoroughly investigated in the past few years either by numerically solvingtight-binding models or be deducing analytical solutions if acylindrical symmetry and inﬁnite-mass boundary conditionsare present [ 27]. However, the properties of coupled QDs are much less understood and call for a detailed, inevitablynumerical study. Recently we investigated coupled grapheneQDs of small radii ( R< 30 nm) utilizing the Green’s function method to calculate the energy spectra [ 28]. The graphene dots have been modeled by a tight-binding Hamiltonian, sincefor very small dots the usage of an effective ﬁeld modelbecomes already questionable. We obtain that, beside thestrong inﬂuence of the boundary for etched graphene dots,the main difference to TI quantum dots lies in the valleydegeneracy unavoidably present in graphene but not at a single-sided topological surface. For instance, if one investigates thespin precession of an Dirac electron in a single QD accordingto an applied perpendicular magnetic ﬁeld, in topologicalinsulators the total angular momentum, as the sum of theorbital and spin momentum ( J z=lz+sz), is conserved due to [H,Jz]=0. This leads to a dynamic transformation between orbital and a spin angular momentum during the precession ina single TI dot, which does not occur in graphene, since thevalley degeneracy exactly cancels out this effect. In this paper we investigate how an electrostatically tunable coupling strength between the TI dots of typical size of aboutR=50 nm and an applied external magnetic ﬁeld inﬂuence the energy spectra of the double-dot system. The tunneling timeis deduced from studying the wave packet dynamics, whichneeds the numerical solution of the time-dependent (2 +1) Dirac equation. For this purpose we use a specially developeddiscretization scheme, which was introduced and discussedin detail in Ref. [ 29]. We study two different scenarios for inducing a coupling between the dots: (i) by conventionalmeans, i.e., by electrostatically reducing the barrier heightin between the dots, and (ii) by coupling the dots via Kleintunneling upon a hole state in the barrier, which can be shiftedby a gate voltage. Especially in the second case we ﬁnd astrong tunability of the coupling strength. We also introducetoy one-dimensional (1D) models to study tunneling of Diracelectrons analytically. Our numerical solutions for coupled 1098-0121/2014/89(7)/075432(8) 075432-1 ©2014 American Physical SocietyCHRISTIAN ERTLER, MARTIN RAITH, AND JAROSLA V FABIAN PHYSICAL REVIEW B 89, 075432 (2014) 2D quantum dots establish quantitatively strong and efﬁcient coupling between the dots. In the Klein tunneling regime weprovide a useful three-level parametric hopping Hamiltonianto describe the conduction and valence band couplings. Ourgoal is to provide the single-electron picture of the tunneling,which could also be used as a starting point to investigate theCoulomb blockade physics. Our paper is organized as follows. The basic idea of a gate- controlled coupling between QDs and qualitative analyticalsolutions are introduced in Sec. I. The numerical investigation of coupled QDs is presented and discussed in Sec. II. Summary and conclusions are given in Sec. III. I. 1D MODEL OF A GATE CONFINED QUANTUM DOT If a uniform mass gap exists throughout the TI surface, QDs can be deﬁned by shifting the energy gap locally byapplying gate voltages as illustrated in Fig. 1. The mass barrier height between the dots becomes electrostatically controllable,allowing for a direct tunability of the coupling strength. Thebarrier can also be shifted upwards in energy as far as a holestate comes into resonance with the ground state of the isolateddots. This leads to an effective strong coupling between thedots via Klein tunneling from the electron states to the holestate, as illustrated in Fig. 2. In order to understand qualitatively how the coupling strength between QDs depends on the barrier height, we ﬁrstinvestigate the transmission probability of a model 1D-massbarrier. We consider two different cases: (i) a mass barrierbetween leads with zero mass, as illustrated in Fig. 3(a), and (ii) a uniform mass gap in the structure with a shiftable regionin the middle, as shown in Fig. 3(b), which directly corresponds to the “conventional” coupling scenario of Fig. 1. For a general 1D structure with an inhomogeneous mass termm(x) and potential V(x) the Dirac-Hamiltonian is H=−iα∂ xσx+m(x)σz+V(x), (1) where α=/planckover2pi1vfandσx,zdenote the Pauli matrices. Let us assume that we can divide the region of interest into subregions Vb Vd1 Vd2ΔE = 2 m0 FIG. 1. (Color online) Schematic band proﬁle of the conduction (blue line) and valence (red line) bands of two coupled topological insulator quantum dots. The uniform band gap /Delta1E=2m0is shifted by applying gate voltages. For electrons a double-dot system is formed with the barrier height being controllable by an external bias Vb. This is conventional coupling as found in semiconductor quantumdots [ 30].VbΔE = 2 m0 FIG. 2. (Color online) Schematic band proﬁle of the conduction (blue line) and valence band (red line) for the Klein-tunnelingscenario: The coupling between the dots is realized via the Klein tunneling upon a hole state, which can be shifted by an external gate voltage V b. in which mandVcan be assumed to be constant. For constant mandVthe eigenfunctions ψ±of left ( −) and right ( +) moving plane waves of energy Eare given by ψ±=/parenleftbigg 1 ±γ/parenrightbigg e±iqx, (2) with q(V,m )=1 α/radicalbig (E−V)2−m2 (3) and γ(V,m )=/radicalbig (E−V)2−m2 (E+V)+m, (4) yielding the general solution ψ=c+ψ++c−ψ−.A tt h e boundary of neighboring subregions iandi+1t h ew a v e function has to be continuous, resulting in the condition ψi(xi)=ψi+1(xi). (5) This continuous connection of the wave functions of the subregions allows us to calculate the transfer matrix Mof the whole system, which connects the amplitudes of the ﬁrst layer m0 dVb d Vb(a) (b) (c) FIG. 3. (Color online) (a) Scheme of a single mass barrier of height m0. A mass barrier (b) or a quantum well (c) of width d is formed for electrons by applying a gate voltage Vbof opposite sign. 075432-2GATE-DEFINED COUPLED QUANTUM DOTS IN . . . PHYSICAL REVIEW B 89, 075432 (2014) C1=(c+ 1,c− 1) with the last, i.e., the most right one CN=MC 1. From the elements of the transfer matrix the transmission function can then be obtained by T(E)=detM \|M22\|2. (6) Mass barrier between massless dots. In the case (i) of a single mass barrier of height m0, which is shifted by the gate voltage Vbthis procedure yields the following result for the transmission function if the electron energies are below thebarrier, i.e., −m 0+Vb<E<m 0+Vb: T(E)=−1+6˜γb2−˜γb4+(1+˜γb2)2cosh(2 d˜qb) 8˜γb2, (7) with ˜qb=−iqb,˜γb=−iγb,γb=γ(Vb,m0), and qb= q(Vb,m0). For energies above the barrier, i.e., for E>m 0+Vb andE<−m0+Vb, the transmission probability is given by T(E)=/bracketleftBigg cos2(dqb)+/parenleftbig 1+γ2 b/parenrightbig2sin2(dqb) 4γ2 b/bracketrightBigg−1 .(8) As expected one obtains an exponential and oscillatory dependence of the transmission function for energies smallerand greater than the barrier height, respectively, as illustratedin Fig. 4. Applying a gate voltage allows one to shift the whole transmission function along the energy axis, which means thatfor a given ﬁxed energy one obtains an exponential dependenceon the applied gate voltage. Note that in contrast to Schr ¨odinger particles the transmission remains ﬁnite even at zero energydue to the ﬁnite group velocity for E=0: T(E=0)=/bracketleftbigg cosh 2/parenleftbiggdm 0 /planckover2pi1vf/parenrightbigg/bracketrightbigg−1 ≈e−2dm0 /planckover2pi1vf. (9) Uniform mass with a gate-controlled barrier. In the case (ii) of a uniform mass region m=m0and a gate induced single mass barrier of height Vb, as shown in Fig. 3(b),t h e transmission function results in T(E)=8γ2 lγ2 b γ4 l+6γ2 lγ2 b+γ4 b−/parenleftbig γ2 l−γ2 b/parenrightbig2cos(2dqb)(10) if the electron’s energy is higher than the barrier ( E>m 0+Vb) withγl=√ E2−m2 0/(E+m0). For electron energies lower −0.3 −0.2 −0.1 0 0.1 0.2 0.300.20.40.60.81 E (eV) T Vb FIG. 4. (Color online) Calculated transmission function T(E)o f a single 1D-mass barrier, as shown in Fig. 3(a). Applying a voltage Vb allows to shift T(E) along the energy axis, which strongly changes the transmission for a ﬁxed energy.0.05 0.1 0.15 0.2 0.2500.20.40.60.81 E (eV) T Vb = 10 meV Vb = 30 meV Vb = 50 meV Vb = 100 meV FIG. 5. Calculated transmission function T(E)o fag a t ei n d u c e d 1D barrier of width d=30 nm, as illustrated in Fig. 3(b), for different applied biases. The mass is set to m0=50 meV . than the barrier m0<E<m 0+Vbthe transmission is given by T(E)=/braceleftbigg1 4/bracketleftbigg 3+cosh(2 d˜qb)+γ4 l+˜γ4 bsinh2(d˜qb) γ2 l˜γ2 b/bracketrightbigg/bracerightbigg−1 . (11) Again one obtains an exponential and oscillatory dependence of the transmission function for energies below and abovethe barrier height, as illustrated in Fig. 5for different barrier heights. Energy spectrum of 1D TI dots. Finally, we calculate the energy spectrum of a single dot of width d, as illustrated in Fig. 3(c). By using the condition det M=0 the eigenenergies of the bounded states are given by E n=±/radicalBigg/parenleftbigg nπα d/parenrightbigg2 +m2 0+Vb. (12) II. COUPLED 2D DOTS A. Numerical solution of the time-dependent 2D Dirac equation Here we provide a numerical investigation of the spectrum of two coupled topological insulator two-dimensional quan-tum dots depending on their coupling strength and externalperpendicular magnetic ﬁelds B. In order to calculate the energy spectra we study the dynamics of wave packets (seeRef. [ 31] for a review of the wave packet method in general). In comparison to a direct numerical diagonalization of theHamiltonian, the wave packet method allows us to investigatelarger systems with a higher number of grid points in areasonable computation time. This is needed since the dotsystems have to be large enough to make an effective theoryactually applicable for describing the carrier dynamics of dotsystems. The energy spectrum is then obtained by a Fouriertransformation of the wave packet autocorrelation functionwith the energy resolution being determined by the totalpropagation time /Delta1E=2π/planckover2pi1/T. However, as a disadvantage compared to exact diagonalization the wave packet methodcan miss some eigenenergy values in the case that thecorresponding amplitudes of the Fourier transformation aresmaller than the numerical signal noise. 075432-3CHRISTIAN ERTLER, MARTIN RAITH, AND JAROSLA V FABIAN PHYSICAL REVIEW B 89, 075432 (2014) In order to obtain the energy spectrum we calculate the local density of states D(E,r)=−1 πIm[G(r,r;E)], (13) which is deﬁned via the diagonal elements of the re- tarded Green’s function G(E). Based on the dynamics of a single wave packet initially centered at rthe retarded Green’s function can be constructed from its autocorrelationfunction C(t), G(r,r;E)≈1 i/planckover2pi1/integraldisplay∞ 0dteiEt//planckover2pi1C(t), (14) with C(t)=/integraldisplay drψ(r,0)∗ψ(r,t). (15) Equation ( 14) becomes exact for a δ-distributed initial state, whereas in the numerical simulations a Gaussian shaped initialstate is used [ 31]. To obtain the correlation function one has to keep track of the wave packet transient, which requires thesolution of the time dependent 2D-Dirac equation ∂ψ(r,t) ∂t=HDψ(r,t). (16) The single-particle Hamiltonian at the surface of TIs can be derived within an effective ﬁeld theory approach [ 1,2], yielding HD=vf{[−i/planckover2pi1∇−eA(r,t)]×ˆz}σ+mz(r)σz−eφ(r,t), (17) where A(r,t) andφ(r,t) denotes the space- and time-dependent vector and electrostatic potential, respectively. The inhomo-geneous mass term m z(r)σzis induced by breaking the TR symmetry at the TI surface, e.g., by proximity of a magneticlayer [ 18,19] or by magnetic doping [ 20–22]. The order of magnitude for the mass-gap /Delta1 gap=2\|mz\|can be expected to be tens of meV . In order to solve numerically the (2 +1) Dirac equation we use a specially developed staggered-grid leap frog scheme,which we introduced and discussed in detail in Ref. [ 29]. The numerical solution of the Dirac equation on a ﬁnitegrid is a more subtle issue than for the nonrelativisticSchr ¨odinger equation. As well known from lattice ﬁeld theory, discretization of the Dirac equation leads to the so-calledfermion-doubling problem, i.e., for large wave vectors a wrongenergy dispersion is revealed. This leads to the doubling of theeigenstates at a ﬁxed energy value. For a longtime propagationit is of great importance to use an almost dispersion-preservingﬁnite-difference scheme, since scattering at spatiotemporalpotentials and at the boundary can introduce higher wavevector components even when one starts with a wave packetwith its wave vector components closely centered at k=0. Moreover, the formulation of proper boundary conditionsis crucial to avoid spurious reﬂections and eventuallyinstability [ 29].40 60 80 100 12001020304050 R (nm)E (meV) E0 Eex Eion EexEion E0 FIG. 6. (Color online) The ground state energy E0(solid line), the excitation energy Eex(dashed line), and the ionization energy Eion(dash-dotted line) of single dot for different radii Rat zero magnetic ﬁeld. B. Energy spectra of single and coupled QDs 1. Energy spectra of single QDs First, we investigate the conﬁnement energy of a single isolated QD as a function of radius Rand the magnetic ﬁeld B. The circular dot potential φd(r0) centered at r0is assumed to be described by the Fermi-Dirac function φd(r0)=Vd˜φd(r0)=VdFD(r0−r), (18) withFD(x)=[1+exp(x/βr)]−1;Vddenotes the potential height. The potential step is smeared on the range of βr= 0.01R. In the following we set the Fermi velocity to vf= 105m/s and the dot potential height is chosen as \|Vd\|= 50 meV . Figure 6shows the ground state or conﬁnement energy E0of the dot, the excitation energy deﬁned as difference of the ground and ﬁrst excited dot state Eex=E1−E0, and the ionization energy Eion=\|Vd\|−E1, as given by the energy difference of the ground state energy to the continuum of thedelocalized states, as a function of different dot radii at zeromagnetic ﬁeld. For radii smaller than about R< 38 nm only a single bound state exists in the QD and, hence, E ex=Eion. As expected, the weaker conﬁnement of the Dirac electrons inlarger dot systems leads to a decreasing of both the ground stateenergy E 0and the excitation energy Eex, as shown in Fig. 6.F o r small radius (here for R< 35 nm) only a single energy level is present and, hence, the ionization energy and the excitationenergy coincide. For increasing dot radius more energy levelsemerge successively. However, the steep slope of the inprinciple continuous line of the conﬁnement energy cannot beresolved by our used grid of the radius Rand appears therefore a sak i n ki nF i g . 6. Note that the typical conﬁnement energy of Dirac electrons is of the order of 10 meV , which is an orderof magnitude higher than in conventional semiconductor dotsof comparable size. The energy spectrum for the lowest QDlevels versus magnetic ﬁeld, which is applied perpendicular tothe TI surface, is plotted in Fig. 7assuming a ﬁxed radius of R=50 nm. A detailed analytical study of the energy spectrum of a single graphene quantum dot in a perpendicular magneticﬁeld is given in Ref. [ 27]. The eigenspectrum of the dot for 075432-4GATE-DEFINED COUPLED QUANTUM DOTS IN . . . PHYSICAL REVIEW B 89, 075432 (2014) 0 1 2 3 401020304050 B (T)E (meV) FIG. 7. Magnetic ﬁeld ( B) dependent energy spectrum of a single QD of radius R=50 nm. The energy levels converge to Landau levels for higher Bﬁelds. B=0 is obtained by solving the implicit equation Jm(kR)=Jm+1(kR), (19) withJmdenoting the Bessel functions of ﬁrst kind of order m andE=/planckover2pi1vfk. Since the total angular momentum Jz= lz+(/planckover2pi1/2)σz, withlzdenoting the orbital momentum operator, commutes with the Hamiltonian ([ H,Jz]=0),mis a good quantum number. Hence, the ground state with l=0 is doubly degenerate according to its spin. For higher magnetic ﬁelds thelevels start to converge to degenerate Landau levels, which aredetermined by the expression [ 27] E m=vf/radicalbig 2e/planckover2pi1B(m+1). (20) As can be seen in Fig. 7at about B≈3.6Tt h eﬁ r s tt w o levels converge numerically, which leads to a sudden kink inthe magnetic ﬁeld dependence of the excitation energy E ex, as indicated by the solid red line in Fig. 8. The magnetic ﬁeld effectively acts as an additional conﬁnement causing an almostlinear enhancement of the conﬁnement energy E 0. 0 1 2 3 4010203040 B (T)E (meV) E0 Eex Eion FIG. 8. (Color online) Magnetic ﬁeld ( B) dependence of the ground state energy E0(solid line), the excitation energy Eex(dashed line), and the ionization energy Eion(dash-dotted line) of single dot of radius R=50 nm. FIG. 9. (Color online) The double-dot potential in the TI surface. The coupling of the two dots is controlled by shifting the barrier potential. 2. Energy spectra of coupled QDs Our double-dot system comprises two circular disks, which are connected by a potential bridge, as shown in Fig. 9. The barrier or bridge potential φb=Vb˜φbis described by a rectangular step function of width wand length d, which is smeared at the boundaries by Fermi-Dirac function ˜φb=FD(−w/2−y)FD(y−w/2)FD(−d−x)FD(x−d). (21) The total potential of the coupled QD system can then be deﬁned by φ(r)=Vd˜φdd+Vbmax(\|˜φb\|−\| ˜φdd\|,0), (22) with ˜φdd=max[ ˜φd(r1),˜φd(r2)] describing the potential of the decoupled double-dot system. For the quantitative simulations we choose the following structure parameters: dot radius R=50 nm, bridge length d=30 nm, bridge width w=40 nm, grid resolution /Delta1x= /Delta1y=1 nm, and an uniform mass term of m0=50 meV . To ensure the stability of the discretization scheme [ 29]t h e Courant-Friedrichs-Lewy (CFL) condition has to be fulﬁlled/Delta1t < min(/Delta1 x,/Delta1y)/vf. We use typically Nt=1.6×105time steps for each wave packet propagation, which leads to anenergy resolution of about /Delta1 E≈0.1 meV in the discrete fast Fourier transformation (FFT), where /Delta1E=2π/planckover2pi1/(Nt/Delta1t). The ﬁnite energy resolution of the ﬁxed energy grid of thediscrete FFT becomes noticeable in the following plots of theenergy spectrum as small discontinuous jumps when externalparameters, such as the barrier voltage, are changed. The dependence of the energy spectrum on the gate- controlled barrier height V bforB=0 is shown in Fig. 10. ForVb=0 the two dots are almost isolated. To realize the “conventional coupling” as illustrated in Fig. 1an e g a t i v e bias has to be applied, which reduces the barrier potential.At around V b=− 40 meV the bonding and antibonding states start to be split in energy by /Delta1due to the increasing coupling between the dots (Fig. 11). From the energy splitting the tunneling time follows as τ=2π/planckover2pi1//Delta1. 075432-5CHRISTIAN ERTLER, MARTIN RAITH, AND JAROSLA V FABIAN PHYSICAL REVIEW B 89, 075432 (2014) −100 −50 0 50 100 150−5051015 Vb (meV)E (meV) H1 H2Δ Δ FIG. 10. Calculated energy spectrum of the double TI QD system versus applied barrier gate voltage for B=0. If a positive bias is applied, the hole states in the barrier region are shifted upwards in energy enabling at some pointthe electrons to hop by Klein tunneling from one QD tothe other via the hole state, as illustrated in Fig. 2.T h e hybridization of the two electron levels and the hole stateinduces an anticrossing of the ﬁrst excited electron state andthe hole state, leading to a strongly tunable excitation energyof maximally /Delta1≈8 meV , giving a typical tunneling time of τ≈0.5 ps. Figure 12shows this anticrossing as a zoom-in of Fig. 10. The main features of Fig. 12can be understand by an effective three-level model. As a starting point we assumethat the direct hopping between the left and right single dotground states \|L/angbracketrightand\|R/angbracketright, respectively, is inhibited and only hopping via the hole state \|H/angbracketrightis possible. Then the effective Hamiltonian in the basis {\|L/angbracketright,\|H/angbracketright,\|R/angbracketright}reads H 0=⎛ ⎝εL it 0 −it∗εHit 0 −it∗εL⎞ ⎠, (23) withεR=εL, and tdenotes the hopping amplitude. The eigenenergies are given by ε2=εL, ε1/3=εL+εH 2∓1 2/radicalbig (εL−εH)2+8t2,(24) −100 −50 0 50 100 150100101 Vb (meV)Δ (meV) FIG. 11. Energy splitting /Delta1of the ﬁrst bonding and antibonding state (as extracted from Fig. 10) versus the applied bias Vb.70 80 90 100 110 120 130−5051015 Vb (meV)E (meV) H2εH H1ε1ε3 (εH+εL)/2ε2 = εL FIG. 12. (Color online) Zoom-in of Fig. 10in the region, where the two dots are coupled via Klein tunneling upon approaching holestateH 1. An effective three-level model reveals that the anticrossing should be symmetric around ( εL+εH)/2. and the (unnormalized) eigenstates result in ϕ(0) 2=(1,0,1), (25) ϕ(0) 1/3=(−1,ξ(−i∓/radicalbig 1+2/ξ2),1), withξ=(εL−εH)/2t. This suggests that one eigenenergy value should remain almost unaffected and that the anticross-ing should be symmetric around ( ε L+εH)/2. As illustrated in Fig. 12this behavior is approximately fulﬁlled by our numerical simulation results. If one introduces an additional weak direct coupling between the dots as described by the Hamiltonian H1=⎛ ⎝00 it1 00 0 −it100⎞ ⎠, (26) perturbation theory yields that the eigenergies of H0are not changed in ﬁrst order. However, now a small component of thehole state of the order of t 1/(ε2−ε1) mixes to the eigenstate ofε2=εL: \|ϕ2/angbracketright=\|ϕ2/angbracketright(0)−2it1 ε2−ε1\|ϕ1/angbracketright(0)−2it1 ε2−ε3\|ϕ3/angbracketright(0). (27) From our numerical data we can extract the voltage dependence of the hopping parameter t. Therefore, we redeﬁne the origin of the coordinate system as the crossing point inwhich ε L=εH, i.e., at the point εL=7.5 meV and V0= 100.82 meV . Then εL=0 by deﬁnition and the bias-dependent hole state is described by the asymptotic linear functionε H(Vb)=kH(Vb−V0)=kH˜V, shown as dashed (red) line in Fig. 12, with kH=0.77 being obtained by linear regression. From the numerical results for the lowest eigenvalue ε1(˜V)w e calculate the hopping parameter, which is given by t(˜V)=/radicalBigg ε2 1−ε1εH 2. (28) Figure 13shows the obtained voltage dependence of the hop- ping parameter. Since the coupling between the electron andhole state vanishes for large voltages, i.e., lim ˜V→±∞t(˜V)=0, 075432-6GATE-DEFINED COUPLED QUANTUM DOTS IN . . . PHYSICAL REVIEW B 89, 075432 (2014) −20 −10 0 10 20 3000.20.40.60.81.01.2 ˜V(meV)t (meV) Gaussian−fit parabolic−fit80 100 120 140Vb (meV) FIG. 13. (Color online) V oltage dependence of the hopping pa- rameter tof the effective three-level model as extracted from the numerical data of the ﬁrst eigenvalue ε1. we ﬁt our numerical results to an Gaussian, as shown in Fig. 13. For comparison also a parabolic ﬁt is provided. The effectivemodel is most suitable in the region for ˜Vbetween 0 and 10 meV , where t≈1 meV is roughly constant. Otherwise the model is to be treated as a convenient parametric ﬁt. A qualitatively different dependence of the energy spectrum on the applied barrier bias V bis found if a strong enough mag- netic ﬁeld is applied, which induces the formation of Landaulevels of magnetic quantum numbers mcorresponding to the total angular momentum J z=lz+/planckover2pi1/2σz, with lzdenoting the orbital momentum. As shown in Fig. 14(forB=4T )t h e ﬁrst hole level almost does not couple to the energy levels ofthe ﬁrst Landau niveaus with m=0. This follows from the fact that due to the Bﬁeld the states are more localized, so that their overlap, on which the hopping parameter essentiallydepends, is exponentially smaller. This is most pronouncedfor the lowest state for m=0 but can be also seen for the ﬁrst excited states ( m=1), where the effect of anticrossing is smaller than for the case of a vanishing magnetic ﬁeld.−100 −50 0 50 100 1503540455055 Vb (meV)E (meV) H1m = 1 m = 0 FIG. 14. (Color online) Energy spectrum of the double-dot system versus applied barrier gate voltage for B=4T . III. CONCLUSIONS We have shown that the coupling between two quantum dots, which are geometrically deﬁned by gate electrodes,can be strongly modulated and controlled by both a gatebias, which brings a hole level into resonance with theelectron states, and a perpendicular magnetic ﬁeld, whichchanges the symmetry properties of the conﬁned states. 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PhysRevB.101.075427.pdf	PHYSICAL REVIEW B 101, 075427 (2020) Nonconventional screening of Coulomb interaction in hexagonal boron nitride nanoribbons A. Montaghemi ,1H. Hadipour,2,F. Bagherpour,2A. Yazdani,1,†and S. Mahdavifar2 1Department of Physics, University of Tarbiat Modares, 14115-111 Tehran, Iran 2Department of Physics, University of Guilan, 41335-1914 Rasht, Iran (Received 1 January 2020; accepted 11 February 2020; published 24 February 2020) Strong excitonic effects is a very subtle issue in pristine hexagonal boron nitride (h-BN) and h-BN nanoribbons (h-BNNRs) due to large band gaps and reduced dimensionality. One of the reasons for such a large excitonbinding energy (as large as 2.5 eV) is weak dielectric screening. Employing ﬁrst-principles calculations inconjunction with the constrained random-phase approximation, we determine the strength of the Coulombmatrix elements for pristine h-BN and h-BNNRs with armchair and zigzag edges. Due to the nonconventionalscreening, the calculated off-site Uparameters for passivated h-BNNRs turn out to be rather sizable. Coulomb interaction is weakly screened at short distances and antiscreened at intermediate distances. Transition fromscreening to antiscreening takes place at a distance as low as 8 Å in narrow passivated h-BNNR. The criticaldistance for the onset of antiscreening in hydrogen-terminated h-BNNRs is longer than in zero-dimensionalmolecules and clusters, but shorter than in graphene nanoribbons and carbon nanotubes. With increasing thewidth of the passivated h-BNNRs from the critical point about 12.6 Å, the antiscreening effect is not observed.For completeness, on-site and long-range Coulomb interactions for metallic nonpassivated zigzag h-BNNRs arealso reported. DOI: 10.1103/PhysRevB.101.075427 I. INTRODUCTION Two-dimensional (2D) systems such as graphene [ 1,2] have become one of the most studied classes of materialsduring the last decade. Graphene is not magnetic and has noband gap, which makes it difﬁcult to use in electronic andspintronic applications. Several methods have been developedto turn graphene into a semiconductor and also to introducemagnetism [ 3–9]. Most of these systematic approaches are not able to be precisely controlled. It is very desirable to ﬁnd 2Dmaterials that are magnetic or have a band gap in the pristineform. Boron and nitrogen atoms with strong covalent bondsin a hexagonal lattice, also called h-BN, is an insulator withan indirect band gap of about 5.0 eV [ 10–23]. The electronic and magnetic properties of h-BN nanoribbons (h-BNNRs)depend on their edge and width. Bare zigzag h-BNNRs (Zh-BNNRs) are magnetic metals [ 24], while hydrogen-passivated structures (Zh-BNNRs:H) are semiconductors with a band gapof 4.5 eV that decreases linearly with increasing ribbon width[25–27]. Both bare armchair h-BNNRs (Ah-BNNRs) and hydrogen-passivated armchair h-BNNRs (Ah-BNNRs:H) aresemiconductors. For example, the band gap of Ah-BNNR:Hof 6.3 Å width is about 4.5 eV [ 28] in the generalized gradient approximation (GGA). Tightly bound excitons with binding energies of about 2.5 eV were observed experimentally in 2D h-BN [ 16,22,29]. This is larger than the corresponding binding energy incarbon-based materials and bulk semiconductors and is ingood agreement with ab initio calculations [ 30]. It is worth hanifhadipour@gmail.com †yazdania@modares.ac.irnoting that the formation of tightly bound excitons in 2D h-BN reveals a signiﬁcantly reduced and nonlocal dielec-tric screening of the Coulomb interaction. Furthermore, the in-corporation of electron-electron correlation effects within theGW approximation gives rise to the large band gap of 7 eV inAh-BNNR:H, which is signiﬁcantly larger than the band gappredicted by the GGA approximation [ 31]. Such a signiﬁcant many-body GW correction to the band gap is associated withlarge Coulomb matrix elements. Also, the exciton bindingenergy for h-BNNRs:H is between 2.1 and 2.4 eV [ 32,33], which is higher than the equivalent graphene nanoribbons(GNRs) [ 34]. Moreover, the reduced dimensionality gives rise to a nonconventional screening of the Coulomb interactionin low-dimensional and ﬁnite-size insulators; that is, it isscreened at short distances and antiscreened at intermediatedistances [ 35–39]. Such a nonconventional screening of the Coulomb interactions occurs in GNRs [ 39,40] and semicon- ducting carbon nanotubes [ 36]. One of the consequences of a nonconventional screening is the large exciton binding energy(as large as 1.5 eV) in GNRs [ 34,41–45]. In the case of metallic h-BNNR with a zigzag edge, mag- netism is a controversial issue and Coulomb interactions playa crucial role to describe the origin of ferromagnetic ordering.Generally, low-dimensional materials have a lower continuityof the density of states and, consequently, have less bandwidththan the bulk. As a result, the ratio of effective Coulombinteraction Uto bandwidth W bis increased, and the strength of the correlation U/Wbbecomes great. One of the consequences of a moderate correlation U/Wb∼1 is inducing magnetism in the carbon-based materials [ 39,46,47]. The formation of tightly bound excitons in semiconducting Ah-BNNR and the existence of ferromagnetism in metallicZh-BNNR motivate us to evaluate the ab initio Coulomb 2469-9950/2020/101(7)/075427(9) 075427-1 ©2020 American Physical SocietyA. MONTAGHEMI et al. PHYSICAL REVIEW B 101, 075427 (2020) interaction parameters in 2D h-BN and h-BNNR for different widths. The effective Coulomb interaction between localizedelectrons plays an important role in constructing a genericsecond-quantized Hamiltonian for BN materials and gives ususeful information for describing the reason behind electronicand magnetic ordering. In this paper, we study the screening of on-site and long- range Coulomb interaction between p zelectrons in passi- vated h-BNNRs and sp3electrons in bare h-BNNRs by employing ab initio calculations in conjunction with the con- strained random-phase approximation [ 48–50] (cRPA) within the full-potential linearized augmented plane-wave (FLAPW)method. Due to the existence of similarities between h-BNand graphene, we compare our results with the recent studyof the graphene nanoribbons [ 39]. Our results show the Hubbard Uparameters for passivated h-BNNR:H with dif- ferent width, slightly smaller than the ones in pristine h-BN.Also, screening in Zh-BNNR:H is stronger than that in Ah-BNNR:H. Due to the nonconventional screening, the transi-tion from screening to antiscreening takes place at a distanceas low as 8 Å in narrow passivated h-BNNR. We found thatthe antiscreening in h-BNNRs occurs in a shorter distancethan observed in GNRs, but longer than in zero-dimensionalmolecules and clusters. In addition, there is a critical widthsuch that if the nanoribbon width is larger than the criticalwidth, the antiscreening effect is not observed. II. COMPUTATIONAL METHOD For the investigation of the hydrogen-passivated h-BNNR with armchair ( Na-Ah-BNNR:H) and zigzag edge ( Nz-Zh- BNNR:H), we use conventional orthorhombic unit cells tosimulate these systems, in which N ais the number of dimer lines across the ribbon width and Nzis the zigzag chain across the ribbon. For comparison, bare Zh-BNNR and pristine 2Dh-BN are also considered. In the armchair group, we con-sider N a=2 (2-Ah-BNNR:H) to Na=11 (11-Ah-BNNR:H) passivated nanoribbons. In the case of the zigzag group,passivated nanoribbons with N z=3 (3-Zh-BNNR:H) to Nz= 9 (9-Zh-BNNR:H) and Nz=7 nonpassivated nanoribbon (7-Zh-BNNR) are considered. Figure 1shows some of these nanoribbons. The unit cells are separated from them by 20 Åin both edge-to-edge and layer-to-layer directions. This vac-uum separation is enough to ensure there is no interactionbetween the ribbon and its periodic images. Also, all atomicpositions and lattice constants are relaxed, so the forces oneach atom are converged until 0.02 eV /Å. A plane-wave basis set with 190 Rydberg cutoff energy has been used. For densityfunctional theory (DFT) calculations, we use the FLAPWmethod as implemented in the FLEUR code [ 51] within the generalized gradient approximation parameterized by Perdew,Burke, and Ernzerhof (PBE) [ 52] for the exchange-correlation energy functional. Dense 24 ×1×1 and 24 ×24×1k- point grids are used for unit cells of h-BNNRs and pristineh-BN, respectively. A linear momentum cutoff of G max=4.5 bohr−1is chosen for the plane waves. The SPEX code [ 53]u s e s DFT results to determine the strength of the partially (fully)screened effective Coulomb Interaction U(W) between local- ized electrons from the ﬁrst-principles cRPA method [ 48]. FIG. 1. h-BN nanoribbons with two types of edges: (a) Na= 8 (8-Ah-BNNR:H) and (b) Na=9 (9-Ah-BNNR:H) are armchair ribbons having different width. Nais the number of dimer lines across the ribbon width. (c)–(e) Zigzag ribbons of Nz=8 (8-Zh-BNNR:H), Nz=9 (9-Zh-BNNR:H), and Nz=7 (7-Zh-BNNR), respectively. Nz is the number of zigzag chains across the ribbon width. In this work, we study partially and fully screened Coulomb interaction parameters calculated with the ab initio cRPA and RPA methods [ 48–50], respectively. The fully screened Coulomb interaction Wis related to the bare Coulomb interaction Vas W(r,r/prime,ω)=/integraldisplay dr/prime/prime/epsilon1−1(r,r/prime/prime,ω)V(r/prime/prime,r/prime), (1) where /epsilon1(r,r/prime/prime,ω) is the dielectric function. In the RPA of the dynamically screened Coulomb in- teraction, the dielectric function is related to the electronpolarizability Pby /epsilon1(r,r /prime,ω)=δ(r−r/prime)−/integraldisplay dr/prime/primeV(r,r/prime/prime)P(r/prime/prime,r/prime,ω),(2) where the polarization function P(r/prime/prime,r/prime,ω) is given by P(r,r/prime,ω) =/summationdisplay σocc/summationdisplay k,munocc/summationdisplay k/prime,m/primeϕσ km(r)ϕσ∗ k/primem/prime(r)ϕσ∗ km(r/prime)ϕσ k/primem/prime(r/prime) ×/bracketleftbigg1 ω−/epsilon1σ k/primem/prime−/epsilon1σ km−iη−1 ω+/epsilon1σ k/primem/prime−/epsilon1σ km−iη/bracketrightbigg . (3) Here, /epsilon1σ kmare single-particle Kohn-Sham eigenvalues ob- tained from DFT and ηis a positive inﬁnitesimal. Further, 075427-2NONCONVENTIONAL SCREENING OF COULOMB … PHYSICAL REVIEW B 101, 075427 (2020) FIG. 2. Nonmagnetic orbital-projected band structure of (a) 8-Zh-BNNR:H, (b) 8-Ah-BNNR:H, and (c) 7-Zh-BNNR. TheFermi level is set to zero energy. theϕσ km(r) are the single-particle Kohn-Sham eigenstates with spinσ, wave number k, and band index m. The tags occ and unocc above the summation symbol indicate that the summation is, respectively, over occupied and unoccupiedstates only. In the cRPA, we separate the full polarization function into the two parts: P lincludes only transitions between the local- ized states, for which the interaction needs to be calculated,andP ris the remainder, P=Pl+Pr. (4) To identify the correlated subspace and understand which orbital should be considered in Pl, we need to look at the states near the Fermi energy for different nanoribbons. Projectedband structures for two passivated systems, 8-Ah-BNNR:Hand 8-Zh-BNNR:H, are depicted in Figs. 2(a) and2(b),r e - spectively. For both systems, the p zorbitals have a signiﬁcant contribution to the band around EFcompared to pxand py orbitals and the corresponding pzbands are disentangled from the rest in a large energy interval. In the case of nonpassivatednanoribbons [see Fig. 2(c)for 7-Zh-BNNR], all p x,py,pz, and s(not shown) orbitals are close to EF. So, sp3orbitals must be considered as a correlated subspace. The partially Coulomb interaction (Hubbard U)i s written as U(ω)=[1−VPr(ω)]−1V. (5) Using maximally localized Wannier functions (MLWFs) at site Rwith orbital index nand spin σ,wσ Rn(r), the matrix elements of the effective Coulomb potential Uat frequency ω Γ X Γ X-6-4-20246E - EF (eV)Wannier-interpolation PBE-DFT (a) 8-Ah-BNNR:H (b) 7-Zh-BNNR FIG. 3. DFT-PBE and Wannier-interpolated band structure of nonmagnetic (a) 8-Ah-BNNR:H and (b) 7-Zh-BNNR. Dashed lines denote the Fermi energy, which is set to zero. in the MLWFs basis are given by Uσ1,σ2 in1,jn3,in2,jn4(ω) =/integraldisplay/integraldisplay drdr/primewσ1∗ in1(r)wσ2∗ jn3(r/prime)U(r,r/prime,ω)wσ2 jn4(r/prime)wσ1 in2(r). (6) The average on-site interaction matrix elements of the effec- tive Coulomb potential are estimated as [ 54] U=1/L/summationdisplay nUσ1,σ2 Rnn:nn(ω=0), (7) and off-site elements are deﬁned as U(R−R/prime)=1/L/summationdisplay nUσ1,σ2 Rnn:R/primenn(ω=0), (8) where Lis the number of localized orbitals. Using the WAN- NIER90 code [ 55],SPEX constructs MLWFs for the porbitals of each atoms (10 states per atom) in all systems. We use18×18×1 and 30 ×1×1k-point grids in the cRPA cal- culations of the h-BN and h-BNNR unit cells, respectively.In Figs. 3(a) and3(b), we have presented the original band structure and Wannier-interpolated bands obtained from thesubspace selected by projecting onto p zorbitals on each atom for 8-Ah-BNNR:H and sp3orbitals for 7-Zh-BNNR, respectively. The overall agreement between DFT-PBE andWannier-interpolated bands is quite good and demonstratesthe validity of the calculated Wannier functions. III. RESULTS AND DISCUSSION As shown in the band structure of Fig. 2, the passivated structures (8-Ah-BNNR:H and 8-Zh-BNNR:H) are semicon-ductors, while the bare zigzag nanoribbons (7-Zh-BNNR)are metallic. Since the electronic screening turns out to bestrongly dependent on the band gap in the single-particlespectrum and dimensionality reduction, we present calculatedvalues of Coulomb interaction parameters in three sections.In Sec. III A , we study the Coulomb interaction parameters 075427-3A. MONTAGHEMI et al. PHYSICAL REVIEW B 101, 075427 (2020) TABLE I. On-site ( U00), nearest-neighbor ( U01), next-nearest- neighbor ( U02), and third-nearest-neighbor ( U03) Coulomb interac- tion parameters (in eV) for 2D h-BN, 3D h-BN, and graphene. The bare V, partially screened (Hubbard U) (cRPA), and fully screened (W) (RPA) parameters are given. The values in the ﬁrst and second rows of 2D h-BN and 3D h-BN for each Coulomb interaction parameter are related to N and B atoms, respectively. 2D h-BN 3D h-BN Graphene Bare cRPA RPA Bare cRPA RPA Bare cRPA RPA U0019.7 10.1 7.7 24.0 9.3 7.6 16.7 8.5 4.5 15.0 8.7 6.7 19.6 8.7 7.5 U01 8.4 5.1 3.8 8.9 3.4 2.7 8.5 4.0 1.5 8.4 5.1 3.8 8.9 3.4 2.7 U02 5.4 3.6 2.7 5.5 2.1 1.7 5.4 2.5 0.95.2 3.5 2.7 5.4 2.0 1.7 U 03 4.7 3.2 2.5 4.4 1.3 1.0 4.7 2.2 0.5 4.7 3.2 2.5 4.4 1.3 1.0 for 2D h-BN and three-dimensional bulk h-BN. In Sec. III B , we consider nonmetallic passivated nanoribbons with differ-ent widths with zigzag and armchair edge. In Sec. III C , we give the results for metallic nonpassivated nanoribbon(7-Zh-BNNR). A. Two-dimensional h-BN and three-dimensional bulk h-BN Considering Table I, we begin with the discussion of the bare V, partially U(cRPA), and fully W(RPA) Coulomb interaction parameters for 2D h-BN and 3D h-BN. The lo-cal (on-site U 00) and nonlocal (off-site U01,U02, and U03) Coulomb matrix elements are also reported. Indexes of00, 01, 02, and 03 are related to the strength of the po-tential that on-site, nearest-neighbor, next-nearest-neighbor,and third-nearest-neighbor parameters are feeling, respec-tively. For comparison, the corresponding results for pristinegraphene are also presented. Due to two inequivalent sub-lattices of h-BN, the corresponding Uvalues for B and N atoms are reported separately. Due to the existence of theband gap in 2D h-BN and 3D h-BN, the calculated on-site U values (Hubbard U) are larger than the corresponding values in graphene. In both sublattices of 2D h-BN, the screeningis very weak and we obtain comparatively large U 00values of 10.1 and 8.7 eV for B and N atoms, respectively. Thecorresponding U 00values of 3D h-BN for B and N atoms are 9.3 and 8.7 eV , respectively, which shows that Hubbard Uis about 1 eV smaller in the N sublattice. Moreover, the nonlocal Uvalues in 2D h-BN are weakly screened and the calculated off-site U01,U02, and U03parameters turn out to be rather sizable. In spite of the existence of the band gapin 3D h-BN, the off-site effective Coulomb interaction dropsfaster than graphene. It shows that Coulomb interaction of2D h-BN is weakly screened at short distances, while it isalmost unscreened at large distances. Screening of Coulombinteraction is closely related to the band gap of the materials.The GW approximation shows the 6 and 5.4 eV band gaps for2D h-BN and 3D h-BN, respectively, which is signiﬁcantlylarger than the 4.5 and 5.0 eV band gap predicted by the GGAapproximation [ 30,56,57]. Such remarkable many-body GW135 791 1 1 3 1 5 1 7 B-N atoms in supercell68106810 U, W (eV) 135 791 1 1 3 1 568106810 U-N U-B W-N W-B(a) 8-Ah-BNNR:H (b) 7-Zh-BNNR:H (c) 9-Ah-BNNR:H (d) 8-Zh-BNNR:H FIG. 4. Calculated partially screened on-site interaction U(blue line) and fully screened Coulomb interaction W(red line) of pzelec- trons for (a) 8-Ah-BNNR:H, (b) 7-Zh-BNNR:H, (c) 9-Ah-BNNR:H,and (d) 8-Zh-BNNR:H. corrections to the band gap are attributed to the large Coulomb interactions. B. Hydrogen-passivated nanoribbons: Armchair and zigzag edge In the following, we show the changes in on-site effec- tive Coulomb interaction (Hubbard U) and fully screened potential ( W) across the passivated nanoribbon’s unit cell in Fig.4. We will consider systems having two different widths in each group, i.e., 8-Ah-BNNR:H and 9-Ah-BNNR:H witharmchair edge and also 7-Zh-BNNR:H and 8-Zh-BNNR:Hwith zigzag edge. It can be observed that Coulomb parametersof h-BNNR:H are slightly smaller than the parameters for2D h-BN (see Table I). For example, the obtained U(W) parameters vary in the range 7.9–10.0 eV (6.2–8.0 eV) for8-Ah-BNNR:H. Similar to h-BN, the UandWparameters for the N atoms (unﬁlled points in Fig. 4) are about 2 eV larger than the corresponding ones for B atoms (ﬁlled pointsin Fig. 4) on the opposite sublattice. Moreover, for the inner atoms in the unit cell, we obtain a substantial reduction in theCoulomb interactions. This reduction in the Coulomb inter-action is more pronounced for zigzag system 7-Zh-BNNR:H.The reduction in UandWfor inner atoms and the difference of Coulomb parameters between the B and N atoms canbe described by the atom-projected density of states (DOS)presented in Fig. 5. Since the p zstates of N atoms are occu- pied, their contribution to the polarization function is small.In other words, the B atoms has a large conduction p zpeak around EF, while this peak is almost absent for N atoms. As a consequence, electronic screening increases signiﬁcantly dueto the contribution of the σ→p zandpz→pztransition and, as a result, gives rise to smaller Coulomb interactionparameters for the B atoms. 075427-4NONCONVENTIONAL SCREENING OF COULOMB … PHYSICAL REVIEW B 101, 075427 (2020) 00.40.8 pz py px s 00.4PDOS (states/eV/cell) 00.4 -6-4 -2 0 2 4 6 E - EF (eV)00.4 -6-4 -2 0 2 4 6 E - EF (eV)(a) 8-Ah-BNNR:H (edge B) (b) 8-Ah-BNNR:H (inner B) (c) 8-Ah-BNNR:H (edge N) (d) 8-Ah-BNNR:H (inner N) (e) 8-Zh-BNNR:H (edge B) (f) 8-Zh-BNNR:H (inner B) (g) 8-Zh-BNNR:H (edge N) (h) 8-Zh-BNNR:H (inner N) FIG. 5. Projected DOS for (a) edge B, (b) inner B, (c) edge N, and (d) inner N of 8-Ah-BNNR; and (e) edge B, (f) inner B, (g) edge N, and (h) inner N of 8-Zh-BNNR:H. It is instructive to compare our results with the recent study of the GNRs. For example, considering the armchair GNRs(AGNRs) with the same width, 8-AGNR:H, the calculated U(W) values turn out to be around 9.2 eV (5.5 eV), with very small variation from atom to atom [ 39]. These values are larger than the corresponding U(W) ones for B atoms and smaller than for N atoms in 8-Ah-BNNR:H. In contrastto 8-Zh-BNNR:H showing a large band gap, the existence ofedge states in GNRs with zigzag edges, 8-ZGNR:H, gives riseto a large contribution to the DOS exactly at E F, and thus the on-site Coulomb interaction reduces to a small value of5.1 eV at the edge. The U−Wdifferences in h-BNNR:H systems vary between 1.5 and 2.0 eV , while in GNR:H, thecorresponding values are around 3.5–4.0 eV . A small contri-bution of the p zstates around EFin both Ah-BNNR:H and Zh-BNNR:H systems results in the small screening through pz→pztransitions; as a consequence, U−Wdifferences become small for all atoms. So far we have only considered the on-site Coulomb inter- actions in h-BNNR:H. In the following, we will discuss theintersite Coulomb interaction parameters for both h-BNNRswith armchair and zigzag edges. In Fig. 6, the bare V, par- tially U, and fully screened Waverage intersite Coulomb interactions as a function of distance rbetween two atoms for 8-Ah-BNNR:H and 8-Zh-BNNR:H are presented. Theresults along the ribbon U(/bardbl)[W(/bardbl)] and across the ribbon U(⊥)[W(⊥)] are shown separately. As shown in Fig. 6, Coulomb interaction is only screened at short distances andis more or less unscreened at long distances. Furthermore,the nonlocal Coulomb interaction U(⊥)[W(⊥)] is slightly larger than the U(/bardbl)[W(/bardbl)]. Considering the central atoms04812 bare U (\|\|) W (\|\|) 051015 bare U (⊥) W (⊥) 03 691 2 1518 r (Å) 04812 U, W (eV) bare U ( \|\|) W (\| \|) 024 6 8 r (Å) 051015 bare U (⊥) W (⊥)(a) 8-Ah-BNNR:H (b) 8-Ah-BNNR:H (c) 8-Zh-BNNR:H (d) 8-Zh-BNNR:H FIG. 6. Bare, partially, and fully screened Coulomb interac- tion for B-N pzelectrons as a function of distance rfor (a), (b) 8-Ah-BNNR:H and (c),(d) 8-Zh-BNNR:H. Here, U(/bardbl)a n d U(⊥) correspond to interactions along the ribbon and across the ribbon, respectively. in 8-Ah-BNNR:H, V−Wbecome negative at an intersite distance of 25 Å, indicating an antiscreening of Coulomb in-teractions in these quasi-one-dimensional systems. Moreover,antiscreening is not seen across the ribbon. Antiscreeningmeans that the fully screened interaction Wis larger than the bare interaction V. This behavior is consistent with the fact that the screening is nonconventional in low-dimensionalsystems, i.e., at short distances, the Coulomb interaction isweakly screened, while at middle distances, it is antiscreened,and, ﬁnally, it is unscreened at long distances. In the case of 8-Ah-BNNR:H, antiscreening takes place between critical intersite distances r c1=25 Å and rc2= 74 Å. The rc1are the critical distances where the transition from screening to antiscreening occurs, and rc2are the crit- ical distances where the transition from antiscreening to un-screening takes place. Furthermore, moving from the centralatoms in the nanoribbons to the edge atoms, the antiscreeningweakens. In Fig. 7, we present the difference V−Was a function of distance rbetween atoms along the ribbon for the edge and central atoms of 9-Ah-BNNR:H and 6-Zh-BNNR:Hseparately. It is noteworthy that the antiscreening phenomenononly occurs for central atoms along the ribbon. From now on,we can only consider the central atoms to evaluate the strengthof antiscreening. To reveal the behavior of the Coulombinteraction at intermediate distance and the occurrence ofantiscreening, we have extended the calculations to muchwider nanoribbons, and we present the V−Wdifference forN a=2 (2-Ah-BNNR:H) to Na=11 (11-Ah-BNNR:H) andNz=3 (3-Zh-BNNR:H) to Nz=9 (9-Zh-BNNR:H) in Fig.8. As expected, the antiscreening region rc2−rc1and the negative values of V−Wtend to be reduced with increas- ing the ribbon width in both armchair and zigzag systems.For example, considering the smallest armchair nanoribbons 075427-5A. MONTAGHEMI et al. PHYSICAL REVIEW B 101, 075427 (2020) 20 40 60 80 r (Å)-0.00500.0050.010.0150.02 V-W (eV)edge N edge B inner B inner N 20 30 40 50-0.0100.010.020.03edge N edge B inner B inner N (a) 9-Ah-BNNR:H (b) 6-Zh-BNNR:H FIG. 7. The difference V−Wbetween bare interaction Vand fully screened interaction Was a function of distance rbetween B-N atoms along the ribbon for the edge and central atoms of (a) 9-Ah- BNNR:H and (b) 6-Zh-BNNR:H. 2-Ah-BNNR:H, antiscreening is observed between intersite distance of rc1=7.8 Å and rc2=92 Å, but antiscreening is not observed in wider armchair nanoribbons 11-Ah-BNNR:H.There is no antiscreening with a width larger than 12.6 Å forboth Ah-BNNR:H and Zh-BNNR:H. Consequently, there is acritical width such that if the nanoribbon width is larger thanthis threshold, the antiscreening effect is not observed. The antiscreening discussed above was conﬁrmed in one- dimensional semiconductors and large molecules [ 35,36]. Simply stated, if one electron is exposed to an electric ﬁeldof another electron, the medium responds by rearranging theother charges in such a way as to weaken the bare interac-tion between the two electrons, which is called screening.Similarly, antiscreening occurs when the medium enhancesthe bare interaction between the two electrons. To understandhow the antiscreening takes place, one can consider that themedium consists of point dipoles leaving around two point FIG. 8. The difference V−Wvalue as a function of distance rfor central atoms of (a) Na=2t o Na=11 Ah-BNNR:H and (b) Nz=3t oNz=9 Zh-BNNR:H.TABLE II. The value of critical distances rc1(transition from screening to antiscreening) and rc2(transition from antiscreening to unscreening) in Å for different one-dimensional systems, zero- dimensional molecules, and clusters. System rc1rc2 system rc1 rc2 2-Ah-BNNR:H 7.8 92 6-AGNR [ 39] 22 115 3-Ah-BNNR:H 8.3 90 7-AGNR [ 39] 22 110 4-Ah-BNNR:H 10 88 8-AGNR [ 39] 35 105 5-Ah-BNNR:H 12.5 86 9-AGNR [ 39]2 3 6 5 6-Ah-BNNR:H 17 85 CNT [ 36]2 0 ≈200 7-Ah-BNNR:H 21 78 C 60[58] 4.0 8-Ah-BNNR:H 25 74 Fe 2O3[59] 2.45 9-Ah-BNNR:H 28 68 Fe 3O4[59] 3.40 3-Zh-BNNR:H 10.5 49 Fe 4O6[59] 2.90 4-Zh-BNNR:H 13.5 45 Nb 4Co [60] 3.90 5-Zh-BNNR:H 16.5 41 Naphtalene [ 35] 2.5 6-Zh-BNNR:H 22 35 Benzene [ 35] 2.0 charges. We can divide point dipoles into two groups: screen- ing dipoles and antiscreening dipoles. Dipoles in the spacebetween the charges result in enhancement, i.e., antiscreening,of the bare interaction, whereas the other surrounding dipolesreduce, i.e., screen, the bare interaction. For one-dimensionalsystems, the ratio of the antiscreening region (space betweenthe charges) to the outside region is signiﬁcant. That iswhy antiscreening occurs in one-dimensional systems suchas carbon nanotubes, large organic molecules, and clusters[35,36,59,60]. As mentioned above, antiscreening is observed in several low-dimensional semiconductors and insulators. In order tocompare our results with them, we present in Table IIthe antiscreening parameters r c1andrc2for all considered materi- als in the literature. The critical distance of antiscreening thatwe found for h-BNNR:H is longer than the zero-dimensionalmolecules r c1=3−4Å[ 35], Fe xOyrc1=2−3Å[ 59], and Nb4Corc1=3.9 Å [ 60], but slightly shorter than the corre- sponding ones for quasi-one-dimensional single-wall carbonnanotubes r c1=20 Å [ 35,36] and AGNR:H rc1=20–30 Å [39]. Even rc1in the quasi-one-dimensional 2-Ah-BNNR:H is comparable to the Nb 4Co cluster. In addition to geometry, the polarizability of the atoms constituting the system plays an important role in the anti-screening since atoms of the systems can be considered asa collection of point dipoles. The polarizability of atoms,which is inversely proportional to the energy difference be-tween occupied and unoccupied states, reduces the magnitudeof the antiscreening contribution and increases the onset ofantiscreening r c1. This describes the smaller critical distance rc1and larger antiscreening contribution for h-BNNR:H com- pared to AGNR systems. The AGNR has more states aroundtheE Fwith respect to h-BNNR:H systems due to the small band gap. A similar discussion holds for a comparison ofantiscreening for edge and inner atoms. As shown in Fig. 6, the antiscreening along the ribbon for edge atoms is weakerthan the inner atoms and the critical distance of antiscreeningr c1at the edge is slightly longer. Then the antiscreening region rc2−rc1is shorter at the edge, which is in good agreement with the behavior of our calculated band gaps. 075427-6NONCONVENTIONAL SCREENING OF COULOMB … PHYSICAL REVIEW B 101, 075427 (2020) 024 68 1 01 21 4 B-N atoms in supercell4681012 U, W (eV) U (B) W (B) U (N) W (N) 048 1 2 16 20 r (Å)02468 U, W (eV)bare W (\|\|) U (\|\|)(a) (b) FIG. 9. (a) Hubbard Uand fully screened interactions Wfor 7-Zh-BNNR. (b) Bare, partially U, and fully Wscreened off-site interactions as a function of distance rfor 7-Zh-BNNR. As we mentioned, for both armchair and zigzag h-BNNR:H, the screening is small and nonconventional. Dueto this nonconventional screening, the calculated nonlocalinteractions turn out to be extremely large. Tightly boundexcitons are the consequence of this reduced dielectric screen-ing of nonlocal Coulomb interactions for low-dimensionalsystems. Such an enhanced excitonic state with bindingenergy about 2.5 eV has been observed experimentally inthe h-BNNR:H [ 32,33], which is in good agreement with ab initio calculations [ 32]. A similar strong excitonic state is observed in 2D semiconductor systems such as graphene [ 37], AGNR [ 34,41,42,45,61,62], ﬂuorographene [ 63], phospho- rene [ 64,65], and transition-metal dichalcogenides [ 66–71] such that these reduced dimensionality systems have a smalldielectric screening of the Coulomb interaction. C. Nonpassivated nanoribbons: Zigzag edge In the following, we will discuss Coulomb interaction parameters for nonpassivated zigzag h-BNNR of Nz=7 (7-Zh-BNNR). Nonpassivated armchair h-BNNRs (Ah-BNNRs) are semiconductors and their Coulomb interactionparameters are similar to passivated Ah-BNNRs:H. So we donot give the results for nonpassivated Ah-BNNRs. Bare h-BNNR with a zigzag edge are magnetic systems [ 24], but spin polarization has a weak inﬂuence on Coulomb interactions ofzigzag systems about 0.2 eV . So, the on-site and off-site U andWvalues presented in Figs. 9(a)and9(b) are for the non- spin-polarized case. Projected non-spin-polarized DOS forinner and edge atoms are also depicted in Fig. 10. Generally, due to the existence of metallic states around E F, Coulomb interaction UandWtend to be reduced at the edge as com- pared to passivated nanoribbons. The UandWparameters for the atoms in the 7-Zh-BNNR show strong variations. Thisvariation can be explained by the fact that as one moves frominner atoms to edge ones, the density of the pstate of the corresponding atoms at E Fincreases, which gives rise to the reduction of UandWparameters for edge atoms. Moreover, as seen in Fig. 10, the nonpassivated zigzag system exhibits more overlap of s,px,py, and pzwith respect to passivated systems. As a consequence, the correlated subspace is sp3in the nonpassivated zigzag h-BNRs. It can be found that thecalculated U(W) values of inner N atoms of 7-Zh-BNNR turn out to be 11.6 eV (9.8 eV). Due to metallic states, Coulomb 00.511.52PDOS(states/eV/cell) 00.20.40.6 -9 -6 -3 0 3 6 E - EF (eV)00.40.81.2 -9 -6 -3 0 3 6 E - EF (eV)00.20.40.6pzpy pxs(a) edge N1 (b) inner N7 (c) edge B14 (d) inner B8 FIG. 10. Orbital-projected DOS for (a) edge N [atom number 1 in Fig. 1(e)], (b) inner N (atom number 7), (c) edge B (atom number 14), and (d) inner B (atom number 8) of 7-Zh-BNNR. interaction is strongly screened for the edge N atoms and U reaches a relatively small value of 5.7 eV . Since sp3orbitals are considered as a correlated subspace, the contribution ofthesp 3→sp3transition gives rise to an enhancement of the electronic polarization, and we ﬁnd small Coulomb Wvalues. Now, we discuss the nonlocal screened Coulomb interac- tion presented in Fig. 9(b). The situation for the nonlocal in- teraction of nonpassivated 7-Zh-BNNR is different comparedto passivated systems. As expected, the large DOS can giverise to a signiﬁcant reduction of nonlocal Coulomb interactioneven at short distance. Nonlocal screened Coulomb interactionof passivated systems preserves at long distance, but for barezigzag nanoribbons, it is fully screened at short separationaround 5 Å. In contrast to the passivated zigzag nanorib-bons, the effective Coulomb interaction in 7-Zh-BNNR ispredominantly local due to the sharp drop of Uversus r,a s shown in Fig. 9(b). Therefore, due to the short-range nature of Coulomb interaction, the zigzag system can be interpreted asa correlated material. The edge states at E Fprovide a metallic phase in the non- spin-polarized calculation and make the system unstable uponlocal electron-electron interaction to induce spin-polarizedstates. Experimental results show that spin-polarized stateswere observed in nonpassivated zigzag nanoribbons [ 19,27]. As we discussed above, the large DOS at the vicinity ofE Fresults in small long-range Coulomb interaction even at short distance. Hence, the on-site Hubbard term is sufﬁcientto describe ferromagnetism in nonpassivated zigzag systems.Due to the Stoner criterion, UD(E F)>1 at edge atoms, and the nonpassivated system 7-Zh-BNNR therefore prefers theferromagnetic ground state. The DOS at the Fermi energyD(E F), magnetic moments, and the Stoner criterion UD(EF) are presented in Table IIIfor all atoms of the 7-Zh-BNNR system. Only atoms located at the edge of the ribbon satisfythe Stoner criterion UD(E F)>1 due to the large D(EF)a tt h e 075427-7A. MONTAGHEMI et al. PHYSICAL REVIEW B 101, 075427 (2020) TABLE III. DOS at Fermi level EF[D(EF)], Stoner criterion UD(EF), and magnetic moment (MM) in μBfor 7-Zh-BNNR nanoribbons. Atom D(EF)( 1/eV) UD(EF)M M ( i n μB) N1 2.5867 14.66 −0.849 B2 0.0302 0.23 0.077N3 0.5113 5.01 −0.130 B4 0.0219 0.21 0.009 N5 0.0688 0.77 −0.015 B6 0.0065 0.07 0.001 N7 0.0087 0.10 −0.002 B8 0.0030 0.03 0.001 N9 0.0028 0.03 −0.001 B10 0.0176 0.18 0.006N11 0.0150 0.17 −0.002 B12 0.1799 1.75 0.061 N13 0.1020 0.97 −0.005 B14 1.8350 11.28 0.842 EFand, as a result, the paramagnetic state is unstable towards the formation of ferromagnetism. Considering the zigzaggraphene nanoribbons 8-ZGNR:H, the calculated magneticmoment turns out to be around 0.35 μ Bat the edge and it becomes rapidly destroyed towards the center of the ribbon[39]. The magnetic moment of edge atoms in 7-Zh-BNNR (0.85μ B) is much larger than that of the GNRs with zigzag edges. The calculated magnetism persists for atoms at thevicinity of the edge and disappears towards the center of theribbon. IV . CONCLUSION We have studied the screening of the on-site and long-range Coulomb interactions for pelectrons of pristine 2D h-BN,bare h-BNNRs, and passivated h-BNNRs by employing ﬁrst- principles calculations in conjunction with the random-phaseapproximation. Screening strongly depends on the ribbonwidth, type of atoms (B or N), and the position of atoms in theunit cell. Our calculated Coulomb matrix elements consider-ably increase the predictive power of the model Hamiltoniansapplied to describe the electronic and magnetic propertiesof h-BN-based materials. We found sizable local UandW parameters for pristine 2D h-BN and passivated h-BNNRswith armchair edge and they are larger than the correspondingvalues in carbon-based materials. Reduced dimensionalityand the presence of the band gap results in a decreased andnonconventional screening of the Coulomb interaction, i.e.,Coulomb interaction is antiscreened at intermediate distancesbetween 8 and 90 Å. This antiscreening becomes weakeras one moves from the center to the edge of passivatedh-BNNRs. Sizable nonlocal parameters agree well with thestrong excitonic effect observed in the experiments. The crit-ical distance for the onset of antiscreening in h-BNNR islonger than in zero-dimensional molecules and clusters, butshorter than in GNRs and carbon nanotubes. Furthermore, theantiscreening effect disappears when the nanoribbon widthis larger than the critical width 12.5 Å. For bare zigzagh-BNNR, we ﬁnd that the interactions turn out to be local,and the nonlocal part is strongly screened due to the metallicstates. The pstates are very well screened, imposing a strong itinerant character of magnetism. We discuss the appearanceof ferromagnetism in 7-Zh-BNNR using the Stoner model. ACKNOWLEDGMENT The authors acknowledge the computational resources provided by the Physics Department of Tarbiat ModarresUniversity. [1] A. K. Geim and K. S. Novoselov, Nat. Mater. 6,183(2007 ). [2] M. I. 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PhysRevB.97.134504.pdf	PHYSICAL REVIEW B 97, 134504 (2018) Quasibound states in short SNS junctions with point defects A. A. Bespalov Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia (Received 13 February 2018; published 6 April 2018) Using the Green functions technique, we study the subgap spectrum of short three-dimensional superconductor– normal metal–superconductor junctions containing one or two point impurities in the normal layer. We ﬁnd thata single nonmagnetic or magnetic defect induces two quasibound Shiba-like states. If the defect is located closeto the junction edge, the energies of these states oscillate as functions of the distance between the impurity andthe edge. In the case of two nonmagnetic impurities, there are generally four quasibound states (two per spinprojection). Their energies oscillate as functions of the distance between the impurities, and reach their asymptoticvalues when this distance becomes much larger than the Fermi wavelength. The contributions of the impurities tothe Josephson current, local density of states, and to the normal-state conductance of the junction are analyzed. DOI: 10.1103/PhysRevB.97.134504 I. INTRODUCTION The effect of disorder on the superconducting correlations has been studied for several decades starting from the seminalpapers by Anderson [ 1] and Abrikosov and Gor’kov [ 2]. In many experiments the measurable quantities such as the critical temperature, electromagnetic response, and the density of states are well determined by their values averaged overdisorder. Still there exists a number of important experimentalsituations when the averaged quantities do not give us theinformation needed for the understanding the properties of aparticular sample. In this case we often need to focus on the properties of an individual impurity or defect and develop an appropriate theoretical description providing the informationnecessary for the experiment interpretation. One can enumeratethe following research directions for which the considerationof an individual impurity is crucial. First, the study of the local electronic characteristics around the individual impurity can help to identify the type of superconducting pairing. Indeed,the scattering process mixes different quasiparticle momentaand the resulting impurity states are sensitive to the momentumdependence of the superconducting order parameter [ 3]. Thus the impurity atom serves in some sense as a probe of the order parameter symmetry. Second, the solution of the scattering problem for a few impurities becomes necessary for rathersmall samples, i.e., at so-called mesoscopic length scales. Inthis regime the transport characteristics strongly ﬂuctuate fromsample to sample and the calculations of ensemble averagescan be irrelevant [ 4]. To sum up, individual impurities are known to produce well-observable local modiﬁcations of the electronic structure in normal and superconducting metals,including spatial oscillations of the density of states [ 5] and localized subgap states in superconductors with conventional[6–8] and unconventional pairing (see Ref. [ 3] for review). It should be also noted that the consideration of the problem with an individual impurity or a few of them allows often toget a deeper insight into the physics of processes which occurin large impurity ensembles. To give just a simple exampleone can recall the Larkin-Ovchinnikov solution [ 9]f o rt h e electronic structure of the superconducting vortex with a singleimpurity atom in the core which allowed one to understand the Landau-Zener mechanism of dissipation accompanying thevortex motion [ 9–11]. In this sense the present paper continues the series of works studying the effect of individual impuritieson the inhomogeneous superconducting states and focuses onthe analysis of the defects in a Josephson junction with a certainnonzero phase difference between the superconducting leads. Quite naturally the effect of impurities strongly depends on the system dimensionality. The simplest one-dimensional(1D) limit when the nonmagnetic defects can be describedby certain potential barriers has been previously consideredin a large number of papers (see Refs. [ 12,13] for review, and also Refs. [ 14–16]). The same can be said about the case of magnetic scatterers/barriers [ 12,13,17–21]. It is a more complicated task to analyze substantially 3D systems,e.g., impurities in bulk materials or wide wires. Indeed,little is known about the inﬂuence of individual impuritieson inhomogeneous superconducting systems. Covaci et al. [22,23] developed an efﬁcient numerical algorithm to solve the Bogoliubov–de Gennes equations self-consistently, andapplied this method to calculate the local density of states andJosephson current in 2D SNS junctions with disorder. Avotinaet al. [24] have studied mesoscopic conductance ﬂuctuations of a SN tunnel junction with a point impurity. Omelyanchouket al. [25] have calculated the stationary current in a Josephson junction with point impurities located in an oriﬁce betweenthe superconducting banks. In this work we present a generalapproach, which allows one to study the electronic structure ofinhomogeneous superconducting systems with one or severalpoint impurities. In particular, we provide an explicit methodto calculate the Green functions of a system with defects usingonly the Green function of a pure system. The main result of the paper is the discovery of a quasibound subgap state at a nonmagnetic impurity located inside thenormal layer of a SNS Josephson junction with a nonzero phasedifference ϕbetween the superconducting leads. This state is somewhat similar to the Yu-Shiba-Rusinov state [ 6–8] induced by a magnetic impurity in a homogeneous superconductor.In an inﬁnite junction the impurity state appears within the 2469-9950/2018/97(13)/134504(17) 134504-1 ©2018 American Physical SocietyA. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) FIG. 1. Subgap density of states of a short, but inﬁnitely wide SNS junction with a nonmagnetic point impurity. The spectrum of a cleanjunction [ 26], shown as a solid line, has a sharp peak at an energy E ∗≈ \|/Delta1\|cos(ϕ/2), where \|/Delta1\|is the gap in the superconducting banks. At energies higher than E∗there is still a nonzero density of states, and in this region the impurity state appears (dashed line, shown strongly exaggerated). continuous subgap spectrum of the system (see Fig. 1), and thus it has a complex energy: E=EA−iE/prime A, such that E/prime A/¯his the decay rate of the state. In a ﬁnite sized junction all subgap statesare localized and the spectrum is discrete. The impurity stateis a superposition Andreev states of the pure junction, whoseenergies are located in a window around E Awith a width E/prime A. This means that, as long as the distance between these energylevels is much smaller than E /prime A, the spatial structure and the decay rate of the impurity state are almost not modiﬁed. We analyze the behavior of the impurity states when the defect (either nonmagnetic or magnetic) is located close to thesurface of a semi-inﬁnite junction. We ﬁnd the oscillationsof the energies of the impurity states as functions of thedistance between the defect and the sample surface. Finally, fortwo nonmagnetic impurities inside the normal region we ﬁndtwo quasibound states (per spin projection), whose energiesstrongly depend on the relative positions of the defects. The structure of the paper is as follows. In Sec. IIour tech- nique of calculating the Green functions in SN systems with apoint impurity is described. We show how the Green functionsof a system with and without defects are related to each other. InSec.IIIwe analyze a short SNS junction with one nonmagnetic or magnetic impurity. The Green functions and the energiesof the impurity states are calculated and the inﬂuence of thesample surface on these states is considered. Section IVis devoted to SNS junctions with two nonmagnetic impurities. Here, we concentrate on the energies of the localized statesonly. In Sec. Vwe discuss how individual defects inﬂuence such measurable characteristics as the Josephson current andnormal-state conductance of the junction. In the conclusion themain results are summarized. II. BASIC EQUATIONS Within the mean-ﬁeld approach, a nonuniform supercon- ducting system with a point impurity can be characterized bythe Hamiltonian H=/summationdisplay αβ/integraldisplay ψ† α(r)[H0(r)δαβ+Vαβ(r−r1)]ψβ(r)d3r +/integraldisplay [/Delta1∗(r)ψ↑(r)ψ↓(r)+/Delta1(r)ψ† ↓(r)ψ† ↑(r)]d3r.(1) Here, ψ† αandψαare the electron creation and annihilation operators, respectively ( α,β=↑,↓are the spin indices), H0(r)=−¯h2∇2 2m−μ, (2) where mis the electron mass and μis the chemical potential, /Delta1(r) is the superconducting order parameter, and the matrix impurity potential Vαβis given by ˆV(r)=U(r)ˆσ0+J(r)ˆσ, (3) where U(r) is an electric potential, J(r) is an exchange ﬁeld, ˆσ0is a unit matrix, and ˆσare the Pauli matrices. We will concern ourselves only with the retarded Green function ˇGE(r,r/prime) of the system—this function contains all the information about single-particle characteristics, such asthe current density and the local density of states. The Greenfunction satisﬁes the Gor’kov equation [ 27]: /braceleftbigg ˆτ 0[H0(r)+U(r−r1)]+ˆτz[J(r−r1)ˆσ−E−i/epsilon1+] +/parenleftbigg 0 −/Delta1(r) /Delta1∗(r)0/parenrightbigg/bracerightbigg ˇGE(r,r/prime)=ˆτ0δ(r−r/prime). (4) Here, ˆτ0and ˆτzare a unit and a Pauli matrix in Nambu space, respectively, Eis the energy, and /epsilon1+is an inﬁnitely small positive quantity. The 4 ×4m a t r i x ˇGEhas the following block structure: ˇGE(r,r/prime)=/parenleftbiggˆGE(r,r/prime) ˆFE(r,r/prime) −ˆF† E(r,r/prime)ˆ¯GE(r,r/prime)/parenrightbigg . (5) For the deﬁnition of the 2 ×2 blocks in terms of the electron ﬁeld operators the reader may refer to Appendix A.N o w ,w e can explicitly write down the local density of states ν(E,r): ν(E,r)=π−1Im[GE↑↑(r,r)+GE↓↓(r,r)]. (6) Our strategy for solving Eq. ( 4) is to relate ˇGE(r,r/prime)t ot h e Green function ˇG(0) E(r,r/prime) of the superconductor without the impurity, i.e., with ˆV(r)=0. The latter Green function can be then determined using well-developed quasiclassical methods(see below). Speaking about the point impurity, we imply that the range of its potential V(r) is much smaller than the Fermi wavelength. Then, it is a predominantly isotropic scatterer, i.e., an effectivesource of spherical waves. In this sense, it acts similar to theδ-function source in the right-hand side of Eq. ( 4). Given this, we seek the solution of Eq. ( 4) in the form ˇG E(r,r/prime)=ˇG(0) E(r,r/prime)+ˇG(0) E(r,r1)ˇA(r1,r/prime), (7) where ˇA(r1,r/prime) is an unknown matrix, which is deﬁned by the behavior of ˇG(0) E(r,r/prime) in the vicinity of the impurity, i.e., r≈r1. The problem of calculating ˇA(r1,r/prime) is relatively simply 134504-2QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) solved in 1D, where we can put ˆV(r)=(Uˆσ0+Jˆσ)δ(r)( 8 ) and determine the Green function from a Dyson equation, as described in Ref. [ 14]. Such a solution is possible because ˇG(0) E(r1,r1) is ﬁnite in 1D; however, this is not the case in higher dimensions. In Appendix Bwe illustrate how to calculate ˇA(r1,r/prime) by a simpler example of a nonmagnetic and nonsuperconducting system. In short, the procedure consistsin separating in Eq. ( 7) the regular part of ˇG E(r,r/prime)a tr=r1 and the ∝\|r−r1\|−1singularity. Then, using basic scattering theory, a relation between these parts can be obtained [seeEq. ( B7)]. This generally yields 16 coupled linear algebraic equations for the components of ˇA(r 1,r/prime). The situation is signiﬁcantly simpliﬁed in our case, since we are dealing withone impurity in a nonmagnetic environment. Then, the Greenfunction of the pure system is proportional to the unit matrix in spin space: ˆG (0) E=G(0) Eˆσ0,ˆF(0) E=F(0) Eˆσ0,ˆ¯G(0) E=¯G(0) Eˆσ0, and ˆF†(0) E=F†(0) Eˆσ0. For brevity, we introduce the notation ˆG(0) E(r,r/prime)=/parenleftBigg G(0) E(r,r/prime)F(0) E(r,r/prime) −F†(0) E(r,r/prime)¯G(0) E(r,r/prime)/parenrightBigg . (9) If we direct the spin quantization axis along the impurity spin [parallel to J(r)], the components of ˇGE(r,r/prime) with spin indices ↑↓and↓↑vanish, and the equations for the components with indices ↑↑and↓↓decouple. For spin “up” components Eq. ( 7)takes the form/parenleftBiggGE↑↑(r,r/prime)FE↑↑(r,r/prime) −F† E↑↑(r,r/prime)¯GE↑↑(r,r/prime)/parenrightBigg =ˆG(0) E(r,r/prime)+ˆG(0) E(r,r1)/parenleftBigg A1↑(r1,r/prime)A3↑(r1,r/prime) A2↑(r1,r/prime)A4↑(r1,r/prime)/parenrightBigg .(10) When writing down conditions of the form ( B7) for the Green functions, it should be born in mind that the pairs of func- tionsGE↑↑,FE↑↑and ¯GE↑↑,F† E↑↑“feel” different scattering potentials, because the exchange term appears with oppositesigns for these pairs in Eq. ( 4). Thus the impurity produces a scattering phase α ↑forGE↑↑andFE↑↑and a scattering phase α↓for¯GE↑↑andF† E↑↑. Applying the expansion ( B7)t o GE↑↑(r,r/prime) andF† E↑↑(r,r/prime) near r=r1, we obtain the relations mA 1↑ 2π¯h2=tanα↑ kF/bracketleftbig G(0) E(r1,r/prime)+A1↑G(0) ER(r1,r1) +A2↑F(0) E(r1,r1)/bracketrightbig , (11) −mA 2↑ 2π¯h2=tanα↓ kF/bracketleftbig F†(0) E(r1,r/prime)+A1↑F†(0) E(r1,r1) −A2↑¯G(0) ER(r1,r1)/bracketrightbig , (12) where kF=(2mμ/ ¯h2)1/2is the Fermi wave number. When writing down the solution of the linear Eqs. ( 11) and ( 12), we will use the relations ¯G(0) E=G(0)∗ −EandF(0) E=F†(0)∗ −E[these follow from Eq. ( 4)]: A1↑(r1,r/prime)=D−1 ↑/braceleftbigg G(0) E(r1,r/prime)/bracketleftbiggmkFcotα↓ 2π¯h2−G(0)∗ −ER(r1,r1)/bracketrightbigg −F†(0)∗ −E(r1,r1)F†(0) E(r1,r/prime)/bracerightbigg , (13) A2↑(r1,r/prime)=−D−1 ↑/braceleftbigg F†(0) E(r1,r/prime)/bracketleftbiggmkFcotα↑ 2π¯h2−G(0) ER(r1,r1)/bracketrightbigg +F†(0) E(r1,r1)G(0) E(r1,r/prime)/bracerightbigg , (14) D↑=/bracketleftbiggmkFcotα↑ 2π¯h2−G(0) ER(r1,r1)/bracketrightbigg/bracketleftbiggmkFcotα↓ 2π¯h2−G(0)∗ −ER(r1,r1)/bracketrightbigg +F†(0) E(r1,r1)F†(0)∗ −E(r1,r1). (15) To determine GE↓↓andF† E↓↓one should simply swap α↑ andα↓in Eqs. ( 11)–(15). The functions FEααand ¯GEααcan be calculated in a similar manner, but we will not write downthe corresponding relations here, since the functions G Eααare sufﬁcient to determine all one-particle characteristics. At this point, it can be seen that the impurity may cause the appearance of additional poles of the Green function at energieswhen the denominator D ↑vanishes. As we know, the poles of the Green function at real and almost real energies deﬁne thelocalized states of the system, and thus the zeros of D ↑(E)m u s t be related to impurity-induced bound or quasibound states. To complete the calculation of the Green functions, the matrix ˆG(0) E(r,r/prime) should be determined. This can be done using quasiclassical methods, which are applicable as longas the superconducting coherence length ξ=¯hv F/(π\|/Delta1\|)i s much larger than the Fermi wavelength λF=2πk−1 F(here, vF=¯hkF/mis the Fermi velocity). A summary of the relevant quasiclassical methods is given in Appendix C. Here, we writedown the most important formula for the Green function with close arguments— \|r−r/prime\|/lessmuchξ: ˆG(0) E(r,r/prime)=imkF 2π¯h2/integraldisplay ˆgE(r/prime,n)eikFn(r−r/prime)d2n 4π +m 2π¯h2cos(kF\|r−r/prime\|) \|r−r/prime\|ˆτ0, (16) where integration goes over a unit sphere, and ˆgE(r,n)=/parenleftbigggE(r,n)fE(r,n) −f† E(r,n)−gE(r,n)/parenrightbigg (17) is the conventional quasiclassical Green function, satisfying the Eilenberger equations [ 27,28]. 134504-3A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) FIG. 2. Inﬁnite SNS junction with a pointlike impurity. III. SHORT SNS JUNCTION WITH ONE IMPURITY A. Green functions Let us apply the theory developed in Sec. IIto a short Josephson SNS junction shown in Fig. 2. Let the order parameters in the left and right superconducting banks be equalto\|/Delta1\|e −iϕ/2and\|/Delta1\|eiϕ/2, respectively. We use a model with an abrupt order parameter proﬁle, which is justiﬁed if thecharacteristic scale of spatial variations of the gap is muchsmaller than the coherence length, like in superconductor-constriction-superconductor weak links [ 29]. However, for a start we consider an inﬁnite system and ﬁnite-size effects willbe discussed in Sec. III D. We assume that the system contains a point defect located at r=r 1in theNlayer at distances of L1andL2from the left and right superconducting banks, respectively. Correspondingly,the total width of the Nlayer is L=L 1+L2andL/lessmuchξ. To calculate the density of states in the presence of the impurity, we need the Green functions of the pure system.These are determined in Appendix C: G(0) ER(r1,r1)≈mkF 4π¯h2/bracketleftbigg cot/parenleftbigg γ(E)+ϕ 2/parenrightbigg +cot/parenleftbigg γ(E)−ϕ 2/parenrightbigg/bracketrightbigg , (18) F†(0) E(r1,r1)≈mkF 4π¯h2/bracketleftbigg sin−1/parenleftbigg γ(E)+ϕ 2/parenrightbigg +sin−1/parenleftbigg γ(E)−ϕ 2/parenrightbigg/bracketrightbigg , (19) where γ(E)=arccos/parenleftbiggE \|/Delta1\|/parenrightbigg . (20) Corrections to Eqs. ( 18) and ( 19) are much smaller than mkF/(4π¯h2) as long as \|sin(γ(E)±ϕ/2)\|∼1. (21) In particular, estimates of the imaginary parts of the Green functions are given in Appendix D.B. Quasilocalized states at a nonmagnetic impurity Now we will focus on the denominator D↑of the Green function to ﬁnd possible localized impurity states. First, weconsider a nonmagnetic impurity, such that α ↑=α↓=α.B y substituting Eqs. ( 18) and ( 19) into Eq. ( 15) we obtain D↑≈m2k2 F 4π2¯h4/bracketleftBigg cot2α+sin2γ(E) sin2γ(E)−sin2ϕ 2/bracketrightBigg . (22) The equation D↑=0 can now be solved with respect to sinγ(E): sin2γ(E)≡1−E2/\|/Delta1\|2=sin2(ϕ/2) cos2α. Thus the Green functions have a pole at an energy approximatelyequal to E≈E A(ϕ)=\|/Delta1\|/radicalbigg 1−sin2ϕ 2cos2α. (23) This corresponds to a spin-degenerate Andreev state localized at the impurity. In fact, as stated above, it is a quasibound(resonant) state, since any quasiparticle in the SNS junctioncan “leak” to inﬁnity along the Nlayer. Hence the energy of the resonance must have an imaginary part, E=E A−iE/prime A, E/prime A>0 (the retarded Green function has poles only in the lower complex half-plane). Formally, this happens due to the functions G(0) ER(r1,r1) andF†(0) E(r1,r1) having imaginary parts. These imaginary parts and the width of the impurity resonanceare estimated in Appendix D. The result for E /prime Ais E/prime A∼L ξ\|/Delta1\|sin2ϕ 2sin4α. (24) Thus, for short Josephson junctions E/prime A/lessmuch/Delta1, so that there is a well-deﬁned resonance. For longer junctions with L/greaterorsimilarξthere is no resonance. For completeness, we will describe the structure of the quasibound state. The electronlike part u(r) and a holelike part v(r) of its wave function satisfy the Bogoliubov–de Gennes equations [ 27]. In a nonmagnetic system these equations are identical to those for the functions GEαα(r,r/prime) andF† Eαα(r,r/prime) with the only difference being the absence of δ(r−r/prime)i nt h e right-hand side. Hence u(r) andv(r)h a v et h ef o r m u(r)=A1G(0) EA(r,r1)+A2F(0) EA(r,r1), (25) v(r)=A1F†(0) EA(r,r1)−A2¯G(0) EA(r,r1), (26) where A1andA2are some constants. The local density of states at the energy EA—ν(EA,r)—is proportional to \|u(r)\|2. Then, according to Eqs. ( C15) and ( C16), for\|r−r1\|/greatermuchλF the envelope of the density of states decays as ν(EA,r)∝\|r−r1\|−2exp⎛ ⎝−2/radicalBig \|/Delta1\|2−E2 A ¯hvF\|r−r1\|⎞ ⎠. (27) Additionally, ν(EA,r) exhibits spatial oscillations with the period λF/2. In this respect, the impurity-induced quasibound state resembles the Shiba state induced by a magnetic impurityin a bulk superconductor [ 6–8]. The main difference from the Shiba state is the strong anisotropy of the ν(E A,r) proﬁle, as can be seen in Fig. 3. 134504-4QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) FIG. 3. Spatial proﬁles of the density of states [Eq. ( 6)] in a plane passing through the zaxis and through the impurity—see Fig. 2.ν(E,r) is measured in units of the density of states in a normal metal— ν0=m/(¯h2λF). In all graphs, α=0.81,L/ξ=1/20, the impurity is located in the center of the Nlayer (and in the center of the pictures), and the size of the area shown is 4 .46λF×4.46λF. The coherence length is assumed much larger than the dimensions of this area, so for the Green function of a clean system Eqs. ( 16), (C9), and ( C10) are used (with the integral evaluated numerically). (a)–(d) Proﬁles of the impurity state for different values of ϕ:E=EA(ϕ). Note the logarithmic color scale, which is used to make several oscillations of ν(EA,r) visible. (e), (f) Proﬁles of ν(E,r) at energies close to the peak of the density of states of a clean junction: E≈\|/Delta1\|cos(ϕ/2). C. Quasilocalized states at a magnetic impurity Here we will generalize the results of the previous section for the case of a magnetic impurity, when α↑/negationslash=α↓.U s i n g Eqs. ( 15), (18), and ( 19), we can write the determinant D↑in the following form: D↑≈m2k2 F[cos(2 /Theta1)−cos(2γ−α↑+α↓)] 4π2¯h4sinα↑sinα↓[cosϕ−cos(2γ)], (28) where /Theta1=arcsin/radicalBigg cosα↑cosα↓sin2ϕ 2+sin2/parenleftbiggα↑−α↓ 2/parenrightbigg .(29)Equating D↑to zero, we obtain sin/parenleftbigg γ+α↓−α↑ 2−/Theta1/parenrightbigg sin/parenleftbigg γ+α↓−α↑ 2+/Theta1/parenrightbigg =0 or γ=α↑−α↓ 2±/Theta1+πn, (30) where nis an integer. To determine the valid values of nwe take into account that γ∈[0,π]. As follows from Eq. ( 29), /vextendsingle/vextendsingle/vextendsingle/vextendsingleα ↑−α↓ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle</Theta1<π 2. 134504-5A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) FIG. 4. Energies of “spin-up” ( ↑) and “spin-down” ( ↓) impurity states in a short SNS junction. In (b) a special case is depicted, when α↑+α↓=0, and there is no gap between the branches of E(ϕ) with the same spin. Hence γcan only take the values γ=α↑−α↓ 2+π 2±/parenleftbiggπ 2−/Theta1/parenrightbigg . (31) This yields the real parts of the energies of quasibound “spin- up” impurity states: E↑=\|/Delta1\|sin/parenleftbiggα↓−α↑ 2±/parenleftbiggπ 2−/Theta1/parenrightbigg/parenrightbigg . (32) For “spin-down” quasiparticles, we need to swap α↑ andα↓: E↓=\|/Delta1\|sin/parenleftbiggα↑−α↓ 2±/parenleftbiggπ 2−/Theta1/parenrightbigg/parenrightbigg . (33) Similar relations have been previously derived for Andreev states in a short Josephson junction with a ferromagneticbarrier [ 30]. As before, the energies of the impurity states have imaginary parts of the order of \|/Delta1\|L/ξ. Equations ( 32) and ( 33) give four values of the energy, two of them being positive and two being negative, exceptfor the case when some values are exactly zero. Of course, theBogoliubov quasiparticles have positive energies; thus thereare generally two quasibound states localized by a magneticimpurity. Let us consider some particular cases. When α ↑=α↓=α we have E↑=±EAandE↓=±EA[Eq. ( 23)], in accordance with Sec. III B. When ϕ=0, we should obtain the energies of Shiba states in a bulk superconductor. Indeed, one can seethat in this case the positive values of E ↑andE↓are\|/Delta1\|and\|/Delta1\|\|cos(α↑−α↓)\|, the latter being the energy of the Shiba state with zero orbital momentum [ 8]. For a nonmagnetic impurity the dependence of EAonϕ is qualitatively similar for all values of the phase α, and has no zero crossings: EA/greaterorequalslant0. For a magnetic impurity qualitatively different behaviors of the bound state energiesvsϕare possible depending on α ↑andα↓. To illustrate this, we ﬁrst calculate the zero crossings of E↑(ϕ) andE↓(ϕ). Using the fact that γ(0)=π/2, we ﬁnd from Eq. ( 31) that at least one branch of both E↑(ϕ) and E↓(ϕ) takes the value zero when sin2ϕ 2=cos(α↑−α↓) cosα↑cosα↓. (34) Hence zero crossings are present if and only if the following conditions are met: cos(α↑−α↓)>0, (35) sinα↑sinα↓<0. (36) At each crossing point one branch of E↑(ϕ) goes from positive to negative with growing ϕ, and one branch of E↓(ϕ) goes from negative to positive, or vice versa. Such a situation is depictedin Figs. 4(a)and4(b). Consider the case when one of the conditions ( 35)o r (36) is violated—both conditions can be violated only in the degenerate case when one of the scattering phases equals±π/2, so that E ↑andE↓do not depend on ϕ. If cos( α↑− α↓)/lessorequalslant0, for all values of ϕthere are two bound states with the same spin. In particular, there are two “spin-up” states for 134504-6QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) FIG. 5. Semi-inﬁnite SNS junction with an impurity. α↓>α↑(E↑/greaterorequalslant0) and two “spin-down” states for α↓<α↑ (E↓/greaterorequalslant0)—see Fig. 4(c). On the contrary, when the condition (36) is violated, we have one spin-up and one spin-down state for all ϕ—see Fig. 4(d). D. Semi-inﬁnite SNS junction In the previous sections we studied the impurity states in an idealized spatially unlimited system, so the obtainedresults cannot be directly applied to real structures, which haveboundaries. In this section, we will analyze how the impuritystates are affected by a ﬂat surface. In particular, we considerhere a SNS junction occupying the half-space y< 0—see Fig. 5. The half-space y> 0 is occupied by vacuum or an insulator. An impurity is located at the point r 1=(x1,−h,z 1) inside the normal layer. We assume that there is an inﬁnitely high potential barrier at the surface of the junction, so that the following boundarycondition should be imposed: ˇG(r,r /prime)\|y=0=0. (37) Then, the Green function without the impurity, which we will denote as ˆG(S) E(r,r/prime), can be determined using the image method: ˆG(S) E(r,r/prime)=ˆG(0) E(r,r/prime)−ˆG(0) E(r,r/prime/prime), (38) where by ˆG(0) E(r,r/prime) we mean the Green function of a pure inﬁnite junction, r/prime=(x/prime,y/prime,z/prime), and r/prime/prime=(x/prime,−y/prime,z/prime). Let us assume that his much smaller than the coherence length ξ. Using Eqs. ( C17) and ( C27), we ﬁnd that the regular part of the Green function is G(S) ER(r1,r1)≡G(0) ER(r1,r1)−G(0) E(r1,r1+2hy0) =G(0) ER(r1,r1)/bracketleftbigg 1−sin(2kFr) 2kFr/bracketrightbigg −mcos(2kFh) 4π¯h2h, (39) where y0is a unit vector directed along the yaxis and G(0) ER(r1,r1) is given by Eq. ( 18). Similarly, F†(S) E(r1,r1)=F†(0) E(r1,r1)/bracketleftbigg 1−sin(2kFr) 2kFr/bracketrightbigg , (40) withF†(0) E(r1,r1) given by Eq. ( 19). If we substitute G(S) ER(r1,r1) andF†(S) E(r1,r1) into Eq. ( 15), after some algebra we may ﬁndFIG. 6. Dependencies of the impurity state energy Eon the distance hbetween the impurity and the surface of the SNS junction forϕ=π. The scattering phases αare positive in graph (a) and negative in graph (b). that the equations for the impurity states take the same form as for an inﬁnite junction with the scattering phases replaced byeffective values α eff↑,↓, which are deﬁned as follows: cotαeff↑,↓=cotα↑,↓+cos(2kFh) 2kFh 1−sin(2kFh) 2kFh. (41) Note that h=0 results in αeff↑,↓=0, i.e., there is effectively no impurity. A remarkable feature of Eq. ( 41) is the oscillating αeff↑,↓vshdependence, which results in the energies of the impurity states also oscillating when his changed. This simple example illustrates the mesoscopic ﬂuctuations in a systemwith one impurity. These ﬂuctuations decay on a length scaleof the order of k −1 F, and hence the inﬂuence of the surface on the energies of the impurity states is small when his much larger than the Fermi wavelength. Typical energy vs hgraphs for a nonmagnetic impurity are shown in Fig. 6. 134504-7A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) FIG. 7. SNS junction with two impurities. IV . BOUND STATES IN A SHORT SNS JUNCTION WITH TWO IMPURITIES In this section we will determine the energies of resonant states in a short SNS junction with two impurities. The systemunder consideration is shown in Fig. 7. Here, one impurity with a scattering phase α 1is located at the origin. The other impurity with a scattering phase α2is located at the point r2= (x2,y2,z2), such that z2/greaterorequalslant0. The vector r2makes an angle β with the zaxis. As before, we can ﬁnd the impurity states in this system by calculating the poles of the Green function. However, even withonly two impurities this method becomes very laborious. Forthis reason, we will use here a different method developed byBeenakker [ 31], which yields the energies of all Andreev states in short nonmagnetic Josephson junctions ( L/lessmuchξ). Theseenergies have the form E i(ϕ)=\|/Delta1\|/radicalbigg 1−Tisin2ϕ 2, (42) where Tiis an eigenvalue of the matrix ˆT=ˆtˆt†, with ˆtbeing the transmission matrix of electrons with the energy μ(E=0) through the normal layer. For the Andreev states that are notaffected by impurities one has ideal transmission, i.e., T i= 1. Hence the impurity-induced states correspond to nonunitvalues of T i. Note that Eq. ( 23) is a particular case of Eq. ( 42) withTi=cos2α. Thus the central result of Sec. III B can be also interpreted as follows: a single nonmagnetic impurity witha scattering phase αaffects one transport channel in the normal layer, reducing the transparency of this channel from unity tocos 2α. The coefﬁcients Tifor the case of two point impurities are calculated in Appendix E. It is shown that all coefﬁcients Tiare equal to unity except for two, which we denote as T1andT2. Hence we have two quasibound states per spin projection. The general expressions for T1andT2are rather cumber- some [Eqs. ( E16), (E17), and ( E19)], but we can analyze some limiting cases. For example, if the impurities are very close toeach other— k Fr2/lessmuch1—they are equivalent to one impurity with a scattering phase α0given by Eq. ( E22). When the distance between the impurities is larger— kFr2/greaterorsimilar 1—the coefﬁcients T1andT2are oscillating functions of both β andr2, as can be seen in Fig. 8. Thus we have another example of sample-to-sample ﬂuctuations of the impurity state energies,somewhat similar to those present in a semi-inﬁnite junction(Sec. III D). FIG. 8. (a) Dependencies of T1,2[Eq. ( E19)] onβfor several values of r2. (b)–(d) Dependencies of T1,2onr2for various values of α1,α2, andβ. It can be seen that with growing r2the transparencies T1,2approach their asymptotic values signiﬁcantly faster when \|α1\|/negationslash=\|α2\|than in the case when \|α1\|=\|α2\|. 134504-8QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) AtkFr2/lessmuch1 one expects that the impurities are more or less independent, so that T1,2≈cos2α1,2. One can prove (see Appendix E) that the differences between T1,T2and their asymptotic values are of the order of ( kFr2)−2when\|α1\|/negationslash=\|α2\| and of the order of ( kFr2)−1when\|α1\|=\|α2\|. V . OBSERV ABLE IMPLICATIONS Let us discuss observable implications of the results above. First, individual impurity states can be observed using theSTM technique [ 32], which has been recently applied for the visualization of Shiba states in a 2D superconductor [ 33]. This technique allows one to determine the energies of the resonantstates as well as their approximate spatial structure. Second, the impurities certainly affect the transport char- acteristics of the Josephson junction. Using the theory ofBeenakker [ 31] for short SNS junctions, we can estimate the inﬂuence of defects on the Josephson current. This current isgiven by I(ϕ)=e\|/Delta1\| 2 2¯hsinϕ/summationdisplay iTi Ei(ϕ)tanh/parenleftbiggEi(ϕ) 2T/parenrightbigg , (43) where eis the elementary charge, Ei(ϕ)i sg i v e nb yE q .( 42), Tis the temperature, and summation goes over all eigenvalues Tiof the matrix ˆtˆt†. Without impurities, the current is I0(ϕ)=e\|/Delta1\|N ¯hsin(ϕ/2) tanh/parenleftbigg\|/Delta1\|cos(ϕ/2) 2T/parenrightbigg , (44) If we add one impurity, it switches one transmission eigenvalue fromTi=1t oTi=cos2α, and the current is modiﬁed by a value δI(ϕ)=e\|/Delta1\| 2¯hsinϕ⎡ ⎢⎣cos2αtanh/parenleftBig\|/Delta1\|√ 1−cos2αsin2(ϕ/2) 2T/parenrightBig /radicalbig 1−cos2αsin2(ϕ/2) −1 cos(ϕ/2)tanh/parenleftbigg\|/Delta1\|cos(ϕ/2) 2T/parenrightbigg⎤ ⎥⎦. (45) As we have seen, two identical impurities separated by a distance much larger than k−1 Fswitch two eigenvalues Ti to cos2α. This fact can be generalized: a sufﬁciently small amount of evenly distributed impurities nin a junction with many channels— N/greatermuch1—should change neigenvalues Ti from unity to cos2α. Here, “sufﬁently small amount” means at leastn/lessmuchN; however, the actual restriction might be stronger and its derivation is beyond the scope of this work. If thedefects are distributed randomly in the Nlayer, of course, there is a chance that the distance between some defects is/lessorsimilark −1 F, and also some defects may be located near the boundary of the junction. However, on the average the number of suchimpurities is much smaller than n, provided that N/greatermuch1 and n/lessmuchN. Thus we may conclude that the average contribution ofnpoint impurities to the Josephson current is nδI(ϕ). Then, the averaged over many samples Josephson current is ¯I(ϕ)=I 0(ϕ)+¯nδI(ϕ), (46) where ¯ndenotes the average number of defects in the Nlayer. If the quantity nobeys a Poisson distribution, the root-mean-square deviation of the current from its average is /radicalBig (I−¯I)2=¯n\|δI(ϕ)\|. (47) In a similar way, we can calculate the normal-state con- ductance Gof the Nlayer. In the presence of nimpurities, according to Landauer’s formula [ 34], the conductance is given by G=2e2 π¯h(N−nsin2α). (48) This can be interpreted as each impurity effectively reducing the cross section of the Nlayer by the scattering cross section σS[Eq. ( B6)]. Now we can also determine the average conduc- tance and its root-mean-square deviation from the average: ¯G=2e2 π¯h(N−¯nsin2α), (49) /radicalBig (G−¯G)2=2e2 π¯h¯nsin2α. (50) Equations ( 47) and ( 50) give the Josephson current and con- ductance ﬂuctuations in the low-disorder limit, complementingBeenakker’s results [ 31] derived in the limit of a diffusive N layer. VI. CONCLUSION In conclusion, we have analyzed the subgap spectral fea- tures due to nonmagnetic and magnetic point impurities ina short clean SNS junction. A single defect induces twoquasibound states, which are somewhat similar to Shiba statesin a uniform superconductor. The energies of the impuritystates are given by Eqs. ( 32) and ( 33) [which simplify to Eq. ( 23) in the case of a nonmagnetic defect], and the widths of the resonances are of the order of \|/Delta1\|L/ξ. If the defect is close to a ﬂat surface of the junction, the impurity state energiesexhibit decaying oscillations as a function of the distance h between the impurity and the sample surface. We have also studied a system with two nonmagnetic im- purities, as shown on Fig. 7. Here, there are two quasilocalized states per spin projection, the energies of which depend ina complex manner on both the length and orientation of thevector r 2connecting the impurities. Finally, we have derived the sample-to-sample ﬂuctuations of the Josephson current and normal-state conductance of thejunction in the pure limit, i.e., in the presence of a relativelysmall amount of impurities. ACKNOWLEDGMENTS I am very grateful to A. S. Mel’nikov for stimulating discussions and for thorough reading of this paper. The workhas been supported by Russian Science Foundation Grant No.17-12-01383 (Secs. IIandIII), Foundation for the advancement of theoretical physics BASIS Grant No. 109 (Sec. IV), and Russian Foundation for Basic Research Grant No. 18-02-00390 (Sec. V). 134504-9A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) APPENDIX A: DEFINITIONS OF THE GREEN FUNCTIONS In this paper we use the following deﬁnition of the real-time retarded Green functions: Gαβ(r,r/prime,t)=i/angbracketleftψα(r,t)ψ† β(r/prime)+ψ† β(r/prime)ψα(r,t)/angbracketrightT,(A1) F† αβ(r,r/prime,t)=/summationdisplay γσyαγ/angbracketleftψ† γ(r,t)ψ† β(r/prime)+ψ† β(r/prime)ψ† γ(r,t)/angbracketrightT, (A2) Fαβ(r,r/prime,t)=/summationdisplay γσyβγ/angbracketleftψα(r,t)ψγ(r/prime)+ψγ(r/prime)ψα(r,t)/angbracketrightT, (A3) ¯Gαβ(r,r/prime,t)=i/summationdisplay γ/epsilon1σyαγσyβ/epsilon1/angbracketleftψ† γ(r,t)ψ/epsilon1(r/prime) +ψ/epsilon1(r/prime)ψ† γ(r,t)/angbracketrightT (A4) fort>0, and Gαβ=F† αβ=Fαβ=¯Gαβ=0f o rt<0. Here, /angbracketleft ···/angbracketrightTdenotes the thermodynamic average and ψα(r,t)i st h e Heisenberg operator: ψα(r,t)=exp(iHt/¯h)ψα(r)e x p (−iHt/¯h). (A5) The matrices with the subscript “ E,” appearing in Eq. ( 5), are the Fourier transforms of the matrices given by Eqs. ( A1)– (A4) with respect to t. Note that our deﬁnitions of the Green functions are somewhat different from those used by Kopnin[27]. In our case, it is convenient to incorporate the ˆ σ ymatrices in the deﬁnitions ( A1)–(A4) to eliminate them in the Gor’kov equation ( 4). APPENDIX B: GREEN FUNCTION OF A NORMAL SYSTEM WITH A POINT IMPURITY In this Appendix we illustrate our method of calculating the Green function in the presence of a point defect by a relativelysimple example of a normal system. The problem of scatteringof a free particle by δ-type potentials has been previously ex- tensively studied in literature. A summary of the main obtainedresults and their strict derivations can be found in Ref. [ 35]. Here, in a somewhat voluntary way we extend the previous re-sults to the case of an inhomogeneous 3D system with a defect. We will determine the retarded Green function G E(r,r/prime)o f the following Schrödinger equation: [H0(r)+U0(r)+V(r−r1)−E−i/epsilon1+]GE(r,r/prime)=δ(r−r/prime), (B1) where H0(r)i sg i v e nb yE q .( 2),V(r) is the potential of a pointlike impurity, and U0(r) is some electric potential not related to the impurity. For simplicity, we put the impurityoutside the range of the potential U 0(r), so that U0(r1)=0. We will assume that the solution G(0) E(r,r/prime)o fE q .( B1) without the impurity [with V(r)=0] is known. It has a regular part and a singularity, which we separate explicitly: G(0) E(r,r/prime)=m 2π¯h2\|r−r/prime\|+G(0) ER(r,r/prime), (B2) where G(0) ER(r,r/prime) is a regular function at all arguments.In the presence of the impurity, assuming that it is small enough so that only spherically symmetric scattering can betaken into account, we can write the solution of Eq. ( B1)i nt h e form G E(r,r/prime)=Gin(r,r/prime)+CGin(r1,r/prime)eik\|r−r1\| k\|r−r1\|, (B3) where the function Gin(r,r/prime) is regular at r=r1and represents an incoming wave, kis the wave number of a free electron, and Cis a coefﬁcient that depends on the impurity potential. By virtue of Hermiticity of the Hamiltonian, a relation betweenthe imaginary part of Cand its modulus exists, namely Im(C)=\|C\| 2, (B4) which is a corollary of the so-called optical theorem. Its derivation can be found in textbooks [ 35,36]. Equation ( B4) implies that Ccan be written as C=eiαsinα, (B5) where αis the energy-dependent scattering phase of the impurity. Note that Cisπperiodic in α, so we may conﬁne αto the range [ −π/2,π/2], with α=±π/2 corresponding to the so-called unitary limit. The opposite Born approximationlimit corresponds to \|α\|/lessmuch1. The scattering cross section of the impurity is given by σ S=4πk−2sin2α. (B6) For the following it is more convenient to rewrite Eq. ( B3)i n the form GE(r,r/prime)=GER(r,r/prime)+GER(r1,r/prime)t a nα k\|r−r1\|, (B7) where GER(r,r/prime)=Gin(r,r/prime)+Gin(r1,r/prime)eiαsinαeik\|r−r/prime\|−1 k\|r−r/prime\| (B8) is another regular function at r=r1. Now, one can see that GE(r,r/prime) andG(0) E(r,r1)h a v et h es a m e \|r−r1\|−1singularity, and hence the solution of Eq. ( B1) can be also written as GE(r,r/prime)=G(0) E(r,r/prime)+G(0) E(r,r1)A(r1,r/prime), (B9) where A(r1,r/prime) is a function to be determined. Indeed, the function given by Eq. ( B9) has the correct asymptotic behavior near the defect [Eq. ( B7)], and also satisﬁes Eq. ( B1) when r andr/primelie outside the range of the impurity potential. To calculate A(r1,r/prime) we note that Eqs. ( B7) and ( B9) should give the same function GE(r,r/prime). After extracting the regular part of the right-hand side of Eq. ( B9)a t r=r1by using Eq. ( B2) with r/prime=r1, we obtain the algebraic equation mA(r1,r/prime) 2π¯h2=tanα kF/bracketleftbig G(0) E(r1,r/prime)+A(r1,r/prime)G(0) ER(r1,r1)/bracketrightbig . (B10) Here, we replace kwith the Fermi wave number kF, assuming that\|E\|/lessmuchμ. After solving Eq. ( B10), we obtain the Green 134504-10QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) function GE(r,r/prime)=G(0) E(r,r/prime)+G(0) E(r,r1)G(0) E(r1,r/prime) mkFcotα 2π¯h2−G(0) ER(r1,r1).(B11) It is important that Eq. ( B11) gives the exact Green function when both randr/primelie outside the impurity. If desired, Born approximations of any order can be deduced from Eq. ( B11). Thus our result should be identical to the result obtainedusing the T-matrix method, which involves the summation ofdiagrams corresponding to Born approximations of all orders.Indeed, the reader may check that the T-matrix method yieldsEq. (B11) in the spatially homogeneous case [when U 0(r)=0] [3]. Finally, we shortly discuss how to generalize the obtained result for a magnetic impurity. The Green function ˆGEand the impurity potential ˆV(r) then will be 2 ×2 matrices in spin space. One typically writes the potential in the form ( 3). If we direct the spin quantization axis along the impurity spin[parallel to J(r)], electrons with spin “up” and spin “down” will have different scattering phases— α ↑andα↓, respectively. Then, the components of the matrix Green function ˆGEwith spin indices ↑↓and↓↑vanish, and the components with indices ↑↑or↓↓are still given by Eq. ( B11), where αis replaced with α↑orα↓, respectively. APPENDIX C: QUASICLASSICAL CALCULATION OF THE GREEN FUNCTIONS IN CLEAN SUPERCONDUCTORS In this Appendix we will recall some old results concerning the quasiclassical methods applied to clean superconductors,and we will derive Eq. ( 16). In the end, expressions for the Green functions with noncoinciding arguments for a short SNSjunction will be given. Gor’kov and Kopnin [ 37] pointed out that in a clean system without potential barriers it is convenient to write the Green function ˆG (0) E(r,r/prime)a tkF\|r−r/prime\|/greatermuch1 in the following form: ˆG(0) E(r,r/prime)=m 2π¯h2R[ˆgE+(r/prime,R,n)eikFR +ˆgE−/parenleftbig r/prime,R,n/parenrightbig e−ikFR], (C1) where R=\|r−r/prime\|andn=(r−r/prime)/Ris a unit vector. In the limitξ/greatermuchk−1 Fa set of quasiclassical equations for ˆg+ Eandˆg− E can be derived. By substituting Eq. ( C1) into Eq. ( 4) we can obtain the Andreev equations [ 37] ∓i¯hvF∂ ∂RˆgE±(r/prime,R,n) +/parenleftbigg −E−/Delta1(r) /Delta1∗(r) E/parenrightbigg ˆgE±(r/prime,R,n)=0, (C2) where nis a unit vector and r=r/prime+Rn. When deriving Eq. ( C2) differentiation with respect to the directions of nhas been neglected, which is justiﬁed when ¯h2 mR2\|δn\|2/lessmuch\|/Delta1\|, (C3)where δnis the characteristic variation scale of ˆg± E(r/prime,R,n) with respect to n. Equation ( C3) can be rewritten as \|δn\|2/greatermuchξ kFR2. (C4) The uniqueness of the solution of Eq. ( C2) is provided by the boundary conditions [ 37] ˆgE±(r/prime,0,n)=1 2[ˆτ0±ˆgE(r/prime,±n)], (C5) where ˆgE(r,n) is the conventional quasiclassical Green func- tion, which satisﬁes the Eilenberger equation [ 27,28] with the normalization condition ˆg2 E=ˆτ0. At small Rthe condition ( C4) is obviously violated. How- ever, a different representation of the Green function maybe obtained for \|r−r /prime\|/lessmuchξ. At such small length scales in the left-hand side of the Gor’kov equation ( 4) all terms can be neglected except for the kinetic energy. Hence the Greenfunction can be written as ˆG(0) E(r,r/prime)=/integraldisplay ˆg/prime E(r/prime,n/prime)eikFn/prime(r−r/prime)d2n/prime+m 2π¯h2eikF\|r−r/prime\| \|r−r/prime\|ˆτ0, (C6) where integration goes over a unit sphere and ˆg/prime E(r/prime,n)i sa function that will be determined shortly. We note that a rangeof distances R/lessmuchξmay exist where Eq. ( C4) is satisﬁed. In that range both Eqs. ( C1) and ( C6) are applicable, and hence should be equivalent to each other. When both k FR/greatermuch1 and Eq. ( C4) are satisﬁed, the integral in Eq. ( C6) can be calculated by the stationary phase method, yielding ˆG(0) E(r,r/prime)=eikFR R/bracketleftbiggm 2π¯h2−2i kFˆg/prime E(r/prime,n)/bracketrightbigg +2πie−ikFR kFRˆg/prime E(r/prime,−n). (C7) Comparing this with Eq. ( C1), we ﬁnd that ˆg/prime E(r/prime,n)=imkF 4π2¯h2[ˆgE+(r/prime,0,n)−ˆτ0] =−imkF 4π2¯h2ˆgE−(r/prime,0,−n). (C8) Finally, taking into account Eq. ( C5), we obtain Eq. ( 16) [38]. Using the approach outlined above, we will now determine the Green functions G(0) E(r,r/prime) andF†(0) E(r,r/prime) for a clean short SNS junction (Fig. 2without the defect). We will assume that the vector r/primelies in the normal layer. For a start, we need the solutions of the Eilenberger equation, which are known[39]: g E(r1,n)=icot/bracketleftbigg(E+i/epsilon1+)L ¯hvF\|nz\|−γ(E+i/epsilon1+)−ϕ 2sgn(nz)/bracketrightbigg , (C9) f† E(r1,n)=iexp/parenleftbigiE(L2−L1) ¯hVFnz/parenrightbig sin/bracketleftbig(E+i/epsilon1+)L ¯hvF\|nz\|−γ(E+i/epsilon1+)−ϕ 2sgn(nz)/bracketrightbig, (C10) 134504-11A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) where γ(E)i sg i v e nb yE q .( 20). We can put here γ(E+ i/epsilon1+)=γ(E), since the term i/epsilon1+L/(¯hvF\|nz\|)i nE q s .( C9) and ( C10) already yields the necessary imaginary part in the denominators. Equations ( C9) and ( C10) are valid for a cleanSNS junction of any length, and they allow one to reproduce the long-known zigzag pattern of the density of states [ 26,40,41]. The Green functions with coinciding arguments are deter- mined using Eq. ( 16): G(0) ER(r1,r1)=−mkF 4π¯h2/integraldisplay1 −1cot/bracketleftbiggEL ¯hvF\|nz\|+i/epsilon1+−γ(E)−ϕ 2sgn(nz)/bracketrightbigg dnz, (C11) F†(0) E(r1,r1)=−mkF 4π¯h2/integraldisplay1 −1exp/parenleftbigiE(L2−L1) ¯hVFnz/parenrightbig dnz sin/bracketleftbigEL ¯hvF\|nz\|+i/epsilon1+−γ(E)−ϕ 2sgn(nz)/bracketrightbig. (C12) We may note that for most of the directions of the vector nthe ratio EL/ (¯hvF\|nz\|) is small and may be neglected (unless \|nz\|/lessorsimilar\|E\|L/¯hvF). Doing so, we obtain Eqs. ( 18) and ( 19). When calculating gE±, for most directions of nwe may additionally disregard the fact that /Delta1(r)=0i nE q .( C2)f o rR< L1,2/\|nz\|(as long as L/\|nz\|/lessmuchξ). Effectively, this means that we can ignore the normal layer, putting L=0. Solving Eq. ( C2) with the boundary conditions ( C5) then yields gE±(r/prime,R,n)≈±iexp/parenleftBig ±iγ(E)+iϕ 2sgn(nz)−√ \|/Delta1\|2−E2 ¯hvFR/parenrightBig 2s i n/parenleftbig −γ(E)∓ϕ 2sgn(nz)/parenrightbig , (C13) f† E±(r/prime,R,n)≈±iexp/parenleftBig −√ \|/Delta1\|2−E2 ¯hvFR/parenrightBig 2s i n/parenleftbig −γ(E)∓ϕ 2sgn(nz)/parenrightbig. (C14) Let us substitute this into Eq. ( C1): G(0) E(r,r/prime)=mi 4π¯h2R/bracketleftBigg e−ikFR−iγ(E) sin/parenleftbig γ(E)−ϕ 2sgn(nz)/parenrightbig−eikFR+iγ(E) sin/parenleftbig γ(E)+ϕ 2sgn(nz)/parenrightbig/bracketrightBigg exp/parenleftBigg iϕ 2sgn(nz)−/radicalbig \|/Delta1\|2−E2 ¯hvFR/parenrightBigg ,(C15) F†(0) E(r,r/prime)=mi 4π¯h2R/bracketleftBigg e−ikFR sin/parenleftbig γ(E)−ϕ 2sgn(nz)/parenrightbig−eikFR sin/parenleftbig γ(E)+ϕ 2sgn(nz)/parenrightbig/bracketrightBigg exp/parenleftBigg −/radicalbig \|/Delta1\|2−E2 ¯hvFR/parenrightBigg . (C16) These relations are valid for kFR/greatermuch1,L/\|nz\|/lessmuchξ, and when Eq. ( 21) holds. To calculate the Green functions for R/lessmuchξ,w e may use Eq. ( 16) together with Eqs. ( C9) and ( C10). In the short-junction limit we may again neglect all terms containing L,L1, andL2to obtain G(0) E(r,r/prime)≈mkF 4π¯h2/bracketleftBig K(kF(r−r/prime)) cot/parenleftBig γ(E)+ϕ 2/parenrightBig +K∗(kF(r−r/prime)) cot/parenleftBig γ(E)−ϕ 2/parenrightBig/bracketrightBig +mcos(kF\|r−r/prime\|) 2π¯h2\|r−r/prime\|, (C17) F†(0) E(r,r/prime)≈mkF 4π¯h2/bracketleftBig K(kF(r−r/prime)) sin−1/parenleftBig γ(E)+ϕ 2/parenrightBig +K∗(kF(r−r/prime)) sin−1/parenleftBig γ(E)−ϕ 2/parenrightBig/bracketrightBig , (C18) where K(R)=/integraldisplay nz>0einRdn 2π. (C19) Here, integration goes over a unit hemisphere. In the ﬁnal part of this appendix we will derive some properties of the function K(R). First, we point out three rather obvious relations: K(0)=1, (C20) \|K(R)\|/lessorequalslant1, (C21) K(−R)=K∗(R). (C22) Now we will transform Eq. ( C19). For this, we introduce an auxiliary coordinate frame ( x/prime,y,z/prime), such that the z/primeaxis is directed along the vector R; see Fig. 9. This vector makes an angle βwith the zaxis, and for now we assume that β/lessorequalslantπ/2. Let θandφbe 134504-12QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) the polar and azimuthal angles in the ( x/prime,y,z/prime) frame, respectively. Then, nz=sinθcosφsinβ+cosθcosβ. Equation ( C19) can be transformed as follows: K(R)=/integraldisplay/integraldisplay cosφ>−cotθcotβsinθeiRcosθdφdθ 2π=/integraldisplayπ 2−β 0sinθeiRcosθdθ+/integraldisplayπ 2+β π 2−βsinθeiRcosθarccos (−cotβcotθ)dθ π =eiR−eiRsinβ iR+/integraldisplaysinβ −sinβeixRarccos/parenleftbigg −x√ 1−x2cotβ/parenrightbiggdx π. (C23) After integration by parts we obtain K(R)=eiR iR+icosβ πR/integraldisplaysinβ −sinβeixRdx (1−x2)/radicalbig sin2β−x2=eiR iR+2icosβ πR/integraldisplaysinβ 0cos(xR)dx (1−x2)/radicalbig sin2β−x2. (C24) It follows from this that Re[K(R)]=sinR R, (C25) Im[K(R)]=2 cosβ πR/integraldisplay1 0cos(xRsinβ)dx√ 1−x2(1−x2sin2β) −cosR R. (C26) Additionally, Eqs. ( C22) and ( C25) yield K(R)=sinR Rwhenβ=π 2. (C27) Moreover, there must be Im[ K(0)]=0[ E q .( C20)], so that 2 cosβ π/integraldisplay1 0dx√ 1−x2(1−x2sin2β)=1 (C28) for any β∈[0,π/2). Using this, we can rewrite Eq. ( C26)a s Im[K(R)]=2 cosβ πR/integraldisplay1 0cos(xRsinβ)−cosR√ 1−x2(1−x2sin2β)dx. (C29) This relation is also valid for β∈[π/2,π], because the integral is ﬁnite and the property ( C22) is satisﬁed. It follows from (C28) and ( C29) that \|Im[K(R)]\|/lessorequalslant4\|cosβ\| πR/integraldisplay1 0dx√ 1−x2(1−x2sin2β)=2 R, (C30) FIG. 9. Coordinate system we use to calculate the function K(R). The unit hemisphere over which we integrate is highlighted in gray.and hence \|K(R)\|/lessorequalslant√ 5 R. (C31) Finally, if we substitute x=sintin Eq. ( C29), we obtain K(R)=sinR R+2icosβ πR ×/integraldisplayπ/2 0cos(Rsinβsint)−cosR 1−sin2tsin2βdt. (C32) This form is particularly convenient for numerical calculations, because the integrand is a limited and continuous function ofβandt. Indeed, /vextendsingle/vextendsingle/vextendsingle/vextendsinglecos(Rsinβsint)−cosR 1−sin2tsin2β/vextendsingle/vextendsingle/vextendsingle/vextendsingle =R 2 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2s i n/parenleftBig R 1−sinβsint 2/parenrightBig R(1−sinβsint)2s i n/parenleftBig R1+sinβsint 2/parenrightBig R(1+sinβsint)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantR 2 2. (C33) This relation allows one to obtain another estimate for Im[K(R)]: \|Im[K(R)]\|/lessorequalslant2\|cosβ\| πR/integraldisplayπ/2 0R2 2dt=\|Rz\| 2. (C34) APPENDIX D: IMAGINARY PARTS OF GREEN FUNCTIONS WITH COINCIDING ARGUMENTS In this Appendix we estimate the imaginary parts of the functions G(0) ER(r1,r1) and F†(0) E(r1,r1), given by Eqs. ( C11) and ( C12), and the width of the impurity resonance, E/prime A.T h e imaginary part of G(0) ER(r1,r1) originates from the poles of the integrand in Eq. ( C11). It can be proven that Im[cot( x+i/epsilon1+)]=−π+∞/summationdisplay n=−∞δ(x−πn). (D1) 134504-13A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) Then Im/bracketleftbig G(0) ER(r1,r1)/bracketrightbig =mkF 4¯h2/integraldisplay1 −1+∞/summationdisplay n=−∞δ/parenleftbiggEL ¯hvF\|nz\|+γ(E)−ϕ 2sgn(nz)/parenrightbigg dnz =mkF\|E\|L 4¯h3vF⎧ ⎪⎨ ⎪⎩/summationdisplay πn>\|E\|L ¯hvF−γ(\|E\|)−ϕ 2/bracketleftBig γ(\|E\|)+ϕ 2+πn/bracketrightBig−2 +/summationdisplay πn>\|E\|L ¯hvF−γ(\|E\|)+ϕ 2/bracketleftBig γ(\|E\|)−ϕ 2+πn/bracketrightBig−2⎫ ⎪⎬ ⎪⎭.(D2) It can be seen that when the condition ( 21) is satisﬁed, then Im/bracketleftbig G(0) ER(r1,r1)/bracketrightbig ∼mkF\|E\|L 4¯h3vF. (D3) The imaginary part of F†(0) E(r1,r1) has a similar contribution due to the poles of the integrand in Eq. ( C12). In addition, there is generally a larger contribution originating from the exponent in the numerator of the integrand. Applying the Taylor expansionto this exponent, we obtain Im/bracketleftbig F†(0) E(r1,r1)/bracketrightbig ≈−mkFE(L2−L1) 4π¯h3vF/braceleftbigg/integraldisplay1 \|E\|L/¯hvFsin−1/parenleftBig −γ(E)−ϕ 2/parenrightBigdnz nz+/integraldisplay\|E\|L/¯hvF −1sin−1/parenleftBig −γ(E)+ϕ 2/parenrightBigdnz nz/bracerightbigg =mkFE(L2−L1) 4π¯h3vFln/parenleftbigg¯hvF \|E\|L/parenrightbigg/bracketleftBigg 1 sin/parenleftbig γ(E)+ϕ 2/parenrightbig−1 sin/parenleftbig γ(E)−ϕ 2/parenrightbig/bracketrightBigg , (D4) assuming that Eq. ( 21) holds, and L2−L1∼L. Thus Im/bracketleftbig F†(0) E(r1,r1)/bracketrightbig ∼mkF\|E\|L 4π¯h3vFln/parenleftbigg¯hvF \|E\|L/parenrightbigg . (D5) Now we will estimate the width of the impurity resonance, E/prime A. First, we note that at real energies the imaginary part of the determinant D↑[Eq. ( 15)] equals Im[D↑]=−2I m/bracketleftbig G(0) ER(r1,r1)/bracketrightbig Re/bracketleftbig G(0) ER(r1,r1)/bracketrightbig −Re/bracketleftbig F†(0) E(r1,r1)/bracketrightbig Im/bracketleftbig F†(0) −E(r1,r1)/bracketrightbig +Re/bracketleftbig F†(0) −E(r1,r1)/bracketrightbig Im/bracketleftbig F†(0) E(r1,r1)/bracketrightbig .(D6) According to Eqs. ( D3) and ( D5), the ﬁrst line of Eq. ( D6) is generally much smaller than the second and third lines;however, the latter two lines almost cancel each other out, asfollows from Eqs. ( 19) and ( D5). Hence Im[D ↑]∼2I m/bracketleftbig G(0) ER(r1,r1)/bracketrightbig Re/bracketleftbig G(0) ER(r1,r1)/bracketrightbig ∼m2k2 F ¯h4\|E\|L ¯hvF, (D7) as long as Eq. ( 21) holds. For a complex energy E=EA−iE/prime A with a small imaginary part we have Im[D↑(EA−iE/prime A)]≈Im[D↑(EA)]−E/prime A∂D↑ ∂E(EA),(D8) where ∂D↑/∂E is determined using Eq. ( 22). Equating Im[ D↑] to zero, we obtain Eq. ( 24). APPENDIX E: ELECTRON TRANSMISSION THROUGH A NORMAL LAYER WITH TWO IMPURITIES In this Appendix we calculate the eigenvalues of the matrix ˆT=ˆtˆt†for the Nlayer with two impurities—see Fig. 7.F i r s t , the transmission matrix ˆtshould be determined. For this, we deﬁne a set of propagating electron modes in the Nlayer. Toobtain a ﬁnite set of such modes, we apply periodic boundary conditions in the xyplane. Then, the orthogonal wave functions of electrons propagating from left to right through a clean N layer are ψ(0) n(r)=/radicalbiggm ¯h2knzS⊥eiknr, (E1) where knz>0,\|kn\|=kF, andS⊥is the cross section of the N layer. The allowed values of the wave vectors knare determined by the boundary conditions in the xyplane. Note that for calculations of scattering matrices we need to normalize thewave functions so that they carry a unit total current: −S ⊥¯h2i 2m/parenleftbigg ψ(0)∗ n∂ψ(0) n ∂z−ψ(0) n∂ψ(0)∗ n ∂z/parenrightbigg =1. (E2) In the presence of impurities electrons are scattered, so that their wave functions become ψn(r)=/radicalbiggm ¯h2knzS⊥/parenleftbigg eiknr+A1eikFr kFr+A2eikF\|r−r2\| kF\|r−r2\|/parenrightbigg . (E3) The amplitudes A1andA2are determined from Eqs. ( B3) and (B5) (with the Green function replaced by the wave function) applied to each impurity. This yields [ 42] A1=D−1 2eiα1sinα1/parenleftbigg 1+eikFr2+iα2+iknr2sinα2 kFr2/parenrightbigg ,(E4) A2=D−1 2eiα2sinα2/parenleftbigg eiknr2+eikFr2+iα1sinα1 kFr2/parenrightbigg ,(E5) D2=1−e2ikFr2+iα1+iα2 k2 Fr2 2sinα1sinα2. (E6) 134504-14QUASIBOUND STATES IN SHORT SNS JUNCTIONS WITH … PHYSICAL REVIEW B 97, 134504 (2018) The transmission matrix is deﬁned as follows: tn/primen=/integraltext/integraltext S⊥ψn(x,y,Z )ψ(0)∗ n/prime(x,y,Z )dxdy /integraltext/integraltext S⊥/vextendsingle/vextendsingleψ(0) n/prime(x,y,Z )/vextendsingle/vextendsingle2dxdy, (E7) where Zis any number larger than z2. Here, the numerator contains two integrals of the form I=/integraldisplay/integraldisplay S⊥eikF\|r−ri\| \|r−ri\|e−ikn/primer/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=Zdxdy, (E8)where ri=0o r ri=r2, and we will denote the components ofrias (xi,yi,zi). To evaluate the integral in Eq. ( E8), we ﬁrst make an inverse Fourier transform: eikF\|r−ri\| \|r−ri\|=/integraldisplayeiq(r−ri) q2−k2 F−i/epsilon1+d3q 2π2. (E9) Using this, integration over xandyin Eq. ( E8) becomes straightforward, and after that integration over qxandqycan be performed. We have then I=2e x p ( −ikn/primezZ−ikn/primexxi−ikn/primeyyi)/integraldisplay+∞ −∞eiqz(Z−zi)dqz q2z−k2 n/primez−i/epsilon1+ =4πiexp(−ikn/primezZ−ikn/primexxi−ikn/primeyyi)R e s qz=kn/primezeiqz(Z−zi) q2z−k2 n/primez=2πik−1 n/primeze−ikn/primeri. (E10) This relation allows one to obtain tn/primen=δn/primen+2πiD−1 2S−1 ⊥√knzkn/primezkF/bracketleftbigg eiα1sinα1/parenleftbigg 1+eikFr2+iα2+iknr2sinα2 kFr2/parenrightbigg +eiα2−ikn/primer2sinα2/parenleftbigg eiknr2+eikFr2+iα1sinα1 kFr2/parenrightbigg/bracketrightbigg .(E11) The Hermitian conjugate to this matrix is deﬁned in the usual way: t† n/primen=t∗ nn/prime. It can be seen that ˆtandˆt†have a common invariant subspace W, which is spanned by the vectors ρandνwith components ρn=/parenleftbigg2π knzkFS⊥/parenrightbigg1/2 ,νn=/parenleftbigg2π knzkFS⊥/parenrightbigg1/2 e−iknr2. (E12) In the orthogonal complement of Wthe operators ˆtandˆt†act as identity operators. Hence the eigenvectors of ˆTcorresponding to nonunit eigenvalues lie in the subspace W. In the basis {ρ,ν}the operators ˆtandˆt†have the matrices ˆt/primeandˆt†/prime, respectively: /parenleftbiggˆtρ ˆtσ/parenrightbigg =/parenleftbigg t/prime 11t/prime 12 t/prime 21t/prime 22/parenrightbigg/parenleftbigg ρ σ/parenrightbigg ,/parenleftbiggˆt†ρ ˆt†σ/parenrightbigg =/parenleftBigg t†/prime 11t†/prime 12 t†/prime 21t†/prime 22/parenrightBigg/parenleftbigg ρ σ/parenrightbigg , (E13) Here, we will illustrate how to calculate one of the components of these matrices. By deﬁnition, ˆtρ=t/prime 11ρ+t/prime 21σ; hence t/prime 11=1+/summationdisplay n2πieiα1sinα1 knzkFS⊥D2/parenleftbigg 1+eikFr2+iα2+iknr2sinα2 kFr2/parenrightbigg =1+ieiα1sinα1 D2/integraldisplay/integraldisplay k⊥<kF/parenleftbigg 1+eikFr2+iα2+ikr2sinα2 kFr2/parenrightbiggd2k⊥ 2πkzkF, (E14) where k⊥is the perpendicular to the zaxis component of kandkz=/radicalBig k2 F−k2 ⊥. To obtain the second line of Eq. ( E14), we went from summation to integration using the common substitution /summationdisplay n→S⊥ (2π)2/integraldisplay d2k⊥. Introducing the unit vector n=k/kF, we can rewrite Eq. ( E14) in the form t/prime 11=1+ieiα1sinα1 D2/integraldisplay nz>0/parenleftbigg 1+eikFr2+iα2+ikFnr2sinα2 kFr2/parenrightbiggd2n 2π=1+ieiα1 D2sinα1/bracketleftbigg 1+eiα2+ikFr2sinα2 kFr2K(kFr2)/bracketrightbigg .(E15) All other components of ˆt/primeandˆt†/primecan be calculated in a similar way, and the result is ˆt/prime=⎛ ⎜⎝1+ieiα1 D2sinα1/bracketleftBig 1+eiα2+ikFr2sinα2 kFr2K(kFr2)/bracketrightBig ieiα1 D2sinα1/bracketleftBig K∗(kFr2)+eiα2+ikFr2sinα2 kFr2/bracketrightBig ieiα2 D2sinα2/bracketleftBig K(kFr2)+eiα1+ikFr2sinα1 kFr2/bracketrightBig 1+ieiα2 D2sinα2/bracketleftBig 1+eiα1+ikFr2sinα1 kFr2K∗(kFr2)/bracketrightBig⎞ ⎟⎠, (E16) 134504-15A. A. BESPALOV PHYSICAL REVIEW B 97, 134504 (2018) ˆt†/prime=⎛ ⎜⎝1−ie−iα1 D∗ 2sinα1/bracketleftBig 1+e−iα2−ikFr2sinα2 kFr2K(kFr2)/bracketrightBig −ie−iα1 D∗ 2sinα1/bracketleftBig K∗(kFr2)+e−iα2−ikFr2sinα2 kFr2/bracketrightBig −ie−iα2 D∗ 2sinα2/bracketleftBig K(kFr2)+e−iα1−ikFr2sinα1 kFr2/bracketrightBig 1−ie−iα2 D∗ 2sinα2/bracketleftBig 1+e−iα1−ikFr2sinα1 kFr2K∗(kFr2)/bracketrightBig⎞ ⎟⎠. (E17) Note that t†/prime n/primen/negationslash=t/prime∗ nn/prime, because the vectors ρandσare not orthogonal. The nonunit eigenvalues of the matrix ˆTare determined by the equation det( ˆt/primeˆt†/prime−ˆ1Ti)=0, or T2 i−Tr(ˆt/primeˆt†/prime)Ti+det(ˆt/primeˆt†/prime)=0. (E18) The solution of this equation is T1,2=Tr(ˆt/primeˆt†/prime)±/radicalbig [Tr(ˆt/primeˆt†/prime)]2−4 det( ˆt/primeˆt†/prime) 2. (E19) Below we will analyze the behavior of T1andT2in several interesting cases. First, let the impurities be very close to eachother: k Fr2/lessmuch1. Then, they can be roughly described as one impurity with some scattering phase α0. Indeed, for r/greatermuchr2 Eq. ( E3) takes the form ψn(r)≈1√knzS⊥/parenleftBigg eiknr+exp/parenleftbig ikF/vextendsingle/vextendsingler−r2 2/vextendsingle/vextendsingle+iα0/parenrightbig kF/vextendsingle/vextendsingler−r2 2/vextendsingle/vextendsinglesinα0/parenrightBigg , (E20) with eiα0sinα0=(cotα0−i)−1≈(A1+A2)e x p/parenleftbigg−iknr2 2/parenrightbigg . (E21) Using the Taylor expansion of exp( ikFr2), we obtain from Eqs. ( E4)–(E6) and ( E21) that tanα0=sin(α1+α2)+2 sinα1sinα2 kFr2 cos(α1−α2)−sinα1sinα2 k2 Fr2 2. (E22) It turns out that α0is a nontrivial function of α1,α2, andr2. When kFr2/lessmuch\|sinα1sinα2\|,E q .( E22) yields α0≈ −2kFr2—the resulting scattering phase is small and does not depend on α1andα2. In the opposite limit, k2 Fr2 2/greatermuch \|sinα1sinα2\|,w eh a v e α0≈α1+α2. On the other hand, when \|α1\|,\|α2\|/lessmuch1,α 1α2≈k2 Fr2 2, (E23)we obtain α0≈π/2, even though both α1andα2are small. When Eq. ( E23) is satisﬁed, the determinant D2is small. This corresponds to a resonant state, which is present even in asystem without superconductivity. The fact that we effectively have one point defect in the limit k Fr2/lessmuch1 means that only one mode in the Nlayer is affected by disorder, which results in T1≈1 andT2≈cos2α0. Let us now place the impurities sufﬁciently far apart from each other, so that kFr2/greaterorsimilar1. Then, T1andT2are oscillating functions of both r2andβ, as can be seen in Fig. 8. Remarkably, the period and amplitude of the oscillations in the T1andT2vs r2dependencies may strongly depend on β, as demonstrated in Figs. 8(c)and8(d). In the limit kFr2/greatermuch1 we expect the two impurities to act independently, so that T1≈cos2α1andT2≈cos2α2when \|α1\|/greaterorequalslant\|α2\|. Indeed, from Eqs. ( E16) and ( E17) we obtain det(ˆt/primeˆt†/prime)=cos2α1cos2α2+O/parenleftbigg1 k2 Fr2 2/parenrightbigg , (E24) Tr(ˆt/primeˆt†/prime)=cos2α1+cos2α2+O/parenleftbigg1 k2 Fr2 2/parenrightbigg , (E25) so that T1=cos2α1+O/parenleftbigg1 k2 Fr2 2/parenrightbigg ,T 2=cos2α2+O/parenleftbigg1 k2 Fr2 2/parenrightbigg (E26) when\|α1\|/negationslash=\|α2\|, and T1=cos2α1+O/parenleftbigg1 kFr2/parenrightbigg ,T 2=cos2α1+O/parenleftbigg1 kFr2/parenrightbigg (E27) when \|α1\|=\|α2\|. In any case, kFr2/greatermuch1 results in \|T1,2−cos2α1,2\|/lessmuch1. Finally, we would like to point out a curious observation: when exactly α1=α2=−kFr2, the two transmission values T1andT2coincide and are equal to T1,2=1−sin2(kFr2) 1−sin2(kFr2) k2 Fr2 2[1−\|K(kFr2)\|2]. (E28) [1] P. Anderson, J. Phys. Chem. Solids 11,26(1959 ). [ 2 ]A .A .A b r i k o s o va n dL .P .G o r ’ k o v ,S o v .P h y s .J E T P 12, 1243 (1961). [3] A. V . Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. 78, 373(2006 ). 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Ishii, Prog. Theor. Phys. 47,1464 (1972 ). [27] N. Kopnin, Theory of Nonequilibrium Superconductivity ,I n t e r - national Series of Monographs on Physics (Clarendon Press,Oxford, 2001).[28] G. Eilenberger, Z. Phys. A: Hadrons Nucl. 214,195 (1968 ). [29] K. K. Likharev, Rev. Mod. Phys. 51,101(1979 ). [30] Y . S. Barash and I. V . Bobkova, Phys. Rev. B 65,144502 (2002 ). [31] C. W. J. Beenakker, P h y s .R e v .L e t t . 67,3836 (1991 ). [32] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod. Phys. 79,353(2007 ). [33] G. C. Menard, S. Guissart, C. Brun, S. Pons, V . S. Stolyarov, F. Debontridder, M. V . Leclerc, E. Janod, L. Cario, D. Roditchev,P. Simon, and T. Cren, Nat. Phys. 11,1013 (2015 ). [34] M. Büttiker, Y . Imry, R. Landauer, and S. Pinhas, P h y s .R e v .B 31,6207 (1985 ). [35] A. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (AMS Chelsea Pub- lishing, Providence, RI, 2005). [36] L. Landau and E. 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PhysRevB.88.195436.pdf	PHYSICAL REVIEW B 88, 195436 (2013) Electronic and atomic structure of Co/Ge nanoislands on the Ge(111) surface D. A. Muzychenko,1,K. Schouteden,2,†and C. Van Haesendonck2 1Faculty of Physics, M.V . Lomonosov Moscow State University, 119991 Moscow, Russia 2Laboratory of Solid-State Physics and Magnetism, KU Leuven, BE-3001 Leuven, Belgium (Received 27 June 2013; revised manuscript received 8 October 2013; published 27 November 2013) We report on a detailed investigation of the electronic and atomic structure of nanometer-size Co/Ge islands obtained by solid-state reactive deposition of Co atoms on a Ge(111) c(2×8) surface. Relying on scanning tunneling microscopy (STM) and spectroscopy (STS) measurements combined with density functionaltheory based calculations, the atomic structure of the Co/Ge(111)√ 13×√ 13R13.9◦surface reconstruction is determined. Real-space STM imaging combined with Fourier-transform analysis reveals the coexistence oftwo inequivalent phases of√ 13×√ 13R13.9◦reconstructed Co/Ge nanoislands that are rotated by +13.9◦ and−13.9◦with respect to the [2 11] direction. STS spectra probe a small band gap that varies within the√ 13×√ 13R13.9◦surface unit cell between 10 and 250 meV, suggesting local metallic behavior. According to the proposed atomic-structure model, each Co/Ge(111)√ 13×√ 13R13.9◦surface unit cell contains one Ge adatom and six Co atoms that are located at hollow sites below the top surface Ge layer and that are stacked in the form of an equilateral triangle. The Ge adatom is located asymmetrically with respect to the Co triangle andoccupies two different, yet physically equivalent, positions, giving rise to two chiral phases of Co/Ge nanoislands.The Co/Ge valence band is dominated by Co atom derived 3 dstates, while states in the conduction band stem from Ge adatom and Ge rest-atom derived states. Analysis of the bonding properties conﬁrms the stability ofthe proposed Co/Ge atomic structure and reveals signiﬁcant charge transfer from Co atoms to Ge rest atoms,suggesting ionic or metallic-covalent interaction. DOI: 10.1103/PhysRevB.88.195436 PACS number(s): 68 .35.−p, 68.47.Fg, 68 .37.Ef, 73.20.At I. INTRODUCTION Advances in nanoelectronics during the last decade stem to a large extent from the undiminished efforts of the scientiﬁccommunity to overcome the difﬁculties encountered in theminiaturization of electronic devices. Currently, the downscal-ing of the conventional Si-based metal-oxide-semiconductor(ﬁeld-effect) transistors 1is reaching its intrinsic limits.2In order to proceed beyond these limits, new devices, that are able to add novel functionalities to Si-based transistors, needto be developed. Moreover, the use of new materials withelectronic properties superior to those of Si is required.For this purpose, Ge offers yet unexploited opportunities 3–5 because of its higher electron and hole mobility6and its compatibility with the Si-based technologies. On the other hand, nanostructured materials that combine a high spin polarization with semiconducting properties, such as dilutedmagnetic semiconductors 7,8and ferromagnet/semiconductor hybrids,9,10are considered as highly promising candidates for future spintronic devices11,12and quantum logic gates13–15that rely on controlled manipulation of the charge and the spin ofelectrons. In this view, an atomistic understanding of the growth of 3dtransition metals on semiconductor surfaces is of utmost importance because the resulting electronic and spin propertiescrucially depend on the precise atomic conﬁgurations. Thegrowth of ferromagnetic materials (Fe, Co, Ni, etc.) on Geleads to the formation of complex germanide compounds at the ferromagnet/semiconductor interface. 16–18The phase evo- lution of, e.g., Co and Ni germanides as a function of the forma-tion temperatures 19–21and their Schottky diode behavior22–25 have already been studied intensively. Furthermore, it has been demonstrated that various metal germanides (NiGe,PdGe, PtGe x, and CoGe 2) also offer perspectives for use asself-aligned contact materials in Ge-based devices because of their low electrical resistance, low formation temperature,and high thermal stability. 26,27Due to the reactive nature of 3dferromagnetic metals, reconstruction of the surface atoms upon deposition can occur even at room temperature.28The growth of ferromagnetic materials on Ge can therefore notbe described by standard epitaxial growth models such as the V olmer-Weber or the Frank–van der Merwe models. Thin-ﬁlm germanide reactions are typically considered to be similar tosilicide reactions. However, when compared to Si, the growthprocess of ferromagnetic materials on Ge surfaces has receivedconsiderably less attention. The (111) surface of Ge is of particular interest because of the various metal-induced reconstructions that may occur upon annealing of the surface. Several experimental studies havealready been devoted to the growth and electronic properties of Mn xGey(Refs. 10and 29–31) and Ag (Refs. 32–36) on Ge(111) c(2×8) surfaces. Mn-doped Ge is considered as a promising diluted magnetic semiconductor, while the Ag-induced√ 3×√ 3R30◦surface reconstruction (hereafter referred to as the√ 3R30 reconstruction)37–40can be used as an effective buffer layer that is suitable for growth of thinferromagnetic (FM) layers. 41,42Direct growth of FM layers on the Ge surface yields complex germanide phases that form a nonmagnetic layer at the interface [e.g., for Co layers with thickness below 3 monolayers (MLs) (Refs. 42and43)], which is detrimental for electron spin injection from the ferromagnetto the semiconductor. Previous studies of the growth of Co on Ag/Ge(111)√ 3R30 revealed the formation of two-dimensional (2D) islandsthat exhibit two different surface reconstructions: Someislands exhibit a√ 13×√ 13R13.9◦(hereafter referred to as√ 13R14◦) reconstruction, while other islands have a(√ 3×√ 3) reconstruction.44The ratio between the 195436-1 1098-0121/2013/88(19)/195436(19) ©2013 American Physical SocietyMUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) probabilities of appearance for both types of islands has been found to depend on the Co coverage and on the annealingtemperature. 45–47However, the precise formation process of the complex Co/Ag/Ge(111) reconstruction and the role playedby the Ag buffer layer is not fully understood. 48A supporting theoretical model for the atomic and electronic structure ofthe√ 13R14◦reconstruction still needs to be developed. Unambiguous interpretation of the reported experimentalresults for the Co/Ag/Ge(111) system obviously requiresdetailed understanding of the more simple growth process ofCo on bare Ge(111) surfaces. Experimental and theoreticalatomic-scale studies of the initial stages of Co adsorption onbare Ge(111) surfaces are, however, still lacking. Here, we report on our combined study based on scan- ning tunneling microscopy (STM) and scanning tunnelingspectroscopy (STS) and on ab initio calculations of the√ 13R14◦reconstructed Co/Ge(111) surface that is obtained by solid-state reactive deposition of Co on the Ge(111) c(2×8) surface and subsequent annealing. By real-space STM imagingas well as Fourier-transform analysis we reveal the existence oftwo structurally inequivalent types of√ 13R14◦reconstructed nanometer-size Co/Ge islands having a different chirality.Relying on voltage-dependent STM imaging, we identiﬁed thetwo surface unit cells (SUCs) of the√ 13R14◦reconstruction and determined an atomic-scale model of the reconstructedsurface. This is veriﬁed in detail by direct comparison withdensity functional theory (DFT) based simulations of STMimages. Each SUC consists of six Co atoms that are located athollow sites below the top surface layer and that are arranged in the form of an equilateral triangle, and of one Ge adatomthat is located above a Ge atom of the second layer. TheGe adatom is located asymmetrically with respect to the“Co triangle” and can occupy two different yet physicallyequivalent sites, yielding two different chiral phases for theCo/Ge√ 13R14◦nanoislands. By means of STS we show that the Co/Ge√ 13R14◦nanoislands exhibit metallic behavior with a vanishing band gap that locally varies within the√ 13R14◦SUC, yielding opportunities for use as Ohmic contact material.26We ﬁnd that ﬁlled-states STM images are dominated by Co atom derived states, while the empty statesmainly originate from the Ge adatoms and from the group ofGe rest atoms. Analysis of the electronic structure and bondingproperties conﬁrms the stability of the proposed Co/Ge atomicstructure. Moreover, our analysis reveals a remarkable chargetransfer from the Co atoms to the Ge rest atoms as well asnearest-neighbor interactions between Co-Ge atomic speciesthat point towards the existence of ionic or metallic-covalentbonding. II. EXPERIMENTAL METHODS STM and STS measurements are performed with a low- temperature ultrahigh vacuum (UHV , base pressure is about4×10 −12mbar) STM setup (Omicron Nanotechnology) at 4.5 K using W tips. The tips are cleaned in situ by repeated ﬂashing well above 1500◦C in order to remove the surface oxide layer and additional contamination. The tip quality isroutinely checked by acquiring atomic-resolution images ofthe “herringbone” reconstruction of the Au(111) surface. 49,50 STM topography imaging is performed in constant currentmode. STS spectra are recorded in the current imaging tunneling spectroscopy (CITS) mode with a grid size of200×200 current-voltage I(V) spectra. Everywhere in the text the tunneling bias voltage V trefers to the sample voltage, while the STM tip is virtually grounded. Image processing isdone by Nanotec WSxM. 51 Ge samples are doped with Pat a doping level of nP∼1018cm−3, resulting in n-type bulk conductivity with resistivity ρbulk∼0.01/Omega1cm. Ge slabs with dimensions 4 × 1.5×0.8m m3and with their long axis aligned along the (111) direction are cleaved52in situ at room temperature in the UHV preparation chamber (base pressure is about 5 ×10−11mbar) of the STM setup. After cleavage, the samples are heatedto 400 ◦C for a few minutes to transform the Ge(111)2 ×1 surface into the c(2×8) reconstructed Ge(111) surface. Next, Co atoms are deposited at a rate of 0 .004±0.001 monolayers (MLs) per second from a high-purity Co rod by meansof electron-beam evaporation. During deposition, the Gesubstrate is kept at 100 ◦C. We take 1 ML coverage equal to 7.22×1014atoms/cm2, i.e., the atomic density of the unreconstructed Ge(111) surface. After Co deposition, thesample is annealed to 300 ◦C–350◦C for about 10 min. III. EXPERIMENTAL RESULTS A. Growth of√ 13×√ 13R13.9◦ reconstructed Co/Ge(111) nanoislands In Fig. 1, we present (a) a typical large-scale and (b) a closeup view STM topography image of the Ge(111) surfaceafter deposition of about 1 4ML of Co as described above. It can be seen in Fig. 1that 2D islands with nanometer size FIG. 1. (Color online) (a) 100 ×100 nm2and (b) 14 ×14 nm2 STM topography images ( Vt=+ 2.0V , It=50 pA) of the Co/Ge(111) surface. The√ 13×√ 13R13.9◦andc(2×8) surface unit cell are indicated by the white solid rhombus and parallelogram, respectively, in (b). Other symbols are explained in the text. 195436-2ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) and with√ 13R14◦surface reconstruction (discussed in more detail in the next section) are formed, which are referred toas Co/Ge nanoislands hereafter. The Co/Ge nanoislands havelateral sizes ranging from a few nanometers up to 50 nmand are randomly distributed across the Ge(111) surface.In the high-resolution empty-states STM image of a Co/Genanoisland in Fig. 1(b) the√ 13R14◦reconstructed nanoisland appears to have a similar height as the surrounding Ge(111)surface. We ﬁnd that the precise height depends on thetunneling voltage V t. For example, the height of a nanoisland with respect to the surrounding Ge(111) surface is h/similarequal10 pm (peak to peak) at Vt=+ 2.0 V and h/similarequal38 pm at Vt=+ 1.5V . Based on an analysis of 10 140 ×140 nm2STM topography images, the total area covered by the Co/Ge nanoislands isfound to be 23% ±7%. Parts of Figs. 1(a) and 1(b) also reveal that the initial long-range c(2×8) reconstruction of the Ge(111) surface surrounding the Co/Ge nanoislands is slightly modiﬁed dueto the Co deposition. The Ge surface atoms now exhibit local(2×2) and c(2×4) reconstructions that reﬂect the intrinsic c(2×8) reconstruction. Within the Chadi and Chiang model, 53 thec(2×8) unit cell is constructed out of a (2 ×2) and a c(2×4) subunit cell with two types of adatoms and rest atoms on top of the Ge(111) with different local environments.54–61 The surface unit cells (SUCs) of the c(2×8) and√ 13R14◦ reconstruction are indicated in Fig. 1(b). Because of the three- fold rotational symmetry of the Ge(111) surface, the [01 1], [110], and [ 101] surface crystallographic directions (rotated by 120◦with respect to each other) can be inferred from STM topography images for the initial c(2×8) reconstruction as well as for the local (2 ×2) and c(2×4) reconstructed regions. This also allows us to determine the projections of thebulk [2 11], [ 112], and [ 121] crystallographic directions on the Ge(111) surface [indicated by red solid arrows in Fig. 1(a)]. Upon more careful inspection of Fig. 1(a) it can be seen that the√ 13R14◦reconstruction of the Co/Ge nanoislands is rotated by an angle αof about +14◦or−14◦with respect to the [2 11] direction. This is illustrated in Fig. 1(a) by the white dotted arrows and the white dashed arrows forthe islands labeled 1 and 2, respectively. The long diagonalof the√ 13R14◦SUC rhombus for nanoisland 1 is rotated clockwise with respect to the [2 11] direction (hereafter referred to as the RIGHT√ 13) orientation, while for nanoisland 2 the long diagonal is rotated anticlockwise (hereafter referred to astheLEFT√ 13 orientation). Both orientations appear with the same probability. Exceptionally, nanoislands can be found thatexhibit both RIGHT√ 13 and LEFT√ 13 orientations [see island labeled 3 in Fig. 1(a)]. It can hence be expected that at full coverage the surface will consist of Co/Ge√ 13R14◦ domains with the two different orientations that are separated by domain boundaries. B. Fourier-transform analysis We investigated the Co/Ge nanoislands and their√ 13R14◦ reconstruction in more detail via Fourier-transform (FT) anal- ysis of the STM topography images. In Fig. 2(a), we present the 2D FT image of the RIGHT√ 13 Co/Ge nanoisland which was already presented in Fig. 1(b).T h e( 2 ×2) and c(2×4) reconstructed regions of the surrounding Ge surface FIG. 2. (Color online) (a) 2D FT image of Fig. 1(b).T h eB r a g g spots indicated by the reciprocal lattice vectors /vectora∗ 1stem from the (1×1) unreconstructed Ge(111) surface, while the spots indicated by the reciprocal lattice vectors /vectorb∗ Roriginate from the RIGHT√ 13 reconstructed Co/Ge nanoislands. (b) 2D FT image of Fig. 1(a).T h e vectors /vectora∗ 2stem from the (2 ×2) unreconstructed Ge(111) surface, while the vectors /vectorb∗ Rand/vectorb∗ La r el i n k e dt ot h e RIGHT√ 13 and LEFT√ 13 reconstructed Co/Ge nanoislands, respectively. (c), (d) Schematic top and side views, respectively, of the atomic structureof the unreconstructed Ge(111) surface. The main crystallographic directions and high-symmetry atom sites are indicated. and the threefold rotational symmetry of the Ge(111) surface give rise to the six outer Bragg spots in Fig. 2(a). These spots correspond to the reciprocal lattice vector set with the length \|/vectora∗ 1\|=1.774 ˚A−1(2π/\|/vectora∗ 1\|=3.54 ˚A) of the unreconstructed Ge(111)(1 ×1) atomic lattice. The six inner Bragg spots [the reciprocal lattice vector set \|/vectorb∗ R\|=0.455 ˚A−1 (2π/\|/vectorb∗ R\|=13.8˚A)] stem from the surface reconstruction of the RIGHT√ 13 type Co/Ge nanoisland. The sizes of the vector sets are related as \|/vectora∗ 1\|=√ 13\|/vectorb∗ R\|and the sets are rotated clockwise over an angle α=+ 13.2◦±1.6◦with respect to each other. The latter value matches well with theangle derived directly from the STM topography image (seeprevious section). Figure 2(b) presents the 2D FT image of the STM topography image in Fig. 1(a). In this case, the Bragg spots corresponding to /vectora ∗ 1can not be observed due to the larger size of the STM image, while the Bragg spots stemming from the 195436-3MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) 2×2 surface reconstruction, i.e., \|/vectora∗ 2\|=2\|/vectora∗ 1\|=0.868 ˚A−1 (2π/\|/vectora∗ 2\|=7.23˚A) can still be discerned [they are also visible in Fig. 2(b)]. Note that the reciprocal vectors /vectora∗ 1and/vectora∗ 2are not rotated with respect to each other, implying they bothcan be used as an orientational reference for other spots.The 12 inner Bragg spots in Fig. 2(b) originate from the presence of Co/Ge nanoislands with both the RIGHT√ 13 andLEFT√ 13 reconstruction in Fig. 1(a). These spots can hence be interpreted in terms of the presence of two subsets of six spots with reciprocal vectors /vectorb∗ Rand/vectorb∗ Lof equal length and with a relative angle of 2 α=29◦±2◦. The latter value is in a good agreement with the theoretical value of 27 .8◦ for the√ 13R14◦reconstruction. The similar intensity of the two subsets of Bragg spots conﬁrms that the RIGHT√ 13 andLEFT√ 13 Co/Ge nanoislands appear with the same probability in the STM images. The fact that two types of√ 13R14◦Co/Ge nanoislands are formed on the Ge(111) surface can be understood byconsidering the atomic-scale model of the Ge(111) surface.Figures 2(c) and2(d) present a sketch of the four surface layers of the unreconstructed Ge(111) surface. The (1 ×1) and (2 ×2) SUCs with (real-space) lattice vectors /vectora 1and/vectora2are indicated in Fig. 2(c). When selecting a SUC, the origin of the vector /vectora1can be (arbitrarily) placed on one of the three different Ge atom sites: On the top atom , in front of the second-layer atom or in front of the fourth-layer atom (hereafter referred to as thehollow site ) [see Figs. 2(c) and 2(d)]. The long diagonal of the SUC rhombus can be chosen along the [2 11], [ 112], or [121] direction. This way, the complete Ge surface can be constructed by repeating the chosen SUC. For a given(1×1) SUC, there are two equivalent possibilities for the “adsorbate SUC” of the√ 13R14◦reconstruction, as indicated in Fig. 2(c). These SUCs have real-space lattice vectors with the length \|/vectorbR\|=√ 13\|/vectora1\|(green) and \|/vectorbL\|=√ 13\|/vectora1\|(red) vectors and are rotated over an angle of +13.9◦and−13.9◦ with respect to the (1 ×1) SUC. Both SUCs are primitive√ 13R14◦unit cells, i.e., they are the simplest unit cells that are able to describe the√ 13R14◦structure. The experimentally observed RIGHT√ 13 and LEFT√ 13 orientations of the reconstructed Co/Ge nanoislands in Fig. 1(a) can hence be linked to these two different√ 13R14◦SUCs. C. Inequivalence of the RIGHT√ 13 and LEFT√ 13 SUCs Although the two SUCs that describe the√ 13R14◦recon- struction appear very similar, they are not fully equivalent.This becomes clear when considering the atomic structureof the unreconstructed Ge(111) surface that “supports” theRIGHT√ 13 and LEFT√ 13 SUCs in Fig. 2(c). It can be seen in Fig. 3that the atomic arrangements within both SUCs is different, i.e., the SUCs exhibit mirror symmetry with respectto each other. The SUCs can hence not be transformed intoeach other by a simple rotation operation, but only by a mirrorreﬂection with respect to the symmetry line indicated in Fig. 3. It is expected that this “inequivalence” of both types of recon-structed Co/Ge nanoislands gives rise to a different appearanceof the islands in STM experiments. Indeed, while both typesof Co/Ge nanoislands appear very similar in the empty-statesregime [threefold rotational symmetry, see Fig. 1(a)], a mirrorFIG. 3. (Color online) (Left) and (right) Schematic top view of the unreconstructed Ge(111) surface area that is “covered” by theLEFT√ 13 and RIGHT√ 13 SUCs of the√ 13R14◦surface reconstruction in Fig. 2(c), respectively. The arrangement of the Ge(111) surface atoms within the LEFT√ 13 and RIGHT√ 13 SUCs exhibits mirror symmetry with respect to each other (see thesequence of atoms indicated by the yellow arrows at the bottom of each SUC). symmetry between the two types can be observed in the ﬁlled-states regime. Figures 4(a) and4(c) present empty-states STM topography images of a RIGHT√ 13 and LEFT√ 13 Co/Ge nanoisland, respectively. The SUC is added for eachimage, together with an “orientational guide” that indicates thehigher (brighter) and lower (darker) region of the√ 13R14◦ SUC by a red and blue triangle, respectively. Figures 4(b) and4(d) present ﬁlled-states STM topography images of the same areas presented in Figs. 4(a) and 4(c), respectively. FIG. 4. (Color online) (a), (c) Empty-states ( Vt=+ 1.5V ,It= 150 pA) and (b), (d) ﬁlled-states ( Vt=− 1.5V ,It=50 pA) STM topography images of a RIGHT√ 13 and LEFT√ 13 reconstructed Co/Ge nanoisland. Image sizes are 15 ×15 nm2for (a), (b) and 10 × 10 nm2for (c), (d). The√ 13×√ 13R13.9◦SUC is indicated by the black rhombus. An “orientational guide” is added to each of the images (see text for more details). 195436-4ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) Determination of the SUCs for the empty- and ﬁlled-states regimes is discussed in more detail in Sec. III D . In the ﬁlled-states regime [Figs. 4(b) and 4(d)], the√ 13R14◦surface reconstruction of the Co/Ge nanoislands is observed as densely packed equilateral triangles with lessbright edges and brighter spots at their corners, as previouslyalso reported by Lin et al. 62This is indicated in Figs. 4(b) and4(d) by the added “orientational guide.” One can see that theRIGHT√ 13 and LEFT√ 13 Co/Ge nanoislands exhibit mirror symmetry with respect to each other (along the verticalaxis). The angle βbetween the long diagonal of the rhombus and the median of the equilateral triangle [see Figs. 4(b) and4(d)] is about ±14 ◦, similar to what was found from our 2D FT analysis (see Sec. III B ) and√ 13R14◦SUCs modeling (Sec. IV). A simple analysis learns that β=0 with respect to the crystallographic directions [2 11]. In other words, we can conclude that the bright triangles in the ﬁlled-states regimehave the same orientation for the RIGHT√ 13 orLEFT√ 13 reconstructed Co/Ge nanoislands and that the three brightcorners of the triangle indicate the [2 11], [ 112], and [ 121] crystallographic directions. It is important to note that the combination of ﬁlled-states and empty-states STM topography imaging allows us tounambiguously determine the phase of the√ 13R14◦surface reconstruction of the Co/Ge nanoislands, without any furtheranalysis of the orientation of the surrounding nanoislandsand without determining the crystallographic directions ofthe surrounding Ge(111) c(2×8) surface (see Sec. III A ). Moreover, from analysis of the√ 13R14◦ﬁlled-states STM topography images, it is possible to determine the [2 11], [ 112], and [ 121] crystallographic directions of the Ge(111) substrate. D. Voltage-dependent STM investigation of the Co/Ge√ 13R14◦ surface reconstruction High-resolution voltage-dependent STM imaging of Co/Ge nanoislands reveals structural details that have not yet beenreported in previous STM-based studies. 44,45,62,63Figure 5 presents a series of empty-states [(a1)–(a5)] and ﬁlled-states[(b1)–(b5)] STM topography images of the√ 13R14◦surface reconstruction of a RIGHT√ 13 Co/Ge nanoisland, recorded at the same location and with the same STM tip for tunnelingvoltages V tranging from +2.0t o−1.5V . In order to exclude possible (thermal) drift effects of the piezo scanner during our voltage-dependent STM imaging,we record “dual-bias STM images.” In particular, a differenttunneling voltage is used for the forward andbackward line scans that govern the forward andbackward STM topography images. In Fig. 5, we also take into account the small (scan- speed-dependent) shift of about 0 .2˚A that was found to exist between both recorded images with respect to each other andhence always use the same scan speed. The tunneling voltagefor the forward image is ﬁxed at V t=2.00 V and is used as a reference for the backward image for which the tunnelingvoltage is varied. This way, the dual-bias STM imaging allowsus to compare exactly the same locations of the√ 13R14◦SUC at different tunneling voltages. The√ 13R14◦SUC, indicated by the dotted rhombus in Fig. 5, has a periodicity of√ 13×√ 13 and a side length of 1 .46±0.04 nm. FIG. 5. (Color online) (a1)–(a5) Empty-states and (b1)– (b5) ﬁlled-states STM topography images of the Co/Ge√ 13×√ 13R13.9◦reconstruction on the Ge(111) surface. Image sizes are 6 .3×6.3n m2. The tunneling voltage Vtis indicated for each image. In the empty-states regime, at voltages above 1 .0V [Figs. 5(a1) and 5(a2)], topography is dominated by bright 195436-5MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) spherical protrusions that are located at the four corners of the SUC rhombus, i.e., there exists one such protrusion for eachSUC. These protrusions appear most pronounced in the 1 .2 to 1.5 V tunneling voltage range [see Fig. 5(a2)]. In addition to the protrusions, a less bright feature can be observed inthe lower half part of the SUC. This feature becomes morepronounced with increasing tunneling voltage, and aroundV t=2.0 V its brightness is similar to that of the protrusions at the rhombus corners. Both the protrusions at the corners andthe extra feature in the lower half of the SUC become less clearwith decreasing tunneling voltage below 1 .0V[ F i g . 5(a3)], and around 0 .8 V the feature in the lower half part of the SUC transforms into a structure consisting of three small brightspots that form an equilateral triangle [Figs. 5(a4) and 5(a5)]. The protrusions at the rhombus corners are no longer visible atthese voltages. In Fig. 5(a4), the equilateral triangle is marked by the dotted symbol labeled 1. This triangle is duplicated atexactly the same location in all of the STM topography imagesin Fig. 5. In the ﬁlled-states regime, at high voltages below −1.0V [Figs. 5(b1)– 5(b3)], the topography is dominated by equi- lateral triangles with less bright edges and brighter corners,similar to the STM topography images in Figs. 4(b) and4(d). As already mentioned in Sec. III C , the angle between the median of the triangle and the long diagonal of the SUCrhombus is about −14 ◦, i.e., the Co/Ge nanoisland is of type RIGHT√ 13. The triangular structure is indicated by a dotted symbol labeled 2 in Fig. 5(b2). This triangular symbol is duplicated at the same location in all of the STM topographyimages in Fig. 5. Note that this triangular feature 2 does not ﬁt completely inside the SUC, i.e., the lower two brightspots lie somewhat outside the SUC. However, it can beseen that one SUC contains (in total) one triangular featurelabeled 2. In particular, it can be seen that the three brightspots, which are indicated in an alternative way by label3i nF i g . 5(b2), ﬁt completely within a single SUC of the√ 13R14◦periodic structure. In the following, we prefer to use the structure labeled 2, as it is more consistent with the atomic-structure model we propose below. Above −1.0 V [Figs. 5(b4) and 5(b5)], the triangular structure 2 becomes less clear and the bright corners fade away. Instead, a ﬁne structure appearswithin the contours of triangle 2 as well as small spots at thecorners of the SUC rhombus. The voltage-dependent STMimages allow us to develop and verify our atomic-structuremodel for the Co/Ge(111)√ 13R14◦surface reconstruction (see Sec. IV). E. Electronic structure of Ge(111) c(2×8) and√ 13R14◦Co/Ge nanoislands The surface electronic structure of bare Ge(111) c(2×8) has already been investigated in great detail using variousexperimental techniques, including angle-resolved photoe-mission spectroscopy (ARPES), inverse photoemission spec-troscopy (IPES), and STS, as well as by ab initio calculations. In Fig. 6, we schematically illustrate the main contributions of electronic states to the surface density of states (SDOS)of the Ge(111) c(2×8) surface that have been reported in literature. Germanium is a semiconductor with a (relatively)large band gap E gof about 0 .74 eV at low temperature,FIG. 6. (Color online) Energy scheme of the main contributions of electronic states to the Ge(111) c(2×8) surface density of states following reported ARPES, IPES, STS, and theoretical studies. allowing clear observation of the Ge(111) c(2×8) surface band gap Esbg=0.49±0.03 eV.64The surface states are mainly localized at adatoms (AAs, emtpy-surface states) andrest atoms (RAs, ﬁlled-surface states). ARPES studies haverevealed that the rest-atom derived states are located near−0.8 eV below the Fermi level E F,65–68in good agreement with STS experiments55,57,60andab initio calculations.58,61,64 The deeper-lying state in the valence band near −1.4 eV below EFis related to back-bond (BB) states of the adatoms and has been observed in ARPES experiments.68A contribution from these BB states to rest atoms has been reported aswell. 58,61Finally, STS55,57,60and IPES69measurements as well as theoretical studies58,64showed that the feature near 0 .7e V above EForiginates from adatom related states. The electronic properties of the Ge(111) c(2×8) surface states are, however, not yet fully understood. In particular, the origin of someaspects of the surface band structure related to the existence oftwo inequivalent “types” of adatoms ( A T,AR) and rest atoms (RT,RR)( R e f . 58) that have a different Ge environment is still a matter of debate. Razado-Colambo et al. found that this inequivalence gives rise to a splitting of the rest-atom bandand adatom band of about 0 .1–0.2e V( s e eF i g . 6) and a partial electron transfer from adatoms to rest atoms. 61In addition, photoemission data by Aarts et al. revealed the existence of a surface band S1 around −0.15 eV below EFthat has a dangling-bond-like character and a (1 ×1) periodicity.68This could not be explained within the existing Ge(111) c(2×8) model and was related to partially ﬁlled adatom danglingbonds. However, recently, it has been shown that the S1 band isactually a combination of a band originating from states belowadatom and rest-atom layers near the center of the Brillouinzone, while the outer part of the S1 band stems from rest-atomrelated states only. 61 The electronic behavior of surface states typically dras- tically changes upon the adsorption of metal atoms. Wetherefore performed STS measurements to investigate the localelectronic structure of the bare Ge(111) c(2×8) surface before 195436-6ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) Co deposition as well as of the Co/Ge√ 13R14◦surface reconstruction and the surrounding uncovered parts of the Gesurface. In order to take into account the effects of the differenttunneling barrier for different tunneling voltages, we relyon the normalized differential conductance ( dI/dV )/(I/V ), which is proportional to the local density of states (LDOS)of the surface, rather than on the differential conductancedI/dV .(dI/dV )/(I/V ) is obtained numerically from the recorded I(V) spectra. However, division by the voltage Vleads to an artiﬁcial divergence around zero bias in the (dI/dV )/(I/V ) spectra. To eliminate the divergence around zero bias, we applied (limited) broadening to the ( I/V ) data values. 70In Fig. 7(c), we present typical ( dI/dV )/(I/V ) spectra recorded at (1) the bare Ge(111) c(2×8) surface prior to Co atom deposition (black solid line), (2) the Ge(111) surface in-between the Co/Ge nanoislands after Co atom deposition (green dashed line), and (3) a√ 13R14◦ reconstructed Co/Ge nanoisland (red dashed-dotted line). The Fermi level EFis located at 0 V. The spectra are obtained by FIG. 7. (Color online) (a) Empty-states and (b) ﬁlled-states STM topography images ( Vt=+ 1.0V ,It=20 pA) of the same RIGHT√ 13 and LEFT√ 13 reconstructed Co/Ge nanoislands on the Ge(111) surface. (c) ( dI/dV )/(I/V ) spectra taken at the Ge(111) c(2×8) surface prior to Co atom deposition (black solid curve) and after Co deposition, i.e., at the√ 13R14◦reconstructed Co/Ge nanoislands (red dashed-dotted curve) and at the defect-free areas with local c(2×4) and (2 ×2) reconstructions in-between the Co/Ge nanoislands (green dashed curve). Inset: logarithm ofI(V) spectra taken on Ge(111) c(2×8) (black solid curve) and on√ 13R14◦reconstructed Co/Ge nanoislands (red dashed-dotted curve).taking the numerical derivative of area-averaged I(V) spectra (including typically 103–104spectra) extracted from a CITS measurement. Figures 7(a) and 7(b) are empty-states and ﬁlled-states STM topography images, respectively, of the sameGe surface with√ 13R14◦nanoislands, on which also the CITS data were recorded. Spectrum (2) in Fig. 7(c) is obtained by averaging CITS I(V) data of defect-free areas with local c(2×4) and (2 ×2) reconstructions that exist in-between the Co/Ge nanoislands, while spectrum (3) in Fig. 7(c) is obtained by averaging CITS data recorded at the three Co/Ge√ 13R14◦ nanoislands that are included in Figs. 7(a) and 7(b). Finally, spectrum (1) in Fig. 7(c) is obtained from another CITS measurement on bare defect-free Ge(111) c(2×8) before Co atom deposition [topography image not included in Figs. 7(a) and7(b)]. The spectrum of the bare Ge(111) c(2×8) surface reveals several pronounced features that can be linked to the statespresented in Fig. 6. The large peak near 0 .65 V (labeled AA) can be attributed to the adatom related empty-surface-statesband. Moreover, it can be seen that this peak exhibits a“shoulder” at its right-hand side, which points towards theexistence of two inequivalent types of adatoms, as alreadydiscussed above. The splitting is found to be about 150 mV[measured in-between A RandAT, see Fig. 7(c)], in agreement with previous theoretical predictions.58,61Features in the spectrum at voltages above the adatom peak can be associatedwith conduction band (CB) states and a higher-lying surface-states band. 60The two CB peaks at 1 .12 and 1 .43 V appear at signiﬁcantly lower voltages (i.e., about 0 .3 V closer to EF) than the two CB peaks previously reported for p-type Ge(111) c(2×8).60Similarly, the valence band (VB) peak in the ﬁlled-states regime near −0.7 V appears at about 0 .3V lower voltage. These shifts can be explained by the fact thatthe Fermi level E Fis closer to the CB for n-type material than forp-type material. Finally, in the ﬁlled-states regime, one can observe another (broad) peak at −1.12 V below EF, which can be associated with the surface rest-atom band (labeledRA). In contrast to the case of the adatom related peak, wedo not resolve a spectral shoulder that can be linked to twodifferent types of rest atoms. This can be explained by the factthat the rest-atom band is rather close to the VB peak, whichhampers resolving the R TandRRpeaks. It should be noted that the rest-atom peak in the STS spectra appears at signiﬁcantlylower energies when compared to earlier ARPES spectra. 68It has recently been shown that the voltage at which the rest-atompeak appears strongly depends on the tunneling current and onthe temperature. This has been related to disorder-inducedstates stemming from (disordered) adatoms that appear atdomain boundaries of the Ge(111) c(2×8) surface, resulting in a nonequilibrium source of positive charge. 60 In order to determine the width and the energy positions of the edges of the surface band gap, we plot the logarithmof the (absolute value of the) tunneling current as a functionof tunneling voltage [see inset of Fig. 7(c)]. This way, the energy band gap is observed as the (noisy) ﬂat (zero-slope)region around V=0. For the bare Ge(111) c(2×8) surface (black solid line), the band gap is located in-between the bulkVB states (below E F) and the surface adatom derived states (above EF). The surface band gap ranges from about −0.45 to 0.15 V and the band gap width is 0 .60 eV. 195436-7MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) In Fig. 7(c), the STS spectrum of the√ 13R14◦recon- structed Co/Ge nanoislands (red dashed-dotted line) differssigniﬁcantly from that of the bare Ge(111) c(2×8) surface (black solid line) as well as from that of the local c(2×4) and (2×2) reconstructed Ge(111) surface (green dashed line). It can be seen that the tunneling spectra of the Ge(111) surfacebefore and after Co atom deposition are rather similar, although the STM topography images of both surfaces appear to be quite different. There occurs only a small shift of the RA andAA peaks of about 0 .1–0.2Vt o w a r d s E F, while the width of the surface band gap remains almost the same. On the other hand, the√ 13R14◦reconstructed Co/Ge nanoislands exhibit a strongly reduced surface band gap when compared to thesurrounding Ge surface. To minimize the (potential) inﬂuenceof nonequilibrium effects in the tunneling spectra, we relied on low-current STS measurements and used identical tunneling voltage and current set-point values for the acquisition of I(V) curves on bare Ge(111) c(2×8) and on Co/Ge√ 13R14◦ reconstructed nanoislands. From the logarithm of the I(V) spectrum taken on a Co/Ge nanoisland [red dashed-dotted linein the inset of Fig. 7(c)], a band gap of 0 .2 eV can be inferred. The Fermi level E Fis located symmetrically in the surface band gap determined by the two surface-states related peaks located at −0.2 and 0 .2 V. Furthermore, we note that careful comparison of the electronic structure of the two RIGHT√ 13 reconstructed Co/Ge nanoislands labeled (1) and (2) inFigs. 7(a) and 7(b) reveals no signiﬁcant differences in the STS spectra. Remarkably, also no signiﬁcant differences are observed between RIGHT√ 13 reconstructed Co/Ge nanois- lands (1,2) and LEFT√ 13 reconstructed Co/Ge nanoislands (3), both in local (non-area-averaged) STS ( dI/dV )/(I/V ) spectra taken at different sites within the√ 13R14◦SUC of different Co/Ge nanoislands, and in area-averaged STS spectra such as those presented in Fig. 7(c). We therefore conclude that the RIGHT√ 13 and LEFT√ 13 reconstructed Co/Ge nanoislands exhibit an identical electronic structure.Finally, we want to note that in the low-current STS mea-surement we did not observe any signiﬁcant dependence ofthe (dI/dV )/(I/V ) spectra on the tunneling current for the Co/Ge√ 13R14◦reconstructed nanoislands. In Fig. 8, we present ( dI/dV )/(I/V ) spectra recorded at three inequivalent locations within the√ 13R14◦SUC indicated in the inset: (1) an adatom site ,( 2 )a“ central” site , and (3) a “ depressed” site . The spectra are the average of 30 to 50 individual spectra recorded at the same locations (1),(2), and (3), but on different Co/Ge nanoislands. It can beseen that the surface band gap depends on the exact location within the√ 13R14◦SUC and varies from below 0 .01 eV (i.e., close to metallic behavior) at location (3), up to about 0 .24 eV at location (1). The band gaps of the adatom site (1) and of thecentral sites (2) are slightly asymmetric with respect to the Fermi level and are enclosed in-between the two surface-statesrelated bands located near −0.25 V ( S 1F) and 0 .2V (S1E). TheS1Epeak appears closer to EFat the central site and the depressed site , while S1Falways appears at the same energy. Another pronounced peak S2Eis observed around 0 .8V a t theadatom site and around 0 .9V a t t h e central site , i.e., around the same energy at which AA states are found on bareGe(111) c(2×8) surfaces [see Fig. 7(c)]. At the depressed site , theS 2Estate is only weakly visible. Instead, an additional state FIG. 8. (Color online) Normalized differential conductance spec- tra taken at the three different locations within the√ 13R14◦SUC that are indicated in the STM topography image in the inset ( Vt=+ 1.8V , It=20 pA). Curves are shifted vertically for clarity. S3Ecan be observed around 1 .6 V. Finally, in the ﬁlled-states regime an additional surface-states peak S2Fis found to exist at the three locations (1), (2), and (3), shifting from −0.5Va t theadatom site to−0.7Va tt h e depressed site . In Fig. 9, we present a series of 2D maps of the tunneling dif- ferential conductance dI/dV (x,y,V ), which is proportional to the LDOS, derived from a CITS measurement at the samelocations as those used for Figs. 7(a) and 7(b). This way, the existence of the low-energy surface states in the Co/Ge√ 13R14◦nanoislands can be visualized using the LDOS maps. One can see that the LDOS near the Fermi level ishigher on the nanoislands when compared to the surroundingarea, in both the empty-states [Fig. 9(a1)] and ﬁlled-states regimes [Fig. 9(b1)]. This is consistent with our STS-based observation of a very small surface band gap at the depressed sites of the√ 13R14◦reconstruction (see Fig. 8), i.e., there is an almost metallic conductivity. At voltages below 100 mVthe main contribution to the LDOS of the nanoislands stemsfrom the tails of the S 1EandS1Fsurface states. At higher voltages, the LDOS at the adatom sites and the central sites increases and the Co/Ge nanoislands can be clearly discernedin the LDOS maps up to V=0.6 V and down to V=− 0.7V for the empty and ﬁlled states, respectively. At more elevatedvoltages, the VB and AA states start to contribute to theLDOS. In the voltage range between 0.6 and 1 .2V , t h e LDOS maps reveal pronounced patterns on the√ 13R14◦ reconstructed Co/Ge nanoislands [Figs. 9(a2) and 9(a3)]. This can be linked to the S2Esurface state that exists within this voltage range (see Fig. 8). Around 0 .7–0.8V ,t h e S2Estate is mainly localized at the adatom site , and for V=0.9–1.0Vi t also appears at the central site . Furthermore, a large amount of surface defects can be resolved on the surrounding Gesurface in-between the nanoislands. This may be related toa (random) embedding of a fraction of the deposited Co atomsinto the Ge(111) surface, which may in turn give rise to theobserved breaking of the (initial) long-range c(2×8) surface reconstruction. From our careful comparison of the LDOSmaps of RIGHT√ 13 and LEFT√ 13 reconstructed Co/Ge nanoislands for a wide range of energies, we can conclude that 195436-8ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) FIG. 9. (Color online) Filled-states (a1–a4) and empty-states (b1– b4) CITS derived LDOS maps at the same location as in Figs. 7(a) and7(b) (the tunneling voltage Vtis indicated for each image). theRIGHT√ 13 and LEFT√ 13 reconstructions exhibit, in spite of their different appearance in STM topography images,identical local electronic properties. IV . COMPUTATIONAL METHODS AND STRUCTURAL MODEL A. Details of the calculations Our theoretical investigation of the Co-induced√ 13×√ 13R13.9◦surface reconstruction of the Ge(111) surface is performed using DFT based calculations71,72within the Perdew-Burke-Ernzerhof73(PBE) generalized gradient approximation74(GGA) exchange-correlation energy functional. Calculations are performed using the SIESTA code,75,76which relies on the expansion of the Kohn-Sham orbitals using a linear combination of pseudoatomicorbitals. The core electrons are implicitly treated by usingnorm-conserving Troullier-Martins pseudopotentials 77with the following electronic conﬁguration of the elements: H = 1s1,G e=(Ar 3d10)4s24p2, and Co =(Ar) 4 s23d7, where the core conﬁgurations are indicated between parentheses.A mesh cutoff of 300 Ry for the grid integration and a splitdouble-zeta basis set without spin polarization are used in allcalculations. Integrals over the Brillouin zone are summed ona6×6×1 Monkhorst-Pack mesh, 78ensuring convergence of the self-consistent ﬁeld iteration process until the changesof total energy are below 0 .1m e V . Our calculations are performed in three stages. First, we start from an unreconstructed Ge(111) slab with√ 13R14◦ surface periodicity. Second, an (arbitrary) amount of so-called “foreign” Co and Ge atoms is added at different locations (based on symmetry considerations) above the Ge(111) surface and the equilibrium conﬁgurations of the Co/Ge system aredetermined using conjugate gradient (CG) geometry optimiza-tion. Third, the electronic structure of a n×nsupercell is calculated and used for the DFT based simulations of STM topography images. The second and third steps are repeateduntil the best match between the experimental and simulated STM images is achieved. In order to model the Ge(111) unreconstructed surface with√ 13R14◦periodic boundary conditions, an extended mono- clinic unit cell with basis vector lengths \|/vectorx\|=a0√ 13/√ 2, \|/vectory\|=a0√ 13/√ 2, and \|/vectorz\|=a04√ 3 is used (see Fig. 10), where a0=5.641 ˚A is the optimized bulk Ge lattice constant for the selected DFT approximation. The angle between the /vectorx=/vectorbRand/vectory=/vectorbRvectors is 60◦and for the RIGHT√ 13 SUC both vectors are rotated over an angle of +13.9◦with respect to the vector /vectora1[i.e., the projection of the bulk lattice vector on the (111) surface, see Fig. 2(c)]. The unit cell consists of a slab of 10 Ge atomic layers [see Fig. 10(b) ] of which one atomic layer is saturated by hydrogen atoms,yielding 130 Ge atoms and 13 H atoms per unit cell. The vacuum space above the Ge surface is 21 .7˚A. Since the bare Ge(111) c(2×8) surface reconstructs from the c(2×8) to the√ 13R14◦reconstruction during the annealing after Co atom deposition, we can not simply estimate the initial amount of Ge adatoms and rest atoms within the new√ 13R14◦SUC. First, the amount of Ge adatoms and rest atoms in the SUCis reduced and becomes fractional upon transition from the old SUC to the new one because of the different size S of the SUCs for both reconstructions, i.e., S c(2×8)/S√ 13= 16 13. Second, during the reconstruction, the Co atoms may “push away” the Ge adatoms that are initially present be- low the Co/Ge nanoislands. Therefore, we start our mod-eling from the bare unreconstructed and unrelaxed Ge(111) surface. Next, Co and Ge atoms are added on top of the (111) surface as illustrated in Fig. 10. The amount of added atoms and their adsorption sites are chosen based on symmetry considerations related to the lattice symmetry of the RIGHT√ 13 SUC and on STM images for the ﬁlled-states and empty-states regimes (see discussion in Sec. III D ). Here, we focus as an example on the RIGHT√ 13 atomic structure, but detailed complementary results were also obtained for the LEFT√ 13 atomic structure. During the CG geometry optimization, the atoms of the six topmost Ge layers are allowed to relax, while 195436-9MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) FIG. 10. (Color online) (a) Top and (b) side views of the ball-and-stick model of the unreconstructed Ge(111) slab for the√ 13R14◦SUC. The model contains 10 Ge layers with vacuum space above the upper surface layer and a H-terminated bottom surfacelayer. The√ 13R14◦unit cell is indicated by bold black dashed lines. The atoms of the ﬁrst and second Ge layers are presented as the larger green and blue spheres, respectively, while atoms belowthe third layer are indicated by the smaller gray-green spheres. The red and yellow spheres indicate the initial (prior to CG geometry optimization) adsorption positions of one Ge adatom (red sphere) and six Co atoms (yellow spheres), respectively. the positions of the atoms of the four lower-lying Ge layers and of the H layer are frozen. The “foreign” atoms and the slab atoms are allowed to relax until all atomic forces acting on the released atoms are smaller than 5 meV /˚A and the remaining numerical error in the total energy is smaller than 10−4eV for each optimization step. Each optimized structure is subjected to an annealing simulation. The annealing allows the system to surmount energy barriers and hence can give rise to another structure with a lower (local-minimum) energy. This way, a fully relaxed equilibrium geometry is obtained for each of thecalculated structures. The face centered cubic (fcc) Ge(111) surface has three types of sites that exhibit threefold rotational symmetry: (1)Sites above the top Ge atoms (green Ge atoms in Fig. 10), (2) sites above Ge atoms of the second layer (blue Ge atoms inFig.10), and (3) hollow sites (i.e., above Ge atoms of the fourthlayer). Since we place the Ge atoms of our Ge(111) model initially at their bulk positions, the surface has a high density ofbroken bonds, i.e., one dangling bond per top Ge atom, whichis energetically unfavorable. The density of these danglingbonds is reduced by introducing “foreign” Ge adatoms. Whenpositioning an adatom above a hollow site or a second-layer Geatom, it will bind to three (nearest-neighbor) Ge surface atomsand only one dangling bond (of the adatom itself) remains.The introduction of Ge adatoms thus reduces the total amountof dangling bonds and hence gives rise to an energeticallymore favorable surface. In the experiments, adatoms areobserved above second-layer Ge atoms only. In order to ﬁndthe energetically most favorable Co/Ge geometry, we use all three types of sites that are available within the√ 13R14◦ SUC (i.e., 13 sites for each type) as the initial coordinates of “foreign” Co and Ge atoms. For the Co atoms, the top Ge atomsites appear to be relevant only when considering bilayer Costructures that are formed at higher Co coverage and, as will bedemonstrated in the following, these sites can be excluded here.For the Ge adatoms the sites above the second-layer Ge atomas well as the hollow sites need to be considered. Adsorptionof Ge adatoms at these sites is found to yield a very similarenergy gain, 79and the possible interaction with Co atoms may imply that the hollow sites are more preferable for the Geadatoms. Our calculations reveal that the Co atoms typicallyrelax towards the hollow sites, regardless of their starting position. Co adsorption at these locations yields a lower total energy when compared to those geometries for which the Coatoms are found to stay above Ge atoms of the second Ge layerafter relaxation. Furthermore, when a Co atom is placed at ahollow site, the top Ge atoms are found to relax upward topositions somewhat above the Co atoms (up to 1 .4˚A). This ﬁnding is in line with our previous work, which demonstratedthat Co atoms deposited on bare Ge(111)2 ×1 surfaces can penetrate down to the fourth Ge subsurface layer. 80,81 If our candidate model system conﬁgurations would all have the same amount of atoms and only a different atomic geometry, the relaxed geometry that matches the√ 13R14◦ reconstruction could simply be determined by comparing the total energies of the different model system conﬁgurationsand ﬁnding the energetically most favorable geometry. Here,however, it is not ap r i o r i known how many “foreign” atoms (i.e., Co atoms and Ge adatoms) contribute to the observed√ 13R14◦reconstruction and need to be included in our model system. Therefore, the total energies of different modelsystem conﬁgurations having a different amount of atomscan not be simply compared. For example, the minimumtotal energy of the Ge model system is that of the (2 ×1) reconstruction without any “foreign” atoms. The amount ofpossible conﬁgurations further increases when consideringpossible penetration of the Co atoms in subsurface Ge layersand the possible formation of bilayer Co structures. The atomic structure that matches the√ 13R14◦reconstruction can therefore only be determined by direct comparison ofDFT based simulations of STM topography images andexperimental STM topography images. Comparison of ﬁlled- states and empty-states topography images reveals that only 1 atomic-structure model out of more than 40 candidates (withvarying bonding geometry and stoichiometry) fully matchesthe experimental observations. 195436-10ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) B. Structural model of Co/Ge(111)√ 13R14◦reconstruction Figure 11presents a ball-and-stick model of the atomic structure that is found to match to the experimentally observedCo/Ge(111)√ 13×√ 13R13.9◦reconstruction. The structure contains one Ge adatom (red sphere) and six Co atoms (yellow spheres). The initial positions (prior to relaxation) of the“foreign” atoms are presented in Fig. 10.T h es i xC oa t o m sa r e located vertically above hollow sites and have an equilateraltriangular arrangement. The Ge adatom is initially locatedasymmetrically with respect to the Co triangle, i.e., verticallyabove a Ge atom of the second layer. In Fig. 11, we present a top and side view of the seven topmost layers after relaxation: sixrelaxed Ge layers and one layer of adsorbed Co and Ge atoms.For the lower-lying Ge layers, we maintain their initial (priorto relaxation) numbering, despite their strong reconstructionupon relaxation. The new top layer thus consists of one Geadatom (red sphere) and the Ge atoms of the initial top layer(green spheres). A more detailed classiﬁcation of the topmostatoms according to their height and electronic properties willbe given below. The Ge atoms of the second layer (bluespheres) also considerably change their position during therelaxation, both laterally in the x,yplane as well as vertically in the zdirection. The nearest neighbors of the Ge adatom, FIG. 11. (Color online) Ball-and-stick model of the RIGHT Co/Ge(111)√ 13R14◦reconstruction. (a) Top and (b) side views of the geometry presented in Fig. 10after relaxation. The basis unit of the√ 13R14◦reconstruction consists of one Ge adatom (red sphere) and six Co atoms (yellow spheres). Green and blue colored spheres represent the Ge atoms of the ﬁrst and secondlayers, respectively, similar to Fig. 10. The smaller gray-green spheres correspond to the bulk Ge atoms. The relaxed Co atoms are all located in-between the ﬁrst and second Ge layers. The SUC isindicated by the dashed black rhombus in (a) and by vertical dashed black lines in (b), respectively. The initial positions of the bulk Ge atoms are indicated by the thin dotted mesh. FIG. 12. (Color online) Top view of the upper two atomic layers of the Co/Ge(111)√ 13R14◦surface reconstruction based on the relaxed atomic structure presented in Fig. 11. The “building units” of the√ 13R14◦reconstruction are indicated by dashed red circles and green dotted triangles (see text for more details). The SUC of the√ 13R14◦reconstruction (indicated by the dashed black rhombus) is rotated over an angle +13.9◦with respect to the [ 121] direction, while the “building unit” indicated by the green dotted triangle is aligned to the crystallographic directions indicated in the upper left corner of the ﬁgure. The ideal fcc(111) surface mesh is indicated by gray thindotted lines in the background. The colors of the different types of surface atoms are the same as the colors in Figs. 10and11. i.e., three top Ge atoms, relax towards each other, while the Ge atoms of the second layer below the adatom move downwards.As a result, the Ge atoms of the third and fourth layers belowthe adatom [see Fig. 11(b) ] move downwards as well. The Co atoms retain their triangular arrangement after relaxation andare all retrieved in-between the ﬁrst and second Ge layers atabout the same zposition [see Fig. 11(b) ]. The positions of the Ge bulk atoms below the fourth atomic layer are found tochange only slightly after relaxation. The relaxed model is visualized alternatively at larger scale in Fig. 12, where only the atoms from the two topmost Ge layers and the additional Co layer are included. This way, it canbe seen that the Ge adatom is positioned symmetrically withrespect to the surrounding Co triangles. Indeed, the Ge adatomis bonded to three top-layer Ge atoms [as is also the case for thec(2×8) reconstructed Ge surface]. 79These four Ge atoms can be considered as a “building unit” of the surface reconstructionand are indicated by a dashed red circle in Fig. 12.T h es i x Co atoms and the surrounding 10 Ge atoms from the topand second Ge layers that interact with the Co atoms formanother “building unit” of the surface reconstruction and areindicated by the green dotted triangle in Fig. 12. Figure 12 illustrates that the triangular building units are aligned withthe crystallographic [ 121], [2 11], and [ 112] directions. The angleβis hence equal to zero for this triangular building unit. Note that the SUC of the√ 13R14◦reconstruction is tilted by an angle α=+ 13.9◦with respect to the [ 121] direction, consistent with the experimentally observed RIGHT√ 13 orientation of the surface reconstruction discussed in Sec. III C . The 10 Ge atoms of the triangular building unit [because of the analogy with the c(2×8) reconstruction again referred to as Ge rest atoms hereafter] relax signiﬁcantly both laterally 195436-11MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) and vertically. Their relaxation is not symmetrical due to the asymmetric position of the Ge adatom with respect to thetriangular building unit. The Ge adatom induces “bending” ofthe side of the triangle formed by four Ge rest atoms adjacentto the Ge adatom. In the lower left part of Fig. 12, the center of the relaxed Ge rest atoms labeled (1)–(3) is highlightedby black-white dotted crosses. A straight dotted line is drawnin-between the two Ge rest atoms (1) located at the two cornersof the triangular building unit. The largest relaxation occursfor the Ge rest atom that is located directly adjacent to theGe adatom, i.e., Ge rest atom (3). Due to the translationalperiodicity of the two building units, the triangular unit issurrounded by Ge adatoms on three sides that all induce anidentical “bending” of the sides of the triangular building unit.This is depicted by the green dotted deformed triangles inFig. 12. The triangular building unit, alone and as well as in combination with the Ge adatom building unit, can hence beconsidered as a chiral structure. The asymmetric position of the Ge adatom with respect to the triangular building block gives rise to the possibility toform either the RIGHT√ 13 or the LEFT√ 13 Co/Ge surface reconstruction. On the fcc Ge(111) surface, the Ge adatomsoccupy two different, yet physically equivalent, positions, asillustrated in the central part of Fig. 13: The Ge adatom can be located both at the left-hand side (green dotted circle) andat the right-hand side (red dotted circle) of the triangularbuilding unit (yellow dotted triangle), yielding a RIGHT√ 13 (green rhombus) and LEFT√ 13 (red rhombus) SUC. The FIG. 13. (Color online) Sketch of the two possible positions of Ge adatoms on the Ge(111)(1 ×1) surface with respect to the triangular building units formed by Co atoms. The Ge adatom (red sphere) building units are indicated by green and red dotted circles enclosing a red colored Ge adatom, while the six Co atoms (yellow spheres) of thetriangular building units are indicated by the yellow dotted triangle. The two positions of the Ge adatom, labeled RIGHT andLEFT , give rise to two different SUCs, i.e., a RIGHT√ 13 SUC (green rhombus) and a LEFT√ 13 SUC (red rhombus). The RIGHT√ 13 andLEFT√ 13 SUC surfaces are repeated twice in the right and left parts of the ﬁgure, respectively.RIGHT√ 13 and LEFT√ 13 SUCs are repeated twice in the right and left parts of Fig. 13, respectively, to illustrate the symmetry of the Ge adatom positions in-between thetriangular building units for both SUCs. Calculations for theLEFT√ 13 SUC (using CG geometry optimization and DFT based simulations of STM topography images) reveal that thegeometry and electronic properties of the LEFT√ 13 surface reconstruction is identical to that of the RIGHT√ 13 surface reconstruction. The bending of the three sides of the triangularbuilding unit of both geometries is the same, yielding the chiralstructures that exhibit mirror symmetry with respect to eachother, as observed in the experiments discussed in Sec. III C .I t is still under question if the triangular Co structures grow ﬁrstand the Ge adatoms arrive afterwards, or that the Ge adatomsgrow ﬁrst and that their positions determine the positions for growth of the triangular Co building blocks. From Fig. 12 it becomes clear that the Co/Ge(111)√ 13R14◦surface reconstruction can be “constructed” using the larger building unit formed bythe smaller Ge adatom and triangular Co/Ge building units(see Fig. 14). As already mentioned above, the Ge rest atoms considerably change their position in the vertical directionupon relaxation, resulting in a different height for each Gerest atom. We can distinguish four types of Ge rest atomsin the triangular Co/Ge building unit: (1) three Ge atoms atcorner positions [Ge (1)], (2) three Ge atoms at side positions [Ge (2)], (3) three Ge atoms at side positions close to the Ge adatom [Ge (3)], and (4) one Ge atom at the center of the FIG. 14. (Color online) (a) Top and (b) side views of a ball-and- stick model of the Ge adatom and triangular Co/Ge building unit of the Co/Ge(111)√ 13R14◦surface reconstruction. In (b) the atomic structure is viewed along the direction indicated by the arrow labeled A in (a). The inset in (a) indicates the relative height of the relaxed Ge rest atoms within the triangular building unit. The numbers on thegray height scale bar indicate the different heights of the Ge rest atoms in (a) and (b). The SUC is indicated in (a) and (b) by a 3D red dashed rhombus (its height in the [111] direction is chosen arbitrarily). 195436-12ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) TABLE I. Calculated distances between the two types of Co atoms and the neighboring Ge rest atoms (labeled in Fig. 14). Atom pair Co–Ge Distance Co(1c)–Ge (1) 2.204 ˚A Co(1c)–Ge (2) 2.365 ˚A Co(1c)–Ge (3) 2.262 ˚A Co(1c)–Ge (21) 2.475 ˚A Co(1c)–Ge (22) 2.430 ˚A Co(2s)–Ge (2) 2.294 ˚A Co(2s)–Ge (3) 2.261 ˚A Co(2s)–Ge (4) 2.471 ˚A Co(2s)–Ge (22) 2.390 ˚A Co(2s)–Ge (23) 2.473 ˚A Co(2s)–Ge (24) 2.504 ˚A triangular building unit [Ge (4)]. The three atoms of types (1)–(3) exhibit identical properties, i.e., they are at the sameheight and have the same bond distances, atom population,and Mulliken overlap population with deviations for eachof these parameters remaining below 0 .05%. The inset in Fig. 14(a) gives the relative height of the relaxed Ge (1)–Ge (4) rest atoms. The Ge (1)atoms have the lowest position within the triangular building unit, while the Ge (2),G e (4), and Ge (3) atoms are located 0 .6, 0.9, and 1 .1˚A above the Ge (1)atoms, respectively. The Co atoms all have the same height, i.e.,below the Ge (2)atoms and 0 .1˚A above the Ge (1)atoms. Based on their identical properties, the six Co atoms can be dividedinto two subgroups: (1) three Co atoms at corner positions[Co (1c)] and (2) three Co atoms at side positions [Co (2s)] [see Fig. 12(a) ]. Similarly, the 12 Ge atoms of the second layer can be divided into 4 subgroups: (1) three Ge atoms bondedto Co (1c),G e (1), and Ge (2)atoms [Ge (21)], (2) three Ge atoms bonded to Co (2s)atoms [Ge (22)], (3) three Ge atoms bonded to Co(1c)atoms [Ge (23)], and (4) three Ge atoms located inside the Co triangle [Ge (24)]. The distances between the different Co–Ge atoms of the relaxed√ 13R14◦model are listed in Table I. The covalent bond length of the Ge adatom is found to be 2.633 ˚A, which is slightly larger (8%) than for typical bulk Ge–Ge bonds (2.439 ˚A) and consistent with previous studies.79,82 C. DFT based modeling of STM topography images and electronic structure of the Co/Ge√ 13R14◦ surface reconstruction To conﬁrm our Co/Ge√ 13R14◦model, we compare the experimental constant current STM topography images to sim-ulated STM topography images. The simulated STM imagesare obtained using the Tersoff-Hamann approximation 83,84at a distance of several angstroms above the position of the topmostsurface atoms. The dependence of the tunneling current I on the tunneling voltage Vbetween the STM tip and the surface is I=2πe ¯h/summationdisplay μ,vf(Eμ)[1−f(Ev+eV)]\|Mμv\|2δ(Eμ−Ev), (1)where f(E) is the Fermi-Dirac distribution function, Mμvis the tunneling matrix element between electronic states ψμof the tip and electronic states ψvof the surface, and Eμ(Ev)a r e the energies of the states ψμ(ψv) in the absence of tunneling. When we assume localized wave functions ψμfor the tip, Mμvis proportional to the amplitude of ψvat position /vectorr0of the probing tip at low temperatures. For small tunneling voltagesV,E q . (1)reduces to I∝/summationdisplay v\|ψv(/vectorr0)\|2δ(Eμ−EF). (2) Following Eq. (2), the tunneling current Iis proportional to the surface LDOS that is probed at position /vectorr0of the tip, integrated over an energy range from EFtoEF+eV.F o ra constant tunneling current Ithe STM tip essentially follows a contour of constant surface LDOS. However, because thesurface wave functions decay exponentially into the vacuumregion, numerical evaluation of ψ v(/vectorr0) (within the DFT based approach these are the Kohn-Sham wave functions of thesurface) for tip-surface distances of the order of severalangstroms poses a signiﬁcant problem for the DFT basedcalculations. For this reason, STM simulations are oftenrestricted to (the vicinity of) the surface, which may yieldincorrect results. In order to tackle this problem, we use the 2DFourier transform of the wave functions ψ v(/vectorr) in combination with spatial extrapolation techniques85to evaluate the surface wave functions ψv(x,y,z 0) in the vacuum region, up to z0=10˚A above the surface. We rely on an experimental z(Vt) spectrum measured on the√ 13R14◦surface to take into account the dependence of the height zon the tunneling voltage Vtin our calculations. This z(Vt) dependence, to which an initial tip-sample distance of 3 ˚A is added, is used to determine the height above the surface at which simulatedSTM images are calculated. For low voltages, i.e., for energiesclose to E F,z(Vt)/similarequal3.5˚A, while for high voltages above 1 V, z(Vt)/greaterorsimilar6.4˚A. This way, we calculate STM topography images for all possible Co/Ge conformations for the ﬁlled-states andempty-states regimes between −3.0 and+3.0 eV. Agreement between theory and experiment is achieved for the entireenergy range only for the relaxed model presented in Fig. 11. In Fig. 15, we present a series of experimental (outer columns, grayscale images) and calculated (inner columns, color scalerhombus) STM constant current topography images for theﬁlled-states (two right columns) and empty-states (two leftcolumns) regimes. Calculated STM topography images areobtained for a 3 ×3 SUC (indicated by the dashed white-blue rhombus) for the RIGHT√ 13 geometry displayed in Fig. 11. Experimental STM topography images in Fig. 15are all recorded at the same location. Initially, the√ 13R14◦SUC rhombus was chosen based on high-bias empty-states STM images (see Sec. III D ) and the corners of the SUC rhombus were positioned on thebright atomic features without any further knowledge ofthe origin of these bright features. Following our proposedmodel, these features stem from the Ge adatoms. We thereforeredeﬁne (using a parallel translation operation) the√ 13R14◦ SUC rhombus in the calculated images with respect to the experimental STM images (white dotted rhombus in Fig. 15). Furthermore, the√ 13R14◦building unit discussed in the 195436-13MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) FIG. 15. (Color online) Experimental empty-states and ﬁlled-states STM topography images (outer columns, grayscale images) and the corresponding simulated STM topography images (inner columns, color scale rhombus) of the Co/Ge(111)√ 13R14◦surface reconstruction. The tunneling voltage Vtis indicated in each of the images. Two√ 13R14◦SUCs are indicated in the simulated images: A red dotted rhombus SUC used for the calculations and a white dotted rhombus SUC inferred from the experimental images (see Sec. III D ). The√ 13R14◦“building unit” (see Fig. 14) is superimposed on each image at exactly the same location. previous section is superimposed on each image at exactly the same location. The calculated STM images nicely reproduce all of the im- portant features of the experimental STM images in Fig. 15.I n the ﬁlled-states regime, perfect agreement between experimentand theory is achieved for the entire tunneling voltage range.For the empty-states regime, perfect agreement is found aswell, except for the lowest tunneling voltages, where smalldifferences can be observed. These small discrepancies can beattributed to the approximations made in the DFT calculations (see discussion below). In the ﬁlled-states regime, at tunneling voltages below −0.9 V, the calculated topography becomes dominated by an equilateral triangular feature that exhibits bright protrusionsabove the Co (1c)atoms (see Fig. 15). Upon more careful analysis, we ﬁnd that this electronic feature originates froma complex contribution of hybridized orbitals of the Co (1c)and Ge(3)atoms to the integrated LDOS probed above the surface. 195436-14ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) FIG. 16. (Color online) Height dependence of high-bias ﬁlled- states calculated DFT based STM topography images. The height above the surface at which the STM images are calculated is(a) 3.5 ˚A, (b) 4.0 ˚A, and (c) 6.2 ˚A. All images are calculated for the same tunneling voltage V t=− 1.6 V. The ﬁrst two surface layers of the Co/Ge√ 13R14◦model are added below the topography images that are made semitransparent. The dependence of these local LDOS maxima on the height above the surface is illustrated in Fig. 16. At lower heights close to the surface, the LDOS is dominated by the topmostatoms, i.e., the Ge adatom and the Ge (3)atoms [see Fig. 16(a) ]. With increasing distance from the surface, their contributionto the integrated LDOS rapidly decays [see Fig. 16(b) ], in contrast to the contribution from the Co (1c)atoms that in turn start to dominate the LDOS at larger distances fromthe surface [see Figs. 16(b) and 16(c) ]. At heights above 4.5 ˚A only states related to the Co (2c)atoms are observed both theoretically and experimentally. At lower voltages inthe ﬁlled-states regime the ﬁne structure of the upper atomswithin the triangular building unit appears, with the Ge atomsbeing more pronounced than the Co atoms (Fig. 15). This ﬁne structure can not be resolved in the experimental STM imagedue to STM tip convolution effects that hamper visualizationof such small corrugations. Nevertheless, the main featuresin the calculated and experimental images are in very goodagreement and the Ge adatom can be clearly traced in bothimages. In order to shed more light on the contribution of Co and Ge rest atoms to the electronic properties, we also performdensity of states (DOS) calculations. Plots of the projectedDOS for different atom sites are presented in Figs. 17(b) and 17(c) , while Fig. 17(a) presents the total DOS and the surface DOS (topmost two Ge atomic layers and Co layer). Inaddition, Fig. 18provides a direct comparison of the experi- mental ( dI/dV )/(I/V ) spectra of the√ 13R14◦reconstructed nanoisland and the theoretical surface DOS of the Co/Ge√ 13R14◦slab. The experimental ( dI/dV )/(I/V ) spectrum is obtained by area averaging CITS data of the locations labeled(1)–(3) in Fig. 8for different Co/Ge√ 13R14◦nanoislands. From comparison of Figs. 17(b) and17(c) , it is clear that the valence band (VB) is dominated by the dstates stemming from the Co atoms, while the spstates of the Ge surface atoms exist in the conduction band. The contribution of the Co atoms to theDOS [see black dashed curve in Fig. 17(b) ] exhibits maxima around −0.8 eV that lie well outside the bulk Ge band gap [Fig. 17(a) ] and below the top of the VB. These maxima are split by 0 .3 eV due to the presence of two types of Co atoms, i.e., a Co (1c)a n daC o (2s)atom. The absolute values of the peaks are the same for the Co (1c)and Co (2s)atom [see red and blue dashed curves in Fig. 17(b) ]. The integrated LDOS above the surface is, however, dominated by the Co (1c)atoms. The Co (2s)FIG. 17. (Color online) (a) Total DOS (black curve) and surface DOS (red dashed curve) for the Co/Ge√ 13R14◦slab. (b) Site projected DOS of a Co (1c)atom (red solid curve), a Co (2s)atom (blue dashed curve), and the sum of both DOSs (black dashed curve). (c) Site projected DOS of all Ge atoms from the top layer (black dashedcurve), partial DOS of a Ge adatom (red solid curve), a Ge (4)atom (blue dotted line), and the sum of the Ge (1),G e (2),a n dG e (3)rest atoms (green dashed line). Curves are shifted vertically for clarity. atoms have a smaller contribution to the LDOS, resulting in less pronounced spots in the triangular building unit. It canhence be concluded from Figs. 17(a) to17(c) that the surface DOS of the VB stems mainly from the Co atoms, in agreementwith the experimental STS measurements (see Fig. 18). The surface state S 2Fcan be directly attributed to Co (1c)and Co (2s) surface atoms, while the surface state S1Fis associated with a different contribution of the Co and Ge atoms. In the empty-states regime at tunneling voltages above 0.7 V, the calculated topography is dominated by the Ge adatom, which exhibits maximum contrast around 1 .3 V (see Fig. 15), and by the Ge (3),G e (2), and Ge (4)rest atoms, which appear as the “central” bright protrusion. The latterfeature becomes more pronounced with increasing tunnelingvoltage and above 1 .8 V it is brighter than the Ge adatom, whose brightness decreases with increasing tunneling voltage(see Fig. 15). At low tunneling voltages, the calculated and experimental images exhibit some discrepancies. In particular, 195436-15MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) FIG. 18. (Color online) Comparison of a typical experimental (dI/dV )/(I/V )s p e c t r u mo fa√ 13R14◦reconstructed nanoisland (black solid curve) and the theoretical surface DOS of the Co/Ge√ 13R14◦slab (red dashed curve). a ﬁne structure related to the Ge (3)and Ge (2)rest atoms is found in the calculated images, while the experimental imagesreveal three broadened spots per SUC at the locations of theCo (2s)atoms. The DOS calculations indicate that the Ge and Co atoms exhibit peaks in the bulk band gap around 0 .3e V ,w h i c h can be considered as surface states [see Figs. 17(b) and17(c) and the surface state S1Ein Fig. 18]. The interplay between these states hence determines the integrated LDOS above thesurface in the empty-states regime at low tunneling voltages.According to our DFT based calculations, the Ge (3)and Ge (2) atoms have a larger contribution to the integrated LDOS, whichis in disagreement with the experimental observations. Thisdisagreement can be explained as follows. First, the threeexperimentally observed spots may indeed be related to Ge (3) and Ge (2)rest atoms. These atoms can form a hybridized state that gives rise to the large broadened spots in theSTM images. This is indicated in the extra SUC rhombusdrawn in the experimental and calculated images in Fig. 15. The three white ellipses drawn in this SUC indicate thecoupling between the Ge (3)and Ge (2)atoms that can give rise to three large broadened spots per SUC. Second, wenote that the calculations do not take into account effectssuch as tip-induced band bending, resonant tunneling, andnonequilibrium effects. 60,86These effects may result in a shift of the energy levels of the surface states, which is evident fromcomparison of the experimental ( dI/dV )/(I/V ) spectrum to the theoretical surface DOS spectrum in Fig. 18. The here encountered discrepancy between theory and experiment canbe related to the well-known fact that DFT is not able to reliablyreproduce the size of the band gap of semiconductor materials.Because doping of the Ge substrate is not included in our DFTbased calculations, our DOS calculations predict the presenceof a pseudogap with a negligible DOS around E Fthat separates the bonding from the antibonding states [see Fig. 17(a) ]. However, our STS measurements indicate a metallic-typeconductivity for the Co/Ge√ 13R14◦nanoislands (see Fig. 18 and also the discussion in Sec. III E ). We therefore believe that inclusion of the tunneling and doping effects in aspin-dependent calculation will resolve the here observeddiscrepancy between theory and experiment at low tunnelingvoltages.The amount of possible chemical bonds of the Co and Ge atoms can be estimated from an analysis of their bondlengths (see Table I). For a single bond between a Co and a Ge atom, the bond length can be estimated as the sum of theCo and Ge covalent radii, which is r c=2.38˚A. Analysis of the calculated Co (1c)–Ge (x)and Co (2s)–Ge (x)distances reveals that all Co–Ge pairs form chemical bonds, since all distancesdiffer from r cby only −7.4% to+3.9%, only the Co (2s)–Ge (24) distance is somewhat larger (5.2%). An underestimation of the bond lengths by a few percent is typical for DFT basedcalculations. The chemical bonding mechanism between theCo and Ge atoms involved in the DFT electronic-structurecalculations can be investigated in more detail by evaluatingthe crystal orbital overlap population/Hamiltonian population(COOP/COHP). 87,88We use an alternative COOP/COHP based approach that allows us to calculate the relevant physicalquantities independent of the choice of the zero of the potentialby relying on the so-called covalent bond energy (ECOV). 89 COOP and ECOV calculations are known to yield similarresults, while the COOP method generally overestimates themagnitude of the antibonding states when deﬁned for a plane-wave basis set. 90Figure 19presents the partial ECOVfor Co (1c)– Ge(x),C o (2s)–Ge (x), and Ge (x)–Ge (x)interactions. Negative, positive, and zero values of ECOV correspond to bonding, antibonding, and nonbonding interactions, respectively. Abovethe Fermi level, the E COVspectra reﬂect a pronounced bonding FIG. 19. (Color online) Chemical bonding in terms of the cova- lent bond energy ECOV.( a )C o (1c)–Ge and Co (2s)–Ge interactions. (b) Ge–Ge bulk (black curve) and Ge (surface atom)–Ge/Co interactions (colored curves). 195436-16ELECTRONIC AND ATOMIC STRUCTURE OF Co/Ge ... PHYSICAL REVIEW B 88, 195436 (2013) TABLE II. Mulliken overlap population and electron population for the Co/Ge√ 13R14◦reconstruction. Co and Ge atoms are numbered according to Fig. 14and Table I. Atom/atom Co (1c) Co(2s) Electron pop. Ge(1) 0.235 3.939 Ge(2) 0.139 0.217 4.210Ge(3) 0.193 0.222 4.123 Ge(4) 0.152 4.351 Ge(21) 0.124 4.181Ge(22) 0.206 4.139 Ge(23) 0.167 4.163 Ge(24) 0.161 4.273Ge(24) 0.077 0.146 4.273 Electron pop. 8.552 8.440 character of the Co–Ge and the Ge–Ge interactions (see Fig. 19). Near −0.8e Vt h e ECOV spectrum indicates an antibonding interaction between the Co (1c)and the neighboring Ge rest atoms. The latter also exhibit antibonding peaks around−0.5 and −1.05 eV. This may be linked to the instability of the Co/Ge system in nonmagnetic conﬁgurations and to theknown thermal instability of the Co/Ge complex. Using Mulliken population analysis, 91we can investigate in more detail the character of the bonds involved in the Co/Ge√ 13R14◦reconstruction. The positive and negative overlap population values indicate bonding and antibonding states,respectively. The partial Mulliken electron orbital overlappopulations of the Co atoms and the electron populationsare listed in Table II. The calculations reveal that signiﬁcant charge redistribution occurs between the Co atoms and theGe surface atoms, i.e., 0.45 \|e\|and 0.56 \|e\|are transferred to the neighboring Ge atoms from Co (1c)and Co (2s), respectively (see Table II). Such densities of transferred electrons indicate a weak bonding with ionic or metallic-covalent character. TheCo (1c)and Co (2s)atoms exhibit ﬁve and six discernible bonds with Mulliken overlap populations, respectively. The overlapof the atomic orbitals is 35% to 58% of that of the Ge–Gebulk covalent bond (approximately 0.4). The Ge adatom has aneutral charge and three covalent bonds with an overlap of 40%to 50% of that of the Ge–Ge bulk covalent bond. There alsoexists a weak interaction between the Co (1c)and Ge (24)atom (see Table II). Because of the very low overlap population (17% of Ge–Ge bulk covalent bond overlap), this interaction shouldnot be considered as a sixth bond. It can be seen in Table II that all values of the bond overlap population are positive,yet relatively small, which indicates that there exist ionic ormetallic-covalent interactions between the populations. Ourﬁndings based on the bond length and the Mulliken overlappopulation analysis are in good agreement with the calculatedelectron charge density distribution ρ(x,y,z ) (data not shown), where zones of high electron localization (the signature ofchemical bonds) between the Co and Ge atoms are found toappear gradually at isosurface values below 0 .06 e/˚A 3.T h e Co(1c)and Ge (24)pair do not exhibit any remarkable electron localization. We therefore conclude that the Co (1c)and Co (2s) atoms form bonds with ﬁve and six neighboring Ge atoms, respectively. We note that our experimental conditions arein the temperature range where solid-state germanide phasessuch as CoGe, 21Co5Ge7,92and CoGe 2(Ref. 93) germanides are known to be formed. Here, however, we do not intend toprovide a similar precise indication of the chemical formula ofthe Co/Ge√ 13R14◦structure because the atomic geometry in our model is a planar system rather than an equilibrium bulkphase such as cubic CoGe, tetragonal Co 5Ge7, or orthorhombic CoGe 2. Interpretation of our planar Co/Ge structure in terms of germanide phases is therefore left as an open question becausefor germanide phases the transition from a surface (planar) to abulk (3D) structure is not well deﬁned. Further analysis of theCo/Ge√ 13R14◦atomic structure model in order to identify the proper analogy with the known germanide bulk phases willbe the subject of our future research. V . SUMMARY Co/Ge(111)√ 13R14◦reconstructed nanoislands are ob- tained by deposition of Co atoms on Ge(111) c(2×8) surfaces and subsequent annealing. Relying on combined STM/STSmeasurements and DFT based calculations, the atomic struc-ture of the Co/Ge(111)√ 13R14◦surface reconstruction is determined. Both our experiments and our calculations demon-strated that the√ 13R14◦reconstruction results from the mixing of Co and Ge without the need to involve a thirdelement such as Ag. 45–47 V oltage-dependent STM imaging reveals the coexistence of two inequivalent structural phases of the Co/Ge√ 13R14◦ surface reconstruction. The RIGHT√ 13 and LEFT√ 13√ 13R14◦reconstructed phases are found to be chiral struc- tures, having a SUC that is rotated over an angle of +13.9◦ and−13.9◦with respect to the [2 11] direction, respectively. STS spectra of the√ 13R14◦reconstructed Co/Ge(111) nanoislands reveal semimetallic behavior and a very smallband gap that locally varies within the SUC between 10and 250 meV. This is consistent with previously reportedmetallic properties of other germanide materials and offerspotential for use as an Ohmic contact material. Spectra ofthe Ge surface surrounding the Co/Ge nanoislands reveal anelectronic structure similar to that of bare Ge(111) c(2×8) surfaces and are used as a reference. Spectra of the√ 13R14◦ reconstructed surface further reveal several additional states, including two surface states at ±0.2 V with respect to EF, i.e., within the projected Ge bulk band gap. LDOS maps of RIGHT√ 13 and LEFT√ 13 reconstructed Co/Ge nanoislands recorded in a wide energy range show that theRIGHT√ 13 and LEFT√ 13 phases exhibit identical local electronic properties. Based on these experimental ﬁndings, a model for the atomic structure of the√ 13R14◦reconstructed Co/Ge(111) surface is obtained. DFT based simulations of the STM imagesfor this model are in perfect agreement with the experimentalSTM images for the entire investigated range of tunnelingvoltages. In our model, each√ 13R14◦SUC contains one Ge adatom located above a Ge atom of the second layer and sixCo atoms located at hollow sites below the Ge surface layer in the form of an equilateral triangle. Ten Ge rest atoms relaxupward due to the relaxation of the Co atoms to a positionwell below the surface. We ﬁnd that the Ge adatom canoccupy two different, yet physically equivalent, sites, locatedasymmetrically with respect to the Co triangle, which gives 195436-17MUZYCHENKO, SCHOUTEDEN, AND V AN HAESENDONCK PHYSICAL REVIEW B 88, 195436 (2013) rise to the RIGHT√ 13 and LEFT√ 13 chiral phases of the Co/Ge nanoislands. The six Co atoms of the SUC can be divided into two subgroups: Atoms at corner positions (Co (1c)) and at side positions (Co (2s)) of the equilateral triangle. Filled-states STM images are dominated by Co (1c)derived states and to some extent also by Co (2s)derived states. The Co (1c)related brighter corners in the STM images are aligned along the [2 11], [ 112], and [ 121] directions. Empty-states STM images are dominated by Ge adatom derived states and by the group of Ge rest atoms.In addition, we ﬁnd that the interplay between the Co (2s)and Ge rest atoms determines the integrated LDOS above the surface at low tunneling voltages in the empty-states regime. Finally, analysis of the covalent bond energy, Mulliken overlap populations, and charge density conﬁrms the sta-bility of the Co/Ge√ 13R14◦structure and moreover re- veals signiﬁcant charge transfer between the Co atoms andGe rest atoms. We conclude that the bonds involved in the Co/Ge√ 13R14◦structure are dominated by nearest- neighbor interactions that have an ionic or metallic-covalentcharacter. DFT based calculations of the Co–Ge bondstrength further reveal that the Co (1c)and Co (2s)atoms form bonds with ﬁve and six neighboring Ge atoms,respectively. 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Chem. 431, 72 (2010). 195436-19
PhysRevB.81.174536.pdf	Pseudogap in underdoped cuprates and spin-density-wave ﬂuctuations Tigran A. Sedrakyan1,2and Andrey V. Chubukov1 1Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA 2Department of Physics, University of Maryland, College Park, Maryland 20742, USA /H20849Received 19 February 2010; revised manuscript received 4 April 2010; published 27 May 2010 /H20850 We analyze fermionic spectral function in the spin-density-wave /H20849SDW /H20850phase of quasi-two-dimensional /H20849quasi-2D /H20850cuprates at small but ﬁnite T. We use a nonperturbative approach and sum up inﬁnite series of thermal self-energy terms, keeping at each order nearly divergent /H20849T/J/H20850/H20841log/H9280/H20841terms, where /H9280is a deviation from a pure 2D, and neglecting regular T/Jcorrections. We show that, as SDW order decreases, the spectral function in the antinodal region acquires peak/hump structure: the coherent peak position scales with SDWorder parameter while the incoherent hump remains roughly at the same scale as at T=0 when SDW order is the strongest. We identify the hump with the pseudogap observed in angle-resolved photoemission spectros-copy and argue that the presence of coherent excitations at low energies gives rise to magneto-oscillations inan applied ﬁeld. We show that the same peak/hump structure appears in the density of states and in opticalconductivity. DOI: 10.1103/PhysRevB.81.174536 PACS number /H20849s/H20850: 71.10.Hf, 75.10.Jm, 74.25.Dw I. INTRODUCTION Understanding of the phase diagram of cuprate supercon- ductors continue to be one of central topics in theoreticalcondensed-matter physics. 1Parent compounds of cuprates are quasi-two-dimensional /H20849quasi-2D /H20850antiferromagnetic in- sulators, heavily overdoped cuprates are Fermi liquids. Inbetween, systems are d-wave superconductors at low T/H11021T c and display the pseudogap behavior at larger Tc/H11021T/H11021T/H11569. How an insulator transforms into a Fermi liquid and what isthe origin of the pseudogap are still the subjects of intensivedebates among researchers. The pseudogap region exists both in underdoped and overdoped cuprates, but the physics evolves substantially be-tween these two limits. For overdoped cuprates, there israther strong evidence 2that the pseudogap region is best de- scribed as a disordered superconductor, when the gap is al-ready developed but the phase coherence is not yet set. 3–6In this doping range, fermions are reasonably well described asstrongly interacting quasiparticles with a large, Luttinger-type underlying Fermi surface /H20849FS/H20850. 7Thed-wave pairing in this doping range most naturally originates from the ex-change of overdamped collective bosonic excitations ofwhich spin-ﬂuctuation-mediated pairing is the keycandidate. 7–11In underdoped cuprates, situation is more com- plex. On one hand, angle-resolved photoemission spectros-copy /H20849ARPES /H20850data taken at low energies /H20849below 50 meV /H20850 and low Twere interpreted as the indication that the under- lying FS still has Luttinger form, and the gap extracted fromthe position of the still visible narrow peak in the spectralfunction has a simple dwave, cos 2 /H9278form over the whole FS, including antinodal region around /H208490,/H9266/H20850and symmetry- related points.12,13On the other hand, ARPES data taken in the antinodal region show that the spectral function in thepseudogap regime develops a broad maximum at around100–200 meV. 13–15The jury is still out16whether the ob- served high-energy hump and low-energy peak are separatefeatures or the peak and the hump describe the same gap,/H9004/H20849k/H20850, which strongly deviates from cos 2 /H9278form with under-doping. The experimental results in Refs. 13–15and17–20 were interpreted both ways. We side with the idea that thepairing gap remains cos 2 /H9278even in underdoped materials, and the hump is a separate feature, associated with Mottphysics. We further take the point of view that the origin ofthe hump is the development of precursors to a Heisenberg-like antiferromagnetically ordered state at half-ﬁlling. 21–28 These precursors are generally termed as spin-density-wave /H20849SDW /H20850precursors though one should keep in mind that the half-ﬁlled state is the strong coupling version of SDW and isbest described by the Heisenberg model with short-range ex-change interaction. The SDW precursor scenario has been wildly discussed in mid-90s, 21,23,29,30and is nearly univer- sally accepted scenario for electron-doped cuprates24,25,31For hole-doped cuprates, it was, however, put aside for a numberof years if favor of non-Fermi-liquid-type scenarios. 32The SDW scenario, however, regained support in the last fewyears, after magneto-oscillation experiments in a ﬁeld of30–60 T detected long-lived Fermi-liquid quasiparticles nearsmall electron and hole FSs. 33Such FS geometry is expected for an SDW ordered state,21and early theory prediction was that a ﬁeld drives the system toward an SDW instability.34 Long-range antiferromagnetic order in applied ﬁeld has beenexplicitly detected in recent neutron-scattering experimentson underdoped YBCO /H20849Ref. 35/H20850./H20849Another widely discussed scenario of quantum oscillations, which we will not considerhere, is a d-wave density-wave order. 36/H20850 In this paper, we analyze the consistency between the de- scription of quantum oscillations andthe pseudogap in un- derdoped cuprates within SDW scenario. The problem is thefollowing: to explain quantum oscillations one has to assumethe existence of small electron pockets. 37,38Such pockets do exist in the SDW scenario near /H208490,/H9266/H20850and symmetry-related points, but they are present only if SDW order /H20855Sz/H20849Q/H20850/H20856 =/H20855Sz/H20856is smaller than a threshold /H20851Q=/H20849/H9266,/H9266/H20850/H20852. For larger /H20855Sz/H20856, only hole pockets around /H20849/H9266/2,/H9266/2/H20850are present, while ex- citations near /H208490,/H9266/H20850have a gap of order 4 t/H11032/H110110.2 eV /H20849see Fig. 2/H20850. Antinodal pseudogap detected in ARPES experi- ments in zero ﬁeld is of the same magnitude.13A ﬁeld ofPHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 1098-0121/2010/81 /H2084917/H20850/174536 /H2084913/H20850 ©2010 The American Physical Society 174536-140–60 T is too small to affect energies of 0.2 eV, hence the same 200 meV pseudogap should be present in the orderedSDW state. 39If this pseudogap is viewed as a precursor to SDW, one could expect that it simply sharpens up in theordered SDW phase and transforms into the true antinodalgap. But then there will be no electron pockets in the SDWphase, in disagreement with magneto-oscillation experi-ments. The pseudogap and quantum oscillations can be rec-onciled within SDW scenario only if the evolution of thespectral function between paramagnetic and SDW states ismore complex than just the sharpening of the pseudogap, andantinodal spectral function in the SDW phase contains bothhigh-energy pseudogap and electron pockets. This coexist-ence also explains ARPES data in the superconductingstate 12,13because pairing of coherent fermions near electron pockets gives rise to a sharp peak in the spectral function atthe gap energy. To address this issue, we consider how fermionic Green’s function G/H20849k, /H9275/H20850evolves within SDW phase, as SDW order gets smaller. We depart from Heisenberg antiferromagnetwith exchange interaction J=4t 2/Uand consider analytically how G/H20849k,/H9275/H20850is affected by thermal ﬂuctuations which in quasi-2D systems destroy long-range order already at T/H11270J. We neglect regular T/Jcorrections but sum up inﬁnite series of self-energy terms which contain powers of /H9252 =/H20849T//H9266J/H20850/H20841log/H9280/H20841, where /H9280is a parameter which measures de- viations from pure two dimensionality and which we willjust use as a lower cutoff of logarithmically divergent 2Dintegrals. In practice, is the ratio of the hoppings along z-axisand in xy plane. 40SDW order disappears when /H9252=/H9252cr =O/H208491/H20850. This yields a set of integral equations for the Green’s function and /H20855Sz/H20856which we obtain and solve. In the terminology of Ref. 41, our computations are valid in the renormalized-classical regime of quasi-2D systems.The idea that, in this regime, at low enough T, one cannot restrict with Eliashberg or ﬂuctuation exchange /H20849FLEX /H20850ap- proximations and has to include self-energy and vertex cor-rections on equal footings has been put forward in Refs. 21 and23. The computational procedure that we are using is similar to eikonal approximation in the scattering theory.Such procedure has been used in the study one-dimensional/H208491D/H20850charge-density wave systems by Sadovskiic 42and others43and has been applied to cuprates in Ref. 22to ana- lyze SDW precursors in the paramagnetic phase /H20849for latest developments, see Ref. 26/H20850. Our computation has one advan- tage over earlier works: in the SDW-ordered state we do notneed to assume that Tis larger than some threshold T 0to restrict with only thermal ﬂuctuations /H20849i.e., with the contri- butions from zero Matsubara frequency /H20850. All we need is a small/H9280such that T/J/H20841log/H9280/H20841=O/H208491/H20850even when T/Jis small. In a paramagnetic phase, eikonal approximation is only validwhen T/H11022T 0, and T0increases as one moves away from the SDW phase. We assume that near SDW boundary T0is small and apply our theory also to a paramagnetic phase. Our re-sults for a paramagnet are in full agreement with Ref. 22. Note that in our theory /H20849and in Ref. 22/H20850, the paramagnetic state is a Fermi liquid at the lowest energies at T=0. W e do not discuss here a possibility that a new, non-Fermi-liquidstate emerges near the region where SDW order is lost. 44We also do not discuss possibilities of more complex spin orderand of open electron Fermi surfaces. Our results are expected to survive if SDW order is incommensurate, with Qstill near /H20849/H9266,/H9266/H20850, but whether our results survive if the system develops a stripe order remains to be seen. We found that the spectral function A/H20849k,/H9275/H20850 =/H208491//H9266/H20850/H20841ImG/H20849k,/H9275/H20850/H20841near /H208490,/H9266/H20850in the SDW state has a peak and a hump. Both originate from a single peak at the value oftheT=0 SDW gap at /H208490, /H9266/H20850. The hump moves little as SDW order decreases and just gets broader, while the peak follows/H20855S z/H20856, shifts to lower energies as SDW order decreases, and vanishes /H9252=/H9252cr, when the system enter the paramagnetic phase. At /H9252/H11350/H9252cr, only the hump /H20849the pseudogap /H20850remains, and the spectral function at antinodal k=kFhas camel-like structure with a minimum at /H9275=0. As /H9252increases further, A/H20849kF,/H9275=0/H20850increases and eventually the spectral function at k=kFdevelops a single peak at /H9275=0, as it should be for a system with a large, Luttinger FS. Rewinding this backward, from a paramagnet to an SDW state, we see that the system ﬁrst develops a pseudogap as aprecursor to SDW. When SDW order sets in, the pseudogapsharpens up, but, in addition, there also appears a true qua-siparticle peak at low energies. The residue of the peak in-creases as /H20855S z/H20856increases. When /H20855Sz/H20856is below the threshold, electron pockets are present, and the spectral function near/H208490, /H9266/H20850has a low-energy coherent peak and a hump at about the same energy as the pseudogap in a paramagnetic phase.When SDW order gets larger, electron pockets eventuallydisappear, peak and hump come closer to each other andmerge when /H20855S z/H20856reaches its maximum. This peak/hump structure also shows up in the density of states and in the optical conductivity /H9268/H20849/H9275/H20850. In the Mott- Heisenberg limit /H208492U/H20855Sz/H20856is larger than free-fermion band- width /H20850the conductivity at T=0 is zero up to a charge-transfer gapU/H110111.7 eV. Once SDW order gets smaller, the peak at Usplits into a hump which slowly shifts to a higher fre- quency, and a peak whose energy scales as 2 U/H20855Sz/H20856. In addi- tion, there appears a metallic Drude component at the small-est frequencies. This behavior is quite consistent with themeasured /H9268/H20849/H9275/H20850in electron-doped cuprates, where SDW phase extends over a substantial doping range.44–47 We also found that, at a ﬁnite T, the system in the pseudogap phase retains the memory about pockets. Thereare no real pockets in the sense that there is no two-peakstructure of the spectral function at zero frequency alongzone diagonal, but we found that, when /H9252/H11350/H9252cr, the spectral weight at /H9275=0 extends almost all the way between the origi- nal FS at kFand the “shadow” FS at k=Q−kF/H20849see Fig. 8/H20850. As/H9252becomes larger, the krange where the spectral weight is ﬁnite shrinks and at large /H9252the spectral function recovers Drude-type structure typical for a metal with a large, Lut-tinger FS. This analysis can be extended into a superconducting state. A system does not need to possess coherent quasipar-ticles to develop a pairing instability, 48,49but fermionic co- herence emerges below the actual Tcmuch in the same way as it emerges in the SDW ordered state. The spectral functionin the antinodal region then displays a coherent supercon-ducting peak and a hump centered at, roughly, the energy ofthe antinodal SDW gap at T=0. This picture is consistent with the data from Refs. 13and18.TIGRAN A. SEDRAKYAN AND ANDREY V. CHUBUKOV PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-2The overall conclusion of our analysis is the phase dia- gram of the cuprates presented in Fig. 1. A similar phase diagram has been proposed in Ref. 52.A t T/HS110050, there is a region where the system displays SDW precursors. In thisregion, magnetic excitations are moderately damped, propa-gating magnons, and magnetically mediated d-wave pairing interaction decrease as doping decreases due to a reductionin the electron-magnon vertex. 51,53,54In this region, the anti- nodal pseudogap, caused by SDW precursors, and a d-wave pairing gap coexist. Outside this region, SDW precursors donot emerge, and arcs and other pseudogap features arecaused by thermal ﬂuctuations of a pairing gap. 5,17,55 We discuss the computational procedure in the next sec- tion, present the results in Sec. IIIand conclusions in Sec. IV. II. COMPUTATIONAL PROCEDURE Our point of departure is the mean-ﬁeld, SDW theory21,56–58of the antiferromagnetically ordered state in the large Uquasi-2D Hubbard model at T=0. We assume t−t/H11032 dispersion in the XY plane /H20851/H9280k=−2t/H20849coskx+cos ky/H20850 −4t/H11032coskxcosky/H20852and weak dispersion along the Zaxis, which we will not keep explicitly in the formulas. Mean-ﬁelddescription neglects quantum ﬂuctuations and is rigorouslyjustiﬁed when the model is extended to 2 S/H112711 fermionic ﬂavors, 59but qualitatively it remains valid even for S=1 /2 /H20849/H20855Sz/H20856becomes 0.32 instead of 0.5 /H20850. Long-range antiferromagnetic order splits the fermionic dispersion into valence and conduction bands, separated byU, and gives rise to a two-pole structure of the bare fermi- onic Green’s function G 0/H20849/H9275,k/H20850=uk2G0c+vk2G0v, /H208491/H20850 where uk,vk=/H20881/H208491/H110074t/H9253k/Ek/H20850/2,/H9253k=/H20849coskx+cos ky/H20850/2, andG0c=1 /H9275−Ekc,G0v=1 /H9275−Ekv. /H208492/H20850 The dispersions of conduction and valence electrons are given by Ekc,v=/H11006Ek−4t/H11032coskxcosky−/H9262, where Ek=/H20851/H900402 +16t2/H9253k2/H208521/2,/H90040=U/H20855Sz/H20856, and/H9262/H11015−/H90040is the chemical poten- tial. The shape of the FS depends on the value of /H90040. We show the evolution of the FS with increasing /H90040in Fig. 2. For small /H90040both hole and electron pockets are present, for larger /H90040only hole pockets remain. At large U/t, which we assume to hold, valence, and conduction bands are well sepa- rated near half-ﬁlling at T=0,uk2/H11015vk2/H110151/2, and the FS only contains hole pockets. The value of /H20855Sz/H20856is determined by the self-consistency condition /H20855Sz/H20856=/H20885d2k /H208492/H9266/H208502ukvk/H20885d/H9275 /H9266nF/H20849/H9275/H20850Im/H20851G0c−G0v/H20852, /H208493/H20850 where both G0candG0vare retarded functions and nF/H20849/H9275/H20850is the Fermi function. At large U, and near half-ﬁlling, /H20855Sz/H20856/H110151/2, and/H90040/H11015U/2. For larger dopings and smaller U,/H20855Sz/H20856is smaller already at T=0. Fermion-fermion interactions in the ordered SDW state can be cast into interactions between fermions and magnons.These interactions are described by the effectiveHamiltonian 21SDW fluct. xT SCPFAF FIG. 1. /H20849Color online /H20850Schematic phase diagram of hole- and electron-doped cuprates. The regions of antiferromagnetism /H20849AF/H20850, superconductivity /H20849SC/H20850, and pairing ﬂuctuations /H20849PF/H20850are shaded. In the PF region, there are vortex excitations and large Nernst signal/H20849Ref. 50/H20850. Precursors to SDW appear at a nonzero Tdue to strong thermal ﬂuctuations. In the region where SDW precursors are al-ready developed, the onset temperature for the pairing increaseswith increasing x/H20849Ref. 51/H20850, in the region with no SDW precursors, it decreases with increasing x/H20849Ref. 7/H20850, the maximum T cis in the area between the two regimes. A similar phase diagram has beenproposed in Ref. 52./Minus3/Minus2/Minus10 1 2 3/Minus3/Minus2/Minus10123 a /Minus3/Minus2/Minus10 1 2 3/Minus3/Minus2/Minus10123 b /Minus3/Minus2/Minus10 1 2 3/Minus3/Minus2/Minus10123 c /Minus3/Minus2/Minus10 1 2 3/Minus3/Minus2/Minus10123 d FIG. 2. /H20849Color online /H20850Evolution of the FS with increasing SDW order. /H20849a/H20850—paramagnetic phase; /H20849b/H20850—SDW order is about to de- velop /H20849/H20855Sz/H20856=0+/H20850. The shadow FS emerges, but the residue of fermi- onic excitations at the shadow FS is 0+;/H20849c/H20850—small /H20855Sz/H20856, both hole and electron pockets are present; /H20849d/H20850—a larger /H20855Sz/H20856, only hole pock- ets around /H20849/H9266/2,/H9266/2/H20850remain.PSEUDOGAP IN UNDERDOPED CUPRATES AND SPIN- … PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-3Hel-mag=/H20858 /H9251,/H9252/H20858 k,q/H20851a/H9251k+a/H9252k+qeq+Vaa/H20849k,q/H20850+b/H9251k+b/H9252k+qeq+Vbb/H20849k,q/H20850 +a/H9251k+b/H9252k+qeq+Vab/H20849k,q/H20850+b/H9251k+a/H9252k+qeq+Vba/H20849k,q/H20850 + H.c. /H20852/H9254/H9251,−/H9252, /H208494/H20850 where operators aandbrepresent conduction- and valence- band fermions, respectively. The vertex functions are givenby V aa,bb/H20849k,q/H20850=U /H208812/H20851/H11006/H20849ukuk+q−vkvk+q/H20850/H9257q+/H20849ukvk+q −vkuk+q/H20850/H9257¯q/H20852, Vab,ba/H20849k,q/H20850=U /H208812/H20851/H20849ukvk+q+vkuk+q/H20850/H9257q/H11007/H20849ukuk+q+vkvk+q/H20850/H9257¯q/H20852 /H208495/H20850 with /H9257q=1 /H208812/H208731−/H9253q 1+/H9253q/H208741/4 ,/H9257¯q=1 /H208812/H208731+/H9253q 1−/H9253q/H208741/4 . /H208496/H20850 At the mean-ﬁeld level, the spectral function A/H20849k,/H9275/H20850has two/H9254-functional peaks at /H9275=EkcandEkv, and the density of states has a gap 2 /H90040=U. Our goal is to analyze how this spectral function gets modiﬁed once we use Eq. /H208494/H20850and com- pute thermal fermionic self-energy and thermal correctionsto/H20855S z/H20856. A. One-loop perturbation theory There are several contributions to fermionic self-energy /H9018/H20849k,/H9275/H20850to one loop order, but the earlier study by Morr and one of us has found21that the dominant one at the lowest T comes from the interaction between valence and conductionfermions mediated by the exchange of low-energy transversespin waves /H20849see Ref. 21for details /H20850. To one-loop order, spin- wave-mediated interaction gives rise to /H9018 1v,c/H20849/H9275,k/H20850=/H9252/H900402Gc,v/H20849/H9275,k/H20850=/H9252/H900402 /H9275−E¯ kcv, /H208497/H20850 where, we remind, /H9252=/H20849T//H9266J/H20850/H20841log/H9280/H20841. The order parameter also acquires a correction proportional to /H9252. To obtain it, one has to substitute the self-energy into the Green’s function andcompute /H20855S z/H20856using Eq. /H208493/H20850, but with the full Ginstead of G0. This yields /H20855Sz/H20856=1 2/H208731−/H9252 2/H20874, /H208498/H20850 i.e.,/H90042=/H900402/H208511−/H9252+O/H20849/H92522/H20850/H20852. The O/H20849/H9252/H20850/H20849 /H20841log/H9280/H20841/H20850correction to /H20855Sz/H20856is in agreement with Mermin-Wagner theorem. How- ever, when we combine self-energies Eq. /H208497/H20850and G0v,cin which /H90040=U/2 is replaced by /H9004=U/H20855Sz/H20856and obtain the new Green’s function, we ﬁnd that O/H20849/H9252/H20850terms cancel out, i.e., to ﬁrst order in /H9252the fermionic Green’s function does not changeG/H20849/H9275,k,/H9252/H20850/H110151 2/H208751 /H9275−Ec−/H90181c+1 /H9275−Ev−/H90181v/H20876 =/H9275¯ /H9275¯2−1 6t2/H9253k2−/H20849/H90042+/H9252/H900402/H20850=G0/H20849/H9275,k/H20850, /H208499/H20850 where /H9275¯=/H9275+4t/H11032coskxcosky+/H9262. This result was obtained in Ref. 21and was interpreted as an indication that the SDW form of G/H20849k,/H9275/H20850may survive even when /H20855Sz/H20856vanishes. How- ever, one-loop result is at best indicative, and we need to goto higher orders to verify what happens with the fermionicGreen’s function when /H9252increases. B. Two-loop corrections As the next step, we obtain two-loop formulas for the self-energy and /H20855Sz/H20856. The two-loop diagrams for the self- energy are the second and third diagrams in Fig. 3. Evaluat- ing them in the same approximation as one-loop diagram, weobtain /H9018 2c/H20849/H9275,k/H20850=2/H92522/H900404 /H20849/H9275−Ekv/H20850/H20849/H9275−Ekc/H208502, /H90182v/H20849/H9275,k/H20850=2/H92522/H900404 /H20849/H9275−Ekv/H208502/H20849/H9275−Ekc/H20850. /H2084910/H20850 Substituting these self-energies into the valence and conduc- tion Green’s functions together with one-loop diagrams, weobtain after a simple algebra G/H20849 /H9275,k/H20850=/H9275¯ 3/H208752 /H9275¯2−1 6t2/H9253k2−/H20849/H90042+2/H900402/H9252/H20850 +1 /H9275¯2−1 6t2/H9253k2−/H20849/H90042−/H900402/H9252/H20850/H20876. /H2084911/H20850 Evaluating /H20855Sz/H20856in the same two-loop approximation we ob- tain /H20855Sz/H20856=1 2/H208731−/H9252 2+5/H92522 8/H20874, /H2084912/H20850 such that /H90042=/H900402/H208511−/H9252+3/H92522/2+O/H20849/H92523/H20850/H20852. Substituting now this/H90042into Eq. /H2084911/H20850we ﬁnd that, up to two-loop order, G/H20849/H9275,k/H20850=/H9275¯ 3/H208752 /H9275¯2−1 6t2/H9253k2−/H900412+1 /H9275¯2−1 6t2/H9253k2−/H900422/H20876, /H2084913/H20850 where /H90041=/H90040/H208491+/H9252/2+5/H92522/8/H20850and/H90042=/H90040/H208491−/H9252+/H92522/4/H20850. We see that the Green’s function splits into two compo- nents. Both have SDW form, but the values of /H9004are different(a)( b) (c) FIG. 3. Three equivalent diagrams for two-loop corrections to the Green’s function. The second and the third diagrams contributeto fermionic self-energy.TIGRAN A. SEDRAKYAN AND ANDREY V. CHUBUKOV PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-4−/H90042decreases with /H9252while/H90041increases. This implies that the peak in the spectral function, originally located at /H9275¯2 =16t2/H9253k2+/H900402, splits into two subpeaks—one shifts to higher /H20841/H9275¯/H20841, another to smaller /H20841/H9275¯/H20841. This is the new trend, not present in the one-loop approximation. This consideration also shows that to understand what happens when /H9252=O/H208491/H20850, one cannot restrict with a few ﬁrst orders in the loop expansion but rather has to sum up inﬁnitenumber of terms. This is what we are going to do next. C. Nonperturbative Green’s function We list several results which can be explicitly veriﬁed by doing loop-expansion order by order in /H9252./H208491/H20850The SDW order parameter /H20855Sz/H20856is given by the same loop expression as in the mean-ﬁeld theory, Eq. /H208493/H20850, but with the full Ginstead ofG0./H208492/H20850The renormalized /H20855Sz/H20856, in turn, appears in the Green’s function through G0v,cin which /H90040=U/2 has to be replaced by /H9004=U/H20855Sz/H20856./H208493/H20850The full Green’s function G/H20849/H9275,k,/H9252/H20850is given by G/H20849/H9275,k,/H9252/H20850=vk2Gv/H20849/H9275,k,/H9252/H20850 +uk2Gc/H20849/H9275,k,/H9252/H20850, where Gv,care the full Green’s functions for valence and conduction fermions and ukandvkare the same as in Eq. /H208491/H20850but with /H9004instead of /H90040./H208494/H20850At loop order nof the perturbation theory there are /H208492n−1/H20850!! equivalent contri- butions to the full Gc,v, each contains /H20851/H9252G0v/H20849/H9275,k/H20850G0c/H20849/H9275,k/H20850/H20852n /H20849see Fig. 4/H20850. Because of the last item, it is advantageous to sum up inﬁnite series of diagrams for the Green’s function ratherthan for the self-energy. We have G trv,c/H20849/H9275,k,/H9252/H20850=G0v,c/H20849/H9275,k/H20850+/H20858 n=1/H11009 /H208492n−1/H20850!!G0v,c/H20849/H9275,k/H20850 /H11003/H20851/H9252/H900402G0v/H20849/H9275,k/H20850G0c/H20849/H9275,k/H20850/H20852n+¯, /H2084914/H20850 where dots stand for nonlogarithmic corrections and subin- dextrimplies that we only considered interaction with trans- verse spin waves. Substituting the expressions for G0v,cand summing up asymptotic series we obtain Gtrv,c/H20849/H9275,k,/H9252/H20850=2 /H90040/H20873/H9266 2/H9252/H208741/2/H9275¯/H11007Ek /H20849/H9275¯2−Ek2/H208501/2exp/H20877−/H9275¯2−Ek2 2/H900402/H9252/H20878 /H20877i+ Erfi/H20875/H20881/H9275¯2−Ek2 2/H900402/H9252/H20876/H20878+¯, /H2084915/H20850 where Erfi /H20849z/H20850=−iErf/H20849iz/H20850is imaginary error function /H20851Erfi /H20849x/H20850is real when xis real and imaginary when xis imaginary /H20852. Observe that Im Gtrv,cvanishes when /H9275¯2/H11021Ek2. To one-loop order, Eq. /H2084915/H20850reduces to G0v,c, but beyond one loop the Green’s functions Gv,cobviously become /H9252de- pendent. The spectral function Atr/H20849/H9275,k,/H9252/H20850=/H9266−1/H20841ImG/H20849/H9275 −i0,k,/H9252/H20850/H20841is readily obtained from Eq. /H2084915/H20850 Atr/H20849/H9275,k,/H9252/H20850=2 /H90040/H208811 2/H9266/H9252e−/H20849/H9275¯2−Ek2/H20850/2/H900402/H9252 /H11003uk2/H20841/H9275¯+Ek/H20841+vk2/H20841/H9275¯−Ek/H20841 /H20849/H9275¯2−Ek2/H208501/2/H9258/H20851/H9275¯2−Ek2/H20852, /H2084916/H20850 where /H9258/H20849x/H20850=1 for x/H110220 and zero otherwise. Substituting Gv,cfrom Eq. /H2084915/H20850into the expression for /H20855Sz/H20856we obtain how the SDW order parameter evolves with /H9252 /H20855Sz/H20856=1 2/H20885d2k /H208492/H9266/H208502/H208812 /H9266/H9252/H9004 /H90040/H20885 −/H11009/H11009 d/H9275¯exp/H20877−/H9275¯2−Ek2 2/H900402/H9252/H20878 /H20881/H9275¯2−Ek2 /H11003nF/H20849/H9275¯−/H9262−4t/H11032coskxcosky/H20850/H9258/H20851/H9275¯2−Ek2/H20852./H2084917/H20850 The remaining unknown parameter /H9262is ﬁxed by the condi- tion on the number of particles in the SDW state60 /H208491−x/H20850 2=/H20885d2kd/H9275 /H208492/H9266/H208502Atr/H20849/H9275,k,/H9252/H20850nF/H20849/H9275/H20850. /H2084918/H20850 These coupled equations were solved numerically. The dependence of /H20855Sz/H20856on/H9252or doping xis quite as expected: /H20855Sz/H20856monotonically decreases as /H9252orxincrease, and vanishes at some particular /H9252crandxcr/H20849see Fig. 6/H20850. The spectral func- tion Atr/H20849/H9275,k,/H9252/H20850has sharp /H9254-functional peaks at /H9252→0, at /H9275¯=/H11006Ek/H20849/H9275=Ekv,c/H20850. At ﬁnite /H9252, quasiparticle peaks transform into branch cuts at /H20841/H9275¯/H20841=Ek, with the width of order /H9252/H11008T, and the spectral weight extends to larger frequencies. Near the branch cut Atr/H20849/H9275,k,/H9252/H20850diverges as 1 //H20881x. We plot Atr/H20849/H9275,k,/H9252/H20850 at a hot spot khs=/H20849kx,/H9266−kx/H20850in Fig. 5. D. Further modiﬁcations of the spectral function On a more careful look, we found that the spectral func- tion given by Eq. /H2084916/H20850have to be further modiﬁed by two reasons. First, in the calculations above we only included theself-energy due to exchange of transverse spin waves, andneglected the self-energy due to exchange of longitudinalspin ﬂuctuations. This is justiﬁed at small /H9252, when longitu- dinal ﬂuctuations are gapped, but when /H9252/H11015/H9252cr, longitudinal ﬂuctuations are nearly gapless and are as important as trans-verse ones. The limiting case when transverse and longitudi-nal spin propagators are identical can be studied within thesame approximation as before, the only difference is thatcombinatoric factors are now /H208492n+1/H20850!!/2 n/H20849Ref. 22/H20850.A sa result, the spectral function becomes+ +G= + + + + + +... .. . + 15 FIG. 4. Diagrammatic series for the Green’s function. Only ther- mal contributions in which a fermion jumps from a valence to aconduction band /H20849and vice versa /H20850and emits /H20849absorbes /H20850a transverse spin wave are included.PSEUDOGAP IN UNDERDOPED CUPRATES AND SPIN- … PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-5Al+tr/H20849/H9275,k,/H9252/H20850=1 /H9004034 /H20881/H9266/H92523exp/H20877−/H20849/H9275¯2−Ek2/H20850 /H900402/H9252/H20878 /H11003uk2/H20841/H9275¯+Ek/H20841+vk2/H20841/H9275¯−Ek/H20841 /H20849/H9275¯2−Ek2/H20850−1 /2/H9258/H20851/H9275¯2−Ek2/H20852, /H2084919/H20850 where subindex l+trimplies that this is a contribution from both longitudinal and transverse spin excitations. One caneasily make sure that A l+tris also fully incoherent at /H9252/H110220, but, in distinction to Atr, it vanishes at /H9275¯=/H11006Ekand has a hump at a frequency which remains of order /H90040for all /H9252 /H11021/H9252crWe plot Al+tr/H20849/H9275,k,/H9252/H20850at a hot spot in Fig. 5. The func- tionAl+tr/H20849/H9275,khs,/H9252/H20850becomes particularly simple at /H9252/H11350/H9252cr Al+tr/H20849/H9275,khs/H20850=/H92752 /H900404 /H20881/H9266/H92523exp/H20877−/H92752 /H9252/H900402/H20878. /H2084920/H20850 For a generic /H9252/H11349/H9252cr, Eq. /H2084919/H20850is the correct result at high energies, larger than the longitudinal gap and Eq. /H2084916/H20850is the correct result at smaller energies. As /H9252approaches /H9252crthe range of applicability of Atr/H20849/H9275,khs/H20850shrinks. The full formula cannot be obtained within eikonal approximation, but it isclear that the actual A/H20849 /H9275,k,/H9252/H20850contains both the branch cut, 1//H20881xsingularity near /H9275=Ekv,cand the hump at a frequency of order/H90040, where the quasiparticle peaks were located at /H9252 =0. To simplify the computational procedure, in Fig. 9below we use for the actual A/H20849/H9275,khs,/H9252/H20850the function Atr/H20849/H9275,khs,/H9252/H20850 up to a frequency where it crosses with Al+tr, and use the function Al+tr/H20849/H9275,khs,/H9252/H20850at larger frequencies. Second, Eqs. /H2084916/H20850and /H2084919/H20850show that fermionic coherence is lost immediately when /H9252becomes nonzero /H20849the pole trans- forms into a branch cut /H20850. Meanwhile, from physics perspec- tive, as long as the system has an SDW order, a Fermi-liquidbehavior near the pocketed FS should be preserved, i.e., a quasiparticle peak with T2logTwidth should survive, albeit with a reduced magnitude. The reason it was lost in the cal-culations above is because we completely neglected regularclassical and quantum corrections to the Green’s function/H20851dots in Eq. /H2084914/H20850/H20852. We veriﬁed that, when these terms are included, only a part of G 0v,cgets involved in the renormal- izations by series of /H9252ncorrections, the other stays intact. This implies that the actual Afullv,c=Z/H9252A0v,c+/H208491−Z/H9252/H20850Av,c. The residue Z/H9252is some number 0 /H11021Z/H9252/H110211a t/H9252=0, where anyway Av,c=A0v,c, it decreases as /H9252increases and vanishes at /H9252 =/H9252cr. For deﬁniteness, we used Z/H9252=0.2 in the panel 0 /H11021/H9252 /H11021/H9252crin Figs. 8and9. These two additions also affect the formula for /H20855Sz/H20856and the equation for /H9262, which become /H20855Sz/H20856=Z/H9252/H20885d2k /H208492/H9266/H208502/H9004 Ek/H20851nF/H20849Ekv/H20850−nF/H20849Ekc/H20850/H20852 +/H208491−Z/H9252/H208501 2/H20885d2k /H208492/H9266/H208502/H208812 /H9266/H9252/H9004 /H90040 /H20885 −/H11009/H11009 d/H9275¯exp/H20877−/H9275¯2−Ek2 2/H900402/H9252/H20878 /H20881/H9275¯2−Ek2 nF/H20849/H9275¯−/H9262−4t/H11032coskxcosky/H20850/H9258/H20851/H9275¯2−Ek2/H20852/H20849 21/H20850 and /H208491−x/H20850 2=/H20885 −/H11009/H11009 d/H9275/H20885d2k /H208492/H9266/H208502nF/H20849/H9275/H20850/H20877Z/H9252 /H9266/H20851vk2/H9254/H20849/H9275−Ekv/H20850 +uk2/H9254/H20849/H9275−Ekc/H20850/H20852+/H208491−Z/H9252/H20850A/H20849/H9275,k/H20850/H20878. /H2084922/H20850 III. RESULTS We solved Eqs. /H2084921/H20850and /H2084922/H20850numerically and plot the dependence of the order parameter /H20855Sz/H20856on/H9252andxin Fig. 6. We used several phenomenological forms of Z/H9252in which Z/H9252cr=0, but found that the functional forms of the spectral functions do not depend on Z/H9252in any substantial way. ForFIG. 5. /H20849Color online /H20850The spectral function A/H20849/H9275,khs,/H9252/H20850at a hot spot. Here and below a.u. stand for arbitary units. Left panel − Atr, obtained by including only transverse spin waves. Right panel−A l+tr, obtained by treating transverse and longitudinal spin excita- tions on equal footings. Dashed lines − /H9252=0.6/H9252cr, solid lines − /H9252 =0.8/H9252cr.Atrhas a branch cut at energy which scales with the mag- nitude of SDW order parameter, while Al+trhas a hump at energy which roughly remains the same as SDW gap at T=0. The actual A/H20849/H9275,khs,/H9252/H20850at/H9252/H11021/H9252crcoincides with Atrat low frequencies, and crosses over to Al+trat frequencies larger than the gap for longitu- dinal ﬂuctuations. As a result, the actual spectral function has apeak at a low energy and a hump at a higher energy.FIG. 6. /H20849Color online /H20850The SDW order parameter /H20855Sz/H20856vs/H9252at a given x=0.05. For simplicity, we set Z/H9252to be a constant /H20849=0.2 /H20850. Inset: /H20855Sz/H20856vsxat a given /H9252=1.TIGRAN A. SEDRAKYAN AND ANDREY V. CHUBUKOV PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-6simplicity, and because below we only present the results for a few/H9252in the SDW state, we set Z/H9252=0.2, independent of the actual /H9252. Also, all cases /H20849even when Z/H9252=0/H20850/H20855Sz/H20856monotoni- cally decreases when either xor/H9252increase and vanishes at some xcrand/H9252cr. In Fig. 7we show the evolution of the electron Fermi surface near /H208490,/H9266/H20850and symmetry-related points. The elec- tron pocket is absent at small /H9252, when SDW order is strong, but appears when /H9252exceeds some critical value. The elec- tron pocket evolves with increasing /H9252and eventually disap- pears when the system looses long-range SDW order. Thisbehavior is qualitatively consistent with the mean-ﬁeld SDWpicture. The results for A full/H20849/H9275,k,/H9252/H20850are presented in Figs. 8–10. In Fig. 8we plot the full spectral function at zero frequency along the diagonal direction in the Brillouin zone. In the twolimits, /H9252=0 and /H9252/H11271/H9252cr, the system possesses sharp quasiparticles—in the ﬁrst case at the two sides of the holepocket, in the second case at the large, Luttinger FS. In be- tween, the spectral function evolves, as /H9252increases, from a well-pronounced two-peak structure to a completely incoher-ent structure at /H9252=/H9252cr, in which the spectral weight at /H9275 =0 is spreaded between the original and the shadow FSs /H20851in reality, Afull/H208490,k,/H9252/H20850spreads outside of this range, but to ﬁnd these tails of the spectral function one has to go beyond theaccuracy of our calculations /H20852. This result implies that the system does retain some memory about SDW pockets evenwhen /H9252=/H9252crand/H9004=0. When /H9252becomes larger than /H9252crand SDW order disappears, the region where Afull/H208490,k,/H9252/H20850/HS110050 progressively shrinks with increasing /H9252toward a single qua- siparticle peak. We emphasize that this evolution ofA full/H208490,k,/H9252/H20850with/H9252is very different from that in the mean- ﬁeld SDW theory, where the spectral function in the SDWstate has two peaks at the two sides of the hole pocket andthe shadow peak just disappears when /H9004vanishes. FIG. 7. /H20849Color online /H20850The appearance and evolution of the electron pocket near /H20849kx,ky/H20850=/H208490,/H9266/H20850and symmetry related points. Electron pocket appears as a single point at /H208490,/H9266/H20850once/H9252increases and reaches a critical value, and evolves with increasing /H9252. From left to right: /H9252//H9252cr=0.8;0.85;0.95. We set x=0.02. FIG. 8. /H20849Color online /H20850The spectral function, Afull/H20849/H9275,k,/H9252/H20850at/H9275=0 along the diagonal /H20849nodal /H20850direction in the Brillouin zone. We used Z=0.2 for the top right panel. Observe that the systems retains a memory about a shadow FS when SDW order disappears at /H9252=/H9252cr/H20849the spectral weight is nonzero in the whole region between the original and the shadow FSs /H20850. We set x=0.05, t=0.32/H90040,t/H11032=−0.2 t. For these parameters, /H9252cr/H110151.PSEUDOGAP IN UNDERDOPED CUPRATES AND SPIN- … PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-7In Fig. 9we show the frequency dependence of Afull/H20849/H9275,khs,/H9252/H20850at a hot spot for a wide range of /H9252. In the two limits, the behavior is again coherent: there are quasiparticle peaks at /H9275¯=/H9275+/H9262−4t/H11032cos2kx=/H11006/H9004 0deep in the SDW phase, and the peak centered at /H9275¯=/H9275=0 deep in the normal phase. In between, the spectral function is again predomi-nantly incoherent. Speciﬁcally, as /H9252increases toward /H9252cr, the quasiparticle peak splits into the peak at /H9275¯=/H9004, and the hump at a larger frequency. The frequency where the peak is lo-cated shifts downwards with decreasing /H9252, while the hump remains roughly at around /H90040, and, on a more careful look, shifts toward somewhat larger frequency. We show this be-havior in more detail in Fig. 10, where we plot the spectral function for a range of /H9252/H11021/H9252cr. The peak in the spectral function is the property of AtrandA0and the hump is the property of Al+tr. We emphasize that the peak and the humpdo coexist in the SDW phase, the ﬁrst describes coherent low-energy excitations, the second describes fully incoherenthigh-energy excitations. Once SDW order disappears at /H9252cr, coherent excitations also disappear, but the hump remainsand disappears only at much larger /H9252. It is quite natural to identify the hump at /H9252/H11022/H9252crwith the pseudogap, and low- energy coherent excitations existing at /H9252/H11021/H9252crwith the building blocks for magneto-oscillations. The implication ofthis result is that the 200 meV pseudogap observed inARPES in zero ﬁeld 13is not an obstacle for observing magneto-oscillations once the system becomes SDW or-dered. A. Density of states The density of states /H20849DOS /H20850in the SDW phase, N/H20849/H9275,/H9252/H20850 =/H20848/H20849d2k/4/H92662/H20850A/H20849/H9275,k,/H9252/H20850, is obtained from Eqs. /H208491/H20850,/H2084916/H20850, and /H2084919/H20850. The expressions for N/H20849/H9275/H20850at ﬁnite /H9252are rather complex fort/H11032/HS110050 but are simpliﬁed for t/H11032=0 which we assume to hold in this section. A ﬁnite t/H11032affects the behavior of the DOS at the smallest frequencies, but not at frequencies /H9275 +/H9262/H11350/H9004, which we are chieﬂy interested in. For free fermions, we then have N0/H20849/H9275,/H9252/H20850=/H9275¯ /H92662/H20885/H20885 −11dudv /H208811−u2/H208811−v2/H9254/H20851/H9275¯2−/H90042−4t2/H20849u+v/H208502/H20852 =1 2/H92662t/H9275¯ /H20881/H9275¯2−/H90042K/H20875/H208811−/H9275¯2−/H90042 16t2/H20876, /H2084923/H20850 where K/H20849x/H20850is the complete elliptic integral of the ﬁrst kind /H20849deﬁned as in Ref. 61/H20850, and for t/H11032=0,/H9275¯=/H9275+/H9262. The DOS vanishes at /H20841/H9275¯/H20841/H11021/H9004, diverges as 1 //H20881xat/H20841/H9275¯/H20841=/H9004+0, monotoni- cally decreases at larger frequencies, and discontinuously drops to zero at the bandwidth, when /H20841/H9275¯/H20841=/H20881/H90042+16t2. The DOS Ntr/H20849/H9275,/H9252/H20850obtained using the spectral function Atrfrom Eq. /H2084916/H20850reduces to a 1D integral FIG. 9. /H20849Color online /H20850The spectral function Afull/H20849/H9275,khs,/H9252/H20850at a hot spot, at various /H9252. The frequency is in units /H9275¯//H90040, where /H9275¯=/H9275 +/H9262−4t/H11032cos2kx. The parameters are the same as in Fig. 8. FIG. 10. /H20849Color online /H20850The spectral function Afull/H20849/H9275,khs,/H9252/H20850at a hot spot, at various 0 /H11021/H9252/H11349/H9252cr. The lowest plot corresponds to /H9252 =/H9252cr, values of /H9252decrease from bottom up. The frequency is in units/H9275¯//H90040. The parameters are the same as in Fig. 8.A s/H9252de- creases, the peak and the hump come closer to each other.TIGRAN A. SEDRAKYAN AND ANDREY V. CHUBUKOV PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-8Ntr/H20849/H9275,/H9252/H20850=/H9275¯ 2/H92665/2t/H20885 0/H20881/H20849/H9275¯2−/H90042/H20850/2/H9252/H900402dze−z2 /H20881/H9275¯2−/H90042−2/H9252/H900402z2 /H11003K/H20875/H208811−/H9275¯2−/H90042−2/H9252/H900402z2 16t2/H20876 /H2084924/H20850 for 0/H11021/H9275¯2−/H90042/H1102116t2and Ntr/H20849/H9275,/H9252/H20850=/H9275¯ 2/H92665/2t/H20885/H20881/H20849/H9275¯2−/H90042−16t2/H20850/2/H9252/H900402/H20881/H20849/H9275¯2−/H90042/H20850/2/H9252/H900402dze−z2 /H20881/H9275¯2−/H90042−2/H9252/H900402z2 /H11003K/H20875/H208811−/H9275¯2−/H90042−2/H9252/H900402z2 16t2/H20876 /H2084925/H20850 for/H9275¯2−/H90042/H1102216t2./H20851The trick how to integrate over 2D mo- menta kxandkyin/H20848d2k//H208494/H92662/H20850Atr/H20849/H9275,k,/H9252/H20850is to introduce u =cos kxandv=cos kyas new variables, use the identity e−/H20849/H9275¯2−Ek2/H20850/2/H9252/H900402 /H20881/H9275¯2−Ek22/H208812/H9252/H900402=/H20885 0/H11009 dze−z2/H9254/H20849/H9275¯2−Ek2−2/H9252/H900402z2/H20850 /H2084926/H20850 and use /H9254function to perform the momentum integration /H20852. The density of states Ntrvanishes at /H20841/H9275¯/H20841/H11021/H9004, diverges loga- rithmically at /H20841/H9275¯/H20841=/H9004+0, and monotonically decreases at larger frequencies. There is no threshold at the bandwidth as Atr/H20849/H9275,k,/H9252/H20850is nonzero everywhere at /H20841/H9275¯/H20841/H11022/H9004, but, indeed, at frequencies larger than the bandwidth our approximationeventually breaks down. For the spectral function A l+trfrom Eq. /H2084919/H20850, we obtain, using the same trick, Nl+tr/H20849/H9275,/H9252/H20850=/H9275¯ /H92665/2t/H20885 0/H20881/H20849/H9275¯2−/H90042/H20850//H9252/H900402dzz2e−z2 /H20881/H9275¯2−/H90042−/H9252/H900402z2 /H11003K/H20875/H208811−/H9275¯2−/H90042−/H9252/H900402z2 16t2/H20876 /H2084927/H20850 for 0/H11021/H9275¯2−/H90042/H1102116t2and Nl+tr/H20849/H9275,/H9252/H20850=/H9275¯ /H92665/2t/H20885/H20881/H20849/H9275¯2−/H90042−16t2/H20850//H9252/H900402/H20881/H20849/H9275¯2−/H90042/H20850//H9252/H900402dzz2e−z2 /H20881/H9275¯2−/H90042−/H9252/H900402z2 /H11003K/H20875/H208811−/H9275¯2−/H90042−/H9252/H900402z2 16t2/H20876 /H2084928/H20850 for/H9275¯2−/H90042/H1102216t2. This spectral function is continuous at /H20841/H9275¯/H20841=/H9004and has a broad maximum at frequencies comparable to/H90040. This behavior is very similar to the one for the spectral function. Again, the transverse-only contribution Ntrhas a peak at the frequency /H9275¯=/H9004, which scales with the order parameter of the SDW phase, while Nl+trwhich treats con- tributions from transverse and longitudinal ﬂuctuations onequal footings /H20849as if longitudinal excitations were massless /H20850 has no features at /H9004, but has a broad maximum at a fre- quency which remains comparable to /H9004 0. Just like the spec- tral function, the actual DOS N/H20849/H9275,/H9252/H20850interpolates between these two terms. At frequencies smaller than the gap for lon-gitudinal spin excitations, N/H20849 /H9275,/H9252/H20850/H11015Ntr/H20849/H9275,/H9252/H20850, while atlarger frequencies N/H20849/H9275,/H9252/H20850gradually approaches Nl+tr/H20849/H9275,/H9252/H20850. We plot the DOS in Fig. 11. We again set N/H20849/H9275,/H9252/H20850 =Ntr/H20849/H9275,/H9252/H20850at frequencies smaller than the crossing point be- tween Ntr/H20849/H9275,/H9252/H20850and Nl+tr/H20849/H9275,/H9252/H20850, and set N/H20849/H9275,/H9252/H20850 =Nl+tr/H20849/H9275,/H9252/H20850at higher frequencies. And we again assumed that the full DOS Nfull/H20849/H9275,/H9252/H20850is the sum of the incoherent N/H20849/H9275,/H9252/H20850with the factor 1− Z/H9252and the coherent, mean-ﬁeld N0/H20849/H9275,/H9252/H20850with the factor Z/H9252. The plots clearly show the same trends in the DOS as we just discussed. At small /H9252, there is a sharp gap 2 /H90040/H11015Ube- tween valence and conduction bands. As /H9252increases, sharp gap decreases, the spectral weight extends to higher frequen-cies, and the DOS develops a hump at an energy comparableto/H9004 0/H20849i.e., the distance between the humps remains U/H20850. The peak in the DOS at the boundary of the sharp gap getssmaller as /H9252increases and the sharp gap decreases. At /H9252 =/H9252crthe gap and the peak disappear, and the spectral func- tion only possesses a hump. As /H9252increases even further, the DOS at /H9275¯=0 increases /H20849and also /H9262gets reduced such that /H9275¯ comes closer to /H9275/H20850, and eventually the DOS at low frequen- cies recovers weakly frequency-dependent form of a metalwith a large FS. This physics is somewhat spoiled in the plots of N full/H20849/H9275,/H9252/H20850by the change in the behavior of Ntr+l/H20849/H9275,/H9252/H20850at the bandwidth /H20849at/H9275¯=/H20881/H90042+16t2/H20850. This change in behavior is seen in Fig. 11as the discontinuity in the frequency deriva- tive of Nfull/H20849/H9275,/H9252/H20850. For large /H9252the hump in the DOS is pre- dominantly the effect of the bandwidth. However, at interme-diate /H9252, the hump is located at a frequency below the bandwidth, as is clearly visible in the top right panel of Fig.11/H20849 /H9252=0.1/H9252cr/H20850and is therefore due to the physics that we described above. B. Optical conductivity The peak/hump structure also shows up in the optical con- ductivity. We computed the conductivity by standard means:by convoluting two full spectral functions A fullusing Kubo formulaFIG. 11. Density of states vs /H9275¯//H90040for different values of /H9252.W e sett/H11032=0 and t=0.32/H90040.PSEUDOGAP IN UNDERDOPED CUPRATES AND SPIN- … PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-9/H9268/H20849/H9275/H20850=/H9266e2/H20885d2k /H208492/H9266/H208502vk2/H20885d/H9024 nF/H20849/H9024/H20850−nF/H20849/H9024+/H9275/H20850 /H9275Afull/H20849/H9024,k,/H9252/H20850Afull/H20849/H9024+/H9275,k,/H9252/H20850, /H2084929/H20850 where vkis fermionic velocity. We used the same computa- tional procedure as before: combined mean-ﬁeld contributiontoA, with the prefactor Z /H9252, and incoherent part of Awith the prefactor 1− Z/H9252, and used Atrfor the incoherent part at small frequencies and Al+trat high frequencies. The results are shown in Fig. 12. We again set t/H11032=0 to simplify the calcula- tions. At small /H9252, the conductivity almost vanishes up to /H9275¯ =2/H90040=U. When /H9252gets larger, the discontinuity at Usplits into a hump which slowly shifts to a larger energy and a peakwhich scales with /H9004and shifts to a smaller frequency as SDW order gets weaker. In addition, there also appears aDrude component at the smallest frequencies. The hump at large /H9252/H11350/H9252cris predominantly the effect of the bandwidth. It appears quite “sharp” in the last panel inFig.12, but this is the consequence of our approximation in which we neglected all regular self-energy terms. Whenthese terms are included, the hump should deﬁnitely getbroader. This behavior of /H9268/H20849/H9275/H20850is quite consistent with the ob- served evolution of /H9268/H20849/H9275/H20850in the SDW phase of electron- doped Nd 2−xCexCuO 4, where the region of SDW-ordered phase extends over a substantial doping range, up to x /H110110.15. Onose et al.45and others46have found that conduc- tivity does have a peak/hump structure at ﬁnite x, the hump shifts toward a higher frequency with increasing x/H20849from a charge-transfer gap of 1.7 eV at x=0 to over 2 eV at x=0.1 /H20850, while the peak shifts downwards as xincreases and wasargued46to scale with the Neel temperature TN. These two features are reproduced in our theory. IV . CONCLUSIONS To summarize, in this paper we obtained fermionic spec- tral function, the density of states, and optical conductivity inthe SDW phase of the cuprates at small but ﬁnite T.W e adopted nonperturbative approach and summed up inﬁniteseries of thermal self-energy terms, keeping at each ordernearly divergent T/J/H20841log /H9280/H20841terms, where /H9280is a deviation from a pure 2D, and neglecting regular T/Jcorrections. We found that, as SDW order decreases, the spectral function inthe antinodal region acquires peak/hump structure: the peakposition scales with the SDW order parameter, while the in-coherent hump remains roughly at the same scale as at T =0, when SDW order is the strongest. We identiﬁed thehump with the pseudogap observed in ARPES experimentsand identiﬁed coherent, Fermi-liquid excitations at low ener-gies as building blocks for magneto-oscillations in an appliedﬁeld. The same peak/hump structure appears in the DOS andin the optical conductivity. The gap in the DOS scales withthe SDW order parameter and disappears when SDW ordervanishes, however, the DOS also develops a hump at anenergy which remains close to U/2, no matter what is value of the SDW order parameter. Optical conductivity at ﬁnite /H9252 has a Drude peak at the lowest frequencies, a peak, whichagain scales with the SDW order parameter and moves tosmaller frequencies as /H9252increases, and a hump which re- mains roughly omegaplus at Uand slightly shifts to higher frequencies as SDW order gets weaker. A more generic result of our study is the phase diagram for the cuprates shown in Fig. 1. A similar phase diagram has been proposed in Ref. 52. At large enough hole doping and small enough temperatures, thermal ﬂuctuations are weakand no SDW precursors appear. In this region, FS is largeand the physics is governed by the interaction between fer-mions and Landau-overdamped spin ﬂuctuations. This inter-action gives rise to a fermionic self-energy which is Fermiliquidlike at the lowest energies, has a non-Fermi-liquidform, /H20849i /H9275/H20850a,a/H110211 at high energies, and displays a marginal Fermi-liquid behavior in the crossover region.7The same interaction with overdamped spin ﬂuctuations gives rise to ad-wave pairing instability. 8,9,48The onset temperature for the d-wave pairing increases as xdecreases and approaches the universal scale of around Tp/H110110.02vF/a, where vF /H110111e V /H11569ais the Fermi energy /H20849Ref. 49/H20850. The actual super- conducting Tcis lower due to ﬂuctuations of the pairing gap. In this regime, the thermal evolution of the FS is entirely dueto thermal effects associated s-wave fermionic damping in- duced by scattering on thermal bosons. 5,17In particular, at a ﬁnite T, the spectral function is peaked at zero frequency in some range of karound the nodal direction, despite that the pairing gap itself has cos 2 /H9278form. At larger Tand smaller x/H110220, thermal ﬂuctuations get stronger and give rise to SDW precursors. These do not im-ply that the FS actually becomes pocketlike, but the spectralfunction in the antinodal region develops a hump at a ﬁnitefrequency, and the low-energy spectral weight progressivelyFIG. 12. Optical conductivity, /H9268/H20849/H9275/H20850, for different /H9252in the SDW phase at x=0.02. For simplicity, we used Z/H9252=0.12 in all plots. At /H9252=0/H9268/H20849/H9275/H20850has a true gap and discontinuity at the onset of scattering between conduction and valence bands. The onset frequency is at /H9275¯ slightly smaller than 2 /H90040=Ubecause xis nonzero and the actual gap/H9004/H11021/H9004 0. At ﬁnite /H9252the discontinuity splits into a hump which slowly moves to a higher frequency and a peak which moves to asmaller energy. We veriﬁed that the peak position scales with /H20855S z/H20856.TIGRAN A. SEDRAKYAN AND ANDREY V. CHUBUKOV PHYSICAL REVIEW B 81, 174536 /H208492010 /H20850 174536-10ﬁlls in the area between the actual FS at kFand the shadow FS at k=/H20849/H9266,/H9266/H20850−kF. This behavior is at least qualitatively consistent with recent ARPES experiments aimed to verifythe existence of the outer side of a pocket. 62We also empha- size that, as long as SDW order is weak, the incoherentpseudogap peak is the dominant feature in the ARPES spec-trum. This last observation is particularly relevant to LBCOnear 1/8 doping, for which there are indications of SDWordering. This SDW order is in any case smaller than theorder at half-ﬁlling, and it is very likely that the dominantARPES intensity is remains the preudogap despite the poten-tial appearance of SDW order. This would be consistent withthe observations in Refs. 20and63. The redistribution of the fermionic spectral weight, in turn, affects the pairing problem: in the presence of SDWprecursors, T cdecreases with decreasing doping51because now the pairing is due to the exchange of propagating mag-nons rather than overdamped spin ﬂuctuations, and theelectron-magnon vertex gets smaller as SDW precursors getstronger. 21,53,54A closely related effect which also reduces T-c is the removal of the low-energy spectral weight due topseudogap opening. 64 The precursors to SDW also give rise to domelike behav- ior of the onset temperature for the pairing in electron-dopedcuprates /H20849left-hand side of Fig. 1/H20850but there the effect is weaker simply because pairing correlations are weaker.47,65 Recently we became aware of a study of thermal SDW ﬂuctuations by Khodas and Tsvelik /H20849Ref. 66/H20850. Their and our results are similar /H20849e.g., both give linear in Twidth of the quasiparticle peaks at low T/H20850, but not identical as we studied isotropic quasi-2D systems with a true long-range SDW or-der below a certain small T/H20849 /H9252/H113491/H20850and exponential behavior of the correlation length at larger T, while Khodas and Tsvelik considered a 2D system with an easy-plane aniso-tropy and put special emphasis to the fact that spin correla-tions decay by a power law at small T. ACKNOWLEDGMENTS We acknowledge helpful discussions with J. C. Campu- zano, L. Glazman, B. Keimer, E. G. Moon, M. Metlitski, M.Norman, M. Rice, M. V. Sadovskii, S. Sachdev, J. Schma-lian, O. Sushkov, A. Tsvelik, Z. Tesanovich, A.-M. S. Trem-blay. We are particularly thankful to M. V. Sadovskii andA.-M. S. Tremblay for careful reading of the manuscript anduseful comments. 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PhysRevB.72.205122.pdf	Gallium stabilization of /H9254-Pu: Density-functional calculations Babak Sadigh and Wilhelm G. Wolfer Lawrence Livermore National Laboratory, University of California, P .O. Box 808, Livermore, California 94550, USA /H20849Received 21 June 2005; published 21 November 2005 /H20850 The/H9251and/H9254phases of plutonium differ in density by 25%. When alloyed with small amounts of gallium, the two phases can be converted directly into each other by a martensitic transformation. It occurs at even lowertemperatures as the Ga content is increased, resulting in what has been called the /H9254stabilization of plutonium. The physical nature of this stabilization has remained unclear, in part because the anomalously low density ofthe /H9254phase has been a mystery. In addition, the atomic size of Ga in these two phases of Pu is perplexing, as it varies by nearly a factor of 3. We show in this paper that the key to the secret behind many of theseanomalies rather lies in the unusual geometry of the /H9251-Pu structure than in the exotic electronic structure of the /H9254-Pu phase. We demonstrate this through extensive density-functional theory calculations that turn out to provide a sufﬁciently accurate model for the structural chemistry of the Pu-Ga alloys. The Ga volumes andtheir heats of solution in the various Pu phases are predicted in good agreement with the experiments. Usingthe results of our supercell calculations in combination with novel geometric arguments, we succeed in ex-plaining the double-C behavior of the martensitic transformation of Pu, as well as the volume collapse due tothe tempering of the /H9251/H11032phase. We propose that these phenomena are manifestations of the temperature- dependent kinetics of ordering in what we call the eighth martensite variant of Ga-containing /H9251-Pu. We also give an account of the microscopic origin of the anomalous variability of the atomic sizes of Ga in Pu. In sodoing, we discover that the measured size of the solute arises from induced volume changes in surrounding Puatoms. We show that this is necessary for a correct interpretation of the EXAFS measurements of Ga-stabilized /H9254-Pu. This new effect may also be signiﬁcant in other alloy systems. DOI: 10.1103/PhysRevB.72.205122 PACS number /H20849s/H20850: 64.10. /H11001h, 71.15.Mb, 75.10.Lp I. INTRODUCTION The light actinide metals /H20849Th to Pu /H20850display an extraordi- nary richness of crystal structures with intricate atomicarrangements. 1Among these metals, plutonium is the most intriguing, as it seems to present all this complexity in oneelement. At ambient pressure, it undergoes a series of sixstructural transformations upon heating from 0 K. 2Many of these transformations induce unexpectedly large changes inproperties, particularly in the density. The least dense Puphase is the /H9254phase, which has a 25% larger volume per atom than the ground-state /H9251-Pu structure. /H9254-Pu has a face- centered cubic /H20849fcc/H20850lattice structure, considered to be the crystal structure that affords the densest atomic packing. Thestructure also confers malleability to Pu, and is therefore ofgreat technological importance. However, since the /H9254-Pu phase is thermodynamically stable only between 592 and724 K, it is alloyed with Al or Ga. The latter has been theelement of choice to extend the stabilty of the /H9254phase to room temperature and below. The release of formerly classi-ﬁed information by Russian scientists has recently led to arevision of the Pu-Ga phase diagram and to our understand-ing of the nature of the /H9254stabilization of Pu.3Furthermore, Hecker et al.4have now summarized and published a great deal of information on the phase transformation in Pu-Gaalloys. A number of anomalies are encountered such as thelarge difference in the partial molar volume of Ga in the /H9251 and/H9254phases, as well as unusual transformation kinetics. In Sec. II, we summarize in greater detail some of the experi-mental observations, many of which are yet to be explained. Our aim in this paper is to ﬁnd out whether electronic structure theory can shed light on the nature of the /H9254stabili-zation and on the anomalies alluded to above. We will focus, in particular, on calculations using the density-functionalTheory /H20849DFT /H20850within the generalized gradient-corrected ap- proximation /H20849GGA /H20850, including spin-polarization when neces- sary. DFT has played an enormous role in our recent under-standing of the structural chemistry of condensed systems.Within this theory the structures, densities, and energies ofalmost all the elements and their compounds have been pre-dicted with stunning accuracy. Its great strength is that it is aparameter-free theory. No ﬁtting to experiments is required.In the past, however, DFT failed spectacularly to even makequalitative predictions of the equilibrium volume or energyof the /H9254-Pu phase.5This failure is attributed to the strong electron correlations in this system. In contrast, the densityand cohesive energy of /H9251-Pu seems to be well reproduced within DFT, suggesting that the electronic structure of thisphase exhibits close to Fermi-liquid behavior. 6 The way to account for the strong electron correlations in /H9254-Pu has generally been done by explicitly including a term representing on-site Coloumb repulsion in the calculations.7,8 The so-called Hubbard Hamiltonian is expected to provide amodel for the strong-coupling regime and thus describe phe- nomena such as high-temperature superconductivity. How-ever, this model is notoriously hard to solve, and variousapproximations /H20849e.g., mean-ﬁeld theories /H20850need to be made. Recently, great excitement was generated by the applicationof the dynamical mean-ﬁeld theory /H20849DMFT /H20850in conjuction with DFT-GGA for the electronic structure of Pu. Impressiveagreement with experiments have been obtained for proper-ties such as the equilibrium volume and the energy as well asthe vibrational properties of /H9254-Pu.8,9In spite of their appeal, these types of calculations at their current formulation sufferPHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 1098-0121/2005/72 /H2084920/H20850/205122 /H2084912/H20850/$23.00 ©2005 The American Physical Society 205122-1from one big drawback, i.e., the strength of the on-site Coloumb repulsion needs to be speciﬁed ad hoc . This is a severe problem since this parameter varies from about 4 eVin the /H9254-Pu phase to only a fraction of that in /H9251-Pu. In light of the fact that the energy difference per atom between /H9254-Pu and /H9251-Pu can be estimated to be less than 0.1 eV, it is hard to see how predictive computational chemistry calcula-tions can be done using this approach. Quite recently, Söderlind showed that including spin/ orbital polarization in the DFT-GGA calculations also canpredict correct energy and volume for /H9254-Pu.10If Södelind’s approach is sound, it offers a key to many of the secrets ofPu chemistry. Södelind’s results were initially received skep-tically by the actinide community, and to date have remainedcontroversial. The main criticism has been that the spin-polarized /H20849SP/H20850calculations predict large stable spin moments on the order of 5 /H9262B, in stark contrast with the magnetic susceptibility measurements, where no trace of any Curie- Weiss behavior is found.11,12It has thus been maintained by the adversaries of the SP-DFT treatment of Pu that it is amere coincidence that the introduction of large magnetic mo-ments happen to expand the fcc lattice to the correct equilib-rium volume. However, in several recent publications, weand others have shown that the SP approximation also repro-duces the energies and volumes of almost all the phases ofPu in a satisfactory way. 13–15 The prediction of large spin magnetic moments is at ﬁrst sight quite troublesome, but upon further reﬂection it is pos-sible to ﬁnd legitimate arguments for their existence. First,due to the localized 5 fcharacter of these moments, Hund’s rule will cause orbital polarization in such a way as to reducethe magnitude of the total measurable magnetic moment peratom from 5 /H9262Bto less than 1 /H9262B. Second, DFT cannot accu- rately predict electronic excitations, especially in exotic sys-tems such as Pu. Hence, effects such as spin ﬂuctuations,Kondo screening, or other novel hybridization effects, arenot properly accounted for, and their contribution to the totalmagnetic moment is missed. Nevertheless, the appearance oflarge spin moments can be viewed as an indication of strongelectron correlations. In previous work, we showed that wecan correlate the magnitude of the spin moments in any ofthe Pu phases with a very simple model of the local atomicconﬁguration around each Pu atom, thus further supportingthe interpretation that the spin moments from the SP-DFTcalculations have a localized character. 16Hence within a more accurate theory such as DMFT, these spin momentsmight appear as ﬂuctuating local moments. In view of thegathering evidence that the SP-DFT approximation can re-produce /H20849at least qualitatively /H20850most of the features of the phase diagram of pure Pu, it is now worthwhile to also applyit to Pu-Ga alloys that offer many unusual phenomena tofurther test this approximation. For this purpose, we ﬁrst review in Sec. II some of the features of the Pu-Ga phase diagram in the region of low Gaconcentration and of the /H9254to/H9251phase transformation. In Sec. III, we discuss the methodology used in this work for thetotal energy calculations of the Pu-Ga system, following withan account of the novel features of the geometry and elec-tronic structure of the /H9251-Pu phase. In Secs. V and VI present calculations for the energies and volumes of substitution ofGa in /H9251-Pu and /H9254-Pu, respectively. We discuss the structure and thermodynamics of dissolution of Ga in the two Puphases and show that the calculations and the experimentsagree quite well. In Sec. VII we describe a unifying modelfor the relaxation of Pu atoms around a Ga impurity in bothPu phases, and explain the breakdown of the Eshelby model.Finally, we conclude with a discussion of the signiﬁcance ofthe results presented in this paper for the /H9254stabilization of Pu. II. REVIEW OF THE EXPERIMENTAL LITERATURE ON Pu-Ga Figure 1 shows the part of the equilibrium phase diagram of the Pu-Ga system that covers the Ga concentrations up to10 at.%, the range where the /H9254stabilization occurs. Up until 1998, a tentative phase diagram as originally published byEllinger was accepted in the West, and the /H9254phase was be- lieved to be stable down to low temperatures within the re-gion bordered by the dotted green lines in Fig. 1. Further-more, the /H9254phase was known to undergo a martensitic transformation, near the lower red line in Fig. 1, that con-verted part but not all of the /H9254phase to a variant of the /H9251 phase. This variant is referred to as the /H9251/H11032phase. It was believed that at Ga concentrations above 3 at.%, the /H9254-phase becomes stable down to 0 K.17It was shown by Timofeeva et al.3and ﬁnally disclosed to the western world, that the /H9254 phase is in fact metastable below a eutectoid temperature ofabout 100 °C. Long-term experiments of phase decomposi-tion /H20849indicated by the circular symbols in Fig. 1 /H20850were re- quired to obtain the actual phase boundary for the /H9254phase, the upper green line. The slow diffusion of Ga preserves the /H9254phase below this line. The equilibrium phases below the eutectoid temperature are the /H9251phase of pure Pu /H20849this phase is known to have almost no solubility for Ga /H20850and the inter- metallic compound Pu 3Ga or the /H9256/H11032phase. To form this two- phase mixture from the metastable /H9254-Pu phase requires ag- gregation of Ga by long-range diffusion, a process that cantake over thousands of years at low temperatures. The Rus-sian scientists managed to speed up the low-temperature ki-netics by ﬁrst transforming some of the /H9254- P ut ot h e /H9251/H11032phase. Annealing for no less than 10 000 hours was then needed todemonstrate phase decomposition into /H9251+Pu 3Ga. A quantitative measure of the degree of stability /H20849or meta- stability /H20850of the /H9254phase is the difference in the Gibbs free energies of the /H9254and/H9251phases. Adler has estimated this quantity18and shown that it increases upon cooling, and eventually becomes positive and so large that a martensitictransformation takes place at a critical temperature, the so-called martensitic start temperature indicated by the lowerred line in Fig. 1. In the case of Ga-stabilized Pu, this trans-formation is of an isothermal kind, meaning that it is timedependent when proceeding at a ﬁxed temperature, and thatthe transformation speed varies over a ﬁnite temperaturerange. 19,20The martensitic start temperature line in Fig. 1 gives the temperature for the shortest transformation time.For temperatures both below and above this line, the trans-formation times are longer. When these temperatures areplotted as a function of their respective transformation timesB. SADIGH AND W. G. WOLFER PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-2to convert a certain fraction of the parent phase /H20849here the /H9254-phase /H20850, a so-called time-temperature-transformation /H20849TTT /H20850 diagram in the shape of a “C” is obtained. For the martensitictransformation of the Pu-Ga system, however, the TTTcurves display a double-C character /H20849see Fig. 2 /H20850, implying either two transformation modes and/or two subvariants ofthe /H9251/H11032phase. This peculiar feature has so far found no con- vincing explanation, but will be addressed below. Isothermal martensitic transformations are usually con- trolled by diffusional processes. What these processes are isnot known in the case of the Pu-Ga system. A further com-plication is that plastic deformation must accompany themartensitic transformation to accommodate the large volumecollapse of about 20% that occurs during the /H9254→/H9251transfor-mation. Since the /H9251phase is hard and brittle, plasticity in the emerging dual phase material becomes increasingly difﬁcultas the martensitic transformation proceeds, and it limits theamount transformed to a volume fraction of less than 30%.The martensitic transformation in the dilute Pu-Ga systemhas been studied quite extensively over the past 30 years.Many unique and puzzling phenomena have been observed.An extensive review of them has ﬁnally been published byHecker et al. 4Below, we mention a few of these anomalies, which will touch upon our results in the later sections. Gallium dissolved in the /H9254-o rt h e /H9251-phase changes the lattice parameters of these structures in proportion to its con-centration. The change in the dilute concentration limit en-ables the determination of the partial molar volume of Ga inPu. The partial molar volumes per Ga atom in the two Puphases are listed in Table I. Curiously they are found to varyby almost a factor of 3. Ga in /H9254-Pu assumes a very small size, while in the /H9251- phase it can either be of the same size as a Pu atom or be almost twice as large. In fact, intermediatevalues are possible, depending on how the /H9251phase is pre- pared from the Ga-containing /H9254-Pu phase. For example, it has been found that the density of the /H9251-phase obtained from the high-temperature part of the double-C curve is largerthan the phase obtained from its low-temperature part. When the /H9251-phase is obtained with the larger partial mo- lar volume for Ga, the unit cell of the /H9251phase is somewhat larger than in the pure /H9251-Pu phase, and it is for this reason that it has sometimes been considered as a separate /H9251/H11032 phase.29The/H9251/H11032phase has the same monoclinic structure as the/H9251phase of pure Pu. However, when tempered /H20849meaning annealed for some time at a temperature close to but belowTABLE I. Partial molar volumes in Ångstrom3. /H9251/H9251 /H11032 /H9254 Pu 20.4 20.4 25.0 Ga 20.4 37.8 10.8 Ga size factor 0% +22.8% −24.4% FIG. 1. /H20849Color /H20850Phase diagram of the Pu-Ga system at ambient pressure. Only Ga concentrations up to 10 at.% is shown. FIG. 2. /H20849Color online /H20850Temperature-time-transformation /H20849TTT /H20850 curves for high-purity Pu-Ga alloys. The transformation follows adouble-C behavior /H20849redrawn from Ref. 20 /H20850. FIG. 3. /H20849Color /H20850The/H9251-Pu structure. The blue and the green planes are related by an inversion symmetry operation. The atomsin each plane are labeled by numbers 1 to 8 according to the stan-dard enumeration /H20849Ref. 28 /H20850in the /H9251structure.GALLIUM STABILIZATION OF /H9254-Pu: DENSITY- … PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-3the reversal temperature /H20850, it is reported to attain a unit cell close to that of the pure /H9251phase. The /H9251/H11032-Pu phase can be converted back to the /H9254phase at and above the reversal tem- perature /H20849the upper red line in Fig. 1 /H20850, which is signiﬁcantly higher than the martensitic start temperature. However, whentempered before, subsequent conversion to the /H9254phase re- quires heating to temperatures higher than if tempering hadnot occurred. The small size of Ga in the /H9254-Pu phase has recently in- spired several experimental studies with EXAFS.22–24From these studies it was found that the interatomic distances be-tween a Ga atom and its nearest-neighbor Pu atoms is short-ened by only 3% to 4% compared to the same distance be-tween Pu atoms, while lattice parameter changes imply ashortening of 10.6% if one were to attribute them only to amisﬁt of the Ga atom in /H9254-Pu. A satisfactory explanation for these observations does not yet exist. In fact, even the very existence of the /H9251/H11032phase as distinct from the /H9251phase is still debated. Based on ﬁrst- principles calculations of Ga in the /H9251- and /H9254-Pu structures, we propose in this paper for the ﬁrst time a rationale for theexistence of the /H9251/H11032phase and for many of the above obser- vations. III. METHODOLOGY As a result of the recent success of spin-polarized DFT, we have carried out a series of ﬁrst-principles electronicstructure calculations of plutonium in the /H9251and the /H9254phases, with Ga in substitutional positions. These DFT calculationswere performed within the generalized gradient approxima-tion /H20849GGA /H20850in the PW91 parametrization, 25using the plane- wave pseudopotential code VASP.26The projector augmented wave method was used to represent the Pu atoms.27No spin- orbit coupling was included, and spin polarization was onlyincorporated for the /H9254-Pu calculations. The calculations included 16 valence electrons for each Pu atom, including the 5 sand 5 psemicore states, and a plane-wave cutoff energy of 23.4 Ry was used. We per-formed calculations in periodic 16- and 32-atom supercellsof /H9251-Pu as well as /H9254-Pu lattices, containing one Ga atom. We fully relaxed the positions of the atoms as well as the latticeparameters of the unit cells. Our calculations sampled theBrillouin zone of the 16-atom /H9251-Pu supercell with 27 irre- ducible k points, and an equivalent k-point mesh for the 32-atom supercell was used. The reciprocal space of the /H9254-Pu lattice was sampled with a k-point mesh equivalent to an 8/H110038/H110038 fcc Monkhorst and Pack mesh. Almost all the calculations in this work are mainly used to give qualitative support for theories that can explain severalpuzzling features of experiments concerning the nature of the /H9254-to-/H9251martensitic transformation of Pu. For this purpose we needed to perform a large number of supercell calculationsof various Pu-Ga conﬁgurations. Hence, including the ﬁnercorrections, such as spin-orbit coupling, would not be fea-sible with current computational resources. However, we be-lieve that the approximations used here are quite adequatefor our purposes. Recently, we demonstrated based on com-prehensive calculations for the /H9251-/H20849Ref. 14 /H20850and/H9254-Pu,13thatresults obtained using the spin-polarized PAW method are in quite good agreement with full-potential all-electron tech-niques including spin-orbit coupling as well as orbital polar-ization. Furthermore, we also showed 14that most of the structural details of /H9251-Pu are quite well reproduced within nonmagnetic GGA. IV. THE /H9251-Pu STRUCTURE The 16-atom unit cell of the /H9251-Pu structure is shown in Fig. 3. This structure with its monoclinic symmetry may beviewed as the stacking of two distorted close-packed planes.The distortions are within, but not perpendicular to, theplanes, resulting in different bond distances between nearestneighbor atoms. The two atomic planes can be translated intoeach other by a shift along the aaxis and by an inversion symmetry operation. As a result, there are only eight non-equivalent lattice sites in this structure. In Fig. 3, we labeleach site with one of the numerals 1–8 according to thestandard enumeration in the /H9251structure.28 Our recent calculations14of pure /H9251-Pu have conﬁrmed that this structure is reasonably well reproduced within thenonmagnetic GGA, when complete relaxation of all the in-teratomic as well as lattice coordinates are carried out. Weshow in Fig . 4 a comparison between the calculated and experimental atomic volumes of Pu in each of the eight dif-ferent sites of the /H9251-crystal structure. These volumes are de- ﬁned by Voronoi cells constructed around each atom. Theexperimental geometry is obtained from the x-ray diffractionresults, 29at room temperature, and thus the extracted Voronoi volumes need to be reduced by 5% to subtract the contribu-tion from thermal expansion. Taking the average of theatomic volumes gives the partial molar volume for Pu in the FIG. 4. /H20849Color online /H20850The Voronoi volumes extracted from the experimental geometry, as well as the DFT-GGA relaxed structureare, shown. The theoretical numbers are, on average, 7% smallerthan the experimental ones. In both structures, Atom No. 8 is about20% large than Atom No. 1.B. SADIGH AND W. G. WOLFER PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-4pure/H9251phase. The calculated volume is 7% smaller than the measured one extrapolated to absolute zero, a discrepancyquite typical of DFT. The /H9251-Pu structure is of course different from fcc due to its low symmetry. However, a further inspection of this struc-ture reveals certain geometric features that are quite unusual.For example, Fig. 4 shows that the largest atom, No. 8, has avolume that is almost 20% larger than the smallest atom, i.e.,No. 1. Note again that the difference in the volume per atombetween the /H9254-Pu and the /H9251-Pu phases does not amount to more than 25%. Hence, one should expect that the propertiesof the felectrons vary quite substantially from site to site in the /H9251-Pu structure. Accordingly, our principal thesis in this paper revolves around the theme that there exists an interplaybetween the f-electron chemistry and the local geometry around each Pu atom, which holds the key to understandingmany of the experimental anomalies reported earlier. Figure5 illustrates one typical manifestation of this relationship.Here we show the density-of-states /H20849DOS /H20850of the felectrons projected on a few of the inequivalent sites of the /H9251-Pu struc- ture. Speciﬁcally, note that the f-DOS at the Fermi level is at a maximum for site 8 with the largest available volume andat a minimum for site 1 with the smallest available volume.This implies that /H20849i/H20850thefelectrons belonging to site 1 are more weakly correlated than the ones belonging to site 8;thus, e.g., for spin-polarized calculations, quite a sizable spinmoment is induced on site 8, while site 1 is practicallynonmagnetic, 14,16and /H20849ii/H20850thefelectrons at site 1 have lower energy than those at site 8. Hence, it costs less energy toremove the felectrons at site 8 than those at site 1. We will in the following sections discuss at length the ramiﬁcationsof the heterogeneity of the electronic structure of /H9251-Pu /H20849Ref. 30/H20850for alloying with Ga. V. GALLIUM IMPURITIES IN THE /H9251-Pu PHASE Using the pure /H9251structure as our reference, alloys were simulated by substituting one Pu atom in the /H9251-unit cell withone Ga atom. Due to the eight nonequivalent sites in the /H9251 lattice, there are then eight distinct substitutional variants of periodic 16-atom unit cells containing 15 Pu atoms and oneGa. 28We denote in the following a dilute Pu 1−xGaxalloy with Ga atoms only appearing in unit cells of the ith substitutional variant /H9251i-/H20849Pu1−xGax/H20850. The substitutional energy of Ga /H20849Ei/H20850at site iis determined after a fully relaxed /H9251structure is achieved. We ﬁnd that each site has a different energy of substitution. Several ofthese energies have also been evaluated for 32 atom cells/H20849one Ga and 31 Pu atoms /H20850, and we have converted all these energies into enthalpies of mixing at absolute zero for thetwo alloy concentrations of 3.125 and 6.25 at.% Ga andpresent them in Table II. All mixing enthalpies turn out to bepositive and increase with the Ga concentration, indicatinglimited or no solubility in the /H9251phase. However, site 8 is by far the most favorable substitutional position for Ga in the /H9251-unit cell. The energy cost to move a Ga atom from this site TABLE II. The mixing enthalpies /H20849in units of kJ/mol /H20850for Ga in /H9251-Pu when placed at one of the eight different lattice sites. For a Ga concentration of 6.67 at.%, each unit cell contains one Ga atom,while for a concentration of 3.225 at.%, every other unit cell hasone Ga atom. 3.225 at.% Ga /H20849fully relaxed /H208506.67 at.% Ga /H20849fully relaxed /H208506.67 at.% Ga /H20849unrelaxed /H20850 Site 1 190.1 207.3 440.0 /H20849kJ/mol /H20850 Site 2 187.4 138.8 379.3Site 3 114.7 332.3Site 4 178.2 220.6 309.9Site 5 112.1 338.0Site 6 56.1 64.3 273.0Site 7 108.9 139.6 305.9Site 8 45.7 74.4 143.5 FIG. 5. /H20849Color online /H20850The densities of states /H20849DOS /H20850of the felectrons projected onto sites 1, 4, and 8. The Fermi level is de-picted with a vertical line. The f-DOS at Fermi level is at a mini- mum for site 1, and at a maximumfor site 8.GALLIUM STABILIZATION OF /H9254-Pu: DENSITY- … PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-5to the least favorable position /H20849site 1 /H20850is as large as 1.5 eV. This is a remarkable difference, indicating that Ga atomshave a strong tendency to order in the /H92518-/H20849Pu1−xGax/H20850con- ﬁguration even if they initially were randomly distributed in the parent /H9254phase. In Table II we have also included the Ga enthalpies of mixing when no internal atomic relaxations areincluded. We see that without relaxations the energy cost ofmoving a Ga atom from site 8 to site 1 becomes as large as3.1 eV. Hence the atomic relaxations do reduce the energy ofsubstitution of Ga in site 1 much more than in site 8. Incontrast to the positive mixing enthalpy for Ga in /H9251-Pu, the heat of formation for the compound Pu 3Ga is calculated to be 14.6 kJ/mol. The equilibrium state is therefore a mixture ofthe pure /H9251-Pu phase and the Pu 3Ga phase in agreement with the phase diagram. Figure 6 shows the partial molar volumes for Ga when placed on each of the 8 different sites in the /H9251structure. It is seen that all the variants have similar partial molar volumesfor Ga, close to 30 Å 3, except for the eighth, which seems to be considerably smaller, i.e., a Ga size of about 20 Å3. This should be compared with the two extreme experimental val-ues given in Table I: The phase obtained immediately afterthe martensitic transformation, the so-called /H9251/H11032-Pu, contains Ga atoms with partial molar volumes of about 35 Å3, while when tempered, the size of the Ga atoms shrink drasticallydown to 20 Å 3. These two experimental values are indicated by two horizontal lines in Fig. 6. Thus, we can for the ﬁrsttime give an explanation for the variability of the Ga sizesin /H9251-Pu, and in so doing we can clarify the nature of the /H9251/H11032phase. This phase is a random mixture of all the /H9251i-/H20849Pu1−xGax/H20850variants, obtained via martensitic transforma- tion from the Ga-containing /H9254-Pu structure. Upon tempering, i.e., heating up to moderate temperatures, modest Ga diffu-sion is facilitated, and thus ordering into the eighth varianttakes place with a subsequent volume collapse of the Gaatoms. In Fig. 6, we see that the experimental results /H20849the two horizontal lines /H20850display a somewhat larger difference thanthe theoretical values. One possible explanation for this could be that the martensitic transformation leads to excessinterstitials in the product phase. We believe, however, thatthe discrepancy in the theoretical and the experimental vol-umes are due to the approximations in our calculations. Ifcorrelation effects, such as spin/orbital polarization and spin-orbit coupling were included, we believe larger volume ex-pansions would take place and better agreement with experi-ments could be obtained. However, the convergence of suchcalculations including atomic relaxations at this point in timeare computationally too expensive. We consider this work bea stepping stone toward such quantitative calculations in thenear future. We cannot rationalize the variability of the molar volume of Ga in terms of the geometry of the /H9251-Pu structure alone. Site 8 has the largest atomic volume and site 1 the smallest inthe pure /H9251phase. But while these sites differ in volume by 2.7 Å3when occupied by Pu, this difference becomes ampli- ﬁed to 12.4 Å3when occupied by Ga. It is also interesting to note that when internal atomic relaxations are suppressed,the volume of Ga varies from 19.5 Å 3in site 8, to 23 Å3in site 1, which is very close to the difference in the size of Puatoms in these sites in the pure /H9251structure. Hence, it be- comes clear that the internal atomic relaxations are respon-sible for the anomalous expansion of Ga atoms in, e.g., site1, while they do not affect Ga in site 8 signiﬁcantly. This isin agreement with the observation above that the internalatomic relaxations reduce the energy for Ga in site 1 muchmore drastically than for Ga in site 8. The random mixture of the variants is only realized when no Ga diffusion is allowed. However, any amount of Gadiffusion will alter the concentrations of the different vari-ants favoring the lower-energy ones. Bearing this in mind,one can explain the curious double-C curve observed in theTTT diagrams of the martensitic /H9254→/H9251transformations. Fig- ure 2 shows the results obtained by Orme et al.20These curves give the fraction /H20849in % /H20850of the metastable /H9254phase that has transformed at a ﬁxed temperature into the /H9251/H11032martensitic phase. The double-C curves indicate that two variants ofthe /H9251/H11032phase form. Our results suggest that the low- temperature part of the double-C curve is simply the /H9251/H11032 phase, i.e., a random mixture of all the eight variants of the /H9251i-/H20849Pu1−xGax/H20850structures. For the high-temperature part, we propose that limited diffusion occurs such that conﬁgurations with higher concentrations of the lower-energy variants, inparticular the eighth, are formed. This is consistent with theexperiments, 21where it is found that the density of the Ga- containing /H9251phase obtained at the upper-C temperatures is higher than the phase that forms at the lower-C temperatures.Figure 7 shows the experimental measurements by Deloffre 21 and our theoretical predictions for the ratio of the volumes ofGa impurities to Pu atoms in tempered /H9251-Pu as well as the high- and low-temperature variants of /H9251/H11032-Pu. It can be seen that Ga impurities are almost twice as large as an average Puatom in /H9251/H11032-Pu obtained at the lower-C temperatures, while they are only 1.5 times larger for the phase obtained at theupper-C temperatures, and when tempered, they furthershrink to the size of an average Pu atom. It is interesting tonote that Deloffre et al. 19also found that in a Pu-1.2 at.% Ga alloy, the low-temperature variant of /H9251/H11032reverts upon heating FIG. 6. /H20849Color online /H20850The partial molar volume of Ga in the different lattice sites of the /H9251structure.B. SADIGH AND W. G. WOLFER PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-6back to the /H9254phase, while the high-temperature variant trans- forms in part ﬁrst to the /H9252, and then to the /H9253phase before it fully converts to /H9254-Pu. This further supports our theory that the high-temperature variant of /H9251/H11032-Pu is energetically more stable than the low-temperature variant. VI. GALLIUM IMPURITIES IN THE /H9254-Pu PHASE The calculations for the /H9254phase require the use of the spin-polarized /H20849SP/H20850DFT. As shown in Ref. 13, the antiferro- magnetic spin conﬁguration results in the lowest energy for /H9254-Pu, and the correct density and bulk modulus. Again, we employ a 32-atom supercell with a face-centered cubic ar-rangement. Calculations are carried out for the pure /H9254phase and then with one Ga atom replacing a Pu atom. Both com-putations are performed until the structure reaches completerelaxation. We ﬁnd that the initial face-centered cubic struc-tures undergo a slight tetragonal distortion upon relaxation toﬁnd a metastable state. It is termed metastable because theminimum energy in both cases is higher than the energies ofthe /H9251or/H9251/H11032structures. However, we ﬁnd that the heat of mixing of Ga in /H9254-Pu is negative. The computed values are −3.02 kJ/mol and −6.39 kJ/mol for alloys with 3.125 at.%and 6.25 at.% of Ga, respectively. In contrast, if one assumesa random distribution of Ga over the possible sites in /H9251-Pu, the/H9251/H11032alloys with 3.125 and 6.25 at.% of Ga have enthalpies higher by 4.1 and 8.9 kJ/mol, respectively. This means thatthe enthalpy of transformation from /H9254to/H9251at absolute zero changes from −22.4 kJ/mol, to −14.6 kJ/mol, and to−5.7 kJ/mol as the Ga content is increased from 0, to 3.125,and ﬁnally to 6.25 at.%, respectively. It is this reduction that leads to the so-called stabilization of the /H9254phase. This is shown in Fig. 8, where we also extrapolate the trend of sta-bilization to higher Ga contents and ﬁnd that for Ga concen-trations equal to or higher than 8 at.%, the enthalpy of trans-formation becomes positive, and hence the /H9254phase becomes thermodynamically stable relative to the /H9251/H11032phase. Of course, the/H9251/H11032phase itself is a metastable phase, and it can decom- pose with time into pure /H9251-Pu and the compound Pu 3Ga/H20849/H9264/H11032/H20850. As already mentioned in the previous section, our theoretical calculations within SP-DFT for Pu 3Ga once again conﬁrm this last point. It is important, however, to point out that /H9254-Pu stabilization as predicted by SP-DFT is only qualita- tively correct. Figure 8 clearly illustrates that our calcula-tions greatly overestimate the enthalpy of transformation forpure Pu. Hence, to obtain quantitative results, more accurateelectronic structure calculations are required. How does the /H9254-Pu structure relax around the Ga impu- rity? Our supercell calculations indicate that the /H9254-Pu lattice contracts around each Ga solute atom, rendering a partialmolar volume of 11 Å 3for Ga, in remarkable agreement with the experimental value listed in Table I. The atomisticdetails of this contraction have recently been measured byEXAFS. 22–24These measurements, after a considerable analysis, provide values of the radial strain or the relativechanges, /H9004R j/Rj, in near-neighbor distances Rj. We show in Fig. 9 both our theoretically predicted and measured radialstrains of the Pu atoms surrounding a Ga atom. The ﬁrstnearest neighbor /H20849j=1/H20850Pu atoms are displaced the most to- ward the Ga atom. Due to the slight tetragonally distorted /H9254 structure, however, the Pu atoms in the ﬁrst shell are dis- placed by different amounts, ranging from 1.5% to 4.2%.These values bracket the experimental results, which, by ne-cessity, are average values. The radial strains for the secondand third nearest neighbors are 1% or less. Again, the theo-retical predictions agree very well with the experimentaldata. FIG. 7. /H20849Color online /H20850A comparison of the ratio of the partial molar volumes of Ga impurities to Pu atoms in /H9251/H11032-Pu obtained by martensitic transformation at lower-C and higher-C temperatures, aswell as tempered. Experimental results are from Ref. 21, and thetheoretical predictions are /H20849i/H20850lower C: equal probability of the Ga impurities to occupy any of the eight inequivalent sites in /H9251-Pu, /H20849ii/H20850 upper C: zero probablity of occupying the highest energy site 1 andtwice higher propbability of occupying site 8, and /H20849iii/H20850tempered: 100% occupation of site 8. FIG. 8. /H20849Color online /H20850Enthalpy of the /H9254-to-/H9251transformation from experiment /H20849open black triangles /H20850, compared with the theoret- ical predictions for the two cases when Ga occupies all the /H9251sites at random /H20849red squares /H20850, and when Ga is only allowed to occupy the lowest energy No. 8 site /H20849blue circles /H20850. The /H9254-Pu stabilization oc- curs at the concentration where each intersects the xaxis.GALLIUM STABILIZATION OF /H9254-Pu: DENSITY- … PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-7The EXAFS measurements provide a direct method for determining the local volume of Ga solute in /H9254-Pu. The mis- ﬁt of the Ga atom is deﬁned in the EXAFS analysis as theaverage radial strain of the ﬁrst nearest-neighbor Pu atoms,which from Fig. 9 can be deduced as −3.5%. If we now viewGa simply as a misﬁtting inclusion, we can perform the elas-tic analysis outlined in the Appendix. From Eq. /H20849A4/H20850we can estimate the local atomic volume of Ga in /H9254-Pu: it is only 11% smaller than the host Pu atoms. This number should becompared with the partial molar volume of the Ga solute,given in Table I, being almost 57% smaller than the Pu host.Hence the misﬁt volume of Ga is only responsible for 19%of the global volume collapse. As reviewed in the Appendix, the conﬁning strain ﬁeld of the host can lead to a local volume for the Ga impurity thatis smaller than its partial molar volume. Eshelby has shownthat for an elastic medium the ratio of the global to the localmisﬁt volumes is a characteristic constant, the so-called Es-helby factor, that is entirely determined by the elastic con-stants of the host material. Hence we need to calculate theEshelby factor for the elastically highly anisotropic /H9254-Pu lat- tice. Using the elastic constants of Ledbetter and Moment31 and the numerical method of Leibfried and Breuer,32we ﬁnd that/H9253E=1.76 /H20849see the Appendix for details /H20850. No such calcu- lation has previously been reported. Using the value for the local misﬁt volume of Ga in /H9254-Pu and the Eshelby factor described above, Eq. /H20849A5/H20850pre- dicts a global volume change due to the Ga inclusion of only−4.62 Å 3, to be compared with the real volume change of −14 Å3. This means that as much as 67% of the partial molar volume of Ga in /H9254-Pu is unaccounted for by Eshelby’s model. Such a dramatic failure of the latter implies thatviewing the Ga impurity in Pu as a misﬁt inclusion in anelastic medium is too simplistic. Apparently other contribu- tions from the electrons are too signiﬁcant to be ignored. How do the electrons reduce the partial molar volume of Ga in /H9254-Pu? The Pu atoms around the Ga impurity are, to a lesser degree, conﬁned by the symmetry of the fcc lattice,and thus they relax around the solute in such a way as toobtain lower energy, in other words become more /H9251-like. They will therefore reduce their sizes. As a result, we pro-pose that 67% of the partial molar volume of Ga in /H9254-Pu arises from the reductions in the size of the host atoms, while19% is due to the misﬁt of the solute, and the remainder of14% arises from elastic strains. Hence the EXAFS data leadus to a very important conclusion, i.e., much of the volumechanges arise from the Pu atoms rather than from elasticdistortions and a misﬁtting Ga atom. We can further quantify our ﬁnding by calculating the Pu-Pu distances of the ﬁrst-shell Pu atoms around the Gaimpurity from our supercell calculations. We ﬁnd that theyreduce by 0.7%. This reduction implies that the averagenearest-neighbor Pu-Pu distances in the Pu-Ga system shoulddecrease with increasing Ga concentration. Recent EXAFSmeasurements 33of the Pu-Pu distances in the dilute ternary Pu-Ce-Ga alloys indicate that the Pu-Pu bond lengths shrinkwith an increasing solute concentration. The small amount ofpublished data, however, does not justify a quantitative valuefor this reduction. Nevertheless, this is a very signiﬁcant re-sult, and further supports our picture of Pu relaxationsaround solutes in /H9254-Pu. Hence, future experimental measure- ments of Pu-Pu bond lengths in dilute binary Pu-Ga alloyswill be valuable in validating our theoretical predictions. VII. THE LOCAL AND GLOBAL VOLUMES OF GALLIUM IMPURITIES IN /H9251- AND /H9254-Pu In the previous section, we explained the large discrep- ancy between the local and global volumes of Ga impuritiesin /H9254-Pu in terms of solute-induced volume changes of the surrounding host atoms. The question is whether this phe-nomenon is in fact even more far-reaching and could explainthe 300% variability of the partial molar volumes of Ga inthe different phases of Pu. To this end, we need to decom-pose the global volume change induced by the solute into itslocal volume plus the Ga-induced volume changes of thesurrounding Pu atoms. We accomplish this by constructing aVoronoi cell around every atom in each of the relaxed /H9251i-/H20849Pu31/32Ga1/32/H20850structures, as well as the Ga-containing /H9254 lattice. For a single component system, the Voronoi cell be- longing to a particular atom is deﬁned to be the set of pointsin space to which no other atom is closer. However, for mul-ticomponent systems a complication arises, i.e., we need toknow the relative sizes of the different species. This intro-duces a certain arbitrariness in the process, but fairly reason-able schemes can be devised. For example, the relative sizescan be determined from the ratio of the atomic volumes ofeach element in its pure ground-state structure. Followingthis scheme for the particular case of Ga and Pu, we ﬁnd thatthe two elements are almost size matched in their groundstates. Hence we apply the usual Voronoi construction forsingle-component systems, i.e., we use bisecting planes be- FIG. 9. /H20849Color online /H20850The radial strain or shortening in nearest- neighbor /H20849NN /H20850distances between Pu atoms around a Ga atom in the /H9254structure. Open symbols are theoretical results; solid symbols are experimental values from EXAFS measurements.B. SADIGH AND W. G. WOLFER PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-8tween neighboring atoms. In this way, we formulate an alter- native /H20849but equivalent /H20850expression for the partial molar vol- ume of Ga when embedded in any of the Pu phases: /H9024¯ Ga/H9266=/H9024Ga/H9266+/H20858 j/HS11005i/H9004/H9024Pu/H9266/H20849Rij/H20850. /H208491/H20850 Above, the superscript denotes the particular crystallo- graphic phase of the Pu-Ga alloy. The ﬁrst term in this equa-tion is the Voronoi cell volume 34around the substitutional Ga atom, and the sum contains all the volume changes of the Puatoms relative to their corresponding Voronoi volumes in thepure phase, be it /H9251or/H9254. The sum can be carried out in any order, but we present our results by adding the contributionsfrom Pu atoms with increasing, but the shortest possible,radial distance from the Ga atom. This tally is plotted in Fig.10 as a function of the radial distance, measured in the ref-erence conﬁguration, namely that of the pure phase. /H20849Almost identical plots are obtained when the radial distances in therelaxed structures are used. /H20850We call this tally the cumulative volume change. Several surprising aspects emerge from thiscumulative representation of the partial molar volume of Ga.First, the Voronoi cell volume of the Ga atom is nearly in-variable with the Pu phase in which it is embedded. Thisbehavior is quite close to the unrelaxed situation. It is alsointeresting to note that the local volume of Ga in Pu is notvery different from its equilibrium volume in its pure groundstate. Second, while the nearest neighbor Pu atoms contrib-ute most to the total volume change, more distant host atomsalso add to or subtract from it. The partial molar volume ofthe Ga solute atom is almost entirely composed of volumechanges induced by it on the surrounding Pu host atoms.Although these volume changes contain the elastic contribu-tions, they are in fact small, as we already demonstrated inthe previous section for the case of Ga in the /H9254phase.As was mentioned above, Voronoi cell construction for a multicomponent system depends on the choice of the relativesizes of the different species. In the case of Ga in Pu, wehave assumed that the two elements are of equal sizes. Thisis a reasonable assumption since they have almost the sameground-state equilibrium volumes. Still our main result inthis section, namely the relatively unchangeable local vol-umes of Ga presented in Fig. 10, can be criticized for beingan artefact of our Voronoi cell construction. In order to further establish the validity of our claims, we will in the following present the change in the average Pu-Pubond lengths in all the different Pu-Ga conﬁgurations studiedin this paper. For an arbitrary lattice, we will deﬁne thisquantity also in terms of Voronoi cells, but it is much lesssensitive to the the Voronoi cell construction. We follow herethe notation in Ref. 16 and deﬁne the average Pu-Pu bondlength for an atom ilocated at R ias b/H20849Ri/H20850=1 2/H20858 jw/H20849Ri,Rj/H20850/H20841Ri−Rj/H20841, /H208492/H20850 where the sum runs over all the neighboring Pu atoms j, that contribute to the Voronoi cell of atom i. The weight factors w/H20849Ri,Rj/H20850, are determined by the relative contribution of the atom jto atom i’s total Voronoi surface area shared with Pu atoms: w/H20849Ri,Rj/H20850=/H9011/H20849Ri,Rj/H20850 /H20858k/H9011/H20849Ri,Rk/H20850. /H208493/H20850 Above /H9011/H20849Ri,Rj/H20850is the area of the Voronoi cell face be- tween atoms iand j, and the sum runs again over all the neighboring Pu atoms k. We now plot the cumulative change in the average Pu-Pu bond length as a function of the dis-tance to the Ga solute, in all the Pu-Ga conﬁgurations in thisstudy; see Fig. 11. We can clearly see that the Ga atom in-duces a reduction of the volume of the surrounding Pu atomsin /H9254-Pu /H20849therefore making them more /H9251-like /H20850, and an in- crease in the size of the surrounding Pu atoms in /H9251-Pu /H20849and therefore making them more /H9254-like /H20850. The exception is again the/H92518-Pu 1−xGax, where Ga can be viewed simply as an in- clusion in an elastic medium. VIII. DISCUSSION In this paper, we have described a comprehensive study where we applied the DFT to explore the structural chemis-try of dilute Pu-Ga alloys in the /H9251-Pu and the /H9254-Pu phases. Based on an unconventional analysis of the experimentaldata, we have recognized that the partial molar volumes ofGa impurities in Pu range from 11 Å 3in/H9254- P ut o3 5Å3in /H9251/H11032-Pu. Our focus in this paper has been to unravel the mi- croscopic origin for this large size variation. Starting with pure Pu, it has been well known for a long time that the average atomic volume of Pu atoms in /H9254-Pu is 25% larger than in /H9251-Pu. However, analyzing the /H9251-Pu struc- ture using the Voronoi decomposition technique, we found aconsiderable dispersion in the sizes of the inequivalent Puatoms in this structure, with the largest atom /H20849No. 8 /H20850almost FIG. 10. /H20849Color online /H20850The partial molar volume of Ga in Pu is the cumulative effect of many volume changes induced in surround-ing Pu atoms. It is shown here as a function of the distance of Puatoms from the nearest Ga atom.GALLIUM STABILIZATION OF /H9254-Pu: DENSITY- … PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-920% bigger than the smallest one /H20849No. 1 /H20850. Using DFT calcu- lations, we showed that this geometric heterogeneity causesstrong dispersion in the energy of the felectrons, with the site 1 felectrons being of much lower energy than the site 8 ones. Following this idea, we further demonstrated through large supercell calculations that it is energetically far morefavorable to replace a Pu atom on site 8 with Ga than one onother sites. Hence Ga atoms have a strong tendency to orderon site 8 in /H9251-Pu, the so-called substitutional variant 8. Perhaps a more unexpected ﬁnding was that the calculated partial molar volume of Ga in /H9251-Pu is close to 30 Å3unless it substitutes for a site 8 Pu atom, in which case it is asize-matched inclusion with a volume of about 20 Å 3. Hence, depending on the concentration of the site 8 Ga im-purities, a range of partial molar volumes from 20 to 30 Å 3 can be obtained. With this in mind, we thus explained the nature of the /H9251/H11032-Pu phase. It is simply the /H9251-Pu structure, where the Pu atoms have been substituted at random with Ga atoms. It isobtained upon the martensitic transformation from the /H9254-Pu phase, since /H20849i/H20850in this phase the Ga atoms are randomly distributed and /H20849ii/H20850such structural transitions do not involve atomic diffusion. Bearing in mind that the substitutional variant 8 is ener- getically more favorable than the /H9251/H11032-Pu phase, and that the former can be obtained from the latter by only a few diffu-sion hops per Ga atom, we can expect that moderate anneal-ing of /H9251/H11032-Pu will result in the eighth variant, and subse- quently to a drastic reduction of the partial molar volume ofGa in this alloy with otherwise little structural change. Thisphenomenon has been observed experimentally, but no ex-planation was offered prior to this work. Along the same lines, we have also been able to explain the double-C behavior of the time-temperature-transformation /H20849TTT /H20850curve of the /H9254-to-/H9251martensitic trans- formation. This is due to the fact that at higher temperatures,larger concentrations of the eighth variant can be obtaineddue to limited Ga diffusion. However, below a certain tem-perature this process stops and the /H9251/H11032phase with a random mixture of all the variants is produced. In light of these re-sults, we can conclude that the key to the understanding ofmany of the properties of the /H9254-to-/H9251martensitic transforma- tion rather lies in the novel geometry of the /H9251phase of Pu than in an exotic electronic structure of /H9254-Pu. In addition to /H9251-Pu, we have also presented in this paper spin-polarized DFT calculations for the heat of solution andthe volume of Ga in /H9254-Pu. We found that in contrast to the large positive mixing enthalpies of Ga in /H9251-Pu, the heat of solution in /H9254-Pu is negative. Based on these calculations we concluded that DFT predicts that Ga concentrations equalto or higher than 8 at.% can stabilize the /H9254-Pu phase with respect to /H9251/H11032-Pu. Furthermore, we found that DFT can accu- rately predict the partial molar volume of Ga in /H9254-Pu /H20849/H1101511 Å3/H20850, as well as Ga-Pu bond lengths, as compared with the recent EXAFS measurements. These results provide evi- dence that DFT can be used as an accurate model for thestructural chemistry of Pu-Ga alloys, predicting 300% vari-ability in the sizes of Ga inclusions in the different Puphases. We have also proposed a theory for the origin of the large variability of Ga sizes in Pu. For this purpose, we introduceda new quantity, i.e., the local volume of a solute. Equippedwith the Voronoi decomposition technique, we calculated thelocal volumes of all the atoms in our various supercells. Sur-prisingly, we found that in spite of the apparently great sen-sitivity of the partial molar volume of the Ga atoms to thegeometry of the surrounding Pu host, their local volumesshow very little variation from the equilibrium volume of Gain its pure state. A further analysis conﬁrmed that the greatvariability in the partial molar volumes of Ga in Pu is mainlydue to the surrounding Pu atoms being affected by the solute.In general, Ga atoms cause the surrounding Pu atoms in the /H9251-Pu phase to enlarge, while they induce those in /H9254-Pu phase to shrink. The accumulated effect is observed when latticeparameter measurements as a function of Ga concentration invarious Pu phases are analyzed. In summary, DFT, with the inclusion of spin polarization for the /H9254structure, proves successful in reproducing all struc- tural and most thermodynamic aspects of Pu-Ga alloys withlow Ga concentrations. In the process of ascertaining its va-lidity for Pu, we have also succeeded in ﬁnally explainingthe nature of the /H9254-stabilization of Pu by small additions of Ga. Furthermore, we have found that solute-induced volumechanges in surrounding host atoms contribute signiﬁcantly tothe partial molar volume or to the misﬁt volume of a solute.While these induced volume changes are individually small,their cumulative effect is substantial. ACKNOWLEDGMENTS We wish to thank Jess Sturgeon for valuable discussions as well as much help preparing the ﬁgures. This work wasperformed under the auspices of the U.S. Department of En-ergy by the University of California Lawrence LivermoreNational Laboratory under Contract No. W-7405-Eng-48. FIG. 11. /H20849Color online /H20850The change in the average Pu-Pu bond distances, deﬁned in Eq. /H208492/H20850, as a function of the distance of Pu atoms from the nearest Ga atom.B. SADIGH AND W. G. WOLFER PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-10APPENDIX: SOLUTE AS AN ELASTIC INCLUSION 1. The atomic size of an impurity in a host In the simplest model for solutes or the point defects in a solid, be it interstitials, substitutional elements, or vacancies,a point defect is viewed as a misﬁtting inclusion whoseatomic volume /H9024 Xdiffers from the atomic volume of the host/H9024H. As a result of this misﬁt, an elastic strain is created in the host material around the point defect. The displace-ment ﬁeld associated with it falls off as 1/ r 2, where ris the radial distance from the defect center. In a ﬁnite solid, thisdisplacement ﬁeld does not vanish on the surface, and thenormal stress component associated with it does not vanisheither, as it should. To evaluate the complete displacement ﬁeld of the point defect requires then the addition of an image ﬁeld so as tosatisfy the stress-free boundary condition. The complete dis-placement ﬁeld results then in a volume change of the entiresolid/H9004V, which is precisely equal to the amount of misﬁt, i.e., /H9004V/V H=/H20849/H9024X−/H9024H/H20850//H9024H/H110153/H9004aH/aH, /H20849A1/H20850 where aHis the lattice parameter of the host. As indicated in this equation, the global volume change /H9004Vis also equal to three times the lattice parameter change induced by the pointdefects in cubic solids. Equation /H20849A1/H20850also deﬁnes the partial molar volume /H9024 Xper solute atom X. For a misﬁtting inclu- sion in an elastic medium, the value of the partial molarvolume is the same as its free volume. The presence of the host’s conﬁning strain ﬁeld modiﬁes the “local” volume made available to the point defect, and itdiffers from the free atomic volume /H9024 X. The conﬁned atomic volume of the point defect, /H9024XHis somewhere between its free value and the host’s atomic volume /H9024H, and it depends on the host’s elastic properties. As Eshelby has shown, the ratio of the global to the local misﬁt volumes is a characteristic constant of the host mate-rial: /H9024 X−/H9024H /H9024XH=/H9253E. /H20849A2/H20850 The Eshelby factor,35/H9253E, is entirely determined by the elastic constants of the host material. For an elastically iso-tropic material, Eshelby has shown that /H9253E=31−/H9263 1+/H9263, /H20849A3/H20850 where /H9263is Poisson’s ratio of the host material. For aniso- tropic materials, the Eshelby factor must be evaluated nu-merically. Before pursuing its dependence on the elastic constants further, we note that it has recently become possible to de- termine the local atomic volume /H9024 XHfrom EXAFS measure- ments of neighbor distances RX/H20849n/H20850, of a solute atom X to thehost atoms in the nth coordination shell. If it is assumed that the nearest neighbor distance RX/H208491/H20850, deﬁnes the size of the local atomic volume, the following relation is obtained: /H9024XH−/H9024H /H9024H/H110153RXH/H208491/H20850−RH/H208491/H20850 RH/H208491/H20850=3/H9004RX/H208491/H20850 RH/H208491/H20850, /H20849A4/H20850 where RH/H208491/H20850is the nearest neighbor distance in the host. Using the above relationships one obtains /H9253E/H9004RX/H208491/H20850 RH/H208491/H20850=/H9004aXH aH. /H20849A5/H20850 We note that this relationship differs from the one em- ployed by Scheuer and Lengeler36by a very large numerical factor that is incorrect. Consequently, their interpretation re-garding local changes in interatomic distances around soluteatoms and global lattice parameter changes is also false. 2. Numerical values for the Eshelby factor The evaluation of strain ﬁelds for defects in anisotropic materials makes use of the elastic Green’s function. Whilethis Green’s function can be represented in analytic form inreciprocal space, in real space analytical forms exist only forisotropic and hexagonal crystals. The strategy adopted foranisotropic materials has been to deﬁne an appropriate iso-tropic approximation and to treat the deviations from isot-ropy by either perturbation theory or by variational methods.Both of these approaches have been studied and utilized byDederichs and Pollman 37as well as by Leibfried and Breuer.32In particular, we shall make use of the results by the latter authors. Their results are correlated to the aniso-tropy constant C A=C11−C12−2C44; /H20849A6/H20850 the bulk modulus, B=/H20849C11+C12/H20850/3; /H20849A7/H20850 and the two Voigt constants: C¯11=C11−2CA/5, /H20849A8/H20850 C¯44=C44+CA/5. /H20849A9/H20850 Leibfried and Breuer have computed the Eshelby factor for many metals and found that it is determined by two con- stants, i.e., C¯11/BandCA/C¯44. We are able to ﬁt their nu- merical results accurately to the following expression: /H9253E=C¯11 B/H208751 + 0.002 458 /H20873CA C¯44/H208744/H20876−1 . /H20849A10 /H20850 Using the elastic constants of Ledbetter and Moment31for /H9254-Pu, we obtain a value of 1.76 for the Eshelby factor of /H9254-Pu.GALLIUM STABILIZATION OF /H9254-Pu: DENSITY- … PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-111S. S. Hecker, Mater. Res. Soc. Symp. Proc. 26, 672 /H208492001 /H20850. 2D. A. Young, Phase Diagrams of the Elements /H20849University of California Press, Berkeley and Los Angeles, 1991 /H20850. 3S. S. Hecker, in Los Alamos Science /H20849Los Alamos National Labo- ratory, Los Alamos, 2000 /H20850. 4S. S. Hecker, D. R. Harbur, and T. G. Zocco, T. B. Massalski Festschrift Volume, Prog. Mater. Sci. 49, 429 /H208492004 /H20850. 5P. Söderlind and C. S. Nash in Advances in Plutonium Chemistry 1967–2000 , edited by D. C. Hoffman /H20849American Nuclear Soci- ety, La Grange Park, IL, 2002 /H20850,p .6 . 6P. Söderlind, J. M. Wills, B. Johansson, and O. Eriksson, Phys. Rev. B 55, 1997 /H208491997 /H20850. 7S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett. 84, 3670 /H208492000 /H20850. 8S. Y. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 /H208492001 /H20850. 9X. Dai, S. Y. Savrasov, G. Kotliar, A. Migliori, H. Ledbetter, and E. Abrahams, Science 300, 953 /H208492003 /H20850; J. Wong, M. Krisch, D. L. Farber, F. Occelli, A. J. Schwartz, T. Chiang, M. Wall, C.Boro, and R. Xu, ibid. 301, 1078 /H208492003 /H20850. 10P. Söderlind, Europhys. Lett. 55, 525 /H208492001 /H20850. 11C. E. Olsen, A. L. Comstock, and T. A. Sandenaw, J. Nucl. Mater. 195, 312 /H208491992 /H20850. 12S. Meot-Reymond and J. M. Fournier, J. Alloys Compd. 232,1 1 9 /H208491996 /H20850. 13P. Söderlind, A. L. Landa, and B. Sadigh, Phys. Rev. B 66, 205109 /H208492002 /H20850. 14B. Sadigh, P. Söderlind, and W. G. Wolfer, Phys. Rev. B R68, 241101 /H208492003 /H20850. 15A. L. Kutepov and S. G. Kutepova, J. Phys.: Condens. Matter 15, 2607 /H208492003 /H20850; Y. Wang and Y. Sun, ibid. 12, L311 /H208492000 /H20850;G . Robert, A. Pasturel, and B. Siberchicot, Phys. Rev. B 68, 075109 /H208492003 /H20850; J. C. Boettger, Int. J. Quantum Chem. 95, 380 /H208492003 /H20850. 16P. Söderlind and B. Sadigh, Phys. Rev. Lett. 92, 185702 /H208492004 /H20850. 17N. T. Chebotarev, E. S. Smotritskaya, M. A. Andrianov, and D. E. Kostyuk in Plutonium 1975 and Other Actinides , edited by H. Blank and R. Lindner /H20849North-Holland, Amsterdam, 1976 /H20850,p .3 7 . 18P. H. Adler, Metall. Trans. A 22A, 2237 /H208491991 /H20850. 19P. Deloffre, J. L. Trufﬁer, and A. Falanga, J. Alloys Compd. 271– 273, 370 /H208491998 /H20850. 20J. T. Orme, M. E. Faiers, and B. J. Ward, in Proc. of 5th Int. Conf.on Plutonium and Other Actinides , edited by H. Blank and R. Lindner /H20849North-Holland, New York, 1976 /H20850, p. 761. 21P. Deloffre, Ph.D thesis, de l’universite Paris XI Orsay, 1997. 22L. E. Cox, R. Martinez, J. H. Nickel, S. D. Conradson, and P. G. Allen, Phys. Rev. B 51, 751 /H208491995 /H20850. 23Ph. Faure, B. Deslandes, D. Bazin, C. Tailland, R. Doukhan, J. M. Fournier, and A. Falaga, J. Alloys Compd. 244, 131 /H208491996 /H20850. 24P. G. Allen, A. I. Henderson, E. R. Sylwester, P. E. A. Turchi, T. H. Shen, G. F. Gallegos, and C. H. Booth, Phys. Rev. B 65, 214107 /H208492002 /H20850. 25J. P. Perdew, Electronic Structure of Solids 1991 , edited by P. Ziesche and H. Eschrig /H20849Akademie-Verlag, Berlin, 1991 /H20850, Vol. 11. 26G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 /H208491996 /H20850. 27G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 /H208491999 /H20850. 28The standard enumeration of the eight inequivalent atoms in the /H9251-Pu structure can be found in W. H. Zachriasen and F. H. Ellinger, Acta Crystallogr. 16, 777 /H208491963 /H20850. 29F. H. Ellinger, C. C. Land, and V. O. Struebing, J. Nucl. Mater. 12, 226 /H208491964 /H20850. 30P. Söderlind, B. Sadigh, and K. T. Moore, Phys. Rev. Lett. 93, 199601 /H208492004 /H20850. 31H. M. Ledbetter and R. L. Moment, Acta Metall. 24, 891 /H208491976 /H20850. 32G. Leibfried and N. Breuer, Point Defects in Metals I /H20849Springer- Verlag, Berlin, 1978 /H20850, pp. 150–155. 33M. Dormeval, N. Baclet, C. Valot, P. Roﬁdal, and J. M. Fournier, J. Alloys Compd. 350,8 6 /H208492003 /H20850. 34The Voronoi cell volume for Ga is not necessarily equal to the atomic volume if the latter is deﬁned according to R. F. W.Bader in Atoms in Molecules: A Quantum Theory /H20849Clarendon, Oxford, 1990 /H20850, pp. 172–175. In the latter deﬁnition, the bound- ary between two neighboring atoms consists of all the pointswhere the gradient of the electron density vanishes. This bound-ary agrees in general with the Voronoi cell boundary betweentwo atoms of the same element, but will be different for differentchemical species. Nevertheless, the cumulative volume will addup to the same partial molar volume irrespective of the deﬁnitionfor atomic volumes. 35J. D. Eshelby, Solid State Phys. 3,7 9 /H208491956 /H20850. 36U. Scheuer and B. Lengeler, Phys. Rev. B 44, 9883 /H208491991 /H20850. 37P. H. Dederichs and J. Pollman, Z. Phys. 255, 315 /H208491972 /H20850.B. SADIGH AND W. G. WOLFER PHYSICAL REVIEW B 72, 205122 /H208492005 /H20850 205122-12
PhysRevB.80.195113.pdf	Spin-charge decoupling and the photoemission line-shape in one-dimensional insulators Valeria Lante and Alberto Parola CNISM and Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, I-22100 Como, Italy /H20849Received 8 July 2009; revised manuscript received 11 September 2009; published 19 November 2009 /H20850 The recent advances in angle-resolved photoemission techniques allowed the unambiguous experimental conﬁrmation of spin-charge decoupling in quasi-one-dimensional /H208491D/H20850Mott insulators. This opportunity stimu- lates a quantitative analysis of the spectral function A/H20849k,/H9275/H20850of prototypical one-dimensional correlated models. Here we combine Bethe Ansatz results, Lanczos diagonalizations, and ﬁeld theoretical approaches to obtainA/H20849k, /H9275/H20850for the 1D Hubbard model as a function of the interaction strength. By introducing a single spinon approximation , an analytic expression is obtained, which shows the location of the singularities and allows, when supplemented by numerical calculations, to obtain an accurate estimate of the spectral weight distributionin the /H20849k, /H9275/H20850plane. Several experimental puzzles on the observed intensities and line-shapes in quasi-1D compounds such as SrCuO 2, ﬁnd a natural explanation in this theoretical framework. DOI: 10.1103/PhysRevB.80.195113 PACS number /H20849s/H20850: 71.10.Fd, 79.60. /H11002i I. INTRODUCTION Since the theoretical prediction of the decoupling of spin and charge excitations in one-dimensional /H208491D/H20850models,1 many experiments have long sought to verify this effect.2 According to the spin-charge separation scenario, the va- cancy /H20849e+/H20850created by removing an electron in a photoemis- sion experiment decays into two collective excitations /H20849or quasiparticles /H20850, known as spinon /H20849s/H20850andholon /H20849h/H20850, carrying spin and charge degrees of freedom respectively. The recent observation of a well-deﬁned two-peak structure in theangle-resolved photoemission spectra /H20849ARPES /H20850of the quasi-1D materials SrCuO 2and Sr 2CuO 3/H20849Refs. 3and4/H20850is deemed a signiﬁcant clue of spin-charge decoupling, con-ﬁrming previous expectations. However, other quasi-one-dimensional materials 5fail to show distinct holon and spinon peaks, casting some doubt onthe interpretation of ARPES experiments based on spin-charge decoupling. A number of puzzling features also sug-gest that more physics, beyond the simple decay e +→s+h,i s involved in the photoemission process: the spectral functionsof SrCuO 2and Sr 2CuO 3reported by Kim et al.3and by Kidd et al.4systematically display broad line-shapes in contrast to the sharp edges expected on the basis of the available calcu-lations on model systems. The spectral intensity also appearsconsiderably weaker in a half of the Brillouin zone, a featureoften ascribed to cross-section effects. 6 A quantitative theoretical understanding of ARPES in low-dimensional systems is important and deserves a carefulinvestigation because ARPES provides a direct experimentalprobe to the single particle excitation spectrum, allowing forreliable estimates of the key parameters governing the phys-ics of strongly correlated electrons: the electron bandwidthand the Coulomb repulsion. Here we will focus on the 1DHubbard model, a simple lattice model deﬁned by just twocoupling constants: the nearest neighbor hopping integral t and the on-site Coulomb repulsion U: H=−t /H20858 i,/H9268/H20851ci+1,/H9268†ci,/H9268+h.c./H20852+U/H20858 ini↑ni↓. /H208491/H20850 Although several other terms, such as next-nearest hopping, further orbital degrees of freedom, temperature, disorder orlattice instabilities, would be necessary in a realistic model of these materials, we believe that an accurate investigationof the simplest Hamiltonians should be performed beforefacing more challenging problems. The theoretical studies aimed at the investigation of the spectral properties of one-dimensional models are eitherfully numerical such as Lanczos diagonalizations 2and den- sity matrix renormalization group /H20849DMRG /H20850techniques,7or are carried out in the limiting cases of inﬁnite8or vanishing9 interaction U/t. In the former case, they suffer from severe ﬁnite size effects, in the latter the interplay between chargeﬂuctuations and strong correlations is not satisfactorily takeninto account. Monte Carlo studies of dynamical properties ofquantum systems are instead hampered by the necessity toperform an analytic continuation to real times. In this paper, we provide the quantitative evaluation of the full spectral function A/H20849k, /H9275/H20850of the 1D Hubbard model at half ﬁlling for intermediate and strong coupling U/t.10A formalism based on the Bethe Ansatz solution,11and supple- mented by Lanczos diagonalizations, is developed and isshown to provide a transparent description of the dynamicalproperties of mobile charges in Mott insulators. From thisanalysis we ﬁnd that the 1D Hubbard model does indeed contain the physics required for a quantitative interpretationof photoemission experiments. In particular: /H20849i/H20850the underly- ing free electron Fermi surface plays a key role in deﬁningthe shape and the intensity of the ARPES signal, up to fairlylarge effective couplings U/t;/H20849ii/H20850the power-law singularities which characterize the spectral function in one dimensiongive rise to intrinsically broad peaks, whose width is propor-tional to the intensity of the line; /H20849iii/H20850ARPES data are ex- tremely sensitive to the Hubbard parameters and allow for adirect determination of the effective coupling constants inquasi 1D materials. As a working example, we apply ourmethod to SrCuO 2, where accurate ARPES data are available,2and we derive reliable estimates for tandU. The plan of the paper is as follows. In Sec. IIwe present and motivate the single spinon approximation which lies atthe basis of our method, deriving the predicted formal struc-ture of the spectral function in one-dimensional models. Sec-tionIIIshows how Lanczos diagonalizations provide a pre- cise quantitative estimate of the quasiparticle weight requiredPHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 1098-0121/2009/80 /H2084919/H20850/195113 /H208498/H20850 ©2009 The American Physical Society 195113-1for the evaluation of the spectral function. Then, in Sec. IV we discuss the weak coupling limit, where a thorough ﬁeldtheoretical analysis is available. The application to the caseof SrCuO 2is performed in Sec. V, while in the Conclusions we brieﬂy discuss the generalization of our method to morecomplex one-dimensional hamiltonians. II. ANALYTICAL STRUCTURE OF THE SPECTRAL FUNCTION The dynamical properties of one /H20849spin down /H20850hole in the half ﬁlled Hubbard model are embodied in the spectral func-tion A/H20849k, /H9275/H20850which, at zero temperature, can be written as A/H20849k,/H9275/H20850=/H20858 /H20853/H20841/H90231/H20856/H20854/H20841/H20855/H90231/H20841ck,↓/H20841/H90230/H20856/H208412/H9254/H20849/H9275−E1+E0/H20850, /H208492/H20850 where /H20841/H90230/H20856is the ground state of the model at zero doping, i.e., when the number of electrons Nequals the number of sites of the lattice L,E0is the corresponding energy and /H20853/H20841/H90231/H20856/H20854represents a complete set of one-hole intermediate states, of energies E1. The whole energy spectrum of the Hubbard hamiltonian /H208491/H20850can be obtained from the Lieb and Wu equations.11In the thermodynamic limit, its structure has been thoroughly investigated in a series of papers byWoynarovich 12,13/H20849see also the comprehensive book by Es- sleret al. Ref. 14/H20850. In summary, the exact excitation spec- trum at half ﬁlling and at arbitrary coupling U/tdepends on two sets of “rapidities” describing the charge and spin de-grees of freedom respectively. The excitation energy is al-ways written as the sum of contributions involving just twoelementary excitations, representing collective quasiparti-cles: “holons” /H20851of momentum k hand energy /H9280h/H20849kh/H20850/H20852and “spinons” /H20851of momentum Q/H33528/H20849/H9266 2,3/H9266 2/H20850and energy /H9280s/H20849Q/H20850/H20852. The simplest physical excitation created by the removal of anelectron of momentum kgives rise to one holon and one spinon satisfying the momentum conservation equation k =k h+Q. The total energy of this state is E1=E0+/H9280h/H20849kh/H20850 +/H9280s/H20849Q/H20850. Besides this suggestive “decay” mechanism of the electron, other excited states also appear in the exact spec-trum: they are either multispinon and multiholon states, orstates involving the creation of double occupancies. 13How- ever, it is remarkable that the full excitation spectrum can bealways expressed in terms of /H9280h/H20849kh/H20850and/H9280s/H20849Q/H20850, showing that spin-charge decoupling holds, in the Hubbard model, at allvalues of U/tand at all energy scales. 14The two quasiparti- cles, holon, and spinon, are both collective excitations in-volving an extensive number of degrees of freedom and canbe approximately related to simple real space pictures of a“hole” and an unpaired spin only in the strong coupling limit,where spin-charge decoupling acquires a more intuitivemeaning. As U→0 the holon and spinon bands reduce to simple analytical forms, 11closely related to the free particle band structure: /H9280h/H20849kh/H20850=4tcos/H20849kh/2/H20850and/H9280s/H20849Q/H20850=2t/H20841cosQ/H20841. While the whole energy spectrum of the Hubbard hamil- tonian is known in detail, the matrix elements appearing inEq. /H208492/H20850are of difﬁcult evaluation. Moreover the summation over the intermediate states formally involves a number ofterms exponentially large in N, making the exact implemen- tation of the deﬁnition /H208492/H20850impractical. Our approach, whichallows for the evaluation of the full spectral function in the thermodynamic limit, is based on the single spinon approxi- mation : i.e., we neglect the contribution to the spectral func- tion coming from all multispinon excited states and all exci-tations with complex rapidities, but we evaluate exactly thematrix elements involving one holon and one spinon. Theaccuracy of this method is tested a posteriori by use of a completeness sum rule and can be estimated of the order offew percents. Such a remarkable performance of the singlespinon approximations is not unusual in one-dimensionalphysics: a known example is provided by the Haldane-Shastry spin model /H20849HSM /H20850, 15where each intermediate state contributing to the dynamical spin correlation is completelyexpressible in terms of eigenstates of the HSM with only twospinons. In this case, only a small O/H20849L/H20850number of eigen- states contribute to the exact dynamical spin correlation function as proved in Ref. 16. Similarly, in our approach, the most relevant intermediate states are expressible is terms ofeigenstates of the Hubbard model with only one spinon andone holon excitations. A ﬁrst clue on the structure of the spectral function in one-dimensional models can be obtained by analyzing theU→/H11009limit, where double occupancies are inhibited and several exact results are available. 8At half ﬁlling /H20849N=L/H20850the Hubbard hamiltonian is mapped onto a Heisenberg Hamil-tonian: each site is singly occupied and the ground state is anondegenerate singlet of zero momentum. 17When a hole of momentum khis created, all the eigenfunctions of the Hub- bard hamiltonian /H20849with periodic boundary conditions /H20850can be written as18 /H20841/H90231/H20856=1 /H20881L/H20858 x,/H20853yi/H20854eikhx/H9278H/H20849y1, ..., yM/H20850/H20841x,/H20853yi/H20854/H20856, /H208493/H20850 where /H20841x,/H20853yi/H20854/H20856represents the conﬁguration of L−1 electrons deﬁned by the positions of the M=L/2 spin up /H20849/H20853yi/H20854/H20850and of the hole /H20849x/H20850. The amplitude /H9278His a generic eigenfunction of the Heisenberg hamiltonian on the “squeezed chain,” i.e., ontheL−1 site ring deﬁned by all the sites occupied by an electron. The intermediate states /H20841/H9023 1/H20856entering the spectral function /H208492/H20850have momentum − krelative to the ground state at half ﬁlling. Due to the factorized form of eigenfunctions/H208493/H20850the total momentum of the state satisﬁes k=k h+Qwhere Qis the momentum of the Heisenberg eigenfunction /H9278H, expressed in integer multiples of 2 /H9266//H20849L−1/H20850in ﬁnite chains. In the thermodynamic limit the energy of the intermediatestate is E 1=E0+/H9280h/H20849kh/H20850+/H9280s/H20849Q/H20850, where, to lowest order in J =4t2/U, the ﬁrst /H20849holon /H20850contribution is just the kinetic en- ergy of a free particle /H20849/H9280h/H20849kh/H20850=2tcoskh/H20850and the second one /H20849spinon /H20850is the energy of the eigenstate /H9278Hreferred to the ground-state energy of the Heisenberg ring of Lsites.19This analysis shows, in an intuitive way, the origin of momentumand energy conservation in the decay process of the vacancyand suggests that, in the U→/H11009limit, the most relevant con- tributions to the sum of intermediate states in Eq. /H208492/H20850come from the lowest-energy eigenstates /H9278Hof the Heisenberg Hamiltonian for the L−1 allowed momenta Q=2/H9266 L−1n,/H20849n =0... L−2/H20850. Accordingly, the sum over an exponentially large set of eigenstates /H20853/H20841/H90231/H20856/H20854in Eq. /H208492/H20850can be /H20849approxi-VALERIA LANTE AND ALBERTO PAROLA PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-2mately /H20850replaced by a sum over L−1single spinon states . This special set of intermediate states /H20841/H90231/H20856, which we argue provides the dominant contribution to the spectral weight foreach spinon momentum Q, will be referred to as /H20841−k,Q/H20856in order to emphasize the two quantum numbers whichuniquely identify them. The single spinon approximation canbe easily tested in the U→/H11009limit 8where it proves ex- tremely accurate. In the next section we will show that itremains fully satisfactory also at ﬁnite coupling. In fact, it isknown 14that the eigenstate structure of the Hubbard model displays a remarkable continuity in U/t, the only singular point being the /H20849trivial /H20850free particle limit U=0. However, when charge ﬂuctuations are allowed for, by lowering thestrength of the on-site repulsion U, the identiﬁcation of the single spinon states /H20841−k,Q/H20856is not easy, because the spinon momentum Qis not a good quantum number any more, al- though it can be still formally deﬁned on the basis of theBethe Ansatz solution of the Hubbard model. 13The key ob- servation, which will be exploited in the next section, is thatthe correct single spinon states can be identiﬁed at ﬁnite U via Lanczos or DMRG calculations by following adiabati- cally the evolution of the Heisenberg states as Uis gradually decreased. Keeping only the single spinon states in the summation of Eq. /H208492/H20850, the spectral function in the thermodynamic limit becomes A/H20849k, /H9275/H20850=/H20885dQ 2/H9266Zk/H20849Q/H20850/H9254/H20851/H9275−/H9280h/H20849k−Q/H20850−/H9280s/H20849Q/H20850/H20852 =1 2/H9266/H20858 Q/H11569Zk/H20849Q/H11569/H20850 /H20841vh/H20849Q/H11569−k/H20850+vs/H20849Q/H11569/H20850/H20841. /H208494/H20850 Where we have deﬁned the quasiparticle weight as the ma- trix element Zk/H20849Q/H20850/H11013lim L→/H11009/H20849L−1/H20850/H20841/H20855−k,Q/H20841ck,↓/H20841/H90230/H20856/H208412. /H208495/H20850 The sum in Eq. /H208494/H20850runs over all the solutions Q/H11569/H20849k,/H9275/H20850of the algebraic equation /H9275=/H9280h/H20849k−Q/H20850+/H9280s/H20849Q/H20850, /H208496/H20850 where /H9280h/H20849kh/H20850/H20851/H9280s/H20849Q/H20850/H20852is the known holon /H20849spinon /H20850excitation energy13andvh/H20849kh/H20850=d/H9280h dkh,/H20851vs/H20849Q/H20850=d/H9280s dQ/H20852the associated velocity. Equation /H208494/H20850is the main result of this work: an explicit and computable expression for the spectral function of one-dimensional models. In the special case of the Hubbardmodel, the Bethe Ansatz solution directly provides spinonand holon dispersions in the thermodynamic limit furthersimplifying the evaluation of the spectral function. Due tothe presence of a spinon Fermi surface, the dispersion rela- tion /H9280s/H20849Q/H20850is deﬁned only in the interval/H9266 2/H11021Q/H110213/H9266 2,20it vanishes at the boundaries and has a single maximum at Q =/H9266, while /H9280h/H20849kh/H20850is an even and periodic function in the whole range − /H9266/H11021kh/H11021/H9266with maximum at k=0.12,13The only missing ingredient in Eq. /H208494/H20850is the quasiparticle weight Zk/H20849Q/H20850which deﬁnes the line-shape and intensity of the spec- tral function. Previous studies21have shown that in spin iso- tropic models such as the Hubbard model, the quasiparticleweight is a regular function with square root singularities at the spinon Fermi surface Q=/H9266/H11006/H9266/2. This implies that A/H20849k,/H9275/H20850has power-law singularities too, whenever either Q/H11569 deﬁned by Eq. /H208496/H20850lies at the spinon Fermi surface, or when the total excitation velocity vh/H20849Q/H11569−k/H20850+vs/H20849Q/H11569/H20850vanishes. In both instances, square root divergences are expected:22in the former case the location of the singularity identiﬁes the ho- lon dispersion via /H208496/H20850/H9275=/H9280h/H20849k+/H9266/H11006/H9266 2/H20850; in the latter case the singularity is trivially due to band structure effects and doesnot necessarily corresponds to a pure spinon contribution asoften assumed. However, at small to moderate interactionsU/t, the holon velocity /H20841 vh/H20849kh/H20850/H20841displays an abrupt drop around kh/H11011/H9266/H20849Ref. 13/H20850placing the band lower edge close to Q/H11011/H9266+k, i.e., at /H9275/H11011/H9280h/H20849/H9266/H20850+/H9280s/H20849/H9266+k/H20850, thereby following the spinon band for 0 /H11021k/H11021/H9266 2. This particular feature of the Hub- bard model dispersion is apparent in the shape of the holonspectrum 12which sharply bends at kh/H11011/H11006/H9266so to display a vanishing charge velocity at band edges. This also agreeswith the “relativistic” form of the holon spectrum predictedby bosonization at weak coupling, 9as reported in Eq. /H208498/H20850. The expected location of the square root singularities of thespectral function in the /H20849k, /H9275/H20850plane is shown for few values of the coupling in Fig. 1. The holon branch /H20849shown as full circles in the ﬁgure /H20850marks precisely the holon excitation spectrum /H9280h/H20849kh/H20850while the location of the singularities due to the band structure /H20849shown as crosses in the ﬁgure /H20850differs from the spinon /H9280s/H20849Q/H20850dispersion by less than 0.1 t. Note also that the curvature of the “spinon branch” displays a signiﬁ-cant dependence on U/t, allowing for a rather precise experi- mental determination of the effective coupling ratio. There-fore we conclude that precise photoemission data, able toidentify the singularities of the spectral function, do providedirect information on both holon and, within a good approxi-mation, also spinon excitations. The full holon bandwidth is always 4 tat all couplings, due to the particle-hole symmetry of the Hubbard model butthe upper and lower branches of the holon band are not sym-metrical at ﬁnite U. This observation is relevant for the cor- rect interpretation of photoemission experiments, because anestimate of the effective hopping integral tis usually per- formed by measuring the half bandwidth of the upper holon branch 23leading to a sizable overestimate of t. In Fig. 2we show the bandwidth Whof the upper holon branch /H20851i.e., /H9280h/H20849/H9266/2/H20850−/H9280h/H20849/H9266/H20850/H20852and the ratio between the spinon and the holon bandwidths Ws/Whas a function of the coupling U/t. Both quantities, which allow for a direct estimate of tandFIG. 1. /H20849Color online /H20850Location of the singularities of the spec- tral function in the thermodynamic limit for three values of Uin the plane /H20849k,E=−/H9275/H20850. The /H20849green /H20850circles correspond to the singularities of the Zk/H20849Q/H20850/H20849holon branch /H20850, while the black stars to the extrema of the excitation spectrum.SPIN-CHARGE DECOUPLING AND THE PHOTOEMISSION … PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-3U/tfrom ARPES, show a remarkable /H20849even nonmonotonic /H20850 dependence on the coupling constants. By comparing theseresults with the dispersion curves for SrCuO 2reported in Ref. 3we can estimate the Hubbard effective coupling con- stants appropriate of this material: t/H110110.53 eV and U/t/H110117. III. QUASIPARTICLE WEIGHT FROM LANCZOS DIAGONALIZATION Unfortunately, the formal Bethe Ansatz solution does not lead to a practical way for the evaluation of the quasiparticleweight /H208495/H20850at arbitrary couplings and therefore we resort to Lanczos diagonalizations in lattices up to L=14 sites. As previously noticed, we ﬁrst have to devise a method to selectthe correct single spinon states at ﬁnite coupling U/t, for these states are not identiﬁed by a good quantum number atﬁnite U. Our method is based on an adiabatic procedure starting from the strong coupling limit. We ﬁrst perform aLanczos diagonalization on the /H20849L−1/H20850site Heisenberg model in the symmetry subspace of total momentum Q, leading to the numerical determination of /H9278Hand of the exact eigen- states of the one-hole Hubbard model for U→/H11009via Eq. /H208493/H20850. In this limit, the single spinon states are indeed the lowestenergy eigenstates at ﬁxed spinon momentum Qand can be easily obtained by Lanczos /H20849or DMRG /H20850technique, while at ﬁnite Uthe relevant intermediate states are not necessarily in the low-excitation energy portion of the Hubbard spectrum.Then we take advantage of the continuity of the one spinonstates between the weak and strong coupling limit by adia-batically lowering the interaction strength Uand performing successive Lanczos diagonalizations for smaller and smallercouplings U n. At the nth step we keep the exact eigenstate having the largest overlap with the eigenstate at the /H20849n −1/H20850th level. In this way we are able to identify the single spinon states down to small values of U/H11011t, each state being uniquely identiﬁed by Q, i.e., by the momentum of the “par- ent” Heisenberg eigenstate. A check on the validity of the single spinon approxima- tion comes from the completeness condition on the interme-diate states: n ↓/H20849k/H20850=/H20855/H90230/H20841ck,↓†ck,↓/H20841/H90230/H20856=/H20858 /H20853/H20841/H90231/H20856/H20854/H20841/H20855/H90231/H20841ck,↓/H20841/H90230/H20856/H208412 /H11350/H20858 Q/H20841/H20855−k,Q/H20841ck,↓/H20841/H90230/H20856/H208412=1 L−1/H20858 QZk/H20849Q/H20850, /H208497/H20850where n↓/H20849k/H20850is the momentum distribution of the down spins at half ﬁlling and the equality holds if and only if the singlespinon states included in the sum via the deﬁnition of thequasiparticle weight Z k/H20849Q/H20850/H208495/H20850exhaust the spectral weight at each k. The amount of violation of this sum rule quantiﬁes the weight of allthe neglected states in the Hilbert space due to the single spinon approximation. In Fig. 3we plot n↓/H20849k/H20850 and1 L−1/H20858QZk/H20849Q/H20850restricted to the one spinon states: the vio- lation of the completeness condition is smaller than 0.01 atallk’s. 24Note how, even at fairly large values of U/t, the momentum distribution is considerably depressed for klarger than the free electron Fermi momentum kF=/H9266 2, strongly re- ducing the spectral weight in the second half of the Brillouinzone. This feature is consistent with the photoemission ex-periments performed with high energy photons. 3,6Con- versely, in the strong coupling limit U→/H11009, the momentum distribution becomes ﬂat, n↓/H20849k/H20850=1 /2, washing out this effect. The dependence of the quasiparticle weight on the strength of the Coulomb repulsion has been investigated andis summarized in Fig. 4for strong and intermediate U/tand for different lattice sizes. The quasiparticle weight has beenevaluated by Lanczos diagonalization on lattices rangingfrom L=6 to L=14 sites. By using standard periodic bound- ary conditions, the total momentum of the state would bequantized in units of 2 /H9266/L, making size scaling impractical. In order to avoid this problem we have adopted skewedboundary conditions: given an arbitrary hole momentum k we choose the ﬂux at the boundary in such a way to match k with the quantization rule. Figure 4reveals an astoundingly negligible size dependence and the expected vanishing of thequasiparticle spectral weight outside the spinon Fermi sur-face, with singularities at the Fermi momenta. While Z k/H20849Q/H20850 is almost independent on kat large U, as expected,8it shows more structure for realistic values of U/t. The further peak /H20849or shoulder /H20850present for k/H11349/H9266 2is indeed reminiscent of the free Fermi nature of the electrons at U=0. In the free particle limit, only one state provides a ﬁnite contribution to the spec- tral function: the holon sits at the bottom of the band /H20849kh =/H9266/H20850and the quasiparticle weight Zk/H20849Q/H20850reduces to a delta function at Q=/H9266+k. When such a form of Zk/H20849Q/H20850is substi- tuted in Eq. /H208494/H20850, the known free particle result is recovered. Remarkably, a remnant of the free particle peak in Zk/H20849Q/H20850is still visible at U=7t, as shown in Fig. 4/H20849b/H20850.FIG. 2. Panel /H20849a/H20850: holon bandwidth Wh=/H9280h/H20849/H9266/2/H20850−/H9280h/H20849/H9266/H20850as a function of U/t. Panel /H20849b/H20850: ratio between the spinon bandwidth Ws=/H9280s/H20849/H9266/H20850andWhas a function of U/t.FIG. 3. /H20849Color online /H20850Momentum distribution n↓/H20849k/H20850/H20849black open circles /H20850and1 L−1/H20858QZk/H20849Q/H20850/H20849red triangles /H20850versus the hole momentum kfrom Lanczos diagonalization, for U=7tin aL=14 ring.VALERIA LANTE AND ALBERTO PAROLA PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-4IV. WEAK COUPLING LIMIT The Green’s function of one-dimensional models has been thoroughly investigated by bosonization methods: while inthe Luttinger liquid regime its asymptotic form is character-ized by power-law tails, 1precisely at half ﬁlling the Green’s function is known to display a more complex behavior due tothe presence of a gap in the holon spectrum. At weak cou-pling, the holon dispersion near the bottom of the bandshows a “relativistic” structure: /H9280h/H20849kh/H20850=/H20881vh2/H9254kh2+m2, /H208498/H20850 where mis the charge gap and /H9254kh=kh/H11006/H9266is the holon mo- mentum measured from the bottom of the band. Note that theholon spectrum /H208498/H20850is shifted by /H9262=U/2 with respect to our previous deﬁnition. In Fig. 5/H20849a/H20850we plot the exact Bethe An- satz spectrum at U=3tand the form /H208498/H20850predicted by bosonization with suitably chosen parameters mandvh. In order to compare the results of our single spinon ap- proximation with the bosonization form, it is convenient tointroduce the single hole Green’s function in imaginary time, G ↓/H20849k,/H9270/H20850=/H20855/H90230/H20841ck,↓†e−/H20849H−/H9262/H20850/H9270ck,↓/H20841/H90230/H20856. /H208499/H20850 According to bosonization,9the Green’s function G↓/H20849k,/H9270/H20850of a hole of momentum close to kF=/H9266/2 acquires a factorized form in real space: G↓R/H20849x,/H9270/H20850/H11013/H20885dk 2/H9266G↓R/H20849k,/H9270/H20850eikx=ei/H9266/2xGh/H20849x,/H9270/H20850Gs/H20849x,/H9270/H20850, /H2084910/H20850 where the superscript Ridentiﬁes the contribution to the Green’s function due to right moving holes. Here, GhandGsjust depend on holon and spinon degrees of freedom, respec- tively. The spinon term is simply given by Gs/H20849x,/H9270/H20850=1 /H20881vs/H9270+ix, /H2084911/H20850 while the holon contribution is predicted, by the form factor approach, to behave as Gh/H20849x,/H9270/H20850=/H9003/H20881m vh/H20885 −/H11009/H11009 d/H9258e/H20851/H9258/2−m/H9270cosh/H9258−imx /vhsinh/H9258/H20852,/H2084912/H20850 with/H9003/H110110.0585....9The question now arises whether our single spinon approximation is consistent with such a factor-ized form. By inserting a complete set of intermediate statesinto the deﬁnition /H208499/H20850and adopting the single spinon ap- proximation, the full Green’s function in imaginary time canbe written as G ↓/H20849k,/H9270/H20850=1 L/H20858 QZk/H20849Q/H20850e−/H20851/H9280h/H20849kh/H20850+/H9280s/H20849Q/H20850/H20852/H9270, /H2084913/H20850 where the momentum conservation relation k=kh+Qis un- derstood and the holon spectrum /H9280h/H20849kh/H20850is now referred to the chemical potential /H9262. Notice that, due to momentum conser- vation, the combined requirements of having k/H11011kF=/H9266 2and kh/H11011−/H9266/H20849i.e., the hole sits near the bottom of the band /H20850force Q/H110113/H9266 2. By substituting the asymptotic forms /H208498/H20850and/H9280s/H208493/H9266 2 −q/H20850=vsqforq/H114070 we get G↓R/H20849x,/H9270/H20850=ei/H9266/2xm vh/H20885 0/H9266dq 2/H9266e−/H20851iqx+vsq/H9270/H20852/H20885 −/H11009/H11009d/H9258 2/H9266Zk/H20849Q/H20850 /H11003cosh/H9258e/H20851−imx /vhsinh/H9258−m/H9270cosh/H9258/H20852, /H2084914/H20850 where we set /H9254kh/H11013−m vhsinh/H9258. This form does indeed factor- ize in a holon and spinon part, as predicted by bosonization, provided the quasiparticle weight does,FIG. 4. /H20849Color online /H20850Panel /H20849a/H20850:Zk/H20849Q/H20850versus spinon momen- tumQforU=100 tand different lattice sizes /H20849/H11003:L=6,/H17009:L=8,/H17039: L=10,/H11569:L=12,/L50098:L=14 /H20850. Open /H20849green /H20850symbols for total momen- tumk=/H9266/4 and full /H20849black /H20850symbols for k=0. Lines are polynomial ﬁt to Lanczos data. Skewed boundary conditions are used in orderto ﬁx the same total momentum of the state − k/H20849relative to half ﬁlling /H20850for all L’s. Panel /H20849b/H20850: same as /H20849a/H20850forU=7t. Panel /H20849c/H20850: binding energy at ﬁxed kreferred to half ﬁlling versus spinon mo- mentum Qin the thermodynamic limit E=E 0−E1forU=100 t. Panel /H20849d/H20850: same as /H20849c/H20850forU=7t.FIG. 5. /H20849Color online /H20850Panel /H20849a/H20850: Holon spectrum /H9280h/H20849kh/H20850of the Hubbard model at U=3tfrom Bethe Ansatz /H20849shifted by /H9262=U/2/H20850 compared to the Lorentz form /H208498/H20850. The ﬁtting parameters are m =0.316 tandvh=2.43 t. Panel /H20849b/H20850: Dimensionless holon quasiparticle weight Z¯h=/H20881m vhZhin the single spinon approximation for U=3tand lattice sizes L=10 /H20849full squares /H20850,L=12 /H20849empty circles /H20850,L=14 /H20849full circles /H20850, compared to the bosonization result /H2084916/H20850/H20849line/H20850.SPIN-CHARGE DECOUPLING AND THE PHOTOEMISSION … PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-5Zk/H20849Q/H20850/H11011Zh/H20849kh/H20850Zs/H20849Q/H20850. /H2084915/H20850 Notice that our approach, being based on a numerical evalu- ation of the quasiparticle weight, does not allow for an inde-pendent demonstration of such a factorized form. We justobserve that the bosonization approach and the single spinonapproximation lead to the same result if we assume that Eq. /H2084915/H20850holds. Following Ref. 14we argue that the factorization of the quasiparticle weight at low energies /H2084915/H20850reﬂects the trivial structure of the holon-spinon scattering matrix in thislimit. As previously noticed, the spinon contribution to the qua- siparticle weight gives rise to the square root divergence at the spinon Fermi surface, with leading behavior Z s/H208493/H9266 2−q/H20850 /H11011q−1 /2forq/H114070 which correctly reproduces Eq. /H2084911/H20850when the ultraviolet cutoff in Eq. /H2084914/H20850is disregarded. Matching Eqs. /H2084912/H20850and /H2084914/H20850then selects a unique form of the holon quasiparticle weight, Zh/H20849kh/H20850=/H208812vh/H20881/H9280h/H20849kh/H20850−vh/H9254kh /H9280h/H20849kh/H20850, /H2084916/H20850 with/H9280h/H20849kh/H20850given by Eq. /H208498/H20850. The scale factor in Eq. /H2084916/H20850has been ﬁxed by evaluating the Green’s function in the /H9270→0 limit, where it coincides with the momentum distribution. InFig.5/H20849b/H20850we compare Eq. /H2084916/H20850with the numerical results for Z h/H20849kh/H20850obtained by Lanczos diagonalization at U=3t.N oﬁ t - ting parameters have been used: In order to obtain Zh/H20849kh/H20850we ﬁrst evaluated Zk/H20849Q/H20850, as discussed in Sec. III, then we di- vided the result by Zs/H20849Q/H20850/H11011q−1 /2evaluated at the spinon mo- mentum Qclosest to the spinon Fermi point QF=3/H9266 2. The two parameters vhand mare independently obtained from the holon spectrum /H20851also shown in Fig. 5/H20849a/H20850/H20852. As usual the Lanc- zos data display a very small size dependence and allow fora precise identiﬁcation of the holon quasiparticle weightZ h/H20849kh/H20850. The agreement between the two expressions is re- markable for /H9254kh/H110220 while some discrepancy is found for negative /H9254kh. Note however that the asymptotic form of the holon quasiparticle weight /H2084916/H20850holds only at low energies and weak coupling, while the comparison shown in Fig. 5is performed for U=3t. The results at lower values of U/tare plagued by severe ﬁnite-size effects: in the U→0 limit, the holon mass mvanishes exponentially and the dimensionless momentum scale m/vhvanishes as well. Therefore, at very weak coupling, the relevant holon momenta are constrainedin an extremely small interval around k h=/H11006/H9266, a range not easily accessible due to the momentum quantization rule inﬁnite Hubbard rings. V. RESULTS FOR SrCuO 2 We are now ready to compare our results for the spectral function of the 1D Hubbard model with precise photoemis-sion data recently obtained for SrCuO 2.3A preliminary study, based on the strong coupling limit of the Hubbard model,pointed out some discrepancies, related to the peak heightsand widths. 3Figure 6shows the singularity loci of the 1D Hubbard model with the parameters t=0.53 eV and U =3.7 eV, together with the ARPES results from Kim et al.3 The nice agreement suggests that this material indeed repre-sents a good experimental realization of the simple one- dimensional Hubbard model. The effects due to interchaincoupling, phonons, ﬁnite temperature, and other perturba-tions appears rather small and mostly limited to the spinonbranch. We remark that the same material has been alreadytheoretically investigated on the basis of the Hubbard and t-J model by several groups 3,25,26leading to different sets of parameters both for the hopping integral 0.3 eV /H11351t /H113510.7 eV and for the Coulomb repulsion 2 eV /H11351U /H113516.5 eV. Our analysis shows that both spin and charge ﬂuc- tuations play a key role in determining the line-shape of thespectral function of the Hubbard model, even at moderatelyhigh values of the coupling U/t. In Fig. 7the spectral function has been plotted versus the binding energy E=− /H9275for three representative values of the total hole momentum k. The experimental line broadening reported in Ref. 3has been also included in the Hubbard model results, leading to a merging of close peaks. The den-sity plot clearly reproduces the overall shape deﬁned by thesingularities of the spectral function shown in Fig. 6.A s expected, most of the spectral weight is indeed concentratedin the ﬁrst half of the Brillouin zone between the holon and thespinon band. Although the relative intensity of the ARPES signal at the two singularities depends on the detailsof the band structure, the power-law nature of the diver-gences implies that the intrinsic width of each peak is alwayscomparable with the separation between the holon and the spinon branch /H9004 /H9275/H11011/H9280h/H20849k+/H9266 2/H20850−/H20851/H9280h/H20849/H9266/H20850+/H9280s/H20849/H9266+k/H20850/H20852. The aver- age intensity can be estimated on the basis of the sum rule/H208497/H20850and scales as /H20849/H9004 /H9275/H20850−1 /2, getting smaller when the two branches separate, as shown both in experiments3and in nu- merical calculations.7 VI. CONCLUSIONS The single spinon approximation, combined to Bethe An- satz results and Lanczos diagonalizations allows to obtainvery accurate results for the dynamic properties of a singlehole in the one-dimensional Hubbard model. The Lehmannrepresentation of the spectral function /H208492/H20850shows that two separate ingredients combine to deﬁne the overall shape ofA/H20849k, /H9275/H20850: the excitation spectrum and the quasiparticle weight. The idea at the basis of our method is to limit the size effectsFIG. 6. /H20849Color online /H20850Locus of singularities of the spectral function for the Hubbard model at U=3.7 eV and t=0.53 eV /H20849open circles /H20850compared to the experimental results by Kim et al. /H20849Ref. 3/H20850/H20849full circles /H20850fork/H11036=0.1.VALERIA LANTE AND ALBERTO PAROLA PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-6that plague numerical results by dealing with these two quan- tities separately: in the 1D Hubbard model the excitationspectrum is given exactly by the Bethe Ansatz equations inthe thermodynamic limit, 11while the quasiparticle weight is obtained, in the single spinon approximation, by Lanczosdiagonalization. Size effects are shown to be negligible andthe accuracy of the approximation can be checked a poste- riori by a frequency sum rule /H208497/H20850. Our expression for the spectral function of the 1D Hubbard model /H208494/H20850is consistent with the structure predicted by bosonization 9at weak cou- pling, provided the quasiparticle weight Zk/H20849Q/H20850factorizes as shown in Eq. /H2084915/H20850. A numerical test carried out at U=3tdoes not show a convincing quantitative agreement with the resultobtained by the form factor approach, 9possibly due to the difﬁculty to achieve the U→0 limit. The extension to ﬁnite doping is straightforward but in principle this approach can be also generalized to other fer-mionic lattice models, in one or more dimensions, providedthe relevant states entering the quasiparticle weight in thestrong coupling limit can be easily classiﬁed. This would bethe case of the extended Hubbard model /H20849i.e., a Hubbard model with nearest neighbor Coulomb repulsion /H20850or in thepresence of lattice dimerization. Clearly, for non integrable models, no analytical information on the excitation spectrumis available and a size scaling on the energy spectrum is also required. A study of such generalizations may be useful tounderstand the role of some perturbation on the spectralfunction of correlated electron models. The speciﬁc example of SrCuO 2shows that our method allows for a direct comparison between theory and ARPESexperiments and for an accurate determination of the Hub-bard parameters which best describe the hole dynamics in thematerial. The spectral function derived here provides a natu-ral explanation of the observed reduction in the spectralweight in a half of the Brillouin zone and of the broad line-shape detected in experiments. Future applications of thismethod to the case of cold atoms in optical traps may help inpointing out the peculiar features of one-dimensional physicsin other experimental realizations of correlated one-dimensional Fermi gases. ACKNOWLEDGMENTS We thank C. Kim and F.H.L Essler for stimulating corre- spondence. valeria.lante@uninsubria.it 1See for instance J. Solyom, Adv. Phys. 28, 201 /H208491979 /H20850. 2C. Kim, A. Y. Matsuura, Z.-X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa, Phys. Rev. Lett. 77, 4054 /H208491996 /H20850. 3B. J. Kim, H. Koh, E. Rotenberg, S.-J. Oh, H. Eisaki, N. Mo- toyama, S. Uchida, T. Tohyama, S. Maekawa, Z.-X. Shen, and C. Kim, Nat. Phys. 2, 397 /H208492006 /H20850. 4T. E. Kidd, T. Valla, P. D. Johnson, K. W. Kim, G. D. Gu, and C. C. Homes, Phys. Rev. B 77, 054503 /H208492008 /H20850. 5M. Hoinkis, M. Sing, S. Glawion, L. Pisani, R. Valenti, S. van Smaalen, M. Klemm, S. Horn, and R. Claessen, Phys. Rev. B 75, 245124 /H208492007 /H20850. 6S. Suga, A. Shigemoto, A. Sekiyama, S. Imada, A. Yamasaki, A. Irizawa, S. Kasai, Y. Saitoh, T. Muro, N. Tomita, K. Nasu, H.Eisaki, and Y. Ueda, Phys. Rev. B 70, 155106 /H208492004 /H20850. 7H. Matsueda, N. Bulut, T. Tohyama, and S. Maekawa, Phys. Rev. B 72, 075136 /H208492005 /H20850.8S. Sorella and A. Parola, J. Phys.: Condens. Matter 4, 3589 /H208491992 /H20850. 9F. H. L. Essler and A. M. Tsvelik, Phys. Rev. B 65, 115117 /H208492002 /H20850. 10The limitation to intermediate and strong coupling is a technical one: at weak coupling size effects become more relevant and theadiabatic procedure employed in this work might fail. 11E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 /H208491968 /H20850. 12F. Woynarovich, J. Phys. C 15,8 5 /H208491982 /H20850. 13F. Woynarovich, J. Phys. C 16, 5293 /H208491983 /H20850. 14F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, The One-Dimensional Hubbard Model /H20849Cambridge University Press, Cambridge, 2005 /H20850. 15F. D. M. Haldane, Phys. Rev. Lett. 60, 635 /H208491988 /H20850. 16F. D. M. Haldane and M. R. Zirnbauer, Phys. Rev. Lett. 71, 4055 /H208491993 /H20850. 17In one dimension and exactly at U=/H11009all spin conﬁgurations are degenerate: however in the U→/H11009limit the spin degeneracy is(b) k/πE/t 0 1/4 3/4 1−3.211.60 (a) FIG. 7. /H20849Color online /H20850Left panels: A/H20849k,/H9275/H20850calculated via Eq. /H208494/H20850for three representative values of the momentum kandU=7t. The binding energy is E=−/H9275. The lower curves are the convolution of the spectral function with a Gaussian with FWHM equal to 0.12 t corresponding to an experimental resolution of 60 meV3. Right panel: density plot of the spectral function on the /H20849k,E/H20850plane with the experimental broadening.SPIN-CHARGE DECOUPLING AND THE PHOTOEMISSION … PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-7lifted and the ground state of the Heisenberg model is singled out. 18M. Ogata and H. Shiba, Phys. Rev. B 41, 2326 /H208491990 /H20850. 19With the notation adopted in this work, at U→/H11009,/H9280s/H20849Q/H20850 =J/H9266 2/H20841cos/H20849Q/H20850/H20841for/H9266 2/H11021Q/H110213/H9266 2. 20Our deﬁnition of the spinon momentum Q, which for U→/H11009 must reduce to the total momentum of the Heisenberg wavefunction in Eq. /H208493/H20850, differs from the one usually adopted in Bethe Ansatz studies /H20849p s/H20850/H20849Ref. 13/H20850:Q=3 2/H9266−ps. 21S. Sorella and A. Parola, Phys. Rev. Lett. 76, 4604 /H208491996 /H20850; Phys. Rev. B 57, 6444 /H208491998 /H20850. 22At the special values of kdeﬁned by the coincidence of the two singularities the critical exponent changes from 1/2 to 3/4.23In experiments the lower branch is not detected because it usu- ally overlaps with other valence bands of the compound. 24The accuracy of the single spinon approximation is similar also at weaker coupling: for U=4tthe completeness condition /H208497/H20850is violated at most by 0.015 at all k’s. 25Z. V. Popovi ć, V. A. Ivanov, M. J. Konstantinovi ć, A. Cantarero, J. Martínez-Pastor, D. Olguín, M. I. Alonso, M. Garriga, O. P.Khuong, A. Vietkin, and V. V. Moshchalkov, Phys. Rev. B 63, 165105 /H208492001 /H20850. 26A. Koitzsch, S. V. Borisenko, J. Geck, V. B. Zabolotnyy, M. Knupfer, J. Fink, P. Ribeiro, B. Büchner and R. Follath, Phys.Rev. B 73, 201101 /H20849R/H20850/H208492006 /H20850.VALERIA LANTE AND ALBERTO PAROLA PHYSICAL REVIEW B 80, 195113 /H208492009 /H20850 195113-8
PhysRevB.42.2893.pdf	PHYSICAL REVIEW B VOLUME 42,NUMBER 5 15AUGUST 1990-I Radiative recombination processes ofthemany-body statesinmultiple quantum wells R.Cingolani andK.Ploog MaxPla-nckIns-titutfurFestkorperforschung, Postfach800665, D70-00Stuttgart 80,FederalRepublicofGermany A.Cingolani,*C.Moro,andM.Ferrara Dipartimento diFisiea,Uniuersita degliStudidiBari,I-70100Bari,Italy (Received 20February 1990) Theradiative recombination processes oftheelectron-hole plasma inaseriesof GaAs/Al„Ga, „Asmultiple-quantum-well (MQW)heterostructures havebeenstudiedbymeansof space-resolved high-excitation-intensity luminescence andoptical-gain spectroscopy. Thespontane- ouselectron-hole plasmaemission dramatically changes depending ontheactualMQWheterostruc- tureconfiguration, whichdetermines thedegreeofopticalconfinement intheepilayer. MQWhet- erostructures consisting of100quantum wellsorgrownonthickbarrierlayers,exhibitasaturation ofthespontaneous emission andhighstimulated-emission eSciency. Heterostructures consisting of fewquantum wellsexhibittheusualhigh-energy broadening oftheluminescence duetotheprogres- sivefillingofthesubbands inthewell.Statistical arguments onthephoton-mode distribution inside theopticalcavityofasemiconductor laserqualitatively accountfortheobserved spectral features. Theresultsofspace-resolved luminescence showthattheemission spectraofhighlyexcitedquan- tumwellsaremostlygivenbythespectral superposition oftheelectron-hale plasmaemission from thecenteroftheexcitedregionandofexcitonic luminescence originating fromthelateralregionof theexcitedspot,wherethecarrier density islower.Thesefindings areexplained byasimple diffusion modeltakingintoaccount thedriftofcarriers intheplasma. Theoptical-gain spectraof suitably designed heterostructures allowustodetermine theband-gap renormalization asafunction ofdensityofthephotogenerated carriers. Additional important information ontheground-level parameters oftheelectron-hole plasma inGaAsquantum wellsisobtained byaline-shape analysis oftheoptical-gain spectra usinganinterband recombination model. I.INTRODUCTION Theexploration ofmultiple-quantum-well (MQW) semiconductor lasershasintroduced important improve- mentsascompared withconventional heterostructure lasers,namely"band-gap engineering" fortunable emis- sionwavelengths, lowerstimulation threshold, andsupe- riorroom-temperature performance. Asaconsequence, agreatdealofefforthasbeendevotedtotheinvestigation oftheinteracting many-body statesinlow-dimensional semiconductor structures, inordertounderstand the physical mechanisms underlying theradiative recombina- tionprocesses inMQW's.'Fromthetheoretical pointof view,thesestudieshaveledtotheidentification ofseveral characteristic features ofthetwo-dimensional (2D) electron-hole plasma(EHP),including theeffectsofthe reduced dimensionality onthescreening oftheCoulomb interaction andontheband-gap renormalization, the effectsofthePauliexclusion principle onthecarrierband fillingandontheexcitonic phase-space filling, andthe largeexcitonic enhancement duetotheincreasing elec- troncorrelation inthepresenceofahigh-density carrier population intheMQW's. Fromtheexperimental point ofview,thecounterparts oftheseeffectshavebeeninves- tigated, withspecialattention paidtoGaAsMQWs,by meansofluminescence (see,e.g.,Ref.4—13)andoptical transmission experiments intheexcite-and-probeconfiguration.''Mostofthephotoluminescence data obtained fromhighlyphotoexcited MQWshaveindicat- edtheprogressive fillingofthehigher-energy quantized states.Thisfillingresultsincharacteristic broadening on thehigh-energy sideoftheemission spectra whenthe densityofphotogenerated electron-hole (e-h)pairsisin- creased.''Inaddition, stimulated emission hasbeen observed atenergies corresponding tothen=1and2 statesinphotopumped quantum-well structures.''" Incontrast totheseobservations, otherexperiments on highlyphotoexcited GaAsMQW's haveshownasharp emission bandarisingabout10meVbelowthefundamen- taln=1heavy-hole exciton transition, whichhasalso beenascribed toanEHPemission.''Thisassignment isconsistent withtheresultsofpump-and-probe absorp- tionexperiments carried outeitherinthepicosecond timedomain' orunderstationary conditions.'The measurements haverevealed distinct absorption —to- optical-gain crossovers (corresponding totheedgeofthe renormalized energygapandatthechemical potential of theEHPabsorption spectrum), occurring atenergies belowthefundamental n=1intersubband opticaltransi- tions. Inthispaperwepresenttheresultsofasystematic in- vestigation oftheEHPradiative recombination in GaAs/Al„Ga, „AsMQW structure under intense quasistationary photoexcitation. Theaimofthisstudyis 2893 1990TheAmerican Physical Society 2894 CINGOLANI, PLOOG, CINGOLANI, MORO, ANDFERRARA 42 twofold: First,tostudytherecombination processes in thedegenerate quasi-2D carriersystem, andsecond,to getdetailed information onthephysical mechanisms un- derlying thestimulated-emission processes inthesehet- erostructures. Wehaveperformed photoluminescence, space-resolved luminescence, andoptical-gain measure- mentsunderstationary high-intensity excitation onaspe- ciallydesigned setofsamples. Ourresultsallowustoes- tablishadirectcorrelation between theconfiguration of theMQWstructure andtheopticalproperties ofthe electron-hole plasma confined inthepotential well. Different manifestations ofthecharacteristic EHP luminescence canbeobserved depending onthedegreeof confinement oftheplasma intheheterostructure. Inad- dition, wedemonstrate thatthecarrier diffusion within thelayerplanestrongly affectsthecharacteristic line shapeoftheEHPluminescence. Atheoretical descrip- tionoftheoptical-gain spectra resulting fromtheoptical amplification ofthespontaneous luminescence quantita- tivelyaccounts fortheobserved spectral features, andit alsoprovides information ontheground-level parameters ofthetwodimensional EHP(i.e.,renormalized energy gap,chemical potential, andcarriertemperature). Thepaperisorganized asfollows: InSec.IIwe presenttheconfiguration andopticalproperties ofthein- vestigated GaAs/Al„Ga& „AsMQWheterostructures andbrieflydescribe theexperimental details. InSec. IIIAwediscusstheEHPspontaneous andstimulated- emission spectra andtheirdependence ontheMQW configuration. Theresultsofthespace-resolved lumines- cencemeasurements arepresented inSec.III8.InSec. IIICwediscusstheoptical-gain —spectroscopy data.A quantitative evaluation oftheground-level parameters of thetwo-dimensional electron-hole plasma intheinvesti- gatedmaterial systemisgivenbymeansofatheoretical analysisoftheoptical-gain spectra. Finally, inSec.IV wesummarize themainresultsanddrawourconclusions. II.EXPERIMENT Theradiative-recombination processes havebeenstud- iedinGaAs/Al„Ga, „AsMQWheterostructures grown bymolecular-beam epitaxy(MBE}onundoped (001}- oriented GaAssubstrate. Theconfigurations ofthesam- plesaresummarized inTableI.TheMQWsamples have nearlyidentical structural parameters, butanincreasingnumberofperiods(N).Ourexperiments showthatthe numberofwellsconstituting theheterostructure strongly affectstheemission properties oftheMQWunderhigh photogeneration rate,anditalsodetermines theperfor- manceofthelaseractionuptoroomtemperature. The investigated setofMQW's with10&N&200alsoin- cludesasamplewhichconsistsof25periods grownona 1-pm-thick A1Q3{jGaQ~AslayerontheGaAsbuffersam- ple(No.6).Thestructural parameters oftheinvestigated samples havebeendetermined independently byhigh- resolution double-crystal x-ray-diffraction measurements. Thesamplespecimens havetypicalsurfacedimensions of10X7mm.Insomecasessmallopticalcavitiesofsize 150pmX2 mmhavebeencleaved. Thisallowed usto reduceself-absorption lossesintheunexcited region,and alsoprovided efFicient feedback fortheoptical amplification ofthespontaneous emission, duetothe highinternal reflectivity ofthecleavedfacets. Theexcellent qualityoftheinvestigated samplehas beenestablished bymeansofphotoluminescence- excitation (PLE)andphotoluminescence (PL)measure- ments.Inallthesamples weobserve sharpexcitonic peakswithtypical fullwidths athalfmaximum (FWHM's) of1.5meUatlowtemperature and8meVat roomtemperature. Insample6anadditional sharpexci- tonpeakfromtheA1Q36GaQ~Asbarrierlayerwasob- servedat625nminthelow-temperature PLEspectrum. Inallthemeasurements theluminescence wasexcited byaN2-laser-pumped dyelaseroperating attheemission wavelength of570nm(i.e.,inthecontinuum oftheab- sorption fortheinvestigated setofsamples} andat10Hz repetition frequency. Themaximum peakpowerdensity obtained bytightlyfocusing thebeamoveraspotof100 pmdiameter wasIn=2MW/cm. Thepulseduration wasabout5ns;therefore allthemeasurements havebeen performed underquasistationary conditions. Thedetec- tionsystem consisted ofa0.6-mmonochromator equipped withafast-response photomultiplier tubeand boxcaramplifier. Twomainconfigurations havebeenadopted forstudy- ingthespontaneous-emission andoptical-gain processes ofthephotogenerated EHPinthequantum wells(QW's), respectively. First,thespontaneous luminescence was collected inthebackward direction fromtheexcitedsam- plesurface. Inparticular, thespatially resolved lumines- cencewasperformed bycollecting thebackscattered luminescence atdifferent distances (d)fromthecenterof TABLEI.Configuration oftheinvestigated GaAs/Al, Ga,„AsMQWsamples.L,andLbarethe wellandbarrierthicknesses, respectively, xistheAlmolefractionoftheAl„Ga&„Asbarrier, andXis thenumberofquantum wells. Sample no. 1(cc643699) 2("6438") 3(cc6432st) 4("6433") 5("6347") 6("6439")L,(nm) 10.6 10.6 10.6 10.6 10.0 10.6Lb(nm) 15.8 15.8 15.8 15.8 15.8 15.810 25 50 100 200 250.36 0.36 0.36 0.36 0.35 0.36Barrier layer no no no no no 1pmAlp36Gap64AS 42 RADIATIVE RECOMBINATION PROCESSES OFTHEMANY-. .. 2895 theexcitedspot.Aspatialresolution ranging between 20 and50pmwasobtained byscanning anenlarged image ofthesamplesurface (magnified byafactorof10)across theentrance slitofthemonochromator bymeansofmi- crometric displacements ofthecollecting lens.Under theseconditions thespatialresolution istherefore con- trolleddirectly bytheslitwidth,which,inturn,depends onthePLefficiency oftheinvestigated sample. Second, theopticalamplification ofthespontaneous emission was studied inaconfiguration withtheincident laserbeam paralleltothe[001]growthaxisandthedetection along theMQWplaneinthe[110]direction. Quantitative mea- surements oftheoptical gainhavebeenperformed by varyingtheexcitedstripelengthonthesampleandusing theone-dimensional optical amplificator approxima- tion.' III.RESULTS ANDDISCUSSION A.Spontaneous andstimulated emission Theresultsofthephotoluminescence measurements elucidate thecharacteristic EHPemission arisingfrom quasi-2D semiconductors underintensephotoexcitation. Theluminescence efficiency growsproportional tothe numberofperiods constituting theGaAs/AI„Ga, „As MQWheterostructure. Weobserve twodifferent mani- festations oftheEHPemission underidentical excitation conditions depending ontheMQWconfiguration. Sam- pleswithmorethan100periods(N)100)exhibit a sharpemission bandpeaked inthelow-energy tailofthe E»ztransition. Thisbandshowsaredshiftasafunction oftheexcitation intensity, and,aboveacertainthres- holdintensity, asharpstimulated emission isobtained (analogous toRefs.7,16,and17).Conversely, GaAs/Al„Ga, „AsMQWsamples withN&100donot showanyseparate EHPemission, butonlytheprogres- sivefillingofthehigh-energy statesintheQW,resulting inthewell-known high-energy emission.''Inthese casestheluminescence ispeaked around theintersub- bandtransition energies anddoesnotshowanyredshift withtheexcitation intensity. Adistinct improvement of theemission performance, resulting inveryefficient stimulated emission atroomtemperature, canbe achieved bygrowing theMQWonathickAl„Ga&„As barrier. Thisconfiguration isrealized insample6,which exhibits byfarthehighestluminescence efficiency ofthe investigated setofsamples. Thecharacteristic luminescences areexemplified inthe spectraofFigs.1(a),1(b),and2.InFig.1(a)weshowthe emission spectraofMQWno.6obtained atdifferent exci- tationintensities. Thespectrawererecorded intheback scattering configuration, andtheyarethusvirtually unaffected bytheself-absorption inthecrystal. Atthe lowestpowerdensity onlythesharpexcitonic E&&&line around799.0nmispresent. Withincreasing excitation intensity anewband(S)arisesinthelow-energy tailof theE»&line.Thisbandgrowssuperlinearly withrespect totheexcitonic line[seetheinsetofFig.1(a)]andshows aredshiftofabout15meVintheintensity rangebetween 0.1IoandIo.Inaddition, weakcontributions totheluminescence spectra fromtheGaAsbuffer(Ebband) andfromthethickAlGa&„Asbarrier layercanbe identified around820and630nm,respectively. These weakfeatures indicate anefficientcarriertrapping inthe MQWactivelayers. Whenwefurtherincrease thein- cidentpowerdensity, theSbandbecomes dominant, as depicted inFig.1(b).Verysimilarspectra havebeenob- tainedfromsamples composed ofalargenumberof QW's,i.e.,withatotalMQWthickness muchlargerthan thepenetration depthoftheexciting radiation (samples 4 and5). Whenwedecrease thenumberoftheQW'sinthesam- pleorremovethebarrierlayer,wenolongerobserve the separateSemission band.Atypicalexampleofthissitu- ationisshowninFig.2,wheretheluminescence spectra fromsample1(10QW's)aredepicted. Although these spectra arerecorded underidentical conditions asthe spectraofFigs.1(a)and1(b)andfromGaAsquantum wellsofidentical well(L)andbarrier(Lb)widths,the emission lineshapesaretotallydifferent. First,astrong emission arisesonthehigh-energy sideoftheE»zband astheexcitation intensity isincreased, whichclearly reflectstheeffectofthePauliexclusion principle onthe photogenerated carrier population. Second, astrong luminescence fromtheGaAsbufferlayerisobserved. Thisclearlyindicates thatalargeportionoftheexciting lightisnotabsorbed bytheMQW,owingtothepenetra- tiondepthoftheradiation beinglongerthantheMQW thickness. Theimpactofthediscussed luminescence features on thestimulated-emission spectraoftheinvestigated sam- plesisexemplified inFigs.3(a)and3(b).Thespectraare obtained intheconfiguration withtheexciting laserbeam perpendicular tothesample edge.InFig.3(a)the stimulated-emission spectraat10Kofsample 5with200 periodsareshownforpumping-power densities ranging from0.002IOtoIo(4kW/cm upto2MW/cm ).Atan excitation intensityofabout0.005Io,theSbandarisesat 802.3nrn(1.545eV),about6meVbelowtheE„zexci- tonicemission, anditbecomes dominant atthehighest excitation levels.TheSlinewidth narrows downto2 meVat0.01IO(20kW/cm ).Whentheexcitation levelis increased, itsemission intensity growsexponentially up to0.05IO(100kW/cm ).Thenitbroadens andsaturates asIoisfurtherincreased upto2MW/cm.Anapprecia- bleredshiftoftheSlineofabout5meVisobserved by increasing theexcitation intensity. Thebuffer-layer emis- sion,locatedat820nm,isobserved onlyatthemaximum powerintensity. Thedependence onthepumpintensity forboththeSandE»&bandsisshownintheinsetof Fig.3(a).Thestimulated emissionSdepends exponen- tiallyontheexcitation intensity, whiletheexcitonic emis- sionexhibitsasublinear dependence. Thestimulated-emission spectraofsample6arevery similartothosemeasured insample5.Inparticular, in sample6thestimulated-emission threshold isabout 1or- derofmagnitude lowerthanthatofsample 5andthe stimulated Sbandispresent evenatverylowexcitation intensity. Thelowest-excitation-intensity thresholds for theoccurrence ofstimulated emission measured inthese twosamples are0.2kW/cm forsample6and11.2 CINGOLANI ,PLOOG, CINGGOLANI, MORO0,ANDFERRARA 42 Z', O vl lOO— Clo- +ol~ UJ Zp p~ pIp0 p / QQlO.l I/Ip AlxGal1.55 I E»nIENERGY (eV) 1.525 150 1 No.6 10K .1Io JD a5Ip I— V) UJI— K3Ip 5Ip 1Ip 630635780 810 790800810 WAVELENGTH810820830 (nm) 1.55 I sample No.6 MT=10K C Io1.53ENERGY (eV) 1.55 0.3Ip1.53 11.55 Ellh1.53 0.03Io O V) LU 795I Ix2.5 800805810I I I 815795800805810 WAV795800805810 FIG.1.(a)Su-ElENNGTH(nm) 10K~)Poo-P o underexcitat'aionintensit0. 1IFobtt 3III0'y)od' p'ngintensitfhbeinba eenmultiplied ba e iebyacon- »&and wn.b)Sameasin(a), 42 NERGY (eV)pRQCESS ESQFTHEMANY- ~~ EGQMBINATIQ IATIVER 2897 ~gpRgy (evI 1.8"-7 I61.5 1Ellh~Eb sample Np. I !T=]OK Ip1565 so~pl~ T=10& (aj1.542 VlZ 10I—zC Szo10&Qt-tf)— Ul UJ1h 10-21 I~~oZ' LLj O x2 X x7O.50&o O,3QI0 O.~0Ip (}.OtIo I I 7OO coo YYAVELENGTH( ](]Owells)obtained un-'sectraofsample F&G.2.Emission P''asthoseofFigl. itationconditions as derthesameexcitaiV)Z LU z O 0~ lUO.O)1o 0.00210 I I I 79279880x]0 x1300 x1000 x5000 I I I 820830 oomtemperature le5at10K,whileatroom P (RT)weobtamj le5. at10KofP 13)odd ii&o0.01I atioitensi oth2 excitation PP PP'g gypo aklsuperlinear. Itro e sstan''htltowards there. alsthe fthedataofFigs.— as Comparison o sconsisting oa'mortantresu p inAl„Ga&„s 11 paewico'' 'R PP Rintheige- ofelectrons an emission intensity r R 1'1 1 tht Wintensity a n ndataraten hcontrary, excion E tnemission. orwit'thaconfining hX1M nd-filling emis- ho b AlGa) h1 ndont eo transition. TheSan thefundarn 1h0 'b thtot1 16 nbothcasesehfo cturemust MWhtotin mainthe minescenceeecr thobd1 b'hexlaine msre-beinvo e 'e-recombina io nor1' ortheSemission, po -1tronscattering p exciton-e ecrWAVELENGTH {nm ENERGY {eV) 1.5511.529 I I sample No.3 T=)OK LRZ: UJ Z O X LUEb 799 822 WAVELENGTH (nm nsectraofsample5(2OO imulated-emission spec inintensities. '1'dbyaconstatac aaremultipie onumping inen'hdd figure. ei S(triangles . ecr forboth edat10Kun er sample3'db aremultiplie y Thespectra are intensities. glviveninthefigure. 2898 CINGOLANI, PLOOG, CINGOLANI, MORO, ANDFERRARA 42 R,(E)[1—exp(EbF)/kT]=N(E— )/t (2) whereAFisthedifference inelectron andholequasi- fermi-levels oftheEHP,andkTisthethermal energyof thecarriers. FromEq.(2)wededucethatthecondition atwhichR„(E)approaches N(E)/t alsoestablishes an upperlimitforthesteady-state Fermi levelofthe electron-hole plasma, thusgivingasaturation ofthe spontaneous emission. Undertheseconditions, anyin- creaseoftheemission rateoccurs inthestimulated- emission channel. Themechanisms discussed explain ourexperimental findings. MQWstructures inwhichopticallossesinthe activelayerarelargedonotshowopticalamplification, andtheEHPspontaneous luminescence exhibits the characteristic band-filling behavior. Thisisthecasefor theMQWsamples withthinactivelayers(samples 1—3),onthebasisofmagneto-optical measurements performed undersimilar excitation conditions. Detailsofthe magneto-optical resultswillbegiveninaforthcoming pa- per20 Thedifferent luminescence properties obtained from MQWheterostructures consisting ofeitherafewora largenumberofwellsand/or havingaconfining barrier layercanbeexplained onthebasisofthephotondistribu- tionintheopticalcavityofasemiconductor-laser struc- turewithcleaved ref(ecting faces.'Thenumberofquan- taMinasinglemodeoftheradiation fieldwithintheac- tivelayerisgivenby R,p(E)M= N(E)/rM—RBt(E) whereR,(E)andR„(E)aretheratesofspontaneous andstimulated transitions atdifferent energiesE,respec- tively,N(E)isthenumberofmodesperunitvolume and unit-energy interval, and1/tisatermrepresenting the photon lossesinamodeofenergy F.duetoself- absorption inthecrystal, transmission, andscattering (Q factorofthemode}.Formodeswithhighlosses(smallQ factor},thevalueofR„(E)canhardlyapproach thatof N(E)/t.Therefore, thedenominator ofEq.(1)islarge, resulting inasmallnumberofquantaofenergyEinthe givenmode.Thiscondition isusuallyachieved whenthe spontaneous luminescence emitted intheactivelayercan beabsorbed bytheunderlying GaAsbufferlayerdueto thepooropticalconfinement ofthestructure. Onthe contrary, formodeswithlowlosses(largeQfactor)even asmallR«rate canapproach theN(E)/t factor,result- inginavanishing denominator forEq.(1),i.e.,inavery largenumberofphotons inthelow-loss mode.Inthis caseasharpstimulated-emission peakappears inthe luminescence spectrum ofthecrystal. Inaphenomeno- logical waythismeans thatalltheradiative- recombination transitions inthecrystaloccurthrough thelasingchannel owingtothelargeQvalueofthecor- responding fractionofmodes.Thelimitfortheexistence ofalargeamountofstimulated emission isgivenwhen theconditionF(E)/t=R„(E) isfulfilled inEq.(1).By usingtheexisting relation between thespontaneous- and stimulated-emission rates,'thiscondition becomesinwhichself-absorption ofthespontaneous luminescence occursintheGaAsbufferlayer(leakages ofphotogen- eratedcarriers intheunderlying substrate insamples whoseactivelayerthickness isshorterthanthepenetra- tiondepthoftheexciting radiation shouldalsobecon- sidered anadditional lossmechanism). Infact,inthese samples saturation ofthespontaneous emission canhard- lyoccur,andthecharacteristic EHPluminescence mani- festsitselfmainlythrough thewell-known band-filling spectra. However, theMQWheterostructures witha thickactivelayerprovide alargeopticalconfinement of theemitted luminescence, whichreduces themodellosses inthecavity, thusresulting inthesharpstimulated- emission band.Infact,intheMQWsamples withalarge numberofwells(samples 4and5)theeffective penetra- tiondepthoftheexciting radiation reasonably involves onlythefirst20periods. Therefore, theunderlying slabs actlikeanopticalconfinement layercharacterized bythe averagerefractive index: n,„=[NnL+(N—1)nbLb]/[NL +(N—1)Lb], (3) where n=3.6andnb=3.6—0.7xaretherefractive indexesofGaAsandAl„Ga~ As,respectively. Inthe caseofsamples 5and4,Eq.(3)results inarefractive- indexdiscontinuity oftheorderof4%,whichissufficient togivesomeopticalconfinement oftheluminescence. In thecaseofsample6,thiseffectisenhanced bythepres- enceofthethickAl„Ga,„Asbarrierlayer,whichpro- videsaveryefficient optical confinement andprevents self-absorption ofthespontaneous luminescence andcar- rierleakageintotheGaAsbufferlayer.Theobservation ofthestimulated Sbandintheluminescence spectra fromthethickMQWconfiguration cantherefore beex- plainedasaconsequence ofthereduction ofthelossesof thespontaneous-emission photons intheMQWactive layer,resulting inthestrongreduction ofthedenomina- torofEq.(1).Suchacondition ismainlyachieved inthe edgeoftherenormalized energygap,wheretheself- absorption isstrongly reduced. Therefore, thestimulated emission isexpected tooccuratanenergybelowthefun- damental E»ztransition andtored-shift withincreasing excitation density. Thisredshiftfollows thedensity- induced band-gap shrinkage, aswedoindeedobservefor theSbandinthespectraofFigs.1(a},1(b),and3(a).In addition, aspreviously discussed, intheseMQW configurations wedonotobserve thecharacteristic band-filling emission lineshapeduetothesaturation of thespontaneous emission. Itisworthnotingthatthisre- sultisconfirmed bytheobservation ofashortening ofthe luminescence decaytimeabovethestimulation threshold. Gobeleta/.haveshownthatinthepresence ofstirnu- latedemission theincreaseoftheexcitation intensity leadstoafurtherincreaseoftheopticalarnplification and toareduction ofthecarrierlifetime, whichindicates the saturation ofthespontaneous emission. B.Space-resolved luminescence Another important featureofthephotolurninescence spectra discussed inthepreceding subsection isthatthe RADIATIVE RECOMBINATION PROCESSES OFTHEMANY-. .. 2899 difFusion ofthephotogenerated carriers alongtheQW basalplanestrongly affectsthelineshapeoftheemission spectra underhighexcitation intensity. Thecareful studyoftheemission spectra bymeansofspatially resolved luminescence measurements allowsustodemon- stratethatthelineshapeoftheEHPluminescence ob- servedathighcarrierdensity isderived fromaspectral superposition ofexcitonic andfree-carrier luminescence originating fromdifferent lateralregionsofthecrystal wheretheactuale-hdensitycanbetotallydifferent. Understationary conditions itisexpected thatthepho- togeneration ofadenseEHPinthecrystalresultsinthe ionization ofexcitons duethescreening oftheCoulomb interaction. Severaltheoretical investigations ofthissub- jectpredictthatatane-h-pair densityoftheorderof10"cmtheexcitons areionized, andthisgivesriseto radiative recombination attheedgeoftherenormalized bandgap.Ontheotherhand,theobserved EHP luminescence spectraclearlyexhibitapeaklocatedatthe exciton transition, evenatveryhighexcitation intensi- ties,whenthephotogenerated e-hdensity isconsiderably largerthan10'2cm.Thiscanbeclearlyobserved not onlyinthespectraofFigs.1and2,butalsoinmanyex- perimental resultspublished previously.'Theuseof spatially resolved luminescence reveals thatthis phenomenon isduetothespectral superposition of different radiative-recombination processes originating in different lateralregionsofthecrystalwheretheactual carrierdensities differstrongly. Thisimportant aspectis exemplified inthespectraofFigs.4and5. InFig.4wedepictthespatially resolved luminescence ofsample6,takenatseveral difFerent distances dfrom thecenteroftheexcitedspot.Thelateralresolution is20 pminthiscase.Asshowninaprevious paper,'the EHPemission canbeobserved onlyinthecenterofthe excitedspot,wheretheactualcarrierdensityisthelarg- est.Bydisplacing thedetection farawayfromthecenter ofthespot,weobserve anoveralldecreaseofthetotalin- tegrated emission intensity andadramatic decrease of theEHPemission, whichreducestoabout10%ofthe d=0valueatadistanceofabout500pmfromthespot center. However, theE»zlinedoesnotshowsignificant changes, even700pmawayfromthespotcenter.From thesefindings wededucethattheexciton luminescence comesfromtheunexcited partofthesamples surround- ingtheexcitation region, wherethecarrier density is strongly reduced. Similareffectsarealsoobserved forthe band-filling luminescence ofMQWsample 2withthe smallnumberofwellsshowninFig.5.Again, weob- serveasharpexcitonic emission inthespace-integrated measurement whichoverlaps theband-filling spectrum originating fromfree-carrier recombination involving different subbands. Byselecting thecenterofthespot withapinholeof200pmdiameter, weobservethesharp decreaseoftheexcitonrecombination, whilenochanges occurinthefree-carrier emission spectrum. Theobservation ofluminescence upto800pmaway fromtheexcitedspot(whosediameter isabout100pm} indicates astrong in-plane EHPexpansion. Many theoretical modelshavebeenproposed toexplainthecar- rierdriftindenseelectron-hole —plasmastates,andre-1.55ENERGY (eV) 1.53 ,d1.51 Ih C C) Lcj I— M UJI—Z ZO V) Z IU I 800 810 WAVELENGTH (nm)I 820 FIG.4.Spatially resolved luminescence ofsample6(25wells andA1036Ga064As barrierlayer)at10Kunderthemaximum excitation intensity recorded atdifferent distances dfromthe excitingspot. 1.83 IENERGY (eV) 1.73 1.63 I I1.53 I Vl ~~C J3 I Cf V) LUz zO V) V) UJsample No.2 T=10K WITHOUT PINHOLE WITHPINHOLEE11h Eb I I I I I I I I I I I I I I I I 660680700720740760780800820 WAVELENGTH (nrn) FIG.5.Emission spectraofsample2(25wells)obtained by collecting theluminescence through apinholeof200pmdiam- eterplacedinthecenteroftheexcitedspot(dashed line),and withoutapinhole(solidline). 2900 CINGOLANI, PLOOG, CINGOLANI, MORO, ANDFERRARA 42 centlyadirectmeasurement oftheexpansion ofa relatively-low-density EHPinGaAsquantum wellshas beenreported. Themainconclusion ofthosestudiesis thatthecarriertransport isthermodiffusive andoccurs alongthelayerplane.Themeasured driftoftheorderof tensofIMmhasbeenmonitored withatime-resolved prob- ingtechnique. Inourexperiments, however, thephoto- generation rateisunderquasistationary conditions and thedetection istimeintegrated. Theobservation ofthe largeEHPexpansion isprobably relatedtosomeaddi- tionaldriftmechanism whichdepends onthee-hdensity. Infact,thepurelydiffusive driftlengthofcarriers in GaAsisoftheorderof&Dr=3. 5Jum(DistheGaAs diffusion coefficient equalto120cm/s,andristhe electron recombination timeoftheorderof1ns),which cannotaccountforourexperimental observation. Apos- siblemechanism whichexplains theobserved strongcar- rierexpansion isincreaseoftheFermipressure inthe nonequilibrium EHP. Undertheseconditions thecon- tinuityequation forthechargedensity n(x,t)inspace andtimeoftheEHPcanbewrittenas dnDdn+dn n dtdx~dx whereg(x)isthecarrier-photogeneration rate, Uisthe driftvelocityoftheplasma, andvistherecombination time.Theanalytical solutionofEq.(4)isaGaussian car- rierdistribution centered atx(t)=ut, withamplitude [g(0)/(2&nDt )]exp(t/r)dec—reasing intime.2Itis therefore evidentthatthisadditional carrier-drift mecha- nismdisplaces thecarrierdistribution atdistance Utfrom thecenteroftheexcitedspot.Thiseffectcanusuallybe neglected atlowcarrierdensity, butitbecomes important attheEHPdensities involved inthepresentexperiments. Undertheseconditions, wecanassumeavelocityofthe orderoftheFermivelocity forbothcarrier species (U)10cm/s).'Theresultofthisroughestimate is thatthecarrierdistribution caneasilybedisplaced by 100pmfromthespotwithinonecarrierlifetime. Anadditional spatialinhomogeneity intheEHPdensi- tycanarisefromintercarrier scattering processes which broaden thecarrier distribution. Theexpansion and broadening ofthecarrierpopulation causesaninhomo- geneous carrierdistribution withadensity profilede- creasing atlargedistances fromtheexciting spot.Atthe external edgeoftheexpanding EHPregion,thecarrier density ishencemuchlo~er.Thiscausesareduction of thescreening andallowsfortheformation ofexcitons at largedvalues.Theabovearguments implythatthespa- tialexpansion andinhomogeneity ofthecarrierdistribu- tionstrongly affecttheluminescence lineshapeofthe electron-hale plasma. Thedescription ofthiseffectmade bymeansofEq.(4)allowsustoexplain qualitatively mostoftheobserved spectral featuresofFigs.4and5.relation existing between theactualconfiguration ofthe MQWheterostructure andtheradiative-recombination processes intheconfined electron-hole plasma. An efficient opticalamplification ofthespontaneous emission caneasilybeachieved inMQWheterostructures havinga largenumberofwellsorgrownonabarrierlayer.Inor- dertoaccount quantitatively fortheopticalamplification capability ofthesamples investigated, wehaveperformed optical-gain measurements byvarying thelengthofthe exciting stripeonthesamplesurface. InFig.6weshowtheunsaturated optical-gain spectra ofMQWsample 5with200periods obtained at10and 300K.Thegainbandsarelocatedalmostattheenergy oftheSband,showing amaximum gvalueofabout1600 and30cm'at10and300K,respectively. Similar optical-gain spectra havebeenobtained fromsample4 (N=100},which shows alessefficient optical arnplification atlowtemperature. Averyefficientoptical gainis,however, observed insample 6(grown onan Al„Ga,„Asbarrier layer),asexpected fromthecon- clusionofSec.IIIA. Theunsaturated optical-gain spectraofsample6ob- tainedat10Kfordifferent photogeneration ratesarede- pictedinFig.7.Theelectron-hole —pairdensities nare calculated fromthewidthofthegainspectraaccording to n=(kT/%co)gmlnI1+exp[(F& EJ)/kT]I— ,(5) J 1.555ENERGY (eV) 1.534 1.4381.431 } } 1600—sample No.5 T=)OK 1200— E 800— Ch40—T=300K 30— 20— 400— 10—wherejindicates thelight-andheavy-hole contributions, EJarethesubband energies, andF&isthequasi-Fermi- leveloftheholepopulation (asimilarformula isvalidfor theelectrons). Themaximum gainvalueispeakedal- mostattheenergyoftheSbandanditreaches5600 cm'at10Kandbecomes 250cm'at300K. Inspection ofthedataofFig.7impliesthattheenergy positionofthemaximum gvalueisalmostindependent of C.Optical-gain spectroscopyI'793797801805858862866 WAVELENGTH (nm} Theresultsofthephotoluminescence measurements discussed inthepreceding subsections evidence thecloseFIG.6.Unsaturated optical-gain spectraofsample 5(200 wells)obtained at10and300K.Thespectrum at300Khas beenmultiplied byaconstant factorindicated inthefigure. 42 RADIATIVE RECOMBINATION PROCESSES OFTHEMANY-. .. 2901 6000— 4000— 2000— 0—-'-, 4000— —2000— C7l 4000— 2000— 0—\ I 1 l I I Isample No.6 5x1P1lcw-2 n=3X1011C~-2////IIIIIII 4X1011c~2 I I I I I I I I I I I(6) I(g,f)=dE;~(E)/2~ 1(E)/4+(~'co —E)' andcomparable withtheexcited stripelength(about100 I'm).InFig.8weshowtheresultsofaleast-squares fit carriedoutontheexperimental unsaturated optical-gain spectrum obtained fromasmalloptical cavity(150 I'mX2mmsurfacesize)cleavedfromsample6.Thegain curvepeaksalmostattheenergyoftheSbandandexhib- itsamaximum gvalueof7860cm'.Theoptical-gain spectrum g(fico)isgivenby I(g',Ace),fico&F, g(%co)=I(g',%co)+g'(fico) fico&F, I I 1.53 1.54 1.55 PHOTON ENERGY (eV) FIG.7.Unsaturated optical-gain spectrum ofsample6(25 wellsandA1Q36GaQ64As barrier layer)obtained at10Kand different carrierdensities n.Neglecting broadening effectsin thespectra, thecrossovers indicate theenergypositions ofthe renormalized energygap(Eginthelow-energy side)andofthe chemical potential (pinthehigh-energy side).Thedashedlines areguidesfortheeyes,indicating thedensity-dependent shiftof thecrossovers. thee-hdensity, whereas thecrossover points(wherethe optical gainchanges toabsorption} shiftinopposite directions. Inparticular, byincreasing thecarrierdensi- ty,thechemical potential p,(thehigh-energy crossover) shiftstohigherenergyandtherenormalized energygapE'(thelow-energy crossover) shiftstolowerenergy.The estimated band-gap shrinkage rangesfrom27rneVat 3X10"cmto36meVat7X10"cm.Thesevalues areinquantitative agreement withtheexperimentalre- sultsofWeberetal.'obtained inpump-and-probe ab- sorption experiments, andtheyareonlyslightly larger thanthetheoretical valuescalculated byHaugetal.and DasSarmaetal.inRef.2.Thepresent resultsindicate thattheband-gap renormalization caneasilybestudied bymeansofsimpleluminescence measurements insam- pleshavingaconfiguration suitableforhighopticalgain. Thisopensupthepossibility ofsystematic experimental investigations ofthewell-width dependence oftheband- gaprenormalization without experimental complications arisingfromtheapplication ofpump-and-probe methods. Moredetailed information ontheground-level parame- tersoftheEHPintheQWcanbeobtained bytheline- shapeanalysisoftheoptical-gain spectra. Thecalcula- tionofthetwo-dimensional EHPluminescence lineshape canbeperformed byusingaband-to-band recombination modelaccounting formomentum conservation. Inaddi- tion,energy- anddensity-dependent lifetime broadening istakenintoaccount. Thequantitative analysis has beencarriedoutontheunsaturated optical-gain spectra measured insample specimens whosedimensions areg'(E}=(A/2)gg(4nmo/f ) XH(EEE—„'E„')(—f„'—f„"),— where Aisaconstant,f„'andf„'aretheoccupation fac- torsofthenthsubband,H(x)istheHeaviside unit-step function, Iisthebroadening parameter, andtheother symbols havetheirusualmeanings. Thecalculation reproduces theexperimental spectrum verywell,with best-fit parameters I=20meVandtherenormalized bandgapE'=1.532eV.Thetotalquasi-Fermi-level of thethree-component electron—heavy-hole and—light-hole population obtained bythefittingprocedure amounts to 17meV,whichcorresponds toacarrier densityof 4.7X10" cmattheeffective temperature of20K. Fromthesedatawederiveaband-gap reduction of29 meVatthee-hdensityof4.7X10"cm,inagreement withtheresultsofFig.7.Thesecalculations demonstrate thattheunsaturated optical gainofthehighlyexcited sample No.6 8000—T=10K 6000— I E4000— 2000— I I I I 1.528 1.536 1.544 1.552 PHOTON ENERGY (eV) FIG.8.Experimental (dots)andtheoretical (solidline)unsa- turated optical-gain spectraofanopticalcavitycleaved from sample6(25wellsandA1Q36GaQ64Asbarrierlayer). 2902 CINGOLANI, PLOOG, CINGOLANI, MORO, ANDFERRARA 42 quantum wellsoriginates fromtheopticalamplification oftheluminescence emitted byinterband transitions (i.e., radiative-recombination processes byfreecarriers inthe EHP).Inaddition, theapplication ofthesimpletheoreti- calmodelofEqs.(6)—(8)allows ustodetermine the ground-level parameters oftheEHPstateintheQW, whichprovides important information onthecarrierdis- tribution responsible forlaseractioninthesemiconduc- tor.Itshouldbenotedthatthisinterband recombination modeldescribes theobserved optical-gain spectra wellin awiderangeofe-hdensities and/orlengthsoftheexcited stripeonthesamplesurface. Deviations fromtheinter- bandrecombination mechanism areobserved onlyatvery largelengthsoftheexcitedstripewhentheopticalgain saturates. IV.CONCLUSIONS Wehavediscussed severalfundamental aspectsofthe radiative-recombination processes occurring inhighlyex- citedquantum wells.Thesystematic investigation ofthe EHPluminescence andoptical gaininasetof GaAs/Al„Ga, „AsMQWheterostructures ofdifferent configuration allows ustodrawthefollowing con- clusions: First,thecharacterized luminescence ofthe electron-hole plasma inMQWsamples depends onthe actualheterostructure configuration. Inparticular, band-filling luminescence causedbytheprogressive popu- lationofhigher-energy subbands inthewellisobserved inMQWsamples consisting ofonlyafewperiods. Con- versely, sharpEHPluminescence occurring belowthe fundamental E»zinterband transition isobserved in MQWsamples consisting of100periods ormoreor grownonathickAlGa&,Asbarrierlayer.Thelatter configuration isalsomostadvantageous inobtaining stimulated emission. Thisapparent discrepancy inthe manifestation oftheEHPluminescence isduetothede- greeofopticalconfinement oftheluminescence insidethe heterostructure. Aqualitative explanation ofthiseffect hasbeengivenintermsofstatistical distribution ofpho- tonmodesinsidetheopticalcavityofasemiconductor laser.InthickMQWheterostructures providing highop- ticalconfinement oftheluminescence, thespontaneous emission saturates atrelatively lowexcitation intensity, resulting insharpstimulated emission atanenergywhere theself-absorption lossesarenegligible (i.e.,aroundthe edgeofthebandgap).Onthecontrary, inthinMQW samples providing lowopticalconfinement thesaturation ofthespontaneous emission hardlyoccurs, resulting inthewell-known band-filling luminescence arisingonthe high-energy sideoftheE»&transition. Thesefindings al- lowustounderstand apparent discrepancies existing in theliterature amongtheresultsofdifferent spectroscopic investigations oftheEHPinMQWheterostructures. Thesecondinteresting resultconcerns thecorrect in- terpretation oftheEHPluminescence lineshapeobtained fromtheexperimental spectra. Ourmeasurements ofthe space-resolved luminescence indicate thatinstandard luminescence experiments theEHPluminescence isgiven bythespectral superposition offree-carrier andexcitonic emission originating indifferent regionsofthecrystal surface, wherethedensityofcarriers istotallydifferent. Inparticular, thefree-carrier luminescence arisesfrom thecenteroftheexcitedspotwherethecarrierdensity is thehighest, whiletheexcitonic emission originates far awayfromthespotcenter,wheretheactualcarrierdensi- tyisreduced bythestrongEHPexpansion. Atthecar- rierdensitiesofourexperiments thisphenomenon canbe described byadiffusion modeltakingintoaccount the carriers' driftinthenonequilibrium plasma. Finally, theground-level parameters oftheelectron- holeplasmaconfined intheMQWheterostructure (renor- malized bandgap,chemical potential, andcarriertem- perature) havebeendetermined bystudying theoptical- gainspectraoftheEHP.Thedependence oftheoptical gainonthee-h—pairdensityprovides quantitative infor- mationontheband-gap renormalization inthepresence ofadensecarrierpopulation. Theground-level parame- tersofthecarrierdistribution havebeendetermined by calculating theoptical-gain lineshapewithaninterband recombination model. Theresultsofthecalculations confirm thatthemainradiative-recombination channel is theinterband free-carrier recombination inthedense electron-hole plasma. ACKNOWLEDGMENTS Wegratefully acknowledge A.Fischer andM.Hauser forexperthelpwithsamplegrowthandR.Muralidharan forpreparation oftheopticalcavities. WethankE.O. Gobelformanyilluminating discussions andsuggestions, andL.Tapferforx-ray-diffraction studies. 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PhysRevB.99.115443.pdf	PHYSICAL REVIEW B 99, 115443 (2019) Graphene: Free electron scattering within an inverted honeycomb lattice Z. M. Abd El-Fattah,1,2,M. A. Kher-Elden,2I. Piquero-Zulaica,3,4F. J. García de Abajo,1,5,†and J. E. Ortega3,4,6,‡ 1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2Physics Department, Faculty of Science, Al-Azhar University, Nasr City, E-11884 Cairo, Egypt 3Centro de Física de Materiales (CSIC-UPV-EHU) and Materials Physics Center (MPC), 20018 San Sebastián, Spain 4Donostia International Physics Center, Paseo Manuel Lardizábal 4, E-20018 Donostia-San Sebastián, Spain 5ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys 23, 08010 Barcelona, Spain 6Departamento de Física Aplicada I, Universidad del País Vasco, 20018 San Sebastián, Spain (Received 13 December 2018; revised manuscript received 28 February 2019; published 29 March 2019) Theoretical progress in graphene physics has largely relied on the application of a simple nearest-neighbor tight-binding model capable of predicting many of the electronic properties of this material. However, importantfeatures that include electron-hole asymmetry and the detailed electronic bands of basic graphene nanostructures(e.g., nanoribbons with different edge terminations) are beyond the capability of such a simple model. Here weshow that a similarly simple plane-wave solution for the one-electron states of an atom-based two-dimensionalpotential landscape, deﬁned by a single ﬁtting parameter (the scattering potential), performs better than thestandard tight-binding model, and levels to density-functional theory in correctly reproducing the detailedπ-band structure of a variety of graphene nanostructures. In particular, our approach identiﬁes the three hierarchies of nonmetallic armchair nanoribbons, as well as the doubly-degenerate ﬂat bands of free-standingzigzag nanoribbons with their energy splitting produced by symmetry breaking. The present simple plane-waveapproach holds great potential for gaining insight into the electronic states and electro-optical properties ofgraphene nanostructures and other two-dimensional materials with intact or gapped Dirac-like dispersions. DOI: 10.1103/PhysRevB.99.115443 I. INTRODUCTION The two-dimensional (2D) honeycomb carbon-atom lattice known as graphene [ 1] is a promising material for applications in optical and electronic devices [ 2–4]. In particular, its unique conical electronic dispersion [ 5,6] and 2D character enable an exceptionally large optical tunability [ 7,8] and conﬁgure a suitable playground for quantum electrodynamics phenom-ena, such as the relativistic Klein tunneling [ 9], as well as a customizable zoo of exotic band structures when decoratedwith defects [ 10], arranged in twisted bilayers [ 11], or lat- erally patterned into ribbons [ 12,13]. Energy-gap engineer- ing in graphene—an essential prerequisite for nanoelectron-ics applications—demands controlled and selective sublatticeperturbations at the atomic scale, such as those one canproduce through chemical doping [ 14,15], electrical gating [16], lateral strain [ 17,18], and substrate-induced sublattice asymmetry [ 19–22]. Graphene nanoribbons (GNRs) have been extensively stud- ied as simple, appealing nanostructures that lead to electronicband features of interest, such as gap openings, due to quan-tum conﬁnement and peculiar edge states that can readilybe tuned through their width, shape, and edge terminations[12,13]. Rapid progress in on-surface chemistry, which allows controlled synthesis of novel graphene-based nanostructures,such as GNRs with complex architectures [ 23–28], combined Corresponding author: z.m.abdelfattah@azhar.edu.eg †Corresponding author: javier.garciadeabajo@nanophotonics.es ‡Corresponding author: enrique.ortega@ehu.eswith the precise mapping of their electronic structures us- ing angle-resolved photoemission spectroscopy (ARPES) andscanning tunneling spectroscopy (STS) [ 29–32], render GNRs as promising candidates for the realization of exotic graphene-based nanodevices [ 33–35]. Theoretical understanding and prediction of extended graphene and GNRs properties has been instrumental in thedevelopment of the ﬁeld. Density-functional theory (DFT)accurately describes their electronic structures, but simplermethods [ 36–39] are preferred because they allow us to gain further physical insight. In particular, following the pioneer-ing work of Wallace [ 39], the tight-binding (TB) model has played a central role in the theoretical description of theelectronic structure of extended graphene, yielding remark-able agreement with DFT calculations. However, noticeablediscrepancies between TB and DFT show up when describingGNRs with either armchair (AGNR) or zigzag (ZGNR) edgeterminations. For example, the widely used nearest-neighborsTB predicts two families of AGNRs, namely semiconductorand metallic, depending on the number of carbon-dimer linesalong the ribbon width ( N a)[13], while three semiconductor categories are obtained from DFT calculations [ 40] in agree- ment with STS experiments [ 41–44]. These discrepancies, which stem from the fact that the nearest-neighbors TB modelrenormalizes long-range interactions and is only strictly appli-cable to extended graphene, can be resolved for nanoribbonsat the expense of including at least third-neighbors hoppings[45,46], in addition to electron-electron interaction in ZGNRs, for example at the level of a mean-ﬁeld Hubbard- Uterm [47,48]. 2469-9950/2019/99(11)/115443(8) 115443-1 ©2019 American Physical SocietyZ. M. ABD EL-FATTAH et al. PHYSICAL REVIEW B 99, 115443 (2019) Both nearest-neighbors TB and nearly-free electron (NFE) models are well-known textbook approaches for band-structure calculations in solids [ 49,50]. Within the NFE frame- work, plane wave expansions (PWEs) of the electron stateshave traditionally played an important role, for example in thedescription of electron scattering in metallic and molecularsuperlattices [ 51–53]. In particular, 2D hexagonal superlat- tices, which are known to exhibit graphenelike band structureswith M-point gap and symmetry-protected degeneracy at the Kpoints [ 54,55], are well described by the PWE approach. Unfortunately, such simple PWEs have not been used forthe description of extended graphene or GNRs, althougha close correspondence between the TB and NFE modelswas demonstrated for the so-called molecular graphene, in which the Shockley surface state conﬁned by a hexagonal COsuperlattice was shown to exhibit a Dirac-like dispersion [ 56]. Here, we demonstrate that the electronic characteristics of πbands in atomic graphene can be ﬁnely reproduced via a simple NFE model with a single ﬁtting parameter, namely thescattering potential. In this context, the graphene non-Bravaishoneycomb lattice is alternatively modeled as a 2D hexagonallattice made of the sixfold symmetric, hexagonally-warpedinner part of the carbon rings, where a sufﬁciently largerepulsive potential V 3is assigned [Fig. 1(a)]. The potential barrier V3in reality delimits the attractive Coulomb potential of each carbon atom ( V1andV2). Perfect agreement with DFT calculations is obtained for the band structure, local density ofstates (LDOS), and constant-energy surfaces (CESs) using anelectron-plane-wave-expansion (EPWE) implementation (seeSec. II). Interestingly, with the same single ﬁtting parameter V 3, the model captures the three categories of AGNRs in decent agreement with DFT. Likewise, the 1D-bulk bandstructure and the nearly-degenerate edge state for ZGNRsare obtained in agreement with TB and DFT without anyinclusion of electron-electron interactions. Additionally, weﬁnd that when the symmetry of the two carbon sublattices isbroken for ZGNRs ( V 1/negationslash=V2), which is a common situation for graphene grown onto different substrates, the edge stateof ZGNRs is split in energy, also without the incorporation ofelectron-electron interactions. We believe that this simpliﬁedpicture can be efﬁciently applied to explore different varietiesof atomic graphenelike extended and ﬁnite structures. II. THEORETICAL FORMALISM A. Effective 2D potential description We simulate the electronic structure of graphene in terms of the one-electron states of a 2D potential landscape, in whicheach carbon atom is represented by a circle ﬁlled with uniformpotential, embedded in a ﬂat interstitial region [see Fig. 1(a)]. We then write the Schrödinger equation as −¯h 2 2meff∇2φ+Vφ=Eφ, (1) where the energy Eis expressed relative to a reference level (e.g., the Dirac point), meffis the effective mass, V(R)i st h e 2D potential as a function of spatial coordinates R=(x,y) along the graphene plane, and φ(R) is the electron wave function. -2-1012 -2 -1 0 1 2 20 15 10 5 0 -5 -10 4 3 2 1 0(b) 012 -2-10 -2 -1 0 1 2(c) FS (d) +1.0 eV -1.0 eV-2.71eV +1.75 eV (a) V2 V1 V3 1.42Å (e) E-EF(eV) k\|\|(Å-1) E-EF(eV)k\|\|x(Å-1)k\|\|y(Å-1) k\|\|y(Å-1) DOS and LDOS (arb. units) FIG. 1. Free-standing graphene. (a) Geometry and scattering potential used in our EPWE calculations. The red and blue circlesrepresent the two carbon sublattices with inner potentials V 1and V2, while white regions deﬁne the hexagonally-warped bases with an inner potential V3. (b) Calculated band structure using EPWE along the /Gamma1MK/Gamma1excursion within the ﬁrst Brillouin zone (BZ), for V3=23 eV and V1=V2=0. The inset is a closeup view at the K point. (c),(d) Simulated CESs taken at (c) 0 eV and at (d) +1e V( t o p half) and −1 eV (bottom half). (e) LDOS calculated at the center of the carbon atoms in each sublattice (red and blue) and total DOS per unit cell (gray). The insets show the 2D LDOS at the boundaries ofthe M-point gap. B. Plave-wave expansion for periodic systems We take Vto be periodic and express it in terms of Fourier components as V(R)=/summationdisplay gVgeig·R, (2) where the sum extends over 2D reciprocal lattice vectors g with coefﬁcients Vgcalculated as an integral over the unit cell (UC), normalized to the unit-cell area A: Vg=1 A/integraldisplay UCd2RV(R)e−ig·R. (3) Using Bloch’s theorem, we anticipate electron wave functions labeled by a band index jand the 2D wave vector k/bardblwithin the ﬁrst Brillouin zone (1BZ): φk/bardblj(R)=1√ NA/summationdisplay gφk/bardblj,gei(k/bardbl+g)·R, (4) where we use the (inﬁnite) number of cells Nfor normaliza- tion purposes. Inserting Eqs. ( 2) and ( 4) into Eq. ( 1), we ﬁnd 115443-2GRAPHENE: FREE ELECTRON SCATTERING WITHIN AN … PHYSICAL REVIEW B 99, 115443 (2019) the linear system of equations Ek/bardbljφk/bardblj,g=/summationdisplay g/prime[(¯h2/2meff)\|k/bardbl+g\|2δg,g/prime+Vg−g/prime]φk/bardblj,g/prime.(5) Because Vg−g/primeis a Hermitian matrix with indices gandg/prime, for each value of k/bardblwe obtain different bands jof real eigenenergies Ek/bardbljand eigenstates of coefﬁcients φk/bardblj,g.W e solve Eq. ( 5) by retaining a ﬁnite number of g’s within a sufﬁciently large distance gmaxto the origin in reciprocal space (i.e.,\|g\|/lessorequalslantgmax). The eigenstates form an orthonormal and complete system, that is, /integraldisplay d2Rφk/bardblj(R)φ∗ k/prime /bardblj/prime(R)=δk/bardblk/prime /bardblδjj/prime, (6a) /summationdisplay k/bardbljφk/bardblj(R)φ∗ k/bardblj(R/prime)=δ(R−R/prime), (6b) provided we impose the normalization condition/summationtext g\|φk/bardblj,g\|2=1. The integral in Eq. ( 6a) is extended over the entire 2D plane. For extended graphene, good convergence (indistinguish- able results within the scale of graphs) is achieved with gmax= 7/√ A, while for nanoribbons gmaxneeds to be increased to 20/√ A. In both cases, the number of plane waves involved in the calculation scales as ∼g2 max. C. LDOS calculation for periodic systems The local density of states (LDOS) is directly calculated from its deﬁnition LDOS( E,R)=2/summationdisplay k/bardblj\|φk/bardblj(R)\|2δ(E−Ek/bardblj), (7) where the factor of 2 accounts for spin degeneracy. In practice, we use the prescription/summationtext k/bardbl→(NA/4π2)/integraltext d2k/bardbland evalu- ate this integral in a dense wave-vector grid by interpolating the eigenstates and eigenenergies within each grid element. D. Calculation of photoemission angular distributions for periodic systems In order to make the calculation feasible and simple, we dismiss the contribution of the normal component of the elec-tron wave function to the photoemission matrix elements, as itshould just introduce a smooth and broad angular dependence,which we represent through a multiplicative coefﬁcient Cin the resulting photoemission intensity. We focus instead onthe contribution of the in-plane wave function of the initialelectron state and further approximate the parallel componentof the photoelectron wave function as a normalized plane wave e ikout /bardbl·R/√ NAof outgoing parallel wave vector kout /bardbl. The doubly-differential angle-resolved photoemission inten-sity corresponding to a binding energy Eand photoelectron wave vector k out /bardblis then given from Fermi’s golden rule as dI(E) d2kout /bardbl=C/summationdisplay k/bardblj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay d 2Rφk/bardblj(R)e−ikout /bardbl·R √ NA/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 δ/parenleftbig E−Ek/bardblj/parenrightbig =C/summationdisplay j,g/vextendsingle/vextendsingleφk/bardblj,g/vextendsingle/vextendsingle2δ/parenleftbig E−Ek/bardblj/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle k/bardbl=kout /bardbl−g, (8)where the sum over gin the last expression is restricted to reciprocal vectors such that k/bardbl=kout /bardbl−glies within the 1BZ. III. RESULTS AND DISCUSSION Figure 1summarizes the electronic characteristics of free- standing graphene, as determined within the EPWE approach.The potential landscape used in the calculations is depicted inFig. 1(a). The red and blue circles deﬁne the position of the carbon atoms, each of radius 0.71 Å (i.e., nearest neighborsare tangentially touching at one point without overlap), andthe white regions stand for the carbon-free interstitial region.The unit cell, enclosing two carbon atoms surrounded by theinterstitial region, is marked by the black dashed lines (shadedarea), with unit-cell vectors of length d=√ 3a0, where a0= 1.42 Å is the C-C bond distance in the material. The band structure presented in Fig. 1(b) is obtained by setting the inter- stitial potential to V3=23 eV and the two equivalent carbon- atom potentials to V1=V2=0, so the potential difference is /Delta1V=23 eV . Furthermore, we take the effective mass meff equal to the electron mass. These parameters successfully re- produce the band structure of free-standing graphene obtainedfrom DFT calculations and experiments [ 57,58]. The Fermi energy E Fis set at the nongapped (see zoom-in) Dirac point, and the /Gamma1-point energy is accordingly found at ∼−8.5e V . The lower and upper edges of the M-point gap are −2.71 eV and+1.75 eV , respectively, while the slope of the linear bands at the Kpoints is ≈6.5 eV Å (i.e., approximately ¯ htimes the Fermi velocity vF≈106m/s), in excellent agreement with literature values [ 59]. We stress that small deviations from the employed values of /Delta1Vandmeffyield noticeable changes in the relative energy position of the bands, the size of the M-point gap, and the degree of electron-hole asymmetry [see Fig. S1 in the Supplemental Material (SM)] [ 60]. In Figs. 1(c) and1(d) we present the simulated photoe- mission intensity of the constant energy surfaces (CESs).The Fermi surface (FS) consists of single spots centered atthe six Kpoints of the BZ (green hexagon), resulting from intact Dirac cones [Fig. 1(c)]. At lower and higher binding energies (e.g., +1 eV and −1 eV), these spots diverge into the characteristic graphene triangular pockets, as shown in theupper (red) and lower (blue) panels of Fig. 1(d), respectively. The trigonal shape of these CESs further conﬁrms that theelectron-hole asymmetry present in (b), which in TB calcu-lations is accounted for by introducing additional hoppingparameters for second and third nearest neighbors [ 6,61,62], is naturally captured by EPWE. Indeed, the different hoppingparameters employed in TB are consistent with the differenteffective potentials (e.g., height ×width) felt by electrons moving across neighboring atoms. Furthermore, the variationof the photoemission intensity within the trigonal pocketsagrees nicely with recent ARPES experiments [ 20,63–65]. The electronic characteristics of graphene, as deduced fromthe total density of states (DOS) and LDOS, are also presentedin Fig. 1(e). The V-like proﬁles at E Fare revealed in both the DOS (gray) and the LDOS evaluated at the two carbon atoms(blue and red), all of them exhibiting a clear electron-holeasymmetry. The onset of the LDOS at ∼−8.5 eV deﬁnes the /Gamma1-point energy, whereas the peaks at −2.71 eV and +1.75 eV signal the borders of the M-point gap. The LDOS at the 115443-3Z. M. ABD EL-FATTAH et al. PHYSICAL REVIEW B 99, 115443 (2019) 1.0 0.5 0.0-8-404E-EF (eV) 4 3 2 1 0 -2.79 eV +1.80 eV -0.69 eV +0.62 eV(a) (b) k\|\| (Å-1) DOS and LDOS (arb. units) FIG. 2. Perturbed graphene. (a) Calculated band structure using EPWE along the /Gamma1MK/Gamma1excursion for V3=23 eV , V1=+1e V ,a n d V2=−1 eV . (b) The corresponding DOS (gray) and LDOS taken at the red and blue circles in (a). The right panel in (b) presents two- dimensional LDOS maps taken at the lower (red) and upper (blue)edges of the M-point and K-point gaps. two carbon atoms are coincident, as they clearly must in pristine graphene. The 2D LDOS maps depicted at insetsand taken at the boundaries of the M-point gap are equally identical, conﬁrming the absence of K-point gaps based on symmetry considerations [ 54,55]. Given the calculations and analysis presented in Fig. 1, the electronic features of a free- standing graphene sheet obtained from experiments and DFT calculations are well reproduced by EPWE. Perturbations induced by a graphene support (i.e., a sub- strate and/or the deposition of adsorbates/dopants) have beenshown to change the electronic properties of graphene indifferent ways. Figure 2presents possible electronic modiﬁca- tions in one of such perturbation schemes, as calculated usingthe present EPWE approach. In Fig. 2(a), we explore the effect of broken symmetry between the two carbon atoms on theelectronic band structure. This is obtained by assigning differ-ent potentials to each of the carbon sublattices [ V 1=−1e V (red) and V2=+1 eV (blue)]. The main modiﬁcation consists of the opening of an energy gap at the Kpoint [ Eg(K)= 1.3 eV], which is a natural consequence of broken symmetry in the potential landscape within the unit cell. Such broken-symmetry-induced gaps have been reported experimentallyfor different graphene structures, such as graphene grown onIr(111) [ 66], Ru(0001) [ 67], hydrogenated-graphene [ 68], and other systems [ 19–22,65]. The large gap induced by broken symmetry shown here is meant only to illustrate the effect,but the actual values of V 1andV2can in principle be tuned in each case to yield the experimental gap. We also note thatthe /Gamma1-point energy is unaltered in this particular example to ﬁrst order, as the average of the potentials V1andV2is zero. The DOS and LDOS spectra presented in (b) preciselyfollow these band structure modiﬁcations. In addition to thepeaks at the borders of the M-point gap, a deviation from the V-shape proﬁles occurs for all spectra, where instead the DOSand LDOS vanish at the energy range spanning the K-point gap boundaries. Particularly relevant is the nonequivalence ofthe LDOS at the two carbon atoms within the unit cell, wherethe weight of the LDOS changes from one carbon (red) to thesecond sublattice (blue) by crossing the energy gap, which is further shown in the 2D LDOS maps presented in Fig. 2(b) (right panels). Additional electronic modiﬁcations, such ashybridization and doping, which still preserve the K-point degeneracy, are brieﬂy presented in Fig. S2 of the SM [ 60]. The calculations and analysis here presented clearly reveal that the electronic structure of graphene could be reproducedthrough a relatively simple 2D potential landscape with spe-ciﬁc values of the potential in different regions of space. Inwhat follows, we explore the applicability of this approach tothe study of both AGNRs and ZGNRs. We employ the samegeometry and potential landscape while varying the ribbonwidth and termination. We make use of the periodic supercellapproach and assume the ribbons to be inﬁnitely extendedalong their axis (i.e., the xdirection), while they are decoupled in the ydirection by separating them by /greaterorequalslant20 Å gaps. The peri- odicity along the ribbon is given by danda=3a 0for ZGNR and AGNR, respectively. Figures 3(a)–3(c) depict the band structure along the ribbon axis for the three different classesof AGNRs with (a) N a=3p,( b ) Na=3p+1, and (c) Na= 3p+2, where pis a positive integer [here, p=1 (left) and 7 (right)]. In contrast to standard TB calculations, the threetypes of ribbons exhibit energy gaps E gsatisfying the relation Eg(3p+1)>Eg(3p)>Eg(3p+2), with sizes following an inverse proportionality with the GNR width [see coloredcurves in Fig. 3(d)] in agreement with DFT calculations. It should be noted that the carbon-carbon distance in the bulkregion and at the ribbon boundaries is ﬁxed here to the samevalue (i.e., structural relaxation is not considered, althoughDFT indicates a ∼3.5% contraction in the bond length at the edges) [ 40], yet the model yields appreciable energy gaps for the 3 p+2 family. We also show that the /Gamma1-point energy, irrespectively of the ribbon family, asymptotically approachesthe /Gamma1point of extended graphene by increasing the width, strictly following the width-dependent /Gamma1point energy (gray curve) obtained by stepping along the Brillouin zone slicesof graphene [see inset in Fig. 3(d)]. Particularly relevant is the shallow dispersion of the bands at −2.6e Vf o r N a=3 (a) and Na=5 (c), which in nearest-neighbor TB exhibit no dispersion [ 13,69]. This constitutes a further indication that the NFE approach naturally incorporates all possible crosstalkand hoppings between neighboring carbon atoms with thescattering potential as a single ﬁtting parameter. In what follows, we check the ability of our model to sim- ulate the photoemission intensity and DOS for nanoribbons. This should serve as guidance for experimentalists to perform proper assignments of speciﬁc GNR bands, which mightbecome problematic for wider ribbons with neighboring dis-persion bands, and in general, due to strong variations inintensity caused by effects related to the photoemission matrixelements. Figure 3(e) presents the simulated photoemission intensity for the 4-AGNR along the ribbon axis. A subtle vari- ation of photoemission intensity for different bands is clear. For example, at k y=0 (left) the frontier of the valence band (VB1) has predominantly-symmetric spectral weight aroundk x=0, while VB2 gains spectral weight over a wider kxrange with an asymmetric photoemission intensity around the top of the band ( ∼1.5Å−1). This distribution of photoemission intensity changes drastically at ky=π/a(center) and ky= 2π/a(right). Indeed, these are all 1D bands, and therefore, 115443-4GRAPHENE: FREE ELECTRON SCATTERING WITHIN AN … PHYSICAL REVIEW B 99, 115443 (2019) ---- ---- ---- -8-6-4-20246 Na = 3 Na = 4 Na = 5 Na = 21 Na = 22 Na = 23 (a) (b) (c) E-EF (eV) 0 π0 π0 π0 π0 π0 π 2.01.51.00.50.0 -0.5-8-6-4-20246 2.01.51.00.50.0 -0.5 2.01.51.00.50.0 -0.5(e) ky = 0 ky= π/a ky=2π/a VB1VB2 kx(Å-1) kx(Å-1) kx(Å-1) CB2 CB1k a E-EF (eV) (f) CB1 CB2 VB1 VB2 3.0 2.0 1.0 0.0(d) -8.5-8.0-7.5-7.0 20 15 10 5Energy gap: Eg (eV) -point energy (eV)Na = 3p Na = 3p+ 1 Na = 3p+ 2 CB2 CB1 FIG. 3. Armchair graphene nanoribbons (AGNRs). (a)–(c) Band structure of Na-AGNR with Na=3 and 21 (a), 4 and 22 (b), and 5 and 23 (c), as obtained using EPWE. (d) Variation of the gap size and /Gamma1-point energy for the three categories of AGNRs. The inset shows the ﬁrst and second BZs of graphene. (e) Simulated photoemission intensity for the 4-AGNR along the ribbon axis ( kxdirection) taken at ky=0 (left), ky=π/a(center), and ky=2π/a(right). (f) LDOS for the 3-AGNR taken at valence- and conduction-band points VB1, VB2, CB1, and CB2 [see labels in (e)]. The corresponding molecular orbitals deduced from DFT calculations are shown to the right (adapted from Ref. [ 71]), and the geometry used in EPWE simulations is shown at the top. not dispersing in the perpendicular direction ky, yet a strong modulation of the photoemission intensity is present for theVB and conduction band (CB) alike [Fig. S3(a), SI] [ 60]. Therefore, constant-energy surfaces (i.e., k xvskymaps) such as the one shown in Fig. S3(d) of the SM [ 60] are essential for a proper assignment of bands. Actually, the simulation of such photoemission intensity maps has recently solved along-standing contradiction between STS and ARPES data forthe 7-AGNR, where the VB2 point was mistakenly assigned inARPES experiments to VB1 [ 30,70]. A further conﬁrmation of the full functionality of our model is provided by compar-ing the calculated 2D LDOS maps to the molecular orbitals obtained from DFT calculations, and in particular, for the 3-AGNR, as shown in Fig. 3(f). The matching between LDOS (left) and DFT orbitals (right) is remarkable [ 71]. The overall agreement with DFT extends even beyond the description ofsimple AGNRs: Complex graphene-based structures such aszigzag [ 71] and heterogeneous ribbons, as well as nanoporous graphene [ 28], are equally well described using our NFE approach (see Fig. S4, SM) [ 60]. Likewise, the electronic structure of ZGNRs could be obtained using our EPWE approach. Figures 4(a) and4(b) present band structure calculations for selected ZGNRs withN a=12 (a) and 6 (b). Their characteristic bulk bands and the energy-degenerate edge states are obtained in agree- ment with TB and DFT calculations when on-site Hubbard potentials and exchange interactions, respectively, are not considered. In (b) the simulated photoemission intensity for the 6-ZGNR is also shown at ky=0, with the edge state exhibiting a high-enough spectral weight as to be probed by average techniques such as ARPES, provided that the ZGNRs aligned on substrates are experimentally available over large areas. As previously discussed, for an extended graphene sheet with symmetry-broken carbon sublattices, a K-point gap opens up (see Fig. 2). Here we demonstrate the effect that this broken symmetry has on the electronic structure of ZGNRs,speciﬁcally on the edge state. By assigning different valuesto the potentials at the two carbons sublattices ( V 1=−1e V andV2=+1 eV), taking the 12-ZGNR as an example, we show that the edge state is split in energy, while the 1D- bulk projected bands are practically unaltered [Fig. 4(c)]. The energy gap between the split edge states is the same asthe size of the K-point gap of the corresponding extended graphene presented in Fig. 2(a). Both bands of the energy-split edge state have reasonable photoemission spectral weights, 115443-5Z. M. ABD EL-FATTAH et al. PHYSICAL REVIEW B 99, 115443 (2019) FIG. 4. Zigzag graphene nanoribbons (ZGNRs). (a-b) Band structure of 12-ZGNR (a) and 6-ZGNR (b) obtained using EPWE.(c,d) Same as (a,b), but assuming that the symmetry of the carbon sublattice is now broken. The simulated photoemission intensity for the 6-ZGNR with an in-plane photoelectron momentum along theribbon and k y=0 is shown in (b,d). (e) DOS for the 6-ZGNR before (gray) and after (red) breaking the symmetry of the carbon sub- lattices. The insets show the atomic structure and two-dimensionalLDOS maps at the edge-state energies. as demonstrated for the 6-ZGNR at ky=0i nF i g . 4(d). Finally, we present in Fig. 4(e) the DOS curves for the free-standing (gray) and symmetry-broken (red) sublatticesin the 6-ZGNR, where energy splitting of the DOS peakatE Fis observed. Although the DOS proﬁle for asymmet- ric ribbons (red) resembles the one reported experimentally (and from DFT) for the 6-ZGNR [ 72], the broken-symmetry- induced gaps could be distinguished from electron-electron-interaction gaps [ 40,73,74] by plotting the 2D spatial distri- bution of the LDOS taken at the energy of the edge state, asshown in insets above the spectra in Fig. 4(e). These insets present the EPWE geometry of the 6-ZGNR and the 2DLDOS maps taken at the energy of the lower (red) and upper (blue) edge states, where the LDOS is clearly localized at one edge, while for fully symmetric ribbons both edges areequally occupied at E F(green). It should be noted that the one- edge localization of the LDOS has not been experimentallyreported, neither for degenerate nor for energy-split edgesstates, although its realization could have potential impact as a switch in 1D conduction channels through gating. We also anticipate that the combination of broken symmetryand electron correlation should produce a clear imbalance inthe LDOS at both edges of the ribbons, in addition to theintrinsic asymmetry produced by the different dispersion ofthe upper and lower edges of the gap and the electron-holeasymmetry. The fact that the extended and ﬁnite graphene character- istics are well captured within the framework of the NFEmodel could have far reaching implications, since some ofthe electronic structure variations and size dependence are notunique to graphene. Similar atomic systems, such as siliceneor boron nitride, could be understood following the sameapproach. Furthermore, nanometer-sized ribbons made fromhexagonal superlattices, such as molecular graphene [ 56]o r metallic superlattices possessing graphenelike band structures[54], should exhibit these types of size-dependent variations. What makes these variations experimentally accessible andpotentially relevant for technology in both graphene and othertypes of 2D crystals is the combination of a large M-point gap (several electronvolts, which for superlattices reduces tojust a few meV) and the steep dispersion near the Kpoints. Finally, this simple NFE description of graphene and itsnanostructures should have a large impact on the efﬁcientsimulation of graphene-based devices and phenomena, suchas negative refraction and super lenses in p-njunctions, using, for example, the complementary electronic boundary-element method (EBEM) solver, which was previously usedto describe similar effects in 2D metallic superlattices[75,76]. IV . CONCLUSIONS We have shown that the electronic structure of the πband in free-standing and perturbed graphene can be reasonablywell described using a simple nearly-free-electron model (theEPWE approach) applied to an inverted honeycomb latticethat is deﬁned by a sufﬁciently large conﬁning potential.With a single ﬁtting parameter (i.e., the scattering barrier)the electronic properties of common graphene nanostructures(armchair and zigzag graphene nanoribbons) are successfullydescribed using this simple model. Our approach simpliﬁesthe exploration of newly emerging artiﬁcial systems with fun-damental and technological interest, such as nanostructured2D materials, topological GNR junctions with peculiar endstates [ 26,27], and artiﬁcial ﬂat band lattices [ 77]. ACKNOWLEDGMENTS This work has been supported in part by the Span- ish MINECO (Grants No. MAT2014-59096-P, No. MAT-2016-78293-C6, No. MAT-2017-88374-P, No. MAT2017-88492-R, and No. SEV2015-0522), the Basque Government(Grant No. IT-1255-19), the European Research Council (Ad-vanced Grant No. 789104-eNANO), the European Commis-sion (Graphene Flagship 696656), the Catalan CERCA Pro-gram, Fundació Privada Cellex, and AGAUR (Grant No. 2017SGR 1651). 115443-6GRAPHENE: FREE ELECTRON SCATTERING WITHIN AN … PHYSICAL REVIEW B 99, 115443 (2019) [1] A. K. Geim and K. S. Novoselov, Nat. Mater. 6,183(2007 ). [2] M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, Nature (London) 474,64(2011 ). [3] L. Vicarelli, M. S. 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PhysRevB.82.235422.pdf	Intrinsic optical anisotropy of [001]-grown short-period InAs/GaSb superlattices L. L. Li,1,2W. Xu,2,3,and F. M. Peeters1,† 1Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium 2Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China 3Department of Physics, Yunnan University, Kunming 650091, China /H20849Received 5 October 2010; published 13 December 2010 /H20850 We theoretically investigate the intrinsic optical anisotropy or polarization induced by the microscopic interface asymmetry /H20849MIA /H20850in no-common-atom /H20849NCA /H20850InAs/GaSb superlattices /H20849SLs /H20850grown along the /H20851001 /H20852 direction. The eight-band K·Pmodel is used to calculate the electronic band structures and incorporates the MIA effect. A Boltzmann equation approach is employed to calculate the optical properties. We found that inNCA InAs/GaSb SLs, the MIA effect causes a large in-plane optical anisotropy for linearly polarized light and the largest anisotropy occurs for light polarized along the /H20851110 /H20852and /H2085111¯0/H20852directions. The relative difference between the optical-absorption coefﬁcient for /H20851110 /H20852-polarized light and that for /H2085111¯0/H20852-polarized light is found to be larger than 50%. The dependence of the in-plane optical anisotropy on temperature, photoexcited carrierdensity, and layer width is examined in detail. This study is important for optical devices which require thepolarization control and selectivity. DOI: 10.1103/PhysRevB.82.235422 PACS number /H20849s/H20850: 78.67.Pt, 73.21.Cd, 78.40.Fy I. INTRODUCTION Many of the optoelectronic devices used in modern opti- cal communication systems require a polarization-independent operation, whereas for laser diodes polarization-dependent mechanisms have been explored that reduce thethreshold current. In recent years optical polarization phe-nomena in vertical-cavity surface-emitting lasers /H20849VCSLs /H20850 have been intensively studied both experimentally andtheoretically. 1–3In addition to VCSLs, some particular III-V semiconductor superlattices /H20849SLs /H20850have also attracted much interest in light of such phenomena.4–6 In the C1A1/C2A2 semiconductor SLs where C1/C2 and A1/A2 denote, respectively, the III-V cation and anion spe- cies and if the host materials C1A1 and C2A2 have no com- mon atoms /H20849NCAs /H20850, i.e., C1/HS11005C2 and A1/HS11005A2, we will term such SLs as NCA SLs. In the other case when the host ma-terials share common atoms, i.e., C1=C2o rA1=A2, we will call them CA SLs. Hence, in NCA SLs, there exists a re-markable peculiarity that chemical bonds formed at the in-terfaces /H20849C1-A2 and C2-A1/H20850are different from those formed in the host materials /H20849C1-A1 and C2-A2/H20850. Typical NCA sys- tems are GaAs/InP and InAs/GaSb SLs while GaAs/AlAsand GaAs/InAs SLs are typical CA SL systems. The differ-ence between the CA and NCA SLs comes from the in-equivalent chemical bonds formed in the host materials andat their interfaces. Owing to this difference, the NCA SLsystem such as GaAs/InP has been intensively studied in thepast decade to explore interface related electronic and opticalproperties. 4–7The most different features exhibited by CA and NCA SLs are seen in their optical properties. It has beenexperimentally veriﬁed that in the GaAs/InP NCA system agiant in-plane anisotropy /H20849or polarization /H20850in the optical ab- sorption of light polarized along the /H20851110 /H20852and /H2085111 ¯0/H20852 directions5,6occurs while CA SLs do not show any in-plane anisotropy of their optical absorption. It is not hard to understand the in-plane absorption aniso- tropy of NCA SLs grown along the /H20851001 /H20852direction. In theCA SLs such as GaAs/AlAs, successive interfaces have the following spatial arrangement of chemical bonds: the ﬁrstinterface has Ga-As and Al-As bonds lying in the /H20849110 /H20850plane while the second has Al-As and Ga-As bonds lying in the /H2084911¯0/H20850plane. As a result, the positive anisotropy induced by different Ga-As and Al-As bonds lying in the /H20849110 /H20850plane is exactly compensated by the negative anisotropy resulting from different Al-As and Ga-As bonds lying in the /H2084911¯0/H20850 plane and thus no in-plane anisotropy occurs. However, thiscompensation does not exist in the NCA SLs such as GaAs/InP: the ﬁrst interface has Ga-As and In-As bonds lying inthe /H20849110 /H20850plane while the second has In-P and Ga-P bonds lying in the /H2084911 ¯0/H20850plane. As a result, two in-plane directions /H20851110 /H20852and /H2085111¯0/H20852are inequivalent and thus exhibits a strong in-plane anisotropy. When linearly polarized light propagates along the /H20851001 /H20852SL growth direction, it will feel different chemical bonds for the /H20851110 /H20852and /H2085111¯0/H20852polarization direc- tions and consequently, in-plane absorption anisotropy is in- duced by /H20851110 /H20852and /H2085111¯0/H20852polarized light. From the group symmetry point of view, CA SLs correspond to the D2d point-group symmetry while NCA SLs correspond to the C2v point-group symmetry. The C2vinterface symmetry in NCA SLs is lower than the D2dinterface symmetry in the CA SLs. Thus, the optical anisotropy induced by the microscopic in-terface asymmetry /H20849MIA /H20850effect is an intrinsic property of the NCA SL system and detailed studies of the in-plane op-tical polarization or anisotropy are necessary. InAs/GaSb NCA SLs have attracted considerable interest over the past two decades due to their unique type-II bandalignment and their application as detectors and lasers withtunable infrared wavelengths. 8,9Most of those studies fo- cused on the electronic10–12and spintronic13,14properties of InAs/GaSb NCA SLs but only few works dealt with the op-tical anisotropy that intrinsically exists in such SL systems.Although the MIA inﬂuence on the in-plane optical polariza-tion /H20849the polarization rate up to 40% /H20850is well described for GaAs/InP NCA structures, 5,6the situation in the InAs/GaSbPHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 1098-0121/2010/82 /H2084923/H20850/235422 /H2084910/H20850 ©2010 The American Physical Society 235422-1NCA SL system is still unclear and is expected to be com- plicated because of a strong conduction-valence-band mixingwhich is absent in the GaAs/InP SL system. In addition, theInAs/GaSb SL system is the best type-II candidate: electronsand holes are, respectively, conﬁned in the InAs and GaSblayers. Due to this fact, it is expected that the optical prop-erties in such type-II NCA SLs should be extremely sensitiveto the MIA effect because they depend completely on theoverlap between the electron and the hole wave functions atthe interface. Thus, one may anticipate that new phenomenarelated to the optical anisotropy could take place in the InAs/GaSb NCA SLs. The InAs/GaSb NCA SL system has been modeled using a variety of theoretical approaches such as the empiricaltight-binding method, 15,16the empirical pseudopotential method,10,11and the standard envelope function approxima- tion /H20849EFA /H20850method.17,18Among these approaches, the EFA method is most widely used because of its greater physical appeal, easier code implementation, and rather good numeri-cal accuracy. However, it has been well known that the stan-dard EFA method fails to distinguish the CA SL system fromthe NCA SL system when considering their optical proper-ties: both the CA and NCA SLs do not show any in-planeoptical polarization in the frame of standard EFA. Therefore,a more accurate approach is needed in which the MIA effectis taken into account in NCA SLs. In the literatures, therewere many proposals for taking this effect into account in theEFA theory. Krebs and V oisin proposed a phenomenologicalmodel 4called the HBFmodel that allows to distinguish the interface chemical bonds that are stacked in backward /H20849B/H20850 and forward /H20849F/H20850directions. General boundary conditions were proposed19for the multiband effective-mass theory in which the hole envelop function boundary conditions aregeneralized and mix the heavy-hole and light-hole spin com-ponents. Szmulowicz et al. 12developed a modiﬁed eight- band EFA formalism that incorporate the MIA effect. It wasrecently shown that a nonsymmetrized eight-band K·P Hamiltonian 20can also model the MIA effect within the Burt-Foreman theory.21More recently, the MIA effect in InAs/GaSb SLs has been modeled successfully within theframework of the eight-band K·Pmodel using a graded and asymmetric interface proﬁle. 22In the present work, we will implement the HBFmodel and investigate the intrinsical op- tical anisotropy in the InAs/GaSb NCA SL system. To calcu-late the electronic band structure and associated wave func-tions, we use an eight-band K·Pmodel that incorporates the MIA effect and solve the resulting EFA Hamiltonian equa-tion using the ﬁnite difference method. With the obtainedeigenvalues and eigenstates obtained, we employ a semiclas-sic Boltzmann equation approach to calculate the opticalproperties of the corresponding SL system such as theoptical-absorption spectra and polarization spectra. Further-more, the dependence of the optical polarization on tempera-ture, photoexcited carrier density, and layer width is exam-ined in detail. Our aim is to understand more deeply thephysical mechanisms of the optical anisotropy or polariza-tion in the InAs/GaSb NCA SLs. This paper is organized as follows. In Sec. II, ﬁrst an eight-band K·Pmodel incorporating the MIA effect is de- veloped to calculate the electronic band structure of InAs/GaSb NCA SLs using the ﬁnite difference method and then a semiclassic Boltzmann equation approach is employed tocalculate the optical properties of the corresponding SL sys-tem. In Sec. III, the numerical results are presented and dis- cussed. Finally, our concluding remarks are summarized inSec. IV. II. THEORETICAL APPROACHES Within an eight-band K·Pformalism, we include the MIA effect for NCA semiconductor SLs grown along /H20851001 /H20852 direction. By imposing the Bloch boundary conditions on thesystem, the resulting EFA Hamiltonian equations can besolved for the SL band structure and associated wave func-tions by the ﬁnite difference method. Subsequently, a semi-classic Boltzmann equation approach is employed to calcu-late the optical properties for the corresponding SL system. A. Band-structure calculation Consider a NCA SL structure constructed by alternating a thin InAs layer with thickness LAand a thin GaSb layer with thickness LBgrown along the /H20851001 /H20852direction. This growth direction is deﬁned as the zaxis. The xandyaxes are along /H20851100 /H20852and /H20851010 /H20852, respectively. The band alignment of such an InAs/GaSb SL is shown in Fig. 1. With the set of the follow- ing basis states: /H20841S↑/H20856,/H20841X↑/H20856,/H20841Y↑/H20856,/H20841Z↑/H20856,/H20841S↓/H20856,/H20841X↓/H20856,/H20841Y↓/H20856,/H20841Z↓/H20856, /H208491/H20850 the eight-band K·PHamiltonian can be written as23 H/H20849K/H20850=HK+HS+HI, /H208492/H20850 where K=/H20849k,kz/H20850=/H20849kx,ky,kz/H20850with kbeing the in-plane wave vector, HKis the dominant K-dependent term, HSis the spin- orbit interaction term, and the term HIdescribes the interface related Hamiltonian. The last term HIincludes the MIA ef- fect. The K-dependent term HKhas the following block- diagonal form:FIG. 1. Illustration of the band alignment for an InAs/GaSb SL grown along the /H20851001 /H20852direction /H20849zaxis /H20850. Here, EcandEvare, re- spectively, the unstrained conduction- and valence-band edges, andL AandLBare, respectively, the InAs and GaSb layer widths.LI, XU, AND PEETERS PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-2HK=/H20875H4K0 0H4K/H20876, /H208493/H20850 where H4K=/H20900Ec+DciPk x iPk y iPk z −iPk xEv/H11032+DxNkxky Nkxkz −iPk yNkxkyEv/H11032+DyNkykz −iPk zNkxkz NkykzEv/H11032+Dz/H20901. /H208494/H20850 Here, Dc=Ac/H20849kx2+ky2+kz2/H20850,Dx=Lkx2+M/H20849ky2+kz2/H20850,Dy=Lky2 +M/H20849kx2+kz2/H20850,Dz=Lkz2+M/H20849kx2+ky2/H20850, and Ev/H11032=Ev−/H9004/3;Ecand Evdenote, respectively, the unstrained conduction- and valence-band edges; /H9004is the spin-orbit split-off energy; P is the interband momentum matrix element deﬁned as P =−i/H20849/H6036/m0/H20850/H20855S/H20841px/H20841X/H20856; and the band parameters Ac,L,M, and N are given by23 Ac=/H60362/2me−2P2/3Eg−P2/3/H20849Eg+/H9004/H20850, /H208495a/H20850 L=− /H20849/H60362/2m0/H20850/H20849/H92531+4/H92532/H20850+P2/Eg, /H208495b/H20850 M=− /H20849/H60362/2m0/H20850/H20849/H92531−2/H92532/H20850, /H208495c/H20850 N=− /H20849/H60362/2m0/H20850/H208496/H92533/H20850+P2/Eg, /H208495d/H20850 where Eg=Ec−Evis the bulk band gap, m0,meare the free and effective electron masses, and /H9253i/H20849i=1,2,3 /H20850are the Lut- tinger parameters. The Hamiltonian due to spin-orbit interaction, HS,i s given by HS=−/H9004 3/H209000 0 0 000 0 0 00 i00 00− 1 0−i0000 0 i 00 0 001 −i0 0 0 0 000 0 0 0 0 0 100 −i0 00 0 i0i0 0 0− 1 −i000 0 0/H20901. /H208496/H20850 The MIA-related term HIhas the following block- diagonal form: HI=/H20875H4I0 0H4I/H20876, /H208497/H20850 where the expression of the microscopic interface Hamil- tonian H4Iwith respect to the basis states /H20841S/H20856,/H20841X/H20856,/H20841Y/H20856, and /H20841Z/H20856 can be found in Ref. 4and has the form H4I=HXY/H209000000 0110 0110 0001/H20901, /H208498/H20850 where HXYis an adjustable parameter characterizing the strength of the interface potential. This 4 /H110034 matrix takesinto account the MIA-induced coupling between one s-like conduction-band state /H20849/H20841S/H20856/H20850and three p-like valence-band states /H20849/H20841X/H20856,/H20841Y/H20856, and /H20841Z/H20856/H20850. Because of the interface symmetry the interface potential does not directly couple theconduction-band states to the valence-band states. 24Hence, one may anticipate that the conduction bands will be lessaffected by the MIA effect than the valence bands in theNCA SLs. The interface potential is short range and is local-ized at the SL interface within about half a monolayer /H20849 /H110112Å /H20850. In the present model, we neglect /H20849i/H20850bulk inversion asym- metry terms resulting from the lack of inversion symmetry inzinc-blende semiconductors because the MIA effect domi-nates in our short-period InAs/GaSb SL structures, 13and /H20849ii/H20850 the strain effect due to the small lattice mismatch betweenInAs and GaSb layers /H20849/H110110.39% /H20850. With the EFA theory, we solve for the SL band structure by transforming the eight-band K·PHamiltonian into a set of eight coupled second-order differential equations by re-placing k zwith − id/dz. Products of kzwith position- dependent material parameters /H20849quadratic polynomials in kz/H20850 are symmetrized17,18before converting the equations to dif- ferential form and the set of differential equations so formu-lated is then Hermitian. The energy dispersions and the cor-responding wave functions for the SL are derived from theSchrödinger equation H/H9023=E/H9023, /H208499/H20850 where /H9023is the multicomponent wave function and Ethe corresponding energy. Usually the SL wave function/H9023 n/H20849k,q,r,z/H20850can be expanded as a linear combination of the basis states U/H9263/H20849r/H20850and the envelope function F/H9263/H20849n,k,q,z/H20850, which reads /H9023n/H20849k,q,r,z/H20850=eik·r/H20858 /H9263=18 U/H9263/H20849r/H20850F/H9263/H20849n,k,q,z/H20850, /H2084910/H20850 where r=/H20849x,y/H20850is the in-plane position vector, n/H20849/H9263/H20850is the subband /H20849basis /H20850index, qis the SL wave vector along the growth direction /H20849zaxis /H20850,U/H9263/H20849r/H20850is the zone-center basis state described by Eq. /H208491/H20850, and F/H9263/H20849n,k,q,z/H20850is the /H9263th component of the nth subband envelope function along the zaxis. By replacing kzwith − id/dzin the K·PHamiltonian, the result- ing eight-band envelope function equation can be written as /H20858 /H9262=18 H/H9262/H9263/H20849k,−id/dz/H20850F/H9262/H20849n,k,q,z/H20850=En/H20849k,q/H20850F/H9263/H20849n,k,q,z/H20850. /H2084911/H20850 Using the ﬁnite difference technique and imposing the Bloch boundary conditions,25one can reduce Eq. /H2084911/H20850to a single eigenvalue equation of the form H·F=E·F, where the Hamiltonian matrix can be diagonalized by standard math-ematical subroutines to obtain the eigenvalues Eand the eigenvectors F. An important advantage of the ﬁnite differ- ence method is that it is quite easy to implement boundaryconditions and arbitrary potentials. Additionally, this methodis also able to describe the tunneling effect with only a fewchanges and is numerically stable in contrast to the transfer-INTRINSIC OPTICAL ANISOTROPY OF /H20851001 /H20852-GROWN … PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-3matrix method26,27which sometimes fails because of grow- ing exponentials. B. Optical-absorption coefﬁcient After obtaining the band structure and the wave functions by solving the EFA Hamiltonian equations described above,the next step is to calculate the optical-matrix element, be-cause the strength of the optical absorption is proportional tothe square of the optical-matrix element. We denote, respec-tively, the initial /H20849valence-band /H20850and ﬁnal /H20849conduction-band /H20850 states involved in the interband optical transition processwith/H9023 n/H20849k,q,r,z/H20850and/H9023n/H11032/H20849k,q,r,z/H20850. For a given light po- larization direction /H9280, the optical-matrix element between such two states can be written as /H9280·Mnn/H11032/H20849k,q/H20850=/H20855/H9023n/H11032/H20849k,q,r,z/H20850/H20841/H9280·p/H20841/H9023n/H20849k,q,r,z/H20850/H20856 =/H9280·/H20858 /H9262/H9263/H20849/H20855F/H9263/H20849n/H11032,k,q,z/H20850/H20841F/H9262/H20849n,k,q,z/H20850/H20856 /H11003/H20855U/H9263/H20849r/H20850/H20841p/H20841U/H9262/H20849r/H20850/H20856 +/H20855F/H9263/H20849n/H11032,k,q,z/H20850/H20841p/H20841F/H9262/H20849n,k,q,z/H20850/H20856/H9254/H9262/H9263/H20850, /H2084912/H20850 where the ﬁrst term gives the interband optical transition strength and the second term, which gives the strength of theintraband optical transition, involves essentially the overlapbetween the same envelope function components for initialand ﬁnal states. Thus, one can anticipate that it also contrib-utes to the interband optical transition strength in InAs/GaSbSL systems, where the conduction-band states are stronglymixed with the valence-band states because of the type-IIband alignment at the InAs/GaSb interfaces. Inserting the optical-matrix element given by Eq. /H2084912/H20850 into Fermi’s golden rule, the electronic transition rate in-duced by direct carrier-photon interaction via absorptionscattering can be obtained as W nn/H11032/H20849k,q/H20850=2/H9266 /H6036/H20873eF0 m0/H9275/H208742 /H20841/H9280·Mnn/H11032/H20849k,q/H20850/H208412/H9254/H20851En/H11032/H20849k,q/H20850 −En/H20849k,q/H20850−/H6036/H9275/H20852, /H2084913/H20850 where F0and/H9275are, respectively, the strength and frequency of the light ﬁeld. In this study, we employ the semiclassic Boltzmann equa- tion as the governing transport equation to study the photoresponse of a SL to the applied light ﬁeld. In case of degen-erate statistics, the Boltzmann equation takes the form /H11509fn/H11032/H20849k,q,t/H20850 /H11509t=/H20858 n/H20858 k,q/H20851Fnn/H11032/H20849k,q,t/H20850−Fn/H11032n/H20849k,q,t/H20850/H20852./H2084914/H20850 Here, Fnn/H11032/H20849k,q,t/H20850=fn/H20849k,q,t/H20850/H208511−fn/H11032/H20849k,q,t/H20850/H20852Wnn/H11032/H20849k,q/H20850, fn/H20849k,q,t/H20850is the momentum distribution function for the elec- tron in a state /H20841n,k,q/H20856, where /H20855R/H20841n,k,q/H20856=/H9023n/H20849k,q,R/H20850and Wnn/H11032/H20849k,q/H20850is the steady-state electronic transition rate. In Eq. /H2084914/H20850, the effect of the light ﬁeld has been included within the time-dependent momentum distribution functions and withinthe electronic transition rates. Thus, to avoid double count-ing, the force term induced by the light ﬁeld does not appear on the left-hand side of the Boltzmann equation. It is knownthat there is no simple and analytical solution for Eq. /H2084914/H20850 with the electronic transition rate given by Eq. /H2084913/H20850. In this work, we apply the usual energy balance-equation approachto solve the Boltzmann equation approximately. 28,29This bal- ance equation can be derived by multiplying/H20858 n/H11032/H20858k,qEn/H11032/H20849k,q/H20850on both sides of Eq. /H2084914/H20850, which reads Pt=/H6036/H9275/H20858 n,n/H11032/H20858 k,qfn/H20849k,q,t/H20850/H208511−fn/H11032/H20849k,q,t/H20850/H20852Wnn/H11032/H20849k,q/H20850,/H2084915/H20850 where Pt=/H11509/H20851/H20858n/H11032/H20858k,qEn/H11032/H20849k,q/H20850fn/H11032/H20849k,q,t/H20850/H20852//H11509tis the electronic energy-transfer rate per cell of the SL. The optical-absorption coefﬁcient can be calculated through30 /H9251=/H92510/H208492/H6036Pt/e2F02/H20850, /H2084916/H20850 where /H92510=e2//H20849/H6036/H92800c/H20881/H9260/H20850,/H9260, and /H92800are, respectively, the di- electric constants of the material layer and the free space,andcis the velocity of light in vacuum. Thus, the optical- absorption coefﬁcient is obtained as /H9251=C/H20858 n,n/H11032/H20858 k,qf/H20851En/H20849k,q/H20850/H20852/H208531−f/H20851En/H11032/H20849k,q/H20850/H20852/H20854/H20841/H9280·Mnn/H11032/H20849k,q/H20850/H208412 /H11003/H9254/H20851En/H11032/H20849k,q/H20850−En/H20849k,q/H20850−/H6036/H9275/H20852, /H2084917/H20850 where C=4/H9266/H6036/H92510//H20849m02/H9275/H20850. Here we used a steady-state energy distribution function such as the Fermi-Dirac function for thedistribution function, i.e., f n/H20849k,q,t/H20850/H11229f/H20851En/H20849k,q/H20850/H20852. The sum- mation in Eq. /H2084917/H20850has to be evaluated numerically. III. NUMERICAL RESULTS AND DISCUSSIONS The input material parameters for InAs and GaSb used in the present study are taken from Ref. 25. The overlap energy between the InAs conduction band and the GaSb valenceband is taken to be 150 meV . It is known that in NCA InAs/GaSb SLs, there are two different kinds of interfaces: the BAinterface GaSb-on-InAs at one side which can be grownInSb-like and the AB interface InAs-on-GaSb at the otherside which can be grown GaAs like. 31Thus, one has two additional parameters HXYBAand HXYABcharacterizing the strength of the InSb-like and GaAs-like interface potentials,which need to be determined in the calculation. Fortunately,we can obtain these two parameters by ﬁtting the experimen- tal data. We allow H XYABandHXYBAto vary such that it reaches the best agreement between the experimental and theoreticalband gaps as shown in Fig. 2, where in the upper panel the InAs width L Ais varied while the GaSb width LBis held constant and in the lower panel the GaSb width LBis varied while the InAs width LAis held constant. The best agreement was obtained for HXYAB=870 meV and HXYBA=490 meV which are of the same magnitude as those used in previous works.12,13HXYAB/HS11005HXYBAis due to the fact that the two inter- faces are not identical. We use a test InAs/GaSb NCA SL to display the main results of our band-structure calculation. The InAs/GaSblayer widths of the test structure are L A/H20849LB/H20850=25 /H2084925/H20850Å. The band structures of this SL calculated without and with theLI, XU, AND PEETERS PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-4MIA effect are shown in Figs. 3/H20849a/H20850and 3/H20849b/H20850, respectively, where d=LA+LBis the SL period. There are only four energy bands, labeled as CB1 for the ﬁrst conduction band, HH1and HH2 for the ﬁrst and second heavy-hole bands, and LH1for the ﬁrst light-hole band. The assignment of the carrier type to the various bands follows from the associated prop-erties of the envelope function at the zone center. ComparingFigs. 3/H20849a/H20850and3/H20849b/H20850, we notice three major differences: /H20849i/H20850at nonzero in-plane wave vector k/H20849 /H20648/H20851100 /H20852direction /H20850, the MIA effect creates a sizable spin slitting for both conduction andvalence bands due to the reduced interface symmetry. How-ever, the splitting is clearly more pronounced in the valencebands than in the conduction band. This is because the con-duction band is less affected by the MIA effect than thevalence bands, which is manifested by the interface Hamil-tonian /H208498/H20850where no direct coupling of conduction band to valence band exists. It appears that the K·Pmatrix elements which couple the electron and the light-hole states play a keyrole in the splitting of the conduction band; /H20849ii/H20850along the q direction /H20849 /H20648/H20851001 /H20852direction /H20850, the MIA effect causes an anti- crossing behavior between the LH1 and HH2 minibandswhich occurs at nonzero qvalue, which is consistent with a previous tight-binding calculation. 16This anticrossing is due to the zone center HH-LH mixing effect induced by the in-terface asymmetry; /H20849iii/H20850at the zone center, the MIA effect results in a signiﬁcant reduction in the band gap. This isbecause a number of valence-band anticrossings induced bythe zone center HH-LH mixing lead to band repulsions, i.e.,a rise of the HH1 band and thus a smaller band gap. The squared envelope functions at the zone center for the same test SL are calculated without and with the MIA effectand are, respectively, shown in Figs. 4/H20849a/H20850and4/H20849b/H20850. A com- mon feature in Figs. 4/H20849a/H20850and 4/H20849b/H20850is that, because of the different effective masses and different quantum size effects,the heavy holes are largely conﬁned in the GaSb layerwhereas the electron /H20849light-hole /H20850envelope functions overlap considerably from one InAs /H20849GaSb /H20850layer to another. Whencomparing the panels /H20849a/H20850and /H20849b/H20850in this ﬁgure, one can see that the CB1, HH1, LH1, and HH2 envelope functions havepeaks at the interface due to the localized interface potential,and what’s more, the peaks in the HH1 and HH2 envelopefunctions are much sharper than those of the CB1 and LH1envelope functions because the heavy holes are morebounded by the interface potential due to their larger effec-tive masses. The sharp peaks of the HH envelope functionsindicate that the interface potential has an ability to bind thehole at the interface. More importantly, due to the abovefeatures of the envelope functions, the overlap between theelectron and the hole states is expected to be considerablyenhanced by the MIA effect, which implies a strong type-IIoptical transition in the InAs/GaSb NCA SL system. For a given light polarization direction /H9280, we ﬁrst deﬁne the squared optical matrix element for the transition betweenthe bands /H9256and/H9257, I/H9256/H9257/H20849/H9280,k,q/H20850=/H20858 nn/H11032/H20841/H9280·Mnn/H11032/H20849k,q/H20850/H208412, /H2084918/H20850 where nruns over the states in the /H9256band and n/H11032runs over the states in the /H9257band. Then, the optical anisotropy with respect to the light polarization directions /H92801and/H92802can be evaluated by the polarization ratioFIG. 3. Band structures of a test InAs/GaSb SL with layer widths LA=LB=25 Å: /H20849a/H20850calculated without the MIA effect and /H20849b/H20850 calculated with the MIA effect.FIG. 2. Band-gap energy as a function of InAs /H20849GaSb /H20850width LA /H20849LB/H20850for a ﬁxed GaSb /H20849InAs /H20850width LB/H20849LA/H20850as indicated. Solid circles are the experimental data /H20849Ref. 32/H20850and solid line represents our theoretical results.INTRINSIC OPTICAL ANISOTROPY OF /H20851001 /H20852-GROWN … PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-5/H9261/H9256/H9257/H20849k,q/H20850=/H20841I/H9256/H9257/H20849/H92802,k,q/H20850−I/H9256/H9257/H20849/H92801,k,q/H20850/H20841 /H20841I/H9256/H9257/H20849/H92802,k,q/H20850+I/H9256/H9257/H20849/H92801,k,q/H20850/H20841. /H2084919/H20850 It was shown previously that the MIA effect due to the low C2vpoint-group symmetry causes an optical anisotropy with respect to the light polarization direction in the NCA SLs.5–7 To observe a pronounced optical anisotropy, the light polar- ization direction is normally set as the /H20851110 /H20852and /H2085111¯0/H20852di- rections in the experiment,5,6which will become clear in the following. In our numerical calculation, we vary /H9280over all the in-plane polarization angles /H9258/H208490/H110112/H9266/H20850and calculate the squared optical matrix elements for the test InAs/GaSb SL asa function of /H9258. Figures 5/H20849a/H20850and5/H20849b/H20850display, respectively, the calculated results for the HH1-CB1 and LH1-CB1 tran-sitions at the zone center without and with the MIA effect.Comparing panels /H20849a/H20850and /H20849b/H20850in this ﬁgure, we notice that /H20849i/H20850 in the absence of the MIA effect, the optical matrix elementsare independent of the light polarization direction, which ismanifested by the isotropic circles in panel /H20849a/H20850; and /H20849ii/H20850with the MIA effect included, the optical matrix elements exhibita strong in-plane anisotropy and have a twofold in-planesymmetry which corresponds to the C 2vpoint-group symme- try of the NCA SL system /H20851see panel /H20849b/H20850/H20852. It is clear that theoptical anisotropy is induced by the MIA effect and the larg- est anisotropy is induced by the light polarized along the /H20851110 /H20852and /H2085111¯0/H20852directions. The reason why this gives the largest optical anisotropy is the nonequivalence of the /H20851110 /H20852 and /H2085111¯0/H20852in-plane directions at two successive interface planes where the chemical bonds are spatially arranged dif-ferently. From the device application point of view, our re-sults provide guidelines for designing optical devices whichrequire the largest optical polarization to achieve an optimalperformance. In the following, when calculating the intrinsicoptical anisotropy in the InAs/GaSb NCA SLs, we will set the light polarized along the /H20851110 /H20852and /H2085111 ¯0/H20852directions as in the experiments5,6because these two directions give rise to the largest optical anisotropy. The optical-matrix elements of the HH1-CB1, LH1-CB1, and HH2-CB1 transitions for the test InAs/GaSb SL as func-tion of the in-plane wave vector kand the wave vector q along the growth direction are shown in Fig. 6, where panels /H20849a/H20850and /H20849b/H20850are, respectively, calculated without and with the MIA effect. As can be seen from Figs. 6/H20849a/H20850and6/H20849b/H20850:/H20849i/H20850since the test SL considered here is thin /H20849L A=LB=25 Å /H20850, the op- tical matrix elements are strongly dependent on both kandq; and /H20849ii/H20850along both the kand the qdirections, the MIA effectFIG. 4. Squared envelope functions at the zone center for the test InAs/GaSb SL with LA=LB=25 Å: /H20849a/H20850calculated without the MIA effect and /H20849b/H20850calculated with the MIA effect. The vertical dashed lines indicate the interface between InAs and GaSb.FIG. 5. Contour plot of angular dependence of the squared op- tical matrix elements at the zone center for the test InAs/GaSb SL:/H20849a/H20850calculated without the MIA effect and /H20849b/H20850calculated with the MIA effect.LI, XU, AND PEETERS PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-6gives rise to a large optical anisotropy in the optical-matrix elements induced by /H20851110 /H20852- and /H2085111¯0/H20852-polarized light. This is attributed to the fact that the MIA effect breaks not only therotational symmetry around the zaxis or in-plane symmetry which causes the k-dependent anisotropy in the optical- matrix elements but also the translational invariance alongthezaxis or the growth symmetry that produces the q-dependent anisotropy in the optical-matrix elements. To calculate the optical-absorption coefﬁcient for the NCA InAs/GaSb SL system, we use a Lorentzian function tobroaden the energy conservation delta function in Eq. /H2084917/H20850. Namely, we take /H9254/H20849x/H20850=/H20849/H9003//H9266/H20850//H20849x2+/H90032/H20850with/H9003being the broadening factor induced by different scattering mecha-nisms. Furthermore, we use a phenomenological formula 33to calculate this broadening factor: /H9003=/H90031+/H90032//H20851exp /H20849/H6036/H92750/kBT/H20850 −1/H20852with/H90031and/H90032being constants and /H92750the optical- phonon frequency. The ﬁrst and the second term in this for-mula represent, respectively, the broadening factors inducedby temperature-independent and temperature-dependent scat-tering mechanisms, i.e., time-independent scattering mecha-nisms mainly refer to the scattering of carriers with impurityand interface roughness and those of temperature-dependentrefer to the scattering of carriers with acoustic-phonons and optical-phonons. As the optical-phonon frequencies for InAsand GaSb is nearly equal, we take /H6036 /H92750/H20849InAs /H20850/H11229/H6036/H92750/H20849GaSb /H20850 =30 meV. Because there are no published values for /H90031and /H90032in the InAs/GaSb SL system, in the present work we take /H90031=4.5 meV and /H90032=44.3 meV. With such two values, the broadening factor /H9003is 5 meV at T=77 K and 25 meV at T=300 K, which is in line with the magnitude of general temperature-dependent absorption linewidths.34,35It is known that in an undoped InAs/GaSb SL, the presence of thelight ﬁeld can pump electrons from the GaSb valance bandinto the InAs conduction band. In general, the density ofphotoexcited carriers depends on the light intensity and fre-quency as well as other scattering and relaxation mecha-nisms. In this study, we assume that the photoexcited carrierdensity N 0in undoped SLs is about 1015–1018cm−3. Figures 7/H20849a/H20850and7/H20849b/H20850show the optical-absorption spectra for the test InAs/GaSb SL calculated without and with theMIA effect, respectively, for a ﬁxed temperature /H20849T=77 K /H20850 and a ﬁxed photoexcited carrier density /H20849N 0=1016cm−3/H20850. The optical-absorption spectra are found to be quite broadand nearly structureless. This is because the thin SL consid-ered here has a strong dispersed subband and miniband struc-ture along the kand the qdirections. Due to the lifting of the parity selection rules in the type-II NCA SL system, theFIG. 6. Squared optical matrix elements for the test InAs/GaSb as functions of the in-plane wave vector kand the growth wave vector q:/H20849a/H20850calculated without the MIA effect and /H20849b/H20850calculated with the MIA effect. The solid and dashed lines represent, respec- tively, the /H20851110 /H20852and /H2085111¯0/H20852light polarization directions.FIG. 7. Optical absorption spectra for the test InAs/GaSb SL: /H20849a/H20850 calculated without the MIA effect and /H20849b/H20850calculated with the MIA effect. The results are calculated at a ﬁxed temperature T=77 K and a ﬁxed photoexcited carrier density N0=1016cm−3. The solid and dashed lines represent, respectively, the /H20851110 /H20852and /H2085111¯0/H20852light polarization directions.INTRINSIC OPTICAL ANISOTROPY OF /H20851001 /H20852-GROWN … PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-7parity-forbidden optical transition between the HH2 and CB1 states with opposite parity becomes allowed, which is not soin type-I CA SL structures, where only the parity-allowedoptical transitions between the HH /H20849LH/H20850and the CB states with the same parities such as the HH1-CB1 and LH1-CB1transitions can take place. However, the strength of the HH2-CB1 optical transition is small due to the very small overlapbetween the HH2 and the CB1 envelope functions at theinterface. Comparing Figs. 7/H20849a/H20850and7/H20849b/H20850, we notice that the MIA effect causes a giant in-plane optical anisotropy in the optical-absorption coefﬁcients for /H20851110 /H20852- and /H2085111 ¯0/H20852-polarized light, which is a direct consequence of the polarized opticalmatrix elements shown in Fig. 6/H20849b/H20850. Because the absorption strength is proportional to the optical matrix elements, it islarger in the presence of the MIA effect than in the absenceof this effect for /H20851110 /H20852-polarized light and the case is re- versed for /H2085111 ¯0/H20852-polarized light. In addition, the absorption cutoff is blueshifted due to the MIA effect because this effectreduces the fundamental band gap. The relative difference or polarization ratio between the two polarized absorption spectra shown in Fig. 7/H20849b/H20850is plot- ted as a function of photon frequency in Fig. 8. From this ﬁgure, we can see that: /H20849i/H20850the polarization spectrum is quite broad /H20849over the whole spectral region /H20850since all the transi- tions from the valence bands /H20849HH1, HH2, and LH1 /H20850to the conduction band /H20849CB1 /H20850contribute to the optical absorption; and /H20849ii/H20850the maximum value of the polarization ratio is close to 55% and the position of this value is located within thespectral region for the HH1-CB1 transitions. This is becausethe largest relative difference between the two polarized op-tical matrix elements, as shown in Fig. 6/H20849b/H20850, occurs for the HH1-CB1 transitions. Thus the highest polarization ratio canbe observed within the spectral region for these transitions. To demonstrate that the optical anisotropy is more sensi- tive to the MIA effect in type-II NCA SLs than in type-INCA SLs, we compared our results for the InAs/GaSb sys-tem with those obtained by Krebs et al. 5for the GaAs/InP system. The polarization spectra in those two systems areremarkably different, which is manifested by the maximumvalue of the polarization ratio /H20849up to 55% vs up to 40% /H20850andby the broadening of the polarization spectrum /H20849about 350 meV vs about 50 meV /H20850. Thus, the optical anisotropy in InAs/ GaSb NCA SLs is more sensitive to the MIA effect becauseit depends essentially on the overlap between the electronand the hole wave functions at the interface. Figure 9displays the temperature dependence of the po- larization spectrum for the test InAs/GaSb SL as a functionof photon frequency. The photoexcited carrier density isﬁxed at N 0=1016cm−3. As can be seen from this ﬁgure, the polarization ratio decreases and the spectral shape broadenswith increasing temperature. The polarization ratio isstrongly temperature dependent at high temperatures /H20849T /H1102277 K /H20850because for this temperature region the main con- tribution to the broadening factor /H9003comes from the scatter- ing of carriers with optic phonons while it is nearly tempera-ture independent at low temperatures /H20849T/H1102177 K /H20850since at this temperature region the broadening is dominated by thescattering of carrier with interface roughness /H20849we assume an impurity-free SL structure /H20850. Thus, we anticipate that the scat- tering mechanisms will greatly affect the polarization spec-trum induced by the MIA effect in the NCA SL system. The dependence of the polarization spectrum on the pho- toexcited carrier density for the test InAs/GaSb is plotted asa function of the photon frequency in Fig. 10. The tempera- ture is ﬁxed at T=77 K. From this ﬁgure, we notice that both the peak position and magnitude of the polarizationspectrum vary signiﬁcantly with the photoexcited carrierdensity. With increasing density, the quasi-Fermi levels forelectrons in the InAs layer and holes in the GaSb layer risesand falls, respectively. As a consequence, the peak position of the polarization spectrum is blueshifted because it reﬂectsthe energy separation between the electron and hole quasi-Fermi levels around which the strongest optical transitionsoccur. In addition, the magnitude of the peak in the polariza-tion spectrum decreases with increasing photoexcited carrierdensity. The reason is that a higher density opens up asmaller number of optical transition channels due to bandﬁlling effects and thus induces a smaller polarization ratio asderived from the corresponding polarized optical-absorptioncoefﬁcients. Therefore, to achieve a larger polarization ratiofor InAs/GaSb NCA SLs, it is necessary to work at relativelylow temperatures and low carrier density.FIG. 8. Relative difference or polarization ratio for Fig. 7/H20849b/H20850. The temperature is ﬁxed at T=77 K and the photoexcited carrier density is ﬁxed at N0=1016cm−3.FIG. 9. Dependence of the polarization ratio for the test InAs/ GaSb SL on the temperature as indicated. The photoexcited carrierdensity is ﬁxed at N 0=1016cm−3.LI, XU, AND PEETERS PHYSICAL REVIEW B 82, 235422 /H208492010 /H20850 235422-8An important feature of InAs/GaSb SLs is that the ab- sorption cutoff and strength can be tuned by simply adjustingthe InAs and/or GaSb layer widths. Thus, one can anticipatethat this feature may have an inﬂuence on the optical-absorption spectrum and thus the polarization spectrum. Fig.11shows the polarization spectrum for different InAs/GaSb layer widths at a ﬁxed temperature T=77 K and a ﬁxed pho- toexcited carrier density N 0=1016cm−3. As mentioned above, the peak position of the polarization spectrum is lo-cated within the HH1-CB1 transition energies. As a result, itis redshifted when increasing InAs/GaSb layer widths be-cause such an increase reduces the energy separation be-tween the electron and the hole conﬁnement energies due toquantum size effect. Due to the fact that in shorter SL struc- tures the percentage of the interface contribution to the opti-cal anisotropy induced by the MIA effect increases signiﬁ-cantly, the peak magnitude of the polarization spectrumincreases with decreasing InAs/GaSb layer widths. In addi-tion, one can observe that in the thicker SL structures /H20849i.e., L A=LB=30, 35 Å /H20850, there are other small peaks in the polar- ization spectra, which result from optical transitions withhigher transition energies. To close this section, we state that the MIA effect on the optical anisotropy in InAs/GaSb NCA SLs as suggested byour theoretical results can be detected in an experiment bymeasuring the optical-absorption coefﬁcient with respect tothe in-plane light polarization direction. The changing of thelight polarization direction can be realized by simply usingan optical polarizer.IV . CONCLUSIONS We have studied theoretically the intrinsic optical aniso- tropy induced by the MIA effect as is present in InAs/GaSbNCA SLs. The eight-band K·Pmodel incorporating the MIA effect was solved using the ﬁnite difference methodwhich gave us the electronic states in such SLs. With theobtained band structure and associated wave functions, weused the Boltzmann equation approach to calculate the opti-cal properties for the corresponding SL system. It was shownthat in the InAs/GaSb NCA SL system, the MIA effectcauses a large optical anisotropy for linearly polarized light and the largest anisotropy occurs for the /H20851110 /H20852and /H2085111 ¯0/H20852 light polarization directions. The giant optical anisotropy wasmanifested by the optical matrix elements and the optical-absorption coefﬁcients. Moreover, we found that the opticalanisotropy and the polarization ratio decreases with increas-ing temperature, photoexcited carrier density and layerwidths of InAs/GaSb NCA SLs. These results could enableus to better understand the physical mechanisms behind theapplication of InAs/GaSb NCA SLs as optical devices whichrequire polarization control and selectivity. ACKNOWLEDGMENTS This work was supported partly by the Flemish Science Foundation /H20849FWO-VL /H20850, the Belgium Science Policy /H20849IAP /H20850, the NSF of China /H20849Grants No. 10664006, No. 10504036, and No. 90503005 /H20850, Special Funds of 973 Project of China /H20849Grant No. 2005CB623603 /H20850, and Knowledge Innovation Program of the Chinese Academy of Sciences. *wenxu_issp@yahoo.cn †francois.peeters@ua.ac.be 1J. H. Ser, Y . G. Ju, J. H. Shin, and Y . H. Lee, Appl. Phys. Lett. 66, 2769 /H208491995 /H20850. 2Y . G. Ju, Y . H. Lee, H. K. Shin, and I. Kim, Appl. Phys. Lett. 71,741 /H208491997 /H20850. 3Y . G. Ju, J. H. Ser, and Y . H. Lee, IEEE J. Quantum Electron. 33, 589 /H208491997 /H20850. 4O. Krebs and P. V oisin, Phys. Rev. Lett. 77, 1829 /H208491996 /H20850. 5O. Krebs, W. Seidel, J. P. André, D. Bertho, C. Jouanin, and P.FIG. 11. Polarization spectrum for different InAs/GaSb layer widths as indicated. 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PhysRevB.79.161407.pdf	Tuning the Josephson current in carbon nanotubes with the Kondo effect A. Eichler,1R. Deblock,2M. Weiss,1C. Karrasch,3V . Meden,3C. Schönenberger,1and H. Bouchiat2 1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland 2Laboratoire de Physique des Solides, CNRS, UMR 8502, Université Paris-Sud, F-91405 Orsay Cedex, France 3Institut für Theoretische Physik A and JARA—Fundamentals of Future Information Technology, RWTH Aachen University, 52056 Aachen, Germany /H20849Received 10 March 2009; published 28 April 2009 /H20850 We investigate the Josephson current in a single wall carbon nanotube connected to superconducting elec- trodes. We focus on the parameter regime in which transport is dominated by Kondo physics. A sizeablesupercurrent is observed for odd number of electrons on the nanotube when the Kondo temperature T Kis sufﬁciently large compared to the superconducting gap. On the other hand when, in the center of the Kondoridge, T Kis slightly smaller than the superconducting gap, the supercurrent is found to be extremely sensitive to the gate voltage VBG. Whereas it is largely suppressed at the center of the ridge, it shows a sharp increase at a ﬁnite value of VBG. This increase can be attributed to a doublet-singlet transition of the spin state of the nanotube island leading to a /H9266shift in the current phase relation. This transition is very sensitive to the asymmetry of the contacts and is in good agreement with theoretical predictions. DOI: 10.1103/PhysRevB.79.161407 PACS number /H20849s/H20850: 72.15.Qm, 73.21. /H11002b, 73.63.Fg, 74.50. /H11001r Metallic single wall carbon nanotubes /H20849SWNTs /H20850have at- tracted a lot of interest as one-dimensional /H208491D/H20850quantum wires combining a low carrier density and a high mobility.Depending on the transparency of the interface between thenanotube and the electrode, the conduction ranges from in-sulating behavior and strong Coulomb blockade at low trans-parency to nearly ballistic transport with conductance closeto 4e 2/hwhen the transparency is high.1,2The intermediate conduction regime is particularly interesting because in thecase of an odd number of electrons on the nanotube astrongly correlated Kondo resonant state can form, where themagnetic moment of the unpaired spin is screened by the spins of the electrons in the leads. 3Moreover, when the car- bon nanotubes /H20849CNTs /H20850are in good contact with supercon- ducting electrodes, it is possible to induce superconductivityand observe supercurrents, as was ﬁrst investigated in un-gated suspended devices. 4,5Proximity induced superconduc- tivity was then explored in gated devices with evidence of astrong modulation of subgap conductance, but in most casesno supercurrent was observed. 6–9More recently, tunable su- percurrents could be detected in the resonant tunneling con-duction regime, where the transmission of the contacts ap-proaches unity. 10,11In this regime, the discrete spectrum of the nanotube is still preserved and the maximum value ofsupercurrent is observed when the Fermi energy of the elec-trodes is at resonance with “the electron in a box states” ofthe nanotube. A superconducting interference device wasfabricated with carbon nanotubes as weak links: supercurrent /H9266phase shifts occurred when the number of electrons in the nanotube dot was changed from odd to even12corresponding to the transition from a magnetic to a nonmagnetic state.Sharp discontinuities in the critical current at this 0− /H9266tran- sition in relation with the even-odd occupation number of thenanotube quantum dot were also observed in single nanotubejunction devices. 13As pointed out in the superconducting quantum interference device /H20849SQUID /H20850experiment12a Jo- sephson current can be observed in the Kondo regime, whenthe Kondo temperature T Kis large compared to the supercon-ducting gap /H9004, conﬁrming theoretical predictions14–19and previous experiments7,20/H20849with no determination of supercur- rents though /H20850. In the present work we explore in detail this competition between Josephson and Kondo physics by monitoring on thesame device as a function of the gate voltage the bias depen-dence of the differential conductance in the normal state andthe Josephson current in the superconducting state. The valueof this current is precisely determined by ﬁtting the data witha theoretical model explicitly including the effect of the elec-tromagnetic environment onto the junction. From the normalstate data, we extract the charging energy Uand the Kondo temperature T Kas a function of the gate voltage and calcu- late the sum of the couplings to the electrodes /H9003=/H9003L+/H9003R from the known form of TK.21The asymmetry /H9003R//H9003Lis de- termined from the value of the conductance on the Kondoresonance for T/H11270T K. The value of this current is precisely determined by ﬁtting the data with a theoretical model ex-plicitly including the effect of the dissipative electromagneticenvironment onto the junction. As expected a supercurrent isobserved /H20849for an odd number of electrons on the nanotube /H20850 when T K/H11022/H9004 and when the asymmetry of the transmission of the electrodes is not too large.14–19On the other hand, when TK/H11021/H9004, the magnetic spin remains unscreened at all tempera- tures leading to a /H9266junction with a very low transmission of Cooper pairs.22We have particularly explored in this Rapid Communication the intermediate regime where the Kondotemperature can be tuned with the gate voltage, on a singleKondo ridge, from a value slightly below the superconduct-ing gap at half ﬁlling to a value T K/H11022/H9004. A sharp increase in the supercurrent is then observed, which is related to thetransition from a magnetic doublet state to a nonmagneticsinglet state of the nanotube island. We compare these resultswith functional renormalization group /H20849FRG /H20850calculations for the single impurity Anderson model 19in a wide region of gate voltage. We grow SWCNTs by chemical vapor deposition on ther- mally oxidized, highly doped silicon wafers. IndividualPHYSICAL REVIEW B 79, 161407 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 1098-0121/2009/79 /H2084916/H20850/161407 /H208494/H20850 ©2009 The American Physical Society 161407-1SWCNTs are located relative to predeﬁned markers and con- tacted to Ti/Al leads using electron-beam lithography /H20851Figs. 1/H20849a/H20850and1/H20849b/H20850/H20852. The Ti/Al leads are superconducting below 1 K. Measurements were done in a dilution refrigerator with abase temperature of T=40 mK. The cryostat was equipped with a three-stage ﬁltering system consisting of LC ﬁlters atroom temperature, resistive microcoax cables, and ﬁnally mi-croceramic capacitors in a shielded metallic box, which alsocontained the samples and was tightly screwed onto the coldﬁnger. As additional ﬁlters against voltage ﬂuctuations, resis-tors of the order of 2 k /H9024were implemented on chip as me- anders in the Au lines connecting the superconducting elec-trodes to the contact pads /H20851Fig. 1/H20849a/H20850/H20852. Measurements were done with a lock-in technique with either an ac voltage of5–10 /H9262V, measuring differential conductance dI/dV,o ra n ac current of 10 pA, measuring differential resistance dV /dI, both as a function of an additional dc-bias voltage or current,respectively. Figure 1/H20849c/H20850shows a schematic view of our de- vice. It consists of a Josephson junction with current I J/H20849/H9278/H20850 parallel to a capacitor CJand a shunt resistance RJ. We ob- tain a rough estimate of CJ/H11015100 aF from the charging en- ergy U=e2/2CJ. The junction resistor RJrepresents the con- tribution of quasiparticles to dissipation at frequencies of theorder of the plasma frequency of the junction /H9275P. The outer capacitance Crepresents the capacitances of the leads to ground within the electromagnetic horizon of the junction,which we estimate as 2 /H9266c//H9275pto some centimeters, depend- ing on RJ. It is therefore mainly determined by the capaci- tance of the metallic leads to the highly doped backgate andcan be estimated to C/H110158 pF. 23The series resistances Rrep- resent the lithographically deﬁned on-chip resistors. Thissituation corresponds to the “extended resistively and capaci-tively shunted junction” /H20849extended RCSJ /H20850model, 10,13,24,25 which we will use later. We ﬁrst characterize the CNT quantum dot with the elec- trodes driven normal by a small magnetic ﬁeld /H20849100 mT /H20850. The color scale plot of dI/dVas a function of VbiasandVBG /H20849the “charge stability diagram” /H20850displays a regular sequence of “Coulomb blockade diamonds” over a wide range of VBG. In other regions this pattern is replaced by a smoother gatedependence characteristic of Fabry-Perot oscillations. 1For further analysis, we concentrate on the gate voltage regionbetween 3 and 4 V olts /H20851see Fig. 2/H20849a/H20850/H20852. It shows a fourfold periodicity in the size of the Coulomb blockade diamonds,26 which indicates a clean nanotube with the twofold orbitaldegeneracy of the electronic states preserved. We extractU=2.5/H110060.3 meV as estimated from the size of Coulomb blockade diamonds with an odd number of electrons. In allstates with odd occupation, the Kondo effect manifests itselfthrough a high conductance region around zero bias, the so-called Kondo ridge. The Kondo temperature T Kcan be esti- mated from the half width of the peaks of these lines whichcan be ﬁtted by Lorentzian curves. 27The Kondo temperature goes through a minimum on the order o f1Ki nt h e center of the ridges and increases on the edges. It is possible to followthis gate dependence along the Kondo ridges A–C /H20851Fig. 2/H20849c/H20850/H20852. The intensity on ridge D is too weak for such an analy- sis.T Kcan be well ﬁtted by the expression predicted by the Bethe Ansatz,21 TK=/H20881U/H9003/2 exp/H20875−/H9266 8U/H9003/H208414/H92802−U2/H20841/H20876, /H208491/H20850 where /H9280is the energy shift measured from the center of the Kondo ridge. Taking U=2.5 meV as determined above, the value of TKat/H9280=0 leads to the characteristic coupling ener- gies/H9003=/H9003R+/H9003Lbetween the electrodes for each Kondo ridge /H20851Fig.2/H20849c/H20850/H20852. The gate voltage VBGdependence of TKyields the ratio/H9251between the electrostatic energy eV BGand the Fermi energy of the nanotube /H9280equal to 20 /H110061. This value agrees source 1 source 2drain 1 drain 2on-chip resistors deviceshighly doped SiSiO2(400 nm)Al (100 nm)SWCNT~550 nm Ti (6 nm)a) b) c) R/2 CJ CV IJ RJ R/2100µm 1µm FIG. 1. /H20849Color online /H20850/H20849a/H20850Scanning electron micrograph of the electrical on-chip environment of the nanotube. /H20849b/H20850Cross section of the nanotube with electrodes and backgate /H20849c/H20850Schematic of the carbon nanotube Josephson junction with its environment accordingto the extended RCSJ model.2.0 1.0 0.0 -1.0 -2.0 G( e2/h) 0.4 -0.40.2 -0.20.0 3.4 3.5 3.6 3.723 4 0.5 1.5 2.5 3.5 G( e2/h)2∆ ∆ -∆ -2∆ 1D C B Aa) b)normal state superconducting stateVbias (mV) VBG(V) VBG(V)Vbias (mV) 3.4 3.5 3.6 3.7 23 4 1 A B C D 0.25 0.75 1.25 3.39 3.40 3.41 3.42 3.43012345TK)K( VBG(V)3.50 3.51 3.52 3.53 3.54 VBG(V)3.59 3.60 3.61 3.62 3.63 VBG(V)U=1 7 ∆ Γ= 2.9∆U=1 7 ∆ Γ= 2.7∆U=1 7 ∆ Γ= 3.2∆c) AB C FIG. 2. /H20849Color online /H20850Color scale plot showing gate and voltage bias dependence of the differential conductance. /H20849a/H20850Normal state. /H20849b/H20850Superconducting state. White lines correspond to vertical sec- tions of Gversus Vbiasin the middle of the four Kondo ridges labeled A–D. /H20849c/H20850Gate dependence of the Kondo temperature ex- tracted on the Kondo ridges far from the degeneracy points. The redlines are ﬁts to Eq. /H208491/H20850; the black line corresponds to T C=1.0 K. U and/H9003are given in units of /H9004.EICHLER et al. PHYSICAL REVIEW B 79, 161407 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 161407-2to within 20% with the value deduced from the normal state conductance data. This ﬁt based on the single level Andersonimpurity model is only valid when U//H9003is sufﬁciently large and /H20841 /H9280/H20841/H11270U. Since for all peaks T/H11270TK, the maximum con- ductance of the ridges yields the asymmetry of the coupling/H9003 R//H9003L, 6.8, 6.2, 2.5, and 70 for A–D, respectively. By switching off the magnetic ﬁeld, we allow the leads to become superconducting. Figure 2/H20849b/H20850shows dI/dVfor the same gate voltage range as in Fig. 2/H20849a/H20850but for a smaller bias voltage range. The BCS-type density of states in the elec-trodes leads to new features in the stability diagram, whichare horizontal lines at V bias=/H110062/H90040and/H11006/H9004 0due to the onset of quasiparticle tunneling and Andreev reﬂection, respec-tively. We can derive the value of the superconducting gap/H9004 0=0.15/H110060.02 meV, which corresponds well to the ex- pected TC/H112291 K for the bilayer Ti/Al. Although they have similar Kondo temperatures in the normal state, the Kondoridges, A–D, show a very different behavior when the elec-trodes are superconducting. The Kondo ridges /H20849A, B, and D /H20850 are suppressed or reduced in amplitude by superconductivity,reﬂecting that T K/H11021TCin the center of the Kondo ridge. Con- trary to ridges A, B, and D that show a minimum at zerobias, there is a strong enhancement of conductance in ridgeC, with Greaching a value roughly four times larger than in the normal state. Note that states A, C, and D show an en-hancement of conductance at V bias=/H9004/eto values larger than the conductance at Vbias=2/H9004/e. This enhancement of the ﬁrst Andreev process is due to the “even-odd” effect in Andreevtransport, which has recently been described in Refs. 9and 28. For a measurement of the supercurrent the device has to be current biased. We simultaneously use ac and dc biaswhile measuring the resulting voltage drop across it. Fromthe ac part, we obtain data on the differential resistance /H20851Fig. 3/H20849a/H20850/H20852. By numerical integration we get I-Vcurves that show a supercurrent branch and a smooth transition to a resistivebranch with higher resistance /H20851Fig. 3/H20849b/H20850/H20852. The transition be- tween the two regimes is not hysteretic, and the supercurrentpart exhibits a nonzero resistance R Sat low bias even if we subtract the contribution of the on-chip resistances R. This behavior is common in mesoscopic Josephson junctions thathave a high normal state resistance on the order of the resis-tance quantum h/e 2. To extract the supercurrent, we use a theory that explicitly includes the effect of the dissipativeelectromagnetic environment onto the junction in the frameof the already mentioned extended RCSJ model. 10,13,24Using the external resistor R/H20849Ref. 29/H20850/H20851Fig.1/H20849c/H20850/H20852and temperature T as input parameters, we can thus extract the critical current Ic and the junction resistance RJ, for every measured backgate voltage, from a ﬁt to I/H20849Vbias/H20850=/H20877IcIm/H20875I1−i/H9257/H20849Ic/H6036/2ekBT/H20850 I−i/H9257/H20849Ic/H6036/2ekBT/H20850/H20876+Vbias Rj/H20878Rj Rj+R,/H208492/H20850 where /H9257=/H6036Vbias /2eRk BTand I/H9251/H20849x/H20850is the modiﬁed Bessel function of complex order /H9251.13The resulting values are plot- ted as a function of VBGin Figs. 3/H20849c/H20850and3/H20849d/H20850. The junction conductance GJ=1 /RJrelates well to the differential conduc- tance GS=1 /RSextracted from the ac part of the current biased data, especially around the resonance degeneracypoints where the conductance is high. The normal state con- ductance GNdeviates from GSmost notably in states B and C, where Kondo physics plays a key role. In state B,G N/H11022GS, whereas the ridge in state C persists in the super- conducting state, resulting in a further enhancement of theconductance. The supercurrent exhibits peaks at the maximum values of the conductance /H20851Fig. 3/H20849d/H20850/H20852. Its behavior between peaks var- ies strongly in states with even and odd occupations. In thefollowing we focus on Kondo ridges B and C. Whereas thesupercurrent on ridge C varies nearly proportionally to theconductance, we observe a sharp drop of supercurrent onridge B as illustrated in Fig. 3/H20849d/H20850/H20849arrows /H20850. This can be un- derstood considering that T Kis smaller on ridge B than on ridge C /H20851see Fig. 2/H20849c/H20850/H20852. As a result the magnetic moment of the excess electron on the nanotube remains unscreened inthe superconducting state near /H9280=0 which is not the case for ridge C. At this stage we compare our experimental ﬁndingsto approximate zero-temperature FRG calculations which al-low for extracting both the stability regions of the screenedsinglet and magnetic unscreened doublet phases and thecomplete supercurrent-phase relation I J/H20849/H9278/H20850within a model of an Anderson impurity coupled to two superconductingelectrodes. 19It was previously observed that the Josephson current cannot be simply described as a single function of theratio T K//H9004.14The relevant parameters are the on site Cou- lomb repulsion energy U, the level position /H9280related to VBG, and the transmission of the electrodes /H9003Rand/H9003Lcompared to the superconducting gap. In Fig. 4we present a comparison between our experimental data to the theoretical predictionsforI c/H20851deﬁned as the maximum value of /H20841IJ/H20849/H9278/H20850/H20841over/H9278/H20852,a sa function of the position in energy of the Anderson impuritylevel, for similar values of theoretical and experimental pa- 4.0 -4.02.0 -2.00.0 3.50 3.55 3.60 01 25 250 dV/dI (k Ω)I (nA) 12 3 4 VBG(V)a)b ) BC 0.01.02.03.04.05.0IC)An( VBG(V)3.50 3.55 3.60c) d) VBG(V)0.02.04.06.0e(G2/h) 3.50 3.55 3.601/RJ GS GNIC-15 -10 -5 0 5 10 1 5-2-1012 3.596 V 3.584 V 3.578 V 3.520 V)An(I Vbias(µV) FIG. 3. /H20849Color online /H20850/H20849a/H20850Color scale plot of the differential resistance as a function of dc-bias current and backgate voltage. /H20849b/H20850 I-Vcharacteristics for different backgate voltages. The red/gray lines are ﬁts using Eq. /H208492/H20850./H20849c/H20850Conductances GJ=1 /RJ/H20849black circles /H20850,GS/H20849blue/dark gray line /H20850, and GN/H20849red/gray line /H20850./H20849d/H20850Gate voltage dependence of the critical current extracted from ﬁts to Eq./H208492/H20850.TUNING THE JOSEPHSON CURRENT IN CARBON … PHYSICAL REVIEW B 79, 161407 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 161407-3rameters. It is remarkable that the gate dependence of the critical current on the Kondo ridges can be qualitatively de-scribed, that is, in particular, the existence of a singlet, dou-blet /H208490/ /H9266/H20850transition on ridge B at nearly the right value of gate voltage when renormalized with the charging energy U /H20849arrows /H20850and the absence of a /H208490//H9266/H20850transition for ridge C corresponding to higher values of TK. One point of disagree- ment between theory and experiment concerns the amplitudeof the critical current in the doublet state /H20849 /H9266junction /H20850region /H20849around VBG=3.65 V /H20850.Icis theoretically found to be re- duced by only a factor of 2 compared to its value in thesinglet region, whereas basically no trace of superconductiv- ity could be detected experimentally. It is known, however,that the /H9266-phase current computed from the approximate FRG is too large compared to numerical renormalization group /H20849NRG /H20850data, which are known to be more accurate but only available at the center of the Kondo ridge.19 Finally, let us emphasize the importance of the asymmetry of the transmission of the electrodes which tends to reduceconsiderably the supercurrent. This is particularly striking forthe data on ridge D for which /H9003 R//H9003L=70, where no super- current could be measured in spite a value of TKon the order of 1 K. This reduction of Icby the asymmetry of contacts is also found in FRG calculations.19It is moreover accompa- nied by a modiﬁcation of the stability regions for the singlet/H20849screened /H20850and doublet /H20849magnetic /H20850states which strongly de- pend on the phase difference between the superconductingelectrodes. The nonmagnetic singlet state is stabilized withrespect to the magnetic doublet state in the vicinity of /H9278=/H9266, which results in a strong modiﬁcation of the current phase relation which unfortunately cannot be checked in asingle I cmeasurement.19 In conclusion, we have shown that it is possible to tune the amplitude of the supercurrent in a carbon nanotube Jo-sephson junction within a Kondo ridge in a very narrowrange of gate voltage. Our data are in good qualitative agree-ment with theoretical ﬁndings and should stimulate new ex-periments where the whole current phase relation ismeasured. 30 We acknowledge fruitful discussions with Y . Avishai, S. Guéron, W. Belzig, A. Levy-Yeyati, T. Novotný, J. Paaske,and B. Röthlisberger. This work was supported by the EU-STREP program HYSWITCH and by the DeutscheForschungsgemeinschaft via FOR 723 /H20849C.K. and V .M. /H20850. 1W. Liang et al. , Nature /H20849London /H20850411, 665 /H208492001 /H20850. 2B. Babi ćand C. Schönenberger, Phys. Rev. B 70, 195408 /H208492004 /H20850. 3D. Goldhaber-Gordon et al. , Nature /H20849London /H20850391, 156 /H208491998 /H20850. 4A. Y . Kasumov et al. , Science 284, 1508 /H208491999 /H20850. 5A. Kasumov et al. , Phys. Rev. B 68, 214521 /H208492003 /H20850. 6A. F. Morpurgo et al. , Science 286, 263 /H208491999 /H20850. 7M. R. Buitelaar et al. , Phys. Rev. Lett. 89, 256801 /H208492002 /H20850;M .R . Buitelaar et al. ,ibid. 91, 057005 /H208492003 /H20850. 8H. I. Jorgensen et al. , Phys. Rev. Lett. 96, 207003 /H208492006 /H20850. 9A. Eichler et al. , Phys. Rev. Lett. 99, 126602 /H208492007 /H20850. 10P. Jarillo-Herrero et al. , Nature /H20849London /H20850439, 953 /H208492006 /H20850. 11Tunable supercurrents through Carbon nanotubes have also been observed in T. Tsuneta et al. , Phys. Rev. Lett. 98, 087002 /H208492007 /H20850; Y . Zhang et al. , Nano Research 1, 145 /H208492008 /H20850; E. Pal- lecchi et al. , Appl. Phys. Lett. 93, 072501 /H208492008 /H20850;G .L i u et al. , Phys. Rev. Lett. 102, 016803 /H208492009 /H20850. 12J.-P. Cleuziou et al. , Nat. Nanotechnol. 1,5 3 /H208492006 /H20850; Note that similar devices were made with semiconducting nanowires in J.A. van Dam et al. , Nature /H20849London /H20850442, 667 /H208492006 /H20850. 13I. Jorgensen et al. , Nano Lett. 7, 2441 /H208492007 /H20850. 14L. I. Glazman and K. A. Matveev, JETP Lett. 49, 659 /H208491989 /H20850. 15A. V . Rozhkov and D. P. Arovas, Phys. Rev. Lett. 82, 2788 /H208491999 /H20850.16M. S. Choi et al. , Phys. Rev. B 70, 020502 /H20849R/H20850/H208492004 /H20850. 17E. Vecino et al. , Phys. Rev. B 68, 035105 /H208492003 /H20850. 18F. Siano and R. Egger, Phys. Rev. Lett. 93, 047002 /H208492004 /H20850. 19C. Karrasch et al. , Phys. Rev. B 77, 024517 /H208492008 /H20850. 20K. Grove-Rasmussen et al. , N. J. Phys. 9, 124 /H208492007 /H20850. 21A. M. Tsvelick and P. B. Wiegmann, Adv. Phys. 32, 453 /H208491983 /H20850; N. E. Bickers, Rev. Mod. Phys. 59, 845 /H208491987 /H20850. 22H. Shiba and T. Soda, Prog. Theor. Phys. 41,2 5 /H208491969 /H20850; J. Zit- tartz and E. Müller-Hartmann, Z. Phys. 232,1 1 /H208491970 /H20850. 23Due to this large value of Cit is possible that the ﬁnite resistivity of the backgate also contributes to dissipation at very high fre-quency. 24V . Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22, 1364 /H208491969 /H20850. 25M. Tinkham, Introduction to Superconductivity /H20849McGraw-Hill, New York, 1996 /H20850. 26M. R. Buitelaar et al. , Phys. Rev. Lett. 88, 156801 /H208492002 /H20850. 27D. Goldhaber-Gordon et al. , Phys. Rev. Lett. 81, 5225 /H208491998 /H20850; Sara M. Cronenwett et al. , Science 281, 540 /H208491998 /H20850. 28T. Sand-Jespersen et al. , Phys. Rev. Lett. 99, 126603 /H208492007 /H20850. 29Note the absence of explicit dependence on the capacitances C and Cjfor an overdamped junction, as it is the case in our experiment /H20849Q=0.03 /H20850. 30M. L. Della Rocca et al. , Phys. Rev. Lett. 99, 127005 /H208492007 /H20850.-0.5 0.0 0.50.000.050.100.15ICI/0 -0.5 0.0 0.50.00.10.2ICI/0 ε/Utheory exp. I C theoryexp. I Ca) b) FIG. 4. /H20849Color online /H20850Comparison with FRG calculations: cal- culated Ic/H20849/H9280/H20850in units of I0=e/H9004//H6036for/H9003=2/H9004,/H9003/U=0.11, /H9003R//H9003L =6 in /H20849a/H20850and/H9003=2/H9004,/H9003/U=0.2, /H9003R//H9003L=3 in /H20849b/H20850. Experimental data for Ic/H20849/H9280/H20850are shown for Kondo ridge B in /H20849a/H20850and Kondo ridge Ci n /H20849b/H20850.EICHLER et al. PHYSICAL REVIEW B 79, 161407 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 161407-4
PhysRevB.75.184406.pdf	Algebraic vortex liquid theory of a quantum antiferromagnet on the kagome lattice S. Ryu,1O. I. Motrunich,2J. Alicea,3and Matthew P. A. Fisher1 1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA 2Department of Physics, California Institute of Technology, Pasadena, California 91125, USA 3Department of Physics, University of California, Santa Barbara, California 93106, USA /H20849Received 13 January 2007; published 7 May 2007 /H20850 There is growing evidence from both experiment and numerical studies that low half-odd integer quantum spins on a kagome lattice with predominant antiferromagnetic near-neighbor interactions do not order mag-netically or break lattice symmetries even at temperatures much lower than the exchange interaction strength.Moreover, there appears to be a plethora of low-energy excitations, predominantly singlets but also spincarrying, which suggests that the putative underlying quantum spin liquid is a gapless “critical spin liquid”rather than a gapped spin liquid with topological order. Here, we develop an effective ﬁeld theory approach forthe spin- 1 2Heisenberg model with easy-plane anisotropy on the kagome lattice. By employing a vortex duality transformation, followed by a fermionization and ﬂux smearing, we obtain access to a gapless yet stable criticalspin liquid phase, which is described by /H208492+1 /H20850-dimensional quantum electrodynamics /H20849QED 3/H20850with an emer- gent SU /H208498/H20850ﬂavor symmetry. The speciﬁc heat, thermal conductivity, and dynamical structure factor are ex- tracted from the effective ﬁeld theory, and contrasted with other theoretical approaches to the kagomeantiferromagnet. DOI: 10.1103/PhysRevB.75.184406 PACS number /H20849s/H20850: 75.10.Jm, 75.40.Gb I. INTRODUCTION Kagome antiferromagnets are among the most extreme examples of frustrated spin systems realized with nearest-neighbor interactions. Both the frustration in each triangularunit and the rather loose “corner-sharing” aggregation ofthese units into the kagome lattice suppress the tendency tomagnetically order. On the classical level, kagome spin sys-tems are known to exhibit rather special properties: nearest-neighbor Ising and XYmodels remain disordered even in the zero-temperature limit, while the O /H208493/H20850system undergoes order-by-disorder into a coplanar spin structure. 1 Quantum kagome antiferromagnets, which are much less understood, provide a fascinating arena for the possible real-ization of spin liquids. Early exact diagonalization and seriesexpansion studies, 2–5as well as further exhaustive numerical works,6–9provide strong evidence for the absence of any magnetic order or other symmetry breaking in the nearest- neighbor spin-1 2system. Moreover, a plethora of low-energy singlet excitations is found, below a small /H20849if nonzero /H20850spin gap. But the precise nature of the putative spin liquid phasein this model has remained elusive. On the experimental front, several quasi-two-dimensional materials with magnetic moments in a kagome arrangementhave been studied, including SrCr 8−xGa4+xO19with Cr3+/H20849S =3/2 /H20850moments,10–14jarosites K M3/H20849OH/H208506/H20849SO 4/H208502with M =Cr3+or Fe3+/H20849S=5/2 /H20850moments,15–17and volborthite Cu3V2O7/H20849OH/H208502·2H 2O with Cu2+/H20849S=1/2 /H20850moments.18Re- cently, herbertsmithite ZnCu 3/H20849OH/H208506Cl2which also has Cu2+ moments on a kagome lattice was synthesized for which no magnetic order is observed down to 50 mK despite the esti-mated exchange constant of 300 K. 19–21The second layer of 3He absorbed on graphoile is also believed to realize the kagome magnet.22,23The suppression of long-range spin cor- relations or other signs of symmetry breaking down to tem-peratures much lower than the characteristic exchange en-ergy scale is manifest in all of these materials, consistent with expectations. Moreover, there is evidence for low- energy excitations, both spin carrying and singlets. But theultimate zero-temperature spin liquid state is often masked inthese systems by magnetic ordering or glassy behavior at thelowest temperatures, perhaps due to additional interactionsor impurities, rendering the experimental study of the spinliquid properties problematic. New materials and other ex-perimental developments are changing this situation, and thequestion of the quantum spin liquid ground state of thekagome antiferromagnet is becoming more prominent. There are two broad classes of spin liquids which have been explored theoretically, both in general terms and for thekagome antiferromagnet in particular. The ﬁrst class com-prise the “topological” spin liquids, which have a gap to allexcitations and have particlelike excitations with fractionalquantum numbers above the gap. Arguably, the simplest to-pological liquids are the so-called Z 2spin liquids, which sup- port a vortex like excitation—a vison—in addition to thespin one-half spinon. For the kagome antiferromagnet,Sachdev 24and more recently Wang and Vishwanath25have employed a Schwinger boson approach to systematically ac-cess several different Z 2spin liquids. For a kagome antifer- romagnet with easy axis anisotropy and further neighbor in-teractions, Balents et al. 26unambiguously established the presence of a Z2spin liquid, obtaining an exact ground state wave function in a particular limit. Quantum dimer modelson the kagome lattice can also support a Z 2topological phase.27Spin liquids with topological order and time- reversal symmetry breaking, the chiral spin liquids which areclosely analogous to fractional quantum Hall states, havebeen found on the kagome lattice 28,29within a fermionic rep- resentation of the spins. But all of these topological liquidsare gapped, and cannot account for the presence of manylow-energy excitations found in the exact diagonalizationstudies and suggested by the experiments.PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 1098-0121/2007/75 /H2084918/H20850/184406 /H2084913/H20850 ©2007 The American Physical Society 184406-1A second class of spin liquids—the “critical” or “alge- braic spin liquids” /H20849ASL /H20850—have gapless singlet and spin car- rying excitations. Although these spin liquids share manyproperties with quantum critical points such as correlationfunctions falling off as power laws with nontrivial expo-nents, they are believed to be stable quantum phases of mat-ter. Within a fermionic representation of the spins, Hastings 30 explored an algebraic spin liquid on the kagome lattice, andmore recently Ran et al. 31have extended his analysis in an attempt to explain the observed properties of herbertsmithiteZnCu 3/H20849OH/H208506Cl2. In this paper, motivated by the experiments on the kagome materials and the numerical studies, we pursue yet adifferent approach to the possible spin liquid state of thekagome antiferromagnet. As detailed below, for a kagomeantiferromagnet with easy-plane anisotropy we ﬁnd evidencefor a new critical spin liquid, an “algebraic vortex liquid”/H20849AV L /H20850phase. Earlier we had introduced and explored the A VL in the context of the triangular XXZ antiferromagnet in Refs. 32–34. Compared to the slave particle techniques for studying frustrated quantum antiferromagnets, the A VL ap-proach is less microscopically faithful to a speciﬁc spinHamiltonian, but has the virtue of being unbiased. Our approach requires the presence of an easy-plane an- isotropy, which for a spin- 1 2system can come from the Dzyaloshinskii-Moriya interaction, although this is usuallyquite small. But for higher half-integer spin systems, spin-orbit coupling allows for a single ion anisotropy D/H20849S z/H208502 which can be appreciable. With D/H110220 this leads to an easy- plane spin character, and from a symmetry stand point ouranalysis should be relevant. Moreover, a very interesting ex-act diagonalization study by Sindzingre 35suggests that an unusual spin liquid phase is also realized for the nearest-neighbor quantum XYantiferromagnet. He ﬁnds a small gap toS z=1 excitations, but below this gap there is a plethora of Sz=0 states, which is reminiscent of the many singlet exci- tations below the triplet gap in the Heisenberg SU /H208492/H20850spin model.6Thus, based on the exact diagonalization studies, the easy-plane anisotropy does not appear to gap out the putativecritical spin liquid of the Heisenberg model, although it willpresumably modify its detailed character. Our approach to easy-plane frustrated quantum antiferro- magnets focuses on vortex defects in the spin conﬁgurations,rather than the spins themselves. In this picture, the magneti- cally ordered phases do not have vortices present in theground state. But there will be gapped vortices, and these canlead to important effects on the spectrum. For example,vortex-antivortex /H20849i.e., roton /H20850excitations can lead to minima in the structure factor at particular wave vectors in the Bril-louin zone, 36–38in analogy to superﬂuid He-4. As we dem- onstrated for the triangular antiferromagnet and explore be-low for the kagome lattice, our approach predicts speciﬁclocations in the Brillouin zone for the roton minima, whichcould perhaps be checked by high order spin-wave expan-sion techniques. On the other hand, the presence of mobilevortex defects in the ground state can destroy magnetic or-der, giving access to various quantum paramagnets. If thevortices themselves condense, the usual result is the breakingof lattice symmetries, such as in a spin Peierls state. But ifthe vortices remain gapless one can access a critical spinliquid phase.For frustrated spin models the usual duality transforma- tion to vortex degrees of freedom does not resolve the geo-metric frustration since the vortices are at ﬁnite density.However, binding 2 /H9266-ﬂux to each vortex, converting them into fermions coupled to a Chern-Simons gauge ﬁeld, fol-lowed by a simple ﬂux-smearing mean-ﬁeld treatment givesa simple way to describe the vortices. The long-range inter-action of vortices actually works to our advantage here sinceit suppresses their density ﬂuctuations and leads essentiallyto incompressibility of the vortex ﬂuid. As demonstrated inthe easy-plane quantum antiferromagnet on the triangularlattice, 33,34including ﬂuctuations about the ﬂux-smeared mean ﬁeld enables one to access a critical spin liquid withgapless vortices. The theory has the structure of a/H208492+1 /H20850-dimensional /H20851/H208492+1 /H20850D/H20852quantum electrodynamics /H20849QED 3/H20850, with relativistic fermionic vortices minimally coupled to a noncompact U /H208491/H20850gauge ﬁeld. The resulting algebraic vortex liquid phase is a critical spin liquid phasethat exhibits neither magnetic nor any other symmetry-breaking order. This approach also allows one to study manycompeting orders in the vicinity of the gapless phase. When applied to the spin- 1 2easy-plane antiferromagnet on the kagome lattice, the duality transformation combined with fermionization and ﬂux smearing also leads to a low-energyeffective QED 3theory with eight ﬂavors of Dirac fermions with an emergent SU /H208498/H20850ﬂavor symmetry. Amusingly, A VL’s with SU /H208492/H20850,32SU/H208494/H20850,33,34and SU /H208496/H20850/H20849Ref. 39/H20850emergent sym- metries were obtained previously for quantum XYantiferro- magnets on the triangular lattice, with integer spin, half-integer spin, and half-integer spin in an applied magneticﬁeld, respectively. QED 3theory with Nﬂavors of Dirac fer- mions is known to realize a stable critical phase for sufﬁ-ciently large N/H11022N c. While numerical attempts to determine Ncare so far inconclusive,40,41an estimate from the large- N expansion suggests Nc/H110114.32It seems very likely that N=8 is large enough, implying the presence of a stable critical spin liquid ground state for the easy-plane spin-1 2quantum anti- ferromagnet on the kagome lattice. Although the algebraic vortex liquid and the algebraic spin liquid30,31obtained for the kagome lattice are accessed in rather different ways, they share a number of commonali-ties, both theoretically and with regard to their experimentalimplications. Both approaches end with a QED 3theory, the former a noncompact gauge ﬁeld theory with fermionic vor-tices carrying an emergent SU /H208498/H20850symmetry, and the latter a compact gauge theory with fermionic spinons with SU /H208494/H20850 symmetry. The noncompact nature of the gauge ﬁeld in theA VL follows from the fact that the S zcomponent of spin appears as the gauge ﬂux in the dualized theory. Hence thestates with zero ﬂux and 2 /H9266ﬂux are physically distinct; moreover, it follows that since the total Szis conserved, so also is the total gauge ﬂux. This should be contrasted withthe slave particle approach, where a compact gauge theoryarises on the lattice and there is no such conservation law.Physically, this means that dynamical monopole operators,which could potentially destabilize the spin liquid and open agap in the excitation spectrum, are not allowed in our low-energy theory for the A VL, while they are allowed in a low-energy description of spin liquids obtained using a slave par-ticle framework.RYU et al. PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-2With regard to experimentally accessible quantities, both theories predict a power law speciﬁc heat C/H11011T2at low tem- peratures, which follows from the linear dispersion of thefermions, and which would dominate the phonon contribu-tion to the speciﬁc heat for magnets with exchange interac-tions signiﬁcantly smaller that the Debye frequency. 42–45Be- cause the spinons couple directly to magnetic ﬁelds, onewould expect the speciﬁc heat in the ASL to be more sensi-tive to an external applied magnetic ﬁeld than the A VLphase. Both theories predict a thermal conductivity whichvanishes as /H9260/H11011T,46,47which is an interesting experimental signature reﬂecting the dynamical mobility of the gaplessexcitations. The momentum resolved dynamical spin structure factor which can be extracted from inelastic neutron experimentscan in principle give very detailed information about the spindynamics. Following the framework developed on the trian-gular lattice, 32–34,39for the kagome A VL we can extract the wave vectors in the magnetic Brillouin zone which have gap-less spin carrying excitations, and ﬁnd 12 of them as shownin Fig. 4. In contrast, the kagome ASL phase is predicted to have gapless spin excitations at only four of these 12 mo-menta. The momentum space location of the gapless excita-tions is perhaps the best way to try and distinguish experi-mentally between these two /H20849and any other /H20850critical spin liquids. The rest of the paper is organized as follows. We start by introducing our model in Sec. II. The low-energy effectiveﬁeld theory is then developed in Sec. III. In Sec. IV , theproperties of the critical spin liquid phase /H20849A VL phase /H20850are discussed, especially the spin excitation /H20849roton spectrum /H20850 /H20849Fig. 5/H20850, the S zdynamical structure factor /H20851Eq. /H2084919/H20850/H20852, and the in-plane dynamical structure factor /H20851Eq. /H2084922/H20850/H20852. We conclude in Sec. V . Some details of the analysis are presented in threeAppendixes. II. EXTENDED KAGOME MODEL Our primary interest is the kagome lattice spin-1 2model with easy-plane anisotropy /H20849quantum XYmodel /H20850, H=1 2/H20858 r,r/H11032/H20849Jr,r/H11032Sr+Sr/H11032−+ H.c. /H20850+/H20858 r,r/H11032Jr,r/H11032zSrzSr/H11032z, /H208491/H20850 with Jr,r/H11032/H11022Jr,r/H11032z. We consider the system with dominant nearest-neighbor exchange J1, but will also allow some second- and third-neighbor exchanges J2and J3. To set the stage, the model with only nearest-neighbor coupling has an extensive degeneracy of classical groundstates, which is lifted by further-neighbor interactions. Forexample, with antiferromagnetic J 2/H110220, the so-called q=0 phase is stabilized /H20849Fig. 1/H20850. On the other hand, for ferromag- netic J2/H110210, the so-called /H208813/H11003/H208813 structure is the classical ground state /H20849Fig. 1/H20850. These ordered phases are also realized in the quantum spin-1 2Heisenberg model for large enough J2,6while the case with J2/J1/H112290 is most challenging as mentioned in the Introduction. The easy-plane spin system can be readily reformulated in terms of vortices. But in order to apply the fermionized vor-tex approach more simply, we consider a wider class of mod- els which includes the nearest-neighbor kagome model. Spe-ciﬁcally, we add one extra site at the center of each hexagon,on which we set an integer spin /H20849Fig. 1/H20850./H20849Technical reasons for doing this are discussed at the end of Sec. III. /H20850In the rotor representation of spins, with a phase /H9272canonically con- jugate to the integer boson number, n/H11011Sz+1/2 on the kagome sites and n/H11011Szat the centers of each hexagon, such an extended model reads H=/H20858 r,r/H11032Jr,r/H11032cos/H20849/H9272r−/H9272r/H11032/H20850+/H20858 rUr/H20849nr−nr0/H208502, /H208492/H20850 where n0=1/2 for the kagome lattice sites, while n0=0 for the added hexagon center sites. We have dropped the Jzterm for simplicity since it amounts to a mere renormalization ofthe interactions in the effective ﬁeld theory that we will de-rive. As shown in Fig. 1, we take the interaction to be Jfor the nearest-neighbor half-integer spins, while the couplingthat connects integer and half-integer spins is J /H11032. When the on-site interaction Uis inﬁnitely large, U→/H11009, there remain two low-energy states realizing the Hilbert space of a S=1 2 spin at the kagome sites, whereas the frozen sector with nr =0 is selected at the hexagon center sites. In the quantum rotor model, we “soften” this constraint and take Uto be ﬁnite. The additional integer-spin degrees of freedom do not spoil any symmetries of the original model, which for the FIG. 1. /H20849Color online /H20850/H20849Top /H20850The quantum spin model with easy- plain anisotropy, Eq. /H208492/H20850, on the kagome lattice /H20849thick line /H20850supple- mented by an extra site at the center of each hexagon. There is ahalf-integer spin for each kagome site whereas integer spins areplaced at each center of hexagons. Four spins in a unit cell arelabeled by j=1,2,3,4. For this triangular extension of the kagome lattice, the dual lattice is the honeycomb lattice indicated in theupper left-hand part. /H20849Bottom /H20850The q=0 state /H20849left-hand side /H20850and the /H208813/H11003/H208813 state /H20849right-hand side /H20850. Positive-negative vorticities /H20849chiralities /H20850for each triangle are denoted by /H11001//H11002.ALGEBRAIC VORTEX LIQUID THEORY OF A QUANTUM … PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-3record include lattice translations, reﬂections, and rotations, as well as XYspin symmetries. If we integrate over the extra degrees of freedom, the model looks very close to the origi-nal one. For example, in the limit when J /H11032/H11270U, we obtain the kagome model with additional ferromagnetic second andthird neighbor exchanges J 2=J3=−/H20849J/H11032/H208502//H208492U/H20850, together with a renormalized nearest-neighbor exchange J1=J −/H20849J/H11032/H208502//H208492U/H20850. Such exchanges would slightly favor the /H208813 /H11003/H208813 state in the classical limit, but as we will see, the spin liquid phase that we obtain by analyzing the extendedkagome model will have enhanced correlations correspond- ing to both the q=0 and/H208813/H11003/H208813 phases that are nearby. Thus the small bias introduced by the added integer-spin sites ap-pears to be not so important for the generic spin liquid phasethat we want to describe. III. FERMIONIZED VORTEX DESCRIPTION The triangular extension of the kagome XYantiferromag- net discussed above is somewhat similar to the triangular lattice spin-1 2XYantiferromagnet studied in Refs. 33and34, except that one-quarter of the sites are occupied by integerspins. The duality transformation for the present model pro-ceeds identically to Refs. 33and34, and turns the original spin Hamiltonian /H208492/H20850into the dual Hamiltonian describing vortices hopping on the honeycomb lattice and interactingwith a gauge ﬁeld residing on the dual lattice links, H=2 /H92662/H20858 xx/H11032Jxx/H11032exx/H110322+U /H208492/H9266/H208502/H20858 r/H20849/H11612/H11003a/H20850r2 −2/H20858 xx/H11032txx/H11032cos/H20849/H9258x−/H9258x/H11032−axx/H11032−axx/H110320/H20850. /H208493/H20850 Here x,x/H11032label sites on the honeycomb lattice /H20849see Fig. 1/H20850, e±i/H9258xis a vortex creation-annihilation operator at site x, and axx/H11032,exx/H11032represent a gauge ﬁeld on the link connecting xand x/H11032.Jxx/H11032is given by Jxx/H11032=J/H20849J/H11032/H20850when the dual link /H20855xx/H11032/H20856 crosses the original lattice link /H20855rr/H11032/H20856with Jrr/H11032=J/H20849J/H11032/H20850. The last term in the dual Hamiltonian represents the vortex hop- ping where the hopping amplitude txx/H11032is given by txx/H11032=t/H20849t/H11032/H20850when the link /H20855xx/H11032/H20856crosses the link /H20855rr/H11032/H20856with Jrr/H11032 =J/H20849J/H11032/H20850. Crudely, we have t/t/H11032/H11011J/H11032/Jsince vortices hop more easily across weak links /H20849Fig. 2/H20850. Finally, the static gauge ﬁeld a0encodes the average ﬂux seen by the vortices when they move; this is described in more detail below. In the dual description, because of the frustration in the spin model, the average density of vortices is one-half persite. On the other hand, the original boson density is viewedas a gauge ﬂux, n r=1 2/H9266/H20849/H11612/H11003a/H20850r=1 2/H9266/H20858 /H20855xx/H11032/H20856around raxx/H11032. /H208494/H20850 Therefore, vortices experience on average /H9266ﬂux going around the triangular lattice sites with half-integer spins, butthey see zero ﬂux going around the sites with integer spins.Thus, one-quarter of the hexagons will have zero ﬂux as seenby the vortices, and this is where the present model departsfrom the considerations in Refs. 33and34.To treat the system of interacting vortices at ﬁnite density, we focus on the two low-energy states with vortex numberN x=0 and Nx=1 at each site and view vortices as hard-core bosons. We can then employ the fermionization as in Refs.33and34and arrive at the following hopping Hamiltonian for fermionized vortices d x: Hferm=−/H20858 xx/H11032/H20849txx/H11032dx†dx/H11032e−i/H20849axx/H11032+axx/H110320+Axx/H11032/H20850+ H.c. /H20850, /H208495/H20850 where we have introduced a Chern-Simons ﬁeld Awhose ﬂux is tied to the vortex density, /H20849/H11612/H11003A/H20850x=2/H9266Nx. Before proceeding with the analysis of the fermionized vortex Hamiltonian, we now point out the technical reasonsfor considering the extended kagome model. If we apply theduality transformation to the kagome model with nearest-neighbor antiferromagnetic coupling J 1only, we would ob- tain a dual vortex theory on the dice lattice, which is the dualof the kagome lattice. In this theory, in order to reproduce therich physics while restricting the vortex Hilbert space at eachsite to something more manageable like the hard-core vorti-ces described earlier, we would need to keep two low-energystates of vortices for each triangle of the kagome lattice /H20849i.e., for each threefold coordinated site of the dice lattice /H20850, whereas there are three such states to keep for each hexagon /H20849i.e., sixfold coordinated dice site /H20850. The latter degrees of free- dom are rather difﬁcult to represent in terms of fermions. Onthe other hand, in the dual treatment of the extended model,such a sixfold coordinated dice lattice site is effectively splitinto six sites. Each such new site now has two low-energystates but the sites are coupled together, and this providessome caricature of the original important vortex states on theproblematic dice sites. This representation now admits stan-dard fermionization, and it was “found” in a rather naturalway without any prior bias regarding how to treat the difﬁ-cult dice sites with three important vortex states. FIG. 2. /H20849Color online /H20850The ﬂux-smeared mean-ﬁeld background for fermionized vortices. Each hexagon is threaded by either zero or /H9266-ﬂux. /H20849Hexagons with zero-ﬂux are speciﬁed by “0” whereas all the other hexagons are pierced by /H9266-ﬂux. /H20850A convenient choice of a gauge is also shown: for links labeled by ↔, we assign the Peierls phase factor e−iaxx/H110320 =ei/H9266. The unit cell in this gauge consists of 16 sites as labeled in the ﬁgure. For links represented by a solid line weassign hopping amplitude t/H20849=1/H20850whereas for links denoted by a broken line we assign hopping t /H11032. We take /H11003in the ﬁgure as an origin.RYU et al. PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-4Flux smearing mean-ﬁeld and low-energy effective theory . The ﬂux attachment implemented above is formally exact,and there have been no approximations so far, although wehave softened the discreteness constraints on the gaugeﬁelds. To arrive at a low-energy effective ﬁeld theory, wenow pick a saddle point conﬁguration of the gauge ﬁeld,incorporating a ﬂux-smeared mean-ﬁeld ansatz, and laterwill include ﬂuctuations around it. That is, we ﬁrst distribute/H20849or “smear” /H20850the 2 /H9266ﬂux attached to each vortex on a dual site to the three /H20849dual /H20850plaquettes surrounding it. Since the vortices are at half-ﬁlling and each hexagon contains sixsites, the total smeared ﬂux through each hexagon is 2 /H9266, which is equivalent to no additional ﬂux. Therefore we as-sume A xx/H11032=0 in the mean ﬁeld. Also, we take axx/H11032=0 on average, including the ﬂuctuations after we identify the im- portant low-energy degrees of freedom for the fermionic vor-tices. Thus, at the mean-ﬁeld level, the original Hamiltonian/H208492/H20850is converted into a problem of fermions hopping on the honeycomb lattice, H ferm,MF =/H20858 xx/H11032/H20849txx/H11032e−iaxx/H110320 dx†dx/H11032+ H.c. /H20850. /H208496/H20850 In the mean ﬁeld, three out of four hexagons are pierced by /H9266-ﬂux, while the remaining hexagons with an integer spin at the center have zero-ﬂux, see Fig. 2. Our choice of the gauge is shown in Fig. 2. The unit cell consists of 16 sites, the area of which is 2 times as large as the physical unit cell of theoriginal kagome spin system. This Hamiltonian can be solved easily by the Fourier transformation. The lattice vectors that translate the unit cellare given by E 1=/H208494,0/H20850,E2=/H20849−1 ,/H208813/H20850, /H208497/H20850 and the ﬁrst Brillouin zone /H20849BZ/H20850is speciﬁed as the Wigner- Seitz cell of the reciprocal lattice vectors G1=/H20873/H9266 2,/H9266 2/H208813/H20874,G2=/H208730,2/H9266 /H208813/H20874. /H208498/H20850 This is shown in Fig. 3. Note that, on the other hand, the physical unit cell is deﬁned by E1phys=E1/2= /H208492,0/H20850, andE2phys=E2=/H20849−1,/H208813/H20850. Thus the physical BZ is determined as the Wigner-Seitz cell of the reciprocal lattice vectors G1phys =/H20849/H9266,/H9266//H208813/H20850and G2phys=/H208490,2/H9266//H208813/H20850, and is also shown in Figs. 3and4. Because of the particle-hole symmetry of the original quantum rotor Hamiltonian, which in terms of the fermion-ized vortices is realized as a vortex particle-hole symmetry/H20849see Appendix B /H20850, the band structure of the ﬂux-smeared mean-ﬁeld Hamiltonian /H208496/H20850is symmetric around zero energy, and consists of eight bands with positive energy and eightwith negative energy. The seventh and eighth bands havenegative energy spectra and are completely degenerate, 48 each having four Fermi points Q1,2,3,4 that touch zero energy. In the same way, the ninth and tenth bands have positiveenergy spectra and are completely degenerate, and each ofthem touches zero energy at the same four Fermi pointsQ 1,2,3,4 thus completing the Dirac nodal spectrum. With the gauge choice speciﬁed in Fig. 2, the locations of the Fermi points in the BZ are Q1=−Q3=/H20873/H9266 12,/H9266 4/H208813/H20874, /H208499/H20850 Q2=−Q4=/H20873/H9266 12,−3/H9266 4/H208813/H20874, /H2084910/H20850 /H20849see Fig. 3/H20850. The low-energy spectrum consists of eight gap- less two-component Dirac fermions with identical Fermi ve-locities given by /H20849here we take t=1 for convenience /H20850 vF=/H208816t/H110322 /H208493+t/H110322/H20850/H208493+2 t/H110322/H20850. /H2084911/H20850 Hence the low-energy effective theory enjoys an emergent SU/H208498/H20850symmetry among eight Dirac cones, which is pro- tected at the kinetic-energy level by the underlying discrete FIG. 3. /H20849Color online /H20850The ﬁrst Brillouin zone for the ﬂux- smeared mean-ﬁeld ansatz and the location of the nodes Q1,. . .,4 . The hexagonal physical Brillouin zone is also presented. Smallhexagons are a guide for the eyes. FIG. 4. /H20849Color online /H20850Summary of the main characterizations of the kagome A VL phase. The low-energy excitations are located atwave vectors 0,±Q,M 1,2,3, and ± P1,2,3 in the physical Brillouin zone of the kagome lattice, where Q=/H20849/H9266/3,/H9266//H208813/H20850;M1=(/H9266/2, −/H9266//H208492/H208813/H20850),M2=(/H9266/2,/H9266//H208492/H208813/H20850),M3=/H208490,/H9266//H208813/H20850;P1=/H20849/H9266/3,0 /H20850, P2=(−/H9266/6,/H9266//H208492/H208813/H20850), and P3=(−/H9266/6,−/H9266//H208492/H208813/H20850). Both Szand S+ exhibit power law correlations at all these momenta, Eqs. /H2084919/H20850and /H2084922/H20850. The Szcorrelations are “enhanced” /H20849/H9257z/H110152.46 /H20850for the subset ±Q,±P1,2,3and not enhanced /H20849/H9257z=3/H20850for the remaining wave vec- tors. On the other hand, all S+correlations are characterized by the same exponent /H9257±/H110153.24.ALGEBRAIC VORTEX LIQUID THEORY OF A QUANTUM … PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-5symmetries in the problem /H20849see Appendix C /H20850. The lattice fermionized vortex operator is conveniently expanded in terms of slowly varying continuum ﬁelds /H9274i/H9251a/H20849x/H20850by dx/H11011/H20858 /H9251=14 /H20858 i=I,II/H20858 a=1,2eiQ/H9251·x/H9278i/H9251a/H20849n/H20850/H9274i/H9251a/H20849x/H20850, /H2084912/H20850 where indices /H9251=1,...,4, i=I,II, and a=1,2, refer to the four Fermi momenta, two sublattices of the honeycomb lat-tice, and two degenerate bands, respectively, while n =1, ... ,16 speciﬁes the site label in the unit cell, cf. Fig. 2. The 16-component wave functions /H9278i/H9251a/H20849n/H20850represent modes at the Fermi points and are speciﬁed in Appendix A. Working with these continuum ﬁelds, we ﬁnd that the low-energy Hamiltonian has the same form vF/H20849−/H92681pˆx +/H92682pˆy/H20850near each Dirac node, where /H92681,2,3are Pauli matrices that act on the “Lorentz” indices i=I,II. Correspondingly, by introducing the gamma matrices in /H208492+1 /H20850D, /H92530=/H92683,/H92531=−/H92682,/H92532=−/H92681, /H9253/H9262/H9253/H9263+/H9253/H9263/H9253/H9262=2/H9254/H9262/H9263,/H9262,/H9263= 0,1,2, /H2084913/H20850 the Euclidean low-energy effective action is given by Seff=/H20885d3x/H20858 /H20849/H9251/H20850=18 /H9274¯/H20849/H9251/H20850/H9253/H9262/H11509/H9262/H9274/H20849/H9251/H20850, /H2084914/H20850 where /H20849/H9251/H20850=/H9251ais a collective index running from 1 to 8; x/H9262=0,1,2represents imaginary time as well as spatial coordi- nates; /H9274¯=/H9274†/H92530; and we have scaled out vF. Having identiﬁed the low-energy fermionic degrees of freedom, we now reinstate gauge ﬁeld ﬂuctuations. The com-plete effective action is that of SU /H208498/H20850QED 3coupled in ad- dition with the Chern-Simons ﬁeld A/H9262and given by Seff=/H20885d3x/H20873/H20858 /H20849/H9251/H20850=18 /H9274¯/H20849/H9251/H20850/H9253/H9262/H20849/H11509/H9262−ia/H9262−iA/H9262/H20850/H9274/H20849/H9251/H20850+L2f+L4f +1 2e2/H20849/H9280/H9262/H9263/H9261/H11509/H9263a/H9261/H208502+i 4/H9266/H9280/H9262/H9263/H9261A/H9262/H11509/H9263A/H9261/H20874. /H2084915/H20850 Here we also included possible fermion bilinears L2fand four fermion interactions L4f.e2/H11011U−1stands for the cou- pling constant of the gauge ﬁeld a/H9262. As discussed in Refs. 33and34, the Chern-Simons ﬁeld A/H9262is irrelevant. The microscopic symmetries of the spin Hamiltonian prohibit fermion mass terms L2ffrom appearing in the effective action /H20849see Appendix C /H20850. Furthermore, four fermion interactions L4fin Eq. /H2084915/H20850are irrelevant when the number of fermion ﬂavors is large enough, N/H11022Nc. Thus, provided Nc/H110218, we have obtained a stable algebraic vortex liquid phase that is described by QED 3with an emergent SU/H208498/H20850symmetry. Using the available theoretical understand- ing of such theories, we will now describe the main proper-ties of the A VL phase, which in terms of the original spins isa gapless spin-liquid phase.IV . PROPERTIES OF THE A VL PHASE A. Roton spectrum First note that the speciﬁc momenta of the Dirac points Eq. /H2084910/H20850and Fig. 3are gauge dependent. We are interested in the physical properties of the system, which are gauge in-variant. One such property is the vortex-antivortex excitationspectrum: By moving a vortex from an occupied state kwith energy E −/H20849k/H20850to an unoccupied state k/H11032with energy E+/H20849k/H11032/H20850, we are creating an excitation that carries energy E+/H20849k/H11032/H20850 −E−/H20849k/H20850. Some care is needed if we want to construct such a state with a deﬁnite momentum in the physical BZ, since we need to consider both k/H11032−kandk/H11032−k+G1. By examining Fig. 3, we ﬁnd that the vortex-antivortex continuum goes to zero at 12 points 0,±Q,M1,2,3,±P1,2,3 in the BZ of the kagome lattice shown in Fig. 4. An accompanying plot of the lower edge of the vortex- antivortex continuum, /H9004rot/H20849p/H20850= min k/H20851E+/H20849k+p/H20850−E−/H20849k/H20850/H20852, /H2084916/H20850 which can be interpreted as a roton excitation energy, is shown in Fig. 5/H20849top panel /H20850along several BZ cuts. Various proximate phases, e.g., magnetically ordered states, will in-stead have gapped vortices. But if such a gap is small, weexpect that the lower edge of the full excitation spectrumwill be dominated by deep roton minima near the same wavevectors /H20849except where the rotons are masked by spin waves /H20850. As a contrasting example, in Fig. 5/H20849bottom panel /H20850we also show the spin-wave spectrum in the q=0 magnetically ordered phase, 49,50assuming this state is stabilized by some means. /H20849Actually, starting from anyof the classically degen- erate ground states of the nearest-neighbor kagome XY model leads to the same linear spin-wave theory if one ro-tates the spin quantization axis suitably for each site. /H20850There are three branches. For the XYantiferromagnet, these are /H92751,2,3sw/H20849p/H20850=SJ/H208814−/H92611,2,3/H20849p/H20850, FIG. 5. /H20849Color online /H20850/H20849Top /H20850Lower edge of the vortex- antivortex continuum along several cuts in the Brillouin zone asshown in Fig. 4for parameter values t /H11032/t=0.5 and t/H11032/t=0.2 from the higher to lower curve. /H20849Bottom /H20850Spin-wave excitations, Eq. /H2084917/H20850, of the kagome XYantiferromagnet with the nearest-neighbor exchange Jonly along the same cuts in the Brillouin zone.RYU et al. PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-6/H92611/H20849p/H20850=−2 , /H92612,3/H20849p/H20850=1/H11007/H208811+8c o s p1cosp2cosp3, /H2084917/H20850 where p1=px,p2=px/2+/H208813py/2, and p3=px/2−/H208813py/2. Clearly, the spin-wave frequency is large and of order J throughout the BZ except near zero momentum. It is possiblethat higher order in 1/ Sspin-wave corrections will lead to roton minima. It would be interesting to explore the possi-bility of such roton minima in the magnetically orderedphase to see if they are coincident with the gapless momentaof the A VL phase as shown in Fig. 4. Such a coincidence between the roton minima obtained with spin-wave perturba-tion theory 36–38and with the A VL approach34was indeed found within the magnetically ordered phase of the square lattice spin-1 2quantum antiferromagnet with a weak second- neighbor exchange across one diagonal of each elementarysquare /H20849a spatially anisotropic triangular lattice /H20850. It would be useful to look for such signatures in the kagome experi-ments. The momenta of the low-energy or gapless excitations capture important lattice- to intermediate-scale physics andprovide one of the simplest characteristics that can be used todistinguish between different spin liquid phases. For ex-ample, in the algebraic spin liquid state proposed by Hast- ings and Ran et al. 30,31for the S=1 2nearest-neighbor Heisen- berg model, the gapless spinon-antispinon excitations occurat four momenta /H208530,M 1,2,3/H20854in the BZ. On the other hand, Sachdev24and Wang and Vishwanath25discuss four different gapped Z2spin liquid phases on the kagome lattice using Schwinger bosons. One of the states has minimum in thegapped spin excitations at 0; one has minima at ± Q; and the two remaining spin liquids have minima at the same 12 mo-menta as the A VL phase, Fig. 4. B.SzSzcorrelator A more formal approach towards characterizing the sys- tem with gauge interactions is to deduce the PSG transfor-mations for a particular mean-ﬁeld state. 54Speciﬁcally, the notion of symmetry is enlarged to also include gauge trans-formations in order to maintain invariance of the mean-ﬁeldHamiltonian under the usual symmetry operations. Some details are presented in Appendix C, where we show the calculated transformation properties of the con- tinuum fermion ﬁelds /H9274¯,/H9274. In particular, with the PSG analy- sis, the low-energy roton excitations can be studied by ana-lyzing fermion bilinears. The S zcomponent of the spin is expressed as a gauge ﬂux in our QED 3theory of vortices. Both the ﬂux induced by the dynamical U /H208491/H20850gauge ﬁeld itself and the ﬂux induced by vortex currents contribute to the SzSzcorrelation. Formally, theSzoperator can be expressed in terms of the gauge ﬂux and appropriate fermion bilinears /H9274¯G/H9274representing the con- tributions by vortex currents,Srz/H11011/H11612/H11003a 2/H9266+/H20858 iAie−iKi·r/H9274¯Gi/H9274+¯, /H2084918/H20850 where the summation over iruns over bilinears /H9274¯Gi/H9274with appropriate symmetry properties so that the right-hand sidetransforms identically to S z. The factor e−iKi·rrepresents the momenta carried by these bilinears, which are the same asthe roton minima shown in Fig. 4, and A iis some amplitude for the bilinear Gi/H20851in fact, one needs to specify four such amplitudes Ai/H20849j/H20850since there are four sites j=1,...,4 in the unit cell, Fig. 1/H20852. Assuming the N=8 QED 3is critical, all correlation func- tions decay algebraically. We then expect to ﬁnd power lawcorrelation at all momenta shown in Fig. 4. As a conse- quence, the singular part of the structure factor at zero tem-perature near such wave vector Kis given by S zz/H20849k=K+q,/H9275/H20850/H11008/H9008/H20849/H92752−q2/H20850 /H20849/H92752−q2/H20850/H208492−/H9257z/H20850/2. /H2084919/H20850 The exponent /H9257zcharacterizes the strength of the Szcorrela- tions, and smaller values correspond to more singular andtherefore more pronounced correlations. In QED 3, the corre- sponding exponent value for the ﬂux /H11612/H11003athat contributes to the Szcorrelation near zero momentum is /H9257z=3, and the same exponent also characterizes fermionic bilinears that areconserved currents, see Appendix C. On the other hand, bi-linears that are not conserved currents have their scaling di-mensions enhanced by the gauge ﬁeld ﬂuctuations as com-pared with the mean ﬁeld. Upon analyzing thetransformation properties of the fermionic bilinears, we con-clude that there are indeed such enhanced contributions to S z near the momenta ± Qand ± P1,2,3 /H20849explicit expressions are given in Appendix C /H20850. The new exponent can be estimated from the large- Ntreatments of the QED 3/H20849Refs. 43and44/H20850to be/H9257z/H110153−128/ /H208493/H92662N/H20850=2.46 for N=8. C.S+S−correlator Since Sz/H11011/H20849/H11612/H11003a/H20850//H208492/H9266/H20850, the spin raising operator S+in the dual theory is realized as an operator that creates 2 /H9266 gauge ﬂux. Following Ref. 34, we treat such ﬂux insertion /H20849“monopole insertion” /H20850classically as a change of the back- ground gauge ﬁeld conﬁguration felt by fermions. By deter-mining quantum numbers carried by the monopole insertionoperators, we can ﬁnd the wave vectors of the dominant S + correlations. Here we only outline the procedure /H20849for details, see Refs. 32and34/H20850. In the presence of 2 /H9266background ﬂux inserted at x=0, each Dirac fermion species /H20849/H9251/H20850=1, ... ,8 has a quasilo- calized zero-energy state near the inserted ﬂux. We will de- note the creation operators for such fermionic zero modes by f/H20849/H9251/H20850†. The corresponding two-component wave functions have the form /H9272/H20849x/H20850/H110111 /H20841x/H20841/H208491 0/H20850, which allows us to relate the transfor- mation properties of f/H20849/H9251/H20850†to those of the ﬁrst component of /H9274/H20849/H9251/H20850†/H20849the latter are summarized in Appendix B /H20850. The monopole creation operators M†are deﬁned by the combination of the ﬂux insertion and the subsequent ﬁllingALGEBRAIC VORTEX LIQUID THEORY OF A QUANTUM … PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-7of four out of eight fermionic zero modes in order to ensure gauge neutrality, M†/H20841DS0/H20856=f/H20849/H9251/H20850†f/H20849/H9252/H20850†f/H20849/H9253/H20850†f/H20849/H9254/H20850†/H20841DS+/H20856, /H2084920/H20850 where /H20841DS0/H20856and /H20841DS+/H20856are the Fermi-Dirac sea in the ab- sence and presence of +2 /H9266ﬂux, respectively. There is a total of 70 distinct such monopoles. The quantum numbers of themonopole operators are then determined by the transforma- tion properties of the operators f /H20849/H9251/H20850†f/H20849/H9252/H20850†f/H20849/H9253/H20850†f/H20849/H9254/H20850†and those of /H20841DS+/H20856relative to /H20841DS0/H20856. Both /H20841DS+/H20856and /H20841DS0/H20856are expected to be eigenstates of the translation operator T/H9254r, and we denote the ratio of the eigen- values as ei/H9016offset ·/H9254r. On the other hand, we can diagonalize the action of T/H9254ronf/H20849/H9251/H20850†f/H20849/H9252/H20850†f/H20849/H9253/H20850†f/H20849/H9254/H20850†to get a set of eigenvalues ei/H9016i·/H9254r,i=1, ... ,70. Thus, the momenta of the monopole op- erators /H20849at zero energy /H20850are determined by /H20853/H9016i/H20854which are offset by the hitherto unknown wave vector /H9016offset. However, once we examine /H20853/H9016i/H20854, we ﬁnd that the offset is in fact uniquely ﬁxed by the requirement that the monopole mo- menta are distributed in the physical BZ in a way that re-spects all lattice symmetries. Monopole momenta determined in this manner are found to be given by the very same wave vectors 0,±Q,M 1,2,3, and ±P1,2,3, as the leading Szcorrelation /H20849Fig. 4/H20850. For the record, we ﬁnd the following monopole multiplicities given in theparentheses: 0/H2084912 monopoles /H20850,±Q/H20849ﬁve each /H20850,M 1,2,3 /H20849eight each /H20850, and ± P1,2,3 /H20849four each /H20850. The monopole insertions which add Sz=1 have overlap with the exact lowest energy spin carrying eigenstates in theA VL. At the lowest energies such eigenstates will be cen-tered around the monopole wave vectors in the Brillouinzone, and we expect that they will disperse with an energygrowing linearly with small deviations in the momenta. Butpresumably any local operator that adds spin-1 will create alinearly dispersing but overdamped “particle” excitation.More formally, we can expand the S +operator in terms of the continuum ﬁelds as follows: Sr+/H11011/H20858 iAiei/H9016i·rMi†, /H2084921/H20850 where/H9016iis the momentum of the monopole M i. The right- hand side has correct transformation properties under thetranslations, but not all of the monopole operators contributetoS +when other symmetries are taken into account. For example, some of the zero momentum monopoles have op-posite eigenvalues under the lattice inversion, and the samehappens for the momenta M 1,2,3. However, given the large monopole multiplicities, it appears very likely that all mo-menta shown in Fig. 4will be present in S +/H20849but we have not performed an analysis of all quantum numbers so far /H20850. The in-plane dynamical structure factor S+−around each /H9016iis thus expected to have the same critical form as Szz, S+−/H20849k=/H9016+q,/H9275/H20850/H11008/H9008/H20849/H92752−q2/H20850 /H20849/H92752−q2/H20850/H208492−/H9257±/H20850/2, /H2084922/H20850 but the characteristic exponent /H9257±is now appropriate for the monopole operators and is expected to be the same for allmomenta. Reference 51calculated this in the large- NQED 3:/H9257mon/H110150.53 N−1. Setting N=8, we estimate /H9257±/H110153.24, which is larger than even the nonenhanced exponent /H9257z=3 entering theSzcorrelations. The above results with /H9257±/H11022/H9257zsuggest that within the A VL phase the system is “closer” to an Ising ordering of Sz than of an in-plane XYordering of S+. This is rather puzzling for an easy-plane spin model, where one would expect XY order to develop more readily than Ising order. This conclu-sion is reminiscent of the classical kagome spin model,where the easy-plane anisotropy present in the XYmodel destroys the zero temperature coplanar order of the Heisen-berg model. However, this interpretation of the inequality /H9257±/H11022/H9257zis perhaps somewhat misleading. Indeed, as we dis- cuss in Appendix C, condensation of any of the enhancedfermionic bilinears that contribute to S zwould drive XYor- der in addition to Szorder—forming a supersolid phase of the bosonic spins. Thus, care must be taken when translatingthe long-distance behavior of the A VL correlators into order-ing tendencies. The exact diagonalization study by Sindzingre 35ﬁnds that in the nearest-neighbor kagome XYantiferromagnet the S+ correlations are larger than the Szcorrelations, which is dif- ferent from the A VL prediction. It would be interesting toexamine this further, perhaps also in a model with further-neighbor exchanges. With regards to such numerical studies,we also want to point out that the discussed dynamical spinstructure factor, Eqs. /H2084919/H20850and /H2084922/H20850, translate into equal-time spin correlations that decay as r −1−/H9257z,±. For the estimated val- ues of /H9257z,±, the decay is rather quick, and could be hard to distinguish from short-range correlations. V . CONCLUSION Using fermionized vortices, we have studied the easy- plane spin-1 2Heisenberg antiferromagnet on the kagome lat- tice and accessed a gapless spin liquid phase. The effectiveﬁeld theory for the resulting algebraic vortex liquid phase is/H208492+1 /H20850D QED 3with an emergent SU /H208498/H20850ﬂavor symmetry. This large number of ﬂavors can be thought to have its origin in many competing ordered states that are intrinsic to the“loose” network of the corner-sharing triangles in thekagome lattice. It is also likely that for this number of Diracfermions the critical QED 3theory is stable against the dy- namical gap formation, which allows us to expect that theA VL phase is stable. The gapless nature of the A VL phase has important ther- modynamic consequences. For example, we predict the spe-ciﬁc heat to behave as C/H11011T 2at low temperatures. Since vortices carry no spin, we expect this contribution to be un-affected by the application of a magnetic ﬁeld. Interestingly,such behavior was observed in SrCr 8−xGa4+xO19and was in- terpreted in terms of singlets dominating the many-bodyspectrum. 8,14Since the vortices are mobile and carry energy, we furthermore predict signiﬁcant thermal conductivity,which would be interesting to measure in such candidategapless spin liquids. The more detailed properties of the kagome A VL phase were studied by the PSG analysis, leading to detailed predic-tions for the spin structure factor. We found that the domi-RYU et al. PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-8nant low-energy spin correlations occur at 12 speciﬁc loca- tions in the BZ shown in Fig. 4. These wave vectors encode important intermediate-scale physics and can be looked forin experiments and numerical studies. They may be also usedto compare and contrast with other theoretical proposals ofspin liquid states. The recent experiments 19–21for the spin-1 2kagome mate- rial ZnCu 3/H20849OH/H208506Cl2observed power law behavior in /H9275of the structure factor S/H20849k,/H9275/H20850in a powder sample. The speciﬁc heat measurements and local magnetic probes also suggest gapless spin liquid physics in this compound. Unfortunately,no detailed momentum-resolved information is available sofar. It should be also noted that our results were derivedassuming easy-plane character of the spin interactions, anddo not apply directly if the material has no signiﬁcant suchspin anisotropy. 52Hopefully, more experiments in the near future will provide detailed microscopic characterization ofthis material, such as the presence-absence of theDzyaloshinskii-Moriya interaction, 53and clarify the appro- priate spin modeling. In the case of the Heisenberg spin sym-metry and nearest-neighbor exchanges, Refs. 30and31offer another candidate critical spin liquid for the kagome lattice,obtained using a slave fermion construction. This algebraicspin liquid state differs signiﬁcantly from the presented A VLphase as discussed in some detail at the end of the Introduc-tion and towards the end of Sec. IV A. Brieﬂy, within theASL the momentum space locations of the gapless spin car-rying excitations are only a small subset of those predictedfor the A VL. Moreover, while both phases support gaplessDirac fermions that contribute a T 2speciﬁc heat, in the A VL phase the fermions /H20849vortices /H20850are spinless in contrast to the fermionic spinons in the ASL. Thus, one would expect thespeciﬁc heat to be more sensitive to an applied magneticﬁeld in the ASL than in the A VL. Contrary to SrCr 8−xGa4+xO19, the experimental data on ZnCu 3/H20849OH/H208506Cl2shows that the speciﬁc heat is affected by the magnetic ﬁeld in the low temperature regime.19However, as pointed out by Ran et al. ,31T/H1101110 K is likely to be an appropriate temperature scale to test several theoretical ap-proaches to this material, since spin liquid physics might bemasked by, for example, impurity effects, Dzyaloshinskii-Moriya coupling, or other complicating spin interactions, atthe lowest temperature scales. /H20849Indeed, the susceptibility data are perhaps consistent with the presence of impurities, al-though it is unclear at this stage if the peculiar temperaturedependence of the speciﬁc heat also has its origins in impu-rities. /H20850The theoretical prediction of the T 2speciﬁc heat is consistent with the experiments for T/H1102210 K. A general outstanding issue is the connection, if any, be- tween the critical spin liquids obtained with slave fermionsand with fermionic vortices. Perhaps our A VL phase corre-sponds to some algebraic spin liquid ansatz, but at the mo-ment this is unclear. ACKNOWLEDGMENTS The authors would like to thank P. Sindzingre for sharing his exact diagonalization results on the spin-1 2XYkagome system. This research was supported by the National ScienceFoundation through Grants No. PHY99-07949 and No. DMR-0529399 /H20851to two of the authors /H20849M.P.A.F. and J.A. /H20850/H20852. APPENDIX A: ZERO MODE WA VE FUNCTIONS AT THE FERMI POINTS In this appendix, the 16-component wave functions /H9278i,/H9251,a/H20849n/H20850/H20849n=1,...,16 /H20850representing zero modes at the Fermi points are presented explicitly. These wave functions are necessary for the PSG analysis. At the node Q1, /H9278I,1,1=ei/H9266/4/H20873/H9273A+Q1 0/H20874,/H9278I,1,2=ei/H9266/12/H20873/H9273A−Q1 0/H20874, /H9278II,1,1=e−i/H9266/4/H208730 /H9273B+Q1/H20874,/H9278II,1,2=e−i/H9266/12/H208730 /H9273B−Q1/H20874;/H20849A1/H20850 at the node Q2, /H9278I,2,1=/H20873/H9273A+Q2 0/H20874,/H9278I,2,2=ei/H9266/6/H20873/H9273A−Q2 0/H20874, /H9278II,2,1=/H208730 /H9273B−Q2/H20874,/H9278II,2,2=e−i/H9266/6/H208730 /H9273B+Q2/H20874; /H20849A2/H20850 at the node Q3, /H9278I,3,1=ei5/H9266/4/H208730 /H9273B−Q3/H20874,/H9278I,3,2=ei13/H9266/12/H208730 /H9273B+Q3/H20874, /H9278II,3,1=e−i/H9266/4/H20873/H9273A+Q3 0/H20874,/H9278II,3,2=e−i/H9266/12/H20873/H9273A−Q3 0/H20874;/H20849A3/H20850 and at the node Q4, /H9278I,4,1=−/H208730 /H9273B+Q4/H20874,/H9278I,4,2=ei7/H9266/6/H208730 /H9273B−Q4/H20874, /H9278II,4,1=/H20873/H9273A+Q4 0/H20874,/H9278II,4,2=e−i/H9266/6/H20873/H9273A−Q4 0/H20874; /H20849A4/H20850 where a convenient choice for the eight-component zero en- ergy wave functions /H9273Aand/H9273Bis /H9273A,sQ/H9251=NA/H208981 /H9251 /H20849xy/H208502/H20851y4+/H20849xy/H208502/H20852 −/H9251/H20849xy/H208502/H20851y4−/H20849xy/H208502/H20852 −xy3+/H9251xy/H20851y4−/H20849xy/H208502/H20852 t/H11032/H20849xy3+/H9251xy3/H20850 −x3y/H20851y4+/H20849xy/H208502/H20852−/H9251xy3 t/H11032x3y/H20853y4x2+y2−/H9251/H20851y4−/H20849xy/H208502/H20852/H20854/H20899, /H20849A5/H20850ALGEBRAIC VORTEX LIQUID THEORY OF A QUANTUM … PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-9/H9273B,sQ/H9251=NB/H208981 /H9252 /H20849xy/H208502/H20851y4+/H20849xy/H208502/H20852 −/H9252/H20849xy/H208502/H20851y4−/H20849xy/H208502/H20852 −t/H11032/H20849xy3+/H9252xy3/H20850 xy/H20851y4+/H20849xy/H208502/H20852+/H9252xy3 t/H11032x3y/H20851/H9252/H20849y4x2−y2/H20850−y4−/H20849xy/H208502/H20852 xy3−/H9252x3y/H20851y4−/H20849xy/H208502/H20852/H20899. /H20849A6/H20850 Here xªeikx/2,yªeiky//H208492/H208813/H20850forQ=/H20849kx,ky/H20850, /H9251=s3/H208812x3y3−/H208491+x6y6/H20850/H20849x4+x4/H20850 2x−2/H208493−x2y6−x2y6/H20850, /H9252=si3/H208812x3y3−/H208491−x6y6/H20850/H20849x4+x4/H20850 2x−2/H208493−x2y6−x2y6/H20850, /H20849A7/H20850 and NAand NBaret/H11032-dependent normalization factors. These zero modes are chosen in such a way that the s= ±1 wave functions are orthogonal to each other for any t/H11032. In order to work with the continuum Dirac ﬁelds Eq. /H2084912/H20850, it is convenient to introduce four sets of the Pauli matriceswhich act on different gradings. In the following, Pauli ma-trices denoted by /H9267/H9260/H20849/H9260=0,1,2,3 /H20850act on italic indices a =1,2 /H20849/H92670is an identity matrix /H20850. Pauli matrices denoted by /H9268/H9260 /H20851and hence /H9253/H9260introduced in Eq. /H2084913/H20850/H20852act on italic indices i =I,II. We separate the node momenta into two groups/H20849Q 1,Q3/H20850and /H20849Q2,Q4/H20850. Then, Pauli matrices denoted by /H9270/H9260 act within a given group, either /H20849Q1,Q3/H20850or/H20849Q2,Q4/H20850. Finally, Pauli matrices denoted by /H9262/H9260act on the space spanned by /H20849Q1,3,Q2,4/H20850. We will also use the notation /H9270±=/H92701±i/H92702. APPENDIX B: PROJECTIVE SYMMETRY GROUP ANALYSIS Any symmetry of the original lattice spin model has a representation in terms of vortices. Unfortunately, upon fer-mionization, time-reversal symmetry transformation be-comes highly nonlocal and we do not know how to imple-ment it in the low-energy effective ﬁeld theory. Since a speciﬁc conﬁguration of the Chern-Simons gauge ﬁeld was picked in the ﬂux smearing mean ﬁeld, we had toﬁx the gauge and work with the enlarged unit cell. The spa-tial symmetries of the original lattice spin model are, how-ever, still maintained if the symmetry transformations arefollowed by subsequent gauge transformations—the originalsymmetries become “projective symmetries” in the effectiveﬁeld theory. The projective symmetry group /H20849PSG /H20850analysis is necessary when we try to make a connection between theoriginal spin model and the effective ﬁeld theory. The original spin system has the following symmetries besides the global U /H208491/H20850:T E1/2, translation by E1/2; TE2, translation by E2; R/H9266, rotation by /H9266around a kagome site; Rx, reflection with respect to yaxis; C, particle hole; T, time reversal. /H20849B1/H20850 We can also include a rotation by /H9266/3 about the honeycomb center but will not consider it here. The ﬁrst four transformations T/H9254r,R/H9258,Rxact on spatial coordinates r=/H20849x,y/H20850,r→r+/H9254r,r→/H20849xcos/H9258 −ysin/H9258,xsin/H9258+ycos/H9258/H20850,r→/H20849−x,y/H20850, respectively. On the other hand, the particle-hole symmetry Cand the time rever- salTact on spin operators as C,Sz→−Sz,Sx±iSy→Sx/H11007iSy, T,S→−S,i→−i. /H20849B2/H20850 Here, the antiunitary nature of the time reversal is reﬂected in its action on the complex number i→−i. Symmetries in terms of the rotor representation can then be deduced as C,n→−n,/H9272→−/H9272, T,n→−n,/H9272→/H9272+/H9266,i→−i. /H20849B3/H20850 Symmetry properties of bosonic and fermionic vortices are deduced from their deﬁning relations. Due to the Chern-Simons ﬂux attachment, the mirror and time-reversal sym-metries are implemented in a nonlocal fashion in the fermi-onized theory. However, if we combine TwithR xandC, the resulting modiﬁed reﬂection R˜xªRxCTcan still be realized locally. We summarize the symmetry properties of the fermi-ons d,d †in Table I. We also introduce a formal fermion time reversal by Tferm,d→d,i→−i. /H20849B4/H20850 The necessary explanations are the same as in Refs. 32and 34and are not repeated here. Finally, the symmetry properties of the slowly varying continuum ﬁelds /H9274¯,/H9274can be deduced from Eq. /H2084912/H20850. As ex- plained, these are realized projectively,54and the symmetry transformations for the continuum fermion ﬁelds are summa-rized in Table II. APPENDIX C: FERMION BILINEARS Once we determine the symmetry properties of the con- tinuum fermion ﬁelds /H9274¯,/H9274, we can discuss symmetry prop- erties of the gauge invariant bilinears /H9274¯G/H9274where Gi sa1 6RYU et al. PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-10/H1100316 matrix. Using the gradings speciﬁed at the end of Ap- pendix A, the bilinears can be conveniently written as /H9274¯/H9268/H9260/H9262/H9261/H9270/H9251/H9267/H9252/H9274, /H20849C1/H20850 where /H9260,/H9261,/H9251,/H9252run from 0 to 3 and hence there are 44 =64/H110034 such bilinears. Among these bilinears, 64 /H110033 bilin- ears with the Lorentz index /H9260=1,2,3 represent U /H208498/H20850cur- rents, /H9274¯/H92681,2,3/H9262/H9261/H9270/H9251/H9267/H9252/H9274. /H20849C2/H20850 On the other hand, there are 64 remaining bilinears with /H9260 =0, /H9274¯/H92680/H9262/H9261/H9270/H9251/H9267/H9252/H9274. /H20849C3/H20850 Since the bilinears /H20849C2/H20850are conserved currents, they maintain their engineering scaling dimensions even in thepresence of the gauge ﬂuctuations. On the other hand, thebilinears of the form /H20849C3/H20850/H20849“enhanced bilinears” /H20850develop anomalous dimensions when we include the gauge ﬂuctua-tions. Let us give some relevant examples of bilinears. The /H208813 /H11003/H208813 magnetically ordered state /H20849shown in Fig. 1/H20850corre- sponds to the staggered charge density wave of vortices,which is obtained by adding the corresponding staggeredchemical potential /H20849i.e., ± /H9254/H9262on the up-down kagome tri- angles /H20850. Using the listed symmetries, it is readily veriﬁed that the following two bilinears in the expression B/H208813/H11003/H208813=/H9251/H9274¯/H92680/H92620/H92703/H92670/H9274+/H9252/H9274¯/H92683/H92623/H92700/H92673/H9274 /H20849C4/H20850 transform in the manner expected of the staggered chemical potential /H20849the coefﬁcients /H9251and/H9252are generically indepen- dent /H20850. This can be also checked explicitly by writing such chemical potential in terms of the continuum ﬁelds obtainingsome t /H11032-dependent coefﬁcients /H9251and/H9252. In the similar vein,theq=0 state /H20849also shown in Fig. 1/H20850corresponds to the uni- form chemical potential for vortices on the up and downkagome triangles, which is realized with the following bilin-ears: B q=0=/H9251/H11032/H9274¯/H92680/H92623/H92703/H92673/H9274+/H9252/H11032/H9274¯/H92683/H92620/H92700/H92670/H9274. /H20849C5/H20850 Note that each B/H208813/H11003/H208813and Bq=0contains an enhanced bilinear of the QED 3theory, and therefore both orders are “present” in the A VL phase as enhanced critical ﬂuctuations. It is insuch sense that the enhanced bilinears encode the potentialnearby orders. With Table IIat hand, it is a simple matter to check that spatial symmetries /H20849translation, rotations, and reﬂection /H20850and particle-hole symmetry prohibit all fermion bilinears from appearing in the effective action /H2084915/H20850, except /H9274¯/H9274and /H9274¯/H92683/H92623/H92703/H92673/H9274. If we are allowed to require the invariance un- derTferm, then these bilinears would be prohibited as well. However, even if we do not use Tferm, we exclude these bilinears from the continuum theory using the following ar- gument from Ref. 34: Consider ﬁrst adding the bilinear /H9274¯/H9274 to the action and analyze the resulting phase for the original spin model. This bilinear opens a gap in the fermion spec-trum, and proceeding as in Ref. 34we conclude that this phase is in fact a chiral spin liquid that breaks the physicaltime reversal. Therefore, if we are interested in a time-reversal invariant spin liquid, we are to exclude this term. The situation with the /H9274¯/H92683/H92623/H92703/H92673/H9274term is less clear since by itself it would lead to small Fermi pockets, and it is thendifﬁcult to deduce the physical state of the original spin sys-tem. In the presence of both terms, depending on their rela-tive magnitude one may have either a gap or Fermi pockets.However, we can plausibly argue that these pockets tend tobe unstable towards a gapped phase that is continuously con-nected to the same chiral phase obtained when the mass termTABLE I. Summary of symmetry transformations for spin, rotor, vortex, and fermionized vortex operators on the lattice. The sign factor /H20849−1/H20850iin the action of the particle-hole transformation on the fermionized vortices, C:d→/H20849−1/H20850id†, is 1 on one of the sublattices of the dual lattice whereas it is −1 on the other. Spin S Rotor /H9272,n C Sy→−Sy,Sz→−Szn→−n,/H9272→−/H9272 T S→−S,i→−in →−n,/H9272→/H9272+/H9266,i→−i R˜xSx→−Sx,i→−i,x→−xn →+n,/H9272→−/H9272+/H9266,i→−i,x→−x V ortex /H9258,Nand gauge ﬁeld a,e Fermion d,d† C a→−a,/H9258→−/H9258,e→−e,N→1−Nd →/H20849−1/H20850id† T a→−a,/H9258→−/H9258,e→e,N→N,i→−i R˜xa→−a,/H9258→−/H9258,e→e,N→N,i→−i,x→−xd →d,x→−x,i→−i TABLE II. Summary of symmetry transformations for continuum fermion ﬁelds /H20851up to unimportant U /H208491/H20850phase factors /H20852. TE1/2 TE2R˜x R/H9266 CT fermi /H9274→ /H92622e−i/H9266/H92703/3/H92671/H9274/H92623ei/H9266/H92703/6/H9274 e+i/H9266/H92703/12/H92703e+i/H9266/H92622/H92671/4/H92673/H9274 /H92683/H92672/H92701ei5/H9266/H92703/12/H92623/H9274 /H92701/H92681/H20851/H9274†/H20852T/H92702/H92682/H9274ALGEBRAIC VORTEX LIQUID THEORY OF A QUANTUM … PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-11/H9274¯/H9274dominates. Since we are primarily interested in the states that are not chiral spin liquid /H20849e.g., states that appear from the A VL description by spontaneously generating some other mass terms like B/H208813/H11003/H208813orBq=0/H20850, we drop the bilinears /H9274¯/H9274 and/H9274¯/H92683/H92623/H92703/H92673/H9274from further considerations. The validity of this assumption and the closely related issue of neglectingirrelevant higher-derivative Chern-Simons terms from the ﬁ-nal A VL action are the main unresolved questions about theA VL approach /H20849see Ref. 34for some discussion /H20850. As another example of the application of the derived PSG, we write explicitly combinations of enhanced bilinears that contribute to S jz/H11011/H20849/H11612/H11003a/H20850j//H208492/H9266/H20850+Gj+Fj+¯, Gj=gj/H20849eiQ·rje−i/H9266/12BQ++ H.c. /H20850, /H20849C6/H20850Fj=fj1/H20849eiP1·rjei5/H9266/12BP1++ H.c. /H20850+fj2/H20849eiP2·rjei5/H9266/12BP2++ H.c. /H20850 +fj3/H20849eiP3·rjei5/H9266/12BP3++ H.c. /H20850. /H20849C7/H20850 Here jrefers to the “basis” labels in the unit cell consisting of four sites in the extended model shown in Fig. 1; the wave vectors are the ones shown in Fig. 4, while the corresponding bilinears are BQ+=/H9274¯/H92622/H9270+/H92670/H9274, BP1+=/H9274¯/H92623/H9270+/H92672/H9274, BP2+ =/H9274¯/H92622/H9270+/H92673/H9274,BP3+=/H9274¯/H92620/H9270+/H92672/H9274. The above is to be interpreted as an expansion of the microscopic operators Sjzdeﬁned on the original spin lattice sites in terms of the continuum ﬁeldsin the theory. 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The effect of gauge ﬂuctuations can be stud- ied systematically for large numbers of fermion ﬂavor Nf./H20849See also discussions in Ref. 31./H20850One way to see how gauge interac- tions affect the speciﬁc heat is to integrate out fermions andderive the propagator for dressed photons, which becomes /H110111//H20881/H20841/H9275/H208412+kx2+ky2/H20849Refs. 43–45/H20850. Calculating the free energy of such photons at ﬁnite temperature, one can then see that thespeciﬁc heat still behaves as /H11011T 2, although the “damped” dy- namics of photons does modify the proportionality constant. 43W. Rantner and X.-G. Wen, Phys. Rev. B 66, 144501 /H208492002 /H20850. 44M. Franz, T. Pereg-Barnea, D. E. Sheehy, and Z. Tesanovic, Phys. Rev. B 68, 024508 /H208492003 /H20850. 45M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev. B 72, 104404 /H208492005 /H20850. 46L. B. Ioffe and G. Kotliar, Phys. Rev. B 42, 10348 /H208491990 /H20850. 47P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 /H208491992 /H20850. 48The double degeneracy that we ﬁnd is reminiscent of that in O. Vafek and A. Melikyan, Phys. Rev. Lett. 96, 167005 /H208492006 /H20850. 49A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 45,2899 /H208491992 /H20850. 50A. Chubukov, Phys. Rev. Lett. 69, 832 /H208491992 /H20850. 51V . Borokhov, A. Kapustin, and X. Wu, J. High Energy Phys. 11, /H208492002 /H20850049. 52For example, Dzyaloshinskii-Moriya interactions can provide the desired easy plane anisotropy. However, typically such interac-tions also break some lattice symmetries, such as inversions. Interms of our effective ﬁeld theory approach, they could thenmodify the set of fermion bilinears allowed by symmetry, whichmay lead to an instability of the critical spin liquid state wediscuss. Nevertheless, there may still be a range of energy scalesover which the spin liquid might provide a reasonable descrip-tion of an experimental system such as herbertsmithite. Theseissues were addressed in Refs. 33and34that discuss geometri- cally frustrated quantum magnets on the triangular lattice, moti-vated by Dzyaloshinskii-Moriya interaction in CsCuCl. 53M. Rigol and R. P. Singh, arXiv:cond-mat/0701087 /H20849unpub- lished /H20850. 54X. G. Wen, Phys. Rev. B 65, 165113 /H208492002 /H20850.ALGEBRAIC VORTEX LIQUID THEORY OF A QUANTUM … PHYSICAL REVIEW B 75, 184406 /H208492007 /H20850 184406-13
PhysRevB.97.134405.pdf	PHYSICAL REVIEW B 97, 134405 (2018) Magnetism of a Co monolayer on Pt(111) capped by overlayers of 5 delements: A spin-model study E. Simon,1,L. Rózsa,2K. Palotás,3,4and L. Szunyogh1,5 1Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary 2Department of Physics, University of Hamburg, D-20355 Hamburg, Germany 3Department of Complex Physical Systems, Institute of Physics, Slovak Academy of Sciences, SK-84511 Bratislava, Slovakia 4MTA-SZTE Reaction Kinetics and Surface Chemistry Research Group, University of Szeged, H-6720 Szeged, Hungary 5MTA-BME Condensed Matter Research Group, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary (Received 21 January 2018; published 9 April 2018) Using ﬁrst-principles calculations, we study the magnetic properties of a Co monolayer on a Pt(111) surface with a capping monolayer of selected 5 delements (Re, Os, Ir, Pt, and Au). First we determine the tensorial exchange interactions and magnetic anisotropies characterizing the Co monolayer for all considered systems.We ﬁnd a close relationship between the magnetic moment of the Co atoms and the nearest-neighbor isotropicexchange interaction, which is attributed to the electronic hybridization between the Co and the capping layers,in the spirit of the Stoner picture of ferromagnetism. The Dzyaloshinskii-Moriya interaction is decreased forall overlayers compared to the uncapped Co/Pt(111) system, while even the sign of the Dzyaloshinskii-Moriyainteraction changes in the case of the Ir overlayer. We conclude that the variation of the Dzyaloshinskii-Moriyainteraction is well correlated with the change of the magnetic anisotropy energy and of the orbital momentanisotropy. The unique inﬂuence of the Ir overlayer on the Dzyaloshinskii-Moriya interaction is traced by scalingthe strength of the spin-orbit coupling of the Ir atoms in Ir/Co/Pt(111) and by changing the Ir concentrationin the Au 1−xIrx/Co/Pt(111) system. Our spin dynamics simulations indicate that the magnetic ground state of Re/Co/Pt(111) thin ﬁlm is a spin spiral with a tilted normal vector, while the other systems are ferromagnetic. DOI: 10.1103/PhysRevB.97.134405 I. INTRODUCTION Owing to promising technological applications, chiral mag- netic structures have become the focus of current experimentaland theoretical research activities [ 1,2]. Chiral magnetism is essentially related to the breaking of space-inversion symme-try, since in this case spin-orbit coupling (SOC) leads to theappearance of the Dzyaloshinskii-Moriya interaction (DMI)[3,4] that lifts the energy degeneracy between noncollinear magnetic states rotating in opposite directions. Noncollinearchiral magnetic structures stabilized by the DMI, such as spinspirals and magnetic skyrmion lattices, have been explored incrystals with bulk inversion asymmetry such as MnSi [ 5–7]. Magnetic thin ﬁlms and multilayers with broken interfacialinversion symmetry represent another class of systems inwhich chiral magnetic structures can emerge. In these systems,magnetic transition-metal thin ﬁlms are placed on heavy metal(e.g., Pt, Ir, W) substrates supplying strong spin-orbit inter-action. For instance, spin spiral ground states were reportedfor Mn monolayers on W(110) [ 8,9] and on W(001) [ 10], spin spirals and skyrmions were detected in the Pd/Fe/Ir(111)bilayer system [ 11–13], while in the case of an Fe monolayer on Ir(111) the formation of a spontaneous magnetic nanoskyrmionlattice has been observed [ 14]. Competing ferromagnetic (FM) and antiferromagnetic (AFM) isotropic exchange couplingsare also capable of stabilizing noncollinear spin structures esimon@phy.bme.huin magnetic thin ﬁlms and nanoislands [ 15–18], while the chirality of these structures is still determined by the DMI. Understanding and controlling the sign and strength of the DMI at metallic interfaces is one of the key tasks inexploring and designing chiral magnetic nanostructures. A large number of experiments has been devoted to the study of the inﬂuence of different nonmagnetic elements on theDMI at magnetic/nonmagnetic metal interfaces [ 19–21], also supported by ﬁrst-principles calculations [ 22]. Recently, it was shown that at 3 d/5dinterfaces the trend for the DMI follows Hund’s ﬁrst rule as the number of valence electrons in themagnetic layer is varied [ 23], while for a Co/Pt bilayer it was studied how the DMI depends on the number of occupied states close to the Fermi energy by resolving the DMI in reciprocalspace [ 24]. It was also demonstrated that the magnetic ground state of an Fe monolayer on 5 dmetal surfaces is strongly inﬂuenced by the electronic properties of the substrate [ 25,26]. Because of the interplay between large spin-orbit coupling andhigh spin polarizability, particular attention has been paid to the inﬂuence of the heavy metal Ir on the DMI. This includes the formation of noncollinear spin structures in ultrathinmagnetic ﬁlms on Ir substrates [ 12–14] and the insertion of Ir into multilayer structures [ 27–30]. It was demonstrated that the insertion of Ir leads to a sign change of the DMI in the Pt/Co/Ir/Pt system [ 20,31], and it was suggested that the Ir/Co/Pt stacking order in magnetic multilayers can lead to an enhancement of the DMI [ 22,27]. Motivated by previous experimental and theoretical inves- tigations, in the present paper we explore the role of selected 2469-9950/2018/97(13)/134405(11) 134405-1 ©2018 American Physical SocietySIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) monatomic 5 d(Re, Os, Ir, Pt, Au) overlayers in inﬂuencing the magnetic properties of a Co monolayer deposited onPt(111). We focus on the investigation of how the electronichybridization with heavy metal capping layers possessingdifferent numbers of valence electrons and different strengthsof the spin-orbit interaction inﬂuences the magnetic propertiesof the Co layer. In Sec. II, the parameters of an extended classical Heisen- berg model are discussed, where the coupling between the spinsis described by tensorial exchange interactions using ﬁrst-principles electronic structure calculations. It is also explainedhow these interactions can be converted to effective or micro-magnetic parameters. In Sec. III A, the modiﬁcations of the Co magnetic moments and of the nearest-neighbor (NN) isotropicexchange coupling between the Co atoms are found to correlatewith the change of the electronic states in the Co and the 5 d overlayers. In Sec. III B, the correlations between the DMI, the magnetic anisotropy energy (MAE), and the orbital momentanisotropy are highlighted. In the case of the Ir/Co/Pt(111)system, we ﬁnd that the DMI in the Co monolayer changes signcompared to Co/Pt(111) and the systems with the other cappinglayers, and we scale the spin-orbit coupling of the Ir layer in order to get a more profound insight into this phenomenon. This investigation is supplemented by investigating the DMIand the MAE in Au 1−xIrx/Co/Pt(111) thin ﬁlms with an alloy overlayer. Finally, in Sec. III C we determine the magnetic ground state of the Co monolayer on the Pt(111) substrate withdifferent capping layers using spin dynamics simulations. Theresults are summarized in Sec. IV. II. COMPUTATIONAL METHODS A. Details of ab initio calculations We performed self-consistent electronic structure calcula- tions for X/Co/Pt(111) ( X=Re, Os, Ir, Pt, Au) ultrathin ﬁlms in terms of the relativistic screened Korringa-Kohn-Rostoker(SKKR) method [ 32,33]. For the case of chemically disordered overlayers, we employed the single-site coherent-potentialapproximation (CPA). We used the local spin-density approx-imation as parametrized by V osko et al. [34] and the atomic sphere approximation with an angular momentum cutoff of/lscript max=2. The energy integrals were performed by sampling 16 points on a semicircle contour in the upper complexenergy semiplane. The layered system treated self-consistentlyconsisted of nine Pt atomic layers, one Co monolayer, one X monolayer, and four layers of vacuum (empty spheres) betweenthe semi-inﬁnite Pt substrate and semi-inﬁnite vacuum. Formodeling the geometry of the thin ﬁlms we used the value a 2D=2.774˚A for the in-plane lattice constant of the Pt(111) surface and fcc growth was assumed for both the Co andthe different overlayers. The distances between the atomiclayers were optimized in terms of V ASP calculations [ 35–37]. Relative to the interlayer distance in bulk Pt, we found aninward relaxation between 5 and 10% for the Co monolayerand between 8 and 15% for the different overlayers. In order to study the magnetic structure in the Co layer we use the generalized classical Heisenberg model H=−1 2/summationdisplay i,j/vectorsiJij/vectorsj+/summationdisplay i/vectorsiK/vectorsi, (1)where /vectorsidenotes the spin vector of unit length at site i,Jijis the exchange coupling tensor [ 38], and Kis the on-site anisotropy matrix. The tensorial exchange coupling can be decomposedinto an isotropic, an antisymmetric, and a traceless symmetriccomponent [ 31]: J ij=JijI+1 2/parenleftbig Jij−JT ij/parenrightbig +/bracketleftbig1 2/parenleftbig Jij+JT ij/parenrightbig −JijI/bracketrightbig . (2) The isotropic part Jij=1 3TrJijrepresents the Heisenberg couplings between the magnetic moments. The antisymmetricpart of the exchange tensor can be identiﬁed with the DMvector: /vectors i1 2/parenleftbig Jij−JT ij/parenrightbig /vectorsj=/vectorDij(/vectorsi×/vectorsj). (3) From the diagonal elements of the traceless symmetric part of the exchange tensor the two-site anisotropy may becalculated. The second term of Eq. ( 1) comprises the on-site anisotropy with the anisotropy matrix K. Note that for the case of C 3v symmetry the studied systems exhibit, the on-site anisotropy matrix can be described by a single parameter, /vectorsiK/vectorsi=K(sz i)2. The effective MAE of the system can be obtained as a sum ofthe two-site and on-site anisotropy contributions as will bediscussed in Sec. II C. Note that the sign convention for J ij, /vectorDij, andKis opposite to Ref. [ 31], from which we include the values for the Co/Pt(111) system without a capping layer forcomparison with the present results. The exchange coupling tensors were determined in terms of the relativistic torque method [ 38,39], based on calculating the energy costs due to inﬁnitesimal rotations of the spins atselected sites with respect to the ferromagnetic state orientedalong different crystallographic directions. For these orienta-tions we considered the out-of-plane ( z) direction and three different in-plane nearest-neighbor directions, being sufﬁcientto produce interaction matrices that respect the C 3vsymmetry of the system. The interaction tensors were determined for allpairs of atoms up to a maximal distance of 5 a 2D, for a total of 90 neighbors including symmetrically equivalent ones. B. Determining the ground state of the system To ﬁnd the magnetic ground state of the Co monolayer, we calculated the energies of ﬂat harmonic spin spiral conﬁgura-tions: /vectors i=/vectore1cos/vectork/vectorRi+/vectore2sin/vectork/vectorRi, (4) where /vectorkdenotes the spin spiral wave vector, /vectore1and/vectore2are unit vectors perpendicular to each other, and /vectorRiis the lattice position of spin /vectorsi. Substituting Eq. ( 4) into Eq. ( 1) yields 1 NESS(/vectork,/vectorn)=−1 2/summationdisplay /vectorRij1 2/parenleftbig TrJij−/vectornJsymm ij/vectorn/parenrightbig cos/vectork/vectorRij −1 2/summationdisplay /vectorRij/vectorDij/vectornsin/vectork/vectorRij+1 2(TrK−/vectornK/vectorn),(5) with/vectorn=/vectore1×/vectore2the normal vector of the spiral, /vectorRij=/vectorRj− /vectorRi, and Jsymm ij=1 2(Jij+JT ij). The ground state conﬁguration was approximated by optimizing Eq. ( 5) with respect to /vectorkand 134405-2MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) /vectorn, and comparing it to the energy of the ferromagnetic state: 1 NEFM(/vectoreFM)=−1 2/summationdisplay /vectorRij/vectoreFMJij/vectoreFM+/vectoreFMK/vectoreFM, (6) which was minimized with respect to the ferromagnetic direc- tion/vectoreFM. Due to the magnetic anisotropy, actual spin spiral conﬁg- urations become distorted compared to the harmonic shapedeﬁned in Eq. ( 4). In order to take this effect into account, we further relaxed the conﬁgurations obtained above usingzero-temperature spin dynamics simulations by numericallysolving the Landau-Lifshitz-Gilbert equation [ 40,41]: ∂/vectors i ∂t=−γ 1+α2/vectorsi×/vectorBeff i−αγ 1+α2/vectorsi×/parenleftbig /vectorsi×/vectorBeff i/parenrightbig ,(7) where αis the Gilbert damping parameter and γ=2μB/¯h is the gyromagnetic ratio. The effective ﬁeld /vectorBeff iis obtained from the generalized Hamiltonian Eq. ( 1)a s /vectorBeff i=−1 m∂H ∂/vectorsi=1 m/summationdisplay j(/negationslash=i)Jij/vectorsj−2 mK/vectorsi. (8) The spin magnetic moment of the Co atom mwas determined from the electronic structure calculations. We used a two-dimensional lattice of 128 ×128 sites populated by classical spins with periodic boundary conditions and considered the fulltensorial exchange interactions and the on-site anisotropy termwhen calculating the effective ﬁeld. In all considered cases wefound that the harmonic model provided a good approximationfor the wave vector and normal vector of the spin spiral orcorrectly determined the ferromagnetic ground state. We alsoperformed simulations initialized in random initial conﬁgura-tions to investigate whether noncoplanar conﬁgurations canemerge in the systems, but found no indication for such abehavior in the absence of external magnetic ﬁeld. C. Effective interaction parameters In order to allow for a comparison between different ab initio calculation methods and experimental results, here we discuss how one can transform between the atomic interactionparameters calculated for many different neighbors used inthis paper, and effective nearest-neighbor interactions andparameters in the micromagnetic model. Complex magnetic textures are often studied in terms of micromagnetic models, where it is assumed that the magneti-zation direction is varying on a length scale much larger thanthe lattice constant, and the spins may be characterized by thecontinuous vector ﬁeld /vectors(/vectorr), the length of which is normalized to 1. In order to describe chiral magnetism, for a magneticmonolayer with C 3vpoint-group symmetry, the energy density is usually expressed as e(/vectors)=J/summationdisplay α=x,y,z(/vector∇sα)2+DwD(/vectors)−K(sz)2, (9) with the linear Lifshitz invariant: wD(/vectors)=sz∂xsx−sx∂xsz+sz∂ysy−sy∂ysz. (10) The relationship between the micromagnetic parameters J,D, and Kand the atomic parameters in Eq. ( 1) may beobtained by calculating the energy of the same type of spin conﬁgurations. Here we will consider spin spiral states withwave vectors along the ydirection: /vectors(/vectorr)=/vectore zcosky+/vectoreysinky, (11) where the plane of the spiral is spanned by the wave-vector direction /vectoreyand the out-of-plane direction /vectorez, corresponding to cycloidal spin spirals. In the micromagnetic model, the averageenergy over the spin spiral reads E micromagnetic =JVak2+DVak−1 2KVa, (12) if it is calculated for the atomic volume Va. For the atomic model one obtains [cf. Eq. ( 5)] Eatomic=−1 2/summationdisplay /vectorRij1 2/parenleftbig Jyy ij+Jzz ij/parenrightbig coskRy ij +1 2/summationdisplay /vectorRijDx ijsinkRy ij+1 2K. (13) Expanding Eq. ( 13) up to second-order terms in kyields Eatomic≈Jeffk2+Deffk+1 2Keff (14) apart from a constant shift in energy, with the effective spin- model parameters deﬁned as Jeff=1 4/summationdisplay jJij/parenleftbig Ry ij/parenrightbig2, (15) Deff=/summationdisplay jDx ijRy ij, (16) Keff=K+1 2/summationdisplay /vectorRij/parenleftbig Jyy ij−Jzz ij/parenrightbig . (17) The effective parameters JeffandDeffare also known as spin stiffness and spiralization, respectively [ 42,43]. The relationship between the micromagnetic and the effectiveparameters is given by J=1 VaJeff,D=1 VaDeff,K=−1 VaKeff. (18) Note that it is possible to deﬁne the atomic volume as Va=√ 3 2a2 2Dtwhere√ 3 2a2 2Dis the area of the in-plane unit cell and tis the ﬁlm thickness. In Ref. [ 22]t h ev a l u eo f t=nlayer√2 3a2Dwas used with nlayerthe number of magnetic atomic layers, corresponding to the ideal interlayer distancein an fcc lattice along the (111) direction. However, thisapproximation becomes problematic when lattice relaxationsare taken into account at the surface, since in this descriptionthe positions of the centers of the atoms are deﬁned instead ofthe thickness of the layers. Therefore, we used the expressionV a=4π 3R3 WS, where RWSis the radius of the atomic spheres used in the SKKR calculations, with RWS≈1.49˚Af o rt h e considered X/Co/Pt(111) systems. The cycloidal spin spiral deﬁned in Eq. ( 11) is called clock- wise or right-handed for k>0, meaning that when looking at the system from the side with the out-of-plane directiontowards the top the spins are rotating clockwise when movingto the right along the modulation direction of the spiral [ 44]. For 134405-3SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) k<0, the spin spiral is called counterclockwise or left-handed. According to Eq. ( 12), the micromagnetic DMI creates an energy difference between the two rotational senses, withD>0 preferring a counterclockwise rotation. Equation ( 16) demonstrates that the micromagnetic parameter is connected tothexcomponent of the atomic DM vector for spin spirals with wave vectors along the ydirection, or the in-plane component D /bardbl ijof the vector for general propagation directions. Note that the magnitude of D/bardbl ijis the same for all neighbors that can be transformed into each other via the C3vsymmetry of the system, while the sign can be deﬁned based on whether thevectors prefer clockwise or counterclockwise rotation of thespins. Note that in the case of C 3vsymmetry /vectorDijalso has a nonvanishing zcomponent, the effect of which on domain walls was investigated in Ref. [ 31]. Finally, we also deﬁne nearest-neighbor atomic interaction parameters JandD, which reproduce the effective parameters in Eqs. ( 15) and ( 16): Jeff=3 4a2 2DJ, D eff=3 2a2DD, (19) where Dis the in-plane component of the nearest-neighbor DM vector with the sign convention discussed above. Instead of performing the direct summations in Eqs. ( 15) and ( 16), we ﬁtted the spin spiral dispersion relation in Eq. ( 13) calculated from all interaction parameters in Eq. ( 1) with an effective nearest-neighbor model containing J,D, andKeff. The ﬁtting was performed in a range that is sufﬁciently largeto avoid numerical problems, but sufﬁciently small that themicromagnetic approximations may still be considered valid,corresponding to \|k\|a 2D/2π/lessorequalslant0.1. We note that this procedure is similar to how the atomic interaction parameters are deter-mined from spin spiral dispersion relations directly obtainedfrom total-energy calculations (see, e.g., Refs. [ 8,12,45]), but we used the spin model containing interaction parametersbetween many neighbors to determine the dispersion relationin the ﬁrst place. We conﬁrmed with spin dynamics simula-tions that in ferromagnetic systems the domain-wall proﬁlescalculated with the full model Hamiltonian ( 1) agree well with the proﬁles that can be calculated analytically from amicromagnetic model with the interaction parameters obtainedusing the above procedure. Nevertheless, we found that not allsystems can be sufﬁciently described by the three parametersused in the micromagnetic model, and this discrepancy canbe attributed to the competition between ferromagnetic andantiferromagnetic isotropic Heisenberg interactions (see Sec.III C for details). In order to support the comparison of our calculated param- eters with corresponding values obtained from experimentsor other theoretical approaches we shall present the micro-magnetic, effective, and nearest-neighbor atomic parameters asdeﬁned above for all considered systems. As an example, in Ta-bleIwe present the comparison between DMI values obtained for the Co monolayer on Pt(111) without a capping layer usingdifferent ab initio calculation methods in Refs. [ 12,22,31,43], similarly to the summary given in Ref. [ 47]. Using the above deﬁnitions, we ﬁnd reasonable agreement between the differenttheoretical descriptions, and all parameters fall into the rangewhere a ferromagnetic ground state is expected based on theexperimental investigations in Ref. [ 47].TABLE I. Nearest-neighbor atomic ( D), effective ( Deff), and mi- cromagnetic ( D) DM coupling obtained in several earlier publications for the Co monolayer on Pt(111). Positive values indicate that the counterclockwise (left-handed) chirality is preferred in the system.For a consistent transformation between the different parameters we used the values a 2D=2.774˚Aa n d RWS=1.44˚A. For Ref. [ 12]w e took into account the different deﬁnition of the atomic interactionparameters compared to Eq. ( 1). For Ref. [ 22] we considered the DMI value for the Co(3)/Pt(3) structure and the correction in Ref. [ 46]. D(meV) Deff(meV ˚A) D(mJ/m2) Ref. [ 31] 2.86 11.90 15.11 Ref. [ 43] 2.72 11.30 14.35 Ref. [ 12] 3.60 14.98 19.02 Ref. [ 22] 3.12 12.98 16.48 III. RESULTS A. Isotropic exchange interactions Figure 1shows the calculated isotropic exchange constants Jijbetween the Co atoms as a function of interatomic distance for the different capping layers (CL) and for the uncapped sys-tem (no CL). According to Eq. ( 1), positive and negative signs of the isotropic exchange parameters refer to FM and AFMcouplings, respectively. For all overlayers the ferromagneticNN interaction is dominating: it is the largest in magnitude forthe Au overlayer, for Pt and Ir a small decrease can be seen,while for Os and Re overlayers it is dramatically reduced. Thesecond- and third-nearest-neighbor couplings are considerablysmaller in magnitude than the NN couplings and the trendfor the different overlayers is also less systematic; e.g., in thecase of Au, Pt, and Os overlayers the second-NN couplingis ferromagnetic, while for Ir and Re it is AFM. Overall, themagnitude of the isotropic interactions decays rapidly with thedistance, becoming negligible beyond the third-NN shell. In Table II, the NN exchange couplings ( J 1) and the spin magnetic moments of Co atoms ( mCo) are summarized for the different overlayers. FIG. 1. Calculated Co-Co isotropic exchange parameters Jijas a function of the interatomic distance and different overlayers, and forthe Co/Pt(111) system without the capping layer (no CL) [ 31]. 134405-4MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) TABLE II. Calculated nearest-neighbor exchange interactions J1 between the Co atoms and the spin magnetic moment of Co mCofor all considered capping layers and for the Co/Pt(111) system without the capping layer (no CL) [ 31]. J1(meV) mCo(μB) Re 5.03 1.04 Os 9.66 1.55Ir 31.73 2.11 Pt 31.55 2.17 Au 37.54 2.10No CL 42.46 2.10 We ﬁnd that capping by 5 doverlayers systematically reduces J1compared to the uncapped case, which can be at- tributed to the hybridization between the Co and the overlayer.The magnetic moment of Co is almost constant for the Au, Pt,and Ir overlayers, while it shows an apparent decrease for Osand Re, similarly to the NN isotropic exchange. This decreaseis, however, much less drastic than for J 1:mCoin the case of the Re overlayer is about half of mCoin the case of the Au layer, while this ratio is about 1 /7f o rJ1. According to the Stoner model of ferromagnetism, the density of states (DOS) of the delectrons of Co at the Fermi level, n(/epsilon1F), in the nonmagnetic phase plays the crucial role in stabilizing spontaneous magnetization: in the case ofIn(/epsilon1 F)>1 (with Ibeing the Stoner parameter) the system becomes ferromagnetic. Hence the observed trends in mCoand J1are governed by the ﬁlling of the 5 dband of the overlayer that inﬂuences the 3 dband of Co via hybridization. In order to trace this effect, in Fig. 2we plot the density of states of thedelectrons in the Co layer and in the overlayer in the nonmagnetic phase, meaning that the exchange-correlation magnetic ﬁeld was set to zero during the density functional theory calculations. Since all the dstates of Au are occupied, the corresponding 5 dband lies well below the Fermi level, 024Re 024Os 024Ir 024Pt −6 −4 −2 0 2 /epsilon1−/epsilon1F(eV)024AuDOS (states/eV) FIG. 2. DOS of delectrons in the Co layer (solid red line) and in the overlayer (dashed blue line) in nonmagnetic X/Co/Pt(111) (X=Re, Os, Ir, Pt, Au) systems.TABLE III. Nearest-neighbor atomic ( J), effective ( Jeff), and micromagnetic ( J) parameters of Co for the isotropic exchange interaction of X/Co/Pt(111) ( X=Re, Os, Ir, Pt, Au) and Co/Pt(111) thin ﬁlms (no CL) [ 31] obtained from the calculated spin-model parameters by the ﬁtting procedure in Sec. II C. J(meV) Jeff(meV ˚A2) J(pJ/m) Re 0.82 4.73 0.56 Os 22.58 130.32 15.48 Ir 6.94 40.05 4.71 Pt 41.89 241.76 27.99Au 49.23 284.12 31.98 No CL 54.40 313.96 39.86 leaving the Co 3 dband localized around the Fermi level, with a large n(/epsilon1F) that explains the strong magnetic moment of Co in this case. Although the 5 dband of Pt is shifted upwards due to the decrease of the band ﬁlling and the hybridization withthe Co dband increases, the large peak in the Co DOS at the Fermi level still pertains, keeping m Coat a high value. This trend remains also in the case of the Ir overlayer, where the3d-5dhybridization further increases and n(/epsilon1 F) of Co clearly decreases, but the magnetic moment of Co is of similar valueas for the Au overlayer. For the cases of Os and Re overlayersthe Co 3 dband gets rather delocalized due to hybridization with the wider 5 dbands and n(/epsilon1 F) is further reduced leading to the observed drop in mCo. Note that a similar dependence of the Co moments on the overlayer was obtained for othersystems [ 48–51]. From the calculated isotropic exchange interactions we obtained the spin stiffness constant ( J eff), the corresponding micromagnetic parameter ( J), and the NN atomic value ( J) for all considered overlayers as described in Sec. II C, and presented them in Table III. Apparently, these values follow the variation of mCoorJ1for Os, Pt, and Au capping layers; however, in the case of Ir and Re they are considerably reduced.The reason for this behavior is the ampliﬁcation of the role ofexchange interactions between farther atoms in J effas follows from Eq. ( 15). From Fig. 1one can see that in the case of the Ir overlayer both the second- and third-NN couplings arenegative (AFM), which drastically reduces the value of J eff. The decrease of the NN coupling is apparently insufﬁcientin itself to explain the very small value of J effin the case of the Re overlayer. However, a detailed investigation ofFig.1shows that the seventh-NN interaction, J 7=−0.39 meV , gives a dominating negative contribution to Jeffdue to the large distance ( d=3.606a2D) and the large number (12) of neighbors in this shell. B. Relativistic spin-model parameters 1. Different capping layers Next, we investigate the in-plane components of Dzyaloshinskii-Moriya interactions between the Co atomswhich are shown in Fig. 3for all capping layers as a function of the distance between the Co atoms, compared to the valuesin the absence of a capping layer [ 31]. The sign changes of the DMI indicate switchings in the preferred rotational sense from 134405-5SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) 4 6 8 10 12 14−1.5−1.0−0.50.00.51.01.52.0D/bardbl ij(meV) clockwisecounterclockwiseRe Os Ir Pt Au no CL FIG. 3. In-plane component of the DM vectors D/bardbl ijas a function of the distance between the Co atoms for different overlayers, and for the Co/Pt(111) system without the capping layer (no CL) [ 31]. shell to shell, analogously to the oscillation between ferro- magnetic and antiferromagnetic isotropic exchange interactioncoefﬁcients. Except for the case of the Au capping layer, the NNDMI is the largest in magnitude; however, the DM vectors formore distant pairs also play an important role. This is somewhatdifferent for the isotropic couplings, J ij,i nF i g . 1, where the NN interaction is much larger in magnitude than the interactionsfor farther shells; therefore, the slow decay with the Co-Codistance is less visible than for the DMI in Fig. 3. To illustrate the overall effect of the overlayers on the DMI, we calculated the NN atomic, effective, and micromagneticDMI coefﬁcients of Co from the ab initio spin-model parame- ters as discussed in Sec. II C. These values are summarized in Table IVfor different capping layers. For comparison, we also included the corresponding values for Co/Pt(111).It is worthwhile to mention that the effective parameters inTable IVfollow exactly the same order for the different capping layers as the in-plane NN DM vectors in Fig. 3, unlike in the case of the isotropic exchange interactions. Regardless ofthe choice of the capping layer, the DMI is shifted towardsthe direction of clockwise rotational sense compared to theuncapped system. For the Pt/Co/Pt(111) system, the DMI isexceptionally weak, which is to be expected since inversionsymmetry is almost restored in this system if we consider that TABLE IV . Nearest-neighbor atomic ( D), effective ( Deff), and micromagnetic ( D) DM coupling of Co obtained from the spin-model parameters for X/Co/Pt(111) thin ﬁlms ( X=Re, Os, Ir, Pt, Au) and for Co/Pt(111) without any capping layer (no CL). D(meV) Deff(meV ˚A) D(mJ/m2) Re 1.82 7.57 8.94 Os 2.58 10.74 12.75 Ir −1.75 −7.28 −8.56 Pt 0.20 0.83 0.96Au 1.50 6.24 7.02 No CL 2.86 11.90 15.11−0.15−0.10−0.05-Δmorb(μB) Co/Pt(111) 05Keﬀ(meV) in-plane out-of-plane Re Os Ir Pt Au010Deﬀ(meV ·) counterclockwise clockwise FIG. 4. Calculated values of orbital moment anisotropy in the Co layer with negative sign −/Delta1m orb,M A E Keff, and effective DMIDeffforX/Co/Pt(111) thin ﬁlms ( X=Re, Os, Ir, Pt, Au). The corresponding parameters for Co/Pt(111) are also illustrated bydashed green lines. generally the interfacial DMI is dominated by the magnetic and nonmagnetic heavy metal layers directly next to each other. We would also like to point out that the Ir capping layer is the only one that switches the sign of the DMI preferring clockwiserotation. This is somewhat unexpected since the Ir layer alsochanged the preferred rotational sense to clockwise when itwas introduced between the Co monolayer and the Pt(111)substrate [ 31], so it should prefer a counterclockwise rotation for the opposite stacking order according to the three-sitemodel of the DMI [ 52]. A possible reason for this effect is that the reduced coordination number of the Ir atoms in the cappinglayer as well as the electrostatic potential barrier at the surfacesigniﬁcantly modify the electronic structure of the cappinglayer compared to the bulk case or when the Ir is inserted belowthe Co layer. This sign change of the DMI in Ir/Co/Pt(111)indicates that ultrathin-ﬁlm systems can display qualitativelydifferent features compared to magnetic multilayers, where theIr/Co/Pt stacking was suggested as a way of enhancing the DMI[27]. The different behavior of Ir as a capping layer and as an inserted layer was recently investigated in Ref. [ 53]. In order to study the dependence of the DMI on the capping layer, we calculated additional quantities determinedby the strength of the spin-orbit coupling, namely, the totalMAE K effand the anisotropy of the orbital moment of Co atoms, /Delta1m orb=m⊥ orb−m/bardbl orb, where the superscripts ⊥and /bardblrefer to calculations performed for a normal-to-plane and an in-plane orientation of the magnetization in the Co layer,respectively. Figure 4shows /Delta1m orbwith a negative sign (top panel), Keff(middle panel), and Deff(bottom panel) for the Co monolayer depending on capping layer. Note that negative andpositive signs of K effrefer to easy-axis and easy-plane types of magnetic anisotropy, respectively. For 3dtransition metals, where the spin-orbit coupling is small compared to the bandwidth, second-order perturbationtheory describes the uniaxial magnetic anisotropy well [ 54]. According to Bruno’s theory, neglecting spin-ﬂop coupling andfor a ﬁlled spin-majority dband, a negative proportionality between the MAE and /Delta1m orbapplies, that was conﬁrmed 134405-6MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) 4 6 8 10 12 14 d−1012D/bardbl ij(meV)counterclockwise clockwiseλ=0.0 λ=0.2 λ=0.4 λ=0.6 λ=0.8 λ=1.0 FIG. 5. In-plane DMI as a function of the distance between the Co atoms for various values of the SOC scaling parameter λin the Ir capping layer of the Ir/Co/Pt(111) system. theoretically and experimentally for Co layers [ 54–58]. From Fig. 4a good qualitative correlation can be inferred between Keffand−/Delta1m orbwith the exception of the Re overlayer. Indeed, due to the large 3 d-5dhybridization, the delocalization of the spin-majority band of Co is increased in the case of theRe overlayer such that the above-mentioned conditions for thesimple proportionality do not apply. From Fig. 4it turns out that the variations of K effand Deffalso correlate well with each other. This is somewhat surprising since, as mentioned above, the MAE is of secondorder in the SOC, while the DM term appears in the ﬁrstorder of the perturbative expansion [ 4]. Compared to the Co/Pt(111) system, the Os capping layer does not modify theDMI signiﬁcantly, but we observe a strong easy-plane MAE.The Re and the Au capping layers decrease the magnitudeofD eff, and the preferred magnetization direction is also in plane. An out-of-plane magnetization was obtained for Ir andPt capping layers, and as discussed above the Ir cappinglayer prefers a clockwise rotation, while in the case of thePt overlayer the DMI is close to zero. 2. Scaling of the spin-orbit coupling in the Ir overlayer To gain further insight into the the sign change of the DMI in the Co monolayer with the Ir capping layer, we artiﬁciallymanipulated the strength of SOC at the Ir atoms. Ebert et al. introduced a continuous scaling of the SOC via the parameter λ within the relativistic KKR formalism [ 59]: calculation without scaling ( λ=1) corresponds to the fully relativistic case, while λ=0 can be identiﬁed with the so-called scalar-relativistic description. Importantly, in the above formalism the scaling ofthe SOC can be used selectively for arbitrary atomic cells. Wethus applied it to the Ir monolayer, while the SOC at all othersites of the system remained unaffected. Figure 5shows D /bardbl ijas a function of the distance between the Co atoms for different scaling parameters. Varying λ has a strong inﬂuence on the NN in-plane DMI: it changescontinuously from preferring counterclockwise ( λ=0) to preferring clockwise ( λ=1) rotational direction, while the−3−2−10Keﬀ(meV) 0.0 0.2 0.4 0.6 0.8 1.0 λ−505Deﬀ(meV ·) FIG. 6. Calculated MAE Keff, and effective DMI Deffas a function of the SOC scaling parameter λin the Ir overlayer of the Ir/Co/Pt(111) system. changes in the other shells are smaller in relative and in absolute terms. In the case of λ=0, the NN in-plane DMI takes a value of 2.32 meV , which means that the NN DMI of the Co/Pt(111)system (1.98 meV) [ 31] is nearly restored in this case. In accordance with the results of ﬁrst-order perturbation theory, Fig. 6illustrates that the variation of the effective DMI is rather linear with λ.F o rλ=0,K effis close to the value of the uncapped Co/Pt(111) system ( −0.20 meV [ 31]) and it increases in magnitude to −3m e Vf o r λ=1. Following the change in the NN in-plane DMI interaction in Fig. 5, the sign of the effective DMI turns from preferring counterclockwiseto preferring clockwise rotation when increasing the strengthof the SOC in the Ir overlayer. On the other hand, at λ=0D eff is somewhat smaller in magnitude than in the case of the uncapped Co/Pt(111) (11 .90 meV ˚A). This indicates that the Ir overlayer inﬂuences the DMI of the system not just due toits strong SOC but also by modifying the electronic states inthe Co monolayer via hybridization. 3. Changing the capping layer composition in Au1−xIrx/Co/Pt(111) Controlling the Ir concentration xin the alloy capping layer Au 1−xIrx(0/lessorequalslantx/lessorequalslant1) represents a transition where the effect of increasing hybridization between the 3 dband of Co and the 5 dband of the capping metal can be traced, as shown in Fig. 2. On the other hand, the strength of the SOC, deﬁned by the operator ξ/vectorL/vectorS, in Au and Ir is roughly the same ( ξ≈600 meV), meaning that the alloying is expected to have a different effect than the scaling of the SOC discussedin the previous section. Thus, we performed calculations ofthe spin-model parameters for x=0.1,0.2,..., 0.9b yu s i n g the CPA for the chemically disordered overlayer. The layerrelaxation was varied as a function of xaccording to Vegard’s law using the calculated layer relaxation of the Au/Co/Pt(111)and Ir/Co/Pt(111) systems. The in-plane components of the DM vectors in the Co monolayer from the ﬁrst to the fourth shell are shown in Fig. 7 as a function of the Ir concentration. When increasing the Ir 134405-7SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) 0.0 0.2 0.4 0.6 0.8 1.0 x(Ir concentration)−1.0−0.50.00.5D/bardbl ij(meV)counterclockwise clockwise D1 D2 D3 D4 FIG. 7. In-plane components of the DM vectors D/bardbl ijof Co from the ﬁrst ( D1) to the fourth shell ( D4), as a function of the Ir concentration ( x)i nt h eA u 1−xIrx/Co/Pt(111) system. concentration, the sign of the ﬁrst-NN and the second-NN D/bardbl ijchanges from positive to negative. The third-NN in-plane DM for the Au/Co/Pt(111) system is negative, it turns positivearound x≈0.1, and it has approximately the same magnitude around 20% Ir concentration as for the pure Ir/Co/Pt(111)layer, with a maximal amplitude at about x=0.5. The sign of the fourth-NN D /bardbl ijis not changed by the alloying, and the magnitude remains nearly constant. The changes of the effective DMI and MAE are shown in Fig. 8as a function of the Ir concentration. Unlike the case where the SOC was scaled (Fig. 6), the variation of KeffandDeffwithxis nonmonotonous, with a maximum of Keffat around 10% and a minimum of Deffat around 90% Ir concentration. The effect of alloying the nonmagnetic heavy metals on the DMI was also investigated recently in Ref. [ 60], where 4-ML Ir xPt1−x/1-ML Co /4-ML Pt and 4-ML Pt xAu1−x/1- FIG. 8. Calculated total MAE Keffand effective DMI Deffin the Co monolayer as a function of Ir concentration ( x)i nt h e Au1−xIrx/Co/Pt(111) system.TABLE V . Obtained magnetic ground states for X/Co/Pt(111) thin ﬁlms ( X=Re, Os, Ir, Pt, Au). KJ/D2Ground state Re −0.20 Tilted SS Os −6.77 In-plane FM Ir 2.27 Out-of-plane FM Pt 436.13 Out-of-plane FMAu −1.46 In-plane FM ML Co /4-ML Pt trilayers were considered along the (111) stacking direction. Similarly to the results presented here,a nonmonotonous dependence on the concentration was re-ported, together with a switching to negative DMI due to thepresence of Ir in the capping layer, although only at smallerIr concentration. In Ref. [ 60], for pure Au or pure Ir capping layers similar values of the DMI were obtained to the uncappedvalues summarized in Table I. As discussed above, in the present calculations the decrease of the DMI due to the Auoverlayer and the sign change due to the Ir overlayer canprobably be attributed to the reduced coordination numberof the atoms if the capping layer is only 1 ML thick. As apossible alternative for obtaining a microscopic understanding,an interesting perturbative model for the DMI in zig-zag chainscan be found in Ref. [ 61], where the dependence of the sign and strength of the DMI on different parameters is reported. C. Magnetic ground states The ground states of the systems were determined by com- bining harmonic spin spiral calculations with spin dynamicssimulations as described in Sec. II C. After scaling out the energy and length scales, the micromagnetic energy density inEq. ( 9) can be described by a single dimensionless parameter KJ/D 2, which governs the formation of the magnetic ground state. As already discussed in earlier publications [ 44,62,63], noncollinear ground states are expected to be formed for −1< KJ/D2<π2 16≈0.62 in this model; the upper limit denotes where magnetic domain walls become energetically favorablein out-of-plane oriented ferromagnets, while the lower limitindicates the instability of the in-plane oriented ferromagneticstate towards the formation of an elliptic conical state. The calculated values are summarized in Table Vfor these systems. For most considered capping layers the parameterKJ/D 2is outside the range where the formation of non- collinear states is expected, and in the simulations we indeedobserved FM ground states. This can be explained eitherby the strong easy-plane (Os) or easy-axis (Ir) anisotropies,the weakness of the DMI for the Pt/Co/Pt(111) system, orthe combination of the above for the Au capping layer. Forthe Re/Co/Pt(111) system, the micromagnetic model predicts[62,63] a cycloidal spin spiral ground state with the normal vector in the plane, just as it was assumed in Eq. ( 11). However, by minimizing Eq. ( 4) with respect to the wave vector /vectorkand the normal vector /vectorn, we obtained a tilted spin spiral state of the form /vectors i=/vectorexcoskRy isin/Phi10−/vectoreysinkRy i+/vectorezcoskRy icos/Phi10. (20) 134405-8MAGNETISM OF A Co MONOLAYER ON Pt(111) CAPPED … PHYSICAL REVIEW B 97, 134405 (2018) xy z xz yΦ0=3 8◦allJij(a) xy z xz yΦ0=0◦NNJ(b) xy z xz yΦ0=5 8◦NNNJ1,J2(c) FIG. 9. The tilted spin spiral ground state found in the Re/Co/Pt(111) system in spin dynamics simulations. The tilting angle /Phi10is deﬁned in Eq. ( 20). (a) Ground state obtained using the full Jij exchange interaction tensors. (b) Ground state obtained with only nearest-neighbor (NN) atomic interaction parameters, J=0.82 meV from Table III, NN DMI, and effective on-site anisotropy. (c) Ground state obtained by performing the ﬁtting procedure discussedin Sec. II C for NN and next-nearest-neighbor (NNN) exchange interactions, J 1=53.46 meV and J2=−18.20 meV, NN DMI, and effective on-site anisotropy. Red and blue colors correspond topositive and negative out-of-plane spin components, respectively. The ground state obtained from the spin dynamics simu- lations is displayed in Fig. 9(a). Although the spiral became slightly distorted due to the anisotropy, we found that it couldstill be relatively well described by Eq. ( 20) using a wavelength ofλ=2π/k≈3.5 nm and a tilting angle of /Phi1 0≈38◦.T h eenergy gain due to the tilting is approximately 0 .04 meV /atom. Note that the tilted spin spiral state is still a cycloidal spiral inthe sense that the wave vector is located in the rotational planeof the spirals, but the normal vector is no longer conﬁned tothe surface plane. This is different from the case of weak DMIin out-of-plane magnetized ﬁlms, where the normal vector ofdomain walls gradually rotates in the surface plane from Néel-type to Bloch-type rotation due to the presence of the magne-tostatic dipolar interaction (see, e.g., Ref. [ 64] ) .I ta l s od i f f e r s from the elliptic conical spin spirals discussed in Refs. [ 62,63] because the tilted spin spiral state has no net magnetization. The formation of such a ground state can be explained by the easy-plane anisotropy preferring an in-plane orientationof the spiral, the DMI preferring a spiral plane perpendicularto the surface, and the simultaneous presence of competingferromagnetic and antiferromagnetic isotropic exchange inter-actions in the system, the latter also leading to the reducedvalue of the effective J effparameter for the Re capping layer in Table II. This is illustrated in Figs. 9(b) and9(c):i nt h e nearest-neighbor atomic model, a vertical cycloidal spin spiralground state is obtained, in agreement with the prediction of themicromagnetic description [ 62,63]. The spin spiral wavelength is also signiﬁcantly shorter, λ≈1.4 nm, due to the inaccuracy of the nearest-neighbor ﬁtting procedure. On the other hand, ifthe ﬁtting is performed with taking nearest- and next-nearest-neighbor isotropic exchange interactions into account, thetilted spin spiral ground state is recovered with λ≈3.8n m and/Phi1 0≈58◦, in reasonable agreement with the full model. IV . SUMMARY AND CONCLUSIONS In conclusion, we examined the X/Co/Pt(111) ( X=Re, Os, Ir, Pt, Au) ultrathin ﬁlms using ﬁrst-principles and spin-modelcalculations. We determined the Co-Co magnetic exchangeinteraction tensors between different pairs of neighbors andthe magnetic anisotropies. From the results of the ab initio calculations we also determined effective and micromagneticspin-model parameters for the Co layers. For the isotropicexchange couplings we found dominant ferromagnetic nearest-neighbor interactions for all systems, which decrease withthed-band ﬁlling of the capping layer. This effect due to the hybridization between the 3 dstates of the Co layer and the 5 dstates of the capping layer can be qualitatively explained within a Stoner picture, which also accounts forthe similarly decreasing magnetic moment. Considering theeffective isotropic couplings of Co, we found signiﬁcantlylower values for Re and Ir overlayers than what would beexpected simply based on the decrease of the nearest-neighborinteractions; this we attributed to competing antiferromagneticcouplings with further neighbors. We also investigated the in-plane Dzyaloshinskii-Moriya interactions of Co, and found it to be weaker for all capping layers compared to the uncapped Co/Pt(111) system. For the Ircapping layer we found a switching from counterclockwise toclockwise rotation, which is unexpected since the same switch- ing can also be observed if the Ir is inserted between the mag- netic layer and the substrate [ 31]. We attributed this effect to the reduced coordination number of Ir atoms and the electrostaticpotential barrier at the surface. We also found a correlation between the effective Dzyaloshinskii-Moriya interactions D eff, 134405-9SIMON, RÓZSA, PALOTÁS, AND SZUNYOGH PHYSICAL REVIEW B 97, 134405 (2018) the effective magnetic anisotropies Keff, and the anisotropy of the orbital moment /Delta1m orb. We further investigated the sign change of the effective Dzyaloshinskii-Moriya interaction ofCo for the Ir capping layer by scaling the strength of thespin-orbit coupling at the Ir sites and by tuning the ﬁlling of the 5dband in a Au 1−xIrx/Co/Pt(111) system. We found a linear dependence of the effective Dzyaloshinskii-Moriya interactionon the spin-orbit coupling strength in agreement with theperturbative description, and a nonmonotonic dependence on the band ﬁlling. Using the spin-model parameters we determined the mag- netic ground state for all considered systems. For Os, Ir, Pt,and Au capping layers we found a ferromagnetic ground state,in agreement with the analytical prediction based on the calcu-lated micromagnetic parameters. For the Re/Co/Pt(111) sys-tem we found a tilted spin spiral ground state, the appearanceof which can only be explained if competing ferromagnetic andantiferromagnetic isotropic exchange interactions are takeninto account alongside the Dzyaloshinskii-Moriya interactionand the easy-plane anisotropy.Our results highlight the importance of ab initio calculations and atomic spin-model simulations in cases where simplermodel descriptions might lead to incomplete conclusions. Thepresent paper may motivate further experimental investigationsin this direction, exploring the sign of the Dzyaloshinskii-Moriya interaction and the role of competing isotropic ex-change interactions in ultrathin-ﬁlm systems. ACKNOWLEDGMENTS The authors would like to thank F. Kloodt-Twesten and M. Mruczkiewicz for insightful discussions. The authors grate-fully acknowledge the ﬁnancial support of the National Re-search, Development, and Innovation Ofﬁce of Hungary underProjects No. K115575, No. PD120917, and No. FK124100,of the Alexander von Humboldt Foundation, of the DeutscheForschungsgemeinschaft via SFB668, of the SASPRO Fel-lowship of the Slovak Academy of Sciences (Project No.1239/02/01), and of the Hungarian State Eötvös Fellowship. [1] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,190 (2008 ). [2] A. Fert, V . Cros, and J. Sampaio, Nat. Nanotechnol. 8,152 (2013 ). [3] I. Dzyaloshinsky, J. Phys. Chem. Solids 4,241(1958 ). [4] T. 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PhysRevB.104.035425.pdf	PHYSICAL REVIEW B 104, 035425 (2021) From nonequilibrium Green’s functions to quantum master equations for the density matrix and out-of-time-order correlators: Steady-state and adiabatic dynamics Bibek Bhandari ,1,2Rosario Fazio,3,4Fabio Taddei,5and Liliana Arrachea6 1NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy 2Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA 3ICTP, Strada Costiera 11, I-34151 Trieste, Italy 4Dipartimento di Fisica, Università di Napoli “Federico II, ” Monte S. Angelo, I-80126 Napoli, Italy 5NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy 6International Center for Advanced Studies, Escuela de Ciencia y Tecnología, Universidad Nacional de San Martín and ICIFI, Avenida 25 de Mayo y Francia, 1650 Buenos Aires, Argentina (Received 8 March 2021; revised 21 May 2021; accepted 30 June 2021; published 20 July 2021) We consider a ﬁnite quantum system under slow driving and weakly coupled to thermal reservoirs at different temperatures. We present a systematic derivation of the quantum master equation for the density matrix andthe out-of-time-order correlators. We start from the microscopic Hamiltonian and we formulate the equationsruling the dynamics of these quantities by recourse to the Schwinger-Keldysh nonequilibrium Green’s functionformalism, performing a perturbative expansion in the coupling between the system and the reservoirs. Wefocus on the adiabatic dynamics, which corresponds to considering the linear response in the ratio between the relaxation time due to the system-reservoir coupling and the time scale associated to the driving. We calculate theparticle and energy ﬂuxes. We illustrate the formalism in the case of a qutrit coupled to bosonic reservoirs andof a pair of interacting quantum dots attached to fermionic reservoirs, also discussing the relevance of coherenteffects. DOI: 10.1103/PhysRevB.104.035425 I. INTRODUCTION The study of heat transport and heat–work conversion in few-level open quantum systems under the action of slowtime-dependent protocols is a subject of active investigationfor some time now. Examples are qubits [ 1–12], harmonic oscillators [ 13–20], and quantum dots [ 21–26] under slow cyclic driving, as well as nanomechanical [ 27–34] and nano- magnetic [ 35,36] degrees of freedom in contact to bosonic or fermionic baths, possibly with a temperature bias. In the context of open systems the concept “adiabatic dy- namics” has been introduced to deﬁne the evolution of slowlydriven systems through time-dependent parameters [ 37–42]. It applies to the nonequilibrium regime where the typicaltime scale of the dynamics of the frozen Hamiltonian for the full setup, including the driven system along with the contact to the reservoirs and the reservoirs themselves ismuch faster than the characteristic time for the changes ofthis Hamiltonian. This motivates a linear-response treatmentwith respect to the rate of change of the time-dependentparameters [ 12,23]. Similar ideas are beyond the adiabatic perturbation theory in closed systems [ 43–45]. A widely used framework to analyze the nonequilibrium dynamics of a few-level system weakly coupled to reservoirsis that based on master equations . The standard approach is the Lindblad formulation [ 46] which has been used to study the dynamics of different systems in the ﬁeld ofcold atoms, optics, quantum information and condensed mat- ter [ 1,4,5,7,8,47–56]. The main strategy of this formulation relies on the equation of motion for the reduced density matrixof the quantum system, with the degrees of freedom of thereservoirs traced away. Another route to derive the masterequation is to calculate the dynamics of the mean values of thematrix elements of the density matrix by treating the couplingbetween the system and the reservoirs in perturbation theorywithin a Schwinger-Keldysh contour. This implies consid-ering a contour that evolves forwards and then backwardswith respect to an initial time t 0→− ∞ . The procedure was introduced in Ref. [ 57] for a metallic island, in Refs. [ 58,59] for a single-level quantum dot in the stationary regime, andextended to time-dependent scenarios in Refs. [ 60–64]. A different formalism was also recently proposed [ 65] and ex- tended to time-dependent systems [ 66]. Here, we present an alternative derivation of the master equations for the density matrix. We rely on the nonequi-librium Green’s function formalism combined with suitableanalytical continuations [ 67–70]. We focus on an adiabatically driven N-level system weakly coupled to thermal reservoirs at different temperatures, see Fig. 1. We derive the master equations for the density matrix upto the adiabatic component.We extend the procedure to the calculation of master equa-tions for out-of-time-order correlators (OTOC), which arecurrently under active investigation in the context of a varietyof physical problems [ 71–76]. OTOCs are considered as good 2469-9950/2021/104(3)/035425(19) 035425-1 ©2021 American Physical SocietyBHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) FIG. 1. A N-level adiabatically driven system in contact with two reservoirs at different temperatures, TL=T+/Delta1TandTR= T−/Delta1T. witnesses of scrambling dynamics in many-body systems. In the systems described by nonintegrable Hamiltonians,OTOC’s are expected to grow as a function of time [ 71–74]. In the systems coupled to thermal baths, they stabilize after sometime [ 75,76] and tend to an asymptotic value. The formalism we describe here enables the analysis of these correlationfunctions in nonequilibrium situations, where the system isunder slow driving and the reservoirs have a thermal or achemical potential bias. The paper is organized as follows. In the next section, we will present the model for a periodically driven quantum sys-tem. There, we will study the dynamics of the density matrixand the OTOCs. In Sec. III, we will perform an adiabatic expansion to obtain full adiabatic master equations for densitymatrix as well as OTOCs. We will also derive the frozen andadiabatic contributions to the charge and energy currents interms of the density matrix. In order to illustrate the generalformulation, in Sec. IV, we put forward two different exam- ples: in the ﬁrst one we shall study a driven qutrit in contactwith bosonic reservoirs, and in the second example we willconsider a driven quantum dot system attached to fermionicreservoirs. Section Vis devoted to summary and conclusions. Some technical details are presented in appendices. II. GENERAL FORMALISM We present here the derivation of the master equation from the nonequilibrium Green’s function formalism combinedwith the analytical continuation procedure known as Langreththeorem [ 67,68]. A. Model We consider a driven quantum system S, which depends on time through a set of time-dependent parameters X(t)= (X1(t),..., XN(t)), described by the Hamiltonian HS(t)≡HS(X(t)). (1) In general, the system Hamiltonian contains one or more subsystems with multiple degrees of freedom, expressed ina convenient basis, which expands the N-dimensional Hilbert state. For example, in Sec. IV, we consider a qutrit (character- ized by the three levels \|0/angbracketright,\|1/angbracketrightand\|2/angbracketright) with time-dependent energies and time-dependent transitions between the differentlevels. We also consider two coupled quantum dots of spinlessfermions with time-dependent gate voltages and tunneling elements, as well as interdot Coulomb interaction. In this case,each quantum dot deﬁnes a subsystem, and the degrees offreedom of each quantum dot are determined by the charge.The corresponding states of the basis are four and read \|s/angbracketright≡ \|0,0/angbracketright,\|1,0/angbracketright,\|0,1/angbracketright,\|1,1/angbracketright. The system S is coupled to a set of N rreservoirs described by the Hamiltonian HB=Nr/summationdisplay α=1/summationdisplay k/epsilon1kαˆb† kαˆbkα, (2) where the operators ˆb† kαandˆbkα(relative to an excitation in the bathαwith momentum k) may satisfy bosonic or fermionic statistics. For the case of bosons, we focus on bosonic excita-tions, like phonons or photons. For the case of fermions, wefocus on electron systems with a ﬁnite chemical potential. Thecontact between the driven system and the baths is given bythe Hamiltonians H (I) C=/summationdisplay s,s/prime/summationdisplay k,αVkαˆπα s,s/prime(ˆb† kα+ˆbkα), H(II) C=/summationdisplay s,s/prime/summationdisplay k,α(Vkαˆb† kαˆπα s,s/prime+H.c.), (3) where Vkαis the coupling strength between the system and reservoir α. The structure of the Hamiltonian H(I) Ccorresponds to changing sin the central system while creating or destroy- ing a quasiparticle in the bath and it is a natural coupling inthe case of reservoirs modeled by harmonic oscillators [ 77]. Instead, H (II) Cimplies the creation of a particle (or quasipar- ticle) in the bath while changing sof the central system. Usually, in the case of fermionic systems and reservoirs, sucha term naturally describes a tunneling process where a fermionis destroyed in the system and created in the reservoir andvice versa. Albeit, that type of coupling is also used in thecase of N-level systems coupled to bosonic reservoirs. In the derivation of the master equations we will consider, for thecase of a bosonic bath, the Hamiltonian H (I) Cand we will indicate how to get from them the corresponding equationsforH (II) C. For fermionic baths we will consider H(II) C.T h e operators ˆ πα s,s/prime=ηα s,s/prime\|s/angbracketright/angbracketlefts/prime\|are deﬁned on the basis \|s/angbracketrightasso- ciated to the degrees of freedom of the system and may berestricted by selection rules and by the Pauli principle in thecase of fermionic systems. For instance, in the case of thetwo coupled quantum dots of spinless fermions that we willanalyze in Sec. IV B , where each quantum dot is connected to one fermionic reservoir through a tunnel coupling, these are ˆπ (1) 0,1=[ˆπ(1) 1,1]†=/summationtext /lscript=0,1η(1) 0,1\|0,/lscript/angbracketright/angbracketleft1,/lscript\|for the quantum dot (1) and ˆ π(2) 0,1=[ˆπ(2) 1,1]†=/summationtext /lscript=0,1η(2) 0,1\|/lscript,0/angbracketright/angbracketleft/lscript,1\|for the quan- tum dot (2). The Hamiltonian for the system at any time tdetermining the value Xof the time-dependent parameters can be diago- nalized by a unitary matrix ˆU(X)a sf o l l o w s : ˜HS(X)=ˆU(X)HS(X)ˆU†(X)=/summationdisplay lεl(X)ˆρll(X),(4) where ˆ ρlj(X)=\|l(X)/angbracketright/angbracketleftj(X)\|is the density matrix ex- pressed in the basis of the instantaneous eigenstates of the 035425-2FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) Hamiltonian, being ˜HS(X)\|l(X)/angbracketright=εl(X)\|l(X)/angbracketright. (5) We stress that this basis depends on time through the time- dependence of the parameters X. We deﬁne ˆ πα=/summationtext s,s/primeˆπα s,s/prime and we express the contact Hamiltonian in the instantaneous basis as follows ˜H(I,II) C(X)=/summationdisplay k,α/summationdisplay l,jVkα[λα,lj(X)ˆb† kαˆρlj+λα,lj(X)ˆρljˆbkα], (6) where, for the case H(I) C,w eh a v e λα,lj(X)=λα,lj(X)=[ˆU(X)ˆπαˆU†(X)]l,j, (7) while for the case H(II) C,w eh a v e λα,lj(X)=[ˆU(X)ˆπαˆU†(X)]l,j, λα,lj(X)=[ˆU(X)ˆπ† αˆU†(X)]l,j. (8) B. Dynamics of the density matrix and of the out-of-time-order correlator (OTOC) The derivation of the equation of motion governing the long-time dynamics of the density matrix and of the out-of-time-order correlator, in the limit of weak coupling to thereservoirs, follows similar lines and we will treat the twocases in parallel. “Long time” refers to the regime beyondthe transient associated to the switching-on of the couplingbetween system and reservoirs. We start by noticing that any observable O, which depends on the degrees of freedom of the system, can be expressed asfollows: O(t)=/summationdisplay l,jOlj(t)ˆρlj, (9) where Olj(t)=/angbracketleftl\|O(t)\|j/angbracketrightare the matrix elements of the operator O(t) in the instantaneous eigenstates basis. The ex- pectation value of this observable at a given time tis /angbracketleftO/angbracketright(t)=Tr[ ˆρtot(t)O(t)]=/summationdisplay ljOlj(t)ρlj(t), (10) where we deﬁne the density matrix as ρlj(t)=Tr[ ˆρtot(t)ˆρlj]. (11) In Eqs. ( 10) and ( 11), ˆρtot(t) is the state of the full system coupled to the baths, which is described by the HamiltonianH(t)=˜H S(t)+HB+˜HC(t). We see that the dynamics of /angbracketleftO/angbracketright(t) is determined by the evolution of the matrix elements of the operator in the basis of the instantaneous eigenstates ofH S(t) and the dynamics of the density matrix ρlj(t). The latter depends on the full Hamiltonian H(t). Changing to the Heisenberg representation with respect to H,ˆρH ij(t)=U†(t,t0)ˆρijU(t,t0), the matrix elements of this operator are written as ρlj(t)=Tr/bracketleftbig ˆρ0ˆρH lj(t)/bracketrightbig , (12) with ˆρtot(t)=U(t,t0)ˆρ0U†(t,t0), ˆρ0being the state at the initial time t0, where U(t,t0)=ˆT{exp−i/¯h/integraltextt t0dt/primeH(t/prime)}is the evolution operator, being ˆTthe time-order operator.We deﬁne the OTOC between observables at time t, rela- tive to a reference time tr, as follows: K(t)=/angbracketleftbig OH A(t)OH B(tr)OH C(t)OH D(tr)/angbracketrightbig . (13) Here OA,OB,OC, and ODare Hermitian operators depend- ing on the degrees of freedom of the system S expressed inthe Heisenberg picture with respect to H. We expand O A(t) andOC(t)a si nE q .( 9). The corresponding matrix elements are denoted, respectively, as OA,lj(t) and OC,lj(t). In this rep- resentation, the OTOC of Eq. ( 13) reads [ 75] K(t)=/summationdisplay lj l/primej/primeOA,lj(t)OC,l/primej/prime(t)Klj,l/primej/prime(t), (14) where we have introduced the OTOC operator ˆKH lj,l/primej/prime(t)=ˆρH lj(t)OH B(tr)ˆρH l/primej/prime(t)OH D(tr) (15) and its corresponding mean value Klj,l/primej/prime(t)=/angbracketleftˆKH lj,l/primej/prime(t)/angbracketright. We now introduce the deﬁnitions of mixed lesser Green’s functions for time correlations between the bath and thedensity /OTOC operators, which we denote with G/G G < lj,kα(t,t/prime)=±i/angbracketleftbig b†H kα(t/prime)ˆρH lj(t)/angbracketrightbig , G< kα,lj(t,t/prime)=±i/angbracketleftbig ˆρH jl(t/prime)bH kα(t)/angbracketrightbig , G< kα;lj l/primej/prime(t,t/prime)=±i/angbracketleftbigˆTK/bracketleftbigˆKH lj,l/primej/prime(t/prime)/bracketrightbig†ˆbH kα(t)/angbracketrightbig , G< lj l/primej/prime;kα(t,t/prime)=±i/angbracketleftbigˆTKˆb†H kα(t/prime)ˆKH lj,l/primej/prime(t)/angbracketrightbig . (16) For fermionic systems, the upper /lower sign applies to many- body states such that \|l/angbracketright,\|j/angbracketright,a sw e l la s \|l/prime/angbracketright,\|j/prime/angbracketright,d i f f e r in odd /even number of particles. For bosonic systems, it corresponds the lower sign. The operator ˆTKdenotes time- ordering along Schwinger-Keldysh contour, which starts at−∞ , evolves forwards towards +∞ and the backwards to −∞ [67,68]. In the expressions for the OTOC, there are four operators at the times tandt r. Hence, this contour in extended in order to include two of these contours [ 73], as explained in Appendix D. Calculating the evolution of ˆ ρH lj(t), taking the mean value with respect to ˆ ρ0as in Eq. ( 12) and introducing the deﬁnitions of the lesser Green’s functions given in Eqs. ( 16), we get d/angbracketleftˆρlj/angbracketright dt =i ¯h[εl(t)−εj(t)]/angbracketleftˆρlj/angbracketright±1 ¯h/summationdisplay k,αVkα ×/bracketleftBigg/summationdisplay mλα,ml(t)G< mj,kα(t,t)−/summationdisplay nλα,jn(t)G< ln,kα(t,t) +/summationdisplay mλα,ml(t)G< kα,jm(t,t)−/summationdisplay nλα,jn(t)G< kα,nl(t,t)/bracketrightBigg , (17) where ±corresponds to fermionic and bosonic reservoirs, respectively, the ﬁrst term in the right-hand side stems from i ¯h/angbracketleft[˜HH S,ˆρH lj]/angbracketrightand we recall that εl(t)≡ε(X(t)). 035425-3BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) Similarly, the equation of motion for the OTOC reads [ 75] ∂/angbracketleftKlj l/primej/prime(t)/angbracketright ∂t=i ¯h(εl/prime(t)−εj/prime(t)+εl(t)−εj(t))/angbracketleftKlj l/primej/prime(t)/angbracketright±1 ¯h/summationdisplay kαVkα/bracketleftBigg/summationdisplay mλα,ml(t)G< mjl/primej/prime;kα(t,t)−/summationdisplay nλα,jn(t)G< lnl/primej/prime;kα(t,t) +/summationdisplay mλα,ml/prime(t)G< lj mj/prime;kα(t,t)−/summationdisplay nλα,j/primen(t)G< lj l/primen;kα(t,t)+/summationdisplay m¯λα,ml(t)G< kα;jmj/primel/prime(t,t)−/summationdisplay n¯λα,jn(t)G< kα;nl j/primel/prime(t,t) +/summationdisplay m¯λα,ml/prime(t)G< kα;jl j/primem(t,t)−/summationdisplay n¯λα,j/primen(t)G< kα;jlnl/prime(t,t)/bracketrightBigg . (18) We now proceed with the line of argument presented in Refs. [ 57–59] to derive the master equations from Eqs. ( 17) and ( 18) based on the expansion of the coupling term Vkα. In our case, we ﬁnd it convenient to deﬁne nonequilibriumGreen’s functions for the operators ˆ ρ l,jand ˆKlj,l/primej/prime, in ad- dition to the ones for the reservoirs and we proceed withthe derivation of the Dyson equation at the lowest orderin the couplings in combination with Langreth theorem.These steps are similar to those followed in the study ofquantum transport for strong coupling between system andreservoirs [ 67,78]. To this end, we introduce the interaction representation with respect to the uncoupled Hamiltonian h= ˜H S(t)+HB. Therefore ˆρH ij(t)=ˆTK/bracketleftbigg exp/braceleftbigg −i/¯h/integraldisplay Kdt/prime˜Hh C(t/prime)/bracerightbigg ˆρh ij(t)/bracketrightbigg , (19) where the superscript hdenotes the interaction representation with respect to hand we recall that ˆTKdenotes time-ordering along the Schwinger-Keldysh contour. Furthermore, we setat the initial time t 0=− ∞ ,ρ0=ρS⊗ρB, where ρS,ρ Bare the density operators of the uncoupled system and reservoirs,respectively. The next step is to evaluate the Green’s functions in Eq. ( 17), up to the ﬁrst order of perturbation theory in V kα. It is convenient to introduce the deﬁnitions /Lambda1α(0) mj(t)=±/summationdisplay kαVkαG< mj,kα(t,t), /Lambda1α(0) mj(t)=±/summationdisplay kαVkαG< kα,mj(t,t). (20) As discussed below Eq. ( 16), the upper /lower sign ap- plies to fermionic /bosonic systems. Using the “Langreth rule” [ 67,68], we obtain the following expressions: /Lambda1α(κ) mj(t)/similarequal±/summationdisplay u,vλα,uv(t)/integraldisplay∞ −∞dt1/parenleftbig gr mj,vu(t,t1) ×/Sigma1<(κ) α(t1,t)+g< mj,vu(t,t1)/Sigma1a(κ) α(t1,t)/parenrightbig ,(21) /Lambda1α(κ) mj(t)/similarequal±/summationdisplay u,vλα,uv(t)/integraldisplay∞ −∞dt1/parenleftbig /Sigma1r(κ) α(t,t1) ×g< uv,mj(t1,t)+/Sigma1<(κ) α(t,t1)ga uv,mj(t1,t)/parenrightbig ,(22) where we have extended the deﬁnition of Eq. ( 20), corresponding to κ=0, to κ=1, by introducing theself-energies /Sigma1r,a,<(κ) α (t,t/prime)=/integraldisplaydω 2πe−iω(t−t/prime)/Sigma1r,a,<(κ) α (ω), (23) which encode the coupling to the baths. In deriving Eqs. ( 21) and ( 22), we consider weak coupling, such that only ﬁrst order terms in system-bath coupling strength are kept and all theother higher order terms are neglected. We can write the lesserself-energies as follows: /Sigma1 <(κ) α(ω)=±inα(ω)ωκ/Gamma1α(ω), (24) which depend on the spectral function /Gamma1α(ω)=− 2Im/bracketleftbig /Sigma1r(0) α(ω)/bracketrightbig =2π/summationdisplay kα\|Vkα\|2δ(ω−/epsilon1kα).(25) Here nα(ω) denotes the Fermi-Dirac or Bose-Einstein distri- bution function for the case of fermionic or bosonic baths,respectively. Importantly, the information on the temperatureand chemical potential of a given reservoir αis only encoded in these functions. Notice that the index κin the previous expressions, denotes the different moments of the spectralfunction. In the previous expressions we have used the deﬁ-nitions of Eqs. ( 7) and ( 8) recalling that they depend on time through X(t). In Eqs. ( 21) and ( 22), the lesser Green’s functions are evaluated with respect to the uncoupled Hamiltonian h: g < lj,vu(t,t/prime)=±i/angbracketleftbig ˆρh uv(t/prime)ˆρh lj(t)/angbracketrightbig , g> lj,vu(t,t/prime)=−i/angbracketleftbig ˆρh lj(t)ˆρh uv(t/prime)/angbracketrightbig , (26) where ˆ ρh jl(t)=[ˆρh lj(t)]†, hence, g> ν,ν/prime(t,t/prime)=± [g< ν/prime,ν(t/prime,t)]∗. The corresponding retarded ones are gr ν,ν/prime(t,t/prime)=θ(t−t/prime)[g> ν,ν/prime(t,t/prime)−g< ν,ν/prime(t,t/prime)], (27) while the advanced Green’s function is given by ga ν,ν/prime(t,t/prime)= [gr ν/prime,ν(t/prime,t)]∗. In the case of the OTOC, we deﬁne /Lambda1α,OTOC lj l/primej/prime(t)=±/summationdisplay kαVkαG< lj l/primej/prime,kα(t,t), /Lambda1α,OTOC lj l/primej/prime(t)=±/summationdisplay kαVkαG< kα,lj l/primej/prime(t,t). (28) The evolution along Keldysh contour can be implemented by considering an augmented contour [ 73], which leads to a generalized Langreth rule, as explained in Appendix D.T h e counterparts to Eqs. ( 21) and ( 22) for the OTOC functions 035425-4FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) read /Lambda1α,OTOC lj l/primej/prime(t)/similarequal±/integraldisplay∞ −∞dt1/summationdisplay u,vλα,uv(t)/bracketleftbig gr lj l/primej/prime,vu(t,t1) ×/Sigma1<(0) α(t1,t)+g< lj l/primej/prime,vu(t,t1)/Sigma1a(0)(t1,t)/bracketrightbig , (29) /Lambda1α,OTOC lj l/primej/prime(t)/similarequal±/integraldisplay∞ −∞dt1/summationdisplay u,vλα,uv(t)/bracketleftbig /Sigma1r(0) α(t,t1) ×g< uv,lj l/primej/prime(t1,t)+/Sigma1<(0) α(t,t1)ga uv,lj l/primej/prime(t1,t)/bracketrightbig , In order to obtain Eqs. ( 29), we ﬁxed tr=t0, the initial time, and then we extended t0to−∞ in the limits of the integrals. The retarded and lesser Green’s functions for the OTOC aredeﬁned as g < lj l/primej/prime,vu(t,t/prime)=±i/angbracketleftˆTKˆρuv(t/prime)ˆρlj(t)OB(tr)ˆρl/primej/prime(t)OD(tr)/angbracketright ±i/angbracketleftˆTKˆρlj(t)OB(tr)ˆρuv(t/prime)ˆρl/primej/prime(t)OD(tr)/angbracketright, (30) g> lj l/primej/prime,vu(t,t/prime)=−i/angbracketleftˆTKˆρlj(t)ˆρuv(t/prime)OB(tr)ˆρl/primej/prime(t)OD(tr)/angbracketright −i/angbracketleftˆTKˆρlj(t)OB(tr)ˆρl/primej/prime(t)ˆρuv(t/prime)OD(tr)/angbracketright, (31) The corresponding retarded Green’s function is gr lj l/primej/prime,vu(t,t/prime)=θ(t−t/prime)[g> lj l/primej/prime,vu(t,t/prime)−g< lj l/primej/prime,vu(t,t/prime)], (32) while the advanced one is given by ga ν,ν/prime(t,t/prime)=[gr ν/prime,ν(t/prime,t)]∗, in complete analogy with Eq. ( 26) C. Dynamics of the particle and energy current between system and baths The time-resolved density matrix ρij(t) fully characterizes the dynamics of the local properties of the system. We arealso interested in evaluating the charge current J (c) α(t)( i n the case of the fermionic reservoirs) as well as the energycurrent J (E) α(t) ﬂowing between the system and the reservoirs. These quantities can also be calculated by recourse to Green’sfunctions as follows: J (c) α(t)=ie ¯h/angbracketleft[H,Nα]/angbracketright=∓e ¯h/summationdisplay k/summationdisplay m,nVkα ×[λα,mn(t)G< mn,kα(t,t)−λα,mn(t)G< kα,nm(t,t)], J(E) α(t)=i ¯h/angbracketleft[H,Hα]/angbracketright=∓1 ¯h/summationdisplay k/summationdisplay m,nVkα/epsilon1kα ×[λα,mn(t)G< mn,kα(t,t)−λα,mn(t)G< kα,nm(t,t)], (33) where the upper sign is for fermionic and lower sign for bosonic reservoirs. Using Eqs. ( 21) and ( 22), we can evaluate these currents to the lowest order in the coupling strength. Theresult is J(c) α=1 ¯h/bracketleftBigg/summationdisplay m,nλα,mn(t)/Lambda1α(0) nm(t)−/summationdisplay m,nλα,mn(t)/Lambda1α(0) mn(t)/bracketrightBigg , J(E) α=1 ¯h/bracketleftBigg/summationdisplay m,nλα,mn(t)/Lambda1α(1) nm(t)−/summationdisplay m,nλα,mn(t)/Lambda1α(1) mn(t)/bracketrightBigg . (34) We see that the coefﬁcients /Lambda1α(0) mn(t) and/Lambda1α(0) mn(t) entering the equation of motion ( 17) for the density matrix also enter the expression for the charge currents. Instead, the energy cur- rents are determined by the coefﬁcients /Lambda1α(1) mn(t) and/Lambda1α(1) mn(t) related to the ﬁrst moment of the spectral function ( κ=1). Notice that the above expressions for the currents are exact upto order V 2 kα. III. ADIABATIC DYNAMICS So far we have not introduced any assumptions regard- ing the nature of the time dependence. Here, we focus onslow (adiabatic) driving, where the rate of change of thetime-dependent parameters is small, which justiﬁes treatingthe dynamics at different orders in these parameters. Moreprecisely, adiabatic driving is the regime where the typicaltimescale τassociated to the driving is much larger than any other timescale associated to the dynamics of the systemcoupled to the baths. A. Green’s function of the isolated system Here, we follow a treatment to evaluate the Green’s func- tions of the isolated system along the line of Refs. [ 12,23], where linear response in the parameters ˙Xwas implemented. We recall that the Green’s function are evaluated with theoperators expressed in the interaction picture with respect toh=˜H S(t)+HB, which, for this particular function, is equiv- alent to the Heisenberg picture with respect to ˜HS(t). We consider the expansion of ˜HS(t/prime) with respect to an “observational time” t, ˜HS(t/prime)=˜Hf S+δ˜HS(t/prime), δ˜HS(t/prime)=∞/summationdisplay n=1(t/prime−t)n n!∂˜HS ∂X·dnX dtn=N/summationdisplay k=1ξk(t/prime)ˆρkk,(35) where ˜Hf Sis the Hamiltonian with the time frozen at tandξk(t/prime)=−/summationtext∞ n=1θ(τad−\|t−t/prime\|)(t/prime−t)n/n!dnXk/dtn, withτad<τ. In the latter expression, we introduce the func- tionθ(τad−\|t−t/prime\|) to indicate that this expansion holds for time differences \|t/prime−t\|, with respect to the observational time t, which are much smaller than the typical time scale τassociated to the time-dependent parameters τad/lessmuchτ. We then change to the interaction representation with re- spect to ˜Hf S. We explain below the procedure followed for the case of the Green’s function g< ij,vu(t1,t2)=−iTr/braceleftbig ˆρ0ˆTK/bracketleftbig e−i ¯h/integraltext Kdt/primeδ˜Hf S(t/prime)ˆρf lj(t+ 1)ˆρf uv(t− 2)/bracketrightbig/bracerightbig , (36) 035425-5BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) where t+ 1andt− 2indicates that the time t1is on the piece of the contour that starts in −∞ , while t2is on the piece of the contour that ends in −∞ . All the operators with the label f are calculated in the Heisenberg representation of the frozen Hamiltonian ˜Hf S. In particular, ˆρf lj(t/prime)=ei ¯ht/prime˜Hf Sˆρlje−i ¯ht/prime˜Hf S=ei ¯h/epsilon1ljt/primeˆρlj (37) with/epsilon1f jbeing the eigenenergies of ˜Hf Sand/epsilon1lj=/epsilon1f l−/epsilon1f j. Evaluating Eq. ( 36) up to linear order in the perturbation ˜Hf S leads to g< lj,vu(t1,t2)/similarequalg<,f lj,vu(t1,t2)+δg<,f lj,vu(t1,t2), (38) where the ﬁrst term is the frozen component and reads g<,f lj,vu(t1,t2)=±iδlv/angbracketleftbig ˆρf uj(t1)/angbracketrightbig ei/epsilon1uv(t2−t1) =±iδlv/angbracketleftbig ˆρf uj(t2)/angbracketrightbig ei/epsilon1jv(t2−t1), (39) while the second term is the correction up to linear order in δ˜Hf S(t/prime) (see calculation in Appendix B) and reads δg<,f lj,vu(t1,t2)=−i ¯hg<,f lj,uv(t1,t2)/bracketleftbigg/integraldisplayt1 −∞dt/primeξj(t/prime) +/integraldisplayt2 t1dt/primeξv(t/prime)−/integraldisplayt2 −∞dt/primeξu(t/prime)/bracketrightbigg . (40) Notice that, in spite of the fact that some of the limits of the integrals are deﬁned to be −∞ , the functions ξj(t/prime) are differ- ent from zero only for \|t−t/prime\|<τ ad. Also notice that these functions, through Eqs. ( 38) and ( 40), enter the deﬁnitions of the functions of Eqs. ( 21) and ( 22) convoluted with the self-energy of the baths, which decay within the relaxationtime due to the coupling to the bath, τ rel=¯h//Gamma1α. Hence, we identify τad/similarequalτrel. Therefore the validity of the present treatment in the de- scription of the ﬁnite system coupled to the bath is restrictedtoτ rel<τ.T h e adiabatic approximation consists in keeping the terms ∝˙Xinδg<,f lj,vu(t1,t2) under the assumption that the changes in X(t) take place within a time scale that is much larger than the typical time scale of the dynamics of the frozensystem, hence τ rel/lessmuchτ. A similar procedure can be followed to evaluate the Green’s functions for the OTOC. In that case, the counterpartof Eq. ( 38)i s g < lj l/primej/prime,vu(t1,t2)/similarequalg<,f lj l/primej/prime,vu(t1,t2)+δg<,f lj l/primej/prime,vu(t1,t2),(41) where the frozen term reads g<,f lj l/primej/prime,vu(t1,t2)=±i/bracketleftbig δlv/angbracketleftbig Kf ujl/primej/prime(t1)/angbracketrightbig +δl/primev/angbracketleftbig Kf lj uj/prime(t1)/angbracketrightbig/bracketrightbig ei/epsilon1uv(t2−t1), (42) while the linear order term in δ˜Hf S(t/prime)i sg i v e nb y δg<,f lj l/primej/prime,vu(t1,t2) =±1 ¯h/integraldisplay∞ −∞dt/prime/parenleftbig/angbracketleftbigˆKf ujl/primej/prime(t1)/angbracketrightbig δvl×[θ(t1−t/prime)ξj/primej,l/primev(t/prime) +θ(t2−t/prime)ξv,u(t/prime)]+/angbracketleftbigˆKf lj uj/prime(t1)/angbracketrightbig ×δvl/prime[θ(t1−t/prime)ξj/primej,lv(t/prime)+θ(t2−t/prime)ξv,u(t/prime)]/parenrightbig ei/epsilon1vu(t1−t2), (43)where ξj/primej,l/primev=ξj/prime+ξj−ξl/prime−ξvandξv,u=ξv−ξu. The lesser functions in Eqs. ( 38) and ( 41) enter the master equations through Eqs. ( 21), (22), and ( 29), respectively. No- tice that while the frozen components of these functions leadto a result ∝V 2 αfor the /Lambda1functions, the adiabatic corrections δg<lead to a higher order correction ∝V2 α˙X. As we will fur- ther discuss below, this term can be neglected in comparisonto others. B. Master equations 1. Density matrix Our aim is to calculate the matrix elements of the density matrix up to linear order in ˙X. Hence, we split them as follows: ρuj(t)≡ρf uj(t)+ρa uj(t). (44) In the previous equations, ρf ujis the solution of the frozen master equation for the density matrix, while ρa uj(t)i st h e corresponding correction ∝˙X. The diagonal and off-diagonal terms of ρujare named, respectively, populations andcoherences and are generally coupled (in what follow we will use the shorthand notation pu=ρuufor the populations). By substituting Eq. ( 36)i n t o Eqs. ( 21) and ( 22), with Eq. ( 39) and the adiabatic approx- imation of Eq. ( 40), the master equation that includes both frozen and adiabatic contributions can be written as dρlj dt=i ¯h[εl(t)−εj(t)]ρlj+/summationdisplay mu,α/bracketleftbig Wju ml,α(t)ρmu(t) +˜Wul jm,α(t)ρum(t)−Wmu jm,α(t)ρlu(t)−˜Wum ml,α(t)ρuj(t)/bracketrightbig , (45) where we introduced the transition rates Wju ml,α(t)=Wju(f) ml,α(t)+δWju ml,α(t). (46) In deriving Eq. ( 45), we have neglected the level renormaliza- tion effects. The explicit expressions for the frozen rates originated in the contribution of Eq. ( 39)a r e Wju,(f) ml,α=λα,ml(t)λα,ju(t)γf α(/epsilon1ju)/2 +λα,ml(t)λα,ju(t)˜γf α(/epsilon1uj)/2, (47) ˜Wul,(f) jm,α=λα,jm(t)λα,ul(t)˜γf α(/epsilon1ul)/2 +λα,jm(t)λα,ul(t)γf α(/epsilon1lu)/2, (48) where γf α(/epsilon1)=¯h−1nα(/epsilon1)/Gamma1α(/epsilon1), ˜γf α(/epsilon1)=¯h−1(1∓nα(/epsilon1))/Gamma1α(/epsilon1). (49) On the other hand, the adiabatic corrections to the transition ratesδWmu lj,α(t), which have their origin in Eq. ( 40), can be evaluated in a similar manner (see Appendix Bfor details). The latter are ∝/Gamma1α˙Xwithin the adiabatic approximation. As mentioned before, in all the calculations leading to the master equations ( 45) we have considered the contact Hamiltonian in Eq. ( 6). Such master equations thus hold for bothH(I) CandH(II) C. In the former case, the quantities λα,ljand 035425-6FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) λα,ljentering the transition rates ( 47) and ( 48) are deﬁned by Eq. ( 7), while in the latter they are deﬁned by Eq. ( 8). We now introduce the following schematic notation for Eq. ( 45): dρ dt=i ¯h/epsilon1ρ+Wρ. (50) Splitting in this equation the density matrix elements and rates into their frozen and adiabatic components as in Eq. ( 44), we can make use of the fact that the frozen component satisﬁes 0=i ¯h/epsilon1ρf+Wfρf, (51) to conclude that the following equation has to be fulﬁlled by the adiabatic components (keeping only linear-order termsin˙X), ∂ρ f ∂X˙X=i ¯h/epsilon1ρa+Wfρa+δWρf. (52) These equations must be supplemented by the normalization of the populations/summationtext lpl=1. Substituting Wffrom Eqs. ( 47) and ( 48) into Eqs. ( 51) and ( 52) gives the frozen and adi- abatic components of the master equations respectively. Forthe frozen case, the master equations obtained are exactly thesame as the ones obtained using the diagrammatic formula-tion introduced in Refs. [ 57–59]. Notice that the term in the left-hand side contains two components. One component isoriginated in the variation with respect to Xof the matrix M= i/epsilon1/¯h+W fentering Eq. ( 51), while the other one is due to the change of the instantaneous eigenstates as Xchanges [instan- taneous eigenvalues and eigenstates are deﬁned in Eq. ( 5)]. The contribution of these two terms in the derivatives of thematrix elements of ρ fwith respect to Xreads ∂ρf l,j ∂X=−/bracketleftbigg M−1∂M ∂Xρf/bracketrightbigg l,j+/summationdisplay l/prime{Al/prime,lρl/prime,j−Aj,l/primeρl,l/prime}, (53)being Al/prime,l=/angbracketleftl/prime\|∂HS ∂X\|l/angbracketright εl−εl/prime,l/negationslash=l/prime. (54) The second term of Eq. ( 53) is equivalent to the contribution of the gauge potential in the moving-frame introduced inthe framework of the adiabatic perturbation theory for closedquantum systems [ 43] (see details in Appendix G). This term does not play any role when the master equation is reducedto a rate equation by taking into account the evolution of thepopulations only [ 12,60–62], but it has been considered in the adiabatic evolution of open quantum systems described by theLindblad master equation [ 7,11]. On the other hand, as already mentioned in Sec. III A ,t h e contribution of the terms collected in δWρ fin the previous equation are effectively higher order in the parameters deﬁn-ing the perturbative treatment. In fact, δWis linear order in the rate amplitude /Gamma1 α(which is in turn second order in the coupling Vkα) times linear order in the adiabatic expansion ˙X. Hence, we neglect these terms in comparison to the ones containing Wf, since these matrix elements are linear in /Gamma1α and 0-th order in the adiabatic expansion. Therefore, keeping only the latter terms and neglecting the former ones, we getρ a∼O(τrel/τ). This reasoning is basically the same as that presented in Ref. [ 60] and the result highlights the fact that the validity of the adiabatic treatment is restricted to variations ofthe driving in a timescale much larger than the relaxation timeof the system with the environment ( τ/greatermuchτ rel). 2. OTOC Similarly, for the OTOC, we introduce the decomposition Klj,l/primej/prime(t)≡Kf lj,l/primej/prime(t)+Ka lj,l/primej/prime(t), (55) where Kf lj,l/primej/primeis the solution of the frozen master equation for the OTOC, while Ka lj,l/primej/primeis the corresponding corrections ∝˙X. The master equation for the OTOC can be derived in a similarmanner as before, by substituting Eqs. ( 41)–(43) into Eqs. ( 29) and ( 30), obtaining dKlj,l/primej/prime(t) dt=i ¯h[εl/prime(t)−εj/prime(t)+εl(t)−εj(t)]Klj,l/primej/prime(t)+/summationdisplay mu,α/bracketleftbig Wju ml,α(t)Kmu,l/primej/prime(t)+Wj/primeu ml,α(t)Kmj,l/primeu(t) +˜Wul jm,α(t)Kum,l/primej/prime(t)+˜Wul/prime jm,αKlm,uj/prime(t)+Wju ml/prime,αKlu,mj/prime(t)+Wj/primeu ml/prime,αKlj m u(t)+˜Wul j/primem,α(t)Kuj,l/primem(t) +˜Wul/prime j/primem,α(t)Klj,um(t)/bracketrightbig −/summationdisplay mu,α/bracketleftbig˜Wum ml,α(t)Kuj,l/primej/prime(t)+˜Wul/prime ml,α(t)Kmj,uj/prime(t)+Wmu jm,αKlu,l/primej/prime+Wj/primeu jm,αKlm,l/primeu +˜Wul ml/prime,α(t)Kuj,mj/prime(t)+˜Wum ml/prime,α(t)Klj,uj/prime(t)+Wju j/primem,α(t)Klu,l/primem(t)+Wmu j/primem,α(t)Klj,l/primeu(t)/bracketrightbig , (56) where the rates Wju ml,α(t) and ˜Wju ml,α(t) are the same appearing in Eq. ( 45) (see the Appendix Ffor details). Using a similar schematic notation as before, we have dK dt=i ¯h/epsilon1OTOCK+WOTOCK. (57) The procedure to formulate the adiabatic master equation for the OTOC is the same as the one for the density matrix. Asbefore, the rates are split into frozen and adiabatic compo- nents whose origin can be traced back to Eqs. ( 42) and ( 43) respectively, WOTOC=WOTOC ,f+δWOTOC. (58) Introducing this decomposition, as well as the one in Eq. ( 55) leads to the master equation for the steady state, describing the 035425-7BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) long-time dynamics (for t→∞ ) of the frozen component, 0=i ¯h/epsilon1OTOC ,fKf+WOTOC ,fKf, (59) which has similar form as the one derived in Ref. [ 75]f o rt h e case of a single reservoir in equilibrium. As already noticedin Ref. [ 75], the master equation for the OTOC is basically the one for two copies of the density matrix. In the case of asingle reservoir and for a system without driving, Eq. ( 57)i s similar to implementing a forward and a backward evolutionwith the master equation for the density matrix as in Ref. [ 76]. The adiabatic component can be calculated from ∂K f ∂X˙X=i ¯h/epsilon1OTOC ,fKa+WOTOC ,fKa+δWOTOCKf,(60) where the last term can be neglected using similar arguments to those presented as in the case of the adiabatic evolutionfor the density matrix. Also in the present case, we must takeinto account the contributions due to the changes of the matrixof Eq. ( 59) and those corresponding to the changes in the eigenstates. The corresponding solutions satisfy /summationdisplay mnKf mm,nn=/angbracketleftOB(tr)OD(tr)/angbracketright=K∞(constant) , (61) for the frozen components, and /summationdisplay mnKa mm,nn=0, (62) for the adiabatic ones. C. Currents Similarly, substituting Eqs. ( 21) and ( 22) in the deﬁnition of the energy currents, we get J(E) α(t)=/summationdisplay m,n,u/bracketleftbig /epsilon1um˜Wum,f mn,α(t)ρun(t)−/epsilon1nuWnu,f mn,α(t)ρmu(t)/bracketrightbig . (63) Similarly, for the charge currents, we obtain J(c) α(t)=e/summationdisplay m,n,u/bracketleftbig˜Vum,f mn,α(t)ρun(t)−Vnu,f mn,α(t)ρmu(t)/bracketrightbig ,(64) where Vju,(f) ml,α=λα,ml(t)λα,ju(t)γf α(/epsilon1ju)/2 −λα,ml(t)λα,ju(t)˜γf α(/epsilon1uj)/2, (65) ˜Vul,(f) jm,α=λα,jm(t)λα,ul(t)˜γf α(/epsilon1ul)/2 −λα,jm(t)λα,ul(t)γf α(/epsilon1lu)/2. (66) Notice that the charge current has been deﬁned only for the fermionic case. Moreover, currents are made up of a frozen and an adi- abatic contributions [ J(E),f α (t) and J(E),a α (t), respectively, for energy currents] coming from the respective terms of thedensity matrix.IV . EXAMPLES The outcome of the previous sections is that the master equations describing the adiabatic dynamics of an open quan-tum system for the density matrix and the OTOC, as well asthe currents, are completely deﬁned by the frozen rates inEqs. ( 47) and ( 48). In order to calculate them, all we need is the unitary transformation diagonalyzing the instantaneousHamiltonian of the system, the spectral function of the bathsand the corresponding Bose-Einstein or Fermi-Dirac distri-bution functions. We illustrate the procedure for two simpleexamples. A. Qutrit We analyze the dynamics of a driven qutrit —a three-level system such as an atom with a ground state and two excitedstates—attached to two bosonic reservoirs. The latter could,for instance, represent two electromagnetic environments towhich the atom is coupled. We consider the following Hamil-tonian for the driven three-level system H S(t)=2/summationdisplay q=0Eq(t)ˆπqq+w(t)( ˆπ12+ˆπ21), (67) where Eq(t), with q=0,1,2, are the energy levels relative to the ground state (0) and the two excited states (1 and 2). Theinter-level coupling parameter w(t) denotes the amplitude for, possibly, time-dependent transitions between the two excitedstates. The consider the bath Hamiltonian described by Eq. ( 2) with N r=2,ˆbkαbeing bosonic operators for reservoir α= L,R. As shown in Fig. 1, we ﬁx the temperature of the two baths as TL=T+/Delta1TandTR=T−/Delta1T. Moreover, we will consider Ohmic baths with linear dissipation relation spectraldensity /Gamma1 α(ε)=ϒαεe−ε//epsilon1 c,withε> 0, (68) where /epsilon1c, is a high frequency cutoff. We assume that the left bath is connected to the qutrit through energy level 1 and rightbath is connected through energy level 2, so that the contactHamiltonian is given by H C=/summationdisplay kVkL(ˆπ10+ˆπ01)(b† kL+bkL) +/summationdisplay kVkR(ˆπ02+ˆπ20)(b† kR+bkR). (69) As detailed in Sec. II, we ﬁrst diagonalize the system Hamil- tonian with a suitable unitary transformation ˆU(t) so that ˜HS(t)=/summationdisplay l=±εl(t)ˆρll+ε0(t)ˆρ00, (70) with the instantaneous eigenstates being \|0/angbracketright,\|−/angbracketright,\|+/angbracketright, being the instantaneous eigenenergies ε0(t)=E0(t) and ε±(t)=/parenleftbiggE1(t)+E2(t) 2/parenrightbigg ±1 2/radicalBig (E1(t)−E2(t))2+4w(t)2. (71) 035425-8FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) 0.0 0.2 0.4 0.6 0.8 ΔT/T0.81.0p−×10−1 p− p0 0.0 0.2 0.4 0.6 0.8 (E1−E2)[kBT]0.00.51.0p−×10−18.08.5 p0×10−1 8.008.25 p0×10−1 FIG. 2. Qutrit: population probabilities in the absence of driving as functions of /Delta1Tfor ﬁxed energy splitting E1−E2=0 (top) and as functions of level splitting for /Delta1T=0.8T(bottom). Green lines refers to p−(with axis on the left) and blue lines refers to p0(with axis on the right). Notice the two scales on the left and right axis are different. Solid (dashed) lines result from the solution of the RE (QME). Parameters values are ϒL=ϒR=0.2,/epsilon1C=100kBT,w= 0.05kBT,E1+E2=5kBT,a n d E0=0. Moreover, the contact Hamiltonian in the instantaneous basis becomes ˜HC=/summationdisplay k,α/summationdisplay l=±Vkα(λα,0l(t)ˆρ0l+λα,l0(t)ˆρl0)(bkα+b† kα), (72) where λL,0+(t)=−λR,0−(t)=cosθ(t)/2, λL,0−(t)=λR,0+(t)=sinθ(t)/2, (73) withθ(t)=tan−1(2w(t) E1(t)−E2(t)) and λα,0l(t)=λα,l0(t). In the present problem, λα,ml(t)=λα,lm(t). 1. Density matrix Given Eqs. ( 71) we immediately have the component of the frozen kernel /epsilon1in Eq. ( 51). On the other hand, given Eqs. ( 73), we readily get the rates deﬁned in Eqs. ( 47) and ( 48), which completes all the information about the elements of the frozenmaster equation in Eq. ( 51). We ﬁrst consider the particular case of the frozen Hamil- tonian, corresponding to ﬁxed values of E landw, and we analyze the effect of coherence by comparing the outcomes ofthe full quantum master equation (QME) with those obtainedfrom the rate equation (RE). The latter corresponds to solvingthe equation for the diagonal elements only. In Fig. 2,w ep l o t the populations p 0(blue lines) and p−(green lines) of the states\|0/angbracketrightand\|−/angbracketright, respectively, as functions of /Delta1T(top panel) and energy level splitting E1−E2(bottom panel). Solid lines0 1 2 3 w[kBT]0123\|¯ρf +−\|×10−3 \|¯ρf +−\| \|¯ρa +−\| 0.00 0.25 0.50 0.75 ΔT/T0.00.51.01.5\|¯ρf +−\|×10−30.00.51.0 \|¯ρa +−\|×10−4 012 \|¯ρa +−\|×10−4 FIG. 3. Qutrit: frozen ( f) and adiabatic ( a) contributions to the period-averaged coherences ¯ ρ+−as functions of the interlevel cou- pling wfor ﬁxed /Delta1T=0.5T(top) and as functions of /Delta1Tfor ﬁxed w=0.5kBT(bottom). Red solid lines refer to the absolute value of the frozen contribution (with axis on the left) and blackdashed lines refer to the absolute value of the adiabatic contribution (with axis on the right). Parameters values are ϒ L=ϒR=0.2,/epsilon1c= 30kBT,E1(t)=2kBT+4kBTcos (/Omega1t+π 2),E0=0a n d E2(t)= 0.5kBTcos (/Omega1t). result from the solution of the RE, while dashed lines from the solution of the QME. The top panel in Fig. 2shows that the effect of coherence on the populations is absent for /Delta1T=0 (where the overall system is at equilibrium) and leads to animportant contribution for large values of /Delta1T. The bottom panel of Fig. 2, highlights the relevant energy scale for which the coherence plays a signiﬁcant role. Concretely, we see thatthis is the case when the level splitting is small compared tok BT. In the ﬁgure, this corresponds to values E1−E2<0.5 kBT, in which range, for this choice of parameters, the gap between bonding and antibonding energy levels /Delta1=ε+−ε− is smaller than the energy scale kB/Delta1T[49]. Let us now assume that the system is driven by mod- ulating the parameters according to the following scheme:E 1(t)+E2(t)=Eav+δεcos(/Omega1t+φ) and E1(t)−E2(t)= Erel+δ¯εcos(/Omega1t), while wis time-independent. In the present case, we solve the adiabatic master equation of Eq. ( 52) along with the frozen one in Eq. ( 51), which also accounts for the gauge potential term. In Fig. 3, we plot the absolute value of the period-averaged frozen and adiabatic contributions to the coherences, ¯ ρf +−and ¯ρa +−, respectively, as functions of w(top panel) and /Delta1T(bottom panel). Red solid lines refer to the frozen contribution (with values on the left axis) and blackdashed lines refer to the adiabatic contribution (with valueon the right axis). The top panel of Fig. 3shows that frozen and adiabatic components of the coherence display differentbehaviors as a function of the inter-level coupling, while their 035425-9BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) 0 20 40 600.00.51.01.52.0×10−1 K00 K0+ K0− K++ K+− K−− 0 20 40 60 t[¯h/k BT]0.00.51.01.52.0×10−1 FIG. 4. Qutrit: time variations of OTOCs for different values of thermal bias /Delta1T=0 (top) and /Delta1T=0.98T(bottom) taking only the diagonal terms of the projection operators and ϒL=0.1,ϒR= 0.2,w=0.2kBT,E0=0,E1−E2=0.3kBT,E1+E2=0.6kBT, /epsilon1c=20kBT, with initial conditions K00(0)=0.2,K0+(0)=0.2, K0−(0)=0.2,K+0(0)=0.2,K++(0)=0.2 and the normalization condition/summationtext ll/primeKll/prime(t)=1. absolute values differ by more than one order of magnitude. In particular, they are zero at w=0, since no coupling is present, and are suppressed at large w, since the gap /Delta1gets larger than the energy scale kB/Delta1T. However, they present a max- imum at different values of w, namely at about w=2kBT, for the frozen component, and at about w=0.2kBT,f o rt h e adiabatic component. The bottom panel of Fig. 3shows that the two components of the coherence ¯ ρ+−behave differently as functions of /Delta1T. While the frozen component vanishes for/Delta1T=0 and thereafter increases, the adiabatic component ﬁrst decreases (starting from a ﬁnite value at /Delta1T=0), reach- ing a minimum at /Delta1T/similarequal0.4Tand thereafter increasing. In summary, in the frozen case, we observe that the ef- fect of coherence becomes signiﬁcant close to degeneracyand in highly nonequilibrium conditions [ 49,79]. Although the coherence due to frozen dynamics is suppressed nearequilibrium conditions, we show that the coherence can beenhanced by adiabatic driving (see Fig. 3). In addition, for small values of interlevel coupling ( w≈0) the coherence in the two cases are comparable. These results can have potentialimpact on theoretical as well as experimental works based onadiabatically driven quantum systems. 2. OTOC The different components of the quantity Kll/prime(t)= Kll,l/primel/prime(t), corresponding to the solution of Eq. ( 57) without driving, are shown in Fig. 4for different values of the initial0.00 0.25 0.50 0.75 1.000.500.751.001.251.50×10−1 K00 K0+ K0− K++ K+− K−− 0.00 0.25 0.50 0.75 1.00 ΔT/T2.53.03.54.0×10−1 p0 p+ p− FIG. 5. Qutrit: steady-state values of OTOCs as a function of thermal bias for the time independent case taking only the diagonal terms of the projection operators. The parameters are the same as in Fig. 4. As a reference, we also show in the bottom panel the populations of the different levels for the same parameters. conditions. Here the gauge term of the master equation ( 60) does not play any role since Kll/primecorrespond to the “rate” com- ponent of the OTOC. Because of the normalization conditionsin Eq. ( 61), we have normalized the operators O(t) such that/summationtext ll/primeKll/prime(t)=1. We recall that OTOCs have been suggested as useful quantities to characterize scrambling dynamics inmany-body systems. In nonintegrable Hamiltonians, OTOC’sare expected to grow as a function of time [ 71–74]. In sys- tems coupled to thermal baths these correlations stabilize aftersome time [ 75,76] and tend to the asymptotic limit determined by Eq. ( 61). This can be appreciated in the evolution shown in Fig. 4. In the bottom panel of the same ﬁgure, we show the evolution of the OTOCs for the same parameters and the sameinitial conditions shown in the top panel, under the presenceof a thermal bias between the two reservoirs. Overall, we seea similar behavior as in the case of equilibrium reservoirsshown in the top panel. However, we can notice that the timeto reach the asymptotic limit is larger and we can see that insome components there is an enhancement with respect to theequilibrium case. In Fig. 5, we can see the behavior of the stationary values solution of Eq. ( 59) for the same parameters as in Fig. 4 in the nonequilibrium regime. For some components we cansee an enhancement of the OTOCs as the temperature bias/Delta1Tincreases. The behavior of K 00,K−−,K++is correlated with the behavior of the populations of the different levels,as shown in the bottom panel of Fig. 5. The reason is that the thermal bias generates relative changes in the populations,which are accounted for by the nonequilibrium features of theOTOCs. 035425-10FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) 0.0 0.2 0.4 0.6 0.8 1.0 ΔT/T−1.0−0.50.00.51.01.5×10−2 K00 K0+ K0− K++ K+− K−− FIG. 6. Qutrit: adiabatic contribution to the OTOCs as a function of thermal bias under adiabatic driving taking only the diagonal termsof the projection operators and ϒ L=0.2,ϒR=0.2,w=0.05kBT, /epsilon1C=20kBT,E0=0,E1(t)+E2(t)=2kBT+2kBTcos (/Omega1t+π 2), E1(t)−E2(t)=0.5kBT+0.5kBTcos (/Omega1t), ¯h/Omega1=0.01kBT, and the normalization condition/summationtext ll/primeKf ll/prime=1. Finally, the effect of driving treated in the framework of the adiabatic approximation is illustrated in Fig. 6. We recall that these are corrections to the frozen components shown inFig. 5. Interestingly, they are positive for the components K 00,K++,K0+, which are the ones that grow with /Delta1Tin the frozen case, while they are negative for the ones that decrease. B. Coupled quantum dots In this section, we analyze now a fermionic driven system consisting of a pair of coupled quantum dots (QDs) [ 21]. For simplicity, we focus on the case with inﬁnite intra-dotCoulomb repulsion, which limits the occupation to, at themost, one electron per quantum dot, and we assume spinlessfermions. Concretely, we consider the following Hamiltonian for a pair of coupled single-level quantum dots: H S(t)=E1(t)ˆa† 1ˆa1+E2(t)ˆa† 2ˆa2 +w(t)( ˆa† 1ˆa2+ˆa† 2ˆa1)+Uˆn1ˆn2, (74) where ˆ ajand ˆa† jare, respectively, the annihilation and creation operators for fermions in the quantum dot j=1,2 and ˆ nj= ˆa† jˆaj. The time-dependent parameters are the QDs’ energy levels E1(t) and E2(t), and the hopping element w(t) between the two QDs, while Uis the inter-dot Coulomb interaction. The bath Hamiltonian is given by Eq. ( 2) with Nr=2,ˆbkαbe- ing fermionic operators for reservoir α=L,R. Moreover, we assume a characterless spectral density, namely /Gamma1α(/epsilon1)=/Gamma1α, independent of energy. The contact Hamiltonian is given by HC=/summationdisplay kVkLˆb† kLˆa1+/summationdisplay kVkRˆb† kRˆa2+H.c., (75) so that each QD is connected only to one reservoir. The Hilbert space of the double-dot system is composed of the following four occupation states: \|0/angbracketright(empty), \|1/angbracketright= ˆa† 1\|0/angbracketright0.00 0.25 0.50 0.75−2−101J(E),a R[kBT]×10−4 RE QME 0.00 0.25 0.50 0.75 ΔT/T−2−101J(E),a R[kBT]×10−4 0 1 2 U[kBT]0.91.0×10−4 FIG. 7. Adiabatically pumped energy current in the right reser- voir averaged over one period relative to the qutrit system (top) andto the coupled QD system (bottom) as a function of a /Delta1T. Solid red (dashed black) lines results from the solution of the RE (QME). Parameters values are w=0.2k BT,/epsilon1c=100kBT,E1(t)=2kBT+ 2kBTcos (/Omega1t−π 2),E2(t)=kBT+kBTcos (/Omega1t), ¯h/Omega1=0.01kBT; for qutrit ϒL=ϒR=0.2 and for coupled quantum dots /Gamma1L=/Gamma1R= 0.2kBT,U=0. In the inset of the lower panel, we plot the adia- batically pumped energy current as a function of interdot Coulomb interaction ( U)f o r/Delta1T=0.8Tand other parameters as in the main panel. The chemical potential of the leads is μ=0. (single occupancy, left QD), \|2/angbracketright= ˆa† 2\|0/angbracketright(single occupancy, right QD) and \|d/angbracketright= ˆa† 1ˆa† 2\|0/angbracketright(double occupancy). The diago- nalyzed system Hamiltonian reads ˜HS(t)=/summationdisplay l=0,±,dεl(t)ˆρll, (76) where ε±(t) are given by Eq. ( 71),ε0=0 and εd(t)=U+ ε1(t)+ε2(t). The contact Hamiltonian becomes ˜HC=/summationdisplay k,α/summationdisplay l=±Vkα[λα,0l(t)ˆb† kαˆρ0l+λα,dl(t)ˆb† kαˆρld+H.c.], (77) where λL,0+(t)=−λR,0−(t)=λR,+d(t)=λL,−d(t)a r et h e same as in Eq. ( 73), while λα,ll/prime(t)=λα,l/primel(t). As before, the full adiabatic master equation for diagonal and off-diagonalterms of the density matrix and the OTOC can be obtainedafter calculating Eqs. ( 47) and ( 48) following Sec. III B . Since the kernel Wis very similar to the qutrit case, the behavior of the matrix elements ρ jlis qualitatively similar to what is shown in Figs. 2and3. 035425-11BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) Our aim now is to calculate the energy currents ﬂowing through the system between the reservoirs as a consequenceof the combined effect of the thermal bias and the ac-driving.We consider a modulation in time of the parameters E 1(t) and E2(t) according to the scheme presented in Sec. IV A , while taking wtime-independent. We notice that, in the adiabatic regime, asymmetric coupling (with respect to the reservoirs) isa necessary condition to obtain a net pumping of energy overa period when /Delta1T=0[12]. In Fig. 7, we plot the adiabati- cally pumped energy current (averaged over a period) ﬂowing into the right reservoir ¯J (E),a R=/Omega1/(2π)/integraltext2π//Omega1 0dt J(E),a R (t)a s a function of the /Delta1Tfor both systems, qutrit and coupled QD, corresponding, respectively, to the upper and lower panels. Itis interesting to analyze here the role of the coherences inevaluating the currents. The solid red lines result from thesolution of the RE, in the absence of coherence effects, whilethe dashed black lines result from the solution of the QME.The fact that the value of ¯J (E),a R is negative (in a range of values of /Delta1T) means that the energy current is exiting the right, cold reservoir, so that the system works as a refriger-ator. Interestingly, in both cases we ﬁnd that the presence ofcoherence decreases the absolute value of the energy current,thus suppressing the refrigeration effect [ 6,79]. The effect is more pronounced for the qutrit than for the coupled QD case.The dependence of adiabatically pumped energy current onthe interdot Coulomb interaction ( U) is shown in the inset of the lower panel of Fig. 3. We observe that, for this driv- ing protocol, the energy current decreases monotonously asa function of U. This is because in the present problem theenergy transport is associated to charge ﬂuctuations betweenthe quantum dots and the reservoirs. For these parameters[chemical potential of the reservoirs μ=0, and E j(t)/greaterorequalslantkBT] the dominant ﬂuctuation is between singly and doubly occu-pancy and the Coulomb blockade mechanism induced by U inhibits the charge and energy transport. V . SUMMARY AND CONCLUSIONS We have presented a derivation of the quantum master equation ruling the adiabatic dynamics of a driven systemweakly coupled to nonequilibrium reservoirs. The formalism applies to any Hamiltonian system with ﬁnite dimension ofits Hilbert space at which a slowly varying time-dependentperturbation is applied and weakly coupled to fermionic orbosonic baths. Our derivation includes the equations for thedynamics of the reduced density matrix of the ﬁnite-size sys-tem, the currents between the system and the reservoirs andthe out-of-time-order correlation (OTOC) functions. We have illustrated the application of the formalism with two examples: a qutrit coupled to two bosonic baths and twocoupled quantum dots attached to fermionic baths. In bothcases, a time-periodic perturbation with low frequencies, con-sistent with the adiabatic regime, and a temperature bias wereconsidered. In the frozen case, we showed the relevance of theoff-diagonal terms of the density matrix (coherences) in thefar-from equilibrium (corresponding to large temperature dif-ferences between reservoirs) and near degeneracy situations.In addition, in the other regime (near equilibrium and for smallvalues of inter-system coupling) when the frozen contribu-tion to off-diagonal terms becomes small, we observed thatthe adiabatic contribution to the off-diagonal terms becomessigniﬁcant. We have also analyzed the steady-state and adiabatic so- lutions of the OTOC. The present formalism may representa useful tool to analyze the dynamics of the OTOC in otherHamiltonian systems coupled to reservoirs in nonequilibriumscenarios. ACKNOWLEDGMENTS We thank Paolo Abiuso, Martí Perarnau-Llobet, Janine Splettstoesser, and Pablo Terrén Allonso for useful discus-sions. L.A. acknowledges support from PIP-2015-CONICET,PICT-2017, PICT-2018, Argentina, Simons-ICTP-Trieste as-sociateship, and the Alexander von Humboldt Foundation,Germany. We thank the support of the CNR-CONICET coop-eration program “Energy conversion in quantum, nanoscale,hybrid devices,” the support of the SNS-WIS joint laboratoryQUANTRA as well as the hospitality of the Dahlem Centerfor Complex Quantum Systems, Berlin and the InternationalCenter for Theoretical Physics, Trieste. APPENDIX A: EVALUATION OF THE MEAN VALUES In this section, we will evaluate the mean values entering in Eq. ( 17). The mean values will be calculated perturbatively up to ﬁrst order in the coupling strength Vkα, starting with the time-ordered correlator i/angbracketleftTKˆb† kα(t/prime)ˆρmj(t)/angbracketright≈/integraldisplay Kdt1/angbracketleftbig TK/bracketleftbig˜HH C(t1)ˆb† kα(t/prime)ˆρmj(t)/bracketrightbig/angbracketrightbig =Vkα/summationdisplay uvλα,uv/integraldisplay Kdt1/angbracketleftTK[ˆρuv(t1)ˆbkα(t1)ˆb† kα(t/prime)ˆρmj(t)]/angbracketright. (A1) We can deform the contour Kinto a pair of contours such that K1goes from −∞ totand back to −∞ andK2from−∞ tot/prime and back to −∞ . We can write/integraldisplay K1=/integraldisplayt −∞+/integraldisplay−∞ t;/integraldisplay K2=/integraldisplayt/prime −∞+/integraldisplay−∞ t/prime, (A2) such that i/angbracketleftˆb† kα(t/prime)ˆρmj(t)/angbracketright≈Vkα/summationdisplay uvλα,uv/bracketleftbigg/bracketleftbigg/integraldisplayt −∞dt1/angbracketleftˆb† kα(t/prime)ˆρmj(t)ˆρuv(t1)ˆbkα(t1)/angbracketright+/integraldisplay−∞ tdt1/angbracketleftˆb† kα(t/prime)ˆρuv(t1)ˆbkα(t1)ˆρmj(t)/angbracketright/bracketrightbigg +/bracketleftbigg/integraldisplayt/prime −∞dt1/angbracketleftˆb† kα(t/prime)ˆρuv(t1)ˆbkα(t1)ˆρmj(t)/angbracketright+/integraldisplay−∞ t/primedt1/angbracketleftˆρuv(t1)ˆbkα(t1)ˆb† kα(t/prime)ˆρmj(t)/angbracketright/bracketrightbigg/bracketrightbigg . (A3) 035425-12FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) For the mixed lesser Green’s function deﬁned in Eq. ( 16), using Wick’s theorem Eq. ( A3) can be rewritten as G< mj,kα(t,t/prime) ≈/integraldisplay∞ −∞dt1Vkα/summationdisplay uvλα,uv(t)/bracketleftbig gr mj,vu(t,t1)g< kα(t1,t/prime) +g< mj,vu(t,t1)ga kα(t1,t/prime)/bracketrightbig , (A4) where the deﬁnition for the system Green’s functions are given in Eqs. ( 26) and ( 27). The lesser and greater Green’s function for the baths are deﬁned as g< kα(t1,t2)=±i/angbracketleftb† kα(t2)bkα(t1)/angbracketright, g> kα(t1,t2)=−i/angbracketleftbkα(t1)b† kα(t1)/angbracketright, (A5) where the upper sign applies to fermionic reservoirs and the lower sign is for bosonic reservoirs. The corresponding re-tarded and advanced Green’s functions can be obtained usinga similar relation as in Eq. ( 27). APPENDIX B: FROZEN AND ADIABATIC COMPONENTS OF LESSER GREEN’S FUNCTION The lesser Green’s function is given by g< lj,vu(t1,t2)=±i/angbracketleftˆρuv(t2)ˆρlj(t1)/angbracketright. (B1) Writing in terms of evolution operators, g< lj,vu(t1,t2)=±i/angbracketleftbigˆTKei/¯h/integraltextt2 t0˜HS(t/prime)dt/primeˆρuv ×e−i/¯h/integraltextt2 t1˜HS(t/prime)dt/primeˆρlje−i/¯h/integraltextt1 t0˜HS(t/prime)dt/prime/angbracketrightbig .(B2) Using ˜HS(t/prime)=˜Hf S+δ˜HS(t/prime) along with Eq. ( 37), we obtain g< lj,vu(t1,t2)=±i/angbracketleftbigˆTKei/¯h/integraltextt2 t0δ˜HS(t/prime)dt/primeˆρf uv(t2) ×e−i/¯h/integraltextt2 t1δ˜HS(t/prime)dt/primeˆρf lj(t1)e−i/¯h/integraltextt1 t0δ˜HS(t/prime)dt/prime/angbracketrightbig . (B3) For the contour shown in Fig. 8, where contour Kgoes from t0→t1→t2→t0, we can write g< lj,vu(t1,t2)=±i/angbracketleftbigˆTKe−i/¯h/integraltext Kδ˜Hf S(t/prime)dt/primeˆρf uv(t− 2)ˆρf lj(t+ 1)/angbracketrightbig ,(B4) where we used δ˜HS≡δ˜Hf Sconsidering the driving to be slow enough. One other simpliﬁcation entailed by slow drivingis that one can Taylor expand the exponential in Eq. ( B4), FIG. 8. The Keldysh contour.obtaining e−i/¯h/integraltext Kδ˜Hf S(t/prime)dt/prime≈1−i/¯h/integraldisplay Kδ˜Hf S(t/prime)dt/prime, (B5) such that g< lj,vu(t1,t2)=±i/angbracketleftbigˆTKˆρf uv(t− 2)ˆρf lj(t+ 1)/angbracketrightbig ±1/¯h/angbracketleftbigg ˆTK/integraldisplay Kδ˜Hf S(t/prime)dt/primeˆρf uv(t− 2)ˆρf lj(t+ 1)/angbracketrightbigg , (B6) where the ﬁrst term on the right-hand side is the frozen con- tribution to the lesser Green’s function, g<,f lj,vu(t1,t2)=±iδlv/angbracketleftbig ˆρf uj(t1)/angbracketrightbig ei/epsilon1uv(t2−t1) =±iδlv/angbracketleftbig ˆρf uj(t2)/angbracketrightbig ei/epsilon1jv(t2−t1), (B7) whereas the second term gives the higher order contributions, δg<,f lj,vu(t1,t2) =± 1/¯hN/summationdisplay k=1/angbracketleftbigg ˆTK/integraldisplay Kξk(t/prime)ˆρf kk(t/prime)dt/primeˆρf uv(t− 2)ˆρf lj(t+ 1)/angbracketrightbigg , (B8) where we used the second equation of Eq. ( 35). Expanding over the Keldysh contour, we get δg<,f lj,vu(t1,t2) =± 1/¯hN/summationdisplay k=1/bracketleftbigg/angbracketleftbigg/integraldisplayt1 t0dt/primeξk(t/prime)ˆρf uv(t2)ˆρf lj(t1)ˆρf kk(t/prime)/angbracketrightbigg +/angbracketleftbigg/integraldisplayt2 t1dt/primeξk(t/prime)ˆρf uv(t2)ˆρf kk(t/prime)ˆρf lj(t1)/angbracketrightbigg +/angbracketleftbigg/integraldisplayt0 t2dt/primeξk(t/prime)ˆρf kk(t/prime)ˆρf uv(t2)ˆρf lj(t1)/angbracketrightbigg . (B9) After some simple calculations, we obtain δg<,f lj,vu(t1,t2)=−i ¯hg<,f lj,uv(t1,t2)/bracketleftbigg/integraldisplayt1 t0dt/primeξj(t/prime) +/integraldisplayt2 t1dt/primeξv(t/prime)+/integraldisplayt0 t2dt/primeξu(t/prime)/bracketrightbigg . (B10) APPENDIX C: CALCULATION OF TRANSITION RATES Using Eqs. ( 39) and substituting in the ﬁrst term of Eq. ( 21), forκ=0 we have for the imaginary part i/integraldisplay dt1Im/bracketleftbig gr,f mj,vu(t,t1)/bracketrightbig /Sigma1<(0) α(t1,t) =±/Gamma1α(/Delta1/epsilon1uv) 2nα(/epsilon1uv)/parenleftbig/angbracketleftbig ˆρf mv/angbracketrightbig tδju±/angbracketleftbig ˆρf uj/angbracketrightbig tδmv/parenrightbig , (C1) which are referred to as dissipation-type terms [ 46]. Similarly, there are also terms of the type, /integraldisplay dt1Re/bracketleftbig gr,f mj,vu(t,t1)/bracketrightbig /Sigma1<(0) α(t1,t) =−i/angbracketleftbig/parenleftbig ˆρf mv/angbracketrightbig tδju±/angbracketleftbig ˆρf uj/angbracketrightbig tδmv/parenrightbig P/integraldisplayd/epsilon1 2πnα(/epsilon1)/Gamma1α(/epsilon1) /epsilon1−/epsilon1uv,(C2) 035425-13BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) which lead to the level renormalization . For some speciﬁc spectral functions, the above integral can be calculated explic-itly [ 49]. Moreover, substituting Eq. ( 39) in the ﬁrst term of Eq. ( 21), forκ=1 the imaginary part is given by i/integraldisplay dt 1Im/bracketleftbig gr,f mj,vu(t,t1)/bracketrightbig /Sigma1<(1) α(t1,t) =±/epsilon1uv/Gamma1α(/epsilon1uv) 2nα(/epsilon1uv)/parenleftbig/angbracketleftbig ˆρf mv/angbracketrightbig tδju±/angbracketleftbig ˆρf uj/angbracketrightbig tδmv/parenrightbig .(C3) Similarly, the real part becomes /integraldisplay dt1Re/bracketleftbig gr,f mj,vu(t,t1)/bracketrightbig /Sigma1<(1) α(t1,t) =−i/angbracketleftbig/parenleftbig ˆρf mv/angbracketrightbig tδju±/angbracketleftbig ˆρf uj/angbracketrightbig tδmv/parenrightbig P/integraldisplayd/epsilon1 2π/epsilon1nα(/epsilon1)/Gamma1α(/epsilon1) /epsilon1−/epsilon1uv. (C4) All other terms in Eq. ( 21) can be similarly evaluated. Sub- stituting the above results in Eq. ( 21) neglecting the effect of lamb shift, for κ=0, we obtain /Lambda1α(0) mj(t)=¯h/summationdisplay u/bracketleftbiggλα,ju(t) 2γα,ujm(t)ρmu −λα,um(t) 2˜γα,jmu(t)ρuj/bracketrightbigg (C5) and /Lambda1α(0) jm(t)=¯h/summationdisplay u/bracketleftbiggλα,ju(t) 2˜¯γα,mju(t)ρmu −λα,um(t) 2¯γα,um j(t)ρuj/bracketrightbigg , (C6) where we have introduced γα,ujm(t)=γf α(/epsilon1ju)+δγα,ujm(t), ˜γα,jmu(t)=˜γf α(/epsilon1um)+δ˜γα,jmu(t), ¯γα,um j(t)=γf α(/epsilon1mu)+δ¯γα,um j(t), ˜¯γα,mju(t)=˜γf α(/epsilon1uj)+δ˜¯γα,mju(t). (C7) The ﬁrst terms on the right-hand side of Eqs. ( C7)a r e the frozen contributions originating from Eq. ( 39) and ex- pressed as γf α(/epsilon1)=¯h−1nα(/epsilon1)/Gamma1(0) α(/epsilon1) and ˜ γf α(/epsilon1)=¯h−1(1∓ nα(/epsilon1))/Gamma1(0) α(/epsilon1). On the other hand, the second terms on the right hand side of Eqs. ( C7) are due to the adiabatic correction given by Eq. ( 40). The contribution due to level renormalization have been neglected. APPENDIX D: EVALUATION OF THE MEAN VALUES FOR THE OTOCs For simplicity, we consider the bosonic case such that the bath and the system degrees of freedom commute. Themean value associated with the lesser Green’s function in theFIG. 9. The augmented Keldysh contour Kfor the OTOC. interaction picture can be written as i/angbracketleftbig b†H kα(t/prime)ˆKlj,hf(t)/angbracketrightbig ≈/integraldisplay Kdt1/angbracketleftTK[˜HCb† kα(t/prime)ˆKlj,hf(t)]/angbracketright =/summationdisplay uvVkα¯λα,uv(t)/integraldisplay Kdt1/angbracketleftTK[ˆρuv(t1)bkα(t1)b† kα(t/prime) ×ˆρlj(t)OB(t0)ˆρhf(t)OD(t0)]/angbracketright, (D1) where tlies in the arm where the contour goes from −∞ to∞andt/primein the arm which goes from ∞to−∞ .T h e calculation of mean values cannot be done in the traditionalKeldysh fashion as the out-of-time-order correlators (OTOC)have an abnormal time ordering. Instead we proceed along theline of argument of Ref. [ 73]. We will consider an augmented Keldysh contour as shown in Fig. 9. In terms of the augmented contour, the lesser Green’s function can be expressed as i/angbracketleftbig b †H kα(t/prime)ˆKH lj,hf(t)/angbracketrightbig =/summationdisplay u,vVkα¯λα,uv(t)/integraldisplay Kdt1/angbracketleftbig TK/bracketleftbig ˆρuv(t1)bkα(t1)b† kα(t/prime) ×ˆρlj(tu)OB/parenleftbig tu 0/parenrightbig ˆρhf(td)OD(td 0)/bracketrightbig/angbracketrightbig . (D2) In the next step, we deform the contour as shown in Fig. 10. FIG. 10. The deformed augmented Keldysh contour such that t andt/primelie in different Keldysh contours. 035425-14FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) The integration over the Keldysh contour can be broken down into 6 different parts ( d−,d+,u−,u+inK1and forward and backward going branch in K2) depending on where t1is pinned. The integral over the Keldysh contour can be expressed as /integraldisplay Kdt1=/integraldisplaytd td 0dtd− 1+/integraldisplayt0 tddtd+ 1+/integraldisplayt/primed t0dtd/prime− 1+/integraldisplaytu 0 t/primeddtd/prime+ 1 +/integraldisplaytu tu 0dtu− 1+/integraldisplayt0 tudtu+ 1+/integraldisplayt/primeu t0dtu/prime− 1+/integraldisplayt0 t/primeudtu/prime+ 1. (D3) We can stretch td/u 0andt0to−∞ , such that i/angbracketleftbig b†H kα(t/prime)ˆKH lj,hf(t)/angbracketrightbig =/summationdisplay u,vVkα¯λα,uv(t)/integraldisplay∞ −∞dt1[θ(t−t1)/angbracketleftˆρlj(t)OB(t0)[ ˆρhf(t),ˆρuv(t1)]OD(t0)/angbracketright/angbracketleftb† kα(t/prime)bkα(t1)/angbracketright +θ(t/prime−t1)/angbracketleftˆρlj(t)OB(t0)ˆρuv(t1)ˆρhf(t)OD(t0)/angbracketright[/angbracketleftb† kα(t/prime)bkα(t1)/angbracketright−/angbracketleft bkα(t1)b† kα(t/prime)/angbracketright] +θ(t−t1)/angbracketleft[ˆρlj(t),ˆρuv(t1)]OB(t0)ˆρhf(t)OD(t0)/angbracketright/angbracketleftb† kα(t/prime)bkα(t1)/angbracketright +θ(t/prime−t1)/angbracketleftˆρuv(t1)ˆρlj(t)OB(t0)ˆρhf(t)OD(t0)/angbracketright[/angbracketleftb† kα(t/prime)bkα(t1)/angbracketright−/angbracketleft bkα(t1)b† kα(t/prime)/angbracketright]]. (D4) In terms of Green’s functions, one can write G< lj hf,kα(t,t/prime) /similarequal/integraldisplay∞ −∞dt1Vkα/summationdisplay u,vλα,uv(t) ×/bracketleftbig gr lj hf,uv(t,t1)g< kα(t1,t/prime)+g< lj hf,uv(t,t1)ga kα(t1,t/prime)/bracketrightbig , (D5) where gr lj hf,vu(t,t/prime) =−iθ(t−t/prime)/angbracketleftˆρlj(t)OB(t0)[ ˆρhf(t),ˆρuv(t/prime)]− ×OD(t0)+[ˆρlj(t),ρuv(t/prime)]−OB(t0)ˆρhf(t)OD(t0)/angbracketright (D6) and g< lj hf,vu(t,t/prime)=−i/angbracketleftˆρuv(t/prime)ˆρlj(t)OB(t0)ˆρhf(t)OD(t0)/angbracketright −i/angbracketleftˆρlj(t)OB(t0)ˆρuv(t/prime)ˆρhf(t)OD(t0)/angbracketright.(D7) In the case with fermionic baths, the commutator changes to anti-commutator and the lesser Green’s function changes by asign. APPENDIX E: FROZEN AND ADIABATIC COMPONENTS OF LESSER GREEN’S FUNCTION FOR THE CASE OF OTOC The lesser Green’s function for the case of OTOC can be separated into frozen and adiabatic components (similar tothe case of density matrix). We now introduce the interaction representation with respect to ˜Hf Sand consider the Green’s function g< lj hf,vu(t1,t2)=g<,f lj hf,uv(t1,t2)+δg<,f lj hf,uv(t1,t2), (E1) where the frozen contribution is g<,f lj hf,vu(t1,t2)=±i/angbracketleftbig ˆρf uv(t− 2)ˆKf lj,hf(t+ 1)/angbracketrightbig ±i/angbracketleftbig ˆρf lj(t+ 1)OB(t0)ˆρf uv(t− 2)ˆρf hf(t+ 1)OD(t0)/angbracketrightbig (E2)and the adiabatic contribution is δg<,f lj hf,vu(t1,t2) =±1 ¯h/angbracketleftbigg Tk/integraldisplay Kdt/primeδ˜Hf S(t/prime)ˆρf uv(t− 2)ˆKf lj,hf(t+ 1)/angbracketrightbigg ±1 ¯h/angbracketleftbigg Tk/integraldisplay Kdt/prime ×δ˜Hf S(t/prime)ˆρf lj(t+ 1)OB(t0)ˆρf uv(t− 2)ˆρf hf(t+ 1)OD(t0)/angbracketrightbigg . (E3) After some calculation using the Keldysh contour in Fig. 9, we obtain δg<,f lj hf,vu(t1,t2) =±1 ¯h/integraldisplay∞ −∞dt/prime/parenleftbig/angbracketleftbigˆKf uj,hf(t1)/angbracketrightbig δvl× ×[θ(t1−t/prime)ξfj,hv(t/prime)+θ(t2−t/prime)ξv,u(t/prime)]+/angbracketleftbigˆKf lj,uf(t1)/angbracketrightbig ×δvh[θ(t1−t/prime)ξfj,lv(t/prime)+θ(t2−t/prime)ξv,u(t/prime)]/parenrightbig ei/epsilon1vu(t1−t2). (E4) APPENDIX F: CALCULATION OF FROZEN TRANSITION RATES FOR THE OTOC MASTER EQUATION We have the following relations for the bath Green’s func- tion gr kα(t,t1)=−iθ(t−t1)e−i/epsilon1kα(t−t1), ga kα(t,t1)=iθ(t1−t)e−i/epsilon1kα(t−t1), g< kα(t,t1)=±inα(/epsilon1kα)e−i/epsilon1kα(t−t1). (F1) Similarly for the system Green’s functions g<,f lj l/primej/prime,vu(t,t1)=±i/bracketleftbig δlv/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +δl/primev/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/bracketrightbig ei/epsilon1vu(t−t1), (F2) g<,f uv,jl j/primel/prime(t,t1)=±i/bracketleftbig δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig +δju/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig/bracketrightbig ei/epsilon1vu(t−t1), (F3) 035425-15BHANDARI, FAZIO, TADDEI, AND ARRACHEA PHYSICAL REVIEW B 104, 035425 (2021) the retarded Green’s function gr,f lj l/primej/prime,vu(t,t1) =−iθ(t−t1)ei/epsilon1vu(t−t1)/parenleftbig δuj/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig ±δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig ±δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/parenrightbig , (F4) and the advanced Green’s function ga,f uv,jl j/primel/prime(t1,t) =iθ(t−t1)ei/epsilon1vu(t−t1)/parenleftbig δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig ±δuj/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig +δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig ±δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig/parenrightbig . (F5) Now we can calculate individual expressions in Eq. ( 24) such as /integraldisplay dt1gr,f lj l/primej/prime,vu(t,t1)g< kα(t1,t) =±i/parenleftbig δuj/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig ±δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig ±δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/parenrightbig nα(/epsilon1kα)/bracketleftbigg1 /epsilon1kα−/epsilon1uv+iη/bracketrightbigg . (F6)Similarly, /integraldisplay dt1g<,f lj l/primej/prime,vu(t,t1)ga kα(t1,t) =∓i/bracketleftbig δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/bracketrightbig/bracketleftbigg1 /epsilon1kα−/epsilon1uv+iη/bracketrightbigg , (F7)/integraldisplay∞ −∞dt1gr kα(t,t1)g<,f uv,jl j/primel/prime(t1,t) =∓i/bracketleftbig δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig +δju/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig/bracketrightbig/bracketleftbigg1 /epsilon1kα−/epsilon1vu+iη/bracketrightbigg , (F8) and/integraldisplay dt1g< kα(t,t1)ga,f uv,jl j/primel/prime(t1,t) =±i/parenleftbig δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig ±δuj/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig +δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig ±δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig/parenrightbig nα(/epsilon1kα)/bracketleftbigg1 /epsilon1kα−/epsilon1vu+iη/bracketrightbigg . (F9) Using the relation 1 /epsilon1kα−/epsilon1mn±iη=P/braceleftbigg1 /epsilon1kα−/epsilon1mn/bracerightbigg ∓iπδ(/epsilon1kα−/epsilon1mn),(F10) and neglecting the principal value (which gives rise of level renormalization effects), we obtain /integraldisplay dt1gr,f lj l/primej/prime,vu(t,t1)/Sigma1<(0) α(t1,t)=±/Gamma1(0) α(/epsilon1uv) 2nα(/epsilon1uv)/parenleftbig δuj/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig ±δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig ±δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/parenrightbig ,(F11) where /Sigma1(0) kα(t1,t)=/summationtext k\|Vkα\|2gkα(t1,t) for all the bath Green’s functions. Similarly, /integraldisplay dt1g<,f lj l/primej/prime,vu(t,t1)/Sigma1a(0) α(t1,t)=∓/Gamma1(0) α(/epsilon1uv) 2/bracketleftbig δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/bracketrightbig , (F12) /integraldisplay dt1/Sigma1r(0) α(t,t1)g<,f uv,jl j/primel/prime(t1,t)=∓/Gamma1(0) α(/epsilon1vu) 2/bracketleftbig δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig +δju/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig/bracketrightbig , (F13) and /integraldisplay dt1/Sigma1<(0) α(t,t1)ga,f uv,jl j/primel/prime(t1,t)=±/Gamma1(0) α(/epsilon1vu) 2nα(/epsilon1vu)/parenleftbig δvl/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig ±δuj/angbracketleftbig Kf lv,l/primej/prime(t)/angbracketrightbig +δvl/prime/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig ±δj/primeu/angbracketleftbig Kf lj,l/primev(t)/angbracketrightbig/parenrightbig .(F14) Substituting Eqs. ( F11), (F12), (F13), and ( F14)i nE q s .( 29) and ( 30), in terms of transition rates, we obtain /Lambda1α,OTOC lj l/primej/prime(t,t)=¯h 2/summationdisplay u/bracketleftbig λα,ju(t)γα(/epsilon1ju)/angbracketleftbig Kf lu,l/primej/prime(t)/angbracketrightbig +λα,j/primeu(t)γα(/epsilon1j/primeu)/angbracketleftbig Kf lj,l/primeu(t)/angbracketrightbig −λα,ul(t)˜γα(/epsilon1ul)/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig −λα,ul/prime(t)˜γα(/epsilon1ul/prime)/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig/bracketrightbig , (F15) /Lambda1α,OTOC jl j/primel/prime(t,t)=−¯h 2/summationdisplay u/bracketleftbig λα,ul(t)γα(/epsilon1lu)/angbracketleftbig Kf uj,l/primej/prime(t)/angbracketrightbig +λα,ul/prime(t)γα(/epsilon1l/primeu)/angbracketleftbig Kf lj,uj/prime(t)/angbracketrightbig −λα,j/primeu(t)˜γα(/epsilon1uj/prime)/angbracketleftbig Kf lj,l/primeu(t)/angbracketrightbig −λα,ju(t)˜γα(/epsilon1uj)/angbracketleftbig Kf lu,l/primej/prime(t)/angbracketrightbig/bracketrightbig . (F16) 035425-16FROM NONEQUILIBRIUM GREEN’S FUNCTIONS TO … PHYSICAL REVIEW B 104, 035425 (2021) Finally, we have ∂Klj,l/primej/prime(t) ∂t=i ¯h[/epsilon1l(t)−/epsilon1j(t)+/epsilon1l/prime(t)−/epsilon1j/prime(t)]Klj,l/primej/prime(t)+1 ¯h/summationdisplay m/bracketleftbig λα,ml(t)/Lambda1α,OTOC mjl/primej/prime(t)−λα,jm(t)/Lambda1α,OTOC lml/primej/prime(t) +λα,ml/prime(t)/Lambda1α,OTOC lj mj/prime(t)−λα,j/primem(t)/Lambda1α,OTOC lj l/primem(t)+λα,ml(t)/Lambda1α,OTOC jmj/primel/prime(t)−λα,jm(t)/Lambda1α,OTOC ml j/primel/prime(t) +λα,ml/prime(t)/Lambda1α,OTOC jl j/primem(t)−λα,j/primem(t)/Lambda1α,OTOC jlml/prime(t)/bracketrightbig . (F17) APPENDIX G: ADIABATIC CHANGE OF THE BASIS OF EIGENSTATES Introducing the notation ∂Xinstead of ∂/∂X,w eh a v e ∂X(\|l/angbracketright/angbracketleftj\|)=\|∂Xl/angbracketright/angbracketleftj\|+\|l/angbracketright/angbracketleft∂Xj\| (G1) Using Eq. ( D3)o fR e f .[ 12], we obtain /angbracketleftl/prime\|∂Xl/angbracketright=/angbracketleftl/prime\|∂XHS\|l/angbracketright εl−εl/prime,l/negationslash=l/prime. (G2) Operating on both sides by/summationtext l/prime\|l/prime/angbracketright, and taking only the contribution originating in the gauge term, we obtain ∂X\|l/angbracketright=/summationdisplay l/primeAl/prime,l\|l/prime/angbracketright. (G3) Similarly, using the other equation in Eq. ( D3)o fR e f .[ 12], we obtain /angbracketleft∂Xj\|l/prime/angbracketright=/angbracketleftj\|∂XHS\|l/prime/angbracketright εj−εl/prime,l/negationslash=l/prime. (G4) Operating on both sides by/summationtext l/prime/angbracketleftl/prime\|, we obtain /angbracketleft∂Xj\|=−/summationdisplay l/primeAj,l/prime/angbracketleftl/prime\| (G5) [1] J. Ren, P. Hänggi, and B. Li, Berry-Phase-Induced Heat Pump- ing and Its Impact on the Fluctuation Theorem, Phys. Rev. Lett. 104, 170601 (2010) . [2] J. E. Avron, A. Elgart, G. M. Graf, and L. 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PhysRevB.91.100502.pdf	RAPID COMMUNICATIONS PHYSICAL REVIEW B 91, 100502(R) (2015) Transition-metal substitutions in iron chalcogenides V . L. Bezusyy, D. J. Gawryluk, A. Malinowski, and Marta Z. Cieplak Institute of Physics, Polish Academy of Sciences, Al. Lotnik ´ow 32/46, 02-668 Warsaw, Poland (Received 18 July 2014; revised manuscript received 12 March 2015; published 20 March 2015) Theab-plane resistivity and Hall effect are studied in Fe 1−yMyTe0.65Se0.35single crystals doped with two transition-metal elements, M=Co or Ni, over a wide doping range, 0 /lessorequalslanty/lessorequalslant0.2. The superconducting transition temperature, Tc, reaches zero for Co at y/similarequal0.14 and for Ni at y/similarequal0.032, while the resistivity at the Tconset increases weakly with Co doping, and strongly with Ni doping. The Hall coefﬁcient RH, positive for y=0, remains so at high temperatures for all y, while it changes sign to negative at low Tfory>0.135 (Co) and y>0.06 (Ni). The analysis based on a two-band model suggests that at high Tresidual hole pockets survive the doping, but holes get localized upon the lowering of T, so that the effect of the electron doping on the transport becomes evident. The suppression of the Tcby Co impurity is related to electron doping, while in the case of the Ni impurity strong electron localization most likely contributes to fast decrease of the Tc. DOI: 10.1103/PhysRevB.91.100502 PACS number(s): 74 .25.F−,74.25.Dw,74.62.Dh,74.70.Xa Substitution of impurities is an efﬁcient method for tuning of the electronic properties and mapping out the phasediagrams of the new compounds. While isovalent substitutionsare expected to be mainly potential scatterers, the heterovalentimpurities usually induce doping of carriers. Many recentstudies focus on iron-based superconductors (IS), in whichpairing mechanism is likely related to spin ﬂuctuations [ 1–4]. Since the IS are multiband materials, the effect of impuritieson the phase diagram may be quite complex. For example, theheterovalent substitution of the transition metals into the Fesite in iron pnictide BaFe 2As2induces not only shift of the Fermi level [ 5–9] but also reconstructs the Fermi surface [ 10]. Both the shift of the chemical potential and loss of the coherentcarrier density are predicted by theory [ 11]. Less is known about substitutions in iron chalcogenides, another class of the IS. Few attempts of the transition-metal doping of single crystals of FeTe 1−xSexhave been reported, and for a limited impurity contents [ 12–15]. The studies are complicated by two types of magnetic corre-lations existing throughout the phase diagram [ 16,17] and by crystal inhomogeneities, such as Fe excess [ 16,18–21] or Fe vacancies [ 22–24]. In this Rapid Communication we report on a comprehensive study of the transport propertiesof FeTe 0.65Se0.35single crystals, doped over a wide doping range by transition-metal elements, Co and Ni [ 25]. Our study reveals that the Fermi surface, which in undoped materialconsists of hole and electron pockets [ 26], evolves dramatically with doping, with some features which are distinctly differentfrom these observed in doped pnictides [ 10]. The inﬂuence of substitutions on superconductivity also shows peculiaritiesdistinct from pnictide superconductors [ 8]. Single crystals of Fe 1−yMyTe0.65Se0.35withM=Co or Ni andyup to 0.2 have been grown using Bridgman’s method and thoroughly characterized [ 12,22–24,27]. The quantitative point analysis, performed by energy-dispersive x-ray (EDX)spectroscopy at many points on each crystal, shows that Coand Ni effectively substitute Fe and their contents are close tothe nominal. The average Fe +Mcontent is 0.99(3) and the Se content is 0.35(2). The study of the magnetic properties, whichwill be described elsewhere [ 30], shows neither localized magnetic moments nor any magnetic order developing as aresult of doping. The resistivities, in-plane ( ρ) and Hall ( ρ xy),were measured by dc and ac four-probe methods, respectively, using a physical property measurement system (QuantumDesign), in the temperature range 2 to 300 K, and in magneticﬁelds up to 9 T. The ac magnetic susceptibility was measuredwith magnetic ﬁeld amplitude of 1 Oe and a frequency of10 kHz in warming mode (ﬁeld orientation has no effect onthe superconducting transition temperature, T c). TheTdependence of the ρ, normalized to resistivity at room temperature, ρ300, is shown in Figs. 1(a) and 1(b) for crystals with different yvalues. In the undoped crystal, y=0, the ρ/ρ 300decreases with Tdecreasing below about 150 K, indicating good metallic character. Figures 2(a) and 2(b) show the Tdependence of the Hall coefﬁcient, RH,f o r the same crystals, in the high- Trange above 20 K, in which the dependence of the ρxyon the magnetic ﬁeld is linear. At y=0 theRHis positive; it rises as Tis lowered down from 300 K to about 55 K, and has a downturn at lower T. Such a behavior, reported before [ 19,21], results from the interplay between different Tdependencies of charge carrier densities and their mobilities in the multiband system. The Tdependencies of ρ/ρ 300for doped crystals acquire low-temperature upturns, which increase progressively with increasing y. These upturns are much larger in the case of Ni doping. The upturns developalso in the R H(T) dependencies at small y[31]. Unexpectedly, in the case of Ni-doped crystals with y>0.06, the ρ/ρ 300at lowTshows nonmonotonic changes with increasing y: ﬁrst saturation, than decrease, and ﬁnally increase at the highestdoping [Fig. 1(b)]. The susceptibility data, shown in Fig. 1(c) for Ni-doped crystals, conﬁrm bulk superconductivity for y<0.03. Fig- ure1(d) displays the T cversus yfor both impurities. Here Tc is deﬁned as the middle point of the transition, and the vertical error bars reﬂect 90% to 10% transition width. The yvalues and the horizontal error bars are the average yand the standard deviations, respectively, measured by the EDX. The Tcis about 14 K at y=0. It is suppressed by both impurities, reaching zero at critical concentrations yc/similarequal0.14 (Co) and yc/similarequal0.032 (Ni). The initial slopes of the Tc(y) dependencies are equal to about −0.9K/at. % (Co) and −3.5K/at. % (Ni). Assuming that the Tcis suppressed by a rigid band shift due to electron doping, one expects roughly twice as fast decrease of the Tcto zero for the Ni than for the Co, because Ni provides twice as 1098-0121/2015/91(10)/100502(5) 100502-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS BEZUSYY , GAWRYLUK, MALINOWSKI, AND CIEPLAK PHYSICAL REVIEW B 91, 100502(R) (2015) FIG. 1. (Color online) ρ/ρ 300vsTfor crystals doped with Co (a) and Ni (b). (c) The ac susceptibility for Ni-doped crystals (the data are not corrected for demagnetization factor). (d) Tcvsy,( e ) ρonvsy, and (f) Tcvsρon,f o ry=0 (brown diamonds), and for Co (green triangles), and Ni impurities (blue circles). In (d)–(f) the dashed lines are guides to the eye. many electrons to the electronic bands. This, however, is not the case. The suppression by the Ni is much more effective;therefore it cannot be explained by electron doping alone. It islikely that deeper impurity potential results in more complexmodiﬁcation of the band structure, or strong localization ofcarriers. FIG. 2. (Color online) RHas a function of Tfor crystals with different y, doped with Co (a) and Ni (b), and as a function of yfor various Tfor Co (c) and Ni (d).The nonlinear ρ(T) dependence prevents easy estimate of scattering rates. Instead, we extract from the data normalized ρ at the onset of superconductivity, ρon/ρ300. To eliminate data scatter due to imperfect estimate of the sample dimensions,we make use of the fact that ρ 300does not show any deﬁnite dependence on y, so we may average it over various yto obtain ρ300. Finally, we deﬁne the quantity ρon≡(ρon/ρ300)ρ300/similarequal ρon, which gives good approximate value of the ρat the Tconset. Figure 1(e) shows the ρon(y) dependence for both impurities. We ﬁnd that the initial increase of ρonwith increasing yis by a factor of 4.5 times larger in the Ni case, indicating much stronger scattering induced by the Ni.Figure 1(f)compares the dependencies of the T conρonfor both impurities. They are distinctly different. At the point atwhich the T creaches zero the increase of ρoninduced by the Ni is about twice as big as the increase induced by the Co. We now turn our attention to RH[Figs. 2(a) and2(b)]. Aside from low- Tupturns at small y, further increase of y reduces the RHat all temperatures, which is consistent with the expectation of electron doping. Eventually, above a certainimpurity content y 0, and below a certain temperature T0,t h e RHchanges sign to negative. The RH(y) dependencies for various Tare collected in Figs. 2(c) and2(d) for Co- and Ni-doped crystals, respectively. It is seen that at 300 K the RH depends weakly on yand remains positive for all yvalues, but signiﬁcant dependencies develop upon cooling. Qualitatively similar behavior is observed for both impurities, with two broad features developing upon lowering of T, the enhanced positive RHaty<y 0and the enhanced negative RHaty>y 0. Both features are stronger in the case of the Ni. The y0at low Tin Ni-doped crystals ( y0/similarequal0.06) is about twice as small in t h eC oc a s e( y0/similarequal0.135), which conﬁrms that electron doping contributes to the behavior. Note also that the low- Tdecrease ofρin Ni-doped crystals occurs for y/greaterorsimilary0, suggesting a relation to electron doping. In Fig. 3(a)we plot the ydependence of the Hall number, nH=1/eRH, for Ni-doped crystals at T=20 K. The nH is positive, hole-dominated at y<y 0, and becomes negative, electron-dominated at y>y 0, with a change of sign at y0= 0.06. Large magnitude of negative nHaty=0.2 suggests that at the largest yand at low Tthe hole contribution becomes small. At all other yboth hole and electron contributions are important. More insights are provided by the dependence of ρxyon the magnetic ﬁeld, μ0H, measured at T=2Kf o ra l lN i - doped crystals, which do not superconduct down to T=2K . As shown in Fig. 3(b), the dependence evolves, from almost linear with positive slope for y=0.032 (y<y 0), to linear with negative slope for y=0.2(y>y 0); it is nonlinear for intermediate y. In the case of y=0.056 (y/similarequaly0)w es e et h e negative slope at low H, and positive slope at large H.I na two-band model in the strong-ﬁeld limit the RHis dependent on the μ0H[32], eRH=μ2 hnh−μ2 ene+μ2 eμ2 h(nh−ne)(μ0H)2 ρ−2+μ2eμ2 h(nh−ne)2(μ0H)2 −−→H→∞1 nh−ne, (1) 100502-2RAPID COMMUNICATIONS TRANSITION-METAL SUBSTITUTIONS IN IRON . . . PHYSICAL REVIEW B 91, 100502(R) (2015) FIG. 3. (Color online) (a) nHvsyfor Ni doping at T=20 K (diamonds), and at T=2K :h i g hﬁ e l d( c i r c l e s )a n dl o wﬁ e l d (squares). Here, nH(carriers/Fe) =0.88×10−22nH(cm−3). (b)ρxyvs μ0HatT=2 K for Ni-doped crystals. (c) T0vsyfor Co (triangles), and Ni (circles); Ni high-ﬁeld data: full red circles. Shaded areas:superconductivity (SC) range. (d) eR H/ρ2at theTconset (full points) and at T=2 K (open points) vs yCoand 2yNi. All lines are guides to the eye. while in the low-ﬁeld limit it is given by eRH=ρ2(μ2 hnh− μ2 ene). Here nhandneare hole and electron concentrations, μhandμeare their mobilities, and ρ=(μhnh+μene)−1is the resistivity at H=0. From the 2 K data we extract the nH in the low-ﬁeld and in the high-ﬁeld limits, and we show it as af u n c t i o no f yin Fig. 3(a). The high-ﬁeld nHchanges sign aty/similarequal0.06, indicating, according to Eq. ( 1), that ne>nhfor y>0.06. On the other hand, the low-ﬁeld nHchanges sign at y/similarequal0.05, when nhis still larger than ne; therefore, we have μe>μhfory>0.05. Arrows indicate in the ﬁgure the y values, at which these changes occur. We summarize the Hall effect data by plotting in Fig. 3(c) theydependence of the temperature T0, at which the RH changes sign. The Ni content is multiplied by 2, to facilitate comparison of the electron doping by the two impurities. Itis seen that the region with R H<0 is restricted to T< T 0 and to y>y 0. At largest doping, 2 yNi/greaterorsimilar0.23, the boundary between positive and negative RHis located at T0≈180 K. The hole-dominated conduction at high Tin heavily doped crystals indicates that hole pockets survive the electron doping,but they are most likely substantially shrunk in comparisonwith the undoped crystal. The shrinking of the hole pockets hasbeen observed by angle-resolved photoemission (ARPES) inCo-doped pnictide BaFe 2As2[10], in which, however, the tran- sport is electron-dominated at all temperatures [ 8]. In the present case one may expect that the hole carriers in theremnant hole pockets are easily localized when temperatureis lowered, which would explain the electron-dominatedconduction below T 0. Interestingly, at slightly smaller doping, 0.12<2yNi<0.23, the T0is reduced to form a plateau at about 140 K. A similar plateau at similar T0is seen in Co-dopedsamples in the doping range 0 .14<y Co<0.2, suggesting common origin of this feature. It may signal some reconstruc-tion of the Fermi surface, such as, for example, vanishing atthe largest yof one of the hole pockets, which exist at the /Gamma1 point in the undoped crystal [ 26]. Combining the data for R Handρ, we obtain in the weak- ﬁeld limit the quantity eRH/ρ2=μ2 hnh−μ2 ene. Figure 3(d) shows the plot of eRH/ρ2versus yCoand 2yNi.T h el o w - Tdata are calculated for all Co- and Ni-doped crystals, either at theonset of superconductivity (when it is present) or at T=2Ki n the case of nonsuperconducting samples. We see three regionswith distinct dependencies of eR H/ρ2ony.A ts m a l l y(labeled A) the eRH/ρ2increases with increasing y, and the increase is larger for Ni impurity. At intermediate doping (region B) thereis a profound decrease of eR H/ρ2until it reaches a negative value, particularly large in the case of the Ni. Finally, atthe largest Ni doping (region C) we observe the decrease ofthe magnitude of negative eR H/ρ2. We now consider the effect of electron doping on carrier concentrations and mobilities, assuming for simplicity thatonly monotonic changes occur. We anticipate that n hdecreases andneincreases with increasing y; also, the μhis expected to decrease due to impurity scattering and shrinking of thehole pockets. On the other hand, the μ eis inﬂuenced by two competing effects, impurity scattering, which reduces μe, and the expansion of the electron pockets, which is likely to increase μe. If the last effect prevails, the eRH/ρ2should be a decreasing function of y. This expectation agrees nicely with the profound decrease of eRH/ρ2observed in the region B. When the increasing μ2 eneexceeds decreasing μ2 hnh,t h eRH changes sign; furthermore, this evolution explains the decrease ofρin the low- Tlimit illustrated in Fig. 1(b). In the region C the hole contribution is substantially reduced, so that μ2 enebecomes the dominant term. Scattering by large density of impurities is likely to reduce the μ2 e, which overweighs the increase of ne, leading to the decrease of the magnitude of eRH/ρ2. In this simple picture the initial increase of eRH/ρ2in the region A, larger for Ni impurity, may be explained only bystrong impurity-induced reduction of μ 2 e, which overweighs the minor increase of neat small doping; moreover, the decrease of μ2 enewould have to be larger than the decrease of μ2 hnh. A conceivable scenario could be the low- Tlocalization of electron carriers in the vicinity of positively chargedimpurity ions, which should be more profound in case of the Niwith larger impurity potential. Such scenario is supported bythe observation of strong increase of ρ onat small yin Ni-doped crystals. Note that theory predicts the coherent carrier densityreduction and localization effects in the case of strong impuritypotential [ 11]. Eventually, with the further increase of yinto the B region the expansion of the electron pockets takes over, andthe electron localization ceases to affect transport. Of course,this reasoning is based on a simple two-band model, whichmay not be entirely appropriate for this multiband system.However, it seems to provide good qualitative explanation ofthe observed behavior. The complex evolution of the multiband system, uncovered in this work, prevents estimating the scattering rates of differ-ent types of carriers on the basis of present experiment, sincethis requires independent evaluation of carrier concentrations 100502-3RAPID COMMUNICATIONS BEZUSYY , GAWRYLUK, MALINOWSKI, AND CIEPLAK PHYSICAL REVIEW B 91, 100502(R) (2015) and mobilities. Therefore, we cannot compare the observed rate of the Tcsuppression to the theoretical predictions of pair-breaking for various gap symmetries, similar to what hasbeen done in the case of doped or irradiated pnictides [ 9,33]. Nevertheless, our results suggest that the electron localizationat small doping levels may play a role in the destruction ofsuperconductivity. As indicated in Fig. 3(c), where shaded areas show superconducting regions, in Co-doped crystals they c,a tw h i c h Tcreaches zero, is comparable to y0. This suggests that the suppression of the Tcis related to the shrinking of hole pockets, similar to the effect identiﬁed by ARPES in Co-dopedBaFe 2As2[10]. On the other hand, in the case of the Ni the ycis much smaller than y0. Instead, it is close to the boundary of the A region, suggesting that electron localization, strongin the Ni case, may contribute, in addition to electron doping,to faster suppression of the T c. This result is different from the behavior reported for Ni-doped BaFe 2As2,f o rw h i c hN i impurity appears to be a much stronger electron scatterer thanthe Co (as in the present case), but the T csuppression seems to be well explained solely by electron doping [ 7,8]. It is intriguing to ask what is the origin of such a strong differencebetween these two materials. It is possible that the answerlies in the persistence of local ( π,0) magnetic ﬂuctuations in FeTe 0.65Se0.35[18], which may enhance incoherent scatteringin the presence of deep Ni-impurity potential. The other possibilities include local lattice distortion around impurity,which may differ depending on the host lattice, or other subtledifference in the evolution of the electronic structure withdoping. In conclusion, the study of FeTe 0.65Se0.35single crystals doped up to high impurity levels with two transition-metalelements, Co and Ni, reveals a change of sign of Hallcoefﬁcient to negative at low temperature, consistent withthe electron doping induced by both impurities. However, theR Hremains positive at high T, suggesting that remnant hole pockets survive the doping, and holes get localized upon thelowering of T. The study suggests also that the suppression of superconductivity is related to electron doping in the caseof Co impurity, while the Ni impurity most likely induces, inaddition, strong electron localization. We would like to thank M. Berkowski, M. Kozłowski, and A. Wi ´sniewski for experimental support and discussions. This research was partially supported by the ERC FunDMSAdvanced Grant (FP7 Ideas) and by the Polish NCS GrantNo. 2011/01/B/ST3/00462, and was partially performed inthe NanoFun laboratories co-ﬁnanced by the ERDF ProjectNanoFun POIG.02.02.00-00-025/09. [1] D. C. Johnston, Adv. Phys. 59,803(2010 ). [2] J. Wen, G. Xu, G. Gu, J. M. Tranquada, and R. J. Birgeneau, Rep. Prog. Phys. 74,124503 (2011 ). [3] P. J. Hirschfeld, M. M. Korhunov, and I. I. Mazin, Rep. Prog. Phys. 74,124508 (2011 ). [ 4 ] G .R .S t e w a r t , Rev. Mod. Phys. 83,1589 (2011 ). [5] P. C. Canﬁeld, S. L. Bud’ko, Ni Ni, J. Q. Yan, and A. Kracher, Phys. Rev. B 80,060501(R) (2009 ). [6] L. Fang, H. Luo, P. Cheng, Z. Wang, Y . Jia, G. Mu, B. Shen, I. I. Mazin, L. Shan, C. Ren, and H. H. Wen, Phys. Rev. B 80, 140508(R) (2009 ). [ 7 ]P .C .C a n ﬁ e l da n dS .L .B u d ’ k o , Annu. Rev. Condens. Matter Phys. 1,27(2010 ). [8] A. Olariu, F. Rullier-Albenque, D. Colson, and A. Forget, Phys. Rev. B 83,054518 (2011 ). [9] J. 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Lett. 108,107002 (2012 ). [21] Y . Sun, T. Taen, T. Yamada, S. Pyon, T. Nishizaki, Z. Shi, and T. Tamegai, P h y s .R e v .B 89,144512 (2014 ). [22] A. Wittlin, P. Aleshkevych, H. Przybyli ´nska, D. J. Gawryluk, P. Dłu ˙zewski, M. Berkowski, R. Pu ´zniak, M. U. Gutowska, and A. Wi ´sniewski, Supercond. Sci. Technol. 25,065019 (2012 ). [23] V . L. Bezusyy, D. J. Gawryluk, M. Berkowski, and M. Z. Cieplak, Acta Phys. Pol., A 121, 816 (2012). [24] V . L. Bezusyy, D. J. Gawryluk, A. Malinowski, M. Berkowski, a n dM .Z .C i e p l a k , Acta Phys. Pol., A 126,76(2014 ); M. Z. Cieplak and V . L. Bezusyy, Philos. Mag. 95,480(2015 ). [25] While the crystals of FeTe 1−xSexwithx=0.5 are optimal for superconductivity, two tetragonal phases with slightly different 100502-4RAPID COMMUNICATIONS TRANSITION-METAL SUBSTITUTIONS IN IRON . . . PHYSICAL REVIEW B 91, 100502(R) (2015) Se content have been reported; see, e.g., B. C. Sales, A. S. Sefat, M. A. McGuire, R. Y . Jin, D. Mandrus, and Y . 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Puzniak, D. J. Gawryluk, and M. Berkowski (unpublished).[31] As discussed in Ref. [ 24], in Co-doped crystals with y<0.04 the upturns in ρandR H, as well as the value of the Tc,m a y vary in different fragments of the same crystal, with all thesequantities remaining strictly correlated; the variations are absentfory>0.04 and for Ni-doped crystals. The likely origin are hexagonal Fe 7(Te-Se) 8inclusions, which enhance incoherent scattering of carriers. In the present study we use Co-dopedsamples with the smallest magnitude of upturn. [32] C. M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, New York, 1972). [33] R. Prozorov, M. Ko ´nczykowski, M. A. Tanatar, A. Thaler, S. L. Bud’ko, P. C. Canﬁeld, V . Mishra, and P. J. Hirschfeld, Phys. Rev. X 4,041032 (2014 ). As demonstrated in this work, the issue is further complicated by the possibility of different ratiosof intra- to interband scattering rates. 100502-5
PhysRevB.83.104415.pdf	PHYSICAL REVIEW B 83, 104415 (2011) Spiral ground state in the quasi-two-dimensional spin-1 2system Cu 2GeO 4 Alexander A. Tsirlin,1,Ronald Zinke,2Johannes Richter,2and Helge Rosner1,† 1Max Planck Institute for Chemical Physics of Solids, N ¨othnitzer Strasse 40, D-01187 Dresden, Germany 2Institute for Theoretical Physics, University of Magdeburg, P .O. Box 4120, D-39016 Magdeburg, Germany (Received 20 August 2010; revised manuscript received 1 October 2010; published 22 March 2011) We apply density functional theory band structure calculations, the coupled cluster method, and exact diagonalization to investigate the microscopic magnetic model of the spin-1 2compound Cu 2GeO 4. The model is quasi-two-dimensional, with uniform spin chains along one direction and frustrated spin chains along theother direction. The coupling along the uniform chains is antiferromagnetic, J/similarequal130 K. The couplings along the frustrated chains are J 1/similarequal− 60 K and J2/similarequal80 K between nearest neighbors and next-nearest neighbors, respectively. The ground state of the quantum model is a spiral, with a reduced sublattice magnetization of0.62μ Ba n dap i t c ha n g l eo f8 4◦, both renormalized by quantum effects. The proposed spiral ground state of Cu2GeO 4opens a way to magnetoelectric effects in this compound. DOI: 10.1103/PhysRevB.83.104415 PACS number(s): 75 .30.Et, 75.10.Jm, 71 .20.Ps, 75.50.Ee I. INTRODUCTION Quantum magnetism is a ﬁeld of fundamental research focused on exotic ground states and nontrivial low-temperatureproperties. 1,2Nevertheless, certain effects in quantum magnets are also relevant for applications. Spin-chain compoundsshow a ballistic regime of heat transport, 3whereas frustrated magnets are capable of a strong magnetocaloric effect.4,5 Additionally, many of the frustrated magnets undergo spiral or, in general, incommensurate ordering and reveal ferro-electricity induced by a magnetic ﬁeld. 6A frustrated spin chain with competing ferromagnetic (FM) nearest-neighbor(J 1) and antiferromagnetic (AFM) next-nearest-neighbor ( J2) couplings is the simplest spin model giving rise to spiralmagnetic correlations at J 2/J1<−1 4(Ref. 7). This model is easily realized experimentally and has a clear structuralfootprint, a chain of edge-sharing Cu X 4plaquettes with X being oxygen,8–11chlorine,12or even nitrogen.13Such chains typically show FM J1due to the nearly 90◦Cu-X-Cu angle and AFMJ2due to the Cu- X-X-Cu superexchange. Indeed, many compounds of this type undergo spiral magnetic ordering andsometimes exhibit magnetic ﬁeld-induced ferroelectricity. 14 However, the detailed microscopic understanding of theseeffects remains challenging, and even the electronic originof ferroelectricity in spin-chain cuprates is vividly debated. 15 Interchain couplings are an important feature of any real material. The couplings between spin chains can modify theground state qualitatively by inducing a long-range order withﬁnite sublattice magnetization. 16,17In the case of frustrated spin chains, such couplings inﬂuence the behavior of dopedsystems 18and play a decisive role for the stability of exotic phases in high magnetic ﬁelds.19,20Regarding magnetoelectric effects, the interchain couplings naturally determine their tem-perature scale by adjusting the magnetic ordering temperature. Theoretical studies of coupled frustrated spin chains remain a challenge owing to the two-dimensional (2D) and frustratednature of the problem. Therefore, experimental benchmarksare especially important. The available frustrated-spin-chaincompounds show relatively weak interchain couplings, 20 while the relevance of the opposite regime with stronglycoupled frustrated spin chains remains unclear. A commonand a somewhat naive picture suggests that leading exchangecouplings should run along the structural chains owing to shorter Cu-Cu distances. 21 In the following, we present a microscopic magnetic model of Cu 2GeO 4. This compound is a unique example of a 2D system of strongly coupled frustrated spin chains. Thecoupling Jbetween the frustrated chains is so strong that the system can be equally viewed as uniform spin chains along J with the frustrated interchain couplings J 1andJ2(see Fig. 1). Both descriptions relate to certain features of the magneticbehavior: while the uniform chain model ﬁts the magneticsusceptibility of Cu 2GeO 4down to T/J/similarequal0.5, the ground state of the 2D model is a spiral, which is typical for thefrustrated J 1-J2spin chains. The crystal structure of Cu 2GeO 4belongs to the spinel type.22Magnetic properties were studied in a relation to the spin-Peierls compound CuGeO 3. The low-dimensional magnetic behavior of Cu 2GeO 4resembles CuGeO 3indeed. However, no signatures of the structural distortion or spingap were found down to 10 K, and the long-range magneticordering at T N=33.1 K is observed instead.23 Yamada et al.23analyzed Cu 2GeO 4using the anisotropic pyrochlore lattice model with two inequivalent exchangecouplings that are J 1andJcin our notation (upper left panel of Fig. 1). This model arises from a straightforward and naive geometrical consideration of the spinel structure, with inequiv-alent couplings driven by the tetragonal distortion of the parentcubic system. At J c/J1/lessmuch1, the anisotropic pyrochlore lattice splits into chains. According to Ref. 23,C u 2GeO 4is close to this limit, with J1=135 K and Jc/J1=0.16. Starykh et al.24 studied the 2D analog of the model theoretically and proposed a quantum-disordered valence-bond-solid ground state. II. BAND STRUCTURE As a derivative of the spinel structure, Cu 2GeO 4might be thought of as a three-dimensional network of CuO 6octahedra. However, this description ignores essential features of theelectronic structure. In oxide compounds, Cu 2+tends to adopt a fourfold coordination (CuO 4plaquette) having dramatic inﬂuence on the orbital ground state and magnetic properties.Such plaquettes can be recognized in Cu 2GeO 4and lead to a 104415-1 1098-0121/2011/83(10)/104415(7) ©2011 American Physical SocietyTSIRLIN, ZINKE, RICHTER, AND ROSNER PHYSICAL REVIEW B 83, 104415 (2011) bb’ b’c aa’ a’J1 J JJ2 Jab J1J1 Jc J2 FIG. 1. (Color online) Top panel: Crystal structure of Cu 2GeO 4 (left) and a single magnetic layer in the abplane (right). Bottom panel: A sketch of the spin spiral with the pitch angle γ(left) and the magnetic model of J1-J2frustrated spin chains coupled by J(right). Circles and dots denote the positions of the Cu atoms. Lines in the top left panel show the anisotropic pyrochlore lattice considered inRef. 23. peculiar superexchange scenario. Four short bonds to oxygen (1.95 ˚A) form the CuO 4plaquettes in the abplane, whereas the two remaining Cu-O bonds are much longer (2.50 ˚A). Edge-sharing CuO 4plaquettes comprise structural chains that run along aorb, with parallel chains forming layers in the abplane (upper right panel of Fig. 1). Equivalent layers with differently directed structural chains alternate along the caxis. In the following, we denote the direction of the structuralchains as b /primeand the perpendicular direction as a/prime, to distinguish those from the crystallographic aandbaxes. GeO 4tetrahedra connect the chains into a three-dimensional (3D) framework(Fig. 1). To evaluate individual exchange couplings, we perform scalar-relativistic density functional theory (DFT) band struc-ture calculations using the FPLO 9.00-33 code.25We apply the local density approximation (LDA) with the exchange-correlation potential by Perdew and Wang 26and use a well- converged kmesh comprising 3350 points in the symmetry- irreducible part of the ﬁrst Brillouin zone. With LDA cal-culations, we are able to identify relevant states and toevaluate hopping parameters t ivia a ﬁt with an effective one-orbital tight-binding (TB) model. The hopping parametersare introduced into a Hubbard model with the effective on-siteCoulomb repulsion potential U eff=4.5e V .27–29In the case of low-lying excitations, the Hubbard model is further reducedto a Heisenberg model under the conditions of half-ﬁlling andstrong correlations ( t i/lessmuchUeff). Then, the AFM parts of the exchange integrals are evaluated as JAFM i=4t2 i/U eff. An alternative way to evaluate the exchange couplings is to treat the strong correlations within DFT, via themean-ﬁeld-like LSDA +Uapproach. We calculate total energies for a set of collinear spin conﬁgurations andmap these energies onto a classical Heisenberg model.Thus, total exchange integrals J iare estimated. In the LSDA +Ucalculations, we use the Coulomb repulsion and exchange parameters Ud=6.5±1 eV and20 0 Energy (eV)4 4 810 030Total Cu GeO40 Ve( SOD)1 FIG. 2. (Color online) LDA density of states for Cu 2GeO 4.T h e Fermi level is at zero energy. Jd=1 eV , respectively.28–30The double-counting-correction (DCC) scheme was set to the around-mean-ﬁeld (AMF)option. The application of the fully-localized-limit (FLL)DCC had little effect on the exchange couplings. The LDA energy spectrum of Cu 2GeO 4is typical for Cu+2oxides. The mixed Cu 3 d-O 2pvalence bands extend down to −8 eV (Fig. 2), with the states near the Fermi level predominantly formed by the Cu dx2-y2orbital (here, xand yalign with the short Cu-O bonds). Germanium orbitals contribute to the bands around −10 eV and show negligi- ble DOS at higher energies. While LDA yields a metallicenergy spectrum due to the underestimation of electroniccorrelations in the Cu 3 dshell, LSDA +Urestores the insulating scenario with the band gap of E g=2.0±0.3e Vf o r Ud=6.5±1e V . The Cu dx2-y2states are represented by four bands crossing the Fermi level and arising from four Cu atoms in the primitivecell of Cu 2GeO 4(Fig. 3). These bands are separated from the rest of the valence bands by a pseudogap. To extracthopping parameters, we construct Wannier functions (WFs)based on the Cu d x2-y2character.31This analysis evidences sizable nearest-neighbor ( t1) and next-nearest-neighbor ( t2) hoppings along the structural chains. However, the hopping 0.0 X M Z0.4 0.4)Ve(ygrenE FIG. 3. (Color online) LDA band structure of Cu 2GeO 4(thin light lines) and the ﬁt of the TB model (thick dark lines). The Fermi level isat zero energy. The kpath is deﬁned as follows: /Gamma1(0,0,0),X(0.5,0,0), M(0.25,0.25,0), and Z(0,0,0.5), where the coordinates are given in units of the reciprocal lattice parameters 4 π/a and 4π/c. 104415-2SPIRAL GROUND STATE IN THE QUASI-TWO- ... PHYSICAL REVIEW B 83, 104415 (2011) TABLE I. Leading exchange couplings in Cu 2GeO 4: hopping parameters tiof the TB model, AFM contributions to the exchange couplings JAFM i=4t2 i/U eff, and the total exchange integrals Jifrom LSDA +Ucalculations with Ud=6.5e V . Cu-Cu distance ti JAFM i Ji (˚A) (meV) (K) (K) J1 2.80 118 144 −60 J2 5.59 82 70 80 J 5.59 115 137 130 Jab 6.25 −37 14 7 Jc 3.07 −11 1 −2 talong a/primeis comparable to t1andt2. Additionally, a weak diagonal hopping in the abplane is found (Table I). The nearest-neighbor hoppings perpendicular to the abplane ( tc) are−11 meV , yielding JAFM c as low as 1 K. The weak dispersion of the bands along /Gamma1-Zalso shows the pronounced two-dimensionality of the system. Introducing the hoppingsinto an effective one-band Hubbard model, we evaluate AFMparts of the exchange integrals J AFM i (Table I). LSDA +Ucalculations modify the LDA-based scenario. We ﬁnd FM nearest-neighbor coupling within the structuralchains, J 1=− 60∓10 K for Ud=6.5±1e V .T h en e x t - nearest-neighbor intrachain coupling J2=80∓20 K and the interchain coupling J=130∓30 K are basically unchanged. Further couplings in the abplane are below 10 K. The interplane coupling becomes FM and remains weak. Thus, weestablish the quasi-2D J-J 1-J2model with a weak interlayer coupling Jc(Fig. 1). The quasi-2D model of Cu 2GeO 4results from the strong tetragonal distortion of the spinel structure. The plaquettedescription (Fig. 1), with the magnetic d x2-y2orbital coplanar to the CuO 4plaquette, clariﬁes the 2D nature of the system. The couplings Jcconnect the plaquettes lying in different planes and therefore remain weak. By contrast, three sizablecouplings in the abplane establish a frustrated spin lattice. Our model is dissimilar to the anisotropic pyrochlore latticeproposed by Yamada et al. 23The pyrochlore spin lattice omits the relevant exchanges JandJ2, and should be discarded. Cu2GeO 4is a frustrated magnet indeed, but the strong frustration is found in the J1-J2chains rather than in the tetrahedral units. The FM nearest-neighbor coupling J1should be referred to the Cu-O-Cu angle of 91 .8◦. The microscopic origin of ferromagnetism is the Hund’s coupling on the oxygen site.8 The next-nearest-neighbor coupling J2is the AFM Cu-O-O- Cu superexchange. Similar values of 50–100 K for \|J1\|andJ2 have been established for the archetype frustrated-spin-chain compounds, such as LiCu 2O2and LiCuVO 4.8,10 Another remark on the structural implementation of the spin model regards the origin of the long-range couplings J andJ2. Since the Ge orbitals weakly contribute to the valence states, both couplings should be assigned to a Cu-O-O-Cusuperexchange. Despite an identical Cu-Cu distance (Table I), a larger Jvalue is caused by the coplanar arrangement of the plaquettes in the adjacent chains. By contrast, thenext-nearest-neighbor plaquettes within the chain ( J 2) lie in different planes due to the buckled chain geometry (Fig. 1).It is worth noting that the abprojections of the Cu 2GeO 4and LiCuVO 4structures are very similar. However, LiCuVO 4is a quasi-1D system with J/lessmuch\|J1\|,J2, while the spin system of Cu2GeO 4is quasi-2D.10 III. MICROSCOPIC MODEL In the following, we explore the ground state and ﬁnite- temperature properties of our model. We ﬁrst consider thepurely 2D regime described by the Hamiltonian H=/summationdisplay n/braceleftbigg/summationdisplay i[J1si,n·si+1,n+J2si,n·si+2,n]/bracerightbigg +/summationdisplay i/summationdisplay nJsi,n·si,n+1, (1) where the index nlabels the structural chains (along b/prime) and idenotes the lattice sites within a chain n. The effect of the interlayer coupling Jcis discussed in Sec. III D . Our model can be viewed as frustrated J1-J2chains (along b/prime) which are uniformly coupled by J(along a/prime). Alternatively, one ﬁnds uniform spin chains along a/primewith frustrated interchain couplings J1andJ2along b/prime. While any of the parent 1D models is rather easy to handle, a rigoroustreatment of their 2D combination is a challenging problem.Below, we apply the Lanczos diagonalization and the coupledcluster method to achieve an accurate description of the groundstate. By contrast, ﬁnite-temperature properties of the quantummodel can only be accessed at high temperatures by a seriesexpansion (HTSE), whereas conventional techniques, such asquantum Monte Carlo or exact diagonalization, fail because ofthe sign problem or ﬁnite-size effects. 32 A. Magnetic susceptibility Since the experimental information on Cu 2GeO 4is re- stricted to the magnetic susceptibility and heat capacity mea-surements in Ref. 23, we discuss thermodynamic properties ﬁrst. The experimental speciﬁc heat contains an unknownphonon contribution; hence the magnetic part cannot beseparated. Therefore, the magnetic susceptibility χ(T) (Fig. 4) remains the only quantity suitable for the comparison betweentheory and experiment. The estimated Curie-Weiss tempera-tureθ/similarequal 1 2(J+J1+J2)=75 K is in good agreement with the experimental value of θ=89 K.23 For a further comparison, we derive the HTSE for our model: χ=NAg2μ2 B 4kBT/parenleftbigg 1+J+J1+J2 2T+J2+J2 1+J2 2 4T2/parenrightbigg−1 ,(2) where NAis Avogadro’s number, μBis Bohr magneton, gis thegfactor, and we used the expressions from Ref. 33up to the third order in temperature. The calculated exchangecouplings (Table I) are in reasonable agreement with the experimental data down to 150 K (Fig. 4). The deviations at lower temperatures are likely related to the divergence ofthe HTSE at T/lessorequalslantJ. To improve the ﬁt at higher temperatures, a slight adjustment of the exchange couplings is required.However, the third-order HTSE contains two J-dependent 104415-3TSIRLIN, ZINKE, RICHTER, AND ROSNER PHYSICAL REVIEW B 83, 104415 (2011) 4816 12 0 200 400 600 800 Temperature (K)Experiment Uniform chain HTSE20 )uClo m/u me 01(4 FIG. 4. (Color online) Fit of the experimental magnetic suscep- tibility data with the uniform chain model (dashed line) and thecomparison to the HTSE of Eq. ( 2) (solid line). Experimental data are from Ref. 23. terms only; hence an unconstrained ﬁt of three exchange parameters is impossible. To access temperatures below 150 K, a simpliﬁcation of the model is required. Since Jexceeds \|J1\|andJ2, the spin lattice is, to a ﬁrst approximation, a set of uniform spin chains alonga /prime. The respective 1D model33ﬁts the experimental magnetic susceptibility down to 70 K with J=140 K and g=2.22. A similar ﬁt with J=135 K has been given in Ref. 23. However, Yamada et al.23erroneously assign the spin chains to the structural chains (in their notation, Jcorresponds to J1). Our DFT calculations show that the uniform spin chains runperpendicular to the structural chains, whereas J 1is FM. Such an intricate situation is not uncommon for low-dimensionalmagnets; see Refs. 34and35for similar examples. Below 70 K, the 1D uniform chain model overestimates the magnetic susceptibility of Cu 2GeO 4. This feature indicates an onset of 2D spin correlations. In contrast to the uniform spin-1 2chain having ﬁnite susceptibility at zero temperature, 2D and3D systems usually develop a long-range order with vanishingsusceptibility at low temperatures. The onset temperatureof 2D spin correlations is a rough measure of interchaincouplings. Indeed, the temperature of 70 K conforms toour estimates of J 1=− 60 K and J2=80 K, the couplings between the uniform spin chains. B. Coupled cluster method The coupled cluster method (CCM) and its application to frustrated spin systems have been previously reviewed inseveral articles; see, e.g., Refs. 21,36–47. Therefore, we give only a brief illustration of the main relevant features of themethod. For more general information on the methodologyof the CCM, see, e.g., Refs. 40and 48, and references therein. The CCM is a universal quantum many-body method. The starting point for a CCM calculation is the choice ofa normalized reference or model state \|/Phi1/angbracketright, together with a complete set of (mutually commuting) multiconﬁgurationalcreation operators {C + L}and the corresponding set of theirHermitian adjoints {CL}. The CCM parametrizations of the ket and bra ground states (GSs) are given by \|/Psi1/angbracketright=eS\|/Phi1/angbracketright,S =/summationdisplay I/negationslash=0SIC+ I; (3) /angbracketleft˜/Psi1\|=/angbracketleft/Phi1\|˜Se−S, ˜S=1+/summationdisplay I/negationslash=0˜SICI. Using /angbracketleft/Phi1\|C+ I=0=CI\|/Phi1/angbracketright∀I/negationslash=0,C+ 0≡1, the com- mutation rules [ C+ L,C+ K]=0=[CL,CK], the orthonor- mality condition /angbracketleft/Phi1\|CIC+ J\|/Phi1/angbracketright=δIJ, and completeness/summationtext IC+ I\|/Phi1/angbracketright/angbracketleft/Phi1\|CI=1=\|/Phi1/angbracketright/angbracketleft/Phi1\|+/summationtext I/negationslash=0C+ I\|/Phi1/angbracketright/angbracketleft/Phi1\|CI,w e get a set of nonlinear and linear equations for the correlationcoefﬁcients S Iand ˜SI, respectively. We choose a reference state corresponding to the classical state of the spin model,i.e., a noncollinear reference state with up-down N ´eel-type correlations along the a /primedirection (uniform Jchains) and with spiral correlations along the b/primedirection (frustrated J1-J2 chains). The spiral correlations are characterized by a pitch angleγ, i.e.,\|/Phi1/angbracketright=\|/Phi1(γ)/angbracketright. In the quantum model, the pitch angle is typically different from the corresponding classical value γcl. Hence, we do not choose the classical result for the pitch angle and rather consider γas a free parameter in the CCM calculation. The value of γhas to be determined by the minimization of the GS energy (in a certain CCM approx-imation, see below) given by E(γ)=/angbracketleft/Phi1(γ)\|e −SHeS\|/Phi1(γ)/angbracketright, i.e., from the dE/dγ \|γ=γqu=0 condition. In order to ﬁnd an appropriate set of creation operators, it is convenient to perform a rotation of the local axes on each ofthe spins so that all spins in the reference state align with thenegative zdirection. This rotation by an appropriate local angle δ i,n=δi,n(γ) of the spin on the lattice site ( i,n) is equivalent to the spin-operator transformation sx i,n=cosδi,nˆsx i,n+sinδi,nˆsz i,n;sy i,n=ˆsy i,n sz i,n=− sinδi,nˆsx i,n+cosδi,nˆsz i,n/bracerightbigg . (4) The reference state and the corresponding creation operators C+ Lare given by \|ˆ/Phi1/angbracketright = \| ↓↓↓↓ ···/angbracketright ;C+ L=ˆs+ i,n,ˆs+ i,nˆs+ j,m,ˆs+ i,nˆs+ j,mˆs+ k,l,..., (5) where the indices ( i,n),(j,m),(k,l),... denote arbitrary lattice sites. This speciﬁed form of the creation operators C+ Land the corresponding reference state \|ˆ/Phi1/angbracketrightimmediately make clear that the general relations listed below Eq. ( 3) are fulﬁlled. In the rotated coordinate frame, the Heisenberg Hamiltonianacquires a dependence on the pitch angle γ(see Ref. 21for more details). The order parameter (sublattice magnetization) in the ro- tated coordinate frame is given by m=− 1/N/summationtext N i,n/angbracketleft˜/Psi1\|sz i,n\|/Psi1/angbracketright. The only approximation of the CCM is the truncation ofthe expansion of the correlation operators Sand ˜S.W eu s e the well-established LSUB nscheme, where all multispin correlations on the lattice with nor fewer contiguous sites are taken into account. In contrast to Ref. 21, which is focused on the J/lessorequalslant\|J 1\|,J2regime, we consider the case of J>\|J1\|,J2.W e also evaluate LSUB napproximations of higher order, up to n=8. In the highest order of approximation, LSUB8, we have 21 124 conﬁgurations, i.e., 21 124 coupled nonlinear 104415-4SPIRAL GROUND STATE IN THE QUASI-TWO- ... PHYSICAL REVIEW B 83, 104415 (2011) equations have to be solved numerically. Moreover, the minimum of E(γ) has to be found numerically to determine the quantum pitch angle γqu. For the numerical calculations, we use the program package CCCM by D. J. J. Farnell and J. Schulenburg.49 Since the LSUB nbecomes exact for n→∞ , the numerical result can be improved by extrapolating the “raw” LSUB ndata ton→∞ using the expression mn=m∞+a/n+b/n2(cf. Refs. 38,40,45, and 48). C. Ground state To ﬁnd the classical ground state of the model given by Eq. ( 1), we write the magnetic energy per lattice site for an arbitrary 2D propagation vector k=(kx,ky): E=1 2[Jcoskx+J1cosky+J2cos(2ky)], (6) where the unit cell of the spin lattice is used.50The energy minimum is found at k=[π,arccos( −J1 4J2)], which corre- sponds to the AFM order along a/primeand the spiral order along b/prime. The classical pitch angle is γ=arccos( −J1 4J2)=79.19◦ and does not depend on J; hence the a/primeandb/primedirections of the spin lattice are fully decoupled. The ordering along a/primeis controlled by J, whereas the ordering along b/primeis controlled by the competing couplings J1andJ2. To check the validity of this result for the quantum case, we use the CCM methodand Lanczos diagonalization. In CCM, we reduce our exchange couplings (Table I)t o J 1=− 1 and J2=4 3and vary J(in Cu 2GeO 4,J=13 6). The CCM results for γandmas a function of Jare shown in Fig. 5. In contrast to the classical pitch angle γcl, the pitch angle of the quantum system ( γqu) slightly depends on J.I nC u 2GeO 4,w e ﬁndγqu=83.9◦, which is about 6% larger than the classical angle, but about 5% smaller than the quantum pitch anglefor the isolated chain, i.e., at J=0. The coupling Jaffects the dimensionality of the system and has a stronger effecton the sublattice magnetization (see Fig. 5). The extrapolated value m ∞has a maximum at J/similarequal− 1.17J1. The calculated 0.45classical LSUB4 Sublat. mag. ( ) in mBSpin-spin correlation SS<>0R LSUB6 LSUB8 m8quantumPitch angle ( / ) 0.4 2 3 Interchain coupling / JJ1 y4 5 1 012340.60.80.420.48 0.40.0 R( 1 ,= y)R (0, )= y0.40.8 FIG. 5. (Color online) Left panel: The pitch angle ( γ) and the sublattice magnetization ( m) calculated by the CCM for J2=4 3\|J1\|. The shaded bar shows the coupling regime of Cu 2GeO 4. Note that the quantum pitch angles γqufor the LSUB napproximations with n=4, 6, and 8 almost coincide. Therefore, the shown LSUB8 curve represents effectively the limit n→∞ . Right panel: Spin-spin correlation /angbracketleftS0SR/angbracketright,R=(x,y) along the J1-J2chains ( b/primeaxis) for a ﬁnite system of N=32=4×8 sites at J2=4 3\|J1\|andJ=13 6\|J1\|.exchange couplings in Cu 2GeO 4lead to m∞∼0.310 (i.e., 0.62 μB). The CCM results are conﬁrmed by the Lanczos diagonaliza- tion data for the spin-spin correlation functions /angbracketleftS0SR/angbracketrightshown in the right panel of Fig. 5. We use a ﬁnite lattice comprising four 8-spin J1-J2chains coupled by J. The correlations between nearest neighbors within the frustrated J1-J2chains [R=(0,1)] are close to zero, whereas the second-neighbor correlations [ R=(0,2)] are AFM. This conforms to the spiral ordering with the pitch angle close to 90◦(neighboring spins are nearly orthogonal). The correlations between the structuralchains [at R=(1,y)] follow the intrachain correlations, yet showing the opposite sign. Thus, the ordering along a /prime is AFM. D. Long-range order The 2D model given by Eq. ( 1) is ordered at zero temperature only. To account for the actual long-range order inCu 2GeO 4below TN=33 K, the interlayer coupling Jcshould be considered. The FM coupling Jcis compatible with J1, yet competing with JandJ2. Assuming a similar ground state with the 2D propagation vector, we ﬁnd that Jcmodiﬁes the energy in Eq. ( 6)b y /Delta1E 3D=Jc 2coskx 4cosky 2. (7) Using DFT estimates of individual exchange couplings (Ta- bleI), we arrive at the classical pitch angle modiﬁed by 0.15%: 79 .07◦vs 79.19◦for the purely 2D model. The classical energy per lattice site is reduced by 0.5% (about0.6 K). This simpliﬁed analysis shows that the interlayercoupling J cis capable of stabilizing the 3D order in Cu 2GeO 4. However, the classical model does not reﬂect all the featuresof the real quantum model that, unfortunately, remains un-feasible for an accurate numerical study. In particular, theordering temperature T Ncannot be determined with sufﬁcient accuracy. IV . DISCUSSION AND SUMMARY Although not obvious at ﬁrst glance, the microscopic magnetic model of Cu 2GeO 4can be deduced from simple qualitative arguments. While the naive geometrical analysisof the crystal structure suggests a 3D pyrochlore-latticemagnetism, a closer look at the crystal structure identiﬁes2D features. In most of the Cu 2+oxides, electronic structure and magnetism are controlled by the arrangement of CuO 4 plaquettes, which are the basic structural entities. Chainsof edge-sharing plaquettes give rise to frustrated J 1-J2spin chains,9–12yet the coplanar arrangement of the plaquettes in the neighboring structural chains induces a strong AFMcoupling along a /prime. Overall, we ﬁnd magnetic layers in the abplane, along with a weak FM interlayer coupling Jc.B y combining the frustration along b/primewith the strong unfrustrated exchange along a/prime,C u 2GeO 4expands the family of cuprates featuring frustrated spin chains. The dearth of the experimental data and the complexity of the 2D frustrated J-J1-J2lattice restrict the opportunities for an experimental veriﬁcation of our microscopic model. 104415-5TSIRLIN, ZINKE, RICHTER, AND ROSNER PHYSICAL REVIEW B 83, 104415 (2011) Nevertheless, the tangible success of DFT in unravel- ing complex spin lattices for a range of transition-metalcompounds 27,35,51,52is a solid justiﬁcation of our results. The 2D nature of the system and the frustrated couplingsalong b /primeare conﬁrmed by qualitative arguments and by a reference to similar Cu2+compounds (see Sec. II). Further on, numerical estimates of individual exchanges conform to theexperimental magnetic susceptibility (Sec. III A ). An ultimate test of the proposed model requires a study of the ground stateby neutron or resonant x-ray scattering. Presently, we notethat our model does predict the long-range magnetic order,in contrast to the strongly anisotropic pyrochlore lattice thatmight have a quantum-disordered valence-bond-solid 24or a gapless spin-liquid53ground state. The experimental data for Cu 2GeO 4and accurate theoret- ical results for the ground state disclose the basic features ofour model. At high temperatures, thermodynamic propertiesare guided by the uniform spin chains along a /prime. The apparent spin-chain behavior is likely related to the partial cancelationof FM J 1and AFM J2in the second-order term for the susceptibility [see Eq. ( 2)]. A further evidence is the perfect ﬁt of the experimental magnetic susceptibility down to 70 K(T/J/similarequal0.5). The frustrated couplings along b /primecome into play at lower temperatures and essentially determine the groundstate. The leading exchange Jdrives AFM ordering along a /prime. The ordering along b/primehas to satisfy the frustrated couplings J1andJ2and is, therefore, a spiral, similar to a single J1-J2frustrated spin chain. The interlayer coupling Jcshould stabilize the long-range order up to TNwithout changing the basic features of the ground state: the collinear AFM orderalonga /primeand the spiral order along b/prime. The highly accurate CCM approach provides reliable information on the ground state of the 2D system. We ﬁnd apitch angle of γ/similarequal84 ◦and a sublattice magnetization close to 0.62 μB.B o t h γandmare renormalized with respect to the classical values and suggest strong quantum ﬂuctuations in thesystem. Enhanced quantum ﬂuctuations should be ascribedto the reduced dimensionality and frustration. The magneticordering temperature T N/J/similarequal0.25 is also suggestive of strong quantum ﬂuctuations. For example, a quasi-2D system ofsquare lattices with a weak interlayer coupling J ⊥/J=0.01 (compare to \|Jc\|/J/similarequal0.01) orders at a higher temperature of TN/J/similarequal0.33 (Ref. 54). The quantum effects in the system could be further probed by an experimental study of the groundstate. An interesting feature of the spiral magnetic order is the possible emergence of electric polarization strongly coupledto the magnetism. 6The direction of the electric polarization depends on the twisting direction of the spiral. In Cu 2GeO 4, the AFM coupling Jleads to opposite twisting directions in the neighboring spirals; hence the polarization is canceled.However, the proposed antiferroelectricity of Cu 2GeO 4does not preclude the strong magnetoelectric coupling and shouldstimulate further experimental investigation of the compound.We also note that the family of 2D frustrated materialsrepresenting the J-J 1-J2model can be further expanded by CuNCN lying in the limit of J/greatermuch\|J1\|,J2.13 In summary, we have shown that the electronic structure of Cu2GeO 4contradicts the previous, empirical-based spin model of the anisotropic pyrochlore lattice. The comprehensivecomputational study discloses the quasi-2D nature of thiscompound and suggests an original 2D spin model comprisingfrustrated and uniform spin chains along the two dimensions.Theoretical results for this model show the robust nature of thespiral ground state that is subject to strong quantum effects,evidenced by the reduced sublattice magnetization of 0.62 μ B and the renormalized pitch angle of about 84◦. ACKNOWLEDGMENTS We are grateful to Oleg Janson and Deepa Kasinathan for fruitful discussions and careful reading of the manuscript. A. T.acknowledges ﬁnancial support from the Alexander vonHumboldt Foundation. J. R. appreciates the funding by theDFG (Project RI 615/16-1). altsirlin@gmail.com †Helge.Rosner@cpfs.mpg.de 1H. T. Diep, eds., Frustrated Spin Systems (World Scientiﬁc, Singapore, 2004). 2U. Schollw ¨ock, J. Richter, D. J. J. Farnell, and R. F. Bishop, eds., Quantum Magnetism (Springer, New York, 2004). 3A. V . Sologubenko, T. Lorenz, H. R. Ott, and A. 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PhysRevB.77.035205.pdf	Nature of polarization in wide-bandgap semiconductor detectors under high-ﬂux irradiation: Application to semi-insulating Cd 1−xZnxTe Derek S. Bale and Csaba Szeles eV PRODUCTS, Compound Semiconductor Group II-VI Inc., Saxonburg, Pennsylvania 16056, USA /H20849Received 7 September 2007; published 14 January 2008 /H20850 In this paper, we theoretically investigate the mechanism of polarization in wide-bandgap semiconductor radiation detectors under high-ﬂux x-ray irradiation. Our general mathematical model of the defect structurewithin the bandgap is a system of balance laws based on carrier transport and defect transition rates, coupledtogether with the Poisson equation for the electric potential. The dynamical system is self-consistently evolvedin time using a high-resolution wave propagation numerical algorithm. Through simulation, we identify andpresent a sequence of dynamics that determines a critical ﬂux of photons above which polarization effectsdominate. Using the experience gained through numerical simulation of the full set of equations, we derive areduced system of conservation laws that describe the dominant dynamics. A multiple scale perturbationanalysis of the reduced system is shown to yield an analytical dependence of the maximum sustainable ﬂux onkey material, detector, and operating parameters. The predicted dependencies are validated for 16 /H1100316 pixel CdZnTe monolithic detector arrays subjected to a high-ﬂux 120 kVp x-ray source. DOI: 10.1103/PhysRevB.77.035205 PACS number /H20849s/H20850: 07.85.Fv, 71.55.Gs I. INTRODUCTION There is growing interest in the potentials of pulse mode CdZnTe detector technology for high-ﬂux high-speed energyselective or hyperspectral x-ray imaging. The energy sensi-tivity provided by CdZnTe opens up a range of intriguingnew potential applications for this detector technology inmedical, industrial, and security imaging and tomography.However, imaging applications typically require photon ﬂuxﬁelds that generate very high count rates within the semi-insulating CdZnTe crystal. In particular, medical computertomography applications represent a large potential marketfor this technology but require detectors capable of handlingcount rates of /H2084920–200 /H20850/H1100310 6counts /mm2s1. One of the greatest challenges in applying pulse mode CdZnTe detector technology to applications requiring suchhigh count rates is avoiding buildup of charge within thecrystal, which collapses the electric ﬁeld and results in cata-strophic device failure /H20849i.e.,polarization /H20850. 1,2Therefore, one must design these devices such that the charge generated bythe x-ray radiation is removed from the device at a sufﬁ-ciently high rate through both drift and recombination. Thechoice of material, detector, and operating parameters isparamount to achieving high charge throughput for devicesbased on CdZnTe or any semiconducting detector material.Clearly, a careful choice of both material and detector designparameters, as well as operating conditions, must be predi-cated on a fundamental understanding of the dependence ofthe onset of polarization on such critical parameters and con-ditions. In Ref. 3,D u et al. modeled the temporal response of CdZnTe under intense irradiation by considering a bandgapwith a single acceptor and donor level. They applied a ﬁniteelement numerical algorithm to the resulting equations anddemonstrated distortions in the electric ﬁeld and carrier con-centrations under high-ﬂux irradiation. In this paper, we de-velop a general mathematical model of the bandgap that in-cludes an arbitrary number of donor and acceptor defects.Our model is a system of nonlinear balance laws based on carrier transport and defect transition rates, coupled togetherwith the Poisson equation for the electric potential. The re-sulting dynamical system is numerically evolved in time us-ing ﬂux-conservative wave propagation algorithms devel-oped for conservation laws with spatially varying ﬂuxfunctions. 4–6Our high-resolution numerical algorithm pre- serves carrier number densities while suppressing numericalinstabilities often caused by large gradients that develop inthe solutions under the extreme conditions encountered insimulations of high-ﬂux x-ray applications. In Sec. II, wepresent the details of both our mathematical model of thebandgap and its numerical solution. Through simulation, we have identiﬁed a sequence of dy- namics that determines a critical ﬂux of photons above whichpolarization effects dominate, resulting in catastrophic de-vice failure. In Sec. III, we illustrate this sequence of eventsthrough a speciﬁc set of example simulations. Using experi-ence gained through a large matrix of such simulations, wehave derived a reduced system of conservation laws that de-scribe the dominant dynamics relevant for a polarizing de-tector. In Sec. IV, we use a multiple scale perturbation analy-sis, for which details can be found in Ref. 7, on the reduced system of equations to derive an analytic expression describ-ing the dependence of the maximum sustainable ﬂux on keymaterial, operating, and detector design parameters. The pre-dicted dependencies on bias voltage and temperature are ex-perimentally validated for 16 /H1100316 pixel CdZnTe monolithic detector arrays in Secs. IV C 1 and IV C 2, respectively. II. MODEL OF THE BANDGAP DEFECT STRUCTURE In this work, we consider a semi-insulating, wide- bandgap semiconductor with MAdistinct acceptor defect lev- els within the bandgap and MDdonor levels. The presence of the energy levels is assumed due to electronic point defectswithin the crystal. Many such defect levels have been mea-PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 1098-0121/2008/77 /H208493/H20850/035205 /H2084916/H20850 ©2008 The American Physical Society 035205-1sured within CdTe and CdZnTe.8Therefore, the number of energy levels associated with acceptor defect levels, MA,a s well as the number of donor levels, MD, are assumed arbi- trary. Each acceptor has total concentration Pland an energy EAlwithin the bandgap, where l/H33528/H208531,2,..., MA/H20854is used to enumerate individual levels. Similarly, donor levels have concentrations denoted by Nkand energy levels by EDk, with k/H33528/H208531,2,..., MD/H20854. The top edge of the valence band is de- ﬁned to be the zero of energy so that the bottom edge of the conduction band is simply the bandgap energy, Ec=Eg/H20849e.g., Eg/H110111.54 eV at room temperature in CdZnTe /H20850. Further, we assume that the semiconductor is nondegenerate so that theFermi energy E Fis far from the band edges relative to the thermal energy kT. A. Carrier transition rates The concentration of free holes in the valence band is denoted by P, while the concentration of trapped holes in the lth acceptor defect is denoted by Pˆl. The free electron con- centration is denoted by Nand that of trapped electrons in thekth donor level is denoted by Nˆk. Equilibrium concentra- tions for the free carriers, denoted by P0andN0, as well as the equilibrium concentrations of trapped carriers, Pˆ 0landNˆ 0k, are governed, as usual, by the position of the Fermi energyunder the charge neutrality condition. 8In what follows, we use these equilibrium values to scale the dynamical systemfor the purpose of numerical, as well as analytical, solution. The attractive potential of ionized traps /H20849i.e., unoccupied donors and occupied acceptors /H20850gives rise to trapping of free electrons onto donor sites and free holes to acceptor sites. Wealso consider detrapping in which a trapped carrier is ther-mally excited back to the energy band. The rates at whichelectrons are trapped and detrapped are given by /H9011 ↓:Dk=/H9268Dk/H9258Nk/H208731−Nˆk Nk/H20874, /H208491a/H20850 /H9011↑:Dk=/H9263Dkexp/H20873−Eg−EDk kT/H20874, /H208491b/H20850 respectively, where /H9268Dkis the capture cross section of the kth defect level, /H9258is the thermal velocity of the carriers, and /H9263Dk is an escape frequency independent of both free and trapped carrier concentrations. It is typical to choose the escape fre- quency /H9263Dkso that the detailed balance of thermal equilibrium is satisﬁed. In general, however, we leave the escape fre-quencies as free parameters due to the fact that the assump-tion of the principle of detailed balance may not be valid forcarrier dynamics driven far from equilibrium. The rates atwhich holes are trapped and detrapped from the lth acceptor are denoted by /H9011 ↓:Al, and/H9011↑:Al, respectively. Their functional forms are similar to Eqs. /H208491a/H20850and /H208491b/H20850, with the electron parameters and concentrations replaced by analogous holequantities. Detrapping at a rate given by Eq. /H208491b/H20850is not the only transition we consider for a trapped carrier. For example, atrapped electron may undergo recombination with a free holein the valence band. The recombination rate is taken to de- pend on the concentration of trapped electrons, while its in-verse transition, consisting of an electron excited from thevalence band to the kth donor level, is taken to be driven by temperature. The rate of electron recombination and its in-verse process are given by /H9020 ↓:Dk=/H9273Dk/H9258Nˆk, /H208492a/H20850 /H9020↑:Dk=/H9262Dkexp/H20873−EDk kT/H20874, /H208492b/H20850 where /H9273Dkis the recombination cross section, and /H9262Dk, like/H9263Dk, is an escape frequency and is assumed to have no depen-dence on the concentrations of free or trapped carriers. Be-cause an occupied donor is neutral, there is no strong Cou-lomb potential driving recombination as there is for electron trapping. Consequently, the recombination cross section, /H9273Dk is assumed to be small compared to the trapping cross sec- tion,/H9273Dk/H11270/H9268Dk. The recombination transition for trapped holes is denoted by /H9020↓:Al, and the rate of its inverse transition is given by /H9020↑:Al. Like the hole rates for trapping and detrap- ping, these rates have functional forms similar to the electronrates in Eqs. /H208492a/H20850and /H208492b/H20850, with the electron parameters and concentrations replaced by analogous hole quantities. It isalso true that the recombination of trapped holes is a weaker process than hole trapping, so we take /H9273Al/H11270/H9268Al. We also point out that, like /H9262Dk,/H9262Alis the escape frequency from the lth acceptor level and can be chosen using detailed balance inthermal equilibrium or a steady state of the dynamical sys-tem. Finally, semi-insulating CdZnTe near equilibrium typi- cally has free carrier concentrations /H1102110 6/cm3,9so direct band-to-band recombination rates are quite low. In steadystates generated by high-ﬂux x-ray sources, however, theconcentrations of free carriers can be many orders of magni-tude larger. In this case, the probability that a free hole willdirectly recombine with a free electron is dramatically in-creased. Direct band-to-band transitions, therefore, are takenas a sink for both free electrons and holes in the form S bb=/H9268bb/H9258/H20849NP−N0P0/H20850, /H208493/H20850 where /H9268bbis the cross section for this transition, and N0and P0are the free electron and hole concentrations in equilib- rium, respectively. B. Dynamical equations Now that we have introduced the transitions that we con- sider in our model, we can write down the system of partialdifferential equations that govern the dynamical response ofa parallel plate detector placed under bias and subjected to anx-ray ﬂux. The bias voltage is considered to be high enoughthat diffusion currents can be neglected relative to drift cur-rents. Under these conditions, the set of dependent variables /H20853N,P,Nˆ k,Pˆl,/H9021/H20854depends on one space dimension Zand time T. The balance laws that include carrier transport, trapping, detrapping, and recombination areDEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-2/H11509 /H11509TN+/H11509 /H11509Z/H20851/H9262eEN /H20852=/H20858 k=1MD /H20851/H9011↑:DkNˆk−/H9011↓:DkN/H20852 +/H20858 l=1MA /H20851/H9020↑:Al/H20849Pl−Pˆl/H20850−/H9020↓:AlN/H20852−Sbb+/H9023, /H208494a/H20850 /H11509 /H11509TP−/H11509 /H11509Z/H20851/H9262hEP /H20852=/H20858 k=1MD /H20851/H9020↑:Dk/H20849Nk−Nˆk/H20850−/H9020↓:DkP/H20852 +/H20858 l=1MA /H20851/H9011↑:AlPˆl−/H9011↓:AlP/H20852−Sbb+/H9023, /H208494b/H20850 /H11509 /H11509TNˆk=/H9011↓:DkN−/H9011↑:DkNˆk+/H9020↑:Dk/H20849Nk−Nˆk/H20850−/H9020↓:DkP,/H208494c/H20850 /H11509 /H11509TPˆl=/H9011↓:AlP−/H9011↑:AlPˆl+/H9020↑:Al/H20849Pl−Pˆl/H20850−/H9020↓:AlN, /H208494d/H20850 where we have introduced the free carrier generation source /H9023/H20849Z,T/H20850and the electric ﬁeld E/H20849Z,T/H20850=−/H11509/H9021//H11509Z. We point out that the free carrier concentrations, namely, Nand P, are affected by transitions into and out of all donor and acceptordefect levels through trapping and recombination processespreviously discussed. This inﬂuence is captured by the sum-mation over all defect levels on the right hand side of Eqs. /H208494a/H20850and /H208494b/H20850. The concentrations of trapped carriers Nˆ kand Pˆl, on the other hand, are only affected by transitions to and from the energy bands, as indicated by the source terms ofEqs. /H208494c/H20850and /H208494d/H20850. Since the free carrier ﬂux depends on the electric ﬁeld, Eqs. /H208494a/H20850–/H208494d/H20850must be coupled to the Poisson equation for the electric potential, /H11509 /H11509Z/H20875/H92800/H11509 /H11509Z/H9021/H20876=q/H20877P−/H20858 l=1MA /H20849Pl−Pˆl/H20850−N+/H20858 k=1MD /H20849Nk−Nˆk/H20850/H20878. /H208494e/H20850 Note that this system of partial differential equations is non- linear in both the source term and the transport ﬂuxes. C. Numerical solution There are two major challenges in developing numerical solutions to Eqs. /H208494a/H20850–/H208494e/H20850when /H9023is an intense source of free carriers due to x rays. The ﬁrst challenge stems from thefact that the evolution of this system has vastly different timescales associated with both the charge transport, and thecharge generation, trapping, and recombination processes.For example, measurements of the electron mobility inCdZnTe typically yield values around 1000 cm 2V−1s−1.8,10 Applied ﬁelds of several thousand V/cm are common and produce transport time scales in the hundred nanosecondrange /H20849/H1101110 −7s/H20850for 5−10 mm thick detectors. On the otherhand, applying the transition rates in Eqs. /H208491a/H20850and /H208491b/H20850with energy levels and other defect parameters consistent with thecomposition-dependent charge transport properties ofCdZnTe /H20849Refs. 8and9/H20850results in shallow donors and accep- tors that can trap and detrap on a tens of picosecond scale/H20849/H1101110 −11s/H20850and deep defect levels that can detrap on a time scale of seconds /H20849/H11011100s/H20850. Therefore, Eqs. /H208494a/H20850–/H208494e/H20850contain potentially 12 orders of magnitude separating the dynamical time scales for these processes, making the system very stiffnumerically. A second challenge originates from the fact that equilib- rium concentrations of free carriers in CdZnTe are typically/H2084910 4−106/H20850cm3.9Therefore, assuming a pair creation energy of /H110114.5–5 eV,11a single 100 keV photon can more than double the concentration of free carriers within a small re-gion of the interaction. Further, in Sec. III, we show thathigh-ﬂux x-ray sources considered here produce interactionsclose enough in time to increase the local concentration offree carriers not by a factor of 2 but by orders of magnitude.As a consequence, carrier concentrations can have large spa-tial gradients associated with their transport that are wellknown to cause numerical instabilities. 5,6In the remainder of this section, therefore, we discuss our choice of scaling andnumerical implementation that enables the accurate andstable integration of Eqs. /H208494a/H20850–/H208494e/H20850under conditions of in- tense x-ray irradiation. 1. Scaling the equations Appropriate scaling of independent, as well as dependent variables, is an important part of the numerical solution ofany system, but due to the very large range of relevant timescales, it is crucial in solving Eqs. /H208494a/H20850–/H208494e/H20850. To that end, we deﬁne nondimensional free carrier densities pandnby scal- ing the physical concentrations PandNby their equilibrium values P 0andN0, so that P=P0p,N=N0n. /H208495a/H20850 Nondimensional trapped carrier concentrations pˆlandnˆkare deﬁned by scaling their dimensional counterparts PˆlandNˆk by the relevant total defect concentration, Pˆl=Plpˆl,Nˆk=Nknˆk. /H208495b/H20850 In addition to scaling the dependent ﬁeld variables, we must choose an appropriate space and time scale that deﬁnes non-dimensional independent variables for space zand time tas Z=Lz,T= /H9270t. /H208496/H20850 In the numerical solutions presented in this paper, we take L to be the detector thickness and the electron transit time asour time scale so that /H9270=/H9270tr=L2//H20849/H9262eV/H20850. Substituting these nondimensional variables into Eqs. /H208494a/H20850–/H208494e/H20850and deﬁning a vector of dependent state variables q/H20849z,t/H20850=/H20851n,p,nˆk,pˆl/H20852T, we can rewrite Eqs. /H208494a/H20850–/H208494d/H20850in vector form so that the dynamical system /H208494a/H20850–/H208494e/H20850can be written as /H11509tq/H20849z,t/H20850+/H11509zf/H20849q,z/H20850=s/arrowdblbothv/H20849q,t/H20850, /H208497a/H20850NATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-3/H11509z2/H9278=/H9253/H20875/H9251/H20873p−/H20858 l=1MA /H9257Al/H208491−pˆl/H20850/H20874−/H20873n−/H20858 k=1MD /H9257Dk/H208491−nˆk/H20850/H20874/H20876. /H208497b/H20850 In the above equations, we have introduced the ﬂux vector f/H20849q,z/H20850and the source vector s/arrowdblbothv/H20849q,t/H20850and deﬁned the nondi- mensional parameters /H9251=P0 N0,/H9257Dk=Nk N0,/H9257Al=Pl P0,/H9253=qL2N0 /H92800V. /H208498/H20850 Note that the dynamical equations for the physical free car- rier concentrations NandP, namely Eqs. /H208494a/H20850and /H208494b/H20850, arethe only equations that contain a transport ﬂux /H20849i.e., spatial derivative /H11509//H11509Z/H20850. Therefore, the ﬂux deﬁned in Eq. /H208497a/H20850has the nondimensional vector form f/H20849q,z/H20850=/H20851e/H20849z/H20850n,−/H9262e/H20849z/H20850p,0, ... ,0 /H20852T, /H208499/H20850 where e/H20849z/H20850=L VE/H20849Z/H20850is the nondimensional electric ﬁeld, and /H9262=/H9262h//H9262e. Finally, in Eq. /H208497a/H20850, we have introduced the band- gap transition source, s/arrowdblbothv/H20849q,t/H20850, which represents a scaled ver- sion of the right hand side of Eqs. /H208494a/H20850–/H208494d/H20850and has the form s/arrowdblbothv=/H9270/H20898/H20858 k=1MD /H20849/H9257Dk/H9011↑:Dknˆk−/H9011↓:Dkn/H20850+/H20858 l=1MA /H20849/H9251/H9257Al/H9020↑:Al/H208491−pˆl/H20850−/H9020↓:Aln/H20850−Sbb+/H9023 /H20858 k=1MD/H20873/H9257Dk /H9251/H9020↑:Dk/H208491−nˆk/H20850−/H9020↓:Dkp/H20874+/H20858 l=1MA /H20849/H9257Al/H9011↑:Alpˆl−/H9011↓:Alp/H20850−Sbb+/H9023 /H208731 /H9257Dk/H20874/H9011↓:Dkn−/H9011˜ ↑:Dknˆk+/H9020↑:Dk/H208491−nˆk/H20850−/H9251 /H9257Dk/H9020↓:Dkp /H208731 /H9257Al/H20874/H9011↓:Alp−/H9011↑:Alpˆl+/H9020↑:Al/H208491−pˆl/H20850−/H208731 /H9251/H9257Al/H20874/H9020↓:Aln/H20899. /H2084910/H20850 System /H208497a/H20850and /H208497b/H20850is in a standard mathematical form for a balance law, and it is this scaled form that we numericallysolve. 2. Numerical implementation We implement a split-scheme approach so that we can apply the appropriate numerical algorithm to both challengespreviously mentioned. The basic split-scheme solution pro-cess over a single time step /H9004tis as follows: /H208491/H20850transport /H20851solve /H11509tq+/H11509zf/H20849q,z/H20850=0 over a half time step /H9004t/2/H20852; /H208492/H20850source /H20851solve /H11509tq=s/arrowdblbothv/H20849q,t/H20850over a full time step /H9004t/H20852; /H208493/H20850transport /H20851solve /H11509tq+/H11509zf/H20849q,z/H20850=0 over a another half time step /H9004t/2/H20852; and /H208494/H20850ﬁeld /H20851solve Poisson Eq. /H208497b/H20850with updated concentra- tions /H20852. The ﬁrst three steps of this process evolve Eq. /H208497a/H20850for- ward in time by a single time step /H9004t. The transport steps neglect the source term and move carrier concentrationsalong characteristics, while the source step neglects the spa-tial derivatives and modiﬁes the concentrations according tothe source term s /arrowdblbothvdeﬁned in Eq. /H2084910/H20850. The fourth step of the solution process assures that the electric ﬁeld is self-consistently evolved in time with the carrier concentrations. Previously, we discussed the fact that in problems involv- ing a high-ﬂux x-ray source, carrier concentrations in Eqs./H208497a/H20850and /H208497b/H20850will develop large gradients during transport.The application of a simple ﬁnite difference numerical scheme to such gradients often introduces spurious oscilla-tions that can, in turn, trigger numerical instabilities. There-fore, during the transport steps of the solution process, wehave implemented a high-resolution wave propagation algo-rithm developed for conservation laws with spatially varyingﬂux functions. 4,5This algorithm is based on the original wave propagation technique developed by LeVeque,6,12and designed to limit spurious numerical oscillations near largegradients while maintaining high accuracy elsewhere. Thealgorithm is ﬂux conservative so that carrier number densi-ties are conserved to machine precision during the transportsteps of the evolution process. Conservation of carriers is adesired numerical property due to the fact that, neglectingthe generation source and losses at boundaries, Eqs. /H208497a/H20850and /H208497b/H20850conserve the total number densities of both electrons and holes up to that lost due to recombination transitions.Finally, a standard Rosenbrock method designed for stiff sys-tems is used during the source step of the solution process. III. SIMULATION RESULTS A large matrix of simulations has been completed in which we varied the number of donors MD, the number of acceptors MA, as well as the individual defect level param- eters such as total concentrations PlandNk, ionization en- ergies EAlandEDk, and trapping cross sections /H9268Aland/H9268Dk.I nDEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-4this matrix, we also varied the operating conditions and de- tector design parameters such as bias voltage V, temperature T, and detector thickness L. The source of photons was taken to be that of an x-ray tube with a differential energy spectrummodeled by 13 1 A/H9004tdn/H9253 de=qFAI 2/H9266r2/H20873Te e−1/H20874exp/H20873−/H20858 i/H9262i/H20849e/H20850/H9254i/H20874, /H2084911/H20850 where Iis the tube current, FAis a constant representing the bremsstrahlung production per incident electron energy, Te =qVtis the incident electron energy proportional to the tube voltage Vt, and eis the energy of the emanating photon. The exponential term describes the attenuation caused by the ith ﬁlter layer with thickness /H9254iand attenuation coefﬁcient /H9262i/H20849e/H20850. In all simulations presented here, the tube was turned on at time t=0, and the voltage and current were held constant until the transient dynamics relaxed to a steady state. Theresulting steady state proﬁles of the electric ﬁeld, carrier con-centrations, carrier lifetimes, and charge induction maps sug-gest a clear sequence of dynamics that serves to unveil thedominant processes that lead to polarization in CdZnTe de-tectors subjected to intense x-ray irradiation. A. Sequence of events in a polarizing detector In order to demonstrate the observed sequence, we present the results for a subset of four simulations for a3 mm thick detector, biased at 300 V. The voltage of thex-ray tube was set at 120 kVp, and the x-ray tube current,though held constant in each simulation, was increased from1040 to 1280 /H9262A in increments of 80 /H9262A for each succes- sive simulation. The bandgap structure used in all four simu-lations includes a shallow donor, a shallow acceptor, a deepdonor, and an intermediate acceptor. The speciﬁc parametervalues for each level are listed in Table I. Our choice of the energy levels and other defect parameters in this example isconsistent with the temperature- and composition-dependentelectrical resistivity and charge transport properties of fullycompensated semi-insulating CdZnTe. 9In what follows, we often imply a photon ﬂux by referring to the x-ray tube cur-rent. This is justiﬁed by measurements under our experimen-tal operating conditions that conﬁrm a photon ﬂux that is linearly proportional to the tube current 14as predicted in the model spectrum of Eq. /H2084911/H20850. An obvious and ever-present feature in the steady state simulation data is a large concentration of positive spacecharge that accumulates near the cathode plane throughwhich the photons enter the detector. An example of such a buildup of positive charge at the cathode is shown in Fig.1/H20849a/H20850. In this ﬁgure, we show the free /H20849dashed curve /H20850, trapped /H20849dot-dashed curve /H20850, and total /H20849solid curve /H20850charge densities in the steady state reached nearly 15 ms after the x-ray tubewas turned on. The fact that the total and trapped curves areon top of one another indicates that the total charge density isdominated by the trapped space charge throughout the detec-tor. Since the holes travel slowly and are trapped quickly, theproﬁle of the positive space charge in Fig. 1/H20849a/H20850closely fol- lows the exponential interaction proﬁle of the x rays inTABLE I. Bandgap structure and defect level parameters. Bandgap defect structure Acceptor levels lEAl /H20849eV/H20850Pl /H20849cm−3/H20850/H9268Al /H20849cm2/H20850/H9273Al /H20849cm2/H20850 gAl 1 0.15 1 /H1100310112/H1100310−133/H1100310−144 2 0.60 1 /H1100310122/H1100310−133/H1100310−144 Donor levels kEDk /H20849eV/H20850Nk /H20849cm−3/H20850/H9268Dk /H20849cm2/H20850/H9273Dk /H20849cm2/H20850gDk 1 1.558 1.075 /H1100310125/H1100310−133/H1100310−142 2 0.802 1 /H1100310115/H1100310−133/H1100310−142 (a)Positive charge build up (b)Cartoon of the charge distribution FIG. 1. /H20849Color online /H20850/H20849a/H20850The free /H20849dashed curve /H20850, trapped /H20849dash-dotted curve /H20850, and total /H20849solid curve /H20850charge densities in steady state. /H20849b/H20850Cartoon of the resulting total charge densities /H20849solid curve /H20850broken down into its positive /H20849dash-dotted curve /H20850and nega- tive /H20849dashed curve /H20850components.NATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-5CdZnTe. The electrons, on the other hand, are fast and only weakly trapped, so the density of trapped electrons is smallerand more spatially uniform. An idealized cartoon of the re-sulting total charge density is shown in Fig. 1/H20849b/H20850. The solid curve represents the total charge density that results from aspatially uniform density of trapped electrons /H20849dashed line /H20850 and an exponentially decreasing density of trapped holes/H20849dot-dashed curve /H20850. Such a massive amount of space charge in the detector has an effect on the electric ﬁeld as shown in Fig. 2/H20849a/H20850.I n this plot, we show the steady state electric ﬁeld resultingfrom the charge density shown in Fig. 1/H20849a/H20850. The horizontal dashed line at e=−1 in the plot represents the reference un- perturbed value of the electric ﬁeld. Note that near the cath-ode, the ﬁeld strength in high-ﬂux steady state is increasedapproximately four to ﬁve times its unperturbed value. Awayfrom the cathode, however, there is a point at which theelectric ﬁeld strength is reduced to a fraction of its unper-turbed value. We denote this point of reduced ﬁeld strengthz=z and call it the ﬁeld pinch point . On the anode side of the pinch point, the ﬁeld strength recovers. The development of the pinch point in the electric ﬁeld is easily explained by considering a density of space chargedistributed as shown in Fig. 1/H20849b/H20850and large enough to exertelectrostatic forces equal in magnitude to those generated by the operating bias. Consider the cartoon in Fig. 2/H20849b/H20850, where the bias voltage is depicted by a negative surface chargedensity on the cathode and a positive charge density on theanode. The exponential distribution of trapped holes and theuniform density of trapped electrons are also depicted aspositive and negative charges. A free electron that is gener-ated near the cathode will be accelerated toward the anodenot only by the bias ﬁeld but also by the electrostatic force ofthe positive charge it has in front of it. This results in anincreased electric ﬁeld near the cathode as shown in Fig.2/H20849a/H20850. At the pinch point, z=z , however, there is enough posi- tive charge behind the electron to generate a force toward thecathode that very nearly balances the force on the electron inthe direction of the anode due to the bias voltage. This forcebalance is manifested as a vanishing electric ﬁeld strength atthe pinch point, z=z . Electrons generated on the anode side of the pinch are once again dominated by the bias voltage,resulting in a ﬁeld strength that recovers. The pinch in the electric ﬁeld has consequences for the electron transport within the crystal. Figure 3/H20849a/H20850shows the resulting transit times for electron clouds as a functionof the depth of interaction for steady state electric ﬁeldsthat result from the four x-ray tube currents I =/H208531040,1120,1200,1280 /H20854 /H9262A. The dotted line in this plot represents the low-ﬂux limit of the electron transit times. In units of the transit time, it has unit value at the cathode andlinearly goes to zero at the anode. As the photon ﬂux isincreased, Fig. 3/H20849a/H20850shows that the transit time is dramati- cally increased for electrons generated on the cathode side ofthe pinch point. In fact, at I=1280 /H9262A, the transit time is nearly 60 times what it is under low-ﬂux conditions. At the same time the electron transit time is increased, the large concentration of both free and trapped charges near thecathode increases the local band-to-band recombination tran-sition rate. This results in a reduced electron lifetime that isdominated by recombination processes instead of trapping tothe deep donor. Figure 3/H20849b/H20850shows the steady state electron lifetime together with its constituents as a function of Zfor the highest x-ray tube current I=1280 /H9262A. The dashed line shows the uniform low-ﬂux lifetime that is typically domi-nated in CdZnTe by electron trapping to the deep donor. Thesteady state lifetime, shown in Fig. 3/H20849b/H20850as a solid curve, is dominated by band-to-band recombination of free electronswith free holes. In fact, the plot shows that the lifetime hasbeen reduced by an order of magnitude near the cathodewhere the transit times are the longest. Longer transit times coupled together with a reduced life- time for the electrons results in a lower charge collectionefﬁciency for events that occur on the cathode side of thepinch, namely, z/H11021z . The induction map for a parallel plate CdZnTe detector in low-ﬂux conditions is shown as a dottedline in Fig. 4/H20849a/H20850. The ﬁgure also shows the collection efﬁ- ciency as a function of interaction depth at the same fourx-ray tube currents I=/H208531040,1120,1200,1280 /H20854 /H9262A. It is clear that as the photon ﬂux is increased, the collection efﬁ- ciency decreases on the cathode side of the pinch point. Thereduced efﬁciency is due to the fact that electrons are likelyto recombine while moving slowly through the pinch point inthe ﬁeld. (a)Steady state electric ﬁeld (b)Cartoon of the ﬁeld strength FIG. 2. /H20849Color online /H20850/H20849a/H20850Steady state proﬁle of the electric ﬁeld strength /H20849solid curve /H20850after 14.88 ms of evolution. /H20849b/H20850Cartoon of the charge distribution necessary to set up such a steady state for theelectric ﬁeld.DEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-6A reduced collection efﬁciency, in turn, suppresses the signal amplitudes and, therefore, events are recorded at lowerenergies. It follows that, as the ﬂux is increased, the entireenergy spectrum shifts to the left as shown by the computedspectra in Fig. 4/H20849b/H20850. This plot shows that as the ﬂux in- creases, a larger fraction of the spectrum, and thereforecounts, lies below a 25 keV low-energy counting threshold.At a high enough ﬂux, the entire spectrum falls below thelow-energy threshold as shown by the solid curve represent-ingI=1280 /H9262A in Fig. 4/H20849b/H20850. The number of counts above the low-energy threshold for each spectra of Fig. 4/H20849a/H20850is shown as a solid square in Fig. 5/H20849a/H20850. The dashed line shows the total counts in the spectrum increasing linearly with x-ray tube current. Note that as the ﬂux is increased, there is a critical ﬂux, denoted by /H9021/H9253, above which counts begin to decrease with an increasing current. Eventually, the detector is para-lyzed, and the counts vanish entirely. Figure 5/H20849b/H20850shows the actual counting response of 256 channels from a 16 /H1100316 pixel CdZnTe monolithic detectorarray as the photon ﬂux is increased by increasing the x-ray tube current from 25 to 400 /H9262A. Both the simulated counts in Fig. 5/H20849a/H20850and the experimentally measured counts in Fig. 5/H20849b/H20850initially increase with increasing ﬂux, but at the critical ﬂux/H9021/H9253, the counts begin to decrease with increasing photon ﬂux. In the following section, we develop a theoretical de-pendence of this critical photon ﬂux on material, detector,and operating parameters. Finally, we point out that Fig. 4/H20849a/H20850shows that for the parallel plate detector simulated here, the interactions thattake place for z/H11021z and, in particular, those very near the cathode have induced signals that beneﬁt from the fact thatthe electrons are able to travel unimpeded to the pinch atz=z . This is evident in the linear rise of the induced signals of Fig. 4/H20849a/H20850asz→0. Of course, many imaging applications do not use planar detectors but make use of monolithic de-tectors patterned with pixel arrays to improve image resolu-tion. Since most of the charge in a pixelated detector is in-duced when the electrons are very near the anode plane /H20849i.e.,(a)Electron time-of-ﬂight (b)Electron lifetime FIG. 3. /H20849a/H20850Proﬁles of the resulting steady state electron time of ﬂights at four increasing ﬂux rates /H20849tube currents /H20850./H20849b/H20850Depthwise proﬁles of the steady state electron lifetime components forI=1280 /H9262A. The total lifetime is shown as a solid curve.(a)Induced ch arge eﬃciency (b)Spectra in polarizing detector FIG. 4. /H20849a/H20850Proﬁles of the resulting charge induction maps at four increasing ﬂux rates /H20849tube currents /H20850./H20849b/H20850Resulting shift of the measured spectrum at the four increasing ﬂux rates /H20849tube currents /H20850.NATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-7after they have passed through the pinch point /H20850, pixelated detectors will be insensitive to this early unimpeded motionof the electrons. As a consequence, pixelated detectors willbe more susceptible to the above-mentioned reduced chargeinduction, and therefore polarization. IV. THEORY In the previous section, we presented a single example from a large matrix of simulations in which we solved thefull set of Eqs. /H208497a/H20850and /H208497b/H20850subjected to an x-ray source with spectrum dn /H9253/dedescribed by Eq. /H2084911/H20850. Not only does this example highlight the sequence of events that take placein a polarizing detector but it also demonstrates the dominantrole played by the hole transport in the polarization se-quence. Speciﬁcally, it is the large buildup of positive charge/H20849i.e., number density of both free and trapped holes /H20850within the detector that begins the sequence by creating a pinchpoint in the electric ﬁeld as shown in Fig. 2/H20849a/H20850. In fact, in the absence of a pinch point, the electric ﬁeld strength is neverlow enough to sufﬁciently reduce the electron transport to alevel that shifts the spectrum below the low-energy thresh-old. In this section, we exploit the dominant role of the hole dynamics to develop an analytic expression that approxi-mately describes the dependence of the maximum sustain-able ﬂux on critical material, operating, and detector designparameters. The analysis we present assumes that polariza-tion is the end result of the creation of a pinch point in theelectric ﬁeld, and the method can be summarized as follows: /H208491/H20850The amount of charge, denoted by Q , that is neces- sary to collapse the electric ﬁeld at a pinch point is calcu-lated. /H208492/H20850The time dependence, Q/H20849T/H20850, of the buildup of positive charge density within the detector is calculated. /H208493/H20850Polarization results when the time-asymptotic limit /H20849i.e., steady state value /H20850of the buildup of positive charge exceeds that necessary to collapse the electric ﬁeld at thepinch point. Mathematically, this is expressed aslim T→/H11009Q/H20849T/H20850=Q. The result of the third step, as we will show, is the desired functional dependence of the maximum sustainable ﬂux /H20849i.e., critical ﬂux /H9021/H9253/H20850on device design and operating parameters. A. Necessary positive charge We begin by considering a photon source with uniform ﬂux/H9021/H9253describing the number of photons per cm2s1that intersect the cathode surface of the detector at right angles.The source is assumed to be monoenergetic with energytaken as the mean value of the x-ray spectrum of Eq. /H2084911/H20850 and denoted by E ¯/H9253. The generation rate of electron-hole pairs within the detector, therefore, is exponentially distributedand has the form /H9023/H20849Z/H20850=/H9021 /H9253E¯/H9253 /H9280czt1 /H9011e−Z//H9011, /H2084912/H20850 where /H9280cztis the pair creation energy for CdZnTe, and /H9011→/H9011/H20849E¯/H9253/H20850=/H9262/H20849E¯/H9253/H20850−1is the characteristic length scale that a photon travels before interacting in CdZnTe /H20849i.e., inverse of thelinear attenuation coefﬁcient /H20850. In order to simplify nota- tion, we deﬁne /H90230=E¯/H9253/H9021/H9253//H9280cztthat represents the number of electron-hole pairs being created per cm2s1. We have shown that since the holes are slow moving and rapidly trapped, the positive buildup of charge due to bothfree and trapped holes can be approximated by the exponen-tial form of the charge generation in Eq. /H2084912/H20850, which was shown as a dot-dashed curve in Fig. 1/H20849b/H20850. Recall that the electrons, on the other hand, are fast and trapped much lessfrequently, so the negative charge density present due toelectrons must be approximated by a small and spatially uni-form charge density as shown in the cartoon of Fig. 1/H20849b/H20850. Therefore, the total charge density built up after Tseconds can be approximated by (a)Resulting counts above threshold (b)Measured counts above threshold FIG. 5. /H20849a/H20850Simulated counts above a 25 keV threshold as the photon ﬂux rate /H20849x-ray tube current /H20850is increased. /H20849b/H20850Measured counts for 256 channels of a polarizing detector as the photon ﬂuxrate /H20849tube current /H20850is increased.DEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-8/H9267/H20849Z/H20850=qT/H902301 /H9011e−Z//H9011−qT/H90230/H9254/L=/H92670/H208731 /H9011e−Z//H9011−/H9254/L/H20874, /H2084913/H20850 where /H9254is a small nondimensional parameter representing the relative number of trapped electrons to holes, and wehave deﬁned /H92670=qT/H90230for convenience of the notation. This approximation to the steady state charge density is shown asa solid curve in the cartoon of Fig. 1/H20849b/H20850. In order to ﬁnd the resulting electric ﬁeld, we turn to the Poisson problem which can be written as d 2/H9278 dZ2=−/H92670 /H92800/H208731 /H9011e−Z//H9011−/H9254/L/H20874, /H2084914a /H20850 /H9278/H20849Z=0 /H20850=−V, /H2084914b /H20850 /H9278/H20849Z=L/H20850=0 . /H2084914c /H20850 Integrating Eq. /H2084914a /H20850twice and applying the boundary con- ditions /H2084914b /H20850and /H2084914c /H20850yields a solution for the electric po- tential within the detector, /H9278/H20849Z/H20850=/H20849Z/L−1 /H20850/H20875V+/H92670 /H92800/H20849/H9254Z−/H9011/H20850/H20876−/H9011/H92670 /H92800/H20875eZ//H9011−Z Le−L//H9011/H20876. /H2084915/H20850 The electric ﬁeld immediately follows as the negative gradi- ent of /H9278, E/H20849Z/H20850=−d dZ/H9278/H20849Z/H20850 =−V L/H208771−/H92670L /H92800V/H20875/H9011 L+/H9254/H208731−2Z L/H20874−e−Z//H9011−/H9011 Le−L//H9011/H20876/H20878. /H2084916/H20850 Now that we have the electric ﬁeld as a function of Z,w ec a n compute the location of the pinch point that will be denotedbyZ. Since the pinch point represents a global minimum of the ﬁeld strength /H20648E/H20849Z/H20850/H20648,Zis located where the derivative of the electric ﬁeld strength vanishes, /H20879dE dZ/H20879 Z=Z=/H92670 /H92800/H208732/H9254 L−1 /H9011e−Z//H9011/H20874=0 . /H2084917/H20850 This can easily be solved for the location of the pinch point, Z=−/H9011ln/H208492/H9254/H9011/L/H20850, /H2084918/H20850 which can be substituted back into Eq. /H2084916/H20850to ﬁnd the strength of the electric ﬁeld at the pinch point. We point outthat having an expression for the strength of the electric ﬁeldatZ takes us nearly to the goal of this section, which is to ﬁnd the amount of charge necessary to collapse the ﬁeld atthe pinch point /H20851i.e., where E/H20849Z/H20850→0/H20852. Prior to writing the resulting expression for the ﬁeld strength, we note that it can be further simpliﬁed using the fact that /H9254/H112701 since trapping is weak for electrons when compared to that of holes. There-fore, substituting Eq. /H2084918/H20850into Eq. /H2084916/H20850, taking the limit as /H9254 vanishes, and setting the result to zero yieldslim /H9254→0E/H20849Z/H20850=−V L/H208771−/H92670/H9011 /H92800V/H208491−e−L//H9011/H20850/H20878=0 , /H2084919/H20850 which deﬁnes the minimum charge density, denoted /H92670, that is necessary to collapse the ﬁeld at the pinch point. Solving this equation for /H92670gives /H92670=/H92800V /H9011/H9252, /H2084920/H20850 where we have deﬁned /H9252=/H208511−exp /H20849−L//H9011/H20850/H20852. Recall that /H92670 represents the amount of charge per unit area in the detector and that the volumetric density of charge is exponentiallydistributed in the Zdirection, so we can express the total charge in the detector as Q=A /H92670/H20885 0L1 /H9011e−Z//H9011dZ=A/H92670/H9252. /H2084921/H20850 Therefore, we can use Eq. /H2084921/H20850to express the minimum total charge necessary to collapse the electric ﬁeld at the pinchpoint, Q=A/H92800V /H9011. /H2084922/H20850 This equation contains the relationship that we have been seeking. As one may expect, Qis proportional to the ap- plied bias voltage since it clearly takes more charge to col-lapse the stronger ﬁeld generated by a larger applied biasvoltage. B. Time dependence of positive charge buildup So far, we have developed an approximation to the mini- mum amount of charge necessary to begin the process ofpolarization /H20849i.e., collapse the ﬁeld at the pinch point /H20850. In this section, we derive an approximation to the time evolution ofthe positive charge buildup in the detector. Once again, weturn to the results of simulation that show that there are twodominant dynamics that govern the time evolution of thecharge that builds up in the detector: /H208491/H20850positive charge is increased due to the x-ray source generating holes and /H208492/H20850positive charge is decreased due to an outgoing ﬂux of holes. Since the levels of trapped holes start off quite low, at the moment the x-ray source is turned on, the increase in chargedue to generation dominates. This causes a buildup of posi-tive charge density. However, as the holes build up near thecathode, more and more of them exit the detector once theyare detrapped. Consequently, there is an asymptotic limit tothe total positive charge in the detector. We begin by calculating the charge buildup with a simple rate equation in Sec. IV B 1. Though this solution is shownto be useful at photon energies for which the detector fullyabsorbs the photons /H20849i.e., for /H9011/H11270L/H20850, we ﬁnd that we must turn to a more sophisticated analysis to accurately describethe charge buildup for high-energy photons. The more accu-rate analysis is presented here in Sec. IV B 2 and is based onNATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-9a multiple scale technique described in detail in Ref. 7. 1. Simple method To calculate this time-asymptotic limit, we begin by writ- ing down the change in total charge during any time interval/H9004T, /H9004Q=Aq/H9023 0/H9004T/H9252−/H20849vh/H9004T/H20850A/H92670//H9011=Aq/H90230/H9004T/H9252−/H20849vh/H9004T/H20850Q /H9011/H9252, /H2084923/H20850 where, in the second equality, we have used Eq. /H2084921/H20850. The ﬁrst term on the right hand side of Eq. /H2084923/H20850represents the increase in charge due to the generation source, and the sec-ond term represents the loss of charge due to the ﬂux of holesout of the detector. Note that the holes are assumed to betraveling at speed vh, which will be discussed in detail shortly. Taking the inﬁnitesimal limit /H9004T→0 yields the rate equation dQ /H20849T/H20850 dT=qA/H90230/H9252−vhQ/H20849T/H20850 /H9011/H9252. /H2084924/H20850 This differential equation has the simple solution Q/H20849T/H20850=Aq/H90230/H92522/H9011 vh/H208751 − exp/H20873−vh /H9011/H9252T/H20874/H20876 /H2084925/H20850 in time. Equation /H2084925/H20850is shown as dot-dashed curves in Fig. 6for both high-, and low-energy photons represented by two values of the parameter /H9011, namely, /H9011/L=/H208531/6,2 /3/H20854. The solution has the characteristics we expect; the charge initially grows quickly and then exponentially asymptotes to a valuethat is proportional to /H9023 0and inversely proportional to the velocity of the holes vh. Figure 6also shows a full numerical solution for this example as circles for low-energy photonswith/H9011/L=1 /6, and squares for high-energy photons with /H9011/L=2 /3. It is clear that when the photons are fully ab- sorbed by the detector, Eq. /H2084925/H20850does quite well. However, when photons are only partially absorbed /H20849i.e., the photon energy is large enough so that /H9011/L/H110111/H20850, this simple solution overestimates the charge buildup with a large error. The reason for this overestimation is clear if we point out that this simple analysis has only assumed an exponentiallydistributed density of holes and does not take into consider-ation the fact that this distribution will change as the holesmove away from the anode plane. Figure 7compares the space-time evolution of the charge density for /H9011/L=2 /3s o that photons are deposited throughout the detector thickness.The full numerical solution is shown in Fig. 7/H20849a/H20850, while the simple analytical result is shown in Fig. 7/H20849b/H20850. Note that the overestimation of charge is due to the error near the anodeplane at z=1. The simple analytical solution neglects the fact that holes are moving away from the anode plane, therebyreducing the charge density there. 2. Multiple scale solution Figures 6and7show that the simple analytical solution in Eq. /H2084925/H20850does not accurately predict the value of the positive FIG. 6. Charge buildup Q/H20849T/H20850predicted by simple analysis /H20849dot-dashed curves /H20850at two values of the parameter /H9011, namely, /H9011/L=/H208531/6,2 /3/H20854. Full numerical solution is shown as circles for /H9011/L=1 /6, and squares for /H9011/L=2 /3. a) b) FIG. 7. Space-time evolution of the buildup of positive charge density. The parameter /H9011/H20849e/H9253/H20850/L=2 /3 so that there are penetrating photons.DEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-10charge within the detector as T→/H11009when the photon energy is high enough that there are interactions throughout the de-tector. In addition to this, it is troubling that the simple so-lution is not capable of decomposing the positive charge into free and trapped carrier constituents, PandPˆ, respectively. Therefore, in order to obtain a more accurate and completedescription of the charge buildup within the detector, we turnto a multiple scale perturbation technique whose details aredeveloped in Ref. 7and brieﬂy summarized here. We begin by reducing the full set of Eqs. /H208497a/H20850and /H208497b/H20850to the relevant system of partial differential equations that de-scribe the transport, trapping, and subsequent detrapping ofholes in the presence of a single, dominant hole trap. Theresulting scaled system of equations can be written as /H11509tp+/H9280/H11509z/H20849ep/H20850=−p+/H9257/H9270h /H9270Dpˆ+/H9274, /H2084926a /H20850 /H11509tpˆ=1 /H9257p−/H9270h /H9270Dpˆ, /H2084926b /H20850 /H11509z2/H9278=−/H9253/H20849p+/H9257pˆ/H20850,e=−/H11509z/H9278, /H2084926c /H20850 where we have used Eqs. /H208495a/H20850and /H208495b/H20850to scale the concen- trations, and /H9274is the nondimensional version of the hole generation source in Eq. /H2084912/H20850with form /H9274=/H927401 /H9261e−z//H9261. /H2084927/H20850 In both system /H2084926a /H20850–/H2084926c /H20850and Eq. /H2084927/H20850, we have introduced /H9261=/H9011/L, as well as the nondimensional source of holes /H92740 =/H20849/H9270h/H9021/H9253E/H9253/H20850//H20849P0L/H9280czt/H20850. The parameter in front of the drift term of Eq. /H2084926a /H20850is a small parameter /H9280=/H9270h/H9262hV/L2/H112701, rep- resenting the fraction of the detector thickness, on average,that a hole travels before trapping. The hole trapping time /H9270h is deﬁned in terms of the trapping rate /H9011↓:A1, so that /H9270h=/H20849/H9011↓:A1/H20850−1/H110151 /H9268A1/H9258P1, /H2084928/H20850 where we have assumed the ionized fraction of the acceptor to be constant. In Eqs. /H2084926a /H20850–/H2084926c /H20850, we have also deﬁned the hole detrapping time /H9270D, which is deﬁned in terms of /H9011↑:A1as /H9270D=/H20849/H9011↑:A1/H20850−1=1 /H9263A1exp/H20873EA1 kT/H20874, /H2084929/H20850 where /H9263A1is the escape frequency typically set by the princi- pal of detailed balance for the system in steady state or inthermal equilibrium. The multiple scale technique takes advantage of the fact that there are two distinct time scales in this problem. Theﬁrst is a fast time scale deﬁned by the rapid trapping of holesand denoted by t 1=t. The second is a slow time scale deﬁnedby the transit time of the holes and is denoted by t2=/H9280t. These time scales are considered independent and deﬁne atwo-scale perturbation expansion for the free and trappedhole concentrations, as well as the electric potential, p/H20849z,t/H20850=1 /H9280p−1/H20849z,t1,t2/H20850+p0/H20849z,t1,t2/H20850+/H9280p1/H20849z,t1,t2/H20850+O/H20849/H92802/H20850, /H2084930a /H20850 pˆ/H20849z,t/H20850=1 /H9280pˆ−1/H20849z,t1,t2/H20850+pˆ0/H20849z,t1,t2/H20850+/H9280pˆ1/H20849z,t1,t2/H20850+O/H20849/H92802/H20850, /H2084930b /H20850 /H9278/H20849z,t/H20850=/H92780/H20849z,t1,t2/H20850+/H9280/H92781/H20849z,t1,t2/H20850+O/H20849/H92802/H20850. /H2084930c /H20850 We point out that the dominant terms of both the free and trapped hole concentrations are not assumed O/H208491/H20850, but taken to be O/H208491 /H9280/H20850. The reason for this is that the small parameter /H9280 is the nondimensional speed at which the holes travel toward the cathode plane /H20851e.g., see the drift term of Eq. /H2084926a /H20850/H20852.I th a s been shown that solutions to drift equations with a continu-ous source such as that in Eq. /H2084926a /H20850depend inversely on the nondimensional carrier speed /H9280.7Such a dependence on the inverse of the speed is exhibited in Eq. /H2084925/H20850previously found by the simple analysis. It follows, therefore, that the domi-nant terms of expansions /H2084930a /H20850–/H2084930c /H20850should be chosen to be inversely proportional to the small parameter /H9280. The multiple scale solution process proceeds by solving successively higher orders of the equations that result fromsubstituting expansions /H2084930a /H20850–/H2084930c /H20850into Eqs. /H2084926a /H20850–/H2084926c /H20850. As the problem is solved at a particular order of the smallparameter /H9280, the degree of freedom that comes from having two independent time scales is used to eliminate seculargrowth at the next order, providing accurate and stable per-turbation solutions over long time scales. 15Details of the multiple scale solution of Eqs. /H2084926a /H20850–/H2084926c /H20850with/H9274given by Eq. /H2084927/H20850are presented elsewhere,7so here, we simply present the dominant solutions for pandpˆ, p/H20849z,t/H20850=/H927401 /H9263/H9280e−z//H9261/H208511−e−/H20849/H9263/H9280//H9261/H20850tH/H20849/H9258−/H20850−e−/H208491//H9261/H20850/H208491−z/H20850H/H20849−/H9258−/H20850/H20852 /H11003/H20851/H9263+/H208491−/H9263/H20850e−t//H208491−/H9263/H20850/H20852+O/H208491/H20850, /H2084931a /H20850 pˆ/H20849z,t/H20850=/H927401−/H9263 /H9257/H9263/H9280e−z//H9261/H208511−e−/H20849/H9263/H9280//H9261/H20850tH/H20849/H9258−/H20850−e−/H208491//H9261/H20850/H208491−z/H20850H/H20849−/H9258−/H20850/H20852 /H11003/H208511−e−t//H208491−/H9263/H20850/H20852+O/H208491/H20850, /H2084931b /H20850 where H/H20849x/H20850is the Heaviside step function, and we have in- troduced the characteristic variable /H9258−=1− z−/H9280/H9263t, with theNATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-11nondimensional parameter /H9263deﬁned by /H9263=/H9270h//H20849/H9270h+/H9270D/H20850. This characteristic variable naturally emerges from the multiple scale solution process and further deﬁnes the reduced effec-tive speed of the holes, veff=L /H9270h/H9280/H9270h /H9270h+/H9270D=/H9270h /H9270h+/H9270D/H9262hE, /H2084932/H20850 ﬁrst found by Zanio et al.16and recently derived by the au- thors using multiple scales.7This reduced effective speed of the holes is a result of the stop and go process that results from multiple cycles of trapping and detrapping during holetransit. Having found solutions /H2084931a /H20850and /H2084931b /H20850representing space-time functions for the free and trapped concentrationof holes, respectively, we compute the total charge densityby simply summing them together. The result is written interms of physical space and time variables /H20849Z,T/H20850in the form/H9267/H20849Z,T/H20850=qP0/H20851p/H20849Z,T/H20850+/H9257pˆ/H20849Z,T/H20850/H20852 =q/H9021/H9253E/H9253 veff/H9280czte−Z//H9011/H209021 − exp/H20873−veff /H9011T/H20874,T/H11349L−Z veff 1 − exp/H20873−L−Z /H9011/H20874,T/H11022L−Z veff,/H20903 /H2084933/H20850 where we have used our deﬁnition of /H92740 =/H20849/H9270h/H9021/H9253E/H9253/H20850//H20849P0L/H9280czt/H20850. This space-time dependence of the buildup of charge density within the detector is compared to the full numerical solution in Fig. 8for/H9011/L=2 /3. The full numerical solution is shown in Fig. 8/H20849a/H20850, and Eq. /H2084933/H20850is plotted in Fig. 8/H20849b/H20850. Note that the multiple scale solution matches the numerical solution very well and has capturedthe fact that the hole density is lower near the anode plane atZ=L. A simple integral of the charge density over the detector thickness gives what we seek in this section, namely, thetime dependence of the buildup of positive charge within thedetector, Q/H20849T/H20850=qA/H9021/H9253E/H9253 veff/H9280czt/H9011/H209021 − exp/H20873−veff /H9011T/H20874−veff /H9011Texp/H20873−L /H9011/H20874,T/H11349L/veff 1−/H208731+L /H9011/H20874exp/H20873−L /H9011/H20874, T/H11022L/veff./H20903/H2084934/H20850 This analytical approximation is shown as a solid curve in Fig.9, where it is compared with a full numerical solution shown as circles at the lower photon energy with /H9011/L =1 /6 and squares at the higher energy with /H9011/L=2 /3. The charge buildup predicted by the simple method is also shownas a dot-dashed curve for comparison of the two analyticalsolutions. It is clear from Fig. 9thatQ/H20849T/H20850derived using the multiple scale perturbation technique accurately predicts the charge buildup for both low- and high-energy photons. As a ﬁnal note, recall that the simple solution method gave no indication whether the excess charge was made offree or trapped holes. The multiple scale solutions /H2084931a /H20850and /H2084931b /H20850, on the other hand, clearly distinguish the constituents of the positive charge. Comparing Eq. /H2084933/H20850with Eq. /H2084931a /H20850,i t is clear that the time-asymptotic fraction of free holes thatcontribute to Qis given by /H9263=/H9270h//H20849/H9270h+/H9270D/H20850and the fraction of trapped holes is given by 1− /H9263=/H9270D//H20849/H9270h+/H9270D/H20850. In real crystals of CdZnTe, however, the acceptor level responsible for the hole trapping is a deep level,17so that /H9270D/H11271/H9270hand/H9263/H112701, meaning that the positive charge is predominantly made oftrapped space charge. C. Maximum sustainable photon ﬂux So far, we have derived Qin Eq. /H2084922/H20850, approximating the minimum charge necessary to create the pinch point in theelectric ﬁeld that is necessary to polarize a detector. In Eq. /H2084934/H20850, we used a multiple scale perturbation technique to de- rive Q/H20849T/H20850describing the time dependence of the buildup of positive charge in a detector subjected to a photon ﬂux /H9021/H9253. We are now in a position to derive the maximum sustainableﬂux above which a detector will polarize, and we denote this critical ﬂux by/H9021 /H9253. The critical ﬂux we seek simply comes from equating the time-asymptotic limit of the total chargeQ/H20849T/H20850to that needed for polarization, Q,o r lim t→/H11009Q/H20849t/H20850=Q. /H2084935/H20850 Using Eqs. /H2084922/H20850and /H2084934/H20850, the above equality can be rewritten as qE¯/H9253/H9021/H9253/H9011 /H9280cztveff/H20873/H9252−L /H9011e−L//H9011/H20874=/H92800V /H9011, /H2084936/H20850 where we have used the fact that we had previously deﬁned /H9252=1−exp /H20849−L//H9011/H20850. Finally, we use Eq. /H2084932/H20850to substitute for veffand solve for /H9021/H9253to getDEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-12/H9021/H9253=/H92800/H9280cztV2 qLE¯/H9253/H90112/H20873/H9252−L /H9011e−L//H9011/H20874−1/H9262h/H9270h /H9270h+/H9270D. /H2084937/H20850 This equation expresses the desired relationship between the maximum sustainable ﬂux and the critical material, operat-ing, and detector design parameters for the CdZnTe device. We point out that using Q/H20849T/H20850in Eq. /H2084925/H20850derived by the simple rate equation of Sec. IV B 1 in the limit on the left side of Eq. /H2084935/H20850yields an approximation to the maximum sustainable ﬂux of the form /H9021 /H9253=/H92800/H9280cztV2 qLE¯/H9253/H20849/H9011/H9252/H208502/H9262h/H9270h /H9270h+/H9270D, /H2084938/H20850 which was previously reported by the authors.18,19Note that for low photon energies such that /H9011/L/H112701 with few photons interacting deep in the detector, we have /H9252→1, and Eqs. /H2084937/H20850and /H2084938/H20850become equivalent. However, for energies such that there are interactions throughout the detector, as onewould expect from the plot in Fig. 9in Sec. IV B 2, the expression for /H9021 /H9253in Eq. /H2084937/H20850is much more accurate.The expression for /H9021/H9253in Eq. /H2084937/H20850provides a theoretical foundation upon which critical material, operating, and de-tector parameters can be chosen when designing a semicon-ductor device applied to high-ﬂux x-ray imaging applica-tions. In the following subsections, we highlight the functional dependence of /H9021 /H9253on a few such design param- eters and validate them using experimentally measured data. 1. Bias voltage dependence The quadratic dependence of the critical ﬂux on the oper- ating bias voltage Vis clear in Eq. /H2084937/H20850. That is, doubling the bias will increase the maximum sustainable ﬂux by a factorof 4 under conditions where all other parameters are heldconstant. Figure 10/H20849a/H20850shows a surface plot of the critical ﬂux /H20851Eq. /H2084937/H20850/H20852as a function of the hole mobility-lifetime product and the bias voltage. The linear dependence of /H9021 /H9253on/H9262h/H9270h, as well as its quadratic dependence on V, is shown in this plot for a 3 mm thick detector at room temperature and EA =0.73 eV. The mean photon energy was assumed to be 60 keV, from which it follows that /H9011/L/H110150.1. Figure 10/H20849b/H20850 shows the same dependencies as a contour map. In order to experimentally validate the quadratic bias de- pendence, we selected ﬁve 3 mm thick 16 /H1100316 pixel CdZnTe monolithic detector arrays with low hole transportthat demonstrated polarizing characteristics. Details of thefabrication process for these devices have been reported inRefs. 14and19. Each detector was temperature stabilized and subjected to a 120 kVp x-ray source, for which the cur-rent was ramped from 10 to 400 /H9262A in increments of 5 /H9262A. At each value of the tube current /H20849photon ﬂux /H20850, the 256 channels of count values were averaged and read out.This process was repeated for ﬁve voltages V /H33528/H20853300,400,500,600,700 /H20854V. All ﬁve detectors gave similar results, and we show the resulting count curves for a single detector in Fig. 11/H20849a/H20850. At each bias voltage, the critical ﬂux was determined by ﬁtting such curves to pick off the tube (a) (b) FIG. 8. Space-time evolution of the buildup of positive charge density. The parameter /H9011/H20849e/H9253/H20850/L=2 /3, so that there are penetrating photons. FIG. 9. Charge buildup Q/H20849T/H20850predicted by multiple scale analy- sis /H20849solid curves /H20850and simple analysis /H20849dot-dashed curves /H20850. Full nu- merical solution is shown as circles for /H9011/L=1 /6 and squares for /H9011/L=2 /3.NATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-13current at which the maximum count values occurred /H20851e.g., see plots in Figs. 5/H20849a/H20850and5/H20849b/H20850/H20852. A log-log plot of the mea- sured values for the same detector is shown in Fig. 11/H20849b/H20850 /H20849solid triangles /H20850, together with a power law ﬁt to the data. The result of the ﬁt shows that the data are best representedby a power law with power p=2.06. This is precisely the quadratic dependence predicted by Eq. /H2084937/H20850. 2. Temperature dependence The dependence of the critical ﬂux on temperature comes through the hole detrapping time /H9270Dthat resides in the de- nominator of Eq. /H2084937/H20850. Speciﬁcally, the temperature depen- dence is evident in the deﬁnition of Eq. /H2084929/H20850, so that /H9021/H9253/H11011exp/H20873−EA kT/H20874, /H2084939/H20850 where EAis the ionization energy of the dominant trap re- sponsible for the hole trapping. Therefore, there is an expo-nential dependence of the maximum sustainable ﬂux on the operating temperature of the detector. Figure 12shows a sur- face and contour plot of /H9021/H9253as a function of both the hole mobility-lifetime product and the temperature. The strongdependence of the critical ﬂux on temperature is evident inthe surface plot of Fig. 12/H20849a/H20850. In order to experimentally validate this exponential tem- perature dependence, we have once again used the same setof detectors described in Sec. IV C 1. In this set of experi-ments, however, we have held the bias voltage ﬁxed at900 V and varied the temperature in the range T /H33528/H2085315,20,25,30,35,40,45 /H20854C. The measured values for /H9021 /H9253* at each temperature for the same detector used in Fig. 11are plotted in Fig. 13 /H20849solid triangles /H20850, together with an exponen- tial ﬁt /H20849dot-dashed curve /H20850. The data are best ﬁtted with an exponential curve with EA=0.76 eV, which corresponds to a deep acceptor level at the middle of the bandgap, as onewould expect. (a)Surface Plot (b)Φ∗ γ= const. contours FIG. 10. /H20849a/H20850Surface and /H20849b/H20850contour plots of the dependence of /H9021/H9253on bias voltage. (a)Voltage dependence of counts (b)maximum sustained ﬂux FIG. 11. Experimentally measured dependence of the maximum sustainable ﬂux on bias voltage.DEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-14V. CONCLUSION We have developed a general mathematical model for the defect structure in the bandgap of CdZnTe that is applicableto any wide-bandgap semiconductor and capable of describ-ing carrier transport and defect transition dynamics in thepresence of a high-ﬂux x-ray source. The model allows foran arbitrary number of both donor and acceptor defectswithin the crystal. The resulting nonlinear system of balancelaws have been numerically solved using ﬂux-conservativewave propagation algorithms developed for conservationlaws with spatially varying ﬂux functions. 4,5The code devel- oped has been applied to a large matrix of simulations, inwhich a parallel plate detector has been subjected to a ﬂux of photons emanating from an x-ray tube. The simulation datahave highlighted the dominant hole dynamics that trigger asequence of events that ultimately result in a reduced chargecollection efﬁciency of the electrons. This reduced efﬁciencycauses count spectra to shift to lower energies, meaning thatfewer counts are above the low-energy counting threshold.As the ﬂux is raised above a critical value, denoted here by /H9021 /H9253, counts begin to decrease as the ﬂux is increased, ulti- mately leading to a polarized detector. The dominant role of the transport of slowly moving holes has been exploited with the application of multiplescale perturbation techniques 7to derive an analytic expres- sion describing /H9021/H9253as a function of critical material, opera- tional, and detector design parameters. The functional depen-dencies of the maximum sustainable ﬂux on bias voltage andtemperature have been validated experimentally using16/H1100316 pixel CdZnTe monolithic detector arrays subjected to a high-ﬂux 120 kVp x-ray source. ACKNOWLEDGMENTS The authors acknowledge fruitful discussions with Michael Prokesch, Stephen Soldner, and David Rundle andare grateful to Jesse Graves and Bradley Hughs for the au-tomation of experimental data collection. This work has beensupported in part by the U.S. Army Armament Research,Development, and Engineering Center /H20849ARDEC /H20850under Con- tract No. DAAE 30-03-C-1171. (a)Surface Plot (b)Φ∗ γ= const. contours FIG. 12. /H20849a/H20850Surface and /H20849b/H20850contour plots of the temperature dependence of /H9021/H9253. FIG. 13. Experimentally measured temperature dependence of the critical ﬂux. Measured values are shown as solid triangles, whilean exponential ﬁt to the data is shown as a dot-dashed curve. Theenergy level of the deep acceptor responsible for the hole trappingthat creates the best ﬁt is E A=0.76 eV, as shown in the plot.NATURE OF POLARIZATION IN WIDE-BANDGAP … PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-15*dbale@ii-vi.com 1A. Jahnke and R. Matz, Med. Phys. 26,3 8 /H208491999 /H20850. 2D. Vartsky, M. Goldberg, Y. Eisen, Y. Shamai, R. Dukhan, P. Siffert, J. M. Koebel, R. Regal, and J. Gerber, Nucl. Instrum.Methods Phys. Res. A 263, 457 /H208491988 /H20850. 3Y. Du, J. LeBlanc, G. E. Possin, B. D. Yanoff, and S. Bogdanov- ich, IEEE Trans. Nucl. Sci. 50, 1031 /H208492003 /H20850. 4D. S. Bale, Ph.D. thesis, University of Washington, 2002. 5D. S. Bale, R. J. LeVeque, S. Mitran, and J. A. Rossmanith, SIAM J. Sci. Comput. /H20849USA /H2085024, 955 /H208492002 /H20850. 6R. J. LeVeque, J. Comput. Phys. 131, 327 /H208491997 /H20850. 7D. S. Bale and C. Szeles /H20849unpublished /H20850. 8T. E. Schlesinger, J. E. Toney, H. Yoon, E. Y. Lee, B. A. Brunett, L. Franks, and R. B. James, Mater. Sci. Eng., R. 32, 103 /H208492001 /H20850. 9M. Prokesch and C. Szeles, Phys. Rev. B 75, 245204 /H208492007 /H20850. 10M. Jung, J. Morel, P. Fougeres, M. Hage-Ali, and P. Siffert, Nucl. Instrum. Methods Phys. Res. A 428,4 5 /H208491999 /H20850. 11Glenn F. Knoll, Radiation Detection and Measurment /H20849Wiley, New York, 2000 /H20850.12R. J. LeVeque, CLAWPACK software, available on the Web at the URL http://www.amath.washington.edu/~claw 13W. J. Iles, National Radiological Protection Board Report No. NRPB-R204, 1987 /H20849unpublished /H20850. 14C. Szeles, S. A. Soldner, S. Vydrin, J. Graves, and D. S. Bale, IEEE Trans. Nucl. Sci. 54, 1350 /H208492007 /H20850. 15J. Kevorkian and J. D. Cole, Multiple Scale and Singular Pertur- bation Methods /H20849Springer-Verlag, New York, 1996 /H20850. 16K. R. Zanio, W. M. Akutagawa, and R. Kikuchi, J. Appl. Phys. 39, 2818 /H208491968 /H20850. 17C. Szeles, Y. Y. Shan, K. G. Lynn, A. R. Moodenbaugh, and E. E. Eissler, Phys. Rev. B 55, 6945 /H208491997 /H20850. 18D. S. Bale and C. Szeles, 16th International Workshop on Room- Temperature Semiconductor X-ray and Gamma-ray Detectors,San Diego, CA, 29 October–4 November 2006 /H20849unpublished /H20850. 19C. Szeles, S. A. Soldner, S. Vydrin, J. Graves, and D. S. Bale, 16th International Workshop on Room-Temperature Semicon-ductor X-ray and Gamma-ray Detectors, October 2006 /H20849unpub- lished /H20850.DEREK S. BALE AND CSABA SZELES PHYSICAL REVIEW B 77, 035205 /H208492008 /H20850 035205-16
PhysRevB.101.201104.pdf	PHYSICAL REVIEW B 101, 201104(R) (2020) Rapid Communications Signature for non-Stoner ferromagnetism in the van der Waals ferromagnet Fe 3GeTe 2 X. Xu,1,Y. W. L i ,2,S. R. Duan,1S. L. Zhang,3Y . J. Chen,1L. Kang,1A. J. Liang,3,4C. Chen,3,4W. Xia,3Y. X u ,1,5,6 P. Malinowski,7X. D. Xu,7,8J.-H. Chu,7G. Li,3Y . F. Guo,3Z. K. Liu,3L. X. Yang ,1,5,†and Y . L. Chen1,2,3,‡ 1State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China 2Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 3School of Physical Science and Technology, ShanghaiTech University and CAS-Shanghai Science Research Center, Shanghai 201210, China 4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 5Frontier Science Center for Quantum Information, Beijing 100084, China 6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan 7Department of Physics, University of Washington, Seattle, Washington 98105, USA 8Department of Materials Science and Engineering, University of Washington, Seattle, Washington 98105, USA (Received 19 April 2019; revised manuscript received 27 February 2020; accepted 21 April 2020; published 11 May 2020) The van der Waals ferromagnet Fe 3GeTe 2has attracted great research attention recently due to its extraor- dinary properties. Here, using high-resolution angle-resolved photoemission spectroscopy, we systematicallyinvestigate the temperature evolution of the electronic structure of bulk Fe 3GeTe 2. We observe largely dispersive energy bands with exchange splitting that are in overall agreement with our density-functional theory calculation.Interestingly, the band dispersions barely change upon heating towards the ferromagnetic transition near 225 K,except for the reduction of quasiparticle coherence, which strongly deviates from the itinerant Stoner model. Wesuggest that the local magnetic moments may play a crucial role in the ferromagnetic ordering and the electronicstructure of Fe 3GeTe 2, which will shed light on the generic understanding of itinerant magnetism in correlated materials. DOI: 10.1103/PhysRevB.101.201104 Magnetism is not only a fascinating quantum phenomenon, but also immensely inﬂuences various emergent proper-ties such as unconventional superconductivity [ 1–3], heavy fermion systems [ 4,5], topological quantum physics [ 6,7], and quantum critical behaviors [ 8,9]. In order to properly describe the magnetism in electronic materials, two paradig-matic frameworks have been established, concentrating ontwo opposing extremes: itinerant and local-moment mag-netism [ 10]. Within the weak-coupling itinerant picture as rep- resented by the well-known Stoner model, the spin-polarizedexchange splitting of electron bands drives the long-rangemagnetic ordering in metallic systems, while the local mag-netic moments take charge of the magnetism mainly in insu-lating materials according to the localized Heisenberg model.However, to distinguish these two mechanisms is usuallychallenging, especially in correlated materials, where the localmagnetic moments, although screened by itinerant electrons,strongly modify the quasiparticle energy bands [ 11]. Such competition between local and itinerant magnetism has beenwell demonstrated by the longstanding debate regarding thenature of magnetic ordering in the cuprate and iron-basedsuperconductors [ 3,12–15]. *These authors contributed equally to this work. †lxyang@tsinghua.edu.cn ‡yulin.chen@physics.ox.ac.ukRecently, Fe 3GeTe 2, as a representative van der Waals fer- romagnet, has been intensively studied due to the realizationof tunable room-temperature ferromagnetism in its thin ﬁlms[16,17]. Besides, bulk Fe 3GeTe 2also exhibits fertile and in- triguing properties, such as an extremely large anomalous Halleffect induced by topological nodal lines [ 18], Kondo lattice physics [ 19], a strongly enhanced electron mass [ 20], and a magnetocaloric effect [ 21]. Although it is widely believed that the ferromagnetism in Fe 3GeTe 2is itinerant in nature [ 22], a local Heisenberg model can likewise properly describe theferromagnetic ordering in Fe 3GeTe 2[17]. There is even a debate regarding whether Fe atoms align ferromagneticallyor antiferromagnetically in Fe 3GeTe 2[23]. These intriguing yet mysterious properties allude to the possible effect oflocal moments in the ferromagnetism of Fe 3GeTe 2. There- fore, Fe 3GeTe 2provides a rare platform to investigate the interplay between the ferromagnetism, electronic structure,and correlation effects. It will be elucidative to investigate theelectronic structure of Fe 3GeTe 2, which is not yet adequate enough. In this Rapid Communication, we systematically inves- tigate the electronic structure of bulk Fe 3GeTe 2and its temperature evolution using high-resolution angle-resolvedphotoemission spectroscopy (ARPES) and an ab initio band- structure calculation. The measured band structure in theferromagnetic state is in overall agreement with our density-functional theory (DFT) calculation after renormalized by a 2469-9950/2020/101(20)/201104(6) 201104-1 ©2020 American Physical SocietyX. XU et al. PHYSICAL REVIEW B 101, 201104(R) (2020) -2.0-1.001.02.0 -0.8-0.4 0.00.40.80.6 0.4 0.2 0 300 200 100 0 280 260 240 220 200 300 200 100 00.16 0.12 0.08 0.04 0 300 200 100 0M (B/Fe) ( cm) H (%) H = xx M (B/Fe) B (T) T (K) T (K) T (K)(a) (b) )d( )c(FC ZFC (c) H =xx(((dd)Tc TcTc FIG. 1. (a) Magnetic moment as a function of magnetic ﬁeld for Fe3GeTe 2measured at 2 K showing the magnetic hysteresis loop. (b) Temperature-dependent magnetization of Fe 3GeTe 2measured under zero-ﬁeld-cooling (ZFC) and ﬁeld-cooling (FC) conditions. (c) Temperature-dependent resistivity of Fe 3GeTe 2. (d) Temperature- dependent anomalous Hall angle of Fe 3GeTe 2. All the data were collected with the magnetic ﬁeld applied along the caxis. The vertical dashed line indicates the Curie temperature ( Tc). factor of 1.6. Interestingly, upon heating towards the Curie temperature ( Tc), we observe a minor change of the shape and position of the band dispersions, which is beyond the expec-tation of the itinerant Stoner model. Therefore, we argue thatthe local magnetic moments are crucial in the ferromagnetismand the electronic structure of Fe 3GeTe 2, although it is a pro- totypical itinerant ferromagnet. Our results provide importantinsights into not only the nature of ferromagnetism and otherproperties of Fe 3GeTe 2, but also a generic understanding of itinerant magnetism in correlated materials. High-quality Fe 3GeTe 2s i n g l ec r y s t a l so fs i z eo f5 ×5× 0.1m m3were synthesized by a chemical transport method with iodine as the transport agent [ 16]. ARPES data were taken with various photon energies at beamline I05 of theDiamond Light Source (DLS) under proposal No. SI20683-1,beamline 9A of the Hiroshima Synchrotron Radiation Cen-ter (HSRC), and beamline 5-2 of the Stanford SynchrotronRadiation Laboratory (SSRL). Scienta R4000 electron an-alyzers are equipped at all three beamlines. The overallenergy and angular resolutions are 15 meV and 0.3°. Theelectronic structures of bulk Fe 3GeTe 2were calculated using DFT with the projected augmented-wave method as imple-mented in the Vienna ab initio simulation package [ 24,25]. The exchange correlation was considered in the Perdew-Burke-Ernzerhof (PBE) approximation [ 26]. The cutoff en- ergy for the plane-wave basis was 400 eV and the recip-rocal space integrations were calculated by summing in a/Gamma1-centered 12 ×12×3 mesh. The convergence of the mesh has also been checked in our calculations. Experimental lat-tice parameters were used with relaxations performed untilthe Feynman-Hellman force on each atom was smaller than0.001 eV /Å. The atomic structure was further optimized by applying a van der Waals (vdW) correction with the DFT-D3method of Grimme [ 27]. -1.2-0.8-0.40 -1.0 0 1.0 K K -1.0 0 1.0M M -1.0 0 1.0KMK k\|\| (Å-1)E - EF (eV) -0.8-0.6-0.4-0.20 7 6 5 4 3 kz (Å-1)E - EF (eV) kx (Å-1)ky (Å-1) 1.0 0 -1.0 1.0 0 -1.0MK High Low -3.0-2.0-1.001.02.0 EX EX spin up spin downEXE - EF (eV)FM NM(a)(b) (c) (d) (e) (f) (g) K M K M FIG. 2. (a) Band structure along ¯/Gamma1¯Ashowing strong kzdispersion in Fe 3GeTe 2measured using photons ranging from 20 to 180 eV . (b) Fermi-surface map obtained by integrating ARPES intensity inan energy window of 20 meV near the Fermi energy ( E F). (c)–(e) Band dispersion along different high-symmetry directions. The black dashed curves are guides to the eyes for the band dispersions. (f),(g) DFT calculation of the band dispersions in ferromagnetic (FM) and nonmagnetic (NM) states. Data in (a) and (b)–(e) were collected using 114 eV photons with linearly (a) horizontal and (b)–(e) verticalpolarizations. Fe3GeTe 2crystallizes into a layered hexagonal structure with the space group of P63/mmc (No. 194). In the ferromag- netic state, the magnetic moments of all the Fe atoms alignalong the caxis below 225 K [ 21–23]. Figure 1shows the magnetic transport measurements on Fe 3GeTe 2with a ﬁeld applied along the caxis. We observe a clear magnetic hys- teresis loop showing the ferromagnetic nature of Fe 3GeTe 2. The magnetic moment saturates to a value of about 1 .6μB/Fe above 0.4 T [Fig. 1(a)]. Figure 1(b) shows the temperature dependence of the magnetization. The Tcof Fe 3GeTe 2is determined to be about 225 K, at which the temperature-dependent resistivity shows an anomaly [Fig. 1(c)], in con- sistence with previous measurements [ 20–22,28]. Below T c, Fe3GeTe 2exhibits a large anomalous Hall effect, as shown by the temperature-dependent anomalous Hall angle in Fig. 1(d), which has been attributed to the topological nodal lines in thesystem [ 18]. 201104-2SIGNATURE FOR NON-STONER FERROMAGNETISM IN … PHYSICAL REVIEW B 101, 201104(R) (2020) -0.4-0.20 0.8 0.4 0 0.8 0.4 0 0.8 0.4 0 0.8 0.4 0 0.8 0.4 0 -0.4-0.20 E - EF)Ve( 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4 -0.4 -0.2 0 0.8 0.4 0 0.8 0.4 0 0.8 0.4 0 0.8 0.4 0 0.8 0.4 0 -0.4 -0.2 0 E - E F ) V e ( 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4 1.6 1.2 0.8 0.4E - EF)Ve( High Low k\|\| (Å-1) 0 -0.2 -0.4 E - EF (eV)0 -0.2 -0.4 E - EF (eV))stinu .bra( ytisnetnI(a) (b) )f( )e( )d( )c()v( )vi( )iii( )ii( )i( )v( )vi( )iii( )ii( )i(230 K 205 K 140 K 80 K 80 K 140 K 220 K 240 K K K M K M K M K M K MK K K K near near 24 K 70 K 100 K 120 K 145 K 150 K 200 K 220 K T (K)0.8 0.6 0.4 0.2 250 200 150 100 50 0Å( MH WF-1) band near 50 meV band near EF TC ) e ( )stinu .bra( aerA kaeP T (K)250 200 150 100500TC near near K C near K 5.2 K 20 K 60 K 100 K 140 K 180 K 220 K 260 K5.2 K 20 K FIG. 3. (a), (b) Temperature evolution of the band dispersions near the (a) ¯/Gamma1and (b) ¯Kpoints. (c), (d) Energy distribution curves (EDCs) integrated in a momentum window of 0 .25 Å−1near the (c) ¯/Gamma1and (d) ¯Kpoints. (e) The peak area of the EDCs in (c) and (d) as a function of temperature. EDC peak area is obtained by integrating the EDC in an energy window of 200 meV . (f) The full width at half maximum (FWHM)of the momentum distribution curves (MDCs) as a function of temperature. The MDCs are integrated in an energy window of 45 meV and ﬁtted with a Lorentzian to extract its FWHM. The orange and blue curves are the guides to the eyes for the temperature evolution of the peak area and MDC width. Data in (a) and (b) were collected with photons at 27 and 114 eV , respectively. Figure 2investigates the electronic structure of Fe 3GeTe 2 in the ferromagnetic state at 10 K. Despite the weak interlayer coupling, we observe a strong kzvariation of the energy bands in Fig. 2(a) (see Supplemental Material [ 29]), suggesting strong interlayer coupling that can play an important rolein the ferromagnetism of Fe 3GeTe 2[17]. Figure 2(b) shows the Fermi surface in the kx−kyplane measured with 114 eV photons. We observe a hexagonal Fermi pocket near ¯/Gamma1, with a complex texture structure inside. There is a small electronpocket near ¯Kand a distribution of blurred spectral weight near ¯M, in consistence with previous ARPES measurements and band-structure calculations [ 19]. Figures 2(c)–2(e) show the band dispersions along the high-symmetry directions. Dueto the multiorbital nature and k zdispersion of the energy bands, the measured Fermi surface and band dispersionsstrongly depend on the photon polarization and photon energy(see the Supplemental Material [ 29]). Using 114 eV photonswith linearly vertical polarization, we observe mainly four bands in an energy range of 1.2 eV below the Fermi energy(E F). The αandγbands contribute to the hole pockets near ¯/Gamma1, while the δband contributes to the small electron pocket near ¯K. Our DFT calculation in Fig. 2(f) suggests that the calcu- lated bands need to be renormalized by a factor of about 1.6 inorder to obtain an overall agreement with the experiment (seeSupplemental Material [ 29]). After renormalization, our cal- culation is consistent with previous dynamical mean-ﬁeld the-ory (DMFT) calculations that suggest a relatively large Hub-bard interaction in Fe sites and conﬁrm the importance of elec-tron correlation in the ferromagnetism of Fe 3GeTe 2[18,20]. By comparing the calculated and measured δband, we obtain an electron effective mass enhancement by a factor of about1.6 (see Supplemental Material [ 29]), and we do not observe mass enhancement more signiﬁcant than this value from theFermi velocity of other bands. This value is much smaller 201104-3X. XU et al. PHYSICAL REVIEW B 101, 201104(R) (2020) than that obtained from the Sommerfeld coefﬁcient, which remains mysterious [ 20,22]. The calculated band structure in the ferromagnetic state shows a large exchange splitting[∼1.5 eV as indicated by the arrows in Fig. 2(f)] compared with the nonmagnetic state [Fig. 2(g)], which is believed to drive the ferromagnetic ordering in Fe 3GeTe 2according to the Stoner mechanism [ 22]. In order to unveil the nature of the ferromagnetism in Fe3GeTe 2, we track the temperature evolution of its electronic structure in Fig. 3. Figures 3(a) and3(b) show the band dis- persions around ¯/Gamma1and ¯Kat selected temperatures measured with 27 eV (horizontally polarized) and 114 eV (verticallypolarized) photons, respectively. We observe different bandsηandβnear ¯/Gamma1due to the k zdispersion and polarization dependence of the energy bands (see Supplemental Material[29]). Both the ηandδbands only slightly shift towards lower binding energies with a spectral weight strongly suppressed athigh temperatures. Figures 3(c)and3(d) show the temperature evolution of the energy distribution curves (EDCs) near ¯/Gamma1and ¯K, respectively. The EDC peak intensity quickly decreases with increased temperature, as presented in Fig. 3(e).I n addition, the ARPES spectra are strongly broadened uponwarming towards T c, suggesting a dramatic enhancement of the disordering level in the system, as estimated by thequickly increased momentum distribution curve (MDC) widthin Fig. 3(f), which will scatter the coherent electron states near E Finto incoherent states and suppress the intensity near EF. Both the suppression of the quasiparticle spectral weight and the increase of MDC width show an anomaly near Tc, suggesting an intimate correlation between the broadening ofthe spectra with the magnetism in the system. Thus, we spec-ulate that the enhanced magnetic ﬂuctuation with increasedtemperature [ 22] dominates the observed spectra broadening, although other effects such as the electron-phonon interactionmay also contribute to the broadening. Notably, we observethe broadening of the energy bands in a large energy range,which is out of the expectation of itinerant spin ﬂuctuationsthat mainly affect the states near E F. Within the Stoner model, a prototypical itinerant ferromag- net is expected to exhibit a temperature-dependent exchangesplitting that disappears above T c[Figs. 2(f)and2(g)]. How- ever, we do not observe a considerable change in the elec-tronic structure with temperature in Fig. 3, as further tracked by MDCs and EDCs in Figs. 4(a) and 4(b). We quantify the band shift /Delta1Eby tracking the temperature evolution of either EDC or MDC peaks ( /Delta1E=/Delta1kd E/dk, where /Delta1k is the MDC peak shift and dE/dkis the dispersion slope) and summarize the result in Fig. 4(c). We also compare the band shift with the temperature-dependent exchange splittingestimated from the magnetic moment that is scaled to half ofthe DFT calculated exchange splitting (about 1 eV taking theband renormalization into account). In a temperature rangeas large as 220 K, we observe a minuscule band shift (about10% of the exchange splitting), which strongly deviates fromthe expectation within the itinerant Stoner model [Fig. 4(c); also see Supplemental Material [ 29]]. On the contrary, our observation ﬁts better to the temperature-independent modelbased on the localized exchange interaction as indicated by thehorizontal dashed line in Fig. 4(c) [30]. Clearly, a completely itinerant mechanism is not capable of explaining our experi- E - EF (eV))stinu .bra( ytisnetnI 0.8 0.4 0 )stinu .bra( ytisnetnI -0.8 -0.4 0 24 K 70 K 100 K 120 K 145 K 150 K 200 K 220 Kk\|\| (Å-1) k\|\| = 0.94 Å-1E - EF =50 meV band band 0.40.20.0 )Ve( tfihs ygrenE 300 250 200 150 100 50 0 T (K) band MDC band EDC Estimated from magnetic momentTC (a) (b) (c) FIG. 4. (a) MDCs integrated in an energy window of 50 meV near−50 meV at selected temperatures. (b) EDCs integrated in a momentum window of 0 .15 Å−1near 0 .94sÅ−1at selected tem- peratures. The gray dashed curves in (a) and (b) are guides to theeyes for the shift of EDC and MDC peaks with temperature. (c) The shift of energy bands as extracted from MDCs and EDCs as a function of temperature. The gray circles are the energy shiftestimated from the magnetic moments in Fe 3GeTe 2measured with neutron scattering in Ref. [ 21]. The gray and green curves are guides to the eyes for the temperature dependence of the energyshift. The temperature-independent dashed line shows the expected temperature evolution of the energy shift within a local-moment ferromagnet. ment. The local magnetic moment, on the other hand, should be taken into account in order to properly describe the ferro-magnetism in Fe 3GeTe 2. Our conclusion is supported by the observation of Kondo lattice behavior and the explanation ofthe ferromagnetism in Fe 3GeTe 2within the Heisenberg model [17,19]. Our results mimic the behaviors of other itinerant fer- romagnets such as SrRuO 3, in which a strong mass 201104-4SIGNATURE FOR NON-STONER FERROMAGNETISM IN … PHYSICAL REVIEW B 101, 201104(R) (2020) enhancement, electron decoherence, and the local character of ferromagnetism indicated by a minor band shift withtemperature were observed [ 31]. However, the underlying microscopic interaction and the impact of local moments onthe electronic structure are different in these two systems.The quasiparticle effective mass is strongly renormalized byan electron-boson interaction in SrRuO 3, as demonstrated by a kink structure in the band dispersion. On the contrary,we did not observe any kink structure in Fe 3GeTe 2and the electron effective mass is only enhanced by a factor of 1.6due to the electronic correlation effect (see SupplementalMaterial [ 29]), although the speciﬁc heat measurement alludes to a dramatic enhancement of the quasiparticle effective mass[20,22]. The electron-electron interaction is more important in the magnetism of Fe 3GeTe 2[20] than in SrRuO 3, since the effective on-site Coulomb interaction between Fe 3 delectrons is much stronger that between Ru 4 delectrons. On the other hand, the persistent exchange splitting in Fe 3GeTe 2above Tcalso resembles the results in the canonical ferromagnets Fe and Ni. Although the exchange splitting in Fe and Nishows a strong temperature dependence, it survives aboveT c[30,32–34], which was attributed to the retained local magnetic moments after the loss of long-range ferromagneticordering [ 31]. Considering the dominant role of Fe 3 dorbitals in the ferromagnetism of Fe 3GeTe 2, it is reasonable that the local magnetic moments play similar roles in Fe 3GeTe 2 and Fe.In conclusion, we have presented a systematic temperature evolution of the electronic structure of the van der Waalsferromagnet Fe 3GeTe 2. We observe substantial electron decoherence in a large energy range and a minor bandshift upon warming towards T c, which are against a weak-coupling itinerant picture. We argue that the localmagnetic moments should be taken into account in order tounderstand the ferromagnetic ordering in the prototypicalitinerant ferromagnet Fe 3GeTe 2. Our results resemble the observations in other prototypical itinerant ferromagneticsystems, which will deepen our generic understanding of themagnetism in condensed materials. We thank M. Arita, K. Shimada, M. Hashimoto, S. W. Jung, and C. Cacho for the experimental support. This workwas supported by the National Natural Science Foundationof China (Grants No. 11774190, No. 11674229, and No.11634009), the National Key R&D Program of China (GrantsNo. 2017YFA0304600 and 2017YFA0305400), and EPSRCPlatform Grant (Grant No. EP/M020517/1). The work at theUniversity of Washington was supported by NSF MRSEC atUW (DMR-1719797) and the Gordon and Betty Moore Foun-dation’s EPiQS Initiative, Grant No. GBMF6759 to J.-H.C.Y .F.G. acknowledges the support from the Shanghai PujiangProgram (Grant No. 17PJ1406200). L.X.Y . acknowledges thesupport from Tsinghua University Initiative Scientiﬁc Re-search Program. [1] X. Chen, P. Dai, T. Xiang, D. Feng, and F.-C. Zhang, Nat. Sci. Rev. 1,371(2014 ). [2] J. Paglione and R. L. Greene, Nat. Phys. 6,645(2010 ). [3] E. Dagotto, Rev. Mod. Phys. 66,763(1994 ). [ 4 ] G .R .S t e w a r t , Rev. Mod. Phys. 56,755(1984 ). [5]Magnetism in Heavy Fermions , edited by H. B. Radousky, Series in Modern Condensed Matter Physics V ol. 11 (WorldScientiﬁc, Singapore, 2000). [6] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83,1057 (2011 ). [7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82,3045 (2010 ). [8] D. A. Sokolov, M. C. Aronson, W. Gannon, and Z. Fisk, Phys. Rev. Lett. 96,116404 (2006 ). [9] H. 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PhysRevB.81.214524.pdf	Compensated electron and hole pockets in an underdoped high- Tcsuperconductor Suchitra E. Sebastian,1N. Harrison,2P. A. Goddard,3M. M. Altarawneh,2C. H. Mielke,2Ruixing Liang,4,5D. A. Bonn,4,5 W. N. Hardy,4,5O. K. Andersen,6and G. G. Lonzarich1 1Cavendish Laboratory, Cambridge University, J. J. Thomson Avenue, Cambridge CB3 OHE, United Kingdom 2Los Alamos National Laboratory, LANL, Los Alamos, New Mexico 87545, USA 3Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom 4Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada V6T 1Z4 5Canadian Institute for Advanced Research, Toronto, Canada M5G 1Z8 6Max-Planck-Institut fuer Feskoerperforschung, Stuttgart, Germany /H20849Received 14 February 2010; published 28 June 2010 /H20850 We report quantum oscillations in the underdoped high-temperature superconductor YBa 2Cu3O6+xover a wide range in magnetic ﬁeld 28 /H11349/H92620H/H1134985 T corresponding to /H1101512 oscillations, enabling the Fermi surface topology to be mapped to high resolution. As earlier reported by Sebastian et al. /H20851Nature /H20849London /H20850454, 200 /H208492008 /H20850/H20852, we ﬁnd a Fermi surface comprising multiple pockets, as revealed by the additional distinct quantum oscillation frequencies and harmonics reported in this work. We ﬁnd the originally reported broad low-frequency Fourier peak at /H11015535 T to be clearly resolved into three separate peaks at /H11015460, /H11015532, and /H11015602 T, in reasonable agreement with the reported frequencies of Audouard et al. /H20851Phys. Rev. Lett. 103, 157003 /H208492009 /H20850/H20852. However, our increased resolution and angle-resolved measurements identify these frequen- cies to originate from two similarly sized pockets with greatly contrasting degrees of interlayer corrugation.The spectrally dominant frequency originates from a pocket /H20849denoted /H9251/H20850that is almost ideally two-dimensional in form /H20849exhibiting negligible interlayer corrugation /H20850. In contrast, the newly resolved weaker adjacent spectral features originate from a deeply corrugated pocket /H20849denoted /H9253/H20850. On comparison with band structure, the d-wave symmetry of the interlayer dispersion locates the minimally corrugated /H9251pocket at the “nodal” point knodal =/H20849/H9266/2,/H9266/2/H20850, and the signiﬁcantly corrugated /H9253pocket at the “antinodal” point kantinodal =/H20849/H9266,0/H20850within the Brillouin zone. The differently corrugated pockets at different locations indicate creation by translationalsymmetry breaking—a spin-density wave has been suggested from the suppression of Zeeman splitting for thespectrally dominant pocket. In a broken-translational symmetry scenario, symmetry points to the nodal /H20849 /H9251/H20850 pocket corresponding to holes, with the weaker antinodal /H20849/H9253/H20850pocket corresponding to electrons—likely re- sponsible for the negative Hall coefﬁcient reported by LeBoeuf et al. /H20851Nature /H20849London /H20850450, 533 /H208492007 /H20850/H20852. Given the similarity in /H9251and/H9253pocket volumes, their opposite carrier type and the previous report of a diverging effective mass in Sebastian et al. /H20851Proc. Nat. Am. Soc. 107, 6175 /H208492010 /H20850/H20852, we discuss the possibility of a secondary Fermi surface instability at low dopings of the excitonic insulator type, associated with themetal-insulator quantum critical point. Its potential involvement in the enhancement of superconducting tran-sition temperatures is also discussed. DOI: 10.1103/PhysRevB.81.214524 PACS number /H20849s/H20850: 74.25.Jb, 74.72. /H11002h, 71.18. /H11001y, 71.35. /H11002y I. INTRODUCTION Different bosonic physics from tightly bound pairs of fer- mions may arise depending on whether the binding takesplace between like particles or between particles and holes. 1 Such bosonic physics can potentially be involved in thephysics of high-temperature superconductivity. On one hand,strongly interacting pairs of like particles with zero momen-tum have been proposed to constitute the fabric of unconven-tional superconductors, 2,3while on the other, electron-hole pairs with ﬁnite momentum could condense into a competingstate with a superlattice, such as a spin-density wave. An-other realization of such particle-hole pairing could occur inthe limit of strong Coulomb coupling, where electron andhole pockets of identical size become susceptible to an exci-tonic insulator instability. 4–6In this study, we use high reso- lution quantum oscillation measurements to show that theunderdoped high temperature superconductor YBa 2Cu3O6+x /H20849Ref.7/H20850contains compensated electron and hole pockets, po- tentially predisposing it to excitonic electron-hole pair con-densation.Quantum oscillations have been measured on underdoped YBa 2Cu3O6+xusing in-plane transport,8,9out-of-plane transport,10torque,11–14and contactless conductivity16–18ex- periments. In this work, the power of quantum oscillations isenhanced by the technique we employ that utilizes interlayerdispersion measurements to enable a momentum-space iden-tiﬁcation of individual Fermi surface sections—details ofwhich are provided in Sec. II. Quantum oscillations are mea- sured on high quality detwinned single crystals ofYBa 2Cu3O6+x/H20849x=0.54,0.56 /H20850/H20849Ref. 7/H20850as a function of mag- netic ﬁeld and of angle. Measurements are made down to /H110151 K with the sample immersed in4He medium, and the contactless conductivity technique used to obtain a highvalue of signal-to-noise ratio. Out-of-plane rotation measure-ments are performed in pulsed magnetic ﬁelds up to 65 T,and two-axis /H20849in-plane and out-of-plane /H20850rotation measure- ments are performed in dc ﬁelds up to 45 T. Additional ex-perimental and analysis details are provided in Appendix A. Our angle-resolved measurements show that the dominant series of low frequency oscillations 8arises not from one, butPHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 Selected for a Viewpoint inPhysics 1098-0121/2010/81 /H2084921/H20850/214524 /H2084917/H20850 ©2010 The American Physical Society 214524-1from two pockets of similar size, the distinct topologies of which point to different positions in the Brillouin zone oncomparison with the interlayer dispersion, which has d-wave symmetry in underdoped YBa 2Cu3O6+x/H20849Refs. 19and20/H20850 /H20851see Fig. 1/H20849a/H20850/H20852. On comparison with the interlayer hopping integrals in YBa 2Cu3O6+x,19,20we show that the prominent frequency corresponding to the /H9251pocket with minimal inter- layer corrugation is likely to represent a section of Fermisurface at the /H20849/H11006 /H9266/2,/H11006/H9266/2/H20850nodal location in the Brillouin zone where holes are expected to nucleate on doping. Incontrast, the more recently measured satellite frequencies 13 of smaller amplitude correspond to a strongly corrugatedpocket /H20849 /H9253/H20850close to the antinodal location in the Brillouin zone, suggestive of electron carriers. II. RESULTS A. Interlayer corrugation High experimental resolution of the quantum oscillation frequency is key to resolving and characterizing the two ad-jacent peaks ﬂanking the larger central peak /H20849shown in Fig. 2/H20850. The weaker spectral features at 460 /H110062 and 603 /H110062T o n either side of the central frequency at 533 /H110062 T are distin- guished on account of their separation being well above thefrequency resolution limit of our current experiment /H20849/H9004F lim /H11015/H208511//H9004/H208491//H92620H/H20850/H20852/H1101542 T /H20850. The three distinct peaks in the Fourier transform /H20849shown in Fig. 2/H20850are observed both with-out /H20849black curve /H20850and with /H20849red curve /H20850a Hann window to reduce diffraction artifacts. Evidence for the three resolved low-frequency oscillations originating from similar volume electron and hole pocketswith different degrees of corrugation /H20851shown schematically in Fig. 1/H20849a/H20850/H20852in addition to the previously resolved high fre- quency oscillation 12is presented in Fig. 3, which shows quantum oscillations and Fourier transforms measured at dif-ferent angles /H9258between the crystalline caxis and H, and in Fig.4where the separately resolved quantum oscillation fre- quencies /H20849F/H9251,F/H9252,F/H9253,neck, and F/H9253,belly /H20850are plotted versus /H9258. The subtle beat pattern modulating the amplitude of the mea-sured oscillations /H20851Fig.3/H20849a/H20850, upper panel of upper inset /H20852and the multiple low frequency peaks in the Fourier transformsuggest two different Fermi surface sections yielding closelyspaced low frequencies. To separate /H9251and/H9253pocket contri- butions, we use three independent analysis methods and ﬁndthem to yield mutually consistent results for a signiﬁcantdifference in corrugation between the /H9251and/H9253pockets. Given the layered character of the cuprate family of high- temperature superconductors, the quasi-two-dimensional na-ture of the Fermi surface needs to be factored into the quan-tum oscillation analysis. Speciﬁcally, for quasi-two-dimensional materials, interlayer corrugation introducesphase smearing that leads to a distinctive beat pattern whichhas been well characterized. 22The location and separation in magnetic ﬁeld of the amplitude zeroes /H20849nodes /H20850constituting the beat pattern are a topologically constrained function of FIG. 1. /H20849Color online /H20850/H20849a/H20850A schematic of the small hole /H20849blue /H20850pocket /H9251and small electron /H20849red/H20850pocket /H9253, illustrating greatly contrasting degrees of corrugation and approximate locations within the sector of the Brillouin zone bounded by /H208490,0/H20850,/H20849/H9266/2,/H9266/2/H20850, and /H20849/H9266,0/H20850, shown for −2/H9266/H11021kz/H110212/H9266. Also shown in semitransparent fashion is the larger hole orbit /H9252/H20851corresponding to a separate Fermi surface section or potentially a magnetic breakdown orbit /H20849Ref.12/H20850/H20852. The degree of corrugation is determined from angle-dependent Fourier transforms shown in Fig. 3and frequencies shown in Fig. 4/H20849a/H20850. On using the Onsager relation, the three pockets collectively yield an effective hole doping of p=11.7/H110060.2%. /H20849b/H20850Estimates of the ratio of the exciton binding energy to kinetic energy Ry/H11569/Ekeand effective exciton separation rsas a function of oxygen concentration x, using the expressions in Appendix B and effective masses from Refs. 16and17, with arrows indicating the appropriate axes. Also shown are the corresponding values of Tc/H20849renormalized /H20850from Ref. 7and different shadings representing the insulating and metallic regimes at low T.SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-2the depth of corrugation and may be used to extract the size of the corrugation. For this purpose, we use an expressionthat explicitly treats the form of phase smearing in layeredsystems 24/H20851Eq. /H208491/H20850/H20852rather than the standard Lifshitz- Kosevich expression21which does not capture this form of interference /H20849see Appendix G /H20850. We perform an unconstrained ﬁt /H20849red line—solid gray on- line/H20850of two corrugated Fermi surface sections to the mea-sured oscillations in the 65 T magnet /H20849black line /H20850for a sample of oxygen concentration x=0.56 at /H9258=0 /H20851Fig. 3/H20849a/H20850, upper inset /H20852. Oscillations are ﬁt to the expression /H9004f=/H20858 i=/H9251,/H9253/H9004fi,0RTRDJ0/H208732/H9266/H9004Fi,/H9258 /H92620Hcos/H9258/H20874cos/H208732/H9266Fi /H92620H/H20874, /H208491/H20850 incorporating a summation over independent /H9251and/H9253corru- gated cylinders.22,24,25The spin splitting factor Rs/H20849Ref. 21/H20850 does not appear in this expression since it has been reportedto be close to unity for the dominant /H9251frequency /H20849Refs. 14 and 15/H20850. Here, /H9004fi,0 is the amplitude, RT =/H208492/H92662kBmi/H11569T/e/H6036/H92620H/H20850/sinh /H208492/H92662kBmi/H11569T/e/H6036/H92620H/H20850is a tem- perature damping factor where mi/H11569is the quasiparticle effec- tive mass obtained from the ﬁt shown in Fig. 3,RD =exp /H20849−/H9003i /H92620H/H20850is a Dingle damping factor where /H9003iis a damping constant,21/H9004Fi,/H9258is the depth of corrugation of sheet “ i”a t angle/H9258, while the Bessel function J0captures the interfer- ence due to phase smearing, with characteristic “neck” and“belly” extremal frequencies F i,neck,belly cos/H9258=Fi/H11006/H9004Fi,/H9258 /H208492/H20850 for each section.22,24,25On performing an unconstrained ﬁt to Eq. /H208491/H20850/H20849shown in Fig. 3/H20849a/H20850upper inset /H20850we obtain corruga- tions /H9004F/H9251,0/H1135111 T and /H9004F/H9253,0=72/H110064 T, while F/H9251,0 =531/H110062 T and F/H9253,0=532/H110062 T are the same within experi- mental uncertainty. These parameters correspond to three fre-quencies F /H9253,neck=460/H110064T , F/H9251,0=531/H110062T /H20849the splitting being unresolvably small /H20850, and F/H9253,belly=604/H110064 T in close agreement with those resolved in the Fourier transform inFig.2. Similar frequencies were recently reported in Ref. 13, in which case the experimental uncertainty was larger due tothe reduced range in 1 /H/H20851see Fig. 2/H20849b/H20850inset /H20852. The analysis presented here is performed on data spanning between /H110159 and 12 measured oscillations. The value of effective hoppingbetween the nearest-neighbor planes of two adjacent unitcells t c/H20849for a tight-binding model assuming a parabolic form of in-plane dispersion /H20850is given by /H9004Fe/H6036/2m*, correspond- ing to a value tc/H9253/H110152.4 meV for the signiﬁcantly warped /H9253 pocket, and tc/H9251/H333550.4 meV for the less warped /H9251pocket. The monotonic ﬁeld dependence of the dominant /H9251com- ponent /H20849dot-dashed green line /H20850identiﬁed in the present ex- periment implies that it can be reliably subtracted from theraw data /H20851on performing an unconstrained ﬁt to Eq. /H208491/H20850/H20852, enabling the residual beat pattern of the /H9253oscillations /H20849purple line/H20850to be extracted /H20851lower panel of the upper inset to Fig. 3/H20849a/H20850/H20852. Distinct nodes are observed in the residual /H9253oscilla- tions. We turn to the expected locations of zeros in the am-plitude due to the interference pattern to conﬁrm that theycoincide with the observed nodes. The values of magneticﬁeld at which amplitude zeros are expected due to interfer-ence are given by /H92620Hn,icos/H9258=8/H9004Fi,/H9258 4n+3. /H208493/H20850 Equation /H208491/H20850/H20849Ref. 24/H20850can be expanded as the interference between neck and belly frequencies Fi−/H9004Fi,/H9258andFi+/H9004Fi,/H9258, respectively. The neck and belly frequencies interfere de-structively whenever they have a relative phase difference FIG. 2. /H20849Color online /H20850/H20849a/H20850Quantum oscillations made using the contactless conductivity method on a sample of YBa 2Cu3O6.54over a broad range of magnetic ﬁelds 28 /H11021/H92620H/H1102185 T at /H110151.5 K, cor- responding to /H1101512 oscillations of the prominent /H9251frequency. The experimental data below and above 65 T are from measurementsmade using the same sample on the same probe in two differentpulsed magnets. /H20849b/H20850A Fourier transform of the quantum oscillations both without /H20849thick black curve /H20850and with /H20849thin red curve /H20850a Hann window applied. The inset shows a comparison of the separation infrequency F /H9253,neck−F/H9251orF/H9251−F/H9253,belly /H20849shown by the horizontal dot- ted line /H20850with the frequency resolution limit /H9004Flim /H11015/H208511//H9004/H208491//H92620H/H20850/H20852 /H20849left axis /H20850, as a function of the number of /H9251oscil- lation periods n/H9251/H20849bottom axis /H20850. The frequency separation exceeds the frequency resolution for n/H9251/H114078, enabling all three frequencies to be clearly resolved in Fourier transforms and ﬁts of the quantumoscillations. All three frequencies are found to be present in a Fou-rier transform only at high enough resolution /H20849i.e. when the number of oscillations exceeds eight /H20850, and irrespective of the choice of win- dow function /H20849e.g. Boxcar, Hann, Blackman /H20850, conﬁrming that they are genuine spectral features.COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-3equal to an odd multiple of /H9266, which occurs at values of the ﬁeld Hngiven by Eq. /H208493/H20850. Using the ﬁt value of /H9004F/H9253,0, the expected nodes for n =2,3,4 are indicated by the vertical red lines in the lower panel of the upper inset to Fig. 3/H20849a/H20850. We ﬁnd these to coin- cide very well with the observed nodes in the residual /H9253 oscillations, establishing that the frequencies F/H9253,neck /H11015460 T and F/H9253,belly /H11015604 T correspond to the neck andbelly extremal frequencies of a single corrugated /H9253cylinder, distinct from the /H9251section. Indeed, these frequencies corre- spond to those that appear in the extracted Fourier transformin Fig. 3/H20849depicted before and after subtraction of the ﬁtted /H9251 oscillation component /H20850and in Fig. 2. In Figs. 3/H20849b/H20850,3/H20849c/H20850, and 4, we extend this analysis to a second sample of oxygen concentration x=0.54, for which oscillations are measured at several different angles /H9258. Fits to FIG. 3. /H20849Color online /H20850/H20849a/H20850Fourier transform before /H20849thin purple /H20850and after /H20849thick magenta /H20850subtracting the dominant /H9251component of an unconstrained ﬁt to Eq. /H208491/H20850/H20849dot-dashed green line in the upper inset /H20850to the quantum oscillations /H20849black line in the upper inset /H20850measured in a sample of YBa 2Cu3O6+xwith x=0.56 at T/H110151.5 K using the contactless conductivity technique /H20849see Appendix A /H20850. The weaker F/H9253,neckand F/H9253,belly frequency peaks in the Fourier transform are clearly visible after the dominant oscillatory component of the oscillations has been subtracted /H20849unlike expected behavior of sidelobes, that would vanish along with the subtraction of the central dominant frequency /H20850.A polynomial is used to ﬁt the background and the waveform is modulated by a Hann window prior to Fourier transformation. The prominentoscillations and harmonics are indicated. Extremal Fermi surface cross sections given by A k=/H208492/H9266e//H6036/H20850Fare obtained from the observed frequencies F/H9251,F/H9253,neck, and F/H9253,bellyand harmonics 2 F/H9251,2F/H9253,neck, and 2 F/H9253,bellyas indicated; here H=/H20841H/H20841. The red line /H20849solid gray online /H20850in the upper inset is an unconstrained ﬁt to Eq. /H208491/H20850, where the dot-dashed green line and the dashed blue lines are the /H9251and/H9253components, respectively. The purple curve /H20849upper inset, lower panel /H20850shows the result of subtracting the /H9251component /H20849dot-dashed green line /H20850of the unconstrained ﬁt to Eq. /H208491/H20850from the raw data, revealing a distinctive beat pattern in the lower panel of the inset. The vertical lines represent the ﬁelds at which nodes occur for n=2,3,4 from Eq. /H208493/H20850using the ﬁt value of /H9004F/H9253,0. The lower inset shows the temperature dependences of the F/H9251,F/H9253,neck, and F/H9253,belly amplitudes /H20849the latter two renormalized to equalize the amplitudes at the highest temperature /H20850, indicating similar effective masses m/H11569for the /H9251and/H9253pockets. A ﬁt to /H9004f=a0RT/H20849Ref. 21/H20850yields m/H9251/H11569=1.56/H110060.05me,m/H9253,neck/H11569=m/H9253,belly/H11569=1.6/H110060.2me /H20849where meis the free electron mass /H20850./H20851/H20849b/H20850and /H20849c/H20850/H20852Examples of Fourier transforms of oscillations measured in a sample with x=0.54 at T /H110151.5 K and different angles /H9258/H20851data in Figs. 4/H20849b/H20850and4/H20849c/H20850/H20852before and after subtracting the ﬁtted /H9251component /H20851see Fig. 4/H20849b/H20850/H20852as described in the text /H20849expanded to show the ﬁner high frequency structure /H20850—further Fourier transforms before and after subtraction are shown in the Appendix D. Note that the higher frequency F/H9252/H20849when observed /H20850is 3.14 /H110060.05 times larger than F/H9251, inconsistent with a harmonic of F/H9251 /H20849Refs. 12and16/H20850.SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-4Eq. /H208491/H20850similar to those in Fig. 3/H20849a/H20850are made to the data in Fig. 4/H20849b/H20850as a function of angle /H20849here/H9004F/H9251,0/H1135110 T, /H9004F/H9253,0 =74/H110064T , F/H9251,0/H11015535/H110062 T, and F/H9253,0/H11015533/H110062T /H20850. Figure 4/H20849c/H20850shows the residual /H9253oscillations on subtracting the dominant ﬁtted /H9251component /H20851dot-dashed green lines in Fig. 4/H20849b/H20850/H20852, and Fig. 4/H20849a/H20850shows the extracted peaks in the Fourier transform /H20851shown in Figs. 3/H20849a/H20850and3/H20849b/H20850and Appendix D /H20852of the subtracted data /H20849hollow symbols /H20850and unsubtracted data /H20849solid symbols /H20850. The angular dependent amplitude nodes ob- served in the residual /H9253oscillations in Fig. 4/H20849c/H20850and frequen- cies in Fig. 4/H20849a/H20850are compared with those expected for a single corrugated cylinder, by using Yamaji’s expression foran angular dependent frequency difference between the neckand belly of a single corrugated cylinder, 23 /H9004Fi,/H9258=/H9004Fi,0J0/H20849k/H20648ctan/H9258/H20850. /H208494/H20850 On substituting the ﬁt value of /H9004F/H9253,0into Eq. /H208494/H20850, the pre- dicted nodes for a single corrugated cylinder in Eq. /H208493/H20850are shown by lines in Fig. 4/H20849c/H20850, and the predicted neck and belly frequencies are shown by black lines in Fig. 4/H20849a/H20850. We see that excellent agreement with the data is obtained in both Figs.4/H20849a/H20850and4/H20849c/H20850independent of any ﬁtting parameters, conﬁrm- ing a single corrugated /H9253Fermi surface section /H20849distinct from /H9251/H20850. The small value of /H9004F/H9251implies that the putative nodes for the/H9251Fermi surface section occur at ﬁelds too low for inter- ference between its neck and belly frequencies to be ob-served in the form of nodes. We can now understand why the /H9251cylinder yields a spectrally dominant peak at F/H9251,0 /H11015532 T in Fig. 3/H20849a/H20850—no signiﬁcant smearing of the quan- tum oscillation phase occurs on averaging over kzfor this section /H20851see diagram in Fig. 1/H20849a/H20850/H20852. The larger corrugation of the/H9253section, by contrast, implies that signiﬁcant phase smearing occurs on averaging over kzfor this section, with phase coherence being achieved only at the extrema F/H9253,neck andF/H9253,belly—accounting for the spectrally weak satellite fre- quencies ﬂanking the central peak in Fig. 3/H20849a/H20850. For a ﬁnal consistency check, we turn to the harmonics 2F/H9251,2F/H9253,neck, and 2 F/H9253,belly, which are observed for both samples /H20849x=0.54 and x=0.56 /H20850in Figs. 3/H20849a/H20850–3/H20849c/H20850/H20851plotted ver- sus/H9258in Fig. 4/H20849a/H20850together with the Yamaji prediction /H20852. Be- cause the harmonics appear in the Fourier transforms of theraw data /H20851Figs. 3/H20849a/H20850–3/H20849c/H20850and Appendix D /H20852, they provide a second estimate of each frequency /H20851Fig.3/H20849a/H20850/H20852that is indepen- dent of any subtraction of the predominant /H9251frequency and therefore independent of the previous ﬁts. A further advan-tage of harmonic detection is that the doubling of the fre-quency yields a twofold increase in the conﬁdence to whichthe separate Fermi surface sections can be resolved. The ab-solute frequency resolution obtained from the interval inmagnetic ﬁeld is therefore /H9004F lim/H11015/H208511//H9004/H208491//H92620H/H20850/H20852/2/H1101526 T. The lack of a peak splitting or deviation of F/H9251and its har- monic from F/H9251=F/H9251/cos/H9258over the entire angular range in Fig.4/H20849a/H20850implies an irresolvably small depth of corrugation /H9004F/H9251,0/H1135113 T, while the values of /H9004F/H9253,/H9258=/H208492F/H9253,belly FIG. 4. /H20849Color online /H20850/H20849a/H20850The product Fcos/H9258/H20849for both fundamental and harmonic—solid symbols for unsubtracted data and hollow symbols for subtracted data /H20850obtained from the Fourier analysis shown in Figs. 3/H20849b/H20850and3/H20849c/H20850and Appendix D plotted versus /H9258.F/H9253,neckand F/H9253,bellyare obtained by Fourier analysis after subtracting the ﬁtted /H9251component /H20851using Eq. /H208491/H20850/H20852. For clarity, the data have been symmetrized with respect to /H11006/H9258: both the swept /H9258data /H20851shown in Fig. 6/H20849a/H20850/H20852and measurements made over extensive range of /H9258in Ref. 14indicate the oscillations to be symmetric in /H11006/H9258, as expected for a detwinned single-phase orthorhombic crystal. Lines indicate the expected /H9258depen- dences according to Eqs. /H208492/H20850and /H208494/H20850, using the ﬁtted parameters in Fig. 3/H20849a/H20850. Consistency with Eq. /H208494/H20850indicates a deeply corrugated /H9253section of Fermi surface /H20849Refs. 22and23/H20850. The harmonics are obtained directly from peaks in Fourier transforms of the raw data—consistency between fundamentals and harmonics conﬁrms our ﬁnding of similarly sized but distinctly different /H9251and/H9253Fermi surface sections. /H20849b/H20850 Examples of quantum oscillations measured in YBa 2Cu3O6+xwith x=0.54 at different angles /H9258, as indicated /H20849where /H9272/H110150/H20850together with the ﬁtted/H9251component /H20851green lines, using Eq. /H208491/H20850/H20852as described in the text. /H20849c/H20850The quantum oscillations measured in YBa 2Cu3O6+xwith x =0.54 at different angles /H9258after subtracting the green ﬁts in /H20849b/H20850, yielding residuals dominated by the /H9253pocket oscillations. Curves are offset according to /H9258/H20849right axis /H20850. Also plotted /H20849red lines /H20850are the predicted values of Hnfrom Eqs. /H208493/H20850and /H208494/H20850using the ﬁt value of /H9004F/H9253,0.COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-5−2F/H9253,neck/H20850/4 coincide well with the expected values from the fundamental analysis /H20851shown in Fig. 4/H20849a/H20850/H20852. The observation of prominent harmonics for both the /H9251and/H9253pockets also enables us to estimate the relative size of the damping termfor each of the electron and hole pockets, which we ﬁnd tobe similar in magnitude: /H9003 /H9251/H11011/H9003/H9253/H11011102T, corresponding to a mean-free path of /H11011200 Å. We therefore ﬁnd the nodal analysis of the oscillations, Fourier transform analysis, Yamaji ﬁts and second harmonicdetection allto yield internally consistent results correspond- ing to two similarly sized /H9251and/H9253sections with signiﬁcantly different degrees of corrugation. Oscillations from two pock-ets with the same degree of corrugation, as expected for sig-niﬁcantly bilayer-split pocket areas in the underdopedcuprates, 13on the other hand, would not be consistent with our data /H20849see Appendix G /H20850. B. Correspondence with equal and opposite electron and hole pockets In the simplest model, our ﬁnding of Fermi surface sec- tions with signiﬁcantly different degrees of corrugation lo-cates them at different locations in the Brillouin zone. Wecompare the different measured interlayer hoppings with cal-culated values in a tight binding model of underdopedYBa 2Cu3O6+x/H20849Ref. 26/H20850to locate the different pockets. As shown in Refs. 19and20the low-energy in-layer dispersion /H20851k/H11013/H20849kx,ky/H20850/H20852of the Cu dx2−y2-like band is /H9255/H20849k/H20850=−4 t/H20875u/H20849k/H20850+2rv/H20849k/H208502 1−2 ru/H20849k/H20850/H20876, /H208495/H20850 where u/H20849k/H20850/H110131 2/H20849coskx+cos ky/H20850,v/H20849k/H20850/H110131 2/H20849coskx−cos ky/H20850,tis the nearest neighbor hopping integral, and ris a “range” parameter due to in-layer hopping via the so-called axialorbital—a hybrid of mostly Cu 4 sand apical oxygen 2 p z. For small r, Eq. /H208495/H20850reduces to the familiar tight-binding form, /H9255/H20849k/H20850= const. − 2 t/H20849coskx+ cos ky/H20850+4t/H11032coskxcosky −2t/H11033/H20849cos 2 kx+ cos 2 ky/H20850, /H208496/H20850 with t/H11032=rt, and t/H11033=1 2rt. Hopping between nearest neighbor planes of two adjacent unit cells in the zdirection /H20849tc/H20850occurs via the axial orbital and therefore depends on /H20849kx,ky/H20850as v/H20849k/H208502 1−2ru/H20849k/H20850, i.e., it has the same symmetry as the superconduct- ing gap, vanishing along the nodal lines and reachingmaxima at the /H20849/H11006 /H9266,0/H20850and /H208490,/H11006/H9266/H20850points. If the layers are stacked on top of each other, the kzdispersion is simply included in Eq. /H208495/H20850by the substitution: r→r+/H20849tc0/t/H20850coskz. For YBa 2Cu3O6+x/H20849x/H110150.5/H20850,t/H11015400 meV, tc0causes rto vary between /H110150.28 and 0.32 and the corresponding Fermi sur- face is shown in Fig. 5/H20849a/H20850. Here the line thickness is the size of the kzdispersion yielded by the perturbation in r. The different interlayer hoppings at different locations in the Bril-louin zone reveals that small pockets at each of these loca-tions created by Fermi surface reconstruction will have dif-ferent corrugations. Accordingly, we proceed to examine thedifference in corrugation expected for each of the multiplepockets throughout the Brillouin zone yielded by a transla-tional symmetry-breaking order parameter.For illustrative purposes, we adopt a model involving a single transformation of /H9255 kbyQs,/H11006=/H20849/H9266/H208511/H110062/H9254/H20852,/H9266/H20850of the helical spin-density or d-density wave type such as that used in Refs. 12and27–29as one possibility, with an incommen- surability /H9254/H110151 16/H110150.06 to match incipient spin order and/or excitations30–32/H20850, and a single variable /H9004sadjusted to have the value 0.625 t, as explained in the next section. The Fermi surface thus obtained comprises equal-sized electron /H20849red/H20850 and hole /H20849blue /H20850pockets at the antinodal and nodal positions, respectively, accompanied by a larger /H20849green /H20850hole pocket at the node /H20851shown in Fig. 5/H20849b/H20850/H20852. Here again, the line thickness is the size of the interlayer hopping. A signiﬁcant differencein corrugation /H20849resulting from the dispersion in k z/H20850is ex- pected for pockets located at the nodal and antinodal loca-tions: the obtained interlayer corrugation for each of thesmaller and larger nodal pockets and the antinodal pocket isof order 14, 45, and 180 T, respectively, being representativeof general translational symmetry-breaking order parametersyielding pockets at the antinodal and nodal regions of theBrillouin zone. 12,27–29,33,34The difference in corrugation thus provides an opportunity for their respective locations to beidentiﬁed in quantum oscillation measurements. We ﬁnd thatthe ratio of observed corrugations for the /H9251and/H9253pocket is comparable to that expected for the nodal and antinodalpockets. The association of the nodes and antinodes with theexpected positions of hole and electron pockets within a bro-ken symmetry Fermi surface, 27–29further identify /H9251as a two- dimensional hole pocket and /H9253as a corrugated electron pocket within this picture. Additionally, bilayer splitting inthe case where bilayers are ferromagnetically coupled is ex-pected to result in a large splitting of frequencies, as opposedto the case of antiferromagnetic coupling between bilayers,in which case the frequency splitting of the bilayers wouldremain small. The lack of a discernible splitting in any of the /H9251,/H9252, and/H9253frequencies is therefore indicative of antiferro- magnetic coupling between bilayers /H20849see Appendix F for a discussion of bilayer splitting /H20850. The observation in inelastic neutron scattering experiments of low energy spin excita-tions and a spin resonance near /H20849 /H9266,/H9266/H20850that antiferromagneti- cally couple bilayers35suggests the association of spin order with pocket formation. For a larger /H9252orbit corresponding to a distinct section of Fermi surface /H20849as opposed to a magnetic breakdown orbit /H20850,12 its character can be identiﬁed in a scenario where /H9251and/H9253 correspond to equal and opposite hole and electron pockets at nodal and antinodal locations. Luttinger’s theorem36/H20851re- quiring a hole concentration of p=10/H110061% /H20849Ref. 7/H20850/H20852and the absence of a discernible splitting of F/H9252from corrugation, suggests the association of the /H9252orbit with a larger “hole- like” pocket located near /H20849/H9266/2,/H9266/2/H20850/H20851see Fig. 1/H20849a/H20850/H20852. The entire hole doping concentration would then be provided bythe /H9252pocket, which acts as a reservoir of charge carriers, while the /H9251and/H9253pockets almost cancel out in carrier concentration. C. Electronic origin of the observed pockets In the simplest case, our observations indicate a broken- translational symmetry order parameter that creates differ-SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-6ently corrugated Fermi surface pockets at different locations in the Brillouin zone—Fermi surface models that yield pock-ets only at the nodal locations would face a considerablechallenge in explaining our observations. Spin ordering involving a broken-translational symmetry has been reported in YBa 2Cu3O6+xsamples with oxygen concentrations x/H110150.45 /H20849Refs. 30and37/H20850and /H110151.0/H20849Ref.38/H20850 in an applied magnetic ﬁeld. Antiferromagnetic order has notyet been observed in samples with oxygen concentrations/H208490.49/H11021x/H113510.8/H20850in which quantum oscillations are observed. 8,12,13,16,17except when 2% of the Cu ions are re- placed by spinless Zn ions in samples with x/H110150.6.39The suppression of Zeeman splitting of the Landau levels hasalso been reported in Refs. 14and 15. A translational symmetry-breaking order parameter involving spin degreesof freedom or alternate means of suppression of the spindegrees of freedom is suggested. In addition, the implicitnear degeneracy of the bilayer-split pocket frequencies /H20851see Fig.4/H20849a/H20850and Appendixes F and G /H20852signal antiferromagneti- cally coupled bilayers. To better understand the underlying origin of the elec- tronic structure, we map the in-plane topology of the corru-gated pocket. The in-plane topology of the warped /H9253pocket is reﬂected in a variation of the in-plane calliper radius k/H20648 /H20849mapped, for example in Ref. 40, see Appendix K /H20850.I no u rexperiment, the calliper radius is mapped by two-axes angle- dependent quantum oscillation measurements where /H9258/H20849the angle between Hand the caxis /H20850is swept while H=/H20841H/H20841re- mains ﬁxed, the sweeps being repeated for many differentorientations /H9278of the in-plane component of the magnetic ﬁeld H/H20648=/H20849Hsin/H9258cos/H9278,Hsin/H9258sin/H9278,0/H20850. The dominant fundamental and harmonic quantum oscillations correspond-ing to the minimally warped /H9251pocket are found to be peri- odic in 1 //H92620Hcos/H9258over a broad angular range /H20849Appendix C/H20850. Because of its low degree of corrugation, the method described here is not applicable for determination of the in-plane Fermi surface topology of the /H9251hole pocket. The sig- niﬁcant corrugation of the /H9253pocket, by contrast, enables variations in k/H20648to be mapped in Fig. 6. The quantum oscil- lation data as a function of /H9258at each value of /H9278is ﬁt to a waveform composed of superimposed /H9251and/H9253oscillations as described in Appendix E, and the calliper radius thereby ex-tracted at various values of /H9278. Our results show a rounded- square in-plane topology of the /H9253pocket. The rounded-square /H20849i.e., nearly circular /H20850topology of the /H9253pocket /H20851shown in Fig. 6/H20849b/H20850/H20852is broadly consistent with an electron pocket predicted by numerous Fermi surface recon-struction models, 12,27–29,33although unconventional models—for example those involving novel quasiparticles—cannot be ruled out. The coexistence of a near circular elec- FIG. 5. /H20849Color online /H20850/H20849a/H20850Schematic unreconstructed Fermi surface depicting the translational vectors Qs,/H11006, the line thickness /H20849obtained by a perturbation in ras described in the text /H20850represents the interlayer dispersion kzresponsible for corrugation and is seen to be greater in the antinodal /H20849/H11006/H9266,0/H20850,/H208490,/H11006/H9266/H20850compared to the nodal /H20849/H11006/H9266/2,/H11006/H9266/2/H20850region. /H20849b/H20850Representative reconstructed Fermi surface according to models used in Refs. 12and27–29/H20849described in the text and Appendixes F and I /H20850. Blue and red lines correspond to the nodal hole pockets and antinodal electron pocket, respectively. The line thickness /H20849obtained by a perturbation in r/H20850represents the interlayer dispersion kzand is signiﬁcantly bigger for the antinodal compared to the nodal pocket. /H20849c/H20850Reconstructed Fermi surface consisting of only a single type of pocket /H20849/H9252/H20850after increasing the coupling /H9004s/H20849neglecting /H9004c/H20850. Antiferromagnetic bilayer coupling /H20849Ref. 19/H20850/H20849shown in Fig. 10of Appendix F /H20850 would yield two slightly different variants of the Fermi surfaces in /H20849b/H20850and /H20849c/H20850, but with very similar pocket areas. /H20849d/H20850Schematic dispersion prior to Fermi surface reconstruction depicted in one dimension. /H20849e/H20850Schematic dispersion of the three pockets. /H20849f/H20850Schematic showing the opening of a gap between the /H9251and/H9253pockets, as expected to result from an excitonic insulator instability. Here the charge potential /H9004cis not included; its effect on the /H9252orbit /H20849or remnants thereof, shown as a dotted line /H20850will depend on its strength and on the speciﬁcs of the superlattice /H20849see Appendix J /H20850. The effects of bilayer splitting are not shown here for clarity, but are discussed in Appendix F.COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-7tron pocket with hole pockets of comparable and/or larger size does not appear consistent with presently proposed“stripe” models involving multiple translations of the elec-tronic bands /H9255 kby ordering vectors of the form Q =/H20849/H9266/H208511/H110062/H9254/H20852,/H9266/H20850where 0 /H11021/H9254/H113511 8.33,34,41We consider an ex- ample of translational symmetry-breaking model that couldyield pockets of the observed size and shape to facilitate anestimation of the size of the order parameter /H9004 sthat breaks translational symmetry. The model we consider involves asingle transformation of /H9255 kbyQs,/H11006=/H20849/H9266/H208511/H110062/H9254/H20852,/H9266/H20850of the helical type described in the earlier section. Adjusting thevalue of a single parameter /H9004 s=0.625 tin such a model yields three pockets of size /H20849to within 2% /H20850, carrier type, corrugation depth, and shape consistent with experiment—see Fig. 5and Appendix I, in conjunction with the experimentally observed constraint of hole doping p/H1101511.7% /H20849see Fig. 5/H20850and/H9254/H110151 16/H20850. The large ratio of /H9004s/tindicates a substantial coupling, con- sistent with the near-circular topology of the electron pocket/H20851the small magnitude of deviation from a circle is seen from Fig. 6/H20849b/H20850/H20852. While band structure calculations indicate t /H11015400 meV, 19t/H11015100 meV provides closer correspondence with the observed effective masses /H20849see Fig. 3, lower inset /H20850, suggesting /H9004s/H1101160 meV. Such a large value of /H9004sis chal- lenging to reconcile with the absence of signatures of longrange order in neutron scattering experiments inYBa 2Cu3O6+xsamples of composition 0.49 /H11021x/H113510.8 /H20849where quantum oscillations are observed /H20850. An unconventional form of broken-translational symmetry-breaking spin order aresuggested, including staggered moments present chieﬂy inthe vortex cores, 38magnetic ﬁeld-stabilized spin-density wave order,37or spin ordering with d-wave pairing symmetry;27,28although unconventional models not involv- ing translational symmetry breaking cannot be ruled out. III. DISCUSSION An open question pertains to the origin of the unexpected similarity in size of the /H9251and/H9253pockets observed in under- doped YBa 2Cu3O6+x. More information on the microscopic nature of the primary instability will assist in addressing thisquestion. Irrespective of their origin, the existence of com-pensated pockets suggests a Fermi surface prone to a second-ary instability 1,5at which the compensated pockets are de- stroyed. A. Excitonic instability The possibility of an “excitonic insulator” instability /H20851at which the compensated /H9253/H20849electron /H20850and/H9251/H20849hole /H20850pockets are destroyed /H20852is raised by their similar volumes /H20849within the ex- FIG. 6. /H20849Color online /H20850/H20849a/H20850Examples of angle-swept measurements made from positive to negative values of /H9258on a sample of YBa 2Cu3O6+xwith x=0.56 at ﬁxed ﬁeld /H92620H=45 T, for different in-plane orientations /H20849/H9278/H20850of the in-plane component of the magnetic ﬁeld H/H20648=/H20849Hsin/H9258cos/H9278,Hsin/H9258sin/H9278,0/H20850. Curves shown correspond to −54 /H11349/H9278/H11349156° /H20849bottom to top /H20850in steps of 15°. The gap in angular data around /H9258=0 is due to a correction made for a small misalignment of the crystal, explained in Appendix A. /H20849b/H20850The/H9278dependence of the caliper radius k/H20648of the/H9253pocket determined from ﬁts /H20851see Appendix E, Eq. /H20849E1/H20850/H20852of superimposed /H9251and/H9253oscillation waveforms to the data in/H20849a/H20850—using values of /H9004F/H9253,0and/H9003/H9251obtained from ﬁts to the ﬁeld swept data. Fit frequencies are given in Appendix E. The full 360° /H9278 rotation is inferred from performing a rotation over 180° in /H9278combined with ﬁts to /H11006/H9258, with only k/H20648allowed to vary. The magenta line is the/H9278dependence of k/H20648expected for the single Qs,/H11006model shown in Fig. 5/H20849b/H20850./H20849c/H20850An example ﬁt /H20849red line—solid gray online /H20850to/H9278=51° /H20849equivalent to 231° /H20850data /H20849black line /H20850, with further ﬁts shown in Appendix E.SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-8perimental uncertainty, both occupying 1.91 /H110060.01% of the Brillouin zone /H20850with effective masses sufﬁciently large /H20851m/H9251/H11569 /H11015m/H9253/H11569/H110151.6meforx=0.56, found from the ﬁt in the lower inset to Fig. 3/H20849a/H20850/H20852to cause strong attraction. Strong coupling between pairs of electrons and holes is suggested by a bind-ing energy /H20849Ry /H11569/H11011430 meV, derived in Appendix B /H20850that is more than ten times larger than their individual kinetic ener-gies /H20849E ke/H1101139 meV /H20850. This also means that the putative exci- ton separation to Bohr radius ratio is larger than unity /H20851i.e., rs/H110153.4 for x=0.54, see Fig. 1/H20849b/H20850/H20852, rendering the system un- stable to an excitonic insulating instability.42Due to the strongly interacting dilute Bose-Einstein condensate /H20849BEC /H20850 limit in which this system is consequently positioned /H20849i.e., rs/H112711/H20850, the excitonic attraction is not contingent on exactly matching pocket shapes or perfect “nesting.” Suggestive of an instability actually taking place is the occurrence of a metal-insulator quantum critical point /H20849QCP /H20850 atxc=6.46 identiﬁed in recent experiments.16,43The three- fold increase in effective mass on reducing x=0.54 to x =0.49 immediately preceding the QCP signals a tenfold in-crease in Ry /E keto/H11011102/H20849see Appendix B /H20850and threefold increase in rsto/H1101110 on approaching the critical doping /H20851shown in Fig. 1/H20849b/H20850/H20852, potentially indicating a further reduc- tion in the effectiveness of screening of the Coulomb attrac-tion, with the ﬁnal trigger for an excitonic instability occur-ring at x c. Strongly correlated systems that support conditions propitious for an excitonic insulator areunusual, 1,4,42given that a single carrier type rather than equal and opposite types typically results close to a Mott insulatingstate. 44This situation appears to be reversed in YBa 2Cu3O6+x, where Fermi surface reconstruction at the ﬁrst instability transforms the single band of carriers with astrong on-site repulsion into two bands containing equal andopposite pockets that are strongly attracted to each other/H20851illustrated in Figs. 5/H20849a/H20850and5/H20849b/H20850/H20852. B. Destruction of compensated pockets A secondary superlattice instability would be anticipated to accompany exciton condensation for x/H11021xc,5destroying the compensated /H9251and/H9253pockets /H20851see Figs. 5/H20849d/H20850and5/H20849e/H20850/H20852. Drawing an analogy with other underdoped cuprates, accom-panying charge order could be expected at wave vectorsQ c=2Qs/H20849Ref. 45/H20850and/or Q/H11036=/H208490,/H9266/2/H20850/H20849Ref. 46/H20850/H20849see Ap- pendix J /H20850. The possibility of the charge plus spin character of the second instability is suggested by the wave vector matchQ c=2Qsbetween the “charge” wave vector Qccharacteriz- ing phonon broadening47and possible superlattice formation,48and the “spin” wave vector Qsassociated with short range spin order30and magnetic excitations.31A ﬁnite charge superlattice potential /H9004cwould reconstruct the re- maining /H9252pocket into smaller pockets or open sheets33—the precise details depending on the choice of wave vector /H20849see Appendix J /H20850. Further, the emergence of signatures of quasistatic mag- netism in neutron scattering30,37and local moment behavior in muon spin rotation experiments49,50is consistent with a strengthened /H9004sfor the region x/H11021xc. A strengthened /H9004s alone could gap the /H9251and/H9253pockets in YBa 2Cu3O6+x/H20851seeFig. 5/H20849c/H20850/H20852, leaving behind only the /H9252hole pockets at the nodes—reminiscent of strong coupling models.51 C. Reconciliation with photoemission experiments Signatures from our measurements that the /H9251pocket from which predominant oscillations arise is of the “hole” carriertype located at the nodes, brings the results of quantum os-cillation measurements close to reconciliation with the re-sults of angle-resolved photoemission spectroscopy/H20849ARPES /H20850experiments. A dichotomy arises at the antinodes, however; while quantum oscillations indicate antinodalFermi surface pockets measured by quantum oscillations, alarge antinodal “pseudogap” has been measured by ARPESmeasurements. An intriguing question concerns whether theﬁnite spectral weight measured by ARPES to be concen-trated at “arcs” of ﬁnite extent centered at the nodal points/H20849/H11006 /H9266/2,/H11006/H9266/2/H20850could conceivably reﬂect the sole remaining parts of the /H9252Fermi surface on annihilation of the equal and opposite /H9251and/H9253hole and electron pockets at the secondary Fermi surface instability as shown in Fig. 5/H20849Ref. 52/H20850—a Fermi surface consisting of arcs is reported in materials withrobust stripe order. 53While such an interpretation is sugges- tive, an additional explanation is still required for the closingof the ARPES pseudogap in the low-temperature and highmagnetic ﬁeld regime relevant to quantum oscillationmeasurements. One possibility suggested by recent models 54,55for the re-emergence of spectral weight at the antinodes in quantumoscillation measurements is that incoherent superconductingpairs destroy the spectral weight under the experimental con-ditions /H20849T/H11407T candB=0/H20850where the normal state is accessed in photoemission experiments, while this spectral weight isrecovered in the “quantum vortex liquid” regime accessed byquantum oscillation measurements. Another contributioncould arise from ﬂuctuations of the excitonic insulating in-stability that cause loss of antinodal spectral weight over arange of oxygen concentrations above x cextending to higher temperatures—similar to the case of NbSe 2and TiSe 2above their charge ordering temperatures.56,57The possible role of density wave reconstruction has also been suggested bytheory 58and by results of ARPES experiments.59,60Alterna- tively, more exotic models such as those involving quantumoscillations from Fermi arcs, 61or novel quasiparticles visible to quantum oscillations and not ARPES have also beenproposed. 62 D. Relevance to pairing Finally, we consider the effect that excitations of the po- tential excitonic insulator instability may have on the quasi-particle pairing mechanism. The unusual occurrence of aFermi surface ideally predisposed to an excitonic insulatorinstability in the same material where high T csuperconduc- tivity occurs could either suggest a link between the phenom-ena, or a coincidence. Similar compensated corrugated cylin-drical Fermi surface pockets are found in other families ofunconventional superconductor that also exhibit unexpect-edly high transition temperatures—such as the Fe pnictides 63 and Pu-based superconductors.64COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-9Theoretical proposals made prior to the advent of high- temperature superconductivity considered conditions underwhich an excitonic mechanism may potentially enhance T c.6 Whether or not such a cooperative mechanism is in play in thed-wave cuprate superconductors, however, is contingent on a better understanding of the relative importance of spinand charge ﬂuctuations in contributing to high T cin these materials. ACKNOWLEDGMENTS This work is supported by the U.S. Department of Energy, the National Science Foundation /H20849including Grant No. PHY05-51164 /H20850, the State of Florida, the Royal Society, Trin- ity College /H20849University of Cambridge /H20850, the EPSRC /H20849U.K. /H20850, and the BES program “Science in 100 T.” The authors thankP. B. Littlewood for theoretical input, B. Ramshaw for dis-cussions, and M. Gordon, A. Paris, D. Rickel, D. Roybal, andC. Swenson for technical assistance. APPENDIX A: EXPERIMENTAL DETAILS Quantum oscillations are measured using the contactless conductivity method65in two pulsed magnets reaching 65 and 85 T, and in a dc magnet reaching 45 T. Platelike singlecrystals of YBa 2Cu3O6+x/H20849Ref. 7/H20850with in-plane dimensions 0.3/H110030.8 mm2are attached to the face of a ﬂat ﬁve turn coil /H20849compensated in the case of pulsed ﬁeld experiments /H20850that forms part of a proximity detector circuit resonating at/H1101122 MHz. 66A change in the sample skin depth /H20849or complex penetration depth in the superconducting state67/H20850leads to a change in the inductance of the coil, which in turn alters theresonance frequency of the circuit. Quantum oscillations areobserved in the in-plane resistivity and hence the skin depth /H20850. To minimize the effects of ﬂux dissipation heating in pulsed magnetic ﬁelds, the sample is immersed in liquid 4He throughout the experiment,68,69and only the oscillations ob- served during the falling ﬁeld are considered for detailedanalysis. On taking such precautions, effective mass esti-mates are obtained that are comparable to those obtained insamples of the same oxygen concentration in static magneticﬁelds. 12 The ﬁxed H/H20849=45 T /H20850angle-swept /H9258measurements per- formed at different in-plane angles /H9278shown in Fig. 6are obtained using a dual-axis rotator. Due to a small misalign-ment of the sample by /H92580/H110157° at/H92780/H1101551°, a minor correc- tion cos/H9258actual= cos/H9258uncorrected /H11003/H20881cos2/H92580+ sin2/H92580sin2/H20849/H9278−/H92780/H20850 is made to obtain the correct /H9258. APPENDIX B: ESTIMATES OF THE EXCITON BINDING ENERGY We estimate the effective “Rydberg” for exciton binding to be4,42 Ry/H11569=/H9262 me1 /H92802Ry /H20849B1/H20850 /H11015430 meV for YBa 2Cu3O6+xsamples of oxygen concentra- tion x=0.54 and 0.56,7where /H9262/H11015m/H11569/2/H110150.8meis the re-duced biexciton mass, Ry /H1101513.6 eV is the hydrogen Ryd- berg, and /H9280/H110155 is the relative permittivity for YBa 2Cu3O6+x.70The kinetic energy is estimated using Eke=e/H6036F m/H11569/H20849B2/H20850 /H1101539 meV for F=535 T and m/H11569=1.6me, yielding a ratio Ry/H11569/Eke/H1101511. The effective Bohr radius of the exciton is estimated to be a/H11569=/H9280me /H9262a0 /H20849B3/H20850 /H110153.3 Å, where a0/H110150.53 Å is the hydrogen Bohr radius. We estimate the exciton density as n=Ak/2/H92662/H110152.6 /H110031017m−2, where Ak=2/H9266eF //H6036/H110155.1/H110031017m−1, yielding an effective exciton separation of rs/H110151 a/H11569/H208811 /H9266n/H20849B4/H20850 from which rs/H110153.4 /H20849here rsis dimensionless /H20850. In estimating Ry/H11569andrs, we have neglected the effects of screening, which could become signiﬁcant should a large /H9252 pocket survive the formation of a superstructure accompany-ing exciton condensation. On the other hand, superstructureformation /H20849see following section /H20850would likely cause signiﬁ- cant reconstruction of the /H9252orbit—the outcome being strongly dependent on the associated ordering vectors andnature of the broken symmetry /H20849see Appendix J /H20850. Screening will also become ineffective if m /H11569globally diverges at the quantum critical point at xc/H110150.46, at which point both Ry/H11569 andrswould diverge while Ekecollapses to zero. When the quasiparticle effective mass changes /H20849as re- ported in YBa 2Cu3O6+xas a function of doping16/H20850, then we can expect the following dependences /H20849shown in Fig. 1of the main text /H20850: Ry/H11569 Eke/H11008/H20849m/H11569/H208502and rs/H11008m/H11569, /H20849B5/H20850 indicating that tuning of the electron correlations is expected to have dramatic consequences for the likelihood of excitonpairing and the applicability of the strong coupling BEClimit. 1,42 APPENDIX C: ORBITAL QUANTIZATION FROM ANGLE-SWEPT MEASUREMENTS The/H9251oscillations corresponding to a minimally warped cylinder dominate the angular dependent oscillations /H20849similar to the ﬁeld dependent oscillations in Figs. 1and3/H20850. Conse- quently a periodicity in 1 /Bcos/H9258is observed in both the fundamental and harmonic oscillations over a broad angularrange /H20849Fig.7/H20850. We note that an angular damping that varies with angle is suggested by the similarly nonmonotonicallyvarying form of the fundamental and harmonic oscillationamplitude /H20849Fig.7/H20850. The weaker /H9253oscillations are extracted by a two cylinder ﬁt /H20851Eq. /H20849E1/H20850/H20852explained further in Appen- dix E.SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-10APPENDIX D: EXTENDED ANALYSIS OF SWEPT MAGNETIC FIELD DATA During Fourier analysis, weaker spectral features can be affected by interference effects between oscillations from aspectrally dominant series of oscillations /H20849nearby in fre- quency /H20850and the window function /H20849a Hann window in the current study /H20850used to modulate the data. Such interference between the /H9251oscillation and the window function affects the resolved /H9253frequencies in the present study. One way to mitigate interference is to subtract ﬁts to a dominant wellcharacterized sequence of oscillations /H20849i.e., the /H9251oscilla- tions /H20850prior to Fourier analysis. This procedure is justiﬁed in the present case owing to the resolved separation betweenthe /H9251and/H9253frequencies in our Fourier analysis /H20849see Figs. 2–4and8/H20850—with the degree of separation growing propor- tionately larger for the harmonics as expected. In Fig. 8we show Fourier transforms of the measured oscillations in samples with x=0.54 before /H20851i.e., Fig. 4/H20849b/H20850of the main text /H20852and after /H20851i.e., Fig. 4/H20849c/H20850of the main text /H20852 subtracting the ﬁt of the /H9251oscillations /H20851Eq. /H208491/H20850in the main text/H20852, a subset of which are shown in Figs. 3/H20849b/H20850and3/H20849c/H20850of the main text. The /H9253frequencies F/H9253,neckandF/H9253,bellyin Fig. 4/H20849a/H20850of the main text are determined with greater accuracy by performing Fourier transformation after subtracting the /H9251os- cillations. APPENDIX E: EXTENDED ANALYSIS OF SWEPT ANGLE DATA When a Fermi surface has an interlayer corrugation, as we identify for the /H9253pocket /H20851depicted in Fig. 1/H20849a/H20850/H20852, angle- dependent measurements provide a means for mapping itsin-plane topology. 22If the degree of corrugation is sufﬁ- ciently deep to yield separately resolved neck and belly fre-quencies in Fourier transforms, 25quantum oscillations then FIG. 7. /H20849Color online /H20850/H20849a/H20850Quantum oscillations measured as a function of the angle of inclination to the crystalline caxis /H20849/H9258/H20850on a sample of YBa 2Cu3O6+x/H20849x/H110050.56 /H20850at a constant ﬁeld /H92620H/H1100545 T and a ﬁxed in-plane angle /H9278atT/H110051.5 K using the contactless conductivity technique. Periodicity in 1 /Bcos/H9258of both the funda- mental /H20849dotted pink lines /H20850and harmonic /H20849solid green lines /H20850oscilla- tions /H20849dominated by the /H9251frequency /H20850are revealed. The gray verti- cal lines label the expected peak positions for fundamental /H9251 quantum oscillations periodic in 1 /Bcos/H9258with periodicity F/H9251, and the dotted vertical cyan lines label the expected peak positions forthe corresponding 2 F /H9251harmonic. The harmonic oscillations are ex- tracted by subtracting the fundamental oscillations ﬁt using Eq. /H20849E1/H20850 from the measured oscillations as a function of /H9258/H20849ﬁt and measure- ments shown in Figs. 6and9/H20850. A residual background has been further subtracted for visual clarity. FIG. 8. /H20849Color online /H20850/H20849a/H20850–/H20849d/H20850Fourier transforms of the oscillations measured in YBa 2Cu3O6+xwith x=0.54 at different angles /H9258before /H20849thin mauve line /H20850and after /H20849thick magenta line with gray shading /H20850subtracting the /H9251component of a ﬁt to Eq. /H208491/H20850of the main text—a subset of which are shown in Figs. 3/H20849b/H20850and3/H20849c/H20850of the main text. The resolved /H9253frequencies are used to construct Fig. 4/H20849a/H20850of the main text. Frequency peaks corresponding to 2 Fharmonic oscillations are observed in the Fourier transform at all measured angles.COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-11provide an accurate means for mapping the in-plane topol- ogy. The in-plane topology can be efﬁciently mapped out bysweeping /H9258in a constant magnetic ﬁeld for many different in-plane orientations /H9278of the magnetic ﬁeld H/H20648 =/H20849Hsin/H9258cos/H9278,Hsin/H9258sin/H9278,0/H20850—requiring the use of a dual-axis rotator /H20849for other examples, see Appendix K /H20850.I n the present experiments on YBa 2Cu3O6+xwith x=0.56 and 0.54, the /H9253electron pocket is found to be deeply corrugated yielding spectral features between 10% and 40% in ampli-tude of the spectral weight associated with the /H9251pocket. A minimal model for ﬁtting is given by /H9004f=/H9004f/H9251,0cos/H208732/H9266F/H9251,0 /H92620Hcos/H9258/H20874exp/H20873−/H9003/H9251 /H92620Hcos/H9258/H20874 +/H9004f/H9253,0cos/H208732/H9266F/H9253,0 /H92620Hcos/H9258/H20874exp/H20873−/H9003/H9253 /H92620Hcos/H9258/H20874 /H11003J0/H208732/H9266/H9004F/H9253,0J0/H20849k/H20648ctan/H9258/H20850 /H92620Hcos/H9258/H20874 /H20849E1/H20850 /H20851similar to Eq. /H208491/H20850but without including a corrugation of /H9251/H20852 that takes into consideration the superposition of /H9251and/H9253 oscillation waveforms in Figs. 6/H20849a/H20850and9/H20849a/H20850. Here /H9004f/H9251,0and /H9004f/H9253,0are amplitude prefactors, /H9003/H9253is a damping factor for the /H9253oscillations, while the remaining parameters are deﬁned in the main text. Changes in the caliper radius of the /H9253pocket k/H20648are as- sumed to be the dominant factor responsible for the subtle /H9278 dependence of the waveform in Fig. 6/H20849a/H20850of the main text and Fig.9/H20849a/H20850. Fourier analysis in Fig. 9/H20849b/H20850of the /H9258-swept oscil- lations in 1 //H92620Hcos/H9258yields a leading frequency F/H9251 /H11015539/H110065 T similar to that obtained on sweeping the ﬁeld atﬁxed angle, conﬁrming the 1 /cos/H9258angular dependence of the dominant /H9251oscillations. Owing to the subtle variation of the waveform with /H9278, parameters /H9004f/H9251,0,/H9004f/H9253,0,F/H9251,0,F/H9253,0, and/H9003/H9253are obtained from ﬁts of Eq. /H20849E1/H20850to the angular data assuming a circular cross section /H20849i.e.,k/H20648=kF, where kF=/H208812eF/H9253,0//H6036is the mean Fermi radius /H20850. The ﬁts yield frequency parameters F/H9251,0 =536/H110065 T and F/H9253,0=535/H110065 T, closely matching those de- termined in magnetic ﬁeld sweeps. These parameters arethen held constant for all /H9278, and a single parameter ﬁt is performed with only k/H20648being allowed to vary to reﬁne the ﬁts for each /H9278. The resulting /H9278dependence of k/H20648—shown in Fig. 6/H20849b/H20850of the main text—yields a rounded-square Fermi surface cross section for the /H9253electron pocket, similar to that calculated using the helical spin-density wave model. Weassume that the in-plane pocket topology accessed is an av-erage between the bilayer-split pocket topologies /H20849such as shown in Fig. 10/H20850. In Fig. 9/H20849a/H20850, we show the /H9278dependence of the actual ﬁts to Eq. /H20849E1/H20850. APPENDIX F: EFFECT OF BILAYER SPLITTING Bilayer splitting in YBa 2Cu3O6+xarises from the bilayer crystalline structure, yielding bonding and antibonding bandsthat lead to two large hole sections of different sizes in theparamagnetic phase /H20851see Fig. 10/H20849a/H20850/H20852. 19Here the width of the line represents the difference in area between bilayer-splitbands, with “+” and “−” representing the sign of this differ-ence. If ordering occurs in which there is antiferromagnetic coupling between bonding and antibonding bands, theHamiltonian factorizes into FIG. 9. /H20849Color online /H20850/H20849a/H20850Swept /H9258oscillations measured in a sample of YBa 2Cu3O6+xwith x=0.56 at constant ﬁeld /H92620H=45 T and different ﬁxed values of /H9278atT/H110151.5 K ﬁt to Eq. /H20849E1/H20850/H20849dashed black lines /H20850as described in Fig. 6. The gaps in the data result from a correction for a sample misalignment of /H110157°/H20849see experimental details /H20850./H20849b/H20850Fourier transforms in 1 //H92620Hcos/H9258of the oscillations shown in /H20849a/H20850, which are dominated by the /H9251pocket. As expected for a nearly ideal two-dimensional Fermi surface section, the dominant /H9251frequency obtained is the same as that from ﬁxed /H9258swept magnetic ﬁeld experiments /H20851i.e., Fig. 6/H20849a/H20850/H20852at all angles of rotation. Periodicity of the oscillations in 1 /Bcos/H9258is shown in Appendix C. The curves correspond from bottom to top to −54 /H11349/H9278/H11349156° in 15° steps.SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-12/H20873/H9255kb/H9004s /H9004s/H9255k+Qa/H20874and/H20873/H9255ka/H9004s /H9004s/H9255k+Qb/H20874. /H20849F1/H20850 The resulting Fermi surface shown in Fig. 10/H20849b/H20850has two different variants—the difference between them being morepronounced for the electron pocket, which is stretched alongeach of the k xandkydirections. The difference in area be- tween each is represented by the thickness of the line, withthe sign of the difference being positive or negative as indi- cated by “+” or “−.” Since the lines bounding each of thepockets contain both positive and negative segments that al-most cancel, the pocket areas of the each bilayer-split vari-ants of the reconstructed Fermi surface are only weaklysplit—i.e., the degeneracy of the pockets is not signiﬁcantlylifted by bilayer splitting /H20849t /H11036/H20850, when the bilayers are antifer- romagnetically coupled. Consequently, different frequencieswould likely not be resolvable in quantum oscillationexperiments 27,28from antiferromagnetically coupled bilayer- split pockets. Were the bilayers instead ferromagnetically coupled, then like bands would be coupled giving rise to signiﬁcant differ-ences in pocket areas between each of the bilayer-split Fermisurfaces. The size of this difference is expected to signiﬁ-cantly exceed the depth of corrugation 19and so would be observable as distinct frequencies in the quantum oscillationdata, and also as well-separated angle-dependent peaks in theFourier transform. APPENDIX G: INCONSISTENCY OF SIMILARLY WARPED POCKETS SCENARIO Here we show that a scenario where the observed multiple frequencies arise from two bilayer-split pockets13is incon- sistent with the experimental data. Figure 11shows the least square error in an unconstrained ﬁt of the measured oscilla-tions to Eq. /H208491/H20850. The Bessel term J 0/H208492/H9266/H9004Fi/B/H20850that modu- lates the quantum oscillation amplitude in Eq. /H208491/H20850/H20849Ref. 24/H20850 arises from phase smearing due to the Onsager phase of aquasi-two-dimensional Fermi surface with depth of corruga-tion/H9004F ithat varies sinusoidally with kz. In the limit 2/H9266/H9004Fi/B/H112711/4, the waveform from each section iis a su- perposition of extremal neck and belly frequencies FIG. 10. /H20849Color online /H20850Calculated Fermi surfaces in which the effects of bilayer splitting are included as a perturbation in r/H20849here r is the hopping “range” parameter discussed in Sec. II B, see Ref. 19/H20850. The effects of corrugation are not shown for clarity, but are the same as in Fig. 5. The difference in area between different bilayer- split variants of the Fermi surface is represented by the thickness ofthe line, where the colors /H20849and “+” and “−” labels /H20850indicate the sign of this difference. /H20849a/H20850shows the translation of the unreconstructed Fermi surface where the coupling between bonding and antibondingbands is represented by different signs for the difference in area/H20849black for “−” and gray for “+” /H20850between /H9255 kand/H9255k+Q./H20849b/H20850shows the reconstructed Fermi surface where the difference in area /H20849red for “+” and pink for “−” for the electron pocket, and blue for “+” andcyan for “−” for the hole pocket /H20850, can be seen to cancel for each pocket. FIG. 11. /H20849Color online /H20850Error analysis of unconstrained oscillation ﬁt to two Fermi surface sections. /H20849a/H20850Least squares error of the unconstrained ﬁt of Eq. /H208491/H20850to the measured oscillations of the sample with oxygen concentration x=0.56 as a function of different ratios between the corrugation of the /H9251and/H9253pockets is shown on the left-hand axis. The corresponding difference between the ﬁt corrugations of the/H9251and/H9253pockets is shown on the right-hand axis. A clear minimum is observed for /H9004F/H9253,0//H9004F/H9251,0/H110156.2, yielding the best ﬁt parameters— while the error is seen to rapidly increase if the corrugations of the two pockets are similar. /H20849b/H20850Example best ﬁt /H20849red line /H20850to the measured oscillations for the scenario where /H9004F/H9253,0/H11015/H9004F/H9251,0. The ﬁt does not yield good agreement, demonstrating the need for two low frequency Fermi surface sections of signiﬁcantly different warpings to explain the measured oscillations.COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-13Fi,neck,belly =Fi/H11007/H9004Fi. For corrugated cylindrical pockets with a small degree of warping, such as the /H9251pocket, the differ- ence in Onsager phase /H208512/H9266/H208492/H9004F/H20850//H92620H/H20852between the neck and belly extremal orbits at /H92620H=65 T is small enough /H20849/H110152 radians for the /H9251pocket /H20850to render inadequate the parabolic approximation assumed in the Lifshitz Kosevichexpression. 21Instead, the corrugated cylinder topology is best represented by Eq. /H208491/H20850in which the amplitude of the neck and belly oscillations is constrained to be identical /H20849i.e., Ci/H20850, and their relative phase difference constrained to be /H9266/2.24 All parameters have been allowed to vary for different ﬁxed ratios of /H9004F/H9253,0and/H9004F/H9251,0, and the resulting least squares error of the best ﬁt obtained as a function of thecorrugation ratio between the two pockets. It is evident thatthe minimum error appears for a signiﬁcantly different cor-rugation of the /H9251and/H9253sections, with a rapidly growing error with increasingly similar corrugations of the /H9251and/H9253Fermi surface sections. Very similar parameters are also obtainedon ﬁtting the quantum oscillations in Ref. 13to Eq. /H208491/H20850 /H20849where a sine term replaces the cosine term for magnetic oscillations /H20850, indicating consistency with two very differently warped pockets. Considering the scenario where bilayer-splitpockets are invoked to explain the multiple low frequencies,the warping of each of the two /H9251and/H9253sections would be required to be the same /H20851we assume that the nonconducting chains in underdoped YBa 2Cu3O6+xdo not hybridize with bonding and antibonding states to create differently warpedbilayer-split sections as in optimally doped YBa 2Cu3O7/H20849Ref. 19/H20850/H20852. The least squares error is near maximum in this case, as demonstrated by the best ﬁt for /H9004F/H9253,0//H9004F/H9251,0/H110151 shown in Fig.11. This disagreement in ﬁt indicates that the multiple observed low frequencies are not explained by similarlywarped pockets, as would be the case for bilayer-split pock-ets originating from a ferromagnetic bilayer coupling. The bilayer splitting which must occur for YBa 2Cu3O6+x is not experimentally resolvable as an additional set of fre- quencies in the Fourier transform. Indications therefore arethat the areas of the bilayer-split pockets remain almost de-generate, as would occur for the antiferromagnetic couplingof bonding and antibonding states, consistent with the low-energy spin excitations observed in inelastic neutron scatter-ing experiments. 35 We can understand from an inspection of the oscillations why the scenario of two equally warped Fermi surface sec-tions does not explain the multiple low frequencies. We de-ﬁne two pockets /H20849labeled “1” and “2” /H20850with neck and belly frequencies F 1,neck and F1,belly, and F2,neck and F2,belly in terms of two median frequencies F1andF2with a depth of corrugation /H9004F, i.e., F1,neck /belly=F1/H11006/H9004F,F2,neck /belly=F2/H11006/H9004F./H20849G1/H20850 where /H9004Fwould have the same magnitude and sign for the two pockets in a ferromagnetically coupled bilayer-splittingmodel. Quantum oscillations result from a superposition of two oscillatory terms/H9004f/H11008/H20875cos/H208732/H9266F1 /H92620Hcos/H9258/H20874J0/H208732/H9266/H9004F /H92620Hcos/H9258/H20874 + cos/H208732/H9266F2 /H92620Hcos/H9258/H20874J0/H208732/H9266/H9004F /H92620Hcos/H9258/H20874/H20876 /H20849G2/H20850 for the two pockets, yielding the four frequencies given above. On combining them, we obtain /H9004f/H11008cos/H20873/H9266/H20849F1+F2/H20850 /H92620Hcos/H9258/H20874/H11003cos/H20873/H9266/H20849F1−F2/H20850 /H92620Hcos/H9258/H20874J0/H208732/H9266/H9004F /H92620Hcos/H9258/H20874. /H20849G3/H20850 Two terms in Eq. /H20849G3/H20850will give rise to nodes: the Bessel term containing /H9004Fwill give rise to nodes at the set of mag- netic ﬁelds given by Eq. /H208493/H20850/H20851Eq. /H208493/H20850in the main text /H20852, while the cosine term containing /H20849F1−F2/H20850/2 will give rise to nodes /H92620Hmcos/H9258=/H20841F1−F2/H20841 2m+1, /H20849G4/H20850 where mis an integer. Since the node terms are multiplica- tive, both sets of nodes would be observed giving rise to asuppression of the total waveform /H20851as shown in the attempted ﬁt in Fig. 11/H20849b/H20850/H20852at the ﬁeld values given by both Eqs. /H208493/H20850and /H20849G4/H20850. The absence of such nodes in the full experimental waveform /H20851Figs. 2/H20849a/H20850and4/H20849b/H20850of the main text /H20852indicates that the multiple observed low frequencies are not a consequenceof two equally warped pockets. APPENDIX H: ADDITIONAL SWEPT ANGLE DATA In Fig. 12we show oscillations in contactless conductiv- ity data measured by sweeping /H9258for constant Hat many different values of /H9278obtained for a single crystal sample of YBa 2Cu3O6+xwith x=0.54. The weak /H9278dependence reﬂects the same behavior observed for x=0.56. FIG. 12. /H20849Color online /H20850Swept /H9258oscillations measured by con- tactless conductivity on a sample of YBa 2Cu3O6+xwith x=0.54 at constant ﬁeld H=45 T and different ﬁxed values of /H9278atT /H110151.5 K.SEBASTIAN et al. PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-14APPENDIX I: BAND-STRUCTURE DETAILS Antiferromagnetic coupling between bilayers couples bonding to antibonding bands, yielding two slightly differentvariants of the Fermi surfaces shown in Figs. 5/H20849b/H20850and5/H20849c/H20850of the main text with very similar areas. For samples with oxygen concentration close to x/H110150.5, ortho-II ordering has the potential to modify the Fermi sur-face, depending on the strength of the relevant potential/H9004 II.27,29For commensurate spin ordering /H20849i.e.,/H9254=0/H20850,i th a s been postulated in Ref. 28that the ortho-II potential /H20849/H9004II/H20850 alone can produce pockets of area comparable to those mea- sured. In the case of incommensurate ordering where /H9254/H114071 16, an additional ortho-II potential is not expected to signiﬁ-cantly change the observed frequencies 28/H20849see Fig. 13/H20850. The inclusion of an additional ortho-II potential can howevermake a small change to the observed frequencies, yieldingimproved agreement between the calculated Fermi surfaceand experiment /H20849Fig.13/H20850. APPENDIX J: POSSIBLE ORDERING RESULTING FROM AN EXCITONIC INSULATOR INSTABILITY While the extended zone representations in Figs. 5/H20849b/H20850and 5/H20849c/H20850of the main text provide a convenient means of visual- izing the three distinct pockets, the actual magnetic Brillouinzone depicted in Fig. 14/H20851obtained on repeatedly folding that in Fig. 5/H20849b/H20850of the main text /H20852must be considered in order to identify possible ordering vectors associated with the exci-tonic insulator instability. Here, we make the simplifying as- sumption that /H9254=1 16/H110150.06. One possible candidate ordering vector is Qc=2Qs =/H20849/H110064/H9254/H9266,0/H20850, corresponding to the charge modulation that can accompany a spin-density wave should it become collin-ear or “stripelike” /H20849as opposed to helical /H20850. In this case Q c does not introduce a new periodicity in Fig. 14, but ratherintroduces additional couplings between /H9255k+nQsand /H9255k+/H20849n/H110062/H20850Qsthat are not present in a purely helical spin-density wave. Such additional couplings will cause gaps to open around the edges of the magnetic Brillouin zone in Fig. 14, leading to the destruction of closed pockets and the creationof open Fermi surface sheets in a manner analogous to those obtained Ref. 33/H20849where /H9254=1 8was considered /H20850. The observa- tion of phonon broadening effects47together with the pos- sible observation of Bragg peaks in Ref. 48could be consis- tent with charge ordering of this type. An alternative possibility is a form of broken-translational symmetry with a characteristic vector Q/H11036/H11015/H208490,/H9266/2/H20850that maps the electron and hole pockets onto each other as in aconventional Fermi surface nesting scenario. In this case, Q /H11036 has a similar periodicity to the purely charge wave vectors identiﬁed in stripe systems33and/or scanning tunneling mi- croscopy experiments.46Although such forms of order are reported in various families of high Tccuprates, they have not been reported in YBa 2Cu3O6+x. FIG. 15. /H20849a/H20850Experimentally measured angular dependence of the dHvA frequencies in LaRhIn 5./H20849b/H20850Band structure calculations of the Fermi surface topology of the /H9251pockets in LaRhIn 5/H20849taken from Ref. 71/H20850. FIG. 13. The left-hand panel shows the Fermi surface calculated for a helical spin-density wave model. Using a hole doping p =11.7% and /H9254/H110151 16/H110150.06 as constraints, adjustment of a single parameter /H9004syields a Fermi surface with frequencies comparable to those observed—i.e., F/H9251,0=575 T, F/H9252,0=1583 T, and F/H9253,0 =517 T. The largest difference from measured area for the case of the/H9252pocket corresponds to 0.2% of the paramagnetic Brillouin zone area. The helical spin-density wave model used is described inRef.12. On including an ortho-II potential /H20849right-hand panel /H20850, using the formalism described in Ref. 29, pockets with frequencies— F /H9251,0=517 T, F/H9252,0=1641 T, and F/H9253,0=515 T—closer to those observed experimentally are obtained in the right-hand panel. The largest difference from measured areafor any one pocket now corresponds to 0.07% of the Brillouin zone. FIG. 14. /H20849Color online /H20850The actual Brillouin zone for Fermi surface reconstruction by Qs=/H20849/H9266/H208511/H110062/H9254/H20852,/H9266/H20850, assuming /H9254=1 16, illus- trating the possible superlattice modulation vectors QcandQ/H11036ac- companying an excitonic insulator instability.COMPENSATED ELECTRON AND HOLE POCKETS IN AN … PHYSICAL REVIEW B 81, 214524 /H208492010 /H20850 214524-15APPENDIX K: ROTATION STUDIES ON OTHER CORRUGATED FERMI SURFACE MATERIALS The rare-earth paramagnet LaRhIn 5/H20849Ref. 71/H20850and the or- ganic superconductor /H9252−/H20849BEDT−TTF /H208502IBr2/H20849Ref. 22/H20850are two examples of strongly correlated materials for which asimilar degree of corrugation enables angle-dependent quan-tum oscillation experiments to access information on the in-plane topology. LaRhIn 5has a section of Fermi surface /H20849labeled /H9251in Fig. 15/H20850with a proportionately similar degree of corrugation to the/H9253pocket in YBa 2Cu3O6+xand a round-square cross section.71de Haas–van Alphen experiments can resolve neck and belly frequencies F/H92511,2andF/H92513, respectively, which be- come degenerate when His rotated by an angle /H9258away from the /H208510,0,1 /H20852axis, corresponding to the ﬁrst Bessel zero in Fcos/H9258/H11015F0/H11006/H9004FJ0/H20849k/H20648c/H11032tan/H9258/H20850. Owing to the rounded- square cross section, the caliper radius is larger along/H208551,1,0 /H20856than along /H208551,0,0 /H20856causing the degeneracy to occur at a smaller angle /H9258when the ﬁeld is rotated toward /H208511,1,0 /H20852 than when it is rotated toward /H208511,0,0 /H20852. 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PhysRevB.96.035108.pdf	PHYSICAL REVIEW B 96, 035108 (2017) One-electron spectra and susceptibilities of the three-dimensional electron gas from self-consistent solutions of Hedin’s equations A. L. Kutepovand G. Kotliar Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08856, USA (Received 15 February 2017; revised manuscript received 24 May 2017; published 6 July 2017) A few approximate schemes to solve the Hedin equations self-consistently introduced in P h y s .R e v .B 94,155101 (2016 ) are explored and tested for the three-dimensional (3D) electron gas at metallic densities. We calculate one-electron spectra, dielectric properties, compressibility, and correlation energy. Considerablereduction in the calculated bandwidth (as compared to the self-consistent GW result) has been found when vertex correction was used for both polarizability and self-energy. Generally, it is advantageous to obtain thediagrammatic representation of polarizability from the deﬁnition of this quantity as a functional derivative ofthe electronic density with respect to the total ﬁeld (external plus induced). For self-energy, the ﬁrst-ordervertex correction seems to be sufﬁcient for the range of densities considered. Whenever it is possible, wecompare the accuracy of our vertex-corrected schemes with the accuracy of the self-consistent quasiparticle GW approximation (QSGW), which is less expensive computationally. We show that the QSGW approach performspoorly and we relate this poor performance with an inaccurate description of the screening in the QSGW method(with an error comprising a factor 2–3 in the physically important range of momenta). DOI: 10.1103/PhysRevB.96.035108 I. INTRODUCTION Many-body perturbation theory (MBPT) diagrammatic approaches offer a path to the solution of the quantummany-body problem of solids, which complements alternativemethods such as QMC (quantum Monte Carlo) and CC (couplecluster). This approach received attention for over half acentury but there is no complete understanding of how thedifferent selection of diagrams performs for different physicalquantities. These insights are important in the search forpredictive ﬁrst-principles methods for correlated solids. Inthis work we investigate these questions in the frameworkof the homogeneous electron gas (HEG) in a neutralizingpositively charged background. The HEG is very useful fortesting, in a simpliﬁed setting, the methods presently beingdeveloped to study the electronic structure of solids for tworeasons: it requires less computational effort and there isa natural benchmark since some properties have also beencalculated using QMC methods [ 1–5]. This model describes the properties of alkali metals well. Most common uses of diagrammatic approaches are based on noninteracting Green functions such as LDA (local densityapproximation) Green’s function. In this work, however, weare interested in self-consistent (sc) diagrammatic approaches.Hence we do not consider ambiguities related to the choice ofnoninteracting Green’s function for reference. We considertwo classes of methods. One, initiated by Hedin [ 6], which carries out a perturbative expansion in the fully self-consistent(renormalized) Green’s function (which obeys the Dyson equa-tion) and screened interaction W. An alternative philosophy (QSGW) uses the lowest order diagrams for polarizabilityand self-energy with Green’s function which is determinedby means of quasiparticle self-consistency condition [ 7]. In this work we perform QSGW calculations using a previously kutepov@physics.rutgers.eduintroduced linearized approach [ 8] that has an advantage of being implementable on Matsubara frequencies. An important cornerstone in the application of self- consistent MBPT-based approximations to the HEG, is thework by Holm and von Barth [ 9], where the authors applied scGW approximation (i.e., lowest order diagram in theperturbation series in terms of GandW) to calculate the total energy and spectra of HEG. Their principal conclusions arethat scGW severely overestimates the bandwidth but gives thetotal energy very close to the QMC results. Thus, the work[9] raised the question of how (if at all) one can get accurate spectra of HEG using scMBPT approach. Progress on thisquestion was made by Shirley [ 10] who showed that if an accurate Wis known (he used QMC input to evaluate W) then the self-consistency in Green’s function Gand the lowest order vertex corrections in self-energy mostly cancel outeach other. This observation justiﬁes, in a certain degree, theso called one-shot G 0Wapproaches (lowest order approach using a noninteracting Green’s function G0) without vertex corrections to self-energy. Takada [ 11,12] used QMC data to parametrize electron-hole four-point irreducible interactionand performed sc vertex-corrected GW calculations with the vertex deﬁned from a condition that Ward identity (WI) issatisﬁed. In this respect, his approach resembles the idea ofQSGW method, where WI is imposed by construction. Theimportance of vertex corrections in electron gas studies wasreported recently in non-self-consistent calculations [ 13,14]. From other studies on the subject, one can mention interestingapplications of quantum chemical methods (ﬁrst of all ofab initio coupled-cluster theory) to studying the spectra [ 15] and correlation energy [ 16,17] of electron gas. In this study we go beyond the earlier diagrammatically inspired works by removing an important limitation related tothe use of Monte Carlo data for parametrization of screenedinteraction or electron-hole four-point irreducible interaction.We examine fully self-consistent (in both GandW) diagram- matic schemes involving diagrams of higher order. We study 2469-9950/2017/96(3)/035108(8) 035108-1 ©2017 American Physical SocietyA. L. KUTEPOV AND G. KOTLIAR PHYSICAL REVIEW B 96, 035108 (2017) relative importance of different diagrams for polarizability and self-energy. The paper begins with a brief presentation of self-consistent schemes we use to solve Hedin’s equations (Sec. II). Section III provides the results obtained and a discussion. The conclusionsare given afterwards. II. METHOD Detailed account of the vertex-corrected schemes we use in this work has been given in Ref. [ 18]. For completeness, below, we brieﬂy repeat the essentials of the approach and point outthe simpliﬁcations in technical implementation in the case ofelectron gas. We solve Hedin’s equations [ 6] self-consistently using different approximations for three-point vertex function /Gamma1. Three-point vertex function enters formally exact expressionsfor polarizability and self-energy (in space-time variables) P(12)=/summationdisplay αGα(13)/Gamma1α(342)Gα(41), (1) /Sigma1α(12)=−Gα(13)/Gamma1α(324)W(41), (2) where the integration/summation over repeated arguments is understood, and αis the spin index. We consider two different types of approximations for /Gamma1. The ﬁrst type consists in expanding vertex function in termsof the screened interaction to a speciﬁed order. Keeping onlyzero-order term ( /Gamma1=1) in both Pand/Sigma1corresponds to the famous GW approximation. We will also consider expansion of the vertex up to the ﬁrst order ( /Gamma1 1=1+WGG ) in both polarizability and self-energy expressions. This approximationis conserving (like GW) as the corresponding Pand/Sigma1 can alternatively be obtained by differentiating the same /Psi1 functional [ 19,20]. The second type of approximation for /Gamma1consists in solving the Bethe-Salpeter equation /Gamma1 α(123)=δ(12)δ(13)+δ/Sigma1α(12) δGβ(45)Gβ(46)/Gamma1β(673)Gβ(75), (3) with a certain approximate expression for the functional derivative /Theta1=δ/Sigma1 δGin (3). We will consider two expressions for the kernel /Theta1in this work. The ﬁrst is obtained by using theGW form for /Sigma1in the functional derivative and neglecting the derivative of the screened interactionδW δG, i.e.,/Theta1=W(we will call the corresponding vertex as /Gamma10 GW). Diagrammatically it corresponds to keeping only the ﬁrst term on the right-handside of Fig. 1. In the second approximate expression for /Theta1 we also are using GW form for self-energy in the functional derivative but we keep the terms up to the second order inWin the derivative δW δG(we will use abbreviation /Gamma1GWfor the corresponding vertex). In this case, the obtained vertexfunction corresponds to keeping all three terms for /Theta1(Fig. 1). It is important to point out that the diagrams resulting from δW δGallow the spin ﬂips (as it is clear from Fig. 1), the importance of which was pointed out in Ref. [ 21]. In the particular case, when GandWhave been found self-consistently with /Sigma1=GW andP=GG,v e r t e x /Gamma1GW yields physical polarizability in scGW approximation (deﬁnedFIG. 1. The approximation for the irreducible four-point vertex function /Theta1. as a functional derivative of electronic density with respect to the total electric ﬁeld). In other cases, when /Gamma1GW is evaluated with /Sigma1and/or Pincluding additional diagrams, the kernel shown in Fig. 1is only an approximation to the derivativeδ/Sigma1 δG, and, as a result, the vertex /Gamma1GW does not provide physical Panymore. Thus, in a search for an optimal approximation, we have to trade between the numberof diagrams included in /Sigma1in the Dyson equation for G and the degree of the “deviation” of polarizability from thephysical one. Another potential problem which can arisewhen higher order diagrams are summed up uncontrollablyis an appearance of negative spectral functions. This issuehas been known since the works by Minnhagen [ 22,23] and a solution (positive-deﬁnite diagrammatic expansion for thespectral function and for the density-response spectrum) wasfound recently in Refs. [ 14,24,25]. Below we demonstrate that our calculated spectral functions are positive. With our four (as speciﬁed above) approximations for the vertex functions (1; /Gamma1 1;/Gamma10 GW;/Gamma1GW) we are able to form different self-consistent schemes for solving Hedin’s equationsby selecting the vertex to be used in polarizability ( 1) and the vertex to be used in self-energy ( 2). As all our vertices are approximate, they do not have to be the same in Pand in /Sigma1. We have tried different combinations and below we will show the results obtained with a reasonable subset of them. Todistinguish the approaches we will use the same notations asthe ones introduced in Ref. [ 18]. For convenience, we have collected them in Table I(slightly modiﬁed Table Ifrom Ref. [ 18]) and we repeat here their deﬁnitions. Scheme A is the scGW approach. It is conserving in Baym-Kadanoff deﬁnition[26], but generally its accuracy is poor when one considers spectral properties of solids [ 8,27,28]. Another conserving sc scheme is scheme B. It uses the same ﬁrst-order vertex/Gamma1 1in both Pand/Sigma1. Scheme C is based on the “physical” TABLE I. Diagrammatic representations of polarizability and self-energy in sc schemes of solving the Hedin equations. Argumentsin square brackets specify GandWwhich are used to evaluate the vertex function. Other details are explained in the main text. Scheme P/Sigma1 A GG GW B G/Gamma1 1[G;W]GG /Gamma1 1[G;W]W C G/Gamma1GW[G;W]G GW D G/Gamma1GW[G;W]G G/Gamma1 1[G;W]W E G/Gamma1GW[G;W]GG /Gamma1 1[G;W]W G G/Gamma10 GW[G;W]GG /Gamma1 1[G;W]W 035108-2ONE-ELECTRON SPECTRA AND SUSCEPTIBILITIES OF . . . PHYSICAL REVIEW B 96, 035108 (2017) polarizability (preserves charge microscopically). In scheme C, we perform the scGW calculation ﬁrst. Underlined Gand Win Table Imean that the corresponding quantities are taken from the scGW run. Then the vertex /Gamma1GW[G;W] is evaluated and it is used to calculate polarizability and correspondingscreened interaction W. We use a bar above the Wto indicate that this quantity is evaluated using GandWfrom the scGW calculation, but it is not equal to Wbecause it includes vertex corrections through the polarizability. This Wis ﬁxed (in scheme C) during the following iterations where only theself-energy /Sigma1=G WandGare renewed. So, scheme C does not include the vertex in /Sigma1explicitly but only through W. Scheme D is similar to scheme C. It also is based on physical polarizability but it uses the ﬁrst-order vertex inself-energy explicitly (skeleton diagram). In scheme D thescreened interaction Wis ﬁxed at the same level as in scheme C, but the ﬁnal iterations involve the renewal of not only Gand /Sigma1, but also /Gamma11. Scheme E is fully self-consistent (both GandW are renewed on every iteration till the end). Scheme E does notpreserve the charge exactly and can be considered as a resultof a trade between the accuracy of self-energy and the degreeof deviation of polarizability from the physical one. SchemeG is similar to scheme E, but with a simpliﬁed Bethe-Salpeterequation for the corresponding vertex /Gamma1 0 GW(the diagrams with spin ﬂips are neglected in the kernel of the Bethe-Salpeterequation). In accordance with the arguments above, we have found that schemes with vertex /Gamma1 GWinPand with vertex of increasing (>1) order in /Sigma1(scheme F in Ref. [ 18]) result in nonphysical polarizability (ﬁrst of all in its improper q→0 behavior) and in the deterioration of the accuracy in calculated properties.We will not consider them further in this work. For 3D electron gas, we solve Hedin’s equations in a periodic cubic box with equidistant 54 ×54×54 mesh. The box contains 729 electrons. We use Matsubara’s formalismwith electronic temperature 1000 K. We do not use plasmonpole approximation and we treat full frequency dependence ofW, as opposite to the often use of its zero frequency limit when solving the Bethe-Salpeter equation for insulators [ 29]o r ,i n the recent paper on the electron gas [ 13]. Detailed formulas, presented in the Appendix of Ref. [ 18] are simpliﬁed for the electron gas considerably by omitting the indexes associatedwith the band states, the mufﬁn-tin orbitals, and the productbasis. III. RESULTS In Table IIwe compare our results for bandwidth with those obtained by Shirley [ 10], who based the calculations (partially) on QMC input. Bandwidth was determined as adifference between the pole in the spectral function at k=0 and chemical potential. If we assume that Shirley’s resultsare close to the exact ones, we can draw certain conclusionsabout our approaches. As one can conclude from Table II, three schemes (D, E, and G) show the best performance withsmall differences between themselves. Common for thesethree schemes are two facts: they all include a solving ofthe Bethe-Salpeter equation for polarizability (but slightlydifferently as it was explained above) and they all applyﬁrst-order vertex correction in self-energy. Scheme C alsoTABLE II. Bandwidths (eV) of the 3D electron gas compared with the results from Ref. [ 10], where the QMC input was partially used. rs 2 345 QSGW 13.48 5.75 3.10 1.92A 13.61 6.08 3.44 2.21 B 12.53 5.51 3.07 1.94C 13.21 5.90 3.28 2.10 D 11.54 5.22 2.85 1.79 E 11.59 5.10 2.78 1.73G 11.79 5.20 2.86 1.80 [10] 11.57(5) 5.04(4) 2.66(4) 1.72(4) involves solving of BSE for polarizability, but it does not use vertex correction to self-energy and, as a result, showsworse performance. Similarly, conserving scheme B, whichapplies ﬁrst-order vertex corrections to the Pand/Sigma1,s h o w s worse performance, because it misses the effects of BSE inW. Nevertheless, scheme B seems to be better than scheme C, demonstrating the importance of vertex corrections inself-energy. Performance of the QSGW approach is slightlybetter than performance of scGW, but it is not competitive withschemes D, E, or G. Thus, for the studied range of densitiesof electron gas, QSGW cannot be considered as a reasonableapproximation in terms of its predictive power. An example of k-resolved spectral functions is shown in Fig. 2. First, we would like to point out that all our calculated spectral functions are positive (they are also positive fornonzero momenta). As one can see there is a well deﬁnedquasiparticle peak near −3 eV . All approaches (excluding QSGW) also show plasmon satellite at higher binding energy.Unfortunately, the exact positions of plasmon satellites arevery sensitive to the quality of analytical continuation whichwe performed using the method of Vidberg and Serene[30]. We checked the accuracy of this method to be rather good to determine the positions of quasiparticle peaks, butwe would give an error bar about 1 eV for the positionsof plasmon satellite peaks. The accuracy in the calculated -15 -10 -5 0 5Spectral function Frequency (eV)QSGW A B C D E G FIG. 2. Spectral function ( k=0, arb. units) of the electron gas forrs=4. 035108-3A. L. KUTEPOV AND G. KOTLIAR PHYSICAL REVIEW B 96, 035108 (2017) positions of plasmon satellite peaks can be improved using more points in frequency summations and time integrationswhen evaluating the higher order diagrams with, however,a corresponding increase in the computation time. Thus,the positions of plasmon satellites following from Fig. 2, namely −8 eV in scheme A, −9e Vi nB , −10.5e Vi nE and G, and −12 eV in C and D, are only preliminary and should be reevaluated in future more elaborate calculations.Experimental plasmon energy (difference between position ofquasiparticle peak and the plasmon satellite) of Na ( r s≈4) is about 6 eV [ 31]. In this respect our preliminary results from scheme B are the best. Scheme A (scGW) underestimatesthe plasmon energy, whereas schemes involving higher orderdiagrams (through the Bethe-Salpeter equation) overestimateit. Whether this overestimation comes from the inaccuracyof analytical continuation or from intrinsic insufﬁciencies ofthe schemes is difﬁcult to conclude at this point. Plasmonsatellites in electron gas have been studied recently usingthe positive-deﬁnite diagrammatic expansion for the spectralfunction [ 14] and the GW+cumulant approach [ 32–35]. Both methods are implemented on real frequency axis and, thus,do not involve analytical continuation as an intermediate stepin evaluating the spectral function. In these works, very goodagreement with the experimental position of plasmon satellitein Na has been achieved. Figure 3presents electron occupations n kobtained for rs=4. Temperature effects are responsible for a slight devia- tion of the QSGW curve from the perfect step function. Otherapproaches also have (in addition to the temperature effects)a correlation-related spectral weight transfer. We compareour results with available QMC data [ 36]. However, it is hard to make this comparison conclusive. First of all, theabove mentioned temperature effects make our calculatedmomentum distribution smoother than it would be at T=0K . Second, QMC data are essentially based on the extrapolationto the thermodynamic limit (the inset in Fig. 1 of Ref. [ 36] shows that the shape of the QMC curve is almost altogetherthe result of an extrapolation). Nevertheless, one can point outthat at k=0 all our schemes (excluding QSGW) show smaller values of n kthan QMC. Close to the Fermi momentum, our vertex corrected schemes seem to be closer to the QMC data 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2nk k/kFQSGW A B C D E G QMC FIG. 3. Electron occupations in the electron gas for rs=4. QMC data are from Ref. [ 36]. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 51/ε qrsQSGW A B C,D E G QMC FIG. 4. Inverse static dielectric function of 3D electron gas, rs=4 .W eu s eaﬁ tt ot h eQ M Cd a t aa sp r o v i d e di nF i g .6o fR e f .[ 4]f o r comparison. Schemes C and D give identical results for this quantity. than scGW result. But it is hardly possible to say which scheme is the best in terms of this physical quantity. In Fig. 4the static (zero-frequency) inverse dielectric function for 3D electron gas is shown for rs=4. As one can see from the graph, we are able to improve the agreementwith QMC data considerably when using the vertices ofincreased complexity. Clearly, the best dielectric function isobtained from the “physical” polarizability (schemes C and D),even if the last corresponds to the scGW approximation(in the sense that the diagrams are evaluated using Gand Wfrom scGW). One can also point out the importance of including the diagrams with spin ﬂips (scheme E resultsin better dielectric function than scheme G does). At thesame time scheme E is worse than schemes C/D whichreﬂects the above mentioned fact about trading an additionaldiagram in self-energy in scheme E violates the requirementof polarizability to be physical. One can also relate theshortcomings of QSGW approach in the one-electron spectrato the poor description of the screening. As one can see fromFig.4, in the physically important range ( q∗r s=0.5–3.0) the inverse dielectric function in QSGW approximation is largerthan the one from QMC data by a factor 2–3. A certain insight on the origin of differences in the dielectric function obtained with approximate methods can be gainedwhen one looks at the static vertex as a function of bosonicmomentum (Fig. 5). It is clear that the range of momenta where the calculated dielectric function shows the largest differences(q∗r s=0–4) correlates very well with the range of momenta where the humps in the vertex function show very differentheights. In Fig. 6we present the calculated correlation energy of the electron gas as a function of r s. It was obtained as the difference between the expectation value of the Hamiltoniancorresponding to the selected level of approximation and theexpectation value of the Hamiltonian in the Hartree-Fockapproximation. In all vertex-corrected schemes the exchange-correlation part was evaluated as a convolution of Green’sfunction and self-energy. Excluding scheme C which missesvertex corrections to self-energy and, as a result, shows ratherdifferent from other schemes behavior, one can conclude 035108-4ONE-ELECTRON SPECTRA AND SUSCEPTIBILITIES OF . . . PHYSICAL REVIEW B 96, 035108 (2017) 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6Static vertex qrsB C,D G FIG. 5. Static vertex ( ν=0) as a function of bosonic momentum qforrs=4. Fermionic frequency and momentum correspond to their values at the Fermi surface. Schemes C and D give identical results for this quantity. that vertex corrections make the correlation energy more negative as compared to the correlation energy obtainedin the scGW approach. Only at r s<1.5 this tendency is reversed. Thus, if we assume that available QMC data areexact, we have to state that vertex corrections systematicallyworsen the scGW result. However, the slope of the curvesseems to get better at least in some of the vertex correctedschemes. As one can see, the deviation from QMC datain fully self-consistent schemes B, E, and G is almost r s independent. One more point related to the issue of correlation energy is its precise value obtained in the scGW approximation. Since itsﬁrst evaluation by Holm and von Barth [ 9], the consensus was that the scGW approximation gives very accurate total energiesof three-dimensional electron gas. However, our calculatedenergies (in scGW) are systematically more negative than theones reported in earlier papers and in QMC studies. To makethis point clearer we present the numbers in Table III.T h e most recent publication by Van Houcke et al. [39] agrees well -0.14-0.12-0.1-0.08-0.06-0.04 0 1 2 3 4 5 6ec (Ry) rsA B C D E G QMC FIG. 6. Correlation energy of 3D electron gas as obtained from conserving approximations. The QMC results are cited fromRef. [ 1].TABLE III. Correlation energy (Ry) of 3D electron gas in scGW approximation compared with the QMC data. Method \rs 1245 scGW [ 37] −0.0901 −0.064 scGW [ 38] −0.1156 −0.0872 −0.061 −0.0538 scGW [ 39] −0.137 −0.0996 −0.0686 −0.060 scGW, present work −0.1358 −0.0988 −0.0677 −0.0591 QMC [ 1] −0.1196 −0.0902 −0.0638 −0.0562 with our results. Our data and the data from Ref. [ 39]a r e almost identical with a discrepancy of about 0.001 Ry or less.Thus, common belief in reliability of scGW total energies forelectron gas needs to be reconsidered. Below we present quantities which show relatively slow convergence with respect to the quality of the momentumdiscretization mesh ( qmesh). They are the renormalization factor Z, the effective mass m ∗/m, and the compressibility κ. Our present computational resources and speciﬁcs of our codedid not allow us to reduce an error bar on these quantities below3%–5%. Still, we believe that the accuracy is good enough tomake certain observations. The quasiparticle renormalizationfactor and the effective electron mass are presented in Figs. 7 and8correspondingly. In order to evaluate them, we used the following formulas (with k Fbeing the Fermi momentum and μbeing the chemical potential): Z=/braceleftBigg 1−∂Im/Sigma1(k,iω ) ∂(iω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle k=kF,ω=μ/bracerightBigg−1 (4) and m∗ m=Z−1 1+m kF∂Re/Sigma1(k,ω) ∂k/vextendsingle/vextendsingle k=kF,ω=μ, (5) 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5Z rsQSGW A B C D E G LFF SJ-DMC SJ-VMC BF-VMC BF-RMC FIG. 7. Quasiparticle renormalization factor Zas a function of rs obtained with different approximations in comparison with QMC results [ 36,40]. Also shown are the results based on local ﬁeld factors (LFF) which were copied from Ref. [ 41]. Abbreviations associated with QMC methods: BF: backﬂow, SJ: Slater-Jastrow, DMC: diffusion Monte Carlo, VMC: variational Monte Carlo, andRMC: reptation Monte Carlo. 035108-5A. L. KUTEPOV AND G. KOTLIAR PHYSICAL REVIEW B 96, 035108 (2017) 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.5 2 2.5 3 3.5 4 4.5 5m*/m rsQSGW A B C D E G FIG. 8. Effective mass of 3D electron gas as a function of rs. correspondingly. We compare our results for Zfactor with available QMC data [ 36,40]. Unfortunately, QMC data involve an extrapolation to the thermodynamic limit and show consid-erable variations for this quantity. We also compare our resultswith the ones based on local ﬁeld factors (LFF) [ 41] which include a certain amount of QMC input. Because of insufﬁcientconvergence of our results and the uncertainty in QMC data itis hard at this point to give a certain conclusion about whichscheme provides the most accurate renormalization factor.If we assume that BF-VMC and BF-RMC results are thebest, then we can state that our schemes D, E, and G arewithin uncertainty of QMC data. QSGW also shows goodperformance for this particular quantity. It is interesting that the results for the effective mass (r s>2) obtained from vertex corrected schemes lie in between the results obtained in scGW and QSGW approximationswhich serve as lower and upper limits correspondingly. Onecan also make an important observation that the r sdependence of the effective mass (for rs>2) is weaker than similar dependence of the Zfactor suggesting that the frequency derivative and momentum derivative in Eq. ( 5) are canceling out considerably. This fact can have certain implicationsbecause there are theories (for example, dynamical meanﬁeld theory) which stress frequency dependence but ignoremomentum dependence of self-energy in localized regime (lowdensity). We also checked how well our calculated vertex functions reﬂect the presence of a pole in the compressibility (at aboutr s=5.25, “dielectric catastrophe” [ 12]). In Fig. 9we compare our calculated compressibilities with the results based on theQMC data. We have obtained the QMC compressibility κas the derivative of the chemical potential μ(= 1 2mk2 f+Vxc) with respect to rs:1 κ=−1 4πr2sdμ drs. In our vertex-corrected schemes, we have evaluated the compressibility from the ratio of two limits of vertex function ( /Gamma1qand/Gamma1ν) and the effective mass m∗/m(see for instance Ref. [ 12]), κ κ0=m∗ m/Gamma1q /Gamma1ν, (6) with two limits of vertex function deﬁned in the Appendix. 0 5 10 15 20 1.5 2 2.5 3 3.5 4 4.5 5κ/κ0 rsB C,D E G QMC FIG. 9. Compressibility of the electron gas. The QMC results have been obtained from the derivatives of the chemical potential with respect to the rs.κ0is the compressibility of the noninteracting electron gas. To make the plot, we used the vertex /Gamma1/Theta1in schemes C, D, and E, and the vertex /Gamma1Win scheme G. It is clear from Fig. 9that the behavior consistent with the presence of a pole in the compressibility can only be obtainedbased on the vertex from schemes C and/or D. In other words,only the vertex corresponding to physical polarizability canbe useful for compressibility evaluation. It is interesting thatadditional self-consistency iterations for vertex function inschemes E and G (as compared to scheme D), which onlyslightly change the one-electron spectra, worsen signiﬁcantlythe quality of the calculated compressibility. Also, it is obviousthat the ﬁrst-order vertex (scheme B) is totally insensitive tothe presence of a pole in compressibility. IV . CONCLUSIONS In conclusion, we have applied the self-consistent diagram- matic approaches based on the Hedin equations to study theproperties of the 3D HEG. We have found that the inclusionof the most important diagrammatic sequences can reproducethe one-electron spectra and dielectric properties of the HEGin the range of metallic densities with good accuracy. For theone-electron spectra, the corrections to polarizability and toself-energy are equally important. For dielectric properties thevertex correction to self-energy is of secondary importance.In all cases, the important conclusion is that the calculation ofpolarizability should follow, as close as possible, its deﬁnitionas a functional derivative of the density with respect to thetotal electric ﬁeld. Our conclusions concerning one-electronspectra of 3D electron gas are similar to the conclusions madeearlier for the spectra of alkali metals and semiconductorsin Refs. [ 18,42], namely, that the best spectra are obtained when the set of diagrams for polarizability is obtained fromBSE, whereas the ﬁrst-order vertex correction is applied tothe self-energy (schemes D, E, and G). Our benchmarksquantiﬁed the inaccuracy the QSGW approximation to predictone-electron spectra of the electron gas at metallic densities(approximately 15% error). We track this inaccuracy to thepoor description of screening in the QSGW approach (with anerror up to a factor 2–3 in the physically important range ofmomenta). Concerning the use of the vertex-corrected schemes 035108-6ONE-ELECTRON SPECTRA AND SUSCEPTIBILITIES OF . . . PHYSICAL REVIEW B 96, 035108 (2017) for the calculation of spectra, one can advocate scheme D, which combines good accuracy and computational efﬁciency(time-consuming BSE has to be solved only once). ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Ofﬁce of Science, Basic Energy Sciences as a partof the Computational Materials Science Program. G.K. wassupported by the Simons Foundation under the Many ElectronProblem collaboration. We thank N. Prokof’ev for usefuldiscussions. APPENDIX: EVALUATION OF TWO LIMITS OF THE VERTEX FUNCTION The two limits of the vertex function, /Gamma1qand/Gamma1νentering Eq. ( 6), were deﬁned as follows. In the momentum-frequency representation, the vertex function [solution of Eq. ( 3)] can be conveniently considered as dependent on fermionic momen-tum+frequncy ( k,ω) and on bosonic momentum+frequency (q,ν), i.e.,/Gamma1(k,ω;q,ν). In these variables, /Gamma1 q=limq→0/Gamma1(k= kF,ω→0;q,ν=0) and /Gamma1ν=limν→0/Gamma1(k=kF,ω→0; q=0,ν), with two vectors, kandq, assumed to be parallel. Quantities /Gamma1qand/Gamma1νare related to the quasiparticle renor- malization factor Zand the Landau Fermi liquid parameter Fs 0:/Gamma1ν=1/Z,/Gamma1 q=1 Z(1+Fs 0). In order to demonstrate that 0 1 2 3 4 5 0 2 4 6 8 10Γ Bosonic Matsubara frequency, ν (eV) FIG. 10. Limiting behavior of the vertex function /Gamma1(kf,π/β ; q=0,ν)a ts m a l l νforrs=4. Symbols ×show the calculated data points (discrete Matsubara’s frequencies), line is drawn for convenience. the above two limits are well deﬁned numerically, we have plotted the vertex function /Gamma1/Theta1from scheme D in Fig. 10. 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PhysRevB.75.054406.pdf	Magnetic ﬁeld dependence of dephasing rate due to diluted Kondo impurities T. Micklitz,1T. A. Costi,2and A. Rosch1 1Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany 2Institut für Festkörperforschung, Forschungszentrum Jülich, 52425 Jülich, Germany /H20849Received 13 October 2006; published 9 February 2007 /H20850 We investigate the dephasing rate, 1/ /H9270/H9272, of weakly disordered electrons due to scattering from diluted dynamical impurities. Our previous result for the weak-localization dephasing rate is generalized from dilutedKondo impurities to arbitrary dynamical defects with typical energy transfer larger than 1/ /H9270/H9272. For magnetic impurities, we study the inﬂuence of magnetic ﬁelds on the dephasing of Aharonov-Bohm oscillations anduniversal conductance ﬂuctuations both analytically and using the numerical renormalization group. Theseresults are compared to recent experiments. DOI: 10.1103/PhysRevB.75.054406 PACS number /H20849s/H20850: 72.15.Lh, 72.15.Qm, 72.15.Rn I. INTRODUCTION Decoherence is the fundamental process leading to a sup- pression of quantum mechanical interference and therefore isindispensable for our understanding of the appearance of theclassical world. The destruction of phase coherence in aquantum system occurs due to interactions with its environ-ment and can be studied, e.g., in mesoscopic metals andsemiconductors where the quantum-mechanical wave natureof the electrons leads to a variety of transport phenomena atlow temperatures. Although the concrete deﬁnition of the dephasing rate, 1/ /H9270/H9272, depends on the experiment used to determine it, the electron-electron interactions are thought to be the dominantmechanism for the destruction of phase coherence in metalswithout dynamical impurities below about 1 K. The dephas-ing rate for interacting electrons in a diffusive environmentwas ﬁrst calculated by Altshuler, Aronov, and Khmelnitsky/H20849AAK /H20850and vanishes at low temperatures, T, with some power of T, depending on the dimensionality of the system. 1 In the last decade several independent groups performed magnetoresistance experiments2–5to probe the inﬂuence of dephasing on weak localization in disordered metallic wires.Irritatingly, a saturation of the dephasing rate, 1/ /H9270/H9272, has been observed at the lowest experimentally accessible tempera-tures. This observation has triggered an intense discussion onthe mechanism responsible for the excess of dephasing. 6–8 The most promising candidates to explain the saturation of 1//H9270/H9272are extremely low concentrations of dynamical impuri- ties, such as atomic two-level systems9,10or magnetic impurities.3–5,11–14This has been corroborated on the one hand by experiments3,5on extremely clean Ag and Au samples where the dephasing rate continues to decrease wellbelow 100 mK and on the other hand by doping studies withmagnetic impurities. 2,3,5,15,16As expected theoretically,12 these experiments show a saturation of 1/ /H9270/H9272above the Kondo temperature, TK, the characteristic scale of screening of the magnetic moment, and a suppression of 1/ /H9270/H9272below this scale. Recent highly controlled experiments,15,16in which a few ppm /H20849parts per million /H20850of Fe ions have been implanted by ion beam lithography into very clean Agsamples, showed that the screening of these Fe ions is sur-prisingly well described 15by the theoretically predicteddephasing rate for spin-1/2 Kondo impurities13down to tem- peratures of 0.1 TK. At the lowest temperature, again a pla- teau in the dephasing rate has been observed proportional tothe number of implanted Fe ions. The origin of this puzzlingbehavior is still unclear but may arise from further dynamicaldefects created during the implantation process or by rare Feions with a different chemical environment and strongly re-duced magnetic screening. An obvious option to study the inﬂuence of magnetic im- purities on the dephasing rate is to measure its dependenceon an externally applied magnetic ﬁeld. The application ofsufﬁciently large magnetic ﬁelds freezes out inelastic spin-ﬂip processes as discussed theoretically in Ref. 12, and there- fore one expects the dephasing rate to return to the valuepredicted by AAK for dephasing induced by Coulomb inter-actions in a diffusive environment. The orbital contributionof the magnetic ﬁeld does, however, destroy the weak-localization /H20849WL /H20850contribution to the magnetoresistance, as the joint propagation of an electron and a hole along time- reversed trajectories /H20849the Cooperon /H20850picks up extra /H20849random /H20850 Aharonov-Bohm phases in the presence of external magneticﬁelds. Measuring the B-dependent dephasing rate in a WL experiment is therefore only possible in strictly one- or two- dimensional systems using magnetic ﬁelds almost exactlyparallel to such a structure, requiring an accurate alignmentof magnetic ﬁelds. Universal conductance ﬂuctuations /H20849UCF /H20850and Aharonov- Bohm /H20849AB /H20850oscillations with a periodicity of h/e,o nt h e other hand, are not suppressed by orbital effects and can beused rather directly to determine the ﬁeld dependence of thedephasing rate. 17 UCFs can be observed as characteristic ﬂuctuations of the conductance as a function of the magnetic ﬁeld. The externalmagnetic ﬁeld enters the metal and changes the pattern of theelectrons wave functions and therefore the conductance in arandom but reproducible way /H20849“magnetoﬁngerprint” /H20850.I nA B experiments performed on mesoscopic rings, these sampleﬂuctuations are further modulated by periodic h/eoscilla- tions resulting from the magnetic ﬂux piercing the ring. BothUCF and AB oscillations rely on the constructive interfer-ence occurring in the collective propagation of electrons andholes traveling along the same path /H20849the diffuson /H20850. These are robust against the breaking of time-reversal invariance/H20849while Cooperon contributions are rapidly suppressed byPHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 1098-0121/2007/75 /H208495/H20850/054406 /H2084910/H20850 ©2007 The American Physical Society 054406-1small ﬁelds /H20850but are sensitive to dephasing by inelastic pro- cesses. Indeed, Benoit et al.18and Pierre et al.5have shown that the amplitude of Aharonov-Bohm oscillations increases by almost an order of magnitude for increasing magneticﬁelds, clearly showing a suppression of dephasing by mag-netic /H20849Zeeman /H20850ﬁelds. This leads to the conclusion that the main mechanism of dephasing in the investigated low-temperature regime is the scattering from magnetic impuri-ties. Previously we have studied the zero-ﬁeld dephasing rate due to diluted Kondo impurities as measured from the WLexperiment. 13We showed that the dephasing rate for all ex- perimentally relevant temperatures is proportional to the in-elastic cross section, 14which itself can be expressed in terms of the Tmatrix describing the scattering of the electrons from a single magnetic impurity. Such a relation has beenproposed previously by Schwab and Eckern 19in the context of UCFs. In this paper we generalize our previous work to arbitrary, diluted impurity scatterers with typical energy transfer largerthan 1/ /H9270/H9272. Furthermore, we supply the magnetic ﬁeld depen- dence of 1/ /H9270/H9272as measured in the AB experiment /H20849and com- pare our results to the AB experiments performed by Pierreand Birge 20/H20850. The outline of this paper is as follows: In Sec. II we discuss the dephasing rate due to generic dynamicalscatterers as measured in the WL experiment. We brieﬂy re-view our previous results 13and generalize them to arbitrary dynamical impurities with typical energy transfer larger than1/ /H9270/H9272. In Sec. III we turn to the dephasing rate as measured from the UCF and the amplitude of the AB oscillations. Webrieﬂy review the main concepts entering the analysis ofthese experiments and discuss how the dephasing rate mea-sured from the UCF differs from that measured in the WLexperiment. The main goal of Sec. III is to give the dephas-ing rate as measured from the amplitude of the AB oscilla-tions. Results for the dephasing rate obtained using the nu-merical renormalization group /H20849NRG /H20850are described in Sec. IV. Section V summarizes with a discussion. II. DEPHASING RATE FROM WEAK-LOCALIZATION CORRECTIONS TO THE CONDUCTIVITY The most accurate way to extract 1/ /H9270/H9272at low magnetic ﬁelds is via the WL corrections to the Drude conductivity,which result from coherent back scattering of an electron-hole pair traveling along time-reversed paths in the disor-dered environment. Technically, the coherent propagation ofthe electron-hole pair is described by the Cooperon, C /H9024/H20849q/H20850, and the WL correction is given by /H9004/H9268WL0=−2e2D /H9266/H20885ddq /H208492/H9266/H20850dC/H9024=0/H20849q/H20850, /H208491/H20850 see Fig. 1. In the absence of dephasing by inelastic processes C/H9024/H20849q/H20850is the bare Cooperon, C/H90240/H20849q/H20850=1 Dq2+i/H9024+1 //H9270B, /H208492/H20850 as diagrammatically depicted in Fig. 2and211 /H9270B=4DeB,d=2 /H20849n/H20648B/H20850, /H208493/H20850 1 /H9270B=D 3/H20849eBL/H11036/H208502,d=1 , 2 /H20849n/H11036B/H20850, /H208494/H20850 is the dephasing rate due to the applied magnetic ﬁeld, B. Here Dis the diffusion constant, dis the dimension of the diffusion process, /H9270denotes the mean scattering time corre- sponding to a mean free path l=vF/H9270,nis a unit vector or- thogonal to the probe in d=2, and pointing along the wire in d=1 and L/H11036is the transverse dimension of the sample. As can be seen from Eq. /H208491/H20850the WL corrections depend on the strength of the applied magnetic ﬁeld, B, and diverge in low dimensions, d=1,2 for B=0, reﬂecting the fact that WL cor- rections in low-dimensional systems may become strong andlead to strong Anderson localization. Taking into account interactions /H20849as, e.g., provided by dy- namical impurities /H20850the bare Cooperon dresses with a mass, 1/ /H9270/H9272, i.e. /H20849if purely exponential decay is guaranteed /H20850, C/H9024=0/H20849q/H20850=1 Dq2+1 //H9270/H9272+1 //H9270B. /H208495/H20850 For weak magnetic ﬁelds /H20849/H9270/H9272/H11270/H9270B/H20850the WL corrections are therefore cut off by 1/ /H9270/H9272, allowing to determine the dephas- ing rate from ﬁtting the magnetoresistance. In this section we study the dephasing rate due to generic diluted, dynamical scatterers with typical energy transferlarger than 1/ /H9270/H9272as measured from WL. To be speciﬁc, we consider a Hamiltonian of the general form FIG. 1. Diagrammatic representation of the Cooperon C/H9024=00/H20849q/H20850 which enters the WL corrections to the Drude conductivity. p±=q/2±pand wavy lines denote current operators. FIG. 2. Bethe-Salpeter equation for the bare Cooperon C0¯/H20849q,/H9024/H20850=/H208491/2/H9266/H9263/H92702/H20850C0/H20849q,/H9024/H20850,p±=q/2±p, and p±/H11032=q/2±p/H11032. Dashed lines denote scattering from static impurities and R,Ade- notes the particle and hole lines, respectively.MICKLITZ, COSTI, AND ROSCH PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-2Himp=Himp0+/H20858 ick/H9268†ck/H11032/H9268/H11032fkk/H11032/H9268/H9268/H11032/H9251Xˆ/H9251ei/H20849k−k/H11032/H20850xi, /H208496/H20850 where Himp0is the Hamiltonian of the isolated impurity, c†,c are creation and annihilation operators of conduction bandelectrons, x idenotes the position of the impurities, and the momentum and spin-dependent function f/H9251parametrizes the coupling to some operator Xˆ/H9251describing transitions of the internal states of the dynamical impurity. Equation /H208496/H20850, e.g., describes the coupling of the conduction band electrons toKondo impurities, two-level systems, etc. In order to ﬁnd thedephasing rate due to such generic impurities one has tocompute the “self energy” or “mass” of the Cooperon gener-ated by the operators of Eq. /H208496/H20850. Two assumptions allow to reduce this problem to that of summing up a simple geometric series. First, we assume thatthe concentration of dynamical impurities n iis small, and, second, that the elastic mean free path lis large. How small niand how large lhas to be, depends on both, the dynamics and the extension of the impurity as brieﬂy discussed belowand—for Kondo impurities—in more detail in Ref. 13 /H20849the inﬂuence of stronger disorder on the distribution of Kondotemperatures has recently been investigated by Kettemannand Mucciolo 22/H20850. Note that in relevant experimental systems in the WL regime2–5the two assumptions are well justiﬁed.13 The ﬁrst observation is that quantum interference corrections to the inelastic scattering rate are small when a diffusingelectron is unlikely to return to the same dynamical impurity/H20849i.e., when the weak localization corrections are weak /H20850. Tech- nically speaking, this is reﬂected in the fact that diagramsmixing scattering from dynamical and static impurities aresuppressed by factors of 1/ N /H11036and a/l/H20851or 1/ /H20849kFl/H20850for a/H110211/kF/H20852, where N/H11036is the number of transverse channels in a quasi one or two dimensional system and ais the typical diameter of the dynamical impurity. This effect and furthersystem dependent factors relevant for the suppression ofquantum interference corrections are discussed in Ref. 13. Only at lowest, experimentally unprobed temperatures, doesthe enhanced infrared singularity, caused by the presence ofextra diffusion modes, overcompensate this phase space sup-pression factor, as discussed in detail in Ref. 13. The small parameter 1/ /H20849k Fl/H20850ora/ltherefore reduces the problem to compute the “mass” of the Cooperon to that of solving the Bethe-Salpeter equation diagrammatically depicted in Fig.3/H20849a/H20850. For small n i, one can furthermore restrict the analysis of the irreducible vertex /H9003to terms linear in nias shown in Fig. 3/H20849b/H20850./H9003can be separated into three distinct contributions: self-energy diagrams /H20851the ﬁrst two terms in Fig. 3/H20849b/H20850/H20852,a n “elastic” vertex correction with no energy transfer betweenupper and lower line /H20849third term /H20850, and an “inelastic” vertex where interaction lines connect the two lines /H20849last term /H20850. Only this inelastic vertex makes the Bethe-Salpeter equationa true integral equation as it mixes frequencies but, fortu-nately, this term can be neglected 1if the typical energy, /H9004E, exchanged between electrons and holes during an interactionprocess greatly exceeds the dephasing rate due to the dy-namical impurities, 1/ /H9270/H9272, i.e.,/H9004E/H9270/H9272/H112711. Physically,1the sup- pression of the inelastic vertex arises as an exchange of en-ergy/H9004Eleads to a phase mismatch of order ei/H9004E/H9270/H9272between electron and hole destroying interference completely for/H9004E /H9270/H9272/H112711. Technically, one can conﬁrm this argument by es- timating corrections to the WL contributions due to the in-elastic vertex, as, e.g., depicted in Fig. 4. 13More importantly, however, the condition /H9004E/H9270/H9272/H112711 always holds for sufﬁ- ciently small concentrations ni, since 1/ /H9270/H9272/H11008ni. In the case of Kondo impurities discussed below, for example, this condi-tion translates to n i/H11270/H9263TK. Restricting to the self-energy and elastic vertex contribu- tions, the Bethe-Salpeter equation is easily solved: Sinceself-energy and elastic vertex contributions conserve the en-ergy of single electron lines, the solution of the reducedBethe-Salpeter equation amounts to a straightforward sum-mation of a geometric series. Setting the center-of-mass fre-quency /H9024to 0 /H20849see Fig. 1and Fig. 3/H20850, the Cooperon is given by C /H9024=0/H20849/H9280,q/H20850=1 Dq2+1 //H9270/H9272/H20849/H9280,T/H20850+1 //H9270B, /H208497/H20850 with the Tand/H9280dependent dephasing rate FIG. 3. /H20849Color online /H20850Bethe-Salpeter equation for the Cooperon C¯in the presence of dilute dynamical impurities to linear order in ni.C¯0is the bare Cooperon in the absence of interactions and /H9003the irreducible vertex obtained by adding self-energy, elastic-, andinelastic-vertex contributions. The crosses with attached dashedlines denote the averaging over impurity positions x i, the squares the inelastic scattering from a single impurity to arbitrary order. FIG. 4. /H20849Color online /H20850Lowest order correction to WL contribu- tions due to inelastic vertex. The Cooperon, C¯, is dressed with a mass resulting from summation of the elastic part of the vertex /H9003, i.e., the self-energy and the elastic vertex contribution.MAGNETIC FIELD DEPENDENCE OF DEPHASING RATE … PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-31 /H9270/H9272/H20849/H9280,T/H20850=2ni /H9266/H9263/H20875/H20885d3p /H208492/H9266/H208503g/H9280/H20849p/H208501 2i/H20851T−p,−pA/H20849/H9280/H20850−TppR/H20849/H9280/H20850/H20852/H20876/H20875 −/H20885d3p /H208492/H9266/H208503/H20885d3p/H11032 /H208492/H9266/H208503g/H9280/H20849p/H20850g/H9280/H20849p/H11032/H20850Tpp/H11032R/H20849/H9280/H20850T−p,−p/H11032A/H20849/H9280/H20850/H20876. /H208498/H20850 Here g/H9280/H20849p/H20850=/H9266/2/H9270 /H20851/H9280/H20849p/H20850−/H9280/H208522+1/4/H92702restricts the electrons momenta, p, and energies, /H9280, to the Fermi-surface, /H9280/H20849p/H20850is the dispersion relation of the conduction band, TA,Rare the advanced and/or retarded Tmatrices, deﬁned by the Green’s function Gxx/H11032/H20849/H9280/H20850=Gxx/H110320/H20849/H9280/H20850+Gx00/H20849/H9280/H20850T/H20849/H9280/H20850G0x0/H11032/H20849/H9280/H20850, and/H9263denotes the den- sity of states per spin. Equation /H208498/H20850generalizes our result of Ref. 13to arbitrary shaped diluted impurities. Notice that also forward scattering processes enter 1/ /H9270/H9272, which do not contribute to the transport scattering rate. We stress that Eq./H208498/H20850is the general result for the dephasing rate for a weakly disordered metal due to a low concentration of generic dy- namical impurities for which the condition /H9004E/H112711/ /H9270/H9272holds. In the opposite limit, /H9004E/H112711//H9270/H9272, vertex corrections become important, as has been discussed earlier on23,24in the context of magnetic impurities. As we assumed that a/H11270l, Eq. /H208498/H20850can be further simpli- ﬁed, 1 /H9270/H9272/H20849/H9280,T/H20850=2ni /H9266/H9263/H20875/H20885 SF/H9280d2p /H208492/H9266/H2085031 /H20841vF/H20849p/H20850/H20841Im/H20851/H9266TppA/H20849/H9280/H20850/H20852 −/H20885 SF/H9280d2p /H208492/H9266/H208503/H20885 SF/H9280d2p/H11032 /H208492/H9266/H2085031 /H20841vF/H20849p/H20850/H208411 /H20841vF/H20849p/H11032/H20850/H20841/H20841/H9266Tpp/H11032R/H20849/H9280/H20850/H208412/H20876, /H208499/H20850 where SF/H9280is the Fermi surface /H20849or more precisely the surface with/H9280k=/H9280/H20850. Here we also assumed a time-reversal invariant system with Tpp/H11032R/H20849/H9280/H20850=T−p/H11032,−pR/H20849/H9280/H20850and employed the identity /H20851Tpp/H11032R/H20849/H9280/H20850/H20852*=Tp/H11032pA/H20849/H9280/H20850. Also Kettemann and Mucciolo have gen- eralized the dephasing rate for a momentum independent T matrix to Eq. /H208499/H20850independently in a recent report.22The dephasing rate given in Eq. /H208499/H20850has a simple interpretation:14 Since the Fermi-surface integrated imaginary part of the T matrix is proportional to the total cross section and /H20841Tpp/H11032R/H208412 /H20849integrated over the Fermi surface /H20850is proportional to the elastic cross section, its difference is, by deﬁnition, propor-tional to the inelastic cross section, /H9268inel, introduced in Ref. 14. Therefore, Eq. /H208499/H20850can be rewritten in the form 1 /H9270/H9272/H20849/H9280/H20850=ni/H20855vF/H20849p/H20850/H9268inel/H20849p,/H9280/H20850/H20856, /H2084910/H20850 where /H20855¯/H20856denotes an angular average weighted by 1/ vF/H20849p/H20850 to take into account that fast electrons are scattered more frequently from elastic impurities. According to Eq. /H2084910/H20850,/H9270/H9272 is nothing but the average time needed /H20849in a semiclassical picture /H20850to scatter from an impurity with cross section /H9268inel. Note that the vanishing of 1/ /H9270/H9272for static impurities is guar- anteed by the optical theorem. From Eq. /H208499/H20850we can read off the dephasing rate for di- luted dynamical isotropic s-wave scatterers13,191 /H9270/H9272/H20849/H9280,T/H20850=2ni /H9266/H9263/H20851/H9266/H9263locIm/H20851TA/H20849/H9280/H20850/H20852−/H20841/H9266/H9263locTR/H20849/H9280/H20850/H208412/H20852,/H2084911/H20850 where /H9263locis the local density of states at the Fermi energy at the site of the impurity which can differ from the thermody-namic density of states entering the prefactor. Note that inthe case of Kondo impurities discussed below /H20849and in Ref. 13/H20850, the combination /H9263locTR/A/H20849/H9280/H20850=f/H20849/H9280/TK,T/TK,B/TK/H20850is an universal dimensionless function of the ratios /H9280/TK,T/TK,B/TK. If the assumptions underlying the deriva- tion of Eq. /H2084911/H20850are valid, one can therefore predict without any free parameter the dephasing rate if the concentration ofspin-1/2 impurities, the Kondo temperature and the thermo-dynamic density of states are known /H20849see, e.g., Ref. 15/H20850. However, one of the assumptions underlying the derivationof the prefactor of Eq. /H2084911/H20850may not be valid in realistic materials: we assumed that the static impurities are com-pletely uncorrelated and local such that electrons are scat-tered uniformly over the Fermi surface. While this should bea good assumption in doped semiconductors, this may not bevalid in metals with complex Fermi surfaces and stronglyvarying Fermi velocities. Under the latter conditions, we ex-pect that the prefactor of Eq. /H2084911/H20850becomes nonuniversal, yielding temperature-independent corrections of order one,which may be important for the interpretation of high-precision experiments. 16,15 In Ref. 13we have calculated the leading corrections to Eq. /H2084911/H20850arising from mixed diagrams involving combined scattering from static and dynamical impurities and from dia-grams including higher processes in n i. We showed that, sup- pressed by the small parameter 1/ /H20849kFl/H20850, their contributions are negligible at all experimentally relevant temperatures, T. Only at the lowest experimentally irrelevant temperatures dothese corrections become important due to infrared singulari-ties of the dressed interaction potential /H20849dressed by coherent backscattering processes /H20850. The estimates of subleading cor- rections presented in Ref. 13can be generalized to extended dynamical impurities by replacing 1/ /H20849k Fl/H20850bya/lfor kFa/H110221. The experimentally measured dephasing rate does not re- solve the dependence on the electrons energy /H9280. We de- scribed in Ref. 13that to allow for a comparison with the /H9280-independent dephasing rate, /H9270/H9272−1/H20849T/H20850, extracted from the WL experiment, the energy-resolved representations of 1/ /H9270/H9272, Eqs. /H208499/H20850–/H2084911/H20850, still require an average over energies accord- ing to 1 /H9270/H9272/H20849T/H20850=/H20902/H20875−/H20885d/H9280fF/H11032/H20849/H9280/H20850/H9270/H9272/H20849/H9280,T/H20850/H208492−d/H20850/2/H208762//H20849d−2/H20850 ,d= 1,3, 1 /H9270exp/H20875/H20885d/H9280fF/H11032/H20849/H9280/H20850ln/H9270/H9272/H20849/H9280,T/H20850 /H9270/H20876,d=2 , −/H20885d/H9280fF/H11032/H20849/H9280/H20850//H9270/H9272/H20849/H9280,T/H20850,/H9270/H9272//H9270B/H112711./H20901 /H2084912/H20850 Here the last line applies to a case where a relatively strong magnetic ﬁeld, B, is present.MICKLITZ, COSTI, AND ROSCH PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-4Specifying to a situation where the coupling of the con- duction band electrons to the /H20849diluted /H20850dynamical impurities is described by the spin-1/2 Kondo effect, the general Hamil-tonian of Eq. /H208496/H20850takes the form H S=J/H20858 iSˆic/H9268†/H20849xi/H20850/H9268/H9268/H9268/H11032c/H9268/H11032/H20849xi/H20850, /H2084913/H20850 where Jis the exchange constant. An external magnetic ﬁeld, B, causes Zeeman splitting, /H9280z, of the conduction band elec- tron spin states, /H9280z=ge/H9262BB, /H2084914/H20850 and couples to the impurity spins according to HB=gS/H9262BB/H20858 iSˆ iz. /H2084915/H20850 ge,gSare the electrons and the magnetic impurities gyromag- netic factors, respectively. As already mentioned in the Intro-duction, measuring the B-ﬁeld dependence of the dephasing rate due to Kondo impurities in a WL experiment is a highlydelicate task. In view of this difﬁculty it is more feasible tomeasure the Bdependence of 1/ /H9270/H9272from the amplitude of the AB oscillations as discussed in the following section. III. DEPHASING RATE FROM UNIVERSAL CONDUCTANCE FLUCTUATIONS AND AHARANOV- BOHM OSCILLATIONS Let us ﬁrst begin with a brief discussion of the universal conductance ﬂuctuations /H20849UCF /H20850and their dependence on 1//H9270/H9272and then turn to the experiment on Aharonov-Bohm rings. To be speciﬁc, we consider a wire of noninteracting elec- trons, scattering elastically from static impurities, and inelas-tically off a low concentration, n i, of Kondo impurities, where the coupling of the conduction band electrons to thedynamical impurities is described by the Hamiltonian equa-tion /H2084913/H20850and the inﬂuence of the magnetic ﬁeld is accounted for by Eqs. /H2084914/H20850and /H2084915/H20850. For wires of length L/H11271L T=/H20881D/T, the ﬂuctuations of the conductance, /H9254g/H9254g, are determined25by /H9254g/H9254g=/H208492e2D/H208502 3/H9266TL4/H20885d/H92801d/H92802fF/H11032/H20849/H92801/H20850fF/H11032/H20849/H92802/H20850 /H11003/H20885dx1dx2/H20841P/H92801,/H92802/H20849x1,x2/H20850/H208412. /H2084916/H20850 Here P/H92801,/H92802/H20849x1,x2/H20850is the amplitude for an electron-hole pair, with energies /H92801,/H92802, respectively, to diffusively travel from x1tox2along the same trajectory /H20849diffuson, see Fig. 5/H20850. The overbar denotes the ensemble average, which is experimen-tally realized by changing the magnetic ﬁeld. Equation /H2084916/H20850 assumes the large ring diameters, L/H11271L /H9272, such that the dephasing rate, 1/ /H9270/H9272, controls the magnitude of the ﬂuctua- tions. Furthermore, we assume temperatures T/H9270/H9272/H112711. Notice that generally there is also a contribution from the Cooperon,which is, however, suppressed already for small magneticﬁelds.We change to momentum representation and separate the two-particle propagator Pinto its spin-singlet and -triplet components, P=/H20858 i=1,...,4 P/H20849i/H20850, where /H20849following the notation of Ref. 12/H20850 P/H92801/H92802/H20849i/H20850/H20849q/H20850=1 Dq2+i/H20849/H92801−/H92802+/H9256i/H9280z/H20850+1 //H9270SO/H20849i/H20850+1 //H9270/H9272,S/H20849i/H20850/H20849/H92801,/H92802,B/H20850. /H2084917/H20850 The modes i=1,2,3 describe the spin triplet state with Sz component equal to 1, −1, and 0, respectively. i=4 denotes the spin singlet channel. The Zeeman splitting enters only thetriplet diffuson with nonvanishing projection S z= ±1, i.e., /H9256i= ±1 for i=1,2 and zero otherwise. 1/ /H9270SO/H20849i/H20850is the spin-orbit scattering rate. 1/ /H9270SO/H20849i/H20850is identical for the three spin triplet diffuson /H20849i=1,2,3 /H20850and zero for the spin singlet mode /H20849i=4/H20850. For strong spin-orbit scattering only the singlet diffu- sion contributes /H20849otherwise 1/ /H9270SO/H20849i/H20850is an additional ﬁtting pa- rameter /H20850. Finally 1/ /H9270/H9272,S/H20849i/H20850is the dephasing rate for the ith diffuson mode due to the presence of diluted magnetic im-purities which has the structure 1 /H9270/H9272,S/H20849i/H20850/H20849/H92801,/H92802/H20850=2ni /H9266/H9263/H20873/H9266/H9263 2i/H20851T/H20849i,a/H20850/H20849/H92802,B/H20850−T/H20849i,b/H20850/H20849/H92801,B/H20850/H20852 −/H20849/H9266/H9263/H208502T/H20849i,c/H20850/H20849/H92801,B/H20850T/H20849i,d/H20850/H20849/H92802,B/H20850/H20874, /H2084918/H20850 where the proper combination of Tmatrices for the various channels can be read off by comparison with Table I. Equa- tion /H2084918/H20850is evaluated from summing up self-energy and elas- tic vertex contributions. Notice that in contrast to the WLexperiment the electron and hole lines /H20849i.e., the inner and outer rings /H20850in Fig. 6represent different measurements. Therefore there are no correlations between dynamical im- TABLE I. Combination of Tmatrices entering the dephasing rates for spin-triplet and spin-singlet diffusons. Sdenotes the total spin and Mitszcomponent. T↑,T↓denotes the Tmatrix for spin-up and spin-down electrons, respectively. i /H20841S,M/H20856 Combinations of Tmatrices 1 S=1,M=1 T1=1 2i/H20849T↓A−T↑R/H20850−T↓RT↑A 2 S=1,M=−1 T2=1 2i/H20849T↑A−T↓R/H20850−T↑RT↓A 3 S=1,M=0 T3=1 2Im/H20849T↑A+T↓A/H20850−1 2/H20849T↑RT↑A−T↓RT↓A/H20850 4 S=0 T4=1 2Im/H20849T↑A+T↓A/H20850−1 2/H20849T↑RT↑A−T↓RT↓A/H20850 FIG. 5. Bethe-Salpeter equation for the bare diffuson D¯0/H20849q,/H9024/H20850 =/H208491/2/H9266/H9263/H92702/H20850D0/H20849q,/H9024/H20850.p±=p±q/2 and p±/H11032=p/H11032±q/2. Dashed lines denote scattering from static impurities and R,Adenotes the par- ticle and hole lines, respectively.MAGNETIC FIELD DEPENDENCE OF DEPHASING RATE … PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-5purities residing on different rings and interaction lines may only be drawn within the same ring. Consequently the inelas-tic vertex contributions do not enter the Bethe-Salpeter equa-tion for the diffuson, see Fig. 7. Notice that there are inelastic vertex contributions, as, e.g., depicted in Fig. 8, which be- come important in the context of electron-electroninteractions. 26It is instructive to compare those to the inelas- tic vertex corrections relevant for WL depicted in Fig. 4.I n the latter case, the sum of the incoming momenta of thevertex is small due to the Cooperon in Fig. 4. Consequently, the inelastic vertex corrections to the WL dephasing rate arenotsuppressed by powers of 1/ /H20849k Fl/H20850but only by powers of 1//H20849/H9004E/H9270/H9272/H20850. In contrast, the relevant momenta in Fig. 8are uncorrelated /H20849i.e., particle and hole are far apart /H20850, leading to an suppression both by powers of 1/ /H20849kFl/H20850and of 1/ /H20849/H9004E/H9270/H9272/H20850.27 We point out the following differences for 1/ /H9270/H9272,Smea- sured from the UCF experiment, Eq. /H2084918/H20850, compared to that found from the WL, Eq. /H2084911/H20850. First, the Tmatrices entering Eq. /H2084918/H20850depend on the spin conﬁguration of the diffuson mode and have acquired a Bdependence due to the coupling of the impurity spin to B, Eq. /H2084915/H20850. Second, 1/ /H9270/H9272,Sdepends on two energies. This results from the fact that in the UCFexperiment electron and hole lines constituting the diffusonare produced in different measurements of the conductance/H20849see Fig. 6/H20850. Therefore their energies are individually aver- aged as can be seen in Eq. /H2084916/H20850. From Eqs. /H2084916/H20850and /H2084917/H20850the amplitude of the UCFs is obtained to be proportional to /H20881/H9270/H9272,S. Especially compared to the AB oscillations, discussed below, the 1/ /H9270/H9272,Sdependence of the UCFs is rather weak. In the following, we will there-fore focus our discussion on AB experiments. Aharonov-Bohm oscillations are measured in a ring geometry,20,28where the conductance oscillates periodically as a function of Bpiercing the ring, and the amplitude of these AB oscillations decrease exponentially with 1/ /H9270/H9272,S. The periodic oscillations result from the change of boundary con-ditions, due to ﬂux lines piercing the ring and can be calcu-lated from /H9254g/H20849/H9278/H20850/H9254g/H20849/H9278+/H9004/H9278/H20850=/H208492e2D/H208502 3/H9266TL4/H20885d/H92801d/H92802fF/H11032/H20849/H92801/H20850fF/H11032/H20849/H92802/H20850 /H11003/H20885dx1dx2/H20841P/H92801,/H92802/H9004/H9278/H20849x1,x2/H20850/H208412. /H2084919/H20850 Here P/H9004/H9278is again given by Eq. /H2084917/H20850but now the continuous qhave to be replaced25by discrete momenta, q=qm/H20849/H9004/H9278/H20850 =2/H9266 L/H20849m+/H9004/H9278 /H92780/H20850, depending on the difference of the magnetic ﬂux during the individual measurements of g. The ﬂuctua- tions are a periodic function in /H9004/H9278//H92780, where /H92780=2/H9266/eis the elementary ﬂux quantum and /H9004/H9278=/H9004BL2//H208494/H9266/H20850. There- fore an expansion in its harmonics can be made,25 /H9254g/H20849/H9278/H20850/H9254g/H20849/H9278+/H9004/H9278/H20850=Ce4 /H92662/H20858 k=0/H11009 Ak/H20849B/H20850cos/H208752/H9266k/H9004/H9278 /H92780/H20876,/H2084920/H20850 where Cis a factor of order 1, depending on the sample geometry in the vicinity of the ring. Restricting to the situa-tion of strong spin-orbit scattering 29where the spin singlet diffuson gives the leading contributions to Eq. /H2084919/H20850, one ﬁnds that /H20849forL/H11271L/H9272/H20850 Ak/H20849B/H20850=/H208492/H9266/H208503D3/2 T2L3/H20885d/H9280e−kL//H20881D/H9270/H9272/H20849/H9280/H20850 cosh4/H20849/H9280/2T/H20850/H20881/H9270/H9272/H20849/H9280/H20850, /H2084921/H20850 where /H20851/H9270/H9272=/H9270/H9272,S/H208494/H20850in the notation of Eq. /H2084918/H20850/H20852 1 /H9270/H9272/H20849/H9280,T,B/H20850=2ni /H9266/H9263/H20849/H9266/H9263Im/H20851T/H208494/H20850A/H20849/H9280,B/H20850/H20852−/H20841/H20849/H9266/H9263/H20850T/H208494/H20850R/H20849/H9280,B/H20850/H208412/H20850. /H2084922/H20850 Here/H9280=/H92801+/H92802and we used that relevant contributions to the integral over energy differences, /H9280¯=/H92801−/H92802, result from ener- gies/H9280¯/H112701//H9270/H9272to eliminate the /H9280¯dependence. Notice that such a reduction to a single energy integral can only be done in aone-dimensional system where the qintegral over the square of the diffuson, Eq. /H2084917/H20850, is dominated by infrared diver- FIG. 6. Diagram giving the main contribution to the UCF. Dashed lines represent coherent impurity scattering of electron /H20849R/H20850 hole /H20849A/H20850pair, i.e., the bare diffuson where interaction due to scat- tering from magnetic impurities is not yet taken into account. FIG. 7. /H20849Color online /H20850/H20849a/H20850Bethe-Salpeter equation for the diffu- son, D¯, in the presence of /H20849dilute /H20850magnetic impurities to linear order in ni.D¯0is the bare diffuson in the absence of interactions. /H20849b/H20850 Diagrammatic representation of the irreducible interaction vertex,/H9003, consisting of the self-energy /H20849represented by the ﬁrst two contri- butions /H20850, the elastic vertex /H20849third contribution /H20850. The inelastic vertex /H20849the fourth contribution /H20850does not enter the diffuson as measured in the UCF. FIG. 8. /H20849Color online /H20850Diagrammatic representation of lowest- order corrections to UCF due to inelastic vertex contributions. No-tice that for a local interaction these are not only suppressed bypowers of 1/ /H20849/H9004E /H9270/H9272/H20850but also small in powers 1/ /H20849kFl/H20850.MICKLITZ, COSTI, AND ROSCH PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-6gences. In a 2- dsystem, e.g., relevant energies extend to /H9280¯/H11011T. For the following, it is convenient to rewrite Eq. /H2084922/H20850 /H20851see also Eq. /H2084910/H20850/H20852as 1 /H9270/H9272=1 /H9270hit/H20855/H9268inel/H20856 /H9268max, /H2084923/H20850 where /H9268max=4/H9266/kF2is the cross section of a unitary scatterer, /H9268inel//H9268maxis the conditional probability of inelastic scatter- ing if an electron hits the impurity, and 1 /H9270hit=2ni /H9266/H9263/H2084924/H20850 describes the typical “hitting rate.” As in the WL experiment discussed above, ﬁts to experi- mental data have to be done with the /H9280-independent dephas- ing rate. For a comparison with experiment we thereforehave to give the /H9280-independent dephasing rate which for k=1 is obtained by solving the equation L/H9272/H20849T,B,L/H20850 Le−L/L/H9272/H20849T,B,L/H20850=3 8T/H20885d/H9280e−L/L/H9272/H20849/H9280,T,B/H20850 cosh4/H20849/H9280/2T/H20850L/H9272/H20849/H9280,T,B/H20850 L, /H2084925/H20850 where L/H9272=/H20881D/H9270/H9272is the dephasing length. Notice that the ac- tually measured dephasing rate depends on the length of thering. It also differs from the WL result due to the differentenergy averages. IV . NUMERICAL RESULTS FOR DEPHASING RATES FROM AHARANOV-BOHM OSCILLATIONS In order to evaluate the dependence of the dephasing rate on magnetic ﬁeld, temperature and ring length from Eq. /H2084927/H20850 we require the T-matrix for the single impurity Kondo modeldeﬁned in Sec. II. By using the equation of motion methodthis can be expressed as 30 T/H9268/H20849/H9275,T,B/H20850=J/H20855Sz/H20856+J2/H20855/H20855Sc/H9251†/H9268/H9251/H9268;S/H9268/H9268/H9251/H11032c/H9251/H11032/H20856/H20856, /H2084926/H20850 where /H20855/H20855¯/H20856/H20856denotes a retarded correlation function and /H9268 are the Pauli spin matrices. We calculate Eq. /H2084926/H20850by apply- ing the numerical renormalization group /H20849NRG /H20850method31for ﬁnite temperature dynamics.32At ﬁnite magnetic ﬁeld, it is also important to use the reduced density matrix33to evaluate the above dynamical quantity. For all calculations presentedhere we used a discretization parameter for the conductionband of /H9011=1.5 and we retained 960 states per NRG iteration. We checked that this number of states was sufﬁcient to main-tain particle-hole symmetry of the spectral densitiesImT ↑/H20849/H9275,T,B/H20850=Im T↓/H20849−/H9275,T,B/H20850at this relatively small value of the discretization parameter. The Friedel sum rule for the T=0 spectral density was satisﬁed to more than 1% accuracy in our calculations. Figure 9shows the numerical evaluation of /H20881A1/H20849B/H20850for various Tat a given length L=10Lhitwhere Lhit=/H20881D/H9270hit. No- tice that, if lengths are measured in units of Lhit, the ampli- tude /H20849A1/H9270hitTK/H208501/2becomes a universal function of B/TKand T/TK. Figure 10gives the magnetic ﬁeld dependence of thedephasing rate at various Tand a ﬁxed ring length L=10Lhit. For large magnetic ﬁelds, B/H11271T,TK, the dephasing rate is expected12to vanish proportional to /H20849T/B/H208502/ln4/H20851B/TK/H20852, consistent with the numerical results /H20849the precise form also depends on L/L/H9272, see below /H20850. Figure 11shows the results for 1/ /H9270/H9272,Sas a function of T for different strengths of the magnetic ﬁeld Bat a ﬁxed ring length L=10Lhit. While the maximal dephasing occurs for T/H11011TKfor small ﬁelds B/H11270TK, it shifts to larger values /H20849T/H11011B/H20850forB/H11271TK. For high temperatures, small ﬁelds and not too large L/L/H9272, see below, T/H11271TK,Bthe dephasing rate is well described by the Nagaoka-Suhl formula,13 1//H9270/H9272/H20849T/H20850=ni 2/H9266/H9263/H926623/4 /H926623/4+ln2T/TK. Figure 12shows the dependence of 1/ /H9270/H9272,S/H20849T/H20850on the ring length, L, at zero magnetic ﬁeld. As pointed out above, the L dependence of the experimentally measured dephasing rateenters through the energy average of Eq. /H2084925/H20850. As can be seen from Fig. 12the dephasing rate only changes by a factor 1/4 on increasing the ring length by a factor 200. We includedcurves for the rather academic cases L=100–1000 L hit/H20849the amplitude is too strongly suppressed to be observed for such FIG. 9. /H20849Color online /H20850Amplitude of the Aharonov-Bohm oscil- lations /H20849in units of /H20851/H208492/H9266/H208503//H20849/H9270hitTK/H20850/H208521/2/H20850as a function of the applied magnetic ﬁeld, B/H20849in units TK,/H9262B/kB=1/H20850, for different temperatures T/H20849in units TK/H20850obtained from NRG calculations for L/Lhit=10. The rapid rise of the oscillation amplitude results from the suppressionof dephasing /H20849see Fig. 10/H20850by polarizing the spins. Here, and in the remaining ﬁgures in this section, the symbols represent the discretevalues of /H20849B,T/H20850at which NRG calculations were carried out. FIG. 10. /H20849Color online /H20850The dephasing rate as a function of the magnetic ﬁeld, B/H20849in units TK/H20850for various temperatures, T, and a ring of length L=10Lhit. Note the rapid suppression of 1/ /H9270/H9272espe- cially for low temperatures.MAGNETIC FIELD DEPENDENCE OF DEPHASING RATE … PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-7large ring lengths /H20850in order to show this weak Ldependence. Figure 12also shows that for temperatures T/H11351TKthe dephasing rate becomes entirely Lindependent. This reﬂects the fact that the energy resolved dephasing rate, 1/ /H9270/H9272,S/H20849/H9280,T/H20850, establishes a deep minimum at /H9280=0 for T/H11351TKand the inte- gral on the right-hand side of Eq. /H2084925/H20850is therefore well ap- proximated by setting /H9280=0. To be precise, in the limit of ring lengths L/H11271L/H9272the integral on the right-hand side of Eq. /H2084925/H20850 /H20849for ﬁxed TandB/H20850is dominated by the saddle points of the function f/H20849/H9280/H20850=L L/H9272/H208494/H20850/H20849/H9280/H20850−l n/H20875L/H9272/H208494/H20850/H20849/H9280/H20850 L/H20876+4l n/H20875cosh/H20873/H9280 2T/H20874/H20876. For temperatures T/H11351max /H20853B,TK/H20854fhas a saddle point at /H9280=0, which for temperatures T/H11407TKbecomes unstable. At very large ring diameters L/Lhit/H11271102, a second saddle point at/H9280=L 2Lhitln2/H20851L/2Lhit/H20852Tstarts to dominate the integral for T/H11407TK. Here Lhit=/H20881D/H9270hitis the diffusive length scale corre- sponding to the time /H9270hit=/H9266/H9263 2niand introduced above. Although this limit is rather academic it is interesting that for such big rings dephasing is dominated by rare events of highly excitedthermal electrons scattering from the magnetic impurities. This originates from the fact that high-energy electrons scat-ter less effectively from Kondo spins, as Kondo renormaliza-tion becomes less effective for /H9280/H11271TK. Inserting this second saddle point into Eq. /H2084925/H20850one ﬁnds that the length depen- dence of 1/ /H9270/H9272for high temperatures follows 1 /H9270/H9272/H20849T,L/H20850/H110111 ln2/H20875L 2Lhitln2/H20851L/2Lhit/H20852/H20876, /H2084927/H20850 explaining the weak suppression of 1/ /H9270/H9272for large ring lengths shown in Fig. 12. V . DISCUSSION AND CONCLUSIONS In this paper we generalized previous results for the dephasing rate due to diluted Kondo impurities as measuredin the weak localization experiment to describe dephasingdue to arbitrary diluted impurities. Furthermore, we investi- gated how magnetic ﬁelds modify the dephasing rate due toKondo spins as can be measured in mesoscopic Aharonov-Bohm rings. We give results for the numerically evaluateddephasing rate as a function of the magnetic ﬁeld, tempera-ture, and the ring length. The inﬂuence of magnetic impurities on dephasing has been studied in a number of magneto-resistance experiments FIG. 11. /H20849Color online /H20850NRG results for the dephasing rate as a function of Tand for different values of B./H9270hit=/H9266/H9263/2nias deﬁned above, Tand Bare given in units of TK/H20849Ref. 34/H20850/H20849we set /H9262B/kB=1/H20850. The values are for a ring of length, L,L=10Lhit, where Lhit=/H20881D/H9270hit. While the maximal dephasing occurs for T/H11011TKfor small ﬁelds B/H11270TK, it shifts to larger values /H20849T/H11011B/H20850forB/H11271TK. FIG. 12. /H20849Color online /H20850Temperature-dependent dephasing rate for various ring lengths L, measured in units Lhit=/H20881D/H9266/H9263/2niand calculated via NRG for B=0. The logarithmic suppression with L for larger Tarises due to the interference of electrons with energies larger than T. FIG. 13. /H20849Color online /H20850Amplitude of the AB oscillations in units of e2/has a function of T/TKforT=40 mK /H20849/H9004/H20850and 100 mK /H20849/H12331/H20850measured by Piere et al. /H20849Refs. 5and 20/H20850, assuming TK=10 mK. Solid and dot-dashed lines are the numerically calcu- lated amplitudes with ﬁtting parameters described in the main text.As for very high magnetic ﬁelds, B/H11271100T K, numerical errors in- crease when the dephasing rate becomes very small, we used anextrapolation of the numerical results, 1/ /H9270/H9272/H110081/B2, in this regime. The saturation of the amplitude at these high ﬁelds arises as thedephasing due to electron-electron interaction dominates. The datais equally well described by the ﬁts used in Refs. 5and20/H20849dashed lines /H20850, see main text. For the solid lines we used the same values for the dephasing rates /H20849 /H9270ee=5.4 ns and 9.9 ns for T=100 mK and T=40 mK, respectively /H20850as in Ref. 5, where /H9270ee/H11008T−2/3was as- sumed. For the dot-dashed curve we use instead /H9270ee=13.5 ns for T=40 mK since one expects theoretically35that/H9270ee/H110081/Tfor L/H11271L/H9278/H20849note, however, that L/H11011L/H9278in this regime explaining the rather weak dependence on /H9270ee/H20850.MICKLITZ, COSTI, AND ROSCH PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-8in Cu, Ag, or Au wires doped with magnetic impurities.2,3,5,15,16More recently, high-precision experi- ments using ion-implanted Fe impurities in Ag wires alloweda quantitative comparison with our theory for spin-1/2 impu-rities, see Refs. 15and16for a critical discussion. Such studies using samples doped with magnetic impurities have,to our knowledge, only been performed in the spin-glassregime 36using magnetic ions with tiny Kondo temperatures. Pierre et al.5,20studied rings made from nominally clean Cu wires. As these wires show a saturation of the dephasing rate/H20849determined from weak localization /H20850at low temperatures /H2084930 mK /H11351T/H113511K /H20850, it was suspected that tiny concentrations of magnetic impurities with Kondo temperatures below 30 mK may be at the origin of the observed saturation. Thispicture could be conﬁrmed as measurements of the ampli-tude of Aharonov-Bohm oscillation displayed a dramatic riseby almost an order of magnitude in moderate magnetic ﬁelds /H20849see Fig. 13/H20850, proving the magnetic origin of the low- B, low- Tdephasing. As neither the concentrations nor the type /H20849s/H20850of magnetic impurities are known, a parameter-free comparison to ourpredictions is not possible for these systems. Assuming Mnimpurities, believed to be characterized by a Kondo tempera-ture of the order of 10 mK, 37and, using the same dephasing rates due to electron-electron interactions as in Ref. 5 /H20849/H9270ee=5.4 ns and 9.9 ns for T=100 mK and T=40 mK, re- spectively /H20850we obtain the ﬁts shown in Fig. 13for a gfactor ofg/H110151.4 and an impurity concentration of 2.7 ppm. Wehave also added a curve at T=40 mK /H20849dot-dashed line /H20850 which uses /H9270ee=13.5 ns /H20849keeping all other parameters iden- tical /H20850to take into account that one expects theoretically35 /H9270ee/H110081/T. The ﬁts and the extracted parameters are not very reliable as can be seen from the observation that the data hasbeen equally well described in Refs. 5and20by the simple perturbative formula/H9270/H9272,S/H20849B=0/H20850 /H9270/H9272,S/H20849B/H20850=g/H9262B/kBT sinh /H20849g/H9262B/kBT/H20850with g=1.08, see Fig.13. For a more meaningful comparison to our results, experi- ments on AB rings, doped with magnetic impurities with ahigher Kondo temperature would be highly desirable. On theone hand, such experiments and their theoretical interpreta-tion can reveal basic dephasing mechanisms in metals, on theother hand, they can be used to obtain insight into the phys-ics of strongly correlated dynamical impurities and their in-teractions. ACKNOWLEDGMENTS We thank Ch. Bäuerle, N. Birge, J. v. Delft, L. Glazman, S. Kettemann, S. Mirlin, J. Mydosh, L. Saminadayar, B. Spi-vak, P. Wölﬂe, and especially, A. Altland for useful discus-sions and N. Birge for sending us his experimental data.Furthermore, we acknowledge ﬁnancial support from theDeutsche Forschungsgemeinschaft through the SFB 608 andTransregio SFB 12 and the NIC Juelich for computing time. 1B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky, J. Phys. C 15, 7367 /H208491982 /H20850. 2P. Mohanty, E. M. Q. Jariwala, and R. A. Webb, Phys. Rev. Lett. 78, 3366 /H208491997 /H20850; P. Mohanty and R. A. Webb, Phys. Rev. B 55, R13452 /H208491997 /H20850. 3F. Schopfer, Ch. Bäuerle, W. Rabaud, and L. Saminadayar, Phys. Rev. Lett. 90, 056801 /H208492003 /H20850. 4C. Van Haesendonck, J. Vranken, and Y. Bruynseraede, Phys. Rev. Lett. 58, 1968 /H208491987 /H20850. 5F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, and N. O. Birge, Phys. Rev. B 68, 085413 /H208492003 /H20850. 6D. S. Golubev and A. D. Zaikin, Phys. Rev. Lett. 81, 1074 /H208491998 /H20850. 7I. L. Aleiner, B. L. Altshuler, and M. E. Gershenson, Waves Ran- dom Media 9, 201 /H208491999 /H20850. 8J. v. Delft, cond-mat/0510563; J. v. Delft, in Fundamental Prob- lems of Mesoscopic Physics , edited by I. V. Lerner et al. /H20849Klu- wer, London, 2004 /H20850, pp. 115–138. 9A. Zawadowski, J. von Delft, and D. C. Ralph, Phys. Rev. Lett. 83, 2632 /H208491999 /H20850. 10Y. Imry, H. Fukuyama, and P. Schwab, Europhys. Lett. 47, 608 /H208491999 /H20850. 11P. Mohanty and R. A. Webb, Phys. Rev. Lett. 84, 4481 /H208492000 /H20850. 12M. G. Vavilov and L. I. Glazman, Phys. Rev. B 67, 115310 /H208492003 /H20850; M. G. Vavilov, L. I. Glazman, A. I. Larkin, ibid. 68, 075119 /H208492003 /H20850. 13T. Micklitz, A. Altland, T. A. Costi, and A. Rosch, Phys. Rev. Lett. 96, 226601 /H208492006 /H20850.14G. Zaránd, L. Borda, J. von Delft, and N. Andrei, Phys. Rev. Lett. 93, 107204 /H208492004 /H20850. 15F. Mallet et al. , Phys. Rev. Lett. 97, 226804 /H208492006 /H20850. 16G. M. Alzoubi and N. O. Birge, Phys. Rev. Lett. 97, 226803 /H208492006 /H20850. 17The h/2e-AB oscillations arise by the same mechanism as the WL and are suppressed when the magnetic ﬁeld penetrates thewire of the sample. 18A. Benoit, D. Mailly, P. Perrier, and P. Nedellec, Superlattices Microstruct. 11, 313 /H208491992 /H20850. 19P. Schwab and U. Eckern, Ann. Phys. 5,5 7 /H208491996 /H20850. 20F. Pierre and N. O. Birge, Phys. Rev. Lett. 89, 206804 /H208492002 /H20850;F . Pierre and N. O. Birge, J. Phys. Soc. Jpn. 72,1 9 /H208492003 /H20850. 21B. L. Altshuler and A. G. Aronov, in Electron-Electron Interac- tion in Disordered Systems , edited by A. L. Efros and M. Pollak /H20849North-Holland, Amsterdam, 1985 /H20850. 22S. Kettemann and E. R. Mucciolo, JETP Lett. 83, 240 /H208492006 /H20850; cond-mat/0609279. 23S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 /H208491980 /H20850. 24V. I. Falko, JETP Lett. 53, 340 /H208491991 /H20850. 25A. G. Aronov and Y. V. Sharvin, Rev. Mod. Phys. 59, 755 /H208491987 /H20850. 26I. L. Aleiner and Ya. M. Blanter, Phys. Rev. B 65, 115317 /H208492004 /H20850. 27Notice that in the AB experiment /H20849considered below /H20850the suppres- sion of inelastic vertex contributions is even larger and propor-tional to 1/ /H20849/H9004Et D/H208502/H112701/H20849Ref. 35/H20850, where tD=L2/D/H11271/H9270/H9272andLis the ring length. This is due to the fact that in this case the typicaltime scale of interfering electrons is t Drather than /H9270/H9272as elec-MAGNETIC FIELD DEPENDENCE OF DEPHASING RATE … PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-9trons contributing the AB oscillations have to circle the ring at least once. 28R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lett. 54, 2696 /H208491985 /H20850. 29In their experiments F. Pierre et al. /H20849Ref. 20/H20850measured spin-orbit lengths of order /H110111/H9262m much smaller than L/H9272. 30T. A. Costi, Phys. Rev. Lett. 85, 1504 /H208492000 /H20850; A. Rosch, T. A. Costi, J. Paaske, and P. Wölﬂe, Phys. Rev. B 68, 014430 /H208492003 /H20850. 31K. G. Wilson, Rev. Mod. Phys. 47, 773 /H208491975 /H20850; H. R. Krishna- murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 /H208491980 /H20850. 32T. A. Costi, A. C. Hewson, and V. Zlati ć, J. Phys.: Condens. Matter 6, 2519 /H208491994 /H20850.33W. Hofstetter, Phys. Rev. Lett. 85, 1508 /H208492000 /H20850. 34We deﬁne and determine the Kondo temperature TKfrom the T =0 suceptiblity /H9273=/H20849g/H9262B/H208502//H208494TK/H20850. 35T. Ludwig and A. D. Mirlin, Phys. Rev. B 69, 193306 /H208492004 /H20850. 36J. Jaroszy ńskiet al. , Phys. Rev. Lett. 75, 3170 /H208491995 /H20850;J . Jaroszy ński, J. Wróbel, G. Karczewski, T. Wojtowicz, and T. Dietl, ibid. 80, 5635 /H208491998 /H20850; G. Neuttiens, C. Strunk, C. Van Haesendonck, and Y. Bruynseraede, Phys. Rev. B 62, 3905 /H208492000 /H20850; A. Benoit, D. Mailly, P. Perrier, and P. Nedellec, Super- lattices Microstruct. 11, 313 /H208491992 /H20850. 37D. K. Wohlleben and B. R. Coles, Magnetism , edited by H. Suhl /H20849Academic, New York, 1973 /H20850, Vol. 5.MICKLITZ, COSTI, AND ROSCH PHYSICAL REVIEW B 75, 054406 /H208492007 /H20850 054406-10
PhysRevB.73.024509.pdf	Searching for a higher superconducting transition temperature in strained MgB 2 Jin-Cheng Zheng and Yimei Zhu Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA /H20849Received 22 August 2005; revised manuscript received 7 November 2005; published 24 January 2006 /H20850 We present a detailed ﬁrst-principles density-functional analysis of the effects of lattice strain on the super- conducting transition temperature, Tc,o fM g B 2, deriving a general rule that governs the enhancement /H20849or suppression /H20850ofTcin strained MgB 2in terms of electronic and phonon contributions. Based on the calculated structural, electronic, vibrational, and superconducting properties of a strained MgB 2superconductor, we show how a higher Tcmight be achieved. Several candidate substrates are suggested for growing MgB 2thin ﬁlms to gain a higher Tc. DOI: 10.1103/PhysRevB.73.024509 PACS number /H20849s/H20850: 74.70.Ad, 71.15.Mb, 74.62.Bf Tailoring the Tcin MgB 2requires a full atomic level un- derstanding of the underlying mechanism of chemical andlattice effects. Immediately after the discovery of supercon-ductivity in MgB 2,1extensive research on chemical substitu- tion was undertaken to understand and hence to improve itsproperties. 2–5In contrast, very few studies focused on lattice effects despite the fact that improving material properties bycontrolling lattice strain is a basic and practical approach inmaterials research. For example, various methods, such asapplying pressures 6,7including mechanical loads, or select- ing different substrates with different lattice constants forﬁlm growth, are commonly employed to investigate the ef-fects of lattice strain on material functionality. Lattice effectsare believed to play an important role in the superconductiv-ity of MgB 2. Although isotropic compression was applied to the crystal lattice of MgB 2to investigate its properties,6,7no research has systematically addressed the effects of strain onsuperconductivity. In fact, various strains exist in epitaxiallygrown MgB 2ﬁlms due to the use of different substrates. Very encouragingly, Pogrebnyakov et al. , reported that the Tcof MgB 2can be increased by as much as about 5% using SiC as the substrate.8Seemingly, a thorough exploration of the ef- fects of both compressive and tensile strains on the supercon-ducting properties of MgB 2would be extremely useful for guiding the way toward a higher Tc. In this paper, we discuss the effects of lattice strain on the electronic /H20849density of states and deformation potential /H20850, vibrational /H20849E2gphonon /H20850, and su- perconducting /H20849Tc/H20850properties in MgB 2using density func- tional theory /H20849DFT /H20850.9 We ﬁrst describe the variation of Tcas a function of strain in a way generalized for low-temperature superconductorsand then specify them for strained MgB 2. The original Mc- Millan formula10forTcof superconductor is given as Tc=/H9052 1.45exp/H20875−1.04 /H208491+/H9261/H20850 /H9261/H208491 − 0.62 /H9262/H20850−/H9262/H20876, /H208491/H20850 where /H9008is Debye temperature, /H9262is the Coulomb pseudo- potential, and /H9261is electron-phonon coupling strength. The Allen-Dynes11,12modiﬁed McMillan formula can be ex- pressed asTc=/H9275log 1.2exp/H20875−1.04 /H208491+/H9261/H20850 /H9261/H208491 − 0.62 /H9262/H20850−/H9262/H20876. /H208492/H20850 The Debye temperature /H9008in Eq. /H208491/H20850and/H9275login Eq. /H208492/H20850are related to phonon frequencies. According to Morel andAnderson, 13/H9262can be expressed as /H9262=/H9262 1+/H9262ln/H20849EF//H9275/H20850, /H208493/H20850 where /H9262=0.5 ln /H20851/H208491+g2/H20850/g2/H20852and g2=/H9266e2NF/kF2. Here, kF =/H208493/H92662Z/V/H208501/3, where Zis the valency and Vis the volume.14 The electron-phonon coupling strength /H9261can be expressed as /H9261=NF/H20855I2/H20856 M/H20855/H92752/H20856=/H9257 M/H20855/H92752/H20856, /H208494/H20850 where NFis density of states /H20849DOS /H20850at Fermi energy level, /H20855I2/H20856is the mean-square electron-ion matrix element, Mis the ionic mass, /H20855/H92752/H20856is the mean-square phonon frequency, and /H9257 is McMillan-Hopﬁeld parameter that includes the electronic components of electron-phonon coupling. In a strained su-perconductor, the relative change of T cas a function of strains can be expressed as /H9004Tc Tc0=/H9004/H9275log /H9275log−/H9251/H9004/H9262 /H92620+/H9252/H9004/H9261 /H92610. /H208495/H20850 All variables with subscript 0 are for a strain-free supercon- ductor, and the strain-free parameters are /H9251=1.04 /H208491+/H92610/H20850/H208491 + 0.62 /H92610/H20850/H92620 /H20851/H92610/H208491 − 0.62 /H92620/H20850−/H92620/H208522 and /H9252=1.04/H92610/H208491 + 0.38 /H92620/H20850 /H20851/H92610/H208491 − 0.62 /H92620/H20850−/H92620/H208522. To achieve higher Tcin terms of modulating the strain, the following condition should be fulﬁlled:PHYSICAL REVIEW B 73, 024509 /H208492006 /H20850 1098-0121/2006/73 /H208492/H20850/024509 /H208496/H20850/$23.00 ©2006 The American Physical Society 024509-1/H9004/H9275log /H9275log−/H9251/H9004/H9262 /H92620+/H9252/H9004/H9261 /H92610/H110220. /H208496/H20850 Equations /H208495/H20850and /H208496/H20850are general ones for low-temperature superconductors where the McMillan-Allen-Dynes formulacan be applied. The above terms can be obtained from DFTcalculations to give T cas a function of strains. Next, we specify how to achieve a higher Tcin strained MgB 2. It is well known that MgB 2is a two-band superconductor.15–18The Tcof MgB 2is derived from both the 2D /H9268and 3D /H9266bands components, but is dominated by the boron 2D /H9268band.15The boron 2D /H9268band and the E2g phonon frequency /H20849boron-boron stretching mode /H20850, as well as the deformation potential associated with this mode, aremainly responsible for the interesting superconducting prop-erties of MgB 2. In a small range /H20849±6% /H20850of strains, the varia- tion of Tcis mainly determined by modulations of the /H9268 band. Therefore, in this paper, we will focus on the variations in the electronic and phonon parts of the /H9268band and their effects on Tcin strained MgB 2. In this case, /H9275logis approximately proportional to the E2g phonon frequency /H9275E2g, and electron-phonon coupling can be expressed19as/H9261/H11008NF/H20849/H9268/H20850/H20841D/H208412/MB/H9275E2g2, where MBis atomic mass of boron, /H20841D/H20841is the deformation potential associated with the boron-boron bond stretching mode, and NF/H20849/H9268/H20850is the density of states of the boron /H9268band at the Fermi level.19In strained MgB 2, a small deviation in /H9004aand/H9004cof the lattice constants a0andc0results in the variation of /H9261,/H9275E2g,/H9262, and Tc. The changes of Tccan be expressed as /H9004Tc Tc0=/H9004/H9275E2g /H9275E2g0−/H9251/H9004/H9262* /H92620+/H9252/H9004/H9261 /H92610, /H208497/H20850 where /H9004/H9261 /H92610=/H9004NF/H20849/H9268/H20850 NF0/H20849/H9268/H20850+2/H9004/H20841D/H20841 /H20841D/H208410−2/H9004/H9275E2g /H9275E2g0. /H208498/H20850 The ﬁrst-principles DFT calculations were made using the full potential augmented plane wave /H20849FPAPW /H20850method implemented in the WIEN2k package,20and the plane-wave pseudopotential method in the ABINIT package,21respec- tively. For both methods, we employed the local density ap-proximation /H20849LDA /H20850suggested by Perdew and Wang, 22and the generalized gradient approximant /H20849GGA /H20850proposed by Perdew, Burke, and Ernzerhof /H20849PBE96 /H2085023to obtain the exchange-correlation potential. The results from the LDAand GGA agree well with each other with difference within2%. For full potential calculations /H20849WIEN2k /H20850, a mufﬁn-tin radius /H20849R MT/H20850of 1.7 bohr was chosen for Mg, and 1.5 bohr for B, and RMTKmaxwas taken to be 8.0. The calculations used the angular momentum expansion up to lmax=10 for the potential and charge density representations. At convergence,the integrated difference between input and output chargedensities /H20849atomic unit /H20850was less than 10 −5. We also employed 3000 kpoints in the Brillouin zone in the calculations. /H20849The convergence of total energy and properties of MgB 2was tested in terms of the number of kpoints. /H20850In pseudopotential calculations /H20849ABINIT /H20850, we employed a very large energy cut-off /H2084940 Hartree /H20850and 10 /H1100310/H1100310 Monkhorst-Pack24k points, and a very small tolerance on the potential V/H20849r/H20850re- sidual /H2084910−18/H20850for self-consistent ﬁeld /H20849SCF /H20850stopping; this ensured good convergence of the results in ground-state and linear-response calculations. Well-tested Troullier-Martinspseudopotentials 25were adopted. With the chosen typical value /H92620=0.15, and /H92610=0.94, as well as our calculated phonon frequency /H20849/H9275E2g/H208500 =579.9 cm−1, we obtain Tc0=39.4 K for bulk MgB 2with the lattice constant a=3.083 Å, c=3.521 Å, agreeing with ex- perimental data.1We note that the changes in superconduct- ing properties /H20849e.g., Tc/H20850as a function of strains are mainly determined by the relative changes of /H9275,/H9261, and/H9262/H20849which are, in turn, determined by the relative changes of /H9275,NF, and D/H20850, while the actual values of /H92620and/H92610are less sensitive to the strain-induced variation of Tcin MgB 2. This can be dem- onstrated by Eqs. /H208497/H20850and /H208498/H20850, in which we express the varia- tion of Tcrelative to Tc0based on the relative changes of /H9275 and/H9261. Other coefﬁcients that are associated with /H9261and in- dependent of /H9275,NF, and Dare cancelled out in the ratio of /H9004/H9261//H92610. We show numerical results later in this work, dem- onstrating that the trends in the superconducting propertiesof MgB 2as a function of strains are not affected signiﬁcantly by the choice of /H92620and/H92610. The change of /H9262in strained MgB 2is rather small and is only about ten times less than the relative changes of /H9261, which is consistent with Chen et al. ’s ﬁndings26for MgB 2 under hydrostatic pressure. Using our calculated value of /H9251/H110110.97, the contribution of the − /H9251/H9004/H9262//H92620term modiﬁes /H9004Tc/Tc0by about 1%–2% in the range of strains described in this work. Hereafter, the term of /H9004/H9262//H92620will not be in- cluded explicitly. We note that the Tcof MgB 2is controlled collectively by three components: phonon frequency /H20849/H9004/H9275//H92750/H20850, density of states /H20851/H9004NF/H20849/H9268/H20850/NF0/H20849/H9268/H20850/H20852, and the deformation potential /H208492/H9004/H20841D/H20841//H20841D/H208410/H20850. By sorting out the individual contributions and deﬁning the electron contribution /H9254e =/H9004NF/H20849/H9268/H20850/NF0/H20849/H9268/H20850+2/H9004/H20841D/H20841//H20841D/H208410, and the phonon contribution /H9254p=/H9004/H9275//H92750, we derive a simple expression, /H9254Tc=/H9004Tc Tc0=/H9252/H9254e+/H208491−2/H9252/H20850/H9254p, /H208499/H20850 where the subscripts eandpindicate the contribution from the electron and phonon component, respectively. For thestrain-free bulk, we obtain the parameter /H9252=2.09; then, /H9254Tc=2.09/H9254e−3.18/H9254p. This clearly demonstrates that the con- tributions from the electron and phonon parts can be consid- ered as two separate competing entities, i.e., /H9254Tcincreases with an increase in /H9254ebut with a decrease in /H9254p, and vice versa. Thus, to improve the Tcin strained MgB 2, the following condition should be fulﬁlled: /H9254Tc/H110220o r /H9254e/H11022/H208492/H9252−1/H20850 /H9252/H9254p/H110151.52/H9254p. /H2084910/H20850 This simple rule indicates the general direction to take for enhancing the Tcof MgB 2: no matter whether the E2gpho-J.-C. ZHENG AND Y. ZHU PHYSICAL REVIEW B 73, 024509 /H208492006 /H20850 024509-2non is softened or strengthened, if the electron contribution, /H9254e, is greater than 1.52 /H9254p,Tccan be raised. So far, we have clariﬁed the underlying mechanism of variation for Tcin terms of the contributions of electrons and phonons. However, in searching for a higher critical tem-perature in strained MgB 2, it is worth looking at their de- tailed behavior as a function of the strains. In Fig. 1, we plot /H9254Tcvs./H9004a/a/H20849−3%–3% /H20850, and/H9004c/c/H20849−3–3% /H20850,i nM g B 2. We found that by increasing /H9004a/aand/H9004c/ctheTccan be raised /H20849solid line counters in Fig. 1 /H20850, as is even more evident from the variation of Tcvs./H9004a/a/H20849or/H9004c/c/H20850with a ﬁxed /H9004c/c/H20849or/H9004a/a/H20850. The most signiﬁcant way of increasing Tcis by increasing the volume, V,o fM g B 2with a ﬁxed c/a. The origin of this enhancement is the decrease in both the phononand electron contributions, namely, /H9254p/H110210 and /H9254e/H110210 with /H9254e/H110221.52/H9254p, as shown in Figs. 1 /H20849c/H20850and 1 /H20849d/H20850. Using the re- sults of our DFT calculations, a simple coupling quadraticﬁtting to DFT data was obtained using/H9254Tc=a1x+a2x2+b1y+b2y2+c1xy, /H2084911/H20850 where x=/H9004a/a, and y=/H9004c/c. The quality of the ﬁt is excel- lent with the following ﬁtting parameters: a1=9.836 028, a2=−0.865 501, b1=2.782 717, b2=−0.642 597, and c1=1.786 229 /H20851Figs. 1 /H20849a/H20850and 1 /H20849b/H20850/H20852. The importance of this ﬁtting formula is that once the strain condition /H20849/H9004a/aand /H9004c/c/H20850in MgB 2is known, /H9254Tccan be predicted directly, and, thus, by using Tc=Tc0·/H208491+/H9254Tc/H20850, where Tc0=39.4 K is the value of unstrained MgB 2,Tccan be obtained without need- ing expensive DFT calculations. We now compare the calculated /H9254Tcboth from DFT cal- culations and the ﬁtting formula Eq. /H2084911/H20850with that from ex- perimental data for two typical cases in MgB 2:/H20849i/H20850bulk ma- terial with uniaxial pressure /H20849Figs. 1 and 2 /H20850and /H20849ii/H20850thin ﬁlms grown on different substrates /H20849Figs. 1 and 3 /H20850. With hydro- static pressure, both the lattice constants aandcare com- pressed /H20849bold red line in Fig. 1 /H20850, and/H9254Tcfalls as the lattice FIG. 1. /H20849Color /H208502-D contour plot of critical temperature and itsassociated electron and phononcontributions in MgB 2as a func- tion of variation of lattice con-stants. /H20849a/H20850Critical temperature /H20849/H9004T c/Tc/H20850obtained from DFT cal- culations. /H20849b/H20850Critical temperature obtained by ﬁtting DFT results us-ing Eq. /H2084911/H20850./H20849c,d/H20850Phonon and electron contributions to /H9004T c/Tc. The colored lines indicate differ-ent variations as a function of/H9004a/aand/H9004c/cin MgB 2/H20849cyan: volume variation Vwith ﬁxed c/a; red: variation of lattice con- stant awith ﬁxed c; green: varia- tion of lattice constant cwith ﬁxed a; yellow: variation of c/a with ﬁxed volume; bold blue line:lattice variation for in-planestrain; bold red line: lattice varia-tion for hydrostatic pressure up to12 GPa /H20850. The solid and dashed counter-lines represent positiveand negative /H9004Tc/Tc, respec- tively. Schematics of MgB 2ﬁlms with in-plane compressive andtensile strains grown on substratesare shown at the bottom right.SEARCHING FOR A HIGHER SUPERCONDUCTING … PHYSICAL REVIEW B 73, 024509 /H208492006 /H20850 024509-3constants decrease. Figure 2 compares the experimentally measured /H9254Tcas a function of applied pressure with the re- sults of DFT calculations and with Eq. /H2084911/H20850; there is excellent agreement between the three. We conclude that applying hy-drostatic pressure always leads to a drop in T c. A higher Tccan be attained in MgB 2thin ﬁlm under bi- axial strain in the a-bbasal plane /H20849in-plane /H20850, as depicted in Fig. 1 /H20849bold blue line /H20850and Fig. 3 /H20849a/H20850. Furthermore, the opti- mized variation in lattice constant, /H9004c/c, determined from minimizing total energy, declines slightly with an increase in/H9004a/awith a negative slope close to −1/3. The calculated /H9254Tc in strained MgB 2from DFT agrees well with the experimen- tal values8using SiC as a substrate /H20851Fig. 3 /H20849a/H20850/H20852. The/H9254Tcpre- dicted from the ﬁtting formula /H20851Eq. /H2084911/H20850/H20852is consistent with the measured /H9254Tcwith an error of about −2% /H20849the same order as the experimental error bar /H20850. Our results are striking be- cause not only do they agree with the experiments, but theyalso reveal the possibility of further improving T cby using large tensile strains /H208511% to 3%, Fig. 3 /H20849a/H20850/H20852. The enhancement ofTcoriginates from the decrease in the phonon /H20849/H9254p/H20850and electron /H20849/H9254e/H20850contributions due to the increase in biaxial strains /H20851see Fig. 3 /H20849c/H20850/H20852that satisfy the condition of Eq. /H2084910/H20850in the 3% range of strains. Our ﬁnding that decreasing the E2g phonon frequency /H20851Fig. 3 /H20849c/H20850/H20852will generate a high Tcin MgB 2agrees well with the experimental observations and FIG. 2. /H20849Color online /H20850Comparison of critical temperature be- tween experiments and the DFT calculation of MgB 2as a function of hydrostatic pressure. The solid line is from the ﬁtting formula/H20851Eq. /H2084911/H20850/H20852. The experimental data is taken from Ref. 7. FIG. 3. /H20849Color online /H20850The critical temperature and othervariables in MgB 2as a function of in-plane biaxial strains. /H20849a/H20850 Critical temperature /H9004Tc/Tc,/H20849b/H20850 lattice variation /H9004c/c,/H20849c/H20850electron /H20851/H9254e=/H9004NF/H20849/H9268/H20850/NF/H20849/H9268/H20850+2/H9004D/D, red squares /H20852and phonon /H20849/H9254p=/H9004/H9275//H9275, blue dots /H20850contribution, and /H20849d/H20850 boron /H9268band density of state /H9004NF/H20849/H9268/H20850/NF/H20849/H9268/H20850/H20849red squares /H20850and deformation potential /H9004D/D/H20849blue dots /H20850.J.-C. ZHENG AND Y. ZHU PHYSICAL REVIEW B 73, 024509 /H208492006 /H20850 024509-4theoretical predictions:8i.e., the softened Raman phonon peak in strained MgB 2gives a higher Tc. Our calculations further demonstrate that the /H9254Tcin MgB 2 increases with rising biaxial strains until reaching its maxi-mum of about 10%, at /H110113% tensile strains; thereafter, it decreases with any further rise in tensile strains and then fallsto a negative value at very high strains /H20849/H110116% or more /H20850. The reason this occurs is that Eq. /H2084910/H20850is not fulﬁlled: thus, al- though both /H9254pand/H9254edecline, /H9254edecreases at a rate faster than 1.52 /H9254p. The fall in Tcat high tensile strains reﬂects the sharp decrease of density of states of the B /H9268band, NF/H20849/H9268/H20850 /H20851Fig. 3 /H20849d/H20850/H20852. Taking into account both /H9254Tcand strained energy in MgB 2, we found there is still plenty of room for raising Tcto higher value: if biaxial strains of about 1%–3% can beachieved, the T ccan be enhanced by about 7%–12% /H20849corre- sponding to Tc=42–44 K /H20850. So, the question is how to in- crease biaxial strains. This problem can be resolved by se-lecting appropriate substrates for growing MgB 2thin ﬁlms that have a slightly larger lattice constant than that of bulk/H20849or unstrained /H20850MgB 2, or applying biaxial tensile stress with a mechanical load in a single crystal. There are several pos-sible substrates: hexagonal SiC, Si 1+xC1−xalloys with a small tunable composition x, to vary lattice constant, AlN, GaN and their alloys Al xGa1−xN, as well as MgB 2doped with Ca /H20849or Mg 1−xCaxB2alloys /H20850. All these materials have a slightly larger biaxial lattice constant than does bulk MgB 2at low temperatures, and, as substrates for MgB 2, they will generate tensile strains in the ﬁlms. In particular, if high-quality MgB 2 ﬁlms can be grown on alloys of Si 1+xC1−x,A l xGa1−xN, or Mg 1−xCaxB2, then, by controlling the composition xof the alloy substrates, the tensile strains, and, thus, Tccan be tuned. For materials that have lattice constants comparablewith /H20849or even smaller than /H20850those of MgB 2at room tempera- ture, for example, 6H–SiC /H20849a=3.0806 Å at 300 K /H20850,27it still is possible to induce tensile strains by cooling the sample tobelow the T cof MgB 2because the thermal coefﬁcient of MgB 2/H208495.4/H1100310−6K−1/H2085028is about twice as large as that of SiC /H208492.77/H1100310−6K−1/H20850,27resulting in a small tensile strain for MgB 2at low temperatures on a SiC substrate. For AlN with its wurtzite structure, its lattice constant /H20849a=3.111 Å /H2085027is slightly larger than that of MgB 2, but its thermal expansion coefﬁcient /H208494.2/H1100310−6K−1/H2085029is less. Therefore, by growing MgB 2on an AlN substrate, the tensile strains in the ﬁlms /H20849/H110111% /H20850will be larger than those /H20849/H110110.5% /H20850of ﬁlms grown on a SiC substrate, and accordingly, the Tcof MgB 2will be enhanced by about 7%–8%. Finally, we discuss the validation of our approach. Al- though we used some approximations /H20849for example, the McMillan-Allen-Dynes formula for Tc, the variation of /H9262in strained MgB 2is neglected /H20850in calculating Tcin the strain- induced modulation of MgB 2, the excellent agreement be- tween the calculated and experimental Tcfor MgB 2in two typical cases /H20849applied hydrostatic pressure and tensile strain /H20850 justiﬁes the validity of this approach and suggests that thesuperconducting properties of MgB 2as a function of strains are mainly dominated by the strain-induced modulation ofthe /H9268band /H20849e.g., density of states, deformation potentials, E2gphonon frequency /H20850. In fact, the McMillan-Allen-Dynesformula has been used for calculating TcMgB 2in many cases, for example, for bulk MgB 2,19,30,31MgB 2under pressure,26,32MgB 2doped with carbon,33or aluminum34and strained MgB 2.8Another concern centers on the effects of the selected values of /H92620and/H92610on the strain-induced modula- tion of Tc. It was found that the trends of superconducting properties of MgB 2as a function of strains are not affected signiﬁcantly by the choice of /H92620and/H92610. We proved this point by using a wide range of values of /H92620and/H92610, with /H92620* being 0.1–0.2, and /H92610being 0.6–1.0, for strain-free MgB 2to reproduce Tc=39.4 K. As clearly demonstrated in Fig. 4, the /H9004Tc/Tc0of strained MgB 2is insensitive to the values of /H92620* and/H92610. It is worth mentioning that our method describes the relative changes of Tcin terms of relative changes of related physical quantities and, therefore, it will have more precisionthan the straight calculation of the absolute value of T cby using the McMillan-Allen-Dynes formula on MgB 2,a na n - isotropic and anharmonic two-band material. In summary, we used ﬁrst-principles DFT calculations to determine the electronic, vibrational, and superconductingproperties of MgB 2under various conditions of strain. We derived a general rule governing the enhancement /H20849or sup- pression /H20850ofTcin strained MgB 2in terms of electron and phonon contributions. Based on this rule, we found that thedirection of increasing T cshould lie in efforts to increase tensile strains /H20849from 0% up to 4% /H20850. The maximum Tcin strained MgB 2may be achieved at tensile strains of 2%–3%. Several candidate substrates for growing MgB 2thin ﬁlms were suggested. We emphasize that our rule is not limited toMgB 2, but is applicable to other conventional BCS supercon- ductors /H20849with different values of /H9252/H20850, especially those having similar lattice structures to MgB 2. Our approach to strain engineering, with the target of enhancing Tcas demonstrated in this work, can be applied to various strain-engineering FIG. 4. /H20849Color /H20850/H9004Tc/Tcas a function of in-plane strain for a wide range of /H92620and/H92610./H20849Color represents different values of /H92620 and/H92610./H20850SEARCHING FOR A HIGHER SUPERCONDUCTING … PHYSICAL REVIEW B 73, 024509 /H208492006 /H20850 024509-5problems in other functional materials. In fact, we have dem- onstrated, by ﬁrst-principles calculations, that half-metallicferromagnetism /H20849an important property for spintronics /H20850can be enhanced in strained zinc-blende structures of MnSb andMnBi. 35ACKNOWLEDGMENTS This work was supported by the U.S. Department of En- ergy, Division of Materials, Ofﬁce of Basic Energy Science,under Contract No. DE-AC02-98CH10886. 1J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Nature /H20849London /H20850410,6 3 /H208492001 /H20850. 2J. S. Slusky, N. Rogado, K. A. Regan, M. A. Hayward, P. Khali- fah, T. He, K. Inumaru, S. M. Loureiro, M. K. Haas, H. W.Zandbergen, and R. J. Cava, Nature /H20849London /H20850410, 343 /H208492001 /H20850. 3S. Y. Zhang, J. Zhang, T. Y. Zhao, C. B. Rong, B. G. Shen, and Z. H. Cheng, J. Convex Anal. 10/H208494/H20850, 335 /H208492001 /H20850. 4W. Mickelson, J. Cumings, W. Q. Han, and A. Zettl, Phys. Rev. B 65, 052505 /H208492002 /H20850. 5J. S. Ahn, Y. J. Kim, M. S. Kim, S. I. 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PhysRevB.76.085433.pdf	Phonon runaway in carbon nanotube quantum dots L. Siddiqui,1A. W. Ghosh,2and S. Datta1 1NSF Network for Computational Nanotechnology, Purdue University, West Lafayette, Indiana 47907, USA 2Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, Virginia 22903, USA /H20849Received 17 September 2006; revised manuscript received 30 April 2007; published 24 August 2007 /H20850 We explore electronic transport in a nanotube quantum dot strongly coupled with vibrations and weakly with leads and the thermal environment. We show that the recent observation of anomalous conductance signaturesin single-walled carbon nanotube quantum dots /H20851B. J. LeRoy et al. , Nature /H20849London /H20850395, 371 /H208492004 /H20850and B. J. LeRoy et al. , Phys. Rev. B 72, 075413 /H208492005 /H20850/H20852can be understood quantitatively in terms of current driven “hot phonons” that are strongly correlated with electrons. Using rate equations in the many-body conﬁgurationspace for the joint electron-phonon distribution, we argue that the variations are indicative of strong electron-phonon coupling requiring an analysis beyond the traditional uncorrelated phonon-assisted transport /H20849Tien- Gordon /H20850approach. DOI: 10.1103/PhysRevB.76.085433 PACS number /H20849s/H20850: 73.23. /H11002b I. INTRODUCTION One of the signiﬁcant challenges in microelectronics is controlling the rapidly increasing thermal budget associatedwith current ﬂow through shrinking devices. Experimental 1–7 and theoretical8–14investigations are revealing intriguing as- pects of the mutual effect of electronic and vibronic modeson each other. Nanoscale vibrations tend to couple stronglywith electronic currents and weakly with their “macroenvi-ronment,” allowing them to be easily driven far from equi-librium. Understanding the dynamics of such electronicallydriven phonon runaway processes is crucial to the evolutionof low-power devices, not to mention novel concepts likemolecular motors and phonon lasers. 15 In this paper, we develop a theoretical treatment of current driven nonequilibrium correlated phonon dynamics in nanos-cale systems, and use this approach to analyze recent experi-ments on single-walled carbon nanotube /H20849SWCNT /H20850quantum dots /H20849QDs /H20850. 1Using a rate equation for correlated transport in the full many-body eigenspace of the coupled electron-phonon–lead-bath system, we explain novel spectroscopicfeatures such as the anomalously large absorption sidebandsarising from phonon runaway in suspended, Coulomb block-aded nanotubes /H20849Fig. 1/H20850. Our model also explains semiquantitative features of the experiment such as the amplitude variation of the Coulombblockaded conductance peaks and their phonon sidebands, asa function of injected current. However, our model predicts alinear variation in phonon population with current, in con- trast with experiments that show a quadratic variation. 1,2We argue that a possible origin of this discrepancy is because ourmodel explicitly incorporates the effect of strong electron-phonon correlation that is characteristic of these experi-ments, while the experimentally extracted variation was ac-complished by employing a traditional Tien-Gordon analysisthat implicitly treats the phonon contribution only through itsmean-ﬁeld oscillating potential acting on the electronicsubsystem. 16 Our paper thus serves a dual purpose: /H20849i/H20850it identiﬁes transport signatures of the individual physical ingredients,such as the electronic energy levels and their Coulomb inter-action that determine the main peak positions in Fig. 1, the phonon modes that determine the positions of their side-bands, and the couplings with the leads and the surroundingbath that determine the heights, respectively, of the mainconductance peaks and their sidebands. /H20849ii/H20850Using general arguments as well as through the comparison between ourtheoretical model and experimental data, we establish theinadequacy of commonly employed “mean-ﬁeld” treatments,such as the Tien-Gordon theory of phonon assistedtunneling, 16in describing strongly correlated systems driven away from equilibrium. While our formalism involves stan-dard techniques employing rate equations within the excita-tion spectrum of a many-body Hamiltonian, our emphasislies in incorporating the various components, particularly theincoherence driven by the dynamics of the bath, and identi-fying speciﬁc signatures of the failure of mean-ﬁeld theory insuspended nanotube dots characterized by strong nonequilib-rium correlations. II. MODEL We use a model Hamiltonian HDfor a quantum dot with on-site energies /H9280i, Coulomb interaction energy Uii/H11032, vibronic modes at energy /H6036/H9275j, and electron-phonon coupling /H9261ij/H20849Fig. 2/H20850. The total Hamiltonian including the contacts /H20849HC/H20850, the phonon bath /H20849HB/H20850, and their couplings with the dot /H20849HDC, HDB/H20850is H=HD+HC+HB+HDC+HDB, /H208491/H20850 HD=Hel+Hph+Hel-ph, /H208492/H20850 Hel=/H20858 i/H9280ici†ci+1 2/H20858 i,i/H11032,i/HS11005i/H11032Uii/H11032nini/H11032, Hph=/H20858 j/H6036/H9275jaj†aj, Hel–ph=/H20858 i,j/H9261ij/H6036/H9275jni/H20849aj†+aj/H20850,PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 1098-0121/2007/76 /H208498/H20850/085433 /H208499/H20850 ©2007 The American Physical Society 085433-1HC=/H20858 k,/H9251/H9280L,R/H9280k/H9251dk/H9251†dk/H9251, /H208493/H20850 HB=/H20858 l/H6036/H9275lbl†bl, /H208494/H20850 HDC=/H20858 i,k,/H9251/H9270ik/H9251/H20849dk/H9251†ci+dk/H9251ci†/H20850, /H208495/H20850 HDB=/H20858 jl/H9260jl/H20849bl†+bl/H20850/H20849aj†+aj/H20850, /H208496/H20850 where c†/H20849c/H20850andd†/H20849d/H20850are the electronic creation /H20849destruc- tion /H20850operators for the dot and the leads, ni=ci†cianda†/H20849a/H20850 andb†/H20849b/H20850are the phonon creation /H20849destruction /H20850operators for the dot and the phonon bath, respectively. /H9270,/H9280k, and/H9260repre- sent, respectively, the dot-contact coupling, the contact bandstructure, and the coupling between the dot vibrations andthe thermal bath.The electron-phonon coupling is eliminated using a stan- dard unitary polaronic transformation H ˜=eSHe−S,17where S =/H20858i,j/H9261ij/H20849aj†−aj/H20850ni. This transformation renormalizes the on- site /H20849/H9280˜i/H20850and Coulomb /H20849U˜ii/H11032/H20850energies for the dot: H˜D=/H20858 i/H9280˜ic˜i†c˜i+1 2/H20858 i,i/H11032U˜ii/H11032n˜in˜i/H11032+/H20858 j/H6036/H9275jaj†aj, /H208497/H20850 /H9280˜i=/H9280i−/H20858 j/H9261ij2/H6036/H9275j, /H208498/H20850 U˜ii/H11032=Uii/H11032−2/H20858 j/H9261ij/H9261i/H11032j, /H208499/H20850 where the “dressed” electron or polaronic annihilation opera- tor c˜i=eScie−S=cX /H2084910/H20850 with X=exp /H20858j/H9261ij/H20849aj−aj†/H20850denoting the generator of the po- laronic shift and a˜j=eSaje−S=ai−/H20858 j/H9261ijni /H2084911/H20850 while preserving the electronic number operator n˜i=ni.A t this point, we will simplify the model by considering only asingle phonon mode, denoted by the index 1. Current ﬂow in this system involves single electron tran- sitions between many-body states /H20841e Nei,k/H20856/H20849kphonons and ith electronic level in the Neelectronic subspace /H20850of the quantum dot, with matrix elements for the electronic destruction op-erator /H20849for a derivation, see the Appendix, Sec. III /H20850: FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic of scanning tunnel micro- scope /H20849STM /H20850measurement on CNT QDs; /H20849b/H20850and /H20849c/H20850Experimental observation and theoretical calculation for CNT QDs at lower andhigher current level /H20851solid line: conductance; dashed line: /H20855N ph/H20856in /H20849c/H20850/H20852. 1,2,...,6 denotes the main Coulomb peaks. The parameters for the calculation are /H9280˜1=/H9280˜2=24 meV, /H9280˜3=/H9280˜4=/H9280˜5=/H9280˜6=54 meV, U˜=27 meV, EF=0,/H6036/H92751=11.5 meV, /H926111=/H926121=¯=/H926161=1.6, /H9003L1 +/H9003L2=7.5/H1100310−6eV,/H9252=5/H1100310−8eV,/H9257=0.6, and T=5 K. FIG. 2. The dot is electrically connected to the left /H20849right /H20850con- tact /H20849with electron tunneling rates /H9003L,R//H6036/H20850and mechanically to the phonon bath /H20849with a phonon escape rate of /H9252//H6036/H20850. The dot has elec- tronic degrees of freedom /H9280iand phonon degrees of freedom /H6036/H9275j /H20849i,j=1,2,3,... /H20850with electron-phonon coupling /H9261ij.SIDDIQUI, GHOSH, AND DATTA PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 085433-2/H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856 =/H20902e−/H9261i12/2/H20849−/H9261i1/H20850k−p/H20881p! k!Lpk−p/H20849/H9261i12/H20850fork/H11350p e−/H9261i12/2/H20849−/H9261i1/H20850p−k/H20881k! p!Lkp−k/H20849/H9261i12/H20850forp/H11350k, /H2084912/H20850 where ci/H20841eNes,k/H20856=/H20841eNe−1r,k/H20856andLpqare the associated La- guerre polynomials. For total injection rates /H9003/H9251//H6036/H20851/H9251 =L/H20849left/H20850/R/H20849right /H20850/H20852 /H20849Fig. 2/H20850, the transition rates are R/H20841eNe−1r,k/H20856→/H20841eNes,p/H20856/H9251=/H20841/H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856/H208412/H9003/H9251 /H6036f/H9251/H20849E/H20849/H20841eNes,p/H20856/H20850 −E/H20849/H20841eNe−1r,k/H20856/H20850−/H9257/H20841q/H20841Vappl/H20850. /H2084913/H20850 fL,Rare the contact Fermi functions with electrochemical po- tentials /H9262L=EF,/H9262R=EF−/H20841q/H20841Vapplat temperatures TL,R, while the electrostatic voltage division factor /H9257represents the frac- tion of the applied bias acting on the levels. Here E/H20849/H20841/H20856/H20850stands for the total energy of the corresponding many-body eigen- state /H20841/H20856of the dot. Finally, the total injection rate /H9003/H9251is given by Fermi’s Golden Rule as /H9003i/H9251=2/H9266/H20858k/H20841/H9270ik/H9251/H208412/H9254/H20849/H9280i−/H9280k/H9251/H20850.W e will use the wide-band approximation to assume /H9003i/H9251=/H9003/H9251. It is worth emphasizing that the formalism for dealing with correlated transport is qualitatively different from itsmean-ﬁeld counterparts, which have traditionally been thepopular way to describe quantum conduction. 18,19For a strongly correlated dot, exclusion principle and charge quan-tization make it difﬁcult to describe transport in terms ofone-electron potentials, and each conducting level must in-stead be written as a difference between two many-bodystates. For instance, the electron afﬁnity /H20849the counterpart of a lowest unoccupied molecular orbital, or LUMO, level /H20850is written as the difference in total energy between the neutraland anionic species. Electron transport at a given energy cor-responds to a many-body transition corresponding to thattransition energy, with partial probability given by the matrixelements described in Eq. /H2084912/H20850. 20,21 In our analysis, electrons enter or leave the dot by emit- ting or absorbing phonons, which in turn are coupled withtheir thermal environment, held at a bath temperature T B, with an escape rate /H9252//H6036/H20849Fig.2/H20850, that maintains a Boltzmann ratio between emission and absorption processes in the con-tact, R /H20841eNes,k/H20856→/H20841eNes,k+1/H20856ph=/H9252 /H6036/H20849k+1/H20850exp/H20873−/H6036/H9275 kBTB/H20874, R/H20841eNes,k/H20856→/H20841eNes,k−1/H20856ph=/H9252 /H6036k. /H2084914/H20850 The state transitions in the joint electron-phonon many- body space /H20849with the loss of some correlation especially be- tween degenerate levels /H20850are described by a master equation describing the rate of populating and depopulating eachmany-body state: 20–23dP /H20841eNes,k/H20856 dt=/H20858 r,Ne/H11032,n/H20851P/H20841eNe/H11032r,n/H20856R/H20841eNe/H11032r,n/H20856→/H20841eNes,k/H20856 −P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNe/H11032r,n/H20856/H20852/H20849 15/H20850 together with the normalization condition for P/H20841eNes,k/H20856, where R=RL+RR+Rph. Solving the rate equations at steady state gives us the current Iand the steady state population of dot electrons /H20855Nel/H20856and phonons /H20855Nph/H20856: I=q/H20858 Ne,Ne/H11032,r,s,k,nsgn /H20849Ne/H11032−Ne/H20850P/H20841eNes,k/H20856R/H20841eNe/H11032r,n/H20856→/H20841eNes,k/H20856L, /H20855Nel/H20856=/H20858 s,Ne,kNeP/H20841eNes,k/H20856, /H20855Nph/H20856=/H20858 s,Ne,kkP /H20841eNes,k/H20856, /H2084916/H20850 where sgn /H20849x/H20850is the signum function. This also gives us the phonon generation rate G/H9251by the current and the phonon extraction rate Xph/H20849see the Appendix /H20850by the bath: G/H9251=/H20858 s,Ne,k/H20858 r,Ne/H11032,n/H20849n−k/H20850P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNe/H11032r,n/H20856/H9251, Xph=/H20858 s,Ne,k/H20858 r,Ne/H11032,n/H20849k−n/H20850P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNe/H11032r,n/H20856ph =/H20873/H9252 /H6036/H20874/H20855Nph/H20856−Npheq Npheq+1, /H2084917/H20850 where Npheqis the equilibrium phonon occupancy at the bath temperature. The rate equations are schematically explained in Fig. 2. The inputs to these equations are the electron tunneling rates/H9003 L,R, the phonon escape rate /H9252, the electronic energy con- ﬁguration /H9280i, the phonon energy /H6036/H9275j, the charging energy U, and the electron-phonon coupling /H9261ij. In our analysis we will set the lead and bath temperatures TL,RandTBto be equal to the ambient temperature. It is important to explain the need for a parametrized theory for correlated transport. The conﬁguration space of amany-electron system scales exponentially with basis size,making it hard to include more predictive ingredients. Thecomplexity of this system should, however, not be confusedwith frequently modeled longer molecules that have weakcorrelations. For those systems, one can greatly simplify thetransport problem with averaged one-electron potentials 18,19 and build ab initio parameter-free theories. The problem here deals with a molecular dot with Hubbard interactionscoupled with leads, phonons, and a thermal bath, and drivenaway from equilibrium by an applied voltage bias. While thiscomplex interacting system is simply too complicated tomodel without a parametrized theory, the parameters we willemploy are not widely adjustable, but are in fact constrainedpretty rigidly by the experimental facts /H20849details follow inPHONON RUNAWAY IN CARBON NANOTUBE QUANTUM DOTS PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 085433-3Sec.III A /H20850. Furthermore, our main conclusions are quite ro- bust with respect to the choice of the parameters. III. NONEQUILIBRIUM PHONON OCCUPATION IN CNT QD A. Peaks in the conductance spectrum due to phonon-assisted tunneling We will now apply our many-body rate equations to ana- lyze recent experiments1,2on phonon-assisted tunneling in suspended SWCNT quantum dots. The experiment showsseveral striking features such as anomalously large phononabsorption peaks at low temperature /H208495K /H20850and a monotonic increase in phonon sideband amplitude with current /H20851Figs. 1/H20849b/H20850and1/H20849c/H20850/H20852. Conductance peaks are observed in groups of 4, suggesting consecutive doubly spin-degenerate electroniclevels in the Coulomb blockade regime. Sidebands are attrib-uted to phonon-assisted tunneling through the radial breath-ing mode /H20849RBM /H20850of the SWCNT, known to predominate at low bias through its effect at the bottom of the conductionband. 24 We will model the dot with a single phonon mode /H20849/H6036/H92751/H20850 corresponding to the RBM and six electronic energy levels— two lower energy degenerate levels /H20849/H9280˜1and/H9280˜2/H20850and four higher energy degenerate ones /H20849/H9280˜3,/H9280˜4,/H9280˜5, and /H9280˜6/H20850. The oc- currence of the singly degenerate and the doubly degeneratediscrete electronic levels in CNT QDs has been observedearlier. 25,26We will assume that the tunneling rate /H9003/H9251//H6036and the phonon escape rate /H9252//H6036are dispersionless, and use the same value for the couplings /H20849/H926111=/H926121=¯=/H926161/H20850of the RBM to all the electronic levels.27 In order to match experimental results, we ﬁnd it neces- sary to consider electron addition levels only, and not elec-tron removal levels lying below the equilibrium Fermienergy. 1,2An analytical estimate of the height of the ﬁrst two conductance peaks, corresponding to two degenerate levels at/H9280˜1,2, explains the justiﬁcation behind this assumption. It can be shown that the height of the ﬁrst two electron removal peaks should be proportional to 2 /H9003L/H9003R//H208492/H9003R+/H9003L/H20850and 2/H9003L/H9003R//H20849/H9003R+/H9003L/H20850−2/H9003L/H9003R//H208492/H9003R+/H9003L/H20850,20assuming that the left contact injects electrons, the temperature is very low, and ignoring phonon sidebands—assumptions which are consis-tent with experimental conditions. However, this predicts, for/H9003 R/lessmuch/H9003L, that the height of the second peak should become zero, in contrast with the experiment /H20851Fig. 1/H20849b/H20850/H20852. The height of the ﬁrst two electron addition peaks, under the same as- sumption, can be shown to be proportional to 2 /H9003L/H9003R//H20849/H9003R +2/H9003L/H20850and 2 /H9003L/H9003R//H20849/H9003R+/H9003L/H20850−2/H9003L/H9003R//H20849/H9003R+2/H9003L/H20850which pre- dicts, for /H9003R/lessmuch/H9003L, that the peak heights should be equal, in agreement with experiment /H20851Fig.1/H20849b/H20850/H20852. This suggests that the observed peaks arise from electron addition rather than re-moval. Next, let us estimate the input parameters. The phonon energy /H6036 /H92751was measured to be 11.5 meV /H20849see Refs. 1and 2/H20850which, together with the separation between the main Coulomb peak and its ﬁrst phonon emission sideband, yieldsa voltage-division factor /H9257/H110150.6. The polaron renormalized charging parameter U˜is estimated to be 30 meV, as ex-tracted from the separation between consecutive Coulomb peaks originating from the same degenerate set of levels.From the estimate of the phonon decay rate and the Qfactor, /H9252was determined to be /H1101110−8eV. In our calculations we varied the tunneling rates /H9003/H9251and electron-phonon coupling constant /H9261i1/H20849i=1,2,...,6 /H20850to match the experimental con- ductance levels. Figures 1/H20849b/H20850and 1/H20849c/H20850show a comparison between the experiment and our calculations with the above parameters.Six levels seem sufﬁcient to capture the essential physics,including the increase in the number of phonons betweenemission and absorption sidepeaks arising from the corre-sponding increase or decrease of phonon occupation at thosebias points /H20851Fig. 1/H20849c/H20850/H20852. The calculated phonon number sig- niﬁcantly exceeds the equilibrium value after each emissionpeak and drops considerably after absorption, indicatingstrongly correlated nonequilibrium phonon dynamics in thissystem. The weak phonon-substrate coupling /H9252for sus- pended tubes leads to a phonon bottleneck whereby the cur-rent emits phonons faster than they are conducted away,leading to anomalous low-temperature absorption peaks thateven exceed the corresponding emission peak heights on oc- casion. Such a phonon bottleneck can also create negativediffernetial resistance due to self-heating /H20851Ref. 30/H20852. B. Variation of conductance with current Experiment1shows a prominent quadratic dependence of the phonon occupancy on current. This variation is extractedindirectly from the observed variation in height of a mainconduction peak and its associated ﬁrst phonon sidepeakwith current /H20849Fig. 3/H20850, by applying a traditional Tien-Gordon model to ﬁt it. 16Note that these reported experimental results0 10 20 30 40 5000.511.5 Current, I [ pA]Cond. [nA /V] 0 2500123 Current, I [pA]No. of phon., 〈Nph〉 main peak: expt. side peak: expt. main peak: theo. first peak: theo FIG. 3. /H20849Color online /H20850Variation of conductance with the current at the second Coulomb peak /H20849atVappl=−0.1 V /H20850and its associated ﬁrst phonon sidepeak /H20849atVappl=−0.125 V /H20850. Inset: Variation of the number of phonons with current at the above-mentioned conduc-tance peaks. The parameters used to generate these results are /H9280˜1 =/H9280˜2=/H9280˜=8 meV, U˜=12 meV, EF=0,/H6036/H92751=12.5 meV, /H926111=/H926121 =2.1, /H9003L=5/H1100310−6eV,/H9003R=0.000 05 /H9003L–0.5/H9003L,/H9252=5/H1100310−10eV, /H9257=0.2, and T=5 K.SIDDIQUI, GHOSH, AND DATTA PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 085433-4are from a different CNT QD sample from the one corre- sponding to Figs. 1/H20849b/H20850and1/H20849c/H20850. So we will use a model with a different conﬁguration of parameters to capture the essen-tial characteristics of the experimental observations. In orderto ascertain the conductance variation with current, we havereplotted the reported normalized conductance,/H20849dI/dV appl/H20850//H20849I/Vappl/H20850vs current, Ito conductance G =dI/dVapplvs current I, using the known bias values Vappl /H20849−0.1 V at main peak and −0.125 V at ﬁrst sidepeak /H20850and current values Iat those peaks obtained from Ref. 1. The experimental observation shows that the main conductancepeak initially increases with current and thereafter tends tosaturate. On the other hand, the conductance at the ﬁrst sidepeak gradually increases with current exponentially withinthe entire current range of the experiment. To explore this variation we further simplify our model to just two degenerate single electronic levels at energy /H9280˜. The simulation produces two Coulomb peaks and their associatedphonon sidepeaks. To mimic the experimental procedure, weincreased the current by varying the tunneling rate /H9003 Rof the STM tip. The calculated current dependences of the conduc-tance at the second main peak /H20849due to the direct tunneling at energy /H9280˜+U˜/H20850and its associated ﬁrst phonon sidepeak at higher energy /H20849/H9280˜+U˜+/H6036/H92751/H20850agree with the experimental trend /H20849Fig. 3/H20850. However, we do not ﬁnd /H20855Nph/H20856varying as I2at the conduction peaks /H20849Fig. 3, inset /H20850, but instead almost linearly within the experimental current range. The disagreement be-tween our result and the claim made in Ref. 1stems from the fundamental difference between our model and the Tien-Gordon model. Unlike our model, the Tien-Gordon modelneglects the correlation between the electronic degrees offreedom and the phonon degrees of freedom by replacing theelectron-phonon coupling Hamiltonian H el-ph with a mean- ﬁeld oscillating potential V=/H20855/H9254Hel-ph//H9254n/H20856which is propor- tional to the average position operator. For a particular pho- non mode of frequency /H9275and in the absence of leads or baths, the position operator and thus the averaged phononpotential is simply proportional to cos /H9275t. Since the Bohr frequency of the electron in turn depends on this oscillatingHamiltonian, the resulting time-evolution operator and thusthe electronic spectral functions end up with independentFourier components whose spectral weights depend onBessel functions, in other words, the Tien-Gordon expres-sion. Crucial to the derivation of the Tien-Gordon model isthe replacement of the electron-phonon coupling Hamil-tonian by its mean-ﬁeld time-dependent component ignoringleads and baths, which eliminates all correlation effects. Thisassumption is clearly inconsistent with our model that showsstrongly correlated phonon dynamics, captured by the di-mensionless electron-phonon coupling constant /H9261that is typically greater than unity, as mentioned in our ﬁgure cap-tions. It is also noteworthy that the Tien-Gordon analysisrequires coherent phonon dynamics. The phonons treated inour model are partly incoherent, because we ignore the off-diagonal terms in our electron-phonon density matrix to fo-cus on occupation probabilities. In addition, the bath imposesthermodynamic averages on the diagonal correlators alone,acting as a physical source of incoherence in this system. It is worth explaining why a single level does not sufﬁce to capture this current variation. We ﬁnd that the conduc-tance of the main Coulomb peak, irrespective of the param- eters chosen, varies linearly with current and does not satu-rate. Since the current prior to the main Coulomb peak iszero, the height of the main conductance peak /H11011I coul/2kBT, where Icoulis the current plateau immediately past this volt- age, the main broadening coming from temperature. This re-sult, consistent with a test simulation with one single elec-tron level /H20849Fig. 4/H20850, also persists for the ﬁrst in a series of Coulomb peaks in a multilevel system for the same reason.In all these examples, the main conductance peak can onlyvary linearly with the current, in contradiction with experi-ment, prompting us to look at higher peaks in a more com-plex, multileveled system. This requires us to adopt a morecomplex model with two single electron levels and to look atthe variation at the second Coulomb peak and its associatedphonon sidepeak. The results reported in the experiment/H20851Figs. 4/H20849a/H20850and 4 /H20849b/H20850of Ref. 1/H20852were later conﬁrmed to in- volve the variation of the second Coulomb peak and its as-sociated phonon sidepeak. 29 We used our doubly degenerate single electron level model to explore the variation of the conductance for currentvalues beyond the experimental current range, to project thecharacteristics under higher STM set current. The results areshown in Fig. 5. The conductance values at both the Cou- lomb peak and the ﬁrst sidepeak show a nontrivial variationwith current. The variation of the conductance at the Cou-lomb peak is compared to that /H20849Fig. 5, inset /H20850in the absence of electron-phonon coupling. We ﬁnd that the coupling ofelectronic and phonon degrees of freedom signiﬁcantly af-fects the Coulomb peak characteristics. In particular, whilethe trends match the experimental data up to the experimen-tal current levels, we notice that at higher currents the mainpeak goes through a kink followed by a rise, while the side-peak amplitude tends to saturate. Finally, we look at the conductance variation with current for a quadruply degenerate single electron level model for aCNT QD that occasionally shows doubly spin degenerateshells. The conductance variation at each Coulomb peak andits associated ﬁrst higher energy phonon sidepeak are shownin Fig. 6. The ﬁrst Coulomb peak varies almost linearly with current /H20849similar to Fig. 4/H20850, as we argued earlier. The conduc-0 10 20 30 40 5000.511.5 Current, I [ pA]Cond.[nA/V]main peak: expt. side peak: expt. main peak: theo. side peak: theo. FIG. 4. /H20849Color online /H20850Simple one single electron level model does not match the experimental result at all. The values of theparameters used are /H9280˜1=0.05 eV, /H6036/H92751=12.5 meV, /H926111=1.7, /H9003L=1 /H1100310−5eV,/H9003R=0.000 05 /H9003L–0.5/H9003L,/H9252=5/H1100310−10eV,/H9257=0.5, and T=5 K.PHONON RUNAWAY IN CARBON NANOTUBE QUANTUM DOTS PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 085433-5tance variations at the other three peaks show the same quali- tative trends as Fig. 5. From the qualitative similarity be- tween Figs. 4–6, we see that the qualitative features of these variations are robust with respect to the values of the param-eters that went into the model. IV. CONCLUSION We have studied strong electron-phonon coupled dynam- ics under nonequilibrium conditions using a many-electronrate equation approach, focusing on recent experiments onsuspended nanotubes. From our calculations we argue thatphonons in weakly coupled systems can be readily driven farfrom equilibrium, leading to anomalous signatures in the cor-responding conductance spectrum and temperature depen-dences. Speciﬁcally, we expect the ﬁrst Coulomb peak andits sideband to have a linear dependence on current, thehigher Coulomb Blockade peak conductances should have anontrivial variation with current with a kink at intermediatevolatge, while the conductance of their corresponding vi- bronic sidebands should saturate /H20849Fig. 6/H20850. Interesting exten- sions of this work would involve studying the imputations ofnonequilibrium phonon dynamics for energy dissipation innanoscale systems, and of maintaining phonon coherence bytuning their decay rates through their mechanical couplingwith the substrate, and through phonon-phonon interactionscontrolled by the inherent nonlinearity of the lattice. ACKNOWLEDGMENTS We would like to thank Cees Dekker and Brian LeRoy for sharing electronic data and for their valuable comments onan earlier version of this manuscript. We would also like toacknowledge NSF Network for Computational Technology/H20849NCN /H20850for providing the computational resources. This work was supported by DURINT and DARPA. APPENDIX In this section we show the derivation of some important analytical results. The derivations in Secs. I and II, which arealready discussed in Refs. 12and17, have been worked out here to maintain continuity. 1. Derivation of Eqs. ( 10) and ( 11) The terms like eSAe−Sare evaluated using eSAe−S=A+/H20851S,A/H20852+/H208491/2 ! /H20850†S,/H20851S,A/H20852‡+¯. /H20849A1/H20850 When A=ciwe have /H20851S,ci/H20852=/H20858 h,j/H9261hj/H20849aj†−aj/H20850/H20851ch†ch,ci/H20852 =/H20858 h,j/H9261hj/H20849aj†−aj/H20850/H20851ch†chci−cich†ch/H20852 =/H20858 h,j/H9261hj/H20849aj†−aj/H20850/H20851−ch†cich−cich†ch/H20852 =/H20858 h,j/H9261hj/H20849aj†−aj/H20850/H20851/H20849cich†−/H9254i,h/H20850ch−cich†ch/H20852 =/H20858 h,j/H9261hj/H20849aj†−aj/H20850/H20851−/H9254i,hch/H20852, /H20851S,ci/H20852=/H20858 i,j/H9261ij/H20849aj−aj†/H20850ci, /H20849A2/H20850 where /H9254is the kronecker delta function. Then, †S,/H20851S,ci/H20852‡=/H20875S,/H20858 i,j/H9261ij/H20849aj†−aj/H20850ci/H20876 =/H20858 i,j/H20851/H9261ij/H20849aj†−aj/H20850/H20852/H20858 h,j/H20851/H9261hj/H20849aj†−aj/H20850/H20852/H20851ch†ch,ci/H20852,0 100 200 300 400 50000.511.522.53 Current, I [pA]Cond. [nA/V] 0 0.5 102040 Current, I [nA]Cond. [nA/V]main peak side peak FIG. 5. /H20849Color online /H20850Variation of conductance with current for current values higher than the experimental range. Inset: The varia-tion of conductance at the Coulomb peak in the absence of electron-phonon coupling. The values of the parameters used are the same asthose of Fig. 3/H20849except that there is no coupling between electrons and phonons for the results in the inset /H20850. The region inside the dotted box is the range of experimental observation in Fig. 3. 0 0.1 0.20102030 Current, I [nA]Cond. [nA /V] 0 0.20510 Current, I [nA]Cond. [nA/V ] 0 0.4 0.80510 Current, I [nA]Cond. [nA /V] 0 0.2012x1 0−9 Current, I [nA]Cond. [nA/V]1st 2nd 3rd 4th FIG. 6. /H20849Color online /H20850Variation of conductance with current for four degenerate single electron levels at the four Coulomb peaks/H20849solid lines /H20850and their associated ﬁrst phonon sidepeaks /H20849dashed lines /H20850. The values of the parameters used are /H9280˜1=/H9280˜2=/H9280˜3=/H9280˜4=/H9280˜ =20 meV, U˜=30 meV, EF=0,/H6036/H92751=12.5 meV, /H926111=/H926121=/H926131 =/H926141=2.1, /H9003L=5/H1100310−6eV, /H9003R=0.000 05 /H9003L–0.5/H9003L,/H9252=5 /H1100310−10eV,/H9257=0.5, and T=5 K.SIDDIQUI, GHOSH, AND DATTA PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 085433-6†S,/H20851S,ci/H20852‡=/H20875/H20858 i,j/H9261ij/H20849aj−aj†/H20850/H208762ci, /H20849A3/H20850 and so on. Putting all these results in Eq. /H20849A1/H20850we ﬁnally get Eq. /H2084910/H20850. When A=ajwe have /H20851S,aj/H20852=/H20858 i,hni/H9261ih/H20851ah†−ah,aj/H20852 =/H20858 i,hni/H9261ih/H20849ahaj−ah†aj−ajah+ajah†/H20850 =/H20858 i,hni/H9261ih/H9254j,h, /H20851S,aj/H20852=/H20858 ini/H9261ij. /H20849A4/H20850 So afterwards, [S,/H20851S,aj/H20852]=/H20851S,[S,/H20851S,aj/H20852]/H20852=¯=0. Putting all these commutators in Eq. /H20849A1/H20850we get Eq. /H2084911/H20850. 2. Derivation of Eqs. ( 7)–(9) Applying the polaronic transformation, H˜D=H˜el+H˜ph+H˜el-ph. /H20849A5/H20850 Now the electronic portion H˜el=eSHele−S, H˜el=/H20858 i/H9280in˜i+1 2/H20858 i,i/H11032,i/HS11005i/H11032Uii/H11032n˜in˜i/H11032. /H20849A6/H20850 The phonon portion H˜ph=eSHphe−S, H˜ph=/H20858 j/H6036/H9275ja˜j†a˜j =/H20858 j/H6036/H9275j/H20873aj†−/H20858 i/H9261ijni/H20874/H20873aj−/H20858 i/H11032/H9261i/H11032jni/H11032/H20874 =/H20858 j/H6036/H9275jaj†aj+/H20858 i,i/H11032,j/H6036/H9275j/H9261ij/H9261i/H11032jnini/H11032 −/H20858 i,j/H6036/H9275j/H9261ijniaj−/H20858 i/H11032,j/H6036/H9275j/H9261i/H11032jni/H11032aj†, H˜ph=/H20858 j/H6036/H9275jaj†aj+/H20858 i,i/H11032,j,i/HS11005i/H11032/H6036/H9275j/H9261ij/H9261i/H11032jn˜in˜i/H11032 +/H20858 i,j/H9261ij2/H6036/H9275jn˜i−/H20858 i,j/H6036/H9275j/H9261ijn˜i/H20849aj†+aj/H20850, /H20849A7/H20850 where we have used nini=ni=n˜i. Now, ﬁnally, the electron- phonon coupling term H˜el−ph=eSHel−phe−S,H˜el−ph=/H20858 i,j/H9261ij/H6036/H9275jn˜i/H20849a˜j†+a˜j/H20850 =/H20858 i,j/H6036/H9275j/H9261ijn˜i/H20873aj†+aj−/H20858 i/H11032/H9261i/H11032jni/H11032†−/H20858 i/H11032/H9261i/H11032jni/H11032/H20874 =/H20858 i,j/H6036/H9275j/H9261ijn˜i/H20849aj†+aj/H20850−2/H20858 i,i/H11032,j/H6036/H9275j/H9261ij/H9261i/H11032jn˜in˜i/H11032, H˜el−ph=/H20858 i,j/H6036/H9275j/H9261ijn˜i/H20849aj†+aj/H20850−2/H20858 i,j/H9261ij2/H6036/H9275jn˜i −2 /H20858 i,i/H11032,j,i/HS11005i/H11032/H6036/H9275j/H9261ij/H9261i/H11032jn˜in˜i/H11032. /H20849A8/H20850 Substituting Eqs. /H20849A6/H20850–/H20849A8/H20850in Eq. /H20849A5/H20850we get the results of Eqs. /H208497/H20850–/H208499/H20850. 3. Derivation of Eq. ( 12) We will start from /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=/H20855eNe−1r,p/H20841ciX/H20841eNes,k/H20856. Now since ci/H20841eNes,p/H20856=/H20841eNe−1r,p/H20856it follows that /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=/H20855eNes,p/H20841X/H20841eNes,k/H20856. For one phonon mode X=exp /H20851/H9261i1/H20849a1−a1†/H20850/H20852. Then /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=/H20855eNes,p/H20841e/H9261i1/H20849a1−a1†/H20850/H20841eNes,k/H20856. Using e/H9261i1/H20849a1−a1†/H20850=e−/H9261i1a1†e/H9261i1a1e−/H9261i12/2: /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=e−/H9261i12/2/H20855eNes,p/H20841e−/H9261i1a1†e/H9261i1a1/H20841eNes,k/H20856. /H20849A9/H20850 By simple algebra, e/H9261i1a1/H20841eNes,k/H20856=/H20858 l=0/H11009/H20849/H9261i1/H20850l l!/H20849a1/H20850l/H20841eNes,k/H20856, e/H9261i1a1/H20841eNes,k/H20856=/H20858 l=0k/H20849/H9261i1/H20850l l!/H20875k! /H20849k−l/H20850!/H208761/2 /H20841eNes,k−l/H20856, /H20849A10 /H20850 and similarly /H20855eNes,p/H20841e−/H9261i1a1†=/H20858 m=0p /H20855eNes,p−m/H20841/H20849−/H9261i1/H20850m m!/H20875p! /H20849p−m/H20850!/H208761/2 . /H20849A11 /H20850 From Eqs. /H20849A9/H20850–/H20849A11 /H20850,PHONON RUNAWAY IN CARBON NANOTUBE QUANTUM DOTS PHYSICAL REVIEW B 76, 085433 /H208492007 /H20850 085433-7/H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=e−/H9261i12/2/H20858 m=0p /H20858 l=0k/H20875/H20849/H9261i1/H20850l/H20849−/H9261i1/H20850m m!l! /H11003/H20849p!k!/H208501/2 /H20853/H20849p−m/H20850!/H20849k−l/H20850!/H208541/2/H9254p−m,k−l/H20876. /H20849A12 /H20850 For the case of k/H11350p, to remove the Kronecker delta func- tion, we substitute l=k−p+min Eq. /H20849A12 /H20850: /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=e−/H9261i12/2/H20858 m=0p/H20875/H20849/H9261i1/H20850k−p+m/H20849−/H9261i1/H20850m m!/H20849k−p+m/H20850! /H11003/H20849p!k!/H208501/2 /H20849p−m/H20850!/H20876 =e−/H9261i12/2/H20858 m=0p/H20849−1/H20850m/H20849/H9261i1/H20850k−p/H20849/H9261i12/H20850m/H20849k!p!/H208501/2 m!/H20849k−p+m/H20850!/H20849p−m/H20850! =e−/H9261i12/2/H20849/H9261i1/H20850k−p/H20881p! k! /H11003/H20858 m=0p/H20849−1/H20850m/H20853/H20849k−p/H20850+p/H20854!/H20849/H9261i12/H20850m m!/H20853/H20849k−p/H20850+m/H20854!/H20849p−m/H20850!. The summed series is nothing but associated Laguerre poly- nomial Lpk−p/H20849/H9261i12/H20850. So ﬁnally, we get /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=e−/H9261i12/2/H20849/H9261i1/H20850k−p/H20881p! k!Lpk−p/H20849/H9261i12/H20850fork/H11350p. Following the exact same procedure, we can show that /H20855eNe−1r,p/H20841c˜i/H20841eNes,k/H20856=e−/H9261i12/2/H20849/H9261i1/H20850p−k/H20881k! p!Lkp−k/H20849/H9261i12/H20850forp/H11350k. 4. Derivation of Eq. ( 17) From Eqs. /H2084914/H20850, we can see that the phonon bath induces a transition between only those two states which have theexact same electronic conﬁguration and phonon number dif- fering by 1. So we can write Xph=/H20858 s,Ne,k/H20858 r,Ne/H11032,n/H20849k−n/H20850P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNe/H11032r,n/H20856ph =/H20858 s,Ne,k/H20849k−k/H110071/H20850P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNes,k±1/H20856ph =/H20858 s,Ne/H20858 k=1/H11009 P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNes,k−1/H20856ph −/H20858 s,Ne/H20858 k=0/H11009 P/H20841eNes,k/H20856R/H20841eNes,k/H20856→/H20841eNes,k+1/H20856ph =/H9252 /H6036/H20858 s,Ne/H20858 k=0/H11009 P/H20841eNes,k/H20856/H20875k−/H20849k+1/H20850exp/H20873−/H6036/H9275 kBTB/H20874/H20876 =/H20873/H9252 /H6036/H20874/H208751 − exp/H20873−/H6036/H9275 kBTB/H20874/H20876/H20858 s,Ne/H20858 k=0/H11009 kP /H20841eNes,k/H20856−/H20873/H9252 /H6036/H20874 /H11003exp/H20873−/H6036/H9275 kBTB/H20874/H20858 s,Ne/H20858 k=0/H11009 P/H20841eNes,k/H20856. So, from Eq. /H2084916/H20850and the normalization condition Xph=/H20873/H9252 /H6036/H20874/H20877/H20855Nph/H20856/H208751 − exp/H20873−/H6036/H9275 kBTB/H20874/H20876− exp/H20873−/H6036/H9275 kBTB/H20874/H20878 =/H20873/H9252 /H6036/H20874/H20875/H20855Nph/H20856/H208731−Npheq Npheq+1/H20874−Npheq Npheq+1/H20876, Xph=/H20873/H9252 /H6036/H20874/H20855Nph/H20856−Npheq Npheq+1, where Npheq=/H20853exp /H20851/H6036/H9275//H20849kBT/H20850/H20852−1/H20854−1. 1B. J. LeRoy, S. G. Lemay, J. Kong, and C. Dekker, Nature /H20849Lon- don /H20850395, 371 /H208492004 /H20850. 2B. J. LeRoy, J. Kong, V . K. Pahilwani, C. Dekker, and S. G. Lemay, Phys. Rev. B 72, 075413 /H208492005 /H20850. 3H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P. Alivisatos, and P. L. McEuen, Nature /H20849London /H20850407,5 7 /H208492000 /H20850. 4W. Wang, T. Lee, I. Kretzschmar, and M. Reed, Nano Lett. 4, 643 /H208492004 /H20850. 5L. H. Yu, Z. K. Keane, J. W. Ciszek, L. Cheng, M. P. Stewart, J. M. Tour, and D. 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PhysRevB.77.073203.pdf	Enhancing hole concentration in AlN by Mg:O codoping: Ab initio study R. Q. Wu, L. Shen, M. Yang, Z. D. Sha, Y. Q. Cai, and Y. P. Feng * Department of Physics, Faculty of Science, National University of Singapore, Singapore 117542, Singapore Z. G. Huang and Q. Y. Wu Department of Physics, Fujian Normal University, Fuzhou 350007, China /H20849Received 10 October 2007; published 15 February 2008 /H20850 Ab initio study based on density functional theory is performed to study the binding energies of Mg acceptors to single oxygen in AlN and the activation energies of the resultant Mg n-O complexes /H20849n= 2 ,3 ,a n d 4/H20850. It is found that such complexes are energetically favored and have activation energies at least 0.23 eV lower than that of single Mg. The lower activation energies originate from the extra states over the valence band topof AlN induced by the passive Mg-O. By comparing to the well-established case of GaN, it is possible tofabricate Mg:O codoped AlN without MgO precipitates. These results suggest the possibility of achievinghigher hole concentration in AlN by Mg:O codoping. DOI: 10.1103/PhysRevB.77.073203 PACS number /H20849s/H20850: 61.72.uj Ultraviolet light-emitting diodes /H20849LEDs /H20850with wave- lengths less than 300 nm are of considerable technological signiﬁcance. They are potential alternatives to the existinggas lasers and mercury lamps in applications such as disin-fection, air and water puriﬁcation, and biomedicine, wherethe latter encounter difﬁculties due to their high operatingvoltages, low efﬁciency, large size, and toxicities. 1They also promise high density optical data storage and high-resolutionphotolithography. 2Wide gap semiconductors such as dia- mond /H20849Eg=5.5 eV /H20850and AlN /H20849Eg=6.2 eV /H20850are the two most studied materials for such LEDs, and AlN is particularly fa- vored due to its direct gap band structure and subsequenthigh light-emission efﬁciency. Diamond as well as AlNbased LEDs have been fabricated recently. 3,4However, in view of their low efﬁciency and high operating voltages,further developments are still required to improve these twoLEDs to the point where they can be used as devices. 5 Fabrication of a homostructured LED requires both p- and n-type doping of a semiconductor. Unfortunately, the asym- metry dopability of wide gap semiconductors makes the fab-rication of LEDs very difﬁcult. The asymmetry dopabilitymeans that a wide gap semiconductor can be either p-o r n-type doped, but not both. 6AlN can be easily n-type doped with Si. However, p-type AlN is of great challenge. The most promising acceptor for AlN is Mg. Yet, the activation energy/H20849E A/H20850of the Mg acceptor in AlN is 0.5 eV.7,8As the ratio of carrier concentration to impurity concentration follows exp/H20849−EA/kBT/H20850, where kBis the Boltzmann constant and Tis the temperature, only a very small fraction /H1101110−8of Mg impurities is activated at room temperature. Since the upperlimit of Mg concentration in AlN is 2 /H11003/H1101110 20cm−3,4the corresponding hole concentration for Mg-doped AlN wouldnot exceed /H1101110 12cm−3. This concentration, however, is still well below that for device applications /H20849which requires a hole concentration of at least 1017cm−3/H20850. Recently, codoping has been proposed and applied to overcome difﬁculties in p-type doping in some wide gap semiconductors,9–12and it has been proved to be an effective method to improve hole concentrations in wide gap semicon-ductors such as GaN and ZnO. Signiﬁcant improvements inhole concentrations have been achieved in these two semi- conductors. In this approach, p-type dopants /H20849D/H20850are incor- porated into the semiconductor along with a small amount ofreactive n-type impurities as codopants /H20849CD/H20850. Then in the host semiconductors, complexes like D-CD, D 2-CD, D 3-CD, and D 4-CD will form and, more often, they have lower ion- ization energies than that of monodopant D /H20849except for the passive D-CD complex /H20850. Thus, the hole concentration may be greatly enhanced. By this approach, the hole concen-tration in ZnO was improved from /H1101110 13cm−3in N-monodoped ZnO to /H110111017cm−3or even higher in /H20849Al, Ga, or In /H20850-N codoped ZnO.13–15Codoping also enhanced the hole concentration in Mg-doped GaN. The conductivity of Mg-doped GaN could be signiﬁcantly enhanced by annealing inan environment with oxygen. 16This enhancement in conduc- tivity was attributed to the decreased activation of Mg accep-tors due to the inclusion of oxygen and a corresponding or-der of magnitude increase of hole concentration /H20849from /H1101110 17to/H110111018cm−3/H20850.17 It is, therefore, reasonable to believe that the hole concen- tration of Mg-doped AlN can be improved using a similarapproach. If the concentration of Mg is equal to that of oxy-gen, then Mg might be completely compensated by the for-mation of passive Mg-O complex. However, if the concen-tration of Mg is two times that of oxygen or even larger, thencomplexes like Mg 2-O, Mg 3-O, and Mg 4-O would be likely to form, and these complexes might have lower activationenergies than a single Mg atom. In this Brief Report, weinvestigate the possibility of the formation of Mg n-O/H20849n=2, 3, and 4 /H20850complexes, where oxygen is the minor codopant, and the activation energies of such complexes by ﬁrst prin-ciples total energy calculations based on density functionaltheory /H20849DFT /H20850. We ﬁrst study the possibility of the formation of Mg n-O/H20849here, nranges from 1 to 4; atomic O occupies N site and atomic Mg the neighboring Al site, respectively /H20850 complexes in AlN. All these complexes under study are as-sumed to be electrically neutral unless otherwise stated. Thisis reasonable since even at an activation energy as low as0.20 eV, only a small portion /H20849/H1101110 −4/H20850of the acceptors are in the charged state. Then the activation energies of complexPHYSICAL REVIEW B 77, 073203 /H208492008 /H20850 1098-0121/2008/77 /H208497/H20850/073203 /H208494/H20850 ©2008 The American Physical Society 073203-1acceptor Mg n-O complexes /H20849n=2, 3, and 4 as Mg-O is pas- sive/H20850are calculated. To explain the results obtained, the den- sity of states /H20849DOS /H20850is given. Most of the calculations are done using a 3 /H110033/H110032 supercell constructed from AlN unit cell using the plane-wave DFT code VASP.18,19The lattice constants of the supercell are kept ﬁxed to avoid effects fromspurious volume expansion. The /H9003-centered 4 /H110034/H110034kmesh is used for irreducible Brillouin zone sampling. The ion-electron interaction is described by Vanderbilt ultrasoftpseudopotentials 20with local density approximation /H20849LDA /H20850 for the exchange-correlation potential. The electron wavefunction is expanded in plane waves with a cutoff energy of400 eV. These parameters ensure a convergence better than1 meV for the total energy. In all the doped supercells,atomic coordinates are fully relaxed using the conjugate-gradient algorithm 21until the maximum force on a single atom is less than 0.03 eV /Å. For AlN in wurtzite structure, the calculated lattice con- stants are 3.08 Å for aand 4.94 Å for c, with the internal parameter being 0.382, in good agreement with experimentalvalues. Based on this, the 3 /H110033/H110032 supercell /H20849Al 36N36/H20850is constructed. Previous calculations on formation energies have shown that O occupies N site in AlN as a deepdonor 22–24and, consequently, in our supercell, one N atom is replaced by an O atom. This O donor may act as an attractioncenter to single Mg atom and form complexes such as Mg-O,Mg 2-O, Mg 3-O, and Mg 4-O with Mg atoms occupying the nearest Al sites to O and without destroying the lattice struc-ture. To study whether single Mg atom will bind to Mg n-O /H20849n=0, 1, 2, and 3, respectively /H20850complexes, we deﬁne the binding energy as the energy required to form the Mg n+1-O complex from well separated Mg dopant and Mg n-O com- plex: /H9004/H20849n/H20850=E/H20849Al36−n−1Mg n+1N35O/H20850+E/H20849Al36N36/H20850 −E/H20849Al35Mg1N36/H20850−E/H20849Al36−nMg nN35O/H20850, /H208491/H20850 where Eis the total energy of the system indicated in paren- theses. A negative /H9004/H20849n+1/H20850suggests that the Mg n+1-O complex is energetically favorable and stable, while a positive /H9004/H20849n/H20850 suggests that Mg n+1-O cannot form. The calculated /H9004/H20849n/H20850are summarized in Table I. As can be seen, single Mg atom will bind to single O atom for a large energy decrease of5.304 eV. This large energy decrease can be attributed to thepassivation of the extra electron of O by Mg. This suggeststhat Mg will be completely compensated by O if their con-centrations are very close. However, the concentration of Mgis larger and, therefore, extra Mg acceptors are present, thenthese extra Mg atoms will bind to the Mg-O complex, form-ing Mg 2-O, Mg 3-O, and Mg 4-O complexes depending on theratio between Mg and O. The energy decrease from single Mg atom binding to Mg-O complex is 0.626 eV, suggestingthat Mg 2-O complex will form provided the amount of Mg exceeds that of O. If there are still extra Mg acceptors avail-able, then Mg 3-O will form with an energy decrease of 0.412 eV, and Mg 4-O complex with an energy decrease of 0.157 eV. The binding of single Mg atom to the O atom inMg n-O complexes can be attributed to the larger electrone- gativity of O than that of N. The local environments aroundMg n-O are given in Fig. 1. There are only minor changes in Mg-O and Al-O bond lengths. This is reasonable as atomicMg and Al are very close in atomic size, and so with O andN. Thus, formation of these Mg n-O complexes does not de- stroy the lattice in AlN. Now we calculate the activation energy of the Mg n-O/H20849n =2, 3, and 4, respectively /H20850complexes, which can be made abundant by appropriate ratio between Mg and O. Themethod proposed by Van de Walle and Neugebauer is appliedfor the activation calculation: 25 EA=Etot/H20851D−/H20852−Etot/H20851D0/H20852−Ev−/H9004V/H20851D/H20852+Ecorr, /H208492/H20850 where EAis the activation energy of the defect /H20849donor /H20850D. Etot/H20851D−/H20852andEtot/H20851D0/H20852are the total energies of the supercell with defect in charged /H20849/H11002/H20850and neutral /H208490/H20850states, respec- tively. Evis the valence band maximum of the bulk semicon- ductor; /H9004V/H20851D/H20852is a correction term to align the reference potential in the charged defected supercell with that of the bulk and it is derived from the electrostatic potential differ-ence between the bulk and that of the defected supercell faraway from defect site. Using a 5 /H110035/H110033 supercell, a value ofTABLE I. Binding energy /H9004/H20849n/H20850for single Mg acceptor attaching to Mg n-O complex and the activation energy EA/H20849Mg n+1-O/H20850of the resultant Mg n+1-O complexes. Unit: eV. n 0123 /H9004/H20849n/H20850−5.304 −0.626 −0.412 −0.157 EA/H20849Mg n+1-O/H20850 0.17 0.14 0.12 FIG. 1. /H20849Color online /H20850Bond-and-stick models of the local envi- ronment around Mg n-O complexes: /H20849a/H20850Mg-O, /H20849b/H20850Mg2-O, /H20849c/H20850 Mg3-O, and /H20849d/H20850Mg4-O. Bond lengths of Mg-O and Al-O are given in parentheses. Unit: Å.BRIEF REPORTS PHYSICAL REVIEW B 77, 073203 /H208492008 /H20850 073203-2−0.15 eV is calculated for both /H9004V/H20851Mg/H20852and/H9004V/H20851Mg2-O/H20852. Calculation obtained using a larger 6 /H110036/H110034 supercell does not result in any signiﬁcant change in these values. Ecorris a correction term for the use of /H9003-included k-mesh sampling for the hexagonal lattice. In practice, this is derived from theenergy difference between the highest occupied level at /H9003 point and other special kpoints /H20849averaged /H20850in the supercell containing the neutral defect D. The calculated E corris 0.26 eV for a supercell containing only Mg defect, and0.03 eV for the supercell containing Mg 2-O complex. The calculated ionization of a single Mg acceptor in AlN is0.40 eV, which is in agreement with previous DFT-LDA cal-culation /H20851/H110110.45 eV /H20849Ref. 26/H20850/H20852and experiments /H20851/H110110.5 eV /H20849Refs. 7and8/H20850/H20852. For the Mg 2-O complex, the calculated EA is only 0.17 eV. Similar calculations are performed for Mg2-O and Mg 3-O complexes, and the obtained results are summarized in Table I. The activation energies of Mg 3-O and Mg 4-O are lower than that of Mg and also of Mg 2-O, suggesting better doping efﬁciency than pure Mg. We cansee that at least a decrease of 0.23 eV in activation energycan be obtained in this codoping approach compared tosingle Mg in AlN. If we apply this decrease to the experi-mental value, then following exp /H20849−E A/kBT/H20850, the hole con- centration can be increased by at least a factor of 103. This is a signiﬁcant improvement, although the total carrier concen-tration is still below the value desired. Other complexesMg 3-O and Mg 4-O have slightly lower activation energies to that of Mg 2-O. Thus, incorporation of some amount of oxy- gen into Mg-doped AlN can improve the hole concentration. The decreased activation energy in the Mg acceptors after attaching to O-Mg complex can be understood from the den-sity of states of the defected supercell as shown in Fig. 2. Although O-Mg complex is passive and cannot accept thehost valence electrons, it induces extra fully occupied statesright on the valence-band maximum /H20849VBM /H20850as indicated by the DOS curve of AlN supercell containing Mg-O complex.In the Mg-O complex, some electrons have higher energythan that of the host valence electrons. Thus, Mg atoms bind-ing to this complex may be activated by electrons from thesecomplex states rather than from the host states. So the acti- vation energy is decreased.For such a codoping, the enthalpy effect should be taken into consideration for experimental study as MgO may ap-pear as a competitive precipitate. In this study, we evaluatethe feasibility of overcoming MgO precipitate by comparingthe enthalpy effect to an established case. The formation en-ergy of a defect is given by E f/H20851D/H20849q/H20850/H20852=Etot/H20851D/H20849q/H20850/H20852−Etot/H20851bulk /H20852−/H9018ini/H9262i+q/H20849Ef+Ev/H20850, /H208493/H20850 where the Edefecttot/H20849q/H20850is the total energy of the supercell con- taining the defect, Etot/H20849bulk /H20850is the total energy of a similar supercell containing the pure crystal, and niis the number of atoms that is involved in the formation of the defect, with /H9262i being the corresponding chemical potentials. Efis the Fermi energy which is set to zero at the valence-band maximum Ev. A low formation energy suggests a high concentration of thedefect without competing precipitates under thermal equilib-rium. In our system, the chemical potentials depend on theexperimental growth conditions, which can be either Al-richor N-rich. Formation of AlN crystal under thermal equilib- rium requires /H9262Al+/H9262N=/H9262Al/H20851bulk /H20852+1 2/H9262N/H20851N2/H20852+/H9004H/H20851AlN /H20852, where /H9004H/H20849AlN /H20850is the formation enthalpy of AlN. In the N-rich condition, which is preferred for incorporating Mg at Al sites, the upper limit of /H9262Nis given by /H9262N/H20851N2/H20852, i.e., the energy of N in a N 2molecular. /H9004H/H20851AlN /H20852is calculated to be −3.58 eV. The formation energy of Mg on Al site at neutral state is Ef/H20851MgAl/H208490/H20850/H20852=1.76 eV, with the solubility limit im- posed by 3 /H9262Mg+2/H9262N=3/H9262Mg/H20851bulk /H20852+2/H9262N/H20851N2/H20852+/H9004H/H20849Mg3N2/H20850. However, if O 2is present in N 2ﬂux, to avoid the formation of MgO precipitate, the upper limit of /H9262Mgis as follows: /H9262Mg+/H9262O=/H9262Mg/H20851bulk /H20852+/H9262O/H20851O2/H20852+/H9004H/H20849MgO /H20850, where /H9004H/H20849MgO /H20850=−6.69 eV from our calculation for rock- salt MgO. Under this constraint, the formation energies of Mg n-O complexes become rather high. The formation energy of Mg 2-O complex is calculated to be 4.81 eV, suggesting an ignorable concentration of Mg 2-O complex in AlN in a growth process under thermodynamic equilibrium. However,this solubility limit can be overcome by proper growth con-ditions, i.e., low growth temperature and high growth rate. 27 For example, in the case of Mg:O codoping of GaN by gas-source epitaxy method, the inclusion of oxygen did not resultin the MgO precipitate in GaN, but it improved the holeconcentration signiﬁcantly. 17,28By calculation, the formation energy of Mg 2-O complex in GaN is 6.31 eV, signiﬁcantly higher than that of AlN /H208494.81 eV /H20850. Thus, in AlN, the solubil- ity limit problem is less severe and can be overcome in com- parison to that of Mg:O codoping of GaN. This suggests thepossibility of Mg:O codoping in AlN without MgO precipi-tate. While in our study on the formation of Mg n-O complexes the activation energies of these complexes and the possibilityof overcoming the MgO precipitate suggest the improvementof hole concentration in AlN, there are still other importantissues remaining unaddressed. The optimum ratio betweenMg and O is very important, but is very difﬁcult to predicttheoretically and can only be determined experimentally. Inview of the extremely low /H1101110 10cm−3hole concentration in0.020.040.060.080.0 -5.6 -4.8 -4 -3.2 -2.4 -1.6 -0.8 0 0.8DOS(states/eV cell)AlNAlN(Mg-O) Ener gy (eV) FIG. 2. /H20849Color online /H20850Total DOS of supercells with pure AlN /H20849curve marked by AlN /H20850and AlN containing one passive Mg-O com- plex /H20851curve marked by AlN /H20849Mg-O /H20850/H20852. The bold arrow indicates the extra states on top of the VBM of pure AlN. The Fermi level of pureAlN is indicated by the vertical dashed line.BRIEF REPORTS PHYSICAL REVIEW B 77, 073203 /H208492008 /H20850 073203-3AlN reported up to date, our proposal of Mg:O codoping of AlN might be a way to improve the hole concentration andrequire further experimental study. To summarize, we have studied the electronic properties of Mg acceptors in AlN at the presence of oxygen by ab initio study. Our calculations suggest the formation of Mg n-O complexes and their lower activation energies com- pared to Mg. Compared to the well-established case of GaN,the MgO precipitate problem can be overcome. Our results suggest that the hole concentration in AlN:Mg can be greatlyenhanced by oxygen codoping. Z.G.H. and Q.Y.W. acknowledge the support of the NSF of China under Grant No. 60676055 and of the National KeyProject for Basic Research of China under Grant No.2005CB623605. *phyfyp@nus.edu.sg 1E. F. Schubert and J. K. Kim, Science 308, 1274 /H208492005 /H20850. 2F. A. Ponce and D. P. Bour, Nature /H20849London /H20850386, 351 /H208491997 /H20850. 3S. Koizumi, K. Watanabe, M. Hasegawa, and H. Kanda, Science 289, 1899 /H208492001 /H20850. 4Y. Taniyasu, M. Kasu, and T. Makimoto, Nature /H20849London /H20850441, 325 /H208492006 /H20850. 5A. 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PhysRevB.73.172410.pdf	Theoretical analysis of highly spin-polarized transport in the iron nitride Fe 4N Satoshi Kokado,1,Nobuhisa Fujima,1Kikuo Harigaya,2Hisashi Shimizu,3and Akimasa Sakuma4 1Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan 2Nanotechnology Research Institute, AIST, Tsukuba 305-8568, Japan 3Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan 4Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan /H20849Received 12 December 2005; revised manuscript received 17 April 2006; published 17 May 2006 /H20850 In order to propose a ferromagnet exhibiting highly spin-polarized transport, we theoretically analyzed the spin polarization ratio of the conductivity of the bulk Fe 4N with a perovskite-type structure, in which N is located at the body center position of the fcc-Fe. The spin polarization ratio is deﬁned by P=/H20849/H9268↑−/H9268↓/H20850//H20849/H9268↑ +/H9268↓/H20850, with /H9268↑/H20849↓/H20850being the conductivity at zero temperature of the up spin /H20849down spin /H20850. The conductivity is obtained by using the Kubo formula and the Slater-Koster tight binding model, where parameters are deter- mined from the least-square ﬁtting of the dispersion curves by the tight binding model to those by the ﬁrstprinciples calculation. In the vicinity of the Fermi energy, /H20841P/H20841takes almost 1.0, indicating a perfectly spin- polarized transport. In addition, by comparing Fe 4N to fcc-Fe /H20849Fe4N0/H20850in the ferromagnetic state with the equilibrium lattice constant of Fe 4N, it is shown that the nonmagnetic atom N plays an important role in increasing /H20841P/H20841. DOI: 10.1103/PhysRevB.73.172410 PACS number /H20849s/H20850: 72.25.Ba Recently, highly efﬁcient spin-electronics devices operat- ing at room temperature have been extensively developed forapplications to the magnetic memory and the magnetic sen-sor. A typical device has ferromagnetic tunnel junctions con-sisting of a ferromagnetic electrode /H20849FME /H20850/insulator/FME, which exhibits a large magnetoresistance /H20849MR /H20850effect. 1–7The efﬁciency of the MR effect is often deﬁned by the MRratio= /H20849R AP−RP/H20850/RAP, with RPandRAPbeing the resistance of the parallel and antiparallel magnetization conﬁgurations of FMEs, respectively. Experimentally, regarding junctionswith an electrode of a half-metallic ferromagnet,Co 2Cr0.6Fe0.4Al/Al-O/CoFe/NiFe/IrMn/Ta junctions ex- hibited a MR ratio of 16% at room temperature,1and Co2MnSi/Al-O/Co 75Fe25had a MR ratio of 70% at room temperature.2Regarding junctions of electrodes with usual ferromagnets, single-crystal Fe /H20849001 /H20850/MgO /H20849001 /H20850/Fe/H20849001 /H20850 junctions exhibited MR ratios of 88% /H20849Ref. 3 /H20850and 180% /H20849Ref. 4 /H20850 at room temperature. Furthermore, CoFeB/MgO/CoFeB junctions achieved MR ratios of 260%/H20849Ref. 5 /H20850and 355% /H20849Ref. 6 /H20850/H20849the world’s highest value /H20850at room temperature, although the crystal structure of CoFeBand the role of the light element B on the spin-polarizedtransport have not been clariﬁed yet. Generally, the MR ratiobecomes large by increasing the spin polarization of the con-duction electron in the FME. In the future, a FME with morehighly spin-polarized electrons, which would result in largerMR ratios, will be strongly desired from the viewpoint of thedevelopment of highly efﬁcient MR devices. Towards a proposal of such a FME, we extracted an idea to obtain electrodes exhibiting the highly spin-polarizedtransport at room temperature. We found that ferromagnetsconsisting of magnetic elements andlight elements , such as CoFeB, might be very useful as the electrodes. We consider Fe as a representative magnetic element in this idea. We deﬁne the spin polarization /H20849SP/H20850ratio about the density of states /H20849DOS /H20850of the bulk system, P DOS /H20849E/H20850,a sPDOS /H20849E/H20850=D↑/H20849E/H20850−D↓/H20849E/H20850 D↑/H20849E/H20850+D↓/H20849E/H20850, /H208491/H20850 with Ds/H20849E/H20850being the DOS of spin s/H20849=↑or↓/H20850for the energy E. Then, /H20841PDOS /H20849EF/H20850/H20841at the Fermi energy EFof fcc-Fe in the ferromagnetic state is evaluated to be about 0.7 /H20851see Fig. 2/H20849c/H20850/H20852, which is about 2.3 times as large as that of bcc-Fe,8 where the most stable state of bcc-Fe /H20849fcc-Fe /H20850is ferromag- netic /H20849not ferromagnetic9/H20850. This may indicate that the highly spin-polarized transport is realized for materials closely re-lated to fcc-Fe. The conductivity, including the velocity ofelectrons, will give an answer to the realizability of suchtransport. We focus on a ferromagnet containing fcc-Fe and a light element Fe 4N with a perovskite-type structure,10–15in which N is located at the body center position of fcc-Fe. This fer-romagnet has a Curie temperature of 761 K. 12It should be noted that studies on the spin-polarized transport of Fe 4N have scarcely been performed so far, although other proper-ties have been experimentally 10–12and theoretically13–15in- vestigated. We are particularly interested in not only the SPratio on the transport of Fe 4N but also the role of the light /H20849or nonmagnetic /H20850element N on the transport. In this paper, we analyzed the SP ratio of the conductivity of Fe 4N in order to elucidate the spin-polarized transport. The conductivity was obtained for each spin and each orbitalof the bulk system using the ﬁrst principles /H20849FP/H20850calculation and the tight binding /H20849TB/H20850model calculation. Consequently, we found that Fe 4N exhibits an extremely highly spin- polarized transport and that N plays an important role in thetransport. A calculation method is introduced to obtain the conduc- tivity and the SP ratio. The method is a combination of /H20849i/H20850the FP calculation, 16/H20849ii/H20850the TB model,17and /H20849iii/H20850the Kubo formula.18Details of each are given below. /H20849i/H20850The FP calculations are performed by the Vienna abPHYSICAL REVIEW B 73, 172410 /H208492006 /H20850 1098-0121/2006/73 /H2084917/H20850/172410 /H208494/H20850 ©2006 The American Physical Society 172410-1initio Simulation Package /H20849VASP /H20850code16based on the spin- polarized density functional theory, where we employ thegeneralized gradient approximation of Perdew and Wang 19 and ultrasoft pseudopotentials to describe the core electrons.The cutoff energy for the plane wave basis is 237.51 eV forFe and 348.10 eV for Fe 4N, and the Monkhorst-Pack set20of 8/H110038/H110038kpoints is used. /H20849ii/H20850The Slater-Koster TB model17is used while taking into account the 3 d,4s, and 4 porbitals for Fe and the 2 sand 2porbitals for N as well as interactions up to the third- nearest-neighbor atoms. The Slater-Koster parameters of po-tential energies and transfer integrals 17are here determined from the least-square ﬁtting of the dispersion curves by theTB model to those by the FP calculation at the equilibriumlattice constant. The ﬁtting is done in energy regions from the lowest energy to E F+5 eV for bcc-Fe and fcc-Fe and from the lowest energy to EF+2 eV for Fe 4N. The number of parameters is 34 for bcc-Fe and fcc-Fe and 87 for Fe 4N. /H20849iii/H20850The Kubo formula18and the Slater-Koster TB model with the determined parameters are used to calculate the con-ductivity at zero temperature. In this calculation, we utilizethe theory given by Tsymbal et al. 21The total conductivity of spin s/H20849=↑or↓/H20850,/H9268s/H20849E/H20850, is written by /H9268s/H20849E/H20850=/H20858 i/H9268i,s/H20849E/H20850, /H208492/H20850 where /H9268i,s/H20849E/H20850is the conductivity of orbital i/H20849=4s,4p,3dor- bitals, and so on /H20850and spin s. This /H9268i,s/H20849E/H20850includes the veloc-ity of electrons and the Green’s function. The Green’s func- tion has a single parameter /H9253in the second-order self-energy due to the weak electron-impurity interaction, where /H9253char- acterizes the degree of electron-impurity scattering. In detail, /H9253is related to the lifetime of the electron of orbital iand spin s,/H9270i,s, via 1/ /H9270i,s=/H208492/H9266//H6036/H20850/H92532Di,s/H20849E/H20850, with Di,s/H20849E/H20850being the partial DOS for iands.21At present, /H9253is set to be 0.5 eV, which was previously chosen so as to reproduce the resis-tivity of copper. 21We also use the diagonal approximation for the self-energy, in which the scattering of electronsdue to impurities is allowed for the same energy levels butforbidden between different energy levels. 21The SP ratio is then defined by P/H20849E/H20850=/H9268↑/H20849E/H20850−/H9268↓/H20849E/H20850 /H9268↑/H20849E/H20850+/H9268↓/H20849E/H20850. /H208493/H20850 We ﬁrst compare the results of bcc-Fe by the present method with the previous ones. The equilibrium lattice con-stant obtained by using the FP calculation is estimated to be2.84 Å with an error of about 1% for the experimental valueof 2.87 Å. 22The total conductivity at EFof the down spin /H9268↓/H20849EF/H20850is larger than /H9268↑/H20849EF/H20850. The SP ratio at EF,P/H20849EF/H20850, there- fore takes a negative value, −0.20, and it qualitatively agrees with the previous result of about −0.26,21which was ob- tained by a similar method combined with the FP calculationwithin the local density approximation. In the following, we investigate the equilibrium lattice FIG. 1. Density of states, conductivities, and spin polarization ratios of Fe 4N:/H20849a/H20850Ds/H20849E/H20850calculated using the FP calculation. /H20849b/H20850Schematic illustration of partial DOSs. The 4 s-4pcomponents are partly covered by the 3 dones /H20849see darkish parts in 3 dcomponents /H20850./H20849c/H20850Ds/H20849E/H20850and PDOS /H20849E/H20850for −1 eV /H33355E−EF/H333551 eV of the TB model and the FP calculation. /H20849d/H20850/H9268i,s/H20849E/H20850,/H9268s/H20849E/H20850, and P/H20849E/H20850for −1 eV /H33355E−EF/H333551 eV. Here, /H92682s,s/H20849E/H20850and/H92682p,s/H20849E/H20850for N are not shown because of their small values.BRIEF REPORTS PHYSICAL REVIEW B 73, 172410 /H208492006 /H20850 172410-2constant Ds/H20849E/H20850, and PDOS /H20849E/H20850for Fe 4N using the FP calcula- tion. The equilibrium lattice constant is evaluated as 3.810 Å, which has an error of less than 1% for an experi-mental value of 3.795 Å. 12As shown in Figs. 1 /H20849a/H20850and 1 /H20849c/H20850, D↓/H20849EF/H20850is higher than D↑/H20849EF/H20850. Partial DOSs /H20849Ref. 14 /H20850sche- matically illustrated in Fig. 1 /H20849b/H20850show that 3 dorbitals are dominant around EF, and 4 sand 4 p/H208494s-4p/H20850orbitals are mainly located in an energy region higher than EF, while each orbital of N atoms mostly exists in an energy regionlower than E F. Furthermore, PDOS /H20849EF/H20850obtained by using Ds/H20849EF/H20850of the FP calculation is evaluated to be −0.6 /H20851see Fig. 1/H20849c/H20850/H20852. Using the TB model with the parameters determined from the ﬁtting of the dispersion curves, we obtain Ds/H20849E/H20850and PDOS /H20849E/H20850for −1 eV /H33355E−EF/H333551 eV. As seen from Fig. 1 /H20849c/H20850, Ds/H20849EF/H20850andPDOS /H20849EF/H20850of the TB model agree well with the respective ones of the FP calculation. With the use of the TB model and the Kubo formula we calculate /H9268i,s/H20849E/H20850,/H9268s/H20849E/H20850, and P/H20849E/H20850for −1 eV /H33355E−EF /H333551 eV. The results are shown in Fig. 1 /H20849d/H20850. For E/H33355EF, /H92683d,↑/H20849E/H20850becomes relatively large owing to the high DOS of the 3 dorbitals of the up spin. For E−EF/H333560.6 eV, the 4 s -4porbitals of the up spin contribute strongly to /H9268↑/H20849E/H20850in spite of their low DOSs because their orbitals have large velocities. For 0 eV /H11021E−EF/H110210.6 eV, each /H9268i,↑/H20849E/H20850has a pronounced valley reﬂecting the low DOS of the up spin. In this energy region, although the DOS of the up spin is actu-ally lower than that of the FP calculation, the qualitative behavior of /H9268i,↑/H20849E/H20850appears to be valid because the DOS of the FP calculation has very few components of the 4 s-4p orbitals and it is low. On the other hand, each /H9268i,↓/H20849E/H20850is almost ﬂat, and the 3 dorbitals of the down spin contribute largely to /H9268↓/H20849E/H20850because of their high DOS. The total con- ductivity at EFof the down spin /H9268↓/H20849EF/H20850is much larger than /H9268↑/H20849EF/H20850. The SP ratio P/H20849EF/H20850therefore takes almost −1.0, in- dicating perfectly spin-polarized transport. The magnitude of the SP ratio /H20841P/H20849EF/H20850/H20841is about 5.0 times as large as that of bcc-Fe. In order to clarify the effect of an N atom on the transport, we investigated Ds/H20849E/H20850,PDOS /H20849E/H20850,/H9268i,s/H20849E/H20850,/H9268s/H20849E/H20850, and P/H20849E/H20850of fcc-Fe /H20849Fe4N0/H20850in the ferromagnetic state with an equilibrium lattice constant of Fe 4N. Regarding Ds/H20849E/H20850around EFob- tained using the FP calculation, D↓/H20849E/H20850is much higher than D↑/H20849E/H20850/H20851see Figs. 2 /H20849a/H20850and 2 /H20849c/H20850/H20852; for further details, low and broad DOSs of the 4 s-4porbitals of the up spin and a high DOS of the 3 dorbitals of the down spin are observed /H20851see Fig. 2 /H20849b/H20850/H20852. The SP ratio of DOS, PDOS /H20849EF/H20850, is evaluated to be about −0.7. Figure 2 /H20849c/H20850also shows Ds/H20849E/H20850and PDOS /H20849E/H20850for −1 eV /H33355E−EF/H333551 eV; these values are obtained using the TB model with the determined parameters. It is found thatD s/H20849EF/H20850andPDOS /H20849EF/H20850of the TB model agree fairly well with the respective ones of the FP calculation. With the use of the TB model, we calculate /H9268i,s/H20849E/H20850and/H9268s/H20849E/H20850/H20851see Fig. 2 /H20849d/H20850/H20852.I n spite of the low DOSs of the 4 s-4porbitals of the up spin, FIG. 2. The same as Fig. 1 for fcc-Fe /H20849Fe4N0/H20850in the ferromagnetic state with the equilibrium lattice constant of Fe 4N: /H20849a/H20850Ds/H20849E/H20850 calculated using the FP calculation. /H20849b/H20850Schematic illustration of partial DOSs. The 4 s-4pcomponents are partly covered by the 3 dones /H20849see darkish parts in 3 dcomponents /H20850./H20849c/H20850Ds/H20849E/H20850andPDOS /H20849E/H20850for −1 eV /H33355E−EF/H333551 eV of the TB model and the FP calculation. /H20849d/H20850/H9268i,s/H20849E/H20850, /H9268s/H20849E/H20850, and P/H20849E/H20850for −1 eV /H33355E−EF/H333551e V .BRIEF REPORTS PHYSICAL REVIEW B 73, 172410 /H208492006 /H20850 172410-3their orbitals contribute strongly to /H9268↑/H20849E/H20850. This tendency originates from the large velocities of the 4 s-4porbitals. Moreover, /H92683d,↓/H20849E/H20850is relatively large reﬂecting the high DOS of the 3 dorbitals of the down spin. The total conductivity at EFof the up spin /H9268↑/H20849EF/H20850then becomes larger than /H9268↓/H20849EF/H20850 even though D↑/H20849EF/H20850is much lower than D↓/H20849EF/H20850. The SP ratio P/H20849EF/H20850has a positive sign, opposite to the sign of PDOS /H20849EF/H20850. This sign of P/H20849EF/H20850is also opposite to that of Fe 4N. The magnitude of the SP ratio /H20841P/H20849EF/H20850/H20841is evaluated to be 0.4, which is between the /H20841P/H20849EF/H20850/H20841of Fe 4N and that of bcc-Fe. On the basis of the above investigations and the evaluated Slater-Koster parameters, we discuss the role of the N atomon the highly spin-polarized transport of Fe 4N using sche- matic illustrations of partial DOSs /H20851see Figs. 1 /H20849b/H20850and 2 /H20849b/H20850/H20852. Note that by merely introducing the nonmagnetic atom N tothe body center position of fcc-Fe, /H20841P/H20849E F/H20850/H20841becomes about 2.5 times as large as that of fcc-Fe and the sign of P/H20849EF/H20850 changes. In the present study, we ﬁnd that, by adding N to fcc-Fe, 4 s-4pbands of Fe are raised to a higher energy re- gion by a large magnitude of transfer integrals between the4s-4porbitals of Fe and the 2 sand 2 porbitals of N, while 3 d bands do not change signiﬁcantly owing to the small magni-tude of the transfer integrals between the 3 dorbitals of Fe and the 2 sand 2 porbitals of N. These behaviors are ex- plained by considering the bonding-antibonding statesformed by the transfer integrals, which correspond to over-laps between both orbitals. A large portion of 4 s-4pbands exists in an energy region higher than 3 dbands, and a small portion of them is located in the energy region of 3 dbands because of hybridizations with 3 dorbitals, and is then spin- polarized there. Hence, comparing Fe 4N with fcc-Fe, theproportion of 4 s-4pbands is small in the vicinity of EF, and, in particular, the proportion of the up spin is extremely small;the 3 dbands of the down spin are dominant there. Therefore, the magnitude of the SP ratio /H20841P/H20849E F/H20850/H20841increases, and P/H20849EF/H20850 becomes negative. Finally, although it would be difﬁcult to derive the general properties of phenomena from the present study alone, itseems possible that various ferromagnets consisting of mag- netic elements and light elements have a large magnitude ofSP ratios according to the same mechanism as that of Fe 4N. In addition, when such ferromagnets are used as the FMEs,FME/insulator/FME junctions may exhibit large MR ratios,although the MR ratios are often inﬂuenced by the interfacialstates and materials of the insulator. In fact, CoFeBelectrodes bring about a very large MR effect inCoFeB/MgO/CoFeB junctions. 5,6 In conclusion, /H20841P/H20849EF/H20850/H20841of Fe 4N was evaluated to be almost 1.0, which was about 5.0 times as large as that of bcc-Fe and about 2.5 times as large as that of fcc-Fe. In comparison withfcc-Fe, it was shown that the large magnitude of the SP ratiooriginated from the contribution to the transport of 3 dbands, which was enhanced by introducing N. We anticipate thatFe 4N will become an electrode with a high efﬁciency of spin injection. Furthermore, various ferromagnets consisting ofmagnetic elements and light elements may exhibit highlyspin-polarized transport due to the present mechanism. The authors thank T. Hoshino of Shizuoka University for useful discussions. One of the authors /H20849S.K. /H20850also thanks members of the nanomaterials theory group, AIST, for valu-able discussions. This work has been supported by a com-petitive Grant program 2005 of Shizuoka University. Electronic address: tskokad@ipc.shizuoka.ac.jp 1K. Inomata, S. Okamura, R. Goto, and N. Tezuka, Jpn. J. Appl. Phys., Part 2 42, L419 /H208492003 /H20850. 2Y. Sakuraba, J. Nakata, M. Oogane, H. Kubota, Y. Ando, A. Sa- kuma, and T. Miyazaki, Jpn. J. Appl. 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PhysRevB.96.155210.pdf	PHYSICAL REVIEW B 96, 155210 (2017) Spatially resolved electronic structure of an isovalent nitrogen center in GaAs R. C. Plantenga,1V . R. Kortan,2T. Kaizu,3Y . Harada,3T. Kita,3M. E. Flatté,1,2and P. M. Koenraad1 1Department of Applied Physics, Eindhoven University of Technology, P .O. Box 513, 5600 MB Eindhoven, The Netherlands 2Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA 3Department of Electrical and Electronic Engineering, Graduate School of Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan (Received 31 May 2016; revised manuscript received 17 July 2017; published 27 October 2017) Small numbers of nitrogen dopants dramatically modify the electronic properties of GaAs, generating spatially localized resonant states within the conduction band, pair and cluster states in the band gap, and very large shiftsin the conduction-band energies with nonlinear concentration dependence. Cross-sectional scanning tunnelingmicroscopy provides the local electronic structure of single nitrogen dopants at the (110) GaAs surface, yieldinghighly anisotropic spatial shapes when the empty states are imaged. Measurements of the resonant states relativeto the GaAs surface states and their spatial extent allow an unambiguous assignment of speciﬁc features tonitrogen atoms at different depths below the cleaved (110) surface. Multiband tight-binding calculations aroundthe resonance energy of nitrogen in the conduction band match the imaged features, verifying that the Green’sfunction method can accurately describe the isolated isovalent nitrogen impurity. The spatial anisotropy isattributed to the tetrahedral symmetry of the bulk lattice and will lead to a directional dependence for theinteraction of nitrogen atoms. Additionally, the voltage dependence of the electronic contrast for two features inthe ﬁlled state imaging suggests these features could be related to a locally modiﬁed surface state. DOI: 10.1103/PhysRevB.96.155210 I. INTRODUCTION In GaAs, the addition of small amounts of nitrogen can have a strong effect on the electronic properties. At concentrationswell below 1% (i.e., <10 19cm−3), nitrogen induces a localized state which gives a strong narrow line in optical measurements,corresponding to an energetic feature 150–180 meV above theconduction-band edge [ 1–5]. Thus, unlike for nitrogen in GaP, the resonance level of a single nitrogen lies in the conduction band of GaAs. Additional narrow lines have been found inthe photoluminescence spectra, and assigned to other statesinvolving the single nitrogen, N-N pairs, and N clusters, severalof which are situated in the band gap [ 6–21]. From the rich spectra [ 3], the appearance of the lines with different growth conﬁgurations [ 10,13] and dependence of the signal on polarization and magnetic ﬁeld [ 17,18,20] it was sus- pected that nitrogen pairs give rise to different electronic levelsdepending on the relative orientation of their two constituents.The resonant states thus obtained are not only interesting assingle-photon sources with many available energies. Resonantstates have also gained interest as probes of localized centersby transport measurements as demonstrated, for example, inRefs. [ 22–24]. The study of dopant complexes plays a major role in quantum spintronics as well, as studies of states beyondthe NV center in diamond, or the divacancy center in siliconcarbide, would beneﬁt from these types of studies. Further alteration of the electronic structure of GaAs can be seen for nitrogen at low alloying concentrations (typically0.1%–2.0%) at which the band gap of GaAsN is reduced upto 600 meV below that of GaAs [ 25–27] despite the band gap of GaN exceeding that of GaAs by a factor of 2. Strongband bowing, common for highly mismatched alloys, has beenattributed to the hybridization of the localized nitrogen stateswith the GaAs conduction band [ 28,29] or the formation of a continuum of localized states forming an impurity band[6,30,31].Multiple theoretical approaches have produced estimates of the energy levels of the single N impurity, N-N pairs, andNc l u s t e r s[ 6,28–43], including tight-binding calculations, em- pirical pseudopotential calculations, band anticrossing models,and density functional theory. The band anticrossing model[28,29] is very successful in explaining the observed trend for band-gap narrowing at low nitrogen content, but is aquasiperiodic theory and thus does not address the spatialstructure of individual nitrogen dopants. Several supercell cal-culations were performed [ 35–38,40–42], of which Ref. [ 42] is the most extensive, including the energy of the single Nlevel and N-N pair levels, the band gap over the full nitrogenconcentration range, and a prediction for the spatial extent of the wave function. A strong directional dependence of the N-pair levels was found here as well. However, even though thesupercells are almost 7 nm in linear size, signiﬁcant ﬁnite-sizeeffects have been found even for these sizes for well-hybridizedstates close to the conduction band or resonances within theconduction band [ 44]. Additionally, ab initio atomistic calculations, like density functional theory calculations, are unable to derive the long-ranged electronic structure that follows from the mixing withthe bulk dispersion, as the dispersion relation is not accuratelyrepresented. Tight-binding calculations with open boundaryconditions, based on the Koster-Slater formalism in termsof Green’s functions, do take the dispersion relation intoaccount and are not inﬂuenced by boundary effects foundin supercell calculations. Although similar calculations havebeen performed for the single nitrogen impurity in GaAs [ 43], no spatial structure was derived and no veriﬁcation of thisspatial structure with experiments was done. Here, we use cross-sectional scanning tunneling mi- croscopy (X-STM) to directly image the local density of states(LDOS) related to the resonant level of the individual nitrogenin the conduction band. X-STM has been applied successfullyin the past to image nitrogen atoms in GaAs as atomically 2469-9950/2017/96(15)/155210(8) 155210-1 ©2017 American Physical SocietyR. C. PLANTENGA et al. PHYSICAL REVIEW B 96, 155210 (2017) sized features and determine the distribution of nitrogen in various structures such as quantum wells [ 45–49]. However, until recently little attention has focused on the imaging of theelectronic state of nitrogen [ 50]. By giving a full description of the appearance of the single nitrogen impurity, we aim not only to develop a robust wayof identifying nitrogen features in more complex systems,but we want to obtain better understanding of the originof the directional dependence of nitrogen pairing states.Furthermore, we want to use the STM as a method to verifythat the Green’s function tight-binding method can give ahigh-quality description of the single nitrogen and as suchcan be applied to calculations as have been presented, forexample, in Ref. [ 43] on band bowing. In those calculations, a linear combination of isolated nitrogen states (LCINS) ismade to make an estimate of the spectral distribution of thenitrogen band. Indeed, our results here show there is a goodmatch between the experimentally observed and calculatedappearance of the single nitrogen and hence conﬁrm that tight-binding calculations following the Green’s function formalismfor the single nitrogen provide a good basis for the LCINS. II. METHOD The sample was grown at 550◦Cb yM B Eu s i n ga1 .1× 1018cm−3n+-doped GaAs wafer as the substrate and nitrogen from a radio-frequency plasma source with ultrapure N 2gas [17,19]. After a 400 nm buffer layer of 1 .0×1018cm−3 n-doped GaAs, 3 nm Al 0.3Ga0.7As marker layers and N layers were alternately grown, starting and ending with anAl 0.3Ga0.7As layer, with GaAs spacer layers in between of at least 35 nm. N layers were deposited by stopping the Gaﬂux and opening the N ﬂux for 2000 s. During the nitridation,the As 2ﬂux was kept at the same ﬂux as during the growth (1.0×10−6Torr). Growth was recommenced 120 s after stopping the N ﬂux. After the last marker layer, a 250 nm GaAscapping layer was grown. The nitridation was monitored byreﬂection high-energy electron diffraction. The X-STM measurements were performed bringing the sample in ultrahigh vacuum (UHV , pressure typically around5×10 −11Torr) and cleaving the sample there, revealing a (110) plane. The sample was then cooled to 77 K. STMtips were made from electrochemically etching a tungstenwire, which was then further sharpened and cleaned by argonsputtering in vacuum. Sample bias was varied per experiment,while currents were kept between 10–50 pA. Images weretaken in constant current mode. Illumination of the samplewas used to create charge carriers in the regions between theAlGaAs barriers. III. RESULTS AND DISCUSSION The position of the nitrogen layer is easily traceable with the help of the AlGaAs marker layers. The marker layers arenot shown in the images here since they are located 40 nmor further away. The nitrogen layer was imaged at varioussample bias voltages (see Fig. 1). At negative sample bias, dark spotlike features and two types of features containingbright contrast (indicated B1 and B2) can be observed [seeFig. 1(a)]. Dark contrast features have been observed for FIG. 1. STM topography images showing a part of the nitrogen layer at (a) −1.90 V , (b) +0.60 V , (c) +1.00 V , and (d) +1.40 V sample bias. The tunnel current was 50 pA at negative sample biasand 30 pA at positive sample bias. The contrast scales are normalized to the amplitude of the atomic corrugation (giving a scale ranging over 29 pm for the lowest contrast picture and 80 pm for the highestcontrast picture). B1 and B2 indicate the features containing bright contrast at −1.90 V in (a). In (b), the features are labeled 0 to 4, corresponding to the layer below the surface in which the nitrogen is located. “d” indicates features related to nitrogen situated deeper below the cleavage surface. nitrogen in X-STM in the past with ﬁlled state imaging [ 45–48] and have been reproduced in theoretical models [ 51,52]. The observation of dark contrast is attributed to a depressionat the surface caused by the shortened bonds between thenitrogen and its neighboring Ga atoms. These dark spots showa variation in intensity due to the variation in depth at whichthe nitrogen atoms are positioned with respect to the surface.The nitrogen atoms located deeper below the surface give riseto less distortion at the surface and thus a weaker dark contrast. The features B1 and B2 always occur around the nitrogen layer. Therefore, we propose that these must be nitrogen relatedas well. Both features show a periodic pattern along the [110]direction, forming barlike contrast in the [001] direction. Theextent of both features differs, with the B1 feature extendingabout three rows in two directions and the B2 extending at leastﬁve rows in both directions. In previous X-STM measurementsat room temperature [ 45–48] these features were not observed. A recent publication on X-STM measurements at 77 K reportsa feature similar to B1 and B2, although the structure of thefeatures was less clearly resolved [ 50]. Discussion of both B1 and B2 will be continued later. When imaging at +0.60 V sample bias [see Fig. 1(b)], various types of bright features can be distinguished having acomplex structure with a strong anisotropy between the [110]and [001] directions. At positive sample bias, these features 155210-2SPATIALLY RESOLVED ELECTRONIC STRUCTURE OF AN . . . PHYSICAL REVIEW B 96, 155210 (2017) are located at the exact positions of the dark and bright features seen at −1.90 V . Similar features have been observed by Ishida et al. [50]. As will be discussed later, these features are related to nitrogen substituting for arsenic atoms in different planesbelow the surface, where the labels 0 to 4 indicate the distancefrom the (110) cleavage surface in number of planes. Unlikethe dark features at −1.90 V , all of these bright features are caused by increased tunneling current due to an enhanced localdensity of states (LDOS) at a resonance energy rather than thetopography. At a higher voltage of +1.00 V [see Fig. 1(c)], the 0 feature develops a strong contrast directed along the [110] direction.The structure of the other features becomes more condensedwhile the anisotropy in the [001] direction is preserved. At+1.40 V [see Fig. 1(d)], the only bright contrast is observed at the 0 feature. The other features with bright contrast at lowerpositive voltages now show a strong dark contrast, that unlikethe localized features at negative voltage spread over multipleatomic positions. At +1.40 V , the observed image deviates strongly from the observed topography at negative voltagesand hence the dark contrast is attributed to an electronic origin.A reduction of LDOS to compensate for the enhancement ofthe LDOS at the nitrogen resonance energy is suspected. Thenitrogen is an isoelectronic impurity and therefore does notintroduce additional density of states when considering theintegral over all energies and space. Hence, a local increase of LDOS at a speciﬁc energy has to be compensated elsewhere in space and energy. Comparing the measurements at positive voltages, we see that the contrast intensity of the features varies with the appliedsample bias. A different voltage dependence is seen per planethe nitrogen is situated in; the features related to the surface(0) hardly show any contrast at +0.60 V [Fig. 1(b)], while for the fourth layer features (4) a clear contrast can be seen. At+1.00 V [Fig. 1(c)], this is reversed. To quantify this behavior the highest contrast associated with this feature, normalized tothe atomic corrugation found on clean GaAs observed at thesame voltage, was determined for each feature for at least sevenvoltages in the range +0.40 to +1.60 V (see Supplemental Material S1 [ 53] and inset Fig. 2). For every depth of the nitrogen below the surface, including the surface nitrogen, aresonance in the voltage-dependent contrast intensity is found.Ishida et al. also suggested that a trend of resonance voltage with depth could be present as they saw a shift in the differentialtunneling conductance spectra when comparing the fourthand seventh layer features [ 50]. A V oigt proﬁle is used to get an estimate of the voltage at which the resonance occurs(see Supplemental Material S1 [ 53]). In Fig. 2, the estimated resonance is plotted against the layer in which the nitrogenresides. A large jump in resonance voltage can be seen whengoing from the surface nitrogen to the ﬁrst layer nitrogen, whilea more gradual drop is observed for deeper lying nitrogen. It isimportant to realize that the energy position of the resonanceswill be inﬂuenced by tip-induced band bending (TIBB). Theelectric ﬁeld following the voltage on the tip pulls up theGaAs bands as well as the nitrogen resonance. The TIBBis strongest at the surface and decays away from the surface,hence, features close to the surface align with the Fermi energyof the tip at higher voltages than features do farther away fromthe surface. We propose that the observed trend seen for theFIG. 2. The voltage of the estimated resonance for the normalized intensity per nitrogen feature from a certain layer. Nitrogen in the top and in four subsequent layers are considered. The error bars show two times the standard error for each ﬁt. The inset shows an example ofthe ﬁt made to estimate the resonance voltage for the contrast of the surface nitrogen (see Supplemental Material S1 for other ﬁts [ 53]). features below the surface is mostly from TIBB, although energy shifts due to deviations from the distortions around an impurity close to the surface with respect to the bulk lattice can not be fully excluded. However, the jump we see for the surfacenitrogen is most likely not mainly as a result of the TIBB.We expect that the surface state has a signiﬁcantly differentenergy caused by the strongly modiﬁed atomic coordinationenvironment at the surface [ 54]. Our resonant voltages differ in absolute value with the STS results of Ishida et al. and Ivanova et al.I nR e f s .[ 49,50]t h e peak position of the single nitrogen level in general showsup between +1.2 and +1.7 V . As illustrated well by the spectroscopy series of Ishida et al. , the observed values in STS are dependent on the tip condition, hence, we are reluctant tocompare the exact values resulting from STS. The differenceobserved with our value is most probably due to the use ofillumination and a more intrinsic environment in our case.Illumination creates charge carriers, partially quenching theTIBB [ 55], while the doping inﬂuences the positioning of the Fermi level. Ishida et al. do not ﬁnd a deﬁned peak in their spectrum for the resonance of the nitrogen in the surface position, whileIvanova et al. indicate that they do not expect a contribution of the surface nitrogen state at all in STS. The resonance thatwe found with constant current imaging at various voltagesfor nitrogen in the surface layer (see inset Fig. 2)s h o w sa peak that is broader than found for most nitrogen in otherlayers. Together with the observation that the resonance ofthe surface nitrogen lies 0.5 V above the next resonance, thiscould be an indication why no peak was observed in earlierSTS measurements; either the peak is smeared out too much,and thus not showing up clearly on top of the regular LDOS,or the peak position is shifted up outside the measured region,or a combination of both. In order to identify the position of the nitrogen atoms in the lattice, it is necessary to determine their position with 155210-3R. C. PLANTENGA et al. PHYSICAL REVIEW B 96, 155210 (2017) FIG. 3. (a) Five nitrogen-related features found at +0.75 V (upper) sample bias, organized by extent in the [001] direction, and their counterparts at −1.90 V (lower). The contrast for the third layer feature at −1.90 V was adjusted to span 17 pm more (the corrugation at −1.90 V spans ∼10 pm) to see the ﬁner structure of the bright contrast. The cross hairs indicate the position of the feature’s center with respect to the surface resonances. Note that a second nitrogen feature is present in the panel depicting the ﬁrst layer feature at −1.90 V . Only the centered feature is associated with the ﬁrst layer. (b) Slice of the surface scanned in the X-STM along a (110) plane, showing in side viewthe positions a nitrogen atom can take when substituting for the arsenic atom. (c) View from the top onto the surface scanned in X-STM, and the two possible in-plane positions for the nitrogen on substitutional sites indicated with a blue dot with dashed circle. The stronger lines and bigger atoms indicate the elevated zigzag rows in the surface. (d), (e) Show the relative position of these projections to the surface resonancesimaged at negative and positive voltage, respectively. respect to the gallium and arsenic atoms on the surface. As can be seen from Fig. 1, the corrugation on the undoped GaAs surface depends on the applied sample bias. At −1.90 and +1.00 V , a two-dimensional (2D) atomic grid can be observed, whereas at +0.60 V stripes directed along [110] are seen and at+1.40 V stripes directed along [001] are seen. The observed stripes are attributed to surface states that arise after the surfacerelaxation and strongly correlate to the position of the surfaceatoms [ 56,57]. We follow labeling of Refs. [ 56,57] in which the various surface states that were calculated were labeled A or C,according to whether the state was mainly related to the anion(A) or cation (C) sites. The maximum contribution of each ofthese surface states lies at a different energy, from which thenumbering is derived. The spatial contribution of these statesis schematically indicated in Fig. 3. Around the bottom of the conduction band the C3 state is found to be dominant, givingrise to stripes which are directed along [001] and centered ontop of the dangling bonds of the surface gallium atoms. At lowpositive voltages we inject into the lower part of the conductionband via the C3 state, therefore, at +0.60 V stripes along [001] are observed. At high positive voltages, like +1.40 V , we inject into the conduction band via the C4 surface resonance state,which is centered on the dangling bonds of gallium as well, but is now directed along [110]. Thus, we see a voltage-dependentcorrugation. The measurement at +1.00 V shows a 2D grid for the corrugation because at this voltage the C3 and C4 statescontribute with similar weight. For the measurements at −1.90 V , a 2D grid is observed as well. This is remarkable because at negative voltages electronsare drawn from the valence band, which lines up with themaxima of the A4 and A5 surface states. The A4 and A5 statesare centered around the arsenic surface atoms and are bothdirected along [110]. As was reported by de Raad et al. [58], due to TIBB it is possible to observe contributions from theC3 mode also at negative voltages when tunneling close to thegap. Therefore, at −1.90 V a 2D grid is formed from the C3 state and the A5 state, whose maximum is located closer to thegap than that of the A4. After taking into account the atomic corrugation for the clean GaAs surface, the nitrogen features can be classiﬁedaccording to their position below the (110) surface. In theimage Fig. 1(b) and many other images, at least ﬁve different contrast varieties can be observed. Figure 3(a) shows these features at +0.75 V . Arranging these features by the extent of 155210-4SPATIALLY RESOLVED ELECTRONIC STRUCTURE OF AN . . . PHYSICAL REVIEW B 96, 155210 (2017) the bright contrast in the [001] direction we see that their centers alternatingly fall on top or in-between the imagedatomic grid [see the top row in Fig. 3(a)]. At−1.90 V , the feature centers show an alteration of position with the imagedgrid as well [see the bottom row in Fig. 3(a)]. Their positions with respect to the surface state along [001] are the same asobserved for the features at +0.75 V . However, the features that are centered on top of the maxima of the surface statealong [110] at +0.75 V fall in between the state directed along [110] at −1.90 V and vice versa. The nitrogen atom normally substitutes for an arsenic atom. Nitrogen atoms positioned at even numbered layers [seeFig.3(b)] will create a depression centered on the position of a surface arsenic atom [see Fig. 3(c)]. Hence, at −1.90 V the even numbered nitrogen atoms will show up on the A5 surfaceresonance in the [110] direction and between the C3 resonancein the [001] direction [see Fig. 3(d)]. Nitrogen atoms on the arsenic positions in the odd numbered layers [see Fig. 3(b)] are not directly imaged, but will cause a distortion distributedover multiple arsenic atoms with the center of the contrastin between the surface arsenic atoms, hence in between thestripes due to the A5 state. From Fig. 3(d) it can also be seen that these odd numbered features fall in line with the Gaatoms along [001] and therefore will be imaged on the C3grid. We conclude that the states labeled with 0, 2, and 4 areindeed related to the even numbered substitutional sites and the features labeled 1 and 3 to the odd numbered substitutional sites shown in Fig. 3(b). The images at +0.75 V have the same C3 surface state making up the rows along [001]. The center of the featureshence have a similar position with respect to the [001] rows.The C4 surface states directed along [110] are not centeredon the surface gallium atoms but centered on the galliumdangling bonds. This places the maximum integrated LDOSof the surface states next to the gallium atoms and between thezigzag rows [ 56,57] [see Fig. 3(e)]. The odd numbered states will coincide more with the rows of the C4 states and the evennumbered sites will fall between them, producing the observedalternation in contrast with depth. The regularly increasingextension of the features combined with the arguments forthe intensity of the dark features at negative voltage and thepositioning of the features on the grid leads to the conclusionthat our labeling of “0” to “4” corresponds to the ordering indistance of the nitrogen atom from the cleavage surface. In order to further investigate the depth dependence of the observed features, tight-binding (TB) calculations, similar tothose in Ref. [ 59], were performed for a single nitrogen atom in an effectively inﬁnite GaAs crystal. Previous calculationsof the spatial structure of nitrogen-related states in GaAshave been performed with density functional theory [ 31,42] with a reduced set of kpoints and a ﬁnite supercell. Here, asp 3d5s∗Hamiltonian [ 60] is used to describe the GaAs crystal and the Koster-Slater method [ 61] is used to include the effect of the nitrogen. This Green’s function method iscomputationally efﬁcient and does not suffer from supercellsize restrictions or boundary effects. The energy of the nitrogenresonance has been set by including an onsite atomic potentialequal to the difference in s- and p-state energies between nitrogen and arsenic. The bonds between the nitrogen and itsnearest neighbors have then effectively been shortened usingHarrison’s d −2scaling law [ 62] to place the nitrogen resonance at 1.68 eV . This energy is deﬁned as the energy from the valence band edge and is derived by adding 165 meV , themiddle of the 150–180 meV range found as the distance withrespect to conduction-band edge for the isolated nitrogen level[1], to the 1.52-eV band gap of GaAs at 0 K [ 63]. Figure 4(b) shows the calculated LDOS in (110) planes at various distances from the center of the nitrogen atom at thisresonance energy. The ﬁrst column is the slice through theplane containing the nitrogen atom, which would correspondwith a nitrogen in the top layer of the sample surface measuredin STM. The second column then shows the slice displacedone atomic plane from the nitrogen atom corresponding to thecontrast measured for the nitrogen in the ﬁrst layer [label 1 in Fig.3(b)] and so forth. The calculations show a barlike feature extending along the [110] direction for the nitrogen in thezeroth/top layer and crosslike features extending in the [001]direction for cuts away from the center with an asymmetrybetween the two lobes. For each layer deeper into the GaAsthe enhanced LDOS cross section expands an additional rowin the [001] direction. Comparing the TB calculations to the measured contrast shows an excellent agreement. The series of Fig. 4(a)shows the same systematic increase of one row of bright contrast in the [110] direction as the calculations show in Fig. 4(b)for each cut one monolayer further away from the nitrogen atom. A strikingresemblance between calculation and measurement is found inthe direction of extension of the nitrogen-related LDOS shapewhich at the surface (or zeroth plane) extends in the [110]direction, while the features from other planes extend in the[001] direction. The best correspondence between measure- ments and calculations is obtained with a tip width of 1.70 ˚A. As can be seen from Figs. 4(c) and4(d), the calculated isosurfaces of state density have highly anisotropic shapes. Thetip width in the calculation has been chosen smaller, 1.13 ˚A, to make the ﬁner features of the isosurface clearer. At a highvalue of the density of states the tetragonal symmetry closeto the nitrogen center is recognizable. The isosurface at lower density further away shows somewhat of a preference for the /angbracketleft110/angbracketrightdirections, similar to what was reported by Virkkala et al. [31], but far less localized. The panels in Fig. 4(b) show the LDOS in parallel (110) planes that either cut throughthe N atom, as is the case for the zeroth plane, or at aninteger number of atomic planes away from the N atom. Inthe zeroth plane, the LDOS is mainly related to the two arms of the 12-fold symmetric electron density that lie in the (110) plane cutting through the N atom. In a plane that is a fewmonolayers away from the N atom the atomic-sized LDOS ismainly due to the arm that is pointing in the [110] direction,i.e., perpendicular to the arms in the zeroth plane. In planesat intermediate distances away from the N atom, the LDOSconsists of the perpendicular [110] arm and four others arms of the 12-fold symmetric state that cut at an angle with the (110) plane. Strongly anisotropic shapes for the LDOS alongthe [001] have been reported for several acceptor impuritieswith levels in the band gap [ 64,65] including Mn [ 66,67]. The anisotropy seen in acceptor states comes from the symmetry ofthe tetrahedral bonds in the cubic lattice and the contributingorbitals, namely, the dorbitals with T 2symmetry and the p 155210-5R. C. PLANTENGA et al. PHYSICAL REVIEW B 96, 155210 (2017) FIG. 4. (a) Measured contrast at +0.75 V for the features in the zeroth/top layer and below. (b) The calculated LDOS for nitrogen atoms in the zeroth/top layer and below, with an assumed STM tip width of 1.70 ˚A. The panels for calculation and measurement are about 3 .6×3.6 nm. (c), (d) Show the calculated isodensity surfaces at 1.68 eV for the difference between the LDOS near the dopant and the background. To view sharper features in this theoretical surface, the STM tip width is made smaller, 1.13 ˚A. (c) Corresponds to an LDOS difference from the background of 0 .12 eV−1˚A−3. (d) Corresponds to an LDOS difference from the background of 0 .012 eV−1˚A−3. orbitals which also have T2symmetry. Although nitrogen is an isoelectronic impurity, the same applies to the bonding ofthe nitrogen in which porbitals are contributing. Therefore, in opposition to Ref. [ 31], we argue that the observed anisotropy of the nitrogen atom is mainly attributed to the symmetry of the surrounding electronic environment and not the strainintroduced into the lattice. The high anisotropy of the nitrogen-related states will have consequences for the interaction between nitrogen centers.As the LDOS is a measure for the amplitude of the wavefunction, the interaction in the directions where the LDOSis high will be stronger than in the directions the LDOS islow. For example, the overlap between two nitrogen atomsspaced 0.5 nm away in the [001] direction is predicted to beless than two nitrogen atoms spaced 1.0 nm away in one ofthe/angbracketleft110/angbracketrightdirections. Furthermore, it is shown that the extent of the nitrogen is much larger than its atomic position, dueto hybridization with the lattice. Interactions beyond nearestneighbors are therefore expected. The larger extent and highdirectional dependence of the energy levels formed from twointeracting nitrogen levels matches earlier calculations [ 42] and observations [ 10,13,17,18,20]. Coming back to the comparison to earlier investigated ac- ceptor states, we note that for acceptor atoms that give rise to alocalized state in the band gap, the spatial integral of the LDOSproduces an integer value. The nitrogen atoms, however, formresonances within the continuum of the conduction band thatlocally increase the density of states around the resonanceenergy. The integral of this region of enhanced LDOS doesnot have to be an integer. The nitrogen atoms have the samenumber of valence states as arsenic atoms, so the increasein the LDOS at the resonance energy has to be compensatedfor through a corresponding reduction at other energies. Thelocal nature of the nitrogen is attributed to its small size andhigh electronegativity. Studies on isoelectronic substitutionalimpurities, like boron, which is also electronegative comparedto gallium, as well as antimony and bismuth, which cause astrong distortion of the lattice, could provide further insightson the formation of these localized isoelectronic states. Aprerequisite to observe the states related to such a center,isoelectronic or not, with X-STM is that the increase in LDOS is localized enough in energy and space to signiﬁcantly changethe tunneling current. The small deviation between our calculations and measure- ments is due to the fact that the calculations are done for abulk system, whereas in the experiment the nitrogen atoms areclose to a semiconductor-vacuum interface. The surface willreconstruct, deforming the layers close by and putting strainon them [ 57]. The slight additional asymmetry along the [001] direction may also be explained by surface strain, as seen forMn acceptors [ 68]. Moreover, the deformation of the lattice around the nitrogen is only taken into account by changing thehopping parameters to the direct neighbors, without actuallychanging the lattice positions of the nearest neighbors oraccounting for changed bond lengths to the next-nearestneighbors. The nitrogen will, however, deform the lattice evenbeyond the ﬁrst neighbors [ 31] and close to the surface this will happen in a spatially anisotropic way [ 52]. These strain arguments are very likely the cause of the observed deviationsand they are consistent with the observations that the featuresassigned to nitrogen further away from the surface are moresymmetric and match the calculations better. The calculated shape of the LDOS does not provide an explanation for the two special bright features, B1 and B2,observed at negative voltages (see Fig. 1). At−2.50 V , the bright contrast of the mixed feature B1 disappears, while thedark contrast remains unaltered (see Supplemental MaterialS2 [53]). The dark contrast is in accordance to what has been reported before for the topographic contrast of a ﬁrst layerfeature [ 48,51,52]. The center is positioned around a point falling in between the A4 /A5 surface states and shows a dark contrast over multiple surface arsenic atoms. This stronglysuggests that the observed contrast at −1.90 V is a mix of the expected topographic contrast and a bright contrast element.Feature B2 is associated with the third layer away from thecleavage surface. The expected topographic contrast for anitrogen in this position is less strong and falls underneaththe strong central bar of the bright pattern. We suggest this asthe reason why at −1.90 V no topographic contrast element is recognizable for this feature. 155210-6SPATIALLY RESOLVED ELECTRONIC STRUCTURE OF AN . . . PHYSICAL REVIEW B 96, 155210 (2017) The B features become less pronounced with more negative voltages (see Supplemental Material S3 [ 53]), whereas the topography should dominate more when tunneling furtherfrom the band gap. Therefore, it is likely that the brightcomponent of the contrast stems from an electronic ratherthan a topographic origin. Noticeably, the bright contrast element only appears with the nitrogen atoms substituting in the odd numbered positions.Tilley et al. [52] calculated that the nitrogen atoms in those positions displace multiple arsenic atoms in the surface. V oltage-dependent measurements on the B2 feature (see Supplemental Material S3 [ 53]) suggest that the visibility of the state is related to the visibility of the C3 surfacestate. At less negative voltages, the C3 surface state as wellas the B2 feature are more pronounced. At a low positivevoltage of +0.45 V where the C3 mode dominates, a feature very similar to the one observed at negative voltages can beobserved (see Supplemental Material S3.2 [ 53]). Therefore, we propose the bright contrast of the B features might berelated to a disturbance of the C3 surface state. This would beconsistent with the larger spread of the B2 feature comparedto the B1 feature. The disturbance of the surface due to thedeeper lying nitrogen atom would be more extended andthat of the shallower nitrogen atom would be more localized.Remarkably, the B2 feature shows a brighter overall contrastthan the B1 feature. IV . CONCLUSIONS We performed X-STM measurements on individual ni- trogen impurities in GaAs layers grown with MBE. Theimpurities were studied at different voltages. At negativevoltages, mainly topographic contrast appeared. At low posi-tive voltages, highly anisotropic bright shapes were observedwhich show voltage-dependent brightness. The resonant tun-nel voltage changes with the layer the nitrogen is situatedin due to TIBB, with exception of the surface nitrogen thatlikely shows an additional increase in energy on account of ahighly disrupted symmetry in bonding with respect to the bulk.The anisotropy of the isolated nitrogen state can be identiﬁedas a cause for the high directional dependence of the levelsstemming from nitrogen pairing. At higher positive voltages, less-deﬁned dark shapes are observed, unlike the topographiccontrast observed at negative voltages. This we relate to adecrease of LDOS compensating the increase of LDOS atlower energy. Using the difference in extent of the observedfeatures at low positive voltages and the atomic corrugationcoming from the voltage-dependent surface states, the featurescan be assigned to nitrogen at different planes below thecleavage surface. TB calculations give similar anisotropicenhanced LDOS at (110) cuts through and next to the nitrogencenter. The excellent match veriﬁes experimentally that the TBcan describe the nitrogen center accurately and can be used as abase for further calculations combining single nitrogen states.These results show that the delocalized anisotropic appearanceof the LDOS originates from the symmetry of the GaAs latticerather than long-ranged strain. Minor deviations between theexperimental and theoretical contrast can still be attributed tothe deformation of the lattice, caused by surface relaxation andthe small nitrogen atom, which is only partially accounted forin these bulk calculations. The hybridization of the nitrogenimpurity center with the bulk dispersion makes the extentof the enhanced LDOS carry much further than the atomicposition of the nitrogen impurity, making interactions plausibleover larger ranges than ﬁrst neighbors expected from simpleatom-to-atom-bonding considerations. 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PhysRevB.97.045308.pdf	PHYSICAL REVIEW B 97, 045308 (2018) Anomalous helicity-dependent photocurrent in the topological insulator (Bi 0.5Sb0.5)2Te3on a GaAs substrate Dong-Xia Qu,1,Xiaoyu Che,2Xufeng Kou,2,†Lei Pan,2Jonathan Crowhurst,1Michael R. Armstrong,1 Jonathan Dubois,1Kang L. Wang,2and George F. Chapline1 1Lawrence Livermore National Laboratory, Livermore, California 94550, USA 2Department of Electrical Engineering, University of California, Los Angeles, California 65409, USA (Received 21 July 2017; published 24 January 2018) The emerging material, topological insulator, has provided new opportunities for spintronic applications, owing to its strong spin-orbit character. Topological insulator based heterostructures that display spin-charge couplingdriven by topology at surfaces have great potential for the realization of novel spintronic devices. Here, we reportthe observation of anomalous photogalvanic effect in (Bi 0.5Sb0.5)2Te3thin ﬁlms grown on GaAs substrate. We demonstrate that the magnitude, direction, and temperature dependence of the helicity-dependent photocurrent(HDPC) can be modulated by the gate voltage. From spatially resolved photocurrent measurements, we showthat the line proﬁle of HDPC in (Bi 0.5Sb0.5)2Te3/GaAs is unaffected by the variation of beam size, in contrast to the photocurrent response measured in a (Bi 0.5Sb0.5)2Te3/mica structure. DOI: 10.1103/PhysRevB.97.045308 I. INTRODUCTION One of the recent advances in spintronics is the development of nonmagnetic systems for a functional spintronic technologywithout the application of magnetic ﬁelds [ 1–3]. The primary driving force of this research direction is the relativistic spin-orbit coupling that links an electron’s momentum to its spinand has led to a variety of new device paradigms, such as spinHall effect transistors [ 2], spin-orbit torque memories [ 4], and broadband terahertz emitters [ 5]. Topological insulator (TI), a new class of quantum state of matter, is a promising material forspintronic devices owing to its exceptionally strong spin-orbitcoupling [ 6–12]. One manifestation of strong spin-orbit coupling in a TI is the circular photogalvanic effect (CPGE). This effect isthe appearance of a DC electrical current under circularlypolarized light as a result of the optical spin orientation offree carriers. Optical experiments so far have focused on thephotogalvanic current measurements in TIs via the inverseEdelstein effect (IEE), in which a circular photocurrent ( J C) arises from the asymmetrical spin accumulation in momentumspace and the spin-momentum locking mechanism [ 13–19]. However, recent theoretical calculations predict that the sur-face photocurrent arising from the IEE is small, suggesting alimited spin-to-charge conversation efﬁciency [ 15]. Therefore, it is interesting to search for new device structures to enhancethe spin current generation. Here, we report that the generationof HDPC in a TI is substrate dependent and gate tunableunder near infrared radiation. Our work suggests that substratematerial may provide an order of freedom to inﬂuence thephoton-spin conversion efﬁciency. Corresponding author: qu2@llnl.gov †Present address: School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China.II. RESULTS A. Experimental details The samples used in this study are 6 ∼10 quintuple layers (QLs) of single-crystalline (Bi 0.5Sb0.5)2Te3(BST) ﬁlms, grown on semi-insulating GaAs (111)B or muscovite mica substrateby molecular beam epitaxy. The growth is monitored in situ by reﬂection high-energy electron diffraction (RHEED) and thethickness of the ﬁlm is inferred from the RHEED oscillationperiods. A 2 nm Al protective layer is deposited immediatelyafter the TI growth. The Bi:Sb ratio is adjusted to ﬁne tunethe Fermi level [ 20–22], so that it lies close to the Dirac point E Dfor 10-QL BST/GaAs [ 20]. The 10-QL BST/GaAs sample isptype, whereas the 6-QL BST/GaAs and 6-QL BST/mica samples are ntype. The thin ﬁlms are photolithographically patterned into Hall bar structures 130 μm long and 20 μm wide, with electrical contacts made from 10 nm Cr and 100 nmAu [Figs. 1(a) and1(b)]. The scanning photocurrent measurements are carried out with a liquid helium cooled probe station (Janis ST-500), in which a three-axis piezo-based nanopositioner (Attocube ANPxyz101/RES) is mounted on the cold ﬁnger in vacuum. The device is mounted onto the nanopositioner and obliquely illuminated with a thulium continuous-wave ﬁber laser beam(2.036μm, IPG Photonics) chopped at a frequency of 238 Hz. The incident angle is θ=30 ◦from the positive zaxis [Fig. 1(a)]. The photon energy (0.609 eV) is chosen to be much smaller than the band gaps of GaAs (1.42 eV) and mica (7.85 eV). Hence, optical absorption primarily occursin BST (Appendix A). To accurately read the temperature of the sample, we anchor a silicon sensor in close proximity to the substrate of the device. The polarization of the linearly polarized excitation beam is modulated by rotating a quartz λ/4 waveplate with a period from right-circularly polarized /Omega1 −, to linearly polarized ↔, to left-circularly polarized /Omega1+, and back to linearly polarized ↔. The beam is consequently focused through an objective with a numerical aperture of 0.24 2469-9950/2018/97(4)/045308(9) 045308-1 ©2018 American Physical SocietyDONG-XIA QU et al. PHYSICAL REVIEW B 97, 045308 (2018) ΩΩ μΩ ΩΩ Ω ΩΩ μΩ+ ΩΩ μΩ ΩΩ Ω ΩΩ μΩ+Ωθ(a) (b)(c) (d) (e) (f) (g) FIG. 1. Sample geometry and results of scanning photocurrent measurements. (a) Schematic of the experimental geometry. (b) The optical image of a representative BST/GaAs sample patterned into a Hall bar structure. The red (black) triangle marks the laser excitation positionin sample G1 (M1) with the corresponding J y/Ishown in panel (c) [panel (d)]. (c),(d) Jy/Iversus photon polarization for sample G1 at y=− 65μm (c) and sample M1 at y=− 36μm (d). The polarization of the light is modulated with a period from right-circularly polarized /Omega1−, to linearly polarized ↔, to left-circularly polarized /Omega1+, and back to linearly polarized ↔. The solid green lines are ﬁts as explained in the text. The solid blue curve is extracted circular photocurrent component JC=Csin 2α. (e),(f) Scanning photocurrent images for sample G1 (e) and M1 (f) under linearly polarized light. Dashed and solid lines indicate the edges of the sample and electrodes. (g) Spatial proﬁles of JC/I along the yaxis for samples G1, M1, and G2 at room temperature. (Mitutoyo, 5X), which focuses the laser beam to a circular spot with a full width at half maximum of 104 μm. The reﬂected light from the sample is imaged using a CCD camera, which allows us to locate the spot position. Electrical transport measurements are conducted using a pre-ampliﬁer (SR570), a lock-in ampliﬁer (SR830), and a digital source meter (Keithley 2400), while illuminating the device at various temperatures. B. Helicity-dependent photocurrent in TI/GaAs and TI/mica We ﬁrst measured the unbiased photocurrent Jyas a function of light polarization performed in the linear regime with a lowlaser intensity I(Appendix B). In Figs. 1(c) and1(d),w ep l o t the raw traces of J y/Iversus photon polarization (black circles) in the 6-QL BST/GaAs sample G1 at position y=− 65μm, corresponding to the edge of the channel [red triangle inFig. 1(b)], and 6-QL BST/mica sample M1 at y=− 36μm [black triangle in Fig. 1(b)], respectively. We set the origin of our coordinate system as the center of the channel. The datacan be well ﬁt to J y(α)=JCsin 2α+JL1sin 4α+JL2cos 4α+JD, (1) where αis the angle between the polarization direction of incident light and the optical axis of the λ/4 waveplate, JC=(J/Omega1+−J/Omega1−)/2 the amplitude of circular photocurrent that is helicity dependent, JL1andJL2the amplitudes of linear photocurrents that depend on the linear polarization of the light,andJ Dthe amplitude of the polarization independent photocur- rent that primarily comes from the thermoelectric effect. In thevicinity of the contacts, the extracted circular photocurrents JCare opposite in sign for samples G1 and M1 (blue curves), whereas their two-dimensional (2D) photocurrent proﬁlesare similar in response to linearly polarized light [Figs. 1(e) and1(f)]. The sign of J Cin M1 agrees with the expectation of IEE-induced photocurrent JIEEon the top surface, whereas JC in G1 near the electrode displays an anomalous opposite sign. For photogalvanic currents arising from the IEE, there is apartial cancellation between the spin currents generated at thetop and bottom surfaces due to their opposite spin helicities.Generally, J IEEon the top surface is expected to be larger than that on the bottom surface due to the absorption of bulk withan absorption coefﬁcient of 2 .25×10 4cm−1, which leads to reduced light intensity reaching the bottom surface [ 13,16,18]. We next map out the one-dimensional spatial proﬁle of JC along the yaxis by repeating light polarization scans at each position in samples G1, M1, and a 10-QL BST/GaAs p-type sample G2 [Fig. 1(g)]. For sample M1, JCis negative and falls off quickly towards the electrode. For sample G1, JCis negative at the center and rapidly changes sign when approaching thecontacts. For sample G2, J Cis positive throughout the channel and reaches a maximum at the BST-electrode interface. In themiddle of the channel, the opposite current response in G1and G2 might be related to the thickness-dependent bindingenergies of E D. Angle resolved photoemission spectroscopy (ARPES) experiments found that EDat 6 QLs is lower by ∼0.1 eV than that of the 10-QL TI ﬁlms [ 23]. It is worth noting that the current electrode may induce an upward band bendingoccurring in both M1 and G1 samples, leading to a Dirac 045308-2ANOMALOUS HELICITY-DEPENDENT PHOTOCURRENT IN … PHYSICAL REVIEW B 97, 045308 (2018) (a) (b) (c) (d) (e) (f) FIG. 2. Gate controllable photocurrent generation in TI/GaAs. (a) Schematic of the device structure with a top gate. (b) Jy/Ias a function of photon polarization and gate voltage VGat the center of the channel in sample S2. The curves are vertically offset for clarity. (c),(d) Extracted photocurrent components as a function of VGfor samples S1 (c) and S2 (d). (e) VGdependence of JC/Iin sample S1 at T=74 and 250 K. (f)VGdependence of JC/Iin sample S2 at T=26 and 150 K. point energy shift relative to Fermi energy EF. The display of maximum peak JCnear the electrodes in both G1 and G2 implies that JCis sensitive to band bending in BST/GaAs. However, interpretation of photocurrent response near thecontact can be complicated by the presence of fringe ﬁeld and laser-induced heating [ 24]. We therefore should focus on the measurements performed at the center of the channel where thelocal fringe ﬁeld and sample heating effects are minimized. JL2 (pA)JL1 (pA)JC (pA)(e) JD(pA)(f) (g) (h)JL1(pA)(c)JC (pA)(a) JD (pA)(b) JL2(pA)(d) FIG. 3. Spatially resolved photocurrents in BST/GaAs and BST/mica samples. Scanning photocurrent images of JC,JD,JL1,a n dJL2in BST/GaAs with a top gate at VG=0 (a)–(d) and BST/mica without a top gate (e)–(h), taken at T=30 K. In (a), JCis uniform under the gate atVG=0 V , whereas it changes sign near the edges of the channel. 045308-3DONG-XIA QU et al. PHYSICAL REVIEW B 97, 045308 (2018) μμ μμ μ μμ μ(a) (b) (c) (d) (e) (f) FIG. 4. Helicity-dependent photocurrent measurements in BST/GaAs and BST/mica as a function of light spot size. (a)–(c) JCalong the xdirection in BST/GaAs (red dots) and BST/mica (green dots), with beam radius r=52,43, and 8 .6μm. (d) Illustration of swirling current generated by a radical spin current via ISHE. Under the inclined incidence of laser beam, photoexcited electrons can possess a zcomponent of spin polarization ( ⊗denotes the spin polarization direction). Due to the spatial Gaussian distribution of the laser intensity, there is a gradient in the spin density, therefore resulting in a diffused spin current ﬂowing in the radial direction (black arrows). Consequently, a vortex charge current (gray circles) arises from the spin current via the lateral ISHE JISHE∝− Js×P, in which the charge current direction (green arrows) is perpendicular to both spin polarization direction ( ⊗) and the spin current direction (black arrows). (e),(f) In BST/mica, JC(x) follows the proﬁle ofdJL1(x) dxin samples M1 and M2 at 296 K. C. Top-gated TI/GaAs structure Bearing this in mind, we then measured the dependence of photocurrent on the gate voltage VG, because gating can tuneEFrelative to ED. Our top-gate device is prepared by fabricating a semitransparent 10 nm Ni electrode on top ofa2 0n mA l 2O3insulated 8-QL BST/GaAs sample [Fig. 2(a)]. Figure 2(b) shows the traces of Jy/Iversus photon polarization at the center of the channel, while stepping VGfrom−3 to 12 V . We ﬁnd that both magnitude and extrema of Jy/I strongly depend on VG, and the polarity of JCcan be completely switched as the external gate ﬁeld is increased. The extracted JC,JD,JL1, andJL2versus VGare summa- rized in Figs. 2(c) and2(d), measured for two samples S1 and S2, in which the applied gate voltage VGchanges JCfrom negative to positive at VG0=2 and 4 V , respectively. The expanded view in Figs. 2(e) and2(f)reveals that JCincreases linearly versus \|VG−VG0\|(VG<V G0) and the slopes decrease as temperature Tis raised from 26 to 250 K. For VG>V G0, JCtends to reach a constant value [Appendix C,F i g s . 8(a) and8(b)].D. Spatially-resolved photocurrent response in BST/GaAs and BST/mica To further characterize the anomalous photogalvanic cur- rent in BST/GaAs, we measured the spatially-resolved pho-tocurrent in BST/GaAs in comparison with BST/mica samples.The 2D images of our top-gate BST/GaAs sample show ahomogeneous photocurrent response under the gate at V G= 0 V [Figs. 3(a)–3(d)]i nJC,JD,JL1, andJL2. For BST/mica, all the photocurrent components display dispersive proﬁlesat the center of the channel [Figs. 3(e)–3(h)]. When the light spot moves up to the area between the two Hall leads, J Creverses sign from negative to positive for x> 20μm [Fig. 3(e)]. The dispersive circular photocurrent in BST/mica might be explained as a result of the spin induced charge currentwhirling around the center of the light spot. Similar effecthas been reported in Al xGa1−xN/GaN heterostructures [ 25]. Under the inclined incidence of a Gaussian laser beam, aninhomogeneous spin density can be excited in TI and hasa nonzero zcomponent. The gradient of the spin density generates a radial spin current J son the plane, which leads to 045308-4ANOMALOUS HELICITY-DEPENDENT PHOTOCURRENT IN … PHYSICAL REVIEW B 97, 045308 (2018) 50 100 150 200 250 0102050 100 150 200 204060 01020 50 100 150 200 250 300-100-50 0 50 100 150 200 25002040∝T-1Temperature (K) JC JD/50Fit amp./I (pA kW-1 cm2) S1VG = 0 V(a) (b) Fit amp./I (pA kW-1 cm2)Temperature (K) S1VG = -3 V JC JD/50 ∝T-1 Temperature (K)(d) G1 JC/I (pA kW-1 cm2) M2 e(c) Beam y-Position ( μm)300 KJC/I (pA kW-1 cm2) G1 61K FIG. 5. Temperature dependent photocurrent measurements. (a),(b) Tdependence of JC/IandJD/Iin sample S1 at VG=0V( a )a n d −3 V (b). (c) JC/Iversus the beam position along the yaxis in sample G1 (6-QL BST/GaAs) at T=61 and 297 K. (d),(e) Tdependence ofJC/Iin BST/GaAs sample G1 (d) and BST/mica sample M2 (e) with the same thickness t=6 nm. The black curves are ﬁt to power-law variation as T−1. a swirly charge current via the inverse spin Hall effect (ISHE) asJISHE∝− Js×P, where Pis the spin angular momenta. Another approach to modulate the lateral spin density distribution is through varying the light spot radius r. We ﬁnd that the line proﬁle of JC/Iremains nearly unchanged with decreasing rin a 10-QL BST/GaAs sample [Figs. 4(a)–4(c) and Appendix D], whereas it varies dramatically in a 6-QL BST/mica sample. It is possible that for BST/mica the in-plane spin transport occurs, leading to a rdependent J C/I that reverses sign by moving the light spot along the xaxis, which is the direction transverse across the channel [Fig. 4(d)]. Additionally, for BST/mica JC(x) closely follows the trend of dJL1(x) dxas shown in Figs. 4(e) and4(f). Since JL1(x) reﬂects the light intensity distribution, the derivative relation between JC andJL1suggests that the dispersive JCproﬁle is caused by the swirly charge current. E. Temperature dependence of photocurrent in TI/GaAs and TI/mica structures We next investigate the dependence of photogalvanic cur- rent with temperature in TI/GaAs and TI/mica structures. Weﬁrst look at how J C/Ivaries as a function of Tat each VG in TI/GaAs. In Figs. 5(a) and5(b),w ep l o t JC/IandJD/I versus Tin sample S1 at VG=0 and −3 V , respectively. In both cases, JD/Idecreases slightly as Tincreases from 60 to 200 K, consistent with previous observation of bulkthermoelectric photocurrent [ 16]. ForV G=0V ,JC/Idisplays a strong dependence on T, which ﬁts well to the power-lawvariation T−α(α=1) from 65 to 270 K [Fig. 5(a)]. By contrast, forVG=− 3V ,JC/Iis nearly Tindependent (Fig. 5(b)). The different T-dependent behavior of JCatVG=0 and −3V implies that they may come from different origins. Moreover, we ﬁnd that the power-law scaling of JCremains valid when the BST thin ﬁlm decreases down to 6 QLs inBST/GaAs, whereas not observed in the BST/mica sample[Figs. 5(d) and5(e)]. Figure 5(c)shows the line scan of J C/Iin sample G1 at T=61 and 297 K. In the middle of the channel, JCdisplays a sign reversal as Tis lowered. We can ﬁt JC versus Tto the power-law scaling JC∝T−α(α=1) between 60 and 235 K [Fig. 5(d)]. In the 6-QL BST/mica sample M2, JCmildly changes with Tbetween 60 and 210 K [Fig. 5(e)], as commonly observed in devices of TI on insulating substrates. III. DISCUSSION To reveal the origin of the anomalous HDPC in BST/GaAs structure through a theoretical approach remains a challenge. Arecent CPGE experiment [ 26] demonstrates that HDPC could reverse sign with a photon excitation of 1.51 eV while tuningthe Fermi energy, due to the asymmetric optical transitionsbetween surface and bulk states. It would be interesting to testwhether this model could explain the different behavior of thespatially resolved HDPC between BST/GaAs and BST/micasamples (Figs. 3and4). Another possible explanation is the interface spin transport from BST to GaAs. Recently, a magnetic ﬁeld-dependentCPGE measurement [ 27] reveals the spin injection from GaAs 045308-5DONG-XIA QU et al. PHYSICAL REVIEW B 97, 045308 (2018) 20 μm -40 -20 0 2020406080Jy(pA) Beam x-Position ( μm)y x -40 -20 0 2002 JC/I (pA kW-1 cm2) Beam x-Position ( μm)(a) (b) (a) (c) FIG. 6. Photocurrent response in the lateral GaAs-TI-GaAs sample. (a) The optical image of the GaAs-BST-GaAs sample with a 5- μm-wide window opened at each edge of the channel. (b) 2D scanning photocurrent image Jyexcited with the linearly polarized light. Dashed and solid lines indicate the edges of the sample and electrodes, respectively. (c),(d) Line scan of Jyat a ﬁxed linear polarization (c) and JC/I(d) as a function of beam spot position x[white dashed line in panel (a)] at room temperature. to TI for a Bi 2Te3/GaAs heterostructure. We speculate that a vertical spin current may also occur at the BST/GaAsinterface and induce a charge current via the ISHE in a TI.However, it should be pointed out that as the electron afﬁnityof BST in MBE-grown BST/GaAs is currently unavailable,the band structure at the interface of BST and GaAs cannot beaccurately determined. Further studies, such as work functionmeasurements at the BST/GaAs interface, tuning the Fermilevel in BST/mica through electrical gating, and extendingmeasurements into low energy optical excitations, could helpto identify the origin of the anomalous photocurrent. In conclusion, we report the observation of anomalous photogalvanic current in BST/GaAs samples. Experimentalevidence for such a substrate dependent anomaly is providedthrough gating experiments, spatially resolved photocurrentmeasurements, and temperature dependent photocurrent mea-surements. We demonstrate that the HDPC in BST/GaAsdisplays different direction and magnitude as a function of gatevoltage, beam size, and temperature compared with those inBST/mica. The results suggest that alternative substrate maylead to a mechanism and possible improvement in the spinphotocurrent generation. We hope our experimental work willprovide important clues for developing a theory of substrate-dependent CPGE in a topological insulator/semiconductorheterostructure. ACKNOWLEDGMENTS We would like to thank Yu Pan, N. Samarth, Yasen Hou, and Dong Yu for insightful discussion. This work was per-formed under the auspices of the US Department of Energyby Lawrence Livermore National Laboratory under ContractNo. DE-AC52-07NA27344. The project was supported by the Laboratory Directed Research and Development (LDRD)programs of LLNL (15-LW-018 and 16-SI-004). APPENDIX A: LATERAL GAAS-TI-GAAS SAMPLE To conﬁrm the measured photocurrent predominantly comes from the topological insulator rather than the GaAssubstrate, we fabricated the lateral GaAs-TI-GaAs controlsample. Figure 6(a) presents the optical image of a typical device. The 6-QL BST channel is connected with the currentelectrodes through the 5- μm-long GaAs substrate to enable the measurement of possible charge transport in the substrate. We ﬁrst characterized the devices with the linearly po- larized light. Figure 6(b) shows the two-dimensional image of the photocurrent J ymeasured at room temperature. The photocurrent displays a polarity reverse across the sample witha maximum at the GaAs/electrode interface, which can beattributed to a thermoelectric current arising from the pho-toexcitation of the in-gap EL 2impurity states in GaAs [ 28–32]. When the laser spot sweeps along the xaxis close to the current electrode [white dotted line, Fig. 6(b)],Jyexhibits a peak at x=0 [Fig. 6(c)] and the extracted JC/Iis on the order of 1p Ak Wc m−2that is 10 ×smaller than the value measured in sample G1 [Fig. 1(g)]. We then conclude that the circular photocurrent response from GaAs substrate is negligible. APPENDIX B: LASER INTENSITY DEPENDENCE OF PHOTOCURRENT In scanning photocurrent measurements, laser heating may cause photocurrent to deviate from the linear dependence onthe laser intensity I. We conﬁrmed that our photocurrent 045308-6ANOMALOUS HELICITY-DEPENDENT PHOTOCURRENT IN … PHYSICAL REVIEW B 97, 045308 (2018) 0.0 0.2 0.4 0.6 0.8 1.0-50050100 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.8 Laser Intensity (kW cm-2) JC JL1 JL2Current (pA) Laser Intensity (kW cm-2)(a) (b) Current (nA) JD FIG. 7. The laser intensity Idependence of photocurrent at the center of the channel. (a) The extracted amplitude JC,JL1,a n dJL2versus laser intensity. (b) The amplitude of the polarization independent photocurrent JDas a function of the laser intensity. The symbols are the experimental data. The solid lines show the linear ﬁt to the data. measurements are performed in the low intensity linear regime. Figure 7shows the laser intensity dependence of the am- plitude JC,JL1,JL2, and JD,ﬁ t t e dt oE q .( 1)i na8 - Q L BST/GaAs sample at T=30 K. All these four photocurrent components increase linearly with the laser intensity up toI=1.1k Wc m −2, below which the laser heating is minimized. We have carried out all the measurements with a low laserintensity I< 0.85 kW cm −2. APPENDIX C: PHOTOCURRENT MAP IN TOP-GATE TI/GAAS DEVICES UNDER POSITIVE GATE VOLTAGE In the top-gate samples S1 and S2, JCvaries linearly as a function of \|VG\|(VG<V G0) [Figs. 2(e) and2(f)]. However, JCis nearly invariant at VG>V G0[Figs. 8(a) and8(b)]. The- oretical studies have shown that surface photogalvanic currentis constant as long as the chemical potential lies between theenergies of the two states involved in the transition [ 15]. In our sample, the Fermi energy E Fincreases by 27 meV as VGis changed from 2 to 11 V . At VG=11 V , the Fermi energy is 150 ∼250 meV , which still lies below the upper transition state, so that surface JIEEis expected to be saturate with increasing VGup to 11 V . In Fig. 8(c), we show that JCbecomes less homogeneous at a large positive gate voltage, as revealed by the 2D scanningphotocurrent image in sample S1 at V G=8 V . Under the gate electrode, JC/Idisplays a weak negative signal that ﬂuctuates in the range of −30∼2p Ak W−1cm2. This implies that there are competing mechanisms. In the gap region between the gateelectrode and current contacts, J Cexhibits similarity to the proﬁle measured at zero-gate voltage [Fig. 4(e)], showing a sign reversal as the laser intensity changes abruptly. The 2Dphotocurrent images are also mapped out for J D,JL1, andJL2 in Figs. 8(d)–8(f). We observe that JDandJL1show similar proﬁles to the current response taken at VG=0 V , whereas JL2reverses direction for opposing gate ﬁeld directions. APPENDIX D: PHOTOCURRENT RESPONSE IN THE 10-QL BST/GAAS SAMPLE In Figs. 9(a) and9(b), we show the line scan of Jyalong the yaxis in p-type 10-QL BST/GaAs sample G2, illuminated by a linearly polarized beam with the spot radius r=52 (a) (b) (c) (d) (e) (f) FIG. 8. (a),(b) The circular photocurrent JC/Iversus gate voltage VGin samples S1 (a) and sample S2 (b). (c)–(f) Scanning photocurrent current images of JC,JD,JL1,a n dJL2in sample S1, measured at T=26 K and VG=8V . 045308-7DONG-XIA QU et al. PHYSICAL REVIEW B 97, 045308 (2018) μΩ+Ω−Ω+Ω−μ μ μ μμ(a) (b) (d)(c) FIG. 9. Photocurrent response in the 10-QL BST/GaAs sample G2 at room temperature. (a) Line scan of Jy/Ialong the yaxis with a beam spot radius r=52μm (half width at half maximum). (b) The trace of photocurrent Jy/Iversus photon polarization at x=− 66μm. The solid green line is the ﬁt to observed data. The solid blue curve is the extracted circular photocurrent component JC. (c) Line scan of Jy/Iversus the yaxis with a beam spot radius r=8.6μm. (d) Line scan of JC/Ialong the yaxis with a beam spot radius r=8.6μm. and 8.6μm, respectively. In both cases, the photocurrent Jy peaks at the channel-contact junctions with a sign reversal across the channel, which arises from the light heating inducedthermoelectric effect. The polarity of the photocurrent corre-sponds to the holelike carriers that is consistent with the Hallmeasurement. Figure 9(c) shows the raw data of J y/Iversus photon polarization (open black circles) at the edge of the channel.The extracted J C/Iversus photon polarization (blue curves)displays positive amplitude. 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PhysRevB.100.075417.pdf	PHYSICAL REVIEW B 100, 075417 (2019) Interplay between in-plane and ﬂexural phonons in electronic transport of two-dimensional semiconductors A. N. Rudenko,1,2,3,A. V . Lugovskoi,2A. Mauri,2Guodong Yu,1,2Shengjun Yuan,1,2,†and M. I. Katsnelson2,3 1Key Laboratory of Artiﬁcial Micro- and Nano-structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China 2Institute for Molecules and Materials, Radboud University, Heijendaalseweg 135, NL-6525 AJ Nijmegen, Netherlands 3Theoretical Physics and Applied Mathematics Department, Ural Federal University, 620002 Ekaterinburg, Russia (Received 29 January 2019; revised manuscript received 24 March 2019; published 12 August 2019) Out-of-plane vibrations are considered as the dominant factor limiting the intrinsic carrier mobility of suspended two-dimensional materials at low carrier concentrations. Anharmonic coupling between in-plane andﬂexural phonon modes is usually excluded from the consideration. Here we present a theory for electron-phononscattering, in which the anharmonic coupling between acoustic phonons is systematically taken into account.Our theory is applied to the typical group V two-dimensional semiconductors: hexagonal phosphorus, arsenic,and antimony. We ﬁnd that the role of the ﬂexural modes is essentially suppressed by their coupling with in-planemodes. At dopings lower than 10 12cm−2the mobility reduction does not exceed 30%, being almost independent of the concentration. Our ﬁndings suggest that compared to in-plane phonons, ﬂexural phonons are considerablyless important in the electronic transport of two-dimensional semiconductors, even at low carrier concentrations. DOI: 10.1103/PhysRevB.100.075417 I. INTRODUCTION Charge carrier transport in two-dimensional (2D) materi- als is different from their three-dimensional analogs [ 1,2]. Among other reasons [ 3,4], this is mainly related to the existence of the ﬂexural phonon modes (out-of-plane vibra-tions), whose energy dispersion is quadratic in wave vectorcompared to the linear dispersion of conventional (in-plane)modes [ 1,5,6]. As the charge carrier scattering on phonons is the main factor limiting the intrinsic mobility, understandingof the interplay between ﬂexural and in-plane phonons is ofvital importance for 2D material’s electronic transport. The contributions of in-plane and ﬂexural phonons are usually considered independently of each other at the levelof the harmonic approximation. Moreover, the role of ﬂexuralphonons is often completely neglected [ 7–9], assuming that their contribution is suppressed by the presence of a substrate[10]. At the same time, ﬂexural phonons are considered to play the dominant role in the electron-phonon scattering offree-standing graphene [ 11–13] and may contribute signiﬁ- cantly in the presence of a dielectric substrate [ 14,15]. Trans- port properties of conventional 2D semiconductors like blackphosphorus are less affected by ﬂexural modes in the regimeof high doping [ 16], whereas at low dopings the situation is unclear. Furthermore, ﬂexural modes could signiﬁcantlyrenormalize the electronic spectrum, leading to the formationof bound states [ 17,18]. The separation between the in-plane and ﬂexural modes becomes undeﬁned at wave vectors q/lessorsimilarq ∗, smaller than the a.rudenko@science.ru.nl †s.yuan@whu.edu.cncharacteristic vector q∗=/radicalbig 3TY/16πκ2, determined by the Young modulus Y, ﬂexural rigidigy κ, and temperature T [6,19,20]. This is a manifestation of the anharmonic cou- pling regime, which is typical to low carrier concentrationswith small Fermi wave vectors, k F/lessorsimilarq∗. In this situation, the electron-phonon interaction for ﬂexural modes divergesand is usually cut off. Several attempts have been madeto systematically account for the anharmonic effects in thetransport properties of graphene [ 13,21–23]. However, only little attention to this problem has been given in the context ofconventional semiconductors [ 24]. Despite the availability of advanced ab initio computational techniques developed to de- scribe electron-phonon scattering [ 25–28], their applicability is limited with respect to the above-mentioned effects. Group V elemental semiconductors is an emerging class of 2D materials with attractive electronic properties [ 29,30]. Hexagonal (A7) 2D phases of phosphorus (P), arsenic (As),and antimony (Sb) are among the most recent materials fabri-cated experimentally [ 31–33]. Apart from being mechanically stable [ 34,35], they are proposed to have excellent electron transport characteristics [ 36,37] and a high degree of gate tunability [ 38–40] due to the buckled crystal structure. Me- chanically, these materials are highly ﬂexible with the ﬂexuralrigidities in the range 0.3–0.8 eV [ 34,41], which is signif- icantly smaller compared to graphene [ 42]. This property allows us to expect a considerable anharmonic coupling inthese materials, and its possible inﬂuence on the electrontransport properties. In this paper, we focus on a theory for phonon-mediated charge carrier scattering in 2D semiconductors taking thecoupling between the acoustic phonons into account. We usethe diagrammatic approach to calculate the anharmonic con-tribution to the electron-phonon scattering rate, ending up at 2469-9950/2019/100(7)/075417(10) 075417-1 ©2019 American Physical SocietyA. N. RUDENKO et al. PHYSICAL REVIEW B 100, 075417 (2019) two plausible approximations. We apply the developed theory to calculate electron and hole mobilities in three representativegroup V isotropic semiconductors: hexagonal monolayers P(also known as blue phosphorene), As (arsenene), and Sb (an-timonene). To this end, we estimate relevant model parametersfrom ﬁrst-principles density functional theory (DFT) calcula-tions. We ﬁnd that although ﬂexural phonons tend to reducethe mobility at concentrations below 10 14cm−2by 15–30%, their effect reaches its maximum at around 1012cm−2.A t lower dopings, the mobility becomes essentially independentof the concentration, being the manifestation of the anhar-monic coupling between the ﬂexural and in-plane phononmodes. The paper is organized as follows. In Sec. II, we present a theory of charge carrier scattering in 2D semiconductors,explicitly considering in-plane and out-of-plane phonons,as well as their anharmonic coupling. In Sec. III,w eu s e ﬁrst-principles calculations to estimate model parameters forhexagonal P, As, and Sb monolayers. The results of numericalcalculations are presented in Sec. IV, where we investigate the interplay between in-plane and ﬂexural phonons in the contextof charge carrier scattering, as well as estimate upper limitsfor the carrier mobility in the materials under consideration.In Sec. V, we summarize our results, and conclude the paper. II. THEORY A. Carrier mobility and relaxation time Carrier mobility of isotropic electron gas can be expressed asμc=σ/ne, where nis the carrier concentration and σis the dc conductivity, which has the form [ 43] σ=e2 2S/summationdisplay kτkv2 k/bracketleftbigg −∂f(εk) ∂εk/bracketrightbigg . (1) Here Sis the sample area, εkis the band energy, vkis the group velocity, τkis the momentum-dependent scattering relaxation time, and f(εk)={1+exp[(εk−μ)/T]}−1is the Fermi-Dirac distribution function. The energy dispersion ofcharge carriers is assumed to have the following form near theband edge: ε k=ε0+¯h2(k−k0)2 2m, (2) where mis the electron ( me) or hole ( mh) effective mass and ε0andk0are the energy and wave vector determining the position of band edges, respectively. The chemical potentialμcan be determined from the carrier concentration nas μ=Tln/bracketleftbigg exp/parenleftbigg2π¯h 2n gvgsmT/parenrightbigg −1/bracketrightbigg , (3) where gvgsis the valley-times-spin-degeneracy factor. In turn, one can recast Eq. ( 1)i nt h ef o r m , σ=gvgse2τ 4π¯h2/braceleftBig μ+2Tln/bracketleftBig 2cosh/parenleftBigμ 2T/parenrightBig/bracketrightBig/bracerightBig , (4) which in the limit T→0 transforms into the standard conduc- tivity formula σ=ne2τ/m. Here we assume the relaxation time to be averaged over thermally broadened Fermi surfacestates τ≡/angbracketleft /angbracketleftτk/angbracketright/angbracketrightFS, which is dependent on the chemical po- tential μand temperature T. The corresponding expression for τcan be obtained from the variational solution of the isotropic Boltzmann transportequation [ 43], yielding 1 τ=2π ¯hS/summationtext kk/prime/bracketleftBig −∂f(εk) ∂εk/bracketrightBig δ(εk−εk/prime)(1−cosθkk/prime)/angbracketleft\|Vkk/prime\|2/angbracketright /summationtext k/bracketleftBig −∂f(εk) ∂εk/bracketrightBig , (5) where k=k/prime+qwith qbeing the scattering wave vec- tor,Vkk/primeis the effective electron-phonon scattering potential deﬁned in Sec. II B, and /angbracketleft.../angbracketright=Tr(e−βH...)/Tr(e−βH) with β=1/Tdenotes thermal averaging over the phonon states with the Hamiltonian Hdeﬁned below. In Eq. ( 5), emission and absorption of phonons are not explicitly involved in theenergy conservation law, which assumes elastic scattering,i.e., ¯ hω kF/lessmuchεFwithωkFbeing the phonon frequency at the Fermi wave vector. This condition is satisﬁed for acousticphonons considered in our study. After some algebra, Eq. ( 5) can be rewritten in a more convenient form as follows: τ −1=m 2π¯h3f(0)/integraldisplay∞ 0dε/bracketleftbigg −∂f(ε) ∂ε/bracketrightbigg/integraldisplay2k 0dqq2/angbracketleft\|Vq\|2/angbracketright k2/radicalbig k2−q2/4, (6) where k=√ 2mε/¯hplays the role of the Fermi wave vector at μ/greatermuchT. We note that Eq. ( 6) is only valid for conventional 2D semiconductors with quadratic dispersion of charge carriers.Deviations from the quadratic dispersion, as well as the pres-ence of a dielectric environment (e.g., substrates), could resultin a considerable modiﬁcation of the energy dependence of thescattering rate (see, e.g., Refs. [ 44–46]). B. Electron-phonon scattering Here we ﬁrst deﬁne the scattering potential. We restrict ourselves to the long-wavelength limit and take both in-planeand out-of-plane (ﬂexural) phonon modes into account. Weassume that electrons interact with phonons through the de-formation potential of the form V(r)=gu αα(r), (7) where uαβ(r)=1 2[∂αuβ(r)+∂βuα(r)+∂αh(r)∂βh(r)] (8) is the strain tensor and gis the coupling constant, whereas uα(r) and h(r) are in-plane and out-of-plane displacement ﬁelds, respectively. The form of the deformation potential in Eq. ( 7) is justiﬁed by the following considerations. In the absence of substratesor external tension, the electronic structure is invariant underrotations of the crystal as a whole in three-dimensional space[47]. Therefore, V(r) must be invariant under the following transformations: u α(r)→uα(r)−δϕαh(r) h(r)→h(r)+δϕα[rα+uα(r)]. (9) To lowest order in ∂αuβ(r) and∂αh(r), this implies that V(r) can only depend on the components of the strain tensor, which 075417-2INTERPLAY BETWEEN IN-PLANE AND FLEXURAL … PHYSICAL REVIEW B 100, 075417 (2019) are scalar quantities under the transformation in Eq. ( 9)[48]. In principle, V(r) can depend on all components of uαβ(r). Here we assume the isotropic form in Eq. ( 7) for simplicity. Notice that, as a consequence of rotational invariance, in-plane and out-of-plane coupling constants in Eq. ( 7) are not independent [ 47,49]. In view of the absence of horizontal mirror ( σ h) symmetry in buckled hexagonal semiconductors ( D3dpoint group) being studied in this work, Eqs. ( 7) and ( 8) require additional justiﬁcation. Particularly, as has been shown in Ref. [ 24], single-phonon processes associated with ﬂexural modes arehighly relevant for non-mirror-symmetric 2D Dirac materialslike silicene and germanene. Such processes correspond toadditional terms in Eq. ( 7) that are linear in out-of-plane displacements, i.e., ∇h. However, in our case those terms are irrelevant for the following reasons. Here we are fo-cused on lightly doped semiconductors with quadratic energydispersion represented by Eq. ( 2). Moreover, the intervalley scattering can be neglected because k F/lessmuchk0. Under these assumptions, it is obvious from in-plane symmetry that ∇h terms can only enter the interacting potential via the transfor-mation k 0→k0+α∇h. However, such terms are forbidden by rotational symmetry in three-dimensional space. This con-sideration does not forbid the appearance of higher-gradientterms in the deformation potential. As can be shown, ∇ 2h terms are forbidden, too, which follows from the combinationof inversion and time-reversal symmetry, particularly becausek 0→− k0. On the other hand, ∇3hand higher terms acquire additional power of phonon wave vector qin comparison to ∇uterms, and since \|q\|/lessorsimilar2kF, such terms have smallness in electron concentration and, therefore, can be neglected. Let us now evaluate the scattering probability /angbracketleft\|Vq\|2/angbracketright.T h e Fourier transform of the scattering potential V(r) is then given by Vq=igqαuα(q)+g 2fαα(q), (10) where uα(q) are the Fourier components of the in-plane displacement ﬁeld uα(r) and fαβ(q)=− S−1 2/summationtext kkα(qβ− kβ)hkhq−kis the Fourier transform of fαβ(r)=∂αh(r)∂βh(r). In the long-wavelength limit, ﬂuctuations of the displacementﬁelds are described by the Hamiltonian [ 19,20,50] H=1 2/integraldisplay dr/braceleftbig κ[∇2h(r)]2+λu2 αα(r)+2Gu2 αβ(r)/bracerightbig ,(11) where λand Gare the Lamé parameters. We assume that ﬂuctuations of the displacement ﬁelds can be described classi-cally. Neglecting quantum effects is fully justiﬁed at not-too-small temperatures, when ¯ hω(k F)/lessmuchT. In the calculation of the thermal average /angbracketleft\|Vq\|2/angbracketright, we can take advantage of the fact that the Hamiltonian in Eq. ( 11) is quadratic with respect to the in-plane displacement ﬁelds uα(r), which allows one to integrate them out exactly [ 19,20,50]. After some algebraic manipulations [ 19,20], we obtain the following expression for the scattering matrix (see the Appendix for details): /angbracketleft\|Vq\|2/angbracketright=g2T Y(1−ν2)+g2(1−ν)2 4/angbracketleftfT(q)fT(−q)/angbracketright,(12) where ν=λ/(λ+2G) is the 2D Poisson ratio and Y= λ(1−ν2)/ν. Being focused on the processes with smallmomentum transfer q=k−k/prime, we ignored the effect of the wave-function overlap between the initial kand ﬁnal k/primestates, which might lead to some overestimation of the scatteringprobability. In Eq. ( 12), the ﬁrst term coincides with the usual contribution of in-plane phonons in the harmonic approxi-mation, whereas the second term describes both out-of-planeand cross interactions generated by anharmonic coupling; f T(q)≡(δαβ−qαqβ/q2)fαβ(q) denotes the transverse part offαβ(q). In what follows, we use two different approxima- tions to calculate the correlation function /angbracketleftfT(q)fT(−q)/angbracketright. 1. Wick decoupling approximation In the ﬁrst approximation, we assume the validity of the Wick theorem, which yields /angbracketlefthkhq−khk/primeh−q−k/prime/angbracketright/similarequal (δk,q+k/prime+δk,−k/prime)/angbracketlefthkh−k/angbracketright/angbracketlefthk−qhq−k/angbracketright[1]. In terms of dia- grams, this assumption corresponds to the neglect of vertexcontributions to the two-particle Green’s function. In thisapproximation, the desired correlation function reads: /angbracketleftf T(q)fT(−q)/angbracketright/similarequal/Pi1(q) ≡2/integraldisplayd2k (2π)2/bracketleftbigg(q×k)2 q2/bracketrightbigg2 G(k)G(q−k),(13) where G(q)≡/angbracketlefthqh−q/angbracketrightis the single-particle correlation func- tion of out-of-plane ﬁelds. The function /Pi1(q) exhibits a crossover from a harmonic behavior for q/greatermuchq∗to an anoma- lous scaling behavior for q/lessmuchq∗, which is controlled by anharmonic interactions. In this work, we addressed the cal-culation of /Pi1(q) within the self-consistent screening approx- imation (SCSA) [ 50,51]. In the limit of large and small wave vectors, /Pi1(q) behaves asymptotically as: /Pi1(q)=/braceleftBigg 3 8πT2 κ2q2, q/greatermuchq∗ C/parenleftbigT κ/parenrightbig2−η/parenleftbigκ Y/parenrightbigηq−2+2η,q/lessmuchq∗, (14) where ηis an universal exponent and Cis a dimensionless constant. The value of ηis close to 0.821 within the SCSA, in the physical case of two-dimensional membranes embeddedin three-dimensional space. Through a numerical solutionof the SCSA equations [ 50,51], we determined /Pi1(q)f o r arbitrary values of the wave vector, including the intermediatescale q/similarequalq ∗and we obtained the value C/similarequal2.60 for the dimensionless amplitude in Eq. ( 14). 2. Screening approximation In the second approximation, we consider the following expression for the correlation function: /angbracketleftfT(q)fT(−q)/angbracketright/similarequal/Pi1(q) 1+Y 4T/Pi1(q). (15) Equation ( 15) can be derived diagrammatically by summing a geometric series of polarization bubble diagrams connectedby bare interaction lines, the same diagrams which determine,within the SCSA, the screened interaction between ﬂexuralphonons [ 50,51]. An expression equivalent to Eq. ( 15)i s reported in Ref. [ 47]. 075417-3A. N. RUDENKO et al. PHYSICAL REVIEW B 100, 075417 (2019) Adopting Eq. ( 15), one arrives at the following asymptotic limits for the correlation function: /angbracketleftfT(q)fT(−q)/angbracketright=/braceleftBigg3 8πT2 κ2q2, q/greatermuchq∗ 4T Y−16 CT2 κ2/parenleftbigTY κ2/parenrightbig−2+ηq2−2η,q/lessmuchq∗ As expected, the two approximations result in the same behavior at q/greatermuchq∗, while the result at q/lessmuchq∗is different. In the Wick decoupling approximation, the electron-phononcoupling /angbracketleft\|V q\|2/angbracketrightscales as 1 /q2−2ηfor small qand it diverges in the limit q→0, although the divergence is slow, because the exponent ηis not very different from 1. By contrast, in the screening approximation /angbracketleft\|Vq\|2/angbracketrightapproaches the constant value 2 g2T Y(1−ν)f o r q→0. This expression ensures that, in the screening approximation, scattering on in-plane phononsalways gives a larger contribution than scattering on out-of-plane phonons. In the discussion below, the two approxima-tions will be considered as the upper and lower limits for therelaxation time. The expressions presented above allow one to estimate the dc conductivity (or mobility) of isotropic 2D electrongas in the presence of elastic scattering on acoustic phonons.Importantly, this approach is equally valid for small carrierconcentrations, where scattering involves the coupling be-tween phonons modes. III. ESTIMATION OF MATERIAL CONSTANTS A. Calculation details To estimate material constants, we use ﬁrst-principles DFT calculations. All calculations have been carried out using theprojected augmented-wave [ 52] formalism as implemented in the Vienna ab initio simulation package ( V ASP )[53–55]. To describe exchange-correlation effects, we employed thegradient-corrected approximation in the parametrization ofPerdew-Burke-Ernzerhof [ 56]. An energy cutoff of 500 eV for the plane waves and the convergence threshold of 10 −9eV were used in all cases. To avoid spurious interactions betweenthe cells, a vacuum slab of 50 Å was added in the directionperpendicular to the 2D sheet. Structural relaxation includingthe optimization of in-plane lattice constants was performeduntil the forces acting on atoms were less than 10 −3eV/Å. Unlike the cases of P and As monolayers, spin-orbit interac-tion in monolayer Sb is important [ 57] and, therefore, has been taken into account. Hexagonal monolayers studied in this work adopt honey- comb structure with vertically displaced (buckled) sublattices.We obtained the following optimized values of the latticeconstant ( a) and buckling parameter: 3.28 Å and 1.24 Å for P, 3.61 Å and 1.40 Å for As, 4.12 Å and 1.64 Å for Sb,respectively. Primitive hexagonal cells were used in most ofthe calculations, for which a Monkhorst-Pack [ 58]/Gamma1-centered (32×32) mesh was adopted to sample the Brillouin zone. To calculate the Poisson ratio and Young modulus, a rectangular (√ 3/2×1)aunit cell was used, together with a (32 ×24) k-point mesh. To induce ﬂexural deformations needed for the calculation of ﬂexural rigidities, we considered rectangu-lar (6√ 3×1)asupercells, such that the deformation period amounts to l=6√ 3a∼40 Å for each system considered. We ﬁrst apply a ﬁeld of 1D sinusoidal out-of-plane deformations-2-1.5-1-0.50 0.51 1.52 Γ ΣhKmh l =0.8(a)Pε-εf (eV)DFT fit -PhΣhPhmh t =3.51(b) Γ Σ Μme l =0.94(c) -PΣ Σ PΣme t =0.17(d) -1.5-1-0.50 0.51 1.5 Γ Σ Mme l =0.51 mh lt =0.08 mh h =0.47(e)As ε-εf (eV) 0me t=0.18(f) -1.5-1-0.50 0.51 1.5 Γ Σ Mme l =0.48 mh=0.11(g)Sbε-εf (eV) -PΣ Σ PΣme t=0.18(h) FIG. 1. Blue: DFT band structure calculated along high- symmetry directions of the Brillouin zone for P [(a)–(d)], As [(e) and (f)], and Sb [(g) and (h)] monolayers. Red: Band edges ﬁtted withinthe effective mass approximation [Eq. ( 2)]. Due to the degeneracy of VBM in case of As (g), two values are given for m h, corresponding to light ( mh lt) and heavy ( mh h) holes. For all three systems considered, two types of electron effective masses are provided: longitudinal ( ml) and transverse ( mt). The same distinction applies to the hole effective mass of P monolayer [(a) and (b)]. in the form h(x,y)=hsin(qx) to each system, where hthe deformation amplitude and q=2π/l. We then perform full structural relaxation for a series of ﬁxed amplitudes h, which allows us to obtain the elastic energies. In the calculationswith rectangular supercells, a (4 ×24)k-point mesh was used. B. Effective masses Electron and hole effective masses are estimated from the band structures by ﬁtting the conduction band minimum(CBM) and valence band maximum (VBM), respectively,using the expression given by Eq. ( 2). The calculated band structure used for the ﬁtting, and the corresponding effectivemasses are given in Fig. 1. In all three systems considered, CBM resides at the /Sigma1point along the /Gamma1–M high symmetry line. The corresponding bands are spin ( g s=2) and valley degenerate ( gv=2). The constant-energy (Fermi) contours at small energies form elliptical pockets [ 41], meaning that the electron effective mass is different in the direction perpen-dicular to /Gamma1–M, denoted in Fig. 1as−P /Sigma1–P/Sigma1. We denote the corresponding masses as longitudinal ( me l) and transverse (me t), as shown in Fig. 1. In practical calculations, we use geometrically averaged effective masses, me=/radicalbigme lme t, with the resulting values listed in Table I. The obtained values overall agree with the literature data [ 36,59,60]. The case of holes in monolayer P is similar to the case of electrons. Since VBM is located at the /Sigma1hpoint along /Gamma1–K symmetry line, the effective mass tensor is anisotropic, andm hcan be calculated as the average of its longitudinal and transverse components. The situation with the hole effectivemasses in As and Sb is different. In these systems, VBM is 075417-4INTERPLAY BETWEEN IN-PLANE AND FLEXURAL … PHYSICAL REVIEW B 100, 075417 (2019) TABLE I. Relevant material constants estimated from ﬁrst prin- ciples for group V hexagonal (A7 phase) 2D semiconductors: Elec- tron (hole) effective mass me(h)(in units of free electron mass), for the case of hole doped arsenene averaged effective mass is given,electron (hole) deformation potential g e(h)(in eV), Young modulus Y(in eV Å−2), Poisson ratio ν, ﬂexural rigidity κ(in eV), and a characteristic wave vector q∗atT=300 K (in Å−1). System memhgeghY νκ q* P 0.40 1.72 2.26 1.34 4.53 0.11 0.70 0.12 As 0.30 0.331.34 6.13 3.07 0.16 0.51 0.14 Sb 0.29 0.11 0.86 5.99 1.86 0.19 0.33 0.17 m=√mdmtr, see Sec. IIIB for details. located at the /Gamma1point, making the Fermi contour isotropic with the direction-independent hole effective masses. Unlikemonolayer Sb, where VBM at the /Gamma1point is doubly degener- ate ( g s=2), VBM of monolayer As is four times degenerate due to the absence of strong spin-orbit coupling. As one cansee, there are two different bands forming the valence band inmonolayer As, meaning that there are two transport channelsfor holes in As. To take them into account, speciﬁcally forAs we deﬁne the transport effective mass, which is the sumof light and heavy hole masses, m h tr=mh lt+mh h. On the other hand, the density of states effective mass is different, mh d=√ mh hmh lt. Given that the conventional electron-phonon contri- bution to the mobility can be expressed through the product mdmtr[61], we employ mh=√ mh dmh trin the calculations of As. In all other systems, mdandmtrare equivalent. C. Elastic properties 1. Flexural rigidities Flexural rigidities κcan be estimated from ﬁrst principles starting from a macroscopic expression for the elastic energydensity of a 2D membrane, Eq. ( 11). In the absence of strain, it reads E elas=κ 2/integraldisplay dr(∇2h)2. (16) The elastic energy Eelascan be directly obtained from DFT calculations as the total energy difference /Delta1Etotbetween the corrugated and ﬂat states of a membrane, Eelas=/Delta1Etot.B y employing the sinusoidal height ﬁeld in the form h(x,y)= hsin(qx), the constant κcan be estimated by a regression ﬁt from /Delta1Etot=κ 41 R2, (17) where Ris the curvature radius of a sine wave at its extrema, given by 1 /R=hq2. In Fig. 2(b), we show the calculated dependencies /Delta1Etotas af u n c t i o no f hq2, which allow us to estimate κfor the systems under consideration (see Table Ifor summary). In all cases the expression given by Eq. ( 17) perfectly ﬁts DFT data. 2. In-plane elastic constants In the absence of ﬂexural deformations, the elastic energy density Eelasof a hexagonal strained 2D material can be−25−20−15−10−505 0 0.2 0.4 0.6 0.8 1(c)CBM shift (meV) Biaxial strain, u (%)DFT fit2gPeu2gAseu2gSbeu\|gPe \|=2.26 eV \|gAse \|=1.34 eV \|gSbe \|=0.86 eV −60−50−40−30−20−100 10 0 0.2 0.4 0.6 0.8 1(d)VBM shift (meV) Biaxial strain, u (%)DFT fit2gPhu 2gAsh u2gSbh u\|gPh \|=1.34 eV \|gAsh \|=6.13 eV \|gSbh \|=5.99 eV0 0.1 0.2 0 0.2 0.4 0.6 0.8 1(a)Elastic energy density (meV/Å2) Uniaxial strain, u (%)DFT fit YSbu2/2YPu2/2 YAsu2/2YP=4.53 eV ⋅Å−2 YAs=3.07 eV ⋅Å−2 YSb=1.86 eV ⋅Å−2 0 0.2 0.4 0.6 0.8 0 0.01 0.02 0.03 0.04 0.05 0.06(b)Elastic energy density (meV/Å2) Curvature, hq2 (1/Å)DFT fit κSbh2q4 / 4 κPh2q4 / 4 κAsh2q4 / 4κP=0.70 eV κAs=0.51 eV κSb=0.33 eV FIG. 2. Elastic energies and band-edge shifts calculated as func- tions of in-plane and out-of-plane deformations for hexagonal mono- layers P, As, and Sb. Panels (a) and (b) are used to determine 2DYoung modulus ( Y) and ﬂexural rigidity ( κ), whereas panels (c) and (d) are used to determine electron and hole deformation potentials (g eandgh). CBM and VBM stand for the conduction band minimum and valence band maximum, respectively. expressed from Eq. ( 11)a s Eelas=Yν 2(1−ν2)u2 αα+Y 2(1+ν)u2 αβ. (18) The Young modulus Ycan be found straightforwardly from DFT calculations by applying a uniaxial deformation in thedesired direction and allowing full relaxation in other direc-tions: E elas=1 2Yu2 αα/vextendsingle/vextendsingle/vextendsingle/vextendsingle σββ=0, (19) where σββis the component of the stress tensor in the relaxed direction. Similarly, the Poisson ratio can be obtained fromthe same set of calculations: ν=−/Delta1l ββ /Delta1lαα/vextendsingle/vextendsingle/vextendsingle/vextendsingle σββ=0, (20) where /Delta1lααand/Delta1lββare the change of lattice constants in the strained and relaxed directions, respectively. In Fig. 2(a), we show the calculated /Delta1Etotas a function of uniaxial strain, which perfectly ﬁts the macroscopic energyexpression given above. This allows us to readily estimate theYoung modulus Yfor the materials under consideration. The calculated Poisson ratios are listed in Table I. D. Deformation potentials 1. Interaction with in plane phonons To estimate the coupling of charge carriers with phonons from DFT calculations, we adopt the concept of the defor-mation potentials [ 62]. Assuming the long-wavelength limit, we quantify linear response of the band edges to externaldeformation. To this end, we apply both tensile and compres-sive biaxial strain and obtain relative band shifts /Delta1E VBMand /Delta1ECBMas shown in Figs. 2(b) and2(c)as a function of strain 075417-5A. N. RUDENKO et al. PHYSICAL REVIEW B 100, 075417 (2019) u=uαα=uββ. The band shifts are calculated relative to the vacuum level Evac, determined for every deformed state, that is /Delta1E=E−Evac, where Eis the energy of the corresponding band edge. The interaction energy Eintis linear in strain and takes the form Eint=2ge(h)u=/braceleftbigg /Delta1ECBM(electrons) /Delta1EVBM(holes).(21) Since the sign of Eintdepends on the strain type (tensile or compressive), the sign of ge(h)is the matter of convention. The coupling constants are given in Fig. 2and also summarized in Table I. While the effective masses, as well as hole defor- mation potentials ( gh) for monolayers As and Sb agree well with the literature data, larger electron deformation potentials(g e) have been reported for these materials in Refs. [ 36,59,60]. The discrepancy can be attributed to a different approachutilized in those works. Particularly, uniaxial strain was usedto quantify the band shifts, while we utilized biaxial strainin our work. The latter approach implies that only diagonalelements of the deformation potential tensor are taken intoaccount in our study [see Eq. ( 7)]. This approximation is ex- pected to quantitatively overestimate the mobilities calculatedbelow. 0 0.1 0.2 0.3 0.4 10101011101210131014(e)Sb electronsτ⊥−1 / (τ\|\|−1+ τ⊥−1) Carrier density, n (cm−2)Wick Screening0 0.1 0.2 0.3 10101011101210131014(c)As electronsτ⊥−1 / (τ\|\|−1+ τ⊥−1) Carrier density, n (cm−2)Wick Screening0 0.1 0.2 0.3 10101011101210131014(a)P electronsτ⊥−1 / (τ\|\|−1+ τ⊥−1) Carrier density, n (cm−2)Wick Screening 0 0.1 0.2 0.3 0.4 0.5 10101011101210131014(f)Sb holesτ⊥−1 / (τ\|\|−1+ τ⊥−1) Carrier density, n (cm−2)Wick Screening0 0.1 0.2 0.3 10101011101210131014(d)As holesτ⊥−1 / (τ\|\|−1+ τ⊥−1) Carrier density, n (cm−2)Wick Screening0 0.1 0.2 10101011101210131014(b)P holesτ⊥−1 / (τ\|\|−1+ τ⊥−1) Carrier density, n (cm−2)Wick Screening FIG. 3. The ratio of carrier scattering rates induced by anhar- monic ﬂexural phonons ( τ−1 ⊥) to the total scattering rate ( τ−1 /bardbl+τ−1 ⊥) calculated as a function of carrier concentration ( n) for different types of dopings in monolayers P, As, and Sb. Red and blue lines correspond to the Wick decoupling and screening approximations,respectively, as discussed in Sec. II. Dashed line represents the average of the two approximations with shaded area showing the uncertainty of the model. In all cases T=300 K.IV . RESULTS AND DISCUSSION We ﬁrst analyze the role of ﬂexural phonons and their anharmonic coupling in the scattering of charge carriers in 2Dsemiconductors P, As, and Sb. To this end, we decompose thetotal scattering rate into the two contributions τ −1=τ−1 /bardbl+ τ−1 ⊥, each of which describes the scattering on pure in-plane modes [ﬁrst term on the right-hand side of Eq. ( 12)] and on ﬂexural modes subject to anharmonic coupling with in-planemodes [second term in the same equation]. In Fig. 3,w es h o w the dependence τ −1 ⊥/τ−1of the carrier concentration both for electron and hole dopings calculated at T=300 K. As long as the effective mass approximation holds, from Eq. ( 6) we have τ−1 /bardbl=mg2T(1−ν2)/¯h3Y, meaning that the doping dependence arises from the τ−1 ⊥term only. In the limit of large concentrations τ−1 ⊥/τ−1→0 because τ−1 ⊥∼1/εF, indicating that ﬂexural phonons are negligible in this regime, which canbe clearly seen from Fig. 3for all the cases considered. At small concentration the behavior is quantitatively different forthe two approximations describing the anharmonic coupling.Nevertheless, the trend is qualitatively similar: The contri-bution of ﬂexural phonons becomes independent or weaklydependent on the doping. This behavior is consistent with q dependence of the scattering matrix /angbracketleft\|V q\|2/angbracketright, which at q/lessmuchq∗ behaves as ∼q−0.36within the Wick decoupling approxima- tion and approaches a constant in the screening approxima-tion. On average, the contribution of ﬂexual phonons at roomtemperature does not exceed 30% of the total scattering ratefor all materials considered. 0 0.1 0.2 0.3 0.4 0.5 50 100 150 200 250 300(e)Sb electronsτ⊥−1 / τ\|\|−1 Temperature, T (K)n=1×1011 cm−2 n=2×1013 cm−20 0.1 0.2 0.3 0.4 0.5 50 100 150 200 250 300(c)As electronsτ⊥−1 / τ\|\|−1 Temperature, T (K)n=1×1011 cm−2 n=2×1013 cm−20 0.1 0.2 0.3 0.4 50 100 150 200 250 300(a)Pelectronsτ⊥−1 / τ\|\|−1 Temperature, T (K)n=1×1011 cm−2 n=2×1013 cm−2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 50 100 150 200 250 300(f)Sb holesτ⊥−1 / τ\|\|−1 Temperature, T (K)n=1×1011 cm−2 n=1×1013 cm−20 0.1 0.2 0.3 0.4 50 100 150 200 250 300(d)As holesτ⊥−1 / τ\|\|−1 Temperature, T (K)n=1×1011 cm−2 n=3×1013 cm−20 0.1 0.2 50 100 150 200 250 300(b)Pholesτ⊥−1 / τ\|\|−1 Temperature, T (K)n=1×1011 cm−2 n=4×1013 cm−2 FIG. 4. Temperature dependence of the ratio between ﬂexural (τ−1 ⊥) and in-plane ( τ−1 /bardbl) scattering rates calculated for different car- rier concentrations ( n) and doping types in P, As, and Sb monolayers. Red, blue, and dashed lines have the same meaning as in Fig. 3. 075417-6INTERPLAY BETWEEN IN-PLANE AND FLEXURAL … PHYSICAL REVIEW B 100, 075417 (2019) 10000 15000 20000 25000 30000 10101011101210131014(e)SbelectronsMobility, μc (cm2V−1s−1) Carrier density, n (cm−2)T=100 K T=300 K 6000 8000 10000 12000 14000 16000 18000 20000 10101011101210131014(c)AselectronsMobility, μc (cm2V−1s−1) Carrier density, n (cm−2)T=100 K T=300 K 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 10101011101210131014(a)PelectronsMobility, μc (cm2V−1s−1) Carrier density, n (cm−2)T=100 K T=300 K 1000 1500 2000 2500 3000 3500 4000 4500 10101011101210131014(f)SbholesMobility, μc (cm2V−1s−1) Carrier density, n (cm−2)T=100 K T=300 K 200 300 400 500 600 700 800 10101011101210131014(d)As holesMobility, μc (cm2V−1s−1) Carrier density, n (cm−2)T=100 K T=300 K 300 400 500 600 700 800 900 10101011101210131014(b)PholesMobility, μc (cm2V−1s−1) Carrier density, n (cm−2)T=100 K T=300 K FIG. 5. Carrier mobility calculated as a function of electron and hole doping in hexagonal monolayers P, As, and Sb for two differenttemperatures. Dashed lines correspond to the case of scattering by in-plane phonons only, while solid lines take both in-plane and ﬂexural phonons into account. Error bars show uncertainty due tothe approximations used to calculate the anharmonic coupling. Let us now examine the temperature dependence of the scattering rate. As has been shown above τ−1 /bardbl∼Tinde- pendently of the carrier concentration. On the other hand,τ −1 ⊥∼T2in the harmonic regime ( q/greatermuchq∗), while it soft- ens signiﬁcantly in the strong anharmonic regime ( q/lessmuch q∗), yielding τ−1 ⊥∼T1.18or∼Tdepending on the approx- imation employed. The temperature dependence calculatedfor materials under consideration is shown in Fig. 4.T h e trend is similar for all the cases considered. In particular,at low temperatures τ −1 ⊥/τ−1 /bardblis small as the electron gas is degenerate ( T/lessmuchεF), resulting in a negligible ﬂexural phonon contribution. At higher temperatures, the behavioressentially depends on the carrier concentration. At smallconcentration around 10 11cm−2,τ−1 ⊥/τ−1 /bardblis almost inde- pendent of the temperature, indicating the strongly anhar-monic regime. At concentrations of the order of 10 13cm−2, the corresponding ratio ∼T, as is expected from the limit q/greatermuchq∗. As a ﬁnal step, we make a quantitative estimate of the carrier mobility, presented in Fig. 5for a wide range of electron and hole doping in P, As, and Sb. At large carrier con-centrations, scattering by in-plane phonons dominates, and themobility approaches its upper limit for a given temperature.As the concentration drops down, the mobility decreases by15–30% for different materials in the carrier density rangefrom 10 12to 1014cm−2. This effect is attributed to the increased role of ﬂexural phonons, whose scattering rate isinversely proportional to the carrier concentration. At lowerdopings, the scattering behavior is modiﬁed by the strong anharmonic coupling between in-plane and ﬂexural phonons.In particular, below 10 12cm−2the mobility behavior is nearly independent of the material and doping type, resulting in aconstant value. Quantitatively, the mobilities are strongly dependent on the carrier type. The electron mobilities are signiﬁcantlyhigher for all the systems considered, which can be attributedto relatively small effective masses and deformation poten-tials, listed in Table I. The lowest values not exceeding 1000 cm 2V−1s−1are obtained for the hole doping of P and As, i.e., materials with relatively massive carriers. It isinteresting to note that the electron-doped hexagonal (blue)phosphorus exhibits mobilities an order of magnitude largerthan its orthorhombic (black) counterpart [ 16]. It should be emphasized that the results presented in Fig. 5represent an upper limit for the intrinsic mobility. A number of impor-tant factors, such as, for instance, optical phonons, dielectricscreening, many-body renormalization, and other effects, dis-cussed, for example in Refs. [ 27] and [ 63], are not taken into account in the present study. This can explain some discrep-ancy with the results obtained from ﬁrst principles [ 28,36,37]. Moreover, under realistic conditions, charge carriers are sub-ject to extrinsic scattering by impurities, defects, and sub-strates, which could further reduce the mobility by orders ofmagnitude. V . CONCLUSION Flexural phonons are the inherent phenomena of 2D ma- terials, which necessarily contributes to the electron-phononscattering. Unlike conventional in-plane modes, whose con-tribution to electronic transport is weakly dependent onthe carrier concentration, scattering on ﬂexural phonons be-comes more effective in the regime of low carrier den-sities. At the same time, this regime is characterized bythe strong anharmonic coupling between in-plane and ﬂex-ural modes, resulting in the modiﬁcation of the scatteringbehavior. Here we developed a theory for the electron-phonon scat- tering in 2D semiconductors taking the anharmonic effectsinto account. We applied this theory to examine the scatter-ing behavior in typical group V hexagonal monolayers. Weshowed that although the scattering rate is affected by theﬂexural phonons, the net effect does not exceed 30%. Com-pared to the harmonic approximation, where the scattering byﬂexural phonons is divergent in the limit of low concentra-tions, the anharmonic coupling suppresses this effect signiﬁ-cantly. For all materials considered, at carrier concentrationsbelow 10 12cm−2ﬂexural phonon contribution turns out to be nearly independent of doping in a wide range of temperatures.Figure 5summarizes our ﬁndings. From the practical point of view, the suppression of ﬂexural phonons (e.g., by a substrate or by strain) is considered tobe one of the ways to enhance the mobility in 2D semi-conductors. Our results show that this approach is highlylimited and cannot lead to signiﬁcant mobility gain. On thetheory side, ﬂexural phonons can be safely excluded from theconsideration in the electronic transport calculations. 075417-7A. N. RUDENKO et al. PHYSICAL REVIEW B 100, 075417 (2019) ACKNOWLEDGMENTS A.V .L. and M.I.K. acknowledge support from the research program “Two-dimensional semiconductor crystals” ProjectNo. 14TWOD01, which is ﬁnanced by the Netherlands Or-ganisation for Scientiﬁc Research (NWO). S.Y . acknowledgessupport from the National Key R&D Program of China (GrantNo. 2018FYA0305800). Numerical calculations presented inthis paper have been partly performed on a supercomputingsystem in the Supercomputing Center of Wuhan University.Computational facilities of Radboud University (TCM /IMM) funded by the FLAG-ERA JTC2017 Project GRANSPORTare also gratefully acknowledged. The work is partiallysupported by the Russian Science Foundation, Grant No.17-72-20041. APPENDIX: DERIV ATION OF ANHARMONIC SCATTERING PROBABILITY Here we provide details on the derivation of Eq. ( 12). Treating the phonon ﬁelds classically, the thermal averageof\|V q\|2is given by the following functional integral over in-plane and out-of-plane displacement ﬁelds: /angbracketleft\|Vq\|2/angbracketright=Z−1/integraldisplay Dh(r)/integraldisplay Du(r)\|Vq\|2e−βH, (A1) where the Boltzmann weight is determined by the Hamilto- nian in Eq. ( 11). The partition function Zis given by: Z=/integraldisplay Dh(r)/integraldisplay Du(r)e−βH. (A2) Before performing the Gaussian integration over in-plane ﬁelds, it is useful to decompose fαβ(q), the Fourier transform offαβ(r)=∂αh(r)∂βh(r), into longitudinal and transverse parts [ 19,20]: fαβ(q)=i[qαφβ(q)+qβφα(q)]+PT αβ(q)fT(q).(A3) Here PT αβ(q)=δαβ−qαqβ/q2is the transverse projector and fT(q)=PT γδ(q)fγδ(q). It is then convenient to change vari- ables in the functional integral [ 19,20] and to integrate overshifted in-plane ﬁelds, deﬁned in Fourier space as: ˜uα(q)≡uα(q)+φα(q)−νi 2qα q2fT(q), (A4) where ν=λ/(λ+2G) is the 2D Poisson ratio. In terms of the variables ˜ uα(q), the Hamiltonian reads H=/summationdisplay q/braceleftbigg1 2κq4\|hq\|2+Y 8fT(q)fT(−q) +1 2[(λ+G)qαqβ+Gq2δαβ]˜uα(qt)˜uβ(−q)/bracerightbigg ,(A5) where Y=λ(1−ν2)/ν. The Fourier transform of the defor- mation potential can be written as Vq=igqα˜uα(q)+g 2(1−ν)fT(q). (A6) The out-of-plane ﬁelds and the in-plane modes ˜ uα(q)a r en o w decoupled in the Hamiltonian. 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PhysRevB.75.205327.pdf	Spatiotemporal dynamics in optically excited quantum wire-dot systems: Capture, escape, and wave-front dynamics D. Reiter, M. Glanemann, V. M. Axt, and T. Kuhn Institut für Festkörpertheorie, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany /H20849Received 14 July 2006; revised manuscript received 17 November 2006; published 18 May 2007 /H20850 Transitions of optically excited carriers between delocalized states in a quantum wire and localized states in a quantum dot are studied on a quantum kinetic level. These transitions are mediated by the emission orabsorption of longitudinal optical phonons. Three different excitation scenarios are considered: The capture ofa traveling wave packet results in occupations of the bound states and, under suitable conditions, in coherentsuperpositions. Both occupations and coherences decay due to thermal escape processes resulting from theabsorption of phonons. A spatially homogeneous excitation in the quantum wire below the threshold for opticalphonon emission leads to capture processes associated with the buildup of a wave front in the carrier densitybetween regions which are already inﬂuenced by the capture and regions where this inﬂuence is not yetpresent. A selective excitation of the quantum dot exciton is at elevated temperatures followed by thermalescape processes with a subsequent spreading of the carriers in the quantum wire. We ﬁnd that for a physicallymeaningful description of the spatiotemporal dynamics a consistent treatment of both diagonal and off-diagonal density matrix elements is essential in all three scenarios. DOI: 10.1103/PhysRevB.75.205327 PACS number /H20849s/H20850: 78.67. /H11002n, 73.63. /H11002b, 72.20.Jv, 72.10.Di I. INTRODUCTION In extended systems quantum mechanical scattering pro- cesses are intrinsically nonlocal in space. At the most basiclevel of theory this is due to the uncertainty between momen-tum and position. When, e.g., homogeneously distributedcarriers interact with a phonon in an extended semiconductorwe have a well-deﬁned momentum transfer in the scatteringprocess but even in principle no information about the posi-tion where the scattering takes place. For spatially inhomo-geneous carrier distributions the quantum kinetic derivationof the scattering term for the Wigner function yields a spatialmemory to comply with momentum-position uncertainty, inaddition to the temporal memory which accounts for energy-time uncertainty. 1The semiclassical Boltzmann scattering term is obtained from the quantum kinetic formula as thelowest order in a gradient expansion. 2This, however, in gen- eral may violate the uncertainty relations. In samples that arespatially structured on a nanometer scale the local environ-ment can have a profound impact on the scattering whichthen acquires a local character. This holds in particular forphonon-induced capture processes, where the carriers per-form transitions between states of different effective dimen-sionality. The ﬁnal state of a successful capture is more lo-calized than the initial state and thus carries informationwhere the scattering has taken place. In this case one shouldexpect that a capture only takes place when the carrier isinitially sufﬁciently close to the localized state and that re-gions far away should not be affected by the capture process.However, if the dynamics of such a system is described bykinetic equations for the occupations of the correspondingstates, any capture process will instantaneously reduce thecarrier density all over the structure. Analogously, the escapeof a carrier from a localized state—e.g., by phononabsorption—will immediately populate the whole extendedcarrier state. Thus, in order to correctly describe the localcharacter of capture and escape processes one has to go be-yond a description in terms of occupations of the eigenstates. There are many important examples for such capture pro- cesses, and calculations of the carrier dynamics have beenperformed on various levels of the theory. The trapping ofcarriers in the quasi-two-dimensional active region from three-dimensional transport states 3–7is a basic ingredient for a quantum well laser. Quantum dots are usually covered by atwo-dimensional wetting layer, and thus carriers generated inthe wetting layer may be scattered into zero-dimensional dotstates. 8–13Also techniques have been developed to fabricate quantum wires with embedded quantum dots,14–16either by cleaved edge overgrowth or by growth on a patterned sub-strate. Such structures support transitions between one-dimensional and zero-dimensional states. 17,18In the present paper we study phonon capture and escape processes inducedby the interaction of the carriers with longitudinal optical/H20849LO/H20850phonons in quantum wire-dot systems. We show that spatial inhomogeneities that arise either from the spatialstructure of the sample or from spatially focusing of the ex-citation lead to a pronounced local character of the scatteringwhich is reﬂected in characteristic spatiotemporal signaturesin the carrier dynamics. As we are dealing here with pro-cesses on short time and length scales, where the fundamen-tal limitations set by the uncertainty relations should matter,a description along the lines of the semiclassical Boltzmannequation reaches its limits. Therefore, all calculations in thepresent paper are performed within the framework of a quan-tum kinetic theory. 2,19,20 Carrier capture processes of a localized traveling wave packet from a semiconductor quantum wire into a quantumdot at zero temperature have been studied in previouspapers. 17,18There we have shown that the combination of ultrashort length and time scales gives rise to pronouncedquantum mechanical features in the capture dynamics suchas the appearance of phonon Rabi oscillations and the cap-ture into superpositions of bound states. 17Both the occupa- tions of the bound states and the quantum mechanical coher-PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 1098-0121/2007/75 /H2084920/H20850/205327 /H208499/H20850 ©2007 The American Physical Society 205327-1ences can be efﬁciently controlled by suitable two-pulse excitations.18In the present paper we will ﬁrst extend these studies involving a spatially localized excitation to highertemperature, where besides phonon emission also phononabsorption processes are possible. Due to these phonon ab-sorption processes, carriers which have been trapped may bereleased again from the quantum dot. Then we will address two other excitation scenarios: a delocalized, spatially homogeneous excitation of the quan-tum wire and a spectrally selective excitation of the quantumdot exciton. The central theme in these two scenarios will bethe interplay between capture and escape processes and thespatiotemporal carrier dynamics in the quantum wire region.While for the modeling of the wave-packet dynamics men-tioned above it is clear that off-diagonal density matrix ele-ments are necessary to describe the localized carrier distri-butions generated by the focused light pulse, in the case ofspatially homogeneous excitation this is less obvious. Here,the generation process essentially leads only to occupationsof the states—i.e., to a diagonal density matrix. However, wewill show that the local character of the capture and escapeprocesses gives rise to off-diagonal density matrix elementsand that a consistent treatment of diagonal and off-diagonaldensity matrix elements is crucial for a physically meaning-ful description of the spatiotemporal carrier dynamics. As in our previous studies we focus on a quantum wire- dot structure as a prototype for structures involving elec-tronic states with different effective dimensionality. How-ever, we expect that in particular the dynamics afterhomogeneous excitation or after selective excitation of thequantum dot will be qualitatively similar in other structureslike, e.g., a quantum dot in a quantum well or a quantumwire in a quantum well. The paper is organized as follows: In Sec. II we brieﬂy summarize the basic ideas of the theoretical approach. InSec. III we show numerical results for three different sce-narios. First we discuss the inﬂuence of the temperature onthe capture of a traveling wave packet. In the second part weanalyze the carrier dynamics and capture after a spatiallyhomogeneous excitation in the quantum wire. In the last sce-nario we study the thermal escape of carriers generated in thequantum dot, which occurs at elevated temperatures. Finally,in Sec. IV we brieﬂy summarize our results and draw someconclusions. II. THEORY In this paper we study on a quantum kinetic level the carrier dynamics in a semiconductor quantum wire with anembedded quantum dot. The semiconductor is describedwithin a two-band model for electrons and holes. Thequantum wire is of cylindrical shape and in the lateraldirections conﬁned by inﬁnitely high potential barriers.In the longitudinal direction periodic boundary conditionsare used. The quantum dot is shaped by a potential V e/h/H20849z/H20850 =−V0e/hsech /H20849z/a/H20850for electrons and holes. While adescribes the width of the potential, V0e/hspeciﬁes the depth where V0e applies to the electrons and V0h=2 3V0eto the holes. In this paper we consider a GaAs quantum wire with 100 nm2crosssection and different quantum dot parameters as speciﬁed below. Electron-hole pairs are generated in this structure bymeans of a short laser pulse. The laser pulse is coupled bythe dipole interaction to interband transitions, and we willdiscuss both localized and delocalized excitations. The carri-ers are coupled to bulklike LO phonons with energy /H6036 /H9275LO =36.4 meV via the Fröhlich interaction. Bulklike, disper- sionless phonon modes are indeed widely used and have pro-vided good results. 21–24By including the Coulomb interac- tion on the mean-ﬁeld level we furthermore account forexcitonic effects. The laser pulse is modeled by an electric ﬁeld which is assumed to be Gaussian in time and space according to E /H20849+/H20850/H20849z,t/H20850=E0exp/H20873−t2 /H92702−/H20849z−z0/H208502 2/H9268z2−i/H9275t/H20874. /H208491/H20850 E0describes the amplitude of the ﬁeld, /H9270the pulse duration, andz0the position of the laser pulse with variance /H9268z. The frequency /H9275=/H20849/H9280gap+/H9004E/H20850//H6036corresponds to the mean excess energy /H9004Eabove the band gap of the quantum wire /H9280gap.B y choosing /H9268zmuch larger than the length of the structure this includes also the case of a spatially homogeneous excitation.We restrict ourselves to rather low excitation densities so wecan treat the phonon system as a bath. At these low excita-tion densities the Coulomb interaction predominantly leadsto excitonic and renormalization effects, which are fully de-scribed on the level of the mean-ﬁeld approximation. Wehave taken care that the densities are low enough such thatan excitation of multiexcition states can be neglected. The calculations are performed in the basis determined by the single-particle potentials for electrons and holes. As hasbeen discussed in Ref. 17, in this representation the numerics is favorable for excitations near the band gap. The details ofthis model can be found in Refs. 1and17. The basic dynamical variables in the density matrix ap- proach to quantum kinetics are the single-particle densitymatrices deﬁned as /H9267n/H11032,ne=/H20855cn/H11032†cn/H20856,/H9267n/H11032,nh=/H20855dn/H11032†dn/H20856,pn/H11032,n=/H20855dn/H11032cn/H20856, /H208492/H20850 where cn†anddn†/H20849cnanddn/H20850describe the creation /H20849annihila- tion /H20850of an electron and a hole in the nth eigenstate of the respective single-particle potential. The lowest few values ofncorrespond to the bound states in the quantum dot. The other values refer to the free states in the quantum wire,where the carriers can move along the zdirection. The diag- onal elements of the density matrix /H9267n,ndescribe the occupa- tions of the single-particle states. The off-diagonal elements /H9267n/H11032,n/H20849n/HS11005n/H11032/H20850of the density matrix describe the coherences between the states. The coherences include the information over the phase relations between the different states. In thediscrete part they describe coherent superpositions whichmay show up as a result of the capture process. In the con-tinuum part they carry the information on the spatial proﬁle,e.g., of a wave packet generated by a localized excitation.Below we will study their role in the case of a spatiallyhomogeneous excitation.REITER et al. PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 205327-2Because of the Coulomb and electron-phonon interac- tions, we are dealing with a many-body problem. Conse-quently, the equations of motion for the single-particle den-sity matrices are not closed. They constitute the starting pointof an inﬁnite hierarchy of equations which for the electron-phonon interaction we truncate on the level of the quantumkinetic second Born approximation. The Coulomb interac-tion is treated on the mean-ﬁeld level to account for exci-tonic effects and the Coulomb enhancement. The corre-sponding equations can be found in Refs. 1and17.I nt h e next section we will analyze the ultrafast dynamics of opti-cally excited carriers for different excitation scenarios anddifferent temperatures. III. RESULTS A. Localized excitation in the quantum wire: Wave-packet capture at ﬁnite temperature In the ﬁrst case we excite the semiconductor with a laser pulse which has an excess energy of /H9004E=20 meV and a pulse duration of /H9270=50 fs centered at z0=−90 nm with /H9268z =10 nm. Due to the symmetrical excitation in kspace, the wave packet splits into two parts moving in opposite longi-tudinal directions of the quantum wire. Because the mass ofthe holes is much higher than the mass of the electrons, theymove much slower and are therefore not of interest for thepresent discussion. Nevertheless, they are fully included inour calculation. When the electrons reach the quantum dot,they can be captured into the dot by the emission of an LOphonon. An extensive analysis of this capture process at zerotemperature is given in Refs. 17and18. In this section we will extend these calculations to ﬁnite temperatures andstudy the inﬂuence of the temperature on the capture process. For zero temperature there are no phonon absorption pro- cesses possible. Because the energy of the electrons issmaller than the LO phonon energy, also no phonon emissionprocesses can occur while the electrons are in the quantumwire. Under these conditions the presence of the electron-phonon interaction is reﬂected only in polaronic dressing ef-fects such as slight renormalizations of the electronic ener-gies and of the effective mass. The phonon interaction is theneffectively switched on only when the wave packet reachesthe quantum dot. Let us ﬁrst consider a quantum dot with parameters V 0e =30 meV and a=4 nm, which has a single bound level at E=−14.4 meV. Because the mean excess energy of the elec- trons is about one LO phonon energy above the energy of thedot state, the capture rate is maximal. The occupation of thebound state as a function of time is shown in Fig. 1for different temperatures T=0, 77, 150, and 300 K. For T =0 K we see a rise of the occupation up to 19% followed bya damped oscillation resulting in a ﬁnal occupation of about16% at t=1.2 ps. The oscillations reﬂect phonon Rabi oscil- lations between the bound state and the continuum states asdiscussed in detail in Ref. 17. For times longer than about 0.9 ps the occupation is approximately constant. At thesetimes the coherent oscillations have decayed and, because ofthe absence of phonons, the trapped electrons cannot leavethe dot anymore. For T=77 K the occupation does not differmuch from the T=0 K case at all times because at this tem- perature we still have T/H11270T LO=/H6036/H9275LO/kB=423 K. If we raise the temperature to T=150 K, the initial occupation still co- incides with its low temperature value but a subsequent de-cay sets in which reduces the dot occupation to about 13% att=1.2 ps. For room temperature T=300 K the decrease is much stronger and only 6% of the occupation is left in thequantum dot at t=1.2 ps. For higher temperatures an inco- herent thermal emission of trapped electrons occurs, leadingto a substantial decrease of the occupation. Electrons escapefrom the quantum dot by absorbing a phonon and can thentravel in both directions away from the quantum dot. Thisincoherent escape process is in clear contrast to the coherentescape process associated with the phonon Rabi oscillationswhich occurs at short times. The Rabi oscillations lead to adirectional emission of follow-up wave packets moving inthe same direction as the initial wave packet. These wavepackets are emitted always when the occupation of the boundstate has a minimum—i.e., when the electronic density ma-trix has rotated back to the continuum states. 17We ﬁnd that this coherent escape is present at all temperatures, but with increasing temperature the thermally activated incoherent es-cape eventually becomes dominant. Let us now turn to a quantum dot with several bound levels. In this case the capture process may result not only inan occupation of the bound states but also in coherent super-positions of these states. As an example of a dot with three bound levels at −30.6 meV, −15.2 meV, and −5.8 meV /H20849V 0e =40 meV and a=10 nm /H20850we discuss the inﬂuence of the temperature on the coherences. In Figs. 2/H20849a/H20850and2/H20849b/H20850the occupations of the three bound levels as well as the totaloccupation of the bound states are shown for T=0 K and T =300 K, respectively. Figures 2/H20849c/H20850and2/H20849d/H20850show the real part of the coherences between the bound levels for the twotemperatures. All quantities have been normalized to the totalnumber of generated electrons. In the zero-temperature casethe total occupation rises up to more than 30% while at roomtemperature T=300 K the occupation reaches only 25% and decreases afterwards due to thermal escape processes. Thecoherences between the discrete dot states build up when theelectrons reach the region of the dot and are oscillating intime with a period corresponding to the energy differencebetween the discrete dot states. While for T=0 K the oscil- lations are essentially undamped, for T=300 K we observe a clear damping of the oscillations.01020304050 0 0.3 0.6 0.9 1.2ρ00(%) t (ps)0K 77 K 150 K 300 K FIG. 1. Occupation of the quantum dot ground state after exci- tation of a traveling wave packet in the quantum wire for differenttemperatures. The quantum dot has a single bound state at−14.4 meV.SPATIOTEMPORAL DYNAMICS IN OPTICALLY EXCITED … PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 205327-3For a quantitative analysis of this damping we have ﬁtted the time dependence of the coherence between the two low-est bound states /H926701by an exponentially damped sinusoidal oscillation. It turns out that this ﬁt reproduces well the cal-culated curves after about 500 fs. The decay rates /H9270−1ob- tained from this ﬁt are shown for different temperatures assymbols in Fig. 3. Obviously we observe a strong increase with increasing temperature. It should be expected that thisdecay of the coherences, like in the case of the occupations,is a result of the thermal emission of the trapped electronsdue to the absorption of phonons, which is proportional tothe phonon occupation. The dashed curve in Fig. 3shows the function /H9270−1=/H92700−1 1 exp/H20873/H6036/H9275LO kBT/H20874−1, /H208493/H20850 where the value of /H92700−1has been obtained by a ﬁt to the extracted decay rates and is given by /H92700−1=4.6 ps−1. Indeed, the temperature dependence of the decay rate is very wellreproduced by the Bose function for LO phonons. B. Delocalized excitation in the quantum wire: Capture and wave-front dynamics After having analyzed the capture of a wave packet which has been generated by a strongly localized excitation, wewill now address the carrier dynamics and capture processes after a spatially homogeneous excitation in the quantumwire. We again choose a laser pulse with a mean excessenergy of 20 meV and a pulse duration of /H9270=50 fs, but now the variance /H9268zof the laser pulse is taken to be much larger than the length of the structure such that we achieve a spa-tially homogeneous excitation. Since coherences between thebound states are only created by spatially narrow wave packets, 25such coherences will be of minor importance in the present case. Therefore we will limit ourselves to the quantum dot with a single bound level at −14.4 meV /H20849V0e =30 meV and a=4 nm /H20850. In order to avoid boundary effects a sufﬁciently large system size has to be taken. To keep theproblem numerically tractable, in this scenario the Coulombmean-ﬁeld terms have been neglected. We have checked that,apart from some slight modiﬁcation of the generated carrierdensity due to the missing Coulomb enhancement, this hasessentially no inﬂuence on the carrier dynamics. Figures 4/H20849a/H20850and 4/H20849b/H20850show the density of the free electrons—i.e., the density calculated without taking into ac-count the bound state in the dot—at lattice temperatures ofT=0 K and T=300 K, respectively, as a function of the spa- tial coordinate zfor different times. At t=0 fs—i.e., at the pulse maximum—the density is indeed homogeneous outsidethe dot region. In the region of the quantum dot the density isreduced because of the orthogonality of the continuum stateswith respect to the bound state. At later times for both tem-peratures we observe on each side of the dot a wave frontbetween a region with low density and a region with highdensity which moves at constant speed away from the dot.This wave front marks the transition between the regionswhere carriers at a given time are already missing due to thecapture into the quantum dot from the region which is stillunaffected by the capture. This interpretation is conﬁrmed bylooking at the corresponding Wigner function—i.e., thequantum analog to the classical distribution function—whichis obtained by a suitable Weyl-Wigner transformation fromthe electron density matrix. 1In contrast to the density pro- ﬁles discussed above now also the bound state is included inthe calculation of the Wigner function. This Wigner function01020304050ρii(%)(a)T =0K (b)T= 300 K (c)T =0K (d)T= 300 Kρ00ρ11ρ22ρtotρ00ρ11ρ22ρtot -10-50510 0 0.2 0.4 0.6 0.8 1 1.2Reρij(%) t (ps)ρ01ρ02ρ12 0 0.2 0.4 0.6 0.8 1 1.2 t( p s )ρ01ρ02ρ12FIG. 2. Density matrix elements in the bound subspace of a quantum dot with three boundstates after excitation of a traveling wave packetin the quantum wire. /H20849a/H20850and /H20849b/H20850Occupation of the ground state /H926700and the ﬁrst /H926711and second /H926722excited states for T=0 K and T=300 K, re- spectively; /H20849c/H20850and /H20849d/H20850real part of the coherences between the three discrete dot states for T=0 K andT=300 K. 01234 0 100 200 300 400 500decay rate (ps-1) T (K) FIG. 3. Symbols: decay rate of the coherence /H926701as a function of temperature extracted from the numerical data; line: ﬁt to a Bosedistribution of LO phonons.REITER et al. PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 205327-4is shown in Fig. 5for two different times t=0.2 ps and t =0.6 ps at both temperatures. At t=0.2 ps, shortly after the excitation is ﬁnished, the Wigner function is essentially sym-metric with respect to both space zand momentum k. The momentum distribution is determined by the spectral proper- ties of the generating laser pulse and is concentrated aroundtwo maxima at ± k maxcorresponding to the excess energy of 20 meV. At t=0.6 ps the captured density can be clearly ob- served at z/H110150. Because of the spatial localization of the bound state the Wigner function extends to rather high k values around z=0. At the maxima of the momentum distri- bution at ± kmaxtwo valleys have built up, for positive mo- menta in the positive zdirection and for negative momenta inthe negative zdirection, both beginning at z=0, where the dot is located. These valleys show up because the electronsare moving in the direction of their momenta. When reachingthe dot those carriers which have an energy about one LO-phonon energy above the bound state are trapped inside thedot and therefore cannot pass the dot region. Thus, behindthe dot these electrons are missing and a valley is formed inthe momentum distribution. If the energy of the carriers doesnot satisfy the resonance condition for capture, the carrierspass the dot. These carriers are responsible for the lowerplateau value in Figs. 4/H20849a/H20850and4/H20849b/H20850. The ﬁnite width of the valleys reﬂects the quantum kinetic energy-time uncertainty.A ﬁnite spread of phonon energies, which is neglected in thepresent paper, would further increase the width of the val-leys, resulting in an additional spread of the velocity of thewave front and a reduction of the plateau value. For not toostrong phonon dispersion we therefore expect only quantita-tive but no qualitative changes of the behavior predicted inthis paper. Interestingly, not only the captured carriers butalso the carriers which are not trapped because they are toofar from the resonance condition for a capture process expe-rience the presence of the dot which results in an increasedgroup velocity around z=0. This leads to a slight structure in the Wigner function along the line z=/H6036kt/m. As a further check of this interpretation we have per- formed calculations with a modiﬁed excess energy of thelaser pulse of 10 meV instead of 20 meV before /H20849not shown /H20850. Indeed we ﬁnd again wave fronts in the density of free carriers moving away from the dot at the same speed asbefore. However, the plateau value of the density betweenthese wave fronts is larger. This again demonstrates that thewave front results from carriers with an energy of /H6036 /H9275LO above the bound state. Therefore its speed depends only on the energy of the bound state. The excess energy determinesthe fraction of carriers which are available for a capture pro-cess and thus the total capture efﬁciency. 18 The main features of the transport and capture dynamics discussed so far are the same for both temperatures 0 K and300 K. This is because on the short length and time scalesand for the excitation conditions chosen here the transport isto a large extent ballistic and therefore independent of tem- perature. However, when looking in detail differences be-tween the two temperatures are clearly observable. First, theWigner function for T=300 K exhibits nonzero values also atkvalues above the generated distribution. These are carri- ers which, after the optical generation, have absorbed a pho-non. Second, when looking at the density of free carriers inthe immediate vicinity of the dot we ﬁnd that, while at zerotemperature /H20851Fig.4/H20849a/H20850/H20852this density is continuously decreas- ing with increasing time, at 300 K /H20851Fig.4/H20849b/H20850/H20852it is increasing with time. Here we observe again the thermal escape of car-riers which previously have been trapped. Like in the case ofthe traveling wave packets discussed above, this thermal es-cape leads to a reduced occupation of the bound state as isshown in Fig. 6where the ground-state occupation is plotted as a function of time for T=0 K /H20849solid line /H20850andT=300 K /H20849dashed line /H20850. While the zero-temperature curve exhibits a linear growth in the full time range shown, the 300- K curvebends down and leads to an occupation which is at t =1.2 ps only about 50% of the corresponding low-temperature value.00.10.20.30.40.50.60.70.80.91 -300 -150 0 150 300 z(nm)0f s 200 fs400 fs 600 fs 800 fs00.10.20.30.40.50.60.70.80.910f s 200 fsdensity o felectrons in free states (normalized )(a)T= 0K (b)T = 300 K (c)T= 0K without OD400 fs 600 fs 800 fs00.10.20.30.40.50.60.70.80.910f s 200 fs400 fs 600 fs 800 fs FIG. 4. Density of the free electrons as a function of the position after a spatially homogeneous excitation of the quantum wire with a50-fs pulse /H20849pulse maximum at t=0/H20850at different times. /H20849a/H20850T=0 K, /H20849b/H20850T=300 K, and /H20849c/H20850T=0 K but without taking into account the off-diagonal elements of the density matrix.SPATIOTEMPORAL DYNAMICS IN OPTICALLY EXCITED … PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 205327-5In the case of a spatially homogeneous excitation the car- rier generation leads to an essentially diagonal electron den-sity matrix; i.e., the eigenstates of the single-particle poten-tial are occupied with a certain probability but there is nearlyno coherence between different eigenstates. When modelingthe carrier capture by semiclassical capture rates 10but also in quantum kinetic calculations12it is then often assumed that off-diagonal elements can be neglected in the carrier dynam-ics. In order to analyze the role of off-diagonal electron den-sity matrix elements in our scenario we have performed cal-culations where we have switched off all these off-diagonalelements. The resulting density proﬁles at different times forT=0 K are shown in Fig. 4/H20849c/H20850. Obviously, the spatiotemporal dynamics of the electrons in the wire region is completelydifferent when off-diagonal elements are neglected. Withoutoff-diagonal elements no spatially inhomogeneous proﬁlescan be present. Therefore, the capture process reduces the density in the free states simultaneously in the whole system.This is in sharp contrast to the picture of a local captureprocess which should result in a trapping of only those car-riers which are close to the dot. This local picture is clearlysupported by the full calculation including off-diagonal ele-ments. The different dynamical behavior in the wire regionalso modiﬁes the occupation of the bound state, as can beseen in Fig. 6/H20849dash-dotted line /H20850. As already mentioned, in the full calculation at T=0 K the occupation grows linearly over the whole time interval shown. The reason is that thewave fronts in Fig. 4/H20849a/H20850have not yet reached the boundary of the system. Therefore, the distribution of carriers arriving atthe dot is not yet affected by previous capture processes andthus, since the excitation density has been chosen sufﬁcientlyweak such that phase-space-ﬁlling effects of the ﬁnal stateare negligible, the effective capture rate is constant over thewhole time interval. In contrast, when neglecting the off-diagonal elements of the density matrix we observe a cleardeviation from the linear growth of the ground-state occupa-tion. As is evident from Fig. 4/H20849c/H20850, since the density of free carriers is reduced in the whole structure, the ﬁnite systemsize may inﬂuence the capture dynamics from the beginning.The reduction of the carriers which are available for the trap-ping process then reduces the effective capture rate, leadingto a bending of the corresponding curve in Fig. 6. C. Excitation of the quantum dot: Thermal escape processes In the previous sections we have seen that at higher tem- peratures a thermal escape of electrons from the bound statesto the continuum states occurs, leading to a reduction of theeffective capture rate and a decrease of both occupations and-0.4-0.200.20.4 -200-1000100200 k (nm-1)(a)T =0 K; t = 0.2 ps z (nm) -0.4-0.200.20.4 -200-1000100200 k (nm-1)( b ) T=0K ;t=0 . 6p s z (nm)-0.4-0.200.20.4 -200-1000100200 k (nm-1)(c)T = 300 K; t = 0.2 ps z (nm ) -0.4-0.200.20.4 -200-1000100200 k (nm-1)(d) T = 300 K; t = 0.6 ps z (nm ) FIG. 5. Wigner function of the electrons after a spatially homogeneous excitation of the quantum wire with a 50-fs pulse /H20849pulse maximum att=0/H20850for the temperatures T=0 K /H20851/H20849a/H20850,/H20849b/H20850/H20852andT=300 K /H20851/H20849c/H20850,/H20849d/H20850/H20852at times t=0.2 ps /H20851/H20849a/H20850,/H20849c/H20850/H20852andt=0.6 ps /H20851/H20849b/H20850,/H20849d/H20850/H20852. 01020 0 0.3 0.6 0.9 1.2ρ00(%) t(ps)T= 0K T=3 0 0K without OD FIG. 6. Occupation of the quantum dot ground state after a spatially homogeneous excitation of the quantum wire with a 50-fspulse /H20849pulse maximum at t=0/H20850. Solid line: T=0 K. Dashed line: T =300 K. Dash-dotted line: T=0 K but without taking into account the off-diagonal elements of the density matrix.REITER et al. PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 205327-6coherences after the capture has ﬁnished. Let us now con- centrate on this thermal escape process by choosing an opti-cal excitation with an excess energy /H9004E=−52 meV corre- sponding to the lowest quantum dot exciton state of thestructure with a single-electron bound state at −14.4 meV.The pulse duration is /H9270=200 fs, such that only the ground- state quantum dot exciton is excited. The excitation is takento be spatially homogeneous; however, in the present casethis is less important since only the overlap of the light ﬁeldwith the quantum dot exciton is relevant. At zero tempera-ture, except for some very small effects resulting from thepolaron dressing process, the electron and hole density ma-trices are essentially constant after the excitation process.Therefore, in the following we will concentrate on the caseofT=300 K. The dynamics of the escape process is presented in Fig. 7/H20849a/H20850which shows the spatially resolved electronic density in real space on a logarithmic scale. At t=0 fs we see the peak of the bound density inside the quantum dot at z=0, where the ground state of the quantum dot was excited and nodensity in the region of the quantum wire outside the dot.With increasing time we ﬁnd electrons at both sides of the dot which are traveling away from it. These electrons haveescaped from the quantum dot by absorbing a phonon. Wecan see a constant outﬂow from the quantum dot and anelectron density extending over an increasingly large regionof the structure. The spatiotemporal behavior of the holes issimilar to that one of the electrons, but the holes move muchslower because of their higher mass /H20849not shown /H20850. This incoherent escape manifests itself in a symmetrical outﬂow of the electrons at both sides of the dot. This is incontrast to the coherent escape associated with phonon Rabioscillations induced by the capture of a traveling wavepacket, where wave packets are emitted only in the directionof the incoming wave packet. 17The Rabi oscillations occur between the initial /H20849free /H20850state of the electron without phonon and the ﬁnal /H20849trapped /H20850state with the emitted phonon. When one Rabi cycle is completed, the electrons are back in theinitial state and have a certain probability to leave the dotregion. However, the initial state has a certain direction givenby the direction of the incoming wave packet which there-fore determines the direction of the emitted wave packets. The occupation of the ground state for the electrons is shown in Fig. 8/H20849solid line /H20850. The occupation is normalized tothe total occupation of all states, which remains constant after the pulse has ﬁnished. During the generation process wesee the occupation rising monotonically and it reaches amaximum of about 80% at about 250 fs. The fact that thisvalue is considerably below 100% shows that thermal escapeis rather efﬁcient already during carrier generation. Later onthe bound-state occupation decreases monotonically due tothe thermal escape processes and the carriers move awayfrom the dot as discussed above. These results obtained from the full quantum kinetic cal- culation with all elements of the density matrix are nowagain compared to calculations where the off-diagonal ele-ments have been neglected. The resulting proﬁles of the car-rier density /H20849on a logarithmic scale /H20850are shown in Fig. 7/H20849b/H20850, and the occupation of the electron ground state is plotted asdashed line in Fig. 8. As in the previous section we ﬁnd again that the spatiotemporal dynamics cannot be reproducedwithout taking into account the off-diagonal elements. Withthis approximation the escape of a carrier from the boundstate due to phonon absorption immediately results in anoccupation in the whole system which is in clear contrast tothe local nature of carrier-phonon interaction. Also whenlooking at the occupation of the bound state /H20849Fig. 8/H20850we observe pronounced differences. After the initial rise due tothe carrier generation process, the dashed line decays muchfaster than the solid line and after about 1 ps it reaches asteady state. This can again be understood from the fact that-180 -90 0 90 180 z(nm)0f selectron density (logarithmic scale ) 50 fs100 fs150 fs200 fs300 fs400 fs500 fs (a) -180 -90 0 90 1800f s z(nm)50 fs100 fs150 fs200 fs300 fs400 fs(b) 500 fs FIG. 7. Electron density as a function of po- sition on a logarithmic scale at different timesafter selective excitation of the quantum dotground state exciton by a 200-fs laser pulse. Thetemperature is T=300 K. /H20849a/H20850Full model includ- ing the off-diagonal density matrix elements and/H20849b/H20850without the off-diagonal elements. 020406080100 -0.3 0 0.3 0.6 0.9 1.2ρ00(%) t (ps)with OD without OD FIG. 8. Occupation of the quantum dot ground state after selec- tive excitation of the quantum dot ground-state exciton by a 200-fslaser pulse. The temperature is T=300 K. Solid curve: full model including the off-diagonal density matrix elements. Dashed curve:without the off-diagonal elements.SPATIOTEMPORAL DYNAMICS IN OPTICALLY EXCITED … PHYSICAL REVIEW B 75, 205327 /H208492007 /H20850 205327-7without off-diagonal density matrix elements the thermal es- cape populates states in the whole structure. We then have adynamical behavior which is similar to that one described bysimple rate equations for the occupations of the states: Afteran initial evolution on a time scale determined by the inversetransition rates a steady state is reached where phonon ab-sorption and emission processes balance each other. Thistime scale is here of the order of a few hundred femtosec-onds. In contrast, in the full calculation no steady state canbe reached before the carriers in the continuum states havereached the boundary of the system. IV . CONCLUSIONS In this paper we have analyzed the spatiotemporal dynam- ics of optically excited carriers associated with transitionsbetween states of different dimensionality. To be speciﬁc, wehave studied a quantum wire with an embedded quantum dotwhere transitions between the states occur due to the emis-sion or absorption of LO phonons. The main subjects of ourpresent analysis have been the role of the temperature andthe dynamics of a wave front resulting from a combinationof local capture and escape processes with the essentiallyballistic motion of carriers with energies below the LO pho-non energy in the quantum wire region. In the case of the capture of a traveling wave packet, which has been studied previously at T=0 K, we ﬁnd that ﬁnite temperatures lead to a decay of the occupations of thebound quantum states as well as of the coherences amongthese states. This decay is due to phonon absorption pro-cesses leading to a thermal emission of carriers which havebeen trapped previously. Indeed, the temperature dependenceof the decay rates is well described by the equilibrium Bosedistribution of the LO phonons. In contrast to the coherentescape associated with phonon Rabi oscillations, which leadsto the emission of follow-up wave packets traveling in thesame direction as the incident wave packet, the thermal es-cape occurs symmetrically into both directions. When the carriers are generated spatially homogeneously in the quantum wire, the capture process leads to the build upof wave fronts in the carrier density proﬁle between the re-gions which are already inﬂuenced by the carrier capture andthose which have not yet experienced this inﬂuence. Thewave fronts move at a velocity given by the group velocityof carriers which are one LO phonon energy above the boundstate, because these carriers are most efﬁciently captured.The plateau value of the density between the wave fronts, onthe other hand, is determined by the excess energy of theexciting laser pulse, because this excess energy determines the total capture efﬁciency. For an accurate description ofthis spatiotemporal dynamics it is important to include off-diagonal density matrix elements of the carrier density ma-trices even in the case of a homogeneous excitation, sincethey contain the information on spatial inhomogeneities aris-ing from the combination of transport and capture. Finally we have studied the thermal escape process after selective excitation of the lowest quantum dot exciton state.The thermal escape leads to a characteristic spatiotemporaldynamics of the ejected carriers in the quantum wire whichmove ballistically away from the quantum dot. In this caseagain a neglect of the off-diagonal elements leads to an un-physical behavior in the description of this dynamics. Butalso the dynamics of the occupation of the bound states ex-hibits pronounced differences when comparing calculationswith and without off-diagonal elements. Without off-diagonal elements a steady state is reached much earlier, be- cause the ejected carriers immediately notice the ﬁnite sys-tem size while in the full calculation they have to reach theboundary by transport. A possible scheme in order to monitor the quantum dot occupation relevant for the capture or escape scenario couldbe a spectrally and temporally resolved pump-probe experi-ment resonant on the quantum dot level. To observe the spa-tiotemporal carrier wave front dynamics in the quantum wireregion a sufﬁciently high spatial resolution is required. Thismight be achieved with near-ﬁeld techniques, 26,27for which spatial resolutions of a few nanometers have already beendemonstrated. 28 In summary, we have shown that a quantum kinetic treat- ment of the interacting carrier-phonon system provides aconsistent description of the ultrafast light-induced dynamicsin structures involving states of different dimensionality bothfor localized and delocalized optical excitation. It avoids thenecessity of preselecting the ﬁnal state of a scattering processwhich is necessary when calculating a semiclassical rate ac-cording to Fermi’s golden rule and therefore is able to de-scribe the capture into superposition states. Furthermore, byincluding the off-diagonal elements of the density matrix,transport and scattering processes are treated on the samefooting and all the fundamental uncertainty relations such asthe uncertainties between time and energy or between posi-tion and momentum are fully accounted for. The full calcu-lations clearly reveal the local character of phonon-inducedcapture and escape processes by affecting only the con-tinuum occupations in the vicinity of the quantum dot. 1M. Herbst, M. Glanemann, V. M. Axt, and T. Kuhn, Phys. Rev. B 67, 195305 /H208492003 /H20850. 2H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors /H20849Springer, Berlin, 1998 /H20850. 3H. Shichijo, R. M. Kolbas, N. Holonyak, R. D. 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Betz, Appl. Phys. Lett. 87, 153113 /H208492005 /H20850. 14G. Schedelbeck, W. Wegscheider, M. Bichler, and G. Abstreiter, Science 278, 1792 /H208491997 /H20850. 15C. Lienau, V. Emiliani, T. Guenther, F. Intonti, T. Elsaesser, R. Nötzel, and K. H. Ploog, Phys. Status Solidi A 178, 471 /H208492000 /H20850. 16W. Wegscheider, G. Schedelbeck, G. Abstreiter, M. Rother, and M. Bichler, Phys. Rev. Lett. 79, 1917 /H208491997 /H20850. 17M. Glanemann, V. M. Axt, and T. Kuhn, Phys. Rev. B 72, 045354 /H208492005 /H20850. 18D. Reiter, M. Glanemann, V. M. Axt, and T. Kuhn, Phys. Rev. B 73, 125334 /H208492006 /H20850. 19V. M. Axt and T. Kuhn, Rep. Prog. Phys. 67, 433 /H208492004 /H20850. 20F. Rossi and T. Kuhn, Rev. Mod. Phys. 74, 895 /H208492002 /H20850.21S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bastard, J. Zeman, A. Lemaître, and J. M. Gérard, Phys. Rev. Lett. 83, 4152 /H208491999 /H20850. 22O. Verzelen, R. Ferreira, and G. Bastard, Phys. Rev. Lett. 88, 146803 /H208492002 /H20850. 23E. A. 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PhysRevB.87.060503.pdf	RAPID COMMUNICATIONS PHYSICAL REVIEW B 87, 060503(R) (2013) ϕ-state and inverted Fraunhofer pattern in nonaligned Josephson junctions Mohammad Alidoustand Jacob Linder† Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (Received 5 June 2012; revised manuscript received 8 January 2013; published 15 February 2013) A generic nonaligned Josephson junction in the presence of an external magnetic ﬁeld is theoretically considered and an unusual ﬂux-dependent current-phase relation (CPR) is revealed. We explain the originof the anomalous CPR via the current density ﬂow induced by the external ﬁeld within a two-dimensionalquasiclassical Keldysh-Usadel framework. In particular, it is demonstrated that nonaligned Josephson junctionscan be utilized to obtain a ground state other than 0 and π, corresponding to a so-called ϕjunction, which is tunable via the external magnetic ﬂux. Furthermore, we show that the standard Fraunhofer central peak of thecritical supercurrent may be inverted into a local minimum solely due to geometrical factors in planar junctions.This yields good consistency with a recent experimental measurement displaying such type of puzzling feature[Keizer et al. ,Nature (London) 439, 825 (2006) ]. DOI: 10.1103/PhysRevB.87.060503 PACS number(s): 74 .50.+r, 74.45.+c, 74.25.Ha, 74 .78.Na A sinusoidal current-phase relation (CPR) and Fraunhofer response of the critical supercurrent through an s-wave Josephson contact exposed to an external magnetic ﬁeldare often considered to be standard characteristics of suchjunctions. 1–4Nevertheless, several theoretical studies have been dedicated to the aim of achieving an experimentallyaccessible situation where the CPR is nonsinusoidal. 5–7In this case, the Josephson ground state may be characterized byan arbitrary superconducting phase difference ϕ, 5–9rather than the so-called 0 and πstates.10The ﬁrst experimental realization of such a ϕjunction was very recently reported in Ref. 11. Recent studies have also pointed to the fact that the conven- tional Fraunhofer pattern in Josephson junctions may be mod-iﬁed by the junction geometry or interfacial pair breaking. 4,8,12 The suppression of the central peak in the interference pattern can also occur in systems consisting of a superpositionof multiple 0- πjunctions. 7,8,12However, there still exists experimentally observed magnetic interference proﬁles thatremain unsettled in terms of a theoretical explanation of thephysical origin. 13,14In particular, Keizer et al.13observed an anomalous interference pattern with a local minimum at zeroﬂux in addition to slowly damped oscillations of the criticalsupercurrent compared to the standard Fraunhofer pattern. Thesetup in Ref. 13consisted of a planar Josephson junction where superconducting leads were deposited on a same sideof a half-metallic ferromagnetic strip which was fully spinpolarized. Figure 1(a) depicts diagrammatically the mentioned experimental setup. To study the system theoretically, Ref. 15 utilized the Eilenberger formalism in a ballistic planar junction,similar to the setup of Ref. 13, while neglecting the orbital motion 16of the quasiparticles. Consequently, an effective spatially dependent superconducting phase difference wasobtained via Ginzburg-Landau theory and substituted intothe Eilenberger equation. An almost /Phi1 0-periodic pattern with nonzero minima of the critical current with respect to externalmagnetic ﬂux was found due to the appearance of secondharmonic (sin 2 ϕ; see also Ref. 5). However, the inverted interference pattern with a local minimum at zero ﬂux wasnot reproduced. In this Rapid Communication, we consider a generic class of Josephson junctions in the presence of an external magneticﬁeld where the position of the superconducting leads relative toeach other is not necessarily aligned (see Fig. 1). The obtained results are derived without recourse to any ansatz — w eh a v ei n - stead utilized a quasiclassical Keldysh-Usadel technique withthe numerical approach in Ref. 12and solved exactly the resul- tant linearized equations of motion for the Green’s function. Asourﬁrstmain result, we unveil that the origin of the unexpected interference pattern in the experiment of Ref. 13lies within the geometry of the setup. In this way, the absence of the standardFraunhofer pattern, which has not been clearly understood, isresolved. In addition to this, we demonstrate as our second main result that the CPR in nonaligned junctions takes on avery unusual feature: it becomes shifted by a term proportionalto the external ﬂux /Phi1, namely, I(ϕ,/Phi1)=I 0(/Phi1)s i n [ϕ+ /Theta1(/Phi1)], where ϕis the superconducting phase difference and /Theta1is a geometry-dependent function. Our investigations reveal that the well-known sinusoidal supercurrent and consequentlythe Fraunhofer pattern manifest only in speciﬁc situations. This result is explained in terms of the current density ﬂowstemming from the orbital effect induced by the magnetic ﬁeld.An interesting consequence of the external magnetic ﬂux-shifted superconducting phase difference is that the groundstate of the system may be tuned via the external ﬁeld so thatthe equilibrium phase difference differs from the conventional0o rπsolutions making a so-called ϕjunction. This might constitute a simpler alternative to realizing a ϕstate compared to the array of superconductor/ferromagnet/superconductor(S/F/S) junctions considered in Ref. 11. In the presence of impurity scattering, i.e., the diffu- sive regime of transport, the quasiparticles’ momentum isintegrated over all directions in the space which leads tothe Usadel equation. Solving the Usadel equation in thepresence of a magnetic ﬁeld allows one to compute the currentdensity ﬂow proﬁle in the junction which is different from theindividual trajectories taken by each quasiparticle. Grazingtrajectories are not well deﬁned in this regime although theyneed to be considered carefully in the clean regime (where theEilenberger equation is valid). 16 The starting point for the analysis is the equation of motion for the Green’s function in the diffusive regime provided bythe Usadel equation: 17 D[ˆ∂,ˇG(x,y,z )[ˆ∂,ˇG(x,y,z )]]+i[εˆρ3,ˇG(x,y,z )]=0,(1) 060503-1 1098-0121/2013/87(6)/060503(5) ©2013 American Physical SocietyRAPID COMMUNICATIONS MOHAMMAD ALIDOUST AND JACOB LINDER PHYSICAL REVIEW B 87, 060503(R) (2013) FIG. 1. (Color online) Diagram of considered setups in this paper. An external magnetic ﬁeld H(not shown) is applied to the junction in thezdirection. The junction lengths and widths are dandW, respec- tively. (a) The planar Josephson junction that has experimentally been studied in, e.g., Ref. 13. The widths of the superconducting leads are a s s u m e dt ob e W1LandW−(W1L+W2L). (b) The usual stacked geometry of a Josephson junction with displaced superconducting leads. The superconducting leads’ sizes are W1LandW2Rat the top and bottom of the junction, respectively. (c) Qualitative view of the current density ﬂow inside the normal strip subject to an external magnetic ﬁeld, which is used to describe the origin of the addressedunusual CPR. where ˆ ρ3is the 4 ×4 Pauli matrix. Here, εis the particles’ energy measured from the Fermi level and Dis medium diffusive constant. The commutator is denoted by [ ...,... ]. In the presence of an external magnetic ﬁeld Hand its vector potential A,ˆ∂≡/vector∇ˆ1−ieA(x,y,z )ˆρ3provided that18 [ˆ∂,ˆG(x,y,z )]=/vector∇ˆG(x,y,z )−ie[A(x,y,z )ˆρ3,ˆG(x,y,z )].(2) The vector potential is an arbitrary quantity except for the restriction /vector∇× A=H. We use the Coulomb gauge /vector∇·A=0 throughout our calculations and assume that the externalmagnetic ﬁeld is oriented in the zdirection, i.e., H=Hˆz (see Fig. 1). Thus, we may use A=−yHˆx. In general, the Usadel equation should be simultaneously solved alongwith the Maxwell equation /vector∇× H=μ 0jin a self-consistent manner to take into account the inﬂuence of screening currents.The experimentally relevant scenario is considered wherethe width of the junction Wis smaller than the Josephson penetration length λ J, allowing us to ignore the screening of the magnetic ﬁeld.4,12,19The Usadel motion equation yields a system of nonlinear coupled complex partial differentialequations that should be supported by suitable boundaryconditions for studying junctions. In our Josephson system,we employ the Kupriyanov-Lukichev boundary conditions atnormal metal/superconductor (N/S) interfaces 20and control the leakage of superconductive correlations into the normalstrip using an interface parameter ζ: ζ{ˆG(x,y,z )ˆ∂ˆG(x,y,z )}·ˆn=[ˆG BCS(ϕ),ˆG(x,y,z )],(3) in which ˆnis a unit vector denoting the perpendicular direction to an interface and ϕis the bulk superconducting macroscopic phase. We deﬁne ζ=RB/RFas the ratio between the resistance of the barrier region and the resistance in thenormal sandwiched strip. The bulk solution for the retardedGreen’s function in an s-wave superconductor is given by 18 gR BCS=coshϑ(ε) andfR BCS=eiϕsinhϑ(ε) in which ϑ(ε)= arctanh( \|/Delta1\|/ε). For a weak proximity effect ( ζ/greatermuch1), the normal and anomalous Green’s functions can be approximatedbygR/similarequal1 and \|fR\|/lessmuch 1, respectively. The current density vector is expressed via the Keldysh block as J(/vectorR,ϕ)=J0/integraldisplay dεTr(ρ3{ˆG(x,y,z )[ˆ∂,ˆG(x,y,z )]}K).(4) Here,J0is a normalization constant proportional to the density of states N0at the Fermi level. The total supercurrent Iis ob- tained by integrating the current density over the interface area of the superconducting banks. The ﬂux penetrating the junction is given by /Phi1=dWH . We also investigate the spatial varia- tion of pair potential inside the normal region calculated via U=U0Tr/braceleftbigg (ˆρ1−iˆρ2)/integraldisplay dεˆτ3ˇGK(x,y,z )/bracerightbigg , (5) where U0=−N0λ/16.18In the presence of an external mag- netic ﬁeld, the resultant differential equations and boundaryconditions have a more complicated coordinate dependencewhich renders an analytical solution virtually impossible.Without any orbital effect, such a solution may be obtained. 9To study the considered Josephson junction we use a collocationﬁnite element numerical method the same as Ref. 12.T h e components of approximate solution are assumed to be linearcombinations of bicubic Hermite basis functions satisfying theboundary conditions. Ultimately, the resultant nonsymmetriclinear algebraic equations are solved via a Jacobi conjugate-gradient method. For more details, see Ref. 21. All lengths and energies are normalized by the superconducting coherentlength ξ Sand superconducting gap at absolute zero /Delta10.T h e barrier resistance ζis ﬁxed at 7 ensuring the validity of weak proximity regime. Temperature and junction width are T= 0.05TcandW=10ξS. We use units such that ¯ h=kB=1. Figure 2illustrates the response of the critical Josephson current in a planar junction to an external magnetic ﬁeld asshown schematically in Fig. 1(a). Various parameter values have been considered in order to make our analysis as general FIG. 2. (Color online) Critical supercurrent as a function of external magnetic ﬂux /Phi1through the normal part of the junction. The corresponding pair potential spatial map is given with ϕ=0. Throughout the paper we have assumed that the junction width isﬁxed at W=10ξ S. The ﬁrst and second columns show the critical current Ic/I0vs normalized external magnetic ﬂux /Phi1//Phi1 0and the corresponding pair potential spatial maps with thicknesses d=ξS and 4ξS, respectively. Each row indicates different values of W1Land W2L, namely, the ﬁrst superconducting lead size and the separation of the superconducting leads, respectively (see Fig. 1). 060503-2RAPID COMMUNICATIONS ϕ-STATE AND INVERTED FRAUNHOFER PATTERN ... PHYSICAL REVIEW B 87, 060503(R) (2013) as possible. To do so, we have considered three scenarios where the superconducting leads have different sizes (ﬁrstrow) and where they have equal sizes with a large (second row)and small (third row) separation distance. Speciﬁcally, the thirdrow is relevant with regard to the experiment in Ref. 13where the size of the electrodes far exceeds the separation distance.As seen, in this case the interference pattern exhibits a localminimum at /Phi1=0 rather than a maximum as in the Fraunhofer case, which is fully consistent with the experimental results inRef. 13. Whereas it was speculated that this minimum might be attributed to a shift in the entire interference curve dueto a ﬁnite sample magnetization in Ref. 13, it is obvious that this is not the case here since the sandwiched stripis not ferromagnetic. Moreover, such a shift would makethe current vs ﬂux curve manifestly asymmetric (see, e.g.,Ref. 22), in contrast to the experimental results of Ref. 13 where the central minimum is ﬂanked by two large peaks,similar to our results. Based on this, it seems reasonable toexplain the deviation from the standard Fraunhofer pattern asa result originating from the combination of a planar geometrywith the size and separation distance of the superconductingelectrodes. The latter fact is seen by considering the secondrow of Fig. 2where the separation distance is large compared to the superconductors: A Fraunhofer-like pattern emerges,although the decay becomes more monotonic as the thicknessdof the normal strip increases. Even columns in both Figs. 2 and3show the pair potential where the superconducting phase difference is zero, ϕ=0, and external magnetic ﬂux is set to /Phi1=4/Phi1 0. As seen, the predicted proximity vortices in Refs. 4 and12vanish for the planar junction geometry. However, as will be discussed further below, they reappear in the speciﬁccase of a stacked geometry [Fig. 1(b)]. It is instructive to contrast these results with the geometry of Fig. 1(b) where the two superconducting leads are connected to the normal strip at opposite edges. This is resemblant tothe experimentally often used stacked geometry. The order offrames (critical current and corresponding pair potential spatialmap) are identical to those in Fig. 3and various lead sizes and locations are investigated. It is seen that the location and sizeof both terminals are vital in terms of determining how thecritical current responds to the external ﬂux. For instance, ourresults reveal that only in the speciﬁc case where the widthsof the leads are sufﬁciently large and connected to oppositeedges precisely in front of each other does one recover a FIG. 4. (Color online) (a) W1L=3ξS,W2L=4ξS,W1R=3ξS, andW2R=4ξS;( b ) W1L=6ξS,W2L=4ξS,W1R=0, and W2R=4ξS; and ﬁnally, (c) W1L=2ξSandW2L=6ξS. The top panels represent the CPRs for various values of /Phi1//Phi1 0=0, 3.92, and 6.28. The current density spatial maps in the bottom row show the results for ϕ=0a n d /Phi1=4/Phi10. The superconducting leads’ sizes are set equal at 4 ξSfor all cases as schematically depicted on top of each column. proximity-induced vortex pattern along with the Fraunhofer curve, i.e., I(ϕ,/Phi1)∝/Phi1−1sin/Phi1sinϕ, which is a special case corresponding to the scenario of Ref. 4. The results for the other scenarios in Fig. 3also show good consistency with previous experimental observations.23 It is worth examining the characteristic length scales and thus the radius of the current circulation in Fig. 4. To illustrate this, we consider for concreteness the simplest case of a wideS/N/S junction subject to a perpendicular magnetic ﬁeld (tosee more details, see Ref. 4). In this particular case, the current density is given by J x(/vectorR,ϕ)=J0xsin(ϕ−2π/Phi1 /Phi10Wy). As seen, the characteristic length scale Lcover which the current density changes upon moving along the yaxis is FIG. 3. (Color online) Critical supercurrent against external magnetic ﬂux and corresponding pair potential spatial maps of standard (stacked) Josephson junctions with displaced superconducting leads including various lead sizes. For the pair potential maps, the superconducting phase difference and external magnetic ﬂux are ﬁxed at ϕ=0a n d/Phi1=4/Phi10, respectively. The junction thickness and width are set to d=2ξSand W=10ξS, respectively. 060503-3RAPID COMMUNICATIONS MOHAMMAD ALIDOUST AND JACOB LINDER PHYSICAL REVIEW B 87, 060503(R) (2013) Lc∼/Phi10W//Phi1 . Thus, for magnetic ﬁelds corresponding to several ﬂux quanta Lccan be smaller than the junction size. With increasing external magnetic ﬂux /Phi1, the current density ﬂow shown in Fig. 4takes on smaller radii. Instead, when decreasing the external ﬂux /Phi1→0,Lc→∞ , which means there exists no current circulation in the system. In other words,the current density spatial map of the system is uniform in theabsence of any external magnetic ﬂux. Having unveiled the origin of the anomalous inverted Fraunhofer response, we now turn to the second main resultof this paper: the possibility to generate a ϕjunction in an S/N/S system with an applied magnetic ﬁeld. In Fig. 4,w e provide the CPR in addition to a spatial map of the currentﬂow in the normal strip for three represented geometries.In (a) the leads are connected opposite to each other; in(b) they are connected antisymmetrically, whereas in (c) theyare connected symmetrically in a planar geometry similarto Ref. 13. It is clear that the CPR remains sinusoidal as a function of the superconducting phase difference ϕin both (a) and (b) independent on the applied ﬂux. However, case (c)is qualitatively different. The generic form of the CPR is nowrevealed as I(ϕ,/Phi1)=I 0(/Phi1)s i n [ϕ+/Theta1(/Phi1)] (6) in which I0(/Phi1) and/Theta1(/Phi1) are geometry-dependent functions of external magnetic ﬂux as seen in Fig. 4. In fact, Eq. (6) holds for all situations considered in Fig. 2where we have demonstrated the CPR is never purely sinusoidal. The standardsinusoidal CPR is recovered only for symmetric situationsrelative to the induced orbital motion by the external magneticﬁeld [see Fig. 1(c)]. This observation has a highly interesting consequence: The anomalous magnetic ﬂux-coupled CPRensures that the ground state of the system may be tuned so thatthe equilibrium phase difference differs from the conventional0o rπsolutions. Instead, a so-called ϕjunction may be realized where the ground-state phase difference ϕis tunable via the external ﬂux. We therefore arrive at a ground state withJosephson energy E Jwhich can be controlled by adjusting the applied external magnetic ﬁeld. The idea of a ϕjunction via a superconducting phase difference shift has been consideredpreviously 6in the context of a noncentrosymmetric normal layer with a Rashba spin-orbit interaction. However, in oursetup the external ﬂux is a well-controlled parameter whichallows for easy tuning of the ground state, as opposed tocontrolling a spin-orbit interaction parameter. Moreover, ourﬁnding is different from Ref. 7where two magnetic junctions, one in 0 state and the other in πstate with different lengths, are connected in parallel and consequently generate an extracosinusoidal term in addition to negative second harmonic.What is then the physical origin of this anomalous CPR? The answer to this question may be obtained by investigatingthe current density ﬂow under the inﬂuence of an externalmagnetic ﬁeld inside the normal strip, as seen in Fig. 4. For zero phase difference ϕ=0, the external magnetic ﬁeld induces a current ﬂow where the orbital paths taken by thequasiparticles move with the same ﬂux in and out of thesuperconducting regions—in effect no net current ﬂow, only in special geometrical conﬁgurations . For instance, both in (a) and (b) the current ﬂow between the superconductors inany part of the normal region is seen to have an antisymmetric,and thus canceling, contribution in a different part of thenormal strip at zero phase difference ϕ=0. In contrast, this is no longer the case in setup (c): There is a netﬂow of current induced by the orbital response due to themagnetic ﬂux, even at ϕ=0. To elucidate this clearly in the current ﬂow, one would have to consider the amplitude ofthe local current as well, but the supercurrent-phase curvesnevertheless demonstrate that this interpretation is correct. Inessence, this is a geometry-dependent effect since it relies onthe positioning of the leads relative to the induced currentﬂow via the applied ﬁeld. Thus, it gives rise to the uniquepossibility to alter the standard CPR so that the groundstate of the system can be adjusted by tuning the externalﬂux. To conclude, we have studied the Josephson critical current and its response to an external magnetic ﬂux in experimentallyfeasible nonaligned junctions. Speciﬁcally, a planar geometrysimilar to a recent experiment 13is considered and it is demonstrated that the observed suppression at zero ﬂux maystem from the junction geometry rather than any intrinsicmagnetization. Moreover, it is shown that a highly unusualsupercurrent-phase difference shift occurs inevitably in a classof nonaligned junctions due to an external magnetic ﬂux.Its precise form is sensitive to the size and location of thesuperconducting leads. Consequently, this offers a route toa tunable junction ground state. The physical origin of thiseffect is traced back to the induced current density ﬂow dueto the presence of an external ﬁeld relative to the positionof the superconducting leads. As an interesting consequence,this type of Josephson junction constitutes an attainable wayof realizing the so-called ϕjunction experimentally. We thank G. Sewell for his valuable instructions in the numerical parts of this work. We also thank F. S. Bergeret,E. Goldobin, J. W. A. Robinson, V . V . Ryazanov, and N. Birgefor useful discussions and comments as well as K. Haltermanfor his generosity regarding the compiler source. phymalidoust@gmail.com †jacob.linder@ntnu.no 1B. D. Josephson, Rev. Mod. Phys. 36, 216 (1964). 2J. M. Rowell, Phys. Rev. Lett. 11, 200 (1963); J. Clarke, Proc. R. Soc. London, Ser. A 308, 447 (1969); S. Nagata, H. C. Yang, and D. K. Finnemore, Phys. Rev. B 25, 6012 (1982); H. C. Yang and D. K. Finnemore, ibid. 30, 1260 (1984).3J. P. Heida, B. J. van Wees, T. M. Klapwijk, and G. 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PhysRevB.83.165124.pdf	PHYSICAL REVIEW B 83, 165124 (2011) Theory of the waterfall phenomenon in cuprate superconductors D. Katagiri,1K. Seki,1R. Eder,1,2and Y . Ohta1 1Department of Physics, Chiba University, Chiba 263-8522, Japan 2Karlsruhe Institut of Technology, Institut f ¨ur Festk ¨orperphysik, D-76021 Karlsruhe, Germany (Received 8 October 2010; revised manuscript received 17 February 2011; published 22 April 2011) Based on exact diagonalization and variational cluster approximation calculations we study the relationship between charge transfer models and the corresponding single band Hubbard models. We present an explanation forthe waterfall phenomenon observed in angle resolved photoemission spectroscopy on cuprate superconductors.The phenomenon is due to the destructive interference between the phases of the O2 porbitals belonging to a given Zhang-Rice singlet and the Bloch phases of the photohole which occurs in certain regions of kspace. It therefore may be viewed as a direct experimental visualization of the Zhang-Rice construction of an effectivesingle band model for the CuO 2plane. DOI: 10.1103/PhysRevB.83.165124 PACS number(s): 74 .72.−h, 79.60.−i, 71.10.Fd I. INTRODUCTION The “waterfall phenomenon” observed in angle resolved photoemission spectroscopy (ARPES) on cuprate supercon-ductors has attracted some attention. This phrase summarizesthe following phenomenology: 1–13for photon energies around 20 eV, where photoholes are created predominantly in O2 p states,14a “quasiparticle band” can be seen dispersing away from the Fermi energy as /Gamma1is approached, but then, along the (1,1) direction roughly at (π 4,π 4), rapidly looses intensity and cannot be resolved anymore. Instead there is a “band” of weakintensity which seems to drop almost vertically in kspace toward higher binding energy, thus creating the impressionof a kink in the quasiparticle band. Up to binding energiesof≈1.0 eV there is a “black region” around the /Gamma1point with no spectral weight at all. Finally, at binding energiesof around 1 .0 eV the “waterfall” seems to merge with one or several bands of high intensity, which often correspond verywell to bands predicted by local density approximation (LDA)calculations. In experiments with photon energies ≈100 eV (where an appreciable fraction of the photoholes is created in Cu3 d states 14) and when the spectra are taken in higher Brillouin zones,7,10however, the low energy quasiparticle band can be resolved all the way to /Gamma1and shows no indication of a kink. This band therefore undoubtedly exists and has no kink so thatthe only possible explanation for the “black region” and theapparent kink seen at low photon energies are matrix elementeffects. This conclusion has, in fact, been reached by Inosovet al. on the basis of their experimental data. 7,10A very similar conclusion was also reached by Zhang et al.13who showed that the kink in the quasiparticle band appears only in the secondderivative of the momentum distribution curves, but is absentin the second derivative of the energy distribution curves whichshows a smooth quasiparticle band instead. They concludedthat the kink is not an intrinsic band dispersion. Some support for this point of view comes from the fact that the waterfall phenomenon is observed over the wholedoping range, from the antiferromagnetic insulator to theFermi-liquid-like overdoped compounds, so that it is obviouslyunrelated to any special features of the electronic structure.Moreover, the spectrum of a hole in an antiferromagneticinsulator is one of the very few reasonably well understoodproblems in connection with cuprate superconductors. For this special problem good agreement between the experiment, 15,16 various approximate calculations,17–24and exact diagonal- ization of small clusters25–28has been found and no theory for hole motion in an antiferromagnet predicts a kink inthe quasiparticle band. Here, in particular, the work ofLeung and Gooding 27should be mentioned who studied the spectral function for a single hole in a 32-site cluster byexact diagonalization and found a well-deﬁned and smoothquasiparticle band in excellent agreement with the results ofthe self-consistent Born approximation. 29,30Moreover, since plasmons with energies below the charge transfer gap are notexpected in the insulating compound and since the couplingto spin excitations is obviously described very well by theself-consistent Born approximation, a hypothetic Bosonicmode can be almost certainly ruled out as an explanation of thewaterfall phenomenon in the undoped compounds and, due tothe near-independence of the phenomenon on the doping level,also for the entire doping range. One type of matrix-element effect which partly explains the nonobservation of the quasiparticle band at /Gamma1has been discussed by Ronning et al. 1At the /Gamma1point in the ﬁrst Brillouin zone the photoelectrons are emitted exactly perpendicular to the CuO 2plane which we take as the ( x-y) plane of the coordinate system. In this situation, and if we neglect smalldeviations from this symmetry in the actual crystal structure,the experimental setup has C 4vsymmetry. The expression for the photocurrent31involves the dipole matrix element /angbracketleftf\|A·p\|i/angbracketright, where Ais the vector potential of the incoming light, pis the momentum operator, \|i/angbracketrightthe initial state, and \|f/angbracketright the ﬁnal state. The state \|f/angbracketrightdiffers from \|i/angbracketrightby (a) the presence of an electron in the so-called Low energy electron diffraction (LEED) state, which far from the surface evolves into a plane wave∝eikzand hence transforms according to the identical representation (b) by the presence of a hole with momentum(0,0) in a Zhang-Rice singlet. Since the Zhang-Rice singlet has aC u 3d x2−y2orbital as its “nucleus”32it has the same symmetry. It follows that \|f/angbracketrightand\|i/angbracketrighthave the same parity under reﬂection in the x-yplane so that the dipole matrix element is zero if Ais in the CuO 2plane. On the other hand \|f/angbracketright and\|i/angbracketrightacquire a relative minus sign under a rotation by π 2around the zaxis so that the dipole matrix element is 165124-1 1098-0121/2011/83(16)/165124(11) ©2011 American Physical SocietyD. KATAGIRI, K. SEKI, R. EDER, AND Y . OHTA PHYSICAL REVIEW B 83, 165124 (2011) zero as well if Ais perpendicular to the CuO 2plane. This means that it is impossible to observe a state with thecharacter of a Zhang-Rice singlet at /Gamma1in normal emission. As Ronning et al. pointed out, however, this argument cannot explain the nonobservation of the quasiparticle band at /Gamma1in higher Brillouin zones, where the photoelectrons are no longeremitted in the direction perpendicular to the surface. Moreover, a similar effect has been observed in angle resolved photoemission spectroscopy studies of the compoundSrCuO 2which contains CuO chains. With a photon energy of 22.4 eV and polarization parallel to the chains, which generates holes in σ-bonding O2 porbital, there is no intensity atk=0( R e f . 33). If the photon energy is increased to 100 eV, however, the spinon-band around k=0 can indeed be resolved.34This behavior is similar to the waterfall effect but in this compound the CuO 2plaquettes are perpendicular to the surface of the crystal. We conclude that there must be asecond mechanism for the extinction of spectral weight around/Gamma1in both, the one- and two-dimensional systems. Various explanations for the phenomenon have been given. Leigh et al. 35developed a theory for the single-band Hubbard model which predicts a Bosonic mode of charge 2 e.T h e ARPES process then would create a superposition of a barehole and an electron accompanied by a charged Boson,leading to a bifurcation of the quasiparticle band which theseauthors identiﬁed with the waterfall phenomenon. Anotherexplanation was given by Basak et al. 36These authors ﬁrst showed that the waterfall phenomenon cannot be reproducedby a calculation of ARPES spectra from LDA band structuresand eigenfunctions alone, a procedure which has otherwisebeen found to be highly successful in describing ARPESspectra of cuprate superconductors. 37,38Instead, these authors obtained good agreement with the experiment by additionallyintroducing a self-energy which describes the coupling to aBosonic mode. The basic mechanism for the waterfall andthe variation of the ARPES spectra with photon energy thenis bilayer splitting which is enhanced by the coupling to theBosonic mode, in particular, the vertical part of the waterfallturns out to be the strongly renormalized bonding combinationof the two single-layer wave functions. This model reproducesthe strong changes of ARPES spectra with photon energyhνin the range 60–80 eV as observed by Inosov et al. 7,10 quite well. On the other hand, Inosov et al. gave a different explanation for this variation, namely the rapid variation of theCu3dphotoemission intensity at the Cu 3 p→3dabsorption threshold at 75 eV. Since the strong variation of photoemissionspectra at the 3 p→3dthreshold is well established for 3 d transition metal oxides 39this appears a plausible explanation. There have been attempts to reproduce a kink in the band structure within the framework of a single-band Hubbardort-Jmodel. 40–42The high-intensity bands observed near /Gamma1at binding energies of ≈1.0 eV thereby are identiﬁed with high energy features observed previously in exactdiagonalization 25–27or quantum Monte Carlo43studies for such models. In the present manuscript we take the point of view that the waterfall phenomenon is a pure matrix-element effect, aspointed out by Inosov et al. 7,10and Zhang et al.13Thereby the crucial point is the nature of the low-energy hole statesas Zhang-Rice singlets (ZRS). Since the phases of the O2 phole “within” a Zhang-Rice singlet correspond to momentum (π,π)( R e f . 32) there is perfect destructive interference with the phases of a p-like photohole with momentum (2 nπ,2mπ) withnandmintegers. The quasiparticle band at /Gamma1therefore can be observed only at photon energies where the cross sectionfor hole creation in dorbitals is large because the destructive interference occurs only for O2 pphotoholes and in higher Brillouin zones where the argument by Ronning et al. 1does not apply. In Sec. IIwe will discuss exact diagonalization results for a one-dimensional charge transfer model. In Sec. IIIwe discuss the spectra of a two-dimensional charge transfer model bythe variational cluster approximation (VCA). In Sec. IVwe discuss the experimental relevance of the binding energy ofthe Zhang-Rice singlet and Sec. Vgives the summary and conclusions. II. ONE-DIMENSIONAL MODEL We study a one-dimensional (1D) charge transfer model by exact diagonalization. We choose a 1D model because we needat least a two-band model and the largest cluster of a two-bandmodel we can study in two dimensions (2D) contains fourCu ions, so that we have virtually no kresolution. As will be seen below, however, a very simple 1D model is sufﬁcient toreproduce the key features of the waterfall phenomenon. To bemore precise, the Hamiltonian reads H=t/summationdisplay i,σ/bracketleftbig d† i,σ/parenleftbig pi+1 2,σ−pi−1 2,σ/parenrightbig +H.c./bracketrightbig +/Delta1/summationdisplay ind,i+U/summationdisplay ind,i,↑nd,i,↓, (1) where d† i,σ(p† j,σ) creates a hole in a d-like orbital at site i(p- like orbital at site j). We choose tas the unit of energy and use the values /Delta1=−4 andU=8 throughout. The crucial feature of the model is that, as in the case for the CuO 2plane, the d-p hybridization integral has an alternating sign. A schematicrepresentation of the model is shown in Fig. 1. Figure 2shows the single particle spectral function for the noninteracting case U=0 and for the strongly correlated case U=8. These are deﬁned as A d(k,ω)=−1 π/IfracturGdd(k,ω+iη), (2) where Gddis thed-like diagonal element of the 2 ×2 single- particle Green’s function [and analogously for Ap(k,ω)]. The Lorentzian broadening η=0.1. In the noninteracting case there are two bands, one with predominant pcharacter and one with predominant dcharacter. Since the matrix element of thep-dhybridization is ∝sin(k 2)t h e r ei sn o p-dhybridization fork=0 and the states have pure pordcharacter. Also, the energy of the p-like peak agrees exactly with that of the p orbital (i.e. zero). +_+_ FIG. 1. (Color online) Possible realization of the model ( 1). 165124-2THEORY OF THE WATERFALL PHENOMENON IN CUPRATE ... PHYSICAL REVIEW B 83, 165124 (2011) FIG. 2. (Color online) Single-particle spectral function for the model ( 1) for the half-ﬁlled system (eight holes in eight unit cells). The ﬁgure combines spectra calculated with different boundary conditions to give an impression of larger systems. The left part of the spectracorresponds to hole creation (i.e., photoemission) the right part to hole annihilation (inverse photoemission). The upper ﬁgure shows the spectra for U=0, the lower one the spectra for U=8. Surprisingly the p-like band can be identiﬁed also in the strongly correlated case, particularly so near k=0. The reason is that the p-like Bloch state with k=0 does not mix with the d-like Bloch state due to parity so that a hole created in this state is unaffected by the Coulomb repulsion on the dsites. Accordingly, the energy of this peak still is zero. This state isanalogous to the “1-eV peaks” in the real cuprate materials. 44 Neark=0 the mixing still is small so that Uis effectively only a weak perturbation. Figure 3shows the intensity of the p-like band as a function of momentum and compares this to the noninteracting case. Close to k=0 the intensity is close to 1e v e nf o r U=8, but for k→πthe intensity in the interacting case is signiﬁcantly reduced. This will be explained below. Whereas the p-like band persists with small change, the upper band of predominant dcharacter disappears completely and is replaced by a spectrum which is very similar to that ofa 1D single-band Hubbard model, deﬁned by H=−t /prime/summationdisplay /angbracketlefti,j/angbracketright(c† i,σcj,σ+H. c.)+U/prime/summationdisplay ini,↑ni,↓. (3) Herec† i,σcreates an electron in the orbital at lattice site i; these sites correspond to the dsites in the two-band model and /angbracketlefti,j/angbracketrightdenotes a summation over all pairs of nearest neighbors. FIG. 3. (Color online) Spectral weight of the p-like band as a function of momentum. The weight is obtained by integrating the spectral weight over a ﬁnite window δaround the center of gravity of the band. The result for U=0 is also shown. The splitting of the d-like band also explains the reduction of the spectral weight of the p-like band near k=πshown in Fig.3. Since the lower Hubbard band is much closer in energy to the p-like band than for U=0 there is stronger mixing between these bands and accordingly a stronger transfer ofp-like weight to the lower Hubbard band. The similarity to the single band Hubbard model spectrum can be seen in Fig. 4which shows a closeup of the low energy region of the photoemission (i.e., hole creation) spectrum andcompares this to the photoemisssion spectrum of a single-bandHubbard model. It turns out that for the above values of t,U, and/Delta1av e r y good match can be obtained by choosing the parameters ofthe single-band Hubbard model to be U /prime/t/prime=10.8 and t/prime= 0.335t. There is a rather obvious one-to-one correspondence between the peaks, the different “holon bands” characteristicfor the spectra of ﬁnite clusters of the one-dimensionalHubbard or t-Jmodel 45can be identiﬁed in both spectra. The main difference occurs at energies Ebetween 0 and 0.5. This is most likely the consequence of mixing and level repulsion between the single-band Hubbard-like bands and thefree-electron-like pband. Moreover one can see a splitting of some of the peaks in the two-band model. The full interacting spectrum therefore can be modeled well by superposing the lower noninteracting pband and a single band Hubbard spectrum. This indicates that the Zhang-Riceconstruction 32for reduction of the low-energy sector of the two-band model to a single band model also works well forthis 1D model. On the other hand, for higher energies thetwo-band model has additional states which correspond to thenonbonding combination of porbitals. Figure 5shows the photoemission spectrum of the two-band model split into its panddcomponents. This demonstrates that the key features 7,10of the waterfall phenomenon can be seen already in this simple 1D model: Near /Gamma1there is nop-like intensity in the single-band Hubbard-like states, instead all p-like intensity resides in the free-electron-like band (this is the huge peak at E=0). The p-like intensity alone therefore shows the black region around k=0 and a 165124-3D. KATAGIRI, K. SEKI, R. EDER, AND Y . OHTA PHYSICAL REVIEW B 83, 165124 (2011) FIG. 4. (Color online) Top: Closeup of the photoemission part of the spectrum in Fig. 2. Bottom: Photoemission spectrum for a half-ﬁlled single-band Hubbard chain with eight sites and U/prime/t/prime= 10.8a n d t/prime=0.335t. The single-band spectrum has been shifted by 0.87t. The ﬁgure combines spectra calculated with periodic and antiperiodic boundary conditions. FIG. 5. (Color online) Closeup of the photoemission part of the spectrum in Fig. 2. The left panel shows the p-like spectrum, the right panel the d-like spectrum.“high-energy-kink” whereby the vertical part of the waterfall is due to the incoherent continuum of the single-particle spectrumof the single-band Hubbard model. In the d-like spectrum, on the other hand, the quasiparticle band can be followedright up to the /Gamma1point. Already this simple model therefore reproduces the key features of the waterfall phenomenon andthe photon energy dependence of the spectra: the waterfalloccurs in the ﬁrst Brillouin zone for all photon energies andin higher Brillouin zones for photon energies where the crosssection for Cu3 dis small. To understand the extinction of p-like weight around /Gamma1we repeat the Zhang-Rice construction 32and consider a single plaquette, which in the 1D model consists of the dorbital and its two nearest-neighbor porbitals, at the dsiteRi. For one hole, the ground state reads /vextendsingle/vextendsingle/Psi11 0,σ/angbracketrightbig =(u1p† i,−,σ+v1d† iσ)\|0/angbracketright, p† i,−,σ=1√ 2/parenleftbig p† i+1 2,σ−p† i−1 2,σ/parenrightbig . (4) The coefﬁcients ( u1,v1) are the ground state (GS) eigenvector of the matrix H1h=/parenleftbigg 0√ 2t√ 2t/Delta1/parenrightbigg . (5) The singlet ground state (i.e., the analog of a Zhang-Rice singlet in the 1D model) of two holes is /vextendsingle/vextendsingle/Psi12 0/angbracketrightbig =/bracketleftbigg u2p† −,↑p† −,↓+v2√ 2(d† ↑p† −,↓+p† −,↑d† ↓) +w2d† ↑d† ↓/bracketrightbigg \|0/angbracketright, and the coefﬁcients ( u2,v2,w2) are the GS eigenvector of the matrix H2h=⎛ ⎝02t 0 2t/Delta1 2t 02t2/Delta1+U⎞ ⎠. (6) In the corresponding single-band model ( 3), the state \|/Psi11 0,σ/angbracketright corresponds to the site ibeing occupied by a spin- σelectron, whereas \|/Psi12 0/angbracketrightcorresponds to a hole at site i. The matrix element of the electron annihilation operator ck,σbetween theses states is eik·Ri. On the other hand, the matrix elements of the operators pk,↑anddk,↑are mp(k)=eik·Ri√ 2isin/parenleftbiggk 2/parenrightbigg/parenleftbigg u1u2+1√ 2v1v2/parenrightbigg , (7) md(k)=eik·Ri/parenleftbigg1√ 2u1v2+v1w2/parenrightbigg . The crucial term here is the factor of isin(k 2) which arises from the overlap of the p-like Bloch state with momentum kand the bonding combination p† i,−,σ. In addition, we have to take into account a shift in energy of /epsilon1=E1h 0−E2h 0. This corresponds to the binding energy of the ZRS and has no counterpart inthe single-band model. In simplest terms, we would thereforeexpect that the photoemission part of single particle spectral 165124-4THEORY OF THE WATERFALL PHENOMENON IN CUPRATE ... PHYSICAL REVIEW B 83, 165124 (2011) functions of the two-band model can be obtained from that of the single band model by Ap,d(k,ω)=\|mp,d(k)\|2A(k,ω+/epsilon1). (8) In this expression several simpliﬁcations have been made: the ZRS is assumed to extend only over one plaquette, which isprobably not correct. This implies that processes where a ZRSin a plaquette around site iis generated by actually creating a hole in a neighboring unit cell are neglected. Moreover theproblem of the overlap of Zhang-Rice singlets in neighboringcells 32is not taken into account either. The energy shift /epsilon1which is necessary to match the two spectra in Fig. 4is found to be 0 .87t, the estimate obtained from the eigenvalues E1h 0andE2h 0is 1.04t(i.e., reasonably close). Figure 6shows the spectra obtained from Eq. ( 8) compared to the actual spectra of the two-band model. Thenumerical values of the prefactors are u 1u2+1√ 2v1v2=0.70 and1√ 2u1v2+v1w2=−0.50. While the agreement is not really perfect, the qualitative trends are reproduced well, particularly so near k=0 where the hybridization with the p-like band is weak. The main differences occur for larger kvalues and may also be due to the fact that the ZRS is not restricted to one plaquette and also the hybridization withthep-like band which is absent in the single-band Hubbard FIG. 6. (Color online) Closeup of the photoemission spectrum in Fig.2(top) compared to the spectra of a single-band Hubbard model corrected according to Eq. ( 8) (bottom). The energy shift /epsilon1is 0.87t rather than 1 .04tas would be obtained from the single-plaquette calculation.model. On the other hand, given the simplicity of the procedure for converting the single-band spectra into two-band spectrathis is not so bad and explains the extinction of p-like intensity around /Gamma1at least qualitatively: this is due to the factor of sin( k 2) in the matrix element mp, which describes the destructive interference between the phase factors of the twoporbitals in the bonding combination p† −,σand in the electron operator p† k,σ. This is, in turn, the consequence of the oscillating sign of the hopping integral in Eq. ( 1) so that the same mechanism should also be effective in the 2D CuO 2 plane. The exact diagonalization results then can be summarized as follows: The oscillating sign of the d-phybridization induces an oscillating sign also in the bonding combinationofporbitals around a given dsite. The phase of the porbitals in the bonding combination therefore “locally” correspondsto a momentum of π. It follows that around k=2nπthere is destructive interference between the phases of the bondingcombination and the phases of the photohole, and it is notpossible to couple to the ZRS by hole creation in porbitals. The ZRS part of the spectrum thus becomes extinct in thep-like spectrum, whereas no such extinction occurs in the d-like spectrum. FIG. 7. (Color online) Low-energy spectral function of the hole-doped two-band model (top) compared to the spectrum of the corresponding single-band model (bottom). In this ﬁgure, thechemical potential is the zero of energy, the part to the left (right) of the chemical potential corresponds to photoemission (inverse photoemission). 165124-5D. KATAGIRI, K. SEKI, R. EDER, AND Y . OHTA PHYSICAL REVIEW B 83, 165124 (2011) To conclude this section we note that none of the above considerations is limited to half-ﬁlling. Figure 7compares the spectra for the two-band model and the single-band modelin the hole-doped case, that means at a hole density of 1 .25. Thereby all parameters are the same as in Fig. 5.A g a i n ,a very good correspondence between the two models exists andagain the extinction of p-like weight around k=0 can be clearly seen. III. TWO-DIMENSIONAL MODEL We now apply the picture gained from the analysis of the one-dimensional two-band model to obtain approximateARPES spectra for the 2D CuO 2plane. Since it is not possible to compute the spectra of a 2D three-band model larger than2×2 unit cells by exact diagonalization we switch to the variational cluster approximation (VCA) 46–48to compute at least approximate spectra. The VCA uses the fact49that the grand canonical potential of an interacting Fermi system can be expressed as a functional of the self-energy /Sigma1(ω) and is stationary with respect to variations of /Sigma1(ω) at the exact self-energy. The VCA then uses ﬁnite clusters, the so-calledreference system, to numerically generate “trial self energies”for an inﬁnite system. This is described in detail in theliterature 46–48and has turned out to be a very successful method to discuss the single-particle spectral functions of correlatedelectron systems. We use this method to calculate spectra forthe three-band Hubbard model H=2t pd/summationdisplay i∈Ld,σ(d† i,σPi,σ+H.c.) +2tppy/summationdisplay α=x/summationdisplay j∈Lα,σ(p† α,j,σYα,j,σ+H.c.) −/Delta1/summationdisplay i∈Ldnd,i+/epsilon1py/summationdisplay α=x/summationdisplay j∈Lα,σp† α,j,σpα,j,σ +Udd/summationdisplay i∈Ldnd,i,↑nd,i,↓+Upp/summationdisplay j∈Lαnp,j,↑np,j,↓.(9) Lddenotes the simple cubic lattice of Cu3 dsites,LxandLy the s.c. lattices of pxandpysites. Moreover P† i,σ=1 2/parenleftbig p† x,i−ˆx 2,σ−p† x,i+ˆx 2,σ−p† y,i−ˆy 2,σ+p† y,i+ˆy 2,σ/parenrightbig ,(10) is the bonding combination of porbitals around the dorbital at site iand Y† α,i,σ=1 2/parenleftbig p† ¯α,i+ˆx 2−ˆy 2,σ−p† ¯α,i+ˆx 2+ˆy 2,σ +p† ¯α,i−ˆx 2+ˆy 2,σ−p† ¯α,i−ˆx 2−ˆy 2,σ/parenrightbig , (11) forα=xdenotes the bonding combination of pyorbitals around a given pxorbital and vice versa for α=y.ˆαdenotes the unit vector in the αdirection. The model is again formulated in hole language (i.e., d† i,σcreates a hole in orbital iand tpd,tpp>0). We choose tpdas the unit of energy, the other parameters are Udd=8,Upp=3,/Delta1=3,/epsilon1p=0,tpp=0.5. We study this model by the VCA, using a cluster with 2 ×2 unit cells (i.e., a square shaped Cu 4O8cluster) with four holes (corresponding to “half ﬁlling”) as the reference system for FIG. 8. (Color online) Top: Sum of p-like and d-like spectrum for the three-band model ( 9) with U=0. Center: d-like spectrum for U=8 as obtained by the VCA. The spectrum has been multiplied by 2.5. Bottom: p-like weight for U=8. The spectra are computed for half-ﬁlling, that means one hole/unit cell, the black line denotesthe respective chemical potential. creating trial self-energies. In the VCA the parameters of the single-electron part of H(/epsilon1p,/Delta1,tpdpandtpd)f o rt h e reference system become effective variational parameters. Inthe present study we varied only the center of gravity of thetwo site-energies /epsilon1 pand−/Delta1, leaving the difference /epsilon1p+/Delta1 unchanged. As shown by Aichhorn et al.50this guarantees the consistency of the electron number obtained by integrating thespectral weight and by differentiating the grand potential withrespect to μ. Since we mainly want to obtain an approximate single-particle spectrum rather than discuss the phase diagramthis should be sufﬁcient. Some results obtained in this wayfor this model have previously been published by Arrigoniet al. 51Figure 8shows the total spectral weight for the case caseU=0a sw e l la st h e p-like and d-like spectral weight for the interacting case, U=8. The p-like spectral weight now is deﬁned as the sum of the two p-like diagonal elements of the total 3 ×3 spectral weight matrix. In the noninteracting case U=0, there are three bands. Switching on the Coulomb repulsion has a very similar effectas for the 1D model: The two lower bands with predominantpcharacter remain essentially unchanged, whereas the partly 165124-6THEORY OF THE WATERFALL PHENOMENON IN CUPRATE ... PHYSICAL REVIEW B 83, 165124 (2011) ﬁlled band of predominant dcharacter is split into an upper and a lower Hubbard band. Comparing the panddspectra in the energy range 0 .5–2.5 the same extinction of p-like spectral weight around (0 ,0) can be seen (i.e., exactly the same behavior as in the 1D model). By analogy with the 1D case we assume that the reason for the extinction of p-like weight, again, is the matrix element between a plane wave of pholes and the bonding combination (10). We again consider a single-plaquette problem. For later reference we include a Coulomb repulsion Updbetween pand dholes on neighboring sites, which is set equal to zero for the time being. The Hamilton matrices for the single and two holeplaquette problems are H 1h=/parenleftbigg−2tpp 2tpd 2tpd /Delta1/parenrightbigg , (12) H2h=⎛ ⎜⎝−4tpp+Upp 42√ 2tpd 0 2√ 2tpd /Delta1−2tpp+Upd 2√ 2tpd 02√ 2tpd 2/Delta1+Udd⎞ ⎟⎠. (13) Thep-pCoulomb repulsion ∝Uppis treated in mean-ﬁeld theory. The ground state eigenvectors of these matrices areagain denoted by ( u 1,v1) and ( u2,v2,w2). To discuss the p-like photoemission spectrum we need the matrix element between the bonding combination ( 10) and a p-like Bloch wave. We write this as mp=−i/bracketleftbigg mxsin/parenleftbiggkx 2/parenrightbigg −mysin/parenleftbiggky 2/parenrightbigg/bracketrightbigg , (14) where mα(α∈x,y) are matrix elements for hole creation in a pαorbital at the origin, which in a real experiment depend on the photon polarization and wave vector of the photoelectrons.In the present calculation A p(k,ω) is obtained by calculating the spectra for creating holes in pxandpyorbitals separately and adding them; accordingly, the k-dependent correction factor in Eq. ( 8) should be replaced by52 m2 p(k)=/parenleftbigg u1u2+1√ 2v1v2/parenrightbigg2 f(k), (15) f(k)=sin2/parenleftbiggkx 2/parenrightbigg +sin2/parenleftbiggky 2/parenrightbigg . To check this we have calculated the photoemission spectrum of a the single-band Hubbard model by the VCA using againa2×2 cluster as reference system. To be consistent with the calculation for the three-band model the single band model ( 3) was augmented by the term /epsilon1/summationtext i,σni,σand/epsilon1w a su s e da st h e sole variational parameter in the VCA. We used a 2 ×2c l u s t e r as reference system to make the spectra of the two models ascomparable as possible. The artiﬁcial supercell structure whichis introduced by the VCA leads to gaps in the calculated bandsdue to Brillouin zone folding. By using the same referencesystem (i.e., the same supercell) these gaps should occur at thesame momenta and be of comparable magnitude if the modelsare equivalent. The parameter values of the single band model are again chosen to obtain a good match with the dispersion of thethree-band model. Thereby in addition to the nearest-neighbor FIG. 9. (Color online) Right part: d-like spectrum of the three- band model (top) compared to the spectrum of the single band Hubbard model multiplied by 0 .5 (bottom). Left part: p-like spectral weight of the three-band model multiplied by 2 (top) compared to thespectrum of the single band Hubbard model multiplied by 0 .5f(k) in Eq. ( 15) (bottom). hopping integral t/primealso hopping integrals t/prime 1between second nearest neighbors and t/prime 2between third nearest neighbors were included. A good match was obtained by using t/prime/tpd=0.38, t/prime 1/t/prime=−0.145,t/prime 2/t/prime=0.118, and U/prime/t/prime=10.3. Figure 9 compares the low-energy hole addition spectrum of the three-band model with the (shifted) photoemission spectrum of thesingle band model. The d-like spectrum of the three-band model should be roughly identical to that of the single bandHubbard model and the two spectra on the right part of theﬁgure are indeed quite similar. There are two main differences:the weakly dispersive band at ≈− 2.0t pdwhich can be seen in the single band model is absent in the spectrum of thethree-band model. This may be a consequence of hybridizationwith the two additional p-like bands in the three-band model. Moreover, the spectrum for the three-band model has a lesssmooth dispersion with additional gaps along (0 ,0)→(π,0). This is likely due to the lower symmetry of the three-bandmodel: in the single band Hubbard model, the 2 ×2c l u s t e r with open boundary conditions is equivalent to a four-site chainwith periodic boundary conditions. For the two-band model asimilar reduction is possible, but with two porbitals in between any two successive dorbitals. The resulting degeneracy of the ligands in the four-site chain may give rise to the additionalband splitting. The left part of the ﬁgure compares the p-like spectrum of the three-band model and the single band modelspectrum corrected by the factor m p(k). Again, the dispersion of the spectral weight along comparable bands is very similarin the two spectra. The energy shift to align the single bandand three-band spectra is /epsilon1=2.00t pd. The estimate obtained from the ground state energies of the matrices ( 12) and ( 13)i s 165124-7D. KATAGIRI, K. SEKI, R. EDER, AND Y . OHTA PHYSICAL REVIEW B 83, 165124 (2011) E1h 0−E2h 0=2.238tpd. The error of ≈10% seems reasonable taking into account the various approximations made. Tomatch the intensities of the single-band and three-band spectrain Fig. 9the single band spectra were multiplied by a factor of 0.5. The estimates for the correction factors from the single-plaquette problem are ( u 1u2+1√ 2v1v2)2=0.56 and (1√ 2u1v2+v1w2)2=0.31. To summarize the discussion so far: the low-energy hole addition spectrum of the three-band model can be obtainedto reasonable approximation from an effective single-bandmodel whereby the p-like spectrum needs to be corrected b yas i m p l e k-dependent factor which originates from the interference between the phases of porbitals in the bonding combinations and the phases in a Bloch state with momentumk. If this correction is done p-like and d-like spectra can be obtained to good approximation from he spectrum of a singleband Hubbard model. We now apply this ﬁnding to compute approximate the spectra for a CuO 2plane, but this time make use of the fact that in a single-band Hubbard model larger clusters can be used as areference system. We again apply the VCA but this time we usea4×4 cluster with periodic boundary conditions as reference system. We used a cluster with two holes corresponding to ahole concentration of 12 .5%. In the VCA calculation we again varied only the site energy of the reference system. Since wemainly want to obtain an approximate single-particle spectrumrather than discuss the phase diagram this should be sufﬁcient.It is important to use as large a cluster as possible for the exactdiagonalization step because only large clusters reproduce theincoherent continua in the spectral function sufﬁciently welland, as will be seen below, these incoherent continua are crucialto explain the experimentally observed spectra. The solutionof a 4 ×4 cluster by exact diagonalization is only possible with periodic boundary conditions, which are not customary inVCA calculations. On the other hand, we do not want to discussthe phase diagram of the Hubbard model, but we mainly wantto obtain the spectral function so this is justiﬁed. In fact, thespectral function obtained by the simpler cluster perturbationtheory 53,54is almost exactly the same as the one obtained by VCA. Having obtained an approximate spectrum for the single- band Hubbard model we again use Eq. ( 8) together with Eq. ( 15) to obtain the d-like and p-like intensities for the three-band model. This is shown in Fig. 10for momenta along the (1,1) direction. It can be seen that the ﬁgure reproduces the waterfall effect quite well. In the d-like spectrum the low-energy quasiparticle band can be seen together with a broad high intensity partat more negative binding energy. Moreover there is appre-ciable incoherent weight around k=(0,0). These incoherent continua are well known from exact diagonalization 25–28and self-consistent Born calculations29,30for the t-Jmodel. Their intensity decreases with increasing distance from k=(0,0). This decrease is due to the coupling of photoholes to spinand charge ﬂuctuations. 24In the p-like spectrum the factor m2 p(k)i nE q .( 15) creates the dark region around k=(0,0). Accordingly, the quasiparticle band seems to disappear at ≈(π 4,π 4). Since the incoherent weight is reduced by the same factor m2 p(k), it disappears at the same momentum which FIG. 10. (Color online) Left: Spectral function of the the single band Hubbard model with U=10,t=1. Right: Spectral function of the the single band Hubbard model multiplied by f(k)[ s e eE q .( 15)]. creates the impression of a “band” which has a kink at ≈(π 4,π 4). At the kink the spectral weight of the band drops sharply. Qualitatively this is exactly what is seen in the ARPES spectra which show the waterfall phenomenon. Thechimney-like appearance of the spectum is due to the fact thatthed-like spectral weight can be approximated by a product: A p(k,ω)=\|mp(k)\|2A(k,ω+/epsilon1). For ﬁxed ωthe second factor, A(k,ω+/epsilon1) decreases with \|k\|, see the left part of Fig. 10, whereas the ﬁrst factor \|mp(k)\|2vanishes at k=(0,0) and increases with \|k\|. Accordingly, the p-like spectral weight must go through a maximum and this maximum corresponds to the apparent vertical part of the band. For a quantitative discussion it would be necessary to take into account alsothe interference between hole creation in Cu3 d x2−y2and O2 p orbitals. This, however, would necessitate to know the relativemagnitude and phase of the respective dipole matrix elements and this is beyond the scope of the present paper. IV . DIRECT MEASUREMENT OF THE BINDING ENERGY OF THE ZRS Lastly we wish to point out that the energy shift /epsilon1(i.e., the binding energy of the ZRS) does indeed have somerelevance for the interpretation of experimental data and has,in a sense, been observed directly. We refer to the results ofMeevasana et al. 8who reported an anomalous enhancement of the noninteracting bandwidth in Bi2201. These authorspointed out that the energy difference between an assumedband bottom at /Gamma1and the Fermi energy seems to be larger than the occupied bandwidth predicted by LDA calculations 165124-8THEORY OF THE WATERFALL PHENOMENON IN CUPRATE ... PHYSICAL REVIEW B 83, 165124 (2011) and concluded that there is an anomalous correlation in- duced band widening rather than the expected correlationnarrowing. We now estimate the position of μwith respect to an “1 eV” peak as an intrinsic reference energy and show thatthe Fermi energy can be obtained quite accurately from asingle-band Hubbard or t-Jmodel by consequent application of the Zhang-Rice construction. 32To begin with, Meevasana et al. observed a band at /Gamma1with downward curvature and a binding energy of ≈− 1 eV relative to the Fermi level at /Gamma1 (this is the band labeled B in Fig. 1 of Ref. 8). The authors point out that this is an umklapp of a band at the Ypoint, or ( π,π)i n a simple cubic 2D model. This is therefore probably the samestate as shown in Fig. 3(c) of Ref. 44(i.e., a state composed of O2pπorbitals which right at ( π,π) has zero hybridization with any of the correlated Cu3 dorbitals). Inspection of Fig. 3(c) of Ref. 44shows that the energy of an electron in this state at (π,π), and accordingly its umklapp at /Gamma1,i sE 1=/epsilon1p+4tpp where /epsilon1pis the orbital energy for O2 pelectrons. Since this is a single-particle-like state, the corresponding binding energyin the photoemission spectrum is E N 0−EN−1 ν=E1. Next, we consider the Fermi energy. The largest part of the energy shift thereby is the binding energy of the ZRS, /epsilon1, which was discussed above. In the matrices ( 12) and ( 13) the energy o fah o l ei na nO 2 porbital (i.e., −/epsilon1p) was chosen as the zero of energy so that we have to change /epsilon1→/epsilon1+/epsilon1p. It remains to add the Fermi energy μHof the single-band Hubbard model itself μ=/epsilon1p+/epsilon1+μH, so that μ−E1=/epsilon1+μH−4tpp. To evaluate /epsilon1we use the parameter set given by Hybertsen et al. ,55in their Table 1. The only exception is the direct oxygen-oxygen hopping. Herewe use the value t pp=0.37 eV which has been extracted directly from experiment in Ref. 44where it also was found to be consistent with previous estimates. We then obtain /epsilon1= 1.82 eV. The Fermi energy of the single-band Hubbard model may be estimated from exact diagonalization results. In a 4 × 4 cluster this was found to be 1 .6tatU/t=8( R e f . 56) and 1.788tatU/t=10 (Ref. 57). Hybertsen et al. estimated U=5.4 eV and t=0.43 eV so that U/t=12.6. We estimate μH≈2t=0.86 eV so that eventually μ−E1=1.2e V .T h e value in Fig. 1of Meevasana et al. is≈1 eV. The agreement is reasonable given the uncertainty about some parameters butit is quite obvious that taking into account the binding energy/epsilon1is indispensable to obtain a correct estimate of the Fermi energy. The value of μ−E 1obviously depends on tppso that small variations from one compound to the other may well explainthe variations observed by Meevasana et al. The Fermi energy of the Hubbard model will change with doping as well, but these changes are a fraction of J≈120 meV. Accordingly, the distance between the free-electron-like state at /Gamma1and the Fermi energy should always be ≈1 eV and this is indeed the case in Bi2201 and Bi2212 (Ref. 8). All in all one can say that the consequent application of the ZRS picture can explain theposition of the Fermi energy relative t oa1e V peak, which forms a natural intrinsic reference energy, quite well.V . SUMMARY AND DISCUSSION In summary, we have investigated the relationship between the single-particle spectra of actual charge-transfer modelsand corresponding “effective” single band Hubbard models.In the noninteracting case U=0 the charge transfer models have several bands and it was found that those bands withpredominant ligand (i.e., pcharacter in the present models) are almost unaffected by the strong correlations. Bands withpredominant dcharacter are split into two Hubbard bands which can be mapped quite well to those of an effective single-band Hubbard model. It turned out that the spectra of pandd electrons to good approximation can be obtained from that ofthe single band Hubbard model by a constant shift in energy,the binding energy of the Zhang-Rice singlet, and, in the caseof the p-like spectrum, by a factor which might be termed the form factor of the Zhang-Rice singlet. These results give a natural explanation for the waterfall phenomenon in terms of a pure matrix element effect, as haspreviously been inferred by Inosov et al. 7,10and Zhang et al.13 from an analysis of their experimental data. Here one has to distinguish between two different effects: the ﬁrst one, pointedout already by Ronning et al. , 1applies only to the special situation of near-normal emission of photoelectrons. Here thevanishing of the dipole matrix element /angbracketleftf\|A·p\|i/angbracketrightmakes it impossible to observe ZRS-derived states. The second matrix element effect is the form factor of the Zhang-Rice singlet mentioned above, which describesthe interference between the phases of the p-like photohole and those of the O2 porbitals “within” a Zhang-Rice singlet, which locally correspond to a state with momentum ( π,π). This makes it impossible to couple to a ZRS-derived state bycreating a p-like photohole with momenta near (2 nπ,2mπ) with integer nandm. These two simple rules explain under which experimental conditions the waterfall is observed or not: in the hole-dopedcompounds and at photon energies where predominantly O2 p- like holes are produced, the quasiparticle band around /Gamma1can be observed neither in the ﬁrst nor in any higher Brillouinzone, and instead the waterfall appears. If photon energieswhere Cu3 dholes are generated are used, the quasiparticle band cannot be observed in the ﬁrst Brillouin Zone, but inhigher Brillouin zones. In this case the waterfall is absent andno kink in the quasiparticle band appears. In the electron-doped compounds the quasiparticles cor- respond to extra electrons in Cu3 d x2−y2orbitals so that the considerations regarding the form factor of the ZRS do notapply. The argument regarding the vanishing of the dipolematrix element /angbracketleftf\|A·p\|i/angbracketright, however, remain unchanged so that the quasiparticle band around /Gamma1cannot be observed in the ﬁrst Brillouin zone either; this has indeed been observedby Ikeda et al. 11and Moritz et al.12It should be possible, however, to observe the full quasiparticle band without a kinkin a higher Brillouin zone. In those cases where the “dark region” around /Gamma1is present, the apparent vertical part of the waterfall corresponds to theincoherent continua in the single-particle spectral functionof the t-Jmodel. Since these continua are formed from states which also correspond to Zhang-Rice singlets, theybecome extinct near (2 nπ,2mπ) as well. Additional evidence 165124-9D. KATAGIRI, K. SEKI, R. EDER, AND Y . OHTA PHYSICAL REVIEW B 83, 165124 (2011) comes from the fact that these band portions show the same dependence on photon polarization as the quasiparticle banditself. 5 Finally, the high intensity bands observed at /Gamma1at binding energies higher than ≈1 eV have no correspondence in a single-band Hubbard or t-Jmodel, otherwise they would not be observed in normal emission, but are precisely thebands of predominant O2 pcharacter which remain unaffected by the strong correlations. They are analogous to the 1-eVpeaks observed at high-symmetry points in cuprates 44and correspond to O2 pderived states which have little or no hybridization with the strongly correlated dorbitals. These states therefore are essentially single-particle states, whichimmediately explains their much higher intensity as comparedto the quasiparticle band. All in all the present theory indicates that the waterfall phenomenon constitutes an experimental proof of the Zhang-Rice construction of a single-band Hubbard or t-Jmodel to describe the low-energy states of the CuO 2planes and, in fact,provides a direct visualization of the energy range in which the states of the real CuO 2plane correspond to those of a single-band Hubbard or t-Jmodel. It shows moreover that the incoherent continua predicted by various calculations for thet-Jor Hubbard model are indeed observable in experiment in that they are responsible for the vertical part of the waterfallsthemselves. ACKNOWLEDGMENTS K.S. acknowledges the JSPS Research Fellowships for Young Scientists. R. E. most gratefully acknowledges thekind hospitality at the Center for Frontier Science, ChibaUniversity. This work was supported, in part, by a Grant-in-Aid for Scientiﬁc Research (Grant No. 22540363) fromthe Ministry of Education, Culture, Sports, Science, andTechnology of Japan. 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PhysRevB.83.014416.pdf	PHYSICAL REVIEW B 83, 014416 (2011) Dependence of magnetism on GdFeO 3distortion in the t2gsystem ARuO 3(A=Sr, Ca) Srimanta Middey,1Priya Mahadevan,2,and D. D. Sarma3 1Centre for Advanced Materials, Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700032, India 2S.N. Bose National Centre for Basic Sciences, JD-Block, Sector III, Salt Lake, Kolkata-700098, India 3Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore-560012, India (Received 1 June 2010; revised manuscript received 12 October 2010; published 18 January 2011) We have examined the stability of the ferromagnetic (FM) state in CaRuO 3and SrRuO 3as a function of the GdFeO 3distortion. Model calculations predict the dependence of the FM transition temperature ( Tc)o nt h e rotation angle θto vary as cos2(2θ)f o reg-electron systems. However, here, we ﬁnd an initial increase and then the expected decrease. Furthermore, a much faster decrease is found than predicted for eg-electron systems. Considering the speciﬁc case of CaRuO 3, a larger deviation of the Ru-O-Ru angle from 180◦in CaRuO 3as compared to SrRuO 3should result in a more reduced bandwidth, thereby making the former more correlated. The absence of long-range magnetic order in the more correlated CaRuO 3is traced to the strong collapse of various exchange interaction strengths that arises primarily from the volume reduction and increased distortionof the RuO 6octahedra network that accompanies the presence of a smaller ion at the A site. DOI: 10.1103/PhysRevB.83.014416 PACS number(s): 75 .10.−b, 75.30.Et, 71.20.−b I. INTRODUCTION As one goes down the periodic table from the 3 dtransition metal oxides to the 4 dtransition metal oxides, one ﬁnds fewer examples of magnetic materials. This is because the wider bands formed by the 4 dcompounds do not allow a local magnetic moment to be sustained in most cases. SrRuO 3is a less commonly encountered example of a 4 doxide which is ferromagnetic. It has a transition temperature Tcof∼160 K with a large magnetic moment of 1.4 μbper formula unit and is metallic down to low temperatures.1In the series Sr1−xCaxRuO 3, when one replaces Sr2+by a smaller cation Ca2+,t h e Tcdecreases monotonically.2For CaRuO 3, the mag- netic ground state at low temperature has been controversial.Some people suggest an antiferromagnetic ordering of thespins at low temperature. 3Some other people suggest absence of any ordering down to the lowest temperature studied.4 The origin of the different magnetic ground states for CaRuO 3and SrRuO 3has been a puzzle. Ca, being a smaller ion, should result in a more distorted perovskite structure thanSrRuO 3. This is indeed what is observed experimentally.5,6 The Ru-O-Ru angle in SrRuO 3is 166.0◦in the ac plane and 170.0◦along the b direction, while it is reduced to 151.1◦in the ac plane and 150.7◦along the b direction for CaRuO 3(both in Pnma settings). The reduction in the Ru-O-Ru angle shouldresult in a more reduced bandwidth for CaRuO 3than SrRuO 3. As one does not expect a signiﬁcant change in U betweenthe two systems, 7these structural distortions should place the U : W ratio for CaRuO 3to be larger than SrRuO 3(i.e., making it more correlated). This would suggest that a magnetic groundstate would be more favorable in CaRuO 3than SrRuO 3. Other effects such as the smaller volume of CaRuO 3compared to SrRuO 3work in increasing the bandwidth for CaRuO 3and weaken the effects of correlation. There are indications fromexperiment, however, of CaRuO 3being more correlated. The enhancement in the electronic speciﬁc heat coefﬁcient overthe band structure value is larger in CaRuO 3than in SrRuO 3. However, as mentioned earlier, there is no long-range magneticorder in CaRuO 3. An additional puzzling aspect is that the magnetic moment is smaller in the more correlated CaRuO 3.The issue of absence of magnetic ordering in CaRuO 3 has been addressed earlier in the literature. First principle electronic structure calculations have been performed withindensity functional theory and have been found to capturethe experimental trends. 8–10However, the microscopic con- siderations that drive this are lost in such calculations. Thedifference in magnetic properties is attributed to the structuraldifferences, primarily the Ru-O-Ru angle. This immediatelyraises the question, how does magnetic stability depend on theGdFeO 3distortion angle in the t2gsystem? In a system where the orbitals at the fermi level are egorbitals, an analytical expression exists. The magnetic stability is found to vary ascos 2(2θ), where (180◦−2θ) is the TM-O-TM angle.11We set out by examining whether a similar dependence in theperovskite ruthenates is responsible for placing CaRuO 3at a point where magnetism gets destabilized. The dependenceof the magnetic stability on angle, we ﬁnd is nonmonotonic.Surprisingly, there is an initial increase in the T cand then a decrease as a function of angle. This is traced to the increasein bond length that takes place when the cubic perovskiteis subject to a GdFeO 3distortion. We have a low-spin system here, with t2gdown-spin levels being ﬁlled up after the up-spin levels are occupied. So with an increase in the bond length,there is an increase in the exchange splitting in the t 2gstate which results in the initial increase in Tc. With larger changes in the angle, the distortion-induced reduction in the effectivehopping between neighboring Ru atoms dominates, and T c decreases. However, the rotation angle corresponding to the observed value is by itself insufﬁcient to destabilize long-rangemagnetic ordering and a part of the destabilization is traced tothe volume reduction that accompanies placing a smaller ionat the A site of the perovskite structure. II. METHODOLOGY The electronic structure of SrRuO 3and CaRuO 3was calcu- lated using V ASP, a plane wave pseudopotential implementa-tion of density functional theory. 12Projected augmented wave (PAW) potentials13were used along with the generalized gradi- ent approximation (GGA) for the exchange. GGA calculations 014416-1 1098-0121/2011/83(1)/014416(5) ©2011 American Physical SocietySRIMANTA MIDDEY , PRIY A MAHADEV AN, AND D. D. SARMA PHYSICAL REVIEW B 83, 014416 (2011) were found to be adequate in getting a correct description of the ground state as GGA +U calculations showed a tendency of overemphasizing magnetism when applied to CaRuO 3. In spite of the fact that the spin-orbit interaction is large in the case ofthe ruthenates, the effect of including spin-orbit interaction wasfound not to change the relative energy differences between thedifferent magnetic states. The experimental crystal structuresfor CaRuO 3and SrRuO 3were taken from Refs. 5,6. The lattice constants were kept ﬁxed at the experimental values, whilethe internal coordinates were optimized to minimum energyconﬁgurations. Total energies for ferromagnetic, nonmagnetic,and different types of antiferromagnetic (A, C, G type) 14spin conﬁgurations were calculated by integrating over a meshof 6×6×6 K points over the complete brillouin zone. In order to understand the experimental trends in magnetism, wecarried out total energy calculations for certain representativesituations. An ideal perovskite unit cell for SrRuO 3was considered with four formula units (√2a×√2a×2a) to treat all magnetic conﬁgurations considered earlier. TheRu-O-Ru angle was ﬁxed at 180 ◦and the lattice constant was varied and the changes in magnetic stabilization energywith volume was examined. We then studied the changes asa function of the Ru-O-Ru angle. Again four formula unitswere considered. The distortion was simulated by rotatingeach successive octahedron clockwise or anticlockwise by anangleθwith respect to the 110 direction. The variation of the magnetic stabilization energies with θhave been mapped on to a classical Heisenberg model of the form −/Sigma1 ijJijei.ej. Here ei,ejare the unit vectors representing the direction of the spin.15This is because our earlier analysis showed that the magnitude of the spin was sensitive to the underlyingmagnetic conﬁguration. 16The exchange interactions Jij entering the Heisenberg Hamiltonian were determined from the differences of energy for different spin conﬁgurations, andincluded up to second neighbor. III. RESULT AND DISCUSSIONS In Table Iwe list the energies for SrRuO 3and CaRuO 3 in different magnetic conﬁgurations for the optimized ex- perimental structure. In SrRuO 3, the ferromagnetic state is strongly stabilized by 25 meV per formula unit compared tothe closest lying antiferromagnetic state which is the A-typeantiferromagnetic state. For CaRuO 3on the other hand, all magnetic conﬁgurations are found to lie close in energy, withinthe error bar of our calculations. In order to investigate whetherthis result had anything speciﬁcally to do with which atom wehad at the A site, we carried out calculations for CaRuO 3in the SrRuO 3relaxed experimental structure and vice versa. As is evident from the result shown in Table I, the magnetic stabiliza- tion energies found are independent of the atoms at the A site.This suggests that the different behavior we ﬁnd for CaRuO 3 and SrRuO 3arises from the differences in the crystal structure. The ﬁrst difference between the two systems arises from the difference in the volume. Ca is a smaller ion than Sr andso occupies a smaller volume. We ﬁrst considered an idealperovskite unit cell of SrRuO 3and examine the stability of magnetic as well as the nonmagnetic state as a function ofvolume. A collapse of all magnetic conﬁgurations was foundat small volumes as shown in Fig. 1(a). The reason for this is because, as the bond length is decreased, the hoppinginteraction strength between Ru dstates and O pstates increases. In the up-spin channel, both the Ru d–Opbonding and antibonding states of t 2gsymmetry are occupied. However, in the down-spin channel, the antibonding states (primarilyRudcharacter) are partially occupied. With an increase in the hopping, one has an enhanced Ru ddown-spin contribution in the occupied states, as a result of which the Ru dexchange splitting decreases. This results in the collapse of differentmagnetic conﬁgurations at smaller volumes. For better visualclarity, we plot the energy difference between the FM as wellas the competing AFM state [inset of Fig. 1(a)] as well as replot Fig. 1(a) with all magnetic energies referenced to the energy of the nonmagnetic conﬁguration [Fig. 1(b)]. The results of Table Isuggest that for a given structure, the magnetic stabilization trends are independent of the choice of atom atthe A site. So, we can infer what is expected for CaRuO 3in the ideal structure, by examining the results for the volumecorresponding to that of CaRuO 3. Indeed the smaller volume of CaRuO 3as compared to SrRuO 3results in decreased magnetic stabilization energies for CaRuO 3. However, the volume change is not enough to entirely destabilize the ferromagneticstate. The volume effect also explains the smaller mag-netic moment found for CaRuO 3compared to SrRuO 3in experiments. In addition we have also considered the CaRuO 3 experimental structure and increased its volume by 5%. Therewe ﬁnd that the ferromagnetic state is stabilized. The small size of the Ca ion at the A site of the perovskite lattice results in a smaller volume for the CaRuO 3unit cell than what one ﬁnds in SrRuO 3. This would result in shorter Ru-O bonds for CaRuO 3. Consequently, there is an increase of Coulomb repulsion between the electrons on Ru and Oatoms. To minimize this, the structure distorts with the RuO 6 octahedra rotating around the 110 direction. To examine T A B L EI .E n e r g i e s( e V )f o rS r R u O 3and CaRuO 3in their experimental structure, as well as SrRuO 3in CaRuO 3 structure and vice versa. SrRuO 3 CaRuO 3 SrRuO 3 CaRuO 3 Spin conﬁguration (in expt. structure) (in expt. structure) (in CaRuO 3structure) (in SrRuO 3structure) Ferromagnetic −139.158 −141.070 −136.721 −140.260 Nonmagnetic −138.979 −141.053 −136.721 −139.963 A-AFM −139.057 −141.071 −136.723 −140.118 C-AFM −139.024 −141.056 −136.721 −140.036 G-AFM −138.984 −141.065 −136.722 −139.986 014416-2DEPENDENCE OF MAGNETISM ON GdFeO 3... PHYSICAL REVIEW B 83, 014416 (2011) FIG. 1. (Color online) (a) The variation in energy with volume for different magnetic conﬁgurations of SrRuO 3. Inset shows the energy difference between FM and competing AFM conﬁguration. (b) Thesame energies plotted with respect to the nonmagnetic energy. In these calculations all atoms are at ideal perovskite lattice positions. the consequences of this effect referred to as the GdFeO 3 distortion we have considered the cubic ideal perovskite unit cell for SrRuO 3and appropriately rotated the RuO 6octahedra. As a consequence of the rotation one has an increase in theRu-O bond length with apical and basal oxygen atoms asshown in Fig. 2(a). This distortion, makes the Ru-O-Ru angle deviate from 180 ◦as shown in Fig. 2(b) as a function of the rotation angle θ. The elongation of the Ru-O bond length as well as the deviation in Ru-O-Ru angle results in a decreasein the one-electron band energy. This is compensated bya reduction in the Coulomb repulsion energy. These twocompeting factors result in the system favoring a distortedstructure, where the total energy is found to be lowest fora rotation angle of 11 ◦for SrRuO 3as shown in Fig. 3(a) and in its inset. This is in reasonable agreement with theexperimentally observed value of 166 ◦for the Ru-O-Ru angle and is independent of the magnetic structure. The FM metallic state derives its stability from an increased bandwidth of the 4 dband. The distortion, however, reduces the effective hopping interaction between the Ru sites, and there-fore the bandwidth. While magnetism gets destabilized at largeθ, we examined the dependence of the stability of the FM state onθ. Model calculations carried out for systems which had e g electrons at the fermi level found a cos22θdependence. This emerges from the dependence of the hopping integral betweennearest neighbor transition metal sites on angle. A conse-quence of the prediction is that T cis expected to the maximumFIG. 2. (Color online) The variation of (a) the Ru-O bond length and (b) the Ru-O-Ru bond angle as a function of the rotation angle θ. A cubic unit cell with a volume of SrRuO 3is considered here, on which GdFeO 3distortions are applied. when the TM-O-TM angle is 180◦and decreases for deviations away from it. The variation in the nearest neighbor exchangeinteraction strengths between Ru atoms in the xyplane ( J 1), Ru atoms in the zdirection ( J1/prime) as well as the second neighbor strengths J2are plotted in Fig. 3(b) as a function of θ.A l m o s t all exchange interactions are FM, with the surprise being the in-crease in J 1/prime,J1forθless than 8◦. The effective J0(related to the Tcup to a multiplative factor) is computed as/summationtext iZiJiwhere Ziis the coordination associated with ithneighbor. An increase from 88.3 to 98 meV is found for the same change in θ.T h i s effect is traced back to Fig. 2where in addition to the decrease in the Ru-O-Ru angle that we ﬁnd in panel (b), we also ﬁnd anincrease in the Ru-O bond lengths as a function of θin panel (a). The latter effect dominates and leads to the increase in theexchange splitting as a function of the θ. For rotation angles greater than 8 ◦we ﬁnd the expected decrease in all exchange interaction strengths. We also examined the dependence of J0 onθin the window that it decreased. The decrease is much faster than the cos22θ[dotted line in the inset of Fig. 3(b)]. At θ=17◦which corresponds to the θof CaRuO 3, we still ﬁnd an FM state to be stabilized. Hence, GdFeO 3distortions alone do not destabilize ferromagnetism in CaRuO 3. We have performed similar calculations as a function of θfor CaRuO 3for the experimental volume for CaRuO 3.T h e total energy shows a minimum at an angle of 17◦[Fig. 4(a)]. The position of the energy minimum is found to be sensitiveboth to the atom at the A site and the volume. This is indeed 014416-3SRIMANTA MIDDEY , PRIY A MAHADEV AN, AND D. D. SARMA PHYSICAL REVIEW B 83, 014416 (2011) (a) (b) FIG. 3. (Color online) (a) The variation in energy as a function the GdFeO 3distortion angle θfor SrRuO 3. The inset shows the magniﬁed view near the energy minima. (b) The different exchange interaction strengths as a function of θand the total exchange interaction strength J0is shown in the inset. The dotted line indicates the behavior of J0=Acos2(2θ). In these calculations we have considered a cubic cell with volume equal to that for SrRuO 3on which GdFeO 3distortions are applied. what one expects based on the discussion of the microscopic interactions that drive the GdFeO 3distortion. The energy difference between the FM state and the competing AFM state(A-AFM) [inset of Fig. 4(a)] indicate that for θ/greaterorequalslant16 ◦the values are within the error bars of our calculation. One ﬁndssimilar trends in J 1,J/prime 1, andJ2[Fig. 4(b)] as found for SrRuO 3 with a reduction in J0to 15 meV from 30 meV found for SrRuO 3in Fig. 3. In addition to the GdFeO 3distortion that we have consid- ered, usually one has a rotation about the zaxis. Considering this we obtained an energy reduction of 86 meV for the lowestenergy of Fig. 4(b) (θ=17 ◦). This is still almost one eV larger than the energy of the experimental structure (Table I). We then carried out a structural optimization and obtained an energyreduction of 0.99 eV , which arose from the displacementof the Ca atom from its ideal position. The origin of thedisplacement of the Ca atoms from their ideal position is easyto understand. With increasing GdFeO 3distortion, the O atoms move closer to the Ca atoms, with a component of the energystabilization arising from increased Ca-O hybridization. 17,18 The decrease in distance between Ca and O, however, increases the electrostatic repulsion between the electrons on Ca and O.The Ca atoms therefore displace and lower the energy of thesystem. The Sr displacements in SrRuO 3are much smaller as it has a smaller GdFeO 3distortion. Thus, the combined effect of the GdFeO 3distortion and volume are primarily responsible for the observed loss of long-FIG. 4. (Color online) (a) The variation in energy as a function of the GdFeO 3distortion angle θfor CaRuO 3with the energy difference between FM and A-AFM spin conﬁguration shown in the inset.(b) The variation of different exchange interaction strengths and J 0 (inset) as a function of θ. Here we have considered a cubic cell with volume of CaRuO 3on which GdFeO 3distortions are applied. range magnetic order in CaRuO 3with the energy difference between FM and A-AFM being within our error bars, and C-AFM and G-AFM lying 14 meV higher. Other distortions bringall magnetic states very close as seen in Table I. The increased distortion reduces the bandwidth and hence makes the systemmore correlated. However, the loss of long-range magneticorder is because of the collapse of all exchange interactionstrengths to very small values for large θas a result of which long-range magnetic order cannot be sustained in CaRuO 3. IV . CONCLUSION We have studied the variation of the ferromagnetic stability as a function of the GdFeO 3distortion in two prototypical t2g systems CaRuO 3and SrRuO 3. An increase in ferromagnetic transition temperature is found initially with distortion, fol-lowed by the expected decrease. The functional form of thedecrease in T ccannot be ﬁt to the expression cos2(2θ) that one encounters for eg-electron systems and exhibits a much faster decrease. The absence of long-range magnetic order inCaRuO 3is traced to the collapse of all exchange interactions to very small values arising primarily from the smaller A ion induced volume reduction and RuO 6octahedra rotation. ACKNOWLEDGMENTS S.M. thanks the Council of Scientiﬁc and Industrial Research, India for fellowship. P.M. and D.D.S. thank theDepartment of Science and Technology, Government of India. 014416-4DEPENDENCE OF MAGNETISM ON GdFeO 3... PHYSICAL REVIEW B 83, 014416 (2011) Corresponding author: priya.mahadevan@gmail.com 1P. B. Allen, H. Berger, O. Chauvet, L. Forro, T. Jarlborg, A. Junod, B. Revaz, and G. Santi, P h y s .R e v .B 53, 4393 (1996). 2G .C a o ,S .M c C a l l ,M .S h e p a r d ,J .E .C r o w ,a n dR .P .G u e r t i n , Phys. Rev. B 56, 321 (1997). 3R. Vidya, P. Ravindran, A. Kjekshus, H. Fjellvag, and B. C. Hauback, J. Solid State Chem. 177, 146 (2004); A. Callaghan, C. W. Moeller, and R. Ward, Inorg. Chem. 5, 1572 (1966). 4P. Khalifah, I. Ohkubo, H. Christen, and D. Mandrus, Phys. Rev. B70, 134426 (2004); Y . S. Lee, J. Yu, J. S. Lee, T. W. Noh, T.-H. Gimm, H.-Y . Choi, and C. B. Eom, ibid.66, 041104(R) (2002). 5H. Nakatsugawa, E. Iguchi, and Y . Oohara, J. Phys. Condens. Matter 14, 415 (2002). 6M. V . Rama Rao, V . G. Sathe, D. Sornadurai, B. Panigrahi, and T. Shripathi, J. Phys. Chem. Solids 62, 797 (2001); International Crystal Structure Database (ICSD) No. 51291. 7P. Mahadevan, F. Aryasetiawan, A. Janotti, and T. Sasaki, Phys. Rev. B 80, 035106 (2009). 8I. I. Mazin and D. J. Singh, Phys. Rev. B 56, 2556 (1997).9G. Santi and T. Jarlborg, J. Phys. Condens. Matter 9, 9563 (1997). 10K. Maiti, P h y s .R e v .B 73, 235110 (2006). 11J. L. Garc `ıa-Mu ˜noz, J. Fontcuberta, M. Suaaidi, and X. Obradors, J. Phys. Condens. Matter 8, L787 (1996). 12G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54, 11169 (1996). 13P. E. Blochl, Phys. Rev. B 50, 17953 (1994); G. Kresse and D. Joubert, ibid.59, 1758 (1999). 14S. Blundell, Magnetism in Condensed Matter (Oxford University Press, Oxford, 2001). 15P. Mahadevan, I. V . Solovyev, and K. Terakura, Phys. Rev. B 60, 11439 (1999). 16S. Middey, P. Mahadevan, and D. D. Sarma, AIP Conf. Proc. 1003 , 148 (2008). 17J. B. Goodenough, Magnetism and the Chemical Bond (Inter- science, New York, 1963); Prog. Solid State Chem. 5, 145 (1971). 18E. Pavarini, S. Biermann, A. Poteryaev, A. I. Lichtenstein, A. Georges, and O. K. Andersen, P h y s .R e v .L e t t . 92, 176403 (2004); E. Pavarini, A. Yamasaki, J. Nuss, and O. K. Andersen, New J. 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PhysRevB.72.205301.pdf	Optical transitions in quantum ring complexes T. Kuroda,1,2T. Mano,1T. Ochiai,1S. Sanguinetti,1,3K. Sakoda,1,4G. Kido,1,5and N. Koguchi1 1Nanomaterials Laboratory, National Institute for Materials Science, Namiki 1-1, Tsukuba 305-0044, Japan 2PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama, Japan 3Dipartimento di Scienza dei Materiali, Universitá di Milano Bicocca, Via Cozzi 53, I-20125 Milano, Italy 4Graduate School of Pure and Applied Science, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8577, Japan 5High Magnetic Field Center, National Institute for Materials Science, Sakura 3-13, Tsukuba 305-0003, Japan /H20849Received 20 April 2005; revised manuscript received 3 August 2005; published 2 November 2005 /H20850 Making use of a droplet-epitaxial technique, we realize nanometer-sized quantum ring complexes, consisting of a well-deﬁned inner ring and an outer ring. Electronic structure inherent in the unique quantum system isanalyzed using a microphotoluminescence technique. One advantage of our growth method is that it presentsthe possibility of varying the ring geometry. Two samples are prepared and studied: a single-wall ring and aconcentric double ring. For both samples, highly efﬁcient photoluminescence emitted from a single quantumstructure is detected. The spectra show discrete resonance lines, which reﬂect the quantized nature of thering-type electronic states. In the concentric double ring, the carrier conﬁnement in the inner ring and that inthe outer ring are identiﬁed distinctly as split lines. The observed spectra are interpreted on the basis of singleelectron effective mass calculations. DOI: 10.1103/PhysRevB.72.205301 PACS number /H20849s/H20850: 78.67.Hc, 73.21.La, 78.55.Cr I. INTRODUCTION Recent progress in nanofabrication technology allows the simulation of novel atomic physical phenomena on an artiﬁ-cial platform, such as presence of /H9254-function-like density of states on quantum dots /H20849QD/H20850,1,2realization of molecular- orbital state on spatially coupled QDs,3,4and formation of nanometer-sized quantum rings,5–8which are nanoscopic analogues of benzene. Among them, fabrication of semicon-ductor quantum rings has triggered strong interest in realiza-tion of quantum topological phenomena, which are expectedin small systems with simply connected geometry. 9,10The Aharonov-Bohm /H20849AB/H20850effect, which engenders so-called per- sistent current ,11has been explored for various types of me- soscopic rings, based on metals12and semiconductors,5,13us- ing magnetic and transport experiments. As an optical, i.e.,noncontact , approach, Lorke et al. ﬁrst observed far-infrared optical response in self-assembled quantum rings, revealinga magneto-induced change in the ground state from angularmomentum l=0 to l=−1, with a ﬂux quantum piercing the interior. 14Later, Bayer et al. reported pronounced AB-type oscillation appearing in the resonance energy of a chargedexciton conﬁned in a single lithographic quantum ring; 15fur- thermore, Ribeiro et al. observed the AB signature in type-II quantum dots, in which a heavy hole travels around a dot.16 Although magneto conductance characteristics have gar- nered considerable attention, optical manifestation of the ABeffect has remained a controversial subject. 17In this regime, both an electron and a hole are excited simultaneously; thenet charge inside a ring decreases to zero. Because of thecharge neutrality, the loop current associated with a magneticﬂux must vanish, engendering the absence of the AB effectfor a tightly bound electron-hole pair /H20849exciton /H20850. Several theo- ries have been proposed that a composite nature of excitonsallows for a nonvanishing AB effect in a small sufﬁcientring, 17and in excited state emissions.18On the other hand, a negative prediction has also been reported in which the ABeffect can hardly take place in more realistic rings with ﬁnite width.19A large difference in trajectories for an electron and a hole is necessary to exhibit the AB effect on neutralexcitation. 20 The present study examines the optical transition in strain-free self-assembled GaAs quantum rings in a zero magnetic ﬁeld. We have recently reported self-production ofnanometer-sized GaAs rings on /H20849Al,Ga /H20850As by means of a droplet epitaxial technique. 7,8Because of their good cyclic symmetry, together with high tunability of the ring size andshape, the present system is expected to open a new route toimplement the AB effect within the optical regime. Both theground-state transition and the excited-state transition areidentiﬁed by single quantum ring photoluminescence /H20849PL/H20850. The spectra are found to be in good agreement with results ofsingle-carrier calculation. The paper is organized as follows. In Sec. II, we brieﬂy explain the sample preparation and the procedure of the op-tical experiment. In Sec. III, we present low temperature PLspectroscopy of single quantum rings. Section IV presentsthe theoretical results; we discuss the experimental data in FIG. 1. /H20849Color online /H20850Atomic force microscope images in 250 /H11003250 nm2area for /H20849a/H20850a GaAs quantum ring /H20849QR/H20850, and /H20849b/H20850a con- centric double ring /H20849DQR /H20850, grown by droplet epitaxy. After ring formation, they are covered by an Al 0.3Ga0.7As layer for optical experimentation.PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 1098-0121/2005/72 /H2084920/H20850/205301 /H208498/H20850/$23.00 ©2005 The American Physical Society 205301-1terms of this model in Sec. V. Our conclusions are summa- rized brieﬂy in Sec. VI. II. EXPERIMENTAL PROCEDURE A. Sample preparation GaAs quantum rings were grown on Al 0.3Ga0.7As using modiﬁed droplet epitaxy.21In this growth, cation /H20849Ga/H20850atoms are supplied solely in the initial stage of growth, producingnanometer-sized droplets of Ga clusters. After formation ofthe Ga droplets, anion atoms /H20849As/H20850are supplied, leading to crystallization of the droplets into GaAs nanocrystals. Incontrast to the other methods to fabricate QDs, such asStranski-Krastanow growth, this technique can producestrain-free quantum dots based on lattice-matched heterosys- tems. In addition to these characteristics, we recently foundthat it has a high controllability of the crystalline shape:When we irradiate the Ga droplets with an As beam of suf-ﬁciently high intensity, typically 2 /H1100310 −4Torr beam equiva- lent pressure /H20849BEP /H20850at 200 °C, the crystalline shape becomes conelike, following the shape of the original droplet.22,23 When we reduce the As intensity to 1 /H1100310−5Torr BEP, the QD becomes ring like, with a well-deﬁned center hole.7Fur- ther reduction of the As ﬂux, down to 2 /H1100310−6Torr BEP, produces the striking formation of unique concentric double-rings: an inner ring and an outer one. 8The rings show a good circular symmetry, whereas small elongation is found along the /H2085101¯1/H20852direction /H208495% for the inner ring and 8% for the outer ring /H20850. Two samples are used in the experiment: a GaAs quantum ring of 40 nm diameter with 15 nm height /H20849abbreviated to QR, hereafter /H20850and concentric double rings consisting of an inner ring of 40 nm diameter and 6 nm height, and an outerring of 80 nm diameter with 5 nm height /H20849abbreviated to DQR /H20850. In the growth of these two rings, the same conditions were applied to the initial deposition of Ga droplets /H208511.75 monolayer /H20849ML/H20850of Ga at 0.05 ML/s to the surface of a Al 0.3Ga0.7As substrate at 300 °C /H20852. Thus, the mean volume for each structure is expected to be equivalent for QR and DQR,whereas its crystalline shape differs drastically. After ringformation, they were capped by an Al 0.3Ga0.7As barrier of 100 nm thickness, following rapid thermal annealing /H20849RTA; 750 °C for 4 min /H20850. The ring shape before capping is charac- terized by atomic force microscopy /H20849AFM /H20850, as shown in Fig. 1/H20851see also the averaged cross section of DQR presented in Fig. 6 /H20849c/H20850/H20852. The ring density is 1.3 /H11003108cm−2for both samples, allowing the capture of the emission from a singlestructure using a micro-objective setup. We would like to stress the difference in growth process between these quantum rings and In /H20849Ga/H20850As rings, whose growth was previously reported. 6,24The ring formation of the latter case is associated with partial capping of a thin GaAslayer on InAs QDs, which were originally made by theStranski-Krastanow method. Subsequent annealing results inthe morphological change from islandlike QDs to ringlikenanocrystals. In contrast, the present rings are formed at thecrystalline stage of GaAs. The ring shape in this case isdetermined by the ﬂux intensity of As beams. After the for-mation of rings, they are capped by a thick /H20849Al,Ga /H20850As layer. Later we apply RTA to improve their optical characteristics.Note that the ﬁnal RTA processing does not modify the nano-crystalline ring shape, according to the negligible interdiffu-sion of Ga and As at a GaAs/ /H20849Al,Ga /H20850As heterointerface at the relevant temperature. 25 B. Optical arrangement In the PL experiment, we used a continuous wave He-Ne laser as an excitation source. The laser emitted 544 nmwavelength light, corresponding to 2.28 eV in energy. Theexcitation beam from the laser was obliquely incident to thesample. It was loosely focused by a conventional lens/H2084930-cm focal length /H20850into an elliptical spot of 0.5 /H110031.4 mm 2. Emission from the sample was collected by an aberration-corrected objective lens of N.A. /H20849numerical aper- ture/H20850=0.55. Combination of the objective lens and a pinhole /H2084950-/H9262m diameter /H20850at the focal plane allowed the capture of emissions inside a small spot of 1.2- /H9262m diameter. For this spot size, 1.3 rings were expected to lie in the focus onaverage. The position of detection was translated laterally sothat single quantum structures were captured individually.For this purpose, we moved the objective lens with submi-crometer precision using piezo transducers. During transla-tion of the spot, the condition of excitation was kept un-changed because the excitation area was sufﬁciently largerthan the area that covered the entire translation. The emissionpassing through the pinhole was introduced into an entranceslit of a polychromator of 32-cm focal length. After beingspectrally dispersed, it was recorded by a charged-coupledevice detector with 0.8-meV resolution. In advance of themicro-objective measurement, we used a conventional PLsetup to observe the spectra of the ring ensemble. All experi-ments were performed at 3.8 K. Before describing experimental results, we mention the carrier dynamics associated with photoexcitation of thepresent condition. For our laser beam, photocarriers are pro-duced initially in the barrier, whose band gap is 1.95 eV.After diffusion, the carriers are captured by the quantumrings. The captured carriers then relax into the lower lyingquantum ring levels where they radiatively recombine. Ourprevious study showed the recombination lifetime inGaAs/ /H20849Al,Ga /H20850As QDs as /H11011400 ps, whereas the characteris- tic time of intradot relaxation was much shorter—less than 30 ps—depending on excitation density. 26,27Because of the rapid intradot relaxation, we can expect that an electron anda hole recombine after they are in quasiequilibrium. Thequantized levels are occupied by carriers according to theFermi distribution. Similar dynamics are expected in the ringsystem. III. EXPERIMENTAL RESULTS A. PL from the ensemble of quantum rings We present the far-ﬁeld PL spectrum of the sample with QR in Fig. 2 /H20849a/H20850. It comprises several spectral components. The sharp line at 1.49 eV is assigned by impurity-relatedemissions from the GaAs substrate. Because of a thin depos-KURODA et al. PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-2ited layer of the sample /H20849500 nm thickness /H20850, the excitation beam is expected to reach the substrate, thereby producingstrong emissions. The broad emission band at 1.544 eV inthe center energy is attributed to recombination of an elec-tron and a hole, which are conﬁned in GaAs QRs. The spec-trum is broadened by 28 meV in full width at half maximum/H20849FWHM /H20850. The line broadening is caused by the size distri- bution of the rings. Several emission components rangingfrom 1.7 eV to 1.95 eV are assigned by recombinations in the/H20849Al,Ga /H20850As barrier. The spectral tail to lower energy suggests the presence of impurities and imperfections in the barrierlayer. Note that the excitation density at this measurement isquite low /H2084950 mW/cm 2/H20850. For that reason, the impurity- related signal should be relatively emphasized. The PL spectrum of the sample with DQR is presented in Fig. 2 /H20849b/H20850. The signals that are related to the GaAs substrate and to the barrier are identical to those of QR. The emissionband at 1.628 eV in center energy originates from recombi-nation of the ensemble of DQRs. It is broadened by 49 meVin FWHM. We ﬁnd that the PL energy of DQR is higher thanthat of QR. The energy shift reﬂects the small height ofDQR. In our rings, stronger conﬁnement is induced along thegrowth direction than in the lateral in-plane direction. Thus,the reduction in height, associated with formation of DQR,enhances their conﬁnement energy, causing a blue shift in thePL spectrum. B. Spectroscopy of a single quantum ring In Fig. 3 we show the PL spectra of three different quan- tum rings, QR- a, QR- b, and QR- c, and their dependence on excitation intensity. In QR- aat low excitation, we ﬁnd a single emission line appearing at 1.569 eV, which resultsfrom recombination of an electron and a hole, both occupy-ing the ground state of the ring. With increasing excitationintensity, a new emission line, indicated by an arrow, emerges at 1.582 eV. Further increase in excitation densitycauses saturation in the intensity of the original line alongwith a nonlinear increase in the new line. Superlinear depen-dence of the emission intensity suggests that the satellite linecomes from the electron-hole recombination from an excitedlevel of the ring. Thus, the energy difference between theground and the excited state in the QR is 13 meV. In addition to the state-ﬁlling feature associated with pho- toinjection, we ﬁnd the ground-state emission being shiftedto low energy. It is a signature of multicarrier effects. In thepresence of multiple carriers inside a ring, their energy levelsare modiﬁed by the Coulomb interaction among carriers. Be-cause the optical transition energy is mainly renormalizedaccording to the exchange correction, the many-carrier ef-fects result in spectral red shift of the emission, depending onthe number of carriers. Similar features have been observedin numerous quantum dot systems including GaAs dots 27and In/H20849Ga/H20850As rings.28Note that the relevant multiplet is not spec- trally resolved in our rings, but it leads to the red shift of thebroad emission spectra. Moreover, the biexcitonic emissionis expected to contribute to the low energy tail of the spectra FIG. 2. Far-ﬁeld emission spectra of the sample with /H20849a/H20850QRs and /H20849b/H20850DQRs at 5 K plotted on a logarithmic scale. The excitation density is 50 mW/cm2. FIG. 3. Emission spectra for a single GaAs QR. Three examples— a,b, and c—are presented. Their respective excitation densities were, from bottom to top, 1, 10, and 30 W/cm2. Spectra are normalized to their maxima and offset for clarity.OPTICAL TRANSITIONS IN QUANTUM RING COMPLEXES PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-3because the biexciton binding energy was found to be /H110110.8 meV in our droplet-epitaxial GaAs dots.27,29At high excitation, we also ﬁnd spectral broadening, which is attrib-uted to an incoherent collision process that occurs amongcarriers. Identical spectral features are found in QR- b. The ground- state emission is observed at 1.537 eV, whereas the excited-state emission appears at 1.546 eV. The energy spacing be-tween these two lines is 9 meV. Although dependence onexcitation density is quite similar for QR- aand QR- b, the energy levels are slightly different because of a small disper-sion of the ring shape and size. We note that the linewidth ofthe ground-state emission is also different between QR- a /H208492.2 meV FWHM /H20850and QR- b/H208493.4 meV /H20850. Moreover, these are considerably large compared with that known for self-assembled QDs. We ascribe the line broadening to spectraldiffusion, i.e., an effect of the local environment that sur-rounds each QR; our samples are expected to contain a rela-tively large density of imperfections and excess dopants, as-sociated with the low-temperature growth of the sample. Itcauses low-frequency ﬂuctuation in the local ﬁeld surround-ing QRs, which is due to the carrier hopping inside the bar-rier. This leads to efﬁcient broadening of the PL spectra,which depend on the local environment of each ring. De-tailed examination of the origin of line broadening is studiedin droplet-epitaxial GaAs quantum dots. 29 In QR- c, we ﬁnd a shoulder structure on the ground-state emission, which suggests the split in the relevant levelcaused by lateral elongation, and/or structural imperfectioninherent in this ring. Broken symmetry in the ring shapecauses degeneracy lift in the energy level, leading to obser-vation of the doublet spectra. 30Apart from this split struc- ture, the same spectral characteristics are found in QR- c. The energy difference from the original line to the satellite isobserved to 11 meV, which is between the value of QR- aand that of QR- b. Spatial dependence of the PL spectrum is shown in Fig. 4, where the position of detection is laterally translated on thesample in steps of 0.32 /H9262m. Figure 4 /H20849a/H20850presents the results obtained at low excitation. They exhibit a single line associ-ated with the ground-state emission from a single QR, de-pending on the position of detection. Lateral broadening ofthe emission is estimated as /H110111.2 /H9262m FWHM, which is con- sistent with the spatial resolution of our micro-objectivesetup. At high excitation, the spectra change into multiplets,as shown in Fig. 4 /H20849b/H20850. They show the same lateral proﬁle with those obtained at low excitation, conﬁrming the multi-plet being emitted from a single QR, and not from multipleQRs with different energies. The energy split from theground-state emission to the ﬁrst excited-state emission isobserved to be 9 meV for this QR. C. Spectroscopy on single concentric double rings We present the PL spectra of two concentric quantum double-rings—DQR- aand DQR- b—in Fig. 5. Similarly to the case of QR, the spectra consist of discrete lines, i.e., amain peak following a satellite one, which is at the highenergy side of the main peak. The former is associated withrecombination of carriers in the ground state, whereas the latter is from the excited states. The energy difference be-tween the ground-state line and the excited-state one is 7.2meV in DQR- aand 8.5 meV in DQR- b. We point out that, in contrast to the QR case, we observe the satellite peak even atthe lowest excitation. For the lowest excitation intensity, wecan estimate the carrier population inside a ring to be lessthan 0.1, according to the carrier capturing cross section de-termined for GaAs QD, which was prepared with the sameepitaxial technique. 27Observation of the excited-state emis- FIG. 4. Position dependence of micro PL spectra in GaAs QR at /H20849a/H208501.2 W/cm2and /H20849b/H2085036 W/cm2. From top to bottom, the position of detection is moved from 0 to 3.8 /H9262m in steps of 0.32 /H9262m. Spec- tra are vertically offset for clarity. FIG. 5. Emission spectra for concentric quantum double-rings, DQR-a and DQR- b. Their respective excitation densities were, from bottom to top, 1, 10, and 30 W/cm2. Spectra are normalized to their maxima and offset for clarity.KURODA et al. PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-4sion suggests reduction of carrier relaxation from the excited level to the ground level. That feature will be discussed later. At high excitation, we ﬁnd that several additional lines are superimposed on the spectra, as shown by the broken arrows.Presence of these contributions implies the presence of ﬁneenergy structures in DQR. IV. CALCULATION FOR SINGLE-CARRIER LEVELS We evaluate the energy levels of the ring in the frame- work of a single-band effective-mass envelope model.31,32In calculation, the actual shape measured by AFM is adopted asthe potential of quantum conﬁnement; for simplicity, the ringis assumed to hold a cylindrical symmetry. The techniqueemployed in this section follows Ref. 31 describing the exactdiagonalization of the effective-mass Hamiltonian. Note that,in our lattice-matched GaAs/ /H20849Al,Ga /H20850As rings, strain effects are negligible. For that reason, the simple effective-mass ap- proach is expected to provide accurate energy levels. Thispresents a contrast to the case of Stranski-Krastanow growndots, where the electronic structure is modiﬁed strongly bycomplex strain effects. 33The versatility of the present method is seen in Refs. 34 and 35, showing good agreementbetween the asymmetric PL lineshape in a GaAs/ /H20849Al,Ga /H20850As QD ensemble and the calculation, taking into account the morphologic distribution of dots. We also notice that the present calculation neglects Cou- lomb interaction between an electron and a hole. Becauseour rings are sufﬁciently small that conﬁnement effects aredominant, the Coulomb interaction can be treated as a con-stant shift in the transition energies, independent of thechoice of an electron state and the hole state. In the follow-ing discussion, we are interested in relative energy shiftsfrom the ground-state transition to the excited-state one, anddependence of the exciton binding energy on relevant/H20849single-carrier /H20850transition should be sufﬁciently below the ex-perimental accuracy. Thus, we restrict ourselves to calculate the single-carrier energy levels, discarding Coulomb correla-tion effects. Our approach to the problem is to enclose the nanocrystal inside a large cylinder of radius R cand height Zc,o nt h e surface of which the wave function vanishes. Care should betaken to set R candZcaway from the ring, so that the eigen- values are almost independent of their choice. Taking intoaccount the rotational symmetry of the Hamiltonian, and forthis boundary condition, the wave function, /H9021 L, where L/H20849=0,±1,…/H20850is the azimuthal quantum number, is expanded in terms of a complete set of the base functions, /H9264i,jL, formed by products of Bessel functions of integer order Land sine functions of z, /H9021L/H20849z,r,/H9258/H20850=/H20858 i,j/H110220Ai,jL/H9264i,jL/H20849z,r,/H9258/H20850, /H208491/H20850 /H9264i,jL/H20849z,r,/H9258/H20850=/H9252iLJL/H20849kiLr/H20850eiL/H9258sin/H20849Kjz/H20850, /H208492/H20850 where kiLRcis the ith zero of the Bessel function of integer order JL/H20849x/H20850,Kj=/H9266j/Zc, and/H9252iLare the appropriate normaliza- tion factors, i.e., /H9252iL=/H208812 /H9266ZcRc22 /H20841JL−1/H20849kiLRc/H20850−JL+1/H20849kiLRc/H20850/H20841. /H208493/H20850 In advance of calculation, we have prepared a Hamiltonian that includes a potential term in cylindrical coordinates.Then, we calculated its matrix elements through numerical integration with /H9264i,jLover randz. Finally, the eigenstates were obtained with a diagonalizing matrix. For the present calcu-lation, we have taken into account 35 Bessels and 35 sinefunctions as the base functions for each value of L, and R c/H20849Zc/H20850=120 /H2084920/H20850nm nm. Material parameters used in calcu- lation are summarized in Table I. FIG. 6. Single-carrier energy levels in /H20849a/H20850QR and /H20849b/H20850DQR. Quantization energies for an electron /H20849a heavy hole /H20850with the three lowest radial quantum numbers, Ne/H20849h/H20850, and various angular momenta /H20849up to 10 /H20850are presented. /H20849c/H20850Cross-sectional imaging of electronic probability density in DQR for Ne= 1 ,2 ,a n d3w i t h L=0. The line represents the potential of conﬁnement used for calculation.OPTICAL TRANSITIONS IN QUANTUM RING COMPLEXES PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-5A series of single-carrier levels of QR is shown in Fig. 6/H20849a/H20850. Because the system has cylindrical symmetry, each car- rier level is speciﬁed by the principal /H20849radial /H20850quantum number38N/H20849=1,2,…/H20850, and an azimuthal quantum number L, corresponding to the angular momentum. Two levels with ± L are degenerated at zero magnetic ﬁeld. In Fig. 6 /H20849a/H20850, the car- rier levels belonging to each radial quantum number arealigned vertically with those of a different angular momen-tum. We ﬁnd that a vertical /H20849L-dependent /H20850sequence of quan- tized levels shows a typical signature of ring-type conﬁne-ment. For an ideal ring with inﬁnitesimal width, beingtreated as a one-dimensional system with translational peri-odicity, the level series is expressed as E L=/H60362 2mR2L2, /H208494/H20850 where Randmrespectively represent the radius of the ring and the carrier mass. We ﬁnd that the bilinear dependence ofthe level series, shown in Eq. /H208494/H20850, is reﬂected clearly in the line sequence in Fig. 6 /H20849a/H20850. The energy levels in DQR are shown in Fig. 6 /H20849b/H20850.A sa result of the smaller height, the quantization energies arelarger than those of QR. The level sequence of N=1 is more densely populated than that of N=2, suggesting a large dif- ference in carrier trajectories between the two levels. Ac-cording to Eq. /H208494/H20850, the situation corresponds to the large ef- fective value of RforN=1. The fact is conﬁrmed by the wave functions shown in Fig. 6 /H20849c/H20850. This ﬁgure illustrates the envelope wave function of an electron with various values ofN. They are of zero angular momentum. We ﬁnd that the electron of N=1 is conﬁned mainly in the outer ring. That of N=2 is in the inner ring, and that of N=3 is situated in both rings. That differential conﬁnement engenders remarkablechanges in their trajectory. The amount of penetration for theelectron of N=1 to the inner ring is found to be /H110110.1, whereas that of N=2 to the outer ring is /H110110.05. V. DISCUSSION We have evaluated the oscillator strength for transitions between each electron-hole /H20849e-h/H20850level to determine a con- sistency between the emission spectra and the theoretical lev- els. The magnitude is proportional to the overlap of corre-sponding /H20849envelope /H20850wave functions. Note that, because of cylindrical symmetry, optical transition is not allowed for anelectron and a hole with different angular momentum. More-over, we have determined the transition strengths for the e-hpair with different Ns as less than 1/10 smaller than that with the same Ns. We can therefore infer that the electron, speciﬁed by a pair of NandL, recombines only with the hole of the same NandL. This selection rule allows us to describe each optical transition simply by /H20849N,L/H20850. A series of transition energies for QR, obtained by the procedure described above, are shown in Fig. 7 /H20849a/H20850. For com- parison, the emission spectra of QR- aare plotted as an ex- ample of experimental data. The main peak and the high-energy satellite in the observed spectra are assignedrespectively by the recombination of the e-hpair in the low- est state, /H20849N,L/H20850=/H208491,0/H20850, and that of the ﬁrst excited radial state, /H208492, 0/H20850. The split between the two transitions is deduced to be 13.1 meV, which agrees with the energy shift obtainedby experiments. Note that the emissions associated with highangular momenta are not present in the data, which suggestsrapid relaxation of angular momentum, whose process isquite faster than transition between radial quantization levelsor recombination between an electron and a hole. A possibleorigin for fast angular momentum relaxation is structuralasymmetry of the ring, which results from elongation, impu-rity, and surface roughness. In this case, angular momentumdoes not represent a good quantum number, and scatteringbetween different Llevels efﬁciently occurs. We show the comparison between the experimental spec- tra of DQR and the results of calculation in Fig. 7 /H20849b/H20850.A si n the case of QR, the main PL peak and the satellite one areexplained respectively in terms of the transition of /H20849N,L/H20850 =/H208491,0/H20850and that of /H208492, 0 /H20850. The energy split deduced from calculation is 8.8 meV, which agrees with the experimentalTABLE I. Material parameters used in the effective mass calcu- lation for the conduction band /H20849CB/H20850and the valence band /H20849VB/H20850 Quantity Units GaAs Al 0.3Ga0.7As CB effective massam00.067 0.093 VB effective mass /H20849heavy hole /H20850am00.51 0.57 CB band offsetbmeV 262 VB band offsetbmeV 195 aReference 36. bReference 37. FIG. 7. /H20849a/H20850A series of optical transition energies in QR, obtained by the calculation. The PL spectrum of QR a at 15 W/cm2is shown in the inset. /H20849b/H20850The energies of optical transitions in DQR, together with the PL spectrum of DQR a at 10 W/cm2for comparison.KURODA et al. PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-6value. It is noteworthy that, in DQR, the wave function of N=1 is localized mainly in the outer ring, whereas that of N=2 is localized in the inner ring. Thus, the two peaks in the observed spectra come from the two rings, which consist of aDQR. In this connection, it is noteworthy that the excited-state emission in our experiment appears even when the car-rier population is less than 1. This presence of the excitedstate emission constitutes direct evidence for the carrier con- ﬁnement into the two rings. Tunneling probability betweenthe inner ring and the outer one is not very large, engender-ing the observation of the excited-state emission. Note alsothat the emission of the outer ring is more intense than thatof the inner ring. The effect is attributable to their differentsurface areas, which affects the efﬁciency of the carrier in-jection from a barrier. Finally, we would like to discuss the validity of the theo- retical treatment which neglects the Coulomb correlation be-tween an electron and a hole. A limitation of the validitytakes place when an exciton binding energy is fairly largecompared to a single-carrier split energy, i.e., 13.1 meV forQR, and 8.8 meV for DQR. In this case the Coulomb corre-lation admixes various single-carrier levels. We can roughlyevaluate the exciton binding energy of the rings in compari-son with that of GaAs/ /H20849Al,Ga /H20850As quantum wells /H20849QWs /H20850. This is because the carrier quantization of our rings is mainly associated with vertical conﬁnement, and the lateral dimen-sion is larger than the exciton Bohr diameter. According tonumerous attempts on the study of GaAs/ /H20849Al,Ga /H20850As QWs, 39 the QW conﬁnement enhances the exciton binding energy from a bulk value of 3.7 meV to /H1135110 meV at /H110116-nm thick QWs. A smaller thickness results in weaker exciton bindingdue to the carrier penetration to a barrier. These results sup-port the exciton binding energy being smaller than, or atmost comparable to, the single-carrier split energy of therings. The observed spectral doublet, therefore, directly re-ﬂects the single-carrier levels. The situation presents a clearcontrast to that of Stranski-Krastanow grown QDs, in whichthe dot dimension is quite smaller than our droplet-epitaxial nanostructures, so that the exciton binding energy reaches/H1101130 meV. VI. CONCLUSIONS We have used a remarkable change in quantum dot shape through droplet epitaxial growth to fabricate semiconductorquantum ring complexes. Electronic structures of the quan-tum rings are identiﬁed using an optical, noncontact ap-proach. In the small ringlike system, carriers are quantizedalong two orthogonal degrees of freedom—radial motion androtational motion. The latter corresponds to angular momen-tum. The optical transition takes place on recombination ofan electron and a heavy hole, which are in the grand state ofthe ring, and in the excited radial state. In concentric doublequantum rings, emission originating from the outer ring andthat from the inner ring are observed distinctly. Results ofeffective-mass calculations well reproduce the emissionspectra applied to a single quantum ring. We believe that the present ring system will contribute to a deeper understanding of quantum interference effects in anonsimply connected geometry. In this connection, we wouldlike to point out that our concentric double rings are a goodcandidate to realize in-plane polarization for carriers, produc-ing a robust Aharonov-Bohm feature in neutral excitonictransition. 20,40Magnetoptical experiments using these quan- tum rings are now in progress. ACKNOWLEDGMENTS We are grateful to J. S. Kim, T. Noda, K. Kuroda, and Professor M. Kawabe for their fruitful discussions. Wewould like to thank K. Kurakami for his experimental assis-tance. T. K. and T. M. acknowledge support of Grant-in-Aidfrom the Ministry of Education, Culture, Sports, Science andTechnology of Japan /H20849Grant Nos. 15710076 and 17710106 /H20850. 1M. Grundmann, J. Christen, N. N. Ledentsov, J. Böhrer, D. Bim- berg, S. S. Ruvimov, P. Werner, U. Richter, U. Gösele, J. Hey-denreich, V. M. Ustinov, A. Yu. Egorov, A. E. Zhukov, P. S.Kop’ev, and Zh. I. Alferov, Phys. Rev. Lett. 74, 4043 /H208491995 /H20850. 2D. Bimberg, M. Grundman, and N. N. Ledentsov, Quantum Dot Heterostructures /H20849John Wiley & Sons, Chichester, 1999 /H20850. 3N. C. van der Vaart, S. F. Godijn, Y. V. Nazarov, C. J. P. M. Harmans, J. E. Mooij, L. W. Molenkamp, and C. T. Foxon, Phys.Rev. Lett. 74, 4702 /H208491995 /H20850. 4M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, and A. Forchel, Science 291, 5503 /H208492001 /H20850. 5D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 /H208491993 /H20850. 6J. M. García, G. Medeiros-Ribeiro, K. Schmidt, T. Ngo, J. L. Feng, A. Lorke, J. Kotthaus, and P. M. Petroff, Appl. Phys. Lett. 71, 2014 /H208491997 /H20850. 7T. Mano and N. Koguchi, J. Cryst. Growth 278, 108 /H208492005 /H20850.8T. Mano, T. Kuroda, S. Sanguinetti, T. Ochiai, T. Tateno, J. Kim, T. Noda, M. Kawabe, K. Sakoda, G. Kido, and N. Koguchi,Nano Lett. 5, 425 /H208492005 /H20850. 9Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 /H208491959 /H20850. 10M. V. Berry, Proc. R. Soc. London A392 ,4 5 /H208491984 /H20850. 11M. Büttiker, Y. Imry, and R. Landauer, Phys. Lett. 96A, 365 /H208491983 /H20850. 12R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lett. 54, 2696 /H208491985 /H20850; L. P. Lévy, G. Dolan, J. Dun- smuir, and H. Bouchiat, ibid. 64, 2074 /H208491990 /H20850; V. Chan- drasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gal-lagher, and A. Kleinsasser, ibid. 67, 3578 /H208491991 /H20850; A. Yacoby, M. Heiblum, D. Mahalu, and H. Shtrikman, ibid. 74, 4047 /H208491995 /H20850. 13A. Fuhrer, S. Lüscher, T. Ihn, T. Heinzel, K. Ensslin, W. Weg-scheider, and M. Bichler, Nature 413, 822 /H208492001 /H20850. 14A. Lorke, R. J. Luyken, A. O. Govorov, J. P. Kotthaus, J. M. Garcia, and P. M. Petroff, Phys. Rev. Lett. 84, 2223 /H208492000 /H20850.OPTICAL TRANSITIONS IN QUANTUM RING COMPLEXES PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-715M. Bayer, M. Korkusinski, P. Hawrylak, T. Gutbrod, M. Michel, and A. Forchel, Phys. Rev. Lett. 90, 186801 /H208492003 /H20850. 16E. Ribeiro, A. O. Govorov, W. Carvalho, Jr., and G. Medeiros- Ribeiro, Phys. Rev. Lett. 92, 126402 /H208492004 /H20850. 17R. A. Römer and M. E. Raikh, Phys. Rev. B 62, 7045 /H208492000 /H20850. 18H. Hu, J. L. Zhu, D. J. Li, and J. J. Xiong, Phys. Rev. B 63, 195307 /H208492001 /H20850. 19J. Song and S. E. Ulloa, Phys. Rev. B 63, 125302 /H208492001 /H20850. 20A. O. Govorov, S. E. Ulloa, K. Karrai, and R. J. Warburton, Phys. Rev. B 66, 081309 /H20849R/H20850/H208492002 /H20850. 21N. Koguchi, S. Takahashi, and T. Chikyow, J. Cryst. Growth 111, 688 /H208491991 /H20850. 22K. Watanabe, N. Koguchi, and Y. Gotoh, Jpn. J. Appl. Phys., Part 239, L79 /H208492000 /H20850. 23K. Watanabe, S. Tsukamoto, Y. Gotoh, and N. Koguchi, J. Cryst. Growth 227-228 , 1073 /H208492001 /H20850. 24D. Granados and J. M. García, Appl. Phys. Lett. 82, 2401 /H208492003 /H20850. 25T. E. Schlesinger and T., Kuech, Appl. Phys. Lett. 49, 519 /H208491986 /H20850. 26S. Sanguinetti, K. Watanabe, T. Tateno, M. Wakaki, N. Koguchi, T. Kuroda, F. Minami, and M. Gurioli, Appl. Phys. Lett. 81, 613 /H208492002 /H20850. 27T. Kuroda, S. Sanguinetti, M. Gurioli, K. Watanabe, F. Minami, and N. Koguchi, Phys. Rev. B 66, 121302 /H20849R/H20850/H208492002 /H20850. 28R. J. Warburton, C. Schäﬂein, D. Haft, F. Bickel, A. Lorke, K. Karrai, J. M. Garcia, W. Schoenfeld, and P. M. Petroff, Nature 405, 926 /H208492000 /H20850.29K. Kuroda, T. Kuroda, K. Sakoda, N. Koguchi, and G. Kido /H20849un- published /H20850. 30In other words, large azimuthal asymmetry induces double- minima potential for carriers. Consequently, two localized levelsare formed in QR- c, leading to observation of the spectral dou- blet. 31J. Y. Marzin and G. Bastard, Solid State Commun. 92, 437 /H208491994 /H20850. 32M. Califano and P. Harrison, Phys. Rev. B 61, 10959 /H208492000 /H20850. 33O. Stier, M. Grundmann, and D. Bimberg, Phys. Rev. B 59, 5688 /H208491999 /H20850. 34S. Sanguinetti, K. Watanabe, T. Kuroda, F. Minami, Y. Gotoh, and N. Koguchi, J. Cryst. Growth 242, 321 /H208492002 /H20850. 35V. Mantovani, S. Sanguinetti, M. Guzzi, E. Grilli, M. Grioli, K. Watanabe, and N. Koguchi, J. Appl. Phys. 96, 4416 /H208492004 /H20850. 36L. Pavesi and M. Guzzi, J. Appl. Phys. 75, 4779 /H208491994 /H20850. 37M. Yamagiwa, N. Sumita, F. Minami, and N. Koguchi, J. Lumin. 108, 379 /H208492004 /H20850. 38The principal quantum number, N, characterizes both of the radial and vertical motions of carriers. However, we describe Nsimply as the radial quantum number, because the vertical component ofelectronic motion is essentially independent of N/H20849/H333553/H20850. This is due to the strong conﬁnement along the growth direction. SeeFig. 6 /H20849c/H20850. 39See, e.g., R. C. Miller and D. A. Kleinman, J. Lumin. 30, 520 /H208491985 /H20850. 40L. G. G. V. Dias da Silva, S. E. Ulloa, and T. V. Shahbazyan, Phys. Rev. B 72, 125327 /H208492005 /H20850.KURODA et al. PHYSICAL REVIEW B 72, 205301 /H208492005 /H20850 205301-8
PhysRevB.100.020101.pdf	PHYSICAL REVIEW B 100, 020101(R) (2019) Rapid Communications High-pressure structure and electronic properties of YbD 2to 34 GPa S. Klotz,1,M. Casula,1K. Komatsu,2S. Machida,3and T. Hattori4 1IMPMC, CNRS UMR 7590, Sorbonne Université, 4 Place Jussieu, F-75252 Paris, France 2Geochemical Research Center, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 3CROSS, Neutron Science and Technology Center, 162-1 Shirakata, Tokai, Ibaraki 319-1106, Japan 4J-PARC Center, Japan Atomic Energy Agency, 2-4 Shirakata, Tokai, Ibaraki 319-1195, Japan (Received 26 September 2018; revised manuscript received 26 March 2019; published 10 July 2019) Ytterbium dihydride (YbH 2) shows a well-known transition at ≈16 GPa from a CaH 2-type structure to a high- pressure (high- P) phase with Yb at hcp sites and unknown H positions. Here, we report its complete structure determination by neutron diffraction at 34 GPa. Hydrogen (deuterium) is located at 2 aand 2 dpositions of space group P63/mmc , thus forming a high-symmetry “collapsed” close-packed lattice. The transition is sluggish and can be seen as a transfer of 1 /2 of the hydrogen atoms from strongly corrugated H layers to interstitial sites of the Yb lattice. We demonstrate by ﬁrst-principles calculations that the transition is related to a change froma completely ﬁlled f-electron conﬁguration to a fractional f-hole ( ≈0.25 h) occupation in the high- Pphase. The f→dcharge transfer closes the gap at the transition and leads to a metallic ground state with a sizable electron-phonon interaction involving out-of-plane vibrational modes of interstitial hydrogen. DOI: 10.1103/PhysRevB.100.020101 There is considerable recent interest in hydrides under high pressure following the discovery of high- Tcsuperconductivity in H 2S compressed to 150 GPa [ 1]. Indeed, theory [ 2] and recent experiments [ 3] give strong indications that supercon- ductivity at elevated temperatures might occur in numeroushydrides, in particular, binary rare-earth hydrides [ 4] with a high hydrogen (H) content, which are unstable at ambientpressure. The case of ytterbium (Yb) hydrides appears tobe interesting for the following reasons: Elemental Yb wasrecently found to be superconducting beyond 80 GPa [ 5]. This is highly unexpected, since Yb is diamagnetic at ambient con-ditions [ 6], and pressure is believed to turn it magnetic through af 14→f13(Yb2+→Yb3+) valence change [ 7] supported by extended x-ray absorption ﬁne structure (EXAFS) [ 8], x-ray absorption near edge structure (XANES) [ 5], and res- onant inelastic x-ray scattering (RIXS) [ 9] data, thus unlikely to be superconducting. This surprising ﬁnding draws attention to its most common hydride, YbH 2. This material is insulating and crystallizes at 0 GPa in the αphase, whose structure is of CaH 2or- thorhombic type (space group Pnma )[10]. Under pressure, it transforms at ≈16−20 GPa into a phase with Yb sites at a hexagonal-close-packed (hcp) lattice [ 11]. The H atom posi- tions are presently unknown. Again, this ﬁrst-order transition(/Delta1V/V=3%) might be driven by a valence change of Yb from 2 +to 3+, as indicated by EXAFS [ 12]. The YbH 2 high-pressure (high- P) phase is stable to at least 60 GPa [ 12], and its electronic properties are unknown. Given this context, it appears timely to determine the full structure of YbH 2, i.e., the H position as well as its electronic properties beyond 16 GPa. Generally speaking, experimentaldata on the H location in high- Pmetal hydride phases are Corresponding author: Stefan.Klotz@upmc.frextremely sparse. Almost all structural information above a few GPa has been obtained by synchrotron x-ray diffractionwhich is blind to H in the presence of heavy atoms. So far,the H positions are simply assumed to be on “favorable”interstitial sites, or deduced from ﬁrst-principles calculations[13–15]. Here, we present high- Pneutron diffraction data to 34 GPa which determine the structure of YbH 2, in particular, the H positions. We use the structural data to determine its elec-tronic properties through ab initio methods, which indicate a semiconducting-to-metal transition concomitant with thestructural transformation. The metallization is driven by an f→dcharge transfer with partial f-hole unbinding. We used deuterated samples (YbD 2) for the well-known fact that H is a strong incoherent scatterer. From the x-raydiffraction data it is clear that deuteration has no signiﬁcantstructural effect, even at high pressure [ 12]. Ytterbium has 70 electrons and hence scatters ≈5000 times stronger than deuterium (D), i.e., an x-ray diffraction pattern is completelydominated by Yb. This is not the case for neutrons: The co-herent neutron cross sections of Yb and D are 19.4 and 5.6 b,i.e., are of the same order of magnitude. Neutron diffraction ismost likely the only technique to solve the problem. The sample was synthesized by heating Yb powder (99 .9% metal purity, fresh ﬁlings with ca. 0.1 mm grain size froma rod purchased from Goodfellow) in 2000 hPa deuteriumatmosphere to 600 ◦C, hence similar to the procedure de- scribed in previous work [ 10]. Neutron diffraction data of ground powder taken at ambient conditions in a vanadium canreveal single-phase material with the expected CaH 2struc- ture, space group Pnma ,a=5.8823(2) Å, b=3.5676(1) Å, c=6.7588(2) Å, and fractional atomic positions x(Yb)= 0.2396(3), z(Yb)=0.1119(1), x(D1)=0.3556(3), z(D1)= 0.4291(3), x(D2)=−0.0305(5), z(D2)=0.6784(3). As ex- pected, the compound is slightly nonstoichiometric with a 2469-9950/2019/100(2)/020101(5) 020101-1 ©2019 American Physical SocietyS. KLOTZ et al. PHYSICAL REVIEW B 100, 020101(R) (2019) FIG. 1. Neutron diffraction patterns of YbD 2at 298 K, in the low-PP n m a phase (upper panel) and the high- PP63/mmc phase (lower panel). The lines are Rietveld ﬁts to the data (dots). Uppertick marks indicate Bragg reﬂections of the sample, and lower tick marks of diamond from the anvils. χ 2=2.05,Rwp=7.95% (top); χ2=2.27,Rwp=8.05% (bottom). Accumulation time is 1 and 2 h, for 4.9 and 34 GPa pressure, respectively. reﬁned D composition of 1.912(4) instead of 2. This is a well-known phenomenon and was observed in all previousstudies on YbD 2[10]. For the sake of simplicity, we will call the sample YbD 2throughout the text. Neutron diffraction measurements were carried out at the high- Pbeamline PLANET [ 16] at MLF, the Japan Proton Accelerator Research Complex (J-PARC), Tokai, Ibaraki,Japan. The three high- Pruns used three types of double-toroidal sintered diamond anvils [ 17,18] with maximum sample volumes of 12, 7, and 3 mm 3, encapsulating TiZr gaskets and a 4:1 methanol-ethanol mixture as thepressure transmitting ﬂuid. All runs applied a VX4-typeParis-Edinburgh load frame [ 17] with the position of the sample maintained to within ±0.1 mm relative to the laboratory frame. The pressure values cited here weredetermined from the equation of state (EOS) of YbD 2reported by the x-ray work [ 12], using the measured ambient-pressure unit cell volume ( V0=141.84 Å3) for the low-pressure (low- P) phase. In the ﬁrst run to 22.6 GPa the sample was temporarily heated to 363 K in each pressure ramp-up to keepthe sample hydrostatic up to 16 GPa. In the runs to 26 and34 GPa compressions were made at 299 K. Figure 1shows diffraction patterns along with Rietveld reﬁnements [ 19]o fY b D 2in the low- Pαphase at 4.9 GPa and in the high- Pphase at 34 GPa. The transition is found to be sluggish in all runs (see Fig. 2), and heating to 363 K in the ﬁrst loading had no signiﬁcant effect on its kinetics. It startsat 20 GPa (approximately consistent with previous data) and FIG. 2. Phase fraction of YbD 2as pressure is increased (up- stroke). Solid circles, open circles, and squares indicate three differ-ent runs to 22.6, 26, and 34 GPa, respectively. Broad lines are guides to the eye. ends slightly above 26 GPa, thus higher than what has been reported from x-ray studies [ 11]. The determination of the high- Pstructure is facilitated by the fact that the heavy-element (Yb) sublattice is known tobe hcp [ 11] and the number of possibilities of incorporating H(D) therein with the required stoichiometry is limited. WithYb placed at the 2 cposition of space group P6 3/mmc ,n i n e conﬁgurations were tested with H(D) on the remaining 2 a,2b, 2d, and 4 fpositions. These include hence the tetrahedral (2 a) and octahedral (4 f) sites which are preferentially occupied by H(D) in various other hcp metals. Wyckoff sites with multi-plicity 6, 12, and 24 would place H(D) on very low-symmetrysites with partial occupancy which hardly can be justiﬁedgiven the high-symmetry environment of the Yb sublattice.Pattern simulations [ 20] show that out of these nine conﬁgu- rations, only one is compatible with the measured diffraction data, and all others give strongly different intensities andhence can be safely excluded. The structure consists of oneH(D) on the 2 aposition (on the octahedral hcp site) and the other on the 2 dsite (see Fig. 3). The reﬁnement of the pattern at 34 GPa gives a=3.5088(2) Å, c=4.6413(3) Å. Apart from lattice constants, the reﬁnements include only a minimalset of parameters, i.e., phase fraction, proﬁle, and preferredorientation. Interestingly, this H(D) conﬁguration was guessed36 years ago [ 10] from purely geometrical arguments applied to the low- Pαphase. Inspection of the αphase along its baxis reveals an interesting relationship with the high- Pphase, and details of the transition mechanism: In the αphase, Yb is on a distorted hcp lattice, the c/bratio at 15 GPa is 1.90 compared to 1.633 for an ideal hcp structure, and there are further smalldisplacements along the orthorhombic aaxis. H(D) in this structure is stored in strongly corrugated layers separated fromthe Yb layers. The shortest H-H distance ( d HH) inside a layer is 2.64 Å. The high- Ptransition renders a relatively irregular arrangement into a highly symmetric crystal, by transferringhalf of all H(D) into planar interstitials of the Yb layers. Bythis mechanism, the in-plane d HHbecomes larger and equal to 3.56 Å, whereas the shortest dHHbetween neighboring planes reduces only slightly to 2.37 Å. This picture is conﬁrmedby phonon calculations in the high- Pphase, where its low- pressure instability is driven by a phonon softening at q=M. The related distortion implies the doubling of the hexagonal 020101-2HIGH-PRESSURE STRUCTURE AND ELECTRONIC … PHYSICAL REVIEW B 100, 020101(R) (2019) FIG. 3. Structure of YbD 2in its low- PP n m a (top) and the high- PP63/mmc phases (bottom). Large symbols are Yb atoms, and small symbols H(D). Different colors for H(D) highlight different layers.Lines are guides to the eye, and arrows refer to the orthorhombic (top) and hexagonal (bottom) axes. unit cell, with a phonon pattern that brings this structure back to the known αphase [ 20]. Such an H arrangement seems to be unique among all hexagonal transition and rare-earth ( R) hydrides where the hydrogen positions are known from neutron diffraction. Inthe well-studied monohydrides FeH and CrH, hydrogen islocated exclusively at the octahedral sites between the hcplayers [ 21,22]. In the hexagonal rare-earth hydrides (all with compositions RH 3) it is located at both octahedral and tetrahe- dral sites, or close to them, i.e., again between the metal planes [23]. High- PYbH 2seems therefore to be the only hydride known up to now adopting a structure of this type. TABLE I. Atomic orbital occupations per YbH 2unit for the α (ambient pressure) and high- P(26 GPa) phases. Orbital symmetry α High- P 4f(Yb) 14.00 13.75 5d(Yb) 1.30 1.80 6s6p(Yb)+1s(H2) 2.70 2.45 FIG. 4. EOS for αand high- Pphases by ab initio calculations with different ﬂavors, and compared with experiment for YbH 2[11]. DFT-GGA calculations with the PBE functional are shown in red and green, for the frozen- fandf-in-valence PAW pseudopotentials, respectively. DFT-GGA calculations with f-in-valence pseudopoten- tial plus Hubbard repulsion (GGA +U) are shown in blue for the α phase. Knowing the structure of YbD 2allows us to derive elec- tronic properties through ﬁrst-principles calculations at ﬁxedexperimental geometries. We carried out density functionaltheory (DFT) calculations within the generalized gradientapproximation (GGA) built in the Perdew-Burke-Ernzerhof(PBE) functional [ 24,25]. We used the plane-wave implemen- tation as coded in the QUANTUM ESPRESSO package [ 26,27]. The key question we address here is about the role of f electrons in driving the structural transition. Hypotheses havebeen made about a possible valence change in YbH 2between theαand high- Pphase, where a tight competition between the atomic 4 f14(5d6s6p)2(2+) and 4 f13(5d6s6p)3(3+) conﬁg- urations could be at play [ 11]. Yb and Eu are the rare-earth elements where divalent and trivalent states are the closest inenergy [ 28], and they are the only ones where the 2 +valence is the most stable in the solid state at ambient pressure [ 29]. To investigate the role of the felectrons in YbH 2,w er a nt w o types of calculations, one with the fmanifold frozen in the Yb pseudopotential, the other with in-valence felectrons. Both pseudopotentials [projector augmented-wave (PAW) type] arefully relativistic and keep the 5 s5psemicore states in valence [30]. The results for the EOS are shown in Fig. 4and com- pared with experimental data for YbH 2[11]. The geometries used in our DFT calculations have both Yb and H positionsdetermined from experiment [ 31]. The frozen- fcalculations poorly reproduce the experimental EOS, with a signiﬁcantpressure discrepancy at the phase transition. The f-in-valence calculations substantially improve the agreement with the ex-periment, signaling the importance of the structural feedbackon the f-band shape. In the αphase, the agreement is further improved by including an Hubbard repulsion term within theDFT+Uscheme [ 32], while in the high- Pphase the agree- ment between theory and experiment is very good alreadyat the DFT level. This points towards an fmanifold more correlated in the αthan in the high- Pphase. The experimental 020101-3S. KLOTZ et al. PHYSICAL REVIEW B 100, 020101(R) (2019) FIG. 5. Band structure and DOS for the αphase at ambient pressure (left panel), and for the high- Pphase at 26 GPa (right panel). In both cases, calculations are done with the fully relativistic PBE functional and f-in-valence PAW pseudopotentials. knowledge of both Yb and H positions gives us a unique chance to assess the validity of the GGA approximationwith f-in-valence electrons for this kind of system. Other widely used DFT approximations, such as the local densityone (LDA), turn out to perform more poorly, as shown byadditional calculations we report in the Supplemental Material(SM) [ 20]. To gain insight into the role played by the felectrons as valence states, we plot in Fig. 5the band structure and the density of states (DOS) at ambient and high pressure forthe approximations reproducing best the experimental EOS,i.e., the DFT and DFT +Ufor the high- Pandαphase, re- spectively. The ambient-pressure phase is a semiconductor, inagreement with experiment, with a gap of about 1 eV betweenthe empty states (mainly of dcharacter) and the narrow f j=7/2 bands. In contrast, the fmanifold in the high- Pphase has a much wider bandwidth, because of strong hybridizationwith the dstates. This is a typical signature of the f-electron delocalization. The fstates are pushed up in energy, and the f 7/2multiplet crosses the Fermi level. The high- Pphase is hence clearly a metal. The fj=7/2-fj=5/2splitting of more than 1 eV is a straightforward manifestation of the strong spin-orbitcoupling (SOC) in YbH 2. While this splitting signiﬁcantly reduces the band gap in the insulating αphase, the fermiology of the metallic phase seems to be only marginally affected bySOC (see SM [ 20]). The f-character change across the transition is also re- ﬂected by the atomic orbital occupation analysis, reported inTable I. One can see that the P6 3/mmc phase is in a mixed valence conﬁguration, through the formation of conducting f7/2holes with a fractional occupation (0.25 per YbH 2unit at 26 GPa). There is also a simultaneous increase of the5doccupation and a slight depletion of the 6 s6pmanifold. In other words, the structural transition is accompanied bya charge transfer towards the dorbitals, and by the partial delocalization of fholes. Both phenomena are responsible fora tighter chemical bond between two neighboring Yb atoms, whose distance is indeed much shorter in the high- Pphase [33]. The hybridization between the 4 fand 5 dorbitals increases thefbandwidth and thereby reduces correlations. This ra- tionalizes the fact that the EOS of the high- Pphase is well described already at the DFT level, while in the αphase, showing very ﬂat felectrons and band-insulating character, the addition of an explicit Hubbard term seems necessary for aquantitative agreement with the experiment. It is interesting tonote that, qualitatively, one can get the same physical picturefor the transition even without U. The only difference is a smaller band gap in the αphase (0.2 eV) and, consequently, a lower transition pressure obtained by standard GGA (seeFig. 4and SM [ 20]). We veriﬁed that the metallicity of this phase is robust against the possible occurrence of magneticorder arising from the f-hole moments [ 34]. Neither a ferro- magnetic nor an interlayer antiferromagnetic order is a stableground state at the DFT level. Given its metallic character, superconductivity in high- P YbH 2appears to be a possibility to explore. 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PhysRevB.99.134434.pdf	PHYSICAL REVIEW B 99, 134434 (2019) Anisotropic magnetization plateaus in Seff=1/2 skew-chain single-crystal Co 2V2O7 L. Yin,1Z. W. Ouyang,1,J. F. Wang,1,†X. Y . Yue,1R. Chen,1Z. Z. He,2Z. X. Wang,1Z. C. Xia,1and Y . Liu3 1Wuhan National High Magnetic Field Center & School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China 2State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences, Fuzhou 350002, People’s Republic of China 3School of Physics and Technology, Wuhan University, Wuhan 430072, People’s Republic of China (Received 5 November 2018; revised manuscript received 15 March 2019; published 24 April 2019) We report anisotropic magnetization plateaus in the skew-chain antiferromagnet Co 2V2O7with Seff=1/2a n d al a r g e gfactor. When the ﬁeld is applied along the easy baxis, a 1 /2-like magnetization plateau is observed within 5 .4∼11.6 T, followed by a 3 /4 plateau at 15.7 T. For the hard aandcaxes, however, there is only a sign of 1/2-like magnetization plateau. These observations are quite different from the isostructural Ni 2V2O7[Z. W. Ouyang et al. ,P h y s .R e v .B 97,144406 (2018 )]. Theoretically, the ﬁrst-principles calculations reveal the large interchain coupling, which leads to classical antiferromagnetic ordering and spin-ﬂop-like transition. With thislarge interchain coupling, the exact diagonalization analysis yields 1 /2a n d3 /4 plateaus, showing the quantum origin of magnetization plateaus. Thus, Co 2V2O7is an interchain-coupled system showing both classical and quantum behaviors. DOI: 10.1103/PhysRevB.99.134434 I. INTRODUCTION Quantum effects in classical antiferromagnets are very interesting. One example of great concern is magnetizationplateau, in which the magnetization corresponds to a fractionof saturation magnetization— M s/n, where nis the period of the spin state [ 1]. Not surprisingly, magnetization plateaus of pure quantum origin can be observed in low-dimensional(D) quantum spin systems like diamond chain [ 2], bond- alternating chain [ 3], ladders [ 4], and spin dimers [ 5], in which the spin number is small ( S/lessorequalslant1) and the long-range antiferromagnetic (AFM) ordering is absent due to quantumﬂuctuations. For the triangular, kagome, and pyrochlore lat-tices, the 1 /3o r1/2 magnetization plateau can occur in both quantum [ 6] and classical systems [ 7–13] with spin number varying from S=1/2t o5/2, because spin conﬁgurations can be stabilized by thermal and/or quantum ﬂuctuations due togeometrical frustration or spin-lattice coupling. Interestingly,the coupled S=1/2 spin clusters [ 14–16] were found to have a coexistence of quantum-mechanical magnetization plateauand AFM long-range ordering. The trimerlike SrMn 3P4O14 (S=5/2) [17] can be considered as an extreme example which exhibits 1 /3 plateau in a classical large-spin system. In most of the chainlike low-D systems exhibiting magneti- zation plateau, the interchain coupling is negligible or at leastnot dominant. Otherwise, the quantum magnetization plateauis not expected to occur. However, we have recently observeda1/2 magnetization plateau and a nematiclike transition in theS=1 skew-chain system Ni 2V2O7[18]. This observation is indeed unusual for a system exhibiting classical AFM zwouyang@mail.hust.edu.cn †jfwang@hust.edu.cnordering and spin-ﬂop transition due to large interchain cou-pling [ 19,20]. There is an isostructural skew-chain antiferromagnet Co 2V2O7[21–23]. As shown in Fig. 1(a),C o 2V2O7has a monoclinic crystal structure (space group P21/c)[21,24] composed of skew chains of edge-sharing octahedra, Co(I)O 6 and Co(II)O 6, running along the caxis of the crystal. The quasi-one-dimensional structure is formed by embedding thecorner-shared nonmagnetic tetrahedrons VO 4[not shown in Fig. 1(a) for clarity] in between the skew chains. It was reported that Co 2V2O7undergoes an AFM ordering at TN= 6K [ 25] and a ﬁeld-induced spin-ﬂop-like transition along thebaxis. Obviously, Co2+ions display three-dimensional (3D) magnetism due to the interchain coupling, similar to theisostructural Ni 2V2O7[19,20]. Very recently, a sign of 1 /2 magnetization plateau has been observed between 7 and 12Ti nC o 2V2O7[26]. Note that this plateau is not as ﬂat as that in Ni 2V2O7[18] even after subtracting the Van Vleck paramagnetic susceptibility. This is probably because of theuse of polycrystalline samples. Also, there is a very tinyanomaly between 15 and 20 T in the derivative dM/dHcurve [26]. All these suggest that consecutive magnetic transitions may occur along the easy axis of Co 2V2O7. In this paper, we present anisotropic magnetization process in a single crystal of Co 2V2O7. Quite different from the case of Ni 2V2O7[18], two fascinating magnetization plateaus–a 1/2-like plateau and a 3 /4 plateau, are observed, depending on crystallographic directions. Our high-ﬁeld magnetizationand electron spin resonance (ESR) data demonstrate the vali-dation of S eff=1/2 description of the magnetic moment with a large gfactor at low temperature. The ﬁrst-principles cal- culations and exact diagonalization are employed to explainthe large interchain coupling and the unusual magnetizationprocess. 2469-9950/2019/99(13)/134434(9) 134434-1 ©2019 American Physical SocietyL. YIN et al. PHYSICAL REVIEW B 99, 134434 (2019) FIG. 1. (a) Skew chain with alternating Co(I)O 6(red) and Co(II)O 6(blue) octahedra along the caxis. There are four Co(I) atoms numbered by 1, 3, 6, and 8 and four Co(II) atoms numbered by 2, 4, 5, and 7 in a unit cell (4 formula unit). The arrows denote the spin orientations of Co ions for the ground-state AFM-1 derived fromthe ﬁrst-principles calculations. (b) Three types of superexchange interactions: J 1—3.053(3) Å (green bond, Co5-O1/O2-Co6), J2— 3.033(5) Å (blue bond, Co6-O3/O4-Co7), and J3—2.972(7) Å (black bond, Co3-O5/O6-Co6) as well as the Co–O–Co bond angles. II. EXPERIMENTAL DETAILS The single crystal of Co 2V2O7was prepared by the pub- lished procedures [ 25]. The polycrystalline powder was syn- thesized by a standard solid-state reaction method. Single-crystal x-ray-diffraction (XRD) data were collected at roomtemperature using the program SHELXL -2016 on an XtaLAB Mini II diffractometer equipped with a Rigaku Mo x-raysource. Powder XRD data were collected using a PANalyticalX’Pert powder x-ray diffractometer with Cu K αradiation. The program FULLPROF was used for the Rietveld reﬁnement [27]. The chemical compositions were checked by an electron probe microanalyzer (EPMA-8050G). Magnetization mea-surements were performed with a superconducting quantuminterference device (SQUID) magnetometer. Speciﬁc heat wasmeasured under 0–7 T using a commercial physical proper-ties measurement system (PPSM). High-ﬁeld magnetizationmeasurements were performed at 0.7–7 K using a homemadepulsed ﬁeld up to 38 T. The pulsed-ﬁeld ESR spectra were col-lected in the ﬁeld-increasing process. The ﬁrst-principles cal-culations were carried out using the self-consistent full poten-tial linearized augmented plane-wave package WIEN 2K[28], FIG. 2. (a) Magnetic susceptibility χ(T) curves measured at 0.1 T. The inset is the corresponding χ−1(T) curves. The dashed lines represent Curie-Weiss ﬁt. (b) Speciﬁc heat curves for H//bmeasured at different ﬁelds. The curves are shifted up by various magnitudesin relation to the 7-T curve. using the general gradient approximation with the Perdew- Burke-Ernzerhof [ 29] parametrization for the exchange cor- relation. The considered spin conﬁgurations are collinear. Toinclude the correlated effect of 3 delectrons, the effective on-site Coulomb interaction parameter was chosen as U eff= U−J=5.0e Vf o rC o2+[30,31], where UandJare on-site Coulomb and exchange interactions, respectively. A mesh of150kpoints was used in the ﬁrst Brillouin zone. The self- consistency was achieved by demanding the convergence oftotal energy to be smaller than 1 meV . III. RESULTS AND DISCUSSION A. Anisotropic magnetization plateaus Figure 2(a) shows the temperature-dependent magnetic susceptibility χ(T) of single-crystal Co 2V2O7measured at 0.1 T. The susceptibility is isotropic at high temperature but ishighly anisotropic at low temperature. Similar to the previousreport [ 25], theχ(T) curve exhibits a sharp cusp at the AFM ordering temperature of T N∼6.0 K when the magnetic ﬁeld is applied along the easy baxis ( H//b). For the hard aandc axes, however, a small increase in susceptibility is seen below 134434-2ANISOTROPIC MAGNETIZATION PLATEAUS IN … PHYSICAL REVIEW B 99, 134434 (2019) TN.T h eχ−1(T) curves shown in the inset of Fig. 2(a)follows the Curie-Weiss law above ∼150 K with a paramagnetic Curie temperature θp=−46,−19, and −20 K for a,b, and caxes, respectively. The corresponding effective moment is 5.6, 5.8.and 5.4μ B/Co2+, close to the previous reports [ 22,25]. Note that below 150 K, the χ−1(T) curves deviate from the Curie- Weiss law, which implies the presence of short-range spincorrelations well above T N. The magnetic phase transitions can be further conﬁrmed by speciﬁc heat measurements. Figure 2(b) shows the C(T) curves for H//bmeasured at different ﬁelds. As expected, a sharpλ-like peak is seen at TN∼6.0 K in the zero-ﬁeld curve. As the magnetic ﬁeld increases, the λ-like peak shifts to the lower temperature, which is a typical feature for classicalantiferromagnet. The vanishing of the λ-like peak above 5 T suggests that the AFM state is suppressed and a new spin stateappears in the high ﬁeld. Figure 3(a) shows the high-ﬁeld magnetization M(H) curves measured at 1.7 K for single-crystal Co 2V2O7.T h e absolute values of magnetization are calibrated with SQUIDdata. The magnetization process is highly anisotropic. All thecurves present complicated ﬁeld-induced magnetic transitions(see, e.g., the dM/dHcurve for the baxis). Note that the high-ﬁeld magnetization increases linearly for all three crys-tallographic directions due to the Van Vleck paramagnetism ofCo 2+originating from the contribution of excited states in the octahedral environment [ 32,33]. From the slope of the M(H) curve, the Van Vleck paramagnetic susceptibility is evaluatedasχ VV=0.022, 0.027, and 0 .010(μB/T)/Co2+fora,b, and caxes, respectively. These values are comparable to those in other Co-based systems exhibiting a magnetization plateauand a similar Van Vleck effect [ 10,34]. Interestingly, an ex- trapolation of the linear part to zero ﬁeld points to the samemagnetization, suggesting that the saturation magnetization isthe same for all three axes. To better examine the anisotropic magnetization process, we plot in Fig. 3(b) theM(H) and the normalized M/M s curves corrected for the Van Vleck term. The saturation magnetization is determined as Ms=2.05μB/Co2+.F o rt h e hard axes, the M(H) curve is composed of two anomalies at ∼6.8 and 18.6 T for H//a(black line) and ∼7.8 and 23.3 T for H//c(blue line), respectively. Between the two anomalies, the magnetization increases smoothly with an upward curvature,signaling the existence of a 1 /2-like magnetization plateau. The case is more complicated when the ﬁeld is appliedalong the easy baxis (red line). In low ﬁeld, a spin-ﬂop-like transition is seen at ∼3.0 T from the maximum of dM/dH [see Fig. 3(a)], similar to the previous report [ 25]. Interest- ingly, between H c1=5.4 T and Hc2=11.6 T, the magneti- zation remains nearly unaltered and a 1 /2-like magnetization plateau is well established. The plateau magnetization is about1.07μ B/Co2+, which is close to 1 /2Ms. With further increas- ing the magnetic ﬁeld from Hc2, there is a fast increase in magnetization followed by an anomaly at Hc3=15.7T . T h e magnetization at Hc3is estimated as 1 .54μB/Co2+, equal to 3/4Ms. Above Hc4=20.2 T, the magnetization reaches saturation. Figure 4shows the raw M(H) curves for H//b measured at 0.7-7 K. As temperature rises, Hc1andHc4move towards lower ﬁeld, whereas Hc2shifts to higher ﬁeld. There is no signiﬁcant shift for Hc3. Above TN=6 K, all anomalies FIG. 3. (a) High-ﬁeld M(H) curves (left) measured at 1.7 K and dM/dHfor the baxis (right). The absolute values of magnetization are calibrated with SQUID data (symbols). The dashed straight lines represent the linear extrapolation of the curves to zero ﬁeld. (b) M(H) (left) and normalized M/Mscurves (right) corrected for the Van Vleck term. The dashed straight lines mark the magnetization between Hc1andHc2,a tHc3, and above Hc4. (c) A comparison of the experimental M(H) curve at 0.7 and 1.7 K for the baxis with the calculated M(H) curve with g=3.95,J1=−17.1K ,J2=−30.1K , andJ3=−13.9 K. The arrows show the ﬁeld direction and possible spin structures at various states. disappear and the curve presents a typical paramagnetic be- havior. The magnetic anisotropy can be described by the molecular-ﬁeld model, in which the spin-ﬂop transition ﬁeld 134434-3L. YIN et al. PHYSICAL REVIEW B 99, 134434 (2019) FIG. 4. Field-increasing and decreasing M(H) curves for H//b measured from 0.7 to 7 K. Hsfand the saturation ﬁeld Hc4are expressed as [ 35] Hsf=/radicalbig (2HE−HA)HA (1) Hc4=2HE−HA (2) With Hsf=3.0 T and Hc4=20.2 T, the exchange ﬁeld is derived to be HE=10.3 T (13.9 K) and the anisotropic ﬁeld HA=0.45 T (0.6 K), which indicates a magnetic anisotropy. Thegfactor is important for theoretical analysis of mag- netization plateaus (see below). Note that effective mag-netic moments of 5 .29μ B/Co2+for polycrystals [ 25] and 5.4–5.8μB/Co2+for our single crystal have been derived from the high-temperature magnetic susceptibility. These val-ues are much larger than theoretical value of 3 .87μ B/Co2+ with S=3/2 and g=2.0, showing a large contribution of orbital moment. This is the case at high temperature. Basedon the crystal-ﬁeld theory, the Co 2+ion located in an octa- hedral environment is in high-spin ( S=3/2) state at room temperature. However, when the temperature is much lowerthan\|λ\|/k B≈250 K ( λis the spin-orbit coupling constant) [32,33], the system can be described by an effective spin of Seff=1/2 with total gfactors of about 13 for three different ﬁeld directions [ 10]. With Seff=1/2 and experimental value ofMs=gμBSeff,t h egfactor is determined as g=4.10, which is much smaller than g=5.2 estimated by high-ﬁeld magne- tization data of powder sample [ 26]. This is obviously due to the difference in saturation magnetization. See Appendixes A andBfor details. The ESR is known to be a precise technique to determine thegvalue. Our high-ﬁeld ESR measurements show that at high temperature well above TN, a single peak is observed and it can be ascribed to the paramagnetic resonance. Thepeak is very broad because of the presence of short-range spin FIG. 5. (a) ESR spectra of single crystal measured at 170 GHz and 40 K. The sharp DPPH line ( g=2.0) is used for a ﬁeld marker. (b) Frequency-ﬁeld ( f-H) relation at 2 K as well as several representative ESR spectra of powder sample. correlations, in line with the result of χ−1(T)i nF i g . 2(a). Figure 5(a) shows the ESR spectra measured at 170 GHz at 40 K along the three axes, where the sharp diphenylpicryl-hydrazyl (DPPH) line ( g=2.0) is for a ﬁeld marker. There is no signiﬁcant change of the resonance ﬁeld for the threedirections and thus the gvalue is derived to be g=5.22. At low temperature, a shift of gfactor is possible due to the AFM ordering. Unfortunately, the ESR spectrumat 2 K ( <T N=6 K) exhibits a multiple-peak structure (not shown here for clarity), which prevents us from determiningthegvalue. However, at low ﬁeld and low frequency, we observed a special AFM resonance mode whose resonanceﬁeld decreases with increasing frequency. The intensity ofthis mode is strong for the powder sample but weak for thetiny single crystal. We plot in Fig. 5(b) the frequency-ﬁeld (f-H) relation at 2 K as well as several representative ESR spectra of powder sample. Clearly, the f-Hrelation is linear and can be described by hf/gμ B=/Delta1−Hwith g=3.95 and zero-ﬁeld gap of /Delta1=270 GHz (i.e., 4.88 T). The value of/Delta1reﬂects the magnitude of HEandHA. As seen from Fig.5(b), extrapolation of the f-Hrelation to high ﬁeld shows that this mode becomes soft at 4.88 T, which is signiﬁcantly 134434-4ANISOTROPIC MAGNETIZATION PLATEAUS IN … PHYSICAL REVIEW B 99, 134434 (2019) larger than the spin-ﬂop transition ﬁeld of Hsf=3.0 T, i.e., (2HE·HA)1/2, but close to the critical ﬁeld for the onset of 1/2 plateau. To clarify the origin of this ESR mode, detailed high-ﬁeld ESR studies are desired. Our high-ﬁeld magnetiza-tion and ESR data unequivocally show that Co 2V2O7indeed has a large gfactor. Thus, the Seff=1/2 description of the magnetic moment is valid at low temperature. The obtainedg=3.95 at 2 K from ESR spectra should be more reliable compared with those estimated by high-ﬁeld magnetizationcalibrated with SQUID data and corrected by subtractingVan Vleck paramagnetism. We also note that a comparablevalue of g=3.84–3.87 was reported in the Co-based quantum magnet Ba 3CoSb 2O9which has Seff=1/2 and exhibits a 1 /3 magnetization plateau at low temperature [ 10]. So far, we have illustrated the presence of a well-deﬁned 1/2-like magnetization plateau and a 3 /4 plateau, depending on crystallographic axes, in the skew-chain antiferromagnetCo 2V2O7which has Seff=1/2 and a large gfactor at low tem- perature. The observation of magnetization plateaus is quitedifferent from the isostructural Ni 2V2O7[18], where a well- deﬁned 1 /2 plateau was observed along all three directions. In Ni2V2O7, there are two easy magnetization directions, aandb axes, and thus two spin-ﬂop-like transitions. The ground-statespins are aligned within the abplane and a spin nematiclike transition appears [ 18]. In Co 2V2O7, however, there is only one easy axis, one spin-ﬂop-like transition, and thus no spinnematiclike phase. As compared with the powder sample of Co 2V2O7exhibit- ing a sign of 1 /2 magnetization plateau within 7–12 T and a very tiny anomaly between 15 and 20 T in the dM/dHcurve [26], the plateaus in single crystal are strongly anisotropic. In particular, the 1 /2-like magnetization for the easy baxis is extremely ﬂat within a more broad ﬁeld range of 5 .4∼11.6T . The 3 /4 plateau appears only along the baxis. Note that the ESR spectra give a precise value of g=3.95 at 2 K for both single-crystal and powder samples [Fig. 5(b)]. This value corresponds to Ms=1.98μB/Co2+. Thus, the saturation magnetization (2 .05μB/Co2+) obtained from single crystal is more reliable than the result (2 .6μB/Co2+) of powder sample [26], which is sample dependent (see Appendix B). These are the advantages of single crystal relative to the powder average. B. Theoretical calculations Since the Seff=1/2 description of the magnetic moment is valid at low temperature, the observed magnetization plateausin Co 2V2O7could be quantum origin. The interpretation of magnetization plateaus relies on information of exchange in-teractions. In the following, we ﬁrst carried out ﬁrst-principlescalculations with the reported structural parameters [ 25]b y considering ﬁve types of collinear spin conﬁgurations. Table I gives the details of each spin conﬁguration in a unit cell(4 formula unit) containing eight atoms, i.e., four Co(I) atomsnumbered by 1, 3, 6, and 8 and four Co(II) atoms numberedby 2, 4, 5, and 7 (see also Fig. 1). As can be seen from Table I, the calculated spin moment is identical for Co(I) and Co(II) and the saturated moment is 2 .7μ B/Co2+, higher than the experimental value of 2 .05μB/Co2+. This discrepancy might be associated with the choice of the Ueffterm and/or the parameter for the exchange correlation. The AFM-1 stateTABLE I. Total energy ( E) relative to the AFM-1 state for the ﬁve various spin conﬁgurations, one ferromagnetic (FM) and four AFM states, in a unit cell (4 f.u.) containing four Co(I) numbered by 1, 3, 6, and 8 and four Co(II) numbered by 2, 4, 5, and 7[see Fig. 1(a)]. The arrows “ ↑”a n d“ ↓” represent the relative spin orientations of Co ions in order of Co1, Co2, …Co8. Conﬁgurations E(meV) MCo(μB) FM (↑↑↑↑↑↑↑↑ ) 12.2 2.70 AFM-1 ( ↑↓↑↓↑↓↑↓ ) 0.0 2.70 AFM-2 ( ↑↑↑↑↓↓↓↓ ) 80.9 2.70 AFM-3 ( ↑↓↑↓↓↑↓↑ ) 163.2 2.70 AFM-4 ( ↑↑↓↓↓↓↑↑ ) 23.7 2.70 is most stable in energy and can be considered as the ground state, similar to the case of Ni 2V2O7[20]. In this state, the exchanges are AFM along the alternating aligned Co(I)-Co(II)-Co(I)-Co(II) chain ( J 1andJ2) and between the nearest- neighboring interchain Co(I)-Co(I) coupling ( J3), which is shown in Fig. 1(a). It should be stressed that the exact values of exchange interactions are not easy to obtain by mappingenergy of different spin conﬁgurations to the isotropic Heisen-berg model Hamiltonian H=−/Sigma1J i,jSiSj(here Jijincludes all the exchanges between sites iand j) because of the complicated 3D magnetism of Co 2V2O7, which was tried in our recent calculations of Ni 2V2O7[20]. However, a qualitative analysis about J1,J2, and J3is possible. Figure 1(b) shows the three superexchange inter- actions and the corresponding bond angles. The intrachainJ 1andJ2are associated with the Co6-O-Co5 and Co6-O- Co7 paths with bond lengths of 3.053(3) Å (green bond)and 3.033(5) Å (blue bond), respectively. The interchain J 3 is via the Co6-O-Co3 path with a bong length of 2.972(7) Å (black bond). The short interchain distance indicates thatJ 3might be comparable to J1andJ2. This is supported by our calculations. Figure 6shows the partial density of states (DOS) of Co 3 dand O 2 pfor the ground-state AFM-1. For Co3 and Co5 atoms, the majority-spin band below Fermi levelis completely occupied while the minority-spin band is onlypartly occupied. The case is reverse for Co6 atom, giving riseto a negative moment. In the valence band mainly composedof Co 3 dand O 2 pstates, the O 2 porbital hybridizes strongly with the Co 3 dorbitals for both Co6-O-Co3 and Co6-O-Co5 superexchange paths. It is therefore inferred that the interchainJ 3should be comparable to the intrachain J1. Furthermore, we plot in Fig. 7(a)the isosurface of valence electron density at the plane determined by Co5, Co7, Co4,and Co2 [see Fig. 1(a)]. A ﬁnite charge distribution crossing the Co-O-Co path is seen, but the distribution via Co6-O-Co3path ( J 3) is relatively weaker than those via Co6-O-Co5 (or Co3-O-Co4) path ( J1) and Co6-O-Co7 (or Co3-O-Co2) path (J2). Note that in this ﬁgure the O atoms associated with the superexchanges are not coplanar. For further comparison,we plot in Figs. 7(b)–7(d) the isosurface of valence electron density in each Co-O-Co plane. Clearly, there is a pronouncedcharge distribution along the Co6-O-Co3 path [Fig. 7(d)], a little bit weaker than the distributions along the Co6-O-Co5 path [Fig. 7(b)] and Co6-O-Co7 path [Fig. 7(c)]. It is 134434-5L. YIN et al. PHYSICAL REVIEW B 99, 134434 (2019) FIG. 6. Partial DOS of Co 3 dand O 2 pfor the ground-state AFM-1. The Fermi level is at zero energy. concluded that J3is relatively weak but comparable to J1 andJ2. With the information that all exchanges are AFM and J3 is comparable to but smaller than J1andJ2, it is possible to simulate the high-ﬁeld M(H) curves. However, the M(H) curves are highly anisotropic and thus anisotropic exchangeinteractions should be taken into account, which complicatesthe simulation. As an approximate treatment, we only at-tempt to simulate the well-deﬁned 1 /2-like magnetization FIG. 7. (a) Isosurface of valence electron density (e Å−3)a tt h e plane determined by Co5, Co7, Co4, and Co2 containing J1,J2, andJ3superexchange paths. (b)–(d) Isosurface of valence electron density for each Co-O-Co plane: J1—Co6-O-Co5, J2—Co6-O-Co7, andJ3—Co6-O-Co3.plateau for the baxis by diagonalizing the Hamiltonian H=−/Sigma1Ji,jSiSj+/Sigma1gμBSizHwith Seff=1/2 and g=3.95 without including anisotropic exchanges. For simplicity, weconsider eight atoms with three interactions J 1,J2, and J3.O u r simulation shows that the best parameters to describe the mag-netization process are J 1=−17.1K ,J2=−30.1 K, and J3= −13.9 K. The averaged value is about −19.6 K, which is com- parable to the Weiss temperatures θpobtained from χ−1(T). Note that similar treatment can be found in diamond-chainCu 3(CO 3)2(OH) 2[2] and triangular-lattice Ba 3CoSb 2O9[10] which exhibit anisotropic 1 /3 magnetization plateaus. The calculated M(H) curve is given in Fig. 3(c), where the experimental curves for the baxis are also plotted for compar- ison. The calculated curve produces a well-deﬁned 1 /2 mag- netization plateau between Hc1andHc2a n da3 /4 plateau at Hc2. Based our simulations, the occurrence of magnetization plateaus is a result of quantum-mechanical discrete energylevels of magnetic eigenstates in spite of the presence of in-terchain exchange. As seen from the geometrical arrangementof Co ions in Fig. 1, the intrachain and interchain interactions could be frustrated with f=θ p/TN∼3[22,25]. It is the quan- tum ﬂuctuations and frustration that results in the occurrenceof plateaus. The possible spin conﬁgurations are shown inFig.3(c), where the classical spin-ﬂop transition coexists with the quantum magnetization plateaus. The ground-state spinsare aligned along the easy baxis with a collinear ↑↓↑↓ spin conﬁguration. Once the spin-ﬂop transition takes place at H sf, the spins become perpendicular to the baxis. As a classical analog, the 1 /2 plateau could correspond to a collinear ↑↑↑↓ conﬁguration and the 3 /4 plateau a ↑↑↑↑↑↑↑↓ conﬁgu- ration, which can be stabilized by frustration and quantumﬂuctuations ( S eff=1/2). The exact spin conﬁgurations call for further experimental studies using neutron scattering andtheoretical calculations containing more Co spins. Finally, we notice a small discrepancy between the ex- perimental and calculated curves in Fig. 3(c), the exact rea- son for which is unknown at this moment. The previouslyreported deviation of the plateau magnetization from theexpected M s/3 in kagome lattice Cu 3V2O7(OH) 2·2H2O and BaCu 3V2O8(OH) 2[36] was later clariﬁed to be due to quality of polycrystalline samples [ 37]. Recently, 1 /2 quantum mag- netization plateau was reported in the breathing pyrochloreBa 3Yb2Zn5O11[38]. The magnetization curve is ﬁeld hys- teretic, depending on the ﬁeld-sweep rate. The plateau mag-netization is signiﬁcantly larger than the calculated curvewith conventional Heisenberg spin-exchange Hamiltonian,but close to the adiabatic simulation result. With this in mind,the slight discrepancy of plateau magnetization from M s/2 seen in Fig. 3(c)may, at least in part, be due to the pulse-ﬁeld magnetization process, which is not completely adiabatic. Asshown in Fig. 3(c), the magnitude of discrepancy is different for 0.7 and 1.7 K curves and both present a signiﬁcant hys-teresis between the H-decreasing and H-increasing processes, which is not intrinsic. IV . CONCLUSIONS In summary, we have demonstrated the anisotropic mag- netization process in Seff=1/2 skew-chain single-crystal Co2V2O7. For the easy baxis, the magnetization curve 134434-6ANISOTROPIC MAGNETIZATION PLATEAUS IN … PHYSICAL REVIEW B 99, 134434 (2019) consists of a spin-ﬂop-like transition at around 3.0 T, a well-established 1 /2-like magnetization plateau within 5.4∼11.6 T ,a n da3 /4 plateau at 15.7 T. For the hard aand caxes, however, only a sign of 1 /2-like plateau is observed, showing a strong magnetic anisotropy. The high-ﬁeld mag-netization and ESR data also demonstrate that Co 2V2O7has an effective spin of Seff=1/2 with a large gvalue. The ﬁrst- principles calculations reveal that the interchain coupling,which leads to classical AFM ordering and spin-ﬂop-liketransition, is comparable to the intrachain couplings. This isfurther conﬁrmed by exact diagonalization analysis, whichyields 1 /2 and 3 /4 plateaus, conﬁrming the quantum origin of plateaus. Thus, there is coexistence of classical and quantummagnetism. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grants No. 11874023, No. 11474110,and No. 11574098) and by the Fundamental Research Fundsfor the Central Universities (Grant No. 2018KFYXKJC010).Z.Z.H is grateful for the Joint Fund of Research UtilizingLarge-scale Scientiﬁc Facilities under cooperative agreementbetween NSFC and CAS (Grant No. U1632159). Z.C.X isgrateful for support from the National Key Research and De-velopment Program of China (Grant No. 2016YFA0401003)and the Natural Science Foundation of China (Grant No.11674115). APPENDIX A: CRYSTAL STRUCTURE AND CHEMICAL COMPOSITION ANALYSIS Single-crystal XRD analysis shows that Co 2V2O7crys- tallizes in the monoclinic crystal system with space groupP21/c[21,24]. The lattice parameters are a=6.584(9) Å, b=8.373(3) Å, c=9.466(6) Å, and β=100.216(4) Å, in good agreement with the previous report [ 25]. The pow- der XRD analysis was performed for the powder sample FIG. 8. Observed and calculated powder XRD patterns and the difference between them. The green vertical bars indicate the ex- pected Bragg reﬂection positions. FIG. 9. Energy-dispersive spectrum measured by an EPM for single-crystal and powder samples. The inset shows a photograph of the single crystal. synthesized by the solid-state reaction method. Figure 8 shows the result of the Rietveld reﬁnement with space groupP21/c. The calculated patterns match well with the observed patterns. No impurity phase is detected, showing the highquality of powder sample. The reﬁned lattice parameters area=6.644(2) Å, b=8.444(5) Å, c=9.554(3) Å, and β= 100.329(2) ◦, slightly larger than those of single crystal. Figure 9shows the energy-dispersive spectrum measured by an electron probe microanalyzer as well as a photographof the single crystal. Table IIlists the details of the chemical composition obtained at two random sites in the samples. Thetrue Co:V:O ratio is quite close to the nominal ratio of 2:2:7for both single-crystal and powder samples, again showing thehigh quality of the samples. TABLE II. The quantitative chemical composition analysis per- formed by an electron probe microanalyzer for single-crystal and powder samples. Co V O Sample Site (at. %) (at. %) (at. %) Total Single crystal Point1 19.192 17.604 63.204 100 Point2 18.432 18.039 63.529 100 Powder Point1 18.336 18.094 63.570 100 Point2 18.711 17.879 63.410 100 134434-7L. YIN et al. PHYSICAL REVIEW B 99, 134434 (2019) APPENDIX B: MAGNETIZATION CURVE OF POWDER SAMPLE Figure 10shows the M(H) curve at 2 K for the pow- der sample using a SQUID magnetometer. As expected, theﬁeld-induced spin-ﬂop-like transition is hard to identify incomparison with the single-crystal data along the easy baxis. If the spin-ﬂop transition ﬁeld is deﬁned as the maximum ofthedM/dH,t h ev a l u eo f H sfis estimated to be 4.3 T, which is larger than 3.0 T for the baxis [Fig. 3(a)]. No magnetic saturation is reached at 7 T. The magnetization at 7 T isabout 1 .27μ B/Co2+. After subtracting the Van Vleck term whose value is unknown for our sample, the magnetization at7 T would be signiﬁcantly smaller than 1 .3μ B/Co2+reported recently [ 26]. Thus, the saturation moment of our sample is smaller than 2 .6μB/Co2+[26], reducing the magnetic differ- ence between powder and single crystal. All these suggest thatthe magnetization is sample dependent. For single crystal, the magnetization at 7 T is, respec- tively, about 0.98, 1.29, and 1 .28μ B/Co2fora,b, and c axes [Fig. 3(a)] without subtraction of the Van Vleck term. The averaged value of 1 .18μB/Co2is still smaller than 1.27μB/Co2+of our powder sample. Keep in mind that our single-crystal and powder samples are of high quality. There FIG. 10. The M(H) curve measured at 2 K for powder sample. is a small difference in lattice parameters (see Appendix A). The magnetic difference between single crystal and powdermight be associated with the difference in microstructures. [1] M. Oshikawa, M. Yamanaka, and I. Afﬂeck, Phys. Rev. Lett. 78,1984 (1997 ). [2] H. Kikuchi, Y . Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, and H. Ohta, Phys. Rev. Lett. 94,227201 (2005 ). [3] Y . Narumi, K. Kindo, M. Hagiwara, H. Nakano, A. Kawaguchi, K. Okunishi, and M. Kohno, Phys. Rev. B 69,174405 (2004 ). [4] T. Sugimoto, M. Mori, T. Tohyama, and S. Maekawa, Phys. 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PhysRevB.70.224503.pdf	Competition between triplet superconductivity and antiferromagnetism in quasi-one-dimensional electron systems Daniel Podolsky, Ehud Altman, Timofey Rostunov, and Eugene Demler Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 4 June 2004; revised manuscript received 6 August 2004; published 3 December 2004 ) We investigate the competition between antiferromagnetism and triplet superconductivity in quasi-one- dimensional electron systems. We show that the two order parameters can be uniﬁed using a SO (4)symmetry and demonstrate the existence of such symmetry in one-dimensional Luttinger liquids of interacting electrons.We argue that approximate SO (4)symmetry remains valid even when interchain hopping is strong enough to turn the system into a strongly anisotropic Fermi liquid. For unitary triplet superconductors SO (4)symmetry requires a ﬁrst order transition between antiferromagnetic and superconducting phases. Analysis of thermalﬂuctuations shows that the transition between the normal and the superconducting phases is weakly ﬁrst order,and the normal to antiferromagnet phase boundary has a tricritical point, with the transition being ﬁrst order inthe vicinity of the superconducting phase. We propose that this phase diagram explains coexistence regionsbetween the superconducting and the antiferromagnetic phases, and between the antiferromagnetic and thenormal phases observed in sTMTSF d 2PF6. For nonunitary triplet superconductors the SO (4)symmetry predicts the existence of a mixed phase of antiferromagnetism and superconductivity. We discuss experimental tests ofthe SO (4)symmetry in neutron scattering and tunneling experiments. DOI: 10.1103/PhysRevB.70.224503 PACS number (s): 74.25.Dw, 71.10.Pm, 74.70.Kn I. INTRODUCTION Quasi-one-dimensional compounds can display a rich va- riety of phases, including spin-Peierls, charge density wave,spin density wave, and superconducting orders. 1–5Due to the large anisotropy in their crystal structure, these materials areoften modeled as a collection of weakly coupled Luttingerliquids. The wealth of phases seen in these compounds isthen attributed to the intrinsic instability of one-dimensionalelectron systems towards the formation of quasi long rangeorder. 6As temperature is lowered, correlations along indi- vidual chains grow, until the coupling between chains stabi-lize true long range order. In the current paper, we follow thisapproach to study the interplay between triplet superconduc-tivity (TSC)and antiferromagnetism (AF)in quasi-one- dimensional electron systems. The starting point of our dis-cussion is an observation that, for weak umklapp scattering,one-dimensional Luttinger liquids at half-ﬁlling have SO (4) symmetry at the boundary between AF and TSC phases.Near this boundary, the two order parameters can be uniﬁedusing SO (4)symmetry, leading to strong constrains on the topology of the phase diagram and on the spectrum of lowenergy collective excitations. Our analysis is motivated by quasi-one-dimensional Bechgaard salts sTMTSF d 2X, and their sulphurated counter- parts sTMTTF d2X. The most well studied material from this family sTMTSF d2PF6is an antiferromagnetic insulator at ambient pressure and becomes a superconductor at high pressure.7–11The symmetry of the superconducting order pa- rameter in sTMTSF d2PF6is not yet fully established,12 but there is strong evidence that electron pairing is spin trip- let: the superconducting Tcis strongly suppressed by disorder;13–17critical magnetic ﬁeld Hc2in the interchain di- rection exceeds the paramagnetic limit;18,19the electron spin susceptibility, obtained from the Knight shift measurements,does not decrease below Tc.20In another material from this family, sTMTSF d2ClO4, superconductivity is stable at ambi- ent pressure and also shows signatures of triplet pairing.21–24 Insulator to superconductor transition as a function of pres- sure has also been found for sTMTSF d2AsF6(Ref. 25 )and sTMTTF d2PF6.26 There are two aspects of the SO (4)symmetry between antiferromagnetism and triplet superconductivity that we ad-dress in this paper. Classical SO(4) symmetry : We consider the possible emergence of the classical (static )symmetry at a ﬁnite tem- perature critical point. We introduce a Ginzburg–Landau(GL)free energy to describe the interaction between the AF and TSC orders, and we study the effects of thermal ﬂuctua-tions through a large Nexpansion and renormalization group (RG)analyses in d=4− eandd=2+ edimensions. For a uni- tary TSC, which we argue to describe Bechgaard salts, weﬁnd a ﬁrst order transition between AF and TSC phases, aﬁrst order transition between AF and normal phases endingin a tricritical point, and a weakly ﬁrst order transition be-tweenTSC and normal phases. For a nonunitaryTSC we ﬁnda mixed phase in which AF and TSC orders are present si-multaneously. We argue that the system is close to having anSO(4)symmetric tetracritical point, but there is a narrow line of direct ﬁrst order transitions between the normal and themixed phase. (For a detailed discussion of the distinction between unitary and nonunitary TSC, see Sec. III A. ) Quantum SO(4) symmetry : We introduce a quantum SO(4)rotor model which encapsulates key features of the competition betweenAF and TSC orders. We use this modelto study collective excitations in the system in variousphases. We argue that the Q-excitation, which gives one of the generators of the SO (4)algebra, should give rise to a sharp resonance in spin polarized neutron scattering in theTSC phase. We further predict that in the case of a unitaryPHYSICAL REVIEW B 70, 224503 (2004 ) 1098-0121/2004/70 (22)/224503 (27)/$22.50 ©2004 The American Physical Society 224503-1TSC the energy of the Q-resonance should decrease to nearly zero at the phase boundary with the AF phase. Such modesoftening is not expected generally near a ﬁrst order transi-tion and would be a unique signature of the enhanced sym-metry at the transition point. Bechgaard salts belong to a class of strongly correlated electron systems displaying proximity of a superconductingstate to some kind of magnetically ordered insulating state.Other examples include the high T ccuprates,27heavy fer- mion superconductors,28,29and in most cases the supercon- ducting (SC)order parameter is spin singlet (sordwave ) and the insulating state has antiferromagnetic or spin densitywave order. Symmetry principles have been introduced tostudy the competition of order parameters in some of thesesystems. In Zhang’s SO (5)theory of high T c superconductivity,30antiferromagnetism and d-wave super- conductivity are treated as components of a ﬁve-dimensionalorder parameter. In addition to the generators of the usualcharge SO (2)and spin SO (3)symmetries, new p-operators are introduced, which rotate superconductivity and antiferro-magnetism into each other. A combination of analytical ap-proximations and numerical results can be used to argue anapproximate SO (5)theory of a class of two-dimensional lat- tice models, such as the Hubbard and the t-Jmodel. 31,32The SO(5)symmetry has also been used to discuss quasi two- dimensional organic k-BEDT-TTF salts.33The uniﬁcation approach based on higher symmetries has been generalizedto several other types of competing states. SO (5)and SO (8) symmetries have been used to classify possible many-bodyground states in electronic ladders. 34,35SO(6)symmetry has been introduced to discuss competing striped phases and su-perconductivity in the cuprates 36SO(4)symmetry has been used to combine s-wave superconductivity and charge den- sity wave orders in the negative UHubbard model,37,38as well as d-wave superconductivity and d-density wave phases.39,40It was also suggested that the SO (5)algebra can be used to combine ferromagnetism and triplet superconduc-tivity in quasi-two-dimensional Sr 2RuO4,41although the ex- istence of microscopic models with such symmetry has notbeen demonstrated. There are several reasons why Bechgaard salts, and sTMTSF d 2PF6in particular, are promising candidates for ex- perimental observation of the emergence of high symmetry from the competition of two orders. The insulator to super-conductor transition in these materials is tuned by pressure,so the entire phase diagram can be explored in a singlesample. This compares favorably to the cuprate supercon-ductors, where the AF/SC transition appears as a function ofdoping and different samples are required to investigate vari-ous regimes. Another important advantage of Bechgaardsalts is that they may be well described by a microscopicLuttinger liquid Hamiltonian, for which we can demonstratethe existence of SO (4)symmetry using standard bosoniza- tion analysis. This is in contrast to the high T ccuprates, in which approximations need to be made in order to even de-ﬁne generators of the SO (5)symmetry. 30,42,43A related issue is the question of quasiparticles in theAF insulating state andin thed-wave superconducting phase. In the former case the quasiparticle spectrum is fully gapped while in the latter casethere are nodal quasiparticles. It is not presently known howthis difference affects a quantum SO (5)symmetry for collec- tive bosonic degrees of freedom. An advantage of the SO (4) symmetry in sTMTSF d 2PF6is that quasiparticles are fully gapped in both the superconducting and the insulating phases. Recent neutron scattering experiments demonstrated the existence of strong AF ﬂuctuations in a triplet supercon-ductor Sr 2RuO4.44–46This material is not quasi one- dimensional, but it has nested pieces of the Fermi surface(see, e.g., Ref. 44 ). Thus, we expect that this material may also show some qualitative features of the competition be-tween AF and TSC discussed in this paper. We note that our approach is phenomenological in nature, since we do not attempt to obtain Luttinger parameters start-ing from microscopic considerations. Instead, we observethat Bechgaard salts remain strongly anisotropic even closeto the AF/TSC phase boundary. Hence, we argue that theLuttinger parameter should be such that individual 1 dchains should be in the vicinity of such phase transition. By startingwith the Luttinger Hamiltonian, we derive the SO (4)symme- try as its immediate consequence. We note, however, that the Luttinger liquid physics is not a necessary requirement forobserving SO (4)symmetry near the AF/TSC phase bound- ary. Several groups have argued that near the TSC phase ofBechgaard salts, the interchain tunneling is sufﬁcient to sup-press Luttinger liquid behavior in favor of a strongly aniso-tropic Fermi liquid. 47,48We will argue below that an approxi- mate classical SO (4)symmetry will be present near the AF/ TSC boundary even if the ordered phases arise from a Fermiliquid state, although we still rely on the assumption thatinterchain hopping of electrons is much smaller than intra-chain hopping (this condition is satisﬁed for Bechgaard salts, see Sec. VIII ). Similarly, we expect that the Qresonance will be present even in a strongly anisotropic Fermi liquid, whoseobservation will verify the approximate quantum SO (4)sym- metry. In this paper, for concreteness, we will concentrate onthe case where the ordered phases emerge from Luttingerliquid behavior on individual chains. This paper is organized as follows. In Sec. II we discuss the Luttinger liquid model for interacting electrons in onedimension. For incommensurate band ﬁlling, we show thatalong the transition line between the TSC and the SDWphases, this model has SO s33SOs4dsymmetry. At half- ﬁlling we argue that, for weak umklapp, this symmetry is reduced to SO (4)symmetry. For quasi-one-dimensional sys- tems such as Bechgaard salts we argue that this SO (4)sym- metry provides a uniﬁed description of AF and TSC orders.In Sec. III we discuss a general GL free energy for the inter-play between magnetism and triplet superconductivity at ﬁ-nite temperatures, and present mean ﬁeld diagrams for theseorders. In Sec. IV we analyze thermal ﬂuctuations using 4− eRG analysis and demonstrate the absence of stable ﬁxed points, which could control multicritical points in the phasediagram. In Sec. V we analyze the case of unitary TSC com-peting with AF by extending the spin SO (3)group to a SOsNdalgebra and using large Nanalysis. In Sec. VI we investigate the interplay of nonunitary TSC and AF using largeNapproach and RG analysis for N.3i n4 − eand 2 +edimensions. We also discuss a physically relevant case of N=3. In Sec. VII we introduce an effective SO (4)quantumPODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-2rotor model that condenses the essential features of the com- petition between the two phases. We use this model to studycollective excitations in various phases. In Sec. VIII we dis-cuss SO (4)symmetry in highly anisotropic Fermi liquids. In Sec. IX we review experimental implications of the SO (4) symmetry for Bechgaard salts. Finally, in Sec. X we summa-rize our results. II. MICROSCOPIC ORIGIN OF THE SYMMETRY A. SO3ˆSO4symmetry at incommensurate ﬁlling Consider a one-dimensional electron gas with the Hamil- tonian: H=H0+H1+H2+H4 H0=o rksser,ks−mdar,ks†ar,ks H1=g1 Loa+,ks†a−,pt†a+,p+qta−,k−qs H2=g2 Loa+,k+qs†a−,p−qt†a−,pta+,ks H4=g4 Loa+,k+qs†a+,p−qt†a+,pta+,ks +g4 Loa−,k+qs†a−,p−qt†a−,pta−,ks. s1d Here,a±,ks†create right/left moving electrons with momenta ±kf+kand spin s, and we assume linearized dispersion of electrons er,ks−m=rvfk. In the Hamiltonian (1)the interac- tion term g1describes backward scattering and terms g2and g4describe forward scattering. For now, we assume that the system has incommensurate ﬁlling, so that umklapp pro-cesses are not allowed. The phase diagram for this systemobtained from the renormalization group analysis has beendiscussed extensively before (see, e.g., Refs. 49 and 50 )and is shown in Fig. 1.For the current discussion, we concentrate on the region of the phase diagram near the transition line between theTSC and the SDW phases at K r=1, i.e., g1=2g2. s2d We demonstrate that on this line the system has a SO s3d 3SOs4dsymmetry that uniﬁes order parameters of the two phases. The total spin operators are deﬁned as Sa=1 2o r,kss8ar,ks†sss8a,ar,ks8, s3d where sss8aare the usual Pauli matrices.These operators form a spin SO (3)algebra fSa,Sbg=ieabgSg. s4d We can also combine the charge operators for right and left movers sr=±d: Qr=1 2o ksSar,ks†ar,ks−1 2D s5d and the operators Qr†=ro kar,k"†ar,−k#†s6d to form two separate isospin SO (3)algebras Jxr=1 2sQr†+Qrd Jyr=1 2isQr†−Qrd Jzr=Qr fJar,Jbr8g=idr,r8eabcJcr. s7d The total isospin group is therefore SO s4disospin=SO s3dR 3SOs3dL. Note that, since spin and isospin operators com- mute, fSa,Jbrg=0, they jointly deﬁne a closed SO s3dspin 3SOs4disospinalgebra. The total spin, Sa, and the total charge, Q++Q−, always commute with the Hamiltonian (1). In addition, due to the absence of umklapp at incommensurate ﬁlling, Q+andQ− are conserved separately. As shown in Appendix A using bosonization, when the condition (2)is satisﬁed, the Qrop- erators also commute with the Hamiltonian. Hence, the sys-tem has full SO s3d3SOs4dsymmetry at the phase boundary between TSC and SDW phases. We emphasize that the SOs3d3SOs4dsymmetry of Luttinger liquids at the SDW/ TSC boundary is generic and does not require ﬁne tuning of the parameters. SO s4d isospininvariance has been discussed in quasi one-dimensional systems with highly anisotropic spin interactions.51,52The Qroperators in Eq. (6)are reminiscent of the hoperators introduced by Yang to study the Hubbard model,37,38but we will show in Sec. III B that the two sets of FIG. 1. Phase diagram for a one-dimensional system of interact- ing spin-1/2 fermions (Ref. 50 ). Here Kr2=s2pvf+2g4+g1 −2g2d/s2pvf+2g4−g1+2g2d. SDW and CDW correspond to spin and charge density wave states, SS and TS to singlet and tripletsuperconducting phases.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-3operators deﬁne different symmetry groups and apply to dif- ferent systems. Spin density wave order away from half-ﬁlling is de- scribed by a complex vector order parameter: Fa=o kss8a+,ks†sss8aa−,ks8. s8d For quasi-one-dimensional systems, the band structure re- stricts the orbital component of the triplet superconductingorder to be Cspd~p x, wherexis the direction parallel to the chains. Thus, the TSC order parameter is also described by a complex vector: Ca†=1 io kss8a+,ks†ssas2dss8a−,−ks8†. s9d The factor of − iis introduced for convenience, − is2 ;s0−1 10d. The four vector order parameters Re F,I m F, ReC, and Im Ccan be combined into a 4 33 matrix, Pˆ=1sReCdxsImCdxsReFdxsImFdx sReCdysImCdysReFdysImFdy sReCdzsImCdzsReFdzsImFdz2.s10d Each column of Pˆtransforms independently as a vector un- der the action of the spin group, fSa,Pb¯bg=ieabgPb¯g. s11d The action of the isospin group on Pˆis easiest to understand in terms of the operators Ia=Ja++Ja−and La=Ja+−Ja−. For a ﬁxed row of Pˆ, the action of the isospin generators in the basis sReCa,ImCa,ReFa,ImFadis represented by Ix=10 000 00−i0 0i00 0 0002Lx=1000 −i 000 0 000 0 i0002 Iy=100−i0 0 000 i000 0 0002Ly=1000 0 000 i 000 0 0−i002.s12d Iz=10i00 −i000 0 000 0 0002Lz=10 000 0 000 00 0 i 00−i02 Once the fourth component is identiﬁed as the “time-like” direction, this is the (Euclidean )Lorentz group, with the Ia acting as rotations and the Laacting as boosts. Hence, the rows of order parameter Pa¯atransform in the vector repre- sentation of the SO s4disospingroup.B. SO(4) symmetry at half ﬁlling In Bechgaard salts sTMTSF d2X, three out of every four conduction states are occupied. At quarter ﬁlling, umklapp processes involving interactions of four electrons are al-lowed. Such interactions are weak, and furthermore are irrel-evant in the RG sense for K r.1/4 (we remind the readers that we are interested in the regime near the SDW/TSCboundary, where K r<1).53On the other hand, due to struc- tural dimerization in Bechgaard salts,54a gap splits the con- duction band into a completely ﬁlled lower band and a half-ﬁlled upper band. Hence, Bechgaard salts are half-ﬁlledsystems. At half ﬁlling, the Hamiltonian (1)must be modi- ﬁed to include two-electron umklapp scattering processes: H 3=g3 2Loa+,k+qs†a+,p−qt†a−,pta−,ks +g3 2Loa−,k+qs†a−,p−qt†a+,pta+,ks. s13d Analysis of the phase diagram of Luttinger liquids at half- ﬁlling reveals that there is still a direct transition betweenAFand TSC orders at K r=1, although this condition now corresponds2tog1−2g2=ug3u.The umklapp term allows scat- tering of two right moving electrons into two left movingones, and vice versa. Thus, it does not commute with theoperator L z=Q+−Q−, which leads to breaking of the SO s3d 3SOs4dsymmetry. To understand the nature of this symme- try breaking, it is useful to rewrite Eq. (13)in the form H3=g3 2Lo qfReFˆsqd·ReFˆs−qd−Im Fˆsqd·ImFˆs−qdg, s14d where Fˆsqdis the SDW order parameter at center of mass momentum 2 kf+q, Fˆsqd=o kss8a+,ks†sss8a−,k−qs8. s15d Equation (14)shows explicitly that umklapp tends to pin the phase of the SDW order parameter at either 0 or p, depend- ing on the sign of g3. This is in agreement with the observa- tion that period two antiferromagnetic order can be describedby a single real Néel vector. We will show in Sec. III B that, whereas the Ginzburg- Landau free energy is no longer SO s3d spin3SOs4disospinsym- metric at half-ﬁlling, to linear order in g3it maintains a SOs4d=SO s3dspin3SOs3disospinsymmetry. The unbroken part of the isospin group, SO s3disospin, is the diagonal subgroup of SOs3dR3SOs3dL, which is generated by the three Iaopera- tors,Ix=1 2sQ†+Qd,Iy=1/2isQ†−Qd,Iz=Q, where Q=1 2o kssa+,ks†a+,ks+a−,ks†a−,ks−1dPODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-4Q†=o ksa+,k"†a+,−k#†−a−,k"†a−,−k#†d. s16d Without loss of generality we consider the case g3,0, where the order parameter for antiferromagnetism is given by thereal part of F, N a=1 2o kss8sa+,ks†sss8aa−,ks8+a−,ks†sss8aa+,ks8d. s17d It is easy to verify that hN,ReC,ImCjtransform as vectors under both spin and isospin SO (3)symmetries. We deﬁne Qˆ=1sReCdxsImCdxNx sReCdysImCdyNy sReCdzsImCdzNz2, s18d Qˆtransforms as a vector under both SO (3)algebras fIa,Qbbg=ieabcQcb fSa,Qbbg=ieabgQbg, s19d so it describes an order parameter that transforms as a (1,1) representation of the SO (4)algebra. Since we are mostly interested in applying our results to Bechgaard salts, we fo-cus mostly on this SO (4)symmetry in Ref. 55, as well as on the remainder of this paper. Unlike the SO s3d3SOs4dsymmetry discussed in the in- commensurate case, the SO (4)symmetry at half-ﬁlling is not a rigorous symmetry of the system. The generators of thisgroup do not commute with the Hamiltonian of the systemexactly. However, the main emphasis of our work is to un-derstand the ﬁnite temperature phase diagram ofsTMTSF d 2PF6. This is obtained from the classical GL free energy, which at the AF/TSC phase boundary has SO (4) symmetry if we retain umklapp processes to linear order ing 3(see discussion in Sec. III B ). In addition, with regards to quantum properties, SO (4)symmetry is a good starting point to study the collective modes of the system when g3is small. The latter assumption is well justiﬁed for sTMTSF d2PF6, since the observed dimerization in this material is less than 1%.54For small g3, modes found assuming SO (4)symmetry will have a ﬁnite overlap with the actual excitations of thesystem. In particular, the quantum numbers of the Qmode discussed in Sec. VII, including charge two and center ofmass momentum 2 k f, are not affected by umklapp. These properties determine which experimental probes couple to Q. We must keep in mind, however, that the explicit breaking ofSO(4)due to higher order corrections in g 3, and also due to interchain coupling, may give a small energy gap and ﬁnitebroadening to Q, even at the AF/TSC phase boundary. We also point out that, from the point of view of Qexcitations, the difference between SO s3d3SOs4dand SO (4)symme- tries corresponds to the question whether +2 k fand −2kfex- citations are the same [SO(4)symmetry at half ﬁlling ]or different [SOs3d3SOs4dsymmetry away from half ﬁlling ]. Thus far in the analysis we have ignored spin-orbit ef- fects. Microwave absorption experiments in sTMTSF d2AsF6 measured56the anisotropy in the exchange couplings to be10−6. This ultimately determines the preferred axes for the Néel vector N(along the baxis of the crystal57)and the spin component of the TSC order C(along either the aorc axis18). However, we do not expect such tiny anisotropy to play a signiﬁcant role in determining the competition be-tween AF and TSC phases. We also point out that NMR experiments in sTMTSF d 2PF6ﬁnd a divergence of T1−1at the Néel temperature that is well-described by the O (3)isotropic Heisenberg model.58Thus, even for critical ﬂuctuations of the AF order parameter, spin anisotropy coming from spin-orbit coupling is unobservably small. Before concluding this section we point out that the iso- spin algebra deﬁned by Eqs. (5)–(7)can also be used to relate charge density wave order and singlet superconductiv-ity in quasi-one-dimensional electron systems. 59This is rel- evant for the lower half of Fig. 1. III. GINZBURG–LANDAU FREE ENERGY The main goal of this section is to investigate conse- quences of the SO (4)symmetry for the true ﬁnite tempera- ture phase transitions, when we need to consider three-dimensional ﬂuctuations of the order parameter. One may beconcerned that by introducing interchain couplings, we willimmediately destroy the SO (4)symmetry. As we will show in Sec. VIII, even in the case where the interchain couplingt bis large enough to make the system into a highly aniso- tropic Fermi liquid, approximate SO (4)symmetry prevails in the Ginzburg-Landau (GL)free energy [see, e.g., Eq. (29)]. This feature of the AF/TSC GL free energy implies that ouranalysis of the phase diagram, based on classical SO (4)sym- metry, is valid even when the normal state is described by ahighly anisotropic Fermi liquid rather than a collection ofweakly coupled Luttinger liquids. A review of the normalstate properties of organic superconductors at low magneticﬁelds is given in Ref. 47, and low temperature transportproperties have been reported recently in Ref. 48. We illustrate the effects of interchain coupling in Fig. 2. As the temperature is reduced on either side of K r=1, the FIG. 2. Proposed phase diagram for weak interchain coupling tb. When interchain coupling is present, tbÞ0, long range order at ﬁnite temperatures becomes possible. In the unitary case, the sec-ond order quantum critical point of a Luttinger liquid becomes aﬁrst order transition betweenAF and unitary TSC.As t bgrows, the AF phase shrinks due to reduced nesting of the Fermi surface.Throughout we assume positive backscattering g 1.0.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-5correlation length along a chain in the appropriate correlation function grows due to intrachain interactions. At some ﬁnitetemperature, before this length diverges, coupling betweenthe chains becomes relevant and a true three-dimensionaltransition can take place. At the crossover between one andthree dimensions, the description of the system in terms of aLuttinger liquid is supplanted by a GLfree energy describingthe interactions of the order parameters in three spatial di-mensions. In this picture, for small enough t bthe presence of a phase boundary between AF and TSC implies that the in-trachain Hamiltonian is close to the SO (4)symmetric point K r=1. An important assumption of our analysis is that pressure varies the value of Krand tunes the transition across the AF/TSC phase boundary. We note that measurements of Kr based on optical conductivity measurements have been car- ried out at ambient pressure, i.e., deep inside the SDW phaseinsTMTSF d 2PF6. For instance, in Refs. 60 and 61, the value Kr=0.23 is obtained. Reference 61 points out that this value ofKrassumes that the dominant umklapp contribution is due to commensurability at quarter ﬁlling. They acknowledgethat if umklapp is dominated by commensurability at half-ﬁlling, these measurements then imply K r=0.925. They also point out that a half-ﬁlled model has the drawback of pre-dicting a gap energy that is too small. However, it is equallydifﬁcult to justify the assumption that quarter ﬁlling com-mensurability is dominant: for K r=0.23, the rate of diver- gence in the RG of commensurability at one quarter is expo-nentially smaller than the corresponding rate forcommensurability at half-ﬁlling. Thus, we feel that the ques-tion of the value of K ris not yet settled. A. Incommensurate ﬁlling At incommensurate ﬁlling, a translation by one lattice constant multiplies the SDW order parameter by a complexphase factor: F!e 2ikf·aF=e2pinF, s20d where nis the ﬁlling fraction of the conduction band. For a completely incommensurate case, when nis an irrational number, the GL free energy must be SO (2)symmetric with respect to the phase of F.62,63In the absence of pinning terms, the most general GL free energy with SO s3dspin 3SOs2dcharge 3SOs2dtranslationis F=1 2u„Cu2+1 2u„Fu2+r1 2uCu2+r2 2uFu2+u1suCu2d2 +u2suFu2d2+u3uC2u2+u4uF2u2+2v1uCu2uFu2 +2v2uF·Cu2+2v3uF·Cu2. s21d Near the phase boundary between SDW and TSC phases, for quasi-one-dimensional systems the form of the free energy isstrongly constrained by the SO s3d spin3SOs4disospinsymme- try. We expect the properties of the system to be well de- scribed by the free energy,F=1 2o a¯a„Pa¯a„Pa¯a+r¯ 2o a¯aPa¯aPa¯a+dr 2o asP1a2+P2a2 −P3a2−P4a2d+u˜1o a¯ab¯bPa¯aPa¯aPb¯bPb¯b +u˜2o a¯ab¯bPa¯aPa¯bPb¯aPb¯b. s22d This is the most general free energy with SO s3dspin 3SOs4disospinsymmetric quartic coefﬁcients, where we have used the order parameter deﬁned in Eq. (10)to display this invariance explicitly.We follow the common assumption thatchanging the external control parameters of the system onlyaffects the quadratic coefﬁcients. Thus, these are allowed tobreak the symmetry and tune the phase transition. For dr Þ0, the symmetry is broken down to SO s3dspin 3SOs2dcharge 3SOs2dtranslation. There is an explicit duality between antiferromagnetism and triplet superconductivity under reversal of the sign of dr in Eq. (22). The mean ﬁeld phase diagram of Eq. (22)de- pends crucially on the sign of u˜2. For negative u˜2, there is a tendency for all vector order parameters to point along acommon axis. The order parameters in this case can be de-scribed by a single real vector times a complex phase, C =e iwnandF=eiun. This is referred to in the3He literature as unitarytriplet superconductivity,64and in magnetism as col- linearspin density order.62On the other hand, for positive u˜2, all vector order parameters tend to be mutually orthogo- nal. The real and imaginary parts of the order parameters canno longer be set to be parallel, Re C3ImCÞ0 and Re F 3ImFÞ0. This is the nonunitary /noncollinear case. The mean-ﬁeld phase diagrams for Eq. (22)foru ˜2negative and positive are shown in Figs. 3 and 4. In principle, the parameters of the GLfree energy Eq. (21) can be obtained from a microscopic Hamiltonian. In Appen-dix B we consider a quasi-one-dimensional electron systemswith weak interactions and obtain a free energy as in Eq. (22) with u ˜1=21zs3d 16p2vfT2 FIG. 3. Mean-ﬁeld phase diagram of Eqs. (22)and (29)for u˜2,0. At half ﬁlling, SDW order reduces to AF order.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-6u˜2=−7zs3d 8p2vfT2. s23d The quadratic coefﬁcients depend on coupling constants in the TSC and SDW channels, and on temperature. They aretypically parameterized in terms of the pressure-dependentmean-ﬁeld transition temperature r 1sT,Pd=aTSCfT−TCsPdg r2sT,Pd=aSDWfT−TNsPdg. s24d The analysis of Appendix B shows that weakly interacting Fermi liquids favor unitary TSC and collinear SDW order, u˜2,0. B. Half-ﬁlling As is well known, the physics of period two antiferromag- netic order is captured by a single real Néel vector. It isinteresting to study how this comes about from the point ofview of the microscopic Luttinger Hamiltonian. As pointedout in Sec. II, the Hamiltonian at half-ﬁlling includes a newcontribution due to umklapp scattering, Eq. (14).This gives a correction to the GL free energy, which to linear order in g 3 can be written as DF=hfsReFd2−sImFd2g, s25d whereh=g3/2L.The new term pins the SDW, and breaks the SOs4disospin=SO s3dR3SOs3dLsymmetry down to its diago- nal subgroup SO s3disospin. In agreement with the Feynman– Hellman theorem, the quartic coefﬁcients derived in Appen- dix B are not modiﬁed to linear order in g3. Therefore, in the linear order in g3, the free energy has SO s4d=SO s3dspin 3SOs3disospinsymmetry. In principle, the higher order con- tributions of the umklapp g3term can break the original SOs4disospinsymmetry all the way down to SO s2dcharge, gen- erated by the total charge Q. The bare value of g3insTMTSF d2PF6is small, since it is proportional to dimerization, which in this compound is very weak.54Assuming that Coulomb interactions are of the orderof the bandwidth, this leads to a bare value of g3of about 0.01. Furthermore, it is not the bare value of g3that enters the GL free energy, but its effective (renormalized )value at the 1d–3dcrossover scale. At high temperatures one- dimensional physics is observed. As temperature is reduced,everywhere on the TSC side of the phase diagram, as well ason theAF/TSC phase boundary, g 3ﬂows to zero.This allows us to approach the critical region from the TSC side, alongwhich the GLfree energy is SO s3d isosymmetric. Even on the AF side of the phase diagram, where g3is relevant, the ﬂow ofg3passes near zero before diverging. Therefore, near the AF/TSC phase boundary, the ﬂow spends a lot of time nearzero, and the eventual upturn of g 3may not be reached for realistic systems, in which the 3 dcoupling may cut off the 1dRG ﬂow at low temperatures. Hence, it is reasonable to take small g3everywhere near the AF/TSC phase boundary, and to consider a model with SO s3disosymmetry. In what follows, we will assume that the umklapp term favors the real part of the SDW, which becomes the Néelorder parameter N: N=Re F. s26d From now on we assume that Im Fis sufﬁciently well gapped, so that it does not need to be included in the analysisof the competition between AF and TSC. This is justiﬁedsince the pinning term Eq. (25)is relevant in the 3 dtheory, and thus any ﬁxed point of the theory will be characterizedby strong pinning. It is useful to consider the relation between our SO (4) symmetry and the SO (4)symmetry introduced by Yang for the Hubbard model. 65The symmetry generators of Yang’s SO(4)is the hoperator: h†=o kck+p"†c−k#†. s27d ksummation goes over the entire Brillouin zone. This opera- tor should be compared to our Qoperators deﬁned in Eq. (6) for incommensurate ﬁlling, and Eq. (16)for half-ﬁlling. Away from half ﬁlling, the difference between the two op-erators is obvious. Qhas momentum equal to the nesting wave vector 2 k f, in contrast to h, which always has center of mass momentum p. This allows us to have SO (4)symmetry for any electron density, in contrast to Yang’s SO (4), which only applies at half-ﬁlling. At half-ﬁlling, however, 2 kf=p, and the only difference is the relative sign between the leftand right moving contributions. This difference is substan-tial. The Néel order parameter transforms as a singletunder the action of h: fh,Ng=0. s28d This should be contrasted with Q, which rotates Ninto the TSC order parameter C, see Eq. (19). The two SO (4)sym- metries thus differ in the order parameters which they unify,and in the microscopic models for which they apply. Yang’sSO(4)applies to the negative UHubbard model, for which singlet SC and CDW are degenerate lowest energy states athalf-ﬁlling. Our SO (4)uniﬁesAF and TSC orders, which are not degenerate for the Hubbard model. As we discussed ear-lier, we expect these to be nearly degenerate phases for half- FIG. 4. Mean-ﬁeld phase diagram of Eqs. (22)and (29)for u˜2.0. At half ﬁlling, SDW order reduces to AF order.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-7ﬁlled systems with small umklapp [e.g., quarter ﬁlled sys- tems with small dimerization, such as sTMTSF 2dPF6and withKrclose to one ]. In analogy with the incommensurate case, at half-ﬁlling we expect that quasi-one-dimensional systems near the AF/TSC phase boundary have a Ginzburg-Landau free energywith SO (4)-symmetric quartic coefﬁcients. The symmetry can be made explicit in terms of the matrix order parameter Qˆ: F=1 2o aa„Qaa„Qaa+rˆ 2o aaQaaQaa+dro asQ3a2−Q1a2 −Q2a2d+u˜1o aabbQaaQaaQbbQbb+u˜2o aabbQaaQabQbaQbb =1 2u„Cu2+1 2s„Nd2+r1 2uCu2+r2 2N2+Su˜1+u˜2 2DsuCu2d2 +u˜2 2uC2u2+2u˜1uCu2N2+2u˜2uC·Nu2+su˜1+u˜2dsN2d2. s29d Changing temperature and some other parameter of the sys- tem[e.g., pressure in sTMTSF 2dPF6]allows to control r1and r2. The SO (4)symmetry is recovered on the line r1=r2. Equation (29)is a special case of the most general free energy with the SO s3d3SOs2dsymmetry of spin and charge rotations,66 F˜=1 2u„Cu2+1 2s„Nd2+r1 2uCu2+r2 2N2+u1suCu2d2 +u2sN2d2+u3uC2u2+2v1uCu2N2+2v2uN·Cu2. s30d Translational symmetry rules out N·C3Cbecause this term has a nonzero wave vector. Similarly, uN3Cu2can be reduced to terms already present in Eq. (30). When the con- ditions r1=r2 u2−u3=u1 u2−2u3=v1 v2=2u3 s31d are satisﬁed, we recover SO (4)symmetry. In addition, if we supplement the conditions (31)by v2=0 s32d there is an even higher SO (9)symmetry, which allows rota- tions between any components of vectors N,R e C, and ImC: F¯=1 2u„Cu2+1 2s„Nd2+r 2suCu2+N2d+u¯suCu2+N2d2. s33dAt half-ﬁlling, there is no distinction between collinear and noncollinear magnetism. However, the sign of u˜2still determines the nature of the triplet superconductivity, as wellas the topology of the mean-ﬁeld phase diagrams. These aresimilar to those displayed for incommensurate ﬁlling, Figs. 3and 4, the only difference being that SDW order is reducedto AF order. The mean-ﬁeld phase diagrams for Eq. (29)for u ˜2negative and positive are shown in Fig. 3. The analysis of Appendix B can be easily modiﬁed to half-ﬁlling, and yieldsaS O (4)symmetric free energy of the form Eq. (29)with coefﬁcients still given by Eq. (23). Thus, weekly interacting Fermi liquids favor the case u ˜2,0. Strong interactions, how- ever, can modify the quartic coefﬁcients Eq. (29), including a possible change of sign of u˜2. In the subsequent discussion we consider both possibilities. It is useful to note that allexperimentally known cases of triplet pairing between fermi- ons, such as 3He,64,67and Sr2RuO4, correspond to the unitary case. Hence, negative u˜2appears more likely. IV. THERMAL FLUCTUATIONS We now consider the free energy (30)and address how ﬂuctuations affect the mean-ﬁeld phase diagram shown inFigs. 3 and 4. For instance, when the quartic coefﬁcients donot lie exactly on the SO (4)symmetric manifold, we will study whether such symmetry appears as we go to longerlength scales and integrate out short wave length ﬂuctua-tions. The possibility of enhanced static symmetry at thecritical point has been discussed previously for several solidstate systems. For easy axis AF in a magnetic ﬁeld, SO (3) symmetry was suggested to appear at the spin ﬂop criticalpoint. 68,69For systems with competing singlet superconduct- ing and antiferromagnetic orders Zhang suggested a staticSO(5)symmetry as the bicritical point. 30,70This SO (5)sym- metry has also been used to study the quasi-two-dimensional k-BEDT-TTF salts.33Yang and Zhang introduced a SO (4) symmetry for the Hubbard model at half-ﬁlling which uniﬁessinglet superconductivity with charge density waveorder. 37,38 To understand the role of ﬂuctuations in models (30)and (29)we use 4− erenormalization group (RG)analysis. For subsequent discussion it is useful to extend the spin SO (3) symmetry of the Eq. (30)to a more general SO (N)symme- try. This is achieved by considering vectors Nand Cas N-component vectors. The RG equations can be derived us- ing the standard approach71 dr1 dl=2r1+8Kd 1+r1fsN+1du1+3u3g+4Kd 1+r2fNv1+2v2g dr2 dl=2r2+8Kd 1+r1fNv1+2v2g+4Kd 1+r2sN+2du2 du1 dl=eu1−Kdfs8N+32du12+32u1u3+32u32+4Nv12+8v1v2 +2v22g du2 dl=eu2−Kdf4sN+8du22+8Nv12+16v1v2+8v22gPODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-8du3 dl=eu3−Kdf8Nu32+48u1u3+2v22g dv1 dl=ev1−Kdfs8N+8du1v1+s4N+8du2v1+16u3v1+16v12 +8u1v2+4v2u2+4v22g dv2 dl=ev2−Kdf8u1v2+8u2v2+16u3v2+32v1v2 +s4N+8dv22gs 34d Heredl=dL/L, where Lis a momentum cutoff, and Kd =...isa surface of a unit sphere in d=4− edimension. For the physically relevant N=3, Eq. (34)has only two ﬁxed points. One is a trivial Gaussian ﬁxed point rh1,2j=uh1,2,3 j=vh1,2j=0 s35d and the other is a SO (9)Heisenberg point rh1,2j=−s3N+2de 6N+16 u1=u2=v1=e s6N+16dKd u3=v2=0. s36d The Gaussian ﬁxed point is completely unstable. The SO (9) Heisenberg point has ﬁve unstable directions [for general N, the Heisenberg point has SO s3Ndsymmetry, but it remains unstable in ﬁve directions for all N.1]. The critical point should have only two unstable directions: rs1,2dshould ﬂow away from the critical point, but all the interaction coefﬁ- cient should ﬂow toward the ﬁxed point. So, neither theGaussian nor the SO (9)Heisenberg ﬁxed points are good candidates for the critical point. In Fig. 5 we show RG ﬂowsin the SO (4)symmetric plane. We ﬁnd two types of runawayﬂows. When we start with u ˜2positive, it continues increas- ing. Foru˜2negative, the RG ﬂow makes it even more nega- tive. In both cases u˜1ﬂows to negative values. In many cases absence of a ﬁxed point in the RG ﬂows implies that we do not have a multicritical point in the phasediagram, but instead ﬂuctuations induce a ﬁrst order phasetransition. Below we discuss consequences of the runawayﬂows in Eq. (34).We point out that two types of the runaway ﬂows in the SO (4)symmetric manifold shown in Fig. 5 cor- respond to unitary su ˜2,0dand nonunitary su˜2.0dTSC. These two cases are considered separately. V. FINITE TEMPERATURE ANALYSIS: UNITARY CASE We consider model Eq. (29)withN-component vectors and with negative u˜2. In the 4− eexpansion there are run- away ﬂows even for large N. Thus, we do not ﬁnd a ﬁxed point that could give a critical point.To understand the phasediagram in this case we employ large Ncalculations in d =3. In the large Nexpansion, all bubble diagrams are summed self-consistently. 72,73The large Napproach for uni- tary triplet superconductors without coupling to magnetic or-der has been discussed previously in Ref. 74 in the context of 3He. Let us start by analyzing the superconducting phase. In the mean-ﬁeld approximation the order parameter factorizesasC=e iun. Hence, we take the average value of the order parameter in the ordered phase to be C0=s0,...,0, sdand separate the longitudinal and transverse components of the ﬂuctuating part dC=sAT+iBT,AL+iBLd. For the Néel order parameter we also separate N=sNT,NLd. Effective masses for AT,BT, andNTare given by rA=r1+4su˜1+u˜2ds2+4su˜1+u˜2dNE 0L d3k s2pd31 k2+rA +4u˜1NE 0L d3k s2pd3S1 k2+rB+1 k2+rND rB=r1+4u˜1s2+4su˜1+u˜2dNE 0L d3k s2pd31 k2+rB +4u˜1NE 0L d3k s2pd3S1 k2+rA+1 k2+rND, rN=r2+4u˜1s2+4su˜1+u˜2dNE 0L d3k s2pd31 k2+rN +4u˜1NE 0L d3k s2pd3S1 k2+rA+1 k2+rBD s37d where Lis the ultraviolet (short distance )cutoff of the free FIG. 5. Renormalization group ﬂow of the SO (4)symmetric theory Eq. (29)ind=4− «dimensions. The sign of u˜2does not change under the ﬂow, and there are no stable ﬁxed points. Instead,there are two types of runaway ﬂow, corresponding to unitarysu ˜2,0dand nonunitary su˜2.0dTSC. The two are separated by a line of SO (9)symmetric theories su˜2=0d.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-9energy in Eq. (29). In writing Eq. (37)we used that in the largeNlimit,u˜h1,2j,1/N,s,˛N, and we neglected terms of the order of 1/ N, including contributions from longitudinal ﬂuctuations.Arequirement of the cancellation of tadpole dia-grams for A Lgives the condition rA=0, as one would expect from the Goldstone theorem. It is convenient to deﬁne pa-rameterr cfrom the condition 0=rc+s12u˜1+4u˜2dNE 0L d3k s2pd31 k2. s38d If we measure r’s with respect to rc tc=r1−rc tN=r2−rc, s39d we can absorb all the cutoff dependence of Eq. (37)into deﬁnitions of tcandtN: 0=tc+4su˜1+u˜2ds2+4u˜1NEd3k s2pd3S1 k2−1 k2+rBD rB=tc+4u˜1s2+4su˜1+u˜2dNES1 k2−1 k2+rBD +4u˜1NEd3k s2pd3S1 k2−1 k2+rND rN=tN+4u˜1s2+4u˜1NEd3k s2pd3S1 k2−1 k2+rBD +4su˜1+u˜2dNES1 k2−1 k2+rND. s40d Integrals are now convergent for large k, so upper limits of integration can be extended to inﬁnity. Solutions to Eq. (40) correspond to extremal points of free energy as a function of s. When both tcandtNare large, there are no solutions toEq.(40). This is a disordered phase. Once we decrease tc sufﬁciently, a single solution appears at tc,Mand splits into two fortc,tc,M. This describes the appearance of the TSC phase as a locally stable state. The point tc,M, where two solutions merge into one and disappear, correspond to theboundary of the local stability region of theTSC phase.As t c is lowered further, at a temperature tc,Lone of the solutions approaches s=0 and then disappears. This is a spinodal point below which a disordered phase is unstable to devel-oping TSC order parameter. The actual ﬁrst order phase tran-sition occurs somewhere between t c,Mandtc,L. Figure 6 shows a phase diagram constructed from the ar- guments presented above, for both TSC and AF phases. Wenote that the mixed phase with simultaneous TSC and AForders is only possible on the t c=tNline. Thus, the ﬁrst order phase transition between two types of ordered phases re-mains even when we include ﬂuctuations. The most interesting feature of this phase diagram is that the transition between the disordered and the antiferromag-netic phases becomes ﬁrst order in the vicinity of the criticalpoint. VI. FINITE TEMPERATURE ANALYSIS: NONUNITARY CASE Mean-ﬁeld calculations for the free energy Eq. (29)with u˜2.0 demonstrated that the TSC phase is nonunitary and there is a mixed phase with both TSC andAF order (see Sec. III and Fig. 4 ). Within mean ﬁeld theory, the mixed phase terminates at a tetracritical point with SO s4d=SO s3d 3SOs3dsymmetry. The goal of this section is to examine how the tetracritical point is affected by thermal ﬂuctuations. To this end we extend the SO (3)spin symmetry to SO sNd and approach the problem with three different methods: A largeNanalysis, a renormalization group calculation in d =4− eand in one in d=2+ e. The physical image that emerges from all of these approaches is that a SO s3d 3SOsNdcritical point exists for sufﬁciently large N, but FIG. 6. Phase diagram of Eq. (29)withu˜2,0 in three dimensions in the large Nlimit including ﬂuctuations. Parameters are u˜1=1/N u˜2=−1/2N.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-10probably does not survive down to the physical N=3. We argue that in this case the tetracritical point is stretched to aline of direct ﬁrst order transition from the normal state tothe mixed phase. A. Large Nphase diagram in three dimensions We consider the model Eq. (29)in three dimensions for largeNand with u˜2.0. We note that the quartic terms give a free energy that is bounded from below for u˜1+u˜2/3.0. Thus, in this section we will always assume that this condi-tion is satisﬁed. In Appendix C we also discuss that when u ˜1+u˜2/3 becomes small, of the order of 1/ N2(in the large N limit both u˜s are of the order of 1/ N), the phase diagram may change qualitatively. In the mixed phase the TSC and AF both have nonzero expectation values and are orthogonal to each other. Hence,in the ordered phase we can choose kCl=s0,...,0, sc,isc,0d kNl=s0,...,0,0,0, sNd. s41d Following the discussion in Sec. V we introduce longitudinal and transverse ﬂuctuations for all order parameters. It is easyto verify that the requirement of cancellation of tadpole dia-grams for longitudinal components implies zero effectivemasses for the transverse components. Shifting r 1andr2as in Eq. (39)we obtain self-consistency conditions for expec- tation values of the order parameters tc+s12u˜1+4u˜2dsc2+4u˜1sN2=0 tc+s4u˜1+4u˜2dsN2+8u˜1sc2=0. s42d These equations can be easily solved and we obtain a phase diagram shown in Fig. 7. We observe that in this case the only effect of ﬂuctuations is to shift the tetracritical point from r1=r2=0tor1=r2=rc.B. Renormalization group analysis in d=4− e. SO3ˆSONﬁxed point As shown in Fig. 5 for N=3, all ﬁxed points with sym- metry SO s4d,SOs3d3SOs3dare unstable within a 4− eex- pansion. In contrast, as Nis increased a ﬁxed point with SOs3d3SOsNdsymmetry appears that is fully stable with respect to changes in the quartic interaction parameters, in- cluding those perturbations that destroy SO s3d3SOsNdsym- metry. Such ﬁxed point exists for Nø33 and has v1=−3e 2KdhsNd v2=e 4Kd1−72hsNd N+7 r1=r2=−2Kdfs3N+2dv1+sN+6dv2g, s43d whereu1,u2, andu3are related to v1andv2by the con- straints (31), and the function hsNd=fN2+8N−65+ sN +7d˛N2−34N+49g−1,1/2N2+Os1/N3dis real only for N.32. The two quadratic parameters r1andr2are relevant, tun- ing the transition on a two-dimensional phase diagram. TheRG ﬂow Eq. (34), linearized about the ﬁxed point Eq. (43), yield two principal directions, s dr1,dr2d~s1,1dassociated with the thermal exponent lt, and sdr1,dr2d~s−1,2 dassoci- ated with the anisotropy exponent lg. From these we ﬁnd the critical exponents 1/n=lt=2− eS1−10 N+28 N2D+OSe N3D, f=lgn=1− eS3 2N+51 2N2D+OSe N3D. s44d Note that the crossover exponent ffor the anisotropy is always less than one. This implies71that the phase bound- aries meet as straight lines at the critical point, and we ﬁndthe same topology of the phase diagram as shown in Fig. 7. C. Renormalization group analysis in d=2+ e. SO3ˆSONﬁxed point The runaway ﬂows in Eq. (34)mean that the system goes to strong coupling. In this limit the magnitudes of the vectorsRe c,I m c, andNhave already developed locally but the directions can still ﬂuctuate on long length scales. For N=3 one of the runaway directions of Eq. (34)corresponds to u3, v2.0 andu1,v1,0. The corresponding strong coupling limit can be described by a triad of vectors that are all mu-tually orthogonal. F=−o kxylhK1e1sxd·e1syd+K2fe2sxd·e2syd+e3sxd·e3sydgj. s45d Heree1,e2, ande3correspond to N,R e C, and Im C, re- spectively. The free energy Eq. (45)has an explicit SO (2) FIG. 7. Phase diagram of the model Eq. (29)withu˜2.0 in three dimensions in the large Nlimit. The four second order lines meet at the tetracritical point at nonzero angles. The same phase diagramappears in the d=4− «analysis for Nø33 and in the d=2+ «analy- sis forNø5.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-11charge symmetry of rotations between e2ande3. The con- tinuum version of this model is given by F=EddxS1 2g1s„e1d2+1 2g2fs„e2d2+s„e3d2gD,s46d wheregi~Ki−1and the constraints ei·ej=dijare implied. Let us now discuss the phase diagram of Eqs. (45)and (46). ForK1,2!0 we have a fully disordered phase. For K1,2!‘we have a fully ordered phase that is a mixture of TSC and AF. When K2=‘the vectors e1ande2are ordered and there is an Ising type transition between TSC and TSC+AF phases. For K 1=‘vectorNis ordered and there is an O(2)transition between the AF and TSC+AF states. For K2=0 there is a Heisenberg transition between the disordered and the AF phases. For K3=0 there is a transition between the fully disordered and the TSC phases. What happens inthe interior of the phase diagram, however, is not clear. When we apply the d−2= eRG analysis to the model Eq. (46)75–79we obtain the ﬂow equations dg1 dl=−eg1+g12sg22+g1g2−g12d 2psg1+g2d2 dg2 dl=−eg2+g12g22 2psg1+g2d2. s47d The ﬂow diagram is shown in Fig. 8. We can see that it lacks the Ising and O (2)phase transi- tions. This is not surprising, since the d−2= eanalysis works well only for the spin-wave excitations of order parameterswithNø3. To shed some light on the phase diagram of Eqs. (45)and (46)we consider the large Ngeneralization of this model.We assume that all vectors e ihaveNcomponentsF=EddxF1 2g1s„e1d2+1 2g2fs„e2d2+s„e3d2g +1 g3fse1·„e2d2+se1·„e3d2g+1 g4se2·„e3d2G s48d The last two terms in Eq. (48)are generated in the RG ﬂow, even if they are absent in the microscopic model (such terms are linearly independent of the ﬁrst two only for N.3). The symmetry breaking pattern of the nonlinear model Eq. (48)is OsNd3Os2d/OsN−3d3Os2ddiag. s49d In order to express the RG equations in a simple form we introduce the variables hi: h1=1 g1, h2=1 g2, h3=1 g1+1 g2+2 g3, h4=2 g2+2 g4. s50d These variables arise naturally in a matrix formulation of the nonlinear model Eq. (48), see Appendix D, where the RG calculation is outlined. To one loop order we ﬁnd the RGﬂow d h1 dl=eh1−1 2pSN−2+h12−h22−h32 h2h3D dh2 dl=eh2−1 2pSN−2+h22−h32−h12 2h1h3−h4 2h2D dh3 dl=eh3−1 2pSN−2+N−3 2h32−h12−h22 h1h2−h4 2h3D dh4 dl=eh4−1 2pSN−3 2h42 h22+h42 2h32D. s51d The conditions g1=g2andg3=g4deﬁne a two- dimensional subspace over which the free energy Eq. (48) has the enhanced symmetry SO s3d3SOsNd. For arbitrary N, the RG equations (51)have SO s3d3SOsNdﬁxed point h1=h2=sN−2−xd/e, h3=h4=xh1, s52d wherex=sN−2+˛N2−5N+5d/sN−1d. Independent of N, this point has one stable direction and one unstable direction within the symmetric plane. The ﬂow in directions perpen-dicular to the symmetric plane depends on the value of N. FIG. 8. Renormalization group ﬂow of nonlinear model Eq. (46), corresponding to Eq. (47).PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-12The case N=4 is special and is discussed inAppendix D. For all other Nø5, the RG ﬂow away from the SO s3d3SOsNd plane has one stable and one unstable direction.Two relevant parameters are therefore necessary to tune the transition, justas we found in the d=4− eanalysis for large N, Eq. (43). Computing the critical exponents nandf, just as we did in the cased=4− e, Eq. (44), we ﬁnd, 1/n=e, f=1−63 16N2−OS1 N3D. s53d Just as before, we ﬁnd that for all ﬁnite N,fis less than one, leading to a phase diagram that is topologically equivalent tothat found in large Nexpansion and in 4− eRG analysis (see Fig. 7 ). What is more, to leading order in 1/ N, expansions about the upper and lower critical dimension, Eqs. (44)and (53), lead to the same critical exponents, 1/n=d−2+OS1 ND f=1−OS1 ND. s54d This supports the fact that for large Nthe SO s3d3SOsNd ﬁxed point changes adiabatically with dimension. D. Phase diagram for N=3 in three dimensions We employed three approaches to study classical ﬂuctua- tions in systems with competing AF and nonunitary TSC:largeNexpansion, d=4− eandd=2+ eRG analyses. When Nis large all three consistently predict a tetracritical point with enhanced SO s3d3SOsNdsymmetry. In the physically relevant case N=3 andd=3 the situation is less clear. For example, expansion from the upper critical dimension sd =4− edshows that such ﬁxed point appears only for Nø33. Expansion from the lower critical dimension sd=2+ edgives aS O s3d3SOsNdﬁxed point in the RG ﬂow for any Nø3, but these ﬁxed points become tetracritical points on the phase diagram only for Nø5. It is possible that in three dimensions even for N=3 there is a tetracritical SO s3d 3SOs3dpoint. The reason why perturbative expansions in dimension fail to see it, is that they work well for small eand extrapolations to d=3 should be treated with caution.79Such a scenario, however, would contradict the results of classicalMonte Carlo simulations in Ref. 80, in which the model (45) has been analyzed for K 1=K2. In that paper the bimodal distribution in the energy histogram has been interpreted as asignature of the ﬁrst order transition. The phase diagram that we propose for the model Eq. (45) and for systems with competing AF and nonunitary TSC ingeneral is shown in Fig. 9. Thermal ﬂuctuations turn a tet-racritical point into a line of direct ﬁrst-order transitions be-tween a disordered and a mixed TSC/AF phase. We expect,however, that the width of such a ﬁrst order line is small andthe transition is very weakly ﬁrst order. We conjecture thatwhen approaching the transition between the normal and the AF/TSC mixed phase, susceptibilities for the AF and TSCorder parameters start increasing as if dominated by theSOs3d3SOs3dtetracritical point. Only very close to the transition line the divergencies are cutoff due to the transition being ﬁrst order. Finally, we note that Eq. (45)withK 1=0 is among a class of closely related models that have been studied extensivelyin the context of frustrated magnetism. 81–91Numerical stud- ies of these models in d=3 dimensions yield nonuniversal critical properties at the boundary of normal and TSCphases, 82–90and even evidence of a ﬁrst order transition.91 The nonperturbative theoretical analysis of Ref. 92 supportsthe latter scenario, claiming that the critical point observed innonlinear sigma models in d=2+ edisappears at dc=2.87 in one such model, being replaced by a weakly ﬁrst order tran-sition above d c.The exact nature of the transition seems to be very strongly model dependent near d=3, and we leave open the possibility that the transition between nonunitary TSCand normal phases is weakly ﬁrst order. VII. QUANTUM SO(4) SYMMETRY The microscopic system that motivated our discussion is an assembly of Luttinger liquids weakly coupled in threedimensions. It is useful to condense this system to a simplereffective quantum model that concentrates on the low energycollective degrees of freedom, such as AF and TSC orderparameters and rotations between them (such a description only applies in the vicinity of the AF/TSC phase boundaryshown in Fig. 1 ). Effective quantum models have been dis- cussedpreviouslyforspinsystems (seeRef.93forareview ), and systems with singlet superconductivity competing eitherwith charge density wave order 37,38,65,94or with antiferromagnetism.30,95–98 A simple form for such an effective model is a SO (4) quantum rotor model: FIG. 9. Phase diagram for systems with competingAF and non- unitary TSC orders described by the model Eq. (45)in three dimen- sions. The tetracritical point in the mean-ﬁeld phase diagram of theGLfree energy in Eq. (29)(see Fig. 4 )is replaced by a line of direct ﬁrst order transitions between a disordered and a mixed TSC/AFphase.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-13Hr=1 2x1o iSi2+1 2x2o iIi2−Jo kijlaaQi,aaQj,aa +u˜1o iababQi,aa2Qi,bb2+u˜2o iababQi,aaQi,abQi,baQi,bb +dro iasQi,za2−Qi,xa2−Qi,ya2d. s55d The model is obtained by coarse graining the original lattice so that, for a half-ﬁlled system, each site of the rotor modelincludes two (or a larger even number, as necessary to in- clude an integer number of spin-triplet Cooper pairs )adja- cent sites along the intrachain direction of the original lattice.By combining the electronic operators that make up eachrotor model site, one can build three local spin and threelocal isospin operators S iandIi,a n daS O (4)tensor order parameter Qi,aa. Following a procedure similar to Ref. 97, one can show that the low energy properties of the systemare given by the rotor commutation relations: fS i,a,Sj,bg=idijeabgSi,g fIi,a,Ij,bg=idijeabcIi,c fSi,a,Qj,abg=idijeabgQj,ag fIi,a,Qj,bag=idijeabcQj,ca fQi,aa,Qj,bbg=0. s56d These relations are analogous to Eqs. (4),(7), and (19).I n Eq.(55)the unit length constraint of the rigid rotor models is replaced by the interaction terms u˜1andu˜2. For drnegative the system favors the AF state and for drpositive the TSC state is preferred. When dr=0 all generators of SO (4)(I =oiIiandS=oiSi)commute with the Hamiltonian Eq. (55) and the system is exactly SO (4)symmetric. We can use Eq. (55)to discuss excitation spectra in vari- ous phases of the system. We choose to orient the AF orderparameter sQ zadin thezdirection so that kQzzl=N. Similarly we take kQxxl=cto describe unitary TSC, and kQxxl=kQyyl =c. for nonunitary TSC. With these choices we can linearize the equations of motion for the ﬂuctuations to obtain: dQj,bb dt=−1 x1o aSj,aeabbkQbbl−1 x2o aIj,aeabbkQbbl, s57d dSj,a dt=J 2o b8,b8eab8b8kQb8b8lo dsQj,b8b8−Qj,b8b8+dd, s58d dIj,a dt=J 2o b8,b8eab8b8kQb8b8lo dsQj,b8b8−Qj,b8b8+dd +4drfeazb8kQb8b8lQj,zb8+eazb8Qj,b8zkQzzlg. s59dThe above equations deﬁne a linear eigenvalue problem for the frequencies of the collective modes and for the sec-ond quantized operators b †skd=o cÞgAcg1Qcgskd+o aAa2Saskd+o aAa3Iaskd, s60d which obey b˙†skd=ifH,bk†g=ivkb†skd. The Fourier trans- forms of lattice operators are deﬁned by Oˆskd =N−1/2ojOˆjeik·xj. In the effective model, neutrons couple to the spin order parameter Qza, so that the low energy scatter- ing intensity of polarized neutrons is given by: xa9sk+2kf,vd=o nuknuQzaskdu0lu2dsv−vn0d.s61d The momentum is shifted by 2 kfbecause a uniform Qzain Eq.(55)corresponds to a SDW order of momentum 2 kfin the microscopic model.The weight associated with a particu-lar collective mode created by b †skdis uk0ubskdQzaskdu0lu2dsv−vkd =uk0ufbskd,Qzaskdgu0lu2dsv−vkd, s62d where we used the fact that bskdannihilates the ground state. The commutator can be calculated using Eq. (56), once the operator content of the mode bskdis determined. We note that forkclose to zero, neutrons couple to Sainstead of Qza, in which case, xa9sk,vd=o nuknuSaskdu0lu2dsv−vn0d. s63d However, in the following, we focus on neutron scattering near 2kf. The nature of collective excitations in the various phases is summarized in Figs. 10–12, and Table I. We now providea detailed analysis of the collective mode spectrum and theassociated neutron scattering intensity in each phase. Acomplementary calculation of the neutron scattering intensityof the Q-excitations in the unitary TSC, based on the micro- scopic model, is given in Ref. 55. A. Antiferromagnet In the AF phase kQzzl=Nand all other order parameters vanish. Then Eqs. (57)–(59)decouple to four independent collective modes. The equations of motion for the pairshQ zx,Syjand hQzy,Sxjyield the usual AF spin waves with linear dispersion reﬂecting broken SO (3)spin symmetry: vAF,Ssk+2kfd=N˛Jz 2x1s1−gkd<˛J 2x1uku,s64d whereJk=Jz/s2x1ds1−gkd,zis the lattice coordination (z =6 for a cubic lattice in three dimensions ),gk=z−1odeikd, anddare bond vectors.Although kandk+2kfare related by reciprocal lattice vectors in the rotor model Eq. (55), they are not in the microscopic Hamiltonian, and the addition of 2 kf to the argument of Eq. (64)serves as a mnemonic for the factPODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-14that the spin mode is centered primarily around 2 kf. Similarly the equations of motion for hQxz,Iyjand hQyz,Ixjdescribe two massive isospin waves: vAF,Isk+2kfd=N˛4udru x2+J 2x2k2. s65d These excitations correspond to rotations between the AF and the TSC states and indicate proximity of the two groundstates. When drgoes to zero, the mass of the isospin waves vanishes reﬂecting an enhanced SO (4)symmetry. The iso- spin modes in the AF phase do not couple to neutrons. B. Unitary triplet superconductor In the unitary TSC sdr.0,u˜2,0dwe choose kQxxl=c, while all other order parameters vanish. Equations (57)and (58)for the pairs hQxy,SzjandhQxz,Syjyield two spin wave modes reﬂecting the broken spin symmetry: vuTSC,Sskd=c˛J 2x1uku. s66d The creation operators for the two spin waves involve the generators SyandSz, respectively. Substitution into Eq. (62)immediately shows that the neutron scattering weight of these modes at 2 kfvanishes. Equations (57)and(59)for the pair hQyx,Izjyields a gap- less phase ﬂuctuation mode reﬂecting the broken charge Us1dsymmetry in the TSC phase: vuTSC, wskd=c˛J 2x2uku. s67d With the inclusion of Coulomb interactions, this mode be- comes massive through the Higgs phenomenon, with a massof the order of the plasma frequency. The creation operatorof this isospin mode involves the generator I z. Substitution into Eq. (62)shows that it does not couple to 2 kfneutrons. Finally the equations of motion for the pair hQzx,Iyjgive the massive Qmode: vuTSC, Qsk+2kfd=c˛J 2x2k2+4dr x2. s68d The operator that creates this mode from the ground state is given by bQ†skd=1 ˛1+x2sJk2/2+4 drd 3f˛x2sJk2/2+4 drdQzxskd+iIyskdg. s69d When substituted into Eq. (62)it gives a neutron scattering intensity at k+2kf: FIG. 10. Collective excitations in various phases: (a)In the AF phase, the excitation spectrum consists of two massless spin wavesand two massive isospin waves. Due to translational symmetrybreaking, 2 k fis a reciprocal lattice vector, and these modes also have nonzero weight at the dashed curves near k=0. The spectrum in the TSC phase contains massless phase and spin modes, as wellas massive Qmodes. For unitaryTSC, there is only one such mode. The nonunitaryTSC (as well as the mixed phase )contains a second, degenerate Qmode, represented by the dotted curve; (b)the SO (4) symmetric point s dr=0d. In a unitary TSC this point corresponds to the transition betweenAF and TSC. It is characterized by four gap-less(Goldstone )modes (two isospin and two spin ). In the nonuni- tary case the SO (4)symmetric point is inside the mixed AF/TSC phase. It supports only three (degenerate )Goldstone modes. This is because the order parameter has a residual SO (3)symmetry. FIG. 11. Spin waves and Qmodes in the mixed phase: (a) dispersions sdr.0dof the mode clearly show mixing between the Qmode and the spin wave. Each of the shown modes is doubly degenerate; (b)neutron scattering intensity of the modes at wave- vectork=2kf+0.1 pas a function of the tuning parameter dr. Note that the weight of the spin wave modes goes to zero at theSO(4)symmetric point dr=0.Qmodes are strongly enhanced near dr=0.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-15uk0ufbQskd,Qzaskdgu0lu2dsv−vuTSC, Qd =daxc2 1+x2sJk2/2+4 drddfv−vnTSC, Qsk+2kfdg. s70d As we approach the point dr=0 with SO (4)symmetry, the gap of the Q-mode vanishes as ˛dr. Hence, the spectrum of the Hamiltonian (55)withu˜2,0 is such that on both sides of the AF/TSC transition we observe mode softening. Modesoftening at the ﬁrst order transition is a property of thehigher symmetry quantum critical points. 30,94,95Exactly at the SO (4)symmetric point dr=0 the system has gapless spin and isospin wave doublets. C. Nonunitary triplet superconductor For the case u˜2.0 the TSC phase is nonunitary. We choose kQxxl=kQyyl=c. There is also a mixed phase where a nonvanishing AF order parameter kQzzl=Nappears in addi- tion to the nonunitary TSC order, considered below. In the pure nonunitary TSC we ﬁnd two spin wave modes hQxz,Syj andhQyz,Sxjwith linear dispersion: vnTSC,Sskd=c˛J 2x1uku. s71d As in the unitary case, these spin waves do not couple to neutrons around 2 kf.The nonunitary TSC also supports two degenerate mas- sive Q-modes hQzx,IyjandhQzy,Ixjwith dispersion vnTSC, Qsk+2kfd=c˛J 2x2k2+4dr x2. s72d These correspond to rotations of the real and the imaginary parts of the TSC order parameter toward the AF. These ex-citations are created by the operators: b Qx†skd=1 ˛1+x2sJk2/2+4 drd 3f˛x2sJk2/2+4 drdQzyskd−iIxskdg bQy†skd=1 ˛1+x2sJk2/2+4 drd 3f˛x2sJk2/2+4 drdQzxskd+iIyskdg.s73d Substituting the bQaoperators in Eq. (62)we ﬁnd the neutron scattering weight near 2 kf: uk0ufbQxskd,Qzaskdgu0lu2dsv−vnTSC, Qd =dayc2 1+x2sJk2/2+4 drddfv−vnTSC, Qsk+2kfdg TABLE I. Symmetry breaking and collective modes. Here uTSC and nuTSC stand for unitary and nonunitary TSC, respectively, nTSC+AF corresponds to a mixed phase of non-unitary TSC and antiferro- magnetism away from the SO (4)symmetric point. Phase Order parameterResidual symmetryGoldstone (massless )modesPseudo-Goldstone (massive )modes AF kQzzl Us1d3Us1dsSz,Izd2sSx,Syd 2sIx,Iyd uTSC kQxxl U(1)sSxd 3sSz,Sy,Izd 1sIyd nuTSC kQxxl=kQyyl U(1)sSz+Izd3sSz,Sy,Sz−Izd 2sIy,Ixd nTSC+AF kQxxl=kQyyl,kQzzlU(1)sSz+Izd3sSz,Sy,Sz−Izd 2sIy,Ixd unitary SO (4) kQxxl Us1d3Us1dsIx,Sxd4sSz,Sy,Iz,Iyd 0 nonunitary SO (4)kQxxl=kQyyl=kQzzlSO(3)sI+Sd 3sI−Sd 0 FIG. 12. Gap of the Qmodes softens toward dr=0 reﬂecting the enhanced SO (4)symmetry at that point: (a)The gap decreases as ˛udru in the case u˜2,0(unitary TSC );(b)in the case u˜2.0 there is a change from ˛udrubehavior in the pure phases to linear decrease at smaller udru, inside the mixed nonunitary TSC and AF phase.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-16uk0ufbQyskd,Qzaskdgu0lu2dsv−vnTSC, Qd =daxc2 1+x2sJk2/2+4 drddfv−vnTSC, Qsk+2kfdg. s74d The phase ﬂuctuation mode in the nonunitary TSC phase differs from its counterpart in the unitary case. Equations(57)–(59)forhQ xy,Qyx,Sz,Izjcannot be decoupled, giving a mode that involves both spin and isospin generators. Due to the residual symmetry generated by Sz+Iz, the mode with Qxy−Qyx, drops out of the spectrum. The remaining excita- tion follows the dispersion vnTSC, wskd=c˛JS1 x1+1 x2Dk. s75d As in the unitary case, Coulomb interactions make this a massive mode, with a mass of the order of the plasma en-ergy. The creation operator of this mode involves the genera- torsIzandSz. Substitution into Eq. (62)gives vanishing neu- tron scattering intensity at 2 kf. D. Mixed phase In the AF/nonunitary TSC mixed phase, the order param- eters form an orthogonal triad c1=cxˆ,c2=cyˆ,N=Nzˆ, i.e., kQxxl=kQyyl=c, and kQzzl=N. It is easy to verify that the phase ﬂuctuation mode remains unchanged, its dispersion given by Eq. (75). However, other modes are complicated due to the fact that Eqs. (57)–(59)couple the coordinates hQxz,Qzx,Sy,Iyjand similarly hQyz,Qzy,Sx,Ixj. Solution of the eigenvalue equations yields two collective modes for each of the above coordinate sets. One is a massive “ Q” mode and the other a gapless spin wave-like mode: vS,Q=˛S2drf x2+Jkr 2xtD7˛S2drf x2+Jkr 2xtD2 −Jk x1x2sJkf2+4drfrd, s76d where xt−1;x1−1+x2−1,Jk;Jz/2s1−gkd,f;c2−N2, and r ;c2+N2. To calculate the spectrum in the mixed phase as a function of the tuning parameter dr, we ﬁnd the values of the order parameters at a given drfrom the mean ﬁeld theory of Eq.(55). Speciﬁcally we use the result c2−N2=dr/u˜2. Fig- ure 11 (a)gives an example of the dispersions obtained for a particular value of drwithin the mixed phase. The asymptotic form of the excitation energies at small wavevectors is given by: vS,˛Jr 2x1uku vQ,˛Jr 2x2k2+4dr2 u˜2x2. s77d Figure 11, demonstrates that the exact dispersion Eq. (76) deviates from these asymptotic forms already at relativelysmall wave vectors.This is due to the strong mixing betweenspin and isospin modes. Due to this mixing, both spin wavesand Qmodes carry some weight in the neutron scattering intensity [see Fig. 11 (b)]. The scattering intensity associated with the spin wave mode vanishes in the vicinity of dr=0. On the other hand the intensity of the Qmodes becomes dramatically enhanced. Another unique feature of the phasewith mixed nonunitaryTSC andAF order is a linear with u dru softening of the Q-excitation gap. Compare this to the ˛dr softening in the unitary TSC (see also Fig. 12 ).The SO (4)symmetric point dr=0 needs special consider- ation. Here N2=c2and the order parameter is invariant under the SO (3)group generated by I+S. This implies that there are only three Goldstone modes at this point. Indeed, a directcalculation at the SO (4)symmetric point gives three degen- erate modes with dispersion: vnSOs4dskd=c˛J 2S1 x1+1 x2Duku. s78d Note that the number of Goldstone modes at the SO (4) point is different in a unitary and nonunitary TSC. In theunitary case the spin and isospin SO (3)symmetries are bro- ken separately with a residual Us1d3Us1dsymmetry of the order parameter local gauge freedom associated with each. This leads to four Goldstone modes. In the nonunitary case,on the other hand, there is residual SO (3)symmetry of the order parameter, corresponding to I+Srotations as discussed above. Consequently, this system has only three Goldstone modes. The gapless spin waves and phase modes that we found away from the SO (4)symmetry are generic to systems that break spin SO (3)and charge U (1)symmetries. However, Q excitations, which can be thought of as massive isospinwaves, are not. Their presence shows the proximity of AFand TSC phases and their softening at the point dr=0 should provide a unique signature of the SO (4)symmetry of the system.COMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-17VIII. SO(4) SYMMETRY IN A STRONGLY ANISOTROPIC FERMI LIQUID Thus far in the analysis we have treated the case of weakly coupled Luttinger liquids, where we showed thatSO(4)symmetry describes the phase diagram and collective modes of a system near the AF/TSC phase transition. How-ever, there is a tendency away from Luttinger behavior aspressure is increased towards the superconducting state, assupported by the observation of ﬁeld-induced SDWphases athigh magnetic ﬁelds, 99–104by optical measurements,105and by low temperature transport experiments.48A review of the normal state of Bechgaard salts at low magnetic ﬁelds isgiven in Ref. 47. In this section, we will consider the effectsof interchain hopping in the extreme case where it is largeenough to destroy all remnants of Luttinger liquid physics,and make the system into a highly anisotropic Fermi liquidinstead.We will see that even in this limit, despite the loss ofnesting, an approximate SO (4)symmetry remains. We begin by looking at the classical SO (4)symmetry of the GL free energy. In order to investigate this symmetry, itis sufﬁcient to consider the quartic GL terms. This followsfrom the fact that our analysis of the phase diagram includesexplicitly anisotropy in the quadratic terms, see Eq. (29), which is tuned to zero by pressure at the AF/TSC transition.As shown inAppendix B, a microscopic derivation of the GLparameters starting from a weakly interacting Fermi liquidleads to the following form for the quartic GL terms: F 4=As2suCu2d2−uC2u2d+BsN2d2 +2sC+DduCu2N2−4DuC·Nu2, s79d where, for a perfectly nested Fermi surface, A=B=C/2=D =7zs3d/16p2vfT2satisfy the SO (4)symmetry conditions (31) A=D, B=D, C/2=D, and the sign of the coefﬁcients corresponds to the unitary TSC case. In the presence of interchain coupling tb, the single electron spectrum becomes jk=−2tacoska−2tbcoskb−m. s80d Although Bechgaard salts are triclinic, and expression (80) applies to rectangular lattices only, it gives a good approxi-mation to the low energy quasiparticle states of the system.Here, we take t a=250 meV, tb=20 meV, as estimated from plasma frequency measurements106and band structure calculations.107In addition, we take m=˛2ta, corresponding to a quarter-ﬁlled band and a nesting vector Q=s2kf,pd <sp/2,pd. We note that, in the Fermi liquid description, dimerization only affects very high energy quasiparticles, and we exclude it from Eq. (80). At ﬁrst glance, interchain hopping seems to have a dev- astating effect on the SO (4)symmetry. The nesting vector Q no longer connects the right and left moving Fermi surfacesexactly. Hence, while the coefﬁcient Ais insensitive to tb, the low temperature divergence in the coefﬁcients B,C, andDis preempted by the loss of nesting. Instead, these coefﬁcients saturate at a temperature of the order of tb2/ta<20 K, chang- ing the ratio A/Bfrom unity at high temperatures to about 10 atTc=1.2 K. However, nesting strongly affects antiferro- magnetism only, and not superconductivity. Hence, its effectson the GL parameters grow in proportion to the number oftimes that each GLparameter multiplies Nin Eq. (79). Thus, most of the effect can be absorbed into the normalization ofthe ﬁeldN. While the ﬁelds NandCcannot be normalized independently in the full GL free energy, as this would change the ratio of gradient terms 1 2fu„Cu2+s„Nd2g, such scaling is allowed when considering the mean-ﬁeld proper- ties of the system. The conditions for SO (4)symmetry at mean-ﬁeld level are then ˛AB=D, sC+Dd=3D. s81d Thus, at the mean-ﬁeld level, SO (4)symmetry is only broken weakly. This is illustrated in Fig. 13, where the left- andright-hand sides of the conditions Eq. (81)are evaluated ex- plicitly. Despite the strong variation in the values of the dif-ferent GL coefﬁcients, the curves shown in Fig. 13 tracesimilar trajectories, indicating the approximate SO (4)sym- metry. At T c=1.2 K, we ﬁnd A=7.0 3104,B=7.3 3103,C =3.1 3104, andD=2.1 3104, leading to ˛AB/D=1.06 and sC+Dd/s3Dd=0.82 (Nwas rescaled by a factor of 1.76 ). Thus, the conditions Eq. (81)deviate from exact SO (4)sym- metry by less than 20% at Tc=1.2 K. FIG. 13. Quartic Ginzburg-Landau coefﬁcients (79)for a strongly anisotropic Fermi liquid, for the choice of parameters ta =250 meV, tb=20 meV. The solid curve shows ˛AB, the dashed curveD, and the dash–dotted curve sC+Dd/3. Although the three curves do not coincide, as would be required by SO (4)symmetry, they trace similar trajectories all the way down to the critical tem-perature T c=1.2 K (vertical dotted line ). At temperatures higher than shown, the three curves converge as nesting is restored.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-18While classical SO (4)symmetry is no longer exact, the phase diagram derived in previous sections does not changein important ways.At the mean-ﬁeld level, the 20% variationfrom the SO (4)conditions Eq. (81)can open a very narrow mixed phase between the TSC and AF phases. Hence, aspressure is varied from theAF phase to the TSC phase, therewill no longer be a discontinuity in the order parameters or inthe density. Instead, these quantities will show a smooth butvery rapid variation as pressure transverses the mixed phase.Therefore, due to the narrowness of the mixed phase, thesystem will be very sensitive to disorder. For realistic sys-tems, which have impurities and crystal defects, the mixedphase will segregate into inhomogeneous regions of AF andTSC, just as found in the coexistence phase in the strictlyﬁrst order case. However, unlike the case of a ﬁrst ordertransition, the inhomogeneous behavior will be apparenteven when the phase diagram is tuned by experimental vari-ables that are intrinsic. We note that the narrowness of themixed phase, and the corresponding sensitivity to disorder, isa direct consequence of the proximity of the system to SO (4) symmetry. We now consider the ﬂuctuation-induced ﬁrst order tran- sition between the AF and normal phases near the SO (4) symmetric point. We note that in order to alter the topologyof the phase diagram, the bare GL parameters must differenough from the SO (4)symmetric values to divert the RG ﬂow near a new critical point. While symmetry can play animportant role in an RG ﬂow, it is difﬁcult to conceive of asituation where reduction of symmetry would lead to soften- ing of the ﬁrst order transitions into second order. Thus, weexpect the ﬁrst order transition between AF and normalphases discussed earlier to still be present. Finally, we notethat the case we consider in this section is extreme, in thatwe look study the system as a weakly interacting, stronglyanisotropic Fermi liquid. This probably gives a strong over-estimate of the magnitude of the breaking of classical SO (4) symmetry in real systems, which are likely to lie between theFermi liquid limit and the weakly coupled Luttinger liquidlimit, where SO (4)is a good symmetry. Before concluding this section, we brieﬂy discuss the ef- fects of interchain coupling on the quantum SO (4)symmetry. One can look for this symmetry by verifying the existence oftheQresonance in a strongly anisotropic Fermi liquid for- malism. Inside the TSC phase, this can be done using RPA type calculations, which include the AF particle–hole and Q resonance particle–particle channels (see, e.g., Ref. 108 ). Re- sults of these calculations will be reported elsewhere. Themain effect of interchain hopping is to ﬁx the transversecomponents of the nesting vector to Q=s2k f,p,pd.Thus, the Qresonance for quasi-one-dimensional systems is a collec- tive mode, whose quantum numbers are spin zero, chargetwo, and wave vector Q.The presence of interchain coupling also introduces broadening of the Qmode, and prevents it from softening all the way down to zero energy at the AF/TSC phase transition. Instead, we expect the minimum en- ergy of the excitation to be of the order of t b2/ta=16 meV. IX. EXPERIMENTAL SIGNATURES OF THE SO(4) SYMMETRY The interplay of AF and SC in the organic material sTMTSF d2PF6has been a subject of active investigation.7–9There is strong experimental evidence supporting that super- conducting order is spin triplet, as discussed in Sec. I. Inaddition, sTMTSF d 2PF6has a quasi-one-dimensional struc- ture, as the anisotropy of electron tunneling along the chains (a), in the planes (b), and perpendicular to the planes (c)is of the order of ta:tb:tc=250:25:1. Hence, sTMTSF d2PF6is a good candidate for comparison with the theoretical model discussed in this paper. Following the discussion in Sec. IIIandAppendix B we expect the triplet order parameter in thismaterial to be unitary. The phase diagram for this case wasobtained in Sec. V. In Ref. 55 we compare this to the experi-mental phase diagram of sTMTSF d 2PF6.10,11One conse- quence of having enhanced symmetry at a phase transition is the suppression of the critical temperature due to ﬂuctuationsof one order parameter into the other. This may contribute tothe drastic drop in T AFas pressure is increased near the AF/ TSC phase boundary in Bechgaard salts.10 The ﬁrst order transition between AF and TSC phases near the critical point, Fig. 6, leads to a regime of frustratedphase separation, with domains of one phase inside the other.The volume fractions of each phase are governed by theMaxwell construction, while the size of individual domainsis determined by the competition between short-range andlong-range parts of the Coulomb interaction. 109If the do- mains are distributed randomly, the total resistance of thesystem may be found using an effective medium approxima-tion. This implies, for example, that the system is supercon-ducting when the TSC phase is beyond the percolatingthreshold. On the other hand, it is possible that the TSCdomains are not distributed uniformly in the system, and aremore favorable on the surface of the sample. In this case, theTSC regions can “short-circuit” the system even before theyreach the percolation condition for the bulk. Transport prop-erties consistent with this scenario of an inhomogeneous sys-tem have been reported in Ref. 10. An interesting direction for exploring competition be- tween AF and TSC phases is to use magnetic ﬁeld experi-ments in the superconducting state near the AF/TSC phaseboundary. Magnetic ﬁeld produces orbital currents thatstrongly suppress electron pairing and leads to a formation ofan Abrikosov vortex lattice. Suppression of the AF order byZeeman effect is much smaller. Thus, we expect magneticﬂuctuations to become strongly enhanced in the mixedstate. 30,62,110–112Since the critical ﬁeld along the caxis is Hc2c=100 mT,18an applied magnetic ﬁeld along the caxis on the order of a few mT can have a strong effect, see Fig. 14.This is in contrast with effects such as ﬁeld induced SDWsand reentrant superconductivity, which require ﬁelds of atleast 5 T for their observation. 1,99–104For pressures close to theAF/TSC phase boundary and for slightly larger magneticﬁelds there may also be a quantum phase transition in whichlong range AF order develops inside the vortex phase. Wenote that strong sensitivity of 1/ T 1to magnetic ﬁelds in the superconducting state of sTMTSF d2PF6have been reported in Ref. 20. Here, increasing the magnetic ﬁeld along the b axis from 12.8 to 232 mT results in a large increase of1/T 1, consistent with the enhancement of antiferromag- netism that we propose. Earlier speciﬁc heat measurementsin Ref. 113 already showed that when the superconductingCOMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-19order in sTMTSF d2ClO4is suppressed by a magnetic ﬁeld, the system goes into a semimetallic state with a suppressed quasiparticle density of states. This is consistent with devel-opingAF order, thus opening a gap in the quasiparticle spec-trum. It may be interesting to study further the enhancementof magnetic order in the mixed state with neutronscattering, 114–117NMR,118–120andmSR121experiments. In Ref. 55 (see also Sec. VII of this paper )we discuss that direct observation of the Qmode in the superconducting phase should be possible through neutron scattering. Themost important feature of the Qresonance, which identiﬁes it as a generator of the SO (4)symmetry, is the pressure de- pendence of the resonance energy inside the TSC phase.When the pressure is reduced and the system is brought to-ward the phase boundary with the AF phase, we predict theenergy of the Qresonance to be dramatically decreased. Mode softening is not expected generically at ﬁrst orderphase transitions and provides a unique signature of theSO(4)quantum symmetry. We note that due to interchain hopping, the center of mass momentum of the Qexcitation in quasi-one-dimensional systems is s2k f,p,pd. Another approach to detect the Qexcitation involves tun- neling experiments with the SSC/ sTMTSF d2ClO4junction shown in Fig. 15 (analogous experiments in the context of p excitations in the high Tccuprates are discussed in Ref. 122 ). A singlet superconductor provides a reservoir of Cooperpairs that can couple to Qpairs in sTMTSF d 2ClO4. One needs to overcome, however, the momentum mismatch be- tween the two types of pairs.Apossible approach is to use anintermediate layer of the quasi-1 dmaterial sTMTTF d 2PF6. This salt is quarter ﬁlled and displays spin-Peierls (SP)order. The modulations of the SP order thus have a periodicity offour TMTTF sites, matching the s2k f,p,pdwave vector ofsTMTSF d2ClO4. The small mismatch between the two wave vectors, due to differences in the lattice constant in these compounds, can be compensated by a parallel magneticﬁeld. 123We expect peaks in the current–voltage characteris- tics of the junction when the voltage bias compensates theenergy difference between Cooper and Qpairs 2eV= vQ. s82d Peaks inIVshould be present even above the superconduct- ing transition temperature of sTMTSF d2ClO4and only re- quire the other material to be superconducting. The choice of sTMTSF d2ClO4is made as this material is likely to be close to the AF/TSC transition at ambient pressure.124This elimi- nates the need for pressure cells, which would make the ex-periments much more difﬁcult. X. SUMMARY The primary purpose of this paper has been to discuss the competition of antiferromagnetism and triplet superconduc-tivity in quasi-one-dimensional systems, such as Bechgaardsalts sTMTSF d 2X. The point of departure of our work is the existence of enhanced symmetry, that uniﬁes the two order parameters, in one-dimensional systems of interacting elec-trons. Analysis of the Luttinger liquid model presented inSec. II showed that the usual charge and spin U s1d3SOs3d symmetry is enhanced to a higher SO s3d3SOs4dsymmetry on the transition line between the two phases for incommen- surate band ﬁlling. For half-ﬁlled systems and weak umklappscattering, the enhanced symmetry group becomes SO (4). Weak coupling between chains, that enables true long rangeorder, is expected to perturb the SO (4)symmetry only slightly. In Secs. III–VI we studied the ﬁnite temperature phase diagram for systems with SO (4)symmetry. For the unitary case, a mean ﬁeld analysis shows that SO (4)symmetry re- quires a direct ﬁrst order transition between TSC and AFphases. In addition, ﬂuctuations of the order parameters turna portion of the boundary between AF and normal phases FIG. 14. The effect of a magnetic ﬁeld on the superconducting state. For points (A)far from the AF/TSC boundary, the magnetic ﬁeld destroys superconductivity leading to a normal state. Forpoints (B)close to the boundary, a magnetic phase is stabilized instead. The double line denotes a ﬁrst order transition, which ex-pands into a AF/TSC coexistence region in the experimental phasediagram. Here we focus on the unitary case, but similar effects canbe seen for the nonunitary case near the AF/TSC mixed phase.Fields of the order of 100 mT are sufﬁcient for a signiﬁcant en-hancement in antiferromagnetism to be observed. This is in contrastwith ﬁeld-induced SDW phases, which require ﬁelds in excess of5T . FIG. 15. Tunneling experiment for detecting the Qexcitation in sTMTSF d2ClO4material.Asinglet superconducting material with a higher transition temperature than sTMTSF d2ClO6provides a reser- voir of Cooper pairs that can couple resonantly to Qpairs. Momen- tum mismatch between the Cooper pairs in SC and Qpairs in sTMTSF d2ClO4is compensated by scattering of electrons in a layer of the SP material sTMTTF d2PF6.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-20into a ﬁrst order transition and also lead to a weakly ﬁrst order transition between TSC and normal phases. For thenonunitary case, SO (4)symmetry requires two second order transitions between TSC and AF phases. We ﬁnd that thesystem is close to having a SO (4)symmetric tetracritical point, but thermal ﬂuctuations stretch this point into a shortline of direct ﬁrst order transitions from the normal state tothe mixed state. Our results have direct implications forquasi-one-dimensional organic superconductors from thesTMTSF d 2X family, which are likely to be unitary triplet superconductors. For example, ﬁrst order transitions between theAF and the TSC phases, and between the Normal and theAF phases explain the AF/TSC and the AF/Normal coexist-ence regions found in the phase diagram of sTMTSF d 2PF6. In Sec. VII we analyze collective excitations in various phases and demonstrate that SO (4)leads to the existence of a new collective mode, the Qexcitation, which describes rota- tions between the AF and the TSC phases. In Sec. IX westudy possible experimental tests of the SO (4)symmetry. We propose that the Qexcitation should be observed as a sharp resonance in spin polarized inelastic neutron scattering ex-periments in the superconducting phase. We predict that theenergy of the peak decreases toward the ﬁrst order phasetransition to AF order. Such softening of modes is not ex-pected in general near a ﬁrst order transition and would be aunique signature of the enhanced symmetry at the transitionpoint. ACKNOWLEDGMENTS The authors thank S. Brown, P. Chaikin, M. Dressel, B.I. Halperin, C. Kilic, S. Sachdev, A. Turner, D.-W. Wang, andS.C. Zhang for useful discussions. This work was supportedby Harvard NSEC. APPENDIXA: DERIVATION OFSO 3ˆSO4SYMMETRY IN LUTTINGER LIQUIDS In this Appendix we demonstrate that along the line g1 =2g2, the Luttinger liquid Hamiltonian (1)has an exact SOs3dspin3SOs4disospinsymmetry. For this, we use bosoniza- tion to write the Q±operators (6)assr=±d: Qr†=rEdxhr"hr# 2pae−rAsxd, sA1d whereAsxd=i˛2ffrsxd+ursxdgandur=pexdx8Prsx8d. Note thatQr†are independent of the spin ﬁelds fsandus. Hence, the spin sector of the bosonized Hamiltonian commutes trivi- ally with Qr†, and we need only keep track of the charge sector Hr=EdxSpurKr 2Pr2+ur 2pKrs]xfrd2D. sA2d Whenever g1=2g2, corresponding to Kr=1, the commutator fHr,Agtakes on a simple form:fHr,Asxdg=˛2urf]xfrsxd+pPrsxdg=−iur]xAsxd, sA3d so that commuting Hrwith an arbitrary function of Asxdis equivalent to taking the derivative with respect to x. For example, fHr,eAg=o n1 n!fHr,Ang =fHr,Ag+1 2AfHr,Ag+fHr,AgAsxd+ﬂ =−iurS]xA+1 2sA]xA+]xAAdD+ﬂ =−iur]xeA. sA4d Hence, FHr,EdxeAsxdG=−iurEdx]xeAsxd=−iurfeAsLd−eAs0dg, sA5d which vanishes if periodic boundary conditions are imposed onfsxdandusxd. Thus, for Kr=1, fH,Q±†g=0, sA6d and the Luttinger liquid has full SO s3dspin3SOs4disospinsym- metry, generated by Q±, the total spin operators Sa, and the charge of left and right movers, Q±=o ksSa±,ks†a±,ks−1 2D. sA7d The enlarged symmetry relies on the independent conserva- tion of total number of right and left movers. This is not agood conservation law, for instance, in the presence of im-purity scattering, dimerization, or umklapp. For general K r, we ﬁnd fH,Q±†g=Kr2−1 2KrEdx˛2urf]xfrsxd−pPrsxdgh±"h±# 2pae7Asxd. sA8d We would like to thank Daw-Wei Wang for helpful dis- cussions on results presented in this section. APPENDIX B: PARAMETERS OF THE GINZBURG- LANDAU FREE ENERGY FOR WEAK INTERACTIONS To extract parameters of the GLfree energy we consider a mean-ﬁeld Hamiltonian H=o kssek−mdaks†aks+C·o kwkaks†a−ks8†sss2dss8 +C·o kwka−ks8aksss2sds8s+F·o kak−kfs†ak+kfs8sss8 +F·o kak+kfs†ak−kfs8sss8sB1dCOMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-21where w k=uku/kgives the sign of k. Integrating out the fer- mions yields an effective action for the order parameterﬁelds. We obtain the fourth order terms: F 4=As2suCu2d2−uC2u2d+Bs2suFu2d2−uF2u2d+2CuCu2uFu2 +2DsuCu2uFu2−uF·Cu2−uF*·Cu2ds B2d where A=1 2bo vnEdk 2pG2s−k,−vndG2sk,vnd B=1 bo vnEdk 2pG2sk,vndGsk+2kf,vndfGsk+2kf,vnd +2Gsk−2kf,vndg C=−1 bo vnEdk 2pG2sk,vndGs−k,−vndfGsk+2kf,vnd +Gsk−2kf,vndg D=1 bo vnEdk 2pGsk,vndGs−k,−vndGsk+2kf,vnd 3Gs−k−2kf,−vndw−kwk+2kf. sB3d For instance, the diagram giving the coefﬁcient Ais shown in Fig. 16. For the Luttinger liquid type model with linearized spec- trum around k=±kfwe obtain A=B=C/2=D. sB4d The relationship among coefﬁcients [Eq.(B4)]implies that the effective GL free energy [Eq. (B2)]is SO s3dspin 3SOs4disospinsymmetric, as expected from the discussion in Sec. II.Fcan thus be parameterized in the form (22), with u˜1=3Aandu˜2=−2A. In the clean limit,A=7zs3d 16p3vfT2, where zs3d=1.202, etc., and vfis the Fermi velocity. Note that, as was pointed out in Sec. III B, to linear order ing3, umklapp does not affect the quartic coefﬁcients of the free energy. For instance, the diagrams in Fig. 17 could con-tribute to the coefﬁcient of the term uF zu2uCzu2. However, although they do not vanish individually, the two add up tozero. This is consistent with the Feynman–Hellman theoremwhich requires that the only corrections to the free energy tolinear order in g 3be given by the expectation value of the perturbation (14). APPENDIX C: NONUNITARY TRIPLET SUPERCONDUCTIVITYAND ANTIFERROMAGNETISM: EXPANSION FROM THE LOWER CRITICAL DIMENSION Here we outline some of the methods used in the RG calculation of the nonlinear model (48)ind=2+ edimen- sions. The RG ﬂow equations of a general nonlinear modelcan be computed using the formalism of Friedan. 75The ﬁelds in such models must satisfy constrains which force them tolie on some target space manifold M. For instance, the usual FIG. 16. Diagram for the coefﬁcient Ain the GL free energy (B2). FIG. 17. Corrections to coefﬁcient of uFzu2uCzu2term in GLfree energy, to linear order in umklapp scattering (dashed line ). The two diagrams add up to zero. Inspection of all such diagrams shows thatthe quartic coefﬁcients of the GL free energy are not modiﬁed tolinear order.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-22nonlinear smodel deals with a single N-component vector with a constrained ﬁxed length, and Min this case is the N−1 dimensional sphere describing the locus of possible values of such vector. A local set of coordinates fican be introduced on a patch of M, in terms of which the free energy becomes F=Edxgijfsxd]mfisxd]mfjsxd. sC1d Unlike the original ﬁelds used to deﬁne the model, the ﬁelds fisxdare unconstrained; all information regarding the origi- nal constrains is contained in the metric gij. The metric also contains the coupling constants of the system. In Friedan’sformalism, the RG ﬂow is thought of as a gradual deforma-tion of the manifold as the short length degrees of freedomare integrated out. The RG equations can be written in acovariant way; to one loop order, ]gij ]l=egij−Rij, sC2d where e=d−2 andRijis the Ricci tensor, which is deter- mined uniquely by the metric. In practice, whenever the manifold is a homogeneous spaceG/H, as in our case, it is simplest to work directly in the tangent space of the manifold, see Ref. 76 for a detaileddiscussion. In terms of the metric on the tangent space, hab, the RG equations become ]hab ]l=ehab−Rab, sC3d where the Ricci tensor in the tangent space is given by Rab=o IcfacIfIbc+o cdhaa2−shcc−hddd2 4hcchddfacdfbcdsC4d in terms of the structure factor constants of the group G fTa,Tbg=fabcTc+fabITI fTI,Tbg=fIbcTc. sC5d Generators labeled by upper case indices are elements of Lie H, while lower case indices denote generators in Lie G-Lie H. In applying expression (C4), we assume that the genera- tors have been chosen so that the structure factor constants(C5)are antisymmetric with respect to exchange of any two indices; such a choice is always possible. Equation (C4)is written in a noncovariant way to make the dependence on thecoupling constants habexplicit, and it shows the advantage of working in tangent space: the Ricci tensor is given di-rectly in terms of the Lie algebra of G. We brieﬂy digress to discuss the Lie algebra of the group SOsNd, which has NsN−1d/2 generators corresponding to inﬁnitesimal rotations in the planes km,m 8l, where the indi- cesmÞm8run through the Nindependent axes. For instance, SOs3dhas three generators, Tx=kyˆ,zˆl,Ty=kzˆ,xˆl, andTz =kxˆ,yˆl. Keeping in mind that km,m8l=−km8,ml(“a clock- wise rotation in the x−yplane is a counterclockwise rotation in they−xplane” ), we introduce a graphical representation:if we draw Npoints on a sheet of paper, an arbitrary genera- torkm,m8lcan be represented by an arrow going from point mto pointm8. The structure factor constants of the Lie al- gebra fkm,m8l,kn,n8lg=dm8nkm,n8l−dm8n8km,nl−dmnkm8,n8l +dmn8km8,nl can be written as a “generalized etensor,” fkm,m8l,kn,n8lg=ekm,m8lkn,n8lkp,p8lkp,p8l, sC6d which has a simple interpretation in terms of the arrows de- scribed above: evanishes unless km,m8l,kn,n8l, and kp,p8l are the edges of a closed triangle. If they do form a closed triangle, count the number of times that the directions of thearrows must be ﬂipped to turn it into an oriented triangle,i.e., one satisfying m 8=n,n8=p, andp8=m. If the number of ﬂips is even, then e=1; otherwise e=−1. With this in mind, inspection of Eq. (C4)shows that, for groups Gbased on SOsNd, wherefabc~eabc, the calculation of the Ricci tensor reduces almost entirely to the counting of triangles. Armed with these tools, consider the nonlinear model Eq. (48), F=EddxS1 2g1s„e1d2+1 2g2fs„e2d2+s„e3d2g +1 g3fse1·„e2d2+se1·„e3d2g+1 g4se2·„e3d2D. sC7d Model (C7)has the symmetry SO sNdof rotations of the N-component vectors, and the symmetry SO s2dof internal rotations between e2ande3. Hence, the symmetry group of Eq.(C7)isG=SO sNd3SOs2d. The order parameter is a triad of mutually orthogonal vectors, F=se1e2e3d, and the ordered phase has residual symmetry H=SO sN−3d 3SOs2ddiag. The generators of Hleave the triad Finvariant, whereas the generators in Lie G-LieHrotate the triad and are in one-to-one correspondence with the spin waves of thesystem. We identify four types of spin waves, corresponding to the following classes of generators: T a1, which leave eh2,3j untouched but rotate e1into one of the remaining N−3 di- rections; Ta2, which leave e1untouched, but rotate either e2 ore3into one of the remaining N−3 directions; Ta3, of rota- tions in either the e1,e2plane or the e1,e3plane; and Ta4, composed of the single generator of rotations in the e2,e3 plane. Each class furnishes an independent irreducible repre- sentation under the action of the group H, leading to four different spin wave velocities, and to four different couplingconstants, h1...h4, hbc=o a1h1dba1dca1+o a2h2dba2dca2+o a3h3dba3dca3 +h4dba4dca4. The RG ﬂow Eqs. (C3)becomeCOMPETITION BETWEEN TRIPLET PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-23dh1 dl=eh1−1 2pSN−2+h12−h22−h32 h2h3D dh2 dl=eh2−1 2pSN−2+h22−h32−h12 2h1h3−h4 2h2D dh3 dl=eh3−1 2pSN−2+N−3 2h32−h12−h22 h1h2−h4 2h3D dh4 dl=eh4−1 2pSN−3 2h42 h22+h42 2h32D. sC8d The ﬁxed SO sNd3SOs3dsymmetric point of Eq. (C8)is described in the body of the text for Nø5. ForN=4, the ﬁxed point is stable with respect to arbitrary perturbationsaway from the SO sNd3SOs3dsymmetric plane [inciden- tally, in this case the ﬁxed point has a larger symmetry than expected, SO s4d3SOs4d]. On the other hand, within the plane, it has one stable and one unstable direction. This sug- gests the RG ﬂows and the phase diagram are shown sche-matically in Fig. 18. Note that we have a whole line of direct transitions be- tween the disordered and the TSC+AF phases. This wholeline is controlled by the point O that has a high SO s4d 3SOs4dsymmetry. This is quite remarkable: a higher sym- metry appears not at a single point but at the whole transition line. APPENDIX D: NONUNITARY TRIPLET SUPERCONDUCTIVITYAND ANTIFERROMAGNETISM: LARGE NANALYSIS In Sec. VI A we pointed out that a free energy in Eq. (29) in the large Nlimit should be considered with care, when u˜1+u˜2/3 is close to zero. Here we assume that u˜2.0, so, this requires negative u˜1. The complications arise when the sys- tem goes outside the basin of attraction of the tetracritical ﬁxed point, and the RG ﬂows carry u˜1to large negative values. As we discuss below, this leads to a ﬁrst order tran-sition which is similar to what was suggested in Ref. 74 for the normal to A 1transition in liquid3He. We take u˜1+u˜2/3=d 4N, sD1d where dis positive and is of order 1/ N. We now extend the calculations presented in Sec. VI A to the next order in 1/ N. For all order parameters we separate expectation values andﬂuctuationsC=sa T+ıbT,sc+aL+ıb1,a0+isc+ıbL,a1+ıb0d N=sNT,N0,N1,sN+NLd. sD2d We can expand Eq. (29)to order 1/ N2and obtain tadpole equations for aLandNL. In addition to the counterterms and loops due to ﬂuctuations of the transverse components, weneed to include ﬂuctuations of the longitudinal components.Note, that loops of longitudinal components may be termi- nated by bubble chains coming from u ˜2saTbTda0vertices. We also need to include diagrams that arise from u˜2saTbTdb1aL vertices. Special attention should be paid to diagonalization of propagators, since the free energy has terms which intro-duce mixing between ﬂuctuating components in Eq. (D2). If we want to absorb the cutoff dependence into renormal- ization of quadratic coefﬁcient [compare to Eq. (38)],w e need to deﬁne the latter relative to r c8=rc−s40u˜1+24u˜2dE 0L d3k s2pd31 k2+s4u˜1+8u˜2dj 2p2logL, sD3d where j=u˜2N 4. sD4d Integrals in tadpole equations cannot be calculated ex- actly. Hence, we expand them in two cases: j2@32u˜2s2and j2!32u˜2s2[s2corresponds to sN2orsC2orssN2+sC2d/2 de- pending on terms in the integrals ]. Also, while solving ﬁnal system of equations, expansions under conditions sC2@sN2, sC2!sN2,sC2<sN2were made. To be concrete, we took the valuesu˜1=−1/ s4Nd+d/s4Ndandu˜2=3/s4Nd. Transition from disordered phase to superconductive and antiferromagnetic phases outside the vicinity of rc8remains of the second order, though transition border shifts such that FIG. 18. RG ﬂows in Eq. (51)forN=4. Point O has a symmetry SOs4d3SOs4d. FIG. 19. Phase diagram for large N under conditions ((D1)). Solid line i s a I order phase transition, dashed—II.PODOLSKY et al. PHYSICAL REVIEW B 70, 224503 (2004 ) 224503-24tN,C=15 16p2Nlog16 3. sD5d In the vicinity of full SO s3d3SOsNdsymmetry line tN =tCwe expanded equations under conditions usN2−sC2u,s2/N, sD6d resulting in ﬁrst order phase transition, limited by boundaries tN,L+2tC,L=0 sD7d and tN,M+2tC,M=C02 4N2d, sD8d whereC0=s1+3˛3/2d/p.Condition (D6)for solution obtained, appears to be valid not only for small deviations from the line of symmetry, butfor entire line of transition. Thus solution [Eqs. (D7)and (D8)]is self-consistent in the entire region, resulting in phase diagram shown in Fig. 19. In comparison with solution of theﬁrst order expansion, boundary of mixed phase becomes aﬁrst order phase transition, and there is no angle betweenNC$NandNC$Cboundaries (which is ~ din ﬁrst order expansion ). Boundary of the basin of attraction of stable ﬁxed point is determined by the validity of expansion fordifferent conditions for j 2. In our case it is Nd,1. 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PhysRevB.81.075430.pdf	Density functional study of oxygen on Cu(100) and Cu(110) surfaces X. Duan,1,O. Warschkow,1,†A. Soon,1B. Delley,2and C. Stampﬂ1 1School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia 2Paul-Scherrer-Institut, WHGA/123, Villigen CH-5232, Switzerland /H20849Received 20 July 2009; revised manuscript received 11 November 2009; published 23 February 2010 /H20850 Using density-functional theory within the generalized gradient approximation, we investigate the interac- tion between atomic oxygen and Cu /H20849100/H20850and Cu /H20849110/H20850surfaces. We consider the adsorption of oxygen at various on-surface and subsurface sites of Cu /H20849100/H20850for coverages of 1/8 to 1 monolayers /H20849ML/H20850. We ﬁnd that oxygen at a coverage of 1/2 ML preferably binds to Cu /H20849100/H20850in a missing-row surface reconstruction, while oxygen adsorption on the nonreconstructed surface is preferred at 1/4 ML coverage consistent with experi-mental results. For Cu /H20849110/H20850, we consider oxygen binding to both nonreconstructed and added-row reconstruc- tions at various coverages. For coverages up to 1/2 ML coverage, the most stable conﬁguration is predicted tobe a p/H208492/H110031/H20850missing-row structure. At higher oxygen exposures, a surface transition to a c/H208496/H110032/H20850added strand conﬁguration with 2/3 ML oxygen coverage occurs. Through surface Gibbs free energies, taking into accounttemperature and oxygen partial pressure, we construct /H20849p,T/H20850surface phase diagrams for O/Cu /H20849100/H20850and O/Cu /H20849110/H20850. On both crystal faces, oxygenated surface structures are stable prior to bulk oxidation. We combine our results with equivalent /H20849p,T/H20850surface free energy data for the O/Cu /H20849111/H20850surface to predict the morphology of copper nanoparticles in an oxygen environment. DOI: 10.1103/PhysRevB.81.075430 PACS number /H20849s/H20850: 68.43.Bc, 81.65.Mq, 68.47.Gh I. INTRODUCTION Copper-based catalysts are of importance to a number of industrial processes including the synthesis of methanol,1the reduction and decomposition of nitrogen oxides,2–4the oxy- dehydrogenation of ammonia,5,6electroless plating,7fuel cell electrodes,8and the treatment of waste water.9In copper catalysis—as for many other metal catalysts—surface oxida-tion and oxidic overlayers are widely believed to play a cru-cial role. 10,11 Oxygen adsorption on Cu /H20849100/H20850has been the subject of numerous studies12–36and a number of distinct oxide recon- structions are known. Low energy electron diffraction /H20849LEED /H20850experiments14,19identiﬁed a /H208492/H208812/H11003/H208812/H20850R45° recon- struction that appears at oxygen coverages above 0.3 ML.This phase is also observed in scanning tunneling micros-copy /H20849STM /H20850and x-ray diffraction /H20849XRD /H20850experiments and was characterized as a missing-row /H20849MR /H20850reconstruction with an oxygen content of 1/2 ML. 16,17,20,23At low coverage /H20849/H110210.3 ML /H20850, STM images show islands with a c/H208492/H110032/H20850struc- ture on the clean Cu /H20849100/H20850surface,23supporting earlier LEED evidence for the existence of such a reconstruction.19Early theoretical work has rationalized the structural properties ofthe O/Cu /H20849100/H20850system in terms of orbital hybridization 15and charge transfer processes.18More recent ab initio studies25–27 have focused on the two observed phases and discuss their formation in terms of the electronic structure. Kangas andco-workers. 29,35contrast the stability of oxygen at surface and subsurface sites for coverages up to 2 ML, ﬁnding stablesubsurface binding at coverages above approximately 3/4ML. Using density-functional theory and density-functional perturbation theory, Bonini et al. 31compare the /H208492/H208812 /H11003/H208812/H20850R45° missing-row reconstructions of O/Cu /H20849100/H20850and O/Ag /H20849100/H20850and ﬁnd similar structural and vibrational prop- erties. An important insight was recently provided by thejoint experimental and theoretical work of Iddir andco-workers, 33explaining the observed /H208492/H208812/H11003/H208812/H20850R45° and c/H208492/H110032/H20850phases as two sides of an entropy-driven order- disorder transition. On Cu /H20849110/H20850, oxygen adsorption results in a major recon- struction of the surface.20Two well-ordered superstructures are known: a /H208492/H110031/H20850phase with an oxygen coverage of 1/2 ML and a c/H208496/H110032/H20850phase with a coverage of 2/3 ML. The oxygen-induced /H208492/H110031/H20850reconstruction has been the subject of several experimental and theoretical studies,15,20,22,37–49 and the added row structure41is now generally accepted. The oxygen-induced c/H208496/H110032/H20850reconstruction is receiving increas- ing attention50–54for its hypothesized role as a precursor phase of bulk oxidation. The transition between the /H208492/H110031/H20850 andc/H208496/H110032/H20850surface phases was directly examined by Sun et al.53using reﬂectance difference spectroscopy which deter- mined a transition temperature of 660 K. In light of the importance of temperature and the avail- ability of oxygen on determining surface reconstruction, weuse the theoretical framework of ab initio atomistic thermodynamics 11,55,56to describe the relative stability of various O/Cu /H20849100/H20850and O/Cu /H20849110/H20850reconstructions in contact with a realistic gaseous environment. This leads us to a com-plete thermodynamic model of the stable surface phase be-tween the oxygen-free surface to the bulk oxide Cu 2O. Im- portantly, the thermodynamical analysis shows that bulkCu 2O formation imposes an upper limit on the oxygen cov- erage of the surface, which, interestingly, is very nearly thesame for the three principal low-index surfaces /H20849100/H20850,/H20849110/H20850, and /H20849111/H20850. The results presented here follow on, and comple- ment, our earlier results for the O/Cu /H20849111/H20850surface. 57This completes our free energy treatment of the three principallow-index surfaces of the oxygen-on-copper system at a con-sistent level of theory allowing free-energy comparisons be-tween the three crystal phases. We use the combined freeenergy data to predict the morphology of copper nanopar-ticles in an oxygen environment.PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 1098-0121/2010/81 /H208497/H20850/075430 /H2084915/H20850 ©2010 The American Physical Society 075430-1Our discussion is organized as follows: In Sec. II, our computational methodology is detailed, including the deﬁni-tions of several calculated quantities and a brief descriptionof the ab initio atomistic thermodynamics approach. Section III A revisits the clean Cu /H20849100/H20850and Cu /H20849110/H20850surfaces, which serves to validate our methodology against previous resultsin the literature. In Sec. III B , we present our calculations for the O/Cu /H20849100/H20850system discussing energetic, structural, and electronic properties. The corresponding results forO/Cu /H20849110/H20850are presented in Sec. III C . In Sec. IVwe com- bine our data into thermodynamic /H20849p,T/H20850-phase diagrams to facilitate comparison with experiment. II. COMPUTATIONAL METHOD All calculations are performed using density-functional theory /H20849DFT /H20850in the generalized gradient approximation /H20849GGA; Perdew, Burke, and Ernzerhof exchange-correlation /H2085058as implemented in the DMol3 software.59The electronic eigenfunctions are expanded in terms of a localized atomic orbital basis set of double nu-merical plus polarization /H20849DNP /H20850quality and a real-space cut- off of 9 Bohr. Our calculations include scalar-relativistic cor-rections. A thermal broadening of 0.1 eV is applied toimprove the convergence of the self-consistent procedure. For bulk copper, we use a 12 /H1100312/H1100312 Monkhorst-Pack grid to integrate over the Brillouin zone of the elementalfcc-unit cell and obtain a calculated lattice constant of3.64 Å, in good agreement with experiment /H208493.61 Å /H20850. 60The calculated bulk modulus and cohesive energy are 136 GPaand 3.45 eV , respectively; also in good agreement with theexperimental values of 137 GPa and 3.49 eV . 60 Periodic slabs are used to model the surfaces of copper. The Cu /H20849100/H20850slab is composed of 9 atomic layers with an approximate thickness of 14.6 Å. The Cu /H20849110/H20850surface is represented using a slab of 11 atomic layers which is ap-proximately 13.0 Å thick. Successive slabs in the three-dimensionally repeated unit cell are separated by a largevacuum region of /H1101530 Å. In all our calculations, oxygen atoms are adsorbed on both surfaces of the slab in order topreserve inversion symmetry. In our geometry optimizations,three atomic layers at the center of the slab are held ﬁxed atthe calculated bulk positions. All other atoms are fully re-laxed until all force components acting on the atoms arebelow 0.015 eV /Å. Test calculations using larger slabs have conﬁrmed that the resultant error to the average oxygen bind-ing energy and the Gibbs free surface energy /H20849see deﬁnitions below /H20850are smaller than 0.01 eV and 0.001 eV /Å 2, respec- tively. Several surface supercells are used to study the effects of oxygen coverage, /H9008, on the electronic and geometric struc- ture of the surface. For the Cu /H20849100/H20850surface we consider coverages of between 1/8 and 1 ML, using /H208491/H110031/H20850,/H208492/H110032/H20850, /H208494/H110034/H20850,/H208492/H208812/H11003/H208812/H20850R45°, /H208492/H208812/H110032/H208812/H20850R45°, and c/H208494/H110036/H20850sur- face unit cells. For the large c/H208494/H110036/H20850unit cell, computational restrictions necessitated a thinner, ﬁve-layer slab, as opposedto the regular nine-layer slab. This increases for this structurethe error in the average oxygen binding energy and the Gibbsfree surface energy to smaller than 0.03 eV and0.002 eV /Å 2, respectively. For Cu /H20849110/H20850, coverages of be- tween 1/8 to 1 ML are achieved using /H208492/H110031/H20850,/H208493/H110031/H20850,/H208494 /H110031/H20850,/H208492/H110032/H20850,/H208493/H110032/H20850,/H208498/H110031/H20850,/H208492/H110034/H20850, and c/H208496/H110032/H20850cells. Brillouin-zone integrations are performed using /H2084912/H1100312 /H110031/H20850and /H208496/H1100312/H110031/H20850Monkhorst-Pack grids for the /H208491/H110031/H20850 surface unit cells of Cu /H20849100/H20850and Cu /H20849110/H20850, respectively. For the larger surface unit cells, correspondingly smaller gridsare used to ensure an equivalent sampling of reciprocalspace. The convergence with respect to k-point density of binding energies and free surface energies to below 0.01 eVand 0.001 eV /Å 2, respectively, was earlier established by Soon et al.61using the same computational model of the Cu surface. Average binding energies per oxygen atom, EbO, are given relative to the clean surface and an isolated oxygen mol- ecule. We calculate EbOas EbO=−1 NO/H20873EO/Cuslab−ECuslab−/H9004NCuECu−NO 2EO2/H20874, /H208491/H20850 where NOis the number of adsorbed oxygen atoms, EO/Cuslabis the total energy of the adsorbate-substrate system, EO2is the energy of an isolated oxygen molecule /H20849see below /H20850, and ECuslab is the energy of the clean surface. /H9004NCuis the difference in the number of Cu atoms between the O/Cu system and theclean surface slab, and E Cuis the energy of a Cu atom in bulk Cu. The term /H9004NCuECuaccounts for differences in the num- ber of Cu atoms, when we compare oxygen adsorption be-tween clean, missing-row, and added-row reconstructions.Average binding energies are deﬁned such that a positivevalue indicates a thermodynamically favorable adsorptionprocess. The energies of an oxygen molecule /H20849E O2/H20850and an isolated oxygen atom are obtained from spin-polarized calculations where we use a real-space basis set cutoff of 20 Bohr. Usingthese two energies, we calculate the molecular binding en-ergy of an oxygen molecule to be 6.08 eV /H20849or 3.04 eV per atom /H20850, somewhat larger than the experimental value of 5.12 eV /H20849or 2.56 eV per atom /H20850. Our overestimation of the O 2 binding energy is inline with previous studies that use DFT in the GGA approximation.47,62For the molecular bond length and the vibrational frequency we obtain 1.22 Å and1544 cm −1, respectively, in good agreement with the experi- mental values of 1.21 Å and 1580 cm−1/H20849Ref. 63/H20850. The effect of the gaseous environment on the relative sta- bility of the considered surface structures is captured by themethod of ab initio atomistic thermodynamics. 11,55,56The availability of oxygen from the environment is representedby the oxygen chemical potential, /H9262O, which is a function of the gas-phase temperature and the oxygen partial pressure, pO2. For a given /H9262O/H20849pO2,T/H20850, the thermodynamically pre- ferred surface is the one with the lowest Gibbs free energy /H9253 which we calculate as /H9004/H9253=1 2A/H20849GO/Cuslab−GCuslab−/H9004NCu/H9262Cu−NO/H9262O/H20850. /H208492/H20850 In this equation, /H9004/H9253is the surface Gibbs free energy rela- tive to the oxygen-free, nonreconstructed surface. The calcu-lated Gibbs free energies of the oxygenated and the oxygen-DUAN et al. PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-2free slab model used to represent the surface are denoted, GO/CuslabandGCuslab, respectively. The term /H9004NCu/H9262Curepresents the cost of exchanging Cu atoms with a reservoir withchemical potential /H9262Cu. In our case, this reservoir is bulk copper with which the surface is assumed to be in equilib-rium; thus, /H9262Cuequals the free energy of a copper atom in bulk. In Eq. /H208492/H20850, the factor1 2Ais the reciprocal of the total /H20849in-plane /H20850surface area of the slab model /H20849i.e., the sum of both sides /H20850. The factor serves to normalize to a common unit area the surface free energies of slabs of varying unit cells. In our calculations, we neglect in GO/CuslabandGCuslabthe free energy contributions due to vibrational motion and thepressure-volume term. 56This allows us to replace these terms with the respective DFT total energies, EO/CuslabandECuslab. It is convenient to combine Eqs. /H208491/H20850and /H208492/H20850to obtain the surface free energy in terms of the average oxygen bindingenergy, i.e., /H9004 /H9253/H110151 2A/H20849−NOEbO−NO/H9004/H9262O/H20850, /H208493/H20850 In this equation, /H9004/H9262Odenotes the chemical potential given relative to the dissociation energy EO2of an isolated O2molecule, i.e., we deﬁne /H9004/H9262O=/H9262O−1 2EO2. /H208494/H20850 In our electronic structure analysis, we consider electron density differences n/H9004/H20849r/H20850=nO/Cu/H20849r/H20850−nCu/H20849r/H20850−nO/H20849r/H20850, /H208495/H20850 where nO/Cu/H20849r/H20850is the total electron density of the substrate- adsorbate system, and nCu/H20849r/H20850andnO/H20849r/H20850are the electron den- sities of the clean substrate and the isolated O ad-layer, re-spectively. The density difference is evaluated and plotted forthe relaxed adsorbate system, which highlights those regionsnear the surface where oxygen adsorption induces a deple-tion or accumulation of the electron density. The surface dipole moment /H20849in Debye /H20850is evaluated using the Helmholtz equation /H9262=A/H9004/H9021 12/H9266/H9008, /H208496/H20850 where Ais the area /H20849in Å2/H20850of a /H208491/H110031/H20850surface unit, and /H9004/H9021 is the work-function change /H20849in eV /H20850relative to the clean surface. The work function is deﬁned as the difference be-tween the electrostatic potential /H20849U/H20850in the middle of the vacuum region and the Fermi energy /H20849E F/H20850, so that, /H9004/H9021=U−EF−/H9021clean, /H208497/H20850 where /H9021cleanis the work function of the clean surface. III. RESULTS A. Clean Cu(100) and Cu(110) surfaces The clean Cu /H20849100/H20850and Cu /H20849110/H20850surfaces are well charac- terized experimentally and theoretically in the literature.64–70 As a point of reference, we relax the atomic positions forboth surfaces and compare our calculated interlayer spacings with previously reported values. In the following, interlayerseparations near the surface are given as percent changes /H9004 ij relative to the respective bulk separations of 1.82 and 1.30 Å in the /H20855100/H20856and /H20855110/H20856directions. For the Cu /H20849100/H20850 surface, the change between the ﬁrst and second atomic layer/H20849this is denoted /H9004 12/H20850is calculated to be −2.3%, correspond- ing to a small contraction relative to bulk. Between the sec-ond and third layer the change is /H9004 23=+1.0%. These values are in good agreement with previous DFT-GGAcalculations 65that report /H900412=−2.6% and /H900423=+0.9% and experimental LEED data68,69with/H900412=−2.1% and /H900423= +0.5%. For the Cu /H20849110/H20850surface, we obtain /H900412=−10.0% and /H900423=+3.0%, consistent with the experimental values of /H900412 =−10.0 /H110062.5% and /H900423=0.0/H110062.5% /H20849Ref. 66/H20850, and theoreti- cal results using the linearized augmented plane wave/H20849FLAPW /H20850method, 67yielding /H900412=−9.7% and /H900423=+3.6%. Our calculated work functions for clean Cu /H20849100/H20850and Cu/H20849110/H20850surfaces are 4.39 and 4.18 eV , respectively, which are in reasonable agreement with the experimental values of4.59 and 4.48 eV . 64Earlier theoretical results include those of Skriver and Rosengaard71who calculated values of 5.26 and 4.48 eV using linear mufﬁn-tin orbitals and the localdensity approximation /H20849LDA /H20850.T a o et al. 65report a work function of 4.49 eV for Cu /H20849100/H20850using DFT-GGA in good agreement with our value of 4.39 eV . We note that the workfunctions of metal surfaces are usually underestimated byDFT-GGA calculations. 67 Our calculated surface energies for Cu /H20849100/H20850and Cu /H20849110/H20850 are 0.64 and 0.97 eV per surface unit cell, respectively.These results correlate well with the FLAPW calculations ofDa Silva et al. 72,73reporting surface energies of 0.60 eV for Cu/H20849100/H20850and 0.90 eV for Cu /H20849110/H20850. The more approximate LDA mufﬁn tin calculations of Skriver and Roosengard71 yield 0.85 and 1.33 eV , respectively. B. Oxygen adsorption on Cu(100) Cu sublattices and O binding sites. We examine oxygen adsorption on three types of Cu /H20849100/H20850surface structures /H20849Fig. 1/H20850. The ﬁrst structure is the simple, nonreconstructed Cu/H20849100/H20850surface shown in Fig. 1/H20849a/H20850which deﬁnes the /H208491/H110031/H20850surface unit cell of this crystal face. On this surface there are three distinct high-symmetry adsorption sites: Thetopsite /H20849T/H20850is located directly on top of a surface Cu atom, thebridge site /H20849B/H20850is a twofold coordinated site between two Cu atoms, and the hollow site /H20849H/H20850is coordinated by four Cu atoms. The second surface structure /H20851Fig. 1/H20849b/H20850/H20852is a missing- row reconstruction with a /H208492/H208812/H11003/H208812/H20850R45° unit cell,14,16,19,20,23which features a chain of Cu atom vacancies in the /H20851011/H20852direction. On this reconstruction the most favor- able adsorption site is the pseudohollow site /H20849pH/H20850, which is similar to the hollow site on the clean surface, but only three-fold coordinated due to the missing row of Cu atoms. Thethird structure prototype is the distributed vacancy recon-struction /H20849not shown in Fig. 1/H20850, which is closely related to the missing-row reconstruction, but is characterized by a nonor-dered, random distribution of Cu vacancies in the surface.The importance of the vacancy order/disorder transition wasDENSITY FUNCTIONAL STUDY OF OXYGEN ON Cu /H20849100/H20850… PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-3highlighted by the work of Iddir et al.33and we will be using their /H208492/H208812/H110032/H208812/H20850R45° quasi-disordered model of this sur- face to study oxygen binding at the pseudohollow sites /H20849pH/H20850. Additionally, for nonreconstructed and missing-row surfaces,we will be exploring subsurface binding sites which are in-dicated in Fig. 1. In the following discussion, we will use preﬁxes N-, MR-, and DV- with our unit cell notation to indicate nonreconstructed, missing row, and distributed va-cancy reconstructions, respectively. Oxygen on the nonreconstructed surface. We ﬁrst con- sider the adsorption of oxygen on the three high-symmetrysites /H20849top, bridge, and hollow /H20850of the nonreconstructed sur- face /H20849N/H20850for a range of coverages. For each structure, we optimize the geometry and calculate the average oxygenbinding energy, the work function change, and the surfacedipole moment. The calculated binding energies are plottedin Fig. 2and are listed for selected structures in Table I. For coverages of 0.5 ML and above, the binding energy de-creases almost linearly with coverage indicating a gradualbuild-up of repulsive interactions between the adsorbed oxy-gen atoms. 28,76The highly coordinated hollow site /H20849red data points /H20850is energetically preferred at all coverages, and top site adsorption is always endothermic /H20849i.e., EbO/H110210/H20850. The bridge site is less stable than the hollow site by approximately 0.8 to0.9 eV . For the energetically preferred hollow site, we haveadditionally investigated a lower coverage of 1/8 ML but ﬁndonly a marginal change in the binding energy relative to the1/4 ML result. This suggests that at these low coverages theinteraction between oxygen adatoms is effectively screened. Included in our nonreconstructed surfaces is a c/H208494/H110036/H20850 structure with a hollow-site oxygen coverage of 1/3 ML.This structure is inspired by earlier work, 77which shows this structure to be thermodynamically stable on the Ag /H20849100/H20850sur- face. For the equivalent Cu /H20849100/H20850structure the stability is not as clear—as shown in Fig. 2, the average oxygen binding energy of the c/H208494/H110036/H20850is very slightly smaller /H20849i.e., less stable /H20850than the linear combination of our N-/H208492/H110032/H208501/4 andN-c/H208492/H110032/H208501/2 ML hollow-site structures. The average oxy- gen binding energies is 1.81 eV for the c/H208494/H110036/H20850, to be com- pared with 1.83 eV for the linear combination of N-/H208492/H110032/H20850 1/4 and N-c/H208492/H110032/H208501/2 ML. This is just within the error of our calculations due to slab thickness effects. For a coverage of 1/2 ML we also consider subsurface binding of oxygen on the nonreconstructed surface. Two siteswere tested: the tetrahedral subsurface site /H20851see Fig. 1/H20849a/H20850/H20852, which is located directly under the surface bridge site, be-tween the ﬁrst and second atomic layers, and the octahedralsite directly under the surface top site in the second layer.Our calculations show that oxygen atoms in these subsurfacesites are much less stable than the on-surface hollow andbridge sites as shown in Fig. 2. Overall, our binding energy results are in qualitative and quantitative agreement with theprevious GGA calculations of Kangas et al. 29 We note in passing that the binding of oxygen to Cu /H20849100/H20850 is signiﬁcantly stronger than to Ag /H20849100/H20850. For 1/2 ML hollow- site adsorption on Ag /H20849100/H20850, Cipriani et al.74and Gajdos ˆ et al.75report GGA binding energies of 0.74 and 0.71 eV , respectively. This is less than half of the 1.71 eV bindingenergy that we ﬁnd for Cu /H20849100/H20850. Oxygen on missing row and disordered vacancy recon- structions. Evidence from STM /H20849Ref. 16/H20850and diffraction 17 experiments suggest the formation of a /H208492/H208812/H11003/H208812/H20850R45° re- construction when the oxygen coverage reaches 1/2 ML.Shown in Fig. 3/H20849b/H20850, this reconstruction is characterized by a missing row of copper atoms with 1/2 ML oxygen atomsadsorbed at the pseudohollow sites /H20849pH/H20850alongside the miss- ing row. In order to evaluate the stability of these missing-row reconstructions, we consider a number of variants of the /H208492 /H208812/H11003/H208812/H20850R45° structure. One variant reconstruction with a FIG. 1. /H20849Color online /H20850Oxygen binding sites on the Cu /H20849100/H20850 surface showing in /H20849a/H20850the nonreconstructed /H20849n/H20850surface, and in /H20849b/H20850 the missing-row /H20849MR /H20850reconstructed surface. Top-layer Cu atoms are white and second layer Cu atoms are gray. The in-plane positionof surface and subsurface binding sites are indicated by red squaresand labeled below the diagram. FIG. 2. /H20849Color online /H20850Calculated average binding energy per oxygen atom on Cu /H20849100/H20850in the on-surface and subsurface sites, for various coverages. The solid lines connecting the calculated bindingenergies are used to guide the eye. The dashed line indicates, as aguide to the eye, the convex hull connection between the 1/4 MLnonreconstructured surface and the 1/2 ML missing-rowreconstruction.DUAN et al. PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-4reduced oxygen coverage /H208491/4 ML /H20850is created using a doubled unit cell, MR- /H208494/H208812/H110032/H208812/H20850R45°, in which only half of the pseudohollow sites are occupied by oxygen. Missing-row structures with increased coverage /H208493/4 and 1 ML /H20850are created in the /H208492/H208812/H11003/H208812/H20850R45° unit cell by placing additional oxygen atoms into subsurface tetrahedral or octahedral sites.The in-plane position of the subsurface sites are labeled asshown in Fig. 1/H20849b/H20850where sT1 and sT2 are tetrahedral sites, and sO1 and sO2 are octahedral sites. For 3/4 ML coverage,two reconstructions were considered: The tetrahedral recon-struction contains oxygen at the sT1 subsurface site, and theoctahedral reconstruction has one oxygen at the sO1 site. Wealso construct tetrahedral and octahedral subsurface recon-structions for 1 ML coverage. The tetrahedral reconstructionhas two oxygen atoms at two sT2 subsurface sites, the octa-hedral reconstructions has one oxygen atom each at sO1 and sO2 sites. We now compare the energetics of the missing-row recon- structions to those of the nonreconstructed surface at thesame coverage /H20849see Fig. 2/H20850. For 1/4 ML coverage, the aver- age oxygen binding energy of the missing-row /H208494 /H208812 /H110032/H208812/H20850R45° structure is 0.36 eV less stable, suggesting that a missing-row reconstruction is not preferred at this low cov-erage. However, this preference reverses when we consider higher coverages. At 1/2 ML, we ﬁnd for the MR- /H208492/H208812 /H11003/H208812/H20850R45° structure an oxygen binding energy that is 0.11 eV more favorable than that of the N-c/H208492/H110032/H20850structure. For 3/4 and 1 ML oxygen coverages, the missing-row reconstruc-tions are also more stable than the nonreconstructed Cu /H20849100/H20850 surface; however, the average oxygen binding energies aresomewhat smaller relative to the 1/2 ML case, probably dueto the less favorable subsurface oxygen binding sites in-volved. Overall, our results are consistent with the STM ex- periments of Fujita et al. 23who report a /H208492/H208812/H11003/H208812/H20850R45° missing-row phase only for higher coverages. Iddir et al.33recently made the convincing argument that the experimentally observed /H208492/H208812/H11003/H208812/H20850R45° to c/H208492/H110032/H20850 transition at 475 K is due to an order-disorder phase transi-tion involving the Cu vacancies that make up the missingrow. They propose that above 475 K, the vacancies in the Cusublattice become disordered, such that the apparent symme-try in the diffraction is determined by the 1/2 ML oxygensublattice with a c/H208492/H110032/H20850unit cell. In our calculation, we use the same 1/2 ML disordered vacancy /H20849DV/H20850structure /H20851Fig. 3/H20849c/H20850/H20852that Iddir et al. 33employ to represent this surface /H20849referred to as a “spaced vacancy structure” in Ref. 33/H20850. This structure has a /H208492/H208812 /H110032/H208812/H20850R45° unit cell and arises by shifting every other Cu atom from a complete row into the missing row, therebyeffectively creating a surface with two half-missing rows.The calculated average oxygen binding energy for this struc-ture is 1.78 eV , which is 0.02 eV less than for the missing-row structure with ordered Cu vacancies. The energy differ-ence per Cu vacancy is 0.058 eV , from which we canestimate, via the conﬁgurational entropy /H20849see Ref. 33for de- tails /H20850, a transition temperature of 300 K between the ordered MR- /H208492 /H208812/H11003/H208812/H20850R45° and the quasi-disordered /H208492/H208812TABLE I. Calculated structural parameters for different O coverage /H20849from 1/8 to 1 ML /H20850on Cu /H20849100/H20850in the hollow site. The values of oxygen adsorbed on the missing-row /H20849MR /H20850reconstructed surface for 1/4 and 1/2 ML are included for comparison. The binding energy, EbO, is calculated with respect to the isolated oxygen molecule. dCu-O indicates the average bond length between oxygen and the nearest copper atom. dO1is the minimum vertical height of O with respect to the surface Cu layer, /H9004ij/H20849i=1,3, j=2,4 /H20850represents percent change relative to the bulk interlayer spacing, between the ith and jth atomic interlayer, where the center- of-mass of the layer is used. Also given are the workfunction change, /H9004/H9021, and the corresponding surface dipole moment, /H9262. StructureN-/H208494/H110034/H20850 1/8 MLN-/H208492/H110032/H20850 1/4 MLMR- /H208494/H208812/H110032/H208812/H20850R45° 1/4 MLN-/H208492/H110032/H20850 1/2 MLMR- /H208492/H208812/H110032/H208812/H20850R45° 1/2 MLN-/H208492/H110032/H20850 3/4 MLN-/H208491/H110031/H20850 1M L dCu-O /H20849Å/H20850 2.01 2.01 1.86 1.96 1.86 1.95 1.89 dO1/H20849Å/H20850 0.81 0.82 0.13 0.71 0.18 0.68 0.51 /H900412/H20849%/H20850 +1.1 +0.6 +6.0 +5.0 +4.4 +14.3 +18.7 /H900423/H20849%/H20850 +0.5 +0.4 +2.2 +1.1 +2.2 +6.0 −1.7 /H900434/H20849%/H20850 +0.6 +0.6 +0.6 +0.6 +1.1 +0.6 +1.1 EbO/H20849eV/H20850 1.85 1.89 1.53 1.71 1.81 1.06 0.60 /H9004/H9021 /H20849eV/H20850 0.37 0.74 0.05 0.92 0.32 1.31 1.51 /H9262/H20849Debye /H20850 0.52 0.52 0.03 0.32 0.11 0.31 0.27 FIG. 3. /H20849Color online /H20850Top views of the favorable oxygen ad- sorption structures on Cu /H20849100/H20850:/H20849a/H208501/4 ML oxygen coverage on a nonreconstructed surface with a /H208492/H110032/H20850unit cell, /H20849b/H20850the missing row /H208492/H208812/H11003/H208812/H20850R45° reconstruction with 1/2 ML coverage, and /H20849c/H20850 the “disordered vacancy” /H208492/H208812/H110032/H208812/H20850R45° structure that becomes stable at high temperature due to conﬁgurational entropy. Largewhite and gray spheres represent top layer Cu atoms and underlyingsubstrate Cu atoms, respectively. Black /H20849red/H20850spheres represent O atoms. The rectangles /H20849blue /H20850indicate the supercells used in the calculations.DENSITY FUNCTIONAL STUDY OF OXYGEN ON Cu /H20849100/H20850… PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-5/H110032/H208812/H20850R45° /H20851i.e., the observed c/H208492/H110032/H20850pattern /H20852. Within the errors of our methodology, this compares reasonably wellwith the results of Iddir et al. 33who report an experimental transition temperature of 475 K. Their DFT-GGA calcula-tions give an energy difference of 0.13 eV /H20849per Cu vacancy /H20850 and a transition temperature of 650 K. While the temperaturevariation is considerable due to the small energy differencesinvolved, it is comforting to see that the experimental tem-perature lies in between the theoretical results of Iddir et al. 33 and the present work. Surface Relaxations. We now consider the atomic struc- ture of various surfaces which are listed in Table I. For the nonreconstructed /H20849N/H20850surface, we see that an increase in the oxygen coverage from 1/4 to 1 ML results in a decrease inthe vertical position, d O1, of oxygen above the surface plane from 0.81 to 0.51 Å. Similarly, the Cu–O bond length de-creases from 2.01 to 1.89 Å. Our interlayer separations arein excellent agreement with the experimentally observed val- ues which range from 0.4 to 0.8 Å depending oncoverage. 21,24Below 1/4 ML coverage, we ﬁnd no appre- ciable change in these properties. Overall, the coverage-dependent changes in the structural parameters match thetrends seen in the binding energy. These trends are consistentwith a reduction in the oxygen-oxygen repulsion with re-duced coverage. The generally lower binding energies foundfor high coverages can thus be understood as due to theincreased O-O repulsion. We believe the hollow site prefer-ence arises, because O-O repulsion is partly screened by theshort Cu–O bond lengths. At low coverage, the electrostaticinteraction between O adatoms is less effective, so that thedistance between the adatoms and the metal surface becomeslarger. Oxygen adsorption results in signiﬁcant relaxations in the ﬁrst and second atomic layers with a concomitant change inthe calculated interlayer spacing. For the clean surface, /H9004 12is negative /H20849−2.3% /H20850corresponding to a small contraction be- tween the ﬁrst and second layers relative to the bulk value.With 1 ML of oxygen at the hollow site, /H9004 12becomes strongly positive /H20849+18.7% /H20850corresponding to a signiﬁcant ex- pansion. With decreasing oxygen coverage, the interlayerspacing gradually decreases toward the clean surface value.The oxygen-induced expansion can be understood as acharge transfer from copper to oxygen atoms resulting adepletion of the electron density between the ﬁrst and secondCu layer and a weakening of the bonding. This is broadlyanalogous to what has been reported for oxygen on palla-dium and other metallic surfaces. 74,78 The obtained structural parameters for the 1/4 and 1/2 ML missing-row reconstructions are listed in Table I. At 1/2 ML coverage, the oxygen atoms are located at a height of 0.18 Åabove the outmost Cu layer, which is comparable to thevalue of 0.28 Å from other DFT-GGA calculations using thepseudopotential method and a plane wave basis. 31The ex- perimental value is 0.17 /H110060.1 Å from a photoelectron dif- fraction study,24and 0.1 Å /H20849Ref. 22/H20850by LEED. The calcu- lated bond lengths to the four coordinating Cu atoms are1.88, 1.86, 1.86 and 2.24 Å, close to the experimental values1.80, 1.83, 1.83, and 2.12 Å, respectively. 22 Electronic structure. We now turn to the electronic prop- erties of the O/Cu /H20849100/H20850system. Figure 4shows the changesin the work function and surface dipole moment as a func- tion of the oxygen coverage. For hollow-site adsorption andthe missing-row reconstructions we can see that the workfunction change /H9004/H9021increases monotonically with coverage. This can be understood in terms of the large surface dipolemoment arising due to partial electron transfer from the sub-strate to the adsorbate; the difference between the electrone-gativity of oxygen /H208493.44 /H20850and copper /H208491.90 /H20850is quite large /H208491.54 /H20850. With increasing coverage, a repulsion builds up among the partially negatively charged O atoms. To reducethis repulsion, there will be partial electron transfer back tothe substrate, giving rise to a decrease in the surface dipolemoment, resulting in a depolarization. The missing-row re-constructed surfaces exhibits similar trend with coverage. Wenote at this point that the work function change forO/Cu /H20849100/H20850behaves differently to what has been reported for the O/Ag /H20849100/H20850system. 75For Ag /H20849100/H20850, the work function in- creases up to 1/2 ML and then decreases due to negativeadsorption heights at high coverage. Negative adsorptionheights do not occur in our Cu /H20849100/H20850structures. This may be due to the 12% smaller lattice constant of Cu compared toAg. Further insights into the electronic structure are gained from the electron density difference distributions and pro-jected density of states /H20849PDOS /H20850. We show in Fig. 5the elec- tron density difference /H20849relative to the clean surface /H20850for a plane perpendicular to the surface for the N-p/H208492/H110032/H208501/4 ML and the MR- /H208492 /H208812/H11003/H208812/H20850R45° structures. We see that the per- turbation caused by O adsorption is mostly localized at theoxygen atom and the topmost layer of Cu atoms. The elec-tron density around the oxygen adatom is enhanced and isdepleted around the Cu atoms. The electron transfer from Cuto the O adatom is reﬂective of the relative electronegativi-ties of the two elements. The projected densities of states presented in Fig. 6for theN-c/H208492/H110032/H208501/2 ML and the MR- /H208492 /H208812/H11003/H208812/H20850R45° 1/2 ML structures, where the O-2 pand Cu-3 dorbitals are shown. It can be seen that there is a hybridization between the O-2 p FIG. 4. /H20849Color online /H20850Calculated work function change, /H9004/H9021, and surface dipole moment, /H9262, as a function of coverage for O on Cu/H20849100/H20850. The solid lines connecting the calculated values are to guide the eye.DUAN et al. PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-6and Cu-3 din the energy range −7 to −5 eV. The greater occupation of hybridized O-2 pand Cu-3 dbonding states compared to antibonding states, reﬂects the relatively strongCu–O bond. The nature of the states in the PDOS is con-ﬁrmed /H20849not shown /H20850by the single particle wave function at the chosen eigenstates at the /H9003point. The eigenvalues corre- sponding to low energies are due to bonding states, whilethose close to the Fermi level are due to antibonding states.Furthermore, surface Cu-3 dstates are narrowed compared to subsurface Cu-3 dstates due to reduced coordination at the surface. For the MR- /H208492 /H208812/H11003/H208812/H20850R45° 1/2 ML structure, the bonding states of O-2 pextend to 8 eV below the Fermi level, and the occupancy of the antibonding states is lower thanthat of the N-c/H208492/H110032/H208501/2 ML system, which indicates the stronger interaction between O and surface Cu atoms in thereconstructed system.C. Oxygen adsorption on Cu(110) Cu sublattices and O binding sites. Being characterized by a rectangular surface unit cell, the Cu /H20849110/H20850surface offers, relative to the square unit cell Cu /H20849100/H20850, a larger number of possibilities for oxygen adsorption. On the nonreconstructedsurface /H20851Fig. 7/H20849a/H20850/H20852, we have ﬁve distinct sites. The hollow site /H20849H/H20850is located above the center of four surface Cu atoms. The long-bridge /H20849LB/H20850and the short-bridge /H20849SB/H20850sites are located between pairs of surface Cu atoms along the /H20851001/H20852 and /H2085111¯0/H20852directions, respectively. The shifted-hollow /H20849shH /H20850 site is pseudo threefold coordinated and located roughly half-way between hollow and short-bridge sites. Oxygen adsorp-tion at the topsite /H20849T/H20850directly above a surface Cu atom has previously been shown to be unstable 47and is not further explored here. We also consider two types of added row /H20849AR/H20850reconstructions /H20851Figs. 7/H20849b/H20850and7/H20849c/H20850/H20852in which rows of Cu atoms are added along the /H20851001/H20852and the /H2085111¯0/H20852directions. We consider these rows at different separations using/H208492/H110031/H20850,/H208494/H110031/H20850, and /H208498/H110031/H20850unit cells for rows in the /H20851001/H20852 direction, and a /H208491/H110032/H20850unit cell for the /H2085111¯0/H20852direction. Added row structures with a /H208491/H110031/H20850periodicity are equiva- lent to a nonreconstructed surface. For the /H20851001/H20852added row structures, we consider oxygen binding at the long-bridge site of the added row as illustrated in Fig. 7/H20849b/H20850. For the /H2085111¯0/H20852 added row, we investigate oxygen binding at short bridge and FIG. 5. /H20849Color online /H20850Difference electron density distribution of O/Cu /H20849100/H20850structures: /H20849a/H20850The nonreconstructured /H208492/H110032/H20850structure with a 1/4 ML coverage of oxygen at the hollow site. /H20849b/H20850The missing row /H208492/H208812/H11003/H208812/H20850R45° reconstruction with a coverage of 1/2 ML. Solid and dashed isolines indicate lines of constant chargeaccumulation and depletion, respectively. The lowest positive con-tour line is at 0.001 electron Bohr −3, while the highest negative contour line corresponds to a value of −0.001 electron Bohr−3. Suc- cessive isolines differ by a factor of 101/3. FIG. 6. /H20849Color online /H20850Projected density of states /H20849PDOS /H20850for the selected O/Cu /H20849100/H20850structures with 1/2 ML oxygen coverage: /H20849a/H20850 The nonreconstructed c/H208492/H110032/H20850and /H20849b/H20850the missing row /H208492/H208812 /H11003/H208812/H20850R45° structure. Energies are given relative to the Fermi level EF. The bulklike Cu-3 dPDOS of atoms in the center of the slab is indicated using a dotted line. FIG. 7. /H20849Color online /H20850Oxygen binding sites on the Cu /H20849110/H20850 surface showing /H20849a/H20850the nonreconstructed /H20849n/H20850surface, /H20849b/H20850the /H20851001/H20852 added row /H20849AR/H20850reconstructed surface with a /H208492/H110031/H20850unit cell, /H20849c/H20850 the/H2085111¯0/H20852added row /H20849AR/H20850surface with a /H208491/H110032/H20850unit cell, and /H20849d/H20850a Cu5Oxadded strand /H20849AS/H20850on the nonreconstructed surface. A close packing of Cu 5O4added strands gives rise to the 2/3 ML c/H208496/H110032/H20850 reconstruction at high oxygen exposures. Top-layer Cu atoms arecolored white and second layer Cu atoms are colored gray. Thein-plane position of surface and subsurface binding sites are indi-cated by red squares and labeled below the diagram.DENSITY FUNCTIONAL STUDY OF OXYGEN ON Cu /H20849100/H20850… PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-7shifted hollow sites /H20851see Fig. 7/H20849c/H20850/H20852. Finally, we consider re- constructions that we describe as added strand structures /H20849AS/H20850in which Cu adatoms are arranged into strands as illus- trated in Fig. 7/H20849d/H20850. On this surface oxygen binds to the two long-bridge sites and also to two threefold sites, forming aCu 5O4-stoichiometric added strand that is the basic building block of an experimentally observed c/H208496/H110032/H20850reconstruction. We will examine several variations in this pattern that differin the relative alignment of nearby strands or the number ofCu atoms within the strand. In our structure notation, we usepreﬁxes N-, AR-, AS- to indicate the Cu-sublattice as nonre- constructed, added row, or added strand, respectively. Anoverview of our calculated average oxygen binding energiesfor these sublattices and a range of oxygen coverages be-tween 1/8 and 1 ML is given in Fig. 8. Oxygen on the nonreconstructed surface. For low cover- age /H208491/8 ML /H20850on the nonreconstructed surface oxygen is preferentially bonded to the shifted-hollow site /H20849labeled N-shH in Fig. 8/H20850, followed in order of decreasing stability by the hollow, the long-bridge, and the short-bridge sites. At 1/8ML, our calculated binding energy of the shifted hollow siteis 1.62 eV , which is in good agreement with the earlier resultof 1.59 eV reported by Liem et al. 47At a coverage of 1/2 ML, the shifted-hollow site is still the preferred structure,however the long-bridge site is now the second most stablestructure. This order is reversed above /H110150.6 ML coverage, and the long-bridge site becomes the preferred binding side.The general trend on the nonreconstructed surface is that at acoverage of 1/4 ML and above, the binding energies begin torapidly decrease, which again reﬂects the increased repulsionbetween an increasing number of oxygen atoms on thesurface. 28,76The long-bridge site is an exception to this trend, showing the least variation in the binding energy with oxy-gen coverage. Added row reconstructions. The average binding energies/H20849Fig. 8/H20850show that added row reconstructions are preferred over most of the oxygen coverage range. The most stablereconstructions are characterized by /H20851001/H20852added rows /H20849blue data points in Fig. 8/H20850in which oxygen atoms occupy all of the long-bridge sites. This creates continuous Cu-O-Cu-O-Cu adatom chains that appear to be particularly stablewhich is evident by the fact that a reduced oxygen coveragedoes not disrupt these chains. Instead the reduced availabilityof oxygen atoms is accommodated by an increased averageseparation between the Cu-O rows, as illustrated in Figs. 9/H20849a/H20850 and9/H20849b/H20850for the 1/4 and 1/2 ML structures, respectively. For the 1/2 ML /H20851001/H20852added row structure, our calculated oxygen binding energy is 2.00 eV , which is in good agree-ment with the previously reported values of 2.11 and 2.03 eVby Liem et al. 47and Frechard et al. ,46respectively. Oxygen adsorption at the short-bridge site of a /H2085111¯0/H20852added row structure /H20851Fig. 7/H20849c/H20850/H20852is considerably less stable, with a calcu- lated binding energy of only 1.29 eV . The shorter bridgecauses the oxygen adatoms to be displaced higher above theadatom row, which presumably results in less effectivescreening and increased repulsion between oxygen atoms. For the stable /H20851001/H20852AR structures, the average oxygen binding energies do not change much when the coveragedecreases below 1/2 ML; for example, 1/4 ML and 1/8 MLcoverages have binding energies of 2.04 and 2.07 eV , to becompared with 2.00 eV for 1/2 ML. This indicates that theCu-O rows are only very weakly repulsive when separatedby more than one lattice unit; however, above 1/2 ML thebinding energies decrease rapidly to 1.71 eV for 2/3 ML and1.27 eV for 1 ML. In this coverage regime, the Cu-O adatomrows are separated by less than two lattice units on average,meaning that a larger number of rows are brought into singlelattice unit separation; the energetics suggests that this anunfavorable arrangement. For a full 1 ML coverage, the sur-face is so densely packed with Cu-O adatom rows that theCu-sublattice is in fact identical to the nonreconstructed sur-face /H20851Fig.7/H20849a/H20850/H20852in which oxygen atoms occupy all of the long bridge sites. FIG. 8. /H20849Color online /H20850Calculated average binding energy of oxygen on Cu /H20849110/H20850for various coverages and conﬁgurations. The dashed line connects the binding energies of the most stable struc-tures and serves to guide the eye. FIG. 9. /H20849Color online /H20850Top views of the favorable oxygen ad- sorption structures on Cu /H20849110/H20850. Shown are /H20849a/H20850the 1/4 ML oxygen coverage added row /H208494/H110031/H20850structure characterized by Cu-O-Cu- O-Cu chains in the /H20851100/H20852direction, /H20849b/H20850the added row /H208492/H110031/H20850struc- ture with 1/2 ML coverage with a closer spacing of the CuO chains,and /H20849c/H20850thec/H208496/H110032/H20850structure with 2/3 ML oxygen composed of Cu 5O4added strands in the /H20851100/H20852direction. Large white and gray circles represent top and second layer Cu atoms, respectively. Smalldark /H20849red/H20850circles represent O atoms. The rectangles indicate the surface unit cells used in the calculations. In the c/H208496/H110032/H20850structure, the nonequivalent Cu atom sites Cu1 and Cu2 are indicated /H20849see Table III/H20850.DUAN et al. PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-8The added strand reconstructions. Experimental evidence52points to the existence of a c/H208496/H110032/H20850reconstruc- tion at oxygen coverages above 2/3 ML. This reconstructionwas attributed by Feidenhans’l et al. 50to the 2/3 ML oxy- genated added strand structure shown in Fig. 10/H20849a/H20850.I no u r calculations, we ﬁnd that this reconstruction has a bindingenergy of 1.87 eV , which is marginally more stable than the 2/3 ML added row structure /H20851Fig. 10/H20849d/H20850,E bO=1.86 eV /H20852. The c/H208496/H110032/H20850reconstruction is a Cu 5O4added strand structure that differs from the /H20851001/H20852added row structures by having addi- tional Cu atoms placed between pairs of Cu-O added rows.These Cu atoms cross link between every other pair of oxy-gen atoms within the two rows, thereby creating a strandstructure that is oriented along the /H20851001/H20852direction and has a width of three lattice units in the /H2085111 ¯0/H20852direction. In the c/H208496/H110032/H20850reconstruction, these strands are closely packed, and the linking Cu atoms are shifted by one /H20851001/H20852lattice unit from one strand to the next. An alternative /H208493/H110032/H20850added strand structure /H20851Fig. 10/H20849b/H20850/H20852is slightly less stable with an average oxygen binding energy of 1.82 eV . This structure issimilar to the c/H208496/H110032/H20850reconstruction but the linking Cu at- oms are not shifted from one strand to the next. In a secondvariant, shown in Fig. 10/H20849c/H20850, we tested the stability of a /H208493 /H110031/H20850added strand reconstruction in which all oxygen atom pairs are cross-linked by Cu atoms. With an average oxygenbinding energy of 1.71 eV , this structure is also less stablethan the c/H208496/H110032/H20850.It is worthwhile to brieﬂy reﬂect on the energetic proxim- ity of the four 2/3 ML structures /H20851Figs. 10/H20849a/H20850–10/H20849d/H20850/H20852consid- ered here and the signiﬁcance of these results in relation toexperiment. Within the computational errors of our proce-dure /H20849/H110150.01 eV in binding energies /H20850, the AS- c/H208496/H110032/H20850and the AR- /H208493/H110031/H20850are very nearly of the same energy. This is to say that the presence or absence of the 1/6 ML bridging Cuatoms has only a very subtle effect on the stability of thesurface. A further increase in the density of bridging Cu at-oms to 1/3 ML, which leads to the AR- /H208493/H110031/H20850structure, produces a signiﬁcant destabilization of 0.15 eV in the aver- age oxygen binding energy. Also signiﬁcant within the mar-gins of error is the 0.05 eV difference in the oxygen bindingenergy between the two added-strand structures c/H208496/H110032/H20850and /H208493/H110032/H20850. These last two ﬁndings, suggest that the positioning of bridging Cu atoms is correlated both within and betweenadjacent strands. This in turn will prompt any bridging Cuatoms present to arrange into some local order. While the amount of bridging Cu atoms between 0 and 1/6 ML is notsigniﬁcantly explained by our energetics, we may speculatethat conﬁgurational entropy would favor structures that con-tain bridging Cu atoms over those that do not. This wouldnot be dissimilar to the mechanism that stabilizes the 1/2 MLdisordered vacancy structure on the /H20849100/H20850surface. 33 Lastly, we also tested the stability of an isolated added strand as shown in Fig. 10/H20849e/H20850with an oxygen coverage 1/3 ML. The calculated average oxygen binding energy is 1.94eV . This is less stable than what we expect /H20849by linear inter- polation /H20850for a 1/3 ML added-row structure /H20849E bO=2.03 eV /H20850. Thus, it appears that the close proximity of /H20849and repulsion between /H20850the Cu-O adatom rows is an important ingredient in making the added strand structures competitive with theadded row structures. Structural Relaxations. Calculated interplanar separations for several relevant surface structures are listed in Table II. The internal coordinates of the two oxygen and two surfaceCu atoms as indicated in Fig. 9/H20849c/H20850are listed in Table III. Our calculated results are in good agreement with the experimen-tal values from low energy ion scattering /H20849LEIS /H20850 measurements, 52improving over earlier effective-medium theory /H20849EMT /H20850calculations.38 Figure 9/H20849b/H20850depicts the AR- /H208492/H110031/H20850structure in which the O atoms occupy the long-bridge sites of every other /H20851001/H20852 copper row, while the other rows are missing. In this struc-ture, the O atoms are bonded to two surface Cu atoms andtwo subsurface Cu atoms, where the Cu–O bond lengths are1.83 and 2.05 Å, respectively. These values are in goodagreement with the various experimental results which are inthe range of 1.81–1.84 and 2.00–2.05 Å /H20849Refs. 22,43,50, and52/H20850and with the ﬁndings of the ﬁrst-principles calcula- tions of Liem et al. 47/H208491.83 and 2.08 Å /H20850and of Frechard et al.46/H208491.83 and 2.05 Å /H20850. The vertical height of O, dO1,i s calculated to be 0.13 Å, which is larger than the LEED val-ues /H110110.03 Å, 22,43but comparable to the value of 0.10 Å obtained by Frechard et al.46where only O atoms and the ﬁrst Cu layer were optimized. Electronic Structure. To elucidate the binding properties of oxygen on the Cu /H20849110/H20850surface we list the change in the work function and surface dipole moment for the most favor-able oxygen structures in Table II. These values are also FIG. 10. /H20849Color online /H20850Variations in added strand reconstruc- tions on Cu /H20849110/H20850./H20849a/H20850The experimental c/H208496/H110032/H20850reconstruction which is composed of Cu 5O4added strands, /H20849b/H20850a/H208493/H110032/H20850variant in which the bridging Cu atoms are not shifted between adjacentstrands, /H20849c/H20850a/H208493/H110031/H20850variant of Cu 3O2stoichiometry, containing a full row of bridging Cu atoms, and /H20849d/H20850, for comparison, a 2/3 ML CuO added row structure. Panel /H20849e/H20850shows a 1/3 ML added strand structure formed by removing every other strand from the c/H208496 /H110032/H20850structure. Large white and gray circles represent top and sec- ond layer Cu atoms, respectively. Small dark /H20849red/H20850circles represent O atoms. The rectangles indicate the surface unit cells used. Notethat the unit cell for the c/H208496/H110032/H20850structure is shifted away from centered to facilitate comparison with the other unit cells.DENSITY FUNCTIONAL STUDY OF OXYGEN ON Cu /H20849100/H20850… PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-9plotted in Fig. 11. It can be seen that as more effective charges /H20849i.e., oxygen atoms /H20850are added to the surface, /H9004/H9021 increases with coverage from 1/8 to 1/2 ML, while the sur-face dipole moment decreases with coverage. We plot the difference electron density for the two stable structures, AR- /H208492/H110031/H20850and AS- c/H208496/H110032/H20850, in Fig. 12. It can be seen that the oxygen atoms appear to be almost coplanarwith adjacent Cu atoms. Due to the large electronegativity ofoxygen, the electron density of the oxygen atoms is en-hanced, while the electron density of the nearest-neighbor Cuatoms is depleted. The slight accumulation of electron den-sity toward the vacuum side of the surface results in an in-ward pointing surface dipole moment and a positive /H9004/H9021.Figure 13shows the PDOS of the O-2 p, Cu-3 dorbitals for the 1/2 ML oxygenated AR- /H208492/H110031/H20850and the 2/3 ML AS-c/H208496/H110032/H20850phases. In both cases, a renormalization of the O-2pstates to lower energies is found. The hybridization of bonding states between the O-2 pand Cu-3 dorbitals is lo- cated mainly in the energy window from −4 to −8 eV. Thecorresponding antibonding states are located at −2 to 2 eV .Furthermore, the surface Cu-3 dstates in Figs. 13are nar- rower than those of the bulk Cu-3 dstates, and this is again due to reduced coordination of surface Cu atoms. IV. DISCUSSION Having surveyed oxygen binding at varying coverages on Cu/H20849100/H20850and Cu /H20849110/H20850, we are now in a position to assess theTABLE II. Calculated structural and electronic parameters for different O coverages /H20849from 0.125 to 0.5 ML /H20850for the missing-row reconstructed Cu /H20849110/H20850surfaces. The binding energy, EbOis calcu- lated with respect to the isolated oxygen molecule. dCu-O indicates the average bond length between oxygen and the nearest copperatom. d O1is deﬁned to be the minimum vertical height of O with respect to the surface Cu layer, /H9004ij/H20849i=1,3, j=2,4 /H20850represents the percent change in the ith and jth metal interlayer distances, where the center-of-mass of the layer is used, relative to the bulk interlayerspacing. /H9004/H9021and /H9262represent the work function change and surface dipole moment, respectively. StructureAR- /H208498/H110031/H20850 1/8 MLAR- /H208494/H110031/H20850 1/4 MLAR- /H208492/H110031/H20850 1/2 MLAS-c/H208496/H110032/H20850 2/3 ML dCu-O /H20849Å/H20850 1.98 1.97 1.97 2.09 dO1/H20849Å/H20850 0.13 0.13 0.13 0.05 /H900412/H20849%/H20850 +14.5 +12.5 +10.8 +23.1 /H900423/H20849%/H20850 −10.0 −6.4 −1.5 −3.9 /H900434/H20849%/H20850 +2.7 +0.5 −3.1 −1.5 EbO/H20849eV/H20850 2.07 2.04 2.00 1.87 /H9004/H9021 /H20849eV/H20850 0.20 0.33 0.56 0.54 /H9262/H20849Debye /H20850 0.40 0.33 0.27 0.20 TABLE III. Atomic positions in Å of nonequivalent atoms of the added-strand c/H208496/H110032/H208502/3 ML reconstruction. The results of this work are compared to the positions obtained by low-energy ion-scattering /H20849LEIS /H20850/H20849Ref. 52/H20850and effective-medium theory /H20849EMT /H20850 /H20849Ref. 32/H20850. In-plane coordinates are given relative to the Cu1 site /H20851see Fig. 9/H20849c/H20850/H20852. Coordinates not listed in the table are zero due to symmetry. The results of the present work use a rectangular unitcell of a=7.2856 Å and c=15.455 Å. Atom This work EMT aLEISb Cu1 z 0.55 1.2 0.45 /H11006.1 Cu2 x 2.29 2.38 2.28 y 1.78 1.805 1.78 /H110060.07 O1 x 1.85 1.36 1.78 z 0.50 0.4 0.4 /H110060.1 O2 x 5.21 5.094 5.2 /H110060.1 z −0.06 −0.2 −0.12 /H110060.1 aReference 70. bReference 57. FIG. 11. /H20849Color online /H20850Calculated work function change, /H9004/H9021, and surface dipole moment, /H9262, as a function of coverage for O on the Cu /H20849110/H20850surface for the various structures considered. Solid lines connect the low-energy added row /H20849AR/H20850and added strand /H20849AS/H20850structures. Filled and open circles indicate the work functions of other nonreconstructed and reconstructed phases, respectively. FIG. 12. /H20849Color online /H20850Difference electron density distribution of O/Cu /H20849110/H20850structures: Shown are /H20849a/H20850the added row /H208492/H110031/H20850struc- ture with a 1/2 ML oxygen coverage and /H20849b/H20850the added strand re- constructed c/H208496/H110032/H20850structure with a 2/3 ML coverage. Dashed isodensity lines represent charge depletion relative to the clean sur-face and the solid isodensity lines depict charge accumulation. Thelowest positive contour line is at 0.001 electron Bohr −3, while the highest negative contour line corresponds to a value of −0.001 elec-tron Bohr −3. Successive isolines differ by a factor of 101/3.DUAN et al. PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-10relative stability of these structures when they are in thermo- dynamic contact with a real gas phase environment. Withinthe framework of ab initio atomistic thermodynamics a sur- face is assumed in thermodynamic equilibrium with both thegas phase above the surface and the bulk elemental solid/H20849Cu/H20850below. The gas phase is represented by the oxygen chemical potential /H9004 /H9262O/H20849p,T/H20850/H20851given here relative to molecu- lar O 2as per Eq. /H208494/H20850/H20852, which in turn is given by the gas phase oxygen partial pressure pand the temperature T. Under these conditions, the preferred surface for a given pandTis the one of lowest surface free energy /H9253and it is this surface that we should expect to observe experimentally at these condi-tions. This type of analysis leads to the phase stability diagrams shown in Fig. 14in which the free energies of the considered surface structures are plotted against the oxygen chemicalpotential /H9004 /H9262O. In these diagrams, free energies are reported as differences /H9004/H9253relative to the clean surface as deﬁned in Eq. /H208493/H20850. Plotted in this way, individual surface structures ap- pear as straight lines with a negative slope /H20849−NO/2A/H20850that is proportional to the oxygen coverage. Critical for this discus-sion are the curves of minimum free energy /H20849colored red in the online version /H20850that form when the free energy lines of the most stable surfaces intersect one another. These pointsof intersection on the /H9004 /H9262Oaxis deﬁne the conditions of a phase transition from one minimum free energy surface toanother. In this way, the /H9004 /H9262Oscales for Cu /H20849100/H20850and Cu /H20849110/H20850 are divided into several distinct segments in which differentclean and oxygenated surface structures are preferred. Thesesegments are indicated in Fig. 14and are labeled by the type of the surface and the oxygen coverage in ML. An upperlimit for /H9004 /H9262Oexists at −1.24 eV. This corresponds to the calculated heat of formation of bulk Cu 2O which deﬁnes the point at which a bulk phase transition from Cu to Cu 2O will occur. In the limit of low oxygen exposure /H20849i.e., large nega- tive/H9004/H9262O/H20850, the clean Cu surface is preferred for both /H20849100/H20850 and /H20849110/H20850crystal faces. In between these two limits, severaloxygenated surface structures are found to be thermodynami- cally stable. We note in passing that at some oxygen potentialabove −1.24 eV a phase transition between bulk Cu 2O and bulk CuO is expected to occur. This transition was not fur-ther explored in this work as it occurs outside the /H9004 /H9262Olimits relevant to surface oxide formation. Starting with the clean Cu /H20849100/H20850surface /H20851see Fig. 14/H20849a/H20850/H20852, an increase in /H9004/H9262Oleads at −1.89 eV to a phase transition to aN-/H208492/H110032/H20850structure with an oxygen coverage of 1/4 ML. This oxygenated surface corresponds to the nonrecon-structed, hollow-site adsorption structure that we found inFig. 2to be preferred for a 1/4 ML coverage. A further in- crease in the oxygen exposure results at /H9004 /H9262O=−1.72 eV in another phase transition from the N-/H208492/H110032/H208501/4 ML structure to the missing row /H208492/H208812/H11003/H208812/H20850R45° reconstruction with a 1/2 ML coverage. This surface phase remains the preferred re-construction until the bulk-oxide limit at /H9004 /H9262O=−1.24 eV is reached. None of the higher coverage structures consideredby us are thermodynamically stable anywhere on the /H9004 /H9262O scale. FIG. 13. /H20849Color online /H20850Projected density of states /H20849PDOS /H20850for selected O/Cu /H20849110/H20850structures showing /H20849a/H20850the added row /H208492/H110031/H20850 structure with 1/2 ML oxygen coverage and /H20849b/H20850the added strand c/H208496/H110032/H20850structure with 2/3 ML coverage. Energies are given relative to the Fermi level EF. The bulklike Cu-3 dPDOS of atoms in the center of the slab is indicated using a dotted line. FIG. 14. /H20849Color online /H20850Calculated surface phase stability dia- gram for /H20849a/H20850the O/Cu /H20849100/H20850system and /H20849b/H20850the O/Cu /H20849110/H20850system as a function of the oxygen chemical potential. A thick /H20849blue /H20850line highlights the minimum free energy curve for each surface. Verticaldashed lines indicate the phase transition points on the /H9004 /H9262Oaxis. A shaded background on the left and right hand side denotes the re-gions of stability of the clean surface and the bulk oxide /H20849Cu 2O/H20850, respectively. In between /H20849without shading /H20850are the surface oxide phases. Note that the free surface energy is reported as /H9004/H9253relative to the respective clean /H20849oxygen-free /H20850surface.DENSITY FUNCTIONAL STUDY OF OXYGEN ON Cu /H20849100/H20850… PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-11In the /H20849p,T/H20850phase diagram /H20851Fig.15/H20849a/H20850/H20852the two /H9004/H9262Oseg- ments corresponding to the stable surface oxides /H20849white background /H20850form a band of roughly 10 to 20 orders of mag- nitude width on the pressure scale and are bounded on thehigh and low pressure side by bulk Cu 2O and the oxygen- free surface, respectively. In the region corresponding to the MR- /H208492/H208812/H11003/H208812/H20850R45° 1/2 ML, we have additionally indicated the 473 K phase boundary33to the 1/2 ML disordered va- cancy structure that is experimentally observed as a c/H208492 /H110032/H20850pattern. This results in a diagram with a total of four distinct surface phases: the clean Cu /H20849100/H20850surface at low oxygen pressure, a nonreconstructed /H208492/H110032/H20850structure with 1/4 ML oxygen atoms at the hollow site at increased pres-sures, and then either the ordered missing row /H20849forT /H11021473 K /H20850or the disordered vacancy c/H208492/H110032/H20850structure /H20849for T/H11022473 K /H20850. While the MR- /H208492/H208812/H11003/H208812/H20850R45° to c/H208492/H110032/H20850tran-sition is well documented experimentally, there does not ap- pear to be any evidence for a 1/4 ML structure with /H208492/H110032/H20850 symmetry /H20849see e.g., the phase diagram Fig. 1 in Ref. 33/H20850. This is not necessarily a conﬂict. At coverages of around 1/4ML /H20849and below /H20850our calculated oxygen binding energies on the nonreconstructed Cu /H20849100/H20850surface are very nearly con- stant with respect to changes in coverage /H20849see Fig. 2/H20850. This implies that the oxygen adatoms in the N-/H208492/H110032/H20850phase are weakly interacting, and plausibly, disordered distribution onthe surface. Thus, the N-/H208492/H110032/H208501/4 ML structure in our phase diagram acts as a “placeholder” for the low-coveragedisordered arrangement of oxygen atoms on the nonrecon-structed surface, which will appear as a /H208491/H110031/H20850structure in experiment. Only when the oxygen exposure increases, andthe surface progresses toward a 1/2 ML coverage, do theoxygen binding energies increase. This causes oxygen atomsto lock into an ordered c/H208492/H110032/H208501/2 ML pattern. Thermody- namically, this structure supports the formation of Cu vacan- cies, which in turn may order into the MR- /H208492 /H208812/H11003/H208812/H20850R45° 1/2 ML phase, or remain disordered in the DV- c/H208492/H110032/H208501/2 ML phase. Looking now at the Cu /H20849110/H20850surface, the phase stability diagram in Fig. 14/H20849b/H20850shows at /H9004/H9262O=−2.07 eV a transition of the oxygen-free surface into the added row /H208498/H110031/H20850struc- ture with a 1/8 ML oxygen coverage. At −1.98 eV, theadded row /H208492/H110031/H20850structure with 1/2 ML oxygen becomes favorable. This structure corresponds to a denser packing ofthe oxygenated added rows. Interestingly, the 1/4 ML addedrow structure is not stable as an intermediate phase betweenthe 1/8 and 1/2 ML added rows, which suggests that therewill be a rather abrupt transition between a low coverage ofoxygen and the AR- /H208492/H110031/H20850reconstruction with 1/2 ML cov- erage. This could, for instance, become apparent in experi-ment in the form of AR- /H208492/H110031/H20850island formation. As the oxygen exposure further increases to /H9004 /H9262O=−1.46 eV, the thermodynamic preference changes from the AR- /H208492/H110031/H20850to the added strand c/H208496/H110032/H20850structure with a 2/3 ML oxygen coverage. The AS- c/H208496/H110032/H20850reconstruction is the stable sur- face phase up to the /H9004/H9262O=−1.24 eV bulk-oxide limit. While higher coverage /H208491M L /H20850structures were considered by us /H20849see Fig. 8/H20850, these do not appear as stable phases in the sta- bility diagram due to the much smaller oxygen binding en-ergies at these coverages. This results in a predicted maxi-mum oxygen coverage of 2/3 ML for the Cu /H20849110/H20850surface. Figure 15translates these /H9004 /H9262Oboundaries into a /H20849p,T/H20850phase diagram. In Fig. 16, the minimum free energy curves of the /H20849100/H20850 and /H20849110/H20850crystal faces are combined with the equivalent data for the /H20849111/H20850face /H20849taken from Ref. 57/H20850to predict the oxygen- dependent morphology of copper nanoparticles using Wulffconstruction. 79This procedure determines the nanoparticle shape by balancing within geometrical constraints the rela-tive fractions of the three contributing crystal faces so as tominimize the particles overall surface free energy. These cal-culations are performed using the programWINXMORPH. 80,81 The predicted nanoparticle shapes at four different oxygen chemical potentials are shown in Figs. 16/H20849b/H20850–16/H20849e/H20850. Under highly reducing conditions at /H9004/H9262O=−2.1 eV, all three crys- tal faces are free of oxygen, and therefore the shape is deter- FIG. 15. /H20849Color online /H20850Stability of oxygen-free, surface-oxide, and bulk-oxide phases in a /H20849p,T/H20850phase diagram for /H20849a/H20850the O/Cu /H20849100/H20850and /H20849b/H20850the O/Cu /H20849110/H20850system, based on the /H9004/H9262Ophase boundaries determined in Fig. 14. The vertical dotted line in panel /H20849a/H20850indicates the experimental /H20849Ref. 33/H20850phase boundary at 475 K above which the Cu vacancies of the MR- /H208492/H208812/H11003/H208812/H20850R45° structure become disordered, giving rise to a disordered vacancy structurethat appears as a c/H208492/H110032/H20850reconstruction.DUAN et al. PHYSICAL REVIEW B 81, 075430 /H208492010 /H20850 075430-12mined by the free energies of the clean Cu surfaces. With Cu/H20849111/H20850being considerably more stable than Cu /H20849110/H20850and Cu/H20849100/H20850, the nanoparticle shape /H20851Fig. 16/H20849b/H20850/H20852maximizes the exposure of the /H20849111/H20850face; only geometric constraints cause some of the next-favored /H20849100/H20850face to be formed. Figure 16/H20849c/H20850shows the nanoparticle morphology at the higher chemical potential of /H9004/H9262O=−1.8 eV at which both the /H20849100/H20850 and /H20849110/H20850faces are oxidized, while the /H20849111/H20850remains oxygen free. Oxidation reduces the free energies of the /H20849100/H20850and /H20849110/H20850surfaces, and at /H9004/H9262O=−1.8 eV, the /H20849110/H20850face is now slightly more stable than the /H20849100/H20850. This change in order manifests itself in the appearance of /H20849110/H20850edges in the nano- particle shape /H20851Fig. 16/H20849c/H20850/H20852.A t/H9004/H9262O=−1.5 eV, all three sur- faces are oxidized and their free energies are very similar. Asa result, all three crystal faces are prominently expressed inthe corresponding particle shape shown in Fig. 16/H20849d/H20850/H20852. Lastly, at /H9004 /H9262O=−1.3 eV, just prior to bulk oxidation, the free energies of the /H20849111/H20850and /H20849100/H20850surfaces are very nearly degenerate, whereas the /H20849110/H20850is slightly set apart. This re- sults in a slight reduction in the /H20849110/H20850surface area in pre- dicted nanoparticle shape /H20851Fig. 16/H20849e/H20850/H20852relative to the /H9004/H9262O= −1.5 eV particle in /H20851Fig. 16/H20849d/H20850/H20852. Overall, the particle shape in the course of oxidation tran- sitions from highly facetted conﬁguration /H20851Fig. 16/H20849b/H20850/H20852to an almost spherical shape /H20851Fig. 16/H20849e/H20850/H20852. This observation can be attributed to the fact that the calculated surface free energiesin the absence of oxygen are much more anisotropic thanthose at the oxygen-rich limit as shown in Fig. 16/H20849a/H20850/H20852. For the latter case, the free energy curves are almost degenerateand, moreover, very nearly parallel. The parallel slope indi-cates that all three faces, despite differing monolayer cover-ages, share very similar surface densities of oxygen atoms/H20849between 0.071 and 0.076 atoms /Å 2using our GGA unit cell dimensions /H20850. The near-degeneracy in the energy in turn suggests that surface free energies at this point are moredetermined by the mutual proximity of oxygen atoms than bythe orientation of the surface. V. SUMMARY AND CONCLUSIONS In summary, we have investigated the chemisorption of O on the Cu /H20849100/H20850and Cu /H20849110/H20850surfaces using density- functional theory for a wide range of atomic conﬁgurationsand coverages, including oxygen adsorbed on ideal and re-constructed surfaces. For the Cu /H20849100/H20850system, oxygen ad- sorption prefers the hollow site at low oxygen exposures. At higher exposures, the /H208492 /H208812/H11003/H208812/H20850R45° missing-row recon- structed structure becomes energetically favored. For theO/Cu /H20849110/H20850system, the /H20851001/H20852Cu-O added row reconstruc- tions are favored at low oxygen exposures, with the AR- /H208492 /H110031/H20850reconstruction prominent under these conditions. At higher oxygen exposures, a transition from the AR- /H208492/H110031/H20850 structure to the c/H208496/H110032/H20850added Cu 5O4-strand structure is pre- dicted to occur. On both crystal faces considered, oxygenatedsurface structures are stable under conditions prior to onsetof bulk oxide Cu 2O formation. Overall, the computational results are broadly consistent with experimental observation,which highlights the ability of DFT free energy calculationsto describe the thermodynamics of surface reconstruction forthis technologically important material. The combinedoxygen-dependent free energies of the /H20849100/H20850,/H20849110/H20850, and /H20849111/H20850surfaces are used to predict the shape of copper nano- particles in contact with an oxygen environment. ACKNOWLEDGMENTS This research is supported by the Australian Research Council /H20849Grant No. DP0770631 /H20850, the Australian Partnership for Advanced Computing /H20849APAC /H20850, and the Australian Centre for Advanced Computing and Communications /H20849ac3/H20850. 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PhysRevB.101.020501.pdf	PHYSICAL REVIEW B 101, 020501(R) (2020) Rapid Communications Editors’ Suggestion Self-doped Mott insulator for parent compounds of nickelate superconductors Guang-Ming Zhang,1,2,Yi-feng Yang,3,4,5,†and Fu-Chun Zhang6,‡ 1State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China 2Frontier Science Center for Quantum Information, Beijing 100084, China 3Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 4School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China 6Kavli Institute for Theoretical Sciences and CAS Center for Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China (Received 29 September 2019; published 7 January 2020) We propose the parent compound of the newly discovered superconducting nickelate Nd 1−xSrxNiO 2as a self-doped Mott insulator, in which the low-density Nd 5 dconduction electrons couple to localized Ni 3 dx2−y2 electrons to form Kondo spin singlets at low temperatures. This proposal is motivated with our analyses of the reported resistivity and Hall coefﬁcient data in the normal state, showing logarithmic temperaturedependence at low temperatures. In the strong Kondo coupling limit, we derive a generalized t-Jmodel with both Kondo singlets and nickel holons moving through the lattice of an otherwise nickel spin-1 /2 background. The antiferromagnetic long-range order is therefore suppressed as observed in experiments. With Sr doping, thenumber of holons on the nickel sites increases, giving rise to the superconductivity and a strange metal phaseanalogous to those in superconducting copper oxides. DOI: 10.1103/PhysRevB.101.020501 Introduction. The recent discovery of superconductivity in Nd0.8Sr0.2NiO 2[1] has stimulated intensive interest in under- standing its pairing mechanism; in particular, its similarityand difference compared to that in cuprate superconductors[2–8]. Despite tremendous efforts over the past 30 years, high- T csuperconductivity (SC) remains one of the most challenging topics in condensed matter physics [ 9–12]. The parent compounds of copper oxides may be described as aMott insulator with antiferromagnetic (AF) long-range order.Superconductivity arises when additional holes are introducedon the oxygen sites in the CuO 2planes upon chemical doping. These holes combine with the 3 dx2−y2spins of Cu ions to form the Zhang-Rice singlets moving through the square lattice ofCu ions by the exchange with their neighboring Cu spins,which leads to an effective two-dimensional t-Jmodel to describe the low-energy physics of the cuprates [ 13]. The AF order is destroyed rapidly by small hole doping, whileat optimal doping, the d-wave SC is established in bulk cuprates [ 14–16]. It has been a long-standing question if these “cuprate-Mott” conditions can be realized in other oxides.Extensive efforts have been made to investigate the nickeloxides both theoretically and experimentally [ 17–26]. Single crystal thin ﬁlms of inﬁnite-layer nickelates were lately synthesized using soft-chemistry topotactic reduction.Superconductivity was reported below 9–15 K in the hole-doped Nd 0.8Sr0.2NiO 2[1]. The nickelate superconductors have similar crystal structure as cuprates, and the monovalent gmzhang@tsinghua.edu.cn †yifeng@iphy.ac.cn ‡fuchun@ucas.ac.cnNi1+ions also possess the same 3 d9conﬁguration as Cu2+ ions. It is therefore thought to be the same as cuprates. However, the parent compound NdNiO 2displays metallic behavior at high temperatures with a resistivity upturn belowabout 70 K, and shows no sign of any magnetic long-rangeorder in the whole measured temperature range [ 27]. Similar results have also been found previously in LaNiO 2[28]. These experimental observations are in sharp contrast with the naiveexpectation of a Mott insulator with AF long-range order forthe parent compounds of nickelates. It is therefore importantto address what is the nature of the parent compounds and howthe AF long-range order is suppressed. Key experimental evidences. Figure 1presents the resistiv- ity and Hall data as functions of temperature for both parentcompounds NdNiO 2and LaNiO 2. Surprisingly, when the data were put on a linear-log scale, we ﬁnd that the resistivity ρup- turn well obeys a logarithmic temperature (ln T) dependence below about 40 K down to 4 K for NdNiO 2and below about 70 K down to 11 K for LaNiO 2. This is clear evidence of magnetic Kondo scattering [ 29,30]. This Kondo scenario is further supported by the Hall effect data in the both compounds. While the Hall coefﬁcient RHex- hibits nonmonotonic temperature dependence, very differentfrom that of the resistivity in the high temperature metallicregime, it shows the same ln Tdependence at low tempera- tures. In the Kondo systems, we have R H∝ρ, due to the inco- herent skew scattering associated with the localized magneticimpurity [ 31,32]. Thus both the resistivity and Hall coefﬁcient support the presence of the magnetic Kondo scattering in theparent compounds of nickelate superconductors. Moreover,at high temperatures where the skew scattering is negligibleand the normal Hall effect dominates, the magnitude of the 2469-9950/2020/101(2)/020501(5) 020501-1 ©2020 American Physical SocietyZHANG, YANG, AND ZHANG PHYSICAL REVIEW B 101, 020501(R) (2020)Resistivity (m Ω cm) Temperature (K)Hall coefficient (x10-3 cm3 C-1)Resistivity (m Ω cm) Temperature (K)Hall coefficient (x10-3 cm3 C-1) FIG. 1. Logarithmic temperature dependence of the resistiv- ity (red) and the Hall coefﬁcient (blue) at low temperatures for (a) NdNiO 2with the experimental data adopted from Ref. [ 1]; (b) LaNiO 2reproduced from Ref. [ 28]. The dashed lines are the ln T ﬁts. Hall coefﬁcient is found to be only about −4×10−3cm3C−1 for NdNiO 2and−3×10−3cm3C−1for LaNiO 2. Both are an order of magnitude higher than those of normal heavy fermionmetals. For example, we have R H≈−3.5×10−4cm3C−1 in all three Ce MIn5compounds ( M=Co, Rh, Ir) at high temperatures [ 33]. This indicates that there are only a few percent of electronlike carriers per unit cell in both NdNiO 2 and LaNiO 2. Therefore, the parent compounds of nickelates belong to a Kondo system with low-density charge carriers. Below we examine the Kondo scenario for NdNiO 2from the microscopic picture. The ﬁrst-principles band structurecalculations [ 34] show that the Nd 5 dorbitals in NdNiO 2are hybridized with the Ni 3 dorbitals, leading to small Fermi pockets of dominantly Nd 5 delectrons in the Brillouin zone. Nd 5 dconduction electrons have a low electron density of nc/lessmuch1 per Ni site, coupling to the localized Ni1+spin-1 /2 of 3dx2−y2orbital to form Kondo spin singlets (doublons) [35]. Here we have considered Ni 3 dx2−y2electrons to be strongly correlated with a large on-site Coulomb repulsion U to disfavor double occupation on the same sites. With this picture in mind, it is attempted to propose a Kondo Hamiltonian to describe the parent compounds ofnickelates. However, unlike the usual Kondo lattice model,the Ni 1+localized spins here are coupled mainly by superex- change interaction through the O 2 porbitals, same as in the cuprates, though the coupling on nickel sites is small. Thus thestarting point should actually be a background lattice of Ni 1+ localized spins with the nearest-neighbor AF Heisenberg su- perexchange coupling and additional local Kondo exchangeswith the itinerant 5 delectrons. FIG. 2. Illustration of the effective model on a two-dimensional square lattice of the NiO 2plane of NdNiO 2. Blue arrow represents Ni spin, which interacts with its neighboring spin antiferromageticallyby coupling J. Orange arrow denotes a Nd 5 delectron, which couples to Ni spin by the Kondo coupling K, to form a Kondo singlet (doublon). Red circle represents Ni 3 d 8conﬁguration, or a holon. t andt∗are the hopping integrals of doublon and holon, respectively. Not shown is the holon-doublon annihilation into two Ni spins. For the parent compound, we have correspondingly 1 −nc electrons per Ni site, or ncNs(Nsas the total number of Ni sites) empty nickel sites (holons) on the NiO 2plane. This introduces a strongly renormalized hopping term of holons.A schematic picture is displayed in Fig. 2. The presence of both the Kondo singlets /doublons and the holons can suppress very efﬁciently the AF long-range order and cause a phasetransition from the Mott insulating state to a metallic state. Ac-tually, as we will show below, an effective low-energy modelHamiltonian can be derived in terms of the doublons, holons,and localized spins, describing a self-doped Mott metallicstate even in the parent LnNiO 2(Ln=La, Nd) compounds. Upon further Sr hole doping, such a low-energy effectivemodel is expected to exhibit d-wave pairing instability as in the usual t-Jmodel. Effective model Hamiltonian. We consider Ni 3 d 8and Nd 5d0as the vacuum, and start with the localized 3 dx2−y2spins on the NiO 2plane that form a two-dimensional quantum Heisenberg model with nearest-neighbor AF superexchangeinteractions, H J=J/summationdisplay /angbracketleftij/angbracketrightSiSj. (1) This is similar to the cuprates, where the superexchange interaction is induced by the O 2 porbitals and the parent compound is a Mott insulating state with AF long-rangeorders. In nickelates, however, we have to further consider theKondo coupling with the Nd or La 5 dconduction electrons. This leads to the following Kondo lattice Hamiltonian: H K=−t/summationdisplay /angbracketleftij/angbracketright,σ(c† iσcjσ+H.c.)+K 2/summationdisplay jα;σσ/primeSα jc† jστα σσ/primecjσ/prime,(2) where tdescribes the effective hopping amplitude of the 5 d itinerant electrons projected on the square lattice sites of theNi 1+ions, and τα(α=x,y,z) are the spin-1 /2 Pauli matrices. We consider a single 5 dorbital for Nd for simplicity. For a low-carrier density system, the average number of conductionelectrons is very small, i.e., N −1 s/summationtext jσ/angbracketleftc† jσcjσ/angbracketright=nc/lessmuch1. 020501-2SELF-DOPED MOTT INSULATOR FOR PARENT … PHYSICAL REVIEW B 101, 020501(R) (2020) In the parent compound LnNiO 2(Ln=L a ,N d ) ,t h et o t a l electron density is 1 per unit cell, hence the total holon densityn h=nc. For Sr-doped compounds, we have δ=nh−nc>0. To describe the doping effect, we introduce the pseudofermionrepresentation for the spin-1 /2 local moments, S + j=f† j↑fj↓,S− j=f† j↓fj↑,Sz j=1 2(f† j↑fj↑−f† j↓fj↓), where fjσis a fermionic operator and denotes a spinon on site j. The holon hopping term between empty nickel sites is then given by Ht∗=−t∗/summationdisplay /angbracketleftij/angbracketright,σ(hif† iσfjσh† j+H.c.), (3) where h† jis the bosonic operator creating a holon on the jsite. In this representation, the Ni 3 dx2−y2electron operator is given bydjσ=h† jfjσwith a local constraint, h† jhj+/summationtext σf† jσfjσ= 1. This is just the slave-boson representation for the con-strained electrons without double occupancy. All together, the total model Hamiltonian for nickelates consists of three terms, H=H J+HK+Ht∗. (4) This model contains several key energy scales. While the electron hopping tmay be roughly estimated from band calculations, the holon hopping t∗is strongly renormalized due to the background AF correlations and thus contributelittle to the transport measurements in the parent compounds.The kinetic energy in the Hamiltonian is therefore relativelysmall due to the small number of charge carriers without Srdoping. The Heisenberg superexchange Jis also expected to be smaller (possibly the order of 10 meV) compared to that(about 100 meV) in cuprates due to the larger charge transferenergy between O 2 pand Ni 3 d x2−y2orbitals. Actually the Heisenberg exchange energy is further reduced in a paramag-netic background. For the Kondo temperature of the value of10 K or 1 meV, which is about one-tenth of the temperatureof resistivity minimum in both LaNiO 2and NdNiO 2, a Kondo coupling of roughly the order of 100 meV would be expectedfor a low-carrier density system with a small electron densityof states [ 36]. Thus for the parent compounds of nickelates, the Kondo coupling is a relatively large energy scale in theabove model Hamiltonian. From these analyses, one may anticipate that the ground state of the nickelate parent compounds may be to some extentcaptured by the large Klimit of the Hamiltonian. The Kondo singlets are then well established between the Ni 3 d x2−y2spins and the 5 dconduction electrons. To explore this possibility, we introduce the doublon operators for the on-site Kondo spinsinglet and triplets: b † j0=1√ 2(f† j↑c† j↓−f† j↓c† j↑); b† j1=f† j↑c† j↑,b† j2=1√ 2(f† j↑c† j↓+f† j↓c† j↑),b† j3=f† j↓c† j↓.The Kondo exchange term is then transformed to K 2/summationdisplay jα;σσ/primeSα jc† jστα σσ/primecjσ/prime=K 43/summationdisplay μ=1b† jμbjμ−3K 4/summationdisplay jb† j0bj0, (5) which describes the doublon formation on each site, namely,the Kondo singlet or triplet pair formed by each conductionelectron with the localized spinon. However, the Kondo tripletcosts a larger energy of Kand is therefore not favored. In addition, there can also be three-electron states with two con-duction electrons and the localized spinon on the same site,e † jσ=f† jσc† j↑c† j↓, and one-electron states with the unpaired spinon only, /tildewidefjσ=(1−nc j)fjσ. So these operators should be used with the constraint h† jhj+3/summationdisplay μ=0b† jμbjμ+/summationdisplay σ(/tildewidef† jσ/tildewidefjσ+e† jσejσ)=1,(6) for each site. These new operators do not commute in a simple way, so for simplicity we should avoid direct operation usingtheir commutation relations. In the large Klimit, following the method used in Refs. [ 37,38], a low-energy effective Hamiltonian can be derived by ﬁrst rewriting the hopping term H tin terms of the new operators bj0,bjμ(μ=1,2,3),ejσ, and/tildewidefjσ and then employing the canonical transformation, Heff= e−SHeS, to eliminate all high-energy terms containing bjμ (μ=1,2,3) and ejσwhile keeping only the on-site dou- blon ( bj0) and unpaired spinons ( /tildewidefjσ). In the inﬁnite- K limit, in particular, the low-energy effective model becomes asimple form H eff=−t∗/summationdisplay /angbracketleftij/angbracketright,σ(hi/tildewidef† iσ/tildewidefjσh† j+H.c.)+J/summationdisplay /angbracketleftij/angbracketright/tildewideSi·/tildewideSj −t 2/summationdisplay /angbracketleftij/angbracketright,σ(b† i0/tildewidefiσ/tildewidef† jσbj0+H.c.), (7) where the spin operators are expressed as /tildewideSα j= /summationtext σσ/prime/tildewidef† jσ1 2τα σσ/prime/tildewidefjσ/primewith a local constraint h† jhj+b† j0bj0+ /summationtext σ/tildewidef† jσ/tildewidefjσ=1. For a large but ﬁnite K, apart from some complicated interactions, an additional term should beincluded: H b=−3 4/parenleftbigg K+t2 K/parenrightbigg/summationdisplay jb† j0bj0+5t2 12K/summationdisplay /angbracketleftij/angbracketrightb† i0bi0b† j0bj0, (8) which could be used to describe the doublon condensation. Discussions . The above effective low-energy Hamiltonian is very similar to the usual t-Jmodel for cuprates [ 13], except that it includes two different types of charge carriers: theKondo singlets (doublons) and the holons on the Ni sites.Their presence can efﬁciently suppress the AF long-rangeorder and bring the phase transition from a Mott insulator to aself-doped Mott metallic state. The effective model thereforedescribes a self-doped Mott insulating state as the parentstate of nickelate superconductors, with possibly an enhancedeffective mass for the charge carriers. It also provides an in-teresting example of holon-doublon excitations for destroyingthe Mott insulator, although the doublons here are associated 020501-3ZHANG, YANG, AND ZHANG PHYSICAL REVIEW B 101, 020501(R) (2020) with the Kondo singlets rather than doubly occupied Ni 3dx2−y2orbitals. At high temperatures, the doublons become deconﬁned, causing incoherent Kondo scattering as observedin experiments. Furthermore, the Sr hole doping reduces the number of electron carriers and thus suppresses the contribution of dou-blons. At large doping, the effective model is then reducedto the usual t-Jmodel. In cuprates, the Cu 3 d x2−y2orbitals and the O 2 porbitals are strongly hybridized. The doped holes sit on the oxygen sites, forming the Zhang-Rice singletswith Cu 2+localized spins. By contrast, the holes in nickelates reside on the Ni ions, leading to a spin zero state or holondue to the much less overlap with the O 2 pband [ 3]. Sr doping hence introduces extra holes on the Ni sites, whichfurther drives the system away from the AF Mott insulatingphase, resembling that in the optimal or overdoped cuprates.However, even at 20% Sr doping, the electron carriers are stillpresent, as manifested by the negative Hall coefﬁcient at hightemperatures in Nd 0.8Sr0.2NiO 2[1]. Since the electron carrier density is reduced with hole doping, the smaller magnitudeofR Hin Nd 0.8Sr0.2NiO 2cannot be explained by a single carrier model but rather indicates a cancellation of electronand hole contributions. The latter grows gradually with Srdoping and eventually becomes dominant at low temperaturesin Nd 0.8Sr0.2NiO 2, causing the sign change of the Hall coefﬁ- cient below about 50 K. Experimentally, with 20% Sr doping in NdNiO 2, super- conductivity also emerges and has the highest transitiontemperature of about 15 K. Interestingly, when ﬁtted with apower-law temperature dependence, ρ∝T α, we notice for this particular sample that the electric resistivity exhibits anon-Fermi-liquid behavior in the normal state. Actually, anexcellent agreement could be obtained with α=1.13±0.02 over a wide range from slightly above the superconductingtransition temperature up to the room temperature. In fact,for all reported samples with high superconducting transitiontemperature, a good power-law ﬁt can always be obtained withα≈1.1–1.3. This is reminiscent of the optimal doped cupratesuperconductors and suggests a similar strange metal phase for the normal state of optimal doped nickelate superconduc-tors. Conclusion and outlook . Our proposed model bridges the Kondo lattice model for heavy fermions and the t-Jmodel for cuprates. However, it is different from both models in thesense that it combines some new physics that is not includedin either of them. Unlike the usual Kondo lattice system, theexchange interaction here between localized spins is producedby the superexchange coupling rather than the Ruderman-Kittel-Kasuya-Yosida coupling. Thus at low carrier density,the magnetic ground state is not ferromagnetic as one wouldexpect for the Kondo lattice. On the other hand, the nickelatesystem indeed exhibits incoherent Kondo scatterings as re-vealed in the transport properties at high temperatures. Unlikecuprates, the presence of strong Kondo coupling could lead toholon-doublon excitations even in the parent compound. Thisself-doping effect suppresses the AF long-range order andproduces the paramagnetic metallic ground state. The parentcompound of nickelate superconductor is therefore describedas a self-doped Mott state. This makes it somehow differentfrom the cuprates but resembles certain organic supercon-ductors under pressure, which reduces the on-site Coulombrepulsion Uand induces a transition from Mott insulator to gossamer superconductor with both holons and doublons [ 39]. Note added. A recent paper [ 40] considers the effect of Hund coupling and crystal ﬁeld splitting in the strongly hole-doped regime. Acknowledgments. The authors would like to acknowl- edge the discussions with Wei-Qiang Chen, Hong-MingWeng, and Yu Li. This work was supported by theNational Key Research and Development Program ofMOST of China (2016YFYA0300300, 2017YFA0302902,2017YFA0303103), the National Natural Science Foundationof China (11774401 and 11674278), the State Key Develop-ment Program for Basic Research of China (2014CB921203and 2015CB921303), and the Strategic Priority Research Pro-gram of CAS (Grant No. XDB28000000). [1] D. Li, K. Lee, B. Y . Wang, M. Osada, S. Crossley, H. R. Lee, Y . Cui, Y . Hikita, and H. Y . Hwang, Nature (London) 572,624 (2019 ). [2] A. S. Botana and M. R. Norman, arXiv:1908.10946 . [3] M. Jiang, M. Berciu, and G. Sawatzky, arXiv:1909.02557 . [4] H. Sakakibara, H. Usui, K. Suzuki, T. Kotani, H. Aoki, and K. Kuroki, arXiv:1909.00060 . [5] M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y . Tseng, X. Feng, M. Osada, E. Been, Y . Hikita, Y .-D. Chuang, Z. Hussain, K. J.Zhou, A. Nag, M. Garcia-Fernandez, M. Rossi, H. Y . Huang,D. J. Huang, Z. X. Shen, T. Schmitt, H. Y . Hwang, B. Moritz, J.Zaanen, T. 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PhysRevB.85.035421.pdf	PHYSICAL REVIEW B 85, 035421 (2012) Ultrafast electron dynamics in GeSi nanostructures S. A. Cavill, A. Potenza, and S. S. Dhesi Diamond Light Source, Chilton, Didcot, Oxfordshire OX11 0DE, United Kingdom (Received 3 October 2011; published 17 January 2012) The relaxation dynamics of photoexcited hot carriers in Ge xSi1−xislands grown on Si(111)-(7 ×7) have been studied with the spatial and temporal resolution of time-resolved two-photon photoemission electron microscopy.The relaxation dynamics of the excited electronic states within the Ge-rich Ge xSi1−xdots and the surrounding Si-rich wetting layer are found to vary signiﬁcantly below the conduction-band minimum. These differences areascribed to faster hot-carrier-diffusion rates for the islands compared to those for the wetting layer. DOI: 10.1103/PhysRevB.85.035421 PACS number(s): 68 .37.Xy, 78 .47.jd I. INTRODUCTION The explosive increase in the speed of nanoscale electronics has resulted in a huge effort to understand semiconductor carrier dynamics on ultrafast timescales. In this respect Ge /Si- based materials are potentially very promising for futuredevice applications due to their enhanced optoelectronic prop-erties, their ease of integration into existing microelectronictechnologies, 1and their ability to self-assemble into nanos- tructures. Quantum conﬁnement offers a pathway to enhancingthe optical performance of Si-based heterostructures, and GeSimaterials are particularly important in this respect becausethere exists a strong relation between morphology and theoptical and transport properties of the material. 2,3Many new quantum-device concepts have recently been reported in theliterature, 4,5but several critical issues remain unresolved. These include morphology, uniformity, and compositional control. The tetragonal deformation of the Ge epilayer, caused by the small lattice mismatch between Si and Ge, allows Ge to initially grow on Si in a layer-by-layer mode. Once a critical thickness is reached, the energy of the system can be lowered by the formation of three-dimensional (3D) islands, providing a simple and promising way to engineer quantum devices. However, the growth of defect-free coherent structures is essential in avoiding hot-carrier traps that affectrecombination pathways and therefore quantum efﬁciencies. On the other hand, compositional homogeneity between individual nanostructures is crucial since small differences can considerably affect the electronic structure. Since these two issues largely determine the optoelectronic properties, it is vital to control and manipulate them. An important step in this direction is to use spectroscopic probes to understandthe changing nature of the electronic structure as defects and composition evolve in the growth of nanostructures. Recent work on relaxation dynamics has concentrated on the use of optical probes, such as time-resolvedphotoluminescence, 6transient reﬂectivity,7or detection of the decay products,8to elucidate the various pathways and processes. Unfortunately, such probes do not directly measurethe hot-carrier electronic distribution and are mainly restrictedto bulk-material properties, rendering them of limited usefor surface nanostructures. Probes which combine ultrafastoptical pulses and surface sensitivity, such as second-harmonicgeneration (SHG), 9ﬁve-wave mixing,10and other nonlinear methods,11have been developed but do not directly accessthe electronic structure. On the other hand, time-resolved two-photon photoelectron spectroscopy (TR-2P-PES)12–16 combines ultrafast time-resolved spectroscopy with surface sensitivity. However, TR-2P-PES cannot resolve the spatiallyvarying electron dynamics of inhomogeneous systems. Apowerful method to overcome this issue is to combine TR-2P-PES with a photoemission electron microscope (PEEM)and use time-resolved two-photon photoemission electronmicroscopy (TR-2P-PEEM). 17–19This combination allows surface morphology and structure to be studied with a res-olution of <30 nm while simultaneously accessing hot-carrier dynamics on ultrafast time scales. Here, we combine TR-2P-PES with the spatial resolution of a PEEM to study ultrafast relaxation dynamics fromGe xSi1−xislands grown on the Si(111)-(7 ×7) surface. The results demonstrate that the relaxation dynamics of the wettinglayer (WL) are distinctly different compared to those of theGe-rich nanostructured islands. The results are interpreted asdifferences in terms of hot-carrier-diffusion efﬁciencies. II. EXPERIMENTAL METHODS The experiments were performed in a UHV system operat- ing at a base pressure of <5×10−10mbar. Atomically clean Si(111)-(7 ×7) surfaces were prepared by extensive degassing ofB-doped ( ρ=0.1/Omega1cm) Si at 600◦C, followed by repeated ﬂash annealing to 1200◦C. The surface crystallinity was veriﬁed by a sharp low-energy electron-diffraction (LEED)pattern characteristic of the (7 ×7) reconstruction. Low- energy electron microscopy (LEEM) of the surface at 41 eVrevealed large terraces ( ∼400 nm). After stabilizing the sample temperature at 560 ◦C, Ge was deposited at a rate of 0.63 ˚A/minute. The growth mode was conﬁrmed to be the Stranski-Krastanov type, i.e., the initial growth was layer-by-layer up to a critical thickness of 3–4 monolayers (MLs),followed by island nucleation. TR-2P-PEEM was performed using the second harmonic (SH) and third harmonic (TH) of a mode-locked Ti:sapphirelaser (repetition rate =83 MHz and pulse length <120 fs). A fraction of the SH (¯ hω=3.1 eV) was used to excite (pump) the ground state into unoccupied intermediate states whilst theTH (¯hω=4.6 eV) was used to photoemit (probe) electrons from both the ground state and excited intermediate states.The probe pulse was delayed with respect to the pump pulseusing a mechanical delay stage. The SH and TH beams werecollinearly focused to a 50 μm spot incident at 73 ◦with respect 035421-1 1098-0121/2012/85(3)/035421(6) ©2012 American Physical SocietyS. A. CA VILL, A. POTENZA, AND S. S. DHESI PHYSICAL REVIEW B 85, 035421 (2012) to the surface normal. The polarization of the pump and probe pulses could be set to an arbitrary state between the sandp polarizations. Photoelectrons emitted over a large angular range (30◦) were imaged using the PEEM. The energy resolution ofthe PEEM depends on the mode of operation. In imaging mode , the resolution is ∼300 meV with a spatial resolution of 30 nm. In order to perform spectroscopy, a series of imageswas collected for varying photoelectron kinetic energies fromwhich an energy-distribution curve (EDC) was generated,allowing spatially resolved TR-2P-PES. In the analyzer mode of operation, the energy-dispersive plane of the PEEM electronanalyzer was used to capture an entire EDC. In this mode thePEEM was used to spatially average over the full ﬁeld of view(FOV) and generated an EDC with a spectral resolution of∼200 meV . III. RESULTS Figure 1shows a LEEM image of a 10-ML ﬁlm of Ge grown on the Si(111)-(7 ×7) surface. At this coverage and growth temperature, large 3D islands are observed on top of the3–4-ML wetting layer. Initially, islands nucleate as truncatedtetrahedra with corners pointing in the /angbracketleft112/angbracketrightdirections 20due to the anisotropy of the growth rate in these directions. Thewetting layer has a (5 ×5) reconstruction whilst the tops of the islands have (7 ×7) reconstructions with Ge-Si intermixing dependent on the growth temperature and base area. 21As the islands evolve with increasing deposition, new facets areadded, and dislocations are introduced to relieve the strainenergy inside each island. Eventually, the largest structuresexhibit a complex-rounded shape with mass depletion atthe island centers. Similar to previous studies, individual3D structures at different stages of evolution coexist on thesurface. 20 FIG. 1. LEEM image of a 10-ML ﬁlm of Ge grown on a Si(111) surface. The ﬁeld of view is 20 μm. The image was recorded with an incident electron energy of 8 eV .FIG. 2. (Color online) The change in photoemitted intensity ( /Delta1I) as a function of /Delta1tandEK. TR-2P-PES of the nanostructured surface was performed in analyzer mode (spatially averaged). Surface-band bending,due to the pinning of the Fermi level at surface states, has itsown dynamical behavior under photoexcitation complicatingthe data. We therefore performed TR-2P-PES under ﬂatbandconditions. We ﬁrst measured an EDC using only probe pulsesand then again with the pump pulses at negative time delaysand different intensities to obtain ﬂatband conditions. Thisprocedure resulted in identical spectra at moderate positivedelays and at negative delays, showing that transient changes inthe surface-band bending are small and that ﬂatband conditionsexist over all time scales. However, a two-photon componentfrom the pump beam was also evident in all spectra measured.As this component was constant for a given pump-pulseintensity, it can be regarded as a background for spectrameasured at positive time delays. The dynamics of the excited-photoelectron distribution are shown in Fig. 2in which a color-scale image maps the change in photoemitted intensity ( /Delta1I) as a function of kinetic energy ( E K) and pump-probe delay ( /Delta1t). A spectrum recorded with /Delta1t=− 2 ps was subtracted from each spectrum to remove the background. In order to highlight importantspectral features in the photoelectron-intensity map, verticalline scans can be extracted to yield time-resolved two-photonphotoelectron spectra for several values of /Delta1tas shown in Fig. 3.A t/Delta1t=− 1 ps [Fig. 3(a)],/Delta1I is essentially zero, conﬁrming that ﬂatband conditions exist for all time scalesand that the two-photon component from the pump is constant.During overlap of the pump and probe pulses ( /Delta1t=0p s ) ,a n increase in spectral weight occurs over a broad range of kinetic FIG. 3. (Color online) TR-2P-PE spectra measured with a p- polarised pump and probe pulses for (a) /Delta1t=− 1p s ,( b ) /Delta1t=0p s , and (c) /Delta1t=1p s . 035421-2ULTRAFAST ELECTRON DYNAMICS IN GeSi ... PHYSICAL REVIEW B 85, 035421 (2012) FIG. 4. (Color online) Transient change in the photoemission intensity measured for (a) EK=1.8e V ,( b ) EK=1.4e V ,a n d (c)EK=0.2e V . energies with a maximum at EK∼0.22 eV (blue arrow) as the pump pulse excites electrons into the unoccupied states.The broad photoemission component shows a fast temporalresponse for energies above E K=1.6 eV with a temporal proﬁle essentially the same as the cross-correlation tracebetween pump and probe pulses. A similar effect was foundpreviously and ascribed to a coherent two-photon process. 16At /Delta1t=1 ps [Fig. 3(c)], two separate peaks can be resolved in the spectra. The low-energy peak with EK∼0.22 eV [blue arrow in Fig. 3(c)] represents transient changes of the temporally occupied surface states. In addition, a peak forms aroundE K∼0.65 eV [green dashed arrow in Fig. 3(c)]. The dynamics of the electrons excited into the unoccupied states can be followed using the photoemission intensityat ﬁxed kinetic energies. Figure 4shows horizontal line scans along the intensity map shown in Fig. 2at several ﬁxed kinetic energies. For a E K>1.6 eV [Fig. 4(a)], the dynamic response shows the same temporal proﬁle as thecross correlation between pump and probe pulses as discussedearlier. However, for 1 .2e V<E K<1.6 eV [Fig. 4(b)], the dynamic response shows a marked difference from thecross-correlation spectrum, implying photoemission from realshort-lived intermediate states. At a E K=0.2 eV [Fig. 4(c)], a different set of dynamics is evident, characterised by anincreasing photoemission intensity for /Delta1t > 500 fs. The temporal response of the photoemission intensity at theseintermediate-state energies therefore shows two time scalesinvolved in the population of the unoccupied states. Initially,the pump beam photoexcites carriers into the intermediatestates, evidenced by the fast initial rise in the TR-2P-PEintensity. However, a second process, which populates stateson much slower time scales, is also present. This can be seenin Fig. 4(c) by the slow increase in photocurrent up to a /Delta1t of 1.5 ps, which continues to rise up to 10 ps after the initialpump pulse. 22For time scales longer than the pump pulse duration, the increase in intensity must involve populationby the decay of carriers out of higher lying states. Thetime-dependent-photoemission intensity consequently reﬂectsa set of complex dynamics involving excitation, scattering, andrecombination processes. TR-2P-PEEM was then performed using the imaging mode (spatially resolving) of the PEEM analyser. Figure 5shows FIG. 5. (Color online) PEEM image of the Ge xSi1−xthin ﬁlm recorded using ¯ hω=4.6 eV light. The FOV is 10 μm. The red (and blue) boxes represents the area from which spectra in Fig. 6are derived for various photoelectron kinetic energies. a PEEM image of the surface, measured with TH light, used for the spatially resolved measurements. A series of imageswas recorded as a function of /Delta1tandE K. Figure 6shows the change in the TR-2P-PEEM intensity ( /Delta1I), determined by integrating over a particular region of interest in Fig. 5.I n Fig. 6the blue curves represent the dynamics of the wetting layer (blue box in Fig. 5) whilst the red curves show the dynamics of the Ge xSi1−xnanoislands (red boxes in Fig. 5). The photoemission intensity at /Delta1t=0 ps for each kinetic energy has been normalized to unity. The dynamics of the islands and wetting layer show consid- erable differences for EK<1.0 eV , i.e., for intermediate-state energies within the band gap. For the islands, the surface-statedynamics ( E K=0.2–0.6 eV) can be characterized by a large increase in intensity due to the pump beam followed by anadditional component, giving rise to a shoulder in the spectra at/Delta1t∼500 fs. These transiently occupied states decay rapidly to near-background intensity within 3 ps. The islands onlyshow a longer relaxation time at E K=0.2e V .T h ew e t t i n g layer, however, shows changes in intensity that resemble thoseshown in Fig. 4(c), recorded using the analyzer mode of the PEEM without spatial resolution. At kinetic energies below0.6 eV , corresponding to photoemission from states within theband gap, two time scales are evident in the carrier dynamics.After the fast initial rise in intensity due to the pump, a slowincrease in intensity over several picoseconds is followed bya slower decay of the photoemission intensity. In order tohighlight the differences in the relaxation time as a function ofE K, the data in Fig. 6are ﬁtted to /Delta1I(t)=[A/w/radicalbig (π/2)] exp {−2[(t−t0)/w]2} +B[1−exp(−t/τ 1)] exp( −t/τ 2), (1) where wandt0are the Gaussian width and center, respectively, andτ1andτ2are the time constants for indirect population 035421-3S. A. CA VILL, A. POTENZA, AND S. S. DHESI PHYSICAL REVIEW B 85, 035421 (2012) FIG. 6. (Color online) Transient changes in the photoemission intensity measured for the islands (red circles) and wetting layer (blue open circles) at kinetic energies of (a) 0.2 eV , (b) 0.4 eV , (c) 0.6 eV , (d) 0.8 eV , (e) 1.0 eV , (f) 1.2 eV , (g) 1.4 eV , (h) 1.6 eV , and (i) 1.8 eV . The solid lines represent ﬁts according to Eq. ( 1). (b) The dashed lines show an example of the ﬁtted Gaussian and exponential terms from Eq. ( 1). into and relaxation out of the electronic states. AandBare constants of proportionality. The ﬁrst term models the directphoton-pulse generation whilst the second term accounts forthe temporal evolution of the indirect population ( τ 1) and decay ( τ2) in the signal after the initial pulse as is shown implicitly in Fig. 6(b). The data from the islands in Fig. 6(i) are used to obtain parameters for the Gaussian generation termas it approximates accurately the cross correlation between thepump and probe pulses. The ﬁtted Gaussian parameters arethen used as constants for all other data. Table Ishows the ﬁtting parameter τ 2(relaxation time) as a function of kinetic energy for the dots and wetting layer. IV. DISCUSSION TheEKof the photoemitted electrons is given by EK=¯hω+ECBM−χ, (2) where ¯ hωis the probe-photon energy, ECBM is the electronic energy relative to the conduction-band minimum (CBM), and TABLE I. Decay parameter ( τ2) from ﬁts to the data in Fig. 6. Kinetic Energy (eV) τ2(ps) [Dots] τ2(ps) [WL] 0.2 3.0 ±0.3 N/A 0.4 0.7 ±0.1 11 ±3 0.6 0.85 ±0.08 0.88 ±0.01 0.8 0.4 ±0.2 0.66 ±0.01 1.0 0.24 ±0.08 0.32 ±0.01χis the electron afﬁnity. Here we assume that the conduction band offset in Ge /Si heterostructures, which depends on the composition and strain, is small and comparable to the energyresolution of the PEEM analyzer. In addition, the electronicstructure of Ge xSi1−xis known to depend strongly on x.23 However, since Ge and Si both have electron afﬁnities of ∼4 eV , the islands and wetting layer are also assumed to have an electron afﬁnity of 4 eV over the whole range ofx. Given the probe energy of ¯ hω=4.6 eV , photoemission from the CBM will result in a peak at a E K∼0.6e V . Therefore, the peak at a EK∼0.65 eV for /Delta1t=1p si n Fig. 3(c) is energetically consistent with photoemission from the CBM. The slight difference in the measured EKis due to the hot-electron distribution near the CBM combined withthe energy resolution of the PEEM electron analyzer. We notethat, given the large angular acceptance of the PEEM and thefact that the CBM can be transferred to near one of the (7 ×7) unit cells as discussed in Ref. 16, momentum requirements also imply that the peak at a E K∼0.65 eV can arise from states close to the CBM. However, it is important to establishif the photoemission process involves a direct transition intoa ﬁnal state ∼4.6 eV above the CBM, a phonon-assisted indirect transition, or a transition into an evanescent ﬁnalstate induced by the surface photoelectric effect. 24In order to resolve this issue, we measured the dependence of theE K∼0.65 eV peak on the probe polarization. Recent studies have shown that, in Si, photoemission from the CBM atthese photon energies can only be generated by p-polarised light. 24A strong polarization dependence would then indicate a surface photoelectric effect that can induce photoemission 035421-4ULTRAFAST ELECTRON DYNAMICS IN GeSi ... PHYSICAL REVIEW B 85, 035421 (2012) by exciting transiently populated electrons at the CBM into ﬁnal states that are evanescent in nature. The results show22 that the EK∼0.65 eV peak decreases in intensity as the probe polarization is rotated from ptos, implying that this feature arises from the surface photoelectric effect.24Photoemission into evanescent ﬁnal states would also explain the low intensityof the E K∼0.65 eV peak, compared to the peak related to emission from the surface states, even though the density ofstates at the CBM should be considerably larger. We next turn our attention to the spatially resolved data of Fig. 6. The electronic properties of both the (7 ×7) and (5 ×5) reconstructed Ge-covered surfaces are found to be qualitativelysimilar to that of the Si(111)-(7 ×7) surface. 25In the absence of detailed electronic-structure calculations for Ge xSi1−x,w e will interpret our results in terms of the known electronicstructure of the Si(111)-(7 ×7) surface. Early TR-2P-PE results from Si(111)-(7 ×7) surfaces found gap-state lifetimes >10 ps 26whilst more recent work demonstrated lower values of the surface-state lifetimes.9,16Our results are consistent with both, c.f., Table I, and indicate that the lifetime is strongly dependent on the intermediate-state energy. However, in Fig. 6 we observe clear differences in the dynamics of the islandsand the wetting layer for E K<0.8 eV , i.e., for intermediate energies that correspond to states close to and below the CBM. For the islands, the TR-2P-PE spectra shown in Fig. 6are dominated by the direct population of surface states by thepump pulse. Photoemission and inverse photoemission 25show that the Ge(7 ×7)-Si(111) surface bands are very similar in energy to those of the pure Si(111) surface. A direct transitionbetween surface bands has been observed on the Si(111)surface 27using a similar photon energy and was assigned to a transition from the S3t oU2 band. Accordingly, it is likely that the origin of the direct population by the pump inour data is the same as that of the Si(111) surface. A secondincrease in the photoemission intensity at delay times greaterthan the pump width and with subpicosecond decay times(τ 2), caused by the competition of decay into and out of these states, is also evident for 0 .4e V<E K<1.6 eV . Only at the lowest EKdoesτ2increase signiﬁcantly (see Table I). This is consistent with a distribution of states that allows carriersentering from the CBM to scatter progressively toward theFermi level via the metallic surface-state bands. In metallicsystems the carrier lifetime scales with the inverse square of theenergy difference of the carrier with the Fermi level 28so that the lifetimes increase as the hot carriers relax toward the Fermilevel. Population feeding of these states by carriers decayingfrom high-lying states seems to be rather small, suggesting aweak coupling between the bulk and the surface states. 9 The wetting layer also shows a fast increase in population on the time scale of the pump beam, but now for kinetic energiescorresponding to states around and below the CBM ( <0.8 eV), the signal is followed by a large and relatively slow rise inintensity on time scales longer than that of the pump. This canonly be due to considerable population feeding from carriersrelaxing out of higher lying states. Evident from the data (Fig. 6and Table I) is that the decay out of these states occurs on longer time scales than does that of the correspondingkinetic energies of the islands. The main reason that TR-2P-PEspectra show such dramatic differences between the islandsand the wetting layer is most likely related to the diffusion ofconduction electrons away from the surface into the bulk afterthe absorption of the (¯ hω=3 eV) pump pulse. Since the absorption depth at the pump energy (¯ hω=3e V ) is 15 nm (Ge) and 122 nm (Si), which is reduced to ∼5n m( G e ) and∼5 nm (Si) at the probe energy (¯ hω=4.6 eV), the hot- carrier-concentration gradient near the surface is much higherafter absorption of the pump for the Ge-rich islands than for theSi-rich wetting layer. In addition, the thickness of the wettinglayer ( ∼3 MLs) implies that the probe beam is predominantly sensitive to the Si substrate whilst for the Ge-rich islands itis only sensitive to the islands. Following a similar analysis, 26 the time for the average carrier to diffuse a distance equal tothe absorption depth of the probe is found to be greater for theSi-rich wetting layer than the Ge-rich islands due to the weakercarrier-concentration gradient. It therefore seems reasonablethat for the wetting layer, the rate at which carriers enter thesurface states via the conduction band will be higher due to theslower rate of diffusion away from the surface. The dynamicsare therefore governed not only by the decay channels at thebottom of the surface band but also by the electron ﬂow into it.Similar effects have also been reported by Tanaka et al. 29who found wavelength-dependence dynamics of the surface statesand CBM related to the wavelength dependence in the carriergradient. V. SUMMARY We have studied the excited-state electron dynamics in GexSi1−xislands using the spatial resolution of PEEM com- bined with two-photon photoemission (2P-PE) spectroscopy.We found that a pronounced peak in the spatially averaged2P-PE can be attributed to photoemission from transiently oc-cupied states near the conduction-band minimum. Therefore,time-resolved 2P-PE can probe hot-electron dynamics fromboth surface and bulk states simultaneously. The spatiallyresolved results reveal that the surface states of the Si-richwetting layer have a different set of dynamics than that ofthe Ge-rich Ge xSi1−xislands. This has been interpreted as a difference in the population feeding of the surface statesbetween the islands and the wetting layer due to the fasterdiffusion of conduction electrons away from the surface forthe former. Future studies involving excitation at differentwavelengths should help to further clarify the distinct roleof carrier-diffusion dynamics in localized structures. ACKNOWLEDGMENT We would like to thank Richard Mott for his technical assistance. 1I. Berbezier, A. Ronda, and A. Portavoce, J. Phys.: Condens. Matter 14, 8283 (2002). 2D. J. Paul, Adv. Mater. (Weinheim, Ger.) 11, 191 (1999).3L. Vescan, M. Goryll, T. Stoica, P. Gartner, K. Grimm, O. Chretien, E. Mateeva, C. Dieker, and B. Holl ¨ander, Appl. Phys. A: Mater. Sci. Process. 71, 423 (2000). 035421-5S. A. CA VILL, A. POTENZA, AND S. S. DHESI PHYSICAL REVIEW B 85, 035421 (2012) 4G. Dehlinger, L. Diehl, U. Gennser, H. Sigg, J. Faist, K. Ensslin, D. Gr ¨utzmacher, and E. M ¨uller, Science 290, 2277 (2000). 5Jifeng Liu, Mark Beals, Andrew Pomerene, Sarah Bernardis, Rong Sun, Jing Cheng, Lionel C. Kimerling, and Jurgen Michel, Nat. Photonics 2, 433 (2008). 6F. Pulizzi, A. J. Kent, A. Patan `e, L. Eaves, and M. Henini, Appl. Phys. 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PhysRevB.70.155419.pdf	Coordination-dependence of hyperﬁne interactions at impurities on fcc metal surfaces. II. Magnetic hyperﬁne ﬁeld V. Bellini,1,S. Cottenier,2,†M. Çakmak,2,3F. Manghi,1and M. Rots2 1INFM-National Research Centre on nanoStructures and bioSystems at Surfaces (S3) and Dipartimento di Fisica, Università di Modena e Reggio Emilia, Via Campi 213/A, I-41100 Modena, Italy 2Instituut voor Kern-en Stralingsfysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium 3Department of Physics, Gazi University, Teknikokullar, Tr-06500 Ankara, Turkey (Received 24 May 2004; published 29 October 2004 ) We present a comparison between ab initio calculations and a high-quality experimental data set (1990– 2002 )of magnetic hyperﬁne ﬁelds of Cd at different sites on Ni surfaces. The experimentally observed parabolic coordination number dependence of this hyperﬁne ﬁeld is veriﬁed as a general trend, but we dem-onstrate that individual cases can signiﬁcantly deviate from it. It is shown that the hyperﬁne ﬁelds of other 5 sp impurities at Ni surfaces have their own, typical coordination number dependence.Amicroscopic explanationfor the different dependencies is given in terms of the details of the s-DOS near the Fermi level. DOI: 10.1103/PhysRevB.70.155419 PACS number (s): 68.47.De, 68.35.Fx, 68.35.Dv, 75.70.Rf I. INTRODUCTION The class of experimental techniques which uses nuclear probe atoms, such as Mössbauer spectroscopy, PerturbedAn-gular Correlation spectroscopy (PAC), and Nuclear Magnetic Resonance (NMR ), plays an important role in the study of the electronic and magnetic properties of materials. 1–6What is most rewarding in these methods is their ability to probesimultaneously both charge symmetry related properties,such as the electric ﬁeld gradients (EFG ), as well as mag- netic properties, such as magnetic hyperﬁne ﬁelds (HFF). While for the EFG commonly accepted guidelines for siteidentiﬁcation on (100)and(111)metal surfaces were derived from experiments, and understood and extended by ab initio calculations, 7no universal trends exist for the HFF. Indeed, the HFF have been found to depend strongly on the type ofprobe atom (magnetic or nonmagnetic )as well as on the elements composing the magnetic host. 8 In this respect, parameter-free ab initioband structure cal- culations have assumed a critical role in providing models tothe experimental community, and assisting in the interpreta-tion of the experimental results. For instance, the generalbehavior of the HFF induced on sp- andd-impurities embed- ded in bulk magnetic materials, especially in Fe and Ni, hasbeen well characterized and the calculated values comparedfairly well with the experimental ones. 9–14Selected calcula- tions have been performed on more exotic systems. To namea few, the HFF on a Cd impurity atom at Fe/Co 15and Fe/Ag16,17interfaces have been studied, focusing on the re- lation between the HFF induced on the radioactive probeatom and the magnetic moment proﬁle across the interface.Similar studies have been performed on magnetic nanoclus-ters embedded in Ag 18and Cu19matrices. In thin layers of fccFe on a Cu substrate, HFF and EFG at the surface and interface have been calculated.20 If, for a few host materials, the dependence of HFF on the atomic number of the impurity atom has been studied indetail, less investigated is the variation in these properties forone speciﬁc probe atom in different environments derivedfrom the same host material. An example of such a study— motivated by the unique capability of nuclear probe tech-niques to follow the diffusion of single atoms—is providedby the experiments of Voigt et al., 5recently extended and completed by Potzger et al.21Those authors put a Cd atom at different low-index Ni-surfaces and at kinks and steps onthose surfaces, and measured the HFF at the Cd probe atoms.Overlooking the now fairly complete data set, Potzger et al. conclude that the HFF at Cd depends more or less paraboli-cally on the numberof Ni-atoms in the ﬁrst nearest neighbor shell (NN-shell )(black dots in Fig. 1 ), and not on the exact positionof those Ni-neighbors (dubbed “symmetry indepen- dence of the HFF” in Ref. 22 ). This is attributed to a gradual change in the local DOS, not further speciﬁed. Our maingoal in the present paper is to assess the validity of thisclaim, to elucidate the possible physical mechanism behind it(Sec. III A ), and to try to extend this “rule” to impurity at- oms other than Cd (Sec. II B ), all this by calculating the HFF’s at Cd in different Ni-environments by ab initiometh- ods. This is the second paper of a series of two. In the ﬁrst FIG. 1. Calculated coordination number (NN)dependence of the Cd hyperﬁne ﬁeld in Ni hosts. Triangles (up and down )refer to simulations with bulk cells, while circles and squares are relative tothe surface cells. Details of the cells are given in the text and inTable I.PHYSICAL REVIEW B 70, 155419 (2004 ) 1098-0121/2004/70 (15)/155419 (8)/$22.50 ©2004 The American Physical Society 70155419-1paper of the series (Ref. 7 )we have focused attention on the site identiﬁcation—by means of EFG calculations—of Cdatoms on low-index fcc metal surfaces [(100),(110), and (111)], and demonstrate how commonly accepted experimen- tal rules are in fact a manifestation of a simple coordinationdependence mechanism. This mechanism was then subse-quently generalized to other 5 spimpurities. Our work fol- lows pioneering cluster calculations by Lindgren et al., 23–26 who investigated HFF and EFG of Cd as an adatom or in a terrace site at (100)and(111)Ni surfaces, and a more sys- tematic work by Mavropoulos et al.27on the HFF for probe atoms belonging to the whole 4 spseries (Cu to Sr ), placed on Ni and Fe (100)surfaces. Independent from our work, some of the questions that will be discussed here were stud-ied very recently by Mavropoulos 28using a different method (full-potential KKR ), a different exchange-correlation func- tional (LDA ), and not considering atomic relaxations. II. METHOD AND DETAILS OF THE CALCULATIONS We have employed a state of the art ﬁrst-principles tech- niques, developed within the Density Functional Theory(DFT ). 29–31Most of the calculations have been performed using the full-potential augmented plane wave+local orbitalssAPW+lo dmethod 31–33as implemented in the WIEN2k package.34We have taken into account atomic relaxations since, as shown in the literature, they induce important ef-fects in close-packed matrices to the HFF’s. 12,13We have speed up such computationally expensive calculations by us-ing a combination of methods. First, the atomic positionswere relaxed using the pseudopotential plane wave VASP code.35,36The all-electron APW+lo code was then used in a second stage to further relax the atoms to their equilibriumpositions (with forces less than 1 mRy /a.u. ), and to calcu- late the HFF. We have simulated sites with different coordi-nation numbers using both bulk cells s23232 periodicity =32 atoms dand surface supercells with slabs with various thicknesses and in-plane periodicities. For a discussion on the convergence of the calculated quantities (EFG and HFF ) with the size of the cells, as well as on the other details andparameters of the calculations, i.e., mufﬁn tin radii, energycut-off, Brillouin zone sampling, etc. , we refer the reader to Sec. II of the ﬁrst paper of this series. 7 Before proceeding with the discussion of the results, it is important to assess the accuracy of our calculated HFF’s. Tothis end, let us look at an experimentally well-known case:Cd in bulk Ni, for which the experimental HFF at 4.2 K is−6.9 T. 37To mimic as good as possible the situation of an isolated impurity in bulk Ni, we have considered two differ-ent supercells: a 2 3232 cell where the Cd impurities are arranged in a simple cubic sublattice (31 Ni and 1 Cd in the primitive cell, Cd–Cd distance of 7.02 Å ),a n da4 3434 cell with Cd arranged in a face centered cubic sublattice (63 Ni and 1 Cd in the primitive cell, Cd–Cd distance of9.93 Å ). The Cd HFF’s attain values of −9.8 T and −9.3 T, respectively, for the 32-atom and 64-atom supercells, whenall Cd and Ni atoms are on ideal fcc positions. When weallow the Ni neighbors to move to new equilibrium positionsas a reaction on the presence of the Cd-impurity, theCd-HFF’s change to −12.0 T (32 atoms, ﬁrst shell of Ni neighbors relaxed )and −11.7 T (64-atoms, 3 shells of Ni neighbors relaxed ). The Cd–Cd distance in both our cells is large compared to the impurity-impurity distance used inprevious supercell studies for different magnetic hosts, forwhich good agreement with experimental HFF’s have beenfound: 5.71 Å for bcc-Fe 13,14and 6.10 Å for fcc-Co.14In a recent paper by Haas,14the Cd HFF’s in a small (16 atoms ) and in a very large (128 atoms )relaxed bulk Ni supercell were reported to be −12.3 and −9.2 T, respectively (in con- trast to our calculations, the supercell lattice constant wasallowed to vary in order to model better the long-range re-laxation ). These values are comparable to our results. We therefore conclude that most of the 5 T discrepancy withexperiment does not have its origin in the artiﬁcial (small ) interaction between the impurity atoms induced by the super-cell technique, and we must attribute the difference with ex-periment mainly to the approximations contained in the cho-sen exchange-correlation functional, i.e., GGA. The 5 Terror is similar to the 4.2 T overestimation of the HFF for Cdin Fe, and is of the same magnitude as was found for all 5thperiod impurities in Fe. 12,13As suggested by Haas,14this is an absolute error that becomes particularly relevant, in pro-portion, in the case of a Ni host, where, due to a lowermagnetization compared to Co and Fe, the impurities feelsmaller HFF’s. Finally, for the 32-atom cell we have done also a calcula- tion including SO-coupling (in contrast to all other calcula- tions reported in this paper ), from which we could obtain the orbital and dipolar contributions to the hyperﬁne ﬁeld (the Ni-moment was put in the [111]direction, as in nature ).They were smaller than 0.01 T, however, as could be expected fora closed shell atom as Cd sfKrg5s 24d10d, such that the so- called Fermi contact term38is the only contribution to the HFF. III. MAGNETIC HYPERFINE FIELDS Two main questions will be addressed in the next sec- tions. First, we want to verify and understand the experimen-tally proposed parabolic dependence of the Cd HFF on thenumber of Ni neighbors in the ﬁrst coordination shell (NN), and we examine whether and how the spatial arrangement ofthese neighbors inﬂuences the HFF.As a second problem, webroaden the scope to the entire 5 spseries, placing 5 spim- purity atoms only in NN=4 (adatom )and NN=8 (terrace ) coordinated sites at the Ni s100dsurface.This will allow us to generalize the behavior observed for Cd to other probe at- oms, and to propose qualitative explanations on the mecha-nisms ruling the observed HFF’s. A. Cd HFF’s on Ni surfaces In this section we focus on Cd probe atoms, placed in terrace and adatom positions at the three low-Miller-indexsurfaces of fccNi. This gives access to 6 differently coordi- nated sites, in addition to the fully coordinated substitutionalbulk site and to a more artiﬁcial bridge site with NN=2. Inorder to test the sensibility of the HFF on the details of theBELLINI et al. PHYSICAL REVIEW B 70, 155419 (2004 ) 155419-2cells, we performed also several calculations by varying the cell size and number of Ni layers in the slab. All the calcu-lated Cd HFF for the differently coordinated systems aresummarized in Table I and Fig. 1. Details on the cell aregiven in the format: (2D cell size, number of Ni layers ). The Cd-Cd distance through the vacuum spacer is given in paren-theses after the HFF values. The long hyphen “—” is used tolabel cases that were not calculated. The NN=2 coordinatedsite has been achieved by placing the Cd atom in a noncrys-tallographic adatom position on Ni s111d. An “ideal unre- laxed position” has therefore no meaning for Cd on this site, which is indicated by the label “n.p.” (“not possible” ). For the fully coordinated bulk site sNN=12 d, both a bulk and a slab calculation are reported, the latter with the Cd placed in the middle layer of the Ni slab. The experimental results ofPotzger 21are reported in the last column. We ﬁrst discuss the unrelaxed calculations, where all the atoms (including Cd )sit at their ideal fccposition. The calculated values predict for the Cd HFF a change in sign for mid-coordination and largepositive values for low coordination, in agreement with the experimental assignments. Changing the size of the slab, byadding Ni layers or by increasing the extension of the cells inthe surface plane, results in some scattered values which lay,except for NN=5 and to a lesser extent also for NN=4,within the aforementioned expected precision s±5 T d. For the NN=5 case, i.e., adatom position at the Ni s110dsurface, the HFF’s are found to attain large negative values for some of the considered cells. We will come back to these puzzlingresults later on. As already discussed before, lattice relaxations are ex- pected in such open systems and might induce importantchanges to the HFF’s. In fact it has been shown recently thattheir inclusion improves the agreement with the experimentaldata, in the case of 5 spand 6spimpurities in bcc bulk Fe. 12,13Due to the larger atomic volume of Cd with respect to Ni, an outward relaxation away from the surface is ex-pected for both terrace and adatom positions, for all low-index surfaces. Our calculations show Cd displacementsTABLE I. Hyperﬁne ﬁelds sTdof Cd in different Ni-environments. Experimental values are taken from Ref. 21. NN is the coordination number, and the “Type of cell” column supplies the cell dimensions, in unitsof the Ni lattice constant. The environment labeled as “bulk” refers to a single Cd inpurity in a 2 3232N i supercell, while in the “bulk-like” environments a number of Ni nearest neighbors have been removed fromthe 2 3232 supercell.All other calculations (where Miller indices are given )have been carried out with slab supercells (see the text for details ). Hyperﬁne ﬁelds (in T) #NN Type of cell Nonrelaxed Relaxed Exp Bulk 12 s23232d −9.8 −12.0 −6.9 (100)Bulk s232,7L d −10.0 −11.7 (111)Terrace 9 s ˛23˛2,7L d−7.4s12.18 Å d−13.7s10.49 Å d−6.6 (100)Terrace 8 s232,5L d−9.8s7.02 Å d−1.9s5.80 Å d−3.5 s232,7L d−8.3s7.02 Å d−6.7s5.82 Å d s˛23˛2,7L d−3.6s13.19 Å d−5.9s11.88 Å d Bulk-like (“random” ) s23232d −9.9 — Bulk-like (terrace ) s23232d −8.2 — bcc-bulk s23232d −8.9 — (110)Terrace 7 s23˛2,7L d−2.9s7.45 Å d1.1s5.66 Å d4.1 s232˛2,9L d−6.8s7.45 Å d — (110)Adatom 5 s23˛2,5L d−4.8s7.45 Å d11.9s6.91 Å d4.3 s23˛2,7L d−14.2 s7.45 Å d−6.5s6.93 Å d s232˛2,7L d−21.7 s7.45 Å d — (100)Adatom 4 s232,5L d−6.5s7.02 Å d−3.6s6.51 Å d7.3 s232,7L d 3.4s7.02 Å d38.7s6.63 Å d s˛23˛2,5L d — 15.9 s9.35 Å d s˛23˛2,7L d2.6s9.67 Å d22.9s9.28 Å d s˛23˛2,9L d — 31.9 s9.35 Å d s˛23˛2,11L d — 25.1 s9.35 Å d Bulk-like (“random” ) s23232d −5.6 — Bulk-like (adatom ) s23232d 8.0 — Bulk-like (free layer ) s23232d −4.2 — bccBulk-like (adatom ) s23232d −9.6 — (111)Adatom 3 s2˛232˛2,5L d24.9s8.11 Å d31.9s7.92 Å d16.0 (111)Adatom 2 s˛23˛2,7L d n.p. 33.4s8.26 Å d—COORDINATION DEPENDENCE OF . II. PHYSICAL REVIEW B 70, 155419 (2004 ) 155419-3from ideal fccpositions towards the vacuum, that range from 0.60 to 0.90 Å for the terrace atom and from 0.09 to 0.26 Åfor the adatom, depending on the surface orientation (Table II). Displacement for the terrace site are larger than for the adatom site: the Cd atom strives for a Cd uNi bond length of about 2.65 Å (an observation taken from our 32-atom bulk calculation ), that is somewhat larger than the ideal fcc Ni-Ni bond length of 2.48 Å. Starting from an adatom posi-tion, this can be realized with less displacement. Minor re-laxations appear also in the Ni atoms around the impurity.Asevident from Table I, the correction due to the relaxation ismostly within 7 T (except for NN=4 with corrections up to 35 T; see later ). The correction is negative for the highest coordintion numbers (NN=9 and 12 )and positive for all others. For a better comparison, the theoretical results for the unrelaxed and relaxed systems are plotted in Fig. 1 togetherwith the experimental results. When more than one valueexists in Table I, the ones relative to the cell with the largestvolume are selected, except for the more sensitive NN=4and NN=5 cases where the values closer to the experimentsare chosen (this will be justiﬁed below ). The chosen HFF’s are given in bold in Table I. As evident from Fig. 1, uponrelaxation of the atoms in the cell the barycenter of the HFFcurve moves towards more positive values for NN .9, in- ducing a sign change already for NN=7 [Cd on a Ni (110) terrace site ]. The largest changes are seen for NN=4 and NN=5, indicating once more a high sensitivity of these en-vironments. No clear overall improvement is found whenrelaxations are included, and the experimental data lay some-how between the relaxed and unrelaxed theoretical curves. In order to investigate the reason for the large variations seen for the NN=4 and NN=5 coordinated sites, it is fruitfulto look at the partial Density Of States (DOS )of Cd with s-symmetry (onlys-electrons contribute to the HFF ). In Fig. 2 the majority and minority s-DOS of a relaxed Cd adatom on Ni (100)is shown in a small energy window in the vicinity of the Fermi energy. The s-DOS shows several structures, among them a pronounced peak which for the majority chan-nel lays right at E Fwhile it remains above EFfor the minor- ity channel. Small variations in the details of the cell (num- ber of layers, 2D cell size, relaxation or not, ... )or in the computational method (LDA/GGA, APW+lo or KKR, ... ) will push this peak in the majority channel below or aboveE F. Since only the majority spin is involved, one has a net change of the sspin magnetic moment, and – because this is roughly proportional to the HFF – also a change of the HFFitself. This is illustrated in Fig. 2 by an LDA-calculation forexactly the same cell as used for GGA. The majority s-peak for LDAis at a slightly higher energy than for GGA, leadingto a reduced sspin moment of 1.05 310 −2mB(2.02310−2mB for GGA ). The corresponding HFF’s are 20.4 T (LDA )and 38.7 T (GGA ). As also in the LDA calculation this s-peak remains at EF, it can be expected that using LDAthroughout will not remove the sensitivity problem for NN=4 or 5,which is indeed what we observe. As a general rule, when-ever such a peak is observed so close to E F, an enhanced sensitivity of the HFF is foreseen, which adds to the innerprecision of the calculations. For NN=4 and NN=5, we doobserve such a peak in the s-DOS, for the other environ- ments this is not the case. This explains the instability andwide scattering of the HFF’s for those two coordination num-bers in Table I (note that this instability does not affect the calculations for the EFG, which is ruled by p- rather than s-electrons ). For impurity elements other than Cd, it might be that such sensitivity does not show up at all, or—if itdoes—it might do so for other values of the coordinationnumber, depending on the details of the s-DOS close to E F. The experimental values in Table I are up to 20 T below thecalculated values for NN ø4.This indicates that in nature the majority s-peak is above E F, while in our calculations it is below. We now turn to the experimental parabolic NN-counting rule from Fig. 1. Both our relaxed and unrelaxed calculationsshow roughly the same trend (Fig. 1 ), and agree with the experimental trend. But how absolutely does this “parabolicrule” hold? Is it really true—as concluded by Potzger et al. 21 —that knowledge of the NN coordination is enough to pre- dict the HFF on Cd in Ni-environments? In order to answerthis question, we exploit the advantage of ab initio calcula- tions that one is not restricted to environments that necessar-ily have to exist in nature. We can easily create artiﬁcialCd-in-Ni-environments with an arbitrary number of nearestneighbors. That allows us to test the NN-counting rule inmore situations than are experimentally accessible. The en-vironments we created are 2 3232 supercells for Cd in bulk Ni, with a given amount of nearest neighbor Ni atoms re-moved (such that vacancies remain ). This removal was done pair by pair, with the requirement that the remaining cell stillhas inversion symmetry (out of the several possibilities for every NN, we calculated only one ). This requirement makes such environments essentially different from the correspond-ing surface environments with the same NN, because theTABLE II. Cd perpendicular relaxation (in Å)from an ideal fcc crystallographic site for the three low-index Ni surfaces. Every- where Cd moves towards the vacuum. Site (100)( 110)( 111) Adatom 0.20 0.26 0.09 Terrace 0.60 0.90 0.84 FIG. 2. Partial s-DOS for relaxed Cd on Ni (100)calculated in the LDA and GGA approximations.BELLINI et al. PHYSICAL REVIEW B 70, 155419 (2004 ) 155419-4latter inevitably do not have inversion symmetry. In a ﬁrst series, we start from the relaxed 32-atom Cd-in-Ni supercellas calculated before (labeled as “semi-relaxed,” because after removal of the Ni atoms no further relaxation is done ),i na second series all atoms are at the ideal Ni bulk positions.Theresults of both series are almost identical, and are reported inFig. 1. Clearly, the same trend as in experiment and as in thesurface calculations is present in the artiﬁcial bulk calcula-tions. This proves that there is a basic truth in the NN-counting rule. On the other hand, the large difference be-tween the bulk and surface calculations—especially for thelow-coordination environments—is a clear sign that contri-butions from higher coordination shells are not negligible. If the requirement of inversion symmetry is removed, we can bridge the gap between these bulk-like cases and thesurface slabs by calculating bulk-like cells where the ﬁrstcoordination shell is exactly the same as on a speciﬁc surfacesite. We did this for NN=8 and NN=4 (Table I, the environ- ment labeled as “random” is the one with inversion symme-try). For NN=8, we could simulate in this way an environ- ment that has exactly the same NN-coordination as the (100) terrace site. If the NN-counting rule was absolutely valid, wewould ﬁnd exactly the same HFF in the random and terrace-like bulk case. The results are indeed quite close: −9.9 T and−8.2 T, which are values that are also not far from the slabcalculation s−8.3 T d. An even more daring test of the count- ing rule is to put Cd at a substitutional site of an unrelaxed hypothetical bccNi 2 3232 supercell (16 atoms ), with a lattice constant chosen such that the Ni-Ni distance (and hence also the unrelaxed Cd-Ni distance )is the same as in thefcccase. Even in this very different kind of environment, the calculated HFF of −8.9 T follows the simple countingrule. All this is different for NN=4. Apart from the randomenvironment s−5.6 T d, we tested a conﬁguration that is iden- tical to the fcc(100)adatom case s8.0 T d, a conﬁguration with all 4 Ni plus Cd in the same plane (a free layer, −4.2 T ), and in a bcccell a conﬁguration that is identical to a bcc (100)adatom s−9.6 T d.These 4 numbers prove that—even in fcc-based environments only—the exact spatial conﬁguration of the Ni neighbors in the ﬁrst coordination shell can beimportant, leading to differences of more than 13 T. It is notsurprising to ﬁnd this effect for NN=4 rather than NN=8,the former being identiﬁed before as a sensitive case. Fromthis analysis we conclude that although the NN-counting rulecertainly indicates a trend, there can be substantial deviationsfrom it for speciﬁc environments. In such sensitive environ-ments, the spatial arrangement of the neighbors is importantas well. There is probably some luck involved that for Cd inNi the behavior in nature is so smooth as experimentallyobserved, and there is no fundamental reason why the datacould not have been considerably more scattered around theparabolic trend. B. HFF’s of the 5 spseries As a last part of this study we now present a survey for the 5spimpurities from Cd to Ba in Ni-environments, to see if and how the NN-counting rule can be extended to otherimpurities. We use the strategy applied by Mavropoulos etal. 27for 4spimpurities on Fe and Ni surfaces, and very recently and independently from this work also for 5 spim- purities on Ni surfaces.28We take the bulk environment sNN=12 dand the terrace sNN=8 dand adatom sNN=4 den- vironments for the (100)surface, and calculate the HFF for the 9 elements from Cd to Ba in those environments.The cellsizes chosen for those calculations are the ones labeled as inTable I as s23232d,(232, 5 L ), and (232, 7 L ), respec- tively. Because we are looking for gross trends now and not for ﬁne details, relaxations were not included. The results forthe 5spHFF’s are displayed in Fig. 3. Exactly the same behavior as Mavropoulos et al.observed for 4 spand 5sp impurities is seen here, which mutually supports the validityof the very different computational methods that were used.In the bulk environment, the HFF starts at about −10 T forCd, strongly increases with increasing atomic number Z, and reaches a maximum near the middle of the series: .46 T for I. Then it decreases again, and at the end of the series turns back to values close to −10 T. This behavior is in agreementwith recent bulk calculations for relaxed 128-atom supercells(see Ref. 14, where also the comparison with experiment is discussed ). When the coordination number is reduced, the main peak of the HFF curve moves to heavier elements andan additional structure—that for NN=8 is more a broadshoulder than a peak—appears at the beginning of the series.For NN=4 two clear structures are evident, and the HFFincreases and decreases twice (with less intense variations than for the bulk )in the course of the 5 spseries. The microscopic origin of Fig. 3 can be understood by a slight extension of arguments given by Mavropoulos et al. 27 for 4spimpurities. Later on, we will derive from them (and test)generalized NN-counting rules for all 5 spimpurities in Ni environments. For a systematic explanation, let us goback to the origin of hyperﬁne ﬁelds in ferromagnets (see Ref. 9 for a detailed and instructive review ).As the hyperﬁne ﬁeld in our cases is dominated by the Fermi contact contri-bution, we have to care about the details of the bond betweenthe 5spstates and its environment (here Ni-3 d). It has been known for a long time 39,40that in the case of such an s-d bond the local s-DOS of the impurity shows a characteristic depression a few eV below the Fermi energy: the “antireso-nance dip” (AR). The position of the AR is mainly deter- mined by the host material sNid, and not by the impurity.The FIG. 3. Hyperﬁne ﬁelds of the entire 5 spseries (Cd!Ba)in bulk Ni (circle, NN=12 ), at a terrace position (squares, NN=8 )and at an adatom position (diamonds, NN=4 )on the Ni (100)surface.COORDINATION DEPENDENCE OF . II. PHYSICAL REVIEW B 70, 155419 (2004 ) 155419-5states below the AR are bonding states; the states above are antibonding.The up and down states are exchange split, suchthat at ﬁrst sight one expects an excess of s-up overs-down, resulting in a positive hyperﬁne ﬁeld. Due to a different s -dhybridization for up and down electrons, however, the number of s-up below AR will be diminished, while the number of s-down will be enhanced.AboveAR, the situation is opposite. 41The ﬁnal result is that the impurity s-moment (and hyperﬁne ﬁeld )will be negative in the beginning of the sp-series, where the effect of the bonding states is dominant (Fig. 4-a-1 ). In the second half of the series, also the anti- bonding states will get ﬁlled, and because they have to besqueezed between AR and the Fermi energy, they have todevelop a sharp peak in the DOS. The exchange splitting ofthis peak is responsible for the large positive HFF at the endof thesp-series (Fig. 4-a-2/3 ), which quickly drops to small and negative values again if also the down antibonding statesare below the Fermi energy (Fig. 4-a-4 ). Mavropoulos et al. have shown by group theoretical arguments that in the caseof reduced point group symmetry for the impurity (as on surfaces ), the antibonding part of the impurity s-DOS is split in two parts. This splitting is more pronounced if the impu-rity is in more non-bulk-like environments, i.e., it is morepronounced for NN=4 than for NN=8. Let us take the NN=4 case with a clear splitting. When going from Cd to Ba,we evolve through the different stages of Fig. 4 (b), which explains the double-peak structure of the HFF for NN=4 inFig. 3. An aspect of Fig. 3 that has not been discussed by Mavropoulos et al.is the physical origin of the coordination number dependence of the HFF for a particular element: whyis, e.g., for In the NN=4 HFF the larger one and the NN=12 the smaller one, while this is reversed for, e.g., Te? Thiswe will explain by the cartoon in Fig. 5. The upper part ofFig. 5 schematically shows the ﬁrst of the two antibondings-peaks of Fig. 4 (b), for any particular impurity. If one low- ers the coordination number of the impurity, the band-widthof these peaks will decrease—an obvious fact, which weclearly observe in our calculations. Figure 5 (a)shows the same situation for 3 typical band-widths: large sNN=12 d, medium sNN=8 d, and small sNN=4 d. The bottom part ofFig. 5 shows the sspin moment derived from Fig. 5 (a)as a function of energy (found by subtracting the integral of the down-peak from the integral of the up-peak, where the inte-grals are made up to the energy under consideration ). Every- thing now depends on where the Fermi energy lies in Fig. 5.If it falls in the region indicated by Cd-In-Sn, Fig. 5 (a)cor- responds to Fig. 4-b-1. At the corresponding energy in Fig.5(b), thes-moment (HFF)for the small band-width sNN =4dis larger than for the medium band-width sNN=8 d, which in turn is larger than for the large band-width sNN =12d. If the Fermi energy falls in the region indicated by Sb-Te, Fig. 5 (a)corresponds to Fig. 4-b-2 and the sequence ofs-moments is reversed.After Te, this story repeats for the second antibonding peak (forI, the Fermi energy will be at the position marked by the vertical arrow, but of course inthe second series of peaks ), but will be increasingly less clear due to the presence of the 6 sstates that start to manifest themselves around the Fermi energy. That is the reason forthe more chaotic evolution for Cs and Ba. Of course, Fig. 5is a cartoon only, and its conclusions should not be taken tooliterally: the real DOS are not Gaussians as used in the car-toon, and therefore the details of the hyperﬁne ﬁeld evolutionmight be different. Nevertheless, it captures the basic mecha-nism. Summarizing, we conclude that the physical mecha- FIG. 4. Cartoons for the majority and minority partial s-DOS of (a)a5spimpurity in bulk Ni and (b)a5spimpurity at a Ni surface (low coordination, e.g., NN=4 ). The vertical line indicates the Fermi energy, the name of the elements indicates for which elementa particular picture is representative.This picture is inspired by Ref.9. FIG. 5. (a)The ﬁrst of the two antibonding s-peaks for majority and minority spin. (b)Thes-moment derived from (a)by subtracting—at a particular energy—the integral of the minoritys-DOS up to that energy from the integral of the majority s-DOS up to that energy. In both (a)and(b)3 typical cases are drawn: high coordination = broad band width (thin full line ), medium coordina- tion = medium band width (thick full line )and low coordination = small band width (dashed line ). The horizontal arrows indicate the region where the Fermi energy falls for the indicated elements (the ﬁrst half of the 5 sp-series ). This story can be repeated with the second of the two antibonding peaks, starting from Ifor which the Fermi energy falls at the place indicated by the vertical arrow.BELLINI et al. PHYSICAL REVIEW B 70, 155419 (2004 ) 155419-6nism behind Fig. 3 can be understood from a combination of three basic features: (i)the double peak structure of the an- tibonding peaks, (ii)the decrease in the band-width—and hence the increase of the peak height—upon reduction of thecoordination number, and (iii)the position of the Fermi en- ergy with respect to the peaks. We now take Fig. 3 as a source of inspiration to extend the parabolic NN-counting rule proposed by Potzger et al. (Ref. 21 )to 5spimpurities other than Cd. It can be seen from Fig. 3 that the Cd-HFF for bulk and NN=8 is almost thesame and negative, while the value for NN=4 is small andpositive: this is the parabolic behavior seen in experiment. Inthe same way, we can then deduce that for In and Sn asimpurities, the HFF should monotonically rise from NN 512 to NN=4 (it is interesting to note that for the experimentally “easily” accessible Mössbauer probe 119Sn, Fig. 3 suggests a linear behavior ). Between Sn and Sb all lines cross, such that for Sb to I the HFF monotonically decreases from NN=12 toNN=4. For Xe to Ba, there is nonmonotonic behavior in-stead. We have checked this deduction by calculating theartiﬁcial bulk-like (nonrelaxed )environments as discussed before, but now for Te and Ba as impurities (both are taken as a representative for the region of monotonic decrease andthe nonmonotonic region, just as Cd is a representative of theregion of monotonic increase ). In these bulk-like cells, we can more easily create environments with NN different from4, 8, and 12. The results are given in Fig. 6. The generaltrends shown in Fig. 4 are the same as the ones which couldbe inferred from the surface calculations of Fig. 3: a mono- tonic (parabolic )increase for Cd, a monotonic decrease for Te, and nonmonotonic behavior for Ba. In light of theseresults we therefore conclude that each of the 5 spimpurities in Ni has its own typical coordination number counting rule. IV. CONCLUSIONS We have undertaken a comparison between ab initio cal- culations and a data set—experimentally collected during thepast 15 years—of magnetic hyperﬁne ﬁelds of Cd at mag-netic metallic fcc surfaces, i.e., Ni. The experimentally sug-gested parabolic-like coordination number dependence forthe HFF of Cd at Ni surfaces is conﬁrmed as being a reliabletrend, but we warn that it is just a trend and not a rigorousrule: sensitive environments exist, for which the spatial ar-rangement of the Ni neighbors considerably inﬂuences theHFF. We have explained in detail the physical mechanismbehind the HFF for all 5 spimpurities at Ni surfaces, by combining knowledge from the literature and new insight. Inparticular we have generalized the parabolic NN-countingrule for Cd to other 5 spimpurities, showing that each impu- rity has its own typical rule, and explaining why this is so.Together with the results on the EFG presented in the ﬁrstpaper of this series, 7we hope to have demonstrated that ab initiocalculations can greatly enhance the physical insight in an experimentally complex problem. ACKNOWLEDGMENTS One of the authors (V.B.)acknowledges a fruitful discus- sion with Dr. K. Potzger. Part of the calculations were per-formed on computer facilities granted by an INFM projectIniziativa Trasversale Calcolo Parallelo at the CINECA su- percomputing center.Another part of the computational workwas performed on computers in Leuven, in the frame ofprojects G.0239.03 of the Fonds voor Wetenschappelijk Onderzoek—Vlaanderen (FWO ), the ConcertedAction of the KULeuven (GOA/2004/02 )and the Inter-University Attrac- tion Pole (IUAP P5/1 ). The authors are indebted to L. Ver- wilst and J. Knudts for their invaluable technical assistanceconcerning the pc-cluster in Leuven. M.C. thanks the Onder- zoeksfonds of the KULeuven for ﬁnancial support (F/02/ 010). Electronic address: Bellini.Valerio@unimore.it †Electronic address: Stefaan.Cottenier@fys.kuleuven.ac.be 1R. Vianden, in Nuclear PhysicsApplication on Materials Science , edited by E. Recknagel and J. C. Soares (Kluwer Academic, New York, 1988 ), p. 239. 2G. Schatz, in Ref. 1, p. 297. 3G. Krausch, R. Fink, K. Jacobs, U. Kohl, J. 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PhysRevB.48.161.pdf	PHYSICAL REVIEW B VOLUME 48,NUMBER 1 1JULY1993-I Hopping transport inthethree-site modelinthepresence ofelectric andmagnetic fields: Rateequation andtransport phenomena forsmallpolarons H.Bottger Institutfu'rTheoretische Physik, Technische Universitat "OttovonGuericke" Magdeburg, Universitatsplatz 2,PSF4120,0-3010Magdeburg, Germany V.V.Bryksin A.F.Io+ePhysico Techn-ical Institute, St.Petersburg, Russia F.Schulz InstitutfiirTheoretische Physik, Technische Universitat "OttovonGuericke" Magdeburg, Universitatsplatz 2,PSF4120,0-3010Magdeburg, Germany (Received 1February 1993) Arateequation fortheelectronic densityatthesitesinthepresenceofelectricandmagnetic fieldshas beenformulated foranystrengthofcoupling withphonons, wherethree-site transitions areincluded. A classification oftheirreducible diagrams isgiven,sothatthetransition probabilities contain also phonon-free processes. Ithasbeenshownthatthesignofthemagnetoresistance changes forjumpsfar fromtheFermilevel,ifonegoesoverfromelectrons toholes.Generally speaking, itisnotjustified to treattheHalleffectandthemagnetoresistance indisordered systems independently withinthepresented formalism. Aconcrete calculation ofthecurrent inordered systems hasbeencarriedoutinthesmall- polaron modelforanystrengthofthefieldsEandH. I.INTRODUCTION Numerous papersboththeoretical andexperimental aredevotedtothestudyofhopping transport inanexter- nalmagnetic field.Thesepaperscanbedividedintotwo groups. Thefirstonedealswiththeinvestigation ofthe Halleffect,i.e.,withphenomena antisymmetric (trans- verse)withrespecttoH.InRef.Iathree-site hopping modelwasformulated, inwhichanonvanishing Hall effectresultsfromquantum-mechanical interferences. In Ref.1ithasbeenshownthattwo-phonon processes pro- ducetheHalleffectinthecaseofweakcoupling with phonons. Thecalculation oftheHalleffectinthesmall- polaron modelonhexagonal crystals wascarriedoutin Ref.2.InRef.3arateequation hasbeenobtained for thedensitymatrixinthesiterepresentation inthepres- enceofmagnetic andelectric fields.Fromthisequation itfollowsthattheHalleffectinthehopping regimein disordered systemscanbedescribed byanequivalent net- workofrandom resistors withextrinsic currents, induced bythemagnetic field.Themethodoftheeffective medi- umhasbeendeveloped forthecalculation oftheHall effectinthethree-site hopping model. Inthesecondgroupofpapersthepointofinterest is focused onthesymmetric withrespecttoHphenomena, i.e.,longitudinal transition phenomena (magnetoresis- tance).Examinations havebeencarriedoutinthemodel ofquantum interference inthecaseofmultisite jumps. Theresultssuggested thepossible occurrence ofanega- tivemagnetoresistance. InRef.6alineardependence of themagnetoresistance onthemagnetic fieldHwasob- tained;inRefs.7and8aquadratic one.Itisimportanttonotethatinthelimiting caseofsufficiently smallHin anisotropic mediathedependence hastobequadratic. Thisfollows frommostgeneral phenomenological con- siderations. Athree-site modelforthemagnetoresistance inthecaseofweakcoupling withphonons wasstudied in Ref.9.Thereitwasshownthatthemagnetoresistance in contrast totheHalleffectarisesfromone-phonon pro- cesses. Therehavealsobeenalargenumberofexperimental investigations ofthequantum interference inmagnetic fieldsinthree-andtwo-dimensional systems.''The studyofthemagnetoresistance inthepresence ofstrong electric fieldswasstartedaswell.' Ontheothersideitfollowsfromtherateequation in thepresenceofamagnetic fieldthattransverse andlon- gitudinal effectsarestrongly coupled. Therefore they mustnotbeconsidered independently ingeneral. How- ever,inthepapers'onlytheHalleffectinthelinearap- proximation withrespecttoHhasbeenstudied. Inthis caseonecanalwaystreatthelimitofsuSciently small magnetic fields,ignoring theinhuence ofHonlongitudi- naleffects.Inthispaperaconsistent theoryofthetran- sitionphenomena inthehopping regimeinthepresence ofelectric andmagnetic fieldsofanystrength hasbeen developed. Three-site jumpsaretakenintoaccount in thistheory. Theobtained rateequation forthedensity matrix wassolvedforthecaseofhexagonal crystals. Thisgivesapossibility tocalculate concrete expressions forthetransverse andlongitudinal currents inthemodel ofsmallpolarons. Asaresultoneobserves theeffectofa p-nanomaly forthemagnetoresistance —thechangeof itssignbythetransition fromholematerials toelectron 0163-1829/93/48(1)/161(11)/$06. 00 161 1993TheAmerican Physical Society 162 H.BOTTGER, V.V.BRYKSIN, ANDF.SCHULZ 48 ones.Thiseffecthasthesamenatureasthewell-known p-nanomalyoftheHalleffectinthethree-site model.' Thep-nanomaly ofthemagnetoresistance canbeob- servedexperimentally bystudying thedependence ofthe longitudinal effectsfortransitions intheimpurity region onthedegreeofcompensation inthecaseofhopping at levelsfarfromtheFermienergy,i.e.,intheregionofan activated dependence oftheelectronic conductivity on thetemperature andnotintheregion,whereMott'slaw holds(Rhopping). II.STARTING HANIILTONIAN ANDRATEEQUATION AFrohlich-like Hamiltonian isusedastheelectron- phonon Hamiltonian. Afterthesmall-polaron canonical transformation' inthepresence ofelectric andmagnetic fieldsofanystrength oneobtainsthefollowing expres- sion:H=gVata+gfico(btqb+i) J~(H)=J expi.[RXR,].eH 2Ac(2) Theinhuence ofthemagnetic fieldistakenintoaccount onlyinthephasefactoroftheresonance integralJ~ Thismeansthateffectswhichleadtoadeformation of thewavefunction inthemagnetic fieldarenotincluded, e.g.,Zeeman effect(dependence ofeonH)andso on.'4.in(1)isthemultiphonon operator+gJ,(H)4, aa mmmm' HereV=c.—eE.R,c—energyoftheelectron at sitemwiththeradiusvectorR,coq—frequency ofa phonon withmomentum q,and 4~=expg[(e—e)yqbqHc].. 2Nq(3) whereyisthedimensionless electron-phonon coupling constant, whichactually candependonmindisordered sys- tems.Wewilldropthisdependence inthefollowing. Nin(3)isthetotalnumberofatomsintheconsidered system. Therateequation forthediagonal components ofthedensitymatrixp(t)withrespecttothesiteindicesinthepres- enceofEandHfields,wherethree-site jumpsaretakenintoaccount, hasbeenobtained inRef.3(seealsoRef.20).In thecaseofsufficiently slowlyvarying intimefieldsEandH(Markovian limit)theequation isgivenby dPm=g[p(1—p)W—p(1—p)W] m& +g[p(1—p)(1—p)W'~'+p(1—p)(1—p)W~'i m~,m2 —(1—p)ppW'"'—(1—p)ppW'"']. (4) Here8'isthetwo-site hopping probability between thesitesm&andm,whichisdescribed bydiagrams withtwo 1 points: W=J(H)J(H)Idtem™P t+ whereV=V—V=e—8—eER,P=IlkT,R=R—R,and 1 1 1 y lcos(coqt)P (t)=expg. [cosqR+cosqR2Nsinhgm/2)~&~4 q q—cosqR—cosqR] Thetwo-site probabilities donotdependonHandaresubjecttothecondition ofdetailed equilibrium: )+mm 1 1 Jistheresonance integral, renormalized bypolaron effects(polaronic narrowing oftheelectronic band), J~(H)=J (H)e ly,l'Sr(m', m)=g (1—cosq.R)coth2N Theelectronic three-site hopping probabilities8"" aredescribed bytwocomplex conjugate diagrams, whichare 12 represented inFigs.1(a)andl(b).Thecorresponding analytical expression reads 48 HOPPING TRANSPORT INTHETHREE-SITE MODEL INTHE... 163 12'3 '1J(H)J(H)J(H)fdt,fdtzexp—(Vt,+Vt2)P t,+ iA 211 2 (9) ImJ(H)J(H)J(H)fdtifdt2exp—(Vt,+Vt2)P t,+t2+ ihP iA m2m1,m1m 22mrn2,m2m1 1 Replacing in(9)m,~mzandt,~t2andtakingintoaccount theidentityJ.(H)=J~(—H),weobtainasymmetry relationship forthethree-site "outgoing" probability: W"(H)=W" (—H). (10) In(4)weneedonlythecombinationW"+W".Becauseofthisweareinterested inthesymmetrical termof 12 21W'" withrespecttoH,i.e.,theproductoftherealpartofthreeJwiththeimaginary partofthetimeintegrals in 12 (9).InRef.3theprobabilities Whavebeendropped, because therewereconsidered onlyantisymmetric termsinHof thethree-site probabilities. Thecorresponding holeprobabilities 8"'"' canbeobtained fromtheelectron onesbyreplacing 12J(H)~—J.(H)andV~—V(Refs.3and20)oralternatively H~—H,multiplying by(—1)andsetting Wenowturntothedescription ofthe"incoming" electronic probabilities. Theholeprobabilities W canbeob- 12 tainedfromtheelectron onesasexplained above.Theincoming electronic probabilities aregivenbyapairofcomplex- conjugate diagrams, whicharerepresented inFigs.2(a)and2(b): W"= 3ImJ(H)J(H)J(H)fdtifdt2exp—(Vt,+V~t2)P~ t,+ iA iA 3ImJ(H)J(H)J(H)fdt,fdt2exp—(Vt,+Vt2)P t,+t2+ ih'P iAp 2'~2~1 22m1m'™22 Comparing (11)with(9)wegetthefollowing symmetry relationship forthesymmetrical partswithrespecttoH: t(e)s (e)sWmmm Wmm m12 21(12) Theantisymrnetric partoftheprobability8"' with 12 respecttothemagnetic fieldcanbeobtained from(11)by extending theintegration overt2from—~to+~and multiplying by—,'.Afterthissteponeobserves thesyrn- metryrelationship: (e)a (e)a~mmm~mmm2 12(13) Equation (12)provides thepossibility toexclude the outgoing probabilities W' fromtherateequation, 12 byexpressing thembyW'.Afterwards itcanreadi- 12~m~ 8"(H)=W"(—H)e~™(14) Thisidentity forthethree-site probabilities istheanalo- gueoftherelationofdetailed balance (7)forthetwo-site probabilities. Fortheholeprobabilities wegetinaccor- dancewiththerulesfortransition fromelectrons toholes theexpressionlybeseenwiththeaidof(14),thattheright-hand sideof (4),summed overallsitesm,yieldszero.Thissumrule expresses thelawoftheconservation ofthenumberof particles intime,g(dpIdt)=0. Wenowreplace in(11)theindicesm~m,.Wethen shiftthevariable ofintegration t,by t&t2—ifiP. —— Afterthesemanipulations weobserve thefollowing syrn- metryrelationship: H.BOTTGER, V.V.BRYKSIN, ANDF.SCHULZ 48 (b) FIG.1.(a)and(b)Three-site "outgoing" probabilities8"".Thewavylinesarephonon lines.Eachofthemcor- 12 responds toafactorP.(t,t2+i—hP/2),wherem&,m3 12™34 (m2,nz4)aretheindicesoftheelectron linesrunning into (awayfrom)theconnected pairofpoints, respectively. Thetimet,liesonthecontour before t2.Afactor +(i/fi)J~(H)exp[(i/iI) V~t)corresponds toeachpointon thecontour, wherem(m')istheindexoftheelectronic line running into(awayfrom)thepointand+(—)holdsforpoints onthelower(upper)branchofthecontour(Ref.20). (15) f=(e +1) (16) theright-hand sideoftherateEq.(4)vanishes. Heref isthebarometric distribution oftheelectrons atthesites inanexternal electric field.Intheabsenceoftheelectric field,i.e.,ifV=s,thefunctionfdescribes thether- modynamical equilibrium distribution oftheelectrons at mi mi (b) FIG.2.(a)and(b)Diagrams, whichdescribe thethree-site "incoming" probabilities8"' 12Therelations ofdetailed balance (7),(14},and(15)lead tothefactthatbypassingoverfromptotheequilibri- umdistribution functionthesitesmwiththeenergy c.andthechemical potential p.Inthepresenceofanelectric fieldsuchacurrentless statecorresponds tothe"fieldeffect"—aninhomogene- ousdistribution oftheelectrons intheareanearthe boundaries. Wediscuss nowtheveryimportant question concern- ingtheconvergence ofthetimeintegrals intheexpres- sionsforthetwo-(5)andthree-(11)siteprobabilities in thelimiting caset~~.AsitwasshowninRef.21,the argument oftheexponential in(6}approaches zeroas (b,co.t)",where b,toisthewidthofdispersion ofthe phonon bandandddenotes thedimension ofthespace. Becauseofthistheintegral overtin(5)diverges atthe upperlimitinordered systems without anelectric field, i.e.,ifallVareequaltozero.Insmall-polaron theory suchadivergence willbecompensated forbysubtracting aonefromP,i.e.,byreplacing PwithP—1inalldia- grams3,is—2iSuchanadding andsubtracting ofone leadstotheoccurrence ofmodified diagrams, inwhich phonon links(wavylines)areremoved inallpossible combinations, i.e.,theP'sarereplaced byone.The laddersummation ofthemodified diagrams resultsina systemofcoupled equations forthediagonal andoff- diagonal components ofthedensitymatrixinthesiterep- resentation. Theneglectoftheoff-diagonal components ofthedensity matrix, which describe quantum- mechanical tunneling between sites,corresponds tothe limitofclassical jumpsbetween thesites.Takinginto account theoff-diagonal components inthetheoryof smallmobility, oneobtainsatunneling contribution to thecurrent inaddition tothehopping one.'' Thesituation changes indisordered systems (andalso inordered onesinthepresenceofanelectric field).Ifone replacesPwiththenumber one,whichcorresponds to theconsideration ofphonon-free transitions between the sites,thetimeintegrals alreadyconverge, ifthedifference oftheenergylevelsatneighboring sitesisnotequalto zero(e.g.,in(5)V%0).Inthiscaseonehastoconsid- 1 ertwotypesofvertical sectionsofthediagrams, which donotcutphonon lines.Thefirstofthesetypescutstwo electron lineswiththeindices mandm',wherem'=m. Thesesections arecalledfreesections. Forthesecond typeofsections rn'Amholdsandwewillcallthemquasi- free.Indisordered systems (orincrystals inthepres- enceofanelectric field)thediagrams whichcontain quasifree sections donotdiverge. Because ofthis,a laddersummation overthesesections cannotbeper- formed,ifoneworksoutatheoryofthehopping conduc- tivity.Theladdersummation onlyoverpowersoffree sections provides aclosedsystemofequations forthedi- agonalcomponents ofthedensitymatrix,i.e.,therate equation ofthedensityofelectrons atthesites.The "price"forsuchapartialsummation overfreesections onlyistheoccurrence ofdiagrams withquasifree sec- tions.Thesediagrams arecontained inthedetermination oftheblocksoftheprobabilities 8'.Particularly inthe diagrams withtwopoints,whichdescribe thetwo-site probability (5),freesections donotexist,becausem,Am holds.ForthatreasonPoccurshereinsteadofP—1asit hasbeeninRef.3.Inthiswayaphonon-free transition rn,~misincluded in(5),whichcanbeobtained byre- 48 HOPPING TRANSPORT INTHETHREE-SITE MODEL INTHE... 165 placingP~1: Ofcoursetheaboverelationship isnonzero onlyinthe caseofidentical energies VandV.However, in 1 disordered systems theeffectofthelevelrepulsion arises iftheenergylevelsofneighboring sitescoincide acciden- tally.'Thisresults inaviolation ofthecondition V=Vduetotheoverlapofthewavefunctions (the 1 effectofthelevelrepulsion isaneffectofhigherorderinJandcantherefore beobtained onlybytakingintoac- counthigher-order diagrams). Reallyweareconcerned witharenormalization oftheatomicenergylevelsbythe interaction between thesites. Inthethree-site diagrams [Figs.1(a),1(b),2(a),and 2(b)]theindicesoftheelectron linesontheupperand lowerbranches ofthecontour alsodonotcoincide. Therefore wegetin(11)afactorPinsteadofP1after- thesurnrnation overpowersoffreesections only.Ac- cordingly theprobabilities contain phonon-free transi- tions.Inthecaseofstrongcoupling withphonons, the phonon-free transitions areknowntogiveasmallcontri- butioncompared tothephonon ones,i.e.,thephonon-—S~freecontribution isexponentially small(-e).How- ever,inthecaseofweakelectron-phonon coupling the situation changes. Herethedisregard ofphonon-free transitions hastoagreaterorlessextentarbitrary char- acter,because aprioriacorresponding parameter of smallness isnotobvious. However, allsystemsforwhich thecontribution offreeandquasifree sections are different, showageneral fundamental property—the presence oflocalization ifthereisnointeraction with phonons. Indisordered systems thecorresponding mech- anismistheAnderson localization, whereas inordered systems inthepresence ofanelectric field,localization alongthedirection ofthefieldoccursduetotheStark quantization. Inthecaseofelectrons localized atsites,it isreasonable toneglect thecontributions whicharise fromphonon-free tunneling between sites.%'ewillpick upthisquestion inanother paper,whichwillbedevoted totheexamination oftherateequation forthecaseof weakcoupling withphonons. Herewenotethatthe phonon-free transitions areimportant forthetransport phenomena intheacregime. Theygovernthephonon- freecontribution totheacconductivity inthetwo-site model. III.HOPPING CONTRIBUTION TOTHECURRENT INORDERED SYSTEMS INTHECASEOF STRONG COUPLING WITHPHONONS (MODEL OFSMALL POLARONS)Aftersomestraightforward transformations weobtain j=j2+j3~ wherej2andj3arethetwo-site andthree-site contribu- tions,respectively, j,=enf(1f)&—RWo (20) j3=enf(lf)g—R[(1f)WO"— ~fW'o"'—.]. m,m' (21) Herendenotes thesiteconcentration ofthelattice.The following relationships forthetransition probabilities in ordered mediums, duetospatialhomogeneity, maybe written mm'~m+mo,m'+mo ~mlm2m ~m)+mO, m2+mo, m+mO Theserelationships havebeentakenintoaccount inEqs. (20)and(21). Thetwo-site contribution doesnotdependonthemag- neticfield.Thiscontribution (foralatticeofcubicsym- metry)hasbeenconsidered inRef.27(seealsoRef.20) andis,inthenearest-neighbor approximation, givenby j2=enf(1f)exp—TE eEgggsinhkT (22) HereJ—:Joistheresonance integral between thenearest neighbors andgisthevectortothenearestneighbor, [y,i' A'co,E,=kTg(1—cosq.g)tanh q [y /(irico)(23) (24) Inthelimiting caseofhightemperature 2kT&fico, whichisthemostinteresting caseforsmallpolarons, we obtainInthecaseofordered systems itispossible toavoid solvingtherateequation (4).Onethenhastoreplacein (17)thetermdp/dtbytheright-hand sideofEq.(4) and,corresponding totheliinitt~~,p(t)byf.Herefistheprobability ofoccupation ofthesitem.This probability doesnotdependonminordered systems: 1 e"~+1 dp(t)j=lim—gR0(17)Thecurrent densityjinelectric andmagnetic fieldsof anystrength isrelatedtothedensitymatrixbytheequa- tionfy,f'Aco,E,~E,=g (1—cosq.g), q g~4E,kT.(23a) (24a) where0denotesthevolumeofthesystem.Relation (22)isvalidinnottoohighelectric fields,if eEa(4E,(a=~g~,latticeconstant). Inallfurthercalcu- 166 H.BOTTGER, V.V.BRYKSIN, ANDF.SCHULZ 48 lations wesuppose thatthiscondition isfulfilled.If eEa=4E,thepolaronic statedissociates intheelectric field,i.e.,theelectron willloseitspolaronic "cavity". Athigherfieldsthetransport isnolongerhopping like. Thiseffectissimilar initsnaturetothecrossover from hopping toviscous motionofpinnedvortices insuper- conductors andJosephson-junction arraysifonein- creasesthecurrent abovethecriticalvalue. Uptonow forthiseffectthereisnorigorous theoretical description. Inthenonlinear regimewithrespecttoE,thecurrent depends ontheanglebetween theelectric fieldandthe axesofsymmetry ofthecrystalevenincubicandhexago- nalcrystals. Thelatteroneswillbeconsidered below. Thedirections ofEandjcoincide onlyinthecaseof motion alongaxesofhighsymmetry. Inthehexagonal structure (seeFig.3}suchdirections aresixfoldaxes, wherewefind eEg eEa .eEaggsinh ~2asinh +sinh2kT 2kT 4kT andforthebisectrix between theseweobtain eEg eEa&3ggsinh ~2&3a sinh2kT 4kT Thetwo-site electric conductivity isisotropic inthe Ohmicregimeandinthecaseofhexagonal crystals we haverepresented as j3=ja+js ~ j,=enf(1—f)gRWo m,m' j,=enf(l—f)(1—2f)gRWo m,m'(26) (27) (28) Herej,andj,aretheantisymmetric andthesymmetric partsofthethree-site current, whichdescribe transverse andlongitudinal effects,respectively, and a (e)a s (e)s Om'm=~om'm ~om'm=~Om'm aretheantisymmetric andsymmetric partsofthethree- siteelectronic probabilities. Equation (27)describes theHallcurrent inthecaseof smallpolarons bytakingintoaccount nonlinear effects withrespecttoEandH.InthelinearinEandHregime theHalleffectforsmallpolarons hasbeenstudied in Refs.2and30. Intheappendix thecalculation ofthethree-site hop- pingprobability ispresented forthemodelofstrong electron-phonon coupling. Fortheantisymmetric partof theprobability fordisordered systems wehaveobtained thefollowing expression: ~(e)a=2JmmJmmJmm1 221 Ig~2g~2+g~2g~2 +g~2(~2I1/2 eEg 3eEa 2kT 2kTXsin.[RXR ]eH W''(EH)=—W"(—E,—H). (25)Wenowturntothestudyofthree-site contribution to thecurrentj3(21),whichdescribes boththeHalleffect andthemagnetoresistance. Theconnection between theprobabilities8"'and8''ismuchmoresimpleinordered systems andhasthe formwhereXexp—o. 12 y i(fico) 2Nsinh%co/2V+V 3kT(29) Because onlytheantisymmetric termoftheprobabilities withrespecttoEcontributes tothecurrent, j3canbe ~mmm12lyql' 2Nsinh(irico&P/2) X[cosh(A'coP/2) —cosh(incog/6) ] X[3—cosq.R—cosqR(30)X[1+cosqR—cosqR—cosqR], —cosqR]. (31) Inthelimiting caseofhightemperature (A'coq&2kT) theseformulas simplifyto FIG.3.Hexagonal structure. Themagnetic fieldisdirected perpendicular tothedrawing planeandtheelectric-field vector lieswithintheplane.Thevectors g&andg2indicate twopossi- bleintermediate statesinathree-site hopping eventforagiven vectorg.Thedashedvectorcorresponds tothevectorg&—g2 [see(35)].o=[E,(mm&}+E, (mm2}+E, (m&m2)],4 m&m2m9kT g"=4kT[E, (mmi)+E.(mm2)—E.(mim2)] ly,l'E,(m,m')=g fico(1—cosqR').4N(32) 48 HOPPING TRANSPORT INTHETHREE-SITE MODEL INTHE... 167 Thesymmetric termofthethree-site probabilities withrespecttoHhas,inthepresenceofstrongelectron-phonon cou- pling,theform (e)s~mmm12 XexpJJJ [E,(mme)+E, (m,mz)—E,(mm,)]g(mm, ) Eg(mm) )m,mT V+ cos[RXR ]2kT 2gmme m(m2(33) iyqi(irzcoq) 2Nsinhfico/2I and aj,=enf(1—f)4'—e''sin ggi2&3eaa' 4AC and /y,/' i)ico,E,(m,m&)=kTQ tanh (1—cosq.R).4kT Werestrict ourselves tohopping between nearest neighbors inthefollowing calculation ofthecurrent. In- sertingin(27)theexpression fortheantisymmetric part ofthethree-site probability 8'ogg(29)weobtainforthe caseofahexagonal crystal(Fig.3),aftersomestraight- forward transformations, anequation forthetransverse (antisymmetric withrespecttoH)contribution tothe current: 3 2f(1f)4rJg(7).&3eHa v'3A'g'2 4A'ceEaXsinh2kT Inthelinearapproximation withrespecttoEandHthis equation turnsintothefamiliar expression oftheHall current inhexagonal crystal,''whichhasapurely transverse character anddoesnotdependontheorienta- tionofthevectorEwithrespecttothecrystallographic axes. Weconsider nowthelongitudinal (symmetric inH) contribution ofthethree-site probabilities tothecurrent j,.Inserting (33)in(28)weobtain inthenearest- neighbor approximation: 2i' RE,gj,=enf(1f)(2f—1)—e eE.gXggcosh sinh2kTeE.(g,—g,) 6kT(35) eH eE-gXcos&3aggsinh44' 2kT(37) where,inaccordance with(30)and(31),wehaveinthe nearestneighbor approximation: 31yql(1—cosq.g)cr(T)= 2Nsinh(A'coP/2) X[cosh(fico+/2) —cosh(A'coP/6)) 43kT' /yq/(fico)cos h(incog/6)(1—cosqg)2NsinhA'co/2(36) =4kTE.. Thevectorg'hastwopossible valuesforagivenvector~ (seeFig.3)andweobserveg,—gzlg,~g,—g2~=a&3. Forordered systems ithasbeen used.,that V=—eE.R Inthepresence ofstrongelectric fieldsthecurrent is onlydirected alongthevectorEXH(i.e.,haspurely transverse character), iftheelectric fieldEisdirected alongaxesofhighsymmetry, exactly asithasbeenfor longitudinal two-site contribution. Inparticular ifthe fieldEisdirected alongasixfoldaxis,whichcoincides withthedirection ofthexaxis,thenj,=0js 3uoH(2f—1)cos 2J(2f—1)cos 2m.E 0(38)wheretheenergygisdetermined byEq.(24)andE,by (23). Themostimportant property oftherelationship (37)is theoccurrence ofap-nanomaly inthemagnetoresistance foratransition fromholetoelectron conductivity within thethree-site hopping model. Reallyifonereplaces in (37)f~1f(i.e.,p~—p—)thesignofj,changes. Actu- allyifp=0(f=—,')thelongitudinal current doesnotde- pendonthemagnetic fieldinthemodelofthree-site hop- ping,sincej,=0.Simultaneously ifoneperforms sucha replacement theHallcurrent doesnotchangeitsdirec- tioninaccordance with(35). Itisinteresting toremarkthatthetemperature andE- fielddependencies ofthetwo-site current (22)andthe symmetric contribution ofthethree-site currentcoincide forsmallpolarons. Actually, thecurrents j2andj,have thesamedirections foranyorientation ofthefieldwith respecttothecrystallographic axesandtheirratiois givenby H.BOTTGER, V.V.BRYKSIN, ANDF.SCHULZ where up=ca/Aisaquantity, whichhasdimensions of themobility;4o=hc/eisthemagnetic-Aux quantum and N=HS isthemagnetic Auxthrough thelatticecellofthe areaS=&3a /4.Inthecaseofnottoohighvaluesof themagnetic fieldandifthelatticeconstant isoftheor- derofsomeangstroms, theparameter 2~C/Cpissmall. Thechangeoftheelectricconductivity Ao(H)duetothe magnetic fieldistherefore verysmall: 2 (12j')—277 (39) oE +o«1. Wenotethatthedimensionless parameterJ/E,isalso smallinthetheoryofsmallpolarons. However, theratiooftheHallcurrenttoj,maynot necessarily besmallinthemodelofsmallpolarons, despite smallvaluesoftheparameter 2m%/4p. The reasonforthisisthedifferent temperature dependence of theHallcurrent andthelongitudinal current withinthis model.Theformer isproportional toexp(4E,/3kT—) andthelattertoexp(E,/—kT).Theirratiointhelinear regimewithrespecttoEinaccordance with(35)and(37) isgivenby JsV'~E/—3k7 exp(E/3kT)8 c 3'upH (40) Theratio(40)canbebothlowerorhigherthanoneand depends strongly onthetemperature, because inthemod- elofsmallpolarons theactivation energyE,forhopping between sitesismuchgreater thankT.Thelatter circumstance—theproportionality oftheHallangleto exp(E,/3kT)——isoneofthemostcharacteristic prop- ertiesofthesmall-polaron model. Nevertheless theexperimental detection ofthedepen- denceofthelongitudinal resistance onthemagnetic field inpolaronic systems isadim.cultproblem. Tosolvethis problem itisfavorable tousematerials withalargelat- ticeconstant orpreferably, superstructures. IV.DISCUSSION Therateequation (4),inwhichthree-site jumpsare takenintoaccount, andtheexpression forthecurrent (17)describe thekineticphenomena inthepresence of electric andmagnetic fieldsofanystrengths. Therate equation hasanexactanalytical solution onlyforordered systems. Inthiscasethesymmetrical W'andthean- tisymmetrical Wcontributions ofthethree-site proba- bilitiestothecurrent areadditive [see(26)—(28)].Here W'andW'represent, respectively, thelongitudinal and transverse contributions tothecurrent. Indisordered systems, Eq.(4)canbesolvedonlynumerically orap- proximately, using,e.g.,theideasofthepercolation theoryofthemethodoftheeffective medium. Inthis case,thecontributions ofWsandW'aredefinitely not additive. Ingeneralitistherefore notpossibletoconsid- ertheHalleffectandthernagnetoresistance independent- ly.Onlyinthelimiting caseofsmallmagnetic fields,if onecanneglectthequadratic corrections withrespecttoH,isitpossible tostudytheHalleffectwithout taking intoaccount thechangeoftheresistance inthepresence ofthemagnetic field.' Theexpressions forWsandW'areverydifferent from eachotherintheorderofmagnitude aswellasintemper- atureandfielddependencies. Inthecaseofstrongcou- plingwithphonons thiscanbeseenbycomparing (29) and(33).Inthecaseofweakcoupling withphonons W' iscausedbyone-phonon transitions, whereasW'isdue totwo-phonon transitions.'Thisresultcanbeobtained fromthegeneral equation forthethree-site probabilities (11).Thislimiting casewillbeconsidered indetailbyus inafollowing paper.Weremarkhere,thatinthecases ofstrongaswellasofweakcoupling withphonons one hasIW'I» IW'I. Theobtained symmetry relationships forthethree-site probabilities arevalidforanystrengthofcoupling with phonons. Theyleadtotheoccurrence ofap-nanomaly ofthekineticcoefficients inamagnetic fieldforjumpsfar fromtheFermilevel,i.e.,iftheelectron doesnotchange toaholeduringthetransition orviceversa:Forthe transition fromhole-like toelectron-like transport, the magnetoresistance changes itssign,whiletheHalleffect doesnotchangeitssign.Thisproperty iscompletely in opposition tothoseproperties whicharetypicalofband transport. Thephysics underlying thisdifference isthat inthecaseofbandtransport themagnetic fieldactson thechargecarriers viatheI.orentzforce,whileinthe caseofhopping transport itactsviathequantum in- terference. Thesignofthemagnetoresistance isdeter- minedbytwofactors: theoccupation numberofthesitesfandthesignoftheresonance integralJ.InthecaseofJ&0,whentheminimum oftheelectron bandislocated inthecenteroftheBrillouin zone,themagnetoresistance isnegative fortheelectronic conductivity (f(—,)andpos- itivefortheholeconductivity (f&—,').IfJ&0theelec- tronsandholesswaptheirplaces. Asalready remarked, theinhuence ofthemagnetic fieldonthetransport inthehopping regimeisconnected tothesmallparameter N/No.HereNoisthemagnetic- Auxquantum andN=HS isthemagnetic Auxthrough theareaofthetriangle, whichisbuiltfromthethreesites (inthecaseofmultisite hopping thisfigureismore difficult). Since8"changes itssignbychanging from electrons toholes,thethree-site contribution canbeob- served, inprinciple, alsoifthemagnetic fieldisequalto zero.Thisfollowsfromthefactthatj,changes itssign duringthetransition fromelectrons toholes,whilej2 doesnotchangeitssign.Becauseofthisthetotalcurrent obtained bytakingintoaccount thethree-site contribu- tionisdifferent forelectrons andholesifallothercondi- tionsarepreserved. Thisdifference canbeobserved, e.g., bychanging thedegreeofcompensation oftheimpurity levels. APPENDIX Wedividethethree-site probability (11)inasymmetric andantisyrnrnetric partwithrespecttothemagnetic field: 48 HOPPING TRANSPORT INTHETHREE-SITE MODEL INTHE... 169 (A2) where I=2f dtfdt2exp—(Vt+Vt2) i'+gIcostoqt&+2iR+Iq~coscoqt] qm1 q2iA+Fq~coscoq t2+qm2 q2 ly,l'I=. [I+cosqR—cosq.R—cosqR].2Nsinhfico/212 1 2(A3) Thequantities IandIqcanbeobtained fromIqbychanging oftheindicesI,m&,andm2. qm1 qm2 Itisfavorable totransform (A3)nowinthefollowing way:multiply theright-hand sidebyJ"dt35(t3—t&t2)— anduseforthe5function theFourierrepresentation. Asaresultweobtainafourfold integral: I=—defoe 'K(—ei)Km m1Vf"dt,expit2i'/2V—co+gIqcostoqt2qm2 q where K(to)=fdtexpicot+QIcoscot=K(—~). q Inaccordance with(A4),theexpression forRe(I)isgivenby ~~mm/2)Vm17712Vmill2Re(I)=e dcoe"~/2' co— K(co).2' fi(A4) (A5) (A6) Theintegral in(A5)oftenappears inthetheoryofsmallpolarons, e.g.,inthecalculation ofthefrequency dependence oftheelectronic conductivity. Itcanbesolvedusingthemethodofsteepest descents. Thesaddlepointtpigis determined bytheequation co=+Ime@sinh(coi)). q 2' gIqcoqcosh(coqri) qexp cubi)+gIqco—sh(coqi) ) qK(co)=Theintegration overthesurrounding areaofthesaddlepointyields '1/2(A7) (AS) Takingintoaccount (AS)weobtainfor(A6)theexpression PV/2 Re(I)=2ire'fdao'gIqco&cosh(coqri) gIqcoqcosh(coqi), )gIqcocosh(coqijz)' q q q gpVm&m&Il+Vmm292Xexpco—g—g&—g2+—1/2 ++[I„cosh(coi))+I cosh(coii,)+I cosh(coij)] whereiI,ij„and riddepend oncoandaredetermined byEq.(A7):ii,byreplacing m—+mz',iI&,byreplacing co~co—V/A;andg2,bym—+I &andco~co—V/A.Theargument oftheexponential intheexpression ofthe 12 2 integral hasamaximum atco=cop.Thequantity copcanbecalculated bysettingthederivative oftheargument inthe 170 H.BOTTGER, V.V.BRYKSIN, ANDF.SCHULZ exponential equaltozero.Takingintoaccount (A7)wegetthefollowing equation forcoo: 2l(~0)+91(~0)+ 92(~0) (A9) Weexpandnowtheargument oftheexponential inthesurrounding areaofthepointco=~0inaserieswithrespecttoco tothesecondorder.Thenweperform theintegration overco.Aftersomemanipulations weobtain —1/2d~odCO0 dCO0dCuo dCO0dCuoRe(I)=2m. + + 9919I291I2 Xexp—(Vil,+Vi))++[I cosh(coqi))+I cosh(coi),)+Icosh(co i))]1 q(A10) Weintroduce nowthenewvariables5=i)—iiip/6,5,=i),—A'p/6,and5&=i)z—Ap/6insteadofil,il„andi)i.Thenwe haveinaccordance with(A9)5(coo)+5, (coo)+5&(coo) =0.Astraightforward examination ofthisequation together with (A7)showsthat5,5„5i-b,V/fit,whereitholdsI=gIcocosh(ficoP/6)andb,Vdenotes thequantitiesV,V Herewesuppose thatcoq5,coq5„and co5z—b,V/iiigql qcoq.Thelattercondition corresponds totherequirement that (B,Vfico)/(E,kT)«1,i.e.,thatthespreadoftheenergylevelsissmallerthanthepolaronic shift.Wenowexpand (A10)inaserieswithrespectto5,5„and 5ztotheorderof(5V)l(E,kT).Weobtainanexpression forRe(I),which coincides with(A10)performing thereplacements: i),i)„i)i—+iit'P/6and dcoo fico+=gIcocoshdg,q-2q Oneobtainsdcoo/di),fromthisequation bysettingmz~m,anddcoo/dilly followsformz~m&.Usingthisexpression forRe(I)inEq.(A2)wegettheantisymmetric contribution tothethree-site probability intheform(29). Wenowturntothecalculation ofIm(I)(A4): ]aIm(I)=—dcoexp—(A'co+V)Eco—V Km Q) 1V 0 OOX—idtz+Im dt'sexpicoti+g Izcoscoqti ifiP/2 0qm2 q Theratiooftheintegrals overtiinthelimitsfrom0tooototheonesfromiftP/2to0isexponentially small,i.e.,isof theorderexp[—gI~(coshftcog/2 1)].We—therefore neglect subsequently theintegral withthelimitsfrom0to oo.Theintegral overtheinterval ift13/2to0hasitslargest contribution nearthepointti=ifit/2if co&gIcosinh(A'coP/2).Weexpandtheargument oftheexponential inaserieswithrespecttotiinthesurround-qm2 ingareaoftz=iftP/2 tofirstorderandperform theintegration overtz.Weobtain Im(I)=—iiiexp[ST(m, m,)+ST(mm, )—ST(mm,)+PV/2] XgIcocoshcoi),gIcocoshcoqi)i q q Xjdco[E,(mme)+E, (m,mz)—E,(mm,)+A'co/2] V 12Xexp—co+ 'g1CO+V i),+g[Icoshcoi),+Icoshcoi)g] (A11) Herewehaveusedthedefinitions ofthequantities STandE,(8)and(32).Thefunctions il,(co)andi)i(co)aredefined bytheequations [see(A7)] V 12=gIcoqsinhcoqi), ; qV 2co+=gIcoslnhco 'gp. q(A12) Theargument oftheexponential in(A11)hasitsmaximum atco=coo,where cooisgivenbytheequation i)&(coo)+rjz(coo)=0. 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PhysRevB.94.054504.pdf	PHYSICAL REVIEW B 94, 054504 (2016) Anomalous oscillatory magnetoresistance in superconductors Milind N. Kunchurand Charles L. Dean Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA Boris I. Ivlev Instituto de F ´ısica, Universidad Aut ´onoma de San Luis Potos ´ı, San Luis Potos ´ı, 78000, Mexico (Received 25 September 2015; revised manuscript received 30 June 2016; published 4 August 2016) We report oscillatory magnetoresistance in various superconducting ﬁlms, with a magnetic-ﬁeld period /Delta1B∼0.1 T that is essentially independent of sample dimensions, temperature, transport current, and the magnitude and orientation of the magnetic ﬁeld, including magnetic ﬁelds oriented parallel to the ﬁlm plane.The characteristics of these oscillations seem hard to reconcile with previously established mechanisms foroscillations in magnetoresistance, suggesting the possibility of another type of physical origin. DOI: 10.1103/PhysRevB.94.054504 I. INTRODUCTION AND BACKGROUND The formation of the superconducting state involves a delicate congruence between the parameters of the electronicstructure and the interactions between the charge carriers andexcitations in the solid. Minute alterations in any of theseconditions can sensitively alter the superconducting state asreﬂected by its transition temperature T cand its resistance Rin the dissipative regime. Rdecreases when the superconducting component strengthens and provides a sensitive probe ofsmall changes in the underlying electronic structure. Arisingfrom ﬂux vortex motion, ﬂuctuations, and percolation betweenresistive and zero-resistance regions, Ris typically a mono- tonically rising function of T, magnetic ﬁeld B, and transport electric current I. In particular, the magnetoresistance is usually positive; however, certain circumstances can lead toan oscillatory magnetoresistance (OMR). The most straightforward source of OMR in supercon- ductors is related to ﬂuxoid quantization and the formationof Abrikosov or Josephson ﬂux vortices; this behavior iswell exempliﬁed by Josephson-junction (JJ) arrays, regularlypatterned hole or wire networks, and other mesoscopic andmultiply connected systems [ 1–24]. For this class of effects, the OMR period is /Delta1B≈/Phi1 0/S, where Sis the effective cross sectional area facing the magnetic ﬁeld that the ﬂuxlinks to, and /Phi1 0=h/2eis the ﬂux quantum. Because of the geometrical basis of the period, such oscillations can beexpected to have a period that is naturally independent of T, B, andI. However, /Delta1Bwould still depend on the orientation ofBsince the projection of the geometrical area onto the plane perpendicular to B, and hence S, will change with angle. What is surprising is that sometimes systems that have not been purposely engineered to have regularly arranged structures also show remarkably periodic OMR. This has been previously reported by other groups in the literature and isfurther explored in the present experiments. Here we presentevidence and arguments that the usual explanations offered forsuch OMR—based on granularity and inhomogeneity causingvortex and junction related effects—are probably not valid Corresponding author: kunchur@sc.edu; http://www.physics. sc.edu/kunchurand some potentially different mechanism might underlie the observed OMR. In the work by Herzog et al. [25] on granular tin wires (grown by thermal evaporation onto cryogenically cooledsubstrates), they observed some OMR which they tentativelycompared to the behavior JJ arrays [ 26]. In a granular superconductor, one would expect that the grain sizes and areasof loops (formed randomly through the coupling of grains)would have spreads in their values so that there would not bea single pronounced period because of averaging. It might beexpected that if the ﬁlm is patterned into a sufﬁciently tinybridge, this volume may enclose so few grains that the limitedaveraging may allow at least some crude oscillatory behavior tosurvive. This was the argument put forth in Ref. [ 25] and they indeed found that although their 2D granular ﬁlms do not showOMR, their very tiny nanobridges (of thicknesses d∼100 nm, widths w=110–200 nm, and lengths l=1–4μm) did show it. In that work, consistent with a random granular system,the oscillations had a weak amplitude and appeared to be asuperposition of multiple periods (the lowest common multipleof these periods appears to be /lessorsimilar0.4T ) . However, even nongranular systems show OMR, such as observed by Wang et al. in nanobridges patterned from high- purity single-crystalline Pb ﬁlms grown by low-temperaturemolecular beam epitaxy (MBE) [ 27]. Their samples 3 and 4, which showed low- BOMR similar to ours, had dimensions ofd∼8n m , w≈300 nm, and l=2 and 10 μm. The OMR periods were /Delta1B=0.13 T and 0.18 T, respectively. These authors do not claim to know the mechanism for theiroscillations in their samples 3 and 4, although they hint at thepossibility of ringlike structures along the lines of the tentativeexplanation offered in Ref. [ 25]. Johansson et al. [28] studied superconducting wires of amorphous indium-oxide made by electron-beam evaporationof a:InO onto a WS 2nanowire suspended across a narrow gap in a substrate. Their sample 99Nb shows an OMR with astrong fundamental period of /Delta1B=0.12 T; this sample has dimensions of d∼30 nm, w≈120 nm, and l=3.4μm. All of these previous works were on wires of very small cross section [ ∼(1–10) ×10 −15m2], focusing mainly on low- temperature conventional superconducting systems (e.g., Sn,Pb, and InO), and measured with the magnetic ﬁeld orientedperpendicular to the ﬁlm plane ( B ⊥). But what emerges 2469-9950/2016/94(5)/054504(8) 054504-1 ©2016 American Physical SocietyKUNCHUR, DEAN, AND IVLEV PHYSICAL REVIEW B 94, 054504 (2016) from these results is that despite the diversity of materials, methods of preparations, and differences in morphology anddimensions, all have OMR fundamental periods within thenarrow range of 0.12–0.18 T. In the present work we investigated OMR in high- temperature cuprate superconducting ﬁlms, with bridges ofmuch more extended sizes (cross sections as large as 1 × 10 −11, four orders of magnitude larger than the previous nanobridges), and have also studied the behavior for magneticﬁelds parallel to the ﬁlm plane ( B /bardbl), in addition to B⊥.W e found that the oscillation period /Delta1B is independent of the orientation of B. While the main focus of the present work is on ﬁlms of the electron-doped inﬁnite-layer Sr 1−xLaxCuO 2 superconductor, we have also observed the oscillations in another electron-doped cuprate Nd 2−xCexCuO 4, in a hole doped cuprate Y 1Ba2Cu3O7, in a conventional supercon- ductor NbTiN, and in the interface between a topologicalinsulator (Bi 2Te3) and a chalcogenide (FeTe). Our observed OMR covered temperatures ranging 4–74 K and sampledimensions covering d=7–250 nm, w=4–50 μm, and l=70–2000 μm. Despite the large ranges in parameters and dimensions in the samples we studied, our OMR periods re-main narrowly clustered within the range of /Delta1B=0.11–0.15 T, comparable to the /Delta1B=0.12, 0.13, and 0.18 T observed in the aforementioned work on low- T csystems. This relative con- stancy of period across such an enormous range of parameters,dimensions, and ﬁeld orientations makes explanations basedon ﬂuxoid quantization and vortices implausible. As we showbelow, other known mechanisms of OMR (based on Fermisurface geometry, spatial modulations in the superconductingstate, etc.) are also incompatible with this OMR phenomenon,thus pointing to the possibility of a fundamentally newunderlying mechanism. II. EXPERIMENTAL DETAILS The main measurements in this study are on c-axis-oriented epitaxial thin ﬁlms of Sr 0.88La0.12CuO 2(SLCO) deposited on heated KTaO 3substrates by rf magnetron sputtering followed by an oxygen reduction step. X-ray diffraction spectra showthe ﬁlms to be epitaxial and highly c-axis-oriented (with a mosaicity of 0 .1 ◦), and single phase with undetectable (<0.1%) impurity phases. SLCO sample A had the following parameters: thickness d=31 nm, width w=13.6μm, length l=512μm, midpoint transition temperature Tc=23.3, and a transition width (10%–90% of normal resistance) of /Delta1Tc≈ 2.5 K. Sample B had d=61 nm, w=4μm,l=100μm, Tc=26.5 K, and /Delta1Tc≈2.5 K. Also included in this study is a c-axis-oriented epitaxial thin ﬁlm of Y 1Ba2Cu3O7(YBCO) deposited on a SrTiO 3substrate by the pulsed-laser-deposition process, with d=50 nm, w=4μm,l=70μm,Tc=78.6 K, and/Delta1Tc≈9 K; and a NbTiN ﬁlm sputtered onto a Si wafer with a 400 nm thick oxide layer, with d=125 nm, w=8μm, l=115μm,Tc=10.46 K, and /Delta1Tc≈0.5 K. SLCO sample A was a four-probe bridge with a straight current path [Fig. 1(b)] patterned with contact photolithography followed by wetetching. SLCO sample B and the YBCO and NbTiN sampleswere four-probe bridges with a folded current path [as shown inFig. 1(a)] and were patterned by projection photolithography and argon-ion milling. The geometries, dimensions, and lead FIG. 1. Schematics of sample patterns (not to scale) and lead arrangements. Actual voltage (V) and current (I) contacts are farremoved ( >1 mm) from the bridges. (a) Four-probe bridge with a folded current path. (b) Four-probe bridge with a straight through current path and voltage contacts on one side. arrangements are quite different for the two SLCO samples, making it unlikely that a particular lead arrangement or overallgeometry is at the root of the observed oscillations. Contactswere made by smearing indium onto contact areas that are farremoved ( >1 mm) from the bridge and then pressing down copper wires with indium pads. In the case of the SLCOsamples, gold dots were deposited in the contact areas priorto the indium treatment. Contact resistances are <1/Omega1(much lower than the actual resistance of the bridge being measured). The cuprate samples were characterized by broad transi- tions, which facilitates the observation of the OMR since thesample remains resistive but well below normal resistance fortheBﬁeld range over which the OMR occurs. The NbTiN sample has a relatively sharp transition but shows OMR if thecurrent is low enough. The cryostat was a Cryomech PT405 pulsed-tube closed- cycle cryocooler, ﬁtted with a 1.2 T GMW 3475-50 water-cooled copper electromagnet. A nonsuperconducting electro-magnet is particularly suited for measurements in low ﬁeldswith numerous closely spaced ﬁeld steps; a superconductingmagnet can add complexity, especially if it goes in and outof persistent mode for every data point. The cryocooler’s coldhead is far removed from the magnet, with a 22 cm longcopper rod protruding from the second-stage heat station intothe magnet poles; this eliminates changes in cooling powerand temperature that may be caused by the magnetic ﬁeld.The sample along with a calibrated cernox sensor (whichserves as the primary thermometer) and a diode temperaturesensor are mounted in close proximity at the end of thiscopper rod (an additional diode sensor on the second-stage heatstation serves as a tertiary indicator). A Hall sensor providesthe primary measurement of B, and the current supplied to the electromagnet serves as a secondary indicator of B.T h e standard active temperature controller was disconnected, sinceit can produce oscillations in Tand add electrical noise. To further avoid temperature variations during the measurementof each R(B) curve, all measurements were conducted at a ﬁxed phase point of the compressor cycle. These specialmeasures and redundancies provided for unusually clean andstable conditions, and highly reliable measurements of B andTfor each data point. The reliability of this system’s measurement of R, and stability of Tover time and against changes in Bwas extensively checked with resistors as “test samples” and through the continuous monitoring of all threethermometers. Figure 2shows the temperature indicated by the cernox thermometer along with a ruthenium-oxide (RuO 2) 054504-2ANOMALOUS OSCILLATORY MAGNETORESISTANCE IN . . . PHYSICAL REVIEW B 94, 054504 (2016) 0.0 0.4 0.8 1.23.743.763.783.80 15.5515.5615.5715.5815.5915.60T (K) B (T) Up Down RRuO2 (kΩ) Up Down FIG. 2. Measurements of the temperature, as indicated by the cernox thermometer, and a ruthenium oxide resistor (mounted as a test sample) show close tracking between temperatures at the sample and theromometer locations and a total temperature variation overtime and magnetic-ﬁeld change of /lessorsimilar20 mK. resistor mounted as a test sample. Like the cernox, RuO 2also has a negative temperature coefﬁcient of resistance and a veryslightly negative magnetoresistance. It can be seen that, despiteoccasional small jumps, long-term drifts, and any possible B dependent shifts, Tstays within a /lessorsimilar20 mK window over the duration of the entire curve, with a short-term stability of∼1 mK. Except where noted, the R(B) data represent four-probe resistance measurements at a constant dc current of I= 12.8μA, taken with current-direction-reversed averaging, i.e., R=(V +−V−)/(I+−I−), the ratio of the difference of the forward and reverse voltages to the corresponding differencein the forward and reverse currents. All data are completelyreversible with respect to changes in I,T, andB. All data also lie in the ohmic response regime (except for the oneset of variable- Icurves). To minimize electrical noise, the dc current source consisted simply of an alkaline battery anda large series resistor, serving as a ballast to hold the currentconstant. (A Hewlett Packard 5532A dc power supply replacedthe battery for the variable- Icurves.) Except where noted, the sample voltage was measured with a Keithley model 2182Ananovoltmeter and other voltages were measured with Keithleymodel 2000 multimeters, with each quantity averaged over∼30 readings (individual readings had integration times of 17 ms). The single conﬁrmatory R(B) curve measured with pulsed signals utilized a Wavetek model 801 pulse generator,in-house built electronics, and a LeCroy LT 322 digital storageoscilloscope. Despite the seemingly long description of the experimental setup, we would like to emphasize that when you comeright down to it, our apparatus is actually very simple andleast subject to interpretation compared to a commercialautomated turn-key measurement system as follows. (1) Ourcurrent source for most of the data is simply a battery witha series resistor, (2) the magnet is not superconducting buta copper-wire electromagnet with an iron core and you canobtain the Bvalue also from the magnet current besides the Hall probe, (3) instead of the constantly varying powersupplied by a traditional automatic temperature controller, we have driven the control heater with a nonﬂuctuating stable dcvoltage resulting in a constant temperature since the coolingpower remains essentially constant, and (4) some of the datawere measured entirely by hand without the use of a computerautomated data acquisition. Thus it will become clear from theextensive tests described below that the observations are freefrom experimental artifacts. III. DATA AND RESULTS Figure 3(a) shows R(B) curves for various ﬁxed Tfor the B⊥orientation ( B⊥ﬁlm plane) in SLCO sample A. There are pronounced oscillations over wide ranges of TandR, superimposed on a steadily rising background magnetoresis-tance that follows the R/R n∼B/B c2ﬂux-ﬂow relation [this linearity is more conspicuous on the linear-linear graph shownin Fig. 4(a)]. The oscillations are not symmetric but have sharper minima in R(B), which we will denote by B X. A graph of BXvs count is very linear [Fig. 4(b)] indicating a high periodicity, and the slope of the straight-line ﬁt yieldsa period of /Delta1B=0.149±0.004 T independent of T;a fast Fourier transform (FFT) of R(B) produces the value /Delta1B=0.154±0.008 T. The oscillations are strongest in the B< 0.5 T range and appear to fade at higher B. Comparing 17.7317.7618.8718.90101001000 0.0 0.2 0.4 0.6 0.8 1.0101001000(b)T (K)T=16.3K T=17.7K T=18.8K T=20.1K(a) T=21.5K T=22.9K T=23.3K T=24.0K T=24.9K T=26.5KT=14.1K T=15.1KR(Ω) (c)T=16K20K21.7K23.3K26.6KR (Ω) B (T) FIG. 3. (a) B⊥magnetoresistance curves in SLCO sample A for various temperatures. Vertical dashed lines pass through the minima BX. (b) Temperature variations for the “17.7 K” and “18.8 K” resistance curves in above panel. (c) B/bardblmagnetoresistance curves in same sample. 054504-3KUNCHUR, DEAN, AND IVLEV PHYSICAL REVIEW B 94, 054504 (2016) 0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0 0123450.00.20.40.6(a) B (T)R(kΩ) (b) Perpendicular B Parallel B Minima countBX (T) FIG. 4. (a) Linear-linear plot of R(B) curves from Fig. 3(a) (SLCO sample A in B⊥). Although the lower Tcurves are now more compressed, this graph better displays the R∝Bgeneral response under the oscillations. The slopes closely reﬂect free ﬂux ﬂow, i.e., dR/dB ≈Rn/Bc2. (b) Magnetic ﬁelds corresponding to minima in R(B) graphs for perpendicular and parallel ﬁelds. The slopes of the straight-line ﬁts yield a period of /Delta1B=0.149±0.0004 T and /Delta1B=0.155±0.0008 T for B⊥andB/bardbl, respectively. curves at different T, the oscillations get weaker and disappear as the normal state is approached, indicating that they are afeature related to the superconducting state and that the normalstate magnetoresistance is itself not oscillatory. In fact, theeffect seems to be most prominent where R/lessmuchR n, i.e., in the foot of the resistive transition. The amplitude of the oscillations is so large (e.g., /Delta1R= 78/Omega1, minimum to maximum, for the 17.7 K curve) that it cannot possibly arise from a variation in T:t h edT/dR ≈ 0.025 K/ /Omega1would imply a /Delta1T=1.9 K, which is drastically higher than the ±15 mK maximum variation of the measured temperatures shown in Fig. 3(b). Figure 3(c) shows similar oscillatory R(B) curves for the B/bardblorientation ( B/bardblﬁlm plane), which have a comparable fractional amplitude superimposed on a less steep backgroundmagnetoresistance (reﬂecting the much higher B c2and lower ﬂux mobility for B/bardbl). The periodicity of /Delta1B=0.155± 0.008 T is the same as for B⊥within error, which calls into question any explanation of the phenomenon based on vorticesand ﬂuxoid quantization. Figure 5(a) shows that the oscillations vanish as Iis increased, indicating that some delicate property of thesuperconducting transition is needed; this action is similar tothe effect of increasing Tseen in Fig. 3(a). These observations establish that the oscillations are not an artifact of the voltagemeasurement, which is indifferent to the I,B, andTof the sample. Could it be that there is something special happen-ing in the apparatus around T∼20 K and R∼100/Omega1, which serendipitously coincides with the transition in thissuperconductor? No, because we have seen these oscillationsin a variety of systems spanning the ranges T=4–74 K andR=0.01–1000 /Omega1, with fractional resistance amplitudes ranging from zero to >50%. Also of signiﬁcance is the fact0.4 0.5 0.6 0.7300400500 0.0 0.2 0.4 0.60801600.0 0.1 0.2 0.3 0.40200400 B (T)T = 18.9K I = 12.8 μAR (Ω) Hall Sensor Magnet CurrentR (Ω)T= 17.9K I = 54 μAT = 18.4K (a) (c)(b)I=12.8 μA I=47 μA I=98 μA I=171 μA I=300 μA I=429 μAR (Ω) FIG. 5. (a) Magnetoresistance curves at various dc currents. Oscillations are most prominent at low I, where the response is ohmic, and disappear at higher I(SLCO sample A in B⊥). (b) The magnetic ﬁeld indicated by both the Hall sensor and electromagnet current produce comparable oscillations. Also these curves were measured by hand in a completely manual fashion without computer intervention.(c) This curve was measured with a pulsed current and still shows oscillations. that the low- IR(B) curves of Fig. 5(a) overlap and show independence of IandVover a sevenfold range. This indicates a perfect ohmic I-Vresponse, whereas the presence of weak links and microwave radiation would have lead to Shapirosteps in the I-Vresponse [ 29]. In fact the entire range of effects related to weak links and noise, is incapable of producing suchR(B) oscillations with a /Delta1Bindependent of Tand geometry. Figure 5(b) shows R(B) curves with the abscissa taken in two ways: one uses the Bmeasured by the Hall sensor and the other is the Bestimated from the electromagnet current. Notwithstanding the small disagreement and offsetinB(expected from the hysteresis and nonlinear response of the magnet’s iron core), the oscillations are reproduced by bothmethods and are therefore not an artifact of the Bmeasurement. Furthermore the curves in Fig. 5(b) were measured with a different voltmeter (a Keithley 2000 multimeter instead ofthe 2182A nanovoltmeter) and were measured completelymanually, to rule out any possible artifacts from a computercontrolled data acquisition system. A ﬁnal test, shown in Fig. 5(c), demonstrates that the oscillations can also be seen in a pulsed transport measurement(involving an entirely different chain of electronics), if care istaken to minimize the introduction of spurious noise (carefulisolation of the pulse generator and ampliﬁers from the mainspower using isolation transformers and capacitors). In lightof these extensive cross-checks using multiple instruments(three thermometers, three voltmeters, three current sources,and two Bmeasurements) we were not able to associate these oscillatory features to any artifact that could be produced 054504-4ANOMALOUS OSCILLATORY MAGNETORESISTANCE IN . . . PHYSICAL REVIEW B 94, 054504 (2016) 0.0 0.2 0.4 0.6 0.8 1.010010000.0 0.2 0.4 0.6 0.8 1.010100 71.0K para74.1K para80.3K para 71.2K perp73.6K perp79.9K perpT=86.5K perpR (Ω) B (T)(b)(a)T=25.5K T=23.7K T=20.8K T=15.6KT=28.8KR (Ω) FIG. 6. (a) B/bardbloscillatory magnetoresistance in SLCO sample B. (b)B/bardblandB⊥oscillatory magnetoresistance in YBCO. by the apparatus, therefore proving that it is an intrinsic phenomenon. Figure 6shows oscillatory R(B) curves for the other SLCO sample B (with very different dimensions) and the YBCOsample. Note that YBCO is a hole-doped cuprate supercon-ductor unlike the electron-doped SLCO, and has a three timeshigher T c. Incredibly, these other samples have periodicities (/Delta1B=0.147±0.007 T and 0 .150±0.004 T, respectively) that are identical to the ﬁrst SLCO sample within theirerror bars. Additionally we have also observed OMR in theNd 2−xCexCuO 4electron-doped cuprate superconductor and in the Bi 2Te3/FeTe topological-insulator/iron-chalcogenide 02468 1 00100200300400500600700 0.0 0.2 0.4 0.60100200300 R (Ω) B (T)R (Ω) T(K) FIG. 7. Resistive transitions of a NbTiN superconducting ﬁlm in parallel magnetic ﬁelds (right to left) of B/bardbl=0, 0.63, and 1.12 T inI=115μA. The inset shows B/bardbloscillatory magnetoresistance in the foot region of the transitions, which only appears at a very low current of I=97 nA; the temperatures of the curves (bottom to top) areT=9.51, 9.57, 9.71, 9.87, 10.00, and 10.14 K.interfacial superconductor with similar /Delta1B periods (0.14 and 0.11 T) despite their different material parameters anddimensions; those results will be described in detail elsewhere.Not all superconducting samples show the phenomenon and itis not known what induces OMR in some samples but not inothers. In some cases the OMR only appears at very low currentdensities j, such as in the NbTiN sample whose data are shown in Fig. 7. Despite being a conventional superconductor with a relatively sharp transition, at a sufﬁciently low j∼10 A/cm 2, OMR with /Delta1B≈0.12 T is discernible despite the low signal-to-noise ratio (a consequence of the low current). IV. DISCUSSION Since this OMR phenomenon is most easily seen within the resistive transition, one might be inclined to associate thephenomenon with inhomogeneity or granularity; however, weshow below that it is more likely that the OMR is fundamen-tally unrelated to the disorder but that a wider transition helpsto make the oscillations more conspicuous: with a perfectlysharp transition, the system would switch too abruptly fromzero resistance to the normal state to show oscillations (eventhe original Little Parks effect would be invisible if thetransition were perfectly sharp as discussed by Tinkham [ 30]). Nevertheless let us explore how granularity and a weak-linkedstructure might produce OMR. Oscillations can arise fromﬂuxoid quantization if the system is multiply connected. Agranular material can potentially have percolating loops thatcould enclose ﬂux; as the enclosed ﬂux changes in steps of/Phi1 0with increasing applied B, the supercurrent Isaround the loop will oscillate to keep the ﬂuxoid /Phi1/primequantized to ensure that the phase is single valued at any point around the loop.In doing so, the order parameter is suppressed when I sis on a maximum. At certain values of TandB, this maximum current may drive some weak links in the loop resistive, leading to aperiodic variation in resistance with a period /Delta1B≈/Phi1 0/S.I f such loops form randomly through coupling of nearby grains,one would expect that the geometry of the coupled structureswould change with TandB. Also one would expect there to be a multiplicity of loop sizes and therefore a superpositionof periods, whereas we ﬁnd only one prominent period in theFFT spectrum that is independent of all parameters. Thereis also the matter of the loop size, which would have tohave the area S=/Phi1 0//Delta1B≈1172nm2transverse to the ﬁeld for our observed /Delta1B≈0.15 T. This dimension is smaller than the ﬁlm thickness in one case, but much larger thandin other cases. How does one address the B /bardblcase when d/lessmuch117 nm? By taking d×las the effective area? The two SLCO samples with identical periods had d×l=34002and 25002nm2, both much larger than the required 1172nm2.F o r the loop theory to work, somehow the length of the bridgewill have to be broken up into segments of length l /primesuch thatd×l/prime≈1172nm2. For our range of thicknesses, this would imply l/prime=59–2700 nm to magically produce the same constant d×l/prime≈1172nm2. There is no evidence to support such a segmentation. Therefore, a possible explanation foroscillations along these lines is probably not valid, as alsonoted by others [ 27,31]. Aside from well connected loops leading to Little Parks like oscillations, Josephson junctions can lead to oscillatory 054504-5KUNCHUR, DEAN, AND IVLEV PHYSICAL REVIEW B 94, 054504 (2016) interference effects and granular systems can surely have junc- tions between grains. A single junction can itself produce OMRbecause of the Fraunhofer pattern in the functional dependencebetween the maximum supercurrent it can carry and the ﬂuxlinking the junction: I m(B)=Im(0)\|sin(π/Phi1//Phi1 0)/(π/Phi1//Phi1 0)\|. Minima in Im(B) will correspond to maxima in R(B) with a periodicity in linked ﬂux of /Phi10. For a junction with parallel faces and a rectangular cross section (of separation Dand length L) transverse to B, the linked ﬂux is /Phi1=BL(D+2λ), where λis the magnetic penetration depth. This leads to the requirement L(D+2λ)≈1172nm2for the observed /Delta1B =0.15 T. However, there are two problems with this: ﬁrst, λvaries with T,s o/Delta1B would not be Tindependent and, second, the linked area and hence /Delta1B would depend on the orientation of B. Furthermore, the neat Fraunhofer pattern becomes replaced by a more general Fourier transform ifthe junction is not rectangular or has a variation in the localcritical current density over the face of the junction. Add tothis the complexity of having a spread in junction sizes andcharacteristics in a random granular system and it becomesclear that the single junction diffraction pattern is not a viablecandidate for the type of OMR discussed in this work. We next consider closed paths containing multiple JJs, taking ﬁrst the simplest case of a two-junction loop in whichthe junctions themselves are small enough to have negligiblesingle-junction diffraction effects. The net current enteringand leaving the loop is the phase sensitive summation ofthe currents through the two junctions. The supercurrentthrough each JJ is given by I s=Icsinγ, where γis the gauge invariant phase difference across the junction and Ic is the JJ critical current (above which resistance appears). Since the net phase difference going around the loop mustbe single valued, we have γ 1−γ2=2π/Phi1//Phi1 0(mod 2 π). Thus the two JJs can carry their maximum critical currents(when γ 1=γ2=π/2) when /Phi1is an integer multiple of /Phi10 once again leading to the periodicity /Delta1B=/Phi10/S; however, if there are more than two JJs in the loop, or if the loops havedifferent areas, or if the junctions themselves have appreciableﬂux linkage to exhibit their own diffraction effects, a singlewell deﬁned periodicity will not exist. Thus a granular system,unless it is so small that it includes only a few grains, will nothave oscillations with a single clearly deﬁned period. Thiswas exactly the observation by Herzog et al. , who found that their granular tin ﬁlms did not display any oscillations unlesspatterned down to extremely small nanowires, and even thencontained a superposition of multiple periods. The epitaxialﬁlms in our work show OMR irrespective of the bridge sizeand ﬁeld orientation. In fact our bridges are seven orders ofmagnitude greater in volume than the nanowires in some ofthese other works, yet we see clean single-period OMR. Suchan extended system will only have ﬂuxoid/vortex based clearOMR if it is purposely patterned with a regular array as inRef. [ 26]. It seems impossible for the bridge to accidentally and coincidentally have loops of 117 2nm2w h e nt h ea r e ao f the bridge facing the ﬁeld is 1011nm2. The formation and motion of vortices—and the interplay between the applied magnetic ﬁeld, thermal excitations, andthe self-ﬁeld of the current—provides yet another basis forOMR. Modulations in the barriers at the sample edges forentry/exit of vortices lead to changes in the static and dynamicphases of the vortex matter. This in turn can lead to alternations in the dissipation even in singly connected superconductingstrips. These ideas were proposed by Anderson and Day-ton [ 32], and elaborated upon by Sochnikov et al. [22] and Berdiyorov et al. [24]. More recently, Berdiyorov et al. [33] performed a careful Ginzburg-Landau study with numericalsimulations of these regimes for the B ⊥case; they found that the qualitative nature and period of the OMR dependedon the strip width and the amount of disorder. Indeed thisvariability of /Delta1B in vortex based OMR was reported in even the earliest experiments by Parks and Mochel [ 34,35]. Thus this class of vortex-based effects cannot explain the nearlyconstant /Delta1B observed in the present work. For Bclose to B /bardbl, rearrangements of the parallel vortex system [ 36] provide another mechanism for oscillations; however, the positions ofthe peaks depend on the ﬁeld tilt angle, whereas the presentoscillations don’t depend on the angle and even occur for B ⊥. Finally, to cover other known origins of OMR, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulated super-conducting state [ 37,38] could result in oscillations if its wave vector q∼2μ BB/(/planckover2pi1vF) (in the microns range) matches some intrinsic spatial periodicity, but this doesn’t match any knownlength scale in our systems, and the /Delta1Bwould be expected to depend on the material and the dimensions. Furthermore thepresent ﬁlms do not have the special characteristics (e.g., clean,appropriate nesting of Fermi surfaces, etc.) that favor FFLOformation. The Shubnikov–de Haas effect [ 39] is another scenario that produces OMR; however, these are periodicin 1/B, rather than in B, and occur at very high values B/greatermuch1 T. All of the above effects depend on the orientation ofB, material parameters (Fermi-surface geometry, Fermi velocity, coherence length, etc.), and/or sample dimensionsand geometry. Thus the previously established mechanismsthat we are aware of cannot straightforwardly explain theobserved nearly universal periodicity. So what other physical mechanisms might underlie the oscillations? In the dissipative state of a superconductor, resis-tance rises—whether the current is ﬂowing through the bulk orpercolating across junctions—when the superconducting stateis weakened, signaled by a reduction in the order parameter /Delta1. Thus minima in R(B) may be assumed to reﬂect values of B where there are peaks in /Delta1./Delta1can rise because of an increase in the strength of the attractive pairing interaction, an increasein the density of states (DOS) at the Fermi level, or a reductionof pair breakers that interfere with the pairing. We are notaware of any mechanism by which Bcould directly alter the fundamental pairing attraction. However, Bcan affect the DOS and it certainly causes pair breaking. A magnetic ﬁeld weakensthe superconducting state in two main ways: through orbitalpair breaking and through Pauli paramagnetism. The former,both in the mixed state and the ﬂuctuation regime [ 40], is highly material, temperature, and ﬁeld-orientation dependent,so we reject the role of this component. The paramagneticeffect, on the other hand, leads to an energy splitting /Delta1E p= ±μB/Delta1B≈± 9μeV (for the observed /Delta1B≈0.15 T) between opposite spins, which is indeed universal and not dependenton the material or other parameters (here μ B=e/planckover2pi1/2mis the Bohr magneton). Normally this would lead to a progressiveweakening of /Delta1and a monotonic rise in Rwith increasing B. However, there are two ways by which /Delta1could be strengthened 054504-6ANOMALOUS OSCILLATORY MAGNETORESISTANCE IN . . . PHYSICAL REVIEW B 94, 054504 (2016) byBas follows. (1) One is if the DOS is a nonmonotonic function of energy, e.g., if the /Delta1Eppushed the Fermi level for one of the spin directions into the vicinity of a peak in the DOS,then the ﬁeld could actually strengthen the superconductingstate. For multiple oscillatory periods, one would need a“comb” in the DOS. (2) The second route is if /Delta1E pcanceled a preexisting pair breaker which had a higher energy forspin-up versus spin-down states, in which case the /Delta1E p would reduce the net pair breaking by bringing the energies of time reversed states into alignment and thereby strengthenthe superconducting state. (An example of a situation wherea spin-up state has a higher intrinsic energy than a spin-downstate, is the 2 S 1/2–2P1/2[equal total angular momentum] pair of states in a singular attractive potential, whose degeneracy islifted because of QED radiative corrections from interactionswith the vacuum; in the context of an atom, this process ismanifested as the familiar Lamb shift, which is of the order of10μeV .) At the present time, it is not clear whether either of these mechanisms have any relevance to the observedphenomenon, or how they would lead to multiple oscillationperiods beyond a single dip in R. The ideas are only suggested as possible directions for exploring explanations. V. SUMMARY AND CONCLUSIONS We have observed periodic oscillations in magnetore- sistance at low ﬁelds ( B∼0–1 T) with a period that is independent of temperature, magnetic ﬁeld, electric transportcurrent, and even the orientation of the magnetic ﬁeld.Moreover, the period doesn’t vary much even across differentmaterials. Collectively, between our measurements and theearlier cited work, a very wide range of materials (from leadto cuprates) are covered. The different ﬁlms were made withalmost every possible method of deposition (from thermaland electron-beam evaporation to MBE) and patterned intobridges/wires by a multitude of techniques (focused ion-beamlithography, shadow masking, resist based lithography withchemical etching as well as ion milling, etc.). The samplesizes span many orders of magnitude for each dimension:d=7–250 nm, w=0.1–50μm, and l=1–2000 μm, and have a variety of geometries and lead arrangements. ThisOMR has been observed at temperatures ranging from 12 mKto 74 K, and these measurements were conducted using avariety of cryogenic systems, which include a closed-cyclecryocooler (the present work), a Quantum Design PhysicalProperties Measurement system [ 27], and even a helium-3 cryostat and dilution refrigerator [ 28]. Yet the periods span a rather narrow range: all three cuprate materials have a periodof 0.15±0.01 T, the conventional low- T cﬁlm has a period of 0.12 T, and our interfacial superconductor has a period of0.11 T. The earlier observations of similar low-ﬁeld periodicOMR in conventional low- T cmaterials also have comparable periods of /Delta1B=0.12, 0.13, and 0.18 T. To our knowledge and understanding, it is hard to reconcile this magnitude andnear universality of period with models invoking vortices,ﬂuxoid quantization, and Josephson-junction type of effects.It seems that the oscillations also cannot originate from Fermi-surface based microscopic phenomena that are known to causeOMR. Thus, while it may still be possible to explain ourobservations through some intricate or exotic modiﬁcationsof the known aforementioned mechanisms, the results alsoindicate the possibility that a potentially new phenomenonmight be operative and some suggestions were made in thetext as to possible directions for pursuing an explanation.We hope that our observations and information will stimulatefurther experimental and theoretical investigations into thisphenomenon. ACKNOWLEDGMENTS The authors would like to thank the following for providing samples, useful discussions, and other assistance: L. Fruchter,Z. Z. Li, M. Liang, J. M. Knight, N. S. Moghaddam, S. Varner,R. A. Webb, K. Stephenson, N. Lu, and M. Geller. This workwas supported by the U. S. Department of Energy, Ofﬁceof Science, Ofﬁce of Basic Energy Sciences, under GrantNo. DE-FG02-99ER45763. B.I.I. acknowledges support fromCONACYT through Grant No. 237439. [1] W. A. Little and R. D. Parks, P h y s .R e v .L e t t . 9,9(1962 ); R. D. Parks and W. A. Little, Phys. Rev. 133,A97 (1964 ). [2] J. Bardeen, P h y s .R e v .L e t t . 7,162(1961 ). [3] R. P. Groff and R. D. Parks, Phys. Rev. 176,567(1968 ). [4] H. J. Fink, A. Lopez, and R. Maynard, Phys. Rev. B 26,5237 (1982 ). [5] J. Simonin, D. Rodrigues, and A. Lopez, P h y s .R e v .L e t t . 49, 944(1982 ). [ 6 ] R .M .A r u t u n i a na n dG .F .Z h a r k o v , J. Low Temp. Phys. 52,409 (1983 ). [7] R. Rammal, T. C. Lubensky, and G. Toulouse, P h y s .R e v .B 27, 2820 (1983 ). [8] J. M. Gordon, A. M. Goldman, and B. Whitehead, Phys. Rev. Lett.59,2311 (1987 ). [9] F. Nori and Q. Niu, Physica B 152,105(1988 ). [10] J. M. Gordon and A. M. 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PhysRevB.75.214411.pdf	MgO/Fe „100 …interface: A study of the electronic structure L. Plucinski, Y. Zhao, and B. Sinkovic Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA E. Vescovo National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973, USA /H20849Received 3 December 2006; revised manuscript received 1 April 2007; published 6 June 2007 /H20850 Epitaxial MgO /H20849100 /H20850ﬁlms, deposited on the Fe /H20849100 /H20850surface by direct evaporation of stoichiometric magne- sium oxide under ultrahigh-vacuum conditions, are examined by spin-polarized valence-band photoemissionand magnetic linear dichroism in core-level spectroscopy. The excellent quality of the MgO ﬁlms prepared inthis way is clearly revealed by relatively small photocurrent intensity observed above the MgO valence-bandmaximum and by spectral features in the photoemission data. No evidence of the formation of an interfacialFeO layer is found in this MgO/Fe /H20849100 /H20850system. Only a structural reordering is promoted by annealing the interface up to 400 °C, as evident by the minute changes in the photoemission spectra. Further annealing to500 °C, however, leads to more substantial changes in the spectra, possibly related to further ordering of theinterface and/or partial uncovering of the Fe /H20849100 /H20850metal due to MgO desorption and/or clustering. The inﬂu- ence of screening effects from the Fe /H20849100 /H20850substrate to the photohole created by the photoemission process in the MgO overlayer is also examined. DOI: 10.1103/PhysRevB.75.214411 PACS number /H20849s/H20850: 75.70. /H11002i, 79.60. /H11002i, 68.35. /H11002p I. INTRODUCTION In recent years, there have been signiﬁcant advances in the preparation of single crystalline magnetic tunnel junc-tions /H20849MTJs /H20850based on MgO /H20849100 /H20850insulating barriers. For ex- ample, the theoretically predicted tunneling magnetoresis-tance /H20849TMR /H20850of several hundred percent 1,2in Fe/MgO/Fe /H20849100 /H20850has been experimentally veriﬁed with val- ues of 180–220 % for room temperature /H20849Refs. 3and4/H20850and higher values are being reported for various other electrodecompositions. 5–7 The exact composition and atomic structure at the barrier/ ferromagnet interface are of primary importance for the per-formance of single crystalline MTJs. In the case ofMgO/Fe /H20849100 /H20850, surface x-ray diffraction measurements 8sug- gest the existence of a partial interfacial FeO layer, which could strongly reduce the Fe spin polarization at the interfaceand, therefore, negatively inﬂuence the performance of theMTJ. 9Other studies10ﬁnd no interfacial FeO layer, and also when the second interface is formed, i.e., Fe is deposited onMgO /H20849100 /H20850, such an oxide layer is clearly absent. 11,12Thus, an FeO layer does not appear to be an intrinsic property ofthe interface. This view is also supported by theoretical cal-culations by Li and Freeman, 13which ﬁnd little interaction between a thin Fe /H20849100 /H20850layer and the MgO /H20849100 /H20850substrate. Similarly, recent total energy calculations of Yu and Kim14 also ﬁnd that no formation of interfacial FeO is preferredunder Mg-rich conditions. Manipulating the composition of the electrodes and im- proving the crystalline structure at the ferromagnet-barrierinterface are keys to fabricating alternative devices withhigher TMR. A considerable effort, theoretical as well asexperimental, is currently underway to investigate this topic.Most notably, recent theoretical calculations suggest placinga thin layer of Au /H20849Ref. 15/H20850or Ag /H20849Ref. 16/H20850in between the ferromagnetic electrode /H20849Fe/H20850and the insulating barrier/H20849MgO /H20850, as an effective way to prevent oxidation of the fer- romagnet, thus preserving its large spin polarization at theinterface. Large enhancements of the TMR effect /H20849up to 1000% /H20850are calculated for these ferromagnet/noble metal/ barrier systems, assuming ideal interfaces. On the otherhand, experimental work indicates that annealing up to400 °C leads to substantial improvements in the magnetore-sistance of the “simple” Fe/MgO/Fe /H20849100 /H20850tunnel junctions prepared at room temperature. 3,6,7,17The characterization of the MgO/Fe interface is, therefore, an important and timelyissue. Spin- and angular-resolved photoemissions are particu- larly suited for characterization of these MTJ systems. In- deed, the electronic structure at the MgO/Fe /H20849100 /H20850interface has already been probed by this technique. Valence-band spectra in the submonolayer and up to 1 ML /H20849monolayer /H20850 MgO coverage have been reported in the photon energyrange between 35 and 60 eV for MgO ﬁlms prepared bydepositing metallic Mg in the presence of an oxygenatmosphere. 18Furthermore, soft x-ray spin-polarized photo- emission spectra, which represent the density of states, of 2ML MgO/Fe /H20849100 /H20850grown on bulk MgO /H20849100 /H20850were presented by Sicot et al. 19,20More recently, a study combining spin- polarized photoemission, diffraction, and microscopy ofMgO grown on another low index Fe surface, Fe /H20849110 /H20850, 21re- vealed the existence of an interfacial FeO layer for theMgO/Fe /H20849110 /H20850interface. However, the MgO ﬁlms in this study were grown by exposing the Fe /H20849110 /H20850surface to Mg vapors in a controlled oxygen atmosphere at room tempera-ture /H20849RT/H20850, 22similar to the method used in Ref. 18. In the present study, valence-band spin-polarized photo- emission spectra and core-level magnetic linear dichroismspectra are presented for the MgO/Fe /H20849100 /H20850system for MgO thicknesses between 0.5 and 10 ML. Moderately high inci- dent photon energy /H20849h /H9263=128 eV /H20850is used to minimize sur- face contributions and simultaneously excite valence bandsPHYSICAL REVIEW B 75, 214411 /H208492007 /H20850 1098-0121/2007/75 /H2084921/H20850/214411 /H208498/H20850 ©2007 The American Physical Society 214411-1and shallow Fe 3 pand Mg 2 pcore levels. The MgO/Fe /H20849100 /H20850junction is prepared at RT by direct evapora- tion of stoichiometric MgO onto the clean Fe /H20849100 /H20850surface under ultrahigh-vacuum conditions /H20849UHV /H20850. In this way, it is less likely to form the FeO layer at the MgO/Fe /H20849100 /H20850inter- face, which we conﬁrm by the combined analysis of valence- band and core-level spectra. Furthermore, in view of reportedpostannealing enhancement of TMR mentioned above, 3,6,7,17 the stability of the MgO/Fe /H20849100 /H20850interface upon annealing has been investigated. Annealing up to 400 °C produces only atomic reordering of the interface as evidenced by the smallmodiﬁcations in the photoemission spectra, mostly a sharp-ening of all spectral features. More substantial modiﬁcationsof the spectra are instead observed by annealing to 500 °C.These higher-temperature changes are possibly related eitherto further ordering of the interface or to the desorption and/orclustering of MgO. Additionally, an interesting dynamic property of the elec- tronic structure of the MgO/Fe interface is discussed. Theresponse of the conduction electrons from the Fe side of theinterface to the photohole creation in the thin MgO layers isobserved as additional screening in photoemission spectra. II. EXPERIMENT Experiments were performed at beamline U5UA at the National Synchrotron Light Source /H20849NSLS /H20850. This beamline is equipped with a planar undulator and a spherical gratingmonochromator. 23The photon beam covers the range 20–200 eV and is highly linearly polarized in the horizontalplane. A commercial hemispherical electron energy analyzer, 24 originally equipped with seven channeltrons, has been modi-ﬁed to host a mini-Mott spin polarimeter, 25leaving three channeltrons for standard high-resolution spin integratedphotoemission. The analyzer was set to the angular-resolvedmode /H20849±1° angular acceptance /H20850. The experiments were per- formed in normal emission with the photon beam incident at45° with respect to the sample normal. Typical overall reso-lution was 150 meV for spin-integrated and 300 meV forspin-polarized spectra. AM o /H20849100 /H20850single crystal was chosen as the substrate. It was mounted on an UHV compatible manipulator with the/H20851001 /H20852direction vertical and cleaned by repeated ﬂashing to 2000 °C. The sample temperature was monitored by a W-Rhthermocouple attached directly to the Mo /H20849100 /H20850crystal. Fe/H20849100 /H20850ﬁlms of 40 Å thickness were grown at RT at the rate of 5 Å/min by electron-beam evaporation. The basepressure in the chamber was 5 /H1100310 −11Torr and rose to about1/H1100310−10Torr during Fe depositions. We are not aware of anyin situ UHV study of Fe growth on Mo /H20849100 /H20850in the coverage region thicker than a few monolayers,26although Mo /H20849100 /H20850single-crystal substrates were successfully used to grow sophisticated Fe/Mo multilayer structures.27Our low- energy electron-diffraction /H20849LEED /H20850observations indicate that above a few monolayers, the growth proceeds epitaxiallywith a relaxed bulk Fe lattice constant. This behavior is es-sentially identical to that found in the similar Fe/W /H20849100 /H20850 system 28/H20849Mo and W have the same bcc structure and very similar lattice constants, aMo=3.147 Å, aW=3.165 Å /H20850. The preparation of atomically ﬂat Fe /H20849100 /H20850ﬁlms requires special precautions. We have found improvements in thephotoemission spectra /H20849sharpening of the features /H20850and LEED patterns of Fe ﬁlms upon annealing the sample to500 °C for approximately 10 min, and such annealedFe/H20849100 /H20850ﬁlms were used for subsequent MgO deposition. Figure 1presents spectra of Fe/Mo /H20849100 /H20850ﬁlms before and after annealing to 500 °C in comparison with the spectrum of clean Mo /H20849100 /H20850. There is no indication of Mo core levels in the Fe/Mo /H20849100 /H20850spectrum after annealing, which excludes the model with 2 ML wetting layer and three-dimensional islands. 29,30 MgO was deposited at the slow rate of approximately 0.2 Å/min from pieces of stoichiometric MgO single crys-tals put into a tungsten crucible and heated by electron-beambombardment. The coverage thicknesses were estimated bycalibrating the evaporator in situ with a quartz balance and FIG. 2. /H20849Color online /H20850LEED patterns for /H20849a/H20850 clean Fe /H20849001 /H20850,/H20849b/H208501 ML MgO/Fe /H20849001 /H20850, and /H20849c/H20850 10 ML MgO/Fe /H20849001 /H20850. FIG. 1. /H20849Color online /H20850Survey photoemission spectra in the re- gion of Mo 4 s,M o 4 p, and Fe 3 dcore levels. The annotations are printed in ﬁgure.PLUCINSKI et al. PHYSICAL REVIEW B 75, 214411 /H208492007 /H20850 214411-2further conﬁrmed by comparisons to previous studies.18 MgO ﬁlms are known to grow pseudomorphically in a layer- by-layer fashion on the Fe /H20849100 /H20850surface up to about 6 ML /H20849Ref. 31/H20850, while the thickness of the barrier in the optimized MTJs is between 5 and 15 ML.3,41 ML of MgO corresponds to approximately 2.2 Å thickness.32The epitaxial growth mode of the MgO /H20849100 /H20850overlayers on Fe /H20849100 /H20850/Mo /H20849100 /H20850is conﬁrmed in the LEED patterns displayed in Fig. 2. While these LEED patterns are not as sharp as those from MgOoverlayers grown on Fe whiskers, 31they do show fourfold satellite spots, surrounding each /H2084910/H20850LEED beam, in good agreement with previous observations. These satellite spotsare clearly visible at high MgO coverage /H20851see LEED picture for 10 ML MgO, Fig. 2/H20849c/H20850/H20852and are related to misﬁt dislocations. 31III. RESULTS AND DISCUSSION A. Valence-band measurements In Fig. 3, a portion of the bulk band structure of Fe is compared with the spin-resolved photoemission spectrummeasured from the clean Fe /H20849100 /H20850surface in normal emission with photon energy h /H9263=128 eV. In normal emission geom- etry, the photoemission experiment from the /H20849100 /H20850surface probes the Fe bands along the /H9003-Hhigh-symmetry line of the bulk Brillouin zone. Assuming, as usual, direct transitionsand free-electron ﬁnal states, only the states intersecting thefree-electron parabola shifted by h /H9263=128 eV /H20849see Fig. 3/H20850are detected in the photoemission spectrum. As one can see fromFig. 3, an excellent agreement between theory and experi- ment is found. The majority bands are located in two regions,close to the Fermi level and at 4 eV binding energy, whilethe energy region in between is ﬁlled by the minority statesat approximately 2.5 eV binding energy. High quality of ourﬁlms is further conﬁrmed by clear band dispersions in offnormal emission at various photon energies /H20849not shown here /H20850. From the band structure shown in Fig. 3, one can see that the Fe 3 dstates are mostly conﬁned within the ﬁrst 4 eV below the Fermi energy, while the emission from MgO ismostly at binding energies above 4 eV. This separation be-tween the valence-band emission of the two materials offers,therefore, the opportunity of examining the modiﬁcations ofthe Fe 3 dstates upon the gradual formation of the Mg/Fe /H20849100 /H20850interface. A representative collection of the valence-band photo- emission spectra recorded as a function of MgO deposition isshown in Fig. 4/H20849a/H20850. For each coverage, both the spin-resolved components and their sum spectra are shown. For low cov-erage /H208490.5, 1, and 2 ML MgO /H20850, very little modiﬁcation is seen in the emission close to E F/H20849between 0 and 3.5 eV /H20850, while the major modiﬁcation is the appearance of the intenseemission centered at approximately 5.5 eV, characteristic of FIG. 3. /H20849Color online /H20850Left panel: bulk band structure of Fe along the /H9003-Hhigh-symmetry direction calculated using WIEN2K /H20849Ref. 49/H20850. Solid and dashed lines represent majority and minority spin bands, respectively. The energy distribution curve appropriateforh /H9263=128 eV /H20849assuming direct transitions, free-electron ﬁnal states, and inner potential V0=10 eV /H20850is superimposed on the band- structure calculation. Right panel: spin-polarized photoemissionspectrum taken at h /H9263=128 eV in normal emission. /H17012, majority spin;/H17008, minority spin. FIG. 4. /H20849Color online /H20850/H20849a/H20850Series of normal emission spin-integrated and spin-polarized spec-tra of MgO/Fe /H20849001 /H20850ath /H9263=128 eV, /H20849b/H20850compari- son of spin-polarized spectra of clean andoxygen-exposed Fe /H20849001 /H20850, and /H20849c/H20850the effect of 400 °C annealing of 1 and 2 ML MgO/Fe ﬁlms.The annotations are printed in ﬁgure; /H17009, majority spin;/H17006, minority spin. MgO-related features show up at binding energies above 4 eV and arepredominantly O 2 pcharacter. Spectrum of 10 ML MgO/Fe was taken at 100 K and all otherspectra were recorded at room temperature.MgO/Fe /H20849100 /H20850INTERFACE: A STUDY OF THE … PHYSICAL REVIEW B 75, 214411 /H208492007 /H20850 214411-3highly hybridized O 2 pand Mg 3 sof the MgO ﬁlm. Clearly, the Fe 3 demission close to EFremains highly spin polarized upon 1 and 2 ML MgO depositions. The MgOfeatures at high binding energy instead are found equallydistributed in the spin-up and spin-down channels, indicatingthat no polarization is induced in the MgO layer. It is difﬁcult to provide unique interpretation of the small changes observed in the region close to E F. We observe simi- lar changes at other photon energies /H20849not shown /H20850, which is consistent with the study reported by Matthes et al.18utiliz- ing photon energies of 30–60 eV. Although still clearlydominated by the highly polarized Fe 3 dstates, this region should also contain some emission due to the likely defectswithin the MgO overlayer. There is a nearly perfect in-planelattice match between MgO /H20849100 /H20850and Fe /H20849100 /H20850; however, a large number of defects can be expected in the thin ﬁlm dueto the large out-of-plane lattice mismatch /H20849interlayer dis- tances in bulk materials: 32Fe, 1.443 Å; MgO, 2.106 Å /H20850. Ev- ery terrace boundary in the Fe substrate is, therefore, asource of defects in the MgO overlayer. The presence ofdefects in the thin ﬁlms is also consistent with the absence ofcharging effects in the photoemission experiments from theseﬁlms. The spectrum obtained from the 10 ML thick MgOﬁlm, shown at the top of Fig. 4/H20849a/H20850, shows feature assigned to residual defects peaks at about 1.7 eV binding energy. Such abroad unpolarized feature located at this binding energy is,therefore, likely to represent the emission from the MgOdefects at low coverage. A spectral feature of this type mayexplain the modiﬁcations observed in the low coverage spec-tra in the low binding-energy region. As previously stated, the MgO emission centered at 5.5 eV binding energy is found equally distributed in thespin-up and spin-down channels, offset only by the differ-ence in their respective backgrounds. Theoretical simulationsof the electronic band structure of Zhang et al. 33for the MgO/Fe system with an FeO interfacial layer ﬁnd that thestrong hybridization between Fe and O in the FeO layer in-duces a sizable magnetic moment /H20849as large as 0.19 /H9262B/H20850on the O site of the FeO. Such a large magnetic moment should bereﬂected in a corresponding large exchange splitting in theO2pband and should be easily detected in the spin-resolved spectra. The experimental results of Fig. 4/H20849a/H20850, showing no difference in shape and binding energy between majority andminority spin spectra for low coverage MgO, constitute,therefore, the ﬁrst strong indication of the absence of inter-facial FeO. It is worth noting that the very small emission in the re- gion close to E Ffor the high coverage regime /H20851note the mul- tiplication factor of 50 in the top spectra in Fig. 4/H20849a/H20850/H20852is a clear indication of the high structural quality of these MgOthick ﬁlms. MgO ﬁlms thermally evaporated under UHVconditions from stoichiometric MgO source material are in-deed known to be nearly stoichiometric or slightly oxygendeﬁcient. 34Theoretical calculations of MgO /H20849100 /H20850oxygen de- ﬁcient surfaces35–37predict wide oxygen vacancy features at 1 eV below the Fermi edge and 2 eV above the valence-bandmaximum /H20849VBM /H20850of MgO. Such features do not necessarily result from bulk vacancies; they may also result from theintrinsic properties of the MgO surface, 35irregularities, and imperfections of the surface.37Only weak vacancy featureswere present in our spectra at high MgO coverage, as judged by the intensity ratio between the oxygen vacancy and themain O 2 pfeature, being below 0.5% in our ﬁlms /H20851from Fig. 4/H20849a/H20850/H20852compared to 5% in the spectra from bulk MgO reported by Tjeng et al. 38Although these bulk spectra were taken with h/H9263=21.2 eV /H20849HeIlamp /H20850, our spectra taken at h/H9263=64 eV /H20849not shown /H20850still show the ratio to be below 1%. We take this as an indication that our MgO ﬁlms are grown with good sto-ichiometry. Another indication comes from the observationof a weak but clear O 2 pfeatures already at very low MgO coverage /H208490.5 ML /H20850and at the binding energy equal to one for thicker ﬁlms. From the above considerations, it is already quite clear that the Fe /H20849100 /H20850surface does not react strongly with the MgO. In order to further address the speciﬁc question con-cerning the formation of an FeO layer at the interface, wehave performed additional experiments intentionally expos-ing the clean Fe surface to small dosages of molecular oxy-gen. The photoemission spectrum for 1 L /H208491L =10 −6Torr s /H20850oxygen exposure is compared with the one from a clean Fe surface in Fig. 4/H20849b/H20850. The effect of 1 L oxy- gen exposure is quite different from the one of the low cov-erage MgO. Already upon 1 L of oxygen exposure, thespin-up and spin-down spectra tend to assume a similarshape in the Fe 3 dregion, suggesting a depolarization of the surface layer as one would expect when forming FeO. 39This different behavior is, therefore, a second indication of theabsence of an FeO layer at the interface. Finally, an additional point can be learned from the valence-band spectra. In Fig. 4/H20849c/H20850, the spin-integrated spectra for 1 and 2 ML MgO/Fe /H20849100 /H20850prepared at room temperature and after annealing to 400 °C are compared. All of the room- temperature spectral features are quite broad and so one doesnot expect major changes from annealing; however, a distinctsharpening of the spectral features is clearly seen, most no-tably in the ﬁne structure in O 2 p/H20849see arrows in Fig. 4/H20850. The sharpening of the spectral features in angular-resolved pho-toemission is usually an indication of atomic ordering andreduction of local inhomogeneities. In Fig. 5, the effects of annealin ga2M LM g O overlayer are further explored. As already noticed, the deposition of 2ML of MgO leaves a strong polarization of the Fe-relatedpart of the spectra /H20849above the VBM of MgO /H20850, although all the spectral features are broadened by the deposition of theMgO. Annealing these MgO ﬁlms up to 400 °C begins tosharpen the Fe 3 dspectral features in the region close to E F and this process is even more visible in the spectrum an- nealed at 500 °C, where the sharp spectral feature of theclean Fe surface is nearly fully recovered /H20849see right panel in Fig.5/H20850. At ﬁrst sight, this behavior seems to be consistent with a further increase of atomic reordering of the MgO interface athigher temperature. A reordering would certainly correspondto a decreased emission from MgO defects in the region ofthe Fe 3 d. Furthermore, the electrons excited in the Fe un- derneath MgO would leave the surface with less scattering ifthe crystalline structure of the interface is improved and thiswould explain both the reappearance of the sharp Fe featuresand the decrease in the intensity ratio of O 2 pto Fe-related parts of spectra.PLUCINSKI et al. PHYSICAL REVIEW B 75, 214411 /H208492007 /H20850 214411-4We cannot, however, exclude also the possibility that the recovering of the sharp Fe 3 dfeature could be due to partial uncovering of the bare Fe surface by clustering and/or de-sorption of the MgO. After all, MgO molecules must havesome degree of mobility at high temperature if reordering isaffected. Furthermore, this hypothesis would also naturallyexplain the decrease in the intensity ratio of O 2 pto Fe- related parts of the spectra in Fig. 5/H20849d/H20850, compared to the constant ratio in Figs. 5/H20849b/H20850and5/H20849c/H20850. Finally, this interpreta- tion would also explain the degradation of the magnetic tun-neling properties experimentally observed for temperaturesjust above 400 °C. 3 In summary, on the important topic of the temperature stability of the MgO/Fe interface, the photoemission data donot offer a complete answer. The two different models pre-sented above would seem to be both consistent with thepresent observations, although they would have quite differ-ent implications for the fabrication of high-quality MTJs. Away to sort out this question would possibly be to performmicroscopy studies on this system. B. Fe 3 pand Mg 2 pcore-level measurements We now turn to the study of the shallow Fe and Mg core levels. As mentioned above, at the photon energy of 128 eV,one can simultaneously excite both the Fe 3 pand the Mg 2 p levels. These core levels are located at binding energies of 53and 51 eV, respectively, and, although quite close, they stilloffer the possibility of monitoring the two different materialsduring the formation of the interface. Furthermore, the Fe 3 p levels exhibit a large magnetic linear dichroism inphotoemission, 40which can be used to monitor the magnetic state of the Fe layer. The core-level data are displayed in Fig.6, where results concerning the growth and annealing of MgO layers on top of Fe /H20849001 /H20850surface are shown, similar tothe study utilizing the valence-band photoemission already discussed. The left panel presents mostly dichroic averagedspectra /H20849continuous lines /H20850, while the right panel shows the corresponding dichroism spectra /H20849open and closed circles /H20850,i n an expanded view emphasizing the smaller region around theFe 3plevels. Comparing the spectra from the clean Fe /H20849100 /H20850surface /H20851Fig.6/H20849a/H20850/H20852to the ones exposed to 1 L of molecular oxygen /H20851Fig.6/H20849b/H20850/H20852, the appearance of a shoulder on the high binding- energy side of the spectrum is evident. This shoulder feature,best seen in the right panel of Fig. 6, is typical of the reaction of oxygen with Fe and indicates the formation of Fe oxide. 41 In agreement with the valence-band observations, this shoul-der does not display any dichroism, indicating the formationof an unpolarized surface oxide upon even 1 L of oxygenexposure to the clean Fe surface. Most importantly, whenMgO is deposited on clean Fe /H20849001 /H20850instead /H20851Fig.6/H20849c/H20850/H20852, there is no indication of this feature in the spectra. On the contrary,the curvature of the relevant part of the core-level spectrumremains clearly positive for 1 ML MgO coverage, which isfurther evidence that no FeO layer is present at theMgO /H20849100 /H20850/Fe /H20849100 /H20850interface. At higher MgO deposition, the Fe 3 pemission becomes rapidly obscured by the much more intense Mg 2 plevels /H20851Figs. 6/H20849c/H20850and6/H20849d/H20850/H20852but the Fe dichroism remains strong, which indicates a highly polarized Fe substrate in contactwith the MgO overlayer. It is also useful to note that whilethe Fe dichroism is very strong, no sign of any dichroism isdetected under the Mg peak /H20849the full dichroic spectra are shown in the left panel for the case 0.5 ML MgO, as anexample /H20850. This again tends to conﬁrm the low interaction between Fe and MgO at the interface. FIG. 5. /H20849Color online /H20850Spin-polarized photoemission spectra at h/H9263=128 eV; /H17009, majority spin; /H17006, minority spin. Left panel: /H20849a/H20850 clean Fe /H20849100 /H20850and /H20849b/H208502M Lo fM g O / F e /H20849100 /H20850/H20849c/H20850annealed to 400 °C for 5 min and /H20849d/H20850subsequently annealed to 500 °C for 5 min. In right panel, selected spectra are magniﬁed in the regionclose to the Fermi edge. Spectra are normalized to the majority spinFe-related part near the Fermi edge. All spectra were taken at roomtemperature. FIG. 6. /H20849Color online /H20850Fe 3pand Mg 2 pcore-level normal emis- sion energy distribution curves at h/H9263=128 eV; /H20849a/H20850clean Fe /H20849001 /H20850, /H20849b/H20850Fe/H20849001 /H20850exposed to 1 L of molecular oxygen, /H20849c/H208500.5 ML MgO/Fe /H20849001 /H20850deposited on clean Fe /H20849001 /H20850, /H20849d/H208501M L MgO/Fe /H20849001 /H20850, and /H20849e/H20850previous ﬁlms annealed to 400 °C for 5 min. In the left panel, the sum of spectra from the sample mag-netized in opposite directions is plotted. The right panel shows thesame spectra as on the left but renormalized to magnify the Fe 3 p contribution. Opened and closed circles represent spectra from thesample magnetized in opposite directions. All spectra were mea-sured at room temperature.MgO/Fe /H20849100 /H20850INTERFACE: A STUDY OF THE … PHYSICAL REVIEW B 75, 214411 /H208492007 /H20850 214411-5For the thicker ﬁlms where no Fe signal remains, there is no additional Mg 2 pline related to metallic Mg. Such a me- tallic feature would be expected to appear at 2 eV lowerbinding energy from the main 2 pline of MgO /H20849Ref. 42/H20850and its complete absence is, therefore, a further conﬁrmation ofthe good stoichiometry of these ﬁlms. In agreement to whatwas found in the valence-band study, when annealingMgO/Fe /H20849100 /H20850interface to 400 °C /H20851Fig.6/H20849e/H20850/H20852, there is little change in the core level with only the shape of the Mg 2 p spectra becomes slightly more symmetrical. However, mostinterestingly, upon annealing the 1 ML MgO/Fe /H20849001 /H20850a small feature, which is barely visible in the RT-deposited MgO, clearly appears at 10.7 eV above the Mg 2 pbinding energy /H20849this is best seen in Fig. 7and further discussed be- low /H20850but, quite independently of the precise interpretation, the observed sharpening of the features indicates again astructural reordering of the MgO layer induced by annealing. Finally, we provide a brief analysis of the effects related to the fast response of the Fe substrate to photoexcitation inthe MgO ﬁlm. Two effects observed are shown in Fig. 6, where a comparison of Mg 2 pspectra of 1 ML MgO and 10 ML MgO ﬁlm on Fe /H20849100 /H20850, both annealed to 400 °C, is made. First, we focus on the interface-induced electron screening asobserved in the Mg 2 pspectra of thin MgO layer in contact with the Fe substrate. Figure 7shows that there is substantial shift of Mg 2 p/H208490.55 eV /H20850to lower binding energy for thin MgO layer. This shift occurs gradually with increasing cov- erage, and was routinely reproduced over many freshly pre-pared MgO/Fe /H20849100 /H20850ﬁlms /H20849not shown /H20850. Such shift is in agreement with previous results from other MgO/metal interfaces. 43,44The interpretation of the energetics of the photoemission process in strongly correlated materials in theimmediate presence of metallic layers, which can rapidly re- spond to the photoemission process, has been presented byDuffy and Stoneham 45and in references therein. More re- cently, related x-ray photoemission experiments performedby Altieri et al. 43,44for the case of the MgO/Ag /H20849100 /H20850inter- face showed a shift of the Mg 2 ppeak on the order of 1 eV with increasing MgO coverage, which was ascribed to imagepotential screening from the underlying metal substrate. Inother words, while the charge relaxation in MgO appears tobe slow on the time scale of the photoemission process, thescreening of the hole created in the photoemission process byunderlying metal is rapid on this time scale. In this way, thepresence of the metal substrate inﬂuences the energy of thephotoelectron emitted from as far as 10 ML away from theinterface. 43In this interpretation, the observed shift can be different for various core-level and valence-band featuresdue to different screening effects. We clearly observe sucheffect in Fig. 7, and it is not due to charging, since no sub- stantial shift between thin and thick MgO layers is observedin the O 2 ppart of the valence band /H20851Fig.4/H20849a/H20850/H20852. This is in contrast to MgO ﬁlms with higher thicknesses /H20849/H3335615 ML /H20850, where charging effects are clearly detected in photoemissionspectra. The absence of charging effects in thinner MgOmight be a result of efﬁcient tunneling from the Fe substrate.The observed interfacial screening reveals the dynamics ofelectronic response at the metal and/or insulator interfacethat could be relevant in tunneling. The second interesting observation that can be made from Fig. 7is an appearance of small feature at about 10.7 eV below the Mg 2 ppeak in the spectrum for 1 ML MgO/Fe /H20849100 /H20850. This feature is completely absent in the clean Fe spectrum and it also disappears for thick MgO ﬁlms. Fur- thermore, we note that this feature is barely visible in theRT-deposited 1 ML MgO ﬁlm /H20851see Fig. 6/H20849d/H20850/H20852, and it is en- hanced signiﬁcantly by annealing to 400 °C /H20851see Fig. 7and also Fig. 6/H20849e/H20850/H20852. It also appears to be diminished in strength for submonolayer MgO coverage, as seen for the 0.5 MLMgO spectrum /H20851Fig. 6/H20849c/H20850/H20852. We suggest that this feature is another example of a response of a conduction electrons ofthe Fe underlayer to the excitation produced in the MgOoverlayer. Full support to this interpretation is given by thefact that Fe surface plasmon peak is reported to be at energyloss of 10–11 eV by previous studies. 46,47Thus, we assign the feature observed at 10.7 eV below the Mg 2 pto a plas- mon excited in the Fe interface by sudden creation of the core hole in the MgO layer across the insulator and/or metalinterface. Such plasmon feature excited across the metaland/or insulator interfaces was reported recently forMgO/Ag system, 48where it is also demonstrated that the exact energy of such plasmon is shifted from the value for aclean surface due to the changes of the electronic structure atthe interface. The energy shift for that system is reported tobe on the order of a fraction of an eV which is fully consis-tent with our observation. In conclusion, the observation ofthe Fe surface plasmon in the Mg 2 pspectrum is another indication of nontrivial electronic dynamics at the metaland/or insulator interface. IV. SUMMARY The electronic structure of epitaxial MgO /H20849100 /H20850ﬁlms grown on Fe /H20849100 /H20850has been investigated by valence-band FIG. 7. /H20849Color online /H20850Core-level and small satellite feature spectra of MgO/Fe /H20849100 /H20850. Full circles, 400 °C 1 ML MgO/Fe; solid line, 400 °C 10 ML MgO/Fe. Spectrum of 1 ML coverage wasmeasured at room temperature and the 10 ML spectrum at 100 K.Renormalized parts of spectra are also shown.PLUCINSKI et al. PHYSICAL REVIEW B 75, 214411 /H208492007 /H20850 214411-6spin-polarized photoemission and shallow core-level spec- troscopy. The deposition of epitaxial MgO overlayers hasbeen realized by direct evaporation of stoichiometric MgOon Fe /H20849100 /H20850at room temperature. The good structural and compositional quality of these MgO ﬁlms was conﬁrmed bythe small photocurrent intensity observed above the valence-band maximum for the 10 ML ﬁlm coverage and by theabsence of a metallic component in the Mg 2 pspectrum. From the valence-band spectra, the Fe 3 p, and the Mg 2 p spectra, no indications of an FeO layer were found at theMgO/Fe /H20849100 /H20850interface prepared in this way. The high spin polarization of the Fe valence band above the VBM of MgO is preserved in the low coverage range and only a smallreduction in the sharpness of the Fe 3 dfeatures is observed. Annealing the interface to 400 °C only partially restores thesharpness of the Fe 3 dfeatures, very probably as a result of structural ordering of the interface. Annealing to 500 °C in-ﬂuences more strongly both the Fe- and MgO-related parts ofthe spectra. The recovery of the full sharpness of the Fe 3 d features could be the result of further and more completeordering of the interface or to uncovering the clean Fe /H20849100 /H20850substrate. No unambiguous interpretation can be offered on this point from our data but a microscopy study should beable to resolve the ambiguity. In addition, we report several interesting electronic prop- erties of the interface by comparing the core-level spectra atvarious MgO coverages. The binding energy of the Mg 2 p levels increases by 0.55 between 1 and 10 ML coverages. At1 ML coverage, a small feature is present at a binding energyof 10.7 eV below the main Mg 2 pline; it is ascribed to a plasmon deexcitation from the underlying Fe substrate. ACKNOWLEDGMENTS We would like to thank the NSLS staff for the technical support on the course of experiments. This work was sup-ported by NSF Grant No. ECS-0300235. 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PhysRevB.71.134502.pdf	Magnetoresistance of junctions made of underdoped YBa 2Cu3Oyseparated by a YBa 2Cu2.6Ga0.4Oybarrier L. Shkedy, G. Koren, and E. Polturak Physics Department, Technion - Israel Institute of Technology, Haifa, 32000, Israel sReceived 24 June 2004; revised manuscript received 11 November 2004; published 11 April 2005 d We report magnetoresistance measurements of ramp-type superconductor-normal-superconductor sSNSd junctions. The junctions consist of underdoped YBa 2Cu3OysYBCO delectrodes separated by a barrier of YBa2Cu2.6Ga0.4Oy. We observe a large positive magnetoresistance, linear in the ﬁeld. We suggest that this unusual magnetoresistance originates in the ﬁeld dependence of the proximity effect. Our results indicate thatin underdoped YBCO-N-YBCO SNSstructures, the proximity effect does not exhibit the anomalously long range found in optimally doped YBCO structures. From our data we obtain the diffusion coefﬁcient andrelaxation time of quasiparticles in underdoped YBCO. DOI: 10.1103/PhysRevB.71.134502 PACS number ssd: 74.45. 1c, 74.25.Ha I. INTRODUCTION In the usual description of the proximity effect, when a superconductor sSdis brought into contact with a normal conductor sNd, the order parameter sOPdin the supercon- ductor is depressed near the interface and superconductivity is induced in N. The pair amplitude induced in Ndecays on a length scale K−1from the interface, called the decay length.1–3InSNSjunctions in which Sis an optimally doped High Temperature Superconductor sHTSC dandNbelongs to the same material family, but is doped to be nonsupercon-ducting, the decay of the pair amplitude in Ntypically takes place over a rather long distance of tens of nm. 4–6In con- trast, if both SandNare underdoped cuprates, the pair amplitude in Nseems to decay over a much shorter distance, on the order of a few nanometers. We haveobserved this effect in underdoped YBa 2Cu3Oy sYBCO dbased junctions7–9having a barrier made of YBa2Cu2.55Fe0.5Oyor a YBa 2Cu2.6Ga0.4Oy.I nSNSjunctions having a barrier much thicker than the decay length, Cooperpairs cannot tunnel through and the junctions exhibit a ﬁniteresistance at all temperatures. Roughly speaking, supercon-ductivity in Nis induced near the two SNinterfaces, while a section of length ,in the middle of the barrier remains nor- mal. This is the type of junction studied in the present work. We are not aware of previous investigations of the prox- imity effect in HTSC under a magnetic ﬁeld. When a mag-netic ﬁeld is applied, superconductivity is reduced and pen-etrates less into the normal conductor. As a result, theproximity effect is ﬁeld dependent. 2If the superconductivity in the barrier is weakened, the length of the normal section inthe junction should increase, and with it the junction’s ﬁniteresistance. As a result, a positive magnetoresistance fMR, deﬁned as MR ;RsHd−Rs0dgshould be observed.We indeed observed such MR, linear in the ﬁeld. An attempt to explain this unusual ﬁeld dependence is the subject of this paper. Besides the ﬁeld dependence of the proximity effect, there are several additional mechanisms that could contribute tothe MR. These include ﬂux ﬂow in the superconductingelectrodes, 3,10normal MR of the barrier material swhich is caused by bending of electron trajectories d,11ﬁeld dependenthopping in the barrier,12,13and resonant tunneling between the electrodes across the barrier.14In the following we show that the contribution of all these processes to the observedMR is insigniﬁcant and we attribute it primarily to a ﬁeld-dependent proximity effect in the barrier. II. EXPERIMENTAL The junctions used in the present study are thin ﬁlm-based ramp junctions of the type that was previously used in ourwork. 6The junctions consist of two underdoped supercon- ducting YBCO electrodes separated by a thin layer ofYBa 2Cu3Ga0.4OysGa-dopedYBCO dbarrier. Ga has no mag- netic properties. The transport current ﬂows in the a-bplane through the Ga-doped YBCO layer. The multistep process ofjunction preparation by laser ablation was describedpreviously. 7Brieﬂy, we ﬁrst deposit a 100 nm thick c-axis-oriented epitaxial YBCO layer onto a s100dSrTiO3 sSTOdsubstrate. This base electrode is then capped by a thick insulating layer of STO. Patterning is done by Ar ionmilling to create shallow angle ramps along a main crystal-lographic direction in the a-bplane. In a second deposition step, the barrier layer, theYBCO cover electrode, and theAuelectrical contacts are deposited, and then patterned to formthe ﬁnal junctions layout. This produces several junctionswith 5 mm width on the wafer. Four terminal-resistance mea- surements of the junctions were carried out as a function oftemperature and magnetic ﬁeld of ł8 Tesla. The ﬁeld was perpendicular to the transport current, which in our geometryﬂows in the a-bplane of the ﬁlms. III. RESULTS AND DISCUSSION Resistance versus temperature sRTdmeasurements of six junctions on the wafer are shown in Fig. 1. In the normal region, the difference in the resistance of the junctions is dueto the different lengths of theYBCO leads. One observes twodistinct superconducting transitions with T conset of 35 K and 53 K, which are attributed to each one of the electrodes.In the oxygen annealing process of underdoped YBCO, theoxygen concentration is kept low and the duration of thePHYSICAL REVIEW B 71, 134502 s2005 d 1098-0121/2005/71 s13d/134502 s6d/$23.00 ©2005 The American Physical Society 134502-1annealing is relatively short. Consequently, the base elec- trode, which is covered by a thick layer of STO, absorbs lessoxygen and its transition temperature is lower. Below about30 K, both electrodes are superconducting and the inset ofFig. 1 shows the low temperature resistance of the junctions,which is due to the barrier. Qualitatively similar behaviorwas observed in edge junctions made of underdoped YBCOseparated by a YBa 2Cu2.55Fe0.5Oybarrier.7,8The scatter of the values between different junctions is typical of our junc-tion preparation process and is probably due to nonuniformi-ties in the local Ga concentration and to variations in thetransparency of the interfaces, most probably resulting fromdamage created by the ion milling of the ramp. The transpar-ency of our junctions can be estimated from measurementsof the critical current described below, which indicate thatthe transparency is low. The temperature dependence of thejunction’s resistance is typically weaker than that of the par-ent material in the form of a ﬁlm, shown in Fig. 2. At lowtemperatures, the absolute resistivity of the junctions is alsomuch smaller than that of a YBa 2Cu2.6Ga0.4Oyﬁlm. One pos- sible interpretation is that the thickness of the barrier s21 nmin this work dis in the range where the material is meso- scopic. Under these conditions, the temperature dependenceof the resistance is expected to be much weaker than that of a macroscopic ﬁlm. 15The differences of the absolute resis- tivities between different junctions may perhaps result fromdifferent interface transparencies, which also affect the con-ductance of the device in the mesoscopic regime. 15 Our main experimental result is shown in Fig. 3, where the measured magnetoresistance MR at low Tis plotted as a function of magnetic ﬁeld normal to the wafer. All junctionsshowed similar behavior. Detailed measurements were doneon three of the six junctions on the wafer. One can see thatall three junctions show a large positive MR, which is linearin the applied ﬁeld. The MR typically reaches a value of,20Vat 8 Tesla, which is larger than the resistance at H =0 by tens of a percent. We consider possible sources for the MR in our junctions. MR originating in the two YBCO electrodes below the su-perconducting transition temperature can result, for instance,from ﬂux motion. This contribution would be linear in theﬁeld. In order to estimate the size of this contribution, weperformed low-temperature MR measurements on bareYBCO microbridges. At temperatures close to T c, ﬂux ﬂow was indeed observed ssee the Appendix d. However, at low temperatures where the junctions of Fig. 3 were measured,no measurable MR was observed in the thin-ﬁlm YBCO mi-crobridges. Therefore, ﬂux ﬂow in the YBCO electrodesdoes not contribute to the MR. We also measured the MR ofjunctions prepared in the same way, but without the barrierlayer. These junctions are referred to as “shorts.” As shownin Fig. 3, under similar bias currents and ﬁelds, the “shorts”did exhibit a small MR of about 0.4 Vat 8 Tesla. The “shorts” show a ﬁnite MR since the interface between thetwo YBCO electrodes is always imperfect. The MR of the“shorts” is smaller than the MR in the junctions by almosttwo orders of magnitude. The interface resistance cannot bedirectly measured. What can be measured is the critical cur-rent density.Typically, the critical current density at low tem-perature of a 60 K YBCO “short” is one order of magnitudesmaller than that of a ﬁlm. This implies that the transparencyof our junctions is low. To summarize this section, the above FIG. 1. sColor online dResistance vs temperature of six junc- tions with 21 nm thick Ga-dopedYBCO barrier. In the normal state,the different resistances of the junctions are due to different lengthsof the YBCO leads. The inset shows the low temperature resistanceof the junctions where both electrodes are superconducting. FIG. 2. sColor online dResistivity vs temperature of 100 nm thick ﬁlm of the Ga-doped YBCO material. Square symbols aremeasured at zero ﬁeld and the circles are measured with 6 Teslaﬁeld applied perpendicular to the ﬁlm. Note that the MR of the ﬁlmis negative, in contrast to the positive MR of our junctions. FIG. 3. sColor online dMagnetoresistance vs ﬁeld of three of the junctions of Fig. 1 at 2 K, and of a “short” junction at 4 K. The“short” resistance is about 0.4 Vat 8 Tesla, which is almost two orders of magnitude smaller than the corresponding MR of the otherjunctions with a barrier. The dashed line is a guide to the eye.SHKEDY, KOREN, AND POLTURAK PHYSICAL REVIEW B 71, 134502 s2005 d 134502-2series of control experiments show that MR in the electrodes isnotthe source of the large MR observed in our junctions. A second potential source for the observed MR in Fig. 3 could be the barrier material itself. We therefore measuredthe MR of the Ga-doped YBCO. Speciﬁcally, we measuredthe resistance versus temperature of microbridges patternedin a thin ﬁlm of this material annealed under the same con-ditions as the junctions in Fig. 3. Figure 2 shows the resis-tivity of these bridges with and without magnetic ﬁeld. Thebarrier material exhibits a clear negativeMR of ,5%a t2K . The sign of this MR is opposite to that of the junctions,which show a large positiveMR.At low temperatures, where the MR of the Ga-doped ﬁlms is largest, the MR contributedby the barrier in the junctions would be at most −8 Vs5% of 160V, as seen in the inset of Fig. 1 d. However, since the sign of the MR of the barrier material itself is negative, thenetspositive dMR of the junctions should be even larger than shown in Fig. 3. Consequently, the properties of the barrier material on its own cannot explain the observed MR of thejunctions. The above-mentioned control experiments clearly show that the MR of our junctions does not originate from thesuperconducting electrodes nor from the normal properties ofthe barrier material.The net MR that we see has a magnitudecharacteristic of the transition of part of the barrier from asuperconducting to a normal state. We therefore examinewhether the MR could originate from the depression of su-perconductivity near the SNinterface of the junction. Before going into a more detailed analysis, we note that our barrier is a mesoscopic section of a Mott insulator sMId, with the conductance of the material in bulk form showingvariable range hopping. 16Its low temperature resistivity, 0.8Vcm, is about three orders of magnitude larger than the maximum resistivity of metals sMott-Ioffe-Regel limit20d. Strictly speaking, our junctions are S/MI/Sjunctions. So, the application of the usual theoretical description of theproximity effect to our junctions is not a priori justiﬁed, since both the de Gennes and Usadel equations are valid onlyfor dirty metals. However, it is an experimental fact thatwhen an MI with resistivity rł1Vcm is in good electrical contact with a superconductor it behaves similarly to ametal. 16,21–23The question of which particular model to use is therefore a matter of choice. In the limit of small inducedpair amplitude in N, which applies to our low-transparency junctions, the de Gennes and Usadel approaches give thesame result. Since the de Gennes approach was traditionallyemployed in all previous and current work on HTSC prox-imity structures, 4,5,24we prefer to follow this route. In any case, the analysis presented below is, nevertheless, useful interms of assigning values to physical quantities, such as thedecay length, which can then be intercompared between dif-ferent experiments. We ﬁrst discuss the MR on the Sside of the SNinterface. In this region, the order parameter is reduced, superconduc-tivity is depressed, pinning is weakened, and ﬂux ﬂow couldoccur despite the low temperature. We now estimate the up-per limit on the contribution of this effect to the MR. Thelow temperature normal state resistivity of YBCO, extrapo-lated from the linear part of the RTplot above the transition, is about 10 −4Vcm. An upper limit on the volume near theinterface in which superconductivity is weakened is 10 j 3A,200 Å 30.5mm2, where Ais the junction cross section.19The normal state resistance of this region is very low, less than 0.1 V. Since the ﬂux ﬂow resistance is a frac- tion of the normal state resistance, it follows that the MR intheSside close to the interface is negligible. Turning now to the Nside of the interface, the resistivity of the barrier material is quite high, 0.8 Vcm at 2 K. A rough estimate done assuming Ohm’s law in the barrier in-dicates tha ta1n m thick slice of the barrier has a resistance ofR,16V. This value is similar to the total MR seen in Fig. 3. In the following, we propose that the observed MR iscaused by changes in the effective penetration of supercon-ductivity into the barrier. In other words, when a magneticﬁeld is applied, the magnitude of the pair amplitude inducedin the barrier is decreased and ,, the effective length of the barrier, which remains normal, increases thus increasing theresistance of the junction. The magnetic ﬁelds used in the present study are small compared to H c2of the 60 K YBCO phase which is 50 T.18 Thus changes in the minigap Ddue to the applied ﬁeld are also small but not negligible. The value of Don theSside near the interface is proportional to Tc, which itself depends on the magnetic ﬁeld due to pair breaking according to3,17 lnSTc Tc0D=CS1 2D−CS1 2+a 2pkTcD, s1d whereTcis the critical temperature under applied ﬁeld and Tc0is the critical temperature at zero ﬁeld. Cis the di- Gamma function deﬁned as Csxd=G8sxd/Gsxdandais the pair-breaking parameter. For a thin ﬁlm under a perpendicu- lar applied ﬁeld a=DSeH/c, whereDSis the diffusion coef- ﬁcient in the superconductor. Because the highest magneticﬁeld we used is small compared to H c2, pair breaking is small and sa/2pkBTcdis a small parameter. In this limit, Eq. s1dreduces to3,17 kBsTc0−Tcd=pa 4. s2d From our RTmeasurements under different ﬁelds we ﬁnd the values of Tc0andTcs65 K at 0 T and 55 K at 7 T, respec- tively d. We can thus calculate the value of a, which is ,1 meV at 7 Tesla. Therefore, a/2pkBTc.1/35, and this justiﬁes the use of Eq. s2d. By assuming a linear scaling between DandTcs2D=bkBTc, with bbeing a constant of about 5 d, we estimate that under a ﬁeld of 7 T the magnitude ofDdecreases by about 15%. The suppression of Dcan therefore be written as dS;DSs0d−DSsHd=pab 8=pb 8DSeH c, s3d where dSis small compared to DSs0d.The spatial dependence of the Din aSNSjunction is shown schematically in Fig. 4. The value of the Don both sides of the interface is related through the standard boundary condition2,3MAGNETORESISTANCE OF JUNCTIONS MADE OF … PHYSICAL REVIEW B 71, 134502 s2005 d 134502-3SDSi NSVSD x=0=SDNi NNVND x=0, s4d where DSiandDNiare the values of the minigap at the Sand Nsides of the SNinterface. NSandNNare the normal state density of states sDOS don theSandNsides of the interface, respectively. Finally, VSandVNare the electron-electron in- teraction on the SandNsides. Assuming the DOS and the electron-electron interaction are ﬁeld independent we obtain DSisHd DNisHd=e=dSi dNi, s5d where e=NSVS/NNVNis a ﬁeld-independent constant and we deﬁne dN;DNs0d−DNsHd.dNi, which is the value at the in- terface, is also a small parameter as dNi/DNis0d!1. Turning to the Nside now, under a magnetic ﬁeld Hap- plied in the cdirection, the spatial dependence of Dis given by the linearized Ginzburg-Landau sGLdequation2 −d2DN dx2+S2eH "cD2 sx0−xd2DN+K2DN=0, s6d wherex0andKare constants. In our experiment, x0−xis limited by 10 nm, which is half the thickness of our junction,the ﬁeldHis less than 8 T, and Kis on the order of a few nanometers. Using these parameters, we estimate that theupper limit of the second term in Eq. s8dis about two orders of magnitude smaller than the last term. In this limit, thesolution of Eq. s8dforDinNexhibits an exponential decay with distance D Nsxd=DNiexps−Kxd. In the dirty limit,2Kis given by K−1=S"DN 2pkBTD1/2 . s7d In this limit, where the cyclotron radius in the magnetic ﬁeld is much larger than the mean-free path, DNis ﬁeld indepen- dent and thus Kdoes not depend on ﬁeld. However, the value ofDat the interface DNiis ﬁeld dependent because it is pinned to the value of the Don theSside at the interface, DSi through Eq. s5d. The pair amplitude induced in the barrier is effectively depressed to zero by thermal ﬂuctuations at somedistance from the interface, and from that distance onwardthe material has a ﬁnite resistance. The natural way to deter- mine this distance is through the condition that the extrapo-lated magnitude of Dthere is of the order of k BT.This length, which we denote by X, depends on the ﬁeld as kBT=DNisHde−KXsHd, s8d whereXsHdis the effective penetration depth of supercon- ductivity into Nwhen a magnetic ﬁeld is applied. Dividing XsHdbyXsH=0dwe ﬁnd DNis0d DNisHd=eKfXs0d−XsHdgs9d and XsHd−Xs0d=1 KlnS1−dNi DNis0dD. s10d Since dNiis a small parameter XsHd−Xs0d,−dNi/KDNis0d. Referring to the schematic model shown in Fig. 4, the ﬁeld- dependent resistance of the barrier is R=r,sHd/A, where ,sHd=L−2XsHdis the length inside the barrier, which is normal. Using Eq. s5dand the relation 2 DS=bkBTc, the mag- netoresistance comes out as MR;RsHd−Rs0d=−2r AfXsHd−Xs0dg=preDS 2cAKS1 kBTcDH. s11d We therefore see that the MR is linear in H, in agreement with the observed behavior in Fig. 2. A rough estimate of the decay length s1/Kdin the under- doped barrier at low temperature can be attempted using the resistivity of the barrier, 0.8 Vcm and the typical resistance of the junctions ,100V. Using these values, we estimate the length of the barrier which remains normal ,sH=0das 6 nm. Taking the thickness of the barrier of 21 nm and assum- ing that the pair amplitude decays to zero over three timesthe decay length s1/Kd, we obtain a value for 1/ Kof about 2.5 nm. 1/ Kcan also be calculated using Eq. s7d, where D N=1 3,NvFN. The mean-free path in the barrier can be esti- mated as the distance between nearest Ga atoms ,N,5Å and the Fermi velocity in the barrier vFN=1.2 3107cm/s was measured in a previous study.16This yields 1/ K .3.5 nm. It appears that both methods of estimating 1/ K give values that are consistent. We note that the decay lengthestimated in underdoped SNSstructure comes out much smaller than in optimally doped ones. 4,5 Using our data we estimate the diffusion coefﬁcient DS and the relaxation time tSof underdoped YBCO. Taking an averageTcof,45 K and an average slope in Fig. 3 of MR/H=2.7 V/Tesla, Eq. s11dyieldsDS,1c m2/s. This is consistent with an independent estimate that can be extractedfrom Eq. s2dand from the relation between aand the diffu- sion coefﬁcient DSwhich yields ,1.7 cm2/s. The relaxation time tSis extracted from the usual relation that connects it with the diffusion coefﬁcient DS=1 3vFS2tS, where vFSis the Fermi velocity of quasiparticles in the superconductor ,2 3107cm/s.25Under these assumptions tSfor YBCO is ,25 fs The value found for tSis of the same order of mag- FIG. 4. Schematic diagram of the junction and the spatial proﬁle ofDsxd.,sHdis the length of the resistive region of the junction. The shaded area shows the region in which superconductivity isweakened on both sides of the interface due to the proximity effect.SHKEDY, KOREN, AND POLTURAK PHYSICAL REVIEW B 71, 134502 s2005 d 134502-4nitude as the recent results of Gedik et al.,25who obtained tS,100 fs, whereas our value of DSis smaller than theirs, DS,20 cm2/s. For completeness, we mention that Abrikosov has pre- dicted another mechanism for linear MR versus Hin superconductors.14He assumed a ﬁeld-dependent resonant tunneling, which yields MR linear in Hat very high mag- netic ﬁelds, when only a few Landau levels are ﬁlled. Whenthe ﬁeld is reduced and the number of ﬁlled Landau levelsincreases, the ﬁeld dependence of the MR changes into aquadratic one. This model could, in principle, explain theobserved linear behavior of our MR results. However, peaksin the density of states due to Landau levels are absent in thedynamic-resistance spectra of our junctions. Moreover, theﬁelds used in our experiment are not high enough to reachthe regime where a low number of Landau levels are ﬁlled.Hence, if this model was applicable to our junctions, weshould have observed a quadratic dependence of the MR onﬁeld, which is not the case. IV. CONCLUSIONS We investigated the resistance of SNSstructures based on underdoped YBCO with a nonmagnetic Ga-doped YBCObarrier as a function of magnetic ﬁeld. We discovered a lin-ear increase of the resistance with the ﬁeld. An extensiveseries of control experiments indicates that this ﬁeld depen-dence does not result from ﬂux ﬂow, which would be theobvious mechanism of MR in a superconductor.Asimpliﬁedanalysis indicates that the effect may well be explained by aﬁeld-dependent proximity effect in the barrier. This explana-tion produces a reasonable estimate of the diffusion coefﬁ-cient and the relaxation time in YBCO. Furthermore, ourestimates indicate that in underdoped YBCO SNSstructures, the superconductivity induced inside the barrier through theproximity effect has a short s,2–3 nm drange, unlike the long-range proximity effect observed in optimally doped YBCO structures. ACKNOWLEDGMENTS We thank Pavel Aronov for the “short” junction data of Fig. 2. This research was supported in part by the IsraelScience Foundation sGrant No. 1565/04 d, the Heinrich HertzMinerva Center for HTSC, the Karl Stoll Chair in advanced materials, and by the Fund for the Promotion of Research atthe Technion. APPENDIX For the sake of comparison to previous work, we also measured the MR of optimally doped YBCO ﬁlms at tem- peratures close to Tc, as shown in the inset of Fig. 5. In this case, the resistance showed a region linear with an appliedﬁeld. In the Bardeen-Stephen model, 3,10,26the resistance re- sulting from ﬂux ﬂow is given by Rflux flow=fH/Hc2sTdg 3RNsTd, whereHis the applied magnetic ﬁeld and RNsTdis the normal state resistance at temperature T, extrapolated from the RTplot close to Tc. Using this model, we extracted the temperature dependence of Hc2nearTc. Our results show good agreement with previous measurements by Kunchur et al.and Ossandon et al.,27,28which are also plotted in Fig. 5. At temperatures much lower than Tc, however, no measur- able MR in the YBCO ﬁlm was observed. Therefore, at lowtemperatures where the junctions of Fig. 3 were measured,ﬂux ﬂow in the YBCO electrodes does not contribute to theMR. This conclusion holds, independent of the oxygen dop-ing level of the YBCO. Electronic address: lior_shk@physics.technion.ac.il 1P. G. deGennes, Rev. Mod. Phys. 36, 225 s1964 d. 2G. Deutscher and P. G. deGennes, in Superconductivity , edited by R. D. Parks sDekker, New York, 1966 d, pp. 1005–1034. 3M. Tinkham, Introduction to Superconductivity , 2nd edition sMcGraw-Hill, New York, 1996 d. 4K. A. Delin and A. W. Kleinsasser, Supercond. Sci. Technol. 9, 227s1996 d. 5E. Polturak, G. Koren, D. Cohen, E. Aharoni, and G. Deutscher, Phys. Rev. Lett. 67, 3038 s1991 d. 6A. Sharoni, I. Asulin, G. Koren, and O. Millo, Phys. Rev. Lett.92, 017003 s2004 d. 7O. Nesher and G. Koren, Appl. Phys. Lett. 74, 3392 s1999 d. 8O. Nesher and G. Koren, Phys. Rev. B 60, 9287 s1999 d. 9G. Koren, L. Shkedy, and E. Polturak, Physica C 403,4 5s2004 d. 10J. Bardeen and M. J. Stephen, Phys. Rev. 140, A1197 s1965 d. 11J. M. Ziman, Electrons and Phonons sOxford University Press, London, 1960 d, Chap. XII. 12B. I. Shklovski and L. Éfros, Sov. Phys. JETP 57s2d, 470 s1983 d. 13I. M. Lifshitz and V. Ya. Kirpichenkov, Sov. Phys. JETP 50s3d, 499s1979 d. 14A. A. Abrikosov, Physica C 317–318, 154 s1999 d. FIG. 5. sColor online dHc2vs temperature of optimally doped YBCO ﬁlm. Hc2was extracted from the slope of the linear part of the MR sinset dusing the Bardeen-Stephen model. Our data ssolid squares dcan be compared to previous measurements by Kunchur et al.and Ossandon tS.27,28MAGNETORESISTANCE OF JUNCTIONS MADE OF … PHYSICAL REVIEW B 71, 134502 s2005 d 134502-515L. I. Glazman and K. A. Matveev, Sov. Phys. JETP 67, 1276 s1988 d. 16L. Shkedy, P. Aronov, G. Koren, and E. Polturak, Phys. Rev. B 69, 132507 s2004 d. 17K. Maki, in Superconductivity , edited by R. D. Parks sDekker, New York, 1966 d, pp. 1035–1105. 18Y. Ando and K. Segawa, Phys. Rev. Lett. 88, 167005 s2002 d. 19I. Lubimova and G. Koren, Phys. Rev. B 68, 224519 s2003 d. 20N. F. Mott, Philos. Mag. 26, 1015 s1972 d; A. F. Ioffe and A. R. Regel, Prog. Semicond. 4, 237 s1960 d; J. H. Mooij, Phys. Status Solidi A 17, 521 s1973 d. 21T. Hashimoto, M. Sagoi, Y. Mizutani, J. Yoshida, and K. Mi- zushima, Appl. Phys. Lett. 60, 1756 s1992 d. 22C. Stozel, M. Siegel, G. Adrian, C. Krimmer, J. Stollner, W.Wilkens, G. Schulz, and H. Adrian, Appl. Phys. Lett. 63, 2970 s1993 d. 23A. Frydman and Z. Ovadyahu, Europhys. Lett. 33, 217 s1996 d. 24I. Bozovic, G. Logvenov, M. Verhoeven, P. Caputo, E. Goldobin, and M. R. Beasley, Phys. Rev. Lett. 93, 157002 s2004 d. 25N. Gedik, J. Orenstein, Ruixing Liang, D. A. Bonn, and W. N. Hardy, Science 300, 1410 s2003 d. 26A. R. Strnad, C. F. Hempstead, and Y. B. Kim, Phys. Rev. Lett. 13, 794 s1964 d. 27M. N. Kunchur, D. K. Christen, and J. M. Phillips, Phys. Rev. Lett.70, 998 s1993 d. 28J. G. Ossandon, J. R. Thompson, D. K. Christen, B. C. Sales, H. R. Kerchner, J. O. Thomson, Y. R. Sun, K. W. Lay, and J. E.Tkaczyk, Phys. Rev. B 45, 12 534 s1992 d.SHKEDY, KOREN, AND POLTURAK PHYSICAL REVIEW B 71, 134502 s2005 d 134502-6
PhysRevB.99.205405.pdf	PHYSICAL REVIEW B 99, 205405 (2019) Raman ﬁngerprint of stacking order in HfS 2-Ca(OH)2heterobilayer M. Yagmurcukardes,1,S. Ozen,2F. Iyikanat,3F. M. Peeters,1and H. Sahin2,4 1Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium 2Department of Photonics, Izmir Institute of Technology, 35430 Izmir, Turkey 3Department of Physics, Izmir Institute of Technology, 35430 Izmir, Turkey 4ICTP-ECAR Eurasian Center for Advanced Research, Izmir Institute of Technology, 35430 Izmir, Turkey (Received 18 January 2019; revised manuscript received 27 February 2019; published 6 May 2019) Using density functional theory-based ﬁrst-principles calculations, we investigate the stacking order depen- dence of the electronic and vibrational properties of HfS 2-Ca(OH) 2heterobilayer structures. It is shown that while the different stacking types exhibit similar electronic and optical properties, they are distinguishable fromeach other in terms of their vibrational properties. Our ﬁndings on the vibrational properties are the following:(i) from the interlayer shear (SM) and layer breathing (LBM) modes we are able to deduce the AB /primestacking order, (ii) in addition, the AB/primestacking type can also be identiﬁed via the phonon softening of EI gandAIII gmodes which harden in the other two stacking types, and (iii) importantly, the ultrahigh frequency regime possessesdistinctive properties from which we can distinguish between all stacking types. Moreover, the differences inoptical and vibrational properties of various stacking types are driven by two physical effects, induced biaxialstrain on the layers and the layer-layer interaction. Our results reveal that with both the phonon frequencies andcorresponding activities, the Raman spectrum possesses distinctive properties for monitoring the stacking typein novel vertical heterostructures constructed by alkaline-earth-metal hydroxides. DOI: 10.1103/PhysRevB.99.205405 I. INTRODUCTION Interest in two-dimensional (2D) ultrathin materials has grown exponentially across materials science thanks to thesuccessful isolation of graphene [ 1,2]. Beyond graphene, other novel 2D monolayer materials, such as monoatomiccrystals of silicon and germanium [ 3–7], transition-metal dichalcogenides (TMDs) [ 8–11], and alkaline-earth-metal hy- droxides (AEMHs) [ 12–15] have been added to the library of 2D materials. Among 2D ultrathin materials, TMDs are an emerging class of materials with electronic properties ranging frommetallic to semiconducting and even to superconductingwhich offer a wide range of opportunities for various appli-cations [ 11,16]. As a member of group IVB TMDs, hafnium disulﬁde (HfS 2) has been predicted to possess higher carrier mobility and higher tunneling current density than those ofMo- and W-dichalcogenides [ 17,18]. The successful fabri- cation of few-layer HfS 2ﬁeld effect transistors (FETs) and observation of the high drain current and mobility have beenrecently reported [ 19,20]. Very recently, Fu et al. reported the synthesis of high-quality atomic layered HfS 2crystals exhibit- ing ultrahigh responsivity (9 orders of magnitude higher thanthat of MoS 2) which is useful for ultrasensitive near-infrared phototransistors [ 21]. On the other hand, the bulk form of AEMHs is reported to be structurally and electronically sen-sitive to external physical effects such as temperature andpressure [ 22–28]. Aierken et al. investigated the thickness- dependent electronic and vibrational properties of Ca(OH) 2 mehmet.yagmurcukardes@uantwerpen.beand reported its robust direct band gap insulating charac-ter with changing thickness [ 12]. We also investigated the optical properties of a heterobilayer structure of monolayerCa(OH) 2combined with GaS layer and reported that their heterostructure can be used as a separator for photoinducedcharge carriers which are located at different layers [ 15]. Thanks to the weak van der Waals interaction between constituent layers, construction of the vertical heterostructuresof ultrathin materials have received considerable attention[29–39]. Jin et al. reported that interlayer electron-phonon interaction in WSe 2-hBN heterostructure plays a crucial role in engineering the electrons and phonons for possible deviceapplications [ 40]. Very recently, Chen et al. reported that electron-phonon interaction can be controlled by the sym-metry of the various 2D materials used in a heterostructureconstructed on top of SiO 2[41]. Similarly, interlayer electron- phonon coupling has been investigated in twisted bilayergraphene and MoS 2systems and its signiﬁcance on the obser- vation of Moire phonons was reported [ 42,43]. It was pointed out that different stacking types result in totally differentproperties in both layered systems and vdW heterostructures. Previous studies have revealed that different types of lay- ered materials possess relatively lower (i.e., ReS 2) or higher (MoS 2, graphene, h-BN, etc. ) energy barriers between dif- ferent stacking types. We reported for bilayer ReS 2that the energy barrier between different stacking orders is negligiblysmall ( ∼8 meV) as compared to that of bilayer MoS 2(∼240 meV) [ 44]. The relatively low energy barrier indicates that in experimental conditions the existence of different stackingtypes is possible and thus, Raman spectrum is crucial todistinguish between them [ 45]. On the other hand, although graphene and monolayer h-BN possess similar structural 2469-9950/2019/99(20)/205405(8) 205405-1 ©2019 American Physical SocietyM. Y AGMURCUKARDES et al. PHYSICAL REVIEW B 99, 205405 (2019) properties and stacking types in their few-layer structures, the response of different stacking types to Raman spectrumhave been reported to be very different [ 46–48]. Notably, similar arguments hold for vdW heterostructures because ofthe various 2D building blocks used as constituents of theheterostructure. Therefore, as a common methodology Ramanspectroscopy is crucial for the determination of the stackingtype in heterostructures. In this study we investigate the response of different stack- ing types to the Raman spectrum in order to distinguish themfrom each other in HfS 2-Ca(OH) 2heterostructure. Particu- larly, HfS 2can be a promising candidate for 2D-based elec- tronic and optoelectronic applications owing to its ultrahighon/off ratio, high carrier mobility, and high tunneling current density as an alternative to other semiconducting TMDs.On the other hand, Ca(OH) 2is known to be an efﬁcient adsorbent for CO 2capturing in energy technologies and also has been reported to be a robust insulator upon changingthickness. As an alternative to well-known 2D insulator hBN, monolayer Ca(OH) 2can be used as encapsulating layer to improve the properties of the 2D materials. Moreover, strongsurface polarizations induced by (OH) −groups may lead to the enhancement of the semiconducting nature of the con-stituent layer. Overall, the construction of a heterostructurecomposed of HfS 2-Ca(OH) 2layers offers a wide range of opportunities for its application in optoelectronic devices. Inthis study, we aim to show how different stacking types canbe monitored via a basic Raman measurement. We proposethat strong O-H bond stretching displays unusual featuresthat help researchers to clearly monitor the stacking type inHfS 2-Ca(OH) 2heterostructure. II. COMPUTATIONAL METHODOLOGY To investigate the structural, electronic, and vibrational properties of monolayers of HfS 2,C a ( O H ) 2, and their het- erobilayer structures, density functional theory (DFT) basedﬁrst-principle calculations were performed as implementedin the Vienna ab initio simulation package (V ASP) [ 49]. The Perdew-Burke-Ernzerhof (PBE) [ 50] form of general- ized gradient approximation (GGA) was adopted to describethe electron exchange and correlation. The van der Waals(vdW) correction to the GGA functional was included byusing the DFT-D2 method of Grimme [ 51]. For the elec- tronic band structure calculations, spin-orbit coupling (SOC)was included with the GGA and Heyd-Scuseria-Ernzerhof(HSE06) [ 52] screened-nonlocal-exchange functional of the generalized Kohn-Sham scheme, respectively, for more accu-rate band gap calculations. The charge transfer between theindividual atoms in the system was determined by the Badertechnique [ 53]. The kinetic energy cutoff for plane-wave expansion was set to 500 eV and the energy was minimized until its variation inthe following steps became 10 −8eV . The Gaussian smearing method was employed for the total energy calculations. Thewidth of the smearing was chosen to be 0.05 eV . TotalHellmann-Feynman forces was taken to be 10 −7eV/Åf o r the structural optimization. 24 ×24×1/Gamma1centered k-point samplings were used in the primitive unit cells. To avoidinteraction between the neighboring layers, a vacuum spaceof 25 Å was implemented in the calculations. For XRD simulations of the crystal structures, the wavelength of copperK-α(1.5406 Å) was considered which is commonly used in XRD experiments. The dielectric function of the heterobilayer was calculated by using the HSE06 functional on top of SOC. Using thedielectric function, the reﬂectivity ( R) was calculated with the following formula: R(w)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ /epsilon1(w)−1√/epsilon1(w)+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (1) The phonon frequencies and the corresponding off- resonant Raman activities were calculated at the /Gamma1point of the Brillouin zone (BZ) using the small displacement method.Each atom in the primitive unit cell was initially distortedby 0.01 Å and the corresponding dynamical matrix wasconstructed. Then, the vibrational modes were determined bya direct diagonalization of the dynamical matrix. The corre-sponding Raman activity of each phonon mode was obtainedfrom the derivative of the macroscopic dielectric tensor byusing the ﬁnite-difference method. The kinetic energy cutofffor plane-wave expansion was increased up to 800 eV with ak-point set of 24 ×24×1 in the case of Raman calculations. III. STRUCTURAL, ELECTRONIC, AND PHONONIC PROPERTIES OF MONOLAYERS HfS 2AND Ca(OH)2 In contrast to monolayers of Mo- and W-dichalcogenides which crystallize in 1 Hphase, monolayer HfS 2crystallizes in 1Tphase in its ground state. In addition, the monolayer of Ca(OH) 2possesses also a 1 Tcrystal structure including hydroxyl groups (OH)−located symmetrically with respect to the Ca atom [see Fig. 1(a)]. In the case of monolayer HfS 2,H f atoms are sandwiched between two layers of S atoms whichcorresponds to the space group P¯3m2 and D 3dpoint group. The optimized in-plane lattice parameters are a=b=3.63 Å anda=b=3.59 Å for HfS 2and Ca(OH) 2, respectively. The Hf-S atomic bond length is 2.55 Å while that of Ca-O is2.37 Å. In addition, the O-H bond length in (OH) −group is 0.97 Å. Moreover, the Bader charge analysis shows thatin monolayer HfS 2, a Hf atom donates ∼1.0e−to a S atom while in Ca(OH) 2each H atom donates 0.6 e−to an O atom in the hydroxyl group and each O atom also receives 0.8 e− from a Ca atom. In addition, the thermionic work function ( /Phi1) of each monolayer, which is the amount of energy requiredto remove a charge carrier located at the Fermi energy tovacuum, are calculated to be 6.19 and 4.82 eV for HfS 2and Ca(OH) 2, respectively (see Table I). The electronic properties of each monolayer crystal are in- vestigated in terms of their electronic band structures throught h ew h o l eB Z .A ss h o w ni nF i g . 1(b), unlike many 1 H-TMDs, 1T-HfS 2possesses an indirect semiconducting character with a band gap of 1.98 eV . The valence band maximum (VBM)and the conduction band minimum (CBM) are located at the/Gamma1and the Mhigh symmetry points, respectively. Moreover, due to the dorbitals of the Hf atom, the spin-orbit splitting is found to be ∼136 meV at the VBM which is relatively smaller than other TMDs ( ∼180 meV in MoS 2and∼400 meV in WS 2)[54]. The symmetry-independent spin-orbit splitting 205405-2RAMAN FINGERPRINT OF STACKING ORDER IN … PHYSICAL REVIEW B 99, 205405 (2019) TABLE I. For the monolayer crystals HfS 2and Ca(OH) 2, the optimized lattice constants a=b, atomic bond lengths in the crystal dX−Y, energy band gaps calculated within SOC on top of GGA EGGA gap,H S Eo nt o po fG G A +SOC EGGA+HSE gap , location of VBM and CBM edges in the BZ, the thermionic work function /Phi1, in-plane and out-of-plane dielectric constants /epsilon1xxand/epsilon1zz, and frequencies of Raman active phonon modes. a=bd X−Y EGGA gap EGGA+HSE gap VBM/CBM /Phi1/epsilon1 xx /epsilon1zz Eg A1g Eg(OH) A1g(OH) (Å) (Å) (eV) (eV) (eV) (cm−1)( c m−1)( c m−1)( c m−1) HfS 2 3.63 2.55 1.19 1.98 /Gamma1/M 6.19 3.09 1.23 255.3 326.4 – – Ca(OH) 2 3.59 2.34 (Ca-O) 3.65 5.16 /Gamma1//Gamma1 4.82 1.33 1.15 360.7 345.3 234.6 3836.1 0.97 (O-H) occurs on top few valence bands which are composed of the px andpyorbitals of a S atom. In contrast, 1 T-Ca(OH) 2is found to be a direct gap insulator with a band gap of 5.16 eV whoseVBM and CBM reside at the /Gamma1point in the BZ [see right panel of Fig. 1(b)]. The valence bands in monolayer Ca(OH) 2 are mainly composed of the O- porbitals while Ca- dorbitals mostly contribute to the conduction bands. A very small spin-orbit splitting energy of 32 meV occurs at the VBM. For the ﬁrst-order off-resonant Raman spectrum, the fre- quencies of phonon modes are calculated at the /Gamma1point of the BZ [see Fig. 1(c)]. Using the vibrational character of each phonon mode, the change of dielectric constantsare used in order to calculate the Raman activity of eachmode. Monolayer HfS 2exhibits nine phonon branches three of which are acoustical phonons. The remaining six opticalbranches consist of two nondegenerate out-of-plane phonons,A 1gandA2uat frequencies 326.4 and 320.0 cm−1, respectively, and four in-plane doubly degenerate phonon modes, Egand Euhaving frequencies 255.3 and 141.0 cm−1, respectively [see Fig. 1(d)]. In contrast to 1 H-TMDs, the Raman activity ofA1gmode is larger than that of Egeven in monolayer HfS 2. Monolayer Ca(OH) 2exhibits twelve optical phonon branches two of which have very high frequencies, 3836.1and 3837.7 cm −1, arising from O-H bond stretching. As shown in Fig. 1(d), the Raman active optical mode with frequency 3836.1 cm−1is attributed to the opposite out-of- plane stretching of O and H atoms in an (OH)−group while the Ca atom remains stationary. In the Raman inactive opticalmode having frequency 3837.7 cm −1, similarly the O and H atoms vibrate in opposite out-of-plane directions while each (OH)−group vibrates in-phase leading to the infrared activity of the phonon mode. Apart from these two high-frequencyoptical modes, another ten optical branches are found to havefrequencies between ∼235–435 cm −1which are the usual phonon modes arising from the 1 Tnature of the structure. The phonon mode having frequency 360.7 cm−1is a doubly degenerate Raman active mode which is attributed to theopposite in-plane vibration of (OH) −groups against each other. The other doubly degenerate Raman active mode iscalculated to have frequency 234.6 cm −1which originates from the opposite in-plane vibration of O and H atoms ina( O H ) −group. In addition to the in-plane Raman active modes, there is only one nondegenerate Raman active modewith frequency 345.3 cm −1and it is attributed to the opposite out-of-plane vibration of each (OH)−group. It is pointed out that the Raman activities of these three phonon modesare much smaller as compared to the high frequency opticalmode.IV . Ca(OH) 2-HfS 2HETEROBILAYER For the heterostructure consisting of two layers, three different stacking orders (i.e., the most symmetric stackingorders) are considered and the corresponding layer-layer bind-ing energies are calculated. Taking the Hf atom as a reference,three stacking types can be deﬁned; the AA stacking (Hf atomresides on top of Ca atom), AB (Hf on top of upper (OH) −), and the AB/prime[Hf on top of lower (OH)−] stacking orders (see Fig. 2). Structural optimizations reveal that all three stacking orders have the same lattice parameters of a=b=3.62 Å which shows the same amount of induced biaxial strain. Asthe heterobilayer structure is constructed, monolayer HfS 2 experiences a compressive biaxial strain of 0.28% whilethe monolayer Ca(OH) 2exhibits a tensile strain of 0.84%. Therefore, the changing electronic and phononic properties ofindividual layers in the heterostructure is attributed not onlyto the layer-layer interaction arising from different stackingorders but also to the induced biaxial strains on each layer.As listed in Table II, the layer-layer binding energies per primitive cell are calculated to be 172, 178, and 117 meVfor AA, AB, and AB /primestacking orders, respectively. Although the AB stacking is the ground state, the binding energies ofthe three stacking orders are very close to each other. It isimportant to analyze the electronic and phononic properties ofthese three stacking orders in order to understand the physicalmechanisms driving the differences between them. The x-ray diffraction patterns of AA, AB, and AB /primestacking orders are calculated and the distinctive parts are presentedin Fig. 2(d) which allow us to identify the type of stacking. In the low-angle regime (between 2 θ7 ◦and 14◦), the XRD patterns of all stacking exhibit three peaks whose relativeintensities are different. It is seen that the intensity of the TABLE II. For the HfS 2-Ca(OH) 2heterobilayer structures; the stacking conﬁguration, the optimized lattice constants a=b, op- timized out-of-plane layer-layer distance dLL, layer-layer binding energy per primitive unit cell Ebind, energy band gaps calculated within SOC on top of GGA EGGA gap, HSE on top of GGA +SOC EGGA+HSE gap , and location of VBM and CBM edges in the BZ. a=bd LL Ebind EGGA gap EGGA+HSE gap VBM/CBM (Å) (Å) (meV) (eV) (eV) AA 3.62 1.88 172 0.14 1.36 /Gamma1/M AB 3.62 1.89 178 0.20 1.32 /Gamma1/M AB/prime3.62 2.55 117 0.03 0.95 /Gamma1/M 205405-3M. Y AGMURCUKARDES et al. PHYSICAL REVIEW B 99, 205405 (2019) -2-10123456 MK M Γ MK MΔSOC=32 meV Γ-2-10123456 ΔSOC=136 meVEnergy (eV)(a) (b) GGA+SOCGGA+SOC+HSEHf SCa O H 200 250 300 350 400 3800 3825 3850x100x70 x300Raman Activity (a.u.) Ω (cm-1)Ca(OH)2 HfS2 A1g Eg EgA1g 234.6 cm-1255.3 cm-1326.4 cm-1 3836.1 cm-1360.7 cm-1345.3 cm-1EgA1g (OH) A1g (OH) A1gA1g Eg (OH)EgEg (OH)(c) (d) FIG. 1. For the monolayers of HfS 2(left panel) and Ca(OH) 2 (right panel), (a) top and side views of the crystal structure (the blue parallelograms show the primitive unit cell of each structure) and (b) electronic band dispersions. In the electronic band dis-persions, the insets display the spin-orbit splitting at the valence band maximum. The Fermi level is set to zero. (c) The calculated Raman spectra and (d) vibrational characteristic of each Ramanactive phonon mode. Ebind = 117 meV Ebind = 178 meV Ebind = 172 meV Hf S Ca O H1.88 Å 2.55 Å 1.89 Å(a)AB' AB AA (b) (c) 010203040506070 AA ABAB'∆Etot (meV/formula) Stacking Position80(e) 27 29 31 33 35 37 71 0 1 1 1 2 1 3 14 89 2θ (degree)Intensity (a.u.)AA ABAB'(d) FIG. 2. Top and side views of (a) AB/prime, (b) AA, and (c) AB stacking conﬁgurations. (d) The calculated x-ray diffractograms ofthe three stacking conﬁgurations. Only the distinctive regime of the diffractograms are shown. (e) The variation of the total energy per unit cell of heterobilayer structure with respect to the sliding of HfS 2 layer on top of Ca(OH) 2along three main directions. The inset shows the sliding directions. peak around 10 .5◦is maximum for AB/primestacking. Moreover, between the 26◦and 37◦there are many peaks which are key to determine the stacking type. The intensity of the peak at2θ35.6 ◦is the maximum for AB stacking order while the intensities of the peaks at 32 .6◦and 34◦are maximum for AA stacking order. Furthermore, the peak at 2 θ31.5◦has approximately the same intensity for AA and AB/primestacking while it is minimum for AB stacking. It appears that XRDpatterns provide an effective way to distinguish the stackingtype of HfS 2-Ca(OH) 2heterobilayer. In addition, we also analyze the energy barrier which is seen by HfS 2layer as it slides on top of the Ca(OH) 2layer. 205405-4RAMAN FINGERPRINT OF STACKING ORDER IN … PHYSICAL REVIEW B 99, 205405 (2019) Energy (eV)0 -1-2 -4 -5-6-7-3 -8-1.11 -4.94 -6.91-6.27(a) Im[ε] 1.0 1.5 2.0 2.5 3.0 3.5 4.0AAAB' AB(c)O HfS -2.0-1.00.01.02.03.04.05.0Energy (eV) M K MΓ(b) Reflectivity 0.00.20.40.60.8 Wavelength (nm)(d) 300 350 450 500 550 400HfS2Ca(OH)2H Energy (eV) FIG. 3. (a) The band alignment between monolayers HfS 2and Ca(OH) 2. (b) The electronic band dispersions of the ground state stacking in heterobilayer structure. The Fermi level is set to zero.For the three different stacking conﬁgurations: (c) the imaginary part of dielectric function and (d) reﬂectivity. Visible light region is presented in the inset. In order to calculate such an energy barrier, we consider three main directions as shown in the inset of Fig. 2(e). When the HfS 2layer slides along the direction denoted by the red curve, all three stacking orders are recovered. For the considereddirections, the maximum energy barrier is calculated to be61 meV which is much smaller than that of in bilayer TMDs(∼240 meV for MoS 2)[44]. Such a smaller energy barrier was also reported for bilayer 1 Tphase of TMDs [ 55]. Notice that the energy barrier between AA and AB stacking ( ∼20 meV) is smaller than the thermal energy at room temperature(25.7 meV). In addition, the transition to AB /primestacking can also be achieved by external effects such as increasing thetemperature. Since our results reveal that different stackingorders can be formed during the experimental procedure, iden-tiﬁcation of electronic, optical, and vibrational characteristicsof different stackings of HfS 2-Ca(OH) 2heterobilayer is of importance. V . MONITORING THE STACKING VIA ELECTRONIC AND OPTICAL PROPERTIES The band alignment of the two monolayer crystals is ﬁgured out by setting the vacuum energy of each monolayerto 0 eV . It is found that the two monolayers form a type-II(staggered type) heterostructure in which the two band edgesoriginate from different individual layers and consequentlythe excited electrons and holes are conﬁned in different layers[see Fig. 3(a)], which form interlayer excitons due to the Coulomb attraction. Such a spatial separation of electronsand holes in interlayer excitons leads to longer lifetimes(in the nanosecond range) [ 56] than intralayer excitons (a few picoseconds) [ 57] that have potential for applications in optoelectronics and photovoltaics. As shown in Fig. 3(b), for the ground state stacking order, AB stacking, the GGA +HSE calculated indirect band gap is found to be 1.32 eV whose VBM resides at the /Gamma1point whilethe CBM is located at the Mpoint. Due to weak vdW in- teraction between individual layers, there is no hybridizationbetween the layers and therefore, the main contribution to theVBM comes from the Ca(OH) 2layer while the states in the vicinity of CBM are from the HfS 2layer. As listed in Table I, while the band gap of AA stacking order is very similar tothat of AB stacking, the calculated band gap of AB /primestacking is lower since the direct interaction between the layers occurthrough the lower S atom and upper (OH) −which contribute to the CBM or VBM. As presented in Figs. 3(c) and3(d), the optical properties of the AA, AB, and AB/primestacking orders are investigated in terms of the imaginary dielectric function and reﬂectivity,respectively. It is seen that the overall trend for the threestacking conﬁgurations are very similar with only smalldifferences. The ﬁrst peak in the imaginary dielectric functionof AA, AB, and AB /primestackings occurs between 2.0 and 2.5 eV , whereas their main peaks reside in the vicinity of 2.8 eV .In order to see visible region optical activity, we plot thereﬂectivity of the heterobilayer [see Fig. 3(d)]. It is seen that the optical reﬂectivity values are very similar in thevisible range. Although it exhibits moderate reﬂectivity in the300–400 nm range, the overall reﬂectivity in the visible regionis low. Therefore, the identiﬁcation of the stacking type inHfS 2-Ca(OH) 2heterobilayer structure is almost impossible from an analysis of the electronic and optical properties. VI. MONITORING THE STACKING VIA RAMAN SPECTRUM Different stacking orders in a layered material or in a vertical heterostructure may exhibit distinctive properties inthe Raman spectrum. In a Raman experiment, it is possibleto distinguish different stacking orders via both the frequencyshift of the phonon modes and the change in the correspondingRaman activities. In this section, we present our results onthe ﬁrst-order off-resonant Raman spectrum for three stackingorders As shown at the bottom panel of Fig. 4, in addition to four characteristic Raman active modes (labeled from I to IV), onenondegenerate out-of-plane and doubly degenerate in-planephonon modes occur in the low-frequency regime. The SMand layer breathing modes LBM are attributed to the rigidin-plane and out-of-plane vibrations of HfS 2and Ca(OH) 2 layers against each other. Previously it was reported by us thatthe SM and LBM phonon modes are important to understandthe layer-layer interaction and thus the stacking type in alayered material [ 58]. For the AB /primestacking conﬁguration, the frequencies of SM and LBM are calculated to be the smallest(8.5 and 36.2 cm −1, respectively) which are consequence of the low binding energy of the layers in the AB/primestacking. However, in AA and AB stacking orders the SM and LBMmodes are found to be at higher frequencies (26.0–46.5 cm −1 and 26.7–49.2 cm−1for AA and AB stacking orders, respectively). Therefore, the AB/primestacking type can be monitored via the peak frequencies of SM and LBM modes.Moreover, distinguishing between AA and AB stackingtypes seems to be feasible through LBM modes with thefrequency difference of 2.7 cm −1which is sufﬁciently large to be detected experimentally. Notably, the peak intensities 205405-5M. Y AGMURCUKARDES et al. PHYSICAL REVIEW B 99, 205405 (2019) 0 20 200 300 400 3750 3800 3850AB' AA ABRaman Activity (a.u.) Ω (cm-1) EgISM LBM EgIVAgIIAgIIIEgI AgIIAgIII EgIV 40 60SM LBM x103x103x104 FIG. 4. Raman spectrum (top ﬁgure) and Raman active modes (bottom ﬁgure) of AB/prime, AA, and AB stackings. of the low-frequency modes are too small as compared to other prominent peaks. Therefore, peak intensities are notconsidered to be distinctive parameters for distinguishingthe stacking type even in experiments due to the dominantRayleigh scattering intensity at zero frequency shift. Apart from the SM and LBM modes, there are four main Raman peaks found in between ∼250–400 cm −1as shown in Fig. 4. We label the in-plane phonon modes as EI g,EIV g and the out-of-plane modes as AII g,AIII g.T h e EI gmode arises from HfS 2layer and it displays a phonon hardening in AA and AB stacking orders (256.6 cm−1in each stacking type) while it displays a phonon softening in AB/primestacking type (251.0 cm−1). It is clear that AB/primestacking can be monitored via the frequency of the EI gphonon mode. In contrast to AA and AB stacking types, in AB/primestacking the frequency shift of the EI gmode is dominated by the layer-layer interaction over the induced strain. Another in-plane Raman active modeE IV garises from the opposite vibrations of (OH)−against each other. It is originally the Egmode of monolayer Ca(OH) 2and it is coupled with that of the HfS 2layer in the heterobilayer structure. The frequency of the EIV gmode is calculated to be 380.4, 384.9, and 388.9 cm−1in AA, AB, and AB/primestacking types, respectively. It is seen that the frequency differencesbetween the three stacking types are considerable and aresufﬁciently large in order to be used to monitor the type of stacking. The other two Raman active modes, labeled as A II gand AIII g, are attributed to the out-of-plane vibration of the atoms from different layers. The AII gmode is a coupled out-of-plane mode in which the main contribution arises from the HfS 2 layer. Contrarily, in the AIII gmode the main contribution to the vibration arises from the Ca(OH) 2layer. The frequen- cies of the AII gmode are calculated to be 328.4, 328.7, and 329.1 cm−1in AA, AB, and AB/primestacking types, respectively. It is obvious that the AII gmode displays phonon hardening in all stacking types as compared to monolayer HfS 2.D u e to the same frequency shift behavior and small frequency differences, monitoring of the stacking type via the AII gmode is not feasible. However, the AIII gmode exhibits phonon hardening in AA and AB stacking (345.6 and 346.9 cm−1, respectively) and it displays phonon softening in AB/primestacking (344.4 cm−1) as compared to its frequency in Ca(OH) 2layer (345.3 cm−1). Although there are small frequency differences ofAIII gmode between various stackings, its phonon softening in AB/primestacking can be used to distinguish the stacking order. There are two Raman active modes having high frequen- cies that arise from the O-H bond stretching. In addition, anadditional Raman active phonon mode appears in the hetero-bilayer structure that is attributed to the opposite out-of-planevibration of O-H atoms in a (OH) −group while each (OH)− also vibrates out-of-phase. Both of the O-H stretching modesdisplay phonon softening as compared to their frequencies inmonolayer Ca(OH) 2. As shown in the top panel of Fig. 4,t h e relative frequencies of the two phonon modes can be used to monitor the type of stacking order. Frequency differenceof the two modes are found to be 3.3, 11.3, and 38.6 cm −1 for AB, AA, and AB/primestacking conﬁgurations, respectively. It is obvious that AB stacking order can be distinguisheddue to the smallest frequency difference while the AB /primestack- ing can be monitored with its largest frequency difference. Moreover, the AB stacking can be also distinguished via the Raman activity ratio of the two modes that is calculatedto be approximately 100 times greater than those for AAand AB /primestacking types. Furthermore, the AB stacking can also be identiﬁed due to the largest Raman activity of thephonon mode having frequency 3829.1 cm −1(approximately one order of magnitude larger). The reason for such rela-tively large activity is the dominant contribution of in-plane dielectric constants to the Raman tensor. The general form of Raman tensor for out-of-plane vibrations in 2D materialsis known to be totally diagonal [ 58]. Calculated in-plane components of the dielectric tensor are found to be slightlylarger for AB stacking (2.90 for AB stacking, 2.86 and 2.80for AA and AB /primestacking types, respectively). The variation of the in-plane dielectric constant with respect to O-H bond stretching is larger in AB stacking type. Our results reveal that the ultrahigh frequency regime of the Raman spectrumcan be used to monitor the type of stacking in HfS 2-Ca(OH) 2 heterobilayer structure in terms of both peak frequencies andRaman activities. Moreover, our ﬁndings show that particu-larly the O-H bond stretchings in layered AEHMs can be usedas important keys for identiﬁcation of stacking type in vdW heterostructures composed of any 2D material and monolayer AEHMs. 205405-6RAMAN FINGERPRINT OF STACKING ORDER IN … PHYSICAL REVIEW B 99, 205405 (2019) VII. CONCLUSIONS In this study we investigated the stacking order dependence of the electronic and vibrational properties of HfS 2-Ca(OH) 2 heterobilayer structures by means of the electronic band dis-persions and the Raman spectra. Electronic band dispersionsand optical spectrum revealed that the band gap varies verylittle between the different stacking types. Analysis of Ramanspectra showed that different stacking types possess distinctcoupling phenomena which can be used to identify the type ofthe stacking. Our ﬁndings on the phononic properties are thefollowing: (i) the SM and LBM modes are able to monitor theAB /primestacking order, (ii) in addition, the AB/primestacking can be identiﬁed via the phonon softening of the EI gandAIII gmodes which harden in the other two stackings, and (iii) importantly,the ultrahigh frequency regime displays distinctive propertieswhich can be used to distinguish between all stacking types.Moreover, the differences in optical and vibrational propertiesof various stacking types were found to be driven by twophysical effects, induced biaxial strain on the layers and the layer-layer interaction. Our results reveal that from both thephonon frequencies and the corresponding Raman activitiesit is possible to distinguish between the different stackingtypes in these novel vertical heterostructures constructed byalkaline-earth-metal hydroxides. ACKNOWLEDGMENTS Computational resources were provided by TUBITAK ULAKBIM, High Performance and Grid Computing Center(TR-Grid e-Infrastructure). H.S. acknowledges ﬁnancial sup-port from the Scientiﬁc and Technological Research Councilof Turkey (TUBITAK) under the Project No. 117F095. H.S.acknowledges support from Turkish Academy of Sciencesunder the GEBIP program. This work is supported by theFlemish Science Foundation (FWO-Vl) by a postdoctoralfellowship (M.Y .). [1] S. K. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, Y . Zhang, S. V . Dubonos, I. Grigorieva, and A. A. Firsov, Science 306,666(2004 ). [2] K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, M. I. Katsnelson, I. V . Grigorieva, S. V . Dubonos, and A. A. 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PhysRevB.76.125204.pdf	Interstitial Fe in Si and its interactions with hydrogen and shallow dopants M. Sanati, N. Gonzalez Szwacki, and S. K. Estreicher * Physics Department, Texas Tech University, Lubbock Texas 79409-1051, USA /H20849Received 19 March 2007; revised manuscript received 20 June 2007; published 11 September 2007 /H20850 The properties of interstitial iron in crystalline silicon and its interactions with hydrogen, shallow acceptors /H20849B, Al, Ga, In, and Tl /H20850, and shallow donors /H20849P and As /H20850are calculated from ﬁrst-principles in periodic supercells. The interactions between the /H20853Fe,B /H20854pair and interstitial hydrogen are also examined. The conﬁgurations, electronic structures, and binding energies are predicted. The relative stability of the trigonal and orthorhombicstructures of the Fe-acceptor pairs are calculated as a function of charge state and temperature. The gap levelsare estimated using the marker method. The vibrational spectra of the complexes containing light impurities /H20849H or B /H20850are predicted. DOI: 10.1103/PhysRevB.76.125204 PACS number /H20849s/H20850: 63.20.Mt, 61.72.Bb I. INTRODUCTION Transition metal /H20849TM /H20850impurities are common and most often undesirable contaminants in both integrated-circuitgrade Si and Si-based photovoltaic materials. Istratov, Hiesl-mair, and Weber have reviewed the properties of Fe and itscomplexes in Si /H20849Ref.1/H20850as well as the possible sources of Fe contamination. 2These reviews underscore how little is known at the atomic level about the chemistry of Fe in Si. Some Fe is always present in the source material. Its solu- bility sharply decreases as the sample cools down from themelting point to room temperature. Thus, the as-grown crys-tal is often supersaturated with Fe. Interstitial iron /H20849Fe i/H20850 readily diffuses and traps within hours at shallow acceptors or TMs, precipitates a various defects including oxides, andmay even form silicides. Many of these defects are magneti-cally and electrically active. More than thirty Fe-related complexes have been detected by electron paramagnetic resonance /H20849EPR /H20850, electron-nuclear double resonance /H20849ENDOR /H20850, deep-level transient spectros- copy /H20849DLTS /H20850or other experimental techniques. In some cases, a speciﬁc center has been identiﬁed. In many cases,the structure of the defect is not conclusively known. A sum-mary of the key results relevant to the present work is asfollows. EPR, 3ENDOR,4and Mössbauer studies5,6show that Fe i resides at the tetrahedral interstitial /H20849T/H20850site. The ﬁtting of /H9252− channeling patterns7following the implantation of the radio- active isotope59Fe implies that Fe becomes substitutional following high-temperature /H20849above 800°C /H20850anneals. At lower temperatures, Fe is near Tsite, where “near” means 0.3–0.8 Å. Such a large displacement and the associatedsymmetry lowering relative to the Tsite is not observed in the EPR and ENDOR data. It is possible that the defect cen-ters formed in the channeling experiments involve native de-fects. The diffusivity of Fe ihas been measured by a number of groups using a range of experimental methods. The activa-tion energies 1span the range 0.49–0.92 eV. The charge state of the diffusing species is not always known with certaintyand trap-limited diffusion could affect some measurements.A ﬁt to all the data over a wide range of temperatures leads 1 to the activation energy for diffusion /H110110.67 eV. Severalauthors8,9ﬁnd that the activation energy for diffusion of Fei+ /H20849/H110110.69 eV /H20850is lower than that of Fei0/H20849/H110110.84 eV /H20850. These val- ues have been debated.10,11Measurements of the formation rates of Fe-acceptor pairs show that the pairing kinetics arelargely independent of the dopant and involve 12an activation energy in the range 0.66–0.68 eV, suggesting that the pro- cess is limited by the diffusion of Fei+. Hall effect and resistivity measurements13,14show the existence of a donor level of Fe i, but iron contamination ofn-type Si does not reduce the concentration of free elec- trons, suggesting that the isolated interstitial has no deepacceptor level. The position of the /H208490/+ /H20850level has been mea- sured to be in the range 0.39–0.45 eV /H20849Ref. 1/H20850from photoionization, 15,16DLTS,14,17–20and the monitoring of the EPR intensities of Fei+and Fei0as a function of the Fermi level.21The latter work conﬁrms that only Fei+and Fei0exist for Fermi levels in the range Ev+0.045 eV to Ec−0.045 eV. Ludwig and Woodbury22and Feher23have proposed a model for the electronic structure of 3 dTM impurities in the Si crystal ﬁeld. The /H20849atomic /H208504selectrons are transferred to the 3 dshell. In tetrahedral symmetry, these electrons popu- late a t2and an elevels /H20849the latter is slightly higher in en- ergy /H20850in accordance with Hund’s rule. In the case of Fei0for example, the 4 s23d6atomic structure becomes 4 s03d8, the eight electrons populate the t2andelevels, and Fei0has spin 1. Similarly, Fei+has spin3 2. These spin states are consistent with the EPR data. Note that since the elevel is above the t2 level, Fei+is an orbital triplet and a Jahn-Teller distortion should result. The EPR lines associated with Fei+are much broader than those associated with Fei0. This could be con- sistent with a dynamic Jahn-Teller effect.24,25 Inp-type material, Fei+readily interacts with /H20849ionized /H20850 shallow acceptors. The trigonal /H20849C3v/H20850/H20853Fe,B /H20854pair was ﬁrst observed by EPR,26,27then ENDOR.28It is a strong recom- bination center with a donor level near Ev+0.11 eV /H20849Refs. 14 and 29–34/H20850and a deep acceptor level near Ec−0.29 eV.20,30,34These values have recently been con- ﬁrmed by temperature-dependent and injection-dependentlifetime spectroscopy data. 35A second acceptor level at Ev +0.074 eV /H20849Refs. 32and33/H20850is tentatively associated with a metastable orthorhombic /H20849C2v/H20850conﬁguration of the pair. Un- der minority carrier injection, the amplitude of the Ec −0.29 eV signal diminishes as a new level at Ec−0.43 eVPHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 1098-0121/2007/76 /H2084912/H20850/125204 /H208499/H20850 ©2007 The American Physical Society 125204-1appears, suggesting that this is the acceptor level of the /H20853Fe,B /H20854pair in the orthorhombic conﬁguration.33The binding energy Eb, deﬁned from Fei++B−→/H20853Fe,B /H208540+Eb,i si nt h e range 0.58 eV /H20849Refs. 36–38/H20850to 0.65 eV.20,39Photolumines- cence /H20849PL/H20850studies40show the existence of optically-active defects which incorporate B and Fe but are distinct from the/H20853Fe,B /H20854pair discussed here. The neutral charge state of the /H20853Fe,Al /H20854pair has been de- tected by EPR in a trigonal /H20849stable /H20850and an orthorhombic /H20849metastable /H20850conﬁguration, 41,42with spin3 2.42,43The positive and negative charge states of the trigonal conﬁguration havealso been detected. 43–45In the dark, the thermally induced activation energies for reorientation46are 0.5 eV for /H20855100 /H20856 →/H20855111 /H20856and 0.6 eV for /H20855111 /H20856→/H20855100 /H20856. One would expect such numbers were Fe to simply hop from the second-nearest to the nearest Tsite to the acceptor and vice versa. Under illumination, these activation energies drop to /H110110.1 eV.47 Following in-diffusion, cooling the sample under illumina- tion produces both DLTS signals, while cooling in the darkenhances the trigonal signal. 46The binding energy in the trigonal conﬁguration is in the range 0.52 /H20849Ref. 46/H20850to 0.70 eV /H20849Ref. 38/H20850and the donor levels46are at Ev+0.20 eV /H20849C3v/H20850andEv+0.13 eV /H20849C2v/H20850. The neutral charge state of the /H20853Fe,Ga /H20854pair has been de- tected by EPR /H20849Refs. 26,42,48, and 49/H20850in both the trigonal /H20849stable /H20850and orthorhombic /H20849metastable /H20850conﬁgurations with spin3 2. The donor levels are at Ev+0.24 eV /H20849C3v/H20850/H20849Refs. 14, 38, and 50/H20850andEv+0.14 eV /H20849C2v/H20850.51The binding energy in the trigonal conﬁguration is Eb=0.47 eV.38 The EPR signal of the neutral /H20853Fe,In /H20854pair was detected in the stable C2v/H20849Refs. 26,52, and 53/H20850and the metastable C3v /H20849Refs. 54and55/H20850conﬁgurations. The latter was achieved following illumination of the sample with 0.5 eV photons.The positive charge state of the pair has also beenobserved. 52,56DLTS studies51show donor levels at Ev +0.15 eV in the orthorhombic and Ev+0.27 eV in the trigo- nal conﬁgurations. Fourier transform infrared absorption/H20849FTIR /H20850experiments 57–59of the excitation spectrum of /H20853Fe,In /H20854 ﬁnd the acceptor levels at Ec−0.39 eV /H20849C2v/H20850and Ec −0.32 eV /H20849C3v/H20850. Finally, the existence of a metastable complex involving Fe and Tl has been inferred from PL bands.60The trigonal symmetry proposed by Sauer and Weber61was later conﬁrmed.62,63The intensities of the PL bands increases with the Fe iconcentration. However, the identiﬁcation of these PL bands with /H20853Fe,Tl /H20854pairs has yet to be conﬁrmed.1,64 Thus, each Fe-acceptor pair has two conﬁgurations with three possible charge states as well as a donor and an accep-tor level in each of them. The Fe pair with B, Al, and Ga isstable in the C 3vconﬁguration, with Fe at the nearest Tsite to the acceptor. In the case of Al and Ga, the C2vconﬁgura- tion with Fe at the second-nearest Tsite is energetically very close. As for /H20853Fe,In /H20854, the C2vconﬁguration has the lowest energy. Chantre et al.46,51have used the ratios of the DLTS inten- sities for various biases during cool down to estimate therelative populations of the trigonal and orthorhombic con-ﬁgurations, and then ﬁt them to a Boltzmann term. The en-ergy differences /H9004Ethey obtained in the 0 and /H11001chargestates are /H9004E/H208490/H20850=0.14,0.13,−0.01 eV and /H9004E/H20849+/H20850 =0.07,0.03,−0.13 eV for Al, Ga, and In, respectively. A positive value favors the trigonal conﬁguration. These smallenergy differences have been compared to various potentialenergy predictions /H20849see Ref. 1/H20850. However, the measured /H9004E’s are not true T=0 K potential energy differences, but rather the differences between free energy minima at the tempera-tureT minbelow which Fe iis not longer able to overcome the energy barrier /H20849/H110110.5–0.6 eV /H20850between the two sites in the time scale commensurate with the cooling down rate. There is no direct evidence for the existence of Fe-donor pairs. A decrease in the Fei0signal with increasing Ps+con- centration has been interpreted as evidence of Fe-P pairing65 with a binding energy of 0.9 eV. However, no change in theEPR spectrum of phosphorus has been observed even after the Fe i0signal disappears following long 200°C anneals.66 Four weak satellites in the EPR spectrum of Fei0at high P or As concentrations exist, but they are weaker than would be expected for close pairs. The formation of Fei−−Ps+pairs has been proposed67but there is no experimental or theoretical evidence of even a single acceptor level of Fe iin the gap. An EPR-active /H20853Fe,P /H20854pair has been detected in irradiated n-type Si, but this complex most probably involves one or morevacancies. 68 Sadoh et al.69performed thermally stimulated capacitance and DLTS measurements in n-type Si samples contaminated with Fe and subsequently etched with H-containing chemi-cals. The presence of a C-H complex conﬁrmed that H pen-etrated into the sample. In addition to the Fe idonor level observed at Ev+0.41 eV, the authors tentatively assigned a new donor level at Ev+0.31 eV to a /H20853FeiH/H20854pair. This signal disappears after annealing at 175°C for 30 min. The early theoretical work on interstitial 3 dTM impuri- ties in Si has focused on explaining trends in the electronicstructure and gap levels using methods that do not allow fordefect geometries to be optimized. Since Fe iis known to have tetrahedral symmetry, it was assumed that the impurityresides at an undisturbed Tsite. Our geometry optimizations /H20849see below /H20850show that this assumption is valid. The scattering X /H9251method in H-terminated clusters,70later enhanced with semi-empirical Hartree-Fock calculations,71have been suc- cessful at predicting many qualitative features of these im-purities. They predicted the donor level of Fe ito be at Ev +0.68 eV. Zunger and co-workers72–76used a self-consistent local density functional /H20849LDF /H20850Green’s function scheme to study trends and predicted the spin-polarized electronic structure ofFe i.74They found its donor level to be at Ev+0.53 eV /H20849Ref. 72/H20850orEv+0.32 eV.74,75Beeler et al.77performed calcula- tions within the spin-unrestricted LDF linear mufﬁn-tin- orbital Green’s function method. They predict a spin3 2for Fei+and a spin 1 for Fei0, in agreement with experiment. They also predict the donor level of Fe ito be at Ev+0.25 eV, close to the DLTS value. Weihrich and Overhof25have calculated in detail the elec- tronic structure and hyperﬁne parameters of Fe iin the/H11001and 0 charge states, using a Dyson’s equation approach to solvethe Kohn-Sham equation within local spin-density-functionaltheory and calculated the Green’s function within a linearSANATI, SZWACKI, AND ESTREICHER PHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 125204-2mufﬁn-tin approach using the atomic-spheres approximation. These calculations do not allow geometry optimizations tobe performed. The authors predict the donor level of Fe ito be at Ev+0.29 eV. The interactions between Fe and shallow dopants have been discussed by several authors. Assali and Leite78use the scattering X/H9251method in small hydrogen-saturated clusters to calculate the electronic structure of the C3v/H20853Fe,B /H20854pair, with- out spin polarization. They predict a strongly covalent Fe-Binteraction. Sugimoto and Seki 79used H-terminated clusters within unrestricted ab initio Hartree-Fock theory, with a combination of minimal and double-zeta basis sets. The ac-ceptor atoms /H20849B, Al, Ga, and In /H20850were assumed to be at unrelaxed substitutional sites. The authors ﬁnd the C 3vcon- ﬁguration to be more stable for B, Al, and Ga, but the C2v one is more stable in the case of In. The population analysis of the /H20853Fe,B /H20854pair shows that Fe ioverlaps only very weakly with B, which argues against the mostly covalent character predicted in Ref. 78. Overhof and Weihrich67used their Green’s function approach25/H20849without lattice relaxations /H20850to study the trigonal and orthorhombic conﬁgurations of the Fe-acceptor pairs.Their calculations predict a mostly ionic bonding between Feand the acceptors. Their calculated binding energies are0.78 eV for /H20853Fe,B /H20854and 0.28 eV for /H20853Fe,Al /H20854. They also calcu- late the relative energies between the two conﬁgurations, butthe predictions are affected by the absence of geometry op-timization. The calculated donor and acceptor levels for theFe-acceptor pairs are in reasonable agreement with experi- ment. The authors also predict a pairing between Fe i−−and Ps+. Zhao et al.80used an ionic model with elastic and electro- static interactions to show that the driving force for the Fei+-As−pairing is electrostatic. They also calculated trends in the position of the donor level and the relative energies of theC 3vandC2vconﬁgurations. The present work differs from the previous theoretical approaches in several ways. First, the host crystal is repre-sented by 64 to 216 atoms periodic supercells. These are notas ideal a representation of the isolated impurity in an other-wise perfect crystal as that provided by Green’s functions,but they allow the use of conjugate gradient geometry opti-mizations. Second, we use a ﬁrst-principles spin-density-functional approach within the generalized gradient approxi-mation /H20849GGA /H20850, with both plane-wave and pseudoatomic basis sets for the single particle states. Such a “ﬁrst-principles” theoretical approach has been successfully used to study a wide range of defects, impuri-ties, pairs, and small complexes in Si and numerous otherhost crystals. 81In this paper, we focus on the properties of Fei, including its interactions with interstitial H, and Fe pairs with shallow substitutional dopants B, Al, Ga, In, Tl, P, andAs. The possibility of interactions between the /H20853Fe,B /H20854pair and interstitial H is also investigated. The geometries, elec-tronic structures, and spin states are calculated for all thepossible charge states of the defects. The binding energies ofthe complexes are predicted. The existence and approximatelocation of donor and acceptor levels are obtained using themarker method. 82The vibrational spectra of the complexes containing light impurities are calculated as well.Note that detailed ﬁrst-principles calculations of the hy- perﬁne parameters of Fe iin the /H11001and 0 charge states and iron-acceptor pairs have been successfully performed.25,67In situations were the impurity can be assume to reside at un-distorted high-symmetry sites—as is the case here /H20849see below /H20850—these calculations are better suited than our pseudo- potential method to reproduce the EPR and ENDOR data.We do not focus on this aspect of the problem. Section II discusses the methodology. Section III contains the results for Fe iand its interactions with H. Section IV deals with Fe pairs with shallow acceptors /H20849B, Al, Ge, In, Tl /H20850 and donors /H20849P, As /H20850. We also consider the possibility that the /H20853Fe,B /H20854pair interacts with interstitial H. A summary and a discussion are in Sec. V. The interactions of Fe with nativedefects and other common impurities will be the subject of alater paper. II. METHODOLOGY The present calculations are carried out using two ﬁrst- principles spin-density-functional packages, VASP /H20849Refs. 83–86/H20850and SIESTA /H20849Refs. 87and88/H20850, within GGA to the exchange-correlation potential /H20849see Ref. 89for VASP and90 forSIESTA /H20850. The reason for repeating a number of calcula- tions with a plane-wave and a local basis set code is to makesure that the predictions are independent of the choice ofbasis set and pseudopotential. Although convergence issometimes more difﬁcult to achieve with SIESTA than VASP, the structures, energy differences between conﬁgurations orspin states, vibrational spectra, and even thermodynamic gaplevels calculated with both codes are in close agreement witheach other and, when available, with experiment. Any sub-stantial discrepancy is noted in the text. The binding energieslisted in the text have been obtained with VASP, the vibra- tional spectra with SIESTA , and gap levels are calculated us- ing projector augmented-wave /H20849PAW /H20850potentials91as imple- mented in VASP. We denote the spin and charge state of a defect XasspinXcharge. In most of our calculations, the host crystal is represented by a 64 host atoms periodic supercell. We also use the 128and 216 host atoms cells in situations where size effects areexpected to be signiﬁcant, that is when calculating the bind-ing energies of Fe-acceptor pairs. The lattice constant of allthe cells is optimized. A 2 /H110032/H110032 Monkhorst-Pack 92mesh is used to sample the Brillouin zone for all the calculationsexcept dynamical matrices and gap levels /H20849see below /H20850. The defect geometries are optimized with a conjugate gradientalgorithm. The VASP calculations use plane-wave basis set and ultra- soft Vanderbilt type pseudopotentials.93In the VASP ap- proach, the solution of the self-consistent Kohn-Sham equa-tions are obtained using an efﬁcient matrix-diagonalizationroutine based on sequential band-by-band residual minimiza-tion method and Pulay-like charge density mixing. 94A plane- wave basis cutoff of 321 eV is used for the ultra-soft pseudo-potential calculations. With PAW potentials, the cutoff is398 eV. The SIESTA calculations use norm-conserving pseudopo- tentials in the Troullier-Martins form95to remove the coreINTERSTITIAL Fe IN Si AND ITS INTERACTIONS … PHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 125204-3regions from the calculations. The basis sets for the valence states are linear combinations of numerical atomicorbitals. 87,96,97In the present calculations, we use double-zeta basis sets /H20849two sets of valence sand p’s/H20850for the ﬁrst and second-row elements /H20849H and B /H20850, and double-zeta polarized basis sets /H20849two sets of valence sandp’s plus one set of d’s/H20850 for elements on the third row and below. The charge densityis projected on a real-space grid with an equivalent cutoff of250 Ryd to calculate the exchange-correlation and Hartreepotentials. This large cutoff is needed to describe the local-ized dstates of iron. The ultrasoft and PAW pseudopotentials are part of the VASP package. The SIESTA Fe pseudopotential has been opti- mized by Izquierdo et al.98,99It includes nonlinear core cor- rections. We use the same orbital populations in VASP as in SIESTA . The SIESTA pseudopotential for other elements have been optimized using the experimental bulk properties of theperfect solids and/or ﬁrst-principles calculations 100as well as vibrational properties of appropriate free molecules or de-fects when experimental data are available. This testing leadsto a ﬁne-tuning of the pseudopotential parameters relative tothe purely atomic ones: small changes in the core radiusand/or orbital populations. Once optimized, we take thesepseudopotentials to be transferable to the defect problems athand. For the heavy elements /H20849Ga, In, Tl /H20850, the semicore d electrons are always included in the valence states. The dynamical matrices are calculated using the force- constant method with k-point sampling restricted to the /H9003 point. Their eigenvalues are the normal-mode frequencies /H9275s. The orthonormal eigenvectors e/H9251is/H20849i=x,y,z/H20850give the relative displacements of the nuclei /H9251for each mode s. A quantitative measure of how localized a speciﬁc mode is on one atom or a group of atoms is provided by a plot of L/H20853/H9251/H208542=/H20849e/H9251xs/H208502 +/H20849e/H9251ys/H208502+/H20849e/H9251zs/H208502vssor/H9275s. Here, /H20853/H9251/H20854may be a single atom /H20849e.g., Fe /H20850or a sum over a group of atoms /H20851e.g., the Si nearest neighbors /H20849NNs /H20850to Fe /H20852. Such a localization plot allows the identiﬁcation of all the local and pseudolocal101vibrational modes in the cell /H20849LVMs and pLVMs, respectively /H20850as well as the resonant modes associated with a speciﬁc defect. Theknowledge of all the normal modes also allows the construc-tion of the phonon density of states g/H20849 /H9275/H20850. We obtain this function by evaluating the dynamical matrix at 100 kpoints in the Brillouin zone of the supercell. Once g/H20849/H9275/H20850is known, the Helmholtz vibrational free energy Fvibis straightforward to calculate.102 The gap levels are estimated using the marker method.82 The calculated ionization energies and electron afﬁnities arescaled to a known marker which we take to be the perfectcrystal. The same scaling is used to determine the donor andacceptor levels of the defect. We choose the perfect crystal asa reference point because there is no single marker that couldbe considered ideal for all the defects studied in the paper.Further, the geometry optimizations show that no substantialdistortion is involved. Our best estimates for the gap levelsare obtained with PAW /H20849Ref. 91/H20850potentials 86with a 3 /H110033 /H110033k-point sampling. The 3 /H110033/H110033SIESTA values are given in some cases for comparison. The PAW potentials are moreaccurate than the ultrasoft ones because the radial cutoffs/H20849core radii /H20850are smaller and the PAW potentials reconstructthe exact valence wave function with all nodes in the core region. The PAW method gives energy differences very closeto the ones obtained with the best full-potential linearizedaugmented-plane-wave method. III. RESULTS A. Fe iandˆFeiH‰ As expected, we ﬁnd that Fe iresides at an undistorted T site in both the /H11001and 0 charge states, with spin3 2and 1, respectively. No Jahn-Teller distortion is apparent in our cal-culations, which is not surprising at this level of theory. The spin 0 and 2 states of Fe i0are 0.74 and 0.69 eV higher than the spin 1 state, respectively. The spin1 2state of Fei+is 0.30 eV above the spin3 2state. We ﬁnd a donor level in the gap at Ev+0.28 eV /H20849SIESTA : 0.37 eV /H20850, but no acceptor level. A population analysis of3/2Fei+shows that the impurity does weakly overlap with its Si neighbors. The overlap popu-lations with each of the four NNs and the six second-NNs are0.17 and 0.12, respectively. These numbers are small butpositive. The spin population in the valence orbitals is 0.016in the 4 s, 0.024 in the 4 p, and 1.140 in the 3 dstates. Thus, 78% of the total spin is localized on 3/2Fei+, and the rest is on the neighboring Si atoms. In the case of1Fei0, the overlap populations are almost the same /H208490.17 and 0.13, respectively /H20850 but the spin populations are 0.013, 0.016, and 0.964, respec- tively. Thus, over 96% of the spin resides on1Fei0. The activation energies for diffusion of3/2Fei+and1Fei0 along T-hexagonal- Tsites are 0.69 and 0.76 eV, respectively. We obtained these values by placing Fe at the hexagonalinterstitial site with both an unrelaxed crystal and a fullyrelaxed one. The former provides an upper bound for theactivation energy /H208490.92 and 0.91 eV in the /H11001and 0 charge states, respectively /H20850and the latter a lower bound /H208490.47 and 0.60 eV in the /H11001and 0 charge states, respectively /H20850. Our ac- tivation energies, obtained by averaging these two values, areclose to the charge-dependent values measured by severalauthors. 8,9 Inp-type material, no reaction involving Fei+and bond- centered hydrogen HBC+is expected because of the long- range Coulomb repulsion. However, for moderate dopinglevels, the following reactions lead to pair formation: 1Fei0+1/2HBC0→1/2/H20853Fei,H/H208540+ 0.82 eV and 1Fei0+0HBC+→1/H20853Fei,H/H20854++ 0.40 eV. The3/2/H20853FeiH/H208540state is 0.26 eV /H20849SIESTA /H20850to 0.30 eV /H20849VASP /H20850 higher than the1/2/H20853FeiH/H208540state. A second interstitial H will not bind to Fe i. Indeed, the trigonal /H20853FeiH2/H208540complex /H20849H- Fe-H, with Fe at the hexagonal interstitial site /H20850is less stable by 0.45 eV than isolated Fe iand an interstitial H 2 molecule.103 The interstitial /H20853FeiH/H20854pair /H20849Fig.1/H20850has trigonal symmetry. Fe is at the hexagonal interstitial site with Fe-H=1.51 Å. Figure 2shows all the vibrational modes localized on Fe and H. The stretch mode frequency /H208491921 cm−1with SIESTA , 1953 cm−1with VASP /H20850is higher than the 1767 cm−1observedSANATI, SZWACKI, AND ESTREICHER PHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 125204-4in free FeH molecules,104indicating that the interstitial pair is compressed by the host crystal. The /H20853FeiH/H20854pair has a deep donor and a deep acceptor level at Ev+0.36 eV /H20849SIESTA : 0.42 eV /H20850and Ec−0.26 eV /H20849SIESTA : 0.30 eV /H20850, respectively. Thus, hydrogen has no pas- sivating effect on Fe i. In the contrary, the formation of the /H20853FeiH/H20854pair results in the appearance of a new and deep ac- ceptor level. A donor level at Ev+0.31 eV, believed to be associated with the interstitial /H20853FeiH/H20854pair, has been reported by thermally stimulated capacitance.69This level anneals out in 30 min at 175 °C, suggesting a relatively weakly boundcomplex. This donor level and annealing behavior are con-sistent with the calculated donor level and binding energy.Note that two additional deep donor levels at E v+0.23 eV andEv+0.38 eV have been reported105following H implan- tation into Si samples contaminated with Fe. It is not knownif these levels are related to isolated Fe ior if native defects are involved.B. Iron pairs with shallow dopants 1. Fe-acceptor pairs We calculated the geometrical conﬁgurations, spin states, electronic structures, and binding energies resulting from the interactions of3/2Fei+with the ionized shallow acceptors0As−, with A=B, Al, Ga, In, and Tl.1/2/H20853Fe,A/H208540is higher in energy than3/2/H20853Fe,A/H208540by several tenths of an eV. In the case of B for example, the energy difference is 0.42 eV with VASP and 0.41 eV with SIESTA . We ﬁnd two competing geometries for each /H20853Fe,A/H20854pair: a trigonal conﬁguration with Fe at /H20849very near /H20850one of the four Tsites nearest to the acceptor, and an orthorhombic conﬁgu- ration with Fe at one of the twelve second-nearest Tsites. Figure 3shows the energy difference between the trigonal and the orthorhombic conﬁgurations in the three possiblecharge states, calculated a T=0 K. In this ﬁgure, /H9004Eis nega- tive if the trigonal conﬁguration is stable and the orthorhom-bic one metastable. The free energy differences 102between the trigonal and the orthorhombic conﬁgurations are plotted vs temperature inFig.4. In the present case, this involves only vibrational free energies since the difference in conﬁgurational entropy aris-ing from the difference in the number of ﬁrst- and second-nearest Tsites can safely be ignored. 102At room temperature, the predicted stable conﬁgurations match the experimentally-observed one in all cases. Figure 5shows the vibrational spectra of the 3/2/H20853Fe,B /H208540 pair in the C3vandC2vconﬁgurations. The isolated Bs−ac- ceptor has a threefold degenerate mode at 641 cm−1 /H20849measured106at 620 cm−1/H20850.I nt h e /H20853Fe,B /H20854pair, the B-related line splits into a doublet and a singlet in C3vsymmetry, and three singlets in C2vsymmetry. Previous calculations involving /H20853Fe,A/H20854pairs have as- sumed that the shallow acceptor resides at an unrelaxed sub- stitutional site and Fe at an undistorted nearest or second-nearest Tsite. Our geometry optimizations show that these assumptions are valid for Al, Ga, and In. All the ionizedacceptors have tetrahedral symmetry and their four Si NNs FIG. 1. /H20849Color online /H20850Fraction of the supercell showing the trigonal1/2/H20853FeiH/H208540pair. Fe /H20849brown or dark grey sphere /H20850is at the hexagonal interstitial site and H /H20849small white sphere /H20850is on the trigo- nal axis. FIG. 2. /H20849Color online /H20850Vibrational spectrum of the interstitial /H20853FeiH/H208540pair in Si. The short arrow shows the calculated /H9003phonon. The plots show L/H20853/H9251/H208542=/H20849e/H9251xs/H208502+/H20849e/H9251ys/H208502+/H20849e/H9251zs/H208502, where /H9251is Fe /H20849solid brown or grey lines /H20850or H /H20849dashed black lines /H20850. The Fe-H stretch mode is at 1921 cm−1and the /H20849degenerate /H20850wag modes are at 594 cm−1.−1 0 1 Char ge state−0.3−0.2−0.10.00.10.2∆E (eV) FeiBFeiAl FeiGaFeiIn FeiTl FIG. 3. /H20849Color online /H20850Calculated potential energy difference between the trigonal and orthorhombic conﬁgurations of the /H20853Fe,A/H20854 pairs, where Astands for B /H20849black circle /H20850,A l /H20849red square /H20850,G a /H20849green diamond /H20850,I n /H20849blue triangle left /H20850, and Tl /H20849purple triangle right /H20850in the three possible charge states of the pairs. /H9004Eis negative if the trigonal conﬁguration is more stable.INTERSTITIAL Fe IN Si AND ITS INTERACTIONS … PHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 125204-5are uniformly displaced inward /H20849negative displacement /H20850or outward /H20849positive displacement /H20850as follows: −0.27 Å for Bs−, +0.05 Å for Als−, +0.03 Å for Gas−, +0.03 Å for Ins−, and +0.16 Å for Tls−. In the trigonal conﬁguration, the Fe- acceptor separation is 2.35 Å for B /H20849that is, Fe is exactly at theTsite/H20850, 2.44 Å for Al, 2.44 Å for Ga, 2.44 Å for In, and 2.62 Å for Tl. The binding energies of the neutral /H20853Fe,A/H20854pairs were obtained by comparing the energies of each pair in its lowest-energy conﬁguration to its dissociation products in different supercells:3/2Fei++0As−→3/2/H20853Fe,A/H208540+Eb. This en- ergy balance has to be considered carefully since we are comparing the total energies of two supercells containing acharged defect /H20849left-hand side /H20850to the total energy of a neutralpair. Hence, one needs to include Madelung energy correc- tions twice, but only on one side of the equation. Since thesize of the correction diminishes as the size of the cellincreases, 107we repeated these calculations in cells contain- ing 64 to 216 host atoms. Figure 6shows the calculated binding energies of the /H20853Fe,B /H20854and /H20853Fe,Ga /H20854pairs vs inverse cell size /H20849the binding energies of the other pairs look similar but crowd the ﬁgure /H20850. The plots of binding energies vs. inverse cell size lead to the following calculated values for Ebextrapolated to inﬁnite cell size: 0.65 eV for /H20853Fe,B /H20854, 0.52 eV for /H20853Fe,Al /H20854, 0.42 eV for/H20853Fe,Ga /H20854, 0.44 eV for /H20853Fe,In /H20854, and 0.55 eV for /H20853Fe,Tl /H20854. The experimental numbers /H20849Table I in Ref. 1/H20850are 0.45–0.65 eV for /H20853Fe,B /H20854, 0.52–0.70 eV for /H20853Fe,Al /H20854, and 0.47 eV for /H20853Fe,Ga /H20854. The experimental binding energies of /H20853Fe,In /H20854and /H20853Fe,Tl /H20854are not reported. The origin of the binding energy is mostly Coulombic. The electrostatic energy gained by placing a +1 charge at2.3–2.6 Å of a −1 charge in Si is of the order of 0.5 eV.However, as in the case of Cu-acceptor pairs, 108the TM im- purity does overlap weakly with the acceptor. Small differ-ences in the Fe- Aand Fe-Si NNoverlap populations and small variations in the amount of lattice relaxation differentiate be-tween the various acceptors. The calculated donor and acceptor levels for the /H20853Fe,A/H20854 pairs in the two conﬁgurations are compared to the measured values in Table I. Since interstitial hydrogen passivates 109shallow acceptors such as B, we considered the possibility that the /H20853Fe,B /H20854pair may interact with H. Various a priori possible conﬁgurations for such a complex have been investigated. They include Hbridging a B-Si bond or an adjacent Si-Si bond, H bound toFe along or off the trigonal axis, or H at the Si antibondingsite of a B-Si bonds. The lowest-energy structure of the 3/2/H20853Fe,B,H /H20854+complex has H bound to Fe in a trigonal Si-B ¯Fe-H conﬁguration. However, this complex is less stable than the dissociated species3/2/H20853Fe,B /H208540and0HBC+by 0.07 eV. The minimum of the potential energy corresponds to Fe ifar away from the passivated /H20853B,H /H20854pair: 3/2/H20853Fe,B /H208540+0HBC+→3/2Fei++0/H20853B,H /H208540+ 0.25 eV.0 150 300 450 60 0 T(K)−0.24−0.18−0.12−0.060.000.060.120.18∆E(eV)FeiAl FeiBFeiInFeiTl FeiGa FIG. 4. /H20849Color online /H20850Calculated free energy differences be- tween the trigonal and orthorhombic conﬁgurations of the 3/2/H20853Fe,A/H208540pairs with A=B, Al, Ga, In, and Tl. /H9004Eis negative if the trigonal conﬁguration is more stable. 0 200 400 6000.00.20.40.60.81.0 0 200 400 600 ω(cm−1)0.00.20.40.60.81.0Lα2<111> <100>651 234531 567588 617225 189 232224 FIG. 5. /H20849Color online /H20850Vibrational spectra of the3/2/H20853Fe,B /H208540pair in the trigonal and orthorhombic conﬁgurations. The plots show L/H20853/H9251/H208542=/H20849e/H9251xs/H208502+/H20849e/H9251ys/H208502+/H20849e/H9251zs/H208502, where /H9251is Fe /H20849solid brown lines /H20850or B /H20849dashed green lines /H208500.00 0.02 0.04 0.06 0.08 0.1 0 1/L (Å−1)−0.8−0.6−0.4−0.20.0Eb(eV)FeiBs FeiGas FIG. 6. Calculated binding energies of the neutral /H20853Fe,B /H20854/H20849open circles /H20850and /H20853Fe,Ga /H20854/H20849open squares /H20850pairs vs inverse supercell size. The solid circle and square are the measured binding energies.SANATI, SZWACKI, AND ESTREICHER PHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 125204-6This suggest that H interacting with the /H20853Fe,B /H20854pair dis- places Fe ileaving the passivated /H20853B,H /H20854pair. This result has been inferred from DLTS measurements of hydrogenatedsamples. 110The disappearance of the /H20853Fe,B /H20854DLTS peak fol- lowing H implantation has also been reported by Kouketsu et al.105In this paper, the authors have interpreted the result in terms of H passivation of the pair. This interpretation is notsupported by the calculated energetics. 2. Fe-donor interactions Inn-type Si, interstitial iron is the neutral1Fei0species which experiences no Coulombic attraction to an ionized P+ or As+shallow donor. The binding energy can only arise from changes in lattice relaxation and distortion, or covalentbonding between the two impurities. Since neither isolatedFe nor the isolated shallow acceptor are characterized bysubstantial atomic rearrangement in their immediate vicinity,one would not anticipate much elastic energy to be gained bypairing. By analogy to the Fe-acceptor pairs, we have examined the trigonal and orthorhombic conﬁgurations for 1/H20853Fe,D/H20854+ pairs /H20849D=P or As /H20850, but ﬁnd that no pair forms. Indeed, the reactions1Fei0+0Ds+→1/H20853Fe,D/H20854++Ebgive slightly negative binding energies in the 64 host atoms cells. Since compa- rable Madelung energy corrections occur on both sides ofthis equation, the binding energies should not vary muchwith cell size. Further, in this case, the free energy contribu-tions will be dominated by a large conﬁguration entropy termsince there are orders of magnitude more T sites with Fe faraway from the donor than adjacent to it. 102This contribution to the free energy favors the dissociated species. IV. SUMMARY AND DISCUSSION First-principles theory is used to calculate the conﬁgura- tions, charge and spin states, electronic structures, bindingand activation energies, and approximate acceptor and donorlevels of isolated Fe i, the /H20853FeiH/H20854pair, all the /H20853Fe,A/H20854pairs /H20849A=B, Al, Ga, In, or Tl /H20850, and of Fe iwith Ds/H20849D=P or As /H20850. The interactions between the /H20853Fe,B /H20854pair and H are also con- sidered.The host crystal is represented by periodic supercells con- taining 64 to 216 host atoms. The Kohn-Sham equations aresolved within GGA with either plane-wave or pseudoatomic basis sets for the valence states. The geometries of the vari-ous defects in all the a priori possible spin and charge states are optimized using conjugate gradients with a 2 /H110032/H110032 k-point sampling. The location of the donor and acceptor levels are obtained with PAW potentials and a 3 /H110033/H110033 sam- pling using the marker method. The marker is the perfectcrystal. The dynamical matrices are obtained using the force-constant approach at the /H9003point. The vibrational spectra of the complexes involving light impurities are predicted. Thefree energy differences between the trigonal and orthorhom-bic conﬁgurations of the /H20853Fe,A/H20854pairs are calculated as a function of temperature. The calculations predict that interstitial iron resides at the Tsite as 3/2Fei+or1Fei0, with a donor level at Ev+0.28 eV. There is no acceptor level. The spin density is localized onthe TM impurity, with nearly 96% of the spin in the 3 d orbitals of 1Fei0, but only 78% in the case of3/2Fei+. The Fe atom overlaps weakly but covalently /H20849positive overlap popu- lation /H20850with its four nearest and six second-nearest Si neigh- bors. The activation energy for diffusion along the trigonalaxis is 0.76 and 0.69 eV in the 0 and /H11001charge state, respec- tively. Fe itraps interstitial H in the /H11001and 0 charge states and forms a trigonal pair, with Fe at the hexagonal interstitialsite. The binding energy relative to bond-centered hydrogenis 0.84 eV in the /H11001charge state and 0.40 eV in the 0 charge state. The /H20853Fe iH/H20854pair has a deep donor /H20849Ev+0.36 eV /H20850and a deep acceptor /H20849Ec−0.26 eV /H20850level in the gap. Thus, H does not passivate Fe ibut forms a pair which is even more elec- trically active. The stable conﬁguration of the /H20853Fe,B /H20854pair is trigonal, that of the /H20853Fe,In /H20854and /H20853Fe,Tl /H20854pairs is orthorhombic, and the two conﬁgurations are nearly degenerate in the case of the/H20853Fe,Al /H20854and /H20853Fe,Ga /H20854pairs. In each conﬁguration, the pairs have a donor and a deep acceptor level. The calculated bind-ing energies are close to the measured values. The binding ofthe pairs is mostly electrostatic in nature. If H is allowed tointeract with the /H20853Fe,B /H20854pair, the lowest-energy conﬁguration has a passivated /H20853B,H /H20854pair and isolated Fe i. We ﬁnd no pairing between Fe iand the shallow donors P and As. One of the key points of the present study is that ﬁrst- principles theory in supercells, including conjugate gradientgeometry optimizations, can be used to study the interactionsof Fe in Si. The calculated properties of Fe iand Fe-acceptor pairs are consistent with experiment: spin states, structures,energetics. The gap levels obtained with PAW potentials andthe perfect crystal as a marker are close to the DLTS values.The calculated vibrational spectra of the /H20853FeH /H20854and /H20853Fe,B /H20854 pairs predict several IR-or Raman-active LVMs. Our best predictions for the donor and acceptor levels are obtained with PAW potentials and a 3 /H110033/H110033k-point sam- pling in a 64 host-atoms supercell. The gap levels predictedwith SIESTA are generally quite close to the PAW ones. The use of a uniform marker, the perfect cell, removes the “semi-empirical” aspect of the method, when different marker de-fects are used to scale the gap levels for different structures.TABLE I. Calculated /H20849PAW /H20850gap levels for the trigonal and orthorhombic conﬁgurations of /H20853Fe,A/H20854pairs with A=B, Al, Ga, In, and Tl. The donor /H208490/+ /H20850levels are given relative to the valence band: Ev+x/H20849eV/H20850; the acceptor /H20849−/0 /H20850levels are given relative to the conduction band: Ec−x/H20849eV/H20850. When available, the experimental numbers are cited in parenthesis. All the references are in Sec. I. Atrigonal orthorhombic Ev+xE c−xEv+xE c−x B 0.11 /H208490.11 /H20850 0.29 /H208490.29 /H20850 0.11 /H208490.07 /H20850 0.45 /H208490.43 /H20850 Al 0.11 /H208490.20 /H20850 0.34 0.06 /H208490.13 /H20850 0.21 Ga 0.10 /H208490.24 /H20850 0.35 0.05 /H208490.14 /H20850 0.30 In 0.28 /H208490.27 /H20850 0.38 /H208490.32 /H20850 0.25 /H208490.15 /H20850 0.24 /H208490.39 /H20850 Tl 0.31 0.20 0.28 0.06INTERSTITIAL Fe IN Si AND ITS INTERACTIONS … PHYSICAL REVIEW B 76, 125204 /H208492007 /H20850 125204-7ACKNOWLEDGMENTS This work of S.K.E. was supported in part by the National Renewable Energy Laboratory and the R.A. Welch Founda-tion Grant No. D-1126. The work of M.S. was supported by a grant from the Advanced Research Program of the State ofTexas. Many thanks to Texas Tech’s High Performance Com- puter Center for generous amounts of computer time. *stefan.estreicher@ttu.edu 1A. A. Istratov, H. Hieslmair, and E. R. Weber, Appl. Phys. A 69, 13/H208491999 /H20850. 2A. A. Istratov, H. 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PhysRevB.97.220501.pdf	PHYSICAL REVIEW B 97, 220501(R) (2018) Rapid Communications Editors’ Suggestion Thermoelectric anisotropy in the iron-based superconductor Ba(Fe 1−xCo x)2As2 Marcin Matusiak,1,Krzysztof Rogacki,1and Thomas Wolf2 1Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland 2Institute of Solid State Physics (IFP), Karlsruhe Institute of Technology, D-76021 Karlsruhe ,Germany (Received 30 March 2018; revised manuscript received 10 May 2018; published 5 June 2018) We report on the in-plane anisotropy of the Seebeck and Nernst coefﬁcients as well as of the electrical resistivity determined for a series of strain-detwinned single crystals of Ba(Fe 1−xCox)2As2. Two underdoped samples (x=0.024, 0.045) exhibiting a transition from the tetragonal paramagnetic phase to the orthorhombic spin density wave (SDW) phase (at Ttr=100 and 60 K, respectively) show an onset of Nernst anisotropy at temperatures above 200 K, which is signiﬁcantly higher than Ttr. In the optimally doped sample ( x=0.06) the transport properties also appear to be in-plane anisotropic below T≈120 K, despite the fact that this particular composition does not show any evidence of long-range magnetic order. However, the anisotropy observed in the optimally dopedcrystal is rather small, and for the Seebeck and Nernst coefﬁcients the difference between values measured alongand across the uniaxial strain has an opposite sign to those observed for underdoped crystals with x=0.024 and 0.045. For these two samples, the insensitivity of the Nernst anisotropy to the SDW transition suggests that theorigin of nematicity might be something other than magnetic. DOI: 10.1103/PhysRevB.97.220501 I. INTRODUCTION The explanation for the unconventional behavior of high- temperature superconductors has turned out to be a challengingtask for the scientiﬁc community [ 1]. One of the early proposed scenarios, which led to the development of both high supercon-ducting critical temperatures as well as a strange metallic phaseabove the superconducting dome, was related to the presenceof a quantum critical point (QCP) [ 2]. It is worth noting that the occurrence of such QCP was reported for both copper-based[3] and iron-based [ 4] superconductors. In the latter group of materials, the quantum transition is suggested to be related toa somewhat enigmatic nematicity [ 4,5], which in condensed matter is deﬁned as the breaking of rotational symmetry of theelectronic system due to correlations rather than the anisotropyof the underlying crystal lattice [ 6]. Recent studies indicate that the relation between nematicity and superconductivity mightbe of a more general nature [ 7]. In this Rapid Communication, we investigate the in-plane anisotropy of the Seebeck and Nernst coefﬁcients, sincethe thermoelectric phenomena are known to be sensitive to thenematic characteristics of an electronic system [ 8–10]. The thermoelectric response measured for a Ba(Fe 1−xCox)2As2 series appears to be signiﬁcantly different between conﬁgu- rations where the thermal gradient is applied along or acrossthe uniaxial strain. This difference is observed in both tetrag-onal and orthorhombic phases. The optimally doped sampleexhibits the smallest anisotropy and its sign is opposite tothat found for underdoped samples. Our results indicate thatthe Dirac cone, presumably present in the electronic structureof the BaFe 2As2parent compound [ 11–13], is effectively eradicated by small cobalt doping. In addition, the determined m.matusiak@int.pan.wroc.plNernst anisotropy provides hints about an origin of the nematicﬂuctuations. II. EXPERIMENT Ba(Fe 1−xCox)2As2crystals were grown by self-ﬂux in glassy carbon or alumina crucibles. Particularly low coolingr a t e so f0 .20−0.61 ◦C/h were applied to minimize the amount of ﬂux inclusions and crystal defects [ 14]. The composition of the crystals was determined by energy dispersive x-rayspectroscopy. The series studied in this work consists of foursamples with cobalt contents of 0.0 at. % (Co0), 2.4 at. %(Co2), 4.5 at. % (Co4), and 6.0 at. % (Co6). For the experiment,square-shaped samples were cut out from as-grown platelikesingle crystals with the edges rotated by 45° in relation to thetetragonal axes. The sides of the square were about 2 −2.5m m and its thickness was 0 .1−0.3 mm. First, the Hall coefﬁcient was measured in unstrained crystals in a magnetic ﬁeld of B=12.5 T. Then a sample was mounted between two clamps made of phosphor bronzeand subjected to a uniaxial pressure applied along its sidesby a beryllium copper spring controlled with a stepper motor.For the resistivity ( ρ) measurements, electrical contacts were placed at the corners of the sample and the orientations of thevoltage and current leads were switched repetitively during theexperiment. This allowed the electrical resistivities ρ aandρb to be determined using the Montgomery method [ 15]. Uniaxial pressure was increased step by step and measurements of theresistivity were repeated until a saturation of the anisotropy,indicating maximal detwinning, was achieved. The maximalpressure determined in this way was used during subsequentthermoelectric experiments. The Seebeck ( S) and Nernst ( ν) coefﬁcients along and across the strain direction were measured in two separateruns with the magnetic ﬁeld (parallel to the caxis) varied 2469-9950/2018/97(22)/220501(5) 220501-1 ©2018 American Physical SocietyMARCIN MATUSIAK, KRZYSZTOF ROGACKI, AND THOMAS WOLF PHYSICAL REVIEW B 97, 220501(R) (2018) FIG. 1. The temperature dependences of the Hall coefﬁcient for the Ba(Fe 1−xCox)2As2series (data for Co0 taken from Ref. [ 18]). Asterisks denote approximate temperatures of the magnetic/structuraltransitions. The top and bottom panels use different vertical scales. from−12.5 to +12.5 T. The temperature difference along the sample was determined using two Cernox thermometers and aconstantan-chromel thermocouple, which was precalibrated inmagnetic ﬁeld, attached to the sample through few-mm-longand 100-µm-thick silver wires. Signal leads were made upof long pairs of 25-µm phosphor bronze wires. More detailsabout the experimental setup are given in the SupplementalMaterial [ 16]. III. RESULTS AND DISCUSSION An expected consequence of the transition from the param- agnetic to the spin density wave (SDW) phase is a changein the electronic transport properties due to Fermi surfacereconstruction [ 17]. One of the quantities likely to be affected by this reconstruction is the Hall coefﬁcient ( R H), which in the Ba(Fe 1−xCox)2As2series undergoes a steplike anomaly presented in Fig. 1. The onset of this anomaly was used to determine the temperature of the transition at Ttr≈140, 100, and 60 K in Co0 [ 18], Co2, and Co4, respectively. TheRH(T) dependence in Co6 exhibits a slight downturn below T≈40 K that might be a sign of the oncoming SDW transition, but instead the system goes directly into the su-perconducting state. The resistive superconducting transitionin Co6 is shown in Fig. 2, which presents the temperature dependences of the electrical resistivity for all samples from theBa(Fe 1−xCox)2As2series detwinned by uniaxial pressure. Two of the samples are superconducting (Co4 with Tc=20.3 K and Co6 with Tc=24.5 K), and three of them show the anomaly connected to the structural/magnetic transition (Co0, Co2,and Co4). As previously reported, the resistivities measuredalong the a(long) and b(short) orthorhombic axes ( ρ aand ρb, respectively) begin to diverge at temperatures signiﬁcantly higher than the temperature of the actual structural transition[19]. Remarkably, in the cobalt-doped samples, ρ a, which in the orthorhombic phase is smaller than ρb, seems to be almost unaffected by the transition. The anomaly at Ttris barely noticeable in ρa(T) for Co2 and Co4, while for Co6, ρa(T)FIG. 2. The temperature dependences of the resistivity for the Ba(Fe 1−xCox)2As2series across ( a) and along ( b) the applied strain. Co2, Co4, and Co6 plots are shifted vertically for the sake of clarity.Asterisks denote approximate temperatures of the magnetic/structural transitions as taken from Fig. 1. stays linear down to the superconducting transition, in contrast toρb(T), which exhibits an upturn below T≈50 K. A possible explanation for this behavior involves the strongly anisotropicscattering of charge carriers [ 20] inﬂuencing only the b-axis electronic transport. This kind of a b-axis-only response is not limited to electrical resistivity but was also observed in the thermoelectricpower ( S), which in EuFe 2(As 1−xPx)2exhibits a sizable anomaly at Ttronly for measurements along the baxis [ 10]. The temperature dependences of the thermoelectric power forBa(Fe 1−xCox)2As2presented in Fig. 3follow the same rule, i.e., anomalies at Ttrin Co2 and Co4 are more pronounced in Sb(T) than in Sa(T), however they are still much smaller than FIG. 3. The temperature dependences of the thermoelectric power for the Ba(Fe 1−xCox)2As2series across (solid points) and along (open points) the applied strain. Co2, Co4, and Co6 plots are shifted vertically for the sake of clarity. Asterisks denote approximatetemperatures of the magnetic/structural transitions. 220501-2THERMOELECTRIC ANISOTROPY IN THE IRON-BASED … PHYSICAL REVIEW B 97, 220501(R) (2018) FIG. 4. The temperature dependences of the Seebeck anisotropy /Delta1S/T =(Sb−Sa)Tfor the Ba(Fe 1−xCox)2As2series. Asterisks de- note approximate temperatures of the magnetic/structural transitions. Open points show the difference between the Sbvalues measured for two different Ba(Fe 0.94Co0.06)2As2samples ( SCo6#1 b−SCo6#2 b )/T providing a test of the reproducibility. those observed in EuFe 2(As 1−xPx)2[10]. Another similarity between our data and those reported by Jiang et al. [10] is that SaandSb, which are normalized in the higher-temperature limit by multiplying by a small (0 .9−1.1) correction factor, begin to diverge at a temperature much higher than Ttr. Figure 4 presents the /Delta1S=Sb−Sadifference, which is divided by T to account for the fact that the thermoelectric power, beinga measure of entropy per charge carrier [ 21], decreases with decreasing temperature and Shas to drop to zero for T→0K . /Delta1Sin Ba(Fe 1−xCox)2As2shows an anomaly at Ttrand in the entire temperature range it is negative for Co2 and Co4,while it is positive in Co6. In Ref. [ 10] the inversion of /Delta1S at the structural transition ( T s) was interpreted as a result of the change in the mechanism causing nematicity—fromscattering for T> T sto orbital polarization for T< T s,b u t we do not observe an equivalent phenomenon at Ttrin our series. Hence, in our opinion, the change of the /Delta1Ssign caused by cobalt doping is a consequence of the multiband structureof the 122 iron pnictides [ 22]. If more than one band takes part in the electronic transport, then the effective coefﬁcientsare the sums of contributions from different bands. For thethermoelectric power this has a form S= /summationtext iSiσi/summationtext iσi, where Siand σiare, respectively, the thermoelectric power and electrical conductivity of a given iband. The cobalt doping supplies the BaFe 2As2system with additional electrons and can alter bothSiandσi. It is worth noting that even if the character of the electrical resistivity anisotropy does not change ( ρ along the strain remains higher than ρacross the strain), the anisotropy of the thermoelectric power still can reverse. Thiscan happen in an electronic system consisting of isotropicand anisotropic bands, when the ratio between respective S coefﬁcients changes disproportionately due to a shift of theFermi level. An analogous scenario was proposed to explainthe anisotropy of the Nernst coefﬁcient ( ν) in the SDW phase of BaFe 2As2and CaFe 2As2[23].FIG. 5. The temperature dependences of the Nernst coefﬁcient in the Ba(Fe 1−xCox)2As2series across (solid points) and along (open points) the applied strain. Co2, Co4, and Co6 plots are shifted vertically for the sake of clarity. Asterisks denote approximate tem-peratures of the magnetic/structural transitions. The top and bottom panels use different vertical scales. As shown in Fig. 5, the magnetic/structural transition in the parent compound Co0 causes the Nernst coefﬁcient torise dramatically in a way similar to the one reported forCaFe 2As2[24] and EuAs 2Fe2[25]. In contrast, a small cobalt substitution is sufﬁcient to invert this anomaly, namely, ν in Co2 (2.4 at. % cobalt content) decreases below Ttrand changes sign to negative below T≈60 K. Such a kind of response of the Nernst effect to cobalt doping was observedin Eu(Fe 1−xCox)2As2[25], where the sudden change of the character of the anomaly was attributed to the suppression ofthe Dirac band contribution, perhaps due to a shift of the Fermilevel or an enhanced scattering of Dirac fermions. Interestingly,unlike the temperature dependences of the resistivity andthermoelectric power, ν(T) measured along the aandbaxes (ν aandνb, respectively) below Ttrlook quite similar for each of the samples. This is likely related to the fact that the Nernstcoefﬁcient measured along the adirection is subject to both a- andb-direction scattering processes, ν xB=−αyx σyy−SxRHσxx (1) (αyxis the off-diagonal element of the Peltier tensor, and σxx andσyyare diagonal elements of the electrical conductivity tensor). The anisotropy of the Nernst coefﬁcient, deﬁned as the difference between νbandνa(/Delta1ν=νb−νa) divided by T,i s plotted versus temperature in Fig. 6. The ﬁrst appearance of the Nernst anisotropy is noticeable at temperatures Tf≈215, 225, 205, and 120 K for Co0, Co2, Co4, and Co6, respectively,which are signiﬁcantly higher than T tr. These values are in good agreement with the onset temperatures of nematic ﬂuctuationsestimated from Raman spectroscopy by Kretzschmar et al. 220501-3MARCIN MATUSIAK, KRZYSZTOF ROGACKI, AND THOMAS WOLF PHYSICAL REVIEW B 97, 220501(R) (2018) FIG. 6. The temperature dependences of the Nernst anisotropy /Delta1ν/T =(νb−νa)/Tfor the Ba(Fe 1−xCox)2As2series. The solid lines represent /Delta1ν(T)/T∼e−cTﬁts. Asterisks denote approxi- mate temperatures of the magnetic/structural transitions. The top and bottom panels use different vertical scales. Open points show the difference between the νbvalues measured for two different Ba(Fe 0.94Co0.06)2As2samples ( νCo6#1 b−νCo6#2 b )/Tproviding a test of the reproducibility. [26]. Another similarity between our results and those reported in Ref. [ 26] is that a sizable nematic behavior is observed for Ba(Fe 1−xCox)2As2in a rather narrow range of Co doping. Namely, Kretzschmar et al. detected a nematic response up tox=0.061 and they called the effect unambiguous only up to x=0.051. Correspondingly, we see that our sample Co6 (x=0.06) is one that behaves differently from Co0, Co2, and Co4, i.e., the anisotropy of the Nernst coefﬁcientin Co6 is small, emerges at a lower temperature, and, mostimportantly, the sign of /Delta1νfor this sample, analogously to /Delta1S, is inverted. Here again, we attribute this inversion to an interplay between different conductivity bands. /Delta1ν/Tdecays exponentially with temperature [ /Delta1ν(T)/T∼e −cT] with the parameter c=3.3×10−2for Co0 and c=1.8×10−2for both Co2 and Co4. Such an exponential temperature depen-dence of the Nernst anisotropy might be surprising, since thenematic susceptibility was reported to obey a Curie-Weisspower law in Ba(Fe 1−xCox)2As2as well as in other optimally doped iron-based superconductors [ 4]. Possibly the most unexpected outcome of the present studies is that we do not observe any anomaly in /Delta1ν/Tat the magnetic/structural transition either in Co2 or in Co4.This means that the presumed scattering processes that causea rise in the anisotropy at T trinρand S, apparently by affecting transport along the baxis, cancel out for the Nernst anisotropy since it consists of contributions from both a- and b-direction transports. Perhaps it is related to the form of /Delta1ν, which can be expressed as a subtraction of two Sondheimercancellations [ 27], one for the adirection, and another for thebdirection, /Delta1ν=[(αxy σxx−SxRHσxx)−(αyx σyy−SyRHσyy)]/B. It is noteworthy that the slope cof the /Delta1ν(T)/T∼e−cT dependences in Co2 and Co4 stays the same in both the paramagnetic and in the SDW phases, which suggests thatthe nematic ﬂuctuations responsible for the high-temperatureanisotropy of the Nernst coefﬁcient are not affected by theformation of the magnetic order. This leads to the question ofwhether the Nernst anisotropy detects the same aspect of ne-maticity as observed in the one-directional electrical resistivityand the thermoelectric power, and, more importantly, whetherthe nematic ﬂuctuations can be of a magnetic origin. Up tonow, several mechanisms were proposed as possibly relevantto nematicity, including orbital [ 28], spin [ 29], or charge [ 30] order. Here, we would like to point at reports indicating that thenematic and magnetic orders are distinct. For instance, studiesof differential elastoresistance point at a nematic quantumcritical point located inside the superconducting dome [ 4]. Results of polarization-resolved Raman spectroscopy studieslink this quantum critical point to a charge quadrupole orderthat was shown to compete with the collinear antiferromagneticorder [ 31]. Another example is an emergence of two types of nematicity that was suggested to take place in the copper-basedsuperconductor YBa 2Cu3Oy[32] In the future, it will be interesting to perform similar anisotropy studies on cobalt-doped CaFe 2As2compounds, in which the magnetic/structural transition seems to be of ﬁrstorder [ 33] and where magnetic ﬂuctuations above T trare absent or at least much weaker [ 23]. IV . SUMMARY We studied the in-plane anisotropy of magnetothermo- electric phenomena in a series of Ba(Fe 1−xCox)2As2single crystals. Uniaxial pressure was applied to the samples inorder to detwin them in the orthorhombic phase and to allowthe detection of macroscopic consequences of the emergingnematicity in the tetragonal phase. For all compositions studiedthe rotational symmetry of the electronic system is brokenmuch above the temperature of the structural/magnetic transi-tion. Moreover, the symmetry is broken even if such a transitionis absent, as in the optimally doped sample Co6. We observestrong thermoelectric anisotropy in the underdoped samples,whereas it is rather small and reversed in Co6. This change inthe sign of anisotropy is attributed to the multiband structure ofBa(Fe 1−xCox)2As2, where the total transport coefﬁcients are composed of contributions from different conduction bandstuned by the cobalt doping. An important, but to some extentunexpected, result is the observation of an exponential temper-ature dependence of the Nernst anisotropy in the underdopedsamples. Furthermore, /Delta1νturned out to be unaffected by the spin density wave transition, which suggests that nematicity inBa(Fe 1−xCox)2As2is distinct from the magnetic order. ACKNOWLEDGMENT This work was supported ﬁnancially by the National Science Centre (Poland) under the Research Grant No.2014/15/B/ST3/00357. 220501-4THERMOELECTRIC ANISOTROPY IN THE IRON-BASED … PHYSICAL REVIEW B 97, 220501(R) (2018) [1] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature supercon-ductivity in copper oxides, Nature (London) 518,179(2015 ). [2] D. van der Marel, H. J. A. Molegraaf, J. Zaanen, Z. Nussinov, F. Carbone, A. Damascelli, H. Eisaki, M. Greven, P. H. Kes, andM. Li, Quantum critical behaviour in a high-T csuperconductor, Nature (London) 425,271(2003 ). [3] S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D. A. Bonn, W. N. Hardy, R.Liang, N. Doiron-Leyraud, L. Taillefer, and C. 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PhysRevB.90.184101.pdf	PHYSICAL REVIEW B 90, 184101 (2014) Coupled experimental and DFT +Uinvestigation of positron lifetimes in UO 2 Julia Wiktor,1,Marie-France Barthe,2G´erald Jomard,1Marc Torrent,3Michel Freyss,1and Marjorie Bertolus1 1CEA, DEN, DEC, Centre de Cadarache, 13108, Saint-Paul-lez-Durance, France 2CNRS/CEMHTI, CNRS UPR 3079/CEMHTI, 45071 Orl ´eans, France 3CEA, DAM, DIF , F-91297, Arpajon, France (Received 27 May 2014; revised manuscript received 20 October 2014; published 4 November 2014) We performed positron annihilation spectroscopy measurements on uranium dioxide irradiated with 45 MeV αparticles. The positron lifetime was measured as a function of the temperature in the 15–300 K range. The experimental results were combined with electronic structure calculations of positron lifetimes of vacanciesand vacancy clusters in UO 2. Neutral and charged defects consisting of from one to six vacancies were studied computationally using the DFT +Umethod to take into account strong correlations between the 5 felectrons of uranium. The two-component density functional theory with two different fully self-consistent schemes was usedto calculate the positron lifetimes. All defects were relaxed taking into account the forces due to the creation ofdefects and the positron localized in the vacancy. The interpretation of the experimental observations in the lightof the DFT +Uresults and the positron trapping model indicates that neutral V U+2VOtrivacancies (bound Schottky defects) are the predominant defects detected in the 45 MeV αirradiated UO 2samples. Our results show that the coupling of a precise experimental work and calculations using carefully chosen assumptions is aneffective method to bring further insight into the subject of irradiation induced defects in UO 2. DOI: 10.1103/PhysRevB.90.184101 PACS number(s): 78 .70.Bj,61.72.jd,71.15.Mb I. INTRODUCTION Uranium dioxide is currently the most widely used fuel material in ﬁssion reactors. During reactor operation, theﬁssion of uranium atoms causes the formation of defects,such as vacancies and vacancy clusters, as well as the creationof ﬁssion products, which induce a signiﬁcant evolution ofthe fuel physical properties. The vacancies can trap insolubleﬁssion products, in particular ﬁssion gases and it is of greatimportance to understand their role in the early stages of theformation of gas bubbles in UO 2. Separate effect experiments and atomic scale modeling can bring invaluable insight intothe elementary mechanisms involved. The necessity of an im-proved understanding of the fuel behavior under irradiation hasalready led to extensive experimental [ 1–12] and theoretical studies on point defects [ 13–23]i nU O 2. One of the experimental techniques permitting the investi- gation of vacancy defects created by irradiation is positronannihilation spectroscopy (PAS). PAS is a nondestructiveexperimental method that allows studying open volume defectsin solids. Vacancies can trap positrons, what is seen through,e.g., changes in the lifetime of positrons in the material. Toidentify the types of defects present in the examined materials,however, comparison with calculated positron lifetimes or withresults of other characterization techniques is required. PASallows one to detect neutral and negative defects only, sincepositive ones have a positron trapping coefﬁcient too smallto be observed. Moreover, the neutral and negative defectscan be distinguished, as the trapping coefﬁcient of negativevacancies decreases with temperature while it is constant forneutral defects [ 24]. Experimentally, several PAS measurements of the positron momentum distribution (Doppler broadening) [ 6–11] and positron lifetimes [ 8,12] have been performed in UO 2.N o julia.wiktor@cea.frcalculations, however, are available to help in the interpretation of the results of these experiments and in the identiﬁcation ofthe defects present in the samples. We present here the resultsof additional positron lifetime measurements as a function oftemperature, as well as the calculations of positron lifetimes inthe two-component density functional theory (TCDFT) [ 25]. Since standard DFT fails to describe the strong correlationsbetween the 5 felectrons of uranium in UO 2, we applied the DFT+Umethod [ 26] to compute the electron structure of the system. We use self-consistent positron lifetime calculationschemes and we take into account the defects relaxations,since it has been shown that this effect can strongly affect thecalculation results in semiconductors and insulators [ 27–34]. This article is organized as follows. In Sec. II, we describe experimental details concerning the samples characteristics,the irradiation conditions, as well as previous analyzes andpresent PAS measurements. We also present brieﬂy thepositron trapping model used for experimental data analysis.In Sec. III, we describe the methods used in the calculations of positron lifetimes and we list the computational details. InSec. IV, we show the results of the PAS measurements in the αirradiated samples. In Sec. V, we present the results of the positron lifetime calculations in UO 2. Finally, in Sec. VI,w e combine the results of experiments and calculations to discussthe nature of the defects created in the samples. Additionally,we compare our calculations results with the experimentalpositron lifetimes observed in UO 2reported in the literature. II. EXPERIMENTAL DETAILS A. Samples The sintered disks of uranium dioxide (0.2 at.%235U) were polished and were then annealed for 24 hours at 1700◦C under Ar/H 2atmosphere containing an appropriate amount of water vapor to preserve their stoichiometric composition.The mean grain size was about 18 μm and the mean O/U ratio 1098-0121/2014/90(18)/184101(11) 184101-1 ©2014 American Physical SocietyJULIA WIKTOR et al. PHYSICAL REVIEW B 90, 184101 (2014) was 2.0051±0.0001 as determined by polarography. The density of the material was 10 .76±0.0 3gc m−3. The disks were 300 −μm thick and 8.2 mm in diameter. B. Irradiation The UO 2disks were irradiated with 45 MeV αparticles at a ﬂuence of 2 ×1016cm−2at 300 K. The damaged layer thickness that can be induced by these particles in UO 2is 347μm, which is more than the thickness of the sample. C. Previous PAS analyzes The same samples were already studied by positron annihilation spectroscopy at room temperature in a previouswork [ 8]. A positron lifetime of 169 ±1 ps was measured in polished and annealed samples. This value can be attributedto the positron lattice lifetime of UO 2, which corresponds to the case when all positrons annihilate in the free state withoutbeing trapped. The samples were then irradiated with electronsandαparticles with different energies at various ﬂuences. No defect was detected after irradiation with 1 MeV electrons. Considering the accepted O and U thresholds dis-placement energy of respectively 20 and 40 eV [ 35], this type of irradiation should create defects in the oxygen sublatticeonly. The oxygen vacancies are predicted to have a positivecharge state [ 14,21,22]i nU O 2close to the stoichiometry, hence cannot be detected by the positrons. This is consistentwith the fact that no defect was seen in PAS. After irradiation of the UO 2disks with 2.5-MeV electrons and 45-MeV αparticles, positron long-lifetime components between 301 ±7 and 307 ±3 ps were detected. Based on the U threshold displacement energy, these components wereattributed to a defect involving the uranium displacement,such as the U monovacancy, the U-O divacancy or thebound Schottky defect. In this earlier study, however, themeasurements on irradiated samples were performed at roomtemperature only, hence no further information about thecharge states of the observed defects could be accessed.Additionally, it is worth noting that the error bars presentedin the previous study (from ±3t o±7 ps) correspond to the statistical accuracy of the ﬁt performed to obtain thelifetime spectra decomposition and not to the actual accuracyof the measurement. Therefore, considering a range of positronlifetimes of 300 ±10−310±5 ps for the previous study would be more reasonable. D. Present PAS measurements In the present study, positron lifetime measurements were performed as a function of temperature in the 15–300 K range.A conventional fast-fast coincidence spectrometer with a timeresolution of 230 ps was used. A 22Na positron source was sandwiched between two identical samples. Approximatelytwo million events were collected for each spectrum. Thelifetime spectra can be expressed as L(t)=R(t)∗/summationdisplay iIiexp/parenleftbigg−t τi/parenrightbigg +BG, (1) where Ris the Gaussian resolution function of the spectrome- ter and BGthe background signal. The spectra were analyzedusing a modiﬁed version of the POSFIT [36] software. After source and background substraction, the data were ﬁtted to asum of exponential lifetime components τ iweighted by the intensities Iiconvoluted with a Gaussian resolution function R,g i v i n g R(t)∗/summationdisplay iIiexp/parenleftbigg−t τi/parenrightbigg . (2) Theoretical average positron lifetime can be calculated as τav.=/summationtext iIiτi. E. Positron trapping model The evolution of the positron annihilation characteristics as a function of the measurement temperature can be analyzedusing a positron trapping model, which has already beendescribed and applied in various studies [ 37–44]. In this model, the description of the positron trapping and annihilation atdifferent states (in the free state in the lattice and in Ndifferent defects) is obtained through solving a set of rate equations [ 31]: dn L dt=−⎛ ⎝λL+/summationdisplay jκj⎞ ⎠nL+/summationdisplay jδjnDj, (3) dnDj dt=κjnL−(λDj+δj)nDj(j=1,..., N ), (4) where nLis the probability of a positron being in the free state,nDjis the probability of being trapped in a given defect. λL,λDj,κjandδjare annihilation, trapping and detrapping rates, respectively. The trapping rate is related to the defectconcentration c jthrough the relation κj=μDjcDj, where μDjis the speciﬁc trapping coefﬁcient. For neutral defects, μDjis temperature independent, while for negative vacancies it varies as T−1/2. A positron can be trapped not only in open volume defects, but also by hydrogenlike Rydberg statesaround negative nonvacancy defects, caused by the long-rangeCoulomb potential. The positron trapping rate at the Rydbergstates also varies as T −1/2[41]. III. CALCULATION METHODS A. Positron lifetime calculations To calculate the positron lifetime, it is necessary to know the distributions of both the positron density n+(r) and the electron density n(r) in the system, as they determine the probability of annihilation. The positron lifetime τdepends on this probability and can be calculated as an inverse of thetrapping rate λ: 1 τ=λ=πcr2 0/integraldisplay R3d3rn+(r)n(r)g(n+,n), (5) where cis the light velocity and r0is the classical radius of an electron. The g(n+,n) term is an enhancement factor taking into account the increase in the electron density at the positronsite caused by the screening of this particle by electrons. Positron and electron densities, needed for the positron lifetime calculation, can be computed using a self-consistentscheme, in the two component density functional theory(TCDFT) [ 25]. Several calculation methods using different 184101-2COUPLED EXPERIMENTAL AND DFT +U. . . PHYSICAL REVIEW B 90, 184101 (2014) parametrization and approximations exist [ 45–47]. In this study we use two methods, one proposed by Giglien, Galli,Gygi, and Car (GGGC) [ 45] and one based on Boro ´nski and Nieminen [ 46] calculation method, with a parametrization by Puska, Seitsonen, and Nieminen (PSN) [ 47]. Both these methods are described in Ref. [ 47] and we will recall here only their main features. In both GGGC and PSN schemes, several self-consistent computation steps are performed. First, the electron densityis calculated, then the density of a positron interacting withthe electrons. Later, the electron density affected by thepositron is recalculated and then these steps are repeated untilconvergence is reached. The two self-consistent schemes use,however, different approximations and parametrizations. TheGGGC scheme uses the LDA electron-positron correlationfunctional parametrized for the positron density tendingto zero (zero-positron-density limit). This functional wasparametrized by Boro ´nski and Nieminen [ 46] using the data provided by Arponen and Pajanne [ 48]. For the enhancement factor g, the form depending only on the electron density, modeled by Boro ´nski and Nieminen, is taken [ 46]. In the PSN scheme, a full LDA electron-positron correlation functionalprovided by Puska, Seitsonen, and Nieminen [ 47]i su s e d . The enhancement factor in this scheme depends on both theelectron and the positron densities. Since the PSN schemeuses a full electron-positron correlation functional it is more suitable for describing localized positrons, e.g., trapped in defects. The GGGC scheme, on the other hand, tends tooverestimate the positron localization [ 45,47]. The enhancement factor gin Eq. ( 5) is used in order to take into account the increase in the electron density at a positronsite due to the screening of the positron by electrons. However,most of the positron calculation schemes were developed tomodel metallic materials and they assume a perfect screeningof the positron by the electrons. In semiconductors andinsulators, corrections have to be used to take into accountthe existence of the gap in the electronic states. Two typesof corrections are available. A semiconductor correction (SC)can be implemented in the enhancement factor as proposed byPuska [ 49]. Alternatively, a gradient correction (GC) proposed by Barbiellini et al. [50] can be implemented in both the en- hancement factor and the electron-positron correlation energy. It is worth noting that the semiconductor correction has already been implemented in both GGGC and PSNschemes [ 32,49], while the gradient correction existed only in the GGGC method. For the purposes of this study, we decidedto implement the gradient correction in the PSN method.Firstly, we implemented the correction, taking an adjustableparameter α=0.22, as proposed by Barbiellini et al. [50], in both the enhancement factor gand the correlation energy. However, it is worth noting that this correction was adaptedto the LDA zero-positron-density limit. Our implementationof the gradient correction in the correlation energy in the PSNmethod led to some instabilities in the convergence cycle.Since the correction proposed by Barbiellini et al. was intended for a simpler formulation of the correlation functional thanthe one used in the PSN scheme, we suppose that the latterrequires a more complex approach. Both Barbiellini et al. in Ref. [ 50] and Kuriplach et al. in their recent work [ 51] showed that the gradient correction has a signiﬁcant inﬂuenceon the enhancement factor while the positron density remains almost unaffected. We decided, therefore, to apply the gradientcorrection on the gfunction only in the PSN scheme, by taking g PSN GGA=1+/parenleftbig gPSN LDA−1/parenrightbig e−α/epsilon1, (6) where /epsilon1is a parameter, /epsilon1=\| ∇ lnn\|2/q2 TF, with 1 /q2 TFbeing the local Thomas-Fermi screening length. We test in Sec. III C the inﬂuence of the choice of the self-consistent scheme and the correction applied tothe enhancement factor gon the calculated lattice positron lifetimes (lifetimes obtained in perfect UO 2cells). It has been shown for several materials [ 27–34] that atomic relaxation effects inﬂuence strongly the positron lifetimes.Therefore, to consider a full relaxation of defects, the forceson atoms due to the positron, electrons and other nucleiwere calculated using the Hellman-Feynman theorem afterconvergence on both the electronic and positronic densitieswas reached. By doing this, we obtained exact forces includingall the contributions from the positron and the electrons anddid not need to add extra Pulay forces as was done in severalTCDFT implementations. B. Computational details Calculations presented in this paper were performed using the ABINIT [52,53] code, which uses pseudopotentials and a plane-wave basis set or projector augmented-wave [ 54]( P A W ) method for the wave function representation. We used the PAWmethod available in the code. The TCDFT was implementedpreviously as a double loop on the electronic and positronicdensities: during each subloop, one of the two densitieswas kept constant while the other was being converged.The TCDFT method has been implemented in ABINIT in an uniﬁed formalism for the positron and the electrons: the wavefunctions of the electrons and the positron in the system areexpressed on the same mixed basis (plane waves and atomicorbitals). The atomic orbital basis must be built with care inorder to be sufﬁciently complete to represent the positronic andelectronic wave functions with the same accuracy. This issuecan be solved by generating atomic data sets with additionalbasis functions and including semicore states, as describedin Ref. [ 32]. The atomic data sets used in the present study were generated by the ATOMPAW code [ 55]. The generalized gradient approximation (GGA) as parametrized by Perdew,Burke, and Ernzerhof (PBE) [ 56] was used to describe the exchange-correlation interactions. Moreover, a Hubbard-liketerm ( U) was added in order to take into account the strong correlations between the 5 felectrons of the uranium atoms. The Liechtenstein scheme [ 57]o ft h eD F T +Umethod was used. The values of the UandJparameters were set to 4.5 and 0.51 eV respectively, in agreement with earlier DFT +Ucalcu- lations [ 26] and the values extracted from experiments [ 58]. To avoid the convergence to one of the metastable states yieldedby the DFT +Umethod and ensure that the ground state was reached, we used the occupation matrix control scheme [ 17]. For calculations of the defects lifetimes, we used su- percells containing 96 atomic sites (2 ×2×2 repetitions of the ﬂuorite unit cell) and 2 ×2×2 Monkhorst-Pack k-point meshes [ 59]. We performed calculations for oxygen and uranium monovacancies (V Oand V U), U-O and U-U 184101-3JULIA WIKTOR et al. PHYSICAL REVIEW B 90, 184101 (2014) divacancies (V U+VOand 2V U), a trivacancy containing one uranium vacancy and two oxygen vacancies (V U+2VO), a 2VU+2VOtetravacancy, and a 2V U+4V0hexavacancy. For all clusters, the vacancies were considered as ﬁrst neighbors.In the case of the V U+2VOtrivacancy, also known as the bound Schottky defect, we considered the three possibleconﬁgurations, with the two oxygen vacancies aligned along(100), (110), and (111) directions. We calculated the positronlifetimes for neutral and charged defects. In the case of theneutral defects, we removed certain atoms to create vacancies.To obtain charged defects, we further added or removed agiven number of electrons in the supercell (e.g., we removedone uranium atom and added four electrons to obtain theV 4− Uvacancy). We veriﬁed that the additional charges were localized at the defect site. Defect charges for which the formalcharge of uranium and oxygen ions in UO 2(O2−and U4+) is conserved were considered since it was shown that thesecharges are the most stable ones for Fermi levels close to themiddle of the band gap [ 14,21,22]. We did not take into account the spin-orbit coupling because of the high computational cost.Extensive investigations of the SOC inﬂuence on the propertiesof actinide compounds [ 60,61] suggest that it does not affect the properties of defects [ 62]. We use a Gaussian smearing of 0.1 eV and a plane- wave cutoff energy of 500 eV . We perform positron lifetimecalculations at the equilibrium volume of UO 2found using the GGA +Umethod, which was chosen based on the results of the tests described in Sec. III C. We ﬁx the lattice parameters to a=b=5.57˚A andc=5.49˚A for perfect UO 2. The distortion in the zaxis is due to the approximate 1 k antiferromagnetic order. Atomic relaxation was taken intoaccount and calculations were stopped when the forces actingon atoms were smaller than 0.03 eV /˚A, which was found to be sufﬁcient to have the positron lifetime converged with aprecision of less than 1 ps. C. Tests of the parameters used in positron lifetime calculations We studied the effect of the parameters used in the calcula- tions on the positron lifetime of perfect UO 2. We compared the results obtained in GGA and GGA +Umethods, while (1) using different cell volumes, (2) taking two types of positronlifetime calculation schemes, (3) applying different correctionsto the enhancement factor g, and (4) considering or neglecting the spin polarization. Results of these tests are presented in Table I.W eu s e dt w o different calculation schemes, PSN and GGGC (see Sec. III A ). GGGC +SC and PSN +SC in Table Irefer to schemesin which the semiconductor correction, based on the one proposed by Puska [ 49] and described in Ref. [ 32], was imple- mented. We took the experimental high-frequency dielectricconstant of UO 2of 5.1 [ 63] in the semiconductor correction. GGGC +GC and PSN +GC refer to schemes in which the gradient correction, proposed by Barbiellini et al. [50], was im- plemented. It is worth noting that in the GGGC +GC scheme this correction is applied on both the enhancement factorand the electron-positron correlation functional, while in thePSN+GC method it is only implemented in the enhancement factor g, as described in Sec. III A . In Table I, we present the lattice positron lifetimes calculated using different volumes.V exp.refers to the experimental volume, corresponding to a lattice parameter of 5.47 ˚A[64].Veq.refers to the equilibrium volume found in calculations using given parameters. Wecan notice that the lifetimes obtained using the semiconductorcorrection, both using the PSN and GGGC schemes, are sys-tematically shorter than the ones calculated with the gradientcorrection. Moreover, the GGGC and PSN schemes yieldsimilar lattice positron lifetimes, both when the semiconductorand the gradient correction is used. Additionally, the resultsobtained using the gradient correction are in better agreementwith the experimental lattice lifetimes obtained for UO 2of 169±1ps [ 8]. It suggests that the schemes using the gradient correction are more suitable for the description of positronlifetimes in uranium dioxide, hence we choose to use them in the present study. We decide to use both GGGC +GC and PSN+GC schemes in our further study of defects positron lifetimes. First, we do it in order to avoid the misinterpretationof experimental results, that could result from possible errorsof one of the calculation methods. Second, since the studies inwhich different self-consistent schemes are used are scarce, wewish to compare those two methods and verify the inﬂuenceof the scheme choice on the defects identiﬁcation. In Table Iwe can also observe that in each scheme, when the experimental volume is considered, we obtain similarresults while using different descriptions of the electronsin the system. No effect of the functional used for theelectron-electron exchange-correlation functional descriptionor of the spin polarization is observed. In particular, it isworth noting that the Uparameter does not affect directly the calculated positron lifetimes. The difference between thelifetimes calculated in GGA and GGA +Umethods is of 1 ps at most, when the experimental volumes are taken. Thelifetimes calculated at the equilibrium volumes found usinggiven method, however, differ more strongly. This is becausepositron lifetime is highly sensitive to the free volume. The bestagreement between the calculated and experimental lifetime is TABLE I. Lattice positron lifetime of UO 2calculated using various computational parameters. GGA GGA +U no spin spin no spin spin Veq.Vexp.Veq.Vexp.Veq.Vexp.Veq.Vexp. PSN+SC 151 ps 157 ps 155 ps 157 ps 152 ps 157 ps 160 ps 156 ps GGGC +SC 149 ps 155 ps 150 ps 154 ps 154 ps 155 ps 158 ps 154 ps PSN+GC 156 ps 162 ps 161 ps 162 ps 158 ps 162 ps 167 ps 162 ps GGGC +GC 157 ps 164 ps 159 ps 164 ps 163 ps 164 ps 168 ps 163 ps 184101-4COUPLED EXPERIMENTAL AND DFT +U. . . PHYSICAL REVIEW B 90, 184101 (2014) reached for the calculation using the GGA +Umethod and spin polarization at the equilibrium volume (168 and 167 psin the GGGC +GC and PSN +GC schemes, respectively, compared to 169 ±1ps measured experimentally). This is, therefore, the set of parameters that we use further in this study. IV . EXPERIMENTAL RESULTS The evolution of the annihilation characteristics as a function of the measurement temperature in 45 MeV α irradiated UO 2disks is presented in Fig. 1. For all measure- ment temperatures two positron lifetimes are extracted fromthe experimental spectrum decomposition. The shortlifetimecomponent τ 1, the long-lifetime component τ2, the average positron lifetime τav.and the intensity I2corresponding to τ2 are represented in Fig. 1. 200210220230240250 280290300310320330 102030405060 0 50 100 150 200 250 300160170180190FitExperimental dataLong component intensity I2(%) Measurement Temperature (K)Average lifetime τav.(ps)Long lifetime τ2(ps) componentShort lifetime τ1(ps) component FIG. 1. (Color online) Evolution of the average positron lifetime τav., short- and long-lifetime components τ1andτ2and the intensity I2, detected in UO 2crystals irradiated with 45 MeV αparticles at a ﬂuence of 2 ×1016cm−2, as a function of the measurement temperature. The solid lines are the ﬁts to the experimental dataobtained from the positron trapping model.It can be seen that τav.increases slightly from approximately 220±5 to 235 ±5 ps when the temperature rises, while τ1andτ2remain stable at about 170 ±5 and 310 ±5p s , respectively. The I2intensity increases when the measurement temperature rises, similarly to the average positron lifetime τav. For all measurement temperatures, the values of τ2are much larger than the lattice lifetime, already determinedin unirradiated UO 2disks [ 8] (169 ±1ps). This indicates positron trapping in vacancy defects. In addition, the short-lifetime component τ 1remains close to the experimental lattice lifetime of UO 2. In the case of materials containing only vacancy defects, if some of the positrons had annihilated in adelocalized state (in the lattice), the short-lifetime componentwould have been shorter than the perfect lattice lifetime sincethe average time spent by the positron in the lattice wouldbe shorter due to trapping in the defects. Thus the values ofτ 1indicate that all the positrons were trapped in vacancies or in negative nonvacancy defects. The short-lifetime componentis still equal to the experimental lattice lifetime at 300 K,which means that the nonvacancy traps are still effective atthis temperature. The nature of these nonvacancy defects willbe discussed further in Sec. VI. The long-lifetime component τ 2changes only slightly as a function of measurement temperature and remains stable at310±5 ps. This positron lifetime is close to the positron long- lifetime components ranging from 300 ±10 to 310 ±5p s observed in the previous study on the same samples [ 8] and 313±19 ps detected in UO 2with 0.2% plutonium weight content [ 12]. The intensity I2corresponding to the long- lifetime component increases slightly when temperature rises,which means that its trapping rate changes only slightly. To determine the nature of the vacancy-type defects detected in the samples, we used a positron trapping model(see Sec. II E). First, we considered models with only two types of positron traps (negative nonvacancy defects andneutral or negative vacancies). Both of them, however, failedto reproduce the experimental data. We concluded, therefore,that at least three types of traps were present in the studiedsamples—negative nonvacancy defects, neutral, and negativevacancies—and that the corresponding model should beused. The solutions of the rate equations used in the modelcontaining three different defect types were obtained byKrause-Rehberg and Leipner [ 41]. The ﬁts to the experimental data obtained using the positron trapping model are presented in Fig. 1(solid lines). Several parameters are needed in the model, some of which mustbe deduced or estimated. For the lattice and the nonvacancydefects we used the same annihilation rate, λ L=λNV=1/τL, withτL=170 ps. We considered the positron binding energy of the nonvacancy defects of at least 0.3 eV , as these trapswere still efﬁcient at 300 K. Since the lifetime spectradecomposition returned only two lifetime components (evenwhen three components decomposition was tested) and we donot observe strong variations of the τ 2lifetime, we suppose that the lifetimes of the neutral and negative vacancies areindistinguishable. We considered τ=310 ps for both of them. As for the trapping coefﬁcients, we used μ V=1×1015s−1 for the neutral vacancies, μV−=4×1016s−1at 20 K for the negative vacancies and μNV=4×1016s−1at 20 K for the nonvacancy defects. The choice of the trapping coefﬁcients 184101-5JULIA WIKTOR et al. PHYSICAL REVIEW B 90, 184101 (2014) was based on the values gathered in Ref. [ 31] and the predicted charge states of the negative defects. The ﬁts presented in Fig. 1were obtained using concen- trations cV=6.5×1019cm−3,cV−=2×1018cm−3, and cNV=1×1019cm−3. It is worth noting that some of the parameters used in the trapping model were only estimated,hence the absolute values of the concentrations cannot beconsidered certain. However, conclusions can be drawn on theproportions between the defects concentrations. The best ﬁtsof the present experimental data were obtained for the neutralvacancies concentration c Vat least 30 times larger than cV−and over six times larger than cNV, which suggests that the neutral vacancies are the predominant positron traps in the examinedUO 2samples. Smaller, but not negligible, concentrations of negative nonvacancy defects and vacancies are also present inthe material. V . CALCULATION RESULTS We performed positron lifetime calculations for several fully relaxed defects in UO 2containing from one to six vacancies in both GGGC +GC and PSN +GC schemes. The results are presented in Table II. For almost all types of defects, we considered two charge states. First, we performedpositron lifetime calculations for neutral defects. Second, wecalculated the lifetimes for vacancies in the charge states thatwere determined as the most stable ones in the stoichiometricmaterial [ 21,22]. Considering the oxygen vacancy, its formal charge (2 +) cannot be detected by PAS. The 2 −charge, however, was found to be stable for Fermi levels lying close tothe middle of the band gap [ 21,22], so we studied it as well. As can be seen in Table II, the differences between the positron lifetimes obtained in the two calculation schemes arelower than 10 ps for almost all considered defects. The biggest TABLE II. Positron lifetimes calculated in GGGC +GC and PSN+GC schemes for fully relaxed neutral and charged defects in UO 2. The lifetimes obtained for the most stable charge state of each defects are marked in bold. Lifetime Lifetime GGGC +GC PSN +GC Charge (ps) (ps) Lattice 168 167V O 0 206 199 VO 2− 203 195 VU 0 295 304 VU 4− 289 293 VU+VO 0 303 306 VU+VO 2− 299 301 VU+2VO(100) 0 301 304 VU+2VO(110) 0 310 313 VU+2VO(111) 0 314 316 2VU 0 313 318 2VU 8− 290 289 2VU+2VO 0 324 339 2VU+2VO 4− 309 319 2VU+4VO 0 323 329 2VU+4VO 2− 341 365differences are found for two large defects, the neutral 2V U+ 2VOtetravacancy (difference of 15 ps) and the 2 −charged 2VU+4VOhexavacancy (difference of 24 ps). It is also worth noting that similar results are yielded by the PSN +GC and GGGC +GC schemes for the stable charge states of the defects up to the 2V Udivacancy, i.e., defects which can most likely be observed in the PAS measurements. For both schemes, we can observe that the lifetimes of the negative defects are almost always shorter than for the neutralones. It is due to both a smaller relaxation and the higherelectronic density in negative defects. However, in the caseof the 2V U+4VOhexavacancy, the lifetime of the negative charge state is longer than the one of the neutral defect in bothcalculation schemes. This is due to the fact that the positronis localized in a different way in these two defects. In theGGGC scheme [Figs. 4(c) and 4(d)], in neutral 2V U+4VO, the positron is localized inside one of the uranium vacancies,while in 2 −charged 2V U+4VOthe majority of its density can be found between two V U. In the PSN scheme [Figs. 5(c) and 5(d)], we ﬁnd a similar localization between two uraniumvacancies in 2 −charged 2V U+4VO. In the case of the neutral hexavacancy, however, the positron density has two maxima,in both uranium vacancies. In both schemes, it can be observed in several instances that different defects have similar positron lifetimes. Forexample, in GGGC +GC, the uranium monovacancy V 4− Uand divacancy 2V8− Uhave lifetimes of 289 and 290 ps, respectively. The PSN +GC scheme yields lifetimes of 293 and 289 ps, respectively. The lifetimes obtained for the (V U+VO)2− divacancy, 299 ps in GGGC +GC and 301 ps in PSN +GC, are also close to these values. Moreover, using the two methodswe calculated positron lifetimes between 301 and 316 psfor the three conﬁgurations of the V U+2VOtrivacancy and lifetimes of 309 ps (GGGC +GC) and 319 ps (PSN +SC) for (2V U+2VO)4−. This can lead to difﬁculties in the defect identiﬁcation in the positron annihilation spectroscopy studies. To understand why different defects have similar positron lifetimes, we plotted the isodensities of the positron local-ized in these systems. We plot the results obtained in theGGGC +GC scheme in Figs. 2and4and in the PSN +GC scheme in Figs. 3and5. It is worth noting that in the PSN +GC scheme we applied the gradient correction on the enhancement factor only, hence there is no effect of this correction on thecalculated densities. First, we can observe that for all defects the GGGC +GC scheme yields more localized positron densities than thePSN scheme, which was expected [ 45,47]. In all defects presented in Figs. 2and4, except the negative hexavacancy, the GGGC scheme ﬁnds the positron localized inside oneuranium vacancy. In these defects, the positron density is onlyslightly affected by the presence of the other vacancies. Thefact that the positron “senses” similar volumes and geometriesin these defects explains why similar lifetimes are obtainedin these cases. The (2V U+4VO)2−hexavacancy is the only defect in which the positron localizes between the two uraniumvacancies. It is reﬂected in the longer lifetime of 341 pscalculated for this cluster in the GGGC +GC scheme. In the PSN scheme, however, we obtain a different positron localization for the clusters containing two uranium vacancies(see Fig. 5). In (2V U)8−and (2V U+4VO)2−the positron is 184101-6COUPLED EXPERIMENTAL AND DFT +U. . . PHYSICAL REVIEW B 90, 184101 (2014) VU (a) (VU)4−VO VU (b) (VU+VO)2−VO VUVO (c) (VU+2V O)0(110) FIG. 2. (Color online) Positron isodensities found in the GGGC +GC scheme (70% of the maximum density—solid, and 30%— transparent), in red, in the neutral defects containing one uranium vacancy. Uranium atoms are presented in gray, oxygen atoms in blue. White spheres represent the oxygen vacancies. Figures were generated using the XCRYSDEN [65,66] program. localized between the two uranium vacancies. However, in the 2−charged hexavacancy the free volume that the positron occupies is much larger than in (2V U)8−, hence we observe a signiﬁcantly longer lifetime for this defect. In both (2V U+ 2VO)4−and (2V U+4VO)0defects, we observe two maxima of the positron density, one in each uranium vacancy. Even though the positron densities yielded by GGGC +GC and PSN +GC differ strongly, using both schemes we obtain very similar positron lifetimes, which are the characteristicsthat we compare with experiments in the present study. Besidethis annihilation characteristic, the momentum distribution ofelectron-positron pairs can be measured through the Dopplerbroadening of the annihilation line using a Dopper broadeningspectrometer [ 67]. This distribution is also a valuable source of information on the nature and chemical environment ofvacancy defects and is complementary to the lifetime. Espe-cially, the annihilation rate with core electrons is more sensitiveto the positron distribution in the defect, hence it shouldbe more affected by the choice of the calculation scheme.Since we are not able to calculate the Doppler broadeningyet, we compare the ratio of the core annihilation fractionsdeduced from the core annihilation rate λ cas proposed by Puska et al. [47] using the expression Rc=(λc/λ)Defect (λc/λ)Lattice, (7)where λis the total annihilation rate. The Rcparameters calculated for several defects in UO 2in their most stable charge states are presented in Table III. It is worth noting that the relative Rcparameter depends on the choice of the core and valence electrons in the PAW data set and it willnot always correspond to the core annihilation parameterWobtained through the Doppler broadening measurements or calculations. From Table III, we can observe that the GGGC +GC scheme yields R cvalues at least twice as small as the PSN +GC scheme does. It is consistent with the results obtained by Puska et al. , who for instance observed that the GGGC scheme yielded values of the relative core annihilationfraction parameter (referred to as the estimated Wparameter in their work) too small by a factor of two for the Ga vacancyin GaAs, when compared with the measured Wparameter. We conclude, therefore, that even tough the lifetimes obtainedusing the two calculation schemes for the majority of defectsin UO 2are similar, the PSN method should be used in the future calculations of Doppler broadening in this material. VI. DISCUSSION In this section, we combine the calculation and experi- mental results to interpret the signals observed by PAS inthe UO 2samples in the present and previous studies. The VU (a) (VU)4−VO VU (b) (VU+VO)2−VO VUVO (c) (VU+2V O)0(110) FIG. 3. (Color online) Positron isodensities found in the PSN +GC scheme (70% of the maximum density—solid, and 30%—transparent), in red, in the neutral defects containing one uranium vacancy. Uranium atoms are presented in gray, oxygen atoms in blue. White spheres represent the oxygen vacancies. Figures were generated using the XCRYSDEN [65,66] program. 184101-7JULIA WIKTOR et al. PHYSICAL REVIEW B 90, 184101 (2014) VUVU (a) (2V U)8−VU2VOVU (b) (2V U+2V O)4− VO VU2VOVU VO (c) (2V U+4V O)0VO VU2VOVU VO (d) (2V U+4V O)2− FIG. 4. (Color online) Positron isodensities found in the GGGC +GC scheme (70% of the maximum density—solid, and 30%—transparent), in red, in the neutral defects containing two uranium vacancies. Uranium atoms are presented in gray, oxygenatoms in blue. White and yellow spheres represent oxygen and uranium vacancies, respectively. Figures were generated using the XCRYSDEN [65,66] program. short-lifetime component τ1detected in the studied UO 2disks remains close to the experimental lattice lifetime of UO 2of 170±5 ps for all measurement temperatures (see Fig. 1). This suggests that part of the positrons annihilate around negativenonvacancy defects in the samples. The samples analyzedin this study were slightly hyper-stoichiometric with O/U = 2.0051±0.0001, which means that excess oxygen atoms were already present in the lattice before the αirradiation. The nature of the point defects in the slightly hyperstoichiometricUO 2+xstructure and their local conﬁgurations have been the object of extensive studies, both experimental [ 1,4,68,69] and theoretical [ 13–15,20,22,70,71]. Depending on the study, different types of defects containing additional oxygen atoms,such as monointerstitials, di-interstitials, split-interstitials orWillis clusters, are proposed as the most stable ones. All thesepossible defects structures were found to be stable in negativecharge states due to the oxygen ions formal charge state of −2. Recently, Wang et al. [71] suggested that the average structure of UO 2+xcan be represented as a combination of all of these defects structures. Therefore we suppose that the short-lifetimecomponent detected in the studied samples corresponds tomixed signals coming from the positron annihilation aroundmonointerstitials and interstitial clusters charged negatively.These could be defects already present in the unirradiatedUO 2discs or created by irradiation. VUVU (a) (2V U)8−VU2VOVU (b) (2V U+2V O)4− VO VU2VOVU VO (c) (2V U+4V O)0VO VU2VOVU VO (d) (2V U+4V O)2− FIG. 5. (Color online) Positron isodensities found in the PSN+GC scheme (70% of the maximum density—solid and 30%—transparent), in red, in the neutral defects containing two uranium vacancies. Uranium atoms are presented in gray, oxygenatoms in blue. White and yellow spheres represent oxygen and uranium vacancies, respectively. Figures were generated using the XCRYSDEN [65,66] program. The long-lifetime component of 310 ±5 ps detected in the samples is close to what we calculated for the neutralV U+2VOtrivacancy (301–314 ps in GGGC +GC and 304– 316 ps in PSN +GC, depending on the conﬁguration) and the 2VU+2VOwith the −4 charge state (309 ps in GGGC +GC and 319 ps in PSN +GC). The analysis of the evolution of the positron annihilation characteristics as a function of TABLE III. Relative core annihilation fraction parameter Rc[see Eq. ( 7)] calculated in the GGGC +GC and PSN +GC schemes for the most stable charge states of defects in UO 2. We also include the result obtained for the neutral oxygen monovacancy. Rc Rc Charge GGGC +GC PSN +GC VO 0 0.39 0.80 VU 4− 0.23 0.53 VU+VO 2− 0.23 0.54 VU+2VO(100) 0 0.23 0.55 VU+2VO(110) 0 0.22 0.55 VU+2VO(111) 0 0.22 0.55 2VU 8− 0.26 0.56 2VU+2VO 4− 0.23 0.58 2VU+4VO 2− 0.09 0.46 184101-8COUPLED EXPERIMENTAL AND DFT +U. . . PHYSICAL REVIEW B 90, 184101 (2014) the measurement temperature based on the trapping model indicates that two types of vacancy defects are present in thesample, the neutral one being predominant. Theoretical studieson the charge states of the defects clusters in UO 2[14,21,22] suggest that the only neutral defect in the material close to thestoichiometry for the Fermi level near the middle of the bandgap is the V U+2VOtrivacancy. We propose, hence, that the bound Schottky defects are the neutral vacancies observed inthe samples. The analysis of the experimental data based onthe positron trapping model implies that negative vacancies arealso detected in the UO 2samples. They should have a positron lifetime close to 310 ps, since only one long componentwas obtained from the lifetime spectra decompositions. Theformation energy calculations of defects in UO 2[22] suggest that several types of negatively charged vacancies can bepresent in the material. The positron lifetime calculationspresented in this work yield values slightly shorter or longerthan 310 ps for various negative defects (from 293 ps forV 4− Uto 319 ps for 2V U+2VOwith the −4 charge state in the PSN +GC scheme). We suppose, hence, that negative uranium monovacancies, U-O divacancies, and 2V U+2VO tetravacancies are present in the examined samples. However, the concentration of all these defects are much smaller thanthe concentration of the Schottky defects. Our results can be compared with the classical molecular dynamics (CMD) study on 10 keV displacement cascades in UO 2by Martin et al. [23]. The authors found that even though initially mostly monovacancies and monointerstitials werecreated, they quickly started to form stoichiometric clusters,such as bound Schottky defects, because of the high oxygenmobility. It is worth noting that the empirical potentials usedin this study favored the neutral defects over the chargedones and the simulations corresponded to a physical time ofapproximately 25 ps. Nevertheless, the present results conﬁrmthe general conclusion of the CMD study. In addition to the studies on positron lifetimes in UO 2, measurements of the Doppler broadening of annihilationradiation have also been performed [ 6–11]. Calculations of the momentum distribution spectra for various defects in UO 2 and their comparison with the experimental data could alsobring additional information on the defects in this material. Toachieve this goal, we are currently implementing the methodallowing the calculation of this annihilation characteristicin the ABINIT code. Since the Doppler broadening is more sensitive to the positron distribution in the defect and thedefects conﬁguration, it is worth keeping in mind thatthe PSN scheme, with a better description of the positronlocalization, should be more suitable for the modeling of theexperimental spectra. Additionally, we observed that the twocalculation schemes yield different positron lifetimes for largedefects. In the present study we did not observe lifetimeslonger than 310 ±5 ps, however, if larger vacancy clusters are present in UO 2in future PAS measurements, the choice of the calculation method can affect their identiﬁcation. VII. CONCLUSIONS We performed PAS measurements in UO 2sintered disks irradiated with 45 MeV αparticles at a ﬂuence of 2 × 1016cm−2. The positron lifetime was measured as functionof temperature in the 15–300 K range. We observed two lifetime components, τ1of 170 ±5 ps and τ2of 310 ±5p s . The short-lifetime component is close to the lattice lifetimefor all measurement temperatures. It means that all positronsannihilated in vacancy defects or around negative nonvacancydefects. These nonvacancy defects are assumed to be negativeoxygen monointerstitials and interstitials clusters. We useda positron trapping model with three types of positrontraps to analyze the evolution of the positron annihilationcharacteristics in function of measurement temperature. Weconcluded that a neutral vacancy was the most predominantpositron trap, while smaller, but not negligible concentrationsof negative vacancies were also present in the material. We calculated the positron lifetimes of neutral and charged fully relaxed vacancies and vacancy clusters in UO 2us- ing two different fully self-consistent calculation schemes(GGGC +GC and PSN +GC), in the DFT +Uformalism. We observed that the parameters used in the electroniccalculations do not affect directly the positron lifetime.However, since the positron lifetime is highly sensitive tothe free volume, there is an effect of the equilibrium volumecorresponding to the method and the parameters used on thelifetimes obtained. We showed that the gradient correctionbetter describes the absolute values of the positron lifetimesin this material. We showed that the PSN and GGGC schemesyielded similar positron lifetime for the majority of studied defects, especially for the stable charge states of defects up to trivacancies. Therefore similar general conclusionscould be drawn by comparing results obtained using bothschemes with the experimental values. However, the choiceof the calculations scheme could affect the experimentsinterpretation if larger defects are present in the material. For several defects, in particular, 2 −charged V U+VO, neutral V U+2VOand 4−charged 2V U+2VO, we calcu- lated similar positron lifetimes. It is due to the fact that foralmost all the studied vacancy clusters the positron is localizedin one uranium vacancy and is only slightly affected by thepresence of the oxygen vacancies or of the second uraniumvacancy. The only cluster having a signiﬁcantly longer positronlifetime (341 ps) is the 2 −charged 2V U+4VOhexavacancy, where the positron is localized between the two uraniumvacancies. Comparison of the results obtained experimentally with the calculated positron lifetimes and the most stable charge statesof the defects in UO 2allowed us to identify the predominant neutral vacancy as the V U+2VOtrivacancy (bound Schottky defect). This result can be further conﬁrmed by calculationsof the momentum distribution spectra for various defectsin UO 2and their comparison with the experimental data. 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PhysRevB.75.195207.pdf	Effects of optical absorption on71Ga optically polarized NMR in semi-insulating GaAs: Measurements and simulations Stacy Mui, Kannan Ramaswamy, and Sophia E. Hayes Department of Chemistry and Center for Materials Innovation, Washington University, St. Louis, Missouri 63130, USA /H20849Received 16 January 2007; revised manuscript received 8 February 2007; published 10 May 2007 /H20850 The intensity and the hyperﬁne shift of optically polarized NMR /H20849OPNMR /H20850signals of71Ga in semi- insulating GaAs have been found to depend on the photon energy and the helicity of light used for opticalpumping. Single-crystal GaAs wafers of two different thicknesses, 400 and 175 /H9262m, were examined. The maximum intensity of the OPNMR signals was observed well below the band gap energy, and this maximumOPNMR signal shifted to higher photon energies for the thinner sample. In the range of photon energies forwhich the maximum OPNMR signals are obtained, there is little or no hyperﬁne shift of the 71Ga OPNMR resonance. Hyperﬁne shifts with the largest magnitude are recorded for photon energies at or above the bandgap. At a given photon energy, asymmetric OPNMR signals were observed, with /H9268+light producing a more intense emissive signal than the corresponding absorptive signal coming from /H9268−light. We developed a model that accounts for optical absorptivity of GaAs in order to simulate the observed OPNMR intensity, the changein the OPNMR maximum for the thinner sample, and the hyperﬁne shift dependence on photon energy. Thismodel also accounts for the asymmetric OPNMR intensity for the two /H9268+and/H9268−helicities of light. The intensity dependence of the OPNMR signals arises as a consequence of two competing factors: the opticalabsorptivity of GaAs which directly impacts nuclear polarization /H20855I Z/H20856, and the number of accessible nuclear spins. The magnitude of the hyperﬁne shift of the OPNMR signals reﬂects the probability of occupation ofoptically relevant defects in the semiconductor, also related to optical absorptivity. Finally, the asymmetry inthe OPNMR signals arises from the sign of electron spin polarization produced in the optical pumping process. DOI: 10.1103/PhysRevB.75.195207 PACS number /H20849s/H20850: 76.60. /H11002k, 31.30.Gs, 78.20.Ci I. INTRODUCTION Lampel experimentally demonstrated in 1968 that it is possible to polarize nuclear spins by orienting the electronspins in a semiconductor through irradiation with light. 1 Since then, there have been investigations in several II-VI,2,3 III-V,4–11and Group IV /H20849Ref. 12/H20850semiconductors where the primary objective has been creating and detecting nuclearspin orientation. These methods, broadly termed “opticallypolarized NMR” /H20849OPNMR /H20850or “laser-enhanced NMR,” ex- amine nuclear spin polarization and its dependence on anumber of variables, such as laser wavelength and polariza-tion, sample temperature, and magnetic-ﬁeld strength. Thisarea of research has garnered considerable interest becauseof the potential to manipulate long-lived nuclear spin degreesof freedom for applications in spin-based electronic devices,spintronics. 13Furthermore, the proposal by Tycko to transfer nuclear spin polarization to structurally complex biologicalmacromolecules, 14and the subsequent experimental realiza- tion of signal enhancement to a chemically bound species byGoehring et al. 15have produced renewed interest in the gen- eration of optically enhanced NMR signals. For these poten-tial applications, it is critically important to understand theorigin and dependence of enhanced nuclear spin-polarizationon experimental conditions. An ongoing line of inquiry in this ﬁeld has addressed the OPNMR signals from semiconductors with respect to theenergy and polarization of the incident light. 3–7,9–11For GaAs in particular, it has been reported that the maximum OPNMRsignal intensity occurs for radiation between 1.49 and1.51 eV, corresponding to photon energies in the range ofapproximately 0.03–0.01 eV below the band gap /H20849E g/H20850.5TheOPNMR resonances obtained with /H9268+and/H9268−light were also observed to be unequal in intensity at a given photon energy.These observations were intriguing because the maximumsignal intensity was expected to occur above the bandedge; 5,10,16however, there was no conclusive explanation for the physical origin of either the intensity proﬁle or its depen- dence on laser helicity. Measurements of the penetrationdepth of the laser into Fe-doped semi-insulating InP weremade by Michal and Tycko using stray ﬁeld imaging. 8These measurements gave the ﬁrst experimental indication of thespatial distribution of OPNMR in a semiconductor. Researchers in the OPNMR arena have tended to investi- gate the most intense resonances, where signal-to-noise ra-tios are maximized, and ostensibly where one might con-clude that the greatest polarizations are being achieved.Despite the experimental success of obtaining substantialOPNMR signals, the intricacies of the optical and electronicprocesses that lead to signal enhancements in OPNMR arenot yet fully understood. In this present study, we have measured and simulated the 71Ga OPNMR signal intensity and hyperﬁne shift depen- dence on the photon energy and helicity of the optical pump-ing source. By investigating two semi-insulating GaAssamples of different thicknesses, the combined roles of thepenetration depth, photon energy, and corresponding prob-ability of laser absorption by the material have been clearlydemonstrated in GaAs. We present analyses of the data thatincorporate optical absorption effects to explain the intensity and hyperﬁne shift dependence of 71Ga OPNMR signals on the photon energy and helicity. From these analyses, a newinterpretation of the OPNMR data emerges that the maxi-mum signal observed is a balance between optical absorptiv-PHYSICAL REVIEW B 75, 195207 /H208492007 /H20850 1098-0121/2007/75 /H2084919/H20850/195207 /H208498/H20850 ©2007 The American Physical Society 195207-1ity/H20849hence, depth in the sample /H20850and the number of accessible nuclear spins. II. EXPERIMENTAL SECTION Experiments were carried out on semi-insulating GaAs /H20849si-GaAs /H20850obtained from ITME, Warszawa, Poland /H20849sample characteristics: growth direction of /H20851100 /H20852, thickness of 400/H9262m, and mobility of 5630 cm2/V s /H20850. For the purposes of this study, another sample thickness was prepared; the400 /H9262m wafer was thinned down by mechanical grinding to a thickness of 175±2 /H9262m using a polishing wheel with SiC polishing paper /H20849600 grit /H20850. The71Ga OPNMR experiments were recorded at a Lar- mor frequency of 61.11 MHz /H20849B0=4.7 T /H20850. The spectra were collected using a pulse sequence consisting of a saturating radio-frequency /H20849rf/H20850train /H20849SAT /H20850, followed by a period of continuous wave laser irradiation, /H9270L/H20849120 s /H20850, a single/H9266 2 pulse, and signal acquisition /H20849ACQ /H20850:SAT– /H9270L–/H9266 2–ACQ. The saturating train consisted of ﬁfty/H9266 2rf pulses separated by 1 ms each. The /H9270Lperiod of 120 s was selected to attenuate effects due to spin diffusion, while still providing sufﬁcientsignal-to-noise ratios for analysis. 17The data acquisition was performed using a Tecmag Apollo console. The experimental procedure used includes recording a reference spectrum at 6 K in a single shot when the targettemperature has been reached, prior to any laser irradiation.We term this spectrum the “Boltzmann” signal /H20849versus the OPNMR signal /H20850in order to record the resonance frequency of the 71Ga nuclei in the sample populated by thermal pro- cesses and not due to coupling to the optically oriented elec- trons or to the laser. The T1times for the71Ga nuclei are extremely long /H20849/H1102220 h /H20850at these temperatures, and these be- come saturated after just a single acquisition. The samples were each mounted in a home-built single- channel transmission line probe inside a continuous-ﬂow cryostat /H20849Janis-200 Supertran /H20850to maintain the sample tem- perature at 6±0.2 K. The mounting procedure and hardwarewere described in detail in a previous publication. 17 A tunable continuous wave Ti:sapphire laser /H20849Spectra Physics 3900S /H20850pumped by a 532 nm solid-state diode laser /H20849Spectra Physics Millenia X /H20850was used as an optical excita- tion source. The wavelength of the Ti:sapphire output wasmeasured with an Ocean Optics HR-2000 spectrometer/H20849resolution of 0.035 nm /H20850. The linearly polarized output from the laser was converted to /H9268+or/H9268−polarized light using a quarter-wave retarder, centered at 825 nm /H20849retardation toler- ance is within/H9261 50for the range used in these experiments /H20850.I n all the experiments presented here, the laser power was keptconstant at 2.5 W/cm 2. The sample was irradiated with the incident beam parallel to the external magnetic ﬁeld, and thebeam waist was 5±1 mm. The laser was blocked only duringthe saturating rf train and between acquisitions; it was onduring the pulse sequence and during acquisition. The OPNMR signal intensity and the hyperﬁne shift were extracted by analyzing the time domain free induction decayusing a Bayesian analysis computer program. 18 III. THEORY Optical excitation in semiconductors is achieved through irradiation with photon energies that allow for promotion ofelectrons out of the valence band into some other excited state. It is therefore important to deﬁne several commonlyused terms, in this context. “Band-gap energy” /H20849E g/H20850is de- ﬁned as the energy difference between the highest valence band energy levels and the lowest-energy conduction bandenergy levels. In contrast the “band edge” is deﬁned as theregion just energetically below the conduction band wherethe onset of optical absorption begins as a consequence ofthe formation of excitonic states due to shallow donors. 19 When a direct gap semiconductor is illuminated with light of photon energies at or above Eg/H20849for GaAs 1.522 eV at 6K /H20850,20the electrons in the valence band are promoted into the conduction band, or for photon energies below Eg, elec- trons are excited into shallow donor levels. If circularly po-larized light is used, the excited electrons can also be spinpolarized. 21Most electrons then relax via radiative recombi- nation with holes in the valence band, with recombinationcenters, or with shallow acceptors. 22Some spin-polarized electrons are trapped near optically relevant defect /H20849ORD /H20850 sites23and consequently couple strongly with nearby nuclear spins. Typically, this coupling is strongest for those nuclearspins within a Bohr exciton radius of the trapped electron. InGaAs, the exciton radius for a localized electron near theconduction band edge is approximately 100 Å. 24 Both localized electrons and delocalized electrons can couple with the nuclear spins through a hyperﬁne interaction.In a previous study by Paravastu et al. , 10there was evidence of nuclear spins coupling to delocalized electron states fromfree excitons. In the present study, where we have observedhyperﬁne shifted OPNMR signals, we have calculated thatfree excitons could only contribute a negligible shift. The Hamiltonian for this interaction, given in Eq. /H208491/H20850,i sa sum of Fermi-contact /H20849H FC/H20850and electron-nuclear dipolar /H20849HD/H20850Hamiltonians:25 H=HFC+HD =16/H9266 3/H60362/H92710/H9253I/H9253S/H20841/H9023/H20849r/H20850/H208412/H20851I+S−+I−S++2IZSZ/H20852 +/H92620/H9253I/H9253S/H60362 4/H9266r3/H208753/H20849I¯·r¯/H20850/H20849S¯·r¯/H20850 r2−I¯·S¯/H20876. /H208491/H20850 In Eq. /H208491/H20850,I¯andS¯are the electron and nuclear spin opera- tors, respectively, /H20841/H9023/H20849r/H20850/H208412is the electron probability density at the nucleus normalized over the sample volume, /H92620is the permeability constant of free space, v0is the volume of the unit cell, and ris the distance between a localized electron and nuclear spin. Recent experiments suggest that the HFCis the dominant term in GaAs,17,26allowing the HDterm to be neglected. The ﬁrst two bilinear spin terms in HFC,/H20849I+S−+I−S+/H20850lead to dynamic nuclear polarization /H20849DNP /H20850, whereas the third term, IZSZ, leads to a frequency shift in the NMR resonance.25DNP occurs because the hyperﬁne coupling that arises from the electron-nuclear interactions is time depen-dent, and the nuclear spins experience a ﬂuctuating magneticﬁeld. 23MUI, RAMASWAMY, AND HAYES PHYSICAL REVIEW B 75, 195207 /H208492007 /H20850 195207-2For excitation by photons with energies in excess of Eg, free electrons are created, some of which become localized.Spin exchange occurs between these free and localized elec-trons, leading to a ﬂuctuating magnetic ﬁeld. 27However, for excitation by photons with energies less than Eg, only shal- low donors become populated. In such a case, a ﬂuctuatingmagnetic ﬁeld could be caused by spin exchange with theelectrons trapped at the neighboring donor centers. 28Another source of ﬂuctuating ﬁelds could be the multiple cycles ofexcitation, deexcitation, and subsequent reexcitation of thesephotogenerated electrons under optical pumping. As stated in the Experimental Section, the OPNMR signal is recorded from a condition that begins with complete satu-ration of the nuclear polarization. As a function of /H9270L, the average zcomponent of nuclear polarization, /H20855Iz/H20849r/H20850/H20856, grows according to Eq. /H208492/H20850,24 /H11509/H20855Iz/H20849r/H20850/H20856 /H11509/H9270L=D/H116122/H20855Iz/H20849r/H20850/H20856−1 T1/H20849r/H20850/H20851/H20855Iz/H20849r/H20850/H20856−I/H11009/H20852/H20849 2/H20850 where Dis the isotropic nuclear spin diffusion coefﬁcient, T1/H20849r/H20850is the electron-nuclear cross-relaxation time,24andI/H11009is the steady-state nuclear spin polarization.16,23The ﬁrst term on the right-hand side of Eq. /H208492/H20850accounts for the diffusion of nuclear polarization away from the ORD, and the secondterm for the buildup of nuclear polarization due to the Fermi-contact interaction. 29The relaxation rate in Eq. /H208492/H20850is given by 1 T1/H20849r/H20850/H11015/H20873−/H92620g0/H9262Bd/H9254/H92710/H9253I 12/H92662a0/H208742 /H9003/H9270cexp/H20873−4r a0/H20874 /H11015/H9024/H92542/H9003/H9270cexp/H20873−4r a0/H20874, /H208493/H20850 where /H9262Bis the Bohr magneton, g0is the free electron g factor, d/H9254is the electronic density at the nuclear isotope /H9254 /H20849d71Ga=5.8/H110031025cm−3/H20850,23a0is the Bohr radius for a local- ized electron in GaAs, /H9003is the probability of occupation of a donor site,23and/H9270cis the electron-nuclear correlation time. We deﬁne /H9024/H9254to be equal to the constants contained in pa- rentheses. To fully analyze the experimental data, it is important to know the magnitude and sign of the hyperﬁne shift in theNMR spectrum, attributable to I ZSZ, due to the average ef- fective magnetic ﬁeld created by the localized electrons. Thehyperﬁne shift in the nuclear Larmor frequency /H20849in hertz /H20850for an isotope /H9254is given by23 /H9254/H9263s/H20849r/H20850=/H9024/H9254/H20855SZ/H20856/H9003exp/H20873−2r a0/H20874, /H208494/H20850 where /H20855SZ/H20856is the average zcomponent of the electron spin polarization.16,23In an earlier study, a shift of the69Ga OP- NMR spectrum has been found as a function of /H9270Lfor exci- tation energies in excess of Eg. We previously presented a quantitative model explaining the dependence of the hyper-ﬁne shift in the spectra with respect to /H9270Lusing a Fermi- contact interaction between localized electrons and surround-ing nuclei, and nuclear spin diffusion. 17The NMR signalarises as a weighted average of nuclei that are located in close proximity to an ORD /H20849therefore experiencing a strong hyperﬁne interaction /H20850and nuclei located remotely /H20849experi- encing a negligible hyperﬁne interaction /H20850. At longer pumping times, the nuclei positioned further from the ORD dominatethe signal, thereby reducing the observed hyperﬁne shift. In this paper, we modify these calculations of /H20855I Z/H20849r/H20850/H20856and /H9254/H9263s/H20849r/H20850by incorporating a photon energy dependence and ir- radiation volume /H20849i.e., the volume of the sample where the absorption of light occurs /H20850dependence due to optical absorp- tion effects, shown explicitly in Sec. V. IV . RESULTS The OPNMR signals exhibit a dependence on photon en- ergy that was not initially predicted by early models of opti-cally polarized phenomena. 16,23Pietrass et al. ,5Kuhns et al. ,7 and Paravastu et al.10observed intense OPNMR signals be- lowEgin semi-insulating GaAs. We report a similar intensity dependence on photon energy. Figure 1/H20849a/H20850shows the71Ga OPNMR signal amplitude /H20849integrated peak area /H20850recorded with respect to the laser photon energy for the two si-GaAssamples: the 400 /H9262m thick /H20849ﬁlled symbols /H20850and the 175 /H9262m thick wafer /H20849open symbols /H20850. The familiar “signature” of ab- FIG. 1. /H20849a/H2085071Ga OPNMR signal intensities in si-GaAs as a function of photon energy for 400- /H9262m-thick /H20849ﬁlled symbols /H20850and 175/H9262m thick /H20849open symbols /H20850samples. Error bars are smaller than the symbols. /H20849b/H2085071Ga hyperﬁne shifts for the 400- /H9262m-thick sample only. Representative error bars are shown. The legend in /H20849a/H20850applies to both parts of the ﬁgure. Regimes I, II, and III, denoted by thevertical lines, are to guide the discussion of the data.EFFECTS OF OPTICAL ABSORPTION ON 71Ga… PHYSICAL REVIEW B 75, 195207 /H208492007 /H20850 195207-3sorptive and emissive OPNMR signals4,5,7,10,11,30—where /H9268+ helicity laser irradiation results in an emissive signal and /H9268− helicity results in an absorptive signal—is observed across nearly this entire range of photon energies. The most intense signal for the 400- /H9262m-thick sample is at /H110151.500 eV for /H9268+irradiation and /H110151.505 eV for /H9268−irradia- tion. Interestingly, the most intense signals for the thinner175 /H9262m sample are at higher photon energies, 1.507 eV, for both/H9268+and/H9268−. These signals are observed at photon ener- gies well below Eg/H20851Eg=1.522 eV, estimated for 6 K /H20849Ref. 20/H20850/H20852. The energy dependence of the signal intensity appears to vanish at a value of /H110151.514 eV for both sample thick- nesses and for both helicities of light. In addition, at a given photon energy, there is an apparent asymmetry in the signal intensity with respect to the helicityof light, evident by the larger emissive /H9268+OPNMR signals as compared to the smaller absorptive /H9268−ones, most apparent below Egin Fig. 1/H20849a/H20850 The hyperﬁne interactions which are responsible for the enhanced nuclear polarization have also been predicted toproduce a shift in the nuclear Larmor frequency. 16Figure 1/H20849b/H20850depicts the dependence of the optically induced hyper- ﬁne71Ga OPNMR shift on laser photon energy for the 400-/H9262m-thick sample. The shifts were measured with respect to the Boltzmann signal recorded at 6 K. The plot depicts thevariation of the shift for both helicities of light, and thequalitative behavior of the shift is found to be similar. The /H9268+ irradiation leads to a positive hyperﬁne shift, whereas the /H9268− irradiation leads to a negative shift arising from the sense ofelectron spin polarization /H20849/H11001or/H11002, depending on the selec- tion rules using circularly polarized light 21/H20850. There is no ap- preciable hyperﬁne shift /H20849meaning /H11021100 Hz /H20850until photon energies in excess of 1.50 eV are used for optical polariza-tion. A dramatic increase in the magnitude of the shift isobserved between 1.500 and 1.514 eV. This shift begins toplateau at /H110151.514 eV±2 meV. For photon energies /H110221.514 eV, the magnitude of the hyperﬁne shift ﬂuctuates about an average value of approximately 340 Hz for /H9268+/H20849and −135 Hz for /H9268−/H20850. The signal-to-noise ratio is much poorer for /H9268−light, leading to greater scatter in the hyperﬁne shift data. The observed hyperﬁne shifts did not exhibit a depen- dence on the laser power used, /H20849operating in a nonlinear excitation regime near saturation, where NMR signal inten-sity did not markedly change with laser power /H20850. Data were recorded with one other /H9270Lvalue /H20849/H9270L=20 s, data not shown /H20850. The shorter /H9270Ltime resulted in a similar trend as that shown in Fig. 1/H20849b/H20850, but as expected, with larger hyperﬁne shift magnitudes.17 V . SIMULATIONS OF OPNMR SIGNALS To incorporate the effects of photon energy on the OP- NMR signal intensity and hyperﬁne shift, we include theenergy-dependent optical absorption coefﬁcient of the mate-rial, /H9251/H20849E/H20850, in our simulations. To account for the decrease of the laser intensity within the material, we utilize the expres- sion: I/H20849x,E/H20850=I0exp /H20851−/H9251/H20849E/H20850x/H20852, where I0is the intensity of light impinging on the surface of the semiconductor, and xis the distance from the irradiated surface.In si-GaAs, /H9251/H20849E/H20850has been experimentally determined previously for a range of temperatures and excitation energies.31Sturge reported that /H9251/H20849E/H20850changes by several or- ders of magnitude /H20849/H11011101to 104cm−1/H20850with excitation at photon energies from approximately 1.49–1.52 eV, exciting the band edge of GaAs up through Eg. This drastic change in /H9251with photon energy can signiﬁcantly inﬂuence the irradia- tion depth. The absorption coefﬁcients used for the modelhave been adapted from Ref. 31and are shown in Fig. 2/H20849a/H20850. As previously stated in Eqs. /H208492/H20850and /H208494/H20850,/H9003inﬂuences both the intensity and the shift of the OPNMR signal. /H9003has been deﬁned previously as the probability of occupation of adonor. 23We extend this original deﬁnition by representing /H9003 as a fraction: the number of photogenerated carriers /H20851n/H20849x,E/H20850/H20852 over the total number of ORD’s /H20849ND/H20850. We calculate n/H20849x,E/H20850 according to the following expression: n/H20849x,E/H20850=/H9251/H20849E/H20850I0exp /H20851−/H9251/H20849E/H20850x/H20852/H9261/H9270e hc, /H208495/H20850 where /H9270eis the carrier lifetime, and /H9261is the wavelength of the laser. Since the total number of ORD’s /H20849ND/H20850cannot ex- plicitly be known, we instead use an approximation that the number of photogenerated carriers produced at a sufﬁcientlyhigh energy above E g/H20849i.e., 1.568 eV in our experiments /H20850is equivalent to the total number of ORD’s at or near the sur- FIG. 2. /H20849a/H20850Photon energy-dependent optical absorption coefﬁ- cients for si-GaAs adapted from Ref. 31used in the simulations. /H20849b/H20850 Simulation of71Ga OPNMR signal intensity as a function of photon energy for the 400 and 175 /H9262m samples. /H20849c/H20850Simulation of71Ga hyperﬁne shifts for the 400 /H9262m sample.MUI, RAMASWAMY, AND HAYES PHYSICAL REVIEW B 75, 195207 /H208492007 /H20850 195207-4face. At this energy where absorption is maximized, /H9003/H20849x =0,E=1.568 eV /H20850would be equal to 1.0, an approximation that was used with our earlier simulations of the evolution of shift with respect to /H9270L.17In other words, there is full satu- ration of the ORD’s throughout the irradiated volume when/H9003/H20849x,E/H20850is maximized. A. Simulation of the relative OPNMR signal intensity The OPNMR signal intensity is proportional to /H20855IZ/H20849r,x,E,/H9270L/H20850/H20856, and /H9003/H20849x,E/H20850enters the calculation through T1/H20849r,x,E/H20850−1. The solution of Eq. /H208492/H20850forD=0 /H20849i.e., neglecting spin diffusion /H20850and at the experimental conditions, /H9270L =120 s, B0=4.7 T, T=6 K, provides us an estimate of aver- age nuclear polarization, /H20855IZ/H20849r,x,E,/H9270L/H20850/H20856=I/H20849I+1/H20850 S/H20849S+1/H20850/H20849/H20855SZ/H20856−S0/H20850/H208771 − exp/H20875−/H9270L T1/H20849r,x,E/H20850/H20876/H20878. /H208496/H20850 In Eq. /H208496/H20850,/H20855SZ/H20856is the average zcomponent of electron spin polarization given by32 /H20855SZ/H20856/H9268/H11007=±0.25 /H208731+/H9270e T1e/H20874+0.5S0 /H208731+T1e /H9270e/H20874, /H208497/H20850 where T1eis the spin lifetime of the photoexcited electrons, andS0is the thermal equilibrium electron polarization, S0= −1 2tanh /H20849g/H9262BB0 2kBT/H20850. The ﬁrst term in /H20855SZ/H20856is the electron spin orientation of the photogenerated carriers, and the second term is the contribution from the thermal electron spin polar-ization of the carriers occupying the states at or just belowthe bottom of the conduction band. The second term cannotbe neglected in our experimental conditions of high magneticﬁeld; at low ﬁeld values, the ﬁrst term in /H20855S Z/H20856dominates. The parameters used in the simulation are /H9270c=10−11s,16I0 =2.5 W/cm2,/H9270e=10 ns,33/H9270e T1e=0.8,17andg*=−0.44.32 The relative OPNMR signal /H20849/H9014/H20850is calculated from the sum of /H20855IZ/H20849r,x,E,/H9270L/H20850/H20856asxis incremented from 0 to L, where Lis the total thickness of the sample: /H9014/H20849r,E,/H9270L/H20850 =/H208480L/H20855IZ/H20849r,x,E,/H9270L/H20850/H20856dx. The results of the simulation are shown in Fig. 2/H20849b/H20850as a function of photon energy. As observed in our experimentaldata, the model predicts that the thinner sample /H20849L =175 /H9262m/H20850has a maximum intensity at higher photon ener- gies than the 400 /H9262m sample. The two sets of data converge at photon energies /H110111.509 eV due to the equivalent penetra- tion depths. The asymmetry in the signal intensity with re-spect to the helicity of light is also captured in our simula-tions. The signal intensity is greater for /H9268+light than for /H9268−. The asymmetry primarily arises due to the contribution ofquasithermal equilibrium electron polarization /H20851see Eq. /H208497/H20850/H20852. Asymmetry of the OPNMR signal intensity with respect tohelicity will be reduced either at high temperatures or atlower magnetic ﬁelds, where the thermal electron spin polar-ization S 0is smaller.B. Simulation of the photon energy-dependent hyperﬁne shifts The other effect of /H9003/H20849x,E/H20850is manifested in the hyperﬁne shift from Eq. /H208494/H20850. The hyperﬁne shift was simulated by es- timating the average nuclear magnetization given as/H20855M Z/H20849r,x,E,/H9270L/H20850/H20856=/H9253/H9254/H6036N/H20849r/H20850/H20855IZ/H20849r,x,E,/H9270L/H20850/H20856, where N/H20849r/H20850is the total number of71Ga nuclei at a distance rfrom the ORD. We previously calculated17N/H20849r/H20850=51r2/H110030.396 in zinc blende GaAs, where 0.396 is the natural abundance of71Ga. The average shift of the OPNMR signal for a given illuminationtime /H20849 /H9270L/H20850and photon energy /H20849E/H20850is the average of /H20855MZ/H20849r,x,E,/H9270L/H20850/H20856weighted by /H9263s/H20849r,x,E,/H9270L/H20850, /H20855/H9263s/H20849E,/H9270L/H20850/H20856=/H20885 jk/H20885 0L /H20855MZ/H20849r,x,E,/H9270L/H20850/H20856/H9263s/H20849r,x,E,/H9270L/H20850dxdr /H20885 jk/H20885 0L /H20855MZ/H20849r,x,E,/H9270L/H20850/H20856dxdr. /H208498/H20850 In Eq. /H208498/H20850, the upper limit for the radial integration kwas set by calculating the largest radius where the relaxation ofnuclear spins is dominated by the Fermi contact interactionfrom T 1/H20849r/H20850. This time can be equated to the expression for three-dimensional Fickian diffusion /H20849D=3000 Å2/s/H20850result- ing in a value of r=770 Å. Nuclei located outside of this radius are relaxed by spin diffusion processes and will there-fore not exhibit any hyperﬁne shift. The lower limit of inte-gration jwas determined by matching the experimentally observed shift values to the simulated shift for 1.568 eV,resulting in a value of /H11011150 Å. The results of the calculation are shown in Fig. 2/H20849c/H20850. The simulated values closely follow the trends in the observed hyperﬁne shifts /H20851Fig.1/H20849b/H20850/H20852remark- ably well. VI. DISCUSSION The physical meaning of these simulations can be inter- preted in terms of how /H9003changes with photon energy; /H9003 emerges as the dominant parameter to predict OPNMR be-havior. Recall that /H9003is the occupation probability of an ORD, a prerequisite for developing nuclear polarization /H20855I Z/H20856. Our departure from existing models is that we have not as- sumed that /H9003is constant as photon energy changes.17,23We have shown in Eq. /H208495/H20850thatn, and hence /H9003, has an intricate dependence on /H9251, photon energy, and the intensity of light. In the discussion that follows, energy regimes I, II, and IIIdenoted on Figs. 1and2will be addressed. We have used the simulation to depict laser intensity, /H9003,/H20855I Z/H20856, and/H9263sas a func- tion of x, the distance from the irradiated surface, for three representative photon energies shown in Figs. 3/H20849a/H20850–3/H20849d/H20850. A. Energy regime I: 1.475–1.500 eV In regime I, the behavior of /H9003is driven by the small value of/H9251/H20849E/H20850. The intensity of light is nearly uniform throughout the entire sample and is not substantially attenuated, as seen in Fig. 3/H20849a/H20850. This causes /H9003to be uniform but small through- out the irradiation volume /H20851Fig.3/H20849b/H20850, inset /H20852, leading to little or no hyperﬁne shift /H20851Fig.3/H20849d/H20850, inset /H20852.EFFECTS OF OPTICAL ABSORPTION ON71Ga… PHYSICAL REVIEW B 75, 195207 /H208492007 /H20850 195207-5The gradual increase in /H9251as photon energy is increased in regime I is reﬂected in the attenuation of the laser intensityas a function of x, a concomitant increase in /H9003, and an in- crease in the OPNMR signal /H9014. These effects can be visual- ized by the progression of curves representing regime I tothose representing regime II in Fig. 3. In this instance, we believe these trends are indicative of light that fully pen-etrates the sample; therefore, the number of nuclei in theirradiation volume stays constant and only /H20855I Z/H20856changes. Al- though in the calculation of /H9014the number of nuclear spins is not explicitly used, this physical quantity is captured throughthe integrated area from 0 to Lof/H20855I Z/H20856./H20849Optical transmission experiments to conﬁrm complete light penetration of the sample are not currently feasible in the experimental appara-tus. /H20850 B. Energy regime II: 1.500–1.514 eV The boundary between regimes I and II is the point at which maximum OPNMR signal intensity has been observedfor the 400 /H9262m sample. This is a unique energy /H208491.500 eV /H20850for this thickness where the product of the number of nuclear spins accessed by the laser and /H20855IZ/H20856is maximized. Using the same argument for the thinned 175 /H9262m sample—where os- tensibly a smaller number of accessible nuclear spins areavailable—one would expect this characteristic energy to behigher, as observed experimentally /H20851Fig. 1/H20849a/H20850/H20852and through simulation /H20851Fig.2/H20849b/H20850/H20852. As photon energy is increased in regime II, there is a drastic change in /H9251, shown in Fig. 2/H20849a/H20850, which results in a signiﬁcant decrease in irradiation depth with a concomitantincrease in /H9003/H20849hence, /H20855I Z/H20856/H20850. These trends can be visualized in Figs. 3/H20849a/H20850–3/H20849c/H20850, when noting the changes going from regime II to III. This increase in /H9003is supported by the hyperﬁne shift dependence, as it grows in magnitude over this range of pho-ton energies shown experimentally in Fig. 1/H20849b/H20850and in the simulation, Fig. 2/H20849c/H20850. In contrast to regime I, where /H20855I Z/H20856determined the total signal observed, now in regime II the OPNMR signal arises from the interplay between /H20855IZ/H20856and irradiation depth. The total observed signal, which is the area under each of the curves in Fig. 3/H20849c/H20850/H20849please note the logarithmic scale /H20850,i s smaller for higher photon energies. It is clear from thisgraphic, despite the increase in nuclear polarization with in-creasing photon energy in regime II, the irradiation volumehas a more dramatic effect on total OPNMR signal. Corre-spondingly, the OPNMR signal intensity decreases over thisrange. C. Energy regime III: 1.514–1.568 eV At the boundary between regimes II and III /H208491.514 eV /H20850, the OPNMR signal has minimum intensity for both helicities of light. The signals observed represent a point where opticalabsorption is maximized /H20849the commonly observed excitonic feature in absorption spectra of GaAs, 8 meV below E g/H20850.A s such, the irradiation depth is exceedingly small for this pho-ton energy, and consequently, very little OPNMR signal in-tensity is observed. At even higher photon energies in regime III, the absorp- tivity of the sample is large /H20849/H1101510 4cm−1/H20850and nearly constant /H20851see Fig. 2/H20849a/H20850/H20852relative to the drastic change in /H9251/H20849E/H20850in re- gime II. With such a large /H9251/H20849E/H20850,/H9003is correspondingly large, representing full or nearly full saturation of ORD’s near the surface /H20849/H1102110/H9262m/H20850, as seen in Fig. 3/H20849b/H20850/H20849on a linear scale, /H9003 approaches 1.0 as xapproaches 0 /H20850. Consequently, the OP- NMR signal intensity is small with such a small irradiationdepth /H20851Fig.2/H20849b/H20850/H20852, but the hyperﬁne shifted signals are large /H20851Fig. 2/H20849c/H20850/H20852. These predicted trends are in good agreement with the experimental data /H20851see Fig. 1/H20849a/H20850and1/H20849b/H20850/H20852. D. Physical interpretation The question remains: what portion of the band structure is being electronically excited by photons over this rangefrom 1.475 to 1.580 eV? This question can be answered byexamining the shift of the OPNMR signals due to hyperﬁnecoupling as a function of photon energy, as shown in Fig.1/H20849b/H20850. Complementary information is provided by the OP- FIG. 3. Simulations of /H20849a/H20850laser intensity, /H20849b/H20850occupation prob- ability of an ORD /H20849/H9003/H20850,/H20849c/H20850average nuclear polarization /H20849/H20855IZ/H20856/H20850, and /H20849d/H20850hyperﬁne shift /H20849/H9263s/H20850as a function of x, the distance from the irradiated surface. Shown are three representative photon energiesof 1.487, 1.504, and 1.568 eV depicting behavior in energy regimesI, II, and III, respectively. The calculations were performed for /H9268+ polarized light. The legend in /H20849a/H20850pertains to all four panels. The insets shown in /H20849b/H20850and /H20849d/H20850are expansions of those plots.MUI, RAMASWAMY, AND HAYES PHYSICAL REVIEW B 75, 195207 /H208492007 /H20850 195207-6NMR signal intensity, an indirect measure of the irradiation volume. There is a reciprocal relationship between OPNMR signal intensities and hyperﬁne shifts. If photon energies at Egor higher are used for optical excitation, large hyperﬁne shiftsare observed, whereas small OPNMR signal intensities re-sult. Photon energies much less than E gresult in little or no hyperﬁne shift, but large OPNMR signals are obtained, pri-marily due to the irradiation volume. Doubling the laser in-tensity in this energy regime below the band edge results indoubled OPNMR signal intensity but with no increase in thehyperﬁne shift. The asymmetry evident in the experimental OPNMR sig- nal intensities and hyperﬁne shifts /H20851Figs. 1/H20849a/H20850and2/H20849b/H20850/H20852arises from two processes contributing to /H20855S Z/H20856, as shown in Eq. /H208497/H20850. As previously discussed, the ﬁrst term on the right-hand side of the equation arises from optical orientation of electrons,where the sign is determined by the selection rules of optical pumping for /H9268+and/H9268−light, while the second term on the right-hand side arises from Boltzmann population of electron spin energy levels. As illustrated in the simulation, if/H9270e T1e/H110111, then the Boltzmann contribution to /H20855SZ/H20856is not negligible and leads to an asymmetry in both the OPNMR signal inten- sity and hyperﬁne shift. However, this asymmetry can bereduced at high temperature or lower magnetic ﬁeld. Our studies have shown that judicious selection of the photon energy used for optical polarization is critical for de-veloping targeted nuclear polarization. For example, if highnuclear polarization is desired for a particular application,photon energies must be selected for high absorptivity, hereatE gor above. If nuclear polarization distributed more uni- formly throughout the sample is desired, photon energies be-low the band edge must be used.VII. CONCLUSION We have developed a physical model to explain the inten- sity dependence of the OPNMR signals as a function of laserphoton energy, used in the optical polarization of si-GaAssemiconductors. This model predicts large OPNMR signalsat energies with low absorption coefﬁcients due to deep pen-etration of the laser light into the sample. At photon energieswith large absorption coefﬁcients /H20849above the band edge /H20850, and therefore shallow irradiation volume, signiﬁcant hyperﬁneshifts in the OPNMR signal become apparent due to strongcoupling between nuclear spins and electrons localized atORD’s. These smaller irradiation volumes also are reﬂectedin less intense OPNMR signals because fewer nuclear spinsare accessible. We have modiﬁed existing optical pumpingmodels to account for a probabilistic interpretation of thepopulation of ORD’s, arising from energy-dependent opticalabsorption coefﬁcient. Several physical insights emerge fromthis new interpretation of OPNMR data: /H208491/H20850The photon en- ergy where the maximum OPNMR signal is achieved is notdue to a particular energy state, but rather is the manifesta-tion of the complex interplay between absorptivity, the ORDdensity, and sample thickness. We have demonstrated this byusing a thinned sample to shift the maximum OPNMR signalto a higher photon energy, thereby illustrating the penetrationdepth dependence. /H208492/H20850The largest hyperﬁne shifts are ob- served when photon energies are at or above E g. ACKNOWLEDGMENTS This material is based on the work supported by the Na- tional Science Foundation under Grant No. CHE-0239560and by the Army Research Ofﬁce under Contract No.W911NF-06-1-0309. We thank Larry Bretthorst for his assis-tance with the Bayesian analysis program. 1G. Lampel, Phys. Rev. Lett. 20, 491 /H208491968 /H20850. 2T. Pietrass and M. Tomaselli, Phys. Rev. B 59, 1986 /H208491999 /H20850. 3I. J. H. Leung and C. A. Michal, Phys. Rev. B 70, 035213 /H208492004 /H20850. 4S. E. Barrett, R. Tycko, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 72, 1368 /H208491994 /H20850. 5T. Pietrass, A. Bifone, T. Room, and E. L. Hahn, Phys. Rev. B 53, 4428 /H208491996 /H20850. 6T. Pietrass, A. Bifone, J. Krüger, and J. A. 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PhysRevB.95.205127.pdf	PHYSICAL REVIEW B 95, 205127 (2017) Electronic structure of FeS J. Miao,1X. H. Niu,1D. F. Xu,1Q. Yao,1Q. Y . Chen,1T. P. Ying,1S. Y . Li,1,2Y . F. Fang,3J. C. Zhang,3 S. Ideta,4K. Tanaka,5B. P. Xie,1,2,D. L. Feng,1,2,†and Fei Chen3,‡ 1State Key Laboratory of Surface Physics, Department of Physics, and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, People’s Republic of China 2Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China 3Materials Genome Institute, Shanghai University, Shanghai 200444, People’s Republic of China 4UVSOR Synchrotron Facility, Institute for Molecular Science, National Institutes of Natural Science, Myodaiji, Okazaki 444-8585, Japan 5UVSOR Synchrotron Facility, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan (Received 19 December 2016; revised manuscript received 19 February 2017; published 18 May 2017) Here we report the electronic structure of FeS, a recently identiﬁed iron-based superconductor. Our high- resolution angle-resolved photoemission spectroscopy studies show two holelike ( αandβ) and two electronlike (ηandδ) Fermi pockets around the Brillouin zone center and corner, respectively, all of which exhibit moderate dispersion along kz. However, a third holelike band ( γ) is not observed, which is expected around the zone center from band calculations and is common in iron-based superconductors. Since this band has the highestrenormalization factor and is known to be the most vulnerable to defects, its absence in our data is likely dueto defect scattering—and yet superconductivity can exist without coherent quasiparticles in the γband. This may help resolve the current controversy on the superconducting gap structure of FeS. Moreover, by comparingtheβbandwidths of various iron chalcogenides, including FeS, FeSe 1−xSx, FeSe, and FeSe 1−xTex,w eﬁ n dt h a t theβbandwidth of FeS is the broadest. However, the band renormalization factor of FeS is still quite large, when compared with the band calculations, which indicates sizable electron correlations. This explains why theunconventional superconductivity can persist over such a broad range of isovalent substitution in FeSe 1−xTex and FeSe 1−xSx. DOI: 10.1103/PhysRevB.95.205127 I. INTRODUCTION Among all the iron-based superconductors, the iron- chalcogenide FeSe has the simplest layered structure yetextraordinarily rich physics [ 1,2]. It undergoes a structural transition around 90 K [ 2], then becomes superconducting near 8 K. Under high pressure, its superconducting temperature ( T c) can be enhanced to 37 K [ 3]. When FeSe was intercalated with potassium by liquid gating [ 4] or dosed with potassium on the surface [ 5], theTccould reach 46 K. Most remarkably, FeSe monolayer ﬁlm grown on SrTiO 3(001) substrate even exhibits Tcas high as 65 K [ 6–8]. Isovalent substitution is a particularly effective way of tuning the properties of FeSe. The Tcof FeSe 1−xTexcan reach 14 K [ 9], while that of FeSe 1−xSxis ﬁrst enhanced by dilute S substitution, then decreases upon further substitution [ 10]. Recently, it was found that FeS, in which Se is totallysubstituted by S, is still superconducting with a T cof about 4.5 K [ 11,12]. During this process, the nematic (orthorhombic) phase transition temperature ( TN) in FeSe is suppressed, and there is no structural or nematic transition in FeS [ 10,13–15]. The presence of superconductivity in FeS is quite un- expected, since S substitution is expected to signiﬁcantlyreduce the electron correlations, which are crucial for theunconventional superconductivity [ 16]. For example, it has been shown that S substitution enhances the bandwidth, andthus reduces the correlation effects in K xFe2−ySe2−zSz, which is not superconducting for z/greaterorequalslant1.4[17]. To address this issue, it bpxie@fudan.edu.cn †dlfeng@fudan.edu.cn ‡chf001@shu.edu.cnis important to examine the electronic structure of FeS. More- over, the gap structure of FeS is currently under debate. Muonspin rotation ( μSR) measurements favor the fully gapped superconductivity coexisting with a low-moment magneticstate at low temperature in polycrystalline FeS [ 18,19]. On the contrary, thermal conductivity and speciﬁc heat measurementssupport the existence of a nodal gap [ 20,21], while scanning tunneling spectroscopy indicates a highly anisotropic or nodalsuperconducting gap structure [ 22]. Furthermore, a theoretical calculation predicted d x2−y2pairing symmetry in FeS [ 23]. A thorough understanding of the electronic structure of FeS willcontribute to these discussions. In this paper, we study the electronic structure of single- crystalline FeS by performing high-resolution angle-resolvedphotoemission spectroscopy (ARPES). There are two elec-tronlike bands around the Brillouin zone (BZ) corner andtwo holelike bands ( αandβ) around the BZ center. The Fermi surface exhibits modest k zvariation. Moreover, we deduce that the absence of the third holelike band with thed xyorbital character, which was discovered in many other iron-based compounds [ 24], is likely due to strong defect scattering. Compared with density functional theory (DFT)calculations [ 25], the αandβbands near the Fermi energy (E F) are renormalized by factors of 3 and 2.5, respectively, indicating sizable electron correlations. By studying therelation between T cand the βbandwidth of various isovalently substituted FeSe compounds, we ﬁnd that the overall evolutionofT cis closely related to the strength of electron correlations. II. EXPERIMENT AND RESULTS So far, FeS single crystals could not be synthesized from elemental Fe and S as for FeTe and FeSe [ 11]. Instead, the 2469-9950/2017/95(20)/205127(6) 205127-1 ©2017 American Physical SocietyJ. MIAO et al. PHYSICAL REVIEW B 95, 205127 (2017) (c) (d) s p -0.4 -0.2 0.0 0.2 0.4 1.6 1.4 1.2 1.0 0.8#1 #1#2 #2 Intensity (arb. units) s pp 1.6 1.4 1.2 1.0 0.8 #1 #1#1 #222 #2#222 -0.3-0.2-0.10.0 -0.3-0.2-0.10.0 -0.4 -0.2 0.0 0.2 0.4p sp s#1 #1#2 #2α β η δβδ αηβ αδ η(a) (e)E - E F (eV) k (A-1) //k (A-1) //EF EF-60 meV -60 meV Μ Γ High LowI (k, E) #2 c+ #1Μ ΓX δ/ηα βΜ ΓX Μ Γ-0.2 -0.1 0.0 Intensity (arb. units) E - E F (eV)Μ Μδ ηs p s(b)p FIG. 1. (a) Photoemission intensity distribution (left panel) integrated over ( EF−10 meV , EF+10 meV) for an FeS single crystal taken with right circular polarized (C +) photons and the corresponding Fermi surface topology (right panel). Here ¯/Gamma1and ¯Mstand for the two-dimensional projected Brillouin zone (BZ) center and corner, respectively. (b) The experimental setup for the polarization-dependentARPES at SSRL. The emission plane is deﬁned by the analyzer slit and the sample surface normal, which is perpendicular to the incident plane. Although p-polarized photon exhibits partial even symmetry with respect to the emission plane, not all the 3 dorbitals with even symmetry could be clearly resolved along the high-symmetry direction due to that the polarization component of p-polarized photon in parallel to the emission plane is along the zaxis. (c) Photoemission intensity distributions of the two cuts no. 1 and no. 2 identiﬁed by solid purple and yellow lines, respectively, in panel (a) taken with s-a n dp-polarized photons as indicated. (d) The momentum distribution curves (MDCs) in the corresponding rectangular areas of panel (c). (e) The energy distribution curves (EDCs) around the BZ corner (cut no. 2) taken with s-a n d p-polarized photons as indicated. The data around the BZ center and corner were taken with 24 eV and 30 eV photons, respectively, at SSRL. FeS single crystals used in our studies were synthesized by deintercalation of K from K 0.8Fe1.6S2precursor with a hydrothermal method [ 20]. Since the starting material is K0.8Fe1.6S2, whose Fe vacancy concentration is known to be quite high, the hydrothermal deintercalation will likely makesamples with Fe 1−xS1−y. Here, the xandyshould be quite close, since the Fe:S ratio is almost 1:1 within the experimentalerror±1% according to the chemical analysis [ 20]. Moreover, our samples exhibit quite high residual resistivity ratios(RRR) of about 40 [ 26], which is better than the typical RRR of FeCh (Ch =Te, Se, S) and close to the one as reported in Refs. [ 27–29]. Therefore, the xandyshould be quite small. This is also consistent with the sizable T c of the sample, since a few percent of defects at the Fe sites would normally kill the superconductivity. Taking FeSe asan example, only 2% Cu doping on the Fe site would fullysuppress the superconductivity [ 30]. Therefore, for simplicity, we still use the nominal chemical formula of FeS hereafter.The photoemission data were collected with a Scienta R4000electron analyzer at Beamline 5-4 of the Stanford SynchrotronRadiation Laboratory (SSRL) and an MBS A-1 analyzer atBeamline 7U of the ultraviolet synchrotron orbital radiationfacility (UVSOR). Both beamlines are equipped with anelliptically polarized undulator which can switch the photonpolarization among horizontal, vertical, and circular modes.The overall energy resolution is set to be 15 meV or better,and the typical angular resolution is 0 .3 ◦. The samples were all cleaved in situ and measured around 10 K under ultrahigh vacuum better than 3 ×10−11torr. Sample aging effects were carefully monitored to ensure they did not cause any artifactsin our analyses and conclusions. The photoemission intensity distribution of FeS around E Fis shown in Fig. 1(a). The spectral weight is mainly located around the BZ center and corner. To clearly resolve theband structure near E F, we measured along the two cuts no. 1 and no. 2 crossing the BZ center and corner with boths- andp-polarized photons as shown in Fig. 1(c), and these two polarizations correspond to odd and even symmetries withrespect to the incident plane deﬁned by the incoming lightand outgoing photoelectron, respectively. Since the emissionplane is perpendicular to the incident plane as shown inFig. 1(b), the odd and even symmetries are nearly inverted for these two planes. The corresponding orbitals, which arevisible in s- andp-polarized photons, respectively, are listed in Table Ion the basis of both experimental studies and theoretical calculations [ 31–33]. According to the momentum distribution curves (MDCs) and the energy distribution curves(EDCs) in Figs. 1(d) and 1(e), two holelike bands ( αand β) are located around the BZ center, and two electronlike bands ( ηandδ) are located around the BZ corner. Based on the polarization of incident photons, αandηexhibit even symmetry, while βandδexhibit odd symmetry. Based on the identiﬁed band dispersions, two circular hole Fermi pocketsaround the BZ center are determined as shown in the rightpanel of Fig. 1(a). Fermi crossings of the ηandδbands are too close to be distinguished, but they would contribute totwo nearly degenerate electron Fermi pockets around the BZ TABLE I. Possibility to detect the 3 dorbitals along two high- symmetry directions with s-a n dp-polarized photons, respectively, in respect to the incident plane by polarization-dependent ARPES experiments and calculations on the ARPES setup shown in Fig. 1(b) [31–33]. High-symmetry 3 dorbitals direction dxz dx2−y2 dz2 dyz dxy /Gamma1−M(s)√√ √ /Gamma1−M(p) Weak Weak√√ √ /Gamma1−X(s)√√ √ √ /Gamma1−X(p)√√ √ √Weak 205127-2ELECTRONIC STRUCTURE OF FeS PHYSICAL REVIEW B 95, 205127 (2017)Intensity (arb. units)2.02.5 0.0 1.0αβηδ 32 eV 30 eV 14 eV16 eV18 eV20 eV22 eV24 eV26 eV28 eV32 eV 30 eV 14 eV16 eV18 eV20 eV22 eV24 eV26 eV28 eV34 eV -0.4 -0.2 0.2 0.4 Μ -0.4 -0.2 0.2 0.4 Μ -0.4 -0.2 0.2 0.4Γ -0.4 -0.2 0.2 0.4Γ k (A-1) //(a) (b) k ( π/a) //2kz (2π/c) ΓZ MA α β η/δEF, s EF, p α β η δΓΜ ΓEF, p EF, sΜ#4 #3 #3#4 FIG. 2. (a) MDCs for the α,β,η,a n dδbands around EFas a function of photon energy, shifted for clarity. The insets indicate the corresponding cuts in the BZ. They are distinguished by using either sorppolarization. (b) The experimental Fermi surface cross section in theZ-/Gamma1-M-Aplane. Fermi surfaces (solid lines) are determined based on the experimental Fermi crossings (points). All data were taken at UVSOR, where the emission plane is parallel to the incident plane. corner that are mutually orthogonal, as predicted by the band calculations on FeS [ 25]. Since the electronic structures of iron-chalcogenides usu- ally contain three holelike bands around the BZ center [ 24], the so-called γband having dxyorbital symmetry is missing here. Among the ﬁve bands near EFin the iron-based superconductors, the γband is by far the most sensitive to defects [ 24], and its renormalization factor in FeSe is around 9[34], which is much higher than those of the other bands. With increased Te or Co concentrations, the γbands of FeSe 1−xTex, NaFe 1−xCoxAs, and LiFe 1−xCoxAs all quickly broaden, indicating the γband becomes more incoherent, and even become featureless in some cases [ 24]. For sulfur- substituted FeSe, γ“disappears” for sulfur content higher than 15% [ 10], i.e., its intensity submerges into the background, and becomes undetectable. Furthermore, the defects on ironsites would cause even stronger scatterings than those onpnictogen /chalcogen sites. Only 2% Cu doping on the Fe site in FeSe would fully suppress the superconductivity [ 30]. As the hydrothermal deintercalation process in the synthesis mightretain some of the Fe vacancies in the FeS layer, it is likelythat 1% or at most 2% defect on the iron site in FeS wouldinduce strong scattering that could both strongly suppress theintensity of γand severely broaden γ, so that no feature could be resolved. A rather ﬂat band around 170 meV below E Fis observed around the BZ center taken with p-polarized photons in Fig.1(c). This ﬂat band in iron-based superconductors usually exhibits dz2orbital character. Its presence here is due to the particular experimental geometry at the Beamline 5-4end-station at SSRL, where p-polarized photons contain a polarization component along the zaxis [ 31]. To illustrate the k zdependence of the Fermi crossings, Fig. 2(a) plots the MDCs along the /Gamma1−Mdirection for these four bands near EF, taken with both s- andp-polarized light and many different photon energies. The Fermi surface crosssection in the /Gamma1−Z−A−Mplane can be traced by the peak positions of the MDCs [Fig. 2(b)]. The Fermi momentum ( k F) ofαﬁrst shrinks and then expands along the /Gamma1−Zdirection, in contrast to that of the βband. Meanwhile, the kFs of bothηandδshrink along the M-Adirection, and they almost always coincide with each other within the experimentalresolution. In general, the Fermi surface of FeS exhibitsa quasi-two-dimensional behavior, consistent with quantumoscillation data [ 35]. Compared with BaFe 2(As 0.7P0.3)2and Ba1−xKxFe2As2, FeS is more two dimensional [ 36]. Based on Luttinger’s theorem, we estimate the electron concentrationto be 0.12 electrons per unit cell through the volume ofthe three-dimensional Fermi surface of the four detectedbands [ 37]. Since FeS should have electron-hole balanced Fermi surface volumes like FeSe [ 38] and the compositions ratio of Fe:S here is almost 1:1, the electron and hole pocketsshould normally have the same total volume. Therefore, thereshould be a missing Fermi pocket with 0.12 holes, which canbe naturally attributed to the γband. As shown in Fig. 1(d), the dispersion of βexhibits parabolic behavior. By ﬁtting its dispersion below E Fby a quadratic curve [Fig. 3(a)], we estimate the position of the βband top to be about 10 meV above EF, and its effective band mass to be−1.74me(meis the free electron mass). Meanwhile, the βband bottom overlaps with the /epsilon1band as shown in both photoemission intensity distribution [Fig. 3(a)] and the EDCs [Fig. 3(b)]. The overlap occurs around 180 meV below EF.I f we take the full width at half maximum of the EDC peaks of /epsilon1 as the error bar, and take the overlapping region as the βband bottom, the βbandwidth is about 190 ±25 meV . Similarly, we estimate the effective band mass of the αband is about −1.47me. Figure 3(c) reproduces the band calculations of FeS from Ref. [ 25]. Although there are subtle differences, the observed band dispersions qualitatively agree with thecalculations, for example, αandβare almost degenerate around the zone center in both the data and the calculations. Bycomparing the band masses obtained in both the data and thecalculations, we obtain a renormalization factor of about 2.5for the βband, and 3 for α, which indicate sizable electronic correlations in FeS. To position FeS in the bigger picture of the so-called “11” series of bulk iron chalcogenide superconductors, photoe-mission intensity distributions of FeSe 1−xTexand FeSe 1−xSx along the /Gamma1−Mdirection and their corresponding EDCs 205127-3J. MIAO et al. PHYSICAL REVIEW B 95, 205127 (2017) -0.3-0.2-0.10.0 1.5 1.0 0.5 0.0 -0.5s -0.3-0.2-0.10.0pE - E F (eV)(a) (c) k ( π/a) //2Γ Μ-0.3-0.2-0.10.0E - E F (eV)k (A-1) // αβ βδ η ε Intensity (arb. units) -0.3 -0.2 -0.1 0.0(b) β ΓΜ E - E F (eV)ε High LowI (k, E) ΓΜ0.0 -1.0-0.5αβγδ/η ε FIG. 3. (a) Photoemission intensity distributions along the /Gamma1− Mdirection taken with s-a n d p-polarized photons, respectively. (b) Corresponding energy distribution curves (EDCs) of data in panel (a) taken with p-polarized photons. (c) Left panel: the band structure of FeS measured by ARPES. Right panel: the calculatedFeS band structure along /Gamma1−Mreproduced from Ref. [ 25]. The calculation of the electronic structure was performed within the local- density approximation (LDA) with the general potential linearizedaugmented plane-wave (LAPW) method, including local orbitals. All the ARPES data were taken at SSRL with 30 eV photons.for various substitutions are collected in Fig. 4. All the data here were taken in the odd geometry in respect to theemission plane, to emphasize βband. Since the βbandwidth can be easily estimated, it is taken as a characterization ofthe correlations in these materials. Figure 4shows that the bandwidth of βincreases monotonically from the tellurium end in FeSe 1−xTexto the sulfur end of FeSe 1−xSx, as expected from the decreasing bond length [ 25]. Consistent with the evolution of the bandwidth, the effective mass of βnear the BZ center, obtained by ﬁtting the parabolic curves in Fig. 4(a), decreases monotonically from the tellurium end in FeSe 1−xTex to the sulfur end of FeSe 1−xSx. Here, the FeS spectra were taken with a different polarization from the other ones. Dueto the particular experimental geometry in Fig. 1(b), the ﬂat band observed in FeS is rather strong and thus blurs the β band. Moreover, different photon energies used for the datacollection also affect the intensity of the spectral featuresaround the Fermi level. In Fig. 5,t h eβbandwidths are plotted onto the phase diagram of FeSe 1−xTexand FeSe 1−xSx. Although more data points are required to make a comprehensive case, the existingdata fall on a line, as a function of substitution. From theFeTe end, bulk superconductivity emerges when the bandwidthexceeds a certain value, and persists all the way to the FeS end.TheT cof FeSe 1−xTexand FeSe 1−xSxis enhanced at ﬁrst, and then generally weakened with increased bandwidth. However,the relation between T cand substitution is not monotonic. The superconductivity strengthens in the lightly S-substitutedregime, likely due to the enhanced ( π,0) spin ﬂuctuations related to the nematic order or to subtle Fermi surface topologyeffects [ 10]. -0.20.0 Intensity (arb. units)E - E F (eV) 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 k (A-1) //(a) (b) -0.2 0.0β -0.2 0.0β E - E F (eV)-0.2 0.0β -0.2 0.0β -0.2 0.0βs s s s p FeSβββββ FeSe 0.3Te0.7 FeSe 0.4Te0.6 FeSe 0.93S0.07 FeSe 0.1Te0.9FeSe 0.1Te0.9 FeSe 0.3Te0.7 FeSe 0.4Te0.6 FeS FeSe 0.93S0.07 High LowI (k, E) Μ ΓΜ ΓΜ ΓΜ ΓΜ Γ FIG. 4. (a) Photoemission intensity distributions along the /Gamma1−Mdirection taken with s-polarized photons for FeSe 0.1Te0.9,F e S e 0.3Te0.7, FeSe 0.4Te0.6,F e S e 0.93S0.07,a n dp-polarized photons for FeS. (b) Corresponding EDCs of the data in panel (a). All the data except those of FeS were extracted from Refs. [ 13,24]. 205127-4ELECTRONIC STRUCTURE OF FeS PHYSICAL REVIEW B 95, 205127 (2017) SC80 60 40 20 0T (K) substitution xFeSe 1-xTex FeSe 1-xSxBandwidth of β (meV) 0.060120180240 0.5 1.0 1.0 0.5100 0BCAF Non-bulk SCNO FIG. 5. Phase diagram of Tcandβbandwidths as a function of the chalcogen content in both Te- and S-substituted FeSe. The supercon- ducting, bicollinear antiferromagnetic, and nematic-ordered phasesare abbreviated as SC, BCAF, and NO, respectively. The bandwidths of FeSe 1−xTexand FeSe 0.93S0.07were taken from Refs. [ 13,24], and the phase diagram, including the Tc’s for FeSe 1−xTexand FeSe 1−xSx were taken from Refs. [ 10,15,39,40]. Two regions of the SC dome marked by black dashed lines are inferred based on interpolation, in the absence of any report on these substitutions. III. DISCUSSIONS It has been shown for almost all the iron-based super- conductors that their phase diagrams can be understoodfrom the bandwidth perspective [ 17,24]. In particular, since isovalent substitution does not alter carrier density or theFermi surface [ 17,38], the strength of electron correlations, which can be represented by the inverse of bandwidth, wouldcontrol the superconductivity of FeSe 1−xTexand FeSe 1−xSx. The measured βbandwidth of FeS is about 40% larger than that of FeSe, which suggests weaker correlation effects in FeSthan in FeSe. On the other hand, the sizable renormalizationfactors of FeS bands indicate that the electrons in FeS stillexperience signiﬁcant interactions amongst themselves andwith bosonic excitations, such as magnons and phonons.This might explain the robustness of superconductivity inFeSe 1−xSx. This is reminiscent of Ba 1−xKxFe2As2, in which strong correlations exist throughout the entire doping range,allowing superconductivity to persist [ 24]. It is also worthwhile to point out that in both FeSe 1−xSx and K xFe2Se2−ySy,Tcdecreases with S substitution, although these two families have different Fermi surface topologies. Thesuperconducting range of βbandwidth is 100–200 meV for RbxFe2Se2−zTezand K xFe2Se2−ySy[17]. Superconductivity in FeSe 1−xTexand FeSe 1−xSxalso emerges for a bandwidth of∼100 meV , but is still not suppressed at 190 meV (FeS end member). Our data clearly imply that the superconductivity in FeS can survive without coherent quasiparticles in the γband. However, the incoherent spectral weight of γmight still contribute to the zero-energy excitations at low temperatures.This could explain the nodal-gap-like behavior observed inthermal conductivity and speciﬁc heat measurements [ 20,21]. On the other hand, because the incoherent γband would not contribute to the superﬂuid response, if the superconductinggaps in the other bands are nodeless, μSR could still observe an overall nodeless-gap behavior for FeS [ 18,19]. Our results may thus help resolve the contradictory reports by these techniqueson the FeS gap structure. IV . CONCLUSIONS In summary, we have studied the electronic structure of superconducting FeS. Two holelike bands and two electronlikebands around the BZ center and corner, respectively, have beenresolved. The third holelike band near /Gamma1(theγband) may be too strongly scattered by defects to be observed, whichmay help resolve the current debate on the gap structure.Thek zdispersions of the Fermi surfaces of FeS exhibit quasi-two-dimensional behavior, and the two electronlikebands are almost degenerate around E Fwithin our resolution. Using the βbandwidth as an indication of correlation strength, we illustrate the evolution of Tcwith electron correlation in FeSe 1−xTexand FeSe 1−xSx, and explain the robustness of superconductivity which still exists in the end member FeS.The observed electronic structure of FeS establishes a concretefoundation for further theoretical calculations and will help inunderstanding its superconducting properties. ACKNOWLEDGMENTS The authors thank Dr. D. H. Lu for the experimental assistance at SSRL. This work is supported in part by theNational Natural Science Foundation of China (Grants No.11604201 and No. 11574194), National Key R&D Programof the MOST of China (Grant No. 2016YFA0300203), OpenProject Program of the State Key Lab of Surface Physics(Grant No. KF2016 _08), Fudan University, Science Challenge Program of China, and Shanghai Municipal Science andTechnology Commission. [1] F. C. Hsu, J. Y . Luo, K. W. Yeh, T. K. Chen, T. W Huang, P. M. Wu, Y . C. Lee, Y . L. Huang, Y . Y . 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PhysRevB.86.104503.pdf	PHYSICAL REVIEW B 86, 104503 (2012) Electronic structure of single-crystalline Sr(Fe 1−xCox)2As2probed by x-ray absorption spectroscopy: Evidence for effectively isovalent substitution of Fe2+by Co2+ M. Merz,1,F. Eilers,1,2Th. Wolf,1P . Nagel,1H. v. L ¨ohneysen,1,3and S. Schuppler1 1Institut f ¨ur Festk ¨orperphysik, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany 2Fakult ¨at f¨ur Physik, Karlsruhe Institute of Technology, 76031 Karlsruhe, Germany 3Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany (Received 20 October 2011; revised manuscript received 17 August 2012; published 4 September 2012) The substitutional dependence of valence and spin-state conﬁgurations of Sr(Fe 1−xCox)2As 2(x=0, 0.05, 0.11, 0.17, and 0.38) is investigated with near-edge x-ray absorption ﬁne structure at the L2,3edges of Fe, Co, and As. The present data provide direct spectroscopic evidence for an effectively isovalent substitution of Fe2+by Co2+, which is in contrast to the widely assumed Co-induced electron-doping effect. Moreover, the data reveal that not only does the Fe valency remain completely unaffected across the entire doping range, but so do the Coand As valencies as well. The data underline a prominent role of the hybridization between (Fe,Co) 3 d xy,dxz, dyzorbitals and As 4 s/4pstates for the band structure in A(Fe 1−xCox)2As 2and suggest that the covalency of the (Fe,Co)-As bond is a key parameter for the interplay between magnetism and superconductivity. DOI: 10.1103/PhysRevB.86.104503 PACS number(s): 74 .70.Xa, 74 .25.Jb, 74.62.Dh, 78 .70.Dm I. INTRODUCTION Since the discovery of superconductivity in iron-based materials1with transition temperatures Tcup to 55 K,2the iron pnictides and chalcogenides have been intensely studied,and the understanding of the physical properties of thesesystems has considerably advanced. As for other materialssuch as heavy-fermion systems and high- T ccuprates, super- conductivity emerges in the vicinity of a magnetic instability.Moreover, similar to the cuprates, the superconducting phaseboundary of most iron pnictides has a domelike shape andsuperconductivity appears when the antiferromagnetic (AFM)phase is signiﬁcantly reduced upon either substitution on thetransition metal and on the pnictide site or upon externalpressure. Even though it has been established that the edge-sharing FeAs 4tetrahedra are the structural key ingredient for the physical properties of the pnictides, the details ofthe superconducting pairing mechanism still remain elusive.Meanwhile, however, many studies strongly suggest thatdistinct nesting properties are important for the magnetic aswell as for the superconducting characteristics. For the stoichiometric parent compound, the good nesting properties, which are closely connected with interband scat-tering between Fermi-surface hole pockets at the /Gamma1point of the Brillouin zone and electron pockets at the Xpoint, are obviously responsible for the development of a spin densitywave (SDW) and AFM order at low temperature. 3–8Upon doping (or upon application of pressure) the shape and sizeof the hole and electron pockets are modiﬁed, 3,7,9the SDW is suppressed, and speciﬁc spin ﬂuctuations associated withclose-to-nesting conditions are assumed to play a decisive role for the superconducting pairing mechanism. Whethermagnetism and superconductivity do indeed coexist in thepnictides is still under intense debate, 10,11as is the related question of whether the same electrons establish AFM orderand are responsible for superconductivity. 6Furthermore, the band structure is very complicated as well in these systemssince it results from a compromise between the covalent Fe-Asbonds (for the tetrahedral coordination the d xy,dxz, anddyzstates are higher in energy than the dx2−y2and the d3z2−r2 states) and the square planar Fe-Fe coordination (where the energy sequence descends as dx2−y2−dxy−dxz,dyz−d3z2−r2 in an ionic picture). As a consequence, all bands with Fe 3dcharacter cross the Fermi energy, thereby reﬂecting the itinerant character of the system. An interesting example is the AFe2As 2system (where Ade- notes Ca, Sr, or Ba) which exhibits two-band superconductivity for the following cases: (i) an external pressure of several GPa is applied to the samples,12–14(ii) As is partially substituted by isovalent P ,15,16(iii) Fe is partially replaced by isovalent Ru,17–19(iv) hole doping achieved by partial replacement of A2+by K+,20,21and (v) Fe is partially substituted by Co, Ni, Co/Cu mixtures, Rh, Pd, Ir, or Pt.22–27Although the valencies of Co, Ni, Cu, Rh, Pd, Ir, and Pt in AFe2As 2are still unclear, “electron doping” is generally assumed for the substitutions under (v). In the framework of the so-called virtual crystalapproach, a rigid-band shift of E Finduced by electron doping is considered.28Density-functional theory (DFT) shows that the main effect of substitution is not on the density of statesbut rather on the relative size of the electron and hole pockets. 5 More recent DFT calculations even indicate that Co does notdope the system with electrons at all but is isovalent to Fe 2+ and solely acts as a random scatterer.29In this case, the effect of Co substitution would rather lead to a topological change of the Fermi surface which destabilizes magnetism in favor ofsuperconductivity. To shed more light on the substitution-dependent changes in the electronic structure of the iron pnictides and to scrutinizethe supposed electron doping by analyzing the valency ofthe relevant chemical elements, we have investigated theSr(Fe 1−xCox)2As 2system ( x=0, 0.05, 0.11, 0.17, and 0.38) with near-edge x-ray absorption ﬁne structure (NEXAFS) attheL 2,3edges of Fe, Co, and As using linearly polarized light.30Since it has been shown that iron pnictide samples may suffer severely from iron oxide contamination,31the OKedge was investigated as well. For completeness the results obtainedat the O Kedges are shown and discussed in Appendix A. 104503-1 1098-0121/2012/86(10)/104503(7) ©2012 American Physical SocietyM. MERZ et al. PHYSICAL REVIEW B 86, 104503 (2012) II. EXPERIMENT All NEXAFS measurements were performed at the In- stitut f ¨ur Festk ¨orperphysik beamline WERA at the ANKA synchrotron light source (Karlsruhe, Germany). All spectrawere taken simultaneously in bulk-sensitive ﬂuorescence yield(FY) and in total electron yield (TEY) on single crystals thatexhibit a mirrorlike shiny surface. Photon energy calibrationto better than 30 meV was ensured by adjusting the Ni L 3 peak position measured on a NiO single crystal before and after each NEXAFS scan to the established peak position.32 The spectral resolution was set to 0.3 eV for the Fe and CoL 2,3edges. While the in-plane spectrum is obtained for a normal- incidence alignment, i.e., for a grazing angle θof 0◦, the out- of-plane spectrum is determined by measuring in a grazing-incidence setup with a grazing angle of 65 ◦. The FY spectra are corrected for the self-absorption and saturation effects inherentto the FY method (as discussed in Appendix B). Utilizing multiplet calculations, the conﬁguration of the correspondingspin and valence states of Fe and Co is determined forthe investigated doping contents. 33Sr(Fe 1−xCox)2As 2single crystals were grown from self-ﬂux in glassy carbon crucibles asdescribed elsewhere. 34,35The composition of the samples was determined using energy dispersive x-ray spectroscopy andwas veriﬁed by the size of the respective background-corrected edge jump in our NEXAFS experiments. Using superconducting quantum interference device (SQUID) magnetometry, the superconducting transition tem-perature was determined to T c≈12 K for the nearly optimally doped x=0.11 sample, to ≈9 K for the slightly overdoped x=0.17 sample, and to ≈7 K for the strongly overdoped x=0.38 sample. No superconducting transition down to 4 K was observed for the undoped x=0 and the underdoped x=0.05 samples. Thus, the studied samples span the entire range from undoped up to highly overdoped. III. RESULTS AND DISCUSSION In Fig. 1the Fe L2,3NEXAFS spectra of Sr(Fe 1−xCox)2As 2 (x=0, 0.05, 0.11, 0.17, and 0.38) measured at 300 K in FY are depicted for (a) normal and (b) grazing incidence.The absorption spectra correspond to ﬁrst order to transitionsof the type Fe 2 p 63d6→Fe 2p53d7. They consist of two manifolds of multiplets located around 708 eV ( L3) and 721 eV (L2) and separated by the spin-orbit splitting of the Fe 2 p core level. In addition, the spectrum of a degraded undopedsample is included in Fig. 1(a) (topmost panel), exhibiting the respective additional peak positions of iron oxide around FIG. 1. (Color online) Comparison of the (a) normal- and (b) grazing-incidence Fe L2,3NEXAFS spectra of Sr(Fe 1−xCox)2As 2(x=0, 0.05, 0.11, 0.17, and 0.38) recorded at 300 K in FY . The spectral shape of both edges is unaffected upon doping. For clarity, the spectra are vertically offset. In addition, the spectrum of a deteriorated iron-oxide-containing sample is included in (a) (topmost panel), thereby exhibitin g the respective peak positions of the iron oxide. For more direct comparison of the doping dependence, the spectra are plotted on top of eachother in the next-to-lowest panel. The multiplet calculations show that the spectra can be described reasonably well for tetrahedrally coordinated Fe 2+. As a representative, in the topmost panel in (b) the anisotropy between normal (line) and grazing incidence (symbols) is depicted for the x=0 sample. 104503-2ELECTRONIC STRUCTURE OF SINGLE-CRYSTALLINE ... PHYSICAL REVIEW B 86, 104503 (2012) FIG. 2. (Color online) Comparison of the (a) normal and (b) grazing incidence Co L2,3NEXAFS spectra of Sr(Fe 1−xCox)2As 2(x=0, 0.05, 0.11, 0.17, and 0.38) taken at 300 K in FY (topmost panel) and in TEY . The spectral shape of both edges is unaffected upon doping. For clarity,the TEY spectra are vertically offset. The multiplet calculations show that the spectrum can be described reasonably well for tetrahedrally coordinated Co 2+. As a representative, in the lowest panel in (b) the anisotropy between normal (dotted line) and grazing incidence (symbols) is depicted for the x=0.17 and the x=0.38 samples. ≈710 and ≈723 eV . It is obvious from the absence of these two oxide-related features in all other spectra in Fig. 1that iron oxide does not play an important role for the bulk propertiesof the investigated series. [The TEY data of most samples (notshown) are similar to the FY spectrum of the degraded samplewhich reﬂects the fact that the surface layer contains someiron oxide phase as well (see also discussion of O Kedge in Appendix A).] It is also evident from the FY data that the spectral shape of both edges remains unaffected upon Fe substitution by Co,as do the respective energetic positions of the onset energyof the L 3andL2edges and of all other spectral features (such as peaks or shoulders). This is an important ﬁndingsince it establishes that the Fe valency is unchanged acrossthe entire doping series. It should be noted that for electrondoping, changes in the spectral shape are expected togetherwith a spectral shift /Delta1E to lower energies which according to multiplet calculations scales in this energy range almostlinearly with the doping content xas/Delta1E≈x×1.5e V . If the assumption of electron doping were valid, 36,37and with the good energy precision in our experiment, it shouldbe possible to observe—at least for the highly doped x= 0.17 and 0.38 samples—a modiﬁed spectral shape and a signiﬁcant energy shift (of ≈0.26 and 0.57 eV , respectively). Such a shift was indeed found in Ref. 38for NEXAFS on electron-doped LaFeAsO 1−xFx—but does not exist here inthe Sr(Fe 1−xCox)2As 2system with partial replacement of Fe by Co. Furthermore, our multiplet calculations show that thespectrum can be described reasonably well for tetrahedrallycoordinated Fe 2+in a high-spin (HS) conﬁguration. To obtain the simulated data, the code developed by Thole, Butler,and Cowan 39–41and maintained and further developed by de Groot42was used to calculate spectra for different values of the crystal-ﬁeld splitting /Delta1CFand of the charge-transfer energy /Delta1c, and by taking the Hund’s rule exchange interaction into account. Charge-transfer effects were included for Fe2+by admixing transitions of the type 2 p63d7L→2p53d8L, where Ldenotes a hole at the arsenic ligand.43Consistent with the TEY spectra in Ref. 31a certain anisotropy between normal and grazing incidence data can, independent of the Co content,be inferred from Fig. 1with the higher density of states found for the in-plane spectra. In Figs. 2(a) and 2(b) the normal- and grazing-incidence Co L 2,3NEXAFS spectra of Sr(Fe 1−xCox)2As 2(x=0.05, 0.11, 0.17, and 0.38) recorded at 300 K in FY and TEY are displayed.As for the Fe L 2,3edge, they consist of two manifolds of multiplets, located around 778 eV ( L3) and 794 eV ( L2). The comparison between the bulk-sensitive FY [topmost panel in(a) and (b)] and the TEY data shows a close resemblancebetween the respective spectra implying the absence of acobalt-oxide surface layer. Due to the much better statisticsof the TEY data we therefore focus on the TEY spectra 104503-3M. MERZ et al. PHYSICAL REVIEW B 86, 104503 (2012) in the following. Similarly to the Fe edge (see above), no concentration-dependent spectral changes or energetic shiftsare observed at the Co L 2,3edge for the x=0.05, 0.11, 0.17, and 0.38 samples. Again a certain anisotropy between normaland grazing incidence data can be attributed to the higherdensity of states found for the in-plane spectra. Compared tothe results at the Fe L 2,3edge, however, the anisotropy between normal and grazing incidence data is slightly reduced. Con-sistent with angle-resolved photoemission spectroscopy, 44this ﬁnding points to an increasingly three-dimensional characterof the electronic structure upon Fe substitution by Co. The factthat the spectral shape of the Co L 2,3spectra is completely independent of x, as is the energetic position of the onset energy of the L3andL2edges and of all other spectral features, unambiguously demonstrates that the Co valency remains thesame throughout the entire investigated doping series. To derive a reliable estimate of the Co valency and the corresponding spin state, we have performed multipletcalculations for Co 2+and Co3+in tetrahedral and square- planar coordination, respectively.43The spectral shape of square-planar coordinated Co2+HS and Co3+low spin (LS) (not shown) does not show any resemblance to ourexperimental data. Hence, this conﬁguration can be excludedright away. Furthermore, a Co 3+LS in tetrahedral coordination is stabilized only for an unusually strong crystal-ﬁeld splittingof/Delta1 CF/greaterorsimilar3.5 eV . Therefore, only the simulated spectra of Co2+and Co3+HS in tetrahedral coordination and Co3+ HS in square-planar coordination are considered in Fig. 2. By aligning the calculated spectra to the energy positionof the main structure of the measured L 3peak position, it is evident that only Co2+in tetrahedral coordination (solid line) reproduces reasonably well the measured spectra forx=0.05, 0.11, 0.17, and 0.38. Even the shoulder at the L 3 edge around 779.5 eV and the small wiggles above 780 eV are adequately described for Co2+in tetrahedral coordination. For Co3+HS in tetrahedral (dotted line) or square-planar coordination (dashed line), additional satellites appear in thecalculated spectrum at ≈776, 777, and 781 eV . Furthermore, the energy position of both the small peak around 782 eVand of the complete L 2edge cannot be reconciled with the experimental data.45It is interesting to note that already the experimental Fe L2,3spectra in Fig. 1are best described by means of a tetrahedrally coordinated Fe atom. It should also bementioned that, in order to properly describe the experimentaldata, the crystal-ﬁeld parameter of the multiplet calculationsincreases from /Delta1 CF≈1.0 eV for FeAs 4to≈1.4e Vf o rC o A s 4, while the charge-transfer energy decreases systematically from/Delta1 c≈3.0e Vt o ≈1.0e V .43Both effects point to a more pronounced covalent character of the Co-As bond. To complement the electronic structure derived from the Fe and Co L2,3edges, the normal- and grazing-incidence As L3NEXAFS spectra of Sr(Fe 1−xCox)2As 2(x=0, 0.05, 0.11, 0.17, and 0.38) taken at 300 K with an energy resolution of1.1 eV are depicted in Fig. 3. These spectra are very similar to the data given in Ref. 31. Although probing the As L 2,3 edge is only an indirect measure of the 4 s/4pstates, it was shown in Ref. 31that the polarization dependence of the As L3spectra allows the determination of the As orbital topology. Consistent with the previous data, the present spectra point to asmall anisotropy between in-plane and out-of-plane states withFIG. 3. (Color online) Comparison of the normal- and grazing- incidence As L3NEXAFS spectra of Sr(Fe 1−xCox)2As 2(x=0, 0.05, 0.11, 0.17, and 0.38) recorded at 300 K in FY . The spectral shape of both edges is almost unaffected upon doping. For clarity, the spectra are vertically offset. A small anisotropy appears for all dopingcontents between normal- and grazing-incidence spectra. For more direct comparison of the doping dependence, the spectra of the x=0 andx=0.38 samples are plotted on top of each other in the lowest panel. a slightly higher density of states found for the out-of-plane spectra. More important, however, is that signiﬁcant changes inthe spectral shape and/or energetic shifts are not observed in thespectra upon substitution of Fe by Co up to almost 40%. This indicates that the As valency remains completely unchanged upon doping as well. Hence neither at the As nor at the Featom is the “extra electron” expected for Co 3+substitution found—lending further support to the notion that Co, insteadof behaving like Co 3+, seems to appear here with a valency of+2. Recent calculations seem to agree,29,46–48and suggest short-range screening as an important factor: in this picture,the ensemble of delectrons in Sr(Fe 1−xCox)2As 2screens the extra positive charge of Co3+on such a short length scale (of the order of the mufﬁn-tin radius) that Co iseffectively Co2+.29,46,47This, of course, would be fully consistent with the present observations. IV . SUMMARY AND CONCLUSIONS In conclusion, our NEXAFS data provide direct spectro- scopic evidence that not only the Fe but also the Co and theAs valencies remain completely unaffected across the entiredoping range. The Fe and Co L 2,3spectra are best described assuming a tetrahedral (Fe,Co)As 4coordination. This ﬁnding underlines a prominent role of the hybridization between(Fe,Co) 3 d xy,dxz,dyzorbitals and As 4 s/4pstates for the Fermi surface in A(Fe 1−xCox)2As 2. According to our experiments in conjunction with the multiplet calculations, Fe as well as Coions retain a valency of +2, which casts serious doubt on the widely assumed electron-doping effect induced by Co.The present data are fully consistent with the calculations ofRefs. 29and 46–48where the effect of Co substitution is described in terms of a (topological 29,46) change of the Fermi 104503-4ELECTRONIC STRUCTURE OF SINGLE-CRYSTALLINE ... PHYSICAL REVIEW B 86, 104503 (2012) surface (probably induced by a Lifshitz transition) which destabilizes the magnetism in favor of superconductivity. Inthese calculations, Co is identiﬁed either as being isovalent to Fe, 29,46,48or as being screened in such a way as to appear isovalent to Fe.47 The current investigation indicates a more pronounced covalent character for the (Fe,Co)-As bond with increasingCo content. Since the corresponding bond length, i.e., thedistance of the As atoms to the (Fe,Co) position, is changed byapplication of external pressure 12–14or upon chemical pressure due to the isovalent substitution of As by P or Fe by Ru,16,18the covalency of the (Fe,Co)-As bond seems to be a key parameterfor the interplay between magnetism and superconductivity inA(Fe 1−xCox)2As 2and for the topology of the Fermi surface as well. The strongly covalent character of the (Fe,Co)-As bondis also consistent with Refs. 16and 18where it was suggested that even the mobility of charge carriers can be signiﬁcantly en-hanced upon the isovalent replacement of As by P or Fe by Ru. ACKNOWLEDGMENTS We are indebted to D. Kronm ¨uller, B. Scheerer, P . Adelmann, A. Assmann, and S. Uebe for their excellenttechnical support and for fruitful discussions. We greatlyappreciate stimulating discussions with F. M. F. de Groot,W. Ku, and G. A. Sawatzky. We gratefully acknowledge theSynchrotron Light Source ANKA Karlsruhe for the provisionof beam time. Part of this work was supported by the GermanScience Foundation (DFG) in the framework of the PriorityProgram SPP1458. APPENDIX A: O K-EDGE NEXAFS OF Sr(Fe 1−xCox)2As2 Since it has been shown that iron pnictide samples may suffer severely from iron oxide contamination,31the OKedge was investigated as well. In Fig. 4the in-plane ( E⊥c)O 1sNEXAFS spectra of Sr(Fe 1−xCox)2As 2(x=0, 0.05, 0.11, FIG. 4. (Color online) O 1 sNEXAFS of Sr(Fe 1−xCox)2As 2(x= 0, 0.05, 0.11, 0.17, and 0.38) with E⊥ct a k e na t3 0 0Ki nF Y . While the degraded (undoped) sample contains a certain but still small amount of iron oxide, all other samples show no indication for a relevant contamination.0.17, and 0.38) recorded at 300 K with an energy resolution of 0.15 eV are compared to the deteriorated sample mentionedin Sec. III. Consistent with previous work, the deteriorated specimen (which is also an undoped sample) clearly displaysa large absorption signal resulting from an oxidic phase,whereas all other samples show no indication for a signiﬁcantcontamination. Together with the results obtained at the Fe L 2,3 edge (see Fig. 1in Sec. III) this phase can unequivocally be ascribed to iron oxide. Since all O K-edge data are measured in the bulk-sensitive FY mode, the small signatures found for thex=0, 0.05, 0.11, 0.17, and 0.38 specimens can be attributed to a small surface contamination which does not affect theinterpretation of the data performed at the other edges. [Due tothe lack of additional peaks in the spectral shape of the Co L 2,3 TEY data for the x=0.05, 0.11, 0.17, and 0.38 specimens (see Fig. 2in Sec. III), a cobalt-oxide surface layer can be excluded in these samples.] Furthermore, the O 1 sto Fe L2,3edge jump ratio is ≈1:1 for the degraded specimen (which is still small compared with typical transition-metal oxides) while it is≈1:50 for all other samples. On the other hand, the observation of a signiﬁcant signal for the degraded specimen unequivocallyimplies that for this sample, oxide is not just limited to thesurface layer but rather is present also in the bulk. This ﬁndingillustrates that great care must be taken when preparing andstudying pnictide samples, especially with surface-sensitivemethods. It also demonstrates that for a degraded sample,cleaving alone would not signiﬁcantly improve the situation. APPENDIX B: CORRECTION OF SELF-ABSORPTION AND SATURATION EFFECTS INHERENT TO FY MEASUREMENTS Finally, we want to brieﬂy address the self-absorption and saturation effects inherent to measurements performed in theFY mode. It is well known that these two effects can stronglyalter the measured intensity of spectral features, especially ifthe relevant chemical element is only moderately diluted in theinvestigated system and if the absorption edge intensity of thiselement is of the order of /greaterorapproxeql15% compared to the background signal of all other elements in this energy range. The Fe L 2,3 edge in Sr(Fe 1−xCox)2As 2is such an example. In the literature, however, it is well established how data which suffer fromthese effects can be corrected quite well. 49–53A corresponding formalism was also used for our spectra. Nevertheless, caremust be taken in the case of the L 2,3edge since the L3and theL2features are separated only by ≈15 eV and must be corrected independently. For instance, to eliminate the self-absorption and saturation effects (SAC) of the measured Fe L 2 edge, the already corrected L3edge must be accommodated in the background of the Fe L2edge together with the contribution of all other chemical elements in the sample. FortheL 2/L 3edge jump ratio the value of the uncorrected spectra was used and, ﬁnally, the total edge jump of the SAC correctedL 2,3spectra was set to 1. Yet we like to point out that even though the correction is indispensable to get the correct peakintensity, the fact that the spectral shape is unchanged upondoping is already evident from the uncorrected data. 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Stavitski and F. M. de Groot, Micron 41, 687 (2010). 43Multiplet parameters (in eV). Fe2+HS tetrahedron: 10 Dq =1.0, /Delta1c=3.0; Co2+tetrahedron: 10 Dq =1.4,/Delta1c=1.0; Co3+HS tetrahedron: 10 Dq =1.9,/Delta1c=2.0; Co3+square: 10 Dq =0.6, Ds=0.5, Dt =0.2,/Delta1c=3.0;Udd,Upd,Tσ,a n dTπwere set for all spin and valence states to 4.0, 5.0, 1.4, and 1.0, respectively. It turnedout that the ratio T σ/Tπhas no signiﬁcant inﬂuence on the spectral shape. The Slater integrals were renormalized to 64% of theirHartree-Fock values to account for the good metallic character of thesystem and the corresponding strong screening effects. Accordingto the experimental resolution and the lifetime broadening, thesimulations have been convoluted with a Gaussian with σ G=0.3e V and with a Lorentzian with σL3=0.2e V(σL2=0.3 eV) for the L3 104503-6ELECTRONIC STRUCTURE OF SINGLE-CRYSTALLINE ... PHYSICAL REVIEW B 86, 104503 (2012) (L2) edge. 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PhysRevB.78.073104.pdf	X-ray imaging of dispersive charge modes in a doped Mott insulator near the antiferromagnet/ superconductor transition Y . W. Li,1D. Qian,1L. Wray,1D. Hsieh,1Y . Xia,1Y . Kaga,2T. Sasagawa,2H. Takagi,2R. S. Markiewicz,3A. Bansil,3 H. Eisaki,4S. Uchida,2and M. Z. Hasan1 1Department of Physics, Joseph Henry Laboratories of Physics, Princeton University, Princeton, New Jersey 08544, USA 2Department of Physics, University of Tokyo, Tokyo 113-8656, Japan 3Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA 4AIST, 1-1-1 Central 2, Umezono, Tsukuba, Ibaraki 305-8568, Japan /H20849Received 8 January 2008; published 15 August 2008 /H20850 Momentum-resolved inelastic resonant x-ray scattering is used to map the doping evolution of bulk elec- tronic modes in the doped Mott insulator class Nd 2−xCexCuO 4. As the doping induced antiferromagnet/ superconductor /H20849AFM/SC /H20850transition is approached, we observe an anisotropic redistribution of the spectral weight of collective excitations over a large energy scale along the /H9003→/H20849/H9266,/H9266/H20850direction, whereas the modes exhibit broadening /H20849/H110111e V /H20850with relatively little softening along /H9003→/H20849/H9266,0/H20850with respect to the parent Mott state /H20849x=0/H20850. Our study reveals a closing of the charge gap in the vicinity of the zone center even though the mode softening and spectral redistribution involve an unusually large energy scale over the full Brillouin zone.The collective behavior of modes in the vicinity of the AFM/SC critical transition is demonstrated. DOI: 10.1103/PhysRevB.78.073104 PACS number /H20849s/H20850: 78.70.Ck, 74.20.Mn, 74.25.Jb, 74.72. /H11002h The evolution of a strongly correlated material with doping—from a Mott insulator to a conducting metal—isone of the most intensively studied issues in modern con-densed matter physics. This fascinating evolution has provento be full of surprises, such as the appearance of high- T c superconductivity, non-Fermi liquid behavior, and nanoscale phase separation.1Mott insulators often exhibit phase transi- tions upon doping, which are signaled or hallmarked by thesoftening or redistribution of the spectral weight of collectivecharge or spin modes. The behavior of spin modes has beenextensively investigated via neutron scattering. 2Although charge excitations near the Brillouin zone /H20849BZ/H20850center can be accessed by optical techniques,3their behavior with momen- tum over the full BZ remains largely unexplored. Here, asdemonstrated in recent experimental 4,5and theoretical studies,6,7inelastic x-ray scattering provides such a unique opportunity. While previous studies have focused largely oneither undoped one-dimensional 8or two-dimensional4,5insu- lators, or hole-doped superconductors,9Mott insulators can be doped with electrons as well. In fact, it appears that withelectron doping, cuprate bands evolve in a much morestraightforward and systematic manner 10–14than with hole doping. A limited previous work15that focused on a super- conductor with a ﬁt to the one-band theory does not provideinsights into the way the collective modes of a nonsupercon-ducting Mott insulator evolve into a superconductor whereinthe high-energy excitations of the Mott insulator are inti-mately connected to the lower-energy physics of the super-conductor. Here, we report a high-resolution study of theevolution of the collective charge excitations of the Mottinsulating state /H20849x=0/H20850with electron doping in approaching the critical antiferromagnet/superconductor /H20849AFM/SC /H20850tran- sition. Our ﬁnding, which was made possible by studying thedoped insulating states, is that as the electron-doping inducedAFM/SC transition is approached from the x=0 Mott side, the system exhibits anisotropic softening of the excitationsover a large energy scale /H20849/H110115e V /H20850along the /H9003→/H20849 /H9266,/H9266/H20850di- rection, whereas the modes exhibit broadening /H20849/H110111e V /H20850with relatively little softening along /H9003→/H20849/H9266,0/H20850. Our results suggest that a multiband Hubbard model is essential to de-scribe the evolution from the x=0 Mott state to the super- conductor since the high-energy and the low-energy excita-tions evolve into one another. Moreover, our results showthat the evolution of the Mott physics of an electron-doped cuprate is dramatically different from that previously re-ported in hole-doped cuprates. 9 The electronic structure of Nd 2−xCexCuO 4/H20849NCCO /H20850has been studied by angle-resolved photoemission /H20849ARPES /H20850and optical spectroscopies. ARPES studies16have found that the electrons directly dope into the bottom of the upper Hubbardband and yield small Fermi surface /H20849FS/H20850pockets, with a crossover around optimal doping to a large FS. Such a sce-nario, wherein the magnetic order remains commensuratewithout signs of “stripe” /H20849charge inhomogeneity /H20850or other phase separation, also describes magnetization 13,17,18and op- tical data.19 The experiments were performed at CMC-CAT beamline 9-ID-B at the Advanced Photon Source. Resonant inelasticx-ray scattering /H20849RIXS /H20850at the copper Kedge allows a large enough momentum transfer to cover several BZs. The scat-tered photon energy was measured by using a diced Ge /H20849733/H20850 crystal analyzer, and the intensity was recorded by a solidstate detector. The overall energy resolution was set to about0.37 eV in order to improve count efﬁciency, enabling us to detect the ﬁne momentum-dependent details of the spectra.All data were taken at room temperature. To avoid possiblepolarization induced artifacts when measuring the in-planeanisotropy, the sample was mounted with the incident polar-ization directed along the caxis /H20851Fig. 1/H20849a/H20850/H20852. The data were collected at several values of the momentum transfer vectorq=k i−kfin the second BZ along the /H20851100/H20852direction and in the fourth BZ along the /H20851110/H20852direction. We carefully set the qzcomponent to zero to avoid contamination of unwanted and spurious ddmodes that mix with the Mott gap excita- tions. We ﬁrst examined the detailed incident energy depen-dence of the loss spectra at several momenta. RepresentativePHYSICAL REVIEW B 78, 073104 /H208492008 /H20850 1098-0121/2008/78 /H208497/H20850/073104 /H208494/H20850 ©2008 The American Physical Society 073104-1data sets are presented in Fig. 1. It is known that the charge- transfer gap excitations resonate near the absorption peaks.9 A similar resonance behavior is seen in NCCO x=0, which is similar to that in the more extensively studiedLa 2−xSrxCuO 4.9However, the scattering intensity at the low- energy branch is about an order of magnitude weaker . Our systematic investigations that are summarized in Fig. 1show that a similar shape of the resonance proﬁle is seen for thelower-energy excitations in the electron-doped system. Ac-cordingly, we employed a photon energy of 8.982 keV /H20849this energy is above the 1 s−/H110223dedge that suppresses crystal- ﬁeld excitations 20/H20850. Figure 2summarizes our RIXS data on NCCO. In the undoped system in Fig. 2/H20849a/H20850, a broad excitation is observed near the zone center around 2 eV with an onset energy of/H110151 eV, which is consistent with the charge-transfer gap found in this compound in optical studies. 21It is seen to lose intensity as it approaches the zone corner along the /H20851/H9266,/H9266/H20852 direction as well as the /H20851/H9266,0/H20852direction /H20851Fig. 2/H20849a/H20850/H20852. Along /H20851/H9266,/H9266/H20852, it merges at the zone corner with a higher excitation band near 5 eV that disperses upward. With doping, thesetwo high-energy branches split at /H20849 /H9266,/H9266/H20850, as seen in Figs. 2/H20849b/H20850and2/H20849c/H20850, but only the lower-energy branch signiﬁcantly softens /H20849i.e., moves to a lower energy /H20850, so that the overdoped system in Fig. 2/H20849c/H20850displays a large excitation gap at the /H20849/H9266,/H9266/H20850point from 2 to 4 eV . In sharp contrast, the aforemen- tioned 2 eV branch uniformly evolves along /H20851/H9266,0/H20852and rap- idly softens near the zone center as it tends to close theexcitation gap with doping. Notably, the zero-loss energy isnot accessible due to the presence of the strong quasielasticpeak /H20851see, e.g., Fig. 1/H20849h/H20850/H20852, making it difﬁcult to extract sig- niﬁcant data below /H110150.4 eV, where the ﬁtting and subtrac- tion of the quasielastic peak leads to uncertainties. Neverthe-less, the softening of the low-energy excitations is evident in the data over the BZ. The preceding observations are further highlighted through the directly measured data curves presented in Fig.3. Doping dependent changes in the individual energy-loss curves are compared in Fig. 3/H20849a/H20850. Along /H20851110/H20852, doping splits the peak in the undoped spectrum around 5 eV at /H20849 /H9266,/H9266/H20850/H20851red uppermost curve on the left side of Fig. 3/H20849a/H20850/H20852into two peaks, and the lower of these peaks rapidly softens with doping. Incontrast, along /H20851100/H20852a monotonic softening of the low- energy excitations is found. However, our work does not ruleout a spectral weight transfer at lower energies. The changesin the positions of various spectral features and some of the 8.988 hv(keV): 8.9848.986 8.982 8.9808.998 hv(keV): 8.996 8.9908.9928.994 [ 10]1 [001] q [110]Polarization x-ray Ener gyLoss (eV)Intensity (arb. units) 0 4 0 4 88.988 8.992 8.980 8.98404 Incident Photon Energy (keV)Energy Loss (eV)Energy Loss (eV)Intensity 04 048 Incident Photon Energy (keV)8.980 8.984 8.988 8.990 8.995 9.00 0Nd Ce Cu O1.91 0.09 4 La Cu O24 (a) (c) (b)(d)(e) (f) (g) (h) (i) q=(1.1 /CID1/CID4/CID5/CID3 q=(1.0 /CID1/CID4/CID5/CID3Nd CuO24La CuO24 FIG. 1. /H20849Color online /H20850Incident energy dependence of electronic excitations. /H20849a/H20850Vertical scattering geometry employed with an x-ray ﬁeld along /H20851001/H20852./H20851/H20849b/H20850and/H20849c/H20850/H20852X-ray energy dependence of inelastic excitations for Nd 2CuO 4and La 2CuO 4. The Qvalues are chosen to minimize the large quasielastic background. /H20851/H20849d/H20850–/H20849g/H20850/H20852Excitations near the Mott gap are seen to be enhanced near the ﬁrst absorptionpeak, not only in the undoped insulator but also in the doped sys-tem. A few energy-loss curves corresponding to the data images in/H20849f/H20850and/H20849g/H20850are shown in /H20849h/H20850and/H20849i/H20850. The red /H20849dark gray /H20850curves show the ﬁts to the quasielastic data.Nd2CuO4Nd1.91Ce0.09CuO4Nd1.86Ce0.14CuO4 <110> <100> <110> <100> <100> <110>46 246 2 (,)/CID1/CID1 (,)/CID1/CID1 (,)/CID1/CID1 (,)/CID10 (,)/CID10 (,)/CID10 ()0,0 q(0,0) ()0,0 3.0 3.3 3.83.5 3.7 2462 462 462 462 462 46 Ener gyLoss (eV)Energy Loss (eV) (b) (c) (a) 0.9 1.0 1.81.3 1.63.0 3.3 3.83.5 3.70.9 1.0 1.81.1 1.30.9 1.0 1.81.3 1.62.8 3.0 3.73.3 3.51.61.1 Intensity (arb. units) FIG. 2. /H20849Color online /H20850Doping evolution of charge excitations in NCCO: /H20849a/H20850x=0/H20849parent Mott state /H20850,/H20849b/H20850x=0.09 /H20849doped AFM /H20850, and /H20849c/H20850x=0.14 /H20849superconductor /H20850. The upper row shows intensity maps in the reduced zone by using a color scheme wherein high intensityis denoted as red /H20849dark gray /H20850and low intensity as blue /H20849black /H20850. The black, gray, and white dots are guides to the eye for the dispersionof low-, medium-, and high-energy excitation branches, respec-tively. The lower row gives a few representative spectra corre-sponding to the images in the upper row. Absolute momentum val-ues/H20849in units of /H9266/ao/H20850are shown attached to various spectra and lie in the fourth BZ along /H20851110/H20852and the second BZ along /H20851100/H20852. 3.7 /CID13.5 /CID13.3 /CID1x2 3.0 /CID1 1.8 /CID11.3 /CID11.1 /CID10.9 /CID1 1.6 /CID1 Ener gyLoss (eV)24 6 24 6Relative Intensity Energy Loss (eV)468 2 0 (0,/CID1/CID4(0,0 /CID4 /CID3/CID4/CID1/CID5/CID1 (0,0 /CID4(0,/CID1/CID4 q[110] [100] 14%9%lead-edge: undopedpeak: / 14% 9% 0% FIG. 3. /H20849Color online /H20850/H20849a/H20850Spectral curves /H20849raw data /H20850along /H20851110/H20852 and /H20851100/H20852for the undoped system /H20851red/H20849dark gray /H20850squares /H20852are compared at various momenta to the corresponding spectra for thedoped system /H20851black circles and green /H20849gray/H20850diamonds /H20852./H20849b/H20850The positions of peaks /H20851marked in Figs. 2/H20849a/H20850–2/H20849c/H20850/H20852based on the center of gravity method /H20849Refs. 4and8/H20850of various spectral features /H20849ﬁlled symbols /H20850and some of the associated leading edges /H20849open symbols /H20850 as a function of momentum and doping are plotted.BRIEF REPORTS PHYSICAL REVIEW B 78, 073104 /H208492008 /H20850 073104-2associated leading edges are plotted in Fig. 3/H20849b/H20850as a function of momentum and doping. The upper branches around 5–6eV are seen to be weakly affected by doping. Despite signiﬁ-cant softening of one of the modes around /H20849 /H9266,/H9266/H20850, the closing of the excitation gap is limited to the region of the zonecenter as one approaches the superconducting phase /H20849x =0.14 /H20850. These results are in clear contrast to the behavior in hole-doped cuprates such as the La 2−xSrxCuO 4or the YBa 2Cu3O7+dseries.9 To interpret the present data in terms of RIXS spectra computed within the framework of a three-band HubbardHamiltonian of NCCO, based on Cu d x2−y2and two O p/H9268 orbitals, we extend the theoretical framework described in Ref. 7by incorporating the doping evolution data made available here. The speciﬁc values of the parameters used inthis work to ﬁt the spectra are t CuO=0.85 eV, tOO= −0.6 eV, /H90040=−0.3 eV, and Up=5.0 eV, where tCuOand tOO are the Cu-O and O-O nearest neighbor hopping parameters, nd/H20849np/H20850is the average electron density on Cu /H20849O/H20850,mdis the average electron magnetization on Cu, and U/H20849Up/H20850is the Cu /H20849O/H20850on-site Coulomb repulsion. The remaining parameter, crucial for ﬁtting the doping evolution of the excitations , the Hubbard U, is taken to be 7.45 eV at x=0 with a weak doping dependence: U=6.69 eV at x=0.09 and U =6.27 eV at x=0.14, so that the effective Udecreases by about 16% over this doping range, presumably reﬂecting theeffects of screening. We note that before proceeding withRIXS computations, we self-consistently determined thechemical potential and the magnetization m don Cu sites at each doping level. The mdvalues so derived were 0.32 at x =0, 0.19 at x=0.09, and 0.12 at x=0.14. Figure 4/H20849top/H20850shows the calculated RIXS intensity maps. The positions of various experimentally observed spectralpeaks /H20849ﬁlled circles /H20850and the leading edge of the low-energy feature /H20849ﬁlled diamonds /H20850are superposed for ease of compari- son. The high-energy peaks involve transitions from the non-bonding O and bonding Cu-O bands to unoccupied states inthe antibonding band, and fall in the same energy range astransitions involving other Cu and O orbitals that are notincluded in the present three-band model. Therefore, it isappropriate to concentrate on the behavior of RIXS peakswithin a few electron volts, which are associated with theantibonding Cu-O band, which is split by antiferromagneticordering. In this energy region, the theoretically predictedchanges in the energies of various features as a function ofmomentum and doping are in reasonable accord with experi-mental observations highlighted in the discussion of Figs. 2 and3above. In particular, the softening of low-energy peaks is well reproduced and the closing of the gap occurs near /H9003. The latter effect can be readily understood. The RIXS tran-sitions involve both intraband and interband contributions. Inthe absence of a gap, only intraband transitions are possible near/H9003, which can only exist near a zero-energy transfer close to the Fermi level. Away from /H9003, intraband transitions can take place at ﬁnite energies, but when a gap opens up,interband transitions become allowed, even at /H9003. 23 Although the computed maps in the top row of Fig. 4are shown unbroadened to highlight spectral features, the theo-retical spectra shown in the bottom panels of Fig. 4/H20851blue /H20849black /H20850lines /H20852have been smoothed to mimic the broadeningof experimental lineshapes. 22The relative intensities of the computed peaks are seen to be in agreement with our experi-mental results in lower energies 24except in the momentum region near the /H20849/H9266,/H9266/H20850point, where both the experimental and the theoretical intensities are weak, but the computedcross section is smaller. A similar behavior is seen in theLa 2−xSrxCuO 4series.9Excitonic effects associated with longer range Coulomb coupling /H20849intersite V/H20850beyond that in- cluded in our model computations could enhance and redis-tribute the spectral weight near zone boundaries. 25The low- energy, gap-edge dispersion relations /H20849Evsq/H20850are in agreement as seen from the top rows of Figs. 4/H20849a/H20850–4/H20849c/H20850.23 In Fig. 5/H20849b/H20850, we plot the doping dependence of the leading-edge gap /H20849edge, ﬁlled squares /H20850and peak position /H20849peak, white diamonds /H20850of the lowest branch at /H9003./H20851Since quasielastic scattering makes it difﬁcult to obtain experimen-tal data right at /H9003, these values are found by extrapolation from nearby points, as shown in Fig. 5/H20849c/H20850./H20852Within the ex- perimental uncertainties, the gap closes in the vicinity of the/H9003point as one approaches the critical doping regime in the phase diagram. For comparison, in Fig. 5/H20849b/H20850, we also show /H9004 m=Um d, which is the antiferromagnetic-correlation gap7,10 based on our present calculation /H20849ﬁlled circles /H20850and the cor- responding single particle spectrum /H20849SPS/H2085010,26results /H20849tri- angles /H20850, which are seen to be in agreement. In general, our analysis suggests that the anisotropic softening involving alarge energy scale observed near the Mott insulating staterequires going beyond the one-band model. In conclusion, we have utilized the unique momentum resolution of x-ray scattering to map the evolution of6 5 4 3 2 1 0 100 50 0 Γ (π,π) (π,0)6 5 4 3 2 1 0 100 50 0 Γ (π,π) (π,0)6 4 2 0ω(eV ) 100 50 0 Γ (π,π) (π,0)x=0 x=0 . 0 9 x = 0.14 (a) (b) (c) 6 4 2 0 6 4 2 0 0 2 4 6 0 2 4 6 0 2 4 6 ω(eV)(π,0) 6 4 2 0 6 4 2 0(π/2,0) (π/2,π/2) (π,π)6 4 2 0 6 4 2 0(π,0) (π/2,0) (π/2,π/2) (π,π) X20 X20(π,0) (π/2,0) (π/2,π/2) (π,π)I (arb. units)X20 FIG. 4. /H20849Color online /H20850Top row: Doping evolution of RIXS spectra obtained within a three-band Hubbard model as a functionof the energy transfer /H9275at momenta qalong/H9003to/H20849/H9266,/H9266/H20850and/H20849/H9266,0/H20850 directions for three different dopings x. The positions in /H9275−qspace of various experimentally observed spectral peaks /H20849ﬁlled circles /H20850 and the leading edge of the low-energy feature /H20849ﬁlled diamonds /H20850are superposed. The computed spectra have not been broadened to re-ﬂect the experimental resolution in order to highlight spectral fea-tures. Bottom row: Comparison of theoretical /H20851blue /H20849black /H20850lines /H20852 and experimental /H20851red/H20849gray/H20850lines /H20852spectra. The three columns refer to the three indicated doping levels x. Each column includes four different qvalues as shown; the experimental spectra are at the closest qvalue measured. The theoretical spectra have been smoothed to reﬂect experimental broadening /H20849Ref. 22/H20850.BRIEF REPORTS PHYSICAL REVIEW B 78, 073104 /H208492008 /H20850 073104-3particle-hole excitations in the electron-doped cuprate from its parent Mott state /H20849x=0/H20850to the doping onset for supercon-ductivity. We observe a nearly degenerate charge excitation mode near the /H20849/H9266,/H9266/H20850point around 5 eV in the Mott insulator, which splits on doping away from half-ﬁlling, with its lower-energy branch anisotropically softening in approaching theAFM/SC critical doping. In contrast, the response near the/H20849 /H9266,0/H20850wave vector exhibits damping with relatively little softening. Our results indicate the importance of multiband-correlation structures in the collective charge dynamics ofMott insulators when electrons are added into the parentMott state, which is in clear contrast to what is observed inhole-doped Mott insulators. We gratefully acknowledge P. W. Anderson, N. P. Ong, and T. Tohyama for discussions and T. Gog and D. Casa forbeamline support. This work is primarily supported by DOE/BES Grant No. DE-FG-02–05ER46200. 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B 70, 174518 /H208492004 /H20850. 19A. Zimmers, L. Shi, D. C. Schmadel, W. M. Fisher, R. L. Greene, H. D. Drew, M. Houseknecht, G. Acbas, M.-H. Kim,M.-H. Yang, J. Cerne, J. Lin, and A. Millis, Phys. Rev. B 76, 064515 /H208492007 /H20850. 20J. W. Seo, K. Yang, D. W. Lee, Y . S. Roh, J. H. Kim, H. Eisaki, H. Ishii, I. Jarrige, Y . Q. Cai, D. L. Feng, and C. Kim, Phys. Rev.B73, 161104 /H20849R/H20850/H208492006 /H20850. 21Y . Tokura, S. Koshihara, T. Arima, H. Takagi, S. Ishibashi, T. Ido, and S. Uchida, Phys. Rev. B 41, 11657 /H208491990 /H20850. 22The spectra were ﬁrst smoothed with a Gaussian of 0.37 eV width to account for the resolution. The spectra were furthersmoothed via folding with a Lorentzian of 0.30 eV width tomimic what appears to be an intrinsic linewidth. 23The branch near 4 eV is weak in the experimental data. Since this branch has a different symmetry than the 2 eV branch, itresonates at a different photon energy /H20849near 8.894 eV /H20850, as evi- dent from the raw data proﬁle in Fig. 1/H20849f/H20850. 24The experimental spectra for energy transfers beyond about 6 eV contain signiﬁcant contributions from bands beyond the three-band model we considered for this work. 25K. Penc and W. Stephan, Phys. Rev. B 62, 12707 /H208492000 /H20850. 26In the one-band model, Uhas a different deﬁnition and the mag- netic gap is /H9004m=2Um d. Also, at x=0, ARPES does not see the upper band giving a lower limit for /H9004m.01 0 200100300e-doping SC AF Doping x (%)200 T emperature (K)(a) 0 5 10 15 200123Energy(eV) Doping x (%)(b) 024 /CID3/CID4/CID1/CID5/CID1 (0,0 /CID4 (0,/CID1/CID4 qEnergy Loss (eV)14%9%Edge 0%Peak / Edge Peak RIXS (c)/CID6m: SPS TNeelexp: FIG. 5. /H20849Color online /H20850/H20849a/H20850Schematic temperature-doping /H20849T-x/H20850 phase diagram of NCCO. /H20849b/H20850The leading-edge gap /H20849ﬁlled squares /H20850 and peak positions /H20849white diamonds /H20850of the lowest-energy excita- tion branch in RIXS data at the /H9003point are compared to antiferromagnetic-correlation gap /H9004mfrom the present calculation /H20849ﬁlled circles /H20850and from SPS /H20849triangles /H20850/H20849Ref. 10/H20850as a function of doping. /H20849c/H20850The excitation modes /H20849branches /H20850are ﬁtted and extrapo- lated to the /H9003point /H208490,0/H20850based on the branch curvatures to extract the data points presented in /H20849b/H20850. The peak positions correspond to markings in Figs. 2/H20849a/H20850–2/H20849c/H20850and the leading edges are extracted from data presented in Fig. 3.BRIEF REPORTS PHYSICAL REVIEW B 78, 073104 /H208492008 /H20850 073104-4
PhysRevB.102.035428.pdf	PHYSICAL REVIEW B 102, 035428 (2020) Spin-resolved photomodulated electronic properties of ferromagnetic kagome lattices Jie Mei and Hengyi Xu* Jiangsu Key Lab on Opto-Electronic Technology, Center for Quantum Transport and Thermal Energy Science, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, People’s Republic of China Xiaoming Zhu Nanjing Institute of Astronomical Optics and Technology, Chinese Academy of Sciences, Nanjing 210042, People’s Republic of China Ning Xu† Department of Physics, Yancheng Institute of Technology, Yancheng 224051, People’s Republic of China (Received 18 March 2020; revised 6 July 2020; accepted 7 July 2020; published 21 July 2020) We study theoretically the band structures and angle-resolved photoemission spectroscopy of both monolayer and bilayer kagome lattices irradiated by variously polarized monochromatic light based on the Floquet-Blochtheory. The polarization of light plays distinctive roles in the different energy regimes of kagome bands. Thelinearly polarized light predominantly affects the ﬂat bands at the bottom of the kagome band, while thecircularly polarized light engenders spin-resolved massive Dirac cones whose spin species can be switchedby the polarization direction. The edge states of kagome ribbons exhibit a spatially anisotropic quantum spinHall effect conforming to the nontrivial topology of kagome bands. In kagome bilayers, we take into accountthe electric ﬁeld component of monochromatic light which results in a band gap in ﬂat bands in the case ofoblique incidence. This shows that the polarized light provides us a versatile tool to manipulate not only thetwo-dimensional and quasi-one-dimensional band structures, but also the anomalous Hall conductivity by eitherthe polarization or the incident plane of monochromatic light. DOI: 10.1103/PhysRevB.102.035428 I. INTRODUCTION There has been a longstanding interest in kagome lattices which are two-dimensional (2D) structures consisting of reg-ular hexagons surrounded by equilateral triangles and viceversa [ 1–3]. Extensive theoretical studies have predicted that the kagome lattice harbors various intriguing electronic prop-erties including the Dirac point analogous to graphene andthe dispersionless ﬂat band with nontrivial topology [ 4–10]. The ﬂat bands emerge in a variety of areas of condensedmatter physics connecting closely phenomena such as mag-netism [ 11], Wigner crystallization [ 12], and superconductiv- ity [13,14]. In kagome lattices, the ﬂat band is particularly interesting owing to its potential applications in realizinga fractional quantum Hall state without Landau levels inlattice systems [ 15]. Very recently, the kagome lattices have been materialized in transition-metal stannides, for example,Fe 3Sn2[16,17] and Co 3Sn2S2[18,19], and their properties have been examined experimentally, which stimulates theenthusiasm in the study of such kagome ferromagnets. Thesematerials possess not only the ferromagnetism splitting thespin-degenerate Dirac point but also ﬁnite spin-orbit interac-tions endowing them with even richer topological phases. *hengyi.xu@njnu.edu.cn †nxu@ycit.cnThe topological ﬂat bands of kagome lattices emerge as a result of the destructive interference of Bloch wave functionsdue to the lattice symmetry [ 20]. On the other hand, engi- neering topological phases has become one of major guid-ing themes in the research of topological matters, in whichthe ultrafast optical manipulation by intensive laser pulsesis the most promising. Theoretical investigations based onthe Floquet theory have revealed various topological Floquetstates, such as Floquet-Anderson insulators [ 21–23], Floquet- Majorana fermions [ 24–26], and Floquet semimetals [ 27,28]. The intense laser ﬁeld has been used to tune the electronic,magnetic, optical, or structural properties of the systems[29–32]. It can drive a trivial state to a topological nontriv- ial state, for example, the Floquet-Chern band [ 21] and the Floquet topological insulator [ 33]. In addition, the photoin- duced topological states in strained black phosphorus [ 34] and graphene [ 35] have been analyzed by ﬁrst-principles calcula- tions combined with Floquet theory. Under the irradiation ofcircularly polarized light (CPL), the strained black phospho-rus exhibits tunable photodressed Floquet-Dirac semimetalstates via changing the direction, intensity, and frequency ofthe incident laser. Also, the CPL can drive nodal line semimet-als into Weyl semimetals accompanied by the emergence of alarge and tunable anomalous Hall conductivity (AHC) [ 36]. In addition, the buckled 2D materials, such as silicene, germanene, and stanene irradiated by linearly and circularlypolarized lasers have been studied theoretically [ 37,38]. For silicene, the analytical expressions of Hall and longitudinal 2469-9950/2020/102(3)/035428(13) 035428-1 ©2020 American Physical SocietyMEI, XU, ZHU, AND XU PHYSICAL REVIEW B 102, 035428 (2020) conductivities under the off-resonant light have been derived based on the linear response theory [ 39]. It has been sug- gested that the peculiar anisotropic chiral edge and singleDirac-cone states emerge with the irradiation of a circularlypolarized light [ 40]. For the purpose of utilization of these topological states, the two-terminal transport behaviors havebeen considered by many different groups [ 41–50]. Compared with the buckled honeycomb structures, the kagome latticesare even more interesting in view of the presence of both theDirac point and the ﬂat band. As a typical kagome latticeimplementation, Fe 3Sn2is composed of two Fe-Sn layers where the iron atoms form a monolayer kagome lattice, orbilayer kagome lattice separated by a tin spacing layer. Themonolayer kagome lattices with the inﬂuence of polarizedlight have been investigated using the Floquet-Bloch theorybased on the tight-binding (TB) Hamiltonian [ 51]. This re- vealed that the circular light is able to open up a gap at thequadratic band touching point and inﬂuences the optical Hallconductivity. In spite of these, the spin-resolved electronicproperties of monolayer and bilayer kagome lattices irradiatedby monochromatic light have not been investigated carefullyto the best our knowledge. In this work, we perform an in-depth study of mono- layer and bilayer kagome lattices irradiated by polarized lightwithin the framework of the TB model including spin-orbitcouplings. Particularly, we take into account the interlayerperiodic potential differences induced by the electric ﬁeldcomponent of the light ﬁeld, which was overlooked in most ofthe previous work. The electric component is important sinceit may break the layer inversion symmetry and facilitate theband gap manipulation [ 52]. We study the 2D band structures of kagome lattices and edge states of quasi-one-dimensional(Q1D) kagome ribbons. Furthermore, the AHC of ferromag-netic kagome lattices irradiated by various incident light arealso investigated. The paper is organized as follows: In Sec. II, we formulate our model and introduce the theoretical methods and formal-ism. The calculation results are presented and discussed inSec. III. Section IVcontains the summary and conclusions. II. THEORETICAL MODEL AND FORMALISM We adopt the realistic layered kagome compound Fe 3Sn2 as the envisaged material under investigation. It comprises two types of Fe-Sn layers with kagome lattices formed bymagnetic iron atoms with nonmagnetic tin atoms at the centerof hexagons as shown in Figs. 1(a) and 1(b). Figure 1(c) shows the ﬁrst-principles band structures of nonmagnetic andferromagnetic Fe 3Sn2by the Vienna Ab initio Simulation package (V ASP) [ 53], both of which contain the Dirac-like bands at the Kpoint and ﬂat bands highlighted by dashed ellipses. The ferromagnetic order originates mainly from theFe orbitals and may be suppressed by doping [ 54]. Thanks to the special kagome structure, the Dirac points are awayfrom the exact Fermi level (zero point) unlike graphene, whichcan be conﬁrmed by the subsequent TB calculations. In thedensity functional theory (DFT) calculations, the presenceof tin atomic layers further modulates the Fermi energy andresults in an upsurge in the number of electronic bands.Practically, the Fermi energy can be controlled by doping αβxy(d)(c)a2 γ1γ1a1a3a0(b) 0E-EF (eV)1.5 -1.5 M K G MDirac-like flatbands K G M(a) spin-polarized spin-unpolarized a 2 γ 1 γ γ γ 1 γ γ a 11111 a 3 a 0 a a FIG. 1. (a) The atomic structure of bulk Fe 3Sn2where the lattice parameters are a=b=5.33 Å, c=19.78 Å. (b) The structure of bilayer kagome lattice under investigation. The upper and lower lay- ers are distinguishable from different colors. The interlayer coupling γ1=0.3γ0occurs between the pairwise neighboring atoms which form an angle of π/3 with respect to the center of equilateral trian- gles [ 16,20]. (c) Band structures of bulk Fe 3Sn2without (left panel) and with (right panel) spin polarization. The Dirac-like crossings and ﬂat bands have been highlighted by dashed ellipses. The red and blue lines represent the bands from majority and minority spins,respectively. (d) A sketch of the kagome lattice irradiated by a beam of monochromatic light. αdenotes the angle of incidence, and βis the angle between the incident plane of the monochromatic light andthex-zplane. without altering the validity of theoretical models. Moreover, the DFT calculations impose a periodicity in the third direc-tion amounting to an inﬁnite number of kagome and tin layerssuch that multiple Dirac-like and ﬂat bands are recognizablealbeit overwhelmed by many other less relevant bands. To grasp the major physics of the kagome structures and make the relevant electronic behaviors more transparent, weconsider only the kagome network formed by iron by omit-ting the bands from nonmagnetic tin atoms to simplify themodel under investigation. It has been shown that the mostimportant phenomena of this material can be well capturedby a single-orbital or two-orbital minimal TB model [ 54,55]. We start with a TB Hamiltonian without Coulomb interactionsand ferromagnetism which will be included subsequentlywithin the framework of mean-ﬁeld theory. The effective TB 035428-2SPIN-RESOLVED PHOTOMODULATED ELECTRONIC … PHYSICAL REVIEW B 102, 035428 (2020) Hamiltonian for kagome monolayers with spin-orbit coupling (SOC) is written as ˆHmonolayer =γ0/summationdisplay /angbracketlefti,j/angbracketrightσc† iσcjσ+iλ/summationdisplay /angbracketleft/angbracketlefti,j/angbracketright/angbracketrightαβνijc† iασz αβcjβ,(1) where c† iσcreates an electron with spin σon the lattice site i./angbracketleftij/angbracketrightand/angbracketleft/angbracketleftij/angbracketright/angbracketrightrepresent the nearest-neighboring (NN) and next-nearest-neighboring (NNN) couplings, respectively. λ represents the spin-orbit interaction with νij=1 if the bond connecting the next-nearest neighbors is counterclockwisewith respect to the zaxis and ν ij=−1 if it is clockwise. σzdenotes the Pauli matrix. For bilayer kagome lattices, the Hamiltonian is given by ˆHbilayer=ˆHmonolayer ⊗τ0+γ1/summationdisplay /angbracketlefti,j/angbracketrightσ(c† iσdjσ+H.c.),(2) where γ1is the interlayer coupling with c† iand djbeing the creation and annihilation operators for upper and lowerkagome layers, respectively. τ 0is the 2 ×2 identity matrix and⊗denotes the direct product. Experimental observations suggest a lattice structure of kagome bilayers in which theupper and lower layers rotate an angle of π/3 mutually along thezaxis as shown in Fig. 1(b), accounting for the weakening of band ﬂatness [ 20]. The energy spectrum of monolayer kagome lattices includes the Dirac bands E 1,2(k)=γ0[−1±/radicalbig 4(cos2k·a1+cos2k·a2+cos2k·a3)−3 ] and the ﬂat band E3=2γ0, in which ajwith j=1,2,3i st h e coordinate vector labeling the atom position in the unit cell.Here we deﬁne a 1=(1,0,0)a0,a2=(1/2,√ 3/2,0)a0, and a3=(−1/2,√ 3/2,0)a0with a0being the lattice constant as indicated in Fig. 1(b).E1,2form the Dirac cone at the K± points. The effective Hamiltonian around the Dirac point reads E(q)=¯hvF\|q\|with q=k−K±and Fermi velocity vF=√ 3a0γ0/¯h[56]. Experimental measurements have shown vF∼2×105ms−1for FeSn [ 17]. This allows us to estimate the hopping energy γ0=¯hvF/√ 3a0≈0.3 eV with a0=0.2664 nm. The monochromatic light is quantiﬁed by the frequency /Omega1 and the maximal vector potential A0. The incident direction of light is described by two parameters αandβwhich are related to the angles with respect to the x-yandx-zplanes, respectively. The 2D kagome lattices are then subject to atime-periodic vector potential of light given by [ 57] A(t)=A 0(sinαcosβsin(/Omega1t+φ)+sinβcos/Omega1t, ×sinαsinβsin(/Omega1t+φ)+cosβcos/Omega1t, ×cosαsin(/Omega1t+φ)), (3) where φ=0 and φ=πcorrespond to respective left- and right-handed circularly polarized light, while φ=π/2 rep- resents linearly polarized (LP) light. For β=0, the inci- dent light is parallel to the x-zplane such that the vector potential becomes A(t)=A0(sinαsin(/Omega1t+φ),cos/Omega1t,cosαsin(/Omega1t+φ)). (4)Ifβ=π/2, the incident light is parallel to the y-zplane with the vector potential A(t)=A0(cos/Omega1t,sinαsin(/Omega1t+φ),cosαsin(/Omega1t+φ)). (5) The vector potential is incorporated into the Hamiltonian bythe Peierls substitution, viz., p→p+eA(t), such that the neighboring couplings are adapted to an additional phase factor eie ¯h/integraltext A(t)·rijdrwith rij=(x,y,z) being the vector con- necting the neighboring sites iandj. Within the framework of the Floquet theory, the monochro- matic light renders the Hamiltonian periodic in time H(t)= H(t+T) with Tbeing the period and the correspond- ing frequency /Omega1=2π/T. The time-dependent Schrödinger equation reads i¯h∂ ∂t\|/Psi1q(t)/angbracketright=H(t)\|/Psi1q(t)/angbracketright;( 6 ) qis a set of quantum numbers. The wave function can be written as \|/Psi1q(t)/angbracketright=e−i/epsilon1qt ¯h\|φq(t)/angbracketrightwith the quasienergy /epsilon1qand\|φq(t)/angbracketright=\|φq(t+T)/angbracketright. By expanding \|φq(t)/angbracketright=/summationtext nexp[−in/Omega1t]\|φn q/angbracketright, we obtain /summationdisplay nHmn/vextendsingle/vextendsingleφn q/angbracketrightbig =(/epsilon1q+m¯h/Omega1)/vextendsingle/vextendsingleφm q/angbracketrightbig , (7) where Hmn≡Hm−n=1 T/integraldisplayT 0dtH(t)ei(m−n)/Omega1t(8) is the matrix form of the Hamiltonian spanned in the Floquet space with m,n=0,±1,±2,.... The hopping energy, for example, the NN term, is modiﬁed to (γ0)n=γ0Jn(eA/¯h)ein/Theta1, (9) where Jnis the ﬁrst-kind Bessel function. A=/radicalBig A2x+A2yand /Theta1=arccos( Ax/A). For the light parallel to the x-zplane with β=0, Ax=A0(xsinαcosφ+zcosαcosφ), (10) Ay=A0(xsinαsinφ+y+zcosαsinφ). (11) To calculate the photomodulated band structures, we write the Floquet Hamiltonian of the monolayer kagome latticerelated to the NN and the SOC in momentum space as follows,respectively (H NN)n=2γ0⎛ ⎝0 Jn(A1)Jn(A2) Jn(A1)0 Jn(A3) Jn(A2)Jn(A3)0⎞ ⎠⊗σ0(12) and (HSOC)n=−2λ⎛ ⎝0 Jn(A+ 23)−Jn(A− 31) −Jn(A+ 23)0 Jn(A+ 12) Jn(A− 31)−Jn(A+ 12)0⎞ ⎠ ×⊗σ3, (13) 035428-3MEI, XU, ZHU, AND XU PHYSICAL REVIEW B 102, 035428 (2020) where we have deﬁned the dressed Bessel functions Jn(Aj)=/braceleftbiggiJn(eaAj/¯h)ein/Theta1jsin(k·aj),for odd n, Jn(eaAj/¯h)ein/Theta1jcos(k·aj),for even n, (14) as well as Jn(A± jk)=/braceleftBigg iJn(eaA± jk/¯h)ein/Theta1± jksin[k·(aj±ak)],oddn, Jn(eaA± jk/¯h)ein/Theta1± jkcos[k·(aj±ak)],even n, (15) where Ajand/Theta1jare calculated with the vector ajaccording to Eqs. ( 10) and ( 11)f o rβ=0, while A± jkand/Theta1± jkare calculated with the vector aj±ak. Evidently, J−n=J∗ nwith the asterisk denoting the complex conjugate. The unit matrixσ 0and Pauli matrix σ3account for the spin degeneracy. To express the Hamiltonian of bilayer kagome lattices related to the interlayer coupling in Floquet space, we needanother dressed Bessel function, J n(Aj)=Jn(eAj/¯h)e x p [ in/Theta1j+ik·dj], (16) where djcontains d1=(a2+a3)/3,d2=(a1−a3)/3, and d3=−(a1+a2)/3. Here Ajand/Theta1jare calculated with the vector dj. So that (Hul)n=γ1⎛ ⎜⎝0 Jn(A1)Jn(A2) Jn(A1)0 Jn(A3) Jn(A2)Jn(A3)0⎞ ⎟⎠⊗σ0,(17) where the superscript stands for the coupling from the upper (“u”) layer to lower (“l”) one. It satisﬁes Hlu=(Hul)†. The the electric ﬁeld induced by the monochromatic light can be obtained by E=−∂A ∂t =−/Omega1A0[sinαcosβcos(/Omega1t+φ)−sinβsin/Omega1t,×sinαsinβcos(/Omega1t+φ)−cosβsin/Omega1t, ×cosαcos(/Omega1t+φ)]. (18) The electric ﬁeld gives rise to a voltage difference along a spe- ciﬁc direction, whose effects may be important for multilayersystems. For the 2D bilayer kagome lattices residing in the x-y plane, this electric ﬁeld generates a periodic voltage differencealong the zdirection in the Floquet space given by (V z)n=−/Omega1A0cosαdz 2(e−iφδn,−1+eiφδn,1), (19) where dzis the distance between layers. Clearly, varying the angle of incidence can change the voltage difference, therebyregulating the Floquet band structures. The inﬂuences of off-resonant light can be well described by an effective Hamiltonian derived from the time-evolutionoperator U(t,t /prime) expressed formally as U(k;t/prime,t)=Texp/bracketleftBigg −i ¯h/integraldisplayt/prime tH(t1)dt1/bracketrightBigg , (20) where Tis the time-ordering operator. The effective Hamilto- nian is then obtained by Heff=i TlnU(T,0), (21) with T=2π//Omega1 . For large driving frequencies /Omega1compared to the energy scale γ0, the system can be depicted approximately by an effective Hamiltonian obtained from [ 47,58,59] Heff(k)=H0+/summationdisplay n/greaterorequalslant1[Hn,H−n] n/Omega1+O/parenleftbigg1 /Omega12/parenrightbigg . (22) The dominant term originates from the commutator [HNN 1,HNN −1]//Omega1and reads 8iγ2 0 /Omega1Im⎛ ⎜⎝0 J1(A2)J∗ 1(A3)J1(A1)J∗ 1(A3) −J1(A2)J∗ 1(A3)0 J1(A1)J∗ 1(A2) −J1(A1)J∗ 1(A3)−J1(A1)J∗ 1(A2)0⎞ ⎟⎠⊗σ0. (23) The most signiﬁcant consequence of the topological bands with nonzero Chern numbers in ferromagnetic kagome lat-tices is the emergence of the AHC. The intrinsic contributionof the AHC is dependent only on the band structures of idealkagome lattices. It can be calculated from the Kubo formula,given the k-dependent nth eigenstates /Psi1 nkand eigenvalues Enkof the Floquet-Bloch Hamiltonian, σxy=e2 ¯h/integraldisplayd2k (2π)2/summationdisplay nfn(k)/Omega1n,xy(k) =e2 ¯h/integraldisplayd2k (2π)2/summationdisplay n/prime/negationslash=nfn2Im/angbracketleft/Psi1nk\|ˆvx\|/Psi1n/primek/angbracketright/angbracketleft/Psi1n/primek\|ˆvy\|/Psi1nk/angbracketright (Enk−En/primek)2, (24)where fnis the Fermi-Dirac distribution function related to thenth band. The Chern number for the nth band is calculated byCn=1 2π/integraltext d2k/Omega1n(k). This shows that AHC is directly linked to the topological properties of the Bloch states andproportional to the integration over the Fermi sea of the Berrycurvature /Omega1 n,xy(k) of each occupied band. The group velocity operator ˆ vx/yis obtained from the effective Hamiltonian ˆv=1 ¯h∇kHeff(k). (25) A pulse of monochromatic light pumps the system into a highly excited nonequilibrium state. It is then followed bya probe pulse of higher-energy photon–ejecting photocurrentwhich is detected experimentally with angle-resolved pho-toemission spectroscopy (ARPES). To connect our calcula- 035428-4SPIN-RESOLVED PHOTOMODULATED ELECTRONIC … PHYSICAL REVIEW B 102, 035428 (2020) G M K X G M K X GNo irradition LP light(a) LowHighγ0 Energy( )4 2 0 -2 (b) FIG. 2. The band structures of the TB model with and without irradiation of LP light (a), and the simulated ARPES spectrum(b) with a LP pump pulse in trapezoidal proﬁle. The probe pulse has a Gaussian proﬁle with a width /Gamma1=4π//Omega1 , whose peak coincides with the middle of the plateau of the pump pulse [ 35]. The embedded white solid line is the photodressed band irradiated by LP light with/Omega1=5,A 0=0.5, and φ=π/2. The angle of incidence is α= π/2 with β=0. Hereafter e=a=¯h=γ0=1i sa s s u m e di nt h e ﬁgure captions. tions with possible experimental measurements, we simulate the time-resolved photocurrent (tr-ARPES) at momentum k which is calculated by [ 60,61] I(k,ω,/Delta1 t)=Im/summationdisplay a/integraldisplay dt1/integraldisplay dt2s(t1,/Delta1t)s(t2,/Delta1t) ×eiω(t1−t2)G<(k;t1,t2), (26) where ¯ hωis the binding energy. s(t,/Delta1t) is the envelop function of probe pulse with /Delta1tbeing the pump-probe delay time. In the calculations, we take a Gaussian envelop functions(t,/Delta1t)=exp[−(t−/Delta1t) 2//Gamma12]/(/Gamma1√π) with a varying width /Gamma1[61]. The lesser Green’s function is given by G<(k;t1,t2)=U(k,t1,0)†G<(k;0,0)U(k,t2,0), (27) where time 0 refers to an initial time where the system is in equilibrium before the pump pulse is turnedon. G <(k;0,0) is related to the initial equilibrium band occupation via the Fermi-Dirac distributionfunction. In the Floquet theory, the lesser Green’sfunction can be expressed alternatively as G <(k;t1,t2)= −i/summationtext q,nexp[−i(/epsilon1q(k)/¯h−n/Omega1)(t1−t2)]\|φn q\|2ρqq(k) with q={j,σ}being a set of quantum numbers including the site jand spin σindices. ρqqis the electron occupation probability related to the Fermi-Dirac function for the closedsystems, /epsilon1 qandφn qare the quasienergy and the Floquet eigenvector for the nth subband, respectively [ 62,63]. It is then clear that the ARPES spectrum is capable of detecting theFloquet subbands. III. RESULTS AND DISCUSSION A. Kagome monolayers The 2D band structures of the kagome lattices without SOC feature the coexistence of the celebrated Dirac pointin the middle and the dispersionless ﬂat band at the bottom.Figure 2(a)shows the band structure with SOC of λ=0.05γ 0 [16] along the G-M-K-X-G line of special points, consistent with Ref. [ 64]. The presence of the SOC opens up a band gap near the GandKpoints giving rise to massive Dirac conesG M K X G M K X G(a) LowHighγ0 Energy( )-2 -4 -66 4 2 0 (b) FIG. 3. (a) The Floquet band structures of the TB model irra- diated by LCP light with /Omega1=3,A0=0.5, and φ=0. The angle of incidence is α=π/2 with β=0. (b) The simulated ARPES spectrum with a LCP pump pulse in trapezoidal proﬁle as well as a Gaussian probe pulse with the same parameters as Fig. 2. around the Kpoint as indicated by the dashed gray line [ 16]. We then expose the ideal kagome lattice to LP light and studyits impact on the band structure. The photodressed band spec-trum deviates slightly from the one free from irradiation andbecomes asymmetric with respect to the Kpoint. In particular, the ﬂat band at the bottom begins rising in energy from the M point, whereas the magnitude of the band gaps at the KandG points are almost unaffected and the spin degeneracy remainsintact since the time-reversal (TR) symmetry persists. To connect the Floquet bands with possible experimental observations, we simulate ARPES which can be used to detectthe occupied states under the Fermi level [ 17,20]. Figure 2(b) displays the simulated ARPES of the light-driven kagome lat-tice compared with the corresponding 2D Floquet band. Thepump pulse in a trapezoidal proﬁle containing 5 cycles is fol-lowed by a Gaussian probe pulse deﬁned in Eq. ( 26) emitting experimentally measurable photocurrents. To demonstrate thewhole band, we have assigned the Fermi level E F=4γ0. It appears that the nonequilibrium band structures from theARPES spectrum reproduce well the photomodulated bandstructures with the irradiation of LP light. The photocurrentpeaks coincide with the massive Dirac cones at the Kpoint as well as the Gpoint where two band branches approach. Particularly, the tilt of the ﬂat band is visible indicating theeffects of the LP light are observable experimentally. Thisﬁnding suggests that LP light degrades the the precise destruc-tive interference of Bloch electrons in kagome lattices suchthat the slope of the ﬂat band increases. Apparently, ARPES is a powerful technique to verify the band structures experimentally. It provides not only thestatic counterpart of photomodulated band structures but alsothe Floquet subbands. We now explore the photodressedelectronic bands modulated by CPL which breaks the time-reversal (TR) symmetry. Figures 3(a) and 3(b) show the Floquet band structures of kagome lattices irradiated by theleft-handed circularly polarized (LCP) light and the simulatedARPES spectrum, respectively. The ARPES spectrum showsthree complete subbands and is consistent with the photomod-ulated band structure. The interplay of Floquet subbands mod-iﬁes the proﬁles of the ﬂat band and Dirac band. We further fo-cus our attention on the static counterpart of the Floquet bandto reveal the light-matter interaction characteristics in moredetail. The LCP light with φ=0 lifts the spin degeneracy 035428-5MEI, XU, ZHU, AND XU PHYSICAL REVIEW B 102, 035428 (2020) No irradiation(a)γ0 Energy( )4 2 0 -2spin up spin down LowHigh G M K X G M K X G (b) FIG. 4. The band structures of the TB model with and without irradiation of LCP light (a), and the simulated ARPES spectrum(b) with a LCP pump pulse in trapezoidal proﬁle. The probe pulse has a Gaussian proﬁle with a width /Gamma1=4π//Omega1 , whose peak coincides with the middle of the plateau of the pump pulse. The embedded white solid line is the photodressed band irradiated by LCP light with/Omega1=5,A 0=0.5, and φ=0. The angle of incidence is α=π/2 withβ=0. particularly in the vicinity of the Kpoint as shown in Fig. 4(a). The inset shows two spin-resolved massive Dirac cones due tothe photomodulated SOC, and the spin-down band apparentlypossesses a narrower band gap compared with the spin-upband. Nonetheless, the LP light does not alter the overallproﬁles of the 2D bands and the band gap size near the G point. It merely shifts the ﬂat band upward leaving its shapeinvariant. Figure 4(b)presents the simulated ARPES spectrum with the LCP pump pulse compared with the effective bandmodulated by LCP light. Apart from the spin-resolved Diraccones which blur the photocurrent maximum in the ARPESspectrum, the major character lies in the robustness of the ﬂatband to the LCP pump pulse. Figure 5(a)shows the band structure of the kagome mono- layer with the irradiation of right-handed circularly polarized(RCP) light. The RCP light moves the ﬂat band upwardslightly, and switches the electronic behaviors of the spin-dependent bands; namely, the spin-down band contrarily ownsa wider band gap at the Kpoint than the spin-up band as magniﬁed in the inset. Analogously, ARPES manifests thestrongest intensities of photocurrents signaling the emergenceof massive Dirac cones and a band gap opening around the No irradiation(a) γ0 Energy( )4 20 -2spin up spin downHigh Low G M K X G M K X G (b) FIG. 5. The band structures of the TB model with and without irradiation of RCP light (a), and the simulated ARPES spectrum(b) with a RCP pump pulse in trapezoidal proﬁle. The probe pulse has a Gaussian proﬁle with a width /Gamma1=4π//Omega1 , whose peak coincides with the middle of the plateau of the pump pulse. The embeddedwhite solid line is the photodressed band irradiated by RCP light with /Omega1=5,A 0=0.5, and φ=π. The angle of incidence is α=π/2 withβ=0.01 2 3 0 123ka ka-2024(a) (b)γ0 Energy( )spin up spin downspin up spin down (c) (d) FIG. 6. The spin-resolved band structures of the kagome ribbons. (a) The static band structure. (b) The photomodulated band structureirradiated by LP light with φ=π/2. The LP light is depicted by the parameters /Omega1=5,A 0=0.5,α=π/2, and β=0. (c), (d) The spatial distributions of the edge-state wave functions with differentFermi energies. The radii of the dots represent the scaled ampli- tudes of the wave functions, and the color corresponds to different Fermi energies. Gpoint, respectively. Evidently, the CPL primarily inﬂuences the electronic behaviors around the Kpoint by breaking TR symmetry, while it has marginal effects on the ﬂat band sinceCPL does not spoil the delicate condition for the destructivephase interference of kagome lattices which matters to the ﬂat-ness of the band. It is known that the dispersionless ﬂat bandrenders superheavy electrons with a tremendously quenchedkinetic energy. Therefore, nonzero on-site Coulomb interac-tions will prevail and naturally result in a rich phenomenologyin kagome structures. The aforementioned calculations revealthat the LP light enables us to manipulate these phenomena,for example, superconductivity and ferromagnetism, exter-nally. The CPL, however, is capable of tuning the spin splittingaround the Dirac points. The strong correlation between thespin and the light polarization stems from the photomodulatedSOC, which can be veriﬁed in ribbon systems. The opening of band gaps around both the Kand G points by SOC hints at nontrivial topology of kagome bands.The calculations of topological invariants of different bandsindicate that the bottom and top bands have Chern number±1, and the middle band has zero [ 64]. The Chern numbers are normally equivalent to the number of quantized edge statesin the systems of restricted dimension. To further reveal thetopological properties of kagome bands under the inﬂuenceof light, we therefore study the edge states of the kagomeribbons. The band structure of the monolayer ribbon withoutirradiation is displayed in Fig. 6(a), which demonstrates the spin-resolved edge states connecting the bulk states withinthe bulk gap owing to the presence of the SOC. Two spinmodes form the nearly linear dispersion relation in separatedvalleys with vanishing gaps. For a ﬁxed Fermi energy, themoving directions of the edge modes lock with a speciﬁcspin orientation characterizing the typical quantum spin Hall(QSH) effect. At the bottom, there also exist spin-resolved 035428-6SPIN-RESOLVED PHOTOMODULATED ELECTRONIC … PHYSICAL REVIEW B 102, 035428 (2020) edge modes above the ﬂat band which distribute only along one of two edges according to the wave function calculations.Under the irradiation of LP light, the overall band structureof Q1D kagome ribbons shrinks in energy slightly leavingthe edge states unchanged as shown in Fig. 6(b). However, the LP light ﬂattens a bunch of modes at the bottom whichare reminiscent of the 2D ﬂat band. Figures 6(c) and6(d) show the spatial distributions of the wave functions of edgestates for different Fermi energies. The edge states underneathare more localized along the boundaries compared with theones in the middle energy regime. As the Fermi energyincreases, the edge states in both cases become more local-ized. Moreover, such helical edge states are immune to bulkdisorder and possibly form dissipationless QSH transport inview of the absence of the available counterpropagating states.The necessary condition for generating no resistance alongthe edges is the absence of bulk states which are contrarilytopologically trivial and sensitive to disorder. Therefore, adissipationless QSH state occurs in the energy gap betweenthe bulk bands corresponding to the case of Fig. 6(c), whereas a dissipative QSH transport may develop within the energyregime slightly above the ﬂat bands corresponding to the caseof Fig. 6(d) where the edge and bulk states intermingle. The group velocity of edge states diminishes as the Fermi energyincreases and vanishes at the band maximum at which theedge state resides at the outermost sites of the boundary andplays a minor role in the longitudinal transport. In contrast to the LP light, the CPL has more prominent effects on the edge states, while it has minor effects on the ﬂatband modes. Panels (a) and (b) of Fig. 7show the Q1D bands irradiated by LCP and RCP light with enlarged edge-stateregime (c) and (d), respectively. This reveals that LCP lightinduces a spin-up band gap around the left valley, whereas thespin-down edge mode at the right valley exhibits a metallic be-havior. The valley anisotropy attains its maximum within theedge-state band gap. By contrast, the RCP light with φ=π interchanges the electronic behaviors of two spins by inducingthe spin-down band gap with a closure of the spin-up one.Within the edge-mode band gap, the spin-down states surviveand distribute separately along the lateral edges such that thesigns of the valley polarization invert. This exotic behavioris substantiated by inspecting the spatial distributions of theedge-state wave functions as shown in Figs. 7(e) and7(f). LCP light delocalizes prominently the spatial distributions ofspin-up edge modes, while it squeezes the spin-down edgemodes to boundaries more severely. This interesting propertyprovides an alternative implementation of spin valves or spinswitches in spintronics. It is worthwhile to note that the above calculations were performed within the single-particle picture. Therefore, therelevant results are valid when the Coulomb interaction isnegligible or small compared with γ 0and can be treated as a perturbation. However, the electron-electron interactionsare important in the realistic kagome materials; in particular,many strong correlation effects are related to the kagome ﬂatbands. Based on the extended Hubbard model, the interaction-driven topological insulators [ 65] fractionally charged topo- logical defects [ 66], and the strongly correlated fermions [67] on the kagome lattice have been studied theoretically. Here we are interested in the magnetism due to the strong-2024 0123 0123)b( )a( 0.20.611.4 0.8 1.2 1.6 2 2.4 0.8 1.2 1.6 2 2.4(c) (d)γ0 Energy( )γ0 Energy( )spin up spin downspin up spin down 0 20 40 60 0 20 40 60 sites sites]Ψ](e)1 3 2 4 13 FIG. 7. The photomodulated band structures of kagome ribbons irradiated by (a) LCP with φ=0 and (b) RCP with φ=πpolarized light. (c) and (d) zoom in on the bulk band gap regimes of (a) and (b), respectively. The CPL is depicted by the following parameters: /Omega1= 5,Ax=Ay=0.5,α=π/2, and β=0. (e), (f) The inﬂuences of LCP light on the wave functions of the edge states on the left valleys (1/circlecopyrtand 2/circlecopyrt) and right valleys ( 3/circlecopyrtand 4/circlecopyrt) at a ﬁxed Fermi energy as indicated by a dashed line in (c). The insets are the corresponding smoothed wave function amplitudes. correlation on the basis of the on-site Hubbard term HU= U/summationtext ini↑ni↓≈U/summationtext ini↑/angbracketleftni↓/angbracketright+ni↓/angbracketleftni↑/angbracketright−/angbracketleft ni↑/angbracketright/angbracketleftni↓/angbracketright.T h ee x - change coupling of the spin-1 /2 Heisenberg antiferromagnet (AFM) is inversely proportional to the Hubbard repulsion as2γ 2 0/\|U\|[68]. However, the competition between the frus- trated local spins and charge transfers by Hund’s rule couplingmay give rise to a ferromagnetic (FM) state [ 69]. The FM ordering determined by the on-site Hubbard coupling withinthe mean-ﬁeld approximation can be described by a term inthe Hamiltonian [ 54], H FM=/summationdisplay iσσ/primem·c† i,σσσσ/primeci,σ, (28) where we simply set m=m(0,0,1) with a strength of m= 0.5γ0. The exact strength of ferromagnetism is determined by the ﬁrst-principle calculations [ 70] which are not relevant to our primary interest in this work, namely, the light-matterinteractions. Here we study the intrinsic anomalous Hall effectinduced by the ferromagnetism and its response to the incidentlight. We concentrate on the CPL cases since the LP lightdoes not affect AHC signiﬁcantly based on our calculations.Figure 8(a) shows the Fermi energy dependence of the AHC 035428-7MEI, XU, ZHU, AND XU PHYSICAL REVIEW B 102, 035428 (2020) $@$@ $@ =0.1, =0=0.3, =0=0.5, =0 =0.1, =0.5 =0.3, =0.5 =0.5, =0.5No irradiation0.51 0 -0.5 -1 -3 -2 -1 0 1 2 3 4G X K M G(a) (b) FIG. 8. (a) Energy dependence of the AHC of kagome mono- layers irradiated by LCP light with various angles of incidence α and different incident planes β.\|m\|=0.5γ0. Here all angles are in units of π. The dashed horizontal lines label the positions of the ﬁrst conductance quantum e2/h. The inset shows the band-resolved Chern numbers with \|m\|=0.05γ0.“±” originates from different spin species. (b) The corresponding 2D band structures with above irradiations for comparison. of kagome lattices irradiated by LCP light for various angles of incidence, accompanied with the corresponding 2D bandstructure shown in Fig. 8(b). The ﬁrst maximum of the AHC which is a remnant of the quantized plateau e 2/happears around EF=−2γ0corresponding to the Chern number Cn= 1 of the ﬂat band as shown in the inset. With increase of theFermi energy, the AHC undergoes the zero point twice be-cause the ﬂat bands for different spins have opposite signs ofChern numbers. In this transition process, the bands at higherenergies are partially occupied so that the AHC smoothlydecreases to zero instead of exhibiting rigid quantized steps.By comparison with the 2D bands, we see that each zeroAHC point corresponds to the energy below which the evennumbers of bands with opposite Chern numbers are fullyoccupied. The impacts of LCP light on the AHC occur pri-marily within an energy interval between −γ 0and 2γ0.T h e AHC grows monotonically as the angle of incidence increasesfromα=0.1πto 0.5πfor the incident plane parallel to the x-zplane ( β=0). For the incident plane parallel to the y-z plane ( β=π/2), the AHC is conversely suppressed by LCP and the suppression becomes more prominent as the angle ofincidence αincreases. When the incident light is turned to RCP light as shown in Fig. 9, the dependencies of the photomodulated AHC and-3 -2 -1 0 1 2 3 4G X K M GNo irradiation =0.1, =0 =0.3, =0 =0.5, =0=0.1, =0.5=0.3, =0.5=0.5, =0.50.51 0 -0.5 -1(a) (b) FIG. 9. (a) Energy dependence of the AHC of kagome mono- layers irradiated by RCP light with various angles of incidence α and different incident planes β.\|m\|=0.5γ0. Here all angles are in units of π. The dashed horizontal lines label the positions of the ﬁrst conductance quantum e2/h. (b) The corresponding 2D band structures with above irradiations for comparison. corresponding band structures on the incident plane of light reverse albeit they share striking similarities in proﬁle withthe case of LCP light. The RCP light restrains the AHC withthe incident plane α=0 light instead of building it up. The dependence of the AHC on the angle of incidence αand the characteristics of partially quantized AHC resemble those ofLCP light. These behaviors indicate the polarized light offersus a versatile tool to tune the AHC by varying either the lightpolarization or the incident plane. B. Kagome bilayers We proceed to study the electronic properties of bilayer kagome lattices irradiated by polarized monochromatic lightin this subsection. The experimental ARPES spectra ofFe 3Sn2show not only a strong intensity around the Gpoint but also a limited ﬂatness of ﬂat bands within a ﬁnite regionof the Brillouin zone deviating from 2D ideal kagome latticeswhere a perfectly ﬂat band extends over the whole Brillouinzone. A strong interlayer hybridization inclines to violatethe precise destructive interference of Bloch electrons. TheTB bilayer model with the conﬁguration shown in Fig. 1(b) derived from the realistic Fe 3Sn2crystal structure accounts for the reduction in band ﬂatness [ 20]. This model constitutes the starting point of our calculations of kagome bilayers. For kagome bilayers, the additional layer degree of free- dom further enriches their electronic behaviors which may not 035428-8SPIN-RESOLVED PHOTOMODULATED ELECTRONIC … PHYSICAL REVIEW B 102, 035428 (2020) (a) LowHigh 5 3 1 -1 -3γ0 Energy( ) G M K X G M K X GNo irradiation LP light (b) FIG. 10. The band structure of the bilayer kagome lattice with interlayer coupling γ1=0.3γ0under an irradiation of LP light (a) and the simulated ARPES spectrum (b) with a LCP pump pulse in trapezoidal proﬁle and a Gaussian probe proﬁle with a width /Gamma1=4π//Omega1 whose peak coincides with the middle of the plateau of the pump pulse. The LP light has parameters /Omega1=5,A0=0.5, and φ=π/2. The angle of incidence is α=π/2 with β=0. arise in the monolayer case. Figure 10(a) presents the 2D band structure of the kagome bilayer irradiated by LP light withα=π/2. The dashed gray lines correspond to the band with- out irradiation. Evidently, the bilayer bands are characterizedby a double splitting of each branch, in particular, the bandsat the bottom which are the the counterparts of the ﬂat bandin monolayers cross each other at the Kpoint producing a pronounced intensity peak in the ARPES spectrum as shownin Fig. 10(b) . The LP light renders the band structure asym- metric about the Kpoint and moves the ﬂat bands upward a bit along the K-X-Gline. The ARPES spectrum exhibits the strongest intensities at the G,M,K,Xpoints with respective energies consistent with the corresponding photomodulatedband structure. Apart from the magnetic ﬁeld component, the periodic- driving electric ﬁeld of monochromatic light may play animportant role in the band-structure modulation by inducinga potential difference between kagome layers. When the LPlight is incident at an angle of α=π/3, the electric ﬁeld component of light becomes nonzero along the zaxis ac- cording to Eq. ( 19). The major difference compared with normal incidence appears around the Kpoint. This produces a band warping around the massive Dirac cone and inducesa band gap in the ﬂat bands at the bottom of the kagomeband [Fig. 11(a) ]. In the simulated ARPES spectrum, the intensity peak due to the crossing of ﬂat bands disappears 5 3 1 -1 -3 LowHigh (a)No irradiation LP light G M K X G M K X Gγ0 Energy( ) (b) FIG. 11. The band structure of the bilayer kagome lattice with in- terlayer coupling γ1=0.3γ0under an irradiation of LP light (a) and the simulated ARPES spectrum (b). /Omega1=6. The angle of incidence isα=0.3π, with other parameters the same as in Fig. 10.G M K X G M K X G(a) LowHighγ0 Energy( )8 4 0 -4 (b) FIG. 12. The band structures of the bilayer TB model with irradiation of LCP light (a) with /Omega1=5,A0=0.5, and φ=0, and the simulated ARPES spectrum (b) with a LCP pump pulse in trapezoidal proﬁle. The probe pulse has a Gaussian proﬁle with a width /Gamma1=4π//Omega1 , whose peak coincides with the middle of the plateau of the pump pulse. The angle of incidence is α=π/2 with β=0. suggesting this effect is experimentally veriﬁable [Fig. 11(b) ]. It is evident that the electric ﬁeld component of light iscritical for the multilayer systems in some circumstances bymanipulating the band gap of the spin-degenerate ﬂat bands. Prior to a detailed study of the static counterpart of the photodressed band, we ﬁrst examine the overall Floquet bandstructure of the 2D bilayer kagome lattice as shown in Fig. 12. The simulated ARPES spectrum is in agreement with theFloquet band calculations, in particular, the characteristicsin the ﬂat band and Dirac bands. Contrary to the LP light,the CPL inﬂuences predominantly the middle band resultingin the spin-resolved massive Dirac cones. In Fig. 13(a) ,t h e spin-down band gap is suppressed by LCP light, while thespin-up band gap is enlarged relatively compared with the gapsize without irradiation analogously to the monolayer case. Atthe same time, the corresponding ARPES spectrum exhibitsthe strongest photocurrent around the crossing point of the ﬂatbands and the Gpoint, as well as the massive Dirac cone at the Kpoint. Furthermore, the RCP light plays an opposite role in the spin manipulation in 2D bands as shown in Fig. 14.T h e spin-up Dirac cone has a much narrower band gap than that ofthe spin-down band. The ARPES spectrum basically reﬂectsthe entire character of 2D bands; however, it is incapable ofdistinguishing the spin-resolved massive Dirac cones. LowHigh G M K X G M K X G5 3 1 -1 -3γ0 Energy( )(a)No irradiation spin up spin down (b) FIG. 13. The band structures of the bilayer kagome with γ1= 0.3γ0under an irradiation of LCP light (a), and the simulated ARPES spectrum (b) with a LCP pump pulse in trapezoidal proﬁle and aGaussian probe proﬁle as before. The interlayer distance d z=4a. The LCP light has parameters /Omega1=5,A0=0.5, and φ=0. The angle of incidence is α=π/2 with β=0. 035428-9MEI, XU, ZHU, AND XU PHYSICAL REVIEW B 102, 035428 (2020) (a) 3 1 -1 -3γ0 Energy( )5 No irradiation spin up spin down LowHigh G M K X G M K X G (b) FIG. 14. The band structures of the bilayer kagome with γ1= 0.3γ0under an irradiation of RCP light (a), and the simulated ARPES spectrum (b) with a RCP pump pulse in trapezoidal proﬁle and a Gaussian probe proﬁle as before. The RCP light has parameters /Omega1=5,A0=0.5, and φ=π. The angle of incidence is α=π/2 withβ=0. On account of the nontrivial 2D kagome bands, we further examine the inﬂuences of polarized light on the kagomebilayer ribbons. Figure 15displays only the ﬂat band modes of bilayer ribbons under the irradiation of LP light sincethe dominant effects occur in this regime. We see that theLP light mainly modiﬁes a bunch of the ﬂat band modesby tuning the band ﬂatness to some extent. Compared withthe monolayer ribbons, the LP light exerts less impact onthe ﬂat band modes of bilayer ribbons. This is because thephotomodulated ﬂat band in monolayer kagome lattices arisesfrom the degradation of the precise destructive interference,while this effect is remarkably diminished in bilayer kagomelattices due to the photoinduced phase cancellation betweentwo layers. Consequently, the ﬂat bands in bilayer kagomeribbons are insensitive to the phase perturbations from LPlight. Moreover, the ﬂat band modes and the related edgestates are also robust to the periodic electric ﬁeld due to thecharacteristics of these modes. In contrast, the LCP and RCP light dominantly inﬂuences the edge modes within the bulk band gap by splitting spinmodes as magniﬁed respectively in Figs. 16(c) and 16(d) . The LCP light leads to the fourfold edge modes originatingfrom the spin and layer degeneracies. The striking distinctionof edge modes between the bilayers and monolayers is theabsence of the band gap and the valley anisotropy indepen-dently of the light polarization. What happens here is that -1 -1.5 -2 -2.5 0 1 2 3ka(a) 0 1 2 3ka(b)γ0 Energy( ) LP light No irradiation FIG. 15. The ﬂat band modes of bilayer kagome lattice with in- terlayer coupling γ1=0.3γ0. (a) The static ﬂat band modes. (b) The photomodulated ﬂat band modes. The LP light has parameters /Omega1= 5,A0=0.5, and φ=π/2. The angle of incidence is α=π/2 with β=0.1.6 1.2 0.8 0.4 0.8 1.2 1.6 2 0.8 1.2 1.6 2 ka ka0 1 2 3 0 1 2 35 3 1 -1 -3(a) (b) (c) (d)γ Energy( )γ Energy( )thgil PCR thgil PCL 0 60 120 0 60 120 sites sites]Ψ]1 3 2 4 14)e( FIG. 16. The photomodulated band structures of kagome bilayer ribbons irradiate by (a) LCP with φ=0 and (b) RCP with φ=π polarized light. (c) and (d) zoom in on the bulk band gap regimes of (a) and (b), respectively. The interlayer coupling γ1=0.3γ0.T h e CPL is depicted by the following parameters: /Omega1=5,Ax=Ay=0.5, α=π/2, and β=0. (e), (f) The inﬂuences of LCP light on the wave functions of the spin-up edge states on the left valleys ( 1/circlecopyrtand 2/circlecopyrt)a n d right valleys ( 3/circlecopyrtand 4/circlecopyrt) at a ﬁxed Fermi energy as indicated by a dashed line in (c). The insets are the corresponding smoothed wave function amplitudes. the LP light enforces additional phase factors to the hopping integrals leading to a spin-dependent energy gap which canbe compensated by opposite spin modes in another layer.These edge modes are valley isotropic and demonstrate theQSH effect characteristics. According to the wave functioncalculations, the spatial distributions of the edge modes aredifferent for a ﬁxed Fermi energy. Generally, the edge statesbetween two valleys are more localized along the upper andlower boundaries. The edge states lying on the two sides ofthe valleys spread into the bulk region more aggressively,leading to a spatially anisotropic QSH effect within the bulkband gap. Figures 16(e) and 16(f) show the spin-up edge states on the two valleys with and without CPL irradiation.The LCP light apparently pushes the edge states on bothvalleys to the boundaries to some extent. The RCP light affectsthe edge modes of the kagome bilayer ribbons in a similarway, but it swaps the spatial and energy distributions of thespin modes. We continue to study the AHC of ferromagnetic bilayer kagome lattices. Because the LP light has no signiﬁcant 035428-10SPIN-RESOLVED PHOTOMODULATED ELECTRONIC … PHYSICAL REVIEW B 102, 035428 (2020) 3 2 0 -1 -2 -3 -3 -2 -1 0 1 2 3 41$@$@$=0.3, =0 =0.4, =0 =0.5, =0\ No irradiation =0.3, =0.5 =0.4, =0.5 =0.5, =0.5\ FIG. 17. Energy dependence of the AHC of kagome bilayers irradiated by LCP light with various angles of incidence αand different incident planes β.\|m\|=0.5γ0. Here all angles are in units of π. The inset shows the Chern numbers with \|m\|=0.01γ0 for better band separation. C=±3 are the total Chern numbers of the two bands underneath with “ ±” corresponding to the band of different spin species. C=±5 are the total Chern numbers of the middle bands with opposite spin orientations. The dashed horizontal lines label the positions of quantized conductance plateaus. effects on the AHC, we focus our attention on the energy dependence of CPL on the AHC. Figure 17shows the AHC of the kagome bilayer irradiated by LCP light, which ex-hibits strong ﬂuctuations and mixes with another with energysmaller than −γ 0because the low-energy bands intersect more frequently than the monolayer case, whereby the Chernnumber for a single band is not well deﬁned. As we sweep theFermi energy continuously, the partially and fully occupiedenergy bands always coexist such that the quantization ofthe AHC is severely suppressed. In the intermediate Fermienergy regime, the AHC is then well separated in energywith respect to the AHC free of irradiation. For the incidentplane parallel to the x-zplane ( β=0), the AHC decreases and nearly vanishes completely. For β=π/2, the magnitudes of the AHC grow regardless of their signs. When the Fermienergy is above the bulk gap, the quantized AHC plateau2e 2/hshows up since the bands are well separated. When the polarization of light is switched to RCP (Fig. 18), the behaviors of the AHC are inverted for different incident planesof light. This implies that the polarization of incident lightplays a similar role in the manipulation of the AHC to thatof its incident plane. For bilayers, the AHC is insensitivetoαcompared with the monolayer case, because the light ﬁeld imposes distinctive phase factors to the hopping energiesfor different layers which may cancel each other due to thespecial stacking form of the upper and lower layers, therebyminimizing the inﬂuences from angles of incidence. By apply-ing the periodic boundary condition perpendicularly, we areable to calculate the electronic properties of inﬁnitely layeredkagome materials with 3D Brillouin zones analogously toDFT calculations. Based on the above results, we expect thatthe qualitative features of inﬁnitely layered kagome latticesresemble those of kagome bilayers, viz., more deformed Dirac=0.3, =0 =0.4, =0 =0.5, =0\No irradiation=0.3, =0.5 =0.4, =0.5 =0.5, =0.5\ -3 -2 -1 0 1 2 3 43 2 0 -1 -2 -31 FIG. 18. Energy dependence of the AHC of kagome bilayers irradiated by RCP light with various angles of incidence αand different incident planes β.\|m\|=0.5γ0. Here all angles are in units ofπ. The dashed horizontal lines label the positions of quantized conductance plateaus. cones and ﬂat bands, due to additional interlayer couplings messing up the kagome band structures. Finally, we would like to discuss the feasibility of the experimental observations of the related phenomena in thiswork. The hopping energy γ 0=300 meV has been estimated through the experimental values of Fermi velocity for FeSn.Based on this value, the interlayer coupling γ 1=90 meV and spin-orbit coupling λ=15 meV are rather reasonable values. The laser amplitude A0=¯h/2ea=371 V /cwith c,a, and ebeing respectively the speed of light, lattice constant, and electron charge. This laser amplitude can be easily achievedwith the present technique. The photon energy ¯ h/Omega1=5γ 0 corresponds to laser light of frequency 3 .6×1014Hz or wavelength 833 nm, which is within the typical frequency orwavelength range of a Ti:sapphire laser. IV . SUMMARY AND CONCLUSIONS We have investigated systematically 2D, Q1D band struc- tures and ARPES spectra of kagome lattices irradiated by LP,LCP, and RCP light based on the Floquet-Bloch theory as wellas the high-frequency approximation. The LP light reducesthe ﬂatness of the ﬂat bands at the bottom of the kagomebands by spoiling the precise destructive phase interferencewith the spin degeneracy. The CPL, however, lifts the TRsymmetry and induces the spin-resolved massive Dirac cones,and further controls them by its polarization. In kagomeribbons, the edge modes belonging to different spins cantransform between the band insulator and the metallic stateby switching the polarization of light. This intriguing propertyis rather advantageous for implementing a spin valve or spinswitch with high on /off ratios in spintronics. For bilayer lattices, the electric ﬁeld component of monochromatic light can play an important role in modifyingthe band ﬂatness at the bottom of the kagome band. The LPlight is able to open up a band gap where two ﬂat bands cross,which can be visualized in the ARPES spectrum. Analogously 035428-11MEI, XU, ZHU, AND XU PHYSICAL REVIEW B 102, 035428 (2020) to monolayers, the 2D bands of bilayers also exhibit the spin-resolved massive Dirac cones tunable by polarization oflight. However, the kagome bilayer ribbons retain fourfoldedge modes due to the spin and layer degeneracies withoutedge-state band gaps in contrast to monolayer ribbons. Fi-nally, the nontrivial topology of kagome bands gives rise tononzero AHC which can be tuned by the polarization, anglesof incidence, and incident plane of light. However, the impactsof incident angles on the bilayer AHC are less prominent thanthose of the monolayers due to the interlayer cancellation of the phase modulations. ACKNOWLEDGMENTS H.X. acknowledges ﬁnancial support from Jiangsu Provin- cial Department of Education through the Shuangchuang teamand Grant No. 164080H00210. [1] A. Mielke, Ferromagnetic ground states for the Hubbard model on line graphs, J. Phys. A: Math. Gen. 24, L73 (1991) . [2] A. Mielke, Ferromagnetism in the Hubbard model on line graphs and further considerations, J. Phys. A: Math. Gen. 24, 3311 (1991) . [3] A. Mielke, Exact ground states for the Hubbard model on the kagome lattice, J. Phys. A: Math. Gen. 25, 4335 (1992) . [4] T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez- Rivera, C. Broholm, and Y . S. 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PhysRevB.76.245304.pdf	Admittance spectroscopy of GeSi-based quantum dot systems: Experiment and theory Xi Li,1W. Xu,2,3,Shihai Cao,1Qijia Cai,1and Fang Lu1,† 1Department of Physics, Surface Physics Laboratory, Fudan University, Shanghai 200433, People’ s Republic of China 2Department of Theoretical Physics, Research School of Physical Sciences & Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia 3Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, People’ s Republic of China /H20849Received 10 July 2007; published 4 December 2007 /H20850 A combined experimental and theoretical study is carried out in examining the important features of the admittance spectroscopy /H20849AS/H20850of self-assembled GeSi quantum dot /H20849QD/H20850systems. In the experimental com- ponent of the study, we measure the dependence of the AS on size of the GeSi QDs. From such measurements,we determine the emission rate and activation energy of the carriers in different QDs. Theoretically, wedevelop a simple approach to understand and reproduce those observed experimentally. It is found bothexperimentally and theoretically that with increasing size of the QDs, the peak of the AS shifts to higher-temperature regime, the activation energy of the carriers increases and the emission rate of the system de-creases. These interesting phenomena can be well explained by the fact that in GeSi-based QD systems, the ASis mainly induced by zero-dimensional to three-dimensional transition through hole interactions with acousticphonons via deformation potential coupling. DOI: 10.1103/PhysRevB.76.245304 PACS number /H20849s/H20850: 73.20.At, 72.10. /H11002d, 81.07.Ta I. INTRODUCTION The state-of-the-art material engineering and nanofabrica- tion techniques have made it possible to realize advancedsemiconductor devices at atomic scales, such as quantumdots /H20849QDs /H20850in which the electron or hole motion is quantized in all spatial directions and the conducting carriers are con-ﬁned within the nanometer distances. Such systems can be-have as artiﬁcial atoms and, therefore, be applied as ad-vanced electronic and optical devices, such as memory chip, 1 quantum computer,2quantum cryptography,3quantum-dot laser,4to mention but a few. At present, both electrical and optical techniques have been widely used in the characteriza-tion and investigation of GaAs-, InGaAs-, and GeSi-basedQD systems. 5In particular, in recent years the admittance spectroscopy /H20849AS/H20850has become a powerful tool in the electrical characterization of InGaAs- /H20849Refs. 6–8/H20850and GeSi-based9,10QD structures. When a QD is subjected to an ac electric ﬁeld, the conducting carriers in the dot can beemitted to the continuum states of the systems and the carri-ers in the continuum states can also be captured into thebound states in the dot with the assistance of electronic scat-tering mechanisms. In such a case, the number of carriers inthe dot alters with the intensity and frequency of the appliedac ﬁeld and, as a result, the conductance and capacitance canbe measured from the system. Hence, the AS measurementcan be applied to determine important device parameterssuch as the activation energy of the carriers, the strength ofthe quantum conﬁnement, the energy level structure, theemission and capture rates, etc. The experimental results ob-tained very recently 6–10have indicated that in InGaAs- and GeSi-based QD systems in which the conducting carriers are,respectively, electrons and holes, the electronic emission rateis about 1 /ms–1 / /H9262s so that the AS can be measured under the action of the ac ﬁelds with frequencies around1 kHz–1 MHz at a temperature range from 50 K to roomtemperature. Consequently, the AS is a simple, conventionaland powerful technique which measures electrically the elec- tronic and transport properties of InGaAs- and GeSi-basedquantum dot systems. In this work, we intend examining experimentally the ef- fect of the size of self-assembled GeSi QDs on features ofthe AS. It is known that the AS in a QD system is mainlyinduced by charge transfer in different states, especially be-tween bound and continuum states, due to the presence of theelectrical driving ﬁelds and electronic scattering mecha-nisms. The variation of the size of the dot changes thestrength of the quantum conﬁnement to the carriers in the dotand, therefore, alters the possibility for electronic transitionin different energy levels. Hence, it is expected that the fea-tures of the AS in QD systems are sensitive to the size of theQD. In this study, the GeSi QDs are grown by the standardtechnique of self-assemble on the basis of molecular-beamepitaxy /H20849MBE /H20850growth. The AS measurements are under- taken by a conventional transport experiment in the presenceof the ac driving ﬁelds. In sharp contrast to quite intensive experimental work in using AS for studying different QD systems, 6–10very little theoretical work regarding the AS in QDs has been reportedso far. At present, there is a lack of corresponding theory in analyzing and understanding the related experimental ﬁnd-ings. To ﬁll this gap, in this study, we develop a tractable andsystematic theoretical approach to calculate the conductanceand capacitance in conjunction with the AS measurement forGeSi-based QD systems. Very recently, we have proposed asimple theory to study the AS in GeSi-based quantum wellsystems. 12This theoretical approach is based on a mass- balance equation derived from the semiclassic Boltzmannequation and the obtained theoretical results agree very wellwith those measured experimentally. 12In the present study, we generalize this approach developed for a GeSi quantumwell to the case of self-assembled GeSi QDs. The validity ofthis model calculation is examined by those obtained fromour current experimental measurements. On the other hand,we can also use those obtained theoretically to understandPHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 1098-0121/2007/76 /H2084924/H20850/245304 /H208499/H20850 ©2007 The American Physical Society 245304-1our experimental ﬁndings and to reproduce those observed experimentally. Thus, this combined experimental and theo-retical study can achieve an in-depth understanding of thefeatures of the AS in GeSi-based QDs. In this paper, the details of the sample growth, device fabrication, experimental setup, and measurement are de-scribed in Sec. II. In Sec. III, we present the theoretical ap-proach to calculate the conductance and capacitance in con-junction with AS measurement in self-assembled GeSi QDs.The experimental and theoretical results obtained from thisstudy are presented and discussed in Sec. IV . The main con-clusions drawn from the present study are summarized inSec. V . II. SAMPLES AND MEASUREMENTS The samples used in the present study are prepared by the techniques of self-assemble on the basis of the MBE growth.The growth of the GeSi islands is achieved in an ultrahigh-vacuum MBE system /H20849Riber Eva-32 /H20850with two electron- beam evaporators as Ge and Si sources. The base pressure ofthe growth system is less than 5 /H1100310 −10Torr. A p-type Si /H20849001/H20850wafer is employed as substrate with a resistivity of about 1–10 /H9024cm. The substrate is chemically cleaned using the Shiraki method.13The protecting oxide layer on the sub- strate is desorbed at 1000 °C for 10 min in the growth cham-ber. The temperature is then lowered to 500 °C and a100-nm-thick Si buffer layer is grown with a growth rate of0.1 nm /s to achieve an epitaxial surface. Afterward, three layers of Ge are deposited and these Ge layers are separatedby 45-nm-thick Si spacer layers. The self-assembled GeSidots are formed using the Stranski-Krastanov growth modeduring which the Si diffuses into the QD layers. During thegrowth of the Si spacer layers, the Boron /H9254doping takes place with the concentration about 1 /H110031016/cm−3. These /H9254-doped layers can provide conducting holes in the QDs. Here, a two-step growth method has been adopted in islandgrowth at a rate of 0.02 nm /s and a temperature at about 500 °C in order to improve the uniformity of the GeSi is-lands. Finally, after depositing a Si cap layer with 320 nmthickness, the substrate temperature is immediately cooleddown to room temperature and the GeSi QD sample isprepared. In general, the size of self-assembled GeSi QDs relies on the growth temperature and the widths of the Ge layer andthe Si spacer layer. Usually, the size of the GeSi QDs in-creases with the growth temperature and with the thicknessof the Ge layer. 14,15In the present study, to investigate the effect of the size of the QDs on the AS, three samples /H20849A, B, and C, as shown in Table I/H20850are prepared at almost the same growth temperature and rate, the same Boron /H9254-doping layer, and the same Si spacer layers but with different widths of theGe layers. Thus, the QD size is mainly determined by thethickness of the Ge layers. The quality and the size of thedots can be examined and determined by the transmissionelectron microscopy /H20849TEM /H20850and atomic force microscopy /H20849AFM /H20850with good accuracy. The cross-section TEM and the AFM images show a uniform spatial distribution of the dots,where the nonuniformity of the QD size is estimated to beless than 10%. The TEM and AFM images also show thatthese QDs have a roughly circular symmetry along the direc-tion perpendicular to the growth direction. The samplegrowth parameters and the sample parameters are shown inTable I. For three samples A, B, and C with the Ge layer widths being, respectively, 13, 15, and 17 Å, the areal den-sity of the QDs, the height of the QDs along the growthdirection, and the QD diameter perpendicular to the growthdirection shown in Table Iare determined by the techniques of TEM and AFM. It can be found that the areal density ofthe QDs decreases with increasing size of the Dots. We notethat the QD sizes and the areal densities of our samples arequite similar to those grown by other groups. 16The Ge con- tent in a GeSi dot shown in Table Iis estimated by the Raman spectra, where the intensities of the Raman scatteringinduced by Si-Si, Si-Ge, and Ge-Ge optic-phonon modes aresensitive to the Ge composition. 17–19From Table I,w es e e that with increasing slightly the thickness of the Ge layer, thesize of the QD increases signiﬁcantly along both the growthdirection and the direction perpendicular to the growth direc-tion, in line with those observed by other groups. 16 To measure the AS, the QD devices are fabricated with a Schottky diode structure, where an Al electrode with the di-ameter of 1 mm without alloying is placed at the front sideand an Ohmic contact is at the back side. In the presentstudy, the admittance is measured along the growth direction,namely, the dc and ac ﬁelds are applied along the growthdirection and the conductance and capacitance are measuredalong this direction as well. The conductance is measured bya commercial HP 4284A LCR meter which can also apply the dc and ac bias voltages to the sample systems. The mea-surements are carried out within a temperature range from80 to 250 K in the presence of the weak ac voltage with afrequency f= /H9275/2/H9266range from 1 kHz to 1 MHz provided by the 4284A LCR meter. The temperature of the measurement is applied by a thermocouple with a Keithley 2400 sourcem-TABLE I. Sample growth parameters /H20849left panel /H20850and the AS peak position and peak half-width obtained experimentally and theoretically /H20849right panel /H20850. Sample No.Ge content in QD xGrowth temperature /H20849°C/H20850Ge layer width /H20849Å/H20850QD diameter /H20849Å/H20850QD height /H20849Å/H20850QD areal density /H20849108cm−2/H20850Peak position /H20849expt. vs theor. /H20850 /H20849K/H20850Peak half-width /H20849expt. vs theor. /H20850 /H20849K/H20850 A 0.75 500 13 130 30 8 166 vs 165 28 vs 20 B 0.75 510 15 200 35 3 184 vs 182 30 vs 22C 0.75 500 17 600 45 1 233 vs 232 31 vs 26LIet al. PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-2eter as the voltage indicator. Both meters are controlled by the computer through the IEEE-488 interfaces. The results of the AS obtained for three samples with different QD diameters and heights are shown in Fig. 1at a ﬁxed ac ﬁeld frequency. With the experimental data mea-sured at different ac ﬁeld frequencies, the emission rate /H9261 E and activation energy Eaof the carriers in the QD can be determined, respectively, by using G=/H9260/H9261E/H92752 /H9261E2+/H92752, /H208491/H20850 and /H9261E=/H9252Texp/H20849−Ea/kBT/H20850, /H208492/H20850 where Gis the measured conductance and /H9260and/H9252are coef- ﬁcients independent on temperature and ac ﬁeld frequency.These equations will be obtained theoretically in the nextsection of the paper. With Eq. /H208491/H20850, the emission rate can be determined simply by one of the following experiments. /H208491/H20850 /H9261 E/H11011/H9275pwith/H9275pbeing the ac ﬁeld frequency at which the peak of the conductance is observed at a ﬁxed temperature inthe measurement or /H208492/H20850/H9261 E/H11011/H9275at a temperature correspond- ing to the peak of the conductance when fis ﬁxed in the measurement. The results of /H9261EandEafor different samples are shown, respectively, in Figs. 2and3. From the above mentioned experimental setup and sample and measurement conﬁgurations, we know that the AS mea-sured experimentally is mainly induced by electronic transi-tion from zero-dimensional /H208490D/H20850/H20849or bound /H20850states in the QD to the three-dimensional /H208493D/H20850/H20849or continuum /H20850states of the sample system. In particular, because the applied ac and dcvoltages in the measurements are along the growth direction,the electronic transition takes place along this direction aswell.In the present study, we also carry out the capacitance- voltage /H20849C-V/H20850measurements for different QD samples. From these results, we can determine the effective carrier concen- tration in a QD along the direction of the lateral conﬁnementthrough the relation 9,11 Nc=C3 e/H9280/H92800A2/H20849−dC /dV/H20850, /H208493/H20850 where C=/H9280/H92800A/Ldis the capacitance with /H9280and/H92800being, respectively, the material and vacuum dielectric constants, A is the area of the electrode, and Ldis the depletion length. This carrier density can be used to determine the ﬁlling fac-tor /H9263of a QD, which is deﬁned as the ratio of the carrier concentration Ncand the areal density of the dots Ndin the direction perpendicular to the growth direction. Using Nd obtained from TEM and AFM images and shown in Table I andNcobtained from the C-Vmeasurements, the ﬁlling fac- tors are about 5–20 for samples A, B, and C shown in TableI. The ﬁlling factor /H9263connects directly to the characteristic frequency of a QD.20FIG. 1. Experimental /H20849upper panel /H20850and theoretical /H20849lower panel /H20850 results for conductance as a function of temperature in GeSi-basedQDs with different dot sizes /H20849the lateral diameters in the xyplane are 13, 20, and 60 nm for, respectively, samples A, B, and C shownin Table I/H20850. The results are obtained by applying an ac ﬁeld with a frequency f=1 MHz.FIG. 2. Experimental /H20849symbols /H20850and theoretical /H20849curves /H20850results of the emission rate for different samples A, B, and C. The experi-mental results are obtained by applying the ac ﬁelds with differentfrequencies. FIG. 3. Experimental /H20849symbols /H20850and theoretical /H20849curves /H20850results for ln /H20849/H9261E/T/H20850as a function of 1 /KBTfor different samples A, B, and C. Here, Eais the activation energy and /H9261Eis the emission rate.ADMITTANCE SPECTROSCOPY OF GeSi-BASED QUANTUM … PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-3III. THEORETICAL APPROACH In conjunction with the AS measurement of a GeSi-based QD system, our theoretical investigation can be undertakenon the basis of following considerations. /H20849i/H20850A dc gate voltage and an ac electric ﬁeld are applied along the growth directionof the QD to measure the capacitance and conductance. Thedc bias plays mainly a role in varying the hole subband struc-ture in the QD, and the ac ﬁeld can be taken as a driving ﬁeldwhich alters the number of holes in the QD in the presence ofthe dc bias. /H20849ii/H20850The AS is a consequence of the carriers going in or out of QD driven by the small ac electric ﬁeld appliedto the QD. /H20849iii/H20850The change of the hole numbers in the QD is mainly due to transitions of holes from bound states /H208490D/H20850to continuum states /H20849emission process /H20850or from continuum states /H208493D/H20850to bound states in the dot /H20849capture process /H20850./H20849iv/H20850 The transition of holes among different states is a conse-quence of the presence of the ac ﬁeld and of the electronicscattering mechanisms. A. Conductance and capacitance In this section, we develop a general theory to study the conductance and capacitance in a QD system in conjunctionwith the AS measurement. For simplicity, we employ thesemiclassic Boltzmann equation as the governing transportequation to calculate the AS induced by charge transfer be-tween the bound and continuum states in a QD. Because theapplied ac ﬁeld in the admittance measurement is time de-pendent and we consider relatively high-temperature andlow-density samples, we can start our calculation from time-dependent Boltzmann equation in nondegenerate statistics.For a hole in the continuum /H208493D/H20850and bound /H208490D/H20850states, the Boltzmann equations can be written, respectively, as /H11509f/H20849/H92513,t/H20850 /H11509t=gs/H20858 /H92510/H20851f/H20849/H92510,t/H20850W/H20849/H92510;/H92513/H20850−f/H20849/H92513,t/H20850W/H20849/H92513;/H92510/H20850/H20852 +gs/H20858 /H92513/H11032/H20851f/H20849/H92513/H11032,t/H20850W/H20849/H92513/H11032;/H92513/H20850−f/H20849/H92513,t/H20850W/H20849/H92513;/H92513/H11032/H20850/H20852 /H208494/H20850 and /H11509f/H20849/H92510,t/H20850 /H11509t=gs/H20858 /H92513/H20851f/H20849/H92513,t/H20850W/H20849/H92513;/H92510/H20850−f/H20849/H92510,t/H20850W/H20849/H92510;/H92513/H20850/H20852 +gs/H20858 /H92510/H11032/H20851f/H20849/H92510/H11032,t/H20850W/H20849/H92510/H11032;/H92510/H20850−f/H20849/H92510,t/H20850W/H20849/H92510;/H92510/H11032/H20850/H20852. /H208495/H20850 Here, /H92510and/H92513are quantum numbers to describe, respec- tively, the bound /H20849or 0D /H20850and continuum /H20849or 3D /H20850hole states, f/H20849/H92510,t/H20850and f/H20849/H92513,t/H20850are time-dependent distribution func- tions, respectively, for a hole at a 0D state /H92510and for a hole at a 3D state /H92513,gs=2 counts for spin degeneracy, and W/H20849/H9251;/H9251/H11032/H20850is the steady-state electronic transition rate for scat- tering of a hole from a state /H9251to a state /H9251/H11032. The ﬁrst term on the right-hand side of Eqs. /H208494/H20850and /H208495/H20850comes from electronic transition between the bound and continuum states, and thesecond term is induced by scattering events within the bound or continuum states. Furthermore, the effect of hole interac-tions with scattering centers has been considered within theelectronic transition rate, the effect of the dc bias has beenincluded within the hole wave function and energy spectrum,and the effects of the ac ﬁeld are implied in the time-dependent hole distribution functions. Thus, to avoid doublecounting, the force terms caused by the ac and dc ﬁelds donot show on the left-hand side of the Boltzmann equation. Ingeneral, there is no simple and analytical solution to Eqs. /H208494/H20850 and /H208495/H20850. In this work, we apply the usual balance-equation approach to solve the problem. 21The advantage of this ap- proach is that one can detour the difﬁculties of solving Bolt-zmann equation directly and the interested physical proper-ties can be calculated approximately on the basis of thestatistical distribution functions. 21For the ﬁrst moment, the mass-balance equation /H20849or rate equation /H20850can be derived by multiplying gs/H20858/H92513andgs/H20858/H92510to both sides of, respectively, Eqs. /H208494/H20850and /H208495/H20850. Thus, we can obtain a rate equation dQ0/H20849t/H20850/dt=Q3/H20849t/H20850/H9261C−Q0/H20849t/H20850/H9261E, /H208496/H20850 and the condition of hole number conservation: dQ0/H20849t/H20850/dt=−dQ3/H20849t/H20850/dt. Here, Q0/H20849t/H20850=gs/H20858/H92510f/H20849/H92510,t/H20850is the hole number in the bound states in the QD, Q3/H20849t/H20850=gsV/H20858/H92513f/H20849/H92513,t/H20850is the hole number in the continuum states in the system with Vbeing the volume of a QD, /H9261C=4 Q3/H20849t/H20850/H20858 /H92510,/H92513f/H20849/H92513,t/H20850W/H20849/H92513;/H92510/H20850/H20849 7/H20850 is the capture rate which measures the strength for transition of holes from the 3D states to the 0D states, and /H9261E=4 Q0/H20849t/H20850/H20858 /H92510,/H92513f/H20849/H92510,t/H20850W/H20849/H92510;/H92513/H20850/H20849 8/H20850 is the emission rate which measures the strength for transi- tion of holes from bound states /H208490D/H20850to continuum states /H208493D/H20850. Equation /H208496/H20850reﬂects a fact that the change of the charge number in the bound states of a QD is induced byelectronic scattering between 0D and 3D states. The condi-tion of hole number conservation implies that the increase inthe hole numbers in the continuum states comes from thedecrease in the hole numbers in the bound states of a QD.Under the action of the ac ﬁeld, a charge transfer can beachieved in the QD system. For example, if holes in the 0Dstates are emitted into the 3D states within the ﬁrst half circleof the ac ﬁeld, holes in the 3D states are captured into the 0Dstates within the second half circle of the ac ﬁeld. Thus, acurrent circuit can be formed due to this kind of changetransfer. Using Eq. /H208496/H20850, the current in the circuit is I/H20849t/H20850=−dQ 0/H20849t/H20850/dt=Q0/H20849t/H20850/H9261E−Q3/H20849t/H20850/H9261C. /H208499/H20850 Under the action of an ac driving ﬁeld /H9254Vt=V0ei/H9275t, with V0 and/H9275being, respectively, the strength and frequency of the ac ﬁeld, the hole number at the 0D states in a QD is thedifference between the mobile hole number /H9254Q0/H20849t/H20850and theLIet al. PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-4emitted hole number in a ﬁeld circle /H9004Q0/H20849t/H20850=/H208480tdtI/H20849t/H20850, namely, Q0/H20849t/H20850=/H9254Q0/H20849t/H20850−/H9004Q0/H20849t/H20850. For the case of a weak ac ﬁeld so that a linear response is achieved, we have /H9254Q0/H20849t/H20850 =/H9260/H9254Vt=/H9260V0ei/H9275tandI/H20849t/H20850=I0ei/H9275t. Here, a coefﬁcient /H9260=/H9254Q0/H20849t/H20850 /H9254Vt=dQ0/H20849t/H20850 dVt=gse/H20858 /H92510/H11509f/H20849E/H92510/H20850 /H11509/H9262t/H11509/H9262t /H11509Vt can be evaluated by assuming that the effect of the ac ﬁeld is mainly on the Fermi energy /H20849or chemical potential /H20850/H9262tof the system, where f/H20849x/H20850is the Fermi-Dirac function and E/H92510is the energy spectrum for a hole in a bound state. We note that for a weak ac ﬁeld which results in a linear response /H11509/H9262t//H11509Vt =e, we have /H9260=2e2 kBT/H20858 /H92510f/H20849E/H92510/H20850/H208511−f/H20849E/H92510/H20850/H20852. /H2084910/H20850 Inserting these results into Eq. /H208499/H20850,w eg e t I/H20849t/H20850=/H20875/H9260V0ei/H9275t−/H20885 0t dtI/H20849t/H20850/H20876/H9261E−Q3/H20849t/H20850/H9261C and, as a result, I˙/H20849t/H20850=/H20851i/H9275/H9260V0ei/H9275t−I/H20849t/H20850/H20852/H9261E−Q˙3/H20849t/H20850/H9261C. /H2084911/H20850 When the system is in equilibrium, the total hole number in the system should be conserved so that Q˙0/H20849t/H20850=−Q˙3/H20849t/H20850=−I/H20849t/H20850. Thus, Eq. /H2084911/H20850can be solved analytically. After using the deﬁnition for conductance G=I0/V0and for capacitance C=−dQ0/H20849t/H20850/dVt=−/H20851dQ0/H20849t/H20850/dt/H20852//H20851dVt/dt/H20852, we obtain G=i/H9260/H9275/H9261E /H9261E+/H9261C+i/H9275andG=R eG=/H9260/H92752/H9261E /H20849/H9261E+/H9261c/H208502+/H92752, /H2084912/H20850 and C=/H9260/H9261E /H9261E+/H9261C+i/H9275andC=R eC=/H9260/H9261E/H20849/H9261E+/H9261C/H20850 /H20849/H9261E+/H9261c/H208502+/H92752. /H2084913/H20850 For case where the measurement is carried out at a rela- tively high-temperature for a relatively low carrier densitysample, we can employ a statistical energy distribution suchas the Maxwillian as the distribution function for a hole inthe QD. Taking f/H20849 /H9251,t/H20850=C/H20849t/H20850e−E/H9251/kBTwhere c/H20849t/H20850is a normal- ization factor determined by Q0/H20849t/H20850=C0/H20849t/H20850gs/H20858/H92510e−E/H92510/kBTfor a 0D hole and Q3/H20849t/H20850=C3/H20849t/H20850Vgs/H20858/H92513e−E/H92513/kBTfor a 3D hole, the capture and emission rates become, respectively, /H9261C=4 A3V/H20858 /H92510,/H92513e−E/H92513/kBTW/H20849/H92513;/H92510/H20850/H20849 14/H20850 and/H9261E=4 A0/H20858 /H92510,/H92513e−E/H92510/kBTW/H20849/H92510;/H92513/H20850, /H2084915/H20850 which are time independent, with E/H9251ibeing the energy spec- trum for a hole at a state iandAi=gs/H20858/H9251ie−E/H9251i/kBT. If the elec- tronic transition rate W/H20849/H9251;/H9251/H11032/H20850is known, we can calculate the emission and capture rates induced by the corresponding transition events and then calculate the conductance andcapacitance. B. GeSi-based quantum dot systems It is known that in a GeSi-based QD, light and heavy holes are conducting carriers. At relatively high tempera-tures, the hole-phonon scattering is the principle mechanismfor relaxation of excited holes in GeSi-based material sys-tems. In this study, we consider a hole-phonon system with aHamiltonian H=H h+Hp+Hh−p, /H2084916/H20850 where Hh=P2/2m+U/H20849x,y,z/H20850is the single-particle Hamil- tonian for a hole, where P=/H20849px,py,pz/H20850with px=−i/H6036/H11509//H11509xbe- ing the momentum operator along the xdirection, mis the effective mass for a hole in the system, and U/H20849x,y,z/H20850is the conﬁnement potential for a hole in the dot, Hp =/H20858Q/H6036/H9275QbQ†bQis the phonon Hamiltonian, where Q =/H20849qx,qy,qz/H20850is the phonon wave vector and bQ†/H20849bQ/H20850is the creation /H20849annihilation /H20850operator for a phonon, and Hh−p =/H20858Q/H20849VQeiQ·RbQ+VQe−iQ·RbQ†/H20850is the hole-phonon interaction Hamiltonian, where R=/H20849x,y,z/H20850andVQis the strength of the hole-phonon coupling. Taking the hole-phonon interaction Hamiltonian as a perturbation, the electronic transition rateinduced by hole-phonon coupling in a QD can be derivedusing Fermi golden rule, which reads W/H20849 /H92510;/H92513/H20850=2/H9266 /H6036/H20858 Q/H20875NQ NQ+1/H20876/H20841/H20855/H92513/H20841eiR·Q/H20841/H92510/H20856/H208412/H20841VQ/H208412 /H11003/H9254/H20849E/H92510−E/H92513±/H6036/H9275Q/H20850/H20849 17/H20850 for transition from a bound state /H20841/H92510/H20856to a continuum state /H20841/H92513/H20856, and W/H20849/H92513;/H92510/H20850=2/H9266 /H6036/H20858 Q/H20875NQ NQ+1/H20876/H20841/H20855/H92510/H20841eiR·Q/H20841/H92513/H20856/H208412/H20841VQ/H208412 /H11003/H9254/H20849E/H92513−E/H92510±/H6036/H9275Q/H20850/H20849 18/H20850 for transition from a continuum state /H20841/H92513/H20856to a bound state /H20841/H92510/H20856. Here, the upper /H20849lower /H20850case refers to absorption /H20849emis- sion /H20850of a phonon, NQ=/H208511−e/H6036/H9275Q/kBT/H20852−1is the phonon occu- pation number, and /H6036/H9275Qis the phonon energy. Furthermore, /H20841/H92510/H20856and /H20841/H92513/H20856are wave functions for a hole at, respectively, bound and continuum states with the corresponding energyspectra E /H92510and E/H92513. They can be determined by a Schrödinger equation regarding to the Hamiltonian Hh. For a self-assembled GeSi QD as described in Sec. II, the ﬁnite thickness along the growth direction /H20849taken along the z direction /H20850is typically much narrower than the lateral exten- sion of the electrostatic conﬁnement along the direction per-ADMITTANCE SPECTROSCOPY OF GeSi-BASED QUANTUM … PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-5pendicular to the growth direction /H20849taken along the xyplane /H20850. This feature has been indicated in Table I. As a result, the conﬁning potential to a hole along the zdirection is much stronger than that along the xyplane. For simplicity, in this study, we model the conﬁnement potential along the growthdirection as a narrow rectangle quantum well. Due to circularsymmetry of the QD in the xyplane, the effective conﬁne- ment has also a roughly circular symmetry in the xyplane. 22 This potential then can be modeled by a simple isotropic harmonic oscillator taken popularly for lateral conﬁnementof a QD along the xyplane. Thus, we can use a parabolic potential for a QD in the direction perpendicular to thegrowth direction. Hence, the wave function and energy spec-trum for a hole at a bound state in the dot can be written,respectively, as /H20841 /H92510/H20856=/H20841/H9263/H20856=eim/H9258 /H208812/H9266RNm/H20849r/H20850/H9274/H9261/H20849z/H20850, /H2084919/H20850 and E/H92510=E/H9263=/H208492N+/H20841m/H20841+1/H20850/H6036/H92750+/H9255/H9261. /H2084920/H20850 Here, r=/H20849x,y/H20850=/H20849r,/H9258/H20850,/H9263=/H20849N,m,/H9261/H20850refers to all quantum numbers in the dot, m=0,±1,±2,.... is the angular quantum number, N=0,1,2,.... is the radial quantum number, /H9261is the quantum number along the growth direction, RmN/H20849r/H20850=1 l0/H208812N! /H20849N+/H20841m/H20841/H20850!e−/H92672/2/H9267/H20841m/H20841LN/H20841m/H20841/H20849/H92672/H20850, where /H9267=r/l0,l0=/H20849/H6036/m/H92750/H208501/2, and LNm/H20849x/H20850is a Laguerre polynomial. The characteristic frequency of a parabolic po- tential in a QD can be evaluated by20 /H927502=2e2/H9263 /H9266/H9280/H92800m/H92783, /H2084921/H20850 where /H9263is the ﬁlling factor of a QD which can be determined experimentally, /H9278is the diameter of a QD along the xyplane, and/H9280and/H92800are, respectively, the material and vacuum di- electric constants. To simplify the analytical and numericalcalculations, we take the usual square-well approximation tomodel the conﬁning potential along the growth direction. Indoing so, we have /H9274/H9261/H20849z/H20850=/H208492/L/H208501/2sin/H20849/H9261/H9266z/L/H20850, /H2084922/H20850 and /H9255/H9261=/H92612/H92662/H92550, /H2084923/H20850 where Lis the height of the QD along the growth direction and/H92550=/H60362//H208492mL2/H20850. Furthermore, considering a parabolic valence-band struc- ture in the GeSi compound, the wave function and energyspectrum for a hole in the continuum states are given, respec-tively, by/H20841 /H92513/H20856=/H20841K/H20856=eiK·RandE/H92513=EK=/H60362K2 2m+U0,/H2084924/H20850 where U0is height of the conﬁning potential along the growth direction and K=/H20849k,kz/H20850=/H20849kx,ky,kz/H20850is the wave vec- tor for a hole in a continuum state. Applying the wave functions and energy spectra for a GeSi-based QD to the electronic transition rate induced byhole-phonon scattering, we have W/H20849 /H9263;K/H20850=2/H9266 /H6036/H20858 Q/H20875NQ NQ+1/H20876/H20841VQ/H208412R/H9261/H20849L/H20841kz+qz/H20841/H20850 /H11003SNm/H20849l0/H20841k+q/H20841/H20850/H9254/H20849E/H9263−EK±/H6036/H9275Q/H20850/H20849 25/H20850 for transition from a bound state /H20841/H9263/H20856to a continuum state /H20841K/H20856, and W/H20849K;/H9263/H20850=2/H9266 /H6036/H20858 Q/H20875NQ NQ+1/H20876/H20841VQ/H208412R/H9261/H20849L/H20841kz−qz/H20841/H20850SNm /H11003/H20849l0/H20841k−q/H20841/H20850/H9254/H20849EK−E/H9263±/H6036/H9275Q/H20850/H20849 26/H20850 for transition from a continuum state /H20841K/H20856to a bound state /H20841/H9263/H20856. Here, k=/H20849kx,ky/H20850,q=/H20849qx,qy/H20850, R/H9261/H20849x/H20850=4L/H20849/H9261/H9266/H2085021−/H20849−1/H20850/H9261cosx /H20851/H20849/H9261/H9266/H208502−x2/H208522, and SNm/H20849y/H20850=4/H9266l02N! /H20849N+/H20841m/H20841/H20850!y2/H20841m/H20841e−y2/H20851LN/H20841m/H20841/H20849y2/H20850/H208522. C. Emission and capture rates In a GeSi-based structure, the hole-phonon scattering is mainly achieved by interaction with acoustic-phonon modesvia deformation-potential coupling. 23In such a case, the strength of the hole-phonon coupling is /H20841VQ/H208412=/H20849/H6036Q/2/H9267/H20850/H20849/H9014L2/uL+/H9014T2/uT/H20850, corresponding to the intravalley scattering,24where /H9267is the density of the material, uLanduTare, respectively, the lon- gitudinal and transverse sound velocities, and /H9014L=/H9014d +/H9014ucos2/H9258and/H9014T=/H9014ucos/H9258sin/H9258, with /H9014dand/H9014ubeing, respectively, the dilatation and uniaxial deformation poten-tials and /H9258an angle to the zaxis /H20849i.e., cos /H9258=qz/Qand sin/H9258=q/Q/H20850. Here, we have included the contributions from both longitudinal and transverse acoustic-phonon couplingsand /H9275Q=uLQor/H9275Q=uTQis the corresponding phonon fre- quency. In the present study, we consider a situation wherethe measurement is carried out at a relatively high-temperature so that /H6036 /H9275Q/lessmuchkBT. For case where the transfer of holes is achieved along the growth direction with a strongconﬁnement, the condition /H20841E K−E/H9263/H20841/greatermuch/H6036/H9275Qcan be satisﬁed. Thus, using Eqs. /H2084914/H20850and /H2084915/H20850, the emission and capture rates induced by hole–acoustic-phonon scattering in a QDare obtained, respectively, asLIet al. PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-6/H9261E=mkBT 2/H92664/H60363/H9267Q0/H20858 /H9263/H9008/H20849E/H9263−U0/H20850e−E/H9263/kBT /H11003/H208850/H11009dqqS Nm/H20849l0q/H20850/H20885 −/H11009/H11009 dqzR/H9261/H20849qz/H20850F−/H20849q,qz/H20850/H20849 27/H20850 and /H9261C=mkBT 2/H92664/H60363/H9267N3V/H20858 /H9263/H9008/H20849E/H9263−U0/H20850e−E/H9263/kBT /H11003/H208850/H11009dqqS Nm/H20849l0q/H20850/H20885 −/H11009/H11009 dqzR/H9261/H20849qz/H20850F+/H20849q,qz/H20850,/H2084928/H20850 where Q0=2/H20858/H9263e−E/H9263/kBTis the hole number in the bound states in a dot, N3=2/H20858Ke−EK/kBTis the hole density in the con- tinuum states in the system, F±/H20849q,qz/H20850=2/H9266a/H9014u2 uL2/H20875/H9014d2 /H9014u2−BA± 128aQ9+C 96Q8/H20876, a=/H208812m/H20849E/H9263−U0/H20850//H60362,Q=/H20881q2+qz2,B=/H20849Q2−a2/H208502B, A±= loga2qz+qzQ2±Q/H20849a2+Q2+2aQ/H110072aqz/H20850 a2qz+qzQ2±Q/H20849a2+Q2−2aQ±2aqz/H20850, B=q2/H2084924a2qz2−1 1a2q2−3q2Q2/H20850X+8/H20849q6−3q2qz4−2qz6/H20850/H9014* +/H208513q4/H20849a2+q2/H20850+3q2qz2/H20849Q2−8a2/H20850+8qz4/H20849a2−Q2/H20850/H20852,/H2084929/H20850 with X=/H20849uL/uT/H208502,/H9014=/H9014d//H9014u, and C=/H208513q4/H20849−3a4+6a2q2+q4/H20850+6/H2084912a4q2−1 9a2q4+1 5q6/H20850qz2 +/H20849−2 4a4−9 2a2q2+ 171 q4/H20850qz4+/H2084940a2+8 4q2/H20850qz6/H20852X* +2 4 /H20851q2+qz2/H208522/H20851q2/H20849a2+q2/H20850+/H20849−2a2+7q2/H20850qz2+6qz4/H20852/H9014* +/H208519a4q4−6a2q6+9q8+/H20849−7 2a4q2+ 114 a2q4+1 8q6/H20850qz2 +/H2084924a4+5 6a2q2+8 1q4/H20850qz4−1 6 /H208494a2−9q2/H20850qz6+7 2qz8/H20852. D. Further considerations In this study, we consider the contributions to the AS from both light and heavy holes in the GeSi QDs. Due to thediffusion of Si into Ge QD during the MBE growth, thecontent of Ge in the GeSi dots can be estimated reasonablyby popularly used methods proposed previously. 25,26We ﬁnd that for samples A, B, and C described in Sec. II, the Gecontent is x=0.75, as shown in Table I. The value of energy band offset on the alignment of valence band along thegrowth direction is 25Ev=U0=0.75 xeV. The effective masses for heavy and light holes and the relative dielectricconstant for material Ge xSi1−xcan be evaluated by the lin- early interruption with the Ge content xthrough26,27/H20875mh ml/H20876/H20882m0=/H20875− 2.8369 − 0.1432/H20876x3+/H208754.6844 0.3618/H20876x2+/H20875− 2.8700− 0.3669/H20876x +/H208750.8956 0.2534/H20876, and /H9280= 11.7 + 4.5 x. In the calculation, we take the width of the quantum well L along the growth direction to be the QD height shown inTable Iand the diameter of the QD /H9278along the xyplane to be the QD diameter shown in Table Ias well. The character- istic frequency /H92750of a QD is evaluated using Eq. /H2084926/H20850in which the QD diameter /H9278and ﬁlling factor /H9263are determined experimentally /H20849see Sec. II /H20850. The volume of a dot is also calculated using Land/H9278. Moreover, other material param- eters taken in the calculations for hole-phonon scattering arelisted in Table II. IV. RESULTS AND DISCUSSIONS In Fig. 1, we show experimental and theoretical results of the admittance spectroscopy for three QD samples with dif-ferent dot sizes at a ﬁxed ac ﬁeld frequency f=1 MHz. With the increasing size of the dot, the peak of the conductanceshifts to higher temperature. The peak positions and half-widths for three samples A, B, and C are shown in Table Ifor experimental and theoretical values. As can be seen, withincreasing the dot size the half-width of the conductancepeak increases. The results obtained from numerical calculations indicate that the emission rate /H9261 Egiven by Eq. /H2084927/H20850is about 2–3 orders of amplitude larger than the capture rate /H9261Cgiven by Eq. /H2084928/H20850. Thus, we can take /H9261E/greatermuch/H9261Cand use Eq. /H208491/H20850to ﬁnd out the emission rate experimentally. In Fig. 2, the experi- mental and theoretical results of the emission rate are plottedas a function of temperature for different samples A, B, andC. To determine experimentally the emission rate for a QDsample, we measure the conductance as a function of tem-perature for different frequencies of the ac ﬁelds. The emis-sion rate /H9261 Eis then the ac ﬁeld frequency /H9275=2/H9266fat the temperature corresponding to the peak of the conductance.Theoretically, /H9261 Eis calculated using Eq. /H2084927/H20850. We see fromTABLE II. Material parameters /H20849Ref. 27–29/H20850. Quantity Symbol Value Unit Longitudinal sound velocityuL 5247 m/s Transverse sound velocityuT 3267 m/s Dilatation deformation potential /H20849Ge/H20850/H9014d 2.0 eV Uniaxial deformation potential /H20849Ge/H20850/H9014u −2.16 eV Density /H20849Ge/H20850 /H9267 5.323 g /cm3ADMITTANCE SPECTROSCOPY OF GeSi-BASED QUANTUM … PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-7Fig. 2that with increasing the QD size the emission rate decreases and with increasing temperature the emission rateincreases rapidly. From Eq. /H2084927/H20850, we see that theoretically the emission rate of a GeSi QD is in a form of /H9261 E=/H9252T/H20858/H9263exp/H20849−E/H9263/kBT/H20850, with /H9252being a coefﬁcient independent on temperature. This im- plies that the emission rate can also be written as /H9261E =/H9252Texp/H20849−Ea/kBT/H20850, with Eabeing the activation energy of the carriers. Thus, using the results shown in Fig. 2, the activation energy can be determined experimentally byArrhenius plot. The Arrhenius plot is shown in Fig. 3for three QD samples A, B, and C, where the correspondingtheoretical results are also presented for comparison. We seethat with increasing the QD size, the activation energyincreases. As shown in Figs. 1–3, the theoretical results for tempera- ture dependence of the conductance and emission rate along with the Arrhenius plot for different samples are quite in linewith those measured experimentally. In samples A, B, and C,the Ge content in the dot is almost the same x/H112290.75 so that the energy barrier for heavy and light holes along the growthdirection U 0is almost the same. In such a case, a larger QD height along the zdirection Lcorresponds to the lower sub- band energies along the growth direction /H20851see Eq. /H2084928/H20850/H20852.I t can be seen from Table Ithat for samples A, B, and C, the QD height along the zdirection is L=30, 35, and 45 Å. Be- cause the AS measured here is induced by electronic transi-tion along the growth direction, the lower hole subband en-ergies imply that the holes in the bound states in a dot needmore energy to overcome the energy barrier U 0to be emitted into the continuum states. Together with the fact that thelower energy subbands are more possibly occupied by holes,with increasing size of the dots the peak of the conductanceshifts to the high-temperature regime, the emission rate de-creases and the activation energy increases. In GeSi-basedQD systems, the electron–acoustic-phonon scattering via de-formation potential coupling is the principal channel forelectronic transition between the bound states in the dot andthe continuum states in the systems. Under the action of anac ﬁeld, holes in the QD can gain the energy from this driv-ing ﬁeld and loss the energy via hole-phonon scattering andvia electronic transition from lower-energy bound states tothe higher-energy continuum states. Because phonon occupa-tion number increases with temperature, the strength of hole-phonon scattering increases with Tand, consequently, the emission rate increases rapidly with increasing temperature. The obtained experimental and theoretical results indicate that the emission rate in GeSi-based QD systems is about1–2 order of amplitude smaller than that in GeSi-based quan-tum wells. 12In a GeSi QD, the conﬁnement of hole motion in all spacial directions implies that the momentum conser-vation law does not hold anymore during a scattering event. Thus, the hole-phonon scattering can only occur among thequantized energy levels so that the effective hole-phononscattering can be reduced in a QD. This is the main physicsreason behind a lower emission rate observed in a QD thanin a quantum well. Furthermore, the theoretical and experi-mental results suggest that in a GeSi QD, the emission rate isin a form /H9261 E=/H9252Texp/H20849−Ea/kBT/H20850, in contrast to /H9261E =/H9252T1/2exp/H20849−Ea/kBT/H20850for a GeSi quantum well.12 It should be noted that in this study, we take a single QD for theoretical modeling. In contrast, the samples used in themeasurements are multidots with different areal densities/H20849see Table I/H20850, although the nonuniformity of the QD size is less than 10% in each sample. We believe this is the mainreason why the results obtained theoretically do not agreefully quantitatively with those measured experimentally.Nevertheless, our theoretical model can reproduce rightlythose observed experimentally with a reasonably goodagreement. V. SUMMARY In this study, we have examined experimentally the effect of the size of self-assembled GeSi QDs on features of admit-tance spectroscopy measured along the growth direction. Wehave found that with increasing the QD size, the peak of theconductance shifts to higher temperature, the emission ratedecreases, and the activation energy increases. In conjunc-tion with these experiments and experimental ﬁndings, wehave developed a simple and systematic theoretical approachto calculate the admittance spectroscopy in GeSi-based QDsystems and obtained the corresponding analytical and nu-merical results. Using this model calculation, we can obtainthe conductance and emission rate of carriers in a QD due tothe presence of the applied ac ﬁeld and electronic scatteringmechanisms. The proposed theoretical modeling is based onthe known sample and material parameters. Thus, the ob-tained theoretical results are in line with those measured ex-perimentally. We have applied the theoretical analysis to un-derstand the experimental ﬁndings. We hope this combinedexperimental and theoretical study can lead to an in-depthunderstanding of electronic properties of GeSi-based quan-tum dot systems. ACKNOWLEDGMENTS This work was supported by special funds from the Major State Basic Research Project, Commission of Science andTechnology of Shanghai, and the National Natural ScienceFoundation of China. One of the authors /H20849W.X. /H20850was sup- ported by the Australian Research Council. *wen105@rsphysse.anu.edu.au †fanglu@fudan.edu.cn 1See, e.g., T. J. Thornton, Rep. Prog. Phys. 58,3 1 1 /H208491995 /H20850. 2See, e.g., D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120/H208491998 /H20850. 3See, e.g., S. N. Molotkov and S. S. Nazin, JETP Lett. 63, 687 /H208491996 /H20850. 4See, e.g., H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y . Sugomito,LIet al. PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-8Appl. Phys. Lett. 69, 3140 /H208491996 /H20850. 5See, e.g., D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures /H20849Wiley, Chichester, 1998 /H20850. 6P. N. Brunkov, A. Patané, A. Levin, L. Eaves, P. C. Main, Y . G. Musikhin, B. V . V olovik, A. E. Zhukov, V . M. Ustinov, and S.G. Konnikov, Phys. Rev. B 65, 085326 /H208492002 /H20850. 7W.-H. Chang, W. Y . Chen, T. M. Hsu, N. T. Yeh, and J. I. Chyi, Phys. Rev. B 66, 195337 /H208492002 /H20850. 8J. F. Chen, R. S. Hsiao, C. K. Wang, J. S. Wang, and J. Y . Chi, J. Appl. Phys. 98, 013716 /H208492005 /H20850. 9Fengying Yuan, Zuimin Jiang, and Fang Lu, Appl. Phys. Lett. 89, 072112 /H208492006 /H20850. 10A. I. Yakimov, A. V . Dvurechenskii, A. I. Nikiforov, A. A. Blosh- kin, A. V . Nenashev, and V . A. V olodin, Phys. Rev. B 73, 115333 /H208492006 /H20850. 11F. Lu, D. Gong, J. Wang, Q. Wang, H. Sun, and X. Wang, Phys. Rev. B 53, 4623 /H208491996 /H20850. 12X. Li, W. Xu, F. Y . Yuan, and F. Lu, Phys. Rev. B 73, 125341 /H208492006 /H20850. 13A. Ishizaka and Y . Shiraki, J. Electrochem. Soc. 133, 666 /H208491986 /H20850. 14Jeff Drucker, IEEE J. Quantum Electron. 38, 975 /H208492002 /H20850. 15A. V . Dvurechenskii, J. V . Smagina, R. Groetzschel, V . A. Zi- novyev, V . A. Armbrister, P. L. Novikov, S. A. Teys, and A. K.Gutakovskii, Surf. Coat. Technol. 196,2 5 /H208492005 /H20850. 16J. Drucker, IEEE J. Quantum Electron. 38, 975 /H208492002 /H20850. 17F. Cerdeira, A. Pinczuk, J. C. Bean, and B. A. Wilson, Appl.Phys. Lett. 45, 1138 /H208491984 /H20850. 18V . I. Mashanov, H.-H. Cheng, C.-H. Chia, and Y .-H. Chang, Physica E /H20849Amsterdam /H2085028, 531 /H208492005 /H20850. 19QiJia Cai, Hao Zhou, and Fang Lu, Appl. Surf. Sci. 253, 4792 /H208492006 /H20850. 20M. Koskinen, M. Manninen, and S. M. Reimann, Phys. Rev. Lett. 79, 1389 /H208491997 /H20850. 21W. Xu, Phys. Rev. B 57, 12939 /H208491998 /H20850; W. Xu, F. M. Peeters, and J. T. Devreese, ibid. 46, 7571 /H208491992 /H20850. 22S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 /H208492002 /H20850. 23H. Mizuno, K. Taniguchi, and C. Hamaguchi, Phys. Rev. B 48, 1512 /H208491993 /H20850. 24C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 /H208491983 /H20850. 25R. Magalhães-Paniago, G. Medeiros-Ribeiro, A. Malachias, S. Kycia, T. I. Kamins, and R. S. Williams, Phys. Rev. B 66, 245312 /H208492002 /H20850. 26Karl Brunner, Rep. Prog. Phys. 65,2 7 /H208492002 /H20850. 27Lianfeng Yang, J. R. Watling, R. C. W. Wilkins, M. Boric, J. R. Barker, A. Asenov, and S. Roy, Semicond. Sci. Technol. 19, 1174 /H208492004 /H20850. 28M. V . Fischettia and S. E. Laux, J. Appl. Phys. 80, 2234 /H208491996 /H20850. 29F. Schafﬂer, in Properties of Advanced Semiconductor Materials: GaN, AlN, InN, BN, SiC, SiGe , edited by M. E. Levinshtein, S. L. Rumyantsev, and M. S. Shur /H20849Wiley, New York, 2001 /H20850, pp. 149–188.ADMITTANCE SPECTROSCOPY OF GeSi-BASED QUANTUM … PHYSICAL REVIEW B 76, 245304 /H208492007 /H20850 245304-9
PhysRevB.86.045205.pdf	PHYSICAL REVIEW B 86, 045205 (2012) Detailed calculation of the thermoelectric ﬁgure of merit in an n-doped SiGe alloy Iorwerth O. Thomas and G. P. Srivastava School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom (Received 31 January 2012; revised manuscript received 27 June 2012; published 16 July 2012) In this study, we develop a detailed numerical approach towards the theoretical calculation of the phonon contribution to the dimensionless ﬁgure of merit ZTthat parameterizes the efﬁciency of the thermoelectric effect, and apply it to the case of an n-doped Si 0.754Ge0.246alloy sample. This is achieved by using accurate lattice dynamical eigensolutions from the application of the density functional perturbation theory, supplemented by asemiempirical approach for crystal anharmonicity. The success of the application of the theoretical method forlattice thermal conductivity, within the single-mode relaxation time scheme, in explaining available experimentaldata is highlighted. Using well-known phenomenological theories, based on the nearly-free electron model,for the behavior of the electronic components of ZT, we are then able to calculate the ﬁgure of merit over a temperature range of 300–1200 K. DOI: 10.1103/PhysRevB.86.045205 PACS number(s): 72 .20.Pa, 63.20.dk, 63 .20.K−,6 6.70.−f I. INTRODUCTION The concept of conversion of electricity into heat and vice versa—the thermoelectric effect—has been considered veryuseful for several decades (see, e.g., Refs. 1–4). This has re- cently been the subject of intense theoretical and experimentalinterest. 5,6This interest has arisen largely since it has been shown experimentally that the efﬁciency of thermoelectricconversion (which is described by the ﬁgure of merit ZT) can be substantially increased with nanostructuring. 5–7In general, existing three-dimensional thermoelectric (TE) materials arefound to exhibit high efﬁciency ( ZT > 0.5) either at low temperatures (such as Bi 2Te3, between 200–500 K) or in an intermediate temperature range (such as PbTe, between 600–900 K), or at high temperatures (such as SiGe alloys, between 800–1300 K). It has also been suggested that it maysoon be possible to engineer thermoelectric structures thatfunction optimally across a wide variety of temperatures, 5 leading to obvious industrial applications. However, in orderto properly engineer nanostructured TE materials, we mustunderstand not only how nanostructuring affects the variousparameters governing thermoelectric efﬁciency but whichparameters produce the greatest effect. At present, it seems thatapproaches that attempt to minimize the phonon contributionto the thermal conductivity show most promise. 5,8However, the theoretical behavior of this quantity is not currently wellunderstood. Over the past several decades, a large number of theoretical investigations have been undertaken in order to assess ZTof semiconductor single crystals. While the electronic propertiesare reasonably well formulated within the nearly-free-electronmodel, numerical calculations that fully account for thetemperature variation of that most basic of quantities, viz., theFermi energy E Fhave not necessarily been made. The quantity that has remained the least well convincingly studied is thelattice contribution to thermal conductivity κ ph. In this respect, four publications deserve to be mentioned. Meddins andParrott 9do not elaborate on any speciﬁc method for evaluating the electrical conductivity σ, Seebeck coefﬁcient Sand the bipolar contribution to thermal conductivity and resort to ahigh-temperature interpolative scheme for the calculation ofκ phfor SiGe alloys. Vining10has adopted a complete treatmentfor the electronic contributions, viz., for σ,S, electronic and bipolar contributions to thermal conductivity κelandκbp, respectively. However, this work does not explicitly considerthe temperature dependence of E F. Moreover, Vining has employed a phenomenologically derived, simple expression for the anharmonic phonon relaxation time, incorporating a low-temperature form of the three-phonon Umklapp scatteringrate, where the three-phonon Normal scattering rate is taken tobe a simple scaling (by an undeclared factor) of the Umk-lapp scattering rate. These low-temperature considerationsshould not be expected to be valid at high temperatures,entailing that an application of that scheme to systems ofreduced dimensionality is unlikely to be useful. Similarly,the more recent work of Minnich et al. 11improves on Vining’s10and Slack and Hussein’s12similar treatment of the electronic parameters of the system, but performs thephonon conductivity calculations using an approach similarto that used by Vining. Slack and Hussein 12themselves use an empirical approach to the phonon conductivity. Hicks and Dresselhaus8have attempted to explicitly include the Fermi energy EFin their formulation of σandS, but did not account for its temperature variation. Although they have discussedthe effects of reducing dimensionality on the various TEcoefﬁcients, in their treatment, κ phis considered to be no more than a temperature-independent adjustable parameter. Thus itseems reasonable to conclude that—to our knowledge—thereis currently no publication that presents a systematic numericalcalculation of the TE coefﬁcients of semiconductors over awide range of temperatures. 13 In this paper, we aim to provide as complete and accurate a theoretical approach as possible to the calculation of the phonon contribution to the thermal conductivity (and hence tothe calculation of ZT) in alloyed systems, which we hope will serve as a basis for future work performed on more complex, nanostructured systems. In the next section of the paper, we shall outline the theoretical background of our calculation. Firstly, we discussthe behavior of the electronic components of ZTusing the well-known expressions based on the nearly free-electronmodel for the electronic band structure. The behavior ofthese components is well understood in bulk materials—our 045205-1 1098-0121/2012/86(4)/045205(16) ©2012 American Physical SocietyIORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) treatment of them will hence be phenomenological and a summary in nature, as it is the calculation of the contributionarising from phonons that is most challenging and which shalltherefore command the majority of our attention. Next, weturn to our theoretical scheme for the computation of thelattice contribution to ZT, viz., the phonon conductivity. One important element in our approach that should be emphasizedis that we make use of an expression for the anharmonic crystalpotential that includes contributions from the optical as well asacoustic modes; this expression is deﬁned in a semiempiricalfashion, and we expect this approach to complement other,recently developed approaches such as those of Refs. 14–16. This will be discussed in more detail in the second section ofthe paper. The third section details our approach for calculating the phonon eigenmodes, frequencies and velocities throughthe use of density functional perturbation theory (DFPT) 17 (reviewed in Ref. 18). In the fourth section, we detail the results of our study, beginning with the results of our ﬁttingof the electronic parameters to the experimental data, andthen, turning to the focus of this paper, we initiate ourdiscussion of the phonon-dependent aspects of the samplewith an examination of the form of the dispersion relationsof the model cell used in the DFTP calculations and howthey are affected by bond relaxation, and a discussion of sometheoretical issues surrounding the notion of a “mean-free path”and its deﬁnition within the context of the study of thermalconductivity. Lastly, we discuss the results of our calculationof the total thermal conductivity and the ﬁgure of merit ZT and how they compare to the experimental results for ZTof sample 7 in Meddins and Parrott, 9before summarizing our conclusions. II. THEORETICAL BACKGROUND As indicated above, a typical measure of the efﬁciency of thermoelectric conversion is given by the dimensionless ﬁgureof merit ZT, deﬁned as follows: ZT=S 2σT κ, (1) where Sis the Seebeck coefﬁcient for the material, σis its electrical conductivity, Tits temperature, and κis its total thermal conductivity. We can treat the thermal conductivity asas u m : κ=κ el+κph, (2) where κelis the electronic and κphis the phonon (lattice) contributions. Obviously, it is through the latter term that thephonon physics of the material affects the ﬁgure of merit.Following previous theoretical works, 8,10–12we will evaluate σ,S, andκelwithin the application of the nearly-free electron model. We present an extensive theoretical approach forcalculation of the lattice thermal conductivity κ ph. A. Electronic components of ZT Firstly, we shall examine the behavior of the components ofZT, which are primarily electronic in nature ( S,σ,κel), as a prelude to examining the more difﬁcult phonon conductivitycontribution κ ph. Since we are more interested in the ﬁnedetail of the latter, our examination of the former is of necessity somewhat brief, being more concerned with suitablephenomenological modeling of the quantities in question asopposed to the minutiae of their physics. To this end, we shalldiscuss the application of our expressions to the relevant datain this section, so as not to detract from the main focus of ourstudy. 1. Temperature variation of Fermi energy Numerically accurate values of the Fermi energy are required at each temperature we wish to calculate the electronictransport coefﬁcients. For the system under study, we use thefollowing expression for the incompletely ionized, extrinsicsemiconductors, 19but replacing the ﬁrst term with one more suited to the case of strongly degenerate semiconductors:20 EF=(3π2Ndon)2/3 2¯h2(2Nval)2/3mdose+kBT 2lnNdon 2AeNval −kBTsinh−1/bracketleftBigg/radicalBigg AeNval 8Ndone−(Ec−Ed)/2kBT/bracketrightBigg .(3) Here, mdos eis the density-of-states effective mass and Nval is the number of valleys in the conduction band. We deﬁne Ae/h=2(mdos/∗ e/hkBT/¯h2)3/2, and the quantity Ndonrepresents the number of donor impurities. 2. Seebeck coefﬁcient We adopt the nearly free electron model and use the following expression for the absolute value of the Seebeckcoefﬁcient: 4 \|S\|=kB e[M(α)−ηR], (4) M(y)=(s+2.5) (s+1.5)Fs+3/2(y) Fs+1/2(y). (5) Here, α=ηR−(Ec/kBT), where ηR=EF/kBTwithEF being the Fermi energy, sis determined by the dominant electron scattering mechanism, and we make use of the Fermiintegrals F a(y)=/integraldisplayxa ex−y+1dx. (6) 3. Electrical conductivity We adopt the nearly-free electron model and write the expression for the conductivity derived in Drabble andGoldsmit, 21generalized to the case of multiple valleys, as σ=Nvale2τ0 3π2m∗c/parenleftbigg2mdos ekBT ¯h2/parenrightbigg/parenleftbigg p+3 2/parenrightbiggp Fp+1/2(α),(7) where Nvalis the number of energy valleys, m∗ cis the conductivity effective mass, and τ0represents the energy- independent portion of the relaxation time τ=τ0(αkBT)p, (8) 045205-2DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) withpbeing a constant depending on the type of scattering. We have considered ionized impurity (imp) scattering, acousticphonon (ac) scattering, and optical phonon (op) scattering asbeing relevant to our work. Expressions for these scatteringrates from each valley are 22 τ−1 imp=Z2e4NiC 16π/epsilon12(2mdos)1/2kBT)3/2α−3/2, (9) τ−1 ac=√ 2E2 Dmdos3/2(kBT)3/2 π¯h4ρc2 Lα1/2, (10) τ−1 op=(2mdos)3/2D2 o 4π¯h3ρω0[(¯n+1)/radicalbig (αkBT−¯hω0) +¯n/radicalbig (αkBT+¯hω0)]. (11) In the above expressions, m∗ cis the conductivity effective mass, ρis the density of the sample, cLthe speed of LA phonons, ED is an acoustic deformation potential, ω0is the frequency of the highest longitudinal optical mode, Dois an optical deformation parameter, /epsilon1is the host dielectric constant, Zeis the impurity charge, Niis the impurity concentration, and Cis a constant that is usually between 1.4 and 2.22The ﬁrst and second terms in the optical scattering expression are contributionsfrom phonon emission and absorption events, respectively.From the above expressions, it is clear that the parameterptakes values 3 /2 for impurity scattering and −1/2f o r acoustic and optical phonon scattering. The conductivity ex-pressions, considering individual scattering mechanisms, thenbecome σ imp=64√ 3Nval/epsilon12/parenleftbig mdos e/parenrightbig2 πm∗cZ2e2NiC/parenleftbiggkBT ¯h/parenrightbigg3 F2(α), (12) σac=2Nvale2¯hρc2 L 3πE2 Dm∗cF0(α), (13) σop=/braceleftbigg σop,ab E/lessorequalslant¯hω0 σop,ab+σop,emE> ¯hω0, (14)σop,ab=4Nvale2ρω0kBT 3πD2o¯nm∗cF0(α), (15) σop,em=4Nvale2ρω0kBT 3πD2o(¯n+1)m∗cF0(α). (16) Using Matthiessen’s rule, we sum the resistivity contribu- tions as follows: σ−1 n=σ−1 ac+σ−1 op+σ−1 imp. (17) We denote the conductivity as σnsince it is the electronic contribution to the conductivity. For our system, and in thetemperature range of interest (i.e., above 300 K), the impurityand optical phonon scattering rates are found, respectively, tobe approximately four and two orders of magnitude smallerthan the acoustic phonon scattering rate. We thus consideredonly the dominant acoustic contribution and neglected theothers. We remark that there also exists a small contribution tothe conductivity from holes, which we denote σ p, and model in a simple fashion described later in this paper. This lastdoes not signiﬁcantly contribute to the overall conductivity,which is dominated by σ n, however, it will prove important when considering the bipolar contribution to the thermalconductivity, which we discuss next. 4. Electronic and bipolar contributions to thermal conductivity The electronic contribution to the thermal conductivity is typically taken to be κel∝σeT, with the Lorentz number Lbeing the constant of proportionality (see, for example, Refs. 19,21, and 23). However, a complication arises when we deal with the behavior of the system at high temperatures,which require an account of the effects of thermally excitedholes. The thermal conductivity contributions arising fromelectrons and holes are not simply additive; in combinationthey give rise to an additional contribution known as thebipolar term, 2,21,23–25which may be quite large even if σh/σn is small. Following the derivation given in Ref. 21,w em a y write the total electronic conductivity (including this term) as κel=k2 B e2/braceleftBigg L(α)σnT+L(β)σpT+σnσpT σ/bracketleftbigg(Ec−Ev) kBT+M(α)+M(β)/bracketrightbigg2/bracerightBigg , (18) L(y)=(s+3.5)(s+1.5)Fs+5/2(y)Fs+1/2(y)−(s+2.5)2Fs+3/2(y)2 (s+1.5)2Fs+1/2(y)2, where we have deﬁned β=(Ev/kBT)−ηR. The hole density can be written as nh≈n2 in/Ndon,19where nin=√AeAhexp[−(Ec−Ev)/2kBT], allowing us to make use of the following simpliﬁed expression: σp=eGexp[−(Ec−Ev)/kBT], (19) withGtreated as an adjustable parameter. B. Phonon contribution to thermal conductivity Here, we move the focus of our discussion to the theoretical heart of our work: the calculation of the lattice (phonon)portion κphof the thermal conductivity κ. Working within the single-mode relaxation time approximation,26we write the following expression for the thermal conductivity: κph=¯h2 3VkBT2/summationdisplay qsc2 s(q)ω2(qs)τ(qs)¯nqs(¯nqs+1) =/summationdisplay qsκqs, (20) where the total volume of the system is V=Ncell/Omega10(Ncell being the number of unit cells in the system and /Omega10being the volume of each unit cell), qbeing the phonon wave vector, s 045205-3IORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) is the label of the phonon branch, cs(q) is the magnitude of the velocity for a given mode qs,ω(qs) is the frequency, and ¯nqsis the Bose-Einstein distribution at equilibrium. The total single-mode relaxation time τ(qs) is given by τ−1(qs)=τ−1 BD(qs)+τ−1 MD(qs)+τ−1 EP(qs)+τ−1 AH(qs),(21) where we have contributions from boundary scattering τBD(qs), mass-defect effects τMD(qs), electron-phonon inter- actions τEP, and anharmonic phonon-phonon interactions τAH. We shall discuss the ﬁrst three contributions separately fromthe latter. 1. Harmonic scattering processes The boundary scattering of phonons is expressed as26 τ−1 BD(qs)=cs(q) L, (22) where Lis a measure of the effective size of the crystallite microstructure in the sample studied in Ref. 9. Mass-defect scattering arises due to the perturbation in crystal potentialfrom the difference between the average mass of the alloy andthe actual mass at a given atomic site of a particular species orisotope. We use the form given in Ref. 27, τ −1 MD(qs)=ω2(qs)gs(ω)PN cell 4π/Gamma1MD, (23) where gs(ω) is the phonon density of states and Pis an adjustable parameter that we have introduced in the mannerof Ref. 28in order to absorb the effects of defects such as impurities and vacancies into τ MD(qs), which are otherwise hard to parameterise. /Gamma1MDdetermines the average effect of species and isotope masses on the relaxation time. We use theformulation in Ref. 29, which incorporates both alloying and isotope effects in a relatively simple fashion, and express it forthe alloy Si xGe1−xas /Gamma1MD=x/parenleftbiggmSi ¯m/parenrightbigg2 /Gamma1IS(Si)+(1−x)/parenleftbiggmGe ¯m/parenrightbigg2 /Gamma1IS(Ge).(24) Here,mSis the average mass of the species S, ¯mis the average mass of the alloy constituents, and /Gamma1IS(S) incorporates the effects of the different isotope masses /Gamma1IS(S)=/summationdisplay ifi/parenleftbiggmi−mS mS/parenrightbigg2 , (25) where miis the mass of the ith isotope and fiits frequency. We use isotope data from Ref. 30for Ge and from Ref. 31 for Si. We parameterize the electron-phonon scattering through a generalization of the expression for medium-high doping usedin Ref. 9: τ −1 EP(qs)=neβ1/vextendsingle/vextendsingleEs def/vextendsingle/vextendsingle2√πα1e−α1 ¯hρcs(q)2. (26) Here,ρis the density of the system, α1=m∗ ecs(qs)2/2kBTand β1=¯hω(qs)/kBT, and we approximate newithNdon.Edefis the deformation potential, expressed as26 Es def=ANat/summationdisplay iˆq·ei qs. (27)In this expression, ei qsis a member of the set of Nateigenvectors of an acoustic phonon mode qs,Natis the number of atoms in the unit cell, and Ais an empirically adjustable parameter that controls the strength of the deformation potential and thusdetermines the magnitude of the electron-phonon scattering.Since this expression is only valid for acoustic phonons, inour numerical calculations, we set it equal to zero when thefrequency corresponding to a mode ( qs) is greater than some cutoff value corresponding to the largest value of an acousticfrequency at the Brillouin zone boundary (in our case, this is170 cm −1, which is the largest value at the X point in the z direction). 2. Anharmonic scattering processes The calculation of the anharmonic contribution to the phonon relaxation time is far from simple. The behavior ofthis aspect of the system is determined by the values of thethird-order force constants, whose effects can be summarizedthrough a number of parameters such as the mode-dependentGr¨uneisen constants ( γ MD), which determine the strength of the scattering for different three-phonon processes.26One might think that the ab initio calculation of the third-order constants using DFPT methods such as those described in Refs. 18 and32or the force-displacement approach of Ref. 33would be sufﬁcient unto themselves, but there are a number of caveatsthat suggest that these approaches may be complemented byothers such as the one employed in this paper. Firstly, Lopuszy ´nski and Majewski 34have shown that in the elastic limit, ab initio calculations carried out via the theory of nonlinear elasticity show that while values of γMDfor longitudinal acoustic (LA) modes are in reasonable agreementwith experiment with discrepancies of around 0.9 to 10%, thedifference between experimental and theoretical values of thetransverse acoustic (TA) modes can be between roughly 84%to 98%. They suggest that this could be an indication that theﬁnal value of γ MDis very sensitive to numerical errors in the third-order elastic constants on which it depends. Somethingsimilar could be true in the case of ab initio calculations performed outside of elastic limit, for in their recent, state-of-the-art calculation Esfarjani, Chen, and Stokes 16employ the method of Ref. 33in order to calculate force-constants ab initio and ﬁnd that the values of the TA modes are again rather different from experiment due to a truncation in the number ofthird-order force constants used (they note that a large numberare needed in order to calculate these modes correctly). Whilewe are not presently aware of any calculations of the variousγ MDfor the method of Ref. 32, on the strength of the foregoing one should perhaps avoid assuming unreservedly that an ab initio DFPT calculation will exactly reproduce the values of quantities that may be very sensitive to numerical error. A further interesting issue is the temperature dependence of theγMDand their constituent force constants; if we examine the experimental behavior of the mode-averaged Gr¨uneisen constant γin Si and Ge in Figs. 61 and 51 on p. 373 and p. 413, respectively, of Ref. 35, we can see a rather strong temperature dependence as Tis increased from 0 K to around 300 K, with γdecreasing from a positive value to a negative minimum before increasing once more and ﬁnally tendingtowards a positive value that is not necessarily identical to the 045205-4DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) value close to T=0 K. This implies that the behavior of the γMDis also temperature dependent, something that can also be inferred from measurements of the temperature dependenceof the linear expansion constants 36from which γMDmay be derived. However, DFT and DFPT calculations are carried outatT=0 K and do not give the behavior of the force constants as temperature is varied. Studies of the lattice heat transportin Si, 14,16Ge,14and SiGe alloys15dogive experimentally compatible values for various temperature ranges (see Ref. 14 for the entirety of the T=0–300 K range, Ref. 16for T=100–500 K, and Ref. 15forT=300–600 K), and that deviations from experiment outside of those ranges may beaccounted for by additional scattering effects as discussed inRef. 16for their results; but these studies contain no discussion of the effects of temperature on the force constants. Oneexception to this is Ward and Broido’s ab initio examination of phonon relaxation times in Si and Ge, 37where the effects of temperature on lattice properties via thermal expansionhave been tested by varying the lattice constant by an amountappropriate for the temperature. They ﬁnd that this gives resultsfor the overall thermal conductivity that are within 1–2% of thevalue calculated using the T=0 K force constants. It should be observed that the assumption that phonon properties areaffected only by thermal expansion as temperature is variedis a product of the quasiharmonic approximation, 18,26,38but since this approximation seems to be assumed valid in manycalculations of this type (including the one in this paper) anyproblems arising from it are unlikely to be unique to theirwork. That the aforementioned calculations work so well in spite of the potential problems we have raised is perhaps due to thecomplexity of the anharmonic contribution to the properties ofthe systems of interest. Simply put, it is difﬁcult to tell how theindividual mode contributions to each allowed three-phononscattering event sum or cancel, since there are so many, and itis possible that much of the detailed variation due to error ortemperature may be counteracted by their weighting as theyenter into the calculation of the relaxation time. It is hard to tellat this point; perhaps more work is needed. This complexityalso renders the calculations somewhat opaque to theoreticalanalysis despite their accuracy, and hence there is still muchneed for an approach to anharmonic and similar effects thatis intermediate between that of a full ab initio approach and traditional long-wavelength limit calculations. In future work, we intend to examine how lattice scattering is modiﬁed due to the properties of various nanostructures and,consequently, how the thermal conductivity (and thereforeZT) are affected by this modiﬁcation. A full calculation would probably work against our aim of understanding themechanisms involved since the complexity of the anharmoniccontributions to systems in question might obscure the majordetails of the underlying mechanisms, and so we choosea semiempirical approach that employs ab initio DFPT elements, which we hope will capture as many of thesefactors as possible. The price we pay is the loss of a certainamount of predictive power regarding a given system; however,the prize we gain is that it becomes relatively simple topredict the effects of changes in structure between systems and account for the mechanisms underlying them, while inaddition (hopefully) attaining good qualitative estimates ofthe magnitude of these effects. However, to begin with, we here test our approach using as detailed a calculation of thelattice contribution to the thermal conductivity as possible fora known system where experimental results are available, notonly for reasons of simple validation, but because it gives anidea of plausible semiempirical parameter ranges suitable forcases of nanostructured systems of similar composition whereappropriate experimental results may not yet be available. The approach we have chosen is developed from the concepts presented in Refs. 26,39, and 40. In brief, we describe the anharmonic phonon-phonon scattering through the use ofa continuum model to treat acoustic as well as optical phononmodes, and express the three-phonon scattering strength usingthemode-averaged Gr¨uneisen’s constant γ, which we treat as a semiempirical adjustable parameter that can also be madetemperature dependent if desired. The relevant expression forphonon anharmonic relaxation time in this approach is τ −1 AH(qs)=π¯h¯γ2 ρV/summationdisplay q/primes/prime,q/prime/primes/prime/prime,G(Bqs,q/primes/prime,q/prime/primes/prime/prime)2 ω(qs)ω(q/primes/prime)ω(q/prime/primes/prime/prime) ×/bracketleftbigg¯nq/primes/prime(¯nq/prime/primes/prime/prime+1) (¯nqs+1)δ(ω(qs)+ω(q/primes/prime) −ω(q/prime/primes/prime/prime))δq+q/prime,q/prime/prime+G+1 2¯nq/primes/prime¯nq/prime/primes/prime/prime ¯nqsδ(ω(qs) −ω(q/primes/prime)−ω(q/prime/primes/prime/prime))δq+G,q/prime+q/prime/prime/bracketrightbigg , (28) where Bi,j,k={/radicalbig ω(i)ω(j)[ω(i)+ω(j)]\|ω/Gamma1(k)−ω(k)\|/c(k) +similar terms with i,j,andkinterchanged }/3!, (29) withω/Gamma1(k) being the frequency at the /Gamma1point (zone center) for mode kandc(k) is the phonon speed for the branch and momentum labeled by k. A zero (nonzero) reciprocal lattice vector Gaccounts for a Normal (an Umklapp) process and ¯γis a mode-averaged rescaled Gr¨unneissen constant. The reasoning behind this generalization is discussed in theAppendix to this paper. We observe that this approach doesnot make use of two of the three approximations examinedin Ref. 37as it uses complete phonon dispersions rather than Debye-type approximations and accounts for the effects ofoptical modes in addition to acoustic modes. It does, however,make use of the third approximation (that of the elasticcontinuum) but we believe that it can provide good qualitativeresults nonetheless. III. NUMERICAL AND COMPUTATIONAL DETAILS A. Electronic parameters for sample under study Before discussing the electronic transport coefﬁcients S, σ, and κel, we ﬁrst note that the system of the present study, sample 7 of Meddins and Parrott’s study,9is a sintered Si0.754Ge0.246n-type doped alloy with P impurities acting as donors. In their study, Meddins and Parrott determinedthe carrier concentration of the sample at room temperatureto be 9 .4×10 25m−3(we take this value to be a reason- able approximation of Ndon) and its density to be around 045205-5IORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) TABLE I. Parameters for the electronic contributions to ZTderived from comparison with experimental data. We list the name of the parameter, its value, the measured quantity with which it is associated, the corresponding equation for that quantity, and the ﬁgures in which the theoretical behavior of that quantity is plotted alongside experimental measurements. A dagger indicates that this quantity was selected rather than directly ﬁtted. Parameter Value Quantity Equation Figure mdos e 0.51me \|S\| (5)via(3) 3 cL 7504 ms−1† σac (13) 4 C 0.0013 † σn (32) 4 ED 10.5eV †11σac (13) 4 mc∗ 0.45me† σac (13) 4 ν 0.4 σn (32) 4 m∗ /bardbl 1.10me ··· ··· ··· m∗ ⊥ 0.35me ··· ··· ··· G 3.2×1023/parenleftbigT 300/parenrightbig2.3m−1V−1s−1κ (2)via(18) and(19) 4 ,10 2860 kg m−3(this is lower than might be expected through a weighted averaging of the masses of the component elementsand could indicate that the sample is somewhat porous). Forour calculations, we will also require the relative locations ofthe conduction band edge E c, the donor energy level Ed, and the valence band edge Ev. Since Meddin and Parrott did not measure these values in their study, we must estimate them.We set E c=0 so that it functions as a reference energy, and take our values for the valence band edges from Fig. 8of Ref. 41and the donor levels from Table 9.1 on page 269 of Ref. 19. Accordingly, for Si, we take Ed=0.045 eV and Ev= Ec−1.170+[4.73×10−4/(636+T)]T2eV; and for Ge, we take Ed=0.012 eV and Ev=Ec−0.7437+[4774 × 10−4/(235+T)]T2eV . We then estimate the corresponding values for the alloy as a weighted average of the values forSi and Ge. This gives us values of E d=−0.037 eV and a set of temperature dependent values for Ev.W et a k e Nval=6, as the composition of our sample places us within the Si-likeregion. 35The value of cLis taken to be 7504 ms−1, which is the speed of the LA mode at the qpoint closest to /Gamma1in the Monkhorst Pack grid used in our phonon calculations. Table I lists the electronic parameters used in this work. B. Technical aspects for calculations of phonon-related quantities Firstly, we replace Brillouin zone integration of required functions with summation over a set of momenta and asso-ciated weighting factors generated through a Monkhorst-Pack(MP) scheme. 42In order to avoid complication in the numerical calculation of velocities of phonons, we chose to use ashifted set of MP points (i.e., a set that does not includesymmetry points in the zone). Secondly, we note that Eq. (28) includes a number of δ-function terms that are difﬁcult to evaluate numerically and must be approximated. We follow theapproximations used in Refs. 39and40and replace the exact momentum conservation condition with an approximation asfollows: q+q /prime±q/prime/prime−G=0→\|qμ+q/prime μ±q/prime/prime μ−Gμ\|/lessorequalslant/Delta1μ, μ=x,y,z, (30) with/Delta1μbeing the absolute value of the smallest momentum division of the MP grid in the μdirection. This replacementis required as the use of the MP grid entails that the exact condition can never be fulﬁlled; an approximation is neededin order to obtain nonzero values of the inverse anharmonicrelaxation time. Previous calculations 39,40have shown that this procedure yields reasonable results. We use a set of 26 G vectors, which produces a stable result in this case. Thirdly,we use the deﬁnition of the Dirac δfunction in terms of a Gaussian function: δ(y)=lim σ→01 σ√πe−y2/σ2, (31) where 0 /lessorequalslanty/lessorequalslant0, the energy conservation condition has been rescaled by ω/Gamma1MAX(the largest zone-center frequency), and we choose a broadening factor σthat is numerically appropriate (for this study, we use σ=0.5). Ifδ(y)<0.01 for a given set of modes, we instead set it to zero, in order to avoidspurious contributions arising from large quantities far fromthe center of the Gaussian. Table IIlists the parameters used in the calculations of phonon-related quantities. C. Calculations of phonon eigensolutions and velocities As discussed in the previous section, for numerical evalua- tion of κph, we require to calculate phonon frequencies {ω(qs)}, eigenvectors {e(qs)}, and velocities {v(qs)}. We obtained these quantities through the application of density functional theoryand linear response theory [the combination known as densityfunctional perturbation theory (DFPT)], 17,18as implemented in the routines incorporated into the QUANTUM ESPRESSO package.43Norm-conserving pseudopotentials utilizing a local density approximation (LDA) to the DFT44were employed in our calculations. TABLE II. The various parameters determining the behavior of κph, their values, and the quantities and equations associated with them. Parameter Value Quantity Equation L 0.2μm τ−1 BD(qs) (22) P 400.0 τ−1 MD(qs) (23) A 0.8 eV τ−1 EP(qs) (26) via(27) ¯γ 0.63 τ−1 AH(qs) (28) 045205-6DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) FIG. 1. (Color online) The Si 0.75Ge0.25cell used in our DFT calculations. Blue (dark) spheres represent Si atoms, and lilac (light)spheres represent Ge atoms. Image generated using XCRYSDEN (see Ref. 47). We simulated the Si 0.75Ge0.25crystal structure by consider- ing the eight-atom cubic unit cell depicted in Fig. 1. We do not expect there to be any signiﬁcant error arising from the slightdiscrepancy in alloy composition, nor from the (inevitably)ordered nature of this alloy model when comparing ourresults with measurements on the Si 0.75Ge0.25disordered alloy studied in Ref. 9. Using a cell dimension of a=5.48˚A consistent with Vegard’s law of the weighted average (i.e. a= 0.75aSi+0.25aGe), we adopted shifted 4 ×4×4, 6×6×6, 8×8×8, 10×10×10 Monkhorst-Pack grids with kinetic energy and density cutoffs of 15.0 and 60.0 Ry, respectively,and performed an electronic calculation while allowing theatomic coordinates in the cell to relax. DPFT calculationswere carried out using the equivalent unshifted grids, andfrom the resulting force constants we calculated the phononeigensolutions for the original shifted grids. From these results,the thermal conductivity matrix was calculated, and κ phwas taken to be the average value of the diagonal components,neglecting the off-diagonal values as artifacts of our model.Comparisons of the thermally averaged mean-free path due toanharmonic scattering for a system with ¯ γ=0.5a tT=100, 600, and 1200 K were made in order to check convergence(see Table III). As can be seen, these results are not very promising, however, if we compare values of κ phfor various grids calculated using the parameters (see Table II) obtained through the ﬁtting procedure used on 10 ×10×10 results and listed in Table IV, we see a very different picture. Here, the convergence is monotonic, and is of the order of less than 30%in the worst case ( T=1200 K) for the ﬁnest grid, which is at the limit of our computational capacity. We suspect thatthis difference in behavior may be due to divergences in thevalue of the anharmonic mean-free path that are present at lowvalues of q; 38these may become troublesome as ﬁner grids (containing ever smaller qvalues) are employed. As Ziman notes in his discussion of the divergence problem in Ref. 38, the cure is the inclusion of additional forms of scattering(usually boundary scattering); since we have (as far as weare aware) accounted for all major sources of scattering inour ﬁnal calculation of κ ph, this may be the reason for the improvement in convergence. The decrease in convergence asTis increased is likely due to the increasingly important role that the anharmonic scattering plays at higher temperatures;however, we should note that in the region of interest its lack ofconvergence is still somewhat tamed by the presence of otherscattering mechanisms. This also indicates that the mean-freepath for anharmonic scattering should not be used alone inorder to estimate convergence; the effects of other scatteringprocesses must be taken into account. There remains only the choice of an appropriate MP grid for our calculation. As implied above, we used the eigensolutionscorresponding to the 10 ×10×10 grid in our calculation ofκ ph; it was felt that this grid presented the best trade-off between convergence and computational effort available. IV . RESULTS A. Electronic properties Figure 2shows the temperature variation of the Fermi energy EFfor the n-doped sample number 7 in Ref. 9with temperature. There is clear evidence that the system showsextrinsic-type behavior in the region of interest. We assumedthat acoustic scattering was dominant and so ﬁxed the valueof the scattering parameter at s=−0.5. We then numerically ﬁtted our expression for \|S\|to the experimental data of Ref. 9 using the algorithms described in (for example) Ref. 45in order to ﬁnd an optimal value of m dos e(displayed in Table I). Since this also required the calculation of EF, we are able to display the behavior of that (see Fig. 2) and of \|S\|(see Fig. 3) with temperature. TABLE III. Convergence of the mean-free path due to anharmonic scattering calculated on successively more ﬁne shifted MP grids for T=100 K and ¯ γ=0.5. Here, /Delta1λTE AHis the relative difference between the value of λTE AHcomputed for the present grid and the value of λTE AH computed for its successor, expressed as the nearest whole percentage of the smallest value. T=100 K T=600 K T=1200 K MP Grid Number of grid points λTE AH(m) /Delta1λTE AH λTEAH(m) /Delta1λTE AH λTEAH(m) /Delta1λTE AH 4×4×41 2 0 .3445×10−7··· 0.3567×10−8··· 0.1748×10−8··· 6×6×63 6 0 .6586×10−6181% 0 .5530×10−7145% 0 .2661×10−7142% 8×8×88 0 0 .3251×10−6100% 0 .3041×10−781% 0 .1485×10−779% 10×10×10 150 0 .8427×10−6159% 0 .7355×10−7141% 0 .3560×10−7139% 045205-7IORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) TABLE IV . Convergence of κphatT=100 K due to anharmonic scattering as MP grids are made successively more ﬁne, using the parameters of Table II. Here, /Delta1κ phis the relative difference between the value of κphcomputed for the present grid and the value of κph computed for its successor, expressed as the nearest whole percentage of the smallest value. T=100 K T=600 K T=1200 K MP grid κph(W m−1K−1) /Delta1κ ph κph(W m−1K−1) /Delta1κ ph κph(W m−1K−1) /Delta1κ ph 4×4×4 3.4321 ··· 1.3251 ··· 0.79718 ··· 6×6×6 4.8023 40% 4.2119 218% 3.8353 381% 8×8×8 3.3913 42% 2.8882 45% 2.3016 67% 10×10×10 3.3795 0.35% 3.3841 17% 2.9478 28% Figure 4shows the temperature variation of σ. As with the Seebeck data, it can be seen from Meddins and Parrott’sexperimental data 9that there is no obvious transition to a region of intrinsic-type conductivity for the temperature rangeof interest. However, in order to properly account for thetemperature dependence of σ n, it was found that multiplication by an ad hoc term was necessary: σn=σacCTν, (32) where Candνare adjustable parameters. The need for these terms may be a result of inelastic scattering processes thatare neglected in our analysis (see Ref. 19); the necessity of including a similar factor has also been noted in arecent theoretical analysis of conductivity data in BiTe-basedsystems. 46 We take E2 Dm∗ c/Candνas the overall adjustable parameters for our ﬁnal ﬁt, obtaining a value of 3286 .90meeV2for the former and 0 .4 for the latter. Choosing ED=10.5e V , consistent with Ref. 11,C=0.013 and m∗ c=0.45me,w e obtain the following values for the parallel and transversemasses by using the equations relating them to m ∗ candmdos e:19 m∗ /bardbl=1.10meandm∗ ⊥=0.35me. It should be observed that m∗ ⊥is slightly more than twice what one would expect from a linear average of the corresponding Si and Ge masses, whereasm ∗ /bardblis more or less what is expected; this is because our m∗ cand mdos evalues are larger than expected. We display the resulting parameters in Table I. 200 400 600 800 1000 1200 Temperature (K)-0.15-0.1-0.0500.05EF (eV) FIG. 2. (Color online) Calculated behavior of EFwith tempera- ture for the system of our study.There is also a small contribution to σfromσh, which can have a large effect on the bipolar contribution to the thermalconductivity. We modelled the hole conductivity using Eq. (19) and obtained the parametrization displayed in Table Ithrough consideration of the total value of the thermal conductivity, asdescribed below. We should note that although Fig. 4displays thesum of the two conductivities, σ h/σnis small enough within the temperature range of the experimental data that theeffect of the inclusion of the hole contribution on the curve isnegligible (at T=1100 K, σ his about 2% of σn, and for lower temperatures it is smaller), supporting our decision to ignoreit when ﬁtting to the experimental conductivity data. Theseassumptions do not hold for extremely high temperatures,however, at T=1200 K, σ his about 8% of σn, and will likely increase; our value for σhere is less trustworthy than for lower temperatures. However, this regime is outside the temperaturerange for which we possess experimental data. B. Phonon dispersion curves and density of states Figures 5and6display the phonon dispersion curves along the Cartesian axes of the system and the density of statesfor the ordered Si 0.75Ge0.25alloy with eight-atom unit cell utilizing a 10 ×10×10 MP grid. Results are presented for both the relaxed and unrelaxed cases. We will note a numberof general features of interest in the relaxed case. Firstly,the dispersions of phonon branches in the [001] directiondiffer from those in the [100] and [010] directions, which 200 400 600 800 1000 1200 Temperature (K)0.00010.000150.00020.000250.00030.000350.0004\|S\| (V K-1)Meddin and Parrott Sample 7 Our Theoretical Calculation FIG. 3. (Color online) Calculated behavior of \|S\|compared with experimental data from Fig. 4of Ref. 9. (Data used with the permission of IOP Publishing Ltd.) 045205-8DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) 200 400 600 800 1000 1200 Temperature (K)20000400006000080000σ (Ω m)Meddin and Parrott Sample 7 Our Theoretical Values FIG. 4. (Color online) Theoretical behavior of σcompared with experimental data taken from Fig. 3 in Ref. 9. (Data used with the permission of IOP Publishing Ltd.) are identical (see Fig. 5). This should not be surprising, as if we refer to Fig. 1, we can see that the positioning of the Ge atoms in the cell distinguishes that direction from the others.From the dispersion relations, we can also see that a pair of“gaps” appear in all three directions between frequencies ofabout 300–350 cm −1and about 400–425 cm−1. By comparison with Fig. 6, we can see that the locations of these gaps roughly correspond to frequencies between approximately310–345 cm −1and approximately 400–428 cm−1where the phonon density of states has fallen to zero. It is thus clearthat two phononic gaps exist for the structure modelled in our work. The presence of such gaps in thin SiGe systems has beennoted and discussed previously. 48,49This is a consequence of our use of an ordered model that resembles a superlatticeof unequal period upon repetition of the unit cell in ourDFPT calculations; strictly speaking, we should use a larger,cubically symmetric cell in our calculations, but we are limitedby our computational resources. In order to overcome this, weassume as a ﬁrst approximation that any macroscopic systemwill consist of randomly oriented cells of this kind, and hencethat the isotropic lattice thermal conductivity can be consideredto be the average of the diagonal components of the thermalconductivity matrix for a single cell. We do not feel that thisapproximation will lead to signiﬁcantly different results fromcalculations carried out using a different choice of cell. Returning to our discussion of the DFPT results, we may support the notion that the ordering of the alloy is the chieffactor in the existence of the phononic gaps by consideringthe phonon dispersion curves for the modelled SiGe alloy along each of the three Cartesian directions. A comparison between the relaxed and unrelaxed results shown in Fig. 5 indicates that the dispersions of low frequency branches(comprising the acoustic modes and optical modes below around 300 cm −1) are relatively unaffected by bond relaxation (there is some small modiﬁcation as 300 cm−1is approached, but nothing signiﬁcant). However, the same cannot be saidof the higher-frequency optical branches, whose behavior isconsiderably modiﬁed. Examination of the phonon density of states in Fig. 6would seem to support this—the unrelaxed and Γ X0100200300400500frequency (cm-1)relaxed unrelaxed Γ X0100200300400500frequency (cm-1) Γ X0100200300400500frequency (cm-1) FIG. 5. (Color online) Comparison of phonon dispersion relations along the [100] (top left), [010] (top right), and [001] (bottom center) directions for the ordered Si 0.75Ge0.25system discussed in the text. 045205-9IORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) 0 100 200 300 400 500 frequenc y (cm-1)00.10.20.30.40.5density of states (states/cm-1 per cell)relaxed unrelaxed FIG. 6. (Color online) Phonon density of states against frequency for the ordered Si 0.75Ge0.25alloy discussed in the text. Note the pair of frequency “gaps” in the vicinity of 325 and 420 cm−1. the relaxed cases are qualitatively similar (though not identical) below around 350 cm−1, which includes the lower of the two frequency gaps. Above this frequency, we can see that there are considerable differences in the locations of the peaks in either case, and the size of the higher frequency gap is slightlyincreased by bond relaxation from around 410–422 cm −1to around 400–428 cm−1. We can therefore conclude that bond relaxation affects the size of one gap in the system, but that it does not give rise to the gaps themselves. C. Phonon mean-free path We now turn to an interesting question that has perhaps not received the consideration that it deserves: the notion of themode-average phonon mean-free path (MFP) λ. The issue in question is the precise form of the mode averaging that one can employ, and it is this which we examine in what follows. One prescription that has been used in calculations of this quantity (for example, in Ref. 50)i sas i m p l et h e r m a l averaging: λ TE=/summationtext qsλqs¯nqs/summationtext qs¯nqs, (33) where λqs=cs(q)τ(qs). Another suggested prescription (see, for example, Refs. 38and 39for theoretical discussions and Ref. 51for a proposed measurement technique) weights the value of λqsat a given mode and momentum with the corresponding speciﬁc heat and velocity: λSH=1 ¯cC/summationdisplay qsCqscs(q)λqs=1 ¯cC/summationdisplay qsκph(qs), (34) where ¯c=/summationtext qscs(q)¯nqs//summationtext qs¯nqsis the mode-averaged ve- locity, Cqs=ω2(qs)¯nqs(¯nqs+1) is the speciﬁc heat for a given mode, C=/summationtext qsCqsis the overall speciﬁc heat, and κph(qs) is the phonon conductivity for a given mode. We tend to prefer this deﬁnition as it relates more directly to themeasured quantity of interest, which is to say the lattice thermalconductivity.500 1000 Temperature (K)10-1410-1210-1010-810-6λ(m)λMAX λTE λSH λIN λLB FIG. 7. (Color online) The temperature dependence of the phonon mean-free path (MFP) calculated using various methods, compared with that of the relaxation length of the mode whose weighted contribution to κphis the greatest ( λMAX). Furthermore, we may also wish to calculate the MFPs arising from different contributions to scattering; in this case,we replace the total τ(qs) in a given equation with the τ(qs) corresponding to the contribution of interest. We will denote the contributions of interest with subscripts as follows:total anharmonic (AH), Normal (N), Umklapp (U), and alsocompare with the values of the following deﬁnition of theMFP: 1 λIN=/summationtext qsτ−1 qsc−1 s(q)¯nqs/summationtext qs¯nqs, (35) and with values of the variationally derived lower bound of the MFP:38 1 λLB=¯c C/summationdisplay qsCqsτ−1(qs) c2s(q). (36) In Fig. 7, we plot the mean-free paths (MFPs) calculated using all these methods and utilising all available phononmodes. We also display λ MAX, which is the MFP associated with the mode whose weighted contribution to κphis the greatest. We can see that there is a large and obvious differencebetween the results of the various methods, with λ TEandλSH being the most similar, λINbeing an order of magnitude smaller than the latter. λLBis smaller still; this is unsurprising since it is the lower bound of the MFP. λMAXis larger than all of these, consistent with the analysis of Ref. 52, which notes that mode-averaged MFPs are likely to greatly underestimate thedegree to which modes with long MFPs contribute to κ phin systems where the effects of defect scattering are strong (asis in fact the case in this alloy—see the next section for moredetails). Because of this, one can immediately rule out λ LB andλINas good deﬁnitions of the MFP; they are far too small. With regards to λSH, as we see here and in Richardson et al. ,52 it also underestimates the contribution from long MFP modes. We ﬁnd that λTEalso offends in this regard (though not quite as badly as λSH). This entails that one must exercise caution when interpreting MFP data—not only can it be misleading in certainsystems, but its value is strongly dependent on the method bywhich it is calculated; in the case of the present system, even 045205-10DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) 500 1000 Temperature (K)10-810-710-610-5λ(m)λAHTE λNTE λUTE λAHSH λNTE λUSH FIG. 8. (Color online) The temperature dependence of the phonon mean-free path (MFP) due to total anharmonic scattering, Umklapp and Normal processes as calculated by thermal averaging (TE) and from the thermal conductivity (SH). ifλSHandλLBdid not likely underestimate the value of the MFP, they do not give the same answer as to its value. We donot know of any strong theoretical reason for preferring onedeﬁnition over the other; however, in a practical sense (as faras thermal conductivity measurements are concerned) we areinclined to prefer λ SH, for its calculation utilises experimental quantities that are readily measured, as in Ref. 51. What is most important, however, is that whichever method one usesis clearly stated. Figures 8and 9display the MFP due to total anhar- monic, Normal, and Umklapp scattering for the above mode-averaging schemes. For all cases, we see that the MFP due toNormal processes is typically larger than that of the MFP dueto Umklapp processes, suggesting that the latter are generallymore important to the overall scattering. We should cautionagainst taking these results as ﬁnal and deﬁnitive, due to thebadly convergent nature of the anharmonic MFP discussed inthe previous section. However, we think they are likely to bequalitatively reliable. 200 400 600 800 1000 1200 Temperature (K)10-1410-1210-1010-8λ(m)λAHIN λNIN λUIN λAHLB λNLB λULB FIG. 9. (Color online) The temperature dependence of the phonon mean-free path (MFP) due to total anharmonic scattering, Umklapp and Normal processes as calculated by thermal averaging of λ−1 qs(IN) and from the variational lower bound formula (LB).200 400 600 800 1000 Temperature (K)3.53.7544.25 κ (W m-1 K-1)Meddin and Parrott Sample 7 Our theoretical calculation Meddin and Parrott theory FIG. 10. (Color online) Comparison of the presently calculated values of the thermal conductivity ( κ) with the theoretical and experimental values read from Fig. 6(g) in Ref. 9. (Data used with the permission of IOP Publishing Ltd.) D. Thermal conductivity Due to the complexity involved in the implementation of the calculation of κ, numerical ﬁtting proved unfeasible, and it was necessary to manually tune the semiadjustable parameters so asto attain results that displayed reasonable agreement with Med-dins and Parrott’s 9measurements of the total thermal conduc- tivityκ. The parameters listed in Table II(and used in Fig. 10) and those concerning σpin Table Iwere obtained by choosing a given parameter set, calculating κphfrom the eigensolutions generated for a 10 ×10×10 MP grid, adding the results to theκelvalues to obtain κ, and then comparing this with the experimental results, and repeating the process with adjustedvalues of the parameters until good agreement was reached. Figure 10displays the resulting values of κalongside the experimental results of and a theoretical calculationfrom Ref. 9. Qualitatively speaking, it can be seen that our calculation is a better match to the data than theirs for theregion in which they give theoretical values aside from thedata point at the lowest temperature, which is somewhatundershot, and that our calculation has a wider range ofvalidity. For temperatures greater than 900 K, where the bipolarcontribution is dominant (and thus the behavior is determinedby our parametrization of σ p), the curve matches the data less well; in fact, the form of the data is such that it madeselecting appropriate values for the behaviour of Gdifﬁcult, which should be kept in mind when considering the accuracyof our calculation in this region. It should be kept in mind thatthe value of the material density ρused for the calculation of κ phis the weighted average value of 3077.5 kg m−3rather than the measured value of 2860 kg m−3found in Ref. 9; but any errors resulting from this difference will be absorbed into thevalues of Aand ¯γ, which control the strength of the scattering processes that exhibit density dependence. Examining the results displayed in Table II, we note that our ﬁt was obtained using a value of L, which is the lower limit of that expected by Ref. 9, suggesting a fairly small crystallite size.Pis fairly large, indicating that we are accounting for scattering from a large number of defects consistent withour discussion of the mean-free path results; it would not be 045205-11IORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) 20 40 60 80 100 Frequenc y (cm-1)0246810κqs/κph(T)T=100.0 T=600.0 T=1200.0 FIG. 11. (Color online) Frequency dependence of κqs(T) nor- malised by κph(T) in our calculation. While κqsis the contribution without the weight factor associated with q, the total contribution κphis computed after summing κqswith the required weight factor associated with q. surprising if the large concentration of impurities in the heavily doped sample were to play a role in this. As a result of the lackof experimental data in the regime where electron-phononscattering has an important effect, the value of Acannot be considered to be precise, however, a ﬁnite but not toolarge value seems to be required in order to properly capturethe behavior of the system in the lower-temperature region.Due to the number of adjustable parameters and the methodwe have had to adopt in order to tune them, we cannot properlysay that this is a uniquely optimal set, but it is certainly a good one, and it seems plausible that any other parameter set thatgives a good ﬁt will be reasonably similar. We also should note that we obtain better agreement with experiment at temperatures in the 700–900 K thanGarg et al. attain for their computation of κfor undoped Si 0.3Ge0.7[presented in Fig. 3(a)of Ref. 15]; this should not be wholly surprising since we are making use of a semiempiricalapproach; this improved agreement may be as much due to ouruse of adjustable parameters as it is to the inherent virtues ofour approach. We now turn to the frequency dependence of κ ph, presented in Fig. 11for three temperatures with frequencies below 100 cm−1(≈3 THz) corresponding to points in momentum space coinciding with a 14 ×14×14 MP grid. The dominant frequencies are all contained within this area, which is consis-tent with the results presented in Fig. 2(a)in the paper by Garg et al. 15for Si 0.5Ge0.5alloys calculated using the virtual crystal approximation, and the “double peak” structure of the two mostdominant modes is reproduced, although the peaks are situatedat higher frequencies in our case. From Fig. 5, we can see that this frequency range is occupied by acoustic modes, entailingthat they contribute the most to κ ph. However, there are some differences. One of these is that in our case the magnitude of thenormalized values does not appear to be as strongly affectedby increases in temperature, and there is a pronounced “spike”in the contribution from a frequency near 70 cm −1. Another is that our values appear to be larger by a factor of around 6.25.20 40 60 80 100 Frequenc y (cm-1)01234%age contribution to κphT=100.0 T=600.0 T=1200.0 FIG. 12. (Color online) Percentage contribution of each fre- quency to κph(T). Note that this includes the weighting arising from our use of MP summation. These differences may both be a function of methodology and of the differing compositions of our materials. However, one must be careful when drawing conclusions from these data as to the relative contributions of variousfrequencies to κ ph. This is because in our actual calculations, we make use of the MP momentum summation scheme, andtherefore each frequency displayed in Fig. 11in fact makes a contribution to κ phthat is weighted according to the q point with which it coincides. We display the percentagecontribution to κ phof each frequency following such weighting in Fig. 12. It is apparent that the general behavior with tempera- ture is consistent with that of Fig. 11. However, we can also see that the weighting of the contributions has drastically alteredour conception of how low frequencies contribute to κ ph,f o r while the overall trend in the degree of contribution is stilldownwards for frequencies greater than 40 cm −1, the contri- butions of a small number are strongly enhanced. For example,the contributions of two frequencies in the region of 45 cm −1 are far greater than would be expected from Fig. 12, in fact con- tributing more than the subdominant frequency of that ﬁgure. There is a caveat with regards to these data, however. We have naively plotted the frequency data without regard fordegeneracy or the manner in which numerical error separatesout what should be degenerate modes, entailing that we may beundercounting the contribution of some frequencies (for exam-ple, those corresponding to transverse acoustic modes). But itwould seem likely that this would not overly affect our generalconclusion, which is that when accounting for the contributionsof various modes to κ phone should take into account the degree to which each mode is weighted by the MP summation scheme,as it is entirely possible that not doing so might mislead as towhich modes are in fact dominant, and to what degree. E. Figure of merit Having calculated the relevant electronic and vibrational contributions, we are now in a position to calculate the ﬁgureof merit ZTfor Sample 7 in Ref. 9.I nF i g . 13, we compare our 045205-12DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) 0 200 400 600 800 1000 1200 Temperature (K)00.20.40.60.8 ZT Meddin and Parrott’s calculation Our Calculation FIG. 13. (Color online) Comparison of the calculation of ZTin the present study with that of Fig. 9 in Ref. 9. (Data used with the permission of IOP Publishing Ltd.) results with the calculation performed by Meddins and Parrott, which is based on their theoretical model detailed in Ref. 9. There is reasonable qualitative agreement between our results and the results presented by Meddins and Parrott.9 However, in quantitative terms, there is a discrepancy ofaround 16% at 1000 K, and at lower temperatures, our resultsare slightly larger than theirs. One factor that may accountfor much of this difference is that Meddins and Parrottdo not appear to include the bipolar contribution to κ elin their analysis, and since it is at high temperatures that thatcontribution is dominant, it is unsurprising that their values ofZTare greater than ours in this region. From our numerical results, we suggest that for the modelled Si 0.75Ge0.25alloy values of ZT > 0.5 can be expected throughout the high- temperature range of around 750–1200 K, with a maximum ofapproximately 0.68 at around 1000 K. A comparison with, and comment on, the work carried out by Minnich et al. , 11is also appropriate. These authors applied a detailed model, similar to that adopted by Vining,10of the electronic behavior and of the phonon contribution to κphto experimental results for n-doped Si 0.8Ge0.2(where Ndonwas obtained by ﬁtting to experimental results) and for n- and p-doped Si 0.7Ge0.3where a number of samples with different Ndonwere available. The results for the n-doped Si 0.7Ge0.3 sample with Ndon=7.3×1025m−3are the most relevant here, since the composition and Ndonof this sample are reasonably close to that of the sample treated in this study. In the fourthpanel of Fig. 1 in Ref. 11, they display ZTfor their n-type Si 0.7Ge0.3samples, and for the sample of interest, it can be seen that the results are qualitatively similar up to aroundT=1000 K, with a peak at a slightly larger value of ZT (closer to the 0.8 of Meddins and Parrot’s results 9than the 0.7 of ours) and a far sharper decrease than we observe at highertemperatures. Some of this difference is likely to arise fromour different approaches, and some of it from the differencesin composition between our samples. However, the qualitativesimilarities are a good sign. Our calculation of ZTin an SiGe alloy includes as complete as possible an account of the phonon scattering rates and theeffects of the bipolar contribution to thermal conductivity. It ishoped that it will provided a benchmark for future theoretical explorations of the thermoelectric efﬁciency of more complexsystems such as nanowires or superlattices. V . SUMMARY AND CONCLUSION In this study, we have examined the theoretical behavior of the thermoelectric ﬁgure of merit ZTand its constituent quantities for the case of a sintered Si 0.754Ge0.246doped with P impurities examined by Meddins and Parrott.9We have established phenomenological models to account for thebehavior of the electronic quantities, and have focused in detailon the mechanisms underlying the phonon contribution to thethermal conductivity κ. In this respect, we have developed a full-scale theory of the lattice thermal conductivity consideringthe role of acoustic as well as optical phonons in Normal andUmklapp three-phonon interactions and included the bipolarcontribution to thermal conductivity. We have made use of density functional methods in order to obtain the required phonon eigensolutions and adetailed calculation of the anharmonic contribution to phononscattering based on a semiempirical model for anharmoniccrystal potential. The subsequent calculation of κ ph+κeshows good agreement with measurements in the entire temperaturerange 300–1100 K. We have also examined the frequencydependence of κ ph, ﬁnding that it is consistent with the previously reported results of Ref. 15apart from some differences in temperature dependence and have discussedsome issues concerning the notion of a mean-free path. From this information, we have calculated the dimension- less ﬁgure of merit ZTand compared it with the (incomplete) calculation in Ref. 9. While, in general, the qualitative behavior of the ZTversus Tcurve in our work is quite similar to that in Ref. 9, with the more complete theoretical treatment we have predicted values of greater than 0.5 in the temperaturerange of around 750 to at least 1200 K with a maximum ofapproximately 0.68 at around 1000 K. In addition to the study of the thermoelectric properties, we have also examined a more conceptual issue: that of the calcu-lation of the phonon mean-free path. We have shown that in theliterature, there exist different prescriptions for evaluation ofphonon mean free path, yielding different results. We suggestthat care must be exercised in comparing results obtained fromdifferent methods of mode averaging procedures. ACKNOWLEDGMENTS This work has been carried out with help of the EPSRC (UK) grant No. EP/H046690/1, and Iowerth O. Thomasis grateful for the ﬁnancial support. The calculations wereperformed using the Intel Nehalem (i7) cluster (ceres) at theUniversity of Exeter. APPENDIX: THE ANHARMONIC INTERACTION TERM While the anharmonic crystal potential involving acoustic phonons has been widely discussed, the contributions dueto optical modes have not been fully considered, althoughKlemens 53and Ridley and Gupta54have given simple presen- tations of the effects of scattering to and from optical modes. 045205-13IORWERTH O. THOMAS AND G. P. SRIV ASTA V A PHYSICAL REVIEW B 86, 045205 (2012) In this work, we have expressed the anharmonic phonon relaxation time due to three-phonon Normal and Umklappprocesses involving acoustic and optical phonon modes, usingEqs. (28) and (29). This form of the anharmonic phonon- phonon interaction was proposed by S. P. Hepplestone 55and was indeed employed in previously published works,40,49but has not yet appeared in print. In this appendix, we shall attemptto justify and present the relevant expression for the cubicanharmocity in crystal potential. We shall begin with following expression for the third-order perturbative term in the elastic continuum potential, taken fromRef. 26: V 3=/radicalBigg ¯h3 8ρ3V/summationdisplay qs,q/primes/prime, q/prime/primes/prime/primeTss/primes/prime/prime qq/primeq/prime/prime(a† qs−a−qs)(a† −q/primes/prime−aq/primes/prime) ×(a† q/prime/primes/prime/prime−a−q/primes/prime)δq+q/prime+q/prime/prime,G, (A1) where Tss/primes/prime/prime qq/primeq/prime/prime=1 3!/radicalBigg qq/primeq/prime/prime cscs/primecs/prime/primeAss/primes/prime/prime qq/primeq/prime/prime, (A2) witha† qs(aqs) being the phonon creation (annihilation) operator for a given mode qs,qbeing the magnitude of the momentum, and csbeing the speed of a phonon belonging to branch s. Note that csis not dependent on qat this point; this expression is derived on the assumption that we are treatingacoustic phonons in the continuum, q→0, limit, and so we can relate frequency and momentum through the dispersion re-lationω(qs)≈c sq. Our aim is to reverse-engineer from (A1) a starting point similar to that used by Klemens56from which we may derive an expression for the anharmonic term that includesaspects of the behavior of optical phonons while remaining trueto the spirit of the original continuum, acoustic approach. Taking the expression for the angularly averaged modulus ofA ss/primes/prime/prime qq/primeq/prime/primeas a guide:26 /vextendsingle/vextendsingleAss/primes/prime/prime qq/primeq/prime/prime/vextendsingle/vextendsingle2=4ρ2 ¯c2γ2c2 sc2 s/primec2 s/prime/prime, (A3) we put Ass/primes/prime/prime qq/primeq/prime/prime=(2ρ/¯c)γcscs/primecs/prime/prime, with ¯cbeing an average phonon speed and γbeing the modulus of the Gr ¨uneissen constant, and so arrive (following a little algebra, and assumingacoustic dispersions at q→0) at the expression T ss/primes/prime/prime qq/primeq/prime/prime=2ργ√ω(qs)ω(q/primes/prime)ω(q/prime/primes/prime/prime)Bss/primes/prime/prime qq/primeq/prime/prime, (A4) Bss/primes/prime/prime qq/primeq/prime/prime=1 3!ω(qs)ω(q/primes/prime)ω(q/prime/primes/prime/prime) ¯c. (A5) Equation (A5) has the same general form as the approximation used by Klemens.56Indeed, if we replace ¯cwith the speedcs/prime/prime, then in the limit of small qand assuming an acoustic dispersion, we obtain Bss/primes/prime/prime qq/primeq/prime/prime=1 3!ω(qs)ω(q/primes/prime)q/prime/prime, (A6) which is very similar to the expression Klemens56has derived in order to motivate his approximation. We shall use this asthe basis for our derivation of the general anharmonic termwith which to treat three-phonon interaction involving bothacoustic and optical modes. 1. Treatment when q/prime/primes/prime/primeis an optical mode We shall begin with the situation when q/prime/primes/prime/primeis an optical mode and consider three-phonon processes of the typesac+ac/arrowrighttophalf/arrowleftbothalfopandac+op/arrowrighttophalf/arrowleftbothalfop. For small values of q /prime/prime, we express ω2 o(q/prime/primes/prime/prime)=ω2 /Gamma1o(s/prime/prime)2−cs/prime/primeoq/prime/prime2. Rearranging this, we may write q/prime/prime=(ω/Gamma1o(s)+ωo(q/prime/primes/prime/prime))1/2\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\|1/2 cso ≈√2ωo(q/prime/primes/prime/prime)\|ω/Gamma1o(s/prime/prime)−ωo(q/prime/primes/prime/prime)\|1/2 cso,asq→0. (A7) Next, we observe that Bss/primes/prime/prime qq/primeq/prime/primeis identical under exchange of the modes qsand q/primes/prime; we account for this in Eq. (A1) by simply multiplying Bss/primes/prime/prime qq/primeq/prime/primeby a factor of 2 and ignoring subsequent terms where those modes are exchanged. Replacing q/prime/primein Bss/primes/prime/prime qq/primeq/prime/primewith this expression, we obtain Bss/primes/prime/prime qq/primeq/prime/prime=1 3ω(qs)ω(q/primes/prime)√2ωo(q/prime/primes/prime/prime)\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\|1/2 cs/prime/primeo. (A8) Instead of following Klemens53or Ridley and Gupta54and assuming that the dominant decay is of the form ωo(q/prime/primes/prime/prime)≈ ω(qs)/2≈ω(q/primes/prime)/2, we shall be more general; examining the energy conservation condition ωo(q/prime/primes/prime/prime)=ω(qs)+ω(q/primes/prime), (A9) we may see that ω(qs)=Bωo(q/prime/primes/prime/prime), (A10) ω(q/primes/prime)=(1−B)ωo(q/prime/primes/prime/prime), (A11) where we consider values of Bbetween zero and unity. From Eq. (A9) , we may write√2ωo(q/prime/primes/prime/prime)=√ 2[ω(qs)+ ω(q/primes/prime)]1/2and so ω(qs)=√Bω(qs)[ω(qs)+ω(q/primes/prime)]1/2, and hence Bss/primes/prime/prime qq/primeq/prime/prime=1 3√2Bω(qs)ω(q/primes/prime)[ω(qs)+ω(q/primes/prime)]\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\|1/2 cs/prime/primeo. (A12) We may also write ω(q/primes/prime) (1−B)≈ωo(q/prime/primes/prime/prime)\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\| \|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\|=A2\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\|, (A13) 045205-14DETAILED CALCULATION OF THE THERMOELECTRIC ... PHYSICAL REVIEW B 86, 045205 (2012) where Ais deﬁned above, and so acquire Bss/primes/prime/prime qq/primeq/prime/prime=A 3√B(1−B)ω(qs)ω(q/primes/prime)[ω(qs)+ω(q/primes/prime)]\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\| cs/prime/primeo, (A14) from which we may obtain Tss/primes/prime/prime qq/primeq/prime/prime=2ρ/Gamma1o√ω(qs)ω(q/primes/prime)ωo(q/prime/primes/prime/prime)1 3!√ω(qs)ω(q/primes/prime)[ω(qs)+ω(q/primes/prime)]\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\| cs/prime/primeo, (A15) where /Gamma1o=2Aγ√2B(1−B). It should be noted that in order for/Gamma1oto have a meaningful value at q=0, we require thatγ√B(1−B)∝\|ω/Gamma1o(s)−ωo(q/prime/primes/prime/prime)\|αasq→0, where α/greaterorequalslant1/2, in order to cancel the zero-tending term in the denominator of A. This is a strong constraint on the behavior ofγ; however, it is an artifact of the form of the term that we must adopt for numerical simplicity. One could considerusing other forms, but then one would have to make useof a frequency window in order to distinguish optical andacoustic terms, a process that is cumbersome and which couldintroduce its own inaccuracies. In this case, we much preferto choose the form given above, which is easy to implement numerically. 2. Treatment when q/prime/primes/prime/primeis an acoustic mode To deal with three-phonon processes of the type ac+ac/arrowrighttophalf/arrowleftbothalf ac, withq/prime/primecorresponding to an acoustic mode, we ﬁrst observe that from the acoustic dispersion relation in the limit of smallq, we may derive q=\|ω /Gamma1a(s)−ωa(qs)\|/csa, since ω/Gamma1a(s)= 0. From Eqs. (A10) and(A11) and energy conservation, we may obtain√ω(qs)ω(q/primes/prime)=√B(1−B)[ω(qs)+ω(q/primes/prime)]. These may be used to derive Tss/primes/prime/prime qq/primeq/prime/prime=2ρ/Gamma1a√ω(qs)ω(q/primes/prime)ωa(q/prime/primes/prime/prime)1 3!√ω(qs)ω(q/primes/prime)[ω(qs)+ω(q/primes/prime)]\|ω/Gamma1a(s)−ωa(q/prime/primes/prime/prime)\| cs/prime/primea, (A16) where /Gamma1a=2γ√B(1−B). This expression for Tis essen- tially the same as in Ref. 26. (Note that the value of Bhere is not necessarily identical to that of Bin the previous section whenq/prime/primewas an optical mode, nor is it necessarily identical with values of Bconsidered for scattering between a different triad of phonon modes.) 3. Form for numerical calculations For ease of calculation, we take /Gamma1a=/Gamma1o=¯γ, by analogy with the use of the mode-averaged Gr ¨uneissen constant in typical computations of this form. Since contributions fromboth acoustic and optical q /prime/primenow share an identical form, we may suppress the indices which distinguish them and write anoverall term: V 3=¯γ/radicalBigg ¯h3 2ρV/summationdisplay qs,q/primes/prime, q/prime/primes/prime/primeBqs,q/primes/prime,q/prime/primes/prime/prime√ω(qs)ω(q/primes/prime)ω(q/prime/primes/prime/prime)δq+q/prime+q/prime/prime,G ×(a† qs−a−qs)(a† −q/primes/prime−aq/primes/prime)(a† q/prime/primes/prime/prime−a−q/primes/prime),(A17) with Bi,j,k={/radicalbig ω(i)ω(j)[ω(i)+ω(j)]\|ω/Gamma1(k)−ω(k)\|/c(k) +similar terms with i,j, andkinterchanged }/3!, (A18)where i,j,klabel phonon modes, and c(k)i sn o wt h e momentum dependent , i.e., phase speed for the mode k; here, we make a generalization to the case where qno longer tends to zero. From this, we may obtain the expressionfor the anharmonic single-mode relaxation time used in ourcalculations—that is, Eq. (28). We should remark that the assumptions made in deriving the above expression are perhaps somewhat crude: unlikeKlemens 53or Ridley and Gupta,54we have not properly considered distinctions in behavior arising from the differentways in which the displacement of optical and acousticvibrations deform the crystal. However, given that Klemens’sexpression 53is equivalent to that of Ridley and Gupta (as the latter have observed54), and that the former’s derivation shows that the differences amount to a rescaling of the opticalGr¨uneissen constant away from the value it would have were it derived assuming acoustic behavior, we should not be tooworried; such a distinction would be ignored by our use ofa mode-averaged form of the rescaled Gr ¨uneissen constant regardless of the ﬁnal form of our expression. 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PhysRevB.76.045217.pdf	Electronic properties of oxygen-deﬁcient and aluminum-doped rutile TiO 2from ﬁrst principles Mazharul M. Islam,1Thomas Bredow,2and Andrea Gerson1 1Applied Centre for Structural and Synchrotron Studies, University of South Australia, Mawson Lakes Campus, Mawson Lakes, Adelaide, South Australia 5095, Australia 2Institut für Physikalische und Theoretische Chemie, Universität Bonn, Wegelerstrasse 12, 53115 Bonn, Germany /H20849Received 4 April 2007; published 20 July 2007 /H20850 The electronic properties of stoichiometric, defective, and aluminum-doped rutile TiO 2have been investi- gated theoretically with periodic quantum-chemical calculations. Theoretical results obtained with the Perdew-Wang density functional method /H20851Phys. Rev. B 45, 13244 /H208491992 /H20850/H20852and with a density functional-Hartree-Fock hybrid method are compared. Occupied defect states are observed in the band gap of rutile due to the presenceof oxygen vacancies, which is in accord with previous theoretical studies and the experimentally observedcoloring. For the investigation of aluminum doping, three different situations have been considered: substitu-tion of a single Ti atom by an Al atom, cosubstitution of Ti by Al and O by Cl, and substitution of two Ti bytwo Al combined with the formation of an O vacancy. In the last two cases, aluminum doping does notintroduce band gap states, and the band gap is even increased compared to undoped rutile. We conclude thatstoichiometric Al doping reduces pigment coloring induced by oxygen vacancies in rutile and also suppressesthe photocatalytic activity of titania pigments. DOI: 10.1103/PhysRevB.76.045217 PACS number /H20849s/H20850: 61.72.Ww, 71.15.Mb, 81.05.Je I. INTRODUCTION Titanium dioxide, TiO 2, has been studied extensively both experimentally and theoretically due to its particular physicaland chemical properties: high refractive index, excellent op-tical transmittance in the visible and near-infrared regions,high dielectric constant, 1and UV induced electron excitation.2It is used in heterogeneous catalysis, as a photocatalyst,2in solar cells for the production of hydrogen and electric energy, as a gas sensor, as white pigment, ascolored ceramic pigment, as a corrosion-protective coating,as an optical coating, and in electronic and electricaldevices. 3It is also used in bone implants due to its high biocompatibility.4In surface science, titanium dioxide is con- sidered as a model system for many metal oxides. Under-standing its properties at fundamental level will help im-prove materials and device performance in many ﬁelds. Titanium dioxide crystallizes in three different modiﬁca- tions: rutile, anatase, and brookite. Rutile is thermodynami-cally the most stable form. It has been chosen as the subjectof this study. The rutile structure has a tetragonal unit cell /H20849space group D 4h14−P42/mnm /H20850containing two titanium and four oxygen atoms. The lattice consists of hexagonal closepacked oxygen atoms, with half of the octahedral spacesﬁlled with titanium atoms /H20849Fig.1/H20850. One additional parameter u, the oxygen fractional coordinate, is necessary to deﬁne the crystal structure. The electronic and optical properties of TiO 2have been investigated experimentally using a wide range of methods:x-ray photoelectron spectroscopy, 5–8x-ray absorption spectroscopy,9–11x-ray emission spectroscopy,12total elec- tron yield /H20849TEY /H20850spectroscopy,13electron-energy-loss spectroscopy,6,14–16ultraviolet photoelectron spectroscopy /H20849UPS /H20850,17resonant ultraviolet photoelectron spectroscopy,18 absorption and photoluminescence spectroscopy,19and wavelength modulated transmission spectroscopy.20During the last years, accurate ab initio and density functionaltheory /H20849DFT /H20850electronic structure calculations have become available for the interpretation of TiO 2spectroscopic data. Theoretical investigations have been performed using the pseudopotential plane wave formalism both for DFT-localdensity approximation /H20849LDA /H20850 5,14,21–26and Hartree-Fock /H20849HF /H2085027approaches. Self-consistent calculations using the lin- ear mufﬁn-tin orbital /H20849LMTO /H20850method,12,13,28–30tight binding model,31–33and extended Hückel molecular-orbital method16 have also been performed to predict the electronic structure of TiO 2. Defects, which may alter the electronic and optical prop- erties of rutile, play an important role in technological appli-cations of TiO 2. Naturally occurring rutile is almost always slightly reduced, leading to a pronounced color change of thecrystal, from transparent to light and dark blue, which isaccompanied by increased electrical conductivity. 34The de- fect structure is assumed to be quite complex with combina-tions of various types of oxygen vacancies and Ti 3+and Ti4+ interstitials.35However, a recent theoretical investigation26 has shown that the formation of oxygen vacancies is ener-getically more favorable than the insertion of titanium atinterstitial sites. The defect structure varies with the oxygenvacancy concentration, which depends on temperature, gaspressure, metal impurities, etc. 36At 1100 °C, an oxygen de- c a FIG. 1. Unit cell of rutile. Black spheres represent the oxygen atoms and light gray spheres represent the titanium atoms.PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 1098-0121/2007/76 /H208494/H20850/045217 /H208499/H20850 ©2007 The American Physical Society 045217-1fect concentration of 0.5% under atmospheric pressure was measured.36,37Several experimental3,17,34,38–41and theoretical26,35,42–48investigations have been performed to understand the role of oxygen vacancies in TiO 2. These re- sults will serve as a basis for comparison with the resultspresented here. Titanium dioxide is well known as a photocatalyst. Its catalytic activity reduces the long-term durability of titaniapigment materials. Upon UV irradiation, electrons /H20849e −/H20850and holes /H20849h+/H20850are created in the conduction band and valence band, respectively.49This process formally reduces titanium ions from Ti4+to Ti3+, causing coloration of the pigment. Commercial TiO 2rutile pigments are always doped with Al2O3to enhance photochemical stability. It is assumed that Al doping creates defects in the rutile lattice that act as trapsfor the photogenerated charges. 49,50The presence of alumi- num is also responsible for reducing particle agglomeration51 and increasing the rate of anatase-to-rutile conversion duringthe chloride process. 51–54In the present theoretical study, we investigate the possible sites of aluminum doping and theelectronic properties of Al-doped rutile. II. COMPUTATIONAL METHODS The electronic and optical properties of TiO 2for stoichio- metric, oxygen-deﬁcient, and Al-doped systems were calcu-lated with periodic supercell models. In order to investigatethe method dependence of the computed properties, two dif-ferent quantum-chemical approaches were used, the Perdew-Wang exchange-correlation functional PWGGA 55,56and the PW1PW DFT-HF hybrid method.57These methods have been applied for calculations of bulk properties of MgO,NiO, CoO, 57Li2B4O7,58B2O3,59and LiO 2,60defect proper- ties of Li 2B4O7,61and electronic properties of Li 2O-B 2O3 compounds.62In those studies, good agreement between cal- culated and experimental bulk properties was observed, inparticular, for the PW1PW hybrid method. The two methods were used as implemented in the crys- talline orbital program CRYSTAL03 .63InCRYSTAL , the Bloch functions are linear combinations of atomic orbitals. Thequality of the atomic basis sets determines the reliability ofthe results. Therefore, we have used high quality atomic ba-sis sets obtained from the literature. For titanium an 86-411 /H20849d31 /H20850G basis and for oxygen an 8-411G basis were used, which have been successfully applied for the investigation ofthe electronic properties of rutile TiO 2ultrathin ﬁlms.64An 88-31Gbasis was used for aluminum, which has been op- timized for alumina.65For chlorine, an 86-311G basis was used, which has been optimized for the structural propertiesinvestigation of NaCl. 66 A primitive unit cell containing two formula units was used as model for stoichiometric rutile. The projected densityof states /H20849PDOS /H20850was calculated using the Fourier-Legendre technique 67with a Monkhorst net68using shrinking factors s=8. In order to minimize direct defect-defect interaction between neighboring cells, we used supercells /H20849Ti16O32, Ti32O64, and Ti 54O108/H20850as models of the defective bulk. III. RESULTS AND DISCUSSION A. Bulk properties of rutile The optimized structure parameters a,c, and u, the cohe- sive energy per TiO 2unitEu, the band gap Eg, and the bulkmodulus B0obtained with PW1PW and PWGGA are given in Table Itogether with the corresponding experimental val- ues. The lattice parameters correspond to an extrapolation ofexperimental x-ray data to T=0 K. 69,70The calculated lattice parameters obtained with the two methods are close to theexperimental values. The largest deviation, +0.9%, is ob-tained for the PWGGA value of a. This is better than the error of −2% obtained by previous DFT-LDA calculations. 71 However, the structural parameters obtained with PWGGAare in good agreement with a previous CRYSTAL -PWGGA study by Muscat et al.72The hybrid method PW1PW gives a slightly better reproduction of experimental structure proper-ties than PWGGA. The experimental value of E ufor rutile is 1915 kJ/mol.70 Theoretical estimates of Euare obtained by subtracting the total energies of the free atoms in their ground states withconverged basis sets from the energy of the periodic system.In this way, temperature effects and contributions from zeropoint energy are neglected. They have been taken into ac-count a posteriori by a frequency calculation with CRYSTAL06 .73The sum of all correction terms is +18 kJ/mol for PWGGA and +20 kJ/mol for PW1PW. The correctedvalue of E uis −1909 kJ/mol at the PW1PW level, which is in excellent agreement with the experimental reference /H20849 −1915 kJ/mol /H20850. The PWGGA method overestimates Eucon- siderably /H20849−2036 kJ/mol /H20850. The experimental value of 216 GPa for the bulk modulus /H20849B0/H20850has been determined under ambient conditions.74A re- cent x-ray spectroscopy study75has obtained a value of 230±20 GPa. We ﬁnd isothermal bulk moduli of 234 and223 GPa with PW1PW and PWGGA, respectively, which arein excellent agreement with the experimental range. Rutile is a semiconductor with a band gap /H20849E g/H20850of 3.03 eV.19In the present study, the band structure was com- puted along the direction that contains the highest number ofhigh-symmetry points within the Brillouin zone, 76namely, Z→A→M→/H9003→Z→R→X→/H9003. The band structure ob- tained with PW1PW is shown in Fig. 2. The calculated Eg values obtained with PW1PW and PWGGA are given in Table I, and the minimal vertical transition /H20849MVT /H20850and mini- mal indirect transition /H20849MIT /H20850energies are given in Table II. Both methods predict a direct gap at /H9003. Experimental in-TABLE I. Comparison of calculated structural parameters a/H20849Å/H20850, c/H20849Å/H20850, and u, binding energy Eu/H20849in parentheses are the temperature corrected values /H20850/H20849kJ/mol /H20850, bulk modulus B0/H20849GPa /H20850, and band gap Eg/H20849eV/H20850with experiment. PWGGA PW1PW Expt. a 4.63 4.59 4.59a c 2.98 2.98 2.96a u 0.305 0.305 0.305a Eu −2054 /H20849−2036 /H20850 −1929 /H20849−1909 /H20850 −1915a B0 223 234 216b Eg 1.90 3.54 3.03c aReference 70. bReference 74. cReference 19.ISLAM, BREDOW, AND GERSON PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 045217-2vestigations using absorption and photoluminescence spectroscopy,19and wavelength modulated transmission spectroscopy20have also determined that rutile is a direct- forbidden-gap semiconductor, i.e., the direct transition is di-pole forbidden. The MVT is along the /H9003-/H9003direction. The MVT value obtained with PW1PW is 3.54 eV, slightly over-estimating the experimental value of 3.03 eV. 19,20The pure DFT-PWGGA approach /H208491.90 eV /H20850underestimates the direct transition energy by 1.13 eV. This is similar to the DFT- LDA-based direct forbidden gaps of 2.0 eV,221.87 eV,25and 1.7 eV.26The gradient corrections therefore do not consider- ably improve the LDA band gap, while the inclusion of exactHartree-Fock exchange leads to an overestimation of E g. In agreement with previous theoretical investigations,22,27 our calculations also predict that rutile has the smallest indi- rect gap along the /H9003-Mdirection. The MIT energies, 3.82 and 2.04 eV with PW1PW and PWGGA, respectively, areclose to the corresponding MVT energies. However, accord-ing to Mathieu et al. , 20the indirect gap shows a nonlinear behavior under /H20851100 /H20852compression, which should include the transition along the Xdirection. It is therefore possible that the indirect transition along the /H9003-Xdirection, the second MIT, has the highest probability although its calculated tran-sition energy is slightly greater than that for the /H9003-Mdirec- tion /H20849Table II/H20850. The DOS calculated with PW1PW is shown in Fig. 3. Thecalculated valence band width is about 6.3 eV, in good agreement with the range of experimental values,5–6 eV. 5–8,12The valence band /H20849VB /H20850is mainly composed of O2pstates with some hybridization with Ti 3 dorbitals. The PDOS of Ti obtained with x-ray photoelectron spectroscopy5shows a splitting of two major peaks by 2.3 eV. In our case, the center-to-center separation of the Tipeaks is 2.2 eV. This is in good agreement with the experi-mental result. According to ligand-ﬁeld theory, 77the pres- ence of two Ti peaks in the VB is a result of the splitting ofTi 3dorbitals into states with t 2gand egsymmetries. t2g states have a lower energy than egstates. These states are mirrored in the structure of the PDOS for O. A similar pic-ture was observed in previous combined theoretical and ex-perimental studies, 5,12and a theoretical DFT-LDA study.22 The t2gand egstates represent the /H9266and/H9268Tidorbitals hybridized with O 2 pstates. Similar to the VB, the conduction band /H20849CB /H20850is also com- posed of both Ti 3 dand O 2 pstates. The bottom of the con- duction band is mainly formed by Ti 3 dstates, their contri- bution being several times higher than that of O 2 pstates. This is in good agreement with the experimental ﬁndingfrom TEY spectroscopy. 13 TABLE II. Minimum vertical electronic transition /H20849MVT /H20850and minimum indirect transition /H20849MIT /H20850ener- gies /H20849eV/H20850. MVT Z-ZA -AM -M /H9003-/H9003 Z-ZR -RX -X PW1PW 6.68 5.50 4.92 3.54 6.68 4.96 5.33 PWGGA 4.60 3.60 3.12 1.90 4.60 3.13 3.39 MIT A-MM -/H9003/H9003 -M /H9003-ZZ -RX -/H9003/H9003 -X PW1PW 4.56 4.77 3.82 5.07 5.53 4.43 4.58 PWGGA 2.74 2.96 2.04 3.14 3.62 2.57 2.69 Energy (eV) FIG. 2. Band structure of rutile TiO 2/H20849PW1PW results /H20850.Ener gy(eV)DOS(arb. units) FIG. 3. Density of states of rutile TiO 2/H20849PW1PW results /H20850.ELECTRONIC PROPERTIES OF OXYGEN-DEFICIENT AND … PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 045217-3B. Oxygen vacancies in rutile The experimentally measured oxygen vacancy concentra- tion in rutile is 0.5% at 1100 °C and near atmosphericpressure. 36,37The enthalpy of formation of oxygen vacancies has been measured as 439 kJ/mol.36An infrared absorption study34measured that neutral O defects produce a peak 1.18 eV below the bottom of the bulk CB. In a UPS study ofthe surface electronic structure, Henrich and Kurtz 40identi- ﬁed an occupied defect state 1.2 eV below the CB due to thepresence of oxygen vacancies. A 1.61 eV peak below theconduction band has been found in a UV spectrum 41and attributed to a positively charged O defect. Chen et al.42investigated oxygen-deﬁcient rutile theoreti- cally. They employed an embedded-cluster numerical dis-crete variation method. A defect peak was observed 0.87 eVbelow CB for neutral defects, and 1.78 eV below CB forpositively charged defects. A similar result was obtained in arecent LMTO study of the neutral O defect in rutile. 47An occupied defect state was found 0.8 eV below the CB bottomedge. The two extra electrons left behind by the oxygen weredistributed on the three Ti atoms surrounding the vacancy.Rutile with charged O defects is reported to have partiallyoccupied defect band states, but their positions relative to theCB bottom are not given. However, the latter investigationcan be regarded as incomplete, since lattice relaxation wasneglected. Another theoretical investigation 35based on a semiempirical self-consistent method found the oxygen va-cancy peak 0.7 eV below CB. At DFT-LDA 26,44and DFT-PWGGA45levels, the gap states are located 0.1 and 0.3 eV below CB, respectively. These differences are consid-erably smaller than the experimental values. 34,38,40 In the present study, a systematic investigation is per- formed for the oxygen vacancy formation energy, Ede/H20849V/H20850, the effect of relaxation, and the optical transition energies of defective rutile. Supercells /H20849Ti16O32,T i 32O64, and Ti 54O108/H20850 were used for the defect calculations. The lowest vacancyconcentration that we studied here is therefore 0.9%, whichis still larger than the experimentally measured value /H208490.5% /H20850. For lower defect concentrations, one has to consider largersupercells. This was not possible due to limited computerresources.1. Defect formation and structural relaxation The defect formation energy of a neutral oxygen vacancy Ede/H20849V/H20850is calculated as Ede/H20849V/H20850=E/H20849TinO2n−1/H20850+ 1/2 E/H20849O2/H20850−E/H20849TinO2n/H20850. /H208493.1 /H20850 Here, E/H20849TinO2n−1/H20850andE/H20849TinO2n/H20850denote the total energies of the supercell with and without defect, respectively. E/H20849O2/H20850is the ground-state energy of the oxygen molecule after optimi- zation. Charged oxygen defects could not be studied with CRYSTAL because the energy and energy gradient expressions are not correctly implemented for charged cells with a com-pensating homogeneous background charge. 73 Ede/H20849V/H20850obtained with PWGGA and PW1PW are presented in Table IIIfor unrelaxed and fully relaxed systems. The basis functions of the oxygen ion were left at the defectposition. Calculations for the triplet state were performedusing the unrestricted Kohn-Sham /H20849UKS /H20850method. The re- moval of oxygen leaves two electrons in the VB that previ-ously occupied O 2 plevels. It is known 34,40,45–47that these unpaired electrons are mainly localized in the 3 dorbitals of the neighboring Ti atoms. Closed-shell singlet state calcula-tions were also performed with the PWGGA method. It wasfound that E de/H20849V/H20850for the singlet state is 37–58 kJ/mol larger than the corresponding value of the triplet state. Therefore, closed-shell singlet state calculations are not considered fur-ther. E de/H20849V/H20850decreases with decreasing defect concentration /H20849i.e., increasing supercell size /H20850. This indicates a long-range repulsive interaction between oxygen vacancies located inneighboring cells. The observed trend can also be due to theeffect of relaxation of the lattice atoms around the vacancy,most probably of electrostatic origin. The movement of theatoms out of their lattice positions due to the presence of thedefect is restricted by the periodic boundary conditions in-troduced on the supercell. This can be best seen by the verysmall relaxation energy /H20849difference between unrelaxed and fully relaxed defect formation energies /H20850of the smallest su- percell /H20849Ti 2O4/H20850, 11 kJ/mol /H20849PW1PW /H20850and 16 kJ/mol /H20849PWGGA /H20850. For the larger supercells /H20849Ti16O32,T i 32O64, and Ti54O108/H20850, the relaxation energy is considerably larger, 85–97 kJ/mol with PWGGA and 52–61 kJ/mol withTABLE III. Formation energy of neutral oxygen vacancies Ede/H20849V/H20850/H20849kJ/mol /H20850as function of the defect concentration c/H20849%/H20850/H20849unrel=unrelaxed, rel=relaxed /H20850. Supercell cEde/H20849V/H20850 Expt.bPWGGA PW1PW Unrel Rel UnrelaRelaUnrel Rel Ti2O4 25.0 630 614 679 651 653 642 Ti16O32 3.1 595 510 637 568 635 583 Ti32O64 1.6 542 445 575 514 Ti54O108 0.9 519 431 530 490 439 aClosed-shell singlet state calculations. bReference 36.ISLAM, BREDOW, AND GERSON PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 045217-4PW1PW. This indicates that relaxation is important for the study of defective systems. The PWGGA method generallygives smaller values of E de/H20849V/H20850than PW1PW, which can be partly traced back to the larger relaxation energy. For the smallest defect concentration /H208490.9% /H20850,Ede/H20849V/H20850is 431 kJ/mol with PWGGA and 490 kJ/mol with PW1PW. This trend is in line with our previous observation for the defect formationenergies in Li 2O /H20849Ref. 60/H20850and Li 2B4O7/H20849Ref. 61/H20850. The PWGGA defect formation energy agrees well with the cal-culated value of 4.44 eV /H20849428 kJ/mol /H20850obtained by Cho et al.at the local spin density approximation level. 26Both DFT values are in better agreement with the experimental value /H20851 439 kJ/mol /H20849Ref. 36/H20850/H20852than the PW1PW method, in contrast to our previous experience for other oxides. However, theconvergence of E de/H20849V/H20850with respect to the supercell size is slow. It is therefore possible that the defect formation energy for larger supercells, which better represent the experimen-tally observed defect concentration /H208490.5% /H20850, is smaller than those reported here. In such case, the PWGGA methodwould underestimate the experimental defect formation en-ergy value, whereas the PW1PW approach would be in betteragreement with the experimental value. The structural relaxation effects for the O vacancy are investigated by measuring the changes of distances of therelaxed atoms from the vacancy site. PW1PW results for therelaxation effects are shown in Table IV. The PWGGA method gives the same trend. Prior to relaxation, the vacancysite is surrounded by two nearest-neighbor /H208491-NN /H20850Ti atoms at a distance of 1.95 Å and by one Ti atom at a distance of1.98 Å /H208492-NN /H20850. Ti 1-NN and 2-NN atoms move away from the vacancy, by 8.2% and 7.1%, respectively. This is reason-able since the Ti ions are positively charged and should,therefore, repel each other as the central oxygen is removed.The trend is in agreement with DFT-LDA investigations per-formed by Cho et al. 26and Ramamoorthy et al. ,44where a relaxation of +0.27–0.30 and +0.1 Å, respectively, was ob-served. The 3-NN and 4-NN O atoms show a small inward relax- ation of 0.4% and 1.4%. The positions of 5-NN and 6-NN Oatoms are virtually unchanged. 2. Optical transition energies of oxygen-deﬁcient rutile The experimental value of the optical transition energy for neutral oxygen-deﬁcient rutile is 1.2 eV,34,40indicating adoubly occupied defect level below the bottom of the con- duction band edge. A charged O defect introduces a partiallyﬁlled defect level at about 1.61 eV below the conductionband. 41The DOS curves for defective Ti 16O31supercell us- ing PW1PW are shown in Fig. 4. We have tested different spin distributions over Ti atoms near the defect site for all the considered supercells. Themost stable structures have been selected for further investi-gation. The two electrons which are trapped in the defectivesupercell are strongly localized in the 3 dorbitals of the 1-NN and 2-NN Ti atoms. The spin density is highest at the two1-NN Ti atoms /H208490.82 a.u. /H20850, and only 0.23 a.u. on the 2-NN Ti atom. The PWGGA method gives a similar spin density dis- tribution as the PW1PW method, with a less pronouncedlocalization on the two nearest Ti atoms. This is in accor-dance with the results of a recent DFT-LDA investigation. 26 Previous theoretical investigations on defective surfaces ofrutile 42,46,48also concluded that the electrons are localized on two ﬁvefold-coordinated titanium atoms, rather than on thedefect site. The same conclusion was also obtainedexperimentally. 3,39 The calculated band gaps of the defective systems as function of the supercell size are presented in Table V. The band gap increases with increasing size of the supercell, i.e.,with decreasing defect concentration. However, its value isnearly converged with the largest considered supercell. Forthe larger supercells, PW1PW /H208490.99–1.06 eV /H20850gives the bestTABLE IV. Distances r/H20849Å/H20850of neighboring atoms from the va- cancy site and changes of the distances /H9004r/H20849%/H20850for the relaxed atoms in Ti 32O64obtained with PW1PW. Atom r Unrelaxed Relaxed/H9004r /H20849%/H20850 Ti/H208492/H20850 r1 1.95 2.11 +8.2 Ti/H208491/H20850 r2 1.98 2.12 +7.1 O/H208491/H20850 r3 2.53 2.52 −0.4 O/H208498/H20850 r4 2.78 2.74 −1.4 O/H208492/H20850 r5 2.98 2.98 0.0 O/H208492/H20850 r6 3.32 3.32 0.0 TABLE V. Calculated values of band gap Eg/H20849eV/H20850for the defec- tive supercells. SupercellEg PWGGA PW1PW Expt.a Ti2O4 0.13 0.51 Ti16O32 0.25 1.06 1.18 Ti32O64 0.21 0.99 aReferences 34and40. DOS(arb. units) Energy (eV) FIG. 4. Total density of states for /H9251and/H9252electrons of rutile TiO 2containing a neutral oxygen vacancy /H20849PW1PW results /H20850.ELECTRONIC PROPERTIES OF OXYGEN-DEFICIENT AND … PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 045217-5reproduction of the experimental optical absorption energy /H20851 1.18 eV /H20849Refs. 34and40/H20850/H20852, whereas PWGGA gives much too small values, 0.21–0.25 eV. Our PWGGA result is inline with previous DFT-LDA studies which gave 0.2 eV/H20849Ref. 26/H20850and 0.3 eV /H20849Ref. 44/H20850, and a PWGGA 45value of 0.1 eV. The underestimation of the optical transition by pure DFT methods can be related to the artiﬁcial self-interaction whichis a consequence of the approximate nature of the exchange-correlation functionals. Moreover, care must be taken in gen-eral when comparing DFT one-particle energies with datafrom optical spectroscopy which correspond to energetic dif-ferences of many-electron states. Quantitative agreementshould therefore not be expected. The hybrid method, on theother hand, was parametrized to reproduce opticaltransitions 57and should therefore better agree with experi- ment. C. Al3+-doped TiO 2 There have been many investigations of the structure of aluminum dopants within the rutile crystal lattice. The resultsare controversial. It is found that Al 3+ions occupy Ti substi- tutional sites,52,78–80interstitial sites,49,81–83a combination of both sites,83–85or Ti substitutional sites in combination with oxygen vacancies.49,82,83,86A UV-visible spectroscopic study49found that the Al3+substitutional doping in combi- nation with oxygen vacancies seems to suppress TiO 2pho- toactivity. On the other hand, the interstitially incorporatedAl 3+has no effect on the photoactivity. In a recent semi- empirical study, Steveson et al.87demonstrated that a single substitution of Ti by Al and two substitutions of Ti by Alcombined with one oxygen vacancy are energetically favor-able over interstitial Al doping. In the present study, substi-tutional Al 3+doping with and without an oxygen vacancy, and with cosubstitution of oxygen by chlorine is examined. 1. Substitutional Al doping AT i 16O32supercell was employed for the defect calcula- tions. One Ti atom was substituted by an Al atom /H20849Fig.5/H20850.I nthis model, the lattice remains neutral but an unpaired elec- tron is introduced. It is treated with the UKS method. It isobserved that the unpaired electron introduced due to theformal substitution of Ti 4+by Al3+is localized on one of the oxygen atoms nearest Al3+. Thus, one oxygen which was formally O2−becomes O−. Initially, the Al is surrounded by four oxygen atoms at a distance of 1.95 Å. After optimiza-tion, the average Al-O distance decreases to 1.92 Å. The electronic properties of the singly Al-doped rutile were investigated by calculating the total density of states/H20849DOS /H20850. The DOS obtained with PW1PW is shown in Fig. 6. There is an unoccupied defect state 2.8 eV above the valenceband maximum which is marked by an arrow. This minorityspin state is composed of oxygen 2 porbitals. The band gap is reduced to 1.6 eV, which would make the oxide colored.This is in contradiction to the experimental observation thatAl-doped rutile is colorless. 49Therefore, we conclude that the nonstoichiometric substitution of Ti4+by Al3+in rutile is not likely. 2. Substitutional Al doping in combination with chloride counterion The formal charge /H20849−1/H20850produced by the substitution of one Ti4+by Al3+can be compensated by a counterion /H20849Cl−/H20850 substituting an oxygen ion /H20849formally O2−/H20850. White titania pig- ments are commonly manufactured via the chloride processwhere titanium tetrachloride reacts with oxygen at tempera-tures between 1300 and 1700 K. 51AlCl 3is also added to increase the rate of anatase-to-rutile transformation.51–54 Therefore, it is assumed that the chloride counterion is in- volved in the photostabilization of commercial rutile pig-ments. In our models, this is realized by replacing one oxy-gen by chlorine in addition to a substitution of Ti by Al. Asthe most likely substitution site, we chose the lattice oxygenthat was reduced to O −after substitution of Ti by Al as de- scribed in the previous section. The distance between Al and Cl is 1.95 Å before relax- ation. After optimization, the Al-Cl distance increases to2.20 Å, corresponding to an outward relaxation of Cl by FIG. 5. Supercell Ti 15AlO 32, doublet state; black spheres repre- sent the oxygen atoms, light gray spheres represent the titaniumatoms, and dark gray sphere represents the aluminum atom. DOS ( arb. units ) Ener gy (eV) FIG. 6. Total density of states for /H9251and/H9252electrons of aluminum-doped TiO 2in the substitutional site /H20849PW1PW results /H20850.ISLAM, BREDOW, AND GERSON PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 045217-613% /H20849PW1PW level /H20850. The PWGGA method gives a similar trend for the relaxation. The DOS /H20849Fig. 7/H20850calculated with PW1PW for a Ti15AlClO 31supercell shows no gap state. As in the undoped system, the VB is mainly composed of oxygen states /H20849with a smaller contribution from titanium states /H20850and the CB is mainly composed of titanium states /H20849with a smaller contribu- tion from oxygen states /H20850. There are very small contributions from aluminum and chlorine states in the valence band. TheFermi energy is unchanged compared to the defect-free situ-ation but the bottom of the conduction band is shifted tohigher energy. Therefore, the band gap increased by 0.18 eV/H20849PW1PW /H20850and 0.09 eV /H20849PWGGA /H20850, compared to undoped rutile /H20849Table VI/H20850. These results indicate that combined sub- stitution of Ti by Al and O by Cl can be responsible for theobserved deactivation of rutile pigments. 3. Substitutional Al doping in combination with an oxygen vacancy Two closest Ti atoms /H20849distance=2.98 Å /H20850were substituted by Al in a Ti 16O32supercell. The resulting formal charge /H20849−2/H20850was compensated by an oxygen vacancy /H20849Fig. 8/H20850at a site bridging the two Al atoms. The distribution of two aluminum atoms and one oxygen vacancy in the supercell was varied and the relative stabilityof various local conﬁgurations was calculated. Instead ofsubstituting only two nearest titanium atoms, two next-nearest titanium atoms /H20849distance=3.57 Å /H20850were also substi- tuted by aluminum atoms. Another test calculation was per- formed by moving the oxygen vacancy at the next-nearestdistance /H208491.98 Å instead of 1.95 Å /H20850. In both cases, the sys- tems are less stable than the ﬁrst considered case where thetwo Al atoms are nearest neighbors of the oxygen vacancy.After optimization, three out of four Al-O distances de-creased from 1.95 to 1.85 Å. One Al-O distance increased to2.20 Å, similar to the Al-Cl distance reported in the previoussection. The DOS of the Ti 14Al2O31supercell calculated with PW1PW is shown in Fig. 9. The Fermi energy is unchanged compared to the undoped system, while the bottom of theconduction band is slightly shifted to higher energies, thusincreasing the band gap /H20849Table VI/H20850. The band gap of rutile decreased due to the presence of oxygen vacancies /H20849Table V/H20850. The oxygen vacancy introduces occupied defect levels in the bulk band gap, leading to apronounced color change of the crystal. In contrast, doping TABLE VI. Comparison of calculated band gap energies Eg /H20849eV/H20850for doped and undoped rutile. MethodEg Ti15AlClO 31 Ti14Al2O31 Undoped PW1PW 3.72 3.65 3.54 PWGGA 1.99 1.95 1.90 DOS (arb. units) Ener gy (eV) FIG. 7. Total density of states for the aluminum-doped TiO 2in combination with chloride counterion /H20849PW1PW results /H20850. FIG. 8. Supercell Ti 14Al2O31; black spheres represent oxygen atoms, light gray spheres represent titanium atoms, and grayspheres represent aluminum atoms. DOS (arb. units) Ener gy (eV) FIG. 9. Total density of states for Al-doped TiO 2in combination with an oxygen vacancy /H20849PW1PW results /H20850.ELECTRONIC PROPERTIES OF OXYGEN-DEFICIENT AND … PHYSICAL REVIEW B 76, 045217 /H208492007 /H20850 045217-7with Al in substitutional sites in combination with an oxygen vacancy does not introduce any defect level in the band gap,since the extra electrons induced by the oxygen vacancy arecompensated by the Al ions. In this way, the rutile band gaphas slightly increased compared to the undoped system.Therefore, aluminum doping does not introduce any color tothe pigment nor does it change the electronic nature of rutile,which remains a semiconductor. 88This is in good agreement with experiments.49,89 IV . SUMMARY AND CONCLUSION The structural, energetic, and electronic properties of sto- ichiometric and defective TiO 2were investigated by means of quantum-chemical calculations and compared to availableexperimental data. The comparison of the optimized latticeparameters with experiment shows that the considered meth-ods give a good reproduction of experimental values. Thehybrid method PW1PW gives reasonable agreement with ex-periment for the calculated energetic and electronic proper-ties, while PWGGA overestimates the binding energy andunderestimates the band gap. The calculated values of theoxygen vacancy formation energy for the largest supercellconsidered here /H20849Ti 54O108/H20850are 431 /H20849PWGGA /H20850and 490 /H20849PW1PW /H20850kJ/mol, which are in the range of experimental enthalpy of oxygen vacancy formation. Relaxation aroundoxygen vacancies is found to be restricted to the nearest-neighbor Ti atoms. The removal of a neutral oxygen atomfrom the rutile lattice leads to the formation of occupieddefect levels below the bottom of the conduction band edge.The extra electrons are mainly localized in 3 dorbitals of the two Ti atoms nearest the vacancy. The band gap is reducedconsiderably compared to that of the nondefective bulk sys-tem. Local d-dtransitions are possible and therefore the pig- ment will appear colored. The best agreement between thecalculated excitation energy and optical spectroscopy wasobtained with the PW1PW approach. The substitution of a single Ti by Al introduces an un- paired electron which is localized on one of the oxygen at-oms closest to Al. An unoccupied peak in the beta ladderappears, which reduces the band gap to 1.6 eV. This wouldmake the system colored, which contradicts the general pur-pose of Al doping and disagrees with experimental observa-tion. Therefore, we conclude that the nonstoichiometric sub-stitution is not likely to occur in the chloride process. Chargecompensation by additional substitution of O by Cl removesthe defect level. In that case, the band gap is even slightlyincreased compared to that in the undoped system. A similarelectronic structure is obtained by substitution of two Ti 4+by Al3+together with the formation of an oxygen vacancy. Therefore, stoichiometric rutile doping with Al is predictedto enhance the photostability of the pigment as compared toO deﬁcient pigments. ACKNOWLEDGMENTS We would like to acknowledge the funding provided by the Australian Research Council /H20849ARC /H20850Linkage grant “Elec- tronic and Optical Properties of Doped Titanium Dioxide”and provision of Visiting Researcher funding by the Univer-sity of South Australia. 1H.-S. Kim, D. C. Gilmer, S. A. Campbell, and D. L. Polla, Appl. Phys. Lett. 69, 3860 /H208491996 /H20850. 2A. Fujishima and K. Honda, Nature /H20849London /H20850238,3 7 /H208491972 /H20850. 3U. Diebold, Surf. Sci. Rep. 48,5 3 /H208492003 /H20850. 4F. H. Jones, Surf. Sci. Rep. 42,7 5 /H208492001 /H20850. 5J. C. Woicik, E. J. Nelson, L. Kronik, M. Jain, J. R. Chelikowsky, D. Heskett, L. E. Berman, and G. S. Herman, Phys. Rev. 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PhysRevB.88.085125.pdf	PHYSICAL REVIEW B 88, 085125 (2013) Mixing between Jeff=1 2and3 2orbitals in Na 2IrO 3: A spectroscopic and density functional calculation study C. H. Sohn,1,2H.-S. Kim,2T. F. Qi,3D. W. Jeong,1,2H. J. Park,1,2H. K. Yoo,1,2H. H. Kim,1J.-Y . Kim,4T. D. Kang,1,2 Deok-Yong Cho,1,2G. Cao,3J. Yu,2S. J. Moon,5,and T. W. Noh1,2,† 1Center for Correlated Electron Systems, Institute for Basic Science, Seoul National University, Seoul 151-747, Korea 2Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea 3Center for Advanced Materials, Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA 4Pohang Accelerator Laboratory, Pohang University of Science and Technology, Pohang 790-784, Korea 5Department of Physics, Hanyang University, Seoul 133-791, Korea (Received 18 September 2012; revised manuscript received 20 July 2013; published 28 August 2013) We investigated the electronic structure of Na 2IrO 3using optical spectroscopy, ﬁrst-principles calculation, and x-ray absorption spectroscopy. We found that the electronic structure of Na 2IrO 3is mainly determined by anisotropic hopping interactions and spin-orbit coupling. Due to the hopping interaction, the orbital characterof the bands near the Fermi level deviates from the spin-orbit coupling-induced J eff=1/2 states. Polarization- dependent O 1 sx-ray absorption spectroscopy showed that the Jeff=1/2 state of an Ir atom can be mixed with theJeff=3/2 state of the neighboring Ir atom. This result implies that mixing between the Jeff=1/2a n d3 /2 states in the valence state should be carefully considered in proposed exotic states of Na 2IrO 3, such as topological insulator and quantum spin liquid states. DOI: 10.1103/PhysRevB.88.085125 PACS number(s): 78 .40.−q, 71.15.Mb, 78 .20.−e, 78.70.Dm Recently, iridates have received much attention as emergent materials for novel quantum phenomena, including spin-orbit coupling (SOC)–driven Mott transition or topologicaleffects. 1–9Most iridates have the Ir4+valence state with t2g5electrons near the Fermi level, EF, and the strong SOC transforms the t2g5states into the J(=L+S) states. The spin-orbital admixture can be represented with effective totalangular momentum J eff=1/2 and Jeff=3/2, as observed in layered perovskite Sr 2IrO 4.1–3TheJeffquantum states of valence electrons will provide a new basis for understandingsof 5dtransition metal oxides, 4–14in which SOC is much stronger than in 3 dtransition metal oxides. Na2IrO 3is a particularly intriguing material whose Jeff states may lead to novel ground states. Na 2IrO 3has edge- sharing IrO 6octahedra that form a honeycomb lattice, where the exchange interaction between two Ir atoms can be highlyanisotropic. Within the J effquantum state, it has been proposed that the magnetic ground state of Na 2IrO 3could be explained in terms of the Kitaev–Heisenberg model.5Many researchers have tried to identify quantum spin liquid,15–19one of the ground states of the Kitaev–Heisenberg model. In addition, thetransfer integral between the next nearest neighboring Ir orbitalvia the oxygen orbitals is complex and spin dependent. Thiscould induce a nonzero Berry phase, leading to a topologicallynontrivial band structure in Na 2IrO 3.4,8It should be noted that the central presumption for the nontrivial ground states inNa 2IrO 3is the full occupation of four Jeff=3/2 states and single electron occupation in the Jeff=1/2 state. However, the orbital characters of the valence bands of Na2IrO 3have not been elucidated. Earlier theoretical work assumed the ideal Jeff=1/2 orbital in a cubic octahedron and suggested the nontrivial electric and magnetic ground statesin Na 2IrO 3.4,5Later, detailed structural studies showed that Ir-O octahedra have a rather signiﬁcant trigonal distortion,which makes the larger O-Ir-O bond angle about 94.5 ◦.17,18 Bhattacharjee et al. pointed out that the crystal ﬁeld due tothe large trigonal distortion could destabilize the Jeff=1/2 states.20However, a recent resonant inelastic x-ray scattering experiment revealed that the trigonal crystal ﬁeld is minimal;thus, the SOC-induced J eff=1/2 orbital scenario is still valid in Na 2IrO 3.21On the other hand, Mazin et al. theorized that the pure Jeff=1/2 atomic-orbital scenario might not be valid in Na 2IrO 3due to the extended nature of the 5 dorbitals and the honeycomb structure.22They suggested that the highly anisotropic hopping interaction in the Ir-O-Ir network causeselectrons to move only within a honeycomb composed of six Irions. This interaction results in quasimolecular orbitals, whichare distinct from the pure J eff=1/2 and 3 /2 states. In these respects, it is important to identify the orbital character ofthe bands near E Fbefore we attempt to identify theoretically suggested nontrivial ground states in Na 2IrO 3. In this paper, we investigate the effects of anisotropic hopping interactions and SOC on the electronic structure ofNa 2IrO 3using optical spectroscopy, ﬁrst-principles calcula- tions, and x-ray absorption spectroscopy (XAS). We observedﬁve clear d-dtransitions in optical conductivity, σ(ω), which can be understood in terms of the Ir t 2gstate splitting due to the anisotropic hopping interaction and SOC. Our polarization-dependent O 1 sXAS data provide experimental evidence that the orbital character of the valence bands of Na 2IrO 3is a mixture of Jeff=1/2 and 3 /2 states. High-quality single-crystal Na 2IrO 3was grown using a self-ﬂux method from off-stoichiometric quantities of IrO 2and Na2CO 3.18We measured the near-normal incident ab-plane reﬂectance spectrum, R(ω), in the energy region between 10 meV and 1 eV at room temperature, and determined theoptical constants between 0.74 and 5 eV using spectroscopicellipsometry. We obtained σ(ω) using Kramers–Kronig anal- ysis. We also measured transmission spectra, T(ω), of thin crystals in the energy region between 10 meV and 1 eV tosee minute spectral features. 23We performed the O 1 sXAS experiment at the 2A beamline of the Pohang Light Source in 085125-1 1098-0121/2013/88(8)/085125(5) ©2013 American Physical SocietyC. H. SOHN et al. PHYSICAL REVIEW B 88, 085125 (2013) FIG. 1. (Color online) (a) Optical conductivity, σ(ω), of Na 2IrO 3. Black circles represent the experimental data, and the solid line is the Lorentz oscillator model ﬁt. The inset of panel (a) shows absorptiondata below 1 eV . Results from DFT calculations: band structures of (b) GGA, (c) GGA +SOC, and (d) GGA +SOC+U. Each arrow in panel (c) indicates a possible optical interband transition between thet 2gorbital states. The yellow and black colors in panel (d) indicate the projected characters of the Jeff=1/2a n d3 /2 orbitals, respectively. the total electron yield mode. We calculated the band structure of Na 2IrO 3using a density functional theoretical (DFT) code, OpenMX,24which is based on the linear combination of pseudoatomic orbital formalism.25We used the Perdew– Burke–Ernzerhof generalized gradient approximation (GGA)function. Details of the calculation method have been reportedelsewhere. 26 Figure 1(a) displays σ(ω)o fN a 2IrO 3. We observe ﬁve peaks in σ(ω) below 2.3 eV , which are marked using alphabetical symbols. Our conductivity data show two peaksbelow 1.0 eV (i.e., peaks A and B), which were not observedin the conductivity data of Ref. 27. These very weak spectral features can be seen more easily in T(ω) thanR(ω). As shown in the inset of Fig. 1(a), the absorption spectra obtained from T(ω) clearly demonstrate the existence of peaks A and B. Because the charge-transfer excitations from the O 2 pto the Ir 5dt 2gstates are located above 2.3 eV ,27we can attribute all ﬁve peaks to the optical transitions between the Ir 5 d orbital states. To obtain more quantitative information on theselow-energy d-dtransitions, we ﬁtted σ(ω) with the Lorentz oscillator model, σ(ω)=/summationdisplay nω2γnSn/parenleftbig ω2n−ω2/parenrightbig2+ω2γ2n(1)TABLE I. Comparison of the energy values of the band gap and interband transitions ( ωn) estimated from the experimental σ(ω) and theoretical band structure. The zigzag magnetic order found in inelastic neutron scattering experiments (Ref. 17) is employed in the band structure calculations. We note that nonmagnetic calculations of GGA +SOC do not produce band gaps, consistent with previous works (Refs. 22and27). Experiment GGA +SOC GGA +SOC+U Band gap 0.32 0.05 0.28 A 0.52 0.30 0.53 B 0.72 0.67 0.86 C 1.32 1.05 1.22D 1.66 1.24 1.39 E 1.98 1.65 1.88 where Sn,ωn, and γnrepresent the strength, resonant fre- quency, and scattering rate of the nth oscillator, respec- tively. The results of the conductivity ﬁtting are summarizedin Table I. The band dispersions from the GGA calculation are shown in Fig. 1(b). All of the calculated bands near E F are very ﬂat and located in three separate energy regions between −2 and +0.5 eV . These band dispersion features result from the formation of the quasimolecular orbitalstates, as pointed out by Mazin et al. 22Namely, in the honeycomb lattice of Na 2IrO 3, the spatial orientations of the Ir 5 dt2gand O 2 porbitals suppress the hopping in one particular direction at each Ir site. The direction of the highlyanisotropic Ir-O-Ir hopping varies from one Ir site to another,which effectively causes electrons to move only in onehexagon, similar to the case of benzene. Such electron motionresults in quasimolecular orbital states. Note that benzenehas well-separated, ﬂat energy levels with singlet, doublet,doublet, and singlet degeneracies. 28According to Mazin et al. , the highly lying singlet and doublet states in Na 2IrO 3 become degenerate due to the O-assisted next nearest neighbor hopping, resulting in three separate energy regions of bands. The effects of the SOC on the electronic structure of Na2IrO 3are illustrated in the result of the GGA +SOC calculation in Fig. 1(c). The SOC splits the energy levels from three to six well-separated energy regions. Note thatin this band structure ﬁve d-dtransitions between the t 2g orbitals are expected, as indicated by the arrows in Fig. 1(c). This is consistent with the experimentally observed ﬁve-peakstructure in σ(ω). The quantitative agreement between the d-dtransition energies of experiment and calculation can be achieved when the on-site Coulomb interaction U=1.0 eV is included [Fig. 1(d)]. As shown in Table I, the GGA +SOC+U calculation produced d-dtransition energies that matched well with the experimental values. These calculation resultsdemonstrate that the ﬁve distinct d-dtransitions in our σ(ω) data can be understood in terms of the combined effect of theformation of the quasimolecular orbital states and SOC in thehoneycomb lattice of Na 2IrO 3. The colors in Fig. 1(d) indicate the projection of the GGA+SOC+Ucalculation results to Jeff=1/2 (yellow color) and 3 /2 (black color) orbitals. One can see that the states near the EFhave orbital characters close to the 085125-2MIXING BETWEEN Jeff=1 2AND3 2... PHYSICAL REVIEW B 88, 085125 (2013) Jeff=1/2 state, and the others exhibit characters close to the Jeff=3/2 state. However, our calculation shows that there should be considerable mixing between Jeff=1/2 and 3 /2 orbitals in every band. For example, unoccupied states andtopmost-occupied states near E FhaveJeff=3/2 projections of about 14% and 30%, respectively. Such orbital mixingscould be important for investigations of numerous proposednovel ground states of Na 2IrO 3. To better understand the orbital characters of the bands near theEF, we performed O 1 sXAS. The O 1 sXAS reﬂects the transition from the O 1 score level to the unoccupied O 2 p states that are hybridized with the Ir 5 dorbitals. Because the Ir 5dorbitals can hybridize with the surrounding 2 pwaves of six oxygen ions only when their point symmetries coincide,we can obtain information on the Ir 5 dorbital characters using polarization-dependent XAS. 29Figure 2(a) shows the experimental geometry. We ﬁxed the incident angle of lightat 70 ◦and changed the light polarization to circumvent any saturation effects of the signals. The peak intensities reﬂect thedirectional O 2 p–Ir 5dhybridization strengths. 30Namely, the σ-polarized light probed (O 2 px/prime,y/prime−Ir 5d), which corresponds to the density of the empty O 2 px/prime,y/primestate hybridized with Ir 5dorbitals. On the other hand, the π-polarized light probed sin270◦(O 2p/prime z−Ir 5d)+cos270◦(O 2px/prime,y/prime−Ir 5d). The primed coordinates z/prime(x/primeandy/prime) are perpendicular (parallel) to the Ir honeycomb plane, and the x,y, andzcoordinates are along the undistorted Ir-O direction, as displayed in Fig. 2(a). Figure 2(b) shows the polarization-dependent O 1 sXAS data for Na 2IrO 3. Three main peaks, labeled α,β, andγ,are FIG. 2. (Color online) (a) Experimental geometry of polarization- dependent O 1 sXAS. The primed coordinate z/prime(x/prime,y/prime) is perpen- dicular (parallel) to the Ir honeycomb plane, and x,y,a n d zare along the undistorted Ir-O direction. σpolarization is parallel to the Ir honeycomb plane, and πpolarization is perpendicular to both incident photon direction and σpolarization. (b) Polarization-dependent O 1 s XAS data. Red triangles and blue circles represent πpolarization and σpolarization data, respectively. The inset of panel (b) shows O 1 s XAS raw data in broad energy region.located at 528.7, 530.8, and 531.5 eV , respectively. Because the features are broad, possible chemical shifts in the O 1 s core hole energy can be neglected. Peak αcan be attributed to the unoccupied t2gorbital states, and peaks βandγare due to the unoccupied egstates. It is interesting to note that whereas the peaks related to egstates ( βandγ)s h o w polarization dependence as well as sizable splitting, the peakassociated with t 2gstates ( α) exhibits negligible polarization dependence. To obtain insight into the orbital character of the t2gstate, we ﬁrst considered the polarization dependence of peaksβandγ, coming from e gstates. As shown in Fig. 2(b), peakβ(γ) is stronger in out-of-plane (in-plane) polarization. The energy splitting of egorbitals and their polarization dependence cannot be explained based on local interactions.The SOC cannot result in energy splitting and polarizationdependences of the e gorbital states, because the egorbital states are insensitive to SOC. The local trigonal distortioncannot explain the energy splitting of the e gstates, either, because egstates provide a good basis for trigonal symmetry. Thus, the energy splitting and polarization dependence of thesetwo peaks implies that a nonlocal interaction affects the orbitalcharacter of Na 2IrO 3signiﬁcantly. Now we consider peak α, which exhibits little polarization dependence. The intensity ratio of peak αinπandσ polarizations is estimated to be 0.88 ±0.07. For comparison, we evaluated the intensity ratio in πandσpolarizations for the pureJeff=1/2 orbital states. We calculated the hybridization between pure atomic Jeff=1/2 orbitals and O 2 porbitals with the inclusion of the reported value of the trigonal structuraldistortion in Na 2IrO 3.18We found that the intensity ratio π/σ should be 1.6 for the highly distorted IrO 6cluster.31This value is higher than the experimental value of 0.88 ±0.07. Therefore, the negligible polarization dependence of peak αin- dicates that the real orbital character of the unoccupied t2gstate should differ from that of the pure atomic Jeff=1/2 orbital. To explain the deviation of the orbital character of the t2gstates from the Jeff=1/2 orbital, we considered the anisotropic hopping interaction.22The left side of Fig. 3(a) shows the Ir honeycomb net structure of Na 2IrO 3. The arrows on the right side of Fig. 3(a) indicate the dominant hopping processes in the Ir 5 din honeycomb structure between two nearest neighboring Ir atoms via O 2 porbitals. Electrons in thedyz(dzx) orbital at one Ir site can hop to the dzx(dyz) orbital of another Ir site via O 2 pz, while electrons in the dxy orbital at one Ir site cannot hop to any orbital at another Ir site via O 2 pz. Due to this anisotropic intersite hopping process, theJeff=1/2 state ( ∓1√ 3[\|dxy,±1/2/angbracketright±\|dyz,∓1/2/angbracketright+ i\|dzx,∓1/2/angbracketright]) can be mixed with the Jeff=3/2 orbital (∓1√ 2[±\|dzx,∓1/2/angbracketright+i\|dyz,∓1/2/angbracketright]). In the same way, the Jeff=1/2 state can be mixed with other Jeff=3/2 states, ∓1√ 2[\|dzx,±1/2/angbracketright+i\|dxy,∓1/2/angbracketright] and ∓1√ 2[\|dyz,±1/2/angbracketright± \|dxy,∓1/2/angbracketright], via O 2 pxand 2py, respectively. The mixed Jeff=1/2 and 3 /2 states explain the small polarization dependence of peak αin our XAS data. Due to the hybridization discussed above, the wavefunction of electrons of Ir ions can be written as /Psi1=/radicalbig 1−\|A\|2\|Jeff=1/2/angbracketright− A\|Jeff=3/2/angbracketright, where Ais a complex mixing coefﬁcient. Using this wavefunction, we calculated the π/σintensity ratio with 085125-3C. H. SOHN et al. PHYSICAL REVIEW B 88, 085125 (2013) FIG. 3. (Color online) (a) Schematic diagram of the hopping process in Na 2IrO 3. The left side of panel (a) shows the Ir honeycomb net structure in Na 2IrO 3. For simplicity, we only show the hopping process in the region surrounded by the black circle, which contains two Ir atoms. Blue and red arrows indicate the dominant hopping process in rectangular Ir-O-Ir chains. (b) Calculated peak intensityratio, π/σ,i nO1 sXAS data for distorted IrO 6octahedra as a function of the portion of Jeff=3/2 in the ground state of /Psi1=/radicalbig 1−\|A\|2\|Jeff=1/2/angbracketright−A\|Jeff=3/2/angbracketright,w h e r e A=Re[A]+ iIm[A] is the complex mixing coefﬁcient. The black lines are contour plots for given values of π/σ, and corresponding values are shown in white. The experimental π/σratio of 0.88 is shown with the red line. the variation in A=Re[A]+iIm[A] [Fig. 3(b)]. The black lines in Fig. 3(b) are contour plots for given values of π/σ. For the pure Jeff=1/2 orbital limit ( A=0),π/σis about 1.6. As \|A\|increases, π/σdecreases; namely, as mixing of theJeff=1/2 and 3 /2 states increases, polarization depen- dence decreases. Using the experimental value π/σ=0.88, the minimum and maximum value of \|A\|2was estimated to beapproximately 0.10 and 0.62, respectively. Thus, the portion of the Jeff=3/2 orbital in the unoccupied Ir 5 dstate near the EFis at least 10%, indicating considerable mixing between theJeff=1/2 and 3 /2 states. Whereas we cannot determine the exact value of \|A\|2solely from the XAS data, weak strength of the peaks A and B in optical conductivity spectra [Fig. 1(a)] further supports that the orbital character should be close tothe minimum value of \|A\| 2.32In addition, the minimum value of\|A\|2is in good agreement with the theoretical estimation for unoccupied states (14%), as mentioned above [Fig. 1(d)]. The observation of sizable mixing between Jeff=1/2 and 3/2 orbitals implies that localized S=1/2 (pseudospin, in that case) Hamiltonian might not be appropriate to account for theground states of Na 2IrO 3. Most of proposed theoretical models that predict exotic ground states of Na 2IrO 3assumed the full- ﬁlledJeff=3/2 orbital and half-ﬁlled Jeff=1/2 orbital.4,5Un- der this circumstance, we can construct S=1/2 Hamiltonian, similar to the 3 dtransition metal oxides system. However, our experimental results clearly showed that the orbital character ofNa 2IrO 3deviates from the localized Jeff=1/2 orbital. Instead, as Mazin et al. pointed out,22our XAS data indicated the delocalized nature of electrons in Na 2IrO 3. Therefore, we insist thatS=1/2 Hamiltonian needs to be modiﬁed to understand the orbital and magnetic ground state of Na 2IrO 3. In conclusion, we found that the anisotropic hopping interaction in the honeycomb Ir lattice contributes to J- state mixing in Na 2IrO 3. Along with the strong SOC effect, J-state mixing explains the experimental observations of the ﬁved-dtransitions in optical conductivity and the negligible polarization dependence of the t2gpeak in the XAS spectra. These ﬁndings suggest that the mixed nature of the Jeff=1/2 and 3/2 states should be taken into account in future studies of novel ground states of Na 2IrO 3. This work was supported by the Institute for Basic Science (IBS) in Korea. S.J.M. was supported by Basic ScienceResearch Program through the National Research Foundationof Korea funded by the Ministry of Science, ICT & FuturePlanning (2012R1A1A1013274). J.Y . and H.S.K. were sup-ported by the NRF through the ARP (R17-2008-033-01000-0). The work at the University of Kentucky was supportedby NSF through grants DMR-0856234 and EPS-0814194.Experiments at Pohang Light Source were supported by MESTand POSTECH. soonjmoon@hanyang.ac.kr †twnoh@snu.ac.kr 1B. J. Kim, Hosub Jin, S. J. Moon, J.-Y . Kim, B.-G. Park, C. S. Leem, Jaejun Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V . Durairaj,G. Cao, and E. Rotenberg, Phys. Rev. Lett. 101, 076402 (2008). 2S. J. Moon, H. Jin, K. W. Kim, W. S. Choi, Y . S. Lee, J. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard, and T. W. Noh, Phys. Rev. Lett. 101, 226402 (2008). 3B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science 323, 1329 (2009). 4A. Shitade, H. Katsura, J. Kune ˇs, X.-L. Qi, S.-C. Zhang, and N. Nagaosa, Phys. Rev. Lett. 102, 256403 (2009).5J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 105, 027204 (2010). 6K. Ishii, I. Jarrige, M. Yoshida, K. Ikeuchi, J. Mizuki, K. Ohashi,T. Takayama, J. Matsuno, and H. Takagi, Phys. Rev. B 83, 115121 (2011). 7I. Kimchi and Y .-Z. You, Phys. Rev. B 84, 180407 (2011). 8C. H. Kim, H. S. Kim, H. Jeong, H. Jin, and J. Yu, Phys. Rev. Lett. 108, 106401 (2012). 9Y . Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012). 10D. Pesin and L. Balents, Nature Physics 6, 376 (2010). 11B.-J. Yang and Y . B. Kim, P h y s .R e v .B 82, 085111 (2010). 085125-4MIXING BETWEEN Jeff=1 2AND3 2... PHYSICAL REVIEW B 88, 085125 (2013) 12X. Wan, A. M. Turner, A. Vishwanath, and S. Y . Savrasov, Phys. Rev. B 83, 205101 (2011). 13A. Go, W. Witczak-Krempa, G. S. Jeon, K. Park, and Y . B. Kim, P h y s .R e v .L e t t . 109, 066401 (2012). 14W. Witczak-Krempa and Y . B. Kim, P h y s .R e v .B 85, 045124 (2012). 15Y . Singh and P. Gegenwart, P h y s .R e v .B 82, 064412 (2010). 16X. Liu, T. Berlijn, W. G. Yin, W. Ku, A. Tsvelik, Y .-J. Kim, H. Gretarsson, Y . Singh, P. Gegenwart, and J. P. Hill, Phys. Rev. B 83, 220403 (2011). 17S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Mazin, S. J. Blundell, P. G. Radaelli, Yogesh Singh, P. Gegenwart, K. R.Choi, S.-W. Cheong, P. J. Baker, C. Stock, and J. Taylor, Phys. Rev. Lett. 108, 127204 (2012). 18F. Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. Fernandez-Baca, R. Custelcean, T. F. Qi, O. B. Korneta, and G. Cao, Phys. Rev. B 85, 180403 (2012). 19J. Chaloupka, G. Jackeli, and G. Khaliullin, P h y s .R e v .L e t t . 110, 097204 (2013). 20S. Bhattacharjee, S. S. Lee, and Y . B. Kim, New J. Phys. 14, 073015 (2012). 21H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin, Y . Singh,S .M a n n i ,P .G e g e n w a r t ,J .K i m ,A .H .S a i d ,D .C a s a ,T .G o g ,M. H. Upton, H. S. Kim, J. Yu, V . M. Katukuri, L. Hozoi, J. vanden Brink, and Y . J. Kim, P h y s .R e v .L e t t . 110, 076402 (2013). 22I. I. Mazin, H. O. Jeschke, K. Foyevtsova, R. Valent ´ı, and D. I. Khomskii, Phys. Rev. Lett. 109, 197201 (2012). 23H. S. Choi, Y . S. Lee, T. W. Noh, E. J. Choi, Y . Bang, and Y . J. Kim, P h y s .R e v .B 60, 4646 (1999).24The DFT code, OpenMX, is available at the web site (http://www.openmx-square.org ) released under the GNU General Public License. 25T. Ozaki, P h y s .R e v .B 67, 155108 (2003). 26H.-S. Kim, C. H. Kim, H. Jeong, H. Jin, and J. Yu, Phys. Rev. B 87, 165117 (2013). 27R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veenstra, J. A.Rosen, Yogesh Singh, P. Gegenwart, D. Stricker, J. N. Hancock,D. van der Marel, I. S. Elﬁmov, and A. Damascelli, P h y s .R e v .L e t t . 109, 266406 (2012). 28M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer, Berlin, Heidelberg, Germany, 2008), Sec. 7.5.3. 29W. B. Wu, D. J. Huang, J. Okamoto, A. Tanaka, H. J. Lin, F. C.Chou, A. Fujimori, and C. T. Chen, P h y s .R e v .L e t t . 94, 146402 (2005). 30J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 31We only considered trigonal distortion itself and ignored the effects of the trigonal crystal ﬁeld because there is theoreticaland experimental evidence of a small trigonal crystal ﬁeld inNa 2IrO 3.21,22 32The optical d-dtransition can be understood as an electron hopping from one Ir site to another Ir site via O 2 porbitals. Because the Ir-O-Ir angle is close to 90◦in Na 2IrO 3, hopping between theJeff=1/2 orbital states of the nearest neighbors is strongly suppressed. Therefore, the weak strength of peaks A and B in opticalconductivity [Fig. 1(a)] implies that the contribution of the J eff= 1/2 orbital to states near the EFis much stronger than that of the Jeff=3/2 orbital. 085125-5
PhysRevB.98.155132.pdf	PHYSICAL REVIEW B 98, 155132 (2018) Efﬁcient vertex parametrization for the constrained functional renormalization group for effective low-energy interactions in multiband systems Carsten Honerkamp* Institute for Theoretical Solid State Physics, RWTH Aachen University, D-52056 Aachen and JARA - Fundamentals of Future Information Technology, Germany (Received 18 May 2018; revised manuscript received 17 September 2018; published 17 October 2018) We describe an efﬁcient approximation for the electron-electron interaction in the determination of the low- energy effective interaction in multiband lattice systems. By using ideas for channel decomposition, form-factorexpansion, and the truncated-unity technique we describe the interaction as arising from the nonlocal and orbital-dependent coupling of particle-hole and particle-particle bilinears formed by ﬁelds residing in the same one ortwo orbitals. This allows us to employ the constrained functional renormalization group (cfRG) with a suitablemomentum and frequency discretization. The approach gives insights into the nonlocal screening of spin andcharge interactions when bands away from the Fermi level are integrated out. Speciﬁcally, we compute theeffective low-energy interactions in the low-energy target band of a three-band model with on-site and nonlocalbare interactions. We show that the cfRG adds important features to the effective target-band interaction thatcannot be found using the constrained random phase approximation (cRPA). DOI: 10.1103/PhysRevB.98.155132 I. INTRODUCTION The constrained random phase approximation (cRPA) [1–4] is a very useful scheme for determining the effective in- teraction in low-energy models for electrons in solids. Startingwith a band structure on a wider energy scale, it allows one totake into account efﬁciently the screening of the interactionsthat act in the low-energy window by the electrons in thebands outside this window. This screening is an importantphysical effect. Accounting for it paves the way for parameter-free calculations [ 4] of properties of correlated materials. As expressed by its naming, the cRPA is an approximatescheme, perturbative in the electron-electron interactions, thatconsists of selecting a certain class of diagrams. It amountsto an inﬁnite-order summation of an appropriately deﬁnedpolarization function. The constraint consists of disallowingcontributions to this polarization that are purely due to thelow-energy bands (also called target bands). This means thatthe allowed contributions are those within the high-energybands only and those between the high-energy and low-energybands. As with any approximation, it is valuable to know how good it is and whether there are relevant corrections terms toit. This holds in particular for the cRPA in its usual realm,where no small parameter like large Nor small q/k Fexists that would render some type of control over this approxima-tion. In two previous works [ 5,6] we have proposed to use a constrained functional renormalization group scheme (thatwe from now on call cfRG) to extend the cRPA and henceto include additional diagrams that are neglected in the cRPAinto the calculation of the effective interactions. The cfRG isan adaptation of the general functional renormalization group *honerkamp@physik.rwth-aachen.deframework for interacting fermions [ 7] to the problem of tailoring effective target-band actions by integrating over thehigh-energy bands in multiband electron systems. We showedthat at least in simple models [ 6], sizable corrections to cRPA can exist (but do not have to—this depends on the model). Inorder to interpret these corrections we analyzed the frequencyand momentum dependence of these terms. However, thesedependencies become quite rich, as in the cfRG the effectiveinteraction of a translationally symmetric model depends onthree momenta and three frequencies, in addition to possiblyfour band or orbital indices. Therefore the cfRG treatmentof more realistic models that embody more deﬁnite mate-rial properties is facing a bottleneck of how this wealth ofinformation can be processed and evaluated efﬁciently. In thisaspect, the cRPA is simpler, as its effective interaction onlydepends on one frequency and one momentum. In addition,its results can be interpreted in terms of effective dielectricfunctions. Regarding the complexity of the momentum and fre- quency dependence, recent fRG literature contains consider-able progress. One important tool has become the channeldecomposition [ 8–11] of the ﬂowing interaction into pair- ing, charge, and spin channels (or linear combinations ofthe latter two). This decomposition can be motivated bothby the structure of second-order perturbation theory in theinteractions as well as by the effective interactions found inthe fRG for usual models. It allows us to read the interactionas mediated by collective bosonlike propagators that couple topairing, charge, or spin bilinears, depending on the respective channel. In addition to that, a form-factor expansion [ 8–11]o f the internal spatial structure of these coupling bilinears wasfound to converge quickly for standard settings [ 12–14]. As a beneﬁt of these reformulations of the momentum structure,the fRG codes can be pushed to much higher momentumresolution and perform well on highly parallel computing 2469-9950/2018/98(15)/155132(18) 155132-1 ©2018 American Physical SocietyCARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) infrastructures. Regarding the frequency structures, the situa- tion is slightly more complex because at least at stronger cou-pling [ 15,16] and in particular contexts [ 17], a simple channel decomposition, as pioneered in zero-dimensional models [ 18] and recently also used in one dimension [ 19,20], can be problematic. Nevertheless, for weaker coupling, the channeldecomposition with some additional tweaking was found tomake sense in standard setups empirically [ 21]. It was also shown that a so-called static channel-coupling approxima-tion for a channel-decomposed one-frequency parametrizationof the frequency dependence is quantitatively reasonable aslong as the initial interaction is not retarded and no strongretardations are generated during the ﬂow. Note also thatform-factor expansions for the frequency dependence arebeing worked out currently [ 22]. These could make the one- frequency parametrizations more robust without increasingthe numerical effort too strongly. Furthermore, efﬁcient treat-ments of the high-frequency asymptotics of the vertex havebeen considered [ 23]. In this work we present a channel-decomposed fRG scheme for the computation of the effective interactions ina low-energy target band when bands away from the Fermilevel are integrated out. Besides restricting the form-factorexpansion to local bilinears, we also choose to concentrateon the orbital-diagonal bilinears. These bilinears are the onesthat interact in the bare action and hence neglecting orbitalnondiagonal bilinears appears to be justiﬁable unless particu-lar evidence for the generation of or interest in new terms ispresent. Of course the formalism can be expended to includethose bilinears as well as nonlocal form factors. Our goal hereis to showcase a numerically fast core scheme with diagonallocal bilinears that, despite the simpliﬁcations, produces moreand interestingly different renormalizations compared to thecRPA. The paper is organized as follows. In Sec. IIwe describe the test model with three orbitals and the multiorbital inter-actions that are considered. Section IIIquickly recapitulates the constrained random phase approximation. In Sec. IVwe lay out the truncated-unity fRG formalism in the contextof the constrained fRG. Readers familiar with the channel-decomposed and form-factor-expanded (or SMfRG [ 11]) fRGformalism can ﬁnd the essential modiﬁcations for the cfRG in Sec. IV B and then jump to the main ﬂow equations ( 45), (46), and ( 47). Section Vcontains the simplifying approximations, which are the main methodical message of this work, andwhich reduce the numerical effort considerably and make thetreatment of multiorbital cases more feasible. Section VA addresses the frequency dependence of the effective inter-action, Sec. VB its momentum dependence, and Secs. VC andVD its orbital structure. In Sec. VIwe explain how we evaluate the resulting target-band interaction, and Sec. VII contains the discussion of the numerical results for the three-orbital test model. In Sec. VIII we conclude with a discussion of the numerical advantages of the proposed approximations. II. MODEL We assume that our model on the wider energy scale of a few tens of eV is spin-rotational symmetric. This assumptioncan also be relaxed (see Ref. [ 24] for a description of channel- decomposed fRG without spin-rotational invariance), but forthe sake of this paper we keep spin rotational invariance aliveas this simpliﬁes the formulas. We further assume that themodel can be formulated in a Wannier basis. The Wannierorbitals within a unit cell shall be indexed by osuch that we have a Hamiltonian of the following type: H=/summationdisplay /vectork,s,s/prime o,o/primehoo/prime(/vectork)c† /vectork,s,oc/vectork,s,o/prime+1 2N/summationdisplay /vectork1,/vectork2,/vectork3,s,s/prime o1,o2,o3,o4Vo1o2o3o4 ×(/vectork1,/vectork2,/vectork3)c† /vectork3,s,o 3c† /vectork4,s/prime,o4c/vectork2,s/prime,o2c/vectork1,s,o 1. (1) The wave vectors /vectorklive in the ﬁrst Brillouin zone of the lattice withNunit cells and periodic boundary conditions. /vectork4is determined by /vectork1+/vectork2−/vectork3modulo reciprocal lattice vectors. The kinetic matrix hoo/prime(/vectork) contains the hopping amplitudes which are obtained from the band structure or from the overlapmatrix elements between the Wannier functions. A. Band structure of three-orbital test model For the testing of the formalism developed below we choose a three-orbital model with a kinetic matrix ˆh=⎛ ⎝/epsilon11−2t11(coskx+cosky) t12 t13 t12 /epsilon12−2t22(coskx+cosky) t23 t13 t23 /epsilon13−2t33(coskx+cosky)⎞ ⎠. (2) Typical parameter choices are (in some sensible energy unit) t11=1,t12=2,t13=0.5,t22=t33=− 0.1,/epsilon11=0,/epsilon12= −/epsilon13=2. This example gives rise to the bands shown in Fig. 1. B. Multiorbital interactions In general the Coulomb interaction expressed in the Wan- nier basis can be quite complex, but in many cases it isreasonable to focus on to a two-center approximation in theorbital indices, i.e., consider nonzero terms where only twophysical orbitals are involved. For these terms we considerthe following standard interactions:(1) Intraorbital density-density interactions of the type V o1o2o3o4(/vectork1,/vectork2,/vectork3)=δo1o3δo2o4Vρ o1o2(/vectork1−/vectork3). (3) Terms like these arise via Fourier transformation from direct matrix elements of the Coulomb interaction with Wannierstates w o(/vectorr−/vectorR), centered at positions /vectorroin the unit cell indexed by Bravais lattice vectors /vectorR, Vρ o1o2(/vectorR)=Vc/integraldisplay d3r/integraldisplay d3r/prime ×\|wo1(/vectorr+/vectorro1)\|2\|wo2(/vectorr/prime+/vectorro2−/vectorR)\|2 \|/vectorr+/vectorro1−/vectorr/prime−/vectorro2+/vectorR\|.(4) 155132-2EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) FIG. 1. Band structure of the three-orbital model described in the text in Eq. ( 2) with parameters t11=1,t12=2,t13=0.5,t22=t33= −0.1,/epsilon11=0,/epsilon12=−/epsilon13=2. The middle band crosses the Fermi level and is called target band. The band ﬁlling of the central band is roughly 40%. Included in this are for /vectorR=0 the on-site intraorbital (for o1= o2) and interorbital (for o1/negationslash=o2) interactions. The nonlocalterms for /vectorR/negationslash=0 will still have some orbital dependence for small /vectorRif the orbitals are not placed at the same center position in the unit cell or have strongly varying spatialextent. However, the Coulomb tail for /vectorRmuch larger than the lattice constant and the /vectorr o, will to good approximation become independent of the orbital index, and only depend onthe distance \|/vectorR\|. Hence, the small- /vectorqpartVρ o1o2(/vectorq) will also be nearly orbital independent. In the model calculations belowwe will include the Coulomb tail by the Hamiltonian H Coul=Vc 2/summationdisplay /vectorR/negationslash=/vectorR/prime o1,o2,σ,σ/primen/vectorR,o1,σn/vectorR/prime,o2,σ/prime \|/vectorR−/vectorR/prime\|e−\|/vectorR−/vectorR/prime\|/λ, (5) assuming /vectorro1=/vectorro2inside the unit cell and /vectorR,/vectorR/primerunning over the unit cell positions. We have added in by hand anexponential decay factor with a screening length λ. This factor regularizes the divergence of the Fourier transform of ( 5) and is hence needed in the numerical implementation. We usuallytakeλ=6 lattice constants. (2) Local intraorbital local density-density and spin- spin interactions are often presented in the form of theso-called Kanamori Hamiltonian (see, e.g., Refs. [ 25,26]), which reads (suppressing the site index for this on-siteinteraction) HU=U/summationdisplay o1no1,↑no1,↓+/summationdisplay o1>o 2 σ,σ/prime(U/prime−δσσ/primeJ)no1,σno2,σ/prime−/summationdisplay o1/negationslash=o2J(c† o1,↓c† o2,↑co2,↓co1,↑+c† o2,↑c† o2,↓co1,↑co2,↓) =U/summationdisplay o1no1,↑no1,↓+/summationdisplay o1/negationslash=o2 σ,σ/primeU/prime 2no1,σno2,σ/prime+J 2/summationdisplay o1/negationslash=o2 σ,σ/prime(c† o1,σc† o2,σ/primeco1,σ/primeco2,σ+c† o2,σc† o2,σ/primeco2,σ/primeco1,σ). (6) HereUandU/primeare intra- and interorbital density-density interaction parameters and Jis the Hund’s rule interaction. In cases with rotational invariance one has U−U/prime=2J. Note that also ( 6) only involves two orbital indices o1ando2, i.e., is an interaction of equal-orbital fermion bilinears. More general local density-density interactions Uoo/primeor spin-spin interactions Joo/primeare also conceivable and do not lead to extra efforts in the approximations discussed below. In general, even if the bare Hamiltonian of the model with the initial wide bandwidth does only contain these two-centerinteraction terms, the renormalization group ﬂow that inte-grates out the high energy bands will generate all sorts of morecomplicated terms that depend on four orbital indices and thatalso exhibit dependencies on three momenta and frequencies.This complexity is hard to deal with for a true multibandsituation with many bands. Below we propose approximationstrategies to reduce the effort. III. CRPA The cRPA [ 1–4] is usually formulated in the band picture and bands bthat do not belong to the target bands nearthe Fermi level are integrated out. Starting with the bare Coulomb repulsion, one computes an effective interactionW cRPA(q) that acts between electron bilinears ¯cb3,k+q,scb1,k,s and ¯cb4,k/prime−q,scb2,k/prime,s, here expressed with fermion Grassmann ﬁelds that depend on frequency-momentum indices k, target- band indices biin the low-energy space, and spin projections sands/prime. The corresponding Bethe-Salpeter equation is shown in Fig. 2(a). For the context of this paper, the orbital representation is more appropriate. Then we can group the orbital indices o1 ando3belonging to external legs with spin s[say on the left side of the rectangular vertex representing the bare Coulombinteraction V o1o2o3o4(/vectorq)i nF i g . 2(a) or3(a)] to a double index and those of the other two legs with spin s/prime,o2ando4,t o another double index. The cRPA effective interaction is, usinghats ˆ to indicate the matrix structure in the two double orbitalindices, ˆW (r)(q)=ˆ/epsilon1−1 r(q)ˆV(/vectorq), (7) with the screening function ˆ/epsilon1r(q)=[ˆ1−ˆV(/vectorq)ˆχ(r)(q)]. (8) 155132-3CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) FIG. 2. (a) Effective cRPA interaction. The empty rectangles denote the bare interaction and the full ones the screened interaction. The spin projection is conserved along the short edge of the rectangle (assuming spin-rotational invariance). (b) Five one-loop diagrams on the right-hand side of the fRG ﬂow equation for the interaction vertex. cfRG takes into account all diagrams. The internal lines carry cutofffunctions and get differentiated with respect to the RG ﬂow parameter in the ﬂow equation. Only taking into account the diagram denoted as RPA amounts to cRPA. This formula is represented diagrammatically on the left side of Fig. 2. It includes the constrained polarization transformed into the orbital frame, ˆχ(r) (o1o2)(o3o4)(q)=2T N/summationdisplay k b1,b2uo1b1(/vectork +/vectorq)u∗ o2b2(/vectork)uo4b2(/vectork)u∗ o3b1(/vectork+/vectorq)Gb1(k +q)Gb2(k)[ 1−/Pi1t(b1,b2)], (9) in which we have indicated by the parentheses like ( o1o2) which orbital indices belong to the same vertex side. Theu ob(/vectork) are the matrix elements of the orbital-to-band trans- formation which diagonalizes the kinetic matrix Hoo/prime(/vectork)i n the bare Hamiltonian ( 1). The projector /Pi1t(b1,b2) is unity if bands b1andb2are both target bands and zero otherwise. The prefactor 2 comes from the spin sum. In contrast withthe full RG, for which the interaction depends on three wavevectors and frequencies, the cRPA just modulates the transfer-wave-vector /vectorqand transfer-frequency ωdependence of the interaction. FIG. 3. (a) Coupling function for the spin-rotational multiorbital case. (b)–(d) The three channels kept in the IOBI approximation of Sec. VC.IV . MULTIORBITAL TRUNCATED-UNITY FUNCTIONAL RENORMALIZATION GROUP Over the last two decades, functional renormalization group methods have been applied to the Hubbard-like fermionlattice model in many different forms and with different de-grees of approximations [ 7,27]. The standard renormalization group formalism based on the Wetterich equation [ 28]f o r the generating functional for one-particle irreducible vertexfunctions prescribes ﬂow equations as a function of a ﬂowingenergy scale, here called /Lambda1. In this work we focus on the one-particle irreducible two-particle interaction vertex withorbital indices and ignore all higher vertices as well as theﬂow of the one-particle vertex, i.e., the self-energy. Let us ﬁrst describe the quantum numbers on which the interaction vertex depends on. We work with combinedMatsubara-frequency/wave vector variables k i=(k0,i,/vectorki) where a fermionic Matsubara frequency k0,iis an odd mul- tiple of πT and/vectorkiis a wave vector in the ﬁrst Brillouin zone. In addition, we have the orbital indices oiwhich label the localized Wannier states that are as basis functions forthe electronic states. Next we assume spin-rotational SU(2)invariance. Then the two-particle vertex can be described bya coupling function V /Lambda1 o1o2o3o4(k1,k2,k3). In this notation [ 29] [see also Fig. 3(a)], the two incoming particles k1,o1and k2,o2of the interaction carry spin projection σandσ/primeand k3,o3denotes the ﬁrst outgoing particle with the same spin projection as the ﬁrst incoming particle with k1,o1, i.e.,σ. The momentum and frequency dependence of two-particlescattering can also be described in terms of Mandelstamvariables s=k 1+k2,t=k3−k1,andu=k4−k1,(10) withk4=k1+k2−k3(modulo lattice). A. Channel decomposition and ﬂow of vertex Based on the models introduced above we make the follow- ing channel-decomposed ansatz [ 7,8,11,12] for the effective interaction at the initial energy scale /Lambda10: S/Lambda10 I=S/Lambda10 D+S/Lambda10 C+S/Lambda10 P, (11) 155132-4EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) with S/Lambda10 D=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4D/Lambda10 o1o2o3o4(k,k/prime;q)[¯ck+q,o 3,σck,o 1,σ][¯ck/prime,o4,σ/primeck/prime+q,o 2,σ/prime] (12) in the density-density channel, S/Lambda10 C=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4C/Lambda10 o1o2o3o4(k,k/prime;q)¯ck/prime,o3,σ¯ck+q,o 4,σ/primeck/prime+q,o 2,σ/primeck,o 1,σ =−T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4C/Lambda10 o1o2o3o4(k,k/prime;q)[¯ck/prime,o3,σck/prime+q,o 2,σ/prime][¯ck+q,o 4,σ/primeck,o 1,σ] (13) in the spin-ﬂip channel, and S/Lambda10 P=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4P/Lambda10 o1o2o3o4(k,k/prime;q)[¯ck,o 3,σ¯c−k+q,o 4,σ/prime][c−k/prime+q,o 2,σ/primeck,o 1,σ] (14) in the pair channel. By the square brackets we group the single fermion ﬁelds into fermion bilinears. In the Dchannel we have equal-spin bilinears, which can be further collected, by summing over the spin and k, to charge bilinears. In the Cchannel, spin- ﬂips from σtoσ/primeare possible, and spin bilinears are obtained upon summation over kand spin, with appropriate Pauli matrices inserted in the spin summations. In the Pchannel, we orbital pair-hopping or pair bilinears. The deﬁnition of the channels follows the singular-mode fRG proposed by Wang et al. [11,30]. Below we discuss how to distribute the bare interactions discussed in Sec. II Bover these three channels. We can also write this interaction in terms of the coupling function that contains all channels, S/Lambda10 I=T 2N/summationdisplay k1,k2,k3V/Lambda10 o1o2o3o4(k1,k2,k3)¯ck3,o3,σ¯ck4,o/prime,σ/primeck2,o2,σ/primeck1,o1,σ, (15) withk4=k1+k2−k3modulo reciprocal lattice. Then V/Lambda10 o1o2o3o4(k1,k2,k3)=D/Lambda10 o1o2o3o4(k1,k4;k3−k1)+C/Lambda10 o1o2o3o4(k1,k3;k3−k2)+P/Lambda10 o1o2o3o4(k1,k3;k1+k2). (16) The channel-decomposed fRG takes advantage of the observation that the wave vector and (in part) also the frequency dependence of the renormalized interaction usually can be understood as a sum of three different channels, which are againof density, spin, and pair type. All of these three channels describe the interaction of a particular type of fermion bilinear anddepend most strongly on a particular collective wave vector qthat upon Fourier transformation to real space or time is related to the space or time distance between the bilinears that interact. The fRG ﬂow is induced by the scale-dependent regulator or cutoff function, here denoted as R /Lambda1(b,k) with/Lambda1as fRG scale. The bare propagator of the system is modiﬁed by the multiplication with this regulator, G(0) o1o2(k)−→G(0),/Lambda1 o1o2(k)=G(0) o1o2(k)R/Lambda1(b,k). (17) Typically [ 7], when integrating over high-energy modes, the regulator is chosen such that low-energy excitations of the system are cut out for initial values of /Lambda1and only included successively when /Lambda1is changed toward its ﬁnal value. Then, when /Lambda1is varied, the ﬂow equation for V/Lambda1 o1o2o3o4(k1,k2,k3) reads in the common level-2 truncation [ 7] d d/Lambda1V/Lambda1 o1o2o3o4(k1,k2,k3)=∂/Lambda1P/Lambda1 o1o2o3o4(k1,k3;s)+∂/Lambda1D/Lambda1 o1o2o3o4(k1,k4;t)+∂/Lambda1C/Lambda1 o1o2o3o4(k1,k3;u), (18) with the one-loop particle-particle contributions ∂/Lambda1P/Lambda1 o1o2o3o4(k1,k3;s) and the two different particle-hole channels ∂/Lambda1D/Lambda1 o1o2o3o4(k1,k4;t) and∂/Lambda1C/Lambda1 o1o2o3o4(k1,k3;u), where ∂/Lambda1P/Lambda1 o1o2o3o4(k1,k3;s)=T N/summationdisplay k o5,o6,o7,o8V/Lambda1 o1o2o5o6(k1,−k1+s,k)∂/Lambda1/bracketleftbig G/Lambda1 o5o7(k)G/Lambda1 o6o8(−k+s)/bracketrightbig V/Lambda1 o7o8o3o4(k,−k+s,k 3), (19) ∂/Lambda1D/Lambda1 o1o2o3o4(k1,k4;t)=− 2T N/summationdisplay k o5,o6,o7,o8V/Lambda1 o1o6o3o5(k1,k+t,k 1+t)∂/Lambda1/bracketleftbig G/Lambda1 o5o7(k)G/Lambda1 o8o6(k+t)/bracketrightbig V/Lambda1 o2o7o4o8(k,k 4+t,k+t) +T N/summationdisplay k o5,o6,o7,o8V/Lambda1 o1o6o3o5(k1,k+t,k 1+t)∂/Lambda1/bracketleftbig G/Lambda1 o5o7(k)G/Lambda1 o8o6(k+t)/bracketrightbig V/Lambda1 o2o7o8o4(k4+t,k,k +t) +T N/summationdisplay k o5,o6,o7,o8V/Lambda1 o1o6o5o3(k+t,k 1,k1+t)∂/Lambda1/bracketleftbig G/Lambda1 o5o7(k)G/Lambda1 o8o6(k+t)/bracketrightbig V/Lambda1 o2o7o4o8(k,k 4+t,k+t),(20) 155132-5CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) ∂/Lambda1C/Lambda1 o1o2o3o4(k1,k3;u)=T N/summationdisplay kV/Lambda1 o1o6o5o4(k1,k+u,k)∂/Lambda1/bracketleftbig G/Lambda1 o5o6(k)G/Lambda1 o8o6(k+u)/bracketrightbig V/Lambda1 o2o7o8o3(k,k 3+u,k 3). (21) In these equations, Nis the number of unit cells in the system which should be sent to inﬁnity toconvert the momentum sums to integrals. The productof the two internal lines in the one-loop diagramscontains the single-particle Green’s function G /Lambda1 oo/prime(k)=/summationtext bR/Lambda1(b,k)uob(k)u∗ o/primeb(k)/[−iω+/epsilon1b(/vectork)+R/Lambda1(b,k)/Sigma1/Lambda1(k)] at RG scale /Lambda1. The sum goes over the bands bof the model. The bare interaction ( 15) constitutes the initial condition for the ﬂow ( 18). The coupling function V/Lambda1 o1o2o3o4(k1,k2,k3) can be interpreted as effective interaction at scale /Lambda1. In the full fRG formalism, the Green’s functions G/Lambda1 oo/prime(k) should contain self-energy corrections. In this work we ignoreself-energy corrections in the integration over the high-energy bands. In general, as this integration does not involve smallenergy denominators in loop diagrams, the self-energy correc-tions should be dominated by Hartree-Fock contributions thatwould mainly lead to a deformed band structure. In order to understand the effects on the effective interactions, we think that is even useful to suppress these band deformation effectsfor the time being. Currently, the self-energy feedback is beingexplored actively in the fRG community and can be added tothe cfRG in future works. At intermediate fRG scales we have the following interac- tion in the effective action: S/Lambda1 I=S/Lambda1 D+S/Lambda1 C+S/Lambda1 P, (22) with S/Lambda1 D=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4D/Lambda1 o1o2o3o4(k,k/prime;q)¯ck+q,o 3,σ¯ck/prime,o4,σ/primeck/prime+q,o 2,σ/primeck,o 1,σ (23) in the density-density channel, S/Lambda1 C=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4C/Lambda1 o1o2o3o4(k,k/prime;q)¯ck/prime,o3,σ¯ck+q,o 4,σ/primeck/prime+q,o 2,σ/primeck,o 1,σ (24) in the spin-ﬂip channel, and S/Lambda1 P=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4P/Lambda1 o1o2o3o4(k,k/prime;q)¯ck,o 3,σ¯c−k+q,o 4,σ/primec−k/prime+q,o 2,σ/primeck,o 1,σ (25) in the pair channel. B. Constrained functional renormalization group In the constrained functional renormalization group (cfRG) [5] the so-called Wick-ordered ﬂow [ 31] is used as approx- imation for the two-particle interaction. Here the one-loopdiagrams of the right-hand side of Eq. ( 18) have one internal line that belongs to the high-energy excitations, which areintegrated out, while the other line can be in the low-energysector, which is targeted for the effective model. In this way,the ﬂow of the interaction sums up all one-loop terms thatcontain at least one high-energy line, i.e., all loop terms thatcannot be captured in the solution of the low-energy effectivemodel. Without self-energy corrections, the ﬂow equation ( 18) for the interaction is basically unchanged up to the differentregulator prescription and an overall minus sign for the right-hand side, so all techniques form the “usual” fRG can beapplied as well. In a momentum-shell variant of the cfRG scheme, the regulator R /Lambda1(b,k) suppresses the modes with \|/epsilon1b(/vectork)\|>/Lambda1, i.e., those above the cutoff. Its /Lambda1-derivative ˙R/Lambda1(b,k) conﬁnes the modes to a momentum shell with \|/epsilon1(/vectork)\|≈/Lambda1. In this workwe however use a simpler, ﬂat cutoff R/Lambda1(b,k)=/Lambda1 forb∈high-energy bands, R/Lambda1(b,k)=1f o r b∈target bands . The second line guarantees that no loops with two target-band lines contribute to the right-hand side of the ﬂow equation,as˙R /Lambda1(b,k)=0f o r bin the target band. The ﬂow goes from/Lambda1=1d o w nt o /Lambda1=0. This adiabatically removes the high-energy bands from the theory. As these bands are awayfrom the Fermi surface, no divergences occur in the ﬂow.Furthermore, different choices for the cutoff function shouldyield quite similar end results. The diagrams for the right-hand side are also shown in Fig. 2(b). As explained Ref. [ 5], the cRPA is obtained by only keeping the ﬁrst term with the −2 in front on the right-hand side of ( 20). C. Form-factor expansion In the channel-decomposed fRG the k,k/primedependence of the three channels P/Lambda1 o1o2o3o4(k,k/prime;q),C/Lambda1 o1o2o3o4(k,k/prime;q), and D/Lambda1 o1o2o3o4(k,k/prime;q) besides the dependence on the collective 155132-6EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) variable qis captured via a form-factor expansion. In the literature (e.g., Refs. [ 8,11,12]), this has so far been done for the wave vector dependence and not for the frequencydependence. Nevertheless, for what follows it is advantageousto lay out the basic formalism for expansions of both. Hence we write, e.g., for the Dchannel, D /Lambda1 o1o2o3o4(k,k/prime;q)=/summationdisplay l,l/prime˜D/Lambda1 o1o2o3o4(l,l/prime;q)˜fl(k)˜f∗ l/prime(k/prime),(26) where the ˜fl(k) form an orthogonal basis system over the Brillouin zone of the lattice and over the Matsubara frequencyaxis, /summationdisplay k˜f∗ l(k)˜fl/prime(k)=δ/vectorl,/vectorl/primeδ(τ−τ/prime). (27) Here we distinguish between spatial components /vectorlof the form-factor index land the Matsubara time τ∈[0,β] with β=1/T. The inversion of ( 26) is then ˜D/Lambda1 o1o2o3o4(l,l/prime;q)=/summationdisplay k,k/primeD/Lambda1 o1o2o3o4(k,k/prime;q)˜f∗ l(k)˜fl/prime(k/prime).(28) A usual approximation for a discrete set of form-factor indices then consists of truncating the form-factor expansion after acertain expansion order l maxsuch that channel propagators ˜D/Lambda1 o1o2o3o4(l,l/prime;q) become lmax×lmaxmatrices for each q.W e can understand the physical meaning of the expansion ( 26)b y reinserting it into ( 23). This leads to an interaction of fermion bilinears that can be transformed back onto the real lattice andto Matsubara time, /summationdisplay k˜fl(k)¯ck+q,o 3,σck,o 1,σ =/summationdisplay /vectorR,/vectorR/prime/integraldisplayβ 0dτ/integraldisplayβ 0dτ/prime¯c/vectorR,o 3,σ(τ)c/vectorR/prime,o1,σ(τ/prime)T N ×/summationdisplay k0,/vectork˜fl(k)eik0(τ−τ/prime)e−i/vectork(/vectorR−/vectorR/prime). (29) Now, a very physical route is to associate bond vectors /vectorb of the Bravais lattice with the spatial part of land write ˜fl(k)=˜f/vectorb(/vectork)fl0(k0) with ˜f/vectorb(/vectork)=/radicalBig 1 Ne−i/vectork/vectorb. Then the wave vector sum in the end of the above expression gives δ/vectorb,/vectorR−/vectorR/prime. Thus the bilinear lives on the bond /vectorb, i.e., its creation and annihilation operators are by the vector /vectorbapart. If the main physics is given by short-bond bilinears, we can truncate theexpansion ( 26)f o rl o w lor/vectorb. The dependence on k 0will determine the τ-τ/primestructure of the bilinear in ( 29). Now we use the simplest approx- imation and only keep k0-independent frequency-form fac- tors ˜f0(k0)=√1/β, i.e., only consider ˜fl(k)=√1/β˜f/vectorb(/vectork). Then the bilinears on the right-hand side above will be atequal times, as we obtain a factor δ(τ−τ /prime)f r o m( 29) when we sum over k0. We can now introduce the instantaneous on-site approximation , which uses exactly those frequency- independent form factors and ignores all bilinears on bondsof ﬁnite length /vectorb/negationslash=0. While it is not obvious that the in- stantaneous on-site approximation is a good approximationto the full ﬂowing interaction in the general case, we notethat the bare interactions are of this type, and that the usual single-channel summations like RPA or ladder summationsfor site-centered spin and charge fermion bilinears are kept inthis approximation. The interaction is still allowed to developa retardation and distance dependence between two bilinears,only the bilinears themselves remain local and instantaneous. For a reason explained below it is useful to rescale the form factors by√ T/N and introduce fl(k)=/radicalbigg N T˜fl(k). (30) For instance, using plane waves this would mean fl(k)= e−i/vectork/vectorbeik0τ. This choice leads to the orthogonality relation T N/summationdisplay kf∗ l(k)fl/prime(k)=δ/vectorl,/vectorl/primeδ(τ−τ/prime) (31) and the resolution of unity T N/summationdisplay lfl(k)fl(k/prime)=δk,k/prime. (32) With the rescaled form factors, using D/Lambda1 o1o2o3o4(l,l/prime;q)= T N˜D/Lambda1 o1o2o3o4(l,l/prime;q), we have the expansion D/Lambda1 o1o2o3o4(k,k/prime;q)=/summationdisplay l,l/primeD/Lambda1 o1o2o3o4(l,l/prime;q)fl(k)f∗ l/prime(k/prime) (33) and the projection D/Lambda1 o1o2o3o4(l,l/prime;q)=T2 N2/summationdisplay k,k/primeD/Lambda1 o1o2o3o4(k,k/prime;q)f∗ l(k)fl/prime(k/prime). (34) Here we see the advantage of the form-factor rescaling: ifD /Lambda1 o1o2o3o4(k,k/prime;q) is just a constant D0, also the expansion co- efﬁcient D/Lambda1 o1o2o3o4(0,0;q) will be D0, without any additional T- andN-dependent prefactors. With these conventions, the charge interaction then becomes in this instantaneous approximation S/Lambda1 D=T 2N/summationdisplay k,k/prime,q,σ,σ /prime o1,o2,o3,o4D/Lambda1 o1o2o3o4(q)¯ck+q,o 3,σ¯ck/prime,o4,σ/primeck/prime+q,o 2,σ/primeck,o 1,σ. (35) Here the l=0 indices in ˜D/Lambda1 o1o2o3o4(q)=˜D/Lambda1 o1o2o3o4(0,0;q) have not been written. We see from this formula thatthe interaction between charge bilinears ¯c k+q,o 3,σck,o 1,σand ¯ck/prime,o4,σ/primeck/prime+q,o 2,σ/primecan still become long ranged in space and Matsubara time, provided that the qdependence develops sufﬁcient structure in ˜D/Lambda1 o1o2o3o4(q). Furthermore, to make the structure clearer it is also useful to stow away the orbitaldependence (here o 1,o3for one bilinear and o2,o4for the other bilinear) in combined indices L=(l,o 1,o3) andL/prime= (l/prime,o2,o4). Then the expansion reads (dropping the ˜s again) D/Lambda1 o1o2o3o4(k,k/prime;q)=/summationdisplay L,L/primeD/Lambda1(L,L/prime;q)fL(k,o 1,o3) ×f∗ L/prime(k/prime,o2,o4), (36) 155132-7CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) where the form factors depend on the l(/prime)content of L(/prime)times indicator functions for speciﬁc orbital combinations, fL=(l,o,˜o)(k,o 1,o3)=fl(k)δo˜o,o 1o3. (37) With these, we can also deﬁne general channel projections of any interaction function V/Lambda1 o1o2o3o4(k,k/prime,k+q)a s ˆDL,L/prime[V/Lambda1](q)=T2 N2/summationdisplay k,k/prime,o,˜o,o/prime,˜o/primeV/Lambda1 o,o/prime,˜o,˜o/prime(k,k/prime,k+q) ×f∗ L(k,o, ˜o)fL/prime(k/prime,o/prime,˜o/prime). (38) In the same vein, we deﬁne projected particle-particle and particle-hole loop matrices ˆL/Lambda1 PP,L,L/prime(q)=T N/summationdisplay k,o,˜o,o/prime,˜o/primef∗ L(k,o, ˜o)Go,o/prime(k)G˜o,˜o/prime ×(−k+q)fL(k,o/prime,˜o/prime) (39) and ˆL/Lambda1 PH,L,L/prime(q)=T N/summationdisplay k,o,˜o,o/prime,˜o/primef∗ L(k,o, ˜o)Go,o/prime(k)G˜o,˜o/prime ×(k+q)fL(k,o/prime,˜o/prime). (40) D. Truncated-unity fRG equations In the so-called truncated-unity fRG (TUfRG) [ 12], one uses the above-described channel decomposition. Further-more, one employs the form factor expansion of the threechannels. Resolutions of unity in momentum space can beexpressed in the form factor basis and inserted into the ﬂowequation for the interaction. If one keeps only a ﬁnite numberof form factors, these unities are truncated, giving rise tothe name TUfRG. This procedure can be understood as arederivation of the singular-mode fRG equations proposed byWang et al. [11,30]. These approaches share numerical bene- ﬁts compared to other channel-decomposed schemes [ 8–10], as by inserting the unities in loop diagrams, the momentumdependence of the coupling functions is decoupled from the loop-momentum integrations. The TUfRG or singular-mode fRG formalism provides three sets of one-loop ﬂow equations, one for D /Lambda1(L,L/prime;q)= ˆD/Lambda1 L,L/prime(q), one for C/Lambda1(L,L/prime;q)=ˆC/Lambda1 L,L/prime(q), and one for P/Lambda1(L,L/prime;q)=ˆP/Lambda1 L,L/prime(q). On the right-hand sides one always has full vertices as the sum over the three channels plus theinitial interactions. By inserting truncated resolutions of unity in the form factors of the type δ k,k/prime=/summationdisplay LfL(k)f∗ L(k/prime), (41) one can cast these equations in the form of TUfRG matrix equations [ 12], d d/Lambda1ˆP/Lambda1(q)=ˆP[V/Lambda1](q)˙ˆL/Lambda1 PP(q)ˆP[V/Lambda1](q), (42) d d/Lambda1ˆC/Lambda1(q)=ˆC[V/Lambda1](q)˙ˆL/Lambda1 PH(q)ˆC[V/Lambda1](q), (43) and d d/Lambda1ˆD/Lambda1(q)=− 2ˆD[V/Lambda1](q)ˆL/Lambda1 PH(q)˙ˆD[V/Lambda1](q)+ˆC[V/Lambda1](q) ×˙ˆL/Lambda1 PH(q)ˆD[V/Lambda1](q)+ˆD[V/Lambda1](q)˙ˆL/Lambda1 PH(q) ×ˆC[V/Lambda1](q). (44) These equations can be integrated numerically. In a previous work [ 12] we have shown how the projections of V/Lambda1can be done on the real lattice in an efﬁcient way and that theTUfRG scheme exhibits a good scalability on parallel com-puters. Nevertheless, for what follows below these tweaks arenot really needed, as the integration over high-energy bandsrequires much less momentum resolution. Alternatively, it may appear advantageous to keep the form factors as simple properties of the Bravais lattice and notto lump together form-factor and orbital indices. Then theTUfRG equation still carry the orbital indices explicitly, withmatrices in the form-factor indices only, d d/Lambda1ˆP/Lambda1 o1o2o3o4(q)=/summationdisplay o5,o6,o7,o8ˆP/bracketleftbig V/Lambda1 o1o2o5o6/bracketrightbig (q)˙ˆL/Lambda1 PP,o5o6o7o8(q)ˆP/bracketleftbig V/Lambda1 o7o8o3o4/bracketrightbig (q), (45) d d/Lambda1ˆC/Lambda1 o1o2o3o4(q)=/summationdisplay o5,o6,o7,o8ˆC/bracketleftbig V/Lambda1 o1o8o5o4/bracketrightbig (q)˙ˆL/Lambda1 PH,o5o6o7o8(q)ˆC/bracketleftbig V/Lambda1 o6o2o3o7/bracketrightbig (q), (46) and d d/Lambda1ˆD/Lambda1 o1o2o3o4(q)=/summationdisplay o5,o6,o7,o8/braceleftbig −2ˆD/bracketleftbig V/Lambda1 o1o8o3o5/bracketrightbig (q)˙ˆL/Lambda1 PH,o5o6o7o8(q)ˆD/bracketleftbig V/Lambda1 o2o6o4o7/bracketrightbig (q) +ˆC/bracketleftbig V/Lambda1 o1o8o5o3/bracketrightbig (q)˙ˆL/Lambda1 PH,o5o6o7o8(q)ˆD/bracketleftbig V/Lambda1 o2o6o4o7/bracketrightbig (q)+ˆD/bracketleftbig V/Lambda1 o1o8o3o5/bracketrightbig (q)˙ˆL/Lambda1 PH,o5o6o7o8(q)ˆC/bracketleftbig V/Lambda1 o2o6o7o4/bracketrightbig (q)/bracerightbig ,(47) with ˆL/Lambda1 PP,o5o6o7o8(q)=T N/summationdisplay kG/Lambda1 o5o7(k)G/Lambda1 o6o8(−k+q) (48) and ˆL/Lambda1 PH,o5o6o7o8(q)=T N/summationdisplay kG/Lambda1 o5o6(k)G/Lambda1 o7o8(k+q). (49) 155132-8EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) V . APPROXIMATION STRATEGIES The fRG equations ( 42), (43), and ( 44)o r( 45), (46), and ( 47) can in principle be solved in general for the full matrix structure with three channel couplings ˆP/Lambda1,ˆC/Lambda1, and ˆD/Lambda1each depending on four orbital indices and a number of form factors. In the one-orbital case this has been done,e.g., in Ref. [ 12], for the one-band Hubbard model with a larger number of form factors. In particular, the convergenceof the results with respect to the length of the form-factorexpansion could be demonstrated. However, for a many-bandproblem the overall effort grows considerably, as the numberof couplings grows with the fourth power of the number ofWannier orbitals considered. Thus, additional approximationsappear useful. We start with an approximation that reducesthe complexity of the frequency and momentum dependence.After that we simplify the orbital structure. A. Instantaneous bilinears and static channel coupling The TUfRG ﬂow equations above were derived for simul- taneous form-factor expansions in wave vector and frequencydependence. We already deﬁned the instantaneous on-siteapproximation in Sec. IV C , where only the zero-bond length and equal-time bilinears are kept, equivalent to constant formfactors in wave vector and frequency. In the TUfRG equationsabove, ( 45), (46), and ( 47), this would require us to project the channel couplings on the instantaneous contribution, bya “ﬂat” (i.e., with constant form factor) summation over all frequencies. This creates the difﬁculty of treating the high-frequency behavior with sufﬁcient precision, which makes theapproach at least more complex and potentially numericallychallenging. Hence, for this work, we will not use frequency-projected couplings and loop diagrams and only used the TUfRGscheme for wave vector dependence of the right-hand sideof the ﬂow equations. We will still restrain the interactionto be between instantaneous bilinears, but use a differentapproach to deal with the frequency summation on the right-hand side of Eqs. ( 45), (46), and ( 47). This approach was laid out previously in the context of one-dimensional [ 19,20] and two-dimensional one-band models [ 21]. There it was argued that one-frequency parametrizations with only the collectiveMandelstam frequency kept [ 18] provide a good approxima- tion. This is indeed equivalent to the instantaneous bilinearapproximation, but there are no frequency-form-factor projec-tions used in the ﬂow equations. To clarify the approximation,we restrict the form-factor expansion and projection to thewave vector content, leading to quantities with ˆ hats like ˆP /Lambda1(k0,k/prime 0;q), and keep the dependence on two fermionic (in front of the semicolon) and one collective bosonic Matsubarafrequency [behind the semicolon in q=(q 0,/vectorq)] explicit. Then the ﬂow equation for the pair-channel coupling wouldread [suppressing the orbital indices for the moment, these aret h es a m ea si nE q .( 45)] d d/Lambda1ˆP/Lambda1(k0,k/prime 0;q)=T/summationdisplay k/prime/prime 0ˆP/Lambda1[P/Lambda1(k,k/prime/prime;q)+C/Lambda1(k,−k/prime/prime;k+k/prime/prime−q)+D/Lambda1(k,−k/prime/prime+q;k/prime/prime−k)] ×˙ˆL/Lambda1 PP(k/prime/prime 0,q)ˆP/Lambda1[P/Lambda1(k/prime/prime,k/prime;q)+C/Lambda1(k/prime/prime,−k/prime;k/prime+k/prime/prime−q)+D/Lambda1(k/prime/prime,−k/prime+q;k/prime−k/prime/prime)]. (50) Here the loop cannot be summed separately over the Matsub- ara frequency k/prime/prime 0as the couplings also depend on k/prime/prime 0.W eh a v e more precisely, with k=(k0,/vectork), ˆL/Lambda1 PP(k0,q)=1 N/summationdisplay /vectorkG/Lambda1(k)G/Lambda1(−k+q). (51) We notice that the summation frequency k/prime/prime 0appears in the third, collective-frequency argument of the pair chan-nel “non-native” channel couplings. Hence, in the sumoverk /prime/prime 0we might have to sum over a sharp peak, if the collective-frequency dependence of any of these couplingsbecomes sharp. This makes the implementation more costly.In Refs. [ 19–21] it is argued that at least for the cases studied it makes sense to pull the couplings out of the k 0sum by setting the collective frequency in the non-native couplings toa certain frequency ¯q. This is chosen to be 0 in Refs. [ 19,20] and 0 or the RG scale in Ref. [ 21]. Here we also use this approximation and we write d d/Lambda1ˆP/Lambda1(q)=ˆP/Lambda1[P/Lambda1(q)+C/Lambda1(¯q)+D/Lambda1(¯q)]T/summationdisplay k/prime/prime 0˙ˆL/Lambda1 PP(k/prime/prime o,q) ˆP/Lambda1[P/Lambda1(q)+C/Lambda1(¯q)+D/Lambda1(¯q)]. (52)In the applications below we choose ¯q=0. This can be called the static channel-coupling approximation. In recentliterature [ 19,20], this choice is also named “coupled-ladder approximation.” For the orbital content, we use the schemedescribed in Eqs. ( 45), (46), and ( 47). B. On-site approximation As next approximation, we truncate the form-factor expan- sion of Eq. ( 33) quite early and only keep the zeroth-order unit-cell-local form factor, which is constant in the Brillouinzone and frequency independent, along the discussion afterEq. ( 29). Then the projections of Eq. ( 34) becoming simply Brillouin zone averages over the momentum D 0,0/bracketleftbig V/Lambda1 o1o2o3o4(k,k/prime;q)/bracketrightbig =1 N2/summationdisplay /vectork,/vectork/primeV/Lambda1 o1o2o3o4(/vectork,/vectork/prime;q).(53) The question is of course whether this quite drastic simpliﬁca- tion is justiﬁable. We argue that the bare interactions describedin Sec. II B reﬂect this unit-cell local or on-site structure, as in the two-center approximation for the Coulomb matrixelements it only consists of interacting on-site charge, spin,and pair bilinears. Most importantly, the standard Kanamoriparametrization in terms of local interaction parameters U 155132-9CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) andU/primefor the intra- and interorbital on-site repulsions and Jfor the Hund’s coupling can also be considered as falling into this class. It is clear that the on-site approximation misses some important physics that could come in the form of interactionsbetween nonlocal bilinears. For instance, d-wave pairing on the square lattice and most other unconventional Cooperpairing examples involve pair bilinears on neighbored sitesand in neighboring unit cells. One might however suspectthat this physics, unless it is already present in the bareinteraction, will only become relevant in the solution of thelow-energy effective model. This is not aimed for here, aswe focus on integrating out the bands away from the Fermilevel. Furthermore, in systems where the orbitals are centerednot at the same but at various places in the unit cell, therestriction of bilinears formed between orbitals only insidethe same unit cell may become questionable. As an example,consider a chain of alternating A and B orbitals centerednext to each other along the chain. Each unit cell containsone A orbital and one B orbital. Then the nearest-neighbordensity-density interaction will have two parts, one where on-site density bilinears with operators a † iaicouple to b† ibi in the same unit cell and one where they couple to b† i−1bi−1 in a neighboring unit cell. In the Dchannel, this nonlocality in the interaction between on-site bilinears is no problem andwell captured by the /vectorqdependence of the channel coupling. However, the projection with unit-cell-local form factor onlyon other channels, CorP, will miss the one half of this interaction that extends out of the unit cell. In such cases, thetruncation has to be adapted to take into account all physicallyequivalent interactions. We note that we could introduce the on-site approxima- tion also directly after writing the channel decomposition inEqs. ( 22) and ( 23)–(25). We prefer to place it in the more general context of the form-factor expansion as this showsthat systematic improvements are possible. In particular, itshould be possible to check numerically if important physicsis missed when the next order in the form-factor expansionis taken into account. For one-band and two-band models,this has been done quite far and convergence in the basistruncation could be reached also at rather low-energy scaleswith very high momentum resolution [ 12–14]. Hence we expect that similar checks can be done at least in speciﬁc casesalso in the multiorbital cfRG case. This in general requiresless momentum resolution due to the higher energy scalesconsidered. C. Intraorbital bilinear approximation With the approximations of the last two Secs. VA and VB, the ﬂowing multiorbital interactions arising from the bare interactions discussed in Sec. II B all involve on-site charge, spin, or pair bilinears formed by two fermionic operators atthe same lattice site, albeit with a possibly different orbitalindex. At the initial scale /Lambda1 0we can write D/Lambda10 o1o2o3o4(l,l/prime;q)=Uδl,0δl/prime,0δo1,oδo3,oδo2,oδo4,o+U/primeδl,0δl/prime,0δo1,oδo3,oδo2,o/primeδo4,o/prime(1−δoo/prime) +Vρoo/prime,/Lambda10(q)δl,0δl/prime,0δo1,oδo3,oδo2,o/primeδo4,o/prime=D/Lambda10 oo/prime(q), (54) [1mm]C/Lambda10 o1o2o3o4(l,l/prime;q)=Jδl,0δl/prime,0δo1,oδo4,oδo2,o/primeδo3,o/prime(1−δoo/prime)=C/Lambda10 oo/prime(q), (55) [1mm]P/Lambda1 o1o2o3o4(l,l/prime;q)=Jδl,0δl/prime,0δo1,oδo2,oδo3,o/primeδo4,o/prime(1−δoo/prime)=P/Lambda10 oo/prime(q). (56) This distribution of the initial interaction is not unique, and below we will discuss and use an alternative choice. Bycomparing with Eqs. ( 12), (13), and ( 14), we ﬁnd that the initial interaction is made from intraorbital bilinears, whereboth constituent fermions carry the same orbital index. Now it appears to be a meaningful approximation to keep only interactions with orbital combinations corresponding tothe terms appearing in the bare interaction, i.e., that involve in-traorbital bilinears that interact. We call this the IOBI (intraor-bital bilinear) approximation. The corresponding couplingsare shown diagrammatically in Figs. 3(b) to 3(d). We will see below numerically that in many cases in the integration overhigh-energy bands, the ﬂow of additional interorbital bilin-ears, which can also be kept in the more general truncation ofSec. VD, will be less important. These restrictions in the IOBI approximation amount to writing D /Lambda1 o1o2o3o4(l,l/prime;q)=D/Lambda1 oo/prime(q)δl,0δl/prime,0δo1,oδo3,oδo2,o/primeδo4,o/prime(57) in the direct and C/Lambda1 o1o2o3o4(l,l/prime;q)=C/Lambda1 oo/prime(q)δl,0δl/prime,0δo1,oδo4,oδo2,o/primeδo3o/prime(58)in the crossed channel. In the pair channel we only consider intraorbital on-site pairs (effectively excluding spin-triplet orany nonlocal pairs), P /Lambda1 o1o2o3o4(l,l/prime;q)=P/Lambda1 oo/prime(q)δl,0δl/prime,0δo1,oδo2,oδo3,o/primeδo4o/prime.(59) This way, the orbital and wave vector structure of the D/Lambda1 oo/prime(q) interaction ﬁts naturally on the particle-hole bubble RPAdiagram on the right-hand side of the TUfRG equations. Theinternal bubble contains a summation over the orbital index.Similarly, the Hund’s rule spin interactions C /Lambda1 oo/prime(q) match the crossed particle-hole channel on the right-hand side of ∂/Lambda1C/Lambda1, and a ladder summation of these diagrams with C/Lambda1 oo/prime(q) alone would maintain its orbital structure. Finally, the pair hoppinginteraction P /Lambda1 oo/prime(q) is kept naturally in the particle-particle channel. In this IOBI ﬂow, no new bilinear interactions terms are generated. It is the distance and frequency dependencebetween the on-site charge, spin, and pair bilinears that getaltered. After some RG steps, e.g., when the screening or nontarget bands have been integrated out, it makes sense to study the 155132-10EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) FIG. 4. The nine coupling functions kept in the two-orbital approximation of Sec. VD. The functions (i), (iv), and (vii) are the ones already considered in the IOBI approximation of Sec. VC. resulting effective interaction. In the IOBI approximation this can be obtained as V/Lambda1 o1o2o3,o4(k1,k2,k3)=P/Lambda1 o1o3(k1+k2)+D/Lambda1 o1o2(k3−k1) +C/Lambda1 o1o2(k3−k2). (60) As laid out in the previous papers on the cfRG [ 5,6], the cRPA would be obtained from the cfRG by only takinginto account the RPA diagram in the Dchannel (see Fig. 2 for the RPA diagram and the other diagrams). In the IOBIapproximation to the cRPA this would mean that only thecharge interactions originating from the nonlocal Coulombinteraction and in the local intra- and interorbital density-density terms UandU /primeare renormalized. The Hund’s rulespin and pair hopping interactions do not ﬂow, as those terms are kept in the CandPchannel, and these channels are not included in the cRPA ﬂow. D. Two-orbital approximation An alternative truncation of the orbital content, still with only local form factors, which remedies the shortcoming ofthe cRPA ﬂow to include the Hund’s rule terms but alsomakes the study of two-orbital interactions more complete,is to consider the enlarged set of nine effective interaction functions D /Lambda1 oo/prime(q),Dx/y,/Lambda1 oo/prime(q),C/Lambda1 oo/prime(q),Cx/y,/Lambda1 oo/prime(q),P/Lambda1 oo/prime(q), and Px/y,/Lambda1 oo/prime(q), or more precisely, D/Lambda1 o1o2o3o4(l,l/prime;q)=δl,0δl/prime,0/bracketleftbig D/Lambda1 oo/prime(q)δo1,oδo3,oδo2,o/primeδo4,o/prime+Dx,/Lambda1 oo/prime(q)δo1,oδo4,oδo2,o/primeδo3,o/prime+Dy,/Lambda1 oo/prime(q)δo1,oδo2,oδo3,o/primeδo4,o/prime/bracketrightbig ,(61) C/Lambda1 o1o2o3o4(l,l/prime;q)=δl,0δl/prime,0/bracketleftbig C/Lambda1 oo/prime(q)δo1,oδo4,oδo2,o/primeδo3o/prime+Cx,/Lambda1 oo/prime(q)δo1,oδo3,oδo2,o/primeδo4o/prime+Cy,/Lambda1 oo/prime(q)δo1,oδo2,oδo3,o/primeδo4o/prime/bracketrightbig ,(62) P/Lambda1 o1o2o3o4(l,l/prime;q)=δl,0δl/prime,0/bracketleftbig P/Lambda1 oo/prime(q)δo1,oδo2,oδo3,o/primeδo4o/prime+Px,/Lambda1 oo/prime(q)δo1,oδo3,oδo2,o/primeδo4o/prime+Py,/Lambda1 oo/prime(q)δo1,oδo4,oδo2,o/primeδo3o/prime/bracketrightbig .. (63) As all these interactions still only depend on two orbital indices, we can call this the two-orbital approximation .T h e coupling functions are shown graphically in Fig. 4.N o w ,f o r a speciﬁc channel D,C,o rPwith momentum/frequency transfer qwe allow for all three possible orbital combinations o1o2o3o4with two pairs of identical orbital indices. Theconstituents of the bilinears that exchange this qare not necessarily in the same orbital, but in the approximation usedhere with just the constant form factor, they are always in thesame unit cell. The larger set of couplings gives more freedom to allow the Hund’s rule terms to ﬂow even if only the D-channel 155132-11CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) ﬂow is considered, i.e., if one restrains the study to cRPA. Namely, using the two-orbital approximation, we can nowassign all terms of the bare interaction to the Dcouplings: The density-density interactions go as before into D /Lambda1 oo/prime(q), while the momentum-independent and nonretarded Hund’s rule spinterms are put into D x,/Lambda1 oo/prime(q) and the Hund’s rule pair hopping terms are assigned to Dy,/Lambda1 oo/prime(q). The initial condition for the C andPcouplings is then zero, but those couplings are generally generated during the ﬂow. We have checked numericallythat the assignment of the Hund’s rule interactions to the C channel gives equivalent results. Below we will see that in the cases studied numerically, the differences between the intraorbital approximation and themore general two-orbital approximation are in may cases onthe quantitative level. VI. EFFECTIVE TARGET-BAND INTERACTIONS We can now employ either cRPA or the IOBI-fRG in order to integrate out the nontarget bands. This yields an effectiveinteraction V /Lambda1 o1o2o3,o4(k1,k2,k3) that can be projected onto thetarget band b=tby V/Lambda1 tttt(k1,k2,k3)=/summationdisplay o1,o2,o3,o4uo1t(/vectork1)uo2t(/vectork2)u∗ o3t(/vectork3)u∗ o4t(/vectork4) ×V/Lambda1 o1o2o3,o4(k1,k2,k3). (64) We can now deﬁne wave-vector-resolved static charge and spin couplings as those couplings that appear in the bubblesummation for the RPA charge susceptibility or in the crossedladder summation for the RPA spin susceptibility. For thecoupling to nonretarded charge source ﬁelds, the appropriatevertex summed in the RPA loop diagrams over the frequencyarguments of the ﬁrst incoming ( k 1) and second outgoing line (k4) with a ﬁxed transfer frequency ωbetween the ﬁrst incoming ( k1) and ﬁrst outgoing line ( k3). In the spin channel, the frequency transfer is between the second incoming ( k2) and and ﬁrst outgoing line ( k3), and the summation is over k1 andk3. For the vertex in the target band, we use the channel couplings obtained by the cfRG ﬂow. More precisely, weuse for the charge coupling in the IOBI approximation withmomentum transfer /vectorqand frequency transfer ω, V/Lambda1 tttt,ch(/vectork1,/vectork2,/vectork3=/vectork1+/vectorq;ω)=/summationdisplay o,o/prime[uo,t(/vectork1)uo/prime,t(/vectork2)u∗ o,t(/vectork1+/vectorq)u∗ o/prime,t(/vectork2−/vectorq)D/Lambda1 oo/prime(/vectorq,ω) +uo,t(/vectork1)uo/prime,t(/vectork2)u∗ o/prime,t(/vectork1+/vectorq)u∗ o,t(/vectork2−/vectorq)C/Lambda1 oo/prime(/vectork2−/vectorq−/vectork1;˜ω) +uo,t(/vectork1)uo,t(/vectork2)u∗ o/prime,t(/vectork1+/vectorq)u∗ o/prime,t(/vectork2−/vectorq)P/Lambda1 oo/prime(/vectork1+/vectork2;˜ω)], (65) and for the spin coupling V/Lambda1 tttt,sp(/vectork1,/vectork2,/vectork3=/vectork2+/vectorq;ω)=/summationdisplay o,o/prime[uo,t(/vectork1)uo/prime,t(/vectork2+/vectorq)u∗ o,t(/vectork2+/vectorq)u∗ o/prime,t(/vectork1−/vectorq)D/Lambda1 oo/prime(/vectork2+/vectorq+/vectork1,˜ω) +uo,t(/vectork1)uo/prime,t(/vectork2)u∗ o/prime,t(/vectork2+/vectorq)u∗ o,t(/vectork1−/vectorq)C/Lambda1 oo/prime(/vectorq,ω) +uo,t(/vectork1)uo,t(/vectork2)u∗ o/prime,t(/vectork2+/vectorq)u∗ o/prime,t(/vectork1−/vectorq)P/Lambda1 oo/prime(/vectork1+/vectork2,˜ω)]. (66) Here the subscript ttttindicates that all legs of the vertex are in the target band. The main contribution to the frequencydependence of the charge channel comes from the “channel-native” coupling D /Lambda1 oo/prime(/vectorq,˜ω) and in the spin channel from the there-native coupling C/Lambda1 oo/prime(/vectorq,˜ω). For the contributions by the non-native couplings, i.e., C/Lambda1 oo/prime(/vectorq,˜ω) and P/Lambda1 oo/prime(/vectorq,˜ω)i nt h e case of the charge coupling and D/Lambda1 oo/prime(/vectorq,˜ω) andP/Lambda1 oo/prime(/vectorq,˜ω)i n the case of the spin coupling, the variables ˜ ωare placeholders for combinations of external leg frequencies and the transferfrequency. This means that the loop sums average over thefrequency argument ˜ ω. The question is then which frequency ˜ωwe should choose here as representative (without having to do the full target band RPA). We could set ˜ ω=0a g a i n . This would be consistent with the static-channel coupling ap-proximation used for the ﬂow of the channels (see Sec. VA). However, as also argued in Ref. [ 21], the one-frequency parametrizations without form-factor expansion are designedto get the leading ﬂow at low frequencies right and cannotbe expected to give good results for higher frequencies ofthe external fermionic legs of the interaction. This can also be observed in the charge and spin couplings deﬁned above.Here, using the static channel coupling approximation, thehigh-ωvalues of the effective interactions would not go back to the bare values, if the zero-frequency values for the non-native couplings are used. This does not make sense, as allrenormalizations are due to loop corrections which vanish forhigh frequencies. Similarly, using frequency averages leadsto the same problem in the high-frequency behavior. A moresuitable approximation is then to set ˜ ω=ω, which is done for (65) and ( 66) in the data discussed below. This guarantees the correct disappearance of perturbative corrections in thehigh-ωlimit and merges with the static-channel coupling approximation in the case of small ω. For the more general two-orbital approximation, the ex- pressions ( 65) and ( 66) are generalized accordingly. Next, in order to obtain expression that only depend on the trans-fer wave vector and frequency, we average over the twoincoming wave vectors, which can again be understood as 155132-12EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) (a) (b) (c) (d) FIG. 5. Comparison of the effective target-band interactions obtained with cRPA and cfRG with the bare target-band interaction, for on-site bare interactions U=6,U/prime=0,J=0, and Vc=0. The upper plots show the real space dependence of the effective interactions along the xdirection on the real lattice, for the charge channel in the left plot and for the spin channel in the right plot. Note that although the bare interaction in the three-orbital model is purely local, the orbital-to-band transformation makes the bare interaction within the central band slightly nonlocal already. The lower plots show the static screening functions in the target band in the charge (left) and spin (right) channel asfunction of the wave vector along the Brillouin zone diagonal. The plots (a) and (b) in the upper half are for the two-orbital approximation of Sec. VD. In the lower plots (c) and (d) we distinguish between the IOBI approximation, called cfRG(3), with just three intraorbital bilinears allowed and the more general two-orbital cfRG with a total of nine interaction channels. For the charge channel the curves lie on top of each other in this case. on-site-bilinear approximation, This leads to V/Lambda1 t,ch(/vectorq,ω)=1 N2/summationdisplay /vectork1,/vectork2V/Lambda1 tttt,ch(/vectork1,/vectork2,/vectork3=/vectork1+/vectorq;ω) (67) and V/Lambda1 t,sp(/vectorq,ω)=1 N2/summationdisplay /vectork1,/vectork2V/Lambda1 tttt,sp(/vectork1,/vectork2,/vectork3=/vectork2+/vectorq;ω).(68) These quantities can be compared between the cRPA and cfRG calculations for V/Lambda1 o1o2o3,o4(k1,k2,k3). For this compari- son it is further useful to analyze the static dielectric chargescreening function /epsilon1 t,ch(/vectorq,ω)=V/Lambda10 t,ch(/vectorq,ω) V/Lambda1 t,ch(/vectorq,ω), (69) which is the ratio between the bare charge interaction in the target band at RG scale /Lambda10and effective charge interaction inthe target band after the screening bands have been integrated out at RG scale /Lambda1. Similarly we deﬁne the spin screening function /epsilon1t,sp(/vectorq,ω)=V/Lambda10 t,sp(/vectorq,ω) V/Lambda1 t,sp(/vectorq,ω). (70) VII. RESULTS IN THREE-BAND MODEL Here we show results for the effective interactions in charge and spin channel as well as for the screening functions.We use the three-orbital model described in Sec. II B and use either cRPA or cfRG to integrate out the two bandsbelow and above the Fermi level. Most data were obtainedusing an 18 ×18 wave vector grid in the Brillouin zone and 18 positive bosonic Matsubara frequencies. These numberscan be increased signiﬁcantly if needed, as the ﬂow part ofthe code scales linearly in the numbers of wave vectors andfrequencies. 155132-13CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) (a) (b) (c) (d) FIG. 6. Same as in Fig. 5, but now for on-site Kanamori interactions U=4.8,U/prime=3.6,J=0.6, and Vc=0. First, let us study the situation for the purely on-site initial interactions using the two approximations with three or ninecoupling functions as in Secs. VC andVD. In Fig. 5we show data for pure intraorbital on-site inter- actions, with U=6,U /prime=J=Vc=0. In the upper left plot we show the spatial dependence of the charge interaction forthe bare case, the cRPA, and the cfRG. The cfRG values in theupper plots are the ones from the two-orbital approximation ofSec. VD. First, we notice that the bare interaction in the target band/lessorsimilar2 is substantially smaller than the value U=6i nt h e orbital picture. This is a common hybridization effect. It alsomeans that our perturbative approach should be qualitativelyreliable, as the target band has a larger width. Furthermore,all perturbative corrections to the effective interactions shouldbe bounded because of the energy gap between target andhigh-energy bands that is present in loop diagrams. Besides this overall reduction, we notice that even the bare interaction has acquired nonlocal contributions. This occursbecause the orbital-to-band transformation is momentum de-pendent. Considering the loop corrections to the target-bandinteractions, the on-site term is seen to be reduced signif-icantly by the cRPA compared to the bare value, but onlyslightly by the cfRG. In contrast with this, the cRPA producesan enhanced nearest-neighbor repulsion, while the cfRG low-ers the nearest-neighbor repulsion compared to the bare inter-action. Similar differences are observed for the effective spininteraction. Notably, the spin nearest-neighbor term in cfRGis repulsive, favoring antiferromagnetic alignment, while it is negative and smaller in magnitude in the bare and cRPAinteractions. This generation of nonlocal AF couplings wasalready observed in Ref. [ 6]. It is possibly understood from the momentum structure of interband particle-hole bubbles,somewhat as a weaker reminiscence of how RKKY interac-tions come about in a metal. Another speciﬁc choice of interaction parameters U=4.8, U /prime=3.6, with J=0.6 is shown in Fig. 6. We see that now the on-site charge interaction in the cRPA is only slightlymore reduced compared to the bare one than it was foundin cfRG. For the nearest-neighbor charge interaction thesituation is again reversed, here the cRPA value is higher thanthe bare one and the cfRG one. Also for the spin interactions,the on-site terms and nearest-neighbor terms are renormalizeddifferently. The nearest-neighbor spin term in cfRG is againpositive. In the lower panels of Figs. 5and6we plot the target-band screening functions in charge and spin channels. Now wealso compare the data for the IOBI approximation with threecoupling functions with that of the two-orbital approximation.For the pure intraorbital on-site repulsion in Fig. 5the two- orbital approximation (thick lines) does not change the resultsof the IOBI (thin lines, if visible), while for the case in Fig. 6, quantitative differences can be observed. The different spatialvariation of the effective interactions in cRPA and both cfRGversions leads to quite different momentum dependencies of 155132-14EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) (a) (b) (c) (d) FIG. 7. Comparison of the effective target-band interactions obtained with cRPA and cfRG with the bare target-band interaction, for nonlocal bare interactions with a screening length of six lattice constants, U=6,U/prime=4.8,J=0.6, and Vc=2. The upper plots show the real space dependence of the effective interactions along the xdirection, for the charge channel in the left plot and for the spin channel in the right plot. The lower plots show the static screening functions in the charge (left) and spin (right) channel as function of the wave vector along the Brillouin zone diagonal. The plots (a) and (b) in the upper half are for the two-orbital approximation. In the lower plots (c) and (d) distinguish between the IOBI approximation, called cfRG(3), with just three intraorbital bilinears allowed and the more general two-orbitalcfRG with a total of nine interactions channels. the screening functions. Most notably, the charge screening in cRPA is far stronger than in cfRG for half of the Brillouinzone at larger /vectorq, in line with the differences at small distances in the upper panel. Also, the spin screening function in thecfRG gets smaller than 1 for large q x=qy→π, i.e., there we have antiscreening or enhancement of a staggered interactioncomponent that can of course also be found in the distancedependence of the real-space spin interaction in the panelabove. Comparable and partially stronger differences at shortdistances and higher wave vectors are found when a longer-ranged bare interaction is used. In Figs. 7and 8we show the same set of data for interaction parameters that include adensity-density repulsion with decay length of six lattice sites.Now in the second data set (Fig. 8) the target-band on-site charge interaction in cfRG is even larger than the bare one,while the cRPA screens it down. For both cases, the spin in-teraction is positive on the nearest-neighbor site again, leadingto increased AF tendencies. The most striking difference isfound in the dielectric or charge screening function for thecase in Fig. 8where /epsilon1 ch(/vectorq) now becomes smaller than 1 in the cfRG for larger qwhile it rises above 1 for the cRPA. Hencethere is signiﬁcant antiscreening for wave vectors near ( π,π ). Similarly to the parameter sets plotted above but even morestrongly, the spin screening function dives below 1 as well for /vectorq→(π,π )i nF i g . 8. Hence, the cfRG static screening properties can be quite different from those in cRPA, at least at small distances of theorder of one lattice spacing. This picture is complemented bythe frequency-dependent screening for which we show someexemplary data in Fig. 9, obtained with the one-frequency parametrization in the static channel-coupling approximation,as described in Sec. VA. Here the frequency dependence of the charge screening in cRPA and cfRG is quite different forwave vectors near M=(π,π ). The antiscreening of the cfRG persists far over a Matsubara frequency range of the orderof the gap between the bands, which is 4 to 5 in our openenergy units. Near /Gamma1the differences are small for the charge screening. Also in the spin channels, cRPA and cfRG give dif-ferent results, again with some antiscreening near /vectorq=(π,π ). In the lower panels of Fig. 9we also show data obtained the more general two-orbital approximation described in Sec. VD. We notice some smaller differences compared to the IOBI, 155132-15CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) (a) (b) (c) (d) FIG. 8. Same as in Fig 7, but with U=6,U/prime=2,J=0, and Vc=2. which maintain the differences between cRPA and cfRG. All qualitative ﬁndings are the same compared to IOBI. Thusthe two-orbital calculation should be more precise becauseit ignores less interactions, but is presumably not needed formost cases to a reasonable ﬁrst estimate of the screening. VIII. DISCUSSION We conclude this presentation of approximation strategies in the cfRG and comparison data by summarizing that thetwo methods cfRG and cRPA show indeed differences thatin certain cases may be strong enough to lead to qualitativelydifferent predictions when the screened interaction are em-ployed in the solution of the low-energy model. For instance,the ratio between on-site and nearest-neighbor interactions often decides on the type of ground state ordering, or in other cases strongly inﬂuences the energy scale for ordering. Inparticular this ratio was found to differ in cfRG and cRPA:the ratio differed by 20% between cRPA and cFRG for theparameters used in Fig. 8, and for bare intraorbital interaction only as in Fig. 5, the effective nearest-neighbor repulsion in cfRG was only a ﬁfth of that in cRPA. Furthermore, additionalspin AF interactions that are generated in the effective model by the cfRG but not by the cRPA can play a role in cases of competing ordering tendencies. Finally, also the frequencydependencies of the screening functions was found to differqualitatively for parts of the momentum space. We plan toextend our studies in order to include the instability analysisthat can directly show the resulting changes in ground state order and energy scales for these to occur. We note that in our three-band model study all screening effects are somewhat small. They are big enough to make thedifferences between different diagrammatic content of cRPAand cfRG visible. However, our computed screening functionsrange between 0.6 and 1.5. So they do not deviate stronglyfrom unity. This is in contrast with the screening values givenin the literature, we can exceed 2 for instance in graphenesystems [ 32] or reach values ∼4–5 in iron superconductors [33]. We hope that our new, numerically lighter IOBI or two- orbital approaches can be applied in a more realistic settingin order to understand if these numerical differences are justcaused by the use of too small bare interactions in our model,or if other factors increase the screening in real materials.Another aspect that should be understood is if the real spacestaggering tendencies found in the spin channel in the cfRGwith the generation of a nearest-neighbor AF component aresomehow linked to band structure of the chosen model. Thedata shown here were for a target-band ﬁlling of ∼60% which should not be a special point. Varying the band ﬁlling of thetarget band by ±15% did not change the results much. On the methodical side, the IOBI or two-orbital approximations are two differently truncated approximationsto the full orbital dependence of the effective interactionsthat both only keep terms that at most depend on two orbitalindices. The usual bare interactions including Hund’s rule 155132-16EFFICIENT VERTEX PARAMETRIZATION FOR THE … PHYSICAL REVIEW B 98, 155132 (2018) (a) (b) (c) (d) FIG. 9. Comparison between cRPA and cfRG data for the Matsubara frequency dependence of the charge [left, (a) and (c)] and spin [right, (b) and (d)] screening functions with the interaction parameters for on-site Kanamori interactions U=6,U/prime=4.8,J=0.6, and Vc=3a s well as a screening length of six lattice sites at T=0.1, for two different wave vectors in the two-dimensional Brillouin zone. terms can already be represented by the IOBI couplings that employs just three functions to represent the orbital structure,momentum, and frequency structure of the interaction. Formost parameter sets, in particular those with J=0, the IOBI and the more general two-orbital approximation that usesnine functions give quite similar results. However, our datashow that for J/negationslash=0 there are some quantitative corrections to the IOBI in the two-orbital approximation such that, ifthe numerical constraints permit, the two-orbital schememay give more precise results. Treating the full orbitaldependence is numerically at least one order of magnitudemore complex. It may however be useful to try to performsome checks, e.g., on the model studied here, to make surethat no other relevant interaction terms depending on morethan two orbital indices get generated. The next aspect thatshould be included and that would add additional precisionwill be self-energy corrections and a better treatment of thefrequency dependence. The latter will possibly involve formfactors also for the frequency dependence. Considering the numerical effort, the IOBI or two-orbital approximations with one-frequency parametrization of thefrequency dependence should be on the same scale (or maybethree or nine times as high) as cRPA or also GW as faras complexity is concerned. One is dealing with effectiveinteractions that depend on one frequency and one momentum and that have a matrix structure in the orbital indices. There-fore, we hope that additional improvements like the inclusionof self-energy corrections will be feasible. Most importantly,the extension to models with many bands, as may be requiredto make useful contributions in ﬁrst-principles theory, shouldbecome realistic. Turned around, our fRG approach can alsobe viewed as a coupled-channel generalization of cRPA orGW that adds magnetic and pairing ﬂuctuations to the chargescreening physics. In the cfRG, the effective interactions are obtained by solv- ing the RG differential equations and not by an inversion of amatrix in orbital space as in the cRPA. As the structures thatdevelop in the ﬂow are not extremely sharp, the integration ofthese coupled differential equations does not pose particularproblems. Note that recently we also reformulated the self-consistent parquet approximation using the language of achannel-decomposed interaction with form-factor expansions[34]. The solution of the parquet equations is to a great deal equivalent (or even more complete) than solving the fRGﬂow equations. By this bridge, we could also reformulate thecfRG as self-consistent constrained parquet approximation.This would then lead to coupled self-consistency equationsthat extend the Bethe-Salpeter-like single-channel equations 155132-17CARSTEN HONERKAMP PHYSICAL REVIEW B 98, 155132 (2018) of the cRPA. Currently we believe that solving the coupled fRG equations is simpler than solving these coupled self-consistency equations, but both approaches may ﬁnd their use. ACKNOWLEDGMENTS We acknowledge discussions with S. Andergassen, S. Biermann, J. Ehrlich, C. Hille, J. Hofmann, M. Kinza,L. Markhof, T. Reckling, A. Tagliavini, and T. Wehling. The German Research Foundation supported this project,in particular via HO2422/10-1 and HO2422/11-1 and theDFG Research Training Group 1995 “Quantum Many-BodyMethods in Condensed Matter Systems.” Furthermore, Iam grateful for hospitality by and discussions with theAustrian SFB ViCom, “Vienna Computational MaterialsLaboratory.” [1] F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein, Phys. Rev. B 70,195104 (2004 ). [2] T. Miyake, F. Aryasetiawan, and M. Imada, Phys. Rev. B 80, 155134 (2009 ). [3] M. Imada and T. Miyake, J. Phys. Soc. Jpn. 79,112001 (2010 ). [4] E. ¸ Sa¸sıo˘glu, C. Friedrich, and S. Blügel, Phys. Rev. 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PhysRevB.86.205429.pdf	PHYSICAL REVIEW B 86, 205429 (2012) Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity Wei Yan, Martijn Wubs, and N. Asger Mortensen* DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark (Received 25 April 2012; revised manuscript received 9 October 2012; published 27 November 2012) We study metamaterials known as hyperbolic media that in the usual local-response approximation exhibit hyperbolic dispersion and an associated broadband singularity in the density of states. Instead, from themore microscopic hydrodynamic Drude theory we derive qualitatively different optical properties of thesemetamaterials, due to the free-electron nonlocal optical response of their metal constituents. We demonstratethat nonlocal response gives rise to a large-wavevector cutoff in the dispersion that is inversely proportional tothe Fermi velocity of the electron gas, but also for small wavevectors we ﬁnd differences for the hyperbolicdispersion. Moreover, the size of the unit cell inﬂuences effective parameters of the metamaterial even in thedeep subwavelength regime. Finally, instead of the broadband supersingularity in the local density of states, wepredict a large but ﬁnite maximal enhancement proportional to the inverse cube of the Fermi velocity. DOI: 10.1103/PhysRevB.86.205429 PACS number(s): 42 .70.Qs, 73 .20.Mf, 78 .67.Pt, 81.05.Xj I. INTRODUCTION Metamaterials, consisting of subwavelength artiﬁcial unit cells, show a great potential in optical applications, such as perfect lenses1and invisibility cloaks.2,3Hyperbolic metama- terials (HMM) are of special interest because of their unusualhyperbolic dispersion curves 4–14that support radiative modes with unbounded wavenumbers. Because of the diverging ra-diative local density of states (LDOS), a point emitter in such amedium would exhibit instantaneous radiative decay. 6–12This broadband “supersingularity”6,9is indeed broadband, since the hyperbolic dispersion does not rely on speciﬁc resonances.The prediction by Jacob et al. 6that hyperbolic media thereby form a new route to enhanced light-matter coupling recentlyfound experimental support. 9,10These measured lifetimes were nonzero, and state-of-the-art theories explain this fromthree parameters: the nonvanishing damping γ, the size aof the unit cell, and the size Dof the emitter. 6–8In particular, as a function of these parameters the radiative LDOS scales asγ −3/2(Ref. 6),a−3(Ref. 7), and D−3(Ref. 8), respectively. Usually unit cells are larger than emitter sizes, which makes a the more important limiting factor to the radiative LDOS. Owing to the great recent progress in nanofabrication techniques, the scale on which metamaterials can be patternedis entering the nanometer regime, where nonlocal responseof the metal becomes important. 15–25For example, nonlocal response can signiﬁcantly blueshift the localized surfaceplasmon polariton (SPP) resonance peak and modify the ﬁeldenhancement of a nanoscale plasmonic structure. 18–22,26 In this paper, we discuss the effects of nonlocal response on the optical properties of hyperbolic metamaterials. It isshown that the nonlocal response gives rise a large-wavevectorcutoff in the dispersion, inversely proportional to the Fermivelocity of the electron gas. In fact, the dispersion in hyperbolicmedia becomes no longer strictly hyperbolic. Accordingly, weidentify a new and fundamental limit on the enhancement of theradiative emission rates of HMMs. In particular, we show thatthe radiative LDOS does not grow arbitrarily large even in theideal limiting case that all three aforementioned parametersγ,a, and Dvanish, since the intrinsic nonlocal response turns the “supersingularity” into a ﬁnite broadband LDOSenhancement. On a more general level, our results illustratethe need, as for metallic nanoparticles, 25,26to take nonlocal response into account in homogenization theories, where thegoal is to predict the effective properties of metamaterials withever decreasing unit cell sizes. The paper is organized as follows: In Sec. II, we introduce the linearized hydrodynamic Drude model within the Thomas-Fermi approximation. In Sec. III, the unusual dispersion curves of the HMMs and their effective parameters are discussed. InSec. IV, to understand better the HMM dispersion found in Sec. III, we discuss the SPP supported by a single metal layer. In Sec. V, we investigate the LDOS of the HMMs, before discussing our results and concluding in Sec. VI. Finally, details of the calculations can be found in Appendices A-C. II. HYDRODYNAMIC DRUDE MODEL We consider a similar multilayer HMM geometry as in the recent experiments by Tumkur et al. ,10see Fig. 1. The unit cell is a subwavelength dielectric-metal bilayer, which is relativelysimple and cheap to fabricate 10and allows analytical analysis. For an effective-medium description of such a metamaterial,a local-response approximation (LRA) is usually employed,i.e., spatial dispersion is neglected. This gives the effectivedispersion relation k 2 z /epsilon1loczz+k2 /bardbl /epsilon1loc /bardbl=ω2 c2, (1) where /epsilon1loc zz=a(ad//epsilon1d+am//epsilon1m)−1,a/epsilon1loc /bardbl=ad/epsilon1d+am/epsilon1m, and k/bardbl=(k2 x+k2 y)1/2. Below the plasma frequency ωp, where /epsilon1m<0 in the Drude model of a pure plasma, the dielectric tensor elements /epsilon1zzand/epsilon1/bardblcan have opposite signs by a proper choice of the ﬁlling factor am/a. Then the dispersion becomes hyperbolic, meaning that an isofrequency contourbecomes a hyperbola rather than the usual ellipse in the ( k z,k/bardbl) plane. The length of this contour diverges, and so does theradiative LDOS. Although we discuss metal-dielectric bilayerstructures, we want to point out that our theory may also beapplied to structures where the metal is replaced by othermaterials with a Drude response. 27 In the present paper, we go beyond the LRA, and dis- cuss the optical properties of the HMMs in the linearized 205429-1 1098-0121/2012/86(20)/205429(8) ©2012 American Physical SocietyWEI Y AN, MARTIJN WUBS, AND N. ASGER MORTENSEN PHYSICAL REVIEW B 86, 205429 (2012) FIG. 1. (Color online) Sketch of a multilayer hyperbolic metama- terial consisting of periodic dielectric-metal bilayers. The dielectric and metal layer thicknesses are adandam, respectively, and their sum equals the period aof the unit cell. Corresponding permittivities are /epsilon1dand/epsilon1m. The red arrow is a dipole emitter located in the middle of a dielectric layer. hydrodynamic Drude model (HDM) within the Thomas-Fermi approximation.15–17In the HDM, the metal supports both the usual divergence-free (“transverse”) and rotation-free(“longitudinal”) waves. Above the plasma frequency bothtypes of waves can propagate. The dispersion k T(ω)o ft h e transverse waves is given by /epsilon1T m(ω)ω2=k2c2while kL(ω)o f the longitudinal waves follows from /epsilon1L m(k,ω)=0, in terms of the dielectric functions /epsilon1T m(ω)=1−ω2 p ω2+iωγ(2a) /epsilon1L m(k,ω)=1−ω2 p ω2+iωγ−β2k2. (2b) Here, γis the Drude damping, ωpis the plasma frequency, and the nonlocal parameter βis equal to/radicalbig 3/5vFwithvF representing the Fermi velocity. While /epsilon1T mis the familiar Drude dielectric function, /epsilon1L mdepends on vFand describes nonlocal response. III. DISPERSION AND EFFECTIVE MATERIAL PARAMETERS To calculate the exact dispersion equation for the inﬁnitely extended HMM, we employ a transfer-matrix method for bothtransverse and longitudinal waves combined. Our method isquite similar to the one developed by Moch ´anet al. , 28but we corrected the additional boundary condition (ABC) that inRef. 28was employed for simplicity. An ABC is required to complement the usual Maxwell boundary conditions, and allboundary conditions together make the solution to the coupledMaxwell and hydrodynamic equations unique. Details how toderive the correct ABC and a consistency check can be foundin Appendix A. For arbitrary unit cell size aand metal and dielectric ﬁlling fractions, we ﬁnd the exact dispersion relation for the inﬁnitez pp \|\| FIG. 2. (Color online) Dispersion curves of the HMM for ω= 0.2ωp, on (a) small and (b) large wavevector intervals. Red curves fora=λF, green curves for a→0, black curves for a→0i nt h e LRA. The unit cell of the HMM is a free-space-Au bilayer with ad=am=a/2. Material parameters for Au: ¯ hω p=8.812 eV, ¯ hγ= 0.0752 eV, and vF=1.39×106m/s. HMM to be cosθb =/braceleftbigg cosθd/bracketleftbigg kLzcosθmsinθl−k/bardbl(wd−wm) zmsinθmcosθl/bracketrightbigg +sinθd/bracketleftbigg k/bardbl(wd−wm) zd(1−cosθmcosθl) −1 2/bracketleftbiggk2 /bardbl kLz(wd−wm)2 zdzm+kLz/parenleftbiggzd zm+zm zd/parenrightbigg/bracketrightbigg ×sinθmsinθl/bracketrightbigg/bracerightbigg/bracketleftbigg kLzsinθl−k/bardbl(wd−wm) zmsinθm/bracketrightbigg−1 , (3) with θb=kza, θ d=kdzad,θ m=kT mzamθl=kL mzam,(4) zd=kdz k0/epsilon1d,w d=k/bardbl k0,z m=kT mz k0/epsilon1Tm,w d=k/bardbl k0/epsilon1Tm,(5) where k2 dz+k2 /bardbl=ω2/epsilon1d/c2,(kT mz)2+k2 /bardbl=ω2/epsilon1T m/c2, and (kL mz)2+k2 /bardbl=k2 Lwithk2 L=(ω2+iγω−ω2 p)β2. This disper- sion equation looks very similar to the one found in Ref. 28, but the essential difference is that the parameter wdhere is wd//epsilon1din Ref. 28. As an example we consider a HMM with free-space-Au bilayer unit cell with ad=am=a/2, and we include the Au Drude loss. Figure 2depicts HMM dispersion curves at ω=0.2ωp. Figure 2(a) shows hyperbolic dispersion in the 205429-2HYPERBOLIC METAMATERIALS: NONLOCAL RESPONSE ... PHYSICAL REVIEW B 86, 205429 (2012) small wavevector regime, whereas Fig. 2(b) zooms out and shows strong deviations from hyperbolic dispersion for largewavevectors. First, in Fig. 2(a) we observe three noncoinciding hyper- bolic dispersion curves in the small- kregion, one for local and two for nonlocal response. This tells us that the HDM isnot just a local theory with a large-wavevector cutoff added,since then the curves for local and for nonlocal response wouldhave coincided for small wavevectors. Furthermore, the twononlocal hyperbolic curves do not coincide, the one for astrongly subwavelength unit cell a=λ Fand the other for a→0. This illustrates that the size of the unit cell affects effective-medium properties, even in the deep subwavelengthlimit, which goes against common wisdom obtained in theLRA. The reason for this is that in the HDM the longitudinalwave in the metal layer has a large vector k L(ω)/greatermuch2π/λ,s o that typically the condition \|kL(ω)\|a/lessmuch1 is not satisﬁed even in the deep subwavelength limit. Thus, the longitudinal wavecan probe the ﬁnite size of the unit cell even though a/lessmuchλ, and this gives rise to the periodicity-dependent dispersion curve ofFig. 2(a). Zooming out, Fig. 2(b) shows that nonlocal response gives rise to closed nonhyperbolic dispersion curves, forboth considered values of a, in stark contrast to the familiar hyperbolic curve in the LRA which is also shown. (Westill call these media hyperbolic because of their hyperbolicsmall-wavevector dispersion.) Both k /bardblandkzare bounded on the curve for a=λF. For smaller values of a,w ed on o t expect the hydrodynamic Drude model to apply,17,22but as we shall see below it is useful to also consider the limit a→0. The curve for a→0 shows a turning point at k/bardbl=kc /bardbl.I nt h e lossless limit, no radiative modes exist above kc /bardbl, as we explain shortly. The wavevector kc /bardblis found to be kc /bardbl=ω β∝ω vF. (6) To analyze the dispersion curves of Fig. 2, we derive the effective material parameters of the HMM by a mean-ﬁeldtheory that can be applied to many geometries. In the limitof vanishing unit-cell size, we obtain the effective materialparameters /epsilon1 nloc zz=/epsilon1d zz,/epsilon1nloc /bardbl=/epsilon1d /bardblk2 L/epsilon1loc /bardbl//epsilon1d /bardbl−k2 /bardbl/epsilon1T m k2 L−k2 /bardbl/epsilon1Tm, (7) where /epsilon1d zzand/epsilon1d /bardblrepresent the effective parameters of the metamaterials when the metal layer is replaced by a free-spacelayer, with /epsilon1 d zz=a(ad//epsilon1d+am)−1, anda/epsilon1d /bardbl=ad/epsilon1d+am.B o t h nonlocal effective material parameters of Eq. (7) differ from the corresponding parameters for local response. Thederivations leading to Eq. (7)are presented in Appendix B. Neglecting loss at ﬁrst, we ﬁnd from Eq. (7)that/epsilon1 nloc /bardblhas a resonance at k/bardbl=kc /bardblwhere both /epsilon1nloc /bardblandkzdiverge. The value of kc /bardblis independent of /epsilon1d(unlike what one would ﬁnd when using the incorrect ABC of Refs. 25and 28). Increasing k/bardblbeyond kc /bardbl,t h e/epsilon1nloc /bardblchanges sign from negative to positive. Since /epsilon1nloc zzis always positive, it follows that no mode exists above kc /bardbl. Thus, nonlocal response gives rise to a large-wavenumber cutoff at k/bardbl=kc /bardbl. With loss, the resonanceis smoothed out and modes exist also above kc /bardbl. However, for k/bardbl→∞ , the corresponding kzapproaches i∞, which shows that such large-wavevector modes are purely evanescent. Thisexplains why the dispersion curves in Fig. 2are closed. We stated in Eq. (7)that unlike in the LRA, in the HDM the effective parameter /epsilon1 nloc zz simply equals the (positive) permittivity /epsilon1d zz. This outcome is ﬁxed for a→0b yt h e continuity of the normal components of the displacementﬁeld and the ABC of Eq. (A1) with/epsilon1 other=1.17In particular, the different boundary conditions explain why the local andnonlocal a→0 curves in Fig. 2(a) exhibit different hyperbolic small-wavevector dispersion. Above the plasma frequency, the HDM and the LRA also exhibit qualitatively different dispersion. In the LRA nohyperbolic dispersion exists for frequencies above the plasmafrequency, not even for small wavevectors, since then both /epsilon1 d and/epsilon1mare positive. By contrast, hyperbolic dispersion can exist in the HDM for ω>ω p, because the effective-medium parameter /epsilon1nloc /bardblgiven in Eq. (7)can assume negative values above ωp. IV . SURFACE PLASMON POLARITON SUPPORTED BY A SINGLE METAL LAYER In Sec. III, it was demonstrated that the dispersion curves in the LRA and HDM differ signiﬁcantly. To understand thisbetter, here we relate these essential differences to the differentproperties of single metal layers in both theories, knowingthat the bulk modes of the HMM result from the couplingof SPPs of neighboring metal layers. So we investigate theSPPs supported by a single metal layer, ﬁrst analytically in thequasistatic limit. With respect to the magnetic ﬁeld, the SPPscan be classiﬁed as even and odd modes. In the HDM, thedispersion relations of the even and odd modes are found tobe tanh/parenleftbiggk spam 2/parenrightbigg =−/epsilon1T m /epsilon1d+ksp/parenleftbig 1−/epsilon1T m/parenrightbig klztanh/parenleftbiggklzam 2/parenrightbigg ,(8a) coth/parenleftbiggkspam 2/parenrightbigg =−/epsilon1T m /epsilon1d+ksp/parenleftbig 1−/epsilon1T m/parenrightbig klzcoth/parenleftbiggklzam 2/parenrightbigg ,(8b) where ksprepresents the SPP wavevector, and klz=(k2 sp− k2 L)1/2. In the limit am→0, the dispersion equation of the even mode has no solution, but the odd mode always has one,even above ω p. Its dispersion is such that ksphaskc /bardblas an upper bound in the limit a→0. So we can now understand that it is this nonlocal “ceiling” for the single-layer SPPwavenumber that leads to a cutoff of k /bardblfor the bulk modes of the metamaterial, as we saw in Fig. 2. In Fig. 3we analyze numerically the effect of retardation on the SPP dispersion of a single Au layer in free space, for localand nonlocal response. With retardation, near the light conealso even-mode solutions exist. Only for nonlocal responsedo we ﬁnd modes above ω p. Again we ﬁnd that nonlocal response gives rise to a forbidden region ksp>kc /bardblfor the odd SPP mode, see Fig. 3(b). By contrast, in Fig. 3(a) for the LRA, both even and odd modes have ﬁnite-frequency solutionswithk spapproaching inﬁnity, which leads to the characteristic 205429-3WEI Y AN, MARTIJN WUBS, AND N. ASGER MORTENSEN PHYSICAL REVIEW B 86, 205429 (2012) pp p sp FIG. 3. (Color online) Dispersion curves of the SPP mode sup- ported by a single lossless Au layer with a thickness amin free space, in the (a) local-response approximation, and (b) the hydrodynamic Drude model. Dashed and solid curves correspond to even and odd modes, respectively, with red curves for am=0.1λp, green for am=0.01λp, and black curves for am→∞ (single-interface SPP). The gray areas are forbidden regions for the SPP modes, with lightcones on the left. hyperbolic curve of the HMM that extends to inﬁnitely large wavevectors. V . LOCAL DENSITY OF STATES The discussed dramatic modiﬁcation of the metamaterial dispersion due to nonlocal response will also strongly affectthe broadband supersingularity known to occur in the local-response LDOS, as we shall see. In general, the LDOS isproportional to the spontaneous-emission rate averaged overall solid angles, and deﬁned as LDOS( r 0,ω)=−2k0 3πcTr{Im[G(r0,r0,ω)]}, (9) where Gis the dyadic Green function of the medium and r0 the position of the emitter. The Green function Gis deﬁned by −∇×∇×G(r,r/prime)+k2 0/integraldisplay dr1/epsilon1(r,r1)G(r1,r/prime)=Iδ(r−r/prime), (10) where Irepresents the unit dyad, and /epsilon1represents the dielectric function, which is a position-dependent delta function for thelocal dielectric medium, and a tensorial nonlocal operatordeﬁned by Eq. (2)for the metal. For the multilayered HMM, G can be decoupled into separate contributions from TM and TEmodes. TM modes support the hyperbolic dispersion curve,and greatly dominate the LDOS, so we will neglect the TEcontribution to the LDOS. If we ﬁrst neglect loss, then only radiative modes contribute to the LDOS. For an electric dipole with moment μ,t h e contribution to the LDOS of a single radiative mode isproportional to \|μ·a k(r0)\|2/\|∇kω\|, where akis the properly normalized mode function.29In the LRA, for the limiting case of a→0, the single-mode contribution to the LDOS scales linearly in kask/bardblandkztend to inﬁnity. This results in a diverging radiative LDOS, the broadband LDOSsupersingularity of hyperbolic media. Let us now consider the LDOS in the HDM instead. If we again take the limit a→0, and let k /bardbltend to kc /bardbland kzto inﬁnity, then this time the single-mode contribution to the LDOS scales as 1 /kz2, which we derived using the effective parameters of Eq. (7). Radiative modes with large wavenumbers are therefore negligibly excited. As a mainresult of this paper, we consequently ﬁnd that in the HDMthe radiative LDOS converges to a ﬁnite value as a→0, even though the integration area in k-space diverges. We ﬁnd thenumerically exact value and its analytical approximation LDOS( ω)=ω 2 6π2β3η, (11) where η=1/radicalbig /epsilon1dzz/integraldisplayπ/2 θ0dθcos2θ+/parenleftbig /epsilon1d /bardbl//epsilon1d zz/parenrightbig [sin2θ−/epsilon1loc /bardbl//epsilon1d /bardbl] /radicalBig sin2θ−/epsilon1loc /bardbl//epsilon1d /bardbl(12) withθ0equal to arcsin( /epsilon1loc /bardbl//epsilon1d /bardbl)f o r/epsilon1loc /bardbl>0 and vanishing otherwise. The derivations leading to Eq. (11) are presented in Appendix C. As illustrated below, Eq. (11) entails that nonlocal response leads to a large upper bound to the radiative LDOS ofthe HMM, proportional to ω 2/v3 F. This exceeds the free-space radiative LDOS approximately by c3/v3 F, which is of order 107 for most metals. When taking metallic Drude loss into account, then the LDOS has contributions both from radiative modes and fromnonradiative quenching, the latter due to loss. For the limitingcase of a→0, we already discussed that /epsilon1 nloc /bardbltends to /epsilon1d /bardbl,s e e Eq.(7). For large wavevectors k/bardblalso the other component /epsilon1nloc zztends to /epsilon1d zz. Thus, to the extent that /epsilon1dis lossless, the evanescent mode with large k/bardbldoes not contribute to the nonradiative LDOS, which therefore stays ﬁnite. As a result,the total LDOS containing both radiative and nonradiativecontributions in the HDM converges as a→0. In the low-loss case, where the radiation LDOS is dominant, Eq. (11) is an accurate expression of the total LDOS, as we verify bynumerically exact simulation below. We calculate the LDOS numerically exactly by merging two methods: the local-response transfer matrix method by Toma ˇs to calculate the Green function of arbitrary multilayer media, 30 and the aforementioned HDM extension of the transfer matrixmethod. 28The details can be found in Appendix D. Figure 4(a) depicts the LDOS enhancement, deﬁned as the ratio between LDOS in the HMM and in free space, as afunction of the periodicity a. Clearly, in the HDM the LDOS 205429-4HYPERBOLIC METAMATERIALS: NONLOCAL RESPONSE ... PHYSICAL REVIEW B 86, 205429 (2012) 11 FIG. 4. (Color online) (a) LDOS versus the periodicity aat the center of the free space layer of the hyperbolic metamaterial forω=0.2ωp.( b )T h e a→0 limiting value of the LDOS in the hydrodynamic Drude model as a function of vF/c. Parameters of the hyperbolic metamaterial as in Fig. 2. converges to a ﬁnite value as a→0. This proves that both the radiative and nonradiative LDOS in the HDM are ﬁnite.By contrast, in the LRA the LDOS diverges as 1 /a 3, where the nonradiative LDOS has a dominant contribution.6,31–33 The HDM ceases to be valid for a<λ F, but the LDOS enhancement value in the limit a→0 is a useful upper bound. We also calculated the LDOS for the lossless case [not shown Fig. 4(a)], and we ﬁnd smaller values for the LDOS owing to the missing nonradiative contribution, but the sametrend for a→0. This proves that nonlocal response rather than loss is responsible for removing the singularity ofthe radiative LDOS. In Fig. 4(b) we compare the limiting LDOS for the numerically exact method in the lossy casewith the lossless analytical approximation of Eq. (11), when artiﬁcially varying the Fermi velocity. The value from theexact method is only larger than that from Eq. (11) by around 6%. We attribute the small difference in LDOS tothe nonradiative LDOS due to the Drude loss. Thus, the lossacts as a small perturbation to the radiative LDOS. VI. DISCUSSION AND CONCLUSIONS For ﬁnite-sized unit cells, small loss gives rises to a regular perturbation of the radiative LDOS, both in the local and inthe nonlocal response theories. However, the theories start todiffer dramatically in the limit of inﬁnitely small unit cells.In particular, in the nonlocal hydrodynamic Drude model thesmall variation of the radiative LDOS with small loss is quitedifferent from the previously found radiative LDOS scalingwith loss in the local theory as γ −3/2for inﬁnitely small unit cells.6 The small increase of the total LDOS due to loss is also quite different from spontaneous-emission rates of a pointemitter inside a homogeneous absorbing medium, where theloss induces nonradiative quenching that can dramaticallydecrease the radiative decay efﬁciency. 6,31–33In this sense, the nonlocal response regularizes the singularity not only ofthe radiative but also of the nonradiative LDOS of a lossyHMM. One can interpret this ﬁnite nonradiative LDOS as dueto a nonlocal screening of the electron scattering loss. 34Ina certain high wavevector region the reverse can also occur, namely the enhancement of the nonradiative LDOS by thenonlocal response, when not only taking Drude loss intoaccount, as we do here, but also electron-hole pair absorption.By neglecting any dielectric response of the metal apart fromthe (hydrodynamic) Drude response, we underestimate thenonradiative LDOS of real metals. However, the importantconclusion that the nonlocal response removes the singularityof nonradiative LDOS is still valid. 34 In conclusion, we have shown that the hydrodynamic Drude model gives closed nonhyperbolic dispersion relationsfor hyperbolic metamaterials, with a fundamental wavevectorcutoff ∝ω/v F. These effective dispersion relations have hy- perbolic limits for small wavevectors, but the precise hyperboladepends on the subwavelength size of the unit cell, contraryto consensus based on the local-response approximation. Weﬁnd that the hydrodynamic model regularizes the broadbandsupersingularity of the radiative LDOS, and provides a largephysical upper bound proportional to ω 2/v3 F. In practice, con- sidering the ﬁnite values of aandD, i.e., the ﬁnite sizes of the unit cell and the emitter, we usually have 1 /a < 1/D < ω/v F. This indicates that the size effects have a dominant role inlimiting the LDOS enhancement. Thus, under an upper boundset up by the nonlocal response, hyperbolic metamaterialshave plenty of room for improvement in boosting light-matterinteractions by decreasing the sizes of the unit-cell and theemitter. ACKNOWLEDGMENTS We thank S. Raza for stimulating discussions. This work was ﬁnancially supported by an H. C. Ørsted Fellowship(W.Y .). APPENDIX A: BOUNDARY CONDITIONS Since the hydrodynamic dynamics allows the excitation of longitudinal waves, the unambiguous solution of the nonlocal-response dynamics requires additional boundary conditions(ABCs), complementing the Maxwell boundary conditions.As is well known, the Maxwell boundary conditions area consequence of Maxwell’s equations themselves, in thesense that the derivation of the boundary conditions onlyinvolves Maxwell’s equations plus mathematics (the Gaussand Stokes theorems). Quite analogously, ABC’s are not amatter of choice but can be derived from the (linearized)hydrodynamic equations, at least for a given equilibriumfree-electron density proﬁle n 0.17,35When assuming a simple zero-to-nonzero step proﬁle of n0at the dielectric-metal interfaces, this unambiguously leads to one and only onerequired ABC, namely the continuity of the normal componentof the free-electron current J. 17,35 Let us now write the relative permittivity of the dielectric medium as /epsilon1d, and the dielectric response of the metal as /epsilon1m(ω). We assume that /epsilon1m(ω) is given by the sum of a nonlocal hydrodynamic Drude free-electron response plus /epsilon1other m(ω), the latter describing the remaining dielectric response of themetal. Since one of the Maxwell boundary conditions is theconservation of the normal component of the displacement 205429-5WEI Y AN, MARTIJN WUBS, AND N. ASGER MORTENSEN PHYSICAL REVIEW B 86, 205429 (2012) ﬁeld, the ABC is equivalent to the condition /epsilon1other mEm·ˆn=/epsilon1dEd·ˆn, (A1) where Em,drepresent the electric ﬁelds in the metal and dielectric, respectively, and ˆnis the unit vector normal to the boundary. From Eq. (A1) , we see that the normal electric ﬁeld is discontinuous across the boundary when /epsilon1other m/negationslash=/epsilon1d.Aj u m p in the electric ﬁeld occurs due to the surface charge producedby polarization of the bound electrons both in the dielectric and in the metal. We discuss the ABC in some detail, because Moch ´an et al. , 28whose pioneering transfer matrix method we employ here, and also recently Cirac `ıet al.25used instead the continuity of the normal component of the electric ﬁeld asthe ABC, E m·ˆn=Ed·ˆn, (A2) or equivalently the continuity of the normal component of the displacement current. There is no derivation of the latter ABCin Refs. 25and28. It happens to be only correct, in agreement with Eq. (A1) ,i f/epsilon1 other m=/epsilon1d, for example in case the dielectric is vacuum ( /epsilon1d=1) and the metal is a pure Drude metal ( /epsilon1other m= 1). Moch ´anet al.28applied their ABC for simplicity and write that they thereby ignore the discontinuity of the electric ﬁeld,due to the accumulation at the surface of bound charges. Ourmain point here is that without additional complication thecorrect ABC can be implemented, and that many physicalpredictions of the hydrodynamic Drude model are sensitive toimplementing the ABC correctly. To understand the ABC physically, recall that in the HDM the dynamics of the free electrons is described by the equationof motion m e/bracketleftbigg∂v ∂t+v·∇v/bracketrightbigg =−∇pdeg n+e(E+v×B),(A3) where pdegis the pressure from the ground state energy of the degenerate quantum Fermi gas, and nis the free-electron density. The pressure force −∇pdeg/n∝−∇n/n drives the free electrons diffusing from the high-density region to thelow-density region. It is this force that prevents the free-electron charge from accumulating on the boundary surface,since the existence of a free-electron surface charge wouldcause an inﬁnitely large pressure force, which is unphysical.The nonexistence of the surface free-electron charge indicatesthat the free-electron current should be continuous across theboundary, as the ABC (A1) indeed describes. By contrast, in the ABC of Eq. (A2) , there exists no surface charges at all. This indicates that the pressure force somehow smearsout not only the surface free-electron charge in the metal butalso the surface polarization charges in both the metal andthe dielectric. However, one cannot expect the smearing outof the surface polarization charges in the HDM, since thepressure force only acts on the free electrons in the metal.In this sense, the ABC of Eq. (A2) is not consistent with the assumed dynamics and thus not physically sound. There is another perhaps simpler argument, a consistency check that conﬁrms that the ABC of Eq. (A2) is more problematic. Assume there is a thin free-space layer withsubwavelength thickness δbetween the nonlocal metal and the local dielectric medium. At the boundary between the metaland free space, the ABC of Eq. (A2) gives E m·ˆn=Ef·ˆn, where Efrepresents the electric ﬁeld in the free-space layer. At the boundary between free space and the dielectric medium,we have E f·ˆn=/epsilon1dEd·ˆnby the standard Maxwell boundary condition of the continuity of the normal component of thedisplacement ﬁeld. In the limit of an inﬁnitely thin free-space middle layer ( δ→0), the three-layer system essentially becomes the two-layer system where the metal and thedielectric medium touch, and for which we ﬁnd E m·ˆn= /epsilon1dEd·ˆnby combining the previous two identities. However, this contradicts with Eq. (A2) for the metal-dielectric interface. Thus, the ABC of Eq. (A2) can not be applied consistently. For the ABC of Eq. (A1) , we obtain instead consistent results when following the above thin-layer argument. APPENDIX B: EFFECTIVE MATERIAL PARAMETERS OF HYPERBOLIC METAMATERIAL When the unit cell has a thickness athat is much smaller than an optical wavelength λ0, then the optical properties of such an inﬁnite multilayer structure can be macroscop-ically described by a diagonal effective dielectric tensor/epsilon1=diag[/epsilon1 /bardbl,/epsilon1/bardbl,/epsilon1zz] with tensor components /epsilon1/bardbl=/angbracketleftDx,y/angbracketright /angbracketleftEx,y/angbracketright,/epsilon1zz=/angbracketleftDz/angbracketright /angbracketleftEz/angbracketright, (B1) and where /angbracketleft.../angbracketrightdenotes spatial averaging over a unit cell. The unit cell can be chosen symmetric, identical for left- and right-traveling waves. Consider a unit cell positioned at−a/2<z<a / 2, with the metal layer at −a m/2<z<a m/2, which is symmetric in z=0. The total ﬁelds in such a unit cell are generated by waves incident both from the left (“l”) andfrom the right “r”, and the average ﬁelds can be split into twoterms, /angbracketleftE/angbracketright=/angbracketleftE/angbracketright l+/angbracketleftE/angbracketrightr. However, by symmetry of the unit cell it follows that /epsilon1=/angbracketleftD/angbracketright//angbracketleftE/angbracketright=/angbracketleftD/angbracketrightl//angbracketleftE/angbracketrightl=/angbracketleftD/angbracketrightr//angbracketleftE/angbracketrightr.T o obtain the effective material parameters, we can simply replacethe average ﬁelds in the periodic structure by the average ﬁeldsin a single unit cell. Before spatially averaging the ﬁelds, we ﬁrst need to ﬁnd them as solutions of Maxwell’s equations. We focus solely onTM-polarized waves since the hyperbolic dispersion occurs forthose waves only. Since k 0am/lessmuch1, we can make the quasistatic approximation, where E=− ∇ φ. Consider an incident electric ﬁeld with the electric potential φ=exp(ik/bardblx−k/bardblz). The electric potential in the whole system can then be written as φ1=exp(ik/bardblx)[exp(−k/bardblz)+rexp(k/bardblz)], φT 2=exp(ik/bardblx)[A1exp(−k/bardblz)+A2exp(k/bardblz)], (B2) φL 2=exp(ik/bardblx)[B1exp(−kLzz)+B2exp(kLzz)], φ3=texp(ik/bardblx)e x p (−k/bardblz), where kLz=√ k2 /bardbl−k2 L. By matching boundary conditions at the two metal-dielectric interfaces, the above equations canbe solved. After obtaining the ﬁeld distributions, we canaverage the ﬁelds from −a/2<z<a / 2 to obtain the effective material parameters using Eq. (B1) . First consider the case in which the metal layers are much thicker than the wavelength of the longitudinal waves(k Lzam/greatermuch1), but where the unit cell is much thinner than 205429-6HYPERBOLIC METAMATERIALS: NONLOCAL RESPONSE ... PHYSICAL REVIEW B 86, 205429 (2012) an optical wavelength. Here we expect an effective (ho- mogenized) description to apply and nonlocal response tobe negligible. Following the above described scheme, theeffective material parameters, to the zeroth-order in the smallparameters k /bardblaand 1/(kLzam), are found to be /epsilon1loc zz=1 fd /epsilon1d+fm /epsilon1Tm,/epsilon1loc /bardbl=fd/epsilon1d+fm/epsilon1T m, (B3) in terms of the ﬁlling factors fd=ad/aandfm=am/a. Indeed, the effective material parameters are just as what onewould ﬁnd in the local response approximation (LRA). Second, we consider the limiting case of a m→0 with kLzam/lessmuch1, where the nonlocal response is extremely strong. As before, we keep the ﬁlling fractions fm,dconstant when taking the limit. The effective material parameters, to zerothorder in both k /bardblaandkLzam, become /epsilon1nloc zz=/epsilon1d zz,/epsilon1nloc /bardbl=/epsilon1d /bardblk2 L/epsilon1loc /bardbl//epsilon1d /bardbl−k2 /bardbl/epsilon1T m k2 L−k2 /bardbl/epsilon1Tm, (B4) Eq.(B4) characterizes how the nonlocal response can modify the effective material parameters to the largest extent. APPENDIX C: LIMITING LDOS OF HYPERBOLIC METAMATERIALS Here we provide the calculation details of the LDOS in the hydrodynamic Drude model in the limit of inﬁnitely small unit cells ( a→0). We employ the effective material parameters derived in Eq. (B4) . Since the by far dominant contribution to the LDOS stems from TM waves, we will neglect the TEcontribution. In kspace, the diagonal components of G,i n an effective medium with material parameters expressed inEq.(B4) for TM polarization, are found to be G k TM,jj=1 k2 /bardbl/epsilon1nloc /bardblk2 x/bracketleftbig 1−k2 /bardbl//parenleftbig k2 0/epsilon1nloc zz/parenrightbig/bracketrightbig k2 0−k2 /bardbl//epsilon1nloczz−k2z//epsilon1nloc /bardblforj=x,y, Gk TM,zz=1 /epsilon1nloczz1−k2 z//parenleftbig k2 0/epsilon1nloc /bardbl/parenrightbig k2 0−k2 /bardbl//epsilon1nloczz−k2z//epsilon1nloc /bardbl. (C1) When inserting these diagonal components for the Green tensor into expression (9)for the LDOS, we obtain lim a→0LDOS =−2k0 3πc1 (2π)3Im/integraldisplay d3k/summationdisplay j=x,y,zGk TM,jj =−k0 6π2cRe⎡ ⎢⎣/integraldisplay∞ 0dk/bardblk/bardbl−k3 /bardbl k2 01−/epsilon1nloc /bardbl//epsilon1nloc zz /epsilon1nloczz/radicalBig k2 0/epsilon1nloc /bardbl−k2 /bardbl/epsilon1nloc /bardbl//epsilon1nloczz⎤ ⎥⎦ ≈k0 6π2cRe⎡ ⎣/integraldisplay∞ 0dk/bardblk2 /bardbl k2 01−/epsilon1nloc /bardbl//epsilon1nloc zz /epsilon1nloczz/radicalBig −/epsilon1nloc /bardbl//epsilon1nloczz⎤ ⎦ =ω2 6π2β3η, (C2) where ηis expressed in Eq. (12). In the derivation, we neglected losses in the metal. To arrive at the second line of Eq. (C2) ,w e use the principal-value identity lim /Delta1↓01 x±i/Delta1=P1 x±iπδ(x). Theﬁnal identity then follows immediately by inserting the a→0 limiting expressions for /epsilon1nloc zzand/epsilon1nloc /bardblgiven in Eq. (B4) . APPENDIX D: GREEN FUNCTION OF HYPERBOLIC METAMATERIAL Consider an emitter positioned in the dielectric layer of the HMM. The HMM can be divided into three regions:(i) the central dielectric layer where the emitter is located;(ii) the left semi-inﬁnite HMM; (iii) the right semi-inﬁniteHMM. The distance between the emitter and the left (right)boundary of the dielectric layer is z l(zr). The Green function Gin the central layer could be separated into two terms G(r,r0)=Gd(r,r0)+Gs(r,r0), (D1) where Gdrepresents the Green function for the emitter in the homogenous dielectric medium, while Gsrepresents the Green function owing to the scattering between central layer and theleft and right semi-inﬁnite HMM. In the plane wave basis, G d is expressed as30 Gd(r,r0)=−δ(z−z0) kd2ˆzˆz/integraldisplay d2k/bardblexp[ik/bardbl·(r/bardbl−r0/bardbl)] +i 8π2/integraldisplay d2k/bardbl[eTEeTE+e± TMeTM±] kz ×exp[ik/bardbl·(r/bardbl−r0/bardbl)+ikz\|z−z0\|], (D2) with eTE=k/bardbl k/bardbl×ˆz, e± TM=k/bardbl±kzˆz kd×eTE, (D3) where k/bardbl=kxˆx+kyˆy,kd=ω√/epsilon1d/c,k2 z+k2 /bardbl=k2 d, and e± TM correspond to z>z 0andz<z 0, respectively. The terms con- taining eTEandeTMrepresent TE and TM waves, respectively. The scattering part Gsof the Green function is expressed as Gs(r,r0)=i 8π2/integraldisplay dk/bardbl1 kzexp[ik/bardbl·(r/bardbl−r0/bardbl)] ×{[r++ TEeTEeTE+r++ TMe+TMe+TM +r+− TEeTEeTE+r+− TMe+TMe−TM]e x p [ikz(z−z0)] +[r−+ TEeTEeTE+r−+ TMe−TMe+TM +r−− TEeTEeTE+r−− TMe−TMe−TM]exp[−ikz(z−z0)]}, (D4) where r±± TE,TMis the reﬂection coefﬁcient, in which the left superscript “ ±” represents the scattering wave in the ±ˆz direction, and the right superscript “ ±” represents the incident wave in the ±ˆzdirection. The r±± TE,TMare found to be r++ TE,TM=r−− TE,TM=R2 TE,TMexp(2ikzad) 1−R2 TE,TMexp(2ikzad), r−+ TE,TM=RTE,TMexp(2ikzzr) 1−R2 TE,TMexp(2ikzad), r+− TE,TM=RTE,TMexp(2ikzzl) 1−R2 TE,TMexp(2ikzad), (D5) 205429-7WEI Y AN, MARTIJN WUBS, AND N. 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PhysRevB.84.235206.pdf	PHYSICAL REVIEW B 84, 235206 (2011) Low-carrier-concentration crystals of the topological insulator Bi 2Te2Se Shuang Jia,1Huiwen Ji,1E. Climent-Pascual,1M. K. Fuccillo,1M. E. Charles,1Jun Xiong,2N. P. Ong,2and R. J. Cava1 1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA (Received 9 November 2011; revised manuscript received 5 December 2011; published 15 December 2011) We report the characterization of Bi 2Te2Se crystals obtained by the modiﬁed Bridgman and Bridgman- Stockbarger crystal-growth techniques. X-ray-diffraction study conﬁrms an ordered Se-Te distribution in theinner and outer chalcogen layers, respectively, with a small amount of mixing. The crystals displaying highresistivity ( >1/Omega1cm) and low carrier concentration ( ∼5×10 16/cm3) at 4 K were found in the central region of the long Bridgman-Stockbarger crystal, which we attribute to very small differences in defect density alongthe length of the crystal rod. Analysis of the temperature-dependent resistivities and Hall coefﬁcients reveals thepossible underlying origins of the donors and acceptors in this phase. DOI: 10.1103/PhysRevB.84.235206 PACS number(s): 72 .20.−i, 73.25.+i, 72.80.Jc, 73.20.At I. INTRODUCTION The narrow-band-gap semiconductors (Bi,Sb) 2(Te,Se) 3 have been studied for over ﬁfty years due to their use as thermoelectric materials.1Recently, these compounds have again come to the forefront in research in condensed-matterphysics, as the prototypical three-dimensional topologicalinsulators (TIs), displaying gapped electronic bulk states andgapless electronic surface states. 2–6The exotic, spin-locked Dirac metallic surface states of Bi 2Se3and Bi 2Te3have been revealed by angle-resolved photoemission spectroscopy(ARPES) studies, 7–9as well as scanning tunneling microscopy (STM) experiments.10Investigating the charge transport char- acteristics of the surface states has, however, proven to bechallenging, because the surface current is usually one or twoorders of magnitude less than the bulk current. 11,12In binary or ternary narrow-band-gap semiconductors ( EG<300 meV),8 the small defect formation energies and the resultant signiﬁcant defect densities often result in relatively large carrier densities∼10 18–19/cm3, leading the bulk conductance to dominate over the surface conductance in transport probe measurements.While previous materials research has primarily focused onheavily doped, low-resistivity (Bi,Sb) 2(Te,Se) 3for optimizing the thermoelectric ﬁgure of merit, the goal of the currentresearch is to achieve highly resistive low bulk carrier-concentration crystals that will facilitate the observation ofthe topological surface-state transport. The original studies of surface transport in TIs in the Bi 2X3family were focused on the binary Bi 2Se3and Bi 2Te3 phases.11–13Near-stoichiometric Bi 2Se3crystals are always heavily-doped n-type materials due to the presence of charged electron-donating selenium vacancies,1though a small amount of Ca substitution for Bi yielded p-type materials.13,14The defect equilibrium can be written as Bi2Se3/arrowrighttophalf/arrowleftbothalf2Bi Bi+3V•• Se+3 2Se2(g)+6e/prime. (1) Near-stoichiometric Bi 2Te3crystals, on the other hand, are p type due to the presence of antisite defects, i.e., Bi on the Tesites, 1,11with the defect equilibrium: Bi2Te3/arrowrighttophalf/arrowleftbothalf2Bi/prime Te+2h•+Te2(g)+TeTe. (2) Although small contributions of surface-state conductance to the total conductivity have been reported in carefully grown,semiconductorlike Bi 2Te3single crystals, the resistivity of the binary compounds has not been reported to exceed15 m/Omega1cm; 11the reported surface conductance of Bi 2Te3 contributes for those samples less than 0 .3% to the total conductance. A transition from p-type to n-type behavior in the Bi2(Te 1−xSex)3solid solution was reported 40 years ago.15,16 Very recent studies have revealed that when xis close to 0 .33, yielding the structurally ordered phase Bi 2Te2Se, crystals show high bulk resistivity at low temperatures, exceeding 1 /Omega1cm. This, along with good topological surface-state mobilities, results in a surface conductance contribution to the total conductivity of up to two orders of magnitude larger than isobserved in Bi 2Te3.17,18In some such Bi 2Te2Se (BTS) crystals, clear Shubnikov-de Hass oscillations due to the surfacestates are observed. 17,18With its much larger surface current contribution, BTS may serve as an ideal model system for thestudy of spin-selected Dirac electron behavior, as well as a starting point for the future development of electronic devices based on topological surface states. This paper describes howsuch crystals can be obtained. The defects in the BTS ternary TI compound are anticipated to be more complex than those found in the binary systems,though the tendency toward Se site vacancies and Bi antisitedefects, as seen in the binary compounds, is expected tobe maintained; the effects of these and possible associateddefects on the carrier density and mobility in the BTSternary compound are still unclear. In this paper, we describethe growth of the crystals of high-quality BTS on whichthe successful surface transport measurements have beenperformed, focusing on the characterization of single-crystalBi 2Te2Se obtained via two different crystal-growth methods. Powder x-ray-diffraction (XRD) characterization of the crystals conﬁrms that laboratory-grown Bi 2Te2Se displays a fully ordered structure in which the Te and Se atomsoccupy their own distinct crystallographic sites with theTe in the outer chalcogen layers and the Se in the innerchalcogen layer—a factor that may be one of the underlyingcauses of the high surface mobility of the BTS phase. Ourelectrical resistivity and Hall measurements reveal that thecrystals’ carrier concentration is highly sensitive to the Se siteoccupancy. As expected for small-band-gap semiconductors,the crystals’ quality from the perspective of the defects 235206-1 1098-0121/2011/84(23)/235206(7) ©2011 American Physical SocietySHUANG JIA et al. PHYSICAL REVIEW B 84, 235206 (2011) and the resultant carrier concentrations is strongly affected by small inhomogeneities in the chemical composition thatoccur during the crystal growth. Realizing this difﬁculty, aBridgman-Stockbarger method was employed to ﬁne-tune thechemical composition of the crystals and optimize the carrierconcentration. We show this method to be a controllable wayfor synthesizing high-resistance low carrier-concentration bulkcrystals of Bi 2Te2Se. The relation between the distribution of the donors and acceptors and the crystal properties is brieﬂydiscussed. II. EXPERIMENT Single crystals of Bi 2Te2Se were prepared by two methods. The ﬁrst method, which we specify as a “modiﬁed Bridgman”method, is similar to that presented in Refs. 14and 17. Five-gram mixtures of high purity (5N) elemental Bi, Te,and Se were sealed in quartz ampoules, and then heatedup to 850 ◦C for 1–2 days followed by cooling to 500◦C at 6–12◦C/h. The samples were then annealed at 500◦C for 3–4 days. The crystals obtained by this method arewithin a large monolithic piece ( ∼1×1×4c m 3) that usually consists of approximately ten grains presenting random crystalorientations. For the powder XRD characterization of thelaboratory-made BTS phase, the samples from the modiﬁedBridgman method were in addition annealed at 400 ◦Cf o r over 2 weeks and then quenched in cold water. These samples were compared to samples that were directly quenched in water from the melt without any annealing to test for therobustness of the Te-Se crystallographic site ordering. Samplesof the related compound Bi 2TeSe 2were similarly prepared for comparison of the structures, and samples for which the Te/Seratio was varied in small increments near the formula Bi 2Te2Se for testing the effects of nonstoichiometry on the electricalproperties were also prepared in this fashion. The samplesfor XRD were pulverized in liquid nitrogen and characterizedby laboratory XRD using graphite monochromated Cu Kα radiation (D8 Focus, Bruker) at room temperature. Rietveldreﬁnements were carried out using the FULLPROF program suite.19 The classic Bridgman-Stockbarger method was employed as the second crystal-growth method. This method allowsfor ﬁne-tuning of the chemical composition of the crystalnear the stoichiometric Bi 2Te2Se composition due to natural variations in stoichiometry along the directionally solidiﬁedcrystal boule. Thirty grams of mixture was sealed in a longinternally carbon-coated quartz ampoule (20 cm long and0.8 cm diameter). This ampoule was tapered at the bottomin order to favor seed selection, and then placed in a verticalfurnace. The temperature proﬁle of the furnace was set toensure that the zone hotter than the melting temperature of theBTS was longer than the length of the liquid. The temperaturegradient at the furnace position crossing the melting pointof the BTS was ∼30 ◦C/cm. The ampoule was then lowered through the hot zone at the speed of 2–4 mm /h. The crystal boules obtained, a characteristic one described in detail in thispaper, were about 14 cm long with fewer than 10 crystals,which were all grown with their abplanes parallel to the long axis of the ampoule. Such a uniform crystal morphologyindicates that the boule is relatively homogeneous on a largescale, with its chemical composition gradually varying along the long axis during the directional solidiﬁcation. The bouleanalyzed was cut to seven pieces of about 2 cm equal lengthfor investigating the electrical properties of different parts(see inset of Fig. 5). The resistance and Hall measurements from 300 to 10 K were performed in the abplane of the crystals in a Quantum Design Physical Property MeasurementSystem (PPMS), and/or in a homemade resistance probe with aKeithley 2000 multimeter. For the Hall-effect measurements,the magnetic ﬁeld was applied perpendicular to that plane.The calculation of the carrier concentration and mobilityis based on single band model for simplicity. Consideringthe samples’ expected variation in defect concentration, weselected multiple pieces from each region for measurements;only representative results are presented in this paper. III. RESULTS Bi2Se3and Bi 2Te3are isostructural ﬁve-layer tetradymite- type compounds, in which the layers stack in the sequenceSe(Te) I-Bi-Se(Te)II-Bi-Se(Te)I(Fig. 1). One crystallographic cell consists of three of these ﬁve-layer units stacked withrhombohedral symmetry, connected via weak van der Waalsbonds between the Se(Te) I-Se(Te)Ilayers in different units. The cleavage occurs between these van der Waals layers,with the exposed Se(Te) Isurface layer hosting the topological surface states. The Se(Te) atoms in site I and site II are bondedto either three or six Bi nearest neighbors, respectively. Sucha difference in coordination of the two sites leads to different Bi 2Se3 Bi 2TeSe 2 Bi 2Te2Se Bi 2Te3Bi BiSe(Te)I Se(Te)ISe(Te)II FIG. 1. (Color online) Upper panel: Schematic crystal structures of Bi 2Se3,B i 2TeSe 2,B i 2Te2Se, and Bi 2Te3. Lower panel: observed (open circles), calculated (solid line), and difference (lower solid line) XRD patterns of Bi 2Te2Se grown via the Bridgman-Stockbarger method. Inset: comparison of the (105) peak of Bi 2Te2Se grown via the Bridgman-Stockbarger method (black), and the modiﬁed Bridgman quenched (red) and annealed (green). 235206-2LOW-CARRIER-CONCENTRATION CRYSTALS OF THE ... PHYSICAL REVIEW B 84, 235206 (2011) TABLE I. Powder XRD reﬁnement results for a ground Bi 2Te2Se crystal grown via the Bridgman-Stockbarger method. S.O.F., site occupancy factor; U, thermal parameter ( ˚A2), all constrained to be equal; RF=7.28;χ2=2.33. a=4.3067(1) ˚A c=30.0509(13) ˚A Atom Wyck. S.O.F. xy z U Bi 6 c 1.00 0 0 0.39681(8) 0.0048(5) SeI6c 0.043(7) 0 0 0.21155(8) 0.0048(5) TeI6c 0.957(7) 0 0 0.21155(8) 0.0048(5) SeII3a 0.915(14) 0 0 0 0.0048(5) TeII3a 0.085(14) 0 0 0 0.0048(5) Se(Te)-Bi bond lengths: the Bi-Se(Te)Ibond length is 7% shorter than the Bi-Se(Te)IIbond length in both Bi 2Se3and Bi2Te3. The longer bond length of the central layer of Se(Te) indicates that Se(Te)Ican be considered as more ionic than Se(Te)II. In order to investigate possible fully ordered ternary composition in the Bi 2(Te 1−xSex)3solid solution, Bi 2Te2Se and Bi 2TeSe 2were characterized by powder XRD (Fig. 1and Table I). Our reﬁnements revealed that Bi 2Te2Se has nearly fully ordered TeIand SeIIlayers, whereas Bi 2TeSe 2has a fully disordered Se(Te)Ilayer (a 50 /50 random mixture of Se and Te) and a fully ordered SeIIlayer (data not shown). When the Te atoms are replaced by the more electronegative Se atomsin the Bi 2(Te 1−xSex)3solid solution, the Se atoms initially ﬁll the central layer for x/lessorequalslant1/3 and then start to replace the TeI atoms in the outside layer. This result is consistent with early XRD experiments.15 Inspection of the XRD patterns of BTS samples grown by different methods reveals different peak shapes (inset ofFig. 1). The diffraction peaks of the BTS samples grown by the modiﬁed Bridgman method are broad and asymmetric witha low angle shoulder. We attribute the asymmetric broadeningto a range of compositions Bi 2Te2±δSe1∓δin the crystals. The XRD pattern of the quenched BTS shows signiﬁcantdouble-peak character, indicating moderate phase separation,as is expected from fast-cooling a solid solution phase. Incomparison, the XRD pattern of BTS grown by the Bridgman-Stockbarger method has a much sharper and more symmetricpeak shape, indicating that it has a more uniform composition.This pattern was reﬁned by using the FULLPROF program based on a model with a 2:1 ratio of Te to Se, as free reﬁnement ofthe composition did not show signiﬁcant deviation from thatformula. As shown in Table I, the sample has near-perfect order of the Se and Te layers, with around 4% disorder on the outerlayers that support the topological surface states. We attributethe imperfect ﬁt to some of the peaks to structural defectsintroduced during the grinding of the very soft BTS material.In contrast, Bi 2TeSe 2shows full disorder on the outer layers, with 50 /50 Te/Se occupancy. Being a nearly fully ordered crystal, Bi 2Te2Se promises relatively low electron scattering that might be present due to general structural disorder at thesurface, which is a Te layer, unlike what would be seen forBi 2(Te,Se) 3at higher selenium contents than Bi 2Te2Se, where the surface layers would be substantially mixed. Hence thiswork focuses on tuning the carrier concentration of Bi 2Te2Se and measuring its general electronic transport properties toallow for its optimization for study of the transport due to thetopological surface states. As will be shown below, the relativehomogeneity of BTS crystals grown via different methods has an impact on their observed electron transport properties. As a starting point for the research, the temperature- dependent resistance (shown as ρ/ρ 300 K ) and Hall coefﬁcient (RH) were measured for stoichiometric Bi 2Te2Se prepared by the modiﬁed Bridgman method. As shown in Fig. 2, our modiﬁed Bridgman growth Bi 2Te2Se sample shows metallic behavior with a small negative RH, leading to a large temperature-independent n-type carrier concentration n=2.6×1019/cm3at 10 K. Such a large carrier concentration and its trivial temperature-independent behavior indicate thatthe material is heavily doped by donors, resulting in a Fermilevel ( E F) in the conduction band, far above the energy gap. FIG. 2. (Color online) (a): The temperature-dependent resistivi- ties of Bi 2Te2+xSe1−x, S1, and S2 present different samples from the same batch. The resistivities for all the samples are 1–10 m /Omega1cm at room temperature. (b) Temperature-dependent Hall coefﬁcients of Bi2Te2+xSe1−xsamples. 235206-3SHUANG JIA et al. PHYSICAL REVIEW B 84, 235206 (2011) ThisEFposition is similar to that seen in metallic Bi 2Se3in the ARPES measurements.8,14 In order to explore the possibility of ﬁnding a low carrier- concentration sample in the vicinity of the p-ntransition point in the Bi 2Te2+xSe1−x, solid solution, the series of samples synthesized near the stoichiometric formula Bi 2Te2Se was characterized. Because nominally stoichiometric BTS is ntype in the modiﬁed Bridgman crystal growths, samples of interestfor reaching the crossover are expected to be rich in Te, where agreater tendency toward p-type behavior is expected (because Bi 2Te3is naturally ptype). Figures 2(a) and 2(b) show ρ(T) andRHfor crystals with x=0.05, 0.1, and 0 .15, respectively. None of them shows typical semiconducting ρ(T) behavior, although two crystals from the x=0.05 and 0 .10 batches show weak negative temperature coefﬁcients for T< 25 K. TheRHfor all the samples vary from −1.0t o−0.2c m3/Ca t 10 K, leading to n-type carrier concentrations from n=6× 1018cm3to 3×1019cm3.T h ep-ntransition in Bi 2Te2+xSe1−x was not found when x/lessorequalslant0.15. This result is consistent with previously reported Seebeck coefﬁcient measurementswhich showed a p-ntransition at x=0.4. 16Recently a high-resistance sample of Bi 2Te1.95Se2.05was reported;20for our crystal-growth methods all such compounds are metallic;this may be due to the fact that majority carrier concentrationsin this regime are strongly affected by very small amounts ofnonstoichiometry, as discussed further below. Tuning the selenium concentration in the selenium-excess or -deﬁcient Bi 2Te2Seyseries near y=1 can induce signiﬁcant changes in the carrier concentration in BTS. Figure 3shows theρ(T) andRHdata for Bi 2Te2Seysamples, where ychanges from 0 .95 to 1 .02. For y> 1, the values of ρ10 K/ρ300 K are all smaller than what is seen in stoichiometric Bi 2Te2Se. The RH value for y=1.02 is∼−0.2c m3/C from 10 to 300 K, leading to a strong metallic behavior. This result indicates that simplyadding Se to stoichiometric Bi 2Te2Se crystal growths cannot compensate for the n-type majority carriers present at the stoichiometric ratio. In contrast, most of the samples for y< 1 show resistivities higher than the stoichiometric material. Thelow-temperature resistivities for these samples are still lowerthan their room-temperature values, however, except for onesample from the batch y=0.98 [inset of Fig. 3(a)], which shows ρ 10 K/ρ300 K∼10. It is worth noting that different crystal samples from one batch show dramatic differencesin resistance when y< 1. Such signiﬁcant sample-to-sample variation for crystals obtained from the modiﬁed Bridgman method presents a difﬁculty for their use in future research on surface-state transport properties. It is well known that a temperature gradient in a Bridgman growth furnace can induce a distribution of the chemicalcomposition for a solid solution phase with a variation inmelting point. 21This mechanism was recently employed in the optical ﬂoating-zone growth of the solid solution ofLa 2−2xSr1+2xMn 2O7for exploring a ﬁne structure of the charge and orbital phases in the diagram near x=0.6.22Motivated by the partial success of the slightly Se-deﬁcient crystals obtainedin the modiﬁed Bridgman growths, a Bridgman-Stockbargermethod was used for growing a crystal at a slightly Se-deﬁcientcomposition, Bi 2Te2Se.995. Figure 4shows the ρ(T) andRH data for different parts of the long Bridgman-Stockbarger crystal boule (see top of Fig. 5). At the growth starting pointFIG. 3. (Color online) (a) Temperature-dependent resistivities of Bi 2Te2Seysamples. S1 and S2 are different samples from the same batch. The resistivities for all the samples are in the range 1–10 m /Omega1cm at room temperature. (b) Temperature-dependent Hall coefﬁcients of Bi 2Te2Seysamples. (GandF), the samples show metallic resistivities and small negative Hall coefﬁcients, which is similar to the metalliccrystals obtained by the modiﬁed Bridgman method. In themiddle part of the boule (parts C, D, and E), the samples showsemiconducting resistivity and much larger absolute values ofR H. The three samples show positive RHbetween 0 .2 and 1c m3/C at 300 K, leading to hole concentrations of p= 0.5–3×1018/cm3.RHsigniﬁcantly increases with decreasing T, leading to a positive maximum at about 70 K for samples C and E, and at about 200 K for sample D. On cooling belowthe maximum, R Hthen decreases with decreasing T, crosses zero, and shows a negative maximum at 10 K for all threesamples, leading to the electron concentration 5 ×10 16/cm3 to 2×1017/cm3at 10 K [Fig. 5(a)]. Given that the typical thickness of the samples is about 0 .02–0.08 mm, this number is still one order of magnitude larger than the estimatedsurface electron contribution ( ∼10 12/cm2), but the surface conductivity is similar in magnitude to the bulk conductivitydue to the high surface mobility. 17,18The observed positive- to-negative transition of RHwith temperature is similar to the previously reported experiments,17,18which clearly reveals the coexistence of two types of carriers in these BTScrystals. The samples turn back to being metallic at the top partof the boule (parts A and B), in which the carrier concentration 235206-4LOW-CARRIER-CONCENTRATION CRYSTALS OF THE ... PHYSICAL REVIEW B 84, 235206 (2011) FIG. 4. (Color online) (a) Temperature-dependent resistivities of Bi2Te2Se.997crystals grown by the Bridgman-Stockbarger method. (b) Temperature-dependent Hall coefﬁcients. Different parts of the crystal boule are designated alphabetically, with Gthe ﬁrst-to-freeze section. Inset: blowup of the Hall coefﬁcient for samples A, B, F,and G. becomes large and negative −1×1019cm3. Although the sample shows metallic or semiconducting behavior at differentpositions along the boule, the properties are homogeneous onthe scale of about 2 cm. The Bridgman-Stockbarger crystalboules made with the starting compositions Bi 2Te2Se1.00and Bi2Te2Se.99do not show semiconducting behavior (data not shown). This indicates that the Bridgman-Stockbarger methodemployed to grow this material induces less than a 1% differ-ence in chemical composition along the boule. We have shownhere that the optimal defect densities can be obtained eitheraccidentally in the crystals obtained by modiﬁed Bridgmangrowth or systematically at the n-to-pcrossover region in crystals obtained by the Bridgman-Stockbarger crystal-growthmethod. The detailed transport properties of the surface statesfor the semiconducting crystals obtained here are reportedelsewhere. 23,24 The carrier concentrations ( n) and mobilities ( μ) at 10 and 300 K for the seven samples at 10 K are plotted in Fig. 5.T h e mobility was calculated by assuming single majority carriersdominating at 10 and 300 K. The metallic samples (A, B,F, and G) show similar nandμvalues at 10 and 300 K, which is expected in heavily doped semiconductors. Thesemiconducting samples (C, D, and E) show μless than 100 cm 2/V s at 10 K, which is similar to previously reported bulk mobility values for BTS.17,24At 300 K, the mobility FIG. 5. (Color online) The carrier density (a) and mobility (b) at 10 (circles) and 300 K (squares) of the Bi 2Te2Se.995boule from the Bridgman-Stockbarger method with respect to the position along the boule. Samples C, D, and E are ptype at 300 K (open squares). of the hole pocket is much less ( <10 cm2/V s). Such small mobility may be due to the large effective hole mass. Thecorrelation between the mobility and carrier concentration forthe semiconducting samples requires further investigation. IV . DISCUSSION AND CONCLUSION Based on our synthesis and characterization of a large number of BTS crystal samples, we can come to some conclu-sions about the relation between the chemical compositionand the carrier distribution. Stoichiometric BTS is alwaysa heavy-doped n-type material showing metallic behavior, while slight tuning of the Te/Se ratio by changing the nominalcomposition at the percent level ( /lessorequalslant15%) does not lead to semiconducting samples. In addition, starting with an excessamount of Se in the BTS growths, in an attempt to suppress theformation of Se vacancies, drives the materials more metallic.A possible explanation for this is that the starting Bi 2Te2Se1+δ will result in a mixture of Bi 2(Te 2−δSeδ)Se with excess Te, because the more electronegative Se atoms tend to bond withBi atoms more strongly than they do with Te atoms. Reducing 235206-5SHUANG JIA et al. PHYSICAL REVIEW B 84, 235206 (2011) the Se starting concentration on the other hand yields higher resistance samples, which are highly inhomogeneous whenobtained by the modiﬁed Bridgman growth method. Thisbehavior indicates that reducing the Se starting concentrationin BTS probably does not introduce the Se vacancies knownto contribute n-type carriers in Bi 2Se3. We postulate instead that many of the Se vacancies that would have been the resultof Se deﬁciency will be ﬁlled by Te atoms, while the excess Biatoms that are present as a consequence will occupy the sitesleft vacant by the displaced Te, leading a p-type doping. We propose a more complex defect equilibrium for BTS: Bi 2Te2Se/arrowrighttophalf/arrowleftbothalfBi/prime Te+h•+(1−x)Te× Se+xV•• Se+2xe/prime +1 2Se2(g)+BiBi+TeTe+x 2Te2(g).(3) Thisn-type carrier compensation mechanism is highly sensi- tive to the stating material ratio and the sample processing,leading to the high electronic inhomogeneity in the crystalsgrown by the modiﬁed Bridgman method. Further study willbe required to determine whether our proposed defect modelfor BTS is correct. With its large temperature gradient at the freezing point, the classical Bridgman-Stockbarger method can introduce acontinuous chemical composition distribution, and in the caseof BTS can therefore allow for the achievement of highlyresistive samples by selecting the appropriate position alongthe boule. The semiconducting samples all manifest a majoritycarrier type changing from ptype to ntype on cooling from 300 to 10 K. A previous study suggested that the acceptors arefrom an impurity band with a gap ∼30 meV from the chemical potential, 17while our recent studies on BTS crystals under pressure reveal a satellite hole pocket ∼50 meV lower than the chemical potential.24Sample D shows an RHmaximum at 200 K, higher in temperature than the current samples C and Eand the samples reported in Refs. 17and24, which indicates that its chemical potential may be closer to the bottom of theconduction band, leading to a larger gap with the hole pocket.It is worth noting that the samples with higher temperaturesfor the R Hmaximum (such as sample D) show a largern-type carrier concentration at 10 K. We believe that the n-type carriers observed at low temperature in our measurements havea minor contribution from the surface metallic state and aremainly from a donor impurity band. This impurity band islikely extended from states due to charged Se vacancies that arehybridized with the conduction band. The chemical potentialbeing closer to the conduction band in the band gap yieldsmore donors at base temperature. Our studies indicate that the BTS samples with small amounts of p-type doping made by the modiﬁed Bridgman or Bridgman-Stockbarger techniques are more likely to besemiconducting than the stoichiometric samples. Given thatthe electrons at the bottom of the conduction band are lighterthan the holes at the top of the valence band, the donors moreeasily form an impurity band near the electron pocket than theacceptors do near the hole pocket. In other words, at equivalentn- andp-type defect concentrations, the Fermi level will be pinned near the isolated p-type energy levels in the band gap, yielding semiconducting samples, whereas the extended n- type impurity band leads Bi 2Te2Se to be metallic for equivalent n-type carrier concentrations. This assumption is consistent with our previous studies on Ca-doped Bi 2Se3.13,14Although the bulk mobility of the semiconducting BTS samples isusually small, ∼100 cm 2/V s, the surface states manifest much higher mobility, leading to the observations of quantumoscillations previously reported. Further optimization of thecrystal growth and doping of BTS is expected to yield excellentcrystals for the study of topological surface-state transport andthe fabrication of surface-state-based electronic devices. ACKNOWLEDGMENTS The authors are thankful for the helpful discussion with N. Ni, M. Bremholm, J. Allred, and K. Baroudi. The crys-tallography work is supported by the NSF MRSEC programthrough Grant No. DMR-0819860. The crystal growth andcharacterization are supported by SPAW AR Grant No. N6601-11-1-4110. Substitutional chemistry work is supported byAFOSR Grant No. FA9550-10-1-0533. 1D. M. Rowe, Handbook of Thermoelectrics , 6th ed. (CRC, Boca Raton, FL, 1995), 2L. Fu and C. L. 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Yazdani, Nature 460, 1106 (2009). 11D.-X. Qu, Y . S. Hor, J. Xiong, R. J. Cava, and N. P. Ong, Science 329, 821 (2010). 12N. P. Butch, K. Kirshenbaum, P. Syers, A. B. Sushkov, G. S. Jenkins, H. D. Drew, and J. Paglione, P h y s .R e v .B 81, 241301 (2010). 13J. G. Checkelsky, Y . S. Hor, M.-H. Liu, D.-X. Qu, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. 103, 246601 (2009). 14Y . S. Hor, A. Richardella, P. Roushan, Y . Xia, J. G. Checkelsky, A. Yazdani, M. Z. Hasan, N. P. Ong, and R. J. Cava, Phys. Rev. B 79, 195208 (2009). 235206-6LOW-CARRIER-CONCENTRATION CRYSTALS OF THE ... PHYSICAL REVIEW B 84, 235206 (2011) 15Seizo Nakajima, J. Phys. Chem. Solids 24, 479 (1963). 16O. Sokolov, S. Skipidarov, N. Duvankov, and G. Shabunina, J. Cryst. Growth 262, 442 (2004). 17Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y . Ando, Phys. Rev. B 82, 241306 (2010). 18J. Xiong, A. C. Petersen, D. Qu, R. J. Cava, and N. P. Ong, e-print arXiv:1101.1315 (to be published). 19J. Rodr ´ıguez-Carvajal and T. Roisnel, FULLPROF ,WINPLOTR , and accompanying programs (2008), [ http://www.ill.eu/sites/ fullprof/index.html ].20Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y . Ando, Phys. Rev. B 84, 165311 (2011). 21W. G. Pfann, in Zone Melting , Wiley Series on the Science and Technology of Materials, 2nd ed. (Wiley, New York, 1966). 22H. Zheng, Q. A. Li, K. E. Gray, and J. F. Mitchell, P h y s .R e v .B 78, 155103 (2008). 23J. Xiong, Y . Luo, Y . Khoo, S. Jia, R. J. Cava, and N. P. Ong, e-printarXiv:1111.6031 . 24Y . Luo, S. Rowley, J. Xiong, S. Jia, R. J. Cava, and N. P. Ong, e-print arXiv:1110.1081 . 235206-7
PhysRevB.94.195118.pdf	PHYSICAL REVIEW B 94, 195118 (2016) Infrared study of the magnetostructural phase transition in correlated CrN J. Ebad-Allah,1,2B. Kugelmann,1F. Rivadulla,3and C. A. Kuntscher1,* 1Experimentalphysik 2, Universit ¨at Augsburg, D-86135 Augsburg, Germany 2Department of Physics, Tanta University, 31527 Tanta, Egypt 3Centro de Investigaci ´on en Qu ´ımica Biol ´oxica e Materiais Moleculares (CIQUS) and Departamento de Qu ´ımica-F ´ısica, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain (Received 26 April 2016; revised manuscript received 23 August 2016; published 9 November 2016) We report on the pressure and temperature dependence of the electronic and vibrational properties of polycrystalline CrN studied by optical transmission and reﬂection measurements over the frequency range0.012–2.48 eV . The optical conductivity spectrum of CrN at ambient conditions shows a phonon mode at≈55 meV with a shoulder at ≈69 meV, a pronounced midinfrared absorption band centered at 123 ±2 meV, and a high-energy absorption band at ≈1.5 eV. The absorption bands are discussed in terms of the charge-transfer insulator picture. Following the reﬂectance spectrum with increasing pressure, the activation of an additionalphonon mode above 0.6 GPa indicates the occurrence of a pressure-induced structural phase transition.Furthermore, the absorption spectrum exhibits signiﬁcant changes in the far-infrared range with decreasingtemperature: The phonon mode shows a sudden broadening followed by a splitting below 270 K. These changesobserved under pressure or while cooling down can be associated with the magnetostructural phase transitionreported previously. DOI: 10.1103/PhysRevB.94.195118 I. INTRODUCTION Transition-metal nitrides provide remarkable properties such as high hardness, mechanical strength, wear and corrosionresistance, high-temperature oxidation resistance, and goodchemical as well as thermal stability [ 1–3]. Furthermore, they have been used in semiconductor devices because of their rel- atively high electrical conductivity [ 1–4]. Among the nitrides, CrN is the only compound which does not have a groundstate of cubic symmetry, making it especially interesting. Atambient conditions, CrN is in a paramagnetic phase witha cubic rock-salt structure similar to other transition-metalnitrides. At the N ´eel temperature T Nwith values in the range 270–286 K reported in the literature [ 5–8], the material undergoes a magnetic phase transition to an ordered antiferro-magnetic phase. The magnetic phase transition is accompaniedby a structural phase transition from the cubic phase to anorthorhombic phase. Recently, it has been reported [ 9] that this magnetostructural phase transition could also be inducedby applying pressure of ≈1 GPa at room temperature. The pressure-induced phase transition at ≈1 GPa was found to be accompanied by a reduction of the bulk modulus of about 25% and a 0.56–0.59% increase in the atomic density [ 9]. The electrical transport studies of CrN revealed discrepant results regarding the temperature dependence of the resistivity. Several transport studies on polycrystalline samples [ 10–12] and thin ﬁlms grown on MgO (001) substrates [ 13] supported the semiconductor scenario in the whole temperature rangebelow 400 K, indicating the existence of a band gap. In con-trast, Constantin et al. observed a change from semiconductor to metallic behavior in the conductivity spectrum at T Non a thin ﬁlm of CrN, while the recent study of Quintela et al. [14] on a thin ﬁlm of CrN reported a metallic-like behavior for the *christine.kuntscher@physik.uni-augsburg.deannealed CrN thin ﬁlm in the temperature range below 400 K,with a steep decrease of the resistivity at T Nassociated with the magnetostructural phase transition. A metallic behavior wasalso observed in other resistivity studies in the temperaturerange below 300 K [ 5,15]. Theoretical works interpreted the observed magnetic, struc- tural, and electronic properties of CrN in terms of chargeordering or Mott-insulating behavior, which is characteristicfor correlated electron systems [ 16]. Theoretical calculations using the local spin-density approximation corrected byHubbard Coulomb terms for the delectrons (LSDA +U) [17] supported the idea that CrN is close to a charge-transfertype insulator. Here, a spin separation of states near theFermi level was reported, leading to opening up a smallcharge gap of less than 1 eV between the N 2 p-type bands and Cr 3 d-type bands. This small energy gap was suggested to be related to the measured optical charge gapof 0.64 eV extracted from the absorbance spectrum [ 13,18]. In contrast, the analysis of the dielectric constant determinedfrom transmission and reﬂection measurements in broadfrequency ranges (0.04–5 eV and 0.1–5 eV , respectively)indicated the presence of a small indirect band gap of 0 .19± 0.46 eV at the Fermi level. This gap was attributed to the formation of local magnetic moments and electron interactioneffects, suggesting that CrN is a Mott-Hubbard-type insulator[19]. Here, we report on the electronic and vibrational prop- erties of polycrystalline CrN investigated by reﬂectance and transmission measurements in a broad frequency range 0.012– 2.48 eV (100–20 000 cm −1). The goal of our optical study was to verify the semiconducting character suggested byelectronic transport measurements, with an energy gap smallerthan 150 meV at ambient conditions [ 13,18–20]. Furthermore, absorption measurements for temperatures between 300 and150 K as well as room-temperature reﬂectance measurementsunder high pressure up to 4.5 GPa were carried out to search for 2469-9950/2016/94(19)/195118(5) 195118-1 ©2016 American Physical SocietyEBAD-ALLAH, KUGELMANN, RIV ADULLA, AND KUNTSCHER PHYSICAL REVIEW B 94, 195118 (2016) signatures of the expected temperature- and pressure-induced magnetostructural transition. II. EXPERIMENT The CrN powder was synthesized by ammonolysis of Cr 3S4 at 800◦C for 10 h; details on the powder preparation are reported in Refs. [ 10,11]. For the temperature-dependent transmittance studies, the CrN powder was mixed with CsI in a certain ratio; herewe used 0.14% of CrN to prepare the CrN/CsI pellet. Themixture was pressed to a sintered pellet of about 50 μm thick. Then pellets of a mixture of CrN/CsI and of pure CsI weremounted on the holder of a cold-ﬁnger cryostat (Cryo VacKonti-Mikro cryostat). CsI was chosen due to its transparent in the infrared frequency range. The actual temperature of the sample was measured by a sensor attached in direct vicinityof the pellets. We measured the intensity of the radiationtransmitted by the CrN pellet, I s(ω); as reference, the intensity Ir(ω) transmitted by a pure CsI pellet was used. The ratio T=Is(ω)/Ir(ω) is a measure of the transmittance of the sample, and the corresponding absorbance was calculatedaccording to A=log 10(1/T). The reﬂectance measurement at ambient conditions was conducted in the frequency range 0.012–2.48 eV (100–20 000cm −1) using a Bruker IFS 66v/S Fourier transform infrared (FT-IR) spectrometer with an infrared microscope (BrukerIR scope II) equipped with a 15 ×magniﬁcation objective. We measured the intensity of the radiation reﬂected from apure CrN pellet, I s(ω); as reference we measured the intensity reﬂected from an Ag mirror, Ir(ω). The ratio R=Is(ω)/Ir(ω) is a measure of the reﬂectance of the sample. The real partof the optical conductivity σ 1was obtained via Kramers Kronig (KK) transformation. Hereby, the reﬂectivity data wereextrapolated to low frequencies based on a Lorentz ﬁt. Forthe high-frequency extrapolation we used the x-ray atomicscattering functions [ 21]. Alternatively, we measured the UV range up to around 5 eV , then used the power law of (1 /ω)t o extrapolate the reﬂectivity data. By this way we estimated theerror bars and found that the high-energy extrapolation affectsthe high-energy features of the obtained optical conductivityonly slightly. Pressure-dependent reﬂectance experiments were carried out at room temperature in the far-infrared frequency range.For the generation of pressures up to 4.5 GPa, a clamp diamond anvil cell (Diacell cryoDAC-Mega) equipped with type IIA diamonds, which are suitable for infrared measurements, wasused. A small piece (lateral dimensions of about 160 μm× 150μm, with a thickness of ∼50μm) was cut from a pure CrN pellet and placed in the hole of a CuBe gasket.Finely ground CsI powder served as quasi-hydrostatic pressuretransmitting medium. The pressure in the diamond anvil cellwas determined in situ by the ruby luminescence method [ 22]. As reference, we used the intensity reﬂected from the CuBegasket inside the DAC. All the pressure-dependent reﬂectancespectra refer to the absolute reﬂectance at the sample-diamondinterface, denoted as R s−d. Further information about the geometry of the reﬂectivity measurements can be found inour earlier publications [ 23,24].FIG. 1. (a) Reﬂectance of a pure CrN pellet over a broad frequency range at ambient conditions, used for the KK analysis to extract the optical conductivity. (b) Lorentz model ﬁt of the corresponding real part σ1of the optical conductivity, obtained by KK analysis, at ambient conditions. III. RESULTS AND DISCUSSION A. Optical spectrum at ambient conditions The ambient-condition reﬂectance spectrum of the pure CrN pellet over a broad frequency range and the correspondingreal part σ 1of the optical conductivity are depicted in Fig. 1. The reﬂectance spectrum consists of several contributions: abroad peak with a shoulder in the far-infrared range followedby a pronounced midinfrared (MIR) absorption band and ahigh-energy absorption band. For a quantitative analysis of theobserved features, we ﬁtted the optical conductivity σ 1with the Lorentz model. The far-infrared peak and its shoulder weredescribed by two Lorentzian functions. Furthermore, both theMIR band and the high-energy band were described by twoLorentzian functions each. Figure 1(b) shows the ﬁt of the σ 1 spectrum with the Lorentz model. The ﬁnite conductivity in the far-infrared range does not give a clear indication of a Drude-type contribution, supporting a semiconducting behavior ofCrN above T N[10–13,25]. From the ﬁtting we obtained the frequency position of the far-infrared peak at ≈55 meV. This peak is related to the transverse-optical (TO) mode predicted at 48 .7± 0.2m e V[ 19]. The shoulder observed at around 69 meV could 195118-2INFRARED STUDY OF THE MAGNETOSTRUCTURAL PHASE . . . PHYSICAL REVIEW B 94, 195118 (2016) be attributed to the predicted longitudinal optical mode located at 75.0±6.8m e V[ 19]. Our results are slightly different from the previously published reﬂectance data, which show a broadphonon mode at around 52 meV [ 19]. The small difference in the position of the phonon mode could be attributed to thethickness, the composition, and the strain of the thin ﬁlms usedin the previous measurements, which affect the phonon modeposition [ 20,26]. The MIR band located at 123 ±2 meV can be attributed to a charge gap of size /Delta1≈100–150 meV, consistent with the activated behavior observed by several resistivity studieswhich found activation energy E avalues in the range 50– 75 meV [ 10–13,25]. Furthermore, we observe a high-energy band located at around 1.5 eV , in good agreement with theelectronic interband transitions occurring at around 1.5 eV asobserved by Zhang et al. [19]. In order to attribute the observed absorption bands to speciﬁc optical excitations, we will refer to the band-structurecalculations of Herwadkar et al. using the LSDA +U approx- imation [ 17]. Here, for the cubic structure a strong depletion in the density of states by the spin separation of states near theFermi level was predicted. A small gap between the N 2 p-type bands and Cr 3 d-type bands was found, suggesting that CrN is likely to be a charge-transfer-type insulator [ 16]. The size of the smallest energy gap was predicted to depend on the valueof the onsite Coulomb repulsion Uand the type of the mag- netic conﬁgurations of the cubic phase: either ferromagnetic(FM) phase or one of the distorted antiferromagnetic (AFM)phases [110] 1and [110] 2. As an example, for U=1e Vt h e band structure of the AFM-[110] 2phase showed the removal of some bands at the /Gamma1point from the region near EF, leading to the opening of an indirect gap smaller than 0.1eV . When increasing Uto 3 eV , both of the AFM-[110] 2and FM phases’ band structures showed the opening of a smalldirect gap slightly less than 1 eV near the /Gamma1point, while for the AFM-[110] 1phase the gap was found to be about 0.4 eV . This small gap was suggested to be opened between the N2p-type valence band and the majority-spin e gbands of Cr, and hence CrN was interpreted in terms of a charge-transfertype insulator, even for small values of U. According to the calculations of Herwadkar et al. , the MIR band observed in our optical data at around 125 meV can be explained in terms of acharge transfer /Delta1f r o mt h eN2 p-type bands to the Cr 3 d-type bands. This would support the transport data, which proposed that CrN is located in the charge transfer insulator regime,close to the itinerant electron limit [ 10,11]. Furthermore, the band structure calculations [ 17] showed another energy gap at around 1.8 eV , opened between the minority spin N 2 p-type valence band and the minority-spin t 2gband at the Xpoint. This could explain the high-energy band at around 1.5 eV observedin our data and the data of Zhang et al. Accordingly, we suggest that both the MIR band and the high-energy band could beexplained in terms of a charge-transfer insulator rather than aMott insulator. A charge gap of similar size was also observedfor other 3 dtransition-metal compounds, like Ni 1−δS, BaTiO 3 and the family of the rare-earth ( R) transition metal oxides (RMO3, withM=Sc, Ti, V , Cr, Mn, Fe, Co, Ni, and Cu), and interpreted in terms of a charge transfer between 2 p-type and 3d-type states [ 27–29].FIG. 2. Temperature-dependent absorbance of a polycrystalline CrN/CsI pellet at ambient pressure. Inset shows the absorbance spectra at room temperature before cooling down and after warming up. B. Temperature-dependent optical spectra Ambient-pressure absorbance spectra of CrN for several selected temperatures are displayed in Fig. 2in the frequency range 0.025–0.868 eV (200–7000 cm−1). The wiggles in the spectra are due to Fabry-Perot interferences due to multiplereﬂections within the sample. As the temperature is lowered,the absorption spectra do not show signiﬁcant spectral changesbelow 0.062 eV (500 cm −1), consistent with the electrical transport measurements which found an activated behaviorfor CrN for the temperature range below 400 K [ 10,11]. Additionally, the MIR absorption band shows only a slight shiftto lower frequencies during cooling down, indicating that thetemperature effect on the activation energy is very small. Themajor changes were observed in the phonon mode range, i.e.,between 0.068 eV (550 cm −1) and 0.083 eV (670 cm−1). With decreasing temperature, the phonon mode shows a suddenbroadening below 270 K, followed by a splitting duringfurther cooling down, as depicted in Fig. 3(a). The changes in the phonon mode with temperature are illustrated by thenormalized difference /Delta1A, deﬁned as /Delta1A(ω,T)=[A(ω,T)−A(ω,300 K)] /A(ω,300 K) .(1) The temperature-dependent absorbance spectra and the corresponding normalized difference spectra /Delta1A(ω,T)a r e shown in Fig. 3in the upper and lower panels, respectively. With decreasing temperature, /Delta1A(ω,T) has a slope of around zero for T> 280 K, while below 280 K the slope starts to change in the frequency range 0.068–0.083 eV (550–670 cm −1). In particular, Fig. 3(c)shows that /Delta1A(ω,T) clearly increases below 270 K during cooling, indicating the splittingof the phonon mode. The broadening and splitting of the phonon mode at T Nare consistent with a symmetry lowering of the crystal structure 195118-3EBAD-ALLAH, KUGELMANN, RIV ADULLA, AND KUNTSCHER PHYSICAL REVIEW B 94, 195118 (2016) 0.50.60.7560 580 600 620 640 660 560 580 600 620 640 660 0.060.12Frequency (cm-1) 260 K 255 K 250 K 200 K 150 K 295 K 286 K 280 K 270 K 265 K Absorbancecooling (a) 260 K 250 K 200 K 150 K(b) 295 K 286 K 275 K 270 K 265 K warmingFrequency (cm-1) (c) cooling (d) warming 560 580 600 620 640 660-0.12-0.060.00 560 580 600 620 640 660 255 K 250 K 200 K 150 K 286 K 280 K 270 K 265 K 260 KΔ A Frequency (cm-1) 260 K 250 K 200 K 150 K 286 K 275 K 270 K 265 K Frequency (cm-1) FIG. 3. The two panels (a) and (b) depict the absorbance of a CrN/CsI pellet as a function of temperature. The two panels (c) and (d) show the normalized difference /Delta1A(ω,T) in the far infrared range, illustrating the evolution of the phonon mode as a function oftemperature above and below the phase transition temperature. Block arrows illustrate the changes during cooling and warming. from a cubic rock-salt structure with space group ( Fm3m)t o an orthorhombic phase with space group ( Pnma )[6,7,30]. These structural changes are accompanied by a magneticphase transition from a paramagnetic phase to an orderedantiferromagnetic phase at T Nas shown by several studies [5–8]. During warming up, /Delta1A(ω,T) is basically ﬂat above 275 K in the frequency range 0.068–0.083 eV (550–670 cm−1) [see Fig. 3(d)], indicating the absence of any broadening or splitting of the phonon mode. The width of the so-obtainedthermal hysteresis amounts to ≈5 K. The thermal hysteresis was observed in earlier studies, but with different values of thewidth depending on the N concentration [ 10–13,25]. Browne et al. [5] reported a hysteresis with a width of ≈3 K, which is close to our result, while Constantine et al. [25] found a higher value of ≈20 K. The thermal hysteresis indicates that the phase transition is of ﬁrst order, as proposed earlier [ 5,25]. Furthermore, we note that the observed temperature-inducedchanges are reversible (see the inset of Fig. 2). C. Pressure-dependent optical spectra The room-temperature reﬂectance spectra of CrN for selected pressures between 0.2 and 4.5 GPa in the frequencyrange 0.02–0.086 eV (180–700 cm −1) are depicted in Fig. 4. At the lowest pressure ( ≈0.2 GPa) a broad phonon mode and a shoulder are observed, consistent with our data at ambientconditions discussed above. The overall reﬂectance increasesmonotonically with increasing pressure, which might indicatea growth of spectral weight within the measured frequencyrange. However, we cannot exclude that the overall increaseis due to pressure-induced changes in the surface ﬂatness,since the CrN pellet is pressed on the diamond anvil. This200 400 6000.00.20.40.6 0.00.20.40.6200 400 600 0.2 GPa Fit 0.6 0.9 1.0 1.8 2.5 4.4 Rs-d Frequency (cm-1) 0.6 GPa 0.7-releasing FIG. 4. Pressure-dependent reﬂectance of a pure CrN pellet at room temperature in the low-frequency range for several selectedpressures between 0.2 and 4.4 GPa. The dashed red line is the ﬁt of the reﬂectance spectrum at 0.2 GPa using the Lorentz model. The two dashed black lines illustrate the splitting of the phonon mode. might also explain the rather high overall reﬂectance after pressure release at 0.7 GPa (see inset of Fig. 4). Nevertheless, the nonreversibility of the pressure-induced effects cannot beexcluded as a possible reason for the discrepancy between thespectra at around 0.6 GPa during pressure increase and afterpressure release. The pressure-dependent phonon spectrum was analyzed by ﬁtting with the Lorentz model. Hereby, the reﬂectance spectrawere ﬁtted according to the Fresnel equation for normal-incidence reﬂectivity taking into account the diamond-sampleinterface R s−d. Further information about Fresnel equation forRs−dis included in an earlier publication [ 31]. The phonon modes are described by two Lorentz functions. As an example,we show in Fig. 4the ﬁt of the reﬂectance spectrum at 0.2 GPa. Above 0.6 GPa an activation of a phonon mode at ≈50 meV occurs, which is directly seen in the reﬂectance spectra (seethe dashed lines in Fig. 4). Therefore, an additional Lorentz term had to be included in the ﬁt. Based on the Lorentz ﬁtting, the phonon frequencies were extracted and are plotted in Fig. 5as a function of pressure. With increasing pressure, the phonon mode at≈55 meV slightly shifts to higher energy, while the shoulder at ≈69 meV broadens. Furthermore, the activation of a phonon mode located at around 50 meV indicates a pressure-inducedsymmetry lowering of the crystal structure above 0.6 GPa. Thisﬁnding supports the structural phase transition observed byRivadulla et al. [9], from the cubic rock salt to an orthorhombic crystal structure at ≈1 GPa. Concomitant with the structural phase transition, a large reduction of the bulk modulus K 0from 340(10) GPa for the cubic rock-salt phase to 243(10) GPa forthe orthorhombic phase was reported; i.e., CrN softens underpressure. IV . SUMMARY In summary, the electronic and vibrational properties of polycrystalline CrN have been investigated by transmission 195118-4INFRARED STUDY OF THE MAGNETOSTRUCTURAL PHASE . . . PHYSICAL REVIEW B 94, 195118 (2016) FIG. 5. Phonon frequencies as a function of pressure, obtained by ﬁtting the reﬂectance spectra with the Lorentz model. and reﬂection measurements in the infrared frequency range for temperatures between 300 and 150 K, and under highpressure up to 4.5 GPa. At ambient conditions, the optical conductivity spectrum shows an optical phonon mode at≈55 meV with a shoulder at ≈69 meV, a MIR absorption band at 123 ±2 meV, and a high-energy absorption band at≈1.5 eV. Both absorption bands can be related to charge transfer excitations, suggesting that CrN is a charge-transfer-type insulator. While cooling down, the phonon mode exhibitsa sudden broadening and a splitting below T N, which con- ﬁrms the occurrence of the expected magnetostructural phasetransition. This magnetostructural phase transition seems tooccur also under pressure according to our pressure-dependentreﬂectance measurements, revealing the activation of a phononmode at 0.6 GPa, in agreement with earlier studies. 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PhysRevB.95.224518.pdf	PHYSICAL REVIEW B 95, 224518 (2017) Pair symmetry conversion in driven multiband superconductors Christopher Triola1,2and Alexander V . Balatsky1,2,3,4 1Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden 2Center for Quantum Materials (CQM), KTH and Nordita, Stockholm, Sweden 3Institute for Materials Science (IMS), Los Alamos National Laboratory, Los Alamos New Mexico 87545, USA 4ETH Institute for Theoretical Studies (ETH-ITS), ETH Zurich, 8092 Zurich, Switzerland (Received 18 April 2017; published 29 June 2017) It was recently shown that odd-frequency superconducting pair amplitudes can be induced in conventional superconductors subjected to a spatially nonuniform time-dependent drive. It has also been shown that, in thepresence of interband scattering, multiband superconductors will possess bulk odd-frequency superconductingpair amplitudes. In this work we build on these previous results to demonstrate that by subjecting a multibandsuperconductor with interband scattering to a time-dependent drive, even-frequency pair amplitudes canbe converted to odd-frequency pair amplitudes and vice versa. We will discuss the physical conditionsunder which these pair symmetry conversions can be achieved and possible experimental signatures of theirpresence. DOI: 10.1103/PhysRevB.95.224518 I. INTRODUCTION Due to the fermionic nature of electrons, the spatial sym- metry ( s-wave, p-wave, d-wave, etc.) of a superconducting gap is intimately related to the spin state (singlet or triplet)of the Cooper pairs making up the condensate. In the limit ofequal-time pairing this relationship is quite simple, even-paritygaps (such as s-wave, or d-wave) correspond to spin singlet states while odd-parity gaps (such as p-wave or f-wave) correspond to spin triplet states. However, if the electrons arepaired at unequal times the superconducting gap could be oddin time or, equivalently, odd in frequency (odd- ω), in which case the condensate could be even in spatial parity and spintriplet or odd in spatial parity and spin singlet. This possibility,originally posited for 3He by Berezinskii [ 1] and then later for superconductivity [ 2], is intriguing both because of the unconventional symmetries that it permits and for the fact thatit represents a class of hidden order, due to the vanishing ofequal time correlations. While some research has been dedicated to the ther- modynamic stability of intrinsically odd- ωphases [ 3–6], a great deal of previous research has been devoted to theidentiﬁcation of heterostructures in which odd- ωpairing could be induced including: ferromagnetic-superconductorheterostructures [ 7–13], topological insulator-superconductor systems [ 14–17], normal metal-superconductor junctions due to broken translation symmetry [ 18–22], two-dimensional bilayers coupled to conventional s-wave superconductors [ 23], and in generic two-dimensional electron gases coupled tosuperconductor thin ﬁlms [ 24]. In addition to theoretical studies, there are experimental indications of the realization of odd-ωpairing at the interface of Nb thin ﬁlms and epitaxial Ho [25]. Furthermore, the concept of odd- ωorder parameters can be generalized to charge and spin density waves [ 26,27] and Majorana fermion pairs [ 28], demonstrating the pervasiveness of the odd- ωclass of ordered states. Additionally, it has been shown that superconductors with multiple bands close to the Fermi level, such as MgB 2 [29–33] and iron-based superconductors [ 34–38], will possess odd-ωpairing in the presence of interband hybridization[39–42]. An advantage of studying odd- ωpairing in multiband superconductors is that these systems do not have to beengineered to generate odd- ωpair amplitudes since interband scattering can arise from disorder or it can be intrinsic tothe system if the Cooper pairs are composed of electronscorresponding to particular orbitals while the quasiparticles ofthe system emerge from a linear combination of these orbitals[40], as is the case in Sr 2RuO 4[42]. Thus, it is expected that bulk odd- ωpairing should be ubiquitous in multiband superconductors. Motivated by the intrinsically dynamical nature of odd- ωcondensates, we recently demonstrated the possibility of inducing odd- ωsuperconducting pair amplitudes in a conventional s-wave superconductor in the presence of a spatially nonuniform and time-dependent external electricﬁeld [ 43]. The purpose of our current work is both to extend this result to the case of driven multiband superconductors andto examine the nature of pair symmetry conversion in thesesystems, establishing a relationship between the symmetry ofthe dynamically generated pair amplitudes and the symmetry of the pair amplitudes in the absence of a drive. Speciﬁcally, we consider a superconductor with two bands close to the Fermilevel, each possessing a conventional intraband s-wave gap, with a ﬁnite interband hybridization so that both even- ωand odd-ωpair amplitudes are present. Then, using perturbation theory, we show that, in the presence of a time-dependentdrive, novel odd- ωpair amplitudes are generated from even- ω amplitudes and novel even- ωamplitudes are generated from odd-ωamplitudes. We also demonstrate that the conditions for this dynamical pair symmetry conversion coincide withthe conditions for the emergence of certain peak structures inthe quasiparticle density of states (DOS). It should be noted that, while a great deal of work has been dedicated to inducing odd- ωpairing in systems with only even- ωpairing, our study examines the inverse effect: inducing even- ωpairing from previously-existing odd- ωpairing. This novel effect offers an additional means to modify the pairing states of existing systems. Furthermore,given that even- ωstates are typically associated with sharp 2469-9950/2017/95(22)/224518(13) 224518-1 ©2017 American Physical SocietyCHRISTOPHER TRIOLA AND ALEXANDER V . BALATSKY PHYSICAL REVIEW B 95, 224518 (2017) spectral features, this effect could point toward new direc- tions for measuring and quantifying odd- ωsuperconducting states. The remainder of this paper is organized as follows. In Sec. II, we establish the model we will use to describe a conventional s-wave singlet superconductor with two bands close to the Fermi level, and review the conditions under whichinterband scattering can lead to odd- ωpairing. In Sec. III, we derive the corrections to the Green’s functions to leadingorder in the drive amplitude, present the conditions for theconversion of even- ωpair amplitudes to odd- ωpair amplitudes and vice versa, and discuss possible signatures in the DOS. InSec. IV, we account for self-consistent corrections to the gap, demonstrating the robustness of the effect. In Sec. V,w eo f f e r concluding remarks. II. MODEL The physical system we wish to consider is a super- conductor in which multiple quasiparticle bands are closeto the Fermi level, as is the case in MgB 2[29–33] and iron-based superconductors [ 34–38]. We assume that the superconductor has an s-wave spin singlet order parameter, /Delta1αβ, with band indices allowing for pairing in both the interband and intraband channels. Additionally, as in previousstudies [ 39–42], we account for a phenomenological interband scattering, which could be caused by disorder or by a mismatchbetween the orbital structure of the quasiparticle bands andthe superconducting order parameter. In this work we willconsider both the case of a two-dimensional (2D) thin ﬁlmsuperconductor and a three-dimensional (3D) superconductor.For concreteness, unless otherwise speciﬁed, all numericalwork will be performed assuming two quasiparticle bands andmodel parameters associated with the two-band superconduc-tor MgB 2. Starting from this system, we will examine the effect of an applied time-dependent drive, which, for concreteness,we assume to be an ac electric ﬁeld, which could be realizedthrough gating (in the case of a 2D superconductor) or usingan RF source. To describe this system we employ the model Hamiltonian: H=H sc+Ht+Hbath+Hmix, (1) where Hscdescribes the undriven multiband superconductor, Htis the time-dependent drive, Hbathdescribes a Fermionic bath held at inverse temperature β, which allows for a phenomenological treatment of dissipation, and Hmixdescribes the coupling between the superconductor and the bath.We will proceed using a two-band superconductor allowing for both interband and intraband pairing: Hsc=/summationdisplay k,σ(ξa,kψ† σ,a,kψσ,a,k+ξb,kψ† σ,b,kψσ,b,k) +/summationdisplay α,β,k/Delta1αβψ† ↑,α,−kψ† ↓,β,k+H.c. +/summationdisplay k,σ/Gamma1ψ† σ,a,kψσ,b,k+H.c., (2) where ξα,k=k2 2mα−μαis the quasiparticle dispersion in bandαwith effective mass mαmeasured from the chem- ical potential μα,ψ† σ,α,k(ψσ,α,k) creates (annihilates) a quasiparticle with spin σin band αwith momentum k, /Delta1αβ≡λ/integraltextddk (2π)d/angbracketleftψ↑,α,−kψ↓,β,k/angbracketrightis the superconducting gap, where dis the dimensionality of the system, and we allow for the possibility of interband scattering with amplitude /Gamma1. With these conventions we write the time-dependent drive as: Ht=/summationdisplay k,σ,α,βUαβ(t)ψ† σ,α,kψσ,β,k. (3) The bath and mixing terms take the form: Hbath=/summationdisplay n,σ,α, k(/epsilon1n−μbath)c† n;σαkcn;σαk Hmix=/summationdisplay k,n,σ,αηnc† n;σαkψσ,α,k+H.c., (4) where /epsilon1ndescribes the energy levels of the Fermionic bath, μbathis the chemical potential of the bath, c† n;σαk(cn;σαk) creates (annihilates) a Fermionic mode with degrees offreedom indexed by n,σ,α, and k, and η nspeciﬁes the amplitude of the coupling between the superconductor andthe bath. From this Hamiltonian we can derive a Dyson equation for the Keldysh Green’s functions describing this system (seeAppendix Afor details): ˆG(k;t 1,t2)=ˆG0(k;t1−t2)+/integraldisplay∞ −∞dtˆG0(k;t1−t) ×/parenleftbiggˆU(t)0 0−ˆU(t)∗/parenrightbigg ⊗ˆρ0ˆG(k;t,t2),(5) where ˆ ρ0is the 2 ×2 identity in Keldysh space, and ˆG0(k;t1− t2) is the Green’s function describing the undriven system written in the Keldysh basis: ˆG0(k;t1−t2)=/parenleftBiggˆGR 0(k;t1−t2)ˆGK 0(k;t1−t2) 0 ˆGA 0(k;t1−t2)/parenrightBigg , (6) where ˆGR 0(k;t1−t2),ˆGA 0(k;t1−t2), and ˆGK 0(k;t1−t2) are the retarded, advanced, and Keldysh Green’s functions, respectively. After integrating out the bath (see Appendix B), the Fourier transform of ˆGR 0(k;t1−t2) is given by: ⎛ ⎜⎝ω+iη−ξa,k −/Gamma1 −/Delta1aa −/Delta1ab −/Gamma1ω +iη−ξb,k −/Delta1ba −/Delta1bb −/Delta1∗ aa −/Delta1∗ ba ω+iη+ξa,k /Gamma1 −/Delta1∗ ab −/Delta1∗ bb /Gamma1ω +iη+ξb,k⎞ ⎟⎠ˆGR 0(k;ω)=1, (7) 224518-2PAIR SYMMETRY CONVERSION IN DRIVEN MULTIBAND . . . PHYSICAL REVIEW B 95, 224518 (2017) where ηis a constant related to the DOS of the bath, but, for our purposes, will be treated as a phenomenological parameter describing quasiparticle dissipation. In equilibrium, the advanced and Keldysh Green’s functions may be obtained from ˆGR 0(k;ω) by: ˆGA 0(k;ω)=ˆGR 0(k;ω)† ˆGK 0(k;ω)=tanh/parenleftbiggβω 2/parenrightbigg/bracketleftbigˆGR 0(k;ω)−ˆGA 0(k;ω)/bracketrightbig , (8) where βis the inverse temperature of the bath. A. Odd-frequency pairing from interband scattering The emergence of odd- ωpairing in multiband superconductors due to interband scattering has previously been studied [ 39,40]. One way to see the emergence of these odd- ωterms is to consider the simple case in which /Delta1ab=/Delta1ba=0 and solve for ˆGR 0(k;ω) using Eq ( 7) which, in the limit of η→0, is given by: ˆGR 0(k;ω)=/parenleftBiggˆGR 0(k;ω)ˆFR 0(k;ω) ˆFR 0(k;ω)ˆGR 0(k;ω)/parenrightBigg , (9) where ˆGR 0(k;ω)=g(k,ω)/parenleftBigg (ω+ξa,k)/parenleftbig ω2−E2 b,k/parenrightbig −/Gamma12(ω−ξb,k)/Gamma1[(ω+ξa,k)(ω+ξb,k)−/Gamma12−/Delta1aa/Delta1bb] /Gamma1[(ω+ξa,k)(ω+ξb,k)−/Gamma12−/Delta1aa/Delta1bb](ω+ξb,k)/parenleftbig ω2−E2 a,k/parenrightbig −/Gamma12(ω−ξa,k)/parenrightBigg , ˆFR 0(k;ω)=g(k,ω)/parenleftBigg /Delta1aa/parenleftbig ω2−E2 b,k/parenrightbig −/Delta1bb/Gamma12/Gamma1[−ω(/Delta1aa−/Delta1bb)+ξa,k/Delta1bb+ξb,k/Delta1aa] /Gamma1[ω(/Delta1aa−/Delta1bb)+ξa,k/Delta1bb+ξb,k/Delta1aa] /Delta1bb/parenleftbig ω2−E2 a,k/parenrightbig −/Delta1aa/Gamma12/parenrightBigg , (10) where we have deﬁned g(k,ω)=1 [ω2−/epsilon1+(k)2][ω2−/epsilon1−(k)2], Eα,k=/radicalBig ξ2 α,k+/Delta12αα, /epsilon1±(k)=/radicaltp/radicalvertex/radicalvertex/radicalbtE2 a,k+E2 b,k 2+/Gamma12±/radicalBigg/parenleftbiggE2 a,k−E2 b,k 2/parenrightbigg2 +/Gamma12(ξa,k+ξb,k)2+/Gamma12(/Delta1aa−/Delta1bb)2. (11) From these expressions one can ﬁndˆGR 0(k;ω) andˆFR 0(k;ω) using the deﬁnitions: GR 0(1; 2)=−iθ(t1−t2)/angbracketleft{ψα1,r1(t1),ψ† α2,r2(t2)}/angbracketright GR 0(1; 2)=−iθ(t1−t2)/angbracketleft{ψ† α1,r1(t1),ψα2,r2(t2)}/angbracketright FR 0(1; 2)=−iθ(t1−t2)/angbracketleft{ψα1,r1(t1),ψα2,r2(t2)}/angbracketright FR 0(1; 2)=−iθ(t1−t2)/angbracketleft{ψ† α1,r1(t1),ψ† α2,r2(t2)}/angbracketright (12) with which one can showˆGR 0(k;ω)=− ˆGR 0(−k;−ω)∗and ˆFR 0(k;ω)=− ˆFR 0(−k;−ω)∗. Notice, that, because g(k,ω)= g(k,−ω), in Eq. ( 10) the interband scattering ( /Gamma1/negationslash=0) has induced a ﬁnite odd- ωinterband pairing in ˆFR 0(k;ω), as shown previously [ 39]. We will now use these expressions to demonstrate that the presence of a time-dependent drive will not only induce similarodd-ωterms but also generate additional even- ωterms as a direct consequence of the odd- ωterms in Eq. ( 10).III. PERTURBATIVE ANALYSIS Iterating the Dyson equation in Keldysh space, Eq. ( 5), one can obtain the components of the Green’s function to linearorder in the drive: ˆG(k;t 1,t2)=ˆG0(k;t1−t2)+/integraldisplay∞ −∞dtˆG0(k;t1−t) ×/parenleftbiggˆU(t)0 0−ˆU(t)∗/parenrightbigg ⊗ˆρ0ˆG0(k;t−t2).(13) Fourier transforming with respect to the relative ( t1−t2) and average [( t1+t2)/2] times we can obtain the linear order corrections in frequency space: ˆG(k;ω,/Omega1)=2πδ(/Omega1)ˆG0(k;ω)+ˆG0/parenleftbig k;ω+/Omega1 2/parenrightbig ×/parenleftbiggˆU(/Omega1)0 0 −ˆU(−/Omega1)∗/parenrightbigg ⊗ˆρ0ˆG0/parenleftbig k;ω−/Omega1 2/parenrightbig . (14) 224518-3CHRISTOPHER TRIOLA AND ALEXANDER V . BALATSKY PHYSICAL REVIEW B 95, 224518 (2017) Focusing on the anomalous part of the Green’s functions, we ﬁnd the terms to linear order in the drive are given by: δˆFR(k;ω,/Omega1)=ˆGR 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU(/Omega1)ˆFR 0/parenleftbig k;ω−/Omega1 2/parenrightbig −ˆFR 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU∗(−/Omega1)ˆGR 0/parenleftbig k;ω−/Omega1 2/parenrightbig δˆFA(k;ω,/Omega1)=ˆGA 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU(/Omega1)ˆFA 0/parenleftbig k;ω−/Omega1 2/parenrightbig −ˆFA 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU∗(−/Omega1)ˆGA 0/parenleftbig k;ω−/Omega1 2/parenrightbig δˆFK(k;ω,/Omega1)=ˆGR 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU(/Omega1)ˆFK 0/parenleftbig k;ω−/Omega1 2/parenrightbig −ˆFR 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU∗(−/Omega1)ˆGK 0/parenleftbig k;ω−/Omega1 2/parenrightbig +ˆGK 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU(/Omega1)ˆFA 0/parenleftbig k;ω−/Omega1 2/parenrightbig −ˆFK 0/parenleftbig k;ω+/Omega1 2/parenrightbigˆU∗(−/Omega1)ˆGA 0/parenleftbig k;ω−/Omega1 2/parenrightbig . (15) To demonstrate the emergence of the even- ωand odd- ω terms we will focus on the retarded components of theanomalous Green’s functions in Eq. ( 15). In general, these corrections, δˆF R(k;ω,/Omega1), could possess terms that are eveninωand terms that are odd in ω. To separate these two possibilities we deﬁne: δˆFe/o(k;ω,/Omega1)=δˆFR(k;ω,/Omega1)±δˆFR(k;−ω,/Omega1) 2.(16) By inserting the expressions for the undriven Green’s func- tions into Eq. ( 15) and evaluating Eq. ( 16), one can study the conditions under which new even- ωpair amplitudes, δˆFe(k;ω,/Omega1), and new odd- ωpair amplitudes, δˆFo(k;ω,/Omega1), will be generated in the presence of the drive. The generalexpressions are quite complicated, therefore we will beginour analysis by studying the simple case in which no odd- ω amplitudes are present in the undriven system. A. Odd-frequency in driven multiband superconductor for /Gamma1=0 In the absence of interband scattering, /Gamma1=0, the anoma- lous Green’s function of the undriven superconductor, Eq. ( 10), possesses only even- ωterms. To see under what conditions the application of a drive will induce odd- ωpairing we substitute Eq. ( 10) into Eqs. ( 15) and ( 16) and we ﬁnd that the odd- ω corrections to the anomalous Green’s function are δFo αβ(k;ω,/Omega1)=−ωUαβ(/Omega1)Aαβ(k,ω,/Omega1 )/braceleftbigg (/Delta1α−/Delta1β)/bracketleftbigg/parenleftbigg ω2+/Omega12 4−E2 α,k/parenrightbigg/parenleftbigg ω2+/Omega12 4−E2 β,k/parenrightbigg −ω2/Omega12/bracketrightbigg +/Omega1/parenleftbig E2 α,k−E2 β,k/parenrightbig/parenleftbigg/Omega1 2/parenleftbig /Delta1α+/Delta1β/parenrightbig +ξα,k/Delta1β+ξβ,k/Delta1α/parenrightbigg/bracerightbigg , (17) where Aαβ(k,ω,/Omega1 )=1 /bracketleftbigg/parenleftbigg ω+/Omega1 2/parenrightbigg2 −ξ2 α,k−/Delta12α/bracketrightbigg/bracketleftbigg/parenleftbigg ω−/Omega1 2/parenrightbigg2 −ξ2 α,k−/Delta12α/bracketrightbigg/bracketleftbigg/parenleftbigg ω+/Omega1 2/parenrightbigg2 −ξ2 β,k−/Delta12 β/bracketrightbigg/bracketleftbigg/parenleftbigg ω−/Omega1 2/parenrightbigg2 −ξ2 β,k−/Delta12 β/bracketrightbigg. (18) From Eq. ( 17) we can see that odd- ωpairing will emerge in the limit of a static drive, Uαβ(/Omega1)=Uαβδ(/Omega1), only if ˆUis off- diagonal in the band index, consistent with previous results formultiband superconductors [ 39–41]. However, when ˆUis time dependent an additional term in Eq. ( 17) emerges, proportional to/Omega1. As with the static case, this term is only nonzero if ˆU(/Omega1) is off diagonal in the band index. However, unlike the staticcase, the dynamical contribution can be nonzero even if thetwo gaps are equal so long as the two bands have differentdispersions. This result is a simple example of the phenomenon of dynamical pair symmetry conversion, whereby even- ωpairing amplitudes are converted to odd- ωamplitudes in the presence of a time-dependent drive. We will now investigate themore general case, in which both even- ωand odd- ωpairing amplitudes are already present before the drive is turned on andthe application of a time-dependent drive converts the odd- ω amplitudes to even- ωamplitudes and vice versa.B. Symmetry conversion in driven multiband superconductor for /Gamma1/negationslash=0 When interband scattering is allowed, /Gamma1/negationslash=0, the anoma- lous Green’s function of the multiband superconductor willpossess both odd- ωterms and even- ωterms, even in the absence of a time-dependent drive. To distinguish betweenthese ambient odd- ωand even- ωcomponents it is useful to deﬁne: ˆF (e/o)(k;ω)=ˆFR 0(k;ω)±ˆFR 0(k;−ω) 2. (19) By substituting Eq. ( 19) into Eqs. ( 15) and ( 16) we can show that the even- ωcorrections to the anomalous Green’s function due to the time-dependent drive are given by: δˆFe(k;ω,/Omega1)=δFe→e(k;ω,/Omega1)+δFo→e(k;ω,/Omega1) (20) 224518-4PAIR SYMMETRY CONVERSION IN DRIVEN MULTIBAND . . . PHYSICAL REVIEW B 95, 224518 (2017) and the odd- ωcorrections are given by: δˆFo(k;ω,/Omega1)=δFo→o(k;ω,/Omega1)+δFe→o(k;ω,/Omega1),(21) where we have isolated the corrections, which preserve frequency parity: δFe→e(k;ω,/Omega1) =/bracketleftbigg ˆGR 0/parenleftbigg k;ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(e)/parenleftbigg k;ω−/Omega1 2/parenrightbigg/bracketrightbigg + +/bracketleftbigg ˆGR 0/parenleftbigg k;−ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(e)/parenleftbigg k;ω+/Omega1 2/parenrightbigg/bracketrightbigg +, δFo→o(k;ω,/Omega1) =/bracketleftbigg ˆGR 0/parenleftbigg k;ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(o)/parenleftbigg k;ω−/Omega1 2/parenrightbigg/bracketrightbigg + +/bracketleftbigg ˆGR 0/parenleftbigg k;−ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(o)/parenleftbigg k;ω+/Omega1 2/parenrightbigg/bracketrightbigg +, (22) and the corrections, which reverse frequency parity: δFe→o(k;ω,/Omega1) =/bracketleftbigg ˆGR 0/parenleftbigg k;ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(e)/parenleftbigg k;ω−/Omega1 2/parenrightbigg/bracketrightbigg − −/bracketleftbigg ˆGR 0/parenleftbigg k;−ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(e)/parenleftbigg k;ω+/Omega1 2/parenrightbigg/bracketrightbigg −, δFo→e(k;ω,/Omega1) =/bracketleftbigg ˆGR 0/parenleftbigg k;ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(o)/parenleftbigg k;ω−/Omega1 2/parenrightbigg/bracketrightbigg − −/bracketleftbigg ˆGR 0/parenleftbigg k;−ω+/Omega1 2/parenrightbigg ˆU(/Omega1),ˆF(o)/parenleftbigg k;ω+/Omega1 2/parenrightbigg/bracketrightbigg −, (23) where, for convenience, we have deﬁned the bracket: [ˆg(ω1)ˆu(ω2),ˆf(ω3)]±≡1 2[ˆg(ω1)ˆu(ω2)ˆf(ω3) ±ˆf(ω3)ˆu(−ω2)∗ˆg(ω1)∗].(24) From Eqs. ( 20)–(23) we can see that the presence of a time-dependent drive will, in general, generate additionaleven-ωand odd- ωterms in the anomalous Green’s function of a multiband superconductor. However, these additional termscould have their origin either from modifying existing corre-lations with the same symmetry or from symmetry conversionof terms with the opposite frequency parity, i.e., even- ωterms generating odd- ωterms or vice versa. To demonstrate that, in general, both symmetry-preserving and symmetry-reversingterms will be nonzero we will now evaluate Eqs. ( 22) and (23), explicitly, using Eqs. ( 10). Assume, for simplicity, that the time-dependent drive takes the form: ˆU(ω)=/parenleftbigg U 0(ω)0 0 U0(ω)/parenrightbigg , (25) where U0(ω) is given by: U0(ω)=2πU 0[δ(ω−/Omega10)+δ(ω+/Omega10)], (26)which corresponds to a drive proportional to cos( /Omega10t)i nt h e time domain. To capture the average time-dependence andrelative frequency-dependence we will work with the Wignertransform of the Green’s functions, deﬁned as: ˆG(k;ω,T)=/integraldisplayd/Omega1 2πe−i/Omega1TˆG(k;ω,/Omega1) (27) and plot these expressions. In Fig 1, we plot the Wigner transform, at T=0, of both the even-ωand odd- ωterms of the anomalous Green’s function, ˆFR(k;ω,T), for a driven multiband superconductor described by Eqs. ( 10) and ( 14) where we have chosen /Delta1aa=2m e V , /Delta1bb=7 meV, and /Gamma1=10 meV. We have used an external drive given by Eq. ( 26) with U0=10 meV, and /Omega10=1m e V . In Fig. 1we have also included plots of the Wigner transforms of both the symmetry-preserving corrections, Eq. ( 22), (green, dashed) and the symmetry-reversing corrections, Eq. ( 23), (red, dash-dotted) to examine the origin of the new contribu-tions. In each plot, in order to show the frequency dependenceat the Fermi surface, we have taken the average value of each function evaluated at the two momenta, \|k\|=k (a) F=√2maμ and\|k\|=k(b) F=√2mbμ. We ﬁrst turn our attention to the intraband components of the anomalous Green’s function, Figs. 1(a) and1(b).N o t i c e that while no new odd- ωintraband terms are present there are two new contributions to the even- ωintraband terms, one contribution coming from the ambient even- ωpairs, and another contribution coming from the ambient odd- ωpairs. These two contributions are most pronounced in the Faa channel in which they yield a net suppression at ω=0 and a net enhancement at ω≈±/Delta1aa. Next we consider the interband components of the anoma- lous Green’s function, Fig. 1(c). Notice a clear enhancement of the odd- ωterms coming from both the ambient even- ω and odd- ωpairs. Additionally, we ﬁnd an enhancement of the even- ωinterband amplitudes at ω≈±/Delta1aaandω≈±/Delta1bb coming from the odd- ωpairs, along with a notable suppression atω=0 coming from the even- ωpairs, similar to the case for the even- ωintraband channels. In Fig. 2, we plot the Wigner transform, at T=π/2/Omega10,o f both the even- ωand odd- ωterms of the anomalous Green’s function, ˆFR(k;ω,T), for a driven multiband superconductor using the same parameters as those appearing in Fig. 1. In contrast to the results at T=0, Fig. 1, we see that at T=π/2/Omega10the drive has very little affect on the even- ω terms, but a rather strong affect on the odd- ωterms. In Figs. 2(a) and 2(b), we see that relatively large intraband odd-ωamplitudes have emerged at ω≈±/Delta1aaandω≈±/Delta1bb for the FaaandFbbchannels, respectively. By examining the red (dash-dotted) and green (dashed) curves we determine thatthese novel odd- ωterms have contributions from both the symmetry-preserving terms and symmetry-reversing terms.However, each contribution can be seen to give rise to distinctpeak structures in these channels. Turning our attention toFig. 2(c), the interband anomalous Green’s function, we can see similar enhancements of the odd- ωamplitude at ω≈±/Delta1 aaandω≈±/Delta1bb. Just as with the intraband channels, the novel interband terms possess both symmetry-preservingand symmetry-reversing contributions. 224518-5CHRISTOPHER TRIOLA AND ALEXANDER V . BALATSKY PHYSICAL REVIEW B 95, 224518 (2017) (a) (b) (c) (d) U0(T) T=0Even-ω Odd-ωOdd-ωOdd-ω Even-ωEven-ω FIG. 1. In the left (right) column we plot the even- ω(odd-ω) terms of the real part of the Wigner transform [deﬁned in Eq. ( 27)] of the anomalous part of the Green’s function in Eq. ( 14),/angbracketleftˆFR(ω,T= 0)/angbracketright, in black (solid), where we have taken the average value of ˆFR(k;ω,T=0) at\|k\|=k(a) Fand\|k\|=k(b) F. In each case we have also plotted the parity-preserving terms, Eqs. ( 22), in green (dashed) and the parity-reversing terms, Eqs. ( 23), in red (dash-dotted). (a) the diagonal component for band- a, (b) the diagonal component for band-b, (c) the interband component. (d) The components of the drive from Eq. ( 26) plotted in the time domain over a full period, the green vertical line denotes the time, T=0, at which all plots in this ﬁgure are evaluated. The parameters used to describe the driven multiband superconductor in this case are: effective masses, ma=0.5˚A−2eV andmb=1˚A−2eV; chemical potentials, μa=μb=2e V ; s-wave gaps,/Delta1aa=2 meV, /Delta1bb=7m e V , /Delta1ab=/Delta1ba=0, consistent with MgB2[44]; interband scattering, /Gamma1=10 meV; dissipation described byη=1 meV; and a drive given by Eq. ( 26) with U0=10 meV, and /Omega10=1 meV (242 GHz). To better understand the time dependence of the pairing amplitudes we have compiled a movie showing the same plotsas in Figs. 1and2over a full period of the drive [ 45]. From this movie we observe that, at generic times during the period, (a) (b) (c) (d) U0(T) T=π/2Ω 0Even-ω Even-ω Even-ω Odd-ωOdd-ωOdd-ω FIG. 2. In the left (right) column we plot the even- ω(odd- ω) terms of the real part of the Wigner transform [deﬁned in Eq. ( 27)] of the anomalous part of the Green’s function in Eq. ( 14), /angbracketleftˆFR(ω,T=π/2/Omega10)/angbracketright, in black (solid), where we have taken the average value of ˆFR(k;ω,T=π/2/Omega10)a t\|k\|=k(a) Fand\|k\|=k(b) F. In each case we have also plotted the parity-preserving terms, Eqs. ( 22), in green (dashed) and the parity-reversing terms, Eqs. ( 23), in red (dash-dotted). (a) the diagonal component for band- a, (b) the diagonal component for band b, (c) the interband component. (d) The components of the drive from Eq. ( 26) plotted in the time domain over a full period, the green vertical line denotes the time, T=π/2/Omega10, at which all plots in this ﬁgure are evaluated. The parameters used to describe the driven multiband superconductor in this case are: effective masses, ma=0.5˚A−2eV and mb=1˚A−2eV; chemical potentials, μa=μb=2 eV; s-wave gaps, /Delta1aa=2m e V , /Delta1bb=7m e V , /Delta1ab=/Delta1ba=0, consistent with MgB2[44]; interband scattering, /Gamma1=10 meV; dissipation described by η=1 meV; and a drive given by Eq. ( 26) with U0=10 meV, and /Omega10=1 meV (242 GHz). contributions to the odd- ωand even- ωpair amplitudes are nonzero. Furthermore, we can see that the corrections to theodd-ωamplitudes are largest exactly when the drive vanishes and smallest exactly when the drive reaches its maximum 224518-6PAIR SYMMETRY CONVERSION IN DRIVEN MULTIBAND . . . PHYSICAL REVIEW B 95, 224518 (2017) amplitude. On the other hand the corrections to the even- ω amplitudes behave in the opposite manner, obtaining theirlargest contribution exactly when the drive is at its maximumamplitude and smallest contribution when the drive vanishes. C. Density of states Now that we have established the possibility of pair symmetry conversion in driven multiband superconductors,we would like to discuss an experimental observable thatmight indicate that such a conversion has occurred. The time-dependent DOS is one such observable, which can be measuredusing scanning tunneling microscopy (STM) [ 46,47]. This quantity can be obtained from the retarded Green’s function by: N d(ω,T)=−1 π/integraldisplayddk (2π)dd/Omega1 2πIm{e−i/Omega1TTrˆGR(k;ω,/Omega1)}, (28) where dis the dimensionality of the system, and ˆGR(k;ω,/Omega1) can be obtained from Eq. ( 14). In Fig. 3,w ep l o t Nd(ω,T) as a function of frequency, ω,f o rd=2, Figs. 3(a) and3(b), andd=3, Figs. 3(c) and 3(d), using the same parameters as in Sec. III B: effective masses, ma=0.5e V−1˚A−2andmb=1e V−1˚A−2; chemical potentials, μa=μb=2e V ; s-wave gaps, /Delta1aa=2m e V , /Delta1bb=7m e V , /Delta1ab=/Delta1ba=0; and interband scattering, /Gamma1=10 meV. However, unlike in Sec. III B w eu s ead i s s i - pation parameter of η=0.1 meV to better highlight the sharp features in the DOS. The black (solid) curves in Figs. 3(a)–3(d) show Nd(ω,T)=N(0) d(ω) without a time-dependent drive, while the green (dashed) and red (dash-dotted) curves showN d(ω,T) in the presence of a drive described by Eq. ( 26) with U0=10 meV, and /Omega10=1 meV (242 GHz) for times T=0 (green, dashed) and T=π/2/Omega10(red, dash-dotted). Notice that, for the range of frequencies considered in Figs. 3(a) and3(c), we see very little difference between the undriven and driven DOS. In each case the dominant featuresare the coherence peaks associated with the gaps at ω≈\|/Delta1 aa\| and\|/Delta1bb\|shifted slightly due to the interband scattering, /Gamma1.I n fact, in Fig. 3(a)(2D DOS) all curves lie directly on top of each other. However, in Fig. 3(c) (3D DOS) the main difference is that for the driven case at T=0 there is a slight suppression of the DOS, which disappears at T=π/2/Omega10consistent with the fact that the drive in Eq. ( 26) vanishes at T=π/2/Omega10.T h i s suppression is a direct consequence of the√ωdependence of the DOS in three dimensions (see Appendix C). In Figs. 3(b) and3(d), we show the same three DOS curves as in Figs. 3(a)and3(c)except plotted over a narrow range of frequencies around the avoided crossing in the quasiparticlespectrum of the superconductor [see inset in Fig. 3(e)] located at: E 0≈/radicalBigg /Gamma12+μ2/parenleftbiggma−mb ma+mb/parenrightbigg2 +ma/Delta12aa+mb/Delta12 bb ma+mb.(29) Notice that for both two dimensions and three dimensions the undriven DOS (black curve) exhibits a slight suppressionaround E 0associated with the depletion of states at the avoided crossing and that the same behavior is exhibitedby the driven DOS at T=π/2/Omega1 0. This feature has been (a) (b) (c) U0(T)T=0T=π/2Ω 0(d) E0E0 Ω0 (e) (f)2D 2D 3D 3D Ω0 FIG. 3. In (a) and (b), the 2D DOS computed using: effective masses, ma=0.5˚A−2eV and mb=1˚A−2eV; chemical poten- tials,μa=μb=2 eV; s-wave gaps, /Delta1aa=2 meV, /Delta1bb=7m e V , /Delta1ab=/Delta1ba=0, consistent with MgB2[44]; interband scattering, /Gamma1=10 meV; dissipation described by η=0.1m e V ; a n d a d r i v e given by Eq. ( 26) with U0=10 meV, and /Omega10=1 meV (242 GHz). In both panels we show the case for no drive in black (solid), and the cases with the drive at times T=0a n d T=π/2/Omega10in green (dashed) and red (dash-dotted), respectively. In (a) we focus on the states near the Fermi surface, in (b) we focus on the range of energiesnear the crossing of the two bands at which we ﬁnd the driven DOS atT=0 possesses two peaks shifted from the avoided crossing at E 0by,±/Omega10/2. In (c) and (d), the 3D DOS plotted for the same parameters as in (a) and (b). Notice that the main difference is that in 3D the driven DOS at T=0 is slightly suppressed relative to the undriven DOS (see inset). In (e) we plot the spectrum of the two bandsuperconductor given by /epsilon1 ±(k)i nE q .( 11). The horizontal gray line denotes the avoided crossing (see inset) at E0,E q .( 29), due to the ﬁnite interband scattering, /Gamma1. In (f) we show the drive from Eq. ( 26) plotted in the time domain over a full period, the green vertical line denotes the beginning of the period at T=0 where the drive has maximum amplitude, while the red line denotes T=π/2/Omega10where the drive amplitude is zero. noted before in multiband superconductors and shares the same origin as the previously discussed odd- ωpair ampli- tudes in multiband superconductors [ 40], i.e., the interband hybridization. However, at T=0 the driven DOS is changed signiﬁcantly at E0with two extrema appearing at E0±/Omega10/2, 224518-7CHRISTOPHER TRIOLA AND ALEXANDER V . BALATSKY PHYSICAL REVIEW B 95, 224518 (2017) similar to the case in superconductors driven by a spatially nonuniform electric ﬁeld [ 43]. The energies associated with these features indicate that their origin can be traced back tothe Floquet bands generated by the periodic drive. However,we note that their appearance requires both the presenceof a drive and ﬁnite interband scattering, necessary andsufﬁcient conditions for the symmetry conversion discussedin Sec. III B. Furthermore, these features can be noticeably enhanced relative to the undriven spectral features at E 0, as can be seen from Figs. 3(b) and3(d). Therefore, we conclude that these peaks offer a potential diagnostic tool for studying pairsymmetry conversion in driven multiband superconductors. IV . SELF-CONSISTENT GAP CALCULATION In the previous sections we have demonstrated the possi- bility of generating both odd- and even-frequency terms in theanomalous Green’s function of a multiband superconductorusing a time-dependent drive to linear order in the drivingamplitude. However, in the above analysis we neglectedcorrections to the gap function, ˆ/Delta1, due to the drive. We will now use the expressions derived in Sec. IIIto analyze these additional terms self-consistently and demonstrate therobustness of the effect. For convenience, in this section wewill focus on the 3D case. Assuming the interaction responsible for the supercon- ducting gap is local in relative time and real space, thetime-dependent gap is given by: /Delta1 αβ(T)=iλ/integraldisplayddk (2π)ddω 2πˆF>(k;ω,T), (30) where ˆF>(k;ω,T) is the Wigner representation of the >anomalous Green’s function, which can be expressed in terms of the retarded, advanced, and Keldysh Green’sfunctions: ˆF >(k;ω,T)=1 2[ˆFR(k;ω,T)−ˆFA(k;ω,T)+ˆFK(k;ω,T)]. (31) This can be expressed in terms of the equilibrium Green’s functions, given by Eq. ( 10), and the corrections due to the drive: ˆF>(k;ω,T) =1 2/braceleftbigˆFR 0(k;ω)−ˆFA 0(k;ω)+ˆFK 0(k;ω) +δˆFR(k;ω,T)−δˆFA(k;ω,T)+δˆFK(k;ω,T)/bracerightbig , (32) where these corrections are given by the Wigner transforms of the expressions in Eq. ( 15). Using Eqs. ( 30) and ( 32) it is straightforward to compute the components of the gap /Delta1αβ(T) at any average time, T, numerically. By inserting these results back into the expres-sions for ˆF >(k;ω,T), recomputing /Delta1αβ(T) and iterating this procedure until the values of /Delta1αβ(T) calculated using Eq. ( 30) match the input values to a precision of our choice we can ﬁndself-consistent solutions for the gap in the presence of a drive. To illustrate that the effect we have discussed in this paper holds even when the gap is allowed to adjust to theapplied time-dependent drive, we have followed the above a (b) (c) (d) U0(T) T=0T=π/2Ω 0Even-ω Even-ω Even-ω Odd-ωOdd-ωOdd-ω(a) FIG. 4. In the left (right) column we plot the even- ω(odd-ω) terms of the real part of the Wigner transform [deﬁned in Eq. ( 27)] of the anomalous part of the Green’s function in Eq. ( 14),/angbracketleftˆFR(ω,T)/angbracketright, w h e r ew eh a v et a k e nt h ea v e r a g ev a l u eo f ˆFR(k;ω,T)a t\|k\|=k(a) Fand \|k\|=k(b) F. In each panel, /angbracketleftˆFR(ω,T)/angbracketrightis calculated self-consistently using Eq. ( 30) for the parameters discussed in the text at times T= 0, green (dashed), and T=π/2/Omega10, red (dash-dotted). Additionally, we note that the self-consistent results for these same parameters but with U0=0 appears as a black curve, which overlaps almost exactly with the T=0 results. (a) the diagonal component for band a, (b) the diagonal component for band b, (c) the interband component. (d) The drive from Eq. ( 26) plotted in the time domain over a full period, the green vertical line denotes the time T=0 while the red line denotes T=π/2/Omega10. self-consistent procedure using a precision of δ=10−5for: effective masses ma=1e V−1˚A−2,mb=1.5e V−1˚A−2; chemical potentials μa=μb=10 eV; dissipation parameter η=50 meV; interband scattering /Gamma1=10 meV; intraband drive amplitude U0=10 meV; drive frequency /Omega10=10 meV (2.4 THz); electron-electron interaction strength λ=1; and approximately zero temperature. For these parameters the 224518-8PAIR SYMMETRY CONVERSION IN DRIVEN MULTIBAND . . . PHYSICAL REVIEW B 95, 224518 (2017) self-consistent gap magnitudes were found at time T= 0:\|/Delta1aa\|≈197.6m e V , \|/Delta1ab\|=\|/Delta1ba\|≈1.495 meV ,\|/Delta1bb\|≈ 1.101 eV and at time T=π/2/Omega10:\|/Delta1aa\|≈197.7m e V , \|/Delta1ab\|=\|/Delta1ba\|≈1.493 meV , \|/Delta1bb\|≈1.102 eV . Additionally, we computed the gaps in the absence of the drive and foundprecise agreement with the magnitudes at time T=π/2/Omega1 0. Notice that the self-consistent magnitudes do not changeappreciably as a function of time T; however, there is a slight suppression of the intraband gaps when the drive isat its maximum and a slight enhancement of the interbandgaps at this same point. Using these self-consistent valuesfor the gaps, we can now examine the frequency-dependentanomalous Greens functions, and determine whether or notthe pair symmetry conversion holds in these cases. In Figs. 4(a)–4(c)we show the even- ωand odd- ωpair am- plitudes computed self-consistently for the above parametersplotted as a function of relative frequency, ω, for two different (a) (b) (c) (d) (e) (f)5×10−5 4×10−5 3×10−5 2×10−5 1×10−5 −1×10−5 FIG. 5. In (a) and (b) we repeat the plots of the even- ω intraband pair amplitudes appearing in Figs. 4(a) and4(b),w h i c h were calculated self-consistently using Eq. ( 30) for the parameters discussed in the text at times T=0, green (dashed), T=π/2/Omega10, red (dash-dotted), and without a drive black (solid). Plotted over this range, the three curves are essentially indistinguishable; however, there a slight differences which we highlight in (c)–(f). In (c)–(f)we show the symmetry preserving (green/dashed) and symmetry reversing (red/solid) contributions to the plots appearing in (a) and (b) calculated using Eqs. ( 22)a n d( 23). In (c) and (e) we show the corrections to the even- ωintraband pairing in band- aat times T=0a n d T=π/2/Omega1 0, respectively; in (d) and (f) we show the corrections to the even- ωintraband pairing in band- bat times T=0 andT=π/2/Omega10, respectively.values of the average time, T:T=0 and T=π/2/Omega10.A s in Figs. 1and2we have taken the average of ˆFR(k;ω,T) at\|k\|=k(a) Fand\|k\|=k(b) F. First, notice that the intraband odd-ωterms are only non-negligible at T=π/2/Omega10where they become larger than either of the interband pairing amplitudes.This conﬁrms that the pair symmetry conversion of even- ω to odd- ωamplitudes holds even when we account for the corrections to the gap. However, notice that we do notsee as dramatic a conversion of odd- ωto even- ωamplitudes as we did for the previous cases considered. This is likelybecause we have restricted ourselves to fairly large equal-timegaps in order to ensure self-consistency in the presence of bothinterband scattering and a time-dependent drive. To better understand how the drive affects the even- ωpair amplitudes, in Figs. 5(c)–5(f) we show both the symmetry preserving (green/dashed) and symmetry reversing (red/solid)corrections to the even- ωintraband pair amplitudes appearing in Figs. 4(a) and4(b), calculated using Eqs. ( 22) and ( 23). Consistent with the results in Sec. III B, we ﬁnd that, in general, both contributions are nonzero. This conﬁrms that the pairsymmetry conversion of odd- ωto even- ωamplitudes holds when the self-consistent corrections to the gap are accountedfor. In Figs. 5(c)and5(d)we show the even- ωcorrections to the intraband pairing in band- aand band- b, respectively, plotted at timeT=0. Notice, as we found earlier, that the contributions coming from pair symmetry conversion (odd →even) are strongest at ω=\|/Delta1 aa\|for band- aandω=\|/Delta1bb\|for band- b. In Figs. 5(e)and5(f)we show the same quantities as Figs. 5(c) and5(d)plotted at time T=π/2/Omega10. As we expect from earlier, we see that, at this time, the symmetry reversing contributionsare signiﬁcantly weakened. V . CONCLUSIONS In this work we considered a model for a two-band superconductor with interband scattering subjected to a time-dependent drive. Working perturbatively, we demonstratedthat, not only can the presence of a time-dependent drivebe used to generate odd-frequency superconducting pairamplitudes, but also that odd-frequency amplitudes generatedfrom the interband scattering can inﬂuence the appearance ofthe even-frequency amplitudes in the presence of a drive. Wehave presented a systematic study of the conversion of odd-frequency pair amplitudes to even-frequency pair amplitudes.We also showed that the appearance of the dynamicallyinduced odd-frequency and even-frequency amplitudes holdseven when the gaps are computed self-consistently. Further-more, by examining the DOS, we found that the conditionsfor this dynamical pair symmetry conversion also gave riseto novel peak structures, offering a potential signature of thephenomenon. Since the derivation of the parity-reversing terms, Eq. ( 23), did not rely on a speciﬁc Hamiltonian or gap symmetrywe conclude that these relations should hold in general.These general relations represent a novel means to controlthe symmetry of Cooper pairs, which could allow for therealization of exotic new superconducting states. Additionally,in light of these results, it would be interesting to study whetheror not a time-dependent external ﬁeld can be used to generatean equal-time gap in an intrinsically odd-frequency supercon- 224518-9CHRISTOPHER TRIOLA AND ALEXANDER V . BALATSKY PHYSICAL REVIEW B 95, 224518 (2017) ductor. Since a key feature of odd-frequency superconductors is the vanishing of an equal-time gap, it is conceivable thatone could use the kind of pair symmetry conversion proposedin this work to generate sharp spectral features, which couldexpose an otherwise hidden order. ACKNOWLEDGMENTS We wish to thank David Abergel, Annica Black-Schaffer, Jorge Cayao, Matthias Geilhufe, Yaron Kedem, Lucia Komen-dová, Sergey Pershoguba, Anna Pertsova, and Enrico Rossi foruseful discussions. This work was supported by US DOE BESE3B7, the European Research Council (ERC) DM-321031,and Dr. Max Rössler, the Walter Haefner Foundation and theETH Zurich Foundation. APPENDIX A: DERIVATION OF EQUATIONS OF MOTION In this appendix, we outline the derivation of the equations of motion describing the Green’s functions in Eq. ( 5). By com- muting the quasiparticle annihilation and creation operators, ψσ,α,kandψ† σ,α,k, with the total Hamiltonian in Eq. ( 1)i ti s straightforward to derive the Heisenberg equations of motionfor these operators: id dtψσ,α,k(t)=ξα,kψσ,α,k(t) +/summationdisplay α/prime[/Gamma1(ˆτ1)αα/prime+Uαα/prime(t)]ψσ,α/prime,k(t) +δσ↑/summationdisplay α/prime/Delta1αα/primeψ† ↓,α/prime,−k(t) −δσ↓/summationdisplay α/prime/Delta1α/primeαψ† ↑,α/prime,−k(t) +/integraldisplay Cdt/prime/Sigma1C(t−t/prime)ψσ,α,k(t/prime)( A 1 ) and id dtψ† σ,α,−k(t)=−ξα,−kψ† σ,α,−k(t) −/summationdisplay α/prime[/Gamma1(ˆτ1)α/primeα+U∗ αα/prime(t)]ψ† σ,α/prime,−k(t) +δσ↓/summationdisplay α/prime/Delta1† αα/primeψ↑,α/prime,k(t) −δσ↑/summationdisplay α/prime/Delta1† α/primeαψ↓,α/prime,k(t) +/integraldisplay Cdt/prime/Sigma1C(t−t/prime)ψ† σ,α,k(t/prime), (A2) where ˆ τiare the Pauli matrices in band space, the contour Cis the standard time contour from the Kadanoff-Baym formalism[48–51] and we have deﬁned the self-energies associated with the presence of the fermionic bath: /Sigma1 C(t−t/prime)=/summationdisplay nη2 nGCbath(n;t−t/prime) /Sigma1C(t−t/prime)=/summationdisplay nη2 nGC bath(n;t−t/prime), (A3)where GC bath(n;t−t/prime)=−i/angbracketleftTCcn;σ,α,k(t)c† n;σ,α,k(t/prime)/angbracketrightandGC bath (n;t−t/prime)=−i/angbracketleftTCc† n;σ,α,−k(t)cn;σ,α,−k(t/prime)/angbracketright are contour- ordered Green’s functions for the free-fermion bath. We then deﬁne the following contour-ordered Green’s functions: GC σ1α1;σ2α2(k;t1,t2)=−i/angbracketleftTCψσ1,α1,k(t1)ψ† σ2,α2,k(t2)/angbracketright GC σ1α1;σ2α2(k;t1,t2)=−i/angbracketleftTCψ† σ1,α1,−k(t1)ψσ2,α2,−k(t2)/angbracketright FC σ1α1;σ2α2(k;t1,t2)=−i/angbracketleftTCψσ1,α1,k(t1)ψσ2,α2,−k(t2)/angbracketright FC σ1α1;σ2α2(k;t1,t2)=−i/angbracketleftTCψ† σ1,α1,−k(t1)ψ† σ2,α2,k(t2)/angbracketright.(A4) Since the Hamiltonian possesses only trivial spin dependence we may restrict our attention to the components: GC α1α2(k;t1,t2)≡GC ↑α1;↑α2(k;t1,t2) GC α1α2(k;t1,t2)≡GC ↓α1;↓α2(k;t1,t2) FC α1α2(k;t1,t2)≡FC ↑α1;↓α2(k;t1,t2) FC α1α2(k;t1,t2)≡FC ↓α1;↑α2(k;t1,t2). (A5) Then, using Eqs. ( A1) and ( A2), one can show that these components satisfy the following equations of motion: /parenleftBigg iˆτ0d dt1−ˆhk−ˆU(t1) −ˆ/Delta1 −ˆ/Delta1†iˆτ0d dt1+ˆh∗ −k+ˆU∗(t1)/parenrightBigg ˆGC(k;t1,t2) −/integraldisplay Cdt/parenleftbiggˆτ0/Sigma1C(t1−t)0 0ˆ τ0/Sigma1C(t1−t)/parenrightbigg ×ˆGC(k;t,t2)=δC(t1,t2), (A6) where ˆ τ0is the identity matrix in band space, ˆhk,ˆ/Delta1, and ˆU(t) are matrices in band space given by: ˆhk=/parenleftbigg ξa,k/Gamma1 /Gamma1ξ b,k/parenrightbigg ˆ/Delta1=/parenleftbigg /Delta1aa/Delta1ab /Delta1ba/Delta1bb/parenrightbigg ˆU(t)=/parenleftbiggUaa(t)Uab(t) Uba(t)Ubb(t)/parenrightbigg (A7) and where we deﬁne ˆGC(k;t1,t2)a s : ˆGC(k;t1,t2)=/parenleftBiggˆGC(k;t1,t2)ˆFC(k;t1,t2) ˆFC (k;t1,t2)ˆGC (k;t1,t2)/parenrightBigg , (A8) where each component is a 2 ×2 matrix in band space. 224518-10PAIR SYMMETRY CONVERSION IN DRIVEN MULTIBAND . . . PHYSICAL REVIEW B 95, 224518 (2017) Using Eq. ( A6) it is straightforward to write the Dyson equation for ˆGC(k;t1,t2): ˆGC(k;t1,t2)=ˆGC 0(k;t1,t2)+/integraldisplay CdtˆGC 0(k;t1,t)/parenleftbiggˆU(t)0 0−ˆU(t)∗/parenrightbigg ˆGC(k;t,t2), (A9) where ˆGC 0(k;t1,t2) is the solution to Eq. ( A6) in the absence of a drive. Assuming that coupling to the bath washes out the correlations between the real and imaginary time contours we may work only with the retarded, advanced, and Keldysh components. Under this assumption we transform Eq. ( A9) to Keldysh space: ˆG(k;t1,t2)=ˆG0(k;t1,t2)+/integraldisplay∞ −∞dtˆG0(k;t1,t)/parenleftbiggˆU(t)0 0−ˆU(t)∗/parenrightbigg ⊗ˆρ0ˆG(k;t,t2), (A10) where ˆ ρ0is the identity in Keldysh space and ˆG(k;t1,t2)=/parenleftbiggˆGR(k;t1,t2)ˆGK(k;t1,t2) 0 ˆGA(k;t1,t2)/parenrightbigg , (A11) where each component may be written as linear combinations of the contour-ordered Green’s functions: ˆGR(k;t1,t2)=1 2/bracketleftbigˆGC 11(k;t1,t2)−ˆGC 12(k;t1,t2)+ˆGC 21(k;t1,t2)−ˆGC 22(k;t1,t2)/bracketrightbig ˆGA(k;t1,t2)=1 2/bracketleftbigˆGC 11(k;t1,t2)+ˆGC 12(k;t1,t2)−ˆGC 21(k;t1,t2)−ˆGC 22(k;t1,t2)/bracketrightbig ˆGK(k;t1,t2)=1 2/bracketleftbigˆGC 11(k;t1,t2)+ˆGC 12(k;t1,t2)+ˆGC 21(k;t1,t2)+ˆGC 22(k;t1,t2)/bracketrightbig , (A12) where ˆGC ij(k;t1,t2) is given by the deﬁnition in Eq. ( A8) with the index i(j) determining on which path of the contour the time argument t1(t2) lies, forward =1 and backward =2 respectively. APPENDIX B: INTEGRATING OUT THE BATH In this appendix we outline the procedure for integrating out the bath and obtaining an expression for the Green’s functions in Eq. ( 7). In the absence of the drive ( Uαβ(t)=0) we can Fourier transform Eq. ( A6) to frequency space to ﬁnd: ˆG0(k;ω)=⎛ ⎜⎜⎜⎝ˆτ 0(ω−/Sigma1R(ω))−ˆhk −ˆ/Delta1 −ˆτ0/Sigma1K(ω)0 −ˆ/Delta1†ˆτ0(ω−/Sigma1R(ω))+ˆh−k 0 −ˆτ0/Sigma1K(ω) 00 ˆ τ0(ω−/Sigma1A(ω))−ˆhk −ˆ/Delta1 00 −ˆ/Delta1†ˆτ0(ω−/Sigma1A(ω))+ˆh−k⎞ ⎟⎟⎟⎠−1 , (B1) where /Sigma1R(ω)=/summationdisplay nη2 nP/parenleftbigg1 ω−(/epsilon1n−μbath)/parenrightbigg −iπ/summationdisplay nη2 nδ[ω−(/epsilon1n−μbath)] /Sigma1R(ω)=/summationdisplay nη2 nP/parenleftbigg1 ω+(/epsilon1n−μbath)/parenrightbigg −iπ/summationdisplay nη2 nδ[ω+(/epsilon1n−μbath)] (B2) /Sigma1A(ω)=/summationdisplay nη2 nP/parenleftbigg1 ω−(/epsilon1n−μbath)/parenrightbigg +iπ/summationdisplay nη2 nδ[ω−(/epsilon1n−μbath)] /Sigma1A(ω)=/summationdisplay nη2 nP/parenleftbigg1 ω+(/epsilon1n−μbath)/parenrightbigg +iπ/summationdisplay nη2 nδ[ω+(/epsilon1n−μbath)] (B3) and /Sigma1K(ω)=−i2πtanh/parenleftbiggβω 2/parenrightbigg/summationdisplay nη2 nδ[ω−(/epsilon1n−μbath)] /Sigma1K(ω)=−i2πtanh/parenleftbiggβω 2/parenrightbigg/summationdisplay nη2 nδ[ω+(/epsilon1n−μbath)]. (B4) 224518-11CHRISTOPHER TRIOLA AND ALEXANDER V . BALATSKY PHYSICAL REVIEW B 95, 224518 (2017) Assuming a featureless bath we approximate η≈π/summationtext nη2 nδ[ω−(/epsilon1n−μbath)] and m≈/summationtext nη2 nP(1 ω−(/epsilon1n−μbath)) in which case Eq. ( B1) simpliﬁes to: ˆG0(k;ω)=⎛ ⎜⎜⎜⎜⎝ˆτ0(ω+iη−m)−ˆhk −ˆ/Delta1 ˆτ0i2 tanh/parenleftbigβω 2/parenrightbig η 0 −ˆ/Delta1†ˆτ0(ω+iη+m)+ˆh−k 0ˆ τ0i2 tanh/parenleftbigβω 2/parenrightbig η 00 ˆ τ0(ω−iη−m)−ˆhk −ˆ/Delta1 00 −ˆ/Delta1†ˆτ0(ω−iη+m)+ˆh−k⎞ ⎟⎟⎟⎟⎠−1 (B5) and, without loss of generality, we account for mby shifting the overall chemical potential appearing in ˆhk. APPENDIX C: DRIVEN DENSITY OF STATES IN dDIMENSIONS In order to illustrate the dependence of the driven DOS on dimension, d, we consider a simple model Hamiltonian describing quasiparticles in one band driven by a time-dependent electric ﬁeld: H=/summationdisplay k[Ek+U(t)]ψ† kψk, (C1) where Ekdescribes the dispersion of the quasiparticles, U(t) is a time-dependent external ﬁeld, and the momentum is summed over a d-dimensional reciprocal space. Following the exact same reasoning leading to Eq. ( 14) one can verify that, to linear order in the drive, the retarded Green’s function describing this system is given by: GR(k;ω,/Omega1)=2πδ(/Omega1)GR 0(k;ω)+GR 0/parenleftbigg k;ω+/Omega1 2/parenrightbigg U(/Omega1)GR 0/parenleftbigg k;ω−/Omega1 2/parenrightbigg , (C2) where GR 0(k;ω)=lim η→01 ω−Ek+iη. (C3) Assuming a drive of the form: U(/Omega1)=2πU 0[δ(/Omega1−/Omega10)+δ(/Omega1+/Omega10)] (C4) we may write the Wigner representation of GR(k;ω,/Omega1), which we deﬁned in Eq. ( 27), as: GR(k;ω,T)=GR 0(k;ω)+2U0cos(/Omega10T)GR 0/parenleftbigg k;ω+/Omega10 2/parenrightbigg GR 0/parenleftbigg k;ω−/Omega10 2/parenrightbigg . (C5) The time-dependent DOS for this system is given by: Nd(ω,T)=−1 π/integraldisplayddk (2π)dImGR(k;ω,T). (C6) Using, the Lorentzian representation of the δfunction we may write this as: Nd(ω,T)=/integraldisplayddk (2π)dδ(ω−Ek)+2U0 /Omega10cos(/Omega10T)/integraldisplayddk (2π)dδ/parenleftbigg ω−/Omega10 2−Ek/parenrightbigg −2U0 /Omega10cos(/Omega10T)/integraldisplayddk (2π)dδ/parenleftbigg ω+/Omega10 2−Ek/parenrightbigg . (C7) Noting that each of these integrals has the same form as the undriven DOS, we can rewrite Eq. ( C7)a s : Nd(ω,T)=N(0) d(ω)+2U0 /Omega10cos(/Omega10T)/bracketleftbigg N(0) d/parenleftbigg ω−/Omega10 2/parenrightbigg −N(0) d/parenleftbigg ω+/Omega10 2/parenrightbigg/bracketrightbigg , (C8) where N(0) d(ω)i st h eD O Si n ddimensions associated with the dispersion Ek. Consider the special case of d=2 andEk=¯h2k2 2m−μ. In this case, N(0) 2(ω) is a constant function of ω, therefore Eq. ( C8)i s constant in Tand unchanged to linear order in U0. This provides insight into why we observe little change in the magnitude of t h e2 DD O Ss h o w ni nF i g s . 3(a)and3(b), the contributions from the Floquet copies cancel at linear order. Now, consider the case of d=3 andEk=¯h2k2 2m−μ. In this case N(0) 3(ω)∝√ω, therefore, unlike the 2D case, the corrections do not cancel. Instead the linear-order corrections provide a net suppression at T=0 since, for/Omega10 2<ω,N(0) 3(ω+/Omega10 2)> N(0) 3(ω−/Omega10 2). However, this suppression will disappear at T=π/2/Omega10due to the vanishing of cosine at this point in the period. 224518-12PAIR SYMMETRY CONVERSION IN DRIVEN MULTIBAND . . . PHYSICAL REVIEW B 95, 224518 (2017) Furthermore, this suppression will turn into an enhancement when the cosine is negative. 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PhysRevB.79.115323.pdf	Local measurement of the entanglement between two quantum-dot qubits Jin Liu,1Zhao-Tan Jiang,1,2,and Bin Shao1 1Department of Physics, Beijing Institute of Technology, Beijing 100081, People’ s Republic of China 2The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy /H20849Received 9 October 2008; revised manuscript received 17 February 2009; published 30 March 2009 /H20850 Using the modiﬁed rate equation the local measurement of the entanglement between two quantum-dot qubits, with two isolated quantum-point-contact detectors, is investigated. It is shown that the measurementprocess will induce the decaying of the two-qubit entanglement, as well as the electron-occupation probabili-ties. Furthermore, we ﬁnd an effective scheme to measure the two-qubit entanglement based on the localmeasurement. It is demonstrated that the entanglement between the two qubits coupled by the strong Coulombinteraction can be fully extracted according to the time-dependent variation rate of the detector current. DOI: 10.1103/PhysRevB.79.115323 PACS number /H20849s/H20850: 73.21.La, 03.65.Ud, 03.67. /H11002a I. INTRODUCTION Quantum measurement /H20849QM /H20850of the entanglement, an im- portant parameter for quantum information processing, is considered to be one of the most crucial steps in the ﬁeld ofquantum information. 1The methods of measuring quantum entanglement are still being extensively investigated boththeoretically and experimentally. In general, based on theworking principles of detectors QM can be classiﬁed intothree kinds: the local measurement, the correlation one, andthe joint one. Intuitively, one would believe that the localmeasurement cannot provide as much information as theother two kinds of measurements because in the local mea-surement scheme the detector only acts on part of the entiresystem. Usually, it is believed that the whole entanglementinformation can be extracted by the method of the joint mea-surement. Therefore, much effort has been made to investi-gate the joint measurement scheme. 2–17Tanamoto and Hu13 showed that the quantum-point-contact /H20849QPC /H20850current can be used for reading out the results of quantum computation andproviding the information about the two-qubit entanglement.To our knowledge, however, little attention has been paid tothe local measurement scheme. In this paper, we design a local measurement scheme based on the quantum-dot /H20849QD /H20850system, as shown in Fig. 1. Each qubit is composed of two QDs /H208490 and 1 /H20850with one extra electron residing in it. When the electron occupies QD 0 or1, the corresponding qubit state is /H208410/H20856or/H208411/H20856. The qubit- i/H20849i =1,2 /H20850states can be detected by measuring the current ﬂow- ing through the nearby QPC i/H20849detector /H20850. Note that this setup can be easily fabricated in two-dimensional electron gasbased on the current experimental nanotechnology. 18Accord- ing to the method proposed by Gurvitz and Prager19we ﬁrst derive the modiﬁed rate equation of the two-qubit systemand then investigate the QM of the entanglement between thetwo qubits numerically. It is found that the electron-occupation probabilities and the entanglement evolve as afunction of time. The mechanism of QM and the inﬂuenceson the qubits induced by QM are further studied in detail byanalyzing the currents ﬂowing through the QPC detectors.The measurement process is found to induce the decays ofboth the electron-occupation probabilities and the entangle-ment. Notably, we demonstrate that the evolution of the en-tanglement can be fully extracted in some case from the time-dependent variation rate of the measured QPC currents.This indicates that in certain cases, the entanglement mea- surement can be accomplished by using the simple localmeasurement scheme alone, rendering the joint measurementunnecessary. The rest of this paper is organized as follows. In Sec. II, we give the Hamiltonian of the two-qubit system and themodiﬁed rate equations. An analytical analysis is performedin Sec. IIIon the time-dependent evolution of the two qubits in the absence of detectors. Then in Sec. IVthe numerical results and discussions about the two qubits coupled by thedifferent coupling strengths are presented. Finally, a conclu-sion is outlined in Sec. V. Qubit-1 Qubit-2 Basis0 1 l r 1Ω 1Ω0 1 l r 2Ω 2ΩQPC1 QPC2 \|10> \|11> \|00> \|01> L FE R FE\|0> ab FIG. 1. /H20849Color online /H20850Schematic illustration of the quantum measurement of the entanglement between two quantum-dot qubitsby two QPC detectors placed above. Each qubit is composed of twoQDs labeled by 0 and 1. The interdot coupling between two QDs ineach qubit is shown by the solid line, and the dashed lines denotethe interdot Coulomb interactions. The quantum entanglement be-tween qubit 1 and qubit 2 can be measured by the currents I 1andI2 ﬂowing through the two detectors QPC1 and QPC2 with the biased voltages V1andV2. The right upper picture shows the Fermi levels EFLandEFRof the detector in the initial vacuum state, and the arrow denotes the electron tunneling from the left lead to the right one.Moreover, shown in the right lower part are the four possible bases/H2084100/H20856,/H2084101/H20856,/H2084110/H20856, and /H2084111/H20856of the two qubits where the ﬁlled circle denotes that there is one electron localized in it.PHYSICAL REVIEW B 79, 115323 /H208492009 /H20850 1098-0121/2009/79 /H2084911/H20850/115323 /H208496/H20850 ©2009 The American Physical Society 115323-1II. FORMULATION The QD two-qubit system, schematically shown in Fig. 1, including two QPC detectors can be expressed by the Hamil-tonian H=H S+H1D+H2D+HI, in which the two-qubit Hamil- tonian is HS=/H20858 i,j/H20849Eijcij†cij+U1i,2jn1in2j/H20850+/H20858 i/H9024i/H20849ci0†ci1+ h.c. /H20850. /H208491/H20850 Here Eijdenotes the ground-state energy of the jth QD in the ith qubit. cij†/H20849cij/H20850is the creation /H20849annihilation /H20850operator of the electron on the energy level Eij, and nij/H11013cij†cijis the corre- sponding particle number operator. /H9024irepresents the interdot coupling in the ith qubit, and U1i,2jdenotes the Coulomb repulsion interaction between the two electrons in the twoqubits. The Hamiltonian of the detector placed above qubit i is H iD=/H20858 lEilcil†cil+/H20858 rEircir†cir+/H20858 l,r/H9024ilr/H20849cil†cir+ h.c. /H20850,/H208492/H20850 while the interaction between the two qubits and the two detectors can be modeled by HI=/H20858 lr/H20849/H9254/H90241lrn10c1l†c1r+/H9254/H90242lrn20c2l†c2r+ h.c. /H20850. /H208493/H20850 Here cil/H20849†/H20850andcir/H20849†/H20850represent the creation /H20849annihilation /H20850opera- tors of the left-lead energy Eiland the right-lead energy Eirin theith detector, respectively. /H9024ilris the coupling between the energy levels EilandEir./H9254/H9024ilrdenotes the coupling variation in the ith detector when the qubit electron enters QD-0 from QD-1. The wave function describing the entire system can beexpressed as 19 /H20841/H9023/H20849t/H20850/H20856=/H20858 ij/H20875bij/H20849t/H20850c1i†c2j†+/H20858 lrbijlr/H20849t/H20850c1i†c2j†c1r†c1l +/H20858 pqbijpq/H20849t/H20850c1i†c2j†c2q†c2p +/H20858 lrpqbijlrpq /H20849t/H20850c1i†c2j†c1r†c2q†c1lc2p+¯/H20876/H208410/H20856. /H208494/H20850 Here, b¯/H20849t/H20850are the amplitudes of the probabilities ﬁnding the system in the states deﬁned by the corresponding creationand annihilation operators. The so-called initial vacuum state/H208410/H20856may be described in the following way: /H20849i/H20850the energy levels of the left and the right leads are ﬁlled up to their Fermi energy levels with E FL/H11271EFRas shown in Fig. 1, and /H20849ii/H20850 four QDs in two qubits are kept empty. Throughout this re-search we consider the transport properties at zero tempera- ture. The quantum evolution of the whole system is de- scribed by the time-dependent Schrödinger equation i/H20841/H9023˙/H20856 =H/H20841/H9023/H20856and the corresponding density matrix is given by /H9268/H20849t/H20850=/H20841/H9023/H20849t/H20850/H20856/H20855/H9023/H20849t/H20850/H20841. In the four-dimensional Fock space com- posed of /H20849a/H20850/H2084100/H20856,/H20849b/H20850/H2084101/H20856,/H20849c/H20850/H2084110/H20856, and /H20849d/H20850/H2084111/H20856/H20849see Fig. 1/H20850, where, for example, /H2084101/H20856denotes the qubit-1 electron occu- pies QD-0 and that of qubit-2 stays at QD-1, we can derivethe differential equations of the reduced density matrix ele-ments according to the procedure proposed by Gurvitz andPrager. 19The diagonal matrix elements are expressed as/H9268˙aa=i/H90241/H20849/H9268ac−/H9268ca/H20850+i/H90242/H20849/H9268ab−/H9268ba/H20850, /H208495a/H20850 /H9268˙bb=i/H90241/H20849/H9268bd−/H9268db/H20850+i/H90242/H20849/H9268ba−/H9268ab/H20850, /H208495b/H20850 /H9268˙cc=i/H90241/H20849/H9268ca−/H9268ac/H20850+i/H90242/H20849/H9268cd−/H9268dc/H20850, /H208495c/H20850 /H9268˙dd=i/H90241/H20849/H9268db−/H9268bd/H20850+i/H90242/H20849/H9268dc−/H9268cd/H20850, /H208495d/H20850 and the nondiagonal ones are20 /H9268˙ab=i/H20849Eba+Uba/H20850/H9268ab+i/H90241/H20849/H9268ad−/H9268cb/H20850+i/H90242/H20849/H9268aa−/H9268bb/H20850 −/H90032d/H9268ab/2, /H208496a/H20850 /H9268˙ac=i/H20849Eca+Uca/H20850/H9268ac+i/H90241/H20849/H9268aa−/H9268cc/H20850+i/H90242/H20849/H9268ad−/H9268bc/H20850 −/H90031d/H9268ac/2, /H208496b/H20850 /H9268˙ad=i/H20849Eda+Uda/H20850/H9268ad+i/H90241/H20849/H9268ab−/H9268cd/H20850+i/H90242/H20849/H9268ac−/H9268bd/H20850 −/H20849/H90031d+/H90032d/H20850/H9268ad/2, /H208496c/H20850 /H9268˙bc=i/H20849Ecb+Ucb/H20850/H9268bc+i/H90241/H20849/H9268ba−/H9268dc/H20850+i/H90242/H20849/H9268bd−/H9268ac/H20850 −/H20849/H90031d+/H90032d/H20850/H9268bc/2, /H208496d/H20850 /H9268˙bd=i/H20849Edb+Udb/H20850/H9268bd+i/H90241/H20849/H9268bb−/H9268dd/H20850+i/H90242/H20849/H9268bc−/H9268ad/H20850 −/H90031d/H9268bd/2, /H208496e/H20850 /H9268˙cd=i/H20849Edc+Udc/H20850/H9268cd+i/H90241/H20849/H9268cb−/H9268ad/H20850+i/H90242/H20849/H9268cc−/H9268dd/H20850 −/H90032d/H9268cd/2. /H208496f/H20850 For convenience we have deﬁned the dephasing rate /H9003id =/H20849/H20881Di−/H20881Di/H11032/H208502, where Di=2/H9266/H9267L/H9267R/H20841/H9024ilr/H208412Vior Di/H11032 =2/H9266/H9267L/H9267R/H20841/H9024ilr/H11032/H208412Viis the transition rate of an electron hopping from the left lead to the right one when the electron stays inQD-0 or QD-1 of qubit i. Here, /H9267Land/H9267Rare the densities of states for the left and right leads, respectively, and Videnotes the voltage bias between the left and right leads of QPC- i. Furthermore, according to the general current formula19 Ii/H20849t/H20850=dQiR/H20849t/H20850/dt=/H20858nn/H20849/H9268˙aan+/H9268˙bbn+/H9268˙ccn+/H9268˙ddn/H20850, with QiR/H20849t/H20850be- ing the total charge in the right lead and /H9268/H9251/H9251nbeing the elec- tron number resolved density matrix element, we can obtainthe current ﬂowing through the two detectors 21 I1=D1/H11032/H20849/H9268aa+/H9268bb/H20850+D1/H20849/H9268cc+/H9268dd/H20850, /H208497a/H20850 I2=D2/H11032/H20849/H9268aa+/H9268cc/H20850+D2/H20849/H9268bb+/H9268dd/H20850. /H208497b/H20850 On the other hand, it has been demonstrated that nonposi- tivity of the partial transposition is a necessary and sufﬁcientcondition for describing the entanglement of a mixed state. 22 For a two-qubit system described by the density operator /H9268, the negativity criterion for the entanglement of the two qu- bits is given by the quantity E=−2 /H20858iui−where the sum is taken over the negative eigenvalues ui−of the partial transpo- sition of the density matrix /H9268. The value of E=1 corresponds to the maximum entanglement between the two qubits whileE=0 indicates that the two qubits are separable. 22,23LIU, JIANG, AND SHAO PHYSICAL REVIEW B 79, 115323 /H208492009 /H20850 115323-2III. DYNAMICS OF COUPLED QUBITS First we discuss the time-dependent properties of the two coupled qubits in the absence of detectors. For the isolatedtwo coupled qubits, the corresponding Hamiltonian can berepresented by H Sin Eq. /H208491/H20850. Using the basis vectors includ- ing /H20841a/H20856,/H20841b/H20856,/H20841c/H20856, and /H20841d/H20856, we can expand the wave function of the two-qubit system as /H20841/H9023/H20849t/H20850/H20856=fa/H20841a/H20856+fb/H20841b/H20856+fc/H20841c/H20856+fd/H20841d/H20856.B y substituting HSand /H20841/H9023/H20849t/H20850/H20856into i/H20841/H9023˙/H20849t/H20850/H20856=HS/H20841/H9023/H20849t/H20850/H20856, we can easily obtain the coupled linear equations if˙a=/H20849/H9255a+Ua/H20850fa+/H90241fc+/H90242fb, /H208498a/H20850 if˙b=/H20849/H9255b+Ub/H20850fb+/H90241fd+/H90242fa, /H208498b/H20850 if˙c=/H20849/H9255c+Uc/H20850fc+/H90241fa+/H90242fd, /H208498c/H20850 if˙d=/H20849/H9255d+Ud/H20850fd+/H90241fb+/H90242fc. /H208498d/H20850 Here we introduce Ua=U00,10,Ub=U00,11,Uc=U01,10, and Ud=U01,11for convenience. In the limit of Ua,b,c,d= 0 ,w ec a n obtain the time-dependent evolution formulation /H20898fa/H20849t/H20850 fb/H20849t/H20850 fc/H20849t/H20850 fd/H20849t/H20850/H20899=1 2/H20898r2r4r3r1 r4r2r1r3 r3r1r2r4 r1r3r4r2/H20899/H20898fa/H208490/H20850 fb/H208490/H20850 fc/H208490/H20850 fd/H208490/H20850/H20899, /H208499/H20850 where we deﬁne r1=cos /H20849/H20841/H9024+/H20841t/H20850−cos /H20849/H20841/H9024−/H20841t/H20850,r2=cos /H20849/H20841/H9024+/H20841t/H20850 +cos /H20849/H20841/H9024−/H20841t/H20850, r3=−isin/H20849/H20841/H9024+/H20841t/H20850−isin/H20849/H20841/H9024−/H20841t/H20850, and r4 =−isin/H20849/H20841/H9024+/H20841t/H20850+isin/H20849/H20841/H9024−/H20841t/H20850with/H9024/H11006/H11013/H90241/H11006/H9024 2and the ini- tial values fa,b,c,d/H208490/H20850. Then we consider the time-dependent evolution in the other limit of Ua,b,c,d→/H11009. We can obtain the relation /H20873fb/H20849t/H20850 fc/H20849t/H20850/H20874=/H20873cos/H9275tisin/H9275t isin/H9275tcos/H9275t/H20874/H20873fb/H208490/H20850 fc/H208490/H20850/H20874, /H2084910/H20850 in the initial conditions fa/H208490/H20850=fd/H208490/H20850=0, and /H20873fa/H20849t/H20850 fd/H20849t/H20850/H20874=/H20873cos/H9275t−isin/H9275t −isin/H9275tcos/H9275t/H20874/H20873fa/H208490/H20850 fd/H208490/H20850/H20874, /H2084911/H20850 in the case of fa/H208490/H20850=fd/H208490/H20850=0. Here /H9275is related to /H90241,2and /H9004Uas/H9275=2/H90241/H90242//H9004Uin the structure designed in Fig. 1. For simplicity, we have assumed that the four QDs are localizedat the four vertices of a rectangle with length aand width b. Consider for example the evolution of the two qubits from apure state /H20841b/H20856/H20849f b=1 and fa,c,d=0/H20850with equal interdot cou- plings /H90241=/H90242=1. From Eq. /H208499/H20850we can ﬁnd that the prob- abilities /H20841fb,c/H208412in the basis vectors /H20841b,c/H20856will evolve in the period of /H9266, while the oscillation periods of the probabilities /H20841fa,d/H208412are/H9266/2. These oscillations with the period of /H9266or/H9266/2 are just a trivial effect induced by the partition of the usualRabi oscillation of each qubit in the basis vectors /H20841a,b,c,d/H20856. In the other limit case where the interdot Coulomb interac-tion is strong enough, the probabilities /H20841f a/H208412=/H20841fd/H208412/H110150 and the probabilities /H20841fb/H208412=cos2/H9275tand /H20841fc/H208412=sin2/H9275t, indicating the oscillation period is /H9266/H9004U//H208492/H90241/H90242/H20850. It can be interpolated that the amplitudes of /H20841fa/H20849t/H20850/H208412and /H20841fd/H20849t/H20850/H208412will decrease tozero with increasing /H9004Uand the oscillation periods of /H20841fb/H20849t/H20850/H208412 and /H20841fc/H20849t/H20850/H208412will become /H9266/H9004U//H208492/H90241/H90242/H20850. This clearly demon- strates that the interdot Coulomb interactions will have aneffect on the oscillation amplitudes and the periods. For ageneral /H9004U, it is difﬁcult to derive analytical results, and therefore a numerical study will be carried out in Sec. IV. IV. NUMERICAL RESULTS In this section we numerically study the dynamics of the two-qubit system based on the modiﬁed rate equations andexplore how to extract the entanglement from the currentsﬂowing through the QPC detector placed nearby. In view ofthe initialization of the qubit states and the Coulomb inter-action between the two electrons in two qubits, here we as-sume that the initial state is chosen to be /H9268bb/H208490/H20850=1 and all the other density matrix elements are kept to be zero at thetime of t=0. This means that the two-qubit system will evolve from a pure state to a mixed one. The Coulomb in-teractions denoted by the dashed lines between the two near- est QDs are chosen to be U 10,20=U11,21=/H208812U10,21=/H208812U11,20 /H11013U, and the interdot couplings /H90241=/H90242=1.0 are used as the energy unit. In our calculation, we choose D1=D2/H11013Dand D1/H11032=D2/H11032/H11013D/H11032=0.9 Dfor clarity and the system temperature T is kept at 0 K. A. Qubit dynamics without detector In order to understand how to measure the two-qubit en- tanglement as well as the measurement-induced inﬂuenceson the qubit information, it is instructive to examine in ad-vance the dynamics of the two-qubit system in the absenceof detectors /H20849D 1,2=0/H20850. First of all, the time-dependent evolu- tions of the electron-occupation probabilities /H9268bband/H9268ccfor the different interdot Coulomb interactions Uare plotted in Fig.2. In the case of U=0, according to the relation in Eq. /H208499/H20850the probabilities /H9268bb/H20849t/H20850=/H20851cos/H208492t/H20850+1/H208522/4 and /H9268cc/H20849t/H20850 =/H20851cos/H208492t/H20850−1/H208522/4 exhibit the oscillations with a period of /H9266. This oscillation, the well-known Rabi oscillation, is attrib-uted to the interdot coupling, which induces the qubit elec-tron to tunnel back and forth between QD0 and QD1. Whenthe Coulomb interaction Ubecomes nonzero, obvious changes appear in the oscillations of /H9268bband/H9268cc. For the moderately strong Coulomb interaction /H20849e.g., U=5 and 10 /H20850 the oscillations exhibit complex patterns, which manifest thecorrelation effect induced by the interdot Coulomb interac-tions. With the further increase in U, the simple pattern of the periodical oscillations is restored for both /H9268bband/H9268cc. This can be easily understood from a physical point of view. Theinterdot Coulomb interactions are inclined to force the two-qubit electrons to occupy the diagonal QDs separated by alarger interdot distance. To see the oscillations clearly, theFourier transforms of the curves of /H9268bbfor different Uare performed in Fig. 2/H20849c/H20850. It is evident that the oscillation for U=0 includes two main components with frequencies f=1 and 2, just as indicated by relation /H208499/H20850. When Ubecomes nonzero, the oscillation pattern changes abruptly and threemain components with different frequencies are observed. AsUincreases further, the amplitudes of the two higher- foscil-LOCAL MEASUREMENT OF THE ENTANGLEMENT BETWEEN … PHYSICAL REVIEW B 79, 115323 /H208492009 /H20850 115323-3lations become smaller and the corresponding frequencies become much higher, while the frequency of the lowest- f oscillation decreases continuously. In the large Ulimit, the two higher- fcomponents diminish and only the lowest fsur- vives. Obviously, the decrease in the frequency of thelowest- foscillation indicates the increase in the oscillation period when Ubecomes strong. It should be emphasized that, in sharp contrast to the periodical oscillation of U=0, this kind of oscillation is closely dependent on the interdotCoulomb interaction U. Furthermore, the oscillations of /H9268aaor/H9268ddcorresponding to the case of Fig. 2are shown in Fig. 3. For U=0, the probability /H9268aa/H11008sin2/H208492t/H20850or/H9268dd/H11008sin2/H208492t/H20850exhibits the oscil- lations with a period of /H9266/2, which is due to the interdot coupling. When the interdot Coulomb interaction becomesnonzero, it is clear that /H9268aaand/H9268ddstill oscillate periodically even in the case of moderately strong U. However, the periodof the oscillation is found to become smaller as Uincreases. Meanwhile, the oscillation amplitude also becomes muchsmaller. When Ubecomes sufﬁciently strong, as it should be, the amplitudes of /H9268aaand/H9268ddare inclined to approach zero, indicating that the two electrons in the two qubits cannot simultaneously occupy the state /H2084100/H20856or/H2084111/H20856anymore. This indicates that the interdot Coulomb interactions prevent thetwo electrons from occupying the nearest QDs. It should beemphasized that the interdot Coulomb interaction Udoes not induce complex oscillation patterns in the /H9268aa/H20849t/H20850and/H9268dd/H20849t/H20850 curves, which are different from the case of /H9268bband/H9268cc. Next, we turn to investigate the entanglement between the two qubits based on the nonpositive partial transpose /H20849NPT /H20850. Figure 4/H20849a/H20850shows the time-dependent evolutions of the quantum entanglement for the different interdot Coulomb in-teractions U. When the Coulomb interaction U=0, the cor- responding entanglement is always zero, indicating that thereis no entanglement between the two qubits. Certainly thisshould be the case since there is no interaction between thetwo qubits. When Ubecomes nonzero, the entanglement ex- hibits many kinds of complex oscillations. For the cases ofthe moderately strong Coulomb interactions /H20849U=5, 10, and 20/H20850the entanglement shows seemingly irregular evolution as the time goes on. However, with further increasing U, the evolution of the entanglement becomes much more regularand exhibits the periodical oscillation. This demonstrates thatentanglement is strongly inﬂuenced by the interaction be-tween the two qubits. Let us consider the details of the en-tanglement for U=40 in Fig. 4/H20849b/H20850. It is evident that the en- tanglement shows the small-amplitude oscillations. Tounderstand what causes this kind of oscillation, the evolutionof /H9268aais also plotted. It is clear that both the entanglement and/H9268aaoscillate in phase.24Therefore, we believe this oscil- lation is mainly caused by the small-amplitude undulationsin /H9268aaand/H9268ddand in /H9268bband/H9268cc/H20851see Fig. 4/H20849c/H20850/H20852. Further- more, the large-amplitude oscillation is examined also, asshown in Fig. 4/H20849c/H20850. It can be seen that when the entanglement takes the largest value of 1 the probabilities of /H9268bband/H9268cc are equal to 0.5. At this moment the two qubits are right in01 0 2 0 3 001234567 012340.00.20.40.60.81.0σ c c σbbProbability t/π10 20 3 0 ft/π (c)(b)AmplitudeFourier Frequency Analysis(a) FIG. 2. /H20849Color online /H20850Time-dependent electron-occupation probabilities /H20849a/H20850/H9268bband /H20849b/H20850/H9268ccwith D=0. From bottom to top, the curves correspond to U=0, 5, 10, 20, 40, 80, and 160, respectively, which are shifted upward by 0, 1, 2, 3, 4, 5, and 6 for clarity. Thecorresponding Fourier transforms of /H9268bb/H20849t/H20850are shown in /H20849c/H20850, and the curves are shifted from bottom to up by 0, 0.1, 0.2, 0.3, 0.4, 0.5, and0.6, respectively. 0123 40.00.61.21.8 t/πPro babili ty FIG. 3. /H20849Color online /H20850Time-dependent electron-occupation probability /H9268aaor/H9268ddwith D=0. From bottom to top, the curves correspond to U=0, 5, 10, 20, 40, 80, and 160, which are shifted upward by 0, 0.3, 0.6, 0.9, 1.2, 1.5, and 1.8, respectively.0 5 10 15 20 2501234567012301 01-0.020.000.020.04 0 8 16 240103690.00.51.0 0.00.51.0 σccσbbσbbσccProbabili ty ProbabilityE E E (d)(c)(b) (a)EσaaProbabili ty t/π t/π t/π t/π FIG. 4. /H20849Color online /H20850/H20849a/H20850Time-dependent entanglement Efor the different interdot Coulomb interactions U. The parameters are the same as those in Fig. 2./H20849b/H20850The entanglement and the corre- sponding probability /H9268aain the case of U=40. The entanglement and the corresponding probabilities /H9268bband/H9268ccin the cases of /H20849c/H20850 U=40 and /H20849d/H20850U=160.LIU, JIANG, AND SHAO PHYSICAL REVIEW B 79, 115323 /H208492009 /H20850 115323-4the maximally entangled Bell state. In addition, it can also be seen that at a certain time the entanglement takes the value of0. This indicates that there is no entanglement between thetwo qubits. By inspecting these positions with E=0, it can be found that only one diagonal element /H20849 /H9268bbor/H9268cc/H20850equals the value of 1. This indicates that the system is in a pure state.Similar phenomena can be found in the large Ulimit /H20851see Fig.4/H20849d/H20850/H20852. However, the small-amplitude oscillations in the entanglement curves disappear. At the same time, the small-amplitude oscillations in the curves of /H9268ii/H20849i=a,b,c,d/H20850also will vanish. This is exactly the reason why the entanglementcurve becomes smooth. B. Entanglement measurement In this section we explore the scheme of extracting the quantum entanglement by means of the currents ﬂowingthrough the two nearby QPC detectors. For convenience,weak measurement condition D /H11032=0.9 Dis still retained to minimize the effect of the decoherence in both the weak/H20849U=5/H20850and strong /H20849U=80 /H20850Coulomb interaction cases. First, the corresponding time-dependent evolutions of the en-tanglement are plotted in Figs. 5/H20849a/H20850and5/H20849b/H20850, respectively. We can see that the entanglement in the case of D/HS110050 will evolve with the same period as that of D=0, indicating that the oscillation period of the entanglement is independent ofthe measurement process. However, the oscillation amplitudedecays when the time tbecomes large and, moreover, will decay more quickly for a larger D. This demonstrates that the measurement process will induce the decay of the entangle-ment. Second, we plot the time-dependent variation rate ofthe corresponding current /H20841/H9004I 1//H9004t/H20841forD/HS110050/H20851see Figs. 5/H20849c/H20850 and5/H20849d/H20850/H20852. Let us pay attention to the strong U=80 case /H20851see Figs. 5/H20849b/H20850and5/H20849d/H20850/H20852. We can see that the current variation rate shows the high-frequency oscillation with a long-periodmodulation oscillation. Very surprisingly, this long-periodmodulation of /H20841/H9004I 1//H9004t/H20841shows a perfect in-phase evolution with the corresponding entanglement. They approach themaximum or minimum position simultaneously. This indi-cates that we can obtain the maximal or minimal entangle- ment information easily from the /H20841/H9004I 1//H9004t/H20841curves. By a fur- ther comparison one can ﬁnd that the extra evolution detailsof the entanglement information and the measured-induceddephasing can also be obtained from /H20841/H9004I 1//H9004t/H20841. Therefore, the entanglement of the two strongly coupled qubits can be ex-tracted based simply on the local measurement. As a conse-quence we have found an effective method to measure thetwo-qubit entanglement information. In the weak Coulomb interaction case of U=5, however, this kind of correspondence between the entanglement and/H20841/H9004I 1//H9004t/H20841can no longer be found and only limited information about the entanglement can be extracted. This indicates thatthis measurement method is not quite valid in the weaklycoupled two-qubit system. One may wonder why it workswell only in the strongly coupled two-qubit case. As a matterof fact its physical picture is particularly intuitive. When theCoulomb interaction is strong enough, the two-qubit elec-trons merely occupy states /H2084101/H20856and /H2084110/H20856with probability occupying state /H2084100/H20856or/H2084111/H20856being zero, which is reduced to the measurement of the single qubit. Hence, if the interactionis strong enough, the entanglement can be measured by alocal measurement. Can the two-qubit entanglement be directly extracted from the QPC currents? In Fig. 6further investigation is conducted to discover the relationship between the currentsI 1,2/H20849t/H20850and the two-qubit entanglement. First, the weak inter- action U=5 case is considered in Fig. 6/H20849a/H20850. By comparing the entanglement and the corresponding currents, one can ﬁndthat although both of them show the seemingly irregular evo-lution as the time goes on, the minimal entanglement posi-tions are exactly at the places where the peaks or the troughsof the currents appear. This correspondence relation betweenthe currents and the entanglement is still correct even at U =80 as clearly shown in Fig. 6/H20849b/H20850. However, it is also difﬁ- cult to extract more information directly from the currents.Therefore, we can only extract partial information of the two-qubit entanglement according to the detector currents inboth the weak and strong Ucases. One may wonder what on earth the detector currents reﬂect. Therefore, we plot thecurves of the current I 1/H20849t/H20850and the probability /H92680=/H9268aa+/H9268bb0 5 10 15 200123U=80 U=5 (b) (a)\|∆I1/∆t\| E S0 10 20 30 4 00123 0 1 02 03 04 00.00.10.2U=80 (d) 0 5 10 15 200.00.30.60.9t/π t/π t/π(c) U=5 t/π FIG. 5. /H20849Color online /H20850Time-dependent entanglement Ewith /H20849a/H20850 U=5 and /H20849b/H20850U=80 for the different dephasing rates D=0, 1, and 2 which are shifted upward by 0, 1, and 2, respectively, from bottomto up. The corresponding time-dependent variation rates of the cur-rents /H20841/H9004I 1//H9004t/H20841are shown in /H20849c/H20850U=5 and /H20849d/H20850U=80 for D=1 and 2. The curves with D=2 are shifted upward by /H20849c/H208500.3 and /H20849d/H208500.1.01 0 1 02 03 04 0 (c)σ000 1 02 03 04 021 1(b) U=80U=80 01 U=5 t/π t/π1.82.02.22.4E(a) U=5Current Current 0 1 02 03 04 01.82.02.22.4(d) 0 1 02 03 04 0 FIG. 6. /H20849Color online /H20850The time-dependent entanglement Eand the corresponding currents I1/H20849t/H20850/H20849label 1 /H20850andI2/H20849t/H20850/H20849label 2 /H20850in the cases of /H20849a/H20850U=5 and /H20849b/H20850U=80. The time-dependent probabilities /H926800and the corresponding current I1/H20849t/H20850in the cases of /H20849c/H20850U=5 and /H20849d/H20850U=80. The parameters used for calculations are chosen to be D=2.LOCAL MEASUREMENT OF THE ENTANGLEMENT BETWEEN … PHYSICAL REVIEW B 79, 115323 /H208492009 /H20850 115323-5which denotes the electron occupying QD0 in qubit 1 in Figs. 6/H20849c/H20850and6/H20849d/H20850, respectively. It is clear that no matter how strong the Coulomb interaction Uis, the current I1/H20849t/H20850 always oscillates in the same way as /H926800does. This veriﬁes that the detector current I1, in nature, completely reﬂects the electron-occupation probability in QD0 of qubit 1. In addi-tion, it should be emphasized that, as estimated in Ref. 21, the oscillation amplitude of the current may be tuned in pArange and can be measured experimentally. V. CONCLUSION In conclusion, quantum measurement of the entanglement between two quantum-dot qubits has been investigated usingthe modiﬁed rate equations. It is found that the measurement process will induce the decaying of the quantum entangle-ment and the electron-occupation probabilities. Especially,our results indicate that when the two qubits are coupled bythe sufﬁciently strong interactions, the entanglement betweentwo qubits can be fully extracted by the local measurementvia the current ﬂowing through the QPC detectors. ACKNOWLEDGMENTS This work is ﬁnancially supported by the NSFC under Grants No. 10811140163 and No. 10604005 and the Excel-lent Young Scholars Research Fund of Beijing Institute ofTechnology /H20849Grant No. 2006Y0713 /H20850. Z.T.J. wishes to thank Jun-Gang Li and Jie Yang for valuable discussions. jiangzhaotan@hotmail.com 1D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information /H20849Springer, Berlin, 2000 /H20850. 2T. Tanamoto, Phys. Rev. A 64, 062306 /H208492001 /H20850;61, 022305 /H208492000 /H20850. 3A. N. Korotkov, Phys. Rev. A 65, 052304 /H208492002 /H20850. 4R. Ruskov and A. N. Korotkov, Phys. Rev. B 67, 241305 /H20849R/H20850 /H208492003 /H20850. 5A. N. Jordan, B. Trauzettel, and G. Burkard, Phys. Rev. B 76, 155324 /H208492007 /H20850. 6R. Ruskov, A. N. Korotkov, and A. Mizel, Phys. Rev. B 73, 085317 /H208492006 /H20850. 7X. B. Wang, J. Q. You, and F. Nori, Phys. Rev. A 77, 062339 /H208492008 /H20850. 8A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Phys. Rev. A 75, 032329 /H208492007 /H20850. 9J. Li and G. Johansson, Phys. Rev. B 75, 085312 /H208492007 /H20850. 10W. J. Mao, D. V. Averin, F. Plastina, and R. Fazio, Phys. Rev. B 71, 085320 /H208492005 /H20850; W. J. Mao, D. V. Averin, R. Ruskov, and A. N. Korotkov, Phys. Rev. Lett. 93, 056803 /H208492004 /H20850. 11C. Hill and J. Ralph, Phys. Rev. A 77, 014305 /H208492008 /H20850. 12A. N. Jordan and M. Buttiker, Phys. Rev. Lett. 95, 220401 /H208492005 /H20850. 13T. Tanamoto and X. Hu, Phys. Rev. B 69, 115301 /H208492004 /H20850;J . Phys.: Condens. Matter 17, 6895 /H208492005 /H20850. 14M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M.Martinis, Science 313, 1423 /H208492006 /H20850. 15Y. Hasegawa, R. Loidl, G. Badurek, S. Filipp, J. Klepp, and H. Rauch, Phys. Rev. A 76, 052108 /H208492007 /H20850.16A. G. Kofman, Q. Zhang, J. M. Martinis, and A. N. Korotkov, Phys. Rev. B 75, 014524 /H208492007 /H20850. 17T. Tanamoto and S. Fujita, Phys. Rev. B 72, 085335 /H208492005 /H20850. 18M. R. Sakr, H. W. Jiang, E. Yablonovitch, and E. T. Croke, Appl. Phys. Lett. 87, 223104 /H208492005 /H20850. 19S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932 /H208491996 /H20850; S. A. Gurvitz, ibid. 56, 15215 /H208491997 /H20850. 20Here, we introduce Eba=E01−E00,Eca=E10−E00,Eda=E11−E00, Ecb=E10−E01,Edb=E11−E01, and Edc=E11−E10; and Uba =U00,11−U00,10,Uca=U01,10−U00,10,Uda=U01,11−U00,10,Ucb =U01,10−U00,11,Udb=U01,11−U00,11, and Udc=U01,11−U01,10. 21S. A. Gurvitz, Phys. Rev. Lett. 85, 812 /H208492000 /H20850; Z. T. Jiang, J. Peng, J. Q. You, and H. Z. Zheng, Phys. Rev. B 65, 153308 /H208492002 /H20850; Z. T. Jiang, J. Q. You, and H. Z. Zheng, J. Appl. Phys. 94, 2142 /H208492003 /H20850; Z. T. Jiang, J. Yang, and Q. Z. Han, J. Phys.: Condens. Matter 20, 075210 /H208492008 /H20850. 22M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223,1 /H208491996 /H20850. 23A. Peres, Phys. Rev. Lett. 77, 1413 /H208491996 /H20850. 24As we know, all four Bell states1 /H208812/H20849/H2084100/H20856/H11006/H2084111/H20856/H20850=1 /H208812/H20849/H20841a/H20856/H11006/H20841d/H20856/H20850and 1 /H208812/H20849/H2084101/H20856/H11006/H2084110/H20856/H20850=1 /H208812/H20849/H20841b/H20856/H11006/H20841c/H20856/H20850are the maximally entangled states and the entanglement is closely related to these four Bell states.Whenever the system evolves into any Bell state, an entangle-ment peak will be observed, which induces the entanglementand the /H9268aaoscillate in phase. When Uis large enough, the two electrons will be inclined to occupy two diagonal QDs withlarger probabilities than those in two nearest QDs. Therefore theprobabilities of /H9268aa,ddbecome zero in the case of U→/H11009. Thus the entanglement related to states /H2084100/H20856and /H2084111/H20856diminishes and only that between /H2084101/H20856and /H2084110/H20856survives. So the small-amplitude oscillations disappear for large U.LIU, JIANG, AND SHAO PHYSICAL REVIEW B 79, 115323 /H208492009 /H20850 115323-6
PhysRevB.79.115126.pdf	Cu2ZnSnS 4as a potential photovoltaic material: A hybrid Hartree-Fock density functional theory study Joachim Paier,1Ryoji Asahi,2Akihiro Nagoya,2and Georg Kresse1 1Faculty of Physics, Universität Wien and Center for Computational Materials Science, Sensengasse 8/12, A-1090, Wien, Austria 2Toyota Central R&D Laboratories, Inc., Nagakute, Aichi 480-1192, Japan /H20849Received 4 December 2008; revised manuscript received 23 February 2009; published 25 March 2009 /H20850 First-principles calculations for the potential photovoltaic material Cu 2ZnSnS 4/H20849CZTS /H20850are presented using density functional theory and the Perdew-Burke-Ernzerhof exchange-correlation functional as well as using theHeyd-Scuseria-Ernzerhof /H20849HSE /H20850hybrid functional. The HSE results compare very favorably to experimental data for the lattice constants and the band gap, as demonstrated for CZTS and selected ternary chalcopyritessuch as CuInS 2, CuInSe 2, CuGaS 2, and CuGaSe 2. Furthermore the HSE band structure is validated using G0W0 quasiparticle calculations. The valence band is found to be made up by an antibonding linear combination of Cu-3 dstates and S-3 pstates, whereas an isolated band made up by Sn-5 sand S-3 pstates dominates the conduction band. In the visible wavelength, the optical properties are determined by transitions from theCu-3 d/S-3pstates into this conduction band. Comparison of the optical spectra calculated in the independent- particle approximation and using time-dependent hybrid functional theory indicates very small excitonic ef- fects. For the structural properties, the kesterite-type structure of I4¯symmetry is predicted to be the most stable one, possibly along with cation disorder within the Cu-Zn layer. The energy differences between structuralmodiﬁcations are well approximated by a simple ionic model. DOI: 10.1103/PhysRevB.79.115126 PACS number /H20849s/H20850: 71.20.Mq, 71.15.Mb, 78.20. /H11002e I. INTRODUCTION To meet the ever increasing demand for energy and to cope with the limited fossil resources available, photovoltaicsolar-energy production will become increasingly important.While the solar cells based on single-crystal silicon or III-Vsemiconductors exhibit already very high efﬁciency, muchcheaper solar cells are required for general and wide spreadapplication. The ternary chalcopyrites, such asCu/H20849In,Ga /H20850Se 2, have attracted much attention in this respect, and the efﬁciency of these materials has already improved upto 20%. 1These materials, however, contain heavy elements, in the former case In, and environmental regulations requirethe long-term reduction of such elements in deposited waste.The quaternary semiconductor Cu 2ZnSnS 4/H20849CZTS /H20850is a rela- tively new photovoltaic material and expected to be interest-ing for environmentally amenable solar cells, as its constitu-ents are nontoxic and abundant in the earth’s crust. 2–11CZTS thin ﬁlms show p-type conductivity, a direct band gap of 1.44–1.51 eV, and high optical absorption/H20849/H110111/H1100310 4cm−1/H20850.5,9,10,12However, the highest conversion ef- ﬁciency of CZTS reported so far is 6.7%,11demanding fur- ther improvement. Although the structure and transport properties of CZTS have been extensively studied experimentally,2–11to the best of our knowledge, there have been only a few ﬁrst-principlesinvestigations; Raulot et al. 13reported chemical trends and defect chemistry among /H20849II-IV /H20850-coupled substitutions for In/Ga in Cu /H20849In,Ga /H20850/H20849S,Se /H208502within the local-density approxi- mation /H20849LDA /H20850. Hence, an atomistic detailed understanding of the binding mechanism and electronic properties of CZTS isstill missing, and to ﬁll this gap is the main objective of thepresent work. Since density functional theory using the local-density approximation or the generalized gradient approxi-mation often severely underestimates the band gap and over-estimates the lattice constants, we have applied a hybrid functional in the present work. Hybrid functionals mix about25% nonlocal Hartree-Fock exchange with 75% semilocalexchange. This usually opens the band gap, and for mostsemiconductors, excellent agreement between theory and ex-periment is found for the band gap. 14As we will show in this work, this also applies to CZTS, and we furthermore conﬁrmthe calculated density of states using highly accurate quasi-particle methods /H20849G 0W0/H20850. An accurate description of the qua- siparticle band structure is a prerequisite for a reliable pre-diction of the optical properties, which are ultimately onekey factor determining the efﬁciency of solar cells. There-fore, we also present results for the optical properties ofCZTS. The work is structured in the following way. After dis- cussing the computational methods employed in this study inSec. II, details about the electronic, structural and optical properties of CZTS are presented in Sec. III. We summarize and conclude in Sec. IV. II. COMPUTATIONAL DETAILS Calculations were performed using the plane-wave pro- jector augmented-wave /H20849PAW /H20850/H20849Refs. 21and22/H20850method ap- plying the semilocal Perdew-Burke-Ernzerhof /H20849PBE /H20850/H20849Ref. 23/H20850exchange-correlation functional and the Heyd-Scuseria- Ernzerhof /H20849HSE /H20850/H20849Ref. 24/H20850hybrid functional as implemented in the Vienna ab initio simulation package /H20849VASP /H20850/H20849Refs. 25 and26/H20850code. The HSE screening parameter was set to a value of 0.2 Å−1.27Details on the implementation and re- sults of the extensive tests of HSE in the framework of thePAW method can be found in the literature /H20849Refs. 28–30/H20850. The PAW data sets with radial cutoffs of 1.2, 1.2, 1.3, and1.0 Å for Zn, Cu, Sn, and S, respectively, were employedPHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 1098-0121/2009/79 /H2084911/H20850/115126 /H208498/H20850 ©2009 The American Physical Society 115126-1using a plane-wave cutoff energy of 350 eV. We also applied theG0W0method31using the HSE eigenvalues and wave functions as initial input for the quasiparticlecalculations. 32–34The optical properties were determined in the independent-particle approximation using PBE and HSE,as well as, using the time-dependent HSE /H20849TD-HSE /H20850 method. 35The last method accounts for the electrostatic in- teraction between electrons and holes, and should give veryaccurate results for the absorption spectrum as well as for thestatic dielectric properties. Brillouin-zone integration was performed on /H9003-centered symmetry reduced Monkhorst-Pack 36meshes using a Gaussian-smearing approach with /H9268=0.01 eV, except for the calculation of total energies and densities of states/H20849DOSs /H20850, as well as PBE/HSE independent-particle spectra. For those calculations, the tetrahedron method with Blöchlcorrections 37was used. The crystal structures were optimized using an 8 /H110038/H110038kmesh for the base-centered-tetragonal /H20849bct/H20850unit cells /H208498 atoms/cell /H20850or an 8 /H110038/H110034kmesh for the twice as large tetragonal unit cells /H2084916 atoms/cell /H20850. HSE structure optimizations have been carried out using a moreapproximate 6 /H110036/H110036a n da6 /H110036/H110034kmesh for the bct and tetragonal unit cells, respectively. In order to check the pertinence of applying the bct unit cells for kesterite /H20849space group I4¯/H20850and stannite /H20849space group I4¯2m/H20850, we repeated the structure optimizations and energy calculations using the larger tetragonal unit cells. PBE struc-tural parameters a 0andc0for the bct and the tetragonal unit cells differed by at most 0.04 Å, whereas the HSE latticeconstants differed by 0.016 Å /H208490.1% /H20850. PBE total energies where reproduced by the bct unit cell within 1 meV, whilethe HSE energies were uniformly shifted by about 150 meV,regarded as sufﬁciently accurate considering the appreciablecomputational work-load. PBE and HSE DOS have been calculated using an 8 /H110038 /H110038o r8 /H110038/H110034kmesh, whereas the G 0W0DOS was ob- tained using 6 /H110036/H110036kpoints. For the optical calculations of the PBE independent-particle spectra, the tetrahedronmethod with 16 /H1100316/H1100316kpoints and 476 conduction bands was used, whereas in the HSE case a total of 44 conductionbands was applied. For the TD-HSE absorption spectra /H20849Sec. III C /H20850, Gaussian smearing /H20849 /H9268=0.01 eV /H20850a n da2 4 /H1100324/H1100324k mesh, mimicked by symmetry adapted shifts of a /H9003-centered 6/H110036/H110036kmesh, was applied /H20849see Ref. 35/H20850. The number of conduction bands involved in the TD-HSE calculations wasset to 28.To illustrate the accuracy of the HSE functional, we have calculated lattice constants and band gaps for the ternarychalcopyrites Cu /H20849In,Ga /H20850/H20849S,Se /H20850 2, which are closely related to CZTS. The results are shown in Table I. They clearly dem- onstrate that the HSE functional systematically improvesupon the PBE lattice constants. Agreement with experimentis typically 1% for all of the considered compounds, al-though a trend toward too large lattice constants is observedfor the compounds containing chalcogenides from the ﬁfthrow /H20849Te/H20850. More importantly, the HSE functional predicts the band gaps reasonably well, while the PBE functional resultsin a signiﬁcant underestimation of the band gap, with incor-rect band ordering /H20849negative band gaps /H20850for CuIn com- pounds. III. RESULTS AND DISCUSSION A. Structural properties Cu2ZnSnS 4crystallizes in the kesterite structure, which is shown in Fig. 1/H20849a/H20850. The most important structural modiﬁca- tion is stannite /H20851Fig.1/H20849b/H20850/H20852, which is observed for the closely related Cu 2FeSnS 4/H20849CFTS /H20850. For details we refer the reader to Ref. 2. Similar to ZnS or ZnO, all cations and anions are located in a tetrahedral bonding environment, with a stackingthat is similar to zincblende. The different structural modiﬁ-cations are related to a different order in the cation sublattice.Kesterite is characterized by alternating cation layers ofCuSn, CuZn, CuSn, and CuZn at z=0, 1/2, 1/2, and 3/4, respectively. The primitive cell is bct, whereas in Fig. 1/H20849a/H20850 the tetragonal supercell is shown. In stannite, ZnSn layersalternate with Cu 2layers, the primitive cell being again bct.2 In addition to kesterite and stannite, we included three struc-tural modiﬁcations of kesterite in our study, which are shownin Figs. 1/H20849c/H20850–1/H20849e/H20850. These modiﬁcations belong to the tetrag- onal space groups P4 ¯2c,P4¯21m, and P2, respectively. All three can be considered to be modiﬁcations of kesterite,where the modiﬁcations are restricted to the exchange of twoions on the cation sublattice. In the structure shown in Fig.1/H20849c/H20850, Cu and Zn atoms in the layer z=1 /4 are exchanged, creating a “stacking” fault with respect to the bct kesteritestructure. In the structure shown in Fig. 1/H20849d/H20850, Cu and Zn atoms are exchanged between two layers to yield completeZn and Cu layers at z=1 /4 and z=3 /4. In the ﬁnal structure Fig.1/H20849e/H20850, a Cu atom at z=1 /2 is exchanged with the Zn atom atz=3 /4. This recovers the stannite structure at the layersTABLE I. Lattice parameters a0andc0/H20849in Å /H20850and band gaps Eg/H20849eV/H20850of some chalcopyrites as obtained using density functional theory /H20849PBE /H20850and hybrid functionals /H20849HSE /H20850compared to experimental values. CuInSe 2 CuInS 2 CuGaSe 2 CuGaS 2 PBE HSE Expt. PBE HSE Expt. PBE HSE Expt. PBE HSE Expt. a0/H20849Å/H20850 5.871 5.834 5.782a5.568 5.537 5.523b5.685 5.637 5.596c5.377 5.357 5.347b c0/H20849Å/H20850 11.79 11.72 11.62a11.25 11.20 11.13b11.22 11.12 11.00c10.61 10.53 10.47b Eg/H20849eV/H20850 −0.35 0.85 1.04d−0.01 1.33 1.53e0.03 1.40 1.68f0.70 2.22 2.43e aReference 15. bReference 16. cReference 17.dReference 18. eReference 19. fReference 20.PAIER et al. PHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 115126-2z=1 /2 and z=3 /4, but maintains the kesterite structure in the other two layers. Although this selection is nonexhaustive, itcovers some important modiﬁcations: cation disorder in theCuZn layer and cation disorder between Cu and Zn layers,which are both difﬁcult to measure experimentally since Cuand Zn posses very similar atomic masses. Furthermore,structural mixtures between kesterite and stannite are cov-ered.Table IIsummarizes the predicted lattice parameters for the ﬁve modiﬁcations shown in Fig. 1. In agreement with experiment, kesterite is the most stable modiﬁcation both for PBE as well as for HSE, but the P4 ¯2cmodiﬁcation is very close in energy. This is in agreement with experimental ob-servations, where a considerable intermixing of the Cu and Zn atoms is observed within the Cu-Zn plane /H20851compare /H20849a/H20850 with /H20849c/H20850in Fig. 1/H20852. The stannite structure is slightly higher in(a) (b) (c) (d) (e) FIG. 1. /H20849Color online /H20850Schematic representations of the /H20849a/H20850kesterite and /H20849b/H20850stannite structures, emphasizing the difference in metal ordering. In addition, schematic representations of three structural modiﬁcations of kesterite are shown /H20849c, d, and e /H20850, which are members of theP4¯2c,P4¯21m, and P2 space group, respectively. Atomic sphere radii are chosen arbitrarily. TABLE II. Lattice parameters a0,b0, and c0/H20849in Å /H20850of modiﬁcations of CZTS as obtained using density functional theory /H20849PBE /H20850and hybrid functionals /H20849HSE /H20850compared to experimental values. Energy differences /H9004E/H20849eV/H20850to the kesterite structure and the band gaps Eg/H20849eV/H20850 are also listed. Type Kesterite Stannite Modiﬁcation Symmetry I4¯I4¯2mP 4¯2cP 4¯21mP 2 Expt. PBE HSE PBE HSE PBE HSE PBE HSE PBE HSE a0/H20849Å/H20850 5.427a5.466 5.448 5.460 5.438 5.466 5.446 5.478 5.464 5.473 5.443 b0/H20849Å/H20850 5.478 5.452 c0/H20849Å/H20850 10.871a10.929 10.889 10.976 10.941 10.929 10.885 10.942 10.857 10.939 10.892 c0/2a0 1.001 1.000 0.999 1.005 1.006 1.000 0.999 0.999 0.993 0.999 1.001 /H9004E/H20849eV/H20850 0.0 0.0 0.046 0.054 0.005 0.012 0.271 0.390 0.195 0.272 Eg/H20849eV/H20850 1.44–1.51b0.096 1.487 −0.030 1.295 0.071 1.458 −0.097 1.206 −0.111 1.073 aReference 2. bReferences 5,9, and 10.Cu2ZnSnS 4AS A POTENTIAL PHOTOVOLTAIC … PHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 115126-3energy /H2084950 meV /H20850, whereas the other two modiﬁcations P4¯21mandP2 are much higher in energy /H20849200–400 meV /H20850. To understand the origin for this particular energy order, we resort to a simple Madelung model, in which we assign“effective ionic charges” to each individual atom, i.e., Cu +, Zn2+,S n3+, and S1.75−. The charges for Sn and S ions are justiﬁed by taking covalent bonding between Sn and S intoconsideration /H20849see Sec. III B /H20850. To account for the strong elec- tronic screening in CZTS, we have divided the Madelungenergies by a factor 10, which is close to the predicted di-electric constant /H20849see Sec. III C /H20850. The results are shown to- gether with the calculated energy differences for PBE andHSE in Fig. 2. It is quite remarkable that despite its exceed- ing simplicity and the neglect of Pauli repulsion, the modelroughly predicts the ab initio energy differences. For in- stance, the near degeneracy of the kesterite and P4 ¯2cmodi- ﬁcation is readily observed, as well as the stability of kester-ite compared to all other structural modiﬁcations. Thisconﬁrms that the usually applied ionic picture for CZTS es-sentially captures the bonding situation correctly; in otherwords, covalency plays only a minor role for the energeticsof CZTS. Table IIalso summarizes the band gaps for all ﬁve modi- ﬁcations. As already mentioned, PBE underestimates theband gaps severely, and resultantly all modiﬁcations but kes- terite and P4 ¯2care predicted to be metals using PBE. The hybrid HSE functional gives certainly a better account forthe band gap, and sizeable band gaps are predicted for allﬁve modiﬁcations with the largest band gaps for kesterite and the P4 ¯2cmodiﬁcation. The most important result is that kesterite and P4¯2cpossess virtually identical band gaps, in- dicating that cation disorder within the Cu-Zn layer willhardly modify the optical properties, but it may reduce mo-bility of carriers due to the potential disorder. B. Electronic properties Sn is generally considered to prefer an oxidation state of +4 in ionic compounds, and it is common practice to regardCZTS as an ionic material associating the following formal valencies with the atoms: Cu+,Z n2+,S n4+, and S2−. For in- stance,57Fe and119Sn Mössbauer spectroscopic and magnetic-susceptibility data collected on CFTS stannite by Eibschütz et al.38suggest Cu2+Fe2+Sn4+S42−. Due to the simi- larity of CFTS stannite with CZTS kesterite6the same argu- ments may apply to the cationic valencies in CZTS kesterite. In order to analyze the chemical bonding, we will concen- trate on the electronic properties of kesterite CZTS. The totaldensity of states, as well as the atom projected density ofstates are shown in Fig. 3using various approximations for the exchange-correlation functional /H20849PBE, HSE /H20850as well as G 0W0. It is observed that different approximations to the functional change the absolute position of the peaks, but thequalitative features are independent of the exchange-correlation functional. Therefore, we will concentrate on theconventional density functional theory results, i.e., PBE ﬁrst. The S-3 sand S-3 pstates are clearly visible at −14 to −13 eV and −6 to −3 eV, respectively. The Zn-3 dstates form a very narrow band located −7 eV below the Fermilevel, and the valence band is clearly made up of Cu-3 d states /H20849− 2t o0e V /H20850. Furthermore, a sizeable hybridization between S-3 pand Cu-3 dis recognized in the energy range between −6 and 0 eV. In fact, the Cu-3 dstates are split into lower e gand higher t2gorbitals in the tetrahedral crystal ﬁeld. The former are nonbonding and appear as a sharp weaklyhybridized peak around −2 eV. On the other hand, the latterstrongly hybridize with S-3 porbitals to create bonding /H20849−6 to −3 eV /H20850and antibonding /H20849−1.5 to 0 eV /H20850states. Since the bonding S-3 p/Cu-3 dand antibonding Cu-3 d/S-3p /H11569linear combinations are fully occupied, the net interaction is Paulirepulsion rather than crystal stability. The charge density cor-responding to the top of the valence band is shown in Fig. 4. The top of the valence band is almost triple degenerated, asexpected from the t 2gcharacter. The charge density is mostly located at the Cu atoms /H2084970% /H20850. However, a sizable contribu- tion of antibonding S-3 sstates /H2084930% /H20850with some small ad- mixture of S-3 pstates is also recognized./PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc /PaintProc I4 I42m P 42c P 421m P2050100150200250300350400∆Eo r ∆Mpot/10 [meV]PBE HSE /PaintProc/PaintProc /PaintProc Mpot(HSE)/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc FIG. 2. /H20849Color online /H20850Total energy differences obtained using PBE /H20849blue circles /H20850and HSE /H20849red hatched squares /H20850, as well as the effective Madelung energy differences, where the energy zero refers to/H20849the most stable /H20850kesterite structure /H20849space group I4¯/H20850. The elec- trostatic Madelung energy has been divided by 10 /H20849see text /H20850.051015 total DOS Zn Cu Sn S 051015 -14 -12 -10 -8 -6 -4 -2 0 2 4 68 10 Ener gy[eV]051015Dens ity o fstates [states /eV]a) b) c)S-3s Sn-5s/S-3pZn-3d S-3pCu-3d Zn-3dSn-5s/S-3p* Sn-5s/S-3p FIG. 3. /H20849Color online /H20850Total and orbital-projected density of states of CZTS kesterite calculated using /H20849a/H20850PBE, /H20849b/H20850HSE, and /H20849c/H20850 G0W0/H20849HSE /H20850/H20849see text /H20850. The origin of energy /H20849vertical line /H20850is set to the valence-band maxima.PAIER et al. PHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 115126-4In PBE, the ﬁrst conduction band is separated by a gap of 0.1 eV from the valence band and contains a single state.This low lying conduction band is separated from the rest ofthe conduction bands by a gap of 0.5 eV. On the basis of theprevious arguments regarding the valency, one would expect that the ﬁrst conduction band is made up of Sn-5 sstates, and a sizeable contribution of Sn-5 sstates /H20849blue/H20850is indeed rec- ognized in the DOS. However, the density of states suggeststhat the yet presented picture is an oversimpliﬁcation: theSn-5 sstates dominate a very narrow band located 8 eV be- low the conduction band. The band is made up of a bondinglinear combination of the Sn-5 sand S-3 porbitals /H20849Sn-5 s/S-3p/H20850. Vice versa, the ﬁrst conduction band is made up of the corresponding antibonding linear combination ofSn-5 sand S-3 p/H20849Sn-5 s/S-3p /H11569/H20850states. To substantiate this covalent picture, the electronic charge density arising fromthe band located at −7 eV /H20849Sn-5 s/S-3p/H20850, and the charge density corresponding to the ﬁrst conduction band/H20849Sn-5 s/S-3p /H11569/H20850are shown in Fig. 5. The left panel conﬁrms that in the Sn-5 s/S-3porbital, a sizeable contribution of S-3pstates is admixed to the dominant spherical Sn-5 sor- bital. The Zn-3 dorbitals are also involved. On the other hand, the ﬁrst conduction band /H20849Sn-5 s/S-3p/H11569/H20850exhibits a nodal plane /H20849zero charge /H20850between the Sn and the S atoms. The resulting charge depletion along the bond causes an in-dentation of the isosurface. One also recognizes that theCu-3 dstates contribute slightly to the conduction band. Covalent bonding also modiﬁes the Born-effective charges /H20849dynamical charges /H20850that describe how much charge moves when an atom is displaced from its ground-state po-sition. In completely ionic compounds, one would expectcharges identical to the formal valencies of each atom: Cu 1+, Zn2+,S n4+, and S2−. For the S atoms, however, we ﬁnd fairly large off-diagonal components and nonisotropic diagonalcomponents /H20849HSE functional: −1.85, −1.55, −1.76, average−1.72 /H20850, conﬁrming the previously discussed non-negligible covalent contribution to the bond. On the other hand, for thecations almost isotropic values of 0.84, 2.1, and 3.2 arefound for Cu, Zn, and Sn, respectively /H20849HSE functional /H20850. The fully ionic character of Cu and Zn is thereby conﬁrmed, andthe Sn ionicity of roughly 3 is in accordance with the previ-ous picture: the occupied Sn-5 s/S-3pband holds a total of two electrons, and due to the hybrid character of this bandone electron is effectively transferred to the Sn atom, yield-ing a covalency closer to +3 than the expected +4. Our results are summarized in a molecular interaction dia- gram shown in Fig. 6. The one-electron energies calculated for atoms are shown in three light shaded boxes divided intoZn/Sn, S, and Cu. The resulting bonding and antibondingstates are shown in between, and the ﬁnal DOS is schemati-cally sketched to the right. A similar bonding structure ap-pears in ternary I-III-VI 2chalcopyrite semiconductors, such as CuInSe 2and CuGaSe 2, where III- s/VI-pbonding states constitute the conduction-band minimum /H20849CBM /H20850as previ- ously reported.39In the case of CZTS, the CBM consists mainly of Sn-5 s/S-3p/H11569states and there is no contribution from Zn-4 sas the result of the much higher atomic energy of Zn-4 sthan that of Sn-5 s. The only not mentioned point yet is that Sn-5 p, Zn-4 s, and Cu-4 sorbitals are formally unoccu- pied in CZTS /H20849ionic picture /H20850. However, the DOS suggests that these orbitals are hybridized with S-3 pcontributing to the states in the valence band in the energy range between −6and −4 eV /H20849marked A in Fig. 6/H20850. The second conduction band is found at energies greater than 2 eV /H20849marked A /H11569in Fig.6/H20850, and it is dominated by these atomic orbitals but also contains some admixture of S-3 pstates. We turn now brieﬂy to the electronic properties calculated using hybrid functionals. The HSE DOS differs qualitativelyvery little from the PBE DOS as shown in Fig. 3. Inclusion of nonlocal exchange increases the overall band width andopens the band gap from 0.1 to 1.5 eV. As for manymaterials, 27the agreement with the experimental band gap is exceptionally good at the HSE level /H20851expt.: 1.44–1.51 eV /H20849Refs. 5,9, and 10/H20850/H20852. The other relevant observation is that the Zn-3 dband has moved toward larger binding energies and is now located below the Sn-5 s/S-3pband, which is a result of the reduced self-interaction for localized orbitals FIG. 4. /H20849Color online /H20850Charge density calculated for the three bands at the top of the valence band at /H9003. Color coding of atoms is similar to Fig. 1. View is along the /H20851100/H20852direction. Similar to Fig. 1, two unit cells are shown, starting with the CuZn layer at the bottom. FIG. 5. /H20849Color online /H20850Charge density corresponding to the Sn-5 s/S-3pstate /H20849left/H20850and ﬁrst conduction band Sn-5 s/S-3p/H11569 /H20849right /H20850. See Fig. 4for details.Cu2ZnSnS 4AS A POTENTIAL PHOTOVOLTAIC … PHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 115126-5using hybrid functionals. Similarly, the S-3 sstates are now located at larger binding energies. We expect that the HSEdescription is more accurate than the PBE one, since PBEnotoriously underestimates the binding energies of localizedelectrons. The important point, however, is that HSE predictsagain a conduction band with Sn-5 s/S-3p /H11569character that is well separated from the valence and other conduction bands. The ﬁnal panel in Fig. 3shows the G0W0results. These results were obtained using the HSE wave functions and one-electron energies as a starting point. This is usually a betterapproximation than simple LDA-based G 0W0.34,35In the present case, the serious underestimation of the band gap inthe LDA yields unrealistic screening properties /H20849see below /H20850 and resultantly much too small band gaps in LDA-basedG 0W0calculations. The HSE-based G0W0quasiparticle ener- gies agree very well with the HSE one-electron energies. Theonly difference is a reduction in the band width of the Cu-3 d band. It is well documented that gradient corrected function-als overestimate the band width of Cu 3 dstates, whereas G 0W0yields a reasonable description.40We therefore expect that the HSE G0W0results are more reliable than HSE, and that the ﬁnal panel approximates the true quasiparticles verywell. Experimental conﬁrmation of this spectrum would cer-tainly be highly desirable. A posteriori , the agreement be- tween the G 0W0quasiparticle spectrum and the HSE band structure also indicates that the correct physics is accountedfor by the hybrid functional. To analyze the electronic and transport properties in more detail, the band structure of CZTS is shown in Fig. 7for PBE and HSE. As already emphasized, the valence bands aremade up of ten Cu-3 dbands /H20849two Cu atoms/unit cell /H20850, eleven S-3pbands, ﬁve Zn-3 dband and the already discussedSn-5 s/S-3pband /H20849giving a total of 12 S-3 prelated bands /H20850. The band gap is direct and occurs at the /H9003point. The con- duction band is made up of the single Sn-5 s/S-3p /H11569band, which has been discussed before. The dispersion of that bandis 1.5 eV with maxima occurring at the X point. Again, it isclearly recognized that this single band is well separatedfrom the rest of the conduction band. For both computationalschemes /H20849HSE and PBE /H20850, the band topology is very similar, as far as the conduction band is concerned, since HSE essen-tially shifts the conduction bands to higher energies /H20849similar to a scissor correction /H20850. C. Optical properties The predicted dielectric functions are shown in Fig. 8for PBE and HSE. We ﬁrst concentrate on the imaginary partand the independent-particle spectra /H20849PBE-IP and HSE-IP /H20850 shown as red and black thin lines, respectively. As one wouldexpect, the PBE-IP and HSE-IP spectra differ only little. Es-sentially, the HSE spectrum is blue shifted and somewhatreduced in intensity. This reduction is required by the f-sum rule, which states that the imaginary part of the dielectricfunction, /H9255 2, observes /H20885 0/H11009 /H92552/H20849/H9275/H20850/H9275d/H9275= constant, independent of the one-particle wave functions. For this re- lationship to hold, a blueshift from /H9275to/H9275+/H9004/H9275must reduce the intensity by a factor/H9275 /H9275+/H9004/H9275. The HSE spectrum can be almost quantitatively obtained by applying this particularscaling relation and a scissor correction of /H9004 /H9275/H110151.2 eV. This semiempirical scaling relation also explains why theﬁrst shoulder in the HSE spectrum at 1.9 eV is much lesspronounced than the ﬁrst shoulder in the PBE spectrum at0.7 eV. The most accurate optical spectrum is obtained using TD- HSE /H20849thick black line /H20850. Compared to the HSE-IP spectrum, TD-HSE includes an electrostatic interaction between the ex-cited particles and created holes. The resultant TD-HSEequation is similar to the Bethe-Salpeter equation /H20849BSE /H20850, but approximates the kernel for the electrostatic interaction be-tween particles and holes by 1/4 of the screened exchange,EfCZTS Cu-S Zn/Sn-S Zn/Sn atomS atomCu atom Sn-5p Sn-5sZn-3dS-3pCu-3d Zn-4sCu-4st2g eg Sn-5s/S-3pSn-3dCu-3d/S-3pCu-3d/S-3p Sn-5s/S-3p AA* FIG. 6. /H20849Color online /H20850Molecular interaction diagram schemati- cally illustrating the atomic one-electron energies /H20849light gray /H20850and the band structure of CZTS /H20849rightmost /H20850, as well as emerging cation-S hybrid bands /H20849e.g., Sn-5 swith S-3 p/H20850. Hybridization of Cu-4 s, Zn-4 s, and Sn-5 pstates results in the bonding linear com- binations marked by A, and antibonding linear combinations A/H11569that make up the second conduction band. Note that the atomic levelsinclude neither any potential shifts due to ionization nor alignmentof the vacuum level with respect to the Fermi energy.-8-404Energy (eV) -8-404 Z Γ XP Γ NZ Γ XP ΓSn-5s/S-3p* Zn-3dSn-5s/S-3pS-3pCu-3d Sn-5s/S-3pZn-3d N FIG. 7. /H20849Color online /H20850PBE /H20849left panel /H20850and HSE /H20849right panel /H20850 band structures of CZTS kesterite.PAIER et al. PHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 115126-6whereas BSE determines the kernel from the dielectric screening properties itself. Since HSE works so well for theone-electron states /H20849see Fig. 3/H20850, we expect it to be accurate for the two-particle spectrum as well. Additionally, Paier et al. 35recently demonstrated that TD-HSE is a fairly good approximation for semiconductors. As a result of the electrostatic particle-hole interaction, intensity is shifted from higher excitation energies to lowerexcitation energies, and high energy peaks become slightlyredshifted. But for CZTS, the changes from the independent-particle approximation to the more accurate interacting ap-proximation /H20849particle-hole interaction /H20850is only quantitative. In particular, a bound exciton, i.e., onset of optical absorp-tion below the energy of the fundamental gap, is not pre-dicted by our calculations. The static dielectric constants could be estimated from the real part of the dielectric function /H9255 1for/H9275→0. In practice we used a ﬁnite ﬁeld approach avoiding a summation overempty states. 35We found values of 6.49 and 6.47 for TD- HSE parallel to aandc, respectively, which implies de facto isotropic screening. The PBE-IP approximation predicts amuch larger value of 10.3, but because of the serious under- estimation of the gap in PBE, this value is expected to bemuch too large. On the basis of the experience for othermaterials, such as Si, ZnO, or ZnS, we expect the presentTD-HSE predictions to be accurate to within 5%. 35To com- pletely determine the dielectric properties, we have also cal-culated the ionic contributions to the static dielectric func-tion. This was done in the simpler PBE approximation, butbecause the ionic contributions depend on the Born-effectivecharges and the vibrational modes only, which are usuallyaccurately predicted using gradient-corrected density func-tionals, we expect this approximation to be very reliable. Theactual calculations were performed using the method re-cently suggested by Wu et al. 41We found contributions of 2.6 and 3.3 parallel to the aandcaxis, respectively. There- fore, the total static dielectric constants are 9.1 and 9.8 par-allel to aandc. To determine which transitions are responsible for the en- ergy range relevant to solar-light absorption, we have calcu-lated the dielectric functions in the independent-particle ap-proximation within PBE considering scattering into the ﬁrstconduction band only, i.e., from Cu-3 d/S-3pstates into the Sn-5 s/S-3p /H11569band. The results are compared to the full spec- trum in Fig. 9. As expected, the resultant spectrum is identi- cal to the full spectrum in PBE up to 2 eV, which covers theﬁrst strong peak located around 3 eV in the HSE spectrum.We furthermore note that transitions into the Sn-5 s/S-3p /H11569 band are also possible between 4 and 6 eV, being related to transitions from the S-3 pstates into the conduction band; nevertheless, in this part of the spectrum, other transitionsdominate. Since the energy region absorbed by solar cells istypically less than 3 eV, 11the most relevant transitions are the electron excitations from the Cu-3 d/S-3pbands into the Sn-5 s/S-3p/H11569band. Crucial for the fairly large absorption co- efﬁcient at this energy, is the sizeable admixture of S-3 p states to both bands. IV. CONCLUSIONS In conclusion, ﬁrst-principles investigations based on the PBE functional and HSE hybrid functional have been carriedout for the potential photovoltaic material CZTS to clarify its024681012 PBE IP HSE IP TD-HSE 02468 1 0 1 2 1 4 Ener gy ( eV)024681012 ε2xxa) b)ε2zz -6-3036912 PBE IP HSE IP TD-HSE 02468 1 0 1 2 1 4 Ener gy(eV)-6-3036912c) d)ε1xx ε1zz FIG. 8. /H20849Color online /H20850Imaginary and real parts of the dielectric function for CZTS kesterite. Panels /H20849a/H20850and/H20849b/H20850show the imaginary parts of the dielectric function using PBE, HSE in the independent-particle /H20849IP/H20850approximation, and TD-HSE for light polarized in a andcdirections, respectively. Similarly, the real parts of the dielec- tric function are shown in panels /H20849c/H20850and /H20849d/H20850.0 2 4 6 8 10 12 14 Ener gy (eV)024681012ε2xx FIG. 9. Imaginary part of the PBE dielectric function for CZTS kesterite. The full line represents the spectrum obtained using 476conduction bands, whereas the dashed line shows the spectrum in-volving transitions into the ﬁrst conduction band only /H20849see Fig. 7/H20850.Cu 2ZnSnS 4AS A POTENTIAL PHOTOVOLTAIC … PHYSICAL REVIEW B 79, 115126 /H208492009 /H20850 115126-7ground-state structure, electronic structure, and optical prop- erties. Our results can be summarized as follows: /H20849i/H20850The ground-state calculations predict that the kesterite structure with the I4¯symmetry is the most stable one, but exchange of Cu and Zn atoms in the CuZn layer costs only very littleenergy. This is consistent with experiment where consider-able intermixing of the Cu and Zn atoms within the Cu-Znplane is observed. /H20849ii/H20850The structural stability is described by a simple ionic picture as demonstrated by equivalent energyordering for the ab initio calculations and a simple Madelung model. Only one strong covalent feature is observed in thedensity of states; the Sn-5 sand S-3 pstates hybridize result- ing in an occupied bonding state about 8 eV below the top ofthe valence band, and an antibonding state making up theconduction band. As a consequence, the ionic charges of Snand S are rather +3 and −1.75 than +4 and −2. /H20849iii/H20850The antibonding Sn-5 s/S-3p /H11569orbital makes up the ﬁrst conduc- tion band. This conduction band is isolated from the remain-ing conduction bands by about 0.2 eV, and the predicted band gap between the conduction and valence band is 1.5 eVusing the hybrid HSE functional. The top of valence band, onthe other hand, is dominated by Cu t 2gorbitals hybridized with S-3 pstates. /H20849iv/H20850The optical-absorption coefﬁcient, which is proportional to the imaginary part of the dielectricfunction, is fairly large in the energy range of visible light.Absorption in this energy range is completely dominated bytransitions from the Cu t 2gorbitals into the Sn-5 s/S-3p/H11569 band. Crucial for the fairly large absorption coefﬁcient, is certainly the admixture of antibonding S-3 p/H11569orbitals to both states. ACKNOWLEDGMENT This work was partially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung within theSTART Grant No. Y218. 1I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Nouﬁ, Prog. 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PhysRevB.98.045403.pdf	PHYSICAL REVIEW B 98, 045403 (2018) Large nonlocality in macroscopic Hall bars made of epitaxial graphene A. Nachawaty,1,2M. Yang,3S. Nanot,1D. Kazazis,4,5R. Yakimova,6W. Escofﬁer,3and B. Jouault1 1Laboratoire Charles Coulomb (L2C), UMR 5221, CNRS-Université de Montpellier, 34095 Montpellier, France 2LPM, EDST, Lebanese University, Tripoli, Lebanon 3LNCMI, Université de Toulouse, CNRS, INSA, Toulouse, France 4Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Sud, Université Paris-Saclay, C2N Marcoussis, 91460 Marcoussis, France 5Laboratory for Micro and Nanotechnology, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland 6Department of Physics, Chemistry, and Biology, Linköping University, SE-58183 Linköping, Sweden (Received 3 April 2018; revised manuscript received 22 May 2018; published 3 July 2018) We report on nonlocal transport in large-scale epitaxial graphene on silicon carbide under an applied external magnetic ﬁeld. The nonlocality is related to the emergence of the quantum Hall regime and persists up tothe millimeter scale. The nonlocal resistance reaches values comparable to the local (Hall and longitudinal)resistances. At moderate magnetic ﬁelds, it is almost independent on the in-plane component of the magneticﬁeld, which suggests that spin currents are not at play. The nonlocality cannot be explained by thermoelectriceffects without assuming extraordinary large Nernst and Ettingshausen coefﬁcients. A model based oncounterpropagating edge states backscattered by the bulk reproduces quite well the experimental data. DOI: 10.1103/PhysRevB.98.045403 I. INTRODUCTION For electronic devices, nonlocal measurements refer to the appearance of voltages across contacts which are far away from the expected path of the charge current. Nonlocality isremarkably useful to unveil various phenomena which wouldbe obfuscated by Ohmic contributions in a local conﬁguration,with voltages measured along the charge ﬂow. Nonlocal volt-ages appear, for instance, in small mesoscopic devices, whenOhm’s law breaks down and phase interference comes into play [ 1]. Nonlocal measurements are also the cornerstone of devices like spin valves, where the signal between excitationand detection is mediated by a neutral spin ﬂow. Nonlocalitycan also be linked to strong distortion of the charge currentpaths, as it happens in the quantum Hall effect (QHE) regime,when a quantizing magnetic ﬁeld forces the charge current toﬂow preferentially along the device edges [ 2]. The remarkable electronic properties of graphene were discovered a bit more than 10 years ago [ 3] and nonlocal measurements revealed them to be a powerful tool to use andstudy the properties of this new material. Spin phenomena ingraphene are of great interest, especially because the smallintrinsic spin-orbit coupling favors very long spin relaxationtimes [ 4]. Various spin valve devices [ 5–7] have been tailored to take advantage of this. Nonlocality in graphene was also studied in Hall bar geometries, without the presence of anymagnetic element and even without magnetic ﬁeld. A strongnonlocality was often detected close to the charge neutralitypoint (CNP), whose origin is still a matter of debate [ 8]. The nonlocal signal has been attributed to the spin Hall effect (SHE)[9,10], thermo(magneto)electric effects [ 11], valley currents [12,13], or even to some unknown physical phenomena [ 14]. In this paper, we are mainly interested in another type of giant nonlocality which is observed in graphene in the QHEregime close to the charge neutrality point (corresponding to a ﬁlling factor ν=0). The nonlocality was originally attributed to the existence of a spin current, triggered by the imbalance ofHall resistivities for spin up and down due to Zeeman splitting[15]. Later, additional experimental evidences demonstrated that this nonlocality is not only due to this so-called Zeemanspin Hall effect (ZSHE) and that other mechanisms comeinto play. First, thermal effects also give rise to a strongnonlocal current. At the excitation point, heat ﬂow is inducedby the Ettingshausen effect, perpendicular to the charge ﬂow.The associated thermal gradient produces in turn a nonlocalvoltage via the Nernst effect [ 11,16]. Second, theory does not predict counterpropagating edge states to be present at theCNP in the QHE (except for very clean samples); nevertheless,assuming their ad hoc existence allows reproducing quite well several experimental results, either in graphene [ 17,18]o ri n two-dimensional HgCdTe-based quantum wells [ 19], which have a very small gap and a band structure quite similar tographene. It is well established that graphene properties are strongly inﬂuenced by the substrate. To date, nonlocal properties ofepitaxial graphene on silicon carbide (G/SiC), under magneticﬁelds close to the CNP, have not been reported. The mainobjective of this paper is to ﬁll this gap, taking advantage ofthe distinct characteristics of the G/SiC devices: the mobilityis low (which prevents the unnecessary complication of aninteraction-induced gap opening at ν=0), the spatial homo- geneity is good (allowing macroscopic devices at the millime-ter scale, with well-controlled and reproduced geometries), andthe QHE is remarkably robust (allowing G/SiC devices to beused as electrical resistance standards [ 20,21]). We studied nonlocality in G/SiC by varying the sample size, the temperature, and by applying a magnetic ﬁeld witha controlled orientation. We found systematically extremely 2469-9950/2018/98(4)/045403(8) 045403-1 ©2018 American Physical SocietyA. NACHAW ATY et al. PHYSICAL REVIEW B 98, 045403 (2018) large and unexpected nonlocal voltages. Our ﬁndings rule out ZSHE as the main origin of the nonlocality. On the contrary, anextended version of the model of McEuen et al. [2], which takes into account both edge and bulk conductivity, describes ourresults surprisingly well, qualitatively and semiquantitatively. II. SAMPLE PREPARATION AND METHODS The G/SiC were grown epitaxially on the Si-terminated face of semi-insulating 4H-SiC at high temperature, T=2000 K. Hall bars of various sizes were patterned on G/SiC usingstandard electron-beam lithography. The samples were thenencapsulated in a bilayer of resists as described in Ref. [ 22]. The resists can be used to lower the Fermi level closer to theDirac point using either UV illumination [ 23] or the corona discharge method [ 24]. Local and nonlocal magnetotransport measurements were made on four G/SiC Hall bars, called G13, G14, G21, and G34. The samples have a length of 420 μm and a width of 100 μm, except for G34, which has a length of 100 μm and a width of 20μm. The Hall bars G13, G14, and G21 originate from the same graphene growth. The Hall bar G34 comes from anothergrowth. The angle of the SiC steps with respect to the Hall baraxis was also carefully checked. This angle is about 30 ◦for G13, G14, and G21, and 0◦for G34. Each sample was prepared close to the charge neutrality point at room temperature using the corona method as de-scribed in our previous work [ 22]. Afterwards, each sample was cooled down to low temperature T=1.7 K several times (up to 13 times for G14) in order to assess the reproducibilityof the nonlocality and to vary the residual doping. At low tem-peratures, all prepared samples had low Hall carrier densitiesn Hwith values in the range nH=− (0.9−3)×1010cm−2, corresponding to a Fermi energy EF/similarequal–20 meV below the CNP. This ensures that the Fermi energy resides in the electron-hole puddles, as the potential ﬂuctuation in G/SiC is of the orderof a few tens of meV [ 25]. In the following, we deﬁne the resistance R ij,kl=Vkl/Iij, where Vklis the ac voltage drop between contacts kandl measured by lock-in technique, and Iijis a low-frequency (∼10 Hz) ac current biased between contacts iandj. III. RESULTS A. Local measurements For all samples, we label the contacts as reported in the inset of Fig. 1(a). Figure 1(a) shows the transverse and longitudinal magnetoresistances for sample G34. We observethat under magnetic ﬁeld, the Hall resistance changes signwhen the magnetic ﬁeld Bincreases, evolving from negative to positive values (from holes to electrons). Furthermore, anunexpected bump is observed in the longitudinal resistance atB/similarequal2 T, approximately at the magnetic ﬁeld for which the Hall resistance cancels. This remarkable behavior was recently reported by our group [ 22] in similar samples. The evolution of the mag- netoresistances cannot be explained by a standard two-ﬂuidDrude model. Charge transfer as a function of magnetic ﬁeld isalmost systematically observed at high ﬁelds in G/SiC samples,because of the quantum capacitive coupling between graphene FIG. 1. Local and nonlocal measurements for sample G34. (a) Transverse and (b) longitudinal magnetoresistances at T=1.7K andI=10 nA. (b), (c) Nonlocal resistances R28,37andR28,46as a function of magnetic ﬁeld. The red dashed lines represent the nonlocalresistance calculated from the Ohmic contribution law, see main text. The inset in (a) is an optical image of one of the largest samples (Hall bar width 100 μm, Hall bar length 420 μm) where the labeling of the contacts, as used in the main text, is reported. and the so-called “dead graphene layer,” as explained by the model developed in Ref. [ 26]. However, this model, in its original form, cannot explain the data observed in Fig. 1(a), as the cancellation of the Hall resistance and the bump of thelongitudinal resistance occur at far too small magnetic ﬁeld.The data suggest that the Fermi energy increases with Bin such a way that the Fermi level evolves from the bottom to thetop of the electron-hole puddles. The exact relation between BandE Fdepends on disorder and cannot be established precisely. In the following, we donot discuss its origin. By contrast, we focus on the remarkableevolution of the nonlocal voltages in the Hall bars preparedclose to the CNP in the presence of a magnetic ﬁeld. B. Nonlocal measurement Figure 1(b) shows nonlocal resistance R28,37as a function of magnetic ﬁeld at T=1.7 K. The solid line represents the experimental data. The nonlocal resistance quickly drops 045403-2LARGE NONLOCALITY IN MACROSCOPIC HALL BARS … PHYSICAL REVIEW B 98, 045403 (2018) at very low magnetic ﬁeld and then starts increasing. The nonlocal resistance reaches values of the order of 1 k /Omega1. This value saturates at higher magnetic ﬁelds correspondingto the electron-doped region. The nonlocal resistance R 28,46 [Fig. 1(c)], further away from the current injection contacts, follows a similar ﬁeld dependence and reaches also a ratherlarge value of /similarequal500/Omega1above B=2T . Similar trends have been observed for the three other samples, as shown, for instance, in Fig. 2(solid lines) for the Hall bar G14. In this case, the nonlocal resistances R 46,37and R46,28reach higher values (10 k /Omega1) than in the Hall bar G34, even if the Hall bar G14 is about 4 times larger. C. Semiclassical current spreading The ﬁrst contribution to consider is how the classical spreading of the charge ﬂow inside the Hall bar gives rise to anonlocal resistance. It follows the equation [ 27] R Ohmic=4 πρxxexp(−πL/W ), (1) where ρxxis the resistivity, Lis the distance separating the current injection from the voltage detection, and Wis the Hall bar width. The dashed lines in Figs. 1(b) and1(c) correspond to the nonlocal resistance calculated using the Ohmic contributionlaw, Eq. ( 1), where the resistivity is estimated from the local resistance: ρ xx/similarequalR15,87. The geometric factor is L/W=1 forR28,37andL/W=2f o rR28,46. The Ohmic contribution corresponds well to the measured nonlocal resistances at B= 0 T and its drop at very low ﬁeld \|B\|<1 T. However, Eq. ( 1) fails to explain how the nonlocal resistances increase at higherBand rapidly underestimate them. D. Zeeman spin Hall effect Nonlocal resistances are often explained by spin diffusion: when a charge current ﬂows, spin currents are created transver-sally via spin Hall effect. These spin currents induce a nonlocalvoltage outside the charge current path detected via the inversespin Hall effect. This charge/spin coupling originates fromintrinsic spin-orbit coupling or extrinsic effects in the case ofgraphene. The spin-induced nonlocal resistance is given by[28] R NL=ρxxW 8λsθ2 SHexp(−L/λs), (2) where θSHis the Hall spin angle and λsis the spin diffusion length. The observed nonlocal resistance can be traced as a function ofLfor different samples and ﬁtted with Eq. ( 2)a saf u n c t i o no f θSHandλs. We could not ﬁnd any reasonable ﬁt of Eq. ( 2)f r o m any of the data. In particular, the nonlocal resistance was muchsmaller (up to ten times) in the smallest sample ( L=20μm, Fig. 1) than in the largest ones ( L=100μm, Fig. 2). And for most of our measurements, see Appendix B,E q .( 2)g i v e s much larger spin diffusion length λ s>200μm than what is reported in the literature [ 6]. Therefore, the observed nonlocal resistance does not seem to originate from spin currents. Equation ( 2) was introduced in Ref. [ 28], where SHE was of spin-orbital origin. However, SHE may have several FIG. 2. (a) Nonlocal resistances R46,37andR46,28as a function of the perpendicular component of the magnetic ﬁeld for severaltilt angles, measured at T=1.7Ka n d I=10 nA on sample G14. (b) Evolution of the nonlocal resistance R 46,21for sample G21, at T=1.3Ka n d I=10μA, with a total magnetic ﬁeld Btots e ta ta n angleθwith respect to the sample plane. (b) Dependence of R46,21 on squared total magnetic ﬁeld B2 totfor different ﬁxed out-of-plane magnetic ﬁelds B⊥=20,25, and 30 T. The green dashed line ﬁts the quadratic dependence on the total magnetic ﬁeld: RNL 46,21=R0+βB2 tot forB⊥=20 T, where β=0.06/Omega1/T2andR0=350/Omega1.F o rB⊥=25 and 30 T, the same quadratic dependence is found. extrinsic origins and Eq. ( 2) will be still valid. Let us assume in the following that SHE is induced by interplay between theZeeman interaction and magnetotransport. This effect, calledthe Zeeman spin Hall effect [ 15,29], is maximized near the 045403-3A. NACHAW ATY et al. PHYSICAL REVIEW B 98, 045403 (2018) charge neutrality point. The magnetic ﬁeld splits the Dirac cone via Zeeman effect and generates electron- and holelikecarriers with opposite spins. The Hall angle is then given by[15] θ SH=1 2ρxx∂ρxy ∂μEz, (3) where μis the chemical potential and EZis the Zeeman splitting. As ZSHE does not impact the validity of Eq. ( 2), introducing Eq. ( 3) into Eq. ( 2) predicts a nonlocal resistance for ZSHE proportional to RNL∝1 ρxx/parenleftbigg∂ρxy ∂μEZ/parenrightbigg2 . (4) The longitudinal and Hall resistances depend mainly on the out-of-plane component of the magnetic ﬁeld B⊥, whereas EZ is proportional to the total magnetic ﬁeld Btot.S oE q .( 4) can be rewritten as RNL=β(B⊥)B2 tot, (5) where βis a function which depends only on B⊥. The relation was ﬁrst noticed in Ref. [ 11] and can be used to check the part of the ZSHE contribution to RNL. Figure 2(a) plots the nonlocal resistances R46,37andR46,28 as a function of the out-of-plane component of the magnetic ﬁeld,B⊥=Btotcosθ, for several tilt angles θ,f o rs a m p l e G14. It is clear that the nonlocal resistances R46,37(B⊥) and R46,28(B⊥) do not depend on the in-plane component B/bardblof the magnetic ﬁeld and Eq. ( 5) cannot ﬁt the results. Nevertheless, the square factor B2 totin Eq. ( 5) could become dominant at high enough magnetic ﬁelds. In order to test thisprediction, we measured the local and nonlocal resistances foranother sample, G21, at very large B. The nonlocal resistance R 46,21is shown in Fig. 2(b) as a function of a pulsed magnetic ﬁeld for different tilt angles. There is a weak dependence ofthe nonlocal resistance on the in-plane magnetic ﬁeld whichappears at magnetic ﬁelds higher than 10 T. In Fig. 2(c) we replotted R 46,21as a function of B2 totforB⊥=20, 25, and 30 T. The data can be ﬁtted by Eq. ( 5), assuming there is an additional constant term: R46,21=R0+βB2 tot. The slope β, equal to 0.06 /Omega1/T2, is slightly smaller (by a factor of 2) than the values reported in Ref. [ 11]. Therefore, we could identify a part of the nonlocal resistance which obeys Eq. ( 5)a n dw h i c hw ea t t r i b u t et oZ S H E .H o w e v e r , the main part of the nonlocal resistance ( R0) does not depend onB/bardbland must have another origin. E. Thermoelectric effects A thermal origin for nonlocal voltages was proposed in Ref. [ 11]. In the presence of a transverse magnetic ﬁeld, the Ettingshausen effect [ 30] and Joule heating generate a heat ﬂow perpendicular to the injected current. This induces a thermalgradient ∂T/∂x which propagates into the Hall bar and is converted into nonlocal voltages by the Nernst effect, which isquantiﬁed by the Nernst coefﬁcient [ 30]S yx=Ey(∂T/∂x ). Joule heating and the Ettingshausen effect have very distinct experimental signatures. Joule heating induces a temperaturegradient proportional to the heating power, ∂T/∂x ∝Q Joule= RI2. Because of this quadratic dependence on the current, FIG. 3. (a) Nonlocal magnetoresistance R28,37measured on sam- ple G34 at T=1.7Ka n d I=1μA (black line). The two-contact resistance R28,28(blue line) used to estimate QJoule is also shown. (b) Measured second harmonic resistance R2f 28,37(blue line) and corresponding estimation of the Nernst coefﬁcient Syxfollowing Eq. ( 6). Joule heating can only appear in the second harmonic of the nonlocal resistance, R2f TE. By contrast, the Ettingshausen effect is proportional to the current, ∂T/∂x ∝QEtt∝TSyxI, and appears directly in the ﬁrst harmonic of the nonlocal resistance,R TE. It is then possible to show that Ettingshausen and Joule signals are proportional, which allows extracting a value ofthe Nernst coefﬁcient [ 11]: S yx=γQ Joule TIRTE R2f TE, (6) where γis a constant which depends on the sample geometry. Figure 3(a) shows the ﬁrst harmonic of the nonlocal re- sistance R28,37measured in sample G34 at I28=1μA,T= 1.7 K, and low frequency f/similarequal10 Hz. The Ohmic contribution R2c=R28,28, used to estimate the Joule heating, is also shown. The second harmonic resistance R2f 28,37is reported in Fig. 3(b). The same ﬁgure also shows Syx(B) calculated using Eq. ( 6). From the Hall bar geometry we estimate γ/similarequal0.3. The Nernst coefﬁcient appears to be asymmetric with B, as expected. However, Syxalso reaches unrealistically high values of more thanSyx/similarequal20 mV/K at \|B\|/similarequal6 T. This is 100 times larger than values typically reported at the charge neutrality point ingraphene [ 31–33]. This suggests that the nonlocality does not originate from thermal effects. It is also worth noting that the2fsignal was unmeasurable at currents lower than 1 μA. This 045403-4LARGE NONLOCALITY IN MACROSCOPIC HALL BARS … PHYSICAL REVIEW B 98, 045403 (2018) FIG. 4. (a) Magnetoresistances R23,64,R13,64,R83,64,a n dR73,64 for the Hall bar G14, at T=1.7Ka n d I=10 nA. The resistances R∞expected for perfectly transmitted counterpropagating edge states are also reported for the four conﬁgurations. (b) Magnetoresistances R1(28),46andR1(28),37for the same Hall bar G14, under the same experimental conditions. indicates that the thermoelectric contribution is weak for the typically used currents of 10 nA. F. Counterpropagating edge states So far, the role of edge states when Landau levels are formed has not been mentioned. Nevertheless, their existencehas been assumed in several similar experiments to explainthe nonlocality [ 17–19]. It was also clearly observed that in clean samples, counterpropagating edge states are formed andlead to an insulating state under perpendicular magnetic ﬁeld,which can start exhibiting well-quantized values when therelative spin contribution is increased under a tilted magneticﬁeld [ 34]. Besides, a recent theory predicts the existence of additional edge states in G/SiC at low ﬁlling factors [ 35]. These edge states appear because the electrostatic potential imposesa charge modulation at the sample edge [ 36]. Figure 4(a) shows four different nonlocal resistances R i3,64 measured in Hall bar G14 for a given corona preparation, where only the current injection contact, i, changes. We ﬁnd that the resistance increases progressively in the following order:R 23,64,R13,64,R83,64, andR73,64. From the Landauer-Büttikerformalism, one gets an estimate for these resistances, assuming that there is an inﬁnite mean-free path for backscatteringbetween the two edge states: R ∞ 23,64=h/4e2,R∞ 13,64=h/2e2, R∞ 83,64=3h/4e2, andR∞ 73,64=h/e2. Thus, the resistance vari- ation from one conﬁguration to the other is in qualitativeagreement with the edge states model. However, this estimate quantitatively gives only the correct order of magnitude for the resistances. A strong ±Basymmetry is also noticeable. An interesting conﬁguration consists ofinjecting current from contact 1, grounding both contacts 2and 8, while the nonlocal resistances R 1(28),37andR1(28),46are measured between contact pairs (3,7) and (4,6), respectively.In such a conﬁguration, the Ettingshausen effect induces aheat ﬂow perpendicular to the main Hall bar axis. Similarly,ZSHE induces a spin current which is also perpendicular tothe Hall bar axis. Moreover, if edge states are responsiblefor the nonlocality and circulate around the sample, then theyshould impose all contacts 3–7 to have the same null potential.Therefore, in all cases, one expects R 1(28),37=R1(28),46=0. Figure 4(b) shows the magnetoresistances R1(28),37and R1(28),46for sample G14. Surprisingly, these two resistances, again, become large when Bincreases. Also, they are mainly antisymmetric with B, while ZSHE and thermal effects are symmetric in B. This conﬁrms that both ZSHE and the thermal effect are of little importance in the appearance of nonlocalresistances. Moreover, this experiment demonstrates that bulkconductivity plays a role in the appearance of the nonlocalityand an edge-to-bulk leakage is present. IV. SUMMARY AND CONCLUSION The main results of this work are summarized in Fig. 5, which reports the maxima of the nonlocal resistances Rmax 28,37 (L/W=1) and Rmax 28,46(L/W=2) for three different Hall bars and various corona preparations. The maxima correspondingto the same Hall bar and the same preparation are linkedby a dotted line. These data can be compared with fourmodels. The solid blue line corresponds to the nonlocalresistance given the deviation of the current ﬂow, as given byEq. ( 1) with ρ xx=10 k/Omega1, the typical transverse resistance measured in the devices. The orange line corresponds tothe spin diffusion model and is given by Eq. ( 2), assuming unrealistically high values θ sh=1 andλ=100μm[18]. The thermoelectric contribution is given by the formula R28,37= (W/w )S2 xyT/dκ xx, where Wis the Hall bar width, wis the width of the lateral probes ( W/w/similarequal5),d=0.3n mi st h e graphene thickness, κxxis the thermal conductivity, and Sxyis the Nernst coefﬁcient. The second nonlocal resistance, locatedfurther away from the injection point, can be estimated as afraction of R 28,37:R28,46/similarequalR28,37/3[11]. From the literature [31,37], taking κxx=1W m−1K−1andSxy=100μV/K, we get the estimate indicated by the red solid line in Fig. 5. These three models severely underestimate the observed nonlocalresistances. At the contrary, assuming perfectly transmittedcounterpropagating edge states in the Hall bars, we obtain thepurple line in Fig. 5, which overestimates the experimental nonlocal resistances. An even better agreement with the datacan be obtained by adapting the model of McEuen et al. [2] to the graphene case (see Appendix A). The model includes indeed another parameter, ρ 0, which describes the edge-to-bulk 045403-5A. NACHAW ATY et al. PHYSICAL REVIEW B 98, 045403 (2018) FIG. 5. (a) Open symbols: maxima of the nonlocal magnetore- sistances Rmax 28,37(L/W=1) and Rmax 28,46(L/W=2), measured at T=1.7 K in the interval –13 T to 13 T (excluding the B=0T peak). The dotted lines link maxima corresponding to the same Hall bar and the same corona preparation. Red dotted lines: Hall bar G14 (ﬁve different corona preparations); black dotted line: Hall bar G13;green dotted line: Hall bar G34. The solid lines are given by the models of current ﬂow (blue line), spin diffusion (orange line), thermoelectric model (red line), and perfectly transmitted edge states (purple line).The color map shows the edge-to-bulk leakage parameter ρ 0as a function of the calculated maxima of nonlocal resistances, following the model presented in Appendix A. leakage when the Fermi energy is exactly at the CNP. The color map in Fig. 5shows the evolution of the maximum of the nonlocal resistances, Rmax NL(ρ0),whenρ0is varied. For small values of ρ0(ρ0=10−3h/e2), the nonlocal resistances corre- spond to what is expected for perfectly transmitted edge states(purple line). As ρ 0increases, the bulk shunts the edge current and the nonlocal resistances decrease. From the ﬁgure, we canget a rough estimate for ρ 0. For the Hall bars G14, G13, and G34 we obtain log10(ρ0e2/h)=(−1.4±0.5),(−0.5±0.6), and (0 ±0.02) respectively. However, these values reﬂect only poorly the real edge-to-bulk leakage, as a complete treatmentshould take into account backscattering between the edgestates. To conclude, we have investigated the local and the nonlocal voltages appearing in epitaxial graphene Hall bars close to theCNP. Very large nonlocal resistances are observed systemat-ically when the QHE regime takes place. In some cases, thenonlocal resistances are so large that they approach h/2e 2and can surpass the local resistances. These high resistances areonly observed when the samples have been prepared to havethe Fermi energy close to the Dirac point at B=0T .T h e y also strongly decrease when the current or the temperatureincreases. A model of edge conduction explains the dataqualitatively and semi-quantitatively.We did not discuss the residual ±Basymmetry in the local and nonlocal magnetoresistances, which can be seen clearly inmost of our measurements, as shown in Figs. 1–4. A similar asymmetry was recently observed and interpreted in Ref. [ 18] in terms of spin-dependent current at the grain boundariesof chemical-vapor-deposited graphene. The asymmetry couldalso be reproduced in our model if it is assumed that the conduc-tion (of either edge or bulk states) depends on spin polarization.However, we do not expect such grain boundaries in our G/SiCsamples, and the in-plane component of the magnetic ﬁeld hasno effect on the nonlocality. As a consequence, this asymmetryneeds further analysis. ACKNOWLEDGMENTS This work has been supported in part by the French Agence Nationale de la Recherche (ANR-16-CE09-0016) and throughthe grant NEXT n ANR-10-LABX-0037 in the frameworkof the “Programme des Investissements d’Avenir”. Part ofthis work was performed at LNCMI under EMFL proposalTSC06-116. APPENDIX A Let us assume that two counterpropagating edges states are present in the samples and are spin degenerate. We assumethat coherence plays no role and that interchannel scatteringcan be weak, which yields to two independent electrochemicalpotentials for the two edge states. These edge states coexistwith a disordered bulk, as shown in Fig. 6(a). We reuse the model proposed by McEuen et al. [2] in which each segment of the Hall bar is modeled by a barrier which backscattersthe edge states through the bulk from one side of the segmentto the other, see Figs. 6(b) and6(c). We neglect interchannel scattering inside graphene. The transmission probability T jof thejth segment can be traced back to an effective resistivity FIG. 6. (a) Simpliﬁed scheme of the N=0 Landau-level struc- ture, with the disordered bulk and the two spin-degenerate edge states.(b) A model of the device where each segment is replaced by a barrier which backscatters separately the two counterpropagating N=0 edge states. (c) Close view of one segment. (d) Energy evolution ofthe effective resistivities ρ eandρhused in the model near the charge neutrality point, ρeρh=(ρ0)2for simplicity. 045403-6LARGE NONLOCALITY IN MACROSCOPIC HALL BARS … PHYSICAL REVIEW B 98, 045403 (2018) FIG. 7. Evolution of the sample resistances as a function of log(ρh/ρ0)∝EF, from strongly backscattered to almost perfectly transmitted edge states: (a) ρ0=h/e2,( b )ρ0=h/e2×10−1;a n d (c)ρ0=h/e2×10−3. The same cases as the experiment in Fig. 4are plotted: R15,34(longitudinal resistance, solid blue line), R15,37(Hall resistance, solid black line), R82,73andR82,64(nonlocal resistances, dashed red and magenta lines), and R1(28),37andR1(28),46(nonlocal resistances, dotted green and dark green lines). by the equation ρN(Lj/Wj)=(h/e2)(1−Tj)/Tj, (A1) where N=(e,h) labels the edge states, LjandWjare the length and width of the jth segment, and ρNis the effective resistivity of the Nth channel only. When the Fermi energy EF increases from the valence band to the conduction band and scans the vicinity of the charge neutrality point, ρhincreases to inﬁnity and ρedecreases to zero. Thus, there must be a Fermi energy E0, where both effective resistivities are equal: ρe(EF=E0)=ρh(EF=E0)=ρ0. The exact relation be- tween ρeandρhis not known. For the sake of simplicity, we assume that ρeρh=(ρ0)2, as shown in Fig. 6(d). Then all resistances can be calculated using the Landauer- Büttiker formalism [ 1,38] as a function of only two parameters, ρhandρ0. Interchannel scattering is taken into account only in the Ohmic contacts. Figure 7shows several calculated local and nonlocal resistances as a function of log( ρh/ρ0), which increases monotonously with EF. The important parameter is FIG. 8. (a) Nonlocal resistances for G14, at T=1.7 K (12th cool- down) and at different ﬁxed negative magnetic ﬁelds. (b) Same as (a) but for positive magnetic ﬁeld. (c) Spin diffusion length, and (d) Hall angle coefﬁcient extracted from the ﬁt using Eq. ( 2). ρ0, which controls if the edges states are decoupled from the bulk (ρ0/lessmuchh/e2) or backscattered ( ρ0/greaterorequalslanth/e2)a tEF=E0. At high ρ0[ρ0=h/e2, panel (a)], all nonlocal resistances are negligible when compared to the local longitudinal resis-tances. At lower ρ 0[ρ0=h/e2×10−1, panel (b)], nonlocal and local resistances become comparable. Finally, at evenlower ρ 0[ρ0=h/e2×10−3, panel (c)], close to EF=E0 backscattering is negligible and all resistances correspond to what is expected for perfectly transmitted edges states. In thislast case, the nonlocal resistances close to the CNP can besigniﬁcantly larger than the local ones. Interestingly, the model also predicts sizable nonlocal resis- tances R 1(28),37andR1(28),46at the onset of the ν=± 2 plateaus, where one edge state is almost perfectly transmitted, whereasthe second edge state has a transmission close to 1 /2. This conﬁguration corresponds precisely to the situation depictedin the original paper of McEuen et al. , when only the upper LL of the ﬁlled conduction band is backscattered through the bulk.From Fig. 4, we ﬁnd experimentally R 1(28),37/similarequal0.15h/e2, R1(28),46/similarequal0.05h/e2, which is of the same order of magnitude than the maximal resistance values obtained from Fig. 7(c): R1(28),37/similarequal0.04h/e2andR1(28),46/similarequal0.02h/e2. 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PhysRevB.91.195146.pdf	PHYSICAL REVIEW B 91, 195146 (2015) Magnetic, electrical, and thermodynamic properties of NpIr: Ambient and high-pressure measurements, and electronic structure calculations H. C. Walker,1,2K. A. McEwen,3J.-C. Griveau,4R. Eloirdi,4P. Amador,4P. Maldonado,5P. M. Oppeneer,5and E. Colineau4 1Deutsches-Elektronen Synchrotron DESY, 22607 Hamburg, Germany 2ISIS Facility, STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, United Kingdom 3London Centre for Nanotechnology, and Department of Physics and Astronomy, University College London, 17-19 Gordon Street, London WC1H 0AH, United Kingdom 4European Commission, Joint Research Centre (JRC), Institute for Transuranium Elements (ITU), Postfach 2340, 76125 Karlsruhe, Germany 5Department of Physics and Astronomy, Uppsala University, P . O. Box 516, SE-75120 Uppsala, Sweden (Received 25 March 2015; revised manuscript received 4 May 2015; published 27 May 2015) We present bulk property measurements of NpIr, a newly synthesized member of the Np-Ir binary phase diagram, which is isostructural to the noncentrosymmetric pressure-induced ferromagnetic superconductorUIr. Magnetic susceptibility, electronic transport properties at ambient and high pressure, and heat capacitymeasurements have been performed for temperatures T=0.55–300 K in a range of magnetic ﬁelds up to 14 T and under pressure up to 17 .3 GPa. These reveal that NpIr is a moderately heavy fermion Kondo system with strong antiferromagnetic interactions, but there is no evidence of any phase transition down to 0 .55 K or at the highest pressure achieved. Experimental results are compared with ab initio calculations of the electronic band structure and lattice heat capacity. An extremely low lattice thermal conductivity is predicted for NpIr attemperatures above 300 K. DOI: 10.1103/PhysRevB.91.195146 PACS number(s): 61 .05.cp,72.15.−v,75.20.Hr,75.40.Cx I. INTRODUCTION The magnetic and electrical properties of the majority of rare earth intermetallics are well explained using thestandard localized moment model of rare earth magnetism,as expounded by Jensen and Mackintosh [ 1], owing to the limited radial extent of the 4 fwave functions. Whereas, upon descending down the periodic table into the 5 factinide series, the f-electron wave function becomes more extended. Therefore, the 5 felectrons have a character intermediate between that of the localized 4 felectrons and the itinerant 3 d electrons of the transition metals, which taken in conjunctionwith the increased difﬁculties in handling such materialsresults in a more limited understanding of magnetism inthe actinides. However, this also contributes to the fact thatactinide systems display a unique complexity, exhibitingvarious interesting properties such as heavy fermion andnon-Fermi liquid behaviors [ 2], multipolar order [ 3–7], and unconventional superconductivity [ 8–14]. UIr has recently been identiﬁed as a ferromagnetic quan- tum critical point pressure-induced superconductor [ 15] and belongs to two different, nonconventional subclasses of su-perconductor: noncentrosymmetric superconductors such asCePt 3Si [16], and ferromagnetic superconductors such as UGe 2[8]. It is thought that the absence of inversion symmetry prohibits spin-triplet ( p-wave) pairing, whilst ferromagnetism prohibits spin-singlet ( s-wave) pairing. Therefore UIr is a particularly interesting system, signiﬁcant for the studyof the interplay between magnetism and superconductivityin strongly correlated electron systems. It is an itinerantferromagnet ( T C=46 K at ambient pressure), with an ordered moment of only 0 .6μB/Ua t o m[ 17]. However, at high temperatures its behavior is better described in a localizedpicture, with an effective moment of 3 .57μ B(consistent with either 5 f2or 5f3) obtained from a Curie-Weiss ﬁtto single crystal high temperature magnetic susceptibility measurements ( T=300–800 K) [ 18]. Studies of isostructural transuranium compounds offer the opportunity to investigate how physical properties evolve as afunction of 5 fshell ﬁlling, with the possibility in the case of NpIr of investigating the signiﬁcance of the itinerant ferromag-netism to the nature of the superconductivity. Here we reporton the synthesis and characterization of NpIr, an isostructuralanalog of UIr, via magnetic susceptibility, resistivity, and heatcapacity measurements. The experiments reveal no magneticordering or superconductivity down to 0 .55 K, in contrast to the development of ferromagnetic order in UIr at T C=46 K, and no evidence of pressure-induced superconductivity. Ab initio calculations reveal the signiﬁcance of the Hubbard +Uterm for the description of the Np 5 felectrons. The calculated phonon dispersions, lattice heat capacity, and latticethermal conductivity are presented. The latter is found to beexceptionally low in the high temperature regime. II. METHODS Polycrystalline samples of NpIr were synthesized at the Institute for Transuranium Elements (ITU) by arc meltingstoichiometric amounts of Np (99 .9%) and Ir (99 .98%) metals under a high purity argon atmosphere (Ar: 6N) in a watercooled copper hearth using a zirconium getter. The samplewas remelted ﬁve times to obtain a good homogeneity and theweight loss was 0 .5%. The as-cast sample was embedded in a tantalum foil and encapsulated in a quartz tube under highvacuum, and then annealed at 873 K for three weeks. X-raypowder diffraction analysis of the samples was performedon a D8 Advance diffractometer with a Bragg-Brentanoconﬁguration, equipped with a Cu x-ray tube (40 kV , 40 mA),a Ge monochromator (111), and a Lynx Eye linear position 1098-0121/2015/91(19)/195146(10) 195146-1 ©2015 American Physical SocietyH. C. WALKER et al. PHYSICAL REVIEW B 91, 195146 (2015) TABLE I. Reﬁned atomic parameters for NpIr (space group P21) at room temperature, with Rwp=10.52 and GoF=2.14. Atom Wyckoff xy z Occ Np1 2a 0.13759 0.00000 0.12635 1 Np2 2a 0.62272 −0.00257 0.63025 1 Np3 2a 0.87689 0.71522 0.38225 1Np4 2a 0.39253 0.71821 0.87415 1 Ir1 2a 0.10633 0.26729 0.10832 1 Ir2 2a 0.62036 0.25888 0.62317 1Ir3 2a −0.12913 0.44782 0.37114 1 Ir4 2a 0.35812 0.45145 0.85769 1 sensitive detector. The powder pattern was recorded at room temperature in step scan mode over a 2 θrange of 10–120◦, with as t e ps i z eo f0 .013◦and a count time of 4 s per step. NpIr was indeed found to be isostructural to UIr, the x-ray diffractionpattern being ﬁtted with a monoclinic structure described bytheP2 1space group, with lattice parameters a=5.5832(6) ˚A, b=10.7368(9) ˚A,c=5.5848(6) ˚A, and β=95.708(5)◦(see Table Ifor reﬁned atomic parameters). Figure 1shows the quality of the reﬁnement. At the lowest 2 θangles there is a broad peak arising from the necessary encapsulation of thesample, but the other features of the data are reproduced bythe reﬁned structure. Magnetic susceptibility and isothermal magnetization mea- surements were performed using a Quantum Design MPMS-7Squid magnetometer on a 89 .9 mg NpIr sample over a temperature range of 2–300 K, in magnetic ﬁelds up to 7 T. The ambient pressure electrical resistivity, magnetoresis- tivity, and Hall effect have been measured in the temperaturerange 1 .8–300 K, and in magnetic ﬁelds up to 14 T, using a Quantum Design PPMS-14T setup, by means of a fourDC probe technique voltage measurement. An NpIr sampleof size ∼1.5×0.4×0.2m m 3was polished on two parallel faces to determine better the form factor. Electrical contacts FIG. 1. (Color online) Rietveld reﬁnement (solid red line) of room temperature x-ray powder diffraction data for NpIr ( ◦) annealed at 873 K. The difference between the calculated and experimental points is shown by the black line which has been offset by −2000 cts /4s for clarity. The vertical tick marks correspond to the Bragg peak positions for the NpIr P21structure shown in the inset.between the sample surfaces and the 50 μm silver wires were ensured by using silver epoxy (Dupont 4929). Finally, eachmounted sample was then encapsulated with Stycast epoxy(1266). For electrical resistivity measurements, the current I was applied in the polished plane. For the magnetoresistivity,Iwas parallel to the voltage direction and parallel to the applied magnetic ﬁeld B. In the Hall conﬁguration, the voltage V Hwas measured perpendicular to the current Iand the applied magnetic ﬁeld B. The Hall resistance ( RH) has been determined by measuring VHunder ﬁelds alternating between +14 and −14 T. The magnetic ﬁeld response VH(B)a tﬁ x e d temperatures has been measured to conﬁrm results obtainedwhen ramping in temperature. For all measurements I=5m A was used. The high-pressure resistance measurements were per- formed by a four-probe DC method in a Bridgman-typeclamped pressure cell, with a solid pressure-transmittingmedium (steatite). Electrical contacts were made with 25 μm diameter platinum wires lightly pressed onto the sample.Before each measurement the cell was loaded and clamped atroom temperature. The exact pressure inside the pressure cellwas determined later by using the pressure dependence of thesuperconducting transition temperature of lead as a manometer[19]. Measurements were performed on a ∼50μg sample of size∼20×50×500μm 3in the temperature range 1 .8–300 K up to 17 .3G P a . Heat capacity measurements on a 5 .6 mg sample of NpIr were performed over the temperature range T=2–250 K, and on a 0.9 mg sample down to 0 .55 K, via the standard relaxation calorimetry method using a Quantum Design PPMS-9 withthe 3He refrigeration insert, after the samples had been coated in Stycast. The data have been corrected for theaddenda and the stycast. Self-heating effects in neptuniummake it difﬁcult to reach lower temperatures. No isostructuralphonon blank exists, so instead, in order to best estimate thephonon contribution, we have followed two strategies. First,we have synthesized orthorhombic ThIr (space group Cmcm ) and cubic LuIr ( Pm¯3m), veriﬁed their structures and phase purities using x-ray powder diffraction, and measured theirheat capacities using the same relaxation method. Second, wehave calculated ab initio the phonon spectrum of NpIr and the lattice contribution to the heat capacity. III. RESULTS AND DISCUSSION A. Magnetic susceptibility The magnetic susceptibility data for NpIr, shown in Fig.2(a), reveal no indication of any magnetic transition above T=2 K. There is no difference in the results for zero-ﬁeld cooling and in-ﬁeld cooling. Between 50 and 300 K the inversesusceptibility may be modeled by a Curie-Weiss law, see theinset to Fig. 2(a), to obtain a Curie-Weiss temperature of −104±1 K, and an effective paramagnetic moment of 3 .01± 0.02μ B/Np. Starting with the Curie-Weiss temperature, such a large negative value is indicative of strong antiferromagneticinteractions. Intriguingly, this is of a similar magnitude to the−300 K obtained from high temperature measurements on UIr [ 18], but in that case, in spite of the antiferromagnetic interactions, it undergoes a ferromagnetic phase transition, 195146-2MAGNETIC, ELECTRICAL, AND THERMODYNAMIC . . . PHYSICAL REVIEW B 91, 195146 (2015) 0 50 100 150 200 250 30000.0050.010.0150.02 T(K)Magnetic susceptibility ( μB/Np.T) 0 100 200 30001002001/χ (Np.T/ μB) T(K) ΘCW=−104 K μeff=3.01 μB/Np(a) −8 −6 −4 −2 0 2 4 6 8−0.1−0.0500.050.10.15 μ0H (T)Magnetisation ( μB/Np) 2 K 10 K 40 K(b) FIG. 2. (Color online) (a) Magnetic susceptibility of NpIr mea- sured with μ0H=1 T. The inset shows a Curie-Weiss ﬁt to the inverse susceptibility. (b) Isothermal magnetization of NpIr measured atT=2,10, and 40 K. highlighting that these compounds sit at the interface between antiferromagnetic and ferromagnetic order. Next we considerthe value for the effective paramagnetic moment, which isinconsistent with either a 5 f 3or a 5 f4conﬁguration in both the Russell-Saunders (3 .62μB/Np and 2 .68μB/Np) and the intermediate coupling schemes (3 .68μB/Np and 2.76μB/Np [ 20]), and suggests, therefore, that possibly the 5felectrons are not fully localized. If alternatively the ﬁt to the inverse susceptibility is only made above T=250 K, then a Curie-Weiss temperature of −155±5 K and an effective paramagnetic moment of 3 .20±0.02μB/Np are obtained, still inconsistent with a localized moment picture. Such resultsare not wholly dissimilar to those for UIr, for which a localizedbehavior was only observed at high temperatures ( T> 300 K) [18]. Regrettably, it was not possible to measure the magnetic susceptibility above room temperature using our experimentalsetup. However, as might be expected given the standardtrend towards increasingly localized electrons on spanningthe actinide series, NpIr is perhaps more localized at lowtemperatures than UIr, as demonstrated by the difference inthe effective moments, obtained by making a Curie-Weissﬁt to the inverse susceptibility over the temperature rangeT=50–100 K, of 1 .67μ B/U[17] and 2 .96±0.01μB/Np for polycrystalline UIr and NpIr, respectively. Figure 2(b) presents the isothermal magnetization data for T=2,10, and 40 K. These reveal no saturation for magnetic ﬁelds up to 7 T, and no evidence of magnetic hysteresis.B. Resistivity The electrical resistivity of an annealed polycrystalline sample of NpIr is shown in Fig. 3(a). Interestingly, the room temperature absolute resistivity of NpIr is ρ=122μ/Omega1cm, which is comparable to the 80 μ/Omega1cm reported for the best quality UIr crystals [ 21]. With decreasing temperature, we observe the presence of a very broad maximum centeredatT max∼150 K, reminiscent of that observed in several neptunium-based systems such as NpCoGa5[22] and NpPd3 [23], which is indicative of a Kondo-type behavior. Below 50 K coherence sets in and the resistivity collapses to only1.95μ/Omega1cm atT=1.8 K. Below 6 K [see inset to Fig. 3(a)], the strong curvature can be modeled by a Fermi liquidbehavior: ρ=ρ 0+AFLT2, with ρ0=1.42±0.02μ/Omega1 cm andAFL=0.209±0.001μ/Omega1cm K−1. This gives a residual resistivity ratio ρRT/ρ0∼90, which approaches the value of 230 reported for the highest quality single crystal UIr [ 21]. Over the temperature range T=5–20 K, the data can be mod- eled by a non-Fermi liquid law: ρ=ρ0+A5/3T5/3withρ0= 0.92±0.07μ/Omega1cm and A5/3=0.412±0.001μ/Omega1cm K−1. Such a variation in the temperature dependence with temper-ature interval may indicate different spin ﬂuctuation regimeswithin NpIr. Figure 3(b) shows the longitudinal magnetoresistivity of NpIr at 14 T over the temperature range T=1.8−50 K, which is similar in shape to that of UAl 2below 20 K [ 24]. For all temperatures up to 50 K the magnetoresistive contributionis positive and, at T=1.8 K, large relative to the resistivity (∼140%). There is no clear evidence of any anomaly down to this temperature that might be associated with a magneticphase transition. The ﬁeld dependence of the isothermalmagnetoresistivity for a range of different temperatures isshown in the inset to Fig. 3(b). For all temperatures mea- sured ( T=1.8,3,5, and 50 K), the magnetoresistivity varied quadratically as a function of the magnetic ﬁeld. Figure 3(c) presents the Hall coefﬁcient ( R H)f o rT= 1.8–75 K for NpIr. RHis slightly enhanced approaching 10 × 10−10m3C−1at 50 K, which is nevertheless rather similar to values for metallic systems, e.g., Cu: 0 .5×10−10m3C−1. Upon decreasing the temperature, RHdecreases almost lin- early, changing sign at 25 K, before reaching a minimumat 7 K, below which R Hstarts to increase again. As RH is the result of the combination of the electron and hole contributions with different carrier velocities and relaxationtimes, our data suggest that the nature and the mobility of thecarriers are changing drastically with temperature. The Hallcoefﬁcient is composed of two terms: R H=R0+RS, where R0is the ordinary Hall coefﬁcient, and RSis the extraordinary or anomalous Hall coefﬁcient. Following Ref. [ 25] we replace RSby a term dependent on the magnetic susceptibility to give RH(T)=R0+R1χ∗(T), (1) where the ﬁrst term R0describes the Hall effect due to the Lorentz motion of the carriers and/or residual skewscattering by defects and impurities, while the second termcomes from skew scattering by Kondo impurities. In thisformula χ ∗is the reduced susceptibility, approximated by χ(T)/C, where Cis the Curie-Weiss constant obtained from the ﬁts to the inverse susceptibility above. Assuming R0and 195146-3H. C. WALKER et al. PHYSICAL REVIEW B 91, 195146 (2015) 0 100 200 300050100150 T (K)Resistivity ( μΩ cm) 0 10 200204060 T (K)ρ (μΩ cm)(a) 0 10 20 30 40 5004080120 T (K)(ρ14T−ρ0T)/ρ0T (%) 0 5 1004080120 B (T)(ρ−ρ0T)/ρ0T (% )(b) 0 20 40 60 80−1001020304050 T (K)RH (10−10 m3C−1) 7 8 9 10 11−100102030 χ* (arb. units)RH (10−10 m3C−1)(c) FIG. 3. (Color online) (a) Electrical resistivity of NpIr, with inset showing ﬁts to a Fermi liquid (solid red line) and non-Fermi liquid (dashed green line) model at low temperatures. (b) Isoﬁeld magnetoresistance of NpIr at 14 T, with inset showing the ﬁeld dependence of the isothermal magnetoresistance for T=1.8 K (o), 3 K ( /square), 5 K ( /triangle), and 50 K ( ⋆). (c) Hall effect measurements of NpIr, with inset showing ﬁt toRH(T)=R0+R1χ∗(T). R1are independent of the temperature, plotting RH(T)a s af u n c t i o no f χ∗(T), as shown in the inset to Fig. 3(c), givesR0=+7.19±0.02×10−9m3C−1, indicating that the ordinary Hall effect is dominated by the hole contribution.A simple one-band model then provides an estimation of8.69±0.03×10 26m−3for the concentration of free holes, giving an upper limit for the actual carrier concentration inNpIr in the normal state. This may then be converted into arough estimate of 0 .05 for the number of free holes per formula unit at high temperature. The global shape of the electrical resistivity of NpIr does not change drastically under pressure up to 12 GPa (Fig. 4). However, with increasing pressures up to ∼12 GPa, the maxi- mum around 150 K, seen in Fig. 3(a), becomes less apparent, but the onset of coherence below this temperature remains.Above ∼12 GPa, the resistivity evolves more smoothly with temperature with a shift of the scattering to higher temperature.At low temperature the resistance can be ﬁtted according toρ=ρ 0+aTn, where the exponent stays essentially constant ∼1.33±0.15 until above 13 GPa, where it ﬂuctuates, as shown 0 100 200050100150200 T(K)Resistance( Ω) 10010210−210−1100 T (K)(R− R0)/(R300−R0)0.16GPa 1.17GPa 2.27GPa 3.33GPa 4.85GPa 5.99GPa 7.9GPa 8.63GPa 10.05GPa 11.67GPa 13.3GPa 15.81GPa 17.27GPa 0 10 20012 Pressure (GPa)Exponent(a) (b) (c)3.33 GPa4.85 GPa8.63 GPa11.67 GPa15.81 GPa17.27 GPa FIG. 4. (Color online) (a) Electrical resistance of NpIr measured as a function of temperature for a selection of applied pressures.(b) Comparison of normalized residual resistance data for NpIr for pressures between 0 .16 GPa and 17 .65 GPa. (c) Temperature exponent extracted from low temperature ﬁts to the resistance ofR=AT n+R0.in Fig. 4(c). No hint of a superconducting transition is detected down to 1 .8 K for any pressure below 17 GPa. One possible explanation for the observed absence of ferromagnetism in NpIr above 1 .8Km a yb ei n f e r r e df r o m a comparison of the forms of the resistance curves ofUIr and NpIr. At ambient pressure, the resistivity of NpIr[Fig. 3(a)] resembles the resistivity of UIr under pressures greater than 2 GPa [ 21], which appear to be unfavorable conditions for ferromagnetism in UIr. The requirements for theappearance of superconductivity in UIr are extremely drastic,and strongly dependent on the pressure transmitting medium[21], but our measurements on NpIr were performed under similar hydrostatic conditions. The superconducting transitiontemperature of UIr is lower than our accessible temperaturerange, so it is possible that NpIr may still superconduct attemperatures below 1 .8 K, but the lack of a ferromagnetic state, which seems to be a prerequisite for nonconventionalsuperconductivity in UIr, potentially makes this less likely. C. Heat capacity As shown in Fig. 5, the heat capacity of NpIr varies smoothly between 2 K and 270 K, with no anomalies whichmight be associated with any phase transition. When a straightline ﬁt is made to C P/Tversus T2forT/lessorequalslant7K ,a si n the inset to Fig. 5, we obtain γ=175±1m J K−2mol−1 for the electronic heat capacity and a Debye temperature /Theta1D=145.7±0.5 K. The electronic heat capacity of NpIr is considerably greater than that for UIr ( γ=49 mJK−2mol−1, Ref. [ 26]), and is indicative of strong electronic correlations. Combining the value of γwith the coefﬁcient AFLfor the quadratic term in the low temperature resistivity obtainedin Sec. III B, gives a Kadowaki-Woods ratio [ 27]A FL/γ2= 0.68±0.01×10−5μ/Omega1cmK2mol2mJ−2, implying that NpIr is a moderately heavy fermion material, which would beconsistent with hybridization causing an effective param-agnetic moment below the free ion value. Such a valuefor the Kadowaki-Woods ratio is comparable with that forUAl 2(0.89×10−5μ/Omega1cmK2mol2mJ−2)[27], in which the low temperature speciﬁc heat is well expressed in termsof spin ﬂuctuations that prevent any magnetic order above1 K. However, the paramagnon upturn present in UAl 2data is absent in that for NpIr. USn 3also displays a similar 195146-4MAGNETIC, ELECTRICAL, AND THERMODYNAMIC . . . PHYSICAL REVIEW B 91, 195146 (2015) 0 50 100 150 200 2500102030405060 T(K)CP (JK−1mol−1) NpIr ThIr LuIr Calc. 0 20 400100200 T2 (K2)CP/T (JK−2mol−1) FIG. 5. (Color online) Heat capacity of NpIr, ThIr, and LuIr, with inset showing a ﬁt to CP/T=γ+βT2at low temperatures. Also shown as a dashed line is the Dulong-Petit limit of 6 Rfor the phonon heat capacity, and the ab initio calculated lattice speciﬁc heat as a solid black line. value of AFL/γ2=0.78×10−5μ/Omega1cmK2mol2mJ−2[27], with a very similar speciﬁc heat value, γ=172 mJK−2mol−1 (Ref. [ 28]) and despite the lack of a paramagnon upturn in the low temperature heat capacity, is also classiﬁed as a spinﬂuctuator system. Figure 5also compares the heat capacity of NpIr with that of LuIr and ThIr, in an attempt to estimate the lattice contribution.Regrettably, neither ThIr nor LuIr, crystallizing in the Cmcm andPm¯3mspace groups, respectively, are isostructural with NpIr. However, we have chosen to use the data for ThIr asa phonon blank, since its molar mass is very similar to thatof NpIr, and its Debye temperature ( /Theta1 D=176.6±0.3K )i s closer than that of LuIr ( /Theta1D=254.7±0.4 K). Furthermore it agrees better with the ab initio calculated (see Sec. IV) phonon heat capacity of NpIr (shown as the black solid line). Hencewe estimate the total electronic (5 f+conduction electron) contribution to the heat capacity of NpIr as C el P=CNpIr P−CNpIr P(phonons) , (2) Cel P=CNpIr P−CThIr P+γThIr·T, where γTh=4.8±0.1m J K−2mol−1is the electronic heat capacity of ThIr. Figure 6(a)displays the total electronic con- tribution to the heat capacity of NpIr divided by temperature,revealing a broad peak centered at T=30 K and a near constant behavior at high temperatures. Integrating the total electronic contribution to the heat capacity of NpIr allows an estimate for the entropy to beobtained, which is shown in Fig. 6(b). The entropy varies smoothly, and by extrapolation to higher temperature appearsto be compatible with the free ion value for Np 3+(Rln 10) or Np4+(Rln 9). Assuming that the 5 fcontribution to the heat capacity is close to saturation by ∼200 K, we can deduce from Fig. 6(a) that the electronic coefﬁcient has an upper limit of γ=20±2m J K−2mol−1at high temperature. Such a considerable difference between the low and high temperatureelectronic heat capacities suggests that the enhanced low0 50 100 150 200 250050100150200250 T(K)CP/T (mJK−2mol−1) 0 50 100 150 200 250051015Δ S (JK−1mol−1) T (K)(a) (b)Rln2Rln3Rln4Rln5Rln6 FIG. 6. (Color online) (a) Total electronic (5 f+conduction elec- trons) contribution to the heat capacity of NpIr divided by temperature obtained from CNpIr P−CThIr P+γThIr·T. (b) The entropy obtained by integrating the data in panel (a) as a function of temperature. temperature γvalue is due to a strong Kondo interaction while localization features are enhanced at higher temperatures. IV .AB INITIO CALCULATIONS A. Methodology Electronic structure calculations were carried out using the Vienna ab initio simulation package (V ASP) [ 29,30], with the generalized gradient approximation (GGA) as the density-functional theory (DFT) exchange-correlation functional, aswell as with its extension to treat strongly correlated electrons,DFT with an additional Hubbard Uterm (DFT +U)[31,32]. Within the GGA +Uapproach, we have used the Dudarev et al. formulation [ 32], where the Hubbard and exchange parameters, UandJ, respectively, are introduced to account for the strong on-site correlations between the neptunium 5 f electrons. This helps to remove the self-interaction error andimproves the description of correlation effects in the open5fshell. We have chosen a Hubbard Uvalue of 4 .0e V and an exchange parameter Jvalue of 0 .6e V ,w h i c ha r e in the range of accepted values for Np and Pu compounds[33,34]. To test the dependence of our results on the Uvalue, we have also performed calculations for U=2a n d3e V , and for U=0 eV , i.e., for the common GGA functional. Further, to deal with the problem of degenerate metastablestates when using the DFT +Umethodology, we have used the occupation matrix control (OMC) method proposed byDorado et al. [35] This method consists of the direct control of the strongly correlated electron occupation matrices. Details ofthe electronic structure calculations can be found in Ref. [ 36]. 195146-5H. C. WALKER et al. PHYSICAL REVIEW B 91, 195146 (2015) We have considered three different magnetic orders: ferro- magnetic (FM), antiferromagnetic (AFM), and paramagnetic(PM) order. In the FM ordered state, we assume that all the Npions have collinear magnetic moments oriented along the c direction. In the AFM ordered state, the Np ions are consideredto be collinear with magnetic moments changing sign fromone Np plane to another. Finally, for the PM ordered state, weadopt the disordered local moments (DLM) approach [ 37,38], which states that paramagnetism can be modeled as a statewhere atomic magnetic moments are randomly oriented (non-collinear magnetism), valid for materials that display a Curie-Weiss paramagnetism, such as NpIr. The DLM approach canbe simpliﬁed by considering only collinear magnetic momentswhen the spin-orbit coupling is not taken into account. Hence,the problem of modeling paramagnetism becomes a problem ofmodeling random distributions of collinear spin components.It can be solved by using special quasirandom structures(SQS) [ 39]. An SQS is a specially designed supercell built of ideal lattice sites to mimic the most relevant pair and multisitecorrelation functions of a completely disordered phase (PMorder in our case). As a PM simulation cell we used an extendedlattice cell of 64 atoms. We note however that on account ofthe large simulation cell needed for the DLM calculations,it was not possible to perform these including the spin-orbitinteraction. B. GGA +Uresults A full structural and atomic-site relaxation has been carried out for NpIr (with U=4 eV and J=0.6e V ) .W eh a v e found, in agreement with experiments, that NpIr crystallizesin a monoclinic structure (described by the P2 1space group). We further found that the calculated lattice parameters a= 5.6072 ˚A,b=10.7829 ˚A,c=5.6088 ˚A, and β=95.708◦are very close to the experimental values (see Sec. II). The relaxed atomic positions for the FM order are given in the Appendix,where again the proximity to the experimental atomic positionscan be observed. The total energy for the three different magnetic orders have been calculated. Although the energy differences are small, wefound that the FM order has the lowest total energy, followedby the PM and AFM orders, having 0.039 and 0.056 eV per formula unit higher total energies, respectively. Theseﬁndings differ from the experimental results which reveal nosign of magnetic order for T> 1.8 K, while the Curie-Weiss temperature implies strong antiferromagnetic interactions. To investigate the origin of the obtained energy order we have investigated the inﬂuence of the Hubbard Uparameter and considered the inﬂuence of the spin-orbit interaction.Performing calculations with Uvalues of 0, 2, and 3 eV did not lead to a change in the relative energy sequence. The FM phasewas always found to have the lowest total energy, followed bythe PM phase, and then by the AFM phase. The relative energydifferences were similar to those found for U=4e V .W e emphasize, however, that these calculations were performedwithout the spin-orbit interaction, as the DLM calculationsare computationally too heavy with spin-orbit interaction. Wehave for the sake of comparison computed a hypotheticalnonmagnetic phase of NpIr (i.e., no moments at all) with thespin-orbit interaction. We ﬁnd its total energy to be higherthan that of both the FM and AFM phases. This indicates thata nonmagnetic state is unlikely and would be an insufﬁcientrepresentation of the PM phase. Thus, in the absence of DLMcalculations with spin-orbit interaction we cannot deﬁnitelystate what the lowest energy magnetic order of NpIr is. Itis however worth noting at this point that isostructural UIr,which also displays antiferromagnetic interactions based onthe Curie-Weiss temperature, in fact orders ferromagneticallybelow 46 K. The local spin moments of the Np ions have alsobeen calculated; these are 3 .78μ B,3.74μB, and 3 .71μBfor the FM, PM, and AFM orders, respectively. The calculated partial densities of states (DOS) for the three different magnetic orders are given in Fig. 7. There are signiﬁcant hybridizations of the Np- fand Ir- delectrons in the energy range of −3t o+3 eV , as can be inferred from the similar DOS structures. The Coulomb Upotential leads to a splitting in the 5 fspectrum (of about 3 eV) which can be clearly seen for the PM and AFM ordered states. To assess the importance of the spin-orbit interaction on the atomic magnetic moments we have carried out an analysis ofits inﬂuence on the FM order. We ﬁnd that the total magneticmoment M, written as the sum of the spin magnetic moment -0.6-0.4-0.20 0.2 0.4 0.6 -6 -4 -2 0 2 4DOS (states/eV/atom)FM Np-f states Ir-d states -6 -4 -2 0 2 4 Energy (eV)PM -6 -4 -2 0 2 4AFM FIG. 7. (Color online) Atom-resolved partial density of states for the three different magnetic orders calculated for NpIr using the GGA +U approach. 195146-6MAGNETIC, ELECTRICAL, AND THERMODYNAMIC . . . PHYSICAL REVIEW B 91, 195146 (2015) 0 0.2 0.4 0.6 0.81 1.2 -6 -4 -2 0 2 4DOS (states/eV) Energy (eV)Np-f states Ir-d states 0 0.2 0.4 0.6 0.81 1.2 -6 -4 -2 0 2 4DOS (states/eV) Energy (eV)Np-f states Ir-d states FIG. 8. (Color online) Top: Total and partial density of states of NpIr calculated using the GGA +Uapproach and including the spin-orbit coupling. Bottom: The same, but computed with the GGAapproach and including the spin-orbit coupling. MSand the orbital magnetic moment MOisM=MS+MO= 3.748μB−0.933μB=2.815μB. This value is very close to the intermediate coupling value for a 5 f4conﬁguration. The total and partial density of states including spin-orbit couplingare shown in Fig. 8(top). C. GGA results To investigate the importance of the Hubbard Uterm, we have performed a similar study using the plain GGA approach.Similarly to the spin-orbit case, we only analyzed the FMorder. For this approximation the lattice parameters are a= 5.4620 ˚A,b=10.5037 ˚A,c=5.4636 ˚A, and β=95.708 ◦. Although they are in good agreement with the experimentalresults, the deviation with respect to these is larger than whencomparing with the GGA +Uresults. The local magnetic moments on the Np ions without accounting for the SOcoupling have a magnitude of 3 .113μ B. If we include the SO coupling the spin magnetic moment drops to 2 .842μBwhile the orbital magnetic moment becomes −2.527μB,g i v i n ga net moment of only 0 .315μB. The calculated density of states are plotted in Fig. 8(bottom), where it can be observed that there is a splitting of the forbitals (into 5 f5 2and 5f7 2)i nt h e case that SO interaction is included. Nonetheless the manifold of 5fstates appears close to the Fermi energy and has itsmaximum at the Fermi energy. Application of the GGA +U method conversely splits the 5 fmanifold of states and leads t oal o w5 fDOS near the Fermi level (Fig. 8, top). D. Phonons properties and thermal conductivity We have calculated the phonons of NpIr using the ﬁnite- displacements method in conjunction with supercells consist-ing of 2 ×2×2 primitive cells (128 atoms). The interatomic forces were calculated with V ASP, adopting as above, theGGA+Uapproach for the electronic structure. The phonon modes were obtained with the phonopy package [ 40]i nt h e quasiharmonic approximation. To enable this approximationthe system volume has been isotropically expanded by 2%from the GGA +Urelaxed volume. The anharmonic effects induced by the volume dependence of phonons frequenciesare explored and the lattice thermal properties such as thelattice speciﬁc heat and the phonon thermal conductivity arecalculated. The ab initio calculated phonon density of states and phonon dispersion curves ω nqare given in Fig. 9. The 16-atom unit cell of NpIr results in 48 phonon modes with a ratherhomogeneous spreading of the bands from 0 to 4 THz. Theatom-projected phonon DOS (right-hand panel) shows that atlow energies the contribution from the Ir atoms is larger thanthat from the Np atoms, while at higher vibrational frequenciesthis behavior is reversed. The lattice heat capacity C pcan be obtained from the Gibbs free energy G(T,p) at constant pressure, Cp= −T(∂2G/∂T2). The Gibbs free energy is obtained from G(T,p)=min V[U(V)+Fphon(T,V)+pV], (3) where U(V) is the volume-dependent electronic total energy andFphonthe phonon free energy, Fphon(T,V)=/integraldisplay∞ 0dωg (ω,V) ×[/planckover2pi1ω/2+kBTln(1−e−/planckover2pi1ω/kBT)],(4) withg(ω,V) the phonon DOS, computed as mentioned above for different volumes. The ab initio calculated Cp(T)o fN p I r 0 0.51 1.52 2.53 3.54 Γ B AE D ZΓ YC ZE (THz) 0 0.5 1 1.5 2 2.5 Phonon DOS/atomNp Ir FIG. 9. (Color online) GGA +Ucalculated phonon dispersions (left panel) and corresponding projected phonon density of states per atom (right panel) of NpIr. The q-point labels in the left panel are those for the standard high-symmetry positions of the monoclinicprimitive Brillouin zone. 195146-7H. C. WALKER et al. PHYSICAL REVIEW B 91, 195146 (2015) is shown in Fig. 5. As can be noted the computed lattice heat capacity is smaller than the measured heat capacity of NpIr.Its temperature dependence corresponds very well with themeasured heat capacity of ThIr. Thus, this conﬁrms that the C p of ThIr can be used as a phonon blank to determine the electronic contribution to the heat capacity of NpIr. Lastly, we investigate the lattice thermal conductivity κL of NpIr. A low thermal conductivity of materials is of interest as this would lead to a high thermoelectric ﬁgure of merit[41]. The thermal conductivity of NpIr is unknown, but its anisotropic, low-symmetry crystal structure suggest that itslattice contribution could be very small. We have computedthe (direction averaged) κ L(T) of NpIr using an approximate solution to the phonon Boltzmann transport equation in therelaxation-time approximation, κ L(T)=1 3/summationdisplay n/integraldisplaydq 8π3v2 nqτnqCnq, (5) where the sum is over all phonon modes, vnqis the group velocity of a given phonon mode, Cnqis the mode heat capacity depending only on the mode frequency ωnqand the temperature, and τnqis the mode dependent relaxation time, which is computed here on the basis of the model ofBjerg et al. [42]. Furthermore, for the determination of the lattice thermal conductivity the Gr ¨uneisen parameter γis a fundamental quantity. It characterizes the relation betweenphonon frequency and crystal volume change and is deﬁned as γ nq=−Vuc ωnq∂ωnq ∂Vuc, (6) where Vucis the unit cell volume. The Gr ¨uneisen parameter provides an estimation of the anharmonicity strength in acompound. The calculated total lattice thermal conductivity of NpIr is shown in Fig. 10. From temperatures of 30 to 100 K an exponential decrease of κ L(T) is observed, which is due to the exponential increase of the phonon-phonon scattering via 0123456789 10 0 100 200 300 400 500 600 700 800 900κ (Wm-1K-1) T(K)κLκaaκbbκcc FIG. 10. (Color online) Calculated lattice thermal conductivity κ of NpIr as a function of temperature. Shown is the total thermal conductivity κLas well as the thermal conductivities along the crystallographic axes. The off-diagonal components of the thermalconductivity are given in the inset.theUmklapp mechanism. For temperatures above 100 K the Umklapp mechanism governs the scattering processes and consequently an intrinsically low thermal conductivity arises.The lattice thermal conductivity at room temperature assumesa value of 0.64 Wm −1K−1and a value of 0.19 Wm−1K−1 at 970 K. Note that these are ultralow values [ 43]; for comparison, recent measurements on orthorhombic SnSecrystals with a very high thermoelectric ﬁgure of merit gaveroom-temperature values between 0.5 and 0.7 Wm −1K−1and values of 0 .23–0.34 Wm−1K−1at 970 K, depending on the crystal axis [ 44]. Very recent ab initio calculations for NaBi predicted ultralow values of about 2 Wm−1K−1at 300 K [45]. NpIr is thus predicted to have a record low lattice thermal conductivity at high temperatures. Apart from a lowlattice thermal conductivity, a high electrical conductivity isdesirable, too, for suitable thermoelectric materials [ 41]. As an intermetallic, NpIr is expected to have a good electricalconductivity and also a considerably larger contribution tothe electronic thermal conductivity than in the chalcogenidesystems. However, it is not currently possible to simplydistinguish between the phonon and electron contributionsexperimentally. Low lattice thermal conductivities can be found for com- pounds with a large molecular weight or a complex, anisotropiccrystal structures [ 41]; both conditions are fulﬁlled for NpIr. In Fig. 10we in addition show the axis-projected thermal conductivities as well as the off-diagonal components (inthe inset). The latter arise because of the low symmetryof the monoclinic structure. The three crystallographic axis-projected thermal conductivities are of similar size in NpIr. To assess the importance of the lattice anharmonicities for the low thermal conductivity the Gr ¨uneisen parameters are evaluated. The calculated q-averaged Gr ¨uneisen parameters projected on the crystallographic axes are: ¯ γ a=2.46, ¯γb= 3.69, and ¯ γc=2.46. As the q-dependent γnqvalues can be negative, their absolute values have been computed. The valuesfor NpIr are large and anisotropic (comparable to those forSnSe, Ref. [ 44]), which provides evidence for substantial lattice anharmonicities that induce heat dissipation and lowvalues of the thermal conductivity. In addition, the Gr ¨uneisen parameter ¯ γ balong the baxis is much larger than those along theaorcaxis. From this we can infer that the phonon modes along the baxis are more strongly anharmonic, and this leads to a weak interatomic bonding and hence a good channel todissipate phonon transport along the baxis. V . CONCLUSIONS In conclusion, a new binary equiatomic Np-Ir intermetallic has been successfully synthesized. Although it is isostructuralwith UIr, it is found to be paramagnetic down to 0 .55 K, despite the presence of possibly antiferromagnetic interactions. Theeffective paramagnetic moment of 3 .20±0.02μ B/Np does not agree well with estimates for a free Np3+or Np4+ ion in either the Russell-Saunders or intermediate coupling schemes, implying some degree of 5 fdelocalization; while heat capacity measurements indicate that NpIr is a moderateheavy fermion system. The form of the electrical resistivity asa function of temperature and a low temperature Sommerfeldcoefﬁcient, which is strongly enhanced relative to that obtained 195146-8MAGNETIC, ELECTRICAL, AND THERMODYNAMIC . . . PHYSICAL REVIEW B 91, 195146 (2015) at high temperatures, both indicate that NpIr should be regarded as a Kondo system. Ab initio calculations reveal that the GGA +Uapproxi- mation provides a better description of the structural andelectronic properties of NpIr than the plain GGA approach.After relaxation, the calculations give the same geometricalstructure as the experimental one; however, the calculationswithout spin-orbit interaction suggest that ferromagnetic orderis energetically the most favorable, followed by the paramag-netic and antiferromagnetic ordered states. This result standsin contrast to the lack of any experimental observations offerromagnetism, and suggest that DLM calculations with spin-orbit interaction are needed to address this issue thoroughly.The absolute value of the local spin magnetic moments onthe Np ions is of the order of 3 .7μ B. However, due to a sizable opposite orbital magnetic moment, when includingthe spin-orbit interaction, the moment drops to 2 .81μ B, which is of a similar magnitude to that extracted frommagnetic susceptibility measurements. The calculated latticeheat capacity of NpIr is in good agreement with the measuredheat capacity of ThIr, which hence can be regarded as a phononblank for NpIr. The lattice thermal conductivity of NpIr ispredicted to be exceptionally low at high temperatures. ACKNOWLEDGMENTS We thank F. Kinnart, D. Bou ¨exi`ere, and G. Pagliosa for their technical support, and R. Caciuffo for useful discussions.The high purity Np metal required for the fabrication of theTABLE II. Calculated atomic positions for NpIr (space group P21). Atom Wyckoff xyz Np1 2a 0.12498 −0.01371 0.12513 Np2 2a 0.62511 −0.01370 0.62492 Np3 2a 0.87507 0.71392 0.37498Np4 2a 0.37500 0.71391 0.87516 Ir1 2a 0.12498 0.25937 0.12504 Ir2 2a 0.62492 0.25939 0.62481Ir3 2a 0.87489 0.44092 0.37484 Ir4 2a 0.37503 0.44089 0.87510 compound was made available through a loan agreement between Lawrence Livermore National Laboratory (LLNL)and ITU, in the framework of a collaboration involving, LLNLLos Alamos National Laboratory, and the U.S. Department ofEnergy. H.C.W. and K.A.M. acknowledge the access to theinfrastructures of JRC-ITU and ﬁnancial support providedby the European Commission within its “Actinide UserLaboratory” program. P.M. and P.M.O. acknowledge supportfrom the Swedish Research Council (Sweden) and the SwedishNational Infrastructure for Computing (SNIC). APPENDIX: OPTIMIZED STRUCTURE OF NpIr In Table IIwe give the GGA +Uoptimized atomic positions of the NpIr compound, which was computed to crystallize inthe monoclinic structure. [1] J. 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PhysRevB.94.115105.pdf	PHYSICAL REVIEW B 94, 115105 (2016) Competing pairing channels in the doped honeycomb lattice Hubbard model Xiao Yan Xu,1Stefan Wessel,2and Zi Yang Meng1 1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Institute for Theoretical Solid State Physics, JARA-FIT and JARA-HPC, RWTH Aachen University, 52056 Aachen, Germany (Received 1 July 2016; revised manuscript received 17 August 2016; published 2 September 2016) Proposals for superconductivity emerging from correlated electrons in the doped Hubbard model on the honeycomb lattice range from chiral d+idsinglet to p+iptriplet pairing, depending on the considered range of doping and interaction strength, as well as the approach used to analyze the pairing instabilities. Here, weconsider these scenarios using large-scale dynamic cluster approximation (DCA) calculations to examine theevolution in the leading pairing symmetry from weak to intermediate coupling strength. These calculationsfocus on doping levels around the van Hove singularity (VHS) and are performed using DCA simulationswith an interaction-expansion continuous-time quantum Monte Carlo cluster solver. We calculated explicitlythe temperature dependence of different uniform superconducting pairing susceptibilities and found a consistentpicture emerging upon gradually increasing the cluster size: while at weak coupling the d+idsinglet pairing dominates close to the VHS ﬁlling, an enhanced tendency towards p-wave triplet pairing upon further increasing the interaction strength is observed. The relevance of these systematic results for existing proposals and ongoingpursuits of odd-parity topological superconductivity are also discussed. DOI: 10.1103/PhysRevB.94.115105 I. INTRODUCTION Many aspects of the fascinating physics of the low- energy Dirac electrons in graphene can be explored based on noninteracting tight-binding models on the honeycomb lattice, in particular close to charge neutrality, where the effects ofthe electronic interactions are delayed to the strong-coupling regime due to a vanishing density of states (DOS) at low energies. At ﬁnite doping, however, the presence of even weakinteractions among the electrons is predicted by several studies to lead to new collective behavior. Of particular recent interest are interaction-driven instabilities towards unconventionalsuperconductivity in doped honeycomb systems [ 1–17]. Some early studies concluded that superconductivity might not be stable with respect to charge or spin order for the basicHubbard model on the honeycomb lattice [ 2,3], and a possible quantum liquid state has been suggested recently for the van Hove singularity (VHS) ﬁlling [ 18]. In most of the recent theoretical studies, however, a general tendency towards some ﬂavor of superconductivity upon doping the honeycomblattice is indeed observed. However, various proposals on the nature of the emerging superconducting state and the stability range of competing pairing channels still lead toa mosaic of different scenarios. Several mean-ﬁeld theory and renormalization-group (RG) calculations predict chiral d+idsinglet superconductivity to emerge in the weak- coupling region upon doping towards or onto the VHS, which corresponds to electronic densities of n=3/4 and 5 /4f o r the Hubbard or related models with explicit spin-exchangeterms or extended interactions [ 4–10]. Variational Monte Carlo simulations [ 19] also showed a chiral d-wave solution over a wide range of doping. The d-wave pairing state in this scenario is related to enhanced antiferromagnetic ﬂuctuations near half ﬁlling as well as the VHS-increased DOS. On the other hand, a recent study using the variational cluster approximation (VCA) and cellular dynamical mean-ﬁeld theory (CDMFT) performed for larger values of the local repulsion found a stable p-wave triplet pairing state for a weak nearest-neighbor repulsion [ 11], with possibly a coexisting Kekul ´e pattern [ 12,13,17]. A possible p+ip pairing state was also reported at low ﬁlling from determinantal quantum Monte Carlo studies; however, the sign problemposes restrictions on the accessible system sizes, interaction strengths, and temperature ranges. In addition, Grassmann ten- sor renormalization calculations have been performed [ 14,15], and in Ref. [ 14], ad+idstate is reported for the t−Jmodel, while for inﬁnite local repulsion a p+ipsuperconducting state, coexisting with ferromagnetic order, has been proposed for the Hubbard model at low doping [ 15]. Hence, despite active pursuits, such deviations among the various proposalsand employed methods show that a consistent picture of possible superconductivity even in the basic Hubbard model on the honeycomb lattice is still lacking, apparently due tocompetition among several possible low-energy states upon varying the doping or interaction strength. It thus appears promising and necessary to examine this problem from theperspective of a method that allows us to tune these parameters over a wide range while accounting for the growing local electronic correlations beyond the weak-coupling regime. Here, we employ such an approach by providing results from large-scale dynamic cluster approximation (DCA) [ 20] calculations, with a focus on pairing susceptibilities to probefor uniform superconducting instabilities. Upon systematically increasing the cluster size, we ﬁnd that a consistent picture starts to emerge for the leading pairing channels on thehoneycomb lattice Hubbard model from small- to medium-sized local interactions: while at weak coupling, chiral d+id singlet pairing dominates close to VHS ﬁlling, when theinteraction becomes stronger, a tendency towards p-wave triplet pairing develops. Our calculations are performed withan interaction-expansion continuous-time quantum Monte Carlo (CT-INT) cluster solver [ 21–24], keeping up to 24 cluster 2469-9950/2016/94(11)/115105(7) 115105-1 ©2016 American Physical SocietyXIAO Y AN XU, STEFAN WESSEL, AND ZI Y ANG MENG PHYSICAL REVIEW B 94, 115105 (2016) sites (see Appendix Afor details on the CT-INT approach). Before discussing our results, we provide details about theconsidered model and the DCA computational framework forcalculating the pairing susceptibilities. II. MODEL AND METHOD The Hubbard model on honeycomb lattice has the Hamil- tonian ˆH=ˆH0+ˆHI, ˆH0=−t/summationdisplay /angbracketlefti,j/angbracketrightσˆc† iσˆcjσ−μ/summationdisplay iσˆniσ, (1) ˆHI=U/summationdisplay i/parenleftbigg ˆni↑−1 2/parenrightbigg/parenleftbigg ˆni↓−1 2/parenrightbigg , where tdenotes the hopping amplitude between nearest- neighbor sites /angbracketlefti,j/angbracketright,μis the chemical potential that controls the electronic density, and ˆniσ=ˆc† iσˆciσis the number operator for spin ﬂavor σon the ith lattice sites. Furthermore, Udenotes the on-site Coulomb repulsion. Longer-ranged interaction willnot be considered here, and at ﬁnite doping, especially closeto the VHS, screening plays an important role and cuts off thelong-ranged tail of the Coulomb potential [ 10]. The DCA maps this original lattice model onto a periodic cluster, embedded into a self-consistently determined bath.Spatial correlations within the cluster are treated explicitly,while those at longer length scales are described at thedynamical mean-ﬁeld level [ 20]. For this work, we have systematically employed three cluster sizes, shown in Fig. 1, withN c=3,4, and 12 unit cells. The lattice D6hsymmetry is enforced for the Nc=4 cluster (see the ﬁgure caption), while theNc=3 and 12 clusters explicitly retain this symmetry. For the largest cluster, we are able to study inverse temperatures uptoβt=40 at a coupling of U=2t. Compared to the widely studied square [ 21,23,25–30] or triangular lattices [ 22,31,32], the DCA formalism needs to be modiﬁed for the honeycomblattice, which is a bipartite lattice with two sites per unitcell. In particular, the single-particle Green’s function and theself-energy for spin ﬂavor σare 2×2 matrices G σ α,β(K,iωn) and/Sigma1σ α,β(K,iωn), with band or orbital indices α,β=1,2. We developed a generic scheme for performing DCA calculationson such more complex lattices, with details provided inAppendixes BandC. III. PAIRING SUSCEPTIBILITIES In order to probe for superconductivity with respect to different pairing channels, we consider appropriate pairingorder parameters in real space, /Delta1 η(i)=/summationdisplay lfη(δl)/parenleftbigˆci↑ˆci+δl↓±ˆci↓ˆci+δl↑/parenrightbig , (2) where ηdenotes the different pairing channels: s,p,d,f,p± ip, andd±id. Here, fη(δl) are form factors that correspond to the pairing symmetry ηand are provided explicitly in Fig. 1; δlindicates the pairing bonds (we restrict ourselves to nearest- neighbor pairings, l=1,2,3), and +and−denote triplet and singlet states, respectively. The possible pairing channels canbe classiﬁed according to the irreducible representations of(a) (d)(c)(b) Nc=3 Nc=12Nc=4 11 1 111 s/f 01 1 -10-1 dxy/px -21 11-21 dx-y/py 1w w1w* w*d±id/p±ip FIG. 1. Real-space clusters with (a) Nc=3, (b) 4, and (c) 12 unit cells along with their corresponding momentum patches within the Brillouin zone. The clusters Nc=3 and 12 already retain the lattice D6hsymmetry, while for the Nc=4 cluster we enforce this symmetry, as shown in (b), where the four outer hollow sites and the two outer black sites are equivalent due to this symmetry. (d) The phase factors for difference nearest-neighbor pairing channels, wheres,d xy,dx2−y2,a n dd±idcorrespond to singlet pairing states, while px,py,p±ipandfare triplet states. w=exp(±i2π/3) here. theD6hpoint group of the honeycomb lattice [ 4,9–11]. The corresponding uniform susceptibility for a pairing channel η is then obtained in the imaginary-time formulation as χη(T)=1 N/integraldisplayβ 0dτ/summationdisplay ij/angbracketleftTτ/Delta1† η(i,τ)/Delta1η(j,0)/angbracketright. (3) Transforming to momentum and frequency space and normal- izing by the form factors, we obtain χη(T)=1 β/summationtext p,p/prime,q=0/angbracketleft/Phi1η(k)\|χ(p,p/prime,q)\|/Phi1η(k/prime)/angbracketright/summationtext k/angbracketleft/Phi1η(k)\|/Phi1η(k)/angbracketright, (4) which contains the form factor, written in vector form as \|/Phi1η(k)/angbracketright=/parenleftBigg/summationdisplay lfη(δl)eik·δl,∓/summationdisplay lfη(δl)e−ik·δl/parenrightBigg† .(5) Here, p=(k,iωn),p/prime=(k/prime,iω/prime n), and q=(q,iνm) are four- momenta containing both momentum and frequency. In thefollowing, we restrict ourselves to uniform pairing states,corresponding to q=0 and ν=0; that is, we focus here on the pairing channel with respect to only the point-groupsymmetry. Directly comparing pairing susceptibilities χ η(T) of different channels is not a practical way to numericallyidentify the leading pairing channel because, usually, thenoninteracting pairing susceptibility masks the interactioneffects in the pairing susceptibility within the temperaturerange where the CT-INT simulation can be performed. Two 115105-2COMPETING PAIRING CHANNELS IN THE DOPED . . . PHYSICAL REVIEW B 94, 115105 (2016) routes can be taken to overcome this issue. One possibility is to perform an eigenvalue analysis of the pairing vertex secularequation [ 23–25,33], which usually requires high-quality data for the irreducible vertices and very low temperatures, such thatthe momentum dependence of the leading eigenvector does,indeed, reﬂect the pairing symmetry of the superconductingground state. In the other approach, one subtracts the decou-pled part of the pairing susceptibility χ 0(the particle-particle bubble) from the interacting one, such that the effective pairingsusceptibility χ eff=χ−χ0, stemming from the electronic correlations, can be extracted [ 3,34]. Here, we adopt the latter scheme and thus extract the effective pairing susceptibilities, χη eff(T)=1 β/summationtext p,p/prime,q=0/angbracketleft/Phi1η(k)\|χ(p,p/prime,q)−χ0(p,p/prime,q)\|/Phi1η(k/prime)/angbracketright/summationtext k/angbracketleft/Phi1η(k)\|/Phi1η(k)/angbracketright. (6) Further details on the calculation of the vertex function and the effective pairing susceptibilities within the DCA frameworkare provided in Appendixes DandE. The latter also discusses the relation to the eigenvalue analysis of the pairing vertex. IV . RESULTS Before discussing interaction effects, it is useful to examine the bare ( U=0) pairing susceptibilities, which are shown in Fig. 2at the VHS density n=3/4. Due to the VHS in the bare DOS, the bare χ(T) diverge logarithmically as T→0, with the strongest divergence exhibited by the p waves. This provides an important background to the pairingsusceptibilities in the interacting case. Hence, at ﬁnite U,t h e decoupled part of the pairing susceptibility χ 0(T) is subtracted in order to make the effective pairing susceptibilities χeff(T) manifest, as mentioned above. Turning then to the interacting case, we calculated both the temperature and the ﬁlling dependence of χefffor different 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4χ(T) T/tdxy dx2-y2 d+id ssinglet px py p+ipftriplet EnergyDOS fit to a(log( b/T))2 FIG. 2. Noninteracting ( U=0) pairing susceptibilities for var- ious channels at the VHS ﬁlling. All channels will diverge at lowtemperature due to the logarithmic divergence of the DOS at the VHS. The p-wave susceptibilities exhibit the strongest increases, as indicated by the ﬁt line. The inset shows the peak in the DOS at theVHS for the density n=3/4( g r a y ) .(c)U=2tU =6t (a) (b) density n-0.05 0 0.1 0.15 0.2 0.5d+id U =2t d+id U =6t p+ip U =2t p+ip U =6t -0.02 0.02 0.06 0.1 0.14 0.18 0 0.1 0.2 T/t 0 0.1 0.2 T/td+id p+ipd+id p+ipχeff 0.75 1 FIG. 3. (a) Effective pairing susceptibilities for Nc=3,U=2t at the VHS density n=3/4. (b) Effective pairing susceptibilities for Nc=3,U=6tat the VHS density. (c) Density dependence of the effective pairing susceptibilities at βt=20 (T/t=0.05). All d-wave channels are degenerate, as are the pwaves. The sandfwaves are not divergent and hence are not shown here. values of Uand cluster sizes Nc. In the following, we present explicitly our results for d- and p-wave pairing, which we observe to be the most dominant channels. InFig. 3(a), the temperature dependence of the effective pairing susceptibilities at U=2tand the VHS ﬁlling are shown for the N c=3 cluster. Here, we identify d+idas the dominant pairing channel. However, for U=6t[Fig. 3(b)], the p+iptriplet channel increases and also becomes positive and furthermore exhibits a tendency to diverge. In order to monitorthis behavior as a function of doping, the dependence of χ eff on the density nfor different interaction strengths at βt=20 is shown in Fig. 3(c). The dome-shaped behavior in the d+idsinglet pairing channel χeffindicates an optimal doping between n=0.75 and 0.85. At U=6t,p+ipalso exhibits a dome-shaped χeffmaximum, even though the amplitude is still lower than for d+id. TheNc=4 cluster results, shown in Fig. 4,a r e ,t oal a r g e extent, similar to the Nc=3 data: For weak interactions ( U= 2tand 4t) and close to the VHS, the dominant pairing channel is also d+id. Upon increasing the interaction strength to U=6t,t h ep-wave effective susceptibility again increases. -0.3-0.2-0.1 0 0.1 0 0.1 0.2 0 0.1 0.2-0.05 0 0.05 0.1 0.15 0.5 0.75 1 density nχeffd+id U=4t d+id U =6t py U=4t py U=6t U=2tU =4t (a) (b) (c)dxy dx2-y2px py d+id p+ip T/t T/t FIG. 4. (a) Effective pairing susceptibilities for Nc=4,U=2t at the VHS density n=3/4. (b) Effective pairing susceptibilities forNc=4,U=4tat the VHS density. Note that the data in (a) and (b) share labels. (c) Density dependence of the effective pairingsusceptibility at βt=20 (T/t=0.05). 115105-3XIAO Y AN XU, STEFAN WESSEL, AND ZI Y ANG MENG PHYSICAL REVIEW B 94, 115105 (2016) However, we also ﬁnd differences between the results for Nc=3 and Nc=4. As shown in Fig. 1(b), the cluster momenta for Nc=4a r et h e /Gamma1point and the Mpoints. In the noninteracting band structure, these four cluster momentaare below the Fermi surface for densities n> 0.75. Thus, theN c=4 cluster does not capture charge ﬂuctuations about (below and above) the Fermi surface for densities n> 0.75. This is reﬂected by the effective pairing susceptibilities. Forexample, for U=4tthe effective paring susceptibility at T/t=0.025 in the ( d+id)-wave channel for N c=3i s 0.154, while for Nc=4, it is 0.074, i.e., about half the Nc=3 value. Similarly, due to the deﬁcits of the Nc=4c l u s t e r , thep-wave channel does not exhibit a tendency to increase forU=4tat the VHS ﬁlling [see Fig. 4(b)]. In Fig. 4(c), we still observe a narrow density regime within which theeffective pairing susceptibility for the p ychannel is positive, but it is smaller than that of the d+idchannel. The observed trends suggest that only upon further increasing the interactionstrength might the p ychannel possibly diverge more rapidly than the d+idchannel. We note that the breaking of the degeneracy among the different p-wave channels is due to ﬁnite-size effect; that is, the corresponding form factors havedifferent gap sizes on the ﬁnite cluster momenta. Within the DCA approach, one must study the systematic behavior upon increasing N cin order to draw conclusions about the thermodynamic limit. As demonstrated by compar-ing the N c=3 and Nc=4 results, it is also important to consider clusters that capture the low-energy ﬂuctuations. Wethus also employed the N c=12 cluster within the DCA/CT- INT framework, which has cluster momenta that include the/Gamma1point, the two Kpoints, the three Mpoints, and six other momenta [see Fig. 1(c)]. It provides a more detailed structure of the pairing symmetry than the N c=3 and 4 clusters. Unfortunately, the minus-sign problem becomes much moresevere for N c=12, and we cannot access large values of Ufor Nc=12. Nevertheless, we can draw interesting observations from the accessible parameter range: In Fig. 5(a),f o rU=t, the dominant pairing channel is still d+idnear the VHS ﬁlling, while in Fig. 5(b), we ﬁnd for U=2tthat the p wave starts to increase at low temperatures. The p-wave -0.06-0.04-0.02 0 0.02 0.04 0.06 0 0.1 0.2 T/tdxy dx2-y2px py d+id p+ip 0 0.1 0.2 T/t-0.05 0 0.05 0.1 0.5 0.75 1 density nχeffd+id U =1t d+id U =2t py U=1t py U=2t U=1tU =2t (a) (b) (c) FIG. 5. (a) Effective pairing susceptibility for Nc=12,U=t at the VHS density n=3/4. (b) Effective pairing susceptibility forNc=12,U=2tat the VHS density. Note that the data in (a) and (b) share labels. (c) Density dependence of the effective pairingsusceptibility at βt=40 (T/t=0.025).channels (in particular py) tend to increase more rapidly upon cooling than the d-wave channels. This trend indicates that the p-wave channels compete strongly with d-wave pairing at this interaction strength, such that in the intermediate interactionrange, there is an enhanced tendency for p-wave triplet pairing to eventually dominate over d-wave singlet pairing in the thermodynamic limit. To analyze this trend in moredetail, the density dependence of the leading effective pairingsusceptibilities is shown in Fig. 5(c). The optimal doping range ford+idpairing is consistent with the N c=3 and 4 results. In addition, the enhancement of the p-wave channels upon increasing the interaction strength is quite pronounced ontheN c=12 cluster. Hence, although the minus-sign problem renders us unable to make a deﬁnitive statement about whetherp-wave triplet pairing will eventually replace d-wave singlet pairing, the available N c=12 data up to U=2tsuggest such a scenario. V . DISCUSSIONS AND CONCLUSIONS Our DCA/CT-INT results are consistent with previous reports that in the weak-coupling limit the dominant pairingis the chiral d+idsinglet channel. However, we ﬁnd upon increasing the interaction strength (e.g., for N c=12,U=2t) a clear tendency towards a competing p-wave triplet pairing. This ﬁnding is consistent with several recent ﬁndings. Forexample, it has been reported [ 11] that in the presence of both on-site interaction and nearest-neighbor repulsion, for a widerange of doping around the VHS, the dominant pairing is ap-wave triplet. In Refs. [ 14,15], where the inﬁnite- Ulimit was considered, a ( p+ip)-wave superconducting ground state was proposed. Our calculation focused on the range of small-and medium-strength interactions and indeed suggested thepossibility that the dominant pairing channel changes fromd+idtopwave upon increasing the interaction strength. For the future, it would be interesting to allow also for inhomogeneous pairing states within the DCA calculations inlight of several recent proposals of superconductivity coexist-ing with Kekul ´e patterns [ 12,13,17]. On a more general note, the effect of Hund’s coupling and spin-orbit coupling could beincluded in the DCA calculations, given the Hund’s-coupling-induced triplet pairing scenario of Ref. [ 35] as well as recent NMR experiments on Cu xBi2Se3, a spin-orbital-coupled topo- logical material with moderate electron correlations, whichsuggest an odd-parity, spin-rotation symmetry-breaking tripletpairing state [ 36]. ACKNOWLEDGMENTS The authors thank H. T. Dang, M. Golor, Z.-C. Gu, C. Honerkamp, and T. Ying for helpful discussions. X.Y .X. andZ.Y .M. are supported by the Ministry of Science and Technol-ogy (MOST) of China under Grant No. 2016YFA0300502,the National Natural Science Foundation of China (NSFCGrants No. 11421092 and No. 11574359), and the NationalThousand-Young-Talents Program of China. X.Y .X. gratefullyacknowledges the hospitality of the Institute for TheoreticalSolid State Physics at RWTH Aachen University and supportfrom the Deutsche Forschungsgemeinschaft (DFG) within theresearch unit FOR 1807. This work was made possible by 115105-4COMPETING PAIRING CHANNELS IN THE DOPED . . . PHYSICAL REVIEW B 94, 115105 (2016) generous allocations of CPU time from the Center for Quantum Simulation Sciences in the Institute of Physics, ChineseAcademy of Sciences, and the National Supercomputer Centerin Tianjin. We also acknowledge computing resources at JSCJ¨ulich and RWTH Aachen University with JARA-HPC. APPENDIX A: INTERACTION EXPANSION The partition function for the CT-INT can be obtained as Z=Tr[e−βˆH0e−/integraltextβ 0dτˆHI(τ)] =/summationdisplay k(−1)k/integraldisplayβ 0dτ1···/integraldisplayβ τk−1dτkTr[e−βˆH0ˆHI(τk)··· ˆHI(τ1)] =/summationdisplay Ck/parenleftbigg −U 2/parenrightbiggk1 k!/productdisplay σ/angbracketleftTτ(ˆn1σ−α1σ)(ˆn2σ−α2σ) ···(ˆnkσ−αkσ)/angbracketright0 =/summationdisplay Ck/parenleftbigg −U 2/parenrightbiggk/productdisplay σdetDσ(k), (A1) where we have written the interaction part ˆHIof the Hamil- tonian (1) in the main text as U/2/summationtext i,s±1/producttext σ[ˆniσ−ασ(s)], withασ(s)=1/2+σs(1/2+0+). The conﬁguration Ck= {[i1,τ1,s1]···[ik,τk,sk]}.T h e Dσ(k) matrix has diag- onal elements Dσ pp(k)=− /angbracketleftTτˆc† ip(τ+ p)ˆcip(τp)/angbracketright0+ασ(si)≡ −g0 0(β)+ασ(si) and off-diagonal elements Dσ pq(k)= −/angbracketleftTτˆc† iq(τ+ q)ˆcip(τp)/angbracketright0≡g0(p,q). The CT-INT solver uses the cluster-excluded Green’s function g0(p,q) as input. APPENDIX B: DCA LOOP The honeycomb lattice has two sites per unit cell, and the DCA self-consistent loop requires more steps for sucha complex lattice than that for simple lattices such as squareor triangular lattices. In this appendix, we describe the generalDCA scheme we have developed for such more complexlattices. We deﬁne the cluster-excluded Green’s function in matrix form g 0(R,τ), whose matrix elements are g0 αβ(R,τ), with R being the distance vector between unit cells and α,βbeing indices for the two sublattices, AandB. The DCA loop starts from a noninteracting cluster self- energy /Sigma1c(K,iωn)=0 or the self-energy obtained from second-order perturbation theory. One then uses the clusterself-energy to approximate the lattice self-energy, and thelattice Green’s function is G latt(k,iωn)=Glatt(K+˜k,iωn) =1 (iωn+μ)1−H0(K+˜k)−/Sigma1c(K,iωn), where H0(k)=/parenleftbigg 0 −t/summationtext3 l=1eik·δl −t/summationtext3 l=1e−ik·δl 0/parenrightbigg (B1) andδlare the three nearest-neighbor vectors, δ1=(0,−1√ 3), δ2=(1 2,1 2√ 3), and δ3=(−1 2,1 2√ 3). Note that for the honey-comb lattice Glatt(k,iωn) and /Sigma1c(K,iωn)a r e2 ×2 matrices with sublattice indices. One then needs to prepare the cluster- excluded Green’s function g0(R,τ) for the CT-INT impurity solver. In the ﬁrst step, we coarse grain the lattice Green’s function ¯Glatt(k,iωn)=1 N˜k/summationdisplay ˜kGlatt(K+˜k,iωn), (B2) withNcbeing the number of the cluster size (the number of unit cells in a cluster) and N˜kbeing the number of ˜kpoints within each Kpatch. Then, by using the Dyson equation, the cluster-excluded Green’s function in ( K,iωn) space can be obtained as g0(K,iωn)=[¯Glatt(k,iωn)−1+/Sigma1c(K,iωn)]−1. (B3) Finally, g0(K,iωn) needs to be transformed to g0(R,τ)t o provide the input for the impurity solver as g0(K,iωn)1−→g0(K,τ)2−→g0 i,j(τ)3−→g0(R,τ). (B4) For simple lattices, g0(K,iωn) and g0(K,τ) are connected by a Fourier transformation. But for more complex lattices, thisrequires more steps. We explain steps 1, 2, and 3 in Eq. ( B4) below. In step 1, we perform an inﬁnite Matsubara frequency summation. In order to ensure numerical precision, we dividethe imaginary-time interval into several ranges and considerthem separately. In the following, n cis the frequency cutoff, and we take nc=1000 in our code. Then, g0(K,τ∈(0,β))=−1 βnc−1/summationdisplay n=−ncg0(K,iωn)e−iωnτ, g0(K,τ=0+)=−1 βnc−1/summationdisplay n=−nc/parenleftbigg g0(K,iωn)−1 iωn/parenrightbigg e−iωnτ −1 β+∞/summationdisplay n=−∞e−iωnτ iωn =−1 βnc−1/summationdisplay n=−ncg0(K,iωn)+1 2, (B5) and the periodic boundary condition in the time axis gives g0(K,τ=β)=1−g0(K,τ=0+), (B6) g0(K,τ(∈[−β,0))=−g0(K,τ+β). (B7) In step 2, which leads from g0(K,τ)t og0 i,α;j,β(τ), we need to perform a modiﬁed Fourier transformation, g0 i,α;j,β(τ)=1 Nc/summationdisplay Kg0 α,β(K,τ)eiK·rie−iK·rj, (B8) using inner-cell coordinates in the phase, i.e., ri=Ri+tα(β), where Riis the unit-cell coordinate and tα(β)is the inner- cell coordinate: tα=(0,0) and tβ=(0,1/√ 3) for the two sublattices. In step 3, leading from g0 i,α;j,β(τ)t og0 α,β(R,τ), we make use of translational symmetry and perform a constrained 115105-5XIAO Y AN XU, STEFAN WESSEL, AND ZI Y ANG MENG PHYSICAL REVIEW B 94, 115105 (2016) summation, g0 αβ(R,τ)=1 Nc/summationdisplay Ri−Rj=Rg0 i,α;j,β(τ). (B9) APPENDIX C: TWO-PARTICLE GREEN’s FUNCTION To calculate correlation functions, we need to evalu- ate the two-particle Green’s functions within the CT-INT,/angbracketleftG σ α1β1(P1,P/prime 1)Gσ α2β2(P2,P/prime 2)/angbracketright, where Gαβ(P,P/prime) is deﬁned as Gαβ(P(K,iωn),P/prime(K/prime,iω/prime n)) =g0 α,β(K,iωn)δK,K/primeδiωn,iω/primen −g0 α,γ(K,iωn)/Gamma1γ,η(K,iωn;K/prime,iω/prime n)g0 η,β(K/prime,iω/prime n),(C1) with /Gamma1α,β(K,iωn;K/prime,iω/prime n) =−T Nc/summationdisplay i,je−iK·rieiωnτiM(k)i,α;j,βe−iω/prime nτjeiK/prime·rj,(C2)where M(k)=D(k)−1andri=Ri+tα(β), where Riis the unit-cell coordinate and tα(β)is the inner-cell coordinate. APPENDIX D: DETAILS ON CALCULATING THE PAIRING SUSCEPTIBILITY In the DCA formalism, we need to distinguish the cluster pairing susceptibility and the lattice pairing susceptibility.The real physical quantities, the lattice susceptibilities ¯ χ,a r e obtained with ¯χ=¯χ 0 1−/Gamma1/prime¯χ0, (D1) where ¯ χ0is the coarse-grained noninteracting susceptibility and/Gamma1/primeis the irreducible vertex. Within the DCA approxima- tion, the irreducible vertex /Gamma1/primein the lattice susceptibility and /Gamma1cin the cluster susceptibility are equivalent once they are coarse grained to the cluster level, so that (¯χ0)−1−(¯χ)−1=/Gamma1/prime=/Gamma1c=/parenleftbig χ0 c/parenrightbig−1−(χc)−1. (D2) The cluster pairing susceptibility matrix χc(P,P/prime,Q=0) is deﬁned as /parenleftBigg /angbracketleftG↑ 11(−P,−P/prime)G↓ 22(P,P/prime)/angbracketright/angbracketleftG↑ 12(−P,−P/prime)G↓ 21(P,P/prime)/angbracketright /angbracketleftG↑ 21(−P,−P/prime)G↓ 12(P,P/prime)/angbracketright/angbracketleftG↑ 22(−P,−P/prime)G↓ 11(P,P/prime)/angbracketright/parenrightBigg , (D3) and the noninteracting pairing susceptibility matrix χ0 c(P,P/prime,Q=0) is deﬁned as /parenleftBigg G↑ 11(−P)G↓ 22(P)G↑ 12(−P)G↓ 21(P) G↑ 21(−P)G↓ 12(P)G↑ 22(−P)G↓ 11(P)/parenrightBigg δP,P/prime. (D4) In the above equations, 1 and 2 denote sublattice indices and P=(K,iω),P/prime=(K/prime,iω/prime), andQ=(Q,iν) are four-vectors of cluster momentum and Matsubara frequency. Based on χc andχ0 c,b yu s i n gE q .( D2), we then get the irreducible vertex /Gamma1c, equivalently, /Gamma1/prime. Then using Eq. ( D1), we obtain the lattice pairing susceptibility ¯ χ. Finally, we get the effective pairing susceptibility from χη eff(T)=1 β/summationdisplay P,P/prime,Q=0/angbracketleft¯/Phi1η(K)\|¯χ(P,P/prime,Q=0) −¯χ0(P,P/prime,Q=0)\|¯/Phi1η(K/prime)/angbracketright/slashbigg/summationdisplay K/angbracketleft¯/Phi1η(K)\|¯/Phi1η(K)/angbracketright, where ¯/Phi1η(K) and ¯/Phi1η(K/prime) are coarse-grained form factors, as shown in Fig. 1(d) in the main text. APPENDIX E: RELATION TO THE PAIRING VERTEX EIGENVALUE ANALYSIS An eigenvalue analysis of the pairing vertex secular equation requires us to solve the eigenvalue problem /Gamma1χ0\|φi/angbracketright=λi\|φi/angbracketright. (E1)In terms of the pairing vertex, the effective pairing suscepti- bility is given as χeff=χ−χ0 =χ0(1−/Gamma1χ0)−1−χ0. (E2) Inserting the diagonal representation of /Gamma1χ0into the above equation, one obtains χeff=χ0/summationdisplay iλi 1−λi\|φi/angbracketright/angbracketleftφi\|. 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PhysRevB.91.121101.pdf	RAPID COMMUNICATIONS PHYSICAL REVIEW B 91, 121101(R) (2015) Dirac semimetals A3Bi (A=Na,K,Rb) as Z2Weyl semimetals E. V . Gorbar,1,2V . A. Miransky,3I. A. Shovkovy,4and P. O. Sukhachov1 1Department of Physics, Taras Shevchenko National Kiev University, Kiev 03680, Ukraine 2Bogolyubov Institute for Theoretical Physics, Kiev 03680, Ukraine 3Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7 4College of Letters and Sciences, Arizona State University, Mesa, Arizona 85212, USA (Received 24 December 2014; revised manuscript received 12 February 2015; published 2 March 2015) We demonstrate that the physical reason for the nontrivial topological properties of Dirac semimetals A3Bi (A=Na,K,Rb) is connected with a discrete symmetry of the low-energy effective Hamiltonian. By making use of this discrete symmetry, we argue that all electron states can be split into two separate sectors of the theory.Each sector describes a Weyl semimetal with a pair of Weyl nodes and broken time-reversal symmetry. The lattersymmetry is not broken in the complete theory because the time-reversal transformation interchanges states fromdifferent sectors. Our ﬁndings are supported by explicit calculations of the Berry curvature. In each sector, theﬁeld lines of the curvature reveal a pair of monopoles of the Berry ﬂux at the positions of Weyl nodes. The Z 2 Weyl semimetal nature is also conﬁrmed by the existence of pairs of surface Fermi arcs, which originate from different sectors of the theory. DOI: 10.1103/PhysRevB.91.121101 PACS number(s): 71 .10.−w,03.65.Vf,71.15.Rf Introduction. Three-dimensional (3D) Dirac semimetals whose conduction and valence bands touch only at discrete(Dirac) points in the Brillouin zone with the electron statesdescribed by the 3D massless Dirac equation are 3D analogs ofgraphene. Historically, bismuth [ 1] was the ﬁrst material where it was shown that its low-energy quasiparticle excitationsnear the Lpoint of the Brillouin zone are described by the 3D Dirac equation with a small mass [ 2]. Since the Dirac point is composed of two Weyl nodes of opposite chiralitywhich overlap in momentum space, it can be gapped out.Therefore, even if the 3D Dirac point is obtained accidentallyby ﬁne tuning the spin-orbit coupling strength or chemicalcomposition, it is, in general, not stable and is difﬁcult tocontrol. It was proposed in Refs. [ 3,4] that an appropriate crystal symmetry can protect and stabilize the 3D Dirac points if twobands which cross each other belong to different irreduciblerepresentations of the discrete crystal rotational symmetry.By using the ﬁrst-principles calculations and effective modelanalysis, A 3Bi (A=Na,K,Rb) and Cd 3As2compounds were identiﬁed in Refs. [ 5,6] as 3D Dirac semimetals protected by crystal symmetry. Various topologically distinct phases canbe realized in these compounds by breaking time-reversaland inversion symmetries. By making use of angle-resolvedphotoemission spectroscopy, a Dirac semimetal band structurewas indeed observed [ 7–9]i nC d 3As2and Na 3Bi, opening the path toward the experimental investigation of the propertiesof 3D Dirac semimetals. For a recent review of 3D Diracsemimetals, see Ref. [ 10]. Closely related to 3D Dirac semimetals are Weyl semimet- als. They were proposed to be realized in pyrochlore iri-dates [ 11], topological heterostructures [ 12], and magnetically doped topological insulators [ 13]. Although not experimen- tally observed yet, Weyl semimetals have been very activelystudied theoretically (for reviews, see Refs. [ 14–16]). A Weyl node is topologically nontrivial because it is a monopole of theBerry ﬂux in momentum space. This is also the reason whyWeyl nodes can appear or annihilate only in pairs [ 17].The simplest way to turn a Dirac semimetal into a Weyl one is to apply an external magnetic ﬁeld, which breakstime-reversal symmetry. This can be realized even in thehigh-energy physics context [ 18]. In fact, the corresponding transition might have already been observed in Bi 1−xSbxfor x≈0.03 [19] and in Cd 3As2[20]. In the case of Bi 1−xSbx [19], the authors measured negative magnetoresistivity at not very large magnetic ﬁelds that might be a ﬁngerprint of aWeyl semimetal phase [ 17,21] (see, however, the discussion in Ref. [ 22]). In the case of Cd 3As2, a magnetic ﬁeld driven splitting of Landau levels consistent with the Weyl phase,time-reversal symmetry breaking, and a nontrivial Berry phasewere detected [ 20]. The existence of surface Fermi arcs [ 11,23–25] is another ﬁngerprint of Weyl semimetals, associated with the nontrivialtopology. Such arcs connect Weyl nodes of opposite chirality.The shape of the arcs depends on the boundary conditionsand can be engineered [ 26]. The two Fermi arcs on opposite surfaces, together with the Fermi surface of bulk states,form a closed Fermi surface. This implies, in particular, thatthe chemical potentials for different chirality quasiparticlesnear distinct Weyl points must be the same in a staticsystem [ 23]. Normally, one would not expect surface Fermi arcs in 3D Dirac semimetals because the Berry ﬂux vanishes forDirac points with vanishing topological charges. However,calculations of Refs. [ 5,6] suggest that Dirac semimetals A 3Bi (A=Na,K,Rb) and Cd 3As2possess nontrivial surface Fermi arcs. This is an indication of a topologically nontrivial nature ofthese Dirac semimetals. In fact, as we argue below, the situationis reminiscent of topological insulators in which there is a Z 2 topological order associated with the time-reversal symmetry [27–31]. This is further supported by the fact that the breaking of time-reversal or inversion symmetry in Dirac semimetalscauses splitting of the surface Fermi arcs into a pair of opensegments resembling the arcs in Weyl semimetals [ 5]. In this Rapid Communication, we explain the reason for the existenceof nontrivial topological properties of the A 3Bi compounds 1098-0121/2015/91(12)/121101(5) 121101-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS GORBAR, MIRANSKY , SHOVKOVY , AND SUKHACHOV PHYSICAL REVIEW B 91, 121101(R) (2015) and shed light on their analytical structure in the low-energy theory. Hamiltonian. Our starting point in the analysis will be the low-energy effective Hamiltonian for electron excitations inA 3Bi (A=Na,K,Rb) derived in Ref. [ 5]. The explicit form of the Hamiltonian is given by H(k)=/epsilon10(k)+H4×4, (1) H4×4=⎛ ⎜⎝M(k)Ak+ 0 B∗(k) Ak−−M(k)B∗(k)0 0 B(k)M(k)−Ak− B(k)0 −Ak+−M(k)⎞ ⎟⎠. (2) Note that the Hamiltonian of the same form is also valid for structure I of Cd 3As2(see Ref. [ 6]). The diagonal elements of the Hamiltonian are given in terms of two quadratic functionsof momenta, /epsilon1 0(k)=C0+C1k2 z+C2(k2 x+k2 y) andM(k)= M0−M1k2 z−M2(k2 x+k2 y). The off-diagonal elements are determined by the functions Ak±andB(k)=αkzk2 +, where k±=kx±iky. The eigenvalues of Hamiltonian ( 1)a r e E(k)=/epsilon10(k)±/radicalbig M2(k)+A2k+k−+\|B(k)\|2. (3) It is easy to check that the square root vanishes at the following two Dirac points, k± 0=(0,0,±√m), where m≡M0/M 1. The function B(k), which can be interpreted as a momentum dependent mass term, also vanishes at the Dirac points. The general considerations of the current study will apply to the compounds A3Bi (A=Na,K,Rb), but our numerical results will be presented for Na 3Bi. By ﬁtting the energy spectrum of the effective Hamiltonian with theab initio calculations for Na 3Bi, the following numerical values of the parameters in the effective model were extracted [ 5]:C0=− 0.063 82 eV, C1=8.7536 eV ˚A2, C2=− 8.4008 eV ˚A2, M0=− 0.08686 eV, M1= −10.6424 eV ˚A2,M2=− 10.3610 eV ˚A2,A=2.4598 eV ˚A, and the lattice constants are a=b=5.448 ˚A,c=9.655 ˚A. Since no speciﬁc value for αwas quoted in Ref. [ 5], we will treat it as a free parameter below. For most of our analysisbelow, however, the actual values of the model parameters arenot very important. We will use them only when presentingsome numerical results. In the simplest case of a vanishing mass function B(k)( o r , equivalently, for α=0), the Hamiltonian H 4×4takes a block diagonal form, H4×4(α=0)≡H+ 2×2⊕H− 2×2. Its upper block is given by H+ 2×2=/parenleftbigg M(k) A(kx+iky) A(kx−iky)−M(k)/parenrightbigg , (4) and has a very transparent physical meaning. It deﬁnes the simplest version of a Weyl semimetal with two Weyl nodeslocated at k ± 0. (The lower block H− 2×2has a similar form, except thatkxis replaced by −kx.) It is well known [ 25,32] that such a Weyl semimetal has the surface Fermi arc in the form ofa straight line connecting Weyl nodes of opposite chiralityatk + 0andk− 0. Because of the sign difference, kx→−kx, the chiralities of the states near the Weyl nodes at k± 0are opposite for the upper and lower block Hamiltonians. Thus,the complete 4 ×4 block diagonal Hamiltonian H 4×4(α=0)describes two superimposed copies of a Weyl semimetal with two pairs of overlapping nodes. The opposite chirality Weylnodes coincide exactly in the momentum space and, thus,effectively give rise to two Dirac points at k ± 0.A tt h es a m e time, because the Weyl nodes come from different blocks,they cannot annihilate and cannot form topologically trivialDirac points. In fact, the corresponding approximate modeldescribes a Z 2Weyl semimetal. The nontrivial topological properties, associated with the underlying Z2Weyl semimetal structure, ensure that the resulting Dirac semimetal possessessurface Fermi arcs. It is easy to show that the existence of the Z 2Weyl semimetal structure in this simplest case is connected withthe continuous symmetry U +(1)×U−(1) of the approximate Hamiltonian H4×4(α=0). This symmetry describes indepen- dent phase transformations of the spinors that correspond tothe block Hamiltonians H + 2×2andH− 2×2, respectively. Symmetries. It is well known that, for B(k)=const, the symmetry U+(1)×U−(1) is broken to its diagonal subgroup Uem(1) that describes the usual charge conservation. However, as we show below, the low-energy Hamiltonian ( 1) with the momentum dependent mass function B(k)=αkzk2 +possesses a new discrete symmetry that protects the Z2Weyl semimetal structure. Before discussing this symmetry, let us start by pointing out that the Hamiltonian ( 1) is invariant under the time-reversal and inversion symmetries, i.e., /Theta1H−k/Theta1−1=Hk(time-reversal symmetry), (5) PH−kP−1=Hk(inversion symmetry), (6) where /Theta1=TK (Kis a complex conjugation) and T=⎛ ⎜⎝00 1 0 00 0 1 −100 0 0−100⎞ ⎟⎠,P=⎛ ⎜⎝1000 0−10 0 0010000 −1⎞ ⎟⎠. (7) Of course, these two symmetries are expected in Diracsemimetals such as A 3Bi, and they do play an important role in understanding their physical properties. The less obvious isthe following symmetry deﬁned by the transformation, UH −kzU−1=Hkz(ud parity) , (8) where matrix Uhas the following block diagonal form, U≡ diag(I2,−I2) andI2is the 2 ×2 unit matrix. We call it the up- down (ud) parity because its eigenstates for B(k)=0i nv i e w of the block-diagonal structure of Hamiltonian ( 2) correspond to bispinors with only two upper or lower nonzero componentsthat describe a Weyl semimetal with a pair of Weyl nodes. Itshould be noted that, for the Hamiltonian to be invariant underthis symmetry, it is crucial that the mass function B(k) changes its sign when k z→−kz[while the functions /epsilon10(k) andM(k) in the diagonal elements do not change their signs]. Werethe mass function momentum independent, such a discretesymmetry would not exist. The existence of the time-reversal ( 5) and ud parity ( 8) symmetries has an important implication that we will nowexplain. The argument relies on the fact that all quasiparticle 121101-2RAPID COMMUNICATIONS DIRAC SEMIMETALS A3Bi (A=Na,K,Rb) as Z2. . . PHYSICAL REVIEW B 91, 121101(R) (2015) states in the low-energy model of a Dirac semimetal naturally split into two separate groups, classiﬁed by the eigenvaluesof the operator U χ=U/Pi1kz.( H e r e , /Pi1kzis the operator that changes the sign of the zcomponent of momentum, kz→−kz.) Taking into account that U2 χ=1, the eigenvalues ofUχare±1. Furthermore, the corresponding eigenstates arenotinvariant under time reversal. This follows from the fact that the operators of time-reversal /Theta1and ud parity Uχ transformations do not commute. This implies that each sector of quasiparticle states with a ﬁxed eigenvalue of Uχdeﬁnes a distinct copy of the Weyl semimetal, for which time reversal isbroken. Of course, the time-reversal symmetry is not brokenin the complete system including both U χsectors. In view of the Uχsymmetry, we can classify the corresponding Dirac semimetal as a Z2Weyl semimetal. The situation resembles that of topological insulators [ 27–31], which are time-reversal invariant due to the Z2topological order parameter. Each Weyl subsystem, described by quasiparticle states with a ﬁxed eigenvalue of Uχ, has well deﬁned Fermi arcs connecting the Weyl nodes at k± 0. These arcs are topologically protected and cannot be removed by small perturbations ofmodel parameters. In our discussion of Fermi arcs, it will be also useful to take into account that there exists yet another discrete symmetrydeﬁned by the following transformation, ˜UH −kx˜U−1=Hkx, (9) where ˜U=⎛ ⎜⎝0010 000110000100⎞ ⎟⎠. (10) Of course, the product of the U χand ˜U/Pi1kxtransformations Uχ˜U/Pi1kx=T/Pi1kx/Pi1kzis also a symmetry of the low-energy Hamiltonian ( 1). Note that the symmetry T/Pi1kx/Pi1kzis related to the time-reversal symmetry if we take into account thatK/Pi1 kyis also the symmetry of Hamiltonian ( 1). Together, the operators Uχ,˜U/Pi1kx, andT/Pi1kx/Pi1kzform a noncommutative discrete group. Eigenstates of Uχ.Since Hamiltonian ( 1) commutes with Uχ, its eigenstates with eigenvalues E(k) given by Eq. ( 3) can be chosen as eigenstates of Uχ, too (alternatively, we can choose the energy eigenstates to be eigenstates of the ˜U/Pi1kxor T/Pi1kx/Pi1kzsymmetries). These eigenstates have the following form: ψ+(k)=N+⎛ ⎜⎜⎝1 E(k)−/epsilon10(k)−M(k) Ak+B(k) Ak+ 0⎞ ⎟⎟⎠, (11) ψ−(k)=N−⎛ ⎜⎜⎝−B∗(k) Ak− 0 1 −E(k)−/epsilon10(k)−M(k) Ak−⎞ ⎟⎟⎠. (12) HereN±are normalization constants and the subscript ± means the eigenvalue of Uχ. It is not difﬁcult to check that ˜U/Pi1kxtransforms ψ+intoψ−and vice versa. Notice thatFIG. 1. (Color online) The projection of the Berry curvature F++(k) (left panel) and F−−(k) (right panel) on the ky=0 plane. the bispinors ψ±in the case with a vanishing mass function, B(k)=0, describe fermions of deﬁnite chirality in the vicinity of the k± 0points. Berry curvature. In order to explicitly reveal the Z2Weyl semimetal structure of A3Bi (A=Na,K,Rb), we calculated the Berry connection and the Berry curvature for each sectordescribed by the ψ ±(k) states. Due to the double degeneracy of the states with the same energy in the present case, the Berrycurvature is a matrix with non-Abelian gauge structure [ 33], A mn(k)≡−i 2[ψ† m(k)(∇kψn(k))−(∇kψ† m(k))ψn(k)], Fmn(k)≡∇k×Amn−iAml×Aln, (13) where m,n,l =± and the summation over lis performed in the last equation. The four components of the Berryconnection A mn(k) deﬁne a U(2) gauge ﬁeld. The Berry curvature components F++(k) and F−−(k) are plotted in Fig. 1. The numerical results are shown for α=50 eV ˚A3and the energy eigenvalue in Eq. ( 3) with the positive sign in front of the square root. (Up to the change of direction of the vectorﬁelds, the plots for the other sign of root look qualitatively thesame.) The results for the diagonal components of the curvature in Fig. 1show that each sector with a deﬁnite eigenvalue of U χcontains a pair of Berry curvature monopoles with charges ±1. Such a dipole structure in the momentum space is an unambiguous signature of a Weyl semimetal in each of thesectors. We would like to emphasize that the presence of the mass function B(k) does not affect the property of the diagonal Berry curvature F ++(k)[ o rF−−(k)] to have nonzero divergencies at the Weyl nodes. Mathematically, the qualitative behavior ofthe curvature in the vicinity of the nodes is preserved becauseB(k) vanishes at k ± 0. Away from the Weyl nodes, on the other hand, the mass function does affect the behavior of theBerry curvature. This is already seen in Fig. 1, where slight distortions of the dipole conﬁgurations become visible. It canbe checked that distortions become much stronger at largervalues of parameter α. We found, however, that the opposite charge monopoles of the Berry ﬂux remain well resolved even forαas large as 250 eV ˚A 3. It is interesting to point out that the off-diagonal compo- nents of the Berry curvature F+−(k) [as well as F−+(k)] are nonzero only because of the nontrivial mass function B(k). The 121101-3RAPID COMMUNICATIONS GORBAR, MIRANSKY , SHOVKOVY , AND SUKHACHOV PHYSICAL REVIEW B 91, 121101(R) (2015) complete implications of this fact remain to be investigated. This task, however, is beyond the scope of the present RapidCommunication. Surface Fermi arcs. The nontrivial topological structure of the ψ +andψ−sectors implies that the A3Bi compounds should have surface Fermi arcs. Previously, the surface Fermiarcs in these 3D Dirac semimetals were studied in Ref. [ 5] by using an iterative method to obtain the surface Green’sfunction of the semi-inﬁnite system [ 34]. The imaginary part of the surface Green’s function makes it possible to determinethe local density of states at the surface. In our study here,we employ the continuum low-energy model and enforceappropriate boundary conditions for the quasiparticle spinorsat the surface of the semimetal. As we will argue, such aconsideration makes the physical properties of the surfaceFermi arc states more transparent. We assume that semimetal is situated at y/greaterorequalslant0 and is inﬁnite in the xandzdirections. The simplest implementation of the boundary condition for the semimetal states on its surfacefollows from the replacement of mwith−˜mand taking the limit ˜m→∞ on the vacuum side of the boundary [ 25]. Taking into account that the Fermi arc states should be localized atthey=0 boundary, we can look for the surface state solution in the following form, /Psi1(r)=/Psi1 1e−p1y+/Psi12e−p2y, (14) where /Psi1ican be chosen as the eigenstates of the Uχsymmetry andpiare the positive (that are necessary for the normalization of the wave function) roots of the characteristic equation /bracketleftbig C2/parenleftbig k2 x−p2/parenrightbig +C1k2 z+C0−E/bracketrightbig2+A2/parenleftbig p2−k2 x/parenrightbig −/bracketleftbig M0−M1k2 z−M2/parenleftbig k2 x−p2/parenrightbig/bracketrightbig2−α2k2 z/parenleftbig p2−k2 x/parenrightbig2=0. (15) The wave function on the vacuum side has a similar form, but with the replacement pi→− ˜pi, where the deﬁnition of ˜piis similar to that of pi,b u tmis replaced by −˜m.( I n the calculation, we take the limit ˜m→∞ , which prevents quasiparticles from escaping into vacuum.) Matching the wave functions across the boundary, we obtain the following equation, (Q+ 1−Q+ 2)(Q− 1−Q− 2)−(T+ 1−T+ 2)(T− 1−T− 2)=0, (16) where Q± i=−C2/parenleftbig k2 x−p2 i/parenrightbig +C1k2 z+C0−E∓Akx M0−M1k2z−M2/parenleftbig k2x−p2 i/parenrightbig −Api,(17) T± i=−αkz(pi±kx)2 M0−M1k2z−M2/parenleftbig k2x−p2 i/parenrightbig −Api. (18) The numerical solutions for the surface Fermi arcs are shown in Fig. 2for several ﬁxed values of the Fermi energy and α=1e V ˚A3. In the special case of E=0, our results are in qualitative agreement with the results obtained in Ref. [ 5]b y using a different method. Other materials. By combining the ﬁrst-principles calcula- tions and effective model analysis, it was recently predicted[35] that the ternary compounds Ba YBi (Y=Au,Ag,Cu) areFIG. 2. (Color online) The surface Fermi arcs for α=1e V ˚A3 and energy E=0,±50,±100 meV . Dirac semimetals. The low-energy effective Hamiltonian of these compounds is similar to that of A3Bi (A=Na,K,Rb), but with a different structure of the mass terms, Hternary=/epsilon10(k)+H/prime 4×4, (19) where H/prime 4×4=⎛ ⎜⎜⎝M(k) Ak+ 0 Bkzk2 + Ak−−M(k)−Bkzk2 + 0 0 −Bkzk2 −M(k) Ak− Bkzk2 − 0 Ak+ −M(k)⎞ ⎟⎟⎠. (20) Since Hamiltonian ( 19) is invariant with respect to the Uχ symmetry transformation, our conclusions remain valid for these compounds. Thus, these Dirac semimetals are Z2Weyl semimetals, too. In conclusion, as we argued in this Rapid Communication, Dirac semimetals A3Bi (A=Na,K,Rb) are, in fact, Z2Weyl semimetals. The conclusion is supported by the existence ofthe ud parity U χthat allows us to split all states into two sectors, with each describing a Weyl semimetal. It is the combinationof both sectors that gives rise to a Z 2Weyl character of the corresponding semimetals. Naturally, the time-reversaland inversion symmetries are preserved in such a theory.The situation might be reminiscent of topological insulators,where the topological order is protected by the time-reversalsymmetry [ 27–31]. The symmetry arguments used in the current study are rather powerful. They suggest that the main conclusions shouldremain unchanged even in the presence of interaction effects,provided the latter do not modify the low-energy spectrumin a qualitative way. A weak disorder [ 36] and a subcritical Coulomb interaction [ 37] are examples of such effects that exist in realistic materials, but are not expected to change ourmain conclusions. The fact that the compounds A 3Bi (A=Na,K,Rb) are Z2Weyl semimetals has important implications. On the one hand, it sheds light on the existence of surface Fermi arcs insuch materials. This is particularly important in view of therecent experimental conﬁrmation of such states in Na 3Bi [38]. 121101-4RAPID COMMUNICATIONS DIRAC SEMIMETALS A3Bi (A=Na,K,Rb) as Z2. . . PHYSICAL REVIEW B 91, 121101(R) (2015) Additionally, it predicts the same types of quantum oscillations as in true Weyl semimetals [ 32], with the period dependent on the thickness of the semimetal slabs. Indeed, when the twoWeyl sectors are protected from mixing by the Z 2symmetry, their contributions will simply superimpose. Furthermore,when the time reversal is broken (e.g., by magnetic impurities),we anticipate that the superposition of two oscillations withnonequal periods will be observed.The work of E.V .G. was supported partially by the Ukrainian State Foundation for Fundamental Research. Thework of V .A.M. was supported by the Natural Sciences andEngineering Research Council of Canada. 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PhysRevB.89.195401.pdf	PHYSICAL REVIEW B 89, 195401 (2014) Interaction of static charges in graphene within Monte Carlo simulation V . V . Braguta,1,2,3,S. N. Valgushev,2,†A. A. Nikolaev,4,‡M. I. Polikarpov,2,§and M. V . Ulybyshev2,5,/bardbl 1Institute for High Energy Physics, Protvino, 142281, Russia 2Institute of Theoretical and Experimental Physics, Moscow, 117218, Russia 3Moscow Institute of Physics and Technology, Institutskii lane 9, Dolgoprudny, Moscow Region, 141700, Russia 4Far Eastern Federal University, School of Biomedicine, Vladivostok, 690950, Russia 5Institute for Theoretical Problems of Microphysics, Moscow State University, Moscow, 119899, Russia (Received 20 July 2013; revised manuscript received 14 April 2014; published 2 May 2014) The study of the interaction potential between static charges within Monte Carlo simulation of graphene is carried out. The numerical simulations are performed in the effective lattice ﬁeld theory with noncompact(3+1)-dimensional Abelian lattice gauge ﬁelds and (2 +1)-dimensional staggered lattice fermions. It is shown that for all considered temperatures the interaction can be well described by the Debye screened potentialcreated by two-dimensional electron-hole excitations. At low temperatures Debye mass m Dplays a role of order parameter of the insulator-semimetal phase transition. In the semimetal phase at high temperaturegraphene effective ﬁeld theory reveals the properties of weakly interacting two-dimensional plasma of fermionexcitations. DOI: 10.1103/PhysRevB.89.195401 PACS number(s): 05 .10.Ln,71.30.+h,72.80.Vp I. INTRODUCTION Graphene is an allotrope of carbon, in which atoms form a two-dimensional honeycomb lattice. Carbon atoms in it arebonded by sp 2bonds and the bond length is about 0.142 nanometers [ 1]. The charge carriers in graphene behave as massless fermions [ 2]. The Fermi velocity of charge carriers is vF≈c 300. Since the Fermi velocity is much smaller than the speed oflight, magnetic and retardation effects in the interactions be-tween charge carriers may be neglected; thus electron-electroninteraction in graphene is well described by the instantaneousCoulomb potential. The effective coupling constant for theCoulomb interaction in graphene ∼ α vF≈2(α=1/137) is large, so this material can be considered as a stronglyinteracting system. In real experiments graphene is put on a substrate. The effective coupling constant for graphene on substrate with the dielectric permittivity /epsilon1is reduced by a factor 2 /(/epsilon1+1). The variation of the dielectric permittivity /epsilon1of substrate changes the effective coupling constant and thus allows one tostudy the properties of graphene in strong and weak couplingregimes. In the weak coupling regime theoretical description of graphene properties based on perturbation theory gives re-liable results. In the strong coupling regime there are noaccurate analytical approaches and Monte Carlo simula-tion is an adequate method to study graphene in strongcoupling. There exists a number of papers where graphene was studied by Monte Carlo method [ 3–6] and insulator-semimetal braguta@itep.ru †valgushev@itep.ru ‡nikolaev.aa@dvfu.ru §Deceased. /bardblulybyshev@goa.bog.msu.ruphase transition was found. At the weak coupling regime graphene is in the semimetal phase. In this phase the conductiv-ity isσ∼α/h and there is no gap in the spectrum of fermionic excitations. The chiral symmetry of graphene is not broken. Atstrong coupling regime graphene is in the insulator phase. Inthis phase the conductivity is considerably suppressed, thereis an energy gap in the spectrum of fermionic excitations, andfermionic chiral condensate /angbracketleft¯ψψ/angbracketrightis not zero. The precise value of the effective coupling constant at which graphenebecomes an insulator was a subject of intensive studies.Initially all lattice calculations predicted phase transition at thedielectric permittivity of substrate /epsilon1∼4. But experimentally an insulator phase was not found even for suspended graphene.In the paper [ 7] it was found that the position of phase transition is under strong inﬂuence of the short-range screening of theCoulomb potential by electrons at other orbitals. In this paperwe do not take into account this effects, so we can reach the correspondence between our results and an experiment only at a qualitative level. In this paper we study the interaction potential between static charges in graphene for various values of the dielectricpermittivity of substrate /epsilon1and the temperature of graphene charge carriers T. 1We present the results of MC simulations of graphene in the framework of the effective ﬁeld model. Thenon-MC calculations of the potential were performed in [ 8] (see also references therein). The paper is organized as follows. In the next section a brief review of the simulation algorithm is given. In the lastsection the results of numerical simulations are presented anddiscussed. In the Appendix we derive the potential of Debyescreening for two-dimensional plasma. 1We discuss phenomena related to electron degrees of freedom and neglect the thermal vibration of the graphene honeycomb lattice. Thuswe can consider the temperatures T∼10 3–104K, at which the real graphene is melted. 1098-0121/2014/89(19)/195401(8) 195401-1 ©2014 American Physical SocietyV. V. B R AG U TA et al. PHYSICAL REVIEW B 89, 195401 (2014) II. LATTICE SIMULATION OF GRAPHENE A. Simulation algorithm The partition function of graphene effective ﬁeld theory can be written as [ 2,9–11] Z=/integraldisplay D¯ψDψDA0exp/bracketleftbigg −1 2/integraldisplay d4x(∂iA0)2 −/integraldisplay d3x¯ψf/parenleftbigg /Gamma10(∂0−igA 0)+/summationdisplay i=1,2/Gamma1i∂i/parenrightbigg ψf/bracketrightbigg ,(1) where A0is the zero component of the vector potential of the 3+1 electromagnetic ﬁeld, /Gamma1μare Euclidean /Gamma1matrices, andψf(f=1,2) are two ﬂavors of Dirac fermions which correspond to two spin components of the nonrelativisticelectrons in graphene, effective constant g 2=4πα/v F2/(/epsilon1+ 1) (/planckover2pi1=c=1 is assumed and α=1/137). The zero component of the vector potential A0satisﬁes the periodic boundary condition in space and time A0(t= 0)=A(t=1/T), where Tis the temperature.2The fermion spinors satisfy periodic boundary condition in space andantiperiodic boundary condition in the time direction ψ f(t= 0)=−ψf(t=1/T). Partition function ( 1) doesn’t depend on the vector part of the gauge potential Ai, since we are working at the leading approximation in vF. The simulation of partition function ( 1) is carried out within the approach developed in [ 3,5]. In order to discretize the fermionic part of the action in ( 1) the staggered fermions [12,13] are used. One ﬂavor of staggered fermions in 2 +1 dimensions corresponds to two ﬂavors of continuum Diracfermions [ 12–14], which makes them especially suitable for simulations of the graphene effective ﬁeld theory. The action for staggered fermions coupled to Abelian lattice gauge ﬁeld is S /Psi1[¯/Psi1x,/Psi1x,θx,μ] =/summationdisplay x,y¯/Psi1xDx,y[θx,μ]/Psi1y =/summationdisplay xδx3,0/parenleftbigg/summationdisplay μ=0,1,2Kμ 2¯/Psi1xαx,μeiθx,μ/Psi1x+ˆμ −/summationdisplay μ=0,1,2Kμ 2¯/Psi1xαx,μe−iθx,μ/Psi1x−ˆμ+m¯/Psi1x/Psi1x/parenrightbigg , (2) where Kμ=1 for the links in spatial directions ( μ=1,2) andKμ=as/atfor the links in time direction ( μ=0),as andatare the spatial and temporal lattice spacings, the lattice coordinates xμ=0...L μ−1(L1=L2=L3=Ls), and x3 is restricted to x3=0 in the fermionic action, ¯/Psi1xis a single- component Grassman-valued ﬁeld, αx,μ=(−1)x0+···+xμ−1, and θx,μare the link variables which are the lattice counterpart of the vector potential Aμ(x). It should be noted that nonzero mass term in ( 2) is necessary in order to ensure the invertibility of the staggered Dirac 2Note that in ( 1)t h etaxis is rescaled by a factor vF(see paper [ 5] for details). So, to get a physical value of the temperature one should multiply Tby a factor vF.operator Dx,y. Physical results are obtained by extrapolation of the expectation values of physical observables to the limitm→0. To discretize the electromagnetic part of partition function (1) the noncompact action is used, S g[θx,μ]=β 2/summationdisplay x3/summationdisplay i=1(θx,0−θx+ˆi,0)2, (3) where the summation is carried out over all four-dimensional (4D) lattice. The constant βis deﬁned as follows: β=vF 4πα/epsilon1+1 2/parenleftbiggas at/parenrightbigg . (4) The factor/epsilon1+1 2takes into account the electrostatic screening for graphene on a substrate. Note that the results that will be obtained in this paper depend on the renormalization of the bare parameters of theaction (in particular, βandK μ). Unfortunately, today there are only a few papers where this question was considered(see, for instance, the paper in [ 15]). The results of the paper in [15] and our results suggest that renormalization effects are not numerically large. It is also important to note thatrenormalization will shift the physical quantities measured inthis paper but it will not affect the most of the quantitativeconclusions. Since the action ( 2) is bilinear in fermionic ﬁelds, they can be integrated out Z=/integraldisplay D¯/Psi1 xD/Psi1xDθx,0 ×exp(−Sg[θx,0]−S/Psi1[¯/Psi1x,/Psi1x,θx,0]) =/integraldisplay Dθx,0exp (−Seff[θx,0]), (5) where Seff[θx,0]=Sg[θx,0]−ln det( D[θx,0]). (6) To generate the gauge ﬁeld conﬁgurations with the statisti- cal weight exp ( −Seff[θx,0]) the standard hybrid Monte Carlo method is used [ 3,12,13]. In order to speed up the simulations we also perform local heatbath updates of the gauge ﬁeldoutside of the graphene plane (at x 3/negationslash=0) between hybrid Monte Carlo updates. Both algorithms satisfy the detailedbalance condition for the weight ( 5)[12,13] and the path integral weight ( 5) is the stationary probability distribution for such a combination of both algorithms. Heatbath updatesare computationally very cheap and signiﬁcantly decrease theautocorrelation time of the algorithm. The temporal lattice spacing a tis equal to the spatial lattice spacing asin isotropic lattice. As was explained before, it is sufﬁcient to introduce only the fourth component ofelectromagnetic vector potential in order to take into accountthe Coulomb interaction between quasiparticles in graphene.This might imply that discretization in temporal direction isparticularly important to get reliable results. In the calculationwe ﬁx the temperature of graphene sample and vary thediscretization in the temporal direction in order to addressthis point in detail. 195401-2INTERACTION OF STATIC CHARGES IN GRAPHENE . . . PHYSICAL REVIEW B 89, 195401 (2014) In the simulation of effective theory ( 1) the lattice spacing asplays a role of ultraviolet cutoff. Exact value of this cutoff is unknown. One can only state that as∼0.142 nanometers, which is the distance between two neighboring carbon atomsin graphene. To estimate dimensional quantities, below wecalculate their values assuming that a s=0.142 nanometers. The values of dimensional quantities obtained in this way canbe considered only as an estimations of their real values. Inaddition, in brackets we put dimensional parameters in termsof∼1/a s. We use a fundamental system of units /planckover2pi1=c=1. The following relation between energy and length units canbe derived in this system: 1 eV ×1n m=5.06×10 −3. Then, for instance, the value of temperature (in electronvolts) for theisotropic ( a s=at) lattice can be calculated as follows: T[eV]=vF/(5.06×10−3×0.142·Nt). (7) Here we take into account the rescaling of time direction (see footnote 2 and the paper in [ 5] for the details). The Fermi velocity should be substituted here also in the system /planckover2pi1=c= 1:vF=1/300. B. Physical observables on the lattice To measure the potential, V(r), between static charges, we calculate the correlator of two Polyakov lines/angbracketleftP γ(0)[Pγ(/vectorr)]+/angbracketright: /angbracketleftPγ(0)[Pγ(/vectorr)]+/angbracketright=aexp/parenleftbigg −V(/vectorr) T/parenrightbigg . (8) where Tis the temperature of the graphene sample and the Polyakov line P(/vectorr)i s P(/vectorr)=exp/parenleftbigg −ie/integraldisplay1/T 0dt A 0(t,/vectorr)/parenrightbigg =Lt−1/productdisplay t=0exp (−iθ(t,/vectorr),0). (9) To suppress statistical errors, we measure the correlator of Polyakov lines in some rational power. Physically this meansthat the interaction potential between static charges ±e·γis considered. We have found that for γ∼0.1 the uncertainty of the calculation is smaller than that in the case of γ=1 (usual Polyakov line). Below the value γ=0.1 is used. In order to illustrate the last statement we considered the following observable: V=1 γ2(log/angbracketleftPγ(0)[Pγ(/vectorr)]γ/angbracketright−log/angbracketleftPγ(0)[Pγ(/vectorr/prime)]γ/angbracketright).(10) Evidently, this expression doesn’t contain self-energy and it is proportional to the difference V(/vectorr)−V(/vectorr/prime). The measurement of theVwas done at the lattice 203×60 for the following set of the parameters: /vectorr=(2,0,0),/vectorr/prime=(3,0,0),/epsilon1=4. Zero value of the third coordinate ( rz=r/prime z=0) means that all static charges are inside the graphene plane. The dependence of the Von the parameter γis shown in Fig. 1. It is evident from this plot that the increase of the γleads to the increase of the uncertainty of the calculation. In addition, the potentials for different γare equal to each other within the uncertainty of the calculation.Similar plots can be drawn for the other set parameters.FIG. 1. Observable Vgiven by Eq. ( 10) as a function of γ. Below we use the following notations: α0=α2 /epsilon1+1(11) is the bare effective charge and αR=α0 /epsilon1R(12) is the effective charge, renormalized due to interaction; /epsilon1Ris effective dielectric permittivity of graphene. III. NUMERICAL RESULTS AND DISCUSSION A. Interaction potential at low temperatures To get the potential between static charges, we measure the correlator of Polyakov lines and ﬁt V(r) by lattice screened Coulomb potential: V(/vectorr)=1 /epsilon1RVC(/vectorr)+c, (13) VC(/vectorr)=−αRπγ2 L3sas/summationdisplay n1,n2,n31/summationtext isin2(pias/2)ei/vectorp·/vectorr, pi=2π Lsasni. (14) After it we determine /epsilon1R.I nf o r m u l a( 14)cis the constant, which parametrizes self-energy contribution to the potential,V C(/vectorr) is the lattice Couloumb potential, which takes into account spatial discretization and ﬁnite volume effects, ni are integers which run in the interval (0 ,Ls−1), and point n1=n2=n3=0 is excluded. First, we discuss the systematic errors due to the temporal discretization. Using the algorithm described above we gen-erated 100 statistically independent gauge ﬁeld conﬁgurationsat the lattices 20 3×Lt,Lt=20,60,120 for a set of values of the dielectric permittivity of substrate /epsilon1∈(1,8). These three lattices correspond to the temperature T=0.23 eV ( asT= 0.000 19) and the ratios as/at=1,3,6 correspondingly. We ﬁt the data for the points which are located in graphene plane and 195401-3V. V. B R AG U TA et al. PHYSICAL REVIEW B 89, 195401 (2014) 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1 2 3 4 5 6 7εR εas/at=1 m=0.01 as/at=3 m=0.01 as/at=6 m=0.01 FIG. 2. (Color online) Dielectric permittivity of graphene /epsilon1Ras a function of the dielectric permittivity of substrate /epsilon1for different as/atatT=0.23 eV ( asT=0.000 19). for the distance between Polyakov loops smaller than Ls/3. For larger distances uncertainties are too large. We have foundan excellent agreement between our data and expression ( 14) (χ 2/DOF∼1 for all /epsilon1). Thus this result conﬁrms that static charges at low temperature in graphene interact via Coulombpotential. The dielectric permittivity of graphene /epsilon1 Ras a function of the dielectric permittivity of substrate /epsilon1for different as/at i ss h o w ni nF i g . 2. From this plot one can see that there is a large difference between the results obtained at as/at=1 andas/at=3. At the same time the results for as/at=3 andas/at=6 are in a reasonable agreement with each other. It seems that at as/at∼3–6 one approaches the continuum limit in the temporal direction. Below as/at=6 discretization scheme is used. Now let us turn to the fermion mass dependence of our results. In Fig. 3/epsilon1Ras a function of the /epsilon1for the fermion masses m=0.005,0.01 is shown. Within the uncertainty of 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2 4 6 8 10εR εm=0.005 m=0.01 FIG. 3. (Color online) Dielectric permittivity of graphene /epsilon1Ras a function of the /epsilon1for the fermion masses m=0.005,0.01 atT= 0.23 eV ( asT=0.000 19). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 1.6αR/vF α0/vFone loop FIG. 4. (Color online) Renormalized charge squared αRas a function of the bare charge squared α0rescaled by the vFand the plot of one loop formula ( 15). The insulator-semimetal phase transition takes place at α0/vF∼0.9. the calculation the results obtained for different masses are compatible to each other. The simulation of the gauge ﬁeldconﬁgurations with the mass m=0.005 is much more time consuming as compared to the mass m=0.01. So, to decrease the time of the calculation, all calculations were carried out forthe fermion mass m=0.01. The ﬁnite-volume effects can be potentially important but any calculations using large latticesare very time consuming. We are planning to study ﬁnite-volume effects in future work. In Fig. 4we show how α Ris renormalized due to the inter- action. In the semimetal phase the effective coupling constantis not large α 0/vF<1 and one can try to apply perturbation theory to describe our data. At one loop approximation thedependence of α Ron the α0for graphene is given by the expression [ 16] αR α0=1 1+π 2α0 vF=1 1+3.42 /epsilon1+1. (15) Figure 4shows that at small α0we have good agreement with perturbation theory. B. Temperature dependence of the interaction potential To study the dependence of the dielectric permittivity /epsilon1R on the temperature, we generated 100 statistically independent gauge ﬁeld conﬁgurations at the lattices 203×Lt,Lt=56, 50, 38, 28, 26, 22, and 18. Temperature in these simula-tions is equal to the following values (the larger L t,t h e smaller T):T=0.50 eV ( asT=0.000 41), T=0.56 eV (asT=0.000 46), T=0.74 eV ( asT=0.000 61), T= 1.00 eV ( asT=0.000 82), T=1.08 eV ( asT=0.000 89), T=1.28 eV ( asT=0.001 05), and T=1.56 eV ( asT= 0.001 28). In Fig. 5the dependence of the /epsilon1Ron the temperature of a graphene sample for different dielectric permittivitiesof substrate /epsilon1is shown. Graphene with /epsilon1=1.8i si nt h e insulator phase. The temperature dependence in this phaseis the weakest as compared to the /epsilon1=4.0 and /epsilon1=7.3 points. This happens since in the insulator phase fermions 195401-4INTERACTION OF STATIC CHARGES IN GRAPHENE . . . PHYSICAL REVIEW B 89, 195401 (2014) 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6εR T(eV)ε=1.8 ε=4.0 ε=7.3 FIG. 5. (Color online) Dependence of the /epsilon1Ron the temperature of graphene sample for the /epsilon1=1.8 (insulator phase), /epsilon1=4.0 (transion region), and /epsilon1=7.3 (semiconductor phase), is shown. have dynamically generated mass. If this mass is larger than the temperature, the production of free charges whichenhances the /epsilon1 Ris suppressed. If the fermions are massless, free charge production is no longer suppressed and the temper-ature dependence of the /epsilon1 Ris stronger. This effect is noticeable in the semimetal phase at /epsilon1=7.3, where quasiparticles are massless. The most rapid temperature dependence takesplace for /epsilon1=4.0, which is in the transition region. In this region graphene is in the insulator phase at low temperature andin the semimetal phase at high temperature which explains themost rapid temperature dependence. In Fig. 6the dependence of/epsilon1 Ron/epsilon1at different temperatures is shown. Formula ( 14) ﬁts data satisfactory ( χ2/DOF∼1–3) for all temperatures. However, the larger the temperature, the largerχ 2/DOF in the semimetal phase. In order to illustrate this, in Table Iwe present the χ2/DOF for the ﬁt of data by the Coulomb potential at different temperatures for /epsilon1=7.8. Similar results can be presented for the other /epsilon1in the semimetal 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7 8 9εR εT=0.23 eV T=0.74 eV T=1.28 eV FIG. 6. (Color online) Dependence of the /epsilon1Ron the /epsilon1at different temperatures obtained from the ﬁtting with Coulomb potentialV C(/vectorr)(14).TABLE I. Values of the χ2/DOF for the ﬁt of data by the Coulomb and Debye potentials at different temperatures for /epsilon1=7.8. T(eV) 0.50 0.56 0.74 1.00 1.08 1.28 1.56 Coulomb potential χ2/DOF 1.1 0.6 0.9 1.3 2.1 1.8 2.6 Debye potential χ2/DOF 0.4 0.5 0.3 0.3 0.3 0.4 0.3 phase. We assume that the worsening of the ﬁtting model can be assigned to the following fact. At sufﬁciently largetemperature graphene contains an equal number of electronsand holes. If one puts electric charge to such media, a nonzerocharge density is created. This charge density leads to somesort of Debye screening in graphene which is not accountedfor in ( 14). In the Appendix the derivation of the Debye screening in graphene is given. It is assumed that the interaction betweenquasiparticles is weak, which is the true only for sufﬁcientlylarge/epsilon1. However, the Debye potential ( A5) without explicit expression for Debye screening mass m D(A6) can be thought of as a modiﬁcation of the Coulomb potential with unknownparameter m D. In this sense formula ( A5) can be applied for all values of /epsilon1and temperature. To carry out the study of the temperature dependence of the interaction potential we replace the lattice Coulomb potentialV C(/vectorr) by the lattice version of Debye screening potential ( A9) in the ﬁtting procedure. Before the modiﬁcations of the poten-tial, the description ( χ 2/DOF∼1–3) of the available data was not as good as it became after the modiﬁcation ( χ2/DOF<1), especially for /epsilon1in the semimetal phase. Table Irepresents this fact for the /epsilon1=7.8. In Fig. 7we plot the /epsilon1Ras a function of the/epsilon1for different temperatures. One can see that contrary to the ﬁtting procedure with Coulomb potential the /epsilon1Rwith Debye screening potential is almost temperature independent. So, theﬁtting with Debye screening potential cancels the temperaturedependence from the dielectric permittivity /epsilon1 Rand encodes 1 1.2 1.4 1.6 1.8 2 2.2 1 2 3 4 5 6 7 8εR εT=0.23 eV T=0.74 eV FIG. 7. (Color online) Dependence of the /epsilon1Ron the /epsilon1at different temperatures obtained from the ﬁtting with Debye screening potentialV D(/vectorr)(A9). 195401-5V. V. B R AG U TA et al. PHYSICAL REVIEW B 89, 195401 (2014) 0 1000 2000 3000 4000 5000 2 4 6 8 10r εT=0.23 eV T=0.74 eV two-dimen. plasma FIG. 8. (Color online) Dependence of the ratio r= (mDα)/(TαR)o nt h e /epsilon1at different temperatures. The line parallel to the /epsilon1axis is the value of the ratio ( mDα)/(TαR)a tt h e approximation of weakly interacting two-dimensional plasma ofquasiparticles. it into Debye mass mD. This conﬁrms that in some region the temperature dependence of the interaction potential resultsfrom the Debye screening. Now let us turn to Debye screening mass. Equation (A8) deﬁnes the Debye mass for two-dimensional plasma of quasiparticles when interaction between quasiparticles isweak. It is not difﬁcult to derive the expression for Debyemassm Dwhich is valid for the interacting quasiparticles. Evidently, if there is no interaction between quasiparticles,m D=0. This means that the expansion of mDstarts for the term proportional to the ∼αR, which determines the strength of the interaction. The second property of the mDis that it disappears if the density of quasiparticles nis zero. So, one concludes that mD∼n/T, where temperature appeared in the denominator for the dimensional reasons.3Now we have the following expression: mD=k(α,T)αRn T, (16) where k(αR,T) is some function which could depend on the αRandT.I nF i g . 8we present the following observable: r=(mDα)/(TαR), which is proportional to the n/T2.T h i s observable allows one to study the density of quasiparticlesin graphene. If the interaction between quasiparticles is weak,the ratio ( m Dα)/(TαR) is equal to r=mDα TαR=8l n 2α v2 F/similarequal3600. (17) In Fig. 8the dependence of the ratio ( mDα)/(TαR)o nt h e /epsilon1at different temperatures is shown. The line parallel to the /epsilon1axis is the value of the ratio ( mDα)/(TαR)(17). Now the following few comments are in order.(i) First let us consider the semimetal phase /epsilon1>5. In this region the ratio ( m Dα)/(TαR) tends to some constant value and this value is by a factor ∼1.5–2.0 smaller than that given by 3The density nin graphene has dimension ∼(energy)2.formula ( 17). The possible source of this disagreement is that in formula ( 17) we used the bare Fermi velocity vF. Evidently one should use the renormalized Fermi velocity vR F, which is beyond the scope of this paper. The vR Fis larger than the vF,s o the inclusion of Fermi velocity renormalization will push theconstant ( 17) to the correct direction. Accounting for this fact one can conclude that within the uncertainty of the calculationin the semimetal phase electron excitations in graphene forma weakly interacting two-dimensional plasma. (ii) Assuming that the difference between constant ( 17) and the position of the plateau in Fig. 8results from Fermi velocity renormalization one can estimate the ratio v R F/vF in the semimetal phase as ∼1.2–1.4. This value is in a reasonable agreement with the results obtained within MonteCarlo simulation of graphene [ 15] and with experiment [ 17]. (iii) It is seen from Fig. 8thatat low temperature Debye massm Dplays a role of order parameter of the insulator- semimetal phase transition. At small dielectric permittivity of substrate, mDequals zero within the accuracy of the calcula- tion, which means that the interaction potential is Coulomb.At/epsilon1∼4–5 Debye mass becomes nonzero, abruptly reaching the regime of the two-dimensional plasma. The interaction inthis region has a form of the Debye potential. Thus the studyof Debye screening mass allows one to determine the positionof the insulator-semimetal phase transition, which takes placeat/epsilon1∼4–5, in accordance with the results of papers [ 3,5]. At large temperatures m Dis not zero for any values of the /epsilon1.I ti s a smoothly rising function of /epsilon1which is saturated at /epsilon1∼4–5. (iv) To understand the behavior of the Debye mass, which is proportional to the density of excitations n, one can use the following model. In the insulator phase /epsilon1<4 the fermion excitation acquires dynamical mass. So, the density of chargedfermion excitations is exponentially suppressed n T2∼exp/parenleftbigg −Mf(g2) T/parenrightbigg . (18) The dynamical fermion mass Mf(g2) depends on the effective coupling constant g2=α0/vF. It is seen from Fig. 8that at temperature T=0.23 eV ( asT=0.000 19) the density is either considerably suppressed or equal to zero, whichimplies that M f(g2)>T. However, at temperature T= 0.74 eV ( asT=0.000 61) the density is no longer suppressed and it is a monotonically rising function of the effectiveconstant, which implies that M f(g2)<T. So, the dynami- cally generated fermion mass in the insulator region can beestimated as M f(g2)∼0.5e V . At the end of this section we should note that because of the smallness of the Fermi velocity vFthe Debye screening radius is rather small. For instance, according to formula ( A6) it is equal to ∼20×distance between carbon atoms for the room temperature and /epsilon1∼5. In conclusion , in this paper we carried out the study of the interaction potential between static charges in grapheneeffective ﬁeld theory within the Monte Carlo simulation fordifferent dielectric permittivities of substrate /epsilon1and various temperatures. To calculate the interaction potential we mea-sured the correlator of Polyakov lines. At low temperaturesthe interaction can be satisfactory described by the Coulombpotential screened by some dielectric permittivity /epsilon1 R.W e 195401-6INTERACTION OF STATIC CHARGES IN GRAPHENE . . . PHYSICAL REVIEW B 89, 195401 (2014) determined the dependence of the /epsilon1Ron the dielectric permit- tivity of substrate. In addition, we determined the dependenceof the renormalized charge squared α Ron the bare one α0 and showed that in the semimetal phase the αRcan be well described by one loop formula. At larger temperatures the interaction potential deviates from Coulomb. The main result of this paper is that for alltemperatures and dielectric permittivities the interaction canbe well described by the Coulomb potential with the Debyescreening by two-dimensional plasma of fermionic excitations.It is shown that at low temperature the Debye mass m Dplays a role of order parameter of the insulator-semimetal phasetransition. At small dielectric permittivity of substrate, m D equals zero within the accuracy of the calculation, which means that the interaction potential is Coulomb. At /epsilon1∼4–5 the Debye mass becomes nonzero, abruptly reaching the regimeof two-dimensional plasma. The interaction in this region isdue to the Debye potential. Thus the study of the Debyescreening mass allows one to determine the position of theinsulator-semimetal phase transition, which takes place at/epsilon1∼4–5. At large temperatures m Dis not zero for any values of the/epsilon1. It is a smoothly rising function of /epsilon1which is saturated at /epsilon1∼4–5. In the semimetal phase for all temperatures studied in this paper the Debye mass can be rather well described by theformula for two-dimensional plasma of fermionic excitations,where the interactions between excitations are accounted forby the renormalization of the charge squared α Rand the Fermi velocity vR F. ACKNOWLEDGMENTS The authors are grateful to Professor Mikhail Zubkov for interesting and useful discussions. The work was supported byGrant No. RFBR-14-02-01261-a, by the Russian Ministry ofScience and Education, under Contract No. 07.514.12.4028,and by the Far Eastern Federal University and FEB RAS,under Grant No. 13-NSC-005. Numerical calculations wereperformed at the ITEP system Graphyn and Stakan (authorsare much obliged to A. V . Barylov, A. A. Golubev, V . A.Kolosov, I. E. Korolko, and M. M. Sokolov for the help), theMVS 100K at Moscow Joint Supercomputer Center, and atSupercomputing Center of the Moscow State University. APPENDIX: DEBYE SCREENING IN GRAPHENE This section is devoted to the derivation of the potential of Debye screening in graphene. An important differencebetween graphene and the usual three-dimensional electro-magnetic plasma is that free charges in graphene are twodimensional. It will be shown below that this property leads tothe change of the exponential screening to power screening. Suppose that positive charge Qis located at the origin of coordinates. It is clear that quasiparticles with positive charge+erepel from the charge Q. The two-dimensional density of positive quasiparticles on graphene plane can be found fromBoltzmann distribution, n +(r)=nexp/parenleftbigg −eϕ(r) T/parenrightbigg , (A1)where nis a density of positive quasiparticles at inﬁnity andϕ(r) is the potential which is created by the charge Q. Analogously, negative quasiparticles attract to the Qand their density on graphene plane n−(r) can be found as follows: n−(r)=nexp/parenleftbiggeϕ(r) T/parenrightbigg , (A2) Evidently, the charge density at distance ris ρ(r)=e[n+(r)−n−(r)]/similarequal−2nαϕ(r) T. (A3) In the last equation it was assumed that eϕ/lessmuchT(e2=α). Taking into account the nonzero charge density ρ(r), one can write the Maxwell equation, −/Delta1ϕ+8πnα Tδ(z)ϕ=4πQδ3(/vectorr). (A4) Note that the δfunction δ(z) in the second term takes into account the fact that the charges are located on the grapheneplane z=0. The solution of the Maxwell equation on the graphene plane can be written as follows: ϕ(r)=Q/integraldisplayd 2p (2π)ei/vectorp·/vectorr \|p\|+mD =Q r/integraldisplay∞ 0dξe−(mDr)ξ (1+ξ2)3/2ξ, (A5) mD T=4πe2n T=2π2 3α v2 F, (A6) where /vectorp=(px,py). It should be noted here that one can use Fermi distribution instead of Boltzmann distributions ( A1), (A2) n±(r)=/integraldisplayd2p (2π)21 exp{[vF\|/vectorp\|±eϕ(r)]/T}+1,(A7) expand them in the ratio eϕ(r)/T, and repeat all the above steps. This leads to the same expression for the potential ϕ(r) (A5) but with different Debye mass mD T=8l n 2α v2 F, (A8) which is 15% smaller than Debye mass in Eq. ( A6). The solution ϕ(r) satisﬁes the following limits: ϕ(r)=/braceleftBiggQ r, (rmD)/lessmuch1, Q r1 (mDr)2,(rmD)/greatermuch1. Thus at large distances Debye screening leads to an ∼1/r3 decrease of the potential. It is rather easy to write a lattice version of the potential ( A5) on the graphene plane, VD(/vectorr)=4πα/summationdisplay n1,n2f(p1,p2) 1+2mD(Lsas)2f(p1,p2)ei/vectorp·/vectorr, f(p1,p2)=1 4L3sas/summationdisplay n31/summationtext isin2(pias/2),p i=2π Lsasni. (A9) In formula ( A9) the integers n1,n2,n3run the values 0,1,..., L s−1, except for the case n1=n2=0. 195401-7V. V. 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Rev. B 83, 081409 (2011 ). [9] K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, Y . Zhang, S. V . Dubonos, I. V . Grigorieva, and A. A. Firsov, Science 306, 666(2004 ). [10] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81,109(2009 ). [11] A. K. Geim and K. S. Novoselov, Nat. Mater. 6,183(2007 ). [12] I. Montvay and G. Muenster, Quantum Fields on a Lattice (Cambridge University Press, Cambridge, UK, 1994). [13] T. DeGrand and C. DeTar, Lattice Methods for Quantum Chromodynamics (World Scientiﬁc, Singapore, 2006). [14] C. Burden and A. N. Burkitt, Eur. Phys. Lett. 3,545(1987 ). [15] J. E. Drut and T. A. L ¨ahde, PoS Lattice2013 , 498 (2013). [16] J. Gonzalez, F. Guinea, and M. A. H. V ozmediano, Nucl. Phys. B424,595(1994 ). [17] G. L. Yu et al. , Proc. Natl. Acad. Sci. USA 110, 3285 (2013). 195401-8
PhysRevB.97.155414.pdf	PHYSICAL REVIEW B 97, 155414 (2018) Bistability and displacement ﬂuctuations in a quantum nanomechanical oscillator R. Avriller, B. Murr, and F. Pistolesi Université Bordeaux, CNRS, LOMA, UMR 5798, F-33405 Talence, France (Received 10 January 2018; revised manuscript received 6 April 2018; published 16 April 2018) Remarkable features have been predicted for the mechanical ﬂuctuations at the bistability transition of a classical oscillator coupled capacitively to a quantum dot [Micchi et al. ,Phys. Rev. Lett. 115,206802 (2015 )]. These results have been obtained in the regime ¯ hω 0/lessmuchkBT/lessmuch¯h/Gamma1,w h e r e ω0,T,a n d/Gamma1are the mechanical resonating frequency, the temperature, and the tunneling rate, respectively. A similar behavior could be expectedin the quantum regime of ¯ h/Gamma1/lessmuchk BT/lessmuch¯hω 0. We thus calculate the energy- and displacement-ﬂuctuation spectra and study their behavior as a function of the electromechanical coupling constant when the system enters theFrank-Condon regime. We ﬁnd that in analogy with the classical case, the energy-ﬂuctuation spectrum and thedisplacement spectrum widths show a maximum for values of the coupling constant at which a mechanicalbistability is established. DOI: 10.1103/PhysRevB.97.155414 I. INTRODUCTION Nanoelectromechanical systems (NEMS) have proved to be devices of great interest, both from fundamental andapplicative points of view [ 1]. A paradigmatic example of such devices is represented by suspended carbon-nanotubemechanical resonators [ 2–5]. Due to their low mass (10 −18g) and high Young modulus (1 TPa), carbon-nanotube mechanicaloscillators are ideal candidates for developing a new generation of ultrasensitive force and mass sensors. A lot of effort was thus devoted in the past decades in order to propose efﬁcientschemes to actuate and detect the mechanical motion of suchdevices. The mixing technique is one of those approaches [ 2,5]. Initially proposed in Ref. [ 2], it enables one to mechanically excite a nanotube quantum dot by applying suitable time-dependent gate and bias voltages. The resulting mechanicaloscillation of the nanotube in the frequency range ω 0/2π≈ 100 MHz–10 GHz [ 6,7] is then transduced toward a mea- surable lower-frequency electronic mixing current. The lattercontains information about both quadratures of the nanotubedisplacement and thus about its mechanical susceptibility. Thistechnique was used to measure tiny variations of the resonancefrequency in real time, upon adsorption of molecules on thesurface of the nanotube [ 8]. This enabled one to perform mass-sensing experiments with a record sensitivity reportedat the yoctogram resolution (proton mass) [ 8] and to detect the backaction of single-electron tunneling events as a measurablesoftening of the mechanical resonance frequency [ 3,4,9,10]. The optimum sensitivity achievable with the mixing techniquewas investigated theoretically in Ref. [ 11] and was shown to arise from a compromise between maximizing the mixingsignal to overcome electronic shot noise and minimizing theadded noise corresponding to electronic backaction. The higher the electromechanical coupling, the higher the achieved sensitivity, thus justifying the goal of reachingthe strong-coupling regime between tunneling electrons andone mechanical degree of freedom of the nanotube. Recentprogress in fabrication techniques was reported that goes alongthat direction [ 12,13] by designing local quantum dots on the surface of the nanotube, with full control of their electricaland mechanical properties. This enables one to probe regimeswhere the height of the tunneling barriers /Gamma1is either smaller or larger than ω 0, as well as to spatially image the excited mechanical mode by changing the location of the quantum dotalong the nanotube direction [ 13]. In those experiments, the electromechanical coupling strength is given by the polaronicenergy scale /epsilon1 P=F2 0/k, with F0the excess of force applied on the oscillator upon tunneling of a single electron, andkthe nanotube spring constant. Typical electromechanical coupling strengths obtained in the experiments of Ref. [ 13]a r e estimated from the softening of the resonance frequency to beof the order of /epsilon1 P≈0.3 K at temperature T=16 K [ 14]. Less invasive and low-noise techniques were recently proposed, theprinciple of which is to extract the oscillator displacementﬂuctuation spectrum S xx(ω) from a measurement of the current ﬂuctuations across the nanotube [ 15]. Large mechanical quality factors Qup to 5 million were reported with this approach [ 16], as well as force-sensing experiments with a resolution up to≈12 zN Hz −1/2[15]. Recently, some of the authors investigated theoretically measurable mechanical properties of a classical and slow sus-pended carbon nanotube [ 14,17], for which ω 0/lessmuchV,T/lessmuch/Gamma1 (in the paper we use the notation that the Planck constant ¯h, the Boltzmann constant kB, and the elementary electron charge eare all set to 1). They showed that entering the strong electromechanical coupling regime has a dramatic impact on the oscillator displacement spectrum Sxx(ω). Upon increasing /epsilon1P//Gamma1, the maximum frequency of the spectrum, ωmax,i s softened toward lower frequencies, while the full width halfmaximum (FWHM) /Delta1ω of the spectral line increases up to a maximum value reached for a critical coupling strength/epsilon1 P=π/Gamma1. At this critical point, the line shape of the spectrum is dominated by a strong frequency noise induced by the dominating quartic nonlinearities of the mechanical oscillator[14]. Universal scaling behavior with bias voltage of both ω maxand/Delta1ω≈ω0(V//Gamma1 )1/4, as well as a universal quality factor Q≈1.7[14], were predicted. Increasing further the 2469-9950/2018/97(15)/155414(13) 155414-1 ©2018 American Physical SocietyR. A VRILLER, B. MURR, AND F. PISTOLESI PHYSICAL REVIEW B 97, 155414 (2018) electromechanical coupling /epsilon1P>π /Gamma1 , the mechanical oscil- lator becomes effectively bistable and the electronic cur-rent across the nanotube is progressively blocked. This phe-nomenon is analogous to the current-blockade transition thatwas predicted for a classical oscillator coupled to incoherenttunneling electrons ( /Gamma1/lessmuchT) when /epsilon1 P>V [18–21]. Interest- ingly, the critical point at which the current blockade occurs coincides with the point at which the dephasing rate due to frequency noise is maximum [ 14] and the mixing technique has a maximum sensitivity [ 22]. The full stability phase diagram for the mechanical oscillator and the corresponding line shapesof the position ﬂuctuation spectra were derived as a function ofbias, gate voltage, and temperature in Ref. [ 17]. This effect can be observed, in principle, in existing samples [ 13], provided they are measured at a temperature of the order of 20 mK. A similar phenomenon, known as the Franck-Condon blockade, has been predicted [ 23–26] and observed [ 27,28]f o r molecular systems in the opposite regime of large resonatingfrequency /Gamma1/lessmuchT/lessmuchω 0, for which the oscillator is close to its quantum ground state. The consequences in electronic trans-port of the Franck-Condon blockade have been investigated indetail, but much less is known about the dynamical propertiesof the mechanical oscillator in this regime [ 29–32]. The aim of the present paper is to investigate if there is a quantum counterpart of the striking behavior of the displace-ment ﬂuctuation spectrum predicted in the classical regime:namely, the existence and measurable manifestations of amechanical bistability and the coupling constant dependenceof the width /Delta1ω of the displacement ﬂuctuation spectrum. Concerning the bistability, it is well known that for strongcoupling, the current is blocked. This means that electronscan no longer tunnel keeping the electronic dot in either theempty or full state. Can one regard this system as a bistableone in a similar manner to the classical system? What isthe relation between the quantum displacement ﬂuctuationspectrum and the appearance of the bistability? One cananticipate that at weak coupling, /Delta1ω exhibits a quadratic dependence on the coupling constant, coming from simpleperturbative arguments, but the strong-coupling limit demandsmore insight since the width may have a different origin:energy dissipation, classical phase ﬂuctuations, and quantumdecoherence. In order to answer these questions, we calculatein the quantum fast oscillator regime the (nonsymmetric)displacement spectrum, the energy-ﬂuctuation spectrum, andthe Wigner distribution for the oscillator. We ﬁnd that the widthof the energy-ﬂuctuation spectrum shows a clear maximumfor the same value of the coupling constant for which theprobability distribution develops a double peak. This canbe interpreted as the onset of the bistability. The energyscale for this transition turns out to be /epsilon1 P=2ωo.T h es a m e energy scale controls the washing out of the bistability asa function of the temperature /epsilon1 P≈Tor the voltage bias /epsilon1P≈V. We present a detailed analytical analysis, indicating that despite the similarity with the classical case, the originof the maximum of the dissipation has a different origin inthe quantum case. The behavior predicted could be observedby detecting ﬁnite frequency current noise through suspendedcarbon nanotubes where electronic transport is coupled eitherto GHz ﬂexural modes [ 6,7] or to THz nanotube breathing modes [ 27]. FIG. 1. Representation of a nanomechanical oscillator with reso- nance frequency ω0. The oscillator is coupled to a quantum dot, de- scribed by a single electronic level of energy ε0. Charge is transferred from the left (right) lead to the dot with a tunneling rate /Gamma1L(/Gamma1R). An externally applied bias voltage Vleads to a difference between the chemical potentials of the electronic reservoirs, μL−μR=V. The organization of the paper is the following. In Sec. II, we introduce the microscopic Hamiltonian describing a me-chanical oscillator coupled to a single-level quantum dot. InSec. III, we derive the generalized master equation with the Born-Markov approximation, which enables one to computethe dynamical properties of the mechanical oscillator. We ﬁndthat for this purpose, it is necessary to compute the evolutionof the off-diagonal elements of the density matrix, even if weare dealing with incoherent transport. The energy and positionﬂuctuation spectra are computed, respectively, in Secs. IVand V. The dissipation and decoherence mechanisms are analyzed in relation to the crossover toward bistability of the mechanicaloscillator. Finally, the bias-voltage dependence of both energyand displacement spectra is shown in Sec. VI. II. THE MECHANICAL SYSTEM We consider a nanomechanical oscillator capacitively cou- pled to a quantum dot (see Fig. 1). We assume that transport is dominated by a single electronic level. Assuming spinlesselectrons, the microscopic Hamiltonian of the full electrome-chanical system is given by H=H 0+/summationdisplay α=L,RHα+HT, (1) H0=[/epsilon10+gω0(a+a†)]d†d+ω0a†a, (2) Hα=/summationdisplay k(εαk−μα)c† αkcαk, (3) HT=/summationdisplay α=L,R/summationdisplay k{tαkc† αkd+t∗ αkd†cαk}, (4) where d†anda†are, respectively, the creation operator for an electron on the dot and a vibron on the mechanical oscillator.The ﬁrst term H 0describes the mechanical oscillator of bare resonance frequency ω0and the single-level quantum dot of energy ε0. The charge operator on the dot nd=d†dcouples linearly to the oscillator displacement operator, x=x0(a+a†), (5) 155414-2BISTABILITY AND DISPLACEMENT FLUCTUATIONS IN … PHYSICAL REVIEW B 97, 155414 (2018) withx0=√1/2mω 0its zero-point motion. The electrome- chanical coupling strength in units of the vibron energy iswritten gω 0, with the excess force acting on the oscillator when one electron is added, F0=gω0/x0. The second term Hαis the Hamiltonian of the α=L(left) and =R(right) free electronic reservoirs, both characterized by an electronic bandstructure ε αkand a chemical potential μα. A voltage bias Vis externally applied, which we will suppose to be equally sharedbetween left and right metallic reservoirs, namely, μ L=V/2 andμR=−V/2. Finally, the last term HTis the tunneling Hamiltonian. It describes charge transfer from the electronicreservoir α=L,R to the quantum dot, with a corresponding tunneling rate /Gamma1 α=2π\|tα\|2ρα. The former is proportional to the hopping term tαk≡tαsupposed to be real and independent of the wave vector kand to the electronic density of states ραevaluated at the Fermi energy (wideband approximation). Note that the relevant energy scale of the problem is thepolaronic energy deﬁned above as /epsilon1 P=F2 0/k=2g2ω0.W e will see that when /epsilon1Pcrosses the other relevant energy scales, as the temperature T, the bias voltage V, or the zero-point motion energy ω0, the strong-coupling effects appear to be relevant. When only ω0matters, one can either use gor /epsilon1P/2ω0=g2as the dimensionless coupling. We will use both in the following since certain expressions and dependences aremore transparent in terms of g 2. We begin by performing the Lang-Firsov unitary transfor- mation [ 33]U=egnd(a−a†)to the Hamiltonian of Eq. ( 1). The transformed Hamiltonian ˜H=UHU†is obtained as ˜H0=˜/epsilon10d†d+ω0a†a, (6) ˜HT=/summationdisplay α=L,Rtα/summationdisplay k{c† αkD+D†cαk}. (7) The meaning of Eq. ( 6) is the following: upon tunneling of a single electron, the quantum dot is excited into a chargedelectronic state. The corresponding excess energy can bepartially released by relaxation of the mechanical oscillatorinto a new equilibrium position, ˜X eq=− 2gx0. The energy of the single-level quantum dot, ˜ /epsilon10=/epsilon10−/epsilon1P/2, is consequently reduced by the polaronic shift. Any explicit term involvingthe electromechanical coupling has thus disappeared from theexpression of ˜H 0, at the price of modifying the tunneling Hamiltonian given by Eq. ( 7). The hopping terms tαbelong- ing to ˜HTare renormalized by the polaron cloud operator Q=eg(a−a†)and incorporated into a redeﬁnition of the dot annihilation operator, D≡dQ. The displacement operator is modiﬁed also by the same transformation and can be written as x→UxU†=X−2gndx0, (8) where X=x0(a+a†) and the dynamics of the operators a andndis now ruled by ˜H. In the following, we consider the regime of electron incoherent transport and quantum oscillator. This regime isachieved when the reservoir temperature Tis larger than the total tunneling rate /Gamma1=/Gamma1 L+/Gamma1R, but smaller than the mechanical frequency ω0. The corresponding hierarchy of frequencies /Gamma1/lessmuchT/lessmuchω0is obtained, for example, for the following realistic values of the parameters: /Gamma1=500 MHz,FIG. 2. Symmetrized energy-ﬂuctuation spectrum Ssym EE(ω)o ft h e mechanical oscillator as a function of frequency ω. Numerical results using the secular approximation developed in Sec. III B . Inset: Relative error in % between the analytical Lorentzian shapesprovided by Eq. ( 33) and the previous numerical curves. Various elec- tromechanical coupling strengths are probed: g 2=0.04,0.4,1.0,5.8. Parameters common to all curves: /Gamma1=0.05ω0,˜ε0=0,T=0.1ω0, andV=0.2ω0. T=50 mK, and ω0/2π=10 GHz. We chose to perform our numerical calculations in Figs. 2and7with those parameters. This gives rise to the well-known Franck-Condon regime ofelectronic transport as studied in Refs. [ 23,24,26,34]. III. MASTER EQUATION A. Born-Markov approximation We deﬁne ρ(t) as the reduced density matrix of the mechanical oscillator and quantum-dot subsystem, obtainedafter tracing out the degrees of freedom of the electronicreservoirs. In the sequential tunneling regime ( /Gamma1/lessmuchT), we derive a generalized master equation ruling the dynamics of thereduced density matrix within the Born-Markov approximation[31,32,35–37], ˙ρ(t)=Lρ(t), (9) where L=L c+Ldand Lcρ=−i[˜H0,ρ], (10) Ldρ=[Dhρ−ρDe,D†]+H.c. (11) The term Lcdescribes the coherent (unitary) evolution of the reduced density matrix induced by the Hamiltonian ˜H0, and Ld describes dissipation and decoherence of the electromechani- cal subsystem due to its weak coupling (tunneling term) to theelectronic bath. It involves the operator D ν=e,hdeﬁned as Dν=/integraldisplay+∞ 0dτCν(−τ)DI(−τ), (12) Cν(τ)=/summationdisplay αk\|tα\|2fνα(εαk)eiεαkτ, (13) with DI(−τ), and the operator Dwritten in interaction representation with respect to ˜H0. The correlation functions 155414-3R. A VRILLER, B. MURR, AND F. PISTOLESI PHYSICAL REVIEW B 97, 155414 (2018) Cν=e,h(τ) for the metallic reservoirs are written in terms of the Fermi-Dirac distributions for electrons, feα(ω)≡ f(ω−μα), and holes, fhα(ω)≡1−f(ω−μα), with f(ω)= {eβω+1}−1. The wideband approximation enables one to obtain a compact expression for the correlation functions Cν(ω)=/integraltext+∞ 0dτCν(τ)e−i(ω−iη)τas Cν(ω)=/summationdisplay α/Gamma1α 2/braceleftbiggisν πRe/Psi1/bracketleftbigg1 2+iβ 2π(ω−μα)/bracketrightbigg +fαν(ω)/bracerightbigg , (14) with/Psi1[ω] the Euler digamma function [ 38], obtained from the Hilbert transform of Fermi distribution functions [ 39] and sν=e(h)=1(−1). The master Eq. ( 9) is ﬁnally projected onto the basis of eigenstates \|q,n/angbracketrightof the Hamiltonian ˜H0, corresponding to q=0,1 charge populating the quantum dot and nvibrons populating the mechanical mode. The eigenvalue associatedto the \|q,n/angbracketrighteigenstate is ε qn=q˜ε0+nω0. The resulting linear equations for the reduced density matrix can be solvednumerically (exact Born-Markov approximation). B. Secular approximation The dynamics of the coupled electromechanical system, as described by Eq. ( 9), is quite complicated. A series of approximations can be derived in order to simplify the masterequation: (i) ﬁrst, by dropping in the dissipative evolution[Eq. ( 11)] terms that can be incorporated into a renormalization of˜H 0(Lamb-shift terms), and (ii) second, by performing a secular approximation, which enables one to separate theevolution of diagonal elements of the density matrix π (q,n)(t)≡ ρ(q,n)(q,n)(t) (populations) from the evolution of off-diagonal terms σ(r,m) (q,n)(t)≡ρ(q,n)(r,m)(t) (coherences). The secular ap- proximation, however, has to be done with some care due tothe equidistance between the energy levels of the mechanicaloscillator [ 40]. We ﬁnally obtain the following set of linear equations describing the dynamics of the damped mechanicaloscillator capacitively coupled to a quantum dot: ˙π (q,n)(t)=/summationdisplay m∈N/braceleftbig /Gamma1(¯q,m) (q,n)π(¯q,m)(t)−/Gamma1(q,n) (¯q,m)π(q,n)(t)/bracerightbig ,(15) ˙σ(r,m) (q,n)(t)=−/bracketleftBigg i/Omega1(r,m) (q,n)+/Lambda1(r,m) (q,n) 2/bracketrightBigg σ(r,m) (q,n)(t) +δq,r/summationdisplay p∈N/Xi1(¯q,p)(¯q,p+m−n) (q,n)(q,m)σ(¯q,p+m−n) (¯q,p) (t),(16) withδq,rthe Kronecker delta and ¯q=1,0 when q=0,1. Equation ( 15) is the Pauli rate equation giving the evolution of populations. The transition rates /Gamma1(q,n) (¯q,m)between the states \|q,n/angbracketrightand\|¯q,m/angbracketrightcoincide with the expressions given by the Fermi golden rule [ 24,26], /Gamma1(0,n) (1,m)=/summationdisplay α/Gamma1α\|Qn,m\|2feα(˜ε0+(m−n)ω0), (17) /Gamma1(1,n) (0,m)=/summationdisplay α/Gamma1α\|Qn,m\|2fhα(˜ε0−(m−n)ω0), (18)withQn,m≡/angbracketleftn\|Q\|m/angbracketrightthe overlap integral between the state of the mechanical oscillator with nvibrons and the state of the displaced mechanical oscillator with mvibrons [ 24,26]. Equation ( 16) provides the evolution of the off-diagonal elements of the density matrix. We introduced the followingquantities: /Omega1(r,m) (q,n)=[(q−r)˜ε0+(n−m)ω0], (19) /Lambda1(r,m) (q,n)=/summationdisplay p∈N/bracketleftbig /Gamma1(q,n) (¯q,p)+/Gamma1(r,m) (¯r,p)/bracketrightbig , (20) with/Omega1(r,m) (q,n)the Bohr frequency associated to the states \|q,n/angbracketright and\|r,m/angbracketright, and /Lambda1(r,m) (q,n)the decay rate that is responsible for the damping of the corresponding off-diagonal element of the density matrix. Finally, the matrix element /Xi1(¯q,p)(¯q,p+m−n) (q,n)(q,m) is associated to the transfer of coherences between the coupleof states {\|q,n/angbracketright,\|q,m/angbracketright}and{\|¯q,p/angbracketright,\|¯q,p+m−n/angbracketright}for the damped mechanical oscillator. It is explicitly given by /Xi1(0,p)(0,p+m−n) (1,n)(1,m) =/summationdisplay α/Gamma1αQ∗ p,nQp+m−n,mfeα/parenleftbig /Omega10,p 1,n/parenrightbig ,(21) /Xi1(1,p)(1,p+m−n) (0,n)(0,m) =/summationdisplay α/Gamma1αQn,pQ∗ m,p+m−nfhα/parenleftbig /Omega10,n 1,p/parenrightbig .(22) The evolution of the off-diagonal elements of the density matrix as described by Eqs. ( 16) was not taken into account in Refs. [ 24,26]. This is due to the fact that they are not needed to compute the average electronic current in the sequentialtunneling regime. However, when dealing with the study of themechanical-oscillator dynamics, these terms are necessary. C. Fluctuation spectrum We wish now to study observable properties characterizing the dynamical state of the mechanical oscillator. For thispurpose, we will investigate the average value ¯A≡/angbracketleftA/angbracketrightas well as the correlation function S AA(t)≡/angbracketleftδA(t)δA(0)/angbracketrightassociated to ﬂuctuations δA(t)=A(t)−¯Aof the observable Aacting on the mechanical oscillator. In the following, Awill stand for either the mechanical energy operator E=ω0nthat is proportional to the phonon-number operator n=a†aor for the position operator as deﬁned in Eq. ( 8). We further introduce the vector ρ(t) made of the matrix elements of the reduced density matrix ρ(t) (including both diagonal and off-diagonal terms). The master Eq. ( 9) can be given the compact form ˙ρ(t)=ˇLρ(t), (23) with ˇLthe superoperator associated to the linear operator L. Assuming a given initial condition for the density matrix ρ(0), we obtain, for ρ(t), ρ(t)=eˇLtρ(0). (24) The stationary density matrix ρstis the solution of the equation ˇLρst=0, from which the average value of the quantum mechanical observable Ais obtained, ¯A=tr(ρstA)≡wtˇAρst, (25) 155414-4BISTABILITY AND DISPLACEMENT FLUCTUATIONS IN … PHYSICAL REVIEW B 97, 155414 (2018) withwtthe null left eigenvector of the ˇLoperator ( wtˇL=0). wtapplied to any vector Areproduces the action of the quan- tum mechanical trace wtA=tr(A). Deﬁning the ﬂuctuation spectrum of AasSAA(ω)=/integraltext+∞ −∞dteiωtSAA(t) and using the quantum regression theorem [ 40,41], we ﬁnally obtain SAA(ω)=− 2Re/braceleftbigg wtδˇA1 (iω−η)ˇId+ˇLδˇAρst/bracerightbigg .(26) In the following, we will consider the symmetrized ﬂuctuation spectrum of the Aoperator, Ssym AA(ω)= [SAA(ω)+SAA(−ω)]/2. IV . ENERGY-FLUCTUATION SPECTRUM A. Dissipation of energy We ﬁrst characterize the dissipation rate γEof the mechanical-oscillator energy. For simplicity, we consider theregime of symmetric tunneling to the leads ( /Gamma1 L=/Gamma1R=/Gamma1), electron-hole symmetric point for the dot-level position(˜ε 0=0), and symmetric bias-voltage drop ( μL=−μR= V/2). In this regime, we ﬁnd that the transition rates in Eqs. ( 17) and ( 18) are equal, namely, /Gamma1(0,n) (1,m)=/Gamma1(1,n) (0,m)≡/Gamma1n→m. This simpliﬁcation enables one to write a rate equation for thephonon distribution π n(t)≡π0,n(t)+π1,n(t)u s i n gE q .( 15), ˙πn(t)=/summationdisplay m∈N,m/negationslash=n{/Gamma1m→nπm(t)−/Gamma1n→mπn(t)}. (27) In the limit of low voltage and temperature ( T,V <ω 0), the transition rates simplify to /Gamma1m→n≈2/Gamma1\|Qm,n\|2θm−n+/Gamma1\|Qn,n\|2δn,m, (28) withθm−n=1i fm>n , andθm−n=0 otherwise. The mean- ing of Eq. ( 28) is that close to equilibrium, only transitions from higher-energy states mto lower-energy ones n<m are allowed. The stationary phonon distribution πst nis thus the one obtained for a mechanical oscillator in its equilibrium quantumground state, namely, π st n=δn,0. In order to ﬁnd the energy relaxation for the mechan- ical oscillator, we consider the time evolution towards thesteady state of a weak ﬂuctuation, π n(t)≈πst n+δπn(t), with \|δπn(t)\|/lessmuch 1. Using Eqs. ( 27) and ( 28), the average vibron population ¯n(t)=/summationtext+∞ n=1nδπn(t) evolves as ˙¯n(t)≈2/Gamma1+∞/summationdisplay n=1n+∞/summationdisplay m=n+1\|Qm,n\|2δπm(t) −2/Gamma1+∞/summationdisplay n=1nn−1/summationdisplay m=0\|Qn,m\|2δπn(t), (29) which is not a closed equation in ¯n(t). However, we remark that in the regime T,V <ω 0, it is very unlikely that high- energy vibrational sidebands are signiﬁcantly excited. We thustruncate the vibron distribution to the ground and ﬁrst excitedstates, δπ n(t)≈δπ0(t)δn,0+δπ1(t)δn,1, such that the average vibron population becomes ¯n(t)≈δπ1(t). This assumption is veriﬁed a posteriori and enables one to rewrite Eq. ( 29)i na closed form, ˙¯n(t)≈−γE¯n(t), (30)FIG. 3. FWHM of the energy-ﬂuctuation spectrum /Delta1ωEas a function of electromechanical coupling g2. Circles: numerical result using the secular approximation developed in Sec. III B . Plain curve: analytical result given by Eq. ( 34). Parameters common to both curves: same as in Fig. 2. with γE=2/Gamma1\|Q1,0\|2=2/Gamma1g2e−g2. (31) Since ¯E(t)=¯n(t)ω0, one can identify γEwith the energy- dissipation rate. Its interpretation is straightforward. Theenergy of the mechanical oscillator is damped due to thetunneling of single electrons on the dot, which happens ona typical timescale given by the inverse electronic tunnelingrate 1//Gamma1. The damping rate is thus proportional to /Gamma1and to the Franck-Condon overlap matrix element, \|Q 01\|2=g2e−g2, which quantiﬁes the probability of a single tunneling electronto lose the energy of the vibrational mode and change thecharge state of the dot. Interestingly, γ Eis a nonmonotonous function of the electromechanical coupling g(see Fig. 3). At low coupling strengths ( g< 1), it is proportional to the square of the electromechanical coupling g2, as provided by perturbation theory. At higher coupling strengths ( g> 1), the damping rate decreases exponentially due to Franck-Condon block-ade: the charge state of the quantum dot becomes frozen,thus prohibiting dissipation to occur through charge ﬂuctu-ations. Finally, the damping rate reaches a maximum value,γ max E=2/Gamma1/e forg=1. B. Energy ﬂuctuations We now consider energy ﬂuctuations of the mechanical oscillator. Consistent with the Born-Markov approximation(see Sec. III A ) and with Eq. ( 30), the time evolution for the mechanical energy E(t) is ruled by the following Langevin equation: ˙E(t)=−γ EE(t)+ξE(t). (32) The ﬂuctuating part of the mechanical energy ξE(t)i so fz e r o average /angbracketleftξE(t)/angbracketright=0 and is δcorrelated in time, /angbracketleftξE(t)ξE(t/prime)/angbracketright= DEδ(t−t/prime). The diffusion coefﬁcient DE=2γE/Delta1n2is re- lated to the dissipation rate γEand to ﬂuctuations of the phonon population /Delta1n2=/angbracketleftn2/angbracketright−¯n2. At thermal equilibrium, we obtain DE=2γEnB, with the Bose distribution nB={eβω 0−1}−1. After Fourier transform, Eq. ( 32) enables one to ﬁnd an 155414-5R. A VRILLER, B. MURR, AND F. PISTOLESI PHYSICAL REVIEW B 97, 155414 (2018) analytical expression for the symmetrized spectrum Ssym EE(ω), Ssym EE(ω)=2γE/Delta1n2 ω2+γ2 E. (33) The energy-ﬂuctuation spectrum is thus a Lorentzian centered around zero frequency with FWHM /Delta1ωEgiven by twice the dissipation rate, /Delta1ωE=2γE=4/Gamma1g2e−g2. (34) The energy-ﬂuctuation spectrum Ssym EE(ω) is presented in Fig. 2, in the regime T=0.1ω0andV=0.2ω0, for which the mechanical oscillator is close to equilibrium. The main curvesare computed numerically using the secular approximationdeveloped in Sec. III B . The inset shows the relative error in % between the analytical Lorentzian shapes provided by Eq. ( 33) and the previous numerical curves. The extraction of theFWHM from the numerical curves is shown as a function of theelectromechanical coupling g 2in Fig. 3(red circles). The plain red curve is obtained from the analytical formula in Eq. ( 34). In both Figs. 2and 3, the excellent agreement (most of the time below 1%) between the numerics and the analytics standsfor a conﬁrmation that the broadening mechanism for energyﬂuctuations is indeed controlled by electronic dissipation, soultimately by tunneling of single electrons in and out of thequantum dot. C. Bistability of the mechanical oscillator In this section, we compute the stationary probability distribution π(x) of the mechanical-oscillator position. The stationary density matrix of the mechanical oscillator coupledto the quantum dot is approximatively diagonal in the basis ofthe eigenstates \|qn/angbracketright, namely, ρ st≈/summationtext q,nπ(q,n)\|qn/angbracketright/angbracketleftqn\|.T h e stationary distribution π(x) is thus approximated by π(x)≈/summationdisplay n∈N{π(0n)\|φn(x)\|2+π(1n)\|φn(x+2gx0)\|2}.(35) In Eq. ( 35),φn(x) is the wave function of the mechanical oscillator’s nth eigenstate, φn(x)=(2π)−1 4 √x02nn!Hn/bracketleftbiggx x0√ 2/bracketrightbigg exp/bracketleftBigg −/parenleftbiggx 2x0/parenrightbigg2/bracketrightBigg ,(36) withHn[x]t h enth Hermite polynomial [ 42]. We present in Fig. 4the probability distribution π(x) obtained with the same parameters as in Sec. IV A ,f o rw h i c h the mechanical oscillator is close to its quantum ground state,n=0. We ﬁnd that at low electromechanical coupling ( g< 1), the probability distribution π(x) has a single peak and the me- chanical oscillator is monostable. At larger couplings ( g> 1), the distribution develops two peaks and the mechanical oscil-lator becomes bistable. The transition between the monostablebehavior and the bistable one happens for g=1, for which the distribution has a very ﬂat top. The mechanism responsible forthis transition is the following. For any value of the couplingstrength g, the mechanical oscillator has two stable equilibrium positions located at x=0 and x=− 2gx 0, for which theFIG. 4. Stationary probability distribution of the oscillator posi- tionπ(x). Various electromechanical coupling strengths are probed: g2=0.04,0.4,1.0,5.8. Parameters common to all curves: /Gamma1=0.05ω0, ˜ε0=0,T=0.1ω0,a n dV=0.2ω0. charge state of the dot is, respectively, frozen at q=0 and q=1. The double-peak structure is resolved whenever the average shift of the equilibrium position /Delta1x=− 2gx0/angbracketleftq/angbracketright≡ −gx0induced by electromechanical coupling overcomes the zero-point quantum ﬂuctuations, −/Delta1x=gx0>x 0.I ti si n - teresting to notice that the transition point ( g=1) coincides with the value of the electromechanical coupling for which thedamping of the mechanical oscillator is maximum (see Fig. 3in Sec. IV B ). We complete the picture of the transition to bistability by showing in Fig. 5the two-dimensional (2D) plots repre- senting the mechanical-oscillator Wigner distribution [ 43,44] deﬁned as W(x,p)= 1 2π/integraltext dy/angbracketleftx+y 2\|ρ\|x−y 2/angbracketrighte−ipy, with p the oscillator momentum expressed in units of p0=√2mω 0. We ﬁnd that the Wigner distribution goes smoothly from asingle-peak distribution at low electromechanical couplingg 2=0.2 towards a double-peak distribution at higher coupling g2=6.0. The critical coupling g2=1 is characterized by a ﬂattened distribution, in agreement with Fig. 4.I ti st ob e noted that no negative contribution to the Wigner distribution isobtained. This is due to the fact that the Wigner distribution ofa harmonic oscillator in its quantum ground state is a Gaussianpositive distribution [ 44]. FIG. 5. Wigner distribution W(x,p) for the mechanical oscillator as a function of the oscillator position xand momentum p.T w o - dimensional maps obtained for various values of the electromechani- cal coupling: g2=0.2,1.0,√ 2,6.0. Parameters common to all panels: /Gamma1=0.05ω0,˜ε0=0,T=0.1ω0,a n dV=0.2ω0. 155414-6BISTABILITY AND DISPLACEMENT FLUCTUATIONS IN … PHYSICAL REVIEW B 97, 155414 (2018) V . DISPLACEMENT-FLUCTUATION SPECTRUM A. Oscillator decoherence time In this section, we investigate the evolution of the average of the Xoperator, X(t)=x0{a(t)+a†(t)}, obtained as X(t)=2x0+∞/summationdisplay n=0√ n+1R e/braceleftbig ρ(mec) nn+1(t)/bracerightbig , (37) withρ(mec) nm(t)=/summationtext q=0,1ρ(qn)(qm)(t) the reduced density matrix of the mechanical oscillator, obtained after tracing out thecharge degrees of freedom of the dot. Note that the physicaldisplacement is given by Eq. ( 8) and also implies the charge operator n d. We will see that the relevant ﬂuctuations of nd are at low frequency, allowing one to regard x≈Xat high frequency, ω≈ω0. We consider the same regime of low voltage and tempera- ture (T,V <ω 0) and symmetric electron-hole point (˜ ε0=0) as in Sec. IV A . Within the same approximation consisting of truncating the oscillator reduced density matrix to, at most, onevibron excitation ( n,m=0,1), the average position is obtained as X(t)≈2x0Re{ρ(mec) 01(t)}. Using Eqs. ( 16) and ( 28), one can show, after some algebra, that in this quasiequilibrium regime, the time evolution of ρ(q) 01(t)≡ρ(q0)(q1)(t) is given by ˙ρ(q) 01(t)≈/braceleftbigg iω0−/Gamma1/bracketleftbigg \|Q10\|2+\|Q00\|2+\|Q11\|2 2/bracketrightbigg/bracerightbigg ρ(q) 01(t) +/Gamma1Q 00Q11ρ(¯q) 01(t). (38) The ﬁrst term in Eq. ( 38) describes the coherent evolution between the states of the same charge q=0,1 and different number of phonons n=0 and m=1. The second (third) term describes the incoherent evolution between the states ofthe same (different) charge q=0,1(¯q=1,0) and different number of phonons n=0 andm=1, due to electromechanical coupling. We deduce from Eq. ( 38) the evolution of the oscillator reduced density matrix, ˙ρ(mec) 01(t)≈{iω0−γX}ρ(mec) 01(t), (39) γX=/Gamma1/braceleftbigg \|Q10\|2+\|Q00\|2+\|Q11\|2 2−Q00Q11/bracerightbigg ,(40) withγXthe decoherence rate of the mechanical oscillator. Equation ( 39) enables one to write the equation for X(t): ¨X(t)+2γX˙X(t)+/parenleftbig ω2 0+γ2 X/parenrightbig X(t)=0, (41) γX=/Gamma1g2/bracketleftbigg 1+g2 2/bracketrightbigg e−g2. (42) Equation ( 41) coincides with the equation of motion of a classical damped harmonic oscillator. Interestingly, the deco-herence rate γ Xas given by Eq. ( 42) does not coincide with the energy-dissipation rate γE/2 obtained in Eq. ( 31). TheFIG. 6. Schematics of the microscopic processes responsible for the decoherence rate γXof the off-diagonal element of the mechanical- oscillator density matrix ρ(mec) 01(t)=/summationtext q=0,1ρ(q) 01(t). (a),(b) Inelastic processes (red dashed lines) responsible for energy dissipation γE. One mechanical vibron is absorbed while the charge state of the quantum dot is modiﬁed. (c)–(f) Elastic processes (red dashed lines)responsible for dephasing γ φ. No mechanical vibron is emitted or absorbed, while the charge state of the quantum dot is modiﬁed. (e) Transfer of coherences (red dashed lines). In all ﬁgures, the redcircles in the charge sector q=0,1 stand for the matrix element ρ (q) 01(t) in Eq. ( 38). It is coupled either to itself or to the matrix element ρ(¯q) 01(t) of the complementary charge sector ¯q=1,0. decoherence rate can also be written as γX=γE 2+γφ, (43) γφ=g2γE 4=/Gamma1 2g4e−g2. (44) The ﬁrst term γE/2i nE q .( 43) gives the standard contribution of the dissipation to the decoherence of the mechanical oscil-lator. The second term γ φis an additional dephasing rate. This term has some interesting consequences. First of all, the deco-herence rate γ Xof the mechanical oscillator is larger than the contribution induced by pure energy dissipation: γX/greaterorequalslantγE/2. Then, γXas a function of g2reaches a maximum for a value of the electromechanical coupling g2=√ 2 that is larger than the value g2=1 for which dissipation is maximal (see Fig. 8). In other words, the maximal decoherence rate is obtained afterentering in the region of bistability of the mechanical oscillator,while the maximal dissipation rate coincides with the frontierbetween the monostable and bistable region (see Figs. 4and3). B. Microscopic mechanism for decoherence The decoherence rate is obtained by the additive contribu- tion of several elementary microscopic processes in Eq. ( 40). The ﬁrst term ∝/Gamma1\|Q10\|2is the degenerate contribution of the processes pictured in Figs. 6(a) and 6(b). Those processes, responsible for energy dissipation γE, are inelastic processes during which one mechanical vibron is absorbed, while thecharge state of the quantum dot is modiﬁed. The second andthird terms, ∝/Gamma1/2(\|Q 00\|2+\|Q11\|2), are purely elastic pro- cesses for which no mechanical vibron is emitted or absorbed, 155414-7R. A VRILLER, B. MURR, AND F. PISTOLESI PHYSICAL REVIEW B 97, 155414 (2018) while the charge state of the quantum dot is modiﬁed. They are presented in Figs. 6(c) and 6(d), respectively. The last terms ∝−/Gamma1Q 00Q11are elastic processes corresponding to the transfer of coherences between pair of states (00),(01) and(10),(11). They are pictured in Fig. 6(e). It is interesting to notice that the dephasing rate γ φin Eq. ( 42) originates entirely from the elastic processes. Those are higher-order terms in the electromechanical coupling. Notethat the standard description of a quantum damped harmonicoscillator [ 35] does not predict a difference between the decoherence rate and half the dissipation rate. This originateshere from the presence of the additional charge degree offreedom. C. Displacement ﬂuctuations It is possible to describe the ﬂuctuations of the variable X by introducing a stochastic force ξX(t) into Eq. ( 41)f o rt h e average of X, ¨X(t)+2γX˙X(t)+/parenleftbig ω2 0+γ2 X/parenrightbig X(t)=ξX(t). (45) This phenomenological description allows taking into account the ﬂuctuating force due to the thermal and off-equilibriumﬂuctuations. We will assume that the correlation function of the force /angbracketleftξ X(t)ξX(t/prime)/angbracketright=DXδ(t−t/prime) satisﬁes the ﬂuctuation- dissipation theorem [ 45]:DX=4ω2 0x2 0γXcoth(βω 0/2). Equivalently, the diffusion coefﬁcient for the ﬂuctuations ofXdeﬁned as DXcan be expressed in terms of its variance DX=4ω2 0γX/Delta1X2, with /Delta1X2=/angbracketleftX2/angbracketright−X2. After Fourier transforming Eq. ( 45), we obtain, in the limit of weak electronic damping ( γX/lessmuchω0), Ssym XX(ω)≈/summationdisplay s=±1γX/Delta1X2 (ω+sω0)2+γ2 X. (46) The symmetrized Xﬂuctuation spectrum is thus a sum of Lorentzians centered at frequencies ω=±ω0.I t sF W H M /Delta1ωXis given by /Delta1ωX=/Delta1ωE 2+/Delta1ωφ=2/Gamma1g2/bracketleftbigg 1+g2 2/bracketrightbigg e−g2,(47) with the contribution of dephasing /Delta1ωφ=2γφ. The displacement-ﬂuctuation spectrum for the oscillator position x=X−2gx0ndreads Sxx(ω)=SXX(ω)+4g2x2 0Sndnd(ω) (48) −2gx0/braceleftbig SXnd(ω)+SndX(ω)/bracerightbig . (49) It is the sum of three terms: (i) the contribution of ther- momechanical noise SXX(ω), (ii) a contribution of charge noise Sndnd(ω) randomly shifting the mechanical-oscillator equilibrium position, and (iii) a contribution associated to cor-relations between the charge state of the dot and the oscillatorposition, S Xnd(ω)+SndX(ω). The symmetrized charge noise contribution can be evaluated with the same methods as derivedin Sec. IV A . We obtain, for the total symmetrized displacement spectrum, S sym xx(ω)≈/summationdisplay s=±1γX/Delta1X2 (ω+sω0)2+γ2 X+2g2x2 0γE ω2+γ2 E,(50)FIG. 7. Fluctuation spectrum of the oscillator displacement Sxx(ω) as a function of frequency ω. (a) Asymmetric spectrum Sxx(ω) computed within the secular approximation developed in Sec. III B . Various electromechanical coupling strengths are probed: g2=0.1,0.4,1.4,4.4. (b) Corresponding symmetrized displacement spectrum Ssym xx(ω) around the phonon-emission peak at ω≈ω0.I n s e t : Relative error in % between the analytical Lorentzian shapes provided by Eq. ( 46) and the previous numerical curves. Parameters common to both panels: /Gamma1=0.05ω0,˜ε0=0,T=0.1ω0,a n dV=0.2ω0. where we neglected the mixed terms Xndsince the two quantities ﬂuctuate at very different frequency scales: nd at low frequencies ω<γ E/lessmuchω0, andXat\|ω−ω0\|/lessmuchγX. Figure 7(a) shows the displacement spectrum Sxx(ω)o ft h e mechanical oscillator as a function of frequency, computednumerically within the secular approximation. The spectrum ofthis quantum noise is strongly asymmetric. It has a main peakatω≈ω 0associated to phonon emission, which dominates the spectrum at low temperature and voltage (only phononemission is possible at low temperature). A secondary peakis observed at ω≈−ω 0associated to phonon absorption. Its height is very weak since phonon absorption is stronglysuppressed for a mechanical oscillator close to its quantummechanical ground state. Finally, a last peak is observedat low frequencies ω≈0, associated to the contribution of charge noise in Eq. ( 48). The symmetrized noise S sym xx(ω) is computed numerically and presented in Fig. 7(b) close to the phonon-emission peak. In the inset is plotted the relativeerror in % between the analytical Lorentzian shapes obtainedwith Eq. ( 46) and the previous numerical curves. The overall agreement between the analytics and the numerics is below 1%. The dependence of the FWHM /Delta1ω xas a function of electromechanical coupling g2is shown in Fig. 8. Here, also, the agreement between the analytical formula in Eq. ( 47) (solid curve) and the numerics (circles) is very good. This validatesthe scenario of decoherence presented in Sec. VB, which results from the combination of dissipation due to inelasticprocesses and dephasing induced by elastic processes. VI. VOLTAGE DEPENDENCE A. Heating of the mechanical oscillator In Sec. V, we studied the dynamical properties of the mechanical oscillator at low voltages and temperatures(T,V <ω 0). In this section, we will unravel the effect of im- posing a bias-voltage larger than the typical vibron frequency, 155414-8BISTABILITY AND DISPLACEMENT FLUCTUATIONS IN … PHYSICAL REVIEW B 97, 155414 (2018) FIG. 8. FWHM of the displacement-ﬂuctuation spectrum /Delta1ωx as a function of electromechanical coupling g2. Circles: numerical result using the secular approximation developed in Sec. III B . Dashed curve: analytical result for the contribution induced by dissipation, /Delta1ω(d) X=/Delta1ωE/2. Solid curve: analytical result including the addi- tional contribution of dephasing /Delta1ωφ(ﬁlled blue sector) as given by Eq. ( 47). Parameters common to both curves: same as in Fig. 7. V/2>ω 0, keeping the temperature of the electronic environ- ment at low values, T/lessmuchω0. The main physical consequence of increasing the bias voltage is to open an additional inelasticchannel each time the bias voltage crosses a multiple of thevibron frequency, V/2>n ω 0, thus modifying the expression for the transition rates in Eq. ( 28)t o /Gamma1m→n≈/Gamma1/summationdisplay α=±\|Qm,n\|2θ/bracketleftbigg αV 2−(n−m)ω0/bracketrightbigg .(51) This gives rise to new possibilities of exciting vibrons in the rate equation ( 27) and thus to heat up the mechanical oscillator. We show in Fig. 9(a) the stationary out-of-equilibrium phonon distribution πnunder a bias voltage V=4.5ω0.I n contrast to Sec. V, where only the ground state of the me- FIG. 9. (a) Stationary distribution of the vibronic population πn. Histograms obtained for various values of the electromechanicalcoupling: /epsilon1 P[ω0]=1.0,5.0,9.1,21.0. Corresponding average phonon population, ¯n=2.29,1.05,0.99,0.42, and effective temperature, Teff[ω0]=2.76,1.49,1.44,0.82. Circle curves: thermal distributions πth nwith effective temperature Teffhaving the same average phonon number ¯n. (b) Corresponding stationary probability distribution of the oscillator position π(x). Parameters common to both panels: /Gamma1=0.05ω0,˜ε0=0,T=0.1ω0,a n dV=4.5ω0.FIG. 10. (a) Effective temperature Teff[ω0] of the mechanical oscillator as a function of bias voltage V, for various values of the electromechanical coupling: /epsilon1P[ω0]=0.4,1.0,2.0,9.0. (b) Same plot as a function of electromechanical coupling /epsilon1P/2, for various values of the bias voltage: V[ω0]=0.2,2.5,4.5,6.5. Parameters common to both panels: /Gamma1=0.05ω0,˜ε0=0,T=0.1ω0. chanical oscillator was signiﬁcantly populated, the phonon distribution now spreads up to high-energy excited vibronicstates. In the regime we investigate, this spreading is interpretedas a bias-induced heating of the mechanical oscillator. In orderto quantify it more precisely, we compared the phonon distri-bution π n[histograms in Fig. 9(a)] computed numerically to an effective thermal distribution πth n[circle curves in Fig. 9(a)] deﬁned as πth n=(1−e−βeffω0)e−nβeffω0, (52) βeff≡1 Teff=1 ω0ln/parenleftbigg1+¯n ¯n/parenrightbigg . (53) The effective temperature Teffin Eq. ( 53) is chosen in such a way as to reproduce the exact average vibron population ¯n computed from the distribution πn. We ﬁnd that for various electromechanical couplings g2=/epsilon1P/2ω0, the vibron distri- bution πnis not far from the ﬁtted thermal distribution πth nof Eq. ( 52). At low /epsilon1P=ω0, the mechanical oscillator is heated above the temperature of the electronic environment, Teff≈ 2.76ω0/greatermuchT=0.1ω0. Upon increasing the electromechanical coupling to /epsilon1P=21.0ω0, the effective temperature decreases down to Teff≈0.82ω0. The obtained effective temperature depends on both voltage Vand electromechanical coupling /epsilon1P [31,46–48], as shown in Figs. 10(a) and10(b) . We ﬁnd that at voltages much lower than the vibron frequency ( V/2/lessmuchω0), the effective temperature converges to the environment tem-perature T eff≈0.1ω0, independently of the coupling strength, as expected for a mechanical oscillator at thermal equilibrium.Upon increasing the bias voltage with V/2>ω 0, the effective temperature Teffbecomes larger than T[48], consistent with Fig. 9(a). The main tendency is a stepwise increase of Teff each time a vibronic sideband is excited. At sufﬁciently high voltage, the stepwise increase of Teffbecomes, on average, linear in Vwith a slope that increases with decreasing /epsilon1P: the smaller the electromechanical coupling, the higher theeffective temperature [ 29]. 155414-9R. A VRILLER, B. MURR, AND F. PISTOLESI PHYSICAL REVIEW B 97, 155414 (2018) Finally, we plot in Fig. 9(b) the stationary probability distribution of the oscillator position π(x), for the same range of parameters as in Fig. 9(a). We ﬁnd that similarly to the quasiequilibrium case (see Fig. 4),π(x) undergoes a transition from a monostable situation (one peak) at low coupling/epsilon1 P=ω0to a bistable situation (two peaks) at sufﬁciently high-coupling strength, /epsilon1P=21.0ω0. However, in contrast to Fig. 4, the intermediate regime ( /epsilon1P≈9.1ω0) is characterized by a multistable situation for which the distribution π(x) develops two minima rather than a single broad maximum.This difference is due to the fact that in this regime, ¯n≈1, so that not only does the ground state of the mechanical oscillator(n=0) contribute signiﬁcantly to Eq. ( 35), but the ﬁrst excited states ( n=1,2) do also. B. Displacement-ﬂuctuation spectrum In this section, we investigate the role of the bias voltage on the displacement-ﬂuctuation spectrum Sxx(ω). In contrast to Sec. V, it is more difﬁcult to obtain analytical insight on the Sxx(ω) curves. This is due to the heating of vibron excitations, which precludes a simple truncation of the master equation[see Eqs. ( 15) and ( 16)] for the vibron mode. One can consider the limit of vanishing damping and decoherence rates γ E,γX→0. In this limit, we compute the correlation function /angbracketleftδA(t)δA(0)/angbracketrightof any operator Ataking into account only the coherent evolution with respect to thefree Hamiltonian ˜H 0in Eq. ( 9). Similarly to Eq. ( 50), the displacement-ﬂuctuation spectrum Sxx(ω) can be approxi- mated as the sum of a thermomechanical noise SXX(ω)p l u sa contribution due to low-frequency charge noise ﬂuctuations ofthe dot, S XX(ω)=Sabs(ω)+Sem(ω)+2πg2x2 0δ(ω), (54) Sabs(ω)≈2πx2 0¯nδ(ω+ω0), (55) Sem(ω)≈2πx2 0(1+¯n)δ(ω−ω0). (56) The thermomechanical noise spectrum in Eq. ( 54) is composed of an absorption noise Sabs(ω) of height proportional to the (voltage- and coupling-dependent) average phonon population ¯nplus an emission noise Sem(ω) of height proportional to 1 +¯n. The ratio between the emission noise and absorption noise,S em(ω)/Sabs(ω), is proportional to (1 +¯n)/¯n=eβeffω0, and is thus related to the oscillator effective temperature Teff[see Eq. ( 53)]. The symmetrized thermomechanical noise is readily obtained as Ssym XX(ω)≈2πx2 0/parenleftbigg ¯n+1 2/parenrightbigg/summationdisplay s=±δ(ω+sω0). (57) Ssym XX(ω) thus has a height that is proportional to the oscillator average mechanical energy. Interestingly, Eq. ( 57) recovers the limits γE,γX→0i nE q .( 46), obtained for the case of an oscillator in the low-bias and -temperature regimes. We present in Fig. 11(a) the displacement-ﬂuctuation spec- trum computed numerically, using either the full Born-Markovresult (solid curve) as developed in Sec. III A or the secular approximation (dashed curve) developed in Sec. III B .I nFIG. 11. Fluctuation spectrum of the oscillator displacement Sxx(ω) as a function of frequency ω. (a) Asymmetric spectrum Sxx(ω) computed numerically. Solid curve: full Born-Markov result as devel- o p e di nS e c . III A . Dashed curve: secular approximation developed in Sec. III B . (b) Same curves for the symmetrized displacement spectrum Ssym xx(ω) computed around the phonon-emission peak at ω≈ω0. Parameters common to both panels: /Gamma1=0.05ω0,˜ε0=0, T=0.1ω0,/epsilon1P=5.0ω0,a n dV=4.5ω0. contrast to Fig. 7, the spectrum now presents a nonvanishing absorption peak at ω≈−ω0. For voltage V=4.5ω0and electromechanical coupling /epsilon1P=5.0ω0, we ﬁnd the computed ratioSem(ω)/Sabs(ω)≈2.0, which is consistent with having heating of the mechanical oscillator, with an average numberof phonons ¯n≈1.0 and an effective temperature T eff≈1.5ω0 [see Fig. 9(a)]. Moreover, we ﬁnd an overall good agreement between the Born-Markov and secular approximation results. Somedifferences emerge in the tails of the three main peaks ofthe spectrum. A zoom onto the symmetrized spectrum closeto the emission peak at ω≈ω 0is plotted in Fig. 11(b) . It is shown there that the Lamb-shift terms generated byEq. ( 14) are responsible for a weak softening of the mechanical mode frequency that is otherwise neglected within the secularapproximation. Finally, we investigate in Fig. 12the dependence of the FWHM /Delta1ω xfor the displacement-ﬂuctuation spectrum with FIG. 12. FWHM of the displacement-ﬂuctuation spectrum /Delta1ωx as a function of electromechanical coupling /epsilon1P/2. Numerical results using the secular approximation developed in Sec. III B . Various volt- age biases are probed: V[ω0]=0.2,2.0,3.0,4.5. Parameters common to all curves, apart from voltage: same as in Fig. 7. 155414-10BISTABILITY AND DISPLACEMENT FLUCTUATIONS IN … PHYSICAL REVIEW B 97, 155414 (2018) FIG. 13. Locus of the points ( /epsilon1P/2,V)o fm a x i m ai nt h eF W H M /Delta1ωmax E(blue triangles) and /Delta1ωmax x(red circles). Numerical results using the secular approximation developed in Sec. III B . Red dashed curve: critical coupling for the current blockade transition in theclassical regime /epsilon1 P/2=V. Blue dashed curve: critical coupling for the current blockade transition induced by ground-state quantum ﬂuctuations /epsilon1P=2.0ω0. Chosen parameters: /Gamma1=0.05ω0,˜ε0=0, andT=0.1ω0. both bias voltage and electromechanical coupling. Upon in- creasing the bias voltage from V=0.2ω0toV=4.5ω0,w e show that the maximum of the FWHM /Delta1ωmax x is shifted toward higher values of /epsilon1P. We attribute this effect to the entering of additional vibronic sidebands into the bias-voltagewindow, which opens new electric channels for decoherenceand dephasing /Delta1ω max φof the mechanical oscillator. The distribution /Delta1ωxas a function of /epsilon1Palso becomes much broader at higher voltages compared to the low-biascase. This implies more sensitivity of the mechanical oscil-lator to decoherence. Indeed, the unavoidable ﬂuctuations inexperimental /epsilon1 Pvalues due to disorder will induce an enhanced inhomogeneous broadening of the spectral line through the ﬂatdependence of /Delta1ω xwith/epsilon1P. C. Phase diagram We summarize our ﬁndings in a phase diagram represented in Fig. 13. The locus of the points ( /epsilon1P/2,V)o fm a x i m a in the FWHM /Delta1ωmax E is plotted with blue triangles. For 0<V/ 2<ω 0, namely, when the mechanical oscillator is close to its quantum ground state, we ﬁnd that the positionof those maxima is independent of voltage and located atvalues of the electromechanical coupling g 2=/epsilon1P/2ω0=1 (blue dashed curve). This is consistent with the results ofSec. IV A , for which the point of maximum energy dissipation coincides with the transition from a monostable mechanicaloscillator (for g 2<1) to a bistable one (for g2>1). Upon increasing voltage above the ﬁrst vibrational sideband ( ω0< V/2<2ω0), the location of the maxima increases toward a larger voltage-independent value /epsilon1P/2ω0≈1.3. Consistently with Sec. VI A , we assign this increased energy-dissipation rate to the opening of new inelastic electronic channels each time avibron sideband ( n) is excited by the bias voltage ( V/2>n ω 0). Finally, the corresponding curve representing the location of the maxima in the FWHM /Delta1ωmax xis presented with red circles. The obtained red curve is always on the right of the previousblue curve. This is consistent with the analysis performed inSec. VA, for which it is shown that the decoherence rate of the mechanical oscillator is larger than the dissipation rate ofenergy because of additional dephasing induced by elasticallytunneling electrons. At low voltages (0 <V/ 2<ω 0), the red curve is voltage independent and pinned at electromechanicalcoupling g 2=/epsilon1P/2ω0=√ 2>1. This coincides with the value of g2maximizing the decoherence rate. Upon increasing voltage to the range ω0<V/ 2<2ω0, we ﬁnd that the locus of maximum decoherence increases in a steplike manner towardsa larger value of the coupling strength, /epsilon1 P/2ω0≈3.3. This corresponds to the entering of a new vibron sideband n=1, which increases both the dissipation rate (through inelastictransitions) and the dephasing rate (through enhanced elastictransitions). Interestingly, we ﬁnd that upon sufﬁciently increasing sufﬁciently the bias voltage, the location of the maxima in theFWHM /Delta1ω max xgets closer to the red dashed curve V=/epsilon1P/2. We give a simple explanation of this phenomenon based on asemiclassical argument (at high voltage, indeed, many phononspopulate the mechanical oscillator, which becomes semiclassi-cal). The argument closely follows the analysis of the current-blockade phenomena in semiclassical mechanical oscillators [18–21]. We use for this the Hamiltonian written in Eq. ( 2). The tunneling electrons on the dot induce a backaction force on themechanical oscillator, /angbracketleftF/angbracketright=−F 0/angbracketleftnd/angbracketright. This backaction force in turn produces a shift of the oscillator equilibrium position,/Delta1X eq=−F0/k/angbracketleftnd/angbracketright. The work performed by the force /angbracketleftF/angbracketright for displacing the equilibrium position of the oscillator by anamount /Delta1X eqcan be interpreted as a reorganization energy of the dot-level position, /Delta1/epsilon1 0=− /angbracketleftF/angbracketright/Delta1Xeq. At half ﬁlling (/angbracketleftnd/angbracketright=1/2), we obtain /Delta1/epsilon1 0=−/epsilon1P/4. If/Delta1/epsilon1 0is smaller than−V/2, namely, that /epsilon1P/2>V , the dot-level position is effectively shifted away from the conduction window and thecurrent is blocked. The critical value for this transition happensat/epsilon1 P/2=V(red dashed curve) and coincides at high voltage with the transition from a monostable to a bistable state of thesemiclassical oscillator. VII. CONCLUSION It is well known that a nanoelectromechanical oscilla- tor in the regime /Gamma1/lessmuchT/lessmuchω0for large coupling constant g2=/epsilon1P/2ω0enters in the so-called Franck-Condon blockade regime. We have shown that the blockade sets in with abehavior similar to what is observed in the semiclassical case,namely, the appearance of a double maximum in the probabilitydistribution for the position of the oscillator. This propertycan be interpreted as a mechanical bistability present also inthe quantum regime, even if one cannot deﬁne an effectivepotential as in the classical case. At T/lessmuchω 0, the transition point can be identiﬁed for /epsilon1P=2ω0(g2=1) (see Fig. 4), while in presence of bias voltage, /epsilon1P/2≈V(see Fig. 9). This is similar to what is found in the classical case for /Gamma1/greatermuchT/greatermuchω0 for which the transition happens at /epsilon1P=π/Gamma1[14,17], with a smoothing given by thermal or nonequilibrium ﬂuctuations.Despite the similarity, the main difference between the tworegimes is that in the classical case, the transition is controlledby the change of the effective potential, while in the quantumcase, the quantum ﬂuctuations are responsible for the disap-pearance of the bistability. 155414-11R. A VRILLER, B. MURR, AND F. PISTOLESI PHYSICAL REVIEW B 97, 155414 (2018) In analogy with the classical case, we have investigated the displacement- and energy-ﬂuctuation spectra. In the case ofa quantum and fast oscillator, the line shape of the spectraremains Lorentzian. Somewhat surprisingly, we ﬁnd that thewidth /Delta1ω of both is not monotonic and that the spectra are maximal exactly at the bistable transition for /Delta1ω Eand at slightly stronger coupling ( /epsilon1P=2√ 2ω0)f o r /Delta1ωx.W e presented a simple analytical analysis valid at low excitationprobability of the oscillator (low TorV) that allows one to understand the origin of these widths. In the weak-couplinglimit, this is simply the lowest nonvanishing order in theperturbative expansion which shows a quadratic behavior. Inthe strong-coupling limit, the suppression of the tunneling dueto the Franck-Condon terms also suppresses dissipation anddecoherence, which can only be mediated by the electrons.Like in the classical case, the width of the displacementspectrum (decoherence rate) is larger than (half) /Delta1ω E,t h e typical dissipation rate. In the quantum case, the origin is notthe nonlinear effective potential, but the elastic transitions thatintroduce decoherence without dissipation. We also investigatethe same quantities as a function of the bias voltage and ﬁndthat the dissipation and decoherence rates increase abruptly each time a new vibrational sideband enters into the conduction window, namely, when V/2 becomes larger than a multiple of the mechanical frequency ω 0. This gives rise to a phase diagram recovering the semiclassical limit for the current-blockadetransition (occurring when /epsilon1 P/greatermuchV)[21] at sufﬁciently high voltages ( V/greatermuchω0). We found that the Wigner distribution of such an oscillator even close to its quantum ground state or tothe threshold for inelastic transitions does not exhibit negativevalues. This is due to the incoherent nature of the electrontunneling in this regime. In conclusion, we have found that the classical picture applies, at least partially, also in the quantum regime. This sce-nario can be observed for high-frequency ﬂexural mechanicaloscillators [ 6,7] or for breathing modes in suspended carbon nanotubes [ 27]. In the case of ﬂexural modes, the observation of the displacement ﬂuctuation spectrum has been demonstrated,for the moment, only for relatively low-frequency modes [ 15]. The method could also be applied to higher frequencies, evenif reaching the strong-coupling limit becomes more difﬁcult.On the other side, for breathing modes, the strong-couplingregime was reached long ago [ 27]. The detection of quantum current noise at high frequency is now possible in carbon nanotubes [ 49], even if this still has not been performed in the case of suspended carbonnanotubes. From the theoretical point of view, other questionsare still open. It would be interesting to extend the presentwork to regimes of higher tunneling rates /Gamma1/T , taking into account corrections induced by the cotunneling of electrons.Addressing the fate of the bistability transition in the regimeof both coherent tunneling of electrons and quantum me-chanical oscillator is still an open theoretical issue, evenif recently a mapping has been established to an effectiveKondo problem in the limit of a slow oscillator in equilibrium[50]. Finally, it would be of interest for future works to investigate the possibility of generating nonclassical states of the mechanical oscillator by parametric driving [ 51]o rb y a suitable coupling of the nanotube mechanical oscillator tosuperconducting electrodes [ 52–54]. These results and per- spectives contribute to show that nontrivial physical behaviorarises from the strong coupling between tunneling electronsand a well-controlled mechanical degree of freedom of theoscillator. ACKNOWLEDGMENT We thank G. 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PhysRevB.85.155452.pdf	PHYSICAL REVIEW B 85, 155452 (2012) Ab initio derivation of electronic low-energy models for C 60and aromatic compounds Yusuke Nomura,1Kazuma Nakamura,1,2and Ryotaro Arita1,2,3 1Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo JP-113-8656, Japan 2JST CREST, 7-3-1 Hongo, Bunkyo-ku, Tokyo JP-113-8656, Japan 3JST PRESTO, Kawaguchi, Saitama JP-332-0012, Japan (Received 7 December 2011; revised manuscript received 26 February 2012; published 25 April 2012) We present a systematic study for understanding the relation between electronic correlation and superconduc- tivity in C 60and aromatic compounds. We derived, from ﬁrst principles, extended Hubbard models for twelve compounds: fcc K 3C60,R b 3C60,C s 3C60(with three different lattice constants), A15 Cs 3C60(with four different lattice constants), doped solid picene, coronene, and phenanthrene. We show that these compounds are stronglycorrelated and have similar energy scales of their bandwidths and interaction parameters. However, they have adifferent trend in the relation between the strength of the electronic correlation and superconducting-transitiontemperature. While the C 60compounds have a positive correlation, the aromatic compounds exhibit a negative correlation. DOI: 10.1103/PhysRevB.85.155452 PACS number(s): 74 .20.Pq, 74 .70.Kn, 74 .70.Wz I. INTRODUCTION Superconductivity in π-electron systems, the history of which dates back to studies on graphite-intercalationcompounds in the 1960s, 1has attracted broad interest in condensed-matter physics. Recently, two seminal discoveriesfor carbon-based superconductors have been reported. One isA15 and/or fcc Cs 3C60, based on a new method to synthesize highly crystalline samples.2–6The observed superconducting- transition temperatures ( Tc) for the high-pressure samples are found to be as high as 38 K for the A15 and 35 K for thefcc compounds. The other is potassium-doped solid picene, 7 which opened a new avenue to various aromatic superconduc- tors for which the maximum Tchas reached 33 K.8–11 The mechanism of superconductivity of these new su- perconductors has not been fully understood. For alkali-doped fullerides, while there are several experimental reportswhich seemingly support the conventional Bardeen-Cooper-Schrieffer (BCS) mechanism, various indications for uncon-ventional superconductivity have been also observed. Forexample, although the positive correlation between T cand the lattice constant found in K- and Rb-doped fullerides has beenunderstood in terms of the standard BCS theory, 12more recent experiments for larger cations have revealed that Tcdoes not necessarily behave monotonically as a function of the latticeparameters. 2–6The fact that the superconducting phase has a domelike shape in the phase diagram is indeed reminiscentof cuprates 13and unconventional organic superconductors such as bis(ethylenedithio)-tetrathiafulvalene (BEDT-TTF).14 In addition, these new C 60superconductors are insulators at ambient pressure,2–6indicating that the superconducting phase resides in the vicinity of an insulating phase. In fact,considering these characteristic features, it has been proposedthat the interplay between the orbital, spin, and lattice degreesof freedom is the origin of the high- T csuperconductivity.15 For aromatic superconductors, following the discovery of K-doped picene (C 22H14),7it has been found that various hydrocarbon compounds such as coronene (C 24H12), phenan- threne (C 14H10), and 1,2:8,9-dibenzopentacene (C 30H18)a l s o exhibit superconductivity.8–11The diversity of hydrocarbon molecules suggests the possibility of new and higher- Tcaromatic superconductors. So far, the electronic structure,16–20 electronic correlations,21,22electron-phonon interactions,23,24 exiton and plasmon properties of the normal state25–28have been studied while studies on superconducting propertiesare still limited. The pairing mechanism is totally an openquestion. 29 There are several similarities between C 60and aromatic superconductors; they are both molecular solids having nar-row bands around the Fermi level, whose energy scalescompete with those of the electron-phonon and electron- electron interactions. The competition among these three factors is a characteristic aspect of carbon-based materials.Ab initio derivations of effective low-energy models for these compounds are important to make the situation transparentand to clarify the origin of their high- T csuperconductivity. By comparing the parameters in the effective models for the C 60 and aromatic superconductors, the differences and similarities are quantitatively identiﬁed and analyzed. Recent methodology for the construction of the elec- tronic model, based on the combination use of the maxi-mally localized Wannier orbital (MLWO) (Ref. 30) and the constrained-random-phase approximation (cRPA), 31extends its applicability range. It does not place limitations on thecharacter of basis orbitals of the effective model whetheratomic or molecular. Indeed, it has already been applied tovarious complex systems such as BEDT-TTF (Refs. 32and33) and zeolites. 34While electronic-interaction parameters of the C60and aromatic superconductors have been estimated by various methods,26,35–38explicit and direct comparison of these systems by the same method has yet to be done. Thus, itis imperative to evaluate the interaction parameters of C 60 and aromatic superconductors by exploiting the state-of-the-art technique and perform a systematic comparison. In thepresent study, we constructed ab initio extended Hubbard models which describe the low-energy electronic structures oftwelve examples of C 60and aromatic compounds. The transfer integrals were given as matrix elements of the Kohn-ShamHamiltonian in the Wannier basis. The interaction parameterswere evaluated by calculating the Wannier-matrix elementsof the screened Coulomb interaction, which is obtained by 155452-1 1098-0121/2012/85(15)/155452(12) ©2012 American Physical SocietyYUSUKE NOMURA, KAZUMA NAKAMURA, AND RYOTARO ARITA PHYSICAL REVIEW B 85, 155452 (2012) cRPA. The estimated correlation strength as the ratio of the interaction energy to the kinetic one is nearly at or beyondunity for the studied materials, indicating that both the C 60and aromatic systems are classiﬁed as strongly correlated electronsystems. On the other hand, we observed a notable differencebetween the two systems; for the C 60system, there exists a positive-correlation regime in the correlation strength andthe experimental T c. In contrast, the aromatic system exhibits negative correlation between these two quantities. This paper is organized as follows. In Sec. II,w es h o wh o w to construct the low-energy models from ab initio calculations. In Sec. III, we show the calculated band structure, MLWOs, transfer integrals, and effective interaction parameters. Wediscuss the material dependence of the derived parametersand relation between the strength of the electronic correlationand superconductivity in Sec. IV. Finally, we give a summary in Sec. V. II. METHODS We derive electronic low-energy models with the combi- nation of MLWOs and the cRPA. This method has widelybeen applied to the derivation of effective models for 3 d transition metals, 39,40their oxides,39organic conductors,32,33 zeolites,34iron-based superconductors,41,42and 5d-transition- metal oxides.43We ﬁrst perform band calculations based on density functional theory (DFT)44,45and choose “target bands” of the effective model. By constructing MLWOs for the targetbands, we calculate transfer integrals and effective interactionsin the effective model. In the calculation of the effectiveinteraction, the screening by electrons besides target-bandelectrons is considered within the cRPA (see below). We apply this scheme to the derivation of effective models, i.e., extended multiorbital Hubbard models, for the C 60and aromatic compounds. The Hamiltonian consists of the transferpartH t, the Coulomb-repulsion part HU, and the exchange- interactions and pair-hopping part HJdeﬁned as H=Ht+HU+HJ, (1) where Ht=/summationdisplay σ/summationdisplay ij/summationdisplay nmtnm(Rij)aσ† inaσ jm, (2) HU=1 2/summationdisplay σρ/summationdisplay ij/summationdisplay nmUnm(Rij)aσ† inaρ† jmaρ jmaσ in, (3) HJ=1 2/summationdisplay σρ/summationdisplay ij/summationdisplay nmJnm(Rij) ×/parenleftbig aσ† inaρ† jmaρ inaσ jm+aσ† inaρ† inaρ jmaσ jm/parenrightbig , (4) withaσ† in(aσ in) being a creation (annihilation) operator of an electron with spin σin the nth MLWO localized at a C 60- or aromatic-hydrocarbon molecule located at Riand where Rij=Ri−Rj. The parameters tnm(Rij) represent an on-site energy ( Rij=0) and hopping integrals ( Rij/negationslash=0), which are described with the translational symmetry as tnm(R)=/angbracketleftφnR\|HKS\|φm0/angbracketright, (5) where \|φnRi/angbracketright=a† in\|0/angbracketrightand HKS is the Kohn-Sham Hamiltonian.46To evaluate effective interaction parameters Unm(R) and Jnm(R), we calculate the screened Coulomb interaction W(r,r/prime) at the low-frequency limit. We ﬁrst calculate the noninteracting-polarization function χ, excluding polarization processes within the target bands. Note that screening by thetarget electrons is considered when we solve the effectivemodels so that we have to avoid double counting of it whenwe derive the effective models. With the resulting χ,t h eW interaction is calculated as W=(1−vχ) −1v, where vis the bare Coulomb interaction v(r,r/prime)=1 \|r−r/prime\|. Once the screened Coulomb interaction W(r,r/prime) is calcu- lated, the matrix elements of Ware obtained as Unm(R)=/angbracketleftφnRφm0\|W\|φnRφm0/angbracketright =/integraldisplay/integraldisplay drdr/primeφ∗ nR(r)φnR(r)W(r,r/prime)φ∗ m0(r/prime)φm0(r/prime) (6) and Jnm(R)=/angbracketleftφnRφm0\|W\|φm0φnR/angbracketright =/integraldisplay/integraldisplay drdr/primeφ∗ nR(r)φm0(r)W(r,r/prime)φ∗ m0(r/prime)φnR(r/prime). (7) For comparison with the cRPA results, we calculate interaction parameters with different levels of screening. One is theunscreened one, i.e., the bare Coulomb interaction, and theother is the fully screened one wherein we calculate χ, including the target-band screening. To distinguish these fromthe cRPA, we denote them as “bare” and “fRPA” (fullyscreened random-phase approximation). III. RESULTS A. Calculation conditions We performed DFT band calculations with the TOKYO AB INITIO PROGRAM PACKAGE ,47based on the pseudopotential- plus-plane-wave framework. We used the generalized-gradient approximation (GGA) exchange-correlation func-tional with the parametrization of Perdew-Burke-Ernzerhof 48 and Troullier-Martins norm-conserving pseudopotentials49in the Kleinman-Bylander representation.50The pseudopoten- tials for alkali metals, K, Rb, and Cs, were supplementedwith partial core correction. 51The cutoff energies for wave functions and charge densities were set to 36 Ry and 144 Ry,respectively, and we employed 5 ×5×5k-point sampling. We conﬁrmed that this condition ensures well convergedresults. The DFT calculations were performed for the following twelve materials: fcc K 3C60,f c cR b 3C60,f c cC s 3C60with three different lattice parameters, A15 Cs 3C60with four different lattice parameters, doped solid picene, coronene,and phenanthrene. The lattice parameters were taken fromthe experiments, and internal coordinates were optimized. 53 In fcc A3C60, the disorder of the orientation of C 60molecules was ignored, so the crystal symmetry is lowered from Fm¯3m toFm¯3. Before presenting the computational results, we summa- rize the basic properties of the compounds studied in thepresent paper. Table Ilists experimental values for the C 60 155452-2Ab INITIO DERIV ATION OF ELECTRONIC LOW- ... PHYSICAL REVIEW B 85, 155452 (2012) TABLE I. Basic properties of fcc and A15 alkali-doped C 60 compounds: the lattice parameter a, corresponding C 603−volume in the solid, and measured superconducting-transition temperature Tcor the N ´eel temperature TN.F o rf c cC s 3C60, the three samples are speciﬁed with the C3− 60volume and correspond to those in the superconducting phases with maximum Tc(Vopt-P SC ), in the vicinity of the metal-insulator transition ( VMIT), and in the antiferromagnetic- insulating phase ( VAFI), respectively. For the A15 structure, we also list another sample with a higher pressure for which Tcis lower than that of Vopt-P SC and is abbreviated to Vhigh-P SC . V olume /C3− 60Pressure Tc(TN) Compound a(˚A) ( ˚A3) (kbar) (K) Ref. fcc-K 14.240 722 0 19 52 fcc-Rb 14.420 750 0 29 52 fcc-Cs( Vopt-P SC ) 14.500 762 7 35 4 fcc-Cs( VMIT) 14.640 784 2 26 4 fcc-Cs( VAFI) 14.762 804 0 (2.2) 4 A15-Cs( Vhigh-P SC ) 11.450 751 15 35 3 A15-Cs( Vopt-P SC ) 11.570 774 7 38 3 A15-Cs( VMIT) 11.650 791 3 32 3 A15-Cs( VAFI) 11.783 818 0 (46) 3 compounds, including the lattice constant a,t h ev o l u m e per C 603−in the solid,57the applied pressure, and the measured superconducting-transition temperature Tcor the N´eel temperature TN.T h eavalue and/or C 603−volume can be controlled by the chemical and external pressures. In thistable, the samples are arranged in the order of increasing latticeconstants. For fcc A 3C60,Tcﬁrst increases and reaches its maximum (35 K) around a=14.500 ˚A. Then, it decreases toTc∼25 K where the system experiences the metal- insulator transition (MIT) and becomes an antiferromagneticinsulator (AFI) for which the N ´eel temperature T Nis around 2.2 K. A similar behavior is observed in the A15 systemwhile T Nis signiﬁcantly higher (46 K). This is because the A15 structure is bipartite and therefore less frustrated.58 Hereafter, we label the nine C 60compounds as fcc-K, fcc-Rb, fcc-Cs( Vopt-P SC ), fcc-Cs( VMIT), fcc-Cs( VAFI), A15-Cs( Vhigh-P SC ), A15-Cs( Vopt-P SC ), A15-Cs( VMIT), and A15-Cs( VAFI). Table IIshows the experimental lattice parameters for undoped solid picene, coronene, and phenanthrene and theT cvalues observed for the doped systems. For doped solid picene, two different Tcvalues (18 K and 7 K) have been observed depending on the preparation conditions.7The superconductivity appears when the system is doped, but thedetails of the crystal structures in the superconducting phaseshave not been determined. Thus in the present study, the bandcalculations for aromatic compounds were performed for the TABLE II. Lattice parameters for pristine solid picene, coronene, and phenanthrene and the superconducting-transition temperaturesT cobserved for their doped systems. Compound a(˚A) b(˚A)c(˚A) β(◦)Tc(K) Refs. picene 8.480 6.154 13.515 90.46 18,7 7,54 coronene 16.094 4.690 10.049 110.79 15 10,55 phenanthrene 8.453 6.175 9.477 98.28 5–6 8,9artiﬁcially charged system for which three negative charges per one hydrocarbon molecule were doped with a uniform,compensating, positive background charge. For geometries,we employed the experimental ones listed in Table II. Hereafter, doped solid picene, coronene, and phenanthrene arereferred to as solid picene 3−, coronene3−, and phenanthrene3−, respectively. B. Band structure and density of states Figure 1shows our calculated GGA band structures for the fcc A3C60(upper ﬁve panels), A15 Cs 3C60(middle four panels), and aromatic compounds (lower three panels). Thesecompounds have common features in their band structures;i.e., we see narrow bands near the Fermi level separatedfrom other bands, being preferable when we choose the targetbands to construct an effective model. In the C 60compounds, there are threefold-degenerate states, which form the so-called“t 1uband” near the Fermi level, and we construct effective models for these bands. For aromatic compounds, the targetbands are made from the two lowest unoccupied molecularorbitals (LUMO and LUMO +1) in an isolated molecule. 16,19,20 It should be noted that unoccupied bands lie above the target bands more densely in the order of solid picene3−, solid coronene3−, and solid phenanthrene3−. Since conduction bands can generate stronger screening when they reside closerto the target bands, we expect a weak repulsive interaction insolid picene 3−compared to the other two. We show in Fig. 2the calculated density of states (DOS) of the t1uband for fcc A3C60[Fig. 2(a)] and A15 Cs 3C60 [Fig. 2(b)]. For both fcc and A15 compounds, the bandwidth Wmonotonically increases as the lattice constant decreases, but the DOS proﬁle does not change drastically. We list thevalues of Win Table III. The bandwidth of the A15 compound (∼0.6 eV) tends to be larger than that of the fcc compound (∼0.4 eV), which is due to the difference in the inter-C 60 contact, i.e., the “hexagon-to-hexagon” conﬁguration for the A15 and “bond-to-bond” one for the fcc compound.59It was found that the bandwidths of the aromatic compounds arenearly 0.5 eV (see Table III). C. Maximally localized Wannier orbitals Figure 3shows a contour plot of one of the MLWOs for the t1ubands of A15-Cs( VAFI). The results of other C 60compounds are almost the same. From this ﬁgure we see that the resultingWannier orbital is well localized at the single C 60molecule. In this plot, we displayed the same orbital along the threedirections; Figs. 3(a),3(b), and 3(c) correspond to the views along the x,y, and zaxes, respectively. We see a node in the center of this orbital for Figs. 3(b) and 3(c); thus, this orbital has p x-like symmetry. Note that the view along the y axis is not identical to the view along the zaxis, which is in contrast with the case of atomic porbitals. We note that the other two py- andpz-like Wannier orbitals are symmetrically equivalent to the presented px-like orbital. We also note that the weight of the Wannier orbitals concentrates in the vicinityof the cage of the C 60molecule and that there is little weight inside it. We next show in Fig. 4a contour plot of two MLWOs of solid phenanthrene3−. In the aromatic compounds, the two 155452-3YUSUKE NOMURA, KAZUMA NAKAMURA, AND RYOTARO ARITA PHYSICAL REVIEW B 85, 155452 (2012) -4-3-2-1 0 1 2 3 4 -4-3-2-1 0 1 2 3 4 -4-3-2-1 0 1 2 3 4fcc-K fcc-Rb fcc-Cs( VSC ) fcc-Cs( VMIT) fcc-Cs( VAFI) Γ X K Γ L KWX solid picene3-solid coronene3-solid phenanthrene3-X R M X Γ R M Γ Energy (eV) Energy (eV) Energy (eV)opt-P A15-Cs( VSC )high- P A15-Cs( VSC )opt-PA15-Cs( VMIT) A15-Cs( VAFI) X R M X Γ R M Γ X R M X Γ R M Γ X R M X Γ R M Γ Γ X K Γ L KWX Γ X K Γ L KWX Γ X K Γ L KWX Γ X K Γ L KWX Γ X A Y Γ Z D E C Z Γ X A Y Γ Z D E C Z Γ X A Y Γ Z D EC Z FIG. 1. (Color online) Calculated ab initio electronic band structure of fcc-K, fcc-Rb, fcc-Cs( Vopt-P SC ), fcc-Cs( VMIT), fcc-Cs( VAFI), A15- Cs(Vopt-P SC ), A15-Cs( Vhigh-P SC ), A15-Cs( VMIT), A15-Cs( VAFI), solid picene3−, solid coronene3−, and solid phenanthrene3−. In the case of aromatic compounds with monoclinic structures, the horizontal axis is labeled by the special points in the Brillouin zone with /Gamma1,X ,A ,Y ,Z ,D ,E , and C, respectively, corresponding to (0, 0, 0), (1 /2, 0, 0), (1 /2, 1/2, 0), (0, 1 /2, 0), (0, 0, 1 /2), (1/2, 0, 1 /2), (1/2, 1/2, 1/2), and (0, 1 /2, 1/2), respectively, in units of ( a∗,b∗,c∗). The interpolated-band dispersion with the derived tight-binding Hamiltonian is depicted as blue dashed lines. basis orbitals of the effective model are not symmetrically equivalent, and therefore, we specify these orbitals as “lower”and “higher” orbitals in terms of the on-site level of theMLWOs. The lower and higher orbitals are shown in Figs. 4(a)and 4(b), respectively. We again see the resulting orbitals are well localized at the single molecules. The MLWOs ofsolid picene 3−and coronene3−are similar to those of undoped systems calculated in Refs. 16and20. TABLE III. Calculated bandwidth Wof the target band and spatial Wannier spread /Omega1for twelve materials: fcc-K, fcc-Rb, fcc-Cs( Vopt-P SC ), fcc-Cs( VMIT), fcc-Cs( VAFI), A15-Cs( Vhigh-P SC ), A15-Cs( Vopt-P SC ), A15-Cs( VMIT), A15-Cs( VAFI), solid picene3−, solid coronene3−, and solid phenanthrene3−. For the aromatic compounds, the two values of /Omega1are listed; the left is the “lower-level” orbital, and the right is the “higher-level” one. Units are given in meV for Wand ˚Af o r/Omega1. fccA3C60 A15 Cs 3C60 Aromatic compounds K Rb Cs( Vopt-P SC ) Cs( VMIT) Cs( VAFI)Vhigh-P SC Vopt-P SC VMIT VAFI picene3−coronene3−phenanthrene3− W 502 454 427 379 341 740 659 614 535 477 447 505 /Omega1 4.28 4.21 4.19 4.14 4.10 4.27 4.20 4.16 4.12 4.08, 4.13 3.64, 3.67 3.20, 3.08 155452-4Ab INITIO DERIV ATION OF ELECTRONIC LOW- ... PHYSICAL REVIEW B 85, 155452 (2012) 0 5 10 15 20 25 30 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 5 10 15 20 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Ener gy (eV)DOS/C60 /spin (states/eV)Energy (eV)DOS/C60 /spin (states/eV)(a) fcc (b) A15RbCs(VSC )opt-PCs(VMIT)Cs(VAFI) K Cs(VAFI) Cs(VSC )high- PCs(VSC )opt-PCs(VMIT)large a small large a small FIG. 2. (Color online) (a) Our calculated density of states (DOS) for the t1uband of fcc-K (red), fcc-Rb (green), fcc-Cs( Vopt-P SC ) (blue), fcc-Cs( VMIT) (purple), and fcc-Cs( VAFI) (light blue). (b) DOS for the t1uband of A15-Cs( Vhigh-P SC ) (red), A15-Cs( Vopt-P SC ) (green), A15-Cs( VMIT) (blue), and A15-Cs( VAFI) (purple). We list in Table IIIour calculated spatial spread /Omega1nof the MLWOs for the 12 materials wherein /Omega1nis deﬁned as /Omega1n=/radicalbig /angbracketleftφn0\|r2\|φn0/angbracketright−\| /angbracketleftφn0\|r\|φn0/angbracketright\|2. (8) In the C 60compounds, the calculated Wannier spread is roughly 4 ˚A, and thus, the estimated effective volume4 3π/Omega13is ∼268 ˚A3. The value is compared with the C 603−volume listed in Table I(∼720–820 ˚A3), clearly indicating the well-localized nature of the MLWO on the single molecule. We see thatthe Wannier spread has a weak positive correlation with thebandwidth W. In the aromatic compounds, the molecular sizes themselves are different from each other, resulting inthe appreciable difference in /Omega1values. Note that /Omega1has no clear correlation with W. D. Transfer integrals Let us move on to transfer integrals. For the C 60compounds, the band dispersion of the target band was found to be wellreproduced only with nearest-neighbor (NN) and next-nearest-neighbor (NNN) transfers. The orbital indices 1, 2, and 3denote p x-,py-, and pz-like orbitals, respectively. On-site energies for the three MLWOs are set to zero. From nowon, “site” means one molecule, and the coordinate of siteRis deﬁned as the center of the molecule. The transfer integrals t nm(R) are represented as 3 ×3 matrices. In the fcc (A15) structure, there are 12 (8) NN sites and 6 (6) NNNsites per molecule. From transfers to the speciﬁc site, othertransfers to the equivalent sites are reproduced by proper (a) (b) (c)View along the x axis View along the z axisView along the y axis FIG. 3. (Color online) Isosurfaces of our calculated px-like maximally localized Wannier orbital of A15-Cs( VAFI) viewed along the (a) xaxis, (b) yaxis, (c) zaxis, drawn by VESTA (Ref. 60). The red surfaces indicate positive isosurfaces, and the blue surfaces indicate negative isosurfaces. symmetry operations. As a representative NN site, we choose R=(Rx,Ry,Rz)=(0.5,0.5,0.0) and (0 .5,0.5,0.5) for the fcc and A15 structures, respectively, where the coordinate is basedon the conventional cell. The transfer matrices to these sitesare ⎛ ⎝F 1F20 F2F30 00 F4⎞ ⎠ and⎛ ⎝A1A2(3)A3(2) A3(2) A1A2(3) A2(3)A3(2) A1⎞ ⎠ (9) for the fcc and A15 structures, respectively. In the A15 structure, the two C 60molecules in the unit cell (denoted as the A- andB-sites) are not equivalent in terms of their orientations. So in the matrix in Eq. (9), we show both the transfers from theA-site to the B-site and from the B-site to the A-site (in parentheses). We choose R=(1,0,0) for a representative NNN site for both structures, and then the transfer matricesare written as ⎛ ⎝F 500 0F60 00 F7⎞ ⎠ and⎛ ⎝A4 00 0A5(6) 0 00 A6(5)⎞ ⎠ (10) 155452-5YUSUKE NOMURA, KAZUMA NAKAMURA, AND RYOTARO ARITA PHYSICAL REVIEW B 85, 155452 (2012) (b)(a) FIG. 4. (Color online) Isosurfaces of maximally localized Wan- nier orbitals of solid phenanthrene3−with (a) lower on-site energy and (b) higher on-site energy, drawn by VESTA (Ref. 60). The red surfaces indicate positive isosurfaces, and the blue surfaces indicate negative isosurfaces. for the fcc and A15 structures, respectively. The transfer matrix for the A15 structure represents the A-Atransfer, and the B-Bone is in parentheses. Table IVshows the value of the parameters from F1toF7for the fcc C 60compounds. We note thatF6/negationslash=F7is due to the lowering of the symmetry from Fm¯3mtoFm¯3. Table Vshows the values of parameters from A1toA6for A15 Cs 3C60. For both systems, the values of the hopping parameters decrease as those of the lattice parametersincrease. We next describe the procedure for obtaining transfers to the other NN sites or NNN sites. First, we consider the NNcase. For the fcc structure, the transfer matrices to the otherﬁve NN sites are written with F 1–F4as follows: ⎛ ⎝F400 0F1F2 0F2F3⎞ ⎠ for R=(0.0,0.5,0.5), ⎛ ⎝F4 00 0F1−F2 0−F2F3⎞ ⎠ for R=(0.0,0.5,−0.5), TABLE IV . Hopping parameters for fcc A3C60in Eqs. (9)and (10). Units are given in 10−4eV . Compound F1 F2 F3 F4 F5F6F7 fcc-K −40−339 421 −187 −94−14−2 fcc-Rb −16−306 392 −159 −75−81 5 fcc-Cs( Vopt-P SC )2 6 −299 372 −120 −60−33 6 fcc-Cs( VMIT)1 5 −267 332 −104 −40 1 30 fcc-Cs( VAFI)1 3 −241 302 −94 −33 1 24TABLE V . Hopping parameters for A15 Cs 3C60in Eqs. (9)and (10). Units are given in 10−4eV . Compound A1 A2A3A4 A5 A6 A15-Cs( Vhigh-P SC )−297 448 67 74 −105 −289 A15-Cs( Vopt-P SC )−262 400 61 74 −97 −239 A15-Cs( VMIT) −239 371 57 76 −89 −212 A15-Cs( VAFI) −206 329 53 73 −79 −180 ⎛ ⎝F30F2 0F40 F20F1⎞ ⎠ for R=(0.5,0.0,0.5), ⎛ ⎝F3 0−F2 0F4 0 −F20F1⎞ ⎠ for R=(−0.5,0.0,0.5), ⎛ ⎝F1−F20 −F2F3 0 00 F4⎞ ⎠ for R=(0.5,−0.5,0.0). For the A15 structure, we have ⎛ ⎝A1 A2(3) −A3(2) A3(2) A1 −A2(3) −A2(3)−A3(2) A1⎞ ⎠ for R=(0.5,0.5,−0.5), ⎛ ⎝A1 −A2(3) A3(2) −A3(2) A1 −A2(3) A2(3) −A3(2) A1⎞ ⎠ for R=(0.5,−0.5,0.5), ⎛ ⎝A1 −A2(3)−A3(2) −A3(2) A1 A2(3) −A2(3) A3(2) A1⎞ ⎠ for R=(0.5,−0.5,−0.5). The remaining transfers to the NN sites are reproduced by the relation tnm(R)=tnm(−R). Similarly, the transfers to the other NNN sites are described as ⎛ ⎝F7(A6(5))0 0 0 F5(A4)0 00 F6(A5(6))⎞ ⎠ for R=(0,1,0), ⎛ ⎝F6(A5(6))0 0 0 F7(A6(5))0 00 F5(A4)⎞ ⎠ for R=(0,0,1), for the fcc (A15) structures, and the remaining NNN transfers are generated according to tnm(R)=tnm(−R). Using these NN- and NNN-transfer parameters, we con- struct the transfer part Htin Eq. (2)of the effective model. The band dispersion for the C 60compounds calculated from the resulting Htis depicted as blue dashed lines in Fig. 1 from which we see that the original GGA band dispersion issatisfactorily reproduced. For the aromatic compounds, since there is no simple symmetry operation, their transfers are difﬁcult to showconcisely. 16,20For the aromatic compounds, we describe only some characteristic features of the transfers. The aromaticcompounds are regarded as the stacking-layered systems, sowe expect a two-dimensional hopping structure. However, inthe present transfer analysis, we found that the anisotropy 155452-6Ab INITIO DERIV ATION OF ELECTRONIC LOW- ... PHYSICAL REVIEW B 85, 155452 (2012) TABLE VI. Comparison between the maximum absolute values of the intralayer transfer t/bardbland interlayer transfer t⊥for aromatic compounds. For the three directions in t/bardbl,s e eF i g . 5. Units are given in meV . Compound tmax /bardbl1tmax /bardbl2tmax /bardbl3tmax ⊥ solid picene3−48 39 59 20 solid coronene3−78 771 4 solid phenanthrene3−49 32 73 36 of the transfers is not so simple. Table VIcompares the maximum absolute values of the intralayer transfers ( t/bardbl) with those of the interlayer transfers ( t⊥) for which the intralayer is deﬁned as the abplane. The intralayer transfers are further decomposed in the three directions and compared with eachother (see Fig. 5for the deﬁnition of the three directions). The anisotropy ( t max ⊥/tmax /bardbl) is not so appreciable for solid picene3−, estimated as 20 /59∼0.34, and for phenanthrene3−as∼0.49. In contrast, the anisotropy of coronene3−is signiﬁcant and is ∼0.16. In the case of coronene3−, the intralayer anisotropy is even strong: tmax /bardbl1/tmax /bardbl2=7/87∼0.08; this system is an almost quasi-one-dimensional chain along the baxis. We note that the original GGA band dispersion is well reproduced byshort-range-transfer hoppings (Fig. 1). E. Effective interaction parameters We performed random-phase-approximation (RPA) calcu- lations to evaluate the screened Coulomb interaction W(r,r/prime)i n Eqs. (6)and(7)in which the dielectric function was expanded in plane waves with an energy cutoff 7.5 Ry for fcc A3C60and aromatic compounds and 5.0 Ry for A15 Cs 3C60. The total number of bands considered in the polarization function wasset to 335 (120 occupied, 3 target, and 212 unoccupied) for fccA 3C60, 670 (240 occupied, 6 target, and 424 unoccupied) for A15 Cs 3C60, 310 (102 occupied, 4 target, and 204 unoccupied) for solid picene3−, 315 (108 occupied, 4 target, and 203 unoccupied) for solid coronene3−, and 270 (66 occupied, 4 target, and 200 unoccupied) for solid phenanthrene3−.T h e Brillouin-zone integral on the wave vector was evaluated bythe generalized-tetrahedron method. 61A problem due to the singularity of the long-wavelength-limit Coulomb interactionin the evaluation of the Wannier-matrix elements, U nm(R)i n Eq. (6)andJnm(R)i nE q . (7), was treated in the manner described in Ref. 62. t1t t ab2 3 FIG. 5. (Color online) Schematic picture of the intralayer transfer t/bardblalong the three directions in the aromatic compounds. The ellipses indicate molecules.The on-site interactions are speciﬁed by U=Unn(0), U/prime=Unm(0), and J=Jnm(0)f o rn/negationslash=m. In the case of C 60 compounds, U,U/prime, andJtake only one value, according to the symmetry. For aromatic compounds, Uis different for two orbitals, so we present two values. We also denote the Coulombrepulsion between the neighboring sites as V. Table VIIshows our calculated interaction parameters U, U /prime,J, andVwith three screening levels (bare, cRPA, and fRPA). We see that the value of the Coulomb repulsiondecreases as the screening processes increase. In the C 60 compounds, the bare value is ∼3.4 eV , and after considering the screening by cRPA, the value is reduced to ∼1e V .B y taking account of the intra-target-band screening by fRPA, thevalue is further reduced to ∼0.1 eV . It should be noted here that the material dependences of the bare values in the fccand A15 structures are small, for example, 3.27 eV for fcc-Kand 3.37 for fcc-Cs( V AFI). The difference is nearly 3%. This difference is ascribed to the difference in the spatial spread oft h eM L W O s( s e eT a b l e III). On the other hand, the material difference in the cRPA values is beyond 20%: 0.82 eV for fcc-Kand 1.07 for fcc-Cs( V AFI). Indeed, this appreciable difference originates from the difference in the macroscopic dielectricconstant deﬁned as /epsilon1 cRPA M=lim Q→0lim ω→01 /epsilon1cRPA GG−1(q,ω), (11) withωbeing the frequency and Q=q+G, where qis a wave vector in the ﬁrst Brillouin zone and Gis a reciprocal lattice vector. We list the values at the bottom of the table. We see thatthe material dependence of /epsilon1 cRPA M is appreciable as 5.6 for fcc-K and 4.4 for fcc-Cs( VAFI), clearly indicating the importance of the screening effect in addition to the spatial Wannier spread. For the aromatic compounds, differences in both the bare interaction and the screening effect contribute to thematerial dependence of the cRPA values. The bare inter- actions are Upicene3− bare ∼Ucoronene3− bare <Uphenanthrene3− bare , and after consideration of the cRPA screening, we obtain Upicene3− cRPA < Ucoronene3− cRPA ∼Uphenanthrene3− cRPA . Especially in picene3−, the dielec- tric constant is markedly high ( ∼12),63making the cRPA- U value appreciably small. We ﬁnally remark on some points. As for the C 60com- pounds, the equality U/prime∼U−2Jholds among effective parameters. This relationship also roughly holds for thearomatic compounds. The present Uvalues of cRPA for the C 60compounds are small compared to the previous estimates ofU(∼1–1.5 eV).35–38For all materials, the cRPA Coulomb interaction decays as 1 /(/epsilon1cRPA Mr) with rbeing the distance between the centers of the MLWOs while the fRPA Coulombinteraction is limited to be short ranged due to the metallicscreening (see Table VII). We note that the fRPA Ugives an opposite trend to the bare and cRPA- Uvalues; for example, in the fcc C 60compounds, the fRPA value slightly decreases from fcc-K to fcc-Cs( VAFI). This is due to the fact that the Coulomb interaction is efﬁciently screened due to theincrease in the density of states accompanied by the decreaseof bandwidth. We also found that, in these systems, theexchange interactions Jare also efﬁciently screened; i.e., J cRPA/Jbare∼0.3. This makes a clear contrast to the case of the inorganic materials as JcRPA/Jbare∼0.8 such as the 155452-7YUSUKE NOMURA, KAZUMA NAKAMURA, AND RYOTARO ARITA PHYSICAL REVIEW B 85, 155452 (2012)TABLE VII. U,U/prime,J,a n dVwith three different screening levels [unscreened (bare), constrained RPA (cRPA), and fully screened RPA (fRPA)] for the twelve compounds: fcc-K, f cc-Rb, fcc-Cs( Vopt-P SC ), fcc-Cs( VMIT), fcc-Cs( VAFI), A15-Cs( Vhigh-P SC ), A15-Cs( Vopt-P SC ), A15-Cs( VMIT), A15-Cs( VAFI), solid picene3−, solid coronene3−, and solid phenanthrene3−. For the aromatic compounds, two values of Uare presented; the left is the lower-level orbital, and the right is the higher-level one. For bare and cRPA U,U/prime,a n dVvalues, the units are given in eV , and Jis given in meV . For fRPA, the units are given in meV . At the bottom of the table, we present our calculated cRPA-macroscopic-dielectric constant /epsilon1cRPA M in Eq. (11). fccA3C60 A15 Cs 3C60 Aromatic compounds Constant K Rb Cs( Vopt-P SC ) Cs( VMIT) Cs( VAFI) Vhigh-P SC Vopt-P SC VMIT VAFI picene3−coronene3−phenanthrene3− Ubare 3.27 3.31 3.32 3.35 3.37 3.36 3.39 3.40 3.42 4.43,4.41 4.64,4.59 5.05,5.17 U/prime bare 3.08 3.11 3.12 3.15 3.17 3.16 3.18 3.20 3.22 3.55 4.33 4.55 Jbare 96 99 100 101 102 97 99 100 101 166 129 275 Vbare 1.31–1.37 1.30–1.35 1.29–1.34 1.28–1.33 1.27–1.32 1.37–1.38 1.36–1.37 1.35–1.36 1.34–1.34 2.08–2.32 2.79–2.84 2.29–2.43 UcRPA 0.82 0.92 0.94 1.02 1.07 0.93 1.02 1.07 1.14 0.73,0.74 1.29,1.26 1.33,1.37 U/prime cRPA 0.76 0.85 0.87 0.94 1.00 0.87 0.95 0.99 1.06 0.58 1.15 1.17 JcRPA 31 34 35 35 36 30 36 36 37 53 58 101 VcRPA 0.24–0.25 0.26–0.27 0.27–0.28 0.28–0.29 0.30 0.30 0.31 0.32 0.34 0.26 0.59–0.60 0.47–0.48 UfRPA 93 91 91 86 83 107 102 99 93 155,151 149,120 166,172 U/prime fRPA 41 39 39 35 32 50 45 42 37 51 53 60 JfRPA 25 26 26 26 25 28 28 28 28 38 39 57 VfRPA 1–3 1–3 1–3 1–3 1–3 2–3 2 2 1–2 1–4 1–4 2 /epsilon1cRPA M 5.6 5.1 4.9 4.6 4.4 4.7 4.4 4.3 4.1 12.0 5.5 6.3 155452-8Ab INITIO DERIV ATION OF ELECTRONIC LOW- ... PHYSICAL REVIEW B 85, 155452 (2012) 3dtransition metals,39its oxides SrVO 3,39and iron-based superconductors.41,42As for the alkali-metal intercalation, since it might cause a substantial change in the band structuresof the aromatic compounds, 16,17,20a careful reexamination for the quantitative values of the effective interactions will bedesired after the determination of the experimental structures,which remain to be explored. IV . DISCUSSION A. Material dependence of effective parameters Let us move on to a comparison of the effective interaction parameters among the 12 compounds. Figure 6summarizes the results of the cRPA calculations: the on-site Coulombrepulsion ¯Uaveraged over the MLWOs derived from the target band, the on-site exchange interaction J, the off-site interaction ¯Vaveraged over the nearest-neighbor sites, and 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.02 0.04 0.06 0.08 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 K Rb Cs (VMIT) Cs (VAFI) picene3- coronene3- phenanthrene3- 0 0.5 1 1.5 2 2.5(a) (b) (d) (e)U (eV) J (eV) V (eV) W (eV) (U-V )/W Cs (VSC ) Cs{ (c) K K K K Rb Rb Rb Rb fcc A3C60 A15 Cs3C60 aromatic compoundspicene3-picene3-picene3-picene3- coronene3- coronene3-coronene3-coronene3- phenanthrene3-phenanthrene3-phenanthrene3- phenanthrene3-opt-PVSC opt-P Cs (VMIT) Cs (VMIT) Cs (VAFI) Cs (VAFI)VMIT VAFI VSC high- P VMITVMITVMIT VMITVMIT VAFI VAFIVAFI VAFIVAFICs{Cs (VSC )opt-PCs (VSC )opt-P VAFIVMITVSC opt-P VSC opt-PVSC opt-PVSC opt-PVSC opt-PVSC opt-PVSC high- PVSC high- PVSC high- PVSC high- P FIG. 6. (Color online) Material dependence of (a) the average of the on-site effective Coulomb repulsion ¯U, (b) the on-site effective exchange interaction J, (c) the average of the off-site effective Coulomb repulsion between neighboring sites ¯V, (d) the bandwidth of the target band W, and (e) the correlation strength ( ¯U−¯V)/W, which are derived within the cRPA method.the ratio ( ¯U−¯V)/Wwhich measures the correlation strength in the system. Note that the net interaction is estimatedas¯U−¯V, based on the analysis of the extended Hubbard model. 64 As for the C 60systems, we see that ¯Uhas appreciable material dependence, ranging from 0.8 eV to 1.1 eV [Fig. 6(a)]. This is ascribed to the differences in the size of the Wannierorbitals and dielectric screening (see Sec. III E ). On the other hand, the material dependence of Jis weak, and the value itself is negligibly small as ∼0.03 eV [Fig. 6(b)]. In general, small Jfavors low-spin states as is observed in experiments. 3–5,65 It is interesting to note that there is a proposal that the Jahn- Teller coupling dominates over the Hund’s rule coupling Jand induces superconductivity with the help of sufﬁciently largeU. 15Compared to J, we found that ¯Vis substantially large as ∼0.3 eV , being as large as ∼25% of ¯U[Fig. 6(c)]. As for W, which measures kinetic energy, we observed a decreasing trendas the lattice constant increases [Fig. 6(d)]. We also note that the energy scale for A15 Cs 3C60is larger than that of fcc A3C60 as mentioned in Sec. III B . The derived correlation strength of (¯U−¯V)/W exhibits a rather simple monotonic increasing behavior [Fig. 6(e)] with the lattice constant increase. The presented ( ¯U−¯V)/W∼1 indicates that C 60superconductors are categorized as strongly correlated electron systems. For the aromatic superconductors, we found that the energy scale of ¯Uis similar to that of the C 60superconductors [Fig. 6(a)]. On the other hand, it is interesting to note that the aromatic superconductors tend to have larger Jand ¯V values [Figs. 6(b) and6(c)]. We can also see that the material dependence of the interaction parameters among the aromaticsuperconductors is also more signiﬁcant since the sizes andshapes of the aromatic molecules are quite different fromeach other. As for W, they are similar for the aromatic and C 60superconductors [Fig. 6(d)]. We found that aromatic superconductors are also in as strongly correlated regimes asthe C 60ones, based on the analysis of the correlation strength [Fig. 6(e)]. B. Relation between electronic correlation and superconductivity Next, let us discuss the relation between electronic correla- tion and superconductivity for the C 60and aromatic supercon- ductors. In Figs. 7(a) and7(b), we plot the superconducting- transition temperature Tcand the N ´eel temperature TNas a function of the volume occupied per fulleride anion ( V0)f o r fccA3C60and A15 Cs 3C60, respectively (see also Table I). To see the relation between the electron correlation and thesuperconductivity, we superimpose a plot of ( ¯U−¯V)/W on the phase diagram. We see that while ( ¯U−¯V)/WandT chave a positive correlation up to V0∼760–770 ˚A3, for larger V0, electron correlation becomes fatal for superconductivity, andthe system eventually becomes an insulator. We note that thecritical value of ( ¯U−¯V)/W for the MIT sample is larger for fccA 3C60(∼1.9) than A15 Cs 3C60(∼1.2). As discussed in Ref. 59, it is important to consider the inﬂuence of the lattice and orbital structure on the MIT.15,66 In Fig. 7(c),w ep l o t Tcand ( ¯U−¯V)/W for the three aromatic superconductors, which show a negative correlation.Therefore, it seems that electronic correlation does not 155452-9YUSUKE NOMURA, KAZUMA NAKAMURA, AND RYOTARO ARITA PHYSICAL REVIEW B 85, 155452 (2012) 0 10 20 30 40 50 720 740 760 780 800 740 760 780 800 820 0 0.5 1 1.5 2 2.5Temperature (K) C603- volume (Å3) C603- volume (Å3)(UV)/W(a) fcc A3C60 (b) A15 Cs3C60 SC SCAFI AFI+SC AFITcTcTN TNKRbCs 0 5 10 15 20 0 0.5 1 1.5 2 2.5 picene3-coronene3-phenanthrene3-Tc (K)(c) aromatic compounds(UV)/W{ FIG. 7. (Color online) Relations between the experimental curves of the superconducting- and magnetic-transition temperatures ( Tc,TN) as functions of the C3− 60volume and the estimated correlation strength ( ¯U−¯V)/W (vertical bar) for (a) fcc A3C60and (b) A15 Cs 3C60. (c) For aromatic compounds, the measured Tcin Table IIand the calculated correlation strength are compared where picene3−=(C22H14)3−, coronene3−=(C24H12)3−, and phenanthrene3−=(C14H10)3−. For the panels (a) and (b), the experimental phase diagrams were taken from Ref. 4for the fcc and Ref. 3for the A15 compounds. favor superconductivity in these aromatic superconductors. Recently, doped 1,2:8,9-dibenzopentacene was found to havea quite high T c∼33 K. Since 1,2:8,9-dibenzopentacene is a bigger molecule than picene, coronene, or phenanthrene,the former interaction is expected to be small comparedto the latter ones, reﬂecting the large Wannier spread ofthe 1,2:8,9-dibenzopentacene molecule. If there is no drasticchange in the bandwidth W, which is probable in terms of the tendency shown in Fig. 6(d), the weakest electronic correlation will be realized in doped 1,2:8,9-dibenzopentacene. This trendis consistent with Fig. 7(c). Regarding the role of the electronic correlation in the C 60 and aromatic superconductors, there are two possibilities.Either, the pairing mechanism in these compounds has acommon root, or these superconductors have completelydifferent pairing glues. If we assume that the aromaticsuperconductors reside in the vicinity of the border betweenthe superconducting and insulating phases, the ﬁrst scenariois (at least partially) explicable in relation to the behaviorin Fig. 7. On the other hand, in the second scenario, the electronic correlation enhances superconductivity of the C 60 compounds and inversely suppresses that of the aromaticcompounds. In order to clarify this issue, experimental studiesto determine the phase diagram for aromatic superconduc-tors against temperature and volume occupied per anionare highly desired. 67Theoretically, microscopic calculations considering both electronic correlation and electron-latticecoupling are needed, which will be an interesting futureproblem. V . SUMMARY To provide insight into the role of electronic corre- lation in C 60and aromatic superconductors, we derived effective models for a wide range of examples: fcc-K, fcc-Rb, fcc-Cs( Vopt-P SC ), fcc-Cs( VMIT), fcc-Cs( VAFI), A15- Cs(Vhigh-P SC ), A15-Cs( Vopt-P SC ), A15-Cs( VAFI), solid picene3−, solid coronene3−, and solid phenanthrene3−. To deﬁne the basis orbital of the effective model, we constructed ML-WOs of isolated bands around the Fermi level. Transferparameters were derived by evaluating the matrix elementsof the Kohn-Sham Hamiltonian between the MLWOs. Thelow-energy electronic structures of the C 60compounds arehighly symmetric and isotropic, so the original GGA band is reproduced with only six or seven parameters. On theother hand, the aromatic compounds have quite anisotropicelectronic structures. To quantify the strength of electronic correlation in these compounds, we estimated the effective interaction parameters,such as U,V, and J, by means of the cRPA method. It was found that in addition to the appreciable reduction ofthe diagonal part of the Coulomb interaction ( UandV), the off-diagonal part Jis also efﬁciently screened. Interestingly, all the C 60and aromatic superconductors studied in the present work have a similar energy scale for the bandwidth and inter-action parameters: W∼0.5e V ,U∼1e V ,J∼0.05 eV , and V∼0.3 eV . This parameter range suggests that these com- pounds are a strongly correlated electron system. However,after examination of the material dependence, we found aclear difference between the C 60and aromatic compounds in the relation between electronic-correlation strength and Tc; i.e., a positive correlation in the C 60system and a negative correlation in the aromatic system. In the present study, we focused on the derivation for the electronic part of the effective model. For a thoroughunderstanding of the low-energy physics for carbon-basedmaterials, however, the derivation of the electron-phononinteraction part is also imperative. The derivation for this partincludes subtle problems concerning the deﬁnition of the basisfor the phonon mode (see Refs. 68and69) and/or the exclusion of the double counting of the screening of the low-energydegree of freedoms, which requires future study. ACKNOWLEDGMENTS We thank Taichi Kosugi for providing us with the optimized structure data of undoped coronene and also for stimulatingdiscussions. This work was supported by Grants-in-Aid forScientiﬁc Research (Grants No. 22740215, No. 22104010,No. 23110708, No. 23340095, and No. 19051016). 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PhysRevB.72.184518.pdf	Doping dependence of coupling between charge carriers and bosonic modes in the normal state of high- Tcsuperconductors H. Saadaoui and M. Azzouz * Laurentian University, Department of Physics and Astronomy, Ramsey Lake Road, Sudbury, Ontario P3E 2C6, Canada /H20849Received 19 September 2005; published 28 November 2005 /H20850 Recently, the doping dependence of the optical-conductivity scattering rate has been used by Hwang, Ti- musk, and Gu to gain some insight in the way the coupling between the charge carriers and the bosonic modesin high- T csuperconductors depends on doping. These authors used the extended Drude analysis, which does not take into account the normal-state pseudogap explicitly. In this work, we calculated the optical conductivitywithin the rotating antiferromagnetism theory, which models explicitly the pseudogap. Then we analyzed theresistivity as a function of temperature Tand doping p. We extracted the scattering rate 1/ /H9270by ﬁtting the La2−xSrxCuO 4resistivity data. We found that for psmaller than a critical value pc,1 //H9270shows a marginal- Fermi-liquid dependence for Tgreater than a p-dependent temperature T/H9267. But for T/H11021T/H9267,1 //H9270deviates down- ward from this law. We attribute this depression to a stronger coupling between the charge carriers and thenormal-state bosonic modes. Both this coupling and T /H9267vanish at pcwhile superconductivity continues to be signiﬁcant well above it. pcis interpreted as a quantum critical point, and T/H9267as the pseudogap temperature T* because it is found to agree with the experimental data on T. We propose that a possible candidate for the bosonic modes may be the spin-wave excitations in the rotating frame of the rotating antiferromagnetic order. DOI: 10.1103/PhysRevB.72.184518 PACS number /H20849s/H20850: 74.25.Fy I. INTRODUCTION In conventional superconductors, the coupling between electrons, which form the charge carriers, and phonons,which are bosonic modes, is responsible forsuperconductivity. 1,2In high- Tcsuperconductors /H20849HTSC’s /H20850, however, the mechanism leading to superconductivity is notyet understood, and most complications arise from thenormal-state pseudogap /H20849PG/H20850. 3The latter is a partial gap that appears in the electron density of states /H20849DOS /H20850below a doping-dependent temperature T, which is greater than the superconducting transition temperature Tc. Using the rotating antiferromagnetism theory /H20849RAFT /H20850,4,5we propose a scenario for the doping dependence of the strength /H9261cbof the coupling of the charge carriers to the normal-state bosonic modes inHTSC’s. This information is obtained by analyzing the resis-tivity data, the quantum critical point /H20849QCP /H20850, 6,7the PG, and the resonance peak in the neutron-scattering experiments.8 Practically, the procedure used here resembles that of theextended Drude analysis /H20849EDA /H20850: 9–11we ﬁt the resistivity data by adjusting the relaxation rate. Several advantages of thepresent approach over the EDA can be highlighted here.First, the PG appears explicitly in the underlying theoryRAFT, but this is not the case in EDA. We think that thestarting theory, upon which the calculation of the transportproperties of HTSC’s is based, is crucial as there is mountingevidence in favor of different symmetries of the PG statebelow T and the state above T. Second, contrary to EDA, the PG vanishes at a QCP in RAFT in agreement with manyexperiments. 7Third, in RAFT the hypothesis of a T-dependent hole density on the copper-oxygen planes is in agreement with the linear Tdependence of the Hall carrier density.12This hypothesis was not made in other approaches like those based on the dynamical-mean-ﬁeld theory/H20849DMFT /H20850, 13on numerical diagonalizations,14or in the densityd-wave /H20849DDW /H20850theory.15Contrary to EDA, however, in the present approach we cannot adjust both the real and imagi-nary parts of self-energy when we ﬁt the resistivity data /H20849see below /H20850. Our main result is the establishment of a clear link between the PG and /H9261 cb’s doping dependence. In addition, a handful of other important results like the calculation of theoptical conductivity in RAFT, the ﬁtting of the resistivitydata of the compound La 2−xSrxCuO 4, and the calculation of the spectral function are also produced. In Sec. II, we review RAFT, and analyze the energy spec- tra along the high-symmetry lines of the Brillouin zone /H20849BZ/H20850. We also discuss the agreement of these spectra with existingangle-resolved-photoemission spectroscopy /H20849ARPES /H20850 data. 16,17In Sec. III, we calculate the Green’s function in the framework of RAFT. Then, in Sec. IV, we calculate the op-tical conductivity. We apply our results to the analysis of theexperimental resistivity data of the material La 2−xSrxCuO 4in Sec. V. In addition to ﬁtting the experimental resistivity databy adjusting the relaxation rate, the doping and temperaturedependence of the relaxation rate is examined in Sec. V aswell. Our results are interpreted in Sec. VI. Conclusions aredrawn in Sec. VII. II. MODEL A. Review of RAFT RAFT is a mean-ﬁeld theory that is based on the concept of the rotating order parameter.4,18In the case of a two- dimensional system at ﬁnite temperature, this concept allowsfor the deﬁnition of a magnetically disordered state that ischaracterized by a local order parameter that has a randomorientation but a nonzero magnitude. In this way rotationalsymmetry is not broken in accordance with the Mermin-Wagner theorem. 19For the HTSC’s, RAFT is implementedPHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 1098-0121/2005/72 /H2084918/H20850/184518 /H2084911/H20850/$23.00 ©2005 The American Physical Society 184518-1on the two-dimensional extended Hubbard model with an on-site Coulomb repulsion energy U, and a phenomenologi- cal positive attraction energy Vfor the electrons residing on adjacent sites. The term Vsimulates the d-wave symmetry of the superconducting order parameter. The Hamiltonian isgiven by H=H 0+U/H20858 rnr↑nr↓−V/H20858 /H20855r,r/H11032/H20856NNnr↑nr/H11032↓, /H208491/H20850 where nr↑/H20849nr↓/H20850is the occupation number operator of an elec- tron at the site position rwith spin up /H20849down /H20850, and H0is the single-particle energy term, which accounts for the kineticenergy and the chemical potential. Here r/H11013/H20849i,j/H20850designates the space coordinates of the lattice sites; iandjare integers. In terms of the creation operator c i,/H9268†and destruction operator ci,/H9268,H0reads as H0=−t/H20858 /H20855r,r/H11032/H20856NN,/H9268cr/H9268†cr/H11032/H9268−t/H11032/H20858 /H20855r,r/H11032/H20856NNN,/H9268cr/H9268†cr/H11032/H9268−/H9262/H20858 r,/H9268cr/H9268†cr/H9268. /H208492/H20850 The kinetic energy takes into account the hopping of the electrons between the nearest-neighbor /H20849NN/H20850and next- nearest-neighbor /H20849NNN /H20850sites with energy amplitudes tand t/H11032, respectively. /H9262is the chemical potential. RAFT has been introduced and explained in detail in Refs. 4,5,12. Here, we brieﬂy summarize it. This theory ischaracterized by two competing orders; the d-wave super- conductivity, which is parametrized by D 0=/H20855ci,j,↑ci+1,j,↓/H20856 =−/H20855ci,j,↑ci,j+1,↓/H20856, and the rotating antiferromagnetic /H20849RAF /H20850or- der modeled by Q, which stands for the magnitude of a ro- tating staggered magnetization. In the present paper, we fo-cus on the nonsuperconducting phases, /H20849the so-called normal state /H20850with D 0=0. The normal-state mean-ﬁeld Hamiltonian takes on the form H=/H20858 k/H9023k†H/H9023k+UNQ2−UNn2, /H208493/H20850 where Nis the total number of the lattice sites, and n =/H20855ni,/H9268/H20856is the average of the occupation operator. Q =/H20841/H20855cr,↑cr,↓†/H20856/H20841is the modulus of the rotating order parameter /H20855cr,↑cr,↓†/H20856/H11013Qei/H9278r, which is shown to give rise to a magneti- zation with a ﬁnite magnitude Qbut a random orientation because the phase /H9278ris allowed to assume any value in /H208510, 2/H9266/H20852. In the RAF state, the phase satisﬁes /H9278r/H11032−/H9278r=/H9266with r and r/H11032labeling nearest-neighbor sites. Mathematically, the RAF order parameter is to some extent similar to the super-conducting order parameter with a ﬁxed magnitude but aﬂuctuating phase /H20849that is responsible for destroying the phase coherence /H20850in the preformed-pairs scenario of HTSC’s. 20 Note that RAFT is rather based on the competing-order sce- nario. Because of the AF correlations the lattice is subdi-vided into two sublattices AandB. The Nambu spinor in Eq. /H208493/H20850is /H9023 k†=/H20849ck↑A†ck↑B†ck↓A†ck↓B†/H20850. /H208494/H20850 The Hamiltonian density Hi sa4/H110034 matrix given byH=/H20898−/H9262/H11032/H20849k/H20850/H9280/H20849k/H20850 QU 0 /H9280/H20849k/H20850−/H9262/H11032/H20849k/H20850 0 −QU QU 0− /H9262/H11032/H20849k/H20850/H9280/H20849k/H20850 0 −QU /H9280/H20849k/H20850−/H9262/H11032/H20849k/H20850/H20899/H208495/H20850 and can be written as H=−/H9262/H11032/H20849k/H20850I+/H9280/H20849k/H20850M+UQN, /H208496/H20850 where M=/H20873/H927010 0/H92701/H20874,N=/H208730/H92703 /H927030/H20874. /H208497/H20850 /H92701and/H92703are Pauli matrices. The energy spectra, found by diagonalizing H, are E±/H20849k/H20850=−/H9262/H11032/H20849k/H20850±Eq/H20849k/H20850, /H208498/H20850 where12 /H9262/H11032/H20849k/H20850=/H9262+4t/H11032coskxcosky−Un, /H208499/H20850 and Eq/H20849k/H20850=/H20881/H92802/H20849k/H20850+/H20849UQ/H208502/H2084910/H20850 is an energy contribution with a gap QU. Here /H9280/H20849k/H20850 =−2t/H20849coskx+cos ky/H20850is the kinetic energy of the electrons that hop to NN lattice sites only. Note that because of the k dependence of /H9262/H11032/H20849k/H20850/H20851Eq. /H208499/H20850/H20852, the energy gap in the spectra E±/H20849k/H20850is not QUbut a gap with a subtle kdependence as we will see later on in Sec. II B 3. This gap is interpreted as the PG of the HTSC’s. To summarize, the PG is kdependent because of the kdependence of /H9262/H11032/H20849k/H20850. Moreover, this k dependence is proportional to the energy t/H11032, which is the energy amplitude of the hopping of the electrons to NNNlattice sites. Thus, according to RAFT, the deviation of thesymmetry of the PG from the s-wave symmetry varies from a HTSC to another depending on the value of t /H11032. This means that the PG may seem to have an s-wave symmetry if t/H11032/tis very small. When D0=0, the parameters Qandnsatisfy the following self-consistent equations:4,5,12 n=1 2N/H20858 knF/H20851E+/H20849k/H20850/H20852+nF/H20851E−/H20849k/H20850/H20852, Q=U 2N/H20858 knF/H20851E+/H20849k/H20850/H20852−nF/H20851E−/H20849k/H20850/H20852 Eq/H20849k/H20850, /H2084911/H20850 where nF/H20849x/H20850=/H208491+e/H9252x/H20850−1is the Fermi-Dirac distribution fac- tor. Here, /H9252=1/kBT, with Tbeing the temperature, and kBthe Boltzmann constant. B. Analysis of the pseudogap within RAFT We analyze in this section the doping dependence of the energy spectrum in order to understand the kdependence of the PG. In the calculations of the rest of Sec. II, we onlyconsider set II of the Hamiltonian parameters /H20849Table I /H20850; the results for set I are very similar to those of set II.H. SAADAOUI AND M. AZZOUZ PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-2To pave the way for the analysis of the energy spectra, thus of the PG, we ﬁrst study the doping and temperaturedependence of the parameter Q. The doping density in RAFT isp=1−2 n, since nis the number of electrons per site and spin. At half ﬁlling, for example, n=1/2 leads to p=0. The hole-doped /H20849electron-doped /H20850systems correspond to p/H110220 /H20849p/H110210/H20850. Even though we are interested only in the normal state with D 0=0 in this work, we will in the following sec- tions II B 1 and II B 2 include also the analysis of the dopingand temperature dependence of the superconducting param-eterD 0in order to highlight the competition between super- conductivity and rotating antiferromagnetism. 1. Doping dependence of Q In Fig. 1 the doping dependence of the parameters Qand D0is reported at the temperature T=0.033 t. For doping in the vicinity of half ﬁlling, D0=0 and Qis large. As doping increases Qdecreases, and D0becomes nonzero for p /H110220.09. D0increases until it reaches a maximum at pQCP /H11015p0/H110150.195 where Qvanishes, then decreases for pgreater than p0and vanishes at p/H110150.38. We checked that this be- havior is valid for several Hamiltonian parameters even atzero temperature. The disappearance of the RAF order nearthe optimal point p 0has been interpreted as a quantum phase transition, and the point pQCPas a QCP.4,5 2. Temperature dependence of Q Next, we address the Tdependence of Q. Figs. 2 /H20849a/H20850and 2/H20849b/H20850show Qand D0vsTfor three values of the doping density. In the neighborhood of half ﬁlling where D0=0,Q behaves as for an ordinary second-order phase transition; i.e.,Qmonotonously decreases as Tincreases, then vanishes at acritical temperature. This is clearly the case for p=0.06 in Fig. 2 /H20849a/H20850. For p=0.14, however, superconductivity enters into competition with RAF. Qstarts to decrease as soon as D 0becomes nonzero below the superconducting transition temperature Tc. In RAFT, Tcis identiﬁed as the temperature where D0vanishes. interestingly, when the competition be- comes stronger, a reentrance behavior sets in for Q; the latter becomes ﬁnite on an interval of temperature with a nonzerolower bound, which is slightly smaller than T cforp=0.20. To highlight the competition between superconductivity andRAF, Fig. 2 /H20849b/H20850shows QandD 0vsTfor the doping densities p=0.14 and p=0.20. 3. Doping dependence of the energy spectra E ±„k… Figures 3 and 4 show the energy spectra E±/H20849k/H20850/H20851Eq. /H208498/H20850/H20852 vskalong the high-symmetry lines for three values of the doping density, and for the temperature T=0.1 t, which is higher than the optimal Tc. In the remainder of this paper, we only consider the temperatures that are greater than the su-perconducting temperature T c. For p=0.04 deep in the un- derdoped regime, a gap clearly exists at /H20849/H9266,0/H20850, in the neigh- borhood of which the spectrum is almost ﬂat. Note that theenergy spectrum E −/H20849k/H20850displays a shallow minimum at /H20849/H9266,0/H20850 forp=0.04. A less pronounced minimum can also be seen for doping p=0.13. For the latter, an energy gap smaller than that realized for p=0.04 exists at /H20849/H9266,0/H20850. In addition, the lower band shifts upward in comparison to the band corre-sponding to p=0.04. In general, the gap at /H20849 /H9266,0/H20850decreases as doping increases, then vanishes for doping densities equal toor greater than p/H110150.2 for set II of the Hamiltonian param-TABLE I. The two sets of the Hamiltonian parameters used in this paper to carry on the numerical calculations are summarized inthis table. The unit of energy is t. Ut /H11032 V Set I 3 t −0.25 tt Set II 2.8 t −0.16 t 0.85t FIG. 1. The doping dependence of QandD0is shown for T =0.033 t. FIG. 2. /H20849a/H20850The temperature dependence of Qis shown for three values of doping. /H20849b/H20850Both QandD0are displayed vs Tfor two values of doping. Set II of the Hamiltonian parameters is used.DOPING DEPENDENCE OF COUPLING BETWEEN … PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-3eters; the spectra for p=0.22 in Fig. 4 clearly show no gap at /H20849/H9266,0/H20850. We turn now to the discussion of the spectra in the vicinity of the point /H20849/H9266/2,/H9266/2/H20850. Figures 3 and 4 show that the spectrum is gapless near this point independently of dop- ing. Indeed, a glance to the right panels of Figs. 3 and 4demonstrates that the lower band E −/H20849k/H20850crosses the zero- energy axis near the point /H20849/H9266/2,/H9266/2/H20850even for the smaller doping density p=0.04. In summary, a gap is found in our spectra at /H20849/H9266,0/H20850, but no gap is observed near /H20849/H9266/2,/H9266/2/H20850.A gap with such features clearly bears some resemblance with a normal-state gap with d-wave symmetry. However, con- trary to a gap with purely d-wave symmetry, the PG here is not identically zero along the diagonal of the BZ, as seen inthe right panel of Fig. 3. Our spectra overall compare well with the experimental ones of Ino et al. , 21Marshall et al.22and King et al.23The ﬂatness of the lower-energy band and the occurrence of theminimum at /H20849 /H9266,0/H20850were observed experimentally by Mar- shall et al. Experimentally, however, unlike what we found Inoet al. reported spectra that are not symmetric at /H20849/H9266,0/H20850. Note that some of Marshall et al. ’s spectra are symmetric at /H20849/H9266,0/H20850. So, maybe more experimental work is needed in order to settle this issue. As shown for p=0.22 in Fig. 4, the bands E+/H20849k/H20850and E−/H20849k/H20850touch when the gap vanishes at the point /H20849/H9266,0/H20850in the overdoped regime. This leads to a continuous transition from the lower band E−/H20849k/H20850to the top one E+/H20849k/H20850. This result is inagreement with the experimental data, which show the emer- gence of a higher-energy band when doping approaches theoptimal point. 22Also, we believe that the sudden appearance of the higher-energy band in experiment well away from halfﬁlling is an indication that it could not be observed deep inthe underdoped regime because the gap by which it is sepa-rated from the lower-energy band is too large. Another relevant feature that needs to be compared with experiment is the bandwidth. Rough estimates of the band-width of the lower-energy band in experiment are between0.2 and 0.4 eV. 21,22For set II, RAFT yields a bandwidth of about 2 tfor the lower-energy band. For La 2−xSrxCuO 4for example, choosing t/H110150.1eV gives a bandwidth of 0.2 eV in good agreement with experiment. Motivated by this good agreement between RAFT and experiment as far as the energy spectra are concerned, andafter having analyzed the mean-ﬁeld parameters and the PG,we will next calculate the optical conductivity. In order to dothis, we ﬁrst need to calculate the Green’s and spectral func-tions. III. RAFT GREEN’S AND SPECTRAL FUNCTIONS The approach we followed in order to calculate the Green’s function in this section, and the optical conductivityin the next section, is similar to the one developed by Kimand Carbotte 24and by Valenzuela, Nicol, and Carbotte25in the case of the DDW theory. The DDW and RAFT ap-proaches are, however, fundamentally different as discussedin Ref. 4, because RAFT is based on rotating spin AF, but theDDW is based on orbital AF. To evaluate the Green’s function G/H20849k,i /H9275n/H20850we use Dys- on’s equation G−1/H20849k,i/H9275n/H20850=i/H9275nI−/H9018/H6018/H20849k,i/H9275n/H20850−H, /H2084912/H20850 where i/H9275n=i/H208492n+1/H20850/H9266//H9252is the Fermionic Matsubara fre- quency, and /H9018/H6018/H20849k,i/H9275n/H20850is the self-energy matrix which will be taken to be diagonal; /H9018/H6018/H20849k,i/H9275n/H20850=/H9018/H20849k,i/H9275n/H20850I, with Ibeing the 4/H110034 identity matrix. Using the Hamiltonian density given by Eq. /H208495/H20850,w eﬁ n d G/H20849k,i/H9275n/H20850=/H20851i/H9275˜n+/H9262/H11032/H20849k/H20850/H20852I+/H9280/H20849k/H20850M+/H20849UQ/H20850N /H20851i/H9275˜n+/H9262/H11032/H20849k/H20850/H208522−/H20851/H92802/H20849k/H20850+/H20849UQ/H208502/H20852,/H2084913/H20850 where i/H9275˜n=i/H9275nI−/H9018/H20849k,i/H9275n/H20850. The imaginary part of the re- tarded Green’s function, −Im G/H20849k,i/H9275n→/H9275+0+/H20850, yields the spectral function A/H20849k,/H9275/H20850=/H20858 s=±Ls/H20849k,/H9275/H20850as/H20849k,/H9275/H20850, /H2084914/H20850 where as/H20849k,/H9275/H20850=I+s/H20873/H9280/H20849k/H20850 EqM+UQ EqN/H20874, /H2084915/H20850 and the Lorentzian weight Lsgiven by FIG. 3. The energy spectra E±/H20849k/H20850are displayed along the sym- metry lines. These spectra are calculated for T=0.1 tand two doping densities p=0.04 and 0.13 in the underdoped regime. FIG. 4. The energy spectra E±/H20849k/H20850are displayed along the sym- metry lines for doping p=0.22 and temperature T=0.1 t.H. SAADAOUI AND M. AZZOUZ PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-4Ls/H20849k,/H9275/H20850=−/H9018/H11033/H20849k,/H9275/H20850 /H20851/H9275+/H9262/H11032/H20849k/H20850−/H9018/H11032/H20849k,/H9275/H20850−sEq/H208522+/H20851/H9018/H11033/H20849k,/H9275/H20850/H208522. /H2084916/H20850 The real and imaginary parts of the self-energy, /H9018/H11032/H20849k,/H9275/H20850and /H9018/H11033/H20849k,/H9275/H20850, respectively, are related by the Kramers-Kronig re- lation. Unfortunately, we do not yet know even an approximate expression of the self-energy. Here, we consider the marginalFermi liquid /H20849MFL /H20850ansatz, which was proposed by Varma and co-workers, 26namely /H9018/H20849/H9275/H20850=2/H9261/H9275lnx /H9275c−i/H9261/H9266x, /H2084917/H20850 where x=max /H20849/H20841/H9275/H20841,T/H20850,/H9261is a parameter that models the cou- pling of the charge carriers to the collective modes in the system /H20849perhaps the spin waves, phonons, etc. /H20850, and/H9275cis a cutoff frequency of the order of 0.5 eV for the HTSC’s. If, inaddition to the MFL contribution, an impuritylike scatteringcontribution /H9257is taken into account, then the relaxation rate 1//H9270is given by 1 2/H9270/H20849T/H20850=−/H9018/H11033=/H9257+/H9266/H9261T /H2084918/H20850 when /H9275=0. We make a generalization of the MFL ansatz by allowing /H9257and/H9261to be temperature and doping dependent. It is within this generalization that we proceed for the calcula-tion of the resistivity, and the ﬁtting of the experimental data.We will show that the original ansatz with temperature anddoping independent /H9257and/H9261is inadequate for ﬁtting the ex- perimental data of the resistivity in the /H20849low- /H20850temperature regime with T/H33355T. IV . CALCULATION OF THE OPTICAL CONDUCTIVITY In the long-wavelength limit where the momentum trans- ferq→0, the real part of the optical conductivity as a func- tion of frequency /H9275can be calculated using Kubo’s formula, namely /H9268/H9263/H9257/H20849/H9275/H20850=−1 /H9275Im/H9016ret/H9263/H9257/H20849/H9275/H20850. /H2084919/H20850 Here, /H9263,/H9257=x,y, and /H9016ret/H9263/H9257/H20849/H9275/H20850=/H9016/H9263/H9257/H20849i/H9275n→/H9275+i0+/H20850is the re- tarded current-current correlation function. In the Matsubara formalism, the correlation function is given by /H9016/H9263/H9257/H20849/H9270/H20850=−1 v/H20855j/H9263†/H20849/H9270/H20850j/H9257/H208490/H20850/H20856, /H2084920/H20850 where vis the volume of the system, /H9270is the Matsubara imaginary time, and j/H9263is the /H9263th component of the current density operator. The Fourier transform of /H9016/H9263/H9257/H20849/H9270/H20850is given by /H9016/H9263/H9257/H20849i/H9275m/H20850=/H20885 0/H9252 d/H9270ei/H9270/H9275m/H9016/H9263/H9257/H20849/H9270/H20850. /H2084921/H20850 In order to calculate /H9016/H9263/H9257we need to calculate the current density operator j/H9263.A. Current density operator Because of the bipartite character of the lattice, the elec- tric current density results from the motion of the electronsbetween NN and NNN sites, as illustrated in Fig. 5. To cal-culate the NN current operator, we consider the square latticeshown in Fig. 5 where the AF sublattices AandBare shown. In real space, the NN current density operator is 27 jNN=it/H20858 i,j,/H9268/H20851xˆ/H20849c2i+1,2 j,/H9268B†c2i,2j,/H9268A−c2i,2j,/H9268A†c2i+1,2 j,/H9268B/H20850 −xˆ/H20849c2i−1,2 j,/H9268B†c2i,2j,/H9268A−c2i,2j,/H9268A†c2i−1,2 j,/H9268B/H20850+yˆ/H20849c2i,2j+1,/H9268B†c2i,2j,/H9268A −c2i,2j,/H9268A†c2i,2j+1,/H9268B/H20850−yˆ/H20849c2i,2j−1,/H9268B†c2i,2j,/H9268A−c2i,2j,/H9268A†c2i,2j−1,/H9268B/H20850/H20852, /H2084922/H20850 where xˆandyˆare the unit vectors along the xaxis and the y axis, respectively. Fourier transforming by using ci,j,/H9268/H9263=1 /H20881N/H20858 keik·ri,jck,/H9268/H9263, /H2084923/H20850 where /H9263=A,Bandri,jis the position of the electron at site /H20849i,j/H20850, yields jNN=/H20858 k,/H9268v1/H20849k/H20850/H20849ck,/H9268A†ck/H9268B+ck/H9268B†ck/H9268A/H20850. /H2084924/H20850 Here v1/H20849k/H20850=/H11509/H9280/H20849k/H20850//H11509k=2t/H20849sinkxxˆ+sin kyyˆ/H20850is the group ve- locity that is associated with the single-electron energy /H9280/H20849k/H20850=−2t/H20849coskx+cos ky/H20850.jNNcan be written in terms of the spinor /H9023kdeﬁned in Eq. /H208494/H20850, and the matrix Mdeﬁned in Eq. /H208497/H20850as jNN=/H20858 kv1/H20849k/H20850/H9023k†M/H9023k. /H2084925/H20850 We emphasize that jNNresults from the hopping of electrons between adjacent sites, which belong to different sublattices. Next, the NNN contribution to the current density opera- tor results from the hopping of the electrons between sitesthat belong to the same sublattice /H20849AorB/H20850. Following the same procedure as for the NN contribution, one ﬁnds FIG. 5. The motion of the electrons between sites AandBis illustrated. The electrons hop to the NN and NNN lattice sites withenergies tandt /H11032, respectively. The lattice sites are labeled using even and odd indices because of the bipartite character of the lat-tice. Solid squares /H20849oval /H20850indicate the sites belonging to the sublat- ticeA/H20849B/H20850.DOPING DEPENDENCE OF COUPLING BETWEEN … PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-5jNNN=/H20858 k,/H9268v2/H20849k/H20850/H20851ck,/H9268A†ck,/H9268A+ck,/H9268B†ck,/H9268B/H20852=/H20858 kv2/H20849k/H20850/H9023k†/H9023k, /H2084926/H20850 where v2/H20849k/H20850=−/H11509/H9262/H11032/H20849k/H20850//H11509k=4t/H11032/H20849sinkxcoskyxˆ +cos kxsinkyyˆ/H20850is the group velocity for the electrons that hop to NNN sites. Adding the current densities jNNandjNNNyields the total electric current density operator j=/H20858 k/H20851v1/H20849k/H20850/H9023k†M/H9023k+v2/H20849k/H20850/H9023k†/H9023k/H20852=/H20858 k,ivi/H20849k/H20850/H9023k†Mi/H9023k, /H2084927/H20850 where M1=MandM2=I. The fact that the matrices M1 andM2are different reﬂects the fact that jNNconnects sub- lattices AandBor vice versa, and jNNNthe same sublattices. Using Eq. /H2084927/H20850, we write the current-current correlation function as /H9016/H9263/H9257/H20849i/H9275m/H20850=/H20885 0/H9252 d/H9270ei/H9275m/H9270/H20858 k,k/H11032,i,jvi/H9263vj/H9257 /H11003/H20855T/H9270/H9023k/H11032†/H20849/H9270/H20850Mi/H9023k/H11032/H20849/H9270/H20850/H9023k†/H208490/H20850Mj/H9023k/H208490/H20850/H20856, /H2084928/H20850 where vi/H9263is the /H9263th component of velocity vi/H20849i=1 or 2 and /H9263=xory/H20850. To calculate this function, we use /H9023k/H11032†/H20849/H9270/H20850Mi/H9023k/H11032/H20849/H9270/H20850=/H20858 l,m/H9023k/H11032l†/H20849/H9270/H20850Mil,m/H9023k/H11032m/H20849/H9270/H20850, /H2084929/H20850 with landmdesignating the rows and columns of the matrix Mi, respectively. The /H9023lis the lth component of the Nambu spinor /H9023k†=/H20849/H9023k1/H9023k2/H9023k3/H9023k4/H20850†/H11013/H20849ck↑Ack↑Bck↓Ack↓B/H20850†. By summing over l,m,l/H11032, and m/H11032Eq. /H2084928/H20850can be rewritten as /H9016/H9263/H9257/H20849i/H9275m/H20850=/H20885 0/H9252 d/H9270ei/H9275m/H9270/H20858 k,k/H11032,i,jvi/H9263vj/H9257 /H11003/H20858 l,m,l/H11032,m/H11032/H20855T/H9270/H9023k/H11032l†/H20849/H9270/H20850Milm/H9023k/H11032m/H20849/H9270/H20850/H9023kl/H11032†/H208490/H20850Mjl/H11032m/H11032 /H11003/H9023km/H11032/H208490/H20850/H20856 /H11013/H20885 0/H9252 d/H9270ei/H9275m/H9270/H20858 k,i,jvi/H9263vj/H9257 /H11003Tr/H20851G/H20849k,/H9270/H20850MiG/H20849k,−/H9270/H20850Mj/H20852. /H2084930/H20850 Using the Fourier transform of the frequency-dependent Green’s function G/H20849k,/H9270/H20850=1 /H9252/H20858 i/H9275ne−i/H9275n/H9270G/H20849k,i/H9275n/H20850/H20849 31/H20850 in Eq. /H2084930/H20850leads to the following expression for the current- current correlation function: /H9016/H9263/H9257/H20849i/H9275m/H20850=1 N/H20858 k,n,i,jvi/H9263vj/H9257Tr/H20851G/H20849k,i/H9275n+i/H9275m/H20850MiG/H20849k,i/H9275n/H20850Mj/H20852. /H2084932/H20850 Using G/H20849k,i/H9275n/H20850=/H20848/H20849d/H9280/2/H9266/H20850/H20851A/H20849k,/H9280/H20850/i/H9275n−/H9280/H20852, and1 /H9252/H20858 n1 i/H9275n+i/H9275m−/H9280/H110321 i/H9275n−/H9280=nF/H20849/H9280/H20850−nF/H20849/H9280/H11032/H20850 /H9280−/H9280/H11032+i/H9275m gives the correlation function in the form /H9016/H9263/H9257/H20849i/H9275m/H20850=1 N/H20858 k,i,j/H20885/H20885d/H9280/H11032 2/H9266d/H9280 2/H9266nF/H20849/H9280/H20850−nF/H20849/H9280/H11032/H20850 /H9280−/H9280/H11032+i/H9275m /H11003vi/H9263vj/H9257Tr/H20851A/H20849k,/H9280/H11032/H20850MiA/H20849k,/H9280/H20850Mj/H20852. /H2084933/H20850 Taking the analytical limit i/H9275m→/H9275+i0+in Eq. /H2084933/H20850,w eg e t the retarded correlation function. Using the identity 1/ /H20849X +i0+/H20850=P/H208491/X/H20850−i/H9266/H9254/H20849X/H20850, integrating over /H9280/H11032, restoring the physical unit of the conductivity for materials with a single CuO 2layer per unit cell, and converting the sum over the momentum kto an integral over kxandky, all together yield the follwoing expression for the real part of the optical con-ductivity: /H9268/H9263/H9257/H20849/H9275/H20850=e2 /H6036c/H20885 RBZdk /H208492/H9266/H208502/H20885 −/H11009/H11009d/H9280 4/H9266nF/H20849/H9280/H20850−nF/H20849/H9280+/H9275/H20850 /H9275 /H11003/H20858 i,jvi/H9263/H20849k/H20850vj/H9257/H20849k/H20850Tr/H20851A/H20849k,/H9275+/H9280/H20850MiA/H20849k,/H9280/H20850Mj/H20852. /H2084934/H20850 Here, cis the c-axis lattice parameter, and except for nF, all the functions in the integrand are now expressed in the unitof either tort −1. RBZ designates the reduced /H20849magnetic /H20850 Brillouin zone. Dividing by a factor 2 and integrating overthe whole BZ is equivalent to integrating over the RBZ. We now focus on the optical conductivity in the normal state along the xaxis. In this case it can be written in the form /H9268/H20849/H9275/H20850=e2 /H6036c/H20885d2k 2/H208492/H9266/H208502/H20885 −/H11009/H11009d/H9280 4/H9266nF/H20849/H9280/H20850−nF/H20849/H9280+/H9275/H20850 /H9275/H20851dt+dt/H11032 +2dtt/H11032/H20852, /H2084935/H20850 where the dfunctions are given in terms of the spectral func- tion components Aij=Aij/H20849k,/H9280/H20850, and Aij/H11032=Aij/H20849k,/H9280+/H9275/H20850by dt=2vx2/H20849A11/H11032A11+A12/H11032A12−A13/H11032A13/H20850, dt/H11032=2vx/H110322/H20849A11/H11032A11+A12/H11032A12+A13/H11032A13/H20850, dtt/H11032=2vxv/H11032x/H20849A11/H11032A12+A12/H11032A11/H20850. /H2084936/H20850 Here vx=2tsinkxandv/H11032x=4t/H11032sinkxcosky. The resistivity is found by taking the limit /H9275→0 of the inverse of the optical conductivity, where lim /H9275→0nF/H20849/H9280/H20850−nF/H20849/H9280+/H9275/H20850 /H9275=−dnF/H20849/H9280/H20850 d/H9280, and the dfunctions in Eq. /H2084936/H20850are rewritten in terms of Aij/H11032 =Aijfor/H9275→0. This givesH. SAADAOUI AND M. AZZOUZ PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-6/H9267−1/H20849T/H20850=e2 /H6036c/H20885d2k 2/H208492/H9266/H208502/H20885 −/H11009/H11009d/H9280 4/H9266/H20875−dnF/H20849/H9280/H20850 d/H9280/H20876/H20851dt+dt/H11032+2dtt/H11032/H20852. /H2084937/H20850 Next, we will focus on the ﬁtting of the experimental resis- tivity data of the material La 2−xSrxCuO 4. V . RESISTIVITY RESULTS A. Fitting parameters First of all, we stress that all the Hamiltonian parameters have a physical meaning, and that there does not exist asingle nonphysical adjustable parameter in the present ap-proach. The number of the mean-ﬁeld parameters in RAFT istwo in the normal state /H20849three in the superconducting state /H20850. Here, in addition to the two mean-ﬁeld parameters Qandn, self-energy /H9018is considered to be a ﬁtting parameter in our ﬁts of the resistivity data. If, for example, impurity scatteringand scattering with some sort of collective modes are con-sidered then an additional two adjustable parameters relatedto/H9018have to be considered. All these parameters have, how- ever, a physical meaning because Qdesignates the rotating AF order proposed by one of the authors for the explanationof the PG behavior, 4,5nis the electron density, and the pa- rameters involved in /H9018are related to the coupling to either the impurities or the collective modes, which could bephonons, spin waves, etc. B. Fitting the resistivity data of La 2−xSrxCuO 4 In RAFT, doping is obtained from the electron density per spin and site nin Eqs. /H2084911/H20850,b y p=1−2 n, and is identiﬁed with xin La 2−xSrxCuO 4. Unlike in the framework of the EDA, it is impossible here to adjust both the real and imagi-nary parts of /H9018/H20849k,i /H9275n/H20850in order to ﬁt the resistivity experi- mental data using Eq. /H2084937/H20850because of the triple integration. In the present work, the imaginary part of the self-energy inthe expression of the resistivity is modeled using an effectivewave vector and frequency independent relaxation ratewithin the MFL ansatz. 26The kindependence is evidently questionable in the light of the fact that this assumption mayseem to contradict the conclusions drawn from ARPES. 16,17 One can argue, however, that because in the transport mea-surements the quantities are an average over the BZ, as anapproximation one may replace the self-energy by ak-averaged transport relaxation rate, and hope that the under- lying theory /H20849RAFT here /H20850contains sufﬁcient kdependence in the energy spectrum. Note that this assumption is similarto the one made in the DMFT, regarding the local characterof the self-energy. 13,28,29 The/H9275independence can be justiﬁed in the framework of the MFL ansatz by the fact that we focus on the resistivity sothat we can let max /H20849 /H9275,T/H20850=Tin the imaginary part of self- energy, Eq. /H2084918/H20850. Thus only the Tdependence of the effective scattering rate 1/ /H9270/H20849T/H20850, which is used as a ﬁtting parameter, is considered, and the real part /H9018/H11032is set to zero for an /H9275inde- pendent self-energy. Interestingly, there is experimentalevidence 30hinting to the constancy of the real part of self-energy for Tc/H11021T/H11021T, a result obtained in the framework of the ARPES, however. Also, in the present approximation,1/ /H9270can be viewed as an average over the BZ and the low- frequency regime. The effective scattering approximation isjustiﬁed in both the Born and unitary impurity scatteringlimits, 24,31but for a complete and fair treatment of HTSC’s, one should eventually consider the most difﬁcult case of a k and/H9275dependent self-energy. Because the results we report here /H20849see below /H20850agree well with the experimental data /H20849for Tfor example /H20850, we believe that our approximation is worth considering. /H9267/H20849T/H20850is calculated numerically for set I and set II of the Hamiltonian parameters /H20849Table I /H20850. Here, t=0.1 eV and the lattice spacing parameter along the caxis is c=13.2 Å. 1. Fitting the resistivity using an MFL law with constant /H9261 and/H9257 Figure 6 shows two of our ﬁts for the La 2−xSrxCuO 4re- sistivity data,32,33where an MFL /H20849Ref. 26 /H20850law for the scat- tering rate 1/ /H9270/H20849T/H20850and set I of the Hamiltonian parameters are used. For doping p/H11013x=0.15, 1/ /H9270=0.4+3.9 Tyields a good ﬁt except for temperatures smaller than about 300 K.Forp=0.2, 1/ /H9270=0.11+3 Tyields a ﬁt that remains acceptable even at low temperatures. We found that as doping ap-proaches the optimal point the temperature below which theﬁt ceases to work decreases. In general, we found that in theunderdoped and overdoped regimes, the MFL linear law forthe scattering rate works well only above a doping dependentthreshold temperature. Next, we will vary empirically 1/ /H9270so that the resistivity data are ﬁtted exactly at all temperatures.In doing so for each of the resistivity curve vs temperature ata given doping, we extract a curve for 1/ /H9270vsTat that dop- ing. This procedure is equivalent to allowing /H9261and/H9257in Eq. /H2084918/H20850acquire a temperature and doping dependence. 2. Empirical curves for 1//H9270vs T We now seek to know how 1/ /H9270should vary as a function of temperature and doping when we perform an exact ﬁt ofthe resistivity data. Figure 7 shows the exact ﬁts of the re- FIG. 6. We display the ﬁt of the La 2−xSrxCuO 4resistivity data /H20849Ref. 32 /H20850made using an MFL law and set I of the Hamiltonian parameters. /H20849a/H208501//H9270=3.9 T+0.4 for x/H11013p=0.15. /H20849b/H208501//H9270=3T+0.11 forx=0.2.DOPING DEPENDENCE OF COUPLING BETWEEN … PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-7sistivity data of Takagi et al.32measured for eight doping densities in the underdoped and overdoped regimes of the material La 2−xSrxCuO 4. Figure 8 displays the curves of 1/ /H9270vsTextracted by performing the ﬁts of Fig. 7 using set I. Figure 9 shows theresults obtained for set II of the Hamiltonian parameters. Asexpected, in the underdoped regime, 1/ /H9270/H20849T/H20850shows an MFL dependence only at high temperatures, but a threshold struc- ture develops below a doping-dependent temperature T/H9267, which marks the onset of the deviation from the MFL linearbehavior. As pincreases the depression seen in 1/ /H9270/H20849T/H20850shifts to lower temperatures and disappears at p/H11013pc/H110150.22 for set I and pc/H110150.2 for set II. It is interesting to note that qualita- tively the same behavior is obtained for the two different setsof the Hamiltonian parameters. This indicates that our resultsare robust against a small change in these parameters. Notethat for both sets, at p c1//H9270is linear down to the lowest temperatures, and almost extrapolates to zero. This meansthat scattering off impurities is not relevant at p c, and only the scattering off the collective modes plays a signiﬁcant roleat the critical doping p c. In the overdoped regime, Fig. 10 for set I and Fig. 11 for set II show 1/ /H9270vsTfor several doping densities. For doping greater than pc, the MFL linear behavior is found to be valid only for T/H11022Tm, where Tmis a doping-dependent temperature below which 1/ /H9270/H20849T/H20850is marked this time by an upward turn. This upward turn means that 1/ /H9270varies as Tmwith m/H110221 forT/H11021Tm. This is perhaps an indication that the Laudau Fermi- liquid behavior is recovered in the overdoped regime. Note,however, that for the Fermi liquid, the electron-electron scat-tering rate behaves as 1/ /H9270/H11011T2. Above Tin the underdoped regime or Tmin the over- doped one, the slopes of 1/ /H9270vsTare practically doping independent. From Eq. /H2084918/H20850, one notes that the slope is re- lated to the coupling constant /H9261of the charge carriers and the collective modes. So, above Tthis coupling is not sensitive to doping, but becomes strongly doping dependent below T. In fact, this coupling is strongly enhanced below Twith underdoping. In the next section, we will interpret our resultsand analyze the doping dependence of this coupling. VI. INTERPRETATION OF THE RESULTS A. Doping dependence of T* The linear Tdependence of 1/ /H9270at high temperatures is consistent with the results of the two-component model for1/ /H9270DvsTfor Bi 2Sr2CaCuO 8+/H9254and YBa 2Cu3O7−y.341//H9270Dis the Drude scattering rate. In general, we expect the present FIG. 7. We display the La 2−xSrxCuO 4resistivity data /H20849Ref. 32 /H20850 that were ﬁtted by adjusting the relaxation rate. Set II of the Hamil-tonian parameters was used here, but identical curves are obtainedfor set I. Obviously, what differs for the two sets is 1/ /H9270. The sym- bols indicate the experimental points for which we carried the cal-culation. /H20849a/H20850Underdoped to optimal regime. /H20849b/H20850Overdoped regime. FIG. 8. The Tdependence of 1/ /H9270used to ﬁt /H9267/H20849T/H20850for La2−xSrxCuO 4in the underdoped regime is shown. Here, set I of the Hamiltonian parameters was used. The dashed lines are linear ﬁtsforT/H11022T /H9267. FIG. 9. The Tdependence of 1/ /H9270used to ﬁt /H9267/H20849T/H20850for La2−xSrxCuO 4in the underdoped regime is shown. Set II of the Hamiltonian parameters was used. The dashed lines are linear ﬁtsforT/H11022T /H9267.H. SAADAOUI AND M. AZZOUZ PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-8approach and EDA to yield similar results only for T/H11022T/H9267in the underdoped regime, and for T/H11022Tmin the overdoped re- gime, because the charge carriers are Drude-like in EDA andRAFT in both these regimes. However, in RAFT the starting point for T/H11021T /H9267in the underdoped phase is a state with RAF,4,5i.e., a state with different local symmetry than above T/H9267. The global rotational symmetry is not broken even below T/H9267. Thus the deviation from linearity can be linked to the appearance of RAF, and the depression in 1/ /H9270indicates a crossover to a scattering regime with a mechanism that in-creases the relaxation time /H9270. To interpret our results, we neglect the complications aris- ing from the commensurability or incommensurability issueof the neutron resonance peak, and the possibility that the PGmay not vanish at a QCP as the Hall data seem to suggest. 3,35 By comparison to the experimental data available on the PG, we interpret T/H9267as the PG temperature32,35–37/H20849T/H9267/H11013T/H20850and Tmusing the data of Naqib et al.38From Figs. 8 and 9, we extract T/H9267and plot it as a function of doping pin Fig. 12 /H20849a/H20850. The values we calculated for Tusing either set I or II are in excellent agreement with the experimental values forLa 2−xSrxCuO 4and YBa 2Cu3O7−y.35,39The PG state appears at low p/H20849forp=0.075, T/H11011800 K /H20850. For both sets, by in- creasing p,Tdrops linearly to about 100 K near the optimal point. Again, we ﬁnd that the doping dependence of Tis robust against a small change in the Hamiltonian parameters.The crossover temperature T mextracted from Figs. 10 and 11 is plotted versus doping in Fig. 12 /H20849b/H20850after shifting down- ward our doping values by 0.055. If not shifted, our valuesdo not agree well with the experimental values of Naqib et al.for Y 1−xCaxBa2/H20849Cu1−yZny/H208503O7−y, but the linear doping de- pendence is well accounted for.38The need to shift doping is due to the fact that the doping-temperature phase diagramcalculated for set I or II does not agree exactly with theLa 2−xSrxCuO 4experimental phase diagram as far as the lo- cation of the dome of superconductivity is concerned.5 In RAFT, the rotating AF order parameter Qvanishes at a critical doping near the optimal point, which has been iden-tiﬁed as a QCP. At this point, Qvanishes as was illustrated earlier in Fig. 1. Also the p-dependent temperature, hereafter called T Q, at which Qvanishes has been previously identiﬁedas the PG temperature T.4,5Here, we rather deﬁne the PG temperature as the temperature below which deviation fromthe MFL behavior appears in the scattering rate. Then, we propose that T Qis the temperature below which the neutron- scattering peak appears,8,40which means that the rotating antiferromagnetism is responsible for the peak observed inthe neutron-scattering experiments. This proposal will be ex-amined further in a forthcoming report. Figure 13 /H20849a/H20850displays both T andTQfor set II of parameters. The interpretation done in terms of these two temperatures is in agreement withthe experimental evidence for the existence of two crossover temperatures. 3TandTQdrop linearly as doping increases, become equal near optimal doping, and vanish simulta-neously at the QCP, p QCP /H11013pc. B. Frequency dependence of the spectral function In this section, we want to get an insight into how the PG appears in the frequency dependent spectral functions. To dothis, we continue to assume that the MFL law in Eq. /H2084918/H20850for the relaxation rate is valid even at nonzero frequency. Wesuppose that max /H20849 /H9275,T/H20850=Tcontinues to be true. This is ob- viously the case for energies /H6036/H9275/H110110 and a ﬁnite temperature, FIG. 10. The Tdependence of 1/ /H9270used to ﬁt /H9267/H20849T/H20850in the over- doped regime of the material La 2−xSrxCuO 4is shown. Set I of the Hamiltonian parameters was used. The dashed lines are linear ﬁtsforT/H11022T m. FIG. 11. The Tdependence of 1/ /H9270used to ﬁt /H9267/H20849T/H20850in the over- doped regime of the material La 2−xSrxCuO 4is shown. Set II of the Hamiltonian parameters was used. The dashed lines are linear ﬁtsforT/H11022T m. FIG. 12. Comparison of RAFT’s values for T/H11013T/H9267andTmwith experiment. /H20849a/H20850Tvs doping. The experimental data are from Ref. 36 for the resistivity /H9267and susceptibility /H9273, and from Ref. 37 for the Hall constant RH./H20849b/H20850We display Tmvs doping shifted by 0.055. The data are from Ref. 38 for Y 1−xCaxBa2/H20849Cu1−yZny/H208503O7−y.DOPING DEPENDENCE OF COUPLING BETWEEN … PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-9i.e., near the Fermi energy. The PG is seen below TQas a depression in the DOS at /H9275=0. The diagonal elements of the spectral function A/H20849k,/H9275/H20850/H20851Eq. /H2084914/H20850/H20852and the corresponding DOS calculated using empirical estimates for 1/ /H9270are shown in Fig. 14. Clearly, the PG is kdependent as the depression inA11/H20849k,w/H20850atk=/H20849/H9266,0/H20850is found at /H9275/H110110, but away from /H9275=0 at k=/H20849/H9266/2,/H9266/2/H20850; the PG opens above the Fermi level in agreement with the spectra plots in Fig. 3. We propose that inverse photoemission experiments could be performed inorder to check if the PG opens above the Fermi energy for/H20849 /H9266/2,/H9266/2/H20850. Our ﬁnding compares well with the experimen- tal results of Norman et al. regarding the opening of the PG asTdecreases near the point k=/H20849/H9266,0/H20850and also regarding their result that no gap was observed on the Fermi surface near the diagonal of the BZ.41As we explained before, it is thekdependence of /H9262/H11032/H20849k/H20850that causes that of the PG. This effect is reminiscent of that of the effective chemical poten- tial on the PG in Ref. 13. C. Doping dependence of the coupling between the charge carriers and the bosonic modes To estimate the pdependence of the coupling of the charge carriers to the normal-state bosonic modes, which, wethink, is responsible for the depression in 1/ /H9270below T,w e calculate the quantity /H9004/H208491//H9270/H20850deﬁned as the difference be- tween the value 1/ /H9270would take if the linear MFL behavior held down to zero temperature and the value 1/ /H9270actually assumes at the lowest temperature. Figure 13 /H20849b/H20850displays /H9004/H208491//H9270/H20850vsp. Following the footsteps of Hwang and co-workers,9we interpret /H9004/H208491//H9270/H20850as resulting from the cou- pling between the charge carriers and some sort of bosons, which appear below T, and write /H9261cb/H11011/H9004/H208491//H9270/H20850./H9261cbde- creases sharply for increasing pand vanishes at pQCPwhere both TandTQvanish as well. Thus we can conclude that not only is the opening of the PG related to the appearance of theRAF with order parameter Q, but also to the appearance of collective bosonic modes that affect the scattering mecha-nism. We propose that T results from the coupling of the charge carriers to the bosonic modes, which are the conse- quence of the pretransitional ﬂuctuations for TQ/H11021T/H11021T. These pretransitional ﬂuctuations can be viewed as due to the fact that above TQ, the phase coherence for the RAF order parameter is possible only for short lattice distances, anddoes not extend over the whole lattice. Below TQ, long-range phase coherence sets in, then leads to the establishment ofthe RAF. As we mentioned earlier, we propose that theneutron-scattering peak results from the establishment of the RAF order for T/H11021T Q. A possible candidate for the bosonic modes may therefore be the spin-wave excitations in the ro-tating frame of the RAF order. 18We plan to check these proposals in the future. VII. CONCLUSIONS In this paper, we calculated the optical conductivity in the framework of the rotating antiferromagnetism theory. We fo-cused our efforts on the resistivity and ﬁtted the experimentaldata of the material La 2−xSrxCuO 4by treating the relaxation rate as a ﬁtting parameter. We analyzed the temperature anddoping dependence of the relaxation rate obtained from theseﬁts. In the underdoped regime, we found that for tempera-tures greater than a doping dependent threshold value therelaxation rate is linear in temperature. For temperaturessmaller than this threshold, the relaxation rate is depressed.The threshold temperature was succesfully identiﬁed withthe experimental PG temperature T . As an interpretation for the depression in the relaxation rate, we proposed that below the PG temperature Tthe charge carriers couple strongly to the normal-state collective/H20849bosonic /H20850modes existing in La 2−xSrxCuO 4. Normally, the PG, being a depression in the single-particle DOS, shouldenhance the resistivity below T , but we believe that the emergence of this coupling below Tcompensates this en- hancement. This leads to no noticeable effect in the experi-mental resistivity, except for a deviation downward aroundthe PG temperature in the underdoped regime. Note, how-ever, that although the bosonic modes appearing below T are responsible for the PG, they cannot /H20849at least solely /H20850be FIG. 13. The doping dependence of Tand TQ, and of /H9261cb /H11011/H9004/H208491//H9270/H20850is shown. /H20849a/H20850TQandTvs doping. /H20849b/H20850/H9004/H208491//H9270/H20850vs doping. Set II of the Hamiltonian parameters was used. The continuous anddashed lines are a guide to the eye only. FIG. 14. The diagonal component of the spectral function for set II of the Hamiltonian parameters is displayed vs /H9275fork=/H20849/H9266,0/H20850in /H20849a/H20850, and k=/H20849/H9266/2,/H9266/2/H20850in/H20849b/H20850. The diagonal DOS is plotted vs /H9275for set II in /H20849c/H20850and set I in /H20849d/H20850.I n /H20849a/H20850–/H20849c/H20850,T/t=0.3, 1/ t/H9270=3.5 /H20849thick line /H20850,T/t=0.1, 1/ t/H9270=1 /H20849thin line /H20850,T/t=0.05, 1/ t/H9270=0.5 /H20849dashed- dotted line /H20850.I n /H20849d/H20850,1 /t/H9270=2, 0.5, and 0.25 for T/t=0.4, 0.1, and 0.05, respectively.H. SAADAOUI AND M. AZZOUZ PHYSICAL REVIEW B 72, 184518 /H208492005 /H20850 184518-10responsible for superconductivity because they stop coupling to the charge carriers at the quantum critical point pQCP, well before superconductivity disappears in the overdoped re-gime. 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PhysRevB.94.024502.pdf	PHYSICAL REVIEW B 94, 024502 (2016) Superconductivity from a conﬁnement transition out of a fractionalized Fermi liquid withZ2topological and Ising-nematic orders Shubhayu Chatterjee,1Yang Qi,2,3Subir Sachdev,1,3and Julia Steinberg1 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Institute for Advanced Study, Tsinghua University, Beijing 100084, China 3Perimeter Institute of Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5 (Received 5 April 2016; published 5 July 2016) The Schwinger boson theory of the frustrated square lattice antiferromagnet yields a stable, gapped Z2spin liquid ground state with time-reversal symmetry, incommensurate spin correlations, and long-range Ising-nematicorder. We obtain an equivalent description of this state using fermionic spinons (the fermionic spinons can beconsidered to be bound states of the bosonic spinons and the visons). Upon doping, the Z 2spin liquid can lead to a fractionalized Fermi liquid (FL) with small Fermi pockets of electronlike quasiparticles, while preserving theZ 2topological and Ising-nematic orders. We describe a Higgs transition out of this deconﬁned metallic state into a conﬁning superconducting state which is almost always of the Fulde-Ferrell-Larkin-Ovchinnikov type, withspatial modulation of the superconducting order. DOI: 10.1103/PhysRevB.94.024502 I. INTRODUCTION TheZ2spin liquid is the simplest gapped quantum state with time-reversal symmetry and bulk anyon excitations [ 1–8]. For application to the cuprate superconductors, an attractive parent Mott insulating state is a Z2spin liquid obtained in the Schwinger boson mean-ﬁeld theory of the squarelattice antiferromagnet with ﬁrst, second, and third neighbor exchange interactions [ 1,9,10]. This is a fully gapped state with incommensurate spin correlations, spinon excitations which carry spin S=1/2, vison excitations which carry Z 2magnetic ﬂux, and long-range Ising nematic order associated with abreaking of square lattice rotation symmetry. Upon doping away from such an insulator with a density of pholes, we can obtain a FL metallic state which inherits the topological order of the Z 2spin liquid, and acquires a Fermi surface of electronlike quasiparticles enclosing a volume associatedwith a density of pfermions [ 11–22]. It was also noted [13] that a Z 2-fractionalized Fermi liquid (FL) metal can undergo a transition into a superconducting state which is concomitant with conﬁnement and the loss of Z2topological order (while preserving the Ising-nematic order). Given therecent experimental evidence for a Fermi-liquid-like metallic state in the underdoped cuprates with a density of ppositively charged carriers [ 23–25], the present paper will investigate the structure of the conﬁning superconducting state which descends from the Z 2-FL state associated with Schwinger boson mean-ﬁeld theory of the square lattice [ 1,9,10]. For insulating Z2spin liquids, the spectrum can be classiﬁed by four separate “topological” or “superselection” sectors, which are conventionally labeled 1, e,m, and/epsilon1[7]. In the Schwinger boson theory, the Schwinger boson itself becomesa bosonic, S=1/2 spinon excitation which we identify as belonging to the esector. The vison, carrying Z 2magnetic ﬂux, is spinless, and we label this as belonging to the msector. A fusion of the bosonic spinon and a vison then leads to a fermionic spinon [ 26], which belongs to the /epsilon1sector. We summarize these, and other, characteristics of insulating Z2 spin liquids in Table I.For a metallic Z2-FL* state, it is convenient to augment the insulating classiﬁcation by counting the charge, Q, of fermionic electronlike quasiparticles: we simply add a spectator electron, c, to each insulator sector, and label the resulting states as 1 c,ec,mc, and/epsilon1c, as shown in Table I.I t is a dynamical question of whether the cparticle will actually form a bound state with the e,m,o r/epsilon1particle, and this needs to be addressed speciﬁcally for each Hamiltonian of interest. Now let us consider a conﬁning phase transition in which theZ2topological order is destroyed. This can happen by the condensation of one of the nontrivial bosonic particles of the Z2-FL* state. From Table I, we observe that there are three distinct possibilities: (1) Condensation of m: this was initially discussed in Refs. [ 2,4]. For the case of insulating antiferromagnets with an odd number of S=1/2 spins per unit cell, the non-trivial space group transformations of the mparticle lead to bond density wave order in the conﬁning phase. The generalization to themetallic Z 2-FL* state was presented recently in Ref. [ 27]. (2) Condensation of e: now we are condensing a boson with S=1/2, and this leads to long-range antiferromagnetic order [28–32]. (3) Condensation of /epsilon1c: this is a boson which carries electromagnetic charge, and so the conﬁning state is asuperconductor [ 13]. This paper will focus on the third possibility listed above: condensation of /epsilon1 c, the bosonic “chargon.” Our speciﬁc interest is in the Schwinger boson Z2spin liquid of Refs. [ 1,9,10]. To study the /epsilon1cstates in this model, we need to consider the fusion of the /epsilon1quasiparticle and the electron (which is in the 1 csector). Thus a key ingredient needed for our analysis will be the projective transformations of the /epsilon1 particle under the symmetry group of the underlying squarelattice antiferromagnet. These transformations are not directlyavailable from the Schwinger boson mean-ﬁeld theory, whichis expressed in terms of the eboson. However, remarkable recent advances [ 33–39] have shown how the projective symmetry group (PSG) of the /epsilon1particle can be computed from a knowledge of the PSG of the eandmparticles. 2469-9950/2016/94(2)/024502(19) 024502-1 ©2016 American Physical SocietyCHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) TABLE I. Table of characteristics of sectors of the spectrum of the Z2-FL* state. The ﬁrst four columns are the familiar sectors of an insulating spin liquid. The value of Sindicates integer or half-integer representations of the SU(2) spin-rotation symmetry. The “mutual semion” row lists the particles which have mutual seminionic statistics with the particle labeling the column. The electromagnetic charge isQ. The last four columns represent Q=1 sectors present in Z2-FL, and these are obtained by adding an electronlike quasiparticle, 1c, to the ﬁrst four sectors. The bottom row denotes the ﬁelds operators used in the present paper to annihilate/create particles in the sectors. 1 em/epsilon1 1c ec mc /epsilon1c S 01 /201 /21 /20 1 /20 Statistics boson boson boson fermion fermion fermion fermion boson Mutual semions m,/epsilon1,mc,/epsilon1c e,/epsilon1,ec,/epsilon1c e,m,ec,mc m,/epsilon1,mc,/epsilon1c e,/epsilon1,ec,/epsilon1c e,m,ec,mc Q 0 000 1 111 Field operator bφf c B Section IIdescribes in detail our computation of the PSG of the /epsilon1excitations of the square lattice Schwinger boson Z2 spin liquid state. These results are then applied in Sec. III to deduce the structure of the superconductor obtained bycondensing /epsilon1 c. II. MAPPING BETWEEN BOSONIC AND FERMIONIC SPIN LIQUIDS ON THE RECTANGULAR LATTICE VIA SYMMETRY FRACTIONALIZATION The Schwinger boson mean-ﬁeld Z2spin liquid described in Refs. [ 1,9,10] spontaneously breaks the C4rotation symme- try of the square lattice, and this nematic order persists in theZ 2-FL. Therefore, we identify the space-group symmetries of the rectangular lattice along with time reversal Tas the symmetries that act projectively on the eandmparticles (bosonic spinons and visons respectively) in the above Ansatz in the Schwinger boson representation (bSR). Below, webrieﬂy describe the idea of symmetry fractionalization [ 33– 39], which enables us to ﬁnd the projective actions of the same symmetries on the /epsilon1particles, or equivalently the spinons in the Abrikosov fermion representation (fSR). We only providea quick summary, and refer the reader to the references abovefor detailed discussions. The key idea behind symmetry fractionalization is that the action of any symmetry on a physical state (which mustnecessarily contain an even number of any anyon in a Z 2spin liquid) can be factorized into local symmetry operations oneach of these anyons. For concreteness, consider the translationoperator T x(Ty), which translates the wave function by one unit in the ˆx(ˆy) direction, and a physical state \|ψ/angbracketrightthat contains twoeparticles at randr/prime. We assume that this operation can be factorized as Tx\|ψ/angbracketright=Te x(r)Te x(r/prime)\|ψ/angbracketright. (1) Since the eparticle is coupled to emergent gauge ﬁelds, Te x(r) is not invariant under gauge transformations. But if we consider a set of operations that combine to the identity,T e xTe y(Te x)−1(Te y)−1for example, then the combined phase that theeparticle picks up is gauge invariant. In a gapped Z2spin liquid, this phase must be ±1. This can be seen by fusing two eparticles, which is a local excitation and therefore can only pick up a trivial phase +1. This also implies that this phase is independent of location of the eparticle as long as translation symmetry is preserved by the spin liquid. Although we chosetheeparticle for illustration, an analogous picture holds for m and/epsilon1particles as well. Generalizing this to other symmetries including internal ones like time reversal T, we can ﬁnd a quantized gauge invariant phase of ±1 for each series of symmetry operations that combine to identity on the physical wave function. Thisphase is ﬁxed for a given anyon in a particular spin liquid, andis also referred to as the symmetry fractionalization quantumnumber. These quantum numbers are universal features of Z 2 spin liquids, and provide a way to characterize topological or- der without parton constructions. However, given a particularparton construction (either in terms of bosons or fermions),we can determine these quantum numbers—we shall illustratehow for the particular bosonic Z 2spin liquid we are interested in. Also, given a set of quantum numbers we can attempt toﬁnd a corresponding spin liquid Ansatz —we again explicitly describe this later when we ﬁnd a fermionic mean-ﬁeld Ansatz . But ﬁrst, we outline how we ﬁnd these quantum numbers forthe fermions from those of the bosons and the visons. In aZ 2spin liquid, the eandmparticle satisfy the following fusion rule [ 7]: e×m=/epsilon1. (2) In other words, we can think of the fermionic spinon ( /epsilon1)a s a bound state of the bosonic spinon ( e) and the vison ( m). Therefore, in most cases, for a set of symmetry operationsOcombining to identity, the phase factor picked up by the fermionic spinon σ /epsilon1 Ois just the product of the phase σe Opicked up by the bosonic spinon and the phase σm Opicked up by the vison. These have been referred to as the trivial fusion rulesin Ref. [ 34]. In certain cases, there is an additional factor of−1 coming from the nontrivial mutual statistics between the spinon and the vison, and these fusion rules are callednontrivial. Once these fusion rules are known, the symmetryfractionalization quantum numbers for the /epsilon1can be calculated from those of eandm. With this preamble, we now outline the procedure to derive the fermionic spin liquid Ansatz corresponding to the bosonic Z 2spin liquid obtained from the J1-J2-J3antiferromagnetic Hamiltonian on the square lattice [ 1,9]. We ﬁrst describe the symmetries of the spin liquid, and list the elementarycombinations for which we need to calculate the symmetryfractionalization quantum numbers. Then we discuss the ideaof PSG for the Schwinger boson spin liquids in general[40], and use it to calculate the aforementioned quantum 024502-2SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) numbers for our bosonic Ansatz . We proceed with analogous derivations of the quantum numbers for the visons [ 41,42] and fermions [ 3,43–45] using PSG techniques. We then derive the nontrivial fusion rules, and use these to relate thebosonic and fermionic symmetry quantum numbers of time-reversal preserving mean-ﬁeld spin liquids on the rectangularlattice. Finally, we ﬁnd the speciﬁc set of quantum numbersfor the fermionic spin liquid of our interest, and ﬁnd anAnsatz consistent with this particular pattern of symmetry fractionalization. A. Symmetries of the spin liquid Consider a mean-ﬁeld Hamiltonian with the following symmetries: global spin rotations, action of the rectangularlattice space group, and time reversal T. Since a mean-ﬁeld spin liquid Ansatz is explicitly invariant under global SU(2) spin rotations, we only need to consider the projective actionsof the other symmetries. Let us deﬁne the lattice pointsr=xˆx+yˆy=(x,y) in a rectangular coordinate system with unit vectors ˆxand ˆy. The space group of the rectangular lattice is then generated by the translations and reﬂections ∈ {T x,Ty,Px,Py}, deﬁned as follows: Tx:(x,y)→(x+1,y), (3a) Ty:(x,y)→(x,y+1), (3b) Px:(x,y)→(−x,y), (3c) Py:(x,y)→(x,−y). (3d) There are algebraic constraints which relate these gen- erators. Below, we present the ﬁnite set of elementarycombinations of these generators that are equivalent to theidentity operator on any physical wave function: T −1 xT−1 yTxTy,P2 x,P2 y,P−1 xTxPxTx,P−1 xT−1 yPxTy, P−1 yT−1 xPyTx,P−1 yTyPyTyandP−1 xP−1 yPxPy.(4a) When we include time reversal T, we also have to consider the following additional operators: T2,T−1 xT−1TxT,T−1 yT−1TyT,P−1 xT−1PxT and P−1 yT−1PyT. (4b) These are the combinations for which we need to calculate the symmetry fractionalization quantum numbers, and allother combinations that lead to identities can be expressedas products of these elementary combinations. B. PSG for bSR 1. Schwinger boson Ansatz The spin operator can be represented in terms of Schwinger bosons operators brαas /vectorSr=1 2b† rα/vectorσαβbrβ, (5) where α=↑,↓. The mean-ﬁeld Hamiltonian is Hb MF=−/summationdisplay rr/prime(Qrr/prime/epsilon1αβb† rαb† r/primeα+H.c.)+/summationdisplay rλr(b† rαbrα−1), (6)where λris a Lagrange multiplier that enforces the single occupancy constraint/summationtext αb† rαbrα=1 on an average and the Qrr/prime=/angbracketleft/epsilon1αβbrαbr/primeβ/angbracketrightare mean-ﬁeld pairing link variables that satisfy Qrr/prime=−Qr/primer. The Schwinger boson SL wave function is \|/Psi1b/angbracketright=PGexp/bracketleftBigg/summationdisplay rr/primeξrr/prime/epsilon1αβb† rαb† r/primeβ/bracketrightBigg \|0/angbracketright, (7) where PGprojects onto states with a single spin, and ξrr/prime= −ξr/primeris obtained by diagonalizing Hb MFvia a Bogoliubov transformation. 2. Gauge freedom, PSG, and algebraic constraints Here, we formally introduce the PSG in the context of the Schwinger bosons, and describe its relation to the symmetryfractionalization quantum numbers. This discussion closelyfollows Ref. [ 40]. In the bSR, consider the following local U(1) transformation of the bosons: b rα→eiφ(r)brα. (8) This leaves all the physical observables unchanged, but the mean-ﬁeld Ansatz undergoes the following transformation to leave the Hamiltonian invariant: Qrr/prime→eiφ(r)+iφ(r/prime)Qrr/prime. (9) Any two mean-ﬁeld Ans¨atze that are related by a local U(1) transformation as described above correspond to thesame physical wave function after projection to single spin-occupancy per site. Therefore, a spin liquid state has aparticular symmetry Xif the corresponding mean-ﬁeld Ansatz is invariant under the symmetry action of Xfollowed by an additional local gauge transformation G X, GX:brα→eiφX(r)brα, GXX:Qrr/prime→exp(i{φX[X(r)]+φX[X(r/prime)]})QX(r)X(r/prime).(10) The set of all such transformations {GXX}that leave the Ansatz invariant form the PSG. Ideally, each PSG element should reﬂect a physical symmetry of the Ansatz .B u ti t turns out that there are certain transformations in the PSGthat are not associated with any physical symmetry, but stillleave the Ansatz invariant. In other words, these are purely local transformations, and correspond to the identity operationX=I. They form a subgroup of the PSG, called the invariant gauge group (IGG) [ 3]. It is natural to associate these members of the PSG with the emergent gauge ﬁeld in the spin liquid.ForZ 2spin liquids, the IGG is therefore Z2, generated by −1. One can now ask how is the IGG related to the Z2symmetry fractionalization quantum numbers? To answer this question,note that elements of the IGG correspond to identity transfor-mations on the Ansatz , and therefore on the physical wave func- tion as well (assuming that the mean-ﬁeld state survives pro-jection). Therefore, for any series of operations that combineto the identity, the corresponding projective operation shouldbe an element of the IGG [for example, for T −1 xTyTxT−1 y=I, we have ( GTxTx)−1(GTyTy)(GTxTx)(GTYTy)−1=± 1]. At the same time, note from Eqs. ( 8) and ( 10) that this projective operation describes the gauge-invariant phase that a single e particle picks up under this set of transformations. Therefore, 024502-3CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) the element of the IGG which we choose for a spin liquid Ansatz is precisely the symmetry fractionalization Z2quantum number for this set of operations. In other words, the symmetryfractionalization quantum numbers determine the particularextension of the physical symmetry group by the IGG that isrealized by a given spin liquid. The algebraic relations between the spatial symmetry operations in a group strongly constrain the possible choicesof gauge transformations G Xassociated with symmetry operations X. Without referring to a particular Ansatz ,w e can use these relations to ﬁnd all possible PSGs for a setof symmetries. Below, we ﬁnd the most general phases φ X consistent with the algebraic constraints on a rectangular lattice with time-reversal symmetry. 3. Solutions to the algebraic PSG We just state the solutions here, and present the derivation in Appendix A. The solutions for the phases φX(modulo 2 π), as deﬁned in Eq. ( 10) can be written down in terms of integers {pi}deﬁned modulo 2, which are precisely the symmetry fractionalization quantum numbers for the eparticles in the spin liquid, φTx(x,y)=0, (11a) φTy(x,y)=p1πx, (11b) φPx(x,y)=p2πx+p4πy+p6 2π, (11c) φPy(x,y)=p3πx+p5πy+p7 2π, (11d) φT(x,y)=p8πx+p9πy. (11e) 4. PSG solutions for the nematic spin liquid Ansatz for the J1-J2-J3model on the square lattice We need to ﬁnd the quantum numbers for the Schwinger boson mean-ﬁeld Ansatz of our interest, which is given by [9,10] Qi,i+ˆx/negationslash=Qi,i+ˆy/negationslash=0,Q i,i+ˆx+ˆy=Qi,i−ˆx+ˆy/negationslash=0, Qi,i+2ˆx/negationslash=0,Q i,i+2ˆy=0. (12) All the mean-ﬁeld variables are real in a particular gauge choice, so time-reversal symmetry is preserved. This statehas nematic order, as the following gauge-invariant observableI=\|Q i,i+ˆx\|2−\|Qi,i+ˆy\|2/negationslash=0. This state has the following so- lution for {pi}, which we can derive (as shown in Appendix B) by using the transformation of the Ansatz under the symmetry operation Xto ﬁx the phases φX(or correspondingly, the integers pi): p1=0,p 2=0,p 3=0,p 4=1,p 5=1,p 6=1, p7=0,p 8=0,p 9=0. (13) C. Vison PSG In this section, we shall derive the vison PSG for the rectangular lattice. To do so, we shall resort to a description ofthe visons by the fully frustrated transverse ﬁeld Ising modelon the dual lattice [ 46]. Denoting the points on the dual lattice FIG. 1. The gauge choice for JRR/primeon the rectangular lattice. The dark and light bonds respectively represent links with JRR/prime=− 1a n d JRR/prime=1. The unit cell is denoted by the blue box, and the sublattice indices by 1 and 2. Dotted blue lines form the original lattice. byR, the vison Hamiltonian is given by H=/summationdisplay RR/primeJRR/primeτz Rτz R/prime−/summationdisplay RhRτx R, (14) where the product of bonds around each elementary plaquette (/square) is negative, given by /productdisplay /squaresgn(JRR/prime)=− 1. (15) Note that this Hamiltonian is invariant under the gauge transformation τz R→ηRτz r,J RR/prime→ηRηR/primeJRR/prime,η R∈{ ± 1}=Z2.(16) For calculating the vison PSG, we make the following gauge choice (depicted in Fig. 1): JR,R+ˆx=(−1)x+y=JR+ˆx,RandJR,R+ˆy=1=JR+ˆy,R. (17) Let us consider the spatial symmetry generators ﬁrst. Since the Hamiltonian is invariant under symmetry transformationsonly up to a gauge transformation, we identify, for eachsymmetry generator Xin the space group of the rectangular lattice, an element G X∈Z2such that GXX[JRR/prime]=JX[R]X[R/prime]GX[X(R)]GX[X(R/prime)]=JRR/prime.(18) Note that all operations are deﬁned with respect to the original lattice. From Fig. 1, we can immediately see what the required gauge transformations are. Since the xbonds change sign under Tx,Ty, and Py, whereas the ybonds are invariant, we must have GTx=GTy=GPy=(−1)X. Further, Pxacts trivially on both the xandybonds, so GPx=1. Now, consider time reversal T. Since the Ising couplings JRR/prime=± 1 are real, these are invariant under T,s oGT=1 as well. With this 024502-4SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) knowledge of additional phases under lattice transformations, we can calculate the symmetry fractionalization quantumnumbers of the visons in a manner analogous to the bosons—we list these in Table IIunder the column σ m O. We comment that these are exactly the quantum numbers one would obtain by thinking of the vison acquiring an extraphase of −1 when it is transported adiabatically with πﬂux per unit cell, corresponding to an odd number of spinons.The results are also consistent with another calculation froma soft-spin formulation of the visons, which we present inAppendix C. D. PSG for fSR 1. Schwinger fermion Ansatz In terms of fermion operators, the spin operator Srcan be written as Sr=1 2f† rα/vectorσαβfrβ. (19) We write down the Hamiltonian in terms of two different mean ﬁelds as follows [ 44]: Hf MF=/summationdisplay rr/prime3 8Jrr/prime/bracketleftbig χrr/primef† r,αfr/prime,α+/Delta1f rr/prime/epsilon1αβf† r,αf† r/prime,β +H.c.−\|χrr/prime\|2−/vextendsingle/vextendsingle/Delta1f rr/prime/vextendsingle/vextendsingle2/bracketrightbig +/summationdisplay ra3 0(f† rαfrα−1) +/bracketleftbig/parenleftbig a1 0+ia2 0/parenrightbig /epsilon1αβf† rαf† rβ+H.c./bracketrightbig , (20) where we have deﬁned the spinon hopping amplitude χrr/primeδαβ and the spinon-pairing amplitude /Delta1f rr/prime/epsilon1αβ, both spin-rotation invariant (and nonzero in general), as follows: /Delta1f rr/prime/epsilon1αβ=− 2/angbracketleftfrαfr/primeβ/angbracketright,/Delta1f rr/prime=/Delta1f r/primer, (21) χrr/primeδαβ=2/angbracketleftf† rαfr/primeβ/angbracketright,χ rr/prime=χ∗ r/primer, (22) and we have also introduced Lagrange multipliers ai 0to enforce single occupancy per site on average. 2. Gauge freedom, PSG, and algebraic constraints In order to see the local SU(2) symmetry of the Hamilto- nian, let us introduce ψr=/parenleftbigg ψ1r ψ2r/parenrightbigg =/parenleftbiggfr↑ f† r↓/parenrightbigg . (23) We also deﬁne a mean-ﬁeld matrix Urr/primeas follows: Urr/prime=/parenleftbigg χ∗ rr/prime/Delta1rr/prime /Delta1∗rr/prime−χrr/prime/parenrightbigg =U† r/primer. (24) In terms of the ψfermions, the single occupancy constraints reduce to /angbracketleftψ† rτlψr/angbracketright=0, so the mean-ﬁeld Hamiltonian can now be written as Hf MF=/summationdisplay rr/prime3 8Jrr/prime/bracketleftbigg1 2Tr(U† rr/primeUrr/prime)−ψ† rUrr/primeψr+H.c.)/bracketrightbigg +/summationdisplay ral 0(r)ψ† rτlψr. (25)Note that Urr/primeis not a member of SU(2) as det( U)<0, butiUrr/prime∈SU(2) up to a normalization constant. Hf MFis explicitly invariant under a local SU(2) gauge transformationW(r): ψ r→W(r)ψ, (26a) Urr/prime→W(r)Urr/primeW†(r/prime). (26b) In general, dynamical SU(2) gauge ﬂuctuations can reduce the gauge group. In particular, in the presence of noncollinearSU(2) ﬂux, the SU(2) gauge bosons become massive and theonly the Z 2gauge structure is unbroken at low energies [ 3,44]. In the following sections, we shall only consider Z2as the IGG, generated by −τ0. Analogous to the bosonic case, we deﬁne the PSG as the set of all transformations (symmetry transformationsfollowed by gauge transformations) that leave the Ansatz U rr/primeinvariant (this will also leave the al 0s invariant as these are self-consistently determined by the Urr/primes). Pure gauge ﬂuctuations, corresponding to the identity element in thephysical symmetry group, make up the IGG. Hence operatorsin the symmetry group that combine to the identity in thephysical group can only be ±τ 0∈IGG in the projective representation. Similar to the bosonic case, this element η= ±Iwill determine the symmetry fractionalization quantum number for the corresponding series of operations. 3. Solutions to the algebraic PSG Algebraic relations between the symmetry group [ 4]e l - ements will lead to a series of conditions for the gaugetransformations G X[r], which are now SU(2) valued. The general solutions (without referring to any Ansatz )a r eg i v e n below in terms of Z2valued variables {η}, and derived in Appendix D: GTx(x,y)=τ0, (27a) GTy(x,y)=(ηTxTy)xτ0, (27b) GPx(x,y)=(ηPxTx)x(ηPxTy)ygPx, gPx∈SU(2),g2 PxηPxτ0, (27c) GPy(x,y)=(ηPyTx)x(ηPyTy)ygPy, gPy∈SU(2),g2 Py=ηPyτ0, (27d) GT(x,y)=(ηTTx)x(ηTTy)ygT, gT∈SU(2),g2 T=ηTτ0, (27e) where the SU(2) matrices are bound by the following con- straints: gPxgTg−1 Pxg−1 T=ηTPxτ0,g PygTg−1 Pyg−1 T=ηTPyτ0, gPxgPyg−1 Pxg−1 Py=ηPxPyτ0. (28) E. Fusion rules We provide a table for trivial and nontrivial fusion rules for Z2spin liquids on the rectangular lattice with time-reversal 024502-5CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) symmetry T, and provide proofs/arguments in Appendix E. Commutation relation Fusion rule T−1 xT−1 yTxTy trivial P2 x nontrivial P2 y nontrivial P−1 xTxPxTx trival P−1 xT−1 yPxTy trival P−1 yT−1 xPyTx trival P−1 yTyPyTy trival P−1 xP−1 yPxPy nontrival T2trival T−1 xT−1TxT trival T−1 yT−1TyT trival P−1 xT−1PxT nontrival P−1 yT−1PyT nontrival(29) F. Fermionic Ansatz 1. General relation between bosonic and fermionic PSGs for rectangular lattice In Table II, we use the anyon fusion rules to relate bosonic symmetry fractionalization quantum number σe Owith the fermionic one σ/epsilon1 OforZ2spin liquids. These are related as follows: σ/epsilon1 O=σt Oσe Oσm O, (30) where we have used the knowledge of the vison quantum number σm O, and the twist factor σt Owhich is −1 for nontrivial fusion rules and +1 otherwise. 2. Speciﬁc fermionic Ansatz Plugging in the values of {pi}for the bosonic Ansatz in Table II, we can ﬁnd the desired values of ηXYsf o rt h e fermionic Ansatz . Doing so and solving the matrix equations (details in Appendix F), we ﬁnd the following solutions for the GXs: GTx(x,y)=τ0, (31a) GTy(x,y)=(−1)xτ0, (31b) GPx(x,y)=τ0, (31c) GPy(x,y)=(−1)x+yiτ3, (31d) GT(x,y)=iτ2. (31e) Now we solve for the allowed nearest-neighbor (NN), next-NN (NNN), and NNNN bonds demanding GXX(Urr/prime)=Urr/primeforTABLE II. Correspondence between bosonic and fermionic Z2 spin liquids on a rectangular lattice with time-reversal symmetry T. Commutation relation σe O σ/epsilon1 O σm Oσt O Relation T−1 xT−1 yTxTy (−1)p1ηTxTy−11 ( −1)p1+1=ηTxTy P−1 xTxPxTx (−1)p2ηPxTy 11 ( −1)p2=ηPxTx P−1 yT−1 xPyTx (−1)p3ηPyTx−11 ( −1)p3+1=ηPyTx P−1 xT−1 yPxTy (−1)p4ηPxTy−11 ( −1)p4+1=ηPxTy P−1 yTyPyTy (−1)p5ηPyTy 11 ( −1)p5=ηPyTy P2 x (−1)p6ηPx 1−1( −1)p6+1=ηPx P2 y (−1)p7ηPy 1−1( −1)p7+1=ηPy P−1 xP−1 yPxPy 1 ηPxPy−1−11 =ηPxPy T2−1 −11 1 1 =1 T−1 xT−1TxT (−1)p8ηTTx 11 ( −1)p8=ηTTx T−1 yT−1TyT (−1)p9ηTTy 11 ( −1)p9=ηTTy P−1 xT−1PxT (−1)p6ηTPx 1−1(−1)p6+1=ηTPx P−1 yT−1PyT (−1)p7ηTPy 1−1(−1)p7+1=ηTPy each bond. The solution is an Ansatz withπﬂux through elementary plaquettes, with real pairing on the NN and NNNbonds, and real hopping on the NNNN bonds: U r,r+ˆx=(−1)y/Delta11xτ1, (32a) Ur,r+ˆy=/Delta11yτ1, (32b) Ur,r+ˆx+ˆy=Ur,r−ˆx+ˆy=(−1)y/Delta12τ1, (32c) Ur,r+2ˆx=−t2xτ3, (32d) Ur,r+2ˆy=−t2yτ3. (32e) We note that this PSG also allows for an on-site chemical potential of the form a3 0τ3, so that the density of fermions can be adjusted. An alternate derivation of the PSG of thisfermionic Ansatz , based on mapping of projected mean-ﬁeld wave functions, is presented in Appendix Gand serves as a consistency check for our results. We can diagonalize the mean-ﬁeld Hamiltonian corre- sponding to this using a two-site unit cell in the ydirection. LetAandBbe the sublattice indices for yeven and odd respectively, and the reduced Brillouin zone (BZ be givenby−π<k x/lessorequalslantπ,−π/2<ky/lessorequalslantπ/2. Since the up-spin and down-spin sectors decouple, we get a pair of degenerate bands.The Hamiltonian can be written in terms of a four-component Nambu-spinor /Psi1 kasH=/summationtext k∈BZ/Psi1† kh(k)/Psi1k, where /Psi1k=⎛ ⎜⎜⎝fkA↑ fkB↑ f† −kA↓ f† −kB↓⎞ ⎟⎟⎠, andh(k)i st h e4 ×4 matrix given below in terms of ε2k=− 2t2xcos(2kx)−2t2ycos(2ky), ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ε 2k 02 /Delta11xcos(kx)2 /Delta11ycos(ky) +4i/Delta12cos(kx)sin(ky) 0 ε2k 2/Delta11ycos(ky) −2/Delta11xcos(kx) −4i/Delta12cos(kx)sin(ky) 2/Delta11xcos(kx)2 /Delta11ycos(ky) −ε2k 0 +4i/Delta12cos(kx)sin(ky) 2/Delta11ycos(ky) −2/Delta11xcos(kx)0 −ε2k −4i/Delta12cos(kx)sin(ky)⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠. (33) 024502-6SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) Diagonalizing this matrix gives us the spinon dispersion, with two doubly degenerate bands, E± k=±/radicalBig [2t2xcos(2kx)+2t2ycos(2ky)]2+4/bracketleftbig /Delta12 1xcos2(kx)+/Delta12 1ycos2(ky)/bracketrightbig +16/Delta12 2cos2(kx)sin2(ky). (34) Both these bands are fully gapped, with the mininum gap occurring at ( kx,ky)=(±π/2,±π/2) for /Delta11x,/Delta11y/greatermuch/Delta12/greatermuch t2x,t2y.E+ kfor typical parameter values is plotted in Fig. 2. Previous PSG studies have investigated fermionic spin liquids with space group symmetries of the square [ 3,44], triangular, and kagome [ 43,45] lattices, whereas we focus on the rectangular lattice. Reference [ 47] discusses projected mean-ﬁeld wave functions of nematic spin liquids on thesquare lattice and their corresponding fermionic versions,but our initial bosonic state does not correspond to any ofthese states (as one can check by calculating ﬂuxes throughtriangular plaquettes). We discuss the connection of theirresults with our work in greater detail in Appendix G. III. SUPERCONDUCTING TRANSITION OF THE FL* So far, we have described the fermionic spinon excitations of the Z2spin liquid. These correspond to states in the /epsilon1 sector of Table I.T h eZ2FL* state has in addition fermionic electronlike gauge-neutral excitations which belong the 1 c sector of Table I. These can be described by some convenient dispersion for electronlike operators ckσ. In the recent analysis of Ref. [ 21], theckσstates were built out of electron orbitals which were centered on the bonds of the square lattice; on theother hand in Ref. [ 17], thec kσwere obtained from electronlike states on the sites of the square lattice. The details of thedispersion and Fermi surface structure of the c kσquasiparticles of the Z2-FL* will not be important here, and so we simply assume they are characterized by some generic dispersion ξk, and can be Fourier transformed to operators crσon the sites of the square lattice. Furthermore, the crσ, being gauge neutral, must have a trivial PSG. Now we are interested in undergoing a conﬁnement transition in which a boson, B,f r o mt h e /epsilon1csector of Table I condenses. Such a boson is obtained by the fusion of the /epsilon1and FIG. 2. Mean-ﬁeld dispersion E+(k) of the fermionic spinons for the parameters ( /Delta11x,/Delta11y,/Delta12,t2x,t2y)=(0.9,1,0.4,0.2,0.2). The other band is not shown for clarity.1cstates of Table I. So we introduce two Bose operators on the sites of the square lattice transforming as B1r∼c† rσfrσ,B 2r∼/epsilon1σσ/primecrσfrσ/prime. (35) Each of these bosonic operators carry a Z2gauge charge of theffermions, and a U(1) charge corresponding to the c fermions. We can then write down an effective Hamiltonianfor the interplay between the /epsilon1,1 c, and/epsilon1csectors of Table I: H=Hc+HMF f−JK 4/summationdisplay r,r/primeB† 1rc† rσfrσ +B† 2r/epsilon1σσ/primecrσfrσ/prime+H.c,where (36) Hc=/summationdisplay k,σξkc† kσckσ,and HMF f=/summationdisplay rr/prime,σχrr/primef† rσfr/primeσ+/summationdisplay rr/prime,αβ/Delta1f rr/prime/epsilon1αβf† rαf† r/primeβ+H.c., where JKis the allowed “Kondo” coupling linking the sectors of Z2FL* together. A large Napproach, based on generalization of SU(2) to SU( N) yields only the term involving B1r[48,49], but we consider a more simplistic mean-ﬁeld approach where both bosons are present. At thetransition, both these bosons condense together [ 13], and this leads to conﬁnement. In the mean-ﬁeld approximation, wereplace B ir=/angbracketleftBir/angbracketrightwhich is nonzero in the conﬁned phase. The conﬁnement transition out of this FL* state leads to asuperconducting state [ 13], because a pairing between the spinons finduces a pairing between the physical cfermions when/angbracketleftB ir/angbracketright/negationslash=0. Further suppression of this superconductivity (by doping/magnetic ﬁeld) will lead to a normal Fermi liquidstate. Since the spin liquid Ansatz breaks lattice symmetries, the conﬁned states can also exhibit a density wave order.In the following subsection, we ﬁrst detail the possiblesuperconducting phases and describe how we obtain them froman effective bosonic Hamiltonian. Possible conﬁned phases On transition out of the FL, we typically ﬁnd that the superconducting phase is of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type [ 50,51]. This is a superconductor with fermion pairing only at ﬁnite momentum Q, i.e., with spatial modulation of the order parameter /Delta1 c(r)∼eiQ·r. It has also been referred to in the literature as a pair density wave(PDW) state [ 52–57]. A PDW is distinct from a state with coexisting superconductivity and charge density wave (CDW)order. In particular, the superconducting order parameter hasno uniform component, i.e., /Delta1 Q=0=0; the Cooper pairs always carry a net momentum Q. In principle we can also have translation symmetry breaking in the particle-hole channel, leading to a generalized chargedensity wave order, often leading to oscillations of chargedensity on the bonds (a bond density wave). FollowingRef. [ 58], let us deﬁne a generalized density wave order 024502-7CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) parameter PQl(k)a s /angbracketleftc† rσcr/primeσ/angbracketright=/summationdisplay Ql/parenleftbigg/integraldisplayd2k 4π2PQl(k)eik·(r−r/prime)/parenrightbigg eiQl·(r+r/prime)/2.(37) When PQl(k) is independent of k, then the order parameter refers to on-site charge density oscillations at momentum Ql. When PQl(k) depends on k, then it denotes charge density oscillations on the bonds, which is also often called a bonddensity wave [ 58]. Note that a PDW at momentum Qtypically leads to a CDW at momentum K=2Q[52]. This can be seen from a Landau- Ginzburg effective Hamiltonian, where a linear term in theCDW order parameter P 2Qof the form of γ/Delta1(/Delta1∗ Q/Delta1−QP2Q+ c.c.) is allowed by symmetry. Therefore, in the phase where/Delta1 Qis condensed, the system can always lower its energy by choosing a nonzero value of P2Q. Explicit computations later will show that boson condensation at ﬁnite momenta can leadto density wave states which have momenta different from2Q PDW. These are therefore states where a PDW coexists along with additional density wave order(s). To ﬁgure out the details of this transition at the level of mean-ﬁeld theory, we ﬁrst write down an effective Hamiltonianfor the bosons H B. This is determined by the PSG of the ffermions, as described in Eq. ( 31). Once we write down the effective Hamiltonian based on the PSG, we can ﬁnd theminima of the boson dispersion at a set of momenta {Q i}, at which the boson will condense on tuning to the phase transition. Across the transition, we can replace Birby the value of the condensate. The spinon pairing /Delta1f rr/primeinduces a pairing /Delta1c rr/primebetween the cfermions, which is given in terms of the boson condensate by (perturbatively, to lowest nonzeroorder in B ir): /epsilon1αβ/Delta1c rr/prime=/angbracketleft/epsilon1αβcrαcr/primeβ/angbracketright∼(B1rB1r/prime+B2rB2r/prime)/angbracketleft/epsilon1αβfrαfr/primeβ/angbracketright =(B1rB1r/prime+B2rB2r/prime)/epsilon1αβ/Delta1f rr/prime. (38) We also want to study if there is some density wave order, present on top of superconductivity or a PDW state. Therefore,in the conﬁned phase we evaluate the order parameter P Q(k) by noting that /angbracketleftc† rσcr/primeσ/angbracketright∼(B∗ 1rB1r/prime+B∗ 2rB2r/prime)/angbracketleftf† rσfr/primeσ/angbracketright. (39) Since each boson is a spin-singlet bound state of the cand fspinon, it has the same spatial symmetry fractionalization quantum numbers as the ffermions. Time reversal Tinter- changes B1randB2rbecause of extra gauge transformation Gτ associated with the fspinon. To deal with both bosons in a compact way, let us deﬁne a two-component spinor as follows: Br=/parenleftbigg B1r B2r/parenrightbigg . (40) The action of the symmetry operations on Bris derived in Appendix H; here we just state the main results. Under any spatial symmetry operation Xs, this column vector just picks up an overall U(1) phase, because the gauge transformationsG Xsfor the ffermions are all diagonal, GXsXs[Br]=eiφXs[Xs(r)]BXs[r],with φTx=0, φTy=πx, φ Px=0,φ Py=π/parenleftbig x+y+1 2/parenrightbig .(41)However, time reversal Tmixes the up and down spinon opera- tors, and imposes extra constraints. We demand GXX(HB)= HBfor all symmetry operations X. Based on this, we can write down an effective Hamiltonian for the bosons as followsconsistent with the PSG. For simplicity, we include only a2×2 hopping matrix T rr/primeup to next-next-nearest neighbors (we neglect pairing of bosons). We ﬁnd that Hb=/summationdisplay rr/primeB† rTrr/primeBr/prime+H.c.,where Trr/prime=Td rr/primeτ0+Tod rr/primeτ1, (42) where TdandTodare the diagonal ( B1→B1orB2→B2) and off-diagonal ( B1↔B2) hopping elements, as described in Appendix H. The diagonal hopping amplitudes are given by Td r,r+ˆx=0,Td r,r+ˆy=iTd y, Td r,r+ˆx+ˆy=Td r,r−ˆx+ˆy=iTd x+y(−1)y, (43) Td r,r+2ˆx=Td 2x,Td r,r+2ˆy=Td 2y, where all the Td αare real. The off-diagonal hopping is also exactly analogous, as the projective U(1) phases for both the B1 andB2bosons are identical. However the overall coefﬁcients Tod αare not ﬁxed by the PSG and generically different from Td α. For simplicity, we ﬁrst set the off-diagonal components Tod α to zero by hand, which implies that we need to study only one boson—let us call that Br. We shall later argue that the resulting superconducting phases are essentially unchanged when oneincludes the off-diagonal components as well. Translationalsymmetry breaking in this gauge choice leads to an enlargedtwo-site unit cell in the ˆydirection. Letting A,B be the sublattice indices (for even/odd y), we deﬁne the Fourier transformed operators as B rα=1√Nc/summationdisplay keik·rαBkα,α=A,B, (44) where Ncis the number of unit cells, and −π<k x/lessorequalslant π,−π/2<ky/lessorequalslantπ/2 deﬁnes the reduced BZ. Let us deﬁne /Psi1† k=(B† kA,B† kB), then we can write HB=/Psi1† khB(k)/Psi1k, where hB(k)=/parenleftbigg ε(k)ξ(k) ξ∗(k)ε(k)/parenrightbigg , ε(k)=T2xcos(2kx)+T2ycos(2ky), (45) ξ(k)=− 2Tysin(ky)+4iTx+ycos(kx)cos(ky). The two bands are therefore given by E±(k)=ε(k)±\|ξ(k)\|=T2xcos(2kx)+T2ycos(2ky) ±2/radicalBig T2ysin2(ky)+4T2 x+ycos2(kx)cos2(ky). (46) In general, the minima of E−(k), which corresponds to the momentum at which the boson condenses, will lie at someincommensurate point. In Fig. 3, we present an approximate phase diagram and look in more details into the different kindsof superconducting phases obtained by condensing the boson.All but one of these phases break time-reversal symmetry T. 024502-8SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) Uniform SC (2) Incommensurate PDW (4) Commensurate PDW (3) 0 2 4 6 8 10 120246810 TyTxy FIG. 3. Phases of the superconductor; phase boundaries are approximate. T2x,T2yare assumed small but nonzero. The number in brackets denotes the subsection in which the phase is discussed.The red dot denotes phase (1), a PDW state with unbroken T.T h e phases are described in detail in the main text. 1.T-invariant PDW First, consider the case where we turn off the imaginary hopping terms, i.e., Ty=Tx+y=0. In this case, the boson hoppings are translationally invariant, and the minima corre-sponds to Q=(0,0). Let the boson condensate at Q=(0,0) be B(r)=B o, we ﬁnd that the nearest-neighbor cfermion pairing amplitude is given by /Delta1c r,r+ˆx=B2 o(−1)y/Delta11x, /Delta1c r,r+ˆy=B2 o/Delta11y. (47) The superconducting phase breaks translation symmetry, therefore we have a PDW state with QPDW=(0,π). Since the bosons condense at zero momentum, the density wave orderparameter can only pick up a nonzero expectation value if the f spinon hoppings themselves break translation symmetry. Thisis not the case for our fermionic Ansatz [described by Eq. ( 32)], and therefore we expect no density wave order in this phase.In fact, one can perturbatively evaluate the renormalizationsof the cfermion hoppings (over and above the ones which are present in H c) as follows: /angbracketleftc† rcr+2ˆx/angbracketright=B2 o(−t2x), /angbracketleftc† rcr+2ˆy/angbracketright=B2 o(−t2y). (48) These are both translation invariant. 2. Translationally invariant SC with broken T Q=(0,0) is also the position of the minima when Ty< Tx+y. However, any nonzero Tx+ywill enlarge the unit cell. The value of the boson condensate is therefore given by B(r)=/parenleftbigg BA(r) BB(r)/parenrightbigg =Bo/parenleftbigg 1 i/parenrightbigg . (49)From the boson condensate at Q=(0,0), we ﬁnd that the nearest-neighbor cfermion pairing amplitude is given by /Delta1c r,r+ˆx=B2 o(−1)y/Delta11x,forr∈A, /Delta1c r,r+ˆx=(iBo)2(−1)y/Delta11x=−B2 o(−1)y/Delta11x,forr∈B,and /Delta1c r,r+ˆy=iB2 o/Delta11y. (50) Noting that the A/B sublattices are deﬁned by even/odd y coordinates, this implies that /Delta1c r,r+ˆx=B2 o/Delta11x. Thus, this su- perconductor does not break translation symmetry. However,it will break necessarily time-reversal symmetry because thereis a relative ibetween the pairing amplitudes along ˆxand ˆy, and the pairing is of the s+id x2−y2type. This state does not have an associated density wave order. Depending on the relative signs of the hoppings, a conden- sate at Q=(π,0) is also possible, and gives a superconducting state with identical features. 3. Commensurate PDW with broken T Next, let us consider the case where the nearest-neighbor hopping dominates, i.e., Ty/greatermuchTx+y,T2x,T2y. In this case, there is a regime where the minima of the boson dispersion liesapproximately at ±Q=(0,±π/2). The boson condensate is given by B(r)=/parenleftbigg B A(r) BB(r)/parenrightbigg =B+/parenleftbigg eiQ·rA eiQ·rB/parenrightbigg +B−/parenleftbigg e−iQ·rA e−iQ·rB/parenrightbigg =B+/parenleftbigg 1 i/parenrightbigg eiQ·rA+B−/parenleftbigg 1 −i/parenrightbigg e−iQ·rA. (51) Using the previously outlined procedure to calculate the superconducting order parameter, we ﬁnd /Delta1c r,r+ˆx=[(B2 ++B2 −)+(−1)y2B+B−]/Delta11x, /Delta1c r,r+ˆy=i(B2 +−B2 −)(−1)y/Delta11y. (52) Both translation symmetry and time-reversal symmetry are explicitly broken by the superconductor, and we have a PDWatQ PDW=(0,π) with s+idx2−y2pairing. Analogous to the ﬁrst PDW phase with unbroken T,w e can evaluate the renormalization of the cfermion hopping amplitudes (suppressing spin indices for simplicity): /angbracketleftc† rcr+2ˆx/angbracketright=[\|B+\|2+\|B−\|2+(−1)y(B+B∗ −+B−B∗ +)](−t2x), /angbracketleftc† rcr+2ˆy/angbracketright=[(\|B+\|2+\|B−\|2)(−1)y +(B+B∗ −+B−B∗ +)](−t2y). (53) The spatially constant parts of the induced hopping amplitudes will just renormalize the bare hopping of the cfermions, but the terms at QCDW=QPDW=(0,π) correspond to a density wave with form factor PQCDW(k)=c1cos(2kx)+c2cos(2ky), w h i c hi so ft h e s/prime+dtype. This is therefore an example of a state where PDW coexists with bond density wave order. 4. Incommensurate PDW with broken T Away from the previous two parameter regimes, the boson b(r) will condense at some generic incommensurate momentum Q=(Qx,Qy). One can carry out an analogous calculation to ﬁnd out the relevant order parameters. Note 024502-9CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) that the boson dispersion is symmetric under k→− k, which implies that there are necessarily a couple of minima at Q and−Q. Assuming no other degenerate minima, the boson condensate is given by B(r)=/parenleftbigg BA+eiQ·rA BB+eiQ·rB/parenrightbigg +/parenleftbigg BA−eiQ·rA BB−eiQ·rB/parenrightbigg . (54) This leads to a PDW at momentum 2 Q+(0,π)a sw e l la s (0,π)f o rt h e cfermions, the latter coming from the inherent translation symmetry breaking of the spinon pairing Ansatz : /Delta1c r,r+ˆx=/bracketleftbig B2 A+ei(2Q·r+Qx)+4BA+BA−cos(Qx) +B2 A−e−i(2Q·r+Qx)/bracketrightbig (−1)y/Delta11x,r∈A =/bracketleftbig B2 B+ei(2Q·r+Qx)+4BB+BB−cos(Qx) +B2 B−e−i(2Q·r+Qx)/bracketrightbig (−1)y/Delta11x,r∈B, /Delta1c r,r+ˆy=[BA+BB+ei(2Q·r+Qy)+BA−BB+eiQy +BA+BB−e−iQy+BA−BB−e−i(2Q·r+Qy)]/Delta11y. (55) An analogous calculation of the density wave order parameter shows that there is an oscillation of charge density on the bondsat momenta QCDW=2Q, /angbracketleftc† rcr+2ˆx/angbracketright∼B2 A/Be2iQ·r(−t2x),r∈A/B, /angbracketleftc† rcr+2ˆy/angbracketright∼B2 A/Be2iQ·r(−t2y),r∈A/B. (56) Therefore, we have an incommensurate PDW coexisting with bond density wave. More generally, boson condensation at two different mo- menta QandQ/primewill lead to a PDW order at KPDW=Q+Q/prime+ (0,π) and (0 ,π), and a bond density wave order at momenta KCDW=Q±Q/primefor our fermionic Ansatz . These are all states with coexisting PDW and density wave order. Note that adensity wave at a different momentum Q DW=Q+Q/prime+K1 is also possible if there is a spinon-hopping term which breaks translation symmetry with momentum K1. In our fermionic Ansatz for the fspin liquid, such a term is absent (up to NNNN) and therefore such a density wave does not exist. We now argue that inclusion of Tod αdoes not change these phases, although it enlarges the phase space and therefore canchange where these show up in the phase space. This canbe explicitly seen from the eigenvalues of the 4 ×4m a t r i x h(k) in momentum space, which are now given by (assuming T d/od 2x=Td/od 2y=Td/od 2 to avoid clutter of notation) E+ k,±=2[cos(2 kx)+cos(2ky)]/parenleftbig Td 2−Tod 2/parenrightbig ±2/radicalBig/parenleftbig Tdy−Tody/parenrightbig2sin2(ky)+4/parenleftbig Td x+y−Tod x+y/parenrightbig2cos2(kx)cos2(ky), E− k,±=2[cos(2 kx)+cos(2ky)]/parenleftbig Td 2+Tod 2/parenrightbig ±2/radicalBig/parenleftbig Tdy+Tody/parenrightbig2sin2(ky)+4/parenleftbig Td x+y+Tod x+y/parenrightbig2cos2(kx)cos2(ky). (57) These are essentially identical to the previous dispersion in Eq. ( 46), with a renormalization of hopping parameters. Therefore, condensates again occur at the same values of Qas described previously, and lead to the same phases. IV . CONCLUSIONS While several recent experiments [ 23,24] have been con- sistent with a FL model for the pseudogap metal at highertemperatures, the most recent Hall effect measurements [ 25] indicate that the FL* model may well extend down to lowtemperatures just below optimal doping. In the light of this, it is useful to catalog the conﬁnement instabilities of the simplest FL* state, the Z 2-FL. The excitations of this state invariably transform nontrivially underglobal symmetries of the model, and so the conﬁnementtransition is then simultaneous with some pattern of symmetry breaking. From Table I, we observe that the Z 2-FL state has three categories of bosonic excitations, and each can then giverise to a distinct conﬁnement transition. The most familiar isthe condensation of the bosonic spinons (column ein Table I), and this leads to spin-density wave order, which is observedin most cuprates at low doping. The second possibility isthe condensation of visons (column min Table I): this was examined recently [ 27], and it was found that bond-density waves similar to recent observations [ 59–61] are a possible outcome. The ﬁnal class of conﬁnement transitions out theZ 2-FL* state was considered in the present paper: this is the condensation of bosonic chargons (column /epsilon1cin Table I).Our main technical challenge in this paper was to compute the projective symmetry group of the fermionic spinons (col-umn/epsilon1in Table I) for a favorable Z 2spin liquid state described by an Ansatz for bosonic spinons [ 1,9,10]. An important feature of the PSG for the fermionic spinons obtained was thattranslational symmetry was realized projectively, with T xTy= −TyTx. After obtaining this PSG, we could then deduce the PSG for the bosonic chargons by fusing the fermionic spinonsto the electron, which has a trivial PSG. The PSG for thebosonic chargons also had T xTy=−TyTx, and this almost always means that the conﬁnement state with condensedchargons will break translational symmetry. Combined withthe pairing of fermionic spinons invariably present in theZ 2-FL* state, such analyses led to the appearance of FFLO, or pair density wave (PDW), superconductivity. And it isworthwhile to note here the recent observation of modulatedsuperconductivity, albeit on a much larger background ofuniform superconductivity [ 62]. In conclusion, we highlight the remarkable fact that the three categories of conﬁnement transitions out of Z 2-FL* allowed by Table I(corresponding to the three columns with bosonic self-statistics) correspond closely to features of thephase diagrams of the cuprates: (i) the condensation of mcan lead to metals with density wave order similar to observations,as discussed recently in Ref. [ 27]; (ii) the condensation of e leads to incommensurate magnetic order found at low doping;(iii) the present paper showed show the condensation of /epsilon1 ccan lead to superconductors with coexisting density wave order, astate observed in recent experiments [ 62]. 024502-10SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) ACKNOWLEDGMENTS We acknowledge helpful conversations with Y . M. Lu and F. Wang. This research was supported by the NSFunder Grant No. DMR-1360789. J.S. was supported by theNational Science Foundation Graduate Research Fellowshipunder Grant No. DGE1144152. Research at Perimeter Institutewas supported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Ministryof Research and Innovation. APPENDIX A: DERIV ATION OF THE BOSONIC PSG To derive the solution, we note a few things. First, if we apply a gauge transformation Gto the Ansatz , then the gauge transformed Ansatz is invariant under GGXXG−1=GGXXG−1X−1X⇒GX→GGXXG−1X−1. (A1) This implies that the phase φXunder a gauge transformation transforms as (except when Xis the antiunitary time-reversal operator) φX(r)→φG(r)+φX(r)−φG[X−1(r)]. (A2) Since we can choose a particular gauge to work in, we shall use this to later simplify our PSG classiﬁcation. Let us ﬁnd the constraints imposed by the structure of the rectangular lattice symmetry group. Consider a stringof space group operators which combine to identity inthe lattice symmetry group. Then in the PSG, these mustcombine to an element of the IGG Z 2, which means it is±1. Therefore, for each such string, we shall deﬁne an integer pn(deﬁned modulo 2) which will denote how thesymmetry fractionalizes in the PSG. It is sufﬁcient to consider the strings in Eqs. ( 4a), because any other string can be reduced to one such string by normal ordering the strings using the same commutation/anticommutation relations. Wecan then use these constraints to ﬁnd the gauge operationsG X, or equivalently their phases φX(r), in terms of the pns. Note that all the following equations for the phases are truemodulo 2 π. For notational convenience, we also introduce discrete lattice derivatives /Delta1 xφX=φX(x+1,y)−φX(x,y) and/Delta1yφX=φX(x,y+1)−φX(x,y). Let us start by looking at the commutation relation between the translations. We have, from Eq. ( 4), /parenleftbig GTxTx/parenrightbig−1/parenleftbig GTyTy/parenrightbig/parenleftbig GTxTx/parenrightbig/parenleftbig GTYTy/parenrightbig−1 =/parenleftbig T−1 xGTxTx/parenrightbig/parenleftbig T−1 xGTyTx/parenrightbig/parenleftbig T−1 xTyGTxT−1 yTx/parenrightbig/parenleftbig G−1 Ty/parenrightbig =± 1=(−1)p1. (A3) Since Y−1GXY:φX(r)→φX[Y(r)], we have the following constraint equation for φTxandφTy: −φTx[Tx(x,y)]+φTy[Tx(x,y)] +φTx/bracketleftbig T−1 yTx(x,y)/bracketrightbig −φTy(x,y)=p1π. (A4) Now we assume we are deﬁning the system on open boundary conditions, so that we can use the gauge freedom in Eq. ( A2) to setφTx(x,y)=0. We also assume, following Ref. [ 40], that we can set φTy(0,y)=0. Then we can write down the solution as /Delta1xφTy(x,y)=p1π⇒φTy(x,y)=p1πx+φTy(0,y)=p1πx. (A5) Now we consider Pxand its commutations with TxandTy. FromGTxTxP−1 xG−1 PxGTxTxGPxPx=± 1=(−1)p2, we get φPx(x,y)−φPx[TxPx(x,y)]+φTx[Px(x,y)]+φPx[Px(x,y)]=p2π⇒/Delta1xφPx=p2π. FromG−1 TyTyP−1 xG−1 PxGTyTyGPxPx=± 1=(−1)p4, we get −φTy[Ty(x,y)]−φPx[PxTy(x,y)]+φTy[TyPx(x,y)]+φPx[Px(x,y)]=p4π ⇒/Delta1yφPx−p1π(−x)+p1π(−x)=p4π⇒/Delta1yφPx=p4π. (A6) Using the above two equations, we can write down φPx(x,y)=p2πx+p4πy+φPx(0,0). (A7) φPx(0,0) is now found out using ( GPxPx)2=± 1=(−1)p6, which implies 2 φPx(0,0)=p6π, φPx(x,y)=p2πx+p4πy+p6 2π. (A8) In an exactly analogous way, we ﬁnd that φPy(x,y)=p3πx+p5πy+p7 2π. (A9) Finally, let us consider time reversal T.F r o mt h e commutations of TwithTxandTy, we ﬁnd the following twoequations: /Delta1xφT=p8π, /Delta1 yφT=p9π. (A10) Solving the above gives us φT(x,y)=p8πx+p9πy+ φT(0,0). The commutations with PxandPydo not yield any new relation. Finally, we note that under a global gaugetransformation G:b rσ→eiθbrσ, due to the antiunitary nature of T,w eh a v e φT(x,y)→φT(x,y)+2θ. We can use this freedom to set θ=−φT(0,0)/2, and we therefore have φT(x,y)=p8πx+p9πy. (A11) Note that this gauge transformation does not affect the φX corresponding to a spatial symmetry X, as these are unitary and follow Eq. ( A2). 024502-11CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) (a) (b) (c) FIG. 4. (a) The original translation invariant Ansatz ; (b) the Ansatz underPx:(x,y)→(−x,y); (c) the Ansatz underPy:(x,y)→(x,−y). The arrow from rtor/primeindicates the orientation for which Qrr/prime>0. APPENDIX B: PSG CORRESPONDING TO THE NEMATIC BOSONIC ANSATZ The phases φXcorresponding to the symmetry operations Xcan be ﬁxed by demanding that the Ansatz remain invariant under GXX. First, we note that the Ansatz itself is translation invariant (see Fig. 4), so both GTxandGTymust be trivial. This implies that our Ansatz is consistent with our trivial gauge choice for GTx, andp1=0. Let us now consider Px. Using translation invariance, we have Px(Qr,r+ˆx)=Qr+ˆx,r=−Qr,r+ˆx. By deﬁnition, GPxPx(Qr,r+ˆx)=Qr,r+ˆx, and this implies that φPx[Px(r)]+ φPx[Px(r+ˆx)]=π, which in turn gives us p2+p6=1. The nearest-neighbor ybond is unaffected by Px, whereas the diagonal bonds are swapped and effectively not affected asthey have the same value in this Ansatz . We get the following equations from demanding that G Xacts trivially on these bonds: p4+p6=0, and p2+p4+p6=0. Solving these we ﬁnd that p2=0,p4=p6=1 (modulo 2). Similarly, acting Pychanges the sign on all bonds except thexbonds, and we have the following equations: p3+ p7=0,p5+p7=1,andp3+p5+p7=1. Solving gives usp3=p7=0,p5=1. The transformations of the Ansatz under reﬂections are schematically described in Table III. Finally, we look at time reversal. Since all the bond variables are real (which we assume is consistent with our gauge choice),we have p 8=p9=0. APPENDIX C: ALTERNATE DERIV ATION OF THE VISON PSG In this section, we present an alternate derivation of the vison PSG, based on the critical modes of the vison asone approaches vison condensation. We assume a soft spinformulation, which is reasonable from coarse graining near a critical point. We replace the Ising variables τ z Rs in the vison Hamiltonian by real ﬁelds φR∈R, and describe the kinetic term by a conjugate momentum πRtoφRand mass m, so that the Hamiltonian becomes Hsoft=1 2/summationdisplay R/parenleftbig π2 R+m2φ2 R/parenrightbig +/summationdisplay RR/primeJRR/primeφRφR/prime. (C1) In our gauge choice (recall Fig. 1), we have a two-site unit cell with primitive vectors a1=ˆx+ˆyanda2=2ˆy(setting lattice spacings =1). Neglecting the kinetic term (which is inessential to the study of vison condensation transitions), theHamiltonian in the momentum space for this extended unitcell is given by H soft=/summationdisplay kH(k),with H(k)=2/parenleftbigg 0 cos ky+isinkx cosky−isinkx 0/parenrightbigg .(C2) Diagonalizing this leads to the following two bands: ω±(k)=± 2/radicalBig cos2ky+sin2kx. (C3) The inequivalent minima of this band structure lie at Q1,2= ±(π/2,0) in the reduced BZ, and the corresponding eigenvec- tors are v1=(−eiπ/4,1)Tandv2=(−e−iπ/4,1)T, where the superscript Tindicates transposition. Later, we shall write out the vison ﬁeld in terms of these soft modes. Now, we analyze the PSG of the visons. Since the Hamiltonian is invariant under symmetry transformations onlyup to a gauge transformation, we identify, for each symmetrygenerator Xin the space group of the rectangular lattice, an ele- mentG X∈Z2such that Jrr/prime=JX[r]X[r/prime]GX[X(r)]GX[X(r/prime)]. These symmetry operations for the rectangular lattice, and TABLE III. Transformation of link variables Qrr/prime. Px Py Q(x,y)→(x+1,y)→Q(x+1,y)→(x,y)=−Q(x,y)→(x+1,y) Q(x,y)→(x+1,y)→Q(x,y+1)→(x+1,y+1)=Q(x,y)→(x+1,y) Q(x,y)→(x,y+1)→Q(x+1,y)→(x+1,y+1)=Q(x,y)→(x,y+1) Q(x,y)→(x,y+1)→Q(x,y+1)→(x,y)=−Q(x,y)→(x,y+1) Q(x,y)→(x+1,y+1)→Q(x+1,y)→(x,y+1)=Q(x,y)→(x+1,y+1) Q(x,y)→(x+1,y+1)→Q(x,y+1)→(x+1,y)=−Q(x,y)→(x+1,y+1) Q(x+1,y)→(x,y+1)→Q(x,y)→(x+1,y+1)=Q(x+1,y)→(x,y+1) Q(x+1,y)→(x,y+1)→Q(x+1,y+1)→(x,y)=−Q(x+1,y)→(x,y+1) 024502-12SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) their associated gauge transformations are listed below. We denote sublattice s=(1,2) at the unit cell r=ma1+na2by (m,n)s. Tx:/braceleftbigg (m,n)1→(m+1,n−1)2 (m,n)2→(m+1,n)1, Ty:/braceleftbigg (m,n)1→(m,n)2 (m,n)2→(m,n+1)1, Px:/braceleftbigg (m,n)1→(−m−1,m+n)2 (m,n)2→(−m−1,m+n+1)1, Py:/braceleftbigg (m,n)1→(m,−n−1)2 (m,n)2→(m,−n−1)1. (C4) The associated gauge transformations can be found out by ﬁguring out appropriate gauge transformations to leave theHamiltonian invariant. As discussed in the main text, alloperations except P xexchange the xbonds with different signs, and hence need a gauge transformation which adds anextra sign to bring the Hamiltonian back to itself. The ybonds are invariant under any of these operations, G Tx(m,n)s=(−1)m, GTy(m,n)s=(−1)m, GPx(m,n)s=1, GPy(m,n)s=(−1)m. (C5) Next, we outline to ﬁnd the general procedure to ﬁnd the representation of the PSG in the order parameter space, andsubsequently apply it to our situation. We ﬁrst deﬁne the orderparameter by expanding the vison ﬁeld in terms of the Nsoft modes as follows: φ s(R)=N/summationdisplay n=1ψnvn seiqn·R. (C6) Here, Ris the unit cell index, s=(1,2) is the sublattice index, Nis the number of soft modes, and the complex number ψn is the vison order parameter corresponding to the nth soft mode at momentum qnwith eigenvector vnofHsoft.N o w ,w e can ﬁgure out how the order parameters ψntransform into each other under different symmetry operations GXXwhich leave the Hamiltonian Hsoftinvariant. This can be found from solving the following equation, which gives us the desiredrepresentation in form of the N×Nmatrix O Xdeﬁned below [with ( R/prime,s/prime)=X(r,s)]: GXX[φs(R)]=N/summationdisplay n=1ψnvn s/primeeiqn·R/primeGX[R/prime,s/prime] =N/summationdisplay n=1ψ/prime nvn seiqn·R =N/summationdisplay n=1/parenleftBiggN/summationdisplay m=1OX,mnψn/parenrightBigg vn seiqn·R.(C7) With nearest-neighbor interactions of the soft spins in the fully frustrated dual Ising model, we earlier found that thereare two minima at Q 1,2=±Q=(±π/2,0) with associatedeigenvectors v1andv2. Since the order parameter φis real, we can write it (in form of a vector with two sublattice indices) /parenleftbigg φ1 φ2/parenrightbigg =ψ/parenleftbigg −eiπ/4 1/parenrightbigg eiQ·R+ψ∗/parenleftbigg −e−iπ/4 1/parenrightbigg e−iQ·R.(C8) We work out the results for Txexplicitly, and just quote the other ones. All of these can be obtained by following thegeneral procedure outlined above. For r=(m,n), we have Q·R=πm/ 2, so we get φ 1(R)=−ψeiπ/4eiπm/ 2−ψ∗e−iπ/4e−iπm/ 2 ⇒GTxTx[φ1(R)]=[ψ(1)eiπ/2(m+1)+ψ∗(1)e−iπ/2(m+1)](−1)m =ψeiπ/2e−iπm/ 2+ψ∗e−iπ/2eiπm/ 2 =−ψ/primeeiπ/4eiπm/ 2−ψ/prime∗e−iπ/4e−iπm/ 2. (C9) Since the above is true for all m,w eh a v e ψ/prime=−ψ∗e−i3π/4= eiπ/4ψ∗. Therefore, in the matrix form, we can write /parenleftbigg ψ/prime ψ/prime∗/parenrightbigg =/parenleftbigg 0 eiπ/4 e−iπ/40/parenrightbigg/parenleftbigg ψ ψ∗/parenrightbigg . (C10) Thus the matrix representation of OTxin the order parameter space (in our chosen gauge) is given by OTx=/parenleftbigg 0 eiπ/4 e−iπ/40/parenrightbigg . (C11) The matrix representations of the other operators are worked out identically; here we just list the results: OTy=/parenleftbigg 0 −e−iπ/4 −eiπ/40/parenrightbigg , (C12) OPx=/parenleftbigg 0 eiπ/4 e−iπ/40/parenrightbigg , (C13) OPy=/parenleftbigg 0 −e−iπ/4 −eiπ/40/parenrightbigg . (C14) The fractionalization of the commutation relations can now be obtained from these matrices: OTxOTyO−1 TxO−1 Ty=− 1, (C15a) OTxOPxOTxO−1 Px=1, (C15b) OTxOPyO−1 TxO−1 Py=− 1, (C15c) OTyOPxO−1 TyO−1 Px=− 1, (C15d) OTyOPyOTyO−1 Py=1, (C15e) OPxOPx=1, (C15f) OPyOPy=1, (C15g) OPxOPyO−1 PxO−1 Py=− 1. (C15h) A more complicated analysis including fourth-nearest- neighbor interactions [ 39] (done on the square lattice, but works for rectangular lattices as well) also leads to matrixrepresentations of the operators with identical crystal symme-try fractionalization. In order to check how the symmetries involving time- reversal fractionalize, we follow Ref. [ 34]. We look at the edge 024502-13CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) modes and require that they are not symmetry protected, or, in other words, we have a gapped boundary. The edge modesof aZ 2spin liquid can always be fermionized with the same number of right and left movers (branch denoted by index n), Ledge,0=/summationdisplay niψ† L,n(∂t−v∂x)ψL,n−iψ† R,n(∂t+v∂x)ψR,n. (C16) In general, we would expect a gapped edge due to backscat- tering terms below, unless these are forbidden by symmetry: Ledge,1=/summationdisplay m,nψ† L,mMm,nψR,n+ψ† L,m/Delta1m,nψR,n+H.c. (C17) The above mass terms correspond to condensing spinons or visons at the edge. Since condensing spin-half spinons wouldbreak SU(2) symmetry, we would need to condense visons toget gapped edges with all symmetries intact. This can onlytake place if the vison PSGs allow a vison condensate at theedge. If the symmetries act nontrivially on the vison ﬁeld φ, then the vison condensate will break the symmetry. Therefore,if we want to preserve the symmetry at the edge with gappededge modes (nonzero mass terms), the symmetries at the edgecannot have a nontrivial action on φ. Consider the square lattice on a cylinder with open boundaries parallel to ˆx. Then the remaining symmetries are T x,Pxand time reversal T. If there are no symmetry-protected gapless edge states on the boundary, then these symmetriesmust act trivially on the visons. Hence, we have O −1 TxO−1 TOTxOT=1,O−1 PxO−1 TOPxOT=1. (C18) We can apply an analogous argument for a cylinder with open boundaries parallel to ˆy, to ﬁnd O−1 TyO−1 TOTyOT=1,O−1 PyO−1 TOPyOT=1. (C19)APPENDIX D: DERIV ATION OF THE FERMIONIC PSG To derive the general solutions to the fermionic PSG, we note that the PSGs of two gauge-transformed Ans¨atze are related (similar to the bosonic case). Recall that the PSG isdeﬁned as the set of all transformations G XXthat leave the Ans¨atze unchanged, GXX(Urr/prime)=GX(UX[r]X[r/prime])=Urr/prime,where GX(Urr/prime)=GX[r]Urr/primeG† X[r/prime]. (D1) Under a local gauge transformation /tildewidestUrr/prime=WrUrr/primeW† r/prime, there- fore GX→/tildewideGX=WrGXW† X(r). (D2) We can use this gauge freedom to choose GTx=τ0.N o w , consider the commutation of TxandTy, /parenleftbig GTxTx/parenrightbig/parenleftbig GTyTy/parenrightbig/parenleftbig GTxTx/parenrightbig−1/parenleftbig GTyTy/parenrightbig−1=ηTxTyτ0 ⇒GTy(r−ˆx)G−1 Ty(r)=ηTxTyτ0.(D3) In an appropriate gauge, we can choose the solution as GTy(x,y)=(ηTxTy)xτ0. This choice of gauge, where both GTx andGTyare proportional τ0, is referred to as the uniform gauge [ 3] as it preserves the translation invariance of SU(2) ﬂux through any loop. Next, consider the commutations of time reversal Twith TxandTy. We ﬁnd that GT(r−ˆx)GT(r)−1=ηTTxτ0, GT(r−ˆy)GT(r)−1=ηTTyτ0. (D4) Hence we can write the solution as GT(x,y)= (ηTTx)x(ηTTy)ygT, where gT∈SU(2). The added constraint G2 T=ηTτ0yields g2 T=ηTτ0. Let us consider the commutations of PxwithTx,Ty, /parenleftbig GPxPx/parenrightbig/parenleftbig GTxTx/parenrightbig/parenleftbig GPxPx/parenrightbig−1/parenleftbig GTxTx/parenrightbig =ηPxTxτ0⇒GPx(r)GPx(r+ˆx)−1=ηPxTxτ0, /parenleftbig GPxPx/parenrightbig/parenleftbig GTyTy/parenrightbig/parenleftbig GPxPx/parenrightbig−1/parenleftbig GTyTy/parenrightbig−1=ηPxTyτ0⇒GPx(r)GPx(r−ˆy)−1=ηPxTyτ0. (D5) T h es o l u t i o ni s GPx(x,y)=(ηPxTx)x(ηPxTy)ygPx, where gPx∈SU(2) satisﬁes g2 Px=ηPxτ0sinceG2 Px=ηPxτ0. Similarly, for Pywe ﬁnd that GPy(x,y)=(ηPyTx)x(ηPyTy)ygPy, where gPy∈SU(2) satisﬁes g2 Py=ηPyτ0sinceG2 Py=ηPyτ0. Finally, we need to look at commutations of PxandPywith time reversal T, and between themselves, /parenleftbig GPxPx/parenrightbig (GTT)/parenleftbig GPxPx/parenrightbig−1(GTT)−1=ηTPxτ0⇒gPxgTg−1 Pxg−1 T=ηTPxτ0, /parenleftbig GPyPy/parenrightbig (GTT)/parenleftbig GPyPy/parenrightbig−1(GTT)−1=ηTPyτ0⇒gPygTg−1 Pyg−1 T=ηTPyτ0, (D6) /parenleftbig GPxPx/parenrightbig/parenleftbig GPyPy/parenrightbig/parenleftbig GPxPx/parenrightbig−1/parenleftbig GPyPy/parenrightbig−1=ηPxPyτ0⇒gPxgPyg−1 Pxg−1 Py=ηPxPyτ0. The full fermionic PSG on a rectangular lattice with time reversal Tis thus given by the following equations, together with the constraints set by Eq. ( D6): GTx(x,y)=τ0, (D7a) GTy(x,y)=/parenleftbig ηTxTy/parenrightbigxτ0, (D7b) GPx(x,y)=/parenleftbig ηPxTx/parenrightbigx/parenleftbig ηPxTy/parenrightbigygPx,g Px∈SU(2),g2 Px=ηPxτ0, (D7c) GPy(x,y)=/parenleftbig ηPyTx/parenrightbigx/parenleftbig ηPyTy/parenrightbigygPy,g Py∈SU(2),g2 Py=ηPyτ0, (D7d) GT(x,y)=/parenleftbig ηTTx/parenrightbigx/parenleftbig ηTTy/parenrightbigygT,g T∈SU(2),g2 T=ηTτ0. (D7e) 024502-14SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) APPENDIX E: TRIVIAL AND NONTRIVIAL FUSION RULES Consider a unitary symmetry operation X2=1 which is realized projectively on the anyons. To detect the symmetryfractionalization corresponding to X, we follow Ref. [ 35]. We actXonce on an excited state containing two anyons, whose positions are swapped by X. The symmetry action on an anyon is accompanied by additional gauge transformations, so wehave X\|a r/angbracketright=Ur\|aX(r)/angbracketright,X\|aX(r)/angbracketright=UX(r)\|ar/angbracketright, ⇒X2\|ar/angbracketright=UrUX(r)\|ar/angbracketright. (E1) Then, the phase factor we get on acting Xtwice is given by UrUX(r), which is nothing but eiφa, the phase corresponding to the anyon a. First, consider acting Xon a physical wave function \|/Psi1/angbracketright=f† rf† X(r)\|G/angbracketright, with two fermionic spinons at randX(r). Assuming that the ground state \|G/angbracketrightis symmetric, we have X\|/Psi1/angbracketright=(Xf† rX−1)(Xf† X(r)X−1)\|G/angbracketright=UrUX(r)f† X(r)f† r\|G/angbracketright =−UrUX(r)\|/Psi1/angbracketright=−eiφf\|/Psi1/angbracketright. (E2) This extra minus sign comes from reordering of the fermionic spinons under X, which is crucially dependent on the statistics of the fermion. Now, the same state can be thought of a pair of bound states of a bosonic spinon and a vison, i.e., \|/Psi1/angbracketright=b† rφ† rb† X(r)φ† X(r)\|G/angbracketright. (E3) Applying Xon this state, there is no fermion reordering sign, and we get X\|/Psi1/angbracketright=eiφbeiφv\|/Psi1/angbracketright. (E4) Hence, comparing the two relations we ﬁnd that in such cases, the fusion rule is nontrivial and carries an extra twist factor of−1, i.e., e iφbeiφv=−eiφf. (E5) For the rectangular lattice, we want to ﬁgure out which symmetry fractionalization quantum numbers have nontrivialfusion rules. First, consider the reﬂections P xandPy, and the inversion I=PxPy. All of these square to identity, implying the relations P2 x=1,P2 y=1, and ( PxPy)2=1h a v e nontrivial fusion rules. Now we use following the algebraicidentity: (P xPy)2=/parenleftbig PxPyP−1 xP−1 y/parenrightbig ·P2 x·P2 y. (E6) Since the PSGs associated with P2 x,P2 y, and ( PxPy)2have nontrivial fusion rules, the fusion rule for PxPyP−1 xP−1 ymust be nontrivial as well. Next, note that the identity P−1 xTxPxTx=1 can also be written as P−2 xY2=1, where Y=PxTx.N o w , P2 xandY2 both have nontrivial fusion rules, so the fusion rule for P−1 xTxPxTx=1 is trivial. Identical arguments show that P−1 yTyPyTy=1 has a trivial fusion rule. Now consider P−1 xT−1 yPxTyand its counterpart x↔y.I n this case, it is sufﬁcient to act on single anyons, and we ﬁndthat the spinon string has cut the vison string an even numberFIG. 5. Crossing of spinon (red blob) strings, represented by dashed red lines, and vison (blue cross) strings, represented by dotted blue lines, under TyPxT−1 yP−1 x. of times under any of these operations, as illustrated in Fig. 5. Therefore, these commutation relations have a trivial fusionrule. An analogous argument shows that T −1 xT−1 yTxTy=1 has a trivial fusion rule. Finally, let us consider time-reversal symmetry. We know that both bosonic and fermionic spinons have half-spin withT 2=− 1, whereas the vison is a spin singlet with T2=1, so the fusion rule for T2must be trivial. To derive the fusion rules of R−1T−1RT, where R= PxorPy, we follow Ref. [ 34]. We ﬁrst consider the antiunitary operator squared ( TR)2. If we act R2on a pair of spinons and visons on the reﬂection axis, the spinon and vison strings cross.This implies that the phase picked up by a bosonic spinonrelative to the bound state of a fermionic spinon and a vison, is±ifor the single reﬂection R. This is offset by the antiunitary time-reversal operator, which complex conjugates the wavefunction. Hence, the net relative phase is ( ±i) ∗×(±i)=1, as illustrated in [ 34]. So, ( TR)2has a trivial fusion rule. Now, we use the algebraic identity (TR)2=(R−1T−1RT)·T2·R2. (E7) Since the PSGs associated with T2and (TR)2have a trivial fusion rule, whereas that of R2obeys a nontrivial fusion rule, the PSGs of R−1T−1RTmust also have a nontrivial fusion rule. Finally, we consider the PSGs of T−1 xT−1TxT.W ea g a i n consider a similar setup as the previous case, with two spinonsand two visons. Under T xfollowed by T−1 x, there is no crossing of the spinon and vison strings—so there is no phase factoracquired by an indvidual bosonic spinon relative to the boundstate of the fermionic spinon and the vison. Therefore, thiscommutation relation has a trivial fusion rule, and so doesT −1 yT−1TyT. APPENDIX F: SOLUTION FOR THE FERMIONIC ANSATZ We need to ﬁnd an Ansatz Urr/primesuch that GXX(Urr/prime)=Urr/prime for all symmetry operations X, where the gauge transformation GXcorresponding to a symmetry operation Xhas been derived from the fusion rules. Note that under time reversal (a slightlymodiﬁed version as described in Ref. [ 3]), we have T(U rr/prime)= −Urr/prime,s ogTmust be nontrivial ( /negationslash=τ0) so that GTT(Urr/prime)= Urr/prime, and therefore we require ηT=− 1 for nonzero 024502-15CHATTERJEE, QI, SACHDEV , AND STEINBERG PHYSICAL REVIEW B 94, 024502 (2016) solutions: GTx(x,y)=τ0, (F1a) GTy(x,y)=(−1)xτ0, (F1b) GPx(x,y)=gPx,g2 Px=τ0, (F1c) GPy(x,y)=(−1)x+ygPy,g2 Py=−τ0, (F1d) GT(x,y)=gT,g2 T=−τ0, (F1e) where the SU(2) matrices gPx,gPy, and gTsatisfy the following (anti)commutation relations: /bracketleftbig gPx,gT/bracketrightbig =/braceleftbig gPy,gT/bracerightbig =/bracketleftbig gPx,gPy/bracketrightbig =0. (F2) In order to work with real hopping and pairing amplitudes in ourAnsatz , we follow Ref. [ 38] and choose gT=iτ2. Since gPxcommutes with both gTandgPy,i fgPyis nontrivial, thengPx=τ0. We assume that this is the case, and choose gPy=iτ3to get the solutions in Eq. ( 31), also listed below: GTx(x,y)=τ0, (F3a) GTy(x,y)=(−1)xτ0, (F3b)GPx(x,y)=τ0, (F3c) GPy(x,y)=(−1)x+yiτ3, (F3d) GT(x,y)=iτ2. (F3e) Note that gPy=iτ3is a gauge choice; we could have as well chosen gPy=iτ1, or any properly normalized linear combination given by gPy=i(cosθτ3+sinθτ1). However, all these choices lead to gauge-equivalent Ans¨atze. Noting that eiθτ2τ1e−iθτ2=cos(2θ)τ1+sin(2θ)τ3, a mean-ﬁeld matrix Urr/primeproportional to τ1can be rotated to τ3by a gauge transformation Wr=eiθτ2withθ=π/2. Therefore, we work with the ﬁrst choice for convenience. First, we note from [ 24] thatiUrr/prime∈SU(2) up to a normal- ization constant in order to preserve spin-rotation symmetry,so we can expand in the basis of Pauli matrices as U rr/prime=3/summationdisplay μ=0αrr/prime μτμ,where iαrr/prime 0,αrr/prime 1,2,3∈R.(F4) GT(Urr/prime)=−Urr/prime⇒{Urr/prime,τ2}=0⇒αrr/prime 2=0 for all bonds /angbracketleftrr/prime/angbracketright. Since the Ansatz (not the spin liquid) must break translational symmetry in the ydirection due to nontrivial GTy, we choose the following forms for the Ansatz (up to third-nearest neighbor): Ur,r+ˆx=ux(−1)y,U r,r+ˆy=uy,U r,r+ˆx+ˆy=(−1)yux+y,U r,r−ˆx+ˆy=(−1)yu−x+y,U r,r+2ˆx=u2x,U r,r+2ˆy=u2y.(F5) Now we just apply the parity relations to each of the bonds in the Ansatz . For the NN bonds GPxPx(Ur,r+ˆx)=Ur,r+ˆx⇒u† x=ux,G PyPy(Ur,r+ˆx)=Ur,r+ˆx⇒τ3uxτ3=−ux, GPxPx(Ur,r+ˆy)=Ur,r+ˆy⇒uy=uy,G PyPy(Ur,r+ˆy)=Ur,r+ˆy⇒τ3u† yτ3=−uy. (F6) Together, these imply that ux=/Delta11xτ1anduy=/Delta11yτ1where both the pairing amplitudes are real. Similarly, we ﬁnd that GPxPx(Ur,r+ˆx+ˆy)=Ur,r+ˆx+ˆy⇒u−x+y=−ux+y,G PyPy(Ur,r+ˆx+ˆy)=Ur,r+ˆx+ˆy⇒τ3u† −x+yτ3=−ux+y. (F7) Together, these imply that for the next-nearest neighbors ux+y=u−x+y=/Delta12τ1. (F8) Analogous calculations show that the next-to-next-nearest neighbors have a hopping term u2x=−t2xτ3,u 2y=−t2yτ3. (F9) One can also check that an on-site chemical potential term proportional to τ3is allowed by the PSG. This Ansatz describes aZ2spin liquid, as it has both hopping and pairing terms for the fermionic spinons in any choice of gauge. Alternately, one can check that the SU(2) ﬂuxes through different loops based at the same point are noncollinear,which also implies that the effective theory has a gaugegroup of Z 2[3,44]. Explicitly, consider the following two loops based at r:LA:r→r+ˆx+ˆy→r+ˆy→rand LB:r→/vectorr+ˆx+ˆy→r−ˆx+ˆy→r. The product of Urr/prime onLAis proportional to τ1, whereas that on LBis proportional toτ3, which clearly point in different directions in SU(2) space.APPENDIX G: ALTERNATIVE DERIV ATION OF THE SPECIFIC FERMIONIC PSG In this Appendix, we present an alternative derivation of the fermionic PSG, which represents the same spin liquid stateas the bosonic PSG in Eq. ( 11) and Appendix B. Instead of calculating the fractional quantum numbers of the fermionicspinon using the ones of the bosonic spinon and the vison,according to the fusion rules, here we derive this by directlymapping the bosonic mean-ﬁeld wave function to a fermionicmean-ﬁeld wave function, using the method introduced in theSupplemental Material of Ref. [ 47]. We start with the Schwinger boson wave function in Eq. ( 7), and we choose the weights to be ξ rr/prime=Qrr/primeon the nearest- neighbor and next-nearest-neighbor bonds, and ξrr/prime=0o n other bonds, where the values of Qrr/primeare shown in Fig. 4(a). With this choice, the wave function in Eq. ( 7) belongs to the phase described by the PSG in Appendix B, because the wave function is invariant under the transformations in Eq. ( 11). We notice that although this wave function is constructed usingthe parameters of the mean-ﬁeld Hamiltonian in Eq. ( 6), it is 024502-16SUPERCONDUCTIVITY FROM A CONFINEMENT . . . PHYSICAL REVIEW B 94, 024502 (2016) not the ground state of that Hamiltonian. However, it belongs to the same spin liquid phase as the ground state of thatHamiltonian. Using the result in the Supplemental Material of Ref. [ 47], we can convert the Schwinger boson wave function to thefollowing Schwinger fermion wave function, \|/Psi1 f(s)/angbracketright=/summationdisplay csδc/productdisplay (rr/prime)∈cζrr/primef† r↑f† r/prime↓\|0/angbracketright, (G1) where cruns over all possible nearest-neighbor and second- nearest-neighbor dimer coverings on the square lattice, ζrr/prime= ζr/primerare weights of the dimers, δccounts the number of dimer crossings in the covering, and each crossing contributes anextra weight factor sto the wave function. With s=− 1, the wave function \|/Psi1 f(s=− 1)/angbracketrightexactly reproduces the Schwinger boson wave function in Eq. ( 7), if for every triangular plaquette p, the fermionic weights ζrr/primesatisﬁes /productdisplay (rr/prime)∈pζrr/prime=−/productdisplay (rr/prime)∈pξrr/prime, (G2) where on the right-hand side, the bonds are oriented in the counterclockwise direction. In other words, in each triangle,the ﬂux in the fermionic model differs from the one in thebosonic model by π. One choice of weights satisfying Eq. ( G2) is the following: ζ r,r+ˆx=(−1)yQ(0,0)→(1,0),ζ r,r+ˆy=Q(0,0)→(0,1), ζr,r+ˆx+ˆy=ζr,r−ˆx+ˆy=(−1)yQ(0,0)→(1,1). (G3) The Schwinger boson wave function can only be mapped to a wave function with a nontrivial weight of s=− 1 for each pair of crossing bonds, which is different from the ordinarySchwinger fermion wave function, \|/Psi1 f(s=+ 1)/angbracketright=/summationdisplay c/productdisplay (rr/prime)∈c/epsilon1αβζrr/primef† rαf† r/primeβ\|0/angbracketright =PGexp/bracketleftBigg/summationdisplay rr/primeζrr/prime/epsilon1αβf† rαf† r/primeβ/bracketrightBigg \|0/angbracketright. (G4) However, assuming that the two wave functions \|/Psi1f(s=± 1)/angbracketright can be smoothly connected by varying sfrom−1t o+1 (along the real axis), the two wave functions belong to the same phase,and the weights in Eq. ( G3) can be used to derive the fermionc PSG that constructs the same phase as the original bosonicPSG. In particular, one can check that the wave function con- structed using the weights in Eq. ( G3) is invariant under the lattice and time-reversal symmetries, if the fermionic spinonoperator f iαtransforms according to the PSG in Eq. ( 31). We notice that this alternative derivation is not rigorous, as it depends on the assumption of the absence of any singularity in\|/Psi1 f(s)/angbracketrightwhensvaries between ±1. Nevertheless, this serves as a consistency check for the results presented in Sec. II, without the explicit usage of the vison PSG and the fusionrules.APPENDIX H: PSG FOR THE SITE BOSONS AND CONSTRAINTS ON HB We derive the transformation of the boson-tuplet Brunder the projective transformations. We ﬁrst focus on spatialsymmetry operations X s, which acts linearly (not projectively) on the cfermion, and therefore all additional projective phase must come from the fspinon. Recall that the ffermion spinor transforms under a gauge transformation GX(r)a s ψ(r)=/parenleftbiggfr↑ f† r↓/parenrightbigg →GX(r)ψ(r). (H1) In our gauge choice, GTx=GPx=τ0, so these will just mapBrto itself. GTy(r)=eiπx≡e−iπximplies that GTyBr= eiπxBr. Finally, we have GPyψr=eiπ(x+y+1/2)τ3ψr=eiπ(x+y+1/2)/parenleftbigg 10 0−1/parenrightbigg/parenleftbiggfr↑ f† r↓/parenrightbigg =eiπ(x+y+1/2)/parenleftbiggfr↑ −f† r↓/parenrightbigg . (H2) Therefore we see that under GPy,frσ→eiπ(x+y+1/2)frσ, and therefore Br→eiπ(x+y+1/2)Br. We conclude that the projec- tive transformation under each spatial symmetry operation Xs can be represented by just a phase φXson each boson, which we have listed in the main text in Eq. ( 41). Finally, we come to time reversal, which acts nontrivially on both the cand the ffermions. Because on an additional gauge transformation GT=iτ2, we now have mixing between the two bosons, GTT[ψ(r)]=/parenleftbigg 01 −10/parenrightbigg/parenleftbiggfr↑ −f† r↓/parenrightbigg =/parenleftbigg −f† r↓ −fr↑/parenrightbigg .(H3) Therefore, we have fr↑→−f† r↓, andfr↓→−f† r↑, under time reversal Tcombined with the gauge transformation GT.F o r the bosons, we ﬁnd that B1r→T(c† r↑)GTT(fr↑)+T(c† r↓)GTT(fr↓) =c† r↓(−f† r↓)+(−c† r↓)(−f† r↑) =/epsilon1βαf† rαc† rβ=B† 2r (H4) and similarly, B2r→b† Br. Imposing time-reversal symmetry on our hopping Hamiltonian in Eq. ( 42) therefore leads to the following constraints: T11 rr/prime=T22 rr/prime,T12 rr/prime=T21 rr/prime. 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PhysRevB.94.245304.pdf	PHYSICAL REVIEW B 94, 245304 (2016) Photoluminescence from ultrathin Ge-rich multiple quantum wells observed up to room temperature: Experiments and modeling T. Wendav,1,I. A. Fischer,2M. Virgilio,3G. Capellini,4,5F. Oliveira,6M. F. Cerqueira,6A. Benedetti,7S. Chiussi,8 P. Zaumseil,4B. Schwartz,9K. Busch,1,10and J. Schulze2 1AG Theoretische Optik & Photonik, Humboldt-Universit ¨at zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany 2Institut f ¨ur Halbleitertechnik, Universit ¨at Stuttgart, Pfaffenwaldring 47, D-70569 Stuttgart, Germany 3Dipartimento di Fisica “E. Fermi”, Universit `a di Pisa, L.go Pontecorvo 3, I-56127 Pisa, Italy 4IHP, Im Technologiepark 25, D-15236 Frankfurt (Oder), Germany 5Dipartimento di Scienze, Universit `a Roma Tre, Viale Marconi 446, I-00146 Roma, Italy 6Centre of Physics, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal 7CACTI, Universidade de Vigo, Campus Universitario, 36310 Vigo, Spain 8Departamento de F ´ısica Aplicada, Universidade de Vigo, Campus Universitario, 36310 Vigo, Spain 9Institut f ¨ur Physik, Brandenburgische Technische Universit ¨at Cottbus-Senftenberg, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany 10Max-Born-Institut, Max-Born-Str. 2A, D-12489 Berlin, Germany (Received 20 September 2016; revised manuscript received 8 November 2016; published 8 December 2016) Employing a low-temperature growth mode, we fabricated ultrathin Si 1−xGex/Si multiple quantum well structures with a well thickness of less than 1.5 nm and a Ge concentration above 60% directly on a Si substrate.We identiﬁed an unusual temperature-dependent blueshift of the photoluminescence (PL) and exceptionally lowthermal quenching. We ﬁnd that this behavior is related to the relative intensities of the no-phonon (NP) peakand a phonon-assisted replica that are the main contributors to the total PL signal. To investigate these aspectsin more detail, we developed a strategy to calculate the PL spectrum employing a self-consistent multivalleyeffective mass model, in combination with second-order perturbation theory. Through our investigation, we ﬁndthat while the phonon-assisted feature decreases with temperature, the NP feature shows a strong increase inthe recombination rate. Besides leading to the observed robustness against thermal quenching, this causes theobserved blueshift of the total PL signal. DOI: 10.1103/PhysRevB.94.245304 I. INTRODUCTION While group IV elemental semiconductors Si and Ge and their Si 1−xGexalloys dominate semiconductor electronics, their use in optoelectronics is limited due to their indirectfundamental band gap and the consequently poor radiativerecombination efﬁciency [ 1]. Therefore, nanostructures based on the SiGe system have been considered as a potentially active material class with improved optical performance [ 2]. In particular, Si 1−xGexquantum wells (QWs) have been the subject of theoretical and experimental investigations[3,4]. Among other techniques, photoluminescence (PL) has proven to be a particularly useful method for studying opticaltransitions in such structures [ 5–8]. Shortly following the ﬁrst PL investigations of SiGe/Si single and multiple QWs by No ¨el et al. [9], the ﬁrst well-resolved low-temperature band edge PL was observed by Sturm et al. [10], who attributed the observed PL spectral features to type II transitions betweenholes that are localized in the Ge-rich well regions andelectrons that are localized in the Si barrier layers occurring atthe barrier-well interface. After the successful demonstrationof modulation based on the quantum-conﬁned Stark effect in strained Ge QWs sandwiched between Ge-rich barrier layers and embedded in PIN diodes [ 11,12], similar SiGe/Ge multiple-QW structures have been used to demonstrate directband gap PL originating from type I transitions within the Ge Corresponding author: wendav@physik.hu-berlin.dewells [ 13]. Finally, there is ongoing interest in designing SiGe QW structures for intraband transitions for applications notonly in QW infrared (IR) photodetectors [ 14–17] but also in quantum cascade laser structures [ 18]. One of the main challenges in utilizing SiGe-based QW structures for optical device applications is the thermal quench-ing of their luminescence. In most experiments on SiGe/SiQWs, PL spectra can only be observed at temperatures below afew tens of degrees Kelvin [ 19,20]. The thermal PL quenching is mainly because of nonradiative recombination centers [ 19] that can originate, e.g., from structural defects such as disloca-tions. Higher growth temperatures have been shown to reducethermal quenching [ 7,19]; however, they also lead to Si and Ge interdiffusion and thus to the broadening of heterointerfaces[21]. Although room temperature PL has been reported in Ge QWs sandwiched between Si 1−xGexbarriers [ 13], those structures have the disadvantage of requiring a relativelythick SiGe virtual substrate to accommodate the large latticemismatch between the Si substrate and the QW structures. Here, we report the room temperature PL emission of ultrathin SiGe wells featuring very high Ge content growndirectly on Si, without using a virtual substrate technology.Our growth strategy consisted of repeatedly depositing 5.5monolayers (ML) of Ge and overgrowing them with Si at lowgrowth temperatures. We ﬁnd that this enables us to fabricatea multiple-QW structure containing 10 SiGe QWs with a Gecontent exceeding 60%. The high quality of the layers isconﬁrmed by a structural analysis based on high-resolutiontransmission electron microscopy (HR-TEM), as well as x-ray 2469-9950/2016/94(24)/245304(10) 245304-1 ©2016 American Physical SocietyT. WENDAV et al. PHYSICAL REVIEW B 94, 245304 (2016) diffraction (XRD). To interpret PL spectra, we developed a strategy to simulate the PL for the sample under investigationbased on a coupled Schr ¨odinger-Poisson description in the effective mass approximation of the electronic states belongingto different near-gap valleys. Even though the PL of variousSiGe/Si QW structures has been extensively investigatedexperimentally, theoretical calculations of the PL spectrumbased on an effective mass approach have not been reportedin the literature. In papers comparing measured PL spectrawith theoretical calculations, the PL transition energies arecommonly computed by taking into account the topmostvalence and bottommost conduction states only [ 22,23]. This can be helpful in cases of low optical excitation andtemperature. However, for higher temperatures and strongerexcitations, the PL shape strongly depends on the ﬁlling ofthe bands due to the optically excited excess carriers andon the temperature-dependent quasi-Fermi distributions of theexcited charge carriers, which leads to the population of higherenergy subband states, as well as to relevant band bendingeffects. In the analysis reported here, we describe the features from indirect recombination between quantum conﬁned statesrelying on second-order perturbation theory, which, to ourknowledge, has not been reported elsewhere in the lit-erature. For second-order perturbative absorption analysis in two-dimensional (2D) semiconductors, see, for instance, Refs. [ 24,25]. Our manuscript is organized as follows. We describe the sample growth and the experimental methods for its analysisin Sec. II. In Sec. III, we introduce the theoretical method used for the calculation of the PL spectra. Sec. IVis divided into two parts. In the ﬁrst part, the experimental results concerningthe structural properties of the sample are described. In thesecond part, we present the PL measurements. We start with theexcitation-density-dependent measurements, and we compareexperimental and simulated spectra to establish the validity ofour method. In a second step, we apply our theoretical methodto the temperature-dependent measurements. We conclude ouranalysis in Sec. Vby discussing the physical consequences of our ﬁndings. II. SAMPLE PREPARATION AND EXPERIMENTAL SETUP The nominally intrinsic Ge multiple-QW sample was fabricated by solid-source molecular beam epitaxy (MBE)on a Si (100) substrate. After thermal desorption of thenative oxide, a 100-nm-thick Si buffer layer was grown at600 °C. A 10 period sequence in which each period consists of 5.5 ML of Ge and 10 nm of Si (Fig. 1) was grown at a constant growth temperature of 350 °C. The Ge layers were grown at a rate of 0.087 ˚A/s, while for the Si spacer layers, a growth rate of 1 ˚A/s was used. A JEOL JEM- 2010F microscope was used for TEM characterization. Ramanscattering experiments were performed at room temperaturein a backscattering geometry on an Alpha300 R confocalRaman microscope (WITec), using a diode-pumped solid-statelaser with a wavelength of 532 nm as an excitation source.The spot size on the sample was roughly 1 .4μm 2, with a power of roughly 4 mW measured close to the externalsample surface. The XRD measurements were carried out FIG. 1. Schematic of SiGe/Si multiple-QW sample stack sequence. with a SmartLab diffractometer from Rigaku using CuK α radiation. Last, microphotoluminescence ( µPL) measurements were carried out at lattice temperatures varying between 80and 300 K using a custom-designed Horiba setup featur-i n ga5 0 ×optical microscope (numerical aperture =0.65), a high-resolution spectrometer optimized for IR measurements(Horiba iHR320), and an extended-InGaAs detector (0.6 to1.1 eV detection range). A 532 nm laser with an outputoptical power between 0.5 and 23 mW was focused on thesample surface, with an excitation power density rangingbetween 3 .2×10 4and 160 ×104Wcm−2. All spectra were collected at normal incidence in backscattering geometry, anda white-body lamp was used to determine the optical responseof the setup used for the calibration. III. NUMERICAL MODEL To better understand the observed PL, we compare exper- imental spectra with numerical simulations. As a ﬁrst step,we compute the electronic states, relying on a multivalleySchr ¨odinger-Poisson code. Spectrally resolved recombina- tion rates, associated with band-to-band indirect transitionsmediated by electron-phonon scattering, are then calculatedin the framework of second-order perturbation theory. Weconsider a 2D carrier interacting with a three-dimensional (3D)bulk phonon bath. Since our ultrathin multiple-QW samplesfeature type II band alignment, the eigenstates are sensitiveto the amount of pump-induced excess carrier density inthe well. However, this quantity cannot be easily estimatedtheoretically. Therefore, in our simulative approach, we choseto phenomenologically relate the excess carrier density to thepump power, introducing a ﬁtting constant. Its value is setby calibrating numerical data for the pump-induced blueshiftof the PL peak against experimental data. This blueshift iscontrolled by the band bending related to electrostatic ﬁelds,caused by the spatial separation of the photoexcited electronsand holes. It follows that its magnitude can be used to indirectlyestimate the amount of pump-induced excess carrier densityin the samples. For the calculation of PL spectra, we are interested in interband radiative transitions involving the quantum-conﬁnedsubband states associated with different near-gap valenceand conduction valleys. In this regard, the type II bandalignment between well and barrier regions is important forthe considered SiGe/Si multiple-QW structure. The type II 245304-2PHOTOLUMINESCENCE FROM ULTRATHIN Ge-RICH . . . PHYSICAL REVIEW B 94, 245304 (2016) band alignment between well and barrier regions results in the spatial separation of photoexcited electrons and holes, as wellas power-dependent band bending effects, which signiﬁcantlyinﬂuence the transition energies. These issues, together withthe splitting and shifts of bands due to the biaxial strainaffecting the SiGe region, have been addressed by solving theSchr ¨odinger-Poisson equation in effective mass approximation for the electronic states belonging to different near-gap valleys. In our model, we have considered heavy hole (HH), light hole (LH), and split-off (SO) bands in the valenceband and /Gamma1 c,L,/Delta12and/Delta14valleys in the conduction band. The SiGe/Si QW composition proﬁle is obtained fromexperimental data, and periodic boundary conditions wereapplied. Strain effects on the band edge and valence masses arecalculated assuming coherent growth and relaxed Si layers asdescribed in Refs. [ 26,27]. For a given pump-induced excess carrier density, valence and conduction quasi-Fermi energiesand band bending are self-consistently evaluated, taking intoaccount the 2D density of states resulting from all consideredvalleys. Following the computation of hole and electron eigenstates, the PL spectrum is calculated. In our SiGe/Si multiple-QWsample, holes are conﬁned in the compressively strained SiGeQW region and are distributed mainly in the HH1 subbandand to a minor extent in the LH1 subband close to the /Gamma1point. However, electrons are mainly localized in the unstrained Si barrier region and are associated with the /Delta1valley. Therefore, a radiative recombination can happen only if the missingmomentum is provided by either phonons (phonon-assistedrecombination) or elastic scattering centers like crystal defects,alloy disorder, or interface roughness (NP recombination).The challenge in calculating the full PL spectrum is toestimate the relative intensity of the phonon-assisted andNP recombination rates. For the phonon-assisted process,the electron-phonon coupling can be estimated using theeffective deformation potential for /Delta1−/Gamma1scattering reported in the literature [ 28,29]; however, we do not have sufﬁcient information on the structural properties of the sample torealistically calculate the elastic scattering rates because ofthe relevant role played by the heterointerface properties.Moreover, from a theoretical point of view, elastic interfaceroughness scattering effects in QW systems are commonlytreated within a perturbative framework, since well thicknessis typically one or two orders of magnitude larger thanthe interface region. However, due to the ultrathin layerthickness of the well region in our investigated samples, thisapproximation cannot be used. Therefore, in this paper, wefocus on the calculation of the phonon-assisted PL contributiononly, avoiding the estimation of the PL intensity ratio betweenthe phonon and the NP features. Upon knowing the peak energyof the phonon-assisted recombination spectrum and the energyof the contributing phonons, we can estimate the energy of theNP feature. To calculate the spectrally resolved emission rate of photons resulting from indirect band-to-band recombination, we usedthe second-order Fermi golden rule: P i→f=2π /planckover2pi1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay m/angbracketlefti\|Hph\|m/angbracketright/angbracketleftm\|Hem\|f/angbracketright E/primem−E/prime f/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 δ(E/prime i−E/prime f),with the energies E/primecorresponding to the total energy of the states, including electronic, phononic, and photoniccontributions. The initial states are the populated subbandsof the /Delta1 2and/Delta14valleys (indicated by /Delta1c). The electronic ﬁnal states are the populated subbands of the HH, LH, and SObands (indicated by /Gamma1 v). For energetic reasons, intermediate electronic states belong to the /Gamma1conduction band (indicated by /Gamma1c) only. Furthermore, due to the large conﬁnement energy of conduction electrons in this band, only the lowest /Gamma1csubband needs to be considered. This leads to the following deﬁnitionof the states: \|i/angbracketright=/vextendsingle/vextendsingle/Psi1(l) /Delta1c(k/bardbl)/angbracketrightbig \|0/angbracketright\|...n ph(q).../angbracketright, \|mem/abs/angbracketright=/vextendsingle/vextendsingle/Psi1(1) /Gamma1c(k/prime /bardbl)/angbracketrightbig \|0/angbracketright\|...(nph(q)±1).../angbracketright, \|fem/abs/angbracketright=/vextendsingle/vextendsingle/Psi1(s) /Gamma1v(k/prime /bardbl)/angbracketrightbig \|0...1ω.../angbracketright\|...(nph(q)±1).../angbracketright, where the factors of the product state represent the electronic, photonic, and phononic states of the system; landsare conduction and valence subband indices, respectively; upperand lower signs refer to phonon emission (em) and absorption(abs), respectively; and \|0···1ω···/angbracketright indicates that a photon with energy /planckover2pi1ωhas been emitted. The electronic part is given by /angbracketleftbig r/vextendsingle/vextendsingle/Psi1 (l) α(k/bardbl)/angbracketrightbig =1√ Aeik/bardbl·ρφ(l) α(z)uα(r), where φ(l) α(z) is the envelope function of subband lof valley α,uαis the related Bloch function, and Ais the area of cross section perpendicular to the growth direction. The interactionbetween electrons and bulk phonons is given by H ph=/summationdisplay q/bardbl,qz/summationdisplay n,m/summationdisplay k/bardblDq/bardbl,qzMn,m(qz)/parenleftbig bq/bardbl,qz+b† −q/bardbl,−qz/parenrightbig ×c† n,k/bardbl+q/bardblcm,k/bardbl, where nandmlabel 2D electronic states. For simplicity, fol- lowing Refs. [ 28,29], we approximate the electron-phonon in- teraction, considering only an effective dispersionless phononbranch at /planckover2pi1ω ph=50 meV. This value has been set equal to the energy separation between the NP and the phonon-assistedspectral features measured experimentally. As one may expect,this energy is close to the transverse optical (TO) phononenergy of bulk Si (58.8 meV) [ 30]a tt h e Xpoint. However, the Si-Ge vibrational mode has comparable energy (50 meV)[30]; hence, it cannot be excluded that interaction with this latter vibrational mode may contribute to the phonon-assistedindirect transition rate. Since our spectral resolution does notallow us to resolve the two phonon channels, and numericallywe are not interested in absolute values, we consider a singlebulklike effective phonon mode in our model. For nonpolarlattices, one can assume D q/bardbl,qz=Deff/radicalBigg /planckover2pi12 2ρV/planckover2pi1ωph, where ρis the mass density, Vis the volume, and Deffis an effective deformation potential. The squared matrix element 245304-3T. WENDAV et al. PHYSICAL REVIEW B 94, 245304 (2016) for the electron-phonon interaction is then given by /vextendsingle/vextendsingle/angbracketlefti\|Hph/vextendsingle/vextendsinglemem/abs/angbracketrightbig/vextendsingle/vextendsingle2 =D2 eff/planckover2pi12 2ρV/planckover2pi1ωph/parenleftbigg nph+1 2±1 2/parenrightbigg/vextendsingle/vextendsingleM(l) /Delta1c(qz)/vextendsingle/vextendsingle2. When considering /Delta1cand/Gamma1cfor the initial and intermediate states, Mn,m(qz) can be written as M(l) /Delta1c(qz)=/integraldisplayL 2 −L 2/parenleftbig φ(l) /Delta1c(z)/parenrightbig∗eiqzzφ(1) /Gamma1c(z)dz, where lis the subband index in the /Delta1cvalley and Lis the single well and barrier length. The electromagnetic interaction is, according to the usual notation, given by Hem=e m0A·p=e m0/radicalBigg /planckover2pi1 2Vωε 0ε(aˆe+a† ˆe)ˆe·p,leading to the following expression of the squared interaction matrix element: \|/angbracketleftmem/abs\|Hem\|fem/abs/angbracketright\|2=/parenleftbigge m0/parenrightbigg2/planckover2pi1 2Vωε 0ε/vextendsingle/vextendsingleI(s) /Gamma1v/vextendsingle/vextendsingle2\|p/Gamma1 cv·ˆe\|2 In the above expressions, I(s) /Gamma1vandp/Gamma1 cvare deﬁned as I(s) /Gamma1v=/integraldisplay+L/2 −L/2/parenleftbig φ(1) /Gamma1c(z)/parenrightbig∗φ(s) /Gamma1v(z)dz, p/Gamma1 cv=/integraldisplay /Omega1u∗ /Gamma1c(r)pu/Gamma1v(r)d3r. The spectrally resolved rate of spontaneous emission of photons via phonon-assisted recombinations can then be cal-culated by summing over all electronic and phononic degreesof freedom and polarization modes related to photons emittednormally to the sample surface. Considering the Fermi distri-butions f e(k/bardbl,i) andfh(k/bardbl,f) for electrons and holes, we obtain Rind sp(/planckover2pi1¯ω)d (/planckover2pi1¯ω)=1 V/summationdisplay ˆe/summationdisplay i,fPi→ffe(k/bardbl,i)fh(k/bardbl,f)G(/planckover2pi1ω)δ(/planckover2pi1ω−/planckover2pi1¯ω)d(/planckover2pi1¯ω) =Find/summationdisplay ˆe/summationdisplay i,f/parenleftbigg nph+1 2±1 2/parenrightbigg/vextendsingle/vextendsingleI(s) /Gamma1v/vextendsingle/vextendsingle2/vextendsingle/vextendsinglep/Gamma1 cv·ˆe/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/summationdisplay qzM(l) /Delta1c(qz)/vextendsingle/vextendsingle/vextendsingle2 ×fe(k/bardbl,i)fh(k/bardbl,f)δ(Ei(k/bardbl,i)−Ef(k/bardbl,f)∓/planckover2pi1ωph−/planckover2pi1¯ω) (Ei(k/bardbl,i)−Em(k/bardbl,f)∓/planckover2pi1ωph)2¯ωd(/planckover2pi1¯ω), where G(/planckover2pi1ω) is the density of states of the electromagnetic ﬁeld,Findis a proportionality constant, and Ei,Em, andEf are the energy of the carriers in the /Delta1,/Gamma1c, and /Gamma1vbands, respectively. To evaluate the above expression, we performeda double integral over the excess energy of the /Delta1 cand/Gamma1vcarri- ers. We calculated the polarization-dependent dipole matrix el-ements p /Gamma1 cvat the/Gamma1point by means of a sp3d5s∗ﬁrst-neighbor tight binding code [ 31] andnphusing Bose-Einstein statistics. The relevant material parameters used to describe the strained SiGe QW alloy were obtained by linear interpolationof the corresponding Si and Ge values, which have been takenfrom the literature (see Table III in Ref. [ 17] and Table I in Ref. [ 31]). The offset at the interface for the energy of the barycenter of the valence HH, LH, and SO bands has beencalculated according to Ref. [ 32]. IV . RESULTS AND DISCUSSION A. Structural analysis As a consequence of the 4.2% lattice mismatch between Ge and Si, pure Ge grown on a Si substrate experiencescompressive strain. In equilibrium conditions, this strain isreleased via the spontaneous formation of Ge dots, on top ofa coherent Ge wetting layer, following the so-called Stranski-Krastanow growth mode. The critical thickness of the wettinglayer at which dot formation occurs is a function of growthtemperature and can exceed several monolayers [ 33]. As can be seen from bright-ﬁeld transmission electron microscopy(BF-TEM) [Fig. 2(a)] and HR-TEM [Fig. 2(b)] images, because of the low growth temperature chosen here, the Gelayer thickness was below the critical thickness necessary forisland formation, and the resulting sample contains SiGe QWs rather than dots [ 34]. The sample is crystalline and no threading dislocations are visible, indicating that the growth strategyemployed enabled us to produce a sequence of well-deﬁnedQW structures. Moreover, the low deposition temperature usedallows a reduction of the Si-Ge intermixing [ 35]. The HR-TEM images allow us to estimate the thickness of the Ge-rich regionsto be in the 1.1 to 1.4 nm range. To obtain more information on the composition and lattice strain of the QWs, we used Raman and XRD analysis. Ramandata of our sample and that of a bulk Si (001) reference areshown in Fig. 2(c). A number of investigations have been devoted to obtaining models for the extraction of composition and strain data fromRaman analysis of SiGe island structures or thin pseudomor-phic ﬁlms [ 36–40]. To isolate the signal coming from the multi- p l eQ W s ,w ef o l l o wR e f s .[ 37,38] and subtract the Si spectrum I Sifrom the sample spectrum IS, i.e.,IS−FISi, where Fis a scaling factor obtained by taking the ratios of the Si peaks at520 cm −1. From the resulting multiple-QW spectrum, we ob- tain for the Ge-Ge and Ge-Si modes the frequencies ωGe−Ge= 303.95 cm−1andωSi−Ge=421.27 cm−1, respectively. From these values, we can estimate the Ge content, x, and the biaxial strain, εxx, in the well region using empirical relationships [ 38]: ωSi−Ge=400+29x−95x2+213x3−170x4+bSi−Geεxx, ωGe−Ge=283+5x+12x2+bGe−Geεxx. For the strain-shift coefﬁcients bGe−GeandbSi−Ge,w e used the empirical relations given in Ref. [ 41], determined 245304-4PHOTOLUMINESCENCE FROM ULTRATHIN Ge-RICH . . . PHYSICAL REVIEW B 94, 245304 (2016) FIG. 2. (a) BF-TEM and (b) HR-TEM cross-section images of the Ge multiple-QW structure. (c) Overlay of c-Si (red) and Ge multiple-QW sample (black) Raman signal. (d) XRD ω-2θscan along the (004) direction of the Ge multiple-QW sample (black) and ﬁt with a multiple square well model (red). experimentally for pseudomorphic Si 1−xGexlayers as a function of Ge content x: bSi−Ge=− 190(15)(x−1)4−555(15)cm−1, bGe−Ge=− 190(15)(x−1)4−460(20)cm−1. We obtained an average Ge content of x=(0.68±0.01) within the QW layer and a compressive strain of εxx= −2.64%. The main errors in this analysis originate not only from the uncertainties associated with the strain-shiftcoefﬁcients but also from the fact that both Ge content andstrain could be inhomogeneously distributed within the QWs. Further insights into the multiple-QW structures were obtained by XRD measurements. Figure 2(d) shows the results of an ω-2θscan along the (004) direction, together with diffraction simulation results. The large number of higherorder superlattice peaks indicates good homogeneity of theQWs within the sample. To simulate the angle-dependentintensity, sharp interfaces were assumed for the Ge-richQWs whose geometrical parameters (well width and Gecontent) were used for data ﬁtting. Good agreement betweenexperiment and simulation was obtained for an average Ge content of x av=0.63 and an average well thickness of wav=(1.2±0.2) nm. Assuming pseudomorphic strain but neglecting material diffusion (i.e., assuming a box proﬁle ofthe Ge content within the well), this analysis yields a total of4.9 ML of Ge deposited, which is close to the nominal valueof 5.5 ML used in the growth process. Summing up all the information acquired from HR-TEM, Raman spectroscopy, and XRD, we can conclude that thegrowth method applied produces well-deﬁned periodic QWstructures with average Ge content in the wells between 63%and 68% and thicknesses in the range of 1.2–1.4 nm. B. Optical properties The samples’ optical properties have been investigated by µPL at different pump power densities and sample tempera- tures. We ﬁrst discuss measurements performed at a constantlattice temperature of T L=80 K and varying the pump power densities [Fig. 3(a)]. The PL intensity increases by a factor of 20 when the excitation density increases from 3 .2×104to (a) 600 700 800 90000.511.5 Photon Energy (meV)PL Intensity (arb. units)0.03 MW /cm2 0.10 MW /cm2 0.22 MW /cm2 0.48 MW /cm2 0.96 MW /cm2 1.60 MW /cm2(b) 00 .511 .55051525354 Excitation Density P (MW/cm2)Peak Separation (meV)600 700 800 900 1000 Photon Energy (meV) PL Intensity(c) 10−1100102103 Excitation Density P (MW/cm2)Integrated PL Intensity (arb. units)Phonon-assisted (Expt.) Phonon-assisted (Fit) FIG. 3. (a) Experimentally observed PL spectra at TL=80 K for different excitation densities. (b) Measured energy separation between phonon-assisted and NP peaks as a function of excitation density; in the inset, the deconvolution of the PL signal taken at 80 K and 1 .60 MWcm−2 with two Gaussian peaks is shown. (c) Experimentally measured integrated PL intensity for the phonon-assisted peak as a function of the excitation power density. 245304-5T. WENDAV et al. PHYSICAL REVIEW B 94, 245304 (2016) 160×104Wcm−2and blueshifts from 740 to 830 meV . As shown in the inset of Fig. 3(b), two distinct features contribute to the PL signal. Gaussian deconvolution analysis indicatesthat their energy separation is ∼51 meV over the entire pump power range. As we suspect that electrons are localized mainlywithin the barrier region due to the type II conﬁnement, we attribute this energy difference to the TO phonon energy of bulk Si [30]. However, as already mentioned above, the Si-Ge vibra- tional mode at the QW interface, whose energy is close to theTO optical phonon of the Si lattice, may contribute to the lowerenergy spectral peak. While the lower energy feature can beattributed to an indirect band-to-band recombination assistedby spontaneous emission of phonons with appropriate mo-mentum, the higher energy must be related to the presence ofan elastic scattering channel (NP), which provides the missingmomentum for the same indirect transition. The observedblueshift can be attributed to pump-induced band bending,which affects the conﬁnement energy in the type II bandalignment of the investigated multiple QWs. In type IIheterostructures, the blueshift of the PL peak energy as afunction of pump power follows a power-law dependence of the form /Delta1E peak∝Pr. After a detailed analysis of the spectra in Fig. 3(a), we determined a scaling exponent r=0.2, in close agreement with values reported in the literature forsimilar type II structures [ 42]. Moreover, the integrated PL intensity I, as a function of pump power, can be described by a power law I∝P m[40]. From Fig. 3(c), where the integrated PL signal related to the phonon-assisted feature isshown as a function of the excitation density, we measure ascaling exponent value m=0.70±0.01, close to the value of 2/3, obtained when the recombination dynamics is dominated by the Auger mechanism. In this case, the generation rate,which is proportional to the pump intensity, scales as thethird power of the excess carrier density δn, which can be obtained by considering that in steady-state conditions it mustbe equal to the Auger recombination rate. It follows that for intrinsic samples I∝P 2/3,since the integrated PL signal is approximately proportional to δn2. As a next step, we use our theoretical model to gain a deeper understanding of the physical processes causing theobserved energy shift and spectral broadening. To this aim,the composition proﬁle of the sample has been modeled as asquare proﬁle with a constant Ge concentration of 68% withinthe QW, which corresponds to the average Ge concentrationmeasured by Raman spectroscopy. The assumption of a squarewell is only an approximation of the Ge distribution within thesample, which is likely to be inﬂuenced by segregation effects.However, determining the position-dependent Ge distributionwithin the QWs would necessitate an experimental techniquewith a subnanometer resolution. In our ultrathin SiGe/Simultiple QWs, the calculated transition energies are quitesensitive to small variations of the well thickness, due to therelevant role played by the conﬁnement in the valence band.As a consequence, in our simulations we tuned, starting fromthe HR-TEM and XRD measurements, the QW thickness ofthe adopted square proﬁle to improve the agreement betweentheoretical and experimental PL peak energies. An effectiveQW thickness of 1.6 nm is only slightly larger than the XRDestimates of (1 .2±0.2) nm and places our theoretical results within 20 meV of the experimentally determined values. Theremaining difference has the order of magnitude of the exciton binding energy, which is not accounted for in our theoreticalmodel. To further justify the introduction of this effectivesquare well thickness, different material parameters, such asband offsets and conﬁnement masses, whose precise valuesare unknown, inﬂuence the calculated transition energies.For instance, increasing the HH mass along the conﬁnementdirection by 10% results in an increase of the transitionenergy by about 10 meV . An effective QW thickness for themultiple-QW square proﬁle can then be regarded, as oftenproposed in the literature, as a way to account for uncertaintiesassociated with the relevant material parameters. Furthermore, the optically excited carrier density δnwithin the QW region depends on the defect density and all other ma-terial parameters controlling the nonradiative recombinationrate, which are largely unknown for our sample. Therefore, toestimate δn, we rely on a phenomenological relation δn=CP 0.37, where Cis a ﬁtting constant. The exponent value of 0.37 has been chosen to reproduce the scaling exponent measuredfor the integrated TO signal m=(0.70±0.01), and as one can expect, it is found to be approximately equal to m/2. The ﬁtting parameter Chas been ﬁxed to reproduce the measured pump-induced energy blueshift. As discussed inthe following, the blueshift is related to band bending effects,which is due to electron and hole spatial separation andwhose magnitude is controlled by the amount of the inducedexcess carrier density. This calibration procedure returnsC=3.5×10 11W−0.75cm0.5, giving an indication that at TL=80 K, the excess carrier density range is 1 .2×1012to 5.0×1012cm−2when the pump power density is varied from 3.2×104to 160 ×104Wcm−2. Numerical values for NP and phonon-assisted peak energy as a function of the pump powerare compared with the experimental data in Fig. 4(a).T h eN P curve has been obtained by upshifting numerical data for thephonon-assisted feature by 58.8 meV . Because of the largephonon energy of Si with respect to kT L, the phonon-assisted signal is related to spontaneous phonon emission only. To elucidate the cause of the observed blueshift, which is a typical signature of type II band alignment [ 43], we compare the band edge electronic states at low and highexcitation densities as shown in Fig. 5. Due to the spatial separation of the excess hole and electrons, which are mainlylocalized in the SiGe and Si regions, respectively, a largerpump power density increases the band bending. The HH1energy is quite insensitive to this effect due to the large values of both the offset and the conﬁnement energy in the valence band. In contrast, the nondegenerate /Delta1 2and/Delta14subband states are more sensitive to the band edge proﬁle, becausethey are localized in the thicker Si region. It follows thatwhen the pump power increases, the conduction conﬁnementenergy becomes larger and, as a result, the PL peak energyblueshifts. A comparison of experimental and simulated PLspectra originating from the phonon-assisted recombinationevaluated at T L=80 K for different pump powers is shown in Fig. 4(b). Although the experimental spectra are slightly broader, probably due to ﬂuctuations in material compositionand QW width, we ﬁnd that the pump-induced increase of the 245304-6PHOTOLUMINESCENCE FROM ULTRATHIN Ge-RICH . . . PHYSICAL REVIEW B 94, 245304 (2016) (a) 10−1100750800850900 Excitation Density (MW/cm2)PL Peak Energy (meV)Phonon (Expt.) Phonon (Theory) NP (Expt.) NP (Theory)(b) 750 800 850 90000.20.40.60.81Expt. Photon Energy (meV)Phonon PL Intensity (arb. units) 750 800 850 90000.20.40.60.81Theory Photon Energy (meV) Recombination Rate (arb. units)0.03 MW /cm2 0.10 MW /cm2 0.48 MW /cm2 0.96 MW /cm2 1.60 MW /cm2 FIG. 4. (a) PL peak energy of the phonon-assisted and NP line according to theory and experiment as a function of the excitation density. (b) Comparison of experimental and simulated phonon-assisted PL at TL=80 K for different pump powers. peak intensity is well reproduced by the model. We stress here that this is not ap r i o r i obvious, because the phenomenological relation between excess carrier and pump density was tunedto reproduce the energy shift, not the PL peak intensity. PL spectra have been also studied at a constant pump power density of 9 .6×10 4Wcm−2with varying TLin the 80–300 K range [Fig. 6(a)]. With increasing temperature, the spectra become broader and the PL peak energy blueshifts by∼30 meV . The PL intensity is not signiﬁcantly quenched up to room temperature, the total integrated intensity being reducedby a factor of 3 only when T Lis increased from 80 to 300 K. Regarding the spectral blueshift, this behavior seems, at ﬁrst,to be in conﬂict with the temperature-driven shrinking of boththe Si and the Ge band gaps. To shed light on this unusualtrend, we again resolve the phonon-assisted and NP feature byGaussian deconvolution. Due to thermal broadening effects, Energy (meV)600700 (a) −40−20 0 20 40-600-400-2000 Position ( ˚A)(b) Δ4 Δ2 −40−20 0 20 40 Position ( ˚A)HH LH FIG. 5. Near-gap subband states calculated for TL=80 K at low (a) and high (b) optical excitation densities. Conﬁnement energies inthe valence band are unchanged, while at high excitation density, the conduction subband states are found at higher energies due to larger band bending.this kind of analysis was signiﬁcant only for TL<200 K. We ﬁnd that when looking at spectrally resolved features,the expected temperature dependence is observed, becauseboth the NP and the phonon-assisted energy peaks decreasemonotonically by roughly 30 meV , as shown in Fig. 7(a). To further explain this ﬁnding, we theoretically investigatedthe PL spectra as a function of temperature. To this aim,the phonon-assisted contribution of the PL spectra for the9.6×10 4Wcm−2excitation density at a given TLhas been calculated by tuning the excess carrier density to reproducethe measured ratio between the integrated phonon-assistedsignal at temperature T Land that at TL/prime=80 K, for which we already know the excess carrier density from the investigationof the excitation density measurements. From this calibrationprocedure, we estimate an excess carrier density decreasefor the 80–200 K temperature variation by a factor of 0.3only. Numerical phonon-assisted and NP peak energies as afunction of temperature are compared with the experimentalcounterpart in Fig. 7(a). It is apparent from Fig. 7(a) that apart from the already-mentioned overestimation of about 20 meV ,the numerical data fully capture the observed trend. The Gaussian ﬁts of the phonon-assisted PL feature at different temperatures are compared with numerical data inFig. 6(b). Although the experimental spectra are broader, presumably due to multiple-QW thickness ﬂuctuation in oursamples, it is remarkable that the ratios among different peakintensities are in good agreement with experiment. This wasnot obvious ap r i o r i , because in the calibration procedure, the integrated intensities were targeted. This indicates thatthe thermal contributions to the broadening are correctlyreproduced by the model. From the above discussion, weconclude that the phonon-assisted PL peak energy redshiftis dominated by the temperature-dependent decrease of theband gap, while the thermal excitation of higher energystates, which in principle could drive a blueshift, plays onlya minor role. The observed blueshift in the total PL spectrumis then to be attributed to a spectral weight shift from thephonon-assisted feature, dominating at low temperature, tothe higher energy NP contribution, which becomes dominantat higher T L. This effect more than compensates for the 245304-7T. WENDAV et al. PHYSICAL REVIEW B 94, 245304 (2016) FIG. 6. (a) Experimental PL spectra at a 0 .10 Wcm−2excitation density for different lattice temperatures. (b) Comparison between phonon-assisted peaks extracted from experiment and theory for different lattice temperatures. band gap shrinkage. Considering the 30 meV redshift of the phonon-assisted and NP peaks predicted by our model andthat their energy is separated by about 58.8 meV , we estimatea blueshift of the total PL spectrum of about 30 meV , whichwell matches the measured value of 20 meV between 80 and200 K. To further consolidate this interpretation, we observethat the measured ratio between the phonon-assisted and theNP-integrated PL signals, ∼10 at T L=80 K, decreases to ∼0.15 at TL=200 K [Fig. 7(b)]. The thermal quenching of the PL is therefore hindered by a strong increase of theNP intensity at higher T L. This behavior can be explained by looking into the temperature dependence of the phonon-assisted and NP features separately. For the phonon-assistedfeature, the coupling between electrons and phonons canbe approximately described as temperature independent overthe investigated temperature range due to the large phononenergy with respect to k T L(spontaneous phonon emission). It follows that the phonon-assisted signal intensity is mainlygoverned by the decreasing excess electron density, related to afaster nondegenerate recombination dynamics with increasing temperature. However, from Fig. 7(b), we infer that the electron-hole coupling of the NP recombination must stronglyincrease with T Lto overcompensate for the lower excess carrier density. We speculate that two mechanisms could beresponsible for this behavior. At higher temperatures, holes and electrons occupy states with higher in-plane momenta. These larger momenta increasethe rate at which charge carriers encounter interface defects.Since these elastic scattering events can provide the missingmomentum for indirect transitions, an increased NP recombi-nation rate is to be expected. An alternative explanation fortheT L-driven increase of the NP electron-hole coupling is that at higher temperature conduction electrons populate higherexcited subbands whose wave function penetrates deeper intothe SiGe region (Fig. 5). This may enhance the interface roughness scattering rate contributing to the NP PL feature. Inthis case, the enhanced contribution from excited conductionsubbands will slightly blueshift the NP feature, resulting in an (a) 100 150 200700750800850 Lattice Temperature TL(K)PL Peak Energy (meV)Phonon (Expt.) Phonon (Theory) NP (Expt.) NP (Theory)(b) 100 150 200050100150 Lattice Temperature TL(K)Integrated PL Intensity (arb. units)NP (Expt.) Phonon (Expt.)(c) 100 150 200404550556065 Lattice Temperature TL(K)Peak Separation (meV) 600 700 800 900 Photon Energy (meV) PL Spectrum FIG. 7. (a) PL peak energies as a function of TLfor theory and experiment. (b) Experimental integrated PL signal as a function of TL. (c) Measured energy separation of phonon-assisted and NP peaks as a function of TL; in the inset, the deconvolution of the PL signal taken at 100 K and 0 .1M Wc m−2with two Gaussian peaks is shown. 245304-8PHOTOLUMINESCENCE FROM ULTRATHIN Ge-RICH . . . PHYSICAL REVIEW B 94, 245304 (2016) increase in energy separation with the phonon-assisted peak. This effect could explain the trend shown in Fig. 7(c). V . CONCLUSIONS The two main challenges in using SiGe-based QW struc- tures for optoelectronic applications are the usually strongthermal quenching of their luminescence and the complexityof manufacturing SiGe QWs with high Ge concentrationsdirectly on a Si substrate. The growth strategy presented hereappears to circumvent both challenges. By employing lowgrowth temperatures and limiting the deposition of Ge to lessthan 5.5 ML, a multiple-QW structure consisting of Si 1−xGex layers with a Ge concentration exceeding 60% can be directly grown on a Si substrate. Furthermore, we ﬁnd that the PL ofthis structure is unusually robust against thermal quenching.Employing an empirical relationship between the opticallyexcited charge carrier density and the excitation density of thePL setup and using a self-consistent effective mass approach,we were able to fully reproduce the phonon-related featuresof the PL spectra. We ﬁnd that while the phonon-relatedPL intensity decreases when the temperature is increased,due to the increase of the nondegenerate recombination rateand as a result of the reduced charge carrier density, theNP-related PL intensity increases. This increase of the NPintensity more than compensates for the decrease in intensity related to the phonon-assisted transitions and leads to a shiftin spectral weight from the phonon-assisted to the NP feature, effectively blueshifting the total PL energy. We conjecturethat the increase in the NP-related recombination rate is dueto the higher in-plane momenta of the charge carriers athigher temperatures, which increases the rate at which chargecarriers encounter elastic scattering centers. Alternatively, thiseffect might be related to an enhanced interface scatteringrate as a result of the occupation of higher energy levels bythe optically excited electrons. The wave functions related tothose higher energy levels penetrate deeper into the interfaceregion between the Si and the SiGe regions, enhancing thecontribution of the interface roughness scattering rate to theNP signal. From the current state of our investigation, we deem that the mechanism limiting the effect of thermal quenching is as-sociated with the type II conﬁnement typical for SiGe/Si QWs.It would be interesting to investigate similar layer structuresin other material systems that exhibit type II conﬁnement. 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PhysRevB.90.035202.pdf	PHYSICAL REVIEW B 90, 035202 (2014) Hole localization, migration, and the formation of peroxide anion in perovskite SrTiO 3 Hungru Chen1and Naoto Umezawa1,2,3,* 1Environmental Remediation Materials Unit, National Institute for Materials Sciences, Ibaraki 305-0044, Japan 2PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan 3TU-NIMS Joint Research Center, School of Materials Science and Engineering, Tianjin University, 92 Weijin Road, Nankai District, Tianjin, People’s Republic of China (Received 7 May 2014; revised manuscript received 25 June 2014; published 10 July 2014) Hybrid density functional calculations are carried out to investigate the behavior of holes in SrTiO 3.A si nm a n y other oxides, it is shown that a hole tend to localize on one oxygen forming an O−anion with a concomitant lattice distortion; therefore a hole polaron. The calculated emission energy from the recombination of the localized holeand a conduction-band electron is about 2.5 eV , in good agreement with experiments. Therefore the localizationof the hole or self-trapping is likely to be responsible for the green photoluminescence at low temperature, whichwas previously attributed to an unknown defect state. Compared to an electron, the calculated hole polaronmobility is three orders of magnitude lower at room temperature. In addition, two O −anions can bind strongly to form an O 22−peroxide anion. No electronic states associated with the O 22−peroxide anion are located inside the band gap or close to the band edges, indicating that it is electronically inactive. We suggest that in additionto the oxygen vacancy, the formation of the O 22−peroxide anion can be an alternative to compensate acceptor doping in SrTiO 3. DOI: 10.1103/PhysRevB.90.035202 PACS number(s): 71 .38.Ht,72.20.Ee,71.20.−b I. INTRODUCTION Strontium titanate SrTiO 3(STO) is a prototypical per- ovskite oxide in which Sr occupies the 12-fold cuboctahedralsite (Asite) and Ti occupies the sixfold octahedral site ( B site). STO is one of the most studied oxides due to itsassociation with a variety of fascinating physical propertiessuch as superconductivity [ 1], ferroelectricity [ 2], and the formation of a two-dimensional electron gas on its surface [ 3], as well as at the interface between STO and LaAlO 3[4]. STO is also known as a promising anode material for the generationof hydrogen fuel from photoelectrolysis [ 5,6]. n-type conductivity can be easily obtained in STO. As a result, the behavior of electrons in STO in a wide range oftemperatures has been intensively studied [ 7–9]. Notably an electron mobility exceeding 30 000 cm 2/V s at low temper- ature has been reported [ 10]. In contrast, the behavior of holes in STO is hitherto unclear. Unlike n-type STO, only af e w p-type STO samples have been reported [ 11,12] and to the best of our knowledge there is no hole mobility in asingle-crystal sample available in the literature. This mightbe rooted in the difﬁculty in introducing hole carriers in oxidematerials because hole carriers are not thermally stable againstthe formation of oxygen vacancies [ 13,14] and the mobility can be measured only when a sample is sufﬁciently conductive.Nonetheless, hole carriers can also be generated by excitations.Since STO functions as a photocatalyst or photoelectrode forphotocatalytic oxidation reactions [ 15,16], there must be holes generated and migrating to the surface, otherwise no surfaceoxidation reaction would occur. Correct theoretical descriptions of holes in oxides have re- cently been discussed and applied to various materials [ 17–19]. However, a detailed theoretical treatment of holes in perovskiteSTO is lacking. In order to gain insight into the behavior of umezawa.naoto@nims.go.jpholes in STO, in this study hybrid density functional theory(DFT) calculations are carrier out to investigate hole-dopedSTO. II. METHODOLOGY All calculations were based on density functional theory [20,21], utilizing the range-separated screened hybrid func- tional formulated by Heyd, Scuseria, and Ernzerhof [ 22] (HSE), which mixes 25% Hartree-Fock exchange with 75%of the Perdew-Burke-Ernzerhof [ 23] (PBE) functional. The inclusion of Hartree-Fock exchange in the HSE functionalcorrects the self-interaction error contained in standard DFTfunctionals and consequently well reproduces experimentalmeasured properties of metal oxides. Interactions betweenthe core and valence electrons were treated by the projectoraugmented wave approach [ 24]. The cutoff energy for plane waves was set at 400 eV . For all cells calculated, the k-point spacing was smaller than 0.05 ˚A −1in the Brillouin zone. Structural optimization including lattice vectors and atomicpositions was performed on all of the cells until the forcesconverged to below 0.03 eV ˚A −1per ion. All calculations were carried out using the Vienna ab initio simulation package (V ASP )[25]. STO adopts a cubic perovskite structure with space group Pm¯3mat room temperature. For comparison hole-doped CaTiO 3(CTO) is also investigated. CaTiO 3adopts an or- thorhombic perovskite structure, as a result of octahedraltilting, with space group Pbnm . Single-hole doping is achieved by removing one electron from a neutral 2√ 2×2√ 2×2S T O cubic supercell and a neutral 2 ×2×1 CTO orthorhombic supercell. Both supercells contain 16 formula units with80 atoms. In STO double-hole doping is also investigated andin this case a 3 ×3×2 supercell containing 18 formula units (90 atoms) is used. The removal of electrons results in chargedcells and therefore the image charge correction proposedby Makov and Payne [ 26] as well as a potential-alignment 1098-0121/2014/90(3)/035202(5) 035202-1 ©2014 American Physical SocietyHUNGRU CHEN AND NAOTO UMEZAWA PHYSICAL REVIEW B 90, 035202 (2014) correction [ 27] are applied to alleviate errors arising from ﬁnite-size effects. It has been demonstrated that the additionof these two corrections is sufﬁcient to reproduce the behaviorof an inﬁnite supercell [ 27]. In the single-hole-doping case in STO, a bigger 3 ×3×3 supercell containing 27 formula units (180 atoms) has also been tested, and it gave no noticeabledifference in the position of hole states and energetics. III. RESULTS AND DISCUSSION When a hole is doped in a supercell by removing one electron, DFT calculations with the HSE functional yield twodistinct stable states, one with a localized hole and the otherwith a delocalized hole. The delocalized hole is obtained whenstarting from the perfect STO geometry. On the other hand, thelocalized hole solution develops if the Ti-O distances around anoxygen in the stating geometry are slightly elongated, whichalso lowers the symmetry of the cell. The delocalized holestate represents a hole residing at the valence-band maximum(VBM) of the host material and simply causes a rigid-bandshift of the Fermi level into the valence band whereas anin-gap state appears if the hole is localized. Figure 1shows the density of states of a STO and a CTO cell containing alocalized hole. The localized hole state is located at about 0.9and 1.3 eV above the VBM in STO and CTO, respectively.From the local geometry and charge density of the localizedin-gap hole states shown in the right panel of Fig. 1, the hole charge density is centered on one single oxygen with a littlecontribution on four neighboring oxygens. The localization of the hole is accompanied by a lattice distortion and therefore the hole can be regarded as being trapped by the potential wellthat the lattice distortion produces, i.e., hole polaron formation[28,29]. The oxygen on which a hole localizes can be formally assigned as O −in the ionic model. The O−anions in STO and CTO are paramagnetic and carry calculated magnetic momentsof 0.569 μ Band 0.627 μB, respectively. A paramagnetic center should be able to be detected by electron paramagneticresonance (EPR) spectroscopy and indeed acceptor-bound O− centers have been observed in STO and CTO [ 30,31]. To investigate the stability of the localized hole relative to delocalization, the hole self-trapping energy EST, deﬁned as [32] EST=Etot(STO :h+ VBM)−Etot(STO :h+ polaron ), where Etot(STO :h+ VBM) is the total energy of the cell containing a delocalized hole and Etot(STO :h+ polaron )i st h e total energy of the cell containing a localized hole, is estimated.Because both cells are charged, a potential-alignment correc-tion [ 27,32] is applied to adjust the calculated total energies. A positive value indicates that hole localization is preferred todelocalization, and a larger absolute value means a strongerlocalization (deeper trap). Furthermore, to investigate theconsequence of hole localization for photoluminescence, theemission energy is calculated by constructing the conﬁgurationcoordinate diagram based on the Franck-Condon principle.Also it is assumed that electrons reside in the conduction-bandminimum (CBM) and the electron-hole interaction is not takeninto account. A schematic illustration of these energies isshown in Fig. 2. The calculated hole self-trapping energies E STare 13 and 252 meV in STO and CTO, respectively. These positivevalues indicate that hole localization is favored rather thandelocalization. The larger self-trapping energy E STin CTO than in STO suggests that the hole is trapped deeper (morelocalized) in CTO. This is consistent with the higher positionof the hole state inside the band gap above the VBM and the stronger magnetic moment on O −in CTO. The calculated emission energies for STO and CTO are 2.51 and 2.69 eV ,respectively, in good agreement with experimentally reportedbroad photoluminescence peaks at low temperature centeredat 2.475 eV [ 33] for STO and 2.7 eV [ 34] for CTO. Therefore these photoluminescence peaks can be well explained by therecombination of electrons in the CBM with self-trappedlocalized holes. FIG. 1. (Color online) Calculated density of states of hole-doped SrTiO 3and CaTiO 3. Charge densities of the in-gap hole state are shown on the right. O denotes the oxygen on which the holes localizes. The dotted green line in the density of states plot indicates the highest occupied energy level. The numbers in the charge density plots are titanium-oxygen bond lengths in ˚A. The isosurfaces shown correspond to the density of 0.01 e /˚A3. 035202-2HOLE LOCALIZATION, MIGRATION, AND THE . . . PHYSICAL REVIEW B 90, 035202 (2014) FIG. 2. (Color online) The conﬁguration coordinate diagram constructed based on the Franck-Condon principle. e−(CBM) denotes a delocalized electron at the conduction-band minimum. h+(VBM) andh+(polaron) denote a delocalized hole at the valence-band maximum and a localized hole, respectively. Next, we estimate the mobility of hole polarons in STO and CTO, which can be characterized by thermal hopping barriersbetween two neighboring oxygen sites. The barrier height isdetermined by calculating the energy proﬁle along a pathwaywith geometries linearly interpolated between the initial andﬁnal polaron states [ 17,35,36], i.e., optimized geometries of a single hole polaron on two neighboring oxygen sites, asshown in Fig. 3(a). The migration of the polaron from one site to the other is clearly visualized in Fig. 3(b). Assuming the adiabatic small-polaron hopping mechanism, the hole mobility(μ) can then be derived from the following equation [ 28,37]: μ=[ea 2ω0/kBT][exp(−Ea/kBT)], where ais the hopping distance, i.e., the distance between nearest-neighboring oxy-gens,ω 0is the longitudinal optical phonon frequency, and Ea is the the barrier height for polaron hopping shown in Fig. 3(a). Taking Ea=66 meV from our calculations, a=2.683 ˚A, and ω0=800 cm−1[38], the hole mobility in SrTiO 3at room temperature (300 K) is 5.09 ×10−3cm2/V s. Similarly, by usingEa=147 meV , a=2.634 ˚A, and ω0=800 cm−1[39], the hole mobility for CaTiO 3is 2.10 ×10−3cm2/Vs . T h e higher barrier height for hole migration and smaller mobility inCTO than in STO are again consistent with the more localizedhole character in CTO. Compared to the electron mobility inSrTiO 3at room temperature, which is about 10 cm2/Vs[7,9], the mobility of the hole in STO is three orders of magnitudelower. (It should be noted that the quoted electron mobility isdeduced from Hall measurements on single-crystal samples.This room-temperature mobility has been interpreted as a largepolaron limited by longitudinal optical phonon scattering [ 7]. The calculated hole mobility is based on the small-polaronadiabatic hopping model. The hopping is phonon assistedand a perfect crystal is assumed, which does not account forinﬂuences from impurities and defects.) The large differencein electron and hole mobilities explains the measured n-type FIG. 3. (Color online) (a) Migration barrier for hole polarons in SrTiO 3and CaTiO 3. (b) The evolution of the spin density of the hole polaron along the migration pathway in SrTiO 3. The isosurface shown corresponds to 0.0002 μB/˚A3. photoconductivity [ 40], despite the same number of electron and hole carriers generated by optical excitation. Also the lowhole mobility means that even if hole carriers can be introducedinto STO, no pronounced p-type conductivity will result. In addition to the hole localization on one single oxygen forming an O −anion, we found that when another hole is introduced in STO, two neighboring O−anions can pair to form an O 22−peroxide anion. Figure 4shows the density of states for a cell containing two separated hole polaronsO −+O−and a cell containing a peroxide O 22−anion resulting from binding of two O−anions. Similarly to the cell containing only one localized hole, in the cell with two separated holepolarons O −+O−, hole states are still present inside the band gap. Conversely, the gap states disappear as a consequenceof the formation of the peroxide O 22−anion. The absence of electronic states related to the peroxide O 22−anion inside the gap or near band edges indicates that it is electronicallyinactive. Figure 5shows charge densities associated with different peaks in the density of states plots. The two separatedholes in the O −+O−case can be clearly seen in Fig. 5(a) 035202-3HUNGRU CHEN AND NAOTO UMEZAWA PHYSICAL REVIEW B 90, 035202 (2014) FIG. 4. (Color online) Density of states for the cell containing two separated holes O−+O−and the cell in which two holes bind together to form an O 22−peroxide anion. The dotted green line indicates the highest occupied energy level. and the binding of two O−anions to form the peroxide O22−anion is shown in Fig. 5(b). The spatial characters and positions of those sharp peaks in the density of states of theO 22−cell are consistent with a molecular orbital diagram of an oxygen-oxygen bond [ 41] and are labeled accordingly. Figure 5(b)also shows that the formation of the peroxide O 22− anion is accompanied by a strong local structural distortion. The optimized oxygen-oxygen distance of the peroxide O 22− anion in STO is 1.48 ˚A, signiﬁcantly shorter than the regular oxygen-oxygen distance of 2.76 ˚A in pristine STO, and remarkably it is very similar to the O-O bond length in peroxide FIG. 5. (Color online) Partial charge densities corresponding to peaks in the density of states plots. (a) The in-gap states in the O−+O− cell. (b) The ﬁve labeled peaks in the O 22−cell. The isosurfaces shown correspond to 0.0002 e/˚A3.compounds, such as 1.45 ˚Ai nH 2O2[42] and 1.49 ˚Ai nB a O 2 [43] and Na 2O2[44]. Despite the large local distortion, the peroxide O 22−anion is found to be signiﬁcantly more stable than two separated O− anions, by /223c496 meV . This means that the energy gain from the oxygen-oxygen bond formation is large enough to surpass theenergy cost of the signiﬁcant geometry distortion. In addition,unlike the O −anion which carries magnetic moment and is visible to electron paramagnetic resonance spectroscopy, theO 22−peroxide anion is diamagnetic. This might make a direct experimental identiﬁcation difﬁcult. Incidentally, it is known that functionals within the local density approximation (LDA) and generalized gradient ap-proximation (GGA) fail to reproduce the localized behaviorof holes in many oxides [ 18,45]. Indeed, the single localized hole and the two separated localized holes solutions in ourcalculations cannot be obtained with the PBE functional.Intriguingly, we found that the O 22−peroxide anion in STO, however, remains stable even in calculations with the PBEfunctional. IV . CONCLUSIONS In summary, it is demonstrated that a hole in STO tends to localize on one single oxygen forming an O−anion with a concomitant lattice distortion; therefore a hole polaron. Thecalculated mobility of the hole polaron is low, at least threeorders of magnitude lower than reported electron mobilitiesat room temperature. The large difference in electron andhole mobilities explains the measured n-type photoconduc- tivity despite the same number of electron and hole carriersgenerated by optical excitation [ 40]. The low hole mobility implies that even if hole carriers can be introduced into STO,no pronounced p-type conductivity will result. Also hole localization on oxygen creates in-gap states and is likely to beresponsible for the green photoluminescence ( /223c2.4 eV) at low temperature, which was previously attributed to an unknowntrap state. Calculated emission energies for STO and CTO arein good agreement with experiments. In addition to the hole localization on one single oxygen f o r m i n ga nO −anion, it is found that if another O−is introduced, there is a strong tendency for the binding of twoO −anions to form an O 22−peroxide anion. Although there is no direct experimental identiﬁcation of the O 22−peroxide anion inside bulk perovskite oxides so far, its existence has alsobeen proposed in other oxide materials with different crystalstructures [ 46–48]. The O 22−peroxide anion in STO in more stable than two separated O−anions by 0.496 eV , indicating strong binding. This would further impede hole transport in thematerial. Furthermore, in addition to the oxygen vacancy theformation of the O 22−peroxide anion would be an alternative to compensate acceptor doping in STO. ACKNOWLEDGMENT This work is partly supported by the Japan Science and Technology Agency (JST) Precursory Research for EmbryonicScience and Technology (PRESTO) program and by the WorldPremier International Research Center Initiative on MaterialsNanoarchitectonics (MANA), MEXT. 035202-4HOLE LOCALIZATION, MIGRATION, AND THE . . . PHYSICAL REVIEW B 90, 035202 (2014) [1] J. F. Schooley, W. R. Hosler, and M. L. Cohen, Phys. Rev. Lett. 12,474(1964 ). [2] J. H. Haeni, P. Irvin, W. Chang, R. Uecker, P. Reiche, Y . L. Li, S. Choudhury, W. Tian, M. E. Hawley, B. Craigo, A. K.Tagantsev, X. Q. Pan, S. K. Streiffer, L. Q. Chen, S. W.Kirchoefer, J. Levy, and D. G. Schlom, Nature (London) 430, 758(2004 ). [3] A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pailhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi,P. Le Fevre, G. 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PhysRevB.94.024505.pdf	PHYSICAL REVIEW B 94, 024505 (2016) Superconductivity in the two-dimensional electron gas induced by high-energy optical phonon mode and large polarization of the SrTiO 3substrate Baruch Rosenstein,1,2,B. Ya. Shapiro,3,†I. Shapiro,3and Dingping Li4,5,‡ 1Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, R.O.C. 2Physics Department, Ariel University, Ariel 40700, Israel 3Physics Department, Bar-Ilan University, 52900 Ramat-Gan, Israel 4School of Physics, Peking University, Beijing 100871, China 5Collaborative Innovation Center of Quantum Matter, Beijing, China (Received 27 January 2016; revised manuscript received 27 April 2016; published 11 July 2016) Pairing in one-atomic-layer-thick two-dimensional electron gas (2DEG) by a single ﬂat band of high-energy longitudinal optical phonons is considered. The polar dielectric SrTiO 3(STO) exhibits such an energetic phonon mode and the 2DEG is created both when one unit cell FeSe layer is grown on its (100) surface and on theinterface with another dielectric like LaAlO 3(LAO). We obtain a quantitative description of both systems solving the gap equation for Tcfor arbitrary Fermi energy /epsilon1F, electron-phonon coupling λ, and the phonon frequency /Omega1, and direct (random-phase approximation) electron-electron repulsion strength α.T h ef o c u si s on the intermediate region between the adiabatic, /epsilon1F>> /Omega1 , and the nonadiabatic, /epsilon1F<< /Omega1 ,r e g i m e s .T h e high-temperature superconductivity in single-unit-cell FeSe/STO is possible due to a combination of threefactors: high-longitudinal-optical phonon frequency, large electron-phonon coupling λ∼0.5, and huge dielectric constant of the substrate suppression the Coulomb repulsion. It is shown that very low density electron gas in theinterfaces is still capable of generating superconductivity of the order of 0.1 K in LAO/STO. DOI: 10.1103/PhysRevB.94.024505 I. INTRODUCTION Single layer of iron selenide (FeSe) grown on a strong polar insulator SrTiO 3(001) (STO) exhibits superconductivity [1–6] at surprisingly high temperatures (70 K to 100 K). This is an order of magnitude larger than the parent bulkmaterial with the superconducting transition temperature [ 7] T cof 8 K. This suggests that the dominant mechanism of creation of the superconductivity in the FeSe layer mightdiffer from that of the bulk FeSe and is caused by inﬂuenceof the STO substrate. To strengthen this point of view, thehigh-resolution angle-resolved photoemission spectroscopy(ARPES) experiments [ 5] and the ultrafast dynamics [ 3] demonstrated the presence of high-energy phonons in STO.The frequency of the oxygen longitudinal optical (LO) modereaches /Omega1≈100 meV . In addition, it turns out that the phonons couple strongly to the electrons in the FeSe layer(the coupling constant was estimated to be [ 3]λ∼0.5, much larger than in the parent material, λ=0.19). The band is ﬂat with only a small momentum transfer to electrons.This identiﬁcation is supported by the earlier ARPES onSTO surface states, which shows a phonon-induced humpat approximately 100 meV away from the main band andthrough inelastic neutron scattering [ 8]. The role of substrate in assisting superconductivity is not limited to generation of phonons. The polar STO has a huge dielectric constant (estimated to be above /epsilon1=1000 on the surface) and hence suppresses Coulomb repulsion inside the FeSe layer. The nature of electronic states within the FeSe layer is by now quite settled experimentally. The Fermi surface of the vortexbar@yahoo.com †shapib@biu.ac.il ‡lidp@pku.edu.cnsingle unit cell (1UC) consists of two electron pockets centeredaround the crystallographic M-point (Brillouin zone corners)with a band bottom below the Fermi level [ 5]/epsilon1 F=60 meV . This means that electrons form a two-dimensional electrongas (2DEG) with small chemical potential. The novelty ofthe superconducting system is that the occupied states areclose to the band edge, very far from the classic case. In bothconventional, Bardeen–Cooper–Schrieffer (BCS) and uncon-ventional superconductors the chemical potential is the largestenergy scale in the problem (even in quasi-2D high- T ccuprates the chemical potential is an order of magnitude higher).The scanning tunneling microscope (STM) experiment [ 6] indicates that the order parameter is gapped (hence no nodes)and, in addition, the quasiparticle interference pattern due tomagnetic and nonmagnetic impurities demonstrates that thereis no sign change of the order parameter between the twoelectron pockets. Hence the in -plane order parameter has thes-wave symmetry across the Fermi surface like conventional superconductors ( s ++in notations adopted for pnictides [ 9]) An early theory [ 10] focused on the screening due to the STO ferroelectric phonons on antiferromagnetic spinﬂuctuations mediated Cooper pairing in parent material FeSe.It suggested that the phonons signiﬁcantly enhance the Cooperpairing and even might change the pairing symmetry. For theelectron-phonon coupling λ∼1 the enhancement was large, although perhaps not enough to explain the experiment. Whenthe interpocket electron-phonon scattering is also strong,opposite-sign pairing will give way to equal-sign pairing. Later[5], it was suggested that the interfacial nature of the coupling assists superconductivity in most channels, including thosemediated by spin ﬂuctuations. Another idea [ 11] is to use both the electron pockets at the Fermi surface band and the “incipient” hole bandbelow it also found in ARPES, namely generalizing to themultiband model. The conclusion was that “a weak bare 2469-9950/2016/94(2)/024505(11) 024505-1 ©2016 American Physical SocietyROSENSTEIN, SHAPIRO, SHAPIRO, AND LI PHYSICAL REVIEW B 94, 024505 (2016) phonon interaction can be used to create a large Tc, even with a spin ﬂuctuation interaction which may be weakened by theincipient band.” The difﬁculty is that the forward scatteringnature of the essential phonon processes then means that LOphonons cannot contribute to the interband interaction. Gorkovconsidered [ 12] polarization on the surface, screening, and the STO surface LO phonon pairing. His conclusion is thatthe LO phonon mediated pairing alone cannot account forsuperconductivity at such high T c. The small chemical potential is typical for the STO systems. Another related superconducting (with much lowerT c) 2DEG system with even much smaller chemical potential is the LaAlO 3(LAO)-STO interface observed earlier [ 13]. The microscopic origin of the superconductivity in the LAO/STOsystem is already quite clear [ 14]. It is the BCS-like s- wave pairing attributed to the same LO phonon modesdiscussed above in context of the 1UC FeSe /STO system. Spin ﬂuctuations seem not to play any role in the pairingleading to superconductivity. The phase diagram of LAO/STOis qualitatively similar to the dome-shaped phase diagram ofthe cuprate superconductors: In the underdoped region, thecritical temperature increases with charge carrier depletion. The theoretical effort to understand the LAO/STO system [15] has resulted in the realization that the Migdal-Eliashberg theory of superconductivity, valid when the phonon frequen-cies are much smaller than the electron Fermi energy, shouldbe generalized. This is not the case for polar crystals likeSTO with sufﬁciently high optical-phonon frequencies, and,consequently, the dielectric function approach proposed longago by Kirzhnits [ 16] and developed in Ref. [ 17] proved to be useful. It was shown that the plasma excitations are importantat larger μ(reduce the electron-phonon coupling) and enable us to explain the nonmonotonic behavior of T cas function of bias that changes chemical potential. In this paper we further develop a theory of super- conductivity in 1UC FeSe/STO and LAO/STO based onthe phononic mechanism, including effects of the screenedCoulomb repulsion. In the ﬁrst stage , a simple model of 2DEG with pairing mediated by a dispersionless LO phononsis proposed with Coulomb repulsion assumed to be completelyscreened by huge polarization of STO ( /epsilon1∼3000 in 1UC FeSe/STO). In this case, the gap equations of the Frohlichmodel can be reduced (without approximations) to an integralequation with one variable only and are solved numerically forarbitrary Fermi energy /epsilon1 F, phonon frequency /Omega1, and electron- phonon coupling λ< 1. An expression for the adiabatic and nonadiabatic limits are derived and results for Tccompare well with experiments on 1UC FeSe/STO. Then, in the second stage , we include the random-phase approximation (RPA) screened Coulomb repulsion (for somewhat smaller valuesof dielectric constants are estimated [ 18]t ob e /epsilon1=186 on the STO side and /epsilon1=24 on the LAO side) and solve a more complicated gap equations numerically (without making useof the Kirzhnits ansatz) for various /epsilon1 Fand Coulomb coupling constant. Both the adiabatic, /epsilon1F>> /Omega1 (conventional BCS), and the nonadiabatic, /epsilon1F<< /Omega1 , cases are considered and compared with the local model studied earlier in the contextof Bose-Einstein Condensation (BEC) physics [ 19–22]. The Coulomb repulsion results in signiﬁcant reduction or evensuppression of superconductivity. A phenomenological modelfor dependence of /epsilon1 Fandλon electric ﬁeld for the LAO/STO is proposed. The paper is organized as follows. The basic 2DEG phonon superconductivity model is introduced in Sec. II.T h e general Gaussian approximation for weak electron-phononinteractions and RPA screening is described in Sec. III.T h e superstrong screening case (neglecting Coulomb repulsionaltogether) case is solved Sec. IV. The same calculation is performed using the Kirzhnits approach in Sec. V. The general case including the RPA screened Coulomb repulsion is inves-tigated numerically in Sec. VI. The phenomenology of 1UC FeSe/STO and LAO/STO and comparison with experimentsare discussed in Sec. VII followed by the Discussion and Summary. Appendices A and B contain the derivation ofGorkov equations and the 2D RPA neutralizing backgroundcontribution, respectively. II. THE LO PHONON MODEL OF 2D PAIRING As mentioned above, various STO systems including 1UC FeSe/STO (medium to low density) and interface LAO/STOthe (very low density) electron gas appears localized in a planeof width of one unit cell (in FeSe layer or on the STO side,respectively). The Hamiltonian of 2D electron gas containsthree parts: H=H e+Hph+He−ph,( 1 ) where He=/integraldisplay rψ† σ/parenleftbigg −/planckover2pi12∇2 2m−μ/parenrightbigg ψσ+1 2/integraldisplay r.r/primen(r)v/parenleftbig r−r/prime/parenrightbig n/parenleftbig r/prime/parenrightbig , (2) andψ† σ,ψσare the creation and annihilation operators in 2D, r=(x,y). The charge density operator is n(r)=ψσ†(r)ψσ(r),( 3 ) andμis the chemical potential (Fermi energy). The electron- electron interactions, not related to the crystalline lattice,are described by potential v(r). The electrostatics on the surface/interface is quite intricate [ 18], and we approximate it by the Coulomb repulsion: v(r)=e 2 /epsilon1r,( 4 ) where /epsilon1is an effective 2D dielectric constant of the system. As mentioned in the Introduction, the effective dielectric constantis huge in STO at low temperatures due to the ionic movements. Crystal vibrations in STO are highly energetic. The optical phonon mode [ 8,14] with frequency near /Omega1=100 meV is most probably associated with pairing attractive electron-electron force is the ferroelectric LO that involves the relativedisplacement of the Ti and O atoms. The high-energy STOoxygen LO phonon band mode is separated from all the otherphonon bands by a substantial energy gap [ 8]. The single branch of the optical phonons described by the bosonic ﬁeld [23]φ(r)=/summationtext k1√ 2(b† ke−ikr+bkeikr). The phonon part of the Hamiltonian therefore is Hph=1 2/integraldisplay r,r/primeφ(r)vph/parenleftbig r−r/prime/parenrightbig φ/parenleftbig r/prime/parenrightbig ,( 5 ) 024505-2SUPERCONDUCTIVITY IN THE TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 94, 024505 (2016) where the phonon energy density vph(r−r/prime) for the nearly ﬂat LO band is approximately local: vph(r)=/planckover2pi1/Omega1δ(r).( 6 ) Experiments demonstrated a substantial electron-phonon cou- pling g. In fact, the collective mode energy is greater or comparable to the width of the electron band. Importantly,the electron-phonon coupling allows only a small momentumtransfer to the electron, H e−ph=g/integraldisplay rn(r)φ(r).( 7 ) Despite the simpliﬁcations, the model is far from being solvable and standard approximations are applied in thefollowing section to obtain the critical temperature of the su-perconductor. Various “bare” parameters like effective masses,/Omega1, and the electron-electron and electron-phonon couplings are renormalized as the interaction effects are accounted for. III. THE PAIRING EQUATIONS A. Matsubara action We use the Matsubara time τ(0<τ< /planckover2pi1/T) formalism [23] with action corresponding to the Hamiltonian Eq. ( 1) (set- ting/planckover2pi1=1),A[ψ,φ]=Ae[ψ]+Aph[φ]+Ae−ph[ψ,φ],with Ae=/integraldisplay r,τψ∗ σ(r,τ)D−1ψσ(r,τ) +1 2/integraldisplay r,r/prime,τn(r,τ)v(r−r/prime)n(r/prime,τ) Aph=1 2/integraldisplay r,r/prime,τφ(r,τ)d−1φ(r/prime,τ); Ae−ph=g/integraldisplay r,τn(r,t)φ(r,t).( 8 ) Here the electron Green’s function is D−1=∂τ−∇2 2m−μ,( 9 ) while that of the phonon ﬁeld is d−1=/parenleftbig −∂2 τ+/Omega12/parenrightbig δ(r−r/prime). (10) In Fourier space the action reads Ae=/summationdisplay pωψσ∗ pωD−1 pωψσ pω +1 2/summationdisplay pωp 1p2ω1ω2vpψσ∗ p1ω1ψσ p1−p,ω 1−ωψρ∗ p2ω2ψρ p2+p,ω 2+ω; Aph=1 2/summationdisplay kωφ∗ kωd−1 ωφkω;Ae−ph =g/summationdisplay pp1ωω 1ψσ∗ p1ω1ψσ p1−p,ω 1−ωφpω (11) with electronic, D−1 p,ω=iω+εp;εp=p2/2m−μ, (12)and optical phonon, d−1 ω=ω2+/Omega12 /Omega12, (13) propagators, respectively. The fermionic Matsubara frequen- cies are ωn=πT(2n+1), while for bosons ωn=2πTn with nbeing an integer. In 2D vp=2πe2 /epsilon1p. (14) The action can be treated with the standard Gaussian approxi- mation. B. The pairing equations The electronic action is obtained by integration of the partition function over the phonon ﬁeld, Ze[ψ]=/integraldisplay φe−A[ψ,φ]=e−Aeff e[ψ]. (15) The Gaussian integral is Aeff e[ψ] =/summationdisplay ωpψσ∗ pωD−1 pωψσ pω +1 2/summationdisplay ωω 1ω2pp1p2Vpωψσ∗ p1−p,ω 1−ωψσ p1ω1ψρ∗ p2ω2ψρ p2−p.ω 2−ω, (16) where Vpω=VRPA pω+Vph ω. The part of the effective electron- electron attraction due to phonons is Vph ω=−g2/Omega12 ω2+/Omega12. (17) To take into account screening, we made the replacement vp→ VRPA pω(the random-phase approximation) in 2D, VRPA pω=vp/bracketleftbigg 1+Nmv p π/parenleftBig 1−x//radicalbig x2+1/parenrightBig/bracketrightbigg−1 , (18) where x=\|ω\|/(vFp) with v2 F=2μ/m . Performing the standard Gaussian approximation aver- aging, see Appendix A, one arrives at the Gorkov equa- tions for the normal, /angbracketleftψ↑I† kωψ↓J qν/angbracketright=δω−νδk−qδIJGkω(I,J= 1,..., N are ﬂavors), and the anomalous, /angbracketleftψ↑I kωψ↓J qν/angbracketright= δω+νδk+qδIJFkω, Greens functions. The result is −/Delta1∗ kωFkω+D∗−1 kωGkω=1 (19) and /Delta1kωGkω=−D−1 kωFkω, (20) where the gap function is deﬁned by /Delta1kω=/summationdisplay p1ω1Vp1−k,ω 1−ωFp1ω1. (21) Near the critical point one can neglect higher orders in /Delta1in Eq. ( 19), resulting in G=D∗. Substituting this into Eq. ( 20), one gets: /summationdisplay pν/vextendsingle/vextendsingleDpν/vextendsingle/vextendsingle2Vp−k,ν−ω/Delta1pν=−/Delta1kω. (22) 024505-3ROSENSTEIN, SHAPIRO, SHAPIRO, AND LI PHYSICAL REVIEW B 94, 024505 (2016) Using the explicit form of the propagator D,Eq. ( 12), the equation takes a ﬁnal form: /summationdisplay pm2NT ω2m+ε2pVp−k,m−n/Delta1pm=−/Delta1kn. (23) C. Simpliﬁcation of the integral equations for critical temperature for the s-wave pairing Transforming to polar coordinates and using rotation invari- ance,/Delta1pν=/Delta1pν,p=\|p\|, and then changing the variables to εp=p2/2m−μ, the electronic part of the kernel of Eq. ( 23) is /integraldisplay/Lambda1−μ ε2=−μmNT π/summationdisplay n21 ω2n2+ε2 2Pε1ε2;n1−n2/Delta1ε2n2=−/Delta1ε1n1. (24) Here/Lambda1is an ultraviolet cutoff of the order of atomic energy scale /planckover2pi12/2ma2with lattice spacing a. The phonon part of the kernel, Pε1,ε2,n=PRPA ε1,ε2,n+Pph n,i s Pph n=−g2/Omega12 ω2n+/Omega12, (25) while in the screened Coulomb part is PRPA ε1,ε2,n =e2 /epsilon1/integraldisplay2π φ=0/braceleftBigg √2(s−rcosφ)+ +2e2 /epsilon1/bracketleftBig 1−\|ωn\|//radicalbig ω2n+4μ(s−rcosφ)/bracketrightBig/bracerightBigg−1 . (26) This formula along with the treatment of the neutralizing background is derived in Appendix B.H e r ew eh a v eu s e d abbreviations s=ε1+ε2+2μ; r=2/radicalbig (ε1+μ)(ε2+μ). (27) To symmetrize the kernel viewed as a matrix, one makes rescaling of the gap function ηεn=1/radicalbig ω2n+ε2/Delta1εn, (28) leading to eigenvalue equation /integraldisplay/Lambda1−μ ε2=−μ/summationdisplay n2Kε1n1;ε2n2ηε2n2=ηε1n1, (29) where the symmetric matrix is Kε1n1;ε2n2=−mNT π1/radicalBig ω2n1+ε2 1/radicalBig ω2n2+ε2 2Pε1ε2,n1−n2. (30) Critical temperature is obtained when the largest eigenvalue of the matrix Kis the unit. This was done numerically by discretizing variable ε. The numerical results for the full model are presented in Sec. IV; however, since the screening of the STO is very strong, we ﬁrst neglect the Coulomb repulsionaltogether. This allows a signiﬁcant simpliﬁcation.IV . SUPERCONDUCTIVITY IN THE LO PHONON MODEL In this case, the theory Eqs. ( 2) and ( 5) has three parameters (in addition to temperature), the optical phonon frequency /Omega1, the electron-phonon coupling g, and chemical potential μ. We ﬁrst relate the bare coupling gto the “binding energy Ec” conventionally determined in the BCS-BEC crossover studies [ 19,21,22]. Then, since this simpliﬁed model will be applied to the 1UC FeSe on STO, one prefers to parametrizethe electron gas via carrier density nrelated to the Fermi energy by /epsilon1 F=π/planckover2pi12n/m instead of chemical potential μ. Following the standard practice, Tcis found by solving the second Gorkov equation [Eq. ( 22)]. This is compared with a simpler Kirzhnits approach applied to the present case inthe next section. To simplify the presentation and withouttoo much loss of generality we take the number of ﬂavorsN=1. A. Binding energy It is customary [ 19,22] to relate the electron-phonon coupling gto the energy of the bound state Eb≡2Ec created by this force in quantum mechanics in vacuum (the two-particle sector of the multiparticle Hilbert space). Weuse the binding energy to estimate the parameter rangein which chemical potential μapproaches the Fermi en- ergy/epsilon1 Fdeﬁned above. In 2D the threshold scattering ma- trix element for total energy Eat zero momentum obeys the integral Lippmann-Schwinger equation for scatteringamplitude: /Gamma1(ω,ν, 2E)=−Vph ω−ν−1 2π/integraldisplay ρVph ω−ρf(ρ,E)/Gamma1(ρ,ν,2E), (31) where f(ρ,E)=1 (2π)2/integraldisplay p1 p2/2m+E+iρ1 p2/2m+E−iρ =m 2π/integraldisplay/Lambda1 ε=E1 ε2+ρ2=m 4\|ρ\|/parenleftbigg 1−2 πarctanE \|ρ\|/parenrightbigg . (32) The equation Eq. ( 31) coincides with the sum of “chain diagrams” at zero chemical potential in the many-body theorywith/Gamma1being the “renormalized coupling” [ 24]. The bound state (there is only one such bound state in 2D) with bindingenergy 2 E cis found as a singularity of /Gamma1(ω,ν, 2E). It occurs at an energy for which the matrix of the linear equation ( 31) has zero eigenvalue, so the eigenvector ψ(ρ) obeys /integraldisplay ρ/bracketleftBig 2πδ(ω−ρ)+Vph ω−ρf(ρ,Ec)/bracketrightBig ψ(ρ)=0. (33) Changing the variables, ψ(ρ)=f(ρ,E)−1/2η(ρ), this equa- tion can be presented as the unit eigenvalue problem mg2 2π/integraldisplay ρK(ω,ρ)η(ρ)=η(ω), (34) 024505-4SUPERCONDUCTIVITY IN THE TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 94, 024505 (2016) FIG. 1. The 2D binding energy per electron of two electrons in the bound state created by the attraction due to LO dispersionless phonon branch with frequency /Omega1. The (bare) coupling strength λis in a wide range, λ∼0–3.5. The essential exact dependence found numerically (dots) is compared with weak coupling (the solid line) and results obtained using the local model (dashed line). with a symmetric kernel K(ω,ρ) =1 4/radicalBigg 1 \|ω\|/parenleftbigg 1−2 πarctanEc \|ω\|/parenrightbigg1 \|ρ\|/parenleftbigg 1−2 πarctanEc \|ρ\|/parenrightbigg ×/Omega12 (ω−ρ)2+/Omega12. (35) It turns out that the unit eigenvalue is the maximal eigenvalue of this positive deﬁnite matrix. The discretized version of Eq. ( 34) was diagonalized numerically. The results are presented inFig. 1. Solution found numerically is well ﬁtted by 2π mg2=1 λ≈1 2sinh−1/bracketleftbigg/Omega1(/Omega1+πEc) πEc(/Omega1+Ec)/bracketrightbigg , (36) where the 2D dimensionless electron-phonon coupling (per spin) is deﬁned as λ=mg2 2π/planckover2pi12. As will be demonstrated in the following subsections, the interesting range of couplings willobey/epsilon1 F>> E cand thus [ 22] we always replace μby/epsilon1F. It has the correct asymptotics at both weak and strong coupling, so Ec /Omega1=1 2s i n h/bracketleftbig2 λ/bracketrightbig⎧ ⎨ ⎩1−sinh/bracketleftbigg2 λ/bracketrightbigg +/radicalBigg/parenleftbigg 1−sinh/bracketleftbigg2 λ/bracketrightbigg/parenrightbigg2 +4 πsinh/bracketleftbigg2 λ/bracketrightbigg⎫ ⎬ ⎭. (37) At weak coupling, Ec//Omega1=2 πe−2/λ<< 1, (38)and hence one can use a local “instantaneous” electron-phonon interaction model, with Eq. ( 25) approximated by Pph n=−g2/Omega12 ω2n+/Omega12≈−g2θ(/Omega1−\|ωn\|), (39) to describe this limit. In the instantaneous model the electron- phonon interaction is assumed to vanish on the scale of /Omega1. In Eq. ( 42), it is clear that in whole range of parameters of interest the ﬁrst term is replaced by π/2. The results for Ec are consistent with the BEC literature [ 22], see the dashed line in Fig. 1. Note that the dimensionless pre-exponential factor in Eq. ( 38) is determined to be2 πwhile our notation for /Lambda1 does not coincide with the Chubukov notation, which is, inour notation, 2 π/Omega1. B. The energy independence of the gap function Equation ( 24) in the limit e2→0i s g2mT 2π/summationdisplay n2/integraldisplay/Lambda1−/epsilon1F ε2=−/epsilon1F1 ω2n2+ε2 2/Omega12 /parenleftbig ωn1−ωn2/parenrightbig2+/Omega12/Delta1ε2n2 =/Delta1ε1n1. (40) Since the left-hand side of the equation is independent of ε2,t h e gap function is independent of energy: /Delta1εn=/Delta1n. Substituting this, one gets a one-dimensional integral equation, λT/summationdisplay n2/Omega12 /parenleftbig ωn1−ωn2/parenrightbig2+/Omega12/Delta1n2/integraldisplay/Lambda1−/epsilon1F ε2=−/epsilon1F1 ω2n2+ε2 2 =λ/summationdisplay n2/Omega12f/parenleftbig ωn2/parenrightbig /parenleftbig ωn1−ωn2/parenrightbig2+/Omega12/Delta1n2=/Delta1n1, (41) where the integral is f(ω)=T \|ω\|/parenleftbigg arctan/Lambda1−/epsilon1F \|ω\|+arctan/bracketleftbigg/epsilon1F \|ω\|/bracketrightbigg/parenrightbigg . (42) Changing of variables, ηn=√f(ωn)/Delta1n, makes the kernel matrix of the integral equation, /summationdisplay n2Kn1n2(T)ηn2=ηn1, (43) symmetric, Kn1n2(T)=λ/radicalBig f/parenleftbig ωn1/parenrightbig f/parenleftbig ωn2/parenrightbig /Omega12 /parenleftbig ωn1−ωn2/parenrightbig2+/Omega12. (44) C. Numerical procedure and results The eigenvalue equation Eq. ( 43) is solved numerically by diagonalizing sufﬁciently large matrix Kn1n2(T). The index −Nω/2<n<N ω/2 with the value Nω=256 used. At this value of Nωthe results are already independent of the UV cutoff /Lambda1. The critical temperature for given λ,/epsilon1Fand/Omega1is determined from the requirement that the largest eigenvalue ofK(T) is 1. The results presented as functions of /epsilon1 Fin Fig. 2 in whole range of /epsilon1Fand Fig. 3for/epsilon1F</Omega1 . 024505-5ROSENSTEIN, SHAPIRO, SHAPIRO, AND LI PHYSICAL REVIEW B 94, 024505 (2016) FIG. 2. The critical temperature of a 2DEG-LO phonon super- conductor (the Coulomb repulsion is assumed to screened out by the substrate). Tcin units of the phonon frequency /Omega1is given as a function of the Fermi energy in whale range of /epsilon1F//Omega1for the dimensionless electron-phonon coupling (from top to bottom): λ=0.5,0.34,0.25. The adiabatic (BCS) limit is a dashed line. The solid line is the result of the local theory. D. Adiabatic and nonadiabatic (local interaction model) limits In the strongly adiabatic situation, /epsilon1F>> /Omega1 , one can take the/epsilon1F→∞ limit in which the matrix simpliﬁes, f(ω)≈πT \|ω\|, KBCS n1n2(T)=λ √\|n1+1/2\|\|n2+1/2\|/braceleftBig/bracketleftbig 2πT /Omega1(n1−n2)/bracketrightbig2+1/bracerightBig. (45) This can be ﬁtted by the phenomenological McMillan-like formula (dashed lines in Fig. 2), Tadiab c(λ)≈0.75/Omega1exp/bracketleftbigg −1 λ/bracketrightbigg . (46) In the opposite strongly nonadiabatic limit, Ec<< /epsilon1 F<< /Omega1 , the local model deﬁned in subsection A can be used. Thegap equation Eq. ( 41) for frequency independent /Delta1 n=/Delta1 FIG. 3. The critical temperature of a 2DEG-LO phonon supercon- ductor in the low temperatures range in units of the phonon frequency/Omega1forλ=0.5,0.34,0.25.Solid line is the result of the local theory.simpliﬁes into λ/summationdisplay/Omega1/(2πTc) n2=−/Omega1/(2πTc)f/parenleftbig ωn2/parenrightbig /Delta1=/Delta1. (47) The solution exists for λTc/summationdisplay/Omega1/(2πTc) n=−/Omega1/(2πTc)1 \|ωn\|/parenleftbiggπ 2+arctan/bracketleftbigg/epsilon1F \|ωn\|/bracketrightbigg/parenrightbigg =1.(48) At low temperatures the sum can be approximated by an integral λ π/integraldisplay/Omega1 ω=πTc1 ω/parenleftBigπ 2+arctan/bracketleftBig/epsilon1F ω/bracketrightBig/parenrightBig =1, (49) one gets the formula Tlocal c(λ)=/radicalbig Ec(λ)/epsilon1F=/radicalbigg 2/Omega1/epsilon1F πexp/bracketleftbigg −1 λ/bracketrightbigg . (50) The curves are given in Fig. 3(dashed lines) and compare well with the simulated result (circles) for λ=0.5,0.34,0.25 (from top to bottom). There exists an alternative approach to such calculations (beyond the Gaussian approximation adopted here), see Ref.[20] in which the correlator at zero chemical potential is subtracted. We do not use it, but very recently Chubukov et al. found [ 22] that for the local instantaneous model the results are identical. It is instructive to compare the direct numericalsimulation with a simpler approximate semianalytic Kirzhnitsmethod that is applied to the model in the following section. V . COMPARISON WITH THE KIRZHNITS ANSATZ A. Application of the Kirzhnits method to the LO phonon model Integral equations in general [Eqs. ( 43)] are very compli- cated and typically approximated by simpler one-dimensionalintegral equations. It was ﬁrst proposed long ago by Kirzhnits[16,17] and later developed for the dielectric function approach to novel superconductors [ 15]. In this section the units of /planckover2pi1=m=/Omega1=1 and physical frequency (not Matsubara) are used. Spectral representation of the dispersionless opticalphonon contribution to inverse dielectric constant is σ(k,E)=/epsilon1 e2λkδ/parenleftbig 1−E2/parenrightbig . (51) The gap equation for the quantity characterizing the anomalous average Fpdeﬁned by Kirzhnits [ 16] reads, /Phi1(p)=−e2 2π/epsilon1/integraldisplay kB(εk) \|p−k\| ×/bracketleftBigg 1−2/integraldisplay/Lambda1 E=0σ(\|p−k\|,E) E+\|εk\|+/vextendsingle/vextendsingleεp/vextendsingle/vextendsingle/bracketrightBigg /Phi1(k), (52) where B(εk)=tanh (εk/2Tc) 2εk. (53) Substituting Eq. ( 53) into Eq. ( 52), and transforming the variable kto the energy, one obtains: /Phi1(p)=λ/integraldisplay/Lambda1−/epsilon1F εk=−/epsilon1FB(εk) 1+\|εk\|+/vextendsingle/vextendsingleεp/vextendsingle/vextendsingle/Phi1(k). (54) 024505-6SUPERCONDUCTIVITY IN THE TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 94, 024505 (2016) FIG. 4. Comparison with the critical temperature of the Kirzhnits ansatz approximation for a wide range of Fermi energies. The brown dots are the same as in Fig. 2forλ=0.5 while the solid line is the result of the instantaneous (local) theory. The Kirzhnitsapproximation T ccalculated numerically is given by blue dots, while the dashed and solid blue lines are the weak-coupling approximation analytic results at leading and the next to leading order, respectively. Symmetrization of the kernel, /Phi1(p)=/radicalbigB(εp)ηp, one ob- tains: λ/integraldisplay/Lambda1−/epsilon1F ε2=−/epsilon1F√B(ε1)B(ε2) 1+\|ε1\|+\|ε2\|η2=η1. (55) This is solved numerically for /epsilon1F=0.5,1,5/Omega1, andλ=0.5 with the ultraviolet cutoff /Lambda1=15/Omega1in the upper limit of integral in Eq. ( 55) with number of values of energy Nε= 4000, so the step is smaller than ( /epsilon1F+/Lambda1)/Nε∼10−2.T h e results are presented in Figs. 4and5. It is possible to obtain a closed analytic expression only at weak coupling. FIG. 5. Comparison with the critical temperature of the Kirzhnits ansatz approximation for a small Fermi energies. The brown dots are the result of numerical solution of the gap equation and are the same as in Fig. 2forλ=0.5 and 0.34, where the solid line is the result of the instantaneous theory. The Kirzhnits approximation Tc, calculated numerically, is given by blue dots, while the dashed and solid blue lines are the weak-coupling approximation analytic results at leadingand the next to leading order, respectively.B. Weak coupling At small coupling the critical temperature can be estimated analytically using the asymptotic theory from Zubarev [ 25]: Tc=2 πeγE/epsilon1Fe−1 λeζ(/epsilon1F,λ), (56) where ζ(/epsilon1F,λ)=/integraldisplay∞ ε=−/epsilon1F1 2\|ε\|/bracketleftbiggφε 1+\|ε\|−/Theta1(/epsilon1F−ε)/bracketrightbigg . (57) Equation determining φε≡ηε/ηε=0for small temperatures is approximated in our case by: φε−λ\|ε\| 2(1+\|ε\|)/integraldisplay∞ ε/prime=−/epsilon1Fφε/prime (1+\|ε\|+\|ε/prime\|)(1+\|ε/prime\|) =1 1+\|ε\|. (58) This is solved iteratively to second order, φε=φ(0) ε+λφ(1) ε, φε=1 1+\|ε\|+λφ(1) ε φ(1) ε=1 2(1+\|ε\|)/braceleftbigg1+2/epsilon1F (1+/epsilon1F)−1 \|ε\|log(1+\|ε\|)2(1+/epsilon1F) 1+\|ε\|+/epsilon1F/bracerightbigg . (59) Substituting this into Eq. ( 57) one obtains ζ(/epsilon1F,λ)=ζ0(/epsilon1F)+λζ1(/epsilon1F)+O/parenleftbig λ2/parenrightbig ζ(0)(/epsilon1F)=/integraldisplay∞ ε=−/epsilon1F1 2\|ε\|/braceleftbigg1 (1+\|ε\|)2−/Theta1(/epsilon1F−ε)/bracerightbigg =−1 2/braceleftbigg1+2/epsilon1F 1+/epsilon1F+log[/epsilon1F(1+/epsilon1F)]/bracerightbigg . (60) The second correction, ζ(1)(/epsilon1F)=/integraldisplay∞ ε=−/epsilon1Fφ(1) ε 2\|ε\|(1+\|ε\|), (61) still can be calculated analytically via hypergeometric function but is cumbersome. It is regular and for λ=0.5 corrects the analytic result shown in Figs. 4and 5as a dotted line into the one (solid line) closer to numerical solution. The formulaworks better for the nonadiabatic regime, Fig. 5, than in the adiabatic limit, Fig. 4. The approximate formula neglecting the second-order correction in the adiabatic regime, /epsilon1 F>/Omega1 ,i s Tc=2 πeγE/epsilon1Fe−1 λexp/bracketleftbig −1−log[/epsilon1F]/bracketrightbig =2 πeγE−1/Omega1e−1 λ≈0.41/Omega1e−1 λ. (62) The coefﬁcient is signiﬁcantly smaller than the ﬁt to the numerical solution, Eq. ( 46). In the opposite nonadiabatic limit Tc=2 πeγE/epsilon1Fe−1 λexp/bracketleftbigg −1 2{1+log[/epsilon1F]}/bracketrightbigg =2 πeγE−1/2/radicalbig /Omega1/epsilon1Fe−1 λ≈0.69/radicalbig /Omega1/epsilon1Fexp/bracketleftbigg −1 λ/bracketrightbigg . (63) 024505-7ROSENSTEIN, SHAPIRO, SHAPIRO, AND LI PHYSICAL REVIEW B 94, 024505 (2016) FIG. 6. Suppression of the critical temperature of a 2DEG phonon superconductor the RPA screened Coulomb repulsion. Tcin units of the phonon frequency /Omega1forλ=0.32 is given as a function of the chemical potential for the following dimensionless effective Coulomb repulsion strength αdeﬁned in Eq. ( 64). From top to bottom: α= 0 (the phonon model, red dots), α=5×10−3(brown dots), α= 10−2(yellow), α=2×10−2(green), α=3×10−2(blue), α=4× 10−2(violet), α=5×10−2(pink), and α=6×10−2(dark red). The curves are well approximated by the interpolating formula, Eqs. ( 65). To conclude the critical temperature in the Kirzhnits approach is generally underestimated by 30% in the adiabaticlimit and is precise in the nonadiabatic limit. Within the rangeof applicability the general tendency is correct. Next we tacklea more complicated model incorporating the effect of thescreened Coulomb repulsion. VI. THE EFFECT OF THE COULOMB REPULSION The eigenvalue equation Eq. ( 29) with the kernel including the RPA dynamically screened Coulomb repulsion, Eq. ( 26), is solved numerically by diagonalizing sufﬁciently large matrixK n1ε1,n2ε2(T). In the presence of moderately screened Coulomb repulsion, to describe the LaAlO 3/STO (LaO/STO) interfaces, the chemical potential is practically equal to the Fermienergy /epsilon1 F. The integral over the angle φin Eq. ( 26) was performed numerically (720 subdivisions). The neutralizing backgroundwas subtracted (the screening is dynamic, so the interactionis generally still long range, see Appendix B). The Matsubara index is in the range −N ω/2<n<N ω/2 with the value Nω= 16 used. The energy cutoff was in the range /Lambda1=3/epsilon1F(for nonadiabatic values /epsilon1F=0.5, 1) and up to /Lambda1=15/epsilon1Fin the adiabatic regime. Number of values of energy Nε=256, so the step is smaller than ( /epsilon1F+/Lambda1)/Nε∼2.4×10−3. Convergence was checked against higher values of /Lambda1,Ne, andNω. The critical temperature for given λ,m,/epsilon1F, and /Omega1is determined from the requirement that the largest eigenvalueofK(T) is 1. The use units in which /planckover2pi1=/Omega1=m=1. In these units the Coulomb couplings become α=e 2m1/2 /epsilon1/Omega11/2/planckover2pi1. (64) For/Omega1=1000 K, m=me,/epsilon1=3000 one gets α=6×10−3. The results presented in Fig. 6in the Coulomb coupling range5×10−3–7×10−2are sufﬁcient for our purposes. One clearly observes the Coulomb suppression that is not homogeneous in/epsilon1 F.A t/epsilon1Fcomparable with /Omega1or slightly smaller (the smallest simulated value is /epsilon1F=0.5/Omega1), one observes that at larger α an approach to the BCS limit is slower. A reasonable interpolation formula for all the values is Tc(/Omega1,/epsilon1F,λ)=0.8/Omega1exp/bracketleftbigg −2 λ−1.2α/Omega1+3/epsilon1F /Omega1+6/epsilon1F/bracketrightbigg .(65) We use this formula to discuss the interface superconductivity in the next section. VII. APPLICATION TO SUPERCONDUCTIVITY IN 1D FESE/STO SUBSTRATE AND RELATED MATERIALS A. 1UC FeSe/STO Based on experiments described in the Introduction, the following parameters should be used in the simple LO modelof Sec. IV. The phonon frequency was estimated by ARPES [ 5] in the /Omega1=80–100 meV range and by the ultrafast dynamics [3]t ob e /Omega1=106 meV . The dimensionless electron-phonon coupling constant was estimated (using a model with a ﬂatphonon spectrum) from the intensity ratios in ARPES [ 5] to be λ=0.5,consistent with λ=0.48 from the ultrafast dynamics [ 3]. The critical temperature estimates were rather scattered and dependent on the method. While the criticaltemperature deduced from the gap in tunneling is T c=70 K, magnetization experiments [ 4] indicate that Tc=85 K and the ultrafast[ 3] dynamics gives Tc=68 K. The temperature was directly measured in transport[ 2] to be 100 K. The Fermi surface [ 5] for the electron pockets is located at /epsilon1F=60 meV . In the simpliﬁed model of Sec. II(neglecting completely the Coulomb repulsion due to the huge dielectric constantof STO) the only parameters determining T careλ,/Omega1, and /epsilon1F. This is presented in Figs. 2and3. Taking /Omega1=100 meV , /epsilon1F=60 meV , one obtains, for λ=0.5,Tc=77 K, see the dotted line in Figs. 2and3. This is within the experimentally possible range. The 2UC FeSe/STO already has three pocketsand resembles the parent material more than 1UC FeSe/STO. B. Interface superconductivity in LAO/STO In this case the dielectric constant is one order of mag- nitude smaller ( /epsilon10=186 on the STO side and /epsilon10=24 on the LAO side, see Ref. [ 18] where accurate electrostatics was considered) than in 1UC FeSe/STO. Consequently, theCoulomb repulsion cannot be neglected, especially in viewof very low T c∼0.2 K. Therefore we have to use the full model of Sec. IV. In this case one takes N=1 and effective mass m=1.65me(where meis the electron mass in a vacuum). Recently [ 14], the electron-phonon coupling and chemical potential were measured by tunneling fromthe underdoped to the overdoped region. Generally, in theunderdoped region, the chemical potential rises linearly withthe gate voltage V g,/epsilon1F(Vg)=μ0(1+ηVg),with the slope η= 1.8×10−3V−1and is saturated in the overdoped region at valueμ0=30 meV . The electron-phonon coupling apparently decreases very slowly, λ=λ0(1−γVg), where λ0=0.28 is the undoped value and γ=1.1×10−4V−1is the slope. Our 024505-8SUPERCONDUCTIVITY IN THE TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 94, 024505 (2016) FIG. 7. Tcas a function on gate voltage Vg. approximate formula Eq. ( 65) in this case gives the dependence Tc/parenleftbig Vg/parenrightbig =0.8/Omega1exp ×/bracketleftBigg −2 λ0/parenleftbig 1−γVg/parenrightbig −1.2α1+3μ0/parenleftbig 1+ηVg/parenrightbig 1+6μ0/parenleftbig 1+ηVg/parenrightbig/bracketrightBigg . (66) Taking a measured value for the LO4 mode /Omega1=99.3m e V lets us estimate the Coulomb repulsion constant as α=e2m1/2 /epsilon1eff/Omega11/2/planckover2pi1=0.09 for/epsilon1eff=200. Substituting these values, one obtains the ﬁt to experimental values of Ref. [ 14], see Fig. 7. Qualitatively, there are two conﬂicting tendencies at play. The reduction of the electron-phonon coupling with Vgreduces Tc, while the increase of /epsilon1F(the charging appears according to experiment only in the underdoped region) increases Tc. The overall effect is that in the underdoped case the secondtendency prevails, while in the overdoped only the ﬁrst exists.This explains the “dome” shape. VIII. DISCUSSION AND SUMMARY Pairing in one atomic layer thick two-dimensional electron gas on a strongly dielectric substrate by a single band ofhigh-energy longitudinal optical phonons is considered indetail. The phonon band is assumed to be nearly dispersionlesswith frequency /Omega1. The polar dielectric SrTiO 3exhibits such an energetic phonon mode and the 2DEG is created both whenone unit cell FeSe layer is grown on its (100) surface andon the interface with another dielectric like LaAlO 3. Both the adiabatic, /epsilon1F>> /Omega1 , and the nonadiabatic, /epsilon1F<< /Omega1 , cases are considered and compare well with conventional weakcoupling BCS and with the local instantaneous interactionmodel (describing the nonadiabatic regime close to the BECcrossover [ 19–22] still assuming that /epsilon1 F>> E c, where 2 Ec is the binding energy, so the pairing is the BCS type rather than BEC), respectively. The focus was, however, on theintermediate region. The reason is that in several novelmaterials this is precisely the case. In particular, in high T cone unit cell FeSe on STO the Fermi energy is a bit smaller than thephonon frequency /epsilon1F=0.65/Omega1. In interface superconductors like LaAlO 3/STO interfaces the ration is smaller /epsilon1F//Omega1∼0.3 still well above the nonadiabatic limit. It turns out that in thecrossover region the critical temperature decreases very slowlyas a function of /epsilon1 F,u pt o /epsilon1F=0.1/Omega1, see Figs. 2and3, and only then drops fast to zero. The critical temperature was calculated within the weak- coupling model of superconductivity. The theory was appliedto two different realizations of such a system: 1UC FeSe/STOand LaAlO 3/STO interfaces. There is a reduction in the nearly instantaneous electron-electron Coulomb repulsion due toscreening by STO, but we believe it is much smaller for theessentially retarded effective electron-electron interaction dueto phonons. This reduction due to STO is thus taken intoaccount by the electron-electron interaction. Similarly, weuse the “dressed” electron and phonon Green functions withparameters (electron density of states, the phonon frequency).The numerical solution of the gap equation at α=0w a s compared with an often-utilized Kirzhnits dielectric approachfor arbitrary ratio /epsilon1 F//Omega1. The validity of the Migdal theorem in the nonadiabatic case was not assumed (discussed recently inRef. [ 26]). This comparison demonstrated excellent agreement between two theories in the nonadiabatic range while in theadiabatic region the Kirzhnits theory gives lower T cthan the numerical solution of the gap equation. We conclude that, despite small electron concentration, very high critical temperatures observed recently are consistent withthe mostly phononic mechanism already due to combinationof two peculiar properties of the system. First, since theoptical phonon frequencies /Omega1are very large and electrons reside in small pockets, /Omega1is larger than /epsilon1 F. Second, due to the huge dielectric constant of STO the Coulomb repulsionis strongly suppressed inside the layer leading to small α. The required value of the electron-phonon coupling in thesuperconducting layer is λ∼0.5 in 1UC FeSe/STO and λ∼0.2i nL A O / S T O .I nl o w - T cLAO/STO the less-suppressed Coulomb repulsion results in signiﬁcant reduction or evensuppression of superconductivity. A phenomenological modelfor dependence of /epsilon1 Fandλon electric ﬁeld for the LAO/STO is proposed. The main insight from this work therefore is that a small value of /epsilon1Fis not an obstacle to achieve Tcof order 0 .1/Omega1as long as λis sufﬁciently large and the Coulomb repulsion is effectively suppressed by polarization of the 3D substrate. Note added . Very recent experiment [ 27] strengthens the assumptions made in the present work about the natureof superconductivity in 1UC FeSe on STO with measuredelectron-phonon coupling in the layer as high as λ=1. ACKNOWLEDGMENTS We are grateful J. Wang, C. Luo, J. J. Lin, M. Lewkowicz, and Y . Dagan for helpful discussions. The work of D.L. andB.R. was supported by NSC of R.O.C. Grant No. 98-2112-M-009-014-MY3 and he MOE ATU program. The work of D.L.was supported by the National Natural Science Foundation ofChina (Grant No. 11274018). 024505-9ROSENSTEIN, SHAPIRO, SHAPIRO, AND LI PHYSICAL REVIEW B 94, 024505 (2016) APPENDIX A: DERIV ATION OF THE PAIRING EQUATIONS We derive the Gorkov equations within the functional approach starting with the effective action [Eq. ( 16)]. The partition function as a functional of sources χσ pωis Z[χ]=/integraldisplay ψexp/bracketleftbigg −Ae[ψ]+/integraldisplay pω/parenleftbig ψσ pωχ∗σ pω+χσ pωψ∗σ pω/parenrightbig/bracketrightbigg . (A1) The free energy, F[χ]=− logZ[χ], deﬁnes the effective action and the “classical ﬁelds” via A(ψ)=F[χ]+/integraldisplay pω/parenleftbig ψσ pωχ∗σ pω+χσ pωψ∗σ pω/parenrightbig ; ψσ pω=δF[χ] δχ∗σpω,ψ∗σ pω=−δF[χ] δχσpω,( A 2 ) where the sources are expressed via the ﬁrst functional derivative of A, χσ pω=−δA[ψ] δψ∗σpω,χ∗σ pω=δA[ψ] δψσpω.( A 3 ) The inverse propagators, the second derivatives, form a Nambu matrix: /Gamma1σρ pωqν=δ2A δψρ qνδψσpω;/Gamma1σρ pωqν=δ2A δψρ qνδψσpω; /Gamma1σ∗ρ pωqν=δ2A δψρ qνδψσ∗pω.( A 4 )Green’s functions also form a Nambu matrix, Gρσ qνpω=/angbracketleftbig ψσ∗ pωψρ qν/angbracketrightbig =−δ2F δχρ∗ qνδχσ∗pω; Gρ∗σ∗ qνpω=−δ2F δχρ qνδχσω; Gρσ∗ qνpω=/angbracketleftbig ψσ pωψρ qν/angbracketrightbig =−δ2F δχρ∗ qνδχσpω.( A 5 ) The two Nambu matrices obey /Gamma1ACGCB=δAB, which con- stitute the Gorkov equations. Let us now calculate /Gamma1. The Gaussian average ﬁrst derivatives, assuming only anomalous averages, are χσ pω=D−1 pωψσ pω−Vp−p2,ω−ω2ψκ∗ p3ω3/angbracketleftbig ψσ p2ω2ψκ p−p2+p3,ω−ω2+ω3/angbracketrightbig . (A6) The second derivatives are /Gamma1σ∗ρ pωqν=δσρδωνδpqD−1 pω; /Gamma1σρ pωqν=Vq−p2,ω−ω1δ−p1−p2+q+pδω−ω1−ω2+ν/angbracketleftbig ψσ∗ p1ω1ψρ∗ p2ω2/angbracketrightbig . (A7) Using the translation symmetry, /angbracketleftbig ψ1 pωψ2 qν/angbracketrightbig =δω+νδp+qFpω, /Gamma1σρ pωqν=/angbracketleftbig ψ1∗ pωψ2 qν/angbracketrightbig =δσρδp+qδω+νD−1 pω,( A 8 ) the equation /Gamma1ACGCB=δABbecomes Eqs. ( 19) and ( 20). APPENDIX B: LONG-RANGE RPA SCREENED COULOMB REPULSION In Eq. ( 26) one detail was not presented: subtraction of the neutralizing background. Since at nonzero frequency the screened repulsion does not become short ranged, the neutralizing background should be taken into account. For our purposes the jelliummodel sufﬁces [ 23]. To this end, one needs the infrared cutoff L. 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PhysRevB.80.161402.pdf	Artiﬁcial atoms in interacting graphene quantum dots Wolfgang Häusler1,2and Reinhold Egger1 1Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany 2Physikalisches Institut, Albert-Ludwigs-Universität, D-79104 Freiburg, Germany /H20849Received 14 September 2009; published 7 October 2009 /H20850 We describe the theory of few Coulomb-correlated electrons in a magnetic quantum dot formed in graphene. While the corresponding nonrelativistic /H20849Schrödinger /H20850problem is well understood, a naive generalization to graphene’s “relativistic” /H20849Dirac-Weyl /H20850spectrum encounters divergencies and is ill deﬁned. We employ Such- er’s projection formalism to overcome these problems. Exact diagonalization results for the two-electronquantum dot, i.e., the artiﬁcial helium atom in graphene, are presented. DOI: 10.1103/PhysRevB.80.161402 PACS number /H20849s/H20850: 73.22. /H11002f, 73.21.La, 78.67.Hc The recent spectacular progress in preparing and usefully employing individual carbon monolayers of graphene1,2con- tinues to stimulate much interest across different scientiﬁccommunities, including material science, applied physics,chemistry, condensed-matter physics and mathematics. Bal-listic electronic motion with quantum coherence extendingover micrometer distances has been achieved in several ex-periments /H20849see, e.g., Ref. 3/H20850. The low-energy physics close to a single Kpoint can then be described by a two-component Dirac-Weyl Hamiltonian, 2,4 H0=vF/H9268·/H20873p−e cA/H20874, /H208491/H20850 suggesting an easily accessible condensed-matter realization of relativistic quantum mechanics. In Eq. /H208491/H20850,/H9268denotes the vector of the ﬁrst two Pauli matrices for the “isospin” encod-ing the two sublattices, the Fermi velocity is vF/H11015106m/s, and we include a static vector potential A/H20849r/H20850describing /H20849pos- sibly inhomogeneous /H20850magnetic ﬁelds. Since graphene’s ef- fective ﬁne-structure constant is /H9251/H110151, present interest has also turned to Coulomb interaction effects.2According to recent Monte Carlo simulations5and analytical arguments,6 sufﬁciently strong interactions may even open a sizeablebulk gap in the Dirac fermion spectrum. Here we study the properties of few Coulomb-correlated electrons conﬁned to a ﬁnite-size quantum dot formed ingraphene. Using electrostatically formed quantum dots insemiconductor devices, such “artiﬁcial atoms” have been in-tensely studied over the past two decades, bothexperimentally 7and theoretically.8In graphene dots formed by electrostatic gating, however, carriers can usually escapedue to the /H20849recently observed 9/H20850Klein tunneling phenomenon, and only quasibound states may appear.10An alternative is to employ lithographically deﬁned quantum dots,11where de- tailed information on ground- and excited-state propertieshas been obtained from transport spectroscopy. Unfortu-nately, the boundary of lithographically fabricated graphenedots is rather disordered and difﬁcult to model. 12On the other hand, suitable and realizable inhomogeneous magneticﬁelds can conﬁne Dirac fermions, 13,14promising to yield tun- able and well-deﬁned magnetic graphene dots. A more challenging difﬁculty to theory arises when trying to generalize Eq. /H208491/H20850to a ﬁrst-quantized many-particle de-scription. The ﬁrst-quantized approach has turned out to be very efﬁcient and convenient for the case of Schrödingerelectrons in semiconductor-based artiﬁcial atoms. 8For the “relativistic” graphene case, the problem arises from the un-boundedness of Eq. /H208491/H20850, in contrast to the corresponding Schrödinger operator /H20849p−e cA/H208502/2m/H11569. While Eq. /H208491/H20850can still be used within effective single-particle approximations suchas the Hartree-Fock approach, 15variational schemes,16or density-functional theory,17the full N-particle problem /H20849for small N/H110221/H20850with Eq. /H208491/H20850for the kinetic part suffers from the so-called “Brown-Ravenhall disease.18,19” Roughly speaking, the unbounded spectrum allows particles to lose arbitraryamounts of energy by transferring their energy in /H20849real /H20850scat- tering events to other particles. The resulting divergent den-sity of states prohibits, for example, the direct use of exactdiagonalization /H20849ED/H20850methods. This difﬁculty of the Dirac equation has been known for half a century. 18To “cure” this “disease,” we follow a proposal by Sucher19and conﬁne the Hilbert space to positive-energy eigenstates through suitablydeﬁned projectors /H20851cf. Eq. /H208495/H20850below /H20852. While we formulate this approach for the magnetic dot only, the general conceptsremain applicable for almost arbitrary graphene dots. Theprojection method then allows, for instance, to apply numeri-cal techniques to the relativistic N-particle problem. In this work, we present ED results for the many-body energy spec-trum of the artiﬁcial helium atom /H20849N=2/H20850in graphene. Let us ﬁrst specify the model discussed here /H20849we set /H6036=1/H20850. In cylinder coordinates, we consider the spherical and parabolic magnetic ﬁeld proﬁle oriented perpendicular to thegraphene plane /H20849with Ain symmetric gauge /H20850, B/H20849r, /H9272/H20850=c e/H9275B2r2ez,A/H20849r,/H9272/H20850=1 4/H9275B2r3/H20898− sin/H9272 cos/H9272 0/H20899. /H208492/H20850 The inverse length scale /H20881/H9275Btunes the ﬁeld inhomogeneity. The dimensionless radial coordinate is /rho1=r/H20881/H9275B, and energy /H20849/H9255/H20850is measured in units of vF/H20881/H9275B./H20851Physical units are recov- ered from /H20881B/H20849/rho1/H20850/Tesla= /H20849vF/H20881/H9275B/26 meV /H20850/rho1./H20852Such mag- netic proﬁles can be generated with reasonable accuracy us-ing suitable lithographically deﬁned ferromagnetic ﬁlmsdeposited on top of the graphene layer after formation of aprotective oxide layer. 20Upper and lower components of eigenspinors /H20841/H9274/H9263/H208490/H20850/H20856to Eq. /H208491/H20850must then differ by one orbitalPHYSICAL REVIEW B 80, 161402 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 1098-0121/2009/80 /H2084916/H20850/161402 /H208494/H20850 ©2009 The American Physical Society 161402-1angular-momentum quantum number /H20849m/H20850due to conserved total angular momentum.13With real functions /H9278m/H20849/rho1/H20850and /H9273m+1/H20849/rho1/H20850, the radial part of /H20841/H9274/H9263/H208490/H20850/H20856 is /H11008/H20849/H9278m/H20849/rho1/H20850,i sgn /H20849/H9255/H20850/H9273m+1/H20849/rho1/H20850/H20850T, where Eq. /H208491/H20850yields the radial equations /H20898−/H9255 /H11509/rho1+m+1 /rho1−/rho13 4 −/H11509/rho1+m /rho1−/rho13 4−/H9255/H20899/H20873/H9278m/H20849/rho1/H20850 /H9273m+1/H20849/rho1/H20850/H20874=0 , /H208493/H20850 which cannot be solved analytically. We here carry out ED calculations, later on including the Coulomb interaction, andthus solve Eq. /H208493/H20850numerically. It is convenient to employ the standard Darwin-Fock states 8as complete orthonormal func- tion set in two dimensions /H208492Ds /H20850to expand /H9278mand/H9273m+1in Eq. /H208493/H20850. To reduce the number nmaxof required basis func- tions /H20849for given m/H20850when approximating the /H20841/H9274/H9263/H208490/H20850/H20856to the de- sired accuracy, we have included a tunable width parameterin the Darwin-Fock states. Figure 1displays the resulting eigenenergies as a function of the orbital angular momentumm. As expected, the spectrum is electron-hole symmetric, and form/H113500, a zero-energy level develops. This zero-energy level is nondispersing /H20849precisely like a quantum Hall level /H20850, despite of the inhomogeneous magnetic ﬁeld which impliesthe nontrivial mdependence of all other energy levels. Note that for the corresponding Schrödinger case with a parabolicmagnetic ﬁeld, the zero-energy level is absent. Next we consider Ninteracting electrons in such a graphene dot. A naive approach is to consider the ﬁrst-quantized Hamiltonian H= vF/H20858 j=1N /H9268j·/H20851pj−A/H20849rj/H20850/H20852+/H20858 i/H11021j/H9251vF /H20841ri−rj/H20841, /H208494/H20850 where the ﬁne-structure constant is /H9251=e2//H20849/H92600vF/H20850. For typical substrate materials, the dielectric constant is /H92600/H110151.4–4.7, resulting in /H9251/H110150.6–2. We mention in passing that the “Wigner molecule” regime21seems out of reach in graphenedots, since both the kinetic and the potential energy show identical scaling when changing the density.22Moreover, we neglect the Zeeman term which is very small in graphene.14 Up to the spin and K-point indices,23many-body spinors then have 2Ncomponents. For the related Schrödinger dot /H20849HS/H20850, conﬁnement of electrons is usually achieved by a para- bolic electrostatic potential,7and the many-particle descrip- tion analogous to Eq. /H208494/H20850simpliﬁes considerably owing to the generalized Kohn theorem.24According to this theorem, HS=Hcm+Hrelseparates into two commuting parts describ- ing center-of-mass /H20849Hcm/H20850and relative /H20849Hrel/H20850motion. Then Hcmis just a 2D harmonic oscillator, while Hrelcontains all Coulomb interaction effects. In addition, Hrelconserves an- gular momentum, as does Hcm. Taking N=2 as example, in effect only a one-dimensional quantum problem for the ra-dial motion of H relremains to be solved. In contrast, Eq. /H208494/H20850 does not beneﬁt from Kohn’s theorem, and only the totalangular momentum remains conserved as dictated by isot- ropy. Therefore, while the additional spinor structure alreadyincreases the rank of the Hamiltonian matrix in the Diraccase by a factor 2 N, the rank grows even more severely be- cause neither Hcmnor a conserved angular momentum of Hrel can be separated off the problem. For N=2 /H20849graphene he- lium /H20850, we needed up to nmax=14 states to reach sufﬁcient accuracy. In addition, contrary to the Schrödinger problem,particles may now exchange relative angular momentum /H9004m through the interaction. Owing to the exponential decay ofCoulomb matrix elements with /H20841/H9004m/H20841, it is sufﬁcient to take /H20841/H9004m/H20841/H113493, yielding an additional factor 7 N−1to the matrix size /H20849for Nparticles /H20850. For N=2, we then need to include 142/H110037/H1100322=5488 product basis states in total. Let us then address the more fundamental difﬁculty aris- ing already for N=2 when naively using Eq. /H208494/H20850. A closely related problem has been pointed out by Brown andRavenhall 18in a relativistic treatment of the helium atom: the Dirac equation analogous to Eq. /H208494/H20850does not possess normal- izable antisymmetric eigenstates in the two-particle Hilbertspace. This failure has its origin in the unbounded spectrumof the Dirac Hamiltonian, which allows for unlimited energyexchange among the particles. As a result, the density oftwo-particle states increases with Hilbert-space dimensionand ultimately diverges. This causes, e.g., divergent contri-butions in second-order perturbation theory. In consequence,neither the true Dirac equation nor the two-component vari-ant /H208491/H20850for graphene allow for naive many-particle generali- zations such as Eq. /H208494/H20850. To overcome this deﬁciency, Sucher 19proposed to restrict the /H20849antisymmetrized /H20850product Hilbert space to the positive-energy eigenspace for each par-ticle by means of a suitable projector /H9011 +, having in mind the original relativistic Dirac problem /H20849e.g., for natural helium /H20850 where a mass gap separates empty particle from ﬁlled anti-particle states. The method is applicable when a ﬁnite-energygap separates the empty states /H20849which can then be occupied by the Nelectrons under consideration /H20850from a ﬁlled sea below the chemical potential /H9262, as long as the interaction strength does not exceed this gap. In a homogeneous 2Dgraphene sheet, weakly interacting fermions are gapless sothat Sucher’s approach does not apply. On the other hand, forﬁnite-size quantum dots, such a gap is generally present. Forthe case shown in Fig. 1, either /H9262=0−or/H9262=0+are interest--8 -6 -4 -2 0 2 4 6 8m-8-6-4-202468ε FIG. 1. /H20849Color online /H20850ED results for the single-particle eigenen- ergies /H9255vs orbital angular momentum min a parabolic magnetic quantum dot in graphene /H20851see Eq. /H208493/H20850/H20852. The/H9255=0 levels are indicated as red ﬁlled circles, other levels are shown as black ﬁlled diamonds.WOLFGANG HÄUSLER AND REINHOLD EGGER PHYSICAL REVIEW B 80, 161402 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 161402-2ing candidates separating a ﬁlled Dirac sea from the levels on which N additional particles may reside. Taking /H9262=0−, they enter the zero-energy level in Fig. 1, a situation closely re- lated to previous ED studies for graphene dots with zigzagboundary 15,16which used merely the zero-energy states as basis. Although such an approximation avoids the Brown-Ravenhall disease, it is not exact anymore. Their results canthen be systematically improved by Sucher’s method uponincluding excited states. To be speciﬁc, we here consider twoelectrons residing above a Dirac sea with /H9262=0+where all negative- and zero-energy states are ﬁlled. Sucher’s projectoris thus expressed as /H9011 +=/H9011+/H208491/H20850/H20002/H9011+/H208492/H20850,/H9011+/H20849j/H20850=/H20858 /H9263/H33528I/H20841/H9274/H9263/H20849j,0/H20850/H20856/H20855/H9274/H9263/H20849j,0/H20850/H20841, /H208495/H20850 where the sum is restricted to strictly positive single-particle energies E/H9263/H33528I/H20849j,0/H20850/H110220/H20849for particle j=1,2 /H20850indexed by /H9263/H33528I, see Fig. 1, with corresponding eigenspinors /H20841/H9274/H9263/H20849j,0/H20850/H20856. With Hin Eq. /H208494/H20850, the projected Hamiltonian HD=/H9011+H/H9011+is well be- haved and exhibits a ﬁnite density of two-particle stateswhich does not increase with Hilbert-space size. Throughoutthe experimentally relevant regime, /H9251/H113512, interactions are not strong enough to induce a breakdown of this projectionapproach. We have carried out EDs of H DforN=2 using a two- particle product basis of Darwin-Fock states. This basis hasthe advantage of allowing to analytically express the matrixelements of the two-particle interaction operator in Eq. /H208494/H20850in terms of ﬁnite sums and products, i.e., no quadratures nortruncations of inﬁnite sums are necessary; for the corre-sponding /H20849lengthy /H20850expressions, see Ref. 25. Their numerical evaluation involves taking small differences of huge num-bers, the latter increasing as n max! with the number of kept Darwin-Fock states. We employed algorithms allowing fornumber manipulations of arbitrary precision and used30-digit accuracy. The resulting energy spectrum of artiﬁcialhelium in a magnetic graphene quantum dot is shown in Fig. 2for conserved total orbital angular momenta M=m 1+m2=−2,−1,0,1. These values include the ground state for /H9251/H110212. All levels rise when increasing the /H20849repulsive /H20850 interaction strength. This also holds true for the hole states/H20849not displayed in Fig. 2/H20850, which, however, cross /H9255=0 only for /H9251/H112712. For /H9251/H110212, interaction matrix elements indeed remain much smaller than the energy difference /H9004/H9255/H112292.981 65 be- tween the lowest two-particle state /H20849forM=−2 at /H9251=0/H20850and the zero-energy level, a posteriori justifying Sucher’s ap- proach here. Figure 2reveals that states with larger total angular momentum M, or higher excited states, tend to in- crease less in energy with /H9251as compared to the M=−2 ground state. This is a consequence of the larger spatial ex-tent of excited-state wave functions, with a reduced Coulombrepulsion between the electrons. Particularly striking is theshallow increase of the lowest M=1 energy level, which even becomes lower in energy than the M=−2 level for /H9251/H114071.6. Approximating this level at /H9251=0 by Darwin-Fock levels, one of the two particles is seen to have m=2 for the lower spinor component, cf. Eq. /H208493/H20850, causing a signiﬁcantly larger spread of this part of the wave function compared to M=−2,−1,0. Figure 2also reveals nontrivial spin physics. In the presence of interactions /H20849/H9251/H110220/H20850, doubly degenerate noninteracting /H20849/H9251=0/H20850energy levels will split into a spin- triplet /H20849S=1/H20850state of lower energy and a spin-singlet /H20849S=0/H20850state of higher energy, in accordance with Hund’s rule. The triplet states are approximately /H20849see below /H20850Zeeman degenerate. Singly degenerate /H9251=0 levels, such as the M=−2 ground state /H20849for small /H9251/H20850, are S=0 states and remain unsplit for /H9251/H110220. Thus we expect singlet-triplet ground-state spin transitions to occur within 0 /H11021/H9251/H110212, as the one seen in Fig. 2at/H9251/H110151.6. Finally, we remark on optical transitions between the many-body energy levels in Fig. 2. For the electrostatically deﬁned parabolic Schrödinger quantum dot, the generalizedKohn theorem implies that Coulomb interactions can never affect optical transitions because the dipole operator /H20858 j=1Nrj acts exclusively on the eigenspace of Hcm. Therefore optical spectra just reﬂect the harmonic excitations of the center-of-mass motion. 8However, in our magnetic graphene dot, Kohn’s theorem is ineffective and optical transitions betweendifferent many-body levels in Fig. 2are possible, thereby allowing to optically probe interaction physics. Note thatmagnetic ﬁelds are usually assumed homogeneous such thatphotons cannot change the total spin Sof the charged many- particle system in electrical dipole transitions. While thiswould prohibit all transitions between states with different S, theinhomogeneous magnetic ﬁeld here /H20849slightly /H20850mixes the S z=0 components of S=0 and S=1 levels. We estimate the amount of this mixing by the variation of the Zeeman energyacross the spatial extent of the wave function compared tothe level separations of H D. As a ﬁrst estimate, compare the Zeeman energy /H9004Zat the maximum of the charge-density distribution with the orbital /H20849Landau /H20850energy /H9004Lat this point, /H9004Z//H9004L=g/H9262BB//H208812ecB /H1122910−5/H20881B/T. While this is small, the Zeeman energy variations can easily exceed orbital levelseparations near spin-singlet-triplet degeneracies, e.g., for /H9251/H112701 or close to level crossings in Fig. 2, resulting in a strong spin mixing. The corresponding transitions are thenoptically allowed.0 0.5 1 1.5α33.544.555.5ε M=-2 M=-1 M=0 M=1 FIG. 2. /H20849Color online /H20850ED results for the energy spectrum of graphene artiﬁcial helium vs ﬁne-structure constant /H9251for different total angular momenta M. States with M=−2 are shown as black solid, M=−1 as blue dashed, M=0 as green dot-dashed, and M=1 as red dotted curves.ARTIFICIAL ATOMS IN INTERACTING GRAPHENE … PHYSICAL REVIEW B 80, 161402 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 161402-3To conclude, we have presented the theory of few inter- acting electrons in a /H20849magnetically conﬁned /H20850graphene quan- tum dot. The low-energy spectrum of graphene suggests thatone can realize relativistic artiﬁcial atoms in this setting.While a naive formulation encounters conceptual difﬁcultiesrelated to the unboundedness of the Dirac Hamiltonian, byvirtue of Sucher’s projection operator approach, a consistentand accurate theory can be given. We have presented EDresults for the energy spectra of artiﬁcial helium, where wepredict singlet-triplet ground-state spin transitions to occur for /H9251/H110212. Moreover, the reported many-body levels can be experimentally probed by optical spectroscopy. We thank H. Siedentop for drawing our attention to Refs. 18and19and acknowledge discussions with A. De Martino, K. Ensslin, and T. Heinzel. This work was supported by theDFG /H20849Grant No. SFB TR/12 /H20850and by the ESF network IN- STANS. 1A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 /H208492007 /H20850. 2A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. Geim, Rev. Mod. Phys. 81, 109 /H208492009 /H20850. 3X. Du, I. Skachko, A. Barker, and E. Y . Andrei, Nat. Nanotech- nol. 3, 491 /H208492008 /H20850; K. I. Bolotin, K. J. Sikes, J. Hone, H. L. Stormer, and P. Kim, Phys. Rev. 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S. Bedell, and A. V . Balatsky, Phys. Rev. B 74, 233405 /H208492006 /H20850. 23K-K/H11032scattering can be neglected for the long-range Coulomb interaction. Assuming that the interaction does not discriminateamong sublattices at long wavelengths, the last term in Eq. /H208494/H20850is diagonal in isospin space and we can map wave functions be-tween the Kpoints. The spectrum is then independent of how particles are distributed over the two Kpoints. 24A. O. Govorov and A. V . Chaplik, JETP Lett. 52,3 1 /H208491990 /H20850. 25V . Halonen, T. Chakraborty, and P. Pietiläinen, Phys. Rev. B 45, 5980 /H208491992 /H20850.WOLFGANG HÄUSLER AND REINHOLD EGGER PHYSICAL REVIEW B 80, 161402 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 161402-4
PhysRevB.73.033203.pdf	Half-metallic ferromagnetism in Cu-doped ZnO: Density functional calculations Lin-Hui Ye,1A. J. Freeman,1and B. Delley2 1Department of Physics & Astronomy, Northwestern University, Evanston, Illinois 60208, USA 2Condensed Matter Theory Group, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland /H20849Received 22 August 2005; revised manuscript received 21 December 2005; published 26 January 2006 /H20850 Half-metallic ferromagnetism in Cu-doped ZnO is predicted by accurate full-potential linearized augmented plane-wave and DMol3calculations based on density functional theory. A net magnetic moment of 1 /H9262Bis found per Cu. At a Cu concentration of 12.5%, total energy calculations show that the ferromagnetic state is43 meV lower than the antiferromagnetic state and is thus predicted to be the ground state with a T cestimated to be about 380 K. The magnetic moments are localized within the CuO 4tetrahedron with ferromagnetic coupling between Cu and O. The electronic states near EFare dominated by strong hybridization between O 2 p and Cu 3 dwhich implies that the Cu-O bond is quite covalent instead of purely ionic. We examine the interplay between the carrier density and the ferromagnetism with N codoping and oxygen vacancies where weﬁnd no apparent relation between them. Oxygen vacancies tend to destroy the ferromagnetism and thereforeshould be avoided during sample fabrication. We found no clustering tendency of the Cu atoms. Since there isno magnetic element in this compound, Cu-doped ZnO appears to be an unambiguous dilute magnetic semi-conductor where ferromagnetic precipitate problems can be avoided. DOI: 10.1103/PhysRevB.73.033203 PACS number /H20849s/H20850: 75.50.Pp, 75.30.Hx The possibility that the electron spin can be manipulated along with the charge degree of freedom in semiconductorshas aroused great enthusiasm for so-called “spintronic” phys-ics and its applications. Dilute magnetic semiconductors/H20849DMS’s /H20850are proposed as spin injection sources because they can be used for seamless spin injection into semiconductors,whereas metal-semiconductor contacts suffer from low spininjection efﬁciency due to interface scattering. Since the pos-sibility of high-temperature ferromagnetism /H20849FM /H20850in DMS’s has been studied by Dietl et al. with the Zener model of ferromagnetism, 1many materials have been found which display room-temperature FM.2Among them, ZnO-based DMS’s have attracted the most attention since they becomeFM when doped with most of the transition-metal elements.The strong FM in these materials, however, challenges ourunderstanding and raises two questions: /H20849i/H20850is the FM really intrinsic and /H20849ii/H20850are there really strong magnetic interactions between well-separated magnetic dopants? Unless we getclear positive answers to these questions, otherwise the use-fulness of the materials is called into question. Unfortunately,as regard to the ﬁrst question, it is found that various mag-netic impurity phases exist in many samples; 2,3to the second question, theoretically it is often found that the dopants oftenhave a clustering tendency, which implies that FM may origi-nate from nonuniform distributions of dopants. Based on accurate ﬁrst-principles electronic structure cal- culations, we propose Cu:ZnO as a high- T cDMS which is free from the FM precipitate problems. We found a net mag-netic moment of 1 /H9262Bper Cu and a half-metallic ground state with 100% carrier polarization that is excellent for spin in-jection applications. T cis estimated to be about 380 K by the Monte Carlo simulation relation.4Our result is in contradic- tion with that of Sato and Katayama-Yoshida,5but agrees with the more recent work of Feng6and Park and Min.7 Following our theoretical predictions,8room-temperature FM in Cu:ZnO has been indeed discovered recently.9Fur- ther, we show that the Cu dopant does not cluster, whichvery likely means that strong FM interactions indeed exist between well-separated dopant atoms. Moreover, wechecked the possible correlation between the FM and holedensity by including N codoping and oxygen vacancies, andwe found that the FM seems not to be related to hole media-tion. Since oxygen vacancies weaken the FM and destroyconductivity, they should be carefully avoided during samplefabrications. The ﬁrst-principles simulations were performed using the full-potential linearized augmented plane-wave /H20849FLAPW /H20850 method 10based on density functional theory. The mufﬁn-tin radii of Cu, Zn, and O were chosen to be 2.3, 2.0, and1.4 a.u., respectively, with the plane-wave basis cutoff/H20841k+G/H20841/H110214.0 a.u. −1and the star function cutoff of 9.0 a.u.−1. Brillouin zone integrations were performed with the specialk-point method over a 5 /H110035/H110033 Monkhorst-Pack mesh. 11 Convergence testings were performed, ﬁrst by increasing the kpoint mesh to 7 /H110037/H110035 and then to 9 /H110039/H110037; we found that the total energy differs only by 10−5hartree. Then we ﬁxed the k-point mesh at 5 /H110035/H110033 and increased the plane- wave basis cutoff to 4.5 a.u.−1and 5.0 a.u.−1; again, we found a similar very small total energy change. Thus our results are well converged. For the exchange correlationfunctional, we employed the local density approximation/H20849LDA /H20850/H20849Hedin-Lunquist form 12/H20850and generalized gradient ap- proximation /H20849GGA /H20850/H20849Perdew-Burke-Ernzerhof form13/H20850,o n pure ZnO. Both functionals gave very good lattice constants.Pure ZnO has a wurzite structure with a=3.25 Å and c/a=1.60. Our LDA results underestimated both quantities by 1.7% and 0.61%, and the GGA overestimated them by0.86% and 0.28%, respectively. For Cu-doped ZnO, we usedthe GGA in our calculations. The supercell employed contains 32 atoms which corre- sponds to a 2 /H110032/H110032 supercell of ZnO. Two doping levels were checked: for x=0.125, two Zn atoms were substituted by Cu, while for x=0.0625 only one Zn is substituted. Most of our results were achieved for x=0.125 and will be com-PHYSICAL REVIEW B 73, 033203 /H208492006 /H20850 1098-0121/2006/73 /H208493/H20850/033203 /H208494/H20850/$23.00 ©2006 The American Physical Society 033203-1pared to the x=0.0625 case where applicable. To make our simulations closest to the dilute doping limit, we put the twoCu atoms on positions with the largest possible distance/H208496.1 Å /H20850between them in the supercell. The preferential sub- stitution of Cu on the Zn site was conﬁrmed by a recent experiment, 14where no well-deﬁned interstitial site for Cu was found. The doped structures were then optimized withrespect to both the lattice constants and the atomic positions.Because the ionic radius of Cu is close to that of Zn, 15one would not expect strong structural relaxation through Cudoping. Indeed, we found that the lattice constants changedby only −0.01 Å for aand +0.02 Å for c. The displacements of the atoms were also very small—typically 0.005 Å inmagnitude. Experimentally, 14Cu was found to occupy al- most the ideal Zn site in the as-planted Cu-doped ZnOsample, until at high temperature /H20849/H11022600 °C /H20850the Cu starts to diffuse. Forx=0.125, total energy comparisons between the FM and antiferromagnetic /H20849AFM /H20850phases show a difference of 43 meV, with the FM state being the ground state. Adapting the Heisenberg model with only nearest-neighbor exchangebetween Cu atoms and the relation T c=0.447 J1from Monte Carlo simulations,4Tcis estimated to be 380 K which is well above room temperature. In Fig. 1, the band structure of the FM Cu-doped ZnO with x=0.125 shows half-metallic behavior with a half- metallic band gap of 0.3 eV; the majority-spin componentis semiconducting while the minority-spin component ismetallic. The carriers are thus 100% polarized which is theideal case for spin injection applications. States near E Fare composed mostly of Cu 3 dbands, hybridized strongly with t h eO2 pbands. Cu and its surrounding four O atoms form a CuO 4tetrahedron; their hybridization dominates the elec- tronic and magnetic properties. The 2 pbands from other O atoms away from the CuO 4-distorted tetrahedron comprise the −1 eV to −5 eV region. Zn 3 dstates are located below EFby more than 5 eV; hence, the Zn2+ion is chemically inactive. Table I displays the magnetic moment distribution in the supercell. For x=0.125, a total magnetic moment of 2 /H9262Bper supercell, or 1 /H9262Bper Cu atom, is found due to the half- metallicity: Cu is polarized with a magnetic moment of0.58 /H9262B, and the four surrounding O atoms in the CuO 4tet- rahedron are polarized with a magnetic moment of 0.04 /H9262B for the top site O and 0.08 /H9262Bfor the other three O in the basal plane. The smaller magnetic moment for the top site Ois due to the distortion of the tetrahedron; the Cu-O bond islonger for the top side O than the three lying in the basalplane. The magnetic moments on Zn and other O atomsaway from the CuO 4tetrahedron are smaller than 0.01 /H9262Band are not listed in Table I. Summing up the magnetic moments,we found about 80% of all magnetic moments are localizedwithin the mufﬁn tins of Cu and O, while only 0.13 /H9262Blocate in the interstitial region. Although most of the magnetic mo-ments are restricted within the CuO 4tetrahedron /H20849as can be seen from Fig. 2 /H20850, the FM coupling between Cu and its four surrounding O atoms may help extend the FM interaction tomore distant atoms, thus providing an enhancement of theFM. The substitution of Zn by Cu is a p-type doping. In TableI we also present the net charge variation in the mufﬁn tins and in the interstitial region. The hole numbers are obtainedby subtracting the number of electrons of Cu-doped ZnOfrom that of the pure ZnO. It is apparent that the substitutionof Zn by Cu actually has a very limited effect on the electronnumbers at the substitutional site. In fact, the charge redistri-bution mostly happens in the interstitial region instead ofinside the mufﬁn tins. Although it was proposed that Custays in the 2+ state when doped into ZnO, 6the hole num- bers in Table I indicate that the chemical valence of Cu isfractional and lies between 1+ and 2+ due to strong Cu-Ocovalency. Since the loss of the electrons on the Cu site is FIG. 1. Band structure of 0.125 Cu-doped ZnO. Half-metallicity is shown for /H20849a/H20850the majority-spin channel to be semiconducting while /H20849b/H20850the minority-spin channel is metallic. The Fermi level is set to 0.0 eVBRIEF REPORTS PHYSICAL REVIEW B 73, 033203 /H208492006 /H20850 033203-2only about 0.19, the Cu chemical valence is actually closer to 1+ than 2+. This may explain why the x-ray absorption spec-troscopy measurements 16found a chemical valence of 1+ for Cu. Because the Cu-O bond in Cu-doped ZnO is largelycovalent /H20849as can also be seen from the strong hybridization between the O 2 pand Cu 3 dbands in Fig. 1 /H20850, a purely ionic description of ZnO:Cu is not appropriate. To test the dependence of our data on the Cu concentra- tion, we repeated the calculations for x=0.0625; the results are also listed in Table I. The total magnetic moment is still1 /H9262Bper Cu and the system keeps its half-metallic character; the mufﬁn-tin magnetic moments and hole numbers are notmuch affected by decreasing the doping level, showing thatthe interplay between dopants is still very small at x=0.125, and so we expect the results to be similar in the dilute dopinglimit. In the interstitial region, the magnetic moment andhole number decrease monotonically with respect to the Cuconcentration. For x=0.125, another comparison is made be- tween FM and AFM in Table I, where the only noticeabledifference is that the magnetic moment in the interstitial re- gion becomes zero, of course. These data suggest that thepolarization of the atoms and the orbital occupancies aremostly determined by the local chemical environment andare not much affected by the presence of another Cu 6.1 Åaway. Since there is no appreciable charge transfer betweenthe two Cu atoms with the addition of the second Cu to thesupercell, the direct d-dcorrelation between Cu atoms may be small which could imply an indirect exchange mechanismbeing responsible for the ferromagnetism. That Cu-doped ZnO has a FM ground state agrees with the prediction by Park and Min: 7with the LDA linear mafﬁn-tin orbital atomic sphere approximation /H20849LMTO- ASA /H20850method they obtained half-metallicity and a Cu mag- netic moment of 0.81 /H9262B, but when taking into account the on-site Coulomb repulsion Uand spin-orbit coupling, the Cu magnetic moment increased to 1.85 /H9262B. The inclusion of U enabled consideration of a strong correlation effect of the Cu 3dstates. However, since there is no experimental evidence to support the strong correlation or any measurement of thevalue of U, the performance of this LDA+ Utreatment is unknown. Further, the authors found that putting in a pos-sible Jahn-Teller /H20849JT/H20850distortion drives the system into a more stable insulating state, quenches the orbital moment,and enhances the Cu spin magnetic moment to 0.99 /H9262B. To test the possibility of a JT distortion, we initialized a random perturbation of the atomic positions and redid thestructural relaxation without assuming any symmetry. Theresult showed no sign of the proposed JT distortion. Thecalculations were then repeated by use of DMol 3, another widely used ﬁrst-principles package,17with the special k-point mesh increased to 9 /H110039/H110037. Again, we did not ﬁnd a structural distortion reported in Ref. 7. A third testing wasperformed by manually rotating the CuO 4tetrahedron as pro- posed in Ref. 7. When the structure was allowed to relax toequilibrium, we found that the distorted CuO 4tetrahedron always rotates back with the three basal plane oxygen atomsreturning to the same zplane. Therefore, the proposed ro- tated CuO 4tetrahedron is found to be unstable in our calcu- lations. Experimentally, however, it is believed that a dy-namical JT effect should be included to successfully explainthe paramagnetic susceptibility of Cu:ZnO. 18This discrep- ancy between theory and experiment may be due to the factthat the experiment measured the paramagnetic phase, whileour calculations consider the ferromagnetic phase. Thestrong Cu-O covalency in our calculations does not supportthe pure ionic description of Cu dopant as a 3 d 9system and so does not favor the JT distortion. Our result is consistentwith that of Luo and Martin 19who performed comparative calculations to check possible JT distortions in Mn:GaAs andMn:GaN. While in Mn:GaN there is indeed a JT distortiondue to the localized feature of the Mn 3 dstates, no appre- ciable effect is found in Mn:GaAs since the Mn 3 dstates strongly hybridize with the As 4 pstates. Another report on the FM in Cu-doped ZnO was made by Feng 6with the semiempirical B3LYP hybrid density func- tional, in which the nonlocal Hartree-Fock exchange ismixed into the GGA. This resulted in a magnetic moment onCu of 0.75 /H9262B, with a total energy difference of 130 meV between FM and AFM at a Cu-Cu distance of 5.2 Å. SimilarTABLE I. Hole /H20849unit: e/H20850and magnetic moment /H20849unit:/H9262B/H20850dis- tributions for the ferromagnetic state with x=0.0625, the ferromag- netic state with x=0.125, and the antiferromagnetic state with x =0.125. FM /H20849x=0.0625 /H20850FM /H20849x=0.125 /H20850AFM /H20849x=0.125 /H20850 Cu O/H20849in CuO 4/H20850:0.18 /H208490.58 /H20850a0.19 /H208490.58 /H20850 0.19 /H208490.56 /H20850 Top site — /H208490.04 /H20850 0.01 /H208490.04 /H20850 0.01 /H208490.04 /H20850 Basal plane — /H208490.09 /H20850 0.01 /H208490.08 /H20850 0.02 /H208490.09 /H20850 Zn and other O — /H20849—/H20850 —/H20849—/H20850 —/H20849—/H20850 Interstitial 0.82 /H208490.05 /H20850 1.63 /H208490.13 /H20850 1.62 /H20849—/H20850 aQuantities outside and inside the parentheses are the hole numbers and magnetic moments, respectively. FIG. 2. Spin density distribution in the x=0.125 FM state of Cu-doped ZnO. Most of the spin density is localized within theCuO 4tetrahedron.BRIEF REPORTS PHYSICAL REVIEW B 73, 033203 /H208492006 /H20850 033203-3to the LDA+ Utreatment in Ref. 17, the nonlocal Hartree- Fock exchange term also included considerations of a pos-sible strong correlation effect. However, the mixing param-eter in the B3LYP hybrid functional is actually not universalbut system dependent. On the other hand, in the work of Satoand Katayama-Yoshida 5by the Korringa-Kohn-Rostoker /H20849KKR /H20850Green function method based on the LDA combined with the coherent potential approximation /H20849CPA /H20850, they re- ported no magnetic moment on Cu and concluded that Cu-doped ZnO is nonmagnetic. The reason for this discrepancyis unknown. It is possible that their simulated doping level/H20849x=0.25 /H20850is too high so that the short-range AFM superex- change interaction becomes stronger and neutralizes the FM interaction. Since the origin of the FM in DMS’s in still under debate, it is interesting to see how a different carrier density affectsthe strength of the magnetic interaction in Cu:ZnO. Wecheck two situations corresponding to increased and de-creased hole density. A higher hole density is simulated bythe substitution of one oxygen by nitrogen, while by creatingone oxygen vacancy the holes are totally eliminated. In bothcases, the related O site locates away from the CuO 4tetra- hedron in order to avoid much disturbance to the Cu-Obonds. The O-to-N substitution increases the total magneticmoments in the supercell to 3 /H9262Bwith the FM state still being the ground state. The total energy difference is 44 meV be-tween FM and AFM states which does not change much byN codoping. On the other hand, by creating an O vacancy,we still got an FM ground state which is substantially weak-ened; the FM and AFM total energy difference drops to26 meV. These studies imply that the FM seems not to berelated to the hole density. In fact, the oxygen vacancy can be more harmful to the FM if located within the CuO 4tetrahedron, since removing one O atom from the CuO 4tetrahedron results in the break- ing of one Cu-O bond, which should have a stronger effecton the magnetic properties. Indeed, the relevant Cu atom isfound to be depolarized, and the system becomes a paramag- netic metal. When we move the O vacancy from outside of the CuO 4tetrahedron to inside, the total energy decreases by 0.7 eV. Therefore, the O vacancy prefers to be attached tothe Cu dopant and tends to destroy the FM. Besides, asn-type defects, O vacancies provide impurity electrons which could neutralize the p-type carriers and destroy the conduc- tivity by Cu doping—an effect called defect compensation.Therefore, it is important in the experiments to keep a highoxygen pressure during sample fabrication in order to avoidthe formation of the O vacancies. Further, we considered the possibility of Cu clustering by varying the Cu-Cu distance. The ﬁrst calculation was doneby moving the two Cu atoms in the supercell from 6.1 Å toa closer distance of 5.7 Å; the total energy is found to in-crease by 68 meV. In our second calculation, the two Cuatoms were placed even closer so that they connect to thesame O atom. A huge total energy increase /H208495.7 eV /H20850is found which deﬁnitely disfavors this structure. We thus conclude that there is no tendency for Cu clustering in Cu-doped ZnO. In summary, our accurate ﬁrst-principles simulations pre- dict Cu-doped ZnO to be a half-metallic dilute magneticsemiconductor. The ferromagnetism emerges from the hy-bridization of the Cu 3 dand O 2 pbands. There is no close relation between the ferromagnetism and hole density. Com-pared to conventional DMS’s, Cu:ZnO does not contain anymagnetic ion; neither is any compound with /H20849Cu,Zn,O /H20850fer- romagnetic. Therefore, Cu:ZnO is free of ferromagnetic pre-cipitate problems. With the unambiguous intrinsic ferromag-netism and the 100% spin polarization of the carriers, weexpect Cu-doped ZnO to be a useful DMS, both in applica-tions and in theoretical studies of the ferromagnetism inDMS’s. This work is supported by DARPA/ONR /H20849Grant No. N00014-02-1-0887 /H20850and the NSF through the Materials Re- search Center of Northwestern University. 1T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, Science 287, 1019 /H208492000 /H20850. 2For a review of recently discovered room-temperature DMS’s, see S. J. Pearton, Y. D. Park, C. R. Abernathy, M. E. Overberg,G. T. Thaler, J. Kim, F. Ren, J. M. Zavada, and R. G. Wilson,Thin Solid Films 447–448 , 493 /H208492004 /H20850, and the references therein. 3G. Lawes, A. S. Risbud, A. P. Ramirez, and R. Seshadri, Phys. Rev. B 71, 045201 /H208492005 /H20850; C. N. R. Rao and F. L. Deepak, J. Mater. Chem. 15, 573 /H208492005 /H20850; S. Kolesnik, B. Dabrowski, and J. Mais, J. Appl. Phys. 95, 2582 /H208492004 /H20850. 4H. T. Diep and H. Kawamura, Phys. Rev. B 40, 7019 /H208491989 /H20850. 5K. Sato and H. Katayama-Yoshida, Jpn. J. Appl. Phys., Part 2 39, L555 /H208492000 /H20850. 6Xiaobing Feng, J. Phys.: Condens. Matter 16, 4251 /H208492004 /H20850. 7M. S. Park and B. I. Min, Phys. Rev. B 68, 224436 /H208492003 /H20850. 8Lin-Hui Ye and A. J. Freeman, Bull. Am. Phys. Soc. 49, L26.14 /H208492004 /H20850. 9D. B. Buchholz, R. P. H. Chang, J. H. Song, and J. B. Ketterson,Appl. Phys. Lett. 87, 082504 /H208492005 /H20850. 10E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Phys. Rev. B 24, 864 /H208491981 /H20850; H. J. F. Jansen and A. J. Freeman, ibid. 30, 561 /H208491984 /H20850. 11H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 /H208491976 /H20850. 12L. Hedin and B. I. Lundquist, J. Phys. C 4, 2064 /H208491971 /H20850. 13J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 /H208491996 /H20850. 14U. Wahl, E. Rita, J. G. Correia, E. Alves, J. C. Soares, and The ISOLDE Collaboration, Phys. Rev. B 69, 012102 /H208492004 /H20850. 15For fourfold coordinates, the ionic radii are Cu 0.6, Cu2+0.57, Zn2+0.6: R. D. Shannon, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 32, 751 /H208491976 /H20850. 16S-J. Han, J. W. Song, C.-H. Yang, S. H. Park, Y. H. Jenong, and K. W. Rhie, Appl. Phys. Lett. 81, 4212 /H208492002 /H20850. 17B. Delley, J. Chem. Phys. 113, 7756 /H208492000 /H20850. 18W. M. Brumage, C. F. Dorman, and C. R. Quade, Phys. Rev. B 63, 104411 /H208492001 /H20850. 19X. Luo and R. M. Martin, Phys. Rev. B 72, 035212 /H208492005 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 033203 /H208492006 /H20850 033203-4
PhysRevB.92.115414.pdf	PHYSICAL REVIEW B 92, 115414 (2015) Coulomb-exchange effects in nanowires with spin splitting due to a radial electric ﬁeld F. S. Gray, T. Kernreiter, M. Governale,and U. Z ¨ulicke† School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, Post Ofﬁce Box 600, Wellington 6140, New Zealand (Received 22 March 2015; revised manuscript received 15 July 2015; published 11 September 2015) We present a theoretical study of Coulomb-exchange interaction for electrons conﬁned in a cylindrical quantum wire and subject to a Rashba-type spin-orbit coupling with radial electric ﬁeld. The effect of spin splitting onthe single-particle band dispersions, the quasiparticle effective mass, and the system’s total exchange energyper particle are discussed. Exchange interaction generally suppresses the quasiparticle effective mass in thelowest nanowire sub-band, and a ﬁnite spin splitting is found to signiﬁcantly increase the magnitude of thequasiparticle-mass suppression (by up to 15% in the experimentally relevant parameter regime). In contrast,spin-orbit coupling causes a modest (1%-level) reduction of the magnitude of the exchange energy per particle.Our results shed light on the interplay of spin-orbit coupling and Coulomb interaction in quantum-conﬁnedsystems, including those that are expected to host exotic quasiparticle excitations. DOI: 10.1103/PhysRevB.92.115414 PACS number(s): 81 .07.Gf,71.70.Ej,71.70.Gm,81.05.Ea I. INTRODUCTION The dimensionality of a many-particle system is a crucial determinator for how importantly interaction effects can shapeits physical properties. Generally, three-dimensional (3D)bulk conductors are less drastically affected by the Coulombinteraction between charge carriers than lower-dimensional,quantum-conﬁned structures such as quasi-two-dimensional(quasi-2D) quantum wells and quasi-one-dimensional (quasi-1D) quantum (nano)wires [ 1]. This is essentially due to phase-space restrictions arising from free motion being onlypossible in fewer than three spatial directions. Furthermore,the exact structure of transverse bound-state wave functionsshapes the density distribution of the conﬁned charge carriersand, thus, turns out to critically inﬂuence Coulombic effects inquantum wells [ 2] and wires [ 3]. Here we explore how another aspect of quantum-conﬁned states, namely their intrinsicspinor structure, modiﬁes the effect of the Coulomb interactionin nanostructured systems. Most low-dimensional conductors are fabricated from semiconductor materials where the coupling between thespin degree of charge carriers and their orbital motion isoften quite strong [ 4]. As a result, quantum conﬁnement can signiﬁcantly affect spin-related properties [ 5]. Such effects are particularly pronounced for valence-band states (i.e.,holes ) because of their peculiar spin-3 /2 character [ 5,6]. In contrast, conduction-band electrons are spin-1 /2 particles and generally subject to weaker spin-orbit couplings thatare due to the bulk inversion asymmetry in the material’scrystallographic unit cell (Dresselhaus [ 7] spin splitting) or the structural inversion asymmetry present in a nanostruc-tured systems (Rashba [ 8,9] spin splitting). The multitude, and often counterintuitive nature, of spin-orbit effects innanostructures has become the focus of recent study, withdeveloping an understanding of the interplay with Coulomb michele.governale@vuw.ac.nz †uli.zuelicke@vuw.ac.nzinteractions being a key question to be addressed. Bulk-hole systems [ 10–13], quantum-well-conﬁned holes [ 14–18], and 2D electron systems subject to Rashba spin splitting[16,19–22] have been considered. The comparatively few studies of Coulomb-interaction effects in spin-orbit-coupledquasi-1D systems [ 23–26] have almost exclusively focused on effective Luttinger-liquid descriptions [ 27] and, in particular, did not investigate the effect of Rashba spin splitting on thetotal exchange energy and exchange-induced quasiparticle-effective-mass renormalization in quantum wires. In this article, we ﬁll precisely this gap and investigate both the exchange energy and effective-mass renormalizationin quantum wires with a Rashba-type spin-orbit coupling.Previous work on the exchange energy of quantum wellsrevealed that spin-orbit coupling has the opposite effect oninteractions in n-type and p-type systems: The exchange energy of a quasi-2D conduction-band electron system isslightly enhanced [ 16,21] due to spin-orbit coupling, whereas the exchange energy for quasi-2D holes is suppressed dueto conﬁnement-induced valence-band mixing [ 18] and the heavy-hole-type Rashba spin splitting [ 16]. The different behavior of conﬁned band electrons and holes warrants moresystematic investigation and, as we will see below, consideringthe quasi-1D case sheds new light on the different ramiﬁcationsof spin-orbit coupling in interacting systems. Our investigationalso reveals that the quasiparticle effective mass is morestrongly suppressed by the exchange interaction in nanowireswith spin splitting. In addition, quantum wires with strong spin-orbit coupling are currently attracting great interest as possible hosts ofexotic quasiparticle excitations such as Majorana [ 28] and fractional [ 29] fermions. Clarifying the effect of interactions in such systems is necessary for a complete understanding ofexperiments aimed at verifying the existence of the unusualquasiparticle excitations. The remainder of this article is organized as follows. We introduce our theoretical model of a Rashba-spin-splitquantum wire in Sec. IIand discuss pertinent properties of the single-particle eigenstates. The formalism for calculatingthe exchange energy for this system is presented in Sec. III, 1098-0121/2015/92(11)/115414(9) 115414-1 ©2015 American Physical SocietyGRAY , KERNREITER, GOVERNALE, AND Z ¨ULICKE PHYSICAL REVIEW B 92, 115414 (2015) together with the results. Among these is the ability to express functional dependencies of the exchange energy perparticle in terms of a universal scaling function, and theenhanced suppression of the density-of-states effective mass.Our ﬁndings are summarized, and related to the existing bodyof knowledge, in Sec. IV. Certain formal details are given in appendices. II. THEORETICAL DESCRIPTION OF RASHBA-SPLIT NANOWIRE STATES In our study, we aim to develop a general understanding of the effect of spin-orbit coupling on exchange-related many-particle corrections in quasi-1D nanowires. Hence, rather thanattempting to describe the detailed electron density proﬁlefor a speciﬁc sample based on a self-consistent Poisson-Schr ¨odinger calculation, we consider a model cylindrical quantum wire with radius Rthat is deﬁned by a hard-wall potential where a constant radial electric ﬁeld E=Eˆrgives rise to a spin-orbit coupling of the Rashba type. In a realsample, such a radially symmetric ﬁeld conﬁguration could begenerated, e.g., via biasing of an external gate that is wrappedaround the wire surface [ 30]. The pragmatic assumption of a constant electric-ﬁeld magnitude is justiﬁed in Appendix A; see especially Fig. 6. For our situation of interest, the noninteracting-electron dynamics in the wire is described bythe Hamiltonian H=H (0)+U(r), where U(r)=/braceleftbigg 0r<R ∞r/greaterorequalslantR, (1) andH(0)is a Rashba-type [ 8,9] single-electron Hamiltonian H(0)=p2 2m∗+αE /planckover2pi1ˆr·(σ×p). (2) Herem∗is the band mass of electrons in the semiconductor material making up the nanowire, αis the material-dependent Rashba spin-orbit-coupling constant, and σ=(σx,σy,σz)T denotes the vector of Pauli matrices. We ﬁnd the conﬁned- electron states in the nanowire by superimposing solutionsof the single-particle Schr ¨odinger equation H (0)ψ=Eψ to satisfy the cylindrical hard-wall boundary condition. The Hamiltonian ( 2) can be conveniently expressed in cylindrical coordinates ( r,ϕ,z )a s H(0)=−/planckover2pi12 2m∗/parenleftbigg∂2 ∂r2+1 r∂ ∂r+1 r2∂2 ∂ϕ2+∂2 ∂z2/parenrightbigg 1 +iαE/bracketleftbigg σz1 r∂ ∂ϕ+i(e−iϕσ+−eiϕσ−)∂ ∂z/bracketrightbigg ,(3) where σ±=(σx±iσy)/2a r et h es p i n - 1 /2 ladder operators. The explicit form of Eq. ( 3) motivates a separation ansatz for the eigenstates of H(0): ψ(r,ϕ,z )=eikz √ Leiνϕe−iσz 2ϕφν,k(r), (4) where φν,k(r) is the radial spinor wave function, ν=±1/2,± 3/2,... is an odd half-integer number, kdenotes the wavenumber associated with the free electron motion in the quantum wire, and Lis the wire length. The resulting radial Schr ¨odinger equation that determines φν,k(r) can be written in dimensionless form as Hν,κχν,κ(/rho1)=εχν,κ(/rho1), with Hν,κ=−/parenleftbigg∂2 ∂/rho12+1 /rho1∂ ∂/rho1/parenrightbigg 1+ˆm2 /rho12−˜ασzˆm /rho1+˜ακσ y+κ21, (5) and the deﬁnitions ˆm=ν1−1 2σz,/rho1=r/R,κ=kR,ε= E/E 0, where E0=/planckover2pi12/(2m∗R2), ˜α=2Rm∗αE//planckover2pi12, and φν,k(r)≡χν,κ(r/R)/R. We employ the sub-band k·pmethod [ 31,32] to ﬁnd the cylindrical-nanowire eigenstates and single-particle sub- band-energy dispersions E(0) nk. Simultaneous invariance under time reversal ( σyH∗ −ν,−κσy=Hν,κ) and spatial inversion (e−iπ 2σzHν,−κeiπ 2σz=Hν,κ) imply that each sub-band is (at least) doubly degenerate [ 33]. The ﬁrst step is to ﬁnd the eigenstates that are associated with the sub-band-edge energies E(0) n0. These states are then used as a basis set for expressing the eigenstates at general k/negationslash=0, with expansion coefﬁcients determined from solving a matrix equation that is equivalentto the Schr ¨odinger equation. The Hamiltonian of Eq. ( 5) is diagonal when κ=0, H ν,0=/parenleftbigg Hν 0 0H−ν/parenrightbigg , (6a) Hν=−/parenleftbigg∂2 ∂/rho12+1 /rho1∂ ∂/rho1/parenrightbigg +/parenleftbig ν−1 2/parenrightbig2 /rho12−˜αν−1 2 /rho1,(6b) and hence the sub-band-edge states are also spin-projection eigenstates of σzwith eigenvalue σ=±1. We can therefore write χν,κ(/rho1)=∞/summationdisplay n/prime=1/parenleftbig c(n/prime↑) ν,κ\|ν,↑,n/prime/angbracketright+c(n/prime↓) ν,κ\|ν,↓,n/prime/angbracketright/parenrightbig ,(7a) with the sub-band-edge basis-state deﬁnitions \|ν,↑,n/prime/angbracketright=F(ε(n/prime) ν,+) ν−1 2(/rho1)/parenleftbigg 1 0/parenrightbigg , (7b) \|ν,↓,n/prime/angbracketright=F(ε(n/prime) ν,−) −ν−1 2(/rho1)/parenleftbigg 0 1/parenrightbigg , (7c) and the functions F(ε(n/prime) ν,σ) σν−1 2(/rho1) being solutions of the radial- conﬁnement problem deﬁned by the Hamiltonian Hσν+ U(/rho1R)/E0with corresponding dimensionless eigenenergies ε(n/prime) ν,σ. We number the sub-band-edge states for ﬁxed νandσ in ascending order of energy, that is, ε(n/prime) ν,σ>ε(n/prime/prime) ν,σwhenn/prime> n/prime/prime. Time-reversal symmetry mandates the Kramers degener- acyε(n/prime) ν,σ=ε(n/prime) −ν,−σ. See Appendix Bfor more mathematical details. 115414-2COULOMB-EXCHANGE EFFECTS IN NANOWIRES WITH . . . PHYSICAL REVIEW B 92, 115414 (2015) −3−2−1 0123 kR6810121416E(0) nk/E0InGaAs (a) ˜α=0 ˜α=1.82 −3−2−1 0123 kR6810121416E(0) nk/E0 InSb(b) ˜α=0 ˜α=1.06 kR1.001.051.101.151.201.25m0(α,k)/m∗ (c) −2 −1 0120.800.850.900.95δE˜α=1.82 ˜α=1.06 FIG. 1. (Color online) Electronic structure of noninteracting electrons in nanowires with spin splitting induced by a radial electric ﬁeld. The solid curves in panel (a) [(b)] show the single-particle energy dispersions of the lowest two sub-bands obtained for a value of ˜ αcorresponding to a recent experimental realization using InGaAs [InSb] as the wire material. To illustrate the effect of spin splitting, the corresponding dispersio ns for ˜α=0 are also plotted as dashed curves. Vertical lines are used to indicate the range of wave numbers for which only the lowest sub-band is occupied. Panel (c) illustrates more quantitatively the effect of spin-orbit coupling on the lowest nanowire-sub-band dispersions. In the upper (lower) panel, the ratios of the density-of-states effective masses (sub-band energies where δE≡[E(0) k(α)−E(0) 0(α)]/[E(0) k(0)−E(0) 0(0)]) for ﬁnite and for zero ˜ αare plotted as functions of wave number. The full single-electron sub-band dispersions can be found from solving the eigenvalue problem ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ε (1) ν,++κ2−i˜ακ I(11) ν ... 0 −i˜ακ I(1n/prime) ν ... i˜ακ/bracketleftbig I(11) ν/bracketrightbig∗ε(1) ν,−+κ2... i ˜ακ/bracketleftbig I(1n/prime) ν/bracketrightbig∗0 ... .................. 0 −i˜ακ I (1n/prime) ν ... ε(n/prime) ν,++κ2−i˜ακ I(n/primen/prime) ν ... i˜ακ/bracketleftbig I(1n/prime) ν/bracketrightbig∗0 ... i ˜ακ/bracketleftbig I(n/primen/prime) ν/bracketrightbig∗ε(n/prime) ν,−+κ2... ..................⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝c (1↑) ν,κ c(1↓) ν,κ ... c(n/prime↑) ν,κ c(n/prime↓) ν,κ ...⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=ε ν(κ)⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝c (1↑) ν,κ c(1↓) ν,κ ... c(n/prime↑) ν,κ c(n/prime↓) ν,κ ...⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (8a) with matrix elements I(nn/prime) ν=2π/integraldisplay1 0d/rho1 /rho1/bracketleftbig F(ε(n) ν,+) ν−1 2(/rho1)/bracketrightbig∗F(ε(n/prime) ν,−) −ν−1 2(/rho1). (8b) For the purpose of this study, we only need to obtain the dispersion of the lowest nanowire sub-band. We ﬁnd that,for realistic values of ˜ α(see, for instance, the examples below), truncation of the eigenvalue problem ( 8a)t ot h e subspace spanned by the states {\|1/2,↑,1/angbracketright,\|1/2,↓,1/angbracketright} and its time-reversed counterpart yields sufﬁciently accurateresults. Hence, in the following, we use the wave functions ψ 1(r,ϕ,z )=eikz √ LR(−isinηkR\|1/2,↑,1/angbracketright +eiϕcosηkR\|1/2,↓,1/angbracketright), (9a) ψ2(r,ϕ,z )=eikz √ LR(−isinηkR\|−1/2,↓,1/angbracketright +e−iϕcosηkR\|−1/2,↑,1/angbracketright) (9b) to describe lowest-sub-band states with the dispersion E(0) 1k≡E(0) 2k=E0/bracketleftbig (kR)2+1 2/parenleftbig ε(1) 1/2,++ε(1) 1/2,−/parenrightbig −1 2/radicalBig/parenleftbig ε(1) 1/2,+−ε(1) 1/2,−/parenrightbig2+/parenleftbig 2˜αkR I(11) 1/2/parenrightbig2/bracketrightbig .(10)The coefﬁcients entering Eqs. ( 9)a r e sinηκ=1√ 2⎛ ⎝1+/vextendsingle/vextendsingleε(1) 1/2,+−ε(1) 1/2,−/vextendsingle/vextendsingle /radicalBig/parenleftbig ε(1) 1/2,+−ε(1) 1/2,−/parenrightbig2+/parenleftbig 2˜ακI(11) 1/2/parenrightbig2⎞ ⎠1 2 , (11a) cosηκ=1√ 2⎛ ⎝1−/vextendsingle/vextendsingleε(1) 1/2,+−ε(1) 1/2,−/vextendsingle/vextendsingle /radicalBig/parenleftbig ε(1) 1/2,+−ε(1) 1/2,−/parenrightbig2+/parenleftbig 2˜ακI(11) 1/2/parenrightbig2⎞ ⎠1 2 . (11b) Figure 1illustrates the noninteracting-electron band struc- ture of nanowires using parameters relevant to recent ex-perimental realizations [ 34]i nR e f .[ 35] (InGaAs material with conduction-band effective mass m ∗=0.037m0, where m0is the electron mass in vacuum, R=300 nm, and αE= 10−11eV m), and Ref. [ 36] (InSb, m∗=0.013m0,R=50 nm, αE=10−10eV m). Within our model, the relevant quantity determining the effect of spin-orbit coupling is ˜ α, which is equal to 1 .82 and 1 .06 for the InGaAs and InSb nanowires, respectively. For comparison, we show also the result for ˜α=0. As the lowest sub-band-edge states have quantum numbers {ν=1/2,↑}and{ν=−1/2,↓}, respectively, their energy is independent of ˜ α, and the spin-orbit coupling only affects the dispersion at ﬁnite k.I nF i g . 1(c), the upper plot 115414-3GRAY , KERNREITER, GOVERNALE, AND Z ¨ULICKE PHYSICAL REVIEW B 92, 115414 (2015) −2 −1 012 kR0.800.850.900.951.00σz ˜α=1.82(InGaAs) ˜α=1.06(InSb) FIG. 2. The magnitude of the expectation value for spin projec- tion parallel to the wire axis, \|/angbracketleftσz/angbracketright\| = /angbracketleftσz/angbracketright1=− /angbracketleftσz/angbracketright2≡2s i n2ηkR− 1, for states from the lowest doubly degenerate ( n=1 and 2) sub-band. shows the ratio of the single-particle density-of-states effective mass, m0(α,k)=/planckover2pi12k ∂E(0) 1k/∂k, (12) of the lowest sub-band with and without spin-orbit coupling for the two values of ˜ α. The lower panel in Fig. 1(c)illustrates the relative change in energy for the lowest-sub-band statesdue to the spin-orbit coupling. As can be seen from the plot,the renormalization of the single-particle effective mass due tospin-orbit coupling can amount to up to 30% (for ˜ α=1.82) and also depends appreciably on the value of the wave vector.In Fig. 2, we show the magnitude of the expectation value of the spin projection along the wire axis for the lowest sub-band as a function of the wave number k. It decreases with increasing k, as the states given in Eqs. ( 9) together with ( 11) become superpositions of ↑and↓states for ﬁnite k. Table I summarizes properties of the three lowest doubly degeneratesub-band edges in the two material systems. Note the ratherlarge energy splitting of the (doubly degenerate) next-to-lowestsubbbands due to the spin-orbit coupling. Without spin-orbitcoupling ( ˜ α=0) the band edge energy of the sub-bands n= 3,..., 6i sE (0) n0/E0≈14.68. III. EFFECT OF SPIN-ORBIT COUPLING ON THE COULOMB-EXCHANGE ENERGY The Coulomb exchange interaction between electrons renormalizes the quasiparticle dispersion of nanowire sub-bands, which is then given by [ 1] E(int) nk=E(0) nk+/Sigma1(X) nk (13)TABLE I. Properties of the three lowest doubly degenerate nanowire sub-band edges obtained for parameters applicable to recent experimental realizations. Sub-band E(0) n0/E0for E(0) n0/E0for Sub-band-edge indexn ˜α=1.82 ˜ α=1.06 (basis) state 1 5.783 5.783 \|+1 2,↑,1/angbracketright 2 5.783 5.783 \|−1 2,↓,1/angbracketright 3 10.87 12.47 \|+3 2,↑,1/angbracketright 4 10.87 12.47 \|−3 2,↓,1/angbracketright 5 18.35 16.85 \|−1 2,↑,1/angbracketright 6 18.35 16.85 \|+1 2,↓,1/angbracketright in terms of the noninteracting sub-band energy dispersion E(0) nk obtained in the previous section and the exchange (Fock) self- energy /Sigma1(X) nk=−/summationdisplay n/prime/integraldisplaydk/prime 2πV(nn/prime) kk/primenF(En/primek/prime). (14) HerenF(E) denotes the Fermi-Dirac distribution function, and V(nn/prime) kk/primeis the matrix element of Coulomb interaction between nanowire-electron states given by V(nn/prime) kk/prime=C/integraldisplay d2r⊥/integraldisplay d2r/prime ⊥/integraldisplayL/2 −L/2dzei(k/prime−k)z /radicalbig z2+\|r⊥−r/prime ⊥\|2 ×ξ† n/primek/prime(r⊥)ξnk(r⊥)ξ† nk(r/prime ⊥)ξn/primek/prime(r/prime ⊥), (15) where C≡e2/(4πε0εr) is the Coulomb-interaction strength, r⊥≡(r,ϕ) denotes the position vector in the coordinates per- pendicular to the wire axis, and ξnk(r⊥)≡eiνϕe−iσz 2ϕφν,k(r) is the transverse spinor part of the wave function in Eq. ( 4). In the following, we consider the zero-temperature limit andthus replace the Fermi-Dirac distribution function by n F(E)≡ /Theta1(EF−E), with /Theta1(E) being the Heaviside step function and EFdenoting the Fermi energy. The condition Enk≡EFdeﬁnes the Fermi wave vectors kFnfor occupied nanowire sub-bands. We now focus on the low-density situation where only statesin the lowest doubly degenerate sub-band are occupied up tothe Fermi wave vector k F=kF1≡kF2. For this situation, we can write /Sigma1(X) nk=−2C R[˜/Lambda1intra(˜α,κ F,κ)+˜/Lambda1inter(˜α,κ F,κ)],(16a) where ˜/Lambda1intra(˜/Lambda1inter) includes contributions arising from the exchange interaction between particles from the same band(from different bands). In the limit L→∞ , we obtain the explicit expressions ˜/Lambda1intra(˜α,κ F,κ)=/integraldisplayκF −κFdκ/prime/integraldisplay1 0d/rho1/rho1/integraldisplay1 0d/rho1/prime/rho1/prime/integraldisplay2π 0d˜ϕK 0(\|κ−κ/prime\|/radicalbig /rho12+/rho1/prime2−2/rho1/rho1/primecos ˜ϕ) ×/bracketleftbig sin2ηκsin2ηκ/prime/vextendsingle/vextendsingleF(ε(1) 1/2,+) 0 (/rho1)/vextendsingle/vextendsingle2/vextendsingle/vextendsingleF(ε(1) 1/2,+) 0 (/rho1/prime)/vextendsingle/vextendsingle2+cos2ηκcos2ηκ/prime/vextendsingle/vextendsingleF(ε(1) 1/2,−) −1(/rho1)/vextendsingle/vextendsingle2/vextendsingle/vextendsingleF(ε(1) 1/2,−) −1(/rho1/prime)/vextendsingle/vextendsingle2 +sinακcosακsinακ/primecosακ/prime/parenleftbig/vextendsingle/vextendsingleF(ε(1) 1/2,+) 0 (/rho1)/vextendsingle/vextendsingle2/vextendsingle/vextendsingleF(ε(1) 1/2,−) −1(/rho1/prime)/vextendsingle/vextendsingle2+/vextendsingle/vextendsingleF(ε(1) 1/2,+) 0 (/rho1/prime)/vextendsingle/vextendsingle2/vextendsingle/vextendsingleF(ε(1) 1/2,−) −1(/rho1)/vextendsingle/vextendsingle2/parenrightbig/bracketrightbig , (16b) 115414-4COULOMB-EXCHANGE EFFECTS IN NANOWIRES WITH . . . PHYSICAL REVIEW B 92, 115414 (2015) ˜/Lambda1inter(˜α,κ F,κ)=/integraldisplayκF −κFdκ/primesin2(ηκ−ηκ/prime)/integraldisplay1 0d/rho1/rho1/integraldisplay1 0d/rho1/prime/rho1/prime/integraldisplay2π 0d˜ϕcos ˜ϕ ×K0(\|κ−κ/prime\|/radicalbig /rho12+/rho1/prime2−2/rho1/rho1/primecos ˜ϕ)F(ε(1) 1/2,+) 0 (/rho1)F(ε(1) 1/2,−) −1(/rho1)F(ε(1) 1/2,+) 0 (/rho1/prime)F(ε(1) 1/2,−) −1(/rho1/prime), (16c) where K0is the modiﬁed Bessel function of the second kind [ 37]. For the numerical evaluation of the intraband contri- bution ( 16b), we employ a modiﬁed quadrature method [ 38], described in greater detail in Appendix C, to deal with the logarithmic singularity encountered when the argument ofK 0(·) approaches zero. The exchange-renormalized density-of-states (quasiparti- cle) effective mass for the lowest sub-band can be calculatedfrom m int(α,k)=/planckover2pi12k ∂E(0) 1k ∂k+∂/Sigma1(0) 1k ∂k. (17) In Fig. 3, we compare the suppression of the quasiparticle ef- fective mass due to the exchange interaction in a nanowire withﬁnite spin-orbit coupling with that of an identical nanowirehaving zero spin-orbit coupling. As can be seen, the presenceof spin-orbit coupling further suppresses the exchange-relatedquasiparticle mass by 10–15% for the parameters used in ourcalculation. Note also the strong wave-vector dependence ofthe exchange-renormalized quasiparticle effective mass. The total exchange energy per particle for the nanowire- electron system is given by [ 1] E X N=1 2ρ/summationdisplay n/integraldisplaydk 2π/Sigma1(X) nknF(Enk), (18) where ρ=N/L is the quasi-1D electron density. Again we focus on the low-density situation where only states in thelowest doubly degenerate sub-band are occupied up to the −2 −1 012 kR0.400.450.500.550.600.650.700.750.80mint(α,k)/m0(α,k) FIG. 3. The ratio of the exchange-renormalized effective quasi- particle mass mint(α,k) to the bare single-particle effective mass m0(α,k) is plotted as a function of wave vector kfor states from the lowest nanowire sub-band assuming a material with dielectric constant εr=12.9,kFR=2.5, and spin-orbit-coupling strength ˜ α= 1.82 ( ˜α=0) as the solid (dashed) curve. For comparison, the dotted curve shows the result obtained under the assumption that the electric ﬁeld strength varies linearly with the radial coordinate near the wire’scenter, as described in Appendix A.Fermi wave vector. For this situation, we can write EX N=−C 2R[/Lambda1intra(˜α,κ F)+/Lambda1inter(˜α,κ F)], (19) where /Lambda1intra(˜α,κ F)=κ−1 F/integraltextκF −κFdκ˜/Lambda1intra(˜α,κ F,κ), and the anal- ogous expression applies for /Lambda1inter. Figure 4illustrates the functional dependences and relative magnitudes of /Lambda1intraand /Lambda1inter. As can be seen, the intraband contribution is generally dominant and weakly dependent on ˜ αvalues considered here. In contrast, the interband contribution changes signiﬁcantly asa function of ˜ α. For quantum wires without spin splitting, i.e., in the case ˜α=0, the exchange energy per particle was found to obey a universal scaling form [ 3,39,40]. Our expression for E X/N given in Eq. ( 19) generalizes these previous results to the case where spin-orbit coupling is ﬁnite. The change in magnitudeof the exchange energy arising from ﬁnite ˜ αcan be quantiﬁed through the relative difference /Delta1 X=EX(˜α/negationslash=0) EX(˜α=0)−1, (20) which is visualized in Fig. 5. For the values of ˜ αthat correspond to recent experimental realizations using InGaAs [ 35] and InSb [ 36], the associated change amounts to a suppression of the exchange-energy magnitude, which can be up to 1.6%. Thisbehavior is markedly different from the case of a 2D electronsystem where Rashba spin splitting has been shown [ 20]t o result in an increase of the exchange energy that is roughly one order of magnitude smaller. Thus the Rashba-type spin-orbitcoupling due to a radial electric ﬁeld in a cylindrical nanowiresystem is more similar to a 2D hole system where the interplaybetween quantum conﬁnement and spin-orbit effects alsoresults in a suppression of the exchange energy [ 18]. 0.00 .51 .01 .52 .02 .5 kFR0.51.01.52.02.53.0Λintra(˜α,kFR) 0.00.51.01.52.02.5 Λinter(˜α,kFR)×10−2 FIG. 4. (Color online) Scaling functions /Lambda1intraand/Lambda1interassoci- ated with the intraband and interband contributions to the exchange energy per particle in cylindrical nanowires with spin-orbit coupling (note the scale of 10−2for the interband contribution). Dashed (solid) curves corresponds to ˜ α=1.06 (1.82). 115414-5GRAY , KERNREITER, GOVERNALE, AND Z ¨ULICKE PHYSICAL REVIEW B 92, 115414 (2015) 0.00 .51 .01 .52 .02 .5 kFR−1.4−1.2−1.0−0.8−0.6−0.4−0.20.0Δx×10−2 ˜α=1.82(InGaAs) ˜α=1.06(InSb) FIG. 5. Relative change /Delta1Xin the magnitude of the exchange energy resulting from a ﬁnite Rashba-type spin-orbit coupling quantiﬁed by parameter ˜ α,a sd e ﬁ n e di nE q .( 20). Note the scale factor of 10−2for the abscissa. The dashed (solid) curve shows the result obtained for ˜ α=1.06 (1.82), which corresponds to a recent experimental realization using InSb (InGaAs) as the wire material. IV . CONCLUSIONS We have studied theoretically the electronic properties of the quasi-1D electron system realized in a cylindrical quantumwire subject to a radially symmetric Rashba-type spin-orbitcoupling. We determined the single-particle states for a hard-wall conﬁnement using sub-band k·ptheory. Focusing on the situation where only the lowest quasi-1D sub-band is occupied,we observed that the corresponding energy dispersion canbe very accurately (to within 0.5% error) calculated from aneffective 2 ×2 Hamiltonian. Taking the material parameters of two experimentally studied nanowire systems (one based onInGaAs and the other on InSb) as input, we have determinedthe inﬂuence of the spin-orbit strength on the lowest quasi-1Dsub-band’s energy dispersion and on the spin projection of itscorresponding eigenstates parallel to the wire axis, ﬁnding bothquantities to be affected by tens of percent due to the presenceof spin-orbit coupling. In particular, the density-of-states effec-tive mass of the noninteracting system turns out to be increasedby 20–25% for parameters applicable to the InSb nanowires. With single-particle states in hand, we calculated the quasiparticle effective mass for the lowest sub-band andfound its exchange-related suppression to be signiﬁcantlylarger in magnitude (by 10–15% for parameters used in ourcalculations) when spin-orbit coupling is ﬁnite. In contrast, themagnitude of the exchange energy per particle is marginallyreduced (by up to 1.6%) by spin-orbit coupling effects. Thuswe ﬁnd that any meaningful discussion of the interplaybetween spin-orbit coupling and exchange interactions in quantum wires needs to be carefully focused on speciﬁc physical quantities, as their relevant parametric dependencescan be quite different, both qualitatively and quantitatively.Furthermore, often the relevance of interaction effects in anelectron system is quantiﬁed in relative terms by a parameterr sthat is related to the ratio of contributions to the total energy arising from interactions and the single-electron dispersion,respectively [ 1]. In the present context, spin splitting causes an increase in the single-particle effective mass of quasi-1Delectrons simultaneously with the suppression of the exchangeenergy. As the relative change in the increase in noninteractingsystem’s effective mass is an order of magnitude larger than the relative decrease of the exchange energy, the relativeimportance of interactions as measured by r sturns out to be enhanced by spin-orbit coupling [ 41]. While we have focused on a speciﬁc conﬁguration of conﬁnement and spin-orbit coupling, our general results andoverall conclusions can be expected to apply also to otherspin-orbit-coupled nanowire systems, e.g., the one consideredin Ref. [ 42]. ACKNOWLEDGMENT We gratefully acknowledge useful discussions with S. Chesi. APPENDIX A: RADIAL ELECTRIC-FIELD PROFILE A proper self-consistent treatment of electrostatic effects generally requires the application of an iterative Schr ¨odinger- Poisson solver method that is speciﬁcally adapted to the samplelayout. An added complication arises from the intricate wayhow the Rashba spin-orbit coupling strength needs to bedetermined from expectation values of the electric ﬁeld takenin a multiband bound state [ 5]. As we intend to focus on the broad implications of spin-orbit coupling in conﬁned systems,we decided to make an assumption about the radial proﬁle ofthe electric ﬁeld entering in the spin-orbit term that enables usto obtain rather general physical insights. Here we show thebasic consistency of this assumption with the electrostatics ofthe bound-state conﬁguration for our system. Application of Gauss’s law using the cylindrical symmetry of the nanowire geometry yields the relation 2πrLE(r)=−e ε0εr/summationdisplay j/summationdisplay \|k\|/lessorequalslantkF/integraldisplayL 0dz/integraldisplay2π 0dϕ/integraldisplayr 0dr/primer/prime ×[ψj(r/prime,ϕ,z)]†ψj(r/prime,ϕ,z), (A1) with the single-particle wave functions ψ1,2(r,ϕ,z )g i v e n in Eq. ( 9). Straightforward calculation yields E(r)= E0[SkFRP0(r/R)+CkFRP1(r/R)], where E0=−Ne/(2πRL ε0εr) is an overall scale containing the number of particles N,Sκ=1 κ/integraltextκ 0dκ/primesin2ηκ/prime, and Cκ=1 κ/integraltextκ 0dκ/primecos2ηκ/primeare weightings of the mixed bound-state contributions for thelowest nanowire sub-band, and P 0(1)(/rho1)=2π /rho1/integraldisplay/rho1 0d/rho1/prime/rho1/prime/bracketleftbig F(ε(1) 1/2,+(−)) 0(1) (/rho1/prime)/bracketrightbig2(A2) are the radial density proﬁles associated with the relevant bound states. The calculated full electric-ﬁeld proﬁle is shown in Fig. 6. Our results from the main paper suggest that generally SkFR≈ 1,CkFR≈0; hence E(r) should be essentially determined by the P0(r/R) contribution. This is indeed observed in the numerical evaluation. Also, as expected from the shape of thedensity proﬁle associated with the m=0 bound-state wave function (cf. Appendix B), the leading behavior at r/R/lessmuch1 is linear. However, over most of the wire’s cross section, theﬁeld proﬁle is quite well approximated by a constant, whichsupports our pragmatic assumption. It is also observed fromdirect calculation that S kFRandCkFRare almost constant in 115414-6COULOMB-EXCHANGE EFFECTS IN NANOWIRES WITH . . . PHYSICAL REVIEW B 92, 115414 (2015) 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.40.81.21.62 rRr FIG. 6. (Color online) The radial proﬁle of the electric-ﬁeld magnitude E(r) associated with our calculated nanowire states is plotted as the solid red (gray) curve. The asymptotically linear behavior for small r/R is captured by the blue (gray) dashed line. The horizontal (cyan [light gray]) band indicates the range of ﬁeld values that lie within 10% of the mean value. The weak variation ofE(r)f o rr/greaterorsimilar0.3Rmotivates our assumption of a constant ﬁeld magnitude for the radially symmetric Rashba term in Eq. ( 2). The scale of the electric ﬁeld is eE0=1.6κFμeVnm−1for the scenario based on InGaAs while it is eE0=41κFμeV nm−1for InSb. the relevant range kFR< 2.5 where only the lowest nanowire sub-band is occupied. In order to conﬁrm that the omission of the linear electric ﬁeld dependence for r/R<∼0.3 will not alter our conclusions, we consider an electric ﬁeld which is modelled by a lineardependence on the radial coordinate up to /rho1</rho1 0and a constant for /rho1>/rho1 0. In terms of the dimensionless Hamiltonian description in Eq. ( 5), this implies the replacement ˜ α→ ˜α[/rho1 /rho10/Theta1(/rho10−/rho1)+/Theta1(/rho1−/rho10)]. Proceeding as in the case of a constant electric ﬁeld, we ﬁnd for the wave functions in the region with /rho1</rho1 0the Bessel-function solutions Jm(/rho1√ε+m˜α//rho1 0). For the region /rho1>/rho1 0, we obtain wave functions which are a superposition of modiﬁed Laguerrefunctions (see Appendix B) and conﬂuent hypergeometric functions of the second kind. Applying the standard matchingconditions at /rho1=/rho1 0to ensure continuity of the wave functions and their products with the velocity operator in the transversedirection determines the unknown coefﬁcients. In this context,it should be noted that the lowest state is independent ofthe electric ﬁeld. Considering the scenario with ˜ α=1.82 as an example and taking into account the hard-wall boundarycondition, we ﬁnd only small changes for the band-edge energies E (0) (3,4)0/E0=11.07 and E(0) (5,6)0/E0=18.22 when /rho10=0.3 (cf. Table I). In Fig. 7we show the real part (dashed curve) and imaginary part (dotted curve) of the sub-bandedge wave function for spin up of the second-excited state\|−1/2,↑,1/angbracketrightand compare this with the corresponding wave function obtained under the assumption of a constant radialelectric-ﬁeld strength. We can therefore conclude that thelinear electric ﬁeld dependence for /rho1</rho1 0changes the relevant wave functions used in our calculations only slightly. For κ/negationslash=0, we ﬁnd that the matrix element I(11) 1/2≈−0.884, while it isI(11) 1/2≈−0.916 with the assumption of a constant electric ﬁeld, yielding only subpercent changes for the dispersions andexchange-related quantities (see, for instance, Fig. 3). 0.00 .20 .40 .60 .81 .0 r/R−2.0−1.5−1.0−0.50.0\|−1 2,↑,1 FIG. 7. Real part (dashed curve) and imaginary part (dotted curve) of the sub-band edge wave function for spin up of the second-excited state obtained for an electric ﬁeld that depends linearlyon the wire’s radius up to /rho1=0.3 and is constant for /rho1> 0.3. For comparison we show the real wave function of the corresponding state (solid curve) obtained under the assumption of a constant electric ﬁeldthroughout. APPENDIX B: SOLUTION OF THE RADIAL-CONFINEMENT PROBLEM The general solution of the differential equations present in the diagonal entries of Eq. ( 5) are power series, given by F(ε(n/prime) ν,±) m(/rho1)=/rho1m/parenleftBigg a0+a1/rho1+∞/summationdisplay n=2an/rho1n/parenrightBigg , (B1) which fulﬁll the relation F(ε(n/prime) ν,±) −m(/rho1)=F(ε(n/prime) ν,∓) +m(/rho1) yielding the eigenstates. Disregarding the ill-behaved and unphysical partin the expansion at the origin, the coefﬁcients of the polyno-mials are determined by the recursion relation n(n±2m)a n+m˜αan−1+ε(n/prime) ν,±an−2=0, (B2) witha1=− ˜αm/(1±2m)a0, where the upper (lower) sign applies to m> 0(m< 0). The coefﬁcient a0is determined by the normalization condition 2 π/integraltext1 0d/rho1/rho1\|F(ε(n/prime) ν,±) m(/rho1)\|2=1. We note that the polynomial with coefﬁcients given by Eq. ( B2) represents a modiﬁed Laguerre function that becomes the standard Bessel function J0(√ ε(n/prime) ν,±/rho1)f o r ˜α=0 and/or m=0. The band-edge energies, ε(n/prime) ν,±are found by imposing hard-wall boundary conditions on the radial wave function; i.e., forr=R, we require F(ε(n/prime) ν,±) m(/rho1=1)=0. (B3) For not too large values of ˜ α, the lowest spin- ↑(↓) sub- band-edge state has ν=1/2(−1/2) total angular momentum. However, as seen from Fig. 8, a level crossing occurs for ˜α≈4.2, beyond which the new lowest spin- ↑(↓) sub-band edge is a state with ν=±3/2(−3/2). The variation of the band-edge energy ε(1) ±3 2,±as a function of ˜ αcan be approximated using standard perturbation theory, yielding ε(1) ±3 2,±=ε0−˜α/integraltext1 0d/rho1J2 1(/rho1√ε0) /integraltext1 0d/rho1/rho1J2 1(/rho1√ε0), (B4) 115414-7GRAY , KERNREITER, GOVERNALE, AND Z ¨ULICKE PHYSICAL REVIEW B 92, 115414 (2015) 012345 ˜α468101214E(n) ν,σE(1) ±3 2,12 E(1) ±1 2,12 FIG. 8. (Color online) Energy eigenvalues of the lowest two doubly degenerate quasi-1D sub-band edges, plotted as a function of the effective Rashba spin-orbit-coupling parameter ˜ α. The blue (gray) dashed curve is an approximation based on Eq. ( B4). where ε0≈14.68 is the band-edge energy of the correspond- ing band for ˜ α=0. APPENDIX C: REGULARIZATION OF THE INTEGRAND FOR CALCULATING THE EXCHANGE ENERGY In the calculation of the exchange energy we have to deal with integrals of the form I=/integraldisplay/integraldisplay dkdk/primeG(k,k/prime)K0(\|k−k/prime\|/radicalbig r2+r/prime2−2rr/primecosϕ), (C1)withG(k,k/prime) being a smooth function of kandk/prime. A logarithmic singularity occurs when the argument of K0(·) vanishes. This happens when either the square root is zero, at /vectorr⊥=/vectorr/prime ⊥,o r when k=k/prime. To regularize the integral for the case where /vectorr⊥=/vectorr/prime ⊥, we add a small amount 0+to the term under the square root. Then by decreasing the value of 0+, we perform a series of calculations until the result for the exchange energydoes not change within a certain tolerance. The situation for k=k /primecan be regularized analytically. To this end, we add to and subtract from Eq. ( C1)t h et e r m /integraldisplay/integraldisplay dkdk/primeG(k,k)l n (\|k−k/prime\|/radicalbig r2+r/prime2−2rr/primecosϕ+0+). (C2) Adding this term to Eq. ( C1) cancels the logarithmic singular- ity. The k/primeintegration of the subtracted term can be performed analytically and Eq. ( C1) becomes I=/integraldisplay dk/braceleftbigg/bracketleftbigg/integraldisplay dk/primeG(k,k/prime)K0(\|k−k/prime\|/radicalbig r2+r/prime2−2rr/primecosϕ) +G(k,k)l n (\|k−k/prime\|/radicalbig r2+r/prime2−2rr/primecosϕ)/bracketrightbigg −G(k,k)/bracketleftbigg kln/parenleftbiggkF+k kF−k/parenrightbigg −2kF+2kFln/parenleftbig/radicalBig k2 F−k2 ×/radicalbig r2+r/prime2−2rr/primecosϕ+0+/parenrightbig/bracketrightbigg/bracerightbigg . (C3) The expression Eq. ( C3) is manifestly ﬁnite for k=k/prime. [1] G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, UK, 2005). [2] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54,437 (1982 ). [3] A. 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Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, UK, 2004). [28] J. Alicea, Rep. Prog. Phys. 75,076501 (2012 ). [29] J. Klinovaja, P. Stano, and D. Loss, Phys. Rev. Lett. 109,236801 (2012 ). [30] K. Storm, G. Nylund, L. Samuelson, and A. P. Micolich, Nano Lett. 12,1(2012 ). [31] D. A. Broido and L. J. Sham, P h y s .R e v .B 31,888(1985 ). [32] S.-R. E. Yang, D. A. Broido, and L. J. Sham, Phys. Rev. B 32, 6630 (1985 ). [33] U. R ¨ossler, Solid-State Theory: An Introduction , 2nd ed. (Springer, Berlin, 2009), pp. 128–129 . 115414-8COULOMB-EXCHANGE EFFECTS IN NANOWIRES WITH . . . PHYSICAL REVIEW B 92, 115414 (2015) [34] Even though the type of Rashba spin splitting realized in these experiments differs from that adopted in our work, thesame bandstructure-related prefactors and orders of magnitudefor the electric ﬁeld apply in both their and our systems. Thismotivates our use of spin-orbit-coupling strengths measured inRefs. [ 35,36]. [35] T. Sch ¨apers, J. 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PhysRevB.73.033202.pdf	High-resolution photoemission study of the temperature-dependent c-fhybridization gap in the Kondo semiconductor YbB 12 Y. Takeda,1,M. Arita,1M. Higashiguchi,2K. Shimada,1H. Namatame,1M. Taniguchi,1,2F. Iga,3and T. Takabatake3 1Hiroshima Synchrotron Radiation Center, Hiroshima University, Kagamiyama 2-313, Higashi-Hiroshima 739-0046, Japan 2Graduate School of Science, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan 3Graduate School of Advanced Sciences of Matter, Hiroshima University, Higashi-Hiroshima 739-8530, Japan /H20849Received 2 November 2005; published 24 January 2006 /H20850 The temperature-dependent metal-to-insulator transition in the Kondo semiconductor YbB 12single crystal has been studied by means of high-resolution photoemission spectroscopy with a tunable photon energy. Anenergy gap in the valence band is gradually formed below T 1/H11011150 K, and at the same time, the Yb 4 f7/2 Kondo peak at 55 meV grows and shifts to a lower binding energy. Below T2/H1101160 K, an additional spectral feature at 15 meV becomes apparent in the Yb 4 fand Yb 5 dderived spectra, indicating a strongly hybridized character. The 15 meV feature in the Yb 4 fderived spectra is intense at the L point of the Brillouin zone and diminishes away from the L point. These results indicate that the energy gap is formed by the hybridizationbetween the Yb 4 fand Yb 5 dstates. DOI: 10.1103/PhysRevB.73.033202 PACS number /H20849s/H20850: 75.20.Hr, 75.30.Mb Among 4 felectron systems, Kondo semiconductors have attracted much interest for their unusual metal-to-insulatortransition /H20849MIT /H20850at a characteristic temperature T .1In the temperature regime T/H11021T/H11021TK/H20849Kondo temperature /H20850, the Kondo semiconductor seems to behave as a metal with alocal moment. As the temperature decreases below T , the local moment gradually disappears, and an energy gap opensat the Fermi level /H20849E F/H20850.1The mechanism of the MIT in Kondo semiconductors is still an unsolved issue. Rare-earth Kondo semiconductors are located at the boundary between the valence ﬂuctuation regime with an itinerant 4 fstate and the Kondo regime with a localized 4 fstate, where unusual physical phenomena are often observed.2YbB 12, with a cu- bic crystal structure, has been regarded as one of the mosttypical Kondo semiconductors, 1,3,4and a detailed study of YbB 12should provide us with a clue to understand physics of the Kondo semiconductors. X-ray absorption spectroscopymeasurement of YbB 12shows that the valence of Yb /H20849v/H20850is close to /H110013/H20849v/H11022+2.95 /H20850.5This suggests that YbB 12is located in the mixed valence regime very close to the Kondo regime and the slight appearance of the Yb2+components should play an important role in the energy gap formation. Transportmeasurements on single-crystal samples indicated that anisotropic gap /H208492/H9004 t=6–8 meV /H20850is formed at a low temperature.3A signiﬁcant reduction of the optical conduc- tivity below /H1101180 K has been reported, which leads to the magnitude of the direct gap being 2 /H9004opt/H1101125 meV.7Since the gap magnitude measured by optical reﬂectivity is largerthan the transport gap /H20849/H9004 opt/H11022/H9004 t/H20850, the gap should be indirect and, therefore, the energy gap probably derives from the hy- bridization between a wide conduction band and a narrow4fband /H20849c−fhybridization /H20850. 7 A photoemission spectroscopy /H20849PES /H20850study, using a He lamp /H20849h/H9263=21.2 eV /H20850, indicated a temperature-dependent re- duction of the spectral intensity at EFon cooling.6It is note- worthy that photoemission spectra of YbB 12taken at h/H9263 =21.2 eV mainly reﬂect the valence bands derived from the Yb 5 da n dB2 spstates.8Although a Yb 4 fspectrum at 30 Kath/H9263=125 eV with energy resolution /H9004E/H1101155 meV has been reported,9,10the details of the temperature dependence of the Yb 4 fstates near EFhas not thus far been clariﬁed. Further- more, there has been no angle-resolved photoemission /H20849ARPES /H20850study on YbB 12, and no direct evidence that the Yb 4fstates are responsible for the energy gap formation. In this paper, we present temperature-dependent ultravio- let PES spectra of a YbB 12single crystal taken at selected two-photon energies, h/H9263=15.8 and 100 eV, with improved energy resolution, /H9004E=5 and 15 meV. The ratio of the Yb5 d/B2pphotoionization cross sections passes through a maxi- mum at h/H9263/H1101116 eV.8On the other hand, the photoemission spectrum taken at h/H9263/H11011100 eV reﬂects the Yb 4 fstate due to the high photoionization cross section compared with otherorbitals in the valence band. 8With an improved energy reso- lution, the temperature-dependent energy gap formation inthe valence bands and in the Yb 4 fstates have been directly revealed. Moreover, a peak at 15 meV was clearly observedat a low temperature for the ﬁrst time. ARPES measurementsclearly indicate that the peak is enhanced near the L point ofthe Brillouin zone. We will discuss a mechanism for the cre-ation of the c-fhybridization gap in YbB 12taking into ac- count the result of the LDA /H11001U calculation,11and inelastic neutron scattering /H20849INS/H20850measurements.12–15 The single crystals of YbB 12and, as a metallic reference LuB 12, were grown by the ﬂoating zone method using an image furnace with four xenon lamps.3The high quality of the YbB 12crystal was conﬁrmed by the fact that its resistiv- ity increased by more than ﬁve orders of magnitude as thetemperature decreased from 300 to 1.3 K. 1,3,6For the tem- perature regime 15 K /H11021T/H1102140 K, the energy gap estimated from the electrical resistivity and Hall coefﬁcients is 68 and90 K, respectively. 3The magnetic susceptibility follows a Curie-Weiss law down to 170 K, passes through a broadmaximum at T max/H1101180 K, and decreases on further cooling.3,4Angle-integrated and angle-resolved high- resolution PES measurements were performed using high-resolution hemispherical electron-energy analyzersPHYSICAL REVIEW B 73, 033202 /H208492006 /H20850 1098-0121/2006/73 /H208493/H20850/033202 /H208494/H20850/$23.00 ©2006 The American Physical Society 033202-1/H20849GAMMADATA-SCIENTA SES200 and SES2002 /H20850installed on beamlines BL-1 /H20849h/H9263=27–300 eV /H2085016and BL-9 /H20849h/H9263 =5–40 eV /H2085017of the electron-storage ring /H20849HiSOR /H20850at Hi- roshima Synchrotron Radiation Center /H20849HSRC /H20850, Hiroshima University. Energy calibration was performed using the AuFermi edge at a low temperature. In order to obtain cleansample surfaces, we cleaved the single-crystal samples in situ. A cleaved /H20849110/H20850surface was used for the ARPES mea- surements. The base pressure was below 1.0 /H1100310 −10Torr. Figure 1 /H20849a/H20850shows temperature-dependent high-resolution PES spectra of YbB 12taken with h/H9263=15.8 eV photons. The spectral intensity at EFgradually decreases on cooling, indi- cating temperature-dependent energy gap formation. This be-havior is similar to that seen in a previous experiment usingHeI radiation 6except that we now see clear spectral features at 15 and 50 meV. These spectral features become apparentdue to stronger Yb 5 dcontribution. The reference system of metallic LuB 12, on the other hand, clearly exhibits a Fermi edge down to 16 K, conﬁrming the importance of the Yb 4 f contribution near EFto the MIT in YbB 12/H20851Fig. 1 /H20849b/H20850/H20852.I n order to derive the spectral density-of-states /H20849spectral DOS /H20850 of the valence states, the photoemission spectra were dividedby a Gaussian-broadened /H20849/H9004E=5 meV /H20850Fermi-Dirac distri- bution function /H20851Fig. 1 /H20849c/H20850/H20852. 6From these spectra, we obtained the spectral intensities at EFand 15 meV as a function of temperature. As Fig. 2 /H20849a/H20850shows, the spectral intensity at EF decreases on cooling below /H11011150 K. The spectral feature at 15 meV, which was not observed in the previous experimentusing a He lamp, 6becomes apparent only below /H1101160 K. Figure 3 /H20849a/H20850exhibits Yb3+and Yb2+4fderived spectral features at h/H9263=100 eV over a wide binding energy region. Figure 3 /H20849b/H20850shows two sharp peaks at 35 meV and 1.3 eV, which are derived from the Yb 4 f7/2and Yb 4 f5/2states in the bulk, respectively. Each of them is accompanied by a broadpeak at 0.9 and 2.2 eV derived from the surface states. 18Thesurface and bulk components are well separated and most of the spectral weight near EFis derived from the bulk compo- nent. It should be noted that the spectral intensity from thebulk component with respect to the surface component ismuch larger in the present data compared with that seen inprevious experiments on the scraped sample surface. 9,10We can see signiﬁcant temperature dependence only in the bulkcomponent. Figure 3 /H20849c/H20850shows the temperature dependence FIG. 1. /H20849Color online /H20850/H20849a/H20850Temperature-dependent high- resolution photoemission spectra of YbB 12taken at h/H9263=15.8 eV. /H20849b/H20850Comparison with LuB 12./H20849c/H20850Photoemission spectra divided by a Gaussian-broadened Fermi-Dirac function. FIG. 2. /H20849Color online /H20850/H20849a/H20850Temperature dependence of spectral intensities of the valence bands at a binding energy of 15 meV andatE F./H20849b/H20850Peak position of the Yb 4 f7/2peak as a function of tem- perature. Below /H11011150 K, the peak is shifted to a lower binding energy by /H1101120 meV. FIG. 3. /H20849Color online /H20850/H20849a/H20850Valence-band spectrum taken at h/H9263 =100 eV at 250 K in a wide energy range. /H20849b/H20850The Yb2+4fderived spectra at 9 and 250 K. /H20849c/H20850Temperature-dependent Yb 4 f7/2spectra near EF. A difference spectrum obtained by subtracting the 120 K spectrum from the 60 K spectrum is plotted. The 15 meV peakappears below /H1101160 K.BRIEF REPORTS PHYSICAL REVIEW B 73, 033202 /H208492006 /H20850 033202-2of the Yb 4 f7/2states near EF. On cooling, the intensity of the Yb 4 f7/2derived structure is increased and this peak is shifted to lower binding energy.19Figure 2 /H20849b/H20850shows the binding energy of the 4 f7/2peak position as a function of temperature. The peak position stays at /H1101155 meV down to /H11011180 K, below which it shifts to reach a value of /H1101135 meV by /H1101160 K, after which it remains almost constant at /H1101135 meV. An enhancement of the Yb 4 f7/2peak at low temperatures was also observed in the Kondo metal YbAl 3,20and was reasonably explained within the scheme of the single-impurity Anderson model /H20849SIAM /H20850. The temperature- dependent enhancement and shift of the Yb 4 f 7/2peak are indicated by a calculation based on the periodic Andersonmodel /H20849PAM /H20850. 21A theoretical investigation of the Seebeck coefﬁcient also supports the notion of a shift of the 4 fenergy level towards EFon cooling.22 Since our energy resolution is improved to 15 meV, a small peak at 15 meV in the Yb 4 fspectrum at 9 K, which was not resolved in the previous photoemissionmeasurements, 9is now clearly resolved in addition to the main peak at /H1101135 meV. Even at 60 K, one can discern a shoulder on the lower binding energy side of the main peakat/H1101135 meV /H20851see a difference spectrum in Fig. 3 /H20849c/H20850/H20852. The appearance of the 15 meV peak structure cannot be ex-plained in the framework of the SIAM, indicating clearlythat we should take into account the periodicity of the 4 f states. Although the energy gap at E Fis obscured by the intense 15 meV peak whose binding energy is comparable tothe energy resolution, as Figs. 1 /H20849a/H20850and 1 /H20849c/H20850show, a corre- sponding spectral feature is also present at 15 meV in the Yb5dspectra. 23We should note that the spectral feature at 15 meV in the Yb 5 dstate also becomes prominent below 60 K /H20851Fig. 2 /H20849a/H20850/H20852. These observations indicate that the 15 meV peak is derived from the hybridization between theYb 4 fand Yb 5 dstates /H20849d−fhybridization /H20850. We thus ﬁnd two characteristic temperatures, namely the onset of the energy gap formation and the start of the Yb4f 7/2peak shift at T1/H11011150 K, and the appearance of the 15 meV structure at a second, T2/H1101160 K. Although there is no anomaly in the physical properties at around T1,w ea s - sume this temperature provides a measurement of the onsetof the gradual crossover from the metallic to insulating stateson cooling. On the other hand, T 2is close to the temperature for the maximum of the magnetic susceptibility /H1101180 K,3,4 and the temperature /H1101150 K from which the electrical resis- tivity increases rapidly.1The observed spectral DOS near EF at 17 K forms a narrow energy gap of /H1102115 meV, visible in the spectrum at the bottom of Fig. 1 /H20849c/H20850, which is close to the value of the transport gap, 2 /H9004t=6–8 meV.3We assume, therefore, that the coherent nature of the Yb 4 fstate mani- fests itself below T2. In order to further examine the nature of the 15 meV peak, we performed ARPES measurements around the Lpoint and away from the L point /H20849on the /H9018line/H20850by rotating the/H20849110/H20850axis as shown in Fig. 4. In this way one can exam- ine energy bands in the /H9003KLUX mirror plane along a line /H20849in green /H20850as indicated in the ﬁgure. In order to improve statis- tics, the spectra were averaged over the shaded region in Fig.4. An inner potential was assumed to be 10 eV. The spec-trum at the L point exhibits the 15 meV peak, but the peak disappears as we move away from the L point. Thus, the15 meV peak is localized in kspace. On the other hand, the peak at /H1101135 meV exists both at the L point and along the /H9018 line, but with much broader width. These observations implythat the 35 meV peak is not spatially delocalized. It is rea-sonable to assume that the 35 meV peak corresponds to theKondo resonance as described by the SIAM, while it is dif-ﬁcult to interpret the 15 meV peak in the ARPES spectrumwithin a localized Kondo picture. 2The present observations strongly suggest again that the 15 meV peak around the Lpoint originates from the 4 fstates which are responsible for thed−fhybridization-gap formation below T 2. Finally we discuss the 15 meV peak in relation to INS measurements.12–15Magnetic excitations were observed at /H1101115,/H1101120, and /H1101137 meV at low temperatures. The peak at 15 meV appeared below /H1101160 K, and became enhanced as temperature decreases.13The intensity of the 15 meV peak showed strong qdependence while that of the peak at /H1101137 meV did not. Recent INS results indicate that the peak at/H1101115 meV in YbB 12is related to the energy gap formation.13–15The 15 meV peak intensity is strongly en- hanced at /H9004q=/H208493/2,3/2,3/2 /H20850.14It is surprising that the tem- perature dependence and the energy of the 15 meV peak in the magnetic response have many similarities with the PESspectral features seen in the present study. Although the physical properties involved in the PES and INS measurements are different, we are tempted by thesesimilarities to compare these two sets of results. If the15 meV peak in the magnetic response corresponds to theindirect gap excitation at a low temperature, 4it is reasonable to interpret the 15 meV peak based on the energy bandsgiven by the LDA /H11001U calculation. 11Since the calculation can explain the speciﬁc heat measurement,11we assume that it gives the correct ground state. The calculation indicates thatthe conduction-band minimum /H20849CBM /H20850exists at the X and W points, and the valence-band maximum /H20849VBM /H20850is at the L point. 11The calculation also reveals that around the L point there exist Yb 4 fderived states with a binding energy of /H1101195 meV. The corresponding peak in the ARPES spectrum FIG. 4. /H20849Color online /H20850Angle resolved photoemission spectra of the Yb 4 f7/2peak at 9 K using h/H9263=100 eV.BRIEF REPORTS PHYSICAL REVIEW B 73, 033202 /H208492006 /H20850 033202-3is located at 15 meV, the energy is smaller than the calcu- lated one due to signiﬁcant electron correlation effects. Theﬂat VBM in the calculation originates in the Yb 4 fand 5 d states, which is consistent with the present PES results. Theelectrons at the VBM around the L point can be excited tothe X points in the CBM, with a momentum transfer of /H9004q =/H208493/2,3/2,3/2 /H20850and an energy transfer corresponding to the indirect energy gap of /H1101115 meV. Thus, the peak at 15 meV in both the magnetic response and PES can be interpretedconsistently within this framework. It has been claimed that the magnetic excitations at 20 and 37 meV should derive from the many-body interactions. 12In the PES spectrum, the peak at /H1101135 meV is broad, which may correspond to the presence of two superimposed peaksat 20 and 37 meV. The two peak structure, is not resolvedsince these peaks are broadened due to the lifetime, and ad-ditionally broadened with the energy resolution /H1101115 meV. In summary, the temperature-dependent energy gap for- mation in both the valence bands and the Yb 4 fstates ofYbB 12have been examined by means of high-resolution PES. An energy gap /H20849/H1102115 meV /H20850is gradually formed in the valence band on cooling. Two characteristic temperatures are found at T1/H11011150 K and at T2/H1101160 K. The 55 meV peak at 250 K is shifted slightly toward lower binding energy by/H1101120 meV on cooling below T 1. The appearance of a 15 meV peak in the Yb 4 fand Yb 5 dstates is clearly observed below T2. The 15 meV peak in the Yb 4 fstate is enhanced near the L point. The present results give direct evidence that thecoherent nature of the Yb 4 fstate plays an important role in the energy gap formation via the d-fhybridization below T 2. The authors thank T. Saso and A. Fujimori for their valu- able discussions. This work was partly supported by a Grant-in-Aid for Scientiﬁc Research /H2084913CE2002 /H20850of MEXT Japan. We thank the Material Science Center, N-BARD, HiroshimaUniversity for supplying liquid helium. The synchrotron ra-diation experiments at HiSOR have been done under the ap-proval of HSRC /H20849Proposal Nos. 03-A-38 and 03-A-39 /H20850. *Synchrotron Radiation Research Center, JAERI/SPring-8, Koto 1-1-1, Sayo-cho, Sayo-gun, Hyogo 679-5148, Japan. Email ad-dress: ytakeda@spring8.or.jp 1T. Takabatake, F. Iga, T. Yoshino, Y. Echizen, K. Katoh, K. Koba- yashi, M. Higa, N. Shimizu, Y. Bando, G. Nakamoto, H. Fujii,K. Izawa, T. Suzuki, T. Fujita, M. Sera, M. Hiroi, K. Maezawa,S. Mock, H. v. Lohneysen, A. Bruckl, K. Neumaier, and K.Andres, J. Magn. Magn. Mater. 177-181 , 277 /H208491998 /H20850. 2T. Kasuya, J. Phys. Soc. Jpn. 65, 2548 /H208491996 /H20850. 3F. Iga, N. Shimizu, and T. Takabatake, J. Magn. Magn. Mater. 177-181 , 337 /H208491998 /H20850. 4F. Iga, S. Hiura, and T. Takabatake, Physica B 259-261 , 312 /H208491999 /H20850. 5P. A. Alekseev, E. V. Nefeodova, U. Staub, J. -M. Mignot, V. N. Lazukov, I. P. Sadikov, L. Soderholm, S. R. Wassermann, Yu. B.Paderno, N. Yu. Shitsevalova, and A. Murani, Phys. Rev. B 63, 064411 /H208492001 /H20850. 6T. Susaki, Y. Takeda, M. Arita, K. Mamiya, A. Fujimori, K. Shi- mada, H. Namatame, M. Taniguchi, N. Shimizu, F. Iga, and T.Takabatake, Phys. Rev. Lett. 82, 992 /H208491999 /H20850. 7H. Okamura, S. Kimura, H. Shinozaki, T. Nanba, F. Iga, N. Shimizu, and T. Takabatake, Phys. Rev. B 58, R7496 /H208491998 /H20850. 8J. J. Yeh and I. Lindau, At. Data Nucl. Data Tables 32,1/H208491985 /H20850. 9T. Susaki, A. Sekiyama, K. Kobayashi, T. Mizokawa, A. Fuji- mori, M. Tsunekawa, T. Muro, T. Matsushita, S. Suga, H. Ishii,T. Hanyu, A. Kimura, H. Namatame, M. Taniguchi, T. Maya-hara, F. Iga, M. Kasaya, and H. Harima, Phys. Rev. Lett. 77, 4269 /H208491996 /H20850. 10T. Susaki, T. Konishi, A. Sekiyama, T. Mizokawa, A. Fujimori, T. Iwasaki, S. Ueda, T. Matsushita, S. Suga, H. Ishii, F. Iga, and M.Kasaya, Phys. Rev. B 56, 13727 /H208491997 /H20850.11T. Saso and H. Harima, J. Phys. Soc. Jpn. 72, 1131 /H208492003 /H20850. 12A. Bouvet, T. Kasuya, M. Bonnet, L. P. Regnault, J. Rossat- Mignod, F. Iga, B. Fåk, and A. Severing, J. Phys.: Condens.Matter 10, 5667 /H208491998 /H20850. 13F. Iga, A. Bouvet, L. P. Regnault, T. Takabatake, A. Hiess, and T. Kasuya, J. Phys. Chem. Solids 60, 1193 /H208491999 /H20850. 14J. M. Mignot, P. A. Alekseev, K. Nemkovski, L. P. Regnault, and F. Iga /H20849unpublished /H20850. 15E. V. Nefeodova, P. A. Alekseev, J.-M. Mignot, V. N. Lazukov, I. P. Sadikov, Yu. B. Paderno, N. Yu. Shitsevalova, and R. S. Ec-cleston, Phys. Rev. B 60, 13507 /H208491999 /H20850. 16K. Shimada, M. Arita, Y. Takea, H. Fujino, K. Kobayashi, T. Narimura, H. Namatame, and M. Taniguchi, Surf. Rev. Lett. 9, 529 /H208492002 /H20850. 17M. Arita, K. Shimada, H. Namatame, and M. Taniguchi, Surf. Rev. Lett. 9, 535 /H208492002 /H20850. 18See for example, S. Hüfner, Photoelectron spectroscopy /H20849Springer-Verlag, Berlin, 2003 /H20850, Chap. 8. 19We have conducted temperature-dependent hard x-ray PES using h/H9263=5.95 keV, and found the similar temperature dependence /H20851Y. Takeda et al. , Physica B 351, 286 /H208492004 /H20850/H20852. 20L. H. Tjeng, S.-J. Oh, E.-J. Cho, H.-J. Lin, C. T. Chen, G.-H. Gweon, J.-H. Park, J. W. Allen, T. Suzuki, M. S. Makivi ć, and D. L. Cox, Phys. Rev. Lett. 71, 1419 /H208491993 /H20850. 21T. Mutou and D. S. Hirashima, J. Phys. Soc. Jpn. 63, 4475 /H208491994 /H20850. 22T. Saso and K. Urasaki, J. Phys. Soc. Jpn. 71, Suppl 288 /H208492002 /H20850. 23Recent photoconductivity measurements by H. Okamura, T. Michizawa, T. Nanba, S. Kimura, F. Iga, and T. Takabatake, J.Phys. Soc. Jpn. 74, 1954 /H208492005 /H20850, show also a structure at 15 meV at low temperature.BRIEF REPORTS PHYSICAL REVIEW B 73, 033202 /H208492006 /H20850 033202-4
PhysRevB.79.035312.pdf	Conductance of the capacitively coupled single-electron transistor with a Tomonaga-Luttinger liquid island in the Coulomb blockade regime Vladimir Bubanja1and Shuichi Iwabuchi2 1Measurement Standards Laboratory of New Zealand, Industrial Research Ltd., P .O. Box 31-310, Lower Hutt 5040, Wellington, New Zealand 2Department of Physics, Graduate School for Interdisciplinary Scientiﬁc Phenomena and Information, Nara Women’ s University, Kitauoya-Nishimachi, Nara 630-8506, Japan /H20849Received 19 August 2008; revised manuscript received 3 December 2008; published 13 January 2009 /H20850 We consider the electron transport in the capacitively coupled single-electron transistor with an ultrasmall Tomonaga-Luttinger liquid island. The charging effects, as well as the Tomonaga-Luttinger liquid nature, aretreated by a self-consistent theory of the Coulomb blockade using the open boundary bosonization technique.Analytical expressions for conductance are derived in the limits of low and high voltages and temperatures, forbulk and edge island contact geometries, and for arbitrary environmental impedance. For an inﬁnite system, weobtain the power law of the conductance with the exponent changed from the usual Tomonaga-Luttingerexponent due to the effects of the electromagnetic environment. For a ﬁnite system, we obtain expressions forthe conductance as a function of voltage near the Coulomb blockade boundary and as a function of temperaturefor low temperatures; these expressions differ from the usual power-law behavior. The results show thepotential for improving the accuracy of single-electron devices such as those used in electrical metrology. DOI: 10.1103/PhysRevB.79.035312 PACS number /H20849s/H20850: 73.23.Hk, 85.35.Gv I. INTRODUCTION Developments in fundamental metrology over recent years have made it feasible to consider introducing a newInternational System /H20849SI/H20850of Units based on a set of exactly deﬁned values of fundamental constants. 1One requirement is the ability to realize the base units in a straightforward fash-ion. For example, by ﬁxing the value of the elementarycharge e, the ampere could be realized from the simple rela- tion I=efby using a single-electron pump that transfers in- dividual electrons at a driving frequency ftraceable to the SI second via an atomic clock. In order to be suitable for prac-tical applications, such a pump should be able to generatecurrents of the order of 1 nA with an accuracy of the order of10 −8. A variety of devices and materials have been investi- gated to satisfy these requirements. These includesemiconducting, 2normal-metal,3and superconducting4de- vices, as well as hybrid devices based on normal-metal-superconductor junctions 5or surface acoustic wave-induced pumping through a carbon nanotube.6 Another important application of single-electron devices in regard to the new SI is the test based on quantum metro-logical triangle 7of the assumed exactness of the expressions for the Josephson constant KJ=h/2eand the von Klitzing constant RK=h/e2. At present, the device used for this purpose8is the so-called R-pump.9This device, as proposed in Ref. 10, is based on the effect of the circuit-impedance- induced power-law suppression of the cotunneling processesthat limit the accuracy of the single-electron-tunnelingdevices. 11The conductance of a capacitively coupled single- electron-tunneling transistor /H20849C-SET /H20850in the Coulomb block- ade /H20849CB /H20850regime behaves as G/H11011V2+2/H9256, where/H9256=R/RK, and Ris the zero-frequency impedance of the environment. In this case, the width of the electrodes and the island are muchlarger than the Fermi wavelength, making a large number oftransverse channels available for the tunneling electron.Since the relevant energies are close to the Fermi level, Lan- dau’s Fermi-liquid /H20849FL/H20850theory is applicable and the only trace of electron-electron interaction is described by thecharging energy of the island. The above power law is ob-tained from the tunnel Hamiltonian of the form H T =/H20858k,lTklck†clei/H9278, where the phase /H9278is a linear combination of Bose operators corresponding to the electromagnetic envi-ronment modes. Using the ﬂuctuation-dissipation theorem,the phase-phase correlation function can be expressed interms of the environmental impedance from whence theabove power law of conductance is derived. On the other hand, when the number of transverse chan- nels in the island is reduced to one, the Fermi-liquid descrip-tion of nearly free quasiparticles is no longer applicable /H20849for a review, see, for example, Ref. 12/H20850. The electron-electron interaction now has a drastic effect, resulting in a charge-spinseparation. However, the fermion ﬁeld /H9023can be represented in terms of collective charge and spin bosonic ﬁelds /H9021as /H9023/H11011e i/H9021. By applying the Baker-Hausdorff formula and the cumulant expansion for bosonic modes, the Green’s functionof the electron can be expressed as a function of the /H9021−/H9021 correlators. Similarly to the environmental effect outlinedabove, this also leads to a power-law dependence of conduc-tance versus voltage G/H11011V /H9251cat large biases /H20849eV/H11271kBT/H20850.I n this case the exponent /H9251cdepends on the interaction param- eters of electrons in a one-dimensional /H208491D /H20850channel. This result applies for an inﬁnite length of the channel and hasbeen observed in carbon nanotube measurements. 13The Green’s function in the case of a ﬁnite system of length Ld can be obtained by conformal mapping of the complex plane onto a cylinder of radius Ld//H9266, which results in a scale- dependent exponent.14This has also been studied numeri- cally in Ref. 15. In the opposite limit of low voltages /H20849eV /H11270kBT/H20850, the conductance of the inﬁnite system also shows power-law behavior G/H11011T/H9251c. This has been conﬁrmed experimentally13but, as the temperature was lowered belowPHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 1098-0121/2009/79 /H208493/H20850/035312 /H208499/H20850 ©2009 The American Physical Society 035312-1the charging energy of the island, a deviation from the power law was observed. This paper considers conductance in thistemperature region, including the environmental effects thatare important in metrology as well as in the physics of theCoulomb blockade. The paper is organized as follows. Section IIdescribes the model and presents the Hamiltonian of the system. SectionIIIderives the expression for the tunneling current. For an inﬁnite system this gives the power law for the conductanceas a function of voltage and temperature, with the normalexponent modiﬁed by the effect of the electromagnetic envi-ronment. For the ﬁnite system, we obtain the analytic resultsfor low voltages and temperatures. Section IVsummarizes the results and concludes with a proposal for possible appli-cations. II. MODEL AND HAMILTONIAN OF THE SYSTEM We consider a voltage-biased Tomonaga-Luttinger liquid /H20849TLL /H20850C-SET connected to an external environment of im- pedance Zi/H20849/H9275/H20850/H20849Fig.1/H20850. The theoretical framework is similar to a C-SET with FL electrodes as reported in Refs. 16and 17.The Hamiltonian of the system is given by H=H0+HT, where H0=HFL+HTLL+Hem, and HTis the tunneling Hamiltonian. The terms in H0describe the Fermi-liquid elec- trodes /H20849HFL/H20850, the Tomonaga-Luttinger liquid island /H20849HTLL /H20850, and the electromagnetic environment /H20849Hem/H20850. A. Fermi-liquid electrodes A description of electrons in all three electrodes can be expressed using the FL model, HFL=/H20858 i=13 /H20858 ks/H9280/H20849i/H20850/H20849k/H20850aks/H20849i/H20850†aks/H20849i/H20850, /H208491/H20850 where/H9280/H20849i/H20850/H20849k/H20850=/H20849/H6036k/H208502//H208492m/H20850, and aks/H20849i/H20850†/H20849aks/H20849i/H20850/H20850are the creation /H20849an- nihilation /H20850operators of the electron with wave vector kandspin s. Indexes i=1,2,3 refer to the left, right, and gate electrodes, respectively. B. Tomonaga-Luttinger liquid island The TLL island12,18–20can be described using the To- monaga model,18which is expressed by the g-ology Hamiltonian21in which only the forward-scattering terms g2 and g4are included. Using the open boundary bosonization technique,22we start with the fermion ﬁeld operator, /H9023/H20849x/H20850=/H20858 s/H9023s/H20849x/H20850=/H20858 s,r=/H11006/H9023rs/H20849x/H20850=/H20858 s,r=/H11006eirkFx/H9274rs/H20849x/H20850, /H208492/H20850 and impose the boundary condition for the island of length Ld,/H9023s/H208490/H20850=/H9023s/H20849Ld/H20850=0, that is, /H9274−s/H208490/H20850=−/H9274+s/H208490/H20850, /H208493/H20850 /H9274−s/H20849Ld/H20850=−e2ikFLd/H9274+s/H20849Ld/H20850. /H208494/H20850 We introduce the chiral boson phases as /H9274rs/H20849x/H20850=/H9257rs /H208812/H9266/H9251ei/H9021rs/H20849x/H20850, /H208495/H20850 /H9021rs/H20849x/H20850=/H9258rs+/H9278rs/H20849x/H20850=r /H208812/H20858 /H9263=/H9267,/H9268s/H9254/H9263/H9268/H9021/H9263/H20849x/H20850+1 /H208812/H20858 /H9263=/H9267,/H9268s/H9254/H9263/H9268/H9008/H9263/H20849x/H20850. /H208496/H20850 Here/H9251is a cut-off parameter, /H9257rsare the Majorana fermion operators satisfying /H20853/H9257rs,/H9257r/H11032s/H11032/H20854=2/H9254rr/H11032/H9254ss/H11032, and/H9263stands for charge /H20849/H9267/H20850and spin /H20849/H9268/H20850degrees of freedom. Taking into ac- count the open boundary condition, the boson phases areexpressed as /H9021 /H9263/H20849x/H20850=Q/H9263+/H9266 LdN/H9263x+i/H20881K/H9263/H20858 n=1/H110091 /H20881nsinn/H9266x Ld/H20851/H9251n/H20849/H9263/H20850†−/H9251n/H20849/H9263/H20850/H20852, /H208497/H20850 /H9008/H9263/H20849x/H20850=−Q˜/H9263−1 /H20881K/H9263/H20858 n=1/H110091 /H20881ncosn/H9266x Ld/H20851/H9251n/H20849/H9263/H20850†+/H9251n/H20849/H9263/H20850/H20852. /H208498/H20850 The Luttinger parameter K/H9263and other quantities appearing in Eqs. /H208497/H20850and /H208498/H20850are deﬁned as K/H9263=/H20881/H9266vF+/H20851g4/H9263−g2/H9263/H20852//H6036 /H9266vF+/H20851g4/H9263+g2/H9263/H20852//H6036, /H208499/H20850 gi/H9263=1 2/H20851gi/H20648+/H20849−1 /H20850/H9254/H9263,/H9268gi/H11036/H20852, /H2084910/H20850 N/H9263=1 /H208812/H20858 rss/H9254/H9263/H9268Nrs=1 /H208812/H20858 krss/H9254/H9263/H9268:ar,k,s†ar,k,s:, /H2084911/H20850 Q/H9263=1 2/H208812/H20858 rsrs/H9254/H9263/H9268/H9258rs, /H2084912/H20850R(1)(1) T R(2)(2) T q TLLTLL FLFL Z2Z1 V2 V1V3FLFL C2Q2 C3Q3C11 Q1I2I FIG. 1. Model of the system: equivalent electrical circuit for TLL C-SET with environmental impedances Zi/H20849/H9275/H20850. TLL island has discrete energy levels with energy spacing /H9280/H9263/H20849/H9263=/H9267,/H9268/H20850./H9267and/H9268 denote bosonic excitations of charge and spin degrees of freedom, respectively. There are two kinds of possible contacts between TLLand external electrodes depending on the fabrication process: bulkcontact /H20849nanotubes deposited over predeﬁned electrodes /H20850and edge contact /H20849evaporating the electrodes over the nanotubes /H20850.VLADIMIR BUBANJA AND SHUICHI IWABUCHI PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-2Q˜/H9263=−1 2/H208812/H20858 rss/H9254/H9263/H9268/H9258rs, /H2084913/H20850 ar,k,s†/H20849ar,k,s/H20850being creation /H20849annihilation /H20850operators of elec- trons in the TLL and : ¯: denotes the normal product. Noticing that the following commutation relations hold be-tween the operators specifying the bosonic excitations and the zero mode /H20851 /H9251n/H20849/H9263/H20850,/H9251n/H11032/H20849/H9263/H11032/H20850†/H20852=i/H9254nn/H11032/H9254/H9263/H9263/H11032,/H20851N/H9263,Q˜/H9263/H11032/H20852=i/H9254/H9263,/H9263/H11032, and /H20851/H9258rs,Nr/H11032s/H11032/H20852=i/H9254r,r/H11032/H9254s,s/H11032, we are led to the bosonic commutation relations between the ﬁeld operators, /H20851/H9021/H9263/H20849x/H20850,/H9016/H9263/H11032/H20849x/H11032/H20850/H20852=i/H9254/H9263/H9263/H11032/H9254/H20849x−x/H11032/H20850, /H2084914/H20850 where we have deﬁned /H9016/H9263/H20849x/H20850=−/H11509x/H9008/H9263/H20849x/H20850//H9266. With regard to the zero mode in this bosonization scheme, the eigenvalues of the operators Nrsand Q˜/H9263are obtained from the open boundary conditions. We ﬁnally have the TLL Hamiltonianas H TLL=/H20858 /H9263=/H9267,/H9268/H20875/H9280/H9263N/H92632+/H20858 n=1/H11009 /H9280¯/H9263/H20849n/H20850/H20873/H9251n/H20849/H9263/H20850†/H9251n/H20849/H9263/H20850+1 2/H20874/H20876,/H2084915/H20850 where v/H9263=/H20877/H20875vF+g4/H9263 /H9266/H6036/H208762 −/H20875g2/H9263 /H9266/H6036/H208762/H208781/2 , /H2084916/H20850 are the excitation velocities for the degree of freedom /H9263, /H9280/H9263=/H6036/H9266 2Ldv/H9263 K/H9263=/H9280¯ 2K/H9263v/H9263 vF=/H9280¯/H9263 2K/H9263, /H2084917/H20850 and /H9280¯/H9263/H20849n/H20850=2K/H9263/H9280/H9263·n=/H9280¯/H9263·n, /H2084918/H20850 vFbeing the Fermi velocity. Note that the ﬁrst and second terms in Eq. /H2084915/H20850specify the zero mode and the bosonic excitations of the TLL, respectively. C. Charging energy and electromagnetic environment In order to describe the electromagnetic energy, we need to determine the phases /H9272icanonically conjugate to the charges Qiof the electrodes /H20849this procedure, known as quan- tum mechanics with constraints was ﬁrst discussed byDirac 23/H20850. In the present case, the condition is that the island charge q=−/H20858 i=13 Qi, /H2084919/H20850 is constant in time in the absence of tunneling, /H20858 i=13 Q˙i=0 . /H2084920/H20850 For simplicity, the environmental impedance is taken as Zi/H20849/H9275/H20850=i/H9275Li. Given that the Lagrangian of the electromag- netic system isLem=/H20858 i=12/H20877Li 2Q˙ i2−Qi2 2Ci+Qi/H20849Vi−Vc/H20850/H20878 −Q32 2C3+Q3/H20849V3−Vc/H20850+/H9261/H20858 i=13 Q˙i, /H2084921/H20850 with a Langrange multiplier /H9261, we can determine the canoni- cally conjugate phases by /H9272i=/H11509Lem /H11509Q˙i. /H2084922/H20850 Following standard procedure, the Hamiltonian of the elec- tromagnetic system is Hem=/H20858 i=12/H20849/H9272i−/H92723/H208502 2Li+/H20858 i=13/H20877Qi2 2Ci−QiVi,c/H20878, /H2084923/H20850 where Vi,cis the voltage applied to the capacitance Ci, eVi,c=e/H20849Vi−Vc/H20850=/H9262i−/H9262c, /H2084924/H20850 expressed in terms of the chemical potentials of the island /H20849/H9262c/H20850, left electrode /H20849/H92621/H20850, right electrode /H20849/H92622/H20850, and the gate electrode /H20849/H92623/H20850. Now we can quantize Hemrequiring /H20851Qi,/H9272i/H11032/H20852=i/H6036/H9254i,i/H11032, /H20851Qi,Qi/H11032/H20852=/H20851/H9272i,/H9272i/H11032/H20852=0 . /H2084925/H20850 Note that matrix K1/2YLK1/2, whose eigenvector is K1/2/H9261, di- agonalizes Hemwith respect to charges /H20849Q→Q/H11032/H20850and phases /H20849/H9272→/H9272/H11032/H20850,17where /H9272=/H20900/H92721 /H92722 /H92723/H20901,Q=/H20900Q1 Q2 Q3/H20901,Vc=/H20900V1,c V2,c V3,c/H20901, /H2084926/H20850 K=/H20900/H9260100 0/H926020 00/H92603/H20901, /H2084927/H20850 YL=/H20900/H51291−10 −/H51291−1 0 /H51292−1−/H51292−1 −/H51291−1−/H51292−1/H51291−1+/H51292−1/H20901, /H2084928/H20850 /H9261=/H20849/H9261ij/H20850,/H9261ij=/H20881/H9260i /H9260i−/H5129i/H20849/H9275j//H9275L/H208502/H20875/H20858 l=13/H9260l /H20853/H9260l−/H5129l/H20849/H9275j//H9275L/H208502/H208542/H20876−1 /2 , /H2084929/H20850 with/H9260i=C/Ci/H20849i=1,2,3 /H20850,/H5129j=Lj/L/H9018/H20849j=1,2 /H20850, and /H51293=0. After straightforward calculations, we ﬁnally obtain Hem=Henv+Hc, /H2084930/H20850 where Henv=/H20858 j=12/H20877/H20849/H9275j//H9275L/H208502 2L/H9018/H9272j/H110322+Qj/H110322 2C−Qj/H11032Vj,c/H11032/H20878, /H2084931/H20850CONDUCTANCE OF THE CAPACITIVELY … PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-3Hc=/H20849q/e−nc/H208502U. /H2084932/H20850 HenvandHcdescribe the electromagnetic environment and island charging effects, respectively. The quantities appear-ing in Eqs. /H2084931/H20850and /H2084932/H20850are deﬁned as /H9275L=1 //H20881L/H9018C, /H2084933/H20850 L/H9018=L1+L2, /H2084934/H20850 nc=−/H20858 j=13 CjVj/e+eVc//H208492U/H20850, /H2084935/H20850U=e2//H208492C/H9018/H20850=EcC/C/H9018, /H2084936/H20850 C/H9018=C1+C2+C3, /H2084937/H20850 C=C1C2//H20849C1+C2/H20850. /H2084938/H20850 Here Uand ncare the charging energy of the island and the noninteger charge offset /H20849induced by the voltage bias condi- tion and the charging energy /H20850, respectively. Note that ncin- cludes the chemical potential of the island and should bedetermined self-consistently by the current continuity condi-tion I 1+I2=0. The electromagnetic environment modes are deﬁned as /H20873/H9275j /H9275L/H208742 =1 2/H20873/H92601+/H92603 /H51291+/H92602+/H92603 /H51292/H20874/H11006/H208811 4/H20873/H92601 /H51291−/H92602 /H51292/H208742 +/H20875/H92603 2/H208731 /H51291+1 /H51292/H20874/H208762 +/H92603 2/H208731 /H51291−1 /H51292/H20874/H20873/H92601 /H51291−/H92602 /H51292/H20874, /H2084939/H20850 where j=1 and j=2 correspond to the plus and minus signs, respectively. Note that the island charge qis the extra /H20849quan- tized /H20850charge which describes the deviation from a neutral island. The nonzero eigenvalue of qresults only from asym- metric tunneling events through the junctions. Therefore, wecan relate qto the zero mode, which describes the change in the number of electrons in the system. Given that q=−/H20858 i=13 Qi=−/H20881/H20849C/H9018/C/H20850Q3/H11032, /H2084940/H20850 and that N/H9267is also the canonical variable conjugate to /H9258rs, which describes the number of electrons in TLL, we are ledto the identities q=−e/H208812/H9004N/H9267, /H20881/H9260i/H9261i3/H92723/H11032=−/H6036 e/H9258rs. /H2084941/H20850 Here we introduced a different variable /H9004N/H9267which describes the change in the charge in the TLL due to tunneling. Inorder to take charging effect into account, we treat /H9004N /H9267as independent variable of N/H9267. Since electron tunneling changes not only the charge but also the spin, it is consistent to in- troduce/H9004N/H9268. The eigenstate of /H9004N/H9263satisﬁes/H9004N/H9263/H20841/H9004N¯/H9263/H20856 =/H9004N¯/H9263/H20841/H9004N¯/H9263/H20856as the charge state of the island /H20841m/H20856satisﬁes q/H20841m/H20856=me /H20841m/H20856. Since /H20851Q3/H11032,/H92723/H11032/H20852=i/H6036, we have /H20851/H20881/H9260i/H9261i3/H92723/H11032,q/H20852=i/H6036, and therefore e/H11006i/H20881/H9260i/H9261i3/H92723/H11032/H20841m/H20856=/H20841m/H110071/H20856. At this stage, we are led to the Hamiltonian of TLL zero mode ﬂuctuations in thepresence of the charging effect,H Z=/H20858 /H9263=/H9267,/H9268/H9280/H9263/H9004N/H92632+Hc=/H20858 /H9263=/H9267,/H9268/H9280/H9263/H9004N/H92632+U/H20849/H208812/H9004N/H9267+nc/H208502, /H2084942/H20850 which speciﬁes the charged state of the TLL island. D. Tunneling Hamiltonian According to the discussion above, the tunneling Hamil- tonian can be written as HT=/H20858 i=1,2/H20853HT/H20849i/H20850+HT/H20849i/H20850†/H20854, /H2084943/H20850 HT/H20849i/H20850=/H20858 rs/H20885dr/H20885 0Ld dxTr/H20849i/H20850/H20849r,x/H20850 /H11003ei/H20858 j=12/H20881/H9260i/H9261ije/H9272j/H11032//H6036e−i/H9258rs/H9023s/H20849i/H20850†/H20849r/H20850/H9023rs/H20849x/H20850, =/H20858 k/H20858 rs/H20885 0Ld dxTkr/H20849i/H20850/H20849x/H20850ei/H20858 j=12/H20881/H9260i/H9261ije/H9272j/H11032//H6036e−i/H9258rsaks/H20849i/H20850†/H9023rs/H20849x/H20850, /H2084944/H20850 where Tr/H20849i/H20850/H20849r,x/H20850is the matrix element of tunneling from po- sition x/H20851with chirality r/H20849=/H11006/H20850/H20852in the island to position rin theith FL electrode, and Tkr/H20849i/H20850/H20849x/H20850=1 /H20881V/H20849i/H20850/H20885dre−ik·rTr/H20849i/H20850/H20849r,x/H20850, /H2084945/H20850 is the Fourier transform of Tr/H20849i/H20850/H20849r,x/H20850with respect to the Fermi-liquid state k. The ﬁeld operator for the ith FL elec- trode can be deﬁned as /H9023/H20849i/H20850/H20849r/H20850=/H20858 s/H9023s/H20849i/H20850/H20849r/H20850=1 /H20881V/H20849i/H20850/H20858 ksaks/H20849i/H20850eik·r, /H2084946/H20850 where V/H20849i/H20850is the volume of the electrode.VLADIMIR BUBANJA AND SHUICHI IWABUCHI PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-4III. TUNNELING CURRENTS The tunneling current Iithrough the ith junction /H20849see Fig. 1/H20850in the lowest order in the tunneling Hamiltonian /H20849see Fig. 2/H20850is given by Ii=ie /H6036/H20885 −/H11009+/H11009 dt/H208831 i/H6036/H20851HT/H20849i/H20850†/H20849t/H20850,HT/H20849i/H20850/H208490/H20850/H20852/H20884exp/H20873ieVi,c /H6036t/H20874, /H2084947/H20850 where we moved to the grand canonical ensemble, namely, HT/H20849i/H20850/H20849t/H20850=eitK0//H6036HT/H20849i/H20850e−itK0//H6036, /H20855¯/H20856=Tr/H20853exp /H20849−/H9252K0/H20850¯/H20854 Tr exp /H20849−/H9252K0/H20850, K0=H0−/H20858 i=1,2,3/H9262i/H20885dri/H9023/H20849i/H20850†/H20849r/H20850/H9023/H20849i/H20850/H20849r/H20850 −/H9262c/H20885 0Ld dx/H9023/H20849x/H20850†/H9023/H20849x/H20850. /H2084948/H20850 By calculating the average in Eq. /H2084947/H20850, we arrive at the result /H20849see Appendix /H20850 Ii=GT0/H20849i/H20850 4e/H20885 −/H11009/H11009 d/H9280tanh/H20875/H9252 2/H20849/H9262c,i−/H9280/H20850/H20876/H20873/H20885 0Ld dx/H20858 sN˜ TLL/H20849i/H20850/H20849/H9280,x/H20850/H20874, /H2084949/H20850 where GT0/H20849i/H20850=/H208494/H9266e2//H6036/H20850/H20841T/H20849i/H20850/H208412NFL/H208490/H20850N1D/H208490/H20850, with N1D/H208490/H20850 =Ld//H20849/H9266/H6036vF/H20850being the density of states per spin in the one-dimensional free-electron system. The normalized local spectral density of TLL N˜ TLL/H20849i/H20850/H20849/H9280,x/H20850is given by N˜ TLL/H20849i/H20850/H20849/H9280,x/H20850=2/H20885 −/H11009/H11009dt 2/H9266/H6036N1D/H208490/H20850ei/H9280t//H6036 /H11003/H20851Fi/H20849env /H20850/H11022/H20849t/H20850Fs/H20849c/H20850/H11022/H20849t/H20850G+s/H20849x,x,t/H20850 +Fi/H20849env /H20850/H11021/H20849t/H20850Fs/H20849c/H20850/H11021/H20849t/H20850G+s/H20849x,x,−t/H20850/H20852. /H2084950/H20850The quantities appearing in the kernel in the above expres- sion are the correlation functions of the phases correspond- ing to the environmental charge Fi/H20849env /H20850/H20849t/H20850, the island charge Fs/H20849c/H20850/H20849t/H20850, and the Green’s function of the electron in the TLL, G+s/H20849x,x,t/H20850. The environmental and island phase correlation functions are given by Fi/H20849env /H20850/H11125/H20849t/H20850= exp/H20877/H20858 j=12 /H9261ij2/H9260i·Ec /H6036/H9275j·J/H11125/H20849/H9275j,t/H20850/H20878, /H2084951/H20850 J/H11125/H20849/H9275,t/H20850= coth/H9252/H6036/H9275 2/H20849cos/H9275t−1 /H20850/H11007isin/H9275t, /H2084952/H20850 and Fs/H20849c/H20850/H11125/H20849t/H20850=/H20858/H9004N¯ /H9267,/H9004N¯ /H9268=−/H11009/H11009e−/H9252E/H20849/H9004N¯ /H9267,/H9004N¯ /H9268/H20850+it//H6036/H20851/H9280/H9267/H20849/H208812/H9004N¯ /H9267/H110071/2/H20850+s/H9280/H9268/H20849/H208812/H9004N¯ /H9268/H11007s/2/H20850+U/H208512/H20849/H208812/H9004N¯ /H9267+/H9254nc/H20850/H110071/H20852/H20852 /H20858/H9004N¯ /H9267,/H9004N¯ /H9268=−/H11009/H11009e−/H9252E/H20849/H9004N¯ /H9267,/H9004N¯ /H9268/H20850, /H2084953/H20850 where E/H20849/H9004N¯/H9267,/H9004N¯/H9268/H20850=/H20858 /H9263/H9280/H9263/H9004N¯ /H92632+U/H20849/H208812/H9004N¯/H9267+nc/H208502, /H2084954/H20850 /H9254nc=nc−/H20851nc+1 /2/H20852, /H2084955/H20850 and /H20851¯/H20852is the Gauss symbol. The exact chiral Green’s function22is given byGrs((x, x'; t )Grs((x', x ; -t )) [ ] s(c)) (c) > >(t)Fs< <(t)F i(env)) (t)Fi(env)) (t)F[ ] [ ]Tr((r,x ))()i [ ]* Tr'((r', x' )()i Gs((r-r'; t )Gs((r-r'; t ) [ ]()i ()>i < FIG. 2. Diagrammatic representation of the tunneling kernel which corresponds to the statistically averaged quantity in Eq. /H2084947/H20850. The bold solid line denotes the exact chiral Green’s function of anelectron in the TLL. The dotted line and the chain line denote thecorrelation functions of phases conjugate to quantized /H20849island /H20850 charge and continuous /H20849environmental /H20850charge, respectively. The thin solid line denotes the lowest-order Green’s functions of anelectron in the FL.CONDUCTANCE OF THE CAPACITIVELY … PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-5G+s/H20849x,y,t/H20850=/H20855/H9274¯ +s†/H20849x,t/H20850/H9274¯+s/H20849y,0/H20850/H20856=/H208831 /H208812/H9266/H9251e−i/H9021+s/H20849x,t/H208501 /H208812/H9266/H9251ei/H9021+s/H20849y,0/H20850/H20884 =1 2/H9266/H9251e−i/H9266/2L/H20849x−y/H20850H/H20849x,y,t/H20850/H20863 /H9263=/H9267,/H9268/H20877/H20851F/H9263/H20849v/H9263t−x+y/H20850/H20852−1 /2/H20849a/H9263+1 /2/H20850/H20851F/H9263/H20849v/H9263t+x−y/H20850/H20852−1 /2/H20849a/H9263−1 /2/H20850 /H11003/H20875/H20841F/H9263/H208492x/H20850F/H9263/H208492y/H20850/H20841 F/H9263/H20849v/H9263t−x−y/H20850F/H9263/H20849v/H9263t+x+y/H20850/H20876b/H9263/2/H20878, /H2084956/H20850 with a/H9263=1 4/H208731 K/H9263+K/H9263/H20874,b/H9263=1 4/H208731 K/H9263−K/H9263/H20874. /H2084957/H20850 The zero mode contribution is H/H20849x,y,t/H20850=/H20863 /H9263=/H9267,/H9268/H20855es/H9254/H9263,/H9268/H20849iu/H9263/H208812N/H9263/H20850/H20856 =/H92772/H20849u/H9267/H20841/H9270/H9267/H20850/H92772/H20849u/H9268/H20841/H9270/H9268/H20850+/H92773/H20849u/H9267/H20841/H9270/H9267/H20850/H92773/H20849u/H9268/H20841/H9270/H9268/H20850 /H92772/H208490/H20841/H9270/H9267/H20850/H92772/H208490/H20841/H9270/H9268/H20850+/H92773/H208490/H20841/H9270/H9267/H20850/H92773/H208490/H20841/H9270/H9268/H20850, /H2084958/H20850 where/H9277n/H20849u/H20841/H9270/H20850are theta functions, and u/H9263=−/H9266 2Ld/H20849v/H9263 K/H9263t+x−y/H20850 and/H9270/H9263=i2/H9252/H9280/H9263//H9266. Function F/H9263/H20849z/H20850appearing in Eq. /H2084956/H20850is given by F/H9263/H20849z/H20850=i2Ld /H9266/H9251sin/H9266z 2Ld/H20863 k=1/H11009/H209001+/H20898sin/H9266z 2Ld sinhk/H9266v/H9263/H9252/H6036 2Ld/H208992 /H20901 =iLd /H9266/H9251/H20875/H9257/H20873iv/H9263/H9252/H6036 2Ld/H20874/H20876−3 /H92771/H20873/H9266z 2Ld/H20879iv/H9263/H9252/H6036 2Ld/H20879/H20874, /H2084959/H20850 where/H9257/H20849/H9270/H20850is the Dedekind eta function. Note that N˜ TLL/H20849i/H20850/H20849/H9280,x/H20850 is modiﬁed by the open boundary condition as well as thecharging effect. This is essential when discussing TLL be-havior in the Coulomb blockade regime. The two cases cor-responding to bulk /H20849with translational invariance /H20850and edge contacts /H20849without translational invariance /H20850are obtained by considering the limits of x/ v/H9263t/H112711 and x/v/H9263t/H112701, respec- tively, in the above expression for the chiral Green’s func-tion. Since I iis a function of chemical potentials of ith FL electrode/H9262iand TLL electrode /H9262c, the tunneling current of the system Iis obtained from I=1 2/H20849I1−I2/H20850/H20849 60/H20850 under the current continuity condition /H20858 i=12 Ii=0 . /H2084961/H20850 We ﬁrst consider the effect of the environment ﬂuctua- tions on an inﬁnite system. When deriving the Hamiltonian/H2084930/H20850we have, for simplicity, considered a purely inductive environment but the approach is also applicable to arbitrarylinear circuit elements. In this case, the derivation can bemade along the lines presented above but with an inﬁnitenumber of oscillators whose spectral density is chosen toreproduce Johnson–Nyquist correlations /H20849see, for example, review articles 24and references therein /H20850. The usual experi- mental setup includes a dissipative environment which isOhmic at low frequencies Z i/H20849/H9275/H20850=Ri/H20849/H9275→0/H20850. In this case Eq. /H2084951/H20850is replaced by exp /H20851Ji/H11125/H20849t/H20850/H20852, where R/H20851Ji/H11125/H20849t/H20850/H20852 /H11011 −2 RK/H20873Cj C/H9018/H208742/H20875/H208731+C3 Cj/H208742 Ri+Rj/H20876 /H11003ln/H20873/H6036/H9275o/H20849i/H20850/H9252 /H9266sinh/H9266/H20841t/H20841 /H6036/H9252/H20874, /H2084962/H20850 for large t, where i,j=1,2 /H20849i/HS11005j/H20850, and where/H9275o/H20849i/H20850is the environment-dependent cut-off frequency given by /H9275o/H20849i/H20850=/H20849Cj+C3/H20850C/H9018 CiCj2/H20851/H208491+C3/Cj/H208502Ri+Rj/H20852. /H2084963/H20850 From Eq. /H2084959/H20850, we have F/H9263/H20849v/H9263t/H20850→iv/H9263/H9252/H6036 /H9251/H9266sinh/H20873/H9266t /H9252/H6036/H20874, /H20849Ld→/H11009/H20850. /H2084964/H20850 For simplicity, let us consider the symmetric system, R1 =R2=R/2,C1=C2=2C,V1=−V2=V/2, and V3=0. By sub- stituting Eq. /H2084962/H20850into Eq. /H2084951/H20850and Eq. /H2084964/H20850into Eq. /H2084956/H20850, and by performing a Fourier transform in Eq. /H2084950/H20850,w eg e t the conductance from Eq. /H2084949/H20850, in the limit/H9252→/H11009, near the CB region /H20849V/H11407e2/C/H9018/H20850, G=GT0/H20849/H9251/H9275o/H20850/H9251cvF 4/H9003/H20849/H9251˜c+1 /H20850v/H9267/H9252/H9267v/H9268/H9252/H9268/H20873eV−e2/C/H9018 2/H6036/H9275o/H20874/H9251˜c , /H2084965/H20850 /H9251˜c=/H9251c+c/H9256,/H9251c=/H20858 /H9263/H9252/H9263−1 ,/H9252/H9263=/H20877a/H9263 /H20849bulk /H20850 a/H9263+b/H9263/H20849edge /H20850,/H20878 /H2084966/H20850 where the environmental parameters are /H9256=R/RKand c =/H20851/H208492C+C3/H208502+/H208492C/H208502/H20852//H208494C+C3/H208502. By increasing the voltage further away from the CB region /H20849V/H11271e2/C/H9018/H20850, we obtain the above formula with /H9251˜c→/H9251c, and the conductance ap- proaches the power law with the usual TLL exponent.VLADIMIR BUBANJA AND SHUICHI IWABUCHI PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-6In the case of the ﬁnite system size, for /H9252→/H11009, we obtain near the CB region /H20849V/H11407e2/C/H9018/H20850, G=GT02/H9253c−1 /H9003/H20849c/H9256/H20850N1D/H208490/H20850/H6036/H9275o/H20873/H9251/H9266 Ld/H20874/H9251c/H20858 n=0/H11009 /H20858 m=0/H11009 F/H20849n,m/H20850, /H2084967/H20850 F/H20849n,m/H20850=1 n!m!/H9003/H20849n+/H9252/H9267/H20850/H9003/H20849m+/H9252/H9268/H20850 /H9003/H20849/H9252/H9267/H20850/H9003/H20849/H9252/H9268/H20850f/H20849V/H20850c/H9256−1e−f/H20849V/H20850/H9008/H20851f/H20849V/H20850/H20852, /H2084968/H20850 f/H20849V/H20850=1 2/H6036/H9275o/H20875eV−2U−/H9280¯/H20858 /H92632/H9252/H9263K/H9263+1 2K/H92632−2/H9280¯/H20873n K/H9267+m K/H9268/H20874/H20876, /H2084969/H20850 /H9253c=/H208770 /H20849bulk /H20850 b/H9267+b/H9268 /H20849edge /H20850./H20878 /H2084970/H20850 The above formula shows that by increasing the electron- electron interaction parameter, the Coulomb gap region ex-tends due to the 1D nature of the island. For a large islandlength, the indexes nand mcan be considered as continuous due to the small energy gap /H9280¯that these indexes multiply. We recover formula /H2084965/H20850by using Stirling’s approximation and performing the integrations. For a short island length, we canretain only the F/H208490,0 /H20850term in Eq. /H2084967/H20850due to the unit step function in Eq. /H2084968/H20850. Away from the CB region, instead of F, we have a set of delta functions which—after integration—reproduce the corresponding long length limit of formula/H2084965/H20850. We consider now the conductance at zero voltage of a ﬁnite system in the high-temperature regime /H20849 /H9280/H9263/H9252/H112701/H20850. By using the asymptotic behavior of elliptic theta functions for/H9270→0,25/H92771/H20849u/H20841e−/H9270/H20850/H110112/H20881/H9266 /H9270exp /H20849−/H92662+/H208492u/H208502 4/H9270/H20850sinh /H20849/H9266u /H9270/H20850, and /H9277j/H20849u/H20841e−/H9270/H20850/H11011/H20881/H9266 /H9270exp /H20849−u2 /H9270/H20850/H208511− /H20849−1/H20850j2 exp /H20849−/H92662 /H9270/H20850cosh /H208492/H9266u /H9270/H20850/H20852forj =2,3, the chiral Green’s function can be written as G+s/H20849x,x,t/H20850=2/H9253c−1 /H9266/H9251exp/H20873−/H9280¯t2 /H9252/H60362/H20858 /H92631−2/H9252/H9263K/H92632 4K/H92633/H20874/H20873/H9251/H92662 Ld/H9280¯/H9252/H20874/H20858/H9263/H9252/H9263 /H11003K/H9267/H9252/H9267K/H9268/H9252/H9268/H20875isinh/H20873/H9266t /H6036/H9252/H20874/H20876/H20858/H9263/H9252/H9263 . /H2084971/H20850 By using Eq. /H2084962/H20850inFi/H20849env /H20850/H11125/H20849t/H20850, approximating the summa- tions in Fs/H20849c/H20850/H11125/H20849t/H20850by integrals and after substituting in Eqs. /H2084950/H20850and /H2084949/H20850we obtain the conductance for /H6036/H9275o/H9252/H112711, G=GT02/H9253c /H9266/H20873/H9251/H92662 Ld/H20874/H9251c K/H9267/H9252/H9267K/H9268/H9252/H9268/H20873/H9266/H9280¯ /H6036/H9275o/H20874c/H9256 /H20849/H9280¯/H9252/H20850−/H9251˜cI/H20849/H9251˜c+1 /H20850, /H2084972/H20850 I/H20849a/H20850=/H20885 0/H11009 dw e−/H9252Uw2cos/H20873/H9266 2a+/H9252Uw/H20874wa−/H20849sinh w/H20850a /H20849sinh w/H20850a+1wa−1. /H2084973/H20850 For higher temperatures /H9252U/H112701, where the Coulomb block- ade is washed out, we obtain the above formula /H2084972/H20850withreplacements c/H9256→0 and/H9251˜c→/H9251c, which is the usual TLL power law.22 On the other hand, in the low-temperature regime where /H20849/H9280/H9263/H9252/H112711/H20850, even without any external impedance, the conduc- tance is exponentially suppressed, G=GT02/H9253c/H20873/H9266/H9251 Ld/H20874/H9251c /H9280¯/H9252exp/H20875−/H20873/H9280¯/H9252/H20858 /H92632/H9252/H9263K/H9263+1 4K/H92632+/H9252U/H20874/H20876. /H2084974/H20850 The conductance decreases monotonously with increasing electron-electron interaction similar to the effect of increas-ing the environment impedance if the island was a FL. 26 IV. SUMMARY Analytic expressions for the tunnel current have been de- rived for the electron transport in a C-SET with a TLL island.For an inﬁnite system and a general electromagnetic environ-ment with dissipative Ohmic impedance at low frequencies,the conductance shows power-law behavior as a function ofvoltage at zero temperature. Near the CB threshold voltage, this power /H20851 /H9251˜cin Eq. /H2084966/H20850/H20852differs from the usual TLL power /H20849/H9251c/H20850as a result of the electromagnetic environment. For high voltages the environmental effects disappear and the usualTLL power-law behavior results with the voltage offset dueto the CB of the island. For a ﬁnite system in the same limitof zero temperature, we obtained the analytic expression forthe conductance versus voltage near the CB region. Thisshows that for a short length case, the power depends onlyon the environment. Zero voltage conductance of a ﬁnite system as a function of temperature is given by the expressions /H2084972/H20850and /H2084973/H20850/H20849for /H9280/H9263/H9252/H112701/H20850, with power being modiﬁed by the environment for low temperatures, compared to the environmental cut-off fre-quency /H20849/H6036 /H9275o/H9252/H112711/H20850, and the usual TLL power law is recov- ered at higher temperatures /H20849/H9252U/H112701/H20850. Conductance is strongly suppressed at low temperatures /H20849/H9280/H9263/H9252/H112711/H20850. The analogous effect of the correlation functions of bosonic modes of the environment and the charge and spinexcitations on conductance is signiﬁcant for SET devices,such as those used in metrology, particularly with regard tothe SI. The accuracy of SET devices is limited by cotunnel-ing and its effect can be reduced by environmental imped-ance or by increasing the electron-electron interaction in theTLL. Therefore, a pump device consisting of TLL islands /H20849or combination of TLL and FL islands /H20850could be used for in- creased accuracy. APPENDIX The expression for tunneling current through ith junction in the lowest order with respect to the tunnel Hamiltonian isderived by evaluating Eq. /H2084947/H20850, I i=ie /H6036/H20885 −/H11009+/H11009 dt/H208831 i/H6036/H20851HT/H20849i/H20850†/H20849t/H20850,HT/H20849i/H20850/H208490/H20850/H20852/H20884exp/H20873ieVi,c /H6036t/H20874. /H20849A1 /H20850 Note that the tunneling kernel becomesCONDUCTANCE OF THE CAPACITIVELY … PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-7Xi/H20849t/H20850=/H20855/H20851HT/H20849i/H20850†/H20849t/H20850,HT/H20849i/H20850/H208490/H20850/H20852/H20856=Fi/H20849env /H20850/H11022/H20849t/H20850Fs/H20849c/H20850/H11022/H9251T/H20849i/H20850/H11022/H20849t/H20850 −Fi/H20849env /H20850/H11021/H20849t/H20850Fs/H20849c/H20850/H11021/H9251T/H20849i/H20850/H11021/H20849t/H20850, /H20849A2 /H20850 with various quantities deﬁned as follows: Fi/H20849env /H20850/H11022/H20849t/H20850=/H20883exp/H20877−ie /H6036/H20858 j=12 /H20881/H9260i/H9261ij/H9272j/H11032/H20849t/H20850/H20878 /H11003exp/H20877ie /H6036/H20858 j=12 /H20881/H9260i/H9261ij/H9272j/H11032/H208490/H20850/H20878/H20884, /H20849A3 /H20850 Fi/H20849env /H20850/H11021/H20849t/H20850=/H20883exp/H20877ie /H6036/H20858 j=12 /H20881/H9260i/H9261ij/H9272j/H11032/H208490/H20850/H20878 /H11003exp/H20877−ie /H6036/H20858 j=12 /H20881/H9260i/H9261ij/H9272j/H11032/H20849t/H20850/H20878/H20884, /H20849A4 /H20850 Fs/H20849c/H20850/H11022=/H20855ei/H9258rs/H20849t/H20850e−i/H9258rs/H208490/H20850/H20856, /H20849A5 /H20850 Fs/H20849c/H20850/H11021=/H20855e−i/H9258rs/H208490/H20850ei/H9258rs/H20849t/H20850/H20856, /H20849A6 /H20850 /H9251T/H20849i/H20850/H11022/H20849t/H20850=/H20858 s/H20858 rr/H11032/H20885 0Ld/H20885 0Ld dxdx /H11032/H20858 kTkr/H20849i/H20850/H11569/H20849x/H20850Tkr/H11032/H20849i/H20850/H20849x/H11032/H20850e−ikF/H20849rx−r/H11032x/H11032/H20850 /H11003Gi/H20849FL/H20850/H11022/H20849k,t/H20850/H20855/H9257rs†/H9257r/H11032s/H11032/H9274¯ rs†/H20849x,t/H20850/H9274¯r/H11032s/H20849x/H11032,0/H20850/H20856, /H20849A7 /H20850 /H9251T/H20849i/H20850/H11021/H20849t/H20850=/H20858 s/H20858 rr/H11032/H20885 0Ld/H20885 0Ld dxdx /H11032/H20858 kTkr/H20849i/H20850/H11569/H20849x/H20850Tkr/H11032/H20849i/H20850/H20849x/H11032/H20850e−ikF/H20849rx−r/H11032x/H11032/H20850 /H11003Gi/H20849FL/H20850/H11021/H20849k,t/H20850/H20855/H9257r/H11032s/H11032/H9257rs†/H9274¯r/H11032s/H20849x/H11032,0/H20850/H9274¯ rs†/H20849x,t/H20850/H20856, /H20849A8 /H20850 /H20873Gi/H20849FL/H20850/H11022/H20849k,t/H20850 Gi/H20849FL/H20850/H11021/H20849k,t/H20850/H20874=/H20873/H20855aks/H20849i/H20850/H20849t/H20850aks/H20849i/H20850†/H208490/H20850/H20856 /H20855aks/H20849i/H20850†/H208490/H20850aks/H20849i/H20850/H20849t/H20850/H20856/H20874, /H20849A9 /H20850 and /H9274¯rs/H20849x/H20850=/H9257rs/H9274rs/H20849x/H20850=1 /H208812/H9266/H9251ei/H9021rs/H20849x/H20850. /H20849A10 /H20850 The correlation functions of phases Fi/H20849env /H20850/H20849t/H20850andFs/H20849c/H20850/H20849t/H20850de- scribe the electromagnetic environment effect and the charg-ing effect in the TLL island, respectively. In the case of F i/H20849env /H20850/H20849t/H20850, it is standard procedure to take the trace of boson correlation function of this type to give Eq. /H2084951/H20850. However, in the case of Fs/H20849c/H20850/H20849t/H20850and bearing in mind thate/H11007i/H9258rs/H20841/H9004N¯/H9263/H20856=/H20879/H9004N¯/H9263/H11007s/H9254/H9263,/H9268 /H208812/H20884, /H20849A11 /H20850 for the eigenstate of q=−e/H208812/H9004N/H9267and noting that the trace should be taken with respect to HZ, we obtain Eq. /H2084953/H20850. In deriving/H9251T/H20849i/H20850/H11125/H20849t/H20850,kdependence of Tkr/H20849i/H20850/H20849x/H20850can be ig- nored as usual. Furthermore, we assume Tkr/H20849i/H20850/H11569/H20849x/H20850Tkr/H11032/H20849i/H20850/H20849x/H11032/H20850/H11011/H20841 T/H20849i/H20850/H208412/H9254/H20849x−x/H11032/H20850, /H20849A12 /H20850 since Tkr/H20849i/H20850/H11569/H20849x/H20850Tkr/H11032/H20849i/H20850/H20849x/H11032/H20850is dominant when x/H11032/H11011x/H110110/H20849fori=1/H20850 orx/H11032/H11011x/H11011Ld/H20849fori=2/H20850. After carrying out a summation over k, we arrive at /H9251T/H20849i/H20850/H11125/H20849t/H20850=−2 i/H6036NFL/H20849i/H20850/H208490/H20850/H20841T/H20849i/H20850/H208412/H9266//H9252/H6036 sinh /H20849/H9266t//H9252/H6036/H20850 /H11003/H20858 s/H20885 0Ld dx G +s/H20849x,x,/H11006t/H20850, /H20849A13 /H20850 where NFL/H20849i/H20850/H208490/H20850is the density of states of ith FL electrode at the Fermi level. When deriving Eq. /H20849A13 /H20850we took r/H11032=r only, since the factor e−ikF/H20849r−r/H11032/H20850xaverages out the integrand over several lattice sites. Using expressions derived above,Eq. /H20849A2 /H20850becomes X i/H20849t/H20850=−i/H6036NFL/H20849i/H20850/H208490/H20850/H20841T/H20849i/H20850/H208412/H9266//H9252/H6036 sinh /H20849/H9266t//H9252/H6036/H20850 /H11003/H20885 −/H11009/H11009 d/H9280e−i/H9280t//H6036/H20875/H20885 0Ld dx/H20858 sN˜ s/H20849i/H20850/H20849/H9280,x/H20850/H20876, /H20849A14 /H20850 where we deﬁned the effective local spectral density as N˜ s/H20849i/H20850/H20849/H9280,x/H20850=2/H20885 −/H11009/H11009dt 2/H9266/H6036ei/H9280t//H6036/H20851Fi/H20849env /H20850/H11022/H20849t/H20850Fs/H20849c/H20850/H11022/H20849t/H20850G+s/H20849x,x,t/H20850 +Fi/H20849env /H20850/H11021/H20849t/H20850Fs/H20849c/H20850/H11021/H20849t/H20850G+s/H20849x,x,−t/H20850/H20852. /H20849A15 /H20850 By substituting Eq. /H20849A14 /H20850into Eq. /H20849A1 /H20850, we obtain the tun- neling current through junction i Ii=GT0/H20849i/H208501 4e/H20885 −/H11009/H11009 d/H9280tanh/H20875/H9252 2/H20849/H9262c,i−/H9280/H20850/H20876/H20873/H20885 0Ld dx/H20858 sN˜ TLL/H20849i/H20850/H20849/H9280,x/H20850/H20874, /H20849A16 /H20850 where GT0/H20849i/H20850=/H208494/H9266e2//H6036/H20850/H20841T/H20849i/H20850/H208412NFL/H20849i/H20850/H208490/H20850N1D/H208490/H20850, N1D/H208490/H20850 =Ld//H20849/H9266/H6036vF/H20850, and N˜ TLL/H20849i/H20850/H20849/H9280,x/H20850=N˜ s/H20849i/H20850/H20849/H9280,x/H20850 N1D/H208490/H20850, /H20849A17 /H20850 is the normalized effective local spectral density of the TLL island.VLADIMIR BUBANJA AND SHUICHI IWABUCHI PHYSICAL REVIEW B 79, 035312 /H208492009 /H20850 035312-81I. 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PhysRevB.103.035108.pdf	PHYSICAL REVIEW B 103, 035108 (2021) Nonequilibrium phases and phase transitions of the XYmodel Tharnier O. Puel ,1,2,Stefano Chesi ,3,4,†Stefan Kirchner,1,2,‡and Pedro Ribeiro5,3,§ 1Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China 2Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China 3Beijing Computational Science Research Center, Beijing 100193, China 4Department of Physics, Beijing Normal University, Beijing 100875, China 5CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal (Received 22 September 2020; accepted 10 December 2020; published 7 January 2021) We obtain the steady-state phase diagram of a transverse-ﬁeld XYspin chain coupled at its ends to magnetic reservoirs held at different magnetic potentials. In the long-time limit, the magnetization bias across the systemgenerates a current-carrying nonequilibrium steady state. We characterize the different nonequilibrium phasesas functions of the chain’s parameters and magnetic potentials, in terms of their correlation functions andentanglement content. The mixed-order transition, previously observed for the case of a transverse-ﬁeld Isingchain, is established to emerge as a generic feature of a wider class of out-of-equilibrium problems. The criticalexponents associated with this universality class are determined analytically. Results are also contrasted withthose obtained in the limit of Markovian reservoirs. Our ﬁndings should prove helpful in establishing theproperties of nonequilibrium phases and phase transitions of extended open quantum systems. DOI: 10.1103/PhysRevB.103.035108 I. INTRODUCTION Quantum matter out of thermal equilibrium has become a central research topic in recent years. An important classof problems deal with nonequilibrium quantum states of sys-tems that are in contact with multiple baths, which in turnare held at speciﬁed thermodynamic potentials. Such statesare not bounded by equilibrium ﬂuctuation relations and thusmay host phases of matter that are impossible to realize inequilibrium. Therefore, phase changes far from equilibriummay exist that lack equilibrium counterparts. Far-from-equilibrium quantum states are routinely realized in mesoscopic solid-state devices [ 1–3] and recently have also become available in cold atomic gas settings [ 4]. Thus it is timely to explore the properties of phases of current-carryingmatter and address the conditions which have to be met fortheir emergence. Nonequilibrium transport across quantum materials dates back to Landauer and Büttiker [ 5], who were motivated by the failure of semiclassical Boltzmann-like approaches to un-derstand phenomena such as the conductance quantizationacross mesoscopic conductors. For noninteracting systems,quantum transport is, by now, well understood [ 6–8]. How- ever, in systems where the physical properties are determinedby the electron-electron interaction, progress has been muchslower. Here, one often has to resort to either approximatemethods or numerically exact techniques [ 9] which, however, tharnier@me.com †stefano.chesi@csrc.ac.cn ‡stefan.kirchner@correlated-matter.com §pedrojgribeiro@tecnico.ulisboa.ptare often restricted to small systems or comparatively high temperatures. Exact analytical results, available for integrablemodels in one dimension, do not typically generalize to opensetups. Moreover, nonthermal steady states in Luttinger liq-uids [ 10–12] seem to be less general than their equilibrium counterparts. Considerable progress has been made in the Markovian case, where the environment lacks memory [ 13–16]. The applicability of the Markovian case is, however, limited toextreme nonequilibrium conditions (e.g., very large bias ortemperature) and is of restricted use for realistic transportsetups [ 17,18]. Other recent developments to study transport include the study of so-called generalized hydrodynamic methods avail-able for integrable systems [ 19,20] and hybrid approaches involving Lindblad dynamics [ 21]. However, these methods are not yet able to describe current-carrying steady states inextended mesoscopic systems. Our recent analysis of the exactly solvable transverse- ﬁeld Ising chain attached to macroscopic reservoirs hasallowed us to study a symmetry-breaking quantum phasetransition in the steady state of an extended nonequilibriumsystem [ 22]. At the equilibrium level, this model can be mapped onto that of noninteracting fermions via a Jordan-Wigner transformation and is thus solvable by elementarymeans. The nonthermal steady state of this model is, however, much richer and allows for a peculiar symmetry-breakingquantum phase transition. In particular, we have shown thistransition to be of a mixed-order (or hybrid) nature, with a dis-continuous order parameter and diverging correlation length.These types of transitions were ﬁrst discussed by Thoulessin 1969 [ 23] in the context of classical spin chains with 2469-9950/2021/103(3)/035108(14) 035108-1 ©2021 American Physical SocietyPUEL, CHESI, KIRCHNER, AND RIBEIRO PHYSICAL REVIEW B 103, 035108 (2021) long-range interactions and have since then been reported in different environments [ 24–28]. Even though realistic systems are only approximately de- scribed by exactly solvable models at best, exact solutions arestill of considerable value. Not only can they be important inunveiling features of novel effects, but they are commonlyinstrumental in benchmarking numerical and approximatemethods. Therefore, exact solutions are particularly helpful insituations where no reliable numerical or approximate meth-ods yet exist, such as in the description of current-carryingsteady states of interacting systems. In this article, we provide a set of exact results of steady- state phases and phase transitions of an XYspin chain in a transverse ﬁeld coupled to magnetic reservoirs held at dif-ferent magnetizations. Our analysis extends and generalizesthe ﬁndings in Ref. [ 22] and points out regimes that are not present in the Ising case. In the Markovian limit, werecover previous results obtained for XYspin chains coupled to free-of-memory reservoirs [ 13,29,30], where an out-of- equilibrium phase transition with spontaneous emergence oflong-range order has been found. The paper is organized as follows. In Sec. II, we deﬁne the out-of-equilibrium model and brieﬂy describe the meth-ods used to solve it. In Sec. III, we describe in detail the nonequilibrium phase diagram based on the energy currentand the properties of the occupation number. The correla-tion functions in the various phases are analysed in Sec. IV, where we also discuss the critical behavior at the mixed-order phase transition and the characteristic oscillations inthez-correlation function. Universal features of the entropy and mutual information are discussed in Sec. V. Finally, we summarize and conclude our work in Sec. VI. II. MODEL AND METHOD A. Hamiltonian and Jordan-Wigner mapping We consider an XY-spin chain of Nsites (labeled by r), exchange coupling J, and coupled to a transverse ﬁeld h. At its ends, i.e., at r=1 and r=N, the chain is coupled to magnetic reservoirs which are kept at zero temperature ( T=0). The Hamiltonian of the chain is given by HC=−J 2N−1/summationdisplay r=1/bracketleftbig(1+γ)σx rσx r+1+(1−γ)σy rσy r+1/bracketrightbig −hN/summationdisplay r=1σz r, (1) where σx,y,z rare the Pauli matrices at site r, andγcontrols the anisotropy. The total Hamiltonian is given by H=HC+/summationdisplay l=L,R(Hl+HC-l), (2) where HlandHC-l, with l=L,R, are, respectively, the Hamil- tonians of the reservoirs and the system-reservoir couplingterms. In the following, we assume that the reservoirs possessbandwidths which are entirely determined by magnetic poten-tialμ l(l=L,R) and which are much larger than the energy scales that characterize the chain. In the wide-band limit, the results become independent of the details of HlandHC-l. For concreteness, we take the FIG. 1. (a) Schematic picture of the XY-model spin chain in contact with magnetic reservoirs and (b) the same system mappedto its fermionic representation, i.e., a triplet superconducting chain of spinless fermions in contact with fermionic reservoirs. reservoirs to be isotropic XYchains, i.e., Hl=−Jl/summationdisplay rl∈/Omega1l/parenleftbig σx rlσx rl+1+σy rlσy rl+1/parenrightbig , (3) with l=L,R, and we have deﬁned /Omega1L≡{ − ∞ ,..., 0}, /Omega1R≡{N+1,...,∞}. Initially, the reservoirs are in an equilibrium Gibbs state, ρl=e−β(Hl−μlMl), where Ml=/summationtext rl∈/Omega1lσz rlis the reservoir magnetization (which is a good quantum number in the absence of system-reservoir coupling,i.e., [H l,Ml]=0). The average value of Mlis set by the magnetic potential μl. For ﬁnite μl, these are non-Markovian reservoirs, with power-law decaying correlations, and a setof gapless magnetic excitations within an energy bandwidthJ l/greatermuchJ,h. The chain-reservoir coupling Hamiltonians are HC-l=−JC-l/parenleftbig σx (rl)Cσx (r)l+σy (r)lσy (rl)C/parenrightbig , (4) with ( rL)C=1, (r)L=0, (rR)C=N, and ( r)R=N+1. A sketch of this system is shown in Fig. 1(a). The full Hamiltonian Hcan be represented in terms of fermions via the so-called Jordan-Wigner (JW) mapping [ 31], σ+ r=eiπ/summationtextr−1 r/prime=1ˆc† r/primeˆcr/primec† r, where ˆ c† r/ˆcrcreates /annihilates a spin- less fermion at site r. The JW-transformed system corresponds to a Kitaev chain [ 32] in contact with two metallic reservoirs of spinless fermions at chemical potentials μL,R, i.e., H=−JN−1/summationdisplay r=1(ˆc† rˆcr+1+γˆc† rˆc† r+1+H.c.)−2hN/summationdisplay r=1ˆc† rˆcr −/summationdisplay l=L,R/bracketleftBigg JC-lˆc† (rl)Cˆc(r)l+Jl/summationdisplay rl∈/Omega1lˆc† rlˆcrl+1+H.c./bracketrightBigg ,(5) where Jγdeﬁnes the superconducting coupling strength and hplays the role of a potential applied on the chain. A sketch of this system is shown in Fig. 1(b). In equilibrium, topo- logically nontrivial phases of the Kitaev chain correspond tomagnetically ordered phases of the original XY-spin model, whereas the topologically trivial cases correspond to dis-ordered phases. With a magnetic bias, the transfer of spinexcitations between the reservoirs was studied rather exten-sively (see, e.g., Refs. [ 33,34]), also considering transport signatures of the topological phase in short junctions [ 35,36]. 035108-2NONEQUILIBRIUM PHASES AND PHASE TRANSITIONS … PHYSICAL REVIEW B 103, 035108 (2021) Here, however, we will be mostly concerned with the bulk properties at N→∞ , and after the reservoirs have been traced out. B. Nonequilibrium Green’s functions As the JW-transformed Hamiltonian is quadratic in its fermionic degrees of freedom, the nonequilibrium system ad-mits an exact solution in terms of single-particle quantities. In the following, we employ the nonequilibrium Green’s func- tion formalism to compute correlation functions and relatedobservables. The procedure is described in the Supplemen-tal Material of Ref. [ 22] and is brieﬂy summarized here for convenience. We start by deﬁning the Nambu vector, ˆ/Psi1 †= (ˆc† 1,..., ˆc† N,ˆc1,..., ˆcN), and the retarded, advanced, and Keldysh components of the Green’s function, given by GR i,j(t−t/prime)=−i/Theta1(t−t/prime)/angbracketleft{ˆ/Psi1i(t),ˆ/Psi1† j(t/prime)}/angbracketright, (6) GA i,j(t−t/prime)=i/Theta1(t/prime−t)/angbracketleft{ˆ/Psi1i(t),ˆ/Psi1† j(t/prime)}/angbracketright, (7) GK i,j(t−t/prime)=−i/angbracketleft[ˆ/Psi1i(t),ˆ/Psi1† j(t/prime)]/angbracketright. (8) Using this notation, the Hamiltonians for the right and left reservoirs and for the chain are given by Hl=1 2ˆ/Psi1†Hlˆ/Psi1, with l=L,R,C. For the chain, HCis a 2 N×2NHermitian ma- trix respecting particle-hole symmetry, i.e., S−1HT CS=−HC, where S=τx⊗1N×Nandτxinterchanges particle and hole spaces. Similar deﬁnitions apply to the degrees of freedomof the right and left reservoirs. The bare retarded and ad-vanced Green’s functions, in the absence of chain-reservoircouplings, are simply given by G R/A l,0(ω)=(ω−Hl±iη)−1. Rewriting the chain-reservoir coupling in the same nota- tion,HC-l=1 2(ˆ/Psi1† (l)Tˆ/Psi1+ˆ/Psi1†T†ˆ/Psi1(l)). The self-energy of the chain, induced by tracing out the reservoirs, is /Sigma1R/A/K=/summationtext l=L,R/Sigma1R/A/K l, where /Sigma1R/A l(ω)=T†GR/A l,0(ω)T, (9) /Sigma1K l(ω)=/bracketleftbig /Sigma1R l(ω)−/Sigma1A l(ω)/bracketrightbig [1−2nF,l(ω)] (10) are the contributions of reservoir l, which obey equilib- rium ﬂuctuation-dissipation relations, and where nF,l(ω)= (eβl(ω−μl)+1)−1is the Fermi function with chemical poten- tialμland inverse temperature βl. The chain steady-state Green’s functions are obtained from the Dyson’s equation, GR/A C(ω)=/bracketleftbig GR/A C,0(ω)−/Sigma1R/A(ω)/bracketrightbig−1, (11) GK C(ω)=GR C(ω)/Sigma1K(ω)GA C(ω). (12) As mentioned above, we consider the case where the band- widths of the reservoirs, Jl=L,R, are much larger than the other energy scales. In this wide-band limit, the coupling to reser-voir lis completely determined by the hybridization energy scale/Gamma1 l=πJ2 C-lDl. Here, Dlis the reservoir’s constant-local density of states. In practice, the wide-band limit yields afrequency-independent retarded self-energy, /Sigma1 R l=i(γl+¯γl), which substantially simpliﬁes subsequent calculations, withγ l=/Gamma1l\|rl/angbracketright/angbracketleftrl\|and ¯γl=/Gamma1l\|¯rl/angbracketright/angbracketleft¯rl\|, and where \|r/angbracketrightand\|¯r/angbracketright≡ S\|r/angbracketrightare single-particle and hole states.In this case, it is convenient to deﬁne the non-Hermitian single-particle operator, K≡HC−i/summationdisplay l=L,R(γl+¯γl), (13) which we assume to be diagonalizable, possessing right and left eigenvectors \|α/angbracketrightand/angbracketleft˜α\|, and associated eigenvalues λα. In terms of these quantities, the retarded Green’s function isgiven by G R(ω)=(ω−K)−1=/summationdisplay α\|α/angbracketright(ω−λα)−1/angbracketleft˜α\|, (14) and the Keldysh Green’s function becomes GK(ω)=−2i/summationdisplay l/summationdisplay αβ\|α/angbracketright/angbracketleftβ\| ×/angbracketleftα/prime\|γl\|β/prime/angbracketright[1−2nF,l(ω)]−/angbracketleftα/prime\|¯γl\|β/prime/angbracketright[1−2nF,l(−ω)] (ω−λα)(ω−¯λβ). (15) Steady-state observables can be obtained from the single- particle correlation function matrix, χ≡/angbracketleftˆ/Psi1ˆ/Psi1†/angbracketright, which is obtained from the Keldysh Green’s function, χ=1 2/bracketleftbigg i/integraldisplaydω 2πGK(ω)+1/bracketrightbigg . (16) The explicit form of χafter performing the integration over frequencies is provided in Eq. ( A1). As the model is quadratic, χencodes all the information about the reduced density ma- trix of the chain, ˆ ρC=trL,R[ˆρ]. This quantity can itself be expressed as the exponential of a quadratic operator, i.e., ˆ ρC= eˆ/Omega1C/Z, where Z=tr[eˆ/Omega1C] and ˆ/Omega1C=1 2ˆ/Psi1†/Omega1Cˆ/Psi1with/Omega1Cbeing a2N×2Nmatrix respecting the particle-hole symmetry con- ditions. ˆ/Omega1Cis related to the single-particle density matrix via χ=(e/Omega1C+1)−1. (17) This relation allows the calculation of mean values of quadratic observables, ˆO=1 2ˆ/Psi1†Oˆ/Psi1, deﬁned by the Hermi- tian and particle-hole symmetric matrix O, /angbracketleftˆO/angbracketright=Tr[ ˆρCˆO]=−1 2tr[O·χ], (18) as well as all higher-order correlation functions. III. PHASE DIAGRAM This section discusses the nonequilibrium phase diagram of the model, as well as the excitations and associated occupa-tion numbers in the various different phases. To contextualizeour ﬁndings, the ﬁrst two sections are devoted to a briefdescription of the equilibrium properties of the XYchain and a review of the nonequilibrium Markovian limit. A. Equilibrium phases The system in equilibrium is more conveniently studied without considering the couplings to the leads and assumingperiodic boundary conditions. After performing the JW trans-formation, the Hamiltonian of the translation-invariant chainis diagonalized in the momentum representation by a suitable 035108-3PUEL, CHESI, KIRCHNER, AND RIBEIRO PHYSICAL REVIEW B 103, 035108 (2021) FIG. 2. (a) Phase diagram h/J×γof the XYmodel in equilib- rium. The black thick lines identify a gap closing. (b) Phase diagram of the out-of-equilibrium XYmodel according to the presence (sensi- tive) or absence (normal) of long-range correlations in the Markovian limit. The vertical dashed line at γ=+1(γ=−1) identiﬁes the Ising model with only XX(YY)-spin-coupling interactions. Bogoliubov transformation, i.e., HC=/summationtext kεk(ˆγ† kˆγk−1/2), where the operators ( ˆ γk,ˆγ† −k)T=eiθkσx(ˆck,ˆc† −k)Tdescribe excitations of energy εk=2J/radicalBig (h/J+cosk)2+(γsink)2, (19) and sin (2 θk)=−2Jγsin(k)/εk. The ground state is characterized by a vanishing number of Bogoliubov excitations, i.e., nk=0, where nk≡/angbracketleftˆγ† kˆγk/angbracketright. (20) For\|h/J\|<1, the ground state is topologically nontrivial with positive and negative anisotropies ( γ> 0o r<0) correspond- ing to opposite signs of the topological invariant, separated bya critical gapless state for γ=0. At\|h/J\|=1, the spectral gap vanishes and the system transitions into a topologicallytrivial phase at large h. A similar phase diagram is obtained in terms of the original spin degrees of freedom. Figure 2(a) illustrates the zero- temperature phase diagram of the equilibrium XY model. Forγ> 0, the system is magnetically ordered along the x direction under a weak transverse ﬁeld h, and possesses a ﬁ- nite magnetization [ 37]φ≡lim hx→0lim L→∞1 N/summationtext r/angbracketleftσx r/angbracketright/negationslash=0, where hxis a symmetry-breaking magnetic ﬁeld along the x direction. A negative anisotropy, i.e., γ< 0, yields a nonvan- ishing magnetization along the ydirection, whereas at γ=0, the system is critical and isotropic for \|h/J\|<1. As the or- dered phases for γ> 0 and <0 are equivalent to each other and related via a simple rotation, only γ> 0 is considered in the subsequent analysis. It is worth recalling that the specialcasesγ=±1 correspond to the transverse-ﬁeld Ising model. A strong hdrives the magnetic phase through a second-order phase transition into a phase of vanishing magnetization, i.e., aphase with φ=0. Near the transition, for \|h/J\|<1, the mag- netization behaves as φ/similarequal/radicalBig 2 1+\|γ\|{γ2[1−(h/J)2]}1/8[37]. The computation of the order parameter, φ, directly from the above deﬁnition is not possible via the JW mapping. In-stead, one considers the two-point correlation functions ( α= x,y,z), C αα r,r/prime=/angbracketleftbig σα rσα r/prime/angbracketrightbig −/angbracketleftbig σα r/angbracketrightbig/angbracketleftbig σα r/prime/angbracketrightbig . (21)For disordered phases in equilibrium, these correlators are expected to show either exponential (EXP) or power-law (PL)decay depending on whether the system is gapped or gapless.In the ordered phase, the system has long-range order (LRO)correlations, e.g., for γ> 0, C xx r,r/prime/similarequalAe−\|r−r/prime\|/ξ+φ2, (22) where ξis the characteristic correlation length and Ais a nu- meric coefﬁcient. This expression allows one to obtain φfrom the correlation function Cxx r,r/prime, which in turn can be computed in terms of a Toeplitz determinant [ 31,38]. In equilibrium, all correlation functions, except Cxx r,r/prime, either vanish or decay exponentially with \|r−r/prime\|. For an open system connected to demagnetized baths, i.e., μL,R=0, the same equilibrium bulk properties as those for the closed system are found. It is natural to expect that bulkproperties pertain for distances greater than ξaway from the leads. Our calculation of fermionic observables follows thatdescribed in Sec. II. The calculation of the spin-spin correla- tion functions is similar to those for the translation-invariantsystem and is given in Appendix Ain terms of the single- particle correlation matrix χ. B. Nonequilibrium phases in the Markovian limit The nonequilibrium features of the XYmodel with Marko- vian reservoirs were ﬁrst reported in Refs. [ 13,39]. This limit can be recovered from the present model by taking \|μR\| or\|μL\|→∞ [17]. The steady-state phase diagram in that limit possesses two distinct phases, one characterized by anexponential decay of all correlation functions with distanceand the other by an algebraic decay of C zz r,r/prime, concomitant with a strong sensitivity to small variations of some controlparameters [ 13]. We dub these regimes as normal and sensitive phases, respectively. Figure 2(b) depicts the phase diagram in this Markovian limit and with its sensitive (white) and nor-mal (light-yellow) phases. The dark-yellow lines mark criticalphase boundaries. The quasiparticle dispersion relation, given by Eq. ( 19), is shown in Figs. 3(a) and3(c). The algebraic correlations are associated to the presence of an inﬂection point in thequasiparticle dispersion which appears for h/J/lessorequalslant\|1−γ 2\|. In the normal region, the extrema of the energy are m1= εk=0=2J\|1+h/J\|andm2=εk=π=2J\|1−h/J\|, while for the sensitive region, m3=2J\|γ\|√ 1+(h/J)2(γ2−1)−1be- comes a global extremum. Figures 3(b) and3(d) show the spectrum of the non-Hermitian single-particle operator K[see Eq. ( 13)] for both normal and sensitive phases. It turns out that the imaginary part of the eigenvalues scales with theinverse system size, Im λ α∝N−1[40]. This is a reﬂection of the fact that for the chain degrees of freedom, the dissipativeeffects of the boundary become less important with increasingsystem size. A key feature of the sensitive region is that forenergies [i.e., Re( λ α)] where four momenta can propagate, the spectrum does not converge to a line with increasingsystem size, but becomes scattered within a ﬁnite area [ 13]. These effects are independent of the Markovian nature ofthe reservoirs and remain for the non-Markovian case as theoperator Kdoes not depend on the chemical potential of the leads. Thus, as explicitly shown below, the normal-sensitive 035108-4NONEQUILIBRIUM PHASES AND PHASE TRANSITIONS … PHYSICAL REVIEW B 103, 035108 (2021) FIG. 3. Quasiparticle dispersion relation of the equilibrium XY model in the (a) normal and (c) sensitive regions, with {γ,h/J}= {1.0,0.2}and{γ,h/J}={ 0.5,0.2}, respectively, which leads to {m1,m2,m3}={ 2.4,1.6,0.97}.±km2are the two momenta with en- ergy m2. (b) and (d) depict the eigenvalues λαof the non-Hermitian single-particle operator Kin the complex plane. transition, reported in Refs. [ 13,39] for the Markovian case, also occurs for ﬁnite values of μLandμR. C. Nonequilibrium phase diagram: Energy current We now present the phase diagram of the nonequilibrium XYmodel and show that the energy current passing through the chain can be used to discriminate between the differentphases. Conservation of energy implies that the steady-state energy current is equal across any cross section along the chain andcan be obtained from χas J e=−1 2tr[Jr·χ], (23) where Jris the single-particle current operator at link (r,r+1) which is explicitly given in Appendix A. We have previously discussed the steady-state energy current in a non-Markovian setting for the particular caseof the transverse-ﬁeld Ising model, i.e., \|γ\|=1, in Ref. [22]. The nonequilibrium phase diagram, as a function of μ LandμR, in the normal phase is qualitatively similar to the Ising case and is reproduced in Fig. 4(a).T w oo f the phases which arise near μL=μRdo not support en- ergy transport, i.e., Je=0: the ordered phase (O) and the nonconducting phases (NC). Other phases may be fur-ther characterized in terms of their energy conductance,i.e.,G L≡∂μLJeandGR≡∂μRJe. The current-saturated (CS) phases are those with Je/negationslash=0 and GR=GR=0. FIG. 4. Non-Markovian phase diagram μL×μRof two illustra- tive settings inside the (a) normal and (b) sensitive regions, following the same set of parameters in Fig. 3. The phases were deﬁned as ordered (O), conducting (C), conducting saturated (CS), and noncon- ducting (NC). In the sensitive case, the phases which acquire a noise in the occupation number are signed with a star∗. The arrows at the corners indicate the Markovian limit, i.e., \|μR\|and\|μL\|→± ∞ . (c) and (d) show the current of energy ( Je) and the conductance (GL≡∂μLJe) computed across the red dotted lines drawn on the phase diagrams (a) and (b), respectively. They arise when one of the reservoir’s chemical potentials is larger than m1, while the other lies inside the quasiparticle excitation gap. The conducting phase (C) is characterized bya nonzero conductance, i.e., G L/negationslash=0 and/orGR/negationslash=0, arising whenever at least one of the chemical potentials lies withinthe quasiparticle excitations band, i.e., \|μ L\|and/or\|μR\|∈ (m1,m2). Figure 4(b) depicts the phase diagram for a generic XY chain. Besides the phases found for γ=1, an analysis of the occupation numbers (see next section) shows that someregions acquire a noiselike behavior. These phases, similar tothe sensitive regions of the Markovian case, are labeled NC ∗, CS∗, and C∗. In Figs. 4(c) and4(d), we show the current Jeand con- ductance GLfor a ﬁxed μRrepresented by the red dashed lines in the phase diagrams. In the sensitive region, the Cphase is crossed by the transition line at μ L=m2, where the conductance becomes nonanalytic. It is worth noting that thenoiselike behavior found in the occupation numbers does notappear in the current of energy. In terms of the JW fermions, the present analysis is similar to that of a transport across a tight-binding model in the sensethat when the chemical potentials cross the dispersion rela-tion, nonanalytic properties of the current appear. However,the existence of anomalous terms in the fermionic Hamilto-nian inhibits a closer comparison with charge transport. D. Occupation numbers In the current-carrying steady-state regime of a Fermi gas, ﬂuctuations in the number of particles were shown to be 035108-5PUEL, CHESI, KIRCHNER, AND RIBEIRO PHYSICAL REVIEW B 103, 035108 (2021) FIG. 5. Distribution of occupations. (a),(b) The momenta kl=L,Rat which the reservoir’s chemical potentials μl=L,Rcross the dispersion relation. (c)–(f) The excitation number nkat different phases inside the normal region. (g)–(l) nkin the sensitive region. km2is such that εkm2=m2, with m2given in Fig. 3(c). intimately related to the entropy of a subsystem [ 41–44]. In analogy, the occupation number of the Bogoliubov ex-citations, given by Eq. ( 20), can be used to describe the properties of the asymptotic steady state away from theboundaries. In the open system setting, n kcan be approxi- mated by numerically computing the Fourier transform χk=/summationtext r∈/Omega1e−ik(r−r0)χr,r0, where /Omega1={r:L/4<r<3L/4}and r0=L/2, followed by a Bogoliubov transformation. In equi- librium, μL=μR=0,nk/similarequal0 as expected, while in a generic out-of-equilibrium situation, nk/negationslash=0. For the Ising model, it was shown that in the CS and NC phases, the nkis a continuous function of k, while in the C phase, it has discontinuities depending on the reser-voir’s chemical potentials [ 22]. These discontinuities happen at the momenta ±k l=L,R, where the chemical potential μl=L,R crosses the dispersion relation; see Fig. 5(a). These results extent straightforwardly to the XYmodel in the normal region; see Figs. 5(c)–5(f). Within the O phase, the system behaves as in equilibrium, i.e., nk/similarequal0. In the sensitive region, there may be two absolute values of momenta, labeled ±kl=L,Rand±k/prime l=L,R, for which each chem- ical potential crosses the dispersion relation, as illustrated inFig. 5(b). Interestingly, we ﬁnd that n khas an intrinsic noise in the sensitive region; see Figs. 5(g)–5(j). The noise appears in phases NC∗,C∗, and CS∗,f o r\|k\|>km2, where εkm2=m2 [see Fig. 3(c)]. In Appendix B, we check that the magnitude of the noise in nkdoes not diminishes with increasing system sizes. Curiously, the noise vanishes along the line μL=−μR, as well as within phases C and CS crossed by this line, as shown in Figs. 5(k) and5(l), and studied in detail in Appendix B. Note that nkis asymmetric upon changing k→− kfor all conducting phases as required to maintain a net energyﬂow through the chain, as ε(k)=ε(−k). Figures 5(c)–5(l)illustrate this feature by showing a larger value of the hybridization, which yields a larger current of energy andconsequently to a more asymmetric n k. IV . CORRELATION FUNCTIONS We now consider in more detail the properties of the spin correlation functions, deﬁned in Eq. ( 21). ForCxx r,r/prime, the generic asymptotic dependence was already given in Eq. ( 20) and is able to signal the presence of long-range order when φ/negationslash=0. We give, in Fig. 6(a), a numerical example of this case. On the other hand, when φ=0, we can extract the correlation length ξfrom the exponential decay of Cxx r,r/prime; see Fig. 6(b). Table I shows a summary of the asymptotic dependence of Cxx r,r/primein the different phases. The typical dependence of φandξon the chemical po- tential of the reservoirs is illustrated in Figs. 6(c) and6(d), showing the remarkable property of a discontinuity in φat the critical point (after extrapolation to the thermodynamic limit)accompanied by a diverging correlation length ξ. Further be- low we will elaborate on the mixed-order transition in moredetail, by also providing an analytical description clarifyingits origin. As shown in Table I, the behavior of C xx r,r/primeis not affected by the transition to the sensitive region. However, earlier studiesshowed how, in the Markovian limit, the CS and NC phasesare characterized by a transition, from short- to long-rangecorrelations, when entering the sensitive region. This behavioris reﬂected by a transition from exponential to power-law de-cay in C zz r,r/prime[13,29]. We observe similar results in the present case, with both nonconducting and saturated phases (CS andNC) showing exponentially decaying correlations in the nor-mal region and power-law decay in the sensitive one (CS ∗and NC∗). Extending the analysis to the highly non-Markovian 035108-6NONEQUILIBRIUM PHASES AND PHASE TRANSITIONS … PHYSICAL REVIEW B 103, 035108 (2021) FIG. 6. (a), (b) The correlation Cxx r,r/primebehavior in the nonequilib- rium phases of Fig. 4(b), with ﬁxed μR=0. Long-range order only appears in the ordered phase (O), as exempliﬁed in (a) at the point\|μ L\|=0.3<m3, while a typical exponential decay appears in all other phases, as exempliﬁed in (b) for the C∗phase at m3<\|μL\|= 1.3<m2. (c),(d) The mixed-order behavior with a discontinuous order parameter φand diverging correlation length ξ, respectively, in the thermodynamic limit. The straight line in (d) is a guide to the critical exponent ν=1/2. All panels share the same legends as in (a). setting, we ﬁnd that long-range correlations also appear in the conducting phase, with a power-law decay in both normal(C) and sensitive (C ∗) regions; see Figs. 7(c) and7(d). These results show that all phases with a noisy excitation numberdistribution possess power-law decaying zzcorrelations. The asymptotic behavior of C zz r,r/primein the various phases is summa- rized in Table I. A. Mixed-order phase transition As mentioned earlier, for γ> 1, the magnetization along thexdirection, φ, is a good order parameter for the broken- symmetry equilibrium phase. In the open system, φcan still be used as the order parameter. However, by changing thechemical potential of the, say, left reservoir, φdrops to zero discontinuously as soon as the system reaches the disorderedphase. Interestingly, this transition shows a mixed-order be-havior where the discontinuity of φis accompanied by a divergence of the correlation length [ 22]. The divergence oc- curs as ξ∝\|μ L±mi\|−νwhen approaching the critical point from the disordered phase ( i=2,3 depending on the values of γandh). The critical exponent is ν=1/2, except for special values of the parameters (see Appendix C). TABLE I. Classiﬁcation of each phase according to the asymp- totic behavior of the correlation functions; the possibilities are exponential (EXP) or power law (PL) decay, and long-range order (LRO). Phase Cxx r,r/prime(γ> 0) Czz r,r/prime OL R O E X P C/C∗EXP PL /PL CS/CS∗EXP EXP /PL NC/NC∗EXP EXP /PL FIG. 7. Correlations Czz r,r/primefor the normal (left column) and sen- sitive (right column) regions; see Fig. 4. We have set (a),(b) μL= μR=0, (c) m2<\|{μL,μR}={ 2.1,−2.1}\|<m1,a n d( d ) m3< \|{μL,μR}={ 1.3,−1.3}\|<m2. These results are summarized in Table I. All panels share the same legends as in (a). We have numerically veriﬁed that this behavior survives away from γ=1, with the same type of dependence of the order parameter and correlation length. In particular, themixed-order transition with ν=1/2 is also present in the sensitive region, e.g., at the transitions between O and C ∗ phases in the phase diagram of Fig. 4(b).W es h o wi nF i g s . 6(c) and6(d) an example of the numerical analysis of the order parameter and correlation length in the sensitive region. Themain difference compared to the normal region is that hereﬁnite-size effects are much stronger, which is consistent withthe sensitive dependence on Nof the rapidity spectrum and occupation numbers; see Figs. 3and5. To derive the value of the critical exponent ν, we consider the explicit form of the correlation function in terms of aToeplitz determinant [ 31,38], /angbracketleftbig σ x rσx r+n/angbracketrightbig =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleD 0 D−1.. D−n+1 D1 D0.. . .. . .. .. . D0 D−1 Dn−1.. D1 D0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (24) where D nis given by Dn=/integraldisplaydk 2πe−ink/radicalBigg 1−(h/J)eik 1−(h/J)e−ik(1−nk−n−k).(25) The asymptotic dependence can be obtained from Szego’s lemma, leading to the following expression for the correlationlength: ξ −1=−1 2π/integraldisplay2π 0ln[1−nk−n−k]dk. (26) Here, the difference from the standard treatment of the transverse-ﬁeld Ising chain [ 31,38] is simply that the occu- pation numbers nkare kept generic, and thus are allowed to assume any nonequilibrium distribution induced by theexternal reservoirs. For example, Eq. ( 26) takes into account that in general n k/negationslash=n−k, as shown by Fig. 5with a large hybridization energy. By substituting the Fermi distribution, 035108-7PUEL, CHESI, KIRCHNER, AND RIBEIRO PHYSICAL REVIEW B 103, 035108 (2021) FIG. 8. (a) The correlation length of the Ising model ( γ=1), computed at weak transverse ﬁeld ( h/J=−0.05) and approaching the critical point m1=1.9 from the disordered phase, with μR=0. The numerical results (dots) agree well with the blue (lower) curve, given by Eq. ( 28). The red (upper) curve is from the equilibrium relation ξ−1=2ρ,w h e r e ρ=/integraltext2π 0dk 2πnkis the density of excitations. (b) Comparison between the numerically computed ρ(upper dots) and the modiﬁed density ρξ(lower dots), deﬁned in Eq. ( 30). The solid curves were obtained by assuming a parabolic dispersion andn k/similarequalnπwithin the occupied region. Eqs. ( 25) and ( 26) recover the known equilibrium expressions at ﬁnite temperature [ 38,45]. Applying Eq. ( 26) to the critical point, we ﬁrst assume, as in Fig. 5(a), that the minimum of the quasiparticle dis- persion occurs at k=π. Furthermore, if μRis inside the gap, the critical point is at μL=m2. Close to the critical point, the only occupied states are in a small range, k∈ [π−/Delta1kL,π+/Delta1kL], around the minimum of εk, and thus we can approximate the quasiparticle dispersion as parabolic,giving /Delta1k L∝√μL−m2. This dependence of /Delta1kLis directly related to the critical exponent ν=1/2. More precisely, /Delta1kL close to the critical point is given by /Delta1kL/similarequal/radicalBigg \|1+h/J\| γ2−h/J−1(μL−m2), (27) and we can set nk/similarequalnk=πin the small integration interval of Eq. ( 26), leading to ξ−1/similarequal−/Delta1kL πln[1−2nπ]. (28) This expression clearly shows how the divergence of ξis due to the shrinking of the region of nonzero occupation. We showin Fig. 8that this theory is accurate by a direct comparison to the numerical results. After having clariﬁed the origin of the critical exponent ν=1/2, it is interesting to compare the behavior of the open chain to the temperature dependence of the equilibrium sys-tem. In the latter case, an ordered phase is only allowed atzero temperature and the order disappears at any arbitrarilysmall temperature T>0. The sudden disappearance of the ordered state is related to the presence of thermal excita-tions and is analogous to the vanishing of φinduced by the nonequilibrium chemical potentials, as soon as either μ Lor μRovercomes the gap. Furthermore, similarly to the nonequi- librium system, the correlation length diverges when T→0. As it turns out, in the low-temperature limit, ξcan be related in a simple way to the density of excitations ρ=/integraltext2π 0dk 2πnk: ξ−1=2ρ(low temperature) . (29)This expression follows immediately from Eq. ( 26) since nk=n−k/lessmuch1 when T→0, and can also be understood by a simple argument in terms of a dilute gas of domain-wallexcitations [ 45]. A naive application of Eq. ( 29) to the nonequilibrium sys- tem is shown in Fig. 8. Although Eq. ( 29) predicts the correct critical exponent ν=1/2, there is a clear disagreement with the numerical results. This failure of Eq. ( 29) can be explained from the nonvanishing value of n karound the minimum of εk(say, k=π): in both the low-temperature limit and the nonequilibrium system, we have ρ→0 at the critical point, which results in a diverging correlation length. However, inthe ﬁrst case, we have n k→0, while for the nonequilibrium system, nπremains ﬁnite and the vanishing of ρis due to the shrinking of /Delta1kL. Instead of the density ρ, we can consider a “modiﬁed” density, ρξ=−1 4π/integraldisplay2π 0ln[1−nk−n−k]dk, (30) which follows naturally from Eq. ( 26) by requiring that a relation similar to the equilibrium system at low temperatureis satisﬁed, ξ −1=2ρξ. It is easy to see that close to the criti- cal point, we have ρξ/similarequal−(ln [1−2nπ]/2nπ)ρ, which differs fromρby a nontrivial multiplicative factor. An interesting exception, discussed more extensively in Appendix C, occurs forJ=h/2, when nπ=0 and the relation between ξand ρis Eq. ( 29), as in equilibrium. At J=h/2, the vanishing ofnkalso affects the value of the critical exponent, which is ν=5/2 instead of 1 /2. The above arguments can be adapted to other parameter regimes. In particular, the discussion is almost unchangedforh/J<min[0,γ 2−1], when the dispersion minimum is atk=0. Instead, the treatment of the sensitive phase with \|γ\|<1 is more delicate. First, as shown in Fig. 5(b),t h e dispersion is characterized by two minima instead of one.More importantly, n kappears to be highly pathological when N→∞ , when the occupation numbers undergo wild oscil- lations. Despite these differences, numerical evaluation of thecritical exponent still gives ν=1/2, and thus we conjecture that a suitable average of n kis well deﬁned, nk=lim /Delta1k→0lim N→∞1 /Delta1k/integraldisplayk+/Delta1k 2 k−/Delta1k 2nk/primedk/prime. (31) Then, the critical exponent would be determined through Eq. ( 28) in a way analogous to the regular case, i.e., the critical exponent would correspond to the shrinking of the occupiedregions around the minima, explaining the persistence of ν= 1/2 in this phase. B.zcorrelations As summarized in Table I,Czz r,r/primedisplays a power-law decay in several of the allowed phases. We focus here on the thenonsensitive region, where this behavior can be understoodfrom the well-known power-law decay of density-densitycorrelations of noninteracting fermions. In fact, through thefermionic mapping, C zz r,r/primeis equivalent to a density-density correlation function, Czz r,r/prime=4(/angbracketleftˆc† rˆcrˆc† r/primeˆcr/prime/angbracketright−/angbracketleft ˆc† rˆcr/angbracketright/angbracketleftˆc† r/primeˆcr/prime/angbracketright). (32) 035108-8NONEQUILIBRIUM PHASES AND PHASE TRANSITIONS … PHYSICAL REVIEW B 103, 035108 (2021) In the C phase, at least one of the reservoirs has its chemical potential within the range of the quasiparticle energy spec-trum. Then, the correlation function is expected to have apower-lay decay with oscillating character, similar to a simpleone-dimensional (1D) Fermi gas where it decays as \|r−r /prime\|−2 and oscillates with wave vector 2 kF, with kFthe Fermi wave vector (see, e.g., Ref. [ 46]). In our case, we express the correlation function through the Bogoliubov excitations ˆ γkof the translational-invariant system (which is appropriate in the thermodynamic limit).The corresponding occupation numbers are deﬁned in Eq. ( 20) and give C zz r,0=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay π −πdk 2πeikr(n−k+nk−1)sin 2θk/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 −/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay π −πdk 2πeikr[(nk+n−k−1)cos 2θk+(nk−n−k)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (33) If, for simplicity, we assume nk/similarequaln−k(which is justiﬁed in the limit of vanishing hybridization energy /Gamma1), the oc- cupation numbers have discontinuities at k=±ki, induced by the left and right reservoirs ( i=L,R). When kir/greatermuch1, we can extract the leading contribution to Eq. ( 33) induced by the discontinuous jumps /Delta1nkiofnk, deﬁned by ∂knk=/summationtext i=L,R/Delta1nki[δ(k+ki)−δ(k−ki)]: Czz r,0/similarequal4 π2r2/bracketleftBigg/parenleftBigg/summationdisplay i=L,R/Delta1nkisin 2θkicoskir/parenrightBigg2 −/parenleftBigg/summationdisplay i=L,R/Delta1nkicos 2θkisinkir/parenrightBigg2/bracketrightBigg , (34) which simpliﬁes to Czz r,0/similarequal4/Delta1n2 kR π2r2/parenleftbig cos2kRr−cos22θkR/parenrightbig , (35) when there is a single Fermi surface (here, induced by i=R). The above expressions display the expected 1 /r2decay and oscillatory dependence. As shown in Fig. 9, we ﬁnd a good agreement between Eq. ( 35) and the numerical results. To better characterize the oscillatory dependence, we have also studied the Fourier transform, Czz k≡1√ N/summationdisplay r−r/primee−ik(r−r/prime)Czz r,r/prime, (36) which is shown in Fig. 10for two representative cases. With a single Fermi surface (left panels), we ﬁnd dominant nonan-alytic features at k=0,2k R, in agreement with Eq. ( 35). We also ﬁnd smaller discontinuities in ∂kCzz kat higher harmonics, k=4kR,6kR, which are not captured by the leading-order ap- proximation given by Eq. ( 35). With two Fermi surfaces (right panels), we ﬁnd the expected singularities at k=0,2kL,R. However, there are additional features at ∂kCzz katk=kL± kR, which are in agreement with Eq. ( 34). As seen there, the correlation function is not simply a sum of i=L,Rcontribu- tions, but involves interference terms between the two Fermisurfaces. FIG. 9. Comparison between numerical results for Czz r,r/prime(dots) and the analytic approximation given by Eq. ( 35) (red curve). The values /Delta1nkR/similarequal0.1a n d kR/similarequal2.85 were extracted numerically from nk, as shown in the inset. We used h=0.2,γ=1,μL=0.5<m2, andμR=1.62/greaterorsimilarm3. Finally, we comment on the power-law dependence of Czz r,r/prime in the sensitive region. Away from the ordered phase, we ﬁnd a power-law decay \|r−r/prime\|−swhere, however, the exponent is generally different from s=2 (we often ﬁnd s<2) and depends on system parameters. This behavior is most likelyrelated to the singular nature of n k, which from our numerical evidence is characterized by a complex pattern of closelyspaced discontinuities (see, e.g., Fig. 5). Such discontinuities will contribute to the square brackets of Eq. ( 34)i naw a y that is difﬁcult to compute explicitly (the summation indexishould become a continuous parameter) and might be able to modify the exponent s. This interpretation is conﬁrmed by the survival of the power-lay decay in the NC ∗and CS∗ regions, where the chemical potentials μL,Rdo not cross the quasiparticle bands, and thus an exponential decay might beexpected. Instead, Figs. 5(g) and5(i)show that a discontinu- ous dependence of n kcan be found in these regions as well, in agreement with the observed power-law dependence of Czz r,r/prime. Finally, the simple discontinuities of Fig. 5(l)result in the regular value s=2. FIG. 10. Within the phase diagram of Fig. 4(a), here (a) and (b) show the Fourier transform of Czz r,r/prime[see Eq. ( 36)] for points inside the C phase, namely, (a) at {μL,μR}={ − 2.9,−2.1}and (b) at {μL,μR}={ − 1.9,−2.1}. (c),(d) Their respective derivatives. In the left column, we considered the case with only nonzero kR, while the right column considers nonzero kLandkR. We have computed these results for a system size of N=500 sites. 035108-9PUEL, CHESI, KIRCHNER, AND RIBEIRO PHYSICAL REVIEW B 103, 035108 (2021) V . ENTROPY AND MUTUAL INFORMATION In this section, we study the entropy of the steady state within the different phases identiﬁed above. For a segment of/lscriptsites in the middle of the chain, the entropy is given by E /lscript=−Tr[ˆρ/lscriptln(ˆρ/lscript)]=−tr[χ/lscriptlnχ/lscript], (37) where ˆ ρ/lscriptis the reduced density matrix and χ/lscriptis the single- particle correlation matrix restricted to the subsystem of /lscript sites. While, for the fermionic system the second equalityfollows from the noninteracting nature of the problem, thisexpression was also shown to hold for the spin chain [ 47]. In the thermodynamic limit ( N→∞ ), the entropy of the segment of a translational-invariant system is expected to obeythe general scaling law [ 48] E /lscript=l0/lscript+c0ln(/lscript)+c1, (38) where l0,c0, and c1are/lscript-independent real constants. For the ground state of gapped systems, l0=c0=0, following the so-called area law, while gapless fermions and spin chainsshow a universal logarithmic behavior with c 0=1/3. This result is a consequence of the violation of the area law in1+1 conformal theories, in which case c 0=c/3, where cis the central charge [ 47,49]. For a nonequilibrium Fermi gas, it was shown that both l0and c0can be nonzero [ 50,51], and that c0depends on the system-reservoir coupling and is a nonanalytic function of the bias [ 51]. The coefﬁcient c0is most easily extracted from the mutual information, I(A,B)≡ E(ˆρA)+E(ˆρB)−E(ˆρA+B), of two adjacent segments Aand Bof size /lscript/2, since I/lscript/similarequalc0ln(/lscript)+c2. (39) For the transverse-ﬁeld Ising model, in Fig. 4(a), all phases, except O, have been shown to have extensive entropy (i.e.,l 0/negationslash=0) [22]. This is due to the presence of a ﬁnite fraction of excitations, which are absent in the ordered phase. In addition,it was found that c 0/negationslash=0 in the C phase due to the presence of discontinuities in nk. In Fig. 11, we show the generalization of the previous results to the XYchain and including the sensitive region of the phase diagram. Figure 11(a) shows l0for both normal and sensitive regions. In both cases, l0follows the expected value [ 51]: l0=/integraldisplaydk 2π−[nklnnk+(1−nk)l n ( 1−nk)]. (40) On the other hand, I/lscriptdoes differ qualitatively in the nor- mal and sensitive regions. As mentioned earlier, logarithmiccorrections come from discontinuities in n k.I f nkhas no discontinuities such as in the O and saturated normal phases,c 0vanishes. Figure 11(b) depicts c0in the normal re- gion. The right inset shows that the leading term of I/lscript in Eq. ( 39) is indeed logarithmic in the large- /lscriptlimit. In principle, for conducting nonsaturated phases in the nor-mal region, c 0can be computed using the Fisher-Hartwing conjecture [ 48]. The presence of noise in nkwithin the sensitive region changes the previous picture. The left inset of Fig. 11(b) shows that I/lscriptis no longer of the form given in Eq. ( 39). It is FIG. 11. (a) Linear coefﬁcient l0, obtained from the entropy scal- ing law, for a range of μLacross the lines depicted in the phase diagrams of Fig. 4. The inset illustrates the typical ﬁtting E/lscript×/lscript, where the value of μLused is denoted by the dotted red line. (b) Sim- ilar results for the logarithmic coefﬁcient c0, obtained from Eq. ( 39), for the same range of μL. The inset on the right illustrates the typical ﬁtting I×ln (/lscript), where the value of μLis denoted by the dotted red line. In the sensitive region (left inset), the mutual information is superlogarithmic. tempting to interpret this result as a collection of discontinu- ities of nkwhose number increases with system size. However, due to the noisy form of nk, it is difﬁcult to give a precise meaning to this picture. Moreover, with the system sizes thatwe could attain, it was not possible to determine the functionalform of this correction with /lscript. Note that a supralogarithmic contribution in I /lscriptis always present in the sensitive region even for the CS∗and NC∗phases. VI. DISCUSSIONS We have studied the steady state of a transverse-ﬁeld XY- spin chain at zero temperature in a nonequilibrium setting bycoupling the ends of the chain to reservoirs which can beheld at different magnetic potentials. Our approach is basedon a Jordan-Wigner mapping and a Keldysh Green’s func-tion treatment of the resulting nonequilibrium interaction-freefermionic system. This allows us to study the steady states of the model as a function of the magnetic potentials including theequilibrium and Markovian limits. For magnetic potentialswhose magnitudes remain smaller than the spectral gap, theequilibrium-ordered state persists and the correlation functionof the order parameter displays long-range order. Away fromthis phase, the order parameter correlations decay exponen-tially. As μ LorμRreaches the spectral gap, the transition from equilibrium to a current-carrying state occurs through 035108-10NONEQUILIBRIUM PHASES AND PHASE TRANSITIONS … PHYSICAL REVIEW B 103, 035108 (2021) a mixed-ordered transition, where the order parameter van- ishes discontinuously while the correlation length diverges.This out-of-equilibrium phenomena was previously observedin Ref. [ 22]. Our present results establish that this behavior is generic to all order /disorder phase transitions of the XY chain. For large \|μ L\|and\|μR\|, we recover the Markovian limit. We identify the two qualitatively different behaviors previ-ously reported using a Lindblad master-equation approach[13,39]. We refer to these as the (i) sensitive region, featuring algebraic decaying correlations in the transverse (i.e., z)d i - rection, and (ii) normal region, where these correlations decayexponentially. Besides the equilibrium and Markovian phases, we iden- tify additional current-carrying phases. Their properties canbe easily understood by studying the quasiparticle excitationnumber n k. By analyzing this quantity, we were able to com- pute the critical exponent of the diverging correlation lengthat the transition, conﬁrming analytically the results of Ref.[22]. The behavior of the transverse correlation function in the normal region is explained in terms of a physical effect, whichis similar to Friedel oscillations in metals, here observed in anonequilibrium setting. For steady-state phases within the sensitive region, n kis noisy, for kbelonging to the intervals of momentum where the dispersion relation allows four propagating modes. Thisnoise cannot be interpreted as a ﬁnite-size feature since n k, within these regions, does not converge to a thermodynamiclimit. Since the transverse correlation function is related tothe Fourier transform of n k, the pathologies of this function explain the nonexponential decay of transverse correlations inthe sensitive region. We have also analyzed the behavior of the entropy of a segment of the steady state with its length. As expected fora mixed state, the steady-state entropy is extensive and fol-lows its predicted semiclassical value. In the normal region,whenever the chemical potential lies within one of the bands,there is a logarithmic component that is reminiscent of thearea-law violation occurring in equilibrium gapless states. Asreported for other nonequilibrium setups [ 51], the logarithmic coefﬁcient depends on the discontinuities of n k. In the sensi- tive region, corrections to the extensive contribution turn outto be superlogarithmic. The analysis presented here generalizes our earlier ﬁndings on the transverse-ﬁeld Ising case to the anisotropic XYchain and, thus, extends the class of spin chains which displayfar-from-equilibrium critical behavior, reﬂected in a divergentcorrelation length, that is absent in equilibrium. Thus, thepresent work suggests that these ﬁndings may reﬂect com-mon features of a wide class of quantum statistical models.A common feature of the models discussed in the presentcontext is their equivalence to interaction-free fermions un-der a Jordan-Wigner transformation. This naturally posesthe question if there exists a ﬁnite region in model spacearound these XYchains where similar nonequilibrium behav- ior ensues, or whether their nonthermal behavior is singular.The Jordan-Wigner transformation is limited to 1D systems.Yet, it would be desirable to understand how the nonequi-librium phases that we have identiﬁed generalize in higherdimensions. FIG. 12. (a) Repeat of the phase diagram of Fig. 4(b). The blue stars mark the parameters used in Fig. 13. (b) The persistent noise in the occupation number, computed for different system sizes, with the set of parameters indicated by the circled blue star on the phase diagram. In equilibrium, interaction-free or quadratic models are commonly associated with ﬁxed point behavior within aﬁeld-theoretic description of criticality. This enables one tocategorize a wide class of systems into universality classes,with respect to the ﬁxed points. Away from equilibrium, sucha categorization is not available. Our results thus offer a van-tage point for the construction of a wider class of models thatshare the same out-of-equilibrium behavior. A better under-standing of the universality of far-from-equilibrium criticalbehavior should prove beneﬁcial for the construction of aﬁeld-theoretic description of quantum critical matter far fromequilibrium. ACKNOWLEDGMENTS We gratefully acknowledge helpful discussions with T. Prosen, V . R. Vieira, and R. Fazio. P. Ribeiro acknowl-edges support by FCT (Grant No. UID/CTM/04540/2019). S.Kirchner acknowledges support by the NSFC (Grant No.11774307) and the National Key R&D Program of the MOSTof China (Grant No. 2016YFA0300202). S.C. acknowledgessupport from the National Key R&D Program of MOST China(Grant No. 2016YFA0301200), the National Science Associ-ation Funds (Grant No. U1930402), and NSFC (Grants No.11974040 and No. 1171101295). 035108-11PUEL, CHESI, KIRCHNER, AND RIBEIRO PHYSICAL REVIEW B 103, 035108 (2021) FIG. 13. Excitation number nkin the sensitive region for different phases. Each plot represents a different set of parameters {μL,μR} marked on the phase diagram in Fig. 12(a) disposed in the same order. We have computed it for a system size of N=500 sites. The vertical dashed lines represent either the momentum km2orkL;s e eF i g s . 3(c),5(a),a n d 5(b). APPENDIX A: METHOD DETAILED The single-particle density matrix in Eq. ( 16) is explicitly given by χ=1 2+/summationdisplay l=L,R/summationdisplay αβ\|α/angbracketright/angbracketleftβ\|/angbracketleft˜α\|[γlIl(λα,λ∗ β) −ˆγlIl(−λα,−λ∗ β)]\|˜β/angbracketright, (A1) where Il(z,z/prime)=−1 πg(z−2ml)−g(z/prime−2ml) z−z/prime , with g(z)= ln{−isgn[Im( z)]z}, and the matrices γlare deﬁned in Eq. ( 13). Here we assumed that Kin Eq. ( 13)i s diagonalizable, having right and left eigenvectors \|α/angbracketrightand/angbracketleft˜α\| with associated eigenvalues λα. The energy drained to the left reservoir is Je= −i/angbracketleft[H,HL]/angbracketright, which equals the steady-state energy current in any cross section along the chain and thus can be obtained asaf u n c t i o no f χ. Explicitly, the energy ﬂow can be obtained as J e=−1 2Tr[Jrχ]∀r, given by Eq. ( 23), with Jr=−2ihJ[(1+S)\|r−1/angbracketright/angbracketleftr\|(1+S)−H.c.].(A2)The linear and nonlinear thermal conductivities, as well as other thermoelectric properties of the chain, are determinedbyJ e. Two-point correlation Let us further analyze the two-point correlation function in Eq. ( 21), which can also be found in terms of χ. To this end, we have extended the equilibrium expressions [ 31] to general nonequilibrium conditions, Cxx r,r/prime=det[i(2χ[r,r/prime]−1)]1 2, (A3) forr>r/prime+1, where χ[r,r/prime]is a 2( r−r/prime) matrix obtained as the restriction of χto the subspace in which PT rr/prime=/summationtextr−1 u=r/prime+1(\|u/angbracketright/angbracketleftu\|+\| ˆu/angbracketright/angbracketleftˆu\|)+\|r+/angbracketright/angbracketleftr+\|+\|r/prime −/angbracketright/angbracketleftr/prime −\|, with \|r±/angbracketright= (\|r/angbracketright±\| ˆr/angbracketright)/√ 2, acts as the identity, and \|r/angbracketrightand\|ˆr/angbracketright≡S\|r/angbracketrightare single-particle and hole states. APPENDIX B: EXCITATION NUMBER The intriguing results of the occupation number, computed in Sec. III D , deserve a more detailed analysis, as follows. As 035108-12NONEQUILIBRIUM PHASES AND PHASE TRANSITIONS … PHYSICAL REVIEW B 103, 035108 (2021) FIG. 14. (a) Correlation length of the Ising model ( γ=1) for the special case h/J=0.5 and approaching the critical point m2=1 from the disordered phase (C), with μR=0. The dots and continuous line compare the numerical result, given by Eq. ( 22), to the analytic result, given by Eq. ( C2), respectively. (b) Density of excitations ρ andρξ,g i v e nb yE q .( 30). The continuous line shows the analytic result for ρ, computed from Eq. ( C2)a sρ=ξ−1/2. In the analytic formulas, we have used the educated guess c=π/5!. discussed in the manuscript, the occupation number shows a noise behavior in some of the phases. The marks in Fig. 12(a) indicate the representative sets ( μLμR) for which the occu- pation numbers are given in Fig. 13, providing the reader with a complete view of all possible cases. Here we have setthe system in the sensitive region with transverse magneticﬁeld h/J=0.2 and anisotropy γ=0.5. The marks are at the points μ L,μR={ ± 2.7,±1.9,±1.3,+0.3o r−0.3}.As discussed in the manuscript, here we clearly see that the noise is present on the phases around the axis μL=μR, while it vanishes around the axis μL=−μR. The noise only appears for \|k\|>km2; see Fig. 3(c). In addition, it is persistent even in the thermodynamic limit, as shown in Fig. 12(b) . APPENDIX C: CORRELATION LENGTH at h/J=0.5 When h/J=0.5, it was observed that nk=π=0[22], and thus the derivation of Eq. ( 28) should be modiﬁed. Interest- ingly, around k=π, we ﬁnd that nkhas a leading term of the form nk/similarequalc(k−π)4, (C1) where the vanishing of the quadratic term and the value of ccould only be obtained numerically. In the limit of a small /Delta1kL,E q .( 26) yields ξ−1/similarequalc π/integraldisplay/Delta1kL −/Delta1kLx4dx=2c 5π/Delta1k5 L. (C2) Finally, from the expression of /Delta1kLin Eq. ( 27), we imme- diately see that the critical exponent is ν=5/2. 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PhysRevB.82.144426.pdf	Bridging frustrated-spin-chain and spin-ladder physics: Quasi-one-dimensional magnetism of BiCu 2PO 6 Alexander A. Tsirlin,1,Ioannis Rousochatzakis,2,†Deepa Kasinathan,1Oleg Janson,1Ramesh Nath,1,3Franziska Weickert,1,4 Christoph Geibel,1Andreas M. Läuchli,2,‡and Helge Rosner1 1Max-Planck-Institut für Chemische Physik fester Stoffe, 01187 Dresden, Germany 2Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany 3Indian Institute of Science Education and Research, Trivandrum 695016, Kerala, India 4Dresden High Magnetic Field Laboratory, Forschungszentrum Dresden-Rossendorf, 01314 Dresden, Germany /H20849Received 12 July 2010; revised manuscript received 24 September 2010; published 18 October 2010 /H20850 We derive and investigate the microscopic model of the quantum magnet BiCu 2PO6using band-structure calculations, magnetic susceptibility and high-ﬁeld magnetization measurements, as well as exact diagonaliza-tion /H20849ED /H20850and density-matrix renormalization group /H20849DMRG /H20850techniques. The resulting quasi-one-dimensional spin model is a two-leg antiferromagnetic ladder with frustrating next-nearest-neighbor couplings along thelegs. The individual couplings are estimated from band-structure calculations and by ﬁtting the magneticsusceptibility with theoretical predictions, obtained using full diagonalizations. The nearest-neighbor leg cou-pling J 1, the rung coupling J4, and one of the next-nearest-neighbor couplings J2amount to 120–150 K while the second next-nearest-neighbor coupling is J2/H11032/H11229J2/2. The spin ladders do not match the structural chains, and although the next-nearest-neighbor interactions J2andJ2/H11032have very similar superexchange pathways, they differ substantially in magnitude due to a tiny difference in the O-O distances and in the arrangement ofnonmagnetic PO 4tetrahedra. An extensive ED study of the proposed model provides the low-energy excitation spectrum and shows that the system is in the strong rung coupling regime. The strong frustration by the next-nearest-neighbor couplings leads to a triplon branch with an incommensurate minimum. This is furthercorroborated by a strong-coupling expansion up to second order in the inter-rung coupling. Based on high-ﬁeldmagnetization measurements, we estimate the spin gap of /H9004/H1122932 K and suggest the likely presence of anti- symmetric Dzyaloshinskii-Moriya anisotropy and interladder coupling J 3. We also provide a tentative descrip- tion of the physics of BiCu 2PO6in magnetic ﬁeld, in the light of the low-energy excitation spectra and numerical calculations based on ED and DMRG. In particular, we raise the possibility for a rich interplaybetween one- and two-component Luttinger liquid phases and a magnetization plateau at 1/2 of the saturationvalue. DOI: 10.1103/PhysRevB.82.144426 PACS number /H20849s/H20850: 75.50. /H11002y, 75.30.Et, 75.10.Jm, 71.20.Ps I. INTRODUCTION One-dimensional /H208491D/H20850spin systems are in the focus of the present-day research due to a range of unusual low-temperature properties governed by quantum effects. The primary 1D spin model is the uniform spin-1 2Heisenberg chain that has a peculiar gapless excitation spectrum.1Nu- merous model compounds and the large set of theoreticaltools in one dimension made extensive comparisons betweenexperiment and theory possible: for example, the universal scaling of spin excitations in the uniform spin- 1 2Heisenberg chain was proposed theoretically and later conﬁrmedexperimentally. 2A number of studies successfully extended the model by including interchain couplings and discussedthe trends for the ordering temperature depending on the to-pology and magnitude of interchain couplings. 3–5 Alterations in the chain topology lead to a dramatic change in the magnetic properties. For example, there areseveral options to switch from the gapless spectrum of the uniform spin-1 2chain to a gapped spectrum. The latter offers an exciting opportunity to close the spin gap by an externalmagnetic ﬁeld and to observe unusual phenomena, such asLuttinger liquid /H20849LL/H20850physics and the Bose-Einstein conden- sation of triplons in the gapless high-ﬁeld phase. 6The sim- plest way to introduce a spin gap into a 1D system is toalternate the exchange couplings along the chain.7Another option is the frustration of the chain by next-nearest-neighbor couplings. 8Finally, several chains can be joined into a spin ladder that shows a spin gap for an even numberof legs. 9Despite the relatively simple chain geometries, such models are rather difﬁcult to realize experimentally. There isstill no experimental observation of the LL phase in the al- ternating spin-1 2chain, and experimental examples of gapped frustrated spin chains are rare.10The quest for spin-ladder systems was more successful. For example, recently a re-markable mapping of high-ﬁeld properties onto the LLmodel in a /H20849C 5H12N/H208502CuBr 4compound was performed.11–13 Combining different features of the modiﬁed chain topol- ogy /H20849alternation, frustration, and coupling into a ladder /H20850, one can achieve further interesting properties. For example, frus-trated spin chains with alternating nearest-neighbor cou-plings are predicted to exhibit a magnetization plateau for acertain range of model parameters. 14However, this predic- tion has never been tested experimentally due to the lack ofproper model compounds. The problems with ﬁnding experi- mental realizations of certain spin models call for an alterna-tive approach: the investigation of complex 1D models,stimulated by real materials. In the following we show that the recently discovered spin- 1 2compound BiCu 2PO6closely corresponds to an interesting quasi-1D spin model combin-PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 1098-0121/2010/82 /H2084914/H20850/144426 /H2084915/H20850 ©2010 The American Physical Society 144426-1ing all the three aforementioned features: frustration, spin- ladder geometry, and alternation of next-nearest-neighbor ex-change couplings. Despite previous experimental and computational studies, 15–17the microscopic model of BiCu 2PO6remains controversial. To resolve this controversy, we apply a rangeof state-of-the-art computational techniques that reveal anaccurate spin model and allow for a precise comparison withthe experimental results. First, we analyze the crystal struc-ture and outline the previous reports in Sec. II. After a brief description of the methods /H20849Sec. III/H20850, we proceed to exten- sive band-structure calculations, derive a consistent spinmodel, and discuss the nontrivial implementation of thismodel in the crystal structure of BiCu 2PO6/H20849Sec. IV/H20850. In Sec. V, we report the magnetic susceptibility and the high-ﬁeld magnetization measurements that challenge the proposedspin model and unambiguously measure the spin gap. Fi-nally, we perform model simulations, investigate the micro-scopic physics of BiCu 2PO6at low energies /H20849Sec. VI/H20850and in the presence of magnetic ﬁeld /H20849Sec. VII /H20850, and conclude our study with a brief discussion and summary in Sec. VIII. II. CRYSTAL STRUCTURE AND MAGNETIC PROPERTIES The crystal structure of BiCu 2PO6/H20849Fig. 1/H20850shows pro- nounced 1D features with complex ribbons running along thebdirection.18Each ribbon is formed by dimers of edge- sharing CuO 4plaquettes. The plaquettes of the neighboring dimers share corners /H20849oxygen sites /H20850while the next-nearest- neighbor dimers are additionally connected by PO 4tetrahe- dra. The spatial arrangement of the magnetic Cu atoms fea-tures both the spin-ladder and frustrated-spin-chaingeometries /H20849see Figs. 1and4/H20850. The stacking of the dimers reminds of the spin ladder with the leg coupling J 1and the rung coupling J3.19Yet, the interactions J1follow a zigzag pattern and form a frustrated spin chain, once the couplingsbetween next-nearest neighbors are considered. The situationis further complicated by the two inequivalent Cu positions,leading to inequivalent next-nearest-neighbor couplings J 2 andJ2/H11032.20 The complex crystal structure of BiCu 2PO6led to a con- troversy regarding the appropriate spin model of this com-pound. Koteswararao et al. 15emphasized the spin-ladder fea- ture of the structural ribbons and considered BiCu 2PO6as a system of J1−J3ladders that are coupled by the inter-ribbon interaction J4. This interpretation prevailed in further studies, focused on the effects of doping.17,21–23However, band- structure calculations, reported by the same authors,15clearly showed sizable next-nearest-neighbor couplings J2and J2/H11032 that would inevitably frustrate the system. Although similar at a ﬁrst glance, Mentré et al.16sug- gested a somewhat different spin model. Using inelastic neu-tron scattering /H20849INS /H20850and band-structure calculations, they showed that the ladders are formed by the couplings J 1and J4while the intraribbon interaction J3is an interladder cou- pling. To ﬁt the INS data, Mentré et al. also had to include the next-nearest-neighbor coupling J2but the difference be- tween J2andJ2/H11032could not be resolved. Experimentally, BiCu 2PO6is a spin-gap material with a singlet ground state /H20849no long-range ordering /H20850. The substitu- tion of Cu by non-magnetic Zn atoms destroys the spin gapand leads to a spin freezing. 17,21These features are fairly general and can be assigned to a range of simple 1D spinmodels /H20849alternating chain, frustrated chain, two-leg ladder /H20850. However, the experimental data cannot be described well byany of these models /H20849see also Sec. V/H20850. The previous reports 15,16evidence the combination of the ladder-type ge- ometry and the frustration by next-nearest-neighbor cou-plings. Yet, the precise way of this combination and, moreimportantly, the resulting physics remain unclear. III. METHODS To evaluate the individual exchange couplings in BiCu 2PO6, we performed scalar-relativistic density- functional theory /H20849DFT /H20850band-structure calculations using the full-potential local-orbital /H20849FPLO /H20850code /H20849version 8.00–31 /H20850.24The calculations were done in the framework of the local /H20849spin /H20850density approximation /H20851L/H20849S/H20850DA /H20852, employing the exchange-correlation potential by Perdew and Wang.25 The symmetry-irreducible part of the ﬁrst Brillouin zone wassampled by a mesh of 512 k-points for the crystallographic unit cell and 64 k-points for the supercells. Superexchange couplings in insulating Cu +2compounds are intimately related to strong electronic correlations thatJ3 b c Cu2aJ4J2’J2 J2 J4J1 J2’J1 Bi Cu1 FIG. 1. /H20849Color online /H20850Crystal structure of BiCu 2PO6with rib- bons comprising CuO 4plaquettes and PO 4tetrahedra /H20849top /H20850and the spin model /H20849bottom /H20850. Open and shaded circles denote the two in- equivalent Cu positions while the larger dark circles label the Biatoms. More details on the structure are shown in Fig. 4. The model is of the spin-ladder type and comprises four inequivalent cou-plings: the leg coupling J 1, the rung coupling J4, and the frustrating next-nearest-neighbor leg couplings J2and J2/H11032. Note that the two legs of the ladder reside on different structural ribbons.TSIRLIN et al. PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-2cannot be properly treated within L /H20849S/H20850DA. To account for the correlation effects, we used two approaches. First, wemapped the half-ﬁlled LDA Cu 3 dbands via an effective one-band tight-binding model onto a Hubbard model. Then,antiferromagnetic /H20849AFM /H20850exchange integrals were derived from the expression of the second-order perturbation theory.This procedure is referred below as the model approach .I n the second /H20849supercell /H20850approach, the correlation effects were treated in a mean-ﬁeld approximation within the band-structure calculations by applying the LSDA+ Umethod. 26 The on-site Coulomb repulsion parameter Udwas varied in the 6–8 eV range27–30while the on-site exchange parameter Jdwas ﬁxed to 1 eV . Total energies for different types of collinear magnetic ordering were obtained within the crystal-lographic unit cell and the two supercells, doubled along theborcdirections. The calculated energies were mapped onto a Heisenberg model, and individual exchange couplings werederived. More details on the computational procedure aregiven in Sec. IV. The resulting spin model was compared to the experimen- tal results from magnetic susceptibility and high-ﬁeld mag-netization measurements. Powder samples of BiCu 2PO6were prepared by ﬁring a stoichiometric mixture of Bi 2O3/H2084999.9% purity /H20850, CuO /H2084999.99% purity /H20850, and NH 4H2PO4/H2084999.9% pu- rity /H20850in air. The mixtures were ﬁrst annealed at 400 °C for 10 h and then at 850 °C for 40 h with one intermediategrinding. The resulting samples were single-phase, as con-ﬁrmed by x-ray diffraction /H20849STOE STADI-P diffractometer, CuK /H92511radiation, transmission geometry /H20850. The magnetic sus- ceptibility was measured in ﬁelds up to 5 T in the tempera-ture range 2–700 K using a Quantum Design MPMS super-conducting quantum interference device magnetometer. High-ﬁeld magnetization measurements were performed at Hochfeld-Magnetlabor Dresden at 1.4 K temperature inﬁelds up to 60 T using a pulsed magnet. Details of the mea-surement technique are given in Ref. 31. The curves mea- sured on increasing and decreasing ﬁeld coincided, indicat-ing the lack of any irreversible effects upon magnetization ofthe sample. Thermodynamic properties of the BiCu 2PO6spin model were calculated by a full diagonalization for ﬁnite latticeswith N=16 and 20 sites and periodic boundary conditions. To obtain the low-energy excitations, we performed exactdiagonalizations /H20849ED /H20850using the Lanczos algorithm that al- lowed to extend the system size up to N=36. The results are well converged with respect to the system size even for N =16 and 20, thus the ﬁnite-size effects for the spin modelunder consideration are relatively small. To obtain the mag-netization process of BiCu 2PO6we have used, in addition to ED, the density matrix renormalization group /H20849DMRG /H20850 /H20849Refs. 32and 33/H20850method with open boundary conditions with up to 128 rungs. Further details are given in Secs.V–VII. IV . DERIV ATION OF THE SPIN MODEL Spin models with exchange couplings derived from DFT have been previously reported in Refs. 15and16. However, the analysis remains incomplete since the two inequivalentnext-nearest-neighbor couplings /H20849between crystallographi- cally different Cu sites /H20850were considered to be equivalent. In the following, we apply two complementary approaches thatevaluate all the relevant exchange integrals and establish themicroscopic model. Additionally, we analyze in detail thestructural features that cause the unusual implementation ofthe ladder-type spin lattice in BiCu 2PO6. A. LDA and model approach Figure 2shows the LDA density of states /H20849DOS /H20850of BiCu 2PO6. The valence band spectrum is formed mainly by copper 3 dand oxygen 2 porbitals, with a sizable contribution from phosphorous 3 porbitals below −3 eV. The states above −0.6 eV are formed by the Cu 3 dx2−y2orbital, in agreement with the expected ligand-ﬁeld splitting.34The shapes and positions of the bands close to the Fermi level/H20849E F/H20850are somewhat different from the Nth-order mufﬁn-tin orbital /H20849NMTO /H20850result of Ref. 15, where the Cu 3 dx2−y2 bands are separated from the lower-lying bands. This differ- ence represents a known shortcoming of the NMTOmethod. 35To check our ﬁndings, we repeated the calculation using the full-potential code WIEN2K . The resulting band structure is in excellent agreement to that from FPLO. Irre-spective of the computational method, the LDA energy spec-trum is metallic due to the underestimation of the correlationeffects in this approximation. Experimentally, the green-colored BiCu 2PO6is a magnetic insulator. The insulating be- havior is readily reproduced by the LSDA+ Ucalculations /H20849see Sec. IV B /H20850. Eight Cu atoms in the crystallographic unit cell of BiCu 2PO6give rise to eight 3 dx2−y2bands /H20849Fig.3/H20850. We ﬁrst ﬁt these bands with a tight-binding model and extract the hop- ping parameters ti/H20849Table I/H20850. The ﬁtting procedure involves Wannier functions /H20849WF /H20850centered on Cu sites.36The applica- tion of the WF technique leads to a reliable ﬁtting despite theslight overlap with the lower-lying bands. We are also able to resolve the couplings J 2andJ2/H11032that correspond to the same-4 -2 0 2 4 Ener gy(eV)010203040DOS ( states /eV/cell)total Cu O PEF FIG. 2. /H20849Color online /H20850Total and site-projected DOS obtained from LDA. The vertical line at zero energy denotes the Fermi levelE F. The bands near the Fermi level primarily comprise Cu and O states. The shading in the plot denotes the Cu 3 dstates while the dashed line represents the O 2 pstates.BRIDGING FRUSTRATED-SPIN-CHAIN AND SPIN- … PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-3Cu-Cu vector /H208490,1,0 /H20850but refer to different Cu sites in the structure /H20849Fig. 1/H20850. The hoppings are in agreement with the apparent features of the band structure. We ﬁnd strong dis-persion along the /H9003-Y,/H9003-Z, X-S, and U-R directions which correspond to the crystallographic bc-plane with the cou- plings J 1,J2,J2/H11032,J3, and J4. The dispersions along the other directions are less pronounced, indicative of a quasi-two- dimensional /H208492D/H20850nature of this system. The hoppings are then introduced into a Hubbard model with the effective on-site Coulomb repulsionU eff=4.5 eV.27–29,37In the limit of strong correlations /H20849ti /H11270Ueff/H20850and in the half-ﬁlling regime, the low-lying excita- tions of the Hubbard model are described by a Heisenberg Hamiltonian comprising AFM exchanges JiAFM=4ti2/Ueff. The resulting JiAFMvalues are listed in Table I. The maxi- mum long-range hoppings tlbeyond t1−t4amount to 30 meV , thus leading to JlAFM/H1102110 K. Since the leading ex- change couplings amount to 150–250 K, the minimal micro-scopic spin model can be restricted to ﬁve interactions: J 1,J2,J2/H11032,J3, and J4.A crucial fact to note at this juncture is the clear differ- ence in the strengths of J2AFMandJ2/H11032AFM. Geometrically, the hopping paths for these exchanges are rather similar /H20849Fig.4/H20850, and this structural feature led the authors of Refs. 15and16 to assume J2=J2/H11032. In our analysis, we ﬁnd that it is essential to treat these two exchanges independently, otherwise theband splittings at the /H9003point would not be reproduced cor- rectly /H20849i.e., one obtains four doubly degenerate bands with J 2=J2/H11032instead of the eight separate bands /H20850. Hence the frus- trating next-nearest-neighbor exchanges “alternate” along the baxis /H20849see Fig. 1/H20850with J2/H11032/H112290.5J2. A detailed analysis of this difference will be given in Sec. IV C . B. LSDA+ U The model approach allows to estimate all the exchange couplings and to select the leading interactions for the mini-mum microscopic model. This is especially important forcomplex compounds with numerous and nontrivial superex-change pathways, like in BiCu 2PO6. On the other hand, the model approach does not account for FM contributions thatare relevant for short-range interactions. 28,38To correct the leading couplings for the FM contributions, we use the su-percell approach. The total exchange integrals, consisting of the FM and AFM contributions, are listed in Table IIfor the physically reasonable range of the U dvalues and for the two double-counting-correction /H20849DCC /H20850schemes. The latter is widely believed to be a minor feature of the LSDA+ U method but our recent studies evidenced a sizable inﬂuenceof the DCC on the exchange integrals in the case of short-range interactions. 38–40 The DCC is an essential part of the LSDA+ Uapproach because a part of the on-site Coulomb repulsion energy iscontained in LSDA and has to be subtracted from the totalenergy, after the explicit /H20849mean-ﬁeld /H20850correction for the on- site Coulomb repulsion is included. The two most commoncorrections are the around-mean-ﬁeld /H20849AMF /H20850/H20849Ref. 41/H20850and the fully-localized-limit /H20849FLL /H20850. 42For spin-1 2magnetic insu- lators, the difference between AMF and FLL was commonlybelieved to be minor. 43By construction, FLL looks more appropriate for the strongly localized regime ti/H11270Ueff.44Yet, both AMF and FLL readily reproduce the insulating state ofBiCu 2PO6. For example, we ﬁnd the band gap Eg/H112292.4 eV and the magnetic moment of 0.81 /H9262BatUd=6 eV in AMF.45FLL yields a somewhat lower gap Eg/H112291.6 eV at the same Udvalue but the gap is readily increased up to 2.1 eV atUd=8 eV. Experimental estimates of Egare presently lacking. However, even the experimental input will hardlyresolve the ambiguity since the U dvalue cannot be estimated precisely. Then, the exchange couplings should be analyzedin more detail. AMF and FLL produce similar estimates for most of the couplings: J 1,J2,J2/H11032, and J4/H20849see Table II/H20850. However, the short-range interaction J3is highly sensitive to the choice of the DCC. AMF suggests J3to be a weak coupling /H20849either FM or AFM, depending on the Udvalue /H20850while FLL ranks J3as one of the leading AFM couplings, comparable to J1andJ2. The FLL values essentially reproduce the previously pub-lished results by Mentré et al. 16that were also obtainedTABLE I. Leading hoppings of the tight-binding model and the resulting AFM exchange couplings. The exchange pathways indi-cated in the ﬁrst column are explicitly depicted in Figs. 1and4. The AFM part of the exchange integral is obtained by mapping thetransfer integrals to an extended Hubbard model and eventually to aHeisenberg model using J iAFM=4ti2/Ueffwith Ueff=4.5 eV. PathsCu–Cu distance /H20849Å/H20850ti /H20849meV /H20850 ExchangeJiAFM /H20849K/H20850 t1 3.21 146 J1 221 t2 5.17 /H20851Cu/H208492/H20850/H20852 110 J2 125 t2/H11032 5.17 /H20851Cu/H208491/H20850/H20852 78 J2/H11032 63 t3 2.89 123 J3 157 t4 4.91 140 J4 203Γ X S Y Γ Z U R T-0.9-0.6-0.300.30.60.9Energy (eV)Cu dx2-y2 Cu WF LDA FIG. 3. /H20849Color online /H20850LDA band structure /H20849thin blue lines /H20850, the WF-based ﬁt of the tight-binding model /H20849bright orange dots /H20850, and the contribution of the Cu dx2−y2orbital /H20849dark purple dots /H20850. The high-symmetry k-path in terms of the reciprocal lattice parameters is as follows: /H9003/H208490,0,0 /H20850,X/H208490.5,0,0 /H20850,S/H208490.5,0.5,0 /H20850,Y/H208490,0.5,0 /H20850,/H9003, Z/H208490,0,0.5 /H20850,U/H208490.5,0,0.5 /H20850,R/H208490.5,0.5,0.5 /H20850, and T/H208490,0.5,0.5 /H20850. The bands are highly dispersive along X−S,Y−/H9003−Z, and U−Rwhich represent the leading interactions within the crystallographic bc plane and the quasi-2D nature of the system.TSIRLIN et al. PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-4within FLL but in a different band structure code. The model approach /H20849Table I/H20850evaluates JiAFM, hence the FM contribu- tions JiFM=Ji−JiAFMcan be calculated. Following this proce- dure, we ﬁnd a simple microscopic argument that supportsthe AMF results with weak J 3. Both J1and J3arise from Cu-O-Cu superexchange with different angles at the oxygenatoms: 112.2° and 92.0°, respectively /H20849see the top left panel of Fig. 4/H20850. According to the Goodenough-Kanamori rules, 46 the nearly 90° superexchange of J3should yield the largest FM contribution. This conclusion conforms to the AMF re- sults with J1FM=−45 K and J3FM=−135 K at Ud=6 eV. The FLL results are opposite, J1FM=−36 K and J3FM=−16 K. As Udis increased up to 8 eV , all the couplings are reduced while the qualitative difference persists: /H20841J3FM/H20841/H11022/H20841J1FM/H20841in AMF but /H20841J3FM/H20841/H11021/H20841J1FM/H20841in FLL. The above considerations suggest the exchange couplings from AMF as a more reliable estimate for BiCu 2PO6. For relevant examples from other compounds with a simplermagnetic behavior, we refer the reader to Sec. IV C . Addi- tionally, we note that computational results for /H9252-Cu 2V2O7 /H20849Ref. 40/H20850and for several other Cu+2-compounds39also prefer AMF. Thus, we further rely on the AMF estimates and con-sider J 3as a weak coupling. The low value of J3compared to J3AFMreduces the 2D J1−J4model, obtained from the model approach, to a quasi-1D model, depicted in the bottom partof Fig. 1. This model basically follows the earlier proposal by Mentré et al. 16We ﬁnd a two-leg spin ladder with the legcoupling J1, the rung coupling J4, and the next-nearest- neighbor frustrating couplings J2andJ2/H11032along the legs. Yet, there are two important differences to be emphasized. First,the two next-nearest-neighbor couplings are inequivalent andfairly different. The J 2coupling connecting the Cu2 sites is twice as large as the coupling J2/H11032between the Cu1 sites /H20849see Tables Iand II/H20850. Second, we can safely establish the quasi-1D nature of the spin model because the J3/J4ratio is below 0.2 /H20849compare to J3/J4=0.55–0.65 in Ref. 16/H20850. Both results are very important for understanding the material. The difference between J2andJ2/H11032clearly alters the spin lattice. The pronounced one-dimensionality allows to simu-late the behavior of the spin model on a quantitative level,despite the presence of the strong frustration that narrows therange of applicable simulation techniques. Before turning tothe experiments and simulations /H20849Sec. V/H20850, we will further discuss the nontrivial implementation of individual exchangecouplings in the crystal structure of BiCu 2PO6and provide further support for the proposed spin model. C. Structural aspects of the magnetic exchange The interactions J1andJ3run between corner-sharing and edge-sharing CuO 4plaquettes, respectively /H20849top left panel of Fig. 4/H20850. This geometry suggests Cu-O-Cu superexchange as the leading mechanism of the coupling and the angle at theoxygen atom as the key structural parameter determining theexchange integral. Following the Goodenough-Kanamorirules, 46we ﬁnd that J3with the Cu-O-Cu angle of 92.0° is weakly AFM or even FM /H20849see Table II/H20850. The pathway of J1 reveals the sizably larger angle of 112.2° and, consequently, a sizable AFM superexchange. A similar superexchange sce-nario is found in the mineral dioptase Cu 6Si6O18·6H 2O /H20849green dioptase /H2085038and, presumably, in its anhydrous counter- part /H20849black dioptase /H20850. The spin lattice of dioptase comprises the AFM coupling Jcbetween corner-sharing CuO 4 plaquettes /H20849the Cu-O-Cu angle amounts to 107.6° and 110.7° for green and black dioptase, respectively /H20850and the FM cou- pling Jdbetween edge-sharing plaquettes /H2084997.4° and 97.3°, respectively /H20850. More speciﬁcally, Jc=78 K and Jd=−37 K in green dioptase.38The nature of the exchange couplings in theCu1 O1 O3O3O4b bO1 O2 O4 Cu22.55 Ao2.63 Ao J2 J1J2’ /c106’/c106112.2o92.0o J3J1 2.75 Ao O2 O1BiJ4 FIG. 4. /H20849Color online /H20850Parts of the crystal structure showing the details of individual superexchange pathways as well as the spin-ladder /H20849top left panel /H20850and the frustrated-spin-chain /H20849right panel /H20850features. The middle panel depicts the difference in the positions of the PO 4 tetrahedra for the couplings J2andJ2/H11032. Curved arrows denote the rotations of the tetrahedra in the ﬁctitious model structures /H20849see text for details /H20850. The right panel shows the difference in the O1-O1 distances for J2andJ2/H11032. TABLE II. Total exchange couplings /H20849in Kelvin /H20850obtained from the LSDA+ Ucalculations. The Udvalue /H20849in electron V olt /H20850denotes the Coulomb repulsion parameter of LSDA+ U. The last column lists the double-counting correction scheme: around-mean-ﬁeld/H20849AMF /H20850or fully-localized-limit /H20849FLL /H20850. U d J1 J2 J2/H11032 J3 J4 6 176 170 90 22 154 AMF 7 145 127 73 −2 113 AMF8 109 99 58 −15 85 AMF6 185 166 93 141 243 FLLBRIDGING FRUSTRATED-SPIN-CHAIN AND SPIN- … PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-5dioptase lattice is conﬁrmed by the magnetic structure that was directly investigated by neutron diffraction.47,48Addi- tionally, our recent computational study of green dioptaseconﬁrms the assignment of the exchange couplings andyields a consistent interpretation for all available experimen-tal data. 38The reference to the closely related superexchange scenario in dioptase should be taken as an additional argu-ment for the weakness of J 3and the resulting quasi-1D char- acter of BiCu 2PO6. In fact, one can ﬁnd further examples supporting the pro- nounced difference between J1and J3. Numerous cuprates with chains of edge-sharing plaquettes are experimental re-alizations of frustrated spin chains with FM nearest-neighborcouplings. Such FM couplings arise from the Cu-O-Cu angleclose to 90° and typically range from −100 to −300 K foroxide compounds /H20849e.g., Li 2CuO 2,L i 2CuZrO 4/H20850.49,50In BiCu 2PO6,J3FMis smaller due to the folded arrangement of the plaquettes. Nevertheless, the pronounced FM contribu-tion reduces the total exchange to a weak coupling, either FM or AFM, despite the sizable AFM contribution of J 3AFM =176 K /H20849cf. Table I/H20850. The leg coupling J1appears for the twisted conﬁguration of corner-sharing plaquettes /H20849see Fig.4/H20850with the Cu-O-Cu angle of 112.2°. A similar conﬁgu- ration is found in AgCuVO 4, where the angle amounts to 112.7°, and a pronounced AFM exchange coupling J /H11229300 K is found.29Thus, our estimates of J1andJ3are in line with the experience regarding other Cu compounds withﬁrmly established microscopic models. All the above arguments support the quasi-1D model with weak J 3. In the following, we use this model as a working hypothesis to interpret the magnetic behavior of BiCu 2PO6. The quasi-1D model captures the essential physics of thematerial, although certain features may require the extensionof the model toward including J 3or anisotropy effects /H20849see Secs. VIandVIII /H20850. Taking J3as a weak interaction, we ﬁnd J4to be the leading coupling along the cdirection. This coupling runs between the CuO 4plaquettes of neighboring ribbons. The bonding between the ribbons arises from Bi cations /H20849bottom left panel of Fig. 4/H20850, yet Bi does not give any sizable contri- bution to the states near the Fermi level. Therefore, we as-sign J 4to the Cu-O-O-Cu superexchange with the double O-O contact of 2.75 Å. Similar couplings between the dis-connected copper plaquettes have been reported for/H20849CuCl /H20850LaNb 2O7and Bi 2CuO 4.39,51,52Due to the large spatial separation of the Cu atoms /H208494.91 Å /H20850, a sufﬁciently strong interaction arises for speciﬁc conﬁgurations of the ligand or-bitals only /H20849see Ref. 51for an instructive example /H20850. This explains the strong inter-ribbon coupling along the cdirec- tion, in contrast to a very weak coupling between the struc-tural ribbons along awhere the shortest Cu-Cu distance is 4.85 Å. Finally, we address the most puzzling feature of BiCu 2PO6, the next-nearest-neighbor couplings J2and J2/H11032. While the other couplings can be tentatively assigned after acareful analysis of the superexchange pathways, the sharp difference between J 2and J2/H11032remains unexpected. The Cu-Cu distances for the two couplings are the same andamount to 5.17 Å, the lattice parameter along the bdirec- tion. On the other hand, J 2andJ2/H11032correspond to different Cupositions and are inequivalent by symmetry . Band-structure calculations within the model and LSDA+ Uapproaches consistently suggest that J2/H11032/J2/H112290.5 /H20849see Tables IandII/H20850. The couplings J2and J2/H11032run between the copper plaquettes, joined by another plaquette via O1 and by a PO 4 tetrahedron via O2 /H20849see the right panel of Fig. 4/H20850. Thus, two different Cu-O-O-Cu channels are available. Despite the verysimilar Cu-O distances and Cu-O-O angles, there is a pro-nounced difference in the O1-O1 distances: 2.55 Å for J 2 /H20851the edge of the Cu1 plaquette /H20852and 2.63 Å for J2/H11032/H20851the edge of the Cu2 plaquette /H20852. The shorter O1-O1 distance should lead to the stronger coupling J2, in agreement with the com- putational result J2/H11032/H11021J2. At ﬁrst glance, the O2 channel looks completely identical, because the O2-O2 distance is con-strained by the edge of the PO 4tetrahedron /H208492.56 Å /H20850. Nev- ertheless, this channel also contributes to the difference be- tween J2andJ2/H11032. To get a deeper insight into the mechanism of the next- nearest-neighbor interactions, we inspect the Wannier func-tions for the Cu1 and Cu2 sites. Each WF comprises aCu 3 d x2−y2orbital along with the /H9268-type porbitals of the neighboring oxygens O1 and O2 /H20849Fig. 5/H20850. We also ﬁnd small O2O1 O1 O1O1 O2 O2 O2 O2 O4 PPO4O2 O3 O3PO4 O3Cu2Cu2 FIG. 5. /H20849Color online /H20850Wannier functions /H20849magnetic orbitals /H20850 centered on Cu2 sites. Each orbital comprises the Cu 3 dx2−y2atomic orbital, large O1 and O2 /H9268pcontributions, and a smaller O3 /H9268p contribution. /c106(deg)140140 145145 150150 155O3 of Cu2 O4 of Cu1 WF coeff.ofO155 1600.010.0250100150 Ji(K)160/c106’(deg) J2’J2 FIG. 6. /H20849Color online /H20850Exchange integrals J2and J2/H11032and the contribution of the second-neighbor oxygens /H20849O3, O4 /H20850to the Wan- nier functions, depending on the position of the PO 4tetrahedron. The dashed vertical line shows the angles in the BiCu 2PO6 structure.TSIRLIN et al. PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-6but signiﬁcant, /H9268-contributions from second-neighbor oxy- gens O3 and O4 for the Cu2 and Cu1 WFs, respectively.These “tail” contributions arise from the speciﬁc orientationof the PO 4tetrahedra: one of the O-O edges aligns along the Cu-O2 bond, i.e., the Cu2-O2-O3 /H20849/H9272/H20850and Cu1-O2-O4 /H20849/H9272/H11032/H20850 angles approach 180°. Indeed, we ﬁnd /H9272=140.4° and /H9272/H11032 =159.1° in agreement with the smaller O3 contribution ofabout 1.0%, compared to 1.7% for O4. Although the tail features of the WFs look tiny, they have a strong effect on the exchange couplings. To probe this, weconstructed ﬁctitious model structures by rotating the PO 4 tetrahedra around the O2-O2 edge. Since the tetrahedra werekept rigid, only the /H9272and/H9272/H11032angles were varied while other geometrical parameters remained constant.53We found that the position of the tetrahedron leads to a dramatic change in the absolute values of J2andJ2/H11032. As the /H9272angle is increased toward 180°, the O3 contribution gets larger, and J2conse- quently decreases /H20849Fig. 6/H20850. The rotation of the tetrahedra by 15° makes J2and J2/H11032equal while the further rotation will switch the system to the J2/H11032/H11022J2regime. The WF of Cu1 and the interaction J2/H11032are less sensitive to the variation in the /H9272/H11032 angle within the studied angle range.54 Our analysis shows that the structural features beyond the CuO 4plaquettes have a sizable effect on the exchange cou- plings in Cu+2compounds. In BiCu 2PO6, the tails of the WFs on the second-neighbor oxygens have 90° orientation andshould then reduce the AFM coupling /H20849see Fig. 5/H20850. This un- expected interference of the magnetic orbitals on the second-neighbor oxygen site is one of the microscopic reasons for the observed difference between J 2andJ2/H11032. It is worth noting that the role of non-magnetic side groups was emphasizedtheoretically long ago 55but is often not taken into account adequately in a quantitative description. Here, we haveshown that the oxygen orbitals play the key role while thephosphorous atom simply “holds” the four oxygens of thetetrahedron together. There is no appreciable phosphorouscontribution at the Fermi level, and its contribution to theWF’s is also minor /H20849below 0.1% /H20850. This general mechanism, involving interacting oxygen atoms, has been recently foundin vanadium phosphates 56and deserves further investigation in the compounds comprising other transition metals. V . EXPERIMENTAL RESULTS A. Magnetic susceptibility The temperature dependence of the magnetic susceptibil- ity is shown in Fig. 7and resembles closely the data from Ref. 15. We ﬁnd a broad maximum at Tmax/H9273/H1122962 K, indica- tive of the predominantly AFM clow-dimensional and/orfrustrated behavior. The sharp decrease in the susceptibility below T max/H9273is a signature of the spin gap. In the low- temperature region, the 0.1 T data show a weak upturn below5 K. This upturn is largely suppressed in the ﬁeld of 5 T andcan therefore be assigned to a paramagnetic contribution ofdefects/impurities. Above 10 K, the susceptibility is ﬁeld-independent in the studied ﬁeld range /H92620H/H113495T . Above 200 K, the system approaches the Curie-Weiss re- gime. In order to improve previous studies,15,16we measured the susceptibility at high temperatures up to 700 K and ﬁttedthe data above 300 K with the expression /H9273=C T+/H9258, /H208491/H20850 where /H9258is the Curie-Weiss temperature and C =NA/H20849g/H9262B/H208502S/H20849S+1/H20850//H208493kB/H20850is the Curie constant. Our ﬁt gives C=0.447 /H208491/H20850emu K /mol Cu, and /H9258=181 /H208491/H20850K/H20849see the inset of Fig. 7/H20850. Fitting the data with an additional temperature- independent /H92730term leads to a small /H92730, therefore, we ne- glect this term in further analysis. We establish the predomi-nant AFM nature of the exchange interactions with an energyscale of about 200 K. The Cvalue corresponds to an effec- tive moment of 1.89 /H208491/H20850 /H9262B, slightly above the ideal spin-1 2 value of 1.73 /H9262Band rather typical for Cu+2compounds.5,28 Since Tmax/H9273//H9258/H112290.3, strong frustration should be expected. For further analysis, we ﬁt the magnetic susceptibility us- ing our microscopic spin model. Koteswararao et al.15have shown that the data do not conform to the model of isolatednonfrustrated spin ladders. The introduction of interladdercouplings does not signiﬁcantly improve the description. 57 Therefore, realistic models with frustrating next-nearest-neighbor couplings have to be considered. Mentré et al. 16 used a frustrated J1−J2−J4spin model and ﬁtted the data with J1/H11229140 K, J2/H112290.5J1, and J4/H112290.4J1/H20849see also Ref. 23/H20850 but this model did not take into account the difference be- tween J2andJ2/H11032. Here we employ the J1−J2−J2/H11032−J4model /H20849Fig. 1/H20850to ﬁt the experimental magnetic susceptibility. This 1D frustratedspin model can be treated by exact diagonalizations for ﬁnitelattices or by renormalization-group techniques. The formerturns out to be appropriate for the present problem due to thesmall ﬁnite-size effects and will be used here for the suscep-tibility ﬁt. The unit cell comprises four inequivalent Cu 2+ ions, hence the number of sites Nin the ﬁnite cluster should be a multiple of four. To ﬁt the experimental data, we ﬁrstapproximate our model by the following set of parameters J 1=J2,J2/H11032=1 2J2, and J4=3 4J1, according to Table II. The simu- lations yield the reduced susceptibility /H9273/H11569, which can be ﬁt- ted to the experimentally obsrved /H9273using200 400 Temperature (K)600Experiment Fit /c1090H=5T1.5 1.0 0.5 0.0/c99(10 emu/mol Cu)/c453 03508001/ (mol Cu/emu)/c99 1600 700 0 FIG. 7. /H20849Color online /H20850Magnetic susceptibility of BiCu 2PO6 measured in the applied ﬁeld /H92620H=5 T and the ﬁt of the 1D spin model with g/H112292.16, J1/H11229140 K, J2=J1,J2/H11032=1 2J1, and J4=3 4J1 /H20849simulation for a ﬁnite lattice with N=20 sites /H20850. The inset shows the Curie-Weiss ﬁt above 300 K.BRIDGING FRUSTRATED-SPIN-CHAIN AND SPIN- … PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-7/H9273=NAg2/H9262B2 J1/H9273/H11569/H208492/H20850 with only two variable parameters: gandJ1. The simulations forN=16 and N=20 sites provide almost identical suscepti- bility curves. Hence, ﬁnite-size effects are negligible and oursimulations yield accurate results for the 1D spin model un-der consideration. Our optimal ﬁts yield J 1/H11229140 K and g/H112292.16. The ﬁtted gvalue is typical for Cu+2compounds29,58and also conforms to the effective magnetic moment of 1.89 /H9262Bwhich leads to g=2.18. The absolute value of J1is in remarkable agreement to the computational estimate of 100–150 K /H20849cf. Table II/H20850. The ﬁt follows the experimental data down to 100 K /H20849see Fig.7/H20850. At lower temperatures, we ﬁnd slight deviations from the experiment. For instance, the position of the susceptibil- ity maximum Tmax/H9273is overestimated and the theoretical curve lies slightly below the experimental data. This shows that ourmodel overestimates the spin gap /H9004. We shall return to this issue below in Sec. VI. We also tried to vary the ratios of exchange integrals and found several ﬁts of similar quality. In particular, the param- eter set from Ref. 16 /H20849J 1/H11229140 K, J2=J2/H11032/H112290.5J1, and J4 /H112290.4J1/H20850is also in agreement with the magnetic susceptibility data and yields a comparable g=2.145. However, this param- eter set does not account for the difference between J2and J2/H11032. Since, as shown above, this difference is evidenced by two different computational approaches and has a clear struc- tural origin, we regard the solution J1=J2,J2/H11032=1 2J2, and J4 =3 4J1as the microscopically justiﬁed parameter set for BiCu 2PO6. B. High-ﬁeld magnetization and the spin gap The low-energy physics of BiCu 2PO6is characterized by the presence of a spin gap /H9004. Previous estimates of /H9004, based on the magnetic speciﬁc heat15,17and the Knight shift,21,22 consistently suggested /H9004/H1122935 K. The INS data revealed a smaller gap of 2 meV /H20849about 23 K /H20850.16The observed discrep- ancy calls for the application of further experimental meth-ods, especially in light of the ambiguity of the speciﬁc heatand Knight shift estimates, which arises from the ﬁtting ex- pressions that depend on the character of the spin excitationsand, in particular, on the dimensionality of the system. High-ﬁeld magnetization data can provide a robust esti- mate of the spin gap. The magnetization process ofBiCu 2PO6is presented in Fig. 8. At low ﬁelds, the magneti- zation shows a weak linear increase with the ﬁeld until /H92620Hc/H1122922 T, where it bends upwards following a much steeper linear increase at higher ﬁelds.59The transition at Hc implies the closing of the spin gap and can be used for the numerical estimate of /H9004. Similar to Ref. 31, we take Hcas the point of the maximum curvature. We ﬁnd /H9004 =g/H9262B/H92620Hc/kB/H1122932 K, in good agreement to the previous estimate /H9004/H1122935 K obtained from the magnetic speciﬁc heat and the Knight shift data.15,17,21,22 The behavior of the magnetization for small ﬁelds needs to be discussed in more detail. In an ideal, SU /H208492/H20850invariant and defect-free gapped system, the magnetization should be zero below Hc/H20849see also Fig. 12below /H20850. Impurities give rise to a ﬁnite magnetization contribution but this should typi-cally saturate around 5 T at the present low temperature of1.4 K. Since the measured magnetization keeps increasing upto 22 T, we conclude that the weak linear ﬁeld dependenceforH/H11021H cis due to the presence of weak anisotropic inter- actions in BiCu 2PO6. One such anisotropy, which is known60 to give rise to a linear magnetization response in the gappedregime of similar ladder systems, is the Dzyaloshinksy-Moriya /H20849DM /H20850anisotropy. 61,62As explained in Ref. 60,a n isolated AFM dimer with a DM energy term of the formD·/H20849S 1/H11003S2/H20850admixes triplet excitations into the singlet ground state, and this gives rise to a uniform magnetizationresponse of the form m u/H11008D/H11003/H20849D/H11003B/H20850even far below the critical ﬁeld. There is also a staggered response in ﬁrst orderinDof the form m s/H11008D/H11003Bwhich can be detected by a local probe, such as NMR experiments. Similar features arise inthe spin-ladder Cu 2/H20849C5H12N2/H208502Cl4compound.60Hence, it is reasonable to expect that the linear response observed forBiCu 2PO6atH/H11021Hcstems from the presence of the DM anisotropy. For completeness, it is worth providing a brief discussion on the main DM vectors, based on the crystal symmetry ofBiCu 2PO6/H20849cf. Fig. 1/H20850. First of all, a DM anisotropy on each rung is allowed by symmetry since each rung comprises twoinequivalent Cu sites and thus the inversion symmetrythrough the middle of each rung is lacking. The translationalinvariance along the baxis /H20849with a period of two rungs /H20850 necessitates that the DM vectors are the same on every sec-ond rung. Furthermore, the fact that the acplane is a reﬂec- tion /H20849i.e., crystallographic mirror /H20850plane 63conﬁnes the DM vectors to the bdirection. There is ﬁnally a screw axis sym- metry along b/H20849translation along bby one rung, followed by aC 2rotation around the baxis /H20850which connects the sites of two consecutive rungs. This last symmetry necessitates thatthe DM vectors on the two consecutive rungs differ in sign.The DM terms are also expected for other, inter-rung cou-plings. Finally, we would like to point out that the measured magnetization data right above H cdo not show any square root singularity /H20849cusp /H20850as is typical for 1D systems with a quadratic branch of magnetic excitations above the ground004 Magnetization (arb. units)81216 10 20 30 Field (T) /c1090H40 50 60/c1090H=2 2TcT= 1.4 K FIG. 8. /H20849Color online /H20850Magnetization curve of BiCu 2PO6mea- sured in pulsed ﬁeld at T=1.4 K. The magnetization values are given in arbitrary units. The arrow shows the critical ﬁeld Hcwhere the spin gap is closed. The solid lines are guide for the eyes.TSIRLIN et al. PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-8state /H20849see also Fig. 12below /H20850. This is probably related to the presence of the DM interactions mentioned above and theinter-ladder coupling J 3, which are both expected to smooth out the singularity. In Sec. VIIbelow, we provide a more detailed theoretical picture for the magnetization process butﬁrst it is essential to understand the nature of the lowestmagnetic excitations in BiCu 2PO6. VI. LOW-ENERGY EXCITATIONS FROM EXACT DIAGONALIZATIONS We have performed an exact diagonalization study of the model Hamiltonian discussed above /H20849but without DM terms /H20850 with parameters J1=J2=1,J2/H11032=0.5, and J4=0.75, using ﬁnite lattices of N=12, 16, 20, 24, 28, 32, and 36 sites with peri- odic boundary conditions along the legs /H20849xaxis /H20850. The model is depicted in the upper panel of Fig. 9. Apart from transla- tions along the legs by 2 a, we also have two discrete spatial symmetries in this model. One is a reﬂection through any ofthe rungs /H20849P y/H20850and the other is a /H9266rotation /H20849C2z/H20850around the zaxis which is perpendicular to the plane of the ladder and passes through the center of a J1−J4rectangle. Instead of the latter, we can take the generator consisting of a translation bya, combined with a reﬂection along the axis crossing the middle of all rungs. A ﬁrst strong insight into physics of this model comes from a simple examination of the ground-state expectationvalues of various local energy terms /H20855s i·sj/H20856. Owing to the spatial symmetries of the problem, there are four inequiva-lent bonds only. These are the bonds associated with the four different exchange couplings J 1,J2,J2/H11032, and J4in each unit cell. The corresponding ground state expectation values, de- noted as e1,e2,e2/H11032, and e4, are provided in Table IIItogether with the total ground-state energy per site E/N=/H208492J1e1+J2e2+J2/H11032e2/H11032+J4e4/H20850/2. The latter shows only small ﬁnite-size variations for N/H1135016, which points to a very short correlation length.64More importantly, we observe a sizably large value for the spin-spin correlations on the rungs, e4/H11229−0.47, which is more than twice the values on the remaining bonds. Thisresult tells us that the system is in the strong rung coupling regime, despite the fact that the leg couplings J 1andJ2are comparable to the rung coupling J4. In Fig. 10, we have superimposed the low-energy excita- tions for each system size as a function of the allowed mo-mentum quantum numbers so that we obtain a clear pictureof the low-energy dispersion of the model. We observe thatthe lowest triplet /H20849total spin S=1/H20850excitations /H20849thick open symbols /H20850form a well-deﬁned /H20849coherent /H20850branch separated from the continuum by a ﬁnite gap for k/H114070.4 /H9266//H208492a/H20850. This branch has an incommensurate minimum at kmin /H112290.8/H9266//H208492a/H20850at/H9004ED/H112290.5J1. In addition to the lowest branch, we also ﬁnd a second branch which is degenerate with theﬁrst at k= /H9266//H208492a/H20850but this shifts quickly to higher energies into the continuum for k/H11021/H9266//H208492a/H20850. Before we discuss the main implications of these results with regard to BiCu 2PO6, we would like to provide a basic microscopic description of the excitation spectrum. To thisend we perform a perturbative expansion around the limit of isolated rungs J 1=J2=J2/H11032=0. We ﬁrst introduce the singlet and triplet states of a single rung with sites 1 and 2 as /H20841s/H20856 =/H20849/H20841↑↓/H20856−/H20841↓↑/H20856/H20850//H208812, /H20841t1/H20856=/H20841↑↑/H20856,/H20841t−1/H20856=/H20841↓↓/H20856, and /H20841t0/H20856=/H20849/H20841↑↓/H20856 +/H20841↓↑/H20856/H20850//H208812. The unperturbed ground state is the product state of singlets on all rungs. Excitations arise by promoting oneor more rungs into triplet states /H20841t m/H20856, with m=/H110061,0. The inter-rung couplings have two effects. The ﬁrst is that theyrenormalize the ground-state energy as well as the energiesin the one-triplon sector. The second is that they induce aﬁnite amplitude for nearest-neighbor and next-nearest-neighbor hoppings of triplons in the one-triplon sector. In-cluding the amplitude from all different processes and ex-ploiting the translational invariance by 2 a, one ﬁnds two separate bands of one-triplon excitations due to the fact thatwe have two rungs per unit cell in the model. Their energiesrelative to the renormalized ground-state energy are given by E /H9251,/H9252/H208492/H20850/H20849k/H20850=Ak/H11006/H20841Bk/H20841, with Ak=J4+12J12+3 /H20849J2+J2/H11032/H208502−4 /H20849J2−J2/H11032/H208502 16J4+/H20875J2+J2/H11032 2 +/H20849J2−J2/H11032/H208502−2J12 8J4/H20876cosk−/H20849J2+J2/H11032/H208502 16J4cos 2 k,a xyJ1 J42J 2J’ FIG. 9. /H20849Color online /H20850Top panel: the actual structure /H20849disregard- ing the buckling /H20850of the present model, and in the lower panel its topologically equivalent version obtained by ﬂipping the two sitesof every second rung.TABLE III. The ground-state expectation values of the four dif- ferent bond strengths /H20855si·sj/H20856per unit cell and the total energy per siteE/Nin units of J1. Ne 1 e2 e2/H11032 e4 E/N 12 −0.21568 −0.17868 −0.17162 −0.42632 −0.50780 16 −0.17466 −0.19499 −0.18411 −0.47136 −0.4949420 −0.16633 −0.21940 −0.20928 −0.46032 −0.5009624 −0.18409 −0.18626 −0.17460 −0.47234 −0.4980028 −0.17210 −0.20336 −0.19237 −0.47127 −0.4986032 −0.17683 −0.19716 −0.18571 −0.47132 −0.49858BRIDGING FRUSTRATED-SPIN-CHAIN AND SPIN- … PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-9Bk=J1 2/H208491+e−ik/H20850−J1/H20849J2+J2/H11032/H20850 8J4/H208491+e−ik+eik+e−2ik/H20850. These second-order dispersions are shown in the lower panel of Fig. 10. Although its prediction for the spin gap is more than twice higher than the exact value /H20849shown in the upper panel /H20850, the second-order perturbation theory captures well the position of the minimum and the overall shape of thedispersion. Next, we would like to comment that the degeneracy of the two branches at k=0 is not a generic feature of the exact dispersions but an accidental feature of the second-order ex-pression for the given values of the exchange integrals. Inhigher orders of perturbation theory or for slightly differentparameter values, this degeneracy will be lifted. In contrast,the degeneracy of the two branches at k= /H9266//H208492a/H20850is a generic feature and persists to all orders as seen in the exact spectra.The reason behind this is the presence of the discrete sym-metry generator mentioned above /H20849translation by afollowed by a reﬂection through the middle of all rungs /H20850. To see this, we may start from the upper panel of Fig. 9and exchange the two sites of every second rung without altering the topologyof the model. This gives the equivalent model shown in thelower panel of Fig. 9which has period aand not 2 a.T o elucidate this point, we may repeat the strong-coupling ex-pansion in this alternative symmetry framework. To this end,we must take into account the extra negative signs that arisefrom the antisymmetry of the singlet rung wave functionwhen ﬂipping the two sites of every second rung. In terms ofthe new momenta, we now obtain a single one-triplon exci-tation band with energy dispersion E/H20849k/H20850=J 4+12J12+3 /H20849J2+J2/H11032/H208502−4 /H20849J2−J2/H11032/H208502 16J4+c1cosk +c2cos 2 k+c3cos 3 k+c4cos 4 k, /H208493/H20850 where c1=−J1+J1/H20849J2+J2/H11032/H20850 4J4,c2=J2+J2/H11032 2+/H20849J2−J2/H11032/H208502 8J4−J12 4J4,c3=J1/H20849J2+J2/H11032/H20850 4J4, andc4=−/H20849J2+J2/H11032/H208502 16J4. This dispersion is shown in the lower panel of Fig. 11. It is clear that by folding this back into the Bril- louin zone /H20851−/H9266/2a,/H9266/2a/H20852we shall obtain the two branches shown before in the lower panel of Fig. 10. It is also evident in this representation that the incommensurate nature of thedispersion arises already in ﬁrst order and is dictated by the frustrated couplings J 2and J2/H11032which appear in the leading term in the above expression for c2. For completeness, we present the exact diagonalization results in the modiﬁed setup /H20849upper panel of Fig. 11/H20850. Again, the overall shape of the lowest dispersion and the position ofthe minimum are in agreement with the prediction of thestrong-coupling expansion shown in the lower panel. An interesting feature which becomes better visible in the representation of Fig. 11is the presence of a number of low- lying singlets for momenta close to k= /H9266/a. These excita- tions can be understood as a singlet bound state of twotriplons. Such excitations could be captured by optical ex-periments, e.g., phonon-assisted infrared absorption. Singletbound states of two triplons have been observed using suchtechniques in cuprate ladders. 65In a broader context, thelow-lying singlet at k=/H9266/acan be considered as a singlet mode going soft at the transition to a dimerized phase withdimers forming along the legs. 66In the model considered, this scenario might occur as the rung coupling J4is reduced further. Let us now discuss the implications of the above ﬁndings for BiCu 2PO6. Taking J1/H11229140 K from the ﬁt of the suscep- tibility we obtain for the spin gap /H9004ED/H112290.5J1/H1122970 K which is almost twice the value obtained from the high-ﬁeld mag-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.811.21.41.61.8 12 16 20 24 28 32 36 11.522.53 E/J1E/J1 ka/( /2 )/c112JJ J J1= = 1, = 0.5, = 0.75 ’22 4 Eka(2)()Ekb(2)() 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.5one-triplon energy dispersion from pert. theory (up to 2nd order) ka/( /2 )/c112 FIG. 10. /H20849Color online /H20850Top: superimposed low-energy disper- sions from exact diagonalizations on systems with N=12, 16, 20, 24, 28, 32, and 36 sites and for the parameters J1=J2=1,J2/H11032=J1/2, and J4=0.75 J1. Empty /H20849black /H20850symbols denote the singlet S=0 states, thick open /H20849blue /H20850symbols denote the S=1 states, and ﬁlled /H20849red /H20850symbols denote the S=2 states. The solid lines are polynomial ﬁts to the visible parts of the lowest one-triplon excitation branches.Bottom: the two one-triplon energy branches predicted fromsecond-order perturbation theory around the strong coupling limit/H20849cf. text /H20850. We emphasize here that the degeneracy of the two bands atk=0 is an accidental feature of the second-order theory for the given values of the exchange parameters while the degeneracy atk= /H9266//H208492a/H20850is a generic feature related to the fact that the model has a period aand not 2 aalong the legs of the ladder /H20849cf. text /H20850.TSIRLIN et al. PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-10netization data, or the value reported by other groups.15,17,21,22Hence, we ﬁnd that the present model of an isolated frustrated ladder overestimates the value of the spingap in BiCu 2PO6, a fact that was already suggested from the behavior of the susceptibility at low temperatures. One wayto account for this discrepancy is to include a ﬁnite interlad-der coupling J 3. Along the lines of the previous perturbative analysis, one ﬁnds that J3gives rise to a ﬁrst-order hopping of triplons along the ydirection. As a result, the two bands attain a common extra dispersion term of the form−/H20849J 3/2/H20850cosky/H20849with kyin units of /H9266divided by the interlad- der distance /H20850. This shifts the minimum of the lowest band down by J3/2. Thus, to account for the 35 K spin gap one would need an interladder coupling on the order of J3 /H1122970 K in this simple approximation. However, in the present regime we expect the perturbative calculation to beonly qualitatively correct, so that a precise determination ofthe interladder coupling either needs to come from moreelaborate theoretical approaches /H20849such as density matrix renormalization group simulations of coupled ladders /H20850or, ul- timately, from inelastic neutron-scattering experiments onsingle crystals. VII. MAGNETIZATION PROCESS FROM ED AND DMRG Here we revisit the magnetization process of BiCu 2PO6, in the light of the physical picture obtained above for thelowest magnetic excitations. To this end, we have employedLanczos diagonalizations up to N=32 sites with periodic boundary conditions, as well as DMRG simulations with uptoL=128 rungs using open boundary conditions. Some rep- resentative magnetization curves are shown in Fig. 12. The results from the two largest clusters treated by DMRG /H20849L =64,128 rungs /H20850converge to a rather smooth magnetization curve. They also give a critical ﬁeld H calmost identical to the one obtained from ED for 32 sites, which further cor-roborates the value of the spin gap /H9004 ED/H112290.5J1given above. To discuss the nature of the magnetization process in more detail, we distinguish three different regimes, namely,the one at low magnetizations above H c, the one at high magnetizations as we approach the saturation ﬁeld Hsat, and the intermediate regime. The low-magnetization regime canbe qualitatively understood on the basis of gradually ﬁllingthe excitation band of Fig. 11 /H20849bottom /H20850with triplons as we ramp up the ﬁeld above H c. One immediate consequence is the presence of a square-root singularity in the magnetizationright above H c/H20849cf. Fig. 12/H20850which is due to the quadratic dispersion above the minimum. Another important ingredientin this consideration is the presence of two incommensurateminima /H20849atk/H11229/H110060.4 /H9266/a/H20850in the triplon dispersion which, given the local hard-core constraint of the triplons, gives fourFermi points. Thus if the four-Fermi-point ﬁx point is indeedstable, the effective low-energy theory of BiCu 2PO6at low magnetizations is a two-component LL. In a similar way, the magnetization process close to the saturation ﬁeld can be understood starting from the fully po-larized state and gradually ﬁlling the one-magnon excitation12 16 20 24 28 32 36 one-triplon energy dispersion from pert. theory (up to 2nd order)JJ J J1= = 1, = 0.5, = 0.75 ’22 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ka/( / )/c11200.20.40.60.811.21.41.61.8 E/J1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ka/( / )/c11211.522.53 E/J1 00.5 FIG. 11. /H20849Color online /H20850Top: same as in Fig. 10but in the sym- metry setup of the lower panel of Fig. 9. The solid line is a poly- nomial ﬁt to the visible part of the lowest one-triplon excitationbranch. Bottom: The one-triplon energy dispersion predicted fromsecond order perturbation theory around the strong coupling limit/H20849cf. text /H20850./PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc 0 0.5 1 1.5 2 2.5 3 3.5 gµBH/J100.20.40.60.81M/MsatL=64 (DMRG) /PaintProc/PaintProc /PaintProc L=128 (DMRG) N=32 (ED) FIG. 12. /H20849Color online /H20850Magnetization curve of BiCu 2PO6as obtained from DMRG and ED.BRIDGING FRUSTRATED-SPIN-CHAIN AND SPIN- … PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-11bands by single spin ﬂips. Using the setup of the lower panel of Fig. 9and setting the energy of the fully polarized state to zero, one obtains two one-magnon bands which are given bythe eigenvalues of the matrix H one-magn =/H20873ukvk vkuk/H11032/H20874 /H208494/H20850 with uk=−/H20849J4/2+J1+J2/H20850+J2cos 2 k,uk/H11032=−/H20849J4/2+J1+J2/H11032/H20850 +J2/H11032cos 2 k, and vk=J4/2+J1cosk. The two one-magnon bands are shown in Fig. 13. As expected, we ﬁnd that each band has two minima at incommensurate wave vectors. Inparticular, the minima of the lowest band sit at k =/H110060.43131 /H9266/a, which are close to the minimum kpoints of the triplon dispersion of Fig. 11/H20849bottom /H20850. The corresponding minimum energy Emin=−3.5643 J1gives the saturation ﬁeld Hsat=3.5643 J1//H20849g/H92620/H9262B/H20850, in agreement with the numerical re- sults of Fig. 12. By gradually ﬁlling the minimum of the lowest one-magnon branch, we describe the magnetizationprocess as we decrease the ﬁeld below H sat. Similar to the low-magnetization regime, the quadratic dispersion aroundthe one-magnon minimum gives rise to a square-root singu-larity right below H satwhich can be seen in our numerical results of Fig. 12. In addition, the presence of two “incom- mensurate” minima /H20849at/H110060.43131 /H9266/a/H20850in the lowest one- magnon branch opens a possibility that the appropriate low-energy effective theory of BiCu 2PO6at high ﬁelds is a two- component LL. It is presently unclear whether the possible two- component LL phases discussed at low and high ﬁelds forma single phase, or whether they are separated by one or moreintervening phases at intermediate magnetizations. Inspect-ing the numerical results displayed in Fig. 12, a plateau might, for instance, occur at M=M sat/2. The phase immedi- ately above the plateau also requires further investigationsince there is a possibility for a one-component LL phasebefore we reach the high-ﬁeld two-component LL phase.Such a rich interplay between one- and two-component LLphases and plateaux is realized in the frustrated antiferro-magnetic J 1−J2Heisenberg chain model /H20849see, e.g., Ref. 67 and references therein /H20850. Testing and conﬁrming the scenario outlined here for the physics of BiCu 2PO6in high magnetic ﬁelds requires a separate and more detailed investigationwhich is, however, beyond the scope of this article. Let us ﬁnally compare to the experimental magnetization data of Fig. 8. Given our earlier estimate of J 1/H11229140 K from the susceptibility ﬁt, we obtain Hsat/H11229345 T, which is much larger than the range of ﬁelds accessible in our experiment/H20849H max=60 T /H20850. Hence, the highest magnetization values re- ported in Fig. 8correspond to less than 10% of Msat.I n contrast to the above theoretical predictions, the measuredmagnetization does not show any square-root singularityright above H c. As we discussed in Sec. VB, this gives evi- dence for interladder coupling J3and/or DM interactions which smooth out the singularity. VIII. DISCUSSION AND CONCLUSIONS Using DFT band-structure calculations, we derived the minimum microscopic model of BiCu 2PO6. This model is based on a two-leg-ladder lattice and comprises four antifer-romagnetic exchange couplings: J 1along the legs, J4along the rungs, and the frustrating next-nearest-neighbor cou- plings J2and J2/H11032along the legs /H20849Fig. 1/H20850. Although such a model does not provide a complete and quantitative descrip-tion of the compound, it is a reasonable compromise betweenthe complexity of the system and the capabilities of present-day numerical simulation techniques for the evaluation ofground-state and ﬁnite-temperature properties of frustratedquantum spin systems. We showed that the ladder geometryleads to strong spin correlations on the rungs, despite thesizable frustration and the weaker rung coupling. This fea-ture might explain why the simple model of the unfrustratedspin ladder reproduces certain properties of BiCu 2PO6, espe- cially the behavior upon the chemical substitution.21–23On the other hand, the reduction to the simple ladder modelcannot be justiﬁed microscopically since the frustrating cou-pling J 2is of the same order as the leg and the rung cou- plings J1andJ4, respectively. In particular, the coupling J2 has an effect on the spin gap. The simple J1−J4two-leg ladder with J4=3 4J1shows a spin gap of about 0.3 J1/H20849Ref. 68/H20850 while in our model the gap amounts to 0.5 J1. Thus, the frus- tration enhances the gap in a spin ladder, similar to a con-ventional frustrated spin chain. 8 We interpret BiCu 2PO6as a system of two-leg ladders with frustrating couplings along the legs. The absolute valuesof individual exchange couplings leave an ambiguity to de-scribe the system as a frustrated spin ladder or as coupledfrustrated spin chains. Indeed, the actual system shows fea-tures of both models. On the one hand, the strongest corre-lations are found on the rungs, as in ordinary ladders. On theother hand, the correlations along the legs are incommensu-rate and lead to the spin gap, being minimal at an incommen-surate position in the Brillouin zone. BiCu 2PO6is a peculiar spin-ladder system interesting for future investigation. One of the exciting branches could behigh-ﬁeld studies above H c. Recent experiments on /H20849C5H12N/H208502CuBr 4evidenced the emergence of the LL physics0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4−3.5−3−2.5−2−1.5−1−0.50 k/(π/a)E/J1one−magnon dispersionsJ1=J2=1, J2’=0.5, J4=0.75 FIG. 13. /H20849Color online /H20850One-magnon energy dispersions ob- tained analytically /H20851see Eq. /H208494/H20850/H20852.TSIRLIN et al. PHYSICAL REVIEW B 82, 144426 /H208492010 /H20850 144426-12in the high-ﬁeld phase of the two-leg spin ladder.11,12 BiCu 2PO6offers an opportunity to explore similar effects in the presence of the frustration, where the incommensurateposition of the gap might lead to a two-component LL orinstabilities thereof at ﬁelds just above H c. Another advan- tage is the relative ease of the chemical substitution that hasstimulated a range of experimental studies on Zn- and Ni-substituted samples. 21–23Here, again, the incommensurate leg spin-spin correlations could inﬂuence the effective inter-action mediated between the impurity-induced localizedspins, and thus lead to hitherto unobserved frustration ef-fects. While working on the minimal microscopic model, one also has to understand its limitations. The main and mostsevere limitation is the reduction to a purely 1D regime byneglecting J 3. In fact, our band structure calculations suggest J3/J4/H110210.2, i.e., /H20841J3/H20841/H1134925 K. If we adjust J3to account for the actual spin gap /H9004/H1122932 K /H112290.2J1, a larger value is ob- tained /H20849see Sec. VI/H20850. Additionally, the shape of the magneti- zation curve with the linear increase right above Hc/H20849Sec. VB /H20850may exclude a purely 1D scenario and point to sizable interladder couplings. Considering all these arguments, weconclude that the interladder coupling J 3is likely relevant for the full picture but its accurate estimate remains a challeng-ing task. Band-structure calculations equally allow for FM orAFM J 3/H20849Table II/H20850. Experimental estimates would require theoretical information on a complex 2D J1−J2−J2/H11032−J3−J4 frustrated spin system with long-range couplings J2andJ2/H11032. Such a system is basically beyond the capabilities of present-day numerical methods. Therefore, the most reasonable ap-proach could be analytical perturbation treatment, based onthe accurate results for the 1D model. We believe that thisapproach will help to clarify the complex magnetic behaviorof BiCu 2PO6and to improve the theoretical estimate of the spin gap with respect to the experimental value /H9004/H1122932 K. The second limitation of our model is the lack of aniso- tropy effects. In particular, the DM interactions scale with J and can be sizable due to the strong isotropic exchange of100–150 K. The DM couplings are allowed for all the bondsof the spin lattice with few restrictions on the arrangement oftheDvectors with respect to the crystal axes /H20849see also Sec. VB /H20850. The comprehensive investigation of the anisotropy ef-fects would require electron spin resonance measurements on single crystals along with sophisticated band structure calcu-lations. Presently, we note that the increase in the magneti-zation below H c/H20849Fig. 8/H20850is a possible signature of the DM couplings. The nonzero Knight shift at low temperatures21,23 may have the same origin. In summary, our study provides a comprehensive descrip- tion of isotropic exchange couplings in the spin-1 2quantum magnet BiCu 2PO6. We interpret this compound as a two-leg spin ladder with frustrating next-nearest-neighbor couplingsalong the legs. The leg coupling /H20849J 1/H20850, the rung coupling /H20849J4/H20850, and one of the next-nearest-neighbor couplings /H20849J2/H20850amount to 120–150 K while the other next-nearest-neighbor coupling J2/H11032is half of J2due to the subtle structural differences be- tween the respective superexchange pathways. The complexcrystal structure of the compound leads to a nontrivial imple-mentation of the spin ladder with two legs residing on dif-ferent structural ribbons. The proposed spin model is a de-rivative of the simple two-leg spin ladder and shows leadingspin correlations on the rungs. Frustrating couplings increasethe spin gap and induce the incommensurate minimum of thetriplon dispersion as well as an exotic behavior in high mag-netic ﬁelds. The effects beyond our spin model include theinterladder coupling and the anisotropy. Experimental datashow possible signatures of these effects and call for furtherinvestigation of BiCu 2PO6by means of inelastic neutron scattering and electron spin resonance measurements onsingle crystals. ACKNOWLEDGMENTS We are grateful to Walter Schnelle for high-temperature susceptibility measurements and for careful reading of themanuscript. We also acknowledge Nicolas Laﬂorencie, ToniShiroka, Markos Skoulatos, and Olivier Mentré for discus-sions and sharing the data prior to publication. A.T. wasfunded by Alexander von Humboldt Foundation. F.W. ac-knowledges the assistance of Yurii Skourski during the high-ﬁeld magnetization measurements and the ﬁnancial supportunder the project M.FE.A.CHPHSM of the Max-Planck So-ciety. Part of this work has been supported by EuroMagNETII under the EC Contract No. 228043. altsirlin@gmail.com †rousocha@pks.mpg.de ‡aml@pks.mpg.de 1G. Müller, H. Thomas, H. Beck, and J. C. Bonner, Phys. Rev. B 24, 1429 /H208491981 /H20850. 2B. Lake, D. A. Tennant, C. 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PhysRevB.84.052506.pdf	PHYSICAL REVIEW B 84, 052506 (2011) Local-moment magnetism in superconducting FeTe 0.35Se0.65as seen via inelastic neutron scattering Zhijun Xu,1,2Jinsheng Wen,1,3Guangyong Xu,1Songxue Chi,4Wei Ku,1Genda Gu,1and J. M. Tranquada1 1Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA 2Department of Physics, City College of New York, New York, New York 10033, USA 3Department of Materials Science and Engineering, Stony Brook University, Stony Brook, New York 11794, USA 4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Received 23 February 2011; published 11 August 2011) The nature of the magnetic correlations in Fe-based superconductors remains a matter of controversy. To address this issue, we use inelastic neutron scattering to characterize the strength and temperature dependence oflow-energy spin ﬂuctuations in FeTe 0.35Se0.65(Tc∼14 K). Integrating magnetic spectral weight for energies up to 12 meV , we ﬁnd a substantial moment ( /angbracketleftM2/angbracketrightLE∼0.07μ2 B/Fe) that shows little change with temperature, from below Tcto 300 K. Such behavior cannot be explained by the response of conduction electrons alone; states much farther from the Fermi energy must have an instantaneous local spin polarization. It raises interesting questionsregarding the formation of the spin gap and resonance peak in the superconducting state. DOI: 10.1103/PhysRevB.84.052506 PACS number(s): 74 .70.Xa, 61 .05.fg, 75.25.−j, 75.30.Fv Antiferromagnetism and superconductivity are common to the phase diagrams of cuprate and Fe-based superconductors,and it is frequently proposed that magnetic correlations are im-portant to the mechanism of electron pairing. 1,2Experiments on various Fe-based superconductors have demonstrated thatmagnetic excitations coexist with, and are modiﬁed by, thesuperconductivity. In particular, the low-energy spin excitationspectrum is modiﬁed by the emergence of a “resonance” peakand spin gap in the superconducting phase. 3–11The magnetic structures and excitations have been extensively studied byneutron scattering in these systems, including the AFe 2As2 (“122”, A=Ba,Sr,Ca) system,3–6,12,13theRFeAsO (“1111”, R=La,Ce,Pr,Nd,Gd,Sm) system,14–19and the FeTe 1−xSex (“11”) system.7–11,20,21Despite some variation in the magnetic structure of the parent compounds, in all known Fe-basedsuperconductors, the “resonance” occurs at the same Q 0∼ (0.5,0.5,0) (in the 2-Fe unit cell unit). At low temperature, the resonance is also accompanied by a well-deﬁned butanisotropic dispersion 10,11,20along the transverse direction, with a spin gap below which there is no spectral weight inthe superconducting state, resembling the spin excitations inmany high T ccuprates.22–25 One essential and currently unsettled issue is the nature of the magnetism in the Fe-based superconductors.2In contrast to the Mott-insulating parent compounds of the cuprates, theparent compounds of all of the Fe-based superconductorsare poor metals. This naturally leads to the suggestion ofitinerant magnetism resulting from the nesting of the Fermisurface, or more generally, enhancement of noninteractingsusceptibility. 26Disregarding the apparent failure of such an itinerant picture in producing the so-called bi-collinearmagnetic structure of Fe 1+yTe,27the spin-ﬂuctuation picture of superconductivity26is qualitatively appealing, and appears to give a natural explanation for the spin resonance and spingap. 28Nevertheless, there are recent theoretical analyses that suggest that there may be a signiﬁcant local-moment characterto the magnetism, as a consequence of Hund’s rule couplingamong Fe 3 delectrons. 29A direct way to test the different the- oretical perspectives is to evaluate the instantaneous momentfrom inelastic magnetic neutron scattering measurements. Thisis the goal of the present work.In this paper, we report an inelastic neutron scattering study on the temperature evolution of the low-energy magneticexcitation of an FeTe 1−xSexsample with x=65%. The magnetic excitations below Tc∼14 K are almost identical to those measured previously on superconducting FeTe 1−xSex samples with x=50%,7–11,20having a spin gap of ∼5m e V and a resonance at ∼7 meV , with anisotropic dispersion along the direction transverse to Q0. On heating to T=25 K, the resonance disappears, with spectral weight moving intothe gap. The Qdependence of the spectrum is still narrow around ¯ hω∼5–6 meV , but appears to disperse outward for energies both above and below, similar to the those observedin the cuprates. 22–24,30With further heating, the spin excitations near the saddle point ( ∼5 meV) start to split in Qand become clearly incommensurate, exhibiting a “waterfall”structure at 100 K and above, similar to the situation inunderdoped YBa 2Cu3O6+x.24However, the integrated spectral weight below ¯ hω=12 meV remains almost unchanged as a function of temperature, indicating a large energy scaleassociated with the stability of the instantaneous magneticmoment. The absolute normalization of the low-energy (LE)weight gives a lower limit (not counting the strong spectralweight at higher energies 8)o f/angbracketleftM2/angbracketrightLE∼0.07μ2 B/Fe. Such a robust and sizable moment is apparently beyond the standardconsideration of the spin-density-wave picture 26and strongly suggests that local-moment magnetism is present in the Fe-based superconductors. 29 The single-crystal sample used in the experiment was grown by a unidirectional solidiﬁcation method with nominal com-position of Fe 0.98Te0.35Se0.65(8.6 g). The bulk susceptibility, measured with a superconducting quantum interference device(SQUID) magnetometer, is shown in in Fig. 1(b), indicating T c∼14K. Neutron scattering experiments were carried out on the triple-axis spectrometer BT-7 located at the NISTCenter for Neutron Research. We used beam collimations ofopen-50 /prime-S-50/prime-open (S =sample) with ﬁxed ﬁnal energy of 14.7 meV and two pyrolytic graphite ﬁlters after the sample.The lattice constants for the sample are a=b=3.81˚A, andc=6.02˚A, using a unit cell containing two Fe atoms. The inelastic scattering measurements have been performedin the ( HK0) scattering plane [Fig. 1(a)]. The data are 052506-1 1098-0121/2011/84(5)/052506(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 84, 052506 (2011) FIG. 1. (Color online) (a) The schematic diagram of the neutron scattering measurements in the ( HK0) zone. Dashed lines denote linear scans performed across Q0=(0.5,0.5,0) in the text. (b) ZFC magnetization measurements by SQUID with a 5 Oe ﬁeld perpendicular to the a-bplane. Tc∼14 K is marked by a dashed line. (c) Constant Qscans at Q0taken at different temperatures: 5 K (red circles), 25 K (blue squares), 100 K (green triangles), and 300 K (black diamonds). Fitted background obtained from constant energyscans has been subtracted from all data sets. described in reciprocal lattice units (r.l.u.) of ( a∗,b∗,c∗)= (2π/a, 2π/b, 2π/c). Absolute normalizations are performed based on measurements of incoherent elastic scattering fromthe sample. Low-energy spin excitations are mainly distributed near theQ 0in-plane wave vector, similar to the case in the 50% Se doped sample.32In Fig. 1(c), we show constant- Qscans at Q0from 4 to 300 K. There is a clear resonance peak for data taken in the superconducting phase ( T=4 K, red circles). When heated above Tc, the resonance peak disappears, and spectral weight starts to ﬁll in the gap below /Delta1∼5m e V . For the normal state, the intensity at Q0appears to peak at around ¯ hω∼10 meV . These results are in good agreement with previous neutron scattering measurements,7,10indicating that further Se doping above the optimal value of 50% doesnot signiﬁcantly alter the low-energy magnetic excitations inthe system. Constant energy scans across Q 0, performed in the trans- verse direction, are plotted in Fig. 2. One can see how the resonance disappears with heating in Figs. 2(c) and 2(d). For ¯hω/lessorequalslant6.5m e V ,F i g s . 2(a)–2(c), we note that the peak on the right side [larger Kside, near (0 .25,0.75,0)] is further out in Q, with respect to Q0, compared to its counterpart on the left (small K) side, and becomes disproportionately strong. This behavior is inconsistent with crystal symmetry,FIG. 2. (Color online) Constant energy scans at (1 −K,K, 0) with different temperatures: 4 K (red circles), 25 K (blue squares), 100 K (green triangles), and 300 K (black diamonds) at different ¯ hω: (a) 3.5 meV , (b) 5 meV , (c) 6.5 meV , (d) 8 meV , (e) 10 meV , and (f)12 meV . A ﬂat ﬁtted background has been subtracted from all data sets. The solid lines are based on the ﬁt described in the text. The error bars represent the square root of the number of counts. which magnetic or simple phonon scattering must follow. The nature of this spurious peak is not entirely known. Itis very likely not associated with magnetic scattering from thesample; its growth with temperature suggests that it arises frommultiscattering processes involving certain phonon modes.Fortunately, it only appears on the large Kside, leaving the smallKside uncontaminated. In our data analysis, we ﬁt the magnetic signal using a double Gaussian function, with twopeaks split symmetrically about Q 0, plus a single Gaussian function for the spurious peak. The ﬁtted magnetic intensitiesare presented as contour maps in Fig. 3. With the spurious peak removed, one can easily see the evolution of the magneticexcitation spectrum with temperature. In the superconducting phase, Fig. 3(a), there is very little spectral weight below 5 meV , while the excitations disperseoutward at higher energies. As a function of temperature[Figs. 3(a)–3(d)], the dispersion at the highest energies changes little, and one can still observe well-deﬁned magneticexcitations at ¯ hω=12 meV up to T=300 K. The temperature effect on the dispersion below the resonance energy is muchmore pronounced. On warming from 4 to 25 K, intensity thatemerges below the gap appears to disperse outward slightly,as shown in Fig. 3(b). Our results are consistent with those in in Ref. 11, where the spectrum is narrowest in Qat the saddle point around 5 meV , and becomes broader for energy transfersboth above and below for T> T c. With further heating, the Qdependence of the spectrum changes most dramatically near the saddle point. At T= 100 K, the lower part of the dispersion clearly moves 052506-2BRIEF REPORTS PHYSICAL REVIEW B 84, 052506 (2011) FIG. 3. (Color online) Contour intensity maps showing the ﬁtted magnetic scattering intensity versus ¯ hωandQat different temperatures: (a) 4 K, (b) 25 K, (c) 100 K, and (d) 300 K. outward from Q0, as shown in Fig. 3(c). The saddle point at 5 meV actually disappears, and the dispersion becomesclearly incommensurate and almost vertical. There is littlechange between 100 and 300 K. In Figs. 4(a) and 4(b), we plot the intensities, integrated along Q=(1−K,K, 0), of the magnetic scattering and the spurious peak. The effect of the resonance in the superconduct-ing phase is observable up to ¯ hω∼10 meV . The plot of the spurious-peak intensity shows signs of temperature activation,and is peaked near 5 meV; in any case, its scale is generallysmall compared to the magnetic signal. The magnetic response in the normal state shows little temperature dependence and the main spectral weight is FIG. 4. (Color online) (a) Q-integrated (integrated only in one dimension, along the transverse direction) magnetic intensity, obtained based on the ﬁt described in the text, plotted vs temperature. (b) Average squared magnetic moment per Fe site vs temperature, forspectral weight integrated over 0 <¯hω < 12 meV . (c) Q-integrated intensity for the spurious peak around (0.25,0.75,0), plotted vs temperature.always located at higher energies (6 meV). Compared to that in the superconducting state, the low-energy spectral weight(below the gap) does appear to increase in the normal statewhen the “resonance” near 6.5 meV disappears, but remainsalmost unchanged with further heating for Tup to 300 K. This is consistent with the system having no static magnetic orderat low temperature, and therefore no shift of spectral weightfrom the elastic channel into those at low-energy transfersupon heating. The lack of temperature dependence for themagnetic excitation spectrum in the normal state for Tbetween 25 and 300 K, suggests a large energy scale associated with themagnetic response. We also note that the dispersion changesfrom a near hour-glass shape at low temperatures ( T=4 K and 25 K) to a “waterfall” shape at high temperatures ( T=100 K and 300 K). This change in dispersion is qualitatively similarto the behavior reported for underdoped YBa 2Cu3O6+x,24but occurs at temperatures well within the normal state, and itsorigin is not entirely understood. Our key result is obtained by integrating the magnetic signal overQand ¯hω. The measured inelastic magnetic scattering intensity is proportional to the dynamic spin correlation func-tionS(Q,ω)=χ /prime/prime(Q,ω)/(1−e−¯hω/k BT), which follows the “sum rule” that/integraltext BZdQ/integraltext+∞ −∞Sαβ(Q,ω)dω=1 3v∗δαβ/angbracketleftM2/angbracketright/g2, where v∗is the volume of the Brillouin zone. By integrating the normalized spectral weight up to ¯ hω∼12 meV , we can obtain a lower bound of the magnetic moment per Fe. FortheQintegration, we assume the peak width along the longitudinal direction is the same as transverse direction andthat the response is uniform along L, based on results from previous measurements. 7,10For energy, we integrated over the interval 0 meV /lessorequalslant¯hω/lessorequalslant12 meV , using the low-energy extrapolation indicated by the dashed lines in Fig. 4(a). From this integral, we obtain a spectral weight of /angbracketleftM2/angbracketrightLE= 0.07(3)μ2 Bper Fe. The temperature dependence of this quantity is negligible, as shown in Fig. 4(c). Similar behavior showing little temperature dependence of the integrated spectral weightis also evidenced in the AFM insulator La 2CuO 433where magnetism is dominated by local moments. The moment we have evaluated is only a small fraction of the total moment per site, considering that previousmeasurements have shown signiﬁcant spectral weight all theway up to a few hundred meV . 8Nevertheless, such a large low-energy magnetic response is already an order of magnitude larger than what is expected from a simple itinerant picture.For example, taking the density of states 34at the Fermi energy of∼1.5 states /eV from a nonmagnetic LDA calculation for FeSe, the corresponding bare susceptibility, or /angbracketleftM2/angbracketrightLEderived from electrons near the Fermi level can be estimated to be no more than of order 0.001 μ2 Bper Fe within an excitation energy range of 12 meV . Even including a mass enhancement factor of2 to 3 as observed by photoemission, 35,36the resulting spectral weight is still at least an order of magnitude smaller than our observation. One could in principle ﬁne-tune the interaction strength to bring the magnon pole to a very low energy toenhance the spectral weight, but then it will surely generatea strong temperature dependence of the spectral weight, 37 in drastic contrast to our observation. The observed lack oftemperature dependence suggests that electronic states over alarge energy range contribute to the effective moment, which is 052506-3BRIEF REPORTS PHYSICAL REVIEW B 84, 052506 (2011) consistent with having a signiﬁcant local moment, as suggested by recent theoretical work.29 We are, of course, not suggesting that the system should behave like an insulator with only a local-moment contri-bution to the magnetism. Apparently the rodlike dispersionwe observe cannot be explained by spin-wave excitationsemerging from a Heisenberg Hamiltonian. Nevertheless, ourresults do suggest that the simplest itinerant picture ofweakly interacting electrons cannot be used to quantitativelyexplain the large, temperature-independent magnet momentin neutron scattering measurements. In addition to states nearthe Fermi surface, states at higher energies will have to beconsidered (nonperturbatively) when magnetism as well assuperconductivity are concerned. This leads to an interesting question. For the itinerant picture, the spin gap and resonance come out naturally fromthe pairing gap for the quasiparticles—although they aresensitive to the symmetry of the order parameter. If themagnetic moments involve states at high binding energies, then one must reconsider the evaluation of the resonance.It is clear that the magnetic correlations are sensitive to thedevelopment of pairing and superconductivity; however, theelectrons involved in the pairing and in the magnetism are notnecessarily identical. Similar issues have been raised in thecase of cuprates. These issues also raise questions concerningthe nature of the pairing mechanism. ACKNOWLEDGMENTS We thank Weiguo Yin and Igor Zaliznyak for use- ful discussions. This work is supported by the Ofﬁce ofBasic Energy Sciences, US Department of Energy underContract No. DE-AC02-98CH10886. J.S.W. and Z.J.X. aresupported by the same source through the Center forEmergent Superconductivity, an Energy Frontier ResearchCenter. 1I. I. Mazin, Nature (London) 464, 183 (2010). 2J. Paglione and R. L. Greene, Nat. Phys. 6, 645 (2010). 3A. D. Christianson et al. ,Nature (London) 456, 930 (2008). 4M. D. Lumsden et al. ,P h y s .R e v .L e t t . 102, 107005 (2009). 5S. Chi et al. ,Phys. Rev. Lett. 102, 107006 (2009). 6D. S. Inosov et al. ,Nat. Phys. 6, 178 (2010). 7Y. Q i u et al. ,P h y s .R e v .L e t t . 103, 067008 (2009). 8M. D. Lumsden et al. ,Nat. Phys. 6, 182 (2010). 9J. Wen, G. Xu, Z. Xu, Z. W. Lin, Q. Li, Y . Chen, S. Chi, G. Gu, and J. M. 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Fujita, andK. Yamada, P h y s .R e v .B 76, 014508 (2007). 31There is no elastic magnetic intensity found near (0 .5,0,0.5), which is the antiferromagnetic ordering wave vector in the parentcompound, and very little spectral weight near (0 .5,0,0) at low energies and low temperature. 32Z. Xu, J. Wen, G. Xu, Q. Jie, Z. Lin, Q. Li, S. Chi, D. K. Singh, andG. Gu, and J. M. Tranquada, P h y s .R e v .B 82, 104525 (2010). 33K. Yamada, K. Kakurai, Y . Endoh, T. R. Thurston, M. A. Kastner, R. J. Birgeneau, G. Shirane, Y . Hidaka, and T. Murakami, Phys. Rev. B 40, 4557 (1989). 34K. W. Lee, V . Pardo, and W. E. Pickett, P h y s .R e v .B 78, 174502 (2008). 35A. Tamai et al. ,Phys. Rev. Lett. 104, 097002 (2010). 36Y. Z h a n g et al. ,P h y s .R e v .B 82, 165113 (2010). 37T. Kariyado and M. Ogata, J. Phys. Soc. Jpn. 78, 043708 (2009). 052506-4
PhysRevB.85.245120.pdf	PHYSICAL REVIEW B 85, 245120 (2012) Functional renormalization group approach to the Ising-nematic quantum critical point of two-dimensional metals Casper Drukier, Lorenz Bartosch, Aldo Isidori, and Peter Kopietz Institut f ¨ur Theoretische Physik, Universit ¨at Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany (Received 16 March 2012; revised manuscript received 1 June 2012; published 19 June 2012) Using functional renormalization group methods, we study an effective low-energy model describing the Ising-nematic quantum critical point in two-dimensional metals. We treat both gapless fermionic and bosonicdegrees of freedom on equal footing and explicitly calculate the momentum- and frequency-dependent effectiveinteraction between the fermions mediated by the bosonic ﬂuctuations. Following earlier work by S.-S. Lee for aone-patch model, Metlitski and Sachdev [ P h y s .R e v .B 82, 075127 (2010) ] recently found within a ﬁeld-theoretical approach that certain three-loop diagrams strongly modify the one-loop results and that the conventional 1 /N expansion breaks down in this problem. We show that the singular three-loop diagrams considered by Metlitskiand Sachdev are included in a rather simple truncation of the functional renormalization group ﬂow equations forthis model involving only irreducible vertices with two and three external legs. Our approximate solution of theseﬂow equations explicitly yields the vertex corrections of this problem and allows us to calculate the anomalousdimension η ψof the fermion ﬁeld. DOI: 10.1103/PhysRevB.85.245120 PACS number(s): 71 .10.Hf, 73 .43.Nq, 71 .27.+a I. INTRODUCTION Inspired by the puzzling normal-state properties of the copper-oxide superconductors, the search for possible non-Fermi liquid states of metals continues to be a central topicin the theory of strongly correlated electrons. An interestingnew clue toward an understanding of the cuprates and othermaterials comes from experiments on a variety of materials,indicating a nematic phase transition. 1–7This is a quantum phase transition which, while preserving translational sym-metry, breaks the lattice rotational symmetry from squareto rectangular; that is, the invariance under rotations of thesystem in the x-yplane by 90 ◦is lost. At the quantum critical point, electrons couple strongly to order parameterﬂuctuations, leading to a destruction of the Fermi liquid state.The resulting distortion of the Fermi surface is also referred toas a Pomeranchuk transition. 8,9 The conventional theoretical approach to quantum critical phenomena is the so-called Hertz-Millis approach, where theinteraction between electrons is decoupled via a (bosonic)Hubbard-Stratonovich transformation and the fermionic de-grees of freedom are integrated out. 10–13However, because in the metallic state the electrons are gapless, this approachusually leads to singular vertices, which, especially in lowdimensions, need to be treated with care. It can therefore beadvantageous not to integrate out the fermions at all. As concerns the nematic phase transition, the most effective scattering processes of electrons take place when the momen-tum of the bosons is locally almost tangential to the Fermisurface. The phase transition can therefore be modeled bycoupling electrons in the vicinity of two patches of a Fermi sur-face to a gapless scalar Ising order parameter ﬁeld. Indeed, thecoupling of gapless fermions to gapless bosonic ﬂuctuationsis known to give rise to non-Fermi liquid behavior. 13,14The corresponding ﬁeld theory is very similar to a nonrelativisticgauge theory, which has been studied intensively, 15–17starting with the important work by Holstein, Norton, and Pincus.15 Such gauge theories have applications in a number of differentproblems, such as the description of the half-ﬁlled Landaulevel, 18the description of spin liquids in terms of spinons which can form a critical spinon Fermi surface,19,20or the description of the instability of a ferromagnetic quantumcritical point. 21Similar gauge theories have also been used to describe fermions on a honeycomb lattice interacting throughan electromagnetic gauge ﬁeld. 22,23In all cases, the low-energy behavior is expected to be described by a scale-invariantscaling theory. The single-particle Green’s function G(ω,k) was calculated 24,25for a spherical Fermi surface within the random phase approximation (RPA), resulting in G(ω+i0+,k)∝1 Aω\|ω\|2/3−ξk. (1.1) Here,ωandkare the frequency and momentum of the electron, Aω=A/primesgn(ω)+iA/prime/primeis a complex constant depending on the sign of ω, andξk=vF(\|k\|−kF), where vFis the Fermi velocity and kFis the Fermi momentum, denotes the single- particle excitation energy. While the static part of the self-energy remains unrenormalized, its dynamic part implies thatboth the renormalized energy and the damping rate of theelectron scale in exactly the same way. Consequently, thereare no well-deﬁned sharp quasiparticles and Landau’s Fermiliquid theory breaks down. For a long time, it was thought that the above scenario holds true when going beyond the RPA. 26It was believed that this can be justiﬁed by considering the limit of large N, where Nis the number of fermion ﬂavors. However, it was recently shown by Lee27,28that, even if one considers only scattering processes in the vicinity of a single patch of theFermi surface, the coupling to a gapless gauge ﬁeld results invery strong correlations such that, even in the large- Nlimit, the theory remains strongly coupled. Subsequently, Metlitskiand Sachdev 29,30considered a more realistic two-patch model where the two patches of the Fermi surface to which a givenbosonic momentum is tangent are retained; they derived ascaling theory and explicitly calculated corrections to thebosonic and fermionic self-energies up to three loops, usingthe one-loop propagator in internal loop integrations. Metlitski 245120-1 1098-0121/2012/85(24)/245120(18) ©2012 American Physical SocietyDRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) (a) (b) FIG. 1. These three-loop diagrams have been identiﬁed by Metlitski and Sachdev (Refs. 29and30) to give singular corrections to the one-loop results for the bosonic and fermionic self-energies.Solid arrows denote the fermionic single-particle Green’s functions within the one-loop approximation, while wavy lines represent the RPA propagator of the bosonic ﬂuctuations. The black dot is the bareinteraction vertex between one boson and two fermion ﬁelds. The diagrams in (a) are three-loop corrections to the fermionic self-energy, while those in (b) represent corrections to the bosonic self-energy ofthe so-called Aslamazov-Larkin type. and Sachdev identiﬁed certain three-loop contributions to the bosonic and fermionic self-energies (see Fig. 1), which give rise to logarithmically divergent corrections to the one-loopRPA results. Exponentiating these logarithmic terms, theyobtained the following expression for the retarded propagatorof the fermions, G(ω+i0 +,k)∝1 [Aω\|ω\|1/z−ξk]1−ηψ/2, (1.2) where ηψis the anomalous dimension of the fermion ﬁeld and zis the fermionic dynamic critical exponent. Using scaling relations for the fermionic and bosonic Green’s functions,Metlitski and Sachdev argued that the fermionic dynamicexponent is given by z=z b/2, where zbis the corresponding bosonic dynamic exponent. Their explicit calculations29show thatzb=3 is not renormalized by ﬂuctuations up to three loops, implying that z=3/2 is correctly given by the one-loop approximation. On the other hand, for the fermionic anomalousdimension, Metlitski and Sachdev obtain the ﬁnite resultη ψ=0.068 at the Ising-nematic transition, whereas ηψ=0 within the one-loop approximation. On a technical level, the reason for the breakdown of the large-Nexpansion can be traced back to the fact that the curvature of the fermion propagator comes with a factor ofN. This in combination with a cancellation of the curvature in a set of planar diagrams eventually leads to the breakdown ofthe large- Nexpansion. In the words of Chubukov, 31there is hidden one-dimensionality in two-dimensional systems. Albeit 1 /Ncannot be used as a control parameter, it was suggested by Mross et al. ,32following earlier work at ﬁnite N by Nayak and Wilczek,33to use zbas a tunable parameter. In this case it is possible to consider the limits N→∞ andzb−2→0 while keeping the product N(zb−2) ﬁnite to bring the calculation under control. Using a tunable zbis a sensible strategy because a nonlocal interaction is not expectedto be renormalized. Extrapolating the results obtained byMross et al. 32to the physically relevant case zb=3 andN=2, one obtains for the anomalous dimension of the fermion ﬁeldη ψ≈0.6 [using our deﬁnition (1.2) ofηψ]. Obviously, thisvalue is much larger than the estimate ηψ≈0.068 by Metlitski and Sachdev.29 Even though the calculations in Refs. 29and32are based on the ﬁeld-theoretical renormalization group, the fact thattwo independent calculations involving different types ofapproximations produce different values for η ψshows that on a quantitative level there are still open questions. Due to thesign problem in quantum Monte Carlo calculations and the factthat dynamical mean-ﬁeld theory can essentially only predictmean-ﬁeld exponents, the number of alternative methods toverify the correctness of the anomalous scaling propertiesof the Ising-nematic transition is limited. In this work, westudy this problem by means of the one-particle irreducibleimplementation of the functional renormalization group (FRG)method, 34–37which is a modern implementation of the Wilsonian renormalization group idea. The ﬂexibility of FRGmethods to deal with systems involving both fermionic andbosonic ﬁelds has already been used by several authors. 38–44 In particular, in Refs. 40,43and44it has been shown that it can be advantageous to introduce a cutoff parameter /Lambda1which regularizes the infrared divergences only in the momentumcarried by the bosonic ﬁeld (the momentum-transfer cutoffscheme). We show in this work that this cutoff schemeis also convenient to study the nematic quantum criticalpoint. The rest of this work is organized as follows. After introducing the model system and deﬁning our notation inSec. II, we give in Sec. IIIthe FRG ﬂow equations for the self-energies and vertex corrections in general form. We alsointroduce the momentum-transfer cutoff scheme and show thatthe singular three-loop diagrams shown in Fig. 1are contained in a rather simple truncation of the hierarchy of FRG ﬂowequations involving only irreducible two-point and three-pointvertices. In Sec. IV, we show how to recover the known one-loop results for the momentum- and frequency-dependentfermionic and bosonic self-energies by integrating the FRGﬂow equations ignoring vertex corrections. In the main part ofthis work, given in Sec. V, we consider the system of FRG ﬂow equations including vertex corrections. We explicitlycalculate the effect of vertex corrections on the value ofthe fermionic anomalous dimension, η ψ, to leading order in the small parameter zb−2. In the concluding Sec. VI, we summarize our results and discuss some open problems.We have added two appendixes with more technical details.In Appendix Awe derive skeleton equations relating the purely bosonic two-point and three-point functions to thefermionic propagators and irreducible vertices. These skeletonequations are useful to close the inﬁnite hierarchy of FRG ﬂowequations. Finally, in Appendix B, we explicitly evaluate the symmetrized fermionic loop with three external bosonic legs(the symmetrized three-loop vertex) for our model system. II. DEFINITION OF THE MODEL We are interested in a minimal model describing the coupling of electrons in the proximity of a two-dimensionalFermi surface to a gapless scalar Bose ﬁeld. In particular,the bosonic ﬁeld can describe the ﬂuctuations of a scalarorder parameter near the onset of a metallic Ising-nematicphase, such as a d-wave nematic state in a two-dimensional 245120-2FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) square lattice, where the point-group symmetry of the lattice is reduced from square to rectangular. However, the gaplessscalar Bose ﬁeld can also describe the ﬂuctuations of anemergent gauge ﬁeld minimally coupled to a two-dimensionalFermi surface. For example, this can be physically realizedin a system where a spin-liquid phase is described in termsof fermionic degrees of freedom (spinons), thereby causingthe emergence of a U(1) gauge symmetry in the system: Thecritical ﬂuctuations near the onset of the spin-liquid phase arethen described by the coupling of the spinon Fermi surface tothe corresponding U(1) gauge ﬁeld. The general form of theaction for our model can be written as S=S ψ+Sφ+Sint, with Sψ=−/integraldisplay K/summationdisplay σ(iω−ξk)¯ψKσψKσ, (2.1) Sφ=1 2/integraldisplay ¯K(ρ0+˜ν0¯k2)φ−¯Kφ¯K, (2.2) Sint=/integraldisplay ¯KˆO¯K[¯ψ,ψ ]φ−¯K, (2.3) where ψandφdenote two-dimensional Fermi and Bose ﬁelds, respectively, and ˆO[¯ψ,ψ ] is a bilinear operator in the fermion ﬁelds which has the same symmetry as the order parameterﬁeldφ.I nE q s . (2.1) –(2.3) ,K=(iω,k) denotes fermionic Matsubara frequency and two-dimensional momentum, while ¯K=(i¯ω,¯k) denotes the corresponding bosonic quantities. The integration symbols are deﬁned by/integraltext K=(βV)−1/summationtext ω/summationtext k, and similarly for the bosonic quantities, where βis the inverse temperature and Vis the volume. Throughout this work it is understood that we eventually take the zero temperaturelimit ( β→∞ ) and the inﬁnite volume limit ( V→∞ ). The index σ=1,..., N labels Ndifferent ﬂavors of the fermion ﬁeld. The fermionic energy dispersion ξ kis deﬁned relative to the Fermi energy /epsilon1F, that is, ξk=/epsilon1k−/epsilon1F, while, in the bosonic dispersion, ρ0plays the role of a mass (or gap) term which measures the distance to the quantum critical point: Atthe quantum critical point, ρ 0=0, such that order parameter ﬂuctuations become gapless. The absence of higher-orderterms in φand gradients of φin the action deﬁning our model can be justiﬁed by a dimensional analysis, which shows thatsuch higher-order terms become irrelevant at the critical point. In the general form given in Eqs. (2.1) –(2.3) , the action of our model is still too complicated to be treated analytically withrenormalization group or ﬁeld-theoretical methods. However,as pointed out by Metlitski and Sachdev 29( s e ea l s oR e f . 45), the relevant critical ﬂuctuations can be described by asimpliﬁed minimal action involving only fermion ﬁelds withmomenta close to two opposite patches on the Fermi surface.The reason is that the most singular scattering processesmediated by a given bosonic mode with momentum ¯kinvolve only fermions lying on patches of the Fermi surface which arealmost tangential to the bosonic momentum ¯k. The situation is shown graphically in Fig. 2, where the label α=±1 denotes the two patches of the Fermi surface which are tangentialto a given bosonic mode with momentum parallel to k ⊥in the ﬁgure. In order to describe the singular behavior of thefermionic and bosonic Green’s functions at the critical pointwe can therefore restrict the general model involving fermionsFIG. 2. The two-patch model considered in this work involves only two types of fermion ﬁelds with momenta close to two oppositepatches on the Fermi surface centered at ±k F. The fermionic momenta are measured locally with respect to ±kF.W ed e ﬁ n e k/bardblas the component of the momentum parallel to the local Fermi surfacenormal and k ⊥as the component orthogonal to the surface normal. on the whole Fermi surface to a so-called two-patch model, characterized by the following Euclidean action: Spatches [¯ψ,ψ,φ ]=S0[¯ψ,ψ ]+S0[φ]+S1[¯ψ,ψ,φ ],(2.4) S0[¯ψ,ψ ]=−/integraldisplay K/summationdisplay α,σ/parenleftbig iω−ξα k/parenrightbig¯ψα Kσψα Kσ, (2.5) S0[φ]=1 2/integraldisplay ¯Kf−1 ¯kφ−¯Kφ¯K, (2.6) S1[¯ψ,ψ,φ ]=/integraldisplay K1/integraldisplay K2/integraldisplay ¯K3/summationdisplay α,σδK1,K2+¯K3 ×/Gamma1¯ψαψαφ 0 (K1;K2;¯K3)¯ψα K1σψα K2σφ¯K3.(2.7) In the above expressions the fermion ﬁelds are now charac- terized by an additional upper index α=±1 labeling the two patches on the Fermi surface centered at the two oppositemomenta k α F=αkF, as shown in Fig. 2. By construction, the momenta of the ﬁelds characterizing the two-patch model areintended to lie in the vicinity of the Fermi momenta k α F, so that \|k/bardbl\|and\|k⊥\|(the momenta relative to the Fermi momenta, as shown in Fig. 2) should be much smaller than \|kF\|.I n particular, one should impose a cutoff, /Lambda1⊥∼kF/Delta1θon the momenta perpendicular to the Fermi surface normal, where/Delta1θis the angular extension of the patch. However, as long as integrals over such momenta turn out to be ultravioletconvergent, we can effectively send this cutoff to inﬁnitywithout affecting the low-energy critical behavior of ourtheory. The energy dispersion relative to the true Fermi energy at patchαis assumed to be of the form ξ α k=/epsilon1kα F+k−/epsilon1kα F=vα Fk/bardbl+k2 ⊥ 2m=αvFk/bardbl+k2 ⊥ 2m, (2.8) where vFis the Fermi velocity, k/bardblis the component of k parallel to the local normal to the Fermi surface, and k⊥is perpendicular to the local Fermi surface normal. In the last 245120-3DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) equality of Eq. (2.8) we have used the fact that the Fermi velocity has opposite sign at the two patches. We normalize thebosonic ﬁeld φ ¯Ksuch that the bare fermion-boson interaction vertex /Gamma1¯ψαψαφ 0 (K1;K2;¯K3) is unity for the model describing the nematic quantum phase transition (which we discuss indetail below) and assumes the values α=±1 for the gauge ﬁeld model, that is, /Gamma1¯ψαψαφ 0 (K1;K2;¯K3)=/Gamma1α 0=/braceleftbigg 1 (nematic model), α(gauge model).(2.9) Finally, the coefﬁcient of the quadratic term in the bosonic part o ft h ea c t i o ni sa s s u m e dt ob eo ft h ef o r m f−1 ¯k=ρ0+ν0/parenleftbigg\|¯k⊥\| 2mvF/parenrightbiggzb−1 , (2.10) where ρ0andν0are dimensionful constants with units of mass. (Recall that in two dimensions, the density of statesalso has units of mass.) We note that, for a nonspherical Fermisurface, mv Fdoes not necessarily equal the Fermi momentum. For simplicity, in this work, we do not keep track of therenormalization of ρ 0by ﬂuctuations, so that we may set ρ0=0 to describe the quantum critical point. We assume that the bosonic dynamic exponent zbis in the range 2 <zb/lessorequalslant3. To construct a sensible limit of large N, the constants ρ0and ν0should be proportional to N. However, according to Mross et al.32the limit of large- Ncan be safely taken only when zb−2 is sent to zero simultaneously, such that the product N(zb−2) remains ﬁnite. Although the FRG, unlike the ﬁeld-theoretical renormaliza- tion group, does not rely on the presence of a small expansionparameter (which indeed is not present in the consideredproblem), it is convenient to express the above action in termsof rescaled dimensionless momenta, frequencies and ﬁelds,to carry out the renormalization group procedure. A generaldiscussion of proper scaling in mixed Fermi-Bose systems canbe found in Ref. 46. Given an arbitrary momentum scale /Lambda1, we deﬁne dimensionless fermionic labels Q=(i/epsilon1,q /bardbl,q⊥)b y setting k/bardbl=/Lambda12 2mvFq/bardbl, (2.11a) k⊥=/Lambda1q⊥, (2.11b) ω=2mv2 F/parenleftbigg/Lambda1 2mvF/parenrightbiggzb /epsilon1=/Lambda12 2m/parenleftbigg/Lambda1 2mvF/parenrightbiggzb−2 /epsilon1.(2.11c) The corresponding bosonic labels ¯Q=(i¯/epsilon1,¯q/bardbl,¯q⊥)a r e deﬁned in precisely the same way: ¯k/bardbl=/Lambda12 2mvF¯q/bardbl, (2.12a) ¯k⊥=/Lambda1¯q⊥, (2.12b) ¯ω=2mv2 F/parenleftbigg/Lambda1 2mvF/parenrightbiggzb ¯/epsilon1=/Lambda12 2m/parenleftbigg/Lambda1 2mvF/parenrightbiggzb−2 ¯/epsilon1.(2.12c) Introducing the rescaled dimensionless ﬁelds ψα Qσ=4m2v3 F/parenleftbigg/Lambda1 2mvF/parenrightbiggzb+5 2 ψα Kσ, (2.13) φ¯Q=4m2v2 F/parenleftbigg/Lambda1 2mvF/parenrightbiggzb+1 φ¯K, (2.14)the Euclidean action of our model can be written as S0[¯ψ,ψ ]=−/integraldisplay Q/summationdisplay α,σ/parenleftbig iζ/Lambda1/epsilon1−ξα q/parenrightbig¯ψα Qσψα Qσ, (2.15) S0[φ]=1 2/integraldisplay ¯Q(r/Lambda1+c0\|¯q⊥\|zb−1)φ−¯Qφ¯Q, (2.16) S1[¯ψ,ψ,φ ]=/integraldisplay Q1/integraldisplay Q2/integraldisplay ¯Q3/summationdisplay α,σδQ1,Q2+¯Q3 ×/Gamma1¯ψαψαφ 0 (Q1;Q2;¯Q3)¯ψα Q1σψα Q2σφ¯Q3,(2.17) where ζ/Lambda1=/parenleftbigg/Lambda1 2mvF/parenrightbiggzb−2 , (2.18a) ξα q=αq/bardbl+q2 ⊥, (2.18b) r/Lambda1=ρ0 2m/parenleftbigg/Lambda1 2mvF/parenrightbigg1−zb , (2.18c) c0=ν0 2m, (2.18d) and the mixed fermion-boson vertex is the same as before, /Gamma1¯ψαψαφ 0 (Q1;Q2;¯Q3)=/Gamma1¯ψαψαφ 0 (K1;K2;¯K3)=/Gamma1α 0.(2.19) If we use the expression ν0=Nm/ (2π) for the density of states of free fermions in two dimensions, we havec 0=N/(4π). Consequently, setting r/Lambda1→0 to describe the quantum critical point, our model does not depend on any freeparameters. Our FRG procedure will generate also higher-order purely bosonic contributions to the effective action, which describe in-teractions between the boson ﬁelds, mediated by the fermions.In an expansion in powers of the ﬁelds, the lowest-orderinteraction process is cubic in the bosonic ﬁelds, S 3[φ]=1 3!/integraldisplay ¯K1/integraldisplay ¯K2/integraldisplay ¯K3δ¯K1+¯K2+¯K3,0 ×/Gamma1φφφ(¯K1,¯K2,¯K3)φ¯K1φ¯K2φ¯K3 =1 3!/integraldisplay ¯Q1/integraldisplay ¯Q2/integraldisplay ¯Q3δ¯Q1+¯Q2+¯Q3,0 ×˜/Gamma1φφφ(¯Q1,¯Q2,¯Q3)φ¯Q1φ¯Q2φ¯Q3, (2.20) with ˜/Gamma1φφφ(¯Q1,¯Q2,¯Q3)=v2 F/parenleftbigg/Lambda1 2mvF/parenrightbigg3−zb /Gamma1φφφ(¯K1,¯K2,¯K3). (2.21) Although for zb<3 this vertex seems to be irrelevant by power counting, it turns out that it has a singular dependenceon the external momenta and frequencies and thereforecannot be neglected. Because, within our bosonic momentum-transfer cutoff scheme all vertices involving only bosonicexternal legs are ﬁnite at the initial renormalization groupscale, 36,40it is crucial to keep track of the FRG ﬂow of the vertex /Gamma1φφφ(¯K1,¯K2,¯K3). In this work, we do this by means of a skeleton equation relating /Gamma1φφφ(¯K1,¯K2,¯K3)t o the symmetrized fermion loop with three external bosoniclegs and renormalized fermionic propagators, as discussed in 245120-4FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) Appendix A. An explicit evaluation of this vertex is given in Appendix B. If we identify /Lambda1with the renormalization group ﬂow parameter which is reduced under the renormalization groupprocedure, the canonical dimensions of all quantities explicitlyappear in the FRG ﬂow equations with the above rescaling.To compare the FRG results with perturbation theory, itis more convenient, however, not to include the canonicaldimensions into the deﬁnition of the vertices. Therefore, wesimply choose /Lambda1=2mv Fin the above expressions, so that ζ/Lambda1→1 and r/Lambda1→r0=ρ0/(2m). Renaming again Q→K, our bare action is then the sum of the following three terms, S0[¯ψ,ψ ]=−/integraldisplay K/summationdisplay α,σ/parenleftbig iω−ξα k/parenrightbig¯ψα Kσψα Kσ,(2.22) S0[φ]=1 2/integraldisplay ¯K(r0+c0\|¯k⊥\|zb−1)φ−¯Kφ¯K, (2.23) S1[¯ψ,ψ,φ ]=/integraldisplay K1/integraldisplay K2/integraldisplay ¯K3/summationdisplay α,σδK1,K2+¯K3 ×/Gamma1¯ψαψαφ 0 (K1;K2;¯K3)¯ψα K1σψα K2σφ¯K3,(2.24) where now ξα k=αk/bardbl+k2 ⊥,r 0=ρ0 2m,c 0=ν0 2m. (2.25) The corresponding Gaussian propagators are Gα 0(K)=1 iω−ξα k=1 iω−αk/bardbl−k2 ⊥, (2.26) F0(¯K)=f¯k=1 r0+c0\|¯k⊥\|zb−1. (2.27) This dimensionless parametrization of our model is what is used in the following sections. III. FRG FLOW EQUATIONS The starting point of our calculation is the Wetterich equation34,35for the coupled Fermi-Bose model deﬁned above, which is an exact FRG ﬂow equation for the generatingfunctional /Gamma1 /Lambda1[¯ψ,ψ,φ ] of the one-line irreducible vertices of our theory. This ﬂow equation describes the exact evolution of/Gamma1 /Lambda1[¯ψ,ψ,φ ] as some (now dimensionless) cutoff parameter /Lambda1 is reduced. By expanding /Gamma1/Lambda1[¯ψ,ψ,φ ] in powers of the ﬁelds, we obtain an inﬁnite hierarchy of coupled integro-differentialequations for the one-line irreducible vertices of our model.This hierarchy is formally exact, but, in practice, furtherapproximations are usually necessary in order to obtain explicitresults for the vertex functions (see Refs. 36and37for recent reviews). Moreover, the proper choice of the cutoff scheme isalso very important. For our effective low-energy model discussed above, it is, in principle, possible to introduce cutoffs in both the bosonicand the fermionic sectors and regularize the inverse Gaussianpropagators as follows: [F 0,/Lambda1(¯K)]−1=f−1 ¯k+¯R/Lambda1(¯K), (3.1) /bracketleftbig Gα 0,/Lambda1(K)/bracketrightbig−1=iω−ξα k−R/Lambda1(K). (3.2) For the calculations in the present problem, we ﬁnd it more convenient to introduce a sharp momentum-transfer cutoffonly in the bosonic sector. Using a similar cutoff procedure, two of us were able to derive the exact scaling behavior of theTomonaga-Luttinger model within an FRG approach. 40We therefore set R/Lambda1(K)=0 in the fermionic sector and choose ¯R/Lambda1(¯K)=f−1 ¯k[/Theta1−1(\|¯k⊥\|−/Lambda1)−1] (3.3) for the boson cutoff. This leads to the cutoff-dependent bare Gaussian propagator F0,/Lambda1(¯K)=/Theta1(\|¯k⊥\|−/Lambda1)f¯k, (3.4) which vanishes for \|¯k⊥\|</Lambda1 and equals f¯kfor\|¯k⊥\|>/Lambda1.A s there is no cutoff function in the fermionic sector, all purelyfermionic loops already have nonvanishing initial values at thebeginning of the ﬂow. We see explicitly below that these arehighly singular and need to be treated with care. The exact hierarchy of FRG ﬂow equations for the one-line irreducible vertices of our model can be obtained as a specialcase of the general hierarchy of FRG ﬂow equations for mixedBose-Fermi theories written down in Refs. 36and40.F o r our purpose, it is sufﬁcient to consider a truncation of thishierarchy which generates, after iteration (apart from manyother diagrams), the important three-loop diagrams identiﬁedby Metlitski and Sachdev, 29shown in Fig. 1. Our truncation is characterized by the following three points. (i) On the right-hand side of the ﬂow equations for the fermionic and bosonic self-energies, retain only contributionsinvolving irreducible vertices with three external legs. (ii) Renormalize all three-legged vertices by triangular diagrams involving all combinations of three-legged vertices. (iii) On the right-hand side of the ﬂow equations for all three-legged vertices, approximate the vertex with one bosonicand two fermionic external legs by its bare value. Let us now explicitly give the corresponding FRG ﬂow equations. The fermionic self-energy /Sigma1 α(K) and bosonic self- energy /Pi1(¯K) satisfy the ﬂow equations ∂/Lambda1/Sigma1α(K)=/integraldisplay ¯K[˙F(¯K)Gα(K+¯K)+F(¯K)˙Gα(K+¯K)] ×/Gamma1¯ψαψαφ(K+¯K;K;¯K)/Gamma1¯ψαψαφ(K;K+¯K;−¯K), (3.5) ∂/Lambda1/Pi1(¯K)=/integraldisplay K/summationdisplay α,σ[˙Gα(K)Gα(K+¯K)+Gα(K)˙Gα(K+¯K)] ×/Gamma1¯ψαψαφ(K+¯K;K;¯K)/Gamma1¯ψαψαφ(K;K+¯K;−¯K) −/integraldisplay ¯K/prime˙F(¯K/prime)F(¯K/prime+¯K)/Gamma1φφφ(¯K,¯K/prime,−¯K−¯K/prime) ×/Gamma1φφφ(−¯K,−¯K/prime,¯K+¯K/prime), (3.6) which are shown graphically in Fig. 3for a general cutoff scheme. Here the scale-dependent bosonic and fermionicpropagators are F(¯K)=1 [F0,/Lambda1(¯K)]−1+/Pi1(¯K), (3.7) Gα(K)=1 /bracketleftbig Gα 0,/Lambda1(K)/bracketrightbig−1−/Sigma1α(K), (3.8) 245120-5DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) (a) (b) FIG. 3. (Color online) (a) Graphical representation of the FRG ﬂow equation (3.5) for the fermionic self-energy /Sigma1α(K), which is represented by a shaded rectangle with one incoming arrowassociated with ψ α Kσand one outgoing arrow associated with ¯ψα Kσ. The shaded triangles represent the three-legged vertex /Gamma1¯ψαψαφ(K+ ¯K;K;¯K) with two fermionic and one bosonic external legs. The boson propagator is represented by a wavy line. The black dot above the rectangle denotes a derivative with respect to the ﬂow parameter, while the slashes attached to the propagators on the right-hand side denote the corresponding single-scale propagators. (b) Graphical representation of the FRG ﬂow equation (3.6) for the bosonic self-energy /Pi1(¯K). The shaded circles on the right-hand side represent the symmetrized bosonic three-point vertex. Note that, in the momentum-transfer cutoff scheme, all diagrams with a slash oninternal fermionic propagators should be omitted. while the corresponding single-scale propagators are ˙F(¯K)=−F2(¯K)∂/Lambda1[F0,/Lambda1(¯K)]−1, (3.9) ˙Gα(K)=−[Gα(K)]2∂/Lambda1/bracketleftbig Gα 0,/Lambda1(K)/bracketrightbig−1. (3.10) Note that in the momentum-transfer cutoff scheme ˙Gα(K)=0 such that we should omit all diagrams involving fermionicsingle-scale propagators. With a sharp cutoff in the bosonictransverse momentum, the full bosonic propagator is F /Lambda1(¯K)=/Theta1(\|¯k⊥\|−/Lambda1) r0+c0\|¯k⊥\|zb−1+/Theta1(\|¯k⊥\|−/Lambda1)/Pi1/Lambda1(¯K),(3.11) while the corresponding single-scale propagator is given by ˙F/Lambda1(¯K)=−δ(\|¯k⊥\|−/Lambda1) r0+c0/Lambda1zb−1+/Pi1/Lambda1(¯K). (3.12) The right-hand sides of Eqs. (3.5) and(3.6) also depend on the one-line irreducible three-point vertex with two fermionicand one bosonic external legs, /Gamma1¯ψαψαφ(K+¯K;K;¯K), and on the one-line irreducible three-point vertex with three bosonicexternal legs, /Gamma1 φφφ(¯K,¯K/prime,−¯K−¯K/prime), where the superscripts indicate the ﬁelds associated with the energy-momentumlabels. Within our truncation, the ﬂow of /Gamma1¯ψαψαφis determined by the following ﬂow equation: ∂/Lambda1/Gamma1¯ψαψαφ(K+¯K;K;¯K) =/integraldisplay ¯K/prime[˙F(¯K/prime)Gα(K+¯K/prime)Gα(K+¯K+¯K/prime) +F(¯K/prime)˙Gα(K+¯K/prime)Gα(K+¯K+¯K/prime) +F(¯K/prime)Gα(K+¯K/prime)˙Gα(K+¯K+¯K/prime)] ×/Gamma1¯ψαψαφ(K+¯K/prime;K;¯K/prime) ×/Gamma1¯ψαψαφ(K+¯K;K+¯K+¯K/prime;−¯K/prime) ×/Gamma1¯ψαψαφ(K+¯K+¯K/prime;K+¯K/prime;¯K)FIG. 4. (Color online) Graphical representation of the FRG ﬂow equation (3.13) for the three-legged vertex /Gamma1¯ψαψαφ(K;K/prime;¯K) with two fermionic and one bosonic external legs. −/integraldisplay ¯K/prime[˙F(¯K/prime)F(¯K+¯K/prime)Gα(K+¯K+¯K/prime) +F(¯K/prime)˙F(¯K+¯K/prime)Gα(K+¯K+¯K/prime) +F(¯K/prime)F(¯K+¯K/prime)˙Gα(K+¯K+¯K/prime)] ×/Gamma1φφφ(−¯K−¯K/prime,¯K,¯K/prime) ×/Gamma1¯ψαψαφ(K+¯K;K+¯K+¯K/prime;−¯K/prime) ×/Gamma1¯ψαψαφ(K+¯K+¯K/prime;K;¯K+¯K/prime). (3.13) A graphical representation of this ﬂow equation is shown in Fig. 4. In the momentum-transfer cutoff scheme, we should omit, again, all terms involving the fermionic single-scalepropagator. Finally, the ﬂow equation for the symmetrizedbosonic three-legged vertex is ∂ /Lambda1/Gamma1φφφ(¯K1,¯K2,−¯K1−¯K2) =/integraldisplay ¯K[˙F(¯K)F(¯K−¯K1)F(¯K+¯K2)+F(¯K)˙F(¯K−¯K1) ×F(¯K+¯K2)+F(¯K)F(¯K−¯K1)˙F(¯K+¯K2)] ×/Gamma1φφφ(¯K1,¯K−¯K1,−¯K)/Gamma1φφφ(¯K2,−¯K−¯K2,¯K) ×/Gamma1φφφ(−¯K1−¯K2,−¯K+¯K1,¯K+¯K2) +/integraldisplay K/summationdisplay α,σ{[˙Gα(K)Gα(K+¯K1)Gα(K+¯K1+¯K2) +Gα(K)˙Gα(K+¯K1)Gα(K+¯K1+¯K2) +Gα(K)Gα(K+¯K1)˙Gα(K+¯K1+¯K2)] ×/Gamma1¯ψαψαφ(K+¯K1;K;¯K1) ×/Gamma1¯ψαψαφ(K+¯K1+¯K2;K+¯K1;¯K2) ×/Gamma1¯ψαψαφ(K;K+¯K1+¯K2;−¯K1−¯K2)+(¯K1↔¯K2)}, (3.14) which is shown graphically in Fig. 5. If we ignore the vertex with three bosonic external legs, the above system of FRG ﬂow equations has already beenderived in Ref. 40(see also Ref. 36). Although the above ﬂow equations involve only one-loop integrations, the iterativesolution of these equations generates higher-loop diagrams. Inparticular, the singular three-loop diagrams shown in Fig. 1 are generated as follows: The Aslamazov-Larkin type ofcontribution to the bosonic self-energy shown in Fig. 1(b) is generated by the last six diagrams on the right-hand sideof the ﬂow equation for /Gamma1 φφφ(¯K1,¯K2,¯K3) shown in Fig. 5 after integrating this ﬂow equation over the ﬂow parameterand substituting the result into the right-hand side of theﬂow equation for /Pi1(¯K) shown in Fig. 3(b). The singular 245120-6FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) FIG. 5. (Color online) Graphical representation of the FRG ﬂow equation (3.14) for the three-legged boson vertex /Gamma1φφφ(¯K1,¯K2,¯K3). fermionic three-loop diagram in Fig. 1(a) is generated by substituting the same contribution to /Gamma1φφφ(¯K1,¯K2,¯K3)i n t o the ﬂow equation for /Gamma1¯ψαψαφ(K;K/prime;¯K) shown in Fig. 4. After integrating the resulting ﬂow equation again over /Lambda1and substituting the resulting vertex correction into the right-handside of the ﬂow equation for /Sigma1 α(K) shown in Fig. 3(a),w e generate the Metlitski-Sachdev diagrams shown in Fig. 1(a). In fact, there are even further contributions; for example,using the renormalized vertex /Gamma1¯ψαψαφ(K;K/prime;¯K) containing the vertex /Gamma1φφφ(¯K1,¯K2,¯K3) as depicted in the second line of Fig. 4also gives rise to an Aslamazov-Larkin contribution when substituted into the diagram on the right-hand side ofFig.3(b). Although Eqs. (3.5) –(3.14) form a closed system of integro-differential equations for the two-point and three-pointfunctions of our model, a direct numerical solution of theseequations seems to be prohibitively difﬁcult, so that furtherapproximations are necessary. As a ﬁrst simpliﬁcation, weuse, below, truncated skeleton equations instead of the FRGﬂow equations (3.6) and (3.14) to determine the bosonic self-energy /Pi1(¯K) and three-point vertex /Gamma1 φφφ(¯K1,¯K2,¯K3). As discussed in Appendix A, the Dyson-Schwinger equa- tions of motion imply exact skeleton equations relating/Pi1(¯K) and/Gamma1 φφφ(¯K1,¯K2,¯K3) to the fermionic propagators, the three-point vertex /Gamma1¯ψαψαφ(K;K/prime;¯K) with two fermionic and one bosonic external legs, and the mixed four-point vertex/Gamma1¯ψψφφ(K;K/prime;¯K1;¯K2) [see Eqs. (A1) and(A3) ]. IV . TRUNCATION WITHOUT VERTEX CORRECTIONS In this section, we show how the known one-loop results for the self-energies can be obtained within our FRG approachif we neglect vertex corrections. Keeping in mind that, in themomentum-transfer cutoff scheme, we do not introduce anycutoff in the fermionic sector, the scale-dependent fermionicpropagator is G α /Lambda1(K)=1 iω−ξα k−/bracketleftbig /Sigma1α /Lambda1(K)−/Sigma1α /Lambda1(0)/bracketrightbig, (4.1) where /Sigma1α /Lambda1(0) is the self-energy at the renormalized ﬂowing Fermi surface. The subtraction of /Sigma1α /Lambda1(0) is necessary because, by assumption, we have expanded the wave vector at therenormalized Fermi surface of the underlying model. Given(a) (b) FIG. 6. (Color online) (a) Graphical representation of the FRG ﬂow equation (4.5) for the fermionic self-energy in the momentum- transfer cutoff scheme in the simplest approximation where the three- legged vertex is approximated by its bare value (represented by a blackdot). (b) Truncated skeleton equation (4.4) for the bosonic self-energy. the cutoff-dependent self-energy /Sigma1/Lambda1(K), we may deﬁne the Fermi surface for a given value of the cutoff parameter /Lambda1via /epsilon1kα F+/Sigma1/Lambda1/parenleftbig i0,kα F/parenrightbig =μ, (4.2) which, for /Lambda1→0, reduces to the deﬁnition of the true Fermi surface. Hence, we may write /epsilon1kα F+k−μ=/epsilon1kα F+k−/epsilon1kα F−/Sigma1/Lambda1/parenleftbig i0,kα F/parenrightbig =ξα k−/Sigma1α /Lambda1(0). (4.3) Following Refs. 43,44and 47, we now use the skeleton equation (A1) to determine the ﬂowing bosonic self-energy. Using the fact that ( /Gamma1α 0)2=1, the scale-dependent bosonic self-energy is thus given by /Pi1/Lambda1(¯K)=1 2/integraldisplay K/summationdisplay α,σGα /Lambda1(K)/bracketleftbig Gα /Lambda1(K+¯K)+Gα /Lambda1(K−¯K)/bracketrightbig , (4.4) while the fermionic self-energy satisﬁes the FRG ﬂow equation ∂/Lambda1/Sigma1α /Lambda1(K)=/integraldisplay ¯K˙F/Lambda1(¯K)Gα /Lambda1(¯K+K). (4.5) A graphical representation of Eqs. (4.4) and(4.5) is shown in Fig.6. Actually, the integral on the right-hand side of Eq. (4.5) is ultraviolet divergent for our model, but since we do not keeptrack of the shape of the renormalized Fermi surface, we caninstead consider the subtracted self-energy, /Delta1 α /Lambda1(K)=/Sigma1α /Lambda1(K)−/Sigma1α /Lambda1(0), (4.6) which appears in our cutoff-dependent fermionic propagator (4.1) . The subtracted self-energy then satisﬁes the FRG ﬂow equation ∂/Lambda1/Delta1α /Lambda1(K)=/integraldisplay ¯K˙F/Lambda1(¯K)/bracketleftbig Gα /Lambda1(¯K+K)−Gα /Lambda1(¯K)/bracketrightbig .(4.7) To obtain an approximate solution of this ﬂow equation, we expand the subtracted self-energy for small momenta and 245120-7DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) frequencies, /Delta1α /Lambda1(K)=−/parenleftbig Z−1 /Lambda1−1/parenrightbig iω+/parenleftbig˜Z−1 /Lambda1−1/parenrightbig ξα k+···,(4.8) where we have used the fact that the k-dependence of the self- energy appears only in the combination ξα k. Because the cutoff, /Lambda1, regularizes the singularity of the bare interaction, the self- energy is analytic for small momenta and frequencies and canhence be expanded into a Taylor series. The correspondinglow-energy approximation for the fermion propagator is G α /Lambda1(K)=1 Z−1 /Lambda1iω−˜Z−1 /Lambda1ξα k=Z/Lambda1 iω−(Z/Lambda1/˜Z/Lambda1)ξα k. (4.9) Substituting Eq. (4.9) into Eq. (4.4) , it is convenient to ﬁrst perform the integration over k/bardbl,29which can be done using the residue theorem, /Pi1/Lambda1(¯K)=N 2Z/Lambda1˜Z/Lambda1/integraldisplaydk⊥ 2π/summationdisplay αiα i¯ω−(Z/Lambda1/˜Z/Lambda1)(α¯k/bardbl+¯k2 ⊥+2¯k⊥k⊥)/integraldisplaydω 2π[/Theta1(α(ω+¯ω))−/Theta1(αω)]+(¯K→− ¯K).(4.10) The integration over ωis now trivial, /integraldisplaydω 2π[/Theta1(α(ω+¯ω))−/Theta1(αω)]=α¯ω 2π, (4.11) so that /Pi1/Lambda1(¯K)=N 4πZ/Lambda1˜Z/Lambda1/integraldisplaydk⊥ 2π/summationdisplay α/bracketleftbiggi¯ω i¯ω−(Z/Lambda1/˜Z/Lambda1)(α¯k/bardbl+¯k2 ⊥+2¯k⊥k⊥)+(¯K→− ¯K)/bracketrightbigg . (4.12) Note that the k⊥integral is still ultraviolet divergent. Following Mross et al. ,32we regularize the divergence by symmetrizing the integrand with respect to k⊥↔−k⊥, so that the expression in the square braces vanishes as 1 /k2 ⊥for large k⊥.T h e k⊥integration can thus again be done using the method of residues, with the result /Pi1/Lambda1(¯ω,¯k⊥)=b/Lambda1\|¯ω\| \|¯k⊥\|, (4.13) where b/Lambda1=N 4π˜Z2 /Lambda1. (4.14) Note that this is independent of ¯k/bardbl, the mathematical reason being that the term α¯k/bardblin the denominator of Eq. (4.12) can be eliminated by means of a simple shift of the integrationvariable k ⊥. We show shortly that at one-loop level, the self- energy /Sigma1α /Lambda1(K) is actually independent of k, so that ˜Z/Lambda1=1 and Eq. (4.13) reduces to /Pi1/Lambda1(¯ω,¯k⊥)=b0\|¯ω\| \|¯k⊥\|, (4.15) where b0=N 4π, (4.16) in agreement with Metlitski and Sachdev.29It should be noted that the bosonic self-energy does not renormalize the exponentz b, which characterizes the momentum dependence of the bare boson propagator.32 Next, we substitute Eq. (4.13) into our expression (3.12) for the bosonic single-scale propagator and obtain ˙F/Lambda1(¯K)≈−δ(\|¯k⊥\|−/Lambda1) r0+c0/Lambda1zb−1+b/Lambda1\|¯ω\|//Lambda1. (4.17)Substituting this expression into the FRG ﬂow equation (4.7) for the subtracted self-energy, we may perform all integrationson the right-hand side and obtain ∂ /Lambda1/Delta1α /Lambda1(K)=isgn(ω) πb/Lambda1/Lambda1 2π˜Z/Lambda1ln/bracketleftbigg 1+b/Lambda1\|ω\|//Lambda1 r0+c0/Lambda1zb−1/bracketrightbigg . (4.18) Relating both Z/Lambda1and ˜Z/Lambda1to ﬂowing anomalous dimensions via η/Lambda1=/Lambda1∂/Lambda1lnZ/Lambda1,˜η/Lambda1=/Lambda1∂/Lambda1ln˜Z/Lambda1, (4.19) the corresponding ﬂowing “frequency” anomalous dimension η/Lambda1is given by36 η/Lambda1=/Lambda1Z/Lambda1lim ω→0∂ ∂(iω)∂/Lambda1/Delta1α /Lambda1(iω,k=0) =/Lambda1Z/Lambda1˜Z/Lambda1 2π2(r0+c0/Lambda1zb−1). (4.20) Keeping in mind that the right-hand side of the ﬂow equa- tion(4.18) for the self-energy depends only on ωand not on k, we conclude that ˜Z/Lambda1=1 in this approximation, so that b/Lambda1=b0. At the quantum critical point, r0=0, our expression (4.20) for the ﬂowing frequency anomalous dimension hence reduces to η/Lambda1=Z/Lambda1 2π2c0/Lambda1zb−2, (4.21) so that Z/Lambda1satisﬁes the ﬂow equation /Lambda1∂/Lambda1Z/Lambda1=η/Lambda1Z/Lambda1=Z2 /Lambda1 2π2c0/Lambda1zb−2. (4.22) 245120-8FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) This differential equation can be easily integrated to give Z/Lambda1=Z/Lambda10 1+Z/Lambda10 2π2c0(zb−2)/parenleftbig /Lambda12−zb−/Lambda12−zb 0/parenrightbig. (4.23) Assuming zb>2, we see that, for /Lambda1→0, the wave-function renormalization factor vanishes as Z/Lambda1∼2π2c0(zb−2)/Lambda1zb−2, (4.24) implying that, at the quantum critical point, the fermion ﬁeld has the frequency anomalous dimension η=lim /Lambda1→0/Lambda1∂lnZ/Lambda1 ∂/Lambda1=zb−2. (4.25) Because, for ˜Z/Lambda1=1, the right-hand side of Eq. (4.18) depends on the ﬂow parameter /Lambda1only via the explicit /Lambda1dependence shown, we can obtain the ﬂowing subtracted self-energy/Delta1 α /Lambda1(K) for all frequencies by simply integrating both sides of Eq. (4.18) over/Lambda1, /Delta1α /Lambda1(K)−/Delta1α /Lambda10(K) =−/integraldisplay/Lambda10 /Lambda1dλ∂λ/Delta1α λ(K) =−isgn(ω) 2π2b0/integraldisplay/Lambda10 /Lambda1dλλ ln/bracketleftbigg 1+b0\|ω\|/λ r0+c0λzb−1/bracketrightbigg .(4.26) Forzb>2, the integral is ultraviolet convergent, so that we may take the limit /Lambda10→∞ , where /Delta1α /Lambda10(K)→0. At the quantum critical point, r0=0, the dependence of the integral onωcan be scaled out and we obtain for /Lambda1→0 lim /Lambda1→0/Delta1α /Lambda1(K)=−isgn(ω)a0 b0/parenleftbiggb0\|ω\| c0/parenrightbigg2/zb , (4.27) where a0=1 2π2zb/integraldisplay∞ 0dxln(1+x) x1+2/zb=1 4πsin(2π/zb),(4.28) in agreement with previous work.24,25,29–32The corresponding one-loop corrected fermion propagator is thus Gα(K)=1 iω−ξα k+isgn(ω)a0 b0/parenleftbigb0\|ω\| c0/parenrightbig2/zb ≈1 isgn(ω)a0 b0/parenleftbigb0\|ω\| c0/parenrightbig2/zb−αk/bardbl−k2 ⊥,(4.29) where we have used the fact that for zb>2 the Matsubara frequency iωis small compared with the contribution from the self-energy at low energies. Comparing Eq. (4.29) with the general scaling form (1.2) , we conclude that z=zb/2 and ηψ=0 within the one-loop approximation. We note that this is in agreement with the one-loop results of previous worksbased on the ﬁeld-theoretical renormalization group. 24,25,29–32 V . VERTEX CORRECTIONS In this section, we use the hierarchy of FRG ﬂow equations given in Sec. IIIto estimate the effect of vertex corrections on the low-frequency behavior of the fermionic self-energy,/Sigma1 α /Lambda1(K). We do not attempt to calculate the entire momentum and frequency dependence of /Sigma1α /Lambda1(K), but focus on the ﬂow ofthe two renormalization constants Z/Lambda1and ˜Z/Lambda1deﬁned via the low-energy expansion (4.8) of the ﬂowing self-energy, and on the associated anomalous dimensions η/Lambda1and ˜η/Lambda1deﬁned via the logarithmic derivatives of Z/Lambda1and ˜Z/Lambda1with respect to the ﬂow parameter [see Eq. (4.19) ]. Note that the deﬁnition (4.9) implies Z−1 /Lambda1=1−lim ω→0∂/Sigma1α /Lambda1(iω,k=0) ∂(iω), (5.1) ˜Z−1 /Lambda1=1+lim k→0∂/Sigma1α /Lambda1(iω=0,k) ∂ξα k, (5.2) while the deﬁnition (4.19) of the ﬂowing anomalous dimen- sions, η/Lambda1and ˜η/Lambda1, allows us to relate these quantities directly to the derivative of the self-energy with respect to the ﬂowparameter η /Lambda1=/Lambda1∂/Lambda1lnZ/Lambda1=Z/Lambda1/Lambda1lim ω→0∂ ∂(iω)∂/Lambda1/Sigma1α /Lambda1(iω,k=0),(5.3) ˜η/Lambda1=/Lambda1∂/Lambda1ln˜Z/Lambda1=− ˜Z/Lambda1/Lambda1lim k→0∂ ∂ξα k∂/Lambda1/Sigma1α /Lambda1(iω=0,k) =− ˜Z/Lambda1/Lambda1lim k→0∂ ∂(αk/bardbl)∂/Lambda1/Sigma1α /Lambda1(iω=0,k), (5.4) where in the last line we have used the fact that the momentum dependence of the self-energy appears only in thecombination ξ α k=αk/bardbl+k2 ⊥. In the one-loop approximation, lim/Lambda1→0˜η/Lambda1=0, but in this section we show that vertex corrections lead to a ﬁnite value of this limit. In Sec. VD we further show that lim /Lambda1→0˜η/Lambda1can be identiﬁed with the anomalous dimension ηψof the fermion ﬁeld deﬁned via Eq.(1.2) ; moreover, we show how to express the fermionic dynamic exponent zin terms of η/Lambda1and ˜η/Lambda1. A. Truncated ﬂow equations and skeleton equations Using the momentum-transfer cutoff scheme in combi- nation with the truncation strategy discussed in the thirdparagraph of Sec. III, we obtain the fermionic self-energy from ∂ /Lambda1/Sigma1α(K)=/integraldisplay ¯K˙F(¯K)Gα(K+¯K)/Gamma1¯ψαψαφ(K+¯K;K;¯K) ×/Gamma1¯ψαψαφ(K;K+¯K;−¯K), (5.5) where the three-point vertex with one bosonic and two fermionic legs is determined by ∂/Lambda1/Gamma1¯ψαψαφ(K+¯K;K;¯K) =/parenleftbig /Gamma1α 0/parenrightbig3/integraldisplay ¯K/prime˙F(¯K/prime)Gα(K+¯K/prime)Gα(K+¯K+¯K/prime) −/parenleftbig /Gamma1α 0/parenrightbig2/integraldisplay ¯K/prime˙F(¯K/prime)F(¯K+¯K/prime)[Gα(K+¯K+¯K/prime) +Gα(K−¯K/prime)]/Gamma1φφφ(¯K,¯K/prime,−¯K−¯K/prime). (5.6) Equation (5.5) can be obtained from the more general ﬂow equation (3.5) by simply omitting the contribution involving the fermionic single-scale propagator, while in Eq. (5.6) we have, in addition, replaced the ﬂowing vertices /Gamma1¯ψαψαφ(K+ ¯K;K;¯K) on the right-hand side of the ﬂow equation by their bare values. The purely bosonic three-legged vertex in the 245120-9DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) (a) (b) (c) FIG. 7. (Color online) (a) Graphical representation of the FRG ﬂow equation (5.5) for the fermionic self-energy in the momentum- transfer cutoff scheme. This ﬂow equation can be obtained from the more general ﬂow equation (3.5) by omitting the contribution involving the fermionic single-scale propagator. (b) Truncated ﬂowequation for the three-legged vertex /Gamma1 ¯ψαψαφ(K+¯K;K;¯K)i nt h e momentum-transfer cutoff scheme, which can be obtained from the more general ﬂow equation (3.13) s h o w ni nF i g . 4by neglecting the contribution from the fermionic single-scale propagator and approximating the vertices /Gamma1¯ψαψαφ(K+¯K;K;¯K) on the right- hand side by the bare vertices. (c) Graphical representation of theapproximate expression (5.7) for the bosonic three-legged vertex used in our calculation. Note that this expression can be obtained from the exact skeleton equation (A3) derived in Appendix Aby neglecting the irreducible four-point vertex and then making the same approximations as in (b). second line of Eq. (5.6) is approximated by /Gamma1φφφ(¯K1,¯K2,−¯K1−¯K2) =N/summationdisplay α/parenleftbig /Gamma1α 0/parenrightbig3/bracketleftbigg/integraldisplay KGα(K)Gα(K+¯K1)Gα(K+¯K1+¯K2) +/integraldisplay KGα(K)Gα(K+¯K1)Gα(K−¯K2)/bracketrightbigg , (5.7) which is derived from the exact skeleton equation by making the same approximations as in the derivation of Eqs. (5.5) and(5.6) . A graphical representation of Eqs. (5.5) –(5.7) is shown in Fig. 7. To obtain the corrections to the one-loop approximation, it is sufﬁcient to approximate all propagatorsin Eqs. (5.5) –(5.7) by the one-loop results given in Sec. IV; that is, we approximate on the right-hand sides G α /Lambda1(K)=Z/Lambda1 iω−Z/Lambda1ξα k, (5.8) F/Lambda1(¯K)=/Theta1(\|¯k⊥\|−/Lambda1) r0+c0\|¯k⊥\|zb−1+/Theta1(\|¯k⊥\|−/Lambda1)b0\|¯ω\|/\|¯k⊥\|,(5.9) ˙F/Lambda1(¯K)=−δ(\|¯k⊥\|−/Lambda1) r0+c0/Lambda1zb−1+b0\|¯ω\|//Lambda1. (5.10) Here,Z/Lambda1is the one-loop result for the ﬂowing wave-function renormalization factor given in Eq. (4.23) and we have used the fact that ˜Z/Lambda1=1 within the one-loop approximation. We thus arrive at a system of ﬂow equations for the momentum-and frequency-dependent two-point and three-point functions.Having determined the right-hand side of the ﬂow equa- tion(5.5) for the self-energy, we can substitute the result into Eqs. (5.3) and (5.4) and obtain for the ﬂowing anomalous dimension related to the frequency dependence of the self-energy η /Lambda1=Z/Lambda1/Lambda1lim K→0∂ ∂(iω)/integraldisplay ¯K˙F/Lambda1(¯K)/bracketleftbig Gα /Lambda1(¯K+K)−Gα /Lambda1(¯K)/bracketrightbig ×/bracketleftbig /Gamma1¯ψαψαφ /Lambda1 (¯K;0 ;¯K)/bracketrightbig2 +2Z/Lambda1/Lambda1/integraldisplay ¯K˙F/Lambda1(¯K)Gα /Lambda1(¯K)/Gamma1¯ψαψαφ /Lambda1 (¯K;0 ;¯K) ×lim K→0∂/Gamma1¯ψαψαφ /Lambda1 (¯K+K;K;¯K) ∂(iω), (5.11) and for the corresponding anomalous dimension associated with the momentum dependence of the self-energy, ˜η/Lambda1=− ˜Z/Lambda1/Lambda1lim K→0∂ ∂(αk/bardbl)/integraldisplay ¯K˙F/Lambda1(¯K)/bracketleftbig Gα /Lambda1(¯K+K)−Gα /Lambda1(¯K)/bracketrightbig ×/bracketleftbig /Gamma1¯ψαψαφ /Lambda1 (¯K;0 ;¯K)/bracketrightbig2 −2˜Z/Lambda1/Lambda1/integraldisplay ¯K˙F/Lambda1(¯K)Gα /Lambda1(¯K)/Gamma1¯ψαψαφ /Lambda1 (¯K;0 ;¯K) ×lim K→0∂/Gamma1¯ψαψαφ /Lambda1 (¯K+K;K;¯K) ∂(αk/bardbl). (5.12) In the ﬁrst terms on the right-hand sides of these expressions we have used the same regularization as in Eq. (4.7) . B. Bosonic three-legged vertex In order to evaluate the self-energy from Eq. (5.5) ,w e should calculate the mixed fermion-boson vertex /Gamma1¯ψαψαφ(K+ ¯K;K;¯K) by integrating the ﬂow equation (5.6) , which in turn depends on the purely bosonic three-legged vertex/Gamma1 φφφ(¯K1,¯K2,−¯K1−¯K2). Fortunately, the integrations in the truncated skeleton equation (5.7) can be carried out exactly in our model, as shown in Appendix B. The result can be written as /Gamma1φφφ(¯K1,¯K2,−¯K1−¯K2) =2!N/summationdisplay α/parenleftbig /Gamma1α 0/parenrightbig3Lα3(¯K1,¯K2,−¯K1−¯K2),(5.13) where the symmetrized fermion loop with three external bosonic legs and one-loop renormalized fermionic propagatorsis given by L α 3(¯K1,¯K2,−¯K1−¯K2) =1 4π/parenleftbigg1 ¯k⊥1+1 ¯k⊥2/parenrightbigg ×¯ω1/Theta1/parenleftbig¯ω1¯k⊥1/parenrightbig +¯ω2/Theta1/parenleftbig¯ω2¯k⊥2/parenrightbig −(¯ω1+¯ω2)/Theta1/parenleftbig¯ω1+¯ω2¯k⊥1+¯k⊥2/parenrightbig /bracketleftbig¯k/bardbl1 ¯k⊥1−¯k/bardbl2 ¯k⊥2−iα Z/parenleftbig¯ω1¯k⊥1−¯ω2¯k⊥2/parenrightbig/bracketrightbig2−(¯k⊥1+¯k⊥2)2. (5.14) Note that this function represents a rather complicated momentum- and frequency-dependent effective interactionbetween the bosonic ﬂuctuations, mediated by the fermions.Obviously, this function cannot simply be approximated by a 245120-10FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) constant, which is assumed to be possible in the Hertz-Millis approach to quantum critical phenomena.10,11 C. Three-legged boson-fermion vertex We have now calculated all functions appearing on the right- hand side of the FRG ﬂow equation (5.6) for the three-legged vertex with one bosonic and two fermionic external legs, so thatwe may next integrate this equation over the ﬂow parameter/Lambda1. Let us begin by evaluating the ﬁrst term on the right-hand side of Eq. (5.6) , ∂ /Lambda1/Gamma1¯ψαψαφ(K+¯K;K;¯K)(1) =/parenleftbig /Gamma1α 0/parenrightbig3/integraldisplay ¯K/prime˙F(¯K/prime)Gα(K+¯K/prime)Gα(K+¯K+¯K/prime), (5.15) corresponding to the ﬁrst diagram on the right-hand side of Fig. 7(b). The integrations in Eq. (5.15) can be explicitlycarried out, with the result /Lambda1∂/Lambda1/Gamma1¯ψαψαφ(K+¯K;K;¯K)(1) =−/parenleftbig /Gamma1α 0/parenrightbig3 (2π)2Z/Lambda1 b0/bracketleftbigg/Lambda12 i¯ω−Z/Lambda1[α¯k/bardbl+¯k2 ⊥+2¯k⊥(k⊥+/Lambda1)] +(/Lambda1→−/Lambda1)/bracketrightbigg ×/bracketleftbigg isgn(ω+¯ω)l n/parenleftbigg 1+b0\|ω+¯ω\| r0/Lambda1+c0/Lambda1zb/parenrightbigg −isgn(ω)l n/parenleftbigg 1+b0\|ω\| r0/Lambda1+c0/Lambda1zb/parenrightbigg/bracketrightbigg . (5.16) For the evaluation of the ﬂowing anomalous dimensions in Eqs. (5.11) and (5.12) , we need only the vertex /Gamma1¯ψαψαφ(¯K;0 ;¯K) at vanishing fermionic energy momentum, as well as the derivatives of /Gamma1¯ψαψαφ(¯K+K;K;¯K) with respect to the components of KatK=0. From Eq. (5.16) ,w es e e that the contribution of the ﬁrst diagram in Fig. 7(b)to the ﬂow of these quantities can be written as /Lambda1∂/Lambda1/Gamma1¯ψαψαφ(¯K;0 ;¯K)(1)=−I(1) /Lambda1/parenleftbigg¯ω /Lambda1zb,¯k/bardbl /Lambda12,¯k⊥ /Lambda1/parenrightbigg , (5.17) /Lambda1∂/Lambda1∂/Gamma1¯ψαψαφ(¯K;K;¯K)(1) ∂(iω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle K=0=−1 /Lambda1zbI(1ω) /Lambda1/parenleftbigg¯ω /Lambda1zb,¯k/bardbl /Lambda12,¯k⊥ /Lambda1/parenrightbigg , (5.18) /Lambda1∂/Lambda1∂/Gamma1¯ψαψαφ(¯K;K;¯K)(1) ∂(αk/bardbl)/vextendsingle/vextendsingle/vextendsingle/vextendsingle K=0=0, (5.19) where we have deﬁned the dimensionless scaling functions I(1) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=γ/Lambda1 (2π)2b0isgn(¯/epsilon1)l n/parenleftbigg 1+b0\|¯/epsilon1\| r/Lambda1+c0/parenrightbigg/bracketleftbigg1 i¯/epsilon1−γ/Lambda1[α¯q/bardbl+¯q2 ⊥+2¯q⊥]+(¯q⊥→− ¯q⊥)/bracketrightbigg , (5.20) and I(1ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=−γ/Lambda1 (2π)2(r/Lambda1+c0)2b0 1+b0 r/Lambda1+c0\|¯/epsilon1\|/bracketleftbigg\|¯/epsilon1\| i¯/epsilon1−γ/Lambda1[α¯q/bardbl+¯q2 ⊥+2¯q⊥]+(¯q⊥→− ¯q⊥)/bracketrightbigg . (5.21) Here,r/Lambda1=r0/Lambda11−zbvanishes at the quantum critical point, and we have introduced the dimensionless parameter γ/Lambda1=Z/Lambda1 /Lambda1zb−2=2π2c0(zb−2) 1−/Lambda1zb−2/bracketleftbig2π2c0(zb−2) Z/Lambda10−/Lambda12−zb 0/bracketrightbig, (5.22) which approaches the limit 2 π2c0(zb−2)∝N(zb−2) for /Lambda1→0. Next, consider the contribution of the last two diagrams on the right-hand side of Fig. 7(b) to the ﬂow of the three-legged boson-fermion vertex, corresponding to the second line in Eq. (5.6) , ∂/Lambda1/Gamma1¯ψαψαφ(K+¯K;K;¯K)(2)=−/parenleftbig /Gamma1α 0/parenrightbig2/integraldisplay ¯K/prime˙F(¯K/prime)F(¯K+¯K/prime)[Gα(K+¯K+¯K/prime)+Gα(K−¯K/prime)]/Gamma1φφφ(¯K,¯K/prime,−¯K−¯K/prime). (5.23) Due to the rather complicated form of the vertex /Gamma1φφφ(¯K,¯K/prime,−¯K−¯K/prime)g i v e ni nE q s . (5.13) and(5.14) , the evaluation of the right-hand side of Eq. (5.23) is quite involved. The ¯k/prime /bardblintegration can still be performed by means of the residue theorem, while the¯k/prime ⊥integration is trivial due to the δfunction in the single-scale propagator. After these integrations, we obtain ∂/Lambda1/Gamma1¯ψαψαφ(K+¯K;K;¯K)(2)=−iαN/Lambda12/parenleftbig /Gamma1α 0/parenrightbig2 (2π)2/integraldisplay∞ −∞d¯ω/prime 2π1 r0+c0/Lambda1zb−1+b0\|¯ω/prime\|//Lambda1/summationdisplay s=±F(¯ω+¯ω/prime,¯k⊥+s/Lambda1)J(¯ω,¯k⊥,¯ω/prime,s/Lambda1) ×/summationdisplay α/prime=±/parenleftbig /Gamma1α/prime 0/parenrightbig3/bracketleftbigg/Theta1(Im(z1))−/Theta1(Im(z3)) (z1−z3)2−z2 4−/Theta1(Im(z2))−/Theta1(Im(z3)) (z2−z3)2−z2 4/bracketrightbigg , (5.24) 245120-11DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) where J(¯ω,¯k⊥,¯ω/prime,¯k/prime ⊥)=/parenleftbigg1 ¯k⊥+1 ¯k/prime ⊥/parenrightbigg/bracketleftbigg ¯ω/Theta1/parenleftbigg¯ω ¯k⊥/parenrightbigg +¯ω/prime/Theta1/parenleftbigg¯ω/prime ¯k/prime ⊥/parenrightbigg −(¯ω+¯ω/prime)/Theta1/parenleftbigg¯ω+¯ω/prime ¯k⊥+¯k/prime ⊥/parenrightbigg/bracketrightbigg , (5.25) and z1=−(k/bardbl+¯k/bardbl)−α[/Lambda1+s(k⊥+¯k⊥)]2+i¯ω/prime+i¯ω+iω Z/Lambda1, (5.26a) z2=k/bardbl+α(/Lambda1−sk⊥)2+αi¯ω/prime−iω Z/Lambda1, (5.26b) z3=/Lambda1/bracketleftbigg¯k/bardbl s¯k⊥−α/prime Z/Lambda1/parenleftbiggi¯ω s¯k⊥−i¯ω/prime /Lambda1/parenrightbigg/bracketrightbigg , (5.26c) z4=/Lambda1(/Lambda1+s¯k⊥). (5.26d) For the calculation of η/Lambda1and ˜η/Lambda1in Eqs. (5.11) and(5.12) we again need only the vertex /Gamma1¯ψαψαφ(¯K;0 ;¯K) and the derivatives of /Gamma1¯ψαψαφ(¯K+K;K;¯K) with respect to the components of the fermionic label KatK=0. In analogy with Eqs. (5.17) –(5.19) , we therefore deﬁne /Lambda1∂/Lambda1/Gamma1¯ψαψαφ(¯K;0 ;¯K)(2)=−I(2) /Lambda1/parenleftbigg¯ω /Lambda1zb,¯k/bardbl /Lambda12,¯k⊥ /Lambda1/parenrightbigg , (5.27) /Lambda1∂/Lambda1∂/Gamma1¯ψαψαφ(¯K;K;¯K)(2) ∂(iω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle K=0=−1 /Lambda1zbI(2ω) /Lambda1/parenleftbigg¯ω /Lambda1zb,¯k/bardbl /Lambda12,¯k⊥ /Lambda1/parenrightbigg , (5.28) /Lambda1∂/Lambda1∂/Gamma1¯ψαψαφ(¯K;K;¯K)(2) ∂(αk/bardbl)/vextendsingle/vextendsingle/vextendsingle/vextendsingle K=0=−1 /Lambda12I(2k) /Lambda1/parenleftbigg¯ω /Lambda1zb,¯k/bardbl /Lambda12,¯k⊥ /Lambda1/parenrightbigg . (5.29) From now on we focus on the Ising-nematic instability and explicitly set /Gamma1α 0=1. In principle, the functions I(2) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥), I(2ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥), andI(2k) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥) can be calculated analytically by performing the remaining frequency integration in Eq. (5.24) . However, the result is rather cumbersome and not very transparent. To simplify the calculation, let us assume that the arguments ¯/epsilon1,¯q/bardbl, and ¯q⊥of these functions are all small compared with unity. Then the resulting expressions simplify and we obtain the approximate expressions I(2) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)≈γ/Lambda1\|¯/epsilon1\|N 2(2π)3(r/Lambda1+c0)2/bracketleftbigg1 \|¯q⊥\|−w−\|¯q⊥\| (\|¯q⊥\|−w)2ln/parenleftbigg\|¯q⊥\| w/parenrightbigg/bracketrightbigg , (5.30) I(2ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)≈γ/Lambda1N (2π)3(r/Lambda1+c0)2/bracketleftBigg αγ/Lambda1¯q/bardbl\|¯/epsilon1\| ¯/epsilon12+γ2 /Lambda1¯q2 /bardbl−isgn(¯/epsilon1)\|¯q⊥\|¯/epsilon12/parenleftbig ¯/epsilon12−γ2 /Lambda1¯q2 /bardbl/parenrightbig /parenleftbig ¯/epsilon12+γ2 /Lambda1¯q2 /bardbl/parenrightbig2/bracketrightBigg , (5.31) I(2k) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)≈isgn(¯/epsilon1)γ2 /Lambda1\|¯q⊥\|N (2π)3(r/Lambda1+c0)2¯/epsilon12/parenleftbig ¯/epsilon12−γ2 /Lambda1¯q2 /bardbl/parenrightbig /parenleftbig ¯/epsilon12+γ2 /Lambda1¯q2 /bardbl/parenrightbig2, (5.32) where, in Eq. (5.30) , the complex quantity wis deﬁned by w=b0 2(r/Lambda1+c0)(\|¯/epsilon1\|−iαγ/Lambda1¯q/bardblsgn(¯/epsilon1)). (5.33) Let us now combine the contributions from all three diagrams on the right-hand side of Fig. 7(b). To be consistent with the approximations made in the derivation of Eqs. (5.30) –(5.32) , we should also expand the right-hand sides of I(1) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥) and I(1ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)i nE q s . (5.20) and(5.21) for small ¯ /epsilon1. We therefore deﬁne I/Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=I(1) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)+I(2) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=γ/Lambda1 (2π)2(r/Lambda1+c0)/braceleftbiggi¯/epsilon1 i¯/epsilon1−γ/Lambda1[α¯q/bardbl+¯q2 ⊥+2¯q⊥]+i¯/epsilon1 i¯/epsilon1−γ/Lambda1[α¯q/bardbl+¯q2 ⊥−2¯q⊥] +N 4π(r/Lambda1+c0)/bracketleftbigg\|¯/epsilon1\| \|¯q⊥\|−w−\|¯q⊥\|\|¯/epsilon1\| (\|¯q⊥\|−w)2ln/parenleftbigg\|¯q⊥\| w/parenrightbigg/bracketrightbigg/bracerightbigg , (5.34a) 245120-12FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) I(ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=I(1ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)+I(2ω) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=γ/Lambda1b/Lambda1 (2π)2(r/Lambda1+c0)2/braceleftbigg −\|¯/epsilon1\| i¯/epsilon1−γ/Lambda1[α¯q/bardbl+¯q2 ⊥+2¯q⊥]−\|¯/epsilon1\| i¯/epsilon1−γ/Lambda1[α¯q/bardbl+¯q2 ⊥−2¯q⊥] +N 2πb/Lambda1/bracketleftbiggαγ/Lambda1¯q/bardbl\|¯/epsilon1\| ¯/epsilon12+γ2 /Lambda1¯q2 /bardbl−\|¯q⊥\|i¯/epsilon1\|¯/epsilon1\|/parenleftbig ¯/epsilon12−γ2 /Lambda1¯q2 /bardbl/parenrightbig /parenleftbig ¯/epsilon12+γ2 /Lambda1¯q2 /bardbl/parenrightbig2/bracketrightbigg/bracerightbigg , (5.34b) I(k) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=I(2k) /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=γ2 /Lambda1\|¯q⊥\|N (2π)3(r/Lambda1+c0)2i¯/epsilon1\|¯/epsilon1\|/parenleftbig ¯/epsilon12−γ2 /Lambda1¯q2 /bardbl/parenrightbig /parenleftbig ¯/epsilon12+γ2 /Lambda1¯q2 /bardbl/parenrightbig2. (5.34c) Finally, we integrate over the ﬂow parameter /Lambda1and obtain the following expressions for the three-legged boson-fermion vertex: /Gamma1¯ψαψαφ(¯K;0 ;¯K)=˜/Gamma1/Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥), (5.35a) ∂/Gamma1¯ψαψαφ(¯K;K;¯K) ∂(iω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle K=0=1 /Lambda1zb˜/Gamma1ω /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥), (5.35b) ∂/Gamma1¯ψαψαφ(¯K;K;¯K) ∂(αk/bardbl)/vextendsingle/vextendsingle/vextendsingle/vextendsingle K=0=1 /Lambda12˜/Gamma1k /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥), (5.35c) where ˜/Gamma1/Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=1+/integraldisplay1 /Lambda1//Lambda1 0ds sI/Lambda1/s(szb¯/epsilon1,s2¯q/bardbl,s¯q⊥), (5.36a) ˜/Gamma1ω /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=/integraldisplay1 /Lambda1//Lambda1 0dsszb−1Iω /Lambda1/s(szb¯/epsilon1,s2¯q/bardbl,s¯q⊥), (5.36b) ˜/Gamma1k /Lambda1(¯/epsilon1,¯q/bardbl,¯q⊥)=/integraldisplay1 /Lambda1//Lambda1 0dssIk /Lambda1/s(szb¯/epsilon1,s2¯q/bardbl,s¯q⊥). (5.36c) Recall that in deriving these expressions we have assumed that \|¯/epsilon1\|/lessorsimilar1. D. Fermionic anomalous dimension and dynamic exponent We are now ready to calculate the anomalous dimensions η=lim/Lambda1→0η/Lambda1and ˜η=lim/Lambda1→0˜η/Lambda1at the quantum critical point. We therefore substitute Eqs. (5.35a) –(5.35c) into our general relations (5.11) and(5.12) for the ﬂowing anomalous dimensions η/Lambda1and ˜η/Lambda1and, introducing the integration variables p=γ/Lambda1α¯q/bardblandy=¯/epsilon1, we obtain for the ﬂowing anomalous dimension associated with the frequency dependence of the self-energy, η/Lambda1=−γ/Lambda1 2πlim ω→0∂ ∂(iω)/integraldisplayλ0 −λ0dy 2π/integraldisplay∞ −∞dp 2π/bracketleftbigg1 r/Lambda1+c0+b0\|y\|/bracketrightbigg/bracketleftbigg1 iy+iω−p−γ/Lambda1−1 iy−p−γ/Lambda1/bracketrightbigg/bracketleftbigg ˜/Gamma1/Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg/bracketrightbigg2 −γ/Lambda1 π/integraldisplayλ0 −λ0dy 2π/integraldisplay∞ −∞dp 2π/bracketleftbigg1 r/Lambda1+c0+b0\|y\|/bracketrightbigg/bracketleftbigg1 iy−p−γ/Lambda1/bracketrightbigg ˜/Gamma1/Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg ˜/Gamma1ω /Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg , (5.37) and for the corresponding anomalous dimension associated with the momentum dependence of the self-energy, ˜η/Lambda1=γ/Lambda1 2πlim k→0∂ ∂k/integraldisplayλ0 −λ0dy 2π/integraldisplay∞ −∞dp 2π/bracketleftbigg1 r/Lambda1+c0+b0\|y\|/bracketrightbigg/bracketleftbigg1 iy−p−k−γ/Lambda1−1 iy−p−γ/Lambda1/bracketrightbigg/bracketleftbigg ˜/Gamma1/Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg/bracketrightbigg2 +1 π/integraldisplayλ0 −λ0dy 2π/integraldisplay∞ −∞dp 2π/bracketleftbigg1 r/Lambda1+c0+b0\|y\|/bracketrightbigg/bracketleftbigg1 iy−p−γ/Lambda1/bracketrightbigg ˜/Gamma1/Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg ˜/Gamma1k /Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg . (5.38) Here,λ0is an ultraviolet cutoff of the order of unity which takes into account that our expressions (5.34a) –(5.34c) , which we use to calculate the vertices in Eqs. (5.36a) –(5.36c) , are valid only for small frequencies. Consider now the limit /Lambda1→0. At the quantum critical point we may then set r/Lambda1→0. For simplicity, we also set c0=b0= N/(4π). To make progress analytically, let us further assume that the parameter γ=lim/Lambda1→0γ/Lambda1=2π2c0(zb−2)=π 2N(zb−2) is small compared with unity. To leading order in zb−2t h esintegrations in Eqs. (5.36a) –(5.36c) can then be carried out analytically, with the result lim /Lambda1→0˜/Gamma1/Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg =1+γ (2π)2c0/braceleftBigg iy iy−p−γln/bracketleftbigg 1−/parenleftbiggiy−p−γ 2γ/parenrightbigg2/bracketrightbigg −iyN 2πc0/bracketleftBiggln/bracketleftbig2isgn(y) iy+p/bracketrightbig 2isgn(y)−(iy+p)+2ln/bracketleftbig 1−iy+p 2isgn(y)/bracketrightbig iy+p/bracketrightBigg/bracerightBigg , (5.39) 245120-13DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) lim /Lambda1→0˜/Gamma1ω /Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg =γ (2π)2c0/braceleftbigg −\|y\| iy−p+N 2πc0/bracketleftbigg\|y\|p y2+p2−iy\|y\|(y2−p2) 3(y2+p2)2/bracketrightbigg/bracerightbigg , (5.40) lim /Lambda1→0˜/Gamma1k /Lambda1/parenleftbigg y,p αγ/Lambda1,1/parenrightbigg =γ2N (2π)3c2 0iy\|y\|(y2−p2) 3(y2+p2)2. (5.41) Note that these vertex functions describe the renormalized effective interaction between two fermions and one boson;clearly, this interaction has a rather complicated dependenceon momenta and frequencies and cannot be approximatedby a constant. If we now substitute Eqs. (5.39) –(5.41) into Eqs. (5.37) and (5.38) , we may perform the pintegration analytically by means of the method of residues. Note thatthe ﬁrst term in the curly braces of Eq. (5.39) and the ﬁrst term in the curly braces of Eq. (5.40) do not contribute to the integrals, because we may close the integration contourin a half plane where the integrand is analytic. These termsarise from the ﬁrst diagram on the right-hand side of Fig. 7(b), so that the three-boson vertex is crucial to obtain the leadingcorrections to the anomalous dimensions. We ﬁnally arrive atthe following estimates for the anomalous dimensions at thenematic quantum critical point, η=z b−2+(zb−2)2 2C(λ0)+O((zb−2)3),(5.42) ˜η=(zb−2)2 2˜C(λ0)+O((zb−2)3), (5.43) where the cutoff functions are given by C(λ0)=/integraldisplayλ0 0dy 1+y/bracketleftbigg7 6+1 1−y+ylny (1−y)2/bracketrightbigg ,(5.44) ˜C(λ0)=/integraldisplayλ0 0dy 1+y/bracketleftbigg1 6+1 1−y+ylny (1−y)2/bracketrightbigg =C(λ0)−ln(1+λ0). (5.45) A plot of the cutoff functions is shown in Fig. 8. Note that these functions depend logarithmically on the ultraviolet cutoff, λ0. This is due to the fact that we have assumed that the rescaledfrequencies and momenta in Eqs. (5.30) –(5.32) are small compared with unity in the evaluation of the vertex-correctiondiagrams shown in Fig. 7(b). By retaining the full momentum and frequency dependence on the right-hand side of the ﬂow FIG. 8. (Color online) Plot of the cutoff functions C(λ0)a n d ˜C(λ0) deﬁned in Eqs. (5.44) and(5.45) .equation (5.24) , one obtains ultraviolet convergent results from our FRG approach. However, because the cutoff dependencein our ﬁnal result [Eqs. (5.42) and(5.43) ] is only logarithmic, we have not attempted to carry out this calculation whichrequires substantial numerical effort. Instead, we make thereasonable cutoff choice λ 0=1 and ﬁnd, within the accuracy of our calculation, that C≈1.27 and ˜C≈0.58. Given our result for the anomalous dimensions ηand ˜η, we may relate these to the anomalous dimension ηψof the fermion ﬁeld and the fermionic dynamic exponent, z. Let us therefore recall that, at the quantum critical point, the retardedsingle-particle Green’s function assumes for small frequenciesand momenta the following scaling form: G α(ω+i0+,k)∝1 /bracketleftbig Aωsgn(ω)\|ω\|1/z−ξα k/bracketrightbig1−ηψ/2,(5.46) where Aωis some dimensionful constant with positive imag- inary part and real part depending on the sign of ω[see Eq.(1.2) ]. Forω=k⊥=0, this implies Gα(i0+,k/bardbl,0)∝k−1+ηψ/2 /bardbl . (5.47) On the other hand, from the deﬁnition of ˜ η/Lambda1in Eq. (4.19) we see that ˜Z/Lambda1∝/Lambda1˜η∝k˜η/2 /bardbl, where we have used the fact that k/bardbl scales as /Lambda12. Hence, Gα(i0+,k/bardbl,0)∝˜Z/Lambda1/k/bardbl∝k−1+˜η/2 /bardbl . (5.48) Comparing this with Eq. (5.47) , we conclude that ηψ=˜η. (5.49) Next, setting k=0i nE q . (5.46) , we ﬁnd Gα(ω+i0+,0)∝ω−(1−ηψ/2)/z. (5.50) On the other hand, from the deﬁnition of η/Lambda1in Eq. (4.19) we infer that Z/Lambda1∝/Lambda1η∝ωη/(2z)(using the fact that ωscales as /Lambda12z), so that Gα(0,ω+i0+)∝Z/Lambda1/ω∝ω−(1−η/(2z)). (5.51) Comparing this with Eq. (5.50) , we see that 1 −η/(2z)= (1−ηψ/2)/z,o r z=1+η−˜η 2. (5.52) Writing Eq. (5.42) asη=zb−2+δηwith δη=(zb−2)2 2C(λ0)+O((zb−2)3), (5.53) 245120-14FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) we see that Eq. (5.52) can also be written as z=zb 2+δη−˜η 2 =zb 2+(zb−2)2 4[C(λ0)−˜C(λ0)]+O((zb−2)3). (5.54) According to the above calculation, the scaling relation z= zb/2 for the fermionic dynamic exponent acquires a correction of order ( zb−2)2, such that the effective theory appears to have two different time scales. However, according to Metlitski andSachdev, 29the strong correlations imply that z=zb/2 should be satisﬁed exactly. This discrepancy might be due to thefact that our truncation of the FRG ﬂow equations introducesapproximations which violate the general scaling theory. Infact, this seems to be true also in the case of the loop expansion.In the previous works of Metlitski and Sachdev 29and Mross et al. ,32the three-loop correction to the fermionic anomalous dimension ηψis calculated via the momentum dependence of the self-energy, and according to scaling theory it is argued thata corresponding correction should appear if one calculates η ψ via the frequency dependence of the self-energy. However, as pointed out in Ref. 31, the correction in the frequency dependence of the self-energy (the term δηin our notation) appears only beyond the three-loop order. The FRG is notbased on an expansion in powers of loops, so that diagramsbeyond three loops are included in our truncation. VI. SUMMARY AND CONCLUSIONS In this work, we have used an FRG approach to calculate the anomalous dimension ηψof the fermion ﬁeld and the fermionic dynamic exponent zof an effective low-energy ﬁeld theory describing the Ising-nematic quantum critical point intwo-dimensional metals. In the limit N(z b−2)/lessmuch1 (where Nis the number of fermion ﬂavors and zbis the bosonic dynamic exponent), we have been able to explicitly calculatethe fermionic anomalous dimension η ψof the system, with the result ηψ≈0.3(zb−2)2+O((zb−2)3). (6.1) If we extrapolate this expression to the physically relevant case, zb=3, we obtain ηψ≈0.3, which is larger than the estimate ηψ≈0.068 given by Metlitski and Sachdev,29but still smaller than the estimate ηψ≈0.6 obtained from an extrapolation of the corresponding expression given by Mross et al.32Given the different types of approximations, it is not surprising thatdifferent values for the exponent η ψare found. This fact shows that the recent calculations done so far are not yet fully undercontrol in the physically relevant case of z b=3 andN=2. It is, however, reassuring that our method also ﬁnds a ﬁnite valueforη ψ. In particular, we note that in the previous calculations based on the ﬁeld-theoretical renormalization group, the valuesof the exponent η ψare obtained within a loop-expansion truncated at the three-loop order, while in our FRG approachthe truncation does not rely on a loop expansion, so thata certain class of diagrams beyond the three-loop level areeffectively resummed to higher order within our truncation.On the other hand, at the lowest order, namely at the one-looplevel in the ﬁeld-theoretical RG and neglecting the vertexcorrections in the FRG, the results obtained using the FRG and other renormalization group methods all coincide withthe well-known results obtained within the RPA. Finally, incontrast to previous works, 29,32we have explicitly shown that both the frequency and the momentum dependence of theself-energy give rise to anomalous corrections to the one-loopresult. While our calculations, in principle, lead to a smallcorrection term to the fermionic dynamic critical exponent, z, the scaling theory of Metlitski and Sachdev 29implies the exact identity z=zb/2. The calculations presented in this work can be extended in several directions. Because our FRG approach does notrely on the smallness of the parameter N(z b−2), with some numerical effort it should be possible to extract the anomalousdimension η ψfor general Nandzb−2 from Eqs. (5.37) and (5.38) . In this case the vertices appearing in these expressions cannot be calculated analytically but must be represented asone-dimensional integrals, so that the evaluation of the anoma-lous dimensions in Eqs. (5.37) and(5.38) requires rather com- plicated numerical integrations, which is beyond the scope ofthis work. In principle, our FRG ﬂow equations also allow us tocalculate the entire frequency dependence of the self-energies,but this seems to be numerically even more expensive. In contrast to the strategy adopted in the Hertz-Millis approach, 10,11in the present problem it is not possible to integrate over the fermionic degrees of freedom to obtainan effective bosonic theory with regular vertices. We havetherefore explicitly retained both bosonic and fermionicdegrees of freedom in our FRG calculation. The FRG approachdeveloped in this work should also be useful to discussother model systems where gapless fermionic and bosonicexcitations are strongly coupled. ACKNOWLEDGMENTS We would like to thank Max Metlitski for useful discus- sions. This work was ﬁnancially supported by the DFG viaFOR 723. APPENDIX A: SKELETON EQUATIONS In Secs. IVandV, we have combined FRG ﬂow equations with skeleton equations for the bosonic self-energy and thebosonic three-legged vertex to obtain a closed system ofequations. In this Appendix, we brieﬂy describe the derivationof these skeleton equations. Skeleton equations relating vertex functions of different order follow from the general Dyson-Schwinger equationfor the generating functional for the connected Green’sfunction, which is a simple consequence of the invarianceof the integration measure of the functional integral underinﬁnitesimal shifts. The derivation of the skeleton equation forthe bosonic self-energy of models of the type considered inthis work has been discussed in detail in Refs. 36,43, so let us here only quote the result, /Pi1(¯K)=/integraldisplay K/summationdisplay α,σGα(K)Gα(K+¯K) ×/Gamma1α 0/Gamma1¯ψαψαφ(K;K+¯K;−¯K). (A1) This exact identity is shown diagrammatically in Fig. 9. 245120-15DRUKIER, BARTOSCH, ISIDORI, AND KOPIETZ PHYSICAL REVIEW B 85, 245120 (2012) To derive the skeleton equation for the three-boson vertex, let us start from the Dyson-Schwinger equation given inEq. (11.27a) of Ref. 36. After taking two successive derivatives with respect to φ ¯K2andφ¯K3, we obtain δ3/Gamma1 δφ¯K1δφ¯K2δφ¯K3=/summationdisplay α,σ/Gamma1α 0/integraldisplay Kδ4Gc δ¯jα Kδjα K+¯K1δφ¯K2δφ¯K3.(A2)Here, /Gamma1[¯ψ,ψ,φ ] is the generating functional of the one-line irreducible vertices, and Gc[¯j,j,J ] is the generating functional of the connected Green’s functions, and is a functional of thesources ¯j,j, andJconjugate to the ﬁelds ψ,¯ψ, andφ.W e now use Eq. (6.82) of Ref. 36and set all external ﬁelds equal to zero. The desired skeleton equation can then be writtenas /Gamma1φφφ(¯K1,¯K2,−¯K1−¯K2)=N/summationdisplay α/Gamma1α 0/bracketleftbigg/integraldisplay KGα(K)Gα(K+¯K1)/Gamma1¯ψαψαφφ(K;K+¯K1;¯K2,−¯K1−¯K2) +/integraldisplay KGα(K)Gα(K+¯K1)Gα(K+¯K1+¯K2)/Gamma1¯ψαψαφ(K+¯K1+¯K2;K+¯K1;¯K2) ×/Gamma1¯ψαψαφ(K;K+¯K1+¯K2;−¯K1−¯K2)+/integraldisplay KGα(K)Gα(K+¯K1)Gα(K−¯K2) ×/Gamma1¯ψαψαφ(K−¯K2;K+¯K1;−¯K1−¯K2)/Gamma1¯ψαψαφ(K;K−¯K2;¯K2)/bracketrightbigg . (A3) A graphical representation of this equation is given in Fig.10. Let us check a known limit of this equation. After re- placing the exact propagators Gα(K) withGα 0(K) and the exact vertices /Gamma1¯ψαψαφ(K1;K2;¯K) with the bare vertices /Gamma1α 0and ne- glecting the contribution involving /Gamma1¯ψαψαφφ(K1;K2;¯K1,¯K2), we obtain from Eq. (A3) /Gamma1φφφ 0(¯K1,¯K2,−¯K1−¯K2) =N/summationdisplay α/parenleftbig /Gamma1α 0/parenrightbig3/bracketleftbigg/integraldisplay KGα 0(K)Gα 0(K+¯K1)Gα 0(K+¯K1+¯K2) +/integraldisplay KGα 0(K)Gα 0(K+¯K1)Gα 0(K−¯K2)/bracketrightbigg , (A4) which is invariant under arbitrary permutations of ¯K1,¯K2, and ¯K3=− ¯K1−¯K2. In fact, in this approximation, the bosonic three legged vertex can be identiﬁed with the symmetrizedclosed fermion loop with three external legs and barepropagators given in Eq. (B1). As concerns Eq. (A3) ,t h e symmetry under permutations of its arguments is less obvious.However, as the left-hand side of Eq. (A2) is symmetric under permutations, and because all manipulations are exact,Eq.(A3) indeed fulﬁlls this symmetry. To obtain the approximation for the bosonic three- loop used in Sec. V, we adopt the same approximation strategy as in the skeleton approximation used for the bosonic FIG. 9. (Color online) Exact skeleton equation relating the bosonic self-energy to the exact fermionic propagators and the three- legged vertex with two fermion and one boson leg [see Eq. (A1) ]. The black dot denotes the bare vertex with two fermionic and onebosonic external leg given in Eq. (2.9) .self-energy: replacing the vertices /Gamma1¯ψαψαφ(K1;K2;¯K) and /Gamma1¯ψαψαφφ(K1;K2;¯K1,¯K2) with the bare vertices /Gamma1α 0and 0, but retaining dressed propagators Gα(K), we arrive at Eqs. (5.13) and(5.14) . APPENDIX B: THE SYMMETRIZED THREE-LOOP In the momentum-transfer cutoff scheme, all irreducible vertices involving only bosonic external legs are ﬁnite at theinitial scale and can be identiﬁed by the symmetrized closedfermion loops with bare fermionic propagators. In particular,the initial condition for the vertex with three external bosonlegs is /Gamma1 φφφ 0(¯K1,¯K2,¯K3)=2!N/summationdisplay α/parenleftbig /Gamma1α 0/parenrightbig3Lα3(−¯K1,−¯K2,−¯K3), (B1) where the symmetrized three-loop Lα 3(¯K1,¯K2,¯K3) can be expressed in terms of the nonsymmetrized three-loop ¯Lα 3(¯K1,¯K2,¯K3), deﬁned by ¯Lα 3(¯K1,¯K2,¯K3)=/integraldisplay KGα 0(K−¯K1)Gα 0(K−¯K2)Gα 0(K−¯K3), (B2) FIG. 10. (Color online) Graph of the skeleton equation (A3) for the three-legged bosonic vertex. 245120-16FUNCTIONAL RENORMALIZATION GROUP APPROACH TO ... PHYSICAL REVIEW B 85, 245120 (2012) as follows, Lα 3(¯K1,¯K2,¯K3)=1 3!/bracketleftbig¯Lα 3(¯K1,¯K1+¯K2,0)+¯Lα 3(¯K2,¯K1+¯K2,0) +¯Lα 3(¯K2,¯K2+¯K3,0)+¯Lα 3(¯K3,¯K2+¯K3,0)+¯Lα 3(¯K3,¯K3+¯K1,0)+¯Lα 3(¯K1,¯K3+¯K1,0)/bracketrightbig . (B3) Actually, taking into account energy-momentum conservation, we may set ¯K3=− ¯K1−¯K2, so that we need Lα 3(¯K1,¯K2,−¯K1−¯K2)=1 3!/bracketleftbig¯Lα 3(¯K1,¯K1+¯K2,0)+¯Lα 3(−¯K1,−¯K1−¯K2,0) +¯Lα 3(¯K2,¯K1+¯K2,0)+¯Lα 3(−¯K2,−¯K1−¯K2,0)+¯Lα 3(¯K1,−¯K2,0)+¯Lα 3(−¯K1,¯K2,0)/bracketrightbig .(B4) For our model, the three-loops can be calculated analytically using the method outlined in the appendix of Ref. 48. Consider ﬁrst the nonsymmetrized three-loop deﬁned in Eq. (B2). To perform the loop integration, we decompose the integrand in partial fractions, then carry out the k/bardblintegration by means of the method of residues, and ﬁnally perform the ωintegration. Using the notation ¯Ki=(i¯ωi,¯ki), the result can be written as ¯Lα 3(¯K1,¯K2,¯K3) =1 2π/integraldisplaydk⊥ 2π3/summationdisplay i=1i¯ωi3/productdisplay j=1 j/negationslash=i1 /Omega1α ij(k⊥)=1 2π/integraldisplaydk⊥ 2π/bracketleftBigg i¯ω1 /Omega1α 12(k⊥)/Omega1α 13(k⊥)+i¯ω2 /Omega1α 23(k⊥)/Omega1α 21(k⊥)+i¯ω3 /Omega1α 31(k⊥)/Omega1α 32(k⊥)/bracketrightBigg , (B5) where we have deﬁned /Omega1α ij(k⊥)=i¯ωi−i¯ωj+ξα k−¯ki−ξα k−¯kj=iωij−α(¯k/bardbli−¯k/bardblj)+(¯k2 ⊥i−¯k2 ⊥j)−2qijk⊥=−2qij[k⊥−kij], (B6) with ωij=¯ωi−¯ωj,q ij=¯k⊥i−¯k⊥j, (B7a) kij=¯k⊥i+¯k⊥j 2+iωij−α(¯k/bardbli−¯k/bardblj) 2qij. (B7b) The remaining k⊥integration in Eq. (B5) can now be done using the residue theorem and we ﬁnally obtain ¯Lα 3(¯K1,¯K2,¯K3) =−1 8π/bracketleftBigg ¯ω1 q12q13/Theta1(Im(k12))−/Theta1(Im(k13)) k12−k13+¯ω2 q23q21/Theta1(Im(k23))−/Theta1(Im(k21)) k23−k21+¯ω3 q31q32/Theta1(Im(k31))−/Theta1(Im(k32)) k31−k32/bracketrightBigg .(B8) Substituting this expression into Eq. (B4) and deﬁning xi=¯ωi ¯k⊥i,si=¯k⊥i ¯k⊥1+¯k⊥2, (B9) we obtain for the symmetrized three-loop, Lα 3(¯K1,¯K2,−¯K1−¯K2)=1 3!2π1 s1s2/bracketleftbig¯k/bardbl1 ¯k⊥1−¯k/bardbl2 ¯k⊥2−iα(x1−x2)/bracketrightbig2−¯k⊥1¯k⊥2[(s1\|x1\|+s2\|x2\|)/Theta1(−x1x2) +(s1\|x1\|−\|s1x1+s2x2\|)/Theta1(−x1(s1x1+s2x2))+(s2\|x2\|−\|s1x1+s2x2\|)/Theta1(−x2(s1x1+s2x2))]. (B10) Alternatively, this expression can be written as Lα 3(¯K1,¯K2,−¯K1−¯K2)=1 3!2π1 s1s2/bracketleftbig¯k/bardbl1 ¯k⊥1−¯k/bardbl2 ¯k⊥2−iα(x1−x2)/bracketrightbig2−¯k⊥1¯k⊥2[(s1x1−s2x2)[/Theta1(x1)−/Theta1(x2)] +(2s1x1+s2x2)[/Theta1(x1)−/Theta1(s1x1+s2x2)]+(2s2x2+s1x1)[/Theta1(x2)−/Theta1(s1x1+s2x2)]] =1 4πs1x1/Theta1(x1)+s2x2/Theta1(x2)−(s1x1+s2x2)/Theta1(s1x1+s2x2) s1s2/bracketleftbig¯k/bardbl1 ¯k⊥1−¯k/bardbl2 ¯k⊥2−iα(x1−x2)/bracketrightbig2−¯k⊥1¯k⊥2 =1 4π/parenleftbigg1 ¯k⊥1+1 ¯k⊥2/parenrightbigg¯ω1/Theta1/parenleftbig¯ω1¯k⊥1/parenrightbig +¯ω2/Theta1/parenleftbig¯ω2¯k⊥2/parenrightbig −(¯ω1+¯ω2)/Theta1/parenleftbig¯ω1+¯ω2¯k⊥1+¯k⊥2/parenrightbig /bracketleftbig¯k/bardbl1 ¯k⊥1−¯k/bardbl2 ¯k⊥2−iα/parenleftbig¯ω1¯k⊥1−¯ω2¯k⊥2/parenrightbig/bracketrightbig2−(¯k⊥1+¯k⊥2)2. 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PhysRevB.102.085113.pdf	PHYSICAL REVIEW B 102, 085113 (2020) Symmetrized decomposition of the Kubo-Bastin formula Varga Bonbien1,and Aurélien Manchon1,2,† 1Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia 2Aix-Marseille Univiversité, CNRS, CINaM, Marseille, France (Received 12 May 2020; revised 24 July 2020; accepted 24 July 2020; published 6 August 2020) The Smrcka-Streda version of Kubo’s linear response formula is widely used in the literature to compute nonequilibrium transport properties of heterostructures. It is particularly useful for the evaluation of intrinsictransport properties associated with the Berry curvature of the Bloch states, such as anomalous and spin Hallcurrents as well as the dampinglike component of the spin-orbit torque. Here we demonstrate in a very generalway that the widely used decomposition of the Kubo-Bastin formula introduced by Smrcka and Streda containsan overlap, which has lead to widespread confusion in the literature regarding the Fermi surface and Fermi seacontributions. To remedy this pathology, we propose a decomposition of the Kubo-Bastin formula based on thepermutation properties of the correlation function and derive a set of formulas, without an overlap, that providesdirect access to the transport effects of interest. We apply these formulas to selected cases and demonstrate thatthe Fermi sea and Fermi surface contributions can be uniquely addressed with our symmetrized approach. DOI: 10.1103/PhysRevB.102.085113 I. INTRODUCTION The seminal work of Kubo [ 1,2] showed that, in the pertur- bative weak-ﬁeld limit, transport coefﬁcients can be expressedas correlation functions of quantum mechanical observableoperators. The resulting Kubo formalism has become a stapleof quantum transport theory calculations and surged in pop-ularity following the realization that applying it to transportphenomena in crystals provides direct access to topologicalinvariants, thereby yielding an explanation for the robustnessof the quantized Hall effect [ 3]. While the original Kubo formula is formally satisfying, realistic calculations with it are rather impractical. Bastin et al. [4] and later Streda and Smrcka [ 5] used Green’s functions to rewrite the Kubo formula and arrived at a result directly ap-plicable to computations in the static limit. Later on, Smrckaand Streda [ 6] further decomposed the Bastin formula into two terms, A I=¯h 2π/integraldisplay dε∂εf(ε)Re{tr[ˆAˆGrˆB(ˆGr−ˆGa)]}, (1) AII=¯h 2π/integraldisplay dεf(ε)Re{tr[ˆAˆGrˆB∂εˆGr−ˆA∂εˆGrˆBˆGr]},(2) Here ˆAis the operator of the perturbation and ˆBis the operator of the observable, ˆGr(a)(ε) is the retarded (advanced) Green’s function of the system and we have suppressed the energyargument in the formulas for brevity, f(ε) is the Fermi-Dirac distribution, and ∂ εindicates an energy derivative. Because of their connection to ∂εf(ε) and f(ε), Eqs. ( 1) and ( 2)w e r e wrongly referred to as Fermi surface andFermi sea terms, respectively. These terms were used by Streda in his famous bonbien.varga@kaust.edu.sa †manchon@cinam.univ-mrs.franalysis of the quantized Hall effect [ 7]. More recently, Crépieux and Bruno [ 8] presented a detailed and widely cited derivation of the Kubo-Bastin and Smrcka-Streda formulasfrom the Kubo formula. The Smrcka-Streda formula has been widely used to com- pute charge and spin Hall currents [ 9,10] as well as spin-orbit torques [ 11]. Whereas a few works use the full Smrcka-Streda formula [ 12–15], most theoretical studies exploit a simpliﬁed version of it, obtained by assuming constant scattering timeand in the weak disorder limit [ 16–18] A /Gamma1 surf→1 π/summationdisplay k,n,m/Gamma12Re[/angbracketleftnk\|ˆB\|mk/angbracketright/angbracketleftmk\|ˆA\|nk/angbracketright] [(εF−εnk)2+/Gamma12][(εF−εmk)2+/Gamma12], (3) A/Gamma1 sea→/summationdisplay k,n/negationslash=mIm[/angbracketleftnk\|ˆB\|mk/angbracketright/angbracketleftmk\|ˆA\|nk/angbracketright] (εnk−εmk)2[f(εnk)−f(εmk)]. (4) Here/Gamma1is the homogeneous broadening and \|nk/angbracketrightis a Bloch state of the crystal. This simpliﬁed version readily attributesA /Gamma1 surfto intraband transitions, yielding a ∼1//Gamma1dependence, andA/Gamma1 seato interband transitions, which are ﬁnite in the clean limit ( /Gamma1→0). In fact, these simpliﬁed formulas elegantly connect the Fermi sea transport contributions to the Berrycurvature of the Bloch states, and, to date, the Berry curvatureformula [Eq. ( 4)] has been widely used to characterize the intrinsic spin Hall effect of bulk materials [ 19–21]. As we mentioned already, this formula is only valid in the clean limitand does not apply in realistic materials where momentumscattering is important. More speciﬁcally, it becomes invalidwhen the broadening /Gamma1is comparable to, or larger than, the local orbital gaps resulting from avoided band crossings,and where Berry curvature is maximized. Indeed, further 2469-9950/2020/102(8)/085113(9) 085113-1 ©2020 American Physical SocietyV ARGA BONBIEN AND AURÉLIEN MANCHON PHYSICAL REVIEW B 102, 085113 (2020) investigations [ 22–24] have addressed the spin Hall effect of metals using the full Smrcka-Streda formula, Eqs. ( 1) and ( 2), showing evidence that the spin Hall conductivity of 5dtransition metals is dominated by AI[24]. Similarly, an inﬂuential work by Sinitsyn et al. [25] demonstrated that in the case of a gapped Dirac cone, spin Hall effect is entirelydue to A Iin the metallic regime, while it is entirely due to AII in the gap. These observations, valid for speciﬁc examples, led to the confusion that AIalways dominate in metals. For instance, some investigations [ 12–14] (including ours) have computed the spin-orbit torque using only AIbased on Kon- taniet al. ’s [24] argument. However, recent calculations have demonstrated that certain transport properties associated withBerry curvature, such as the dampinglike torque in magneticheterostructures, have contributions from both A IandAII [26–28]( s e ea l s oR e f .[ 29]). This suggests that attributing purely Fermi surface origin to AIand purely Fermi sea origin toAIIis incorrect. In this paper we ﬁrst show in a very general way that the AI-AIIdecomposition of the Kubo-Bastin formula introduced by Smrcka and Streda contains an overlap, and there appearsto be widespread confusion regarding this aspect in the lit-erature. This overlap was hinted at for the special case of atwo-dimensional Dirac material by Sinitsyn et al. [30], but the fact that Smrcka-Streda and many subsequent authors unjusti-ﬁably neglected a subtle term relating to position operators incertain versions of the Smrcka-Streda formula responsible forgeometric effects, went unmentioned [ 7,8,31]. This subtlety is unnoticeable for simple models—such as the quadraticmagnetic Rashba gas—when A IIis vanishingly small away from the avoided band crossing, since the neglected geometricterm exactly cancels out Streda’s orbital sea term [ 7]i n A IIwhich, due to the overlap, also appears in the AIterm. However, in the general case, AIIis non-negligible [ 29,32] and thus, Smrcka and Streda’s decomposition of the Kubo-Bastinformula into A I,AIIdoes not lend itself to a proper analysis of different physical effects. To remedy this we proposea decomposition of the Kubo-Bastin formula based on thepermutation properties of the correlator and derive a set offormulas without an overlap, which provides direct access tothe intrinsic geometric effects. II. THE KUBO-BASTIN FORMULA AND THE SMRCKA-STREDA DECOMPOSITION The Kubo-Bastin formula for the electrical conductivity σklin the static limit as obtained from the Kubo formula is (Eq. (A9) of Ref. [ 8]) σkl=−¯h 2π/integraldisplay dεf(ε)t r ( ( ˆjk∂εˆGrˆjl−ˆjl∂εˆGaˆjk)(ˆGr−ˆGa)), (5) where ˆjk,ˆjlare electric charge current operators in the k,l∈ {x,y,z}directions, ˆGr(a)(ε)=limη→01/(ε−ˆH0±iη)i st h e retarded (advanced) Green’s function corresponding to theequilibrium Hamiltonian ˆH 0, and∂εˆGr(a)(ε) is the derivative of the Green’s function with respect to its energy argument,which we have suppressed in the formula for brevity. We alsotook the sample volume to be unity, however this can be rein-stated at any step of the calculations. Across the manuscript, ˆ···denotes an operator and tr( ···) is the trace operation. Splitting ( 5) into two halves, integrating one of them by parts and combining it with the other half yields the Smrcka-Streda decomposition of the Kubo-Bastin formula (Eq. (A10)of Ref. [ 8]) with σ kl=σI kl+σII kl, where σI kl=¯h 4π/integraldisplay dε∂εf(ε)t r ( ( ˆjkˆGrˆjl−ˆjlˆGaˆjk(ˆGr−ˆGa)) (6) and σII kl=¯h 4π/integraldisplay dεf(ε)t r (ˆjkˆGr(ε)ˆjl∂εˆGr−ˆjk∂εˆGrˆjlˆGr +ˆjl∂εˆGaˆjkˆGa−ˆjlˆGaˆjk∂εˆGa). (7) Integrating ( 7) by parts shall not yield any surface terms, thus we might naively conclude that this term describes effectsresulting purely from the Fermi sea. However, this is not thecase, since there is signiﬁcant overlap between σ I klandσII kl. Indeed, manipulating ( 6) and ( 7) we arrive at (details in the Appendix) σI kl=σsurf kl+σol kl, (8a) σII kl=σsea kl−σol kl, (8b) where σsurf kl=¯h 4π/integraldisplay dε∂εf(ε)t r (ˆjk(ˆGr−ˆGa)ˆjl(ˆGr−ˆGa)),(9) σsea kl=−¯h 4π/integraldisplay dεf(ε)t r ({ˆjk(∂εˆGr+∂εˆGa)ˆjl −ˆjl(∂εˆGr+∂εˆGa)ˆjk}(ˆGr−ˆGa)), (10) and the overlap term is σol kl=¯h 8π/integraldisplay dε∂εf(ε)t r ({ˆjk(ˆGr+ˆGa)ˆjl −ˆjl(ˆGr+ˆGa)ˆjk}(ˆGr−ˆGa)). (11) Upon closer inspection we note that σsurf klis symmetric, whereas σsea klalong with σol klare antisymmetric under the exchange of operators ˆjkand ˆjl. Furthermore, in the special case of k=l,σsurf kkcan be recognized as the Kubo-Greenwood formula for the diagonal conductivity. The separation of σI into symmetric and antisymmetric parts yielding σsurfandσol is already present in the literature [ 32,33], however it was not realized that the antisymmetric part σolis an overlap and gets exactly canceled when considering an appropriate separationofσ IIintoσseaandσol, as it is done here. In order to gain some understanding of σsea klandσol klwe use the expressions ˆjk=−ie/¯h[ˆG−1,ˆxk], where ˆ xkis the position operator and ∂εˆGr(a)=−(ˆGr(a))2. In the clean limit (ˆGrˆG−1→ˆ1,ˆGaˆG−1→ˆ1), the overlap term σol klfrom ( 11) becomes σol kl→ie 4π/integraldisplay dε∂εf(ε)t r ( ( ˆGr−ˆGa)(ˆxkˆjl−ˆxlˆjk)),(12) 085113-2SYMMETRIZED DECOMPOSITION OF THE KUBO-BASTIN … PHYSICAL REVIEW B 102, 085113 (2020) whereas σsea klfrom ( 10) simpliﬁes to σsea kl→−e2 2π¯h/integraldisplay dεf(ε)t r ( ( ˆGr−ˆGa)[ˆxk,ˆxl]). (13) We recognize ( 12) as Streda’s orbital sea term [ 7]. However, contrary to the original derivation in Ref. [ 6]a sw e l la si nt h e rederivation in Ref. [ 8] (see also Refs. [ 31,34–36]), this is not equivalent to σII klbut, as seen in Eq. ( 8b), is an overlap term which has no overall effect since it gets canceled out. Indeed,looking at Appendix A of Ref. [ 8], we see that their ˜ σ Ifrom Eq. (A11) is the same as our σIin Eq. ( 6), but their ˜ σIIin Eq. (A12), which should be the total σIIis only our overlap term−σol. In other words, something was “lost” while go- ing from the general term σII—the second integral in their Eq. (A10) and our Eq. ( 7)—to the “simpliﬁed” or orbital sea term that is their Eq. (A12) and what we call the “overlap”term in Eq. ( 12). What was lost is precisely σ sea, expressed in Eq. ( 10) and in the clean limit as Eq. ( 13), due to the fact that the position operators were assumed to commute. However,the latter is not necessarily true, since the weighting with theFermi-Dirac distribution projects the total space of states tothe ﬁlled states, and such terms containing noncommutingposition operators are responsible for certain geometric effectssuch as those stemming from the Berry curvature [ 37–39]. In order to see this explicitly, consider expanding the trace of Eq. ( 13) in the basis of Bloch states {\|nk/angbracketright}. Making use of the identity ˆG r(ε)−ˆGa(ε)=−i2πδ(ε−ˆH0), we have −e2 2π¯h/integraldisplay dεf(ε)t r ( ( ˆGr−ˆGa)[ˆxk,ˆxl]) =ie2 ¯h/summationdisplay k,nf(εnk)/angbracketleftnk\|[ˆxk,ˆxl]\|nk/angbracketright =ie2 ¯h/summationdisplay k,n/negationslash=m[f(εnk)−f(εmk)]/angbracketleftnk\|ˆxk\|mk/angbracketright/angbracketleftmk\|ˆxl\|nk/angbracketright. (14) The matrix elements of the current operator ˆjk= ie/¯h[ˆH0,ˆxk]i nt h e {\|nk/angbracketright}basis are /angbracketleftnk\|ˆjk\|mk/angbracketright=ie ¯h(εnk−εmk)/angbracketleftnk\|ˆxk\|mk/angbracketright, (15) and we can rearrange this to express the matrix elements of the position operator with those of the current operator. Pluggingthe result into the ﬁnal expression of ( 14) we arrive at −e 2 2π¯h/integraldisplay dεf(ε)t r ( ( ˆGr−ˆGa)[ˆxk,ˆxl]) =i¯h/summationdisplay k,n/negationslash=m(f(εnk)−f(εmk))/angbracketleftnk\|ˆjk\|mk/angbracketright/angbracketleftmk\|ˆjl\|nk/angbracketright (εnk−εmk)2,(16) which is the widely used Kubo formula for the intrinsic, Berry curvature contribution to the anomalous conductivity. It is thusapparent that the commutator of the position operators cannot be neglected and, in fact, yields the intrinsic contribution.Furthermore, the commutator in question is deeply connectedto the geometry of the Bloch states and we refer the readerto Appendix A of Ref. [ 37] for a broader discussion of this aspect.Physically, the noncommutation of the position operators can be interpreted as an uncertainty relation between thepositions in different directions. This means that we cannotmeasure the exact position, since this would require the si-multaneous diagonalization of the position operators in thedifferent directions, which is not possible due to them notcommuting (see Ref. [ 40] for further elaboration of this point). This loss of independence between the positions in differingdirections is realized in the intrinsic contribution to the Halleffect, since, heuristically, an electric ﬁeld E xapplied in direction xcouples to the position operator ˆ xin the same direction as ˆHE∼− ˆxExleading to an anomalous velocity ˆvy∼[ˆHE,ˆy]∼[ˆx,ˆy]Exin direction y. A crucial question that arises then is, how come that even though Streda’s orbital sea term—what we call the overlapterm—from ( 12) has no overall effect and the geometric term (13) has been neglected in the literature, it is still possible to obtain proper results including Berry curvature effects forcertain cases? In order to answer this question, consider thecase of a vanishing σ IIterm: σII kl=0, such as for the 2D metallic Dirac gas, or quadratic magnetic Rashba gas [ 9,41]. From ( 8a) and ( 8b)w eh a v e σII kl=σsea kl−σol kl=0⇒σsea kl= σol klgiving σI kl=σsurf kl+σol kl=σsurf kl+σsea kl. Thus we see that for the particular case of a vanishing σIIterm, Streda’s orbital sea term ( 12) is exactly equal to the geometric term σsea kl and consequently describes Berry curvature effects. This is an advantage in the zero temperature case, since ∂εf(ε)→ −δ(ε−εF)a sT→0, meaning that we can simply evaluate the Green’s functions in ( 12) at the Fermi energy and there is no need for a complete energy integration, as would berequired for ( 10)o r( 13). III. THE PERMUTATION DECOMPOSITION Once we exclude pathological toy models from our inves- tigations, such as the quadratic Rashba gas mentioned above,and turn our focus to real materials, the general sea term σ II kl is strictly nonvanishing [ 29,32], consequently we propose not to consider the conventional Smrcka-Streda decompositionσ kl=σI kl+σII kl=(σsurf kl+σol kl)+(σsea kl−σol kl) with the over- lap term in any capacity. Rather, we offer a new one, thepermutation decomposition: σ kl=σsurf kl+σsea kl, (17) where σsurf klandσsea klare expressed in Eqs. ( 9) and ( 10), respectively. As brieﬂy mentioned above, σsurf klis symmetric, whereas σsea klis antisymmetric under the exchange of ˆjkand ˆjl.D u et o σsurf klandσsea klbeing in different permutation classes they cannot overlap, and so they can be derived directlyfrom the Bastin formula in Eq. ( 5) by decomposing the latter into symmetric and antisymmetric terms with respect to thepermutation of ˆj kand ˆjl, effectively foregoing the need to go through the Smrcka-Streda decomposition and all subsequentanalysis. To see the direct derivation explicitly, we ﬁrst symmetrize (5) σ kl=1 2(σkl+σlk)+1 2(σkl−σlk). (18) It is important to add that although the notation suggests symmetrizing the cartesian indices of the conductivity tensor, 085113-3V ARGA BONBIEN AND AURÉLIEN MANCHON PHYSICAL REVIEW B 102, 085113 (2020) we are in fact exchanging the operators themselves. In the given case, these are equivalent since the two current operators ˆjk,ˆjlonly differ in their direction. The distinction is, however, crucial for other cases, such as the spin response to an electricﬁeld, where the two operators under consideration are not thesame, but are in fact ˆ s k,ˆjl, where ˆ skis the spin operator in the kdirection, instead of ˆjk,ˆjl. The symmetric part becomes σsurf kl=−¯h 2π/integraldisplay dεf(ε)1 2tr({ˆjk(∂εˆGr−∂εˆGa)ˆjl +ˆjl(∂εˆGr−∂εˆGa)ˆjk}(ˆGr−ˆGa)) =¯h 4π/integraldisplay dε∂εf(ε)t r (ˆjk(ˆGr−ˆGa)ˆjl(ˆGr−ˆGa)),(19) where we used integration by parts and the cyclicity of the trace to arrive at the second equality from the ﬁrst. We can rec-ognize σ surf klfrom Eq. ( 9). The antisymmetric part σsea kl, written as Eq. ( 10), is obtained directly from the antisymmetrization of (5) without any intermediate steps. The terms arrived at in this way carry a physical interpre- tation. Consider the clean limit ( /Gamma1→0). In this case, σsurf kl vanishes as is seen by using ˆjk=−ie/¯h[ˆG−1,ˆxk]i n( 9) and so is purely extrinsic. On the other hand, σsea kldoes not vanish, reduces to ( 13), and so is an intrinsic contribution. In the general case of a material with impurities the intrinsic contri-bution thus arises purely from σ sea kl, which can be very helpful when trying to extract information from experimental resultsby comparing them to numerical calculations performed usingthe permutation decomposition. A further utility of decomposing the Kubo formula into permutation classes is the possibility of dealing with distinctphysical effects arising as higher order responses in a straight-forward manner. This has been completed for second and thirdorder responses and is currently under preparation. IV . APPLICATION TO HALL EFFECTS, SPIN CURRENTS, AND SPIN-ORBIT TORQUE In this section we compute the transport properties of three illustrative systems using the two different decompositions ofthe Kubo-Bastin formula, the Smrcka-Streda decomposition, A I=¯h 2π/integraldisplay dε∂εf(ε)Re{tr[ˆAˆGrˆB(ˆGr−ˆGa)]},(20) AII=¯h 2π/integraldisplay dεf(ε)Re{tr[ˆAˆGrˆB∂εˆGr−ˆA∂εˆGrˆBˆGr]},(21) and our permutation decomposition, Asurf=¯h 4π/integraldisplay dε∂εf(ε)Re{tr[ˆA(ˆGr−ˆGa)ˆB(ˆGr−ˆGa)]}, (22) Asea=¯h 2π/integraldisplay dεf(ε)Re{tr[ˆA(ˆGr−ˆGa)ˆB(∂εˆGr+∂εˆGa)]}. (23) As discussed in the previous section, it is clear that AI+ AII=Asurf+Asea. Now we would like to show how the new separation can speciﬁcally distinguish between extrinsic and FIG. 1. Energy dependence of (a) longitudinal conductivity, (b) ﬁeldlike torque, (c) transverse conductivity, and (d) dampinglike torque in the two-dimensional magnetic Rashba gas. The solid red (blue) curve refers to the AI(AII) contribution, whereas the black curve is their sum AI+AII. The black (red) dots refer to Asurf(Asea). The inset of (b) shows the band structure of the magnetic Rashba gas. The dashed horizontal line indicates the position of the avoidedband crossing and the dotted line stands for the maximum energy taken in this calculation. The conductivity is in /Omega1 −1m−1and the spin conductivity is in (¯ h/2e)/Omega1−1m−1. intrinsic phenomena. To do so, we consider nonequilibrium transport (i) in the magnetic Rashba gas, (ii) in a multi-orbital tight-binding model of a ferromagnet/normal metalheterostructure, and (iii) in a noncollinear antiferromagnet. A. Magnetic Rashba gas Let us ﬁrst consider the canonical magnetic Rashba gas regularized on a square lattice and described by the Hamil-tonian ˆH=−2t(cosk x+cosky)+/Delta1ˆσz +tR(ˆσxsinky−ˆσysinkx). (24) Here tis the nearest-neighbor hopping, tRis the Rashba parameter, and /Delta1is the s-dexchange. This model has been central to the investigation of the anomalous Hall effect[42,43] and spin-orbit torque [ 44,45]. Here we do not consider the vertex correction since our interest is to illustrate the supe-riority of our permutation decomposition of the Kubo-Bastinformula. The Green’s function is simply given by ˆG r(a)(ε)= (ε−ˆH±i/Gamma1)−1,/Gamma1being the homogeneous broadening com- ing from short-range (deltalike) impurities. In this section wecompute the nonequilibrium properties induced by the electricﬁeld ( ˆA=−eˆj x), with particular focus on the longitudinal conductivity ( ˆB=−eˆjx), the transverse conductivity ( ˆB= −eˆjy), the ﬁeldlike torque ( ˆB=−/Delta1ˆσy), and the dampinglike torque ( ˆB=/Delta1ˆσx). The conductance of the two-dimensional electron gas is in /Omega1−1and the spin torque is expressed in terms of an effective spin conductivity (¯ h/2e)/Omega1−1m−1. Finally, for the parameters we take tR=2.4t,/Delta1=0.2t, and/Gamma1=0.1t. Figure 1reports the (a) longitudinal and (b) transverse Hall conductivities as well as the torque components, (b) ﬁeldlikeand (d) dampinglike, as a function of the energy. In this 085113-4SYMMETRIZED DECOMPOSITION OF THE KUBO-BASTIN … PHYSICAL REVIEW B 102, 085113 (2020) ﬁgure and the ones following, the AIandAIIcontributions of the Smrcka-Streda formula are represented with red andblue solid lines, while the Fermi surface ( A surf) and Fermi sea (Asea) contributions of our permutation decomposition of the Kubo-Bastin formula are represented by black and red dots,respectively. The black line represents the sum A I+AII.I n the case of transport properties only involving the Fermi sur-face, such as the longitudinal conductivity [Fig. 1(a)] and the ﬁeldlike torque [Fig. 1(b)],A II=Asea=0 and AI=Asurf,s o using either the conventional Smrcka-Streda decomposition orour permutation decomposition is equivalent. The transport properties involving Fermi sea are more interesting to consider. Indeed, as discussed in the previoussection, it clearly appears that when using the conventionalSmrcka-Streda formula, both A I(red) and AII(blue) contri- butions are equally important. In fact, the variations of AII can be readily correlated with the band structure displayed in the inset of Fig. 1(b).T h e AIIcurve exhibits two peaks, one close to the bottom of the lowest band, where the dispersionis quite ﬂat (around −2.5t), and one when the Fermi level lies in the local gap corresponding to the avoided crossing ofthe two bands [dashed line in the inset of Fig. 1(b)]. Away from this local gap, A IIvanishes. This is an important obser- vation because it indicates that the overlap contribution of theSmrcka-Streda formula is peaked close to locally ﬂat bands,irrespective whether it is geometrically trivial (around −2.5t) or nontrivial (around −2t). When summing A IandAII,t h e complex structure of AIIclose to the bottom of the lowest band compensates AIexactly, so that the total contribution AI+AII=Aseahas a much simpler overall structure and is peaked only at the local (geometrically nontrivial) gap, whichillustrates the Berry curvature origin of this contribution. Thissimple calculation points out the dramatic need to computeboth A IandAIIcontributions to obtain correct Fermi sea contributions such as dampinglike torque and anomalous Halleffect, whereas A seacontains these contributions in itself. B. Transition metal bilayer The previous calculation shows that the contribution of AII becomes particularly crucial when crossing local ﬂat bands . Nonetheless, one might argue that this sensitivity is due tothe simplicity of the Rashba model that only involves twobands of opposite chirality. To generalize these results, wenow move on to a more complex system, a metallic bilayermade of two transition metal slabs and modeled using amultiorbital tight-binding model within the Slater-Koster two-center approximation. This model has been discussed in detailin Refs. [ 14,46] and here we only summarize its main features. The structure consists of two adjacent transition metal layerswith bcc crystal structure and equal lattice parameter. Thetendorbitals are included and the tight-binding parameters are extracted from Ref. [ 47]. Importantly, we consider atomic (Russell-Saunders) spin-orbit coupling, so that bulk and inter-facial spin-orbit coupled transport are modeled in a realisticmanner. Concretely, the Hamiltonian of the metallic bilayerreads ˆH=ˆH F+ˆHN+ˆHFN, (25)where ˆHF,Nare the Hamiltonians of the isolated ferromag- netic (F) and nonmagnetic (N) slabs, whereas ˆHFNaccounts for the interfacial hybridization between these two slabs.The Hamiltonian of a given slab μ(μ=F,N) is composed ofnsquare monolayers stacked on top of each other. The Hamiltonian of the ith monolayer of slab μis ˆH i μ=/summationdisplay α,βˆc† iβ(ˆti α,β(k)+ξi soˆL·ˆσ+/Delta1i αδαβˆσ·m)ˆciα,(26) where αandβrefer to two atomic dorbitals ( xy,yz,zx, x2-y2,2z2) and ˆ ciα(ˆc† iα) is the annihilation (creation) operator in the orbital ⊗spin basis. ˆti α,β(k) is the hopping energy of monolayer ibetween orbitals αandβand computed using the Slater-Koster parametrization, ξi sois the spin-orbit coupling energy, and /Delta1i αis the exchange parameter for a given orbital α.ˆLand ˆσare the orbital and spin angular momentum operators. The total Hamiltonian of slab μ, composed of n monolayers, reads ˆHμ=n/summationdisplay i=1ˆHi μ+/summationdisplay i,α,β/bracketleftbig ˆc† i+1βˆti,i+1 α,β(k)ˆciα+c.c./bracketrightbig +/summationdisplay i,α,β/bracketleftbig ˆc† i+2βˆti,i+2 α,β(k)ˆciα+c.c./bracketrightbig , (27) where the second (third) term on the right-hand side accounts for the coupling between monolayers iandi+1(iandi+ 2) and c.c. refers to the complex conjugate. ˆHFNis built in a similar manner. The parameters adopted for this model are theones of bulk Fe and W computed in Ref. [ 47]. Further details of this model can be found in Ref. [ 14]. Figure 2reports the same transport properties as Fig. 1, i.e., (a) longitudinal conductivity (i.e., the two-dimensionalconductance divided by the thickness of the bilayer), as wellas (b) the ﬁeldlike torque as a function of the energy. Again,we ﬁnd that Fermi surface properties are well described bythe surface terms when using either the conventional Smrcka-Streda or our permutation decomposition of the Kubo-Bastinformula [Figs. 2(a) and 2(b)]. Nonetheless, the Fermi sea properties displayed in Fig. 3exhibit a much richer behavior. The considerably more complex band structure of the multi-orbital model (e.g., see Fig. 4 in Ref. [ 14]) possesses a high density of ﬂat band regions which results in highly oscillatingA IandAIIcontributions, in both transverse conductivity [Fig. 3(a)], and dampinglike torque [Fig. 3(b)]. These oscil- lations are partially washed out when summing both contribu-tions [Figs. 3(c) and3(d)] so that the remaining oscillations are only associated to the local Berry curvature of the bandstructure. These results agree with our recent work where wedemonstrated, using a similar multiband model for topolog-ical insulator/antiferromagnet heterostructures, that both A I andAIIcontributions are necessary to obtain the appropriate magnitude of the damping torque, particularly in the regionsdisplaying avoided band crossing [ 26]. Figure 3clearly shows that both contributions should be accounted for when comput-ing dampinglike torque and anomalous transport. Taking onlyA Iinto account like in Refs. [ 12,14] is insufﬁcient. 085113-5V ARGA BONBIEN AND AURÉLIEN MANCHON PHYSICAL REVIEW B 102, 085113 (2020) FIG. 2. Energy dependence of (a) longitudinal conductivity and (b) ﬁeldlike torque in the multiorbital transition metal bilayer model.The solid red curve refers to the A Iand the black dots refer to Asurf. The conductivity is in /Omega1−1m−1and the spin conductivity is in (¯h/2e)/Omega1−1m−1. C. Noncollinear antiferromagnet We conclude this investigation by considering one last system of interest: a noncollinear antiferromagnet displayinganomalous transverse spin currents even in the absence ofspin-orbit coupling. As a matter of fact, the transport ofspin and charge in noncollinear antiferromagnets has been FIG. 3. Energy dependence of (a) and (b) transverse conductivity and (c) and (d) dampinglike torque in the multiorbital transition metal bilayer model. The solid red (blue) curve refers to the AI (AII) contribution, the black curve is their sum AI+AII, and the red dots refer to Asea. The conductivity is in /Omega1−1m−1and the spin conductivity is in (¯ h/2e)/Omega1−1m−1. FIG. 4. Angular dependence of (a) in-plane and (b) out-of-plane spin Hall effect in the noncollinear antiferromagnetic Kagome lattice model. The solid red (blue) curve refers to the AI(AII) contribution, the black curve is their sum AI+AII, and the black (red) dots refer toAsurf(Asea). The inset displays the angle made by the applied electric ﬁeld with respect to the crystal axes. The spin conductivityis in (¯ h/2e)/Omega1 −1. the object of intense scrutiny recently, as anomalous Hall as well as magnetic spin Hall effects have been predicted[17,48,49] and observed [ 50–52]. We test our permutation decomposition on an ideal Kagome lattice with 120 ◦magnetic moment conﬁguration, as depicted in the inset of Fig. 4.T h e model is the same as Ref. [ 48], and the Hamiltonian reads ˆH=t/summationdisplay /angbracketleftiα,jβ/angbracketrightˆc† jβˆciα+/Delta1/summationdisplay iˆc† iαˆσ·mαˆciα. (28) Here tis the nearest neighbor hopping, and /Delta1is the s-d exchange. The indices α,β refer to the different magnetic sublattices of a magnetic unit cell, and i,jrefer to different unit cells. In this work we set /Delta1=1.7t. Such a system displays two types of transverse spin currents [ 17,53], even in the absence of spin-orbit coupling: one spin current σz s possesses a polarization perpendicular to the plane, and the otherσ/bardbl shas a polarization in-plane and normal to the applied electric ﬁeld. We refer to the former as perpendicular spin Hallcurrent and the latter is called in-plane spin Hall current. We compute in Fig. 4the (a) in-plane and (b) perpendicular spin conductivities as a function of the angle of the electricﬁeld with respect to the crystal lattice directions. We obtainthat the in-plane spin current is purely a Fermi surface term,corresponding to the “magnetic spin Hall effect” predictedby Zelezný et al. [17] and observed by Kimata et al. [52]. This spin current strongly depends on the orientation of theelectric ﬁeld with respect to the crystallographic axes. In 085113-6SYMMETRIZED DECOMPOSITION OF THE KUBO-BASTIN … PHYSICAL REVIEW B 102, 085113 (2020) contrast, the perpendicular spin current shows a weak angular dependence and is purely given by the Fermi sea contribution[53]. Again, the A IIcontribution is small but nonzero. The reduced magnitude of AIIcompared to AIis due to the fact that the Fermi level is taken away from the avoided bandcrossing in this particular case. V . CONCLUSION We have shown that the widely used Smrcka-Streda de- composition of the celebrated Kubo-Bastin formula possessesan overlap that makes it inappropriate to distinguish betweenFermi sea and Fermi surface contributions to transport co-efﬁcients. This is particularly crucial in multiband systemspossessing a high density of locally ﬂat bands and avoidedband crossings. As a matter of fact, whereas intrinsic (Berry-curvature induced) transport properties are dominated bygeometrically nontrivial avoided band crossings, the overlapis enhanced close to any (trivial and nontrivial) locally ﬂatbands, as illustrated in the case of the magnetic Rashba gas.Therefore, the Smrcka-Streda decomposition of the Kubo-Bastin formula can lead to an incorrect estimation of the intrinsic transport properties. To remedy this difﬁculty, wedemonstrated that the Kubo formula can be decomposed intosymmetric and antisymmetric parts, which gives direct accessto Fermi surface and Fermi sea contributions. The superior-ity of this permutation decomposition over Smrcka-Streda’s,apart from its apparent conceptual clarity, has been illustratedby computing the extrinsic and intrinsic transport coefﬁcientsof three selected systems. This observation has substantialimpact on quantum transport calculations, especially whenconsidering Berry curvature induced mechanisms such asHall conductance and torques, since it provides a neat wayof separating the intrinsic part of these anomalous transporteffects from Fermi surface related effects, removing spuriouseffects stemming from local trivial band ﬂatness. ACKNOWLEDGMENT This work was supported by the King Abdullah University of Science and Technology (KAUST) through the award OSR-2017-CRG6-3390 from the Ofﬁce of Sponsored Research(OSR). APPENDIX: DERIVATION OF THE OVERLAP TERM TheσIterm ( 6) can be handled in a simple way by separating it into symmetric and antisymmetric permutations of ˆjkandˆjl as follows: σI kl=¯h 4π/integraldisplay dε∂εf(ε)t r ( ( ˆjkˆGrˆjl−ˆjlˆGaˆjk)(ˆGr−ˆGa)) =¯h 8π/integraldisplay dε∂εf(ε)t r ( ( ˆjk(ˆGr−ˆGa)ˆjl+ˆjl(ˆGr−ˆGa)ˆjk)(ˆGr−ˆGa)) +¯h 8π/integraldisplay dε∂εf(ε)t r ( ( ˆjk(ˆGr+ˆGa)ˆjl−ˆjl(ˆGr+ˆGa)ˆjk)(ˆGr−ˆGa)) =¯h 4π/integraldisplay dε∂εf(ε)t r (ˆjk(ˆGr−ˆGa)ˆjl(ˆGr−ˆGa))+¯h 8π/integraldisplay dε∂εf(ε)t r ( ( ˆjk(ˆGr+ˆGa)ˆjl −ˆjl(ˆGr+ˆGa)ˆjk)(ˆGr−ˆGa))=σsurf kl+σol kl. (A1) It is clear that σsurfis symmetric and σolis antisymmetric in the exchange of ˆjkandˆjl.T h eσIIterm ( 7) is more complicated, requiring the following manipulations: σII kl=¯h 4π/integraldisplay dεf(ε)t r (ˆjkˆGrˆjl∂εˆGr−ˆjk∂εˆGrˆjlˆGr+ˆjl∂εˆGaˆjkˆGa−ˆjlˆGaˆjk∂εˆGa) =1 2¯h 4π/integraldisplay dεf(ε)t r (ˆjkˆGrˆjl∂εˆGr−ˆjk∂εˆGrˆjlˆGr+ˆjl∂εˆGaˆjkˆGa−ˆjlˆGaˆjk∂εˆGa) +1 2¯h 4π/integraldisplay dεf(ε)t r (ˆjkˆGrˆjl∂εˆGr−ˆjk∂εˆGrˆjlˆGr+ˆjl∂εˆGaˆjkˆGa−ˆjlˆGaˆjk∂εˆGa) =1 2¯h 4π/integraldisplay dεf(ε)t r (ˆjk(ˆGr−ˆGa)ˆjl(∂εˆGr+∂εˆGa)−ˆjk(∂εˆGr+∂εˆGa)ˆjl(ˆGr−ˆGa)) +1 2¯h 4π/integraldisplay dεf(ε)t r (ˆjk(ˆGr+ˆGa)ˆjl(∂εˆGr−∂εˆGa)−ˆjk(∂εˆGr−∂εˆGa)ˆjl(ˆGr+ˆGa)). (A2) 085113-7V ARGA BONBIEN AND AURÉLIEN MANCHON PHYSICAL REVIEW B 102, 085113 (2020) Looking at the terms after the last equality in ( A2), we integrate by parts the second term and combine the result with the ﬁrst term. Some straightforward algebra yields σII kl=¯h 4π/integraldisplay dεf(ε)t r (ˆjk(ˆGr−ˆGa)ˆjl(∂εˆGr+∂εˆGa)−ˆjk(∂εˆGr+∂εˆGa)ˆjl(ˆGr−ˆGa)) −1 2¯h 4π/integraldisplay dε∂εf(ε)t r (ˆjk(ˆGr+ˆGa)ˆjl(ˆGr−ˆGa)−ˆjk(ˆGr−ˆGa)ˆjl(ˆGr+ˆGa)) =−¯h 4π/integraldisplay dεf(ε)t r ( ( ˆjk(∂εˆGr+∂εˆGa)ˆjl−ˆjl(∂εˆGr+∂εˆGa)ˆjk)(ˆGr−ˆGa)) −¯h 8π/integraldisplay dε∂εf(ε)t r ( ( ˆjk(ˆGr+ˆGa)ˆjl−ˆjl(ˆGr+ˆGa)ˆjk)(ˆGr−ˆGa)) =σsea kl−σol kl. (A3) [1] R. Kubo, Can. J. Phys. 34, 1274 (1956) . [2] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957) . [3] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982) . [4] A. Bastin, C. Lewiner, O. Betbedermatibet, and P. Nozieres, J. Phys. Chem. Solids 32, 1811 (1971) . [5] P. St ˇreda and L. Smr ˇcka, Phys. Status Solidi B 70, 537 (1975) . [6] L. Smrcka and P. Streda, J. Phys. C 10, 2153 (1977) . [7] P. Streda, J. Phys. C 15, L717 (1982) . [8] A. Crépieux and P. Bruno, Phys. Rev. B 64, 014416 (2001) . [9] N. Nagaosa, J. Sinova, S. Onoda, A. H. 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PhysRevB.66.184516.pdf	Theory of the de Haas –van Alphen effect in type-II superconductors Kouji Yasui and Takafumi Kita * Division of Physics, Hokkaido University, Sapporo 060-0810, Japan ~Received 19 June 2002; revised manuscript received 7 August 2002; published 27 November 2002 ! Theories of quasiparticle spectra and the de Haas–vanAlphen ~dHvA !oscillation in type-II superconductors are developed based on the Bogoliubov–de Gennes equations for vortex-lattice states. As the pair potentialgrows through the superconducting transition, each degenerate Landau level in the normal state splits intoquasiparticle bands in the magnetic Brillouin zone. This brings Landau-level broadening, which in turn leadsto the extra dHvA oscillation damping in the vortex state. We perform extensive numerical calculations forthree-dimensional systems with various gap structures. It is thereby shown that ~i!this Landau-level broaden- ing is directly connected with the average gap at H50 along each Fermi-surface orbit perpendicular to the ﬁeldH,~ii!the extra dHvA oscillation attenuation is caused by the broadening around each extremal orbit. These results imply that the dHvA experiment can be a unique probe to detect band- and/or angle-dependentgap amplitudes.We derive an analytic expression for the extra damping based on the second-order perturbationwith respect to the pair potential for the Luttinger-Ward thermodynamic potential. This formula reproduces allour numerical results excellently, and is used to estimate band-speciﬁc gap amplitudes from available data onNbSe 2,N b3Sn, and YNi2B2C. The obtained value for YNi2B2C is fairly different from the one through a speciﬁc-heat measurement, indicating presence of gap anisotropy in this material. DOI: 10.1103/PhysRevB.66.184516 PACS number ~s!: 74.60.Ec, 74.25.Jb, 74.25.Ha I. INTRODUCTION The de Haas–van Alphen ~dHvA !experiment on normal metals has been a unique and powerful tool to probe theirFermi surfaces. 1,2The main purpose of this paper is to estab- lish theoretically that it can even be used to detect detailedgap structures of type-II superconductors. Back in 1976, Graebner and Robbins discovered the dHvA oscillation in 2H-NbSe 2persisting down through the superconducting upper critical ﬁeld Hc2.3It was after 15 years later when O ¯nukiet al.ﬁrst reconﬁrmed it.4Since then, however, a considerable number of materials have beenfound to display the dHvA oscillation in the vortex state. They include A15 superconductors V 3Si~Refs. 5,6 !and Nb3Sn,7a borocarbide superconductor YNi 2B2C,8,9heavy- fermion superconductors CeRu 2,10URu2Si2,11,12UPd2Al3,13 and CeCoIn 5,14and an organic superconductor k-(BEDT-TTF) 2Cu(NCS) 2;15see Refs. 16 and 17 for a re- cent review. The basic features of the oscillation are summa-rized as follows: ~i!the dHvAfrequencies remain unchanged through the transition, ~ii!the oscillation amplitude experi- ence an extra attenuation, ~iii!the cyclotron mass does not change except for strongly correlated heavy fermion sys-tems. It is somewhat surprising that the dHvA oscillation is ob- servable even in superconductors without a well-deﬁnedFermi surface. Many theories have been presented to explainthe persistent oscillation and the extra damping, 18–33which may be classiﬁed into three categories. The ﬁrst approach applies a Bohr-Sommerfeld semiclas- sical quantization to either the Brandt-Pesch-Tewordt34 ~BPT!Green’s function near Hc2,18the electron number Nat H50,19or Dyson’s equation at H50,20for obtaining the oscillatory behavior of the magnetization.As may be seen bythe diversity of the applications, however, there is no uniquesemiclassical quantization scheme for quasiparticles in su-perconductors, and the validity of the procedure is not clear. This category includes Maki’s theory, 18which was later re- produced by Wasserman and Springford21by treating the BTPself-energy as the extra broadening factor in the normal-state thermodynamic potential and then following Dingle’s procedure. 35However, the BTP self-energy itself is obtained within the quasiclassical approximation without the Landau-level structure so that this approximation may also be ques-tionable. The second approach relies on some approximate analytic solutions for the Bogoliubov–de Gennes ~BdG!equations or the equivalent Gor’kov equations, such as averaging overvortex lines, 22,23the diagonal-pairing approximation,24or the Ginzburg-Landau ~GL!expansion for the free energy with respect to the order parameter.25,27,26However, quantitative estimations of those approximations are yet to be carried out.For example, although Stephen reproduced Maki’s formulabased on the BdG equations, 22he used two approximations of averaging over vortex lines and replacing a sum over dis-crete Landau levels by an integral; the formula derived withthese approximations may be quantitatively questionable.Asfor the GL expansion, it necessarily integrates out the quasi-particle degrees of freedom so that the physical origin of theextra oscillation damping may be obscured in the GL ap-proach. The third approach solves the BdG equations numerically without approximations. 31,32Norman et al.31thereby ex- tracted an analytic formula for the dHvAoscillation dampingthrough a ﬁtting to their numerical data. 31However, it is a two-dimensional calculation for the isotropic s-wave pairing where the number of Landau levels below the Fermi level is NF;10 atHc2. As may be realized from the appearance of the quantized Hall effect, two-dimensional systems in highmagnetic ﬁelds may be qualitatively different from three-dimensional systems. Thus, the obtained formula may not beappropriate for describing real three-dimensional materialsPHYSICAL REVIEW B 66, 184516 ~2002! 0163-1829/2002/66 ~18!/184516 ~16!/$20.00 ©2002 The American Physical Society 66184516-1withNF@1. On the other hand, another calculation by Miller and Gyo ¨rffy for a two-dimensional lattice model32 corresponds to the low-ﬁeld limit near Hc1~Ref. 36 !and may not be suitable to explain the experiments. Moreover,calculations for lattice models have a ﬂaw that they cannotyield continuous magnetic oscillation due to the commensu- rability condition between the underlying lattice and the vor-tex lattice. Notice ﬁnally that most of the above theories consider only the isotropic s-wave pairing. Especially, no numerical studies have been performed yet for anisotropic pairings orthree-dimensional systems. With these observations, we will perform both numerical and analytic calculations for three-dimensional BdG equa-tions with various gap structures. They can be solved efﬁ-ciently by the Landau-level-expansion ~LLX!method, which was formulated for vortex-lattice states of arbitrary pairingsymmetry 37and used successfully to compare low-energy quasiparticle spectra between s- andd-wave pairings.38We will thereby clarify how the discrete Landau levels experi-ence quantitative changes as the pair potential grows below H c2. Another purpose is to ﬁnd out the connection between the gap anisotropy and the extra dHvA amplitude attenua-tion. Terashima et al. 9reported a dHvA experiment on YNi2B2C where an oscillation is observed to persist down to a ﬁeld ;0.2Hc2. On the other hand, a speciﬁc-heat experi- ment at H50 shows a power-law behavior }T3at low temperatures,39indicating the existence of gap anisotropy in this material. Indeed, Izawa et al.40recently reported pres- ence of four point nodes in the gap based on a thermal-conductivity measurement. Miyake 19argued that point or line nodes along the extremal orbit may weaken the damp-ing, and proposed to use the dHvAeffect as a probe to detectgap anisotropy. Vavilov and Mineev 29also considered the dHvA oscillation in unconventional superconductors andconcluded that the oscillation does not experience any appre-ciable damping through the superconducting transition if aline node is located exactly at the extremal orbit. However,their formula has been derived with several unknown ap-proximations so that its quantitative validity is also question-able. We examine the connection between the gap anisotropyand extra oscillation damping in full detail and present aquantitative theory on the issue. This paper is organized as follows. Section II provides a formulation to solve the BdG equations for vortex-latticestates. Section III shows calculated quasiparticle spectra andthe dHvA oscillation for a couple of two-dimensional sys-tems. We thereby clarify basic features of the quasiparticlespectra as well as the origin of the extra dHvA oscillationdamping in the vortex state. Section IV presents the mainresults of the paper. It is demonstrated, based on three-dimensional numerical calculations for various gap struc- tures, that the gap anisotropy at H50 can be detectable from the dHvA oscillation in the vortex state. We also derive ananalytic formula for the extra dHvA oscillation damping us-ing the second-order perturbation expansion with respect tothe pair potential. Section V presents estimations of the en- ergy gap for NbSe 2,N b3Sn, and YNi 2B2C, using the ana- lytic formula obtained in Sec. IV D. Section VI concludesthe paper with a brief summary. In Appendix A, we derive a convenient expression for the thermodynamic potential. Ap-pendix B summarizes the expressions of basis functions andoverlap integrals used in the numerical calculations. A BriefReport of the contents was already presented in Ref. 41. II. FORMULATION A. Bogoliubov –de Gennes equations Throughout the paper we will rely on the mean-ﬁeld BdG equations, which obtain the quasiparticle wave functions us andvswith a positive eigenvalue Es.0b y Edr2FH~r1,r2! D~r1,r2! D†~r1,r2!2H~r1,r2!GFus~r2! 2vs~r2!G 5EsFus~r1! 2vs~r1!G. ~1! Here Dis the pair potential and Hdenotes the normal-state Hamiltonian in the magnetic ﬁeld; both are 2 32 matrices to describe the spin degrees of freedom. The dagger denotesHermitian conjugate in both the coordinate and spin vari- ables as @D†(r1,r2)#s1s25Ds2s1(r2,r1) with sj5",#. With this deﬁnition, we can see immediately that the matrix in Eq. ~1!is Hermitian. We adopt the free-particle Hamiltonian for H: H~r1,r2!5d~r12r2!HF2i\21e cA~r2!G2 2me2«FJ1, ~2! whereme,2e(e.0),c, and «Fare the electron mass, the electron charge, the light velocity, and the chemical poten-tial, respectively. We will not consider the spin paramagnet-ism throughout. We also neglect the spatial variation of the magnetic ﬁeld as appropriate for the relevant high- kmateri- als. Then, the vector potential Acan be expressed using the symmetric gauge as A~r!521 2Bzˆ3r, ~3! whereBdenotes the average ﬂux density, and we have cho- sen the ﬁeld along 2zˆfor convenience. The pair potential in turn is given with respect to the quasiparticle wave functions as D~r1,r2!5V~r12r2!F~r1,r2!, ~4! whereVdenotes the interaction and Fis the order parameter deﬁned by F~r1,r2![( s@us~r1!vsT~r2!2vs~r1!usT~r2!#1 2tanhEs 2kBT, ~5!KOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-2withTthe temperature and superscript Tdenoting the trans- pose. It is shown inAppendixAthat the thermodynamic poten- tial corresponding to Eqs. ~1!–~4!is given by V52kBT( sln~11e2Es/kBT!2( sEsEuvs~r!u2dr 21 2EETrD†~r1,r2!F~r2,r1!dr1dr2, ~6! where Tr denotes taking trace over spin variables. This ex- pression will be useful to obtain an analytic expression forthe extra dHvAamplitude attenuation in the vortex state.Themagnetization is then calculated by M52 ]~V/V! ]B, ~7! where Vis the volume of the system. B. Vortex lattices and basic vectors Solving the above equations for general nonuniform sys- tems is a formidable task. For lattice states, however, it canbe reduced into a numerically tractable problem using thesymmetry that they are periodic with a single ﬂux quantum f0[hc/2eper unit cell. We hence deﬁne a pair of basic vectors by a1[~a1x,a1y,0!,~a13a2!zˆ5f0 B5plB2, a2[~0,a2,0!, ~8! wherelB[A\c/eBis the magnetic length. The basic vectors of the corresponding reciprocal lattice are then deﬁned by b1[2~a23zˆ!/lB2, b2[2~zˆ3a1!/lB2. ~9! We now introduce magnetic Bloch vectors for quasiparticle eigenstates by37 k[m1 Nfb11m2 Nfb2S2Nf 4,mj<Nf 4D ~10! and those for the center-of-mass coordinate by q[m1 Nfb11m2 Nfb2S2Nf 2,mj<Nf 2D, ~11! where Nfis an even integer with Nf2denoting the number of ﬂux quanta in the system. Notice that qcovers an area four times as large as that of k. C. Landau-level-expansion LLXmethod It has been shown37that the pair potential of the conven- tional Abrikosov lattice can be expanded in two ways with respect to ( r1,r2) and (R,r)[@(r11r2)/2,r12r2#asD~r1,r2!5( ka( N1N2DN1N2(kpz)cN1ka~r1!cN2q2ka~r2!eipz(z12z2) L 5Nf A2( Nc( Nrmpz~21!NrD¯ Nrpz(Ncm)cNcq(c)~R! 3cNrm(r)~r!eipzz L. ~12! Here cNkais a quasiparticle basis function with Ndenoting the Landau level, kdeﬁned by Eq. ~10!, and a~51,2!sig- nifying signifying twofold degeneracy of every orbital state. On the other hand, cNcq(c)andcNrm(r)are basis functions for the center-of-mass and relative coordinates, respectively, with NcandNrdenoting the corresponding Landau levels, qde- ﬁned by Eq. ~11!, andman eigenvalue for the relative orbital angular momentum operator lˆz. The quantities pzandLare, respectively, the wave number and the system length alongthezdirection parallel to the magnetic ﬁeld; we adopt a notation of using pas a wave vector in zero ﬁeld to distin- guish it from the two-dimensional magnetic Bloch vector k perpendicular to the ﬁeld. As noted in Ref. 42, an arbitrary singleqsufﬁces to describe the conventional Abrikosov lat- tices due to the broken translational symmetry of the vortexlattice. Then the ﬁrst expansion of Eq. ~12!tells us that, by choosing q50, we get an complete analogy with the uniform system in that the pairing occurs between ( k,p z) and ( 2k, 2pz). Finally, the two expansion coefﬁcients DN1N2(kpz)and D¯ Nrpz(Ncm)are connected by DN1N2(kpz)5Nf 2( NcNr^N1N2uNcNr&( m^2k2qum1Nr& 3~21!NrD¯ Nrpz(Ncm), ~13a! D¯ Nrpz(Ncm)52 Nf( N1N2^NcNruN1N2&( k^m1Nru2k2q& 3~21!NrDN1N2(kpz), ~13b! where ^N1N2uNcNr&and^2k2qum1Nr&are elements of unitary matrices for the basis change, i.e., overlap integrals. Their explicit expressions together with those for cNka, cNcq(c), and cNrm(r)are given in Appendix B. Agreat advantage of using Eq. ~12!is that it enables us to transform Eq. ~1!into a numerically tractable problem, as mentioned before. Indeed, by expanding the quasiparticlewave functions as u ~r!5( Nkapzus~N!cNka~r!eipzz AL, ~14a! v~r!5( Nkapzvs~N!cNq2ka~r!e2ipzz AL, ~14b!THEORY OF THE de HAAS–van ALPHEN EFFECT IN. . . PHYSICAL REVIEW B 66, 184516 ~2002! 184516-3Eq.~1!is reduced to a separate matrix eigenvalue problem for eachkapzand the eigenstate is labeled by s5nkapzs with nandsdenoting the quasiparticle band and its spin, respectively. Explicitly, Eq. ~1!becomes ( N2FHN1N2(pz)DN1N2(kpz) DN1N2(kpz)†2HN1N2(pz)GFus~N2! 2vs~N2!G5EsFus~N1! 2vs~N1!G, ~15! where DN1N2(kpz)is given by Eq. ~13a!andHN1N2(pz)is diagonal as HN1N2(pz)5dN1N2F~N111 2!\vB1\2pz2 2me2«FG1, ~16! with vB[eB/Mcthe cyclotron frequency. The self-consistency equation ~4!are also simpliﬁed greatly. Let us deﬁne Vpp8[EV~r!e2i(p2p8)rd3r 54p( l50‘ ( m52ll V¯l~p,p8!Ylm~pˆ!Ylm~pˆ8!,~17! whereYlm(pˆ)[Qlm(up)(1/A2p)exp(imwp) is the spherical harmonic.43We also expand both DandFin terms of the center of mass and relative coordinates as the last line of Eq. ~12!. Then Eq. ~4!is transformed into an equation for the expansion coefﬁcients of each ( Nc,m)a s D¯ Nrpz(Ncm)51 2plB2L( lNr8pz8V¯l~p,p8!Qlm~up!Qlm~up8!F¯ Nr8pz8(Ncm) ~18! withp5ANr/lB21pz2andup5tan21(ANr/lB/pz). Let us further assume that a single lis relevant in Eq. ~17! and take the corresponding V¯l(p,p8) in a separable form as V¯l~p,p8!5VlWl~j!Wl~j8!, ~19! whereWl(j) is some cutoff function with respect to j [\2p2/2me2«Fsatisfying Wl(0)51. Then can rewrite Eq. ~18!as D¯ Nrpz(Ncm)5D˜(Ncm)Wl~j!Qlm~up!, ~20a! with j5\2(Nr/lB21pz2)/2me2«Fand D˜(Ncm)5Vl 2plB2L( Nr8pz8Wl~j8!Qlm~up8!F¯ Nr8pz8(Ncm).~20b! Thus, we only need a self-consistent solution for a set of discrete parameters $D˜(Ncm)(T,B)%through Eqs. ~15!and ~20!using Eq. ~13!. It has been shown44,42that retaining a few Nc’s, e.g.,Nc 50,6,12 for the hexagonal lattice, is sufﬁcient to describe the vortex lattices of H0.05Hc2. Thus, the original problem of obtaining self-consistency for D(r1,r2) at all space points isnow reduced to the one for a few expansion coefﬁcients $D˜(Ncm)(T,B)%. This situation is analogous to the zero-ﬁeld case where a single parameter D0(T) speciﬁes the pair po- tential. The linearized self-consistency equation is obtained by substituting into Eq. ~20b!the expression of F¯ Nrpz(Ncm)linear in the pair potential37 F¯ Nrpz(Ncm)521 2( N1N2^NcNruN1N2& 3tanh~j1/2T!1tanh~j2/2T! j11j2 3( n~21!n^N1N2uNc1nNr2n&D¯ Nr2npz(Nc1nm1n). ~21! Equation ~20!with Eq. ~21!determines the mean ﬁeld Hc2(T), i.e.,Tc(H). If we use the asymptotic expression ~B6!for the overlap integral ^NcNruN1N2&and replace the sum over N1by the integral over x[(N1 2N2)/A2(Nc1Nr), we reproduce the smooth quasiclassical Hc2quasi(T) obtained, for example, for the s-wave pairing by Helfand and Werthamer.45 Finally, Eq. ~4!for two-dimensional systems can be trans- formed similarly. It is also obtained from Eqs. ~17!–~20!by replacing Vl(p,p8)!V(m)(p,p8), extending the summation overmin Eq. ~17!from 2‘to‘, and ﬁnally restricting the summation over landpzonly tol50 andpz50, respec- tively. III. TWO-DIMENSIONAL CALCULATIONS We ﬁrst consider a couple of two-dimensional models and perform fully self-consistent calculations. Our purposes inthis section are summarized as follows: ~i!to clarify the es- sential features of the results by self-consistent calculations,~ii!to see whether point nodes in the gap really enhances the dHvA signals as Miyake claims. 19 A. Models The one-particle Hamiltonian ~2!yields an isotropic Fermi surface speciﬁed by a unit vector pˆ 5(coswp, sinwp). As for the pairing interaction ~17!,w e adopt the following models: Vp8p85HV0W~j!W~j8!, V2W~j!~pˆx22pˆy2!W~j8!~pˆx822pˆy82!,~22! where W~j!5expF21 2Sj \vDD4G ~23! is a smooth cutoff function with vDa cutoff frequency. The second model of Eq. ~22!is beyond the original isotropicKOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-4interaction ~17!, but it is convenient for the above mentioned purposes. In zero ﬁeld, the two interactions yield the s- and dx22y2-wave gaps as Dp5HD0W~j!is2: swave, D0W~j!~pˆx22pˆy2!is2:dx22y2wave,~24! respectively.The corresponding D¯ Nr(Ncm)of Eq. ~20!forBizˆis given by D¯ Nr(Ncm)5HD˜(Nc)W~j!dm0is2, D˜(Nc)W~j!dm21dm22 2is2~25a! with j[\2Nr/2melB22«Fand D˜(Nc)55V0 4plB2( Nr8Wl~j8!F¯ Nr8(Nc,0) V2 4plB2( Nr8Wl~j8!F¯ Nr8(Nc,2)1F¯ Nr8(Nc,22) 2.~25b! Here we have adopted a normalization for D˜(Nc)different from Eq. ~20!so that D˜(Nc)acquires a direct correspondence to the maximum gap D0in Eq. ~24!; the factor1 2in the second case stems from cos2 wpin Eq. ~24!. We have chosen Vlin Eq. ~22!as gl[2N~0!Vl50.5, ~26! whereN(0)5me/2p\2is the density of states per spin at the Fermi level. Another important parameter is the zero-temperature coherence length deﬁned by j0[\vF/D0, ~27! with vFthe Fermi velocity. We have adopted pFj055 for our calculations. The above two quantities ﬁx our models completely; the cutoff \vDin Eq. ~23!andTc(B50) are calculated using the gap equation. It should be noted that choosing pFj0also determines the following quantities: ~i!the ratio \vHc2quasi/kBTc, where \vHc2quasiis the zero-temperature cyclotron energy at the qua- siclassical upper critical ﬁeld Hc2quasi,~ii!the number NFof the Landau levels below the Fermi level at Hc2quasi. Indeed,using the usual cutoff model W(j)5u(\vD2ujpu) with u the step function, we obtain the following results for the s-wave pairing: \vHc2quasi/kBTc56.28/pFj0, NF[«F/\vHc2quasi50.140 ~pFj0!2. ~28! To reproduce the experimental situation \vHc2quasi/kBTc51 ;3 within the present model, we should go into the quantum limitpFj056;2, but we then have NF55;1. In real ma- terials, however, \vHc2quasi/kBTcandpFj0are apparently in- dependent parameters due to effects not covered by the free- particle model such as the energy band structure. Indeed, pFj0is of the order of 30 (NbSe 2) or even larger, whereas \vHc2quasi/kBTc51;3; see Table I below. Also, NF@1 for those materials. The failures to describe these situations are among the main difﬁculties of the free-particle model of Eq.~2!. Motivated by these observations, we also perform another calculations with much more Landau levels below the Fermilevel. This is achieved by including the effect of the banddispersion. Speciﬁcally, we apply the Onsager-Lifshiz ~OL! quantization scheme to Hof Eq. ~1!, i.e., the procedure which has been very successful for describing the dHvA os- cillations in the normal state.1,2Given the density of states per spinN(«) and the average ﬂux density B, theNth Lan- dau level «N(N50,1,2,...)i sdetermined by 2SN11 2D\2 lB254p\2E 0«NN~«8!d«8. ~29! We ﬁx HN1N2(pz)of Eq. ~15!in this way assuming it is diagonal. In contrast, we use the same expression for DN1N2(kpz)of Eq. ~15! as the free-particle case. Finally, we choose j5«Nr/22«Fin Eq.~25!based on the consideration of the free-particle model.37With these prescriptions together with the transfor- mation ~13!, the coupled equations ~15!and~25!are deﬁned unambiguously. We adopt the model density of states N~«!5me 2p\2S11aG «21G2D, ~30! TABLE I. Parameters characterizing three superconductors displaying the dHvAoscillation in the vortex state, together with values of the average gap along the extremal orbit A^uDpu2&EOestimated by Eq. ~39!. Here the symbol aandgare band indices. These values of A^uDpu2&EOare to be compared with the band- and/or angle-averaged quantity D(0) extracted from a speciﬁc-heat experiment ~Ref. 39 !,a far-infrared measurement ~Ref. 55 !, a tunneling experiment ~Ref. 56 !, or a Raman-scattering experiment ~Ref. 57 !. Compound Tc~K! Hc2(T50)~T! mb/me \vHc2~K!\ vHc2/kBTcA^uDpu2&EO ~meV!D(0)~meV! NbSe27.2~Ref. 16 !8.01 ( u568.6°) ~Ref. 16 !0.61 ( a)~Ref. 16 !17.6~Ref. 16 !2.20~Ref. 16 !1.1 ( 60.04) 1.1 ~Refs. 55,56 ! Nb3Sn 18.3 ~Ref. 7 !19.7 (Hic)~Ref. 7 !1.10 ( g)~Ref. 7 !24.1~Ref. 7 !1.31~Ref. 7 !3.2 ( 60.19) 3.2 ~Ref. 57 ! YNi2B2C 14.5 ~Ref. 9 !8.8 (Hic)~Ref. 9 !0.35 ( a)~Ref. 9 !33.8~Ref. 9 !2.33~Ref. 9 !1.5 ( 60.28) 2.5 ~Ref. 39 !THEORY OF THE de HAAS–van ALPHEN EFFECT IN. . . PHYSICAL REVIEW B 66, 184516 ~2002! 184516-5and choose the numerical constants ( a,G)5(2.1,2.7) and (5.0,1.0) for the s- andd-wave models of Eq. ~24!, respec- tively.We also use Eq. ~26!for the pairing interaction and ﬁx \vD50.5«Fin Eq. ~23!. We thereby obtain \vHc2quasi’kBTc atT50 andNF;30 atH5Hc2quasi, which describe the ex- perimental situation much better than the free-particle model. B. Numerical procedures Coupled equations ~15!and~25!are solved iteratively with the help of the transformation ~13!to obtain self- consistent $D˜(Nc)%’s and the corresponding quasiparticle eigenstates. The hexagonal ~square !vortex lattice is assumed for thes-wave (dx22y2-wave !model, as expected theoreti- cally in high magnetic ﬁelds46,47and observed recently in La1.83Sr0.17CuO41d.48It should be noted, however, that the precise lattice structure is not important for the theory of the dHvA oscillation in superconductors. We set q51 2(b11b2) in the relevant equations so that a core of the pair potential is located at the origin R50.42An advantage of this choice is that the corresponding quasiparticle energies have the rota- tional symmetry of the hexagonal ~square !lattice around k 50; other choices would shift the rotation axis from the origin. We then perform calculations of Eqs. ~15!and~25! for a set of discrete k’s deﬁned by Eq. ~10!, where Nfis chosen as a multiple of 12 to include all the high-symmetrypoints G,M, andK(G,X, andM) of the hexagonal ~square ! lattice. Three different values N f512,24,36 are used to see the size dependence, and it has been checked that the resultsdo not differ for the three cases. The hexagonal ~square ! symmetry enables us to restrict the summation over kinto approximately 1/12 (1/8) area of the Brillouin zone. Thus, the calculations can be reduced greatly with due care on thedegeneracy of high-symmetry points. Finally, the obtainedeigenvalues and eigenstates are substituted into Eq. ~6!to calculate the magnetization by Eq. ~7!. All the calculations are performed at T50.1T c. C. Results The above self-consistent procedure is known to give rise to oscillatory singular behaviors in both Hc2andD˜(Nc)in the ﬁeld range where the dHvAoscillation persists.49–52Figure 1 displaysHc2(T) calculated self-consistently for the s- and d-wave models; it is normalized by the quasiclassical upper critical ﬁeld Hc2quasi(T50). An oscillatory behavior sets inaroundT&0.2Tc, andHc2deviates substantially from the smooth Helfand-Werthamer behavior45predicted by the qua- siclassical theory. The number of the Landau levels below the Fermi level is NF;10 around Hc2quasi(0),which is con- siderably smaller than those for the real materials. Figure 2 shows D˜(Nc)as a function of BatT50.1Tcfor thes-wave hexagonal lattice ~ﬁrst column !and thed-wave square lattice ~second column !; they are real and ﬁnite only for Nc 50,6,12,... (0,4,8,...) f o r t h e hexagonal ~square ! lattice,44,42as already mentioned.We observe that D˜(Nc)’s are also singular, and the dominant D˜(0)component cannot be described by the square-root behavior near Hc2quasi(0) ex- pected from the quasiclassical theory. However, those singu- lar behaviors disappear gradually as Bdecreases. Figure 3 displays the quasiparticle energies in the mag- netic Brillouin zone for the s-wave hexagonal lattice ~ﬁrst column !and thed-wave square lattice ~second column !at B/Hc2quasi(0)50.8, 0.5, and 0.1. At B/Hc2quasi(0)50.8, we al- ready observe large dispersion for E&2D0where the pair potential is effective. In contrast, the ﬂat Landau-level struc- ture remains for E2D0where the pair potential vanishes in the present cutoff model of Eq. ~23!. Thus, the dispersion is caused clearly by the scattering from the growing pair poten-tial, and as will be discussed below, it is the origin of theextra dHvA oscillation damping in the vortex state. We also notice that, for B/H c2quasi(0)0.5, almost no qualitative dif- FIG. 1. The upper critical ﬁeld Hc2as a function of Tfor~a!s wave and ~b!dwave of Eq. ~24!. HerepFj055, andHc2is nor- malized by the quasiclassical upper critical ﬁeld Hc2quasi(T50). FIG. 2. The expansion coefﬁcients D˜(Nc)in Eq. ~25!as a func- tion ofBfor theswave ~ﬁrst column !and thedwave ~second column !withT50.1TcandpFj055. The dotted lines in the ﬁrst row signify the square-root behavior expected from the quasiclassi-cal theory.KOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-6ference can be seen between the s- andd-wave cases. At a lower ﬁeld of B/Hc2quasi(0)50.1, however, a marked differ- ence grows around E&D0. Thes-wave energy bands of E &0.7D0are ﬂat and occur in pairs with the level spacing of the order of D02/«F. As already pointed out by Norman et al.,31these corresponds to the bound core states of an iso- lated vortex with little tunneling probability between adja-cent cores. In contrast, the d-wave bands in the same region are densely packed with large dispersion, indicating the ex-tended nature of the corresponding quasiparticle wave func-tions. From this comparison, we conclude that no boundstates exist for the d-wave model even in the zero-ﬁeld limitof an isolated vortex, in agreement with the result of Franz and Tesˇanovic´. 53This difference in the low-energy disper- sion at low ﬁelds was already reported in Ref. 38. Figure 4 shows oscillatory part of the magnetization Mosc calculated numerically by Eq. ~7!, where curves of the cor- responding normal state are also plotted for comparison. The damping starts from above Hc2quasi(0) where D˜(0)is already ﬁnite as in Fig. 2, and develops rapidly as D˜(0)grows in decreasing B. Thus, the mean-ﬁeld theory predicts that the dHvAoscillation comes together with the oscillatory singular behaviors in Hc2andD˜(Nc). Combined with the energy dis- persion given in Fig. 3, we are now able to attribute theorigin of the extra damping unambiguously to the Landau-level broadening due to the pair potential. The oscillationsare rather irregular in both the s-wave and d-wave cases, in accordance with the singular behaviors of D (Nc)in Fig. 2.We also see no qualitative difference between the two cases. However, the free-particle model has several inappropri- ate points as discussed already around Eq. ~28!. For example, the number of Landau levels below the Fermi level NFis necessarily NF;10 atHc2quasiforpFj055, which is much smaller than the values of the materials displaying the dHvA oscillation. Hence the above numerical results may not besufﬁcient to say anything quantitative about the dHvA at-tenuation or the differences between the s- andd-wave cases. We have thus performed another calculations for the model described around Eqs. ~29!and~30!where \ vHc2quasi/kBTc;1 andNF;30 atB5Hc2quasi(0). Figure 5 shows the ﬁeld dependence of the expansion coefﬁcients D˜(Nc)calculated self-consistently for the s-wave hexagonal lattice ~ﬁrst column !and thed-wave square lattice ~second column !. Singular oscillatory behaviors are manifest in both cases as in the case of the quadratic dispersion, whichoriginate from the singular density of states of Landau levels. 52For example, the dominant D˜(0)component have a nonzero value from above Hc2quasiand deviates substantially from the quasiclassical square-root behavior ~dotted lines !. Figure 6 displays the corresponding oscillatory part of the magnetization Mosccalculated numerically by Eq. ~7!, where normal-state results ~dotted lines !are also plotted for com- parison. The main features are summarized as follows: ~i! The oscillations are seen to decrease from above the quasi- classicalHc2quasi, due to the reentrant behavior of D˜(Nc),t ob e FIG. 3. Quasiparticle dispersion in the magnetic Brillouin zone for thes-wave hexagonal lattice ~ﬁrst column !and thed-wave square lattice ~second column !. HerepFj055,T50.1Tc, and B/Hc2quasi(0) is equal to 0.8, 0.5, and 0.1 from top to bottom, respec- tively FIG. 4. Oscillatory part of magnetization Moscfor~a!theswave and~b!thedwave, over 0.8 <Hc2quasi(0)/B<2.0@1.2>B/Hc2quasi(0) >0.5#withT50.1TcandpFj055. The dotted lines are the curves of the corresponding normal state.THEORY OF THE de HAAS–van ALPHEN EFFECT IN. . . PHYSICAL REVIEW B 66, 184516 ~2002! 184516-7reduced considerably around B;0.8Hc2quasi, i.e.,Hc2quasi/B ;1.25. However, they do not disappear completely in lower ﬁelds. ~ii!This extra attenuation is due to the broadening of the Landau levels caused by the pair potential, as in the case of the quadratic dispersion. Indeed, we have obtained quasi-particle spectra similar to those of Fig. 3. ~iii!The period of the oscillation remains unchanged above ;0.8H c2quasi, but some irregularity appears in lower ﬁelds. These features are in agreement with the results by Norman et al.31~iv!Little difference can be seen between the s- andd-wave attenua- tions.D. Summary of Two-Dimensional Calculations Let us summarize results and conclusions from our two- dimensional calculations. ~i!Combining Figs. 3 and 4, we are now able to attribute the origin of the extra dHvA oscil-lation damping unambiguously to the Landau-level broaden-ing due to the scattering by the pair potential. ~ii!As may be realized from Fig. 6, presence of point nodes along the ex-tremal orbit does not weaken the attenuation, contrary to thestatement by Miyake. 19This fact suggests that the attenua- tion is determined by the average gap along the extremalorbit. ~iii!The mean-ﬁeld theory predicts that the dHvA os- cillation comes together with the oscillatory behaviors in H c2 andD˜(Nc). This will be so in three dimensional models where Hc2(T) also shows an oscillatory behavior.52However, such singular behaviors of Hc2have never been identiﬁed deﬁ- nitely in any materials displaying the dHvA oscillation, and reportedHc2curves show more or less the smooth quasiclas- sical behavior. This discrepancy between the mean-ﬁeldtheory and the dHvA experiments remains a puzzle to beresolved in the future. ~iv!The oscillation attenuates consid- erably around B;0.8H c2quasi, i.e.,Hc2quasi/B;1.25, although we have set \vHc2quasi/kBTc;1 andNF@1a tHc2quasi. Thus, the two dimensional models fail to explain the experiment by Terashima et al.9which shows a persistent oscillation down to 0.2Hc2. In addition, the models cannot say anything about whether presence of a line node along the extremal orbit weakens the attenuation. ~v!The approximation of retaining onlyD˜(0)works excellently for calculating Mosc. Indeed, we have checked that the curves of Moscthereby obtained are almost indistinguishable from those of Fig. 6. ~vi!The dis- crepancy mentioned in ~iii!above suggests that we should rather use D˜(0)obtained quasiclassically to reproduce the smooth behaviors of Hc2in real materials. Figure 7 plots curves of Mosccalculated using quasiclassical D˜(0), i.e., the dotted lines of Fig. 5. The oscillations are seen more regularthan those of Fig. 6, but the amplitudes attenuate almost similarly and are reduced considerably around H c2quasi/B ;1.25. We hence realize that using the quasiclassical D˜(0) sufﬁces for the theory of the oscillation damping. This state- ment is especially true in the low-ﬁeld region ;0.2Hc2quasi where D˜(0)approaches to the quasiclassical behavior, as may be realized from Fig. 5. FIG. 5. The expansion coefﬁcients D˜(Nc)in Eq. ~25!as a func- tion ofBfor theswave ~ﬁrst column !and thedwave ~second column !. HereT50.1Tc, and the nonquadratic dispersion given by Eq.~30!is used. The dotted lines in the ﬁrst row signify the square- root behavior expected from the quasiclassical theory. FIG. 6. Oscillatory part of magnetization Moscfor~a!theswave and ~b!thedwave, over 0.8 &Hc2quasi(0)/B<1.7, i.e., 1.25 B/Hc2quasi(0)>0.59. Here T50.1Tc, and a nonquadratic disper- sion given by Eq. ~30!is used.The dotted lines are the curves of the corresponding normal state. FIG. 7. Oscillatory part of magnetization Moscfor~a!theswave and~b!thedwave. The difference from Fig. 7 lies in the use of quasiclassical D(0)’s given by the dotted lines in Fig. 5.KOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-8IV. THREE-DIMENSIONAL CALCULATIONS Having clariﬁed basic features of the dHvAoscillation for two-dimensional models as well as the mechanism of theextra oscillation damping, we proceed to consider three-dimensional models with various gap structures which aremore relevant to real materials. Our purposes in this sectionare summarized as follows: ~i!to clarify the connection be- tween the extra dHvA oscillation damping and the gap an-isotropy by numerical calculations, ~ii!to obtain an analytic formula for the extra oscillation damping, ~iii!to estimate the gap magnitudes of various materials using the obtained ana-lytic formula. A. Model The one-particle Hamiltonian ~2!yields a spherical Fermi surface in the normal state.As for the pairing interaction, weconsider three different models: V p8p85HV0W~j!W~j8!, V2W~j!~pˆx22pˆy2!W~j8!~pˆ2x82pˆy82!, V1W~j!pˆcˆW~j8!pˆ8cˆ.~31! HereW(j) is a cutoff function given by Eq. ~23!,pˆ [(sinupcoswp, sinupsinwp, cosup) speciﬁes a point on the Fermi surface, and cˆ[(sinuc,0,cos uc) denotes the direction of the crystal caxis, in the coordinate frame where Bizˆ. Again the latter two models of Eq. ~31!are beyond the origi- nal spherical interaction ~17!, but they are convenient for the abovementioned purposes. In zero ﬁeld, Eq. ~31!yield Dp5HD0W~j!is2: swave, D0W~j!~pˆx22pˆy2!is2:dx22y2wave, D0W~j!pˆcˆis3s2:pzwave,~32! which denote the isotropic s-wave state, a three-dimensional dx22y2-wave state with four point nodes in the extremal orbit perpendicular to B, and the p-wave polar state with a line node perpendicular to cˆ, respectively. The corresponding D¯ Nrpz(Ncm)in Eq. ~18!can be written as D¯ Nrpz(Ncm)55D˜~Nc!W~j!dm0is2, D˜(Nc)W~j!sin2updm21dm22 2is2, D˜(Nc)W~j!Fcosupcosucdm0 1sinupsinucdm11dm21 2Gis3s2, ~33a! where j[\2(Nr/lB21pz2)/2me2«F, up [tan21(ANr/lB/pz), andD˜(Nc)is deﬁned byD˜(Nc)5ƒV0 4plB2( Nr8pz8Wl~j8!F¯ Nr8pz8(Nc,0), V2 4plB2( Nr8pz8Wl~j8!sin2up8F¯ Nr8pz8(Nc,2)1F¯ Nr8pz8(Nc,22) 2, V1 4plB2( Nr8pz8Wl~j8!Fcosup8cosucF¯ Nr8pz8(Nc,0) 1sinup8sinucF¯ Nr8pz8(Nc,1)1F¯ Nr8pz8(Nc,21) 2G. ~33b! Here we have adopted a normalization for D˜(Nc)different from Eq. ~20!so that this quantity acquires a direct corre- spondence to the maximum gap D0in Eq. ~32!.The factors1 2 in the second and third cases stem from cos2 wpand cos wpin Eq.~32!, respectively. The coefﬁcients D(Nc)5D(Nc)(B,T) in Eq. ~33a!com- pletely specify the pair potential, as already mentioned.Based on the reasoning given in Sec. III D ~vi!, we here adopt a quasiclassical D˜(Nc)rather than the fully self-consistent one.Then the dominant D˜(0)nearHc2follows the mean-ﬁeld square-root behavior to an excellent approximation D˜(0)5a~12B/Hc2!1/2. ~34! See the dotted lines in Figs. 2 and 5, for example. In addi- tion, other components D˜(Nc.0)can be neglected for the rel- evant region B0.1Hc2, as pointed out in Sec. III D ~v!.W e hence use the lowest-Landau-level approximation of retain- ing only D˜(0). The coefﬁcient a5a(T) in Eq. ~34!is deter- mined by requiring that the maximum of 1 VEdRUEdrD~r1,r2!e2ipr/\U2 ~35! be equal to D02(12B/Hc2), where D0(T) denotes the maxi- mum gap obtained from the weak-coupling theory. This pro- cedure yields a’A0.5, ~36! for all the three cases of Eq. ~33a!. Substituting Eq. ~34!into Eq.~13a!with the choice \vD;10D0(T50), the off- diagonal elements of Eq. ~15!are ﬁxed completely. The above non-self-consistent procedure has another ad- vantage that we can choose \vB5Hc2and«Fin Eq. ~16! independently. We have set \vHc25kBTcatT50, ~37! in accordance with \vHc2/kBTc51–3 and NF@1 for rel- evant materials ~seeTable I below !.Also, we have chosen «F in such a way that there are about 50 Landau levels below «F for the extremal orbit at Hc2. Now, the matrix elements ofTHEORY OF THE de HAAS–van ALPHEN EFFECT IN. . . PHYSICAL REVIEW B 66, 184516 ~2002! 184516-9Eq.~15!are speciﬁed completely. Hexagonal, square, and hexagonal lattices are assumed for the three cases of Eq.~33a!, respectively. B. Numerical procedures The wave vector pzof 0<pz<1.2pFis discretized into ;1000 points with an equal interval. For each of them, we have diagonalized Eq. ~15!with the same procedure as de- scribed in Sec. III B.The obtained results are substituted intoEq.~6!to calculate the magnetization by Eq. ~7!. All the calculations are performed at T50. C. Numerical results We ﬁrst focus on the dHvA oscillation of the s-wave model in Eq. ~32!to clarify its basic features. Using our numerical data, we also test the applicability of various the-oretical formulas presented so far. Figure 8 ~a!presents oscillation of the s-wave magnetiza- tion~blue line !as compared with the normal-state one ~sky- blue line !. With \ vHc25kBTc, the oscillation is observed to persist down to a fairly low ﬁeld of Hc2/B&1.8, i.e., B 0.55Hc2, which is smaller than 0.8 Hc2around which \vB becomes equal to the spatial average of the energy gap, Eq. ~35!. This might be partly because the dispersion is smallerwithin the extremal orbit, as shown quasiclassically by Brandtet al.34Indeed, Fig. 9 calculated at B50.968Hc2 demonstrates that the dispersion for pz50 is smaller than that forpz50.9pF. This tendency remains in the high-ﬁeld region of B0.5Hc2. It has become conventional to express this extra attenua- tion in the vortex state by introducing an additional factor Rs for the dHvA oscillation amplitude Rs5expS2p vBtsD5expS22p2kBTD \vBD, ~38! FIG. 9. Quasiparticle dispersion in the magnetic Brillouin zone for thes-wave model at B50.968Hc2.~a!pz50;~b!pz50.9pF. FIG. 10. ~Color !~a!The oscillatory part of the magnetization Moscin the vortex state ~blue line !as compared with the normal- state one ~sky-blue line !for thed-wave model of Eq. ~32!atT 50.~b!The corresponding Dingle plot ~points with error bars !as compared with the theoretical prediction ~39!. FIG. 11. ~Color !Left ﬁgures: the oscillatory part of the magne- tizationMoscin the vortex state ~blue lines !as compared with the normal-state one ~sky-blue lines !for thep-wave model of Eq. ~32! atT50. The crystal caxis, which is perpendicular to the nodal plane, is tilted from the ﬁeld Bbyuc50~top!,uc5p/6~second !, anduc5p/4~bottom !. Right ﬁgures: the corresponding Dingle plots ~points with error bars !compared with Eq. ~39!. FIG. 8. ~Color !~a!The oscillatory part of the magnetization Moscin the vortex state ~blue line !as compared with the normal- state one ~sky-blue line !for thes-wave model of Eq. ~32!atT 50.~b!The corresponding Dingle plot ~points with error bars !as compared with various theoretical predictions; see text for details.KOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-10where the parameters tsandTDare directly connected with the extra Landau-level broadening Gsin the vortex state as Gs5\/2ts5pkBTD. The points with error bars in Fig. 8 ~b! shows ln Rsas a function of 1/ B, i.e., the Dingle plot, ob- tained by numerical differentiation. This extra damping at high ﬁelds shows the behavior }12B/Hc2in the logarith- mic scale, but irregularity sets in around 0.55 Hc2where the oscillation disappears. We attribute this irregularity to theeffect of the bound-state formation in the core region. The lines in Fig. 8 ~b!are predictions from various theo- retical formulas. Maki’s formula 18reproduces the correct functional behavior }12B/Hc2at high ﬁelds, but the pref- actor is seen too large. The NMA formula,31deduced from the two-dimensional self-consistent numerical results with NF;10 atHc2, predicts a more rapid attenuation incompat- ible with our numerical data. One reason for this discrepancymay originate from the fact that their numerical data with N F;10 atHc2are not appropriate for obtaining an analytic formula by ﬁtting. Another may be attributed to the differ-ence in dimensions. Indeed, the dHvA oscillation in threedimensions differs from that in two dimensions on the point that some ﬁnite region dpzaround the extremal orbit is rel- evant. Most of the Landau levels in the region do not satisfy the particle-hole symmetry with respect to «F, so that the effect of the pair potential becomes smaller than that in twodimensions. Another theory by Dukan and Tes ˇanovic´, 24 which would predict Rs50 in the clean limit of T50, is also inconsistent with the data. The red line in Fig. 8 ~b!is due to our formula for the extra Dingle temperature kBTD50.5G˜^uDpu2&EOmbc pe\12B/Hc2 B, ~39! which will be derived in Sec. IV D based on the second- order perturbation with respect to the pair potential. Here ^uDpu2&EOdenotes the average gap along the extremal orbit at B50, andmbis the band mass. The numerical constant 0.5 stems from Eq. ~36!, and G˜is a dimensionless quantity char- acterizing the Landau-level broadening due to the pair poten- tial. This unknown parameter G˜is determined by a best ﬁt to thes-wave numerical data, i.e., the points with error bars in Fig. 8 ~b!. This procedure yields G˜50.125. We observe in Fig. 8 ~b!that Eq. ~39!, which predicts the dependence }12B/Hc2for lnRs, agrees with the numerical results. This formula will be seen below to reproduce othernumerical data excellently without any adjustable param-eters. Adifference of Eq. ~39!from Maki’s formula 18lies in the prefactor where the Fermi velocity vFis absent. Indeed, a dimensional analysis on the second-order perturbation tellsus that the Landau-level broadening in the vor- tex state should be of order uD˜(0)(B)u2/\vB, where D˜(0)(B)‘A^uDpu2&EO(12B/Hc2) is essentially the average gap along the extremal orbit. This leads to Eq. ~39!except for the numerical constant.We now turn our attention to see how the presence of point nodes affect the dHvA oscillation. Figure 10 ~a!shows the oscillation of the d-wave magnetization ~blue line !as compared with the normal-state one ~sky-blue line !. Al- though the d-wave gap in Eq. ~32!has four point nodes on the Fermi surface along the extremal orbit, the damping isseen strong and not much different from the s-wave case. From this fact, we may conclude that it is the average gapalong the extremal orbit which is relevant for the extra dHvAoscillation damping. Figure 10 ~b!presents the corresponding Dingle plot ~points with error bars !, which is compared with the prediction of Eq. ~39!. The formula with the average gap ^uDpu2&EOreproduces the numerical result for Hc2/B&1.8 excellently without adjustable parameters, thereby providinga strong support for the above statement. Thisd-wave result is in disagreement with Miyake’s theory that point nodes in the extremal orbit should weakenthe attenuation. 19Indeed, Miyake’s theory is based on a semiclassical quantization for the expression of the electron numberNeatB50. Neither his starting point Ne(B50) nor the use of the semiclassical quantization may be justiﬁed for describing the dHvA oscillation observed mainly near Hc2. We ﬁnally consider the p-wave model with a line node in Eq.~32!to double check the applicability of Eq. ~39!. The left ﬁgures in Fig. 11 display the dHvA oscillation for theline-node model, where the crystal caxis is tilted from the magnetic-ﬁeld direction by uc50~top!,uc5p/6~second !, anduc5p/4~bottom !. The damping is seen weakest in the top ﬁgure where the gap vanishes exactly along the extremalorbit, supporting the prediction ofVavilov and Mineev on thepolar state. 29However, it increases gradually as ﬁnite gap opens along the orbit for uc50!p/4. These results indicate conclusively that the average gap along the extremal orbit isrelevant for the extra dHvA attenuation. However, the non-zero extra damping in the top ﬁgure implies that not only theextremal orbit alone but some ﬁnite region around it contrib-utes to the extra damping. Theoretically, this corresponds tothe fact that we have to perform the Fresnel integral 2‘‘exp@2i(A2pNFpz/pF)2#dpzfor obtaining the LK for- mula in the normal state. Our data show that this off- extremal-orbit contribution cannot be neglected in the casewhere the gap vanishes exactly and completely at the ex-tremal orbit. However, this off-extremal contribution is ex-pected to become less important where the ﬁnite gap ispresent along the extremal orbit. The right ﬁgures in Fig. 11show the corresponding Dingle plot ~points with error bars !, which is compared with the prediction of Eq. ~39!. Except for the weak damping of uc50 due to the off-extremal-orbit contribution, the formula is observed to reproduce the nu-merical results excellently. D. Analytic formula We here derive the analytic formula ~39!for the extra dHvAoscillation damping based on the second-order pertur-bation expansion with respect to the pair potential. This is anappropriate approach to see how the signal changes through H c2, which has also been adopted by most of the existing theories.18,21–29However, we here carry out a rigorous per-THEORY OF THE de HAAS–van ALPHEN EFFECT IN. . . PHYSICAL REVIEW B 66, 184516 ~2002! 184516-11turbation expansion retaining the Landau-level structure without recourse to any quasiclassical approximations norany semiclassical quantization schemes. The difference fromStephen’s result will be discussed in detail. We start from the thermodynamic potential given by Eq. ~6!, which is an exact expression within the mean-ﬁeld ap- proximation. Since the last term 2 1 2TrDFmay be express- ible solely with respect to the pair potential, it can be ne-glected for the present model ~34!to consider the oscillatory part.We then calculate E sandvsin the ﬁrst two terms by the second-order perturbation with respect to D. They are ob- tained as ENkapzs5ujNpzsu1hNkpz(1)sgn~jNpzs!, ~40! EuvNkapzs~r!u2dr5u~2jNpzs!1hNkpz(2)sgn~jNpzs!, ~41! where uis the step function and hNkpz(n)is deﬁned by using Eq.~13a!as hNkpz(n)[( N8uDNN8(kpz)u2 ~jNpz1jN82pz!n. ~42! The ﬁrst terms on the right-hand side of Eqs. ~40!and~41! are just the normal-state results. The second terms, on theother hand, denote ﬁnite quasiparticle dispersion in the mag-netic Brillouin zone and smearing of the Fermi surface, re-spectively, due to the scattering by the pair potential. Using Eqs. ~13a!and~33a!, let us express this hNkpz(n)in terms of D˜(0)(B) and the cyclotron energy \vBof the extremal orbit as hNkpz(n)5uD˜(0)~B!u2 ~\vB!nh˜Nkpz(n). ~43! The quantity h˜Nkpz(n)thus deﬁned is dimensionless, and we realize that the main Bdependence in Eq. ~43!lies in the prefactor uD˜(0)(B)u2/(\vB)n. Considering the case uc50 for simplicity and putting W(j)51 as appropriate for NF@1, thish˜Nkpz(n)is given by h˜Nkpz(n)5Nf2 4( N8mm8u^NN8u0N1N8&u2^N1N81mu2k2q& @N1N822~NF1d!#n 3^2k2quN1N81m8&3Hdm0dm80, dm,62dm8,62sin4up, dm0dm80cos2up, ~44! where the quantity d5d(B,pz)(udu,1/2) speciﬁes the location of «Fbetween two closest Landau levels, and the overlap integrals are deﬁned by Eqs. ~B4!and~B7!. Thecorresponding normalized density of states DNpz(n)~h˜![2 Nf2( kad~h˜2h˜Nkpz(n)!, ~45! will play a central role in the following. Indeed, it describes broadening of the Landau levels due to ﬁnite quasiparticledispersion in the Magnetic Brillouin zone which originatesfrom the scattering by the pair potential. Substituting Eqs. ~40!and~41!into Eq. ~6!, we ﬁnd that the terms containing hNkpz(2)may be neglected due to the can- cellation between the particle and hole contributions. The remaining term can be transformed with the standard procedure.1We thereby obtain for the ﬁrst harmonic of V/V the expression V1 V52kBT 2p2lB2( sE 21/2‘ dNcos~2pN!E 2‘‘ dpzE 2‘‘ dh˜ 3DNpz(1)~h˜!ln@11e2[jNpzs1h˜uD˜(0)(B)u2/\vB]/kBT#.~46! It should be noted that the pzintegral in Eq. ~46!passes through the regions where d50 in Eq. ~44!. For those points, the second-order perturbation is not justiﬁed, and we shouldresort to a special technique based on the degenerate pertur-bation theory. 63However, those are points of measure zero in the integration, and the main contribution certainly comes from the regions where we can use Eq. ~40!. Notice that d 521/8 at the maximal orbit when the magnetization takes a local maximum. Now,DNpz(1)(h˜) in Eq. ~46!h a sa(N,pz) dependence. However, it may be replaced by a representative one D¯l(1)(h˜) to be placed outside the Nandpzintegrals, where the recov- ered index lspeciﬁes the s-,d-, orp-wave case of Eq. ~44!.I t may also be acceptable to use a Lorenzian for it: D¯l(1)~h˜!5G˜l p~h˜21G˜l2!. ~47! Replacing DNpz(1)(h˜) in Eq. ~46!by Eq. ~47!and carrying out the integration, we ﬁnally obtain an expression for the mag- netization which carries an extra damping factor: Rs~B![E 2‘‘ D¯l(1)~h˜!exp@22pih˜uD˜(0)~B!u2/~\vB!2#dh˜ 5exp@22pG˜luD˜(0)~B!u2/~\vB!2#. ~48! Comparing this with Eq. ~38!, we conclude that the super- conductivity gives rise to an extra Dingle temperature of kBTD[G˜luD˜(0)(B)u2/p\vB, or equivalently, the extra scat- tering rate ts21[2pkBTD/\. There seems to be no analytic way to estimate G˜s,s ow e ﬁx it through the best ﬁt to the s-wave numerical data of Fig. 8~b!. Using Eq. ~34!witha250.5D02, the procedure yields G˜s50.125, as mentioned before. It is also clear from Figs. 10 and 11 as well as from Eq. ~44!that the average gapKOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-12around the extremal orbit is relevant for the extra attenua- tion. We hence put a2G˜l50.5^uDpu2&EOG˜s. We thereby ob- tain Eq. ~39!, which yields excellent ﬁts to the d- andp-wave numerical data without any adjustable parameters, as seen inFigs. 10 and 11. Equation ~39!has an advantage that we can trace the ori- gin of the extra dHvAdamping deﬁnitely to the growing pairpotential, which brings ﬁnite quasiparticle dispersion in themagnetic Brillouin zone as Eq. ~44!, and the corresponding Landau-level broadening as Eq. ~45!. Moreover, Eq. ~44! tells us that this broadening near H c2is closely connected with the zero-ﬁeld gap structure given by Eq. ~32!. We ﬁnally compare Eq. ~39!with other theories. As al- ready mentioned, the main difference of Eq. ~39!from Ma- ki’s formula lies in the absence of vF}A«Fin the denomi- nator. Since Maki’s formula is reproduced by Stephen22 based also on the BdG equations, we should clarify wherethe difference originates from. It can be attributed toStephen’s averaging over vortex lines, 22which corresponds to putting Nf2 4^N1N81mu2k2q&^2k2quN1N81m8&!1~49! in Eq. ~44!. Indeed, if we calculate the discrete sum of Eq. ~44!with Eq. ~49!ford.0, we ﬁnd a behavior h˜Nkpz(1) }N21/2. If we further use the quasiclassical approximation ~B6!and replace the sum over N8by an integral, we repro- duce Stephen’s result for the self-energy which is propor- tional toN21/2}«F21/2. However, one can check numerically that the approximation ~49!for Eq. ~44!yields a poor result. Indeed, h˜Nkpz(1)has noN21/2dependence but distributed ran- domly with oscillation as a function of N. It is also found numerically that the variance of Eq. ~45!depends little on the values of NFandN(;NF), as expected; for example, it is ;0.2 for d521/8 for the s-wave pairing.The absence of the factor «F21/2is also supported by a dimensional analysis on the second-order perturbation; it tells us that the Landau- level broadening in the vortex state should be of order uD˜(0)(B)u2/\vB, where D˜(0)(B)}A^uDpu2&EO(12B/Hc2)i s essentially the average gap along the extremal orbit. Thisleads to Eq. ~39!except for the numerical constant. Vavilov and Mineev 28,29derived a damping factor which is different from Eq. ~38!and given for the s-wave pairing by RsVM’12ApNF lnNFHc22B Hc2. ~50! The second factor originates from their analytic expression forD2~i.e.,D˜(0)of this paper !derived from the self- consistency equation for the pair potential D2’8pNF lnNFGimp2Hc22B Hc2, ~51! where Gimpdenotes the normal-state level broadening due to the impurity scattering. However, they replaced in the deri-vation every sum over Landau levels by an integral using Eq.~B6!. It hence follows that Eq. ~51!is essentially a quasiclas- sical expression which should be compared with the exact numerical results of Eilenberger. 64Since «Fis absent in the quasiclassical Eilenberger equation65given in the dimension- less form, however, there is no possibility that the depen- dence D2}NF/lnNFappears. It hence follows that their ex- pression ~51!may be questionable. Apart from this fact, the damping mechanism they proposed is completely differentfrom the present one. Indeed, the second term of Eq. ~50!is directly connected with the decrease in the total density of statesN(E50) obtained by averaging over the discrete Landau-level structure as well as over the space points.Thus,the extra damping in their interpretation is caused by thedecrease of the quasiclassical density of states. On the otherhand, our formula ~39!attributes the extra damping to the Landau-level broadening due to the scattering by the pairpotential. We believe that our formula provides a correct andphysically transparent understanding for the extra oscillationdamping in the vortex state. V. ESTIMATION OF ENERGY GAP Our calculations in Sec. IV C have clariﬁed that ~i!the gap anisotropy can be detected by measuring the extra dHvAoscillation damping in the vortex state, and ~ii!Eq.~39!is particularly useful for this purpose. Using the formula, weﬁnally provide quantitative estimations of the average gapalong the extremal orbit for several materials displaying thedHvA oscillation in the vortex state. Table I summarizes pa-rameters describing three relevant materials. These materials commonly have fairly high T c’s, and the ratio \vHc2/kBTc ranges from 1 to 3. These features seem to be basic condi- tions for observing the dHvA oscillation in the vortex state. The values for A^uDpu2&EOare obtained by applying Eq. ~39! to the observed dHvA attenuation in the vortex state. In do- ing so, we have adopted as mbin Eq. ~39!the values from dHvA experiments rather than those from band calculations,as indicated by the theory of Luttinger. 54For comparison, we have also listed the values D(0) estimated by a speciﬁc-heat experiment,39a far-infrared measurement,55a tunneling experiment,56or a Raman-scattering experiment.57Thus, D(0) is expected to represent band- and/or angle-averaged energy gap. As seen in Table I, the two quantities coincide excellently for NbSe 2and Nb 3Sn, indicating uniformly opened gap in these materials. On the other hand, A^uDpu2&EO51.5 for the aband of YNi 2B2C is considerably smaller than D(0)52.5 from a speciﬁc-heat experiment.39 This fact implies that YNi 2B2C have large band- and/or angle-dependent gap anisotropy. Indeed, Bintley et al.58have recently carried out a detailed dHvA experiment on this ma- terial, rotating the ﬁeld direction and observing the extra at-tenuation. They have reported a large angle dependence ofthe attenuation magnitude. They have also pointed out thattheir result is in agreement with the model with point nodespresented by Izawa et al. 40based on a thermal-conductivity measurement.THEORY OF THE de HAAS–van ALPHEN EFFECT IN. . . PHYSICAL REVIEW B 66, 184516 ~2002! 184516-13VI. SUMMARY We have carried out three-dimensional numerical calcula- tions on the dHvA oscillation in the vortex state for variousgap structures. We have thereby clariﬁed the relation be-tween gap anisotropy and persistence of the oscillation. Wehave also derived an analytic formula for the extra dHvAattenuation in the vortex state. Our main results are given by Figs. 8–11 and Eq. ~39!. Those ﬁgures indicate clearly that the extra dHvA attenua-tion in the vortex state is directly connected with the average gap along the extremal orbit at B50. The derived formula ~39!has been shown to reproduce the numerical results ex- cellently. Our theory attributes the origin of the extra dHvAdamping to the Landau-level broadening caused by the pairpotential. Hence the periodicity of the vortex lattice assumedhere is almost irrelevant, and the theory is applicable also tothe cases with irregularity such as a random array of vortices.Using Eq. ~39!, we have estimated average gap amplitudes along the extremal orbit for NbSe 2,N b3Sn, and YNi 2B2C. The results indicate presence of large gap anisotropy in YNi2B2C. Thus, we have shown explicitly that the dHvA effect in the vortex state can be a powerful tool to probe the averagegap along the extremal orbit. Our results imply that, by ro-tating the ﬁeld direction and observing the attenuation am-plitude, we can obtain unique information on the band-and/or angle-dependent gap structure. Such an experiment has recently been performed on UPd 2Al3by Inada et al.13 and on YNi 2B2C by Bintley et al.,58and the latter group indeed has detected large gap anisotropy in the abplane. Equation ~39!will be useful in similar experiments for esti- mating band- and/or angle-dependent gap amplitudes. ACKNOWLEDGMENTS We are grateful to F. J. Ohkawa for enlightening discus- sions. This research was supported by a Grant-in-Aid forScientiﬁc Research from the Ministry of Education, Culture,Sports, Science, and Technology of Japan. APPENDIX A: THERMODYNAMIC POTENTIAL The Luttinger-Ward thermodynamic potential correspond- ing to Eq. ~1!is given by59 V52kBT 2( nTrlnFH2i«n D D†2H2i«nG 3Fei«n010 0e2i«n01G21 2TrD†F, ~A1! where we have adopted a compact notation of using x[rs withDs1s2(r1,r2)!D(x1,x2), etc., andTr also implies both integration and summation over rands, respectively. The quantity «n/\denotes the Matsubara frequency, and 0 1is an inﬁnitesimal positive constant. Now, Eq. ~1!tells us that the ﬁrst matrix in Eq. ~A1!can be diagonalized as59FH~x,x8! D~x,x8! D†~x,x8!2H~x,x8!G 5( sFus~x!2vs~x! vs~x!2us~x!GFEs0 02EsG 3Fus~x8! vs~x8! 2vs~x8!2us~x8!G. ~A2! Substituting Eq. ~A2!into Eq. ~A1!, the ﬁrst term on the right-hand side becomes 2kBT 2( nsEdx$@uus~x!u2ezn011uvs~x!u2e2zn01# 3ln~Es2zn!1@uvs~x!u2ezn011uus~x!u2e2zn01# 3ln~2Es2zn!%, ~A3! withzn[i«n. The summation over nare then transformed with a standard technique60into a contour integral just above and below the real axis, using f(z)[(ez/kBT11)21andf (2z) for the terms with ezn01ande2zn01, respectively. Con- sidering the poles inside the two contours and using @uus(x)u21uvs(x)u2#dx51, we obtain Eq. ~6!. APPENDIX B: BASIS FUNCTIONS AND OVERLAP INTEGRALS We here present explicit expressions for the quantities ap- pearing in Eqs. ~12!and~13!; see Ref. 37 for their detailed derivations. It should be noted that we here adopt the sym- metric gauge ~3!which is more convenient than the Landau gauge used in Ref. 37. Hence there is an extra factor due tothe gauge transformation in every expression of the basis functions, such as e 2ixy/2lB2in Eq. ~B1!below. The basis function cNka(N50,1,2,...; a51,2) in Eq. ~12!is deﬁned by cNka~r!5( n52Nf/211Nf/2 ei[ky(y1lB2kx/2)1na1x(y1lB2kx2na1y/2)/lB2] 3e2ixy/2lB22(x2lB2ky2na1x)2/2lB21i(a21)np 3Aa1x/lB 2NN!ApSHNSx2lB2ky2na1x lBD, ~B1! where S[plB2Nf2, andHN(x)[ex2(2d/dx)Ne2x2is the Hermite polynomial.61The basis function cNq(c)for the center- of-mass coordinates is obtained from Eq. ~B1!by putting k !q,a51, andlB!lc[lB/A2a sKOUJI YASUI AND TAKAFUMI KITA PHYSICAL REVIEW B 66, 184516 ~2002! 184516-14cNq(c)~r!5( n52Nf/211Nf/2 ei[qy(y1lc2qx/2)1na1x(y1lc2qx2na1y/2)/lc2] 3e2ixy/2lc22(x2lc2qy2na1x)2/2lc2 3Aa1x/lc 2NN!ApSHNSx2lc2qy2na1x lcD. ~B2! Finally, the basis function cNmfor the relative coordinates, which is conveniently chosen as an eigenstate of the orbital angular momentum operator lˆz, is given by cNm(r)~r!5~21!N A2plrAN! ~N1m!!zme2uzu2/2LN(m)~uzu2!, ~B3! where z[(x1iy)/A2lrwithlr[A2lB, andLN(m)(x) [(1/N!)exx2m(d/dx)Ne2xxN1mis the generalized La- guerre polynomial satisfying N1m>0.61 We next provide expressions of the overlap integrals in Eq.~13!. The ﬁrst one, which was obtaind by Rajagopal and Ryan,62is given by ^NcNruN1N2& [dN11N2,Nc1NrAN1!N2!Nc!Nr! 2N11N2 3( n5max(0,Nc2N2)min(N1,Nc)~21!N22Nc1n n!~N12n!,~Nc2n!!~N22Nc1n!!, ~B4!which satisﬁes ^NcNruN1N2&5~21!Nr^NcNruN2N1&, ~B5a! ( N1N2^Nc8Nr8uN1N2&^N1N2uNcNr&5dNc8NcdNr8Nr. ~B5b! The asymptotic expression of Eq. ~B4!forNc!N1,N2is given by37 ^NcNruN1N2&’dN11N2,Nc1Nr~21!N2F2 p~Nc1Nr!G1/4 3e2x2/2HNc~x! A2NcNc!, ~B6! withx[(N12N2)/A2(Nc1Nr). This expression forms the basis for quasiclassical approximations. The second overlap integral is given by ^qum1Nr&5A2plr~21!m1Nr@cm1Nrq(r)~0!#,~B7! where cm1Nrq(r)is another basis function of the relative coor- dinates deﬁned by cNq(r)~r!5( n52Nf/411Nf/4 ei[qy(y1lr2qx/4)/2 12na1x(y1lr2qx/22na1y)/lr2] 3e2ixy/2lr22(x2lr2qy/222na1x)2/2lr2 3A2a1x/lr 2NN!ApSHNSx2lr2qy/222na1x lrD.~B8! *Electronic address: kita@phys.sci.hokudai.ac.jp. URL: http:// phys.sci.hokudai.ac.jp/k ˜ita/index-e.html 1I. M. Lifshitz and A. M. Kosevich, Zh. E´ksp. Teor. Fiz. 29, 730 ~1955!@Sov. Phys. JETP 2, 636 ~1956!#. 2D. Shoenberg: Magnetic Oscillations in Metals ~Cambridge Uni- versity Press, Cambridge, 1984 !. 3J. E. Graebner and M. Robbins, Phys. Rev. Lett. 36, 422 ~1976!. 4Y. O¯nuki, I. Umehara, T. Ebihara, N. Nagai, and K. Takita, J. Phys. Soc. Jpn. 61, 692 ~1992!. 5F. M. Mueller, D. H. Lowndes,Y. K. Chang,A. J.Arko, and R. S. List, Phys. Rev. Lett. 68, 3928 ~1992!. 6R. Corcoran, N. Harrison, S. M. Hayden, P. Meeson, M. Spring- ford, and P. J. van der Wel, Phys. Rev. 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PhysRevB.99.205115.pdf	PHYSICAL REVIEW B 99, 205115 (2019) Charge control of blockade of Cooper pair tunneling in highly disordered TiN nanowires in an inductive environment S. E. de Graaf,1,R. Shaikhaidarov,2,3T. Lindström,1A. Ya. Tzalenchuk,1,2and O. V . Astaﬁev1,2,3 1National physical laboratory, Hampton Road, Teddington, TW11 0LW, United Kingdom 2Royal Holloway, University of London, Egham, TW20 0EX, United Kingdom 3Moscow Institute of Physics and Technology, Dolgoprudny, Russia (Received 29 October 2018; revised manuscript received 13 March 2019; published 10 May 2019) A central problem in understanding the superconductor-insulator transition in disordered superconductors is that the properties of grains and intergrain medium cannot be independently studied. Here we demonstratean approach to the study of strongly disordered superconducting ﬁlms by relying on the stochastic natureof the disorder probed by electrostatic gating in a restricted geometry. Charge tuning and magnetotransportmeasurements in quasihomogeneous TiN nanowires embedded in a superinductor environment allow us toclassify different devices and distinguish between spontaneously formed Coulomb islands (with typical blockadevoltage in the mV range) and homogeneous wires showing behavior indicative of coherent quantum phase slips(with signiﬁcantly smaller blockade voltage). DOI: 10.1103/PhysRevB.99.205115 I. INTRODUCTION The superconductor-insulator transition (SIT) in disor- dered superconducting thin ﬁlms is gaining increasing atten-tion for its relevance in a range of applications in supercon-ducting and quantum electronics [ 1–3] and electrical metrol- ogy [ 4]. The SIT occurs as a result of Anderson localization or as Coulomb interaction breaks global phase coherence, thelatter often described by the proliferation of quantum phaseslips (QPS) forming a normal matrix around spontaneouslyformed localized regions of superconductivity [ 5–7]. The disorder [ 8,9] and magnetic ﬁeld [ 10–13] driven SIT have been extensively studied in thin 2D ﬁlms and more recentlynanopatterned devices [ 4,10,14–17] of highly disordered su- perconductors such as InO, NbN, NbTiN, and TiN. Nanowiresof these materials are of particular interest in the view oftheir proposed applications in quantum devices and electricalmetrology [ 15,18,19]. While radio-frequency measurements have demonstrated the coherence of QPS in a range of materials [ 20–24], it re- mains a signiﬁcant challenge in DC measurements. To ensurelarge quantum phase ﬂuctuations it is essential to embed suchnanowires in a high impedance environment. The resultinginsulating state due to blockade of Cooper-pair tunnelingoccurs below a critical voltage V c=2πEs/2e, due to the QPS rateEs/h. The most straightforward implementation of a high impedance environment, a resistive environment, is generallyincompatible with DC currents due to Joule heating [ 18]. Alternatively, embedding the QPS element in a superinductor[3,6,25,26] ensures the required dynamics [ 19,27]. In this work we perform magnetotransport studies of TiN nanowires of width 2–3 ξ 0, where ξ0is the BCS coherence sdg@npl.co.uklength, embedded in a high impedance inductive environment. The presented devices were selected from a larger set tohighlight distinctively different regimes observed. All deviceswere designed to be identical and made from the same super-conducting ﬁlm. From the dependence of the voltage blockadeon magnetic ﬁeld and electrostatic gate we ﬁnd that nanowiresof similar dimensions fall into signiﬁcantly different regimes.We focus on three different devices: Larger blockade voltages(in the mV range) can be attributed to single-electron transis-tors (SETs) with a behavior consistent with the grain size ofthe ﬁlm. In contrast, wires with signiﬁcantly smaller V cshow a qualitatively different behavior; the gate modulation of theblockade is very weak, and V cis suppressed at large magnetic ﬁelds. We discuss this latter scenario in terms of QPS ina homogeneous wire. A third device shows an intermediateregime, supporting these conclusions. Our results provideadditional insight into recent experiments in somewhat widerTiN nanowires closer to the SIT [ 14] and shed further light on the nature of the SIT in nanopatterned TiN ﬁlms, enabled bycharge control. II. EXPERIMENT The devices were made from the same TiN thin ﬁlm de- posited using plasma-assisted atomic-layer deposition (ALD)on a high resistivity Si(100) substrate using the same methodand equipment as described in Ref. [ 28]. Further fabrication and measurement details are outlined in the SupplementalMaterial [ 29]. The device design is shown in Fig. 1(a). Figure 1(b) shows the overall measurement circuit and Fig.1(c) illustrates the different regimes found in the devices presented. We note that this is a very simpliﬁed picture usefulfor the classiﬁcation of regimes, the real microstructure islikely much more complicated. 2469-9950/2019/99(20)/205115(7) 205115-1 ©2019 American Physical SocietyS. E. DE GRAAF et al. PHYSICAL REVIEW B 99, 205115 (2019) L L Cg Vg+ -+ - Vb A A : B : C : =(a) (b) (c) FIG. 1. (a) Scanning electron micrograph of a typical device. Scale bars are 1 μm and 100 nm for the meanders and the closeup of the constriction, respectively. (b) Equivalent circuit with the various device regimes explored is represented by a square and clariﬁed in (c) with a simpliﬁed schematic of each of the presented devices A–C. Each nanowire is approximately 20–30 nm wide and 100 nm long and is symmetrically embedded in a wider mean-dering nanowire of width 100 nm and length ≈50μm(N /square= 480 squares on each side, respectively). The footprint ofeach meandered superinductor is approximately 3 ×3μm 2, giving it a self-capacitance Cm≈0.2 fF. From the measured normal state resistance we ﬁnd a sheet resistance of R/square= 3.0k/Omega1which translates to a sheet kinetic inductance Lk,/square= ¯hR/square/π/Delta1=4.4 nH, using the BCS gap /Delta1=1.76kBTcand measured critical temperature Tc=1.0 K. This results in an impedance of the circuit in the superconducting state, as seenby the nanowire, of Z m=2/radicalbig N/squareLk,/square/Cm≈200 k/Omega1, larger than the superconducting resistance quantum R2q=h/4e2≈ 6.5k/Omega1. In the normal state each meander has a resistance of Rm=1.45 M/Omega1/greatermuchRq=h/e2. III. RESULTS A. Material properties From isotherms of R(B), we ﬁnd the diffusion constant D=3.6×10−5m2s−1and a coherence length in the dirty limitξ0(T=0)≈12 nm (see Supplemental Material for de- tails), in good agreement with previous ﬁndings in similarﬁlms [ 30]. The disorder induced SIT and the competition between superconductivity and Coulomb repulsion in theseﬁlms [ 28] results in a reduction of T cas the ﬁlm thickness is reduced. In contrast to distinctly granular ﬁlms the observedsuperconducting transition remains sharp (for our ﬁlm T c= 1.0 K and δTc=0.4 K), a signature of average homogeneous disorder. Such a superconductor can be regarded as granu-lar on length scales smaller than the critical dimension forsupporting superconductivity in a single isolated grain [ 5] b<b crit=(ν/Delta1)−1/3≈8 nm, where b3is the grain volume andν=(e2DR/squared)−1≈13 eV−1nm−3is the density of states FIG. 2. Conductance of device A versus back-gate voltage, mea- sured in the superconducting state at 10 mK and 600 mT (perpendic-ular ﬁeld). The behavior remains the same above the critical ﬁeld when superconductivity is suppressed (see Supplemental Material [29]). The inset shows a very simpliﬁed sketch of a possible micro- scopic device geometry; circles represent grains. (DOS) at the Fermi level. For a grain level spacing δ>/Delta1 a single grain is too small to support superconductivity, and theobserved superconductivity is a result of intergrain coherence.Nevertheless, superconducting ﬂuctuations will suppress theDOS near the Fermi level due to electron-phonon coupling[5]. For the typical grain size b≈4n m[ 28] we obtain an average grain level spacing δ/k B=(νb3kB)−1∼14 K/greatermuch/Delta1. B. Large blockade devices The nature of these devices is most clearly elucidated from the conductance in response to a substrate back-gatevoltage, shown for one device in Fig. 2(device A). The gate dielectric is provided by the 300 μm Si substrate. A gap modulation with a period of ∼180 mV is observed. This is the typical behavior of a (superconducting) SET. The amplitudeof oscillations in V c∼4 mV could not be due to QPS as this would require a pair of pointlike QPS junctions, each with Vc exceeding /Delta1/eby one order of magnitude. For the device in Fig. 2we ﬁnd the total capacitance C/Sigma1=e/max( Vc(Vg))= 40 aF, a charging energy Ec/kB=e2/2C/Sigma1kB=23 K, and a gate capacitance to the effective island Cg=e/δVg≈0.9a F , consistent with that of an island of area ≈20 nm2in our back- gate geometry. This size agrees well with the observed grainsize in these ﬁlms [ 28], and it follows that the tunnel junction capacitance C J∼20 aF is given by an insulating gap between grains of 1–2 nm ( εr=110 [ 31,32]). Similar values for Cg andC/Sigma1were found for all devices showing a larger blockade voltage Vc∼1–5 mV /greatermuch/Delta1/e(we have characterized three such devices). C. Small blockade We now turn to a device (B) with strongly connected grains (as will become apparent in the following discussion), 205115-2CHARGE CONTROL OF BLOCKADE OF COOPER PAIR … PHYSICAL REVIEW B 99, 205115 (2019) FIG. 3. Small blockade (device B). (a) Zero bias resistance as a function of temperature with ﬁt to theory of quantum corrections for the conductivity. The inset shows the normalized residual resistance from the ﬁts in the region highlighted by the dashed circle. (b) Differential conductance measured as a function of applied substrate back-gate voltage Vg. Dashed lines indicate the gate voltages in (c) and (d). (c),(d) Differential conductance measured at a gate voltage of −2.2Va n d +3.0 V , respectively, versus applied perpendicular magnetic ﬁeld. Vertical lines at low ﬁeld and large bias are due to phase-slip centers in the inductive meanders. The inset shows a very simpliﬁed schematic sketch of the device. We attribute the small offset in bias voltage to our biasing circuitry. (e) Example of two IV curves showing the twoextremes with a blockade voltage (black) and a superconducting branch (red). A resistance of 16 k /Omega1has been subtracted to highlight the superconducting branch. (f) and (g) Selected cross sections taken from (c) and (d), respectively. showing no initial voltage blockade. The data is presented in Fig.3. Despite the narrow width of meanders ( wm=100 nm) they are still in the limit of 2D conductivity for all rele-vant temperatures. From the saturation in resistivity at highmagnetic ﬁelds we estimate that the electronic temperaturereaches ∼100 mK, corresponding to a maximum thermal length L th=√D¯h/kBT=52 nm <wm. Quantum corrections to the total conductivity Gabove Tcof a disordered 2D ﬁlm is described by the Aronov-Altsuler (AA) electron-electron in-teraction, weak localization, and superconducting ﬂuctuationsfrom Maki-Thompson (MT), DOS and Aslamazov-Larkin(AL) corrections. We also add a contribution G CB(T)= −cln [gEc/max ( T,/Gamma1)] from granularity and single electron charging effects of (some of) the grains [ 33,34], and the normal Drude term Gn. The total conductivity is given by (see Supplemental Material [ 29] for full details) [ 13]:G= Gn+GAA+GMT+GDOS+GAL+GCB≡Gq+GCB. The R(T) dependence of device B with ﬁts to Gis shown in Fig. 3(a). The broadened change in slope of the resistance at≈32 K can be interpreted as originating from a few poorly connected grains with a distribution of sizes (in the otherwisehomogeneous ﬁlm) freezing out due to Coulomb blockade(CB). These poorly connected grains are located in the widermeanders; the nanowire itself is homogeneous and no block-ade is seen in the current-voltage (IV) curves. From the saturation of the CB term we extract an average grain level broadening /Gamma1/k B=gδ/kB=32 K giving an average intergrain dimensionless conductanceg=G/(2e2/h)=2.4. Taking the single grain charging en- ergy from device A, Ec/kB=23 K, as a representation of the average grain charging energy we ﬁnd an exceptionallygood ﬁt to R(T) [Fig. 3(a)]. For comparison we also show the calculated resistance excluding the G CBterm. From this we conclude that the ﬁlm is on average homogeneous withstrongly connected grains with b<b crit, however, a smaller number of grains are weakly coupled which stochasticallyaffects the transport characteristics of nanopatterned devices. We now explore this device (B) in an applied magnetic and electric ﬁeld. The response to a gate voltage, shown inFig.3(b), is much weaker than what is to be expected from a single grain dominating the transport (device A). For negativegate voltages a gap emerges of maximum size V c∼40μV. The gate response occurs at voltages two orders of magnitude larger than for the SET-like devices. Next, by ﬁxing the gate voltage at −2.2 V and measuring Vcas a function of applied magnetic ﬁeld [Figs. 3(c) and (f)] we observe that Vcis increasing as we approach Bc2⊥. Above Bc2⊥Vcbecomes negligibly small. On the contrary, for positive gate voltages we observe an increasing zero-biasconductance, shown in Figs. 3(d) and3(g). IV curves for the two extremes are shown in Fig. 3(e) for comparison. This is reminiscent of a ‘critical current’ corresponding to ≈1n A . This should be put in relation to the critical current of the100 nm wide meanders which is I c≈5 nA. Assuming the same critical current density throughout the whole device thistranslates to a nanowire width of about 20 nm, as expected. 205115-3S. E. DE GRAAF et al. PHYSICAL REVIEW B 99, 205115 (2019) This excess critical current is suppressed in an applied mag- netic ﬁeld [Figs. 3(d) and3(g)]. If we further attribute the residual resistance below the critical current of the nanowire tocontact /lead series resistance ( ∼16 k/Omega1) we ﬁnd a resistance of the nanowire itself of 13.7 k /Omega1, corresponding to 4.6 R /square, or a 100 nm long nanowire of width 22 nm. We note thatthe series resistance could not be due to phase diffusionas the junction capacitance required to sustain the observedsupercurrent of 1 nA must be in excess of 80 fF ( E c<EJ= /Phi10Ic/2π), in which case residual resistance due to phase diffusion becomes negligible. We also note that slowly repeating the measurement in Fig. 3(b) multiple times (not shown) yields the same global dependence. However, the smaller features vary in positionand intensity between measurements. Charge relaxation ona time scale of several minutes is also observed, consistentwith the behavior of charge traps in the substrate, or in theremaining hydrogen silsesquioxane (HSQ) resist, near thenanowire. The weak gate dependence of devices B could be an indication of QPS interference, as the induced charge alongthe nanowire is expected to result in interference of QPS am-plitudes, suppressing the phase-slip rate due to the Aharonov-Casher (AC) effect [ 24]. The relevant scale for charge local- ization deep in the phase-slip regime is on the order of ξ=√ ξ0l≈2 nm, where lis the mean free path [ 35]. This length scale is consistent with the much weaker gate dependence,as compared to the charging of a granule (device A). Weinterpret these results as the emergence of a few strongerphase slip centers along the wire. The gate dependence arisesdue to one or more larger segments of the wire being enclosedby these stronger phase slip centers. This can be understoodin analogy to the simplest such implementation: the chargequantum interference device [ 24]. The elementary building blocks, corresponding to sketch B in Fig. 1(c),w o u l db ea gateable grain (or cluster of well-connected grains) connectedto the rest of the wire via two QPS junctions, being regionsof enhanced phase slip rate. Due to the nontrivial gate depen-dence we are likely dealing with a more complicated geometrythan what is schematically sketched in Fig. 1(c) and in the inset of Fig. 3(c). If we assume that we are able to arrange complete de- structive interference of QPS amplitudes ( V c=0) using the gate, this situation should be the dual to a Josephson junctionwhere the critical current is suppressed; no voltage gap de-velops since phase coherence is maintained, and the junctioninstead turns dissipative. In a high impedance environment(E s/greatermuchEL=/Phi12 0/2N/squareLk,/square) charge is the well deﬁned quan- tum variable. Suppressing the phase-slip rate is not sufﬁcientto establish the required phase coherence to observe a ‘criticalcurrent’ branch. However, for an intermediate impedance, asin our case, both regimes would still be accessible for a QPSwire, which we argue is the case of device B. Phase slip rate From the measured critical voltage below Bc2we can calculate the phase-slip rate Es=2eVc/2π, for devices B (and C, discussed below) which ranges between 3 and 17 GHz(30÷220μV), on the same order as obtained in coherentmeasurements of QPS qubits [ 20–23], and in good agreement with the theoretical expectations for the phase-slip rate inshort wires [ 20,21,36] E s=/Delta1/radicalBigg L ξR2q Rξexp/parenleftbigg −aR2q Rξ/parenrightbigg , (1) where a≈0.36 for a diffusive conductor [ 37], and Rξis the normal state resistance of a wire segment of length ξ. We thus expect Es≈5 GHz ≈60μV for device B given the previously estimated wire dimensions. Charge control of theQPS rate allows us to tune the nanowire through the SIT viathe AC effect. The blockade in device B also fulﬁls an important criteria for attributing the blockade to QPS: V c<2/Delta1≈290μV. The relatively small blockade is an indication of a largekinetic capacitance due to quantum phase slips [ 14,38]C k= 2e/2πVc=e2/π2Es=0.1÷0.9 fF, much larger than any achievable geometric capacitance of any part of the homoge-neous device and between any grains. D. Intermediate device We now turn to device C (Fig. 4), showing a behavior very similar to that reported in Ref. [ 14]. There it was suggested that the peak in conductivity above Bc2⊥is due to the order parameter inside grains persisting to magnetic ﬁelds muchhigher than those that suppress intergrain couplings, whichallowed for good agreement between experiment and theoryby a duality transformation applied to the theory of transportin over-damped small Josephson junctions [ 39]. Our data for two magnetic ﬁeld orientations is shown in Figs. 4(a)and4(b), respectively. A smeared voltage gap of V c≈200μV is seen at B=0, and increasing the ﬁeld reveals an oscillatory behavior of the gap persisting well above the suppression of the criticalcurrent, strikingly similar to Ref. [ 14]. We also note that these oscillations vary between cool downs from room temperature[(c.f. Fig. 4(e)], as would be expected from a bosonic SIT and spontaneously formed electronic inhomogeneity [ 40], and that they are perfectly symmetric with respect to ﬁeld orientation: V c(+B)=Vc(−B). This particular device was only possible to gate to ±300 mV after which the gate dielectric started to leak, but only a very weak ( <5%) variation in Vcwas observed in this range (not shown), similar to device B. While the zero ﬁeld data for the two datasets in Figs. 4(a) and4(b) are very similar, for even modest ﬁelds the blockade becomes much sharper for the perpendicular ﬁeld orientation.This onset is consistent with the expected vortex entry ﬁeldin the superinductor B ⊥=/Phi10√2d/πw/(2πλLξ)≈30 mT, and the effect can be attributed to quasiparticle trappingby vortex cores [ 41] cooling the nanowire. In Fig. 4(c) we compare two IV curves for the two ﬁeld orientations where theapplied ﬁeld is such that the critical current of the meandersis suppressed to the same value. Interestingly even aboveB c2the gap remains much sharper for the perpendicular ﬁeld orientation, a behavior signiﬁcantly different from the ﬁeld-induced parity effects in insulating Josephson junction chains[42]. Similar behavior was instead seen in wider TiN [ 43] and InO ﬁlms [ 44] which can be phenomenologically described by anisotropic orbital effects competing with the isotropic 205115-4CHARGE CONTROL OF BLOCKADE OF COOPER PAIR … PHYSICAL REVIEW B 99, 205115 (2019) FIG. 4. Intermediate blockade (device C). (a) Differential conductance measured as a function of parallel magnetic ﬁeld and (b) and (c) perpendicular ﬁeld in two consecutive cool downs, respectively. The inset shows a very simpliﬁed sketch of the likely device geometry. (d)Comparison of IV curves for magnetic ﬁelds where critical currents of meanders have been suppressed equally. (e) Extracted blockade voltage V c(solid markers) and normal state resistance (hollow markers) from the data in (b) (purple) and (c) (orange). Zeeman effect, governing the percolation of superconductivity [45]. Peak in didv in the normal state and gap modulation The peaks in ∂I/∂VatV=±Vcremains above Bc2.T h e absence of SET-like behavior could indicate the presenceof a single tunnel junction as opposed to QPS. The wellestablished P(E) theory [ 46] describing tunnel junctions in high impedance environments fails to reproduce the peakin conductance at the gap edge under the assumption of aconstant DOS in the normal state. The required nonlinearDOS at the Fermi level is expected from disordered super-conductors even in the normal state near the SIT even wellabove T candBc2[9,40,47], supporting the debated notion that superconducting ﬂuctuations persist in grains well above Bc2. However, the smearing of the gap above Bc2/bardbland a smaller zero bias resistance compared to Bc2⊥could not be explained within this scenario alone. An onset of back bending of anIV curve above V ccould be a signature of the so-called Bloch nose due to coherent charge oscillations. However,such features could also be attributed to overheating due toweak electron-phonon coupling resulting in poor dissipationof the Joule heating in the resistive state [ 48,49]. These effects could not be directly distinguished in a typical IV trace,however, overheating is not expected to yield the observedgate and magnetic ﬁeld dependence of V c. In particular, theBmodulation of the gap could be attributed to the isotropic Zeeman splitting of energy levels in the grains [ 33,50]. Their low and broadened DOS will thus vary in B, changing the conductivity of the nanowire, in analogy with an electrostaticgate. The difference in gap modulation between 4(b) and 4(c) could arise due to aging and /or the spontaneous for- mation of different percolation networks of superconductingislands, uncorrelated from the underlying metallurgical ﬁlmmorphology, in the different cool downs, separated by sixmonths in time. This picture is also supported by the observedmagnetoconductance of device A which shows a similar, butweaker, modulating behavior of the gap (see SupplementalMaterial [ 29]). The absence of a strong gate effect would thus imply that in sample C we are most likely dealing with a singlewell-developed tunnel junction between the granular, but con-tinuous leads, a statistically plausible scenario in a conﬁnedgeometry, falling in-between devices A and B. The DOSin this system may still be subject to local superconductingﬂuctuations. IV . CONCLUSIONS To conclude, we have shown transport measurements on narrow TiN nanowires embedded in a high impedance su-perinductor environment. By studying the behavior both inmagnetic ﬁeld and as a function of applied gate voltages 205115-5S. E. DE GRAAF et al. PHYSICAL REVIEW B 99, 205115 (2019) we are able to identify nanowires exhibiting several regimes; both incoherent Coulomb blockade in single isolated grainsin the ﬁlm as well as new interesting physics in wires thatappear to be much more homogeneous and show indica-tions of coherent quantum phase slips controlled by elec-trostatic gate. Our work highlights the stochastic nature ofthese on average homogeneously disordered nanowires whereisolated grains may be present that can inﬂuence devicephysics.ACKNOWLEDGMENTS We thank Y . Nazarov, A. V . Danilov, and I. Rungger for fruit-ful discussions, T. M. Klapwijk for providing the TiN ﬁlms,and T. Hönigl-Decrinis for assistance with sample prepara-tion. Samples were fabricated in the nanofabrication facilitiesof the department of microtechnology and nanoscience atChalmers university and at Royal Holloway university ofLondon. This work was supported by the UK Department ofBusiness, Energy and Industrial Strategy (BEIS). [1] J. T. Peltonen, P. 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PhysRevB.89.094513.pdf	PHYSICAL REVIEW B 89, 094513 (2014) Dissipation signatures of the normal and superﬂuid phases in torsion pendulum experiments with3He in aerogel N. Zhelev,1R. G. Bennett,1E. N. Smith,1J. Pollanen,2W. P. Halperin,2and J. M. Parpia1 1Laboratory of Atomic & Solid State Physics, Cornell University, Ithaca, New York, 14853 USA 2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA (Received 25 September 2013; revised manuscript received 19 February 2014; published 13 March 2014) We present data for the energy dissipation factor Q−1over a broad temperature range at various pressures of a torsion pendulum setup used to study3He conﬁned in a 98% open silica aerogel. Values for Q−1above Tcare temperature independent and have weak pressure dependence. Below Tc, a deliberate axial compression of the aerogel by 10% widens the range of metastability for a superﬂuid equal spin pairing (ESP) state; weobserve this ESP phase on cooling and the B phase on warming over an extended temperature region. Whilethe dissipation for the B phase tends to zero as T →0,Q −1exhibits a peak value greater than that at Tcat intermediate temperatures. Values for Q−1in the ESP phase are consistently higher than in the B phase and are proportional to ρs/ρuntil the ESP to B phase transition is attained. We apply a viscoelastic collision-drag model, which couples the motion of the helium and the aerogel through a frictional relaxation time τf. We conclude that unless τfis an order of magnitude larger than expected, an additional mechanism to dissipate energy not captured in the collision-drag model and related to the emergence of the superﬂuid order must exist. The extradissipation below T cis possibly associated with mutual friction between the superﬂuid phases and the clamped normal ﬂuid. The pressure dependence of the measured dissipation in both superﬂuid phases is likely related tothe pressure dependence of the gap structure of the “dirty” superﬂuid. The large dissipation in the ESP state isconsistent with the phase being the A or the Polar with the order parameter nodes oriented in the plane of the celland perpendicular to the aerogel anisotropy axis. DOI: 10.1103/PhysRevB.89.094513 PACS number(s): 67 .30.eh,67.30.ht I. INTRODUCTION Unconventionally paired Fermi systems exhibit strong sensitivity in their transport properties to the presence of even a small degree of nonmagnetic impurities [ 1–6]. For the oth- erwise pure superﬂuid3He, an elastic scattering mechanism, in addition to the inelastic two-particle scattering processes, isprovided by porous silica aerogel “impurities” [ 7–10]. Since the discovery of superﬂuidity of 3He in aerogel [ 11,12], the analogy of this so-called “dirty” Fermi superﬂuid with “dirty”unconventional superconductors has been investigated in theliterature. Transport measurements in the normal Fermi liquid(spin [ 13,14], thermal conductivity [ 15,16], and viscosity [ 17]) reveal a crossover from an intrinsic inelastic quasiparticle-quasiparticle (qp-qp) scattering rate at high temperatures to aquasiparticle-impurity dominated relaxation mechanism whenthe temperature is lowered. In the 3He-aerogel composite system, the3He is always on the order of the zero-temperature coherence length awayfrom the aerogel strands. The zero-temperature coherencelength is deﬁned to be ξ 0=/planckover2pi1vF/2πkBTc. It is expected that the superﬂuid order parameter is suppressed and surfacebound states exist near macroscopic surfaces and domainwalls [ 18–20]. However, the aerogel strands do not act as conventional surfaces—else superﬂuidity would be entirely suppressed. Instead, scattering from the aerogel leads to asuppression of the superﬂuid gap. We expect a spectrum oflow energy excitations inside the gap to appear, which couldlead to a gapless superﬂuid state in which the density of statesis ﬁnite around the entire Fermi surface [ 9]. Evidence for such states exists in thermal conductivity [ 21] and heat capacity [ 22] measurements as T→0, but the exact proﬁle for the density of states of the 3He in aerogel system and its dependence on strong coupling effects is still not fully understood.In order to probe the dynamics of the aerogel embedded ﬂuid, we have placed the experimental cell in the head ofa torsion pendulum. We track the frequency and the qualityfactor ( Q) of the pendulum as the temperature is changed. Observing the frequency shift has proved instrumental instudying the effects of disorder at the onset of superﬂuidtransition [ 11,23]. However, due to the close spacing between the aerogel strands (of the order of 50 nm), even the smallimpurity limited viscosity of the normal state 3He would be sufﬁcient to clamp the ﬂuid at the audio frequencies (2.1 kHz)corresponding to the torsional resonant mode we employ. Inorder to probe the transport properties, we cannot rely on thefrequency shift data alone. Instead, in this paper, we focus onthe energy dissipation factor ( Q −1) of the pendulum, which should be sensitive to the Fermi surface excitations discussedin the previous paragraph. The aerogel sample is deliberately compressed along the pendulum axis by 10%. It is generally accepted that theaerogel anisotropy due to the axial compression should favorthe anisotropic, equal spin pairing (ESP) superﬂuid 3He-A phase [ 24,25]. Previously, we would have also expected that the /lscriptvector would preferentially align along the axis of compression. However, recent pulsed NMR tip anglemeasurements on axially compressed aerogel at moderatemagnetic ﬁelds (both along and perpendicular to the strainaxis) show that the /lscriptvector tends to be oriented in the plane of the cell and perpendicular to the strain axis regardless of thedirection of the magnetic ﬁeld [ 26]. Recent theoretical results [27] also point to the possibility of a Polar phase (also an ESP phase) with a line of nodes away from the strain axis.In an earlier work, we observed that the superﬂuid fractionin the ESP phase is less than that in the B phase [ 23]. If /lscriptin the A phase (nodal direction in the Polar phase) was 1098-0121/2014/89(9)/094513(11) 094513-1 ©2014 American Physical SocietyN. ZHELEV et al. PHYSICAL REVIEW B 89, 094513 (2014) aligned perpendicular to the ﬂow, we would instead observe the superﬂuid fraction in the A phase exceeding that in theB phase [ 28,29]. Thus consistent with the equal spin pairing state realized in this experiment is either an A phase with /lscript randomly oriented along the plane of the cell or a Polar phase.Lacking NMR data to identify the phase at zero magnetic ﬁeld,we refer to the intervening phase as ESP rather than the A/Polarphase. The metastable ESP phase is supercooled to temperatures well below the equilibrium ESP to B phase boundary. Onthe other hand, after completion of the ESP to B transitionby further cooling the cell, the superﬂuid B phase persistson warming and the ESP phase only reappears in a regionof small temperature width very close to T c. This results in a signiﬁcant range of temperatures over which we have ESPphase on cooling and B phase on warming, and allows us tomake a direct comparison of the properties ( ρ s,Q−1)o ft h e two superﬂuid phases. In the following sections, we brieﬂy outline experimental details, and present the experimental data. Then we discuss amodel for the energy dissipation factor of the torsion pendulumarising from the normal state ﬂuid. Finally, we discuss the databelow T c, where we observe additional dissipation intrinsic to the superﬂuid. We relate our data to the presented theoreticalmodel and propose a possible mechanism that could accountfor the observed behavior. II. EXPERIMENTAL SETUP A. Construction of the torsion pendulum The torsion pendulum consisted of two hollow beryllium- copper torsion rods: an upper one with a 1.27-mm outerdiameter, and a lower one with an outer diameter of 1.22 mm.A 1-mm diameter hole, bored through both torsion rods, servedas a ﬁll line for the ﬂuid into the pendulum’s head. An epoxyjoint coupled the pendulum “head” to the upper torsion rod. Two magnesium “wings,” electrically insulated from the rest of the pendulum, were attached to the cylindrical massat the junction of the two torsion rods. Each of the wingswas maintained at 100-V bias with respect to closely spacedadjacent electrodes. A function generator was connected toone of the electrodes to drive the pendulum capacitively. Theresulting motion of the pendulum induced a small ac voltagein the second electrode, which was ampliﬁed and sensed by adual phase lock-in ampliﬁer. The pendulum can be excited at two torsion resonance modes: a symmetric mode in which the wings and the headof the pendulum move in phase, and an antisymmetric modewhen they move out of phase. The latter mode provides greatersensitivity to motion of the ﬂuid in the head and lower noiseand was thus selected for this experiment. The 98% open silica aerogel was grown directly into a pillbox shaped stainless steel cavity consisting of a tightlyﬁtted lid, a base and a shim inserted between them. Moreinformation about the physical properties and method ofgrowth of aerogel can be found in Ref. [ 30]. The aerogel was compressed by 10% along its main axis by removing the shim and pressing the lid onto the base, bringingthe height of the cell to 400 μm. The height was chosen to be FIG. 1. (Color online) (a) A schematic for the torsion pendulum setup. (b) A cross-section of the torsion pendulum head. The cell in which the aerogel was grown and compressed is shown with itsdimensions. Indicated are also the regions of bulk ﬂuid in the cell. Note that the gap between the cell and the epoxy cast around the cell’s periphery is greatly exaggerated. small enough to couple the aerogel well to the walls (though aerogel displacement relative to the walls of the cell still needsto be considered), but large enough to ensure ﬁne resolution indetermining the fraction of superﬂuid. The moment of inertiaof the torsion head and aerogel ﬁlled cell is calculated to be0.064 g cm 2and that of the helium at saturated vapor pressure −5.85×10−5gc m2, or about 1 part in 103of the inertia of the head. The steel cavity was dry ﬁtted into an already hardened epoxy cast in order to reduce possible contamination of theaerogel by any epoxy penetrating through holes in the stainlesspillbox. Despite careful machining of the epoxy cast, thereappeared to be empty regions around the periphery of the celloccupied by 3He not embedded in the aerogel (bulk ﬂuid). In addition, we need to consider the bulk ﬂuid within the ﬁllline inside the upper torsion rod in our subsequent analysis.Appendix Adescribes how we modeled the contribution coming from these two regions. A schematic of the torsion pendulum setup along with a detailed sketch of the head of the pendulum is shown atFig. 1. The locations of the inferred bulk ﬂuid regions are also indicated. B. Thermometry and data acquisition Thermometry was provided through a3He melting curve thermometer (MCT), which had a∼30 min (dependent on the temperature) time constant with respect to the3He in the aerogel. A quartz tuning fork immersed in the3He ﬂuid allowed for a more immediate reading of the temperatureof the 3He in the torsion head. The fork was swept through resonance every ten minutes, and its frequency and qualityfactor recorded. These values were calibrated against the MCTand provide secondary thermometry for the experiment. After acquiring the resonance curve at a ﬁxed temperature (T 0), we ﬁtted the resonance and established the quality factor at this temperature Q(T0). We also plotted the values of the 094513-2DISSIPATION SIGNATURES OF THE NORMAL AND . . . PHYSICAL REVIEW B 89, 094513 (2014) quadrature ( Y) and in-phase ( X) components of the signal against each other. They formed a circle with a diameter equalto the signal amplitude at resonance ( A), centered at ( A/2, 0). Provided that we drove the pendulum within ±0.1 radian of phase error with respect to resonance, we could deduce theresonant frequency and Qof the pendulum using the following relationships: f res(T)=fdrive(T)/bracketleftbigg 1+Y(T) 2Q(T)X(T)/bracketrightbigg , (1) Q(T)=Vdrive(T0) Vdrive(T)Q(T0) A(T0)X2(T)+Y2(T) 2X(T), (2) where Q(T0),A(T0) were determined from the sweep. To avoid driving the pendulum at a level away from its linear behavior,the driving amplitude was also adjusted when it deviated morethan 3% from its target value. The amplitude of motion of thependulum’s wings was of the order of a few angstroms leadingto a peak velocity of about a few μm/s. The typical noise in data obtained in this manner was 2 .5×10 −9for the inferred resonant frequency and 2 .5×10−3for the quality factor. The additional energy dissipated by the ﬂuid is determined by subtracting the empty cell value for Q−1(T)f r o mt h e values of Q−1(T) when ﬁlled with3He. The superﬂuid fraction of the ﬂuid [ ρs/ρ(P,T )] can be found through the relative reduction in the moment of inertia of the torsionpendulum head as the temperature is lowered below T cfor the ﬂuid in the aerogel. From knowledge of the period[p(P,T )=f −1 res(P,T )] of the torsion pendulum when the ﬂuid is fully locked to it [ p0(P)] and the period when the cell is empty ( pempty ), we can deﬁne the superﬂuid fraction as ρs/ρ(P,T )=[p0(P)−p(P,T )]/[p0(P)−pempty ]. To map the temperature dependence of the empty cell values for the period and dissipation, we took points at discretetemperatures between 100 and 1 mK, waiting for a few hoursto reach equilibrium between points, before any 3He wasintroduced in the cell. Plots for the empty cell data can be found in Ref. [ 31]. Both the pendulum’s period and the quality factor change very little below 5 mK. We attribute this to thetime dependent heat release from the epoxy [ 32]. By plotting the data versus log 10(T), we extrapolate the empty cell data below 5 mK. We assign the uncertainty for Q−1in the empty cell to be ∼1×10−6. The relative uncertainty in the empty cell period below 5 mK is also estimated to be ∼1×10−6,o r 1 part in 1000 of ( p0−pempty ). NearTc, the viscosity of bulk normal3He ensures that even at high pressures, the3He is well locked to any cavities smaller in size than ∼100μm at kilohertz frequencies. Thus the period of the pendulum at Tcwould be p0(P) apart from a correction due to the ﬂuid in the ﬁll line, which can beaccurately calculated (see Appendix A). III. DATA A. Normal state Figure 2summarizes the data for the energy dissipation factor due to3He ﬂuid versus temperature in the normal state at four widely spaced pressures: 0.14, 2.6, 15.7, and 25.7 bar.In each of these measurements, we changed the temperature indiscrete steps and waited until the signal for the frequency andQof the pendulum reached equilibrium. The wait time varied with temperature and was of order two hours or less. Thecalculated dissipation from the bulk ﬂuid regions is shown assolid and broken lines. Subtracting this contribution from thedata taken in the normal state, we observe a residual dissipationof∼(2.4±0.6)×10 −6that we attribute to the3He liquid in the aerogel. The uncertainty arises mainly from the need toinfer the geometries of the bulk ﬂuid regions. It is important tonote that the dissipation does not have an obvious temperaturedependence and any pressure dependence cannot be discernedfrom the plot in Fig. 2. FIG. 2. (Color online) Experimental data for Q−1vs temperature for four pressures after empty cell data are subtracted (open (blue) circles). Shown also are the ﬁts for the bulk ﬂuid contribution to the Q−1for two components: bulk ﬂuid contained in the ﬁll line [solid (black) line] and the bulk ﬂuid around the periphery of the cell, modeled as a channel of thickness 28 μm [dashed (red) line]. After subtracting off the two bulk ﬂuid contributions, the dissipation due to the3He and aerogel combination is shown as the open (black) triangles. The dissipation of ∼2.4×10−6is essentially temperature and pressure independent within the scatter in the normal state data ( ∼±0.6×10−6). 094513-3N. ZHELEV et al. PHYSICAL REVIEW B 89, 094513 (2014) FIG. 3. (Color online) Data for Q−1vsρs/ρat six different pressures are plotted. We note slow mode resonance crossings for ρs/ρ < 0.015. ESP-phase (cooling-blue solid circles) and B-phase (warming-red open triangles) coexistence regions are shown in the insets at lower pressures.The dissipation is larger in the ESP-phase compared to the B phase. The higher dissipation associated with the ESP phase is especially evident at higher pressures and close to the ESP to B transition. Bulk ﬂuid contributions have been subtracted, assuming bulk B phase. The discontinuity in the 31.9-bar data on cooling is due to the bulk A to B transition. B. Superﬂuid state of 3He in aerogel Data were taken on both cooling and warming in the superﬂuid state at a number of different pressures, maintain-ing a constant cooling (warming) rate ( ∼30μK/hr). From knowledge of the bulk superﬂuid fraction and viscosity, we subtracted contributions due to the bulk regions in order todetermine the corresponding values for the 3He in the aerogel. In particular, we note that bulk ﬂuid Q−1is at most ∼4×10−6 nearTcand rapidly decreases at lower temperatures, and is thus unable to account for the measured dissipation. A summary of the data for Q−1versus ( ρs/ρ)aerogel at six different pressures is shown in Fig. 3. Standing wave modes of the fourth-sound like “slow mode” (in which the superﬂuid moves out of phase with the normalﬂuid; normal ﬂuid is clamped to the nonrigid aerogel) crossthe torsional oscillator frequency, as the slow mode’s velocityevolves, between 0 /lessorequalslant(ρ s/ρ)aerogel/lessorequalslant0.015. We can identify the resonance crossings in Fig. 3as a number of closely spaced “loops.” These resonance effects [ 33–37] will be ignored in our subsequent discussions. The superﬂuid transition temperatures and precise phase diagram for this sample were identiﬁed in our previouspublication [ 23]. Below the superﬂuid transition, we enter thesuperﬂuid ESP phase on cooling. At lower temperatures, we observe a continuous phase transition between the ESP and theB phases (extended over a temperature interval of ∼70μK). It is thought that this width is due to the strong pinning ofthe phase interface by the aerogel. On warming, we stay inthe B phase until just below the critical temperature. Thereappearance of the ESP phase is very pressure dependent.This strongly hysteretic behavior allows us to probe ESP andB phase properties over an extended temperature window,especially at elevated pressures. The pressure dependence of Q −1against ( ρs/ρ)aerogel and (1−T/T c) is shown in Fig. 4. In the B phase, we observe a broad peak in the dissipation [Fig. 4(a)]. Below Tcthe dissipation rises, even though the impurity limited normal ﬂuid viscosity should be constant. The dissipation in the ESP phase rises even faster than in the B phase [Fig. 4(b)]. This is in sharp contrast with experiments in the bulk, where the viscosity isseen to drop sharply below T cand scale as e−/Delta1/k BTin the ﬁnite size regime [ 38].Q−1(T) scales well with ρs/ρand not (1−T/T c), as shown in Figs. 4(a) and 4(c). Since ρs/ρ∝ /Delta12, this implies that Q−1and the energy gap /Delta1are related. The anomalous dissipation of the ESP phase scales almost linearly with ρs/ρ[Fig. 4(b)], and exceeds the corresponding 094513-4DISSIPATION SIGNATURES OF THE NORMAL AND . . . PHYSICAL REVIEW B 89, 094513 (2014) FIG. 4. (Color online) (a) A plot of Q−1in the B phase vs ρs/ρfor all the data sets in Fig. 3combined in one plot. We note the consistent pressure dependence of Q−1withρs/ρ. Strong coupling effects enhance the anomalous superﬂuid dissipation. (b) A plot of Q−1in the ESP phase vs ρs/ρfor all the data. The pressure dependence of Q−1is seen to arise mainly due to the larger extent in temperature of the ESP phase at high pressure; the Q−1scales well with ρs/ρ. Discontinuities in the data are due to the bulk A →B transition on cooling. (c) and (d) Plots ofQ−1vs 1−T/Taerogel c in the B and ESP phases, respectively. Much of the scaling behavior is lost in this view compared to that seen in (a) and (b). value of Q−1at the same ρs/ρ(andT/T c) in the B phase. As pressure is increased, Q−1measured in the ESP phase rises considerably above the values for Q−1in the B phase. This effect is emphasized further since the width of temperatureregion in which the ESP phase is stable on cooling increases with pressure. IV . COMPARISON TO EXPERIMENTS WITH UNCOMPRESSED AEROGEL Previous torsion pendulum experiments with uncompressed aerogel [ 11,39] used signiﬁcantly larger aerogel samples and had much lower Q’s. Thus no direct comparison with previous torsion pendulum experiments can be made. However, we canrelate the Q −1reported here (in the superﬂuid state) to the Q of the slow mode of3He in uncompressed aerogel samples in the ESP (Ref. [ 40]) and the B (Ref. [ 41]) phases . Results for the B-phase ultrasound dissipation can also be found inRef. [ 55]. The qualitative behaviors described in these references (increased ESP- and B-phase dissipation as thetemperature is lowered near T c,Q−1 ESP>Q−1 B) are similar to what we observe. Thus apart from allowing the ESP stateto persist on cooling to much lower temperatures, the aerogelcompression is probably not a signiﬁcant factor in the observedresults. V . COLLISION DRAG MODEL IN A TORSION PENDULUM GEOMETRY A starting point in the model for the dynamics of the helium- aerogel system is to map out the angular velocity proﬁles ofthe ﬂuid and the aerogel across the ﬂow channel. We expectt h eﬂ u i dt ob ei naD r u d eﬂ o wr e g i m e[ 10,17], where theangular velocity of the ﬂuid with respect to the aerogel is constant across the channel, with the exception of a smallregion of size δ d=/radicalbig(ητf/ρ) away from the edges [ 17,41]. The frictional relaxation time τfis related to the friction force per unit volume coupling the helium with the aerogel matrix[42–44]: F(v l,va)=ρ τf(vl−va), (3) where vlandvaare, respectively, the velocities of the normal 3He and the aerogel. The frictional force can be related to the average change of momentum a quasiparticle experiencesupon scattering from an aerogel impurity. In Ref. [ 44],τ fis given by τf=˜τ/parenleftBig 1−ρ0s ρ/parenrightBig/bracketleftBig 1+F1s 3/parenleftBig 1−ρ0s ρ/parenrightBig/bracketrightBig (4) withF1 sbeing a Landau parameter. The bare superﬂuid density, ρ0 sis related to the measured superﬂuid density stripped of Fermi-liquid effects through: 1−ρs ρ=m∗ m1−ρ0 s ρ 1+F1s/parenleftBig 1−ρ0s ρ/parenrightBig. (5) In the normal state, ˜ τis the transport relaxation time equal to the quasiparticle mean free path divided by theFermi velocity [ 45]. In the superﬂuid state, however, ˜ τis temperature dependent. Reference [ 44] deﬁnes ˜ τin terms of integrals of quasiclassical Keldysh Green’s functions, but nodirect relationship between ˜ τand conventional experimental observables in the superﬂuid state is shown. Instead, the valuesfor ˜τare numerically calculated in the different scattering 094513-5N. ZHELEV et al. PHYSICAL REVIEW B 89, 094513 (2014) limits for various degrees of Tcsuppression by the aerogel. It is evident from the plots for ˜ τ(T) given in in Ref. [ 44] that ˜τin the superﬂuid state could be somewhat larger that ˜ τin the normal state, before ˜ τ(T) eventually approaches zero as T→0. The quasiparticle mean free path can be estimated from the suppression of the superﬂuid transition as discussed inRef. [ 23] using a model proposed by Abrikosov and Gorkov in Ref. [ 46] and reﬁned into the isotropic inhomogeneous scat- tering model (IISM) described in Refs. [ 7,47–49]. Assuming a Fermi velocity of 30 m /s, the value of the normal state τ f inferred from the 155-nm mean free path used to ﬁt the Tc suppression is ≈5×10−9s. The sound velocity in the aerogel sample is expected to be in the range of c∼30–50 m /s[33,50]. For a frequency of 2.1 kHz, we expect a compressional sound mode wavelengthof a few millimeters, an order of magnitude larger than theheight of the cell. Yet, there will be a small displacement ofthe aerogel in the interior of the cell relative to the motion of theadjacent wall. The normal helium is well locked to the aerogel;the aerogel and helium form a composite medium exhibiting avelocity proﬁle largely determined by the viscoelasticity of theaerogel. Through numerical calculations, we predict a∼1% difference in the angular velocity in the middle of the cell andthe wall. This angular velocity proﬁle gives rise to dissipationin the cell. In addition, there is a small velocity difference between the entrained ﬂuid and the aerogel itself that arises due to the ﬁnite value of τ f. To solve for the angular velocity proﬁles of the helium and the aerogel, we write the Navier-Stokes and wave equations,coupled by the collision drag force: ρ˙/Omega1 l=η∂2/Omega1l ∂z2−ρ τf(/Omega1l−/Omega1a), (6) ρa˙/Omega1a=iμ ω∂2/Omega1a ∂z2+ρ τf(/Omega1l−/Omega1a), (7) where /Omega1l(z) and/Omega1a(z) are the angular velocity proﬁles of the helium liquid and the aerogel across the channel, respectively.The shear modulus of the aerogel is μ, which we can deduce from the aerogel sound velocity. The viscosity of the heliumηequal to that of the bulk liquid at high temperatures, but reaches an impurity limited value at about 10 mK, leading toη/lessorsimilar0.01 Poise. Having solved for the angular velocity proﬁles, we ﬁnd the induced torque on the walls of the cell due to the motion of thehelium liquid ( N l) and the aerogel ( Na). With the assumption that the angular velocities of the liquid and the aerogel at thewalls are equal to the angular velocity of the cell wall, i.e.,/Omega1 l(a)(±z/2)=˙θ, we obtain Nl=−πR4η/parenleftbigg∂/Omega1l ∂z/parenrightbigg z=h/2,Na=−iπR4μ ω/parenleftbigg∂/Omega1a ∂z/parenrightbigg z=h/2. The empty cell Q−1shows a nonzero value when extrap- olated to T=0. Yet, a purely elastic aerogel should not be dissipative. A previous iteration of this experiment used anaerogel sample (grown in a different process) with a height of≈4 mm. The otherwise identical torsion pendulum containing that sample had a Q≈100×lower than the one describedhere. We can expect a h 2dependence of the dissipation, with hbeing the height of the cell. Furthermore, there have been a number of experiments on silica aerogels (though on samplesdenser compared to ours and at room temperature [ 51–53]), that report a complex elastic modulus, which would lead todissipation effects associated with the plastic deformations ofthe aerogel. We write the shear modulus of the aerogel asμ=μ re−iμim. Accounting for the complex shear modulus, we obtain Q−1(T)=−Re(Na+Nl) I0ω˙θ ≈Ia I0/bracketleftbigg 1+ρn(T) ρρ ρa/bracketrightbigg2ρaω3h2 12μ2re/bracketleftBig η(T)+μim ω/bracketrightBig +ρn(T) ρIl I0ωτf. (8) More details about the exact solution to the equations of motion and how we derive the result for Q−1can be found in Appendix B. VI. DISCUSSION There are three terms in Eq. ( 8) that contribute to the normal state dissipation. The ﬁrst one is proportional to the normal ﬂuid viscosity η(T) and is due to the aerogel ﬂexure modifying the angular velocity proﬁle of the liquid and causing extradissipation. Using η∼0.01 Poise, this term accounts for a contribution to Q −1of the order of ∼10−8. In order to match the experimental value of Q−1=2.4×10−6, we need ηto be two orders of magnitude larger, which we consider unphysical. The third term in Eq. ( 8) contains contributions to Q−1 arising from the frictional relaxation time τf. For this term to have a large enough contribution to match the experimentaldata for Q −1, we need τf∼10−7s. However, the quasiparticle mean free path in a 98% open aerogel has been shown tobe/lessorsimilar200 nm [ 13,15,21–23,54]. Assuming a Fermi velocity of 30 m/s and effective mass m ∗/m∼3–5, we ﬁnd that τfabove Tccan at most be a few nanoseconds. We suggest that the large temperature independent normal state dissipation could be due to the intrinsic dissipative natureof the aerogel, characterized by the ratio μ im/μ2 re. The reason we are sensitive to the aerogel intrinsic dissipation term is thelow-resonant frequency of the torsion pendulum. Since thisterm depends on μ im/ω[Eq. (8)], its contribution would be less signiﬁcant at the higher frequencies employed in ultrasoundattenuation experiments [ 17,55]. To obtain Q −1of the order of 10−6, we need μim/μre∼0.1. Such a large loss tangent could be due to the fractal nature of the aerogel or could be relatedto the expected presence of a few monolayers of solid 3He on the surface of the aerogel strands. Figure 5shows the pressure dependent values for Q−1at Tbulk c with the contributions from bulk ﬂuid and the empty cell subtracted plotted versus [1 +ρ(P,Tbulk c)/ρa]2−1[ s e e Eq. (8)]. The transition temperature as a function of pressureis well known as are the density and the viscosity of the bulkﬂuid at T callowing us to accurately subtract the bulk ﬂuid contributions and reveal the pressure dependence of Q−1in the normal state. If our assumption that the main contribution 094513-6DISSIPATION SIGNATURES OF THE NORMAL AND . . . PHYSICAL REVIEW B 89, 094513 (2014) FIG. 5. (Color online) A plot of Q−1measured at Tcfor 0.14, 3, 15.2, 18.5, 20.1, 21.9, 24.3, 25.7, 27.5, and 29.1 bar (pressure increases as we go from left to right) with the bulk ﬂuid and empty cell dissipation subtracted versus (1 +ρs/ρaerogel )2−1. A linear regression line is shown, with a slope of ∼2.4×10−7andyintercept of∼3.1×10−7. to the dissipation in the normal state comes from the lossy aerogel, we would expect a linear relationship. A linear ﬁt tothe data is shown in Fig. 5, providing an evidence in support of this model. The yintercept of ∼3.1×10 −7could be due to the uncertainty of the empty cell data. Assuming that energy dissipation of the torsion pendulum due to the interaction of the normal state excitations and theaerogel scales as [1 +(1− ρs ρ)ρ ρa]2, then such a contribution will decrease as the cell is cooled below T cand deeper in the superﬂuid state. This cannot explain the dissipation wemeasure in both the ESP and B phase superﬂuid states. We subtract the normal ﬂuid contribution (using parameters from the linear ﬁt in Fig. 5) and consider the residual dissipation. If we allow its origin to be due to the ρn(T) ρIl I0ωτf(T) term, we can plot the so-inferred τf(T) as a function of the temperature. Figure 6shows this for the 29.1-bar data along with the data from Ref. [ 41]. We ﬁnd good agreement between the two experiments, implying that the observeddissipation in the 50-kHz sound attenuation experiment and FIG. 6. (Color online) A plot for τfas a function of temperature assuming that the τfterm in Eq. ( 8) is responsible for all of the extra dissipation we observe in the B phase. The data plotted are for 29.1 bar. We also show the data from Ref. [ 41], which are deduced in a similar way.the torsion pendulum Q−1in the superﬂuid B phase probably have a similar origin. As discussed in the previous section, the relaxation time ˜ τin Eq. ( 4) can be shown to increase as we enter the superﬂuid state due to the rapid opening of thesuperﬂuid gap, before ˜ τeventually diminishes to near zero at extremely low temperatures [ 44]. In addition, the denominator in Eq. ( 4) should also decrease as ρ s/ρgrows. These two effects combined could produce a temperature dependence ofτ fwith a similar shape to what we observe in Fig. 6. We can expect an enhancement of τfin the superﬂuid state up to a factor of ten compared to its value at Tc. However, in order to produce a peak τfof order 0.15 μs, we need τf(Tc)/greaterorsimilar10 ns, a value which is higher than the few nanoseconds that wouldbe consistent with the T csuppression measurements. Thus temperature variation of the frictional relaxation time cannotsolely produce the observed data. Therefore we concludethat there is an additional mechanism to dissipate energy notcaptured in the collision-drag model presented in Sec. IVand related to the emergence of the superﬂuid order. One way superﬂuid currents can dissipate energy is through interactions with bound states pinned to the boundary with thenormal ﬂuid at the vortex cores [ 56]. This leads to a mutual friction term, which can be shown to be proportional to [ 57] ρ s ρρn ρ(vs−vn). Such a term would produce a peak in the dissipation similar to what we observe in our data for the B superﬂuid phase.However, no evidence for vortex states has been found in ourexperiment. The velocity amplitude of the superﬂuid current issmall, much smaller than the velocities the ﬂuid is driven at intypical experiments observing vorticity [ 57,58]. We also do not detect a noticeable increase in Q −1as we drive the pendulum harder. While the vortex dynamics model may not be applica-ble to our experiment, one can imagine that regions of normalﬂuid with the size of a typical vortex core (coherence length)exist, bound to denser regions of the aerogel. Such bound stateswill allow for lower energy excitations to interact with the su-perﬂuid ﬂow and provide a mechanism for energy dissipation. An object (in this case an aerogel strand) moving through bulk superﬂuid with velocity vshould feel a force that scales as e −/Delta1/k bTv,a ss h o w ni nR e f .[ 59]. Assuming that the nodes of the ESP state order parameter tend to orient in the plane of the ﬂow,then we would expect that the ESP state should be associatedwith higher dissipation than the B phase. However, thisargument does not explain the different functional dependenceof the dissipation in the ESP phase in terms of ρ s/ρcompared to the dissipation in the B phase. Experiments with samples of aerogel attached to vibrating wire resonators immersed in3He show that ﬂow tends orient the ESP state orbital texture along the ﬂow [ 60]. Such an effect is clearly demonstrated for velocities signiﬁcantly larger thanthe velocities of the ﬂuid in our experiment, but alignment ofthe/lscriptvector is possibly realized also at lower velocities, albeit with a smaller magnitude. Changing the direction of the /lscript vector will damp the ﬂow due to the orbital viscosity of the su-perﬂuid [ 61,62] and manifests itself as the extra dissipation of the pendulum observed in the ESP state. A similar (but smaller)effect has been shown for the B phase if the order parameteris slightly anisotropic [ 63]. Further, a previous experiment 094513-7N. ZHELEV et al. PHYSICAL REVIEW B 89, 094513 (2014) studying superﬂuid ﬂow through a small oriﬁce (18- μm diameter) shows large dissipation in the A phase, linearlyincreases with velocity until a critical velocity is attained [ 64]. Finally, we note that the pressure dependence of the observed dissipation could be related to the degree of gapsuppression in both ESP and B superﬂuid phases. Dissipationis higher at high pressures, where the gap suppression is lesssevere, and lower at lower pressures where the superﬂuid gaptends to be less pronounced and the density of states at lowerenergies increases. VII. CONCLUSION We presented torsion pendulum Q−1data for a compressed aerogel sample ﬁlled with3He in both normal and superﬂuid states. We developed a model for the normal ﬂuid dynamics asembedded in the viscoelastic aerogel. We assert that frictionalrelaxation time is not large enough to account for either normalor superﬂuid Q −1data. Instead, we propose that dissipation features of the data below the superﬂuid transition originatefrom the superﬂuid state. ACKNOWLEDGMENTS We thank S. Higashitani and J. Sauls for fruitful discussions. We acknowledge support from the NSF under DMR-1202991at Cornell University and DMR-1103625 at NorthwesternUniversity. APPENDIX A: BULK FLUID CONTRIBUTION We expect the normal state helium liquid to be well locked to the strands of the aerogel. In the normal state, any changein the resonant frequency compared to that of a cell with afully locked ﬂuid should originate from the bulklike ﬂuidregions of the cell. Figure 7shows data for the fraction of the moment of inertia not coupled to the walls of thecell at the four experimental pressures that were shown inFig. 2(0.14, 2.6, 15.2, and 25.7 bar). The decoupled ﬂuid fraction and dissipation show temperature dependent behaviorcharacteristic of two distinct bulk ﬂuid regions (two peaksin the normal state dissipation data, two “shoulders” in thenormal state decoupled fraction data). The effective length and diameter of the ﬁll line in the torsion rod and the cast epoxy cell are 6 and 1 mm, respectively.The bulk ﬂuid column amounts to 0.8% of the inertia of theﬂuid in the cell and is designated as bulk ﬂuid region 1. In orderto calculate the contribution to dissipation and period shiftcoming from the ﬂuid in the ﬁll line, we start by calculatingthe angular velocity proﬁle /Omega1 θ(r) by using the Navier-Stokes equation in a tall cylindrical geometry, which leads to ∂2/Omega1 ∂r2+3 r∂/Omega1 ∂r+iωρ η/Omega1=0( A 1 ) with/Omega1(radius of the cylinder) =/Omega1cell. Solving for /Omega1we ﬁnd the torque exerted by the ﬂuid: N=2πR3hη/parenleftbigg∂/Omega1 ∂r/parenrightbigg r=R=β1+iωβ 2, (A2) FIG. 7. (Color online) The fraction of ﬂuid decoupled from the pendulum vs temperature for four pressures after background subtraction (open circles). Also shown are the ﬁts for the bulk ﬂuidcontribution for two components: region 1, ﬂuid in the ﬁll line, a 1-mm diameter, 6-mm long cylinder comprising 0.8% of the total ﬂuid moment of inertia [solid (black) line], and region 2, ﬂuid at theperiphery of the cell, modeled as a cavity of height 28 μm[ d a s h e d (red) line] comprising 3.2% of the moment of inertia. The dash-dotted (green) line shows the sum of the contributions from the two bulk ﬂuidcomponents. where β1contributes to the damping of the pendulum and β2 to the moment of inertia. Temperature dependence of these values is determined by the temperature dependence of theviscosity of the ﬂuid, η(T). NearT c, we expect the normal state bulk viscosity to scale asT−2. Above T> 10 mK the viscosity deviates from the Fermi liquid T−2behavior and we use the following relations between the thermal conductivity ( κ), heat capacity ( CV), and the viscosity ( η) to calculate higher temperature values for η: κ=1 3CVv2 Fτκ, (A3) η=1 5m∗ mρv2 Fτη, (A4) CV=m∗π2kB /planckover2pi12/parenleftbiggV 3π2N/parenrightbigg2/3 RT. (A5) 094513-8DISSIPATION SIGNATURES OF THE NORMAL AND . . . PHYSICAL REVIEW B 89, 094513 (2014) FIG. 8. (Color online) Values of viscosity in the normal state at the four experimental pressures. Assuming that density and molar volume do not change in the temperature range 1–100 mK, and assuming τη∝τκ, we can infer that η∝κ. To ﬁnd the exact values for the viscosity in the normal state, we use the values for η(Tc)g i v e ni nR e f s .[ 65,66], andκ(Tc)i nR e f .[ 67] and divide the two values to ﬁnd the proportionality factor. We then multiply κ(T) from Ref. [ 67] by this factor for each of the pressures we are interested andwe ﬁnd η(T) up to 100 mK. The values for the viscosity for the four experimental pressures we used to calculate bulkﬂuid contribution in the normal state are shown in Fig. 8.I nt h e superﬂuid state, experimental values for the superﬂuid fractionare taken from Ref. [ 68] and for the viscosity from Ref. [ 38] Numerically solving Eq. ( A1), we can calculate the contribution from the bulk ﬂuid in the ﬁll line. Thiscontribution is shown with a solid (black) line in Figs. 2and7. It is evident in Fig. 7that there is bulk ﬂuid within the cell we have not yet accounted for. The steel cavity containing the aerogel was dry ﬁtted in the epoxy cast to prevent epoxy running in. We believe thisresulted in small pockets of bulk ﬂuid existing around theperiphery of the cell. While we cannot do an exact calculationfor the effects of these regions the same way as we did forthe ﬂuid in the torsion rod, we can still use the uncoupledmoment of inertia data (Fig. 7) to estimate the contribution to the pendulum’s dissipation. We assume that the relationshipbetween the real and the imaginary part of the torque arisingfrom the cell periphery bulk ﬂuid is the same as that of auniform thickness ﬁlm encompassing all of the cell. For a thinﬁlm of ﬂuid with a thickness hand inertial contribution I per, the torque exerted is N=β1+iωβ 2, with β1=ωIperδ hsin(h/δ)−sinh(h/δ) cos(h/δ)+cosh(h/δ), (A6) β2=Iperδ hsin(h/δ)+sinh(h/δ) cos(h/δ)+cosh(h/δ), (A7)where δ=√2η/ρω is the viscous penetration depth of the ﬂuid. Fitting to the dissipation data in Fig. 2, we ﬁnd h= 28μm and Iper=0.032If, where Ifis the moment of inertia of all the helium in the torsion pendulum head. These valuesare consistent with our expectations. The accuracy to which theepoxy cast and stainless steel cell are machined is within one-thousand of an inch, i.e., 25 μm, and a ﬁlm of that thickness around all of the cell surface amounts to 0 .05I f. Since the bulk ﬂuid is more likely coming from a few separate regions aroundthe periphery, rather than from a continuous ﬁlm, we wouldexpect that I per/lessorsimilar0.05If. We also use these values and the viscosity of3He to obtain the fraction of decoupled ﬂuid from the periphery (region 2) which we plot as the dashed (red) linein Fig. 6. At the lowest experimental pressures (0.14, 2.6, and 4 bar), the liquid in the aerogel does not transition to a superﬂuidstate. At these pressures, the resonance period shift below T c originates from the bulk ﬂuid regions. In addition to the bulk ﬂuid decoupling, we observe fourth sound resonance crossingseffects, which occur at speciﬁc values of the sound velocity andtherefore ρ s/ρbulk. We obtain a good ﬁt to these data using the model described in this appendix, which gives an independentconﬁrmation that bulk ﬂuid effects are fully accounted for.More information about these effects can be found in thesupplementary material of Ref. [ 23]. APPENDIX B: DYNAMICS OF NORMAL3HE IN AEROGEL We start by rewriting Eqs. ( 6) and ( 7)a s ∂2/Omega1a ∂z2+aa/Omega1a−ba/Omega1l=0, (B1) ∂2/Omega1l ∂z2+al/Omega1l−bl/Omega1a=0, (B2) where we have deﬁned the coefﬁcients aandbas aa=iρω μba=aa/parenleftbigg 1−iωτFρa ρ/parenrightbigg , (B3) al=−ρ ητF,b l=al(1−iωτF). (B4) Solving the coupled differential equations, we arrive at /Omega1a(z)=/bracketleftbiggD−al−aa 2−ba 2Dcos(k1z) cos(k1h/2) +D+al−aa 2+ba 2Dcos(k2z) cos(k2h/2)/bracketrightbigg ˙θ, (B5) /Omega1l(z)=/bracketleftbiggD−aa−al 2−bl 2Dcos(k1z) cos(k1h/2) +D+aa−al 2+bl 2Dcos(k2z) cos(k2h/2)/bracketrightbigg ˙θ, (B6) where D=√ (al−aa 2)2+blbaandk1,2=√al+aa 2±D. To obtain a qualitative picture of the angular velocity proﬁles, we can explore the fact that ωτf/lessmuch1 andηω/μ /lessmuch1. The coefﬁcients ( D±al−aa 2±ba,l)/2Dand the values of k1,2 in Eqs. ( B5) and ( B6) are approximated to the lowest order. 094513-9N. ZHELEV et al. PHYSICAL REVIEW B 89, 094513 (2014) This approximation gives us the zdependence of the angular velocity of the aerogel, /Omega1a(z), and that of the ﬂuid /Omega1l(z): /Omega1a(z)≈˙θcos/bracketleftbig/radicalBig (ρ+ρa)ω2 μz/bracketrightbig cos/bracketleftbig/radicalBig (ρ+ρa)ω2 μh 2/bracketrightbig, (B7) /Omega1l(z)≈˙θ⎧ ⎨ ⎩cos/bracketleftbig/radicalBig (ρ+ρa)ω2 μz/bracketrightbig cos/bracketleftbig/radicalBig (ρ+ρa)ω2 μh 2/bracketrightbig−iωτfcosh/bracketleftbigz δd/bracketrightbig cosh/bracketleftbigh 2δd/bracketrightbig⎫ ⎬ ⎭,(B8) where δd=/radicalbigητf/ρ/lessmuchhis the “dirty” ﬂuid penetration depth, i.e., the length scale over which the velocity of thehelium ﬂuid deviates from the Drude ﬂow regime with respectto the aerogel velocity. We observe that the shape of bothaerogel and ﬂuid velocity proﬁles is largely set by the elasticmodulus of the aerogel, μ. The relative velocity difference between the aerogel and the helium ﬂuid is of the order ofωτ f/lessmuch1 of the total velocity. Equations ( B7) and ( B8) present a qualitative picture for the differences in the velocities of the ﬂow and the aerogel, but weneed to include higher-order terms in the expressions above toestimate the dissipation factors associated with the aerogel andthe ﬂuid in the cell. Importantly, we also allow the possibilityof the elastic modulus of the aerogel to be a complex number,μ=μ re−iμim, with μim/μre/lessmuch1. Then for k1,2we have k1≈/radicalBigg (ρ+ρa)ω2 μre ×/braceleftbigg 1+i/bracketleftbiggωη 2μre+μim μre+ωτf 2ρ (ρ+ρa)/bracketrightbigg/bracerightbigg ,(B9) k2≈i δd/bracketleftbigg 1−i/parenleftbiggωη 2μre+ωτf 2/parenrightbigg/bracketrightbigg . (B10) As for the coefﬁcients in Eqs. ( B5) and ( B6): C1=D−al−aa 2−ba 2D≈1, (B11) C2=D+al−aa 2+ba 2D≈(ωτf)/parenleftbiggηω μre/parenrightbigg , (B12) C3=D−aa−al 2−bl 2D≈1+iωτf, (B13) C4=D+aa−al 2+bl 2D≈−iωτf/parenleftbigg 1+iηω μreρ+ρa ρ/parenrightbigg .(B14)The expressions for the induced torque by the aerogel ( Na) and the helium liquid ( Nl) can be written as Na=iωIa2μ ρaω2h/bracketleftbigg C1k1tan/parenleftbigg k1h 2/parenrightbigg +C2k2tan/parenleftbigg k2h 2/parenrightbigg/bracketrightbigg ˙θ, (B15) Nl=Il2η ρh/bracketleftbigg C3k1tan/parenleftbigg k1h 2/parenrightbigg +C4k2tan/parenleftbigg k2h 2/parenrightbigg/bracketrightbigg ˙θ.(B16) Further, the expressions for the tangents can be approximated as tan/parenleftbigg k1h 2/parenrightbigg ≈k1h 2+/parenleftBig (ρ+ρa)ω2h 4μre/parenrightBig3/2 3−/parenleftBig (ρ+ρa)ω2h 4μre/parenrightBig+i/bracketleftBig (ρ+ρa)ω2h 4μre/bracketrightBig3/2 1−/bracketleftBig (ρ+ρa)ω2h 4μre/bracketrightBig ×/parenleftbiggηω μre+μim μreωτf 2ρ ρ+ρa/parenrightbigg , (B17) tan/parenleftbigg k2h 2/parenrightbigg ≈i, (B18) where we used the following relation: tan(α+iβ)≈α/parenleftbigg 1+α2 3−3α2/parenrightbigg +iβ/parenleftbiggα2 1−α2/parenrightbigg ,(B19) w h i c hi st r u ei nt h ec a s eo f β/lessmuchαandα/lessorsimilar0.1. 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PhysRevB.45.9497.pdf	PHYSICAL REVIEW B VOLUME 45,NUMBER 16 15APRIL1992-II Hole-quasiparticle resonant polaron coupling inquantum wellscontaining highdensities offreecarriers P.E.Simmonds Department ofPhysics, University ofWollongongW, ollongong, NSW2500,Australia M.S.Skolnick Department ofPhysics. University ofSheffield,ShieldS37RH,UnitedKingdom T.A.Fisher Department ofPhysics, University ofWollongong, Wollongong, NSW2500,Australia K.J.Nash RoyalSignals andRadarEstablishment, St.Andre~'s Road,Malvern, Worcestershire, W'RI43PS,UnitedKingdom R.S.Smith GECHirstResearch Centre, Hembley, Middlesex, HA97PP,UnitedKingdom (Received 10October1991) Strongresonant polaron coupling phenomena inthemagnetophotoluminescence (magneto-PL) spec- traofmodulation-doped quantum wellsarereported. Thepolaron interactions arisebetween hole quasiparticles inotherwise filledelectron Landau levels,andLOphononsofthequantum-well materi- al.Thistypeofmany-body polaron coupling phenomenon isexpected tobeauniversal featureofthe magneto-PL spectraofquantum wellswithelectron Fermienergygreater thantheLO-phonon energy ofthewellmaterial. Newresonant polaron coupling (RPC) phenomena in themagneto-optical spectraofquantum wells(QW's) containing highdensities offreecarriers arereported. Thephenomena haveamany-body character, sincethe resonances arisefromtheFrohlich interaction between holequasiparticles inotherwise filledelectron Landau lev- els(LL's)andtheLO-phonon modesoftheQWmaterial. Itisshownthatacommon requirement fortheobserva- tionofRPCinthemagnetophotoluminescence (mag- neto-PL) ofQW'sisthatthecarrierdensity shouldbeof suScient magnitude thattheelectron Fermienergy isof theorderof,orgreaterthan,theLO-phonon energyofthe QWmaterial. RPChasbeenmuchstudied inboththree-dimen- sional'2(3D)andquasi-2D (Refs.3-6)systems, princi- pallyincyclotron resonance (CR)experiments. Insuch cases,theRPCisusually observed asastrongcyclotron massenhancement, theresultofelectron-LO-phonon in- teractions whichleadtopronounced anticrossing between unoccupied LL's(LLindexN,.&0)andtheN„=Oplus one-phonon state.'"'Inthepresentwork,theresonant coupling isobserved asaninteraction between LO-phonon satellites ofrecombination involving electrons inupperLL states(LLindexN„)2)andPLtransitions fromthe W,,=OLL.''Largevaluesfortheresonant polaron in- teraction strength, incomparison withCRexperiments, areobserved. Thisisattributed totheabsenceofLLoc- cupation eAects"'fortheholequasiparticles. Themagneto-PL experiments werecarried outona seriesofasymmetric, modulation-doped AI,,Gai—,As- In„Ga~—,As-GaAs (y=0.23,x=0.1)strained-layerQW'swithhighn,valuesranging from(0.8-1.6)x10' cm2andwellwidthsof150-250 A.Zero-field results arediscussed elsewhere.''Theparticular sample inves- tigated indetailhasaQWwidthof150Aand n,=I.6X.10'cm(EF=56meV),determined from Shubnikov-de Haasmeasurements. Magneto-PL mea- surements werecarriedoutat4.2Kinmagnetic fields(B) upto10T. ThePLspectrum atB=0isshown inFig.1(a).Itis composed oftwobands,E~~andE2~,arisingfromrecom- binationofelectrons inthen=1andn=2subbands, re- spectively, withphotocreated holesthermalized inthe lowestheavy-hole subband closetok=0.Thespectraex- tendfromthebottomofthen=1subband atE~uptoan energycorresponding toEF.Theh,k&=0in-plane wave- vectorselection ruleisbroken bydisorder (orholelocali- zation)effects''givingrisetotheE~~transitions, ob- servedwithstrongly decreasing strength abovethen=1 subband edge. Eq~transitions areseensincetheEFof56(+1)meV isveryclosetotheE2E~subband sep—aration(56~ 1 meV), withanupperlimittothen=2population of 5x10'cmdeduced fromtheabsenceofanyLLsplit- tingoftheEpiPL.TheE2ipeakis—3timesstronger thanEiiprincipally becauseofthelargeroverlapofE2 electrons withthephotocreated holesthanthatforEi electrons.' Inmagnetic fieldtheEiiPLbreaksupintoaseriesof LLpeaks,labeled(N„,O)~,asshown inFigs.1(b)-I(e), arisingfromrecombination ofelectrons inLLs(index N„)withholesinthelowestvalence-band LL.The 9497 1992TheAmerican Physical Society 9498 SIMMONDS, SKOLNICK, FISHER, NASH, ANDSMITH QWw=15nm AlGaAs-InGaAs-GaAs quantum well Y&pIt1X n,=16~10'cm,T=4K 1460 1-440,0), ~~C (e) I COz' LUI-z UJOz(d) OM UJz O(c) Z CL (b) (a)— 1.39I 1.438.75T 7.7T 7.2T 6.7T O.OT 1.47) 1.420,0)) C) CL 1400— 1.380—/// 9r /~1 //"/r/r///r ////r 0f$ rr ////////--0 //j L /// 10I i[ I ) 0 2 4 6 8 MAGNETIC FIELD(T) FIG.2.PLenergies oftransitions inFig. Iasafunction o magnetic field.TheWl.transitions areoneLO-phonon energy belowthe(N,„O)ipeaks.Theresonant coupling between tee3 and4isatellites and(0,0)iisclearlyvisible. Triangles denote Etnsitions andsolidcirclesdenote lV,, transitions. ra.' LO 0 b1denoteweaktransitions. Nofeatures at opensymos areobserved, PHOTON ENERGY (eV) FIG.I.PLspectraasafunctionofmagnetic field.(a)n=I andn=2subbandE)iandE2)recombination isobserved at B=O. In(b)-(e) splitting intoLLs(N,,O)iisobserved. LO- phonon satellites Niof(N„O)i areobserved withresonant enhancement ofintensity astheyapproach (0,0)i. hN=0selection ruleisbroken bydisorder forthesame reason athtnonvertical transitions occuratzeroel.' N,)I) Theenergyspacings ofthehigherLLtransitions ( aregivenbytheelectron cyclotron energy hro,.,where ro,.=eB/m, withm,,=0..069mo.'TheEqipeakshows srotrongoscillations inintensity witheldseeFigs. rtedb I(b)-l(e)], ofverysimilar formtothosereported y Chenandco-workers.''Theinterpretation oftheseos- cillations willbediscussed indetailelsewhere. Forthepresent paper,themostimportant information iscontained inthelow-energy peakslabeled NI=1&,21, 3I,....TheseNIsatellites ariseveryclosetotheGaAs LO-phonon energyofhoot~=36. 7meVbelowthecorre- sponding (N,,O)itransitions. TheenergyoftheGaAs- likemodeoftheQW,whichcontains only9%In,willcor- responpondcloselytothisvalue.ThePLtransition energies arepoelttedasafunction offieldinFig.2.Te satellite Landau fanisreproduced bytheNlLO-phonon saeielinesdisplaced tolowerenergy byAcomia. Verymarked resonant anticrossing between thelower-lying N,„O ] transitions andtheNllinesisobserved around fieldsof 4.4,5.5,and7.3Tgivenbythe"magnetophonon" reso- nancecondition AN,,6co,.=Acot~.'"h,N,,isthediA'erence inLandau indexbetween theparentLLoftheJVIsatellite andtheN,,levelwithwhichresonance occurs. Intheregionofresonance, clearexchange ofoscillator strength occursbetween NIand(N,„O)i,asseeninFigs. I(b)-1(e). Awayfromresonance thesatellite 3tforex- ample, has&l%ofthe(0,0)iintensity. As3Iap- proaches (0,0)ifromlowerenergy itsintensity increases ~~uniat1tresonance wherethetwotransitions areattheir minimum separa iseparation [AE=4.4meVfor3Iwith is.Icand4]. theintensities areapproximately equal[Figs. I(c)an Beyond thisfieldthelowercomponent ispinnedtotheen- ergyoftheunperturbed (0,0)itransition whilethe3I character istransferred totheuppercomponent anditsin- tensitydecreases. Nearresonance, theNIsatesatellite inten- sitiesfarexceedthoseoftheweakparent(N,.,Oizero- honontransitions. Thepinning behavior andexchange ofoscillator strength arecharacteristic ofhybridizing statesingeneral andspecifically ofthecoupledLL-LO- phononstatesinvolved inRPC." HOLE-QUASIPARTICLE RESONANT POLARON COUPLING IN... 9499 Almostidentical resonance behavior isfoundinasimi- larsample whichhasslightly lowern,(1.45X10' cm EF=48meV)andonlyoneoccupied electron subband. Twofurtherasymmetric QW'sofwidths200and250A withE~between 35and40meValsogavequalitatively similarresonances. Inaddition, strongresonant enhance- mentofLOsatellites hasbeenseeninan(In,Ga)As-InP QW(n,=.10'cm,width100A).' Unlikeconventional CRmeasurements, inthepresent experiments boththeparentLLoftheNIsatellite andthe N,,levelinvolved intheresonance are,generally, fullyoc- cupied.Theresonant coupling occurs inthefinalstatesof thePLtransitions between quasiholes (unoccupied elec- tronstates) whichoccurintheotherwise filledelectron LL'sasaresultofrecombination. Theinteracting states areshownschematically inFig.3andconsistof(i)one "hole" intheN,.=0LLoftheFermiseaand(ii)one "hole" intheN,&1LL,plusoneLOphononemitted in thePLprocess. HereN,,&1istheparentLLofthe resonating Niphononsatellite. Thetwostatesarecou- pledtogether bytheFrohlich hole-LO-phonon interac- tion.Theresonance condition /tN„Itru„=hruLo willbe fulfilled inthemagneto-PL ofallsystems wheretheelec- tronFermienergyEpis~hruLosuchthattheenergysep- arationbetween/lied LL'scanbetunedthrough theen- ergyregionofAcoLg. InFig.4,thenormalized relative intensities oftheNL satellites [NNIintensity divided bytheNI.plus(0,0)~in- tensities] areplottedagainst theestimated unperturbed transition energyseparations. Therelative NIintensities ofFig.4areobtained fromtheobserved peakintensities, aftersubtraction ofthe(0,0)~contribution, assuming con- stantPLlinewidths withfield.Theaccuracy islimitedto approximately +30% intherelative intensities, particu- larlyclosetoresonance because ofthedifficulty in separating thecontribution fromtwocloselyoverlapping lines. Closetoresonance, thesatellite intensities aredescribed quitewellintermsofthehybridized (0,0)~and1VIfinal hole+ LOphonon N=0 hole1I Fnefgyof fina(state FIG.3.Schematic diagram showing theresonating quasihole energy levelsinvolved inthefinalstatesofthePLtransitions. Thefinalstateswithout andwithemission ofanLOphononare shownontheleft-andright-hand sidesofthefigure,respective- ly.0.5— L 0.4— Z', UJ 0.3— UJ 0.2 UJ~o.i-2—o 3—0 4—It I\ I I I I It I I I~lI I IOII / / / // ~O~ Q s 1 s 1 s 1 states,treatedasatwo-level coupled system. Thisisa marked simplification ofRPCtheory,"'''butdescribes behavior similartothatpredicted byperturbation theory closetoresonance."'Usingthevalue /tE/2=2. 2meV (fromFig.2)fortheelectron-phonon mixingpotential, thevariation ofintensities withenergyseparation, sho~n bythedashedcurveinFig.4,isobtained. Thisprovides a reasonable fittotheexperimental points inFig.4,and demonstrates theexpected exchange ofoscillator strength between twointeracting levelswhichanticross atreso- 77nance.— Itisinstructive todrawcomparisons oftheLL-LO- phonon coupling phenomena observed inmagneto-PL withprevious studies inCR. Inpure2D,theelectron- LO-phonon interaction isexpected tobeenhanced relative to3D,butduetothenonzero spatialextentoftheelectron wavefunction normaltothelayersandtoscreening, the electron-phonon coupling strength inrealdopedhetero- junctions (HJ's)(andQW's) isreduced tovaluescloseto orlessthanthoseobserved in3D."''Themostde- tailedstudiesofresonant polaron eA'ects inGaAs- Ga~—,Al,-AsHJ'shavebeencarriedoutbyLangerak et at.'Theseworkers studied thecyclotron massenhance- mentsclosetoresonance forn,.from0.8to5.4x10" cm.ThevaluesforCRmassenhancements canbeex- pressed intermsofthesplitting (AE),assumed sym- rnetric,between theupperandlowerpolaron branches at resonance. Averystrongreduction ofh,Efrom-5.0 meVat3.4x10''crn,toaboutafactorof2smallerat 5.4x10'' cm-',wasobserved inRef.5.ThevalueofhE at3.4X10'icrn—2isclosetothatfoundat4X10''crn bySigg,Wyder, andPerenboom. Atstillhigherdensi- ties(9X10" cm),Ziesmann, Heitmann, andChang foundnoevidence forpolaron coupling atresonance from CRina200-AInAsQW. Theseresultsforthemagnitude oftheRPCinCRwith increasing n,.areinstrongcontrast tothepresentPLre- sults,wherehE=4.4meVisobserved forthe31.-(0,0)~—12-8—4048 ENERGY SEPARATION (meV) Ffo.4.PlotofNI.satellite intensity divided bysumoflV(. and(0,0)~intensities asafunction ofunperturbed energysepa- rationbetween thetwotransitions. Thedashedcurveisafittoa modeloftwointeracting stateswithmixingpotential AE/2=2.2 meV. 9500 SIMMONDS, SKOLNICK, FISHER, NASH,ANDSMITH resonance, atthemuchhigherdensityof1.6x10' cm Thecontrast between magneto-PL (resonance withN„=3 LL)andCR(resonance withN„= ILL)isevengreater whenaccount istakenofthelikelyscalingofh,Ewith Landau index. AF,hasbeencalculated tovaryasN, (Ref.21)leading topredicted hEvalues3i=2times smaller formagneto-PL inthepresentexperiment (reso- nancewithN,.=3LL)thanforCR(resonance with N,,=ILL). Theverylargestrengthoftheresonant polaroninterac- tionfortheholequasiparticles inPL,bycomparison with CR,canbeunderstood qualitatively byconsideration of theroleofLLoccupation effects."'InCRsuchoccu- pationeffectsareveryimportant. Theresonant coupling arisesbetween theN,,=lLLandtheN,,=O,1-LO- phononstate.AstheN,,=0population increases withn, (inRef.5,LLfillingv=0.67at3.4&&10''cm,v=1at 5.4&10'' cm-'atresonance), thepolaroninteraction will beincreasingly quenched, duetotheoccupation ofthe necessary scattering statesinN,=0.Atcomplete filling, theresonant contribution totheCRmassenhancement is absent.'Goodagreement hasbeenfoundinRef.5with calculations inwhichtheoccupation effectisthedom- inantmany-particle factor. Bycontrast inmagneto-PL suchoccupation effectsare notimportant, sincethescattering oftheholesoccursbe-tweenotherwise filledLL's;allstateswithintheresonat- inglevelsareavailable forthequasihole scattering since theholelevelsareunoccupied. Coulomb screening alone actstoreducethestrengthofthepolaron interactions. Extrapolation ofthepredictions foriJF.fora100-AQW inRef.10ton,=1.6x10'cm,including onlythen, dependence ofthescreening (nooccupation effects), leads toreasonable agreement withthevalueofhE=4.4meV observed inmagneto-PL. However, amoredetailed analysis ofthepresent polaron phenomena clearlyre- quiresatheoretical treatment specifically directed to- wardsthepresentcaseofhole-quasiparticle-LO-phonon coupling. Inconclusion, newresonant polaron coupling phenome- nahavebeenobserved inthePLspectraoffilledLandau levelsinmodulation-doped QW's.Theresultsareinter- preted intermsofhole-LO-phonon coupling inthefinal stateoftheFermisea.Suchhole-quasiparticle- LO- phonon coupling isexpected tobeauniversal featureof themagneto-PL spectraofallQW'swithelectron Fermi energyequaltoorgreaterthanAntg. Wearegrateful toL.Swierkowski forveryhelpfulsug- gestions andtoC.J.G.M.Langerak andR.J.Nicholas for informative discussions. Oneofus(P.E.S.)acknowledges financial support fromRSRE. E.J.Johnson andD.M.Larsen, Phys.Rev,Lett.16,655 (1966). -'G.Lindemann, R.Lassnig, W.Seidenbusch, andE.Gornik, Phys.Rev.828,4693(1983). M.Horst,U.Merkt, andJ.P.Kotthaus„Phys. Rev.Lett.50, 754(1983). 4M.Ziesmann, D.Heitmann, andL.L.Chang, Phys.Rev.B35, 4541(1987). -C.J.G.M.Langerak,J.Singleton, P,J.vanderWel,J.A.A. J.Perenboom, D.J.Barnes,R.J.Nicholas, M.A.Hopkins, andC.T.Foxon,Phys.Rev.83$,13133(1988). "H.Sigg,P.Wyder, andJ.A.A.J.Perenboom, Phys.Rev.B31, 5253(1985). RPCbetween LL'shasalsobeenstudied recently inresonant tunneling structures. G.S.Boebinger, A.F.J.Levi,R. Schmitt-Rink, A.Passner, L.N.PfeiA'er, andK.W.West, Phys.Rev.Lett.65,235(1990). "D.M.Larsen, Phys.Rev.830,4595(1984). "R.Lassnig,Surf.Sci.170,549(1986). 'Xiaoguang Wu,F.M.Peeters, andJ.T.Devreese, Phys.Rev. 840,4090(1989);Phys.StatusSolidi(b)143,581(1987). ''Nonresonant satellites havebeendiscussed byK.J.Nash,M. S.Skolnick, P.A.Claxton, andJ.S.Roberts, Phys.Rev.B39, 5558(1989). Weakfeatures inB=0PLspectra wereattri- butedtoLOsatellites ofrecombination nearEFbyC.Col- vard,N.Nouri,H.Lee,andD.Ackley, Phys.Rev.B39,8033 (1989). '-'P.B.Kirby,J.A.Constable, andR.S.Smith,Phys.Rev.8 40,3013(1989). 'M.S.Skolnick, D.M.Whittaker, P.E.Simmonds, T.A.Fish- er,M.K.Saker,J.M.Rorison, P.B.Kirby,andC.R.H.White,Phys.Rev.843,7354(1991). 'M.S.Skolnick, K.J,Nash,S.J.Bass,P.E.Simmonds, and M.J.Kane,SolidStateCommun. 67,637(1988). ~-E.D.Jones,S.K.Lyo,I.J.Fritze,J.F.Klem,J.E.Schirber, C.P.Tigges, andT.J.Drummond, Appl.Phys.Lett.54,2227 (1989). '"W.Chen,M.Fritze,A.V.Nurmikko, D.Ackley,C.Colvard, andH.Lee,Phys.Rev.Lett.64,2434(1990). '7M.S.Skolnick, P.E.Simmonds, andT.A.Fisher,Phys.Rev. Lett.66,963(1991);W.Chen,M.Fritze,andA.V.Nurmik- ko,Phys.Rev.Lett.66,964(1991). '"P.G.Harper,J.W.Hodby, andR.A.Stradling, Rep.Prog. Phys.36,I(1973). '"M.S.Skolnick andM.K.Saker(unpublished). -'OS.DasSarma andA.Madhukar, Phys.Rev.B22,2823 (1980);S.DasSarma,ibid27,2590(1983.). -"F.M.Peeters andJ.T.Devreese, Phys.Rev.B3l,3689 (1985). '—'Suchanexchange ofoscillator strength atresonance, of course, ischaracteristic ofanytwo-level coupled system, in- dependent ofthenatureofthemixingpotential. Thegoodfit totheexperimental resultsofFig.4doesnotimplysupport foranyspecific modelforthehole-LO-phonon coupling. -'~Reference 10showsthatthereduction ofAEbyoccupation effects isconsiderably largerthanthatduetoCoulomb screening, forpartially filledLL's. -'Weestimate avalueofhE=3.8meVatn,=1.6x10" cm fromtheresultsofRef.10withnooccupation effects,after making allowance fortheJV,,dependence ofAEof hEa-lV,,-''(Ref.21).Intheabsenceofscreening, thecorre- sponding valuewouldbeAE=5.7meV(Ref.10).
PhysRevB.86.155130.pdf	PHYSICAL REVIEW B 86, 155130 (2012) Imaginary-time quantum many-body theory out of equilibrium: Formal equivalence to Keldysh real-time theory and calculation of static properties Jong E. Han* Department of Physics, State University of New York at Buffalo, Buffalo, New York 14260, USA Andreas Dirks and Thomas Pruschke Department of Physics, University of G ¨ottingen, D-37077 G ¨ottingen, Germany (Received 3 May 2012; published 17 October 2012) We discuss the formal relationship between the real-time Keldysh and imaginary-time theory for nonequi- librium in quantum dot systems. The latter can be reformulated using the recently proposed Matsubara-voltageapproach. We establish general conditions for correct analytic continuation procedure on physical observables,and apply the technique to the calculation of static quantities in steady-state nonequilibrium for a quantum dotsubject to a ﬁnite bias voltage and external magnetic ﬁeld. Limitations of the Matsubara voltage approach arealso pointed out. DOI: 10.1103/PhysRevB.86.155130 PACS number(s): 73 .63.Kv, 72 .10.Bg, 72 .10.Di, 72 .15.Qm I. INTRODUCTION Experimental investigation of solids is in most cases concerned with observation of static or dynamic properties in a weakly perturbed macroscopic system. Therefore, standardtechniques from equilibrium statistical mechanics are usuallysufﬁcient, possibly supplemented by linear-response theoriesto account for transport. Equilibrium statistical mechanics isbased on the Gibbsian approach where the statistical densitymatrix of a state sat energy E sand particle number Nsis given by the Boltzmann factor e−β(Es−μNs)with inverse temperature β=1/kBTand the chemical potential μ. The big success in the theoretical description of quantum systems in thermalequilibrium is based on the fact that both the thermal averageand time evolution are based on the same operator, and onecan use the concept of Wick rotation to formulate a theorywhich actually condenses both types of dynamics into a single complex Matsubara frequency theory. The advances in experimental methods over the past two decades have, however, opened the access to studies, wheretime dependencies on the scale of internal time scales becomevisible, 1or where mesoscopic systems can be driven out of thermal equilibrium in a controlled way and various propertiescan be experimentally observed, 2–5both in steady- and time- dependent states. Therefore, one pressing question to modern quantum many-body theory is how one can describe genericnonequilibrium situations in macroscopic or mesoscopic sys-tems. For the latter the paradigms are the single-electronquantum dot and nanowires, where a tremendous amount ofdata on transport or transient response has been collected overthe past 10 years. 6,7 Out-of-equilibrium many-body theory is an emerging ﬁeld which poses an extreme challenge. There are manyattempts to use existing theoretical approaches, the mostpopular being the ones based on the Keldysh formulationof perturbation theory. 8In particular, the growing interest in transport through mesoscopic systems triggered a variety ofapplications of this technique; for example, direct perturbationtheory with respect to different zeroth order Hamiltonians, 9–11 functional renormalization group methods,12,13real-timediagrammatic approaches,14or direct numerical evaluation of the real-time propagators.15–21There are many other ideas, for example, based on the concept of inﬁnitesimalunitary transformations. 22A comprehensive overview can, for example, be found in Refs. 23and24. An early attempt to formulate an out-of-equilibrium version of statistical mechanics for steady-state properties of generalquantum many-body systems is due to Zubarev, 25who tried to construct a time-independent density matrix formalism bysolving the equation of motion within the scattering stateformalism. This approach has later been revisited by Hershﬁeldin the context of transport through quantum dot systems. 26 The main problem with these, in principle exact formulations,is that they cannot be readily applied because they requirethe solution of the Lippmann-Schwinger equation 27for the scattering states, which amounts to knowing the full solutionitself. Some efforts have been made to directly implementHershﬁeld’s density matrix within ﬁnite-order perturbationtheories, 28–30but they have proven quite cumbersome to be extended to inﬁnite orders. There have been other attemptsto tackle this problem by utilizing advanced nonperturbativetools of quantum many-body theory like Bethe ansatz 31or an extension of Wilson’s numerical renormalization techniques.32 However, the former approach could only be applied to a veryspeciﬁc model, while the latter may lack a thorough foundation regarding the proper steady-state limit. 33 In the present manuscript we focus on a different way to extend the theoretical framework of equilibrium quantum me-chanics to steady-state nonequilibrium for quantum impuritymodels via an imaginary-time theory. We especially discussthe possibility to deform the complex time contour for physicalobservables in equilibrium to the Keldysh contour appropriatefor nonequilibrium, as proposed by Doyon and Andrei. 34 One fundamental problem that arises in any such attemptstems from the fact that the nonequilibrium steady-stateBoltzmann factor and the time-evolution operator now havea fundamentally different structure, and thus a straightforwardWick rotation is not possible. As an alternative procedure,we show that, by introduction of Matsubara voltage , 35the problem of the dual operators can be resolved and a consistent 155130-1 1098-0121/2012/86(15)/155130(19) ©2012 American Physical SocietyJONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) theory for steady-state nonequilibrium based on auxiliary statistical mechanical problems formulated. As the ﬁrst step we need to properly deﬁne in what sense we achieve a steady state in a quantum impurity model. Thisis done in Sec. IItogether with a discussion of the general structure for Keldysh perturbation theory, the problem ofanalytical continuation, and the idea of the Matsubara voltageformulation. The equivalence of the Keldysh real time and theMatsubata voltage perturbation theory for the steady state willbe shown in Sec. IIIfor the single-impurity Anderson model. In Sec.IVwe derive expressions for calculating static observables on the impurity via an analytical continuation procedure fromthe Matsubara voltage description. As summary, Sec. Vwill conclude the paper. Since many details are rather technical and not really necessary to understand the main line of argument, we includedthem in a series appendices, which will be referred to whennecessary. II. MANY-BODY THEORY OFF EQUILIBRIUM A. Convergence to steady-state nonequilibrium To establish a steady-state nonequilibrium, one requires the system to be in the inﬁnite-size limit. In mesoscopic systems,such as quantum dots, this requirement means that the size ofthe reservoirs Lshould be the largest scale and this limit should be taken before any others. The time t Wfor the wake of the perturbation occurring in the quantum dot region to reach theedge of the reservoir with the Fermi velocity v F(tW=L/vF) should be greater than any time scale used for the turn-on of theperturbation or measurements. This ensures that the reﬂectedwave does not interfere with the formation of the steady stateand its measurements. Alternatively, the reciprocal v F/Lalso represents the level spacing of the continuum states, whichsets the smallest energy scale in the model. As in conventional many-body theory, we start with a perturbation which we turn on inﬁnitesimally slow with a rateη −1as ˆV(t)=ˆVeηt(1) for the time interval t∈[−T,0], where Tis some initial time which eventually will be sent to inﬁnity. For t>0, the perturbation remains constant at the full strength, ˆV(t)=ˆV. The above discussions lead to the relation between the threeenergy scales (we set ¯ h=1), v F L/lessmuch1 T/lessmuchη. (2) In his original proposal,9Hershﬁeld assumed the presence of an external relaxation process to derive the time-independentdensity matrix in the limit T→∞ . Recently Doyon and Andrei 34have shown that for mesoscopic systems inﬁnite reservoirs provide a relaxation process and any assumptionof an additional external relaxation source is not necessary.This suggests that we can do away with the adiabatic factore ηtin a time-dependent theory as long as the limit L→∞ is taken ﬁrst. Here we show through an explicit calculationthat the adiabatic factor e ηtis not necessary for the steady state if local measurements are made near the quantum dot,36 henceforth abbreviated as QD.Our model system consists of a QD connected to two fermionic reservoirs labeled by α=L,R (or±1, respectively, when the reservoir index is taken numerically). We include thesingle-particle tunneling between the leads and the QD into thenoninteracting part of the Hamiltonian, which then becomesthe resonant level model (RLM), ˆH 0=/summationdisplay αkσ/epsilon1αkσc† αkσcαkσ+/epsilon1d/summationdisplay σd† σdσ −/summationdisplay αkσtα√ /Omega1(d† σcαkσ+H.c.). (3) Here,c† αkσis the creation operator of conduction electrons for the reservoir αwith energy /epsilon1αkσat the continuum index kand spinσ;d† σcreates an electron on the QD orbital and tαis the tunneling integral. /Omega1is the normalization due to the volume of the reservoirs. This Hamiltonian can be diagonalized by the scattering state operators ψ† αkσgiven by the formal Lippmann- Schwinger operator equation, ψ† αkσ=c† αkσ−tα√ /Omega11 /epsilon1αkσ−L0+i0+d† σ, (4) with the Liouville operator acting on the operator space as L0O=[ˆH0,O]. This equation can be easily solved as ψ† αkσ=c† αkσ−tα√ /Omega1gd(/epsilon1αkσ)d† σ +/summationdisplay α/primek/primeσtαtα/prime /Omega1gd(/epsilon1αkσ)c† α/primek/primeσ /epsilon1αkσ−/epsilon1α/primek/primeσ+i0+, (5) with the bare retarded Green’s function for the QD, gd(ω)= (ω−/epsilon1d+i/Gamma1)−1. Here, /Gamma1=π(t2 L+t2 R)N(0) is the hybridiza- tion broadening, and we assume for simplicity a ﬂat densityof states (DOS) N(0) for both reservoirs. With this simple DOS, we suppress the reservoir and spin indices in /epsilon1 αkσunless necessary. In the Hamiltonian (3), we represent the QD by a single level under the assumption that the QD level spacing is large enoughthat the interlevel transition does not alter strong correlationphysics of single-level QD transport in a fundamental way.As the QD becomes large, the multiorbital nature of the QDbecomes important and we need to introduce the interlevelphysics. Orbital-ﬂuctuation physics of a QD is an importantproblem for inelastic transport process, but the full many-bodytreatment of such physics has so far been quite limited. 37,38 The range of validity of the imaginary-time theory has been discussed in Ref. 39regarding the level-connectivity, and we limit our discussion here to single-orbital QD with on-siteCoulomb interaction, namely, the Anderson impurity model. According to Hershﬁeld, 9the nonequilibrium steady state created by a shift of chemical potential on the source (drain)reservoir by /Phi1/2(−/Phi1/2) can be described by a density matrix, ˆρ 0=exp[−β(ˆH0−/Phi1ˆY0)], (6) with the so-called Yoperator deﬁned as ˆY0=1 2/summationdisplay kσ(ψ† LkσψLkσ−ψ† RkσψRkσ). (7) Since ˆY0is diagonal in the eigenoperator basis, [ ˆH0,ˆY0]=0, and ˆρ0is time independent. It is important to realize that the 155130-2IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) convergence factor i0+in the denominator of the Lippmann- Schwinger equation determines that the one particle states c† αkσ originate from the inﬁnite past inside the reservoir of inﬁnite size. Thus the limit L→∞ has already been taken implicitly before the perturbation is turned on. B. Real-time theory for open system In addition to the noninteracting part H0, the full Hamilto- nianHof the system will in general also contain an interaction we will denote as ˆVin the following. For a general observable ˆA, we deﬁne its nonequilibrium expectation value as lim T→∞/angbracketleftˆA(T)/angbracketright= lim T→∞Tr(eiˆHTˆAe−iˆHTˆρ0) Tr ˆρ0, (8) where ˆAhas been evolved with the full Hamiltonian ˆH during the time interval −T< t< 0. Unlike Eq. (1), here we take ˆV(t)=ˆVfor−T< t< 0. Deﬁning the time-dependent operator ˆA(t) in the Heisenberg picture, ˆA(t)=eiˆHtˆAe−iˆHt, ˆA(t) satisﬁesd dtˆA(t)=i[ˆH,ˆA(t)] and ˆA(t)=ˆA+i/integraldisplayt 0dt/prime[ˆH,ˆA(t/prime)]. (9) One can now form the average with respect to ˆ ρ0, to obtain /angbracketleftˆA(T)/angbracketright=/angbracketleft ˆA/angbracketright0+i/integraldisplay0 −Tdt/prime/angbracketleft[ˆH,ˆA(t/prime)]/angbracketright0 =/angbracketleftˆA/angbracketright0+i/integraldisplay0 −Tdt/prime/angbracketleft[ˆV,ˆA(t/prime)]/angbracketright0. (10) For the existence of a well-deﬁned limit /angbracketleftˆA(∞)/angbracketright, one must show that36 /integraldisplay0 −∞dt/angbracketleft[ˆV,ˆA(t)]/angbracketright0<+∞. (11) To this end one argues that as long as ˆVand ˆAare operators local to the quantum dot,40the time evolution of ˆA(t) will decay as electrons travel away and the integral is ﬁnite. To make the argument concrete, we consider as example the usual on-site Coulomb interaction, ˆH=ˆH0+ˆVwith ˆV=Und↑nd↓, (12) and ˆH0deﬁned in Eq. (3), and measure the current through the dot, ˆA=ˆI. With the requirement that the current through theL/R leads, IL/R, is the same, the current operator ˆIcan be symmetrized as ˆI=(t2 RˆIL+t2 LˆIR)/(t2 L+t2 R) and /angbracketleftˆI/angbracketright=−itLtR√ /Omega1/parenleftbig t2 L+t2 R/parenrightbig/summationdisplay kσ[/angbracketleftd† σ(tRcLkσ−tLcRkσ)/angbracketright−H.c.] =tLtR t2 L+t2 Ri /Omega1/summationdisplay kk/prime(g∗ d(k)−gd(k/prime)) ×/bracketleftbig tLtR/angbracketleftψ† LkψLk/prime−ψ† RkψRk/prime/angbracketright −/parenleftbig t2 L−t2 R/parenrightbig /angbracketleftψ† LkψRk/prime+ψ† RkψLk/prime/angbracketright/bracketrightbig . (13) We evaluate Eq. (10) using Wick’s theorem. Due to the commutator inside the expectation value, only connected con-tractions between any ˆVandˆI(t) will contribute. Therefore any nonvanishing Wick’s contractions must have an even numberof contractions connecting ˆVand ˆI(t) and contain factors of/angbracketleftψ αkσ(0)ψ† αkσ(t)/angbracketright0or/angbracketleftψ† αkσ(0)ψαkσ(t)/angbracketright0. More speciﬁcally, the ﬁrst-order perturbation involves factors like /angbracketleft[ˆV,ˆI0(t)]/angbracketright0∝1 /Omega12/summationdisplay kk/prime[g∗ d(k)−gd(k/prime)]gd(k)g∗ d(k/prime) ×[fL(k)−fL(k/prime)]e−i(/epsilon1k−/epsilon1/prime k)t+··· .(14) Herefα(k)=[1+eβ(/epsilon1k−α/Phi1/2)]−1(α=L,R or+1,−1, re- spectively) is the Fermi-Dirac function within the αreservoir. Summation over the continuum variables k,k/primeleads to terms of the form, /angbracketleftd†(t)d(0)/angbracketright=1 /Omega1/summationdisplay kαt2 α\|gd(k)\|2fα(k)e−i/epsilon1kt /lessorequalslant1 /Omega1/summationdisplay kt2 α\|gd(k)\|2e−i/epsilon1kt∝e−/Gamma1\|t\|. (15) Note that the inequality holds both for equilibrium and nonequilibrium. Therefore, the following expression, /angbracketleft[ˆV(sk),[...,[ˆV(s1),ˆI0(t)]...]/angbracketright0∝e−/Gamma1·min{\|s1−t\|,...,\|sk−t\|}, (16) holds to any order of the perturbative expansion in V, and the integral over t,E q . (11), becomes convergent. This shows that the steady-state limit of the nonequilibrium is well deﬁneddue to the built-in exponential time dependence e −/Gamma1\|t\|and the physics is invariant regardless of the adiabatic factor eηtin Eq.(1). We stress here that the above conclusion on the adiabatic rateηholds on the condition that the many-body interaction ˆV and the observable ˆAare short ranged from the QD. Generally, the two different limits of adiabatic ( η/greatermuch1/T) and sudden (η/lessmuch1/T) switching of interaction lead to different global quantum states. However, the main difference in the wavefunctions in the two limits is located at the front of thepropagating wave from the QD region and local observables near the QD reach the same steady-state values. As pointed outby Doyon and Andrei, 34the inﬁnite reservoirs [Eq. (2)] absorb excess energy in the switching process and carry it away fromthe QD. We caution that, although the convergence factor e ηtis not necessary for a time-dependent theory, such adiabaticfactor should be treated carefully in a time-independent theory,like the steady-state nonequilibrium. Such a situation arisesin particular when we perform a Fourier transformation torepresent a steady-state quantity in a spectral representationwith sinusoidal basis. For instance, let us express a steady-statequantity Aas an integral over a time-dependent function F(t), A=/integraldisplay 0 −∞F(t)dt, (17) where the integral is absolutely convergent without any adiabatic factor eηt. We write F(t) in a spectral representation as F(t)=/integraldisplay∞ −∞dω 2π˜F(ω)e−iωt, (18) 155130-3JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) with the Fourier component ˜F(ω), and the quantity Abecomes A=/integraldisplay0 −∞dt/bracketleftbigg/integraldisplay∞ −∞dω 2π˜F(ω)e−iωt/bracketrightbigg . (19) If we now want to express Avia a spectral representation, we need to change the order of integrals. However, e−iωtis an oscillatory function and we have to insert a regularizationfactor e ηtto unambiguously allow the integral exchange. Then, A=/integraldisplay∞ −∞dω 2π˜F(ω)/bracketleftbigg/integraldisplay0 −∞dte−i(ω+iη)t/bracketrightbigg =/integraldisplay∞ −∞dω 2πi˜F(ω) ω+iη, (20) where the limit η→0 has to be taken after the integral has been evaluated. Thus, the regularization factor iηappears explicitly in time- independent theories. A possible way to avoid it is to use animaginary-time formulation, which is built on a ﬁnite contourcut off by a ﬁnite temperature and therefore does not needsuch a regularization factor. It is thus one of our goals toclarify under what conditions a regularization is not necessaryand justify the use of an imaginary-time theory. C. Conventional analytic continuation In this subsection, we discuss conventional arguments of the analytic continuation of a real-time theory to an imaginary-time theory. We furthermore illustrate why such deformationof time contour fails for a steady-state nonequilibrium, closelyfollowing the argument by Doyon and Andrei. 34 In equilibrium, the thermal average of an observable ˆAis given as /angbracketleftˆA/angbracketright= lim T→∞TrS(0,−T)ˆρ0S(−T,0)ˆA TrS(0,−T)ˆρ0S(−T,0), (21) with the time-evolution operator S(t1,t2)=e−itH(t1−t2)with the full Hamiltonian ˆHand the noninteracting density matrix ˆ ρ0= e−βˆH0. We consider that the limit T→∞ exists as discussed in the previous section. In the interaction picture with ˆVI(t)= eitˆH0ˆVe−itˆH0, the above relation can be rewritten as /angbracketleftˆA/angbracketright= lim T→∞TrSI(0,−T)ˆρ0SI(−T,0)ˆA TrSI(0,−T)ˆρ0SI(−T,0), (22) with SI(t2,t1)=Texp/bracketleftbigg −i/integraldisplayt2 t1dsˆVI(s)/bracketrightbigg , (23) with the time-ordering operator Tdeﬁned as the time moving in the direction from the right argument t1to the left argument t2. Using the relation, SI(b,a)=e−icH 0SI(b+c,a+c)eicH 0, (24) one can write SI(0,−T)ˆρ0=ˆρ0SI(−iβ,−iβ−T), (25)t=0 t=−iβt=−T t=−T−iβt=0 t=−iβ(a) (b) FIG. 1. (a) Keldysh contour for real-time diagrammatics. If the time evolution along the dashed line does not contribute an extra factor, the whole contour can be deformed to one along the imaginary time from t=−iβtot=0 as shown in (b). in a similar manner as Ref. 34. Then /angbracketleftˆA/angbracketrightis written as /angbracketleftˆA/angbracketright=lim T→∞Tr ˆρ0SI(−iβ,−iβ−T)SI(−T,0)ˆA Tr ˆρ0SI(−iβ,−iβ−T)SI(−T,0) =lim T→∞/angbracketleftSI(−iβ,−iβ−T)SI(−T,0)ˆA/angbracketright0 /angbracketleftSI(−iβ,−iβ−T)SI(−T,0)/angbracketright0. (26) If we can insert the factor SI(−iβ−T,−T) [denoted as a dashed line in Fig. 1(a)] between SI(−iβ,−iβ−T) and SI(−T,0), one can close the time contour and analytically continue to the contour along the imaginary time (0 ,−iβ) [Fig. 1(b)]. Using the Wick’s theorem and the linked-cluster theorem, the perturbation terms contributing to /angbracketleftˆA/angbracketrightare of the type, /angbracketleftVI(s1)VI(s2)...V I(sn)ˆA(0)/angbracketright0,connected , (27) where the time s=0f o r ˆAand the interaction times {s1,..., s n}are all interconnected by Wick’s contractions. When the interaction ˆVand the observable ˆAare operators local to the QD, one can use the relation Eq. (15).W e consider a case that one of skin/angbracketleftVI(s1)...V I(sn)ˆA/angbracketright0,con belongs in the interval [ −T,−iβ−T]. In its connected Wick’s contractions the operators in ˆAmay be eventually linked to skvia a forward sequence {s/prime 0=0,..., s/prime p−1,s/prime p=sk} and a backward sequence {s/prime/prime 0=sk,..., s/prime/prime q−1,s/prime/prime q=0}.F o rt h e forward sequence {s/prime 0=0,..., s/prime p−1}with the times on the real axis, we can use Eq. (15), e−/Gamma1/summationtextp−1 n=1\|s/prime n−s/prime n−1\|∼e−/Gamma1max{\|s/prime 1\|,...,\|s/prime p−1\|}. (28) Similar expression holds for the backward sequence. For the last term involving sk∈[−T,−iβ−T], we have a con- traction of /angbracketleftd(s/prime/prime 1)d†(sk)/angbracketright/angbracketleftd(sk)d†(s/prime p−1)/angbracketright.F o r−β< Im(sk)< 0, the two factors remain ﬁnite and give a contribution proportional to e−/Gamma1(\|T+s/prime p−1\|+\|T+s/prime/prime 1\|). Therefore, when one of the interaction events occurs on the contour [ −T,−iβ−T], the corresponding term becomes exponentially small. Whentraced with local operator ˆA, the factorization of the time contour 34holds SI(−iβ,−iβ−T)SI(−T,0)→SI(−iβ,0). (29) This shows that the Wick rotation between real-time and imaginary-time theory is valid in equilibrium and /angbracketleftˆA/angbracketright=/angbracketleftSI(−iβ,0)ˆA/angbracketright0 /angbracketleftSI(−iβ,0)/angbracketright0. (30) Next we ask whether the same argument can be extended to the steady-state nonequilibrium with the initial density matrix at time t=−Tgiven by ˆ ρ0=e−β(ˆH0−/Phi1ˆY0). In order to move 155130-4IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) (a ( ) b)( c)ˆA ˆA ˆAρ0 t1 t2 s1 s2 s3 FIG. 2. (a) Keldysh contour in forward direction. Crosses mark interaction points ˆVand the dot an observable ˆA. (b) Reversed series of scattering points. (c) Backward Keldysh contour with scattering events equivalent to (a) if ˆAis written in terms of QD operators. ˆρ0in Eq. (21) to the leftmost position in the trace, we write ˆH=ˆH/Phi1+ˆV/Phi1with ˆH/Phi1 0=ˆH0−/Phi1ˆY0and ˆV/Phi1=ˆV+/Phi1ˆY0. Deﬁning V/Phi1 I(t)=eitˆH/Phi1 0ˆV/Phi1e−itˆH/Phi1 0, we can utilize the same argument as before to write /angbracketleftˆA/angbracketright= lim T→∞/angbracketleftS/Phi1 I(−iβ,−iβ−T)S/Phi1 I(−T,0)ˆA/angbracketright0 /angbracketleftS/Phi1 I(−iβ,−iβ−T)S/Phi1 I(−T,0)/angbracketright0.(31) However, unlike in equilibrium, we cannot use Eq. (15) for a contraction containing V/Phi1 I(s) since ˆV/Phi1=ˆV+/Phi1ˆY0contains spatially extended operators c† αkσcα/primek/primeσ/primewith contributions well away from the QD. Furthermore, V/Phi1 I(s)=eisˆH/Phi1 0ˆVe−isˆH/Phi1 0+ /Phi1ˆY0with a constant of motion ˆY0with respect to ˆH/Phi1 0, and V/Phi1 I(s) would never lead to an exponential decay for the interactions occurring on the dashed contour in Fig. 1(a). This shows that a straightforward analytic continuation of thenonequilibrium Keldysh contour to an imaginary-time one isnot possible. D. Matsubara voltage Recently, one of the authors and Heary35proposed that, by introducing a Matsubara term to the source-drain voltage,one can extend the equilibrium formalism such that the per-turbation expansion of the imaginary-time Green function canbe mapped to the Keldysh real-time theory. The unperturbedHamiltonian is written as ˆK 0(iϕm)=ˆH0+(iϕm−/Phi1)ˆY0, (32) with the Matsubara voltage ϕm=4πm/β with integer m.W e take the many-body interaction ˆVas perturbation. The noninteracting Hamiltonian appears in the perturbative expansion in two ways: ﬁrst in the thermal factors e−βˆK0, and second in the time evolution e−τˆK0for the imaginary-time variable τ∈[0,β). The main trick of this formalism is that in the thermal factor iϕm-dependence drops out as follows. Since [ ˆH0,ˆY0]=09,e−βˆK0=e−β(ˆH0−/Phi1ˆY0)e−iϕmβˆY0. Since, with respect to the noninteracting scattering state basis, ˆY0is diagonal and has (half)-integer eigenvalues, e−iϕmβˆY0=1, and we have the important identity, e−βˆK0(iϕm)=e−β(ˆH0−/Phi1ˆY0)=ˆρ0. (33) Therefore, the equivalence of the imaginary-time and real- time formalism crucially rests on how the double analyticcontinuation iϕ m−/Phi1→0 and τ→itis performed. Since theiϕmdependence in the thermal factor completely drops out, the analytic continuation only concerns the time evolution. For τ∈[0,β),e−iϕmτˆY0/negationslash=1, and iϕmdependence does not drop out. Thus, one could argue that as iϕm−/Phi1→0 and τ→it are taken in that order, e−τ[ˆH0+(iϕm−/Phi1)ˆY0]→e−τˆH0→e−itˆH0. (34)However, as we will point out in detail later, integrals over interaction times may create energy denominators of the type(K n−Km)−1in the perturbation expansions, with Knbeing thenth eigenvalue of ˆK0. In such cases, the details of the path in the complex plane, along which the analytic continuation /epsilon1ϕ≡ iϕm−/Phi1→±i0+is taken, become relevant. On the other hand, in the real-time theory, the convergence factor iηin the energy denominators determines what poles should be chosen. III. PERTURBATION EXPANSION A. Real-time expansion In this section, we investigate under what conditions the role of the regularization factor ηof the time-independent real-time theory becomes unimportant. We assume that a perturbationexpansion of Eq. (22)exists. To better illustrate the mathemat- ical structure we choose the ﬁfth-order contribution (as shownin Fig. 2) and introduce a spectral representation with respect to the noninteracting scattering state basis. For the particulartime ordering considered in Fig. 2(a), the expression reads S a=(−i)5Tr/bracketleftbigg/integraldisplay−∞ 0ds3/integraldisplays3 0ds2/integraldisplays2 0ds1 ×ˆVI(s3)ˆVI(s2)ˆVI(s1)ˆA ×/integraldisplay0 −∞dt2/integraldisplay0 t2dt1ˆVI(t1)ˆVI(t2)ˆρ0/bracketrightbigg . (35) Here we use the notation for intermediate times such that tiare for the forward contour ( −∞ → 0, upper time contour) and sifor the backward (lower) contour. We redeﬁne the time as t/prime 1=t1,t/prime 2=t2−t1,t/prime i=ti−ti−1, etc., and the upper part of the Keldysh contour becomes /integraldisplay0 −∞dt2/integraldisplay0 t2dt1ˆVI(t1)ˆVI(t2) =/integraldisplay0 −∞dt/prime 2/integraldisplay0 −∞dt/prime 1ˆVI(t/prime 1)ˆVI(t/prime 1+t/prime 2) (36) =/integraldisplay0 −∞dt/prime 2/integraldisplay0 −∞dt/prime 1eiH0t/prime 1ˆVeiH0t/prime 2ˆV. For a spectral representation with respect to energy eigenstates, we introduce the convergence factor eη(t/prime 1+t/prime 2)for the reasons discussed in Sec. II A. Then with respect to the noninteracting scattering-state Fock basis \|n/angbracketrightand\|p/angbracketright, (−i)2/angbracketleftp\|/integraldisplay0 −∞dt2/integraldisplay0 t2dt1ˆVI(t1)ˆVI(t2)\|n/angbracketright =/summationdisplay qVpqVqn (En−Ep+iη)(En−Eq+iη). (37) 155130-5JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) One can do the same for the lower part of the Keldysh contour, (−i)3/angbracketleftn\|/integraldisplay−∞ 0ds3/integraldisplays3 0ds2/integraldisplays2 0ds1ˆVI(s3)ˆVI(s2)ˆVI(s1)\|l/angbracketright =/summationdisplay mkVnmVmkVkl (En−Em−iη)(En−Ek−iη)(En−El−iη). (38) Therefore the above expression Sacan be written as Sa=/summationdisplay nmklpqVnmVmkVkl (En−Em−iη)(En−Ek−iη)(En−El−iη) ×AlpVpqVqn (En−Ep+iη)(En−Eq+iη)ρn. (39) Note that all energy denominators consist of one energy anchored at\|n/angbracketrightwhere ˆ ρ0acts at t=− ∞ and the other energy of intermediate states \|m,k,l,p,q /angbracketright. For the forward contour, the state \|n/angbracketrightcontributes the energy En+iηin the energy denominator, and En−iηfor the backward contour. We now consider a counter-time-ordering as depicted in Fig. 2(b) where the number of scattering events on the lower and upper branches are swapped. After an explicit calculationby applying the same rules as before, one gets S b=/summationdisplay nmklpqVnqVqp (En−Eq−iη)(En−Ep−iη) ×AplVlkVkmVmn (En−El+iη)(En−Ek+iη)(En−Em+iη)ρn. (40) Starting with the state \|n/angbracketright, the numerator VnqVqpAplVlkVkmVmnρnin Eq. (40) represents the reversed process of ρnVnmVmkVklAlpVpqVqnin Eq. (39). The factor ρnVnmVmkVklAlpVpqVqnis understood as the amplitude of the following process: Sa:\|n/angbracketrightˆV−→\|q/angbracketrightˆV−→\|p/angbracketrightˆA−→\|l/angbracketrightˆV−→\|k/angbracketrightˆV−→\|m/angbracketrightˆV−→\|n/angbracketright.(41) The many-body interaction can be written in terms of four scattering state operators as ˆV=/summationtextv1234ψ† 1ψ† 2ψ3ψ4. With the on-site Coulomb interaction, ˆV=U/summationdisplay {α,k}t1t2t3t4g∗ 1g2g∗ 3g4ψ† 1↑ψ2↑ψ† 3↓ψ4↓, (42) where the shorthand notations ti=tαi/√ /Omega1,gi=gd(ki), and ψ† iσ=ψ† αikiσhave been used. Note that any creation of a particle ψ† iis associated with the factor tig∗ i, and the annihilation ψjwithtjgj. For the observable ˆAwe consider a one-body operator ˆA=/summationtexta12ψ† 1ψ2for simplicity. The operator ˆVcreates up to two particle-hole pairs of type ψ, and for a nonzero matrix element /angbracketleftn\|V\|m/angbracketright,\|n/angbracketright, and\|m/angbracketrightdiffer only by up to one particle-hole pair per spin channel. Thus,in the above process Eq. (41), which starts and ends with \|n/angbracketright, the product of creation operators ψ† αkσmust match that of annihilation operators ψαkσ. Therefore, the matrix element for the process Eq. (41) must be of the form, Sa:\|t1g1\|2\|t2g2\|2...tigiaijtjg∗ j. (43)Similarly, the process for Sbterm, Sb:\|n/angbracketrightˆV−→\|m/angbracketrightˆV−→\|k/angbracketrightˆV−→\|l/angbracketrightˆA−→\|p/angbracketrightˆV−→\|q/angbracketrightˆV−→\|n/angbracketright (44) must contain the same set of {ψ†,ψ}with the same states, only in the reversed order. The matrix element for the process thenbecomes S b:\|t1g1\|2\|t2g2\|2...tjgjajitig∗ i. (45) If the operator ˆAsatisﬁes the following property, gd(ki)aij[gd(kj)]∗=gd(kj)aji[gd(ki)]∗, (46) the matrix elements for counter-contours (a) and (b) match, that is, VnmVmkVklAlpVpqVqn=VnqVqpAplVlkVkmVmn.(47) With this condition, Sa(η)=Sb(−η), and Sa+Sb, inside the expression for /angbracketleftˆA/angbracketright, is independent of the sign of ηand has a well-deﬁned limit of η→± 0. The above argument can be repeated for any order of the perturbation expansion, that is,the use of a spectral representation is permitted and the resultindependent of the convergence factor ηprovided that the contour has itself as the counter-contour, S a(η)=Sa(−η). Which of the physically interesting operators do satisfy the above condition Eq. (46), respectively, (47)? It is easy to see that it is true for any operator ˆAwhich is a simple function of ndσ=d† σdσ. The occupation number operator can be expressed in terms of ψ† αkσas ˆndσ=/summationdisplay kk/prime,αα/primetαtα/prime /Omega1g∗ d(/epsilon1k)gd(/epsilon1/prime k)ψ† αkσψα/primek/primeσ, (48) and Eq. (46) is satisﬁed. A general two-body operator, ˆA=/summationdisplay 1234a1234ψ† 1ψ† 2ψ3ψ4, also falls into this class if it satisﬁes gd(ki)gd(kj)aijnm[gd(kn)gd(km)]∗ =gd(kn)gd(km)anmij[gd(ki)gd(kj)]∗. (49) Unfortunately, the current operator Eq. (13) does not satisfy the condition Eq. (46), and a direct analytic continuation is not available, as we will discuss shortly. Therefore, we haveto resort to the Meir-Wingreen formula, 41which relates the current to the spectral function. We have so far ignored coinciding energy denominators in the perturbation expansion leading to overlapping δfunctions. For the sake of simplicity we consider a second-order con-tribution from Eq. (22). By expanding it into different time orderings, we obtain /integraldisplay T 0dt1/integraldisplayT t1dt2ˆρ0ˆVI(t2)ˆVI(t1)ˆA +/integraldisplay0 Tdt1/integraldisplayT 0dt2ˆVI(t1)ˆρ0ˆVI(t2)ˆA +/integraldisplay0 Tdt1/integraldisplayt1 Tdt2ˆVI(t1)ˆVI(t2)ˆρ0ˆA. (50) 155130-6IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) We now introduce the convergence factor eηtand take T→∞ to obtain the expression, /summationdisplay nml/bracketleftbiggρn (En−Em+iη)(En−El+iη) +ρm (Em−En+iη)(Em−El−iη)(51) +ρl (El−En−iη)(El−Em−iη)/bracketrightbigg VnmVmlAln, which needs precaution when the two energies in the denomi- nators become equal, because the contribution will be a productof two δfunctions with the same argument. One must be careful when one performs the limit T→∞ . To see this let us go back to the time-dependent description. By keeping Tﬁnite, contributions of the form δ(E n−Em)2will actually amount to terms proportional to T2from the integrals. Combining all three integrals we obtain the coefﬁcient to the T2term (i.e., δ2 term) proportional to /summationdisplay nml(ρn−2ρm+ρl)VnmVmlAln ×δ(En−Em)δ(Em−El). (52) In equilibrium ρn=ρm=ρlforEn=Em=Eland this term vanishes identically. The argument can be easily extended toarbitrary orders in the perturbation expansion. In the case of nonequilibrium the situation is more complex. Here we discuss in detail what happens to Eq. (52).W e consider the case \|n/angbracketright/negationslash=\|m/angbracketright/negationslash=\|l/angbracketright, while E n=Em=El. Suppressing the δfunctions, Eq. (52) has the form, e−βEn(eβ/Phi1Y 0n−2eβ/Phi1Y 0m+eβ/Phi1Y 0l)VnmVmlAln. In the matrix element VnmVmlAln, the transition \|n/angbracketright→ \|m/angbracketright→\|l/angbracketright→\|n/angbracketrightinvolves a certain series of particle-hole excitations. For instance, \|n/angbracketright→\|m/angbracketrightis given by an exchange of two particle-hole pairs, ψ† α1k1σψα2k2σψ† α3k3σ/primeψα4k4σ/primeinˆV, and similarly for \|m/angbracketright→\|l/angbracketrightand\|l/angbracketright→\|n/angbracketright. However, since any creation of ψ† αkσshould be matched by ψαkσonly up to six indices are independent. Given a particular set of thesix indices of wave vectors and spins {k 1σ1,k2σ2,..., k 6σ6}, different permutations of the above six pairs of {ψ† kiσi,ψkiσi} inˆVˆVˆAdetermines the matrix element VnmVmlAln.N o w , we sum over all possible combinations of reservoir indices{α 1,..., α 6}(while keeping the kindices unchanged) for the all 12 {ψ†,ψ}operators. The matrix element VnmVmlAln∝/producttext i=1,6t2 αi\|g(/epsilon1ki)\|2. Since the product of \|g(/epsilon1ki)\|2is invariant, we collect all possible reservoir weights in/producttext i=1,6t2 αieβ/Phi1Y 0{n,m,l} and each of the three sums in Eq. (52) become the same, that is, the whole contribution vanishes. A detailed discussion ofthe mathematics can be found in Appendix A. In summary, if the observable ˆAsatisﬁes Eq. (46),t h e energy integration in the perturbation expansion can beinterpreted as principal valued, similarly to equilibrium. InAppendix B, we provide as an example the fourth-order contribution to the QD electron self-energy and show explicitlythat the above properties are satisﬁed. Since the structuresappearing in higher order are of the same type as discussedabove, we may actually infer that this property holds in any order of the perturbation expansion. B. Imaginary-time expansion Unlike the real-time theory, the imaginary-time description is formulated on a ﬁnite time interval of [0 ,β), and there is no need for a convergence factor eηt. Therefore, the energy integrals appearing in the equilibrium theory are alwaysprincipal-value integrals, which we conﬁrmed in the previousSec. II C. In nonequilibrium, with the imaginary-time effective Hamiltonian ˆK(iϕ m)=ˆH0+/epsilon1ϕˆY0+ˆV(/epsilon1ϕ=iϕm−/Phi1), the thermal average is deﬁned as /angbracketleftA/angbracketright=Tre−βˆKA Tre−βˆK. (53) The Boltzmann factor can be expanded as e−βˆK=e−βˆK0Tτexp/bracketleftbigg −/integraldisplayβ 0dτVI(τ)/bracketrightbigg , (54) withVI(τ)=eτˆK0ˆVe−τˆK0ˆVandTτdenoting the time-ordering operator for τ∈[0→β]. We consider a second-order expan- sion to understand its mathematical structure, Tre−βˆK0/integraldisplayβ 0dτ/integraldisplayτ 0dτ/primeVI(τ)VI(τ/prime)ˆA =/integraldisplayβ 0dτ/integraldisplayτ 0dτ/prime/summationdisplay nmlρneτ(Kn−Km)Vnmeτ/prime(Km−Kl)VmlAln =/summationdisplay nml/bracketleftbiggρn (Kn−Km)(Kn−Kl)+ρm (Km−Kl)(Km−Kn) +ρl (Kl−Kn)(Kl−Km)/bracketrightbigg VnmVmlAln. (55) This expression has the same mathematical structure as in the real-time theory, Eq. (51). Even though we considered only one time ordering in the imaginary-time theory, the upper and lower integral limits in/integraltextβ 0dτ/integraltextτ 0dτ/primecombine to create the same permutation of terms as in the real-time theory.35 We have seen earlier that, in the real-time theory, energy denominators can be interpreted as principal valued sinceallδ-function contributions from the energy poles vanish. Therefore, if we interpret the energy denominators as principalvalued as iϕ m→/Phi1, 1 Kn−Km→P/parenleftbigg1 En−Em/parenrightbigg , (56) the terms in the imaginary-time theory indeed match those of the real-time approach. In Sec. IV A1 , we calculate the double occupancy from the continuous-time quantum Monte Carlo method,42and numerically verify that the analytic continuation procedureoutlined so far works accurately in all orders of perturbationtheory as well as for the resummed perturbation series. C. Single-particle self-energy The analytic properties discussed so far can be used to examine the single-particle self-energy for the Anderson 155130-7JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) impurity model. To illustrate how the imaginary-time theory is applied by using the conventional diagrammatic technique,we compute the electron self-energy in second order of theCoulomb interaction. The noninteracting Green’s function canbe easily obtained as G 0(iωn,iϕm)=/summationdisplay αk\|/angbracketleftψαkσ\|dσ/angbracketright\|2 iωn−α 2/epsilon1ϕ−/epsilon1k(57) =/summationdisplay α/integraldisplay d/epsilon1(/Gamma1α//Gamma1)A0(/epsilon1) iωn−α 2/epsilon1ϕ−/epsilon1, (58) with/Gamma1α=πt2 αN(0) as the hybridization broadening from the αreservoir. Using the standard imaginary-time perturbation theory for the second order of Coulomb interaction,35 /Sigma1(2)(iωn,/epsilon1ϕ) =U2 β2/summationdisplay m,lG0(iωn+iωm−iωl)G0(iωm)G0(iωl),(59) which we rewrite as /Sigma1(2)(iωn,/epsilon1ϕ)=/summationdisplay γ/integraldisplay d/epsilon1σγ(/epsilon1) iωn−γ 2/epsilon1ϕ−/epsilon1, (60) with the spectral function, σγ(ω)=U2/summationdisplay α1−α2+α3=γ/bracketleftBigg3/productdisplay i=1/integraldisplay d/epsilon1i/Gamma1αi /Gamma1A0(/epsilon1i)/bracketrightBigg [f1(1−f2)f3 +(1−f1)f2(1−f3)]δ(ω−/epsilon11+/epsilon12−/epsilon13), (61) for the γ-branch cuts ( γ=±1,±3), where A0(/epsilon1)=/Gamma1/π (/epsilon1−/epsilon10)2+/Gamma12 denotes the noninteracting spectral function of the QD level andfα=[1+e−β(/epsilon1−α/Phi1/2)]−1the Fermi-Dirac factor for the αth reservoir. The branch index γis the sum of reservoir indices for the three Green’s functions [Eq. (59)] representing a particle dressed by a particle-hole pair. The identity, Eq. (33), manifests in the Fermi-Dirac factor in f/parenleftbig /epsilon1k+α 2/epsilon1ϕ/parenrightbig =f/parenleftbig /epsilon1k−α 2/Phi1/parenrightbig . (62) Summation over γfor the self-energy spectral function, Eq.(61), leads to the identical retarded self-energy spectral function in the real-time theory corresponding to the samediagram. Therefore the procedure of /epsilon1 ϕ=iϕm−/Phi1→0 followed by iωn→ω+iηresults in the correct retarded self-energy. Recently, it has been proposed39that an inclusion of higher- order contributions will mainly modify the spectral functionσ γ(/epsilon1), leading to a /epsilon1ϕdependence like /Sigma1(iωn,/epsilon1ϕ)=/summationdisplay γ/integraldisplay d/epsilon1σγ(/epsilon1,/epsilon1ϕ) iωn−γ 2/epsilon1ϕ−/epsilon1. (63) Based on this expression, one can ﬁt43σγ(/epsilon1,/epsilon1ϕ)t ot h e numerical single-particle self-energy generated from quantumMonte Carlo calculations. 44,45However, in order to establish the existence of an analytic continuation limit of the imaginary-time self-energy, one should ﬁrst show that the real-timeself-energy possesses the analytic property discussed in the previous section, namely that the energy poles are principalvalued. The rather lengthy and technical argument is providedin Appendix Afor the fourth-order self-energy diagrams. It can be shown explicitly that contributions involving products of δ functions with identical argument vanish identically, resultingin the necessary analytic properties discussed in the previoussection. Again, investigating the general structures appearing in the perturbation expansion of the self-energy, we are conﬁdentthat this property indeed holds in any order and also survivesthe resummation of the series. The latter aspect, however,cannot be proven rigorously, but is strongly supported by thenumerical evidence from our Monte Carlo simulations. In a recent work by Dirks et al. 46and an accompanying paper to this work, a general analytic continuation approachbased on the multivariable complex function theory andits double analytic continuation of ( iω n,iϕm) have been systematically studied. D. Forward and backward steady state We have seen in Sec. III A that we need Eq. (47) for any sequence of matrix elements in order to establish theequivalence of the real- and imaginary-time theory. In orderto close the formal discussions, let us re-examine the complexconjugate of the matrix elements in relation to the forward-and backward-in-time propagation of scattering state densitymatrix. Assume that we propagate a noninteracting density matrix ρ 0=exp[−β(H0−/Phi1Y 0)] from the initial time t=−Tto the present in the forward direction. Then, according to Gell-Mann and Goldberger47, we obtain ˆρout=η/integraldisplay∞ 0e−iLT(eiL0Tˆρ0)e−ηTdT =η/integraldisplay∞ 0e−iLTˆρ0e−ηTdT=η η+iLˆρ0 =ˆρ0+1 −L+iηLVˆρ0, (64) withLVthe Liouvillian representing the interaction parts not contained in L0.ˆρoutis the fully interacting density matrix at t=0 and ˆρ0noninteracting density matrix at t=0. The meaning of the above equation is that we unwind anoninteracting density matrix to a remote time t=−Tand re-evolve it with full interaction to the present time. By takingthe average over the remote time T, we ﬁlter out transient oscillations. Alternatively, we can also consider a backward propagation of density matrix evolving from the remote future by writing ˆρ in=η/integraldisplay∞ 0eiLT(e−iL0Tˆρ0)e−ηTdT =ˆρ0+1 −L−iηLVˆρ0. (65) If we initially choose ˆ ρ0as the density matrix of a quantum dot system of disconnected dot and reservoirs, LV=Lt+LU with both the hopping to the leads and the Coulomb interaction 155130-8IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) on the dot, and we then consider the construction of scattering states as a two-step process. We ﬁrst construct the scatteringstates with respect to the hopping, and then with respect to theCoulomb interaction. After the ﬁrst step, the scattering statesbecome 29 ψ† αkσ, out=c† αkσ+t√ /Omega1gd(k)d† σ+··· , (66) ψ† αkσ, in=c† αkσ+t√ /Omega1gd(k)∗d† σ+··· , (67) and we can construct respective scattering-state density matri- ces ˆρ0t,outand ˆρ0t,in. The coefﬁcients appearing in front of the dot operators d† σ,dσ, etc., for the out- and in-scattering states are the complex conjugate of each other. As the second step, the matrix elements of the interaction ˆV=Und↑nd↓, written in terms of ψαkσ,{out,in}basis, are complex conjugate to each other, that is, Vnm=V∗ ˜n˜m(with the tilde denoting the in-scattering basis). We can now repeatthe arguments from Sec. III A for the backward propagation of the density matrix as shown in Fig. 2(c)and ﬁnd S c=/summationdisplay nmklpqV˜n˜qV˜q˜p (En−Eq+iη)(En−Ep+iη) ×A˜p˜lV˜l˜kV˜k˜mV˜m˜n (En−El−iη)(En−Ek−iη)(En−Em−iη)ρ˜n. For observables satisfying Anm=A∗ ˜n˜m, this expression be- comes identical to Sain Eq. (39). The same argument holds in any order of the perturbation expansion, and we haveTrˆAˆρ out=TrˆAˆρinand/angbracketleftˆA/angbracketright=1 2(/angbracketleftˆA/angbracketrightout+/angbracketleftˆA/angbracketrightin). Therefore, from Eqs. (64) and(65),w eh a v e /angbracketleftˆA/angbracketright=/angbracketleft ˆA/angbracketright0+/angbracketleftbigg ˆA1 2/parenleftbigg1 −L+iη+1 −L−iη/parenrightbigg LVˆρ0/angbracketrightbigg =/angbracketleftˆA/angbracketright0+/angbracketleftbigg ˆAP/parenleftbigg1 −L/parenrightbigg LVˆρ0/angbracketrightbigg , (68) that is, the conditions for replacing the energy denominators by their principal values, as discussed in Sec. III A , correspond to a measurement protocol where the observable ˆAhas the same expectation values with respect to the forward- and backward-propagating density matrices. It is interesting to note that the forward and backward density matrices, Eqs. (64) and (65), have different signs in the time-evolution operator and are related by a timereversal (or more appropriately motion reversal 48), where the coefﬁcients to {d† σ,dσ,c† αkσ,cαkσ}are complex conjugates between ˆ ρinand ˆρout.F o r ˆA=d† σdσ, its expectation value is not affected by the motion reversal. The same can be saidfor magnetization ˆA=n d↑−nd↓. However, expectation value for current deﬁned as ˆIL=itL(c† Lσdσ−d† σcLσ) is asymmetric with respect to ˆ ρin,outwith the motion-reversal property, /angbracketleftc† Lσdσ/angbracketrightin=/angbracketleftc† Lσdσ/angbracketright∗ out, etc., and Eq. (68) cannot be applied. As discussed in Sec. III A below Eq. (47), the same conclusion resulted regarding direct evaluation for ˆA=ndσ, but not for the current observable ˆIL,R. It is interesting to note that, in the Gell-Mann and Low theorem,49the symmetry between the forward and backward propagation of a ground state has been used to deform theKeldysh contour to a straight-line contour. Our work can be interpreted as an analogy to nonequilibrium steady state withlimited scope, namely, that the theory applies for scatteringproblems (i.e., quantum dots coupled to open systems) and thatthe forward-backward symmetry has a meaning with respectto the expectation values of motion-reversal symmetric localobservables. IV . STATIC EXPECTATION VALUES A. Theoretical background We have shown that steady-state expectation values of certain local observables ˆAcan be obtained from analytical continuation of expectation values calculated within theimaginary-time Matsubara-voltage formalism. As long as weknow the analytic structure of these objects, this can be doneeasily. However, for a model with true two-particle interac-tions, one eventually has to resort to numerical evaluations, andan analytical continuation in general requires a more involvedcomputational technique. We therefore want to provide in thefollowing a representation which allows the use of standardtools from equilibrium many-body theory. A numerical method gives /angbracketleftˆA/angbracketright(iϕ m) and let /angbracketleftˆA/angbracketright(zϕ) be its analytic continuation. We may write formally, /angbracketleftˆA/angbracketright(zϕ)=/angbracketleftˆA/angbracketrightconst+χA(zϕ), (69) where the part χA(z) is holomorphic in the upper and lower half plane, with singularities only on the real axis. If one canfurthermore show that the zχ A(z) is nonsingular in the limit z→∞ , one can ﬁnally infer that a spectral representation with respect to the jump function on the real axis exists andhence /angbracketleftˆA/angbracketright(iϕ m)=/angbracketleftˆA/angbracketrightconst+/integraldisplay/rho1A(ϕ) (iϕm−/Phi1)−ϕdϕ. (70) Note that the latter property is not necessarily guaranteed and has to be proven individually for each observable. Once the validity of the representation (70) is established, one only needs to obtain the “spectral function” /rho1A(ϕ). One ev- ident method to calculate the Matsubara voltage data /angbracketleftˆA/angbracketright(iϕm) for the observable ˆAwith respect to the effective system with non-Hermitian Hamiltonian at Matsubara voltage iϕmis via a QMC simulation.46For such data with statistical noise, one then typically employs a maximum-entropy approach(MaxEnt). 50The implementation of a MaxEnt estimator for the physical expectation value is rather straightforward. Thevalues for different iϕ mare truly statistically independent, and only the variance and correlation between imaginary and realparts of a single iϕ mvalue play a role. However, one still needs accurate and unbiased measurements of imaginary-voltagedata over a large range of ϕ m.46This latter requirement makes the use of a continuous-time quantum Monte Carlo(CT-QMC) 42algorithm mandatory. In particular, the necessary estimation of the constant offset /angbracketleftˆA/angbracketrightconstin Eq. (70)is possible only with CT-QMC, because at present no direct measurementalgorithm for this quantity is available and one must determineit from the tail of /angbracketleftˆA/angbracketright(iϕ m) by ﬁtting it to /angbracketleftˆA/angbracketright(iϕm)m→∞→/angbracketleft ˆA/angbracketrightconst+cA iϕm+˜cA (iϕm)2+··· .(71) 155130-9JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) In practice, a weighted least-square ﬁt yields reliable values and error bars for /angbracketleftˆA/angbracketrightconst. Via Gaussian error propagation it is then possible to incorporate the uncertainty of /angbracketleftˆA/angbracketrightconstinto the covariance matrix of the quantity /angbracketleftˆA/angbracketright(iϕm)−/angbracketleftˆA/angbracketrightconst.51 In general, the spectral function /rho1A(ϕ) needs not to be positive semideﬁnite, or show any symmetry relations withrespect to ϕ. Since on the other hand the MaxEnt method is only applicable for the inference of positive deﬁnite functions,a shift function /rho1 shift(ϕ) of the spectral function /rho1A(ϕ) has to be introduced, which makes the to-be-inferred /rho1/prime A(ϕ)=/rho1A(ϕ)− /rho1shift(ϕ) positive. We also employ a symmetry condition, /rho1shift(ϕ)=/rho1shift(−ϕ), (72) because with respect to this choice, the physical result, /angbracketleftˆA/angbracketrightphys=1 2/summationdisplay α=±1/angbracketleftˆA/angbracketright(/Phi1+αiη)=/angbracketleftˆA/angbracketrightconst−P/integraldisplay dϕ/rho1A(ϕ) ϕ, (73) is robust. In the following we want to prove that the double occupancy or magnetization obey this constraint (i.e., have arepresentation), where /angbracketleftˆA/angbracketright constis a real number, and /rho1A(ϕ)∈R is a real-valued spectral function. 1. Double occupancy The double occupancy in the Matsubara-voltage represen- tation is deﬁned as D(iϕm):=/angbracketleftnd,↑nd,↓/angbracketrightK(iϕm), (74) where the expectation value is taken with respect to the mth effective equilibrium system. We will ﬁrst show that the representation (70) holds for the double occupancy, that is, that we have, indeed, D(iϕm)=D0+/integraldisplay dϕ/rho1D(ϕ) iϕm−/Phi1−ϕ. (75) We restrict the discussion to the case of particle-hole symmetry and symmetric coupling to the leads, /Gamma1L=/Gamma1R. Within the Matsubara-voltage approach, one can—for ﬁxed iϕm— employ the standard techniques of equilibrium many-bodytheory and obtains the standard result, 52 D(iϕm)=/angbracketleftn↑/angbracketright/angbracketleftn↓/angbracketright+1 βU/summationdisplay ωn/Sigma1(iϕm;iωn)G(iϕm;iωn)eiωnη. (76) Due to particle-hole symmetry, we have /angbracketleftn↑/angbracketright/angbracketleftn↓/angbracketright=1/4. Furthermore, from the discussion in Sec. III C we can infer that at least the Green’s function decays like 1 /iϕmand hence allows for the existence of a spectral representation (75),a s long as there is only a single branch cut at Im zϕ=0. The real valuedness of spectral function and constant offset remain to be shown. The general relation G(−iϕm,−iωn)∗= G(iϕm,iωn) holds for Green’s function and self-energy. Insert- ing this into Eq. (76), we ﬁnd D(−iϕm)∗=D(iϕm). (77) Consequently, the real part of D(iϕm)−D(−iϕm) vanishes. Using the symmetric coupling to the leads, we have aninvariance of the Green’s function and self-energy under(iϕ m−/Phi1)↔− (iϕm−/Phi1). As a result, D0is an actual constant which is obtained for both upper and lower half plane.Due to the symmetry of Im D(iϕ m),D0is real. By inserting the representation (75) into Eq. (77) we also see that /rho1D(ω)i s real valued. For example, let us consider the equilibrium setup (i.e., /Phi1=0). At half ﬁlling and symmetric coupling to the leads, the function, ReD/Phi1=0(iϕm)=ReD/Phi1=0(−iϕm), (78) ImD/Phi1=0(iϕm)≡0. (79) This is compatible with a conventional bosonic spectral representation, D/Phi1=0(iϕm)=/integraldisplay dϕ/rho1D(ϕ) iϕm−ϕ+D0, (80) with an antisymmetric spectral function, /rho1D(ϕ)=−/rho1D(−ϕ);/rho1D(ϕ> 0)<0, (81) and the offset D0>0. Equation (79) is not evident for asymmetric couplings or off particle-hole symmetry, becausehereG 0(iϕm,iτ) is not real. 2. Magnetic susceptibility An observable which is much more sensitive to the Kondo effect is the magnetization M:=(/angbracketleftn↑/angbracketright−/angbracketleftn↓/angbracketright) in the presence of a magnetic ﬁeld Bin thezdirection, respectively, the mag- netic susceptibility χ=M/B of the quantum dot, because it directly probes the spin degree of freedom of the dot electrons.In equilibrium, a strong dependence on the temperature isobserved, on the scale of the Kondo temperature. 53 As for the double occupancy, the validity of a spectral representation, M(iϕm)=M0+/integraldisplay dϕ/rho1M(ϕ) iϕm−/Phi1−ϕ, (82) can readily be conﬁrmed. Starting from the symmetry G(−iϕm,−iωn)∗=G(iϕm,iωn), one can again show that M(−iϕm)∗=M(iϕm), and the same arguments apply con- cerning the interchange ( iϕm−/Phi1)↔− (iϕm−/Phi1). B. Numerical effective-equilibrium data Let us now turn to the discussion of actual numerical data for magnetization and double occupancy from the quantum MonteCarlo simulations. As the ﬁrst step, we analyze these data withrespect to the auxiliary variable ϕ m, and want to argue that they have a physical interpretation with respect to the actualvoltage /Phi1. In particular, the convergence of the numerical procedures described below implies full consistency of theMatsubara-voltage formalism with regard to the numericaldata. We ﬁnd that effective-equilibrium data come along with characteristic energy scales which—after analyticcontinuation—may translate almost directly into energy scaleswith respect to the actual source-drain voltage /Phi1. It is there- fore worthwhile to discuss the dependence of the effective-equilibrium expectation values as a function of ϕ mfor given physical parameters β,U, and/Phi1. 155130-10IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) ϕΓϕΦ=0Φ=0.5Γ Φ=2.0ΓΦ=6.0Γ (a)U=3 Γ , βΓ=1 0ϕ ΓϕΦ=0.25ΓΦ=0.5ΓΦ=2.0ΓΦ=6.0Γ (b)U=5 Γ , βΓ=2 0 ϕΓϕΦ=0.25ΓΦ=0.5ΓΦ=2.0ΓΦ=6.0Γ (c)U=8 Γ , βΓ=2 0ϕ Γϕ Φ=0.25ΓΦ=0.5ΓΦ=2.0ΓΦ=6.0Γ (d)U=1 0 Γ , βΓ=2 0 FIG. 3. (Color online) Real part of the effective-equilibrium double occupancy as a function of the Matsubara voltage ϕmat several values of interaction strength Uand bias voltage /Phi1. a. Dependence on /Phi1.The ﬁrst thing to notice is that the dependence of the shape of the curves M(iϕm) andD(iϕm)o n /Phi1, as shown in Figs. 3and 4(a), is rather moderate: For the examples considered, we do not observe any new characteristicenergy scales with respect to the Matsubara voltage ϕ m emerging or disappearing as a function of the physical voltage /Phi1. The most striking inﬂuence of /Phi1is a change of the offset of the curves D0andM0. The offset is changed monotonically as a function of /Phi1and cannot explain features such as dips and peaks which are found in the analytically continued data(cf. next section). This is the very reason of our claim thatlow- to intermediate-energy scales with respect to ϕ mrather directly translate into low- to intermediate-energy scales withrespect to /Phi1, although ϕ mhas no direct physical meaning itself. Let us discuss the data plotted in Figs. 3and 4(a) in more detail. In Fig. 3, effective-equilibrium double occupancy curves are shown over a wide range of values of the physicalvoltage and Coulomb interaction. Each curve exhibits a dipatϕ m=0. As already pointed out above, the dependence on /Phi1is rather mild, except for the offset. The same behavior is observed for the magnetization in Fig. 4(a)(i.e., the voltage /Phi1merely introduces an overall shift and a moderate smoothening of the structures). b. Limiting behavior ϕm→± ∞ .For each Uand/Phi1a different limit D0is obtained as ϕm→∞ . If the values β, U,/Phi1, and in particular ϕmare large, the effective-equilibrium QMC simulations start to suffer from a signiﬁcant signproblem. This may result in particularly noisy tails suchas the ones for the data with largest /Phi1in Fig. 3(d). In these cases, the estimate of D 0is subject to much uncertainty and limits the statistical accuracy of physical expectation values. c. Dependence on U.AsUis increased, the depth of the dips in the double occupancy curves also increases.On the other hand, neither the width nor the shape changesigniﬁcantly. In particular, the emergence of a Kondo scaleT Kcannot be inferred from these data. Interestingly, for small U, the relative contribution of the constant term D0 is large compared to the height of the peak which emerges around ϕm≈0. As the interaction increases, the central peak becomes more pronounced, and the physical expectation valueincreasingly depends on the structure of the peak. For the magnetization in Fig. 4(b), a similar picture seems to emerge at ﬁrst glance, namely a strong increase of the offset 155130-11JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) ϕ Γϕ Φ=0 Φ=0.2Γ Φ=0.5Γ (a)U=8 Γ , βΓ = 40, μBB=0.02Γϕ ΓϕΓ Γ Γ Γ Γ Γ Γ Γ (b)βΓ = 40, μBB=0.02Γ,eΦ=0 .5Γ FIG. 4. (Color online) Real part of the effective-equilibrium magnetization as a function of the Matsubara voltage. M0withUtogether with a more pronounced peak structure at ϕm=0. The strong increase of both is readily understood as with increasing Uthe system forms a local moment which is aligns with the external ﬁeld. d. Kondo effect. Up to now there seems to be no evidence whatsoever for the presence of the Kondo scale TKin the data presented so far. On the other hand, the generation of thismany-body scale is usually considered as a crucial test for anymethod proposed for studying the Anderson impurity model.As already pointed out, it is quite apparent from the data inFig. 3thatT Kobviously does not appear to be relevant for this quantity; a fact that is already well known in equilibrium.There the scale T Kshows up only in a very indirect way as renormalization of the zero temperature value, respectively,the scale regulating the approach to it. 54 The situation is different for the magnetization. Here, the Kondo scale plays a crucial role53as it determines the ﬁeld strength necessary to break up the Kondo singlet. Hence it must show up in the magnetization; in particular, one must actually expect a scaling behavior with TKfor small enough ﬁelds. Let us therefore plot the magnetization as a functionof Matsubara voltage in the form M(ϕ m/TK) for values of U beyond the weak-coupling regime for ﬁelds and voltages muchsmaller that the corresponding equilibrium Kondo scales. Theresult is shown in Fig. 5. Evidently, the width of the peak in the effective-equilibrium magnetization data is nicely scalingwith the equilibrium Kondo temperature, that is, for differentvalues of Uthe peak structure is essentially left invariant at ﬁxed values of B,/Phi1, andT. C. Results for real voltages In this section we will introduce the MaxEnt procedure used to infer the spectral functions /rho1D(ϕ) and /rho1M(ϕ)f r o m the effective-equilibrium QMC data. Based on this analyticalcontinuation, we then will discuss the physical results obtainedfrom the auxiliary Matsubara voltage data. 1. MaxEnt procedure Based on the effective-equilibrium data and the exact relation (70), it is in principle possible to uniquely reconstructthe spectral function /rho1A(ϕ) and the offset /angbracketleftˆA/angbracketrightconst.T h i s is almost completely analogous to the conventional Wickrotation. However, because in practice a ﬁnite set of data is considered, the inversion of Eq. (70) is no longer unique. On top of this, the quantum Monte Carlo data are not exact butmerely Gaussian random variables. One may easily verify thatthe noise associated with the variables is ampliﬁed by theinversion of Eq. (70). As a consequence, it will always be possible to ﬁnd qualitatively very different functions /rho1 A(ϕ) which are in agreement with the QMC data. In particular,these functions will yield physically different predictions viaEq.(73). The problem to obtain physical results from the effective-equilibrium data is thus ill-posed . Since essentially the same integral Eq. (70) also re- lates imaginary-time and real-time properties of conventionalGreen’s functions, this issue is well known to the community. 50 Although no solution to the problem can be provided, Bayesianinference provides a framework to systematically incorporateap r i o r i information about a quantity into an estimate. The estimate is most likely with regard to the prior informationat hand. The resulting method is called maximum entropy(MaxEnt). 50 Let us consider the situation in which the offset /angbracketleftˆA/angbracketrightconst has already been determined via a least-square ﬁt. Via error propagation it has been possible to determine the covariancematrix of the quantity /angbracketleftˆA/angbracketright−/angbracketleft ˆA/angbracketright const [i.e., the imaginary- voltage values of the quantity χA(zϕ)i nE q . (69)]. The remaining task of the MaxEnt is to infer the spectral function/rho1 A(ϕ). Let us furthermore assume that the data have been sufﬁciently transformed with a shift function, such that thefunction, /rho1 /prime A(ϕ)=/rho1A(ϕ)−/rho1shift(ϕ), (83) is positive (see Sec. IV A ). The default model for /rho1/prime A(ϕ) is then a positive deﬁnite function which in principle should contain features whichdetermine in particular the high-energy behavior, if known. 50 In the case of Green’s functions, perturbation theory orhigher-temperature solutions often give good default models. 50 155130-12IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) ϕϕ (a)U=5 Γϕϕ (b)U=8 Γϕϕ (c)U= 10Γ FIG. 5. (Color online) Kondo scaling analysis of effective-equilibrium magnetization data at μBB=TK/2,e/Phi1=TK/4. The analysis makes use of the equilibrium Kondo temperatures kBTK(U=5/Gamma1)≈1 10/Gamma1,kBTK(U=8/Gamma1)≈1 20/Gamma1,kBTK(U=10/Gamma1)≈1 40/Gamma1. The latter ratios are chosen to be approximately identical to the results of Haldane’s scaling formula.55 In our case, apart from that we used a shift function to construct the positive spectrum, nothing is known about the function, soa ﬂat default model is preferable. As consequence, we usethe shift function itself as the default model in the actualcomputation. For simplicity, let us call the to-be-inferredspectrum /rho1(ϕ) and the default model /rho1 def(ϕ). On the one hand, the default model gives rise to a relative entropy,50 S=/integraldisplay dϕ/bracketleftbigg /rho1(ϕ)−/rho1def(ϕ)−/rho1(ϕ)l o g/rho1(ϕ) /rho1def(ϕ)/bracketrightbigg , of the spectral function. On the other hand, the (transformed) effective-equilibrium simulation data with mean values ¯aiand covariance Cijyield the measure, χ2=1 2NQMC/summationdisplay i,j(¯ai−yi)C−1 ij(¯aj−yj), (84) for the quality of the ﬁt. Here yiare the ﬁt values which result from transforming the considered /rho1(ϕ) to the data space, and NQMCis the number of QMC data points ¯ai. Within the MaxEnt it follows that a functional Q=χ2−αSmust be minimized, where α> 0 is some hyperparameter.50 In order to determine α, there are several methods, for ex- ample, the “historic” and the “classic” MaxEnt.50The former extracts information from the Monte Carlo data up to the pointat which the χ 2=NQMC, that is, the MaxEnt regularization parameter is ﬁxed to the value at which χ2=NQMC. The latter (“classic” MaxEnt) extracts information from QMC data to alarger extent. Based on the probability distribution impliedby the default model and maximum-likelihood functionals, aposterior probability of the MaxEnt regularization parameter α is maximized. Because information from the default model isagain incorporated rather explicitly, this strategy is particularlygood for default models which are close to the actual solution.A rather general feature of “classic” MaxEnt appears to bethat the χ 2value of the inferred estimate is generally much smaller than the “historic” value of NQMC. Our feeling is that this aspect makes the “classic” estimate more sensitive tostatistical ﬂuctuations and vulnerable for overﬁtting, but onthe same side, the estimate is less biased. A similar increasein ﬂuctuations was pointed out in a recent study. 56At least if Bayesian evidence coming from the data is weak, the “historic”MaxEnt, on the other hand is more biased towards the default model value, since its estimate is more conservative with regardto the χ 2. In our case, the default-model estimate is given by the constant offset D0, because our default models are chosen to be even functions with respect to ϕ. As shift functions, wide Gaussians with width σ=200 3/Gamma1 were used, that is, /rho1shift(ϕ)=λ·e−ϕ2/2σ2. (85) The amplitude of the functions was varied in such a way that positive functions could be inferred. The different valuesfor differently scaled functions give rise to a certain intervalof expectation values, which will be plotted as a result, inthe following. An example for the set of inferred functionsobtained for a single nonequilibrium system is shown in Fig. 6. The left panel shows the actually performed MaxEnt for theshifted spectral functions, using “historic” MaxEnt. Resultingfrom a ﬂat default model for the function /rho1 D(ϕ), the shift function acts as the default model here. In this case, choosinga parameter λ< 0.01 yields artifacts in the physical solutions, because the negative regions of /rho1(ϕ) cannot be represented any more. The corresponding actual spectral functions /rho1(ϕ), obtained by subtracting the shift function (85) from the data in the left panel, are shown in the right panel of Fig. 6.T h e ﬂat default model represents our lack of prior informationabout the solution and the preference of a smooth solutionin case of uncertainty. In general, the different realizationsof a ﬂat default model with the shift functions yields almostbut not exactly the same spectral functions. In case of limitedQMC data quality, it is well known 50that the usage of a ﬂat default model yields less accurate spectra than an appropriatelyconstructed more informative default model. For example, inthe case of conventional equilibrium spectral functions ofFermi or Bose systems, a default model should preferablyobtain the correct low-order moments, which can often becomputed exactly. It can thus be expected that quantities thatare calculated from the spectra inferred using the ﬂat defaultmodel are biased towards a certain value. Nevertheless, anincrease in data quality will eventually reduce the bias of theestimated quantity. We also expect that the precision of ourmethod can be increased by the development of default modelswhich contain additional information like moments. However,at present this type of information is not yet available. 155130-13JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) ϕ/Γλ=0.01 λ=0.02 λ=0.04 λ=0.08 λ=0.16 λ=0.16 (a) inferred MaxEnt spectraD(ϕ)ϕ/Γλ=0.01 λ=0.02 λ=0.04 λ=0.08 λ=0.16 (b) resulting spectral functions D(ϕ) FIG. 6. (Color online) MaxEnt inference process for the double occupancy. Parameters are U=5,e/Phi1=0.25/Gamma1,β=20/Gamma1−1. Due to lack of prior knowledge, we use a ﬂat default model, that is, the shift function /rho1shift(ϕ); see Eq. (85). Remember that the actual spectral function /rho1D(ϕ) was shifted to a positive one, /rho1/prime D(ϕ), via Eq. (83). One ﬁnds that the different equivalent ways of imposing a ﬂat default model for /rho1D(ϕ) yield practically the same spectral function. Nevertheless, computing the physical value (73) yields values which are distributed over a certain range. This range is displayed as error bars in the results, plots Figs. 7and8. In order to obtain a rough estimate on the error of a physical estimate, we will plot the intervals which are generated bycomputing the estimates for different values of λ. Typically, a range from λ=0.01 to λ=0.16 is imposed, unless the negative regions of /rho1(ϕ) cannot be represented. For the magnetic susceptibility, the same strategy is used. 2. Double occupancy We will now discuss the analytically continued data of the double occupancy and compare it with respect to zero-temperature second-order perturbation theory. 57In Fig. 7 we show double occupancy data for different values of the Φ / Γ〈↑↓〉 Γ, β=20Γ−1Γ, β=20Γ−1Γ, β=10Γ−1 FIG. 7. (Color online) Double occupancy as a function of the bias voltage at different values of U, as compared to second-order perturbation theory. In addition, the dashed lines show the temperature dependence of /angbracketleftn↑n↓/angbracketrightin equlibrium as obtained by NRG, assuming e/Phi1=kBT(see text).Coulomb interaction computed with the two different MaxEnt estimators. The complementary behavior of the two estimators may be well observed in Fig. 7. In the large-bias limit, in which the perturbation theory may be expected to be correct, the classicestimator is closer, and the historic estimate is systematicallytoo high. This is in agreement with our expectation that thehistoric estimate will be biased from above in case of ratherweak Bayesian evidence from QMC data, because the ill-posedcontinuation problem is particularly severe at high energies. 50 Apart from some ﬂuctuations in the “classic” estimator, thesame curves are predicted for small voltages. It is importantto note that error bars in the ﬁgures do not denote statisticalerrors (which cannot be estimated), but the range of valueswhich a given set of symmetric default models generates. As compared to the second-order perturbation theory, we ﬁnd that both methods agree perfectly for interaction strengthU=3/Gamma1. In addition, both methods predict a minimum in the double occupancy at voltage e/Phi1≈2/Gamma1which slowly shifts to larger values of /Phi1and becomes increasingly distinguished as the interaction is increased. There is, however, a cleardifference concerning the magnitude of this minimum, whichappears much more pronounced in the QMC data as in theperturbation theory. Note that this seems to be the case forboth MaxEnt estimators. At present the origin of the deviationis not clear. One of the issues related to the /Phi1dependence of stationary nonequilibrium quantities is to what extent they can be mappedonto an effective equilibrium temperature dependence. Tohave an idea whether this mapping works, we includedin Fig. 7also the corresponding curves for /angbracketleftn ↑n↓/angbracketright(T)a s obtained from an NRG equilibrium calculation, assuminge/Phi1=k BT. Quite apparently, the values at /Phi1→0 nicely coincide, which also tells us that the Matsubara-voltage QMCreproduces the proper low bias results even for strong coupling.Note that perturbation theory here deviates systematicallywith increasing U. However, the dependence of /angbracketleftn ↑n↓/angbracketright(/Phi1) 155130-14IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) Φ/〈↑〉 − 〈↓〉)/Β [1/Γ] FIG. 8. (Color online) Magnetic susceptibility as a function of bias voltage in the Kondo regime U=8/Gamma1atμBB=kBTK/2,T= TK/2. The dot-dashed line represents an equilibrium NRG calculation forT/greaterorequalslantTK/2, rescaled in both magnitude and temperature to match the low-bias behavior of historic MaxEnt (see inset). The double-dot- dash curve ﬁnally is a ﬁt of historic MaxEnt to some scaling function (see text). cannot be mapped even qualitatively onto /angbracketleftn↑n↓/angbracketright(T)b ya simple ansatz /Phi1ˆ=α·Twith some value αfor any of the Uvalues considered here. From this observation we would thus conclude that such a mapping is—at least for the simplestpossible quantity—not appropriate. 3. Magnetic susceptibility Similarly, the magnetic susceptibility may be computed as a function of the bias voltage by analytical continuation ofthe QMC data. As an example, we show the result for U= 8/Gamma1at the temperature T=T K/2 and magnetic ﬁeld μBB= kBTK/2i nF i g . 8. When we compare our continuation results at/Phi1→0 to the exact low-bias limit (i.e., the equilibrium value, displayed as a cross in Fig. 8), the historic MaxEnt is again more strongly biased than the classic MaxEnt (i.e.,the deviation from the equilibrium value is stronger). Withinsufﬁcient QMC information, the outcome is more biasedtowards the ﬂat default model and from Eq. (73) the integral vanishes in such a limit. The constant offset M 0lies below the actual physical limit, and therefore, as QMC quality improves,our estimate approaches the correct limit from below. Again,the classic MaxEnt is subject to stronger ﬂuctuations. In physical terms, the decay in magnetic susceptibility is because of the destruction of the Kondo effect due tothe decoherence introduced by the bias voltage. This is inprinciple similar to the equilibrium behavior found as afunction of temperature. 53The scale on which the decay of the magnetization takes place appears to be alreadyvisible within the imaginary-voltage data shown in Fig. 5(b). Apparently, this is due to the rather weak voltage dependenceof imaginary-voltage data [cf. Fig. 4(a)]. V oltages above 10k BTKwere not accessible to the MaxEnt, due to a strong sign problem occurring for the QMC simulations of theeffective-equilibrium systems associated with the high- ϕm tails. We again may compare the voltage dependence of the stationary nonequilibrium magnetization to the temperaturedependence in equilibrium. Since we here are at a ﬁnitetemperature T=T K/2, hence the magnetization is smaller than the value at T=0, the natural thing to look at is the curve M(T)·[M(TK/2)/M(0)] and rescale temperature with an appropriate factor. The result is shown as a dot-dashed linein Fig. 8. Although one can reach a reasonable match for low voltages, a signiﬁcant deviation occurs already at moderatebias. Thus there does not seem to exist a simple mapping/Phi1→Twhich will bring the curves to overlap (i.e., it again seems doubtful that one can describe the effect of ﬁnite biasvoltage by an effective temperature scale, at least beyond smallbias voltages of the order of the Kondo scale). On the other hand, a rather good account for all data can be achieved by the very simple ansatz, m(/Phi1) B≈a B1 ˜/Phi12√ b2+˜/Phi12+c, where ˜/Phi1:=/Phi1/(2TK). The result of this ﬁt with a=0.52, b≈2, and c≈3 is shown as the double-dot-dash curve in Fig. 8. Note that this formula gives the right behavior in the two limits /Phi1→0, vizM/B∝1−c˜/Phi12with some numerical constant c, and /Phi1→∞ ,v i z M/B∝1//Phi1. From scaling analysis12one would expect that, in particular, for large bias, additional logarithmic corrections appear. Due to the limiteddata space available we are of course not able to resolve those;furthermore, it is not clear if these logarithmic corrections willactually be visible in the intermediate coupling regime studiedhere, due to residual charge ﬂuctuations. We therefore view theabove formula as a reasonable description in the regime of bias,temperature, and ﬁeld of the order of the Kondo temperaturefor the intermediate coupling regime of the SIAM. V . SUMMARY The present paper presents a detailed study on how the imaginary-voltage formalism proposed in Ref. 35relates to Keldysh theory. Using series resummations, we are able toshow up to all orders that static expectation values of observ-ables, which satisfy certain symmetry relations with respectto the Keldysh contour, map exactly onto the correspondingexpressions in Keldysh perturbation theory. In particular, itwas pointed out that in order to obtain a physical expectationvalue, the limiting process iϕ m→/Phi1has to be taken as principal value. This prescription ensures, that one generatesthe principal-value integrals which emerge in the properreal-time theory. For dynamical correlation functions, this wasshown explicitly up to fourth order of perturbation theory. As one important novel result of the present paper we were able to provide an exact spectral representation forstatic expectation values similar to a Lehmann representation.Based on the representation, using unbiased numerical datafrom continuous-time quantum Monte Carlo simulations,we found that the evaluation of the limiting procedure asprincipal-value expression does indeed give real numbers asphysical expectation values. Consequently, the theory is found 155130-15JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) to be fully consistent in this respect beyond the perturbation arguments given. The double occupancy as a function ofbias voltage computed this way shows features similar tostraightforward second-order perturbation theory, but we ﬁndthem to be more pronounced. For the magnetic susceptibilitywe were able to give numerical estimates on the destructionof the Kondo effect. A comparison to equilibrium NRGshows that the dependence on bias voltage for both, thedouble occupancy and the magnetic susceptibility, cannot beexplained by a simple effective-temperature interpretation. ACKNOWLEDGMENTS We acknowledge valuable discussions with M. Jarrell, J. Freericks, F. B. Anders, S. Schmitt, K. Sch ¨onhammer, and A. Schiller. A.D. acknowledges ﬁnancial support from theGerman Academic Exchance Service (DAAD) through thePPP exchange program, and J.H. acknowledges the NationalScience Foundation (Grant No. DMR-0907150), T.P. theGerman Science Foundation through Grant No. SFB 602. A.D.and T.P. also acknowledges computer support from the Nord-deutscher Verbund f ¨ur Hoch- und H ¨ochstleistungsrechnen (HLRN), the Gesellschaft f ¨ur wissenschaftliche Datenverar- beitung G ¨ottingen (GWDG), and the GOEGRID initiative of the University of G ¨ottingen. Parts of the implementation are based on the ALPS 1 .3library.58 APPENDIX A: CANCELLATION OF OVERLAPPING δFUNCTIONS IN EQ. (52) With a set of {ψ† αikiσi,ψαikiσi;i=1,..., 6}appearing for the matrix elements in Eq. (52), we categorize the thermal factor eβ/Phi1Y 0{n,m,l}as follows. (i) If Y0n=Y0m=Y0l,E q . (52) vanishes. (ii) If only one of Y0n,Y0m,Y0lis different from others, (Y0n,Y0m,Y0l)∈{(Y0,Y0,Y0+1),(Y0+1,Y0,Y0),(Y0,Y0+1,Y0), (Y0,Y0,Y0+2),(Y0+2,Y0,Y0),(Y0,Y0+2,Y0)}for some reference value Y0. If we take the case of (Y0n,Y0m,Y0l)=(Y0,Y0,Y0+1), the terms contributing for the matrix elements Vnm,Vml, and Alnare from ψ† ˜α1˜k1ψ† ˜α2˜k2ψ˜α3˜k3ψ˜α4˜k4,ψ† Rk1ψLk2ψ† ˜α5˜k5ψ˜α6˜k6, and ψ† Lk2ψRk1ψ† ˜α7˜k7ψ˜α8˜k8, respectively, where ( ˜k1,..., ˜k8)i s some permutation of ( k3,k3,k4,k4,..., k 6,k6). The reservoir indices should be chosen such that ˜ α5=˜α6and ˜α7=˜α8, and (˜α1,˜α2,˜α3,˜α4) should satisfy Y0n=Y0m.T h e ˜ αiindices are summed over for L/R . Then the term in Eq. (52) becomes proportional to (tLtR)2/parenleftbig t2 L+t2 R/parenrightbig4/productdisplay i=1,6\|g(ki)\|2eβ/Phi1Y 0(1−2+eβ/Phi1). For other combinations of ( Y0n,Y0m,Y0l)=(Y0,Y0+ 1,Y0),(Y0+1,Y0,Y0) the thermal factor becomes (1−2eβ/Phi1+1) and ( eβ/Phi1−2+1), respectively, and all three contributions sum up to zero. With thecase of ( Y 0,Y0,Y0+2), the contribution becomes (tLtR)4(t2 L+t2 R)2/producttext i=1,6\|g(ki)\|2eβ/Phi1Y 0(1−2+e2β/Phi1). The other terms have factors of (1 −2e2β/Phi1+1),(e2β/Phi1−2+1), and these sum up to zero again. (iii) When all of Y0n,Y0m,Y0lare different, ( Y0n,Y0m,Y0l) is a permutation of ( Y0,Y0+1,Y0+2). Since ˆV,ˆAare at most two-particle operators the difference of Yvalues betweenstates cannot be greater than two. If ( Y0n,Y0m,Y0l)=(Y0,Y0+ 1,Y0+2), the factor in Eq. (52) becomes proportional to (tLtR)4/parenleftbig t2 L+t2 R/parenrightbig2/productdisplay i=1,6\|g(ki)\|2eβ/Phi1Y 0(1−2eβ/Phi1+e2β/Phi1). Permuting ( Y0,Y0+1,Y0+2) the sum of the thermal factors can easily be shown to be zero. APPENDIX B: FOURTH-ORDER EXPANSION OF ELECTRON SELF-ENERGY We investigate the energy-pole structure in the real-time perturbation expansion to verify that the δ-function residue disappears and the energy denominators can be interpreted asprincipal valued. In the following we consider the perturbationexpansion for the self-energy in the fourth order of theCoulomb parameter U,/Sigma1 > (4)(t,0) according to the time or- derings along the Keldysh contour [Figs. 9(a)–9(d)]. Different types of time orderings will be considered shortly. These timeorderings have one of the intermediate times (marked as across) within a ﬁnite time interval ﬁxed by time at 0 and t. Given a time ordering, a particular Wick’s contraction shouldbe chosen. The chosen Wick’s contraction is according to thediagrams in (g) and (h) which correspond to the most nontrivialvertex correction. (a) (b) (e)(d)t0 0t, τ s1 s2123 4 567 (g)(c) 0 τs1 s2 β(f)s2s1 s1s2 s2 s1s2 s1 s1 s2 5 1 s2s1 7 326 4 (h)t, τ FIG. 9. (a)–(d) Real-time Keldysh contour for self-energy /Sigma1> (4)(t,0) in the fourth-order perturbation when one intermediate time is in the ﬁnite interval [0 ,t] and the other time in along the contour stretching to −∞. The Wick’s contraction is taken as shown in (g) and (h). The dummy label of (g) is used for time orderings (a), (c),(e) and the label (h) is used for (b) (d), (f). The cross represents the intermediate times s 1ands2for interaction, in addition to the creation/annihilation points 0 and t. 155130-16IMAGINARY-TIME QUANTUM MANY-BODY THEORY OUT ... PHYSICAL REVIEW B 86, 155130 (2012) We can evaluate each contribution as follows. Sa=f1f2¯f3¯f4¯f5f6¯f7/integraldisplay0 −∞ds1/integraldisplayt 0ds2e−i(/epsilon11−/epsilon14−/epsilon15+/epsilon16−iη)s1−i(−/epsilon12+/epsilon13+/epsilon14−/epsilon17)s2−i(/epsilon15−/epsilon16+/epsilon17)t, (B1) Sb=¯f1f2¯f3f4f5¯f6¯f7/integraldisplay0 −∞ds2/integraldisplayt 0ds1e−i(/epsilon12−/epsilon13−/epsilon14+/epsilon17)s1−i(−/epsilon11+/epsilon14+/epsilon15−/epsilon16−iη)s2−i(/epsilon11−/epsilon12+/epsilon13)t, (B2) Sc=¯f1f2¯f3f4f5¯f6¯f7/integraldisplay−∞ tds1/integraldisplayt 0ds2e−i(/epsilon11−/epsilon14−/epsilon15+/epsilon16−iη)s1−i(−/epsilon12+/epsilon13+/epsilon14−/epsilon17)s2−i(/epsilon15−/epsilon16+/epsilon17)t, (B3) Sd=f1f2¯f3¯f4¯f5f6¯f7/integraldisplay−∞ tds2/integraldisplayt 0ds1e−i(/epsilon12−/epsilon13−/epsilon14+/epsilon17)s1−i(−/epsilon11+/epsilon14+/epsilon15−/epsilon16−iη)s2−i(/epsilon11−/epsilon12+/epsilon13)t. (B4) In these shorthand notations (as discussed in the main text), we omitted the expression U4[/producttext i/integraltext d/epsilon1i\|gd(/epsilon1i)\|2] which is common to allSiterms. fi=[1+eβ(/epsilon1i−αi/Phi1/2)]−1and ¯fi=1−fi. After some algebra, we get Sa+Sd=−2f1f2¯f3¯f4¯f5f6¯f7 (−/epsilon12+/epsilon13+/epsilon14−/epsilon17)(/epsilon11−/epsilon14−/epsilon15+/epsilon16)[e−i(−/epsilon12+/epsilon13+/epsilon14+/epsilon15−/epsilon16)t−e−i(/epsilon15−/epsilon16+/epsilon17)t]. (B5) The exponential terms cancel each other at the energy poles and (/epsilon12−/epsilon13−/epsilon14+/epsilon17)−1and (/epsilon11−/epsilon14−/epsilon15+/epsilon16)−1give a well-deﬁned principal-valued integral. This is typical behaviorsince an integral within a ﬁnite interval (0 ,t) does not need the convergence factor e ηtand, accordingly, principal-valued integral is enough. The same can be said for the combination Sb+Sc. Now, we take the imaginary-time contours in Figs. 9(e) and9(f). After straightforward calculations, we have (˜ /epsilon1i= /epsilon1i−αi/epsilon1ϕ/2), Se=f1f2¯f3¯f4¯f5f6¯f7 ×e−(−˜/epsilon12+˜/epsilon13+˜/epsilon14+˜/epsilon15−˜/epsilon16)τ−e−(˜/epsilon15−˜/epsilon16+˜/epsilon17)τ (˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16)(−˜/epsilon12+˜/epsilon13+˜/epsilon14−˜/epsilon17)(B6) −¯f1f2¯f3f4f5¯f6¯f7 ×e−(˜/epsilon11−˜/epsilon12+˜/epsilon13)τ−e−(˜/epsilon11−˜/epsilon14+˜/epsilon17)τ (˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16)(−˜/epsilon12+˜/epsilon13+˜/epsilon14−˜/epsilon17).(B7) Here (B6) corresponds to Saof(B1) and(B7) toScof(B3). Similarly for Sf, Sf=¯f1f2¯f3f4f5¯f6¯f7 ×e−(˜/epsilon11−˜/epsilon14+˜/epsilon17)τ−e−(˜/epsilon11−˜/epsilon12+˜/epsilon13)τ (˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16)(−˜/epsilon12+˜/epsilon13+˜/epsilon14−˜/epsilon17)(B8) −f1f2¯f3¯f4¯f5f6¯f7 ×e−(˜/epsilon15−˜/epsilon16+˜/epsilon17)τ−e−(−˜/epsilon12+˜/epsilon13+˜/epsilon14+˜/epsilon15−˜/epsilon16)τ (˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16)(−˜/epsilon12+˜/epsilon13+˜/epsilon14−˜/epsilon17).(B9)At the energy poles for /epsilon1ϕ→iη,Sfbecomes identical to Se. Similarly to the real-time diagrams, ( −˜/epsilon12+˜/epsilon13+˜/epsilon14−˜/epsilon17)−1 has a well-deﬁned principal-value integral regardless of the sign of η. Therefore for diagrams Sa−Sfwe have correct analytic continuation of imaginary-time results to those of thereal time via 1 ˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16→P/parenleftbigg1 /epsilon11−/epsilon14−/epsilon15+/epsilon16/parenrightbigg .(B10) In Fig. 10, we consider the remaining time orderings with the two intermediate interaction points extending to inﬁnity.These are harder to deal with, as we discuss below, since the s1 s2 s2 s1 (a) s2 s1(c)(b) (d) s1 s2 (e) s1s2 (f) s2s1 s2 s1(h) (g)s2 s1 FIG. 10. Different time ordering with two intermediate interac- tion events extend to inﬁnity. (a), (d), (e), (g) use the label in Fig. 1(g) and (b), (c), (f), (h) use the label in Fig. 1(h). 155130-17JONG E. HAN, ANDREAS DIRKS, AND THOMAS PRUSCHKE PHYSICAL REVIEW B 86, 155130 (2012) energy poles may overlap. Da=f1¯f2f3¯f4¯f5f6¯f70/integraldisplay/integraldisplay −∞ds1ds2e−i(/epsilon11−/epsilon14−/epsilon15+/epsilon16−iη)s1−i(/epsilon11−/epsilon12+/epsilon13−/epsilon15+/epsilon16−/epsilon17−iη)s2−i(/epsilon15−/epsilon16+/epsilon17)t, (B11) Db=− ¯f1f2¯f3f4f5¯f6f7/integraldisplay e−i(−/epsilon11+/epsilon12−/epsilon13+/epsilon15−/epsilon16+/epsilon17−iη)s1−i(−/epsilon11+/epsilon14+/epsilon15−/epsilon16−iη)s2−i(/epsilon11−/epsilon12+/epsilon13)t, (B12) Dc=f1¯f2f3¯f4¯f5f6¯f7/integraldisplay e−i(−/epsilon11+/epsilon12−/epsilon13+/epsilon15−/epsilon16+/epsilon17−iη)s1−i(−/epsilon11+/epsilon14+/epsilon15−/epsilon16−iη)s2−i(/epsilon15−/epsilon16+/epsilon17)t, (B13) Dd=− ¯f1f2¯f3f4f5¯f6f7/integraldisplay e−i(/epsilon11−/epsilon14−/epsilon15+/epsilon16−iη)s1−i(/epsilon11−/epsilon12+/epsilon13−/epsilon15+/epsilon16−/epsilon17−iη)s2−i(/epsilon11−/epsilon12+/epsilon13)t, (B14) De=− ¯f1¯f2f3f4f5¯f6¯f7/integraldisplay e−i(/epsilon11−/epsilon14−/epsilon15+/epsilon16−iη)s1−i(−/epsilon12+/epsilon13+/epsilon14−/epsilon17−iη)s2−i(/epsilon11−/epsilon14+/epsilon17)t, (B15) Df=f1f2¯f3¯f4¯f5f6f7/integraldisplay e−i(/epsilon12−/epsilon13−/epsilon14+/epsilon17−iη)s1−i(−/epsilon11+/epsilon14+/epsilon15−/epsilon16−iη)s2−i(−/epsilon12+/epsilon13+/epsilon14+/epsilon15−/epsilon16)t. (B16) After integrals over s1ands2it is easy to see that Da(iη)=Dc(−iη) andDb(iη)=Dd(−iη). ForDeandDf, we can swap the dummy indices as 1 ↔7, 2↔6, and 3 ↔5, and it becomes De(iη)=De(−iη) andDf(iη)=Df(−iη). Therefore, we obtain the desired result as (B10) , /summationdisplay k=a,···,fDk(iη)=/summationdisplay kDk(−iη)=/summationdisplay kPDk(±iη). (B17) In deriving these relations, no assumptions of L/R and particle-hole symmetry have been used. One can rewrite Daas Da=f1¯f2f3¯f4¯f5f6¯f7e−i(/epsilon15−/epsilon16+/epsilon17)t /epsilon12−/epsilon13−/epsilon14+/epsilon17/bracketleftbigg1 /epsilon11−/epsilon12+/epsilon13−/epsilon15+/epsilon16−/epsilon17−iη−1 /epsilon11−/epsilon14−/epsilon15+/epsilon16−iη/bracketrightbigg . (B18) Here the +iηin the denominator will be canceled by Dcand all fractions can be written as principal valued, unless the poles coincide. We can now turn to the imaginary-time diagrams Figs. 10(g) and10(h) . Dg=f1¯f2f3¯f4¯f5f6¯f7 −(˜/epsilon12−˜/epsilon13−˜/epsilon14+˜/epsilon17)/parenleftbigg −1 ˜/epsilon11−˜/epsilon12+˜/epsilon13−˜/epsilon15+˜/epsilon16−˜/epsilon17+1 ˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16/parenrightbigg e−(˜/epsilon15−˜/epsilon16+˜/epsilon17)τ −¯f1f2¯f3¯f4f5¯f6f7 (˜/epsilon12−˜/epsilon13−˜/epsilon14+˜/epsilon17)e−(˜/epsilon11−˜/epsilon12+˜/epsilon13)τ (˜/epsilon11−˜/epsilon12+˜/epsilon13−˜/epsilon15+˜/epsilon16−˜/epsilon17)+¯f1¯f2f3f4f5¯f6¯f7 (˜/epsilon12−˜/epsilon13−˜/epsilon14+˜/epsilon17)e−(˜/epsilon11−˜/epsilon14+˜/epsilon17)τ (˜/epsilon11−˜/epsilon14−˜/epsilon15+˜/epsilon16). (B19) After swapping 1 ↔7, 2↔6, and 3 ↔5, the ﬁrst two terms correspond to DaandDcfor/epsilon1ϕ→iηand the third term to De. Using a similar technique in (B18) , we can decouple the product of energy denominators to a sum of simple poles of /epsilon1ϕand then by taking the limit Eq. (B10) , all energy denominators become principal valued, unless poles coincide. Now we deal with the case when the δfunctions overlap. As discussed in Sec. III A , the double- δterms manifest as terms proportional to T2.T h et e r m s Da,Dc, andDehave double- δterms canceled among themselves. At the energy poles /epsilon11−/epsilon14−/epsilon15+/epsilon16=0 and/epsilon12−/epsilon13−/epsilon14+/epsilon17=0, Da=Dc∝f1¯f2f3¯f4¯f5f6¯f7T2 2e−i(/epsilon15−/epsilon16+/epsilon17)t. (B20) ForDe,w eﬁ r s tr e w r i t e /integraldisplayT tds1=/integraldisplayT 0ds1+/integraldisplay0 tds1, (B21) and note that the second integral with a ﬁnite interval should not contribute a δfunction. So as long as double δis concerned, we only consider the ﬁrst interval, De∝− ¯f1¯f2f3f4f5¯f6¯f7T2e−i(/epsilon15−/epsilon16+/epsilon17)t→−f1¯f2f3¯f4¯f5f6¯f7T2e−i(/epsilon15−/epsilon16+/epsilon17)t, (B22) where at the last step the dummy indices are swapped as 1 ↔5 and 4 ↔6. Therefore, the double- δterms disappear in Da+Dc+De. 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PhysRevB.103.165139.pdf	PHYSICAL REVIEW B 103, 165139 (2021) Interacting spin-3 2fermions in a Luttinger semimetal: Competing phases and their selection in the global phase diagram András L. Szabó ,1Roderich Moessner,1and Bitan Roy1,2,* 1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany 2Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA (Received 22 January 2019; revised 22 November 2020; accepted 13 April 2021; published 27 April 2021) We compute the effects of electronic interactions on gapless spin-3 /2 excitations that in a noninteracting system emerge at a biquadratic touching of Kramers degenerate valence and conduction bands in three dimen-sions, also known as a Luttinger semimetal. This model can describe the low-energy physics of HgTe, gray-Sn,227 pyrochlore iridates, and half-Heuslers. For the sake of concreteness, we only consider the short-rangecomponents of the Coulomb interaction (extended Hubbard-like model). By combining mean-ﬁeld analysis witha renormalization group (RG) calculation (controlled by a “small” parameter /epsilon1,w h e r e /epsilon1=d−2), we construct multiple cuts of the global phase diagram of interacting spin-3 /2 fermions at zero and ﬁnite temperature and chemical doping. Such phase diagrams display a rich conﬂuence of competing orders, among which rotationalsymmetry breaking nematic insulators and time-reversal symmetry breaking magnetic orders (supporting Weylquasiparticles) are the prominent candidates for excitonic phases. We also show that even repulsive interactionscan be conducive to both mundane s-wave and topological d-wave pairings. The reconstructed band structure (within the mean-ﬁeld approximation) inside the ordered phases allows us to organize them according to theenergy (entropy) gain in the following (reverse) order: s-wave pairing, nematic phases, magnetic orders, and d-wave pairings, at zero chemical doping. However, the paired states are energetically superior over the excitonic ones for ﬁnite doping. The phase diagrams obtained from the RG analysis show that for sufﬁciently stronginteractions, an ordered phase with higher energy (entropy) gain is realized at low (high) temperature. In addition,we establish a “selection rule” between the interaction channels and the resulting ordered phases, suggestingthat repulsive short-range interactions in the magnetic (nematic) channels are conducive to the nucleation ofd-wave ( s-wave) pairing among spin-3 /2 fermions. We believe that the proposed methodology can shed light on the global phase diagram of various two and three-dimensional interacting multiband systems, such as Diracmaterials, doped topological insulators, and the like. DOI: 10.1103/PhysRevB.103.165139 I. INTRODUCTION The discovery of new phases of matter, and the study of the transitions between them, form the core of condensedmatter and materials physics [ 1]. Much experimental inge- nuity is devoted to realising and controlling different tuningparameters in the laboratory–such as temperature, pressure,magnetic ﬁeld, chemical composition—to drive a system fromone phase to the other [ 1]. Most simply, the appearance of different phases can often be appreciated from the competitionbetween energy and entropy. For example, with decreasingtemperature transitions from water vapor to liquid to ice areintimately tied with the reduction of entropy or gain in en-ergy. Similarly, in a metal, entropy of gapless excitations onthe Fermi surface is exchanged for condensation energy as asuperconducting gap opens [ 2]. *Corresponding author: bitan.roy@lehigh.edu Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Openaccess publication funded by the Max Planck Society.With increasing complexity of quantum materials attained in recent decades, the richness of the global phase diagramof strongly correlated materials has ampliﬁed enormously.And various prototypical representatives, such as cuprates,pnictides, heavy fermion compounds display concurrence ofcompeting orders, among which spin- and charge-densitywave, superconductivity, nematicity are the most prominentones. Besides establishing the existence of—and, hopefully,eventually utilizing in technological applications—these phe-nomena, an obvious challenge is to discover any simplifying perspective, or at least heuristic classiﬁcation scheme, for predicting and classifying them: do there exist any organisingprinciples among multiple competing orders that can shedlight on the global phase diagram of strongly correlated ma-terials? Restricting ourselves to a speciﬁc but remarkably richmetallic system, we here give a partially afﬁrmative answerto this question. We study a collection of strongly interactingspin-3 /2 fermions in three dimensions that in the normal phase display a biquadratic touching of Kramers degeneratevalence and conduction bands at an isolated point in the Bril-louin zone, see Fig. 1. This system is also known as Luttinger semimetal [ 3,4]. Such peculiar quasiparticle excitations can be found for example in HgTe [ 5], gray-Sn [ 6,7], 227 pyrochlore iridates 2469-9950/2021/103(16)/165139(32) 165139-1 Published by the American Physical SocietySZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) FIG. 1. A schematic phase diagram of noninteracting Luttinger fermions. The red dot represents the Luttinger semimetal ﬁxed point separating electron- and hole-doped Fermi liquid of spin-3 /2 fermions. The shaded sector corresponds to the quantum critical regime associated with the z=2 ﬁxed point in d=3 (the red dot), controlling scaling properties in this regime (see Sec. II A). Above a nonuniversal high energy cutoff E/Lambda1(red dashed line), spin-3 /2 fermions lose their jurisdiction. The crossover temperature T∗(black dashed lines) above which critical Luttinger fermions are operativeis given by T ∗∼\|n\|z/d, with z=2a n d d=3, where nis the carrier density measured from the biquadratic band touching point. In this work we demonstrate the role of electron-electron interactions on thequantum critical regime (the shaded region), at ﬁnite temperature and chemical doping (see Sec. II Bfor summary of our results). (Ln 2Ir2O7, where Ln is the lanthanide element) [ 8–12] and half-Heusler compounds (such as LnPtBi, LnPdBi) [ 13–15]. Biquadratic band touching has recently been observed in thenormal state of Pr 2Ir2O7[10] and Nd 2Ir2O7[11] via an- gle resolved photo emission spectroscopy (ARPES). Whilemost of the iridium-based oxides support all-in all-out ar-rangement of 4 dIr electrons [ 16–24], a singular member of this family, namely Pr 2Ir2O7, possibly resides at the brink of a metallic spin-ice or 3-in 1-out ordering and supportsa large anomalous Hall conductivity [ 25–29]. While these materials harbor various competing magnetic orders (due tocomparable Hubbard and spin-orbit interactions), smallnessof the Fermi surface prohibits the onset of superconductivityat the lowest achievable temperature. On the other hand, half-Heusler compounds accommodate both antiferromagentismand unconventional superconductivity [ 30–39]. A transition between them can be triggered by tuning the de Gennes factor[37]. Moreover, the superconducting YPtBi (with transition temperature T c=0.78K) supports gapless BdG quasiparti- cles at low temperatures [ 38]. Therefore, besides its genuine fundamental importance, our quest focuses on a timely issuedue to growing material relevance of interacting spin-3 /2 fermions, which has triggered a recent surge of theoreticalworks [ 17–22,28,29,40–61].FIG. 2. Hierarchy of various dominant orders in an interacting Luttinger semimetal according to the gain of the condensation energy (/Delta1F) and entropy ( /Delta1S) inside the ordered phases. The condensation energy (entropy) gain increases in the direction of the /Delta1F(/Delta1S)a r - row. If an order-parameter matrix ( M) anticommutes (commutes) with NM anti(NM com) number of matrices appearing in the Luttinger model [see Eq. ( 1)], then /Delta1F∼NM antiand/Delta1S∼NM com(qualitatively), see Sec. IV D for detailed discussion. At ﬁnite chemical doping superconducting orders are always energetically superior over the excitonic ones. In the global phase diagram, phases with higher gainin condensation energy (entropy) appear at low (high) temperature, see, for example, Figs. 3–5. We also display the scaling of density of states [ /rho1(E)] at low-energies in the presence of both point and line nodes. Notice that with increasing N M anti(NM com) the stiffness of the spectral gap (amount of gapless quasiparticles), determining the scaling of the DoS at low-energy, increases. The effects of electronic interactions on spin-3 /2 fermions are addressed within the framework of an extended Hubbard-like model, composed of only the short-range components ofrepulsive Coulomb interactions. 1Here we aim to construct a minimal interacting model that captures the competi-tion among various experimentally observed ordered states,e.g. magnetic and superconducting, in different compounds,where the normal state quasiparticles are described by three-dimensional spin-3 /2 Luttinger fermions at low energies. We show that an extended Hubbard model, composed of all symmetry allowed momentum-independent four-fermion in-teractions serves this purpose, at least qualitatively. Due to the vanishing density of states (DoS) in a Lut- tinger system [namely, /rho1(E)∼√ E], sufﬁciently weak short- range interactions are irrelevant perturbations. Therefore any 1In this work, we neglect the long-range tail of the Coulomb inter- action. When the chemical potential is pinned at the band touching point, long-range Coulomb interaction can give rise to an infraredstable non-Fermi liquid ﬁxed point [ 40,62,63], which may however be unstable toward the formation of an excitonic nematic phase [41,43,64]. By contrast, at ﬁnite temperature due to the thermal screening, yielding ﬁnite mass to photon, it becomes short-ranged in nature [ 65]. Otherwise, purely long-range interaction in doped Lut- tinger systems yields s-wave and p-wave pairings [ 66,67]. However, these phases have not been observed in any Luttinger material so far. 165139-2INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) FIG. 3. Various cuts of the ﬁnite temperature ( t) phase diagram of an interacting isotropic Luttinger semimetal. The interaction couplings (gi) are measured in units of /epsilon1( a l s oi nF i g s . 4–7and16). However, nature of the order states in all cuts of the phase diagram do not depend on the value of /epsilon1>0. (a) and (b) respectively depict onset of T2gandEgnematic orders at higher temperatures as we tune interactions in these two channels. Recall while the s-wave superconductor yields fully gapped spectra, both nematic phases produce anisotropic gaps [see Sec. IV C ]. Hence, the energy-entropy competition [see Sec. IV D ]f a v o r s s-wave pairing (nematic phases) at low (high) temperature. When we tune the strength of (c) A2uand (d) T1umagnetic interactions, EgandT2gnematic orders set in at low temperature, respectively, and corresponding magnetic orders nucleate only at higher temperature, since both magnetic orders produce gapless Weyl fermions (less gain of condensation energy, but higher entropy). See Sec. VI C for details of the renormalization group analysis at ﬁnite t. The white regions represent Luttinger semimetal without any ordering. The gray shaded region to the ordered phase (see footnote 2), and its boundaries with Luttinger semimetal, occupied by various ordered phases are shown in different colors. ordering takes place at ﬁnite coupling. We here employ a renormalization group (RG) analysis to construct various cutsof the global phase diagram at zero as well as ﬁnite chemicaldoping (see Figs. 3–5), and combine it with mean-ﬁeld analy- sis to gain insight into the organizing principle among distinctbroken-symmetry phases (BSPs). 2A gist of our ﬁndings can be summarized as follows. 2All cuts of the global phase diagram are obtained from a RG analysis, accounting for interaction effects on Luttinger fermions.Such an RG analysis can only predict the phase boundaries between the Luttinger semimetal and various BSPs in an unbiased fashion. However, our RG analysis is not applicable deep inside a BSP(depicted as gray shaded regions in Figs. 3–7and16) and cannot capture order-order transitions. When a cut of the phase diagram accommodates two competing phases, as in Fig. 3–6,a n d 16, inside the ordered phase typically there exists a regime of coexistence (as the corresponding order parameters never fully commute with each(1) By computing the reconstructed band structure (within the mean-ﬁeld approximation) we organize dominant BSPsaccording to the condensation energy and entropy gains. Re-sults are summarized in Fig. 2. We note that while the stiffness (uniform or anisotropic) of the band gap measures the con-densation energy gain, the amount of gapless quasiparticles(resulting in power-law scaling of DoS at low energies) mea-sures the entropy inside the ordered phase. All cuts of theglobal phase diagram (obtained from a RG analysis) show thatthe low (high) temperature phase yields larger condensationenergy (entropy) gain, see Fig. 3. (2) The quasiparticle spectra inside the s-wave paired state and two nematic phases (belonging to the T 2gandEgrepre- sentations of the cubic or Ohpoint group) are fully gapped. other, see Sec. II B3 and Sec. VI E), with two pure phases on either side of it. Such coexistence can be captured by standard mean-ﬁeldanalysis, which goes beyond the scope of the present discussion. 165139-3SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) FIG. 4. Realization of s-wave pairing from repulsive electronic interactions in the (a) T2gand (b) Egnematic channels, for ﬁnite μ. In the presence of chemical doping the s-wave pairing occupies a larger portion of the phase diagrams and the threshold couplings for the onset of two nematic orders get pushed toward stronger couplings [compare with Figs. 3(a)and3(b)]. Hence, nematic interactions favor the onset of s-wave pairing in an isotropic Luttinger system ( α=π 4). Here coupling constants are measured in units of /epsilon1. The gray shaded regions represent ordered states (see footnote 2), and its boundarieswith Luttinger metal, occupied by various ordered phases are shown in different colors. Hence, nucleation of these three phases leads to the maximal gain of condensation energy, and they appear as the dominantBSPs at zero temperature, as shown in Fig. 3. (3) At ﬁnite temperature condensation energy gain com- petes with the entropy, and phases with higher entropy arerealized at higher temperatures. Onset of any nematicity re-sults in an anisotropic gap, in contrast to the situation withans-wave pairing. Thus former orderings are endowed with larger (smaller) entropy (condensation energy), and are foundat higher temperatures in Figs. 3(a)and3(b). By contrast, the dominant magnetic orders (belonging to the A 2uandT1urepre- sentations) produce gapless Weyl quasiparticles and result in/rho1(E)∼\|E\| 2scaling of the DoS at low energies. Hence, these two magnetic orders carry larger entropy than the nematicphases or the s-wave pairing, and can only be found at ﬁ- nite temperature, see Figs. 3(c)and3(d). Luttinger semimetal (LSM)- A 2umagnetic order (results from the all-in all-out state in pyrochlore lattice [ 8,20]) transition at ﬁnite temperature is consistent with the experimental observation in Nd 2Ir2O7 [16], while the T1umagnetic ordering (yielding a 3-in 1-out ordering in pyrochlore lattice and supporting anomalous Halleffect [ 28]) can be germane for Pr 2Ir2O7[25–27].3 (4) Local four fermion interactions in the nematic channels are conducive for s-wave pairing (at zero and ﬁnite chemi- cal doping), see Fig. 3(a),3(b), and 4, whereas short-range magnetic interactions give birth to topological d-wave pairing (only at ﬁnite chemical doping), see Fig. 5. We further elabo- rate such an emergent “selection rule” in Sec. II B. Conﬂuence of magnetic order and d-wave pairing (resulting in gapless BdG quasiparticles, found in YPtBi [ 38]) is in (qualitative) agreement with the global phase diagram of LnPdBi [ 37]. The theoretical approach outlined in this work is quite general and can be extended to address the effects of elec- 3ARPES measurements in Nd 2Ir2O7[11]a n dP r 2Ir2O7[10] suggest that the LSM in these compound is isotropic.FIG. 5. Top row: Onset of d-wave pairing (belonging to the Eg representation) at low temperature from magnetic interaction in the A2uchannel, namely, g3, in the presence of ﬁnite chemical doping μ[cf. Fig. 3(c) forμ=0]. The A2umagnetic order gets pushed toward stronger coupling with increasing μ. (Bottom row) Similar phase diagrams depicting the onset of d-wave pairing (belonging to theT2grepresentation) from strong repulsive magnetic interaction in theT1uchannel, when μ> 0 [compare it with Fig. 3(d) forμ=0]. With increasing chemical potential the onset of the T1umagnetic order takes place at stronger coupling. Hence, magnetic interactions are conducive for the nucleation of d-wave pairings in correlated Luttinger metal. Here coupling constants are measured in units of /epsilon1and the gray shaded regions represent ordered states (see footnote 2). Note that the shape of the phase boundaries in the lower panelsuggests that the LSM-ordered phase transition at low temperatures and ﬁnite chemical doping is possibly ﬁrst order in nature. Even though the RG methodology is tailored to capture continuous tran-sitions, the possibility of a ﬁrst-order transition is extremely sparse. The boundaries between Luttinger metal and various ordered phases are shown in different colors. tronic interactions in various strongly correlated multiband systems, among which two-dimensional Dirac and quadraticfermions (respectively relevant for monolayer and bilayergraphene) [ 68,69], three-dimensional doped topological, crys- talline and Kondo insulators [ 70–72], Weyl materials [ 73], twisted bilayer graphene [ 74–76], are the most prominent and experimentally pertinent ones. The proposed organiza-tion principle (Sec. II B 1 ) and the selection rule (Sec. II B 3 ) among competing orders are consistent with recently reportedconﬂuence between (1) charge-density-wave and f-wave su- perconductor [ 77] and (2) quantum spin Hall insulator and s-wave superconductor [ 78] in doped monolayer graphene, re- spectively demonstrated from nonperturbative functional RGand quantum Monte Carlo simulations. We discuss these casesin Appendix A. In the future, we will systematically study these systems in details. Even though competing orders incorrelated insulators and standard Fermi liquids (such as theone realized in a square lattice system) have been investigatedin terms of the intertwinded orders [ 79,80], the organizing 165139-4INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) principle (see Sec. II B 1 ) and selection rules (see Sec. II B 3 ) for gapless fermionic topological multiband systems with orwithout strong spin-orbit coupling remains vastly unexploredso far. Outline The rest of the paper is organized as follows. In the next section we present an extended summary of our main results.The low-energy description of the Luttinger model and itssymmetry properties are discussed in Sec. III. In Sec. IV, we discuss the reconstructed band structure inside variousexcitonic and superconducting phases. In Sec. V,w ei n t r o - duce the interacting model for spin-3 /2 fermions and analyze the propensity toward the formation of various orderingswithin a mean-ﬁeld approximation. Section VIis devoted to a renormalization group (RG) analysis of interacting Luttingerfermions at zero and ﬁnite temperatures and chemical doping.We summarize the main results and highlight some futureoutlooks in Sec. VII. Additional technical details are relegated to appendices. II. EXTENDED SUMMARY Our starting point is a collection of spin-3 /2 fermions for which the normal state is described by a biquadractic touchingof Kramers degenerate valence and conduction bands. Thecorresponding Hamiltonian operator is [ 3,4] ˆh L(k)=−k2 2m/parenleftBigg cosα3/summationdisplay j=1ˆdj/Gamma1j+sinα5/summationdisplay j=4ˆdj/Gamma1j/parenrightBigg −μ, (1) where ˆdjsa r eﬁ v e d-wave harmonics in three dimensions, /Gamma1js are ﬁve mutually anticommuting four-dimensional Hermitianmatrices, and m/[cosα(sinα)] is the effective mass for gap- less excitations in the T 2g(Eg) orbitals in a cubic environment. Momentum kand chemical potential μare measured from the band touching point. Additional details of this model arediscussed in Sec. IIIand Appendix D. The mass anisotropy parameter ( α) lies within the range 0 /lessorequalslantα/lessorequalslant π 2[28]. But, for the sake of concreteness we restrict our focus on the isotropicsystem with α= π 4. For discussion on the evolution of phase diagrams with varying α, see Secs. VA andVI, and Figs. 6 and16. The LSM is realized as an unstable ﬁxed point at μ=0, the red dot in Fig. 1. This ﬁxed point is character- ized by the dynamic scaling exponent z=2, determining the relative scaling between energy and momentum according toE∼\|k\| z. The chemical potential is a relevant perturbation at this ﬁxed point, with the scaling dimension [ μ]=2. Hence, the correlation length exponent at this ﬁxed point is ν=1/2. Therefore the LSM can be envisioned as a quantum criticalpoint (QCP) separating electron-doped (for μ> 0, the brown region) and hole-doped (for μ< 0, the green region) Fermi liquid phases, as shown in Fig. 1. Our discussion is focused on the quantum critical regime (the shaded region). The crossover temperature ( T ∗) separating the quantum critical regime accommodating gapless spin-3 /2 excitations from the Fermi liquid phases can be estimated in the followingway: T∗∼¯h2 2m×1 ξ2∼¯h2 2m\|n\|2/3, (2) where ξis a characteristic length scale and \|n\|∼\|μ\|/ξdis the carrier density. Two critical exponents ( zandν) and the dimensionality of the system ( d=3) determine the scaling of various thermodynamic (such as speciﬁc heat, compressibil-ity) and transport (such as dynamic conductivity) quantities inthis regime. A. Critical scaling in noninteracting system The free-energy density (up to an unimportant temperature (T) independent constant) inside the critical regime is given by (setting kB=1) f=T5/2(2m)3 2 4π3 2/bracketleftbig Li 5 2(−eμ/T)+Li 5 2(−e−μ/T)/bracketrightbig ,(3) where Li represents the polylogarithimic function. The spe- ciﬁc heat of this system is given by CV=−T∂2f ∂T2≈T3/2(2m)3 2 32π3 2/bracketleftbigg 15a−bμ2 T2/bracketrightbigg , (4) forμ/T/lessmuch1 (ensuring that the system resides inside the critical regime), where a≈1.7244 and b≈0.6049. From the above expression of the free-energy density we can alsoextract the scaling of compressibility ( κ), given by κ=−∂ 2f ∂μ2≈√ T(2m)3 2 2π3 2/parenleftbigg b+6cμ2 T2/parenrightbigg , (5) where c≈0.00989. Therefore the presence of ﬁnite chem- ical doping does not alter the leading power-law scaling ofphysical observables, such as C V∼T3/2,κ∼√ T, but only provides subleading corrections, which are suppressed by a parametrically small quantity μ/T.4Also note that CV/T κ≈5.37611 (6) is a universal ratio, capturing the signature of a z=2 quantum critical point in d=3. A detailed derivation of this analysis is shown in Appendix B. Qualitatively similar sub-leading corrections are also found in the scaling of the dynamic con-ductivity, which we discuss now. Gauge invariance mandates that the conductivity ( σ)m u s t scale as σ∼ξ 2−d. Hence for a collection of z=2 quasi- particles (such as the Luttinger fermions), σ∼√ Tor√ω in three spatial dimensions. Indeed we ﬁnd that the Drude(Dr) component of the dynamic conductivity in the Luttingersystem is given by (see Appendix C) σ Dr(ω,T)=e2δ/parenleftbiggω T/parenrightbigg√ mT F Dr/parenleftbiggμ T/parenrightbigg , (7) 4The scaling of speciﬁc heat and compressibility is determined by the dimensionality ( d) and dynamic scaling exponent ( z) according toCV∼T1+d/zandκ∼Td/z, respectively. 165139-5SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) FIG. 6. Speciﬁc cuts of the phase diagram of an interacting, but anisotropic Luttinger metal ( α/negationslash=π 4), showing the appearance of various superconducting phases at ﬁnite chemical doping ( \|μ\|>0) from repulsive electron-electron interactions, accommodating only excitonic orders forμ=0. (a) and (b) show appearance of a conventional s-wave pairing (blue lines) from strong repulsive interaction in the T2gandEgnematic channels for α=1.5a n d α=0.1, respectively [see also Figs. 16(a) and16(b) ]. The phase boundaries of two nematic phases with Luttinger semimetal are respectively denoted by red and golden yellow lines. Nucleation of various topological d-wave pairings, belonging to the Eg (dark yellow lines) and T2g(dark green lines) representations, from strong repulsive magnetic interactions in the A2uandT1uchannels, are respectively shown in (c) and (d) for α=1.5 and 0.1 [see also Figs. 16(c) and16(d) ]. The phase boundaries of A2uandT1umagnetic phases with Luttinger semimetal are shown by purple and magenta lines. Due to a large separation of the interaction strength g4required for any ordering at μ=0a n d\|μ\|>0, we display the μ=0 cut of the phase diagram from panel (d) in Fig. 7. The region at weaker interaction and higher temperature is occupied by correlated Luttinger metal (white regions), without any long-range ordering. In (a) and (b), 100 gi→giand in (c) 10 gi→gi. Throughout the coupling constants are measured in units of /epsilon1and the ordered states are displayed as the gray shaded regions (see footnote 2). where FDr(x) is a monotonically increasing universal function of its argument [see Eq. ( C6) and Fig. 21] and ωis the frequency. On the other hand, the interband (IB) componentof the optical conductivity reads as σ IB(ω,T)=e2√mω/summationdisplay τ=±tanh/parenleftbiggω+2τμ 4T/parenrightbigg . (8) Hence, interband component of the optical conductivity van- ishes as√ωasω→0 and the LSM can be identiﬁed as a power-law insulator . Therefore even when the chemical doping is ﬁnite there exists a wide quantum critical regime, shown in Fig. 1, where the scaling of thermodynamic and transport quantities areessentially governed by z=2 quasiparticles, and the chem- ical potential provides only sub-leading corrections. Nextwe highlight the imprint of ﬁnite temperature and chemical doping on the global phase diagram of interacting spin-3 /2 fermions. B. Electron-electron interactions in a Luttinger semimetal In this work, we compute the effects of electron-electron interactions on Luttinger fermions, occupying the criticalregime of the noninteracting ﬁxed point, see Fig. 1.I nt h i s regime any short-range or local four-fermion interaction ( λ)i s anirrelevant perturbation, since [λ]=z−d=2−3=−1, due to the vanishing DoS, namely, /rho1(E)∼√ E.W eu s eaR G analysis, tailored to address the effects of electronic interac-tions on Luttinger fermions in d=3, constituting a z=2 165139-6INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) band structure, to arrive at various cuts of the global phase diagram of this system. If, on the other hand, temperatureand chemical doping are such that the system resides insidea Fermi liquid phase, the notion of z=2 nodal quasiparticles becomes moot and our RG analysis loses its jurisdiction. 5 Furthermore, we augment the RG analysis with an organiz-ing principle based on the competition between energy andentropy. To this end we rely on the computation of the recon-structed band structure inside the ordered phases within themean-ﬁeld approximation. Subsequently, we also promote a“selection rule” among neighboring phases in the global phasediagram, originating purely from their algebraic or symmetryproperties. For the sake of simplicity we concentrate on theisotropic system ( α= π 4) in the following three subsections. Nonetheless, our results hold (at least qualitatively) for anyarbitrary value of α, as summarized in the last section. 1. Organizing principle: emergent topology and energy entropy Let us ﬁrst promote an organizing principle among BSPs according to their contribution to the energy and entropy gainand anticipate their presence in the global phase diagram.In what follows we highlight the reconstructed band struc-ture inside the dominant ordered phases within a mean-ﬁeldor Hartree-Fock approximation, which by construction un-dermines the ordered parameter ﬂuctuations. The emergentband topology is computed by diagonalizing an effectivesingle-particle Hamiltonian, composed of the noninteractingLuttinger Hamiltonian and corresponding order parameter,see Sec. IV C for details. Perhaps it is natural to anticipate that at zero temperature strong electronic interactions favor the phases that producethe largest spectral gap, as the onset of these ordered statesoffers maximal gain of condensation energy. In a LSM thereare three candidate BSPs that yield fully gapped quasiparticlespectra: (a) an s-wave superconductor, producing a uniform and isotropic gap, and (b) two nematic orders (belonging totheT 2gandEgrepresentations), producing anisotropic gaps. As shown in Fig. 3, only these three phases can be found in an isotropic and interacting LSM at zero temperature. The energy-entropy competition, leading to an organizing principle among competing phases at ﬁnite temperature, canbe appreciated from the scaling of the DoS or the stiffness (uniform or anisotropic) of the spectral gap. As mentioned above, the s-wave pairing and nematic orders respectively produce uniform and anisotropic gaps, whereas two mag-netic orders, belonging to the A 2uand T1urepresentations, respectively produce eight [ 20] and two [ 28] isolated sim- ple Weyl nodes, around which the DoS vanishes as /rho1(E)∼ \|E\|2for sufﬁciently low energies. On the other hand, each copy of the d-wave pairings accommodates two nodal loops 5Note that in the presence of a Fermi surface the DoS is con- stant and the interaction coupling λis dimensionless. Consequently a Fermi liquid becomes unstable toward the formation of a su-perconductor (often non- s-wave) even in the presence of repulsive electron-electron interactions, following the spirit of the Kohn- Luttinger mechanism [ 81–86] and the superconducting transition temperature ( T c) mimics the BCS-scaling law Tc∼exp(−1/λ).for which the low-energy DoS scales as /rho1(E)∼\|E\|(see Sec. IV D )[50,52,54,55,61]. Since we are interested in en- ergy or temperature scales much smaller than the ultravioletcutoff or bandwidth ( \|E\|/lessmuch1), the structure of the spectral gap (isotropic or anisotropic) and power-law scaling of DoScarry sufﬁcient information to organize the ordered phasesaccording to their contribution to condensation energy andentropy, summarized in Fig. 2. In brief, existence of more gapless points (resulting in higher DoS near E=0) yields larger entropy, while a more uniform gap leads to highergain in condensation energy. From various cuts of the globalphase diagram at ﬁnite temperature, see Fig. 3, we note the following common feature: among competing orders, the one with maximal gain in condensation energy appears at lowtemperature, while the phase with higher entropy is realized athigher temperature , in accordance with the general principle of energy-entropy competition. Since the DoS in a LSM scalesas/rho1(E)∼√ E(maximal entropy), it can always be found at sufﬁciently high (weak) enough temperature (interactions). A similar conclusion can also be arrived at when the chem- ical potential is placed away from the band touching point. Atﬁnite chemical doping all particle-hole (two nematic and twomagnetic) orders produce a Fermi surface (according to theLuttinger theorem [ 87]) and hence a ﬁnite DoS. By contrast, any superconducting order at ﬁnite doping maximally gapsthe Fermi surface. Therefore, at ﬁnite chemical doping, super-conducting orders are energetically superior to the excitonicorders, and they can be realized at sufﬁciently low temperatureeven in the presence of repulsive electronic interactions. Con-comitantly, the particle-hole orders are pushed to the highertemperature and interaction regime, see Figs. 4and5. Even though we gain valuable insights into the organiza- tion of various BSPs in the global phase diagram of stronglyinteracting spin-3 /2 fermions from the competition between energy and entropy inside the ordered phases (guided byemergent topology of reconstructed band structure), the phasediagrams shown in Figs. 3–5are obtained from an unbiased RG analysis, which systematically accounts for quantum ﬂuc-tuations beyond the saddle point or mean-ﬁeld approximation.Next we highlight the key aspects of the RG analysis. 2. Methodology: renormalization group The RG analysis we pursue in this work is controlled by a “small” parameter /epsilon1, measuring the deviation from the lower critical two spatial dimensions ( d=2) of the theory, where local four-fermion interactions are marginal , with /epsilon1= d−z=d−2, and hence [ λ]=−/epsilon1. Both temperature and chemical potential (bearing the scaling dimension of energy)arerelevant perturbations at the z=2 ﬁxed point, with scaling dimension [ T]=[μ]=z=2. The RG ﬂow equations are cast in terms of the dimensionless coupling constants ( g is), temperature ( t) and chemical potential ( ˜ μ), deﬁned as gi=mλi/Lambda1/epsilon1 4(2π)3,t=2mT /Lambda12,˜μ=2mμ /Lambda12, (9) which can be inferred from their bare scaling dimensions. Here/Lambda1is the ultraviolet momentum cutoff. 165139-7SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) The leading order RG analysis can be summarized in terms of the following set of coupled ﬂow equations dt d/lscript=zt,dμ d/lscript=zμ, dgi d/lscript=−/epsilon1gi+/summationdisplay j,kgjgkHjk(α,t,μ), (10) where /lscriptis the logarithm of the RG scale. For notational simplicity we take ˜ μ→μin the above ﬂow equations, and Hjk(α,t,μ) are functions of the mass anisotropy parameter α,tandμ. The RG ﬂow equations for gis are obtained by systematically accounting for quantum corrections to thequadratic order in the g is. The relevant Feynman diagrams are shown in Fig. 14. A more detailed discussion of the RG analysis is presented in Sec. VIand Appendix F. Some salient features of the RG analysis are the followings. (1) Temperature ( t) and chemical potential ( μ) provide two infrared cutoffs [ 88–90], respectively given by /lscriptt ∗=1 zln/parenleftbigg1 t(0)/parenrightbigg ,/lscriptμ ∗=1 zln/parenleftbigg1 μ(0)/parenrightbigg (11) for the ﬂow of quartic couplings {gi}, where t(0) and μ(0) represent the bare values ( <1). Ultimately /lscript∗=min (/lscriptt ∗,/lscriptμ ∗) stops the ﬂow of four-fermion interactions. At zero tempera-ture and chemical doping, the system is devoid of any suchnatural infrared cutoff, implying /lscript ∗→∞ . (2) Any weak local four-fermion interaction is an irrelevant perturbation and all orderings (realized when gi(/lscript∗)→∞ ) take place at ﬁnite coupling gi∼/epsilon1through quantum phase transitions (QPTs). Such QPTs are controlled by quantumcritical points (QCPs) and all transitions are continuous innature. The universality class of the transition is determinedby two critical exponents, given by ν −1=/epsilon1+O(/epsilon12) and z=2+O(/epsilon1), (12) and for the physically relevant situation /epsilon1=1. Using the RG analysis we arrive at various cuts of the global phase diagram of interacting spin-3 /2 fermions at (1) zero chemical doping (see Fig. 3) and (2) for ﬁnite- μ(see Figs. 4and5). In all these cuts of the phase diagram (as well as the one shown in Figs. 6,7, and 16) coupling constants are always measured in units of /epsilon1and most importantly the nature of the ordered states is completely impervious to theexact value of /epsilon1. The universality class of the QPT leaves its signature on the scaling of the transition temperature ( t c). Note that tc∼\|δi\|νz [91,92], where δi=(gi−g∗ i)/g∗ iis the reduced distance from the critical point, located at g∗ i. Hence, tc∼\|δ\|2forν=1 andz=2, obtained from the leading order /epsilon1expansion, af- ter setting /epsilon1=1, irrespective of the choice of the coupling constant and the resulting BSP (see Fig. 17). We discuss this issue in detail in Sec. VI C . Even though ultimately we are interested in three-dimensional interacting Luttinger materialsfor which /epsilon1=1, the RG methodology employed here follows the general spirit of the /epsilon1expansion, succinctly employed in the past for bosonic /Phi1 4theory and fermionic Gross-Neveu model for which as well /epsilon1=1[93,94].FIG. 7. The phase diagram of an interacting Luttinger semimetal forα=0.1a n d μ=0, obtained by tuning the strength of the mag- netic interaction in the T1uchannel ( g4), measured in units of /epsilon1. The shaded (white) region represents the ordered phase (Luttinger semimetal). 3. Competing orders and selection rule The correspondence between a given interaction coupling and the resulting phases can be appreciated by formulatingthe whole theory in the basis of an eight component Nambu-doubled spinor /Psi1 Nam, introduced in Sec. III C. In this basis the Luttinger Hamiltonian ˆhL(k)→η3ˆhL(k). Pauli matrices {ημ}operate on the Nambu or particle-hole index. Any four- fermion interaction takes the form gI(/Psi1† NamˆI/Psi1Nam)2and an order parameter ( /Delta1O) couples to a fermion bilinear according to/Delta1O(/Psi1† NamˆO/Psi1† Nam), where ˆI and ˆO are eight-dimensional Hermitian matrices. We argue that when gIis sufﬁciently strong, it can support only two types of ordered phases, for which6 either (1) ˆO≡ˆIo r( 2 ) {ˆO,ˆI}=0. (13) This outcome can be appreciated in the following way. When an interaction coupling gIdiverges toward +∞ un- der coarse graining (indicating onset of a BSP), it providespositive scaling dimension to an order parameter ﬁeld /Delta1 Oonly when one of the above two conditions is satisﬁed. We sub-stantiate this argument by considering the relevant Feynmandiagrams (see Fig. 15) in Sec. VI E. Even though we arrive at this “selection rule” among competing orders from a leadingorder RG calculation, such a simple argument relies on inter-nal symmetries among competing orders (breaking differentsymmetries) and is expected to hold at the nonperturbativelevel. We now support this claim by focusing on a speciﬁcexample. Let us choose a particular four-fermion interaction (in the T 2gnematic channel) g13/summationdisplay j=1(/Psi1† Namη3/Gamma1j/Psi1Nam)2. 6IfˆOa n d ˆI are multicomponent vectors of MOandMIelements, re- spectively, then condition (2) is satisﬁed when at least ⌈MO 2⌉matrices anticommute with ⌈MI 2⌉matrices, where ⌈ ···⌉ is the ceiling function. 165139-8INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) From the phase diagrams shown in Figs. 3(a) for zero and ﬁnite temperature and 4(left) for ﬁnite chemical doping, we ﬁnd that when this coupling constant is sufﬁciently strong, itsupports two distinct BSPs. (1) A nematic order following the T 2grepresentation, for which ˆO=η3{/Gamma11,/Gamma12,/Gamma13}. In this case selection rule (1) from Eq. ( 13) is satisﬁed, since ˆI=η3{/Gamma11,/Gamma12,/Gamma13}, and hence ˆO≡ˆI. (2) An s-wave superconductor following the trivial A1g representation, for which ˆO={η1,η2}/Gamma10, where /Gamma10is a four- dimensional identity matrix. The onset of s-wave pairing follows from selection rule (2) in Eq. ( 13), since {ˆO,ˆI}=0. Moreover we realize that the T2gnematic order and the s-wave pairing together constitute an O(5) vector, {η3/Gamma11,η3/Gamma12,η3/Gamma13,η1/Gamma10,η2/Gamma10}, of ﬁve mutually anticom- muting matrices, reﬂecting the enlarged internal symmetrybetween these two orders. Following the same spirit, we arriveat the following observations. (1) Four fermion interaction in the E gnematic channel (g2) supports (a) a nematic order transforming under the Eg representation [satisfying selection rule (1)] and (b) an s-wave superconductor [satisfying selection rule (2)], as shown inFigs. 3(b) and4(right). One can construct an O(4) vector by combining these two order parameters [see Eq. ( 44)]. (2) Four fermion interaction in the A 2uchannel ( g3) supports (a) a magnetic order transforming under the A2urep- resentation [satisfying selection rule (1)], and (b) Egnematic order and d-wave pairings [satisfying selection rule (2)], as shown in Figs. 3(c) and 5(top). Notice, we can construct multiple copies of composite SU(2) order parameters, by com-bining the A 2uorder with Egnematic or d-wave pairing, see Figs. 18(a) and19. (3) Four fermion interaction in the T1umagnetic channel (g4) supports (a) a magnetic order transforming under the T1urepresentation [satisfying selection rule (1)], and (b) T2g nematic and d-wave pairings [satisfying selection rule (2)], as shown in Figs. 3(d) and5(bottom). Combining the magnetic order with T2gnematic or d-wave pairing we can construct multiple copies of composite SU(2) vector, see Figs. 18(b) , 18(c) , and 20. A more detailed discussion supporting these scenarios is presented in Sec. VI E. Therefore combining the energy- entropy competition (obtained from the reconstructed bandtopology within a mean-ﬁeld approximation) and an unbi-ased (controlled by a small parameter /epsilon1) RG analysis with the selection rule, we gain valuable insights into the natureof broken symmetry phases, competing orders and quantumcritical phenomena in the global phase diagram of stronglyinteracting spin-3 /2 fermions. 4. Anisotropic Luttinger semimetal So far, we centered our focus on the isotropic system [realized for α=π 4in Eq. ( 1)]. Note that for α=π 4the system enjoys an enlarged (but artiﬁcial) SO(3) rotationalsymmetry. However, in a cubic environment α/negationslash= π 4in gen- eral. Nonetheless, all the central results we quoted in the lastthree subsections hold (at least qualitatively) for any arbitraryvalue of α:0/lessorequalslantα/lessorequalslant π 2. The discussion on the role of the mass anisotropy parameter αon the global phase diagram ofinteracting spin-3 /2 fermions is rather technical, which we address in depth in Secs. VA,VI A , and VI D . We here only quote some key results, which nicely corroborate with the restof the discussion from this section. (1) We identify the mass anisotropy parameter as a valuable nonthermal tuning parameter, and for suitable choices of thisparameter one can ﬁnd (a) T 2gnematic and A2umagnetic order respectively for strong enough g1and g3couplings (when α→π 2), see Figs. 6(a) and 6(c),( b ) Egnematic and T1u magnetic orders for strong enough g2andg4(when α→0), as shown in Figs. 6(b) and7at zero temperature and chemical doping, respectively. These outcomes are in agreement withselection rule (1). (2) At ﬁnite chemical doping (a) an s-wave pairing emerges from repulsive electronic interaction in the T 2gchannel [see Fig. 6(a)]a sw e l la s Egchannel [see Fig. 6(b)], (b) d-wave pairings belonging to the EgandT2grepresentations respec- tively appear for repulsive interaction in the A2uchannel [see Fig.6(c)] and T1uchannel [see Fig. 6(d)]. These outcomes are in accordance with selection rule (2), as we argued previouslyfor an isotropic Luttinger system. We now proceed to a detailed discussion on each compo- nent of our work, starting from the noninteracting Luttingermodel. III. LUTTINGER MODEL We begin the discussion with the Luttinger model describ- ing a biquadratic touching of Kramers degenerate valence andconduction bands at an isolated point (here chosen to be the/Gamma1=(0,0,0) point, for convenience) in the Brillouin zone. In this section, we ﬁrst present the low-energy Hamiltonian anddiscuss its symmetry properties (Sec. III A ). Subsequently, we introduce the corresponding imaginary time ( τ) or Euclidean action and the notion of the renormalization group (RG)scaling (Sec. III B). Finally, we deﬁne an eight-component Nambu-doubled basis that allows us to capture all, includingboth particle-hole or excitonic and particle-particle or super-conducting, orders within a uniﬁed framework (Sec. III C). A. Hamiltonian and symmetries The Hamiltonian operator describing a biquadratic touch- ing of Kramers degenerate valence and conduction bands inthree dimensions is given by [ 3,4] ˆh L(k)=−k2/parenleftBigg3/summationdisplay j=1ˆdj(ˆk) 2m1/Gamma1j+5/summationdisplay j=4ˆdj(ˆk) 2m2/Gamma1j/parenrightBigg −μ/Gamma1 0, (14) where /Gamma10is the four-dimensional identity matrix. Chemical potential μand momenta kare measured from the band touch- ing point. Here, ˆd(ˆk) is a ﬁve-dimensional unit vector that transforms under the l=2 representation under the orbital SO(3) rotations. Hence ˆd(ˆk) is constructed from the d-wave form factors or spherical harmonics Ym l=2(θ,φ), as shown in Appendix D. The corresponding four-component spinor basis is given by /Psi1/latticetop k=/parenleftbig ck,+3 2,ck,+1 2,ck,−1 2,ck,−3 2/parenrightbig , (15) 165139-9SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) where ck,jis the fermionic annihilation operator with mo- menta kand spin projection j=±3/2 and ±1/2. The ﬁve mutually anticommuting /Gamma1matrices are deﬁned as /Gamma11=κ3σ2,/Gamma1 2=κ3σ1,/Gamma1 3=κ3σ0, /Gamma14=κ1σ0,/Gamma1 5=κ3σ3. (16) Two sets of two-dimensional Pauli matrices {κν}and{σν} respectively operate on the sign (sgn[ j]) and magnitude ( \|j\|∈ {1/2,3/2}) of the spin projections, where ν=0,1,2,3. To close the Clifford algebra of all four-dimensional Hermi-tian matrices we also deﬁne ten commutators according to/Gamma1 jk=[/Gamma1j,/Gamma1k]/(2i), with j>kand j,k=1,..., 5. All six- teen four-dimensional matrices can be expressed in termsof the products of spin-3 /2 matrices ( J), as also shown in Appendix D. The energy spectra for Luttinger fermions are given by ±E s(k)−μ, where for s=±1 Es(k)=k2 2m/radicaltp/radicalvertex/radicalvertex/radicalbtcos2α3/summationdisplay j=1ˆd2 j+sin2α5/summationdisplay j=4ˆd2 j, (17) reﬂecting the quadratic band touching for μ=0, which is protected by the cubic symmetry. For brevity we drop theexplicit dependence of {ˆd j}onˆk. Notice that the independence of Es(k)o n smanifests the Kramers degeneracy of the valence and conduction bands,ensured by (1) the time-reversal ( T) and (2) the parity or inversion ( P) symmetries. Speciﬁcally, under the reversal of time, k→− kand/Psi1 k→/Gamma11/Gamma13/Psi1−kand hence T=/Gamma11/Gamma13K, where Kis the complex conjugation, yielding T2=−1( r e - ﬂecting Kramers degeneracy of bands). Under the inversionP:k→− kand/Psi1 k→/Psi1−k. The “average” mass mand the mass anisotropy parameter αare respectively given by [ 28] m=m1m2 m1+m2,α=tan−1/parenleftbiggm2 m1/parenrightbigg . (18) Note that {ˆdj}forj=1,2,3 and j=4,5 belong to the T2g (three component) and Eg(two component) representations of the cubic or octahedral ( Oh) point group, and m1andm2 are effective masses in these two orbitals, respectively. The mass anisotropy parameter αallows us to smoothly interpolate between (1) the m1→∞ limit when the dispersion of the T2g orbital becomes ﬂat, yielding α→0 and (2) m2→∞ when theEgorbital becomes nondispersive, leading to α→π 2.F o r α=π 4orm1=m2, the Luttinger model enjoys an enlarged spherical symmetry. Any α/negationslash=π 4captures a quadrupolar dis- tortion in the system (still preserving the cubic symmetry). In what follows, we treat αas anonthermal tuning parameter to explore the territory of interacting Luttinger fermions. The connection between the spin projections ( j=±3/2 and±1/2) and the bands can be appreciated most econom- ically by taking k=(0,0,k). For such a speciﬁc choice of momentum axis, the Luttinger Hamiltonian takes a diagonal form, given by ˆhL(kˆz)=k2 2m2Diag.[−1,1,1,−1]−μ. (19)From the above expression we can immediately infer that the pseudospin projections on the valence and conduction bandsare respectively \|j\|=3/2 and 1 /2. B. Lagrangian and scaling The imaginary time ( τ) Euclidean action corresponding to the noninteracting Luttinger model is given by S0=/integraldisplay dτddx/Psi1†(τ,x)ˆhL(k→− i∇)/Psi1(τ,x). (20) The action remains invariant under the following rescaling of space-(imaginary)time coordinates and the fermionic ﬁeld x→e/lscriptx,τ→ez/lscriptτ, /Psi1 →e−d/lscript/2/Psi1, (21) where zis the dynamic scaling exponent, measuring the rel- ative scaling between energy and momentum according toE(k)∼\|k\| z. For Luttinger fermions, z=2. The parameter /lscript is the logarithm of the RG scale. In what follows in Secs. V and VI, we use the above scaling ansatz while addressing the effects of electronic interactions in this system. Underthe above rescaling of parameters, the temperature ( T) and chemical potential ( μ) scale as T→e −z/lscriptT,μ→e−z/lscriptμ. (22) Therefore the scaling dimension of these two quantities is [T]=[μ]=z(same as that of energy). Throughout, we use thenatural unit , in which ¯ h=kB=1. C. Nambu doubling To facilitate the forthcoming discussion we here introduce an eight-component Nambu-doubled spinor basis (suitable tocapture both exitonic and superconducting orders within auniﬁed framework) according to /Psi1 Nam=/bracketleftBigg/Psi1k /Gamma11/Gamma13(/Psi1† −k)/latticetop/bracketrightBigg , (23) where /Psi1kis a four-component spinor, see Eq. ( 15). In the lower block of /Psi1Nam we absorb the unitary part of the time-reversal operator T, ensuring that the eight-component Nambu spinor ( /Psi1Nam) transforms the same way as the orig- inal four component spinor /Psi1kunder the SU(2) pseudospin rotation. In this basis, the eight-dimensional Luttinger Hamil-tonian takes a simple form ˆh Nam L(k)=η3ˆhL(k), (24) and the time-reversal operator becomes TNam=η0/Gamma11/Gamma13K. The newly introduced set of Pauli matrices {ην}operates on the Nambu or particle-hole indices, with ν=0,1,2,3. Therefore, by construction while the excitonic orders assume block-diagonal form, all superconducting orders are block-off- diagonal in the Nambu subspace. Note that ˆhNam L(k)commutes with the number operator ˆN=η3/Gamma10. IV . BROKEN SYMMETRY PHASES Next we discuss possible BSPs in this system. We intro- duce various possible excitonic and superconducting orders inthe Nambu basis ( /Psi1 Nam) in two subsequent sections. Finally, 165139-10INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) we discuss the reconstructed band structure and emergent topology inside the ordered phases. A. Particle-hole or excitonic orders The effective single-particle Hamiltonian in the presence of all possible momentum-independent or local or intraunit cellexcitonic orders is given by H exc local=/integraldisplay d3r/parenleftbig /Psi1† Namˆhexc local/Psi1Nam/parenrightbig , (25) where ˆhexc local=Density/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright /Delta10η3/Gamma10+Nematic/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright η3/bracketleftBigg3/summationdisplay j=1/Delta1j 1/Gamma1j+5/summationdisplay j=4/Delta1j 2/Gamma1j/bracketrightBigg +η0/bracketleftBigg /Delta13/Gamma145+3/summationdisplay j=1/Delta1j 4/Gamma145/Gamma1j+3/summationdisplay j=15/summationdisplay k=4/Delta1jk 5/Gamma1jk/bracketrightBigg /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright Magnetic. (26) The ordered phases can be classiﬁed according to their transformation under the cubic ( Oh) point group symmetry. Regular fermionic density ( /Delta10) does not break any sym- metry (hence does not correspond to any ordering) andtransforms under the trivial A 1grepresentation. A three- component nematic order parameter, constituted by /Delta11= (/Delta11 1,/Delta121,/Delta131), transforms under the T2grepresentation. By contrast, a two-component nematic order transforming un-der the E grepresentation is captured by /Delta12=(/Delta11 2,/Delta122). Both of them break only the cubic symmetry, but preservetime-reversal and inversion symmetries. The ordered phaserepresents either a time-reversal invariant insulator or a Diracsemimetal, about which more in a moment [see Sec. IV C ]. Since ﬁve /Gamma1matrices transform as components of a rank-2 tensor under SO(3) rotations, the two nematic phases repre-sentquadrupolar orders, see Appendix D. All ordered phases shown in the second line of Eq. ( 26) break time-reversal symmetry and represent different mag-netic orders. For example, /Delta1 3corresponds to an octupolar order (since /Gamma145∼JxJyJz), transforming under the singlet A2u representation. In a pyrochlore lattice of 227 iridates such an ordered phase represents the “ all-in all-out ” arrangement of electronic spin between two adjacent corner-shared tetrahedra[8,20]. By contrast, “ two-in two-out ”o r“ spin-ice ” magnetic orderings on a pyrochlore lattice are represented by a three-component vector /Delta1 4=(/Delta11 4,/Delta124,/Delta134) (accounting for six possible two-in two-out arrangements in a single tetrahedron).Since /Gamma1 45/Gamma1j∼7Jj−4J3 jsuch an ordered phase contains a linear superposition of dipolar and octupolar moments, andtransforms under the T 1urepresentation [ 28]. Any other mag- netic ordering can be represented by a six component vector /Delta1jk 5with j=1,2,3 and k=4,5. No physical realization of such multicomponent magnetic ordering in any material iscurrently known, and we do not delve into the discussion onsuch ordering for the rest of the paper.B. Particle-particle or superconducting orders The effective single particle Hamiltonian in the presence of all possible momentum-independent or local or intraunit cellsuperconducting orders reads [ 50,52–54] Hpair local=/integraldisplay d3r/parenleftbig /Psi1† Namˆhpair local/Psi1Nam/parenrightbig , (27) where ˆhpair local=(η1cosφ+η2sinφ)/bracketleftBiggs−wave/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright /Delta1p A1g/Gamma10 +3/summationdisplay j=1/Delta1p,j T2g/Gamma1j+5/summationdisplay j=4/Delta1p,j Eg/Gamma1j /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright d−wave/bracketrightBigg , (28) andφis the global U(1) superconducting phase. Any pairing proportional to η1(η2) preserves (breaks) time-reversal sym- metry (recall that the time-reversal operator in the Nambubasis is T Nam=η0/Gamma11/Gamma13K). Here, /Delta1p A1gis the amplitude of thes-wave pairing, transforming under the A1grepresentation. Thes-wave pairing breaks only the global U(1) symmetry, but preserves the cubic symmetry. On the other hand, /Delta1p,j T2gcap- tures the amplitude of three d-wave pairings (for j=1,2,3) transforming under the T2grepresentation, and /Delta1p,j Egforj= 4,5 represents the amplitude of two d-wave pairings belong- ing to the Egrepresentation. Notice {/Gamma1j,j=1,..., 5}can be expressed in terms of the product of two spin-3 /2 matrices, and all ﬁve d-wave pairings break the cubic symmetry, while introducing a lattice distortion or electronic nematicity in thesystem. Hence, they stand as representatives of quadrupolar nematic superconductors . C. Reconstructed band structure and emergent topology Next we consider the reconstructed band structure inside different BSPs which provides valuable information regardingthe emergent topology inside ordered phases. The onset ofany ordering discussed in the previous sections destabilizesthe biquadratic touching and gives rise to either gapped orgapless quasiparticles (see below). Furthermore, this exercisewill allow us to appreciate the energy-entropy competitionamong different orderings [see Sec. IV D ], which ultimately plays a decisive role in the organization of various phases inthe global phase diagram of interacting Luttinger fermions. (1)T 2gnematicity. The three component order parameter for the T2gnematic phase gives birth to gapless quasiparticles for the following four conﬁgurations: /Delta11=\|/Delta11\|√ 3/braceleftBig (+,+,+)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 1,(−,−,+)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 2,(+,−,−)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 3,(−,+,−)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 4/bracerightBig . (29) These four phase lockings are respectively shown as blue, red, green and black points in Fig. 8(a). The gapless phase cor- responds to a topological Dirac semimetal (since nematicity preserves the Kramers degeneracy of valence and conductionbands), similar to the ones recently found in Cd 3As2[95] and Na 3Bi [96]. The DoS in a Dirac semimetal vanishes as 165139-11SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) FIG. 8. (a) Four phase lockings (blue, red, green, and black dots) of the three component T2gnematic order, see Eq. ( 29), that yield gapless phase (topological Dirac semimetal) inside the ordered state.Corresponding locations of two Dirac points in momentum space are shown in panel (b). For any other generic phase locking, the ordered phase is a time-reversal symmetry preserving insulator. Dirac pointsare located along the body-diagonals ( C 3vaxes in a cubic system). /rho1(E)∼\|E\|2. The Dirac points are located along the body diagonals (the C3vaxes) of a cubic system and respectively placed at k=±/braceleftBig (1,1,1)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 1,(1,1,−1)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 2,(1,−1,1)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 3,(1,−1,−1)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 4/bracerightBig k0, (30) where k0=[2m1/Delta11/3]1/2, as shown in Fig. 8(b). For any other phase locking within the T2gsector the system becomes aninsulator . The spectral gap in the insulating phase is anisotropic and it is energetically superior over the gapless Dirac semimetal phase. (2)Egnematicity. The two component Egnematic order is most conveniently described in terms of the followingparametrization /Delta1 2=\|/Delta12\|√ 2/parenleftbig sinφEg,cosφEg/parenrightbig , (31) where φEgis the internal angle in the order-parameter space. Only for φEg=/braceleftBig 0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 1,2π/3/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 2,4π/3/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright 3/bracerightBig (32) the quasiparticle spectra are gapless, as shown in Fig. 9(a), and the ordered phase represents a topological Dirac semimetal. Speciﬁcally, for φEg=0,2π/3,4π/3, the Dirac points are respectively located on kz,kxand kyaxes (the C4vaxes), see Fig. 9(b), and the separation of two Dirac points is given by 2 k0, where k0=(2m2/Delta12/√ 2)1/2. For any other phase locking within the Egsector, the system becomes an insulator. Recently, it was shown that the gapless phasesin both T 2gand Egnematic phases correspond to higher- order Dirac semimetals supporting one-dimensional hingemodes, whereas the insulating phases accommodate a mixed topology [ 97]. (3)A 2umagnet. In the presence of an octupolar A2uor- dering, the twofold degeneracy of the valence and conduction FIG. 9. (a) Three possible phase lockings (shown by blue, black, and red dots) of two-component Egnematic order, see Eq. ( 32), that give rise to topological Dirac semimetal inside the ordered phase.The Dirac points inside the E gnematic phase are located on the kz,kx, andkyaxes ( C4vaxes in a cubic system), as shown in (b), in contrast to the situation inside the T2gnematic phase, see Fig. 8(b).F o ra n y other phase locking the system is an insulator. bands gets lifted and a pair of Kramers nondegenerate bands touch each other at the following eight points in the Brillouin zone [see Fig. 10(a) ] k=(±1,±1,±1)k0, (33) where k0=√2m1/Delta13/3. They represent simple Weyl points, which act as source (4 of them) and sink (4 of them) ofAbelian Berry curvature of unit strength. However, due toan octupolar arrangement of the Weyl nodes, the net Berrycurvature through any high-symmetry plane is precisely zero and this phase does not support any anomalous Hall effect.The DoS at low energies then scales as /rho1(E)∼\|E\| 2[8,20,28]. (4)T1umagnet. For each component of T1umagnetic order (represented by the matrix operator /Gamma145/Gamma1jwith j=1,2,3) the ordered phase supports two Weyl nodes along one of theC4vaxes and a nodalloop in the corresponding basal plane. For example, when /angbracketleft/Psi1†/Gamma145/Gamma13/Psi1/angbracketright≡/Delta13 4/negationslash=0t h el e f t and right chiral Weyl nodes are located at (0 ,0,±k0) where k0=√2m2/Delta14and a nodal loop is found in the kx−kyplane, as shown in Fig. 10(b) . Similarly, for j=1 and 2 the Weyl nodes are separated along the kxandkyaxes, and the nodal loops are respectively found in the ky-kzand kx-kzplanes. Due to the presence of two Weyl nodes, each conﬁgurationof two-in two-out magnetic order supports a ﬁnite anomalousHall effect in the plane perpendicular to the separation of theWeyl nodes. However, any triplet magnetic order, represented by/Delta1 4=\|/Delta14\|(±1,±1,±1)/√ 3, gets rid of the nodal loop and supports only twoWeyl nodes along one of the body- diagonals ( C3vaxes). Hence, triplet T1umagnetic orders are energetically favored over their uniaxial counterparts [ 28].7 (5)A1gors-wave pairing. Notice that the matrix operator representing an s-wave pairing fully anticommutes with the 7The low energy DoS in the presence of a nodal loop and two point nodes (due to a uniaxial T1uorder) is dominated by the former and scales as /rho1(E)∼\|E\|, while in a triplet T1ustate the DoS scales as /rho1(E)∼\|E\|2(due to the point nodes). Hence, formation of the triplet ordering causes the power-law suppression of the DoS and increasesthe condensation energy gain. 165139-12INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) FIG. 10. (a) Location of eight Weyl nodes in the presence of A2uoctupolar order, which in a pyrochlore lattice corresponds to the all-in all-out magnetic order. Here, red and gray dots respectively correspond to the source ( +) and sink ( −) of Abelian Berry curva- ture, with monopole charge ±1. (b) Nodal structure in the presence of a uniaxial T1uorder (when its moment points along ˆ z), supporting two Weyl nodes separated along ˆkz(red and gray dots) and a nodal loop (dark yellow ring) in the kx-kyplane. When the moment of the uniaxial T1uorder points along ˆ xand ˆydirection, the Weyl nodes are respectively separated along the kxandkyaxes, and the nodal-loops are found in the ky-kzandkx-kzplane. By contrast, inside a triplet T1uphase, only two Weyl nodes are found along one of the body diagonals. Luttinger Hamiltonian (for any value of α) and thus corre- sponds to a mass for Luttinger fermions. The quasiparticle spectra inside the paired state is fully gapped, but the phaseis topologically trivial . (6)T 2gpairing. Three d-wave pairings, proportional to /Gamma11, /Gamma12and/Gamma13matrices, belong to the T2grepresentation and respectively possess the symmetry of dyz,dxz, and dxypairings. Each component supports two nodal loops in the orderedphase, as shown in the ﬁrst three rows of Table I[50,54]. Two nodal loops for the /Gamma1 3ordxypairing are shown in Fig. 11(a) . The two nodal loops for /Gamma11ordyzand/Gamma12ordxzpairings can respectively be obtained by rotating the ones shown for dxy pairing byπ 2, with respect to the kyandkxaxes. (7)Egpairing. Egpairings proportional to /Gamma14and/Gamma15ma- trices respectively possess the symmetry of dx2−y2andd3z2−r2 pairings and each of them supports two nodal loops, as shown in the last two rows of Table I[50,54]. Note that two nodal loops for the dx2−y2pairing can be realized by rotating the ones for the dxypairing byπ 4about the kzaxis. However, two nodal loops for the d3z2−r2pairing, shown in Fig. 11(b) , cannot be rotated into the ones for dx2−y2pairing. Therefore despite the fact that the d3z2−r2anddx2−y2pairings belong to the same Eg representation, they are not energetically degenerate [ 54,98]. Since the radius of the nodal loops for the d3z2−r2pairing is the smallest , this paired state is the energetically most favorable among ﬁve d-wave pairings.8 8Even though d+idtype, such as dx2−y2+id3z2−r2, pairing can eliminate nodal loops from the quasiparticle spectra in favor of pointnodes around which /rho1(E)∼\|E\| 2in a single band Fermi liquid [ 98], the strong interband coupling causes inﬂation of such nodes in doped LSM and yields Fermi surface of BdG quasiparticles, leading to aTABLE I. The structure of two nodal loops in the presence of ﬁve individual d-wave pairings, belonging to the T2gandEgrepre- sentations, where k2 ⊥=k2 x+k2 y. We display the symmetry of each d-wave pairing in the proximity to the Fermi surface (realized on the conduction or valence band) in the last column. Note that two nodal loops for dxy,dxz,dyz,a n d dx2−y2parings can be rotated into each other, while those in the presence of d3z2−r2pairing are disconnected from the remaining ones, see Fig. 11. For the sake of simplicity, we here assume m1=m2=m, for which the nodal loops are circular in shape. For m1/negationslash=m2, the nodal loops become elliptic . Here, /Delta1is the amplitude of d-wave pairings. Pairing IREP. Equations for nodal loops Symmetry /Gamma11 T2g k2 x+k2 y=2m/Delta1,k2 x+k2 z=2m/Delta1 dyz /Gamma12 T2g k2 x+k2 y=2m/Delta1,k2 z+k2 y=2m/Delta1 dxz /Gamma13 T2g k2 y+k2 z=2m/Delta1,k2 x+k2 z=2m/Delta1 dxy /Gamma14 Eg k2 z+k2 ⊥=2m/Delta1,kx=±ky dx2−y2 /Gamma15 Eg k2 ⊥=4m/Delta1/3,kz=±k⊥/√ 2 d3z2−r2 D. Energy and entropy inside ordered phases From the computation of the reconstructed band structure, we can gain insight into the condensation energy ( /Delta1F) and entropy ( /Delta1S) inside the ordered phases. While the stiffness of the spectral gap measures the gain of condensation energy, thescaling of the DoS at low-energies (due to gapless quasipar-ticles) measures the entropy. Recall that the s-wave pairing yields fully gapped spectra (isotropic), while the nematicorders produce either an anisotropic gap or gapless quasipar-ticles. Hence, the former ordering is associated with higher(lower) gain in condensation energy (entropy). On the otherhand, the DoS vanishes as /rho1(E)∼\|E\|and\|E\| 2respectively in the presence of a nodal-loop and Dirac or Weyl points.We found that the A 2umagnetic order gives birth to eight Weyl nodes, while only two Weyl nodes can be found insidethetriplet T 1umagnetic order. By contrast, all ﬁve d-wave pairings are accompanied by two nodal loops (see Table I). Therefore we can organize these ordered phases according totheir contribution to (a) condensation energy and (b) entropygain, as shown in Fig. 2. The LSM, on the other hand, accom- modates the largest amount of gapless fermionic excitations constant DoS at lowest energy, followed by /rho1(E)∼\|E\|2a higher energies [ 51]. Presently, it is not very clear between (a) individual d-wave pairings and (b) d+idtype pairings, which one is energet- ically more advantageous. However, based on the power-law scaling of DoS, we expect individual d-wave pairings to be energetically fa- vored over d+idtype pairings at least when the interband coupling is strong, which is the case when pairing results from pure Hubbard- like repulsive interactions. This conclusion is in accordance with the organizing principle discussed in Sec. II B 1 and the energy-entropy argument, summarized in Fig. 2.T h e\|E\|-linear DoS for individual d-wave parings (stemming from the underlying nodal loops) results in aT-linear scaling of the penetration depth, as observed in YPtBi [38]. 165139-13SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) FIG. 11. Structure of two nodal loops in the presence of an underlying (a) dxyand (b) d3z2−r2pairings. These two pairings are respectively represented by /Gamma13and/Gamma15matrices for the Luttinger fermions. Note that the nodal loops for dyz(/Gamma11),dxz(/Gamma12),dx2−y2(/Gamma14) pairings can be obtained by rotating the ones shown here for dxy pairing about suitable momentum axes. However, the two nodal loops for the d3z2−r2pairing are disjoint from the remaining ones (note these two nodal loops do not cross each other), see Sec. IV C and Table I. near zero energy, where the DoS vanishes as /rho1(E)∼√ E. Hence, the LSM is endowed with largest entropy.9 We now present a simple prescription to estimate (at least qualitatively) the gain of condensation energy or en-tropy. Let us assume that the eight-dimensional Hermitianmatrix M, entering the deﬁnition of an order parameter as /Delta1 M≡/angbracketleft/Psi1† NamM/Psi1Nam/angbracketrightrespectively anticommutes and com- mutes with NM antiandNM comm number of matrices appearing in the Luttinger Hamiltonian ˆhNam L(k) (when μ=0), and thus NM anti+NM comm=5. (34) Therefore, for the s-wave pairing, two nematic orders, two magnetic orders and ﬁve d-wave pairings NM anti=5,4,2,1, while NM comm=0,1,3,4, respectively. Then for any ordered phase /Delta1F∼NM anti,/Delta1 S∼NM comm. (35) This correspondence can be anchored from a simple example. Let us choose s-wave pairing, for which NM anti=5,NM comm= 0 and chemical potential, for which NM anti=0,NM comm=5, as two perturbations in a LSM. While the s-wave pairing yields an isotropic gap, a ﬁnite chemical doping creates aFermi surface (producing a constant DoS). Consequently, thes-wave pairing (chemical doping) is accompanied by larger gain of condensation energy (entropy). Therefore the anal-ysis of reconstructed band structure and emergent topologyinside BSPs allows us to organize them according to the 9Such an organization of ordered phases according to their con- tributions to the gain of condensation energy and entropy is purelybased on the power-law dependence of low-energy DoS or the stiff- ness (isotropic or anisotropic) of the spectral gap. This procedure, however, cannot distinguish two phases with similar scaling of theDoS, such as between A 2uand triplet T1umagnetic orders (producing Weyl nodes), or the stiffness of the spectral gap, such as between T2gand Egnematic orders (producing anisotropic gaps). A more microscopic analysis is needed to resolve these situations.gain of condensation energy and entropy. The RG analysis at ﬁnite temperature (in the regime where the dimensionlesstemperature t=2mT//Lambda1 2/lessmuch1) captures such energy-entropy competition, which we discuss in Sec. VI C , see also Fig. 3. The hierarchy of the energy and entropy gains inside the ordered phases changes when the chemical potential is placedaway from the band touching point (i.e., μ/negationslash=0). Since any pairing operator anticommutes with the number operator ( ˆN= η 3/Gamma10), superconducting orders maximally gap (either fully by the s-wave pairing or partially by the individual d-wave pairings) the Fermi surface. By contrast, any excitonic or-der always gives birth to a Fermi surface, according to theLuttinger theorem [ 87]. Hence, at ﬁnite doping all super- conductors are energetically superior over the particle-holeorders, while excitonic orders are accompanied by largerentropy (due to presence of a Fermi surface). The energy-entropy competition at ﬁnite- μis also captured by the RG analysis, discussed in Sec. VI D , leading to the phase diagrams shown in Figs. 4–6. V . ELECTRON-ELECTRON INTERACTIONS Next we proceed to demonstrate the onset of various BSPs, discussed in the previous section, triggered by repulsive (at the bare level) electron-electron interactions. As mentionedearlier we will focus only on the local or short-range part ofCoulomb interaction and neglect its long-range tail. For thesake of concreteness, we assume that the local interactions aredensity-density in nature. Any generic local density-densityinteraction (such as the ones appearing in an extended Hub-bard model, for example) can be captured by sixquartic terms and the corresponding interacting Hamiltonian reads H int=−/bracketleftBigg λ0(/Psi1†/Psi1)2+λ13/summationdisplay j=1(/Psi1†/Gamma1j/Psi1)2 +λ25/summationdisplay j=4(/Psi1†/Gamma1j/Psi1)2+λ3(/Psi1†/Gamma145/Psi1)2 +λ43/summationdisplay j=1(/Psi1†/Gamma1j/Gamma145/Psi1)2+λ53/summationdisplay j=15/summationdisplay k=4(/Psi1†/Gamma1jk/Psi1)2/bracketrightBigg .(36) In this notation, λj>0 corresponds to repulsive interaction. However, all four-fermion interactions are notlinearly inde- pendent due to the existence of Fierz identity among sixteen four-dimensional Hermitian matrices, closing a U(4) Cliffordalgebra (see Appendix E)[99–101]. It turns out that any generic local interaction can be expressed in terms of onlythree quartic terms, and we conveniently (without any loss of generality) choose them to be λ 0,λ1andλ2. Following the Fierz relations we can express local quartic terms pro-portional to λ 3,λ4,λ5as linear combinations of above three, see Eq. ( E4). Whenever we generate four-fermion interactions proportional to λ3,4,5during the coarse graining (discussed in Sec. VI), they can immediately be expressed in terms of λ0,1,2, and the interacting model deﬁned in terms of λ0,1,2 [see Eq. ( 37) below] always remains closed under the RG procedure to any order in the perturbation theory. 165139-14INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) FIG. 12. The Feynamn diagram contributing to the leading order bare susceptibility ( χM), given in Eq. ( 39), for zero external mo- mentum and frequency. The reddashed ( black solid) lines represent the order parameter (fermionic) ﬁelds. The two vertices ( green dots) are accompanied by the appropriate eight-dimensional Hermitianorder-parameter matrix M, appearing in the corresponding fermion bilinear /Psi1 † NamM/Psi1Nam. The imaginary time Euclidean action for the interacting system is given by Sint=S0−/integraldisplay dτddr/bracketleftBigg λ0(/Psi1†/Psi1)2+λ13/summationdisplay j=1(/Psi1†/Gamma1j/Psi1)2 +λ25/summationdisplay j=4(/Psi1†/Gamma1j/Psi1)2/bracketrightBigg , (37) where /Psi1≡/Psi1(τ,r) and /Psi1†≡/Psi1†(τ,r). Under the rescal- ing of space-time(imaginary) coordinates and the fermionicﬁelds, see Eq. ( 21), the local four-fermion interaction scales as λ j→e(d−z)/lscriptλj. (38) Hence the scaling dimension of quartic couplings is [ λj]= z−d.F o r z=2 and d=3, [λj]=−1, and any weak local interaction is an irrelevant perturbation and leaves the Lut- tinger fermions unaffected. Therefore any ordering sets in atan intermediate strength of coupling through QPT. In Sec. VI, we demonstrate appearances of various BSPs using a RG anal-ysis, controlled via an /epsilon1expansion, where /epsilon1=d−2, about thelower-critical two spatial dimension of this theory. Within the framework of the /epsilon1expansion, such QPTs take place at a critical interaction strength λ ∗ j∼/epsilon1, and in three spatial dimensions ( d=3)/epsilon1=1. Before proceeding to the RG anal- ysis, we seek to gain some insight into the propensity towardvarious orderings by computing the corresponding mean-ﬁeldsusceptibility for a wide range of the mass anisotropy param-eter (α). Readers interested in the RG analysis may skip the following discussion and directly go to Sec. VI. A. Mean-ﬁeld susceptibility To gain insights into the propensity toward the formation of various orderings, we ﬁrst compute the bare mean-ﬁeld sus-ceptibility ( χ M) of all possible symmetry allowed fermionic bilinears /Psi1† NamM/Psi1Nam, where Mis an eight-dimensional Her- mitian matrix (see Sec. IV). For simplicity, we set μ=0. For zero external momentum and frequency, this quantity is givenby χ M=−1 2/integraldisplayd3k (2π)3/summationdisplay iωnTr[MG k(iωn)MG k(iωn)]. (39)FIG. 13. Bare mean-ﬁeld susceptibility [see Eq. ( 39)] for zero external momentum and frequency (see Fig. 12for the relevant Feynman diagram) for various orderings at zero temperature andchemical doping, as a function of α[parametrizing the anisotropy between the mass parameters in the T 2gandEgorbitals]. Here, χ is measured in units of m/Lambda1.F o rα=π 4, two nematic orders and two d-wave pairings (belonging to the T2gandEgrepresentations) possess equal susceptibilities, and so do two magnetic orders (within the A2u andT1urepresentations). The A1gs-wave pairing always possesses the largest susceptibility (for any α)a si tr e p r e s e n t sa mass for spin- 3/2 fermions. Susceptibilities for the s-wave pairing, T2gnematic andA2umagnetic orders display exact degeneracy as α→π 2,w h e n all of them become mass (see Sec. VA2 ). On the other hand, as α→0t h e s-wave pairing and Egnematicity become mass and their bare susceptibilities are degenerate and largest (see Sec. VA3 ). For detailed discussion consult Sec. VA. The relevant Feynman diagram is shown in Fig. 12and the “−” sign arises from the fermion bubble. Here Gk(iωn)i s the fermionic Green’s function in the Nambu doubled basis.The factor of 1 /2 takes care of the artiﬁcial Nambu doubling. Results are displayed in Fig. 13. Next we discuss the scaling ofχ Min different channels for a few speciﬁc values of the mass anisotropy parameter. 1. Isotropic Luttinger semimetal ( α=π 4) Forα=π 4, the effective masses for the T2gandEgorbitals are equal (i.e., m1=m2) and the system enjoys an enlarged spherical symmetry. Since each one of the ﬁve /Gamma1matrices (representing two nematic orders) anticommutes with fourmatrices and commutes with one matrix appearing in theLuttinger Hamiltonian, two nematic orders belonging to theT 2g(red curve) and Eg(orange curve) representations possess equal susceptibility. On the other hand, all ten commuta- tors (representing various magnetic orders) anticommute withthree and commute with two matrices appearing in this model.Hence, magnetic orders in the A 2u(purple curve) and T1u (magenta curve) channels also possess equal susceptibility. Two copies of the d-wave pairing, transforming under the T2g (dark green curve) and Eg(dark yellow curve) representations, have degenerate susceptibilities, as all ﬁve d-wave pairing ma- trices commute with four matrices and anticommute with onlyone matrix appearing in the Luttinger model. As the s-wave pairing fully anticommutes with the Luttinger Hamiltonian, 165139-15SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) it always possesses the largest susceptibility (blue curve) for anyα. Susceptibilities for different BSPs (characterized by the fermion bilinear /Psi1† NamM/Psi1Nam)∼NM anti, the number of matrices inˆhNam L(k) anticommuting with M. This simple correspon- dence is operative irrespective of the choice of α. Note that in an isotropic system NM anti=5,4,3 and 1 for the s-wave pairing, nematicity, magnetic orders and d-wave pairings, re- spectively, hence χM/parenleftbigg α=π 4/parenrightbigg :s−wave>nematic >magnetic >d-wave. Computation of the bare susceptibility suggests a strong propensity toward the formation of s-wave pairing and two nematic orders in the world of interacting spin-3 /2 fermions with isotropic dispersion. The magnetic orders and d-wave pairings are expected to be suppressed near α=π 4,a tl e a s t at zero temperature (see Figs. 3and16). We also note that the gain in free energy (see Sec. IV D ) and the mean-ﬁeld susceptibility follow the same hierarchy /Delta1F,χM∼NM anti. 2. Anisotropic Luttinger semimetal near α=π 2 When the effective mass in the T2gorbital becomes sufﬁ- ciently large, the Luttinger model simpliﬁes to lim α→π 2ˆhNam L(k)=−η3k2 2m25/summationdisplay j=4/Gamma1jˆdj(ˆk). (40) This Hamiltonian possesses an emergent SU(2) ⊗U(1) chi- ral symmetry, where {/Gamma145/Gamma1j}with j=1,2,3 are the three generators of an SU(2) rotation, whereas a U(1) rotation isgenerated by ˆN=η 3/Gamma10, the number operator. In this limit, the Hamiltonian is similar to the one for spinless fermionsin Bernal-stacked bilayer graphene, which altogether supportssixmasses, given by 10 Mπ 2=⎧ ⎪⎪⎨ ⎪⎪⎩SO(5) vector/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright η3(/Gamma11,/Gamma12,/Gamma13)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T2gnematic,(η1,η2)/Gamma10/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright s-wave,η 0/Gamma145/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright A2umagnet⎫ ⎪⎪⎬ ⎪⎪⎭. (41) Note that three components of the T 2gnematicity break the continuous SU(2) chiral symmetry, while the s-wave pairing breaks the global U(1) symmetry. On the other hand, the A2u magnet transforms as a scalar under the chiral rotation and breaks only time-reversal symmetry. These three mass orderspossess the largest and equal susceptibilities as α→ π 2,s e e Fig. 13. Therefore repulsive interactions favor two excitonic masses for zero [see Figs. 16(a) and16(c) ] and s-wave pairing [see Fig. 6(a)] for ﬁnite chemical doping. Also note that each member of the following vector M/prime π 2=⎧ ⎪⎪⎨ ⎪⎪⎩multiplet of SO(3) vectors/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright η3(/Gamma14,/Gamma15)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Egnematic,(η1,η2)(/Gamma14,/Gamma15)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Egd-wave⎫ ⎪⎪⎬ ⎪⎪⎭. (42) 10When an order parameter matrix fully anticommutes with the noninteracting Hamiltonian, we coin it as mass order.anticommutes and commutes with one matrix appearing in limα→π 2ˆhNam L(k). Hence, Egnematicity and d-wave pairing have identical susceptibilities as α→π 2,b u t χM/primeπ 2<χMπ 2. As a result such an anisotropic system can accommo- date an Egd-wave pairing at ﬁnite chemical doping, see Fig.6(c). However, T1umagnet and T2gd-wave superconduc- tor have exactly zero susceptibility as they fully commute with limα→π 2ˆhNam L(k). Hence, onset of these two orders is unlikely whenα≈π 2. 3. Anisotropic Luttinger semimetal near α=0 Finally, we compare the susceptibility for various orders when the mass of the Egorbital becomes sufﬁciently large. The Luttinger Hamiltonian then takes the form11 lim α→0ˆhNam L(k)=−η3k2 2m13/summationdisplay j=1/Gamma1jˆdj(ˆk), (43) which possesses an emergent U(1) ⊗U(1) symmetry, gen- erated by η0/Gamma145andη3/Gamma10. The mass orders in this limiting scenario constitute the following vector M0=⎧ ⎪⎪⎨ ⎪⎪⎩SO(4) vector/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright η3(/Gamma14,/Gamma15)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Egnematic,(η1,η2)/Gamma10/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright s-wave⎫ ⎪⎪⎬ ⎪⎪⎭. (44) and consequently the Egnematicity and s-wave pairing ac- quire an identical and the largest susceptibility as α→0, see Fig. 13. Hence, a competition between these two ordered phases can be anticipated near α=0 [see Figs. 6(b) and 16(b) ]. Any order parameter from the following vector anti- commutes with two matrices and commutes with one matrixappearing in Eq. ( 43) M /prime 0=⎧ ⎪⎪⎨ ⎪⎪⎩multiplet of SO(3) vectors/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright η3(/Gamma11,/Gamma12,/Gamma13)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T2gnematic,η 0/Gamma145(/Gamma11,/Gamma12,/Gamma13)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright T1umagnet⎫ ⎪⎪⎬ ⎪⎪⎭, (45) and they also possess degenerate susceptibilities, but χ M/prime 0<χM0. As a result, a competition between T1umagnet and T2gne- maticity can also be observed around α=0, see Fig. 16(d) . On the other hand, the T2gd-wave pairing matrices anti- commute with one matrix and commute with two matricesappearing in Eq. ( 43) and its susceptibility is smaller than the orders appearing in M 0andM/prime 0. Nonetheless, when as- sisted by ﬁnite chemical doping, the T2gd-wave pairing can be realized even for repulsive magnetic interaction in the T1u channel, as shown in Fig. 6(d). Finally, we note that A2umag- net and Egd-wave pairing fully commute with lim α→0ˆh(k) 11This Hamiltonian is quite similar to the one for three-dimensional massless Dirac fermions, with the crucial difference that for the Dirac Hamiltonian ˆdj(ˆk)∼kj, while for the Luttinger Hamiltonian ˆdj(ˆk)∼\|/epsilon1jlm\|ˆklˆkm,w h e r e j,l,m=1,2,3. 165139-16INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) FIG. 14. (a) Bare four-fermion interaction vertex ( /Psi1†M/Psi1)2,a n d (b) self-energy correction due to four-fermion interaction. Note con-tribution from Feymann diagram (b) is ﬁnite only when the chemical potential ( μ) is ﬁnite and renormalizes μ(see Sec. VI). Feynman diagrams [(c) and (f)] yield corrections to the bare interaction vertexto the leading order in the /epsilon1expansion, where /epsilon1=d−2. Here, solid lines represent fermions, and MandNare four-dimensional Hermitian matrices. While the red lines in (b)-(f) correspond to thefast modes (living within a thin Wilsonian shell /Lambda1e −/lscript<\|k\|</Lambda1, where /Lambda1is the ultraviolet momentum cutoff), the black lines are the slow modes with \|k\|</Lambda1 e−/lscript. Recall /lscriptis the logarithm of the renormalization group scale. and possess zero susceptibility. Hence, onset of these two orders around α=0 is unlikely. Note that various matrices appearing in Mπ 2,M/prime π 2,M0,M/prime 0 can form composite order parameters and the enlarged sym- metries among distinct orderings are displayed in Eqs. ( 41), (42), (44), and ( 45). Such enlargement of order-parameter vectors plays an important role in determining the conﬂuenceof competing orders, which we discuss in Sec. VI E. VI. RENORMALIZATION GROUP ANALYSIS After gaining insight into the propensity toward various orderings in the Luttinger system, next we seek to investi-gate the onset of different BSPs and the competition amongthem within the framework of an unbiased RG analysis. Thiswill allow us to go beyond the mean-ﬁeld analysis, presentedin the last section, and systematically incorporate ﬂuctua-tions. In what follows we here restrict ourselves to the leadingorder in the /epsilon1expansion, where /epsilon1=d−2, and account for corrections to the bare interaction vertices ( λ js) to quadratic order in the coupling constants. The relevant Feynman di-agrams are shown in Fig. 14. After performing summation over fermionic Matsubara frequencies ω n=(2n+1)πTwith −∞/lessorequalslantn/lessorequalslant∞, we integrate out a thin Wilsonian shell /Lambda1e−/lscript<\|k\|</Lambda1 to arrive at the following RG ﬂow equations βgi=−/epsilon1gi+2/summationdisplay j=0g2 jHi jj(α,t,μ)+2/summationdisplay j,k=0/prime gjgkHi jk(α,t,μ), (46) fori=0,1,2, where βX≡dX/d/lscript, in terms of dimensionless quantities deﬁned in Eq. ( 9). The prime symbol in the summation indicates that j>k. For notational clarity, we take ˜μ→μ, and the functions Hi jk(α,t,μ)f o r i,j,k=0,1,2FIG. 15. (a) The bare vertex associated with the source term /Psi1† NamM/Psi1Nam. The leading order renormalization of such vertices arises from Feynman diagrams (b) and (c), yielding the RG ﬂow of the source terms, displayed in Eq. ( 49). Here, wavy lines stand for the source ﬁeld, while solid lines for fermions, and the dashed lines for the interaction vertex. The black (red) solid lines represent slow (fast) modes. are shown in Appendix F. Due to their lengthy expressions (which are not very instructive in particular) we here onlydisplay the schematic form of the ﬂow equations. Bothtemperature and chemical potential also ﬂow under coarsegraining [see ﬁrst line of Eq. ( 10)] as relevant perturbations with bare scaling dimensions [ t]=[μ]=z, where z=2f o r the Luttinger system. These two ﬂow equations can then besolved, respectively yielding t(/lscript)=t(0)e z/lscript,μ (/lscript)=μ(0)ez/lscript, (47) where t(0) and μ(0) are the bare values. We supply these solutions to Eq. ( 46) to ﬁnd the phase diagram of interacting Luttinger fermions using the following prescription.12 While the divergence of at least one of the quartic cou- plings (i.e., gi→+ ∞ ) under coarse graining indicates the onset of a BSP, to unambiguously determine the pattern ofsymmetry breaking we also account for the leading orderRG ﬂow of all source terms, associated with different BSPs,discussed in Sec. IV. The effective action in the presence of all symmetry allowed fermionic bilinears [see Eqs. ( 25)–(28)] is given by S s=/integraldisplay dτd3r/Psi1† Nam/parenleftbigˆhexc local+ˆhpair local/parenrightbig /Psi1Nam, (48) with/Psi1† Nam≡/Psi1† Nam(τ,x) and/Psi1Nam≡/Psi1Nam(τ,x)a st w oi n - dependent Grassmann variables. Relevant Feynman diagramsare shown in Fig. 15. The resulting RG ﬂow equations take the following schematic form dln/Delta1 i d/lscript−2=2/summationdisplay j=0Fj i(α,t,μ)gj. (49) See Appendix Ffor explicit form of these ﬂow equations. The quantities appearing on the right-hand side of each equation,represent the scaling dimension of the corresponding order parameter. We simultaneously run the ﬂow of the quartic couplings (g js) and the source terms ( /Delta1js). When at least one of the 12We should note that Feynman diagram (b) in Fig. 14provides in- teraction driven corrections (linear in gi)t oμ.H o w e v e r ,t om a i n t a i n the order by order correction to quartic interactions, we neglect such corrections in Eq. ( 46) within the framework of the leading order /epsilon1 expansion. 165139-17SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) FIG. 16. Speciﬁc cuts of the global phase diagram of an interacting Luttinger semimetal at zero temperature, obtained from an RG analysis (see Sec. VI A ). Here αis the mass anisotropy parameter. For α=π 4, the system possesses an enlarged spherical symmetry, whereas for α=0(π 2)t h e Eg(T2g) orbital possess inﬁnite mass (see Sec. III). Here g1(g2) captures the strength of repulsive interactions in the T2g(Eg) nematic channels. Magnetic interactions in the A2u(T1u) channel is denoted by g3(g4). Notice that even in the absence of a Fermi surface, pure repulsive nematic interactions are conducive to s-wave pairing among spin-3 /2 fermions at zero temperature [see (a) and (b)]. In all the panels, the coupling constants are measured in units of /epsilon1. The ordered states correspond to the gray shaded regions (see footnote 2), and its boundaries with the Luttinger semimetal (white regions), occupied by distinct broken symmetry phases are identiﬁed by different colors. quartic couplings diverges and ﬂows toward →+ ∞ (thus indicating onset of a BSP), we identify the source term (say/Delta1 j) that diverges toward →+ ∞ fastest (assuming a possible scenario when more than one source term diverge toward+∞). The BSP is then characterized by the order parame- ter/Delta1 j/negationslash=0. We use this strategy to determine various cuts of the global phase diagram of spin-3 /2 Luttinger fermions, displayed in Fig. 16(fort=μ=0), Fig. 3(forμ=0, but ﬁnite- t), and Figs. 4–6(for ﬁnite- tand ﬁnite- μ). Next we discuss these cases in three subsequent sections. Unless the coupling constants are ﬁne tuned, only one of them diverges fastest toward +∞. The selection rule among the competing orders, discussed in Sec. II B3 (see also Sec. VI E for details), is then determined in terms of the fastest diverging coupling constant. Appearance of various orderedphases in all the cuts of the global phase diagram are thenconsistent with the selection rule, even in the presence ofmultiple running coupling constants.A. Quantum criticality in Luttinger semimetal We ﬁrst discuss the effects of electronic interactions on a LSM (i.e., when the chemical potential is pinned at the bandtouching point) at zero temperature. The RG ﬂow equationsforμ=0 and t=0 can be derived by taking the limit μ→0 and then t→0i nE q .( 46), suggesting that weak interactions are irrelevant perturbations and any ordering takes place atﬁnite coupling g i∼/epsilon1through a QPT. Next we discuss the following three cases separately (i) isotropic Luttinger system(α= π 4), large mass for (ii) the T2gorbital ( α→π 2) and (iii) theEgorbital ( α→0). Such systematic analysis will allow us to anchor our anticipations regarding the nature of BSPsfrom the mean-ﬁeld susceptibility, discussed in Sec. VA.N o t e att=μ=0, the system is devoid of any natural infrared cutoff as /lscript t ∗,/lscriptμ ∗→∞ . Hence, we run the ﬂows of quartic couplings up to an RG time /lscript∗→∞ to determine the stability of LSM and the ﬂows of the source terms to pin the pattern ofsymmetry breaking. 165139-18INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) TABLE II. Locations of quantum critical points (QCPs) possess- ing one unstable direction and bicritical points (BCPs) possessing two unstable directions for three speciﬁc choices of the mass anisotropy parameter α[see Eq. ( 18)]. Note for α=π 4the Luttinger model possesses an emergent spherical symmetry, while for α→ 0(π 2) the band dispersion in the Eg(T2g) orbitals becomes almost ﬂat. Forα=π 4the coupled RG ﬂow equations support only one QCP, denoted by QCP1π 4, while for α→π 2andα→0 we ﬁnd two QCPs, respectively denoted by QCPj π 2and QCPj 0forj=1,2, see also Ref. [ 44]. For these two limiting cases the ﬂow equations also support one BCP, respectively identiﬁed as BCP π 2and BCP 0. Existence of such BCP in the presence of two QCPs is necessary to ensure the continuity of the RG ﬂow trajectories and separate the basins ofattraction of two QCPs. All coupling constants at the ﬁxed points are measured in units of /epsilon1,w h e r e /epsilon1=d−2 measures the deviation from the lower critical two spatial dimensions, where all local quarticinteractions are marginal . Note that QCP 1π 4,Q C P1π 2,Q C P1 0represent the same QCP, only its location shifts as we tune α. By contrast, QCP2π 2and QCP2 0are solely introduced by mass anisotropy and bear no analog to the ﬁxed points found around α=π/4. Besides the above QCPs and the BCPs, there always exists a trivial Gaussian ﬁxed point, representing the noninteracting Luttinger semimetal, at (g∗ 0,g∗ 1,g∗ 2)=(0,0,0), endowed with three stable directions. See Appendix Gfor details. α=π 4α=1.5 α=0.04 Coup. QCP1π 4QCP1π 2QCP2π 2BCP π 2QCP1 0QCP20BCP 0 g∗ 0×1032.24 1.49 −2.93 −1.58 1.95 −5.98 −5.60 g∗ 1×1032.03 1.47 −2.84 −1.33 1.71 4.80 4.97 g∗ 2×1032.03 1.26 3.76 5.54 1.94 −5.87 −5.48 1. Isotropic Luttinger semimetal ( α=π 4) Forα=π 4, the coupled RG ﬂow equations support only one QCP, reported in Table IIand identiﬁed as QCP1 π 4, besides the trivial (and fully stable) Gaussian ﬁxed point at g∗ 0=g∗ 1= g∗ 2=0 (representing the stable LSM). This QCP controls QPTs from LSM to various BSPs (depending on the interac-tion channel). To gain insight into the nature of the candidatecompeting BSPs, we compute the scaling dimensions for allfermion bilinears at this QCP. The results are summarized inTable III. Note that the s-wave pairing has the largest scaling dimension at this QCP, while two nematic orders possess de-generate but second largest (and positive) scaling dimensions.However, the rest of the fermion bilinears possess negative scaling dimensions. Notice that the scaling dimensions fordifferent orders at this QCP follow the same hierarchy as themean-ﬁeld susceptibilities, discussed in Sec. VA1 . The phase diagrams in an interacting LSM are displayed in Fig. 3for various interaction channels. Around α= π 4strong repulsive nematic interactions ( g1andg2) favor s-wave pairing even in the absence of a Fermi surface (since μ=0), see Figs. 3(a) and3(b). On the other hand, strong magnetic interactions in the A2u(g3) and T1u(g4) channels respectively support T2g[see Fig. 3(d)] and Eg[see Fig. 3(c)] nematicities. However, we could not ﬁnd any magnetic ordering or d-wave pairing in the very close vicinity to α=π 4, at least when t=0, see Fig. 16. Therefore the computation of mean-ﬁeld susceptibilities and scaling dimensions of fermion bilinears, inTABLE III. Scaling dimensions (in units of /epsilon1) of various source (Sr) terms or fermion bilinears at different ﬁxed points (reported in Table II), obtained by substituting ﬁxed point values of the coupling constants g∗ 0,g∗ 1andg∗ 2on the right-hand side of the corresponding ﬂow equation [see Eq. ( 49)] at t=0a n d μ=0. At each QCP the largest scaling dimension is shown in bold, while the secondlargest ones are shown in italic. At the two magnetic QCPs (QCP 2π 2 and QCP2 0), the largest scaling dimensions for the superconducting channel (namely, d-wave pairings) are shown in blue. α=π 4α=1.5 α=0.04 Sr QCP1π 4QCP1π 2QCP2π 2BCP π 2QCP1 0QCP20BCP 0 /Delta10 000000 0 /Delta11 0.426 0.531 −0.238 0.378 0.353 0.614 0.656 /Delta12 0.426 0.275 0.325 0.678 0.539 −0.153 −0.070 /Delta13−0.076 −0.204 1.147 1.006 −0.004 −0.024 −0.024 /Delta14−0.076 −0.007 −0.021 −0.030 −0.134 0.731 0.699 /Delta15−0.076 −0.094 0.182 0.092 −0.062 0.020 0.011 /Delta1p A1g0.551 0.545 −0.254 0.355 0.547 −0.166 −0.083 /Delta1p T2g−0.034 −0.004 −0.011 −0.016 −0.059 0.016 0.006 /Delta1p Eg−0.034 −0.088 0.169 0.073 −0.002 −0.012 −0.012 corroboration with our unbiased RG calculation, show that the strongly interacting isotropic LSM becomes unstable towardthe formation of two nematic orders and s-wave pairing at the lowest temperature. 2. Anisotropic Luttinger semimetal: α→π 2 Next we turn our focus to the vicinity of α=π 2.T h e coupled ﬂow equations then support two QCPs (denoted by QCP1 π 2and QCP2 π 2) and onebicritical point (denoted by BCP π 2), see Table II. The BCP possesses two unstable di- rections. Notice QCP1 π 2is the same as QCP1 π 4, only shifted toward weaker coupling, which can be veriﬁed from the fact that the signs of the coupling constants and scaling dimensionsfor all fermion bilinears are identical at these two QCPs (see Tables IIandIII). On the other hand, QCP 2 π 2is new and bears no resemblance to any ﬁxed points we found for α=π 4. This QCP is induced by the mass anisotropy of Luttinger fermions. At this QCP, theA 2umagnetic ( Egnematic) order possesses the largest (second largest) scaling dimension, see Table III. Therefore, when repulsive interaction in the A2uchannel dominates among various ﬁnite range components of the Coulomb interaction,the Luttinger semimetal can display a competition betweenthese two orders as we approach α= π 2from an isotropic system, see Fig. 16(c) [consult also Sec. VI E]. Among three possible local pairings the Egd-wave su- perconductor possesses the largest (and positive) scaling dimension at this QCP. Hence, the emergence of this pairedstate can be anticipated when the LSM is doped away from thecharge-neutrality point around α= π 2, see Fig. 6(c) (consult Sec. VI D for details). 3. Anisotropic Luttinger semimetal: α→0 Finally, we approach the opposite limit, when the mass in theEgorbital becomes sufﬁciently large, i.e., m2/greatermuchm1or 165139-19SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) equivalently α→0. In this regime the RG ﬂow equations support two QCPs (denoted by QCP1 0and QCP20) and a BCP (denoted by BCP 0), see Table II. Note that QCP1 0is similar to QCP1 π 4, only shifted toward weaker coupling. By contrast, QCP2 0is induced by the mass anisotropy. At this QCP, the T1u magnetic ( T2gnematic) order possesses the largest (second largest) scaling dimension, see Table III. Therefore, when repulsive interaction in the T1uchannel dominates we expect a strong competition between these two orderings, as onetunes toward α→0 starting from an isotropic system, see Fig.16(d) . Also note that among three local pairings, the d-wave one transforming under the T 2grepresentation possesses the largest (and positive) scaling dimension. Hence, around α=0w e expect onset of this paired state when the chemical potentialis placed away from the band touching point, see Fig. 6(d) and discussion in Sec. VI D . Finally, we comment on the role of the BCPs possessing two unstable directions, see Table II. Note that a BCP can only be found when there exists two QCPs in the three-dimensionalcoupling constant space ( g 0,g1,g2). The existence of a BCP separates the basin of attraction of two QCPs and ensurescontinuity of the RG ﬂow trajectories in coupling constantspace. B. Universality class and critical exponents All QCPs, listed in Table II, are characterized by only one unstable direction or positive eigenvalue of the corresponding stability matrix (see Appendix G), deﬁned as Mij(g0,g1,g2)=d dgjβgi. (50) To the leading order in the /epsilon1expansion the positive eigenvalue at all QCPs is exactly /epsilon1, which in turn determines the correla- tion length exponent (ν) according to ν−1=/epsilon1+O(/epsilon12). (51) For the physically relevant situation, /epsilon1=1 and we obtain ν= 1. The fact that νis the same at all QCPs is, however, only an artifact of the leading order /epsilon1expansion. Generically νis expected to be distinct at different QCPs, once we account forhigher order corrections in /epsilon1. Since to the leading order in the /epsilon1expansion, local inter- actions do not yield any correction to fermion self-energy, thedynamic scaling exponent (z) at all interacting QCPs is z=2+O(/epsilon1). (52) Together the correlation length and dynamic scaling expo- nents determine the universality class of all continuous QPTsfrom a LSM to various BSPs. In the next section, we willdiscuss the imprint of these two exponents on the scalingof the transition temperature ( t c) associated with the ﬁnite temperature order-disorder transitions. C. Interacting Luttinger semimetal at ﬁnite temperature Even though our RG analysis in the previous section was performed at zero temperature, one can still ﬁnd the imprint ofvarious interacting QCPs at ﬁnite temperatures. The RG ﬂowequations at ﬁnite temperature can be derived from Eq. ( 46)b y taking the μ→0 limit in H l jk(α,t,μ). Recall that tempera- ture introduces a natural infrared cutoff for the ﬂow equations /lscriptt ∗[see Eq. ( 10)]. Physically such an infrared cutoff corre- sponds to a scenario when the renormalized temperature t(/lscript) becomes comparable to the ultraviolet energy E/Lambda1=/Lambda12/(2m) (see Fig. 1), beyond which the notion of quadratically dis- persing fermions becomes moot and the ﬂow equations fromEq. ( 46) lose their jurisdiction. To capture the effects of electronic interactions in a LSM at ﬁnite- t, we run three quartic coupling constants up to a scale /lscript/lessorequalslant/lscript t ∗. Now depending on the bare strength of interactions, two situations arise (a)g(/lscriptt ∗)<1,or (b) g(/lscriptt ∗)>1, respectively representing a disordered LSM (without any long-range ordering) or onset of a BSP at ﬁnite temperature.Hence, for a given strength of interaction g>g ∗, where g∗is the requisite critical strength of interaction for a BSP at t=0, we always ﬁnd a temperature tcabove (below) which the BSP disappears (appears). We identify tcas the critical or transi- tion temperature . None of the coupling constants diverge for t>tc. All ordered phases can display true long-range order at ﬁnite tin three dimensions and tccorresponds to a genuine transition temperature. General scaling theory suggests that the transition temper- ature scales as [ 91,92] tc∼δνz, (53) forδ/lessmuch1, where δ=(g−g∗)/g∗is the reduced distance from a QCP, located at g=g∗(say). Hence, for interacting LSM, tc∼δ2for/epsilon1=1 (i.e., the prediction from the lead- ing order /epsilon1expansion). The scaling of critical or transition temperature for various choices of coupling constants andresulting BSPs, and different choices of the mass anisotropyparameter α,a r es h o w ni nF i g . 17, indicating a fairly good agreement with the ﬁeld theoretic prediction t c∼δ2around all QCPs, reported in Table II. 1. Phase diagrams at ﬁnite temperature Besides the scaling of the transition temperature, we also investigate the phase diagram of an interacting LSM at ﬁ-nite temperature, allowing us to demonstrate the competitionbetween condensation energy gain and entropy. For concrete-ness, we focus on the isotropic system ( α= π 4), where this competition is most pronounced. As argued in Sec. IV D ,t h e onset of s-wave pairing leads to the maximal gain in conden- sation energy, while the two nematic orders produce higherentropy in comparison to the former. Two speciﬁc cuts of theglobal phase diagram, Figs. 3(a) and3(b), show that while s-wave pairing is realized at low temperature, nematicities set in at higher temperature as we increase the strength of nematicinteractions ( g 1andg2) in the system. By contrast, when we tune the magnetic interactions (namely g3,g4), an isotropic LSM becomes unstable in favor of two nematic orders at t=0. Such an outcome can be substantiated from the simple picture of condensation energygain, as A 2uand T1umagnetic orders accommodate Weyl nodes (yielding more entropy), while nematicities produce 165139-20INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) FIG. 17. Scaling of dimensionless transition temperature ( tc) with the reduced distance ( δj=(gj−g∗ j)/g∗ j) from the quantum critical point, located at gj=g∗ jforj=1,2,3,4, see Sec. VI C . The mass anisotropy parameter αand nature of the ordered phase for δj>0 in each panel are quoted explicitly. Note that tc∼δ2 jforδ/lessmuch1, in agreement with ﬁeld theoretic prediction. The QCPs controlling the Luttinger semimetal to ordered states quantum phase transitions at t=0 are quoted in each panel. Red dots represent numerically obtained transition temperature and the blue lines correspond to least-square ﬁt of tc∼δ2 j. anisotropic spectral gap (leading to higher gain of condensa- tion energy), see Sec. IV D . As we tune the strength of the A2u(T1u) magnetic interaction, Eg(T2g) nematic order sets in at lower and A2u(T1u) magnet at higher temperature, see Figs. 3(c)and3(d). The LSM is endowed with the largest entropy in the global phase diagram of interacting spin-3 /2 fermions, since /rho1(E)∼√ E, in comparison to any BSP. Consequently, the requisite strength of interactions for any ordering increases with in-creasing temperature, irrespective of the nature of the BSP,see Fig. 3. Therefore our RG analysis for an interacting LSM at ﬁnite temperature corroborates the energy-entropy competi-tion picture and substantiates the following outcome: ordered phases providing larger condensation energy gain are found atlower temperature, while at higher temperature, phases withlarger entropy are favored . This observation is also consistent with the notion of reconstructed band structure and emergenttopology, discussed in Sec. IV C . Therefore the phase diagram of an interacting LSM at ﬁnite temperature is guided by topo-logical structure (gapped or nodal) of competing BSPs. We close this section by answering the following question: why do we ﬁnd two magnetic orders in an isotropic LSM atﬁnite temperature, since this system supports only one QCP,see Table II, where all the magnetic orders bear negative scaling dimension (see Table III)? Note that any QCP can only be accessed at t=0, whereas ﬁnite- tintroduces an infrared cutoff ( /lscript t ∗) for the RG ﬂow of the quartic couplings, and thus prohibits a direct access to any QCP. Hence, at ﬁnite- t, when magnetic interactions are sufﬁciently strong, the system canbypass the basin of attraction of QCP 1 π 4and nucleate magnetic phases at moderately high temperature. D. RG analysis in Luttinger metal Finally we proceed to the RG analysis when the chemical potential ( μ) is placed away from the band touching point. The chemical potential introduces yet another infrared cutoff/lscriptμ ∗[see Eq. ( 11)], suggesting that the RG ﬂow equations of the three quartic coupling constants should be stopped whenthe renoramlized chemical potential μ(/lscript) reaches the scale of the band-width E/Lambda1=/Lambda12/(2m). At ﬁnite temperature and chemical doping, two infrared scales compete and the smaller one/lscript∗(say), given by /lscript∗=min.(/lscriptμ ∗,/lscriptt ∗) (54) determines the ultimate infrared cutoff for the RG ﬂow of gj. Now depending on the bare strength of the coupling constantsone of the following two situations arises: (a)g j(/lscript∗)>1o r( b ) gj(/lscript∗)<1. While (a) indicates onset of a BSP, (b) represents a stable Luttinger metal. The resulting phase diagrams for variouschoices of chemical potential, temperature, coupling constantsand the mass anisotropy parameter ( α) are shown in Figs. 4–6. We note that in an isotropic system and for strong enough nematic interactions an s-wave pairing can be realized even for zero chemical doping [see Figs. 3(a) and 3(b)]. With increasing doping the s-wave pairing occupies larger portion of the phase diagram, while two nematic phases get pushedtoward stronger coupling, see Fig. 4. However, the d-wave pairings do not set in for zero chemical doping. Nonetheless,when the magnetic interactions in the A 2uandT1uchannels are strong, the presence of a Fermi surface is conducive tothe nucleation of d-wave pairings, belonging to the E gandT2g representations, respectively, see Fig. 5. Now we focus on the anisotropic system. We chose the mass anisotropy parameter αsuch that at zero chemical dop- ing the repulsive electronic interactions accommodate eithernematic or magnetic orders, see Fig. 16. Speciﬁcally for α= 1.5 (close to π 2) the system enters into the T2gnematic [see Fig. 6(a)]o r A2umagnetic [see Fig. 6(c)] phase, while for α=0.1 (close to 0) we ﬁnd Egnematic [see Fig. 6(b)]o rT1u magnetic (see Fig. 7) order. For such speciﬁc choices of α, the QPTs into T2gnematic, Egnematic, A2umagnetic and T1u magnetic orders are respectively controlled by QCP1 0,Q C P1 π 2, QCP2 π 2and QCP2 0(see Sec. VI A and Table II). At QPT1 0 and QPT1 π 2,t h e s-wave pairing possesses the largest scaling 165139-21SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) dimension among all possible local pairings (see Table III). Hence, in the presence of ﬁnite chemical doping, repulsiveinteractions in the nematic channels become conducive tos-wave pairing, as shown in Figs. 6(a) and6(b). On the other hand, at QPT 2 π 2(QCP2 0), the d-wave pairing belonging to the Eg(T2g) representation possesses the largest scaling dimen- sion. Therefore repulsive interactions in the A2u(g3) and T1u (g4) magnetic channels become conducive to the nucleation ofEg(forα=1.5) and T2g(forα=0.1)d-wave pairings, respectively, as shown in Figs. 6(c)and6(d). We conclude that nematic and magnetic interactions among spin-3 /2 Luttinger fermions are respectively conducive to the s-wave and d-wavepairings . Otherwise, at ﬁnite chemical doping the excitonic orderings set in only for stronger couplings. Such a genericfeature is also consistent with the energy-entropy competi-tion picture as the superconducting phases maximally gap theFermi surface (yielding optimal gain of condensation energy),while exitonic orders are accompanied by a Fermi surfacewith constant DoS (producing more entropy). E. Competing orders and selection rule So far we have presented an extensive analysis of the role of electronic interactions among spin-3 /2 fermions. We showed multiple cuts of the global phase diagram at zeroand ﬁnite temperature and chemical doping (see Figs. 3–5). In addition, we also addressed the imprint of the quadrupolardeformation (or the mass anisotropy parameter α) on a such phase diagram (see Figs. 6,7, and 16). Altogether we unearth a rich conﬂuence of competing orders in this system. In thiscontext an important question arises quite naturally: Is there a selection rule among short-range interactions in differentchannels (such as nematic and magnetic) and various orderedstates (such as s- and d-wave pairings, magnetic phases andnematic orders) for interacting spin-3 /2 fermions? In this section, we attempt to provide an afﬁrmative (at least partially)answer to this question. To this end it is convenient to express the quartic terms ap- pearing in H int[see Eq. ( 36)] in the Nambu doubled basis. For concreteness, we focus on the relevant interaction channels,namely λ 1,2,3,4. In the Nambu basis, these four quartic terms take the form ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩/summationtext 3 j=1(/Psi1†/Gamma1j/Psi1)2 /summationtext5 j=4(/Psi1†/Gamma1j/Psi1)2 (/Psi1†/Gamma145/Psi1)2 /summationtext3 j=1(/Psi1†/Gamma1j/Gamma145/Psi1)2⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ →1 2⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩/summationtext 3 j=1(/Psi1† Namη3/Gamma1j/Psi1Nam)2 /summationtext5 j=4(/Psi1† Namη3/Gamma1j/Psi1Nam)2 (/Psi1† Namη0/Gamma145/Psi1Nam)2 /summationtext3 j=1(/Psi1† Namη0/Gamma1j/Gamma145/Psi1Nam)2⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭. (55) The factor of 1 /2 takes care of the artiﬁcial Nambu doubling. The above question can then be reformulated in the followingway. When at least one of the coupling constants, say λ j, diverges toward +∞, what is the nature of the resulting BSP; or which one of the source terms, appearing in Eq. ( 48),(a) (b) (c) FIG. 18. Schematic representations of SU(2) symmetry among (a)Egnematic and A2umagnetic orders and [(b) and (c)] components ofT2gnematic and T1umagnetic orders (with i/negationslash=j/negationslash=k=1,2,3). Three vertices of each triangle are occupied by the order parameter matrices (in red), represented in the Nambu doubled basis ( /Psi1Nam). Three sides of each triangle represent SU(2) rotations. The genera- tors of SU(2) rotations (also in Nambu doubled basis) are shown in blue. simultaneously diverges toward +∞? This is our quest in this section. When the coupling constant λof the quartic interaction λ(/Psi1† NamMInt Nam/Psi1Nam)2diverges toward +∞ it nucleates a BSP, characterized by the order parameter /angbracketleft/Psi1† NamMor/Psi1Nam/angbracketright, only if one of the following two conditions is satisﬁed (1)Mor=MInt Nam or (2)/braceleftbig Mor,MInt Nam/bracerightbig =0. (56) If, on the other hand, MInt NamandMorhave more than one com- ponent, then selection rule (2) requires a slight modiﬁcation,see footnote 6. The above selection rule can be justiﬁed from the Feynman diagrams shown in Fig. 15. A source term diverges toward +∞ (indicating onset of a BSP) only when the net contribu- tion from diagrams (b) and (c) is positive . Feynman diagram (b) gives nonzero and positive contribution only when condi- tion (1) from Eq. ( 56) is satisﬁed (due to the Trarising from the fermion bubble). Even though diagram (c) then yields anegative contribution, all togther they still produce a positive deﬁnite quantity, since the contribution from (b) dominates over (c), as the former one involves Tr. Hence, when condi- tion (1) is satisﬁed, interaction λcan enhance the propensity toward the formation of a BSP with M or=MInt Nam. On the other hand, when condition (1) is not satisﬁed, only Feynman diagram (c) contributes [due to Trinvolved in (b)]. The contribution from this diagram is positive only when condition (2) is satisﬁed. By contrast, when Mor/negationslash=MInt Nam,j and [ Mor,MInt Nam,j]=0 the net contribution from (b) and (c) isnegative . Interaction coupling λthen does not support such an ordered phase. All phase diagrams presented in this worksupport one of these two selection rules (discussed below). 165139-22INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) (a) (b) FIG. 19. Schematic representation of SU(2) symmetry between theA2umagnetic order and d-wave pairing belonging to the Eg representation. Notations are the same as in Fig. 18.I n( a ) , j=1,2, whereas in (b), j=4,5. The above selection rule can be phrased in a slightly differ- ent fashion. Two ordered phases respectively breaking O(M1) andO(M2) symmetries13can reside next to each other (when we tune the interaction strength in a particular channel, ortemperature or chemical potential) if they constitute an O(M) symmetric composite order parameter, where M 1,M2/lessorequalslantM/lessorequalslantM1+M2. (57) We invite the readers to verify the equivalence between Eqs. ( 56) and ( 57). Note that at the transition between two competing phases the corresponding order parameters do notneed to be equal, as the emergent fermionic quasiparticlespectra inside the competing ordered phases, dictating thecondensation energy gains inside the ordered phases, are typi-cally distinct (see Sec. IV C ). Furthermore, as the neighboring competing orders always mutually anticommute (at least par-tially), a coexistence between them with the two pure phaseson either side of it is a very generic situation (see footnote 2),as in the coexistence regime two orders can then be rotatedinto each other. Such a coexistence between two compet-ing phases can be demonstrated from standard mean-ﬁeld orGinzburg-Landau theory. Next, we discuss some prototypicalexamples to support our claims. (1) The T 2gnematic order [an O(3) order parameter] and s-wave pairing [an O(2) order parameter] constitute an O(5) vector [see Eq. ( 41)], and these two ordered phases reside next to each other, see Figs. 3(a),4(left), 6(a), and 16(a) , when we tune the strength of g1. (2) The E2gnematicity (an O(2) order parameter) and s- wave pairing constitute an O(4) composite order parameter [see Eq. ( 44)], and Figs. 3(a),4(right), 6(b), and 16(b) display a conﬂuence of these two ordered phases, as one tunes theinteraction g 2. (3) The Egnematicity and A2umagnet [described by an O(1) or Z2order parameter] form an O(3) vector [see Fig.18(a) ], and these two ordered phases often (in particular, when g3is tuned) reside next to each other, see Figs. 3(c) and16(c) . (4) One can construct multiple copies of O(3) composite order parameters by combining the components of T2gne- matic and T1u[an O(3) order parameter] magnetic orders [see 13In this notation, a Z2symmetry breaking order parameter is de- noted by an O(1) vector.(a) (b) (c) FIG. 20. Schematic representation of SU(2) symmetry between the components of the T1umagnetic order and T2gd-wave pairing, with i/negationslash=j/negationslash=k=1,2,3 [for (a)–(c)] and m=1,2 in (a) and (c). Notations are the same as in Fig. 18. Figs. 18(b) and18(c) ], and these two phases can be realized by tuning the quartic interaction g4, see Figs 3(c)and16(d) . (5)A2umagnetic order and Egd-wave pairing (an O(2) order parameter) can be combined to form O(3) vectors (see Fig.19). When the chemical potential is ﬁnite and we tune the strength of g3, the system accommodates a paired (magnetic) state at low (high) temperature, see Figs. 5(top) and 6(c). (6) Finally, note that multiple copies of an O(3) vector can be formed by combining the components of T1umagnetic order and T2gd-wave pairing [an O(2) order parameter], see Fig. 20. These two phases reside next to each other at ﬁnite chemical doping when we tune the quartic coupling g4,s e e Figs. 5(bottom) and 6(d). From the matrix representations of all quartic interac- tions [see Eq. ( 55)] and order parameters [see Eqs. ( 26) and (28)] the readers can convince themselves that the above examples are in agreement with our proposed selection rulefrom Eqs. ( 56) and ( 57). It is admitted that we arrive at the conclusion from a leading order RG analysis. However, thealternative version of the selection rule [see Eq. ( 57)] solely relies on the internal symmetry among competing orders. Wetherefore believe that our proposed selection rule is ultimatelynonperturbative in nature, which can be tested in numericalexperiments, for example. VII. SUMMARY AND DISCUSSIONS To summarize we have presented a comprehensive anal- ysis on the role of electron-electron interactions in athree-dimensional Luttinger system, describing a biquadratictouching of Kramers degenerate valence and conductionbands (in the absence of chemical doping) of effective spin-3/2 fermions at an isolated point in the Brillouin zone. This model can succinctly capture the low-energy physics of HgTe[5], gray-Sn [ 6,7], 227 pyrochlore iridates [ 8–12], and half- Heuslers [ 13–15]. For concreteness, we focused only on the short-range components of the Coulomb interaction (such 165139-23SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) as the ones appearing in an extended Hubbard model), and neglected its long-range tail. Due to the vanishing densityof states, namely /rho1(E)∼√ E, sufﬁciently weak, but generic local four-fermion interactions are irrelevant perturbations inthis system. Consequently, any ordering sets in at an in-termediate coupling through quantum phase transitions. Wehere address the instability of interacting Luttinger fermionsat ﬁnite coupling within the framework of a renormaliza-tion group analysis, controlled by a “ small ” parameter /epsilon1, where /epsilon1=d−2. Notice that in two spatial dimensions a biquadratic band touching (similar to the situation in Bernal-stacked bilayer graphene) yields a constant density of states, and local four-fermion interactions are marginal ind=2. Hence, our renormalization group analysis is performed aboutthelower-critical two spatial dimensions. In this framework all quantum phase transitions from a disordered Luttingersemimetal to any ordered phase take place at g ∗ j∼/epsilon1, where gjis the dimensionless coupling constant, and for three di- mensions /epsilon1=1. We here restrict ourselves to repulsive (at the bare level) electron-electron interactions and present multiplecuts of the global phase diagram at zero and ﬁnite temper-ature (see Figs. 3and16) and ﬁnite chemical doping (see Figs. 4–6). Using the renormalization group analysis, we show that in an isotropic system an s-wave pairing and two nematic orders are the prominent candidates for a broken symmetry phaseat zero temperature and chemical doping. While the s-wave pairing only breaks the global U(1) symmetry and producesa uniform mass gap, two nematic orders, transforming undertheT 2gandEgrepresentations, produce lattice distortion along theC3vandC4vaxes, respectively. Such ordered phases can describe either time-reversal symmetry preserving insulatorsor topological Dirac semimetals. However, a collection ofstrongly correlated gapless spin-3 /2 fermions do not show any noticeable propensity toward the nucleation of any mag-netic order or d-wave pairings in an isotropic system at least when t=0. This is so, because the magnetic orders ( d-wave pairings) produce gapless Weyl nodes (nodal loops) aroundwhich the density of states vanishes as /rho1(E)∼\|E\| 2(\|E\|), while the former three orders support gapped spectra. Hence,the magnetic orders and d-wave pairings are energetically in- ferior to s-wave pairing and electronic nematicities. However, with increasing temperature, one ﬁnds a smooth crossoverfrom nematic to magnetic phases, as shown in Fig. 3. Hence, energetically superior orders are found at low tempera-tures, whereas at higher temperature broken symmetry phasespossess larger entropy . This energy-entropy competition is discussed in Sec. IV D and summarized in Fig. 2. We identify that the mass anisotropy ( α), measuring the quadrupolar distortion in the Luttinger system, can be a usefulnonthermal tuning parameter to further explore the territoryof strongly interacting spin-3 /2 fermions [ 28,44]. In par- ticular, we ﬁnd that strong quadrupolar distortions can beconducive for various magnetic orderings even at zero tem-perature. Speciﬁcally, when the electronic dispersion alongtheC 3vaxes becomes almost ﬂat (realized as α→π 2)t h e singlet A2umagnetic order stabilizes at t=0, see Figs. 6(c) and16(c) . On the other hand, when Luttinger fermions are almost nondispersive along the C4vaxes (realized when α→0), the system becomes susceptible toward the formation of a triplet T1umagnetic order, see Figs. 7and16(d) . For these two limiting scenarios, the above two magnetic orders become (al-most) mass [see Secs. VA2 andVA3 ], and their nucleation becomes energetically beneﬁcial even at zero temperature. Irrespective of these details, we realize that all quantum phase transitions from the Luttinger semimetal to symmetry-breaking phases are continuous and controlled by variouscritical points, see Sec. VI A and Table II. To the leading order in the /epsilon1expansion, the universality class of these transitions is characterized by the (a) correlation length exponent ν −1= /epsilon1and (b) dynamic scaling exponent z=2 (see Sec. VI B ). The presence of such quantum critical points manifests itselfthrough the scaling of the transition temperature t c∼\|δ\|νz, yielding tc∼\|δ\|2ford=3o r/epsilon1=1, see Fig. 17, where δ is the reduced distance from the critical point. Finally, we introduce (chemical) doping as another non- thermal tuning parameter to map out the global phase diagramof an interacting Luttinger metal, see Sec. VI D . Since any paired state maximally gaps the Fermi surface, its appearanceat the lowest temperature is quite natural, at least when \|μ\|> 0. By contrast, excitonic phases (insulators or semimetals)become metallike (possessing a ﬁnite density of states) at ﬁ-nite chemical doping, according to the Luttinger theorem [ 87]. Therefore particle-hole orders are accompanied by higher en-tropy due to the presence of a Fermi surface (with a constantdensity of states). To demonstrate the energy-entropy com-petition in a metallic system, we choose the mass anisotropyparameter αsuch that at μ=0 the system only supports excitonic orders. Upon raising (lowering) the chemical po-tential to the conduction (valence) band, we observe that asuperconducting order develops at low temperature and theexcitonic order gets pushed toward higher temperature andstronger interactions, see Figs. 5and6. Therefore the overall structure of the global phase diagram is compatible with theenergy-entropy competition, dictated by the emergent bandtopology of competing broken symmetry phases. Furthermore, we also identify a deﬁnite “selection rule” among competing phases [see Sec. VI E]. From multiple cuts of the global phase diagram of a collection of strongly inter-acting spin-3 /2 fermions, we ﬁnd that two phases can reside in close vicinity of each other if the order parameters describingtwo distinct phases can be combined to form a compositeorder parameter. In other words, two ordered phases, re-spectively described by O(M 1) and O(M2) symmetric order parameters [see footnote 13], can reside next to each otheronly if one can construct an O(M) symmetric composite- vector, where M 1,M2/lessorequalslantM/lessorequalslantM1+M2, from the elements of two individual order parameters. This is so because whenan interaction favors O(M 1) symmetry breaking order, it also enhances the scaling dimension of an O(M2) symmetry break- ing order and vice-versa, when the above selection rule issatisﬁed. Therefore, by tuning a suitable parameter (such astemperature, mass anisotropy parameter, chemical potential)one can induce a transition between two competing phases.As an immediate outcome of this selection rule we realizethat while repulsive interactions (short-range) in the nematicchannels are conducive to s-wave pairing [see Figs. 3(a),3(b), 4,6(a), and 6(b)], magnetic interactions favor nucleation of 165139-24INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) d-wave pairings [see Figs. 5,6(c), and 6(d)] among Luttinger fermions. A few speciﬁc cuts of the global phase diagram corroborate with ones extracted experimentally in Ln 2Ir2O7[16] and half- Heusler compounds [ 37]. For exmaple, a ﬁnite temperature phase transition from A2uor all-in all-out magnetic order to a Luttinger semimetal has been observed in majority of 227pyrochlore iridates (except for Ln =Pr) [ 16], and Fig. 3(c) qualitatively captures this phenomena (for strong enough g 3). By contrast, Pr 2Ir2O7supports a large anomalous Hall effect below 1 .5K [ 25–27]. Note that a triplet T1uor 3-in 1-out magnetic order supports anomalous Hall effect due to thepresence of Weyl nodes in the ordered state (see Sec. IV C and Ref. [ 28]) and the phase diagram from Fig. 3(d) shows the appearance of T 1umagnetic order at ﬁnite temperature for strong enough interaction ( g4). It is worth recalling that ARPES measurements strongly suggests that isotropic Lut-tinger semimetal describes the normal states of both Nd 2Ir2O7 and Pr 2Ir2O7[10,11]. On the other hand, half-Heusler com- pounds LnPdBi display a conﬂuence of magnetic order andsuperconductivity [ 37] and the phase diagrams shown in Figs. 5,6(c), and 6(d) capture such competition (at least qualitative). This connection can be further substantiated froma recent penetration depth ( /Delta1λ) measurement in YPtBi [ 38], suggesting /Delta1λ∼T/T c(roughly) at low enough temperature, where Tc=0.78 K is the superconducting transition temper- ature. Such a T-linear dependence of the penetration depth can result from gapless BdG fermions yielding /rho1(E)∼\|E\|. Any d-wave pairing, producing two nodal loops (see Table I), is therefore a natural candidate for the paired state in half-Heuslers (see also Refs. [ 50,52,54,55]). It is admitted that more microscopic analysis is needed to gain further insightsinto the global phase diagram of strongly interacting spin-3/2 fermions in various materials, which we leave for future investigation. The energy-entropy competition and the proposed selec- tion rule among competing orders provide valuable insightsinto the overall structure of the global phase diagram ofstrongly interacting spin-3 /2 fermions. Various cuts of the phase diagram, which we exposed by pursuing an unbi-ased renormalization group analysis, corroborate (at leastqualitatively) the former two approaches. These approachesare not limited to interacting spin-3 /2 fermions living in three dimensions. The methodology is applicable for a largeset of strongly interacting multiband systems, among whichtwo-dimensional Dirac semimetals, doped Bernal-stacked bi-layer graphene (supporting biquadratic band touchings in twodimensions) [ 68,69,89], twisted bilayer graphene near the so called magic angle [ 74–76], three-dimensional Dirac or Weyl materials [ 73], doped topological (Kondo-)insulators (described by massive Dirac fermions) [ 70–72], and nodal loop metals [ 102,103] are the prominent and experimentally pertinent ones. In the future we will systematically study thesesystems, which should allow us to gain further insights intothe global phase diagram of correlated materials, appreciatethe role of emergent topology inside various broken symmetryphases, and search for possible routes to realize unconven-tional high temperature superconductors.ACKNOWLEDGMENTS This work was in part supported by Deutsche Forschungs- gemeinschaft under grant SFB 1143 and a start-up grant fromLehigh University (B.R.). APPENDIX A: ORGANIZING PRINCIPLE AND SELECTION RULE IN DOPED GRAPHENE The low-energy Dirac excitations in monolayer graphene are captured by a sixteen component Nambu-doubled spinor/Psi1 /latticetop Nam=(/Psi1k,iσ2τ1/Psi1⋆ −k), where /Psi1/latticetop k=(/Psi1k,↑,/Psi1 k,↓),/Psi1/latticetop k,σ= (/Psi1K+k,σ,/Psi1−K+k,σ), and /Psi1/latticetop τK+k,σ=(uτK+k,σ,vτK+k,σ). (A1) Here, uτK+k,σand vτK+k,σare fermion annihilation opera- tors with momentum kon the sublattices A and B of the honeycomb lattice, respectively, with the Fourier componentsnear two inequivalent valleys at τKwithτ=±, and spin projections σ=↑,↓. The two sets of Pauli matrices {σ μ} and{τμ}respectively operate on the spin and valley indices. In this Nambu-doubled basis the sixteen-dimensional DiracHamiltonian reads as ˆh Nam D=v(η3σ0τ3α1kx−η3σ0τ0α2ky). (A2) Two sets of Pauli matrices {ημ}and{αμ}respectively operate on the Nambu or particle-hole and sublattice indices, and kis measured from the respective valley. In this basis, the dominant component of the nearest- neighbor local four-fermion interactions takes the form (/Psi1† Namη3σ0τ0α3/Psi1Nam)2, and therefore ˆI=η3σ0τ0α3(see Sec. II B 3 ). It supports charge-density-wave and spin-triplet f-wave pairing respectively at zero and ﬁnite chemical dop- ing, as recently found from a nonperturbative functionalRG analysis [ 77]. These two ordered states are respec- tively described by the fermion bilinears /Psi1 † NamˆOCDW/Psi1Namand /Psi1† NamˆOf-wave/Psi1Nam, with ˆOCDW=η3σ0τ0α3,ˆOf-wave=ηjστ3α0, (A3) where j=1,2, and σ=(σ1,σ2,σ3). Notice that nucleation of these two ordered states for nearest-neighbor interactionsis consistent with the selection rule shown in Eq. ( 13), as (1)ˆI≡ˆO CDW and (2) {ˆI,ˆOf-wave}=0. Furthermore, the fact that by tuning chemical doping one can induce a transitionfrom charge-density-wave to f-wave pairing is also consistent with the other selection rule that {ˆO CDW,ˆOf-wave}=0 and these two order parameters can be combined to form an O(4)supervector {η 3σ0τ0α3,ηjστ3α0}forj=1 or 2, see Sec. VI E. On the other hand, the dominant component of the next- nearest-neighbor repulsion on honeycomb lattice is captured by the four-fermion term ( /Psi1† Namη3στ3α3/Psi1Nam)2, and therefor ˆI=η3στ3α3(see Sec. II B 3 ). Quantum Monte Carlo simula- tion (nonperturbative) shows that such interaction respectivelysupports a quantum spin Hall insulator and spin-singlets-wave pairing for zero and ﬁnite chemical doping, respec- tively [ 78]. These two ordered phases are described by the fermion bilinears /Psi1 † NamˆOQSHI/Psi1Namand/Psi1† NamˆOs-wave/Psi1Nam,r e - 165139-25SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) spectively, with ˆOQSHI=η3στ3α3,ˆOs-wave=ηjσ0τ3α0, (A4) where j=1,2. Nucleation of these two ordered states is also compatible with the selection rule from Eq. ( 13), since (1)ˆI≡ˆOQSHI and (2) {ˆI,ˆOs-wave}=0. The fact that these two phases are neighbors as one tunes the chemical dop-ing in the system is consistent with the fact that theycan be be combined together to form an O(5) supervector{η 3στ3α3,η1σ0τ3α0,η2σ0τ3α0}and{ˆOQSHI,ˆOs-wave}=0( s e e Sec. VI E). Therefore our proposed selection rule among interaction channel and competing ordered phases is also operative inother correlated systems, such as monolayer graphene [ 77,78]. Most importantly, these two nonperturbative and unbiasednumerical works strongly suggest the nonperturbative natureof the selection rule, which we derive here by exploiting theinternal symmetry among the competing orders. Also the factthat these two numerical analyses found insulators (supercon-ductors) at zero (ﬁnite) chemical doping, is also compatiblewith the organizing principle from Sec. II B 1 , based on the energy and entropy argument. For detailed RG analysis, sup-porting these claims see Ref. [ 104]. APPENDIX B: SPECIFIC HEAT AND COMPRESSIBILITY This Appendix is devoted to present the scaling of speciﬁc heat ( CV) and compressibility ( κ) in a noninteracting Luttinger system, when the chemical potential ( μ) is placed away from the band touching point. We begin with the expression forthe free-energy density in this system, given by (after settingk B=1) f=−DT/summationdisplay τ=±/integraldisplayd3k (2π)3ln/bracketleftbigg 2 cosh/parenleftbiggEτ k 2T/parenrightbigg/bracketrightbigg , (B1) where Dis the degeneracy of the valence and conduction band, hence D=2, and Eτ k=k2 2m+τμ. (B2) The chemical potential and momentum ( k) are measured from the band touching point. One can rewrite the above expressionfor the free energy density as f=−D 2/summationdisplay τ=±/integraldisplayd3k (2π)3Eτ k −DT/summationdisplay τ=±/integraldisplayd3k (2π)3ln/bracketleftbigg 1+exp/parenleftbigg −Eτ k T/parenrightbigg/bracketrightbigg .(B3) The ﬁrst term is independent of temperature and not important for the thermodynamic properties of the system. So we focusonly on the second term. After proper rescaling of variables,the free-energy density becomes f=−DT 5/2(2m)3/2 4π2/summationdisplay τ=±/integraldisplay∞ 0dy√yln(1+e−y−τ˜μ), (B4)leading to Eq. ( 3), where ˜ μ=μ/T, and Li s(z) represents the polylogarithm function of order sand argument z.F o r ˜μ/lessmuch1, the free-energy density reads as f=−DT5/2(2m)3/2 8π3/2/parenleftbigg a+bμ2 T2+cμ4 T4+ ···/parenrightbigg ,(B5) where a=1 2(4−√ 2)ζ/parenleftbigg5 2/parenrightbigg ≈1.7344, b=(1−√ 2)ζ/parenleftbigg1 2/parenrightbigg ≈0.6049, c=1 12(1−4√ 2)ζ/parenleftbigg −3 2/parenrightbigg ≈0.00989 . (B6) From the above expression of the free-energy density we arrive the expression for speciﬁc heat [see Eq. ( 4)] and com- pressibility [see Eq. ( 5)]. From these two expressions, we ﬁnally arrive at the following universal ratio: CV/T κ=15(4−√ 2) 16(1−√ 2)×ζ(5/2) ζ(1/2)≈5.37611 , (B7) reported in Eq. ( 6). This number is a characteristic of a z=2 scale invariant ﬁxed point in d=3. APPENDIX C: DYNAMIC CONDUCTIVITY This Appendix is devoted to disclose some key steps of the computation of the dynamic conductivity in Luttinger system.We focus on an isotropic system (for the sake of simplicity)and explicitly compute the zcomponent of the conductivity (σ zz). Due to the cubic symmetry σzz=σxx=σyy≡σ(say). To this end we use the Kubo formula and compute the polar-ization bubble as a function of external (Matsubara) frequency /Pi1 zz(iωn)=−e2 β/summationdisplay m/integraldisplay kTr[ˆjzG0(ipm,k)ˆjz ×G0(ipm+iωn,k)], (C1) where eis the electronic charge, βis the inverse temperature and the current operator along the zdirection is given by ˆjz=−1 2m[√ 3ky/Gamma11+√ 3kx/Gamma12+2kz/Gamma15]. (C2) In the spectral representation, the noninteracting Greens func- tion reads as G0(iωn,k)=/integraldisplay∞ −∞d/epsilon1 2πA(/epsilon1,k) iωn−/epsilon1, (C3) where A(/epsilon1,k)=π/parenleftBigg 1+5/summationdisplay j=1/Gamma1jˆdj/parenrightBigg δ/parenleftbigg ω+μ−k2 2m/parenrightbigg +π/parenleftBigg 1−5/summationdisplay j=1/Gamma1jˆdj/parenrightBigg δ/parenleftbigg ω+μ+k2 2m/parenrightbigg .(C4) 165139-26INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) FIG. 21. Scaling of the universal function FDr(x) with its argu- ment x, appearing in the expression for the Drude conductivity of Luttinger fermions, see Eq. ( 7). Components of ˆdare shown in Eq. ( D1). After performing the analytic continuation iω→ω+iη, we ﬁnd the dynamic conductivity using the Kubo formula σzz(ω)=/Ifractur/Pi1zz(iω→ω+iη) ω. (C5) From the above expression we obtain both Drude and inter- band components of the dynamic conductivity, respectivelyshown in Eqs. ( 7) and ( 8). The universal function F Dr(x) appearing in the expression of the Drude component in Eq. ( 7) is given by FDr(x)=32 3/integraldisplay∞ 0du u3/2/summationdisplay τ=±sech2/parenleftbigg u+τμ 2T/parenrightbigg . (C6) The scaling of this function is shown in Fig. 21. APPENDIX D: DETAILS OF LUTTINGER MODEL In this Appendix, we present some essential details of the Luttinger model, introduced in Sec. III. The components of a ﬁve-dimensional unit vector ˆd(ˆk), appearing in the Luttinger Hamiltonian [see Eq. ( 14)], are given by ˆd1=i/bracketleftbig Y1 2+Y−1 2/bracketrightbig √ 2=√ 3 2sin 2θsinφ=√ 3ˆkyˆkz, ˆd2=/bracketleftbig Y−1 2+Y1 2/bracketrightbig √ 2=√ 3 2sin 2θcosφ=√ 3ˆkxˆkz, ˆd3=i/bracketleftbig Y−2 2+Y−2 2/bracketrightbig √ 2=√ 3 2sin2θsin 2φ=√ 3ˆkyˆkx, ˆd4=/bracketleftbig Y−2 2+Y2 2/bracketrightbig √ 2=√ 3 2sin2θcos 2φ=√ 3 2/bracketleftbigˆk2 x−ˆk2 y/bracketrightbig , ˆd5=Y0 2=1 2(3 cos2θ−1)=1 2/bracketleftbig 2ˆk2 z−ˆk2 x−ˆk2 y/bracketrightbig , (D1) where Ym l≡Ym l(θ,φ),ˆdj≡ˆdj(ˆk), and θandφare the polar and azimuthal angles in the momentum space, respec-tively. Five mutually anticommuting /Gamma1matrices are [see alsoEq. ( 16)] /Gamma1 1=1√ 3{Jy,Jz},/Gamma12=1√ 3{Jx,Jz},/Gamma13=1√ 3{Jx,Jy}, /Gamma14=1√ 3/bracketleftbig J2 x−J2 y/bracketrightbig ,/Gamma15=1 3/bracketleftbig 2J2 z−J2 x−J2 y/bracketrightbig , (D2) where {A,B}≡AB+BA, and Jare three spin-3 /2 matrices. Both ˆdand/Gamma1transform as vectors under the cubic point group (Oh), and their scalar product yields the Luttinger Hamilto- nian, an A1gquantity. On the other hand, ten commutators /Gamma1jk=[/Gamma1j,/Gamma1k]/(2i) with j>kcan be expressed in terms of the products of odd number of spin-3 /2 matrices as follows: /Gamma145=−2√ 3[JxJyJz+JzJyJx],/Gamma112=1 3/bracketleftbig 7Jz−4J3 z/bracketrightbig , /Gamma113=−1 3/bracketleftbig 7Jy−4J3 y/bracketrightbig ,/Gamma123=1 3/bracketleftbig 7Jx−4J3 x/bracketrightbig , /Gamma134=−1 6/bracketleftbig 13Jz−4J3 z/bracketrightbig ,/Gamma135=1√ 3/braceleftbig Jz,J2 x−J2 y/bracerightbig , /Gamma114=1 12/bracketleftbig 13Jx−4J3 x/bracketrightbig +1 2/braceleftbig Jx,J2 y−J2 z/bracerightbig , (D3) /Gamma115=1 4√ 3/bracketleftbig 13Jx−4J3 x/bracketrightbig −1 2√ 3/braceleftbig Jx,J2 y−J2 z/bracerightbig , /Gamma124=1 12/bracketleftbig 13Jy−4J3 y/bracketrightbig −1 2/braceleftbig Jy,J2 z−J2 x/bracerightbig , /Gamma125=−1 4√ 3/bracketleftbig 13Jy−4J3 y/bracketrightbig −1 2√ 3/braceleftbig Jy,J2 z−J2 x/bracerightbig . The four-dimensional identity matrix can be written as /Gamma10=4 15/parenleftbig J2 x+J2 y+J2 z/parenrightbig . (D4) APPENDIX E: FIERZ IDENTITY In this Appendix, we present the Fierz reduction of the number of linearly independent quartic terms for the inter-acting Luttinger system. To perform this exercise for genericlocal density-density interactions, we introduce a sixcompo- nent vector X /latticetop=/bracketleftBigg (/Psi1†/Gamma10/Psi1)2,3/summationdisplay j=1(/Psi1†/Gamma1j/Psi1)2,5/summationdisplay j=4(/Psi1†/Gamma1j/Psi1)2, (/Psi1†/Gamma145/Psi1)2,3/summationdisplay j=1(/Psi1†/Gamma1j/Gamma145/Psi1)2,3/summationdisplay j=15/summationdisplay k=4(/Psi1†/Gamma1jk/Psi1)2/bracketrightBigg , (E1) constituted by the quartic terms appearing in Hint,s e eE q .( 36). The Fierz identity allows us to rewrite each quartic term ap-pearing in Xas a linear combination of the remaining ones according to [/Psi1 †(x)M/Psi1(x)][/Psi1†(y)N/Psi1(y)] =−1 16(TrM/Gamma1aN/Gamma1b)[/Psi1†(x)/Gamma1b/Psi1(y)][/Psi1†(y)/Gamma1a/Psi1(x)], (E2) 165139-27SZABÓ, MOESSNER, AND ROY PHYSICAL REVIEW B 103, 165139 (2021) where Mand Nare four-dimensional Hermitian matrices and for local interactions x=y. The set of sixteen matrices {/Gamma1a,a=1,..., 16}closes the U(4) Clifford algebra of four- dimensional matrices, and we choose /Gamma1† a=/Gamma1a=(/Gamma1a)−1.T h e above Fierz constraint then takes a compact form FX=0, where Fis the Fierz matrix given by F=⎡ ⎢⎢⎢⎢⎢⎢⎢⎣5 11111 33 −33 −11 2−24 −22 0 11 −15 1 −1 3−13 3 3 −1 62 0 −6−24⎤ ⎥⎥⎥⎥⎥⎥⎥⎦. (E3) The rank of Fis 3. Hence, the number of lienarly independent quartic terms is 3 =6 (dimensionality of F) –3 (rank of F). We chose four-fermion interactions proportional to λ 0,λ1and λ3asthree linearly independent quartic terms. The remaining three quartic terms can then be expressed in terms of linearcombinations of the above three according to (/Psi1 †/Gamma145/Psi1)2=−1 2(/Psi1†/Psi1)2−1 23/summationdisplay j=1(/Psi1†/Gamma1j/Psi1)2 +1 25/summationdisplay j=4(/Psi1†/Gamma1j/Psi1)2, 3/summationdisplay j=1(/Psi1†/Gamma1j/Gamma145/Psi1)2=−3 2(/Psi1†/Psi1)2+1 23/summationdisplay j=1(/Psi1†/Gamma1j/Psi1)2 −3 25/summationdisplay j=4(/Psi1†/Gamma1j/Psi1)2, 3/summationdisplay j=15/summationdisplay k=4(/Psi1†/Gamma1jk/Psi1)2=−3(/Psi1†/Psi1)2−3/summationdisplay j=1(/Psi1†/Gamma1j/Psi1)2.(E4)TABLE IV . Three eigenvalues (EVs) of the stability matrix (SM) M(g0,g1,g2), deﬁned in Eq. ( G1) at various ﬁxed points (FPs) lo- cated at ( g∗ 0,g∗ 1,g∗ 2), see Table II. The trivial Guassian ﬁxed point is found for any arbitrary value of α. FPs ( g∗ 0,g∗ 1,g∗ 2)×103EVs of SM Gaussian (0,0,0) ( −1,−1,−1)/epsilon1 QCP1π 4(2.24,2.03,2.03)/epsilon1 (1,−0.870,−0.704)/epsilon1 QCP1π 2(1.49,1.47,1.26)/epsilon1 (−1.075,1,−0.654)/epsilon1 QCP2π 2(−2.93,−2.84,3.76)/epsilon1 (−1.169,1,−0.933)/epsilon1 BCP π 2(−1.58,−1.33,5.54)/epsilon1 (1.271,1,−0.527)/epsilon1 QCP1 0 (1.95,1.71,1.94)/epsilon1 (1,−0.889,−0.642)/epsilon1 QCP2 0 (−5.98,4.80,−5.87)/epsilon1 (−1.131,1,−0.134)/epsilon1 BCP 0 (−5.60,4.97,−5.48)/epsilon1 (−1.075,1,0.132)/epsilon1 Therefore, during the RG analysis whenever we generate any one of the above three quartic terms we can immediatelyexpress them in terms of the ones proportional to λ 0,λ1and λ2. Therefore the interacting model [see Eq. ( 37)] remains closed under coarse garining to any order in the perturbationtheory. APPENDIX F: DETAILS OF RG FLOW EQUATIONS The RG ﬂow equations, displayed in Eq. ( 46), are expressed in terms of the functions Hj km(α,t,μ), where j,k,m=0,1,2. For brevity, we here drop the explicit de- pendence of these functions on α,tandμ. The functions appearing in the βfunction of g0are given by H0 00=5 4[−f0−˜f0−4fg−6ft],H0 11=5 4[−6˜f0−6fg+6˜fg−3ft+3˜ft], H0 22=5 4[f0−3˜f0−3ft+3˜ft],H0 01=5 4[6f0+6fg+6˜fg+18ft+12˜ft], H0 02=5 2[2f0+5fg+3ft+3˜fg+3˜ft],H0 12=15 2[−f0−˜f0−fg+˜fg−2ft+2˜ft]. (F1) The six functions appearing in the RG ﬂow equation for g1are given by H1 00=5 4(ft+˜ft),H1 11=5 4(−5f0+˜f0+10fg+2˜fg+10ft+10˜ft),H1 22=5 4(−f0−˜f0+3ft+˜ft), H1 01=5 2(2f0−˜f0−3fg+˜fg−˜ft),H1 02=−5 2(fg−˜fg+ft−˜ft),H1 12=−5 2(3f0+˜f0−7fg−˜fg−2˜ft). (F2) Finally, the ﬂow equation for g2is expressed in terms of the following six functions H2 00=5 4(fg+˜fg),H2 11=5 4(−3f0−3˜f0+3fg+3˜fg+3ft−3˜ft),H2 22=5 4(−3f0+˜f0+4fg+4˜fg+9ft+3˜ft), H2 01=5 4(−6ft+6˜ft),H2 02=5 2(2f0−˜f0+fg−˜fg−3ft),H2 12=15 2(−f0−fg+˜fg+4ft+˜ft). (F3) Note that in the above expression fj≡fj(α,t,μ) and ˜fj=˜fj(α,t,μ)s for j=0,t,g, and f0(α,t,μ)=−1 2/integraldisplay d/Omega11 2t/summationdisplay τ=±1/bracketleftbigg sech2/parenleftbiggf+τμ 2t/parenrightbigg +2t ftanh/parenleftbiggf+τμ 2t/parenrightbigg/bracketrightbigg , (F4) ft(α,t,μ)=−1 2/integraldisplay d/Omega1cos2α 2t/summationdisplay τ=±1/bracketleftbigg1 f2sech2/parenleftbiggf+τμ 2t/parenrightbigg −2t f3tanh/parenleftbiggf+τμ 2t/parenrightbigg/bracketrightbiggˆd2 1+ˆd2 2+ˆd2 3 3, (F5) 165139-28INTERACTING SPIN-3 2FERMIONS IN A … PHYSICAL REVIEW B 103, 165139 (2021) fg(α,t,μ)=−1 2/integraldisplay d/Omega1sin2α 2t/summationdisplay τ=±1/bracketleftbigg1 f2sech2/parenleftbiggf+τμ 2t/parenrightbigg −2t f3tanh/parenleftbiggf+τμ 2t/parenrightbigg/bracketrightbiggˆd2 4+ˆd2 5 2, (F6) ˜f0(α,t,μ)=−1 2/integraldisplay d/Omega11 μ/parenleftBig 1−/parenleftbigμ f/parenrightbig2/parenrightBig/summationdisplay τ=±1tanh/parenleftbiggf+τμ 2t/parenrightbigg/bracketleftbigg τ+μ f−2τ/parenleftbiggμ f/parenrightbigg2/bracketrightbigg , (F7) ˜ft(α,t,μ)=1 2/integraldisplay d/Omega1cos2α f2μ/parenleftBig 1−/parenleftbigμ f/parenrightbig2/parenrightBig/summationdisplay τ=±1tanh/parenleftbiggf+τμ 2t/parenrightbigg/bracketleftbigg −τ+μ f/bracketrightbiggˆd2 1+ˆd2 2+ˆd2 3 3, (F8) ˜fg(α,t,μ)=1 2/integraldisplay d/Omega1sin2α f2μ/parenleftBig 1−/parenleftbigμ f/parenrightbig2/parenrightBig/summationdisplay τ=±1tanh/parenleftbiggf+τμ 2t/parenrightbigg/bracketleftbigg −τ+μ f/bracketrightbiggˆd2 4+ˆd2 5 2, (F9) where d/Omega1=sinθdθdφ,f≡f(α,θ,φ ) and f(α,θ,φ )=/radicalBig cos2α/parenleftbigˆd2 1+ˆd2 2+ˆd2 3/parenrightbig +sin2α/parenleftbigˆd2 4+ˆd2 5/parenrightbig . (F10) The deﬁnition of ˆdis already provided in Eq. ( D1). In terms of fjs and ˜fjs, we can express the leading order RG ﬂow equations for all source terms [see Eq. ( 48)] as dln/Delta10 dl−2=−5 4(f0+2fg+3ft)(3g0−3g1−2g2), dln/Delta11 dl−2=5 4(f0−2fg−ft)(g0−5g1−2g2), dln/Delta12 dl−2=5 4(f0−3ft)(g0−3g1−4g2), dln/Delta13 dl−2=5 4(f0−2fg+3ft)(g0+3g1−2g2), dln/Delta14 dl−2=5 4(f0+2fg−ft)(g0−g1+2g2), dln/Delta15 dl−2=5 4(f0+ft)(g0+g1), dln/Delta1p A1g dl−2=−5 4(˜f0−2˜fg−3˜ft)(g0+3g1+2g2), dln/Delta1p T2g dl−2=−5 4(˜f0+2˜fg+˜ft)(g0−g1−2g2), dln/Delta1p Eg dl−2=−5 4(˜f0+3˜ft)(g0−3g1). (F11) These ﬂow equations are schematically shown in Eq. ( 49). APPENDIX G: STABILITY MATRIX ANALYSIS To gain insight into the local structure of the RG ﬂow trajectories in the close proximity to any ﬁxed point, we per-form the stability analysis around them. To proceed with this analysis we introduce a 3 ×3 stability matrix M(g 0,g1,g2)a sfollows: M(g0,g1,g2)=⎡ ⎢⎢⎢⎣dβg0 dg0dβg1 dg0dβg2 dg0dβg0 dg1dβg1 dg1dβg2 dg1dβg0 dg2dβg1 dg2dβg2 dg2⎤ ⎥⎥⎥⎦. (G1) We classify the ﬁxed points, located at ( g∗ 0,g∗ 1,g∗ 2), accord- ing to the number of positive and negative eignevalues ofM(g ∗ 0,g∗ 1,g∗ 2). (1) There exists only one ﬁxed point at ( g∗ 0,g∗ 1,g∗ 2)= (0,0,0) with three negative eigenvalues ( −1,−1,−1)/epsilon1of the stability matrix. This ﬁxed point, referred as Gaussian in Table IV, is stable from all directions and represents the noninteracting LSM for sufﬁciently weak but generic short-range interactions. (2) Fixed points possessing one positive and two negative eigenvalues for the stability matrix are the QCPs. Such ﬁxedpoints control QPTs from LSM to BSPs. Since such QPTscan be triggered by tuning only one parameter (the interactionstrength along the relevant direction, determined by the eigen-vector associated with the positive eigenvalue of the stabilitymatrix), they are continuous in nature. Concomitantly, a singleparameter scaling emerges in the vicinity of all QCPs. Thepositive eigenvalue of the stability matrix also determines thecorrelation length exponent ( ν) at the QCPs (see Sec. VI B ). (3) Fixed points with two positive and one negative eigen- values for the stability matrix are the BCPs. Typically BCPsare found when there exists more than one QCP in the multidi-mensional coupling constant space. For example, only aroundα= π 2and 0, when the coupled RG ﬂow equations support two QCPs, there exists one BCP (respectively denoted byBCP π 2and BCP 0). Existence of such a BCP is necessary to ensure the continuity of the RG ﬂow trajectories. A BCP alsoseparates the basin of attractions of two QCPs. 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PhysRevB.100.024515.pdf	PHYSICAL REVIEW B 100, 024515 (2019) Observation of plasmon-phonons in a metamaterial superconductor using inelastic neutron scattering Vera N. Smolyaninova,1Jeffrey W. Lynn,2Nicholas P. Butch,2Heather Chen-Mayer,2Joseph C. Prestigiacomo,3 M. S. Osofsky,3and Igor I. Smolyaninov4,5 1Department of Physics Astronomy and Geosciences, Towson University, 8000 York Rd., Towson, Maryland 21252, USA 2NIST Center for Neutron Research, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899–6102, USA 3Naval Research Laboratory, Washington, DC 20375, USA 4Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA 5Saltenna LLC, 1751 Pinnacle Drive #600 McLean, Virginia 22102, USA (Received 25 August 2018; revised manuscript received 24 May 2019; published 24 July 2019) A metamaterial approach is capable of drastically increasing the critical temperature, Tc, of composite metal-dielectric superconductors as demonstrated by the tripling of Tct h a tw a so b s e r v e di nb u l kA l - A l 2O3core- shell metamaterials. A theoretical model based on the Maxwell-Garnett approximation provides a microscopicexplanation of this effect in terms of electron-electron pairing mediated by a hybrid plasmon-phonon excitation.We report an observation of this excitation in Al-Al 2O3core-shell metamaterials using inelastic neutron scattering. This result provides support for this mechanism of superconductivity in metamaterials. DOI: 10.1103/PhysRevB.100.024515 I. INTRODUCTION Recent theoretical [ 1–3] and experimental [ 4–6] work has demonstrated that many tools developed in electromagnetic metamaterial research can be used to engineer artiﬁcial meta-material superconductors having improved superconductingproperties. This connection between electromagnetic metama- terials and superconductivity research stems from the fact that superconducting properties of a material may be expressedvia its effective dielectric response function ε eff(q,ω), and the critical temperature, Tc, of a superconductor is deﬁned by the behavior of εeff−1(q,ω) near its poles [ 7]. In conventional superconductors these poles are deﬁned by the dispersion law,ω(q),of phonons that mediate electron-electron pairing. Recently, we have demonstrated a considerable enhancementof the attractive electron-electron interaction in such metama-terial scenarios as epsilon near zero (ENZ) [ 8] and hyperbolic metamaterials [ 9]. In both cases the inverse dielectric response function of the metamaterial may exhibit additional polescompared to the parent superconductor. The most striking ex- ample of successful metamaterial superconductor engineering was the observation of tripling of the critical temperature T c in Al-Al 2O3ENZ core-shell metamaterials compared to bulk aluminum [ 5] .T h ef o r m a t i o no ft h e s eb u l kA l - A l 2O3samples enable studies that require large sample volumes, such as neutron scattering, which can reveal phonon spectral featuresthat are not accessible in thin ﬁlms. Here, we report on the use of inelastic neutron scattering to provide experimental evidence of an excitation that doesnot exist in pure Al and corresponds to the metamaterialpole of the engineered inverse dielectric response functionresponsible for the T cenhancement in this material. We also identify the microscopic physical origin of this additionalmetamaterial pole as coming from a hybrid plasmon-phononmode that arises in a composite metal-dielectric metamaterial. The hybrid character of this mode enables efﬁcient inelasticneutron scattering from its phonon component, which is ob-served in the experiment. The direct observation of plasmon-phonon modes in Al-Al 2O3ENZ core-shell metamaterials using inelastic neutron scattering provides strong supportof this plasmon-phonon mechanism of superconductivity incomposite metal-dielectric ENZ metamaterials. II. THEORETICAL MODEL The theoretical model of superconductivity in ENZ meta- materials is based on the paper by Kirzhnits et al. [7], which demonstrated that within the framework of macroscopic elec-trodynamics the electron-electron interaction in a supercon-ductor may be expressed in the form of an effective Coulombpotential V(/vectorq,ω)=4πe 2 q2εeff(/vectorq,ω)=VC εeff(/vectorq,ω), (1) where VC=4πe2/q2is the Fourier-transformed Coulomb potential in vacuum, and εeff(q,ω) is the linear dielectric response function of the superconductor treated as an effectivemedium. The critical temperature of a superconductor in theweak coupling limit is typically calculated as T c=θexp/parenleftbigg −1 λeff/parenrightbigg , (2) where θis the characteristic temperature for a bosonic mode mediating electron pairing (such as the Debye temperature θD in the standard Bardeen–Cooper–Schrieffer theory) [ 10]. The dimensionless coupling constant λeffis deﬁned by V(q,ω)= VCε−1(q,ω) and the density of states ν(see, for example, 2469-9950/2019/100(2)/024515(7) 024515-1 ©2019 American Physical SocietyVERA N. SMOLYANINOV A et al. PHYSICAL REVIEW B 100, 024515 (2019) Ref. [ 11]): λeff=−2 πν/integraldisplay∞ 0dω ω/angbracketleftVCImε−1(/vectorq,ω)/angbracketright, (3) where VCis the unscreened Coulomb repulsion, and the angle brackets denote average over the Fermi surface. Now thisformalism will be applied to a composite metal-dielectricmetamaterial. Following Ref. [ 12], a simpliﬁed dielectric response func- tion of a metal may be written as ε m(q,ω)=/parenleftbigg 1+k2 q2/parenrightbigg/parenleftbigg 1−/Omega12(q) ω2+iω/Gamma1/parenrightbigg , (4) where kis the inverse Thomas-Fermi screening radius, /Omega1(q) is the dispersion law of a phonon mode, and /Gamma1is the corre- sponding damping rate. The zero of the dielectric responsefunction of the bulk metal [which occurs at ω=/Omega1(q) where ε mchanges sign] maximizes the electron-electron pairing interaction given by Eq. ( 1). This simpliﬁed consideration of εm(q,ω) has been justiﬁed in [ 3]. A superconducting metallic “matrix” with dielectric “in- clusions”, which forms a composite metal-dielectric meta-material, is now considered. We will assume that the per-mittivity ε dof the dielectric does not depend on ( q,ω) and stays positive and constant. According to the Maxwell-Garnettapproximation [ 13], mixing of nanoparticles of a supercon- ducting “matrix” with dielectric “inclusions” (described bythe dielectric constants ε mandεd, respectively) results in an effective medium with a dielectric constant εeff, which may be obtained as /parenleftbiggεeff−εm εeff+2εm/parenrightbigg =(1−n)/parenleftbiggεd−εm εd+2εm/parenrightbigg , (5) where nis the volume fraction of metal (0 /lessorequalslantn/lessorequalslant1). The explicit expression for εeff−1may be written as ε−1 eff=n (3−2n)1 εm+9(1−n) 2n(3−2n)1 (εm+(3−2n)εd/2n). (6) For a given value of the metal volume fraction n,the ENZ conditions ( εeff≈0) may be obtained around εm≈0( a tt h e phonon frequency ω=/Omega1(q) of the superconducting metal), and around εm≈−3−2n 2nεd. (7) According to Eq. ( 4), the additional pole of the inverse dielectric response function described by Eq. ( 7) occurs at some frequency ω</Omega1 (q) close to the phonon resonance. Indeed, Eqs. ( 4) and ( 7) allow us to predict the location of the additional pole with respect to the phonon frequency /Omega1(q). Neglecting the unknown value of /Gamma1in Eq. ( 4), we obtain εm(q,ω)≈/parenleftbigg 1+k2 q2/parenrightbigg/parenleftbigg 1−/Omega12(q) ω2/parenrightbigg ≈−3−2n 2nεd.(8)0.4 0.5 0.6 0.7 0.8 0.9 1.00.40.50.60.7 ratio nexperimentally observed peak FIG. 1. Ratio of the plasmon-phonon and the phonon frequencies calculated as a function of metal volume fraction nin the Al-Al 2O3 metamaterial using Eq. ( 9). The arrow indicates the frequency of an additional peak observed in the neutron scattering experiment. This leads to the following expression for the ratio of the additional pole and the regular phonon frequencies: ω /Omega1≈/parenleftBigg 1+(3−2n)εd 2n/parenleftbig 1+k2 q2/parenrightbig/parenrightBigg−1/2 . (9) The dielectric constant for Al 2O3in the long wavelength infrared range is εd∼2.25. Using the ratio of the inverse Thomas-Fermi radius to the Fermi radius for aluminumk/k F=1.17 tabulated in Ref. [ 12], and assuming q=2kF,t h e ω//Omega1 ratio given by Eq. ( 9) may be calculated as a function of n. This function is shown in Fig. 1. Based on these estimates, we may expect that the additional pole must be located inthe 0.4–0.7 /Omega1range depending on the volume fraction of aluminum in the metamaterial. III. EXPERIMENTAL RESULTS To search for the existence of this additional excitation, which should occur at frequencies lower than the phononfrequencies of aluminum, an inelastic neutron scattering ex-periment was performed. To prepare the samples, commercial(U.S. Research Nanomaterials) 18-nm diameter aluminumnanoparticles were oxidized under ambient conditions. Uponoxidation, an approximately 2-nm-thick layer of Al 2O3is formed on the surface of aluminum nanoparticles [ 14]. The particles were subsequently compressed into dense pelletsin a hydraulic press. A scanning electron microscopy imageof such a core-shell Al-Al 2O3compressed sample is shown in Fig. 2(a). The superconducting transition of this bulk metamaterial sample was measured to be Tc=3.7 K, which is more than three times higher than Tc=1.2Ko fb u l k aluminum [Fig. 2(b)]. The zero-ﬁeld cooled magnetization data shown in Fig. 2(b) were measured using a magnetic property measurement system superconducting quantum in-terference device magnetometer in a magnetic ﬁeld of 1 mT. 024515-2OBSERV ATION OF PLASMON-PHONONS IN A … PHYSICAL REVIEW B 100, 024515 (2019) (a) (b) (b) 1234567 -0.00015-0.00010-0.000050.00000M (emu/g)T (K) FIG. 2. (a) Scanning electron microscope image of the Al-Al 2O3 metamaterial. (b) Temperature dependence of zero-ﬁeld-cooled mag- netization per unit mass for oxidized Al-Al 2O3core-shell metamate- rial samples made of 18-nm aluminum nanoparticles. The observedonset of superconductivity at ∼3.7 K in the oxidized sample is 3.25 times larger than T c=1.2 K of bulk aluminum [ 5]. The inset shows photo of the compressed metamaterial sample (note: 1 emu = 10−3Am2). Note that the superconducting coherence length in aluminum (which roughly corresponds to the size of a Cooper pair)is known to be very large ( ξ=1600 nm, see, for example, Ref. [ 12]). The superconducting coherence length in the alu- minum based hyperbolic metamaterial superconductors hasbeen experimentally measured in Ref. [ 6]. The measured value appears to be ξ=181 nm, which is smaller than ξin pure aluminum. The coherence length for our Al-Al 2O3ENZ core-shell metamaterial was determined from a measurementof the critical ﬁeld, H c2(Fig. 3). Although pure Al is a type I superconductor, the granular samples exhibit type IIbehavior. Thus, the coherence length, as determined by ξ=√ φ0/2πHc2, is 105 nm for the measured Hc2of 300 G. It is important to note that this experimentally measured value ofthe superconducting coherence length is much larger than the18-nm aluminum grain size, thus conﬁrming the considerationof our samples as “superconducting metamaterials”. The inelastic neutron scattering experiments were per- formed at the NIST Center for Neutron Research using triple-axis spectrometers BT-7 [ 15] and BT-4 and the Disk Chop- per Spectrometer [ 16]. For the measurements on BT-7 we0 100 200 300 400-0.0006-0.0004-0.00020.0000M (emu/g) B (G) FIG. 3. Magnetization as a function of magnetic ﬁeld for the core-shell Al-Al 2O3metamaterial at T=1.75 K. The paramagnetic background is subtracted from the data. employed horizontal focusing with a ﬁxed ﬁnal energy Ef= 14.7 meV to increase the range of the wave vector integration. A pyrolytic graphite ﬁlter was used in the scattered beam and avelocity selector in the incident beam to suppress higher-order wavelength contaminations and reduce background [ 15]. Data were collected in the energy range from 3 to 43 meV at 5 K employing a closed cycle helium refrigerator. The high energy data were collected on BT-4 Filter Analyzer NeutronSpectrometer using the Cu(220) monochromator. Data were obtained from 36 to 250 meV at 78 K in a liquid nitrogen cryostat. Background data were obtained with the sample lifted out of the beam area. The highest resolution data were collected on the Disk Chopper Spectrometer [ 16] in a pumped 4He cryostat. Data were collected at 1.5 and 6 K, below and above the Tcof 3.7 K, respectively, with an incident energy of 13 meV . In subtracting the 6-K data from the 1.5-K data, we found a small but noticeable increase in the scattering in the superconducting state over a broad energy range from justabove the elastic line to ≈8 meV . This behavior is typical for inelastic scattering experiments with superconductors. The results from the inelastic neutron scattering measure- ments at T=5 K are shown in Fig. 4. The sample was ther- mally treated at 110 °C in vacuum before the measurements toremove possible water adsorbed on the particles. Data above 40 meV are dominated by scattering from hydrogen in the form of OH [ 17], which may come from a small amount of AlOOH. Prompt γ-ray neutron activation analysis [ 18,19] showed a hydrogen content of 12 at. % H in the sample. Theneutron scattering cross section for H is 80.27 compared to Al of 1.5, which is why the H can contribute to the scattering at high energies even though the amount of H is relatively small. Figure 4(b) shows averaged data for energies below 43 meV , taken at wave vectors Q=4, 4.25, and 4 .5Å −1, which are proportional to the generalized phonon densityof states (PDOS), which is the phonon density-of-statesweighted by the neutron cross sections. The PDOS forbulk aluminum at T=10 K (black circles) [ 20]i ss h o w nf o r comparison. Two peaks in the aluminum PDOS corresponding 024515-3VERA N. SMOLYANINOV A et al. PHYSICAL REVIEW B 100, 024515 (2019) (a) 01 0 2 0 3 0 4 0 5 00100200300400 Metamaterial AlIntensity (Counts) E (meV)(b) 0 50 100 150 200 2500100200300400500600700800Intensity(counts) E (meV) FIG. 4. (a) Inelastic neutron scattering data obtained for the core-shell Al-Al 2O3metamaterial at energies below 250 meV . The lower-energy data were obtained on BT-7 and the high energy data were taken on BT-4 FANS. Data above 40 meV are dominated by scattering from hydrogen in the form of OH, which may comefrom a small amount of AlOOH in the metamaterial. (b) Inelastic neutron scattering data for energies below 43 meV taken on BT-7, averaged over measurements taken at wave vectors Q=4, 4.25, and 4.5Å−1. Uncertainties originate from counting statistics and corre- spond to one standard deviation. The data were taken at T=5K . The averaged dependence is proportional to the generalized phonon density of states (PDOS). The DOS for bulk aluminum at T=10 K (black circles) [ 20] is shown for comparison. Two peaks that are indicated with black arrows in the aluminum PDOS correspond to the van Hove peaks for the transverse and longitudinal acoustical phonons, respectively, at the Brillouin zone boundaries. They arealso present in the core-shell Al-Al 2O3metamaterial. The additional peak at around 15 meV (indicated with a blue arrow) is not present in pure aluminum. It corresponds to the hybrid plasmon-phonons ofthe metamaterial. to transverse and longitudinal acoustical phonons are also present in core-shell Al-Al 2O3metamaterial, which are indicated with black arrows in Fig. 4(b). However, there is an additional peak at around 15 meV (indicated with blue arrow),100200300400 Q = 4 A-1Intensity (Counts)100200300 Q = 4.25 A-1 0 1 02 03 04 05 0100200300400 Q = 4.5 A-1 E (meV) FIG. 5. Inelastic neutron scattering data for energies below 43 meV taken on BT-7 for separate neutron wave vectors Q=4, 4.25, and 4 .5Å−1, respectively, as marked in each plot. The addi- tional peak at around 15 meV (indicated with arrows) is not present in pure aluminum. which is not present in pure aluminum. This peak cannot be attributed to the aluminum oxide, since aluminum oxide hasa stiffer lattice [ 21], and will have a peak in the PDOS at energies higher than that for aluminum. It also cannot beattributed to the effect of 12% hydroxide impurities. Due tothe much smaller mass of hydrogen atoms compared to alu-minum, the energy range of inelastic neutron scattering peaksassociated with the O-H bonds must be located at much higherenergies. The considerable enhancement of inelastic neutronscattering at energies above 40 meV is indeed observed in theexperimental results presented in Fig. 4(a). Consequently, 024515-4OBSERV ATION OF PLASMON-PHONONS IN A … PHYSICAL REVIEW B 100, 024515 (2019) the inelastic neutron scattering features near 15 meV cannot be affected by hydroxide impurities. Therefore, we haveobserved an additional contribution to the PDOS which waspredicted by our metamaterial model at energies within thespan of phonon energies of aluminum. The dispersion of thisadditional excitation may be evaluated based on the individualinelastic neutron scattering data (Fig. 5) taken on BT-7 at neutron wave vectors Q=4, 4.25, and 4 .5Å −1, respectively, as marked in each plot. The additional peak at around 15 meV(indicated with arrows) exhibits very weak dispersion, whichis consistent with the hypothesized van Hove character of thispeak. This peak is also consistent with unexplained extra spectral weight seen at low energy in the α 2F(ω) extracted from the tunneling conductance of granular Al ﬁlms that is absent inpure Al ﬁlms [ 22]. Since this quantity is a direct measure of the boson mechanism responsible for superconductivity,it is reasonable to infer that the observed peak reﬂects acontribution to mechanism responsible for the enhanced T c in our metamaterial samples. We should also recall that in 1968, Cohen and Abeles observed that the superconductivetransition temperature, T c, of thin ﬁlms of granular aluminum (Al grains coated with Al 2O3) was signiﬁcantly higher than that of bulk Al ( ∼1.2K )[ 23]. Several mechanisms were pro- posed to explain this enhancement but none has proven to besatisfactory. Our observations provide potential explanationfor the enhanced T cin granular aluminum thin ﬁlms. IV . DISCUSSION Now we discuss the microscopic physical origin of these additional excitations. According to Eq. ( 7), at n=0.75 the resonant conditions occur at εm=−εd, which corresponds to the dispersion law of surface plasmons propagating alongplanar metal-dielectric interfaces [ 24]: k=ω c/parenleftbiggεmεd εm+εd/parenrightbigg1/2 . (10) For nonplanar interfaces of aluminum nanoparticles the plas- mon resonance occurs at slightly different points. However,Eqs. ( 7) and ( 10) clearly relate the additional pole of the inverse dielectric response function ε eff−1of the metamate- rial to the plasmon resonance at metal-dielectric interfaces.Since this additional plasmon resonance of the metamaterialstructure may exist only in the vicinity of the metal phonon atω=/Omega1(q) (where ε mchanges sign), the proper name of this resonant excitation should probably be chosen as a “hybrid”plasmon-phonon resonance. Unlike regular plasmons in met-als (which typically occur in the visible and infrared frequencyranges) the hybrid plasmon-phonon mode in metal-dielectriccomposite metamaterials occurs near the frequency rangeof conventional phonons. This means that hybrid plasmonphonons are coupled to lattice vibrations via the lattice polar-izability, which enables their detection via inelastic neutronscattering. Based on Eqs. ( 2) and ( 3), it is clear that the existence of an additional plasmon-phonon pole of the inverse dielectricresponse function ε eff−1of the metamaterial may lead to increased Tc. The microscopic physical origin of this effect(a) 0.0 0.2 0.4 0.6 0.8 1.001234567Tc (K) n experiment plasmon-phonon pole phonon pole combined bulk Al bulk Al2O3nc (b) FIG. 6. (a) Schematic view of the surface plasmon resonance geometry: an electron e 2located next to the interface between two media with dielectric permittivities ε1andε2, interacts with electron e1and its image e1/prime. Resonant conditions are obtained at ε1=−ε2. (b) Plots of the theoretically calculated values of Tcas a function of metal volume fraction n, which would originate from either phonon [Eq. ( 16), black line] or plasmon-phonon [Eq. ( 14), red line] pole of the inverse dielectric response function of the Al-Al 2O3core-shell metamaterial in the absence of each other. Blue dashed line shows the predicted behavior in the presence of both poles. The experimentallymeasured data points from Ref. [ 5] are shown for comparison on the same plot. The vertical dashed line corresponds to the assumed value ofn cr. may be illustrated by Fig. 6(a). As pointed out in Ref. [ 25], plasmon-mediated pairing of electrons may be understood interms of an image charge-mediated Coulomb interaction. Letus consider two electrons located next to a planar interfacebetween two media with dielectric permittivities ε 1andε2,a s shown in Fig. 6(a). The ﬁeld acting on the charge e2in the medium ε1atz>0 is obtained as a superposition of ﬁelds produced by the charge e1and its image e1’[26]. As a result, the effective Coulomb potential may be obtained as V=e ε1/parenleftbigge r1−e r2/parenleftbiggε2−ε1 ε2+ε1/parenrightbigg/parenrightbigg , (11) which may be simpliﬁed as V=2e2 r(ε2+ε1), (12) 024515-5VERA N. SMOLYANINOV A et al. PHYSICAL REVIEW B 100, 024515 (2019) if both charges are located very close to the interface, so thatr1=r2=r.T h eε2=−ε1condition, which maximizes the electron pairing interaction, corresponds to the dispersionlaw of surface plasmons propagating along the interface [seeEq. ( 10)]. The values of λ effdue to each pole in Eq. ( 6) may be evaluated in terms of λmfor the bulk metal, assuming that νin Eq. ( 3) is proportional to n,which is justiﬁed by the fact that there are no free charges in the dielectric phase ofthe metamaterial. We assume that Im( ε d)∼0 and neglect the dispersion of Im( εm) in this simpliﬁed estimate (the detailed calculations may be found in [ 3]). Near the plasmon-phonon pole Im/parenleftbig ε−1 eff/parenrightbig ≈9(1−n) 2n(3−2n)ε/prime/primem, (13) where ε/prime/prime m=Im(εm). Therefore, using Eq. ( 3), we may obtain the expression for λeffas a function of λmandnfor the plasmon-phonon pole: λeff≈9(1−n) 2(3−2n)λm, (14) where λmfor the parent superconductor is determined by Eq. ( 3)i nt h e n→1 limit. On the other hand, near the regular phonon pole the inverse dielectric response function of themetamaterial behaves as Im/parenleftbig ε −1 eff/parenrightbig ≈n (3−2n)εm/prime/prime. (15) Therefore, near this pole λeff≈n2 (3−2n)λm. (16) Assuming the known values Tcbulk=1.2 K and θD=428 K of bulk aluminum [ 27], Eq. ( 2) results in λm=0.17, which corresponds to the weak coupling limit. Let us plot the hypo-thetical values of T cas a function of metal volume fraction n, which would originate from either Eq. ( 14)o rE q .( 16)i nt h e absence of each other. The corresponding values calculated as Tc=TCbulk exp/parenleftbigg1 λm−1 λeff/parenrightbigg (17) are shown in Fig. 6(b). The vertical dashed line corresponds to the assumed critical value of the metal volume fractionn crat which the additional plasmon-phonon pole of εeff−1 disappears as n→0 (since according to Eq. ( 4) the magnitude ofεmis limited). The blue dashed line shows the predicted behavior in the presence of both poles [ 3]. The experimentally measured data points from [ 5] are shown for comparison on the same plot. The match between the experimentallymeasured values of enhanced T cand the theoretical curve obtained based on Eq. ( 17) is impressive, given the fact thatour model does not contain any free parameters. It implies that the plasmon-phonon mediated electron pairing is the physicalmechanism responsible for the tripling of T cin the Al-Al 2O3 core-shell metamaterial superconductor. At ﬁrst glance, the strength of the observed additional peak at ∼15 meV in the inelastic neutron scattering data in Fig.4(b) appears to be quite weak. In fact, the observation of this peak requires quite strong electron-plasmon coupling asdiscussed below. For the case of conventional electron-phononsuperconductors the interaction is very strongly dependent onwhich part of the Brillouin zone is being studied. Typicallysuch anomalies are only observed when measuring singlecrystals, even in systems that exhibit strong phonon anomalies(which is not the case for pure Al). Furthermore, one has toknow where to look. A good example is YNi 2B2C[28], which has an enormous anomaly at one particular wave vector. Inthat case, the Fermi wave vector is revealed by the incom-mensurate magnetic ordering in the related ErNi 2B2C and HoNi 2B2C magnetically ordered superconductors [ 29], so one knows where to look for a large electron-phonon interaction.On the other hand, when measuring the (generalized) phonondensity of states (GPDOS) such as for the present nanoparticlesystem, any phonon anomaly gets averaged with all the restof the phonons and most often only a very small anomaly,or no anomaly, is seen. That is certainly the case for purealuminum. Therefore, the only way one would see any effectin the described system is if there is a strong coupling whenthe plasmon and phonon dispersions cross, since for inelasticneutron scattering one only sees the effects through the changein the phonons (since neutrons do not “see” the plasmons orelectrons directly). It would also be inappropriate to compare the size of the observed anomaly in the GPDOS with the van Hovesingularity for the transverse acoustic phonons in the phonondensity of states around 20 meV , where these phonons becomedispersionless at the zone boundary. Typically, it is difﬁcultto see any effect in a phonon density of states measurementsince the effect only occurs in a small region of ( Q,E), no matter how strong the coupling is, as the GPDOS averageseverything. The fact that we do see an anomaly is interestingand requires quite strong coupling. ACKNOWLEDGMENTS This work was supported in part by the DARPA Award No. W911NF-17-1-0348 (71740-PH-DRP) “MetamaterialSuperconductors”, by ONR through the NRL Basic Re-search Program and Awards No. N00014-18-1-2681 and No.N00014-18-1-2653, and National Science Foundation (NSF)MRI Award No. 1626326. 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PhysRevB.62.5461.pdf	Raman investigation of damage caused by deep ion implantation in diamond J. O. Orwa, K. W. Nugent, D. N. Jamieson, and S. Prawer * School of Physics, University of Melbourne, Parkville, Victoria, 3052, Australia ~Received 9 December 1999 ! Raman microscopy has been employed to investigate the nature of damage created when natural type-IIa diamond is irradiated with MeV alpha particles. Three features appear in the Raman spectrum due to damage,viz.,~i!the ﬁrst-order diamond Raman line is broadened and downshifted, ~ii!broad features appear which are a measure of the vibrational density of states of ion-beam-amorphized diamond, and ~iii!the damage causes the appearance of sharp defect-induced Raman peaks at 1490 and 1630 cm 21. For damage below an amorphization threshold, a linear relationship exists between the full width at half maximum and frequency shift, which showsthat these are Kramers-Kronig related. The annealing behavior of the sharp Raman feature at 1490 cm 21 suggests that this peak is associated with vacancies with an activation energy for annealing of 4.06 eV, while the 1630-cm21peak is due to an interstitial related defect with an activation energy of 1.2 eV. For sub-MeV ion irradiation, damage beyond the critical amorphization level usually leads to relaxation of the diamondstructure to graphite upon thermal annealing. However, for MeV ion irradiation, it was found that annealing,even when the ion induced damage level is well above the amorphization threshold, could restore the originaldiamond structure. We attribute this result to the high internal pressure the damaged layer is subjected to whichdoes not allow relaxation to graphitically bonded structures. I. INTRODUCTION Diamond, because of its unique physical and chemical properties,1is an ideal material for specialized applications in electronics compared to widely used materials such as silicon. Diamond is a wide bandgap semiconductor, and thusits electrical behavior at room temperature largely resemblesan insulator. As with most semiconductors, the electronicproperties in diamond can be improved by adding dopants.An attractive method for doping diamond is ionimplantation. 2The drawback is that ion implantation intro- duces defects which must be repaired to realize the envis-aged superior electronic properties. Residual defects are re-sponsible for trapping and compensation which degrade theelectrical behavior and limit realization of electronic gradedoped diamond ~particularly ntype!. Raman spectroscopy has long been used to study carbons because of its ability to sensitively distinguish carbonallotropes. 3,4In a recent report,5we showed that Raman spectroscopy is extremely sensitive to the damage created byion irradiation. One particular defect which gives rise to apeak at 1490 cm 21in the Raman spectrum of ion-irradiated diamond was found to be strongly correlated with the com-pensation of carriers in p-type diamond. When this defect was removed by thermal annealing, very high mobility p-type diamond ~hole mobility 5600 cm 2/Vs with a low compensation ratio of about 5% !could be fabricated. Hence there is a strong motivation for studying the evolution andthermal annealing of defects in diamond as part of the over-all effort to optimize diamond for electronic applications. This paper presents a detailed Raman investigation of MeV radiation damage in diamond. We use cross-sectionalRaman microscopy ~described in Sec. II !to study the defects created as a function of damage density and annealing tem-perature. The motivation for using MeV ions is to create adamage layer deeply buried below the diamond surface witha relatively undamaged conﬁning cap which keeps the dam- aged region under high pressure and thus restrains it fromrelaxing to graphite. The use of inert helium ions ensures thatno chemical interaction occurs between the implanted spe-cies and the substrate so that any features observed arepurely due to damage. A. Ion-beam-induced damage in diamond Energetic ions incident on the surface of a solid will pen- etrate and come to rest at a distance determined by the ionenergy, ion mass, and properties of the target material. Thepassage of an ion creates a collision cascade, which is welldescribed by simulation programs such as transport of ions inmatter ~ TRIM!code.6This passage breaks many bonds, cre- ating many defects along the ion track, which can aggregate:thus the damaged material can be rich in both point and morecomplex defects such as dislocations. Diamond, like othercovalently bonded semiconductors, will amorphize once thedamage level exceeds a critical density. For Si and Ge, ther-mal annealing results in recrystallization of the lattice. How-ever, the situation is more complicated for diamond becauseits structure is metastable with respect to graphite, and trans-formation to the latter is a common occurrence during irra-diation or following post-implantation heat treatment. The processes of ion beam modiﬁcation and thermal an- nealing of diamond have been extensively reviewed. 7–11The macroscopic manifestations of the damage, which are asso- ciated with the transformation from sp3tosp2bonding, in- clude very large increases in electrical conductivity, chemi-cal etchability, increased Rutherford backscattering spectroscopy ~RBS!dechanneling and reduction in xmin, and discoloration. Most studies have focused on ion-beam-induced damage close to the surface such as created by sub-MeV light ions. For this near surface damage, a critical dam- age level of 1 310 22vaccm23,12referred to as Dc, has beenPHYSICAL REVIEW B 1 SEPTEMBER 2000-I VOLUME 62, NUMBER 9 PRB 62 0163-1829/2000/62 ~9!/5461 ~12!/$15.00 5461 ©2000 The American Physical Societyidentiﬁed above which the diamond structure amorphizes and cannot be restored by annealing. It is not establishedwhether this critical level applies to deeply buried damage aswell. While the macroscopic changes which occur in ion- irradiated diamond are known, the microscopic nature of thedamage is not clear, partially due to the absence of detailedtransmission electron microscopy ~TEM !studies of ion- irradiated diamond. One attempt at a microscopic model 13 based on the model of Morehead and Crowder envisages thatthe passage of each ion leaves a ‘‘trail’’ of damaged spheresof average radius r. In the case of ion irradiation of silicon, these spheres are primarily comprised of amorphous silicon.When this model is applied to diamond, each sphere is en-visaged to consist of partially graphitized material. The elec-trical conductivity and size of each of these spheres dependlargely on the implantation temperature. When the concen-tration of these spheres reaches a sufﬁcient level, a connec-tive pathway may be formed between them resulting in avery sharp rise in conductivity as is observed. 14The damage level at which this overlap occurs is also very close to thedamage level above which sub-MeV ion-irradiated diamondcollapses to graphite upon thermal annealing and the damagelevel at which RBS and other measurements indicate that thediamond lattice has been ‘‘amorphized.’’ Despite the successof this model, no direct evidence has been produced to con-ﬁrm the existence of islands of amorphized and graphitizedclusters in ion-beam-irradiated diamond. B. Raman spectroscopy of ion-irradiated materials The effect of ion implantation on covalent materials is characterized by Raman spectroscopy using the phonon con-ﬁnement model, 15which has been used to explain the skew- ing and downward shift of the ﬁrst-order Raman line as thedamage level increases. Normally, momentum conservationlimits the Raman scattering to the zone center phonon ~i.e., q50). However, if the phonons are conﬁned in space by microscrystallite boundaries or defects, there will be uncer- tainty in the phonon momentum, allowing phonons with q .0 to contribute to the Raman signal. For example, skewing of the ﬁrst-order Raman peak in silicon at 520 cm 21, indica- tive of phonon conﬁnement, has been observed for doses as small as 1 31011Sicm22. The exact shape of the Raman line depends on the details of the phonon dispersion curve closeto the zone center. At damage levels approaching or exceeding the amor- phization limit, the Raman spectrum reﬂects the vibrationaldensity of states ~VDOS !integrated over all modes and all frequencies. As the translational symmetry, which character-izes the crystalline structure, is lost, the optical vibrations arelocalized instead of extending in the total periodic crystal. Ifthe coherence length between optical vibrations is too small,the Raman spectrum is basically the broadened phonon den-sity of states. ~For silicon this occurs when the cluster size is of the order of 10 interatomic spacings or less. !The Raman intensity can then be expressed as 16 I~v!5( b~Cb/v!@11n~v!#gb~v!, ~1!wheregb(v) is the VDOS, 1 1n(v) is the thermal popula- tion of the initial states, and the constant Cbdepends on the bandb. Fora-Si a broad, but well-deﬁned peak is observed at about 480 cm21in excellent agreement with the VDOS as measured by neutron scattering. For silicon damaged at anintermediate level, the Raman spectrum consists of a mixtureof a skewed Lorentzian crystalline silicon peak and an amor-phous silicon peak. This is consistent with the Morehead-Crowder model in which amorphous Si clusters are created along the ion track. The ratio of the intensity of the a-Si peak to the c-Si peak can then be used to deﬁne an amor- phous fraction. In contrast with Si, as will be seen below, the Raman signal shows evidence for the presence of amorphized dia-mond for even the lowest damage levels studied. Further-more, for higher doses, despite the presence of the amor-phous component, no corresponding skewing of the ﬁrst-order diamond line is observed even when down shifted byup to 50 cm 21. In addition, very sharp peaks appear in the Raman spectrum at 1490 and 1630 cm21, which, in some cases, have a full width at half maximum ~FWHM !less than that of the ﬁrst-order diamond line. C. Annealing kinetics Defect annealing can result from one or a combination of processes. Simple defects can form defect complexes bycombination whereas complex defects can dissociate to formsimpler defects. In each case the concentration of the originaldefect diminishes as manifested by a decrease in the intensityof the monitored signal. Each process that results in the an-nealing of a defect is characterized by an activation energy,which the defect must have to initiate the annealing. Theactivation energy is easily extracted from the equations thatdescribe the relevant annealing kinetics. If a material consisting of Ndefects per unit volume is subjected to annealing, the rate of change in the number ofdefects is given by the differential equation dN dt52raNa, ~2! where ais the order of the reaction: rais a temperature- dependent quantity called the rate constant and is related tothe diffusion coefﬁcient of the defect and, hence, the tem-perature. The rate constant is expressed as r a5AexpS2Ea kBTD, ~3! so that Eq. ~2!can be solved for a51 and a52 to obtain N~t!5N0exp~2r1t! ~4! and N~t!5N0 11N0r2t, ~5! respectively. In the preceding three equations, Ais a constant which contains the vibrational frequency associated with the process17andN0is the defect concentration at time t50.5462 PRB 62 J. O. ORWA, K. W. NUGENT, D. N. JAMIESON, AND S. PRAWERTo obtain the activation energy, one may perform isother- mal or isochronal annealing. We performed isochronal an- nealing for which, in the case of a51, Eqs. ~2!and~3!can be combined and solved to yield17,18 N5N0expF2AkBT2 EaexpS2Ea kBTDG, ~6! whereAhas the same meaning as in Eq. ~3!. By treating Ea andAas adjustable parameters, experimental data on the intensity of the various peaks as a function of the annealingtemperature can be ﬁtted to Eq. ~6!to obtain the activation energy. II. EXPERIMENT The sample studied was a natural type-IIa diamond slab of dimensions (3 3330.5) mm from Drukker Corporation, Netherlands. All faces of the diamond were polished with themain face a ~100!plane and the edges parallel to the ~010! and~001!crystal planes. The edge was implanted with 3-MeV alpha particles at room temperature in the random orientation to a dose of 1 310 17cm22. The beam was deliv- ered by the 5U pelletron accelerator at The University ofMelbourne. The longitudinal range and range straggling of the helium ions was determined using TRIMasRp6DRp 55.8860.25mm. The as-implanted diamond was oriented such that the @100#direction was parallel to the incident laser beam. Thus the laser beam was perpendicular to the original implantationdirection, allowing cross-sectional sampling of the damagedregion. A schematic diagram of the implantation and Ramananalysis geometry employed is presented elsewhere. 19,21The incident beam polarization was varied from Eii@010#to Eii@001#, while the polarization of the scattered beam, Es, was ﬁxed parallel to @001#. Under these conditions, the ﬁrst- order diamond Raman line at around 1332 cm21is allowed by the selection rules when Eiis parallel to @010#~i.e., the polarization of the input beam is perpendicular to the direc-tion of the ﬁxed polarization analyzer !and forbidden when E iis parallel to @001#~i.e., the input beam polarization di- rection is parallel to the analyzer direction !. The spectra ob- tained in the above conﬁgurations will be referred to as ‘‘dia-mond allowed’’ and ‘‘diamond forbidden.’’ The 514.5-nm line of an argon ion laser was the main excitation wavelength used although the 488-nm line wasalso used to distinguish between Raman and luminescencebands as well as to identify any resonance enhancement. Thebeam was focused to a 1- mm spot using a 3100 Olympus microscope objective. The backscattered light was collectedthrough the same objective and was analyzed by a DILORXYconfocal micro-Raman spectrometer with optical multi- channel collection using a charge coupled device ~CCD!ar- ray detector. Using a confocal aperture of 120 mm, a cylindrical region of about 1 mm diameter and 2 mm height was sampled in a typical measurement. Use of the cross-sectional Ramananalysis geometry described earlier allowed the Raman spec-tra to be recorded at different depths below the implantationsurface. Since the damage level increases with ion penetra-tion depth, the analysis described provided a convenient andaccurate way of proﬁling damage compared to the usualtechnique of directing the analysis beam along the original implantation direction. According to the selection rules the diamond line at 1332 cm 21should not be visible at all in the ‘‘diamond- forbidden’’ conﬁguration. A small breakthrough in the dia-mond line was, however, observed in this orientation and isattributed to deviation from ideal backscatter geometry of thescattered light due to the large numerical aperture of theobjective used. Correction for the breakthrough in the twoanalysis orientations was achieved through the equations N TFS5NMFS2~0.065 333NMAS! ~7! and NTAS5NMAS2NTFS, ~8! where ‘‘TFS’’ and ‘‘TAS’’ refer to true forbidden spectra and true allowed spectra, while ‘‘MFS’’ and ‘‘MAS’’ aremeasured forbidden spectra and measured allowed spectra,respectively. The factor of 3 in Eq. ~7!takes into account the difference in integration time between spectra collected inthe two conﬁgurations, while the 0.065 is the ratio of theintensity of the ﬁrst-order diamond line in ‘‘diamond-forbidden’’ orientation of a pristine diamond to that in‘‘diamond-allowed’’ conﬁguration. The sample was thensubjected t o 1 h annealing in ﬂowing argon at successively higher temperatures between 373 and 1473 K. Each anneal-ing was followed by the set of Raman measurements de-scribed above for the as-implanted sample. III. EXPERIMENTAL RESULTS AND DISCUSSION In this three-part presentation, we begin by studying the damage-induced structural modiﬁcation of the diamond byfollowing the changes in the ﬁrst-order diamond line widthand frequency as a function of depth. These changes occur asa result of the transition from an intact diamond structurewith single isolated point defects close to the implantationsurface to a disordered material near the end of range. Thesecond part identiﬁes the nature of the damage created fol-lowing implantation and studies its subsequent annealing be-havior by monitoring the intensity of the defect Raman peaksas a function of temperature. Finally, we discuss the anneal-ing behavior of diamond, which has been completely amor-phized by the ion irradiation, and show that under certainconditions this material will anneal back to diamond and notgraphitize. A. Structure of the zone center diamond peak as a function of the implantation depth Undamaged diamond displays a single ﬁrst-order Raman line at 1332 cm21and no other peaks in the range between 0 and 2000 cm21. When the diamond is damaged such as by ion beam irradiation, several other peaks appear in thisrange, most notably at 1490 and 1630 cm 21. The Raman spectra obtained in the ‘‘diamond-allowed’’ and ‘‘diamond-forbidden’’ conﬁgurations are displayed in Figs. 1 and 2. Thespectra of Fig. 1 have not been corrected for the 6.5% break-through, while those of Fig. 2 have been corrected to empha-size the damage peaks. The skewed appearance of the ﬁrst-order diamond peak observed in Fig. 1 disappears once thePRB 62 5463 RAMAN INVESTIGATION OF DAMAGE CAUSED B Y...spectra are corrected so that the peak can be ﬁtted with justone Lorentzian as shown in the inset to Fig. 3 for a spectrum taken at a depth of 2 mm below the implanted surface. Figure 3 shows the FWHM and peak position extracted from suchﬁts as a function of depth. The decrease in the intensity ofthe ﬁrst-order diamond peak with depth ~damage !, evident in Fig. 1, is accompanied by a shift to lower wave numbers anda smooth linewidth broadening. The peak width varied from13.3 cm 21at a depth of 0.5 mm to 87.8 cm21at a depth of 5.0mm. Over the same depth range, the peak position varied from 1325.8 to 1297.7 cm21. These parameters are compared with a peak width of 3.09 cm21centered at 1332.03 cm21for pristine diamond obtained under the same measurement con-ditions. Despite these large increases in peak width and shift with damage, the symmetric nature of the peak persisted for alldepths below the surface except at the very end of rangewhere the Raman spectrum is that of an amorphous materialand no ﬁrst-order peak can be observed. That the highlyshifted and enormously broadened peak was still the ﬁrst-order diamond peak was ascertained by the polarization be-havior of the peak, which was similar to the behavior of thepeak in undamaged diamond. The symmetry of the diamond line indicates that there is little or no phonon conﬁnement. The results suggest that theincrease in the FWHM is due to a decrease in phonon life-time as a result of scattering from the ion-induced defects.When a phonon decays, the zone center unperturbed fre- quency v0undergoes a complex shift, Dv1iG. The real part redshifts the normal mode frequency and the imaginarypart, which is the reciprocal of the phonon lifetime ~usually referred to as the linewidth !, increases. In the anharmonic approximation ~which has been found to be applicable to diamond !the real and imaginary shifts should be linearly related to each other via Kramers-Kronig relationships. 20 Thus as the level of damage increases one expects the pho-non lifetime to decrease, the FWHM to therefore increase,and~via Kramers-Kronig relationships !the peak position to be redshifted. Indeed, when the FWHM is plotted against thepeak position, as shown in Fig. 4, a linear relationship existsover a wide range of frequencies ~1331 to ;1312 cm 21!and corresponding linewidths ~4t o ;45 cm21!. Outside this range, the FWHM and peak shifts are still Kramers-Kronigrelated, but the slope of the line changes. This change inslope corresponds to the damage level at which amorphiza-tion of the diamond lattice occurs. The linear dependence ofthe FWHM on peak shift indicates that the increase in theFWHM is a direct measure of the decrease in phonon life-time and should be linearly related to defect density, pro-vided the defects are isolated. Plotting the FWHM and peak position versus the damage production as estimated by TRIMcan further elucidate the role of damage. The projected end of range of 5.88 mm pre- dicted by TRIMis about 0.5 mm deeper than the depth at which maximum damage occurs as determined by experi-ment from the ﬁrst-order diamond line shift and from opticalmicroscopy. Therefore, in constructing Fig. 5, the TRIMpro- ﬁle has been adjusted so that the peak damage coincides withthe maximum peak shift at the same depth. It was also nec-essary to average the TRIMdata over 0.5- mm intervals to match the spatial resolution of the Raman measurements. FIG. 1. ‘‘Diamond-allowed’’ spectra showing the ﬁrst-order diamond peak and the damage peaks located at ;1450, 1490, 1630, and 1680 cm21. The spectra have been vertically displaced by about 250 units for clarity. The spectrum, which corresponds to the depthof 5.5 mm, represents the zero value for the traces. FIG. 2. ‘‘Diamond-forbidden’’ spectra showing much more pro- nounced damage peaks compared to corresponding spectra of Fig. 1obtained in the ‘‘diamond-allowed’’ conﬁguration. No base line hasbeen subtracted, but the traces have been vertically displaced by afactor of about 200 for clarity.5464 PRB 62 J. O. ORWA, K. W. NUGENT, D. N. JAMIESON, AND S. PRAWERFigure 5 shows that both the peak shift and FWHM exhibit a linear dependence on damage below a damage level of 131022vaccm23. Above this damage level, the FWHM and peak position begin to saturate. This suggests that at thisdamage level the defects overlap to create a continuousamorphous layer. In Fig. 2 the Raman spectrum correspond-ing to this damage level ~at 4.5 mm!shows no evidence of the presence of a ﬁrst-order diamond line and is typical ofthe Raman spectrum from an amorphized material. As mentioned earlier, for sub-MeV ion irradiation, there exists a critical damage level of about 1 310 22vaccm23 above which the diamond structure amorphizes and cannot be restored by annealing. Annealing, under such circum-stances, results in the collapse of the matrix to the morestable form of carbon, viz., graphite. The difference betweensub-MeV and MeV ion damage is that for the latter, thedamage is deeply buried underneath a large, relatively intact,diamond cap layer which keeps the damaged region underhigh pressure. The results of Fig. 5 suggest that a critical damage level of about 1 310 22vaccm23for amorphization is also applicable to deep ion implantation. However, asshown below, the high pressure, which exists due to the largeconstraining cap, prevents the amorphized layer from relax-ing to graphite upon thermal annealing. If indeed amorphous regions were formed along each ion track, one would expect to observe a signal from these re-gions, which closely follows the VDOS as per Eq. ~1!. Fig- ure 6 shows normalized Raman spectra taken in the forbid-den orientation ~these spectra have not been corrected for polarization leakage and hence show a low-intensity ﬁrst-order diamond line in some of the spectra close to the surfacewhere the damage level is lowest !. In addition to the various sharp peaks, the Raman spectrum of radiation-damaged dia-mond is characterized by broad skewed peaks centered atabout 350 and 1245 cm 21. The shapes of these broad fea- tures conform closely to the VDOS of diamond.21An esti- FIG. 3. Variation of the ﬁrst-order diamond Raman peak posi- tion and FWHM with depth for 2 MeV, 1 31017cm22carbon im- planted, but unannealed diamond. The error for both the FWHMand peak position was of the order of the size of the symbols usedto designate the data points. The inset is the ﬁrst-order diamondRaman peak at ;1330 cm 21taken at a depth of 2 mm below the surface. The solid squares in the inset are data points, while thesolid line is a single Lorentzian ﬁt to the data points. The dottedcurve is a spectrum from pristine diamond, shown here for com-parison. The pristine diamond spectrum has been divided by 8000to ﬁt into the scale of the damaged diamond spectrum. FIG. 4. Variation of the ﬁrst-order diamond peak FWHM with peak position. The two lines correspond to regions with differentlevels of damage. The line with smaller linewidths corresponds tolow levels of damage where the defects occur as isolated pointdefects. The other line represents higher levels of damage where thedefects begin to overlap. FIG. 5. First order diamond peak position and FWHM vs dam- age. The damage data were obtained from TRIMwithEd550eV.PRB 62 5465 RAMAN INVESTIGATION OF DAMAGE CAUSED B Y...mate of the intensity of this amorphous component can be obtained by measuring the step height at 1245 cm21with respect to the background intensity at 1400 cm21. Figure 6 shows that, in the region of low damage where the damageoccurs as isolated point defects, and away from the end ofrange, the amorphous peak coexists with the zone centerphonon. Figure 7 is a plot the amorphous fraction normalizedto unity at the maximum damage level ~the inset of the same ﬁgure shows how the amorphous intensity was estimated !. This amorphous fraction is the ratio of the amorphous inten-sity to that of the ﬁrst-order diamond line in the irradiatedregion. Indeed, Fig. 7 shows that the amorphous fractionincreases linearly with damage up to a damage level of ;1310 22vaccm23. Parameters such as the ﬁrst-order dia- mond peak shift and FWHM which vary linearly with dam-age should also therefore vary linearly with the amorphousfraction as is indeed demonstrated in Fig. 8. B. Defect-related peaks Figure 2 shows the presence of sharp peaks at 1490 and 1630 cm21following ion irradiation. A number of smaller peaks are also observed at 1450, 1680, and 1800 cm21.T o extract quantitative information from the spectra, the peaksat 1490 and 1630 cm 21were ﬁtted with a combination of Lorentzians and Gaussians. Figure 9 shows that the peakpositions shift to lower wave numbers with increasing dam-age in an almost linear fashion below a damage level of about 1 310 22vaccm23, above which the peak shifts satu- rate. The inset in Fig. 9 shows the relationship between theFWHM and peak shift, which shows that the peak shifts arelinearly related to the FWHM, consistent with a Kramers- Kronig relationship. Although the smaller peaks mentionedwere not ﬁtted, they showed behavior with damage and an-nealing temperature that was similar to the 1490-cm 21peak as can be seen from Figs. 6 and 15. The 1490-cm21peak and the smaller peaks thus appear to belong to one family andmay be caused by similar defects. The intensity of the 1490- and 1630-cm 21peaks, normal- ized with respect to the intensity of the diamond line in the FIG. 6. Normalized Raman spectra as a function of depth ~dam- age!showing the growth of the amorphous peak with increasing damage. FIG. 7. Amorphous intensity vs TRIMdamage showing that amorphous intensity is linear with damage up to a damage level ofaround 1 310 22vac/cm3. The double arrow in the inset shows how the amorphous intensity was measured. FIG. 8. The ﬁrst-order diamond Raman peak shift and FWHM vs normalized amorphous intensity.5466 PRB 62 J. O. ORWA, K. W. NUGENT, D. N. JAMIESON, AND S. PRAWER‘‘diamond-allowed’’ conﬁguration, varied with damage. However, the behavior was different for the two peaks. InFig. 10 the intensity of the 1630-cm 21peak increases lin- early with damage up to about 1 31022vaccm23, above which it saturates. By contrast, the intensity of the1490-cm 21peak jumps quickly to a maximum at a damage level of only ;431021vaccm23and stays fairly constant in the less damaged region close to the implanted surface,decreasing sharply when the damage level exceeds 7310 21vaccm23. Above a damage level of;631022vaccm23, the peak disappears. The inset of Fig. 10, where the TRIMdamage has been plotted on a semiloga- rithmic scale, clearly shows the behavior of these peaks. Thedifferent dependence of the peaks on the level of damagemakes it clear that the defects which give rise to the 1490-and 1630-cm 21peaks have different origins. Initial clues to the origins of these peaks can be obtained from their positions. Theoretical calculations on the VDOSof diamond 22show that no states occur above 1400 cm21for sp3-bonded carbon structures. The position of the 1490-cm21peak is roughly midway between the position of the vibrations due to the singly bonded diamond line at 1332cm 21and the graphitic double-bond modes at 1580 cm21. Thus the defect responsible for this peak is likely to consistof conjugated double and single bonds. On the other hand,the 1630-cm 21peak most likely originates from some kind of CvC bond. Theoretical calculations22also suggest that modes above 1400 cm21are strongly localized. The ob- served sharpness of the peaks indicates that they originatefrom well-deﬁned local rather than extended defects. C. Annealing behavior Annealing was performed for 1 h ~this was sufﬁcient time for annealing to reach equilibrium at each temperature !in ﬂowing argon at several temperatures in the range 373–1473K. In Fig. 11 the position of the ﬁrst-order diamond line andFWHM are plotted as a function of annealing temperaturefor a point 2 mm below the surface ~which corresponds to an initial defect density of 3.5 31021vaccm23). A progressive upward shift in the ﬁrst-order diamond Raman line positionand a corresponding linewidth narrowing is observed as thetemperature increases. After annealing at 1473 K, the ﬁrst-order diamond line is at 1330.2 cm 21, while the FWHM is 6 cm21. These values are to be compared with a shift of 1332.03 cm21and a FWHM of 3.09 cm21for pristine dia- mond, which indicates that some residual damage still re-mains at this annealing temperature. Figure 12 plots the peakposition as a function of the FWHM for the temperature FIG. 9. The 1490-cm21~open squares !and 1630-cm21~solid squares !peak positions vs damage showing that the intensity of the peaks is linear with damage only for vacancy concentrations below1310 22cm23. The linear relationship between the defect peak po- sitions and FWHM shown in the inset shows that the two areKramers-Kronig related like the ﬁrst-order diamond line. FIG. 10. The 1490- and 1630-cm21peak intensities vs TRIM damage. ( I/Id) in the vertical axis is the ratio of the intensity of the defect peak to the ratio of the intensity of the ﬁrst-order diamondline measured at the same point. The inset shows the same plot withthexaxis replaced by a logarithmic scale, while the other axes remain the same. FIG. 11. The ﬁrst-order diamond peak width and peak position averaged over the ﬁrst 2 mm below the surface plotted against an- nealing temperature showing that the two are correlated.PRB 62 5467 RAMAN INVESTIGATION OF DAMAGE CAUSED B Y...range 300–1473 K, which is well described by a single straight line. The existence of a simple linear dependencebetween the peak position and the FWHM indicates that thedefects being removed by thermal annealing are those re-sponsible for the scattering and decay of the zone centerphonons as per the Kramers-Kronig relationship. In Fig. 13 the amorphous fraction is plotted as a function of annealing temperature for both an initial damage level of 3.5310 21vaccm23~i.e., a point 2 mm below the surface !and a damage level of 1.1 31023vaccm23@i.e., at a point near the end of range ~EOR!#. As expected for a point close to the surface with a defect density well below Dc, the amorphous fraction decreases as a function of annealing temperature.After annealing at 1473 K, the amorphous fraction is reduced to about 10% of its initial value. These results show thatannealing at up to 1473 K removes most of the damage, butis not enough to completely restore the diamond lattice. Theremaining amorphous component is consistent with the ob-servation that both the diamond peak shift and FWHM arenot restored to their preimplantation values after annealing atup to 1473 K. The results for the annealing of the EOR damage ~i.e., a defect level of 1.1 310 23vaccm23) is remarkable. Despite the damage level being more than 10 times Dcnear the end of range, the material nevertheless anneals back to diamond,and the amorphous fraction reduces to nearly zero for an-nealing temperatures in excess of 1300 K. It is seen fromFig. 13 that when the near-surface amorphous intensity ismultiplied by a constant, the curve nearly overlays that forthe end of range intensity, implying that the annealing be-havior for both regions is the same. It is interesting to notethat the EOR damage appears to have annealed somewhatmore completely than the surface damage. Figure 14 shows the intensity of the 1630- and 1490-cm 21 peaks as a function of annealing temperature for an initial defect density of 3.5 31021vaccm23~taken at a depth of 2 mm below the implantation surface !. The smooth curves are theoretical ﬁts to the data points obtained by the method ofleast squares. The annealing behavior of the 1490-cm 21peak was described by the differential equation of the form of Eq.~2!and the data points could be ﬁtted with one decaying exponential similar to Eq. ~6!. The value of the activation energy for defect annealing, E a, obtained from the ﬁtting was 4.06 eV. The non-monotonic dependence of the 1630-cm21peak intensity as a function of temperature indicates that two pro-cesses are involved: one in which the defect is formed~which occurs at about 650 K !and the other in which the defect is annealed ~which occurs at about 1000 K !. Because FIG. 12. Variation of the ﬁrst-order diamond Raman peak posi- tion with FWHM in the temperature range 300–1470 K. FIG. 13. Average amorphous intensity over the ﬁrst 2 mm below implantation surface and at the end of range vs annealing tempera-ture. The decrease in the intensity with annealing temperatureshows that the diamond lattice is gradually restored as the sample isannealed to higher temperatures. Also shown is a plot of the near-surface intensity multiplied by a factor of 3.5. The similarity of thetwo curves indicates that the annealing behavior is similar for theend of range and near-surface damage. FIG. 14. Relative intensity of the 1490-cm21~open triangles ! and 1630-cm21~solid squares !damage peaks vs annealing tempera- ture at 2 mm below the implantation surface. The smooth curves are least-squares ﬁt to the data points ~see text for details !.5468 PRB 62 J. O. ORWA, K. W. NUGENT, D. N. JAMIESON, AND S. PRAWERof the initial growth observed in the annealing of the 1630-cm21peak, an additional term to Eq. ~2!was required to describe the observed behavior. Thus we used the differ-ential equation dN dt5ra~Nf2N!2rbN. ~9! Equation ~9!is a ﬁrst-order differential equation, which can be solved as outlined in Sec. IC with ra 5A1exp(2E1/kBT) andrb5A2exp(2E2/kBT). HereNfin Eq.~9!is the maximum concentration of the defect respon- sible for 1630-cm21peak after a complete annealing of the unstable species, which aggregate to create it. Since Nrefers to the concentration of the defect at any time, the quantity (Nf2N) represents the concentration, at any time, of the unstable species. By ﬁtting the data points to the solution toEq.~9!, the activation energy for the growth of the defect was 0.55 eV, while a value of 1.22 eV was obtained for theannealing process. Once again, the 1490- and 1630-cm 21 peaks showing such markedly different annealing behavior is a clear indication that the defects which give rise to them arenot of the same origin. According to a recent publication by Prawer and Nugent, 23a defect which ﬁts the behavior of the 1630-cm21 peak is the dumbbell defect or the ^100&split interstitial, which consists of an isolated sp2-bonded carbon pair occu- pying the position of one carbon atom in a normal diamondlattice. The dumbbell defect is the dominant stable defectwhich forms during ion irradiation. 24The assignment of the 1630-cm21peak to the dumbbell defect is supported by the theoretical calculations by Drabold et al.25of the VDOS of a ta-C sample with 10% sp2-bonded carbon content. In addi- tion to revealing the existence of paired three fold-coordinated defects in the amorphous network, the calcula-tions showed sharp localized modes at around 1600 cm 21 due to these paired sp2-bonded carbon atoms. Furthermore, unpublished molecular dynamics results by Rosenblum andPrawer 26show that the introduction of the split interstitial into an otherwise perfect diamond lattice gives rise to a peakin the VDOS at 1630 cm 21. The observed annealing behavior is also consistent with the assignment of the 1630-cm21peak to the dumbbell de- fect. The lower activation energy of 0.55 eV corresponds tothe energy that the carbon self-interstitials require for mobil-ity and to combine to form the split interstitial. 27The higher activation energy corresponds to the energy required for themore stable split interstitial to migrate to sinks and annihilateor the energy required to break up the split interstitial backinto self-interstitials. The initial rise in the intensity of the1630-cm 21defect peak suggests that the defect is formed from simpler defects which become mobile at lower tem-peratures and which combine to form the relatively stablesplit interstitial. The nonzero intensity of the peak in theas-implanted sample indicates that some of the defect isformed during the irradiation process. This can be explainedby possible dynamic annealing which would not be unex-pected under the implantation conditions employed, espe-cially along the ion tracks. It is possible that the carbon self-interstitials, which are produced in large numbers during theirradiation process, are the building blocks for this defect.This view is supported by observing that an increase in dam- age in the unannealed sample towards the end of range isaccompanied by an increase in the intensity of this peak. Theincrease in the intensity of the peak with damage is consis-tent with an increased availability of self-interstitials to formthe defect. The bulk of these self-interstitials are createdaway from the regions of high thermal energy around the iontracks and thus lack the energy to migrate and form the splitinterstitial. Upon heating to around 650 K, they become mo-bile and combine to form the split interstitial. At higher tem- peratures, when vacancies become mobile, the defect annihi-lates at the vacancy sites and disappears. To speculate further on the structure of the defects that give rise to the 1490- and 1630-cm 21peaks, a comparison is made with damage studies using absorption and electronparamagnetic resonance ~EPR!techniques. Following radia- tion damage in type-II diamonds, the GR absorption series isobserved. The lowest-energy peak of the series, the GR1, hasa zero phonon line at 1.673 eV and is due to a neutral va-cancy with an activation energy of 2.3 eV. 28In addition, following irradiation, absorption bands with zero phononlines at 4.582 and 1.86 eV, among others, are observed, as isa vibrational peak at 1570 cm 21. The band at 4.582 eV, referred to as the 5RL center, and the 1570-cm21vibrational peak result from a pair of carbon atoms, which ﬁt the structure of a split interstitial.29,30The 5RL center and the 1570-cm21peak grow in intensity with increasing annealing temperature up to about 770 K, afterwhich they diminish, annealing out in the range 920–1170K. 30–32The annealing behavior of these structures follows closely the results of the present study for the 1630-cm21 Raman peak and may be explained similarly, lending strong support to our assignment of the 1630-cm21peak to a split interstitial. The 1.86-eV absorption band is thought to be due to an interstitial with activation energy for annealing of 1.68 eV.Comparisons with EPR show that the 1.86-eV line correlateswith the strength of the R2 EPR band, 33the properties of which have recently been shown31to be consistent with those for a ^001&split interstitial. R2 anneals out in the range 570–720 K.34Another EPR band, the R1, anneals out in the temperature range 600–700 K with an activation energy of0.6 eV. The temperatures at which the R1 and R2 bandsanneal out are too low compared to 1000 K, at which the1630-cm 21Raman peak, thought to be interstitial related, anneals out. They do, however, correspond to the tempera-ture at which the 1630-cm 21Raman peak intensity increases as a function of temperature. It may be, then, that the defects,which correspond to R1 and R2, combine to give rise to thedumbbell defect. As mentioned above, the energy of the 1490-cm 21peak lies midway between that corresponding to single sp3-bonded carbon and sp2-type double bonds, thus suggest- ing that the defect corresponds to a mixture of conjugateddouble and single bonds. The sharpness of the peak indicatesthat the defect responsible is very small and well deﬁned.The most likely defect is a single vacancy or divacancy. Inthe vicinity of a vacancy, regions of mixed single and doublebonds are formed. Rosenblum 26has shown that, when a va- cancy is introduced into an otherwise pristine diamond lat-tice, new vibrational modes appear at 1470 cm 21. EPR dataPRB 62 5469 RAMAN INVESTIGATION OF DAMAGE CAUSED B Y...have also identiﬁed a vacancy-related center in radiation- damaged diamond. The center, referred to as R4/W6, annealsout at 1200 K and is though t to be a nearest-neighbor diva-cancy in the neutral charge state. 35This annealing tempera- ture is very close to 1223 K where the 1490-cm21Raman peak observed in the present study anneals. Thus the defectgiving rise to the 1490-cm 21peak may be composed of two nearest-neighbor vacancies. D. Annealing of diamond irradiated well above Dc Figure 15 shows the Raman spectra recorded as a function of annealing temperature for a cross-sectional point close to the end of range where the initial damage level is 1.1310 23vac cm23. The spectra were recorded in the ‘‘diamond-forbidden’’ orientation, but without correction, sothat the ﬁrst-order diamond line is visible because of polar-ization breakthrough. In some of the spectra a very sharpﬁrst-order diamond line can be observed at 1330 cm 21, which originates from the undamaged region beyond the endof range and is ignored in this discussion. The spectrum ofthe unannealed ~303 K !sample is typical of that of amor- phous material. No ﬁrst-order diamond line is visible and thespectrum closely follows the VDOS of diamond.The most dramatic and unexpected observation is the near-complete annealing of the diamond as is evidenced bythe disappearance at the highest annealing temperatures ~see the inset to Fig. 15 !of all the defect peaks, the near restora- tion of the ﬁrst-order diamond line to its preimplantationFWHM and position, and the near reduction to zero of theamorphous fraction ~see Fig. 13 !. We note that annealing did not result in graphitization even at the end of range where the damage far exceeded D c. This may be due to the large pres- sures to which the damage structures are subjected whichconstrain them from relaxing to graphite. Some damage stillremains as is evidenced by the fact that the diamond FHWMis;6c m 21and the peak position is ;1328 cm21~c.f. 3.01 and 1332.03 cm21for pristine diamond !. It is conceivable that annealing to temperatures higher than the ;1500 K em- ployed here will result in total repair of the diamond latticefollowing deep damage. In addition to the features already discussed, in the Raman spectra of Fig. 15, broad peaks are observed at 250, 480, and750 cm 21. The peaks at 250 and 480 cm21appear at around 1100 K, have their highest intensity at 1220 K, and decreaseagain at 1300 K. The peak at 750 cm 21has its maximum intensity at about 873 K. ~These features are also observed for damage levels well below Dc, but are less pro- nounced. !We cannot at present assign these peaks to par- ticular defect structures; however, by changing the excitationwavelength from 514 to 488 nm, we were able to conﬁrmthat these peaks originate in Raman scattering and are notdue to ﬂuorescence. In Fig. 15, the 1490-cm 21peak and its associates, the 1430-, 1680-, and 1800-cm21peaks are virtually absent for the as-implanted sample ~this is consistent with the depen- dence of this peak’s intensity on damage as shown in Fig.10!. Between 400 and 600 K, only a small showing is re- corded. From 600 K, the peaks gradually grow peaking ataround 1000 K, after which they again begin to decrease inintensity for higher-temperature anneals. This observationand the damage-dependent behavior shown in Fig. 10 sug-gest that the vibrational modes arising from this vacancy-related defect are easily quenched in the vicinity of extensivedamage. As damage is removed by annealing at progres-sively higher temperatures, there is a corresponding increasein the intensity of the peak because the removal of damage inthe vicinity reduces the degree of quenching. However,above about 1100 K, the defect that gives rise to this peakbecomes mobile and quickly anneals out, as is reﬂected inthe diminishing size of the peak in Fig. 15. IV. SUMMARY AND CONCLUSIONS ~i!The critical dose for amorphization, Dc, of dia- mond damaged by deep ion implantation is about 131022vaccm23, the same value previously determined for shallow ion damage. The difference between shallow anddeeply buried damage is that for the latter damage levels aboveD ccan be annealed, whereas in the former case an- nealing results in relaxation to graphite. ~ii!The annealing of the 1490-cm21peak obeys ﬁrst- order kinetics and is characterized by one ﬁrst-order processwith an activation energy of 4.06 eV. ~iii!The annealing of the 1630-cm 21peak follows ﬁrst- FIG. 15. Normalized Raman spectra acquired by the 514.5-nm line of an argon ion laser at the end of range of a diamond sampleimplanted with 1-MeV helium ions. Despite the end of range dam-age being well above the amorphization threshold, the inset showsthat all the damage features have disappeared and only the ﬁrst-order diamond Raman peak is left after annealing at 1470 K.5470 PRB 62 J. O. ORWA, K. W. NUGENT, D. N. JAMIESON, AND S. PRAWERorder kinetics characterized by two ﬁrst-order processes with activation energies of 0.55 and 1.22 eV. The lower activationenergy is attributed to the migration energy of simpler de-fects which combine to form the defect responsible for the1630-cm 21peak, while the higher activation energy corre- sponds to the migration energy of the defect that causes the1630-cm 21peak. ~iv!Based on the location of the defect peaks, their an- nealing behavior, and theoretical models, the 1490-cm21 peak is assigned to a vacancy or divacancy with conjugated single and double bonds, while the 1630-cm21peak is attrib- uted to the ^100&split interstitial defect. ~v!The peak width and peak shift of both the ﬁrst-order Raman line and the defect peaks are linearly related. Theincrease in width is due to a linear decrease in phonon life-time with damage below the amorphization limit due to in-creased scattering and decay of the phonons. The shift tolower wave numbers of the peak positions is Kramers-Kronig related to the increase in linewidth. The ﬁrst-orderdiamond peak symmetry suggests that the phonon decay isvia simple scattering from point defects rather than phononconﬁnement. ~vi!The Raman spectrum shows signals which originate from amorphous zones even for the lowest damage levelsstudied, which are concomitant with signals from the ﬁrst-order diamond line and from the divacancy and split intersti-tial. Note that the shape of the amorphous component, asseen from Fig. 6, does not change with increased damage.This implies that amorphous clusters and damaged, but stillessentially single-crystalline, material coexist in the irradi-ated volume. The picture to emerge is then as follows: Ra-diation damage creates tiny amorphized regions along theion tracks which are isolated and far apart for low levels ofdamage. Interspersed between the amorphous clusters arepoint defects in an otherwise largely damage-free diamond lattice. As the damage increases, the amorphized regionsgrow in size and in number. These changes are reﬂected inan increased intensity of the amorphous fraction with dam-age. At sufﬁciently high damage levels, the amorphized clus-ters overlap and the whole damaged region amorphizes. Inthis picture, the Raman spectra obtained in the ‘‘diamond-allowed’’ conﬁguration come mainly from the regions be-tween the amorphous clusters, which explains why the sym-metry in the ﬁrst-order diamond line persists even withincreased damage. The ‘‘diamond-forbidden’’ spectra are aresult of scattering from the amorphized damage clusters andthe point defects. The Raman spectrum thus corresponds tothe broadened diamond VDOS, although it retains its dia-mond characteristics with a cutoff frequency slightly higherthan the zone center diamond mode. ~vii!In conclusion, the ﬁndings of this research hold im- portant implications for the realization of chemical doping ofdiamond using ion implantation. Far greater care must be taken if shallow doping is intended to ensure that D cis not exceeded. Because it is possible to anneal damage in excess ofDcfor deeply buried damage, the latter offers some ﬂex- ibility and would be ideal in cases where heavy doping isrequired. Our present and future work will now focus onapplying the knowledge gained here to more practical sys-tems such as correlating the intensity of the defects withconductivity of doped diamond samples. ACKNOWLEDGMENTS The authors gratefully acknowledge stimulating discus- sions with Professor R. Besserman of the Solid State Insti-tute and Physics Department, Technion-Israel Institute ofTechnology, Haifa, Israel. *Author to whom correspondence should be addressed. Email ad- dress: s.prawer@physics.unimelb.edu.au 1The Properties of Natural and Synthetic Diamond , edited by J. E. Field ~Academic, New York, 1992 !. 2J. F. Prins, Mater. Sci. Rep. 7, 271 ~1992!. 3D. S. Knight and W. B. White, J. Mater. Res. 4, 385 ~1989!. 4W. 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PhysRevB.101.075433.pdf	PHYSICAL REVIEW B 101, 075433 (2020) Impact of epitaxial strain on the topological-nontopological phase diagram and semimetallic behavior of InAs /GaSb composite quantum wells H. Irie ,T. Akiho, F. Couëdo ,†R. Ohana , K. Suzuki ,‡K. Onomitsu, and K. Muraki NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan (Received 8 November 2019; revised manuscript received 26 January 2020; accepted 10 February 2020; published 27 February 2020) We study the inﬂuence of epitaxial strain on the electronic properties of InAs /GaSb composite quantum wells (CQWs), host structures for quantum spin Hall insulators, by transport measurements and eight-band k·p calculations. Using different substrates and buffer layer structures for crystal growth, we prepare two types ofsamples with vastly different strain conditions. CQWs with a nearly strain-free GaSb layer exhibit a resistancepeak at the charge neutrality point that reﬂects the opening of a topological gap in the band-inverted regime. Incontrast, for CQWs with 0.50% biaxial tensile strain in the GaSb layer, semimetallic behavior indicating a gapclosure is found for the same degree of band inversion. Additionally, with the tensile strain, the boundary betweenthe topological and nontopological regimes is located at a larger InAs thickness. Eight-band k·pcalculations reveal that tensile strain in GaSb not only shifts the phase boundary but also signiﬁcantly modiﬁes the bandstructure, which can result in the closure of an indirect gap and make the system semimetallic even in thetopological regime. Our results thus provide a global picture of the topological-nontopological phase diagram asa function of layer thicknesses and strain. DOI: 10.1103/PhysRevB.101.075433 I. INTRODUCTION InAs/GaSb composite quantum wells (CQWs), in which electrons and holes are separately conﬁned in the InAs andGaSb wells, have drawn renewed attention as a host structurefor quantum spin Hall insulators (QSHIs), or two-dimensionaltopological insulators [ 1–10]. A CQW, usually ﬂanked by Al xGa1−xSb barriers ( x=0.5–1.0), is characterized by a broken-gap type-II band alignment where the conduction-band bottom of InAs is located below the valence-band topof GaSb. When the thicknesses of the InAs and GaSb layersare such that the quantum conﬁnement is not too strong,the CQW has an inverted band structure; namely, the ﬁrstelectron subband lies below the ﬁrst heavy-hole subband atthe Brillouin-zone center. In the absence of coupling betweenelectrons and holes, the system becomes a semimetal, with thein-plane dispersion curves of the electron and hole subbandsintersecting each other at a ﬁnite momentum. However, inthe presence of ﬁnite coupling between the electron and holewave functions, a hybridization gap of a few meV opens atthe band crossing point. This energy gap, possessing a naturedistinct from that of normal semiconductors, gives rise totopologically protected gapless states at the sample edges [ 1]. hiroshi.irie.ke@hco.ntt.co.jp †Present address: Laboratoire National de Métrologie et d’Essais (LNE) Quantum Electrical Metrology Department, Avenue Roger Hennequin, 78197 Trappes, France. ‡Present address: Fukuoka Institute of Technology, Fukuoka 811- 0295, Japan.Although the existence of the hybridization gap in InAs/GaSb CQWs with the inverted band order was con- ﬁrmed in early experiments [ 11,12], subsequent transport measurements have consistently shown that the conductivityremains ﬁnite at low temperatures even when the Fermilevel is adjusted to the middle of the gap [ 2,13]. Since this residual bulk conductivity is in most cases comparable toor greater than the conductance e 2/hexpected for the edge states ( eis the elementary charge and his Planck’s constant), it represents an important issue in exploring and exploitingthe exotic properties of the topological edge states. Whiledisorder-induced in-gap states are a likely cause, experimentsshow that the residual conduction remains strong even in high-quality samples [ 7,9] and tends to weaken in more disordered samples [ 5,14]. It has been argued on the basis of quantum transport theory that the low-temperature conductivity in thehybridization gap of an electron-hole coupled system hasan intrinsic lower limit determined by the degree of bandinversion and the magnitude of the hybridization gap [ 15]. However, as noted in many studies, the semimetallic behaviorof the conductivity, represented by the absence of activatedtemperature dependence, suggests an effective closure of theband gap, with its origin being discussed in terms of theanisotropy, or warping, of the valence band [ 6,12,16]. Another factor of particular notice is the epitaxial strain that arises from the lattice mismatch among constituentmaterials of heterostructures, which has recently attractedinterest as a useful tool to engineer the band structureof QSHIs. Speciﬁcally, a lattice-mismatched system ofInAs/In xGa1−xSb CQWs, in which the In xGa1−xSb layer is under compressive epitaxial strain, has been shown to havean enlarged hybridization gap and thus exhibits signiﬁcantly 2469-9950/2020/101(7)/075433(12) 075433-1 ©2020 American Physical SocietyH. IRIE et al. PHYSICAL REVIEW B 101, 075433 (2020) reduced residual conduction [ 17–19]. These results remind us of the need to fully take into account the strain effectsalso in the conventional InAs /GaSb/AlSb system [ 20,21], termed “the 6.1-Å family” [ 22], that is generally thought of as being approximately lattice matched. Indeed, Zakharovaet al. have calculated the band structures of InAs /GaSb CQWs pseudomorphically grown on InAs and GaSb substrates andcompared them with that for the unstrained case [ 20]. They showed that strain strongly affects the order of the levels atthe Brillouin-zone center, subband dispersion, and magnitudeof the hybridization gap. By noting that the band anisotropydepends on strain, they also showed that, when the InAswell is wide, the band gap becomes negative and indirect(i.e., semimetallic) for a structure grown on a GaSb substrate,whereas it is positive and direct for the same structure grownon an InAs substrate. On the experimental side, externallyapplied hydrostatic pressure [ 23] and uniaxial strain [ 24]h a v e been shown to induce measurable changes in the electronicproperties of the InAs /GaSb system. In this paper, we study the inﬂuence of epitaxial strain on the electronic properties of InAs /GaSb CQWs via mag- netotransport measurements and theoretical calculations. Inparticular, we focus on the effects of biaxial tensile strain inthe GaSb layer, which become relevant in some situations.By utilizing different substrates and buffer layer structures forcrystal growth, a tensile strain of up to ∼0.50% ( ∼1.13%) is exerted on the GaSb (InAs) layer via the lattice-constantmismatch between the quantum-well (QW) layers and theunderlying buffer layer. Analyzing the magnetotransport databased on an electrostatic capacitor model of the CQWsenables us to distinguish the inverted and noninverted bandalignment. A comparison of samples with varying QW thick-ness and different strain conditions reveals that, for the sameGaSb thickness, the tensile strain shifts the boundary betweenthe inverted and noninverted regimes to a larger InAs thick-ness. For a thick InAs well, a CQW with tensile-strainedGaSb shows semimetallic behavior indicating a gap closure,whereas an unstrained-GaSb CQW with a similar degree ofband inversion shows a gap. Eight-band k·pcalculations also reveal that the tensile strain makes the phase diagram as afunction of InAs and GaSb layer thicknesses essentially differ-ent from that known for the unstrained case. Our results willbe useful to better understand previous reports on InAs /GaSb CQWs, where the strain effects have often been overlooked,and appropriately design future experiments. II. EXPERIMENT A. Heterostructure design We study two types of CQWs with different buffer layer structures grown by molecular-beam epitaxy. The ﬁrst type,hereafter denoted as the “tensile-strained-GaSb” CQW, wasformed on a thick (800 nm) AlSb buffer layer grown on aGaAs substrate [Fig. 1(a)]. Because of the large lattice mis- match ( ∼8%) between GaAs ( a GaAs=5.653 25 Å) and AlSb (aAlSb=6.1355 Å), a nearly complete strain relaxation occurs in the AlSb layer near the AlSb /GaAs interface. Since the lattice constants of both InAs ( aInAs=6.0583 Å) and GaSb (aGaSb=6.0959 Å) are smaller than aAlSb, both the InAs and FIG. 1. (a), (b) Heterostructures using two types of buffer layer sequence referred to as (a) tensile-strained- and (b) unstrained-GaSb CQWs in the main text. (c) HAADF-STEM of a tensile-strained- GaSb CQW. The intensity plot (right panel) taken along the magentabar on the enlarged image (middle panel) is used to determine the layer number. (d) Conduction- and valence-band edges for (left) tensile-strained- and (right) unstrained-GaSb CQWs. Dashed linesrepresent band edges of InAs and GaSb under no strain. GaSb layers comprising the CQW are tensile strained. For the growth on GaAs substrates, the use of a thick (800 nm) AlSbbuffer layer was essential to reduce the threading dislocationdensity and obtain CQWs of reasonable quality [ 13]. The second type, denoted as the “unstrained-GaSb” CQW, wasformed on a thin (50 nm) AlSb buffer layer grown on aGaSb substrate [Fig. 1(b)]. Because of the small thickness of the AlSb buffer layer and the rather small lattice mismatch(∼0.6%) between GaSb and AlSb, the AlSb buffer layer remains fully strained, with its in-plane lattice constant equaltoa GaSb. Accordingly, the GaSb layer of the CQW formed on it is unstrained. The InAs layer, on the other hand, is tensilestrained, due to the ∼0.6% lattice mismatch between InAs and GaSb. In both types of CQWs, the thickness d GaSb of the GaSb layer comprising the CQW was ﬁxed at 7 .3 nm, which cor- responds to 24 monolayers (MLs). The thickness dInAsof the InAs was varied from 9.1 to 11 .8 nm (30 to 39 MLs) for the 075433-2IMPACT OF EPITAXIAL STRAIN ON THE TOPOLOGICAL- … PHYSICAL REVIEW B 101, 075433 (2020) TABLE I. In-plane lattice constant of AlSb buffer layer ( abuffer) measured by HRXRD and resultant strain in GaSb and InAs QW layers ( εGaSbandεInAs).abuffer is obtained as the average of RSM mea- surements using orthogonal incident angles ([110] and [1 10]). The relaxation ratio is deﬁned as ( abuffer−asub)/(aAlSb−asub), where asub is the lattice constant of the substrate, i.e., aGaAsfor tensile-strained- GaSb samples and aGaSbfor unstrained-GaSb samples, respectively. Relaxation Sample abuffer (Å) ratio (%) εGaSb(%)εInAs(%) Tensile-strained GaSb 6.1266 98.15 0.50 1.13 Unstrained GaSb 6.0972 3.3 0.02 0.64 tensile-strained-GaSb CQWs and from 9.1 to 10 .6n m( 3 0t o 35 MLs) for the unstrained-GaSb CQWs. We calibrated thelayer thickness by counting the number of atomic layers inhigh-angle annular dark-ﬁeld scanning transmission electronmicroscopy (HAADF-STEM) images with atomic resolution[Fig. 1(c)]. The magnitudes of strain in the CQWs were evaluated us- ing two-dimensional reciprocal space mapping (RSM) of thehigh-resolution x-ray diffraction (HRXRD) (see Appendix A for details). The in-plane lattice constant, relaxation ratio, andstrain calculated from the mean value of the measurements for[110] and [1 10] directions are summarized in Table I.I nt h e tensile-strained-GaSb samples, the InAs and GaSb layers inthe CQWs are 1.13 and 0.50% tensile strained, respectively.For the unstrained-GaSb CQWs, the InAs layer is 0.64%tensile strained, while the GaSb layer is nearly unstrained. Figure 1(d) shows the band-edge alignment of InAs and GaSb for the cases of tensile-strained- and unstrained-GaSbCQWs. The dashed lines represent the positions of the bandedges for the case where no strain is taken into account. Withno strain, the conduction-band bottom of InAs is 0.14 eVlower than the valence-band top of GaSb. With tensile strainin the InAs layer, its conduction-band bottom shifts to lowerenergy. With tensile strain in the GaSb layer, its valence-band top is split into heavy-hole (HH) and light-hole (LH)bands. It is worth noting that the tensile strain lowers theHH band with respect to the LH band. The energy differencebetween the bulk band edges of the HH and conduction bandsbarely depends on the strain. However, as we will elaboratein Sec. III, the band overlap between the electron and HH subbands in a CQW depends on the strain, due to the mixingbetween the electron and LH subbands. B. Magnetotransport and equivalent-circuit analysis Here, we describe the procedure to characterize the elec- tronic properties of CQWs. We employed Hall-bar deviceswith a length and width of 180 and 50 μm, respectively. For this large device size, the contribution of edge conduction isnegligible for most of the cases studied here, so the mea-sured resistance reﬂects the bulk property. We measured theelectronic properties at 2 K under a perpendicular magneticﬁeld Bup to 14 T using the standard lock-in technique. The Hall-bar devices are ﬁtted with a front gate, which we use totune the Fermi level across the charge neutrality point (CNP).The gate insulator was 40-nm-thick atomic-layer-depositedaluminum oxide. The front-gate voltage V FGwas swept in the range −1.5V/lessorequalslantVFG/lessorequalslant3.5V . T h e l o w e s t VFGwas limited to −1.5 V, because hysteresis occurs at VFG<−1.5 V, which shifts the device characteristics. All the data presented in thispaper were taken with the substrate (back gate) kept at 0 V . Inthis subsection, we outline our analysis using data taken froma tensile-strained-GaSb CQW with d InAs=11.8n m . Figure 2(a)shows the dependence of the longitudinal resis- tance RxxatB=0 and 9 T and the Hall resistance RxyatB= 9 T on the front-gate voltage VFG.A t B=0T , Rxxexhibits a single peak at VFG=−0.72 V without any additional features. As we will show below, this peak is not located at the CNPand therefore not a manifestation of an energy gap opening.R xychanges sign at VFG=−0.4 V, demonstrating that the majority carrier type changes from holes to electrons withincreasing V FG. The fact that the sign change occurs through Rxy=0 (instead of diverging to ±∞) indicates that electrons and holes coexist over a range of VFG.A tB=9 T, several Rxy plateaus along with Rxxdips due to quantum Hall effects are observed in both the electron- and hole-dominant regions. Figure 2(b) shows a color plot of Rxxas a function of VFG andB. Shubnikov–de Haas (SdH) oscillations in the low- B regime evolve into quantum Hall effects with vanishing Rxx in the high- Bregime. We deduced the carrier densities as a function of VFGby fast Fourier transform (FFT) analysis of the SdH oscillations at each VFGwith respect to 1 /B.T h e FFT power spectra are shown in the top panel of Fig. 2(c) as a color plot. The vertical axis is the carrier density nSdH, which is related to the frequency f1/BasnSdH=gs(e/h)f1/B, where gsis the spin degeneracy. Throughout this paper, we take gs=2, which gives results consistent with those obtained from Rxy. The FFT analysis identiﬁes only one frequency (i.e., one density nSdH) at each VFG, which is plotted as solid circles in the middle panel of Fig. 2(c).nSdHﬁrst decreases with VFG atVFG/lessorequalslant−0.9 V and then starts to increase at VFG/greaterorequalslant−0.1V . This VFGdependence conﬁrms that nSdHrepresents the density of majority carriers at each VFG, i.e., hole density nhatVFG/lessorequalslant −0.9 V and electron density neatVFG/greaterorequalslant−0.1V . We determine both neandnhat each VFGby analyzing the VFGdependence of nSdHusing the equivalent-circuit model il- lustrated in the inset of Fig. 2(c)(top panel) [ 7,17]. In addition to the geometrical capacitances that represent the couplings tothe front and back gates ( C FandCB) and between the InAs and GaSb QW layers ( CM), the model includes the quantum capacitances of the QW layers— CInAs=e2m∗ e,InAs/π¯h2and CGaSb=e2m∗ h,GaSb/π¯h2with m∗ e,InAs (m∗ h,GaSb) the effective mass of electrons in InAs (holes in GaSb) and ¯ h=h/2π.F o r simplicity, we neglect the energy dependence of the effectivemasses and the hybridization between the electron and holebands. With this circuit model, n eandnhare obtained as the charges stored in CInAsandCGaSb, respectively. We chose the m∗ e,InAsandm∗ h,GaSbvalues that give the best ﬁt to the observed VFGdependence of nSdH. (See Appendix Bfor the details of the ﬁtting procedure and the parameters used in the analysis.) In the middle panel of Fig. 2(c), we plot the calculated ne andnh(solid lines) along with the net carrier density deﬁned asnnet=\|ne−nh\|(dashed line). The model reproduces the VFGdependence of nSdHover the entire range, including the slope change at VFG=2.2 V. As shown by the magenta solid line, nhdecreases to 0 at VFG=2.2 V, from which the slope 075433-3H. IRIE et al. PHYSICAL REVIEW B 101, 075433 (2020) 30 20 10 0Rxx (k) 3 2 1 0 -1 VFG (V)40 30 20 10 0Rxx (k) -20-10010Rxy (k) B = 0 TB = 9 T2 345678 -2 14 12 10 8 6 4 2 0B (T) 3 2 1 0 -1 VFG (V) 101103105 2 -234 56Rxx () 20 15 10 5 0 20 10 03 2 1 0 -120 15 10 5 0 ne nh nnet nSdH 103104105106 VFG (V)Rxx (k) nSdH (1015 m-2) Density (1015 m-2) CNPVFGVBGCFCMCB CInAs CGaSb(a) )c( )b( ne > 0, nh > 0 nh = 0 ne = 0 FIG. 2. Transport properties of tensile-strained-GaSb CQW with dInAs=11.8n ma t T=2 K. (a) Longitudinal resistance RxxatB=0a n d 9 T, and Hall resistance RxyatB=9 T. Landau-level ﬁlling factor derived from the Rxyplateau values is shown. (b) Color plot of Rxxas a function of VFGandB. Filling factor derived from Rxyis shown. (c) Top: Fast Fourier transform (FFT) spectra of Rxx(1/B) performed at each VFG. The vertical axis is the oscillation frequency multiplied by 2 e/hto associate the peak position with the carrier density. The inset shows the equivalent circuit used for analyzing the density change as a function of VFG. Middle: VFGdependence of carrier density. The solid symbols represent the density obtained from the FFT spectra ( nSdH). The open symbols represent the difference between nSdHand the linear ﬁtting of nSdHin the deep electron regime. The lines are the electron density ne(cyan), the hole density nh(magenta), and the net carrier density nnet (black dashed), all calculated from the equivalent-circuit model. Bottom: RxxvsVFGatB=0 T [same data as in (a)] replotted for comparison with the circuit-model analysis of the carrier density change with VFG. change in nSdHis understood as arising from the onset of the hole-band occupation. This, in turn, allows one to locatethe boundary between the single-carrier regime ( n e>0 and nh=0) and the electron-hole coexistence regime ( ne,nh> 0). For ne>nh, the hole density can be expressed as nh= ne−nnet. We experimentally deduced the hole density in the coexistence regime by substituting neandnnetin this equation with the measured nSdHand its linear ﬁt in the single-carrier regime extrapolated to the coexistence regime, respectively.The latter is justiﬁed because the slope of n netremains un- changed in the single-carrier and coexistence regimes. Thehole density deduced in this way, shown as open symbolsin the middle panel of Fig. 2(c), agrees with the calculated n h(magenta solid line). Similarly, by extrapolating nSdHin the coexistence regime to 0, the onset of the electron-bandoccupation is found to be at V FG=−0.74 V. The coexistence of electrons and holes over a ﬁnite VFGrange (from −0.74 to 2.2 V) demonstrates that this CQW has an inverted bandstructure. One can locate the CNP at V FG=−0.27 V where neandnhcross. The density ncross(=2.1×1015m−2)a tt h i s crossing point provides a quantitative measure of the degreeof band inversion. n net(=\|ne−nh\|) shows a V-shaped VFG dependence, with a constant slope of 5 .2×1015m−2V−1, which reﬂects mostly the geometrical capacitance to the frontgate. Turning to the R xxvsVFGcurve at B=0 T, which we replot in the bottom panel of Fig. 2(c) for comparison, we notice that the Rxxpeak is not located at the CNP ( VFG=−0.27 V). Rather, the peak position is close to the onset of the electron-band occupation ( V FG=−0.74 V). This is reasonable as theelectrons have higher mobility than holes. The classical two- carrier-model analysis of magnetoconductance using the ne andnhvalues obtained above (not shown) gives a mobility ratio of 4 near the CNP. Importantly, Rxxshows no feature at the CNP. As we discuss in Sec. III, this is due to the semimetallic band structure caused by the tensile strain in theGaSb layer. C. Effects of InAs thickness and buffer layer structure 1. d InAsdependence of tensile-strained-GaSb CQWs Using the procedure outlined in the previous subsection, we ﬁrst examine the dInAs dependence of tensile-strained- GaSb CQWs. Figure 3(a) shows the B=0 sheet resistivity ρxxof the tensile-strained-GaSb CQWs with different dInAs.I n all samples, ρxxexhibits a single peak, with the height mono- tonically increasing with decreasing dInAs. We performed magnetotransport measurements and analyses for CQWs withd InAs=9.1, 10, and 10 .9 nm, similarly to what we did for the CQW with dInAs=11.8 nm shown in Fig. 2. Figure 3(c) compiles the results for the four CQWs, where we plot nSdH vsVFGobtained from the FFT analysis (solid symbols) along with ne(solid lines) and nh(dashed lines) calculated using the equivalent-circuit model (parameters are summarized inAppendix B). The hole density deduced from the SdH data following the procedure described in the previous subsectionis shown as open symbols. While the analysis used the SdHdata taken over a wide V FGrange up to 3.5 V as in Fig. 2(c), we only show the results for VFG/lessorequalslant1.85 V in Fig. 3(c) to highlight the behavior near the CNP. 075433-4IMPACT OF EPITAXIAL STRAIN ON THE TOPOLOGICAL- … PHYSICAL REVIEW B 101, 075433 (2020) 10 8 6 4 2 0 1.5 1.0 0.5 0.0 -0.5 -1.010 8 6 4 2 0 1.5 1.0 0.5 0.0 -0.5 -1.0102103104105106107 1.5 1.0 0.5 0.0 -0.5 -1.0102103104105106107 1.5 1.0 0.5 0.0 -0.5 -1.0)b( )a( dInAs 9.1 nm 10 nm 10.9 nm 11.8 nm 9.1 nm 10.6 nmxx (xx ( 01( ytisneD51m 2-) 01( ytisneD51m 2-) VFG (V) VFG (V)dInAs )d( )c(bSaG deniartsnU bSaG deniarts-elisneT FIG. 3. Impacts of the InAs layer thickness and buffer layer structures on transport properties of InAs /GaSb CQWs. (a),(b) Sheet resistivity as a function of VFGfor (a) tensile-strained-GaSb and (b) unstrained-GaSb CQWs measured at T=2K and B=0 T. The arrows represent the position of the charge neutrality point determined from the analysis shown in (c) and (d). (c), (d) Carrier density obtained from the FFT analysis of the SdH oscillations. Solid symbols represent the peak position ( nSdH) of the FFT spectra. Open symbols are the difference between nSdHand the linear ﬁt of nSdHextrapolated from the deep electron regime. Open symbols with a center dot in (d) are obtained from the magnetic ﬁeld position of the ν=−2 quantum Hall state. The solid (dashed) lines are electron (hole) density calculated from the equivalent-circuit model. Like the CQW with dInAs=11.8n m , nSdHvsVFGof the one with dInAs=10.9 nm shows a slope change at VFG= 0.9 V due to hole-band occupation, indicating an inverted band structure. The CQWs with dInAs=10.9 and 11 .8n m have ncross of 1.0 and 2 .1×1015m−2, respectively. The smaller ncrossfordInAs=10.9 nm indicates a shallower band inversion that results from the stronger quantum conﬁne-ment of electrons in InAs. In contrast, the n SdHvsVFGfor dInAs=9.1 and 10 nm exhibits a simple V shape, with no slope change, indicating no electron-hole coexistence. Thisindicates that these CQWs are in the noninverted regime, asa result of the even stronger quantum conﬁnement in the InAsQW. The equivalent-circuit analysis also provides the position of the CNP, which is marked by arrows in Fig. 3(a).I nt h e CQWs with d InAs=9.1 and 10 nm, the ρxxpeak position roughly coincides with the CNP, as expected for the nor-mal semiconducting gap. For d InAs=10 nm, the ρxxpeak is slightly shifted to the hole regime, presumably due to thelarge mobility difference between electrons and holes. Forthe CQW with d InAs=10.9 nm, despite its inverted band structure, the ρxxpeak position coincides with the CNP. This contrasts with the one with dInAs=11.8 nm, indicating a gap opening. 2. d InAsdependence of unstrained-GaSb CQWs Next, we examine the properties of the unstrained-GaSb CQWs, which are shown in Figs. 3(b) and3(d). The carrierdensity plot in Fig. 3(d) demonstrates that the two studied samples ( dInAs=9.1 and 10 .6 nm) are both in the band- inverted regime and thus have nonzero ncross[25]. Note that the tensile-strained-GaSb CQWs with similar InAs thickness(d InAs=9.1 and 10 .0 nm) are noninverted, which indicates that the band overlap is larger in the unstrained-GaSb CQWs.This becomes clear by plotting n cross as a function of dInAs (Fig. 4). The larger ncross for the same dInAsdemonstrates a larger degree of band inversion in the unstrained-GaSbCQWs. Accordingly, the boundary between inverted ( n cross> 0) and noninverted ( ncross=0) regimes is located at a larger dInAsin the tensile-strained-GaSb CQWs. Turning back to the transport data in Fig. 3(b), we ﬁnd that in both samples the position of the ρxxpeak matches the CNP. We note in particular that this holds even in the deeply invertedd InAs=10.6 nm CQW with large ncross of 2.4×1015m−2. This shows that the observed ρxxpeak originates from an energy gap, thus corroborating the opening of a hybridizationgap. This contrasts with the case of tensile-strained-GaSbCQW with d InAs=11.8 nm, where a semimetallic behavior is observed for the similar ncrossof 2.1×1015m−2. Further- more, the CQW with dInAs=10.6 nm has a satellite dip (or a shoulder) on both sides of the ρxxpeak [Fig. 3(b)]. Similar features have previously been reported for high-quality CQWsin the inverted regime [ 2,7,9,26,27] and have been interpreted to be due to the Van Hove singularity in the density of states atthe hybridization gap edges [ 2,16]. The opening and absence of the hybridization gap in unstrained- and tensile-strained-GaSb CQWs with a similar degree of band inversion, revealed 075433-5H. IRIE et al. PHYSICAL REVIEW B 101, 075433 (2020) 12 11 10 9 8 dInAs (nm)3 2 1 0nssorc01( 51m 2-) Tensile-strained GaSb Unstrained GaSb FIG. 4. Dependence of ncrossondInAsfor the tensile-strained- and unstrained-GaSb CQWs, determined from the data in Figs. 3(c) and 3(d). Data points with ncross=0 represent CQWs in the noninverted regime. by the transport data in Fig. 3, are shown to be consistent with the band calculation in the next section. III. k ·p CALCULATION A. Impact of tensile strain on the band structure To understand the difference in the electronic proper- ties between tensile-strained- and unstrained-GaSb CQWs,we performed theoretical calculations based on the eight-band k·pHamiltonian with strain effects taken into account [28,29] (material parameters are from Ref. [ 30]). For sim- plicity, we use the axial approximation [ 31], neglecting the band anisotropy. Furthermore, we assume a ﬂat potential, ne-glecting the Hartree potential due to interlayer charge transfer.Since the latter is known to be important for a quantitativelyaccurate description of the band structure in InAs /GaSb CQWs [ 21,32], our calculations should be taken as providing a qualitative guide. The inﬂuence of the above approximationswill be discussed later. In Figs. 5(a) and5(c), we compare the band dispersions calculated for CQWs with the same layer thicknesses ( d InAs= 9 nm and dGaSb=7.3 nm) but grown pseudomorphically on (a) GaSb and (c) AlSb, which correspond to εGaSb=0 and 0.76%, respectively. The energies are plotted as a functionof the magnitude k /bardblof the in-plane wave vector. We use the line color to represent the character of the bands at each k/bardbl; namely, red, green, and blue indicate that the band predom-inantly has an electron (E), light-hole (LH), and heavy-hole(HH) character, respectively. For the unstrained-GaSb case [Fig. 5(a)], the character of the bands at the Brillouin-zone center ( k /bardbl=0) is identiﬁed as HH1, E1, HH2, LH1, and HH3 in descending order of energy.(We hereafter label the bands according to their character atk /bardbl=0. Note that the actual band character varies with k/bardbland is therefore different from that at k/bardbl=0.) The inverted order of E1 and HH1, with the hybridization gap opening at aroundk /bardbl=0.15 nm−1where E1 and HH1 bands anticross, shows that the system is in the topological regime. Indeed, the bandstructure in Fig. 5(a) is typical of InAs /GaSb CQWs with an inverted band ordering [ 1]. In contrast, the band structure(a) (b) (c)0.25 0.20 0.15 0.10 0.05 0.00 -0.05)Ve( ygrenE 0.4 0.2 0.0 k\|\| (nm-1)1.0 0.5 x (Al)0.4 0.2 k\|\| (nm-1)ECEHHELH LH1HH2E1HH1 HH3 FIG. 5. (a), (c) Band dispersion of CQWs pseudomorphically grown on (a) GaSb and (c) AlSb substrates. Layer thicknesses are dInAs=9n ma n d dGaSb=7.3 nm. The color of the lines denotes the band character; i.e., red, green, and blue represent an electron, light- hole, and heavy-hole character, respectively. (b) Energy positions of bulk band edges and subbands as a function of the lattice constantrepresented by Al fraction xof Al xGa1−xSb substrate. Solid lines represent the band edge of conduction ( EC), light-hole ( ELH), and heavy-hole ( EHH) bands in the bulk. Broken lines represent the subband levels at the zone center of E1, HH1, HH2, and LH1. The energy is measured from the conduction-band edge of InAs without strain. for the tensile-strained-GaSb case, shown in Fig. 5(c),i s obviously different. The band located at the top has an Echaracter at large k /bardbl, but acquires more of an LH character at small k/bardbl. (We use the term “band” to represent the pair of bands which are spin split at ﬁnite k/bardblbut become degenerate atk/bardbl=0.) The HH1 band is located below it, which indicates that, despite the same layer thicknesses, the band order haschanged by the tensile strain. Note that the top of the HH1band at k /bardbl=0.4n m−1(where the band has more of a LH character) is located above the bottom of the upper band.Consequently, the system is a semimetal, even though thesebands are separated by a gap at each k /bardbl[33]. To see how the band crossover is caused by tensile strain, we plot the energy levels of the bands at k/bardbl=0i nF i g . 5(b) (dashed lines) as a function of Al composition x(0/lessorequalslantx/lessorequalslant1) of a virtual Al xGa1−xSb substrate that would produce εGaSbof 0–0.76% ( εInAsof 0.52–1.28%). The bulk band edges of InAs and GaSb are shown as thick solid lines. With increasing x,t h e conduction-band edge of InAs ( EC) shifts to lower energy as the tensile strain exerted on the InAs layer decreases its bandgap via the deformation potential [ 28,29]. The valence-band edge of GaSb, which is fourfold degenerate at x=0 (i.e., ε GaSb=0), splits into LH and HH bands, with their energies, ELHandEHH, shifting upward and downward, respectively. While HH1 and HH2 (and HH3) levels follow the xdepen- dence of EHH, other levels, which have an E and LH character atx=0, do not show a direct correspondence with either ECorELH. We note that in heterostructures, HH bands are completely decoupled from other bands at k/bardbl=0,whereas 075433-6IMPACT OF EPITAXIAL STRAIN ON THE TOPOLOGICAL- … PHYSICAL REVIEW B 101, 075433 (2020) )e( )d( )c( )b( )a( )j( )i( )h( )g( )f(0.4 0.2 0.0 k\|\| (nm-1)0.4 0.2 0.0 k\|\| (nm-1)0.4 0.2 0.0 k\|\| (nm-1)0.20 0.15 0.10 0.05)Ve( ygrenE 0.4 0.2 0.0 k\|\| (nm-1) 0.20 0.15 0.10 0.05 0.00)Ve( ygrenE0.20 0.15 0.10 0.05 12 11 10 9 8 7 dInAs (nm) 0.20 0.15 0.10 0.05 0.00dInAs = 7.2 nm dInAs = 7.7 nm dInAs = 9 nm dInAs = 11 nm dInAs = 7.5 nm dInAs = 8.3 nm dInAs = 9.7 nm dInAs = 11 nm=dc =dc =dcdc dcdcE1 HH1 HH2 LH1 E1 HH1 LH1 HH2'' 0.4 0.2 0.0 k\|\| (nm-1)0.4 0.2 0.0 k\|\| (nm-1)0.4 0.2 0.0 k\|\| (nm-1)0.4 0.2 0.0 k\|\| (nm-1)12 11 10 9 8 7 dInAs (nm) FIG. 6. (a)–(d) Band dispersion of CQWs pseudomorphically grown on a GaSb substrate for different dInAswith dGaSbﬁxed at 7.3 nm. The grey shaded area represents the energy gap. The color of the lines denotes the band character in the same way as that in Fig. 5. (e) Energy levels at the Brillouin-zone center of E1, HH1, HH2, and LH1 as a function of dInAs. (f)–(i) Band dispersion of CQWs grown on AlSb substrate for different dInAswith dGaSbﬁxed at 7.3 nm. (j) Subband levels at the zone center as a function of dInAs. ﬁnite coupling exists between E and LH bands even at k/bardbl=0 [34]. In addition, a tensile strain shifts ECdownward and ELH upward, which would bring the two levels identiﬁed as E1 and LH1 at x=0 closer together if they are to follow EC andELH, respectively. Consequently, E1 and LH1 levels are strongly mixed with increasing xand, by x=1, they almost swap their characters. Because of this E1-LH1 mixing andthe opposite xdependence of E CandELH, the energies of the E1-LH1 mixed levels become barely dependent on x, leading to the level crossing with HH1 at x=0.8 [Fig. 5(b)]. As seen in Fig. 5(c), at ﬁnite k/bardblthe HH1 band mixes with the E1-LH1 band and loses the HH1 character with increasing k/bardbl. Thus, the upturn of the HH1 band around k/bardbl=0.4n m−1(and the resultant semimetallic band structure) can be understood toarise from the HH-LH band character crossover and the tensilestrain that lifts LH-like bands. B. InAs thickness dependence N e x t ,w et u r nt ot h e dInAsdependence of the band disper- sion. Figure 6shows how the band structure of CQWs grownon (a)–(d) GaSb and (f)–(i) AlSb substrates evolve when dInAs is varied with dGaSbﬁxed ( =7.3 nm). As already shown in the previous subsection, the CQW grown on a GaSb substrate is inthe topological regime at d InAs=9.0 nm, which is reproduced here as Fig. 6(c). There, the E1 band is located below the HH1 band, with a hybridization gap opening at k/bardbl/negationslash=0. As dInAsis decreased, the increased quantum conﬁnement in the InAs layer raises the E1 level at k/bardbl=0 and, at a critical InAs thickness dc(=7.7 nm), E1 coincides with HH1, where the system becomes gapless [Fig. 6(b)]. Upon further decreasing dInAs, the system enters the noninverted regime, where E1 is located above HH1, separated by a normal gap at k/bardbl=0 [Fig. 6(a)]. In Fig. 6(e), the energy levels at k/bardbl=0 are plotted as a function of dInAs, which makes it clear that the topological phase transition occurs as a result of the level crossing be-tween E1 and HH1 at d InAs=dc. Figure 5(e) also reveals an anticrossing between E1 and LH1 [ 35], which explains why the E1 band acquires an LH character near k/bardbl=0 when dInAs increases to 11 nm [Fig. 6(d)]. When CQWs are pseudomorphically grown on AlSb sub- strates and the GaSb layer is tensile strained, the dInAs 075433-7H. IRIE et al. PHYSICAL REVIEW B 101, 075433 (2020) dependence of the band structure becomes signiﬁcantly dif- ferent [Figs. 6(f)–6(i)]. For the thinnest dInAs(=7.5n m ) ,t h e system is in the noninverted regime with E1 located aboveHH1 [Fig. 6(f)]. Although the dispersion looks similar to that of thin CQWs on GaSb substrates, a notable difference isthat the band gap is now indirect; i.e., it forms between thebottom of E1 at k /bardbl=0 and the maximum of the HH1 band atk/bardbl=0.4n m−1(where the latter has an LH character). As dInAsincreases, this indirect band gap closes when the E1 band moves down, and its bottom at k/bardbl=0 coincides with the maximum of the HH1 band at k/bardbl/negationslash=0 [Fig. 6(g)]. We denote thisdInAs(=8.3 nm) at which the indirect gap closes as d/prime c, and distinguish it from dcat which a direct gap closes as a result of band touching. For dInAs>d/prime c, E1 continues to lower and the indirect gap becomes negative, making the system asemimetal as we discussed in the previous section for d InAs= 9.0n m . A s dInAsincreases further, the E1 and HH1 bands touch at k/bardbl=0 [Fig. 6(h)], and then their order becomes inverted for dInAs>dc[Fig. 6(i)]. Even though there is no band touching for dInAs>dc, the system remains semimetallic because the LH-like band around k/bardbl=0.4n m−1stays higher up in energy. The evolution of the energy levels at k/bardbl=0 as a function of dInAs[Fig. 6(j)] is similar to that for CQWs on GaSb substrate, except that E1 is strongly mixed withLH1 over a wide range of d InAsincluding dInAs=dc.T h e topological phase transition, which must be accompanied by aclosure of a direct gap (i.e., band touching), is found to occuratd InAs=dc. We see that at dInAs=dcHH1 is crossed by a level having an LH character. It is important to note thatthis topological transition at d InAs=dcis pre-empted by a transition to a semimetal at dInAs=d/prime cand could therefore be masked in transport measurements. The above line of argument based on the band calcula- tion as a function of dInAsand strain is consistent with the results of the transport experiments presented in the preced-ing section. The resistivity peak observed at the CNP ofthe unstrained-GaSb CQWs in the inverted regime indicatesthe opening of a hybridization gap. The absence of such aresistivity peak at the CNP in the deeply inverted CQW withtensile-strained GaSb suggests a semimetallic band struc-ture. Concerning the critical InAs thickness at which theband inversion takes place, our experimental results indicated c<9.1 nm and 10 /lessorequalslantdc<10.9 nm for the unstrained- and tensile-strained-GaSb CQWs, respectively (Fig. 4). Although these estimates for dcare greater than the calculated ones (7.7 and 9.7 nm, respectively), the trend that tensile strainshifts the critical thickness to larger d InAsis consistent. We note that our calculation underestimates dcbecause it neglects the Hartree potential arising from the electron transfer fromGaSb to InAs [ 21,32]. We emphasize, however, that the above discussion remains valid for understanding the impact oftensile strain and d InAsat the qualitative level. C.dGaSb-dInAsphase diagram Our discussion so far has been conﬁned to CQWs with aﬁ x e d dGaSb (=7.3 nm). In the following, we examine the impact of dGaSb, another key parameter that dictates the band inversion and the size of the hybridization gap in the in-verted regime [ 1,36]. We calculated the energy gap \|/Delta1\|as (a) (b)10 9 8 7 6 5 4dbSaG) mn( 12 11 10 9 8 7 dInAs (nm) 60 40 20 0 e m( )V 10 9 8 7 6 5 4dbSaG) mn( 12 11 10 9 8 7 dInAs (nm) 60 40 20 0 m( )Ve2 4 6 11.5dcdc dcdc '' FIG. 7. Calculation of the band gap \|/Delta1\|as a function of dInAsand dGaSb for CQWs grown on (a) GaSb and (b) AlSb substrates. The color maps show /Delta1, with its positive and negative signs assigned to indicate a normal semiconducting gap in the noninverted regime and hybridization gap in the inverted regime, respectively. The greenarea indicating a gap closure over a wide parameter space represents a semimetallic phase. The solid and dashed lines indicate the critical thicknesses d candd/prime c, respectively. The former separates the topo- logical and nontopological regimes while the latter delineates the boundary of the semimetallic region. The dash-dotted lines represent the contour line of the gap in the topological regime, with the labels s h o w i n gt h eg a ps i z e \|/Delta1\|in meV . af u n c t i o no f dInAsand dGaSb. Figures 7(a) and 7(b) show the results for CQWs pseudomorphically grown on GaSb andAlSb substrates, respectively. These color maps show /Delta1,t o which we assign positive and negative signs in the noninverted(d InAs<dc) and inverted ( dInAs>dc) regimes, respectively, to distinguish the two gapped phases with \|/Delta1\|>0. The green color represents regions with /Delta1=0, i.e., where the system is semimetallic. The overall feature of the topological-nontopological phase diagram in Fig. 7(a) is similar to those in Refs. [ 1,36]. The phase boundary is deﬁned by the line indicating dcat each dGaSb, where the direct gap closes as a result of E1-HH1 band touching and /Delta1changes sign [ 37]. In Fig. 7,w ea l s os h o w in a contour plot the size of the gap \|/Delta1\|in the topological regime. For a ﬁxed dGaSb,\|/Delta1\|takes its maximum at dInAs slightly larger than dcand then decreases slowly upon further increasing dInAs as a result of the reduced wave function overlap between E1 and HH1. For a similar reason, the largestgap (∼6 meV) is attained at small d GaSb<6.5 nm. When both dInAsand dGaSb are large, the system enters a semimetallic 075433-8IMPACT OF EPITAXIAL STRAIN ON THE TOPOLOGICAL- … PHYSICAL REVIEW B 101, 075433 (2020) phase, where the strong LH character of the E1 band leads to an indirect gap closure similar to that shown in Fig. 6(g) [38]. In contrast, the phase diagram for CQWs on AlSb substrate is signiﬁcantly different [Fig. 7(b)]. Now the semimetallic phase prevails over a wide region, taking up a large portionof the d InAs-dGaSb space that corresponded to the topological phase for CQWs on GaSb substrate. As already explained, thishappens because the tensile strain in the GaSb layer raisesthe energies of bands having an LH character, shifting thethickness d /prime cat which the indirect gap closes to smaller dInAs. The topology of each band, on the other hand, is expected tochange at d InAs=dc, where the direct gap closes as a result of the level crossing between HH1 and E1 /LH1. As shown by the black solid line in Fig. 7(b),dcﬁrst decreases with increasing dGaSb, but then starts to increase for dGaSb>6.5n m [39]. Consequently, for dGaSb>5.3 nm, the semiconductor- to-topological-insulator transition expected at dInAs=dcis pre-empted by a semiconductor-to-semimetal transition atd InAs=d/prime c(<dc) as we have seen in Figs. 6(f)–6(i). There- fore, despite the inverted band order and the direct-gap open-ing at d InAs>dc, the system is a semimetal. Accordingly, a topological insulating phase exists only for dGaSb<5.3n m , where a strong quantum conﬁnement of holes pushes downthe LH1 band well below the HH1 band at k /bardbl=0, making the band structure similar to that on the GaSb substrate. Note,however, that only a tiny gap, much smaller than in Fig. 7(a), can be attained in this case. IV . DISCUSSION The phase diagram in Fig. 7(b) suggests that, for the tensile-strained-GaSb CQW with dGaSb=7.3 nm used in our experiment, a topological insulating phase does not exist,having been taken over by a semimetallic phase. As alreadypointed out, our calculations neglect the Hartree potentialdue to interlayer charge transfer, which leads to the overes-timation of band inversion. Accordingly, it is possible thatin reality the semimetallic phase occupies a smaller portionof the d InAs-dGaSb plane. As shown in Fig. 3(c), the tensile- strained-GaSb CQW with dInAs=10.9 nm clearly shows band inversion with ncross=1.0×1015m−2, yet exhibits a resistivity peak reaching 63 k /Omega1//squarenear the CNP. Such a resistivity peak near the CNP is not expected from the simpletwo-carrier model, suggesting an opening of an energy gap.The semiconductor-semimetal direct transition suggested inFig.7(b) would thus be relevant in CQWs with a wider GaSb well. We add that our calculations neglect the in-plane bandanisotropy due to the valence-band warping, which could bethe dominant mechanism for the semimetallic behavior whenthe E-LH mixing is weak, i.e., in CQWs with an unstrained ornarrow GaSb well. Finally, we mention the relation between our results and those of Tiemann et al. [24]. The focus of Ref. [ 24]i so n the effects of externally applied uniaxial strain, whereas wefocus on the biaxial epitaxial strain due to lattice mismatch.Nevertheless, the tight-binding calculations in Ref. [ 24]h a v e shown that, even without strain, a deeply inverted CQWwith 15-nm-thick InAs and 8-nm-thick GaSb is semimetallic,whereas an energy gap opens for a CQW with a narrower12-nm-thick InAs well. We note that their calculations assumeInAs/GaSb CQWs pseudomorphically grown on GaSb so that the GaSb is unstrained in the absence of externally applieduniaxial strain. In contrast, we consider a case where theGaSb well is under biaxial tensile strain, a situation that canoccur when CQWs are grown on a thick Al(Ga)Sb bufferlayer. On the experimental side, Ref. [ 24] used a piezo device to externally apply uniaxial strain, the magnitude of whichwas 0.03% at maximum. To study the effects of epitaxialbiaxial strain, we exploited different substrates and bufferlayer structures, where the magnitude of the strain is muchlarger, i.e., up to 0.50% (1.13%) for GaSb (InAs). V . SUMMARY We studied the inﬂuence of the epitaxial strain on the electronic properties of InAs /GaSb CQWs by magnetotrans- port measurements and eight-band k·pcalculations. We have shown by both experiment and calculation that the tensilestrain in the GaSb layer shifts the topological-nontopologicalphase boundary to a wider InAs well width. In addition, ourstudy reveals an adverse effect of tensile strain, namely theclosing of the bulk gap resulting from the enhanced mixingof light-hole states into the heavy-hole band, which couldexplain why this system often behaves as a semimetal. Ourresults thus give an insight into the heterostructure design fora robust QSHI state in InAs /GaSb CQWs, corroborating the importance of strain engineering as recently demonstrated forInAs/In xGa1−xSb CQWs with compressive strain [ 17–19]. ACKNOWLEDGMENTS The authors thank H. Murofushi for device fabrication. This work was supported by JSPS KAKENHI Grants No.JP15H05854 and No. JP26287068. H.I. and T.A. contributed equally to this work. APPENDIX A: STRAIN EVALUATION METHOD We evaluated the magnitudes of strain in the CQWs from the two-dimensional reciprocal space mapping (RSM)of the high-resolution x-ray diffraction (HRXRD) at roomtemperature using GaAs(224) and (2 24). A clear 224 AlSb peak originating from the AlSb buffer was observed for bothtensile-strained- and unstrained-GaSb samples. As shown inFig.8(a), in the tensile-strained-GaSb sample, the reciprocal- lattice point of the AlSb buffer layer is located on the dashedline connecting the origin and the 224 GaAs peak of the GaAs substrate, indicating that the buffer layer is almost fully re-laxed. The position of the 224 AlSbpeak provides a relaxation ratio of 98%, which is deﬁned as ( abuffer−aGaAs)/(aAlSb− aGaAs), where abuffer is the measured lattice constant of the AlSb buffer layer. In contrast, in the unstrained-GaSb sample,the reciprocal-lattice point of the thin AlSb buffer layer isvertically aligned with that of the GaSb substrate [Fig. 8(b)], indicating that the AlSb buffer layer is pseudomorphicallystrained. APPENDIX B: EQUIVALENT-CIRCUIT MODEL To describe the VFGdependence of electron and hole densi- ties, we use an equivalent-circuit model that takes into account 075433-9H. IRIE et al. PHYSICAL REVIEW B 101, 075433 (2020) (a) Tensile-strained GaSb (b) Unstrained GaSb 4.0 3.9 3.8 3.7l]100[ 2.1 2.0 1.9 1.8 h [110]4.00 3.98 3.96 3.94l]100[ 2.05 2.00 1.95 h [110]224 224224 224GaAs AlSbGaSb AlSb FIG. 8. Reciprocal space mapping of (a) tensile-strained- and (b) unstrained-GaSb CQWs. The dashed line represents the expected peak position for the fully relaxed AlSb. the charge transfer and internal electric ﬁeld in the QW layers. As shown in Fig. 9(a), the equivalent circuit consists of three geometrical capacitances, which represent the couplings tothe front and back gates ( C FandCB) and between the InAs and GaSb QW layers ( CM), and two quantum capacitances of the QW layers ( CInAs=e2m∗ e,InAs/π¯h2, where m∗ e,InAsis the electron effective mass of InAs, and CGaSb=e2m∗ h,GaSb/π¯h2, (a) Two-carrier regime ( Ve > 0, Vh < 0) (b) Single-carrier regime (electron only)VFG VBGCF CM CB CInAs CGaSbVe Vh VFG VBGCF CM CB CInAsVe (c) Single-carrier regime (hole only) VFG VBGCF CM CB CGaSbVh FIG. 9. Equivalent circuit for InAs /GaSb CQWs in (a) two- carrier regime, (b) single-carrier regime of electrons, and (c) single- carrier regime of holes.where m∗ h,GaSb is the hole effective mass of GaSb). V oltage biases are applied through the front gate ( VFG) and the back gate ( VBG), while the QW layers are grounded. The voltages VeandVhin Fig. 9(a) are related to the densities of electrons in InAs and holes in GaSb as ne=CInAsVe/\|e\|and nh= −CGaSbVh/\|e\|, respectively. Accordingly, electrons and holes exist when Ve>0 and Vh<0, respectively. First, we consider the two-carrier (TC) regime, i.e., where Ve>0 and Vh<0. The change in densities /Delta1ne,TCand /Delta1nh,TCin response to the change in gate voltages /Delta1VFGand /Delta1VBGcan be written as /bracketleftbigg /Delta1ne,TC /Delta1nh,TC/bracketrightbigg =1 \|e\|detA/bracketleftbigg CInAs CInAs −CGaSb −CGaSb/bracketrightbigg A/bracketleftbigg CF/Delta1VFG CB/Delta1VBG/bracketrightbigg , where A=/bracketleftbigg CM+CB+CGaSb CM CM CM+CF+CInAs/bracketrightbigg . These equations are modiﬁed to be suitable for ﬁtting experi- mental data [Fig. 2(c)]; ne,TC(VFG)=1 \|e\|CInAs(CM+CB+CGaSb) detACF(VFG−VCNP) +ncross, nh,TC(VFG)=−1 \|e\|CGaSbCM detACF(VFG−VCNP)+ncross, nnet,TC(VFG)=\|ne,TC(VFG)−nh,TC(VFG)\|, in which two ﬁtting parameters, VCNPandncross, are introduced in such a way that ne,TC=nh,TC=ncrossatVFG=VCNPfor a ﬁxed VBG(=0 V). The equations above are valid only for the situation with Ve>0 and Vh<0 (i.e., ne,TC>0 and nh,TC> 0). Once we have Ve/lessorequalslant0o r Vh/greaterorequalslant0 in the studied VFGrange, the equivalent circuit needs to be replaced with that for thesingle-carrier (SC) regime hosting only electrons [Fig. 9(b)] or holes [Fig. 9(c)]. For the electron SC regime, the electron density can be written as n e,SC(VFG)=1 \|e\|CInAs CF+CInAs+C/prime BCF(VFG−VFG1) +ne,TC(VFG1), where C/prime B=(1 CM+1 CB)−1andVFG1is the front-gate voltage at the boundary between the TC and electron SC regimes.Similarly, in the hole SC regime, the hole density can bewritten as n h,SC(VFG)=−1 \|e\|CGaSb CB+CGaSb+C/prime FC/prime F(VFG−VFG2) +nh,TC(VFG2), where C/prime F=(1 CF+1 CM)−1andVFG2is the front-gate voltage at the boundary between the TC and hole SC regimes. Bydeﬁnition, the net carrier density in the electron (hole) SCregimes equals n e,SC(VFG)[nh,SC(VFG)]. Next, we describe how to determine the parameters used in our analysis [Figs. 3(c) and3(d)] . Table IIshows the values 075433-10IMPACT OF EPITAXIAL STRAIN ON THE TOPOLOGICAL- … PHYSICAL REVIEW B 101, 075433 (2020) TABLE II. Parameters used in the equivalent-circuit model. N.I. means that the sample is in the noninverted regime. Sample dInAs(nm) dGaSb(nm) CF(nF/mm2) CB(nF/mm2) me,InAs(m0) mh,GaSb(m0) ncross(1/m2) VCNP(V) Tensile strained 11.8 7.3 0.84 0.12 0.09 0.1 2 .1×1015−0.27 Tensile strained 10.9 7.3 0.75 0.12 0.09 0.1 1 .0×1015−0.25 Tensile strained 10.0 7.3 0.75 0.12 0.09 0.1 N.I. −0.11 Tensile strained 9.1 7.3 0.75 0.12 0.09 0.1 N.I. −0.08 Unstrained 10.6 7.3 0.82 1.3 0.06 0.1 2 .4×1015−0.37 Unstrained 9.1 7.3 0.88 1.3 0.09 0.1 0 .6×1015−0.30 used in Figs. 3(c) and 3(d). For inverted samples, m∗ e,InAs andm∗ h,GaSb were chosen so that the slope of the calculated ne,TC(VFG) matches that of experimental nSdH(VFG)i nt h e two-carrier regime. m∗ e,InAswas obtained as (0 .06–0.09)m0, where m0is the free-electron mass. These values are heavier than that at the band edge of bulk InAs (0 .024m0) and those reported for InAs /GaSb CQWs (0 .04m0[7], 0.032m0[40]). We note that the latter was measured in the deep electronregime. The larger m ∗ e,InAs we obtained in the coexistence regime is thought to be affected more strongly by the hy-bridization of electron and hole wave functions. On the otherhand, m ∗ h,GaSb cannot be accurately determined because the calculated ne,TC(VFG)[ o r nh,TC(VFG)] is insensitive to m∗ h,GaSb,unless m∗ h,GaSb takes an unreasonably small value. Therefore, we tentatively used m∗ h,GaSb=0.1m0in our analysis, which is close to the values reported for InAs /GaSb CQWs (0 .09m0 [7], 0.136m0[40]).VCNPandncrosswere determined so that the overall behavior matches the experimental data. Using them ∗ e,InAsandm∗ h,GaSb values above, the band overlap can be es- timated as Eg0=¯h2πncross(m∗ e,InAs+m∗ h,GaSb)/m∗ e,InAsm∗ h,GaSb [3]. The estimated Eg0ranges from 3 to 15 meV for ncross= (0.6−2.4)×1015m−2, which is reasonable in comparison with the result of k·pcalculation that gives the same ncross. For noninverted samples, neither m∗ e,InAsnorm∗ h,GaSb can be determined, and thus we tentatively used m∗ e,InAs=0.09m0 andm∗ h,GaSb=0.1m0. [1] C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, Quantum Spin Hall Effect in Inverted Type-II Semiconductors,P h y s .R e v .L e t t . 100,236601 (2008 ). [2] I. Knez, R. R. Du, and G. Sullivan, Finite conductivity in mesoscopic Hall bars of inverted InAs/GaSb quantum wells,P h y s .R e v .B 81,201301(R) (2010 ). [3] I. Knez, R.-R. Du, and G. Sullivan, Evidence for Helical Edge Modes in Inverted InAs/GaSb Quantum Wells, Phys. Rev. Lett. 107,136603 (2011 ). [4] K. Suzuki, Y . Harada, K. Onomitsu, and K. Muraki, Edge chan- nel transport in the InAs/GaSb topological insulating phase,P h y s .R e v .B 87,235311 (2013 ). [5] L. Du, I. Knez, G. Sullivan, and R.-R. Du, Robust Helical Edge Transport in Gated InAs/GaSb Bilayers, Phys. Rev. Lett. 114, 096802 (2015 ). [6] K. Suzuki, Y . Harada, K. Onomitsu, and K. Muraki, Gate- controlled semimetal-topological insulator transition in anInAs/GaSb heterostructure, P h y s .R e v .B 91,245309 (2015 ). [7] F. Qu, A. J. A. Beukman, S. Nadj-Perge, M. Wimmer, B.-M. M. Nguyen, W. Yi, J. Thorp, M. Sokolich, A. A. Kiselev, M. J.Manfra, C. M. Marcus, and L. P. Kouwenhoven, Electric andMagnetic Tuning Between the Trivial and Topological Phases in InAs/GaSb Double Quantum Wells, P h y s .R e v .L e t t . 115, 036803 (2015 ). [8] F. Couëdo, H. Irie, K. Suzuki, K. Onomitsu, and K. Muraki, Single-edge transport in an InAs/GaSb quantum spin Hall insu-lator, P h y s .R e v .B 94,035301 (2016 ). [9] M. Karalic, S. Mueller, C. Mittag, K. Pakrouski, Q. S. Wu, A. A. Soluyanov, M. Troyer, T. Tschirky, W. Wegscheider, K.Ensslin, and T. Ihn, Experimental signatures of the invertedphase in InAs/GaSb coupled quantum wells, Phys. Rev. B 94, 241402(R) (2016 ).[10] F. Nichele, A. N. Pal, P. Pietsch, T. Ihn, K. Ensslin, C. Charpentier, and W. 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Wegscheider, Suppression ofbulk conductivity in InAs/GaSb broken gap composite quantumwells, Appl. Phys. Lett. 103,112102 (2013 ). [15] Y Naveh and B Laikhtman, Magnetotransport of coupled electron-holes, Europhys. Lett. 55,545(2001 ). [16] S. de-Leon, L. D. Shvartsman, and B. Laikhtman, Band struc- ture of coupled InAs/GaSb quantum wells, Phys. Rev. B 60, 1861 (1999 ). [17] T. Akiho, F. Couëdo, H. Irie, K. Suzuki, K. Onomitsu, and K. Muraki, Engineering quantum spin Hall insulators by strained-layer heterostructures, Appl. Phys. Lett. 109,192105 (2016 ). [18] L. Du, T. Li, W. Lou, X. Wu, X. Liu, Z. Han, C. Zhang, G. Sullivan, A. Ikhlassi, K. Chang, and R.-R. Du, TuningEdge States in Strained-Layer InAs/GaInSb Quantum Spin HallInsulators, P h y s .R e v .L e t t . 119,056803 (2017 ). 075433-11H. IRIE et al. PHYSICAL REVIEW B 101, 075433 (2020) [19] T. Li, P. Wang, G. Sullivan, X. Lin, and R.-R. 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Tiemann, S. Mueller, Q.-S. Wu, T. Tschirky, K. Ensslin, W. Wegscheider, M. Troyer, A. A. Soluyanov, and T. Ihn, Impact ofstrain on the electronic properties of InAs/GaSb quantum wellsystems, P h y s .R e v .B 95,115108 (2017 ). [25] As seen in Fig. 3(d), the FFT analysis of the SdH oscillations in the hole-dominant regime yields unexpectedly low hole density(solid symbols) which cannot be explained by the equivalent-circuit model. To circumvent this issue, we determined the holedensity from the magnetic ﬁeld position of the ρ xxminimum corresponding to the ν=−2 quantum Hall state. The hole density determined in this way [open symbols with a centerdot in Fig. 3(d)] agrees with the behavior expected from the equivalent-circuit model. The reason why the SdH oscillationsyield a smaller hole density is unknown at present. [26] B.-M. Nguyen, W. Yi, R. Noah, J. Thorp, and M. Sokolich, High mobility back-gated InAs/GaSb double quantum well grown onGaSb substrate, Appl. Phys. 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[33] The semimetallic band structure in Fig. 5(c) is similar to those in wide HgTe/CdTe QWs grown on a fully relaxed CdTebuffer layer, where the HgTe well is under 0.3% biaxial tensilestrain. The coexistence of electrons and holes in such samples,revealed in transport measurements, has been interpreted tobe due to the semimetallic band structure [Z. D. Kvon, E. B.Olshanetsky, D. A. Kozlov, N. N. Mikhailov, and S. A. Dvoret-skii, Two-dimensional electron-hole system in a HgTe-basedquantum well, JETP Lett. 87,502(2008 ); E. B. Olshanetsky, Z. D. Kvon, N. N. Mikhailov, E. G. Novik, I. O. Parm, andS. A. Dvoretsky, Two-dimensional semimetal in HgTe-basedquantum wells with surface orientation (100), Solid State Com- mun. 152,265 (2012 ); P. Leubner, L. Lunczer, C. Brüne, H. Buhmann, and L. W. Molenkamp, Strain Engineering of theBand Gap of HgTe Quantum Wells Using Superlattice VirtualSubstrates, Phys. Rev. Lett. 117,086403 (2016 )]. [34] A. Zakharova, S. T. Yen, and K. A. Chao, Hybridization of elec- tron, light-hole, and heavy-hole states in InAs/GaSb quantumwells, P h y s .R e v .B 64,235332 ( 2001 ). [35] Recently, this E1-LH1 crossover has been shown to impact the position of the Dirac point in the edge-state dispersion relativeto the bulk energy gap [ 36]. [36] R. Skolasinski, D. I. Pikulin, J. Alicea, and M. Wimmer, Robust helical edge transport in quantum spin Hall quantum wells,Phys. Rev. B 98,201404(R) (2018 ). [37] The exact position of the phase boundary depends on the details of the calculation. A tensile strain in the InAs layer shifts thephase boundary to smaller d InAs, whereas the Hartree potential due to interlayer charge transfer tends to counteract it [ 21]. The former (latter) is neglected in Ref. [ 1] (our calculation), which implies that the resultant dcis overestimated (underestimated). [38] This semimetallic phase has been identiﬁed in the dInAs-dGaSb phase diagram in Ref. [ 36]. [39] The latter behavior stems from the strongly mixed character of t h eE 1 / L H 1b a n da t k/bardbl=0. That is, without E1/LH1 mixing, the LH1 level rises more quickly than the HH1 level with increasingd GaSb. To keep the E1/LH1 level aligned with the HH1 level, the E1 contribution to the energy must be reduced by increasingd InAs. [40] X. Mu, G. Sullivan, and R. R. Du, Effective g-factors of carriers in inverted InAs/GaSb bilayers, Appl. Phys. Lett. 108,012101 (2016 ). 075433-12
PhysRevB.81.235210.pdf	Structural and paramagnetic properties of dilute Ga 1−xMn xN Wiktor Stefanowicz,1,2Dariusz Sztenkiel,2Bogdan Faina,3Andreas Grois,3Mauro Rovezzi,3,4Thibaut Devillers,3 Francesco d’Acapito,5Andrea Navarro-Quezada,3Tian Li,3Rafał Jakieła,2Maciej Sawicki,2,Tomasz Dietl,2,6,†and Alberta Bonanni3,‡ 1Laboratory of Magnetism, Bialystok University, ul. Lipowa 41, 15-424 Bialystok, Poland 2Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, PL-02-668 Warszawa, Poland 3Institut für Halbleiter- und Festkörperphysik, Johannes Kepler University, Altenbergerstr. 69, A-4040 Linz, Austria 4Italian Collaborating Research Group, BM08 “GILDA,” ESRF, BP 220, F-38043 Grenoble, France 5Consiglio Nazionale delle Ricerche, IOM-OGG, c/o ESRF GILDA CRG, BP 220, F-38043 Grenoble, France 6Institute of Theoretical Physics, University of Warsaw, PL-00-681 Warszawa, Poland /H20849Received 23 December 2009; published 10 June 2010 /H20850 Systematic investigations of the structural and magnetic properties of single crystal Ga 1−xMn xN ﬁlms grown by metal organic vapor phase epitaxy are presented. High-resolution transmission electron microscopy, syn-chrotron x-ray diffraction, and extended x-ray absorption ﬁne structure studies do not reveal any crystallo-graphic phase separation and indicate that Mn occupies Ga-substitutional sites in the Mn concentration rangeup to 1%. The magnetic properties as a function of temperature, magnetic ﬁeld and its orientation with respectto the caxis of the wurtzite structure can be quantitatively described by the paramagnetic theory of an ensemble of noninteracting Mn 3+ions in the relevant crystal ﬁeld, a conclusion consistent with the x-ray absorption near edge structure analysis. A negligible contribution of Mn in the 2+ charge state points to a lowconcentration of residual donors in the studied ﬁlms. Studies on modulation-doped p-type Ga 1−xMn xN//H20849Ga,Al /H20850N:Mg heterostructures do not reproduce the high-temperature robust ferromagnetism reported recently for this system. DOI: 10.1103/PhysRevB.81.235210 PACS number /H20849s/H20850: 75.50.Pp, 75.10.Dg, 75.70.Ak, 75.30.Gw I. INTRODUCTION The search for magnetic semiconductors with a Curie temperature TCabove room temperature /H20849RT/H20850is currently one of the major challenges in semiconductor spintronics.1–3 In single-phase samples the highest Curie temperatures re- ported are /H11011190 K for /H20849Ga, Mn /H20850As.4,5The magnetic order- ing in these materials is interpreted in terms of the p-dZener model.1,6This model assumes that dilute magnetic semicon- ductors /H20849DMSs /H20850are random alloys, where a fraction of the host cations is substitutionally replaced by magnetic ions—hereafter with magnetic ions we intend transition metalions—and the indirect magnetic coupling is provided by de-localized or weakly localized carriers /H20849sp-dexchange inter- actions /H20850. The authors adopted the Zener approach within the virtual-crystal /H20849VCA /H20850and molecular-ﬁeld /H20849MFA /H20850approxima- tions with a proper description of the valence band structurein zinc-blende and wurtzite /H20849wz/H20850DMSs. The model takes into account the strong spin-orbit and the k·pcouplings in the valence band as well as the inﬂuence of strain on theband density of states. This approach describes qualitatively,and often quantitatively the thermodynamic, micromagnetic,transport, and spectroscopic properties of DMSs with delo-calized holes. 3,7 Experimental data for Ga 1−xMn xN reveal an astonishingly wide spectrum of magnetic properties: some groups ﬁnd hightemperature ferromagnetism 8–10with TCup to 940 K;10how- ever, other detects only a paramagnetic response and theirresults show that the spin-spin coupling is dominated by an-tiferromagnetic interactions. Generally, the origin of the fer-romagnetic response in Mn doped GaN is not clear and twobasic approaches to this issue have emerged, namely, /H20849i/H20850methods based on the mean-ﬁeld Zener model 1—according to this insight, in the absence of delocalized or weakly local-ized holes, no ferromagnetism is expected for randomly dis-tributed diluted spins. Indeed, recent studies of /H20849Ga,Mn /H20850N indicate that in samples containing up to 6% of diluted Mn,holes are strongly localized and, accordingly, T Cbelow 10 K is experimentally revealed.11,12Higher values of TCcould be obtained providing that efﬁcient methods of hole doping willbe elaborated for nitride DMSs. Surprisingly, however, elec-tric ﬁeld controlled RT ferromagnetism has been recentlyreported in Ga 1−xMn xN layers, with a Mn content as low as x/H110150.25%.13These results /H20849TC/H11407300 K /H20850cannot be explained in the context of the p-dZener model, where the Curie tem- perature increases linearly with the Mn concentration and forx/H110210.5% T Cshould not exceed 60 K; /H20849ii/H20850several studies14–16 acknowledge the /H20849likely /H20850presence of secondary phases— originating from the low solubility of magnetic ions inGaN—as being responsible for the observation of ferromag-netism. It has been found that the aggregation of magneticions leads either to crystallographic phase separation, i.e., tothe precipitation of a magnetic compound, nanoclusters of anelemental ferromagnet, or to the chemical phase separationinto regions with, respectively, high and low concentrations of magnetic cations, formed without distortion of the crystal-lographic structure. It has been proposed recently that theaggregation of magnetic ions can be controlled by varyingtheir valence /H20849i.e., by tuning the Fermi level /H20850. Particularly relevant in this context are data for /H20849Zn,Cr /H20850Te, 17/H20849Ga,Fe /H20850N,18 and also /H20849Ga,Mn /H20850N,17,19,20where a strict correlation between codoping, magnetic properties, and magnetic ion distributionhas been put into evidence. There is generally a close relation between the ion ar- rangement and the magnetic response of a magneticallyPHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 1098-0121/2010/81 /H2084923/H20850/235210 /H2084914/H20850 ©2010 The American Physical Society 235210-1doped semiconductor. Speciﬁcally, depending on different preparation techniques and parameters, coherently embeddedmagnetic nanocrystals /H20851such as wz-MnN in GaN /H20849Refs. 21 and22/H20850/H20852or precipitates /H20851such as, e.g., MnGa or Mn 4N/H20852might in fact give the major contribution to the total magnetic mo-ment of the investigated samples. In particular, randomly dis-tributed localized spins may account for the paramagneticcomponent of the magnetization, whereas regions with ahigh local density of magnetic cations are presumably re-sponsible for ferromagnetic features. 23In the case of low concentrations of the magnetic impurity, it is often exceed-ingly challenging to categorically identify the origin of theferromagnetic signatures. Up to very recently, in most of the reports the observation of ferromagnetism or ferromagneticlike behavior with appar-ent Curie temperatures near or above RT, has been discussedprimarily or even solely based on magnetic hysteresis mea-surements. However, indirect means such as superconducting quantum interference device /H20849SQUID /H20850magnetometry mea- surements or even the presence of the anomalous or extraor-dinary Hall effect, may be not sufﬁcient for a conclusivestatement and to verify a single-phase system. Therefore, acareful and thorough characterization of the systems at thenanoscale is required. This can only be achieved through aprecise correlation of the measured magnetic properties withadvanced material characterization methods, such as syn-chrotron x-ray diffraction /H20849SXRD /H20850, synchrotron based ex- tended x-ray absorption ﬁne structure /H20849EXAFS /H20850and ad- vanced element-speciﬁc microscopy techniques, suitable forthe detection of a crystallographic and/or chemical phaseseparation. The present work is devoted to a comprehensive study of the Ga 1−xMn xN/H20849x/H113491%/H20850fabricated by metalorganic vapor phase epitaxy /H20849MOVPE /H20850, which was also employed by other authors.13,19A careful online control of the growth process is carried out, which is followed by an extended investigationof the structural, optical, and magnetic properties in order toshed new light onto the mechanisms responsible for the mag-netic response of the considered system. Particular attentionis devoted to avoid the contamination of the SQUID magne-tometry signal with spurious effects and, thus, to the reliabledetermination of the magnetic properties. Experimental pro-cedures involving SXRD, high-resolution transmission elec-tron microscopy /H20849HRTEM /H20850, EXAFS, and x-ray absorption near-edge spectroscopy /H20849XANES /H20850are employed to probe the possible presence of secondary phases, precipitates or nano-clusters, as well as the chemical phase separation. Moreover,we extensively analyze the properties of single magnetic-impurity states in the nitride host. The understanding of thislimit is crucial when considering the most recent suggestionsfor the controlled incorporation of the magnetic ions andconsequently of the magnetic response through Fermi levelengineering. By combining the different complementarycharacterization techniques we establish that randomly dis-tributed Mn ions with a concentration x/H110211% generate a paramagnetic response down to at leas t2Ki nG a 1−xMn xN. In view of our ﬁndings, the room temperature ferromag-netism observed in this Mn concentration range 13,19,20,24,25 has to be assigned to a nonrandom distribution of transition metal impurities in GaN. We emphasize that in all reportedworks on /H20849Ga,Mn /H20850N fabricated by MOVPE the Mn concen- tration was well below 5%. The paper is organized as follows: in the next section we give a summary of the fabrication details, in situ monitoring of the employed MOVPE process and an abridged overviewof the characterization techniques, together with a table list-ing the principal properties and parameters characterizing the/H20849Ga,Mn /H20850N-based samples considered. In Sec. IV, the results of the structural analysis of the layers by SXRD, HRTEM,and EXAFS are reported. These measurements prove a uni-form distribution of the Mn ions in the Ga sublattice of GaN.Section Vis devoted to the determination of the Mn concen- tration and of the charge and electronic state of the magneticions. In Sec. VI, we give the experimental magnetization characteristics of the system obtained from SQUID measure-ments, and interpret the databased on the group theoreticalmodel for Mn 3+ions taking into account the trigonal crystal ﬁeld, the Jahn-Teller distortion and the spin-orbit coupling.Finally, conclusions and outlook stemming from our workare summarized in Sec. VII. II. GROWTH PROCEDURE The wz- /H20849Ga,Mn /H20850N epilayers here considered are fabri- cated by MOVPE in an AIXTRON 200 RF horizontal reac-tor. All structures have been deposited on c-plane sapphire substrates with trimethylgallium /H20849TMGa /H20850,N H 3, and MeCp 2Mn /H20849bis-methylciclopentadienyl-manganese /H20850as pre- cursors for, respectively, Ga, N, and Mn, and with H 2as carrier gas. The growth process has been carried out accord-ing to a well established procedure 26consisting of: substrate nitridation, low temperature /H20849540 °C /H20850deposition of a GaN nucleation layer /H20849NL/H20850, annealing of the NL under NH 3until recrystallization, and the growth of a /H110111/H9262m thick device- quality GaN buffer at 1030 °C. On top of these structures,Mn doped GaN layers /H20849200–700 nm /H20850at 850 °C, at constant TMGa and different—over the samples series—MeCp 2Mn ﬂow rates ranging from 25 to 490 sccm /H20849standard cubic cen- timeters per minute /H20850have been grown. The nominal Mn con- tent in subsequently grown samples has been alternativelyswitched from low to high and, vice versa , to minimize long term memory effects due to the presence of residual Mn inthe reactor. During the whole growth process the sampleshave been continuously rotated in order to promote the depo-sition homogeneity and in situ and on line ellipsometry is employed for the real time control over the entire fabricationprocess. The p-type superlattices have been grown according to the optimized procedure already reported. 27Our MOVPE system is equipped with an in situ Isa Jobin Yvon ellipsom- eter that allows both spectroscopic /H20849variation of the optical parameters as a function of the radiation wavelength /H20850and kinetic /H20849ellipsometric angles vs time /H20850measurements28,29in the energy range of 1.5–5.5 eV . In Table I, the considered /H20849Ga,Mn /H20850N samples are listed together with their speciﬁc pa- rameters. III. EXPERIMENTAL TECHNIQUES A. HRTEM experimental HRTEM studies have been carried out on cross-sectional samples prepared by standard mechanical polishing followedSTEFANOWICZ et al. PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-2by Ar+ion milling, under a 4° angle at 4 kV for less than 2 h. The ion polishing has been performed in a Gatan 691 PIPSsystem. The specimens were investigated using a JEOL 2011Fast TEM microscope operated at 200 kV equipped with aGatan charge-coupled device camera. The setup is capable ofan ultimate point-to-point resolution of 0.19 nm, with thepossibility to image lattice fringes with a 0.14 nm resolution.The chemical analysis has been accomplished with an Ox-ford Inca energy dispersive x-ray spectroscopy /H20849EDS /H20850sys- tem. B. HRXRD and SXRD experimental High-resolution x-ray diffraction /H20849HRXRD /H20850rocking curves are routinely acquired on each sample with a PhilipsXRD HR1 vertical diffractometer with a Cu K /H9251x-ray source working at a wavelength of 0.15406 nm /H20849/H110118 keV /H20850.A monochromator with a Ge /H20849440/H20850crystal conﬁguration is used to collimate the beam, that is diffracted and collected by aXe-gas detector. Angular /H20849 /H9275/H20850and radial /H9275/2/H9258scans have been collected along the growth direction for the /H20849002/H20850GaN reﬂex, in order to gain information on the crystal quality ofthe samples from the full width at half maximum /H20849FWHM /H20850 of the diffraction peak.Though being aware that, if great care is exercised, also conventional XRD may allow to detect small embeddedclusters /H20849like in the reported case of Co in ZnO /H20850, 30–32we performed SXRD measurements that gave us the possibilityto additionally carry out in situ annealing experiments. The experiments have been carried out at the beamline BM20/H20849Rossendorf Beam Line /H20850of the European Synchrotron Ra- diation Facility /H20849ESRF /H20850in Grenoble, France. Radial coplanar scans in the 2 /H9258range from 20° to 60° were acquired at a photon energy of 10 keV . The beamline is equipped with adouble-crystal Si /H20849111/H20850monochromator with two collimating/ focusing mirrors /H20849Si and Pt coating /H20850for rejection of higher harmonics, allowing an acquisition energy range from 6 to33 keV . The measurements are performed using a heavy-dutysix-circle Huber diffractometer, that is the system is suitablefor/H20849heavy /H20850user-speciﬁc environments /H20849e.g., in our case a Be dome for the annealing experiments was required /H20850. C. EXAFS and XANES experimental The x-ray absorption ﬁne structure /H20849XAFS /H20850measurements at the Mn Kedge /H208496539 eV /H20850have been performed at the GILDA Italian collaborating research group beamline/H20849BM08 /H20850of the ESRF in Grenoble. 33The monochromator isTABLE I. Data related to the investigated Ga 1−xMn xN. The following values are listed: the MeCp 2Mn ﬂow rate employed to grow the Mn-doped layers, the FWHM of the /H208490002 /H20850reﬂex from GaN determined by ex situ HRXRD, the Mn3+concentration as obtained from magnetization data, the total Mn content from SIMS measurements and the thickness of each /H20849Ga,Mn /H20850N layer. Letters A and B denote the two different growth series SampleMeMnCp 2ﬂow rate /H20849SCCM /H20850Thickness of /H20849Ga,Mn /H20850N /H20849nm/H20850FWHM /H20849arc sec /H20850Mn3+conc. SQUID /H208491020cm−3/H20850Mn conc. SIMS /H208491020cm−3/H20850 000B 0 470 /H110210.06 025A 25 450 242 0.28 0.3050A 50 400 267 0.8 0.6100A 100 400 243 0.8 100B 100 520 0.27 125A 125 400 267 0.6 0.5150A 150 400 247 1.0 0.7175A 175 400 251 2.2 200B 200 500 0.9 225A 225 370 263 1.6 1.1250A 250 370 243 1.4275A 275 400 256 1.6300A 300 400 272 1.4 1.3 300B 300 520 1.4 325A 325 400 269 2.2350A 350 370 273 2.2375A 375 400 284 2.5 1.9400A 400 370 265 2.6 400B 400 500 2.0 475A 475 700 2.7490A 490 700 3.8 2.4 490B 490 470 2.7STRUCTURAL AND PARAMAGNETIC PROPERTIES OF … PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-3equipped with a pair of Si /H20849311/H20850crystals and run in dynamical focusing mode.34Harmonics rejection is achieved through a pair of Pd-coated mirrors with an estimated cutoff of 18 keV .Data are collected in the ﬂuorescence mode using a 13-element hyperpure Ge detector and normalized by measuringthe incident beam with an ion chamber ﬁlled with nitrogengas. In order to minimize the effects of coherent scatteringfrom the substrate, the samples are mounted on a dedicatedsample holder for grazing-incidence geometry; 35measure- ments are carried out at room temperature with an incidenceangle of 1° and with the polarization vector parallel to thesample surface /H20849E/H11036c/H20850. For each sample the integration time for each energy point and the number of acquired spectra arechosen in order to collect /H1101510 6counts on the ﬁnal averaged spectrum. Bragg diffraction peaks are eliminated by selectingthe elements of the ﬂuorescence detector or by manually de-glitching the affected spectra. In addition, before and aftereach measurement a metallic Mn reference foil is measuredin transmission mode to check the stability of the energyscale and to provide an accurate calibration. Considering thepresent optics setup an energy resolution of /H110150.2 eV is ob- tained at 6539 eV . In our context, the EXAFS signal /H9273/H20849k/H20850is extracted from the absorption raw data, /H9262/H20849E/H20850, with the VIPER program36em- ploying a smoothing spline algorithm and choosing the en-ergy edge value /H20849E 0/H20850at the half height of the energy step /H20849Sec. VC/H20850. The quantitative analysis is carried out with the IFEFFIT /ARTEMIS programs37,38in the frame of the atomic model described below. Theoretical EXAFS signals are com-puted with the FEFF8 code39using mufﬁn tin potentials and the Hedin-Lunqvist approximation for their energy-dependent part. In order to reduce the correlations betweenvariables, the minimum set of free-ﬁtting parameters used in the analysis are /H9004E 0/H20849correction to the energy edge /H20850,S02/H20849am- plitude reduction factor /H20850,/H9004R0and/H9004R1/H20849lattice expansion factors, respectively, for the ﬁrst Mn-N coordination shell distances and all other upper distances /H20850, and /H9268i2Debye- Waller factor for the ith coordination shell around the ab- sorber plus a correlated Debye model40for multiple scatter- ing paths with a ﬁtted Debye temperature of 470 /H2084950/H20850K.41 D. SQUID experimental The magnetic properties have been investigated in a Quantum Design MPMS XL 5 SQUID magnetometer be-tween 1.85 and 400 K and up to 5 T. For magnetic studies thesamples are typically cut into /H208495/H110035/H20850mm 2pieces, and both in- and out-of-plane orientations are probed. The /H20849Ga,Mn /H20850N layers are grown on 330 /H9262m thick sapphire substrates so that the TM-doped overlayers constitute only a tiny fractionof the volume investigated, and due to the substantial mag-netic dilution their magnetic moment is very small to smallwhen compared to the diamagnetic signal of the substrate.Therefore, a simple subtraction of a diamagnetic componentoriginating from the sapphire substrate and linear with theﬁeld only exposes the resulting data to various artifacts re-lated to the SQUID system and to arrangement of the mea-surements, as already discussed in Refs. 28,42, and 43.I n order to circumvent this issue, the magnetic data presented inthis paper are obtained after subtracting the magnetic re- sponse of a sapphire substrate with dimensions equivalent tothose of the investigated sample, independently measured onthe same holders and according to the same experimentalprocedure. This method, in particular, eliminates a spuriousmagnetic contribution that is due to the sapphire substrateand is not linear with the ﬁeld and, moreover, depends on thetemperature. As exempliﬁed in Fig. 1this extra m/H20849T,H/H20850con- stitutes a nontrivial and quite sizable contribution to the sig-nal of interest. Additionally—as shown in the inset to thisﬁgure—the sapphire itself may convey a ferromagnetic re-sponse to the signal at the lowest temperatures. We have alsomade sure that this method is adequate to eliminate anotherweak and ferromagneticlike contribution appearing in thedata after subtracting only the compensation linear in theﬁeld. This fault is caused by an inaccuracy in the value of themagnetic ﬁeld as reported by the SQUID system, which as-sumes that the ﬁeld acting on the sample is strictly propor-tional to the current sent to the superconducting coil, anddisregards the magnet remanence due to the ﬂux pinninginside the superconducting windings. 44This remanence in our 5 T system is as high as −15 Oe after the ﬁeld has beenrisen to H/H11022+1 T and results in a zero ﬁeld magnetic mo- ment of +2 /H1100310 −7emu for our typical sapphire substrate. Although the value is small, it linearly scales with the massof the substrate and it exceeds the magnitude of the signalexpected from a submicrometer thin layer of a DMS ﬁlm. E. SIMS experimental The overall Mn concentration in the epilayers has been evaluated via secondary-ion mass spectroscopy /H20849SIMS /H20850. The FIG. 1. /H20849Color online /H20850Magnetic response at 2, 15, and 200 K of a typical /H208495/H110035/H110030.3/H20850mm3sapphire substrate measured for both in-plane /H20849darker shade squares /H20850and out-of-plane /H20849lighter shade circles /H20850conﬁgurations after the application of a correction linear in the magnetic ﬁeld and proportional to the magnetic susceptibility ofsapphire at 200 K. The magnetic moment obtained in this way at 2K reaches a value that would give a magnetization of /H110111e m u /cm 3 for/H20849a typical /H20850200 nm thick layer. Inset: m/H20849H/H20850at 2 K for the same sapphire sample, but without correction. The axes labels are thesame as in the main panel. This ferromagneticlike signal is isotro-pic, decreases with temperature and vanishes above 15 K.STEFANOWICZ et al. PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-4SIMS analysis is performed by employing cesium ions as the primary beam with the energy set to 5.5 keV and the beamcurrent kept at 150 nA. The raster size is /H20849150/H11003150/H20850 /H9262m2 and the secondary ions are collected from a central region of 60/H9262m in a diameter. The Mn concentration is derived from MnCs+ species, and the matrix signal NCs+ was taken asreference. Mn implanted GaN is used as a calibration stan-dard. IV . STRUCTURAL PROPERTIES As already underlined, it is necessary to ascertain if the investigated material contains any secondary phases, nano-clusters, or precipitates. In this context, it has been realizedrecently 18,21,28,45that the limited solubility of transition met- als in semiconductors can lead to a chemical decompositionof the alloy, i.e., the formation of regions with the samecrystal structure of the semiconductor host, but with, respec-tively, high and low concentrations of magnetic constituents.In this work the structural properties of the system are ana-lyzed by SXRD, HRTEM, and EXAFS. A. HRXRD and SXRD—Results In order to verify the homogeneity of the grown /H20849Ga,Mn /H20850N layers, conventional XRD measurements have been routinely performed. From rocking curves around theGaN /H20849002/H20850diffraction peak, the crystal quality from the FWHM is veriﬁed and we obtain values in the range from240 to 290 arc sec, indicating a high degree of crystal per-fection of the layers. For the /H20849Ga,Mn /H20850N/H20849002/H20850diffraction peak, we observe a shift to lower angles with increasing Mnconcentration in the acquired /H9275/2/H9258scans. This shift points to an increment in the c-lattice constant, as it has been also reported by others.46,47Apart from the diffraction peak shift, no evidence for second phases is observed in the XRD mea-surements. These results have been conﬁrmed by the SXRDdiffraction spectra reported in Fig. 2/H20849a/H20850, where no crystallo- graphic phase separation is detected over a broad range ofMn concentrations. The lattice parameters are determined by averaging the values for the two symmetric SXRD diffractions /H20849004/H20850and /H20849006/H20850for the cparameter, and one asymmetric diffraction /H20849104/H20850for the aparameter. The variation of the lattice param- eters with increasing incorporation of Mn is presented in Fig.2/H20849b/H20850. To obtain further information on the solubility of Mn in our /H20849Ga,Mn /H20850N layers, in situ annealing experiments have been carried out at the ESRF BM20 beamline. Sample 400Awas annealed up to 900 °C in N-rich atmosphere at a pres-sure of 200 mbar to compensate the nitrogen loss duringannealing. Several radial scans have been acquired upon in-creasing the sample temperature in subsequent 100 °C steps,and realignment was performed after reaching each tempera-ture. The diffraction curves upon annealing are shown in Fig.3, and no additional diffraction peaks related to the formation of secondary phases have been detected over the whole pro-cess. This leads us to conclude that the considered /H20849Ga,Mn /H20850N grown with Mn concentration below the solubility limit atthe given deposition conditions is stable in the dilute phase upon annealing over a considerable thermal range. This be-havior is to be contrasted with the one reported for dilute/H20849Ga,Mn /H20850As, where annealing at elevated temperatures pro- vokes the formation of either hexagonal or zinc-blendeMnAs nanocrystals. 48,49a) b)cps FIG. 2. /H20849Color online /H20850/H20849a/H20850SXRD spectra for /H20849Ga,Mn /H20850N samples showing no presence of secondary phases over a broad range ofconcentration of the magnetic ions. /H20849b/H20850Lattice parameters vs Mn concentration. Values for an undoped GaN layer are added forreference [[[cps] FIG. 3. /H20849Color online /H20850In situ SXRD spectra upon annealing of sample No. 400A at different temperatures: no formation of second-ary phases is detected up to an annealing temperature of 900 °C.STRUCTURAL AND PARAMAGNETIC PROPERTIES OF … PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-5B. HRTEM results HRTEM has been carried out on all /H20849Ga,Mn /H20850N layers un- der consideration and independently of the Mn concentrationno evidence of crystallographic phase separation could befound. This is also conﬁrmed by selected area electron dif-fraction /H20849SAED /H20850patterns /H20849not shown /H20850recorded on different areas of each sample, where no satellite diffraction spot apartfrom wurtzite GaN are detected. In Fig. 4, an example of the HRTEM images acquired along the /H20849a/H20850/H20851101 ¯0/H20852and /H20849b/H20850 /H20851112¯0/H20852zone axis, respectively is given. Through measure- ments previously reported and carried out with the same mi-croscope, we have been able to discriminate in /H20849Ga,Fe /H20850N different phases of Fe-rich nanocrystals as small as 3 nm indiameter and also to detect mass contrast indicating the localaggregation of Fe ions. 18,50The HRTEM images in Fig. 4,i n contrast to the case of phase separated /H20849Ga,Fe /H20850N, strongly suggest that the /H20849Ga,Mn /H20850N ﬁlm here studied are in the dilute state The EDS spectra collected on the /H20849Ga,Mn /H20850N layers pro- vide signiﬁcant signatures of the presence of Mn, as evi-denced in Fig. 5. The EDS detector and the software we used here identify the Mn elements automatically, and are sensi-tive to Mn concentrations as low as 0.1% /H20849at. % /H20850. The Mn concentration for sample 300A and reported in Fig. 5is found to be 0.18% /H20849at. % /H20850. C. EXAFS results EXAFS /H20849Ref. 51/H20850is a well-established tool in the study of semiconductor heterostructures and nanostructures52and hasproven its power as a chemically sensitive local probe for the site identiﬁcation and valence state of Mn and Fe dopants inGaN DMS. 53–57The crystallinity of the ﬁlms and the optimal signal to noise ratio of the collected spectra are demonstratedby the large number of atomic shells visible and reproducibleby the ﬁts below 8 Å in the Fourier-transformed spectra re-ported in Fig. 6for the two representative samples 100A and 490A, respectively. In addition, the homogeneous Mn incor-poration along the layer thickness is tested by measuring theMn ﬂuorescence yield /H20849at a ﬁxed energy of 6700 eV /H20850as a function of the incidence angle /H20849not shown /H20850. The EXAFS response of these two samples is qualitatively equivalent, asevidenced in Fig. 6, and this is conﬁrmed by the quantitative analysis. The best ﬁts are obtained by employing a substitu-tional model of one Mn at a Ga site /H20849Mn Ga/H20850in a wurtzite GaN crystal /H20849using the lattice parameters previously found by SXRD /H20850. The possible presence of additional phases in the sample as octahedral or tetrahedral interstitials /H20849MnIO,MnIT/H20850 in GaN or Mn 3GaN clusters58has been checked by carrying out ﬁts with a two phases model. The fraction of the Mn Gais found to be 98 /H208494/H20850% for the pair Mn Ga-Mn 3GaN, 99 /H208493/H20850% for the pair Mn Ga-MnIO, and 97 /H208493/H20850% for the pair Mn Ga-MnIT, re- spectively. With these results we can safely rule out the oc- FIG. 4. HRTEM images: /H20849a/H20850along /H20851101¯0/H20852and /H20849b/H20850along the /H20851112¯0/H20852zone axes. FIG. 5. EDS spectrum of sample 300A, with the identiﬁcation of the Mn peaks /H20851L/H9251/H208490.636 keV /H20850,K/H9251/H208495.895 keV /H20850, and K/H9252/H208496.492 keV/H20850/H20852.1 2 3 4 5 6 7 8Magnitude of the FT, \| χ(R)\| [Å−3] Distance, R[Å] − without phase correctionRmin RmaxMnIT MnIO Mn3GaN(b)2 4 6 8 10 12EXAFS signal, k2χ(k) Photoelectron wavevector, k[Å−1]kmin kmax (a) #490A Fit(MnGa) #100A 1 2 3 4 5 6(c)MnGaMn3GaNMnIO MnIT FIG. 6. /H20849Color online /H20850k2-weighted EXAFS signal /H20849a/H20850for samples 100A /H20849circles /H20850and 490A /H20849diamonds /H20850with relative best ﬁts /H20849solid line /H20850in the region /H20851Rmin-Rmax/H20852and/H20849b/H20850amplitude of the Fou- rier transforms /H20849FTs/H20850carried out in the range /H20851kmin-kmax/H20852by an Hanning window /H20849slope parameter dk=1/H20850; the vertical lines indicate the position of the main peaks in the FT of the MnIT,M nIO, and Mn 3GaN additional structures. Their possible presence would be promptly detected as they fall in a region free from other peaks. FEFF8 simulations, /H20849c/H20850, for the tested theoretical models as described in the text.STEFANOWICZ et al. PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-6currence of phases other than Mn Ga, at least above 5% level. The local structure parameters found for the measured samples are equivalent within the error bars /H20849reported on the last digit within parentheses /H20850and averaged values are given for simplicity. The value of the amplitude reduction factor S02=0.95 /H208495/H20850demonstrates the good agreement with the theo- retical coordination numbers for Mn Ga/H20849considering the in- plane polarization /H20850and the correction to the energy edge /H9004E0=−7/H208491/H20850eV supports the XANES analysis /H20849Sec. VC/H20850. With respect to the lattice parameters previously found bySXRD, the long range distortion ﬁts within the error /H20849/H9004R 1 =0.1 /H208492/H20850%/H20850, while the Mn-N nearest neighbors have a /H9004R0 =2.5 /H208495/H20850%/H20849expansion to 1.99 /H208491/H20850Å/H20850, in line with previously reported experimental results54–56and recent ab initio calculations.59Finally, all the evaluated /H9268i2attest around the average value of 8 /H208492/H20850/H1100310−3Å−2, conﬁrming the high crys- tallinity of the layers. V . PROPERTIES OF HOMOGENEOUS SINGLE-PHASE (GA,MN)N Thus, SXRD, HRTEM, and XAFS experiments have con- ﬁrmed the wurtzite structure of the samples, the absence ofsecondary phases, and the location of Mn in the Ga sublatticeof the wurtzite GaN crystal. Furthermore, the samples havebeen investigated to determine the actual Mn concentrationand the charge state of the magnetic ions. A. Determination of the Mn concentration The depth proﬁling capabilities of SIMS provide not only an accurate analysis of the /H20849Ga,Mn /H20850N layers composition, but allow also monitoring the changes in composition alongthe sample depth. The SIMS depth proﬁles reported in Figs.7/H20849a/H20850and7/H20849b/H20850give evidence that the distribution of the Mn concentration n Mnin the investigated ﬁlms is essentially uni- form over the doped layers, independent of the magnetic ionscontent as well as that the interface between the /H20849Ga,Mn /H20850N overlayer and the GaN buffer layer is sharp. This is con-ﬁrmed by EDS studies, which with the sensitivity around0.1% at. do not provide any evidence for Mn diffusion intothe buffer. The determined total Mn concentration increaseswith increasing MeCp 2Mn ﬂow rate and the correspondingnMnvalues for the considered samples can be found in Table I. B. Energy levels introduced by Mn impurities The character of the paramagnetic response of DMS de- pends crucially on the magnetic ion conﬁguration. In III-Vsemiconductors, Mn in the impurity limit substitutes the cat-ion site giving three electrons to the crystal bond. Dependingon the compensation ratio, Mn can exist in three differentcharge states and electron conﬁgurations, namely, /H20849i/H20850ionized acceptor Mn 2+, with ﬁve electrons localized in the Mn d shell. The electronic conﬁguration of Mn2+isd5, and the ground level of the ion at zero magnetic ﬁeld is a degeneratemultiplet with vanishing orbital momentum /H20849L=0, S=5 /2/H20850. The magnetic moment of the ion results solely from the spin,and its magnetic contribution can be described by a standardBrillouin function for any orientation of the magnetic ﬁeld.The neutral conﬁguration of Mn 3+/H20849S=2, L=2/H20850can be real- ized in two ways: /H20849ii/H20850by substitutional manganese d4with four electrons tightly bound in the Mn dshell; /H20849iii/H20850Mn2+ +hole /H20849d5+hole /H20850with ﬁve electrons in the Mn dshell and a bound hole localized on neighboring anions. C. XANES results The XANES spectra allow to determine the redox state of the probed species and give information on the structure ofthe surroundings of the absorbing atom. 60Basically, the near edge region resembles the density of those empty states, thatare accessible via optical transitions from the Mn 1 sshell. The goal of our XANES analysis is to determine the va- lence state of Mn and to conﬁrm the Mn Gaincorporation, in comparison to the ﬁndings and analysis carried out previ-ously for molecular beam epitaxy /H20849MBE /H20850-grown /H20849Ga,Mn /H20850N, and interpreted in terms of Mn 3+/H20849Refs. 11and61/H20850or Mn2+ /H20849Ref. 62/H20850. In order to assign the Mn valence state, ﬁrst of all we proceed with a comparison of the position of the absorp-tion Mn Kedge to reference compounds, such as Mn-based oxides since we do not have available data on Mn nitrides.This procedure was already adopted by other groups 55,62but its reliability could be questionable; to clarify this point ab initio calculations are also performed. As shown in Fig. 8, the XANES spectra determined for two samples differing in Mn concentrations /H20849100A and 490A /H20850are identical, conﬁrming a conclusion from the SQUID data on the independence of the Mn charge state ofthe Mn concentration. In Fig. 8/H20849a/H20850three spectra collected in transmission mode from commercial powders of MnO,Mn 2O3, and MnO 2, with Mn-valence states 2+, 3+, and 4+, respectively, are used as reference. As seen, with the increas-ing charge state, the edge moves to a higher energy, as theaccumulated positive charge shifts downwards in energymore the 1 sMn shell than the valence states, in agreement with the Haldane-Anderson rule. Usually, the edge position is taken at the ﬁrst inﬂection point of the plot, but in the present case /H20849since the oxide spectra exhibit a broad peak that modiﬁes the slope at theedge /H20850a better estimate of the edge position is obtained by considering the energy of the half step height of the back-FIG. 7. /H20849Color online /H20850SIMS depth proﬁles of Mn, C, O, and H for the samples: /H20849a/H20850150Aand/H20849b/H20850375A.STRUCTURAL AND PARAMAGNETIC PROPERTIES OF … PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-7ground function. In both investigated samples this lies at 6550.0 /H208495/H20850eV . For the oxides, their half-height energies are determined to be 6545.7 /H208495/H20850, 6550.3 /H208495/H20850, and 6553.3 /H208495/H20850eV for MnO, Mn 2O3, and MnO 2, respectively. This would strongly suggest that we deal with Mn3+, in line with the SQUID results /H20849Sec. VI A /H20850. On the other hand, taking the position of the inﬂection points, the determined charge state would be2+, as reported in Ref. 62. This demonstrates that, in this case, relying only on the edge position to determine the va-lence state is prone to error and strongly depends on the local surrounding of the probed species. 63 To clarify this point, we look at the pre-edge peaks of the XANES lines /H20851Figs. 8/H20849b/H20850and8/H20849c/H20850/H20852. In both probed samples, there are two deﬁned peaks below the absorption edge,which we label A 1andA2, while the edge itself shows two shoulders, A3andA4. In Table IIthe results of Gaussian ﬁts performed by using an arctan-function as baseline, are re-ported. Similar ﬁndings were previously interpreted 61,64as indicative of the Mn3+charge state. The peaks A1and A2 correspond to the transitions to Mn 3 d-4phybrid states, while A3andA4end in the GaN higher conduction bands at positions with a high density of 4 pstates. Due to the tetra- hedral environment, the Mn 3 dlevels split in two nearly de- generate eand three nearly degenerate t2levels for each spin direction. The actual position of those states with respect tothe GaN band structure is still a matter of debate, but fromabsorption 65,66and photoluminescence67measurements it is known that for the majority spin carriers in Mn3+, the elev- els lie around 1.4 eV below the t2levels of Mn incorporated substitutionally in GaN, and the t2level, i.e., the Mn3+/Mn2+ state is located about 1.8 eV above the valence band. An interpretation of simulations applied to x-ray absorptionspectra is given in Refs. 61and68, and states that, due to crystal ﬁeld effects, the 3 dand 4 pstates can hybridize, mak- ing transitions from the 1 slevel to the t 2levels dipole al- lowed, while the interaction of the elevels with the 4 por- bitals is much weaker and cannot be seen in K-edge XANES. In view of the above discussion we explain the physical mechanism beyond the observed data considering possibleﬁnal states of the transitions from the 1 sMn shell. The ﬁnal state corresponding to the A 1peak is Mn2+, i.e., a6A1state /H208496Sfor the spherical symmetry /H20850, consisting of e2↑andt23↑one electron levels. The A2peak can be interpreted as a crystal ﬁeld multiplet derived from the4Gstate consisting of e2↑t22↑t2↓, and lying about 2.5 eV higher than the A1state. Apart from what reported in literature, a reason why A1and A2are assigned to localized Mn-states is that from the pre- vious EXAFS analysis /H20849Sec. IV C /H20850we obtain an absorption edge value of 6543 /H208491/H20850eV , between the energies of the A2and A3peaks, meaning that electrons excited to A1andA2cannot backscatter at the surrounding atoms, and they are thus lo-calized. This assignment gives a valuable information,namely, that there is an empty state in the majority-spin t 2 level conﬁrming that most of the incorporated Mn-ions are6530 6540 6550 6560 6570 6580 6590 6600 6610Norm. Absorption Coefficient − µ(E) Energy [eV](b) #100A #490A Mn(3d4) Mn(3d5)6535 6540 6545 6550 6555 6560 6565 6570Norm. µ(E)(a)#100A #490A MnO Mn2O3MnO2 6537 6540 6543 6546 6549(c) A1A2A3A4 Bkg Fit FIG. 8. /H20849Color online /H20850Normalized XANES spectra of the samples 100A and 490A /H20849points /H20850compared with: /H20849a/H20850the reference manganese oxides /H20849MnO, Mn 2O3, and MnO 2/H20850—the chosen edge positions are highlighted by vertical lines; /H20849b/H20850ab initio absorption spectra /H20849without convolution /H20850for Mn Gain the 3 d4and 3 d5elec- tronic conﬁgurations. Inset /H20849c/H20850: method used to extract the results of Table II, focusing the near-edge region for sample No. 490A with the baseline /H20849Bkg/H20850, the relative ﬁt and its components /H20849A1-A4/H20850. TABLE II. Position P, integrated intensity Iand full width at half maximum Wof the Gaussians ﬁtted to the peaks before and at the absorption edge. The background function is used to normalize the spectra. 100A 490A P /H20849eV/H20850 IW /H20849eV/H20850P /H20849eV/H20850 IW /H20849eV/H20850 /H110060.2 /H110060.05 /H110060.1 /H110060.2 /H110060.05 /H110060.1 A1 6538.9 0.18 1.5 6538.9 0.18 1.3 A2 6540.8 0.32 1.6 6540.8 0.33 1.5 A3 6545.8 0.70 3.6 6545.7 0.64 3.3 A4 6548.7 0.28 2.1 6548.4 0.22 2.0STEFANOWICZ et al. PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-8really in the 3+ valence state, in agreement with the conclu- sions of Refs. 11and61. The model explains also the pres- ence of only one pre-edge peak in the case of /H20849Ga,Mn /H20850As and p-/H20849Zn,Mn /H20850Te.61In those systems we deal with Mn2+and de- localized holes, so that the ﬁnal state of the relevant transi-tions corresponds to the Mn d 6level, involving only one spin orientation. On the other hand, the XANES data do not pro-vide information on the radius of the hole localization in/H20849Ga,Mn /H20850N, in other words, whether the Mn 3+conﬁguration corresponds to the d4or rather to the d5+h situation, where the relevant t2hole state is partly built from the neighboring anion wave functions owing to a strong p-dhybridization. We also have simulated the Mn GaK-edge absorption spec- tra in a Ga 95Mn 1N96cluster /H20849a4a/H110034a/H110033csupercell, corre- sponding to 1% Mn concentration /H20850focusing the attention on the Mn electronic conﬁguration: 3 d4and 3 d5. The calculation is conducted within the multiple-scattering approach imple-mented in FDMNES /H20849Ref. 69/H20850using mufﬁn-tin potentials, the Hedin-Lunqvist approximation for their energy-dependentpart, a self-consistent potential calculation 70for enhancing the accuracy in the determination of the Fermi energy andthe in-plane polarization /H20849E/H11036c/H20850. Despite it is common prac- tice to report convoluted spectra to mimic the experimentalresolution, we ﬁnd out that this procedure can arbitrarychange the layout of the pre-edge peaks and for this reason itis preferred to show non-convoluted data /H20851Fig.8/H20849b/H20850/H20852. Regard- ing the ﬁne structure of the simulated spectra, we have agood agreement with experimental data, conﬁrming theMn Gaincorporation as found by the EXAFS analysis /H20849Sec. IV C /H20850. On the other hand, the simulated pre-edge features need a further investigation: the experimental intensity of A1 andA2and the position of A3are not properly reproduced. This could be due to some neglected effects in the employedformalism, as explained in Ref. 61, where the two peak structure was reproduced theoretically within a more elabo-rated model. VI. MAGNETIC PROPERTIES A. SQUID results We investigate both the temperature dependence of the magnetization Mat a constant ﬁeld M/H20849T/H20850and the sample response to the variation of the external ﬁeld at a constanttemperature M/H20849H/H20850. The same experimental routine is re- peated for both in-plane and out-of-plane conﬁgurations, thatis with magnetic ﬁeld applied perpendicular and parallel tothe hexagonal caxis, respectively. In Fig. 9representative low temperature M/H20849H/H20850data for both orientations are re- ported. We note that these curves exhibit a paramagnetic be-havior with a pronounced anisotropy with respect to the c axis of the crystal. This indicates a nonspherical Mn ionconﬁguration, expected for a L/HS110050 state. At the same time we report an absence of any ferromagneticlike featuresthat—on the other hand—are typical for /H20849Ga,Fe /H20850N layers 28at these concentrations of the magnetic ions, supporting the ab-sence of crystallographic phase separation in our layers, assuggested by the SXRD and HRTEM studies. The same ﬁnd-ing additionally indicates that both chemical phase separa-tion /H20849spinodal decomposition /H20850and medium-to-long range fer-romagnetic spin-spin coupling are also absent in this dilute layers. The latter allows us to treat the Mn ions as completelynoninteracting, at least in the ﬁrst approximation. The solidlines in Figs. 9and10represent ﬁts to our experiential data on the paramagnetic response of noninteracting Mn 3+ions /H20849L=2, S=2/H20850with the trigonal crystal ﬁeld of the wurzite GaN structure and the Jahn-Teller distortion taken into ac-count /H20849details in Sec. VI B /H20850. The overall match validates our approach, which, in turn, is consistent with previousﬁndings 71,72that without an intentional codoping, or when the stoichiometry of GaN:Mn is maintained, Mn is occupy-ing only the neutral Mn 3+acceptor state. Interestingly, all theoretical lines in Figs. 9and10are calculated employing only one set of crystal ﬁeld parameters /H20849as listed in Table III/H20850 having the Mn3+concentration nMn3+as the only adjustable parameter for each individual layer. In Fig. 11, the nMn3+FIG. 9. /H20849Color online /H20850Magnetization measurements at 1.85, 5, a n d1 5Ko fG a 1−xMn xN as a function of the magnetic ﬁeld applied parallel /H20849closed circles /H20850and perpendicular /H20849open squares /H20850to the GaN wurtzite caxis. The solid lines show the magnetization curves calculated according to the group theoretical model for noninteract-ing Mn 3+ions in wz-GaN. FIG. 10. /H20849Color online /H20850Temperature dependence of the magne- tization Mfor sample 375A /H20849points /H20850atH=10 kOe. The solid lines represent the magnetization calculated within the group theoreticalmodel of noninteracting Mn 3+ions in wz-GaN.STRUCTURAL AND PARAMAGNETIC PROPERTIES OF … PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-9values as a function of the manganese precursor ﬂow rate are given together with the total Mn content xMnas determined by SIMS. However, there are hints that the interaction between Mn spins may play a role for x/H114070.6%. In Fig. 12theM/H20849H/H20850 normalized at high ﬁeld /H20849H=50 kOe /H20850to their in-plane val- ues, are reported. The fact that the shape of their magnetiza-tion curves is independent of xforx/H113510.6% means that the interactions between Mn ions are unimportant for these dilu-tions. On the other hand, the M/H20849H/H20850for a layer with x =0.9% /H20849490A /H20850secedes markedly from the curves for samples with x/H113510.6%, indicating that supposedly ferromag- netic Mn-Mn coupling starts to emerge with increasing rela-tive number of Mn nearest neighbors in the layers. Neverthe-less, due to the generally low Mn concentration in theconsidered samples, no conclusive statement about thestrength of the magnetic couplings can be drawn from ourmagnetization data. Interestingly, depending on the very na-ture of the Mn centers both ferromagnetic and/or antiferro-magnetic d-dinteractions can emerge in /H20849Ga,Mn /H20850N. The presence of Mn 2+ions essentially leads to antiferromagnetic superexchange, as in II-Mn-VI DMS, where independentlyof the electrical doping, the position of the Mn d-band guar- antees its 3 d 5conﬁguration. Signiﬁcantly, the same antifer- romagnetic d-dordering and paramagnetic behavior typical forS=5 /2o fM n2+was reported in n-type bulk /H20849Ga,Mn /H20850N samples containing as much as 9% of Mn.73On the other hand, calculations for Mn3+within the density-functional theory point to ferromagnetic coupling74,75and, experimen- tally, a Curie temperature TC/H112298Kwas observed in single- phase Ga 1−xMn xN with x/H112296% and the majority of Mn atoms in the Mn3+charge state.11,76Our experimental data seems to support these ﬁndings and to extend their validity toward thevery diluted limit. Finally, we remark that the carrier-mediated ferromagnetism can be excluded at this stage dueto the insulating character of the samples, conﬁrmed by room temperature four probe resistance measurements and consis-tent with the mid-gap location of the Mn acceptor level. The observations presented here point to an uniqueness of Mn in GaN. The fact that Ga 1−xMn xN with x/H113511% is para- magnetic without even nanometer-scale ordering should becontrasted with GaN doped with other TM ions. Dependingon the growth conditions, the TM solubility limit is ratherlow and typically, except for Mn, it is difﬁcult to introducemore than 1% of magnetic impurities into randomly distrib-uted substitutional sites. For example, the solubility limit ofFe in GaN has been shown to be x/H110150.4% at optimized growth conditions /H20849see Ref. 28/H20850, but signatures of a nano- scale ferromagnetic coupling are observed basically for anydilution. 28The relatively large solubility limit of Mn in GaN, in turn, has a remarkable signiﬁcance in the search for long-range coupling mediated by itinerant carriers. 1,6Not only it lets foresee a high concentration of substitutional Mn—important for the long-range ordering—but it can ensure thatthe effects brought about by carriers are not masked by sig-nals from nanocrystals with different phases. B. Magnetism of Mn3+ions—Theory The Mn concentration in our samples is then x/H113511% as evaluated by means of various characterization techniques/H20849Sec. VA/H20850, implying that most of the Mn ions /H20849/H1135090% /H20850have no nearest magnetic neighbors. Therefore, the model thatconsiders the Mn ions as single, noninteracting magneticcenters should provide a reasonable picture. To describe theMn 3+ion we follow the group theoretical model developed for Cr2+ion by Vallin77,78and then successfully used for a Mn-doped hexagonal GaN semiconductor.79,80It should be pointed out that symmetry considerations cannot discrimi-nate between the d 5+hole and d4many-electron conﬁgura- tions of the Mn ions, therefore the presented model should beapplicable to both conﬁgurations. Through this section, the FIG. 11. /H20849Color online /H20850Mn concentration nMnobtained from magnetization measurements /H20849circles—series A, diamonds—series B/H20850and SIMS /H20849squares—series A /H20850as a function of the Mn precursor ﬂow rate.FIG. 12. /H20849Color online /H20850Magnetization curves at T=1.85 K of ﬁve Ga 1−xMn xN samples normalized with respect to their in plane magnetization at H=50 kOe.TABLE III. Parameters of the group theoretical model used to calculate the magnetization of Ga 1−xMn xN. All values are in meV . B4 B20B40 B˜ 20 B˜ 40 /H9261TT /H9261TE 11.44 4.2 −0.56 −5.1 −1.02 5.0 10.0STEFANOWICZ et al. PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-10capital letters Ti/H20849i=1,2 /H20850,Edenote the irreducible represen- tations of the point group for the multielectron conﬁgurationsin contrast to the single electron states indicated by the smallletters d,e, and t 2. We consider a Mn ion that in its free state is in the elec- tronic conﬁguration d5s2of the outer shells. When substitut- ing for the group III /H20849s2p/H20850cation site, Mn gives three of its electrons to the crystal bond and assumes the Mn3+conﬁgu- ration. In a tetrahedral crystal ﬁeld, the relevant levels areﬁvefold degenerate with respect to the projection of the or-bital momentum and are split by this ﬁeld and by hybridiza-tion with the host orbitals into two sublevels eandt 2with different energies. In the tetrahedral case the estates lie lower than the t2states. This fact can be understood by ana- lyzing the electron density distribution of the t2xy,t2yz,t2zx, and ex2−y2,ez2levels. The density of the t2state extends along the direction toward the N ligand anions, while the eorbital has a larger amplitude in the direction maximizing the distanceto the N ion and due to the negative charge of the N anions,thet 2energy increases. However, the relevant, i.e., the up- permost t2state may actually originate from orbitals of neighboring anions, pull out from the valence band by the p-dhybridization.81If the system has several localized elec- trons, they successfully occupy the levels from the bottom,according to Hund’s ﬁrst rule, and keep their spins parallel.By considering the full orbital and spin moments, the Mn 3+ center can be described through the following set of quantumnumbers /H20849Lm LSmS/H20850with L=2 and S=2. However, we under- line that this procedure can be used only if the intra-atomicexchange /H9004 exinteraction is larger than the splitting between theeand t2states /H9004CF=Et2−Ee/H20849/H9004ex/H11022/H9004 CF/H20850. After this, the effect of the host crystal is taken into account as a perturba- tion such as in the single electron problem. One forms ﬁrst2L+1 wave functions for the n-electron system determined by the Hund’s rule, calculates the matrix elements for thesestates and determines the energy level structure. In this way,the impurity ions states are found and classiﬁed according tothe irreducible representations of the crystal point group andcharacterized by the set /H20849/H9003MSm S/H20850of quantum numbers, with Mthe number of the line of an irreducible representation /H9003 =A1,A2,E,T1,T2of the corresponding point group. In the case of a Mn3+/H20849L=2, S=2/H20850ion in a tetrahedral environment the ground state corresponds to the5T2/H20849e2t22/H20850conﬁguration with two electrons in the eand two electrons in the t2level. The ground state is threefold degenerate, since there are threepossibilities to choose two orbitals from three t 2orbitals. The ﬁrst excited state for the Mn3+ion is5E/H20849e1t23/H20850/H20849see Ref. 82/H20850. The energy structure of a single ion in Mn3+charge state can be described by the Hamiltonian H=HCF+HJT+HTR+HSO+HB, /H208491/H20850 where HCF=−2 /3B4/H20849Oˆ 40−20/H208812Oˆ 43/H20850gives the effect of a host having tetrahedral Tdsymmetry, HJT=B˜ 20/H9008ˆ 40+B˜ 40/H9008ˆ 42is the static Jahn-Teller distortion of the tetragonal symmetry, HTR=B20Oˆ 40+B40Oˆ 42represents the trigonal distortion along the GaN hexagonal c-axis, that lowers the symmetry to C3V, HSO=/H9261LˆSˆcorresponds to the spin-orbit interaction and HB=/H9262B/H20849Lˆ+2Sˆ/H20850Bis the Zeeman term describing the effect of an external magnetic ﬁeld. Here /H9008ˆ,Oˆare Stevens equivalent operators for a tetragonal distortion along one of the cubicaxes /H20851100/H20852and trigonal axis /H20851111/H20852 /H20648c/H20849in a hexagonal lattice /H20850 andBqp,B˜ qp,/H9261TT, and/H9261TEare parameters of the group theo- retical model. As starting values we have used the parametersreported for Mn 3+in GaN:Mn,Mg,80which describe well the magneto-optical data on the intracenter absorption related tothe neutral Mn acceptor in GaN. Remarkably, only a notice- able modiﬁcation /H20849about 10% /H20850of/H9261 TTandB20has been nec- essary in order to reproduce our magnetic data /H20849the remain- ing parameters are within 3% of their previously determinedvalues. /H20850Actually, the model with the parameter values col- lected in Table IIIdescribes both the magnetization M/H20849H/H20850 and its crystalline anisotropy /H20849Figs. 9and10/H20850as well as the position and the ﬁeld-induced splitting of optical lines. 80 The ground state of the Mn3+ion is an orbital and spin quintet5Dwith L=2 and S=2. The term HCFsplits the5D ground state into two terms of symmetry5Eand5T2/H20849ground term /H20850. The5E−5T2splitting is /H9004CF=120 B4. The nonspherical Mn3+ion undergoes further Jahn-Teller distortion, that low- ers the local symmetry and splits the ground term5T2into an orbital singlet5Band an higher located orbital doublet5E. The trigonal ﬁeld splits the5Eterm into two orbital singlets and slightly decreases the energy of the5Borbital singlet. The spin-orbital term yields further splitting of the spin or-bitals. Finally, an external magnetic ﬁeld lifts all of the re-maining degeneracies. For the crystal under consideration, there are three Jahn- Teller directions: /H20851100/H20852,/H20851010/H20852, and /H20851001/H20852/H20849centers A, B, and C, respectively /H20850. 79,80It should be pointed out that the mag- netic anisotropy of the Mn3+system originates from different distributions of nonequivalent Jahn-Teller centers in the twoorientations of the magnetic ﬁeld and the hexagonal axialﬁeld H TRalong the caxis. This picture of Mn in GaN em- phasizing the importance of the Jahn-Teller effect, which lowers the local symmetry and splits the ground term5T2 into an orbital singlet and a doublet, is in agreement with a recent ab initio study employing a hybrid exchange potential.59 The energy level scheme of the Mn3+ion is calculated through a numerical diagonalization of the full 25 /H1100325 Hamiltonian Eq. /H208491/H20850matrix. The average magnetic moment of the Mn ion m=L+2S/H20849in units of /H9262B/H20850can be obtained according to the formula /H20855m/H20856=Z−1/H20849ZA/H20855m/H20856A+ZB/H20855m/H20856B+ZC/H20855m/H20856C/H20850, /H208492/H20850 with Zi/H20849i=A,B,o rC/H20850being the partition function of the ith center, Z=ZA+ZB+ZCand /H20855m/H20856i=/H20858 j=1N /H20855/H9272j/H20841Lˆ+2Sˆ/H20841/H9272j/H20856exp/H20849−Eji/kBT/H20850 /H20858 j=1N exp/H20849−Eji/kBT/H20850, /H208493/H20850 where Ejiand/H9272jare the jth energy level and the eigenstate of the Mn3+ionith center, respectively. As already mentioned,STRUCTURAL AND PARAMAGNETIC PROPERTIES OF … PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-11the Mn concentration in our samples is relatively small x /H113511%. Therefore, the model assuming a system of single Mn ions provides a reasonable description of the magnetic be-havior. The macroscopic magnetization M, shown in Figs. 9 and10, can then be expressed in the form M= /H9262B/H20855m/H20856nMn, /H208494/H20850 where nMn=NMn /Vis the Mn concentration and NMnthe total number of Mn ions in a volume V. C. Search for hole-mediated ferromagnetism As already mentioned, according to the theoretical predic- tions within the p-dZener model,1,6RT ferromagnetism is expected in single-phase /H20849Ga,Mn /H20850N and related compounds, provided that a sufﬁciently high concentration of both sub-stitutional magnetic impurities /H20849near 5% or above /H20850and valence-band holes will be realized. The latter condition is amore severe one, as the high-binding energy of Mn acceptorsin the strong coupling limit leads to hole localization. Surprisingly, RT ferromagnetism in p-type Ga 1−xMn xN with a Mn content as low as x/H110150.25% was recently reported.13The investigated modulation-doped structure con- sisted of a /H20849Ga,Mn /H20850N//H20849Al,Ga /H20850N:Mg/GaN:Si /H20849i-p-n/H20850 multilayer and a correlation between the ferromagnetism ofthe/H20849Ga,Mn /H20850N ﬁlm at 300 K and the concentration of holes accumulated at the /H20849Ga,Mn /H20850N//H20849Al,Ga /H20850N:Mg interface was shown. The interfacial hole density was controlled by anexternal gate voltage applied across the p-njunction of the structure, and a suppression of the FM features—already ex-isting without the gate bias—took place for a moderate gatevoltage applied. Apart from a high value of T C, a puzzling aspect of the experimental results is the large magnitude ofthe spontaneous magnetization, 75 /H9262emu /cm2.13Since the holes are expected to be accumulated in a region with athickness of the order of 1 nm, the reported magnetic mo-ment is about two orders of magnitude larger than the oneexpected for ferromagnetism originating from an interfacialregion in /H20849Ga,Mn /H20850N with x=0.25%. Nevertheless, we have decided to check the viability of this approach that not only seemed to result in high tempera-ture FM in GaN:TM, but also allowed the all-electrical con-trol of FM. Thus, we have combined the p-type doping pro- cedures we previously optimized 27with the growth of the dilute /H20849Ga,Mn /H20850N presented in this work to carefully repro- duce the corresponding structure.13The desired architecture of the investigated sample is conﬁrmed by SIMS proﬁling/H20849see Fig. 13/H20850indicating the formation of well deﬁned /H20849Ga,Mn /H20850N//H20849Al,Ga /H20850N:Mg and /H20849Al,Ga /H20850N:Mg/GaN:Si inter- faces. However, as shown in Fig. 14, no clear evidence of a ferromagnetic-like response is seen within our present ex-perimental resolution of /H110150.3 /H9262emu /cm2. To strengthen the point, we note here that the maximum error bar of our results/H20849/H110150.7 /H9262emu /cm2/H20850corresponds to about 1/100 of the satu- ration magnetization reported in the assessed experiment.While the absence of a ferromagnetic response at the level ofour sensitivity is to be expected, the presence of a largeferromagnetic signal found in Ref. 13in a nominally identi- cal structure is surprising. Without a careful structural char-acterization of the sample studied in Ref. 13by methods similar to those we have employed in the case of our layers,the origin of differences in magnetic properties between thetwo structures remains unclear. VII. SUMMARY In this paper, we have investigated Ga 1−xMn xN ﬁlms grown by MOVPE with manganese concentration x/H113511%. A set of experimental methods, including SXRD, HRTEM, andEXAFS, has been employed to determine the structural prop-erties of the studied material. These measurements reveal theabsence of crystallographic phase separation and a Ga-substitutional position of Mn in GaN. The ﬁndings demon-strate that the solubility of Mn in GaN is much greater thanthe one of Cr /H20849Ref. 83/H20850and Fe /H20849Ref. 18/H20850in GaN grown under the same conditions. Nevertheless, for the attained Mn con-centrations and owing to the absence of band carriers, theMn spins remain uncoupled. Accordingly, pertinent magneticproperties as a function of temperature, magnetic ﬁeld and itsorientation with respect to the caxis of the wurtzite structure can be adequately described by the paramagnetic theory ofan ensemble of noninteracting Mn ions in the relevant crystalﬁeld. Our SQUID and XANES results point to the 3+ con-ﬁguration of Mn in GaN. However, the collected informationFIG. 13. /H20849Color online /H20850SIMS depth proﬁles of our /H20849Ga,Mn /H20850N/ /H20849Al,Ga /H20850N:Mg/GaN:Si /H20849i-p-n/H20850structure. FIG. 14. /H20849Color online /H20850Room temperature magnetic signal from the/H20849Ga,Mn /H20850N//H20849Al,Ga /H20850N:Mg/GaN:Si structure. For completeness, results of both in-plane and out-of-plane orientations are shown.Diamagnetic and paramagnetic contributions have beencompensated.STEFANOWICZ et al. PHYSICAL REVIEW B 81, 235210 /H208492010 /H20850 235210-12cannot tell between d4andd5+hmodels of the Mn3+state, that is on the degree of hole localization on the Mn ions. A negligible contribution of Mn in the 2+ charge state indicatesa low concentration of residual donors in the investigatedﬁlms. Our studies on modulation doped p-type Ga 1−xMn xN//H20849Ga,Al /H20850N:Mg heterostructures do not repro- duce the high temperature robust ferromagnetism reportedrecently for this system. 13 ACKNOWLEDGMENTS The work was supported by the FunDMS Advanced Grantof the European Research Council within the “Ideas” 7th Framework Programme of the EC, and by the AustrianFonds zur Förderung der wissenschaftlichen Forschung/H20849Grants No. P18942, No. P20065, and No. N107-NAN /H20850.W e also acknowledge H. Ohno and F. Matsukura for valuablediscussions, G. Bauer and R. T. Lechner for their contribu-tion to the XRD measurements as well as the support of thestaff at the Rossendorf Beamline /H20849BM20 /H20850and at the Italian Collaborating Research Group at the European SynchrotronRadiation Facility in Grenoble. mikes@ifpan.edu.pl †dietl@ifpan.edu.pl ‡alberta.bonanni@jku.at 1T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, Sci- ence 287, 1019 /H208492000 /H20850. 2I. Žuti ć, J. 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PhysRevB.73.054426.pdf	Ferromagnetic resonance linewidths in ultrathin structures: A theoretical study of spin pumping A. T. Costa,1Roberto Bechara Muniz,2and D. L. Mills3 1Departamento de Ciências Exatas, Universidade Federal de Lavras, 37200-000 Lavras, M. G. Brazil 2Instituto de Física, Universidade Federal de Fluminense, 24210-340 Niterói, R. J. Brazil 3Department of Physics and Astronomy, University of California, Irvine, California 92697, USA /H20849Received 4 November 2005; published 16 February 2006 /H20850 We present theoretical studies of the spin pumping contribution to the ferromagnetic resonance linewidth for various ultrathin ﬁlm ferromagnetic structures. We consider the isolated ﬁlm on a substrate, with Fe onAu/H20849100 /H20850and Fe on W /H20849110 /H20850as examples. We explore as well the linewidth from this mechanism for the optical and acoustical collective modes of FM/Cu N/FM/Cu /H20849100 /H20850structures. The calculations employ a realistic electronic structure, with self-consistent ground states generated from the empirical tight binding method, withnine bands for each material in the structure. The spin excitations are generated through use of the randomphase approximation applied to the system, including the semi-inﬁnite substrate on which the structure isgrown. We calculate the frequency response of the system directly by examining the spectral density associatedwith collective modes whose wave vector parallel to the surface is zero. Linewidths with origin in leakage ofspin angular momentum from the adsorbed structure to the semi-inﬁnite substrate may be extracted from theseresults. We discuss a number of issues, including the relationship between the interﬁlm coupling calculatedadiabatically for trilayers, and that extracted from the /H20849dynamical /H20850spin wave spectrum. We obtain excellent agreement with experimental data, within the framework of calculations with no adjustable parameters. DOI: 10.1103/PhysRevB.73.054426 PACS number /H20849s/H20850: 76.50. /H11001g, 75.75. /H11001a, 72.25.Mk I. INTRODUCTORY REMARKS The damping of spin motions in nanoscale ferromagnetic structures has been a topic explored actively in recent years.The interest focuses on the 3 dtransition metal ferromagnets and their alloys, since one may now synthesize diverse ultr-asmall magnetic structures of very high quality from them. Itis the case as well devices are fabricated from these materi- als, in which ferromagnetism is realized even well aboveroom temperature. It is very clear that one encounters damp-ing mechanisms not present in the bulk crystalline form ofthe constituents. That this is so has been evident for manyyears now. 1The primary question then centers on the origin of the damping mechanisms operational in nanomagneticstructures. Of course, the electronic structure of such entities may differ substantially from that in the bulk crystalline matter,by virtue of distortions of the lattice associated with the mis-match in lattice constant between the ﬁlm and the substrate.In addition, a large fraction of the moment bearing ions sit atinterfaces or on surfaces, and this will inﬂuence the elec-tronic structure as well. Hence, if the Gilbert damping con-stant Gis used as a measure of the damping found for long wavelength spin motions, one may expect intrinsic differ-ences between the values of Gappropriate to nanoscale structures, and that appropriate to bulk materials. It is now very clear from diverse experimental studies that the damping mechanisms operative in ultrathin ferromagnetsdo not have their origin only in differences between the elec-tronic structure of bulk and nanoscale matter. Evidently dis-tinct mechanisms are present. For instance, an earlier analy-sis of data on ferromagnetic resonance /H20849FMR /H20850linewidths show a dependence on growth conditions whose inﬂuence onelectronic structure is surely rather indirect. 1A few years ago, it was argued2that a mechanism referred to as twomagnon damping can be activated by defects on the surface or at interfaces. The density and character of such defects isclearly inﬂuenced by the manner in which the ﬁlm is grown.The theory of two magnon damping makes explicit predic-tions regarding the frequency variation of the linewidth andits dependence on the orientation of the magnetization. 2,3 Also, Rezende and his colleagues have demonstrated that itaccounts quantitatively for the strong wave vector depen-dence found upon comparing linewidths measured in FMR,and those measured in Brillouin light scattering /H20849BLS /H20850. 4We refer the reader to a review article which describes the theoryof two magnon damping in ultrathin ferromagnetic ﬁlms, andthe experimental evidence for its presence. 5 Two magnon damping is an extrinsic mechanism acti- vated by defects on or within an ultrathin ﬁlm. In the recentliterature, experimental evidence has been presented for thepresence of a mechanism operative in ultrathin ferromagneticﬁlm structures which is intrinsic in character. 6It should be remarked that Berger7and Slonczewski8predicted this mechanism should be present in ultrathin ﬁlms in advance ofthe experiments. Suppose we consider an ultrathin ﬁlm of aferromagnetic metal placed on a metallic substrate. The mag- netic moments in an ultrathin ferromagnetic ﬁlm are embed-ded in a sea of conducting electrons which, of course alsohave spins and magnetic moments. As the magnetic momentsare excited in a FMR or BLS experiment, they precess co-herently. Angular momentum is transferred to the conductionelectrons, and this is transported across the interface betweenthe ﬁlm and the substrate in the form of a spin current. Thus,spin angular momentum is lost from the ferromagnetic ﬁlm,and the spin motions are damped as a consequence. Thissource of damping is referred to as the spin pumping mecha-nism. As the ferromagnetic ﬁlm is made progressivelythicker, the spin pumping contribution to the linewidth de-creases roughly inversely with the ﬁlm thickness, accordingPHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 1098-0121/2006/73 /H208495/H20850/054426 /H2084910/H20850/$23.00 ©2006 The American Physical Society 054426-1to theory. This behavior is found in the experimental data. By now, numerous theoretical descriptions have been given for the spin pumping contribution to the linewidth. Inthe early papers cited in the previous paragraphs 7,8a simple phenomenological picture of the ferromagnetic metal and thesurrounding materials is employed. The magnetic moments are described as localized moments, with S/H6023/H20849l/H6023/H20850the spin angu- lar momentum associated with lattice site l/H6023. Such local mo- ments are assumed to be coupled to a conduction electronbath, modeled as free electrons in quantum wells. These in-teract with the local moments via the classical sd exchange interaction − JS /H6023/H20849l/H6023/H20850·/H9268/H6023, with /H9268/H6023the conduction electron spin. Within such a framework, general features of the spin pump- ing mechanism may be elucidated, but quantitative predic-tions for speciﬁc materials are not contained in such simpledescriptions. Within such a framework, Simanek and Heinrich 9intro- duced an elegant picture of the origin of the spin pump-ing contribution to the linewidth. A ferromagnetic ﬁlmwith static magnetization placed in contact with a non-magnetic metal induces spin oscillations in the nonmagneticmaterial. These are the Ruderman, Kittel, Kasuya, Yosida/H20849RKKY /H20850oscillations. If the magnetization of the ferromagnet precesses at a ﬁnite frequency, the RKKY oscillations do notquite follow the magnetization of the substrate. In essence,the RKKY coupling is frequency dependent. If the frequencydependent RKKY interaction is expanded in powers of thefrequency, the lowest order linear term in the effective equa-tion of motion of the magnetization provides a contributionto the effective damping constant felt by the ferromagneticspins. In Ref. 9, only a single layer of localized spins wasconsidered, embedded within an electron gas. One of thepresent authors has elaborated on this basic scheme by con-sidering Nlayers of local moments to simulate a multilayer ﬁlm, with conduction electrons described by a quantum wellpicture. 10A prediction which emerges from this model is that the spin pumping contribution to the linewidth does not falloff simply like 1/ D, with Dthe thickness of the ferromag- netic ﬁlm. Quantum oscillations are superimposed on this1/Dfalloff. Further discussion of this approach has been presented by Simanek. 11 A rather different approach has been taken by Tserk- ovnyak and his colleagues.12These authors use a one elec- tron picture, in which the electron wave functions extendover the entire structure considered, the ferromagnetic ﬁlmand the nonmagnetic metals which are in contact with it.Spin precession within the ferromagnetic ﬁlm is introducedby virtue of a postulated form for a time dependent densitymatrix. The ﬂow of spin angular momentum out of the fer-romagnetic ﬁlm in the presence of the spin precession isdescribed in terms of the transmission and reﬂection coefﬁ-cients associated with appropriate scattering states. Recentlya study by this group has appeared where density functionalelectronic structure calculations are employed to calculatespin pumping contributions to the linewidth for speciﬁc ma-terial combinations. 13The agreement with experimental data which follows from this analysis is very good, though wenote that the calculations explore an ultrathin ferromagneticﬁlm bounded on both sides by a semi-inﬁnite nonmagneticmetal, a structure quite different than employed in the actual experiments. The authors conclude that the oscillations foundwithin the quantum well model of Ref. 10 are in fact verymodest in amplitude when a proper electronic structure isemployed. We remark that in the series of calculations whichmotivate the present paper, we ﬁnd similar results. In this paper, we address the calculation of spin pumping linewidths for various ultrathin ﬁlm/substrate combinationsby a method quite different than found in the studies citedabove. We calculate directly the full frequency spectrum ofthe response of diverse structures to applied microwaveﬁelds whose wave vector parallel to the surfaces of the struc-tures is zero, within the framework of a method which pro-vides a realistic description of the electronic structure of thesample of interest. We extract linewidths by ﬁtting the linesin the absorption spectra with a Lorentzian, very much asdone in experimental analyses of actual data. As remarkedabove, we note that the calculations presented by the authorsof Refs. 12 and 13 assume the ferromagnetic ﬁlm to bebounded on both sides by nonmagnetic metals of semi- inﬁnite extent, since the reﬂection and transmission coefﬁ-cients required are found from scattering states associatedwith electrons reﬂected from and transmitted through the ul-trathin ferromagnetic ﬁlm. Our method allows us to addressthe more realistic case of an ultrathin ﬁlm adsorbed on asemi-inﬁnite substrate, bounded by vacuum /H20849or if we wish a capping layer /H20850on the other side. As we shall demonstrate below, we can also calculate the linewidths of trilayers ad-sorbed on a semi-inﬁnite substrate, so we can study the line-width of both the acoustic and optical spin wave modes.Such structures have been explored by the Baberschkegroup. 14While, for reasons discussed below, it is difﬁcult to make a detailed comparison between our results and theirdata, we do see features in our results rather similar to thosefound experimentally. Also, we obtain an excellent accountof the linewidths reported in Ref. 6 for ultrathin ﬁlms of Feadsorbed on the Au /H20849100 /H20850surface. We have also examined other ﬁlm/substrate combinations with the aim of exploring the inﬂuence of the electronic structure of the substrate andthe local atomic geometry at the interface on the spin pump-ing contribution to the linewidth. The calculations reported here employ the formalism and methodology we developed earlier 15which was employed in our successful quantitative account16of the frequencies, line- widths and line shapes associated with large wave vectorspin waves excited in spin polarized electron energy loss/H20849SPEELS /H20850studies of Co ﬁlms adsorbed on the Cu /H20849100 /H20850 surface. 17The very large linewidths found in these experi- ments have their origin in the same process which enters thespin pumping contribution to the FMR linewidth: in theseitinerant materials, the collective spin wave mode is dampedby virtue of the transfer of angular momentum to band elec-trons. The spin pumping FMR linewidth then may be viewedas the zero wave vector limit of the large linewidths found inthe SPEELS data. It is the case, then, that this contributionlinewidth has a strong dependence on wave vector parallel tothe surface, for wave vector variations on the scale of thesurface Brillouin zone. It is also now the case that our meth-odology has led to quantitative accounts of both the nature ofthe spin excitation spectrum, 18and the damping rates associ-COSTA, MUNIZ, AND MILLS PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-2ated with these modes throughout the entire surface Brillouin zone. In Sec. II of this paper, we discuss our approach and com- ment on aspects of the numerical computations. Our resultsare discussed in Sec. III and Sec. IV is devoted to concludingremarks. II. COMMENTS ON THE ANALYSIS The approach we have employed has been discussed in detail in our earlier publications,15,16so we summarize it only brieﬂy here. The most detailed discussion is to be foundin the ﬁrst reference cited in Ref. 15. As remarked in Sec. I,we employ an empirical tight binding description of the elec-tronic structure of the various constituents in the samples wemodel. Nine bands are included for each material. These arethe four spbands supplemented by the dband complex. Fer- romagnetism in the relevant ﬁlms is driven by intratomic, onsite Coulomb interactions that operate within the dshell. We refer to the model Hamiltonian as a generalized Hubbardmodel. A description of the ground state is generated in meanﬁeld theory, allowing the magnitude of the moments in eachplane of each ferromagnetic ﬁlm to vary. In an early publi-cation, we have shown that our approach provides results forthe magnetic moment proﬁle of the ﬁlm and layer dependentpartial wave density of states in very good accord with thosegenerated by full density functional calculations. 19 We then generate a description of the frequency spectrum of spin ﬂuctuations in the ﬁlm through calculating the appro-priate wave vector frequency dependent transverse suscepti-bility. This is done through use of the random phase approxi-mation /H20849RPA /H20850of many body theory, which is “conserving approximation.” The quantity we generate by this means is written /H9273+,−/H20849Q/H6023/H20648,/H9024;l/H11036,l/H11036/H11032/H20850. It has the following physical meaning. Suppose the magnetizations of the various ferro- magnetic ﬁlms in the structure are parallel to the zaxis. Then apply a circularly polarized, space and time dependent ﬁeld in the xyplane of the form h/H6023/H20849l/H6023/H20648,l/H11036/H20850=h/H20849l/H11036/H20850/H20853xˆ+iyˆ/H20854exp/H20851iQ/H6023/H20648·l/H6023/H20648 −i/H9024t/H20852. Here a lattice site in the structure is denoted by l/H6023=l/H6023/H20648 +nˆl/H11036, where l/H6023/H20648lies in the plane parallel to the ﬁlm surfaces, and nˆis a vector perpendicular to the surfaces. The wave vector Q/H6023/H20648lies in the two dimensional surface Brillouin zone of the structure, and xˆ,yˆrepresent Cartesian unit vectors in thexyplane. In the present paper, since we are interested in the ferromagnetic resonance response, all calculations are conﬁned to zero wave vector, Q/H6023/H20648/H110130. The applied magnetic ﬁeld just described can have arbitrary variation in the direc- tion normal to the ﬁlm surfaces. If /H20855S/H6023+/H20849l/H6023/H20648,l/H11036;t/H20850/H20856is the expec- tation value of the transverse component of spin at the site indicated, then linear response theory provides us with therelation /H20855S/H6023+/H20849l/H6023/H20648,l/H11036;t/H20850/H20856=/H20875/H20858 l/H11036/H11032/H9273+,−/H20849Q/H6023/H20648,/H9024;l/H11036,l/H11036/H11032/H20850h/H20849l/H11036/H11032/H20850/H20876exp/H20851i/H20849Q/H6023/H20648·l/H6023/H20648 −i/H9024t/H20850/H20852. /H208491/H20850 Of central interest to us is the spectral density functionA/H20849Q/H6023/H20648,/H9024;l/H11036/H20850=1 /H9266Im/H20851/H9273+,−/H20849Q/H6023/H20648,/H9024;l/H11036,l/H11036/H11032/H20850/H20852. /H208492/H20850 It follows from the ﬂuctuation dissipation theorem that this quantity, considered as a function of /H9024, provides us with the frequency spectrum of spin ﬂuctuations of wave vector Q/H6023/H20648on layer l/H11036. Suppose for the moment we consider a ﬁlm of N layers which may be modeled by the Heisenberg Hamil-tonian, which envisions a localized spin situated on each lattice site. Then for each choice of Q/H6023/H20648, we have exactly N spin wave modes, each with inﬁnite lifetime. Let /H9024a/H20849Q/H6023/H20648/H20850be the frequency of one such mode, and let ea/H20849Q/H6023/H20648;l/H11036/H20850be the associated eigenvector. For this model, which we note is in- appropriate for the itinerant electron ferromagnetic ﬁlmsconsidered here, 15,16,18one may show that A/H20849Q/H6023/H20648,/H9024;l/H11036/H20850=/H20858 a/H20841ea/H20849Q/H6023/H20648,l/H11036/H20850/H208412/H9254/H20851/H9024−/H9024a/H20849Q/H6023/H20648/H20850/H20852. /H208493/H20850 The form in Eq. /H208493/H20850provides one with an understanding of the information contained in the spectral density function. Of course, it would be highly desirable to base the analy- sis of the spin dynamics of systems such as we explorethrough use of a state of the art density functional descriptionof the ground state, combined with a time dependent densityfunctional analysis /H20849TDDFA /H20850of the spin ﬂuctuation spec- trum. At the time of this writing, one cannot carry out studiesof the spin ﬂuctuation spectrum within the TDDFA for sys-tems as large as we have studied in the past 15,16and that we explore in the present paper. The demands on computationtime are prohibitive. We believe it is essential to employfully semi-inﬁnite substrates for proper calculations of spinwave linewidths with origin in decay of the collective exci-tations to the Stoner excitation manifold. One must have atrue continuum of ﬁnal states for a proper description of thelinewidth. All of our calculations employ a full semi-inﬁnitesubstrate, unless otherwise indicated. The virtue of our multi-band Hubbard model is that once the irreducible particle holepropagator is computed, inversion of the RPA integral equa-tion is straightforward and fast, by virtue of the fact that theparticle-hole vertex is separable in momentum space withinour framework. TDDFA calculations employ an ab initio generated vertex function which requires the numerical solu-tion of a full integral equation, once the irreducible particlehole propagator is in hand. One can handle only rather smallsystems within this framework, at present. If one describes our treatment of the ground state and then the RPA description of the spin dynamics by Feynman dia-grams, it is the case that we include the same set of diagramsas incorporated into the full density functional theory, so in acertain sense one may view what we do as a simpliﬁed ver-sion of time dependent density functional theory. However,technical simpliﬁcations such as those described in the pre-vious paragraph enable us to address very large systems. Wenote that our calculations involve no adjustable parameters,since all of our input parameters are taken from the appro-priate literature. A rather detailed discussion of how we pro-ceed is contained in Ref. 19, and considerable detail is foundin the ﬁrst paper cited in Ref. 15.FERROMAGNETIC RESONANCE LINEWIDTHS IN ¼ PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-3In spite of the remarks above, it is the case that the nu- merical work involved in the calculation of the dynamicalsusceptibility is extremely demanding from the computa-tional point of view. As discussed in detail in the ﬁrst papercited in Ref. 15, to calculate the irreducible particle holepropagator it is necessary to calculate an energy integralwhose integrand is itself a two dimensional integral over thefull surface Brillouin zone. We have also shown that verycareful attention must be paid to convergence of the Bril-louin zone integrations for physically reasonable results to emerge from /H9273+,−/H20849Q/H6023/H20648,/H9024;l/H11036,l/H11036/H11032/H20850. We refer the reader to Fig. 6 of the ﬁrst paper in Ref. 15 and the associated discussion in the text. We ﬁnd that 4224 points in the surface Brillouinzone must be employed to insure proper convergence. Theirreducible particle hole propagator is a matrix structure,whose size is the number of layers in the ﬁlm, N, times thenumber of orbitals included in the calculation. Thus, for us toanalyze spin ﬂuctuations in a ten layer ferromagnetic ﬁlm,we must perform the Brillouin zone integration just de-scribed for a 90 /H1100390 matrix. For ﬁlms deposited on semi- inﬁnite substrates, of course we require the single particleGreen’s function for the semi-inﬁnite structure. This part ofthe computation also consumes a considerable fraction of thetotal computing time. Fortunately, the problem in question can be easily adapted to run in parallel on a cluster of workstations. We adopted avery straightforward parallelization strategy, in which wespread 64 points of energy integration among 32 processors.For interprocess communication, we use the MPICHimplementation 20of the message passing interface /H20849MPI /H20850.21 The interprocess communication overhead for this speciﬁc problem is minimal, so that the execution time scalesroughly linearly with the number of processors. This allowsus to calculate the dynamic susceptibility of relatively thickferromagnetic ﬁlms /H20849/H1101115 layers /H20850on a fully semi-inﬁnite substrate, in a relatively short time. A computation of thefrequency response for one ﬁlm can be performed in lessthan 48 h. Typically to generate the absorption spectra shownin Sec. III, we use 100 frequencies. III. RESULTS AND DISCUSSION In our previous studies of spin waves in various ultrathin ﬁlm/substrate combinations, our emphasis was on excitationswith relatively large wave vectors. The excitation energies ofsuch modes are very large compared to the Zeeman energy,which describes the interaction of the spins in the systemwith externally applied ﬁeld. Thus the external ﬁeld wastaken to be zero in all of our previous studies. Here, when weexamine the modes whose wave vector is identically zero,with emphasis on the modes seen in FMR experiments, quiteclearly we must incorporate the externally applied magneticﬁeld. In the results that follow, we have immersed all theelectrons in a spatially uniform Zeeman ﬁeld, and we denotethe precession frequency in this Zeeman ﬁeld by /H9024 0. We should say a few words about how the strength of the Zeeman ﬁeld has been chosen. First of all, the energy scaleused in our electronic structure is a few electron volts, orequivalently the Rydberg. The spin excitations we studied inour earlier papers had energies compatible with this scale, in the range of 0.01–0.3 eV /H20849Refs. 15 and 16 /H20850depending on the wave vector we chose to study. If we choose our appliedZeeman ﬁeld to be in the range actually used in laboratoryexperiments, then the energy scale of the FMR mode is sosmall /H20849/H1101110 −5eV/H20850that an enormous amount of computation time would be required to produce calculations with an en- ergy grid so ﬁne as to allow us to generate accurate proﬁlesfor these very low energy modes. It is the case, however, thatso long as /H9024 0is very small compared to the typical elec- tronic energy scales, the linewidths scale linearly with reso-nance frequency. We remark that we have checked carefullythat this is so, in our early studies. So long as we remain inthe regime where the linewidth scales in this linear manner,we may choose an unphysically large value for /H9024 0, and scale the computed linewidths appropriately when we compare ourresults with data. What we choose to do instead is always toplot the ratio /H9004/H9024 0//H90240, with /H9004/H9024 0the linewidth determined as discussed below. This is independent of applied ﬁeld, in theregime where the linearly scaling holds. In the calculationsto be presented in this paper, we have chosen /H9024 0=2 /H1100310−3Ry=27 meV. In Fig. 1 /H20849a/H20850, we show the absorption spectrum for the FMR mode /H20849the lowest lying spin wave mode of the ﬁlm, with wave vector parallel to the ﬁlm surface identically equalto zero /H20850of a two layer Co ﬁlm adsorbed on the Cu /H20849100 /H20850 surface. One sees the resonance line, with peak very close to /H9024 0. In principle, there can be a gshift associated with this mode even in the absence of spin orbit coupling, sincestrictly speaking the amplitude of the spin motion in thedirection normal to the surface is not perfectly uniform. Inour earlier studies of the electron spin resonance response ofisolated magnetic adatoms adsorbed on the Cu /H20849100 /H20850surface, we found such gshifts to be quite large. 22Here we ﬁnd that the peak of the absorption line coincides exactly with /H90240,s o far as we can tell. There is another idealization we have made that must be mentioned. In an actual sample, where a ferromagnetic ﬁlmis on top of a nonmagnetic substrate such as Cu, when thespins precess in the ferromagnet, dynamic dipolar ﬁelds aregenerated by the spin motion. Thus, for a ﬁlm magnetizedparallel to its surfaces, the FMR frequency is shifted from/H9024 0=/H9253H0to/H9253/H20851H0/H20849H0+4/H9266Ms/H20850/H208521/2, where /H9253is the gyromag- netic ratio for the ferromagnetic ﬁlm, H0is the applied Zee- man ﬁeld, and Msis the saturation magnetization. It is also the case that the gyromagnetic ratio for the ferromagneticﬁlm will differ from that of the conduction electrons in thesubstrate. Thus, in an actual sample, the precession fre-quency of the spins in the ultrathin ferromagnetic ﬁlm willdiffer from that for electrons in the substrate. We have ex-plored the inﬂuence of this effect by varying the ratio of theZeeman frequency of the spins in the ferromagnet, to that ofthe spins in the substrate by as much as a factor of 2. We ﬁndthat our linewidths, normalized to the resonance frequency inthe ferromagnet, are insensitive to this difference. On physi-cal grounds one expects this. In microscopic language, thelinewidths we calculate have their origin on the transfer ofangular momentum from the coherently precessing momentsin the ferromagnet to the bath of itinerant band electrons. Ifwe think of this as a form of Landau damping, where theCOSTA, MUNIZ, AND MILLS PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-4spin wave decays to the bath of Stoner excitations of the ﬁlm/substrate complex /H20849spin triplet particle hole excitations /H20850, a shift in the precession frequency of the ferromagnetic ﬁlmrelative to that of the substrate electrons is compensated by avery tiny shift in the wave vector transfer involved in thedecay process, on the scale of the relevant two dimensionalBrillouin zone. Note that wave vector components normal tothe surface are not conserved for the systems we study. Thus,in Fig. 1 /H20849a/H20850and in the results to be discussed below, all electrons in the system are subjected to the same externalﬁeld, and are assumed to have the same gyromagnetic ratio,with dipolar ﬁelds in the ferromagnet ignored. In Fig. 1 /H20849b/H20850, we show calculations of the of a Co 2Cu2Co2 trilayer adsorbed on a semi-inﬁnite Cu /H20849100 /H20850surface. One now sees two modes, the acoustical and the optical spin wave mode of the ferromagnetic bilayer. It is clear fromvisual inspection that the acoustical mode is narrower thanthe FMR line shown in Fig. 1 /H20849a/H20850, while the optical mode is substantially broader, a pattern seen in experimental studiesof such systems. 14It should be noted that in Fig. 1 /H20849b/H20850,i ti s not quite the FMR absorption spectrum that is displayed. Fortwo such identical ﬁlms, the antisymmetric optical spin wavemode has no net transverse magnetic moment, and thuswould be silent in a FMR spectrum. In actual FMR experi- ments where both modes are observed, the two ferromag-netic ﬁlms are inequivalent. What we display in Fig. 1 /H20849b/H20850is the spectral density function A/H20849Q /H6023/H20648=0,/H9024;l/H20850, where we have taken l/H11036=1, where the 1 refers to the outermost layer of the outermost ﬁlm. The physical interpretation of this spectraldensity function is that it describes the frequency spectrumof the ﬂuctuations of the spins in the atomic layer indicated,at zero wave vector parallel to the surface. The acoustic andoptical spin wave both leave their signature on this responsefunction. One ﬁnal remark is that our procedure for determining the linewidths discussed below is as follows. For each structureof interest, we calculate a spectrum such as that illustrated inFigs. 1 /H20849a/H20850and 1 /H20849b/H20850. Then we ﬁt the curves to appropriate Lorentizians, and from this we extract a linewidth. The quan-tity/H9004/H9024is the half width at half maximum. The ﬁrst case we discuss is that of Fe ﬁlms adsorbed on the Au /H20849100 /H20850surface. This is the system studied by Urban, Woltersdorf, and Heinrich. 6 In Fig. 2, we show our calculated linewidths as open circles, for Fe ﬁlm thicknesses in the range of 2–10 ML. Thesolid line is a best power law ﬁt to the data, which turns outto be 1/ /H20849N Fe/H208500.98law, with NFethe number of Fe layers. Vari- ous authors7,8,12have argued that the spin pumping linewidth should fall off inversely with the thickness of the ferromag-netic ﬁlm, so these result are consistent with this picture. Ourcalculations show this behavior nicely. When ﬁtting such asmall number of points, the difference between 0.98 andunity is not signiﬁcant. For small thicknesses, we see oscil-lations of modest amplitude around the 1/ N Felaw. These are quantum oscillations such as those discussed in Ref. 10,though as discussed by the authors of Ref. 13 the simplequantum well model exaggerates their amplitude. The solid circles in Fig. 2 are data taken from Ref. 6. Theory and experiment agree very nicely. Unfortunately it isdifﬁcult for us to carry out calculations for ﬁlms muchthicker than ten layers for the FMR mode. In our integrations FIG. 1. We plot the spectral density function A/H20849Q/H6023/H20648=0,/H9024;l/H11036/H20850for two cases: /H20849a/H20850a two layer Co ﬁlm adsorbed on the Cu /H20849100 /H20850surface and /H20849b/H20850a trilayer consisting of two Co ﬁlms separated by two layers of Cu, with the complex adsorbed on the Cu /H20849100 /H20850surface. Each of the Co ﬁlms has two layers. The Zeeman energy has been taken tobe/H9024 0=2/H1100310−3Ry, or 27.2 meV. Here l/H11036is chosen to be the in- nermost atomic layer of the magnetic structure. FIG. 2. For an Fe ﬁlm on the Au /H20849100 /H20850substrate, we plot the ration /H9004/H9024//H90240as a function of NFe, the number of Fe layers. The results of the calculations are given as open circles, and the solidline is the best power law ﬁt to the data, which is 1/ /H20849N Fe/H208500.98. The solid circles are taken from measurements reported in Ref. 6.FERROMAGNETIC RESONANCE LINEWIDTHS IN ¼ PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-5over energy, a small imaginary part is added to energy de- nominators, and we must always keep this small compared tothe linewidth. We must decrease this as the ﬁlm gets thicker, and at the same time we must increase the density of pointsin our integration grids to generate a reliable, converged lineshape uninﬂuenced by the numerical procedures. It is interesting to inquire how the linewidth depends on the electronic structure of the substrate. To explore this, wehave carried out calculations for an Fe ﬁlm deposited on theW/H20849110 /H20850surface. While we are unaware of any FMR studies of this system, the magnetism associated with the Fe/W /H20849110 /H20850system has been extensively studied in the litera- ture. Our results are given in Fig. 3. The straight line is a best ﬁt to a power law dependence on the Fe ﬁlm thickness, andin this case we ﬁnd 1/ /H20849N Fe/H208500.81, just a bit slower that a simple variation inversely proportional to the ﬁlm thickness. How- ever, we again see quantum oscillations about a simplepower law ﬁt, at smaller thickness, so a simple power law isa bit of an oversimpliﬁcation. Surprisingly, in our view, thenumerical values of the linewidths displayed in Fig. 3 are notso very different than those in Fig. 2. We had expected largerlinewidths, because of the large density of states in the sub-strate associated with the unﬁlled dbands of W. It should be pointed out, however, that the connectivity of an Fe ﬁlm islarger when it is adsorbed on Au /H20849100 /H20850than when it is ad- sorbed on W /H20849110 /H20850. An Fe atom at the interface with the Au/H20849100 /H20850surface is connected to four nearest neighbor Au atoms, whereas it is coupled to just two nearest neighbor W atoms in the Fe/W /H20849110 /H20850interface. One virtue of our empiri- cal tight binding approach is that we can probe the physics responsible for results such as these by artiﬁcially selectingparameters. We turn to such studies next, to obtain insightinto this result. It is interesting to compare the calculated linewidths for an Fe ﬁlm adsorbed on W /H20849110 /H20850, and for an identical ﬁlm adsorbed on a bcc Cu crystal, which we can simulate with our approach. Here the connectivity to the substrates is thesame. Of course, in the ﬁrst case the Fermi level intersectsthedband complex of the substrate, whereas in the secondcase the d bands are well below the Fermi level. For the cases of a two layer Fe ﬁlm on W /H20849110 /H20850we see from Fig. 3 that/H9004/H9024//H9024 0assumes the value 1.8 /H1100310−2, whereas for the Fe ﬁlm on Cu /H20849110 /H20850, the line is actually a bit broader, with /H9004/H9024//H90240=2.7/H1100310−2. Even though the two numbers are rather close to each other, it would seem that the physics whichunderlies the transfer of spin angular momentum across theinterface is rather different in the two cases. In Fig. 4, weshow calculations which illustrate this. Let us ﬁrst look at theright panel, the case of Fe on W /H20849110 /H20850. The solid curve is the spectral density calculated with use of the full electronic structure. The dashed curve is a calculation in which the d-d hopping terms are set to zero across the interface. The linenarrows very substantially when the d-dhopping is shut off, as we see. Evidently for this case, direct communication be-tween the delectrons in the Fe ﬁlm and the W substrate plays a central role in the transfer of spin angular momentum. Theelectrons at the W Fermi surface have strong dcharacter, and the electrons in the Fe with strong spcharacter play a minor role in the damping process. This view is reinforced by the dot-dash curve in the right hand panel of Fig. 4, where the sp hopping terms are set to zero across the interface. We seelittle change in the spectral density. The situation is verydifferent for the case of Fe on Cu /H20849110 /H20850, as we see from the left hand panel in Fig. 4. We see here that communication between both the sp-like and the d-like portions of the elec- tronic wave function across the interface play an importantrole. Even though the 3 dbands of Cu are completely ﬁlled, there is dcharacter admixed in the electron wave functions in the vicinity of the Fermi energy, while at the same timecoupling through the spcharacter of the wave function enters as well. Thus, while the ﬁnal numbers for the spin pumpingcontributions to the linewidth are rather similar for these twovery different substrates, the physical picture which underliesthe transfer of spin angular momentum across the interface israther different. In view of this discussion, it would be ofgreat interest to see experimental FMR linewidth studies forthe case of Fe on W /H20849110 /H20850. FIG. 3. The ratio /H9004/H9024//H90240, for an Fe ﬁlm adsorbed on the W/H20849110 /H20850surface. The straight line is a best power law ﬁt to the calculations, and gives a fall off with ﬁlm thickness of 1/ /H20849NFe/H208500.81. Clearly quantum oscillations such as discussed in Ref. 10 arepresent for small ﬁlm thicknesses, so the power law ﬁt is anoversimpliﬁcation. FIG. 4. For a two layer Fe ﬁlm on Cu /H20849110 /H20850/H20849left panel /H20850and for such a ﬁlm on W /H20849110 /H20850/H20849right panel /H20850, we show how the linewidth is inﬂuenced by various coupling across the interface. In both cases,the solid curve shows the spectral density function generated by acomplete calculation. The dashed curve has the d-dhopping terms across the interface set to zero, whereas the dot dash curve has thesp-sphopping terms turned off.COSTA, MUNIZ, AND MILLS PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-6We turn next to a discussion of our calculation of line- widths of the FMR modes of ferromagnetic/nonmagneticmetal/ferromagnetic ﬁlm combinations adsorbed on semi-inﬁnite substrates. We have already seen that such structuresproduce two features in the absorption spectrum, one froman acoustic spin wave mode, and one from an optical mode.The splitting between these two modes is controlled by thewell known interﬁlm exchange coupling transmitted throughthe nonmagnetic spacer layer. As remarked above, if one hasa sample with two identical ferromagnetic ﬁlms, only theacoustical mode is active in an FMR experiment, since theoptical mode possesses no net transverse magnetic moment.In samples fabricated from ultrathin ﬁlms, the thickness ofthe trilayer is small compared to the microwave skin depth,so to excellent approximation the ﬁlms are illuminated by aspatially uniform magnetic ﬁeld. We note that in Brillouin light scattering, one may observe both modes, since the op-tical skin depth is in the range of 10–20 nm for materials ofcurrent interest. Here, the exciting optical ﬁeld is spatiallynonuniform, on the length scale of the structure. It is thecase, though, that in any real sample, the two ﬁlms will al-ways be inequivalent even if their thicknesses are the same.For instance, both interfaces of the innermost ﬁlm are incontact with nonmagnetic metals, with the substrate on oneside and the spacer layer on the other. The outer ﬁlm hasvacuum on one side, and the spacer layer on the other.Hence, the anisotropy ﬁelds will be different in each ﬁlm,and this will render them inequivalent. As discussed below,we have simulated such effects by applying different externalmagnetic ﬁelds to each ﬁlm. First, however, we present ourresults for the case where the two ferromagnetic ﬁlms areregarded as identical in character. Before we turn to our results, we remark on how we pro- ceed with our determination of the linewidths for the opticaland the acoustical mode of the trilayer structure. In Fig. 1 /H20849b/H20850, we show the spectral density function for a case where theCu spacer layer is sufﬁciently thin that the two modes can beresolved easily in the spectral density associated with oneselected layer. As the Cu spacer is made thicker, the splittingbetween the two modes decreases and if we plot the spectralfunction illustrated in Fig. 1 /H20849b/H20850, we simply see a single asym- metric line, with the optical mode buried in the wing of theacoustical mode. We have proceeded as follows. For eachfrequency, we form a two by two matrix with elements /H927311=/H20858 l/H11036,l/H11036/H11032/H335281Im/H20851/H9273+,−/H208490,/H9024;l/H11036,l/H11036/H11032/H20850/H20852, /H927312=/H20858 l/H11036/H335281,l/H11036/H11032/H335282Im/H20851/H9273+,−/H208490,/H9024;l/H11036,l/H11036/H11032/H20850/H20852 and so on. In the deﬁnition of /H927311, the sums on both l/H11036and l/H11036/H11032range over the atomic planes in the outermost ﬁlm, while in/H927312the sum on l/H11036ranges over the atomic planes of the outermost ﬁlm, and the sum over l/H11036/H11032ranges over the atomic planes of the innermost ﬁlm. Diagonalization of this matrixat each frequency leads us to two eigenvectors, one withacoustical character and one with optical character. If wegenerate plots of the spectral densities of these two charac-teristic motions, in one we see only the acoustical feature,and in the second we see only the optical feature. We illus-trate this in Fig. 5. In Fig. 5 /H20849a/H20850we show the spectral densityfunction associated with spin ﬂuctuations in the innermost layer of the inner ﬁlm of a Co 2Cu4Co2/Cu /H20849100 /H20850structure. The optical mode appears in the wing of the acoustical mode, and one cannot reliably extract a width for the mode fromthis response function. In Fig. 5 /H20849b/H20850we see how one may isolate the acoustical mode by the procedure just describe,and in Fig. 5 /H20849c/H20850we show the optical mode. We remark that for the samples studied experimentally in Ref. 3, the two ferromagnetic ﬁlms in the trilayer have verydifferent resonance frequencies when taken in isolation, sothe issue of discriminating between the two modes did notarise. Save for one measurement discussed below, the trilayerstructures studied in Ref. 3 may be viewed as two oscillatorswhose resonance frequencies differ by an amount muchlarger than their linewidths, coupled weakly by the interﬁlmexchange coupling mediated by the spacer layer. In our earlier studies of large wave vector spin waves and their dispersion 16we have compared the dispersion relation which emerges from our dynamical theory with that calcu-lated on the basis of adiabatic theory, where exchange cou-plings between spins are calculated within quasistatic theory,and a Heisenberg Hamiltonian constructed from these is usedto generate a spin wave dispersion curve. We found substan-tial differences between the two cases. The strong dampingpresent at large wave vectors leads to shifts in the frequencyof the spin wave. This effect, which softens the modes, is notcontained in calculations based on an adiabatic description ofenergy changes associated with rotations of the moments.One can inquire if the effective interﬁlm exchange couplingsdeduced from the splitting between the acoustical and opticalspin wave modes of the trilayer are the same as those de-scribed by adiabatic theory. We provide a comparison which addresses this issue in Fig. 6. The solid line shows the dependence of the interﬁlmcoupling strength on N, the number of Cu layers between the two Co 2ﬁlms, calculated adiabatically using the method in FIG. 5. For a Co 2Cu4Co2/Cu /H20849100 /H20850structure, we show /H20849a/H20850the spectral density function A/H208490,/H9024;l/H11036/H20850, for the case where l/H11036is cho- sen to be the innermost layer of the inner ﬁlm. Then in /H20849b/H20850we show the acoustical mode proﬁle, where the mode has been isolated bythe procedure described in the text. In /H20849c/H20850, we display the optical mode.FERROMAGNETIC RESONANCE LINEWIDTHS IN ¼ PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-7Ref. 23. The dashed line provides the results deduced from the splitting between the two spin wave modes of the struc-ture. The two agree rather well, though there are indeedsmall differences between the two. It is evident that we ﬁnd the wide of the optical mode of the trilayer signiﬁcantly larger than that of the acousticmode. This is consistent with the results reported in Ref. 14,as noted earlier. Very interesting to us is the data in Fig. 6 ofRef. 14, where the linewidth measured for these two modesis given as a function of the number of Cu layers between thetwo ferromagnetic ﬁlms. One sees very clear quantum oscil-lations in these two linewidths. In our Fig. 7, we show cal-culations of the variation of the acoustic and optic modelinewidths with N, the number of Cu spacer layers in our Co 2CuNCo2/Cu /H20849100 /H20850structure. The calculations reproduce the features found in the data strikingly well. For instance, in our calculations we see a peak in the linewidth of the acous-tic mode at six atomic layers of Cu and a second somewhatsmaller peak at nine layers. Both features appear in the data,indeed with the peak at six layers the most prominent. Alsovery much as in the data, for the optical mode we see thepeak at six layers in our calculations, while the peak at ninelayers is suppressed strongly. In our calculations, the opticalmode linewidth, reckoned relative to that of the acousticmode, is larger than that in the data. At the peak at six layers,we ﬁnd the width of the optical mode larger than that of theacoustical mode by a factor of roughly 5, whereas in the datathis ratio is a bit less than a factor of 2. A direct comparisonof this ratio with the data is not so relevant in our minds,since the resonance frequencies of our two Co 2ﬁlms /H20849each taken in isolation /H20850are identical, whereas that of the Co 1.8 ﬁlm and the Ni 9ﬁlm used in Ref. 14 are different. Some of the data in Ref. 14 has been taken on Ni8CuNNi9/Cu /H20849100 /H20850samples. The resonance frequencies of the Ni 8and Ni 9ﬁlms differ substantially when the externally applied ﬁeld is in plane, presumably because of the nearproximity of these Ni ﬁlms to the spin reorientation transi-tion. However, the authors of Ref. 14 were able to sweep theresonance frequency of one ﬁlm through the second by vary-ing the angle of the external applied magnetic ﬁeld withrespect to the plane of the ﬁlms. They argue that the line-widths of the two modes approach each other when the reso- nance frequencies coincide. This is illustrated in their Fig. 4,for a sample with N=12. For such a thick copper spacer layer, the interﬁlm exchange coupling is negligible. We can simulate this with our Co 2CuNCo2/Cu /H20849100 /H20850struc- tures by applying a different magnetic ﬁeld to each ﬁlm. In Fig. 8 we present results of a study where the resonancefrequency of the inner ﬁlm is /H9024 0and that of the outer ﬁlm is /H9261/H9024 0, where /H9261is varied from 1 to 2. By varying /H9261,w ec a n study how the linewidth of each mode behaves if the twoﬁlms initially are detuned /H20849/H9261=2/H20850, and then brought to the point where the isolated ﬁlm resonance frequencies coincide /H20849/H9261=1/H20850. Our calculations are carried out for N=10, where the interﬁlm exchange coupling is very weak. We presume the authors of Ref. 14 have ﬁtted their FMR spectra to a sum of two Lorentizians in order to extract thewidth of the two modes. We have attempted a similar proce-dure with simulations of the FMR absorption spectrum of thestructure. However, we ﬁnd that when /H9261is fairly close to unity, we cannot reliably extract linewidths for the twomodes by ﬁtting the overall spectral density in this manner.In Fig. 8 /H20849a/H20850, the solid line shows the absorption spectrum of the trilayer for the case where /H9261=1.2, and we see the high frequency mode as a shoulder on the high frequency side ofthe line. Very clearly one cannot determine the width of thetwo modes from such an absorption spectrum, since the twomodes are not resolved. The solid line is the function FIG. 6. For a Co 2CuNCo2/Cu /H20849100 /H20850structure, we compare inter- ﬁlm couplings deduced from our dynamic calculations of the split-ting between the optical and acoustic mode of the trilayer /H20849open circles, dashed line /H20850with that calculated by adiabatic theory /H20849solid squares, solid line /H20850. FIG. 7. For a Co 2CuNCo2/Cu /H20849100 /H20850trilayer, we show the varia- tion with Nof the linewidth for /H20849a/H20850the acoustic spin wave mode of the structure, and /H20849b/H20850the optical spin wave mode of the structure.COSTA, MUNIZ, AND MILLS PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-8A/H20849/H9024/H20850=/H20858 l/H11036/H11032=14 /H20858 l/H11036=14 Im/H20851/H9273+,−/H208490,/H9024;l/H11036,l/H11036/H11032/H20850/H20852. This gives the absorption spectrum of the trilayer, if it is illuminated by a spatially uniform microwave ﬁeld. Also inFig. 8 /H20849a/H20850, we give the absorption spectrum for the outer ﬁlm by displaying A 1/H20849/H9024/H20850, deﬁned in a manner similar to A/H20849/H9024/H20850, except the spatial sums are now conﬁned to the outer ﬁlm. Similarly, A2/H20849/H9024/H20850is the absorption spectrum of the inner ﬁlm. We have found that we can only extract linewidths meaning- fully from our simulations if we isolate the optical andacoustical modes by the procedure outlined earlier in thissection. Of course, this cannot be done from experimentalFMR spectra. In Fig. 8 /H20849b/H20850, we show the variation of the linewidths of the two modes with /H9261. We see that /H9261is de- creased from the value of 2 toward unity, the optical modelinewidth indeed approaches that of the acoustical mode, butthen when /H9261comes close to 1, the linewidths diverge. The acoustic mode narrows, and the optical mode broadens, as we have seen for the calculations presented earlier where /H9261 was taken to be unity.IV. FINAL COMMENTS In this paper, through analysis of spectral densities calcu- lated at zero wave vector, we have extracted linewidths ofthe FMR modes of the structures we have explored. Theorigin of these linewidths, which are built into the theoretical methodology we have presented in the earlier paper, is in thespin pumping mechanism discussed elsewhere in the litera-ture. We have seen that we obtain an excellent account of thelinewidths reported in Ref. 6 for Fe ﬁlms on Au /H20849100 /H20850, and we also obtain a very good description of the quantum oscil- lations in linewidth reported in Ref. 14, for both the acous-tical and optical modes of ferromagnetic trilayers grown onthe Cu /H20849100 /H20850surface. These calculations employ the same theoretical methodol- ogy that we have developed and applied earlier to the analy-sis of the SPLEED data reported in Ref. 17. In the SPLEEDmeasurements, of course, it is large wave vector spin excita-tions that are studied. In Ref. 16, we obtained an excellentaccount of the dispersion of the single, broad asymmetricfeature seen in SPEELS, along with an excellent account ofboth its width and asymmetric line shape. Thus, the calcula-tions presented in the present paper show that our methodaccounts very nicely for the damping of spin motions inthese ultrathin ﬁlms throughout the two dimensional Bril-louin zone, from the zero wave vector excitations explored inthe FMR experiments to the large wave vectors probed bySPEELS. It is interesting to offer a physical viewpoint of the FMR linewidths calculated here that is somewhat different /H20849but complementary to /H20850that set forward in papers devoted spe- ciﬁcally to FMR linewidths. First, suppose we were to con-sider the FMR linewidth in an inﬁnitely extended, three di-mensional crystal described by a model such as we use here,in which electron energy bands result from interactions ofelectrons with the crystal potential, and ferromagnetism isdriven by Coulomb interactions between the electrons. TheFMR mode is the inﬁnite wavelength, or zero wave vectormode of the system. Since such a Hamiltonian is form in-variant under rigid rotations of the spins, the Goldstone theo-rem requires the linewidth of the zero wave vector mode tobe identically zero. This is a rigorous statement, independentof approximations used in any speciﬁc calculation to studyspin waves. Our mean ﬁeld description of the ground statecombined with the RPA description of the spin dynamics, isa conserving approximation and thus all of our conclusionsare compatible with the Goldstone theorem. At zero ﬁeld wewill then ﬁnd a width of identically zero, for the zero wavevector spin wave of the inﬁnitely extended crystal. In realmaterials, of course, there is a ﬁnite width to the FMR mode,described phenomenologically by the damping term in thewell known Landau Lifschitz Gilbert equation. The origin ofthe linewidth in real materials is the spin orbit interaction,since its introduction into the Hamiltonian produces a struc-ture no longer invariant under rigid body rotations of allspins in the system. Now when we place a thin ferromagnetic ﬁlm on a non magnetic substrate, as in the systems considered here, trans-lational symmetry normal to the surface is broken, and onlythe components of wave vector parallel to the surface remain FIG. 8. This ﬁgure gives information on the spectrum of a Co2Cu10Co2/Cu /H20849100 /H20850structure, where the frequency of the outer ﬁlm is /H9261/H90240and that of the inner ﬁlm is /H90240.I n /H20849a/H20850, for the case where /H9261is 1.2, we show with the solid line the spectral density function A/H20849/H9024/H20850deﬁned in the text. The dashed line is the spectral function A1/H20849/H9024/H20850and the dot dash line is A2/H20849/H9024/H20850.I n /H20849b/H20850we give the linewidth of the acoustical and optical modes of the trilayer as afunction of the parameter /H9261.FERROMAGNETIC RESONANCE LINEWIDTHS IN ¼ PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-9good quantum numbers. The lowest lying mode with Q/H6023/H20648=0 is the FMR mode. This is not a uniform mode of the system, byvirtue of the breakdown of translational symmetry normal to the surfaces; the amplitude of the spin motion in the non-magnetic substrate differs from that in the ferromagneticﬁlm, for instance. Hence, even in the absence of spin orbitcoupling, one realizes a ﬁnite linewidth for this mode. This isa way of viewing the “spin pumping” contribution to thelinewidth. Now in the limit that the thickness of the ferro-magnetic ﬁlm becomes inﬁnite, the FMR mode of the ﬁlmmust evolve into the uniform mode of the inﬁnite crystalwhose linewidth must vanish in the absence of spin orbitcoupling. Hence, the spin pumping contribution to the line-width falls to zero, with increasing ferromagnetic ﬁlm thick-ness.Other questions will be interesting to explore. Our method will allow us to explore the wave vector dependence of thelinewidth, for example. Studies of this issue and other ques-tions are underway presently. ACKNOWLEDGMENTS During the course of this investigation, we have enjoyed stimulating conversations with Professor B. Heinrich andwith Professor K. Baberschke. This research was supportedby the U. S. Department of Energy, through Grant No. DE-FG03-84ER-45083. A.T.C. and R.B.M. also received supportfrom the CNPq, Brazil. A.T.C. also acknowledges the use ofcomputational facilities of the Laboratory for ScientiﬁcComputation/UFLA. 1For instance, see the discussion of linewidths presented by B. Heinrich, of Ultrathin Magnetic Structures II , edited by B. Hei- nrich and J. A. C. Bland /H20849Springer Verlag, Heidelberg, 1994 /H20850,p . 195. 2R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 /H208491999 /H20850; J. Appl. Phys. 87, 5455 /H208492000 /H20850. 3J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spoddig, R. Meckenstock, J. Pelzl, Z. Frait, and D. L. Mills, Phys. Rev. B 68, 060102 /H20849R/H20850/H208492003 /H20850. 4A. Stebarski, M. B. Maple, A. Wrona, and A. Winiaska, Phys. Rev. B 63, 214416 /H208492001 /H20850. 5D. L. Mills and S. Rezende, Spin Dynamics in Conﬁned Magnetic Structures II /H20849Springer-Verlag, Heidelberg, 2002 /H20850,p .2 7 . 6The ﬁrst experimental study was reported by R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 /H208492001 /H20850. 7L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 8J. Slonczewski, J. Magn. Magn. Mater. 195, L261 /H208491999 /H20850. 9E. Simanek and B. Heinrich, Phys. Rev. B 67, 144418 /H208492003 /H20850. 10D. L. Mills, Phys. Rev. B 68, 014419 /H208492003 /H20850. 11E. Simanek, Phys. Rev. B 68, 224403 /H208492003 /H20850. 12Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 /H208492002 /H20850; Phys. Rev. B 66, 224403 /H208492002 /H20850. 13M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 /H208492005 /H20850. 14K. Lenz, T. Tolinski, J. Lindner, E. Kosubek, and K. Baberschke, Phys. Rev. B 69, 144422 /H208492004 /H20850. 15R. B. Muniz and D. L. Mills, Phys. Rev. B 66, 174417 /H208492002 /H20850;A . T. Costa, R. B. Muniz, and D. L. Mills, J. Phys.: Condens.Matter 15, S495 /H208492003 /H20850; Phys. Rev. B 68, 224435 /H208492003 /H20850; Phys. Rev. B 70, 54406 /H208492004 /H20850. 16A. T. Costa, R. B. Muniz, and D. L. Mills, Phys. Rev. B 70, 54406 /H208492004 /H20850. We direct the reader to Fig. 3 for a comparison between experiment and the theoretically calculated linewidthand lineshape of large wave vector spin waves, for an eight layer Co ﬁlm on Cu /H20849100 /H20850. 17R. Vollmer, M. Etzkorn, P. S. Anil Kumar, H. Ibach, and J. Kir- schner, Phys. Rev. Lett. 91, 147201 /H208492003 /H20850. 18Quite a number of authors use an adiabatic approach to calculate effective exchange couplings between nearby spins in ultrathinferromagnets, then generate spin wave spectra utilizing aHeisenberg Hamiltonian appropriate to localized spins. In thispicture, for a given wave vector in the surface Brillouin zone, one realizes Nspin wave modes, each with inﬁnite lifetime, with Nthe number of layers in the ﬁlm. The calculations reported in Refs. 15 and 16 show that the adiabatic approximation breaksdown qualitatively for ultrathin ferromagnetic ﬁlms, and the pic-ture of spin excitations just described is qualitatively incorrect.Instead, through most of the surface Brillouin zone, save for thenear vicinity of the point, our theory predicted that one has asingle, broad feature in the frequency spectrum of spin ﬂuctua-tions, which displays dispersion with wave vector similar to thatassociated with a single spin wave branch. This picture is con-ﬁrmed by the data reported in Ref. 17, and in Ref. 16 we dem-onstrate our approach provides a fully quantitative account ofthe data. An explicit comparison between adiabatic or Heisen-berg like calculations, and our full dynamical calculations seeFigs. 4 and 5 of Ref. 16. 19H. Tang, M. Plihal, and D. L. Mills, J. Magn. Magn. Mater. 187, 23/H208491998 /H20850. 20MPIC is a portable implementation of MPI; see http://www- unix.mcs.anl.gov/mpi/mich. 21MPI is a message passing interface standard; see http://www- unix.mcs.anl.gov/mpi. 22R. B. Muniz and D. L. Mills, Phys. Rev. B 68, 224414 /H208492003 /H20850. 23J. Mathon, M. Villeret, A. Umerski, R. B. Muniz, J. d’Albuquerque e Castro, and D. M. Edwards, Phys. Rev. B 56, 11797 /H208491997 /H20850.COSTA, MUNIZ, AND MILLS PHYSICAL REVIEW B 73, 054426 /H208492006 /H20850 054426-10

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