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PhysRevB.93.155301.pdf	PHYSICAL REVIEW B 93, 155301 (2016) Magneto-optical spectroscopy of single charge-tunable InAs/GaAs quantum dots emitting at telecom wavelengths Luca Sapienza,1,Rima Al-Khuzheyri,2Adetunmise Dada,2Andrew Grifﬁths,3Edmund Clarke,3and Brian D. Gerardot2 1Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom 2Institute of Photonics and Quantum Sciences, SUPA, Heriot-Watt University, Edinburgh, United Kingdom 3EPSRC National Centre for III-V Technologies, University of Shefﬁeld, United Kingdom (Received 20 January 2016; published 1 April 2016) We report on the optical properties of single InAs/GaAs quantum dots emitting near the telecommunication O band, probed via Coulomb blockade and nonresonant photoluminescence spectroscopy, in the presence ofexternal electric and magnetic ﬁelds. We extract the physical properties of the electron and hole wave functions,including the conﬁnement energies, interaction energies, wave-function lengths, and gfactors. For excitons, we measure the permanent dipole moment, polarizability, diamagnetic coefﬁcient, and Zeeman splitting. The carriersare determined to be in the strong conﬁnement regime. Large range electric ﬁeld tunability, up to 7 meV , is demon-strated for excitons. We observe a large reduction, up to one order of magnitude, in the diamagnetic coefﬁcientwhen rotating the magnetic ﬁeld from Faraday to V oigt geometry due to the unique dot morphology. The completespectroscopic characterization of the fundamental properties of long-wavelength dot-in-a-well structures providesinsight for the applicability of quantum technologies based on quantum dots emitting at telecom wavelengths. DOI: 10.1103/PhysRevB.93.155301 I. INTRODUCTION Single quantum dots grown by molecular beam epitaxy are one of the most promising sources of nonclassical light due totheir stable and sharp emission lines and easy integration on achip via the mature III-V semiconductor fabrication technol-ogy. In particular, InAs quantum dots emitting at wavelengthsaround 950 nm have proved to be pure sources of singleindistinguishable photons [ 1,2] and entangled photons [ 3,4], and a powerful platform for spin initialization and manipula-tion, and spin-photon and remote spin entanglement [ 5]. To encode information in single photons and transmit it over longdistances, sources of quantum light emitting at the so-calledtelecommunication wavelengths are most desirable. Advancesin the development of superconducting detectors operatingat cryogenic temperatures [ 6] allow detection efﬁciencies exceeding 90% [ 7], making single-photon experiments and technologies eminently feasible. The growing interest in theﬁeld of long-wavelength quantum dots is demonstrated byrecent achievements such as the demonstration of brightsources of indistinguishable photons [ 8], interference of photons emitted by dissimilar sources [ 9], entangled photon pair generation [ 10], and exciton ﬁne-structure splitting ma- nipulation [ 11] in the telecom wavelength band. However, the growth and fundamental characterization of quantum dots emitting at telecom wavelengths is lessmature compared to emitters at wavelengths <1μm. The longer emission wavelength can be achieved by growingquantum dots in a quantum well (the so-called dot-in-a-wellor DWELL structures [ 12]), a technique that partially relaxes l.sapienza@soton.ac.uk Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License . Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.the strain accumulated during the Stranski-Krastanow growth, resulting in larger quantum dot dimensions. The InGaAsquantum well provides local strain relief and also preservesthe quantum dot composition and height during growth byreducing the out-diffusion of In during capping of the dotlayer [ 13,14]. A larger physical conﬁning potential implies a reduced energy separation between the quantum dotconﬁned states and radiative electron-hole recombinationsfor InAs/GaAs quantum dots can reach wavelengths around1.3μm both at room temperature—for example, for quantum dot lasers [ 15]—and at low temperature for single quantum dot applications [ 16]. Besides the technological interest associated with lower transmission losses, telecom-wavelength quantumdots present interesting fundamental properties due to verydifferent conﬁnement of electron and hole wave functions andpotentially larger oscillator strengths compared to shorter-wavelength quantum dots. In order to translate the more devel-oped technology of 950-nm-band quantum dots towards longerwavelengths, the fundamental properties of the emitters needto be further understood. To this end, the application of externalelectric and magnetic ﬁelds as well as Coulomb blockade isa means to characterize the electron and hole wave functionsand the Coulomb interactions between carriers. Since quantumdots emitting around 1300 nm are physically larger and havea higher In composition than shorter-wavelength quantumdots, the different composition and morphology can result ina different wave function extension and electron-hole overlap,impacting their fundamental response to applied ﬁelds. In thisdirection, analysis of the emission properties of quantum dotsemitting at wavelengths >1.2μm in the presence of an external magnetic ﬁeld [ 17–19] and of 1300 nm quantum dots in the presence of external strain [ 11] have been reported. However, the full characterization of the fundamental properties ofquantum dots allowing direct comparison of emitters at 950nm and at telecom wavelengths is still incomplete. Here, we report on magneto-optical studies of the emission properties of single telecom-wavelength quantum dots grownwithin a charge-tunable structure. We extract the physical 2469-9950/2016/93(15)/155301(6) 155301-1 Published by the American Physical SocietyLUCA SAPIENZA et al. PHYSICAL REVIEW B 93, 155301 (2016) properties of the electron and hole wave functions, including the conﬁnement energies, interaction energies, wave-functionlengths, and gfactors. For excitons, we measure the permanent dipole moment, polarizability, diamagnetic coefﬁcient, andZeeman splitting. II. EXPERIMENTAL DETAILS The sample investigated was grown by molecular beam epitaxy. The DWELL layer was grown at 500◦C by initial deposition of 1 nm In 0.18Ga0.82As, followed by a deposition of nominally 1.8 monolayers (ML) of InAs to form the quantumdots, at a growth rate of 0.016 ML s −1. Sample rotation was stopped during growth of the quantum dot layer to providea variation in InAs coverage across the wafer, resulting in avariation in quantum dot density across the wafer. The quantumdot layer was subsequently capped by 6 nm In 0.18Ga0.82As and a further 4 nm GaAs at 500◦C, before the substrate temperature was raised to 580◦C for growth of the remaining structure. A cross-section transmission electron microscopy (TEM) imageof a DWELL layer grown under similar conditions is shownin Ref. [ 11]. Analysis of TEM images indicates the dots have a base width of 20–30 nm and a relatively large capped heightof around 8–10 nm, preserved due to reduced out-diffusionof In during capping by the InGaAs layer. A sketch of theenergy diagram of the ﬁeld effect structure with a single layerof DWELL quantum dots is shown in Fig. 1(a). By applying a voltage between the semitransparent NiCr Schottky gate onthe sample surface and the doped ( n +) GaAs layer (Ohmic contact), discrete charging of the dots with single electrons canbe achieved [ 20]. The sample is placed in a cryostat at ∼4K and, using a microscope in confocal geometry, a ﬁber-couplednonresonant (830-nm-wavelength) laser is used to excite theemitters. A zirconia super-solid immersion lens is positionedon the surface of the sample to reduce the excitation spot (andtherefore be able to excite single emitters in the relativelyhigh density sample) and increase the collection efﬁciency ofthe photoluminescence signal [ 21]. The emission from single quantum dots is then coupled to a single-mode ﬁber and sentinto a grating spectrometer equipped with an InGaAs arraydetector for spectral characterization. An external magneticﬁeld can then be applied to the sample either parallel ororthogonal to the growth axis. III. ELECTRIC FIELD DEPENDENCE OF SINGLE QUANTUM DOT CONFINED STATES Examples of photoluminescence spectra acquired as a function of applied voltage are shown in Fig. 1(b). Distinct emission lines are visible and can be attributed to singleexciton ( X 0) and negatively charged exciton ( X1−andX2−) recombinations. Discrete jumps in the emission wavelength,visible when varying the applied voltage, are the signature of the Coulomb blockade effect occurring when an extra electron is added to the quantum dot bound states [ 20]. We observe a partial coexistence of excitonic lines of differentcharged states of the same quantum dot around the transitionvoltages due to the comparable rates of electron tunnelinginto the quantum dot from the Fermi sea and excitonrecombination [see Fig. 1(b)][22]. By applying a perturbative Coulomb blockade model [ 23] to single quantum dots, we can0 100 200 300 400 500-20246 (a)Energy (eV) Position (nm) (b) Bext z=0T X4-X3- X0X1-X0X2- X1- X2- (c) Bext z=9T -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 Voltage (V)1278127612741272 1277127512731271Wavelength (nm) FIG. 1. (a) Schematic of the energy diagram of the charge-tunable structure under study, including a single layer of telecom wavelengthquantum dots (QDs) grown in a quantum well. E findicates the Fermi energy level. (b), (c) Photoluminescence spectra collected as a function of the applied gate voltage Vunder nonresonant excitation (λ=830 nm) at T=4 K. The external magnetic ﬁeld Bapplied along the quantum dot growth axis zi s0Ti n( b )a n d9Ti n( c ) . The emission lines corresponding to the different charge states of a single quantum are labeled accordingly ( X0=neutral exciton, X1−= negatively charged exciton, etc.). extract important physical parameters including the electron- electron (electron-hole) interaction energy, the electron (hole) conﬁnement energy, and the effective lengths of the electron(hole) wave functions. These results are shown in Fig. 2.W e ﬁnd that the Coulomb interaction energies are much smallerthan the conﬁnement energies and they can, therefore, betreated as perturbations. Thus, we consider the trapped carriersto be in the so-called strong conﬁnement regime. Compared to typical 950 nm quantum dots [ 24,25], we derive wave-function effective lengths and conﬁnement energies about a factor 2larger and carrier-carrier interaction energies about a factor2 smaller. This is compatible with a larger quantum dotphysical size. It is worth noting that engineering of larger wave-function lengths could lead to larger oscillator strengths for theexcitonic transitions, relevant for quantum electrodynamics experiments with single quantum dots. The larger conﬁnement energies have important consequences in the tunability of thetransition energies of the quantum dot conﬁned states. Whilethe transition energy of quantum dots emitting below 1 μm can be tuned by about 1 meV in typical ﬁeld-effect transistordevices [ 24], the telecom wavelength quantum dots under study show a tunability up to 7 meV . This can be explainedby the larger electron and hole conﬁnement energies which 155301-2MAGNETO-OPTICAL SPECTROSCOPY OF SINGLE . . . PHYSICAL REVIEW B 93, 155301 (2016) 9121518Eee(meV) 9121518 Eeh(meV) 180210240270300Ec(meV) 180210240270300 Ev(meV) 1165 11701275 1280 1285 1290 1295591317 Wavelength (nm)le(nm) 591317 lh(nm) FIG. 2. The Coulomb blockade model (see main text) is applied to extract the electron-electron and electron-hole interaction energies (Ess eeandEss eh, circles and triangles respectively), as well as the electron and hole conﬁnement energies ( ECandEV, squares and triangles respectively) and the electron and hole wavefunction extension ( le andlh, squares and triangles respectively). The wavelength on the x-axis corresponds to the emission wavelength for the X0line in the middle of the emission tuning range. enable larger electric ﬁelds to be applied before the charges tunnel out of the conﬁning potential. This larger wavelength tunability is important for potential applications such as the mutual tuning of the emission lines with respect to opticalcavities for quantum optics experiments, or cancellation of theﬁne-structure splitting to create sources of entangled photonpairs by applying an external electric ﬁeld [ 26]. Measurements of the ﬁne-structure splitting of quantum dots, from the samegrowth and within the same structure as the ones presented in this work, were reported in Ref. [ 11]. Further, such large conﬁnement energies yield carriers more decoupled from theelectron reservoirs and have potential for reduced impact ofphonon-induced dephasing. When we apply an external biasV, exciton transition energies ( E PL) experience a Stark shift, following the relation EPL=E0−pF+βF2, where the electric ﬁeld F=− (Vg−V0)/dis a function of the Schottky barrier height V0and the distance dbetween the back gate and sample surface, pis the permanent dipole moment, and β is the polarizability. Given the structure of the sample understudy [see Fig. 1(a)], we use V g=0.62 V and d=400 nm, and ﬁt the exciton lines with quadratic functions, as shownin Fig. 3(a), to extract the permanent dipole moment pand the polarizability β. We observe permanent dipole moments withp/e values (where eis the electron charge) ranging from FIG. 3. (a) The energies of different charged states from a single quantum dot as a function of the applied electric ﬁeld. Here the multiparticle Coulomb interaction energies have been subtracted. The solid black line is a parabolic ﬁt to the data. (b) Permanent dipolemoments p, measured from several single quantum dots, plotted as a function of polarizability β. The solid red line is a linear ﬁt with slope 3 .2±0.2n m/[meV/(kV/cm) 2] and the error bars are obtained from the errors in the ﬁts. −0.5t o−3.0 nm, values similar to those reported for quantum dots emitting around 950 nm [ 24], indicating that the electron and hole wave functions are centered within the quantumdot. The observed polarizabilities [ranging between −0.5 and −1.2μeV/(kV/cm) 2] are slightly smaller than the values reported for shorter wavelength quantum dots [ 24], which can be again explained by the stronger conﬁnement of the carriersin larger telecom wavelength quantum dots. The negative signof the polarizability implies that the hole is conﬁned nearthe base of the dot, while the electron wave function, givenits lighter effective mass, is delocalized over the quantumdot. The polarizability and the permanent dipole moment, as expected [ 24], are linearly related, as shown in Fig. 3(b). IV . INVESTIGATION OF THE QUANTUM DOT CONFINED STATES IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD To further investigate the properties of the bound-state wave functions of the telecom-wavelength quantum dots 155301-3LUCA SAPIENZA et al. PHYSICAL REVIEW B 93, 155301 (2016) under study, we apply an external magnetic ﬁeld Bin the Faraday ( Bparallel to the quantum dot growth direction) and in the V oigt ( Borthogonal to the quantum dot growth direction) conﬁgurations. Examples of photoluminescencespectra collected at a ﬁeld of 9 T in the Faraday geometry, whenvarying the electric ﬁeld applied to the charge-tunable structureare shown in Fig. 1(c). A clear splitting of each excitonic transition is visible; this will be analyzed and discussed inmore detail in the following sections. A. Diamagnetic coefﬁcients of excitons in Faraday and Voigt geometries We ﬁrst consider the Faraday geometry, where we apply magnetic ﬁelds up to 9 T parallel to the quantum dot growthaxis and collect photoluminescence spectra as a function ofthe voltage applied to the charge-tunable structure. Examplesof the spectra collected for the negatively charged exciton ares h o w ni nF i g . 4(a): the energy of the emission lines experiences the so-called diamagnetic shift in the presence of an externalmagnetic ﬁeld, following the expression E=αB 2, where α is the diamagnetic coefﬁcient. The values of αobtained from individual quantum dots are shown in Table I. The diamagnetic coefﬁcient is related to the exciton binding and conﬁnementenergies, and therefore to the microscopic properties of eachspeciﬁc quantum dot. For the telecom-wavelength quantumdots under study, we generally observe a modest (about 10%)increase of αbetween the neutral exciton and the charged exciton (see Table I). In the weak conﬁnement regime, due to electron-electron interaction the addition of a second electronto a neutral exciton can reduce αby up to a factor 2 [ 27]. The fact that we do not observe a reduction, but rather a modestincrease, in αfurther validates that the carriers are in the strong conﬁnement regime in the quantum dots under study,as reported also in Ref. [ 19]. We observe signiﬁcant differences between the electron and hole wave function extents, asexpected due to the difference in conﬁnement potentials andeffective masses. By applying the external magnetic ﬁeld orthogonal to the quantum dot growth axis (V oigt conﬁguration), the rotationalsymmetry of the wave functions is broken. This results in themixing of the originally bright and dark neutral exciton states,with the latter becoming visible in the photoluminescencespectra [ 28,29]. If we consider quantum dots in the 950 nm emission range, the difference between the diamagneticcoefﬁcients measured in Faraday and V oigt conﬁgurationreaches values up to about a factor 3 [ 30]. Interestingly, for the telecom wavelength quantum dots under study, the diamagnetic coefﬁcient is one order of magnitude smaller inthe V oigt conﬁguration compared to the results obtained inthe Faraday conﬁguration. As the diamagnetic coefﬁcient isa measure of the effect of conﬁnement, this striking resultconﬁrms the unique morphology of the DWELL quantum dotscompared to typical self-assembled quantum dots emittingnear 950 nm. As αis an order of magnitude larger for applied ﬁelds in plane versus out of plane, we conclude thatthe conﬁnement in the growth direction is less for DWELLquantum dots as expected.1273 1274 1275 1276(a) 9T 8T7T6T 5T 4T3T2T 1T 0TNormalized PL intensity Wavelength (nm)0.9735 0.973 0.9725 0.972Energy (eV) 02468 1 00.97240.97280.97320.9736(b)Energy (eV) Magnetic field (T) 02468 1 00.95960.95980.96000.9602 E4E3E2E1(c) H VEnergy (eV) Magnetic Field (T) FIG. 4. (a) Faraday conﬁguration: Normalized photolumines- cence (PL) spectra (shifted for clarity) of a negatively charged exciton state from a single quantum dot, collected under nonresonantexcitation at a temperature of 4 K, for applied magnetic ﬁelds ranging from 0 to 9 T (with 0.5 T increments). The solid lines are Lorentzian ﬁts to the data. (b) Energy position of the peaks, as found from the Lorentzian ﬁts of panel (a), plotted as a function of the applied magnetic ﬁeld. The solid lines are quadratic ﬁts. (c)V oigt conﬁguration: Energy position of the negatively charged exciton peaks, as found from the Lorentzian ﬁts of the spectra collected for two orthogonal polarizations [horizontal (H) and vertical (V)], plottedas a function of the applied magnetic ﬁeld. The error bars (often within the symbol size) in panels (b) and (c) are obtained from the Lorentzian ﬁts of the photoluminescence peaks. 155301-4MAGNETO-OPTICAL SPECTROSCOPY OF SINGLE . . . PHYSICAL REVIEW B 93, 155301 (2016) TABLE I. Diamagnetic coefﬁcients α,gfactors and electron and hole gfactors ( geandgh, respectively) extracted from single quantum dot photoluminescence spectra collected under external magnetic ﬁeld applied in the Faraday and V oigt conﬁgurations. λX0corresponds to the emission wavelength of the excitonic line in the middle of the emission tuning range. Faraday conﬁguration λX0(nm) Excitonic state \|g\| α(μeV/T2) 1274 X00.73±0.02 13 .48±0.09 X1−0.63±0.01 14 .83±0.07 1281 X00.77±0.01 17 .01±0.20 X1−0.89±0.01 18 .20±0.31 1282 X00.36±0.02 14 .25±0.09 X1−0.98±0.01 14 .72±0.04 V oigt conﬁguration λX0(nm) Excitonic state gh ge α(μeV/T2) 1290 X0−0.17±0.01 −0.77±0.01 2 .15±0.12 X1−−0.49±0.06 −0.91±0.07 5 .30±0.47 1292 X1−−0.24±0.01 −1.04±0.01 2 .50±0.05 1300 X0−0.21±0.02 −0.87±0.04 0 .93±0.30 X1−−0.36±0.02 −0.80±0.03 1 .83±0.26 1310 X1−−0.26±0.04 −0.89±0.04 1 .82±0.19 1317 X1−−0.19±0.05 −0.72±0.03 0 .63±0.14 B. Zeeman splitting and gfactors Examples of the spectra collected for the negatively charged exciton in the Faraday conﬁguration are shown inFig.4(a): the emission line in the presence of the magnetic ﬁeld is split by the so-called Zeeman splitting [see Fig. 4(b)] with a magnitude /Delta1E given by /Delta1E=gμ BB, where μBis the Bohr magneton and gis the Land ´e factor. The Zeeman splittings that we measure range between about 10 and 40 μeV/T, considerably smaller than for 950 nm quantum dots (thatwere reported to be 120 ±30μeV/T[27]). This is consistent with theoretical calculations showing that an increase in thequantum dot size implies a reduction of the gfactor [ 31]. By applying the external magnetic ﬁeld orthogonal to the quantum dot growth axis (V oigt conﬁguration), the rotational symmetry of the wave functions is broken. As shown in Fig. 4(c), the single quantum dot negatively charged exciton line splits into four separate contributions ( E 1,E2,E3,E4), as a result of the Zeeman splitting of the bright and dark states.Thes-shell electron and hole gfactors can be determined from the exciton energies: by ﬁtting the energies of each transition, one can determine g eμBB=E1−E3=E2−E4 andghμBB=E1−E2=E3−E4[32]. The values that we extract from these measurements are plotted in Table I. We attribute the variations in the values of the gfactors for different quantum dots to be due to the dependence of gon the quantum dot shape [ 33,34]. The magnitudes of the gfactors measured for the negatively charged states are consistently higher than those measured forthe neutral exciton, in accordance with previous reports [ 32]. The electron gfactors are two to four times larger than hole g factors since the electron wave functions are less conﬁned thanthe hole ones and are therefore more sensitive to the externalmagnetic ﬁeld. From the measurements shown in Fig. 4(c), one can also see that the four transitions are linearly polarized and,as expected, two of the four transitions disappear when polar-ization is resolved into horizontal and vertical components.V . CONCLUSIONS In summary, we have fully characterized the exciton and carrier properties of single quantum dots emitting at telecomwavelengths (near 1.3 μm) under applied electric and magnetic ﬁelds. Via the Coulomb-blockade model, we extract theelectron and hole wave-function lengths as well as multi-particle Coulomb interaction and conﬁnement energies. Theresults are consistent with a strong-conﬁnement picture.The conﬁnement energies of these quantum dots are foundto be a factor of 2 larger than 950 nm quantum dots. Due tothe larger conﬁnement energies, the excitonic transitions canbe tuned over a larger range than 950-nm-band quantum dotsin comparable devices. Additionally, the deeper conﬁnementholds promise for better decoupled spins from Fermi orphonon reservoirs. With applied external magnetic ﬁelds weextract the Zeeman and diamagnetic coefﬁcients as well aselectron and hole gfactors. Compared to 950 nm quantum dots, the Zeeman splittings are signiﬁcantly smaller and thediamagnetic coefﬁcient shows a drastic shift when changingfrom the Faraday to the V oigt conﬁguration due to theunique DWELL morphology. These results give insights intothe fundamental properties of telecom wavelength quantumdots. 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PhysRevB.85.195432.pdf	PHYSICAL REVIEW B 85, 195432 (2012) Magnetotransport through graphene nanoribbons at high magnetic ﬁelds S. Minke,1,S. H. Jhang,1J. Wurm,2Y . Skourski,3J. Wosnitza,3C. Strunk,1D. Weiss,1K. Richter,2and J. Eroms1,† 1Institute of Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany 2Institute of Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 3Dresden High Magnetic Field Laboratory, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany (Received 17 November 2011; revised manuscript received 15 February 2012; published 15 May 2012) We have investigated the magnetoresistance of lithographically prepared single-layer graphene nanoribbons in pulsed, perpendicular magnetic ﬁelds up to 60 T and performed corresponding transport simulations using atight-binding model and several types of disorder. In experiment, at high carrier densities we observe Shubnikov-deHaas oscillations and the quantum Hall effect, while at low densities the oscillations disappear and an initiallynegative magnetoresistance becomes strongly positive at high magnetic ﬁelds. The strong resistance increase atvery high ﬁelds and low-carrier densities is tentatively ascribed to a ﬁeld-induced insulating state in the bulkgraphene leads. Comparing numerical results and experiment, we demonstrate that at least edge disorder andbulk short-range impurities are important in our samples. DOI: 10.1103/PhysRevB.85.195432 PACS number(s): 72 .80.Vp, 73 .22.Pr, 73.43.Qt I. INTRODUCTION For the application of graphene in nanoelectronics one has to understand the behavior of graphene nanostructures,in particular graphene nanoribbons (GNRs). They weretheoretically predicted to show either metallic or insulatingbehavior around the charge neutrality point, depending ontheir crystallographic orientation. In experiment, however,GNRs always exhibit an insulating state close to the chargeneutrality point (CNP), 1which is dominated by disorder rather than a conﬁnement-induced gap in the spectrum.2,3A clear proof of conductance quantization only appeared very recentlyin ultraclean suspended nanoribbons. 4Furthermore, in clean zigzag edges, a magnetic state has been predicted,5,6but so far it has remained elusive in transport experiments. Atpresent, therefore, the behavior of GNRs is mainly governedby extrinsic defects rather than their intrinsic properties, andinformation on the nature of those defects is highly desired. In previous experiments, large disorder was attributed to cause strong localization effects which inﬂuence themagnetoconductance. 7Poumirol et al. report a large positive magnetoconductance and explain this by simulations whichtake into account different types of disorder. They afﬁrm thequalitative behavior, but the computed conductance remainslarger than the experimental ones. Also, an unambiguousseparation of bulk and edge disorder was not possible. 8 Here we present magnetotransport measurements on GNRs inmagnetic ﬁelds of up to 60 T and corresponding tight-bindingsimulations with several types of realistic bulk and edgedisorder. By considering the magnetoconductance close to theDirac point and at high densities, we observe characteristicsignatures of bulk and edge disorder and can disentangle theircontributions to transport in GNRs. II. EXPERIMENTAL DETAILS Single-layer graphene is deposited on a highly doped silicon wafer with a 300 nm thick SiO 2layer by conven- tional exfoliation. The graphene nanoribbons were deﬁned byelectron-beam lithography and oxygen plasma reactive ionetching. For the transport measurements, palladium contactswere attached to the GNRs. A scanning electron micrograph of the sample discussed here is shown in Fig. 1(a). The dc magnetotransport measurements with 10 mV dc bias weredone in pulsed perpendicular magnetic ﬁelds at temperaturesbetween 1.8 and 125 K. Typical pulse durations were rangingfrom 100 to 300 ms. During the pulse the current through theGNR was converted to a voltage signal by a current-to-voltageampliﬁer and recorded by a high-speed oscilloscope and datarecorder. In total two single-layer nanoribbons have beenmeasured which show similar behavior. Here we focus on datafrom one device. Figure 1(b) shows the resistance Rof the nanoribbon as a function of backgate voltage V bgatT=25 K and zero magnetic ﬁeld. The sharp peak at Vbg=VCNP= −4.4 V indicates the charge neutrality point. After patterning, the hole mobility μof the ribbons is about 590 cm2/Vsa t Vbg=− 15 V .9Figure 1(c) shows a magnetoresistance curve taken at high carrier density.10A quantum Hall plateau at ν=611and Shubnikov-de Haas oscillations for ν=10 and 14 are observed. Signatures of Hall states were already foundin previous experiment. 12From the zero-ﬁeld mobility and the condition μB/greatermuch1 we would not expect to observe quantum Hall features at ν=14, at 13 T. This is already an indication that the high ﬁeld changes the impact of disorder on transportin our sample. III. DENSITY AND TEMPERATURE DEPENDENCE Let us now consider the density and temperature depen- dence of the magnetoresistance in more detail. First, we willfocus on the transport properties at gate voltages close tothe CNP. For all temperatures we tuned the backgate voltagesuch that the samples remained as close as possible to theCNP. In Fig. 2(a) the magnetoresistance is plotted for various temperatures ranging from 1.8 to 125 K. For all temperaturesa resistance decrease is observed for ﬁelds up to about 20T, so that the ribbon crosses over from a highly resistivestate to a metallic regime. Subsequently, it is followed bya prominent resistance increase. The divergent form of thelatter increase suggests that the nanoribbon approaches aﬁeld-induced insulating state. 195432-1 1098-0121/2012/85(19)/195432(4) ©2012 American Physical SocietyS. MINKE et al. PHYSICAL REVIEW B 85, 195432 (2012) 0 1 02 03 04 05 06 020406080 R( kΩ) B(T )T= 25K-20 -10 0 10 200.10.20.30.40.5R( MΩ) Vbg(V)(b) (a) 1mµ v=6 10 14(c)T= 25KPd contact GNRetched lines graphene leadgraphene lead FIG. 1. (Color online) (a) Scanning electron microscope image of a typical sample. The length of the GNRs is 1 μm, the width is 70 nm. In the upper part of the image a palladium contact is visible. (b) Two-terminal resistance as a function of VbgatT=25 K and zero magnetic ﬁeld. (c) Magnetoresistance trace at Vbg=− 20 V , showing quantum Hall features at ν=6,10, and 14. In order to better comprehend the observed behavior, we studied the magnetoresistance for different gate voltagesranging from −4.8 to −13.7 V at T=25 K. As one can see in Fig. 2(b), the observed divergence of the resistance at very high ﬁelds only appears for gate voltages close to theCNP ( \|V bg−VCNP\|<9 V). At higher densities [see Fig. 2(c)] we observe weak localization at ﬁelds up to 1 T, a fairlyconstant resistance up to about 20 T, and then pronouncedresistance oscillations. These oscillations can be identiﬁed asShubnikov-de Haas (SdH) oscillations, which can be assignedto Hall-plateau values of single-layer graphene ( ν=2 and 6). The capacitive coupling C gof the nanoribbon to the backgate, which strongly depends on the ribbon dimensions, 0.010.1110 125K25K 64K14K7K3K1.8KR( MΩ) 0 1 02 03 04 05 06 00.00.10.20.3 -6,2VG( 2 e2/h) B( T )-15,6V0.010.11 -13,7V-12V-9,2V -8,0V-6,2V-4,8V (CNP)R( MΩ) 0 1 02 03 04 05 0204060 -12V-19V -15,6V -13,7VR( kΩ) B( T )(a) (b) ν=6(c) (d)T= 25K T= 25K T= 25Kat CNP ν=2 FIG. 2. (Color) (a) Magnetoresistance of the GNR for various temperatures at the charge neutrality point. (b) Magnetoresistance for different gate voltages close to the CNP and (c) further away from the CNP at T=25 K. The arrows and the numbers indicate the corresponding ﬁlling factors νof the quantum Hall state ν=2a n d 6. (d) Conductance as a function of magnetic ﬁeld for Vbg=− 15.6 and−6.2 V .was calculated using a ﬁnite-element model, yielding Cg= 576 aF /μm2for a 70 nm wide GNR. Plotting the fan diagram of the minima of the SdH oscillations gives acoupling C gof 560 aF /μm2, which matches the calculated value well. Therefore, the carrier density is estimated as n≈ 3.5×1015m−2×(Vbg−VCNP) and the Fermi-energy scales asEF≈69 meV ×/radicalbig\|Vbg−VCNP\|, where VbgandVCNPare given in volts. For easier comparison to the numerical calculations, Fig. 2(d) shows the conductance Gas a function of magnetic ﬁeld for two different carrier densities representative for thelow- and high-carrier-density regime. The high-carrier-densityconductance ( V bg=− 15.6 V) shows the oscillating behavior as described before, the low-density trace ( Vbg=− 6.2V ) exhibits ﬁrst a conductance increase followed by a conductancedecrease. In the following we discuss the observed behaviorwith the help of numerical simulations. IV . NUMERICAL TRANSPORT SIMULATIONS The experimental data in Fig. 2will give us important insight into the nature of the defects relevant in our GNRs.Speciﬁcally, in this section we will focus on the visibility ofthe SdH oscillations, the positive magnetoconductance at low-carrier densities and ﬁelds up to about 20 T, and the rather highzero-ﬁeld resistance at both low- and high-carrier densities.To this end, we have performed numerical magnetotransportsimulations of (armchair) graphene nanoribbons with realisticsizes ( L=320 nm, W∼25 nm). Since Ohmic scaling is not applicable at those length scales 13we do not expect a full quantitative match between theory and experiment.However, the qualitative behavior will be well reproducedby the simulations since the system size is of the sameorder as the experimental samples. We used the well-knowngraphene tight-binding Hamiltonian in nearest neighbor (n.n.)approximation, H=/summationdisplay i,jn.n.tijc† icj, (1) where for ﬁnite magnetic ﬁeld the corresponding hopping inte- gral is given by tij=−texp[ie/¯h/integraltextxj xidsA(x)], with constant t≈2.7 eV and the vector potential A(x). The conductance was then computed using an adaptive recursive Green-functionmethod, capable of treating arbitrarily shaped systems. 14 To appropriately describe the experimental situation, we considered different types of disorder. Since the fabricationprocess certainly leads to disordered edges, we also took thisinto account in the numerical simulations. To this end, we cut“chunks” of about 4 nm out of the graphene lattice at randompositions close to the edge, which simulates the large-scaleedge roughness that occurs due to e-beam resist roughnessand the random nature of reactive ion etching. Additionally, weaccounted for edge roughness on a smaller scale of a few latticeconstants using a model introduced in Ref. 15: About 10% of the edge atoms are randomly removed and subsequentlydangling bonds are additionally removed. This procedure wasrepeated 5 times to yield an edge roughness of a few latticeconstants. The numerical results, however, showed that bothtypes of disorder yield similar results. In the following, in 195432-2MAGNETOTRANSPORT THROUGH GRAPHENE NANORIBBONS ... PHYSICAL REVIEW B 85, 195432 (2012) 0.00.51.01.52.02.53.0 EF= 92meVEF= 226meVG( 2 e2/h) 0.00.51.01.52.02.53.0 92meV226meVG( 2 e2/h) 02 04 06 08 00.00.20.40.60.81.0 92meV 226meV B (T) G( 2e 2/h ) 0 2 04 06 08 0 1 0 00.00.20.40.60.81.0 226meV92meVG( 2 e2/h) B (T)(b) (d) (c)(a) FIG. 3. (Color online) Magnetoconductance of armchair GNRs (L=320 nm, W∼25 nm) calculated numerically, using tight- binding simulations14and different disorder models. (a) Edge disorder (cf. text, inset: a close up of the ribbon edge with disorder).(b) Long-range Gaussian disorder (puddles, cf. text). (c) Short-range impurities. We used Gaussian disorder with a decay length of ∼0.44 nm. The height of the individual Gaussian potentials is randomly distributed within the interval [ −δ,δ] with δ=0.1tand the impurity density is p=15%. (d) Edge disorder and short-range Gaussian disorder. Here δ=0.09tandp=8%. the case of edge disorder, both mechanisms will always be included. In addition to the edge disorder, we studied two types of bulk potential disorder. On the one hand, we modeledso-called electron-hole puddles, that is, long-range potentialﬂuctuations due to charged impurities trapped beneath thegraphene ribbon in the silicon-oxide substrate. Second, wealso consider shorter-ranged impurity potentials, that can arisedue to adsorbates, defects, or charged impurities. In bothcases, we add Gaussian on-site potentials to the tight-bindingHamiltonian (1). For the puddles we use Gaussians with a decay length of ∼8.5 nm and a total height of ∼80 meV , which is comparable to the experimentally determined values. 16The impurities were modeled by Gaussians with a decay length of∼0.44 nm. 17 In Fig. 3we present our numerical results for magneto- transport through disordered nanoribbons at relatively high(E F≈226 meV) and lower ( EF≈92 meV) carrier densities, corresponding to the Fermi energies of the experimental datain Fig. 2(d). First, we consider ribbons with edge disorder only [Fig. 3(a)]. We ﬁnd that while the zero-ﬁeld conductance for low densities is comparable to the experiment, this is notthe case for the high-density result. Upon increasing the ﬁeld,the wave functions become more localized close to the edges.Without bulk disorder, backscattering is strongly suppressed,so that calculations yield nearly perfect quantum Hall plateausfor all densities already at moderate ﬁelds, in contrast to theexperimental ﬁndings. This means that edge disorder alonecannot explain the experiment. Considering only long-rangeGaussian disorder [Fig. 3(b)], we ﬁnd that the puddles are rather effective scatterers at low density, while they affectGonly little at high densities. Simulations where only the short-range impurities are taken into account [Fig. 3(c)], showthat indeed for strong enough scattering potentials, the zero- ﬁeld conductance can be very close to the experimental data.However, such strong bulk disorder leads to backscatteringeven for very high magnetic ﬁeld, so that at high-carrierdensity no SdH oscillations can be observed. This implies thatindeed a combination of bulk and edge disorder is necessary to describe the high-ﬁeld experiments. In Fig. 3(d) we show the results for ribbons with disordered edges and short-rangebulk disorder. In this case, the experimental ﬁndings for lowand moderate ﬁeld are reproduced semiquantitatively. For lowdensity we ﬁnd a strong increase of Gdue to the formation of edge channels, while clear SdH oscillations are obtainedat higher densities. The zero-ﬁeld conductance ﬁts well withthe experiment. In contrast, in simulations that additionallyinclude the long-range puddles, the difference in the zero-ﬁeldconductance for high and low densities is much too high, thuswe conclude that puddles are not the dominant scatterers in oursamples. We note that beyond our disorder model interactioneffects may further inﬂuence the measured conductance. V . HIGH FIELD INSULATING STATE AT LOW DENSITIES We now turn our attention to the sample properties at high magnetic ﬁelds near the CNP. As shown in Fig. 2(a), the resistance at low temperatures initially decreases with B and then diverges steeply by several orders of magnitude forB> 20 T. While the initial negative magnetoresistance at low densities is explained in the previous section by the formationof edge channels related to the zero-energy Landau level (LL)in graphene, a crossover to a divergent resistance for B> 20 T requires another transport mechanism. The zero-energy statein bulk graphene has been investigated by several researchgroups, and a strong increase in Rat the CNP and intense magnetic ﬁelds has been observed, resulting in a B-dependent LL splitting 18,19and eventually a strongly insulating state,20,21 the exact nature of which is still under debate.22 Adopting a simple model involving the opening of a ﬁeld- dependent spin gap,18we can ﬁt the temperature dependence ofRforT/greaterorequalslant14 K in an Arrhenius plot for distinct magnetic- ﬁeld values (inset of Fig. 4). In Fig. 4energy gaps /Delta1are 0 10 20 30 40 50 600102030405060 Data fit, following Ref. [23] linear fit(K) B( T )0.02 0.04 0.06 0.080.11 1/T (1/K)B( T ) 55 45 35R( M )50 K 25 K 12.5 K FIG. 4. (Color online) Energy gaps /Delta1extracted from the slope of the Arrhenius plot for T/greaterorequalslant14 K (inset). The (red) dotted line ﬁts the Zeeman splitting /Delta1=(gμBB)/kB−8.9 K, with the Bohr magneton μB, the Boltzmann constant kB, and a gyromagnetic factor of g= 1.73. The (blue) continuous line is a ﬁt following Ref. 23(cf. text). 195432-3S. MINKE et al. PHYSICAL REVIEW B 85, 195432 (2012) extracted from linear ﬁts to the Arrhenius plot. The gap /Delta1 shows a linear dependence on B(Fig. 4), consistent with spin splitting of the zero-energy LL, with the gyromagneticfactor g=1.73. However, another origin of the gap can also be considered. Following for example Ref. 23, we can ﬁt /Delta1∝C·(B−B c)0.5withBc≈29 T and C≈11, see Fig. 4, suggesting a chiral symmetry breaking transition. Comparingthese different models we conclude that both mechanisms arecompatible with our data, but the exact nature of the gapcannot be determined experimentally. For lower temperatures(T/lessorequalslant7 K), however, the resistance diverges strongly with B, and a simple activated behavior can no longer explain ourdata. This divergent behavior of Rin our GNRs resembles a ﬁeld-induced transition to a strongly insulating state reportedin bulk graphene at low T. 20,21In cleaner samples the transition to the insulating state occurred at signiﬁcantly lower ﬁelds. Given the sample geometry displayed in Fig. 1(a), we note that (bulk) graphene leads are attached to the GNR. Sinceour GNRs, after patterning, have lower mobility than the bulkgraphene leads, the ﬁeld required for the B-induced insulating state is also expected to be higher. Therefore, the observeddivergent Rat very high Band low densities is tentatively attributed to the leads: when we apply high Bﬁelds the leads become insulating and mask the electron transport in the GNR.VI. CONCLUSIONS In conclusion, we have performed transport experiments in graphene nanoribbons in pulsed high magnetic ﬁelds andcorresponding transport simulations, based on a tight-bindingmodel. This allows us to separate the contributions of differentdisorder types to magnetotransport. At least a combinationof edge disorder and short-range bulk impurities is neededto reproduce the experimental results semiquantitatively.The short-range bulk disorder is responsible for the partialsuppression of the quantum Hall effect, while the edgedisorder, together with the bulk disorder, provides sufﬁcientbackscattering to explain the observed high resistance atzero ﬁeld for all carrier densities. Additionally, we observea magnetic-ﬁeld-induced insulating state at very low den-sities, which presumably originates from the bulk grapheneleads. ACKNOWLEDGMENTS We would like to thank B. Raquet for helpful discussions. This research was supported by the Deutsche Forschungsge-meinschaft within GRK 1570 and by EuroMagNET under theEU Contract No. 228043. n´ee S. Schmidmeier. †jonathan.eroms@physik.uni-regensburg.de 1M. Y . Han, B. ¨Ozyilmaz, Y . Zhang, and P. Kim, Phys. Rev. Lett. 98, 206805 (2007). 2C. Stampfer, J. G ¨uttinger, S. Hellm ¨uller, F. Molitor, K. Ensslin, and T. Ihn, P h y s .R e v .L e t t . 102, 056403 (2009). 3P. Gallagher, K. Todd, and D. Goldhaber-Gordon, P h y s .R e v .B 81, 115409 (2010). 4N. Tombros, A. Veligura, J. Junesch, M. H. D. Guimar ˜aes, I. J. Vera-Marun, H. T. Jonkman, and B. J. van Wees, Nat. Phys. 7, 697 (2011). 5M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 (1996). 6Y . W. Son, M. L. Cohen, and S. G. Louie, Nature (London) 444, 347 (2006). 7J. B. Oostinga, B. Sacep ´e, M. F. Craciun, and A. F. Morpurgo, Phys. Rev. B 81, 193408 (2010). 8J.-M. Poumirol, A. Cresti, S. Roche, W. Escofﬁer, M. Goiran, X. Wang, X. Li, H. Dai, and B. Raquet, Phys. Rev. B 82, 041413 (2010). 9This value does not change signiﬁcantly if a contact resistance ofup to 4 k /Omega1is taken into account. Our palladium contacts usually have a contact resistance of 1 k /Omega1or less. 10Compared to Figs. 1(b) and2, these data were taken after thermal cycling where the CNP had shifted by about 1 V , but the mobilityremained unchanged. 11Here the resistance value exceeds the expected value of 4.3 k /Omega1 since it contains a series contribution of the Pd contacts andthe bulk graphene leads, which are also in the quantum Hall regime. 12R. Ribeiro, J.-M. Poumirol, A. Cresti, W. Escofﬁer, M. Goiran,J.-M. Broto, S. Roche, and B. Raquet, Phys. Rev. Lett. 107, 086601 (2011). 13G. Y . Xu, C. M. Torres, J. S. Tang, J. W. Bai, E. B. Song, Y . Huang,X. F. Duan, Y . G. Zhang, and K. L. Wang, Nano Lett. 11, 1082 (2011). 14M. Wimmer and K. Richter, J. Comput. Phys. 228, 8548 (2009). 15E. R. Mucciolo, A. H. Castro Neto, and C. H. Lewenkopf, Phys. Rev. B 79, 075407 (2009). 16J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacoby, Nat. Phys. 4, 144 (2008). 17A. Castellanos-Gomez, R. H. Smit, N. Agra ¨ıt, and G. Rubio- Bollinger, Carbon 50, 932 (2012). 18A. J. M. Giesbers, L. A. Ponomarenko, K. S. Novoselov, A. K. Geim, M. I. Katsnelson, J. C. Maan, and U. Zeitler, Phys. Rev. B 80, 201403 (2009). 19L. Zhang, Y . Zhang, M. Khodas, T. Valla, and I. A. Zaliznyak, Phys. Rev. Lett. 105, 046804 (2010). 20J. G. Checkelsky, L. Li, and N. P. Ong, Phys. Rev. Lett. 100, 206801 (2008). 21J. G. Checkelsky, L. Li, and N. P. Ong, P h y s .R e v .B 79, 115434 (2009). 22S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011). 23D. V . Khveshchenko, P h y s .R e v .L e t t . 87, 206401 (2001). 195432-4
PhysRevB.91.205115.pdf	PHYSICAL REVIEW B 91, 205115 (2015) Signatures of nematic quantum critical ﬂuctuations in the Raman spectra of lightly doped cuprates S. Caprara,1,2M. Colonna,1C. Di Castro,1,2R. Hackl,3B. Muschler,3,L. Tassini,3,†and M. Grilli1,2 1Dipartimento di Fisica, Universit `a di Roma Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy 2Istituto dei Sistemi Complessi CNR and CNISM Unit `a di Roma Sapienza 3Walther Meissner Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany (Received 16 January 2015; revised manuscript received 25 March 2015; published 18 May 2015) We consider the lightly doped cuprates Y 0.97Ca0.03BaCuO 6.05and La 2−xSrxCuO 4(withx=0.02, 0.04), where the presence of a ﬂuctuating nematic state has often been proposed as a precursor of the stripe or, more genericallycharge density wave phase, which sets in at higher doping. We phenomenologically assume quantum criticallongitudinal and transverse nematic, and charge-ordering ﬂuctuations, and investigate their effects in the Ramanspectra. We ﬁnd that the longitudinal nematic ﬂuctuations peaked at zero transferred momentum account wellfor the anomalous Raman absorption observed in these systems in the B 2gchannel, while the absence of such an effect in the B1gchannel may be due to the overall suppression of Raman response at low frequencies, associated with the pseudogap. While in Y 0.97Ca0.03BaCuO 6.05the low-frequency line shape is fully accounted for by longitudinal nematic collective modes alone, in La 2−xSrxCuO 4, also charge-ordering modes with ﬁnite characteristic wave vector are needed to reproduce the shoulders observed in the Raman response. This differentinvolvement of the nearly critical modes in the two materials suggests a different evolution of the nematic stateat very low doping into the nearly charge-ordered state at higher doping. DOI: 10.1103/PhysRevB.91.205115 PACS number(s): 74 .72.−h,74.25.nd,75.25.Dk,74.40.Kb I. INTRODUCTION Growing experimental and theoretical evidence indicates that (stripelike) charge ordering (CO) [ 1–3], possibly re- lated to a hidden charge-density-wave quantum critical point near optimal doping [ 4–8], plays a role in determining the unconventional properties of superconducting cuprates. Charge ordered textures were assessed by neutron scatteringexperiments in La cuprates, codoped with Nd [ 9–11], Ba [ 12], or Eu [ 13], and conﬁrmed also by soft resonant x-ray scattering [ 14,15]. The occurrence of stripelike charge- and spin-density waves in other cuprates is supported by the similarities of the noncodoped and codoped La cuprates inthe spin channel, e.g., the doping dependence of the low- energy incommensurability [ 16], and the high-energy magnon spectra in La 2−xBaxCuO 4[17], La 2−xSrxCuO 4(LSCO) [ 18], and YBaCuO 6+p[19,20]. These features are well described in terms of striped ground states [ 21–23]. CO in cuprates, possibly with ﬂuctuating character, was also conﬁrmed by EXAFS [ 24], NMR experiments [ 25,26], scanning tunnel- ing spectroscopy [ 27–29], and resonant x-ray measurements [30–33]. A recent theoretical analysis of Raman spectra in LSCO [ 34] showed that nearly critical spin and charge ﬂuctuations coexist at intermediate and high doping. This coexistence also accounts [ 35] for the speciﬁc momentum, en- ergy and doping dependence of the single-particle anomalies, the so-called kinks and waterfalls, observed in photoemission spectra [ 36]. The above facts, support the occurrence and relevance of (ﬂuctuating) stripes in cuprates and raise the question abouttheir precursors at very low doping [ 37]. The experimental Present address: Zoller & Fr ¨ohlich GmbH, Simoniusstrasse 22, 88239 Wangen im Allg ¨au, Germany. †Present address: MBDA, Hagenauer Forst 27, 86529 Schroben- hausen, Germany.evidence of rotational symmetry breaking [ 20,38–41] points towards nematic order, although it is not yet clear whetherthis order arises from a melted stripe state [ 42], from incipient unidirectional ﬂuctuating stripes [ 43], or from an unrelated d-wave-type nematic order which preserves translational sym- metry [ 44]. On the theoretical side, it was recently proposed that a ferronematic state occurs at very low doping, formed bystripe segments without positional order [ 45]. These segments are oriented because they sustain a vortex and an antivortexof the antiferromagnetic order at their extremes, and breakrotational and inversion symmetry. This phase has no orderin the charge sector, but induces incommensurate peaks inexcellent agreement with experiments in LSCO [ 46]. Recent Monte Carlo calculations [ 47] showed that, lowering the temperature, the ferronematic state turns into a ferrosmecticstate, where the segments have a typical lateral distance /lscript c, corresponding to CO with a characteristic wave vector qc (with\|qc\|∼1//lscriptc). The segments thus appear as the natural precursors of stripes. It is therefore important to assess nematic order in cuprates. The aim of the present work is to identify the signatures ofnematic ﬂuctuations in Raman scattering. This is a bulk (nearlysurface-insensitive) probe and measures a response functionanalogous to that of optical conductivity [ 48]. However, while the latter averages over the Brillouin zone (BZ), differentpolarizations of the incoming and outgoing photons weightdifferent parts of the BZ in Raman scattering [ 49], introducing speciﬁc form factors. It turns out that the so-called B 1gandB2g channels are the most relevant to extract the contributions of collective modes (CMs) in cuprates. We already investigatedhow these form factors can be exploited to identify thecontributions of different (e.g., charge and spin) critical CMs,based on their different ﬁnite wave vectors [ 34,50,53–55]. There are two classes of CM contributions. In one class, theCMs dress the fermion quasiparticles, introducing self-energyand vertex corrections, which affect the Raman spectra upto substantial fractions of eV [ 34,53,54]. In the other class, 1098-0121/2015/91(20)/205115(11) 205115-1 ©2015 American Physical SocietyS. CAPRARA et al. PHYSICAL REVIEW B 91, 205115 (2015) ω ωνωνν νε+ν ε ε+ω+ν ε+ν εεω ωε+ν εε+ω+ν ε+ω+νε+ν εν ν ω ω ε + + ν ε ε + ν ε ε ω ω ε + FIG. 1. Diagrammatic representation of the Raman response due to the excitation of two CMs. The grey dots represent the γB1gor γB2gform factors. The solid lines are the propagators of the fermion quasiparticles in the fermionic loops, the wavy lines represent the NCM or CO CM propagators [Eqs. ( 2)–(4)], which are coupled to the quasiparticles by the coupling functions gλ(k,q) (solid dots). the excitation of pairs of CMs [ 50], reminiscent of the Aslamazov-Larkin (AL) paraconductive ﬂuctuations near themetal-superconductor transition (see Fig. 1), affects mainly the low-frequency part of the spectrum and produces ananomalous absorption up to few hundreds of cm −1, as indeed observed, e.g., in LSCO [ 56]. The analysis for LSCO [ 50] was based on CO CMs with ﬁnite wave vector qc, while the role of spin CMs was ruled out by symmetry arguments.At moderate doping, the value q c≈(±π/2,0),(0,±π/2) was deduced from inelastic neutron scattering as the double ofthe wave vector of spin incommensuration [ 16], within the stripe scheme (we use hereafter a square unit cell on the CuO 2 planes, with lattice spacing a=1). By symmetry arguments, and in agreement with experiments, ﬂuctuations with suchq cgive rise to an anomalous absorption in the B1gchannel only. A rotated qc≈2π(±2x,±2x) occurs for x< 0.05 [46], making the anomalous Raman absorption show in the B2g channel only, consistent with the experiments. However, a similar anomalous absorption in the B2gchannel is observed in Y 1−yCayBa2Cu3O6+x(YBCO) for doping p(x,y) between 0.01 and 0.06 [ 52]. Recent measurements [ 38] do not support the rotation of the spin modulation vector in YBCO, at leastdown to p=0.05, and the extrapolation of the available data indicates that spin incommensuration disappears forp≈0.02−0.03, while CO seems to disappear for p< 0.08 [33]. Thus, if only CO ﬂuctuations were to play a role, the anomalous peak observed in YBCO in the B 2gchannel would be unexplained. Furthermore, CO CMs yield in LSCO spectrathat are fully satisfactory at x=0.1, but less convincing at x=0.02, where the experimental line shape seems to have a composite character, with a main peak accompanied bya shoulder at slightly higher frequencies. This suggests thepresence of two CMs contributing to the anomalous absorptionin the B 2gchannel at low doping in LSCO and raises the question about the nature of the additional CM. The uncertainsituation with YBCO and the compositeness of the LSCO spectra call for a critical revision of the results of Ref. [ 50]. The above mentioned evidences for nematic order make it natural to inquire whether the anomalous Raman absorptionobserved in underdoped cuprates might be due to nematicﬂuctuations (not considered in Ref. [ 50]), possibly mixed with CO ﬂuctuations (in LSCO). Therefore, within the same formalscheme of Ref. [ 50], we include here the contribution of nematic ﬂuctuations. We ﬁnd indeed that at low doping theobserved anomalous absorption can be due to the excitationof long-wavelength overdamped nematic ﬂuctuations withlongitudinal character, whose strong dynamics is apt to repro-duce the observed line shape. While in strongly underdopedYBCO this is enough, in LSCO, a secondary CM with ﬁnitecharacteristic wave vector, which we identify with the COCM, is needed to better represent the line shape. The dopingdependence of the line shape in LSCO indicates that there is anevolution from a dominating NCM towards a major relevanceof the CO CM, upon increasing doping. The scheme of the paper is the following. In Sec. II,w e introduce a phenomenological model of fermion quasiparticlescoupled to nearly critical CO CMs and NCMs in underdopedcuprates. Then, we proceed with the theoretical calculationof the Raman response due to these CMs (Secs. II A,II B, and II C). In Sec. III, we compare the theoretical results with available Raman spectra for underdoped YBCO andLSCO. Section IVcontains our ﬁnal remarks and conclusions. Appendix Acontains some details about the calculations of the Feynman diagrams involved in the anomalous Ramanresponse. Details of the ﬁtting procedure are found inAppendix B, while a discussion on the role of the pseudogap in the fermionic spectrum is found in Appendix C. II. THE FERMION-COLLECTIVE MODE MODEL AND THE RAMAN RESPONSE A. The fermion-collective mode model We consider a phenomenological model where, similarly to the electron-phonon coupling, electrons are coupled to NCMsor CO CMs. This approach relies on the presence of fermionquasiparticles. This assumption, which is natural in the metal-lic phase of cuprates, is still justiﬁed in the strongly underdopedphase, where angle resolved photoemission [ 57,58] and trans- port experiments [ 59,60] highlight the presence of fermionic low-energy states (the so-called Fermi arcs) with a substantialmobility, indicating that fermion quasiparticles still survive inthis “difﬁcult habitat.” Thus we adopt the Hamiltonian H=/summationdisplay k,σξkc† kσckσ+/summationdisplay k,q,σ/summationdisplay λgλ(k,q)c† k+qσckσ/Phi1λ −q,(1) where c† kσ(ckσ) creates (annihilates) a fermion quasiparticle with momentum kand spin projection σ, andξkis the fermion dispersion on the CuO 2planes of LSCO or YBCO (measured with respect to the chemical potential). Its speciﬁc form israther immaterial for our analysis, once the generic shapeof the Fermi surface of cuprates is taken into account. Theindex λlabels transverse ( λ=t) or longitudinal ( λ=/lscript) nematic ﬂuctuations [ 61,62], and charge ﬂuctuations ( λ=c), represented by the boson ﬁelds /Phi1 λ. The quasiparticles couple 205115-2SIGNATURES OF NEMATIC QUANTUM CRITICAL . . . PHYSICAL REVIEW B 91, 205115 (2015) to NCMs via gλ(k,q)≡gλdλ k,q, with d/lscript k,q=cos(2ϕk,q) and dt k,q=sin(2ϕk,q), where ϕk,qis the angle between kandq (see, e.g., Eqs. ( 2) and ( 3)i nR e f .[ 62]). The CO CM has instead a ﬁnite characteristic wave vector qcand couples to the fermion quasiparticle via a weakly momentum dependentcoupling g c(k,q)≈gc(i.e.,dc k,q≈1). We assume that these CMs are near an instability and their propagators take the standard Gaussian form, validwithin a Landau-Wilson approach, and already adopted formodels of fermion quasiparticles coupled to nearly criticalcharge [ 4] and spin [ 63,64] CMs in cuprates. As customary in quantum critical phenomena, different damping processesmay lead to different dynamical critical exponents z, relating the divergent correlation length ξand time scale τ∝ξ z.I nt h e case of the nematic instability, a multiscale criticality occursdue to the different dynamics of transverse and longitudinalﬂuctuations [ 62,65]. The longitudinal ﬂuctuations are Landau- overdamped, and decay in particle-hole pairs acquiring adynamical exponent z /lscript=3, and their propagator is D/lscript(q,ωn)=−1 m/lscript+c/lscript\|q\|2+\|ωn\|/\|q\|+ω2n//Omega1/lscript,(2) where ωnis a boson Matsubara frequency and wave vectors q are henceforth assumed dimensionless and measured in unitsof inverse lattice spacing a −1(when needed, conventional units are restored in our formulas by replacing qwithaq). Apart from the term ∝ω2 n, this propagator is the same as that in Eq. (2.14) of Ref. [ 62]. Transverse ﬂuctuations have instead zt=2, and their propagator is (see, e.g., Eq. (2.15) in Ref. [ 62]) Dt(q,ωn)=−1 mt+ct\|q\|2+\|ωn\|+ω2n/(/Omega1t\|q\|2).(3) Both propagators, in the static limit ( ωn=0), are peaked at q=0. Similarly, the nearly critical CO CM has a dynamical critical index zc=2, with propagator (see, e.g., Eq. ( 1)i n Ref. [ 51]o rE q .( 2)i nR e f .[ 34]) Dc(q,ωn)=−1 mc+cc\|q−qc\|2+\|ωn\|+ω2n//Omega1c, (4) peaked at a ﬁnite wave vector qc(actually, at the whole star of equivalent wave vectors). This circumstance allows to reabsorba factor \|q c\|2in the deﬁnition of the parameter /Omega1c, and marks the difference with respect to the propagator of the transverseNCMs, Eq. ( 3). In the doping regime we are considering, q cis directed along the diagonals of the BZ in LSCO with x< 0.05 [ 16]. According to the discussion in Sec. I,w e consider instead that CO is absent in YBCO with p≈0.015. In Eqs. ( 2)–(4), the parameters cλset the curvature at the bottom of the CM dispersions, whereas the parameters /Omega1λ set high-frequency cutoffs. The low-frequency scales mλare proportional to the inverse squared correlation lengths ξ−2 λ, thus being the relevant parameters that measure the distancefrom criticality. B. The fermionic loop in the Raman response Our theoretical analysis is based on the calculation of the Raman response represented by the Feynman diagrams ofFig. 1(more details are given in Appendix A). The ﬁrst step is to calculate the sum of the fermionic loops with attacheddirect and crossed boson lines (see top and bottom diagramsin Fig. 1): /Lambda1 λη i(q,νl,ωm)=T/summationdisplay k,nγi(k)gλ(k,q)gη(k,−q) ×[G(k+q,/epsilon1n−ωm)+G(k+q,/epsilon1n +ωm+νl)]G(k,/epsilon1n)G(k,/epsilon1n+νl), (5) where Tis the temperature, i=B1g,B2glabels the form factors, γB1g(k)=cos(ky)−cos(kx) and γB2g(k)= sin(kx)s i n (ky)[49],νlis the external Matsubara frequency which, once analytically continued, represents the frequencyshift between the incoming and the scattered photons, ω m is the Matsubara frequency of one of the boson propagators in Fig. 1(the other carries ωm+νl),/epsilon1nis the fermion frequency to be summed over in the fermionic loop,andG(k,/epsilon1 n)=(i/epsilon1n−ξk)−1is the fermion quasiparticle propagator. In Eq. ( 5), we exploited the parity of G(k,/epsilon1n), γi(k), and gλ(k,q)gη(k,−q) with respect to k. The dependence of the loop on the CM indexes λandη is diagonal: the CO CM cannot mix with the NCMs, havinga ﬁnite characteristic wave vector, and the /lscriptandtNCMs cannot mix, because the product of g /lscript(k,q) and gt(k,−q), each depending only on the angle between kandqand having a different parity, averages to zero when summed with respecttok. This fact entails a selection rule stating that the two NCMs attached to the same fermionic loop must be either longitudinalor transverse. The average over the Fermi surface of twocouplings with the same NCM yields a result that is weaklydependent on qand can be safely approximated to a constant that can be reabsorbed in the deﬁnition of the dimensionalcoupling g λ. Thus /Lambda1λη i(q,νl,ωm)≡g2 λδλη/Lambda1i(q,νl,ωm). Summing over the fermion frequencies, one obtains the general expression /Lambda1i(q,νl,ωm)=2/summationdisplay kγi(k)/Delta1fk/bracketleftbig /Delta1ξ2 k−ωm(ωm+νl)/bracketrightbig /parenleftbig /Delta1ξ2 k+ω2m/parenrightbig/bracketleftbig /Delta1ξ2 k+(ωm+νl)2/bracketrightbig, where /Delta1fk≡f(ξk+q)−f(ξk),/Delta1ξk≡ξk+q−ξk, andf(z)≡ (ez/T+1)−1is the Fermi function. The next steps are different in the case of NCMs (with characteristic wave vectors q≈0) and of CO CMs (with ﬁnite characteristic wave vectors qc), and will be dealt with in Secs. II B 1 andII B 2 , respectively. 1. The fermionic loop for NCMs To proceed with the calculation of the fermionic loop in the case of NCMs, we consider that the main features of the bosonpropagators ( 2) and ( 3) are their poles at small momenta q= \|q\|and even smaller frequencies, because of their dynamics withz /lscript=3(ω∼q3)o rzt=2(ω∼q2). Thus, expanding the above result for small frequencies and keeping the lowest orderin the Matsubara frequencies ω mandωm+νl, one obtains /Lambda1i(q,νl,ωm)≈2/summationdisplay kγi(k)/Delta1fk /Delta1ξ2 k≈2/summationdisplay kγi(k) /Delta1ξk∂f(ξk) ∂ξk. The summation on kcan be transformed into a two- dimensional integral, yielding /Lambda1i(q)≈2M (2π)2/integraldisplay/integraldisplay dkdθδ(k−kF)γi(k,θ) /Delta1ξk, (6) 205115-3S. CAPRARA et al. PHYSICAL REVIEW B 91, 205115 (2015) where θis the angle between the wave vector kand the xaxis in reciprocal space, and Mis the quasiparticle effective mass. By noticing that the form factor γi(k), calculated on the Fermi surface, depends weakly on k=\|k\|while it substantially depends on θ, one can write γB1g(k,θ)≈cos(2θ)≡γB1g(θ) andγB2g(k,θ)≈sin(2θ)≡γB2g(θ). When expanding /Delta1ξk one has to keep track of the inverse band curvature M (otherwise the integral vanishes). The limit \|q\|→ 0 can then be taken, and the ﬁnal result is that the fermionic loopdepends only on the angle φbetween qand the xaxis. This dependence can be made explicit observing that thedenominator /Delta1ξ kin Eq. ( 6) depends on the cosine of the angle θ−φbetween kand q. Shifting the variable θ−φ→θ, one is left with γi(θ+φ) in the numerator. Expanding, one has γB1g(θ+φ)=γB1g(θ)γB1g(φ)−γB2g(θ)γB2g(φ) and γB2g(θ+φ)=γB2g(θ)γB1g(φ)+γB1g(θ)γB2g(φ). The inte- gral with respect to θof the terms with γB2g(θ) vanishes by symmetry. Thus we ﬁnally obtain /Lambda1i(φ)≈M2 πk2 Fγi(φ). (7) This result, which is crucial in our development, implies that the original form factor γi(θ) coupling the fermion quasiparticles to the incoming and outgoing photons in theRaman vertex, in the integrated form of the loops, is translatedinto a direct coupling of the photons to the NCMs with thesame form factor γ i(φ). 2. The fermionic loop for the CO CMs The fermionic loop for the CO CMs has been calculated in Ref. [ 50], and we recall here the main results. The main difference with respect to the calculation of Sec. II B 1 , is that the propagator ( 4) is peaked at ﬁnite wave vectors qc. Then, the sum over kin Eq. ( 5) is now dominated by the neighborhood of the points along the Fermi surface where ξk=ξk+qc, i.e., the so-called hot spots (HS). The result is /Lambda1i(qc)≈1 2π2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleW+ W−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay HSγi,HS v2 HSsinαHS, (8) where W±are the upper and lower cutoffs for the linearized band dispersion at the hot spot, while γi,HSandvHSare, respectively, the Raman form factor and the Fermi velocityatk=k HS, andαHSis the angle between the Fermi velocities at the two hot spots connected by the given qc.F o r qcalong high-symmetry directions (i.e., the axes and the diagonals) ofthe BZ, the moduli of the Fermi velocities in k HSandkHS+qc are equal. As pointed out in Ref. [ 50], summing over kat ﬁxed qc, various different hot spots are visited, where, due to the above-mentioned symmetry, the form factors can havepairwise equal magnitude and equal or opposite signs. As aconsequence, the terms in the above hot-spot summation canadd or cancel each other. This induces a “selection rule” which,in the case pertinent to the strongly underdoped LSCO, whereq cis short and directed along the ( ±1,±1) directions, leads to ﬁniteB2gvertex loops, while the B1gvertex loops vanish by symmetry.C. The Raman response Few considerations are now in order. First of all, the NCM propagators, Eqs. ( 2) and ( 3), do not depend on the angleφand therefore the product of the two fermionic loops entering the diagrams of Fig. 1only introduces a multiplicative constant factor, which can be enclosed in the overall intensityof the Raman response. However, we emphasize that theφintegration, to be performed when the summation over qis carried out, introduces an important selection rule; the fermionic loops with attached Raman vertices, enter pairwisein the response diagrams of Fig. 1and must both be of the same symmetry, B 1gorB2g. Similarly, the CO propagator ( 4) depends only on the magnitude of the deviation of qfrom qc. In this case, the Raman response is given by a ﬁrst summationon all the possible q cof [/Lambda1i(qc)]2and an internal integral over \|q−qc\|of two CO CM propagators. Thus both for the two nematic CMs and for the CO CM, the sum of the two diagrams of Fig. 1reads χi,λ(νl)=Ki,λT/summationdisplay n/integraldisplay¯q 0dqq ¯q2Dλ(q,ωn)Dλ(q,ωn+νl), where ¯q∼1 is the momentum cutoff, its precise value being re-absorbable in a multiplicative rescaling of the parameters ofthe CM propagator, q=\|q\|for the NCMs, and q=\|q−q c\| for the CO CMs. The factor Ki,λcomes from the product of two fermionic loops and is proportional to g4 λ(each loop/Lambda1icarrying two fermion-CM coupling constants). For NCMs, Ki,λ≡M4g4 λ/angbracketleft[γi(φ)]2/angbracketright/(πk2 F)2, with λ=/lscriptort, and /angbracketleft[γi(φ)]2/angbracketrightis the angular average of the square of the function γi(φ) that appears in Eq. ( 7), whereas for the CO CM we have [cf. Eq. ( 8)]Ki,c=g4 c/summationtext qc[/Lambda1i(qc)]2, that vanishes in the B1g(B2g) channel for diagonal (vertical/horizontal) qc.T h i s “selection rule” is the only place where the ﬁnite wave vectorof the CO CM plays a role within our nearly critical theory ofRaman absorption. This selection rule is instead absent in thecase of the NCMs, which are peaked at q=0. The analytic continuation to real frequencies iν l→ω+ iδand the use of the spectral representation of the boson propagators ﬁnally yield the Raman response χ/prime/prime i,λ(ω)=Ai,λ/integraldisplay+∞ −∞dz[b(z−)−b(z+)]/integraldisplay1 0dq ×2qFλ(z+,q)Fλ(z−,q), (9) where b(z)≡(ez/T−1)−1is the Bose function, and we performed the customary symmetrization z→z−ω 2≡z− andz+ω→z+ω 2≡z+, which makes explicit the fact that Eq. ( 9) is an odd function of ω. The constant multiplicative prefactors, including those transforming the Raman suscepti-bility into the measured Raman response, are reabsorbed in theparameters A i,λ. Unfortunately, a fully analytical expression forAi,λcannot be given, the Raman response being affected by resonance effects that prevent even order-of-magnitudeestimates. However, whenever we studied the contributionof critical CMs in situations where the prefactors can beexplicitly calculated, like optical conductivity [ 34,51], or angle resolved photoemission spectra [ 35], we always found that the dimensionless coupling constants are of order one, in a regimeof moderate coupling. 205115-4SIGNATURES OF NEMATIC QUANTUM CRITICAL . . . PHYSICAL REVIEW B 91, 205115 (2015) 0 100 200 300 ω [cm-1]01234χ" [a.u.]0 50 100 ω [cm-1]0123χ" [a.u.]L-NCM CO T-NCM FIG. 2. (Color online) Schematic representation of the AL-like Raman response of the three different CMs: longitudinal L-NCM(black curve), transverse (T) NCM (green curve), and CO CM (red curve). (Inset) AL-like Raman response from different CMs, but taken with the same nearly critical set of parameters ( m λ=17 cm−1,/Omega1λ= 70 cm−1,cλ=3.16). The amplitudes are instead rescaled by factors of order one to bring all responses to a common maximal height for easier comparison. The coloring of the lines is the same as in the mainpanel. The spectral density of the longitudinal NCMs is F/lscript(z,q)=z q/parenleftbig m/lscript+c/lscriptq2−z2 /Omega1/lscript/parenrightbig2+z2 q2, while the spectral density of the transverse NCMs is Ft(z,q)=z /parenleftbig mt+ctq2−z2 /Omega1tq2/parenrightbig2+z2. Finally, for the CO CMs, we ﬁnd Fc(z,q)=z /parenleftbig mc+ccq2−z2 /Omega1c/parenrightbig2+z2. The anomalous peak of the Raman response, both in LSCO and YBCO, is strongly temperature dependent, it shrinks andsoftens upon reducing T. This behavior is naturally encoded in the temperature dependence of the mass m λof the CMs. In general, the low-frequency scale mλcontrols the slope of the Raman response, while the scales ω1∼√mλ/Omega1λand ω2∼√(mλ+cλ)/Omega1λset the frequency window over which the spectral function of the corresponding CM is sizable. However,the different dynamical properties and values of the parametersof the CMs mirror into different shapes of the AL-like Ramanresponses, which are schematically represented in Fig. 2.W e point out that the curves displayed in this panel do not exhaustall the possible regimes of parameters, and only representthe corresponding CM in the regime where, after a thoroughanalysis, they were found to better reproduce the variousfeatures of the Raman response. The inset of the same ﬁgurereports instead the behavior of Raman absorption spectra (fromthe AL processes) due to the various CMs. In this inset,while we rescale the height to bring all responses to thesame maximal height, we use the same nearly critical set ofparameters to highlight the differences arising purely from thedifferent form of the propagators and dynamical critical indexz. Apparently, the shape of the spectra is quite similar, but the behavior upon changing the mass is different on a quantitativelevel. In particular we found that the z=2 propagators shift the position of the maxima upon reducing mmore rapidly than the NCM z=3 propagator. Since we apply a strict ﬁtting protocol (see below in Sec. III), which ﬁxes all parameters and follows the temperature evolutions of the main peaks by onlychanging m, these different behavior affects in a substantial way the accuracy of the ﬁts. An inspection of Fig. 3 in Ref. [ 50] shows that the ﬁts with a z=2 CO-CM are not very accurate at low temperatures. Instead, we will see in the next section thatthez=3 NCM does a much better job within the adopted strict ﬁtting protocol and therefore it will be considered henceforthas the primary (i.e., most critical) CM. The additional shoulderin the spectra of LSCO, is instead better reproduced by the COcurve in Fig. 2than by the broader T-NCM curve, when both CMs are taken in the regime of parameters apt to describe thisspectral feature. Therefore, at these doping levels, the CO CMacts as the secondary CM in LSCO. III. RESULTS A. Raman absorption in Y 0.97Ca0.03Ba2Cu 3O6.05 An anomalous Raman absorption at low frequencies, up to few hundreds of cm−1, is experimentally found in theB2gchannel in lightly doped YBCO with p/lessorequalslant0.05 [52]. Since, however, the whole spectra also display broad absorptions up to electronic energy scales, we ﬁrst extractthe speciﬁc anomalous low-frequency contributions. To thispurpose, we subtract from the low-temperature spectra thespectra obtained at the highest measured temperature. Thissubtraction is delicate because at temperatures below about150–200 K, the spectra are characterized by the formationof a pseudogap over a frequency range of several hundredsof cm −1, which reduces the electronic background. Then, the simple subtraction leads to regions of negative absorptions,which are obviously meaningless. In Appendix B, we provide the detailed procedure adopted to circumvent this drawback. InFig. 3, the data, processed according to the previous procedure, are shown for p≈0.015. The experimental line shape clearly resembles the L-NCM spectrum in Fig. 2, which is narrow due to the z /lscript=3 damped dynamics of the corresponding CM, whose temperaturedependence is ruled by the mass m /lscript. Indeed, the data in Fig. 3are best ﬁtted with the only contribution of longitudinal NCMs. In the spirit of our nearly-critical approach, we onlyadjust their mass m /lscript(T), while keeping all other parameters (i.e., the high-frequency cutoffs of the CM propagator, the c/lscript coefﬁcients, and the overall intensity coefﬁcient A/lscript)ﬁ x e da ta l l temperatures. This strict procedure was already successfullyadopted in Ref. [ 50] and seems to us the most suitable to pinpoint the quantum nearly-critical character of the collectiveexcitations responsible for the anomalous Raman absorption.The ﬁts with this restricted procedure turn out to be quite good.Of course, they could be further improved if this constrainedprocedure were relaxed. The ﬁts reproduce well the lineshapes and the strong temperature dependence of the peak,encoded in the rapid decrease of the mass with temperature,a ss h o w ni nt h ei n s e to fF i g . 3. From this inset, it is evident thatm /lscript(T) decreases with T. Its linear extrapolation starting 205115-5S. CAPRARA et al. PHYSICAL REVIEW B 91, 205115 (2015) 0 50 100 150 200 250 300 ω [cm-1]00.511.522.53χ" [a.u.]T=55K T=86K T=127K T=190K T=254K T=283K 050100 150 200 250 300 T [K]050100150m [cm-1] FIG. 3. (Color online) Subtracted experimental Raman absorp- tion spectra in the B2gchannel, at various temperatures, for YBCO atp≈0.015 (symbols). The theoretical ﬁts (solid lines) consider the contribution of the longitudinal NCM only. The ﬁtting parameters arec /lscript=0.63 cm−1,/Omega1/lscript=110 cm−1,A/lscript=5.0 (a.u.). The inset reports the temperature dependence of the mass of the longitudinal NCM (black circles). from high temperature should vanish at some ﬁnite critical temperature for the onset of nematicity ( ≈125 K), if static order would occur. However, at lower temperatures, the massseems instead to saturate, likely indicating that nematic orderstays short-ranged and dynamic. B. Raman absorption in La 2−xSrxCuO 4 Figures 4and 5report the experimental Raman spectra in the B2gchannel, for LSCO samples at doping x=0.02 0 50 100 150 200 250 300 ω [cm-1]01234χ" [a.u.]T=35K T=88K T=125K T=182K T=255K 050100150200250 T [K]050100150m [cm-1] FIG. 4. (Color online) Subtracted experimental Raman absorp- tion spectra in the B2gchannel, at various temperatures, for LSCO at x=0.02 (symbols). The theoretical ﬁts (solid lines) consider the contribution of the longitudinal NCM and of the CO CM, withc /lscript=3.16 cm−1,cc=333 cm−1,/Omega1/lscript=70 cm−1,A/lscript=8.3 (a.u.). The other ﬁtting parameters are reported in Fig. 6. The inset reports the temperature dependence of the mass of the longitudinal NCM(black circles) and of the CO CM (red squares).0 50 100 150 200 250 300 ω [cm-1]01234χ" [a.u.]0 100 200 300 T [K]050100150m [cm-1] T=301KT=252KT=207KT=169KT=154KT=137KT=105KT=52K FIG. 5. (Color online) Subtracted experimental Raman absorp- tion spectra in the B2gchannel, at various temperatures, for LSCO at x=0.04 (symbols). The theoretical ﬁts (solid lines) consider the contribution of the longitudinal NCM and of the CO CM, with c/lscript=3.16 cm−1,cc=333 cm−1,/Omega1/lscript=50 cm−1.A/lscript=7.14 (a.u.). The other ﬁtting parameters are reported in Fig. 6. The inset reports the temperature dependence of the mass of the longitudinal NCM (black circles) and of the CO CM (red squares). and 0 .04 and various temperatures. The raw data were again processed according to the procedure described in theAppendix B. As mentioned above, the anomalous Raman absorption observed in LSCO is characterized by a line shapethat is more complex than in YBCO, and displays a peculiarshoulder or, at low T, even a secondary peak, see Fig. 5. The anomalous peak and the shoulder (or secondary peak)both depend on temperature, but their frequency and intensityare not simply related by constant multiplicative factors. Theshoulder (or secondary peak) becomes stronger with increasingdoping. This indicates that the excitations responsible for thisabsorption have a distinct dynamics. Again these absorptions are described by the AL-like pro- cesses (direct and crossed, see Fig. 1). Owing to the selection rules found in Sec. II, the response due to two (or more) CMs is the sum of the responses associated with each individualCM. As already mentioned, our thorough analysis showed thatthe primary anomalous absorption should be attributed to thelongitudinal NCM, which has the stronger dynamical behavior.Within our context, the transverse NCM and the CO CMare the two candidates for the shoulder (or secondary peak).Looking at the line shape of the two CMs reported in Fig. 2, it is easy to convince oneself that the best choice for a goodﬁt is the CO CM, due to its much more pronounced peakedform at intermediate frequency. We also attempted a ﬁt withthe transverse NCM. At x=0.02, we obtained a reasonable ﬁt taking a very large and almost temperature independentCM mass, which is hardly compatible with our assumption ofnearly critical CMs. Moreover, at x=0.04, when the shoulder evolves into a secondary peak, the attempt failed completely.Thus we ruled out a contribution of transverse NCMs. Again, having attributed the main peak to the more critical longitudinal NCM, we describe the low-frequency side of thespectra by only adjusting the mass m /lscript(T) of this excitation, 205115-6SIGNATURES OF NEMATIC QUANTUM CRITICAL . . . PHYSICAL REVIEW B 91, 205115 (2015) 0 100 200 300 T [K]020406080100 Ωc[cm-1] 0 100 200 300 T [K]0100200300 Ac[a.u.] FIG. 6. (Color online) (a) High-frequency cutoff /Omega1cfor the CO CM for a sample at x=0.02 (black empty squares) and at x=0.04 (red ﬁlled squares). (b) Amplitude coefﬁcients Acfor the CO CM for a sample at x=0.02 (black empty squares) and at x=0.04 (red ﬁlled squares). while keeping all other parameters of this mode (i.e., the high- frequency cutoff of the CM propagator, the c/lscriptcoefﬁcient, and the overall intensity coefﬁcient A/lscript) ﬁxed at all temperatures, within the temperature range considered here. Thus we obtainthe marked temperature dependence of the longitudinal NCMmass, which is reported in the insets of Figs. 4and 5.O n the other hand, the complete quantitative agreement betweendata and theoretical ﬁts is only obtainable by adjusting morefreely the secondary CO CM. This mode is therefore allowedto vary its parameters with T, as reported in Fig. 6.T h e temperature dependence of the CO CM parameters /Omega1 cand Ac∝g4 clikely reﬂects an increasing damping and a decreasing coupling to the fermion quasiparticles with increasing T.O f course, the estimates and variations of these parameters maybe quantitatively affected if the constraint of T-independent parameters for the longitudinal NCM (but for its mass m /lscript) were relaxed. Furthermore, we cannot exclude that staticnematic order has eventually occurred, e.g., in the sample withx=0.02 at the lowest temperature. In this case our analysis, which is only valid above the critical temperature, should bemodiﬁed to deal with a broken-symmetry phase. This mightreﬂect in a reduction of the primary peak, due to the freezing ofNCM ﬂuctuations, and could be the cause of the non monotonicbehavior of the peak height as a function of T, observed in the sample with x=0.02. To asses the occurrence of static nematic order at low temperature, a systematic experimentalinvestigation in this temperature regime is needed. IV . DISCUSSION AND CONCLUSIONS Our analysis showed that the anomalous Raman absorption observed in underdoped cuprates can be interpreted in termsof direct excitation of nearly critical CMs (see Fig. 1). The strong temperature dependence of the mass (i.e., inverse squarecorrelation length) of the “primary” CM, identiﬁed as thelongitudinal NCM (with dynamical critical index, z /lscript=3), captures the correspondingly strong variation of the spectra.This CM alone fully accounts for the spectra of YBCO. InLSCO, instead, a distinct “secondary” CM, with different FIG. 7. (Color online) Schematic comparison of the theoretical expectations and the experimental observation of an anomalousRaman absorption. The theoretically involved CMs are indicated with N in the case of the NCM, while for CO we also report the direction of the characteristic wave vector, as established by inelasticneutron scattering. The related symbols only appear in the box where they are expected to contribute on the basis of symmetry arguments. The experimental observation of an anomalous Raman absorption is depicted as a green case in the column of the corresponding channel. Red cases indicate instead the lack of anomalous Raman absorptionin experiments. Our remarks and possible indications (in boldface) are contained in the comment boxes. dynamical critical index z=2, is needed to reproduce the composite line shape. Within the two candidates considered inour scheme (transverse NCM and CO CM), our ﬁts indicatethat the CO CM is the most suitable. For symmetry reasons, the secondary CO CM cannot occur in all channels: the ﬁrst two rows in the sketch ofFig. 7summarize the ﬁndings of Ref. [ 50]i nL S C Oa sf a r as CO is concerned. The correct CO (i.e., with ﬁnite q c in the direction compatible with inelastic neutron scattering experiments) appears as an observed absorption (green case)only in the theoretically predicted channel. Two questions still remain to be answered, in order to complete the scheme of Fig. 7. First of all, the NCMs would equally contribute to the B 1gandB2gchannels. Therefore they would not only add to the CO ﬂuctuations that give absorptionin the B 1gchannel at larger doping ( x> 0.05) in LSCO, but would also give rise to absorption in the B2gchannel. Since this is not observed (the corresponding box is red in Fig. 7), we infer that NCMs disappear in LSCO at x> 0.05 (see the comment box in the ﬁrst row of Fig. 7). This is consistent with the observation of an increasingly stronger stripe orderat higher doping [ 50], where CO CM alone [along the (1,0) and (0,1) directions of the BZ] accounted for the anomalousRaman absorption at x=0.10 and 0 .12. The second related question is: if the NCMs are present and contribute to the absorption in B 2gat low doping both in LSCO and YBCO, why are they not visible in the (forthem allowed) B 1gchannel? As yet, we do not have a deﬁnite answer. We argue that the strong pseudogap occurring in 205115-7S. CAPRARA et al. PHYSICAL REVIEW B 91, 205115 (2015) FIG. 8. (Color online) Schematic evolution of the nematic (blue region) and stripe CO (green region) phases in underdoped cuprates with doping (disregarding superconductivity). The pseudogap region,where the Raman response in the B 1gchannel is expected to be suppressed, is highlighted in red. Upon increasing doping, the nematic phase evolves into a CO phase, which in turn vanishes at a COquantum critical point around optimal doping. The orientation of the segments and/or stripes may change with cuprate family and doping. lightly doped cuprates at T< 200 K could play a key role in suppressing the B1gabsorption. Speciﬁcally, the B1gform factors select the quasiparticles in the fermionic loops of Fig. 1 precisely from the BZ regions where the pseudogap is largest.Therefore, only the quasiparticles in the remaining Fermi arcs,mostly weighted by the B 2gform factors, remain to couple the Raman photons with the NCMs. In Appendix C, we obtained a numerical estimate of this suppression, ﬁnding indeed that itcan be substantial. Based on the above discussion, we can draw the schematic “phase diagram” for underdoped cuprates reported in Fig. 8. Despite its speculative character, it is compatible with varioustheoretical and experimental ﬁndings, and accounts for theassessed relevance of nematic order in cuprates [ 20,38–41]. It also complies with the proposal of a nematic order resultingfrom the melting of stripes [ 42] or of a nematic or smectic phase in strongly underdoped LSCO (and possibly YBCO) [ 47], arising from the aggregation of doped charges in shortsegments (blue region). The orientation of these segmentsbreaks the lattice C 4rotational symmetry preparing the route to CO at higher doping, when the segments merge into stripes(green region). The ﬂuctuating character of the nematic phaseshould give rise to nearly critical ﬂuctuations of the form ofEqs. ( 2) and ( 3). On the other hand, CO ﬂuctuations become prominent by increasing doping and appear in the B 1gchannel above x=0.05. In LSCO these ﬂuctuations are present (in the diagonal directions of the BZ) also at x< 0.05 and contribute to the B2gabsorption, but the tendency of CO ﬂuctuations to become more relevant at larger doping is clearly visibleby comparing Figs. 4and 5. At the same time, the insets of Figs. 4and5also display an increase of the low-temperature limit of the correlation length of the CO CM upon increasingdoping. Hence nematic and CO ﬂuctuations coexist in veryunderdoped LSCO, the predominance shifting from nematicto CO with increasing doping. This indicates a continuousevolution from the nematic (charge segment) phase to the stripe phase where charge and spin degrees of freedom are tightlybound, yielding a deﬁnite relation between spin and chargeincommensurabilities (typical of the stripe phase). We reliedon this relation to implement our symmetry-based selectionrules for LSCO. On the other hand, our ﬁnding that NCMsalone are relevant in YBCO at very low doping supports theidea that oriented charge segments may occur in this materialas well, accounting for the order-parameter-like disappearanceof the incommensurability in the spin response with increasingtemperature [ 45,47], as observed in Refs. [ 20,38]. The lack of CO ﬂuctuations at low doping and the opposite doping depen-dence of the charge and spin characteristic wave vectors [ 33] indicate a nematic-to-CO switching different from that inLSCO. However, both materials seem to eventually evolve intoa charge-density-wave phase ending into a quantum criticalpoint around optimal doping, as theoretically proposed [ 4] and recently observed [ 33]. ACKNOWLEDGMENTS S.C. and M.G. acknowledge ﬁnancial support form the Sapienza Universit `a di Roma, Project Awards C26H13KZS9. The work in Garching was supported by the DFG via theResearch Unit FOR 538 (Grant No. HA2071/3) and theCollaborative Research Center TRR80. APPENDIX A: DETAILS ON THE CALCULATION OF THE RAMAN RESPONSE Inspection of Fig. 1shows that the diagrammatic structure of the Raman response due to the excitation of two CMsinvolves two inequivalent fermionic loops (on the left ofthe two diagrams), which multiply two CM propagators andthe fermionic loop on the right of the diagrams. Calling /Lambda1 λη i(q,νl,ωm) the sum of the two different fermionic loops (frequencies and momenta are those displayed in Fig. 1), we can write the expression for the sum of the two diagrams as χλη ij(νl)=T/summationdisplay q,m/Lambda1λη i(q,νl,ωm) ×Dλ(q,ωm)Dη(q,ωm+νl)Lλη j(q,νl,ωm), where i,j=B1g,B2g,λ,η=/lscript,t,c , andLλη j(q,νl,ωm) stands for the fermionic loop in the right part of the diagrams. Theabove expression can be made symmetric also with respect tothe latter fermionic loop, observing that the integrated k /primecan be changed into −k/prime, and the fermionic Matsubara frequency /epsilon1/prime n can be shifted to /epsilon1/prime n−νl(frequencies and momenta are those displayed in the fermionic loop on the right of both diagramsin Fig. 1). Then, exploiting the parity of the CM propagators with respect to both momentum and frequency arguments, wecan take q→− qandω m→−ωm,νl→−νl. Summing the two equivalent expressions and dividing by 2 we are ﬁnallyled to calculate χλη ij(νl)=T 2/summationdisplay q,m/Lambda1λη i(q,νl,ωm) ×Dλ(q,ωm)Dη(q,ωm+νl)/Lambda1λη j(q,νl,ωm). 205115-8SIGNATURES OF NEMATIC QUANTUM CRITICAL . . . PHYSICAL REVIEW B 91, 205115 (2015) This expression has the formal structure of a Raman response where to CMs are directly excited by the scattered electromagnetic radiation, and /Lambda1λη i(q,νl,ωm) plays the role of an effective Raman vertex, resulting from the sum ofthe fermionic loops with attached direct and crossed bosonlines in Fig. 1. In Sec. II B, we discuss the calculation of the effective vertex /Lambda1 λη i(q,νl,ωm). This calculation is further specialized to the cases of NCMs and CO CMs in Secs. II B 1 andII B 2 , respectively. APPENDIX B: FITTING PROCEDURE OF THE ANOMALOUS RAMAN ABSORPTION To ﬁt the anomalous contribution of the Raman absorption due to two virtual CMs, as represented in Fig. 1, one needs to subtract the regular part of the spectra arising, e.g., from thedressed quasiparticles. However, the subtraction procedure hasto face the problem of pseudogap formation occurring in theunderdoped regime; at substantially high temperatures (above300 K), there is no pseudogap, which instead sets in below200 K. The anomalous absorption peak we are interested instarts to appear on top of the (pseudogapped) broad absorptionspectra at lower T. Thus, when the nonpseudogapped spectra atT≈300 K are subtracted from the low-temperature pseudogapped ones, a negative differential absorption is foundover the frequency range of the pseudogap. Although this is notcrucial for the qualitative description of the anomalous peaks,to get quantitatively more precise ﬁts we exploit the fact that thepseudogap sets in rather rapidly and, once established, dependsonly very weakly on T. Therefore we add a smooth parabolic contribution χ /prime/prime b=ω(/Omega1MAX−ω)/[B(T)]2, with ωin cm−1 andTin K, just designed to cancel the negative part at each temperature. For YBCO, we take /Omega1MAX=1000, B(55)= 760, B(86)=860, B(127)=1100, while at T=190, 254,and 282 K no compensation is needed because the 0 500 1000 ω [cm-1]-1012345χ" [a.u.] FIG. 9. (Color online) Subtraction procedure on the raw Raman data of a LSCO sample at x=0.04 and T=52 K (blue curve and symbols). The red curve and symbols correspond to the raw data atT=331 K. Once the latter are subtracted from the former, the purple curve and symbols are obtained. To eliminate the negative absorption, the parabola χ/prime/prime b=ω(1000 −ω)/6002, with ωin cm−1, is added to ﬁnally yield the absorption reported with the black curveand symbols.pseudogap is not open. For LSCO at x=0.02, we take /Omega1MAX=800,B(35)=500,B(88)=500,B(125)=600, and B(182)=1000, while at T=255 K, again no compensation is needed because the pseudogap is not open. The sameprocedure is carried out at x=0.04, with /Omega1 MAX=1000, B(52)=600,B(105)=600,B(137)=800,B(154)=1000, B(169)=800,B(207)=1500, B(252)=2000, B(301)= 2000. Figure 9exempliﬁes the procedure for LSCO with x=0.04 atT=52 K. The blue curve represents the raw data, from which we subtract the red data at T=331 K, obtaining the purple curve with unphysical negative absorption. Thepseudogap effect is then corrected by the addition of thesmooth parabolic contribution, leading to the ﬁnal black curve.Once these differential spectra are thus brought to have azero background we proceed to ﬁt the strongly T-dependent anomalous peaks. APPENDIX C: PSEUDOGAP, FERMI ARCS, AND RAMAN RESPONSE SUPPRESSION The strongly underdoped phase of cuprates is characterized by the presence of a pseudogap that strongly suppresses thelow-energy electronic degrees of freedom. In particular, theelectronic states in the so-called antinodal regions of the BZ,around ( ±π,0),(0,±π), are gapped, while the so-called nodal states, along the ( ±1,±1) directions, survive giving rise to Fermi arcs which shrink upon lowering temperature anddoping. In this appendix, we investigate the effects of thissuppression of the low-energy electronic states on the couplingbetween the Raman vertices and the NCMs. Indeed, thefermionic loops entering the diagrams of Fig. 1involve the integration over fermionic degrees of freedom coupled tothe nearly critical CMs, with the low-energy fermions being themost effective in coupling to the low-energy CMs. Thereforethe opening of gaps in the electronic spectra naturally entailsa substantial reduction of the overall response of the CM.However, the Raman vertices γ i(k) weight differently the fermionic states along the Fermi surface and it is thereforequite natural that the fermionic loops are differently suppresseddepending on the channel. To estimate this effect is the aim ofthis Appendix. More speciﬁcally, we will consider the NCMsonly, because the CO modes in the very underdoped LSCOwere shown in Ref. [ 50] to be only visible in the B 2gchannel. So it would be meaningless to compare the pseudogap effectsin the two Raman channels. On the contrary, the NCM are sin-gular at q≈0 and therefore should give a strong contribution to the Raman response in both channels. This Appendix willinstead demonstrate that the interplay of Raman vertices andmomentum dependence of the pseudogap strongly suppressthe loop in the B 1gchannel in comparison to the B2gcase. We adopt the simplifying assumption of a circular Fermi surface and we start from the expression for the fermionicloop, Eq. ( 6). Since the NCMs are singular at small q,w e expand in this limit the quantity /Delta1ξ k≡ξk−ξk+q. Expanding up to order q2the denominator and exploiting the δfunction to perform the integral along the radial momentum variable,one obtains /Lambda1 i≈2M2 (2π)2/integraldisplay2π 0dθF(θ)γi(θ) 1−q2 2k2 F+cos(2θ−2φ),(C1) 205115-9S. CAPRARA et al. PHYSICAL REVIEW B 91, 205115 (2015) where φis the angle between qand the xaxis in reciprocal space. At this stage, we have phenomenologically introduceda function F(θ)=/summationdisplay n1 1+e{[θ−(2n−1)π/4]2−/Theta12 M}//Delta12 θ, (C2) withn=1,2,3,4, which simulates the effect of the pseudogap on the Fermi surface of the 1 −4 quadrants. The parameter /Theta1Mtunes the length of the residual arc on the Fermi surface. Speciﬁcally, this function leaves the states near the diagonaluntouched, while for /Theta1 M<π / 4 it rather sharply suppresses the integration in the gapped antinodal regions for θ’s far from the nodal direction θ=π/4( f o r /Theta1M=π/4 one recovers the full ungapped Fermi surface). This essentially restricts theintegration in Eq. ( C1) to the angles of a Fermi arc allowing to explore the different action of the Fermi surface shrinking onthe value of the fermionic loop. The parameter /Delta1 θmeasures how rapidly the pseudogap is switched on and off along theFS, and we take it to be much smaller than π/4. Figure 10displays the square of the fermionic loops in the two Raman channels as a function of the angle φbetween the boson transferred momentum qand the xaxis. The calculation clearly shows the increasingly strong suppression of the B 1g fermionic loop (solid curves) upon reducing the length of the Fermi arcs. The suppression is much less pronouncedin the B 2gfermionic loop (dashed lines). These results are rather natural because the pseudogap suppresses the states that0 φ0(B1g, B2g loop vertices)21/4 1/2 3/4 1/1 π/4 π/2 FIG. 10. (Color online) Square of the fermionic loops as a func- tion of the angle φcalculated according to Eq. ( C1)i nt h e B1g channel (solid lines) and in the B2gchannel (dashed lines). The F(θ) function [Eq. ( C2)] has been set to produce arcs shrinking the Fermi surface by a factor 1 ( /Theta1M=π/4, whole Fermi surface, black curves), 0.75 ( /Theta1M=3π/16, red curves), 0.5 ( /Theta1M=π/8, green curves), and 0.25 ( /Theta1M=π/16, blue curves). 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PhysRevB.90.184102.pdf	PHYSICAL REVIEW B 90, 184102 (2014) Structural stability and thermodynamics of CrN magnetic phases from ab initio calculations and experiment Liangcai Zhou,1,Fritz K ¨ormann,2David Holec,3,4Matthias Bartosik,1,4Blazej Grabowski,2 J¨org Neugebauer,2and Paul H. Mayrhofer1,4 1Institute of Materials Science and Technology, Vienna University of Technology, A-1040 Vienna, Austria 2Max-Planck-Institut f ¨ur Eisenforschung GmbH, D-40237 D ¨usseldorf, Germany 3Department of Physical Metallurgy and Materials Testing, Montanuniversit ¨at Leoben, A-8700 Leoben, Austria 4Christian Doppler Laboratory for Application Oriented Coating Development at the Institute of Materials Science and Technology, Vienna University of Technology, A-1040 Vienna, Austria (Received 12 August 2014; revised manuscript received 2 October 2014; published 5 November 2014) The dynamical and thermodynamic phase stabilities of the stoichiometric compound CrN including different structural and magnetic conﬁgurations are comprehensively investigated using a ﬁrst-principles density functionaltheory (DFT) plus U(DFT+U) approach in conjunction with experimental measurements of the thermal expansion. Comparing DFT and DFT +Uresults with experimental data reveals that the treatment of electron correlations using methods beyond standard DFT is crucial. The nonmagnetic face-centered cubic B1-CrNphase is both elastically and dynamically unstable, even under high pressure, while CrN phases with nonzerolocal magnetic moments are predicted to be dynamically stable within the framework of the DFT +Uscheme. Furthermore, the impact of different treatments for the exchange-correlation (xc)-functional is investigated bycarrying out all computations employing the local density approximation and generalized gradient approximation.To address ﬁnite-temperature properties, both magnetic and vibrational contributions to the free energy havebeen computed employing our recently developed spin-space averaging method. The calculated phase transitiontemperature between low-temperature antiferromagnetic and high-temperature paramagnetic (PM) CrN variantsis in excellent agreement with experimental values and reveals the strong impact of the choice of the xc-functional.The temperature-dependent linear thermal expansion coefﬁcient of CrN is experimentally determined by the wafercurvature method from a reactive magnetron sputter deposited single-phase B1-CrN thin ﬁlm with dense ﬁlmmorphology. A good agreement is found between experimental and ab initio calculated linear thermal expansion coefﬁcients of PM B1-CrN. Other thermodynamic properties, such as the speciﬁc heat capacity, have beencomputed as well and compared to previous experimental data. DOI: 10.1103/PhysRevB.90.184102 PACS number(s): 64 .60.−i,63.20.dk,63.50.Gh,65.40.−b I. INTRODUCTION Transition metal nitrides have attracted much interest due to their excellent performance in applications such as hardprotective coatings on cutting tools, diffusion barriers, andwear resistant electrical contacts [ 1–6]. Among the transition metal nitrides, chromium nitride (CrN) is valued especially forits good wear, corrosion, and oxidation resistance [ 3,4,6]. There are many experimental reports on CrN due to its wide applicability in the industrial area [ 3,7–17]. At room temperature, it adopts a paramagnetic (PM) cubic B1 (NaCl prototype, space group Fm¯3m) structure with lattice constant a=4.14˚A[12]. Upon cooling below the N ´eel temperature (T N=200–287 K) [ 7,12,15,18], a simultaneous structural and magnetic phase transition to an antiferromagnetic orthorhom-bic (AFM Ortho; space group Pnma ) phase induced by magnetic stress takes place, which is accompanied by adiscontinuous volume reduction of /223c0.59% [ 7]. The AFM Ortho-CrN phase shows a small structural distortion fromthe underlying cubic B1 lattice characterized by the angleα≈88.3 ◦, and the antiferromagnetic ordering consists of alternating double (110) planes of Cr atoms with spin up andspin down, respectively [ 12]. Regarding theoretical investigations, CrN has received great attention mainly due to its electronic and magnetic zhlc1985@gmail.comproperties [ 19–26], while only a few papers have been devoted to its phase stability [ 22,27]. Most theoretical studies assumed CrN in the nonmagnetic rather than in the PM state[16,28–32], while experiments clearly reveal nonvanishing local magnetic moments even above the N ´eel temperature [33]. The reason for this serious approximation is largely related to the challenges in describing the dynamic magneticdisorder in the PM state. Recently, Alling et al. [22] employed the special quasirandom structures (SQS) approach [ 34]t o simulate the magnetic disorder in PM B1-CrN and concludedthat the magnetic disorder together with strong correlationeffects in the PM B1-CrN phase largely inﬂuence the Gibbsenergy. Including the magnetic entropy, a phase transitionbetween the high-temperature PM B1-CrN phase and the low-temperature AFM ground state with a distorted orthorhombic structure was predicted at 498 K [ 22]. In their treatment, however, the vibrational contribution to the free energy wasnot taken into account, despite the known signiﬁcant inﬂuenceon thermodynamic properties [ 35]. An improvement was provided in Ref. [ 36] where Alling et al. included vibrational contributions via the disordered-local-moment molecular-dynamics-simulations in conjunction with the temperature-dependent effective potential (DLM-MD-TDEP) method. Theunderlying density functional theory (DFT) calculations werebased on the local density approximation (LDA) with an onsiteCoulomb interaction term U. An improved phase transition temperature of 381 K has been found, which is, however, still 1098-0121/2014/90(18)/184102(12) 184102-1 ©2014 American Physical SocietyLIANGCAI ZHOU et al. PHYSICAL REVIEW B 90, 184102 (2014) /223c100 K higher than the experimental observations. Despite the recent progress, many questions are still unanswered. A decisive issue is, for instance, the impact of the underlying exchange-correlation (xc)-functional on the thermodynamicstability, which is so far not known. Recently, we havedeveloped an alternative method to compute the vibrationalGibbs energy contributions based on a spin-space averaging(SSA) method [ 37]. Our method does not require computation- ally demanding molecular dynamics simulations and can betherefore efﬁciently employed to scan the inﬂuence of differentxc-functionals as will be presented in this paper. When discussing phase stability, it is important to distin- guish energetic (meta)stability, mechanical, and dynamical stability. The ﬁrst one is related to the Gibbs free energy G, which predicts the phase with the lowest Gto be the most stable one, all other phases being metastable or even unstable. Regarding mechanical stability, the Born-Huang criteria [ 38] for elastic constants have to be fulﬁlled. The dynamical stabil-ity considers the complete vibrational spectrum of a material;a material is (dynamically) stable when no imaginary phonon frequencies exist. In other words, the Born-Huang mechanical stability criteria provide anecessary condition for the dynami- cal stability, but not a sufﬁcient one. For instance, by explicitly computing phonon dispersions, imaginary frequencies can occur at the Brillouin zone boundary (indicating dynamical instability), even if the Born-Huang criteria are fulﬁlled [ 39]. It is therefore necessary to perform explicit phonon calculationsto evaluate the dynamical stability. Such an evaluation of the dynamical stability is computationally much more expensive than calculating the full tensor of elastic constants (in particularfor systems with large numbers of atoms in their unit cells).Consequently, explicit phonon computations have been so far mostly neglected, and we therefore present here a complete study of CrN phase stabilities. The nonmagnetic (NM) B1-CrN phase has been reported to be energetically unfavorable with respect to other spin arrangements with nonzero local magnetic moments [ 22]. It is however not clear whether NM B1-CrN is dynamically unstable with respect to mechanical distortions, i.e., whether the elastic constants fulﬁll criteria for mechanical stability [ 38] and/or if there are any imaginary frequencies in its phonon dispersion. Another question of interest is whether the NM B1-CrN can be stabilized, e.g., by high pressure, and thus represents a potential metastable state. The literature data based on DFT suggest its mechanical stability [ 28,29,31,32], but an explicit evaluation via the full phonon spectrum analysis will be presented in this paper. As mentioned above, another important and so far unre- solved issue is the impact of strong electron correlation effectson the stability of CrN phases. For example, it has beenargued that LDA +Uis more suitable than the generalized gradient approximation (GGA) [ 22], but a comprehensive study employing GGA +U, which might turn out to be even more appropriate, is so far missing. Additionally, the impactof the previously proposed value for the onsite Coulombinteraction parameter Uon the vibrational free energy has not been discussed so far as well [ 36]. The aims of this paper can be hence summarized as follows: (i) To comprehensively investigate the energetic, mechani-cal, and dynamical stability of CrN including NM, AFM,ferromagnetic (FM), and PM conﬁgurations; (ii) to elucidate the impact of the chosen xc-functional approximation and U parameter; and (iii) to compute the thermodynamic stabilityof the different variants. First, elastic constants are calculatedand used to evaluate the mechanical stability based on theBorn-Huang mechanical stability criteria. As discussed above,these stability criteria might not be sufﬁcient. Therefore,in a second step, phonon calculations are performed toprove the dynamical stability. Eventually, Gibbs free energiesbased on the SSA formalism [ 37], including vibrational and magnetic contributions, are computed to predict the phasetransition in CrN employing GGA, GGA +U, and LDA +U approaches, and the transition temperatures thus obtained arecompared with available experimental results. Furthermore,thermodynamic properties such as lattice thermal expansioncoefﬁcient and speciﬁc heat capacity of the high-temperaturePM B1-CrN are presented and, where applicable, comparedto experimental data. In particular, we compare our DFTcomputed linear thermal expansion coefﬁcient with a set ofnewly determined experimental data obtained by the wafercurvature method and from a single-phase face-centered cubicB1-CrN thin ﬁlm. II. METHODS A. Static DFT calculations As discussed above, unlike other early transition metal nitrides, CrN requires extra efforts due to the nonzero localmagnetic moments of the Cr atoms. In this paper, the SQSapproach [ 22,34,40] is used to model the PM state by employing 2 ×2×2 cubic supercells with 32 Cr and 32 N atoms and randomly distributing the spin-up and spin-down moments on the Cr-atoms. In order to provide a completepicture of phase stabilities, we consider in this paper alsoB1-CrN with FM and AFM ordering consisting of alternatingsingle (001) planes of Cr atoms with spin up and spin down inaddition to the NM B1, PM B1, and AFM Ortho structures ofCrN (cf. the crystal structures in Fig. 1). All ﬁrst-principles calculations are based on DFT as implemented in the Vienna Ab initio Simulation Package ( V ASP )[41,42]. The ion-electron interactions are described by the projector-augmented wave (PAW) method [ 43] with a plane wave energy cutoff of 500 eV . The semicore pstates are treated as valence for Cr (3 p63d54s1), while there are 5 valence electrons for N (2 s22p3). In order to take into account the strong onsite Coulomb interaction ( U) caused by the localized 3delectrons of Cr, the LDA and the GGA plus a Hubbard U- term method is adopted within the framework of the Dudarevformulation [ 44,45]. Here, only the difference U-J, with J being the screened exchange energy, determines the materialproperties. Alling et al. [22]t e s t e d U-Jin the range from 0 to 6 eV in their previous LDA +Ustudy and found U-J=3e V to yield an optimal description of the structural and electronicproperties of AFM and PM-CrN. Apart from structural andelectronic properties, the inclusion of the Uparameter is also decisive for the correct description of the magnetismin CrN. This is exempliﬁed in Fig. 1, where we show that a too small value of the U-Jparameter might even yield a wrong magnetic ground state for CrN. The total energy wasevaluated for a number of structures as function of U-Jranging 184102-2STRUCTURAL STABILITY AND THERMODYNAMICS OF . . . PHYSICAL REVIEW B 90, 184102 (2014) 01234 5 UJ (eV)-0.20.00.20.40.60.81.0 Used U J(b) GGA+U AFM OrthoAFM B1FM B1NM BhNM B1 2 2.5 3 3.5 40.00.1 01234 56 UJ (eV)-0.20.00.20.40.60.81.0Energy (eV/atom)NM B1 NM Bh FM B1 AFM B1 AFM Ortho 2 2.5 3 3.5 40.00.1NM B1 NM Bh FM B1 AFM B1 AFM Ortho(a) LDA+U FIG. 1. (Color online) T=0 K total energy of the stoichiometric compound CrN in different structural and magnetic conﬁgurations as a function of the U-Jterm in the (a) LDA +Uand (b) GGA +Uschemes. The energy of the AFM Ortho-CrN phase is used as reference. The vertical dash-dotted line indicates the U-Jused in this paper. The crystal structures to the right show the various investigated atomic and magnetic arrangements with the blue (white) balls indicating Cr (N) atoms. The insets show the total energies of FM B1-CrN and AFM B1-CrN with respect to AFM Ortho-CrN phase. from 0 to 6 eV . For the sake of clarity, only a few structures are listed. It becomes obvious that both standard LDA andGGA-PBE [ 46] predict the NM Bh-CrN phase (TaN prototype, space group C2mm) to be the ground state of CrN, which disagrees with experimental observations of AFM Ortho-CrN[7]. The energy difference between AFM Ortho-CrN and AFM B1-CrN becomes smaller as the U-Jvalue increases. Our results suggest that the value of U-Jshould be larger than 1.5 eV (LDA) or 0.25 eV (GGA) in order to obtain AFMOrtho-CrN as the ground state. An accurate determination oftheUparameter is difﬁcult from both experiment as well as theory, as discussed in detail by Alling et al. in Ref. [ 22]. In order to allow a thorough comparison with the previous results,we adopted the value of U-J=3 eV also for the present DFT+Ucalculations. To demonstrate the impact of the onsite Coulomb interaction on the vibrational properties, wefurthermore applied the conventional GGA-PBE (without theUterm) for the computation of elastic constants and phonon properties. Eventually, the impact on the phase transitiontemperature for choosing an even larger value for U-J, i.e., U-J=4 eV, is discussed in terms of total and magnetic energy contributions. The energy convergence criterion for electronic self- consistency was set to 0.1 meV /atom. The Monkhorst-Pack scheme [ 47] was used to construct k-meshes of 3 ×3×3 (128-atom supercells), 3 ×4×5 (96-atom supercells), 5 × 5×5 (64-atom supercells), 12 ×12×12 (8-atom cells), and 21 ×21×21 (2-atom cells). The elastic constants were calculated using the stress-strain approach discussed in detailin a previous paper [ 40]. B. Phonon calculations The phonon calculations for NM, FM, and AFM B1-CrN were performed employing 2 ×2×2 supercells consisting of 64 atoms constructed from a conventional face-centeredcubic cell with 8 atoms. For AFM Ortho-CrN, 2 ×3×2 supercells with 96 atoms were created from its unit cell with8 atoms. Supercell size convergence tests were performed forNM and FM B1-CrN using up to 4 ×4×4 supercells (based on primitive cells with 2 atoms) containing 128 atoms. The64-atom SQS adopted to simulate the PM state in B1-CrN wasalso used for the corresponding phonon calculations. Note thatthe computation of vibrational properties for PM materials isa tremendous task due to the delicate coupling of magneticand atomic degrees of freedom [ 48]. We use here a recently developed SSA procedure [ 37], which allows the computation of forces in PM materials. Within the SSA approach, the SSAf o r c eo na na t o m jis given as the gradient on the SSA free energy surface F SSAas [37] FSSA j=−∂FSSA ∂Rj=/summationdisplay mpmFHF j({Rj},σm), (1) where FHF j({Rj},σm) are the Hellmann-Feynman forces for an individual magnetic conﬁguration σmandpm= exp[−EBO({Rj},σm) kBT] denotes the Boltzmann weights with EBO the Born-Oppenheimer energy surface. As discussed in Ref. [ 37], the summation over different magnetic conﬁgu- rations σmin the above equation can be expressed by a sum over lattice symmetry operations. In this paper, weemploy such a summation over lattice-symmetry equivalentforces. After applying the symmetry operations, these forcescorrespond to locally inequivalent magnetic conﬁgurations andallow one to perform the SSA procedure based on a singlemagnetic SQS structure. In the PM regime, it is typicallysufﬁcient to restrict the sum in Eq. ( 1) to completely disordered conﬁgurations, since they dominate the partition sum (i.e., theweights p m). Usually about 50–100 magnetic conﬁgurations yield converged forces [ 37]. Note that the treatment of mag- netic partially disordered conﬁgurations requires advancedsampling techniques, which are beyond the scope of this 184102-3LIANGCAI ZHOU et al. PHYSICAL REVIEW B 90, 184102 (2014) paper and will be discussed elsewhere [ 49]. In this paper, the magnetic disordered conﬁgurations mare constructed by all 3NCartesian (positive and negative) displacements for the given SQS supercell, resulting in 384 magnetic conﬁgurations. For the purpose of comparison, we additionally employed theconventional method for computing PM phonons [ 50], i.e., we allow atomic relaxations in the PM state which results invirtual displacements at T=0 K. The consequences of this approach and the necessity to employ the SSA technique willbe discussed later. Also, the impact of choosing a differentSQS will be discussed. The ﬁnite-displacement method im-plemented in the phonopy [ 51] combined with V ASP was used to calculate the real-space force constants and correspondingphonon properties. The residual forces (background forces)in the unperturbed supercell were subtracted from the forcesets of the displaced structures. The summation over latticesymmetry equivalent forces according to the SSA scheme [ 37] has been carried out by the Phonopy software [ 52]. In order to benchmark the direct-force constant method, the NM B1-CrN,FM B1-CrN, AFM B1-CrN, and AFM Ortho-CrN real-spaceforce constants were additionally calculated within the densityfunctional perturbation theory framework [ 53] as implemented in the V ASP code. The phonon frequencies were subsequently calculated using the Phonopy code. The phonon densitiesof states (DOS) from both methods (direct-force constantand perturbation theory) coincide without any noticeable differences (0.5 and 2 meV /atom difference in the free energy evaluated using both methods at 0 and 1000 K, respectively).Anharmonic phonon-phonon interactions which can becomeimportant close to the melting point [ 54] are not relevant for the present paper. We will show this explicitly in Sec. III C by comparing our calculations with the molecular dynamicssimulations performed in Ref. [ 36], which by default include anharmonicity. When LDA +Uor GGA +Uis used to treat the strong cor- relation effects, a small band gap opens for AFM Ortho-CrNand PM B1-CrN. Consequently, the dipole-dipole interactionsincluding properties of the Born effective charge tensor anddielectric tensor, which result in a longitudinal-transversaloptical phonon branches (LO-TO) splitting [ 53], should be taken into account during the phonon calculations. For thatpurpose, the dipole-dipole interactions were calculated fromthe linear response method within the density functionalperturbation theory framework as implemented in V ASP at the /Gamma1point of reciprocal space. The contribution of nonanalytical term corrections to the dynamical matrix developed by Wanget al. [55] was considered by the following formula: /Phi1jk αβ(M,P )=φjk αβ(M,P )+1 N4πe2 V[qZ∗(j)]α[qZ∗(k)]β qε∞q,(2) where φjk αβis the contribution from short-range interactions based on supercell, Nis the number of primitive unit cells in the supercell, Vis the volume of the primitive unit cell,qis the wave vector, αandβare the Cartesian axes, Z∗(j) is the Born effective charge tensor of the jth atom in the primitive unit cell, and ε∞is the high-frequency static dielectric tensor, i.e., the contribution to the dielec-tric permittivity tensor from the electronic polarization. Inprinciple, the Born effective charge matrix Z ∗should fulﬁllthe symmetries of the underlying crystal, i.e., for the given caseZαβ(i)=0f o r α/negationslash=β(off-diagonal elements), and Zαα(Cr)=−Zαα(N)≡Z(due to charge neutrality [ 56]). However, due to the broken (magnetic) symmetries of the givenmagnetic random structure, the Born effective charge matrixelements Z /vectorσ γδ(j) for each individual magnetic conﬁguration /vectorσdo not fulﬁll these conditions. In the spirit of the SSA, we therefore average over the crystal symmetry equivalent matrix elements, i.e., Zsym αβ(i)=/summationtext γδ,jSβδ αγZ−→σ γδ(j), where Sβδ αγ denote the crystal lattice symmetry operations (translation and rotational operations). This summation is equivalent to thesummation over different magnetic snapshots (see Eqs. (5)and (6) in Ref. [ 37]). It is similar for the dielectric tensor, i.e.,ε sym αβ=/summationtext γδSβδ αγε−→σ γδ, where ε≡εsym ααandεsym αβ=0f o r α/negationslash=β. The derived values for ε=15.4/17.3 and Z= 4.1/4.2 (GGA +U/LDA+U) are in fair agreement with the experimental values of 22 ±2 and 4 .4±0.9[23], which have been employed in the previous work by Shulumba et al. [36]. C. Thermodynamic properties Once the phonon DOS is obtained, the vibrational energy and its effect on thermal properties can be directly evaluated. Incombination with ﬁrst-principles calculations, the Helmholtzfree energy F(V,T) is the most convenient choice for a thermodynamic potential, since it is a natural function of V andT F(V,T)=E 0(V)+Fel(V,T)+Fvib(V,T)+Fmag(V,T), (3) where E0(V) is the internal energy at 0 K obtained from the equation of state [ 57],Fel(V,T) and Fvib(V,T)a r et h e thermal electronic and lattice contributions to the free energy,respectively. Further details on the ab initio calculations of F el(V,T) and Fvib(V,T) are discussed in Ref. [ 58] and references therein. Here, Fmag(V,T) is the magnetic free energy which in this paper is considered within the mean-ﬁeldapproximation as [ 59] F mag≈−kBTln(M(T,V)+1), (4) where M(T,V) is the magnitude of the local magnetic moment (in units of μB) and kBis the Boltzmann constant. In this paper, we used the averaged magnitude of the local magneticmoment M(T=0K,V=V 0) at 0 K at the ground state volume V0to predict the magnetic entropy. The local magnetic moments derived from the magnetically disordered conﬁgu-rations naturally include the implicit temperature-dependenteffect of magnetic disorder. Furthermore, our test calculationsrevealed that the inclusion of the volume dependence hasno signiﬁcant impact on our main results and conclusionsand will be therefore not considered in the following. Theinclusion of explicit temperature-dependent effects on the localmagnetic moments in complex alloys is still in its infancy.Progress for pure elements such as Fe and Ni by meansof band structure models [ 60], dynamical mean-ﬁeld theory [61], and extended Heisenberg models in combination with constrained spin-DFT and coherent potential approximation(CPA) [ 62] provides in this respect promising future routes. The average magnetic moments based on GGA, GGA +U, 184102-4STRUCTURAL STABILITY AND THERMODYNAMICS OF . . . PHYSICAL REVIEW B 90, 184102 (2014) and LDA +Uschemes, which in the following are used in Eq. ( 4), are 2 .48,2.90,and 2.83μB, respectively, agreeing well with previous theoretical values [ 22,27]. Consistent with the SSA treatment of phonons in the PM state, the above mean-ﬁeld expression does not include magnetic short-range ordercontributions or noncollinear magnetic structures. It charac-terizes therefore the scenario of uncorrelated magnetic spinmoments. It has been shown earlier [ 63] that, for completely uncorrelated magnetic systems, noncollinear and collinearpictures provide the same thermodynamic descriptions. Wetherefore adopt the same approach as recently employed inRef. [ 36], i.e., Eq. ( 4) above. Nevertheless, short-range order as well as noncollinear structures are likely relevant for theenergetics close to the N ´eel temperature, as suggested by recent magnetic cluster expansion techniques [ 64–66]. Its incorporation would be very valuable but requires techniquesbeyond the present scope of this paper, such as the magneticcluster expansion or quantum Monte Carlo methods (see, e.g.,Ref. [ 59] for a recent overview of the different techniques), and will be therefore left for a future contribution. D. Experimental details Since no experimental study, to our knowledge, has been devoted so far to the lattice thermal expansion coefﬁcient α, this property has been measured for this paper. A single-phase face-centered cubic CrN thin ﬁlm was deposited onSi (100) substrates (7 ×21 mm 2) using reactive magnetron sputtering. The deposition was carried out at a depositiontemperature of 743 K in an Ar and N 2gas atmosphere of a total pressure of 0.4 Pa and a constant Ar /N2ﬂow ratio of 2/3. A target power of 250 W and a 3" Cr target (purity 99.9%) were used. During the deposition, a bias potentialof−70 V was applied to the substrates to ensure dense ﬁlm morphology. The ﬁlm thickness was measured usingcross-sectional scanning electron microscopy. The biaxialstress in CrN was recorded as a function of temperature usingthe wafer-curvature method [ 67]. The wafer-curvature systemwas operating with an array of parallel laser beams and a position sensitive charge-coupled device detector. The samplewas heated by a ceramic heating plate with a constant heatingrate of 5 K /min from room temperature to 518 K under vacuum conditions of /lessorequalslant10 −4mbar. The maximum temperature was chosen to be clearly below the deposition temperature toavoid thermally activated processes in the ﬁlm (e.g., recoveryof deposition-induced defects) and the substrate and thus toguarantee pure thermoelastic behavior. The ﬁlm stress wasdeduced from the sample curvature 1 /Raccording to σ=Mh 2 6Rtf, (5) withM∼180 GPa being the biaxial modulus of the substrate, h=380μm the substrate, and tf=1.50μm the ﬁlm thick- nesses. Due to the mismatch in the coefﬁcients of thermalexpansion of CrN and Si, the temperature change results inthe formation of thermal stresses in the ﬁlm. When the elasticmoduli are taken as constants in the given temperature range,the linear thermal expansion coefﬁcient αof CrN thin ﬁlm is related to that of substrate (Si) as α(T)=α si(001) (T)+1−vCrN ECrN·dσ dT, (6) withECrN=330 GPa, vCrN=0.22 [68,69], and the thermal expansion coefﬁcient of substrate (Si) is expressed as [ 70] αsi(001) (T)=(3.725{1−exp[−5.88×10−3(T−124) +5.548×10−4T]})×10−6(Tin K).(7) III. RESULTS AND DISCUSSION A. Elastic constants and mechanical stability The elastic constants of the different studied structures as calculated by the GGA, GGA +U, and LDA +Uschemes are listed in Table I. It can be seen that most of the elastic constants derived from GGA and GGA +Uare smaller than TABLE I. T=0 K elastic constants in GPa of CrN within the framework of the GGA, GGA +U,a n dL D A +Umethods including a comparison with experimental values. Structure C11 C22 C33 C12 C13 C23 C44 C55 C66 GGA 580 210 8 NM B1 GGA +U 477 266 −120 LDA+U 641 260 −59 GGA 348 117 74 FM B1 GGA +U 508 108 156 LDA+U 589 128 162 GGA 535 535 567 126 86 86 150 94 94 AFM B1 GGA +U 555 555 389 98 66 66 166 129 129 LDA+U 696 696 722 113 76 76 174 124 124 GGA 516 115 116 GGA+U 538 88 143 PM B1 LDA +U 649 99 145 Experiment [ 68,69] 540 27 88 GGA 439 529 495 195 110 114 221 125 103 AFM Ortho GGA +U 444 524 497 169 84 92 223 151 137 LDA+U 503 580 626 228 87 102 275 155 137 184102-5LIANGCAI ZHOU et al. PHYSICAL REVIEW B 90, 184102 (2014) those obtained from the LDA +Uscheme. This is consistent with the fact that GGA frequently overestimates latticeparameters leading to an underestimation of binding energies,elastic properties, and phonon frequencies for most materials[71]. In contrast, LDA (or LDA +U) tends to overestimate binding as expressed by smaller lattice parameters andhigher elastic constants and phonon frequencies when compared with experimental or GGA and GGA +Uvalues. A comparison with the limited available experimental datareveals good agreement for the C 11component (in particular for GGA +U), while C12andC44seem to be overestimated by all the LDA +U, GGA, and GGA +Utheoretical predictions. However, it might be that the substantially too lowexperimental C ijconstants are a consequence of the fact that the measurements have been performed on a polycrystalline sample at room temperature, i.e., the measured elastic constants may be inﬂuenced by soft grain boundaries [ 69]. The mechanical stability of any crystal requires the strain energy to be positive, which implies that the whole set ofelastic constants C ijmust satisfy the Born-Huang stability criterion [ 38]. Using this requirement, Table Ishows that CrN is mechanically stable within the GGA scheme for all magnetic and structural conﬁgurations studied here in consistency with previous results [ 28,29,32]. In contrast, we observe that C44derived from the LDA +Uand GGA +U schemes suggest NM B1-CrN to be mechanically unstable(C 44<0). This result underlines the importance of strong correlation effects, which were not considered in previouselastic constant calculations using standard LDA or GGA approaches. Finally, we observe AFM B1-CrN to show a small tetragonal distortion, which is reﬂected in the number ofnonequivalent elastic constants: C 11,C12,C13,C33,C44, and C66in agreement with previous results [ 28]. B. Dynamical stability As discussed above, a necessary condition for a structure to be dynamically stable is that it is stable against all possible small perturbations of its atomic structure, i.e., that all phonon frequencies are real. The GGA, GGA +U, and LDA +U phonon spectra of NM B1-CrN are presented in Fig. 2(a). Imaginary acoustic branches at the XandWpoints of the Brillouin zone suggest an internal instability of NM B1-CrN.These phonon anomalies are due to the high electron DOSat the Fermi level of NM B1-CrN [ 19], which causes the existence of soft phonon modes at these points. We evaluated the phonon DOS of NM B1-CrN for several pressures up to 900 GPa to check whether applying external pressure caneliminate these imaginary phonons. For the sake of clarity, onlythe representative LDA +Uresults are presented in Fig. 2(b), yielding the following tendency: NM B1-CrN is dynamicallyunstable even under high pressures, indicating the NM B1-CrNcannot be stabilized by high pressure. In order to check the phase stability of CrN for various magnetic states, the phonon spectra of CrN from GGA,GGA+U, and LDA +Uwith ordered magnetic states are presented in Fig. 3. The comparison of the phonon spectra of FM B1-CrN, AFM B1-CrN, and AFM Ortho-CrN fromGGA, and GGA +Uor LDA +Ureveals that the phonon spectra are signiﬁcantly shifted to higher frequencies for thelatter approximations. The GGA and GGA +Ubased phonon-10-505101520 GGA GGA+U LDA+U Γ X Γ LX W LFrequency (THz)(a) ImaginaryNM B1 -20 -10 0 10 20 30 Phonon DOS Frequency (THz)P=0 GPa P=300 GPa P=900 GPa(b) LDA+U Imaginary FIG. 2. (Color online) (a) Phonon spectra of NM B1-CrN calcu- lated using GGA, GGA +U,a n dL D A +U, and (b) phonon DOS as a function of pressure for NM B1-CrN from the LDA +Uscheme. The gray shaded region highlights imaginary frequencies. frequencies are always lower than those derived from the LDA+Uscheme, a trend already reﬂected by the softer elastic constants when GGA or GGA +Uis used. For FM and AFM B1-CrN, when no Ucorrection is used, a softening tendency for the phonon branch around the Xpoint is observed, and for AFM B1-CrN, phonon frequencies even decrease to negativevalues. The dynamical instability, not observed via the elasticconstants calculations, indicates once more the necessity ofexplicit phonon calculations for the stability analysis. Thisphonon anomaly observed for the GGA calculations originatesfrom the high electron DOS at the Fermi level. When the onsiteCoulomb repulsion Uin the DFT +Uscheme is switched on, the electron DOS at Fermi level signiﬁcantly decreases for theFM B1-CrN, although no gap opens [ 21]. Here, one should note that despite the fact that the DFT +Uapproximation does not result in a gap opening at the Fermi level for NM and FMB1-CrN, as shown in Refs. [ 19,21], it signiﬁcantly increases the stability of FM and AFM B1-CrN as measured by phononfrequencies around the Xpoint. No imaginary frequencies in the phonon dispersion curves are observed independent ofwhether the GGA +Uor LDA +Uscheme is used, implying 184102-6STRUCTURAL STABILITY AND THERMODYNAMICS OF . . . PHYSICAL REVIEW B 90, 184102 (2014) 05101520Frequency (THz)GGA GGA+U LDA+U FM B1 Γ X Γ LX W L(a) 05101520Frequency (THz)GGA GGA+U LDA+U AFM B1 Γ X Γ LX W L(b) 05101520Frequency (THz)GGA GGA+U LDA+U AFM Ortho Γ XS Y Γ Z(c) FIG. 3. (Color online) Phonon spectra of (a) FM B1-CrN, (b) AFM B1-CrN, and (c) AFM Ortho-CrN calculated using theGGA, GGA +U,a n dL D A +Uschemes. that FM B1-CrN, AFM B1-CrN, and AFM Ortho-CrN are all dynamically stable and represent hence potential (meta)stablephases. In contrast to the ordered magnetic states of FM B1-CrN, AFM B1-CrN, and AFM Ortho-CrN, the magnetic state inPM B1-CrN is disordered. As a ﬁrst step, we allow foratomic relaxations at 0 K. Since at ﬁnite temperatures the0 5 10 15 20Phonon DOS Frequency (THz) fully relaxed unrelaxed (a) 0 5 10 15 20Phonon DOS Frequency (THz) SQS-1 SQS-2 (b) FIG. 4. (Color online) Phonon DOS from the LDA +Uscheme (a) for fully relaxed and unrelaxed SQS and (b) for two SQSs with different SRO parameters. magnetic ﬂuctuations are usually faster compared to the atomic motion, these displacements can be considered asartiﬁcial. In order to test the effect of such artiﬁcial staticdisplacements, the atomic positions are ﬁxed to the idealB1 sites, and cell shape remains cubic (we refer to thisstructure as “unrelaxed” hereafter). The phonon DOS corre-sponding to unrelaxed and fully relaxed SQS are presented inFig. 4(a). Clearly, the phonon DOS of the unrelaxed SQS exhibits some different features than that of the fully relaxedone, which means that the artiﬁcial static displacements havesome impact on the phonon calculations. A closer inspection ofthe phonon dispersion of the unrelaxed structure reveals someimaginary phonon modes at the /Gamma1point. In order to elucidate if these imaginary frequencies are physical, we employ inthe following the recently developed SSA method, Eq. ( 1), to compute the phonon frequencies in the PM regime. Infact, no imaginary phonon frequencies are obtained (see thered dashed lines in Fig. 5) revealing the applicability of the method and the fact that the imaginary phonon modes along 184102-7LIANGCAI ZHOU et al. PHYSICAL REVIEW B 90, 184102 (2014) 05101520Frequency (THz)LDA+U GGA+U GGA Γ XW K Γ LU W L K FIG. 5. (Color online) Phonon spectra of PM B1-CrN simulated with a 64-atom SQS from the GGA, GGA +U,a n dL D A +U schemes in combination with SSA approach, together with exper- iments at the /Gamma1point from Raman and infrared measurements (open circles [ 23]). the acoustic dispersions are indeed caused by the artiﬁcial T=0 K relaxations. In order to further analyze the impact of artiﬁcial T=0 K relaxations on the speciﬁc SQS structure, we have performed similar calculations for a second SQS with a different spin arrangement. Figure 4(b) shows that the phonon DOS for the two SQS almost fully coincide, suggestingthat a particular spin arrangement and consequently a speciﬁcspin-induced relaxation of atoms at T=0 K in the SQS of PM B1-CrN does not signiﬁcantly affect the phonon DOS andhence thermodynamic properties. Since the SSA computedphonon DOS does not show any imaginary frequencies anddoes not require any (unphysical) T=0 K relaxations, we will proceed with the SSA obtained phonon DOS andthermodynamic properties for the following thermodynamicanalysis. As discussed above, supercell convergence tests forNM and AFM variants have revealed that a 2 ×2×2 supercell is sufﬁcient for an accurate description within the consideredtemperature range. This is in agreement with the ﬁndings ofShulumba et al. [36], who reported convergence with respect to the supercell size for PM B1-CrN for a supercell with 64atoms (2 ×2×2), which we will employ in the following. Figure 5presents the phonon spectra of PM B1-CrN derived from GGA, GGA +U, and LDA +Uschemes employing the SSA method and in comparison with available experimentaldata (open circles [ 23]) at the /Gamma1point. It demonstrates that the transverse branches derived from DFT +Uare signiﬁcantly shifted to higher values, and the LO-TO splitting happensat the /Gamma1point, when comparing to the result from GGA scheme. The obtained dispersion from DFT +Uis in excellent agreement with experimental data (blue circles), especially theresults derived from the GGA +Uscheme are very close to experimental data. The accurately predicted transverse andlongitudinal optical phonon frequencies at the /Gamma1point further validate the reliability of the SSA scheme. The phonon spectraderived from GGA, GGA +U, and LDA +Uin combination with the SSA scheme conﬁrm that PM B1-CrN is dynamicallystable.0 200 400 600 800 1000-200-150-100-50050100Fvib(meV/atom) Temperature ( K)SSA DLM-MD-TDEP FIG. 6. (Color online) Comparison of the vibrational free ener- gies for AFM Ortho-CrN (blue lines) and PM B1-CrN (black lines) under the framework of LDA +U. The solid lines denote the results from the SSA approach in this paper and dotted lines denote theDLM-MD-TDEP method [ 36]. C. Thermodynamic stability and thermodynamic properties of CrN The free energy has been evaluated according to the treatment described in Sec. IIC, employing in particular the SSA method for the PM phases. Due to the opening of asmall gap in AFM Ortho-CrN and PM B1-CrN as obtainedfrom GGA +Uand LDA +U[24], the thermal electronic contribution F el(V,T) to the Helmholtz free energy can be ignored for AFM Ortho-CrN and PM B1-CrN. First, we compare the SSA results for the energies to previous results from the DLM-MD-TDEP method [ 36]t o verify the reliability of our method and also to check theinﬂuence of anharmonic effects on the vibrational free energy.The comparison between this paper and the results from theDLM-MD-TDEP method [ 36] is presented in Fig. 6.A s the latter method is based on MD simulations, anharmoniccontributions are implicitly included. This allows us to evaluatethe importance of anharmonic effects for the consideredtemperature regime. Figure 6demonstrates that the vibrational free energies of PM B1-CrN from both methods (SSA andDLM-MD-TDEP) fully coincide with each other and that thereis no noticeable difference even at high temperatures. Thelargest deviation of about 5 meV /atom is observed for AFM Ortho-CrN at 1000 K. Keeping in mind that the phase transitionunder consideration occurs at room temperature, we obtain anexcellent agreement with the previous work in the relevanttemperature regime, revealing that anharmonic contributionsare not decisive for the considered transition. Next we concentrate on the impact of the xc-functional. The differences of harmonic vibrational free energies as a functionof temperature between AFM Ortho-CrN and PM B1-CrN inthe GGA, GGA +U, and LDA +Uschemes are presented in Fig. 7(a). For all considered scenarios, the vibrational contribution to the free energy favors the PM B1-CrN phase.This contribution becomes smaller when the strong correlationeffects are taken into account by using the Hubbard CoulombtermU. 184102-8STRUCTURAL STABILITY AND THERMODYNAMICS OF . . . PHYSICAL REVIEW B 90, 184102 (2014) 0 200 400 600 800 1000-70-60-50-40-30-20-100FPM B1 vib-FAFM Ortho vib(meV/f.u.) Temperature ( K)GGA GGA+U LDA+U(a) 0 100 200 300 400 500 600 700 800 900 1000-19.2-19.0-18.8-18.6-18.4-18.2-17.2-17.0-16.8(b) GGA+U LDA+UHelmholtz free energy (eV/f.u.) Temperature ( K)GGA TGGA s =428 KTLDA+U s =370 KTExp N ~280 K TGGA+U s =300 K PM B1 without -TSmag PM B1 with -TSmag AFM Ortho FIG. 7. (Color online) (a) The differences of vibrational free energy between AFM Ortho-CrN and PM B1-CrN in the GGA, GGA+U,a n dL D A +Uschemes and (b) calculated phase transition temperature from Helmholtz free energies of AFM Ortho-CrN and PM B1-CrN by taking lattice vibrational and magnetic contributions into account under the framework of GGA, GGA +U,a n dL D A +U schemes. The dotted lines denote the Helmholtz free energies of PM B1-CrN without magnetic contributions. We now include the magnetic entropy, Eq. ( 4), into our free energy computations. The results including both vibrationalas well as the magnetic entropy terms evaluated usingGGA, GGA +U, and LDA +Uschemes, are presented in Fig. 7(b) as a function of temperature for AFM Ortho-CrNand PM B1-CrN phases. The critical temperature for the structural transition between AFM Ortho and PM B1 CrN isT s=428 K for GGA, Ts=300 K for GGA +U, andTs= 370 K for the LDA +Uscheme. The excellent agreement of the latter with the previous theoretical results [ 36] reveals once more the reliability of our SSA approach. A particularlyimportant conclusion is that the GGA +Uvalue turns out to be signiﬁcantly closer to the experimental value comparedto the previous LDA +Uestimations. This improvement of GGA+Ucompared to the LDA +Udata is remarkable and of similar magnitude and importance as the inclusion ofvibrational contributions [ 22,36]. The inclusion of vibrational free energy contributions is considerable and shifts the transition temperature, e.g., forGGA, by almost 500 K [ 22]. The inclusion of magnetic entropy turns out to be similarly important. To elucidate theimpact of magnetic contributions, we included in Fig. 7(b) also the free energy curves without the magnetic entropy termS magin Eq. ( 4) (dotted lines). Consequently, AFM Ortho-CrN becomes energetically preferred over the PM B1-CrN phase inthe whole investigated temperature range. In agreement withprevious papers [ 22,36], we can therefore conclude that both magnetic and vibrational contributions are equally importantto accurately predict the transition temperature. The remainingdifference between predicted and experimental values for thephase transition temperature may be related to the approximate treatment of magnetism (neglected noncollinear magnetic conﬁgurations, magnetic entropy extracted from mean-ﬁeldapproximation) or to the inherent DFT approximation, i.e., thetreatment of the xc-functional. In order to shed some light on the impact of the U-J parameter on the phase transition, we brieﬂy discuss theexpected changes upon choosing a larger U-Jparameter. In Fig. 8, the impact of the U-Jparameter on the phase transition temperature is presented. The transition temper-ature is evaluated based on the total and magnetic energycontributions. The computationally demanding vibrationalcontribution discussed above is indicated by the arrows atU-J=0 and U-J=3 eV . As can be seen from Fig. 8,t h e 0123456100200300400500600700800Etotal-TSmag Etotal+Fvib-TSmagPhase transition temperature (K) U-J (eV)vibrations ~44KGGA+U FIG. 8. (Color online) Impact on phase transition temperature within GGA +Ufor different Uparameters. 184102-9LIANGCAI ZHOU et al. PHYSICAL REVIEW B 90, 184102 (2014) phase transition temperature is less affected by changes in U-J at larger U-Jvalues. This can be traced back to compensating total energy and magnetic entropy contributions. Changing theU-Jparameter, e.g., from 3 to 4 eV only slightly shifts the theoretical phase transition temperature by /223c44 K, i.e., even closer to the experimental one. In Fig. 7(a), it is shown that an increase of the U-Jparameter [from 0 eV corresponding to GGA to 3 eV corresponding to GGA +Uin Fig. 7(a)] reduces the vibrational contribution to the phase transitiontemperature, which is also indicated by the arrows in Fig. 8. This suggests that the vibrational contribution for U-J=4e V , which is not explicitly evaluated here due to the heavyadditional computational costs, will likely further decrease thephase transition temperature, although presumably to muchless extent. Eventually, only a full U-J=4 eV calculation taking into account all contributions consistently (includingvibrations) can conﬁrm this assumption. The phase transition from AFM Ortho-CrN to PM B1- CrN induced by temperature can be alternatively triggered bypressure. In order to investigate the pressure effect on the phasetransition, the Gibbs energy is computed via the Legendretransformation as G(p,T)=F(V p,T)+pVp, (8) where Fis the Helmholtz free energy, Eq. ( 3), evaluated within the framework of the quasiharmonic approach at severalvolumes including the magnetic entropy contributions fromEq. ( 4) for PM B1-CrN. In Eq. ( 8),pis the pressure, and V p is the volume corresponding to this pressure. Using the Gibbs energies of AFM Ortho-CrN and PM B1-CrN based on theGGA+Uand LDA +Uschemes, the p-Tphase diagram was derived (Fig. 9). The transition temperature increases with pressure in agreement with experimental and theoretical 0 5 10 15 20 25 Pressure (GPa)02004006008001000Temperature (K)GGA+ULDA+Umag only vibrations Experiment AFM OrthoPM B1 FIG. 9. (Color online) Calculated pressure-temperature phase di- agram of CrN based on the GGA +U(black line) and LDA +U (blue lines) schemes compared to experiment (open circles [ 12]a n d squares [ 15]). The solid lines represent the phase diagram based on the relevant spectrum of excitations: total energy, vibrational, and magnetic entropy contributions. For LDA +U, the dashed line shows the phase diagram if only the total energy and magnetic excitationsare included to emphasize the impact of vibrational entropy.0 200 400 600 800 1000024681012Linear thermal expansion coefficient (10-6K-1) Temperature ( K)PM B1 LDA+UGGA+U Experiment FIG. 10. (Color online) Linear thermal expansion coefﬁcient α(T) of PM B1-CrN computed within the GGA +U(black solid line) and LDA +U(blue dashed line) schemes compared to experimental data (open squares) as obtained in this paper. results reported in literature. The transition temperature at p=0 GPa obtained from experiment [ 7,12,15]v a r i e sf r o m 200 to 287 K, showing an uncertainty of up to 87 K. Within this error bar, our theoretical predictions based on the GGA +U method are in excellent agreement with the experimental workand yielding a signiﬁcant improvement as compared with thepreviously reported theoretical results. It should be noted thatthe transition temperatures at p=0 GPa are approximately 10 K lower than the results shown in Fig. 7. This is due to the fact that the free energy in Eq. ( 3) is evaluated within the quasiharmonic approximation, while the results presented inFig.7are obtained from the harmonic approach. This implies that the quasiharmonic contribution (i.e., caused by thermalexpansion) is not as important as, for example, the strongcorrelation effect or the magnetic entropy contribution withinthe considered temperature-pressure regime. On the other hand, thermal expansion is a key thermo- dynamic property of PM B1-CrN when it is applied as ahigh-temperature coating material. If compared to experiment,the derivative of the linear thermal expansion coefﬁcient α(T) provides an even more sensitive measurement of the accuracyachievable by the theoretical predictions. The calculated α(T) of PM B1-CrN is compared to our experimental data in Fig. 10. The results are in good agreement around room temperature,and the discrepancy increases with raising temperature. It wasrecently demonstrated that α(T) of CrN strongly depends on the grain size [ 72]. Depending on the thin ﬁlm microstructure, α(T) varies in the range of 6 .7×10 −6/K and 9 .8×10−6/K, which for this material somewhat complicates a distinct com-parison with our predictions. Considering these complications,the overall agreement with experiment is very reasonable. Finally, in Fig. 11, we plot the speciﬁc heat capacity C P(T) at constant (ambient) pressure as a function of temperaturepredicted from our GGA +Uand LDA +Ucalculations together with available experimental data from the literature[73]. The GGA +Uvalues are slightly closer to the ex- perimental data. This is consistent with the ﬁndings for the 184102-10STRUCTURAL STABILITY AND THERMODYNAMICS OF . . . PHYSICAL REVIEW B 90, 184102 (2014) 0 200 400 600 800 1000 1200 1400051015202530 GGA+U LDA+UCP(J/mole-atom.K) Temperature ( K)PM B1 FIG. 11. (Color online) Heat capacities of PM B1-CrN in the GGA+U(black solid line) and LDA +U(blue dashed line) schemes compared to experimental values (open triangles [ 73]). phase transition temperature discussed above; however, the differences between LDA +Uand GGA +Uare only minor. The literature data exhibit a peak near the phase transitiontemperature ( /223c280 K), due to the N ´eel transition, which has not been accounted for in this paper. This peak originatesfrom the magnetic phase transition and can be resolved onlywhen the magnetic contribution is exactly evaluated by, forexample, the Heisenberg model for spin interactions [ 74]. Nevertheless, our predictions are reasonably accurate from0 K to about 200 K and also above 600 K. IV . CONCLUSIONS The phase stability of different structural and magnetic conﬁgurations of stoichiometric CrN is studied systematicallyby ﬁrst-principles calculations based on the GGA, GGA +U, and LDA +Uschemes. In combination with our recently developed SSA procedure, the phonon contributions in PMmaterials are computed. A comparison of the three xcapproximations demonstrates that strong correlation effectshave a signiﬁcant impact on the mechanical and phase stabilityof CrN. The elastic constants and phonon spectra show thatthe NM B1-CrN phase is dynamically unstable even under high pressures, due to the high electron DOS at the Fermilevel. The (meta)stability of the FM and AFM B1-CrN phaseis signiﬁcantly improved when strong correlation effects areconsidered using the DFT +Uapproach. Including the vibrational, electronic, and magnetic free energy contributions, the results of our LDA +U-SSA based approach agree well with previous LDA +U-MD simulations. By performing ﬁnite-temperature GGA +Usimulations for CrN, we show that the treatment of the xc-functional, i.e.,GGA+Uversus LDA +U, is decisive to predict accurate phase transition temperatures in CrN. In particular, we ﬁndthat GGA +Usigniﬁcantly improves the transition tempera- ture compared to previous LDA +Upredictions. The phase transition between AFM Ortho-CrN and the PM B1-CrN phaseis predicted to be 293 K at ambient pressure, being in excellentagreement with the experimental value of 200–287 K. The im-pact of the xc-functional is similar in magnitude to the impactof vibrational contributions to the phase stability. The linearthermal expansion coefﬁcient α(T) and the heat capacity C p of PM B1-CrN as a function of temperature are obtained from experimental measurements and ab initio calculations. The comparison between the experimental results and predictionsfor these thermodynamic properties reveals good agreementand further conﬁrms the reliability of our theoretical method. ACKNOWLEDGMENTS The ﬁnancial support by the START Program (Y371) of the Austrian Science Fund (FWF) and the Austrian FederalMinistry of Economy, Family, and Youth and the NationalFoundation for Research, Technology, and Development isgratefully acknowledged. Funding by the European Re-search Council under the EU’s 7th Framework Programme(FP7/2007–2013)/ERC Grant Agreement No. 290998 and bythe Collaborative Research Center SFB 761 “Stahl- ab initio ” of the Deutsche Forschungsgemeinschaft is also gratefullyacknowledged. First-principles calculations were carried outpartially on the cluster supported by the ComputationalMaterials Design Department at the Max-Planck-Institut f ¨ur Eisenforschung GmbH in D ¨usseldorf and the Vienna Scientiﬁc Cluster (VSC). 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PhysRevB.77.241201.pdf	Magnetic interactions of Cr-Cr and Co-Co impurity pairs in ZnO within a band-gap corrected density functional approach Stephan Lany, Hannes Raebiger, and Alex Zunger National Renewable Energy Laboratory, Golden, Colorado 80401, USA /H20849Received 20 February 2008; revised manuscript received 29 April 2008; published 3 June 2008 /H20850 The well-known “band-gap” problem in approximate density functionals is manifested mainly in an overly low energy of the conduction band /H20849CB/H20850. As a consequence, the localized gap states of 3 dimpurities states in wide-gap oxides such as ZnO occur often incorrectly as resonances inside the CB, leading to a spurioustransfer of electrons from the impurity state into the CB of the host, and to a physically misleading descriptionof the magnetic 3 d-3dinteractions. A correct description requires that the magnetic coupling of the impurity pairs be self-consistently determined in the presence of a correctly positioned CB /H20849with respect to the 3 d states /H20850, which we achieve here through the addition of empirical nonlocal external potentials to the standard density functional Hamiltonian. After this correction, both Co and Cr form occupied localized states in the gapand empty resonances low inside the CB. In otherwise undoped ZnO, Co and Cr remain paramagnetic, butelectron-doping instigates strong ferromagnetic coupling when the resonant states become partially occupied. DOI: 10.1103/PhysRevB.77.241201 PACS number /H20849s/H20850: 75.30.Hx, 75.50.Pp, 71.15.Mb 3dimpurities generally tend to render semiconductors in- sulating due to deep levels inside the gap,1in particular in wide-gap systems such as diluted magnetic oxides /H20849DMO /H20850.2 For the purpose of spintronics, however, it is desired to achieve spin-polarized electrons in a conductive state closeto the conduction-band minimum /H20849CBM /H20850. 3While ferromag- netic signatures are frequently observed in 3 ddoped ZnO and other wide-gap oxides, the origin and nature of the fer-romagnetism /H20849FM/H20850remains enigmatic: For example, carrier /H20849electron /H20850mediated magnetism has been assumed in ZnO:Co on the basis of the correlation of magnetism with Aldonor doping, 4or with the O 2partial pressure controlling the conductivity.5Other interpretations involve the formation of nanoclusters of the naturally magnetic Co metal,6or uncom- pensated spins at the interface between paramagnetic Co-poor and antiferromagnetic /H20849AFM /H20850Co-rich phases of /H20849Zn,Co /H20850O, formed due to spinodal decomposition. 7Even more perplexing questions about the nature of ferromag-netism in DMO have been raised by recent reports showingthat the 3 dsublattice remains paramagnetic even though the sample as a whole appears ferromagnetic, as observed, e.g.,in ZnO:Co /H20849Ref. 8/H20850and ZnO:Cu, 9and the observation that magnetism in polycrystalline thin-ﬁlm ZnO occurs evenwithout transition-metal doping. 10 In the present study, we address theoretically the possibil- ity of transition-metal /H20849TM/H20850induced ferromagnetism in single-crystal ZnO /H20849note that this type of magnetism may be overshadowed in polycrystalline thin ﬁlms or nanocrystalsby a poorly understood magnetism that is independent of TMdoping 10/H20850. In light of the unclear experimental situation, nu- merous theoretical studies emerged on 3 dimpurities in ox- ides, using mostly the local-density or generalized-gradientapproximations /H20849LDA or GGA /H20850to density functional theory /H20849DFT /H20850. 11–13However, certain oxides such as ZnO or In 2O3 have a large electron afﬁnity /H20849low CBM energy /H20850that is fur- ther exaggerated in LDA/GGA calculations where the noto-rious band-gap underestimation /H20849e.g., in ZnO, E g=0.73 eV in GGA compared to 3.4 eV in experiment /H20850is mainly due to a too low energy of the CBM.14As shown below, these sys- tematic errors cause the highest occupied level of most 3 dimpurities in ZnO to incorrectly appear as resonances inside the LDA/GGA host conduction band, leading to a spuriouscharge transfer from the 3 dimpurity level into the host bands, whereas a deep impurity level inside the gap is ex-pected from experiment. 3,15 The need for a self-consistent band-gap correction . The magnetic 3 d-3dpair interaction energies in ZnO:3 dmust be determined in the presence of corrected host band energies/H20849relative to the impurity levels /H20850, so that the correct descrip- tion of the orbital and spin conﬁguration of the impurity isrecovered during the self-consistent calculation. It is nowrecognized that Hubbard- Ucorrections 16to LDA or GGA signiﬁcantly improve the description of the TM- dstates in magnetic semiconductors,12,13,17but as these corrections in general do not sufﬁciently open the band gap,18they cannot remove the spurious hybridization of the 3 dorbitals with the host conduction band. The need for both a self-consistentcorrection of the host band-edge energies and for an efﬁcientcomputational scheme capable of calculating total-energydifferences and atomic relaxations in fairly large supercells isthe central challenge for the description of ferromagnetism in3ddoped wide-gap oxides. Due to these simultaneous re- quirements, accurate ab initio methods that avoid the band- gap problem, such as GW calculations, 14are currently not practical. We achieve here a self-consistent band-gap correction by adding to the standard GGA+ UHamiltonian empirical non- local external potentials /H20849NLEP /H20850/H9004V/H9251,lNLEPthat depend on the atomic type /H20849/H9251/H20850and the angular momentum /H20849l/H20850. This ap- proach follows the spirit of the method of Christensen,19but here we use angular-momentum-dependent /H20849“nonlocal” /H20850 potentials20that allow for more ﬂexibility in ﬁtting experi- mental band-structure properties. The NLEP correction isimplemented into the projector augmented wave /H20849PAW /H20850 formalism 21within the V ASP code22/H20849see below /H20850. The host- crystal NLEP parameters /H9004VZn,s= +9.4 eV, /H9004VZn,p= −1.2 eV, /H9004VO,s=−6.4 eV, and /H9004VO,p=−2.0 eV are obtained by ﬁtting to target properties taken from experiment23and GW calculations,14as summarized in Table I./H20849Note that negative values of /H9004Vimply an attractive potential, and posi-PHYSICAL REVIEW B 77, 241201 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 1098-0121/2008/77 /H2084924/H20850/241201 /H208494/H20850 ©2008 The American Physical Society 241201-1tive values imply a repulsive potential for the respective l component. /H20850The main contribution to the band-gap correc- tion comes from the repulsive Zn- spotential correction, in accord with the GW ﬁnding14that most of the correction occurs through an upward shift of the conduction band,which has strong Zn- scharacter. 24Other methods with simi- lar capabilities of a self-consistent band-gap correction thathave been applied to DMO are hybrid-DFT /H20849Ref. 25/H20850and approximate self-interaction correction /H20849SIC /H20850methods. 26,27 We will compare our results to those methods below. For the conventional GGA+ UHamiltonian, we use the GGA parametrization of Ref. 28and the rotationally invari- ant “+ U” formulation of Ref. 16/H20849b/H20850. The Hubbard- Uparam- eters for the TM- dorbitals are determined according to Ref. 18such that the thermochemically correct relative stability of the different oxide stoichiometries /H20849e.g., CoO vs Co 3O4/H20850is obtained. Thus, we use U=2.6, 3.9, 3.5, 2.8, and 3.4 eV for Cr, Mn, Fe, Co, and Ni, respectively, where the exchangeparameter is set to the typical value of J=1 eV. 16As dis- cussed in Ref. 18, these values are considerably smaller than the respective values that would reproduce the experimentalband gaps of the TM oxides, which, however, should not beexpected from the GGA+ Umethod. The larger value U =7 eV for Zn /H20849d 10/H20850/H20849Ref. 29/H20850compared to the TM reﬂects its deeper and more localized semicore d10shell. The following results are obtained in supercells of 72 atoms, using an en-ergy cutoff of 440 eV and a /H9003-centered 4 /H110034/H110034kmesh for Brillouin-zone integration. In the calculations with additionalelectron doping, we apply the general methodology forcharged supercells, as described in Ref. 30. We test the present /H20849GGA+ U+/H20850NLEP methodology by predicting defect properties that were not included in theﬁtting of the empirical parameters, namely the optical-absorption energies of several 3 dimpurities. By studying photoinduced changes in the electron paramagnetic reso-nance /H20849EPR /H20850spectrum, Jiang et al. 15concluded that light with h/H9263=1.96 eV was able to excite electrons from the va- lence band of ZnO into the gap levels of the ionized /H20849singly positively charged /H20850transition metals for Mn /H20849+III /H20850,C o /H20849+III /H20850, and Ni /H20849+III /H20850, but not for Fe /H20849+III /H20850. Thus, we calculated, ac- cording to the description given in Ref. 29, the optical /H20849ver-tical /H20850transition energy /H9255O/H20849+/0;h/H20850, which is deﬁned as the threshold photon energy required for the excitation TMZn+→TMZn0+h/H20849oxidation states: TMZn+III→TMZn+II+h/H20850.A s seen in Table II, the NLEP approach reproduces the experi- mental observations. For completeness, we also give in TableIIthe calculated thermodynamic /H20849thermal /H20850transition levels /H9255/H20849+/0/H20850/H20849see, e.g., Ref. 30/H20850. Finally, we note that our /H9255/H20849+/0/H20850=E VBM+0.31 eV transition energy for Co Znlies con- siderably lower in the gap than the respective 2.9 eV levelfound in Ref. 26, where the band-gap correction was achieved by treating the Zn- dshell as frozen-core electrons, and where self-interaction corrections were applied only toTABLE I. Target properties used for the ﬁt of the NLEP poten- tials. Band-structure parameters: The energy of the CBM, theconduction-band effective mass, the energy of the conduction bandat the Lpoint /H20849from the GW calculation of Ref. 14/H20850, and the Zn- d band energies /H20849all energies with respect to the VBM /H20850. Structural parameters: the unit cell volume, the c/aratio, and the displacement parameter u. GGA NLEP target E C/H20849/H9003/H20850/H20849eV/H20850 0.73 3.23 3.44 /H20849expt. /H20850 m*/me 0.19 0.47 0.28 /H20849expt. /H20850 EC/H20849L/H20850/H20849eV/H20850 5.64 6.43 7.40 /H20849GW /H20850 Zn-dband /H20849eV/H20850 −4.8 −7.0 −8.8 to −7.5 /H20849expt. /H20850 V olume /H20849A3/H20850 49.75 45.02 47.61 /H20849expt. /H20850 c/a 1.613 1.575 1.602 /H20849expt. /H20850 u 0.379 0.386 0.383 /H20849expt. /H20850TABLE II. The calculated /H9255O/H20849+/0;h/H20850excitation energy for the optical /H20849vertical /H20850transition TMZn+→TMZn0+h, compared with the conclusions-obtained from photo-EPR experiments /H20849Ref. 15/H20850using 633 nm light /H20849h/H9263=1.96 eV /H20850. The respective thermodynamic /H20849re- laxed /H20850/H9255/H20849+/0/H20850transition levels are also given. All numbers in eV . Mn Fe Co Ni NLEP /H9255O/H20849+/0;h/H20850= 1.20 2.83 0.96 1.69 Expt. /H20849Ref. 15/H20850/H9255O/H20849+/0;h/H20850 /H113491.96 /H110221.96 /H113491.96 /H113491.96 NLEP /H9255/H20849+/0/H20850−EVBM= 0.48 1.96 0.31 0.67 3.1 d+5e-1.9c0.13 d+5e-2(a) ZnO:Co (b) ZnO:Crd+5t- e-2 d- e+2t+2m= µB 3.3 e+2t+1.3c0.73.6 e+2t+1.6c0.44 e+2t+2m= µBGGA NLEP GGA+U e+2a+0CB VB 3 d+5e-2 FIG. 1. /H20849Color online /H20850Orbital and spin conﬁguration of /H20849a/H20850CoZn and /H20849b/H20850CrZnin ZnO and the resulting magnetic moment min GGA, in GGA+ U/H20849for Zn- d, Co- d, and Cr- d/H20850, and in the fully gap- corrected NLEP method. Charge transfer from occupied TM- d states into the host conduction band /H20849e.g., e−2→e−1.9c0.1for Co Znin GGA /H20850is indicated by open arrow symbols. Dashed lines in /H20849b/H20850 indicate the average level energy before the Jahn-Teller splitting.LANY , RAEBIGER, AND ZUNGER PHYSICAL REVIEW B 77, 241201 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 241201-2the Co- dshell. A /H9255/H20849+/0/H20850level higher than EVBM+1.96 eV is, however, inconsistent with the conclusion obtained from thephoto-EPR experiments. 15 Orbital and spin conﬁguration of single Co and Cr impu- rities . Figure 1shows the calculated orbital and spin conﬁgu- ration of single, charge-neutral Co Znand Cr Znimpurities in ZnO at the levels of uncorrected GGA, of GGA+ Uand of fully gap-corrected NLEP. For Co Znin the GGA description /H20851Fig. 1/H20849a/H20850/H20852, the doubly occupied e−2minority-spin level occurs as a resonance inside the/H20849uncorrected /H20850GGA conduction band, leading to a charge transfer of 0.1 einto the ZnO host conduction band for the 72-atom supercell. Accordingly, Co Zneffectively forms an e−1.9c0.1conﬁguration with a noninteger total magnetic mo- ment m=3.1/H9262B/Co at this Co concentration /H208491021cm−3/H20850. Us- ing the GGA+ Udescription for the Zn- dorbitals of the ZnO host, the band gap is increased mostly by lowering the en-ergy of the valence-band maximum. 29The simultaneous ap- plication of Uto the Co- dstates increases the splitting be- tween occupied e−and the unoccupied t−minority-spin levels /H20849the symmetry labels eand trefer to the approximate local tetrahedral symmetry /H20850. Thus, in GGA+ U, the e−level occurs correctly inside the gap, leading to an integer mo-ment, but the unoccupied t −level creates a resonance that is still far too high above the CBM /H20851Fig. 1/H20849a/H20850/H20852. After additional application of the NLEP correction, which recovers the cor-rect magnitude of the band gap mostly by raising the energyof the conduction bands, the unoccupied resonance of the t − level of Co Znlies close to the CBM at about EC+0.5 eV /H20851Fig. 1/H20849a/H20850/H20852. Comparing our NLEP result to recent band-gap cor- rected hybrid-DFT /H20849Ref. 25/H20850and SIC /H20849Ref. 27/H20850calculations, we ﬁnd that the t−resonance occurs slightly higher at EC +1 eV in SIC,27and considerably higher above EC+2 eV in hybrid-DFT.25The proximity of the t−level to the CBM will turn out to be important when considering theaddition of electrons via n-type doping /H20849see below /H20850. For Cr Znimpurities /H20851Fig. 1/H20849b/H20850/H20852in the GGA description, the resonance of the occupied majority-spin t+2level lies deep inside the conduction band in GGA, at about EC+1.2 eV, leading to a large charge transfer of 0.7 einto the host con- duction band and to an electron conﬁguration e+2t+1.3c0.7. Accordingly, we ﬁnd a noninteger total magnetic momentm=3.3 /H9262Bper supercell, much smaller than the expected 4 /H9262B /H20851Fig. 1/H20849b/H20850/H20852. As expected from the partial occupancy of the t+ level, there exists a Jahn-Teller effect, manifested by splitting of the e+and t+levels by /H110110.2 eV /H20851not shown in Fig. 1/H20849b/H20850/H20852. In the GGA+ Udescription, the Jahn-Teller effect is strongly enhanced, and we observe the splitting of the t+level into three sublevels, spread by 1.4 eV. The lower-energy e+level is now split by 0.3 eV due to breaking of the C3vsymmetry. Since one occupied sublevel lies still inside the GGA+ U conduction band, there is again a charge transfer to the hostconduction band leading to a noninteger moment of m=3.6 /H9262Band an effective e+2t+1.6c0.4conﬁguration. The spu- rious charge transfer into the host band is avoided only afterfull correction of the band gap in NLEP, where the correct e +2t+2conﬁguration and the integer moment of 4 /H9262Bof Cr /H20849+II/H20850 are recovered. Since the nominal t+2conﬁguration is realized in this case, the Jahn-Teller effect leads to a different atomicstructure than in GGA+ U, such that the splittingt +2→e+2+a+0/H20851cf. Fig. 1/H20849b/H20850/H20852does not lift the degeneracy of the esymmetries /H20849theC3vsymmetry of wurtzite is preserved /H20850. Thus, in contrast to the case of Co Zn, where the correct or- bital and spin conﬁguration is obtained already at theGGA+ Ulevel, for Cr Znthe correct electron conﬁguration and atomic structure of Cr Znare obtained only after the full band-gap correction in NLEP. Magnetic Co-Co and Cr-Cr pair interactions . We now compare the FM stabilization energies /H9004EFM=EFM−EAFM for Co-Co and Cr-Cr pairs in 72-atom supercells considering uncorrected GGA and fully band-gap corrected NLEP. For Co-Co pairs /H20851Fig. 2/H20849a/H20850/H20852, both GGA and NLEP predict rather small differences /H20841/H9004EFM/H20841/H110210.05 eV between the FM and AFM states, similar to the case of the uncorrected LDA/H20849Ref. 11/H20850and the gap-corrected LDA+SIC calculations of Ref. 27. We next study the pair interactions in the presence of additional electrons that can be supplied in ZnO through n-type doping. 4Whereas in the uncorrected GGA calculation the addition of 1 eper Co-Co pair /H208491021cm−3doping level /H20850 does not signiﬁcantly affect the FM coupling energies, in theNLEP calculation electron doping induces a strong FM inter-action between close pairs /H20851Fig.2/H20849a/H20850/H20852, showing that the band- gap correction is essential to obtain ferromagnetism inelectron-doped ZnO:Co. This FM coupling occurs when theresonant t −level of Co Znbecomes partially occupied at high doping levels, conforming with the general expectation17that partial occupancy of spin-polarized orbitals promotes ferro-magnetism. In the recent SIC calculation of Ref. 27, where the t −resonance occurs at somewhat higher energy /H20849see above /H20850, FM coupling of Co-Co would require higher electron concentrations than in the present work. It was found in Ref.27, however, that pairing of Co Znwith O vacancies lowers the t−level, allowing for long-range ferromagnetism at achievable electron densities. In contrast, in an uncorrectedGGA or in a GGA+ Ucalculation, no partial occupation of thet −level is achieved at realistic doping levels,11because due to the too low CBM energy /H20851Fig. 1/H20849a/H20850/H20852, the additional electrons populate the host conduction band instead of the t− defect level of Co Zn.(a) ZnO:(Co-Co) (b) ZnO:(Cr-Cr) dCo-Co(Å) dCr-Cr(Å)gap-corrected (NLEP)gap-corrected (NLEP)uncorrected (GGA) uncorrected (GGA)FM stabil. energy per pair ∆EFM(eV) FIG. 2. /H20849Color online /H20850The ferromagnetic stabilization energy /H9004EFM=EFM−EAFMin eV for /H20849a/H20850the Co-Co and /H20849b/H20850the Cr-Cr pairs in a 72-atom ZnO supercell, as a function of the pair distance d. Results are given for the uncorrected GGA and for the gap-corrected NLEP methods, and for different levels of additional elec-tron doping up to 1 eper TM pair /H20849/H1101110 21cm−3/H20850.MAGNETIC INTERACTIONS OF Cr-Cr AND Co-Co … PHYSICAL REVIEW B 77, 241201 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 241201-3For Cr-Cr pairs /H20851Fig. 2/H20849b/H20850/H20852, the uncorrected GGA calcula- tion erroneously predicts a strong and long-range FM cou-pling between Cr pairs even when no additional electrons aresupplied /H20851Fig. 2/H20849b/H20850/H20852. This prediction of magnetism originates from the spurious partial occupancy of the t +level of Cr being resonant inside the uncorrected GGA conduction band/H20851see Fig. 1/H20849b/H20850/H20852. In contrast, in the band-gap corrected NLEP calculation, this partial occupancy is removed by the Jahn- Teller effect /H20851Fig. 1/H20849b/H20850/H20852, and the ensuing FM coupling ener- gies/H9004E FMbecome small /H20851Fig. 2/H20849b/H20850/H20852. When electrons are added through doping, the unoccupied Jahn-Teller split a+ state in the conduction band becomes partially occupied, leading to strong and long-ranged FM coupling that in-creases with the amount of doping /H20851Fig. 2/H20849b/H20850/H20852. Thus, also in ZnO:Cr, the supply of additional electrons is essential forFM coupling. Technical description of the NLEP implementation .I nt h e PAW method, 21,22the all-electron /H20849AE/H20850wave functions /H9274AE are reconstructed from the pseudo- /H20849PS/H20850wave functions /H9274PS by means of a linear transformation, /H20841/H9274AE/H20856=/H20841/H9274PS/H20856+/H20858 i/H20849/H20841/H9278iAE/H20856−/H20841/H9278iPS/H20856/H20850/H20855/H20841pi/H20841/H9274PS/H20856, using a set of projector functions pi. Here, the index icom- prises the individual atomic sites, the angular-momentumquantum numbers l,mand the reference energies /H20849usually two per l/H20850, which are used in the atomic reference calculation to construct the partial waves /H9278iAEand/H9278iPS, and the pseudo- potential. Similarly, the AE potential operator is obtained inthe PAW method as 21,22VAE=VPS+/H20858 i,j/H20841pi/H20856/H20849/H20855/H9278iAE/H20841VAE,1/H20841/H9278jAE/H20856−/H20855/H9278iPS/H20841VPS,1/H20841/H9278jPS/H20856/H20850/H20855pj/H20841, where VAE,VPSare the “global” effective Kohn-Sham poten- tials /H20849ionic+Hartree+exchange correlation /H20850, and VAE,1,VPS,1 are their respective one-center expansions within the aug- mentation spheres. The NLEP potentials /H9004V/H9251,lNLEPfor the atomic types /H9251and the angular momenta lare added to the AE one-center potential, VAE,1→VAE,1+/H9004V/H9251,lNLEP/H9254l,l/H20849i/H20850/H9254l/H20849i/H20850,l/H20849j/H20850, where l/H20849i/H20850and l/H20849j/H20850are the lsubindices within iand j. Conclusions . Due to the band-gap problem exhibited by the LDA and GGA functionals, and their “+ U” extensions, these methods may predict the absence of FM couplingwhere such coupling is expected to exist /H20849e.g., ZnO:Co +electron doping /H20850, or may predict FM coupling where such coupling should not exist /H20849e.g., ZnO:Cr /H20850. For the correct description of magnetism in wide-gap oxides such as ZnO, itis essential to recover the correct band-edge energies of thehost in a self-consistent manner. Determining ferromagneticcoupling energies for Co-Co and Cr-Cr pairs in ZnO within afully band-gap corrected method using empirical nonlocalexternal potentials, we ﬁnd that both Co and Cr show para-magnetic behavior in the absence of additional carriers, butferromagnetic coupling occurs when sufﬁcient additionalelectrons are supplied such that the initially unoccupied reso-nant defect levels of Co and Cr inside the conduction bandbecome partially occupied. This work was funded by the DARPA PROM program and the U.S. Department of Energy, Ofﬁce of Energy Efﬁ-ciency and Renewable Energy, under Contract No. DE-AC36-99GO10337 to NREL. 1V . F. Masterov, Fiz. Tekh. Poluprovodn. /H20849S.-Peterburg /H2085018,3 /H208491984 /H20850/H20851Sov. Phys. Semicond. 18,1/H208491984 /H20850/H20852. 2J. Osorio-Guillén, S. Lany, and A. Zunger, Phys. Rev. Lett. 100, 036601 /H208492008 /H20850. 3K. R. Kittilstved, W. K. Liu, and D. R. Gamelin, Nat. Mater. 5, 291 /H208492006 /H20850. 4A. J. Behan et al. , Phys. Rev. Lett. 100, 047206 /H208492008 /H20850. 5K. R. Kittilstved et al. , Phys. Rev. Lett. 97, 037203 /H208492006 /H20850. 6K. Rode et al. , Appl. Phys. Lett. 92, 012509 /H208492008 /H20850. 7T. Dietl et al. , Phys. Rev. B 76, 155312 /H208492007 /H20850. 8A. Barla et al. , Phys. Rev. B 76, 125201 /H208492007 /H20850. 9D. J. Keavney et al. , Appl. Phys. Lett. 91, 012501 /H208492007 /H20850. 10N. H. Hong, J. Sakai, and V . Brizé, J. Phys.: Condens. Matter 19, 036219 /H208492007 /H20850. 11E. C. Lee et al. , Phys. Rev. B 69, 085205 /H208492004 /H20850; K. Sato et al. , Phys. Status Solidi B 229, 673 /H208492002 /H20850. Using the GGA and LDA functionals for ZnO:Co, these authors found signiﬁcantFM coupling only at very high concentrations of Co impuritiesand additional electrons /H20849up to 25% or 10 22cm−3/H20850compared to the present work /H20849up to 3% or 1021cm−3/H20850. At such unrealisti- cally high Fermi levels, the t−level of Co Znmay be partially occupied despite its high energy at EC+2 eV in GGA /H20851see Fig. 1/H20849a/H20850/H20852. 12P. Gopal and N. A. Spaldin, Phys. Rev. B 74, 094418 /H208492006 /H20850. 13T. Chanier et al. , Phys. Rev. B 73, 134418 /H208492006 /H20850. 14M. Usuda et al. , Phys. Rev. B 66, 125101 /H208492002 /H20850.15Y . Jiang, N. C. Giles, and L. E. Halliburton, J. Appl. Phys. 101, 093706 /H208492007 /H20850. 16V . I. Anisimov et al. , Phys. Rev. B 48, 16 929 /H208491993 /H20850;A .I . 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Pemmaraju et al. , arXiv:0801.4945v1. 28J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 /H208491996 /H20850. 29S. Lany and A. Zunger, Phys. Rev. B 72, 035215 /H208492005 /H20850. 30C. Persson et al. , Phys. Rev. B 72, 035211 /H208492005 /H20850.LANY , RAEBIGER, AND ZUNGER PHYSICAL REVIEW B 77, 241201 /H20849R/H20850/H208492008 /H20850RAPID COMMUNICATIONS 241201-4
PhysRevB.73.195205.pdf	Self-interaction-corrected pseudopotentials for silicon carbide Björn Baumeier, Peter Krüger, and Johannes Pollmann Institut für Festkörpertheorie, Universität Münster, D-48149 Münster, Germany /H20849Received 25 May 2005; revised manuscript received 23 February 2006; published 16 May 2006 /H20850 We report electronic and structural properties of cubic and hexagonal 3C-, 2H-, 4H-, and 6H-SiC bulk crystals and of the C-terminated SiC /H20849001 /H20850-c/H208492/H110032/H20850surface as resulting from density functional theory /H20849DFT /H20850 within local density approximation /H20849LDA /H20850. In particular, we employ newly constructed nonlocal, norm- conserving pseudopotentials which incorporate self-interaction corrections. Results obtained with usualpseudopotentials show the typical LDA shortcomings, most noticeably the systematic underestimate of theband gap. These problems are attributed to an unphysical self-interaction inherent in the common DFT-LDA.We describe the construction of appropriate self-interaction-corrected pseudopotentials for Si and C atoms andshow how they can be transferred to the SiC solid by adequate modiﬁcations. It is in the very nature of ourpseudopotentials that they cause no additional computational effort, as compared to usual pseudopotentials instandard LDA calculations. To test their transferability to different crystal structures we apply these pseudo-potentials to both cubic and hexagonal polytypes of SiC. The resulting energy gaps are in excellent agreementwith experimental data and the bulk band structures are in most gratifying agreement with the results ofconsiderably more elaborate quasiparticle calculations. Structural properties of the different polytypes arefound in excellent agreement with experiment, as well, not showing the usual LDA underestimate of latticeconstants and overestimate of bulk moduli. Also the electronic structure of SiC /H20849001 /H20850-c/H208492/H110032/H20850, calculated to exemplify the usefulness of the pseudopotentials for surfaces, shows improved agreement with experiment ascompared to the respective surface band structure obtained within standard LDA. DOI: 10.1103/PhysRevB.73.195205 PACS number /H20849s/H20850: 71.15.Mb, 71.20.Nr I. INTRODUCTION Since the advent of semiconductor technology, Si and silicon-based materials have played a vital role in the devel-opment of modern semiconductor devices. At present, how-ever, the physical limits of such devices exclusively based onSi are gradually reached, e.g., the maximum operating tem-perature of approximately 200 °C which severely limits the applicability of such devices for process control or data log-ging in many relevant high temperature processes. A moreintensive use of silicon carbide compounds is expected toovercome some of these limitations as SiC has a number offavorable properties, among those a high operating tempera-ture /H20849approximately 800 °C /H20850and high mechanical stability. 1 From a microscopic point of view, SiC is a very unique material.2In contrast to homopolar elemental semiconduc- tors, like Si or Ge, it is the only existing heteropolargroup-IV compound. Its heteropolarity gives rise to consid-erably ionic Si uC bonds. Another interesting aspect is the polytypism of SiC. 3Different polytypes are characterized by the stacking sequence of their constituent Si-C double layersalong a certain direction. There are more than 200 knownpolytypes, with the cubic 3C-SiC and the hexagonal 2H-SiC being the most extreme. All other hexagonal andrhombohedral polytypes show combinations of cubic andhexagonal stacking sequences. The respective band-gap en-ergies range from 2.4 eV in 3C-SiC to 3.3 eV in 2H-SiC. For applications of SiC in opto- and microelectronic de- vices a precise knowledge of its electronic properties is es-sential. From a theoretical point of view, density functionaltheory using the local density approximation has been estab-lished as an extremely useful ab initio method to calculate these properties. However, standard LDA calculations typi-cally underestimate critical band structure data, like the band gap or the valence bandwidth. In order to remedy these deﬁciencies in the description of electronic properties, several improvements have been devel-oped. For example, quasiparticle approaches based on theGW approximation 4,5/H20849GWA /H20850, which treat one-particle exci- tations using electron Green functions, have been particu-larly successful in this regard. 6–9Compared to standard LDA, however, the numerical effort for GWA calculations isconsiderably higher. This is particularly true when systemswith broken translational symmetry are described by largeunit cells containing many atoms. In such cases GWA calcu-lations become extraordinarily demanding computationally. The systematic deviations of DFT-LDA results from ex- perimental data can primarily be traced back to unphysicalself-interactions inherent in LDA, as has been shown by Per-dew and Zunger. 10The authors applied a self-interaction cor- rection /H20849SIC /H20850to atomic systems and were able to overcome the shortcomings of the LDA to a large extent. These correc-tions are state dependent, however, so that a direct transfer ofthis approach to bulk solids is computationally very demand-ing. Nevertheless, Svane and Gunnarson 11–14have performed respective calculations for transition metals using a SIC en-ergy functional, allowing the system to minimize its totalenergy by forming delocalized, as well as localized states.The authors observed that localization minimizes the totalenergy. Further results of SIC calculations have been re-ported by Szotek, Temmerman, and Winter 15–17for high- Tc superconductors and by Arai and Fujiwara18for transition- metal oxides. All these results indicate that the main effect ofself-interaction correction originates from localized atomicstates. This ﬁnding leads us to expect that the introduction ofatomic and hence localized self-interaction corrections intoPHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 1098-0121/2006/73 /H2084919/H20850/195205 /H2084912/H20850 ©2006 The American Physical Society 195205-1state-of-the-art nonlocal, norm-conserving pseudopotentials will approximate the results of full SIC calculations at leastto a signiﬁcant extent. The idea of incorporating corrections for self-interaction approximately has previously been implemented by variousgroups in different approaches. First, Rieger and V ogl 19have reported respective calculations for bulk Si, Ge, Sn, andGaAs. While the authors found signiﬁcant effects in the de-scription of strongly bound core levels, improvements ob-tained for the gaps of these s,pbonded semiconductors have only been marginal. Later on, some of the present auth-ors 20–22have successfully applied a related approach to II-VI semiconductors and group-III nitrides accounting for self-interaction and relaxation corrections /H20849SIRC /H20850in a solid by modiﬁed atomic SIC and SIRC pseudopotentials. In the latterwork, the relaxation corrections turned out to be of particularimportance for the semicore dbands in these compounds. Inspired by this previous work, Filippetti and Spaldin 23have more recently extended and modiﬁed the approach and ap-plied it not only to a II-VI compound and a group-III nitridebut also to a number of transition metal and manganese ox-ides. Their pseudo-SIC approach turned out to work verywell for the latter materials, as well. The materials, studiedby V ogel et al. 20–22and Filipetti and Spaldin23are all char- acterized by localized semicore dstates on which SIC and SIRC have a very pronounced effect. In this paper, we construct self-interaction-corrected pseudopotentials for the ionic compound semiconductor sili-con carbide and investigate their usefulness. It was not obvi-ousa priori that the SIC approach leads to quantitative im- provements for silicon carbide polytypes, as well, since SiCis a s,pbonded semiconductor and does not have highly localized semicore dstates, to begin with. Nevertheless, we ﬁnd that an appropriate inclusion of self-interaction correc-tions does improve the description of the bulk electronic andstructural properties of SiC polytypes very signiﬁcantly, in-deed. The description of an exemplary SiC surface showsnoticeable improvements, as well. Relaxation correctionshave only a very minor inﬂuence on the band structure of thepolytypes and have been ignored, therefore, for simplicity ofour approach. The paper is organized as follows: First, the principles of the construction of SIC pseudopotentials for Si and C aresummarized in Sec. II using cubic 3C-SiC as the prototype example for a ﬁrst application. For this polytype there is thelargest set of experimental and theoretical electronic struc-ture data available in the literature for comparison. Next weaddress structural properties of cubic and hexagonal SiCpolytypes in Sec. III. The results of our electronic structurecalculations using SIC pseudopotentials for the hexagonalpolytypes are then presented in Sec. IV and discussed incomparison with standard LDA results, as well as with GWAresults and experiment. Finally, the SiC /H20849001 /H20850-c/H208492/H110032/H20850sur- face is brieﬂy addressed in Sec. V . A short summary con- cludes the paper. II. CONSTRUCTION OF SIC PSEUDOPOTENTIALS In this section, we outline the construction of self- interaction-corrected pseudopotentials and discuss their ap-plication in calculations of electronic properties of cubic 3C-SiC, as a prototype example. A. Standard pseudopotentials For reference sake, we ﬁrst very brieﬂy address the stan- dard pseudopotentials which we use in our accompanyingLDA calculations. As is well known, electrons from innercore states do not inﬂuence chemical bonding in bulk crys-tals. Therefore, electronic structure calculations can be re-stricted to the valence electrons accounting for the effects ofthe core electrons by introducing ionic pseudopotentials. Thestarting point for constructing usual state-of-the-art ab initio pseudopotentials are all-electron LDA calculations for re-spective atoms. There are several conditions that have to befulﬁlled in the construction process, most notably, and alsomost intuitively, that the all-electron eigenvalues for theatomic valence states are reproduced by the pseudopo-tentials. 24–26One characteristic feature of such ionic pseudo- potentials is their dependence on angular momentum as Vˆps=/H20858 lVlpsPˆl, /H208491/H20850 where Pˆlis a projection operator on angular momentum eigenstates Pˆl=/H20858 m/H20841lm/H20856/H20855lm/H20841. /H208492/H20850 These ionic pseudopotentials are semilocal, i.e., nonlocal with respect to the spherical angles /H9277and/H9272but local with respect to the radial coordinate r, within a chosen core re- gion. They can be separated into a local and a nonlocal partas Vˆ ps=Vˆ locps+Vˆ nlocps/H208493/H20850 with Vˆ nlocps=/H20858 l/H9004VlpsPˆl. /H208494/H20850 For practical purposes, it has proven useful to represent the above semilocal pseudopotentials in a fully separableform as proposed by Kleinman and Bylander. 27 In our standard LDA reference calculations we use the nonlocal, norm-conserving ab initio pseudopotentials con- structed according to the prescription of Hamann.26In all calculations to follow we employ the exchange-correlationpotential of Ceperley and Alder, 28as parametrized by Perdew and Zunger.10As basis sets we use Gaussian orbitals with appropriately determined decay constants.29In the following construction and ﬁrst exemplary application of SIC pseudo-potentials we use 3C-SiC as a reference. This cubic modiﬁ- cation of SiC crystallizes in the zinc-blende structure, with alattice constant of 4.36 Å. Within standard LDA we obtainthe band structure shown in Fig. 1 in direct comparison witha number of experimental data points. It exhibits a heteropo-lar or ionic band gap between the lowest C 2 s-derived band and the three higher s,p-like valence bands as is typical for an ionic compound semiconductor. The total width of theBAUMEIER, KRÜGER, AND POLLMANN PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-2LDA valence bands is 15.29 eV. 3C-SiC has an indirect op- tical gap between the /H9003and the Xpoint. The calculated LDA gap energy of 1.29 eV underestimates the experimentalvalue 30of 2.42 eV by about 45%, as is typical for standard LDA. In addition, the calculated conduction bands show sig-niﬁcant kdependent deviations from the data points. To the best of our knowledge there are no experimental data avail-able in the literature on the low-lying C 2 sband. B. SIC pseudopotentials The LDA shortcomings of the band structure in Fig. 1 occur in spite of the fact that the employed standard pseudo-potentials reproduce by construction the atomic all-electronLDA term values exactly as is shown in Table I, where boththe all-electron and the pseudopotential eigenvalues aregiven. This raises the question how reliable the all-electronLDA results are with respect to experiment. To this end, theexperimental ionization energies E /H9251expare given for Si and C atoms32in Table I, as well. If one interprets the eigenvalues /H9280/H9251LDAas excitation energies, which is usually done, it be- comes obvious that they deviate strongly by some 50% fromthe experimental data. In particular, the measured energy dif-ference between the C 2 pand Si 3 pterm values of 3.2 eV is strongly underestimated by the respective energy differenceof the LDA term values amounting to 1.2 eV, only. Perdewand Zunger 10have attributed this type of shortcomings in atomic systems to an unphysical self-interaction contained inLDA and have proposed a method to introduce self-interaction corrections of the energy functional, which can bewritten as E SIC=ELDA−/H20858 /H9251occ /H20853ECoul/H20851/rho1/H9251/H20852+ExcLDA/H20851/rho1/H9251/H20852/H20854. /H208495/H20850 Minimization of the energy according to Eq. /H208495/H20850yields the equivalent to the Kohn-Sham equations /H20853−/H116122+V/H9251,effSIC/H20849r/H20850/H20854/H9278/H9251SIC/H20849r/H20850=/H9280/H9251SIC/H9278/H9251SIC/H20849r/H20850. /H208496/H20850 Within pseudopotential framework the orbital-dependent self-interaction corrected effective potential reads V/H9251,effSIC/H20849/H20851/rho1/H20852,/H20851/rho1/H9251/H20852,r/H20850=V/H9251ps+VCoul/H20849/H20851/rho1/H20852,r/H20850+VxcLDA/H20849/H20851/rho1/H20852,r/H20850 +V/H9251SIC/H20849/H20851/rho1/H9251/H20852,r/H20850/H20849 7/H20850 and V/H9251SIC/H20849/H20851/rho1/H9251/H20852,r/H20850=−VCoul/H20849/H20851/rho1/H9251/H20852,r/H20850−VxcLDA/H20849/H20851/rho1/H9251/H20852,r/H20850. /H208498/H20850 Here /rho1and/rho1/H9251are the atomic valence and orbital charge densities, respectively. The solution of Eq. /H208496/H20850for Si and C pseudoatoms yields the SIC term values /H9280/H9251ps,SICgiven in Table I. While there is no exact agreement between the SIC termvalues and the experimental ionization energies, the devia-tions from the latter have been reduced dramatically. Forexample, the energy difference between the C 2 pand Si 3 p term values resulting from the SIC calculation as 3.7 eV is inmuch closer agreement with the experimental value of3.2 eV than the energy difference between the respectiveLDA term values of 1.2 eV. Exact agreement was not to beexpected, anyway, since we have solved Eq. /H208496/H20850without in- cluding spin polarization because it is insigniﬁcant for theSiC solid, to be addressed below. Comparing the term valuesresulting from the all-electron or pseudopotential LDA cal-culations with those resulting from the pseudopotential SICcalculations, we ﬁrst note a pronounced absolute shift of theSIC term values with respect to the LDA term values. Muchmore importantly, however, the term values resulting fromthe SIC calculations show prominent relative shifts with re-spect to one another as compared to the LDA term values.These have very signiﬁcant bearing on the outcome of elec-tronic structure calculations for solids since the atomic SICterm values of the interacting atoms in the solid occur atlargely different relative positions from the start, as com-pared to the respective LDA term values. So the solid stateinteraction of the different atoms is strongly inﬂuencedthereby giving rise to changes in the energy positions anddispersions of the bulk bands. TABLE I. Atomic term values /H20849in eV /H20850for C and Si atoms as resulting from nonspinpolarized LDA and SIC calculations. For ref-erence we show both the all-electron and pseudopotential term val-ues resulting in LDA, as well as the energy shifts /H9004 /H9280/H9251=/H9280/H9251ps,SIC −/H9280/H9251ps,LDAof the eigenvalues due to self-interaction correction. E/H9251exp/H9280/H9251ae,LDA/H9280/H9251ps,LDA/H9280/H9251ps,SIC/H9004/H9280/H9251 C2s −13.7 −13.7 −19.7 −6.0 C2p −11.3a−5.4 −5.4 −11.1 −5.7 Si 3s −10.9 −10.9 −15.1 −4.2 Si 3p −8.1a−4.2 −4.2 −7.4 −3.2 aFrom Ref. 32. FIG. 1. LDA band structure of 3C-SiC along high-symmetry lines of the Brillouin zone. The dashed line indicates the experimen-tal gap of 2.42 eV /H20849Ref. 30 /H20850. Open circles show wave vector- resolved photoemission data from Ref. 31. The full dots are derivedfrom optical data. For the respective references, see Table II.SELF-INTERACTION-CORRECTED PSEUDOPOTENTIALS ¼ PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-3The atomic SIC pseudopotentials for Si and C ions are deﬁned according to Eq. /H208497/H20850by V/H9251ps,SIC/H20849/H20851/rho1/H9251/H20852,r/H20850ªV/H9251ps/H20849r/H20850+V/H9251SIC/H20849/H20851/rho1/H9251/H20852,r/H20850. /H208499/H20850 Next, we have to modify these atomic SIC pseudopoten- tials such that they can meaningfully be applied to solids./H20849For details, see Ref. 21. /H20850They feature an asymptotic −2/ r tail originating from the Coulomb potential V /H9251SIC/H20849/H20851/rho1/H9251/H20852,r/H20850. Such long-range tails would cause an unphysical overlap of the SIC potential contributions, which are introduced as trulyatomic properties in our approach, after all, from differentatomic sites. To reduce the overlap of the ﬁnal correctionpotentials in the solid we refer all correction potentials rela-tive to the energetically highest atomic state and cut off the−2/rtails appropriately. The energetically highest atomic state is Si 3 pin the case of SiC. So we rigidly shift all correction potentials accordingly by the same value V shift ª/H9280Si3pLDA−/H9280Si3pSIC=3.2 eV /H20849see/H9004/H9280/H9251for Si 3 pin Table I /H20850. Note that the relative distances of the term values, as resultingfrom the atomic SIC calculations, are not changed thereby sothat the physics of the atomic levels remains to be describedmuch more rigorously from the start than by the usual LDAterm values. Actually, if the atomic self-interaction correc- tions V /H9251SIC/H20849/H20851/rho1/H9251/H20852,r/H20850would directly be applied in a solid state calculation allstates would experience a strong SIC correc- tion. However, delocalized states are only weakly affected byself-interaction corrections, if at all /H20849see, e.g., Refs. 11–18 /H20850. This is especially true for atomic states that contribute to theconduction bands of a semiconductor. These are usually thehighest atomic valence states. We therefore refer all correc- tion potentials relative to the Si 3 pstate. This shift of all atomic SIC potentials by the same amount does not changethe relative distances between the atomic SIC levels but re-duces the overlap of the ﬁnal potentials in the solid substan-tially /H20849see, e.g., Fig. 3 in Ref. 21 /H20850. By this modiﬁcation, the inﬂuence of the Si 3 pself-interaction correction is reduced to a large extent in accord with the fact that delocalizedconduction-band states themselves do not experience a sig-niﬁcant self-interaction. The changes in the band structureare predominantly brought about by the SIC contributions tot h eC2 s,C2 p, and Si 3 spseudopotentials. The −2/ rtails of the radial components of the correction terms V /H9251SIC/H20849/H20851/rho1/H9251/H20852,r/H20850 are then cut off at suitable radii r/H9251which we deﬁne by the condition that the pseudopotentials with the SIC contribu-tions cutoff at r /H9251reproduce the atomic SIC term values within 10−2Ry. For the valence states of the Si and C atoms the above criteria yields the radii 3.84 and 4.36 a.u. for C 2 s and 2 p, and 4.72 and 5.87 a.u. for Si 3 sand 3 p, respectively. The cutoff is actually achieved on a short length scale bymultiplying the correction terms with the smooth function f/H20849x /H9251/H20850=exp /H20849−x/H92517/H20850with x/H9251=r/r/H9251to avoid problems in their Fourier representation. The respectively modiﬁed self-interaction correction con- tributions can now be used in the calculations for the solid.For the valence states of a given ion they are uniquely speci- ﬁed by the angular momentum quantum number l. They can therefore be written as V lSIC/H20849r/H20850+Vshiftmultiplied by the pro- jector on the angular momentum eigenstates and by theabove cutoff function and can simply be added to the nonlo- cal part of the usual pseudopotentials Vˆps,SIC=Vˆ locps+Vˆ nlocps,SIC/H2084910/H20850 with Vˆ nlocps,SIC=Vˆ nlocps+Vˆ nlocSIC=/H20858 l/H9004VlpsPˆl+/H20858 l/H9004VlSICPˆl/H2084911/H20850 and /H9004VlSIC/H20849r/H20850=/H20853VlSIC/H20849r/H20850+Vshift/H20854f/H20849xl/H20850/H20849 12/H20850 with xl=r/rl/H11013r/r/H9251. The nonlocal SIC contributions to the ionic pseudopoten- tials can now be represented in the fully separable Kleinman-Bylander form Vˆ nlocSIC=/H20858 l,m/H20841/H9278l,mSIC/H9004VlSIC/H20856/H20855/H9278l,mSIC/H9004VlSIC/H20841 /H20855/H9278l,mSIC/H20841/H9004VlSIC/H20841/H9278l,mSIC/H20856/H2084913/H20850 just as ordinary nonlocal pseudopotentials. The l,mvalues entering Eq. /H2084913/H20850are uniquely deﬁned by the orbital indices /H9251for each ion. We conclude this discussion of the construction of SIC pseudopotentials for the solid by noting that we fully incor-porate the SIC corrections according to Eq. /H2084912/H20850in our cal- culations. If one would try, on the contrary, to explicitly in-corporate the actual occupation of each band state one wouldhave to construct the SIC pseudopotentials iteratively for theself-consistently changing occupation of the band states. Thiswould necessitate an additional inner self-consistency loopfor each n,kwhich obviously would render the calculations extremely demanding. Filippetti and Spaldin 23have consid- ered this alternative. Due to the extremely heavy numericalload involved, however, they do not take the occupation ofeach particular band state explicitly into account but only a k space average of the band-state occupations. In addition,they do not construct their pseudopotentials iteratively foreach average band occupation anew but construct them onceand for all and then weight them by the average band occu-pation. Using this pragmatic way, the calculations becomefeasible again in spite of the fact that the actual occupationsof the band states are taken into account at least on average.From a general formal point of view this might be somewhatbetter than the consideration of the band-state occupations inour approach. Yet, the actual results of Filippetti and Spaldinfor ZnO and GaN are very similar to our previous results 20–22 so that no conclusive answer as to which approach is bettercan easily be inferred at present. The SIC pseudopotentials according to Eqs. /H2084910/H20850–/H2084913/H20850for the silicon carbide solid can now readily be employed in ausual LDA code causing no additional computational effortas compared to a standard LDA calculation. Employing thesepseudopotentials for Si and C we obtain the SIC band struc-ture shown in Fig. 2. Compared to the LDA band structure,the fundamental band gap has increased to 2.46 eV and isnow in very gratifying agreement with experiment. At thesame time, the total width of the valence bands has increasedto 17.18 eV. The broadening of the SIC valence bands, ascompared to the LDA valence bands, mainly originates fromBAUMEIER, KRÜGER, AND POLLMANN PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-4the lowering of the C 2 sband relative to the higher s,p valence bands due to its stronger self-interaction correction,as already evidenced by the /H9004 /H9280/H9251value in Table I which is largest for C 2 s. The dispersion of the measured valence bands along the /H9003-Xline is very well described. In particular, the energy of the highest occupied X5vstate, which is ob- served at −3.60 eV in experiment,34is much more accurately described in SIC than in standard LDA /H20849cf. Fig. 1 /H20850. Most importantly, the SIC approach does not only yield a verygood description of the valence bands and the band gap butalso a very accurate description of the experimental data forthe conduction bands. In Table II we have summarized band-structure energies for 3C-SiC resulting from our LDA and SIC calculations, aswell as theoretical results from two different GWAcalculations 8,9and experimental results30,33–35for 3C-SiC. The LDA results show the typical shortcomings discussedabove underestimating all conduction-band energies consid-erably. The SIC results are in very good agreement with themajority of the experimental data. The LDA band-gap prob-lem seems to have largely been overcome by including SIC,at least in this case of 3C-SiC. The overall width of thevalence bands resulting from the SIC calculation is largerthan that resulting from the GWA calculations of Rohlﬁnget al. 8but is close to that in the GWA results of Wenzien et al.9To date there are no experimental data on the total valence bandwidth to compare with. Comparing the GWAresults of Wenzien et al. 9with our SIC results, the GWA results from Ref. 8 and the experimental data it appears thatthe former band-structure energies result in the upper con-duction bands signiﬁcantly higher than all other values. Weemphasize this fact already at this point since for the hex-agonal SiC polytypes to be discussed below we have onlythe results of Ref. 9 to compare with. To further evidence the above difference we summarize in Table III critical point transition energies as resulting fromthe different calculations in comparison with experimental data. As is most obvious, the LDA values fall far short of allmeasured transition energies due to the LDA band-gap prob-lem. On the contrary, most of the SIC results and the quasi-particle results from Ref. 8 are in very good accord with theexperimental data. The quasiparticle results from Ref. 9overestimate the transition energies for the reason mentionedabove whenever ﬁnal states in the higher conduction bandsare involved. III. STRUCTURAL PROPERTIES We now address the question whether the SIC approach yields satisfying results for structural properties, as well.TABLE II. Calculated band-structure energies /H20849in eV /H20850at high- symmetry points for 3C-SiC in comparison with the results ofquasiparticle calculations by Rohlﬁng et al. /H20849Ref. 8 /H20850/H20849QPR /H20850and Wenzien et al. /H20849Ref. 9 /H20850/H20849QPW /H20850and experiment. 3C LDA SIC QPR QPW Exp /H9003 1v −15.29 −17.18 −16.44 −17.31 /H900315v 0.00 0.00 0.00 0.00 0.00 /H90031c 6.25 7.35 7.35 8.29 7.59a /H900315c 7.10 8.45 8.35 9.09 8.74a X1v −10.25 −10.96 −11.24 −11.82 X3v −7.79 −8.95 −8.64 −8.53 X5v −3.13 −3.55 −3.62 −3.49 −3.60b X1c 1.29 2.46 2.34 2.59 2.42c X3c 4.07 5.32 5.59 5.77 5.50b L1v −11.72 −12.79 −12.75 −13.39 L1v −8.49 −9.58 −9.42 −9.39 L3v −1.04 −1.17 −1.21 −1.13 −1.16b L1c 5.24 6.46 6.53 7.22 6.34d L3c 7.07 8.41 8.57 8.94 8.50b aFrom Ref. 33. bFrom Ref. 34. cFrom Ref. 30. dFrom Ref. 35. TABLE III. Calculated critical point transition energies /H20849in eV /H20850 in 3C-SiC in comparison with respective results of quasiparticlecalculations by Rohlﬁng et al. /H20849Ref. 8 /H20850/H20849QPR /H20850and Wenzien et al. /H20849Ref. 9 /H20850/H20849QPW /H20850and with various values derived from experimental data. 3C LDA SIC QPR QPW Exp aExpb /H90031c-/H900315v 6.25 7.35 7.35 8.29 7.59 7.4 /H900315c-/H900315v 7.10 8.45 8.35 9.09 8.74 9.0±0.2 X1c-X5v 4.42 6.05 5.96 6.08 6.02 5.8 X3c-X5v 7.21 8.91 9.21 9.26 9.10 8.3±0.1 L1c-L3v 6.29 7.63 7.74 8.35 7.50 7.5 L3c-L3v 8.11 9.58 9.78 10.07 9.66 9.4 aDerived from the experimental data in Table II. bFrom Ref. 35. FIG. 2. SIC band structure of 3C-SiC along high-symmetry lines of the Brillouin zone. For further details, see caption of Fig. 1.SELF-INTERACTION-CORRECTED PSEUDOPOTENTIALS ¼ PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-5Here we discuss all four SiC polytypes considered in our work. Structural parameters of solids such as lattice con-stants or bulk moduli usually result in good agreement withexperiment from LDA calculations. Lattice constants are un-derestimated in the order of 1% and bulk moduli are overes-timated often by a somewhat larger percentage. In general,SIC potentials are attractive causing the electrons to be stron-ger localized around the atomic nuclei. This gives rise to anincreased screening of the atomic nuclei leading to an in-crease in the lattice constants and a decrease in the bulkmoduli. Therefore we expect these quantities to result fromour approach in even better agreement with the data thanfrom usual LDA calculations. To determine these parameters we have to calculate the total energy of the system which is a ground-state property.The SIC pseudopotentials allow for an accurate descriptionof the occupied valence bands, as noted above, and shouldlead to very good total energies, therefore. In the frameworkof pseudopotential theory the total energy within the fullSIC-LDA approach /H20851Eq. /H208495/H20850/H20852can be written as E SIC=/H20858 /H9251occ /H9280/H9251SIC+/H9004E1+/H9004E2+Eion, /H2084914/H20850 with /H9004E1=/H20885/H20873−1 2VCoul/H20849/H20851/rho1˜/H20852,r/H20850+/H9280xcLDA/H20849/H20851/rho1˜/H20852,r/H20850 −VxcLDA/H20849/H20851/rho1˜/H20852,r/H20850/H20874/rho1˜/H20849r/H20850d3r /H2084915/H20850 and /H9004E2=/H20858 /H9251occ/H20885/H208731 2VCoul/H20849/H20851/rho1˜/H9251/H20852,r/H20850−/H9280xcLDA/H20849/H20851/rho1˜/H9251/H20852,r/H20850 +VxcLDA/H20849/H20851/rho1˜/H9251/H20852,r/H20850/H20874/rho1˜/H9251/H20849r/H20850d3r. /H2084916/H20850 Here, /rho1˜and/rho1˜/H9251are the valence and orbital charge densi- ties in the solid, respectively, and Eionis the ion-ion interac- tion energy. The terms /H9004E1+/H9004E2account for double count- ing that occurs when the SIC eigenvalues /H9280/H9251SICare simply summed up. The term /H9004E1is the usual term accounting for double counting within standard LDA. In order to evaluate the term /H9004E2, we rewrite it as /H9004E2=/H20858 /H9251occ/H20885„VCoul/H20849/H20851/rho1˜/H9251/H20852,r/H20850+VxcLDA/H20849/H20851/rho1˜/H9251/H20852,r/H20850…/rho1˜/H9251/H20849r/H20850d3r −/H20858 /H9251occ/H20885/H208731 2VCoul/H20849/H20851/rho1˜/H9251/H20852,r/H20850+/H9280xcLDA/H20849/H20851/rho1˜/H9251/H20852,r/H20850/H20874/rho1˜/H9251/H20849r/H20850d3r. /H2084917/H20850 Except for the sign, the term in parantheses in the ﬁrst line is the solid state analog to the SIC contribution in the atomiceffective potential of the Kohn-Sham equations as deﬁned inEq. /H208498/H20850while the integral in the second line is the Hartreeexchange-correlation energy E HXC/H20851/rho1˜/H9251/H20852of the orbital charge density /rho1˜/H9251./H9004E2then reads /H9004E2=−/H20858 /H9251occ/H20885V/H9251SIC/H20849/H20851/rho1˜/H9251/H20852,r/H20850/rho1˜/H9251/H20849r/H20850d3r−/H20858 /H9251occ EHXC/H20851/rho1˜/H9251/H20852. /H2084918/H20850 In the SIC pseudopotential approach, we only calculate the valence charge densities /rho1˜/H20849r/H20850for the solid by solving the Kohn-Sham equations but not the orbital charge densities /rho1˜/H9251. Therefore, we resort in the same way as in the constructionof the SIC pseudopotentials to the modiﬁed SIC pseudopo- tentials /H9004V /H9251SICas deﬁned in Eq. /H2084912/H20850andEHXCas functions of the atomic charge densities /rho1/H9251and approximate /H9004E2corre- spondingly. Projecting the solid-state wave functions onto the localized atomic one-particle orbitals /H9278/H9251SIC,/H9004E2can be approximated by21 /H9004E2/H11015−/H20858 n,k/H20855/H9274n,k/H20841Vˆ nlocSIC/H20841/H9274n,k/H20856−/H20858 /H9251occ EHXC/H20851/rho1/H9251/H20852/H20849 19/H20850 with Vˆ nlocSICaccording to Eq. /H2084911/H20850. EHXC/H20851/rho1/H9251/H20852is then an atomic property which is constant in the solid and drops out when derivatives of the total energy are calculated. Using Eq. /H2084914/H20850with the above approximation for /H9004E2we evaluate the total energy of the investigated systems for anumber of unit cell volumes around its minimum and deter-mine the lattice constants and bulk moduli. For comparisonwe have also calculated these quantities within standardLDA. The results for the cubic and hexagonal 3C,2H,4H, and 6H polytypes are summarized in Table IV. The agreement ofthe structure parameters with the experimental values is ex-cellent. The lattice constants are underestimated by only0.3%, at most, while the bulk modulus is underestimated by0.9% for 3C-SiC and overestimated by 0.4% for 2H-SiC. The agreement of our SIC results with experiment is signiﬁ-cantly better than that of the standard LDA results whichTABLE IV . Calculated lattice constants aandc/H20849in Å /H20850and bulk moduli B/H20849in Mbar /H20850of the four investigated SiC polytypes in com- parison with experiment /H20849Ref. 37 /H20850. LDA SIC Exp 3C a 4.30 4.35 4.36 B 2.32 2.22 2.24 2H a 3.04 3.07 3.08 c 4.99 5.04 5.05 B 2.33 2.24 2.23 4H a 3.04 3.07 3.07 c 9.95 10.06 10.05 B 2.34 2.23 6H a 3.04 3.07 3.07 c 14.92 15.07 15.08 B 2.33 2.24BAUMEIER, KRÜGER, AND POLLMANN PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-6underestimate the lattice constants up to 1.4% and overesti- mate the bulk moduli up to 4.5%. The lattice constants andbulk moduli thus result from the SIC calculations about onepercent larger and about ﬁve percent smaller, respectively,than from LDA. This is due to a stronger increase in thelocalization of the carbon states, as compared to the Si states,by SIC since the former experience a stronger downwardshift in energy by self-interaction correction than the latter/H20849cf. the /H9004 /H9280/H9251values in Table I and the resulting increase in valence-band width within SIC as evidenced in Fig. 2 and inthe third column of Table II /H20850. This stronger localization of the C states, as compared to the Si states, gives rise to aweakening of the Si uC bonds which leads to larger lattice constants, as compared to LDA. By the same token, the lat-tice becomes “weaker” so that the bulk moduli show a de-crease in the SIC results, as compared to LDA. This behaviorwas also observed in other approximate SIC results 19as well as in the results of full SIC calculations.14,17 IV. HEXAGONAL POLYTYPES Now we address the question whether the very same SIC pseudopotentials used above to calculate the band structureof cubic 3C-SiC work equally well for the band structure of other SiC lattices. To this end, we consider the most commonhexagonal 2H,4H, and 6H polytypes in the following. Figure 3 shows a two-dimensional representation of the stacking sequences of these three hexagonal polytypes alongthe /H208510001 /H20852direction. To ease the comparison, we have ex- tended all plots along the /H208510001 /H20852direction to six Si-C double layers, with the actual lengths of the unit cell marked by the hexagonal lattice constants c. The purely hexagonal 2H-SiC exhibits a stacking sequence ABAB , in contrast to ABCB for 4H-SiC and ABCACB for 6H-SiC. Electronic properties are being inﬂuenced by the stacking sequence andthe related hexagonality of the crystals. The 2H polytype hasthe largest and the 6H polytype has the smallest hexagonalitywhile the cubic 3C-SiC has no hexagonality at all. Choyke et al. 36have found in experiment that there is a linear de- pendence between the width of the fundamental gap andthe hexagonality of the polytypes. The purely hexagonal2H-SiC has the largest while cubic 3C-SiC has the smallest energy gap. The position of the conduction band minimum inkspace and the band splitting at the top of the valence bands are affected by hexagonality, as well. The experimental lattice constants of 2H-SiC are a =3.08 Å and c=5.05 Å. 37Our calculated lattice constants /H20849see Table IV /H20850are very close to these values. The calculated band gap energies for 2H-SiC, as resulting from our LDAand SIC calculations are compared in Table V with the re-sults of quasiparticle calculations and with experiment. Theelectronic band structure of 2H-SiC as resulting from ourSIC calculations is shown in the left panel of Fig. 4. Respec-tive band-structure energies resulting from our LDA and SICcalculations are summarized in Table VI in comparison withthe GWA results from Ref. 9. Experimental data for2H-SiC are very scarce, the only known quantity seems to bethe width of the fundamental gap of 3.33 eV, 37with the minimum of the conduction bands at the Kpoint of the hex- agonal Brillouin zone. Our band gap of 3.33 eV calculatedwith the SIC pseudopotentials happens to exactly agree withthe experimental value showing a very signiﬁcant improve-ment as compared to the LDA result of 2.12 eV. Since thereare four ions per unit cell in 2H-SiC the band structurefeatures eight valence bands. Contrary to cubic 3C-SiC,for which the upper valence band is triply degenerate at the/H9003point, hexagonal 2H-SiC features a splitting of the top of the valence bands by 0.14 eV. This is attributed to the hex-agonal crystal ﬁeld which gives rise to doubly degenerate TABLE V . Calculated band-gap energies /H20849in eV /H20850of the four investigated SiC polytypes in comparison with the results of quasi-particle calculations by Rohlﬁng et al. /H20849Ref. 8 /H20850/H20849QPR /H20850and Wenzien et al. /H20849Ref. 9 /H20850/H20849QPW /H20850and with experiment. LDA SIC QPR QPW Exp. 3C 1.29 2.46 2.34 2.59 2.42 a 2H 2.12 3.33 3.68 3.33b 4H 2.14 3.30 3.56 3.26b 6H 1.94 3.08 3.25 3.02b aFrom Ref. 30. bFrom Ref. 37. FIG. 3. Stacking sequences in hexagonal polytypes of SiC in /H208510001 /H20852direction. Side views of six Si-C double layers are shown in each casefor better comparison.SELF-INTERACTION-CORRECTED PSEUDOPOTENTIALS ¼ PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-7states with pxandpysymmetry and a single pzlike state. The valence-band width of 17.35 eV, resulting within SIC, is1.9 eV larger than that resulting in LDA. Note that it is closeto the valence band width of 17.18 eV resulting from ourSIC calculations for 3C-SiC. This is, like in the case of3C-SiC, mostly caused by a strong lowering of the C 2 sband which is most noticeably around the /H9003point. Due to the lack of further experimental data we can only compare our resultswith the GWA results of Ref. 9. The agreement of the SICresults with the GWA results is quite good, in particular forband-structure energies around the fundamental gap and withrespect to the valence-band width. But also in this case theGWA calculations yield higher band-structure energies fur- ther up in the conduction bands as was already the case for3C-SiC /H20849see Table II /H20850. Similarly satisfying results follow for 4H-SiC, which crystallizes with the hexagonal lattice constants 37a=3.07 Å andc=10.05 Å. Also in this case our calculated lattice con- stants are in excellent agreement with these values /H20849see Table IV/H20850. The gap energies resulting from our LDA and SIC cal- culations are compared to GWA results9and experiment in Table V. The SIC band structure is shown in the middlepanel of Fig. 4 and respective band-structure energies arecompared with GWA results from Ref. 9 in Table VII. Also TABLE VI. Calculated band-structure energies at high- symmetry points of the Brillouin zone for 2H-SiC /H20849in eV /H20850in com- parison with the results of quasiparticle calculations by Wenzienet al. /H20849Ref. 9 /H20850/H20849QPW /H20850. 2H LDA SIC QPW /H9003 1v −15.45 −17.35 −17.39 /H90036v 0.00 0.00 0.00 /H90031c 4.60 5.79 6.66 K2v −3.79 −4.22 −4.12 K2c 2.12 3.33 3.68 H3v −1.73 −1.93 −1.83 H3c 4.92 6.17 6.86 A5,6v −0.71 −0.77 −0.75 A1,3c 5.70 6.94 7.81 M4v −1.18 −1.30 −1.13 M1c 2.59 3.84 4.28 L1,2,3,4 v −2.32 −2.59 −2.30 L1,3c 3.16 4.39 4.85TABLE VII. Calculated band-structure energies at high- symmetry points of the Brillouin zone for 4H-SiC /H20849in eV /H20850in com- parison with the results of quasiparticle calculations by Wenzienet al. /H20849Ref. 9 /H20850/H20849QPW /H20850. 4H LDA SIC QPW /H9003 1v −15.45 −17.38 −17.30 /H90036v 0.00 0.00 0.00 /H90031c 5.00 6.20 6.92 K2v −1.66 −1.86 −1.85 K2c 3.84 5.02 5.45 H3v −2.45 −2.72 −2.68 H3c 3.10 4.30 4.68 A5,6v −0.21 −0.22 −0.20 A1,3c 5.21 6.41 7.14 M4v −1.11 −1.24 −1.23 M1c 2.14 3.30 3.56 L1,2,3,4 v −1.54 −1.71 −1.68 L1,3c 2.53 3.72 4.06 FIG. 4. Band structures of the hexagonal 2H-, 4H-, and 6H-SiC polytypes as resulting from SIC calculations. The respective experi- mental energy gaps are indicated for reference.BAUMEIER, KRÜGER, AND POLLMANN PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-8for this polytype the band gap of 3.30 eV, calculated with the SIC pseudopotentials, is in very good agreement with theexperimental gap of 3.26 eV /H20849see also Table V /H20850. The LDA gap of only 2.14 eV strongly underestimates the measuredgap, as usual. In 4H-SiC there are eight inequivalent ions perunit cell so that sixteen valence bands result. They are sepa-rated from the conduction bands by the fundamental gapwhich occurs in this case between the /H9003andMpoints. The splitting of the upper valence bands at the /H9003point by 0.08 eV is smaller than in 2H-SiC. This is not surprisingsince 4H-SiC has a smaller hexagonality than 2H-SiC.Hence the crystal ﬁeld is less pronounced. The total valence-band width of 4H-SiC results from our SIC calculations as17.38 eV and is very close to the respective value for the 2Hpolytype. As was the case for 2H-SiC, our SIC band-structure energies for 4H-SiC are in very gratifying agree-ment with most of the GWA results of Ref. 9 near the gap-energy region. In the higher conduction bands similardeviations as noted above for the 3C and 2H polytypes occurin this case, as well. Finally, we address 6H-SiC. The measured hexagonal lat- tice constants are 37a=3.07 Å and c=15.08 Å. Our calcu- lated lattice constants are basically identical with these val-ues /H20849see Table IV /H20850. The band structure calculated using the SIC approach is shown in the right panel of Fig. 4 and acomparison of our calculated band-structure energies withthe GWA results of Ref. 9 is given in Table VIII. As in theother cases above, the band gap of 3.08 eV, calculated usingthe SIC approach, closely agrees with the experimentalvalue 37of 3.02 eV /H20849see also Table V /H20850while the respective LDA gap of 1.94 eV is again much too small. In 6H-SiCthere are twelve inequivalent ions per unit cell so thattwenty-four valence bands result. Their total width of17.35 eV is basically identical to those of the other two hex-agonal polytypes. Due to the further reduced hexagonality ofthe crystal ﬁeld, the /H9003point splitting of the upper valence bands is only 0.06 eV and thus less pronounced than in both2H- and 4H-SiC. The band structure of 6H-SiC has one particularly intriguing feature. Unlike the cases of the 2Hand 4H polytypes, the exact position of the conduction-bandminimum has been a matter of dispute. 9,38,39Standard LDA calculations yield the conduction-band minimum at a kpoint along the L-Mline. Our SIC calculations, however, yield the minimum at the Mpoint as in 4H-SiC, albeit that the lowest conduction band is very ﬂat along the L-Mline. This might be viewed as an indication that it actually does not occuralong the L-Mdirection. Comparing our SIC results in Table VIII with the GWA results of Ref. 9 very similar conclusionscan be drawn as in the case of the 2H and 4H polytypes. As noted above, there are no experimental data on the valence-band width of the 3C, 2H, and 4H polytypes of SiC.For 6H-SiC, however, King et al. 40have performed x-ray photoemission spectroscopy measurements which are espe-cially useful for assessing the lower valence bands. When wecompare the density of states for 6H-SiC resulting from ourSIC pseudopotential calculations /H20849not shown for brevity sake /H20850with the measured spectrum we ﬁnd good agreement for the peaks originating from the lowest C 2 sband and the following C2 p-Si3sbands, in particular. From this agree- ment we infer that our calculated valence-band widths for allfour polytypes seem to be realistic. In summary, the SIC pseudopotentials turn out to yield very reliable band-structure energies also for all three con-sidered hexagonal SiC polytypes. In particular, the band gapsof all four polytypes considered resulting from the SIC cal-culations /H20849see Table V /H20850are in excellent agreement with ex- periment so that the usual LDA shortcomings in describinggap energies seem to be conquerable entirely at least for theSiC polytypes by taking self-interaction corrections into ac-count.TABLE VIII. Calculated band-structure energies at high- symmetry points of the Brillouin zone for 6H-SiC /H20849in eV /H20850in com- parison with the results of quasiparticle calculations by Wenzienet al. /H20849Ref. 9 /H20850/H20849QPW /H20850. 6H LDA SIC QPW /H9003 1v −15.42 −17.35 −17.28 /H90036v 0.00 0.00 0.00 /H90031c 5.10 6.30 6.95 K2v −2.06 −2.30 −2.31 K2c 3.35 4.54 4.88 H3v −2.26 −2.48 −2.49 H3c 3.54 4.71 5.06 A5,6v −0.10 −0.10 −0.09 A1,3c 5.17 6.37 7.02 M4v −1.09 −1.22 −1.40 M1c 1.94 3.08 3.25 L1,2,3,4 v −1.30 −1.45 −1.63 L1,3c 1.98 3.15 3.36 FIG. 5. /H20849Color online /H20850Top and side view of the BDM of the C-terminated SiC /H20849001 /H20850-c/H208492/H110032/H20850surface. Top layer carbon atoms in the C wC surface dimers are shown by small black dots. Third layer C atoms are depicted by small gray /H20849dark gray /H20850circles. Sec- ond and fourth layer Si atoms are shown by large dark ocher /H20849dark gray /H20850and large light ocher /H20849light gray /H20850circles, respectively.SELF-INTERACTION-CORRECTED PSEUDOPOTENTIALS ¼ PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-9V . 3C-SiC „100 …-c„2Ã2…SURFACE Finally, to explore the usefulness of the SIC pseudopoten- tials for surfaces, we brieﬂy address their application to theC-terminated 3C-SiC /H20849001 /H20850-c/H208492/H110032/H20850surface. In particular, there are angle-resolved photoelectron spectroscopy /H20849ARPES /H20850and angle-resolved inverse photoelectron spectros- copy /H20849ARIPES /H20850data available for comparison. On the basis of a whole body of experimental data and recent ab initio DFT calculations there is now general agreement on the bridging-dimer model /H20849BDM /H20850of the 3C-SiC /H20849001 /H20850-c/H208492/H110032/H20850surface. 41Top and side views of the BDM, as resulting from our structure optimization42are shown in Fig. 5. Triple-bonded C wC dimers in the top layer form the main building blocks of this reconstruction /H20849see Fig. 5 /H20850. We have calculated the surface electronic structure of the BDM employing both standard LDA as well as the SICpseudopotentials from Sec. II. To describe the surface we usethe supercell approach with ten atomic layers /H20849one H, four Si, and ﬁve C layers /H20850per supercell. The H layer saturates the C bottom layer of the SiC slab in each supercell to avoidspurious surface states from the bottom layer. The surface band structure resulting from our LDA calcu- lation is shown in Fig. 6. It basically agrees with the respec-tive surface band structure which we have reported inRef. 42. Minor differences are due to a number of differencesin technical details of the two calculations. 43We have labeled the most pronounced surface state bands in Figs. 6 and 7according to Ref. 42. The T 1band originates from bonding states of the C wC surface dimers while the T1band origi- nates from the respective antibonding states /H20849cf. respective charge densities in Ref. 42 /H20850. The T2andT3bands originate from antibonding surface states, as well. Note that the lattertwo bands coincide with the projected bulk bands of SiCalong the /H9003-S /H11032and/H9003-S symmetry lines in the LDA surface band structure.The surface band structure resulting from our SIC calcu- lation is shown in Fig. 7. It shows the same topology of themost salient surface state bands as the LDA surface bandstructure in Fig. 6. There are signiﬁcant differences to benoted, however. First and foremost the SIC approach yieldsan appropriate projected bulk band structure and a realisticprojected gap energy region, in particular, at last. The T 1 surface band results slightly higher in energy relative to the projected bulk valence bands than in LDA. The T1band results in the SIC surface band structure throughout mostparts of the surface Brillouin zone 0.4 eV higher in energythan in the LDA surface band structure. Note, in particular,that it has moved up in energy by about 1 eV close to the /H9003 point along the /H9003-S /H11032line where it becomes resonant with the projected Si-derived conduction bands. The T3band, which is Si-derived to a considerable extent, is about 0.7 eV higherin energy in the SIC results than in the LDA results. Yet, itremains to be a band of localized surface states within theprojected gap also along most of the /H9003-S /H11032and/H9003-S symmetry lines. This is due to the fact that the projected bulk conduc-tion bands have shifted up in energy by more than 1 eV ascompared to the projected LDA bulk band structure in con-sequence of the realistic description of the bulk conductionbands within the SIC approach. We have included in Figs. 6and 7 experimental ARPES and ARIPES data for compari-son. In the ARPES experiments, the measured occupied valence-band states have been referred to the extrinsic Fermilevel of the samples used but the doping has not been givenin Ref. 44. We have, therefore, aligned the top of the mea-sured bands to the top of the projected bulk valence bands in Figs. 6 and 7. A number of valence-band surface states fromthe SIC calculations, most noticeably the T 1dangling-bond band, result in very satisfying agreement with the ARPESdata. 44It might well be that some of the valence-band fea- tures observed in experiment are bulk derived since there isno counterpart at all for these features in the calculated sur-face band structure. The same good overall agreement in thevalence bands could also be achieved with the LDA results ifthe experimental ARPES data were aligned, in view of thelack of knowledge of their absolute energy position, with theT 1band of the LDA surface band structure at the S /H11032point, as was done in Ref. 42. FIG. 6. Surface band structure of the BDM of the C-terminated SiC /H20849001 /H20850-c/H208492/H110032/H20850surface as resulting from standard LDA calcula- tions. The gray-shaded areas show the projected bulk band struc-ture. Surface states and resonances are indicated by thick and thinlines. The thick lines refer to pronounced surface states or reso-nances which are predominatly localized on the ﬁrst two surfacelayers. ARPES data from Ref. 44 and ARIPES data from Ref. 45show measured valence and conduction band states, respectively.ARPES data have not been reported along the S /H11032-M-S line, to date, and ARIPES data have only been measured along the S- /H9003-M line. FIG. 7. Surface band structure of the BDM of the C-terminated SiC /H20849001 /H20850-c/H208492/H110032/H20850surface as resulting from SIC calculations. For further details, see caption of Fig. 6.BAUMEIER, KRÜGER, AND POLLMANN PHYSICAL REVIEW B 73, 195205 /H208492006 /H20850 195205-10Also the ARIPES data have been referred to the extrinsic Fermi level of the samples used in Ref. 45. In this case theFermi level position with respect to the valence band maxi-mum has been inferred from other literature data on equallydoped samples to be located 1.5 eV above the top of thevalence bands. If this assignment is correct we can refer theARIPES data to the top of the valence bands, as is done inFigs. 6 and 7 without the need of any rigid relative shift. Comparing the two ﬁgures it becomes obvious that the low-est empty surface-state band resulting from LDA deviatesmore strongly from the lowest band determined in ARIPES,actually by 1.3 eV, while this deviation is reduced to 0.9 eVin the SIC surface band structure. In general we note fromthe comparison that some of the dispersions of the ARIPESdata /H20849even if the lowest measured empty band was aligned with the calculated T 1band /H20850cannot be reconciled with the theoretical results, neither with the LDA nor the SIC surfaceband structure. We conclude from this comparison that the surface band structure of 3C-SiC /H20849001 /H20850-c/H208492/H110032/H20850, calculated within the SIC approach, shows general improvements over the standard LDA surface band structure concerning the projected bulkband structure and the projected gap, in particular, the abso-lute energy positions of empty surface-state bands, the char-acter of localized surface states /H20849most noticeably the band T 3/H20850and the antibonding T1band which is in somewhat better agreement with experiment. Certainly, these improvementsare less impressive than those for the bulk band structures ofthe SiC polytypes discussed above. The fact that the upward shift of the T 1band resulting within SIC, as compared to LDA, is relatively small /H20849only 0.4 eV /H20850ought largely to be due to the fact that the occupied T1and the empty T1bands both originate from the triple-bonded C wC surface dimers and thus are mainly derived from bulk states in the uppervalence bands. These are not inﬂuenced dramatically by SIC,as we have seen in Sec. II, so that the improvements in thecalculated band gap and conduction bands of 3C-SiC do not fully affect the T 1band position by the same upward shift in energy. To the best of our knowledge, there are no GWAresults for this surface available in the literature, to date,which could be used for further comparison. A better ex-ample for showing pronounced SIC effects on empty surfacestates would certainly be the relaxed cubic 3C-SiC /H20849110 /H20850-/H208491 /H110031/H20850surface which features an occupied C-derived dangling- bond band near the top of the valence bands and an emptySi-derived dangling-bond band near the bottom of the con- duction bands. 46So the latter can be expected to show a similar upward shift in energy as the bulk conduction bands/H20849mainly Si-derived /H20850when calculated within the SIC ap- proach. Nevertheless we refrained from selecting that ex-ample since there are no experimental surface spectroscopydata available in the literature on 3C-SiC /H20849110 /H20850-/H208491/H110031/H20850. VI. SUMMARY In this paper we have shown how atomic self-interaction corrections can be incorporated in the nonlocal part of ionicSi and C pseudopotentials to be used in bulk and surfacecalculations. Within DFT calculations we have applied theseSIC pseudopotentials to the most commonly considered cu-bic and hexagonal polytypes of silicon carbide and haveshown that the typical LDA shortcomings in the descriptionof the electronic band structure of these polytypes can almostentirely be overcome. From the comparison of our resultswith experimental data and other theoretical results from theliterature we arrive at the conclusion that SIC pseudopoten-tials are most suitable for electronic structure calculations.Our results have been achieved without any extra computa-tional effort compared to standard LDA calculations, muchin contrast to GWA calculations. In particular in view of thisfact, the reached agreement with literature data from experi-ment and GWA calculations is highly satisfactory and em-phasizes that our approach to account for self-interaction cor-rections is a powerful tool for a more accurate description ofthe electronic properties of 3C-, 2H-, 4H-, and 6H-SiC bulk crystals. In addition, we have found that structural pa-rameters, such as lattice constants and bulk moduli, derivedfrom total energies calculated employing the SIC pseudopo-tentials, result in excellent agreement with experiment. Fi-nally, we have shown for an exemplary case that the SICapproach also yields a number of general improvements inthe description of surface electronic states, as compared tostandard LDA. 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PhysRevB.77.195320.pdf	Adiabatic charge and spin pumping through quantum dots with ferromagnetic leads Janine Splettstoesser Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland Michele Governale and Jürgen König Institut für Theoretische Physik III, Ruhr-Universität Bochum, D-44780 Bochum, Germany Theoretische Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany /H20849Received 4 February 2008; revised manuscript received 8 April 2008; published 22 May 2008 /H20850 We study the adiabatic pumping of electrons through quantum dots attached to ferromagnetic leads. Hereby, we make use of a real-time diagrammatic technique in the adiabatic limit that takes the strong Coulombinteraction in the dot into account. We analyze the degree of spin polarization of electrons pumped from aferromagnet through the dot to a nonmagnetic lead /H20849N-dot-F /H20850as well as the dependence of the pumped charge on the relative leads’ magnetization orientations for a spin-valve /H20849F-dot-F /H20850structure. For the former case, we ﬁnd that, depending on the relative coupling strength to the leads, spin and charge can, on average, be pumpedin opposite directions. For the latter case, we ﬁnd an angular dependence of the pumped charge, whichbecomes more and more anharmonic for large spin polarization in the leads. DOI: 10.1103/PhysRevB.77.195320 PACS number /H20849s/H20850: 72.25.Mk, 73.23.Hk, 85.75. /H11002d I. INTRODUCTION Charge and spin transport through a nanoscale conductor can be obtained, in the absence of a transport voltage, byperiodically varying in time some of its parameters. If thetime dependence of the system is slow compared to its char-acteristic response time, we refer to this transport mechanismas adiabatic pumping. This particular regime allows us to study the properties of a system that is slightly out of equi-librium due to an explicit time dependence of its parameters.Numerous works have studied mesoscopic pumps boththeoretically 1–5as well as experimentally.6–10The established framework to calculate the pumped charge through a meso-scopic scatterer is based on the dynamical scatteringapproach. 1,11This approach can be applied when the Cou- lomb interaction can be neglected or treated within the Har-tree approximation. Recently, the interest in including theeffects of Coulomb interaction beyond the Hartree level tothe problem of adiabatic pumping has arisen. 12–20 Spin-dependent transport through nanostructures has re- cently attracted a lot of interest. A model example is aquantum-dot spin valve, which consists of an interacting/H20849single-level /H20850quantum dot attached to two ferromagnetic leads /H20849F-dot-F /H20850/H20849see Fig. 1/H20850. The leads have, in general, non- collinear magnetization directions and different polarizationstrengths. Transport through a quantum-dot spin valve withnoncollinear leads has been extensively studied in the dclimit. 21,22In magnetic multilayers, the tunneling current chieﬂy depends on the relative orientation of the magnetiza-tion of the ferromagnetic layers. 23,24The situation is more complex in a quantum-dot spin valve due to the interplay ofthe lead magnetization, the Coulomb interaction, the non-equilibrium spin accumulation, and the quantum ﬂuctuations.In particular, a ﬁnite spin accumulation is generated on thedot, which plays an important role in determining chargetransport. In the present paper, we combine the ideas of adiabatic pumping and spin-dependent transport through interactingnanostructures. We consider two scenarios. First, we focuson the situation when only one of the two leads is ferromag- netic /H20849N-dot-F /H20850, for which we study the spin pumped into the nonmagnetic lead. Spin pumping in systems where the spindegeneracy is lifted by a magnetic ﬁeld has been the subjectof several studies. 10,14,18,25,26Furthermore, spin pumping by means of electrical gating only was predicted in a systemwith Rashba spin-orbit coupling. 27Several aspects of a non- interacting spin pump based on ferromagnets were studied inRef. 28. Spin pumping through an interface between a ferro- magnet and a nonmagnetic metal has been investigated inRef. 29, wherein the pumping cycle is realized by exploiting the precession of the magnetization of the ferromagnet. Inour setup, we are interested in spin pumping obtained byperiodically varying the properties of the scattering region,such as the dot level position and the tunneling strength tothe left and the right leads, but leaving the lead properties,as, e.g., their magnetizations, constant in time. A particularintriguing result for the system under consideration is that,depending on the relative coupling strength between dot andleads, spin and charge can, on average, be transported inopposite directions. Second, we consider the case that both leads are spin polarized /H20849F-dot-F /H20850. We study the inﬂuence of the spin po- larizations of the leads on the pumped charge and on theaverage spin accumulated on the quantum dot during apumping cycle. As a result, we ﬁnd that also the pumpedcharge displays the spin-valve effect, i.e., a dependence on dotF F,N ˆnRx ˆnL ˆnRφˆnL yz FIG. 1. Schematic illustration of the N-dot-F or F-dot-F setup. The magnetization directions and the polarization strengths of theleft and right leads can, in general, differ from each other /H20849left /H20850. Sketch of the coordinate system and the magnetization directions ofthe leads /H20849right /H20850.PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 1098-0121/2008/77 /H2084919/H20850/195320 /H208499/H20850 ©2008 The American Physical Society 195320-1the relative angle between magnetization directions of the leads. For stronger spin polarization of the leads, the pumpedcharge becomes a more and more anharmonic function of therelative angle. In order to calculate the charge and the spin pumped through the dot, we use a real-time diagrammatic techniquein the adiabatic limit 19,30and perform a rigorous perturbation expansion in the tunnel coupling to the leads. We considerthe system in the regime of weak coupling, taking into ac-count only ﬁrst-order processes in the tunnel-couplingstrengths. II. MODEL AND FORMALISM A. Hamiltonian We consider a single-level quantum dot contacted by tun- nel barriers to two ferromagnetic leads with different spinpolarization axes, as shown in Fig. 1. For ﬁnite spin polar- ization in both leads /H20849F-dot-F /H20850, the system is called a quantum-dot spin valve. The limit of one normal and oneferromagnetic lead /H20849N-dot-F /H20850is included by setting the spin polarization of one lead to zero. The total Hamiltonian of thesystem can be written as H=H dot+/H20858 /H9251=L,RHlead,/H9251+Htunnel. /H208491/H20850 It consists of the Hamilton operators for the dot, for the left /H20849L/H20850and right /H20849R/H20850leads, and for the electron tunneling be- tween the dot and the leads. The single-level quantum dot isdescribed by the Hamiltonian H dot=/H20858 /H9268=↑,↓/H9280/H9268/H20849t/H20850n/H9268+Un↑n↓, /H208492/H20850 where the operator d/H9268†/H20849d/H9268/H20850creates /H20849annihilates /H20850an electron with spin /H9268=↑,↓on the dot and n/H9268=d/H9268†d/H9268is the number operator for electrons with spin /H9268. The strength of the Cou- lomb interaction between electrons on the dot is denoted byU, which can be arbitrarily large. The energy level /H9280/H9268/H20849t/H20850=/H9280¯/H9268+/H9254/H9280/H9268/H20849t/H20850of the dot can vary in time. In the follow- ing, we assume the dot level to be spin degenerate, i.e., /H9280↑/H20849t/H20850=/H9280↓/H20849t/H20850=/H9280/H20849t/H20850. The Hamiltonian of the lead /H9251, with /H9251=L,R, is given by Hlead,/H9251=/H20858 k/H9268/H9280/H9251k/H9268c/H9251k/H9268†c/H9251k/H9268. /H208493/H20850 We choose the spin quantization axis of lead /H9251along the direction of its magnetization, nˆ/H9251. The spin /H9268of an electron in lead /H9251can take the values /H9268=/H11006, where /H20849+/H20850refers to the majority spin and /H20849−/H20850to the minority spin of this lead. We choose a coordinate system with the three basis vectors, eˆx, eˆy, and eˆz, which point along nˆL+nˆR,nˆL−nˆR, and nˆR/H11003nˆL, respectively, in analogy to the deﬁnition in Ref. 21. The angle between the spin quantization axis of the left lead nˆL and the spin quantization axis of the right lead nˆRis given by /H9278. We show a sketch of the coordinate system and of the magnetization directions of the leads in Fig. 1. As the spin quantization axis of the dot, we take the zaxis of this coor- dinate system. With this choice, the tunneling HamiltonianreadsH tunnel =VL/H20849t/H20850 /H208812/H20858 k/H20851cLk+†/H20849ei/H9278/4d↑+e−i/H9278/4d↓/H20850 +cLk−†/H20849−ei/H9278/4d↑+e−i/H9278/4d↓/H20850/H20852 +VR/H20849t/H20850 /H208812/H20858 k/H20851cRk+†/H20849e−i/H9278/4d↑+ei/H9278/4d↓/H20850 +cRk−†/H20849−e−i/H9278/4d↑+ei/H9278/4d↓/H20850/H20852+ H.c. /H208494/H20850 The tunnel matrix elements, VL/H20849t/H20850and VR/H20849t/H20850, can be both time dependent. The generalized tunnel rates are deﬁned by /H9003/H9251/H20849t,t/H11032/H20850=1 2/H20858/H9268=/H110062/H9266V/H9251/H11569/H20849t/H20850V/H9251/H20849t/H11032/H20850/H9267/H9251,/H9268=1 2/H20858/H9268=/H11006/H9003/H9251,/H9268/H20849t,t/H11032/H20850 and /H9003/H9251/H20849t/H20850=/H9003/H9251/H20849t,t/H20850. Here, /H9267/H9251,/H9268is the density of states of the spin species /H9268in lead /H9251, which is supposed to be constant. The spin polarization of lead /H9251is deﬁned as p/H9251=/H9267/H9251+−/H9267/H9251− /H9267/H9251++/H9267/H9251−, /H208495/H20850 and it can take values between 0 and 1. B. Real-time diagrammatic approach The Hilbert space of the single-level quantum dot has four dimensions, and it is spanned by the states /H9273=0,↑,↓,d /H20849empty dot, singly occupied dot with spin up, singly occu- pied dot with spin down, and doubly occupied dot /H20850.O nt h e other hand, the noninteracting leads attached to the dot havea large number of degrees of freedom and act as baths.Hence, we can trace them out to obtain an effective descrip-tion of the quantum dot. The dot dynamics are fully de-scribed by its reduced density matrix /H9267dotwith matrix ele- ments P/H92732/H92731=/H20855/H92732/H20841/H9267dot/H20841/H92731/H20856. We introduce also the notation P/H9273=P/H9273/H9273for the diagonal matrix elements /H20849probabilities /H20850. The time evolution of the reduced density matrix is governed bythe generalized master equation, d dtP/H92732/H92731/H20849t/H20850=−i /H6036/H20849E/H92731−E/H92732/H20850P/H92732/H92731/H20849t/H20850 +/H20858 /H92731/H11032,/H92732/H11032/H20885 −/H11009t dt/H11032W/H92732/H92732/H11032/H92731/H92731/H11032/H20849t,t/H11032/H20850P/H92732/H11032/H92731/H11032/H20849t/H11032/H20850. /H208496/H20850 The kernel W/H92732/H92732/H11032/H92731/H92731/H11032/H20849t,t/H11032/H20850connects the states /H92731/H11032and/H92732/H11032at time t/H11032with the states /H92731and/H92732at time t. It is useful to deﬁne the vector of the average occupation probabilitiesP=/H20849P 0,P1,Pd/H20850=/H20849P0,P↑+P↓,Pd/H20850and the spin expectation value, in units of /H6036,S=/H20849Sx,Sy,Sz/H20850, whose components are given by Sx=P↓↑+P↑↓ 2,Sy=iP↓↑−P↑↓ 2,Sz=P↑−P↓ 2. /H208497/H20850 Since we consider spin-degenerate dot levels, E↑=E↓,w ec a n drop the ﬁrst term on the right-hand side of Eq. /H208496/H20850. We are concerned with adiabatic pumping, where no transport voltage is applied across the dot. The leads aretherefore described by the same Fermi function f/H20849 /H9275/H20850.A sa consequence, the instantaneous current through the dot van-SPLETTSTOESSER, GOVERNALE, AND KÖNIG PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 195320-2ishes, and we need to consider the ﬁrst adiabatic correction in order to obtain the pumping current through the dot. Weperform an adiabatic expansion along the lines of Ref. 19. We start by performing a Taylor expansion around the time t ofP/H20849t /H11032/H20850appearing inside the integral on the right-hand side of the generalized master equation, d dtP/H92732/H92731/H20849t/H20850=/H20858 /H92731/H11032,/H92732/H11032/H20885 −/H11009t dt/H11032W/H92732/H92732/H11032/H92731/H92731/H11032/H20849t,t/H11032/H20850/H20875P/H92732/H11032/H92731/H11032/H20849t/H20850+/H20849t/H11032−t/H20850d dtP/H92732/H11032/H92731/H11032/H20849t/H20850/H20876. /H208498/H20850 This expansion is justiﬁed by the fact that the response time of the system is much smaller than the time scale of theparameter variation in time. We then expand the kernel itselfas W /H92732/H92732/H11032/H92731/H92731/H11032/H20849t,t/H11032/H20850→ /H20849W/H92732/H92732/H11032/H92731/H92731/H11032/H20850t/H20849i/H20850/H20849t−t/H11032/H20850+/H20849W/H92732/H92732/H11032/H92731/H92731/H11032/H20850t/H20849a/H20850/H20849t−t/H11032/H20850. /H208499/H20850 The superscript i/H20849a/H20850denotes the instantaneous contribution /H20849its adiabatic correction /H20850. The instantaneous contribution cor- responds to freezing all parameters to their values at time t, i.e.,X/H20849/H9270/H20850→X/H20849t/H20850. The adiabatic correction is obtained by lin- earizing the time dependence of the parameters, i.e.,X/H20849 /H9270/H20850→X/H20849t/H20850+/H20841/H20849/H9270−t/H20850d/d/H9270X/H20849/H9270/H20850/H20841/H9270=t, and retaining only ﬁrst- order terms in the time derivatives. Finally, we perform theadiabatic expansion of the elements of the reduced densitymatrix, P /H92732/H92731/H20849t/H20850→ /H20849P/H92732/H92731/H20850t/H20849i/H20850+/H20849P/H92732/H92731/H20850t/H20849a/H20850. /H2084910/H20850 The subscript tin Eqs. /H208499/H20850and /H2084910/H20850denotes the time with respect to which the adiabatic expansion is performed. Thistime tparametrically enters the respective quantities; both the instantaneous and the adiabatic correction to the kernelare functions of the time difference /H20849t−t /H11032/H20850. At this stage, it is convenient to introduce the zero-frequency Laplace trans- form of the kernel as /H20849W/H92732/H92732/H11032/H92731/H92731/H11032/H20850t/H20849i/a/H20850=/H20848−/H11009tdt/H11032/H20849W/H92732/H92732/H11032/H92731/H92731/H11032/H20850t/H20849i/a/H20850/H20849t−t/H11032/H20850.I n order to evaluate the kernel of the master equation, we per- form, on top of the adiabatic expansion, a perturbation ex-pansion in the tunnel coupling /H9003. In the following, we take into account processes in ﬁrst order in the tunnel coupling.This approach is valid in the weak-coupling limit, i.e.,k BT/H11271/H9003. At the same time, the condition for adiabaticity, /H9003/H11271/H9024 , needs to be fulﬁlled. The instantaneous occupation probabilities and their adiabatic corrections obey the equa-tions 0=/H20858 /H92731/H11032,/H92732/H11032/H20849W/H92732/H92732/H11032/H92731/H92731/H11032/H20850t/H20849i,1/H20850/H20849P/H92732/H11032/H92731/H11032/H20850t/H20849i,0/H20850, /H2084911/H20850 d dt/H20849P/H92732/H92731/H20850t/H20849i,0/H20850=/H20858 /H92731/H11032,/H92732/H11032/H20849W/H92732/H92732/H11032/H92731/H92731/H11032/H20850t/H20849i,1/H20850/H20849P/H92732/H11032/H92731/H11032/H20850t/H20849a,−1 /H20850. /H2084912/H20850 The number in the superscripts designates the order in the perturbation expansion in the tunnel coupling. The fact thatwe ﬁnd elements of the reduced density matrix in minus ﬁrstorder in the tunnel coupling is consistent with our perturba-tive scheme, as those terms are proportional to /H9024//H9003, which is small in the adiabatic limit. The evaluation of the matrixelements of the kernel is done by using a real-time diagram-matic technique, which was developed in Ref. 30, extended to systems containing ferromagnetic leads in Ref. 21, and extended to adiabatic pumping in Ref. 19. The adiabatic cor- rection to the matrix elements of the kernel does not appearin lowest order in the tunnel coupling, as considered in thispaper. The equations for the instantaneous probabilities andfor the adiabatic correction to the probabilities can be sum-marized as /H6036d dt/H20898P0 P1 Pd/H20899=/H9003/H20898−2f/H20849/H9280/H20850 1−f/H20849/H9280/H20850 0 2f/H20849/H9280/H20850−/H208511−f/H20849/H9280/H20850+f/H20849/H9280+U/H20850/H20852 2/H208511−f/H20849/H9280+U/H20850/H20852 0 f/H20849/H9280+U/H20850 −2 /H208511−f/H20849/H9280+U/H20850/H20852/H20899/H20898P0 P1 Pd/H20899+/H208981−f/H20849/H9280/H20850 −/H208511−f/H20849/H9280/H20850−f/H20849/H9280+U/H20850/H20852 −f/H20849/H9280+U/H20850/H20899/H20858 /H92512/H9003/H9251S·p/H9251./H2084913/H20850 Similarly, the equations for the expectation value of the spin read /H6036d dtS=/H20875f/H20849/H9280/H20850P0−1 2/H208511−f/H20849/H9280/H20850−f/H20849/H9280+U/H20850/H20852P1 −/H208511−f/H20849/H9280+U/H20850/H20852Pd/H20876/H20858 /H9251/H9003/H9251p/H9251 −/H9003/H208511−f/H20849/H9280/H20850+f/H20849/H9280+U/H20850/H20852S+S/H20858 /H9251B/H9251, /H2084914/H20850 where we introduced the notation p/H9251=p/H9251nˆ/H9251. The interaction- induced exchange ﬁeld or effective Bﬁeld appearing in Eq. /H2084914/H20850is given by the principal-value integral,B/H9251=/H9003/H9251p/H9251/H20885 Pd/H9275 /H9266/H208731−f/H20849/H9275/H20850 /H9275−/H9280+f/H20849/H9275/H20850 /H9275−/H9280−U/H20874. /H2084915/H20850 The instantaneous elements of the reduced density matrix are obtained by setting the left-hand side of Eqs. /H2084913/H20850and /H2084914/H20850to zero and by assigning the superscripts /H20849i,0/H20850to the vectors PandSon the right-hand side of the equations. The ﬁrst adiabatic corrections to the elements of thereduced density matrix are obtained by assigning to thevectors PandSthe superscripts /H20849i,0/H20850on the left-hand side of the equations and the superscripts /H20849a,−1 /H20850on the right-hand side.ADIABATIC CHARGE AND SPIN PUMPING THROUGH … PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 195320-3III. RESULTS Starting from the master equation, we compute both the instantaneous matrix elements of the reduced density matrixand their ﬁrst adiabatic correction. These are needed as aninput for calculating the spin and charge currents. A. Dot occupation and spin The fact that no transport voltage is applied to the system has important consequences on the occupation probabilitiesand the expectation value of the spin. To lowest order in thetunnel-coupling strength /H9003, the instantaneous occupation probabilities are given by their equilibrium values, i.e., bythe Boltzmann factors of the respective states, P /H9273/H20849i,0/H20850=e−/H9252E/H9273/H20849t/H20850 Z, /H2084916/H20850 where /H9252=1 /kBTis the inverse temperature, E/H9273/H20849t/H20850is the en- ergy of the dot state /H9273, and Zis the partition function. The spin expectation value vanishes, i.e.,31 S/H20849i,0/H20850=0 . /H2084917/H20850 When considering only ﬁrst-order tunneling processes, the adiabatic correction to the reduced density matrix is lin-ear in /H9024//H9003. While the spin polarization of the leads has no inﬂuence on the instantaneous probabilities, the situation isdifferent for the adiabatic correction, which reads P /H20849a,−1 /H20850=−dP/H20849i,0/H20850 dt/H9270relQ/H20849t/H20850/H9003/H20849t/H208502 /H90032/H20849t/H20850−/H20875/H20858 /H9251p/H9251/H9003/H9251/H20849t/H20850/H208762, /H2084918/H20850 with the charge relaxation time given by /H9270relQ/H20849t/H20850with /H20851/H9270relQ/H20849t/H20850/H20852−1=/H9003/H20849t/H20850/H208531+f/H20851/H9280/H20849t/H20850/H20852−f/H20851/H9280/H20849t/H20850+U/H20852/H20854 /H20849the derivation of the expression for /H9270relQis given in the Appendix /H20850. For vanishing polarization in the two leads, this result coincides with thatobtained for an N-dot-N system, 19,32 P/H20849a,−1 /H20850=−dP/H20849i,0/H20850 dt1 /H9003/H20849t/H208501 1+f/H20851/H9280/H20849t/H20850/H20852−f/H20851/H9280/H20849t/H20850+U/H20852. /H2084919/H20850 We ﬁnd nonvanishing contributions to the off-diagonal terms of the reduced density matrix, which vanish for zero polar-ization in the leads. The spin expectation value reads S /H20849a,−1 /H20850=1 2/H11509/H20855n/H20856/H20849i,0/H20850/H20849t/H20850 /H11509t/H9270relS/H20849t/H20850/H9003/H20849t/H20850/H20858 /H9251p/H9251/H9003/H9251/H20849t/H20850 /H90032/H20849t/H20850−/H20875/H20858 /H9251p/H9251/H9003/H9251/H20849t/H20850/H208762,/H2084920/H20850 where the spin relaxation time /H9270relS/H20849t/H20850is given by /H20851/H9270relS/H20849t/H20850/H20852−1 =/H9003/H20849t/H20850/H208531−f/H20851/H9280/H20849t/H20850/H20852+f/H20851/H9280/H20849t/H20850+U/H20852/H20854 /H20849the derivation of the expres- sion for /H9270relSis given in the Appendix /H20850. Notice that the ﬁrst adiabatic correction is the leading contribution to the expec-tation value of the dot spin. Furthermore, a time-dependentdot spin can be accumulated only by varying in time theoccupation of the dot. The adiabatic correction to the spincomponent is parallel to the exchange ﬁeld, which was intro-duced in Eq. /H2084915/H20850. Therefore, no precession of the spin around this ﬁeld takes place. This is different from the caseof a time independent but biased spin valve. 21 Finally, we remark that the limit where both leads are fully polarized along the same magnetization axis /H9278=0 and pL=pR=1 is ill deﬁned; in fact, in this case, the lifetime of a minority spin in the dot diverges and, consequently, in orderfor the adiabatic expansion to hold, the pumping frequencyneeds to be zero. B. Pumping current The results for the dot occupation probabilities and the expectation value of the spin on the dot serve to calculate thepumping current. By using a similar approach as for the gen-eralized master equation, we write the current into the leftlead as I L/H20849t/H20850=−e/H20885 −/H11009t dt/H11032/H20858 /H92731,/H92732,/H92731/H11032,/H92732/H11032/H20849W/H92731/H92731/H11032/H92732/H92732/H11032/H20850L/H20849t,t/H11032/H20850P/H92731/H11032/H92732/H11032/H20849t/H11032/H20850, /H2084921/H20850 where /H20849W/H92731/H92731/H11032/H92732/H92732/H11032/H20850L/H20849t,t/H11032/H20850=/H20858qq/H20849W/H92731/H92731/H11032/H92732/H92732/H11032/H20850Lq/H20849t,t/H11032/H20850, and /H20849W/H92731/H92731/H11032/H92732/H92732/H11032/H20850Lq/H20849t,t/H11032/H20850 is the sum of all processes, which describes transitions where the difference of the number of electrons entering and leav-ing the left lead is equal to the integer number q. We compute the ﬁrst-order adiabatic correction to the cur- rent including only ﬁrst-order tunneling processes. We ﬁnd I L/H20849a,0/H20850/H20849t/H20850=−e/H20858 /H92731,/H92732,/H92731/H11032,/H92732/H11032/H20849W/H92731/H92731/H11032/H92732/H92732/H11032/H20850tL/H20849i,1/H20850/H20849P/H92731/H11032/H92732/H11032/H20850t/H20849a,−1 /H20850. In the following, we suppress the superscript /H20849a,0/H20850for the current, since the instantaneous current is always zero and IL/H20849a,0/H20850/H20849t/H20850is therefore the dominant contribution. The current is of zeroth order in the tunnel coupling and proportional to thepumping frequency /H9024. To this order in the tunnel-coupling strengths, the pumped current is nonvanishing only if the dotlevel position is one of the pumping parameters, since /H20849P /H92731/H11032/H92732/H11032/H20850/H20849a,−1 /H20850is proportional to the time derivative of the dot level position. We ﬁnd for the pumping current IL=−e/H11509/H20855n/H20856/H20849i,0/H20850 /H11509t/H9003L/H20849t/H20850 /H9003/H20849t/H20850−e/H11509/H20855n/H20856/H20849i,0/H20850 /H11509t/H9003L/H20849t/H20850/H9003R/H20849t/H20850 /H9003/H20849t/H208502/H20849pR−pL/H20850/H9266/H20849t/H20850 1−/H9266/H20849t/H208502, /H2084922/H20850 where we have deﬁned the quantity /H9266/H20849t/H20850=/H20858/H9251/H9003/H9251/H20849t/H20850p/H9251//H9003/H20849t/H20850 =/H20858/H9251/H9266/H9251/H20849t/H20850, which depends on time via /H9003/H9251/H20849t/H20850. The pumped current consists of two terms of different origin: the ﬁrst oneis independent of the lead polarizations and can be inter-preted as arising from a peristaltic mechanism; 19the second term depends on the polarizations and can be seen as arisingfrom the relaxation of the accumulated spin on the dot. Thelatter contribution due to spin relaxation can be either posi-tive or negative depending on the polarization strengths, po-larization directions, and tunnel coupling to the differentleads. This means that the time-resolved current can be en-hanced with respect to the nonmagnetic case, in contrast tothe spin-valve effect in a time-independent system, whichalways leads to a current suppression . Similarly, we will see later that also the charge pumped through the dot per periodSPLETTSTOESSER, GOVERNALE, AND KÖNIG PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 195320-4is always suppressed due to the polarization of the leads. Therefore, this inverse spin-valve effect is observable only inthe time-resolved current response. The effect could be ex-perimentally investigated by means of time-resolved mea-surements or by rectifying the current response. Reduction orenhancement of the current can be achieved by tuning thetunnel coupling or the lead magnetizations. C. N-dot-F: Spin pumping We now turn our attention to spin pumping in a setup where only one of the leads, the right one for the sake ofdeﬁniteness, is ferromagnetic. We calculate the spin pumpedin the unpolarized left lead. The instantaneous contribution to the reduced density ma- trix is independent of the polarization and therefore remainsunchanged. The ﬁrst adiabatic correction for the occupationprobabilities and the dot spin are given by Eqs. /H2084918/H20850and /H2084920/H20850, respectively, with p L=0. The charge current through such an N-dot-F system can be directly obtained from Eq. /H2084922/H20850by setting pL=0 and it readsIL=−e/H11509/H20855n/H20856/H20849i,0/H20850 /H11509t/H9003/H20849t/H20850/H9003L/H20849t/H20850 /H9003/H20849t/H208502−pR2/H9003R/H20849t/H208502. /H2084923/H20850 For calculating the spin current, we chose a global spin- quantization axis parallel to the magnetization of the rightlead. In this basis, the reduced density matrix does not haveany off-diagonal terms. For the spin current, we ﬁnd I LS=/H6036 2/H11509/H20855n/H20856/H20849i,0/H20850 /H11509t/H9003R/H20849t/H20850pR/H9003L/H20849t/H20850 /H9003/H20849t/H208502−pR2/H9003R/H20849t/H208502. /H2084924/H20850 The ratio of the time-resolved spin and charge currents, Eqs. /H2084923/H20850and /H2084924/H20850, reads ILS IL/H20882/H20875/H6036/2 −e/H20876=pR/H9003R/H20849t/H20850 /H9003/H20849t/H20850. /H2084925/H20850 This ratio is, in general, time dependent. The time-resolved spin current is smaller than the time-resolved particle currentat any time t. The ratio of these two currents is always posi- tive, which implies that spin and charge ﬂow in the samedirection, as expected. The situation is different for the spin/H20849in units of /H6036/2/H20850and the charge /H20849in units of − e/H20850pumped per period . We calculate the pumped charge and spin for the following two choices of pumping parameters, /H20853/H9003 L,/H9280/H20854or /H20853/H9003R,/H9280/H20854, in bilinear response, i.e., we calculate the pumped charge and spin per inﬁnitesimal area in parameter space.The result for the ratio of the pumped spin /H20849in units of /H6036/2/H20850 per period, N S, and the pumped charge /H20849in units of − e/H20850per period, N, reads NS N=−pR1+pR2/H20849/H9003¯R//H9003¯/H208502−2 /H20849/H9003¯R//H9003¯/H20850 1+pR2/H20849/H9003¯R//H9003¯/H208502−2 /H20849/H9003¯R//H9003¯/H20850pR2. /H2084926/H20850 It turns out that the efﬁciency of the spin pump does not depend on which pair of pumping parameters one chooses.In Fig. 2/H20849a/H20850, we plot the ratio of pumped spin to pumped charge as a function of the relative tunnel-coupling strength /H9003¯R//H9003¯, where the bar indicates time-averaged quantities, for different values of the polarization of the right lead. Theabsolute value of the ratio is maximally equal to one in the0 0.2 0.4 0.6 0.8 1 ΓR/Γ-1-0.500.51NS/NpR=0.99 pR=0.6 pR=0.3 pR=0(a) 0 0.2 0.4 0.6 0.8 1 ΓR/Γ00.20.40.60.81GS/GpR=0.99 pR=0.6 pR=0.3 pR=0(b) FIG. 2. /H20849a/H20850Ratio of the pumped spin per period /H20849in units of /H6036/2/H20850 to the pumped charge per period /H20849in units of − e/H20850as a function of the relative tunnel-coupling strength /H9003¯R//H9003¯for different polarizations of the right lead. /H20849b/H20850Ratio between the linear dc spin conductance and the linear dc conductance as a function of the relative tunnel- coupling strength /H9003¯R//H9003¯for different polarizations of the right lead.01234 Ωt/π-0.01-0.00500.0050.01IIL/(-eΩ) IS L/(hΩ/4π) FIG. 3. Time-resolved spin and charge currents as a function of time. The values of the parameters used for this plot are pR=0.99, /H9003¯L=/H9003¯R,/H9280¯=−/H9003¯,U=10/H9003¯,/H20841/H9254/H9003L/H20841//H9003¯=/H20841/H9254/H9280/H20841//H9003¯=0.1, and kBT=/H9003¯.ADIABATIC CHARGE AND SPIN PUMPING THROUGH … PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 195320-5case of full polarization of the right lead. For pR/H110211, this ratio goes from − pRfor vanishing /H9003¯RtopRfor vanishing /H9003¯L changing its sign for /H9003¯R /H9003¯=1 pR2/H208491−/H208811−pR2/H20850. /H2084927/H20850 This is a very intriguing result, which implies that the respec- tive direction in which spin and charge are pumped dependson the coupling to left and right leads. The average pumped charge and the average pumped spin can have opposite signs, while the time-resolved spin andcharge currents ﬂow in the same direction at any instant of time due to the fact that the ratio of the time-resolved cur-rents /H20851Eq. /H2084925/H20850/H20852is itself time dependent. To elucidate this, in Fig. 3, we plot the time-resolved spin and charge currents as a function of time for a conﬁguration, where the pumpedspin and charge per period have different signs. Note that thecharge current has a positive average and the spin current has a negative average, while both currents ﬂow in the samedirection at any time. We now contrast the results for the pumped spin and charge to the dc transport properties of the N-dot-F system.For the spin and charge currents, we ﬁnd IS I/H20882/H20875/H6036/2 −e/H20876=/H208511−fL/H20849/H9280/H20850+fL/H20849/H9280+U/H20850/H20852/H9003LpR /H208511−fL/H20849/H9280/H20850+fL/H20849/H9280+U/H20850/H20852/H9003L+/H208511−fR/H20849/H9280/H20850+fR/H20849/H9280+U/H20850/H20852/H9003R/H208491−pR2/H20850, /H2084928/H20850 which, in the linear response regime, yields for the ratio of the spin to the charge conductance GS G/H20882/H20875/H6036/2 −e/H20876=/H9003LpR /H9003L+/H9003R/H208491−pR2/H20850. /H2084929/H20850 The linear conductance ratio is shown in Fig. 2/H20849b/H20850. Its behav- ior is completely different from that obtained by pumping.First, the spin polarization decreases as a function of /H9003 R//H9003 and, second, it stays always positive. Finally, we consider the spin accumulated on the dot in one pumping period. We ﬁnd two different results, dependingon whether /H20853/H9003 L,/H9280/H20854or/H20853/H9003R,/H9280/H20854are the pumping parameter, /H20855S/H20849a,−1 /H20850/H20856T/H20849/H20853/H9003L,/H9280/H20854/H20850=−/H9257 /H9003¯/H11509/H20855n¯/H20856 /H11509/H9280¯/H9270¯relS/H9266¯R /H208491−/H9266¯R2/H208502, /H2084930/H20850 /H20855S/H20849a,−1 /H20850/H20856T/H20849/H20853/H9003R,/H9280/H20854/H20850=/H9257 2/H9003¯/H11509/H20855n¯/H20856 /H11509/H9280¯/H9270¯relS/H20873/H9003¯L /H9003¯−/H9003¯R /H9003¯/H20874+/H9266¯R2 /H208491−/H9266¯R2/H208502pR, /H2084931/H20850 where the area of the cycle in parameter space /H9257is deﬁned as /H9257=/H208480Tdt/H11509/H9280 /H11509t/H9254/H9003Land/H9257=/H208480Tdt/H11509/H9280 /H11509t/H9254/H9003Rfor the ﬁrst and second equations, respectively. In the case of pumping with /H20853/H9003R,/H9280/H20854, i.e., when the coupling to the ferromagnetic lead is time de-pendent, the average spin changes sign at the same values of /H9003¯R//H9003¯at which the ratio ILS/ILchanges its sign. On the con- trary, when pumping with /H9003Land/H9280, the average spin polar- ization of the dot does not change sign as a function of /H9003¯R//H9003¯, while the ratio ILS/ILstill does. D. F-dot-F: Spin-valve effect We now consider the spin-valve setup with both leads having arbitrary spin polarizations. We compute the numberof pumped charges per period, N=−1 e/H208480TdtIL/H20849t/H20850, in bilinear response in the pumping parameters. For the pumping cycle deﬁned by /H9280/H20849t/H20850=/H9280¯+/H9254/H9280/H20849t/H20850and/H9003L/H20849t/H20850=/H9003¯L+/H9254/H9003L/H20849t/H20850, the number of pumped charges per period reads N=/H9257/H11509/H20855n¯/H20856/H20849i,0/H20850 /H11509/H9280¯/H11509 /H11509/H9003¯L/H20898/H9003¯L/H20858 /H9251/H9003¯/H9251−/H9003¯LpL/H20858 /H9251/H9003¯/H9251p/H9251 /H9003¯2−/H20873/H20858 /H9251/H9003¯/H9251p/H9251/H208742/H20899, /H2084932/H20850 where /H9257=/H208480Tdt/H11509/H9280 /H11509t/H9254/H9003Lis the area of the cycle in parameter space. Notice that the charge number in Eq. /H2084932/H20850is a product of two terms, where one contains the effects of interactionsand another one the effects of the leads’ magnetizations. In the following, we show the results for the case that both leads have the same spin polarization strength. Thiscorresponds to the experimentally relevant situation that bothleads are realized with the same ferromagnetic material. InFig.4, we show the pumped charge as a function of the level position for different values of the angle between the direc-tions of the magnetizations of the two leads. The pumped charge shows a peak when the energy /H9280¯or/H9280¯+Uis close to the Fermi energy, similar to pumping through a quantum dotcontacted to two nonmagnetic leads /H20849N-dot-N /H20850. 19As far as the dependence on the angle between the magnetization ofthe two leads /H9278is concerned, for /H9278/H20678/H208510,/H9266/H20852, the charge is monotonically suppressed for increasing /H9278until a minimum is reached for /H9278=/H9266as in the usual dc spin-valve effect. The full/H9278dependence of the pumped charge is shown in Fig. 5, where we plot N/H20849/H9278/H20850/N/H20849/H9278=0 /H20850. This result does not depend on the value of the level position and of the interaction strength, since the dependence on /H9280¯andUcancels out when we divide byN/H20849/H9278=0 /H20850. We notice that the suppression of charge pump- ing is stronger for higher lead polarizations. Furthermore, themore the lead polarization is increasing, the stronger the be-havior of the pumped charge as a function of the angle de-viates from a cosine law.SPLETTSTOESSER, GOVERNALE, AND KÖNIG PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 195320-6In Fig. 6, we show the pumped charge as a function of the lead polarization strengths for different values of the anglebetween the magnetization directions. This plot conﬁrms thatthe pumped charge decreases for increasing spin polarizationof the leads. The charge suppression is strongest when /H9278is near to /H9266. Independent of the angle between the polarization axis of left and right leads, the pumped charge goes to zerofor fully polarized leads. It is important to point out that thislast property depends on the order in which limits are taken,since the two limits /H9278→0 and pL=pR→1 do not commute, as was already mentioned in Sec. III A. In fact, comparingFigs. 5and6, we notice that in the ﬁrst case the charge is maximal for /H9278=0 even for the polarization increasing toward one, while in the second case, for pL=pR=1, the charge is maximally suppressed even for /H9278going to zero. The spin, which is accumulated on the dot during one pumping cycle, is given by/H20855S/H20856T/H20849a,−1 /H20850=/H9257 2/H11509/H20855n¯/H20856/H20849i,0/H20850 /H11509/H9280/H9270¯relS/H20900/H9003¯pL /H9003¯2−/H20873/H20858 /H9251p/H9251/H9003¯/H9251/H208742 −2/H9003¯2−/H9003¯pL/H20858 /H9251p/H9251/H9003¯/H9251 /H20875/H9003¯2−/H20873/H20858 /H9251p/H9251/H9003¯/H9251/H208742/H208762·/H20858 /H9251p/H9251/H9003¯/H9251/H20901, /H2084933/H20850 where the pumping parameters are chosen to be /H9003Land/H9280. The result for pumping with /H9003Rand/H9280is easily obtained by swapping the indices LandR. Depending on the spin polar- ization of the leads and on the values of tunnel-couplingstrengths, the average spin on the dot can point along anydirection in the plane containing the magnetizations of theleads. IV. CONCLUSIONS We have investigated adiabatic pumping through a single- level quantum dot with ferromagnetic leads in the regime ofweak tunnel coupling between dot and leads by means of areal-time diagrammatic approach. In the case that only onelead is ferromagnetic, we have computed the spin injected inthe nonmagnetic lead by pumping. We have found that, de-pending on the relative strength of the tunnel coupling to theleads, spin and charge can be pumped, on average, in oppo-site directions. For the case when both leads are polarized,we have found a suppression of the pumped charge by meansof the spin-valve effect and determined the average spin ac-cumulated on the dot during one pumping cycle. ACKNOWLEDGMENTS We would like to thank M. Büttiker for useful discussions. We acknowledge ﬁnancial support from the EU via theSTREP project SUBTLE and from the DFG via ContractsNo. SPP 1285 and No. SFB 491.-15 -10 -5 0 5 ε/Γ00.020.040.06 N[ -η/Γ2]φ=0 φ=π/4 φ=π/2 φ=π FIG. 4. Pumped charge as a function of the average level posi- tion/H9280¯for different values of the angle between the magnetizations. The polarizations in the leads are pL=pR=0.8. 0 π 2π 3π 4π φ00.250.50.751N/N( φ=0) p=0.4 p=0.6 p=0.9 FIG. 5. Pumped charge as a function of the angle between the magnetizations of the leads /H9278for different polarization strengths pL=pR=p. This result does not depend on the level position and the interaction strength.0 0.2 0.4 0.6 0.8 1p00.20.40.60.81N/N( φ=0) φ=π/10 φ=π/4 φ=π/2 φ=π FIG. 6. Pumped charge as a function of the polarization strength p=pL=pRfor different values of the angle between the magnetiza- tions. This result does not depend on the level position and theinteraction strength.ADIABATIC CHARGE AND SPIN PUMPING THROUGH … PHYSICAL REVIEW B 77, 195320 /H208492008 /H20850 195320-7APPENDIX: RELAXATION TIMES In this appendix, we calculate the spin and charge relax- ation times. In order to calculate the spin relaxation time, weconsider the case when the charge on the dot is in equilib-rium and the occupation probabilities are therefore given bythe Boltzmann factors. Then, Eq. /H2084914/H20850simpliﬁes to dS dt=−/H9003/H208511−f/H20849/H9280/H20850+f/H20849/H9280+U/H20850/H20852S, /H20849A1 /H20850 where we also made use of the fact that the spin is always parallel to the exchange ﬁeld. The spin relaxation time istherefore given by /H9270relS=1 /H90031 1−f/H20849/H9280/H20850+f/H20849/H9280+U/H20850. /H20849A2 /H20850 In order to calculate the charge relaxation time, we consider Eq. /H2084913/H20850, where we take the spin in equilibrium, such that S=0. 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PhysRevB.83.035207.pdf	PHYSICAL REVIEW B 83, 035207 (2011) Comparison of the defective pyrochlore and ilmenite polymorphs of AgSbO 3using GGA and hybrid DFT Jeremy P. Allen,M. Kristin Nilsson, David O. Scanlon, and Graeme W. Watson† School of Chemistry and CRANN, Trinity College Dublin, Dublin 2, Ireland (Received 20 October 2010; published 24 January 2011) Silver antimonate, AgSbO 3, in both its defective pyrochlore and ilmenite structural polymorphs, has been suggested as a possible candidate mixed metal oxide for use in the photocatalytic splitting of water in visible light.In this study, we report electronic-structure calculations, using both standard and hybrid density-functional-theoryapproaches, on both structural forms of AgSbO 3to fully characterize the band structure and composition of the valence and conduction bands. Analysis of conduction properties and optical absorption is also used to comparethe predicted properties of the two materials. Results show that the valence band is dominated by O 2 pand Ag 4dstates, whereas the conduction band is composed mainly of Ag and Sb 5 sstates. Band-edge effective-mass calculations indicate the materials operate via an n-type mechanism, with conduction properties being comparable for the two materials. The fundamental and optical band gaps are also predicted to be compatible with visiblelight adsorption. DOI: 10.1103/PhysRevB.83.035207 PACS number(s): 31 .15.−p, 71.20.Mq, 71 .15.Mb I. INTRODUCTION Over the past decade, photocatalysis has received a great deal of attention.1–3This has mainly been in an effort to achieve an efﬁcient use of solar radiation to help combat issues relatingto both energy production, such as the formation of H 2from the splitting of water, and environmental concerns, such as the degradation of organic pollutants.1,4,5 The search for new or improved photocatalysts is never straightforward, as certain requirements in the electronicstructure are needed. For example, a number of condi-tions are required for an efﬁcient water-splitting material.Not only is the size of the band gap of importance, but also the positions of the band edges. The conduction- band minimum (CBM) must have a potential more neg-ative than that of the H +/H2redox potential [0 V ver- sus normal hydrogen electrode (NHE)]. In addition, thevalence-band maximum (VBM) must have a more positivepotential than the redox potential of O 2/H2O(+1.23) eV . This also provides the requirement that the very minimum theoretical band gap for a water-splitting material is 1 .23 eV . To generate an effective photocatalyst that is driven by visiblelight, a band gap of less than 3 .0 eV is also required. 1 The ﬁrst reported semiconductor for use in solar hydrogen production was anatase TiO 2,6and, as such, it has spawned a vast amount of research.1,7–9TiO 2is also highly stable and cost-effective, however, as anatase TiO 2has a band gap of 3.2e V ,4it cannot operate efﬁciently under visible light illumination. One approach to improve the efﬁciency of TiO 2has been through doping with nonmetallic elements, such as N and C,or metallic elements, for example Cr and V , but this approachonly yields limited improvement. 10–14Although doping offers one way to enhance the properties, investigations of alternative systems can also be instructive. One such alternative is mixedmetal oxides, which have been shown to possess promisingphotocatalytic properties. 1,15–18 Most mixed metal oxide photocatalysts typically contain at least two different metal cations, one of low valence (I-II) anda second of higher valence (III-VI). The higher valence metal cation possesses either d0, such as Ti(IV), Nb(V), or W(VI), ord10electronic conﬁgurations, such as Ga(III), Sn(IV), or Sb(V). This gives rise to conduction bands (CB) that arecomposed mainly of dors/pstates, respectively. As s/p electrons are less localized than delectrons, they are believed to give rise to a more dispersive conduction band with higherelectron mobility and higher photocatalytic activity. 19,20The low valence cation, often an alkali or alkaline earth metalelement, has little inﬂuence on the top of the valence band(VB), giving rise to an O 2 pdominated VB. However, many of the mixed metal oxides with these compositions, such asCa 2Nb2O7, NaTaO 3, and NaSbO 3, have a band gap greater than 3 eV, making them only responsive to ultraviolet (UV)rather than visible light. To make them usable as a visible lightphotocatalyst, some kind of modiﬁcation or band engineeringis required. 1By choosing a low valence cation that has orbitals that will mix with the O 2 pstates, such as Ag(I) 4 dor Pb(II) 6sstates, the energy of the valence band can be raised and the band gap decreased. An example of such a material is thephotocatalyst AgSbO 3.21,22 AgSbO 3has two main polymorphs, with defective pyrochlore23and ilmenite24structures. The ilmenite is metastable, only forming through low-temperature ionexchange from the isostructural NaSbO 3, and will undergo a phase transition to the defective pyrochlore under heattreatment. 24,25The reported optical band gaps are 2.6 (Ref. 22) and 2.4–2 .5 eV (Refs. 21and25) for the defective pyrochlore and ilmenite structures, respectively. The evolution of O 2, via the photocatalytic splitting of water, in the defective pyrochlore structure has been studiedby Kako et al. 22The results of this study showed it to have a greater performance than WO 3, which is known to be a good photocatalyst for O 2evolution in the presence of Ag(I) but inactive for H 2evolution.26Although the defective pyrochlore was untested for H 2evolution, this suggests that it could have a possible role as a water splitter. The authorsalso considered its activity toward the degradation of organicmolecules, through the oxidation of 2-propanol, with results 035207-1 1098-0121/2011/83(3)/035207(8) ©2011 American Physical SocietyALLEN, NILSSON, SCANLON, AND WATSON PHYSICAL REVIEW B 83, 035207 (2011) suggesting that AgSbO 3in a defective pyrochlore structure has a strong enough oxidizing potential to decompose organiccompounds. Singh and Uma 21also looked at the potential for AgSbO 3 to decompose organic molecules, studying both the ilmeniteand defective pyrochlore polymorphs. Their results suggestedthat the ilmenite is superior to the defective pyrochlore forthe degradation of the organic dyes and 2-chlorophenol undervisible light. The defective pyrochlore structure showed eitherreduced activity, or, for the 2-chlorophenol, a lack of signiﬁcantactivity at all. However, they did suggest that this lower activitycould be related to varying stoichiometries in the samples, ast h ew o r ko fK a k o et al. 22showed that deviations away from ideal stoichiometries had a signiﬁcant effect on the reactivity,with Ag 1.00SbO 3>Ag1.02SbO 3>Ag0.99SbO 3. The reduction in reactivity with stoichiometry for the defective pyrochlore AgSbO 3suggests that any defects present in the material will cause a reduction in the photocatalyticproperties. For the Ag-deﬁcient Ag 0.99SbO 3, Ag vacancies are suggested to act as centers of recombination between thephotogenerated holes and electrons, 22,27in a similar manner to that observed for AgTaO 3.28For hyperstoichiometries, Ag1.02SbO 3, the silver excess is manifested as metallic silver, which Kako et al. reasoned would have a shielding effect on the surface of the material, reducing both the number ofactive sites for O 2evolution and the amount of visible light it could adsorb.22Wang et al.29have also studied the effect of varying the Ag /Sb ratio on the photocatalytic properties. Conversely, they reported that an increase in the amount ofAg to Sb caused a reduction in the optical band gap and anincrease in the photocatalytic activity, which they attributed tothe formation of Sb(III) in the sample. Kako and Ye 25suggested that the photocatalytic properties of AgSbO 3can be improved by preparing samples with mixed phases of the defective pyrochlore and ilmenite. Their resultsshowed a greater activity than both a TiO 2photocatalyst and the single-phase ilmenite material for the decomposition ofacetylaldehyde to CO 2. They claim that the cause of this increased photocatalytic activity is a synergistic effect betweenthe two phases, which occurs as the band edges of the ilmenitepolymorph lay within those of the defective pyrochlore. Although the primary focus on AgSbO 3has been for its utilization in photocatalysis, the defective pyrochlore has alsobeen investigated for use as an Ag(I) ion conductor, 30ann-type thermoelectric material,31,32and as a transparent conducting oxide (TCO).23,33 Density-functional-theory (DFT) calculations have been previously employed by Kako et al.22and Mizoguchi et al.23 to consider the electronic structure of the defective pyrochlore form of AgSbO 3. Both studies showed that the composition of the VB and CB are as expected, with the top of the VBconsisting of a mixture of Ag 4 dand O 2 pstates and the bottom of the CB dominated by Ag and Sb 5 sstates. The composition of the VB has also been conﬁrmed through UVphotoemission spectroscopy (UPS). 34As expected for these computational approaches, the calculated band gaps for thedefective pyrochlore structures are signiﬁcantly underesti-mated, with reported values of 0.1 (Ref. 23) and 0 .4e V . 22 The aim of this study is to provide a characterization of the electronic structures of both the defective pyrochlore andilmenite forms using hybrid-DFT, which is expected to not only give a better structural representation but also to signiﬁ-cantly improve the calculated band gap. 35–38Calculations have also been carried out using the standard generalized-gradientapproximation (GGA), allowing for a direct comparison to bemade with the hybrid-DFT method. In addition, the calculationof band structures, optical absorption and the hole effectivemasses at both the VBM and CBM allows a quantiﬁcation ofthe conduction properties of these materials. This not onlyallows comparisons to be made between the two differentstructures, but also a discussion of the suitability of the twomaterials for n-type water splitting. II. COMPUTATIONAL METHODS The calculations described in this study were all performed using the periodic DFT code V ASP ,39,40which uses a plane- wave basis set to describe the valence electrons. The projector-augmented-wave (PAW) 41,42method was used to describe the interactions between the cores (Ag: [Kr], Sb: [Kr], and O:[He]) and valence electrons. Two methods of treating theexchange and correlation were used in this study to allow acomparison to be made of their effectiveness. The ﬁrst methodused the standard GGA approach with the Perdew-Burke-Ernzerhof (PBE) 43functional. The second approach was that of Heyd, Scuzeria, and Ernzerhof (HSE06),44,45which uses a screened hybrid functional and includes a percentage ofexact Fock exchange. The HSE06 methodology is identicalto that described elsewhere, where the percentage of exactnonlocal Fock exchange added to the PBE functional is 25%and the long- and short-range parts of the functional arepartitioned by a screening of ω=0.11 bohr −1.38,46Although hybrid functionals are more computationally demanding, theyare often found to give better approximations of band gaps insemiconductor systems and improved structural data. 35,38,47–63 The bulk equilibrium lattice parameters were determined by performing structural optimizations at a series of volumes. Ineach of these calculations, the atomic positions, lattice vectors,and cell angles were allowed to relax while the total cellvolume was held ﬁxed. The resulting energy-volume curveswere then ﬁtted to the Murnaghan equation of state to obtainthe equilibrium bulk cell volume. 64This approach avoids the problems of Pulay stress and changes in basis set thataccompany volume changes in plane-wave calculations. Thetwo polymorphs were modeled using their primitive unit cells,for which a /Gamma1-centered 4 ×4×4k-point mesh was found to be sufﬁcient for both materials. A plane-wave cutoff of 500 eVwas used for the PBE calculations but reduced to 400 eVfor the HSE06 due to the high computational cost. 46For all calculations, the structures were deemed to be converged whenthe forces on all the atoms were less than 0 .01 eV ˚A −1. The optical-absorption spectra, as well as the opti- cal transition matrix, was calculated within the transver-sal approximation, 65using an increased k-point mesh of 6×6×6. This approach sums all direct VB to CB transi- tions to determine the optical absorption, thereby ignoringboth indirect and intraband transitions. 66As only single- particle transitions are included, any electron-hole correlationswould require higher-order electronic-structure methods. 67,68 035207-2COMPARISON OF THE DEFECTIVE PYROCHLORE AND ... PHYSICAL REVIEW B 83, 035207 (2011) However, this approach has been shown to provide reasonable optical-absorption spectra.16,38,46,56,69 Structural ﬁgures have been generated using the VESTA package.70 III. RESULTS AND DISCUSSION A. Defective pyrochlore structure The defective cubic pyrochlore structure, space group Fd3m, is the most common form adopted by AgSbO 3.A typical cubic pyrochlore structure has a general formula ofA 2B2O6X, where Xis typically O, F, or OH.71The structure is composed of a corner-sharing BO6octahedra network, with the larger Acations possessing eightfold coordination, approaching a hexagonal bipyramid, to six O and twoXanions. This AX 2sublattice forms a channel network through the structure. The defective pyrochlore structure, exhibited byAgSbO 3, differs from that of a typical cubic pyrochlore in that theXanions are absent. This gives rise to sixfold-coordinated Ag ions with a distorted octahedral geometry, approximating aﬂattened trigonal antiprism. 72The structure is shown in Fig. 1, with a comparison of the calculated structural parameters tothe experimental structure of Mizoguchi et al. 23provided in Table I. As can be seen, there is good agreement between the calculated and experimental structures, with the HSE06functional providing a better ﬁt to experiment as expected. The calculated total and partial (ion decomposed) electronic densities of states (EDOS and PEDOS, respectively) for thedefective pyrochlore structure are shown in Fig. 2. The EDOS can be broadly separated into four regions, with the VBcomprising regions I–III and region IV representing the CB. The HSE06 calculation gives rise to a widening of the band gap and a small expansion of the VB in comparison to thePBE calculation, therefore giving slightly different widths forthe different regions of the EDOS. However, similarities in thepeak structure and composition are seen between the differentmethods. The O 2 pstates are seen throughout all regions in the EDOS for both methods, although the contributions from thecations are seen to differ between regions. The cation states ab c FIG. 1. (Color online) Schematic showing the optimized AgSbO 3 defective pyrochlore structure. Silver, antimony, and oxygen atoms are colored blue (light gray), purple (medium gray), and red (darkgray), respectively. The antimony atoms are also shown as polyhedra in the main image. Coordination environments of the AgO 6and SbO 6 octahedra are also shown.TABLE I. Comparison between experimental lattice constants and bond lengths for AgSbO 3in the defective pyrochlore structure with those calculated using PBE and HSE06 methods. Percentage changes from experiment are given in parentheses, and all values arein˚A except the cell volume, which has units of ˚A 3. Property PBE HSE06 Experiment23 a 10.43(1.6) 10 .31(0.4) 10 .27 V olume 1134 .62(6.0) 1095 .91(2.4) 1083 .21 Ag-O 2 .59(1.6) 2 .58(1.2) 2 .55 Sb-O 2 .01(1.5) 1 .97(−0.5) 1 .98 Ag-Ag 3 .69(1.7) 3 .65(0.6) 3 .63 Ag-Sb 3 .69(1.7) 3 .65(0.6) 3 .63 Sb-Sb 3 .69(1.7) 3 .65(0.6) 3 .63 in regions I and II are primarily composed of Sb 5 sand 5 p states, respectively, with a small amount of Sb 4 dstates seen in region II. Region III, however, is a mixture of Ag and Sb 4 d states mixing with the O 2 p, with the Ag states dominating. The CB, region IV , is comprised of Ag 5 s,S b5 sand p, and O 2 p. Agreement is seen between the calculated PEDOS and experimental photoemission spectra.34The UPS data of Yasukawa et al. indicate four peaks in the upper valence band. Working back from the Fermi energy, the ﬁrst three peaks weredesignated as being composed of Ag 4 dand O 2 pstates, with the peak at the top of the VB having a signiﬁcant contributionfrom the Ag 4 dstates. The fourth peak was described as consisting of mixed O 2 pand cationic sandpstates. This description is qualitatively similar to the calculated PEDOS, with Ag 4 dand O 2 pstates dominating the upper valence band. The calculated PEDOS also suggests that the cationic s p states d states s states(a) (b) (i) (ii) (iii) (iv)(i) (ii) (iii) (iv)II VIII II II VIII IITotal EDOS Ag PEDOS (x2) O PEDOS (x3)-10 -8 -6 -4 -2 0 24 6 -10 -8 -6 -4 -2 0 2 4 6 Ener gy (eV)-10 -8 -6 -4 -2 0 24 6 -10 -8 -6 -4 -2 0 2 4 6 Energy (eV)Sb PEDOS (x15)Total EDOS Ag PEDOS (x2) Sb PEDOS (x15) O PEDOS (x3) FIG. 2. (Color online) Electronic density of states for AgSbO 3in a defective pyrochlore structure, split into (i) the total EDOS, (ii) Ag PEDOS, (iii) Sb PEDOS, and (c) O PEDOS, as calculated using the (a) PBE and (b) HSE06 method. The s,p,a n d dstates are colored blue (dot-dash), red (solid), and green (dash), respectively. Vertical dashed gray lines represent divisions between different regions in the DOS. The Fermi energy has also been set to 0 eV. 035207-3ALLEN, NILSSON, SCANLON, AND WATSON PHYSICAL REVIEW B 83, 035207 (2011) LXEnergy (eV) W(a)Energy (eV)(c)(b) (d)-4-202468 ΓΓ-4-202468 LX W ΓΓ LX W ΓΓ-4-202468 LX W ΓΓ-4-202468 FIG. 3. (Color online) Band structure of the defective pyrochlore form of AgSbO 3calculated using (a) the PBE approach and (b)–(d) the HSE06 method. The HSE06-determined band structures are shown in fatband format with the contributions to the bands from the(b) Ag 4 d(green), (c) Sb 5 s(blue), and (d) O 2 p(red) states indicated. The Fermi level is set to 0 eV, as indicated by the horizontal dashed gray line. andpstates described for the fourth peak originate from the Sb cations, with very little contribution seen for the Ag sand pstates in the valence band. The calculated band structures for the defective pyrochlore, along the space group high-symmetry lines from Bradley andCracknell, 73are given in Fig. 3. The VBM is located at Land extends in the Lto/Gamma1direction for both methods, whereas the CBM is observed at the /Gamma1point, leading to indirect band gaps of 0.08 and 2 .09 eV for PBE and HSE06, respectively. The smallest direct band gaps, however, are only slightly larger at0.09 and 2 .11 eV for PBE and HSE06, respectively, and are observed at the /Gamma1point. For the HSE06-predicted band structure, fatband analysis has been conducted to show the contribution from the Ag 4 d, Sb 5s, and O 2 pstates, Figs. 3(b)–3(d), respectively. As seen in the PEDOS, the top of the VB is dominated by mixed Ag 4 dand O2pstates, with very little contribution from the Sb 5 sstates. The bottom of the CB is comprised of mainly Sb 5 sand O 2pstates. The contributions from Ag 5 sstates to the bottom of CB, not shown, are comparable to those of the O 2 pstates. The extent of the Sb 5 sand O 2 pcontributions to the lowest bands in the CB are also seen to be phase-dependent. For example,at the /Gamma1point, the primary component is seen to be from the Sb 5 sstates, however this is reversed at the W point, where O2pstates dominate. Although not shown, fatband analysis of the PBE-predicted band structures yields similar results. Thisanalysis is qualitatively similar to a previous GGA study byMizoguchi et al. , 23although they report a greater mixing of Ag 5 sstates at the CBM with the Perdew-Wang functional. B. Ilmenite structure The ilmenite structure has a rhombohedral unit cell, space group R3H, and consists of distorted octahedral AgO 6and SbO 6units, with the former showing the greatest distortion, ex- hibiting a trigonal antiprism polyhedron. The SbO 6octahedra form edge-sharing sheets in the abplane, as do the distorted AgO 6octahedra, with the layers alternating between Ag and Sb in the cdirection, as shown in Fig. 4. The distorted AgO 6 octahedra connect to the SbO 6octahedra by face-sharing on one side in the cdirection and corner-sharing in the other, with this arrangement alternating across the layer in the abplane. Comparisons between the calculated and experimental structures are given in Table II. As expected, the HSE06 method gives rise to the closest ﬁt to experiment, with cellvectors and bond lengths within 2 .1 %, whereas the PBE method predicts values within 3 .0 %. However, the clattice vector is seen to be expanded relative to the aandbvectors for both methods. This is more apparent in the HSE06-minimizedstructure as the PBE-calculated vectors also possess the typicaloverestimation seen for PBE calculations. Although the sourceof this expansion is not apparent, it may be a result of the failureof DFT (and hybrid-DFT) to account for van der Waals forces,as previously seen in the expansion of layered materials, suchas SnO (Refs. 74and75) and V 2O5.76,77 The calculated EDOS and PEDOS for the ilmenite structure are shown in Fig. 5. As with the defective pyrochlore structure, the EDOS can be broadly characterized by four regions, with abc FIG. 4. (Color online) Schematic showing the optimized AgSbO 3 ilmenite structure. Silver, antimony, and oxygen atoms are colored blue (light gray), purple (medium gray), and red (dark gray),respectively. The antimony atoms are also shown as polyhedra in the main image. Coordination environments of the AgO 6and SbO 6 octahedra are also shown for reference. 035207-4COMPARISON OF THE DEFECTIVE PYROCHLORE AND ... PHYSICAL REVIEW B 83, 035207 (2011) TABLE II. Comparison between experimental lattice constants and bond lengths for AgSbO 3in the ilmenite structure with those calculated using PBE and HSE06 methods. Percentage changes from experiment are given in parentheses, and all values are in ˚A except the cell volume, which has units of ˚A3. Property PBE HSE06 Experiment24 a 5.42(1.7) 5 .35(0.4) 5 .33 b 5.42(1.7) 5 .35(0.4) 5 .33 c 17.05(2.1) 16 .99(1.7) 16 .70 V olume 500 .87(5.6) 486 .30(2.5) 474 .43 Sb-O 2 .02(2.5) 1 .99(1.0) 1 .97 2.04(3.0) 2 .01(1.5) 1 .98 Ag-O 2 .41(0.0) 2 .41(0.0) 2 .41 2.75(1.5) 2 .75(1.5) 2 .71 Ag-Ag 3 .28(2.2) 3 .24(0.9) 3 .21 Ag-Sb 3 .40(2.7) 3 .38(2.1) 3 .31 Sb-Sb 3 .13(1.6) 3 .09(0.3) 3 .08 the VB spanning regions I–III and region IV representing the CB. As expected, both the PBE- and HSE06-predicteddensities of states have a number of similarities in both the peakstructures and their compositions, with the primary differencebeing an increased band gap with the HSE06 approach. TheO2pstates are evident throughout the VB, with the different regions being characterized by the mixing with different Agand Sb states. The cation states are very similar to those seen forthe defective pyrochlore structure, with regions I and II beingcomposed primarily of Sb 5 sand 5 p, respectively. Region II also has a minor contribution from the Sb 4 dstates. Region III, however, is a mixture of Ag and Sb 4 dstates with the O 2 p. p states d states s states(a) (b) (i) (ii) (iii) (iv)(i) (ii) (iii) (iv)II V III II I IV III IITotal EDOS O PEDOS (x3)Total EDOS Ag PEDOS (x2) Sb PEDOS (x15) O PEDOS (x3)-10 -8 -6 -4 -2 0 24 6 -10 -8 -6 -4 -2 0 2 4 6 Energy (eV)-10 -8 -6 0 6 -10 -8 -6 -4 -2 0 2 4 6 Energy (eV)Sb PEDOS (x15)Ag PEDOS (x2)-4 -2 2 4 FIG. 5. (Color online) Electronic density of states for AgSbO 3in an ilmenite structure, split into (i) the total EDOS, (ii) Ag PEDOS, (iii) Sb PEDOS, and (c) O PEDOS, as calculated using the (a) PBE and (b) HSE06 method. The s,p,a n d dstates are colored blue (dot-dash), red (solid), and green (dash), respectively. Vertical dashed gray lines represent divisions between different regions in the DOS. The Fermi energy has also been set to 0 eV.F Γ Z Γ L-4-202468 F Γ Z Γ L-4-202468Energy (eV)(a) F Γ Z Γ L-4-202468Energy (eV)(c)(b) -4-202468 FZ L(d) FIG. 6. (Color online) Band structure of the ilmenite form of AgSbO 3calculated using (a) the PBE approach and (b)–(d) the HSE06 method. The HSE06-determined band structures are shownin a fatband format with the contributions to the bands from the (b) Ag 4 d(green), (c) Sb 5 s(blue), and (d) O 2 p(red) states indicated. The Fermi level is set to 0 eV, as indicated by the horizontal dashedgray line. The CB, region IV , is comprised of mixed Sb 5 sandpwith O2pstates. Overall, the EDOS for the two different AgSbO 3structures share a number of similarities, with the basic composition ofboth the VB and CB being qualitatively the same. The band structures of the ilmenite form of AgSbO 3,u s i n g both PBE and HSE06 methods, are given in Fig. 6. Although both methods give rise to a similar band structure, the majordifference is in the opening up of the band gap with the HSE06method. The positions of both the VBM and CBM are the samefor both methods, located at the /Gamma1point, with direct band gaps of 0.13 and 1 .92 eV for the PBE and HSE06 methods, respectively. The HSE06-predicted band structure is shown in a fatband format, displaying the contributions from the Ag 4 d,S b5 s, and O 2 pstates in Figs. 6(b)–6(d), respectively. As expected from the PEDOS, the top of the VB is dominated by mixedAg 4 dand O 2 pstates, with little contribution from the Sb 5sstates. The bottom of the CB, however, is comprised of mainly Sb 5 sand O 2 pstates. The contribution of the Ag 5 s states, not shown, to the lowest-energy band is similar to thoseseen for the Sb 5 sstates. A phase-dependent mixing of the Sb 5sand O 2 pstates is observed in the lowest bands in the CB. For the lowest energy band, the O 2 pcontribution is seen to decrease signiﬁcantly on approaching the /Gamma1point, whereas for the second lowest energy band in the CB, a similar case is seenfor the Sb 5 scontribution around the Zpoint. Although not 035207-5ALLEN, NILSSON, SCANLON, AND WATSON PHYSICAL REVIEW B 83, 035207 (2011)Optical Absorption α2 (arb. units) Photon Energy (eV)1.01.0 1.21.2 1.41.4 1.61.6 1.81.8 2.02.0 2.22.2 2.42.4 2.62.6 2.82.8 3.03.0 3.23.2 3.43.4 3.63.6 3.83.8 4.04.0 Defective pyrochlore (calculated) Ilmenite (calculated)Defective pyrochlore (experimental) Ilmenite (experimental) Extrapolation FIG. 7. (Color online) Calculated optical absorption ( α2)o ft h e two forms of AgSbO 3summed over all possible direct VB to CB tran- sitions. Solid lines represent calculated absorption using the HSE06 method with the extrapolation used to determine the calculated optical band gap shown by the dotted lines. The experimental optical bandgaps are given by the dashed lines. 21,22 shown, fatband analysis of the PBE-predicted band structures yields similar results. C. Optical absorption As the experimental optical band gaps of 2.6 and 2 .5e V f o r the defective pyrochlore21and ilmenite22structures, respec- tively, are determined by optical absorption, it is instructive tocompare these directly to the calculated optical absorption,rather than the fundamental band gaps. The Tauc relationstates that E g∝α2, therefore, by extrapolating α2, we can determine the value of the optical band gap. Figure 7shows the plots of optical absorption (in terms of α2) for both AgSbO 3structures calculated using HSE06, with comparison to experiment.21,22The calculated optical band gaps for the defective pyrochlore and ilmenite forms are seen to be equalto 2.44 and 2 .24 eV, respectively. Both the magnitudes and the relative order are consistent with experiment. The calculatedoptical band gaps, however, are larger than the fundamentalband gaps of 2.09 and 1 .92 eV for the defective pyrochlore and ilmenite, respectively. For both materials, adsorption between the VBM and CBM at the /Gamma1point is forbidden, which is the location of the direct fundamental band gap. The onset of adsorption forthe defective pyrochlore is therefore seen along the /Gamma1-X high-symmetry line, and along /Gamma1-Zfor the ilmenite structure. This observation is also noted for a range of oxide materialsthat show a symmetry-forbidden fundamental band gap, suchas CuBO 2,46SrCu 2O2,78In2O3,79and Cu 2O.80 D. Band-edge effective masses To allow a comparison of the electronic conductivity of the two structures, the electron and hole effective masses at theTABLE III. Band-edge effective masses for the VBM and CBM of the defective pyrochlore and ilmenite forms of AgSbO 3, calculated with the HSE06 method. Note: For the ilmenite, the /Gamma1-Ldirection is the same as the [010]. Defective pyrochlore [001] [010] [100] L-/Gamma1L -W/Gamma1 -L/Gamma1 -X VBM 0.85 0.85 0.85 11.79 1.03 CBM 0.26 0.26 0.26 0.26 0.26 Ilmenite [001] [010] [100] /Gamma1-F/Gamma1 -Z VBM 4.30 2.51 4.30 18.17 1.55 CBM 0.27 0.27 0.27 0.28 0.29 CBM and VBM, respectively, can be determined. The effective mass ( m∗) is calculated by 1 m∗(E)=1 ¯h2kdE dk, (1) where E(k) is the band-edge energy as a function of wave vector k, obtained directly from the calculation.81The bands at the top of the VB for both structures are clearly not parabolic,therefore AgSbO 3is not expected to be well described under a typical semiconductor effective-mass approximation. How-ever, as the bottom CB bands for the two structures are seen tobe more parabolic in nature, the calculated electron effectivemasses should have a higher degree of accuracy than the holeeffective masses. However, the calculated effective masses willserve as an approximate guide allowing comparisons to bedrawn, as has been previously done for Cu MO 2(where M= Al, Sc, Y , Cr, and B),82(Cu 2S2)(Sr 3Sc2O5),38and In 2O3,83 indicating the relative abilities of the two structures for n-or p-type conductivity. The calculated effective masses at the VBM and CBM for the defective pyrochlore and ilmenitestructures are given in Table III. Effective masses have been calculated in the [001], [010], and [100] directions, as well asalong the special directions shown in the previously mentionedband structures. As can be seen, the results show that in terms of the n-type conductivity, both structures have very similar properties. Theelectron effective masses of the CBM are also smaller thanthe hole effective masses of the VBM for both structure types,indicating that the n-type ability of the materials will be greater than the p-type ability. The p-type properties differ between the structure types though, with the VBM hole effective massessuggesting that the defective pyrochlore structure will possessbetter p-type properties. However, neither material is predicted to exhibit strong p-type properties. Effective-mass calculations have been used recently to describe In 2O3, which is an industrial standard n-type TCO.83 The effective masses at the VBM and CBM were calculated as being 16 meand 0.24 me, respectively, and can be considered as being indicative of poor p-type and good n-type ability. The CBM electron effective masses of both AgSbO 3polymorphs are comparable to In 2O3, albeit slightly larger, suggesting that AgSbO 3has strong n-type properties. Although the calculated hole effective masses of the VB are more approximate, dueto its nonparabolic nature, the p-type properties will also be 035207-6COMPARISON OF THE DEFECTIVE PYROCHLORE AND ... PHYSICAL REVIEW B 83, 035207 (2011) signiﬁcantly better than In 2O3.C u 2O is the parent compound of a range of p-type delafossite TCO’s with general formula CuMO2, where Mis typically a group 3 or 13 metal. The ex- perimental hole effective mass of the VB of Cu 2Oi s0 . 5 6 me,84 whereas the calculated value for the delafossite CuBO 2is 0.45 me.82In comparison, our calculated VBM hole effective masses suggests that the defective pyrochlore may also exhibitreasonable p-type properties. IV . CONCLUSION In conclusion, this study has used both PBE and HSE06 DFT approaches to model the electronic structure of thedefective pyrochlore and ilmenite forms of AgSbO 3.A s expected, the HSE06 method affords a structure that has amuch better ﬁt to experiment, in terms of both the unit-celldimensions and the bond lengths. Despite the different structures, the orbital composition of the density of states and band structures for the two materialsis similar, with orbital contributions to the bands in agreementwith experiment 34and previous calculation.23,25The top of the VB is composed primarily of Ag 4 dand O 2 pstates. For the ilmenite polymorph, the O 2 pstates in the uppermost bands are slightly diminished, with respect to the defectivepyrochlore, which may give rise to the increased dispersion inthese bands. For the bottom of the CB, the composition is seento be mainly mixed Ag 5 s,S b5 s, and O 2 pstates. However, there is a difference in the phase dependence of contributionsto the lowest-energy conduction band. Both materials show areduction in the O 2 pstates at the CBM, which increases in directions away from this point. For the defective pyrochlore,t h eS b5 scontribution also has an inversely proportional relationship to those of the O 2 p, reaching a maximum at the CBM.Agreement between experiment and the HSE06-calculated band gaps is also seen. For the defective pyrochlore (ilmenite)structure, the HSE06 approach predicts an indirect (direct) fun-damental band gap of 2 .09 eV (1 .92 eV) and an optical band gap of 2 .44 eV (2 .24 eV), which compares well with the ex- perimental value of 2 .6e V ( R e f . 22) [2.4–2 .5e V ( R e f s . 21and 25)]. The calculated band gaps are also consistent with their use in photocatalytic splitting of water in visible light, whichrequires a band gap of between 1.23 and 3 .00 eV. The magni- tudes of these optical band gaps, however, exclude the use ofAgSbO 3as a TCO, which previous studies have suggested. The two materials are also seen to have comparable effective masses for the CB, which would give rise to similar n-type properties. As the values of the effective masses are onlyslightly bigger than those calculated for In 2O3,83their n-type properties are predicted to be good, which is a consequenceof the strong dispersion seen at the bottom of the CB dueto Sb 5 sand O 2 pinteractions. In contrast to these results, experiment has suggested different reactivities for the twopolymorphs. 21,29However, this has been linked to differing stoichiometries rather than an inherent property of the purematerials. The defective pyrochlore is predicted to have betterp-type ability than the ilmenite, and, while neither polymorph is predicted to be a strong p-type material, the defective pyrochlore may show reasonable p-type properties. ACKNOWLEDGMENTS This work was supported by Science Foundation Ireland through the Principal Investigators program (PI Grants No.06/IN.1/I92 and No. 06/IN.1/I92/EC07). Calculations wereperformed on the IITAC and Lonsdale supercomputers asmaintained by TCHPC, and the Stokes supercomputer asmaintained by ICHEC. allenje@tcd.ie †watsong@tcd.ie 1A. Kudo and Y . Miseki, Chem. Soc. Rev. 38, 253 (2009). 2D. Ravelli, D. Dondi, M. Fagnoni, and A. Albini, C h e m .S o c .R e v . 38, 1999 (2009). 3U. I. Gaya and A. H. Abdullah, J. Photochem. 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PhysRevB.71.134527.pdf	Physics of cuprates with the two-band Hubbard model: The validity of the one-band Hubbard model A. Macridin and M. Jarrell University of Cincinnati, Cincinnati, Ohio, 45221, USA Th. Maier Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA G. A. Sawatzky University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada V6T 1Z1 sReceived 3 November 2004; published 29 April 2005 d We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respectto electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen inone-particle spectral functions and in the charge, spin, and d-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhangand Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, wehave found similar low-energy physics provided that the next-nearest-neighbor hopping term t 8has a signiﬁ- cant value st8/t<0.3d. The parameter t8is the main culprit for the electron-hole asymmetry. However, a signiﬁcant value of t8cannot be provided in a strict Zhang and Rice fPhys. Rev. B 37, R3759 s1988 d;41, 7243 s1990 dgpicturewheretheextraholesaddedintothesystembindtotheexistingCuholesforminglocalsinglets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of t8, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band t -t8-UHubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromag- netism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describephysical properties involving energy scales larger than about 0.5 eV. DOI: 10.1103/PhysRevB.71.134527 PACS number ssd: 74.25.Dw, 71.10.Hf, 71.10.Fd, 74.25.Jb I. INTRODUCTION The theory of the cuprate high-temperature superconduct- ors remains one of the most important and daunting prob-lems in condensed-matter physics. The high T ccuprate su- perconductors are layered materials with relatively complexstructures and chemical composition. They are highly corre-lated, with an effective bandwidth roughly equal to the ef-fective local Coulomb interaction. The short-range correla-tions are known to play a paramount role in these materials.Therefore, the dynamical cluster approximation 1sDCA d, which treats short-range correlations explicitly and the long-range physics at the mean-ﬁeld level, is an ideal tool for theinvestigation of these systems. Acommon characteristic all cuprate materials share is the presence of quasi-two-dimensional CuO 2planes. These planes are commonly believed to contain the low-energyphysics. However, the full complexity of the orbital chemis-try of just the CuO 2planes and the strong Coulomb repulsion on the Cu ions would lead to models that are very difﬁcult tostudy with conventional techniques. The cuprates are characterized by a very rich, but also, in many respects, very intriguing physics. The undoped materi-als are antiferromagnetic sAFM dinsulators with a gap of approximatively 2 eV. Upon doping the AFM is destroyedand the system becomes superconducting sSCd. At small doping, in the proximity of the AFM phase, the normal statephysics cannot be described in terms of Fermi-liquid theory and is characterized by the presence of a pseudogap. An essential demand of every successful theory is to capture allthese fundamental features at the same time. Experimental data show that the phase diagram and other physical characteristics, such as the density of states sDOS d near the Fermi level of the hole- and electron-doped materi-als, are very different. 2–4There could be many reasons for this asymmetry. The electron- and hole-doped materials arephysically different, and apart from the CuO 2planes, they contain different elements and chemical structures. Thesestructural and compositional differences can inﬂuence thelow-energy physics. Therefore in this paper, we use DCA toaddress whether the physics of a pure CuO 2plane contains this asymmetry or if the origin of the asymmetry in realmaterials comes from other inﬂuences. Different models for describing the physics of a CuO 2 plane were proposed by various authors. Photoemission ex-periments in the insulating parent material show that the ﬁrstelectron-removal states have primarily oxygen character;whereas, the ﬁrst electron-addition states have dcharacter, already suggesting a strong asymmetry. This places thesematerials in the charge-transfer gap region of the Zaanen-Sawatzky-Allen scheme. 5Early on, considering the ligand ﬁeld symmetry and band-structure calculations,6–8it was re- alized that the most important degrees of freedom are the Cud x2−y2, which couple with the in-plane O porbitals. There-PHYSICAL REVIEW B 71, 134527 s2005 d 1098-0121/2005/71 s13d/134527 s13d/$23.00 ©2005 The American Physical Society 134527-1fore, one of the ﬁrst models proposed to describe the physics of highTcmaterials was the so-called three-band Hubbard model presented by Varma et al.9and Emery et al.,10which considers explicitly both the oxygen psand the cooper dx2−y2 orbitals. In fact, because the direct oxygen-oxygen hopping is neglected, only the combination of oxygen orbitals withx 2−y2symmetry couples with the dorbitals, and the above- proposed three-band model reduces to a two-band model. However Zhang and Rice sZRd11argue that the low- energy physics of the hole-doped superconductors can be described by a single-band model. Starting from the two-band model, Zhang and Rice claim that an extra hole addedinto the oxygen band binds strongly with a hole on the Cu,forming an on-site singlet. This singlet state, which has zerospin can be thought as moving through the lattice like a holein an antiferromagnetic background. Consequently, the phys-ics can be described by a one-band t-Jmodel. Pertinent criticism to these simpliﬁed models were raised by various authors. With respect to Cu degrees of freedom,Eskeset al. 12stressed the possible importance of the other d orbitals, showing that they should be explicitly consideredwhen physics, which implies excitations with energy largerthan <1 eV, is involved. However, these criticisms do not concern us for the present study because we are interestedonly in physics at energies lower than <0.5 eV. Investigating the relative importance of various param- eters describing the CuO 2planes it was realized early on that, in addition to the Cu on-site Coulomb repulsion sUdd <8e V dand Cu-O hopping integral stpd<1.3–1.5 eV d, the O-O hopping integrals result in a large O 2 pbandwidth sW <5e V d, indicating that these should be included explicitly in any theory.12–14Therefore, using the DCA technique as a means of including all these most important parameters andbands, we address two major problems in this paper: thephysics of the CuO 2plane sincluding a detailed study of the electron-hole asymmetry dand the reduction of the multiband model to a single-band model. Regarding the reduction to a one-band model, one of the most serious criticisms to ZR theory is the neglect of the O2pband structure. 15,16The natural tendency of the ﬁnite oxy- gen bandwidth is to delocalize and destabilize the ZR sin-glets. The question arises whether the low-energy states si.e., the ZR singlets dare still well separated from the higher- energy states si.e., the nonbonding oxygen states d. Otherwise, the reduction to a single-band model, which neglects thesehigh-energy states, is not possible. This problem was previ-ously considered by Eskes and Sawatzky 16within an impu- rity calculation approach, but there, unlike in the DCA ap-proach, both the spatial correlation effects and the dispersionof the low-energy states were neglected. Another important objection to ZR theory was raised by Emery and Reiter 17and regards the nature of the low-energy states. Are these states real singlets that can be mapped ontoholes, or does the hole on the O bind into a more compli-cated state that involves more than one Cu hole? Choosing aparticular solvable example, which considers the Cu spinsarranged ferromagnetically, they showed that the low-energystates are, in fact, an admixture of the Zhang-Rice singletsand the corresponding triplets. This implies a nonzero valuefor the oxygen spin and destroys the equivalence of these states to holes. However, it is not clear if the situation issimilar in the cuprates, i.e., if the ZR singlet-ZR triplet ad-mixture is signiﬁcant. But the merit of Emery and Reiter is toemphasize that the fact that, as a consequence of the strongCu-O hybridization low-energy states well separated fromthe nonbonding oxygen band states appear, does not neces-sarily mean that the physics can be reduced to a single-bandmodel. The third problem we address regarding the reduction to a single-band model is the estimation of the single-band pa-rameters. We note that different approximations result in dif-ferent values of the parameters. Especially the magnitude ofthe next-nearest-neighbor hopping is very dependent of theinitial assumptions. For example, if we assume that the holeaddition low-energy states are genuine ZR singlets, i.e.,bound states between a Cu hole and a orthogonal Wannieroxygen orbital, we obtain a negligible next-nearest-neighbor hopping. 18On the other hand, if we consider the low-energy states to be plaquette singlets, i.e., bound states between a Cuhole and a hole on the state formed by the four oxygensaround the Cu, the value of the next-nearest-neighbor hop-ping is signiﬁcant. 19Of course, because of the nonorthogo- nality of the plaquette states, the plaquette singlets are notgenuine singlets and, therefore, they cannot be rigorouslymapped into holes. However, because their overlap with thelocal singlets is large s96%d, 11,17it is still possible that this approximation is good. Our calculations show that a multiband model and a single-band t-t8-UHubbard model with a signiﬁcant value of the next-nearest-neighbor hopping exhibit a similar low-energy physics. The essential parameter needed for theagreement is the next-nearest-neighbor hopping, t 8. This pa- rameter is also the main culprit for the observed electron-hole asymmetry. However, as mentioned above, the largevalue oft 8cannot be obtained in a strict ZR picture.Thus our results also implicitly indicate that the multiband model can-not be rigorously reduced to a single-band model. Therefore,besides showing the similarities between the two models, wealso point out their signiﬁcant differences in this paper. The ﬁnal conclusion is that a single-band t-t 8-UHubbard model, with a signiﬁcant value of t8, captures the basic phys- ics of the cuprates and thus is suitable to describe the AFM,pseudogap, and SC physics together with the relevant asym-metries observed in the phase diagram, in the one-particlespectra and in the two-particle response functions. However,we believe that it is not suitable for a quantitative material-speciﬁc analysis, for describing the higher-energy spectro-scopic features as in optical spectroscopy or resonant inelas-tic x-ray scattering, or for studying more subtle featuresrelated to the ﬁnite value of the spin on the oxygen. This paper is organized as follows. In Sec. II the two-band Hubbard model and the DCA technique is introduced. Ourtwo-band model takes fully into account the oxygen disper-sion and considers only the oxygen degrees of freedom thatcouple directly to the Cu d x2−y2orbitals. The results of the DCA calculation applied to the two-band Hamiltonian are presented in Sec. III. The possible reduction of the two-bandmodel to a single-band model, together with a detailed analy-sis of the single-band t-t 8-UHubbard model, is addressed inMACRIDIN et al. PHYSICAL REVIEW B 71, 134527 s2005 d 134527-2Sec. IV. A discussion regarding the similarities and the dif- ferences between the two-band and single-band models isgiven in Sec. V.The conclusions of our study are reviewed inSec. VI. II. FORMALISM A. The model Hamiltonian Band-structure calculations,14,20cluster calculation,12 photoemission,12and other experiments show that the rel- evant Cu degrees of freedom are the dx2−y2orbitals, which couple with the in-plane pxandpyO orbitals. All these de- grees of freedom result in a ﬁve-band sfour oxygen and one copper band dHamiltonian, in general. We have studied the ﬁve-band model, in detail,21and have found that due to the strong Cu-O hybridization, only the oxygen degrees of free-dom, which couple directly with Cu, are relevant for thelow-energy physics. Consequently, to a very good approxi-mation, the ﬁve-band model can be reduced to a two-bandmodel. The two-band model contains one Cu d x2−y2correlated band and one oxygen band. At every site the oxygen states are obtained by taking a linear combination with x2−y2sym- metry of the four O psorbitals, which form a plaquette around the Cu ion. These are the only oxygen states thathybridize directly with Cu. However, it should be mentionedthat these plaquette states are not orthogonal, two neighbor-ing states sharing a common oxygen atom. An orthogonalbasis can be obtained by the procedure described in the origi-nal ZR paper. 11First, applying a Fourier transform, transla- tional invariant sBloch dstates are constructed. The Bloch states are orthogonal but not normalized, so they should bemultiplied by a normalization factor bskd=fsin2skx/2d+sin2sky/2g−1/2. s1d After normalization a complete and orthonormal set of oxy- gen states is obtained. In this basis the two-band Hubbard Hamiltonian can be written as H=o k,sEskdcks†cks+Eddks†dks+Vskdscks†dks+H.c. d +Uo indi"ndi#. s2d We work in the hole representation, and dks†scks†drepresents the creation operator of a Cu sOdhole with spin sand mo- mentumk. The O-band dispersion and the Cu-O hybridiza- tion are given by Eskd=Ep−8tppb2skdsin2skx/2dsin2sky/2ds3d Vskd=2tpdb−1skds 4d withtppbeing the O-O hopping integral. The last term in Eq. s2drepresents the Coulomb repulsion between two holes on the same dorbital.We choose the commonly accepted values of the parameters, based on the band-structure calculations ofMcMahan et al. 20and Hybertsen et al.14Because of the lowdensity of oxygen holes s25–30% d, we treat the Coulomb repulsion on Osgiven by Uppdand the repulsion between nearest-neighbor Cu and O holes sgiven byUpddat the mean- ﬁeld level as a reasonable approximation. The effect will be an increases of our estimation for D=Ep−EdbyUpsn¯p/2d +Updsn¯d−n¯pd, wheren¯dandn¯pare the average occupation of Cu and, respectively, O bands. A choice of Upp=6 eV,Upd =1.3 eV, and n¯p=0.3 results in a increase of Dby 1.3 eV. To conclude, we take in Eq. s2d,tpd=1.3 eV, tpp=0.65 eV, D =4.8 eV, and U=8.8 eV. B. DCA technique The DCA is an extension of the dynamical mean ﬁeld theory22sDMFT d. The DMFT maps the lattice problem to an impurity-embedded self-consistently in a host and thereforeneglects spatial correlations. The DCA maps the lattice to aﬁnite-sized periodic cluster embedded in a host. Nonlocalcorrelations up to the cluster size are treated explicitly, whilethe physics on longer length scales is treated at the mean-ﬁeld level. Here we calculate the properties of the embeddedcluster with a quantum Monte Carlo sQMC dalgorithm. The cluster self-energy is used to calculate the properties of thehost, and this procedure is repeated until a self-consistentconvergent solution is reached. The self-energy and vertexfunctions of the cluster are then used to calculate latticequantities. Below we give a brief description of DMFT andits generalization to DCA. In DMFT, the self-energy can be obtained by neglecting the momentum conservation at the interaction vertices of thegenerating functional and its derivatives. Formally, forHubbard-like models, 23this is done by replacing the Laue function D=o rN e−isk1+k2−k3−k4dr=Ndk1+k2,k3+k4, s5d responsible for momentum conservation ssee Fig. 1 d, with24 DDMFT=1. s6d This is equivalent to replacing the Green’s function used in the calculation of the self-energy diagrams, with FIG. 1. Vertex interactions, which enter in the calculation of the self-energy. In DMFT the momentum conservation is completelyneglected. In DCA the momentum conservation is partiallyconsidered.PHYSICS OF CUPRATES WITH THE TWO-BAND … PHYSICAL REVIEW B 71, 134527 s2005 d 134527-3GDMFT sivd=1 No kGsk,ivd, s7d i.e., the “impurity” Green’s function GDMFTis obtained as the average of the lattice Green’s function over the entire Bril-louin zone sBZd. The DMFT algorithm is the following. sid One starts with a guess for the self-energy S DMFT, which, for instance, can be zero or a perturbation theory result. Thelattice Green’s function is then Gsk,i vd=fiv−eskd−SDMFT sivdg−1s8d siidThe impurity Green’s function is obtained using Eq. s7d, and the impurity excluded Green’s function as G0−1sivd=GDMFT−1sivd+SDMFT sivd. s9d Such a problem is reduced to an impurity embedded in a host; the impurity excluded Green’s function containing thefull information about the hybridization of the impurity withthe host. siiidThe “embedded impurity” problem is solved using techniques such as QMC, exact diagonalization, renor-malization group, etc., 22and the impurity Green’s function GDMFTis obtained. The resulting self-energy is SDMFT sivd=G0−1sivd−GDMFT−1sivd. s10d This self-energy is used again as a input for step sid, and the procedure is repeated until the convergence is reached. In DCA, the momentum conservation at the internal ver- tices of the irreducible quantities is partially restored. TheBZ is split into N ccoarse-graining cells each equivalent to the Wigner-Seitz cell of the superlattice formed by tiling thelattice with the cluster ssee Fig. 2 for N c=4d.The momentum transferred between the cells, i.e., the momentum larger thanthe cell length, is conserved. On the other hand, the conser-vation of the momentum within the cell, i.e., the momentumsmaller than the cell length, is neglected. Formally, this isdone by approximating the Laue function withD DCA=NcdK1+K2,K3+K4, s11d where the K1,K2,…K4label the cell centers. The Green’s function used in the calculation of the self-energy is then GDCAsK,ivd=Nc No k˜GsK+k˜,ivd, s12d where the k˜summation is taken over the cell centered on K. The DCA algorithm is very similar with the DMFT one,containing the same steps. The difference is that now, theself-energy is partially momentum dependent, and theproblem does not reduce to an impurity embedded in ahost, but to a cluster with periodic boundary conditions em-bedded in a host. The Green’s functions in Eqs. s9dands10d slabeled now with the DCAsubscript instead of DMFT dwill beKdependent, as it is the self-energy S DCAsK,ivd.W e solve the cluster-embedded-in-a-host problem with a Hirsch-Fye-type25QMC algorithm. A detailed description of the QMC-DCA algorithm is given in Ref. 26. Neglecting the conservation of small momentum fk,DK=s2p/NddNcgin the calculation of the self-energy is equivalent with neglecting long-ranged correlations sL .p/DKd, according to Nyquist theorem. Therefore this technique is ideal for the problems where short-range corre- lations are predominant, such as the high- Tcmaterials. For simplicity, the above discussion about DMFT and DCA was done by assuming a single-band Hubbard model.In the two-band model the oxygen degrees of freedom arenot correlated, and therefore they are not included explicitlyin the cluster. Their effect is fully contained in the cluster-host hybridization function and in the host of Green’s func-tions. The Green’s function G DCA, which enters in the calcu- lation of the self-energy, is obtained by coarse-graining thelattice Green’s function describing the dorbitals, i.e., G DCAsK,ivd=Nc No k˜GddsK+k˜,ivd, s13d where Gddsk,ivd=Fiv−Ed−Vpd2skd iv−Eskd−SDCAsK,ivdG−1 . s14d By comparing Eq. s14dto Eq. s8done can see that in Gdd there is a term resulting from the hybridization of the dand porbitals. Here we consider a 2 32 cluster of Cu ions, which we believe to be large enough to capture the essential physics ofHubbard-type models. The 2 32 cluster will result in a coarse-graining of the BZ in four cells, as shown in Fig. 2. III. TWO-BAND HUBBARD MODEL RESULTS The undoped materials have one hole per CuO unit. For tpd=0 the DOS is given by the dashed line in Fig. 3 and the hole addition states would be of pure O character. When theCu-O hybridization t pdis switched on, the extra holes added FIG. 2. Coarse-graining of the Brillouin Zone in four cells sNc =4daroundK=s0,0d,s0,pd,sp,0d, and sp,pd.MACRIDIN et al. PHYSICAL REVIEW B 71, 134527 s2005 d 134527-4to the oxygen band will scatter with the Cu spins and bound states will appear at the bottom of the oxygen band. This isillustrated by the solid line, which plots the partial dDOS that was obtained using the maximum entropy methodsMEM d 27for the analytic continuation of the QMC data to real frequencies. It can be noted that the ﬁrst hole additionstates have a strongly mixed dandpcharacter sthedchar- acter in the spectrum is large now dand an energy pushed well below the edge of the initial nonbonding oxygen band.Therefore only these states are relevant for the low-energyphysics. 28In the ZR theory these low-energy states, which appear as a consequence of the strong Cu-O hybridization,are considered to be local singlets that move through thelattice like holes in an AFM background. Consequently, theclaim is that the physics can be described by a one-bandt-Jmodel. In order to determine the phase diagram we calculate a large number of susceptibilities that are relevant for spin,charge, and superconducting ordering, both at the center andthe corner of the BZ. For example, the Néel and SC criticaltemperatures, T Nand respectively Tcin the phase diagram presented in Fig. 4 are determined from the divergence of thecorresponding susceptibilities. The pseudogap crossovertemperature T* is obtained from the maximum in the uni- form magnetic susceptibility when accompanied by a sup-pression of spectral weight in the DOS. Similar to what wasfound in the single-band Hubbard model, 29we ﬁndAFM and d-wave SC for both electron- and hole-doped regimes. How- ever, the electron-hole symmetry is broken. In the electron-doped case AFM persists to a much larger doping. On thecontrary, SC disappears at a smaller critical doping. 30These features of the phase diagram are in qualitative agreement with the experimental ﬁndings.2 The one-electron spectral functions, as measured with photoemission, are also different. Our 2 32 cluster divides the BZ into four cells around K=s0,0d,s0,pd,sp,0d, and sp,pdssee Fig. 2 dand approximates the lattice self-energy by a constant SsK,vdwithin a cell. Because of this coarse- graining, a comparison with ARPES is not possible, apart from gross features. However, as the phase diagram shows,we believe that even our small cluster captures much of thephysics of the cuprates. Here we want to stress the differencebetween the electron- and hole-doped cases within our 232 cluster approximation. In Fig. 5 sadand 5 sbdwe show the totaldstates DOS and the dcoarse-grained Kdependent DOS fwhich would correspond to the average over all kbe- longing to a coarse-graining cell of the single particle spectraAsk, vdgfor the hole- and, respectively, for the electron- doped case, at 5% doping. The total DOS looks qualitatively similar, and at the chemical potential, we see in both cases adepletion of states, which indicates the presence of thepseudogap. The Kdependent DOS is very different. The im- portant feature that we want to stress is the location of thepseudogap in the BZ. In the hole-doped case, the pseudogapappears around s0, pd. For the electron-doped case we do not detect any suppression of states around s0,pdeven though the pseudogap is clearly present in the total DOS. These features are in agreement with the photoemission experi-ments. The hole-doped materials show Fermi pockets arounds p/2,p/2dand gapped states around s0,pd.3For the electron-doped materials the photoemission spectra4exhibit a gap near sp/2,p/2dand Fermi surface pockets around s0,pd. With the present cluster size the DCA cannot deter- mine where in kspace the pseudogap is, but it is interesting that it is not at s0,pd. The presence of the pseudogap at sp/2,p/2dfor the electron-doped system can only be checked by increasing the cluster size, and this work is in progress. The electron- and the hole-doped susceptibility functions are also different both for the divergence temperatures andthe temperature and doping dependence. In Fig. 6 we showthe uniform spin and charge susceptibilities versus tempera-ture at 5% and 10% doping. A common feature for all casesis the existence of a characteristic temperature T* below FIG. 3. Two-band Hubbard model DOS at 0% doping.The solid line is the dpart of the DOS calculated at T=685 K. The value of the parameters is tpd=1.3 eV, tpp=0.65 eV, D=4.8 eV, and U =8.8 eV. The dashed line shows the DOS when tpd=0. The chemi- cal potential m=0. FIG. 4. Two-band Hubbard model phase diagram. FIG. 5. Total dDOS and coarse-grained Kdependent dDOS at 5% doping: sadhole-doping case and sbdelectron-doping case.PHYSICS OF CUPRATES WITH THE TWO-BAND … PHYSICAL REVIEW B 71, 134527 s2005 d 134527-5which the spin response is suppressed and the charge re- sponse is enhanced. T* corresponds to the pseudogap sseen in the DOS donset temperature. The suppression of the spin excitations below T* was also seen in NMR experiments31 and it was associated with the pseudogap. Besides these common features the electron- and the hole-doped suscepti-bilities behave differently. Generally, the maximum value ofthe spin susceptibility increases with hole ﬁlling. This meansthat in the hole-doped case, the spin susceptibility at thepseudogap temperature is strongly increasing with dopingunlike in the electron-doped case, where it decreases upondoping. At the same doping the hole-doped spin susceptibil-ity is much larger than the electron-doped one. Another in-teresting feature is the very strong increase of the chargesusceptibility for the electron-doped case in the underdopedregion s5% doping d, suggesting a tendency toward phase separation. 32 Asymmetric behavior can also be noted in Fig. 7, where we plot the inverseof thed-wave-paring susceptibility. AboveTcthe pairing susceptibility increases with doping in the electron-doped case and remains more or less constant inthe hole-doped case. Because of the large Cu-O hybridization the system is strongly covalent. For example, in the undoped regime theCu occupation number is only <73%. The fact that the cu- prates are strongly covalent was also observed in NMRmeasurements. 33We note that the system exhibits a slightly doping-dependent covalency. This is shown in Fig. 8 sad,where the Cu occupation number versus hole density is plot- ted. A constant covalency, equal to the one in the undopedregime si.e., 0.73 Cu holes and 0.27 O holes per site d, would correspond to the dashed line. It can be noted that, for theelectron-doped regime, the Cu hole occupation number isdecreasing faster than the hole concentration, which indi-cates an increasing covalency with increasing electron dop-ing.This happens because at large electron doping, i.e., whenthe hole-ﬁlling of the CuO 2plane is small, the effective hy- bridization is a result of a large Vskdin the BZ.34Increasing the number of holes, the BZ starts to ﬁll up and a smaller Vskdwill be responsible for the hybridization, and, conse- quently, the covalency decreases. For the hole-doped regime, the extra holes go primarily on the oxygen band, and there-fore we do not have a direct measure of the covalency. In Fig. 8 sadthe unscreened moment on the Cu orbitals is shown. It is deﬁned as m2=ksndi"−ndi#d2l=nd−2kndi"ndi#l. s15d The difference between ndandm2is a measure of the double occupancy with holes on Cu sites. In the electron-doped re-gime the double occupancy is very small, but it increasessubstantially in the hole-doped regime, which indicates thatthe low-energy hole addition states contain double-occupiedCu conﬁgurations in a signiﬁcant measure. In Fig. 8 sbdthe screened moment on Cu, deﬁned as T xlocal=T No iE 0b kSi−stdSi+s0dldt, s16d whereSiis Cu spin operator at site i, is shown. The main effect of the extra holes is to screen the spins on the Cu sites.The screening starts to be effective below temperatures ofabout <0.5 eV snot shown d. In the Zhang-Rice 11scenario an extra hole perfectly screens one spin on Cu forming astrongly bound on-site singlet, which would contain a sig-niﬁcant amount of the double-occupied Cu conﬁguration. So,our results do not contradict the ZR theory, but also do notexclude other scenarios where the extra holes form morecomplicated bound states that involve more than one Cuspin. Quantitative analysis based on the amount of screeningas function of hole doping cannot give an answer to thevalidity of the ZR assumption because, aside from the FIG. 6. Uniform spin xspinsupper part dand charge xchslower partdsusceptibilities vs temperature for different hole densities. nin the legend represents the number of holes per unit cell. FIG. 7. Inverse of the d-wave-pairing susceptibility xSC−1vs tem- perature for different hole densities. FIG. 8. sadThe Cu occupation number nd, the unscreened Cu moment m2fEq.s15dgvs hole ﬁlling. sbdThe screened Cu moment TxlocalfEq.s16dgvs hole ﬁlling.MACRIDIN et al. PHYSICAL REVIEW B 71, 134527 s2005 d 134527-6screening due to the oxygen holes, there are also nonlocal processes that contribute to the screening of Cu momentssfor example, a possibility is the formation of intersite spin singlets associated with the resonance valence bond sce-nario d. IV. REDUCTION TO SINGLE-BAND HUBBARD MODEL Concluding that the electron-hole asymmetry is an intrin- sic property of the CuO 2plane, we next address the cause of this asymmetry and the possible reduction to a one-bandmodel. In Sec. III we showed that, because of the Cu-O hybrid- ization, the addition of holes results in the formation of low-energy states, with an energy well below s<1e V dthe initial oxygen band ssee Fig. 3 d. The reduction to a one-band model is based on the ZR claim that these states are singlets, i.e.,spinless entities that can be regarded as holes moving in anantiferromagnetic background. Because of the Monte Carlonature of our calculation, which does not provide a wavefunction for the ground state, we cannot directly determinethe exact nature of these states. The most we can do is tocompare the results of a two-band Hubbard model calcula-tion to those of a one-band Hubbard model and, based on thesimilarities and differences that we might ﬁnd, to decideabout the validity of the single-band approach. A. Zhang and Rice11approximation and derivation of the effective parameters In order to compare the two- and one-band models, we should ﬁrst get an idea about the possible single-band modeleffective parameters. We discuss here two different ap-proaches for calculating these parameters, both based on theassumption that the low-energy states are localized and closeto the ZR-proposed singlets. 1. Cell-perturbation theory The cell-perturbation theory18assumes that the ZR map- ping isstrictlytrue and therefore the low-energy states are genuine local singlets. Here and everywhere in the paper bylocal we refer to the oxygen orthogonal Wannier states, which are different from the non orthogonal plaquette statesaround the Cu ions. To deduce the one-band model parameters we work in the site representation. We can Fourier transform Eq. s2dand write it as H=H 0+H1, s17d whereH0=o iH0i=SiSsfE0cis†cis+Eddis†dis+V0scis†dis+H.c dg +Undi"ndi#. s18d Hereirepresents the site index. The oxygen operators ci describe the orthogonal Wannier states. The ZR assumption implies that H0is responsible for the formation of the low- energy states slocal singlets d, andH1will determine the hop- ping parameters. Therefore the cell-perturbation theory pro-vides a means to determine the one-band parametersprovided that the ZR assumption is correct. Elaborate calcu-lations along this line were done in Ref. 18 for a variety ofmultiband parameters. In a ﬁrst-order approximation in H 1, the effective Uis given by Ueff=E2+E0−2E1, s19d whereE2,E1, andE0represent the energies of the two si.e., the ZR singlet d, one si.e., the bonding state d, and, respec- tively, zero-hole states of Eq. s18d.An important point is that H1introduces three types of hoppings. If we denote with u2il,u1il, and u0il, the lowest energy states of H0icorrespond- ing to two, one, and, respectively, zero holes, we have thefollowing hopping integrals: t ijh=k2i,1juH1u1i,2jl, s20d tije=k0i,1juH1u1i,0jl, s21d tijJ=k1i,1juH1u0i,2jl, s22d wherethfEq.s20dgdescribes the hopping of the ZR singlet, tefEq.s21dgis the hopping of the electron, and tJproduces the exchange interaction J=4tJ2/Ueff. s23d The cell-perturbation theory applied to our model gives the parameters shown in the ﬁrst row of Table. I. We want to point out two things. First, the reduced Hamil- tonian in the cell-perturbation theory is a t-t8-Jmodel, H=−to ki,jlbˆ i†bˆj−t8o kki,jllbˆ i†bˆj+Jo ki,jlSiSj, s24d with different hopping parameters for the electron- and the hole-doped regions and with a value of the exchange inter-action not determined by the quasiparticle’s hopping st hor ted, but, as it is shown in Eq. s23d,b ytJ. Therefore, a com- parison with a one-band Hubbard model, should be donecautiously. Second, we want to stress that the value of the next-nearest-neighbor hopping terms st e8andth8dis very small compared to the nearest-neighbor terms. The reason is thatthe initial oxygen-oxygen hybridization t ppresults in an ef-TABLE I. First row: parameters calculated using cell-perturbation theory, and second row: parameters calculated using cluster diagonal- ization sin eV d. cell perturbation U=3.04J=0.25 J8<0th=0.477 te=−0.35 tJ=0.433 th8=−0.03 te8=−0.016 tJ8=−0.003 cluster calculation J=0.192 J8=0.012 th=0.452 te=−0.323 th8=−0.169 te8=0.078PHYSICS OF CUPRATES WITH THE TWO-BAND … PHYSICAL REVIEW B 71, 134527 s2005 d 134527-7fective hopping term comparable in magnitude to the one resulting from the copper-oxygen hybridization, but with adifferent sign. This was also remarked in Ref. 18 and turnsout to be an important observation for our ﬁnal conclusions. 2. Cluster calculation The other approach used for determining the parameters of the one-band model is based on a cluster calculation. Inorder to estimate the nearest-neighbor hopping, the next-nearest-neighbor hopping, and the exchange terms, Eskes et al. 19considered two clusters, CuO 7swhich contains two nearest-neighbor Cu ions dand, respectively, CuO 8swhich contains two next-nearest-neighbor Cu ions d. The exchange term is determined as the energy difference between the sin-glet and the triplet state of two holes on a cluster. For threeholes on a cluster, the two energetically lowest states can bevery well s98%dapproximated with the bonding and anti- bonding states of a plaquette ZRsinglethopping between the two cells. Therefore, the differences between these two lev-els is two times the ZR singlet hopping t h. In an analogous way, considering only one hole on a cluster, the electronhoppingt eis determined. Using the cluster approach, our two-band model results in the effective parameters shown inthe second row of the Table I. 3. Comparison of the two approaches It can be immediately noted that the two approaches pro- duce different parameters, especially regarding the value ofthe next-nearest-neighbor hoppings. In the cluster calculationwe obtain signiﬁcant next-nearest-neighbor hoping terms, ut e8u/uteu=0.22 and uth8u/uthu=0.37 with different signs for the hole- and, respectively, electron-doped case sth8k0,te8l0d. The reason for the discrepancy between the two ap- proaches is that, unlike the cell-perturbation method, whichconsiders local singlets, the cluster approach considers sin-glets between a Cu hole and an oxygen state formed on theplaquette around the Cu ion. Since the oxygen plaquettestates are nonorthogonal, it is possible to write them as alinear combination of many orthogonal oxygen states at dif-ferent sites, i.e., the plaquette singlets are nonlocal states sin the orthogonal base d. At ﬁrst glance this nonlocality seems irrelevant sthe overlap of the local oxygen states with the plaquette states 11,17is about 96% d, but apparently it turns out to inﬂuence the value of the next-nearest-neighbor hoppingof the reduced Hamiltonian considerably. It is worth pointing out that, in the cluster approach, the large value of the next-nearest-neighbor hopping terms re-sults solely from the ﬁnite oxygen dispersion and the lack ofhopping between the copper and the next-nearest-neighboroxygen plaquette state. On the other hand, in the cell-perturbation theory a copper next-nearest-neighbor oxygen-hopping term is present. It results in an effective next-nearest-neighbor hopping with a sign different from the oneproduced via oxygen-oxygen hopping. B. Possible reasons for the reduction to fail We believe that a comparison between the two-band Hub- bard model and a single-band Hubbard model should bedone with extreme caution. We want to stress the possible problems here. First, the reduction based on the ZR approximation, which results in a single-band t-Jsort-t8-Jdmodel assumes the strong-coupling limit, i.e., a ratio Ueff/t@8sthe two- dimensional bandwidth is W=8td. The low-energy density of states of the two-band model shown in Figs. 3 and 5 indi-cates a bandwidth of the order of the gap, showing that weare rather at the intermediate coupling than at strong cou-pling. In the cell-perturbation theory we get U eff/tJ=7.02, which also suggests intermediate-coupling physics. There-fore, the question to be asked is whether the intermediatecoupling regime, characterized by an effective repulsion ofthe same order of magnitude as the bandwidth, can still bewell approximated by a second-order perturbation reducedt-Jmodel. Second, considering the previous objection, one may think that a reduction to the single-band Hubbard model inthe intermediate coupling regime, rather than to a t-Jmodel, is more appropriate. However, serious problems arise fromthe fact that, in the ZR theory the nature of the antiferromag-netic correlations is different from that in the single-bandHubbard model, i.e., it is not directly related to the quasipar-ticle sZR singlet or electron dhopping. Therefore, unless both the two- and one-band Hubbard models can be reduced to at-Jmodel, a comparison between them does not make much sense. Nevertheless, we believe that even when the effectiverepulsion is comparable to the bandwidth the second-orderperturbation theory, which produces the t-Jmodel, can be used successfully. We are going to discuss this at the begin-ning of Sec. IV C. Third, the nonlocality of the low-energy states sin the sense discussed in Sec. IV A 3 dcan have very serious con- sequences beyond determining the value of the hopping pa-rameters, making the single-band approach to fail com-pletely. C.t-t8-UHubbard model results Thet-Jmodel results as a low-energy effective Hamil- tonian from the Hubbard model by projecting out the doublyoccupied states. Therefore, the double occupancy of the siteorbitals constitutes a measure of the validity of this approxi-mation. In Fig. 9 we plot the double occupancy of the siteorbitals for different values of the ratio U/t. It can be noted that forU/tø6, the double occupancy is always smaller than 6%. 35This indicates that, even in the intermediate coupling regime, the low-energy physics of the one-band Hubbardmodel can be well described by a t-Jmodel. Even if, it is more natural to compare the two-band model with at-t 8-Jsor at-t8-J-J8dmodel, this turns out, from our perspective, to be rather inconvenient because of the techni-cal difﬁculties encountered by the QMC when applied tosuch models. Therefore, we proceed by comparing the two-band model with a t-t 8-UHubbard model, focusing on the qualitative features rather than on a quantitative comparison.In the strong-coupling limit, the t-t 8-Umodel reduces to a t-t8-J-J8model, with the constraint J8=J3st8/td2.Therefore, it is reasonable to assume that if the value of st8/td2is not tooMACRIDIN et al. PHYSICAL REVIEW B 71, 134527 s2005 d 134527-8large and the reduction of the two-band model to a single- band model is valid, the two models should exhibit similarphysics. Assuming that the reduction to a one-band model in the spirit of the ZR approximation is possible, we should expectfrom Table. I the hopping parameters to be different in thehole- and electron-doped regions. On the other hand, theexchange interaction, J=4t 2 U, s25d should be the same. Therefore, we study the single-band t-t8-UHubbard model and address the following questions: sidHow do the system properties depend on the ratio t/J?siidWhat is the role of the next-nearest-neighbor hoping t’? 1. t/J dependence The values of the parameters in Table I show that, in general, the ratio ut/Juis larger in the hole-doped regime than in the electron-doped case. In order to address the electron-hole asymmetry observed in the two-band model, in this sec-tion we study the properties of the single-band Hubbardmodel as a function of t/J, by keeping Jfgiven by Eq. s25dg constant and varying the hopping t. The next-nearest- neighbor hopping t 8is set to zero. With respect to antiferromagnetism, with increasing tthe Néel temperature at small doping and the critical dopingwhere the antiferromagnetism disappear decrease. For ex- ample, at 5% doping, the antiferromagnetic susceptibility isdiverging only for the small value of tshown in Fig. 10. Assuming that the hole-doped regime is characterized by alarger value of t/J, this feature is in agreement with the two-band model asymmetric behavior ssee Fig. 4 d. The uniform spin susceptibility is shown in Fig. 11. One can note that an increase of tresults in an increase of T* and a decrease of the spin susceptibility at T. This together with the behavior of the susceptibility as a function of doping is incontrast to what was observed in the two-band model ssee Fig. 6 dwhere the spin susceptibility is larger in the hole- doped case and an increase sdecrease dwith doping of the susceptibility at T for the hole- and electron-doped regimes is found. The behavior of the d-wave-pairing susceptibility as a function of tis shown in Fig. 12. The critical temperature increases with increasing tsthe increase of T cis about 10% of the increase of td, as can be seen in Fig. 12 sad. This in- crease is much too large to be in agreement with the two-band-model results even if, actually, for the two-band modelwe obtained a hole-doped T clarger, with about 20 K, than the electron-doped one.36By extrapolating the inverse of the d-wave-pairing susceptibility at 28% doping fsee Fig. 12 sbdg, it can be concluded that an increase of tresults in an increase FIG. 10. Antiferromagnetic susceptibility at 5% doping, for three different values of t, whenJis constant. FIG. 11. Spin and charge susceptibilities at 5% sblack dand 10% sgrayddoping for t=0.37 eV scircle dandt=0.52 eV ssquare d. FIG. 9. The relative double occupancy of the orbitals, kn"n#l/n, vs hole ﬁlling nfor different values of the ratio U/tof the single- band Hubbard model. FIG. 12. Inverse of d-wave-pairing susceptibility vs temperature for different hole densities and hopping parameters. Inset sadThe critical temperature vs tat 5% scircle dand 10% ssquares ddoping. Inset sbdInverse of d-wave-pairing susceptibility vs temperature at 28% doping, for t=0.37 eV scircles dandt=0.52 eV sdiamonds d.PHYSICS OF CUPRATES WITH THE TWO-BAND … PHYSICAL REVIEW B 71, 134527 s2005 d 134527-9of the critical doping where SC disappears. We also note that, at small doping and above Tc, a large tsuppresses the pairing correlations.These features are in agreement with theasymmetry of the two-band-model phase diagram. Neverthe-less, we note that, above T cand for both values of t,b y increasing the doping the pairing correlations increase, too.This behavior is characteristic in the electron-doped regimeof the two-band model, but cannot explain the hole-dopedregime where the pairing does not depend on the doping ssee Fig. 7 d. The other difference between the two-band and the single-band Hubbard model is the value of the SC suscepti-bility critical exponent g, which is much smaller in the two- band model. The density of states and the K-dependent DOS for the one-band t-UHubbard model at 5% doping is shown in Fig. 13. The one-particle spectra exhibit a pseudogap in the totalDOS and in the K-dependent DOS at s0, pdpoint in BZ, similar to the hole-doped spectra of the two-band Hubbard model.The single-band t-UHubbard Hamiltonian is particle- hole symmetric and therefore cannot explain the one-particlespectra in the electron-doped regime of the two-band Hub-bard model. At the end of this section we conclude the following: A single-band t-UHubbard model si.e.,t 8=0dwith a larger value of the hopping parameter for the hole-doped regimecannotexplain the electron-hole asymmetries observed in the two-band Hubbard model, especially the ones that character-ize the one-particle spectral functions and the susceptibilityfunctions. 2. t8dependence In this section we study the role of the next-nearest- neighbor hopping t8in the single-band Hubbard model H=−to ki,jlbi†bj−t8o kki,jllbi†bj+Uo ini"ni#. s26d We choose the following parameters, U=3.6 eV, t =−0.45 eV, and t8=0.15 eV. These parameters are close to the ones in Table. I, resulting in J=0.22 eV and J8 =0.02 eV. As for the two-band Hubbard model, we work in the hole representation, deﬁned as the one where the ﬁlling 1+ dcor- responds to a hole doping d.Values of the ﬁlling smaller thanone correspond to the electron-doped regime. We keep the sign oft8always positive. In order to avoid confusion we want to point out that in a t-Jmodel the ﬁlling is always smaller than one.Therefore, in order to describe the electron-and hole-doped regimes one has to employ the hole and, respectively, the electron representation. Accordingly, thesign oft 8has to be chosen negative in the hole-doped regime and positive in the electron-doped case.37 In Fig. 14 the phase diagram of the t-t8-Umodel is shown with a solid line. For comparison, the phase diagram of t-U Hubbard model si.e.,t8=0 case d, which is symmetric with respect to hole and electron doping, is shown with a dashedline. At half ﬁlling, t 8introduces an effective antiferromag- netic exchange J8=4t82/Ubetween the same sublattice spins and subsequently frustrates the lattice. However, at ﬁniteelectron doping, t 8favors the antiferromagnetism, making it persist up to a larger doping. On the other hand, in the hole-doped case, the antiferromagnetism is always suppressed byt 8. With respect to superconductivity, the presence of t8re- sults in a smaller slarger dcritical electron sholeddoping at which the superconductivity disappears. The asymmetry in-troduced by t 8is in agreement with the one observed in the two-band model phase diagram. We ﬁnd that t8has no major inﬂuence on the maximum superconductivity critical tem- peratureTcmax. The uniform spin and charge susceptibilities are shown in Fig. 15. The spin susceptibility at the pseudogap temperatureT* is strongly increasing with doping for the hole-doped case, and an opposite effect is seen for the electron-dopedcase. The downturn at T* in the spin susceptibility is much sharper for the hole-doped regime, indicating a fast transitionto the pseudogap physics.All these features are in very goodqualitative agreement with the ones corresponding to thetwo-band Hubbard model. Because of the similarity with thetwo-band model, it is also worth mentioning that in theelectron-doped regime the charge susceptibility is stronglyincreased below T* in the underdoped region. Thed-wave-paring susceptibilities shown in Fig. 16 ex- hibit asymmetric features, also in a qualitative agreementwith those in the two-band model. In the electron-doped re-gime, by increasing the doping, the pairing correlationsaboveT cincrease. In the hole-doped regime close to Tc, the pairing correlations do not signiﬁcantly depend on the dop-ing. However, contrary to the two-band model behavior, at FIG. 13. Single-band t-UHubbard model total and K-dependent DOS at 5% doping. J=0.22 eV, t=0.45 eV FIG. 14. t-t8-UHubbard model ssolid line dandt-UHubbard model sdashed line dphase diagrams for t=−0.45 eV, U=3.6 eV. For thet-t8-UHubbard model t8/t=−0.3.MACRIDIN et al. PHYSICAL REVIEW B 71, 134527 s2005 d 134527-10larger temperature, an increase of pairing correlations with doping is observed.The magnitude of this increase is smallerthan in the electron-doped case and a larger value of t 8se.g., t8<0.4t, not shown dwill reduce it further, improving the resemblance with the two-band model. In Fig. 17 we present the DOS of the t-t8-UHubbard model at 5% doping. The one-particle spectral functions re-semble the corresponding two-band Hubbard model ones.The presence of the t 8parameter is responsible for the loca- tion of the pseudogap in the BZ. The necessity of the t8in explaining the measured angle- resolved photoemission spectroscopy sARPES dline shape and the electron-hole asymmetry was realized early on.38,39 Representing hoppings in the same sublattice, this parameteris not severely renormalized by the AFM background and,consequently, its inﬂuence turns out to be important. Exactdiagonalization results 39of at-t8-Jmodel are in agreement with ours. The t8-hopping process lowers the kinetic energy and moves the quasiparticle position from sp/2,p/2dto s0,pdin the electron-doped case. The Néel-like conﬁgura- tions, which do not hinder this process, are stabilized. In the hole-doped case the t8hopping does not lower the kinetic energy of quasiparticles and it is not energetically favorable,therefore leading to a suppression of AFM at all dopings. The main conclusion of this section is that a one-band t-t 8-UHubbard model describes qualitatively well the phys- icssi.e., the phase diagram, the one-particle spectra, and the two-particle response functions dof the two-band Hubbardmodel, provided a signiﬁcant value of the next-nearest- neighbor hopping st8/t<0.2−0.5 d, is considered. However, besides all these similarities there are also some important differences that we emphasize in Sec. V. V. DISCUSSION In general, the deduction of an effective low-energy Hamiltonian implies two steps. First, deﬁning the low-energystates, and second, projecting the resolvent operator, GsEd =sE−Hd −1, on the subspace spanned by these low-energy states.40The inverse of the projected operator can be written asE−HeffsEd, whereHeffis the low-energy Hamiltonian.41 This procedure is equivalent to ﬁnding an Hamiltonian that produces the same one-, two-, three-particle, etc., spectralfunctions on the energy range considered to be “low energy.” Rigorously, in order to prove that the one-band model is the effective Hamiltonian, which describes the two-bandHubbard model in low-energy physics, we should comparenot only the one- and two-particle spectra, but also allhigher-order correlation functions. However, we believe thatthe comparison of only the one- and two-particle spectralfunctions is compelling enough, especially since the experi-mental information is also obtained by measuring the re-sponse functions behavior sand in almost all cases the two- or one-particle operators; as in photoemission, are involved d. It is also true that a comparison of the dynamic susceptibili-ties would be required, but with our quantum Monte Carlobased algorithm the calculation of these quantities for thetwo-band model is extremely computational resource con-suming and has not been done yet. However partial informa-tion about the relevant excited states is contained in the tem-perature behavior of the static susceptibilities. The main conclusion of Sec. IV is that a t-t 8-UHubbard model describes qualitatively well the physics of the two-band Hubbard model, but only if a substantial next-nearest-neighbor hopping is considered. However, the calculation inSec. IV A 1 sﬁrst row of Table I dand the more rigorous results by Jefferson et al., 18show that in a strict ZR picture the next-nearest-neighbor hopping is negligible. Therefore itis difﬁcult to explain the two-band Hubbard model physicsassuming the formation of local ZR singlets. For hole-dopedsystems, a signiﬁcant value of t 8can be obtained only if the FIG. 17. sadt-t8-Utotal DOS and coarse-grained K-dependent DOS at 5% doping for t=−0.45 eV, t8/t=−0.3,U=3.6 eV. sadhole- doping case and sbdelectron-doping case. FIG. 15. t-t8-UHubbard model. Uniform spin xspinsupper part d and charge xchslower part dsusceptibilities vs temperature for dif- ferent hole densities. FIG. 16. t-t8-UHubbard model. Inverse of the d-wave-pairing susceptibility xSC−1vs temperature for different hole densities.PHYSICS OF CUPRATES WITH THE TWO-BAND … PHYSICAL REVIEW B 71, 134527 s2005 d 134527-11extra holes form nonlocal bound states with the existing Cu holes, presumably something close to the plaquette singlets.Of course we have no reasons to discard other states spreadover even more oxygen sites, which can result in a magni-tude of the hopping parameters different sprobably not too much dfrom the one obtained by cluster calculation ssecond row of Table I d. In the electron-doped systems, the doping- dependent covalency shown in Fig. 8 sadclearly indicates that the hybridization of the Cu with the O states at different sitesis important. A doping-dependent covalency should also im-ply doping-dependent parameters. The cluster calculation, which allows the formation of nonlocal splaquette dlow-energy states, unlike the cell- perturbation approach sor strict ZR d, provides a value of the hopping parameters that qualitatively captures the physics ofthe two-band model. However, we do not believe that ﬁndingthe exact value of the one-band Hubbard model parameters isa relevant or even a well-addressed problem, because thenonlocality of the low-energy states implies that the twomodels are not equivalent. Aside from the similarities be-tween the two-band and t-t 8-UHubbard models discussed in Sec. IV C 2 we also ﬁnd some differences. For example, one important difference can be observed in thed-wave-pairing susceptibility sFigs. 7 and 16 d.I nt h e two-band Hubbard model the critical exponent g, which de- ﬁnes the divergence of the susceptibility at Tc, is much smaller saround <0.4 at ﬁnite hole doping dthan the one characteristic to the one-band model saround <0.6d, indicat- ing larger ﬂuctuations.42,43 Both the cell perturbation and cluster calculation provide a larger nearest-neighbor hopping tfor the hole-doped re- gion.According to the analysis presented in Sec. IV C 1, thisshould result in both larger T* andT c. However, the two- band model results do not indicate that this is the case, therespective critical temperatures being not very different inthe electron- and hole-doped regimes. Based on our comparison we can draw the following con- clusions. The one-band Hubbard model retains much of thetwo-band Hubbard model physics, but a signiﬁcant next-nearest-neighbor hopping st 8/t<0.3dshould be provided. If the purpose of the investigation is the study of the basic physics, such as the SC mechanism, the proximity of AFM,SC, and pseudogap, we believe that a one-band t-t 8-UHub- bard model should be good enough. On the other hand, if thepurpose is to describe more subtle features, such as the onesthat may result from the ﬁnite value of the spin correlationon oxygen, or if a quantitative material-speciﬁc calculation isdesired, the single-band model approach fails. Obviouslyalso the single-band model should not be used to describespectral features at energies above 0.5 eV, such as the optical,electron energy loss, and inelastic x-ray-scattering results. VI. SUMMARYAND CONCLUSIONS In this paper we use the DCA to calculate the properties of the two-band Hubbard model. The 2 32-site cluster phase diagram resembles the generic phase diagram of the cupratesand exhibits electron-hole asymmetry. We also ﬁnd asym-metric features for the one-particle spectral functions and forthe relevant susceptibility functions.These characteristics are in qualitative agreement with experimental ﬁndings. We address the validity of the single-band Hamiltonian as the effective low-energy model for the cuprates. We discussthe possible problems that may cause the failure of the re-duction from two-band to one-band and also show that, de-pending on the approximations involved, the value of theone-band Hubbard parameters sespecially the next-nearest- neighbor hopping dcan be signiﬁcantly different. We use DCA to study the role of the different parameters in the single-band t-t 8-UHubbard model and compare the phase diagram, the one-particle, and two-particle responsefunctions to those corresponding to the two-band Hubbardmodel. We conclude that the two models exhibit similar low-energy physics provided that a signiﬁcant next-nearest-neighbor hopping t 8is considered. The parameter t8is also the main culprit for the electron-hole asymmetry of the cu-prates. The large value of t 8needed for a qualitative agreement between the two models cannot be obtained in a strict ZRpicture, where the extra holes form local singlets with theexisting Cu holes. Plaquette singlets, which in the oxygenWannier representation are not local, and presumably otherspatially extended states can provide a larger value of t 8. The doping-dependent covalency in the electron-doped case alsoindicates that the nonlocal Cu-O hybridization is important.However, the formation of nonlocal low-energy states alsoimplies that they are not real singlets and, consequently, can-not be rigorously mapped into holes, and therefore the twomodels are not equivalent. We also point out some differences between the two mod- els. In the two-band Hubbard model the ﬂuctuations in thed-wave-pairing channel above T cis much stronger. The de- duction of the parameters both in cell perturbation and clus-ter approach results in a larger nearest-neighbor hopping tfor the hole-doped regime. However the critical temperatures T* andT cin the two-band Hubbard model are approximatively the same in both regimes, quite different from what shouldbe expected. The conclusion is that a single-band Hubbard model with a signiﬁcant value of the next-nearest-neighbor hoppingst 8/t<0.3dcaptures the basic physics of the two-band Hub- bard model, including the proximity of antiferromagnetism, superconductivity, and pseudogap and explaining theelectron-hole asymmetry seen in the phase diagram, one-particle, and two-particle spectral functions. However, thesingle-band Hubbard model is not entirely equivalent to thetwo-band Hubbard model and we believe that it is not suit-able for quantitative material-speciﬁc studies or for describ-ing more subtle features that may result from the nonlocalityof the low-energy states. It is also not suitable to describephysics, which implies excitations with energy scales largerthan <0.5 eV. ACKNOWLEDGMENTS We thank F.C. Zhang and Paul Kent for useful discus- sions. The work was supported by NSF Grant No. DMR-0073308, by CMSN Grant No. DOE DE-FG02-04ER46129MACRIDIN et al. PHYSICAL REVIEW B 71, 134527 s2005 d 134527-12and by the Netherlands Foundation for Fundamental Re- search on Matter sFOM dwith ﬁnancial support from the Netherlands Organization for Scientiﬁc Research sNWO d and the Spinoza Prize Program of NWO. The computationwas performed at the Pittsburgh Supercomputer Center, theCenter for Computational Sciences at the Oak Ridge Na-tional Laboratory, and the Ohio Supercomputer Center. Part of this research was performed by T. M. as a Eugene P.Wigner Fellow and staff member at the Oak Ridge NationalLaboratory, managed by UT-Battelle, LLC, for the U.S. De-partment of Energy under Contract No. DE-AC05-00OR22725. 1M. H. Hettler, A. N. Tahvildar-Zadeh, M. 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Macridin et al.,sunpublished d. 33M. Takigawa, et al.Physica C 162–164, 853 s1989 d; M. Taki- gawaHigh-Temperature Supercon-ductivity , edited by K. Be- dell, D. Coffey, D. E. Meltzer, D. Pines, and J. R. SchriffersAddison-Wesley, Redwood city, CA, 1990 d,p .2 3 6 34The Cu-O hybridization fEq.s4dgis strongly kdependent, its value taking values from 2 ˛2tpdatsp,pdpoint to zero at s0, 0d point in the BZ. 35We also ﬁnd snot shown dthat with decreasing U/tbeyond this value, the double occupancy increases fast, being about 15% forU/t=4. 36Nevertheless we should take the necessary precautions saying that this value is of the order of the error bar ssee Fig. 4 d. 37Under electron-hole representation change both tandt8change sign. A change of tsign has no physical consequences aside from a translation with sp,pdin the BZ corresponding to a canonical transformation which changes the sign of site orbitalson one sublattice. An equivalent statement is that the particle-hole transformation, as deﬁned in Ref. 44, change the sign of t 8 but not of t. 38A. Nazarenko, K. J. E. Vos, S. Haas, E. Dagotto, and R. J. Good- ing, Phys. Rev. B 51, 8676 s1995 d; R. Eder, Y. Ohta, G. A. Sawatzky, ibid.55, R3414 s1997 d; P. W. Leung, B. O. Wells, and R. J. Gooding, ibid.56, 6320 s1997 d; O. P. Sushkov, G. A. Sawatzky, R. Eder, and H. Eskes, ibid.56, 11 769 s1997 d 39T. Tohyama and S. Maekawa, Phys. Rev. B 64, 212505 s2001 d; Supercond. Sci. Technol. 13, R17 s2000 d. 40A. Auerbach, Strongly Interacting Electrons and Quantum Mag- netism,sNew York, Springer-Verlag, 1994 d,p .2 5 . 41Heffcan be considered independent of Eif the high-energy and the low-energy states are well separated. 42The deviation of gfrom the mean-ﬁeld value 1 is a measure of ﬂuctuation. 43S. Moukouri and M. Jarrell, Phys. Rev. Lett. 87, 167010 s2001 d. 44Eduardo Fradkin, Field theories of condensed matter systems sAddison-Wesley, Redwood City, CA, 1991 d, pp. 9–10.PHYSICS OF CUPRATES WITH THE TWO-BAND … PHYSICAL REVIEW B 71, 134527 s2005 d 134527-13
PhysRevB.78.165401.pdf	Nonequilibrium-induced metal-superconductor quantum phase transition in graphene So Takei1and Yong Baek Kim1,2 1Department of Physics, The University of Toronto, Toronto, Ontario M5S 1A7, Canada 2School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea /H20849Received 30 April 2008; revised manuscript received 12 September 2008; published 2 October 2008 /H20850 We study the effects of dissipation and time-independent nonequilibrium drive on an open superconducting graphene. In particular, we investigate how dissipation and nonequilibrium effects modify the semi-metal-BCSquantum phase transition that occurs at half ﬁlling in equilibrium graphene with attractive interactions. Oursystem consists of a graphene sheet sandwiched by two semi-inﬁnite three-dimensional Fermi-liquid reser-voirs, which act both as a particle pump/sink and a source of decoherence. A steady-state charge current isestablished in the system by equilibrating the two reservoirs at different but constant chemical potentials. Thegraphene sheet is described using the attractive Hubbard model in which the interaction is decoupled in thes-wave channel. The nonequilibrium BCS superconductivity in graphene is formulated using the Keldysh path-integral formalism, and we obtain generalized gap and number density equations valid for both zero andﬁnite voltages. The behavior of the gap is discussed as a function of both attractive interaction strength andelectron densities for various graphene-reservoir couplings and voltages. We discuss how tracing out thedissipative environment /H20849with or without voltage /H20850leads to decoherence of Cooper pairs in the graphene sheet, hence, to a general suppression of the gap order parameter at all densities. For weak enough attractiveinteractions we show that the gap vanishes even for electron densities away from half ﬁlling and illustrate thepossibility of a dissipation-induced metal-superconductor quantum phase transition. We ﬁnd that the applica-tion of small voltages does not alter the essential features of the gap as compared to the case when the systemis subject to dissipation alone /H20849i.e., zero voltage /H20850. The possibility of tuning the system through the metal- superconductor quantum critical point using voltage is presented. DOI: 10.1103/PhysRevB.78.165401 PACS number /H20849s/H20850: 03.65.Yz, 64.70.Tg, 74.78. /H11002w I. INTRODUCTION The landmark experimental realization of an isolated graphite monolayer, or graphene,1,2has sparked intense the- oretical and experimental interest in the material over the lastfew years. 3,4A source of interest in the study of graphene is the unique properties of its charge carriers. At low energies,these charge carriers mimic relativistic particles and are mostnaturally described by the /H208492+1 /H20850-dimensional Dirac equation with an effective speed of light c/H11011 vF−1/H11003106ms−1. The fact that graphene is an excellent condensed-matter analog of/H208492+1 /H20850-dimensional quantum electrodynamics /H20849QED /H20850has been known to theorists for over 20 years. 5–7However, it was not until the spectacular experimental realization of iso-lated graphene that experimentalists began observing signa-tures of the QED-type spectrum in their laboratories. Conse-quences of graphene’s unique electronic properties have beenrevealed in the context of anomalous integer quantum-Halleffect 8,9and minimum quantum conductivity in the limit of vanishing carrier concentrations.8 In addition to its importance in fundamental physics, graphene is expected to make a signiﬁcant impact in theworld of nanoscale electronics. Research efforts in devel-oping graphene-based electronics have been fueled by astrong anticipation that it may supplement the silicon-basedtechnology which is nearing its limits. 3Graphene is a promising material for future nanoelectronics because ofits exceptional carrier mobility which remains robustly highfor a large range of temperatures, electric-ﬁeld-inducedconcentrations, 1,2,8,9and chemical doping.10Indeed, recent experiments have explored the possibilities of in-planegraphene heterostructures by engineering arbitrary spatial density variation using local gates.11–13The application of local-gate techniques to graphene marks an important ﬁrststep on the road toward graphene-based electronics. From a theoretical point of view we realize that graphene nanoelectronics requires a theoretical understanding of opennonequilibrium graphene. Naturally, graphene in nanocir-cuits is subject to decoherence effects due to its coupling toexternal leads via tunnel junctions. Furthermore, a nonequi-librium treatment of graphene becomes necessary when acharge current is driven through it. To this date, effects ofdissipation and nonequilibrium drive on graphene electronicproperties have not been addressed. The focus of this paperis to show a theoretical framework in which these effects canbe studied and illustrate how they give rise to striking inﬂu-ences on the equilibrium properties of graphene. This work considers dissipation and nonequilibrium ef- fects on superconducting graphene. Beside the possibility ofsuperconductivity in graphene by proximity effect, 14some works suggested the potential of achieving plasmon-mediated singlet superconductivity in graphene. 15,16Several groups have investigated the equilibrium mean-ﬁeld theoryof superconductivity in graphene using the attractive Hub- bard model on the honeycomb lattice. Uchoa and CastroNeto 15studied spin singlet superconductivity in graphene at various ﬁllings by considering both the usual s-wave pairing as well as pairing with p+iporbital symmetry permitted by graphene’s honeycomb lattice structure. Zhao andParamekanti 17examined the possibility of s-wave supercon- ductivity on the honeycomb lattice. Both works show that /H20849in the absence of p-wave pairing /H20850half-ﬁlled graphene displays a semi-metal-superconductor quantum critical point at a ﬁ-PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 1098-0121/2008/78 /H2084916/H20850/165401 /H2084914/H20850 ©2008 The American Physical Society 165401-1nite critical attractive interaction strength uc. Away from half ﬁlling, the system exhibits Cooper instability at any ﬁnite u and thus undergoes the usual BCS-BEC /H20849Bose-Einstein con- densate /H20850crossover as uis increased. The difﬁculty in achiev- ing superconductivity at half ﬁlling is a result of the vanish-ing density of states at the Dirac point and the absence ofelectron screening. In this work, the superconducting graphene sheet is sub- jected to dissipation and nonequilibrium drive by coupling itto two semi-inﬁnite particle reservoirs via tunnel junctions.The geometry of the system is shown in Fig. 1. While the two reservoirs are independently held in thermal and chemi-cal equilibriums at all times, an out-of-plane steady-state cur-rent through graphene is established by equilibrating the res-ervoirs at two different but constant chemical potentials. Theleads act as inﬁnite reservoirs and are assumed to be held ata common temperature Tat all times. Nonequilibrium theory of BCS superconductivity is formulated using the Keldyshpath-integral formalism, and the resulting nonequilibriummean-ﬁeld equations are used to investigate the gap behaviorat and near half ﬁlling for various attractive interactionstrengths. The zero-temperature gap phase diagram in theparameter space of ﬁlling nand the interaction strength uis particularly interesting due to the survival of the semimetal-lic phase at half ﬁlling. The main goal of this work is toinvestigate the fate of this phase in the presence of dissipa-tion and nonequilibrium current, and our results can be di-rectly compared to the gap phase diagram in Fig. 2of the work of Zhao and Paramekanti. 17 Our main results are now qualitatively summarized. We ﬁnd that the gap is generally suppressed in the presence ofthe leads. As this paper will discuss in detail, the key tounderstanding our ﬁndings is to notice that the dissipation ofelectrons into the leads acts as a pair-breaking mechanismfor the Cooper pairs in the central graphene sheet. Thismechanism, hence the suppression, is present at both zeroand ﬁnite voltages and for all electron densities. As a conse-quence, the Fermi-liquid ground state of the system remainsstable against Cooper pairing up to some density-dependentﬁnite attractive interaction strength u c/H20849n/H20850at all densities.With respect to the gap phase diagram, dissipation gives rise to a ﬁnite region around half ﬁlling in which the gap van-ishes /H20849see Fig. 5/H20850. From these results, we infer that dissipa- tion induces a metal-superconductor quantum phase transi-tion at all ﬁllings, for which the tuning parameter is theattractive interaction strength u. The qualitative behavior of the gap is not greatly different in both the zero and ﬁnitevoltage cases as long as the voltage is small, i.e., V/H11270/H9003, where/H9003denotes the average tunneling rate of electrons be- tween graphene and the two leads. However, we stress thepossibility of tuning the system across the dissipation-induced metal-superconductor quantum phase transition us-ing voltage. The signiﬁcance lies in the fact that voltageintroduces a different means of tuning the system across thetransition in addition to a more difﬁcult approach of adjust-ing the attractive interaction strength. This paper is organized as follows. In Sec. II, we intro- duce the Hamiltonian which models our heterostructure. Themean-ﬁeld treatment of the model is formulated on theKeldysh contour in Sec. III. In Sec. III B, the nonequilibriumgap and number density equations will be derived. The re-sults are presented in Sec. IV. The effects of dissipation inthe absence of voltage is discussed in Sec. IV A while theﬁnite voltage effects are included in Sec. IV B. We concludein Sec. V. II. MODEL The lead-graphene-lead heterostructure considered in this work is shown in Fig. 1. Graphene is located on the z=0 plane, and each of its sites is labeled using two coordinatesr i=/H20849xi,yi,zi/H110130/H20850. The semi-inﬁnite metallic leads extend from both sides of the graphene sheet for z/H110220 and z/H110210. We assume that the leads are separated from graphene by thininsulating barriers and the tunneling of electrons througheach of the barriers can be described by phenomenologicaltunneling parameters. Full translational symmetry is presentalong the planes parallel to the xyplane for z/HS110050 while onlyV right lead µRleftl e a d µL Honeycomb lattice laye r zxy FIG. 1. A schematic of the system considered. Chemical poten- tial mismatch in the two leads will lead to a charge current parallelto the zaxis.e1e2 t t′ a FIG. 2. Graphene honeycomb lattice. e1ande2are the unit-cell basis vectors of graphene with lattice constant /H208813a/H110152.46 Å /H20849a /H110151.42 Å /H20850. A unit cell contains two carbon atoms belonging to the two sublattices A/H20849white circles /H20850andB/H20849black circles /H20850. All nearest- and next-nearest-neighbor hopping matrix elements are − tand − t/H11032, respectively.SO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-2the discrete translational symmetry of the graphene lattice is present at z=0. The leads are assumed to be in thermal equi- librium with their continuum of states occupied according tothe Fermi-Dirac distribution f /H9251/H20849/H9275/H20850=/H208531+exp /H20851/H9252/H20849/H9275−/H9262/H9251/H20850/H20852/H20854−1, where/H9251=L/H20849left /H20850and R/H20849right /H20850label the leads. An electric potential bias is set up in the out-of-plane direction by tuningthe chemical potentials of the leads to different values. The Hamiltonian consists of three parts, H=H sys+Hres+Hsys-res . /H208491/H20850 The central graphene sheet is modeled using the attractive Hubbard model on the honeycomb lattice. The kinetic term isa tight-binding description for the /H9266orbitals of carbon that includes nearest- and next-nearest-neighbor hopping pro-cesses. The on-site interaction strength is parametrized by U. The Hamiltonian for the layer is H sys=−t/H20858 /H20855i,j/H20856,/H9268/H20849ci,/H9268†cj,/H9268+ H.c. /H20850−t/H11032/H20858 /H20855/H20855i,j/H20856/H20856,/H9268/H20849ci,/H9268†cj,/H9268+ H.c. /H20850 −U/H20858 ici,↑†ci,↓†ci,↓ci,↑. /H208492/H20850 ci,/H9268†/H20849ci,/H9268/H20850creates /H20849annihilates /H20850electrons on site riof the graphene honeycomb lattice with spin /H9268/H20849/H9268=↑,↓/H20850.Uis as- sumed positive due to attractive interaction, and tandt/H11032are the nearest- and next-nearest-neighbor hopping parameters,respectively. Speciﬁc values for tand t /H11032have been estimated18by comparing a tight-binding description to ﬁrst- principles calculations. Following their estimates, we take t =2.7 eV and ﬁx t/H11032/t=0.04. The honeycomb lattice can be described in terms of two interpenetrating triangular sublattices AandB/H20849see Fig. 2/H20850. Each unit cell is composed of two atoms, each one of types A andB. Primitive translation vectors, e1ande2, are e1=/H20849/H208813,0 /H20850e2=/H20849−/H208813/2,3 /2/H20850e3=e1+e2, /H208493/H20850 where they are expressed in units of a, which is the distance between two nearest carbon atoms. Any Aatom is connected to its nearest neighbors on the Blattice by three vectors, d1=/H208490,1 /H20850, d2=/H20849−/H208813/2,− 1 /2/H20850, d3=/H20849/H208813/2,− 1 /2/H20850. /H208494/H20850 In momentum space, the kinetic term reads HsysK=1 N/H9004/H20858 k,/H9268/H20849ak,/H9268†bk,/H9268†/H20850/H20873/H9261kgk/H11569 gk/H9261k/H20874/H20873ak,/H9268 bk,/H9268/H20874, /H208495/H20850 where /H9261k=−t/H11032/H20873/H20858 i=13 eik·ei+ c.c./H20874, /H208496/H20850gk=−t/H20858 i=13 eik·di. /H208497/H20850 Components of the pseudospinor, ak,/H9268†andbk,/H9268†, describe qua- siparticles that belong to sublattices AandB, respectively. Here, N/H9004denotes the number of lattice sites in a triangular sublattice. N=2N/H9004will denote the total number of sites on the honeycomb lattice. Coupling between leads and the graphene sheet is mod- eled using the following Hamiltonian: Hsys-res =/H20885dkz 2/H9266/H20858 /H9251=L,R/H20858 i,/H9268/H9256/H9251/H20849Ci,/H9268,/H9251,kz†ci,/H9268+ H.c. /H20850. /H208498/H20850 /H9256/H9251is a phenomenological tunneling matrix that describes the tunneling of an electron between site ion the graphene sheet and an adjacent site on lead /H9251/H20849see Fig. 3/H20850. We only consider lead-graphene tunneling processes in which /H20849x,y/H20850coordi- nates of the electron in the initial and ﬁnal states are thesame. This assumption simpliﬁes various computationalsteps without altering the qualitative features of the ﬁnal re- sults. C i,/H9268,/H9251,kz†creates an electron in lead /H9251at coordinates /H20849xi,yi/H20850with spin /H9268and longitudinal momentum kz. We as- sume here that the tunneling parameters are independent offrequency and momentum but maintain their lead depen-dence in order to describe possible asymmetries in the lead-layer couplings. In momentum space, the tunneling Hamil-tonian in Eq. /H208498/H20850becomes H sys-res =/H20858 /H9251/H9256/H9251/H20885dkz 2/H92661 N/H9004/H20858 k,/H9268 /H11003/H20849Ak,kz,/H9268,/H9251†ak,/H9268+Bk,kz,/H9268,/H9251†bk,/H9268+ H.c. /H20850. /H208499/H20850 The in-plane momentum, k, is the component of momentum parallel to the graphene plane and the out-of-plane momen- tum, kz, is its component normal to the plane. Ak,kz,/H9268,/H9251† /H20849Bk,kz,/H9268,/H9251†/H20850corresponds to an electron mode propagating in “sublattice A/H20849B/H20850” in lead/H9251with spin/H9268and wave vector k. Although the full in-plane translational symmetry of theLead α Barrier Graphene sheet FIG. 3. A diagram illustrating the type of tunneling processes that are considered in this work. The diagram is an edge-on view ofthe interface between the graphene sheet and a lead. The only tun-neling events that are allowed are those in which the /H20849x,y/H20850coordi- nates of electrons remain unaltered. Thus, while the lower two pro-cesses in the diagram are allowed, tunneling of the type shown atthe top is disallowed.NONEQUILIBRIUM-INDUCED METAL-SUPERCONDUCTOR … PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-3leads implies that kcan take on any value in R2, the tunnel- ing assumption /H20849see Fig. 3/H20850tells us that the only modes that tunnel are those with kvalues that are the allowed modes of the triangular sublattices in the graphene sheet. All other in-consequential modes can eventually be integrated out in thepath-integral sense and will merely contribute a multiplica-tive factor in front of the partition function. Therefore, wewill not consider these modes further. Because graphene is an atomically thin two-dimensional material, an electron may tunnel from one lead to the otherwithout scattering within the graphene sheet. However, weexpect the amplitude of this direct tunneling between theleads to be smaller in comparison to the considered lead-layer coupling since the former involves tunneling throughtwo tunnel barriers as opposed to one. For this reason, directtunneling processes will not be considered in this work. Both leads are assumed to be Fermi liquids, H res=/H20858 /H9251,/H90111 N/H9004/H20885dkz 2/H9266/H20858 k,kz,/H9268/H9280k,kz /H11003/H20849Ak,kz,/H9268,/H9251†Ak,kz,/H9268,/H9251+Bk,kz,/H9268,/H9251†Bk,kz,/H9268,/H9251/H20850, /H2084910/H20850 with a separable dispersion /H9280k,kz=/H9280k+/H9280kz=/H20841k/H208412 2me+kz2 2me. /H2084911/H20850 Beside their role as a particle pump/sink, the leads play an important role as a heat sink. An important assumption wemake is that any heat generated in the interacting region dueto the application of a transverse electric ﬁeld is efﬁcientlydissipated into the leads so as to prevent build up of heat inthe region. This is a well-justiﬁed assumption because theleads are assumed to be inﬁnite and the interacting regionhas a thin proﬁle. In equilibrium /H20849 /H9262res=/H9262R=/H9262L/H20850, the central system is ex- pected to reach chemical equilibrium with the reservoirs inthe long-time limit so that /H9262sys=/H9262res. In the out-of- equilibrium case, the system is coupled to two reservoirs thatare not in chemical equilibrium. Therefore, although theelectron distribution in the interacting system reaches a staticform in the long-time limit, it is in no way expected to havean equilibrium form due to constant inﬂux /H20849outﬂux /H20850of par- ticles from /H20849into /H20850the leads. III. KELDYSH PATH INTEGRAL FORMULATION In this section, we formulate a theory of nonequilibrium BCS superconductivity in graphene using the Keldyshfunctional-integral formalism. The theory is ﬁrst expressed interms of a Keldysh partition function using coherent states ofﬁelds deﬁned on the time-loop Keldysh contour C. Following a Hubbard-Stratonovic decoupling of the quartic interactionterm in the pair channel, a BCS theory for superconductinggraphene is obtained by assuming a static homogeneous gapintegrating out both leads and graphene electrons and ex-tremizing the effective action with respect to the gap. Theresulting mean-ﬁeld equations, which are a nonequilibriumgeneralization of the corresponding equilibrium equations, 17 are analyzed in the remainder of this paper.The starting Keldysh generating functional reads ZK=/H20885D/H20853a,a¯,b,b¯,A,A¯,B,B¯/H20854eiSK, /H2084912/H20850 where SK=SsysK+SresK+Ssys-resK. /H2084913/H20850 If we introduce four-component spinors deﬁned in Nambu- sublattice space for both graphene and leads electrons, /H9278k/H20849t/H20850/H11013/H20898ak,↑/H20849t/H20850 a¯−k,↓/H20849t/H20850 bk,↑/H20849t/H20850 b¯−k,↓/H20849t/H20850/H20899, /H2084914/H20850 /H9021k,kz,/H9251/H20849t/H20850/H11013/H20898Ak,kz,↑,/H9251/H20849t/H20850 −A¯−k,−kz,↓,/H9251/H20849t/H20850 Bk,kz,↑,/H9251/H20849t/H20850 −B¯−k,−kz,↓,/H9251/H20849t/H20850/H20899, /H2084915/H20850 the actions in Eq. /H2084913/H20850become SsysK=/H20885 Cdt1 N/H9004/H20858 k/H9278¯k/H20849t/H20850/H20851i/H11509t−/H9261k/H9270zN−gk/H9270zN/H9270−/H9011−gk/H11569/H9270zN/H9270+/H9011/H20852/H9278k/H20849t/H20850 +U/H20885 Cdt/H20858 i/H20851a¯i,↑/H20849t/H20850a¯i,↓/H20849t/H20850ai,↓/H20849t/H20850ai,↑/H20849t/H20850 +b¯i,↑/H20849t/H20850b¯i,↓/H20849t/H20850bi,↓/H20849t/H20850bi,↑/H20849t/H20850/H20852, /H2084916/H20850 SresK=/H20885 Cdt/H20885dkz 2/H9266/H20858 /H92511 N/H9004/H20858 k/H9021¯k,kz,/H9251/H20849t/H20850/H20849i/H11509t−/H9280k,kz/H9270zN/H20850/H9021k,kz,/H9251/H20849t/H20850, /H2084917/H20850 and Ssys-resK=/H20885 Cdt/H20885dkz 2/H9266/H20858 /H9251/H9256/H92511 N/H9004/H20858 k /H11003/H20851/H9021¯k,kz,/H9251/H20849t/H20850/H9278k/H20849t/H20850+/H9278¯k/H20849t/H20850/H9021k,kz,/H9251/H20849t/H20850/H20852. /H2084918/H20850 /H9270/H11006/H9263are 2/H110032 matrices given by /H9270/H11006/H9263=1 2/H20849/H9270x/H9263/H11006i/H9270y/H9263/H20850, /H2084919/H20850 where/H9270x,y,z/H9263are Pauli matrices. Superscript /H9263indicates the space in which the matrices act; /H9011/H20849N/H20850denotes sublattice /H20849Nambu /H20850space. The quartic interaction term in Eq. /H2084916/H20850is decoupled using Hubbard-Stratonovic ﬁelds /H9004iA/H20849t/H20850and/H9004iB/H20849t/H20850. In the BCS mean-ﬁeld approximation, where this ﬁeld is assumed static and homogeneous /H20849i.e.,/H9004iA/H20849t/H20850=/H9004iB/H20849t/H20850/H11013/H9004/H20850, the resulting action of the system readsSO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-4SsysK=/H20885 Cdt1 N/H9004/H20858 k/H9278¯k/H20849t/H20850/H20851i/H11509t−/H9261k/H9270zN−gk/H9270zN/H9270−/H9011−gk/H11569/H9270zN/H9270+/H9011 +U/H9004/H9270+N+U/H9004/H11569/H9270−N/H20852/H9278k/H20849t/H20850−2U/H20841/H9004/H208412. /H2084920/H20850 The self-consistency condition for the gap is /H9004=/H20855ai,↓ai,↑/H20856/H20849t/H20850=/H20855bi,↓bi,↑/H20856/H20849t/H20850. /H2084921/H20850 The time-loop contour integral is carried out by ﬁrst splitting every ﬁeld into two components, labeled as “+” and “−,”which reside on the forward and the backward parts of thetime contour, respectively. 19–21The continuous action then becomes SK=/H20885 −/H11009/H11009 dt/H20851L+/H20849t/H20850−L−/H20849t/H20850/H20852, /H2084922/H20850 where L/H11006/H20849t/H20850is the Lagrangian corresponding to the action deﬁned in Eq. /H2084913/H20850written in terms of + /H20849−/H20850ﬁelds. When time-ordered products of Heisenberg ﬁelds in the theory areconstructed on the Keldysh contour, we obtain four Green’sfunctions, iG T/H20849t,t/H11032/H20850=/H20855/H9020+/H20849t/H20850/H9020¯+/H20849t/H11032/H20850/H20856, iGT˜/H20849t,t/H11032/H20850=/H20855/H9020−/H20849t/H20850/H9020¯−/H20849t/H11032/H20850/H20856, iG/H11021/H20849t,t/H11032/H20850=/H20855/H9020+/H20849t/H20850/H9020¯−/H20849t/H11032/H20850/H20856, iG/H11022/H20849t,t/H11032/H20850=/H20855/H9020−/H20849t/H20850/H9020¯+/H20849t/H11032/H20850/H20856. Because these Green’s functions are not linearly indepen- dent, a linear transformation of the ﬁelds from the Kadanoff-Baym basis /H20849+,− /H20850to the Keldysh basis /H20849clandqfor bosons; 1 and 2 for fermions /H20850is commonly performed. For bosons, the barred ﬁelds are related to the unbarred ﬁelds simply bycomplex conjugation, and thus, the transformation is identi-cal for both, /H20873/H9020cl /H9020q/H20874=1 /H208812/H2087311 1− 1/H20874/H20873/H9020+ /H9020−/H20874. /H2084923/H20850 For fermions, unbarred ﬁelds are transformed in the same manner as Eq. /H2084923/H20850. For barred ﬁelds, we choose a different transformation,19 /H20873/H9020¯1 /H9020¯2/H20874=1 /H208812/H208731− 1 11/H20874/H20873/H9020¯+ /H9020¯−/H20874. /H2084924/H20850 In order to express the Keldysh action /H20851Eq. /H2084922/H20850/H20852in the Keldysh basis it is now appropriate to deﬁne eight-component spinors for graphene and leads electrons deﬁnedin the Nambu-sublattice-Keldysh space. Since we are inter-ested in steady-state properties of the system, it is useful toﬁrst Fourier transform the ﬁelds into frequency space. Wedeﬁne the eight-component spinors as/H9274k/H11013/H20898ak,↑1 a¯−k,↓1 bk,↑1 b¯ k,↓1 ak,↑2 a¯−k,↓2 bk,↑2 b¯ −k,↓2/H20899/H9023k,kz,/H9251/H11013/H20898Ak,kz,↑,/H92511 −A¯ −k,−kz,↓,/H92511 Bk,kz,↑,/H92511 −B¯ −k,−kz,↓,/H92511 Ak,kz,↑,/H92512 −A¯ −k,−kz,↓,/H92512 Bk,kz,↑,/H92512 −B¯ −k,−kz,↓,/H92512/H20899, /H2084925/H20850 where k/H11013/H20849k,/H9275/H20850is the energy-momentum three vector. The action /H20851Eq. /H2084922/H20850/H20852then becomes SsysK=/H20885 k/H9274¯k/H20853/H20851g0R/H20849k/H20850/H9270↑N−g0R/H20849−k/H20850/H9270↓N/H20852/H9270↑K/H20851g0A/H20849k/H20850/H9270↑N−g0A/H20849−k/H20850/H9270↓N/H20852/H9270↓K +g0K/H20849k/H20850/H9270↑N/H9270+K+g0K/H20849k/H20850/H9270↓N/H9270−K−gk/H9270zN/H9270−/H9011−gk/H11569/H9270zN/H9270+/H9011 +U/H20851/H9004q/H9270+N+/H9004q/H11569/H9270−N+/H20849/H9004cl/H9270+N+/H9004cl/H11569/H9270−N/H20850/H9270xK/H20852/H20854/H9274k −2U/H20851/H9004cl/H11569/H9004q+/H9004q/H11569/H9004cl/H20852, /H2084926/H20850 SresK=/H20885 k/H20885dkz 2/H9266/H20858 /H9251/H9023¯k,kz,/H9251/H20853/H20851g˜/H9251R/H20849k/H20850/H9270↑N−g˜/H9251R/H20849−k/H20850/H9270↓N/H20852/H9270↑K /H11003/H20851g˜/H9251A/H20849k/H20850/H9270↑N−g˜/H9251A/H20849−k/H20850/H9270↓N/H20852/H9270↓K +g˜/H9251K/H20849k/H20850/H9270↑N/H9270+K+g˜/H9251K/H20849k/H20850/H9270↓N/H9270−K/H20854/H9023k,kz,/H9251, /H2084927/H20850 and Ssys-resK=/H20885 k/H20885dkz 2/H9266/H20858 /H9251/H9256/H9251/H20851/H9023¯k,kz,/H9251/H9274k+/H9274¯k/H9023k,kz,/H9251/H20852. /H2084928/H20850 Here, /H20848k/H110131 N/H9004/H20858k/H20848d/H9275 2/H9266, and/H9270↑,↓are 2/H110032 matrices deﬁned by /H9270↑,↓=/H2087310 00/H20874,/H208730001/H20874. /H2084929/H20850 Superscript Kon various /H9270matrices indicates that they act in Keldysh space. g0R,A,K/H20849k/H20850denote inverse retarded, advanced, and Keldysh Green’s functions for noninteracting electrons in the graphene sheet while g˜/H9251R,A,K/H20849k/H20850are the corresponding Green’s functions for lead /H9251. For the graphene sheet, they are given by g0R/H20849k/H20850=/H9275−/H9261k+i/H9254=g0A/H11569/H20849k/H20850, /H2084930/H20850 g0K/H20849k/H20850=2i/H9254K/H20849/H9275/H20850. /H2084931/H20850 Here, K/H20849/H9275/H20850/H110131+2nF/H20849/H9275/H20850where nF/H20849/H9275/H20850is the usual Fermi- Dirac distribution function. /H9254is an inﬁnitesimal regulariza- tion parameter. For the noninteracting case, g0Kmerely serves as a regularization for the Keldysh functional integral. Be-cause a ﬁnite self-energy term is anticipated from the cou- pling of graphene electrons to the leads, g 0Kcan be safely omitted here /H20849i.e.,g0K/H20849k/H20850/H110150/H20850.19NONEQUILIBRIUM-INDUCED METAL-SUPERCONDUCTOR … PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-5A. Integrating out the leads We now integrate out the leads degrees of freedom in order to obtain an effective theory only in terms of ﬁeldsdeﬁned on the graphene sheet. The inverse retarded, ad- vanced, and Keldysh Green’s functions for the leads, g ˜/H9251R,A,K, are those corresponding to free fermions, and because theleads are always in thermal and chemical equilibrium, theKeldysh Green’s function is strictly related to the retardedand advanced Green’s functions via the ﬂuctuation-dissipation theorem /H20849FDT /H20850. They are given by g ˜/H9251R/H20849k/H20850=/H9275−/H9280k,kz+i/H9254=g/H9251A/H11569/H20849k/H20850, /H2084932/H20850 g˜/H9251K/H20849k/H20850=2i/H9254tanh/H20873/H9275−/H9262/H9251 2T/H20874. /H2084933/H20850 Upon integrating over the leads, the resulting self-energy ac- tion becomes S/H9018=/H20885 k/H9274¯k/H20853−/H9018R/H20849k/H20850/H9270zN/H9270↑K−/H9018A/H20849k/H20850/H9270zN/H9270↓K−/H9018K/H20849k/H20850/H9270↑N/H9270+K −/H9018K/H20849k/H20850/H9270↓N/H9270−K/H20854/H9274k, /H2084934/H20850 where /H9018R/H20849k/H20850=/H20858 /H9251/H20885dkz 2/H9266/H9256/H92512 /H9275−/H9280k−/H9280kz+i/H9254 =−i/H20858 /H9251/H9266/H9267t/H92512=−i/H9003=/H9018A/H11569/H20849k/H20850/H20849 35/H20850 and /H9018K/H20849k/H20850=−2/H9266i/H20858 /H9251/H20885dkz 2/H9266/H9256/H92512tanh/H20873/H9275−/H9262/H9251 2T/H20874/H9254/H20849/H9275−/H9280k−/H9280kz/H20850 =−2 i/H20858 /H9251/H9003/H9251tanh/H20873/H9275−/H9262/H9251 2T/H20874. /H2084936/H20850 Here,/H9003/H9251/H11013/H9266/H9267t/H92512measures the effective coupling strength be- tween the layer and leads, and /H9003=/H9003L+/H9003R./H9267is the lead den- sity of states to tunnel into the layer assumed to be constant.The frequency-independent damping coefﬁcient, /H9003, and the vanishing real energy shift that result from our assumptionsindicate that the bath is treated as an Ohmic environment. 22 Combining the actions in Eqs. /H2084926/H20850and /H2084934/H20850, we obtain the dressed inverse Green’s functions for electrons in thegraphene sheet, g R/H20849k/H20850=/H9275−/H9261k+i/H9003=gA/H11569/H20849k/H20850, /H2084937/H20850 gK/H20849k/H20850=2i/H20858 /H9251/H9003/H9251tanh/H20873/H9275−/H9262/H9251 2T/H20874. /H2084938/H20850 The negative imaginary part of /H9018R/H20849k/H20850leads to an irreversible damping in the time-dependent Green’s function GR/H20849k,t/H20850. The damping term formally describes decoherence sufferedby a propagating electron wave due to incoherent escape andinjection of electrons into and from the leads. At this point, it is convenient to shift the energy scale so that all energies are measured with respect to /H9262=/H20849/H9262L +/H9262R/H20850/2. This is equivalent to the following mapping: /H9275→/H9275−/H9262, /H9261k→/H9261k−/H9262, /H9262/H9251→V/H9251/2, where VL,R=/H11006VandV/H11013/H9262L−/H9262R. We assume V/H110220. Follow- ing this choice the inverse retarded Green’s function /H20851Eq. /H2084937/H20850/H20852remains invariant while Eq. /H2084938/H20850becomes gK/H20849k/H20850=2i/H20858 /H9251/H9003/H9251tanh/H20873/H9275−V/H9251/2 2T/H20874. /H2084939/H20850 Using the dressed inverse Green’s functions deﬁned in Eqs. /H2084937/H20850and /H2084939/H20850, the effective action for the graphene sheet is SsysK,eff=/H20885 k/H9274¯kGk−1/H9274k−2U/H20851/H9004cl/H11569/H9004q+/H9004q/H11569/H9004cl/H20852, /H2084940/H20850 where the inverse Green’s function matrix Gk−1is now given by Gk−1=/H20898gR/H20849k/H20850/H9004q −gk/H115690 gK/H20849k/H20850/H9004cl 0 0 /H9004q/H11569−gR/H20849−k/H208500 gk/H11569/H9004cl/H11569 000 −gk 0 gR/H20849k/H20850/H9004q 0 0 gK/H20849k/H20850/H9004cl 0 gk/H9004q/H11569−gR/H20849−k/H20850 00 /H9004cl/H115690 0/H9004cl 0 0 gA/H20849k/H20850/H9004q −gk/H115690 /H9004cl/H11569gK/H20849k/H20850 00 /H9004q/H11569−gA/H20849−k/H20850 0 gk/H11569 000 /H9004cl −gk 0 gA/H20849k/H20850/H9004q 0 0/H9004cl/H11569gK/H20849K/H20850 0 gk/H9004q/H11569−gA/H20849−k/H20850/H20899. /H2084941/H20850SO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-6B. Mean-ﬁeld equations In closed equilibrium, solutions to the mean-ﬁeld gap and number equations on the honeycomb lattice have shown thatwhile graphene exhibits a BCS-BEC crossover behavioraway from the Dirac point for increasing attractive interac-tion strength, u, superconductivity in graphene at half ﬁlling requires a ﬁnite attractive interaction. 15,17In this section, we derive the main results of our work which are the mean-ﬁeldgap and number equations in the presence of leads and volt-age. Solving these equations will allow us to study the ef-fects of dissipation and nonequilibrium current on the gap asa function of attractive interaction strength uand ﬁlling n and compare the results to the equilibrium calculations. Webegin by obtaining an effective theory for the s-wave order parameter alone by integrating out the graphene electrons.From Eq. /H2084940/H20850, we obtain iS effK/H20849/H9004,/H9004/H11569/H20850=Trln/H20851−iGk−1/H20852−2iU/H20849/H9004cl/H11569/H9004q+ c.c. /H20850. /H2084942/H20850 1. Gap equation The saddle-point analysis of the effective Keldysh action, Eq. /H2084942/H20850, proceeds by taking functional derivatives of the action with respect to either of the two order-parameter ﬁeldsdeﬁned in the Keldysh space. Extremizing with respect to theclassical component and ﬁxing the quantum component tozero yield a trivial relation which we do not pursue further.On the other hand, one may extremize with respect to thequantum component together with ﬁxing the quantum com-ponent to zero /H20849i.e.,/H9004 q=0/H20850,/H20879/H11509SeffK /H11509/H9004q/H11569/H20879 /H9004cl=/H9004,/H9004q=0=0 . /H2084943/H20850 As we will show below, this yields a self-consistent equation for/H9004cl/H11013/H9004. In the equilibrium limit, this equation reduces to the expected BCS gap equation. We therefore interpret theobtained nonequilibrium self-consistent equation for /H9004to be the nonequilibrium analog of the equilibrium gap equation. Difﬁculties in nonequilibrium mean-ﬁeld analyses arise in general because the associated equations possess richerstructure than the equilibrium counterparts, and one is oftenleft with a series of possible solutions with no basis of know-ing which of these solutions are relevant for the subsequentanalysis. A resolution to this problem has been proposed inRef. 23for a model quantum dot system where features in the steady-state density matrix is used to select out the rel-evant solutions. From applying this analysis to the case of anextended system in a previous work, 21we believe that the “classical”19saddle-point solution /H20849i.e.,/H9004q=0/H20850is in general a unique solution to the nonequilibrium mean-ﬁeld equationsfor extended systems. In light of this observation, nonclassi-cal saddle points with /H9004 q/HS110050 are not studied in this work. Equation /H2084943/H20850yields 0=/H20879/H11509SeffK /H11509/H9004q/H11569/H20879 /H9004q=0,/H9004cl=/H9004=−iTr/H20877/H20879/H9270−N Gk−1/H20879 /H9004q=0,/H9004cl=/H9004/H20878−2/H9004 U. /H2084944/H20850 This equation leads to the generalized nonequilibrium gap equation, 2/H9004 U=/H20885 k4/H9004/H9275/H20858/H9251/H9003/H9251tanh/H20873/H9275−V/H9251/2 2T/H20874/H20853/H20851/H20849/H9275+Ek/H208502+/H90032/H20852/H20851/H20849/H9275−Ek/H208502+/H90032/H20852+4/H9261k2/H20841gk/H208412/H20854 /H20853/H20851/H9275−E+/H20849k/H20850/H208522+/H90032/H20854/H20853/H20851/H9275−E−/H20849k/H20850/H208522+/H90032/H20854/H20853/H20851/H9275+E+/H20849k/H20850/H208522+/H90032/H20854/H20853/H20851/H9275+E−/H20849k/H20850/H208522+/H90032/H20854. /H2084945/H20850 The spectra of the two bands are given by E/H11006/H20849k/H20850=/H20881/H9264/H110062/H20849k/H20850+/H90042/H9264/H11006/H20849k/H20850=/H9261k/H11006/H20841gk/H20841, /H2084946/H20850 andEk=/H20881/H9261k2+/H20841gk/H208412+/H90042. After scaling all energies by band- width tand evaluating the /H9275integral we obtain 1 u=1 2/H9266N/H20858 k/H20853Fv/H20851/H9014+/H20849k/H20850/H20852+Fv/H20851/H9014−/H20849k/H20850/H20852/H20854, /H2084947/H20850 where Fv/H20849x/H20850/H110131 x/H20900tan−1/H20898v 2+x /H9253/H20899− tan−1/H20898v 2−x /H9253/H20899/H20901, and/H9014/H11006/H20849k/H20850=E/H11006/H20849k/H20850 t,u=U t,/H9253/H9251=/H9003/H9251 t,v=V t. /H9253=/H9253L+/H9253Rdenotes the sum of lead-graphene tunneling rates scaled by t. Equation /H2084947/H20850is the BCS gap equation in the presence of leads /H20849/H9253/H20850and voltage /H20849v/H20850and is the nonequilib- rium generalization of Eq. 2 in Ref. 17. Indeed when one takes the limit as /H9253→0 and v→0 in Eq. /H2084947/H20850, the equilib- rium gap equation is recovered. At low energies, excitations in graphene at or near half ﬁlling are concentrated near two inequivalent Fermi points atthe corners of the hexagonal Brillouin zone. In the vicinity ofthese points, we have /H9261 k/H110153t/H11032−/H9262/H11013m /H20841g/H11006K+k/H20841/H11015vF/H20841k/H20841, /H2084948/H20850 where vF=3t/2 is the Fermi velocity and /H11006K =/H20849/H110064/H9266/3/H208813,0 /H20850are the locations of the inequivalent FermiNONEQUILIBRIUM-INDUCED METAL-SUPERCONDUCTOR … PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-7points. Within this approximation, the quasiparticle disper- sions,/H9014/H11006/H20849k/H20850, become /H9264/H11006/H20849k/H20850/H11015m/H11006/H9280/H9014/H11006/H20849k/H20850/H11015/H20881/H9264/H110062+/H90042, /H2084949/H20850 where/H9280=vF/H20841k/H20841. Noting that the area per lattice site is A/N =3/H208813/4 the conversion from ksummation to /H9280integral is given by 1 N/H20858 k=3/H208813 4/H9266vF2/H20885 0D /H9280d/H9280. /H2084950/H20850 The energy cutoff, set by conserving the total number of states in the Brillouin zone, is D=/H20881/H208813/H9266/H110152.33 in units of t. In the continuum limit, the gap equation then becomes 1 u=3/H208813 8/H92662vF2/H20885 0D /H9280d/H9280/H20853Fv/H20851/H9014+/H20849k/H20850/H20852+Fv/H20851/H9014−/H20849k/H20850/H20852/H20854. /H2084951/H20850 2. Number density equation In equilibrium, the number density is computed using a thermodynamic relation /H11509FMF //H11509/H9262=−Ne. Out of equilibrium,the relation does not hold and the particle density, n, must be extracted from one of the four Kadanoff-Baym Green’s func-tions, G /H11021, using24,25 n=−i 4/H20858 /H9268,/H9011/H20885 kG/H9268,/H9011/H11021/H20849k/H20850. /H2084952/H20850 /H9268labels the electron spin and /H9011/H33528/H20853A,B/H20854labels the sublattice in which it propagates. In terms of Keldysh Green’sfunctions, 19 n=−i 4/H20858 /H9268,/H9011/H20885 k/H20851G/H9268,/H9011K/H20849K/H20850−G/H9268,/H9011R/H20849K/H20850+G/H9268,/H9011A/H20849K/H20850/H20852, /H2084953/H20850 where GR,A,K/H20849k/H20850are the retarded, advanced, and Keldysh Green’s functions for the graphene electrons. These Green’sfunctions can be obtained by inverting the matrix, G −1/H20849k/H20850,i n Eq. /H2084941/H20850. We ﬁnd that the form of the Green’s functions is independent of spin and sublattice, and the resulting numberequation reads n=4/H9253 N/H20858 k/H20885d/H9275 2/H9266/H208511−F/H20849/H9275,v/H20850/H20852/H20849c6/H92756+c5/H92755+c4/H92754+c3/H92753+c2/H92752+c1/H9275+c0/H20850 /H20851/H20849/H9275+/H9014+/H208502+/H92532/H20852/H20851/H20849/H9275+/H9014−/H208502+/H92532/H20852/H20851/H20849/H9275−/H9014+/H208502+/H92532/H20852/H20851/H20849/H9275−/H9014−/H208502+/H92532/H20852. /H2084954/H20850 F/H20849/H9275,v/H20850is the zero-temperature nonequilibrium electron distribution and is given by F/H20849/H9275,v/H20850=/H20858 /H9251/H9253/H9251 /H9253sgn /H20849/H9275−v/H9251/H20850=/H9253L /H9253sgn/H20873/H9275−v 2/H20874+/H9253R /H9253sgn/H20873/H9275+v 2/H20874. /H2084955/H20850 An exact evaluation of the /H9275integral in Eq. /H2084954/H20850is difﬁcult. However, it can be done in the limit where the applied bias is assumed small compared to the bandwidth and the dampling coefﬁcient, i.e., v/H11270min /H208531,/H9253/H20854. Computing the integral up to quadratic order in vthe number density yields n=3/H208813 4/H9266vF2/H20885 0D /H9280d/H9280c0/H2084910/H92532+/H9014+2+/H9014−2/H20850+/H20849/H92532+/H9014+2/H20850/H20849/H92532+/H9014−2/H20850/H208512c2+c4/H208492/H92532+/H9014+2+/H9014−2/H20850+c6/H2084910/H92534+6/H92532/H9014−2+/H9014−4+6/H92532/H9014+2+/H9014+4/H20850/H20852 /H20849/H92532+/H9014+2/H20850/H20849/H92532+/H9014−2/H20850/H2085116/H92534+/H9014+2/H208498/H92532+/H9014+2−/H9014−2/H20850+/H9014−2/H208498/H92532+/H9014−2−/H9014+2/H20850/H20852 −2 /H9266/H20851/H20849/H9014+2−/H9014−2/H208503+8/H92532/H9014+2/H208492/H92532+/H9014+2/H20850−8/H92532/H9014−2/H208492/H92532+/H9014−2/H20850/H20852/H20902tan−1/H20873/H9014+ /H9253/H20874 /H9014+/H20853c1/H20849/H9014+2−/H9014−2−4/H92532/H20850+c3/H20851/H9014+4+/H92532/H9014−2+4/H92534−/H9014+2/H20849/H9014−2 −3/H92532/H20850/H20852+c5/H20851/H9014+6+6/H92532/H9014+4+9/H92534/H9014+2−/H9014−2/H20849/H92534−6/H92532/H9014+2+/H9014+4/H20850−4/H92536/H20852/H20854−/H20849−↔+/H20850+/H9253ln/H20873/H92532+/H9014+2 /H92532+/H9014−2/H20874/H208512c1+c3/H20849/H9014+2+/H9014−2+2/H92532/H20850 +2c5/H20849/H9014+2/H9014−2−/H92532/H9014+2−/H92532/H9014−2−3/H92534/H20850/H20852/H20903+/H208492x−1/H208502/H9253c0 /H9266/H20849/H92532+/H9014+2/H208502/H20849/H92532+/H9014−2/H208502v+/H9253c1 2/H9266/H20849/H92532+/H9014+2/H208502/H20849/H92532+/H9014−2/H208502v2, /H2084956/H20850 where x=/H9253L//H9253and/H9014/H11006are given by Eq. /H2084949/H20850. The coefﬁcients c0,..., c6are dependent on /H9014/H11006,/H9264/H11006, and/H9253and are deﬁned asSO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-8c6=1 , c5=/H9264++/H9264−, c4=3/H92532−/H9014+2+/H9014−2 2, c3=2/H20851/H9264+/H20849/H92532−/H9014−2/H20850+/H9264−/H20849/H92532−/H9014+2/H20850/H20852, c2=3/H92534+/H20849/H9014+2−/H9014−2/H208502+/H92532/H20849/H9014−2+/H9014+2/H20850−/H9014−4 2−/H9014+4 2, c1=/H9264−/H20849/H92534+2/H92532/H9014+2+/H9014+4/H20850+/H9264+/H20849/H92534+2/H92532/H9014−2+/H9014−4/H20850, c0=1 2/H20849/H9014−2+/H92532/H20850/H20849/H9014+2+/H92532/H20850/H20849/H9014+2+/H9014−2+2/H92532/H20850. /H2084957/H20850 It can be easily veriﬁed that in the limit of /H9253→0 and v→0, Eq. /H2084956/H20850reduces to the equilibrium number equation /H20849cf. Eq. 3 in Ref. 17/H20850. The mean-ﬁeld equations in Eqs. /H2084951/H20850and /H2084956/H20850 are the central results of this work. These equations will beanalyzed in the remainder of the paper. IV . RESULTS Our main focus will be on obtaining and analyzing gap phase diagrams in the parameter space of interaction strength/H20849u/H20850and number density /H20849n/H20850for various leads-graphene cou- plings /H20849 /H9253L,/H9253R/H20850and external biases /H20849v/H20850. A previous work on closed equilibrium graphene17revealed that at half ﬁlling, the superconducting instability of the semimetallic phase re-quires a critical attractive interaction strength u c, and thus, the gap vanishes up to uc. Away from half ﬁlling, the metallic phase is immediately unstable to superconductivity for arbi-trarily weak attractive interaction strength. As a result, thegap remains ﬁnite for any ﬁnite uand the system displays a typical BCS-BEC crossover behavior. In this section wequantitatively discuss the effects of dissipation and nonequi-librium current on the gap phase diagram by numericallysolving the generalized mean-ﬁeld equations /H20851Eqs. /H2084951/H20850and /H2084956/H20850/H20852. Sections IV A 1 and IV A 2 will show that a dramatic modiﬁcation to the phase diagram is observed by the merecoupling of graphene to its environment even in the absenceof nonequilibrium current. We ﬁnd that the effects of externalbiases in addition to dissipation do not substantially alter thequalitative features of the phase diagram from the case inwhich the system is subject to dissipation alone. However, asSec. IV B will discuss, the application of an external biasleads to shifts in the metallic region surrounding half ﬁllingwhich result from voltage-induced changes in the grapheneelectron density. The results presented here are applicable tothe case of small biases /H20849 v/H11270min /H208531,/H9253/H20854/H20850; effects of large bi- ases are not considered here. A. Finite lead-layer coupling /H9253Å0 with zero voltage ( v=0) First, we begin with the case in which the lead-graphene- lead heterostructure is in thermodynamic equilibrium. In par-ticular, this is the situation where /H9262L=/H9262R=/H9262res, and in the long-time limit /H9262sys=/H9262resis maintained. Here, electron- tunneling processes between the central graphene system andthe leads are providing a mechanism for decoherence for theparticles in the system /H20849 /H9253/HS110050/H20850, but an external bias that ex- plicitly breaks time-reversal symmetry of the heterostructureis absent /H20849 v=0/H20850. Consider the case where the central graphene sheet is in a superconducting phase. Because of itscoupling to the leads one can envisage a situation in whichan electron that constitutes a Cooper pair escapes into theleads. Because the leads are assumed to be inﬁnite the elec-tron that has escaped the system is completely lost in theleads and as a consequence looses its coherence with itsformer partner. Although a different electron may enter thesystem from a lead within a time scale of /H9270tun/H110111//H9003, the electron will not necessarily pair with the widowed electron since it completely lacks coherence to do so. Because dissi-pation effectively acts as a pair-breaking mechanism we ex-pect a suppression of the gap throughout the entire region ofthe phase diagram. Figure 5plots the gap phase diagrams for various leads- graphene coupling strengths /H20849 /H9253/H20850. Figure 5/H20849a/H20850corresponds to the closed equilibrium case which has been obtainedpreviously. 17Figures 5/H20849b/H20850and5/H20849c/H20850display the behavior of the gap as /H9253is increased. It is apparent from these plots that the suppressed region in the gap /H20849dark blue region /H20850grows as /H9253is strengthened. Regions of large gap values corresponding to the region with large ualso display an overall suppression in the gap as /H9253is increased. The qualitative features of the diagrams are consistent with the expectation describedabove. Let us now discuss the results more quantitatively. 1. Half ﬁlling (n=1) For the closed equilibrium case at half ﬁlling /H20849/H9253=v=0 and n=1/H20850the semi-metal-superconductor transition is possible mainly because the divergent nature of the integral on theright-hand side of Eq. /H2084951/H20850is cured by particle-hole symme- try. When the integral is convergent, it is clear that a solutionto the gap equation does not exist for small uwhere u −1 becomes larger than the integral. The value of the critical interaction parameter at which the transition occurs can beeasily quantiﬁed. At half ﬁlling the number equation, Eq./H2084956/H20850, is satisﬁed by m=3t /H11032−/H9262=0, and thus, at the critical point /H20849n=1,/H9004=0, and m=0/H20850the gap equation reads 1 uc=3/H208813 4/H9266vF2/H20885 0D d/H9280=1 2.33. /H2084958/H20850 For any u/H11021ucthe equations cannot be solved with any real /H9004and the system enters the semimetallic phase. In the pres- ence of dissipation /H20849/H9253/H110220/H20850the number equation is still solved bym=0 at half ﬁlling, and the gap equation at the critical point yields 1 uc=3/H208813 2/H92662vF2/H20885 0D d/H9280tan−1/H20873/H9280 /H9253/H20874 =3/H208813D 2/H92662vF2/H20875tan−1/H20849/H9253D−1/H20850−/H9253D 2ln/H208491+/H9253D−2/H20850/H20876, /H2084959/H20850 where the reduced coupling strength is given by /H9253D=/H9253/D.NONEQUILIBRIUM-INDUCED METAL-SUPERCONDUCTOR … PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-9The integral on the right-hand side of Eq. /H2084959/H20850is convergent, and thus, tells us that the semi-metal-superconductor transi-tion exists in the presence of dissipation at half ﬁlling. Thebehavior of u cas a function of /H9253Dis plotted in Fig. 4. We see that the value of ucincreases as /H9253is increased. This is con- sistent with the above considerations from which we expectthat a larger interaction parameter is necessary to achievepairing since leads-induced decoherence generally sup-presses superconductivity. The phenomenon can also be ob-served in Fig. 5where the apex of the blue region shifts right for larger /H9253. The plots show that at /H9253=0ucconverges to the closed equilibrium value of uc/H110112.33 as predicted by previ- ous calculations. 2. Away from half ﬁlling (nÅ1) In the closed equilibrium case away from half ﬁlling, m /HS110050 and the critical-point condition becomes1 uc=3/H208813 4/H9266vF2/H20885 0D /H9280d/H9280/H208751 /H20841m+/H9280/H20841+1 /H20841m−/H9280/H20841/H20876=/H11009. /H2084960/H20850 The divergence of the integral results in a solution with /H9004 /H110220 for any small u/H110220. This gives uc=0 implying that Coo- per instability occurs for any ﬁnite uaway from half ﬁlling. Let us now investigate how this is modiﬁed when /H9253is ﬁnite. What is notable in Fig. 5is the expansion of the blue region, where the gap is small, as /H9253is increased. The ques- tion is whether or not the typical BCS-BEC crossover behav-ior observed in the closed equilibrium case is a correct physi-cal picture away from half ﬁlling for ﬁnite /H9253. The external baths acting as a pair-breaking mechanism make the issuesubtle. The pair-breaking perturbation in a superconductorwith magnetic impurities has been shown 26,27to strongly suppress the transition temperature of the superconductor.Therefore, when such perturbation is strong enough the gapmay vanish completely and gives rise to a metal-superconductor quantum phase transition at ﬁnite doping.The question of whether or not the gap vanishes away fromhalf ﬁlling depends on the convergence of the integral in thegap equation. At v=0, the generalized gap equation becomes 1 u/H11008/H20885 0D /H9280d/H9280/H208751 /H9014+tan−1/H20873/H9014+ /H9253/H20874+/H20849+→−/H20850/H20876. /H2084961/H20850 We see that for any m/H20849i.e., regardless of being at half ﬁlling or not /H20850, the integral is convergent because for any small/H9014/H11006, which is the source of divergence, the arctangent factor nulliﬁes the divergence. This implies a ﬁnite ucat both half ﬁlling and away from half ﬁlling. Consequently, the sys-tem should undergo a superconductor-to-metal phase transi-tion as the interaction parameter is lowered. Notice that theanalysis above infers that the system will eventually enter themetallic phase as uis decreased for any density. Figure 6explicitly shows regions in the gap phase dia- gram where the gap equation lacks a solution with any posi-tive/H9004. The diagrams are plotted for the same values of /H9253as in Fig. 5. The black regions are where the gap equation is solutionless and represent a /H20849semi /H20850metallic phase. Clearly, as /H9253is increased, the metallic region expands. We ﬁnd that the superconducting /H20849white /H20850and metallic /H20849black /H20850regions are separated by a second-order phase transition. The emergence of this dissipation-induced metal- superconductor quantum phase transition is not a peculiarconsequence of the relativistic nature of the quasiparticles ingraphene. A similar result is obtained for an ordinary BCSsystem with Schrödinger fermions. In this case, a single-band form of the gap equation in Eq. /H2084947/H20850obtained with the dispersion in the formula replaced by the usual quadraticform, 1 U=1 N/H20858 k1 /H9266Ektan−1/H20873Ek /H9003/H20874, /H2084962/H20850 where2345 0 0.1 0.2 0.3 0.4uc γD FIG. 4. /H20849Color online /H20850The plot of critical coupling ucas a func- tion of reduced leads-graphene coupling /H9253D=/H9253/D. Interaction stren gth u [in units of t](c) 1 2 3 4 50.80.911.102Electron Density n(b) 0.80.911.1(a) 0.80.911.11.2 FIG. 5. /H20849Color online /H20850Plots of the BCS gap, /H9004, in the parameter space of attractive interaction strength uand electron density n. The three diagrams correspond to different values of leads-graphenecoupling strengths. In /H20849a/H20850, the system is closed, i.e., /H9253=0, while in /H20849b/H20850and /H20849c/H20850,/H9253=0.1 and 0.2, respectively. As the coupling is in- creased, the blue region in the phase diagram where the gap is smallgrows. In parts of the blue regions in /H20849b/H20850and /H20849c/H20850the gap is zero even for n/HS110051, indicating that a metal-superconductor quantum phase transition emerges in the presence of dissipation.SO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-10Ek=/H20881/H20849/H9280k−/H9262/H208502+/H90042/H9280k=k2 2m. At/H9004=0, where the integral is maximized, we get 1 Uc=Am 2/H92662N/H20885 /H9262−/H9011/H9262+/H9011d/H9280 /H20841/H9280−/H9262/H20841tan−1/H20873/H20841/H9280−/H9262/H20841 /H9003/H20874 =Am 2/H92662N/H20885 −/H9011/H9011dx /H20841x/H20841tan−1/H20873/H20841x/H20841 /H9003/H20874. /H2084963/H20850 This integral is convergent for any ﬁnite /H9003/H110220. But at/H9003=0 the integral diverges signifying that Cooper instability occursfor any ﬁnite attractive U. B. Effect of voltage ( vÅ0) So far, we have discussed the effect of leads-induced dis- sipation on the gap phase diagram in the absence of voltage.We now consider the effects of driving an out-of-planecharge current through the superconducting graphene sheet.Here, we are limited to a regime of small voltages, speciﬁ-cally v/H11270min /H208531,/H9253/H20854. As mentioned before, we assume v /H11008/H9262L−/H9262R/H110220 and allow asymmetric couplings of the lead- layer couplings /H9253Land/H9253R. In the absence of current /H20849v=0/H20850, the gap equation depends only on the sum of these couplings /H9253=/H9253L+/H9253R. But Eq. /H2084956/H20850shows that in the presence of current /H20849v/HS110050/H20850the number density depends on these couplings inde- pendently, and depending on the relative strengths of thesecouplings the dominant correction term may change sign.The main qualitative modiﬁcations to the gap phase diagramin the presence of ﬁnite voltage reﬂect the inﬂuence of thiscorrection term. In the small voltage regime and for /H9253/H110211, the dominant correction term gives a correction of order /H9253v/H112701 to thenumber density, which is of order unity. Because the modi- ﬁcations to the gap phase diagram due to voltage are ex-pected to be small we present a cartoon representation ofhow it affects the boundary of the metallic region /H20849black region in Fig. 6/H20850. This is shown in Fig. 7. Modiﬁcations to the metallic region of the phase diagram are plotted here for /H9253L/H11022/H9253Rin Fig. 7/H20849a/H20850and/H9253L/H11021/H9253Rin Fig. 7/H20849b/H20850. The plots reveal that the metallic region /H20849also the dark blue regions in Fig. 5/H20850 shifts vertically away from half ﬁlling. For /H9253L/H11022/H9253Rthe apex shifts up while for /H9253L/H11021/H9253Rit shifts down. Given that /H9262 =/H20849/H9262L+/H9262R/H20850/2 and v/H110220, the lowest-order voltage correction in Eq. /H2084956/H20850tells us that the number density increases or de- creases depending on the asymmetry of the lead couplings. If /H9253L/H11022/H9253R,nincreases, while if /H9253L/H11021/H9253R,ndecreases. The gap equation yields the largest value of ucgiven by/H9253andvwhen m=0. Thus, the above observation tells us that for /H9253L/H11022/H9253R, m=0 is achieved not at half ﬁlling as in the equilibrium case but at n/H110221. This shifts the apex upward. The opposite oc- curs for/H9253L/H11021/H9253R. The nonequilibrium gap equation is conver- gent for all /H9262; thus, a metallic phase is once again expected at all densities. V . CONCLUSION In conclusion, we have theoretically studied the effects of dissipation and nonequilibrium drive on the properties of su-perconducting graphene. An external steady-state currentwas perpendicularly driven through the graphene sheet byattaching it to two leads which were equilibrated at two con-stant but different chemical potentials. The mean-ﬁeld BCStheory of superconductivity on graphene was extended to thenonequilibrium situation by formulating the theory on theKeldysh contour. After obtaining nonequilibrium gap andnumber density equations we studied the BCS gap as a func-tion of attractive interaction strength uand electron density n for various lead-graphene coupling strengths /H9253and voltagesInteraction strength u [in units of t](c) 1 2 3 4 50.80.911.1Electron Density n(b) 0.80.911.1(a) 0.80.911.11.2 FIG. 6. The dark areas above show regions in the phase diagram where the gap equation lacks a solution for any ﬁnite /H9004; the gap vanishes in these regions. As in Fig. 5, the system is closed for plot /H20849a/H20850while/H9253=0.1 and 0.2 in plots /H20849b/H20850and /H20849c/H20850, respectively.1234 511n u/t(a) (b) FIG. 7. A cartoon plot showing the effect of voltage on the boundary of the metallic region. The dashed lines in both plotsdenote the boundary at v=0. The shaded area is the metallic region after a steady-state bias is applied. In both plots, the applied voltageis v=0.1. However, /H9253L/H11022/H9253Rin/H20849a/H20850while/H9253L/H11021/H9253Rin/H20849b/H20850. Essentially, the voltage-induced modiﬁcation is to shift the metallic region tohigher values in density or to lower values depending on the polar-ity of the voltage and the lead-coupling asymmetry.NONEQUILIBRIUM-INDUCED METAL-SUPERCONDUCTOR … PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-11v. We have shown that dissipation results in a suppression of the BCS gap at both zero and ﬁnite voltages. We argued thatthe coupling of the graphene sheet to external baths acts as apair-breaking mechanism because it causes an electron thatconstitutes a Cooper pair to escape into the leads. Once anelectron leaves the scattering region, it looses coherence withits time-reversed partner and the destruction of the Cooperpair entails. A quantitative understanding of why the gap is signiﬁ- cantly suppressed by dissipation can be gained by observinghow dissipation affects the gap equation. Recall that the BCSgap equation for an ordinary superconductor 28in closed equilibrium is given by /H9004=uTN /H208490/H20850/H20858 n/H9004 /H20881/H9275n2+/H90042. /H2084964/H20850 N/H208490/H20850is the density of states at the Fermi energy, and u/H110220i s the attractive interaction strength. A general result for theseordinary superconductors is that the gap equation /H20851Eq. /H2084964/H20850/H20852, and hence the gap, is unaffected by time-reversal-invariantperturbations. Take, for example, the inﬂuence of nonmag-netic impurities on the superconducting state. The gap equa-tion obtained after invoking disorder averaging and the Bornapproximation reads /H9004=uTN ˜/H208490/H20850/H20858 n/H9004˜ /H20881/H9275˜n2+e/H9004˜2, /H2084965/H20850 where/H9275˜and/H9004˜are frequency and order parameter renormal- ized by the perturbation,29–31andN˜/H208490/H20850is the density of states in the presence of the perturbation. The essential point is that /H9275˜and/H9004˜are related to their unrenormalized counterparts by a common factor /H9257=/H9257/H20849/H9275n,/H9004/H20850, i.e., /H9275˜=/H9257/H9275, /H9004˜=/H9257/H9004. Because this factor /H9257cancels out in Eq. /H2084965/H20850, the gap equa- tion remains invariant and leads to the result that the gap isunaffected by nonmagnetic impurities. 32 Imagine now that a pure ordinary superconductor is coupled to an external bath in equilibrium. The Nambu-Gorkov equations can be straightforwardly derived for thiscase, /H20851i /H9275n+isgn /H20849/H9275n/H20850/H9003−/H9264k/H20852G+/H9004F=1 , /H2084966/H20850 /H20851i/H9275n+isgn /H20849/H9275n/H20850/H9003+/H9264k/H20852F+/H9004G=0 , /H2084967/H20850 where the ordinary and anomalous Green’s functions are given byG/H20849k,/H9275n/H20850=−/H20885 0/H9252 d/H9270/H20855T/H9270ck,↑/H20849/H9270/H20850ck,↑†/H208490/H20850/H20856ei/H9275n/H9270, F/H20849k,/H9275n/H20850=−/H20885 0/H9252 d/H9270/H20855T/H9270ck,↑/H20849/H9270/H20850c−k,↓/H208490/H20850/H20856ei/H9275n/H9270. We immediately see from Eqs. /H2084966/H20850and /H2084967/H20850that/H9275and/H9004 scale asymmetrically, namely, /H9275˜=/H9257/H9275,/H9004˜=/H9004;/H9257=1+/H9003 /H20841/H9275n/H20841. /H2084968/H20850 Here,/H9003is the rate at which electrons decay into the bath. The asymmetry in the renormalization of /H9275and/H9004/H20851Eq. /H2084968/H20850/H20852 greatly affects the gap equation, Eq. /H2084965/H20850, and shows how dissipation can affect the gap signiﬁcantly. This is consistentwith the qualitative argument given above. We believe the observed features in the gap phase dia- gram /H20849Figs. 5and6/H20850to be robust even in the presence of ﬂuctuation effects. Indeed, ﬂuctuation effects are expected tobe important only near the critical point. 21,33Renormaliza- tion group treatment of these ﬂuctuations in the vicinity ofthe metal-superconductor quantum critical point is a topicthat we are currently addressing. Results from this work mayshift the boundaries of the transition and modify scalingproperties near the transition. However, the gap phase dia-gram presented in this work is nevertheless expected to bevalid at a qualitative level. We also expect the phenomenon of dissipation-induced suppression of the gap to occur in cases where the geometryof the system has more relevance to actual experimental set-ups. In particular, we have veriﬁed in equilibrium that thephenomenon persists even for a graphene sheet placed on topof a single substrate with which it exchanges particles. In thecontext of itinerant electron magnets, 21,33nonequilibrium renormalization-group analysis showed that the critical prop-erties of the system near its ferromagnet-paramagnet quan-tum critical point are impervious to the change in geometry uv u0 cu∗v(u∗) SCM etal FIG. 8. A plot of ucvsvfor a ﬁxed /H9262. The plot line separates the metallic and superconducting phases of our system. Adjusting /H9262 will tune the location of uc0on the xaxis but the general shape of the curve is not modiﬁed.SO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-12with which the nonequilibrium drive is applied to the system. In light of these works and our analysis, we expect our quali-tative results to hold both in and out of equilibrium evenwhen the system geometry is altered to the more experimen-tally accessible conﬁguration mentioned above. The emergence of the metal-superconductor quantum phase transition in the graphene subsystem at both zero andﬁnite voltages gives rise to the possibility of inducing thephase transition using external bias. While ﬁxing the average chemical potential /H9262to some value, vcan be changed by adjusting/H9262Land/H9262Rsymmetrically about /H9262.ucis obtained from the gap equation in this situation by ﬁxing /H9004=0 and/H9262 to some value. Figure 8shows a generic plot of ucas a function of voltage. If the interaction strength, u, of the sys- tem is at u=u/H11569, then for v/H11021v/H20849u/H11569/H20850the system will be metallic. However, when vis increased and passes v=v/H20849u/H11569/H20850, the sys- tem will become superconducting. uc0can be tuned by adjust- ing the average chemical potential /H9262. It is clear from Eq. /H2084956/H20850 that when the average chemical potential /H9262is ﬁxed, the elec- tron density can change as a function of voltage. This voltage-induced metal-superconductor quantum phase transition in open nonequilibrium graphene is possible when uc0in Fig. 8is less than the attractive interaction strength u/H11569so that voltage can be increased to drive the sys- tem from the superconducting phase to the metallic phase.An estimate for u/H11569can be made within the weak-coupling limit using15 u/H11569=uc/H20849/H9253=0/H20850 1−m D/H208751+l n/H20873Tc/H9266 1.154 m/H20874/H20876. /H2084969/H20850 Equation /H2084969/H20850was obtained in the context of closed equilib- rium graphene and may not be an accurate estimate for u/H11569in the presence of external leads. Nevertheless, we use this es-timate in conjunction with the assumption of weak lead-layercoupling /H9253/H110110.001. Adjusting the average chemical potential of the leads at m/H110110.2tand estimating the critical tempera- ture from those of graphite intercalated compounds,34,35i.e., Tc/H1101110 K, we obtain u/H11569/H110151.56tanduc0/H110151.47t. The system then is initially in the superconducting phase and enters themetallic phase with the application of voltage. ACKNOWLEDGMENTS The authors would like to thank Michael Lawler, Eun-Ah Kim, Erhai Zhao, Arun Paramekanti, and Ilya Vekhter forhelpful discussions. This research was supported by NSERC/H20849S.T. /H20850, The Canada Research Chair program, Canadian Insti- tute for Advanced Research, and Korea Research Foundationthrough KRF-2005-070-C00044 /H20849Y.B.K. /H20850. 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Sci-ence 306, 666 /H208492004 /H20850. 2K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khot- kevich, S. V. Morozov, and A. K. Geim, Proc. Natl. Acad. Sci.U.S.A. 102, 10451 /H208492005 /H20850. 3A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 /H208492007 /H20850, and references therein. 4A. H. Castro Neto, F. Guinea, N. M. R. Peres, and A. K. Geim, arXiv:0709.1163, Rev. Mod. Phys. /H20849to be published /H20850. 5G. W. Semenoff, Phys. Rev. Lett. 53, 2449 /H208491984 /H20850. 6E. Fradkin, Phys. Rev. B 33, 3263 /H208491986 /H20850. 7F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 /H208491988 /H20850. 8K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,Nature /H20849London /H20850438, 197 /H208492005 /H20850. 9Y. Zhang, J. W. Tan, H. L. Stormer, and P. Kim, Nature /H20849London /H20850 438, 201 /H208492005 /H20850. 10F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov, Nat. Mater. 6, 652 /H208492007 /H20850. 11B. Özyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov, and P. Kim, Phys. Rev. Lett. 99, 166804 /H208492007 /H20850. 12M. C. Lemme, T. J. Echtermeyer, M. Baus, and H. Kurz, IEEE Electron Device Lett. 28, 282 /H208492007 /H20850. 13B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, Phys. Rev. Lett. 98, 236803 /H208492007 /H20850. 14H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen, and A. F. Morpurgo, Nature /H20849London /H20850446,5 6 /H208492007 /H20850.15B. Uchoa and A. H. Castro Neto, Phys. Rev. Lett. 98, 146801 /H208492007 /H20850. 16A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nat. Phys. 3,3 6 /H208492007 /H20850. 17E. Zhao and A. Paramekanti, Phys. Rev. Lett. 97, 230404 /H208492006 /H20850. 18S. Reich, J. Maultzsch, C. Thomsen, and P. Ordejón, Phys. Rev. B66, 035412 /H208492002 /H20850. 19A. Kamenev, arXiv:cond-mat/0412296 /H20849unpublished /H20850. 20S. Takei and Y. B. Kim, Phys. Rev. B 76, 115304 /H208492007 /H20850. 21A. Mitra, S. Takei, Y. B. Kim, and A. J. Millis, Phys. Rev. Lett. 97, 236808 /H208492006 /H20850. 22U. Weiss, Quantum Dissipative Systems /H20849World Scientiﬁc, Sin- gapore, 1999 /H20850. 23A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. Lett. 94, 076404 /H208492005 /H20850. 24L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics /H20849Benjamin, Reading, 1962 /H20850, Chap. 2. 25G. D. Mahan, Many-Particle Physics /H20849Plenum, New York, 1990 /H20850, Chap. 3. 26A. A. Abrikosov and L. P. Gorkov, Zh. Eksp. Teor. Fiz. 39, 1781 /H208491960 /H20850/H20851Sov. Phys. JETP 12, 1243 /H208491961 /H20850/H20852. 27B. T. Matthias, H. Suhl, and E. Corenzwit, Phys. Rev. Lett. 1,9 2 /H208491958 /H20850; J. Phys. Chem. Solids 13, 156 /H208491959 /H20850. 28Here, we are considering an ordinary Fermi liquid /H20849with qua- dratic dispersion relation /H20850in the BCS superconducting phase. 29See K. Maki, Superconductivity , edited by R. D. Parks /H20849Dekker, New York, 1969 /H20850, Vol. II, and references therin. 30M. Crisan, Theory of Superconductivity /H20849World Scientiﬁc, NewNONEQUILIBRIUM-INDUCED METAL-SUPERCONDUCTOR … PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-13Jersey, 1989 /H20850Chap. 30.16. 31A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics /H20849Dover, New York, 1963 /H20850, Chap. 39.3. 32A. A. Abrikosov and L. P. Gorkov, Sov. Phys. JETP 8, 1090 /H208491959 /H20850.33A. Mitra and A. J. Millis, Phys. Rev. B 77, 220404 /H20849R/H20850/H208492008 /H20850. 34G. Lamura, M. Aurino, G. Cifariello, E. Di Gennaro, A. Andre- one, N. Emery, C. Hrold, J. F. March, and P. Lagrange, Phys.Rev. Lett. 96, 107008 /H208492006 /H20850. 35I. I. Mazin, Phys. Rev. Lett. 95, 227001 /H208492005 /H20850.SO TAKEI AND YONG BAEK KIM PHYSICAL REVIEW B 78, 165401 /H208492008 /H20850 165401-14
PhysRevB.93.125142.pdf	PHYSICAL REVIEW B 93, 125142 (2016) Transport in a one-dimensional hyperconductor Eugeniu Plamadeala,1Michael Mulligan,2and Chetan Nayak1,3 1Department of Physics, University of California, Santa Barbara, California 93106, USA 2Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA 3Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, California 93106-6105, USA (Received 9 December 2015; revised manuscript received 8 March 2016; published 29 March 2016) We deﬁne a “hyperconductor” to be a material whose electrical and thermal dc conductivities are inﬁnite at zero temperature and ﬁnite at any nonzero temperature. The low-temperature behavior of a hyperconductoris controlled by a quantum critical phase of interacting electrons that is stable to all potentially gap-generatinginteractions and potentially localizing disorder. In this paper, we compute the low-temperature dc and ac electricaland thermal conductivities in a one-dimensional hyperconductor, studied previously by the present authors, inthe presence of both disorder and umklapp scattering. We identify the conditions under which the transportcoefﬁcients are ﬁnite, which allows us to exhibit examples of violations of the Wiedemann-Franz law. Thetemperature dependence of the electrical conductivity, which is characterized by the parameter /Delta1 X, is a power law,σ∝1/T1−2(2−/Delta1X)when/Delta1X/greaterorequalslant2, down to zero temperature when the Fermi surface is commensurate with the lattice. There is a surface in parameter space along which /Delta1X=2a n d /Delta1X≈2 for small deviations from this surface. In the generic (incommensurate) case with weak disorder, such scaling is seen at high temperatures,followed by an exponential increase of the conductivity ln σ∼1/Tat intermediate temperatures and, ﬁnally, σ∝1/T 2−2(2−/Delta1X)at the lowest temperatures. In both cases, the thermal conductivity diverges at low temperatures. DOI: 10.1103/PhysRevB.93.125142 I. INTRODUCTION A. Goal of this paper In this paper, we study transport in the one-dimensional non-Fermi liquid introduced in Ref. [ 1]. This metallic phase is very different from a Fermi liquid: in addition to anoma-lous single-electron properties, it is a perfect metal at zerotemperature, with inﬁnite dc conductivity even in the presenceof impurities, unlike a Fermi liquid. We call such a material a “hyperconductor,” to distinguish it from a superconductor, since a hyperconductor does not have a Meissner effect atzero temperature; its electrical conductivity is ﬁnite at anynonzero temperature; and its thermal conductivity divergesas the temperature approaches zero. The goal of this paperis to compute the temperature and frequency dependence ofthe electrical and thermal conductivity of a hyperconductorat low temperature. The temperature dependence of theconductivities is characterized by the parameter /Delta1 Xand depends on whether the Fermi surface is commensurate withthe lattice. In the commensurate case, both the electrical σ and thermal κconductivities behave as a power law: σ,κ∝ 1/T 1−2(2−/Delta1X)with the special case /Delta1X=2 occurring along a surface in parameter space. This constitutes a violation of the Wiedemann-Franz “law,” which states that the ratio κ/σT is constant, and is due to differing relaxation mechanisms ofthe electrical and thermal currents. In the incommensuratecase, there is a range of temperatures over which both σ andκdiverge exponentially, although with differing algebraic prefactors, as T→0; at the lowest temperatures, σ∝κ/T∝ 1/T 2−2(2−/Delta1X). The above temperature dependencies reﬂect the non-Fermi liquid physics of this hyperconductor. As a concreteand well controlled example of transport in a non-Fermi liquid,these results may shine light on general principles regardingnon-Fermi liquids and transport in strongly-correlated electronsystems.B. General remarks about metallic transport Transport provides one of the most important characteriza- tions of a physical system. It is often said that the dc electricalconductivity is the ﬁrst property to be measured when a newmaterial is investigated. However, this is usually followed bynoting that it is often the last property to be understood,highlighting the subtle nature of transport properties, whencompared with thermodynamic ones [ 2]. This is one of the difﬁculties involved in understanding metallic states whoselow-temperature behavior is not controlled by the Fermi liquidﬁxed point but by some other ﬁxed point—generally called a “non-Fermi liquid.” Experimental systems that are candidate non-Fermi liquid metals have primarily been identiﬁed by theoccurrence of dc conductivity exhibiting unusual temperaturedependence. Perhaps the most famous example is the normalstate of the cuprate high-temperature superconductors [ 3,4] around optimal doping, where the dc electrical conductivityσ∼1/Tover a large range of temperatures T. It is difﬁcult to construct models that show such behavior; non-Fermiliquids [ 5–26] (e.g., fermion-gauge ﬁeld systems) often have more pronounced anomalies in single-particle properties, butmore conventional behavior in transport [ 27]. (See Refs. [ 28] and [ 29] for two counterexamples.) The rate at which the conductivity of a metal approaches its zero-temperature value is determined by the availablerelaxation mechanisms, which are, in turn, reﬂective of thenature of the zero-temperature metallic state. In a cleanFermi liquid, umklapp scattering provides the leading low-temperature momentum-relaxation mechanism and results inthe familiar contribution, δρ xx(T)∝T2, in spatial dimen- sions D> 1[30,31], to the dc electrical resistivity [ 32]. In 3D, an electron-phonon interaction contributes δρxx∝T5 below the Debye temperature, while ρxx(T)∝Tis found above the Debye temperature [ 31]. Similar behavior is found for the scattering of electrons by other collective bosonic 2469-9950/2016/93(12)/125142(18) 125142-1 ©2016 American Physical SocietyPLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) modes. However, at the lowest temperatures, which is in- evitably below the Debye temperature or its analogues forother collective bosonic modes, the resistivity vanishes fasterthan linearly in almost all theoretical models. One way to understand this is as follows. In a metal, the resistivity generally vanishes at low temperatures asρ∼1/τ tr, where τtris the decay rate for the current, usually called the transport lifetime. On dimensional grounds, 1 /τtr∝ (gT−/Delta1g)2T, where gis the coupling constant that dominates the relaxation of the current and /Delta1gis its scaling dimension. [For umklapp-dominated relaxation, gis the strength of umklapp scattering process and /Delta1gis its scaling dimension, with/Delta1g=2−/Delta1XifXis the umklapp scattering operator speciﬁed in Eq. ( 24). For disorder-dominated relaxation, g2is the variance of the disorder and 2 /Delta1gis its scaling dimension, with 2 /Delta1g=3−2/Delta1XifXis the operator that is coupled to disorder in Eq. ( 27).] If the coupling gis an irrelevant perturbation, /Delta1g<0, (including the case of a marginally irrelevant perturbation) at the zero-temperature metallic ﬁxedpoint, then the resistivity vanishes faster than linearly with T, which is the usual case. If, on the other hand, gis a relevant or marginally relevant perturbation, /Delta1 g>0, then the ﬁxed point is not stable, and the ultimate low-temperature behavior isdetermined by some other ﬁxed point. Hence ρ∝Tcan only occur in a model that contains a strictly marginal operator,/Delta1 g=0, that relaxes the current. This, in turn, implies that an observed ρ∝Tis either an intermediate temperature behavior that does not survive to the lowest of temperatures,as in the case of electron-phonon scattering above the Debyetemperature, or it is a consequence of physical processesencapsulated by a strictly marginal operator. See Refs. [ 33–35] for related scaling arguments. The 23-channel Luttinger liquid parameter regime that was called the “asymmetric shorter Leech liquid” in Ref. [ 1] has many such marginal operators. This model is a 1Dhyperconductor, in the sense deﬁned above: its electricaland thermal conductivities diverge at zero temperature inthe presence of arbitrary (perturbative) electron-electron anddisorder-mediated interactions. However, the temperature andfrequency dependence of these transport coefﬁcients is inter-esting because of the presence of these marginal operators.The purpose of this paper is to explore this dependence. In the presence of conservation laws, there is an important caveat to the scaling considerations given above [ 36–42]. Some theoretical models may have conservation laws thatprevent the electrical and/or thermal currents from fullyrelaxing, thereby leading to inﬁnite conductivities. Some careis required in these cases, since approximate calculations oftransport relaxation times τ trmay give ﬁnite answers due to the failure of these approximations to properly account forthese conservation laws. An additional complication is thatthe Fermi momentum k Fand the reciprocal lattice vectors G enter into (pseudo)-momentum conservation for low-energyexcitations. As a result, these momentum scales, which arenominally short-distance or ultraviolet scales, may enter intothe low-temperature, low-frequency response [ 43]. Conserva- tion laws, together with these momentum scales, may conspireto modify the simple scaling form 1 /τ tr∝(gT−/Delta1g)2Tto 1/τtr∝(gT−/Delta1g)2Tf(p/T ), where f(x) is a scaling function that could have, for instance, the asymptotic form f(x)∼e−xfor large xandpis some characteristic momentum (e.g., a combination of the Fermi momentum and reciprocal latticevectors) that is relevant to the relaxation of the current. Onepossible consequence is that the Wiedemann-Franz law maybe implied by scaling, but need not be realized because ofsymmetry considerations. C. Organization of this paper The remainder of this paper is organized as follows. In Sec. II, we review the construction of the hyperconductor of Ref. [ 1]. In Sec. III, we discuss the relation between conservation laws and dissipative transport with an eye towardsthe application to the hyperconductor phases. In Sec. IV, we calculate the electrical and thermal conductivities of thehyperconductor at both commensurate and incommensurateﬁlling for a pure system with umklapp scattering and a weaklydisordered system. The memory matrix formalism provides the calculational tool of this section. We conclude and outline future plans in Sec. V. We include three appendices that provide details for the calculations underlying the resultspresented in Sec. IV. II. REVIEW OF THE 1D HYPERCONDUCTOR In this section, we give a highly condensed review of the derivation of the hyperconductor of Ref. [ 1] in order to establish notation that is used in the remainder of thispaper. For the most part in this paper, when we use the term,hyperconductor, we speciﬁcally have in mind the examplepreviously called the 1D asymmetric shorter Leech liquid,however, we emphasize that the notion is more general andwe are merely studying one particular realization. The readerinterested in the details of this construction is directed toRef. [ 1]. The 1D hyperconductor that is the subject of this paper obtains from the low-energy effective theory of a particularinteracting model of electrons in a 1D quantum wire. Wecan regard the bands with different values of the transversemomentum, as well as the two spin states of the electron, asseparate channels. The simplest example then, and the onewe will study in this paper has N=23 channels of spinless fermions /Psi1 I. At low energies, the nonrelativistic fermions can be lin- earized into a theory of N=23 channels of chiral linearly- dispersing spinless (Dirac) fermions, with a left and a rightmovers in each channel. Their complete action is given by S lin=S0+Sint, (1) S0=/integraldisplay t,x[ψ† R,Ii(∂t+vI∂x)ψR,I+ψ† L,Ii(∂t−vI∂x)ψL,I], (2) Sint=/integraldisplay t,x(UI,Jψ† R,IψR,Iψ† R,JψR,J +UI+N,J+Nψ† L,IψL,Iψ† L,JψL,J +2UI,J+Nψ† R,IψR,Iψ† L,JψL,J), (3) 125142-2TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) where the operator ψ† R,I(ψ† L,I) creates a right-moving (left- moving) fermion excitation about the Fermi point kF,I(−kF,I) in channel I=1,..., N and we have used the short-hand/integraltext t,x≡/integraltext dtdx . The velocity of the Ith channel of fermions is vI. It is important to keep in mind that the linear regime only includes momenta smaller than some cutoff /Lambda1, where /Lambda1/lessmuchkF As the real symmetric matrix UI,JforI,J=1,..., 2N specifying the density-density interaction is varied, the systemexplores the parameter space of a 23-channel Luttinger liquid.As discussed in Ref. [ 1], there is an open set of U I,Jfor which all potentially gap-opening or potentially localizingperturbations to Eq. ( 1) are irrelevant; this entire parameter regime is the hyperconductor phase. The calculations ofRef. [ 1] that establish the existence of this phase as well as the following transport calculations rely on the bosonicrepresentation of Eq. ( 1): S b=1 4π/integraldisplay t,x(KIJ∂tφI∂xφJ−VIJ∂xφI∂xφJ), (4) withK=Kferm=−IN⊕IN,VIJ=vIδIJ+UIJ,INthe N×Nidentity matrix, and I,J=1,..., 2Nin Eq. ( 4). The operators ψ† I,R=1√ 2πaeiφIγIandψ† I,L=1√ 2πae−iφI+NγI+N create, respectively, right- and left-moving fermions in theIth channel; ais a short-distance cutoff, and the Klein factors γIsatisfy γJγK=−γKγJforJ/negationslash=K.T h e bosonic ﬁelds satisfy the equal-time commutation relations [φI(x),/Pi1J(y)]=iδI,Jδ(x−y), where the canonical momenta /Pi1I=1 2πKIJ∂xφJ. (The index on the ﬁelds /Psi1I,R/L runs from 1,..., N , while the index on the bosonic ﬁelds φIruns from 1,... 2N.) The hyperconductor construction is based on the observa- tion that under an SL(2N,Z) basis change, φI≡WIJ˜φJ,i ti s possible to transform Kto the Gram matrix ˜K=WTKW= −˜KR⊕˜KLof a signature ( N,N ) lattice of the form −˜/Lambda1R⊕ ˜/Lambda1L, where ˜/Lambda1R,˜/Lambda1Lare positive-deﬁnite unimodular [ 44] N-dimensional lattices. For N/greaterorequalslant23, there exist nonroot positive-deﬁnite unimodular lattices—i.e., lattices such thatall vectors vin the lattice satisfy \|v\| 2>2—and there exist matrices Wthat transform Kfermto the corresponding Gram matrices. If, in this basis, ˜V=WTVW is block diagonal (i.e., does not mix right movers and left movers), then allpotentially gap opening or localizing operators cos( ˜m I˜φI)a r e irrelevant when ˜/Lambda1Ror˜/Lambda1Lis nonroot, where ˜mJ=mIWIJ. Stability persists for a small but ﬁnite range of values ofany parameters in the model (i.e., away from block diagonal ˜V), including the chemical potentials in each channel, the velocities, and all the inter-channel and inter-spin interactions.In the hyperconductor phase considered in this paper, ˜/Lambda1 R is the so-called shorter Leech lattice, the unique nonroot unimodular integral lattice in 23 dimensions, while ˜/Lambda1LisZ23, the ordinary hypercubic lattice, which is nota nonroot lattice. This phase was called the asymmetric shorter Leech liquid . (See Refs. [ 45,46] for a fuller discussion of the mathematical technology underlying the hyperconductor construction.) For simplicity, we perform the calculations in this paper using an interaction matrix ˜VIJin the transformed basis that is simply proportional to the positive-deﬁned matrix ˜KR⊕˜KL, so that all of the eigenmodes have equal velocities v.W e similarly assume, for simplicity, that kF,I=kFfor all I.The salient feature of the asymmetric shorter Leech hyperconductor that is relevant to this paper is the existenceof a large number of marginal backscattering operators ofthe form cos ( ˜m I˜φI) when ˜V=WTVW is block diagonal and ˜/Lambda1Rand ˜/Lambda1Lare, respectively, the shorter Leech lattice andZ23. In conformal-ﬁeld theory [ 47] (CFT) terminology, these operators have different right and left scaling dimensions(/Delta1 R,/Delta1L)=(3 2,1 2). If ˜Vis moved slightly away from block diagonal, then the scaling dimensions of any such operatorwill be shifted to ( /Delta1 R,/Delta1L)=(3 2+y,1 2+y), where ywill depend on the particular operator in question. For blockdiagonal ˜V, these scaling dimensions are protected by their chirality: their RG equations do not contain higher-orderterms [ 48]. (See Appendix Dfor a review of this argument.) As a result, transport coefﬁcients exhibit anomalous power-lawdependence all the way to zero temperature. For block diagonal ˜V, this is manifested as dc electrical resistivity ρ DC∝Tall the way to zero temperature. III. SYMMETRY AND TRANSPORT In this section, we describe some of the complications associated with computing the transport properties of a 23-channel Luttinger liquid. Most of the material in this sectionhas been described elsewhere (see below for references) but,for the sake of completeness, we give a review of transportthat is tailored to the application of the formalism described inthe next section. The reader that is interested primarily in ourresults may wish to skip this rather technical section on a ﬁrstreading of this paper. A. Conservation laws The conservation of total electrical charge and total energy, d dtQ=d dtH=0, (5) (where QandHare the total electrical charge and energy operators) make it possible for those quantities to diffuse,thereby leading to ﬁnite electrical and thermal conductivities.If, however, the charge or energy currents , respectively, J eor JT, were conserved, d dtJe=0o rd dtJT=0, (6) then the electrical or thermal conductivity would be inﬁnite. Even if the charge and energy currents were not themselvesconserved, the electrical or thermal conductivity would stillbe inﬁnite, if there were some other conserved quantities withnonzero “overlap” [in a sense to be made precise in Eq. ( 29)] with the charge or energy current. Hence ﬁnite conductivitiesonly occur when the corresponding currents have no overlapwith any conserved quantities [ 38,49,50]. In addition to the total charge and energy there are other globally conserved quantities (we will interchangeable callthem charges) for the ﬁxed point action of a hyperconductorin Eq. ( 4). There are 47 conservation laws at the asymmetric shorter Leech ﬁxed point that are important for transport: thecharges of the right and left movers in each channel as wellas the total energy [ 51]. We now discuss these conservation laws, as well as the relaxation mechanisms due to irrelevant 125142-3PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) perturbations of the ﬁxed point that are required to make these conductivities ﬁnite. Continuous translation symmetry of the parent nonrelativis- tic theory, whose low-energy effects are captured by Slin,g i v e s a globally conserved charge (total momentum), here writtenin fermionic language: P=P 0+PD, (7) P0=kF/summationdisplay I/parenleftbig NR I−NL I/parenrightbig , (8) PD=/integraldisplay x[ψ† R,I(i∂xψR,I)+ψ† L,I(i∂xψL,I)], (9) where NR I,NL Iare, respectively, the number operators of the right-moving and left-moving Dirac fermions in channel I: NR,L I=/integraldisplay xψ† R/L,IψR/L,I. (10) PD, as suggestively named, is the momentum of a Dirac fermion theory also described by Slin, but where ψ† R,I(ψ† L,I) creates a right-moving (left-moving) fermion about zeromomentum instead of the Fermi point k F,I(−kF,I). From the perspective of the low-energy theory, the total momentumoperator Parises from two separately conserved emergent symmetries of S lin: the ﬁrst is generated by a chiral rotation of the right- and left-moving fermions by the “angle” kF, while the second is generated by continuous translations inthe linearized Dirac theory. P 0accounts for the large momenta ∼kF, while PDaccounts for deviations from the Fermi surface. These expressions can be rewritten in bosonic form: NR I=1 2π/integraldisplay x∂xφI, (11) NL I=1 2π/integraldisplay x∂xφN+I, (12) and PD=1 4π/integraldisplay xKIJ∂xφI∂xφJ. (13) The fermionic and bosonic expressions for P=P0+PDare the integrals over all space of the component Ttxof the energy-momentum tensor derived via Noether’s theorem from,respectively, the fermionic Eq. ( 1) and bosonic Eq. ( 4)f o r m s of the effective action. The ﬁxed point action S bhas emergent U(1)N L×U(1)N R chiral symmetries ( φI→φI+cI) generated by the charges QR/L I: QR,L I=eNR/L I. (14) The continuity equation for each chiral charge and the equations of motion for the bosonic ﬁelds allow us to obtainthe corresponding currents: J e R,I=e 2πVIJ/integraldisplay x∂xφJ, (15) Je L,I=−e 2πVN+I,J/integraldisplay x∂xφJ. (16)The total electrical and thermal currents are then given by Je=N/summationdisplay I=1/parenleftbig Je R,I+Je L,I/parenrightbig , (17) JT=−1 4π2N/summationdisplay I,J,L=1VIJKIIVIL/integraldisplay x∂xφJ∂xφL, (18) where the Hamiltonian, H=1 4π/integraldisplay xVIJ∂xφI∂xφJ, (19) and corresponding thermal continuity equation gives JT.W e study the case when all of the eigenvalues of VIJare the same, so that the Dirac momentum PDis equal to the thermal current JT. Particle-hole symmetry breaking band-curvature effects couple the electrical and thermal currents to one another. Forcompleteness, we give, in fermionic form, the correspondingcorrections to the expressions for the currents: δJ e=ge mPD, (20) δJT=g m/summationdisplay I/integraldisplay x[(∂tψ† R,I)∂xψR,I+(∂xψ† R,I)∂tψR,I +(∂tψ† L,I)∂xψL,I+(∂xψ† L,I)∂tψL,I]. (21) In an operator formalism, the time derivative of the fermion operator above is computed by taking the commutator of thefermion operator with the Hamiltonian H. If the fermions have quadratic dispersion, so that there are no higher-ordercorrections to these expressions for the currents, the actionis Galilean-invariant. The band curvature corrected electricalcurrent then gives the expected relation between the totalelectrical current and total momentum, J e+δJe=e mP. Band curvature effects that do not break particle-hole symmetryintroduce corrections to J ethat are odd in the φIand corrections to JTthat are even in the φI. These and other corrections due to band curvature are interesting and deservefurther study (see Ref. [ 52] for a review), however, we focus upon the linearly dispersing regime in this paper. To summarize, the ﬁxed point action S bhas 47 individually conserved quantities, QR,L I andPD, that generally have nonzero overlap with the electrical and thermal currents. Onelinear combination of these conserved quantities, the total electrical charge Q=/summationtext(Q R, I+QL I), will always [ 53] remain conserved, but it has no overlap with either the electrical orthermal currents and so it does not prevent their decay. Theother 46 conservation laws must be broken in order for thesystem to have ﬁnite electrical and thermal conductivities. B. Relaxation mechanisms To see the relation between the conductivity and conserva- tion laws, it is helpful to consider the most general expressionfor the real part of the optical conductivity [ 41]: σ /prime(ω,T )=2πD(T)δ(ω)+σreg(ω,T ), (22) where D(T) is the so-called Drude weight. If D(T)i sﬁ n i t e , it signals that the dc conductivity is inﬁnite. Using Mazur’s 125142-4TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) inequality [ 49,50], Zotos, Naef, and Prelovsek pointed out in Ref. [ 38] the following implication of conserved charges for electrical charge transport: D(T)/greaterorequalslant1 2LT/summationtext k/angbracketleftJeQk/angbracketright2 /angbracketleftbig Q2 k/angbracketrightbig, (23) where Lis the length of the system. The angled brackets denote the thermodynamic average and the right-hand side of Eq. ( 23) is independent of time because the Qkare conserved quantities. This inequality says that in the presence of conserved chargesQ kwhich have nonzero overlap with Je, the electrical current does not completely relax, and the system has dissipationlesscharge ﬂow even at ﬁnite temperature T. [See Eq. ( 29)f o r an equivalent notion of an “overlap,” which is the one thatwe adopt in this paper.] A similar inequality and conclusionapplies for the thermal current J T. It follows that to fully relax the electrical and thermal currents a system must break all conservation laws, apart fromthe conservation of total charge and total energy, which havevanishing overlap with the electrical and thermal currents. Atzero temperature and zero frequency, the ﬁxed point theoryS bdetermines the response of the system. Since this theory has the 47 conservation laws described above, it has inﬁniteconductivity. Note that, in a time-reversal invariant 23-channelLuttinger liquid, we would only need to break 24 conservationlaws since the time-reversal symmetric conserved quantitieswould ordinarily have vanishing overlap with the electricalcurrent; but the asymmetric Leech liquid hyperconductor isnot time-reversal invariant. At ﬁnite temperature and frequency, irrelevant perturba- tions can have an effect on the response functions of the system.The bulk of this paper is a discussion of the effects of suchperturbations. In particular, we answer two questions. Whichoperators can relax the currents? Which are the most importantones? In order to break the conservation of the Dirac momentum P Dand the chiral electrical currents {Je R/L,I}, we need to include physical processes that (1) break continuous trans-lation symmetry with respect to the low-energy effectivetheory S band (2) break particle number conservation within each channel, but (3) conserve total charge and energy.Umklapp scattering at incommensurate ﬁllings and disorderbreak continuous momentum conservation and generally breakthe conservation of the chiral currents in individual channels,and so we focus on them here. Umklapp processes scatter some number of right movers into left movers so that the total momentum change is areciprocal lattice vector. The most general umklapp term is speciﬁed by a vector of integers m (α) I,I=1,..., 2N: Hu=/summationdisplay αHu α =/summationdisplay α(hu α+H.c.) =−/summationdisplay αλα/integraldisplay x/parenleftbigg1 a2eim(α) IkF,Ix−ip(α)Gxeim(α) JφJ+H.c./parenrightbigg , (24)where λαis the coupling constant, Gis a basis vector of the reciprocal lattice, ais a short-distance cutoff [ 54], and the Einstein summation convention is employed. Here, theoperator Xto which we referred in our general remarks in Sec. IBisX=e im(α) JφJ. The most important umklapp processes at low energies are those for which the corresponding operators X=eim(α) JφJhave the lowest scaling dimension. In the asymmetric shorter Leech hyperconductor studied in thispaper, such operators have scaling dimension ( /Delta1 R,/Delta1L)= (3/2,1/2), so they are marginal. The integer p(α)is the “order” of the umklapp process, or the number of Brillouin zone foldings after which the momentum m(α) IkF,Iis again in the ﬁrst Brillouin zone. Thus p(α)is actually ﬁxed by m(α) IkF,I,b u t we will retain it as a formally free parameter. At commensurate ﬁlling, there is always a p(α)such that m(α) IkF,I=p(α)G, but we work more generally. Without loss of generality, we may take the difference m(α) IkF,I−p(α)G∈[0,2π) where the lattice constant has been set to unity. Charge conservationis maintained by requiring equal numbers of creation and annihilation operators:/summationtext N I=1m(α) I=/summationtextN I=1m(α) N+I. While any single umklapp process Hu αmight break the conservation of individual currents (e.g., [ Hu α,Je R/L,I ]/negationslash=0), a linear combination of currents might still be conserved [ 40]. (The linear combination corresponding to total charge isalways conserved, however, it has no overlap with the totalelectrical current.) That is why our model generally requires at least 46 carefully chosen umklapp processes, i.e., m (α) Ivectors to break all conservation laws. Such a requirement is notunreasonable. In the spirit of effective-ﬁeld theory, we expectall operators consistent with symmetry to be present in thelow-energy effective action. We simply focus on the minimalset of scattering processes that dominate the low-energyphysics. See the accompanying MATHEMATICA ﬁle for explicit expressions of the m(α) Ithat we choose to study [ 55]. To study whether some linear combination (other than the total charge) aIJIwithJI=Je R,IforI=1,...N andJe I= Je L,I−NforI=N+1,..., 2Nis also conserved, we compute the equal-time commutators: /bracketleftbig Hu α,aIJe I/bracketrightbig =iaIbα Ihuα+H.c., (25) where the vectors bα Iare deﬁned by bα I=(eλαsgn(N−I)sgn(N−J)VIJ)m(α) J, (26) and we deﬁne sgn( X)=+ 1f o r X/greaterorequalslant0 and sgn( X)=− 1 forX< 0. We ask whether there exist solutions aI=/vectora∈ R2N−{0}, such that ∀α,aIbα I=0. All umklapp operators preserve total U(1) electrical charge, therefore the vectors m(α) I specifying them can span at most a 2 N−1 dimensional space. The linear equations, aIbα I=0, say that /vectorais orthogonal to this space. It follows that when the number of linearly independentumklapp terms N U(α=1,..., N U) equals 2 N−1,/vectoralies in the one-dimensional space corresponding to total charge, andso no nontrivial conserved linear combination of the currentsexists. Disorder can also relax the electrical and thermal currents by violating conservation laws. A generic disorder-mediated 125142-5PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) backscattering term takes the form Hdis=/summationdisplay αλdis αHdis α =/summationdisplay αλdis α/integraldisplay x/bracketleftbigg ξα(x)1 a2eim(α) IφI+H.c./bracketrightbigg , (27) where αindexes the various backscattering terms speciﬁed by m(α) I∈Z. At low temperatures, the most important backscat- tering processes are again due to the dimension (3 2,1 2) operators eim(α) IφIintroduced in Eq. ( 24). However, due to randomness in ξα(x), their effect is weaker than that of uniform umklapp terms. [In the general remarks in Sec. IB, the operator X=eim(α) IφIin Eq. ( 27).] For simplicity, we will take all the couplings λdis α=λdis equal and ξα(x)ξ∗ β(x/prime)=δαβDδ(x−x/prime) withξα(x)=0, where the overline denotes disorder averaging. Then, we use thereplica trick to integrate out the disorder, thereby obtainingthe following term in the replicated action: S dis−avg=(λdis)2D/summationdisplay A,B/summationdisplay α/integraldisplay t,t/prime/integraldisplay x1 a4eim(α) I(φA I(t)−φB I(t/prime)).(28) For a dimension (3 2,1 2) operator eim(α) IφI, the coupling ( λdis)2D of the interaction in the replicated theory has scaling dimensionequal to −1. Hence the interaction is irrelevant and its effects are formally subleading compared to the uniform umklappterms considered above. However, in the commensurate case,umklapp terms commute with P D; disorder is the leading effect that violates conservation of PD, thereby leading to ﬁnite thermal conductivity. Meanwhile, in the incommensuratecase, the effects of uniform umklapp terms are exponentiallysuppressed at low temperatures, and disorder becomes theleading effect that relaxes both electrical and thermal currentsat low temperatures. In summary, for a pure system at commensurate ﬁlling, the Dirac momentum P Dis not relaxed, however, there is no overlap between the chiral electrical currents Je Iand PDwhen particle-hole symmetry is preserved. Thus we need 45 umklapp operators to relax the electrical current.When particle-hole symmetry is broken by band-curvaturecorrections at commensurate ﬁlling, /angbracketleftJ ePD/angbracketright/negationslash=0, so both the electrical and thermal conductivities diverge. When theﬁlling is incommensurate or disorder is present, particle-holesymmetry is broken, so there is generally an overlap betweenthe electrical currents and the Dirac momentum. However,P Ddoes not generally commute with an umklapp process at incommensurate ﬁlling or a disorder-mediated scatteringinteraction, thereby allowing momentum relaxation. In thiscase, both the electrical and thermal transport coefﬁcients canbe ﬁnite in the presence of 46 scattering interactions. Theadditional interaction arises from the additional conservedcharge P D. To see this, one must generalize the previous argument by writing the commutator in Eq. ( 25)a sat o t a l derivative. C. Memory matrix The details of the memory matrix formalism can be found in Refs. [ 40,56–59]; we merely observe that it is well-suitedfor computing transport coefﬁcients in the hydrodynamic regime: when there are globally conserved quantities (energy,electrical charge) that propagate diffusively. Unlike a directapplication of the Kubo formulas it makes the role of theseconservation laws transparent. In essence, it is a reorganizationof the perturbative expansion of the current-current correlationfunctions of interest [ 41]. We choose as a complete basis of conserved quantities the set {Q p}={Je R,1,...Je R,N,Je L,1,...Je L,N−1,PD}.Je L,Ncan be excluded because the total charge is always conserved, so acorrelation function involving J L Ncan be obtained from an expression involving the other currents. There is a notion ofa symmetric inner product on the vector space of conservedquantities provided by the static susceptibility matrix: ˆχ pq=(Qp\|Qq) ≡1 LGR QpQq(ω=0). (29) The retarded Green’s functions GR QpQq(ω) are calculated at temperature T(left implicit in the deﬁnitions below) and eval- uated at real frequency ω. (Recall that there is no momentum dependence in the static susceptibility matrix ˆ χpqbecause the conserved charges are obtained by integrating densities overall space.) Thus the static susceptibility may be used to deﬁnethe notion of an ‘overlap’ between two conserved quantities.Note that the real-time thermodynamic correlation functionsinvolved in Mazur’s inequality Eq. ( 23) are nonzero if and only if the corresponding static susceptibilities are nonzero. The memory matrix ˆM(ω) has contributions from each separate umklapp and disorder-mediated scattering process,both labeled by α. We schematically write this as ˆM(ω)=/summationdisplay α(λ2 αˆMu α(ω)+(λdis α)2DˆMdis α(ω)), (30) (ˆMu)pq α=1 L/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ω−/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ω=0 iω, (31) (ˆMdis)pq α=1 L/angbracketleftbig Fdis p,α;Fdis q,α/angbracketrightbig ω−/angbracketleftbig Fdis p,α;Fdis q,α/angbracketrightbig ω=0 iω. (32) Here,Fu q,α=i λα[Hu α,Qq],Fdis q,α=i λdisα√ D[Hdis α,Qq], andQqis a conserved charge (either Je R/L,I orPD)./angbracketleftFu p,α;Fu q,α/angbracketrightωand /angbracketleftFdis p,α;Fdis q,α/angbracketrightωare retarded ﬁnite-temperature Green’s functions evaluated to leading order using Sbin Eq. ( 4).λαandλdis α parametrize the umklapp scattering and coupling to disorder, respectively, and Dis the disorder variance of Gaussian- correlated disorder. As mentioned above, we take λα=λ andλdis α=λdisfor all αfor simplicity. ˆMucontains the contributions to the memory matrix from umklapp scattering,while ˆM discontains the contributions from the disorder- mediated interaction. We stress that the form of the memorymatrix given above is correct to leading order in the scatteringinteraction. (See Refs. [ 40,56–59] for further discussion.) The label αalso speciﬁes the momentum mismatch of an incommensurate scattering process, /Delta1k α≡m(α) IkF,I−p(α)G∈[0,2π), (33) for unit lattice constant, and the vector of integers m(α) Ithat deﬁnes the umklapp process. The vectors m(α) I, in turn, help 125142-6TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) determine, along with the matrix VIJ, the right and left scaling dimensions ( /Delta1R,/Delta1L) of the operators entering scattering interactions in Eqs. ( 24) and ( 27). Recall that we choose to take the Fermi vectors in all channels to be equal, kF,I=kF. The conductivities associated to the various charges Qpare encoded in the matrix ˆσ(ω)=ˆχ(ˆN+ˆM(ω)−iωˆχ)−1ˆχ, (34) where (ˆN)pq≡ˆχp˙q=/parenleftBigg Qp,i/bracketleftBigg/summationdisplay α(Hu α+Hdis α),Qq/bracketrightBigg/parenrightBigg/parenrightBigg .(35) We show in Appendix Cthat, at least to quadratic order in the umklapp λand disorder λdiscouplings, ˆN=0. The electrical conductivity σis determined by the (2 N− 1)×(2N−1) submatrix ˆ σJe I,Je J. The thermoelectric conduc- tivity ˜ αis determined by the (2 N−1)-dimensional vector ˆσJe I,PD/T. The thermal conductivity κ=ˆσPD,PD T−˜α2T σ.F o r commensurate ﬁllings and in the disorder-dominated regime,the thermoelectric conductivity can be ignored to leading orderso that the thermal conductivity is equal to the P D−PD component of ˆ σ. IV . HYPERCONDUCTOR TRANSPORT We now assemble the conductivity matrix ˆ σ. The ﬁrst ingredient is the static susceptibility matrix, which takes thefollowing form: ˆχ Je IJe J=e2 4πsgn(N−I)sgn(N−J)VIJ, (36) ˆχJe IPD=0, (37) ˆχPDPD=Nπ2T2 6, (38) where there is no sum over IandJand we have computed to zeroth order in any perturbation to Sb. See Appendix A for details on the calculation of the static susceptibilty matrixand the auxiliary Mathematica ﬁle for the explicit expressionforV IJ. See Appendix Bfor details on the evaluation of the memory matrix elements. In the following two sections, we study the contribu- tions to the conductivity in systems at commensurate andincommensurate ﬁllings in the presence of both umklappscattering and disorder. For the most part, we focus upon thedecoupled surface subspace within the hyperconductor phase,however, we provide the more general expressions for the dcconductivities where appropriate. A. Commensurate ﬁllings If the electron ﬁlling is commensurate with the lattice, kFdivided by the reciprocal lattice basis vector is a rational fraction, and so the momentum mismatch /Delta1kαin any umklapp scattering process may vanish. Umklapp scattering interac-tions with /Delta1k α=0 provide the dominant contribution to the electrical conductivity matrix. Thus we consider Sbtogetherwith 45 umklapp terms, all with /Delta1k(α) p=0. As argued earlier, the most important umklapps are those with total scaling dimension ( /Delta1R,/Delta1L)=(3/2,1/2). 1. Direct current conductivity We ﬁrst note that Fu PD,αvanishes when /Delta1k(α)=0, along with all the memory matrix elements involving it. This tells usthat the dynamics of the electrical current-carrying excitationsdecouple from the thermal carriers (with P Dremaining conserved) at commensurate ﬁllings without disorder. Incomputing the electrical conductivity, it is sufﬁcient to choose{J e I}as the complete basis of hydrodynamic modes. The conservation of PDin the linearly-dispersing regime also implies that the thermal conductivity κis inﬁnite in a pure system since ( PD\|JT)/negationslash=0. At commensurate ﬁllings, disorder is the leading effect that causes ﬁnite thermal conductivity, aswe discuss. To obtain the dc conductivity at commensurate ﬁllings, we need the memory matrix elements obtained in Appendix B2a: (ˆM u)Je IJe Jα(T)=π4 32UJe I,αUJe J,αT, (39) where the ﬁnite, nonzero coefﬁcients, UJe I,αUJe J,α∝e2are deﬁned in Eq. ( B10). This immediately gives the dc electrical conductivity σ(T)∝e2 λ21 T. (40) As promised, the electrical resistivity vanishes linearly in tem- perature. Note that the dimensionless proportionality constantsin Eq. ( 40) and in subsequent conductivity formulas are ﬁnite and nonzero [ 60]. We have neglected band curvature terms in the preceding and subsequent calculations by working with the linearizedaction in Eq. ( 4). Their inclusion does not lead to ﬁnite thermal conductivity since any nonoscillatory term will commutewithP D. However, particle-hole symmetry-breaking band curvature terms will mix PDandJe I, thereby leading to inﬁnite electrical conductivity so long as PDis conserved. Disorder, on the other hand, does cause PDto decay. While it gives a subleading contribution to the electrical conductivityin the commensurate case—disorder contributes the O(T 2) correction in Eq. (B25) to the dc electrical memory matrixelements—it is the leading contribution to the relaxation rateof the thermal conductivity: κ(T)∝/parenleftbigg1 D(λdis)2/parenrightbigg1 T, (41) where we have used the static susceptibility matrix in Eq. ( 38), the disorder memory matrix elements in Eq. ( B27), and the fact thatκTis equal to the PD−PDcomponent of the conductivity tensor ˆ σwhen the thermoelectric coefﬁcient vanishes (to leading order). Equations ( 40) and ( 41) constitute a violation of the Wiedemann-Franz “law.” Marginal umklapp scattering isthe leading low-temperature relaxation mechanism for the 125142-7PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) electrical current, while O(1) irrelevant disorder is the leading relaxation mechanism for the thermal current at commensurateﬁllings. In this case, the Lorentz ratio L=κ σT∝λ2 e2D(λdis)21 T(42) diverges as T→0. Remaining within the hyperconductor phase, but departing from the decoupled surface, the exponents for the electrical andthermal conductivities will generally be modiﬁed to the form:σ∝1/T 1−2(2−/Delta1X)andκ∝1/T1−2(2−/Delta1X), where deviations of/Delta1Xfrom 2 encode the shift of the scaling dimensions of the scattering processes away from marginality. 2. Alternating current conductivity The ac conductivities at commensurate ﬁllings are found similarly. From Appendix B2a, (ˆMu)Je IJe Jα(ω)=UJe I,αUJe J,α/bracketleftbiggπ2 32ω+iπ 16ωln(a2ω)/bracketrightbigg ,(43) where a2is proportional to the short-distance cutoff a. Therefore the ac electrical conductivity at T/lessmuchωtakes the form σ(ω)∝e2 iω(c1+c2ln(a2ω))+c3ω, (44) for constants c1,c2, andc3. The ﬁnite contribution to the real part of the electrical ac resistivity has given the Drude peakﬁnite width. Disorder is required for ﬁnite ac thermal conductivity. Using the memory matrix element in Eq. ( B27), we ﬁnd κ(T/ω/lessmuch1)∝T 3 ic4ωT2+c5Dω4, (45) for constants c4andc5. B. Incommensurate ﬁllings When the ﬁlling is incommensurate, there is no scattering process for which /Delta1kα=0. In this case, both the electrical and thermal conductivities are generally ﬁnite and so weuse the charge basis {Q p}={Je R,1,...Je R,N,Je L,1,...Je L,N−1,PD}. Band-curvature corrections contribute subleading terms to thetemperature dependence and will not be considered. The/Delta1k αassociated to the 46 umklapp scattering processes deﬁned by the m(α) Ivectors are all generally different from one another. Nevertheless, we set /Delta1kα=/Delta1kfor all αin the presentation of the results below. 1. Direct current conductivity The memory matrix elements for umklapp scattering at incommensurate ﬁlling is provided in Appendix B2b whose results we quote below. Inﬁnitesimally close to commensurateﬁlling, ω/lessorequalslant/Delta1k/lessmuchT, we may borrow our previous results computed precisely at commensurate ﬁlling with the under-standing that /Delta1k/negationslash=0 in the expression for F u PD,αin Eq. ( B6). The leading contribution to the electrical conductivity isunchanged from Eq. ( 40). However, the thermal conductivity is now ﬁnite even in the absence of disorder, κ(T)∝T2 λ2/Delta1k2. (46) As expected, the thermal conductivity is divergent as com- mensurability is restored, /Delta1k→0. The Lorentz ratio is a decreasing function of T2in the regime /Delta1k/lessmuchTas the temperature is decreased. As the temperature is lowered, we enter the regime T/lessmuch/Delta1k in which the dc electrical and thermal memory matrix elementstake the asymptotic low-temperature form: (ˆM u)pq α(T)=π2 32Up,αUq,α/Delta1k2 Te−/Delta1k 2T. (47) The resulting dc electrical and thermal conductivities for T/lessmuch /Delta1kare σ(T)∝e2 λ2T /Delta1k2e/Delta1k 2T, κ(T)∝1 λ2T4 /Delta1k4e/Delta1k 2T. (48) In this case, the Lorentz ratio, L∝T2 e2/Delta1k2, (49) vanishes as T→0 in the absence of disorder. If we had considered instead a more generic model in which the Fermimomenta were not identical, the /Delta1kwould then no longer be same. This would imply that the leading contribution tothe memory matrix in Eq. ( 47) would be dominated by the contribution with minimal /Delta1k. Disorder, if present, eventually dominates the low- temperature transport. The disorder dc electrical and thermalmemory matrix elements derived in Appendix B3: (ˆM dis)Je IJe Iα=2π3 3˜UJe I,α˜UJe J,αT2, (50) (ˆMdis)Je IPD α=0, (51) (ˆMdis)PDPD α=8π5 5˜UPD,α˜UPD,αT4, (52) where the coefﬁcients ˜Up,α˜Uq,αare deﬁned in Eq. ( B18). For generic, perturbative values of the couplings, the disorder-dominated regime occurs when the exponentially vanishingcontribution to the memory matrix in Eq. ( 47) is overcome by the disorder-dominated contribution above. The resultingelectrical and thermal conductivities in the presence of disorderfor temperatures T/lessmuch/Delta1kare σ(T)∝e 2 D(λdis)21 T2, (53) κ(T)∝1 D(λdis)21 T. Away from the decoupled surface, the low-temperature results will be modiﬁed as follows: σ=κ/T∝1/T2−2(2−/Delta1X).I nt h i s 125142-8TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) regime, the Lorentz ratio, L∝1 e2, (54) is constant, although the gapless metallic phase is certainly nota Fermi liquid. The Wiedemann-Franz law is satisﬁed at the lowest of temperatures for incommensurate ﬁllingsbecause disorder is the dominant relaxation mechanism atincommensurate ﬁllings for both the electrical and thermal currents. 2. Alternating current conductivity The ac conductivity at incommensurate ﬁlling follows straightforwardly from the previous analysis. For T/lessorequalslant/Delta1k/lessmuch ω, the ac electrical conductivity is unchanged from the previous result in Eq. ( 44). In fact, the real part of the ac electrical resistivities can be found from inversion of thedc electrical conductivities in Sec. IVB1 by the replacement T→ωin all algebraic prefactors and so we shall not write them out explicitly. Let us now concentrate on the real part of the ac thermal conductivities. For T/lessmuch/Delta1k/lessmuchω, κ(ω)∝1 λ2T3 /Delta1k2ω. (55) ForT< ω /lessmuch/Delta1kwithT/lessmuch(/Delta1k2/ω)e x p (ω−/Delta1k 2T) and in the absence of disorder the thermal conductivity is dominated byincommensurate umklapp scattering, κ(ω)∝1 λ2T3ω /Delta1k4e/Delta1k−ω 2T, (56) w h e r ew eu s e dE q .( B17). Notice the divergent thermal conductivity as T→0. Finally, in the disorder-dominated regime with T2/lessmuchDω3, κ(ω)∝1 DT3 ω4. (57) V . CONCLUSIONS In this paper, we have determined the dc and ac electrical and thermal conductivity of the one-dimensional hypercon-ductor phase introduced in Ref. [ 1] in the presence of umklapp and disorder-mediated scattering. For instance, wehave shown that this metallic phase exhibits a dc conductivityσ∼1/T 1−2(2−/Delta1X)down to T=0 without ﬁne-tuning at commensurate ﬁllings, thereby manifesting the non-Fermiliquid nature of the phase. In addition, we have discussedthe relation between conservation laws and transport whichhas allowed us to provide examples of violations of theWiedemann-Franz law. As a simple example, the thermalconductivity is only ﬁnite in the presence of disorder, whilethe electrical conductivity can be ﬁnite in a pure system atcommensurate ﬁlling with only umklapp scattering. Moregenerally, we have seen how differing relaxation mechanismsfor the electrical and thermal currents can result in violationsof the Wiedemann-Franz law. The power-law σ∼1/Tobtains along the “decoupled sur- face” of the hyperconductor when the interactions determinedby˜V IJ—see Sec. II—are block diagonal at commensurateﬁllings. On this surface, /Delta1X=2. The hyperconductor phase survives within a ﬁnite window off the decoupled surface bythe addition of off-diagonal terms to ˜V IJmixing right-moving and left-moving hyperconductor excitations. Departing fromthe decoupled surface, but remaining within the hypercon-ductor phase, the relaxation of the current is controlled by46 umklapp scattering operators with conformal dimensions( 3 2+δ,1 2+δ) so that /Delta1X=2+2δ, with δdetermined by the distance from the decoupled surface. The conductivitywill generally behave σ∼1/T 1−2(2−/Delta1X)with/Delta1X>2d o w n toT=0. For /Delta1X<2, the zero-temperature perfect metal ﬁxed point is unstable. However, the relevant perturbations arechiral and, therefore, cannot open a gap. At low temperatures,they may strongly renormalize the velocities, shift the Fermimomenta, or otherwise modify the ground state (withoutopening a gap) in such a manner that the dangerous processescan no longer occur. In the marginal case, /Delta1 X=2, such an instability presumably occurs at sufﬁciently large marginalcoupling. The large marginal coupling limit of this hyperconductor regime is an interesting testing ground for Hartnoll’s recentlyconjectured [ 61] lower bound on the diffusion constant, D/greaterorequalslant /planckover2pi1v 2 F/(kBT). This bound applies to systems in the “incoherent” metallic regime where there is no overlap between theelectrical current and momentum operator. If satisﬁed, thislower bound implies an upper bound on the coefﬁcient of the linear in temperature dc electrical resistivity that we found at commensurate ﬁllings. The distinction between a hyperconductor and a supercon- ductor is that a hyperconductor does not have long-rangedorder [ 62]. This distinction is not apparent in zero-temperature electrical transport, which is inﬁnite in both cases. (It doesmanifest itself in the differential tunneling conductance, whichvanishes algebraically with voltage in the hyperconductor butis strongly suppressed at voltages below the energy gap in asuperconductor—it would be zero but for Andreev reﬂection.)However, the difference between a hyperconductor and asuperconductor is clearer in low-temperature transport. Ina superconductor, the electrical resistivity vanishes for alltemperatures below the critical temperature, but in a hypercon-ductor, the resistivity increases smoothly, with the temperaturedependence described above. In the incommensurate case,the resitivity is exponentially small in temperature over a wide range of temperatures, has the feature of very small (albeit not vanishing) resistivity without the threat of a suddenlarge jump at a critical temperature. While a superconductorconducts electrical current without dissipation even in thepresence of impurities for T< T c, a hyperconductor has nonzero resistivity for T> 0, but strongly suppressed—in the hyperconductor studied here, the impurity contribution issuppressed by a factor ( T/T F)2/Delta1X−2with/Delta1X/greaterorequalslant2. Meanwhile, a hyperconductor has radically different thermal transport thana superconductor. In a superconductor, thermal currents areonly carried by excited quasiparticles and phonons. Therefore,the thermal conductivity divided by the temperature vanisheswith decreasing temperature. In particular, the electroniccontribution to the thermal conductivity of an s-wave super- conductor has activated form. In a hyperconductor, on the other hand, the thermal conductivity diverges as a power-law at the lowest temperatures and diverges exponentially with 125142-9PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) inverse temperature over a wide range of temperatures. Thus, the hyperconductor phase, though neither a superconductornor a superﬂuid, has an electrical conductivity that approachesthat of the former and a thermal conductivity that approachesthat of the latter. In the future, we plan to understand the 2D metallic phase that emerges from an array of hyperconductor wires. Thiswire array should exhibit diffusive ﬁnite-temperature transportboth along and transverse to the wires and be stable toweak disorder. This paper makes clear the reason why ﬁniteconductivities obtain along the wires. To understand the lattertwo statements, we need only observe that such an array formsa sort of “chiral transverse Fermi liquid” in the sense thatonly half of the Fermi surface excitations can hop betweenwires at the lowest of energies, reminiscent of the chiralmetals studied in Refs. [ 63–65] (see Ref. [ 66] for related work). In these works [ 63–65], it was found that a collection of wires, each hosting a chiral Fermi liquid (obtained fromthe edge excitations of a collection of integer quantum Hallsystems layered in a transverse direction), exhibits diffusivetransport transverse to the wires and does not localize. Oneimportant difference between these constructions and the2D hyperconductor is the diffusive, as opposed to ballistic,ﬁnite-temperature transport exhibited by the hyperconductoralong the wires. ACKNOWLEDGMENTS We thank S. Hartnoll, S.-S. Lee, S. Raghu, E. Shimshoni, and B. Ware for enjoyable and helpful discussions. We alsothank S. Hartnoll and S. Kivelson for comments on an earlydraft of the manuscript. M.M. acknowledges the support of theJohn Templeton Foundation. C.N. has been partially supportedby AFOSR under grant FA9550-10-1-0524. APPENDIX A: STATIC SUSCEPTIBILITY MATRIX The static susceptibility matrix ˆ χpq=1 LGR QpQq(ω=0) where the conserved charges QpandQpof the action Sb involved in the retarded Green’s function GR QpQqare either one of the chiral electrical currents, Je I=e 2πsgn(N−I)/integraldisplay xVIJ∂xφJ =e 2πsgn(N−I)/integraldisplay xVIJOJa∂xXa, (A1) or the Dirac momentum, PD=−1 4π/integraldisplay xsgn(N−I)∂xφI∂xφI =−1 4πsgn(N−a)/integraldisplay x∂xXa∂xXa. (A2) In the above equations, x∈(−L,L) with the length of the system L→∞ ,s g n (Z)=+ 1f o rZ/greaterorequalslant0 and sgn( Z)=− 1 forZ< 0, and Je I=Je R,IforI=1,..., N andJe I=Je L,N−I forI=N+1,..., 2NwithN=23. Note that Iis not summed over on the right-hand side of Eq. ( A1). We have introduced the ﬁelds φI=OIaXawithOIa∈SO(23,23) that diagonalize the action Sb, tuned via the interaction matrix VIJto the asymmetric Leech liquid point, Sb=1 4π/integraldisplay t,x[sgn(N−I)∂tφI∂xφI−VIJ∂xφI∂xφJ] =1 4π/integraldisplay t,x[sgn(N−a)∂tXa∂xXa−v∂xXa∂xXa].(A3) Henceforth, we set the velocity v=1. To isolate the leading temperature and frequency dependence of the conductivity,we need only compute the static susceptibility with respecttoS b. The bosonic action Sbenjoys the particle-hole symmetry φI→−φI,Xa→−Xa. Thus, the retarded Green’s functions GR Je IPD=0 when computed with respect to Sband so we focus upon the Je I−Je JorPD−PDstatic susceptibilities. Scattering interactions at incommensurate ﬁllings, interactionsmediated by disorder, and higher-derivative band structurecorrections to S bgenerally break particle-hole symmetry and, thus, induce a nonzero overlap between the electricalcurrents and the momentum. We ignore such overlaps as theycontribute higher-order corrections to the conductivity thanthat to which we choose to work. At commensurate ﬁllings andin the absence of higher-derivative corrections, particle-holesymmetry is preserved. To compute the retarded correlator, we exploit the relation G R QpQq(ω)=GE QpQp(iωE→ω+iδ) with the inﬁnitesimal δ> 0 between the retarded Green’s function and the Euclidean Green’s function at Euclidean frequency ωE. The frequency ωof the retarded correlator has been analytically continued to the upper-half plane. We shall often simply set δ=0 without mention. Thus, the static susceptibility ˆ χpq=1 LGE QpQp(ωE= 0). We begin with the Je I−Je Jcomponents of the static susceptibility, ˆχJe IJe J≡1 Llim ωE→0/integraldisplay τeiωEτ/angbracketleftbig Je I(τ)Je J(0)/angbracketrightbig =e2Mab IJ 4π2Llim ωE→0/integraldisplay τ,x,yeiωEτ/angbracketleftX/prime a(τ,x)X/prime b(0,y)/angbracketright,(A4) where X/prime(τ,x)≡∂xX(τ,x), Mab IJ=sgn(N−I)sgn(N−J)VIKVJLOKaOLb =sgn(N−I)sgn(N−J)(O−1)aI(O−1)bJ, (A5) the Euclidean time τ∈[0,1/T] and the brackets denote the thermal average at temperature T. In simplifying Eq. ( A5), we have made use of the identity OIaVIJOJb=δab. Because Sbis diagonal when expressed in terms of the Xaﬁelds, the only nonzero correlators in Eq. ( A4) occur when a=band we obtain the well-known result [ 47] /angbracketleftX/prime a(τ,x)X/prime b(0,0)/angbracketright=− δab/bracketleftBigg πT sinh/parenleftbig πT(x−sgnaiτ)/parenrightbig/bracketrightBigg2 , (A6) where we have used the shorthand, sgna=sgn(N−a). It will be convenient to calculate a slightly more general Fouriertransform than Eq. ( A4) by replacing the exponent in Eq. ( A6), 125142-10TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) 2→2hwithhassumed to be half-integral. Thus we consider 1 L/integraldisplay τ,x,yeiωEτ/bracketleftbiggπT sinh(πT(x−y−sgnaiτ))/bracketrightbigg2h =−(πT)2h 2L/integraldisplay x+,x−,τeiωEτ 1 [sinh (πT(x−−sgnaiτ))]2h =−π2h(2T)2h−1/integraldisplay x−esgna2πTx −1 2πi/contintegraldisplay \|ζ\|=1ζωEτ 2πT+h−1 (ζ−esgna2πTx −)2h =−T2h−1 2ωE(2π)2h (2h−1)!2h−1/productdisplay i=1/parenleftbiggωE 2πT+h−i/parenrightbigg . (A7) In the ﬁrst line, we made the change of variables, x±=x±y and then integrated over x+; in the second line, we made the change of variable ζ=exp(2πTiτ ), performed the contour integration about the circle \|ζ\|=1, and then integrated over x−. Thus we ﬁnd for the current-current static susceptibility: ˆχJe IJe J=e2 4π2N/summationdisplay a=1Maa IJ =e2 4πsgn(N−I)sgn(N−J)VIJ, (A8) where I,J are not summed over and we used the relation (O−1)T.(O−1)=V. Following an analogous procedure, we now calculate the PD−PDstatic susceptibility, ˆχPDPD≡1 Llim ωE→0/integraldisplay τeiωEτ/angbracketleftPD(τ)PD(0)/angbracketright =2 16π2Lsgn(N−a)sgn(N−b) ×/integraldisplay τ,x,yeiωEτ/angbracketleftX/prime a(τ,x)X/prime b(0,y)/angbracketright2 =1 8π2L/integraldisplay τ,x,yeiωEτ/angbracketleftX/prime a(τ,x)X/prime a(0,y)/angbracketright2,(A9) where we used Wick’s theorem in going from the ﬁrst to the second line and the fact that the only nonzero correlators occurwhena=bin going from the second to the third line. We may now borrow the general result in Eq. ( A7) by setting h=2t o conclude that ˆχ PDPD=Nπ2T2 6. (A10) APPENDIX B: MEMORY MATRIX ELEMENTS Recall the deﬁnition of the memory matrix reviewed Sec. III C , which we repeat here for convenience. The memory matrix ˆM(ω) (the temperature dependence is left implicit) is deﬁned as follows: ˆM(ω)=/summationdisplay α(λ2 αˆMu α(ω)+(λdis α)2DˆMdis α(ω)), (B1) (ˆMu)pq α=1 L/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ω−/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ω=0 iω, (B2) (ˆMdis)pq α=1 L/angbracketleftbig Fdis p,α;Fdis q,α/angbracketrightbig ω−/angbracketleftbig Fdis p,α;Fdis q,α/angbracketrightbig ω=0 iω. (B3)Here,Fu q,α=i λα[Hu α,Qq],Fdis q,α=i λdisα√ D[Hdis α,Qq], andQqis a conserved charge (either Je R/L,I orPD)./angbracketleftFu p,α;Fu q,α/angbracketrightωand /angbracketleftFdis p,α;Fdis q,α/angbracketrightωare retarded ﬁnite-temperature Green’s functions evaluated using Sb.λαandλdis αparametrize the umklapp scattering and coupling to disorder, respectively, and Dis the disorder variance of the Gaussian-correlated disorder, ξα(x)= 0,ξα(x)ξ∗ β(y)=Dδαβδ(x−y). For simplicity, we take λα=λ andλdis α=λdisfor all α.ˆMucontains the contributions to the memory matrix from umklapp scattering, while ˆMdiscontains the contributions from the disorder-mediated interaction. Westress that the form of the memory matrix given above iscorrect to leading order in the scattering interaction. (SeeRefs. [ 40,56–59] for further discussion.) 1. Evaluation of the Fu,dis p,α To compute the Fu,dis p,α commutators, we make use of the equal-time commutators: /bracketleftbigg eim(α) JφJ(x),φ/prime I(y) 2π/bracketrightbigg =m(α) IsgnIδ(x−y)eim(α) JφJ(x). (B4) We ﬁnd for the commutators Fu p,αof theQpwith the umklapp scattering operators: Fu Je I,α=− 2esgn(N−I)sgn(N−J)VIJm(α) J ×/integraldisplay x1 a2sin/parenleftbig /Delta1kαx+m(α) KφK/parenrightbig , (B5) Fu PD,α=2/Delta1kα/integraldisplay x1 a2sin/parenleftbig /Delta1kαx+m(α) KφK/parenrightbig , (B6) where the momentum mismatch /Delta1kα≡/summationtext Im(α) IkF−p(α)G, Gis a basis vector for the reciprocal lattice, and we have taken the Fermi momenta in all channels to be equal. Recall that ais a short-distance cutoff. We see that the Dirac momentum PD commutes with the umklapp operators when /Delta1kα=0, i.e., when the translation symmetry of the low-energy effectivetheory is preserved. The result for [ H u α,PD] is found, using the integration by parts, /integraldisplay xei/Delta1kαxm(α) K 2/braceleftbig φ/prime K,eim(α) LφL/bracerightbig ≡−i/integraldisplay xei/Delta1kαx∂xeim(α) LφL =−/Delta1kα/integraldisplay xei/Delta1kαx+im(α) LφL,(B7) where we have dropped the boundary term and have deﬁned the derivative operator on the right-hand side of the ﬁrst 125142-11PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) line via a symmetric ordering prescription: ∂xexp(im(α) IφI)≡ i 2m(α) J(∂xφJexp(im(α) IφI)+exp(im(α) IφI)∂xφJ). The commutators Fdis p,α of the Qpwith the disorder- mediated interactions are computed in a similar fashion: Fdis Je I,α=ie√ Dsgn(N−I)sgn(N−J)VIJm(α) J ×/integraldisplay x/bracketleftbigg ξα(x)1 a2eim(α) KφK−H.c./bracketrightbigg , (B8) Fdis PD,α=−1√ Dv2/integraldisplay x/bracketleftbigg (∂xξα(x))1 a2eim(α) KφK+H.c./bracketrightbigg .(B9) We see that the umklapp commutators in Eqs. ( B5) and ( B6) may be obtained from the disorder commutators in Eqs. ( B8) and ( B9) by substituting ξα(x)=exp(i/Delta1kαx). 2. Evaluation of the ( ˆMu)pq α We begin with the evaluation of the retarded two-point correlation functions /angbracketleftFu p,α;Fu q,β/angbracketrightω. To leading order in the umklapp (and disorder) perturbations, these correlators areonly nonzero when α=βbecause of the linear independence of the m(α) Iso we set α=βin the remainder. Also, notice that/angbracketleftFu p,α;Fdis q,β/angbracketrightω=0 because the disorder we study has zero mean, ξα(x)=0. We simplify the following expressions by introducing the coefﬁcients: UJe I,α=− 2esgn(N−I)sgn(N−J)VIJm(α) J, (B10) UPD,α=2v2/Delta1kα. We see that UPD,α=0 for commensurate ﬁllings when /Delta1kα= 0 because translation invariance in the low-energy effectivetheory S lin(interpreted as Dirac fermions created about zero-momentum) is preserved, resulting in divergent thermalconductivity. Just as in Appendix A, we compute the retarded correlators by Fourier transforming the Euclidean real-spacecorrelation functions and then analytically continuing theMatsubara frequencies ω Eto real frequencies ωby way of the formula GR Fup,αFuq,α(ω)=GE Fup,αFuq,α(iωE→ω+iδ)≡ /angbracketleftFu p,α;Fu q,α/angbracketrightωE→−iω+δ. Thus the Fourier transformed Euclidean correlation func- tions take the form 1 L/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ωE=Up,αUq,α L1 a4/integraldisplay x,y,τeiωEτ/angbracketleftbig sin/parenleftbig /Delta1kαx+m(α) KφK(τ,x)/parenrightbig sin/parenleftbig /Delta1kαy+m(α) LφL(0,y)/parenrightbig/angbracketrightbig =Up,αUq,α 4L/integraldisplay x,y,τeiωEτ/bracketleftbigg ei/Delta1kα(x−y)/angbracketleftbiggeim(α) KφK(τ,x) a2e−im(α) LφL(0,y) a2/angbracketrightbigg +e−i/Delta1kα(x−y)/angbracketleftbigge−im(α) KφK(τ,x) a2eim(α) LφL(0,y) a2/angbracketrightbigg/bracketrightbigg =Up,αUq,α 2L/integraldisplay x,y,τeiωEτcos(/Delta1kα(x−y))(πT)4 sinh3[πT((x−y)−iτ)] sinh[ πT((x−y)+iτ)], (B11) where x,y∈(−L,L) with L→∞ andτ∈[0,1/T]. The ﬁrst equality follows from direct substitution of Eqs. ( B5) and ( B6); for the second equality, we have only retained the nonzero terms in the product; for the third equality, we have used the standard thermal real-space Euclidean two-point function of a dimension ( /Delta1R,/Delta1L)=(3/2,1/2) vertex operator1 α2exp(im(α) JφJ)[47]. It is a great simpliﬁcation of the calculation that all vertex operators considered have the same scaling dimension. If only afraction of the operators necessary to relax the currents had dimension (3 /2,1/2) and the remaining operators were of higher dimension, it would be straightforward to calculate their effects by methods similar to those presented here. These operatorswould give subleading contributions to the memory matrix leading to slower relaxation of some conserved currents. As a resultthese operators would give the dominant contributions to the matrix of conductivities. Similar to Appendix A, we evaluate Eq. ( B11) by ﬁrst making the change of variables x ±=x±yandξ=e2πiTτ. We assume a short-distance cutoff 0 <a< \|x−y\|. The integral over x+factors out, canceling the 1 /Lprefactor, and we are left with the following integral to evaluate: 1 L/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ωE=− 4π4T3Up,αUq,α/integraldisplay x−e−2πTx −cos(/Delta1kαx−)1 2πi/contintegraldisplay \|ζ\|=1ζωE 2πT+1 (ζ−e−2πTx −)3(ζ−e2πTx −) =π2TUp,αUq,α 4/integraldisplay∞ adx−e−ωEx−cos(/Delta1kαx−) sinh3(2πTx −)/bracketleftbig 4π2T2+ω2 Esinh2(2πTx −)+πTω Esinh(4 πTx −)/bracketrightbig .(B12) Next, we Wick rotate, ωE→−iω+δ,E q .( B12) to obtain the retarded Green’s function 1 L/angbracketleftbig Fu p,α;Fu q,α/angbracketrightbig ω=π2TUp,αUq,α 4/integraldisplay∞ adx−e−δx−+iωx−cos(/Delta1kαx−) sinh3(2πTx −)[4π2T2 +(−iω+δ)2sinh2(2πTx −)+πT(−iω+δ)s i n h ( 4 πTx −)]. (B13) The remaining integral in Eq. ( B12) can be evaluated ex- actly to obtain the memory matrix elements ( ˆMu)pq αdeﬁned inEq. ( B2). The exact expression is rather complicated and so we shall examine it in various low-frequency and low-temperature 125142-12TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) limits for both commensurate and incommensurate ﬁllings. To study the low-frequency and low-temperature behavior of(ˆM u)pq α, we ﬁrst perform two expansions. First, we expand the result as the short-distance cutoff a→0, keeping only the singular and ﬁnite nonzero terms. Any a→0 singularities are a reﬂection of the short-distance divergences of the correlationfunction. Second, we expand to linear order in δ, however, we ﬁnd it sufﬁcient to study the resulting expression at δ=0 as the real part of the memory matrix is generally nonzero atﬁniteωand ﬁnite T. (a) Commensurate ﬁllings For commensurate ﬁllings, we set /Delta1kα=0. For ω/T/lessmuch1, the expression for the memory matrix element at commensu-rate ﬁllings has the following behavior: (ˆM u)pq α/parenleftbiggω T/lessmuch1/parenrightbigg =Up,αUq,α/bracketleftbiggπ4 32T+iπω 16ln(a1T)/bracketrightbigg , (B14) where we have dropped all O(δ) terms and absorbed all constants via a redeﬁnition of the cutoff a→a1. We shall make these multiplicative redeﬁnitions of the short-distancecutoff a→a iin each of the following expressions. In the opposite regime when T/ω/lessmuch1, we ﬁnd the following expression for the memory matrix elements at commensurateﬁlling: (ˆM u)pq α/parenleftbiggT ω/lessmuch1/parenrightbigg =Up,αUq,α/bracketleftbiggπ2 32ω+iπ 16ωln(a2ω)/bracketrightbigg , (B15) where a1/negationslash=a2. (b) Incommensurate ﬁllings When the ﬁlling is incommensurate, /Delta1kα/negationslash=0. We shall study the memory matrix for frequencies and temperaturesω,T/lessmuch/Delta1k α. Forω/T/lessmuch1, the expression for the memory matrix ele- ments at incommensurate ﬁllings have the following behavior, (ˆMu)pq α/parenleftbiggω T/lessmuch1/parenrightbigg =Up,αUq,α/bracketleftbiggπ2 32/parenleftbigg(/Delta1kα)2 T+4π2T/parenrightbigg e−/Delta1kα 2T +iπω 16ln(a3/Delta1kα)/bracketrightbigg , (B16)where we have only retained the leading term present for T→0. Precisely at T=0 (but ﬁrst ω→0), the real part of the ( ˆMu)pq α(ω T/lessmuch1) vanishes when /Delta1kα/negationslash=0 and we obtain a purely imaginary memory matrix which implies a ﬁnite Drude weight. When T/ω/lessmuch1, the incommensurate memory matrix takes the form (ˆMu)pq α/parenleftbiggT ω/lessmuch1/parenrightbigg =Up,αUq,α/parenleftbiggπ2 16/parenleftbigg(/Delta1kα)2 ω+ω/parenrightbigg eω−/Delta1kα 2T +iπ 32/braceleftbigg ωln/bracketleftbig a2 4/parenleftbig (/Delta1kα)2−ω2/parenrightbig/bracketrightbig +(/Delta1kα)2 ωln/parenleftbigg 1−ω2 (/Delta1kα)2/parenrightbigg/bracerightbigg/parenrightbigg . (B17) While we have studied the memory matrix for incommen- surate ﬁllings in the limit ω,T/lessmuch/Delta1kα, we have checked that the initial expression obtained before taking the low-frequencyor low-temperature limits reverts to the commensurate valuesby taking /Delta1k α=0. 3. Evaluation of the ( ˆMdis)pq α Because the same vertex operators are used in both the umklapp and disorder-mediated interactions, the calculationof the disorder memory matrix elements ( ˆM dis)pq αwill be very similar to that of the previous section. We begin withthe evaluation of the retarded two-point correlation functions/angbracketleftF dis p,α;Fdis q,α/angbracketrightω, which we determine by analytically continuing the Euclidean correlator /angbracketleftFdis p,α;Fdis q,α/angbracketrightωE. We again simplify the ensuing expressions by introducing the coefﬁcients ˜UJe I,α=iesgn(N−I)sgn(N−J)VIJm(α) J, (B18) ˜UPD,α=−v2, that occur in the disorder commutators in Eqs. ( B8) and ( B9). Unlike the correlators of the commutators involved in the umklapp calculation, we need to examine each set of corre-lators/angbracketleftF dis Je I,α;Fdis Je J,α/angbracketrightωE,/angbracketleftFdis Je I,α;Fdis PD,α/angbracketrightωE, and/angbracketleftFdis PD,α;Fdis PD,α/angbracketrightωE in turn. First consider 1 L/angbracketleftbig Fdis Je I,α;Fdis Je J,α/angbracketrightbig ωE=iω+δ=−(πT)4˜UJe I,α˜UJe J,α LD/integraldisplay x,y,τeiωEτ ξα(x)ξ∗ α(y)+ξ∗ α(x)ξα(y) sinh3[(πT((x−y)−iτ)]sinh [πT((x−y)+iτ)] =π2T˜UJe I,α˜UJe J,α 4LD/integraldisplay x+/integraldisplay∞ adx−e(−δ+iω)x− sinh3(2πTx −)[ξα(x)ξ∗ α(y)+ξ∗ α(x)ξα(y)] ×[4π2T2+(−iω+δ)2sinh2(2πTx −)+πT(−iω+δ) sinh(4 πTx −)], (B19) where x±=x±yand we have performed identical manipulations to those explained in the previous section to evaluate Eqs. ( B11), (B12), and ( B13). To explicitly evaluate the integrals over x+andx−in Eq. ( B19), we must choose a form for the functions ξα(x) parameterizing the disorder. As we have discussed, we have chosen to consider zero-mean Gaussian-correlated disorder, ξ(x)=0,ξα(x)ξ∗α(y)= Dδ(x−y). To make contact with the pure system calculation of umklapp scattering at incommensurate ﬁllings, we comment 125142-13PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) that this form of the disorder may be obtained by choosing a disorder potential, ξα(x)=/integraltext /Delta1pα˜ξ(/Delta1pα)ei/Delta1pαxwith ˜ξ(/Delta1pα)=1. We see that incommensurate ﬁllings can be understood as a particular disorder realization with ˜ξ(/Delta1pα)=δ(/Delta1pα−/Delta1kα). Before integrating over x+andx−in Eq. ( B19), we ﬁrst disorder average. This allows us to again factor out the x+integral to cancel the 1 /Lprefactor and also to replace the product of disorder potentials ξα(x) inside the ﬁrst brackets by 2 Dδ(x−y), where the δ(x−y) is understood to evaluate all terms containing x−=a, the short-distance cutoff. We ﬁnd 1 L/angbracketleftbig Fdis Je I,α;Fdis Je J,α/angbracketrightbig ω=π2T˜UJe I,α˜UJe J,α 2e(−δ+iω)a sinh3(2πTa )[4π2T2+(−iω+δ)2sinh2(2πTa )+πT(−iω+δ) sinh(4 πTa )].(B20) Next, consider1 L/angbracketleftFdis Je I,α;Fdis PD,α/angbracketrightω. The calculation of this correlator is identical to the previous one except that the overall coefﬁcient now involves the ˜UJe I,α˜UPD,αand the product of disorder potentials in the ﬁrst line of Eq. ( B19) is replaced: ξα(x)ξ∗ α(y)+ξ∗ α(x)ξα(y)→ξα(x)∂yξ∗ α(y)−ξ∗ α(x)∂yξα(y)=∂y(ξα(x)ξ∗ α(y)−ξ∗ α(x)ξα(y)). (B21) Upon disorder averaging, the term in the parentheses in Eq. ( B21) vanishes. Thus we ﬁnd 1 L/angbracketleftbig Fdis Je I,α;Fdis PD,α/angbracketrightbig ω=0. (B22) There is no overlap to leading order in the disorder-variance Dbetween the electrical and thermal currents. Finally, we evaluate1 L/angbracketleftFdis PD,α;Fdis PD,α/angbracketrightωby replacing in Eq. ( B19): ˜UJe I,α˜UJe J,α→˜UPD,α˜UPD,α,ξ α(x)ξ∗ α(y)+ξ∗ α(x)ξα(y)→∂xξα(x)∂yξ∗ α(y)+H.c. (B23) Disorder averaging, performing the integration by parts with respect to ∂x/y=∂x+±∂x−, discarding all boundary terms, and evaluating x−=a, we ﬁnd 1 L/angbracketleftbig Fdis PD,α;Fdis PD,α/angbracketrightbig ω=π2T˜UPD˜UPD 2∂x−∂x−/braceleftbigge(−δ+iω)x− sinh3(2πTx −)[4π2T2+(−iω+δ)2sinh2(2πTx −) +πT(−iω+δ) sinh(4 πTx −)]/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle x−=a. (B24) Equipped with the above correlation functions, we may now evaluate the memory matrix elements ( ˆMdis)Je IJe Iα and (ˆMdis)PDPDα . As before, we determine these memory matrix el- ements by expanding about the limit a→0 and subsequently expanding about δ=0. It is sufﬁcient to set δ=0. In summary, we ﬁnd (ˆMdis)Je IJe Iα=˜UJe I,α˜UJe J,α/parenleftbigg2π3 3T2+π 6ω2−i3π 24ω a/parenrightbigg , (B25) (ˆMdis)Je IPD α=0, (B26) (ˆMdis)PDPD α=˜UPD,α˜UPD,α/parenleftbigg8π5 5T4+2π3 3T2ω2 +π 15ω4+iπ 15ω a3/parenrightbigg . (B27) We notice that the logarithmic singularities that occurred in the umklapp memory matrix elements for a=0a r e replaced by power-law singularities. Such singularities reﬂectthe short-distance divergences as correlation function insertionpoints become coincident. They only occur in the imaginarypart of the memory matrix elements at ﬁnite frequencies.Our prescription is to remove such power-law divergencesby hand and concentrate on the real parts of the memorymatrix elements that determine the long-wavelength responseof the system. This prescription leads to agreement with relatedcomputations [ 67,68] studying the tunneling conductance between quantum wires at a single point contact. APPENDIX C: ˆNMATRIX In this Appendix, we show that ˆN=0 to quadratic order in the umklapp λand disorder λdiscouplings using rather general considerations. Recall the deﬁnition (ˆN)pq≡ˆχp˙q=/parenleftbigg Qp,i/bracketleftbigg/summationdisplay α(Hu α+Hdis α),Qq/bracketrightbigg/parenrightbigg/parenrightbigg .(C1) 1. Umklapp contributions First, consider the contribution to ˆNfrom umklapp pro- cesses Hu α. Observe that i[Hu α,Qq]=λFu q,αandi[Hdis α,Qq]=√ Dλdis, where Qq∈{Je I,PD}, so that by using the deﬁnition of the static susceptibility and conventions in Appendix A: (ˆN)pq=λ Llim ωE→0/integraldisplay τeiωEτ/angbracketleftQp(τ)Fu q,α(0)/angbracketright, (C2) and likewise for the disorder contribution studied momentarily where the bracket denotes the Euclidean correlation functionat temperature T. At leading order in λ, the above two- point function /angbracketleftQ p(τ)Fu q,α(0)/angbracketrightvanishes when computed with respect to Sb; more speciﬁcally, /angbracketleft∂xφI(τ,x)eim(α) JφJ(0,y)/angbracketright= 0 and /angbracketleft∂xφI(τ,x)∂xφI(τ,x)eim(α) JφJ(0,y)/angbracketright=0 when computed 125142-14TRANSPORT IN A ONE-DIMENSIONAL HYPERCONDUCTOR PHYSICAL REVIEW B 93, 125142 (2016) with respect to Sb. At quadratic order, λ2, there is the correction δ(ˆN)pq=λ2 Llim ωE→0/integraldisplay τ,τ/prime,zeiωEτ/angbracketleftQp(τ)Fu q,α(0)Hu α(τ/prime,z)/angbracketright. (C3) The above correlation function, computed with respect to Sbfactorizes, into the sum of two three-point functions: λ2 L/integraldisplay τ,τ/prime,zeiωEτ/angbracketleftQp(τ)Fu q,α(0)Hu α(τ/prime,z)/angbracketright∝iλ2(πT)5 L/integraldisplay τ,τ/prime,x,y,zeiωEτ/bracketleftbiggC1e−i/Delta1kαXzy−C2ei/Delta1kαXzy sinh(πT(Xzy+iτ/prime))/bracketrightbigg ×1 sinhh(πT(Xxy−iτ))sinhh(πT(Xxz−iτ+iτ/prime))sinh3−h(πT(Xzy−iτ/prime)), (C4) for constants C1=(−1)hC2(whose precise magnitude will not be required) equal to the operator product coefﬁcients for the fusion, Qpexp(im(α) IφI)∼exp(im(α) IφI), and h=1 whenQp=Je Iandh=2 when Qp=PD. Above, we have introduced the “difference coordinates” Xxy=x−y,Xxz= x−z,Xzy=z−y.A tωE=0, we notice that the integrand is odd under the reﬂection of all spatial and temporal coordinatesfollowed by the shifts, τ,τ /prime→τ−1/T,τ/prime−1/T. Therefore the integral is zero at ωE=0 and the quadratic contribution toˆNfrom umklapp processes vanishes. 2. Disorder contributions Next, consider the contributions to ˆNfrom disorder- mediated processes Hdis α. The term linear in λdisagain vanishes for the same reason as before. At quadratic order, we considerEq. ( C3) with the superscript u replaced by dis. The form of the resulting three-point function is very similar to that whichappears in Eq. ( C4). The difference is due to the disorder ξ α appearing in the disorder commutators Eqs. ( B8) and ( B9) andHdis α.F o rFdis q,α=Fdis Je I,α, we disorder average and insert δ(y−z) into integrand in Eq. ( C4)a tωE=0: when Qp=Je I, the three-point function vanishes using the above reﬂection andtranslation argument; when Q p=PD, the three-point function vanishes identically after setting y=zand using C1=C2 forh=2. For Fdis q,α=Fdis PD,I, we disorder average, replace the relative minus sign between C1andC2by (+1), and insert ∂yδ(y−z) into the integrand in Eq. ( C4): when Qp=Je I, the integrand vanishes identically similar to PDbefore; when Qp=PD, we may again apply the reﬂection and translation argument to conclude that the integral vanishes at ωE=0. Thus we may safely ignore the ˆNmatrix in our transport calculations. APPENDIX D: EXACT MARGINALITY ALONG THE “DECOUPLED SURFACE” In this Appendix, we argue perturbatively for the exact marginality, along the decoupled surface, of the dimension(/Delta1 R,/Delta1L)=(3/2,1/2) operators used to relax the electrical and thermal currents. Our argument strictly applies in thescaling limit in which only classically marginal and relevantinteractions are retained in the low-energy effective theorywith irrelevant interactions being set to zero.Recall from Sec. IIthat the decoupled surface is a subspace within the hyperconductor phase in which the interactionmatrix ˜V IJis block diagonal. The scaling dimensions of operators are independent of ˜VIJwhen the theory lies on the decoupled surface; however, scaling dimensions varycontinuously with ˜V IJupon departing from the decoupled surface. We consider the collection of operators Oα=cos (m(α) IφI) with scaling dimension equal to (3 /2,1/2) along the decoupled surface whose coupling constants we denote by gα. These operators are exactly marginal if their beta function βgα vanishes to all orders in the couplings of the theory, ˙gα=βgα, (D1) where the dot denotes a variation of the coupling with respect to the renormalization group scale. We will understand thecontributions to β gαas arising from corrections to scaling (i.e., conformal perturbation theory) of the zero-temperature two-point function /angbracketleftO α(z,¯z)Oα(0)/angbracketright∼z−1¯z−3, (D2) forz=x+iτ,¯z=x−iτcomputed with respect to the ﬁxed point action Sbin Eq. ( 4)[47]. First, observe that Oαhas unit spin, /Delta1R−/Delta1L, under the SO(2) =U(1) rotation symmetry of the Euclidean theory. When the action is perturbed, Sb→Sb+gα/integraltext Oα, the SO(2) symmetry is broken. We may view gαas a spurion—a “ﬁeld” that transforms oppositely to the operator it multiplies so thatthe product is an SO(2) singlet—of this broken rotationalsymmetry. This means that g αmay be understood to have spin−1. With this understanding, we may constrain the form ofβgα. The left-hand side of Eq. ( D1) is linear in gαand so the equality implies that βgαalso carries spin −1. Thus, we must determine what spin-1 combination of operatorscould possibly contribute to β gα[48]. Working in the scaling limit where all irrelevant operators are ignored allows usto disregard any contribution from high-dimension operatorswith negative spin. There are no marginal spin-( −1) operators because the lowest scaling dimension of a right-moving vertexoperator is equal to 3 /2. There do exist spin-( −1) relevant and spin-(−2) marginal operators, which are quadratic and quartic in the fermions of the left-moving sector along with marginalspin-0, i.e., dimension (1 ,1) operators, and spin-2 operators in addition to the marginal O αoperators. Perturbations by 125142-15PLAMADEALA, MULLIGAN, AND NAY AK PHYSICAL REVIEW B 93, 125142 (2016) spin-(−1) operators can be absorbed by a ﬁeld redeﬁnition of the left-moving fermion sector and so we ignore suchdeformations. A general contribution to the O αtwo-point function contains N−2spin-(−2) insertions, N0spin-0 insertions, N2 spin-2 insertions, and NOβOβinsertions. Note that we are collectively referring to all additional insertions of the Oβ operators as NOβ. In order for βOαto carry spin equal to −1, we require the number of insertions of various operators tosatisfy: 2N −2−NOβ−2N2=− 1. (D3) ThusNOβshould be odd. All operators in the left-moving sector can be built from products of the fermion operators and their spatial derivatives.Since the left-moving sector is describable in terms ofinteracting chiral fermions, fermion parity constrains anynonzero contribution to the O αtwo-point function to contain an even number of left-moving fermion operators: 4N−2+NOβ+2N2+2N0∈2Z. (D4) The ﬁrst contribution to the left-hand side of Eq. ( D4) assumes an operator quartic in the fermion operators. An operator thatis only quadratic with a single spatial derivative acting on one of the fermions might also contribute. However, this has noeffect on the conclusion that the parity of the left-hand sidemust be even. Equations ( D3) and ( D4) are not consistent with one another: the former requires N Oβto be odd, while the latter requires NOβto be even. The only resolution is that the Oα operators are exactly marginal in the scaling limit and so βgα=0. There is likewise no renormalization of the Luttinger liquid parameters of Sbdue to the spin-1 Oαoperators. Exact marginality of the dimension (3 /2,1/2) operators and the Luttinger parameters along the decoupled surface isa consequence of the chirality or spin-1 nature of the O α operators, which is ultimately due to the asymmetric nature of the left-moving and right-moving excitations in the asym-metric shorter Leech liquid underlying the hyperconductorstudied in this paper. The de-coupled renormalization groupequations described above should be contrasted with thoseof the Kosterlitz-Thouless transition that involve a dimension(1,1) vertex operator and the Luttinger parameter [ 57]. It is this difference that results in the logarithmic corrections toscaling in the expressions for the conductivities in the work ofGiamarchi on transport in a 1D Luttinger liquid [ 36]. 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PhysRevB.76.214415.pdf	Disorder and temperature dependence of the anomalous Hall effect in thin ferromagnetic ﬁlms: Microscopic model K. A. Muttalib * Department of Physics, University of Florida, P .O. Box 118440, Gainesville, Florida 32611-8440, USA P. Wölﬂe† ITKM, Universität Karlsruhe, D-76128 Karlsruhe, Germany and INT, Forschungzentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany /H20849Received 23 May 2007; published 19 December 2007 /H20850 We consider the anomalous Hall /H20849AH/H20850effect in thin disordered ferromagnetic ﬁlms. Using a microscopic model of electrons in a random potential of identical impurities including spin-orbit coupling, we develop ageneral formulation for strong, ﬁnite range impurity scattering. Explicit calculations are done within a shortrange but strong impurity scattering to obtain AH conductivities for both the skew scattering and side-jumpmechanisms. We also evaluate quantum corrections due to interactions and weak localization effects. We showthat for arbitrary strength of the impurity scattering, the electron-electron interaction correction to the AHconductivity vanishes exactly due to general symmetry reasons. On the other hand, we ﬁnd that our explicitevaluation of the weak localization corrections within the strong, short-range impurity scattering model canexplain the experimentally observed logarithmic temperature dependences in disordered ferromagnetic Feﬁlms. DOI: 10.1103/PhysRevB.76.214415 PACS number /H20849s/H20850: 75.50.Cc, 73.20.Fz, 72.10.Fk, 72.15.Rn I. INTRODUCTION It has been recognized since the 1950s /H20849Ref. 1/H20850that a Hall effect can exist in ferromagnetic metals even in the absenceof an external magnetic ﬁeld, hence the name anomalousHall effect /H20849AHE /H20850. There are several different mechanisms that might be responsible for the AHE observed in thin fer-romagnetic ﬁlms, namely, the skew scattering 2and side-jump mechanisms3as well as Berry phase contributions.4All such mechanisms depend on the spin-orbit interaction induced bythe impurities and on the spontaneous magnetization in aferromagnet which breaks the time reversal invariance andtherefore gives rise to the AHE. For a disordered ferromag-netic ﬁlm, AH conductivity due to the skew scattering andside-jump mechanisms have been theoretically consideredusing a variety of methods within weak, short-range impurityscattering. 5–9However, a systematic calculation, starting from a microscopic Hamiltonian, of the longitudinal as wellas the AH conductivities for different mechanisms for strongimpurity scattering has been lacking. Recently, the effects ofstrong, short-range impurity scattering on the longitudinaland Hall conductivities were considered for skew scatteringas well as side-jump mechanisms, 10but quantum corrections, namely, electron-electron /H20849e-e/H20850interaction corrections11or weak localization /H20849WL /H20850effects,12were not included. Earlier experiments13have shown logarithmic tempera- ture dependences of the longitudinal as well as Hall resis-tances highlighting the importance of such quantum correc-tions. However, the results were consistent with, and wereinterpreted as, vanishing interaction contributions to the AHconductivity, obtained theoretically within a weak impurity scattering model 9and the absence of any weak localization effects. Recent experiments, on the other hand, clearly showa nonvanishing contribution to the total quantum correctionto the AH conductivity, 14which can arise in principle eitherfrom an interaction correction due to strong impurity scatter- ing or from a weak localization effect, or from a combinationof both. It has been commonly believed that weak localiza-tion effects in ferromagnetic ﬁlms would be cut off by the presence of large internal magnetic ﬁeld among others,which suggests that the interaction corrections to the AHconductivity need to be revisited for strong impurity scatter-ing as a source of difference between the two experiments. In this paper, we systematically develop a general formu- lation for the AHE for strong, ﬁnite range impurity scatter- ings starting from a microscopic model of electrons in a ran-dom potential of impurities including spin-orbit coupling.This generalizes an earlier work 6which considered weak, short-range impurity scattering only and did not includequantum corrections. We show on very general symmetrygrounds that quantum correction to the AH conductivity dueto/H20849e-e/H20850interaction effects vanish exactly, which shows that the previous weak scattering results 9remain valid for arbi- trary strengths of the impurity scattering. This forces us toconsider the weak localization effects 8as the only remaining source of the logarithmic temperature dependence in theabove experiments despite the presence of large internalmagnetic ﬁelds and spin-orbit scatterings in these ferromag-netic ﬁlms. As we show below, the temperature independentcutoff of the weak localization effects in strongly disorderedsystems can be ineffective at higher temperatures if a tem-perature dependent contribution dominates the phase relax-ation rate. It turns out that while the contribution from thee-e interaction to the phase relaxation rate is indeed too smallfor WL effects to be observed, a much larger contribution isobtained from scattering off spin waves, 15which should al- low the observation of the WL effects within a reasonabletemperature range. We ﬁnd that the effects of strong impurityscatterings on the WL effects can be evaluated to obtain avery simple result, namely, that the ratio of the WL correc-PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 1098-0121/2007/76 /H2084921/H20850/214415 /H2084915/H20850 ©2007 The American Physical Society 214415-1tions to the AH to the longitudinal conductivity can be writ- ten simply in terms of the eigenvalues of the impurity aver-aged particle-hole scattering amplitude for zero momentumtransfer. This result, taken together with contributions to theAH conductivity from both the skew scattering and side-jump mechanisms calculated within the same microscopicmodel, can explain both the earlier as well as the recentexperiments on the disorder and temperature dependences ofthe AH conductivities of ultrathin Fe ﬁlms 14mentioned above. This last result has been reported without details incombination with the recent experiment in a short letter. 14 The paper is organized in the following way. A micro- scopic model Hamiltonian is introduced in Sec. II and a gen-eral formulation in two dimensions for strong, ﬁnite rangeimpurity scatterings is developed in Sec. III. Section IV re-views the results on the conductivity tensor in the absence ofinteractions. In Secs. V and VI, we consider the e-e interac-tion corrections and the weak localization corrections, re-spectively, to both longitudinal and AH conductivities withinthe general strong, ﬁnite range impurity scattering formula-tion. We then consider the special case of a short range, butstill strong, impurity scattering model in Sec. VII. In Sec.VIII, we collect all the results and compare them with recentexperiments. Section IX summarizes the paper. For the sakeof completeness, we include models of small and large anglescatterings in the Appendix. II. HAMILTONIAN The single particle Hamiltonian of a conduction electron in a ferromagnetic disordered metal, including spin-orbit in-teraction induced by the disorder potential V dis/H20849r/H20850, is given in its simplest form by /H20849throughout the paper, we use units with /H6036=kB=1/H20850 H1=/H20875−/H116122 2m+Vdis/H20849r/H20850/H20876/H9254/H9268/H9268/H11032−M/H9270/H9268/H9268/H11032z −i/H9261c2 /H208494/H9266/H208502/H20851/H9270/H9268/H9268/H11032·/H20849/H11612Vdis/H11003/H11612/H20850/H20852, /H208492.1/H20850 where/H9261c=2/H9266 mcis the Compton wavelength of the electron and Mis the Zeeman energy splitting caused by the ferromag- netic polarization. Here, H1i sa2/H110032 matrix in spin space with/H9268,/H9268/H11032=↑,↓being spin indices and /H9270is the vector of Pauli matrices. The above model is only a crude approxima-tion of the band structure of Fe, which has been determinedby several authors /H20849see, e.g., Ref. 16/H20850. We model the energy band crossing the Fermi surface by a single isotropic band.As will be discussed below, the quantum corrections to theconductivity exhibit certain qualitative features, which donot depend sensitively on the details of the band structure.The disordered potential in Eq. /H208492.1/H20850will be modeled as randomly placed identical impurities, V dis/H20849r/H20850=/H20858jV/H20849r−Rj/H20850. We will later average over the impurity positions Rj. The matrix elements of H1in the plane wave /H20849or Bloch state /H20850representation are given by/H20855k/H11032/H9268/H11032/H20841H1/H20841k/H9268/H20856=/H20885d2re−ik/H11032·rH1e−ik·r =/H20873k2 2m−M/H9268/H20874/H9254kk/H11032/H9254/H9268/H9268/H11032+/H20858 jV/H20849k−k/H11032/H20850 /H11003ei/H20849k−k/H11032/H20850·Rj+Vso/H20849k/H11032/H9268/H11032;k/H9268/H20850, /H208492.2/H20850 where V/H20849k−k/H11032/H20850is the Fourier transform of the single impu- rity potential and the spin-orbit interaction part is given by Vso/H20849k/H11032/H9268/H11032;k/H9268/H20850 =−i/H9261c2 /H208494/H9266/H208502/H20858 jV/H20849k−k/H11032/H20850e/H20851i/H20849k−k/H11032/H20850·Rj/H20852/H9270/H9268/H9268/H11032·/H20849k/H11003k/H11032/H20850. /H208492.3/H20850 Here, we have used −i/H20885d2rexp/H20849−ik/H11032·r/H20850/H20849/H11612Vdis/H11003/H11612/H20850exp/H20849−ik·r/H20850 =−i/H20885d2r/H20885d2q /H208492/H9266/H208502ei/H20849k−k/H11032−q/H20850·r/H20849−iq/H20850V/H20849q/H20850/H11003/H20849ik/H20850 =−iV/H20849k−k/H11032/H20850/H20849k/H11003k/H11032/H20850. /H208492.4/H20850 The many-body Hamiltonian is given in terms of electron creation and annihilation operators ck/H9268+,ck/H9268as H=/H20858 k/H9268/H20849/H9255k−M/H9268/H20850ck/H9268+ck/H9268+/H20858 k/H9268,k/H11032/H9268/H11032/H20858 jV/H20849k−k/H11032/H20850 /H11003ei/H20849k−k/H11032/H20850·Rj/H20853/H9254/H9268/H9268/H11032−ig¯so/H9270/H9268/H9268/H11032·/H20849kˆ/H11003kˆ/H11032/H20850/H20854ck/H11032/H9268/H11032+ck/H9268, /H208492.5/H20850 where we have deﬁned a dimensionless spin-orbit coupling constant g¯so/H11013/H9261c2kF2 /H208494/H9266/H208502,kˆ/H11013k//H20841k/H20841. Note: An estimate of the spin- orbit coupling constant g¯so, using a typical Fermi wave num- berkF, shows that it is rather small, of order 10−4. However, in transition metal compounds, the coupling is substantiallyenhanced by interband mixing effects, 3so that the renorm- alized coupling constant gsois of order unity: gso /H11011csoEso//H9004Ed, where Eso/H110110.1 eV is a measure for the atom- ic spin-orbit energy, /H9004Ed/H110110.5 eV is a typical energy split- ting of dbands, and the constant cso/H110115. In the following, we will replace g¯soby the phenomenological spin-dependent pa- rameter g/H9268. III. IMPURITY SCATTERING: GENERAL FORMULATION In this section, we will develop a general formulation for strong, ﬁnite range impurity scattering in two dimensions using standard ﬁeld theory techniques at ﬁnite temperature.17 For simplicity, we will need to make approximations forshort-range impurity scattering later. However, keeping theformulation general as long as possible will allow us, e.g., tocheck if the anisotropic scattering can have a large impact onour ﬁnal results. The repeated scattering of an electron off a single impu- rity may be described symbolically in terms of the scatteringamplitude f k/H9268,k/H11032/H9268/H11032asK. A. MUTTALIB AND P. WÖLFLE PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-2f=V+VGV +VGVGV +¯, /H208493.1/H20850 where Gis the single particle Green’s function Gk/H9268/H20849i/H9275n/H20850=/H20851i/H9275n−/H9255k/H9268−/H9018k/H9268/H20849/H9275n/H20850/H20852−1, /H208493.2/H20850 with the single particle self-energy /H9018k/H9268/H20849i/H9275n/H20850. Here,/H9275n =/H9266T/H208492n+1/H20850is the fermion Matsubara frequency with Tbe- ing the temperature and N/H9268is the density of states at the Fermi level of spin species /H9268./H20849We use units of temperature such that Boltzmann’s constant is equal to unity. /H20850Vis the bare interaction with one impurity at R=0and includes the spin-orbit scattering Vk,k/H11032;/H9268=V/H20849k−k/H11032/H20850/H208511−ig/H9268/H9270/H9268/H9268z/H20849kˆ/H11003kˆ/H11032/H20850/H20852, /H208493.3/H20850 where we have used the fact that Vis diagonal in spin space. In the case of ﬁnite range, or even long-range correlatedscattering potentials, we may still use the model of indi-vidual impurities or scattering centers, but now of ﬁnite spa-tial extension. This is reasonable as long as the scatteringcenters do not overlap too much. If they overlap, a morestatistical description in terms of correlators of the impuritypotential should be used. Within our model, the nonlocalcharacter of scattering is described in terms of the momen-tum dependence of the Fourier transform of the potential of asingle impurity /H20849assuming only one type of impurity /H20850V/H20849k −k /H11032/H20850, which for an isotropic system depends only on the angle/H9258between kandk/H11032,V=V/H20849/H9258/H20850=V/H20849−/H9258/H20850. In two dimen- sions, we may expand Vin terms of eigenfunctions /H9273m/H20849kˆ/H20850 =eim/H9278, where/H9278is the polar angle of vector k,kˆ=k//H20841k/H20841. Add- ing the skew scattering potential, we may write Vk,k/H11032/H9268=/H20858 mVm/H9268/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850, /H208493.4/H20850 where Vm/H9268is a sum of the normal and skew scattering parts Vm/H9268=Vmns+Vm/H9268ss. /H208493.5/H20850 Time reversal invariance and rotation symmetry in the case of potential scattering impliy V−mns=/H20849Vmns/H20850=Vmns. /H208493.6/H20850 Equation /H208493.3/H20850then yields Vm/H9268ss=1 2g/H9268/H9270/H9268/H9268z/H20849Vm−1ns−Vm+1ns/H20850. /H208493.7/H20850 A. Scattering amplitude For Vdiagonal in spin space, the scattering amplitude fk/H9268,k/H11032/H9268/H11032=/H9254/H9268,/H9268/H11032fk,k/H11032/H9268obeys the integral equation fk,k/H11032/H9268s=Vk,k/H11032/H9268+/H20858 k1Gk1/H9268/H20849i/H9275n/H20850Vk,k1/H9268fk1,k/H11032/H9268s =Vk,k/H11032/H9268−is/H9266N/H9268/H20855Vk,k1/H9268fk1,k/H11032/H9268s/H20856k1, /H208493.8/H20850 where s/H11013sign /H20849/H9275n/H20850and /H20855¯/H20856k1denotes averaging over the direction of wave vector k1. Deﬁning the dimensionless po- tential V¯m/H9268/H11013/H9266N/H9268Vm/H9268and the dimensionless scatteringamplitude f¯k/H9268,k/H11032/H9268/H11032/H11013/H9266N/H9268fk/H9268,k/H11032/H9268/H11032and expanding f¯k/H9268,k/H11032/H9268 =/H20858mf¯m/H9268/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850,w eﬁ n d f¯ m/H9268s=V¯m/H9268 1+isV¯m/H9268. /H208493.9/H20850 For notational simplicity, we will always use a bar on a symbol to represent the corresponding dimensionless quan-tity. B. Single particle relaxation rate The single particle relaxation rate /H9270/H9268is given by the imaginary part of the self-energy, 1 2/H9270/H9268/H11013−sIm/H9018k/H9268/H20849i/H9275n/H20850=−snimpIm/H20849fk/H9268,k/H9268s/H20850=nimp /H9266N/H9268/H9253/H9268, /H208493.10 /H20850 where/H9253/H9268is a dimensionless parameter characterizing the scattering strength, /H9253/H9268/H11013−s/H20858mIm/H20849f¯m/H9268/H20850=/H20858mV¯ m/H92682 1+V¯m/H92682, and N/H9268is the density of states at the Fermi energy of spin species /H9268. Note that V¯m/H9268are all real. C. Particle-hole propagator The particle-hole propagator /H9003kk/H11032/H20849q;i/H9280n,i/H9280n−i/H9024m/H20850is an important ingredient of vertex corrections of any kind. Here, k+q/2,k−q/2 are the initial momenta, k/H11032+q/2,k/H11032−q/2 the ﬁnal momenta, and /H9280n,/H9280n−/H9024mare the Matsubara frequencies of the particle and the hole lines, respectively. In terms of theparticle-hole scattering amplitude t k,k/H11032/H20849q;i/H9280n,i/H9024m/H20850,/H9003satisﬁes the following Bethe-Salpeter equation /H20851we have deﬁned di- mensionless quantities /H9003¯,t¯by multiplying both with a factor /H208492/H9266N/H9268/H9270/H9268/H20850/H20852: /H9003¯kk/H11032/H20849q;i/H9280n,i/H9024m/H20850=t¯kk/H11032/H20849q;i/H9280n,i/H9024m/H20850+/H208492/H9266N/H9268/H9270/H9268/H20850−1 /H11003/H20858 k1t¯kk1/H20849q;i/H9280n,i/H9024m/H20850 /H11003Gk1+q/2,/H9268/H20849i/H9280n/H20850Gk1−q/2,/H9268/H20849i/H9280n−i/H9024m/H20850 /H11003/H9003¯k1k/H11032/H20849q;i/H9280n,i/H9024m/H20850. /H208493.11 /H20850 The /H20849dimensionless /H20850impurity averaged particle-hole scatter- ing amplitude t¯/H20849we consider only the case of equal spin of particle and hole /H20850is given in terms of the /H20849dimensionless /H20850 scattering amplitudes f¯by the equation t¯kk/H11032ss/H11032/H20849q;i/H9280n,i/H9024m/H20850=2/H9270/H9268nimp /H9266N/H9268f¯ k+q/2,/H9268;k/H11032+q/2/H9268s/H20849i/H9280n/H20850 /H11003f¯ k/H11032−q/2,/H9268;k−q/2,/H9268s/H11032/H20849i/H9280n−i/H9024m/H20850./H208493.12 /H20850 We will later need the limit of small q,q/H11270kF, of this expres- sion, t¯kk/H11032ss/H11032/H20849q;i/H9280n,i/H9024m/H20850=t¯kk/H11032ss/H11032/H20849q=0/H20850+/H9004t¯kk/H11032ss/H11032/H20849q/H20850. /H208493.13 /H20850 It is useful to represent the operator t¯kk/H11032/H20849q=0/H20850in terms of its eigenvalues /H9261m. Assuming isotropic band structure, theDISORDER AND TEMPERATURE DEPENDENCE OF THE … PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-3eigenfunctions /H9273m/H20849kˆ/H20850=exp /H20849im/H9272/H20850are those of the angular mo- mentum operator component Lz. The eigenvalue equation is /H20855t¯kk/H11032/H20849q=0/H20850/H9273m/H20849kˆ/H11032/H20850/H20856k/H11032=/H9261m/H9273m/H20849kˆ/H20850. /H208493.14 /H20850 The operator t¯kk/H11032+−/H20849q=0/H20850may be represented as t¯kk/H11032+−/H20849q=0/H20850=/H20858 m/H9261m/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850, t¯kk/H11032−+/H20849q=0/H20850=/H20851t¯k/H11032k+−/H20849q=0/H20850/H20852. /H208493.15 /H20850 In general, using the deﬁnitions tk,k/H11032/H9268ss/H11032=nimp /H20849/H9266N/H9268/H208502f¯ k,k/H11032/H9268sf¯ k/H11032,k/H9268s/H11032=/H208492/H9266N/H9268/H9270/H9268/H20850−1t¯k,k/H11032/H9268ss/H11032, t¯k,k/H11032/H9268ss/H11032=/H20858 mt¯m/H9268ss/H11032/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850, /H208493.16 /H20850 we have t¯m/H9268ss/H11032=/H9253/H9268−1/H20858 m/H11032f¯ m/H11032/H9268sf¯ m/H11032−m,/H9268s/H11032. /H208493.17 /H20850 We will consider /H9004t¯kk/H11032/H20849q/H20850for the special case of strong short- range impurity scatterings in Sec. VII A. The energy integral over the product of Green’s functions in the integral equation for /H9003kk/H11032may be done ﬁrst, after ex- panding the G’s in/H9024mandq, /H20885d/H92551Gk1+q/2,/H9268/H20849i/H9280n/H20850Gk1−q/2,/H9268/H20849i/H9280n−i/H9024m/H20850 =2/H9266/H9270/H208511+i/H9270/H20849i/H9024m−q·vk1/H20850−/H92702/H20849q·vk1/H208502/H20852,/H208493.18 /H20850 with/H9280n/H110220 and/H9280n−/H9024m/H110210, where q·vk=qvF/H20849qˆ·kˆ/H20850. Expand- ing/H9003¯kk/H11032and t¯kk/H11032in terms of eigenfunctions /H9273m/H20849kˆ/H20850,/H9003¯kk/H11032 =/H20858m/H9003¯mm/H11032/H9273m/H20849kˆ/H20850/H9273m/H11032/H20849kˆ/H11032/H20850and using t˜m/H9268+−/H11013/H9261m, one obtains /H20849s/H11032=−s/H20850 /H9003¯ mm/H11032ss/H11032=/H9261m/H9254mm/H11032+/H9261m/H20877/H208511−/H9270/H20849/H20841/H9024n/H20841+D0q2/H20850/H20852/H9003¯ mm/H11032ss/H11032 −i 2vFq/H9270s/H20851/H9003¯ m−1,m/H11032ss/H11032/H92731/H20849qˆ/H20850+/H9003¯ m+1,m/H11032ss/H11032/H92731/H20849qˆ/H20850/H20852 −1 4/H20849vFq/H9270/H208502/H20851/H9003¯ m−2,m/H11032ss/H11032/H92732/H20849qˆ/H20850+/H9003¯ m+2,m/H11032ss/H11032/H92732/H20849qˆ/H20850/H20852/H20878. /H208493.19 /H20850 Form=m/H11032/HS110050, the solution is /H9003¯mm=/H9261m 1−/H9261m+O/H20849q/H20850/H11013/H9261˜m+O/H20849q/H20850, /H208493.20 /H20850 where we have deﬁned /H9261˜m/H11013/H9261m//H208491−/H9261m/H20850. The/H9261m/H20849and there- fore/H9261˜m/H20850are complex valued and depend on the spin projec- tion/H9268. Using conventional notation, we will denote the real and imaginary parts of /H9261mby/H9261m/H11032and/H9261m/H11033, respectively, andsimilarly the real and imaginary parts of /H9261˜mby/H9261˜ m/H11032and/H9261˜ m/H11033, respectively. The case m=0 needs special consideration because par- ticle number conservation causes /H9003¯00to have a pole in the limit/H9024n,q→0, here expressed by /H92610=1. Solving the above equation for /H9003¯00in lowest order in q, one ﬁnds /H9003¯00=1//H9270 /H20841/H9024m/H20841+Dq2, /H208493.21 /H20850 where the renormalized diffusion constant is deﬁned as D=D0/H208491+/H9261˜ 1/H11032/H20850,D0=1 2vF2/H9270, /H9261˜ 1/H11032/H11013Re/H9261˜1=1 2/H20849/H9261˜1+/H9261˜−1/H20850. /H208493.22 /H20850 This is found by solving the following equations for small vFq/H9270/H20849s/H11032=−s/H20850: /H9003¯ 00ss/H11032=1+ /H208511−/H9270/H20849/H20841/H9024m/H20841+D0q2/H20850/H20852/H9003¯ 00ss/H11032 −i 2vFq/H9270s/H20851/H9003˜ −1,0ss/H11032/H92731/H20849qˆ/H20850+/H9003¯ 1,0ss/H11032/H92731/H20849qˆ/H20850/H20852, /H9003¯ −1,0ss/H11032=/H9261−1/H20875/H9003¯ −1,0ss/H11032−i 2vFq/H9270s/H9003¯ 0,0ss/H11032/H92731/H20849qˆ/H20850/H20876, /H9003¯ 1,0ss/H11032=/H92611/H20875/H9003¯ 1,0ss/H11032−i 2vFq/H9270s/H9003¯ 0,0ss/H11032/H9273−1/H20849qˆ/H20850/H20876. /H208493.23 /H20850 Substituting /H9003¯ ±1,0ss/H11032into the equation for /H9003¯ 00ss/H11032, one ﬁnds /H9003¯ 00ss/H11032/H20877/H20841/H9024m/H20841+D0q2/H208751+1 2/H20849/H9261˜1+/H9261˜−1/H20850/H20876/H20878=1 /H9270. /H208493.24 /H20850 The leading singular dependence on kˆ/H11032is obtained from /H9003¯ 0,±1ss/H11032=/H208511−/H9270/H20849/H20841/H9024m/H20841+D0q2/H20850/H20852/H9003¯ 0,±1ss/H11032 −i 2vFq/H9270s/H20851/H9003¯ −1,±1ss/H11032/H92731/H20849qˆ/H20850+/H9003¯ 1,±1ss/H11032/H92731/H20849qˆ/H20850/H20852, /H9003¯ −1,1ss/H11032=−i 2/H9261˜−1vFq/H9270s/H9003¯ 0,1ss/H11032/H92731/H20849qˆ/H20850, /H9003¯ 1,1ss/H11032=/H9261˜1−i 2/H9261˜1vFq/H9270s/H9003¯ 0,1ss/H11032/H92731/H20849qˆ/H20850. /H208493.25 /H20850 The complete particle-hole propagator in the regime vFq/H9270/H110211 is given by /H9003¯kk/H11032=1 /H9270/H9253k/H9253˜k/H11032 /H20841/H9024m/H20841+Dq2+/H20858 m/HS110050/H9261˜m/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850, /H208493.26 /H20850 withK. A. MUTTALIB AND P. WÖLFLE PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-4/H9253k=1−i 2vFq/H9270s/H20858 m=±1/H9261˜m/H9273m/H20849kˆ/H20850/H9273m/H20849qˆ/H20850 =1−i 2vF/H9270s/H20858 m=±1/H9261˜m/H9273m/H20849kˆ/H20850q−m /H208493.27 /H20850 and /H9253˜k=1−i 2vF/H9270s/H20858 m=±1/H9261˜m/H9273m/H20849kˆ/H20850qm. /H208493.28 /H20850 The vertex corrections of the density Tkand current vertices jk/H9251andj˜k/H9251/H20849for the incoming and outgoing current /H20850are ob- tained by Tk/H20849q/H20850/H110131+ /H20855/H9003¯kk/H11032/H20856k/H11032=1+1//H9270 /H20841/H9024m/H20841+Dq2/H9253k /H208493.29 /H20850 and jk/H9251/H20849q/H20850=vk/H9251+/H20855vk/H11032/H9251/H9003¯k/H11032k/H20856k/H11032 =vk/H9251+/H20858 m=±1/H9261˜m/H9273m/H20849kˆ/H20850/H20855vk/H11032/H9251/H9273m/H20849kˆ/H11032/H20850/H20856k/H11032 +/H20855vk/H11032/H9251/H9253k/H11032/H20856k/H110321//H9270 /H20841/H9024m/H20841+Dq2/H9253˜k, j˜k/H9251/H20849q/H20850=vk/H9251+/H20855vk/H11032/H9251/H9003¯kk/H11032/H20856k/H11032 =vk/H9251+/H20858 m=±1/H9261˜m/H9273m/H20849kˆ/H20850/H20855vk/H11032/H9251/H9273m/H20849kˆ/H11032/H20850/H20856k/H11032 +/H20855vk/H11032/H9251/H9253˜k/H20856k/H110321//H9270 /H20841/H9024m/H20841+Dq2/H9253k. /H208493.30 /H20850 Note that j˜k/H9251/HS11005/H20849jk/H9251/H20850, as the eigenvalues /H9261˜mare in general complex valued. Using /H9273−1/H20849kˆ/H20850/H92731/H20849kˆ/H11032/H20850+/H92731/H20849kˆ/H20850/H9273−1/H20849kˆ/H11032/H20850=2/H20849kˆ·kˆ/H11032/H20850, /H9273−1/H20849kˆ/H20850/H92731/H20849kˆ/H11032/H20850−/H92731/H20849kˆ/H20850/H9273−1/H20849kˆ/H11032/H20850=2i/H20849kˆ/H11003kˆ/H11032/H20850/H20849 3.31 /H20850 and /H20855kˆ /H9251/H11032/H20849kˆ·kˆ/H11032/H20850/H20856=1 2kˆ/H9251,/H20855kˆ /H9251/H11032/H20849kˆ/H11003kˆ/H11032/H20850/H20856=−1 2/H20849eˆ/H9251/H11003kˆ/H20850 /H208493.32 /H20850 and deﬁning jk/H9251/H11013jk/H9251/H20849q=0/H20850,j˜k/H9251/H11013j˜k/H9251/H20849q=0/H20850, we have jk/H9251=vF/H20851/H208491+/H9261˜ 1/H11032/H20850kˆ/H9251+/H9261˜ 1/H11033/H20849eˆ/H9251/H11003kˆ/H20850z/H20852, j˜k/H9251=vF/H20851/H208491+/H9261˜ 1/H11032/H20850kˆ/H9251−/H9261˜ 1/H11033/H20849eˆ/H9251/H11003kˆ/H20850z/H20852. /H208493.33 /H20850 More explicitly, for /H9251=x,y, the incoming and outgoing cur- rent vertices jandj˜have the forms jkx=vF/H20851/H208491+/H9261˜ 1/H11032/H20850kˆx+/H9261˜ 1/H11033kˆy/H20852=1 2vF/H20851/H208491+/H9261˜ 1/H20850kˆ++/H208491+/H9261˜1/H20850kˆ−/H20852,jky=vF/H20851/H208491+/H9261˜ 1/H11032/H20850kˆy−/H9261˜ 1/H11033kˆx/H20852=−i1 2vF/H20851/H208491+/H9261˜ 1/H20850kˆ+−/H208491+/H9261˜1/H20850kˆ−/H20852, j˜kx=vF/H20851/H208491+/H9261˜ 1/H11032/H20850kˆx−/H9261˜ 1/H11033kˆy/H20852=1 2vF/H20851/H208491+/H9261˜1/H20850kˆ++/H208491+/H9261˜ 1/H20850kˆ−/H20852, j˜ky=vF/H20851/H208491+/H9261˜ 1/H11032/H20850kˆy+/H9261˜ 1/H11033kˆx/H20852=−i1 2vF/H20851/H208491+/H9261˜1/H20850kˆ+−/H208491+/H9261˜ 1/H20850kˆ−/H20852, /H208493.34 /H20850 where we have deﬁned k±=kx±iky. D. Particle-particle propagator The integral equation for the particle-particle propagator or Cooperon reads /H20849again multiplying the Cooperon Cand the particle-particle scattering amplitude tpby the factor 2/H9266N/H9268/H9270/H9268to deﬁne dimensionless Cooperon C¯and dimension- less particle-particle scattering amplitude tp/H20850 C¯kk/H11032/H20849Q;i/H9280n,i/H9024m/H20850=t¯kk/H11032p/H20849Q;i/H9280n,i/H9024m/H20850 +/H208492/H9266N/H9268/H9270/H9268/H20850−1/H20858 k1t¯kk1p/H20849Q;i/H9280n,i/H9024m/H20850Gk1,/H9268/H20849i/H9280n/H20850 /H11003GQ−k1,/H9268/H20849i/H9280n−i/H9024m/H20850C¯k1k/H11032/H20849q;i/H9280n,i/H9024m/H20850, /H208493.35 /H20850 tk,k/H11032/H9268p,ss/H11032=nimp /H20849/H9266N/H9268/H208502f¯ k,k/H11032/H9268sf¯ −k,−k/H11032,/H9268s/H11032=/H208492/H9266N/H9268/H9270/H9268/H20850−1/H9253/H9268−1f¯ k,k/H11032/H9268sf¯ −k,−k/H11032,/H9268s/H11032, /H208493.36 /H20850 t¯k,k/H11032/H9268p,ss/H11032=2/H9266N/H9268/H9270/H9268tk,k/H11032/H9268p,ss/H11032=/H20858 mt¯m/H9268p,ss/H11032/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850,/H208493.37 /H20850 t¯m/H9268p,ss/H11032=/H9253/H9268−1/H20858 m/H11032f¯ m/H11032/H9268sf¯ m−m/H11032,/H9268s/H11032. /H208493.38 /H20850 If rotation invariance or time reversal invariance is broken, t¯0/H9268p,ss/H11032=/H9253/H9268−1/H9253/H9268p/HS110051, where/H9253/H9268p=/H20858m/H11032f¯ m/H11032/H9268sf¯ −m/H11032,/H9268s/H11032. The energy integral over the product of Green’s functions in the integral equation for Ckk/H11032may be done ﬁrst, after ex- panding the G’s in/H9024mandQ, /H20885d/H92551Gk1,/H9268/H20849i/H9280n/H20850GQ−k1,/H9268/H20849i/H9280n−i/H9024m/H20850 =2/H9266/H9270/H208511+i/H9270/H20849i/H9024m−Q·vk1/H20850−/H92702/H20849Q·vk1/H208502/H20852, /H208493.39 /H20850 with/H9280n/H110220,/H9280n−/H9024m/H110210, where Q·vk=QvF/H20849Qˆ·kˆ/H20850. Expanding C¯kk/H11032and t¯kk/H11032pin terms of eigenfunctions /H9273m/H20849kˆ/H20850,C¯kk/H11032 =/H20858mC¯mm/H11032/H9273m/H20849kˆ/H20850/H9273m/H11032/H20849kˆ/H11032/H20850and denoting t˜m/H9268p,+−=/H9261mp, one obtains /H20849s/H11032=−s/H20850DISORDER AND TEMPERATURE DEPENDENCE OF THE … PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-5C¯mm/H11032=/H9261mp/H20877/H9254mm/H11032+/H208511−/H9270/H20849/H20841/H9024n/H20841+D0Q2/H20850/H20852C¯mm/H11032 −i 2vFQ/H9270/H20851C¯m−1,m/H11032/H92731/H20849Qˆ/H20850+C¯m+1,m/H11032/H92731/H20849Qˆ/H20850/H20852 −1 4/H20849vFQ/H9270/H208502/H20851C¯m−2,m/H11032/H92732/H20849Qˆ/H20850+C¯m+2,m/H11032/H92732/H20849Qˆ/H20850/H20852/H20878. /H208493.40 /H20850 The m=m/H11032=0 component of C¯mm/H11032obeys the equation /H20851/H20849/H9270/H9272so/H20850−1+/H20841/H9024n/H20841+D0Q2/H20852C˜00 =/H9270−1−i 2vFQ/H20851C¯−1,0/H92731/H20849Qˆ/H20850+C¯1,0/H92731/H20849Qˆ/H20850/H20852+O/H20849Q2/H20850, /H208493.41 /H20850 where /H20849/H9270/H9272so/H20850−1is the phase relaxation rate contributed by spin- orbit interaction processes, /H20849/H9270/H9272so/H20850−1=/H9270−1/H20851/H20849/H92610p/H20850−1−1/H20852. /H208493.42 /H20850 Using C¯±1,0=/H9261±1p/H20875C¯±1,0−i 2vFQ/H9270C¯0,0/H9273±1/H20849qˆ/H20850/H20876, /H208493.43 /H20850 the Cooperon is found as C¯kk/H11032=1 /H9270/H9253kp/H9253˜k/H11032p /H20841/H9024m/H20841+DpQ2+/H9270/H9272−1+/H20858 m/HS110050/H9261˜ mp/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850, /H9261˜ mp=/H9261mp 1−/H9261mp, /H208493.44 /H20850 with /H9253kp=1−i 2vFQ/H9270/H20858 m=±1/H9261˜ mp/H9273m/H20849kˆ/H20850/H9273m/H20849Qˆ/H20850 =1− i/H9270/H20858 m=±1/H9261˜ mp/H9273m/H20849kˆ/H20850/H20855Q·vk/H11032/H9273m/H20849kˆ/H11032/H20850/H20856 /H20849 3.45 /H20850 and /H9253˜kp=1− i/H9270s/H20858 m=±1/H9261˜ mp/H9273m/H20849kˆ/H20850/H20855q·vk/H11032/H9273m/H20849kˆ/H11032/H20850/H20856. /H208493.46 /H20850 Here, the diffusion coefﬁcient Dpis in general different from the one in the p-h channel, Dp=D0/H208751+1 2/H20849/H9261˜ 1p+/H9261˜ −1p/H20850/H20876/HS11005D, /H208493.47 /H20850 the difference being proportional to the spin-orbit coupling g/H9268. IV . CONDUCTIVITY TENSOR IN THE ABSENCE OF INTERACTION As mentioned before, there are three mechanisms contrib- uting to the anomalous Hall conductivity, namely, the skewscattering, the side-jump, and the Berry phase mechanisms. In this section, we will write down the generic formulationsfor evaluating these contributions within the diagrammaticperturbation theory. The contribution to the conductivity /H9268/H9251/H9252 will be given in terms of a correlation function L/H9251/H9252, deﬁned as17 /H9268/H9251/H9252=e2/H20858 /H9024→0lim1 i/H9024mL/H9251/H9252, /H208494.1/H20850 where L/H9251/H9252=/H20858nL/H9251/H9252dnis a sum of the different relevant diagrams dn. We will take the current to be along the xdirection, so the longitudinal conductivity will correspond to /H9251=/H9252=x, while the /H20849anomalous /H20850Hall conductivity will be given by the off- diagonal part /H9251=x,/H9252=y. Note that /H9268yx=−/H9268xy. A. Skew scattering contribution The skew scattering contribution to the conductivity ten- sor/H9268/H9251/H9252in lowest order in 1 //H9255F/H9270is given by the bubble diagram dressed by vertex corrections given by the correla-tion function L /H9251/H9252=T/H20858 /H9280n/H20858 k,/H9268Gk/H9268/H20849i/H9280n/H20850Gk/H9268/H20849i/H9280n−i/H9024m/H20850vk/H9251j˜k/H9252/H9268. /H208494.2/H20850 The energy integration over GGis nonzero only if the poles are on opposite sides of the real axis, requiring 0 /H33355/H9280n/H33355/H9024 m /H20849we assume /H9024m/H110220/H20850, and yields 2 /H9266N/H9268/H9270/H9268, and the summation on/H9280ngives/H9024m//H208492/H9266T/H20850. Substituting j˜k/H9252/H9268from Eq. /H208493.34 /H20850into the Kubo formula, the conductivity tensor follows as /H9268/H9251/H9252ss=/H20858 /H92681 2vF2/H9270/H9268N/H9268/H208731+/H9261˜ 1/H11032/H9261˜ 1/H11033 −/H9261˜ 1/H110331+/H9261˜ 1/H11032/H20874. /H208494.3/H20850 Deﬁning the tensor of diffusion coefﬁcients D/H9251/H9252/H9268as D/H9251/H9251/H9268=1 2vF2/H9270/H9268tr, Dxy/H9268=D/H9251/H9251/H9268/H20851/H9261˜ 1/H11033//H208491+/H9261˜ 1/H11032/H20850/H20852=−Dyx/H9268, /H208494.4/H20850 where /H9270/H9268tr/H11013/H9270/H9268/H208491+/H9261˜ 1/H11032/H20850/H20849 4.5/H20850 is the momentum relaxation time, we may write /H9268/H9251/H9252ss=/H20858 /H9268N/H9268D/H9251/H9252/H9268. /H208494.6/H20850 From the deﬁnition /H9261˜m=/H9261m//H208491−/H9261m/H20850, we obtain the following identities: 1+/H9261˜1=1 1−/H92611,1 +/H9261˜ 1/H11032=1−/H92611/H11032 /H208411−/H92611/H208412,/H9261˜ 1/H11033=/H92611/H11033 /H208411−/H92611/H208412. /H208494.7/H20850 B. Side-jump contribution The side-jump contribution has been ﬁrst calculated by Berger.3It arises because the trajectory of a wave packetK. A. MUTTALIB AND P. WÖLFLE PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-6scattered by an impurity is shifted sidewise due to the spin- orbit interaction /H20849“side jump” /H20850. This effect may be calculated in a straightforward way18by observing that the side jump leads to an additional term in the particle velocity due to thespin-orbit interaction. Indeed, the quantum mechanical ve-locity obtained from the Heisenberg equation of motion forthe position operator has two terms, v=d dtr=−i/H20851r,H1/H20852=p m+1 4m2c2/H20849/H9270/H11003/H11612Vdis/H20850. /H208494.8/H20850 The Bloch state matrix elements of vare given by /H20855k/H11032/H9268/H11032/H20841v/H20841k/H9268/H20856=k m/H9254kk/H11032/H9254/H9268/H9268/H11032−ig/H9268 2m/H9255F/H20858 jV/H20849k−k/H11032/H20850 /H11003ei/H20849k−k/H11032/H20850·Rj/H20853/H9270/H9268/H9268/H11032/H11003/H20849k−k/H11032/H20850/H20854. /H208494.9/H20850 For strong impurity scattering, there are six diagrams that contribute to the current correlation function, four of type /H20849a/H20850 and two of type /H20849b/H20850, shown in Fig. 1. For example, contribu- tions from diagrams of Figs. 1/H20849a/H20850and1/H20849b/H20850give Lxy1a=−inimpg /H9280FT/H20858 kk/H11032V2Gk+Gk/H11032+Gk−/H20875/H9270/H11003k−k/H11032 2m/H20876 xfk/H11032k+j˜ky, Lxy1b=−inimpg /H9280FT/H20858 kk/H11032V2Gk/H11032+Gk1+Gk1−Gk−/H20875/H9270/H11003k−k/H11032 2m/H20876 x /H11003fk/H11032k1+fk1k−j˜k1y. /H208494.10 /H20850 These were evaluated within the short-range strong impurity scattering model in Ref. 10. We will later use the results reported there. C. Berry phase contribution In general, Berry phase contributions can arise when there is an anomalous velocity term, as in the case of the side-jump contribution given by Eq. /H208494.8/H20850. In principle, such terms can also arise in the presence of a periodic potentialand spin-orbit interaction leading to ﬁnite Berry curvatures. 4 It has been found that the intrinsic Berry curvature contribu-tions to the AH conductivity for bulk ferromagnetic metalscan be large in magnitude. 19Analogous contributions for thin ﬁlm ferromagnets have not been obtained yet. Such contri-butions depend on the details of the band structure and arebeyond the scope of the present work. On the other hand, the focus of the current work is on the disorder and temperaturedependence of the AH conductivity in which the Berry con-tributions are qualitatively similar to the side-jump contribu-tions /H20849both arise from an additional velocity term due to spin-orbit interactions /H20850. Therefore, the effects of Berry con- tributions can be included in a phenomenological way, whilecomparing with experiments, by considering a larger side-jump contribution to the total AH conductivity. V . INTERACTION CORRECTIONS TO THE CONDUCTIVITY The e-e interaction corrections to the conductivity will be calculated in ﬁrst order in the screened Coulomb interaction.It may therefore be represented as an integral over a kernelK/H20849q,i /H9275l/H20850multiplied by the screened Coulomb interaction Vc/H20849q,i/H9275l/H20850, /H9254/H9268I=T/H20858 /H9275l/H20885dq2K/H20849q,i/H9275l/H20850Vc/H20849q,i/H9275l/H20850. /H208495.1/H20850 Gauge invariance requires that /H9254/H9268should be invariant against an energy shift of the interaction potential, V/H20849r/H20850 →V/H20849r/H20850+C, which only leads to a constant term in the total Hamiltonian. In Fourier space, the transformation is V/H20849q/H20850 →V/H20849q/H20850+C/H9254/H20849q/H20850, which requires the kernel to vanish in the limit q→0.20/H20851Even more general, since V/H20849q/H20850is an electric potential, a gauge transformation of the above form, but with arbitrary time dependence C=C/H20849t/H20850, does not change the physical ﬁelds. /H20852We will see below that this gauge invari- ance, together with an additional mirror symmetry, will im-pose a strong constraint on the interaction corrections to theHall conductivity. A. Coulomb interaction renormalized by diffusion The Coulomb interaction Vc/H20849q,/H9275l/H20850is renormalized by dif- fusion processes. The bare screened interaction is given by Vc/H20849q,i/H9275l/H20850=VB/H20849q/H20850//H208511+VB/H20849q/H20850/H9016/H20849q,i/H9275l/H20850/H20852, /H208495.2/H20850 where VB/H20849q/H20850=4/H9266e2/q2in three dimensions and VB/H20849q/H20850 =2/H9266e2/qin two dimensions, and the polarization function is given by12 /H9016/H20849q,i/H9275l/H20850=dn d/H9262Dq2 /H20841/H9275l/H20841+Dq2. /H208495.3/H20850 In two dimensions, one therefore ﬁnds Vc/H20849q,i/H9275l/H20850=2/H9266e2 q/H20841/H9275l/H20841+Dq2 /H20841/H9275l/H20841+Dq2+DqK 2→/H20873dn d/H9262/H20874−1 ,/H208495.4/H20850 in the limit /H9275l=0,q→0. Note that in a ferromagnet, an ad- ditional effective electron-electron interaction arises by ex-change of spin-wave excitations. We do not consider thisinteraction here because it is small, of order /H20849J/ /H9280F/H208502, where J is the exchange energy /H20849see Sec. VI B /H20850. B. Singular contributions for skew scattering The diagrams for the correlation functions L/H9251/H9252deﬁned in Eq. /H208494.1/H20850can have up to three diffusion poles.21The gauge(a) (b) FIG. 1. Diagrams for side-jump contributions. Solid lines are impurity averaged Green’s functions. Shaded triangles with dashedlines represent impurity scattering amplitudes while the dotted linefrom a vertex denotes spin-orbit term in the velocity operator. Theshaded vertex represents vertex corrections to the current densityoperator.DISORDER AND TEMPERATURE DEPENDENCE OF THE … PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-7invariance argument presented above suggests that the rel- evant contributions to K/H20849q,i/H9275l/H20850should have a factor of q2, which cancels one of the diffusion poles. Therefore, only diagrams with three diffusion poles shown in Fig. 2contrib- ute. For example, contribution from diagram /H20849a/H20850of Fig. 2is given by L/H9251/H92522a=−T/H20858 /H9280nT/H20858 /H9275l/H20858 k,k/H11032,qGk2/H20849/H9280n/H20850Gk−q/H20849/H9280−/H9275/H20850 /H11003Gk/H11032−q/H20849/H9280−/H9275/H20850Gk/H11032/H20849/H9280n/H20850Gk/H11032/H20849/H9280n−/H9024/H20850 /H11003V/H20849q,/H9275l/H20850/H20851/H9008/H20849/H9280/H20850/H9008/H20849/H9024/H20850/H9008/H20849/H9275−/H9280/H20850 /H11003Tk+−/H20849q,/H9275/H20850Tk/H11032−+/H20849−q,−/H9275/H20850/H9003k/H11032k+−/H20849q,/H9275−/H9024/H20850 +/H9008/H20849−/H9280/H20850/H9008/H20849/H9024−/H9280/H20850/H9008/H20849/H9280−/H9275/H20850Tk−+/H20849q,/H9275/H20850 /H11003Tk/H11032+−/H20849−q,−/H9275/H20850/H9003kk/H11032+−/H20849−q,/H9275+/H9024/H20850/H20852vk/H9251vk/H11032/H9252./H208495.5/H20850 Using only the singular parts /H9003kk/H11032+−/H20849q,/H9024/H20850=/H9253k/H20849q/H20850/H9253˜k/H11032/H20849q/H20850 /H20841/H9024/H20841+Dq2, /H9003kk/H11032−+/H20849q,/H9275/H20850=/H9003k/H11032k+−/H20849−q,−/H9275/H20850/H20849 5.6/H20850 and Tk+−/H20849q,/H9275/H20850=/H9253k/H20849q/H20850 /H20841/H9275/H20841+Dq2,Tk−+/H20849q,/H9275/H20850=/H9253˜k/H20849−q,/H20850 /H20841/H9275/H20841+Dq2/H208495.7/H20850 and deﬁning Dq/H20849/H9275l,/H9024m/H20850=V/H20849q,/H9275l/H20850 /H20849/H20841/H9275l/H20841+Dq2/H208502/H20849/H20841/H9275l−/H9024m/H20841+Dq2/H20850, /H208495.8/H20850 one gets L/H9251/H92522a=/H20858 /H9268/H20849−2/H9266iN0/H92702/H208502/H20858 q/H20875T/H20858 /H9275l/H11022/H9024m/H20849/H9275l−/H9024m/H20850 /H11003/H20855vk/H9251/H9253k/H20849q/H20850/H9253˜k/H20849q/H20850/H9264k/H20849q/H20850/H20856k/H20855vk/H11032/H9252/H9253˜k/H11032/H20849q/H20850/H9253k/H11032/H20849q/H20850/H9264k/H11032/H20849q/H20850/H20856k/H11032 +T/H20858 /H9275l/H110210/H20841/H9275l/H20841/H20855vk/H9251/H9253˜k/H20849−q/H20850/H9253k/H20849−q/H20850/H9264k/H20849q/H20850/H20856k /H11003/H20855vk/H11032/H9252/H9253k/H11032/H20849−q/H20850/H9253˜k/H11032/H20849−q/H20850/H9264k/H11032/H20849−q/H20850/H20856k/H11032/H208761 2/H9266Dq/H20849/H9275l,/H9024m/H20850, /H208495.9/H20850where we have expanded the Green’s functions for small q and deﬁned the factor /H9264k/H110131−2 i/H9270/H20849q·vk/H20850. /H208495.10 /H20850 Note that/H9253˜k/H20849−q,−/H9024/H20850=/H9253˜k/H20849q,/H9024/H20850. The leading terms in qare the linear in qterms in the products /H9253/H9253˜/H9264, /H9253k/H20849±q/H20850/H9253˜k/H20849±q/H20850/H9264k/H20849q/H20850 =1/H110072i/H9270/H20849q·vk/H20850/H11007i 2vF/H9270/H20858 m=±1/H20851/H9261˜m+/H9261˜ m/H20852/H9273m/H20849kˆ/H20850q−m. /H208495.11 /H20850 The/H9261˜’s combine to /H9261˜ m/H11032=/H9261˜ −m/H11032, which may be pulled in front of themsummation. Observe that vF/H20858 m=±1/H9273m/H20849kˆ/H20850q−m=2/H20849q·vk/H20850. /H208495.12 /H20850 Therefore, quite generally, /H20855vkx/H9253k/H20849q/H20850/H9253˜k/H20849q/H20850/H9264k/H20849q/H20850/H20856k=−ivF2/H9270qx/H208491+/H9261˜ 1/H11032/H20850. /H208495.13 /H20850 C. Corrections to longitudinal conductivity within skew scattering model For contributions from diagram /H20849a/H20850of Fig. 2to the longi- tudinal conductivity, each of the two angular averages /H20849in each term /H20850in Eq. /H208495.9/H20850with/H9251=/H9252=xgives a factor propor- tional to qx/H20851see Eq. /H208495.13 /H20850/H20852, the product yielding qx2. Diagram /H20849b/H20850also has the same combination. This yields, for the four diagrams /H20849a/H20850,/H20849a/H11032/H20850,/H20849b/H20850, and /H20849b/H11032/H20850, the total contribution /H20849Lxx2a =Lxx2a/H11032;Lxx2b=Lxx2b/H11032/H20850, Lxx2a+2a/H11032+2b+2b/H11032 =1 2/H9266/H20858 /H9268/H208492/H9266N/H9268/H92702/H208502/H20849vF2/H9270/H208502/H208491+/H9261˜ 1/H11032/H208502/H20858 qq2/H9023/H20849q,/H9024m/H20850, /H208495.14 /H20850 where we have deﬁned /H9023/H20849q,/H9024m/H20850=T/H20858 /H9275l/H110220/H9275l/H20851D/H20849−/H9275l,/H9024m/H20850−D/H20849−/H9275l−/H9024m,/H9024m/H20850/H20852 =T/H20875/H20858 0/H11021/H9275l/H11021/H9024m/H9275l+/H20858 /H9275l/H11022/H9024m/H9024m/H20876D/H20849−/H9275l,/H9024m/H20850. /H208495.15 /H20850 The sum over qconverted to an integral yields /H20858 qq2/H9023/H20849q,/H9024m/H20850=1 4/H9266e2 D2/H9260/H9024/H208731+l n/H9275c 2/H9266T/H20874, /H208495.16 /H20850 where/H9260/H110132/H9266e2/H20858/H9268N/H9268is the screening length. The exchange interaction correction to the longitudinal conductivity is then given by /H9254/H9268xxex=e2 /H9024mLxx=−e2 2/H92662ln/H9275c T, /H208495.17 /H20850 where we used D/H9268=D0/H9268/H208491+/H9261˜ 1/H9268/H11032/H20850. Note that the correction /H9254/H9268xxis independent of the scattering strength.(a) (b) FIG. 2. Diagrams for interaction corrections. Solid lines are im- purity averaged Green’s functions, wavy lines denote screened Cou-lomb interactions, and dashed lines denote diffusion poles. Thereare two diagrams of type /H20849a/H20850and two of type /H20849b/H20850.K. A. MUTTALIB AND P. WÖLFLE PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-8D. Corrections to Hall conductivity within skew scattering model For/H9251=x,/H9252=y, the two angular averages in Eq. /H208495.9/H20850are proportional to qxandqy, respectively, so that the angular q integral yields zero. This is true for all four diagrams /H20849a/H20850, /H20849a/H11032/H20850,/H20849b/H20850, and b /H11032. Thus, the total correction to the Hall con- ductivity Lxywithin the skew scattering model is zero. Note that the results are true for arbitrary strength as well as ﬁniterange and anisotropy of the impurity scattering. Note that the result that the angular average /H20851Eq. /H208495.13 /H20850/H20852is proportional to q xis a special consequence of the fact that Eq. /H208495.9/H20850contains the combination /H9253k/H9253˜k. This particular com- bination is proportional to q·vk, as shown in Eq. /H208495.12 /H20850, which results in Eq. /H208495.13 /H20850. This is true for the class of dia- grams considered here. This leads to the obvious question ifthere are other diagrams where the angular average is over adifferent combination of /H9253k’s leading to a nonzero contribu- tion to Lxy. It turns out that, indeed, there are such terms with less than three diffusion poles, but that there is a deeperreason why the total interaction correction to the Hall con- ductivity must always vanish in the ﬁrst order in Coulombinteraction. In this case, the interaction correction has theform Eq. /H208495.1/H20850and the kernel must be proportional to q 2as mentioned before. In addition, we have the following sym-metry properties for the Hall conductivity with respect to asign change of the magnetization /H20849magnetic ﬁeld /H20850and a mir- ror reﬂection from the yzplane x→−x/H20849or from the xzplane y→−y/H20850, which follow from the invariance of the Hamil- tonian under a simultaneous transformation B→−Band x →−x/H20849ory→−y/H20850, /H9268xy/H20849B/H20850=−/H9268xy/H20849−B/H20850, /H9268xy/H20849B;x/H20850=/H9268xy/H20849−B;−x/H20850=−/H9268xy/H20849B;−x/H20850, /H208495.18 /H20850 which means that the Kernel must be proportional to qxqyto preserve the mirror symmetry. Thus, even though individualdiagrams do contribute, the total sum of all diagrams of agiven class must cancel to yield vanishing contribution to theHall conductivity. Note that the above argument remainsvalid for the side-jump contributions as well. Therefore, wehave, quite generally, /H9254/H9268xyI=0 . /H208495.19 /H20850 This generalizes the results of Ref. 9where this result was ﬁrst obtained within a skew scattering model with short-range and weak impurity scattering. Note that the above arguments do not imply that the weak localization correction to the Hall conductivity must alsovanish because the WL contributions do not have the formEq. /H208495.1/H20850and the gauge invariance arguments do not apply. E. Corrections to conductivity within side-jump model We have already argued that the e-e interaction correc- tions to the Hall conductivity due to side-jump scatteringmust vanish on very general symmetry grounds. The corre-sponding corrections to the longitudinal conductivity are ofcourse ﬁnite. However, these contributions are proportionalto the spin-orbit coupling and therefore are much smaller than the corrections due to normal scattering obtained above.We will therefore neglect such contributions. F. Hartree terms Equation /H208495.17 /H20850should be corrected by including dia- grams of the Hartree type. This leads to the total interactioncorrection in two dimensions, 11 /H9254/H9268xxI=−e2 2/H92662/H208731−3 4F˜/H9268/H20874ln/H9275c T, /H208495.20 /H20850 where F˜/H9268=8/H208491+F/2/H20850ln/H208491+F/2/H20850/F−4 /H208495.21 /H20850 and F=1 v/H20849q=0/H20850/H20885d/H9258 2/H9266v/H20849q=2kFsin/H9258/2/H20850. /H208495.22 /H20850 As we will discuss later, experiments suggest an approximate cancellation between the exchange and Hartree terms, whichwill imply that the quantity h xx/H11013/H208731−3 4F˜/H20874 /H208495.23 /H20850 can be very small. VI. WEAK LOCALIZATION CORRECTION TO CONDUCTIVITY As pointed out before, the weak localization contributions cannot be written as an integral over a kernel, as in Eq. /H208495.1/H20850 for the Coulomb interaction. Therefore, although the mirrorsymmetry is still preserved, the total contribution to the Hallconductivity need not be zero. A. Cooperon contributions The weak localization correction to the current-current correlator is obtained from diagrams shown in Fig. 3, with(a) (b) (c)( d) FIG. 3. Diagrams for weak localization corrections. Solid lines are impurity averaged Green’s functions and broken lines are impu-rity scattering amplitudes. Shaded cross is the Cooperon and shadedvertices are vertex corrections to the current density operator. Thereare two diagrams of type /H20849b/H20850and four diagrams of type /H20849c/H20850.DISORDER AND TEMPERATURE DEPENDENCE OF THE … PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-9one Cooperon propagator connecting the upper and lower lines of the conductivity bubble. The frequency arguments ofthe upper /H20849particle /H20850line and the lower /H20849hole /H20850line have oppo- site signs. The current vertices are dressed. For example, thecontribution of diagram /H20849a/H20850of Fig. 3to the current correla- tion function is L /H9251/H92523a=/H20858 /H9268T/H20858 /H9280n/H20858 k,k/H11032,QGk/H9268/H20849i/H9280n/H20850Gk/H9268/H20849i/H9280n−i/H9024m/H20850Gk/H11032/H9268/H20849i/H9280n/H20850 /H11003Gk/H11032/H9268/H20849i/H9280n−i/H9024m/H20850jk/H9251/H9268j˜ k/H11032/H9252/H9268/H208492/H9266N/H9268/H9270/H9268/H20850−1C¯kk/H11032/H20849Q;i/H9280n,i/H9024m/H20850. /H208496.1/H20850 Here, the momentum Q=k+k/H11032can be taken to be small, as forQ→0 the Cooperon is strongly peaked. Consequently, one may take k/H11032/H11015−kin the arguments of the Green’s func- tions and of the current vertex, i.e., j˜ k/H11032/H9252/H9268/H11015−j˜k/H9252/H9268. Then, L/H9251/H92523a=− /H20849/H9024m/2/H9266/H20850/H20858 /H9268/H208494/H9266N/H9268/H9270/H92683/H20850 /H11003/H208492/H9266N/H9268/H9270/H9268/H20850−1/H20855jk/H9251/H9268j˜k/H9252/H9268/H20856k/H20858 QC¯k,−k/H20849Q/H20850. /H208496.2/H20850 The Cooperon contribution is given by /H9021/H11013/H20858 QC¯k,−k/H20849Q/H20850=/H20885 0QcQdQ 2/H92661//H9270 /H20841/H9024m/H20841+DpQ2+/H9270/H9272−1 =/H208494/H9266/H9270/H9268Dp/H20850−1ln/H20849/H9270/H9272//H9270/H9268/H20850, /H208496.3/H20850 leading to a logarithmic temperature dependence through /H9270/H9272/H20849T/H20850. Similarly, contributions from the two diagrams of type /H20849b/H20850can be evaluated to give L/H9251/H92523b=nimp/H20858 /H9268T/H20858 /H9280n/H20877/H20858 k/H20851Gk/H9268/H20849i/H9280n/H20850/H208522Gk/H9268/H20849i/H9280n−i/H9024m/H20850/H208782 /H11003jk/H9251/H9268j˜k/H9252/H9268fk,−k/H11032/H9268+f−k,k/H11032/H9268+/H9021 =nimp/H9024m 2/H9266/H20858 /H9268/H20849−2/H9266iN/H9268/H9270/H92682/H208502/H208492/H9266N/H9268/H9270/H9268/H20850−1 /H11003/H20849/H9266N/H9268/H20850−2/H20855jk/H9251/H9268j˜k/H9252/H9268f¯ k,−k/H11032/H9268+f¯ −k,k/H11032/H9268+/H20856k/H9021, L/H9251/H92523b/H11032=nimp/H9024m 2/H9266/H20858 /H9268/H208492/H9266iN/H9268/H9270/H92682/H208502/H208492/H9266N/H9268/H9270/H9268/H20850−1 /H11003/H20849/H9266N/H9268/H20850−2/H20855jk/H9251/H9268j˜k/H9252/H9268f¯ k/H11032,−k/H9268−f¯ −k/H11032,k/H9268−/H20856k/H9021, /H208496.4/H20850 so that L/H9251/H92523b+3b/H11032=nimp/H9024m 2/H9266/H20858 /H9268/H20849−2/H9266iN/H9268/H9270/H92682/H208502/H208492/H9266N/H9268/H9270/H9268/H20850−1 /H11003/H20849/H9266N/H9268/H20850−2/H20849vF2/H9253/H9268/H20850−1 /H11003/H9021 /H20855jk/H9251/H9268j˜ k/H11032/H9252/H9268/H20851f¯ k,−k/H11032/H9268+f¯ −k,k/H11032/H9268++f¯ k/H11032,−k/H9268−f¯ −k/H11032,k/H9268−/H20852/H20856k. /H208496.5/H20850 In a similar fashion, the total contributions from all diagrams can then be written asL/H9251/H9252WL=−/H9024m 4/H92662/H20858 /H9268/H20849D/H9268/D/H9268p/H20850J/H9251/H9252ln/H20849/H9270/H9272//H9270/H9268/H20850, J/H9251/H9252=J1/H9251/H9252+J2/H9251/H9252+4iJ3/H9251/H9252−4J5/H9251/H9252, /H208496.6/H20850 where J1/H9251/H9252=2 vF/H92682/H20855jk/H9251/H9268j˜k/H9252/H9268/H20856, J2/H9251/H9252=/H20849vF2/H9253/H9268/H20850−1/H20855jk/H9251/H9268j˜ k/H11032/H9252/H9268/H20851f¯ k,−k/H11032/H9268+f¯ −k,k/H11032/H9268++f¯ k/H11032,−k/H9268−f¯ −k/H11032,k/H9268−/H20852/H20856k, J3/H9251/H9252=/H20849vF2/H9253/H9268/H20850−1/H20855jk/H9251/H9268j˜ k/H11032/H9252/H9268/H20851f¯ k,−k/H11032/H9268+f¯ −k1,k/H11032/H9268+f¯ k1,k/H9268− −f¯ −k/H11032,k/H9268−f¯ k/H11032,−k1/H9268−f¯ k,k1/H9268+/H20852/H20856k,k/H11032,k1, J5/H9251/H9252=/H20849vF2/H9253/H9268/H20850−1/H20855jk/H9251/H9268j˜ k/H11032/H9252/H9268f¯ k,k2/H9268+f¯ −k1,k/H11032/H9268+f¯ k/H11032,−k2/H9268−f¯ k1,k/H9268−/H20856k,k/H11032,k1,k2. /H208496.7/H20850 Here, J1/H9251/H9252corresponds to the contribution from diagram /H20849a/H20850 of Fig. 3,J2/H9251/H9252is a sum of contributions from the two dia- grams of type /H20849b/H20850,J3/H9251/H9252is a sum of contributions from two diagrams of type /H20849c/H20850/H20851the other two of type /H20849c/H20850gives J4/H9251/H9252 =J3/H9251/H9252/H20852, and J5/H9251/H9252is a contribution from diagram /H20849d/H20850.I nt h e above, we have used the relation /H20849nimp //H9266N/H9268/H20850=1 //H208492/H9253/H9268/H9270/H9268/H20850. B. Phase relaxation rate The Cooperon contribution depends on the phase relax- ation rate /H9270/H9272−1, which grows linearly with temperature T.I n general, this may be cut off by spin-ﬂip scattering /H9270s,b y spin-orbit scattering /H9270so, or by a magnetic ﬁeld characterized by/H9275H, all of which are independent of temperature. There- fore, a logarithmic temperature dependence in the conductiv-ity requires that the phase relaxation rate satisﬁes the in-equality max /H208491/ /H9270s,1//H9270so,/H9275H/H20850/H112701//H9270/H9272/H112701//H9270tr. /H208496.8/H20850 The contribution to /H9270/H9278from e-e interaction is given by 1//H9270/H9272=T /H9280F/H9270trln/H9280F/H9270tr 2. /H208496.9/H20850 This is typically too small to satisfy the above inequality in thin ferromagnetic ﬁlms where, in particular, the internalmagnetic ﬁeld B incan be estimated to give rise to /H9275H =4/H20849/H9280F/H9270tr/H20850/H20849eBin/mc/H20850which can be large. A much larger con- tribution is obtained from scattering off spin waves in such systems,15which is given by 1//H9270/H9272=4/H9266TJ2 /H9280F/H9004g, /H208496.10 /H20850 where Jis the exchange energy of the selectrons and /H9004gis the spin-wave gap. As estimated in Ref. 14, with this contri- bution to the phase relaxation rate, the inequality /H20851Eq. /H208496.8/H20850/H20852 can be satisﬁed within experimentally accessible disorderand temperature ranges where the WL effects can be ob-served.K. A. MUTTALIB AND P. WÖLFLE PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-10VII. STRONG SHORT-RANGE IMPURITY SCATTERING The results of the previous section can in principle be used to obtain the weak localization corrections to both lon-gitudinal and Hall conductivities. However, the algebra getsfairly involved without contributing extra insight into theproblem. Since higher angular momentum components areexpected to be smaller, we will consider the dominant con-tribution that arises from a short-range impurity model andshow in the Appendix how effects of ﬁnite range anisotropicscattering can be included within model calculations. On theother hand, we will keep the calculations valid for arbitrarystrength of the impurity scattering. A. Scattering amplitude, relaxation rate, and particle-hole and particle-particle propagators These were already obtained for short-range strong impu- rity scatterings in Ref. 10and we will simply quote the re- sults. The scattering amplitude is given by f¯k/H9268,k/H11032/H9268/H11032=w˜ /H20881w−i/H9270/H9268/H9268z/H20849kˆ/H11003kˆ/H11032/H208502u˜ /H20881u−is/H9275n/H20851w˜+2u˜/H20849kˆ·kˆ/H11032/H20850/H20852. /H208497.1/H20850 Here, we deﬁned w˜=w//H208491+w/H20850and u˜=u//H208491+u/H20850, where w =/H20849/H9266N/H9268V/H208502andu=/H20849g/H9268/2/H208502w, and all quantities depend on the spin orientation /H9268/H20849suppressed here and in the following, except in the ﬁnal expressions involving spin summation /H20850.I n terms of the angular momentum components of f¯deﬁned in Eq. /H208493.9/H20850,f¯ ms, we have from Eq. /H208497.1/H20850, f¯ kk/H11032s=f¯ 0s+f¯ 1skˆ+kˆ −/H11032+f¯ −1skˆ−kˆ +/H11032,f¯ 0s=w˜ /H20881w−isw˜, f¯ ±1s=−isu˜±/H9270/H9268/H9268zu˜ /H20881u,f¯ ms=0 , /H20841m/H20841/H110221. /H208497.2/H20850 Using Eq. /H208497.2/H20850, the single particle relaxation rate given by Eq. /H208493.10 /H20850becomes 1 2/H9270/H9268=nimp /H9266N/H9268/H20849w˜+2u˜/H20850. /H208497.3/H20850 One observes that1 2/H9270/H9268is proportional to the Fermi energy, the average number of impurities per electron, and the dimen- sionless factor /H20849w˜+2u˜/H20850, expressing the effective scattering strength per impurity. Eigenvalues of the particle-hole scat- tering amplitude t¯kk/H11032+−are obtained to be /H92610=1 ,/H9261−m=/H9261m, /H92611=2w˜u˜/H20849w˜+2u˜/H20850−1/H208731+is1 /H20881u/H9270/H9268/H9268z/H20874, /H92612=u˜2 u/H20849w˜+2u˜/H20850−1/H20849u−1+2 is/H20881u/H9270/H9268/H9268z/H20850, /H208497.4/H20850 while for t¯kk/H11032++one obtains /H20851with t¯kk/H11032ss/H11013/H20858 m/H9264m/H9273m/H20849kˆ/H20850/H9273m/H20849kˆ/H11032/H20850/H20852/H92640=/H20849w˜+2u˜/H20850−1/H20875w˜ 1+w/H208491−w−2is/H20881w/H20850+2u˜1−u 1+u/H20876, /H92641=−2 w˜u˜/H20849w˜+2u˜/H20850−1/H208731+is1 /H20881w/H20874, /H92642=−u˜/H20849w˜+2u˜/H20850−1. /H208497.5/H20850 It may be shown that /H9004t¯kk/H11032/H20849q/H20850deﬁned in Eq. /H208493.13 /H20850gives rise to small corrections to the diffusion coefﬁcient, of order /H208491//H9255F/H9270/H20850, and hence may be dropped. Eigenvalues of the particle-particle scattering amplitude t¯kk/H11032p,+−are obtained to be /H92610p=/H20851w˜−2u˜/H208491−2 u˜/H20850/H20852//H20849w˜+2u˜/H20850, /H9261±1p=/H208732w˜u˜±2w˜ /H20881wu˜ /H20881u/H9270/H9268/H9268z/H20874/H20882 /H20849w˜+2u˜/H20850, /H9261±2p=u˜//H20849w˜+2u˜/H20850. /H208497.6/H20850 We observe that /H92610p/HS110051 if skew scattering is present, as it violates time reversal symmetry. The phase relaxation rate /H20849/H9270/H9272so/H20850−1deﬁned in Eq. /H208493.42 /H20850is given by /H20849/H9270/H9272so/H20850−1=/H9270−14u˜/H208491−u˜/H20850//H20851w˜−2u˜/H208491−2 u˜/H20850/H20852, /H208497.7/H20850 which is positive for not too large spin-orbit scattering, u /H11351w/2o r g/H9268/H113511. B. Hall conductivity The conductivity tensor due to skew scattering was al- ready evaluated in Sec. IV A for general strong ﬁnite rangeimpurity scattering in terms of the eigenvalues of theparticle-hole propagator /H9261. In particular, it gives /H9268xyss /H9268xxss=/H92611/H11033 1−/H92611/H11032. /H208497.8/H20850 For short-range scattering, Eq. /H208497.4/H20850gives explicit expres- sions for the eigenvalues in terms of the scattering potentials.The side-jump contribution was already evaluated in Ref. 10 and we quote the result, /H9268xysj=e2 2/H9266/H20858 /H9268/H9270/H9268/H9268zg/H9268w˜ w˜+2u˜/H208491+/H9261˜ 1/H11032/H20850 1+u/H208497.9/H20850 Using Eq. /H208497.4/H20850, this yields, in the small u/H11270w/H112701 limit, /H9268xysj=e2 2/H9266/H20858 /H9268/H9270/H9268/H9268zg/H92681 1−/H92611/H11032. /H208497.10 /H20850 C. Weak localization correction Evaluation of J/H9251/H9252deﬁned in Sec. VI /H20851Eqs. /H208496.6/H20850and /H208496.7/H20850/H20852 in the present short-range /H20849but arbitrary scattering strength /H20850 model givesDISORDER AND TEMPERATURE DEPENDENCE OF THE … PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-11J1xx=/H208491+/H9261˜ 1/H11032/H208502−/H20849/H9261˜ 1/H11033/H208502,J1xy=2/H9261˜ 1/H11033/H208491+/H9261˜ 1/H11032/H20850, J2xx=/H20851/H92611/H11032J1xx−/H92611/H11033J1xy/H20852,J2xy=/H20851/H92611/H11033J1xx+/H92611/H11032J1xy/H20852, J3xx=i 2/H208512u˜/H92611/H11032J1xx−/H208492u˜+1/H20850/H92611/H11033J1xy/H20852, J3xy=i 2/H20851/H208492u˜+1/H20850/H92611/H11033J1xx+2u˜/H92611/H11032J1xy/H20852, J5xx=−1 2/H20851/H208492u˜−1/H20850/H92611/H11032J1xx−2u˜/H92611/H11033J1xy/H20852, J5xy=−1 2/H208512u˜/H92611/H11033J1xx+/H208492u˜−1/H20850/H92611/H11032J1xy/H20852, iJ3/H9251/H9252−J5/H9251/H9252=−1 2J2/H9251/H9252,J/H9251/H9252=J1/H9251/H9252−J2/H9251/H9252. /H208497.11 /H20850 We may combine this into the compact expression Jxx=R e /H20853/H9011/H20854,Jxy=I m /H20853/H9011/H20854,/H9011=1 1−/H92611. /H208497.12 /H20850 Note that the ﬁnal result for J/H9251/H9252contains detailed effects of the potentials only through the eigenvalues /H9261. This sug- gests that the results may be more general than the short-range potentials used in the calculations. Also, as we will show in the Appendix, /H9261 1/H11032may approach unity in the limit of extreme forward scattering. In any case, for the short-range impurity scattering model considered above, we then have contributions from weak lo-calization corrections given by /H9254/H9268xxWL=−e2 4/H92662/H20858 /H9268/H20849D/H9268/Dp/H20850ln/H20849/H9270/H9272//H9270/H9268/H20850, /H9254/H9268xyWL /H9254/H9268xxWL=Im/H20849/H9011/H20850 Re/H20849/H9011/H20850=/H92611/H11033 1−/H92611/H11032. /H208497.13 /H20850 VIII. COMPARISON WITH EXPERIMENTS Experiments measure the longitudinal and Hall resis- tances R/H9251/H9252as functions of both sheet resistance and tempera- ture. In order to compare, we obtain the normalized relativeconductances deﬁned as /H9004 N/H9268/H9251/H9252/H110131 L00R0/H9254/H9268/H9251/H9252 /H9268/H9251/H9252, /H208498.1/H20850 where L00/H11013e2/2/H92662andR0=1 //H9268xx. As shown above, a loga- rithmic temperature dependence in these quantities can ariseeither from interaction corrections or from weak localizationcorrections. However, although two separate groups haveseen such logarithmic temperature dependences, 13,14the pref- actors seem to be more universal for /H9004N/H9268xx, independent ofsheet resistance R0or sample preparation for a range of R0, but clearly disorder and sample dependent for /H9004N/H9268xyin the same range of R0. In this section, we collect all our results above to obtain the total contribution to /H9004N/H9268/H9251/H9252from all pos- sible mechanisms considered above. As used in the text, su-perscripts ssandsjwill refer to the skew scattering and side jump mechanisms, and IandWLwill refer to the interaction and weak localization corrections, respectively. While the re- sults for /H9268/H9251/H9252ssand/H9254/H9268xyIare valid for ﬁnite range strong impu- rity scatterings, others are evaluated within a short-rangestrong impurity scattering model. We have also assumed thatthe spin-orbit coupling is weak. The conductivities due to skew and side jump scatterings are /H9268xxss=/H20858 /H92681 2vF/H92682N/H9268/H9270tr,/H9268xxsj/H11270/H9268xxss, /H9268xyss=/H9268xxss/H92611/H11033 1−/H92611/H11032,/H9268xysj=e2 2/H9266/H20858 /H9268/H9270/H9268/H9268zg/H9268/H208491−/H92611/H11032/H20850 /H208411−/H92611/H208412./H208498.2/H20850 Quantum corrections to the conductivities due to Coulomb interaction and weak localization effects leading to a loga-rithmic temperature dependence are /H9254/H9268xxss,I=L00hxxln/H20849T/H9270/H20850,/H9254/H9268xxss,WL=L00ln/H20849T/H9270/H20850, /H9254/H9268xyss,I=0 ,/H9254/H9268xyss,WL=/H9254/H9268xxss,WL/H92611/H11033 1−/H92611/H11032, /H9254/H9268xysj,I=0 ,/H9254/H9268xxsj,I/H11270/H9254/H9268xxss,I, /H9254/H9268/H9251/H9252sj,WL/H11270/H9254/H9268xyss,WL. /H208498.3/H20850 The total conductivities and quantum corrections are simply /H9268xx=/H9268xxss,/H9268xy=/H9268xyss+/H9268xysj, /H9254/H9268xx=/H9254/H9268xxss,I+/H9254/H9268xxss,WL,/H9254/H9268xy=/H9254/H9268xyWL. /H208498.4/H20850 Using these results, we obtain /H9004N/H9268xx=/H9268xxss L00/H9254/H9268xxss,I+/H9254/H9268xxss,WL /H9268xxss=/H208491+hxx/H20850ln/H20849T/H9270/H20850, /H9004N/H9268xy=/H9268xxss L00/H9254/H9268xyss,WL /H9268xyss+/H9268xysj=1 /H208491+rxy/H20850ln/H20849T/H9270/H20850, /H208498.5/H20850 where hxxdeﬁned in Eq. /H208495.23 /H20850is the exchange plus Hartree interaction contribution to the longitudinal conductivity andwe have deﬁned r xy/H11013/H9268xysj /H9268xyss/H208498.6/H20850 as the ratio of side-jump to skew scattering contributions to the Hall conductivity. Note that rxyis a nonuniversal quan- tity. As shown in Ref. 14, all current experiments can be understood if hxx/H112701 and rxyis sample dependent and is al- lowed to vary with disorder. In particular, this means thatK. A. MUTTALIB AND P. WÖLFLE PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-12while the skew scattering and side-jump mechanisms both contribute to the AH conductivity, the side-jump contribu-tions to the longitudinal conductivity as well as to the weaklocalization corrections to the conductivity tensor are muchsmaller than the corresponding skew scattering contributionswhen the spin-orbit coupling is weak. IX. SUMMARY AND CONCLUSION We develop a systematic general formulation for the AHE for strong, ﬁnite range impurity scattering starting from amicroscopic model of electrons in a random potential of im-purities including spin-orbit coupling. In particular, we con-sider quantum corrections to the AH conductivity, observedin different experiments on disordered thin ferromagneticﬁlms with apparently different results. General symmetry ar-guments presented here show that the e-e interaction correc-tions must vanish exactly, which then implies that there mustbe weak localization corrections in these ferromagnetic ﬁlmsdespite the presence of large internal magnetic ﬁelds. Our evaluations of the WL effects within a short range but strong impurity scattering lead to the normalized relativeconductances given by Eq. /H208498.5/H20850, where the spin-orbit cou- pling has been assumed to be weak. These results are con-sistent with all experimental observations, where the differ-ence between different experiments arise due to differentcontributions from skew scattering vs side-jump mechanism. In this paper, we have only brieﬂy mentioned the Berry phase effects. A systematic study of the Berry phase contri-butions to the AHE will be reported elsewhere. ACKNOWLEDGMENTS We thank A. Hebard, R. Misra, and P. Mitra for useful discussions on the experimental data on the Fe ﬁlm. Thiswork has been supported by the DFG-Center for FunctionalNanostructures at the Karlsruhe Institute of Technology/H20849KIT /H20850. APPENDIX: LONG-RANGE CORRELATED POTENTIALS For completeness, here we consider models to incorporate possible effects of small and large angle scattering. 1. Model of small angle scattering Long-range correlated potentials will scatter electrons predominantly by a small angle /H9258/H11270/H9266. A simple model is provided by a Gaussian dependence V/H20849k−k/H11032/H20850=V/H20849/H9258/H20850=4/H20881/H9266V0/H92580−1e−/H20849/H9258//H92580/H208502, /H20849A1/H20850 where/H92580/H11270/H9266. The angular momentum components of V/H20849/H9258/H20850 are given by Vmns=/H20885 0/H9266d/H9258 2/H9266V/H20849/H9258/H20850=V0e−m2/H925802/4. /H20849A2/H20850 In the limit of weak scattering, we have f¯m/H9268=V¯m/H9268and then/H9253/H9268=/H20858 m/H20841V¯m/H9268/H208412=/H20849/H9266N/H9268V0/H208502/H208812/H9266//H92580. /H20849A3/H20850 Neglecting skew scattering for the moment, we ﬁnd t¯1/H9268+,−=/H9253/H9268−1/H20849/H9266N/H9268V0/H208502/H20858 me−/H925802/4/H20851m2+/H20849m−1/H208502/H20852=e−/H925802/8./H20849A4/H20850 It follows that 1− t¯1/H9268+,−/H11015/H925802/8/H112701 and therefore the diffusion coefﬁcient is enhanced by a factor D/D0=/H20849/H925802/8/H20850−1. /H20849A5/H20850 2. Model of strong backscattering It is well known that the scattering of conduction elec- trons in amorphous metals can be anomalous in the sensethat the transport relaxation time is smaller than the singleparticle relaxation time. This is due to the fact that the atomicstructure is characterized by ﬁnite range order. The pair cor-relation function shows enhanced peaks corresponding to thenearest neighbor, next nearest neighbor, etc., shell. In otherwords, the system shows crystalline order over a certain usu-ally short distance. As a consequence, electrons are sufferingBragg scattering by large angles. The scattering cross sectionfor large angles is larger than that for small angles. Conse-quently, the angular average of the cross section /H9268/H20849/H9258/H20850, weighted with the factor /H208491−cos/H9258/H20850, appearing in the expres- sion for the transport relaxation rate is larger than the uni- form average in the single particle transport rate. In the caseof polycrystalline material, we expect a similar effect. The scattering potential V/H20849r/H20850of a crystallite or a small grain of amorphous metal will show oscillating behavior in real space reﬂecting the nearly regular arrangement of atoms,and its Fourier transform will show a peak at a ﬁnite momen-tum q=2 /H9266/acorresponding to the spatial period a, which will be equal or close to the lattice constant of the crystallinephase. The width of the peak will be determined by the rangeof the short-range order or the size of the crystallites. This isin contrast to a usual impurity potential whose Fourier trans-form has a peak at q=0 and a width corresponding to the range of the potential. In terms of the angular momentumcomponents V lof the scattering potential, a peak in V/H20849q/H20850 implies that some of the Vlwill be negative. In particular, the component /H92611of the tmatrix tkk/H11032determining the transport relaxation rate will be negative. Let us consider a simple model of a crystallite of size L. Its scattering potential seen by a conduction electron of thematrix /H20849assumed to be isotropic, as appropriate for an amor- phous system /H20850is something like V 1/H20849x/H20850=V0cos/H208492/H9266x/a/H20850/H9258/H20849L/2− /H20841x/H20841/H20850 =V0S1/H20849x/H20850one dimension, V2/H20849x,y/H20850=V0S1/H20849x/H20850S1/H20849y/H20850two dimensions. /H20849A6/H20850 The Fourier transform of S1/H20849x/H20850is given byDISORDER AND TEMPERATURE DEPENDENCE OF THE … PHYSICAL REVIEW B 76, 214415 /H208492007 /H20850 214415-13S1/H20849k/H20850=L 2/H20851Kcos/H20849K/H20850sin/H20849/H9260/H20850−/H9260sin/H20849K/H20850cos/H20849/H9260/H20850/H20852 /H20851K2−/H92602/H20852, /H20849A7/H20850 where K=kL /2,/H9260=/H9266L/a.S1/H20849k/H20850increases linearly with kat small k, has maximum at k/H110152/H9266/a, and decreases as 1 /kfor large k. We may model this behavior by V2/H20849k/H20850=V0kk0 k02−k2, /H20849A8/H20850 where k0=2/H9266/a. Using the relation of the transferred mo- mentum k=kf−kito the scattering angle /H9278,k2=2kF2/H208491 −cos/H9278/H20850, where /H20841kf,i/H20841=kF,w eg e t V2/H20849/H9278/H20850=V¯/H208811 − cos/H9278 /H9257+ cos/H9278, /H20849A9/H20850 where V¯=V0//H20849kF/H208812/H20850,/H9257=k02/2kF2−1. The angular momentum components Vlmay be calculated as Vl=/H20885 02/H9266d/H9278 2/H9266cos/H20849l/H9278/H20850V2/H20849/H9278/H20850. /H20849A10 /H20850 In particular, we ﬁnd V0=2 /H9266V¯/H20849/H9257−1/H20850−1 /2arctan/H208812 /H9257−1/H110220, V1=2 /H9266V¯/H20877−/H9257 /H20881/H9257−1arctan/H208812 /H9257−1+/H208812/H20878/H333550. /H20849A11 /H20850 In the limit /H9257→1, the ratio of the l=1 and l=0 components is given by V1/V0=−/H9257. We may estimate /H9257by assuming Z electrons in a unit cell of area a2resulting in kF2=2/H9266Z/a2and therefore/H9257=2/H9266/Z−1. For Z/H110152,5 appropriate for a mixture of Fe2+and Fe3+, one ﬁnds /H9257/H110151.5 and then V1/V0/H11015−0.6. In the following, we will take the Vlto be given parameters, which may be negative. In order to keep the calculation simple, we will neglect all angular momentum components with /H20841l/H20841/H333562. Deﬁning dimen- sionless quantities V¯l=/H9266N/H9268Vlas before, the dimensionless scattering amplitudes are given by f¯ 0s=V0//H208491+isV¯0/H20850,f¯ ±1,/H9268s=V¯±1,/H9268//H208491+isV¯±1,/H9268/H20850,V¯±1,/H9268=V1±/H20881u/H9270/H9268/H9268z. /H20849A12 /H20850 Assuming weak spin-orbit scattering, we may expand in /H20881u, f¯ ±1,/H9268s=V1 1+isV¯1±/H208491+isV¯1/H208502/H20881u/H9270/H9268/H9268z. /H20849A13 /H20850 The normalization factor /H92530entering the expression for the relaxation rate is obtained as /H92530=w 1+w+2w1 1+w1+O/H20849/H20881u/H20850, /H20849A14 /H20850 where w=V02,w1=V12. The eigenvalue /H92611oftkk/H11032is found as /H92611=1 /H92530/H208752V0V1/H208491+V0V1/H20850 /H208491+w/H20850/H208491+w1/H20850+2i/H20881u/H9270/H9268/H9268zV0V0/H208491−w1/H20850−2V1 /H208491+w/H20850/H208491+w1/H208502/H20876. /H20849A15 /H20850 Analyzing this expression, one ﬁnds that the largest negative values of /H92611are reached for weak scattering, V0,V1/H112701, when /H92611=2V0 w+w1/H20851V1+i/H20849V0−2V1/H20850/H20881u/H9270/H9268/H9268z/H20852. /H20849A16 /H20850 The minimum of /H92611/H11032is obtained if V1/V0=−1 //H208812, where/H92611/H11032 =−1 //H208812. Let us now consider diagram w2, which is determined by the parameter J2/H9251/H9252, given by J2xx=−/H92530−1/H20851/H208491+/H9261˜1/H208502/H20849f¯ 0+f¯ +1,/H9268++f¯ 0−f¯ −1,/H9268−/H20850+ c.c. /H20852, b1/H11013f¯ 0+f¯ +1,/H9268++f¯ 0−f¯ −1,/H9268−/H20849A17 /H20850 =2V0 /H208491+w/H20850/H208491+w1/H20850 /H11003/H20875V1/H208491−V0V1/H20850−i/H20881u/H9270/H9268/H9268zV0/H208491−w1/H20850+2V1 /H208491+w1/H20850/H20876./H20849A18 /H20850 In the weak scattering limit, we have /H92521/H11013b1//H92530=2V0 w+w1/H20851V1−i/H208492V1+V0/H20850/H20881u/H9270/H9268/H9268z/H20852,/H20849A19 /H20850 which differs from /H92611only by the sign of the term V¯0in the imaginary part, i.e., /H92521/H11032=/H92611/H11032. muttalib@phys.uﬂ.edu †woelﬂe@tkm.uni-karlsruhe.de 1R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 /H208491954 /H20850;W . Kohn and J. M. Luttinger, ibid. 108, 590 /H208491957 /H20850. 2J. Smit, Physica /H20849Amsterdam /H2085021, 877 /H208491955 /H20850; Phys. Rev. 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PhysRevB.98.035109.pdf	PHYSICAL REVIEW B 98, 035109 (2018) Equilibrium and real-time properties of the spin correlation function in the two-impurity Kondo model Benedikt Lechtenberg1and Frithjof B. Anders2 1Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2Lehrstuhl für Theoretische Physik II, Technische Universität Dortmund, 44221 Dortmund,Germany (Received 11 May 2018; revised manuscript received 20 June 2018; published 6 July 2018) We investigate the equilibrium and real-time properties of the spin-correlation function /angbracketleft/vectorS1/vectorS2/angbracketrightin the two- impurity Kondo model for different distances Rbetween the two-impurity spins. It is shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect governs the amplitude of/angbracketleft/vectorS 1/vectorS2/angbracketright. For distances Rexceeding the Kondo length scale, the Kondo effect also has a profound effect on the sign of the correlation function. For ferromagnetic Heisenberg couplings Jbetween the impurities and the conduction band, the Kondo effect is absent and the correlation function only decays for distances beyond a certain lengthscale introduced by ﬁnite temperature. The real-time dynamics after a sudden quench of the system reveals thatcorrelations propagate through the conduction band with Fermi velocity. We identify two distinct timescales forthe long-time behavior, which reﬂects that for small Jthe system is driven by the RKKY interaction while for largeJthe Kondo effect dominates. Interestingly, we ﬁnd that at certain distances a one-dimensional dispersion obeying /epsilon1(k)=/epsilon1(−k) may lead to a local parity conservation of the impurities such that /angbracketleft/vectorS 1/vectorS2/angbracketrightbecomes a conserved quantity for long times and does not decay to its equilibrium value. DOI: 10.1103/PhysRevB.98.035109 I. INTRODUCTION Quantum impurity systems are promising candidates for the realization of solid-state-based quantum bits [ 1–5]. The perspective of combining traditional electronics with novelspintronics devices leads to an intense research of controllingand switching magnetic properties of such systems. Mag-netic properties of adatoms on surfaces [ 6–11] or magnetic molecules [ 12–19] might serve as the smallest building blocks for such devices. From a theoretical perspective, the two-impurity Kondo model (TIKM) [ 20–24] constitutes an important but simple system which embodies the competition of interactions be-tween two localized magnetic moments with those betweenthe impurities and the conduction band. The TIKM has beenviewed as a paradigm model for the formation of two differentsinglet phases separated by a quantum critical point (QCP): aRuderman-Kittel-Kasuya-Yosida (RKKY) [ 25–27] interaction induced singlet and a Kondo singlet [ 28]. This quantum critical point investigated by Jones and Varma (see Refs. [ 20,21,29]), however, turned out to be unstable against particle-hole (PH)symmetry breaking [ 24]. The two different singlet phases are adiabatically connected by a continuous variation of thescattering phase. This led to the conclusion that for ﬁnite dis- tances between the impurities no QCP exists, and the original ﬁnding is just a consequence of unphysical approximations[23] which is generically replaced by a crossover regime [30]. Only recently, it has been shown [ 31] that for certain dispersions and distances between the impurities the TIKMexhibits a QCP between two orthogonal ground states withdifferent degeneracy. In this paper, we examine the equilibrium as well as nonequilibrium properties of the spin-correlation function/angbracketleft/vectorS 1/vectorS2/angbracketright(R) for different distances Rbetween both impu- rity spins using the numerical renormalization group (NRG)[32,33] and its extension to the nonequilibrium dynamics, the time-dependent NRG (TD-NRG) [ 34,35]. Previously, the spa- tial dependence of the equilibrium properties has been mainlystudied using a simpliﬁed density of states (DOS) [ 20,21,36] that suppresses the antiferromagnetic (AFM) correlations [ 24]. In this paper, we include the full energy dependency of theeven- and odd-parity conduction-band DOSs that properlyencode the ferromagnetic (FM) as well as the antiferromag-netic contributions to the RKKY interaction. This approachgenerates the correct RKKY interaction and does not requireadding an artiﬁcial spin-spin interaction to account for thisterm [ 20,21,36]. In order to set the stage for the investigation of the nonequi- librium quench dynamics, we present results for the impurity spin-spin-correlation function /angbracketleft/vectorS 1/vectorS2/angbracketright(R). For an isotropic dispersion in one dimension, we ﬁnd that the amplitude of/angbracketleft/vectorS 1/vectorS2/angbracketright(R) is completely governed by the ratio between the distance of the impurities and the Kondo length scale R/ξ K. ξK=vF/TKis often referred to as the size of the Kondo screening cloud where vFdenotes the Fermi velocity of the metallic host and TKdenotes the Kondo temperature. For small distances R<ξ Kand vanishing temperature, steplike oscillations between ferromagnetic and antiferromagnetic cor-relations can be observed for /angbracketleft/vectorS 1/vectorS2/angbracketright(R) due to the RKKY interaction. Interestingly, at large distances R/greaterorequalslantξKthe fer- romagnetic correlations vanish and only small antiferromag-netic correlations between the impurities are found. Theseweak antiferromagnetic correlations are related to the PH symmetry breaking in the two parity channels and vanish for R→∞ . 2469-9950/2018/98(3)/035109(13) 035109-1 ©2018 American Physical SocietyBENEDIKT LECHTENBERG AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 98, 035109 (2018) For a ferromagnetic coupling between the impurities and the conduction band, the Kondo effect is absent, and a constantamplitude for the correlations is observed at zero temperature even for R→∞ . A ﬁnite temperature introduces a new length scale beyond which correlations are exponentially suppressed. The time dynamics of the correlation function /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) is examined after a quench in the coupling strength betweenthe impurities and the conduction band, starting from initiallydecoupled impurities. Experimentally, such quenches can berealized with strong laser light [ 37]. We have identiﬁed two distinct timescales characterizing the long-time behavior: The RKKY interaction drives the dynamics for small Kondo cou-pling whereas a timescale ∝1/√ TKindicates that the physics is dominated by the Kondo effect at large Kondo coupling. The correlation function approaches its equilibrium value in the steady state for most distances. For special R, however, it remains almost constant although the RKKY interaction reaches a ferromagnetic maximum for those distances. Focus- ing on a dispersion of an inversion symmetric one-dimensional(1D) lattice, parity conduction-band states decouple from theimpurities at low temperatures, thus enforcing a local impurityparity conservation such that /angbracketleft/vectorS 1/vectorS2/angbracketrightbecomes a conserved quantity for long times. We combined the time-dependent correlation functions for different but ﬁxed distances into a two-dimensional (2D) spatial-temporal picture of the real-time dynamics. It allowsfor better visualization of the the propagation of correlations.Starting from a distance around k FR/π=0.5 a ferromagnetic correlation emerges which afterwards propagates with theFermi velocity v F, deﬁning a light cone [ 38,39], through such a ﬁctitious two-impurity Kondo system with variable impurity distance R. II. MODEL AND METHODS A. Mapping the model onto an effective two-band model While Wilson’s original NRG approach [ 32] was only designed to solve the thermodynamics of one localized impu-rity, the NRG was later successfully extended by Jones andVarma (see Refs. [ 12,20,21,24,36,39,40]) to two impurities separated by a distance R. For this purpose the conduction band is divided into two bands, one with even-parity andone with odd-parity symmetry, the effective DOSs of whichincorporated the spatial extension. In the following, we brieﬂysummarize this procedure for the TIKM. The Hamiltonian of the TIKM can be separated into three partsH=H c+Hint+Hd.Hccontains the conduction band Hc=/summationtext /vectork,σ/epsilon1/vectorkc† /vectork,σc/vectork,σwhere c† /vectork,σcreates an electron with spin σand momentum /vectork. The interaction between the conduction band and the impurities is given by Hint=J[/vectorS1/vectorsc(/vectorR1)+/vectorS2/vectorsc(/vectorR2)], (1) where the impurity /vectorSilocated at position /vectorRiis coupled via the effective Heisenberg coupling Jto the unit-cell volume averaged conduction electron spin /vectorsc(/vectorr)=Vu/vectors(/vectorr). Here, /vectors(/vectorr)i s the conduction-band spin density operator expanded in planarwaves: /vectors(/vectorr)=1 21 NVu/summationdisplay σσ/prime/summationdisplay /vectork/vectork/primec† /vectorkσ[/vectorσ]σσ/primec/vectork/primeσ/primeei(/vectork/prime−/vectork)/vectorr, (2) withNbeing the number of unit cells in the volume V,Vu= V/N the volume of such a unit cell, /vectorka momentum vector, and/vectorσa vector of the Pauli matrices. In the following, we set the origin of the coordinate system in the middle of the twoimpurities such that /vectorR 1=/vectorR/2 and /vectorR2=−/vectorR/2. HDcomprises all contribution acting only on the impurities Hd=K/vectorS1/vectorS2, (3) with the direct Heisenberg interaction Kbetween two-impurity spins. Unless stated otherwise, we use K=0 throughout this paper. Instead, the correlations between the two-impurity spins are caused by the indirect Heisenberg interaction KRKKY∝J2 which is mediated by the conduction-band electrons [ 25–27]. Exploiting the symmetry [ 12,20,21,24,36,39–41], the con- duction electron band is mapped onto the two distance andenergy dependent orthogonal even-parity ( e) and odd-parity (o) eigenstate ﬁeld operators: c σ,e/o(/epsilon1)=/summationdisplay /vectorkδ(/epsilon1−/epsilon1/vectork)c/vectork,σ(e+i/vectork/vectorR/2±e−i/vectork/vectorR/2) Ne/o(/epsilon1,/vectorR)√Nρc(/epsilon1).(4) Hereρc(/epsilon1) is the DOS of the original conduction band and the dimensionless normalization functions are deﬁned as N2 e(/epsilon1,/vectorR)=4 Nρc(/epsilon1)/summationdisplay /vectorkδ(/epsilon1−/epsilon1/vectork) cos2/parenleftBigg/vectork/vectorR 2/parenrightBigg ,(5a) N2 o(/epsilon1,/vectorR)=4 Nρc(/epsilon1)/summationdisplay /vectorkδ(/epsilon1−/epsilon1/vectork)s i n2/parenleftBigg/vectork/vectorR 2/parenrightBigg (5b) such that cσ,e/o(/epsilon1) fulﬁll the standard anticommutator relation {cσ,p(/epsilon1),c† σ/prime,p/prime(/epsilon1/prime)}=δσ,σ/primeδp,p/primeδ(/epsilon1−/epsilon1/prime). With these even- and odd-parity conduction-band states the interaction part of theHamiltonian reads H int=J 8/integraldisplay/integraldisplay d/epsilon1 d/epsilon1/prime/radicalbig ρc(/epsilon1)ρc(/epsilon1/prime)/summationdisplay σσ/prime/vectorσσσ/prime ×/braceleftBigg (/vectorS1+/vectorS2)/summationdisplay p[Np(/epsilon1,R)Np(/epsilon1/prime,R)c† σ,p(/epsilon1)cσ/prime,p(/epsilon1/prime)] +(/vectorS1−/vectorS2)Ne(/epsilon1,R)No(/epsilon1/prime,R)[c† σ,e(/epsilon1)cσ/prime,o(/epsilon1/prime)+H.c.]/bracerightBigg . (6) It is important to note that due to the energy dependent factors Np(/epsilon1,R) the model will generally be particle-hole asymmetric even if the original conduction band and, therefore, the orig-inal DOS ρ c(/epsilon1) are particle-hole symmetric. For Ne(/epsilon1,R)/negationslash= No(/epsilon1,R) this asymmetry will generate potential scattering terms that are different for the even and odd conduction bandsand lead to the destruction of the Jones and Varma QCP (seeRefs. [ 24,42]). Up until now we have not speciﬁed the dispersion of the conduction band. Unless stated otherwise, we will use a 1D 035109-2EQUILIBRIUM AND REAL-TIME PROPERTIES OF THE … PHYSICAL REVIEW B 98, 035109 (2018) 00.511.522.5 −1 −0.50 0 .51[N1D e/o(R)]2/2 Deven, kFR=π odd,kFR=πeven, kFR=2π odd,kFR=2π FIG. 1. Normalization functions of Eq. ( 7) for a linear dispersion in one dimension for two different distances kFR=π(red) and kFR=2π(blue). For these distances either the even (solid) or the odd (dashed) normalization function exhibits a pseudogap at /epsilon1=0. linear dispersion /epsilon1(k)=vF(\|k\|−kF) throughout this paper which yields for the normalization functions [ 39,40] /bracketleftbig N1D e/o(/epsilon1,R)/bracketrightbig2ρc(/epsilon1)=2ρc(/epsilon1)/braceleftbigg 1±cos/bracketleftbigg kFR/parenleftbigg 1+/epsilon1 D/parenrightbigg/bracketrightbigg/bracerightbigg , (7) with the half bandwidth D=vFkF.[N1D e/o(/epsilon1,R)]2are plotted for the two different distances kFR/π=1a n d2i nF i g . 1.N o t e that one of the normalization functions exhibits a pseudogapat the Fermi energy /epsilon1=0 for distances k FR/π=n, withn= 0,1,2,... as a consequence of the dispersion /epsilon1(k)=/epsilon1(−k) [31] employed here. This can also be seen from the deﬁnitions in Eq. ( 7). It has been pointed out that the absence of the screening in one of the parity channels leads to the breakdownof the two-stage Kondo screening process and the emergenceof a new kind of quantum critical point in the TIKM [ 31]. This has also a profound effect on the time dynamics of the TIKM. In addition to the emergence of the pseudogap, both nor- malization functions are particle-hole symmetric for thesespecial distances and, thus, lead to a completely particle-holesymmetric model. B. Nonequilibrium dynamics and the TD-NRG In order to calculate the real-time dynamics of the TIKM, we employ the TD-NRG, which is an extension of the standardNRG. The TD-NRG [ 34,35] is designed to calculate the full nonequilibrium dynamics of a quantum impurity system aftera sudden quench: H(t)=H 0/Theta1(−t)+Hf/Theta1(t). For this purpose, the initial state of the system is described by the density operator ρ0=e−βH 0 Tr[e−βH 0], (8) until at time t=0 the system is suddenly quenched. After- wards, the system is characterized by the Hamiltonian Hfand the time evolution of the density operator is given by ρ(t/greaterorequalslant0)=e−itHfρ0eitHf. (9) By means of the TD-NRG the time-dependent expectation valueO(t) of a general local operator Oshould be calculated.In this paper, the local operator is given by the spin-correlation function of both impurities O=/angbracketleft/vectorS1/vectorS2/angbracketright. The time evolution of such local operators can be written as [34,35] /angbracketleftO/angbracketright(t)=N/summationdisplay mtrun/summationdisplay r,seit(Em r−Em s)Om r,sρred s,r(m), (10) where Em randEm sare the NRG eigenenergies of the Hamilto- nianHfat iteration m/lessorequalslantN,Om r,sis the matrix representation ofOat that iteration, and ρred s,r(m) is the reduced density matrix deﬁned as ρred s,r(m)=/summationdisplay e/angbracketlefts,e;m\|ρ0\|r,e;m/angbracketright, (11) in which the environment is traced out. In Eq. ( 10) the restricted sums over randsrequire that at least one of these states is discarded at iteration m. The temperature TN∝/Lambda1−N/2of the TD-NRG calculation is deﬁned by the length of the NRGWilson chain Nand enters Eq. ( 8). Here, /Lambda1> 1 denotes the Wilson discretization parameter. The TD-NRG comprises two simultaneous NRG runs: one for the initial Hamiltonian H 0in order to compute the initial density operator ρ0of the system in Eq. ( 8) and one for Hfto obtain the approximate eigenbasis governing the time evolution in Eq. ( 10). This approach has also been extended to multiple quenches [43], time evolution of spectral functions [ 44], and steady-state currents at ﬁnite bias [ 45–47]. The only error of this method originates from the representation of the bath continuum bya ﬁnite-size Wilson chain [ 32]. This error is essentially well understood [ 48,49] and may lead to artiﬁcial oscillations and slight deviations of the long time value from the exact result.These can be reduced by using an increased number of NRGz-tricks [ 50]. III. EQUILIBRIUM A. Antiferromagnetic coupling J Two characteristic length scales have been identiﬁed [ 39,40] in the TIKM with an antiferromagnetic JforT=0: the inverse Fermi momentum 1 /kFand the Kondo length scale ξK=vF/TKwith the Kondo temperature TK=√ρJe−1/ρJ. The length scale 1 /kFdeﬁnes the oscillations of the RKKY interaction and its envelope. As ξKchanges exponentially with the Kondo coupling J, we use different Jto examine the different distances R<ξ KandR>ξ K. The impurity spin-correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R)i ss h o w n in conjunction with the RKKY interaction (dashed line) inFig. 2(a) for different couplings J.F o rR/lessmuchξ K(small Kondo couplings J) one can clearly observe oscillations between ferromagnetic and antiferromagnetic correlations caused bythe RKKY interaction. For an effective ferromagnetic RKKYinteraction, the impurity spins align parallel while for anantiferromagnetic interaction they align antiparallel. For these small distances R/lessmuchξ Kthe impurity spins are located inside the respective screening cloud of the otherimpurity and are not completely screened by the conductionelectrons. Therefore, the Kondo effect has almost no effect on/angbracketleft/vectorS 1/vectorS2/angbracketright(R), which can be seen by a comparison with Fig. 4(a) 035109-3BENEDIKT LECHTENBERG AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 98, 035109 (2018) (a) −1−0.8−0.6−0.4−0.200.20.4 01234567S1·S2 kFR/πRKKY ρJ=0.10 ρJ=0.15ρJ=0.20 ρJ=0.30 ρJ=0.35ρJ=0.40 ρJ=0.45 ρJ=0.50 (b) −0.0500.050.10.150.20.25 10−210−1100101102103−0.0500.050.10.150.20.25 10−1100101102103S1·S2 R/ξ KρJ=0.15 ρJ=0.20 ρJ=0.25ρJ=0.30 ρJ=0.35 ρJ=0.40 R FIG. 2. (a) The /angbracketleft/vectorS1/vectorS2/angbracketright(R) correlation function plotted against the distance kFR/π for different antiferromagnetic couplings J. The red dashed line depicts the 1D RKKY interaction ∝1/Rin arbitrary units. Note that for distances R≈ξK(large Jvalues) the ferromagnetic correlations around kFR/π=nbegin to vanish. ForR/greatermuchξKonly at exactly kFR/π=nferromagnetic correlations persist. (b) Correlation function for the distances kFR/π=(n+0.11) and different couplings Jplotted against the rescaled distance R/ξ K. The inset shows the same data vs the distance R. showing /angbracketleft/vectorS1/vectorS2/angbracketright(R) for ferromagnetic couplings Jwhere the Kondo effect is absent. Note that /angbracketleft/vectorS1/vectorS2/angbracketright(R) does not decay but instead exhibits steplike oscillations with a constant amplitude since it reﬂectsthe ground-state properties of two free impurity spins which arecoupled via a Heisenberg interaction. Even for an inﬁnitesimalsmall effective Heisenberg interaction between the impurityspins, the spins align completely parallel or antiparallel atT=0. This behavior is modiﬁed for larger couplings J, where R/lessmuchξ Kis not valid anymore due to an increasing Kondo temperature. Upon lowering the temperatures, the spins beginto align parallel or antiparallel until the Kondo temperatureT Kis reached, at which the impurities are screened by the conduction electrons and, hence, the correlation functiondoes not change anymore. The exact value of the correlationfunction depends on the ratio between the RKKY interactionand the Kondo temperature K RKKY/TK[20].Furthermore, one should note that when the distance ap- proaches the Kondo length scale, R∼ξK, large JandR in Fig. 2(a), the Kondo effect leads to a drastic departure from the conventional RKKY interaction [ 51]. While for smallJandRthe position of the sign change of the RKKY interaction agrees with the position of the sign change of thecorrelation function, the latter is shifted towards the integerdistances k FR/π=nwith increasing coupling Jand distance R. The interval in the vicinity of the distances kFR/π=n where we observe ferromagnetic correlation, therefore, shrinksand antiferromagnetic correlations between the impurity spinsemerge instead [ 22]. Sinceξ Kexponentially depends on the Kondo coupling J, the precise distance Rat which the ferromagnetic correlations disappear is also Jdependent. Therefore, one almost only ob- serves antiferromagnetic correlations between the impuritiesforR/greatermuchξ K. In order to review the inﬂuence of the Kondo effect on the ferromagnetic correlations, we calculated /angbracketleft/vectorS1/vectorS2/angbracketright(R)a tt h e distances kFR/π=(n+0.11) where we expect a ﬁnite FM RKKY interaction. The results are shown in Fig. 2(b) plotted as a function of the rescaled distance R/ξ Kand as a function ofRin the inset. The crossover from FM to AFM is governed by the Kondo effect and occurs once the distance exceedsR∼0.53ξ K. Based on the observed universality, we can understand this surprising sign change of the spin-spin-correlation functionwithin the strong-coupling limit. For J→∞ , a Kondo sin- glet is formed locally at each impurity site, and the localconduction-band electron is antiparallel to the local spin. Inthis case, the system consists of two Kondo singlets whichare decoupled from the remaining Fermi sea with two missingelectrons. In the generic case, however, the two bound statesin the even-odd basis are subject to the potential scatteringterms emerging from the particle-hole asymmetry (see Sec.II A). These scattering terms are different for the even and odd conduction-band channel [ 24,52] and, hence, generate a hopping term between the bound states in the real-space basis. This hopping term evokes an antiferromagnetic interaction so that the two bound conduction electron spins arrange inopposite orientation inducing an AF correlation between theimpurity spins as observed in Fig. 2(b) forR/ξ K>1. A word is in order to justify the choice kFR/π=(n+0.11) as generic distance. kFR/π=nleads to a different physics [31] for a linear dispersion in one dimension considered here for two reasons: At ﬁrst, one of the two parity conduction bandsdevelops a pseudogap DOS at low temperatures, as depictedin Fig. 1, and does not participate in the screening any more. Second, at k FR/π=nthe system is perfectly particle-hole symmetric and the above-mentioned additional hopping termbetween the bound conduction electrons does not appear.Consequently, the system is equivalent to the physics at R=0 [12] for these distances and ferromagnetic correlations remain for all integers n. A similar behavior has also been observed in the single impurity Kondo model (SIKM) for the correlation function/angbracketleft/vectorS/vectors(R)/angbracketrightwhich measures the correlations between the impurity spin and the spin density of the conduction band in distance R to the impurity [ 39]. The ferromagnetic correlations located at k FR/π=(n+0.5) vanish for distances R>ξ Kand instead 035109-4EQUILIBRIUM AND REAL-TIME PROPERTIES OF THE … PHYSICAL REVIEW B 98, 035109 (2018) 10−610−510−410−310−210−1100 0.001 0 .01 0 .1 1 10 100−0.4−0.20.00.20.4 01234567S1·S2 R/ξKρJ=0.15 ρJ=0.20 ρJ=0.25 ρJ=0.30 ρJ=0.35 ρJ=0.40 ∝1/R2S1·s(R)kFR/πkFR/π ρJ=0.15 ρJ=0.20 ρJ=0.30ρJ=0.40 ρJ=0.45 ρJ=0.50 FIG. 3. The envelope \|/angbracketleft/vectorS1/vectorS2/angbracketright(R)\|of the impurity spin-correlation function on a double logarithmic scale plotted against the rescaled distance R/ξ K. The rescaling leads to a universal behavior. For large distances R/greatermuchξKa1/R2decrease is observed. The inset shows the correlation function /angbracketleft/vectorS/vectors(R)/angbracketrightbetween an impurity spin /vectorSand the conduction-band spin density /vectors(R) at distance Rfrom the impurity. also antiferromagnetic correlations appear in accordance with theoretical predictions [ 53,54]. The inset of Fig. 3shows the same correlation function /angbracketleft/vectorS/vectors(R)/angbracketrightfor the TIKM measuring the correlation between an impurity spin and the conduction-band spin density at theposition of the second impurity located a distance Rfrom the ﬁrst impurity. To counteract the decay, the correlationfunction has been rescaled with the distance Rfor a better prospect. In comparison to the correlation function for theSIKM, the second impurity leads to a π/2 phase shift such that now the antiferromagnetic correlations around k FR/π=n instead of the ferromagnetic ones around kFR/π=(n+0.5) vanish. Consequently, the ferromagnetic correlations betweenthe impurity spins /angbracketleft/vectorS 1/vectorS2/angbracketright(R) at the distances kFR/π=n also have to vanish since the RKKY interaction between theimpurity spins is mediated by the conduction band. Figure 3depicts the envelope of /angbracketleft/vectorS 1/vectorS2/angbracketright(R) measured at the distances kFR/π=(n+0.5). The universal behavior of the envelope function is revealed by plotting the data as afunction of the dimensionless distance R/ξ K. This shows that the amplitude of the correlation function is completelygoverned by the distance dependent RKKY interaction and theKondo effect. For large distances R/greatermuchξ Ka∝1/R2behavior, indicated by the solid line, is observed. At these large distancesthe impurities are located outside of the Kondo screening cloudof the respective other almost completely screened impurity,therefore the ∝1/Rdecay of the RKKY interaction in one dimension is enhanced to a ∝1/R 2decay. The same ∝1/R2 behavior for R/greatermuchξKhas also been found for the correlation between an impurity spin and the conduction-band spin density/angbracketleft/vectorS/vectors(R)/angbracketrightin the SIKM [ 39]. B. Ferromagnetic coupling Jand ﬁnite temperatures So far, we have only investigated the TIKM for an antifer- romagnetic coupling Jwhere the Kondo effect is present. We now extend our discussion also to ferromagnetic J.(a) −1−0.8−0.6−0.4−0.200.20.4 01234567S1·S2 kFR/πRKKY ρJ=−0.1ρJ=−0.2 ρJ=−0.3ρJ=−0.5 (b) −0.8−0.6−0.4−0.200.2 10−610−510−410−310−210−11000.040.050.060.070.080.090.100.110.120.13 0.10 .20 .30 .40 .5S1·S2 T/DρJ=−0.10ρJ=−0.15ρJ=−0.20ρJ=−0.25ρJ=−0.30ρJ=−0.35ρJ=−0.40ρJ=−0.45ρJ=−0.50Sis(ri) ρJ FIG. 4. (a) The /angbracketleft/vectorS1/vectorS2/angbracketright(R) correlation function vs the distance kFR/π for different ferromagnetic couplings J. The red dashed lines depicts the 1D RKKY interaction ∝1/Rin arbitrary units. (b) Temperature-dependent correlation function for different cou- plings and the two different distances kFR/π=1.0 (solid lines), where the RKKY interaction is ferromagnetic, and kFR/π=0.5 (dashed lines), where the RKKY interaction is antiferromagnetic.The inset shows the ﬁxed-point value of the correlation between an impurity spin and the spin density of the conduction electrons at the position of this impurity /angbracketleft/vectorS i/vectors(ri)/angbracketrightfor different couplings Jand the two distances kFR/π=1.0 (red solid line) and kFR/π=0.5 (blue dashed line). The correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R) for ferromagnetic cou- plings as well as the RKKY interaction (dashed line) is depictedin Fig. 4(a). Since the Kondo effect is absent, there is no screening of the local moments with increasing Jin contrast to AFMJshown in Fig. 2(a). The correlation function preserves its steplike oscillations even for very large ferromagneticcouplings. Similar to the case for antiferromagnetic J, we observe that the modulus of /angbracketleft/vectorS 1/vectorS2/angbracketright(R) is reduced with increasing \|J\|.H o w - ever, the decrease is much weaker than for antiferromagnetic J. This reduction cannot be caused by a screening of the impurityand, therefore, must have a different origin. Figure 4(b) depicts the temperature-dependent correlation function for different ferromagnetic couplings Jand for the distance k FR/π=1.0 (solid lines), where the RKKY 035109-5BENEDIKT LECHTENBERG AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 98, 035109 (2018) 10−610−510−410−310−210−1100 0.1 1 10 100 1000 10000T/D=0.0001 T/D=0.001T/D=0S1·S2 kFR/πρJ=−0.15 ρJ=−0.20 ρJ=−0.25 ρJ=−0.30 ρJ=−0.35 ρJ=−0.40 ∝1/x1.30 ∝1/x1.22 FIG. 5. The envelope of /angbracketleft/vectorS1/vectorS2/angbracketright(R) for ferromagnetic couplings depicted on a double logarithmic scale. For T=0, the amplitude of the correlation function remains constant for all distances. For the ﬁnite temperatures T/D=0.001 and 0.0001 a power-law decay is observed when the RKKY interaction is smaller than the temperature Twhich turns over into an exponential decay once the length scale ξT=vF/Tis reached. interaction is ferromagnetic, as well as for kFR/π=0.5 (dashed lines), where the RKKY interaction is antiferro-magnetic. For both regimes we observe (i) a reduction ofthe modulus of the spin-correlation function with increasingKondo coupling Jand (ii) a simultaneous increase of the crossover temperature from two uncorrelated spins at hightemperatures to spin correlation in the ﬁxed point. For a ferromagnetic coupling J< 0, the effective coupling in the NRG renormalization ﬂow is renormalized to zeroJ eff→0f o rT→0[55]. As soon as the effective coupling is zero, a ﬁxed point is reached and, consequently, the correlationfunction reaches its ﬁxed-point value. Note, however, that theoperator content of the renormalized operators is important:The larger the Kondo coupling, the smaller the fraction ofthe original spin that contributes to the effective spin degreeof freedom that decouples from the conduction band. This isdemonstrated in the inset of Fig. 4(b), which shows the ﬁxed- point value of the correlation between an impurity spin and thespin density of the conduction electrons at the position of thisimpurity /angbracketleft/vectorS i/vectors(ri)/angbracketrightfor different couplings J. The correlations remain ﬁnite in the ﬁxed point even if the effective couplingis renormalized to zero since only a part of the impurity spinsdecouples. The fraction of the impurity spins which remainscoupled to the conduction band is the larger the larger Jis. Therefore, for a ﬁnite coupling to the conduction band the effective decoupled spins are reduced in the renormalizationﬂow until the ﬁxed point is reached where J eff=0, which is the origin of the reduction of \|/angbracketleft/vectorS1/vectorS2/angbracketright(R)\|for FM and AFM RKKY couplings. Note, however, that a clearly noticeable reduction of the amplitude occurs only for very large ferromagneticcouplings J. An increasing crossover scale to the ﬁxed point with increasing Jis not observed in the SIKM and can, therefore, be ascribed to a growing RKKY interaction ∝J 2since it is theonly additional effect in a TIKM with ferromagnetic couplings. We have also checked that an increasing direct Heisenberginteraction between the impurity spins, as given in Eq. ( 3), with vanishing RKKY interaction ( R→∞ ) has the same effect as a ﬁnite indirect RKKY interaction and also leads to an increasingcrossover scale. It is already known that the RKKY interactionhas a profound effect on the renormalization ﬂow of the TIKMfor antiferromagnetic Kondo couplings [ 56]. Figure 5shows the envelope of the correlation function for different ferromagnetic couplings and different temperatures.As can be seen, for zero temperature T/D=0, the amplitude is almost constant even for R→∞ . This changes for the ﬁnite temperatures T/D=0.001 and 0.0001 where a power-law decay is observed as soon as the energy scale of the RKKYinteraction is smaller than the temperature. However, the ﬁnitetemperature also introduces a new length scale ξ T=vF/T beyond which the correlation function decays exponentially.The same ﬁnite temperature behavior has also been found inthe SIKM for the correlation between the impurity spin andthe spin density of the conduction band at a distance Rfrom the impurity [ 40]. IV . REAL-TIME DYNAMICS OF THE TIKM A. Spin-spin-correlation function after a quench The discussion of the equilibrium properties in the previous section sets the stage for the investigation of the real-timedynamics of the time-dependent spin-spin-correlation function/angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) after a quench of the system. We focus on anti- ferromagnetic Kondo couplings J> 0 and set the coupling of the impurities to the conduction band initially to zeroJ=0 such that the impurities are completely decoupled from the band. At time t=0 the coupling is switched on to a ﬁnite antiferromagnetic value J> 0 and the time-dependent behavior of /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) is calculated using the TD-NRG by evaluating Eq. ( 10). Figure 6(a) shows /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) for times up to tD=106 after such a quench for three different distances. The RKKY interaction is antiferromagnetic at the distance kFR/π=0.51 and ferromagnetic for the distances kFR/π=1.00 and 1.11. As can be seen, the correlation function behaves very differ-ently for the three different distances, even for the two distancesat which the RKKY interaction is ferromagnetic and a similarbehavior is expected. A ferromagnetic correlation emerges for small times for all distances the origin of which is caused by a ferromagneticwave propagating through the system as we will show later. Forthe distance k FR/π=0.51, the correlation function becomes antiferromagnetic only at longer times and approaches itsequilibrium value. Note the log timescale in Fig. 6(a).T h e equilibrium value of about /angbracketleft/vectorS 1/vectorS2/angbracketright(kFR/π=0.51)≈− 0.42 is, however, not completely reached. For strong antiferromag-netic interactions the two impurity spins form a singlet andthus decouple from the conduction band [ 20,21]. Without any additional relaxation mechanism, this decoupling prevents thecorrelation function from reaching its equilibrium value. The RKKY interaction has a ferromagnetic maximum for k FR/π=1.00. Strikingly, the correlation function changes only for short times and remains almost constant after the ﬁrst 035109-6EQUILIBRIUM AND REAL-TIME PROPERTIES OF THE … PHYSICAL REVIEW B 98, 035109 (2018) (a) −0.3−0.25−0.2−0.15−0.1−0.0500.050.10.15 10−210−1100101102103104105106S1·S2 t·DkFR/π =0.51 kFR/π =1.00 kFR/π =1.11 (b) −0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.0500.050.1 10−210−1100101102103104105106S1·S2 t·DkFR/π =0.51 kFR/π =1.00 kFR/π =1.11 FIG. 6. (a) The long-time behavior of /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) after a quench in the coupling from ρJ=0 to 0.2 for the three different distances kFR/π=0.51, 1.00, and 1.11. (b) Time dynamics of /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) after a quench in magnetic ﬁelds applied to the impurities from H1= −H2=10DtoH1=H2=0 for the same distances as in (a). NRG parameters: λ=3, Ns=2000, and Nz=32. ferromagnetic maximum. This surprising behavior is related to the property of the dispersion, i.e., /epsilon1(k)=/epsilon1(\|k\|)[31]. At special distances kFR/π=n, we observe that the impurity correlation function /angbracketleft/vectorS1/vectorS2/angbracketrightbecomes a conserved quantity resulting in a ﬁxed value for /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) for long times. In order to understand this effect, one has to examine the energy dependent normalization functions between theimpurities and the conduction band. For a 1D symmetricdispersion, either the even or the odd normalization function inEq. ( 7) exhibits a pseudogap at the Fermi energy /epsilon1=0f o rt h e distances k FR/π=n, with n=0,1,2,... ( s e ea l s oF i g . 1). Due to the pseudogap either Ne(0,R)=0o rNo(0,R)=0 always vanishes at the Fermi energy for these special distances.This also leads to the fact that the last term of the Hamiltonianin Eq. ( 6) proportional to ∝(/vectorS 1−/vectorS2)Ne(/epsilon1,R)No(/epsilon1/prime,R) always vanishes on low-energy scales for the distances kFR/π=n. This term is, however, responsible for the correlation functionto smoothly evolve from a spin triplet to a singlet value or viceversa since it mixes electrons from the even and odd conductionband via impurity scattering processes. In a parity-symmetricTIKM the global parity remains conserved; however, the local impurity parity and the parity in the conduction bands maychange. Once this term vanishes, the band mixing is suppressedand, therefore, the local impurity parity becomes a conservedquantity at low-energy scales. Consequently, the correlationfunction /angbracketleft/vectorS 1/vectorS2/angbracketright(kFR/π=n,t) is ﬁxed for long times due to parity symmetry. Note that this effect is not necessarily restricted to 1D dispersions. Generally, a dispersion is needed where at certaindistances either the even or the odd normalization functionin Eq. ( 5b) vanishes or, at least, almost vanishes for small temperatures inducing a local parity conservation. At the distance k FR/π=1.11 the RKKY interaction is also ferromagnetic, but the effective density of states does notexhibit a pseudogap. /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) approaches its ferromagnetic equilibrium value, as expected. However, the equilibrium valueof/angbracketleft/vectorS 1/vectorS2/angbracketright(R)≈0.2 is not completely reached. Although qualitatively the results remain unchanged, the long-time limit of /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) slightly depends on the dis- cretization parameter /Lambda1of the NRG for times tD > 1000. In order to demonstrate that the characteristic difference in the real-time dynamics of the correlation function is not onlyrestricted to quenches in the coupling J,F i g . 6(b) shows the be- havior of /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) after a quench in magnetic ﬁelds applied to the impurity spins from H1=−H2=10DtoH1=H2= 0. Since the impurity spins are initially antiparallel aligned, thecorrelation function starts from /angbracketleft/vectorS 1/vectorS2/angbracketright(R,0)=− 0.25 att=0. As can be seen, the behavior is very similar to Fig. 6(a) such that for the distances kFR/π=0.51 and 1.11 the correlation function approaches its equilibrium value while for kFR/π= 1.00 it remains close to its initial value. Also note that the initial condition with antiparallel aligned impurity spins is notparity symmetric; however, the Hamiltonian driving the timedynamics is parity symmetric and, therefore, leads to a localimpurity parity conservation for the distance k FR/π=1.00 for long times. B. Short-time behavior After presenting the real-time dynamics for all timescales in the previous section, we now discuss the short-time behaviorin more detail. For zero initial correlation function /angbracketleft/vectorS 1/vectorS2/angbracketright(R,0)=0, the ﬁrst- and second-order contributions in a perturbation ex-pansion in Jvanish so that the ﬁrst nonvanishing order is ∝J 3(for details see the Appendix). The impurity correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) calculated with the TD-NRG is depicted in Fig. 7(a) for the distance kFR=0.51πand different antifer- romagnetic couplings J. By rescaling the results with 1 /J3we demonstrate a perfect agreement with the scaling predictionof the perturbation theory which becomes exact in the limitt→0. Around this distance a ferromagnetic correlation develops where the peak position is only dependent on the reciprocalbandwidth, and, therefore, related to the Fermi velocity. Wewill show below that this peak will be linearly dependent onthe distance Rbetween the impurities and is related to the information spread between the two impurities. The inset of Fig. 7(a) shows the correlation function for the same distance and couplings plotted against the rescaled 035109-7BENEDIKT LECHTENBERG AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 98, 035109 (2018) (a) −0.0500.050.10.150.20.250.30.350.4 0123456−0.004−0.0020.0000.0020.0040.0060.0080.0100.012 0.01 .02 .03 .04 .05 .06 .0S1·S2/J3 t·DρJ=0.0500 ρJ=0.0625 ρJ=0.0750 ρJ=0.0875 ρJ=0.1125 ρJ=0.1375 ρJ=0.1500 S1·S2 t·J (b) −0.15−0.1−0.0500.050.10.150.20.250.3 02468 1 0 1 2 1 4S1·S2/J3 t·DρJ=0.075 ρJ=0.100 ρJ=0.125ρJ=0.150 ρJ=0.200 ρJ=0.250 FIG. 7. (a) The short-time behavior of the spin-correlation func- tion of the TIKM rescaled with 1 /J3for different couplings J and the ﬁxed distance kFR=0.51π. The inset depicts /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) against the rescaled time tJ. Note that, due to the rescaling with J, the zero crossings from positive to negative correlations at tJ≈5 approximately coincide for all J. (b) Short-time behavior for the distance kFR=1.00πand different couplings Jplotted against tD. For this distance the zero crossings from positive to negative correlations at tD≈7 coincide without any rescaling of the time. NRG parameters: λ=3, Ns=2000, Nz=16. timetJ. For the increase of the correlation function at times tJ < 1 again a universal short-time behavior is found. We can, therefore, conclude that the initial buildup of the ferromagneticwave is proportional to ∝(tJ) 3. For the distances kFR/π=n+0.5 the equilibrium correla- tion function is antiferromagnetic since the RKKY interactionreaches its largest antiferromagnetic amplitude during eachoscillation cycle. However, /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) remains ferromagnetic for a relatively long time before it later approaches its antifer-romagnetic long-time value. The inset of Fig. 7(a) reveals that the timescale of this ferromagnetic range is given by 1 /Jsince for the rescaled time tJthe zero crossing from ferromagnetic to antiferromagnetic correlations is approximately tJ≈5f o r all couplings J.(a) −0.35−0.3−0.25−0.2−0.15−0.1−0.0500.050.1 10−210−1100101102103104105106S1·S2(R,t ) t·DρJ=0.150 ρJ=0.175 ρJ=0.200 ρJ=0.225 ρJ=0.250 ρJ=0.275 ρJ=0.300 ρJ=0.325 ρJ=0.350 ρJ=0.375 ρJ=0.400 ρJ=0.425 ρJ=0.450 ρJ=0.475 ρJ=0.500 (b) −0.200.20.40.60.811.2 10−310−210−1100101102103104105f0.51(t) t/tcor 0.51ρJ=0.150 ρJ=0.175 ρJ=0.200 ρJ=0.225 ρJ=0.250 ρJ=0.275 ρJ=0.300 ρJ=0.450 ρJ=0.475 ρJ=0.500 ρJ=0.525 ρJ=0.550 ρJ=0.575 FIG. 8. (a) /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) for the distance kFR/π=0.51 and differ- ent couplings J. (b) The reduced correlation function f0.51(t) plotted against the rescaled time t/tcor 0.51. NRG parameters: λ=6, Ns=2000, Nz=32, and a TD-NRG damping α=0.2. Figure 7(b) depicts the rescaled correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R,t)/J3for the distance kFR=1.00πand different couplings. As before, a universal buildup of the ferromagneticwave can be observed. For this distance, however, the sign change from ferro- magnetic to antiferromagnetic correlations is governed by theinverse bandwidth Dthat is proportional to the Fermi velocity. Because of the decoupling of one effective band, the localparity is dynamically conserved. The energy scale, and conse-quently also the timescale, of the decoupling are deﬁned by thedistance between the impurities and the bandwidth of the con-duction band D. Therefore, due to the pseudogap formation at the distances k FR/π=n, the relevant timescale is given by D. C. Long-time behavior We now turn to the investigation of the long-time behav- ior for different couplings J. Figure 8(a) depicts the time- dependent correlation function for the distance kFR/π=0.51 and different couplings J. The correlation function reaches its long-time value /angbracketleft/vectorS1/vectorS2/angbracketright(R,t→∞ ) faster the stronger the coupling to the conduction band Jis. Furthermore, the long- time value \|/angbracketleft/vectorS1/vectorS2/angbracketright(R,t→∞ )\|is reduced with increasing J, 035109-8EQUILIBRIUM AND REAL-TIME PROPERTIES OF THE … PHYSICAL REVIEW B 98, 035109 (2018) 12345678910 0.10.15 0 .20.25 0 .30.35 0 .40.45 0 .50.55 0 .6tcor 0.51/t∗ ρJt∗=1/J4.1 t∗=1/√TK FIG. 9. The rescaled timescales tcor 0.51J4.13(red line) and tcor 0.51√TK (blue line) plotted against ρJ. which coincides with the behavior observed in the equilibrium model [see Fig. 2(a)]. In order to identify a coupling dependent timescale on which the correlation function decreases and approaches its long-timevalue, we introduce the reduced correlation function f 0.51(t)=/angbracketleft/vectorS1/vectorS2/angbracketright(kFR/π=0.51,t)−/angbracketleft/vectorS1/vectorS2/angbracketrightmin /angbracketleft/vectorS1/vectorS2/angbracketrightmax−/angbracketleft/vectorS1/vectorS2/angbracketrightmin, (12) where /angbracketleft/vectorS1/vectorS2/angbracketrightmaxis the maximum ferromagnetic value and /angbracketleft/vectorS1/vectorS2/angbracketrightminis the value of the minimum after the decrease [ 57]. We use this function to deﬁne the coupling dependent timescalet cor 0.51by the condition f0.51(tcor 0.51)=0.25. Figure 8(b) shows the reduced correlation function f0.51(t) plotted versus the rescaled timet/tcor 0.51for different couplings J. We identify two distinct universal behaviors: one for small couplings ρJ < 0.3 (solid lines) and one for larger couplings ρJ > 0.45 (dashes lines). While for small couplings Jthe RKKY interaction drives the physics, for larger couplings the Kondo effect becomesdominant. This is in accordance with the equilibrium physicsdiscussed before. For small couplings the inverse timescale 1 /t cor 0.51shows a power-law dependence 1 /tcor 0.51∝J4.1, which is very close to K2 RKKY∝J4. In contrast, for larger couplings we observe an exponential dependency on Jwhich agrees very well with√TK. In order to visualize the two different dependencies of the timescale tcor 0.51,F i g . 9shows the rescaled timescale tcor 0.51J4.1 (red line) and tcor 0.51√TK(blue line) plotted against ρJ. While for small couplings tcor 0.51J4.1is almost constant, it starts to increase forρJ > 0.3. On the other hand, for large couplings ρJ > 0.4, the curve tcor 0.51√TKis almost constant. This quantiﬁes that the crossover between an RKKY dominated physics for small J and a Kondo driven physics for large Jis also found in the characteristic timescales of the nonequilibrium dynamics. Figure 10(a) shows the long-time behavior of the correlation function for different couplings and the distance kFR/π= 1.11. Since the RKKY interaction is ferromagnetic for this distance, the correlation function increases after the ferromag-netic wave has passed. We observe that the correlation functionreaches its long-time value faster with increasing coupling(a) −0.0200.020.040.060.080.10.120.14 10−210−1100101102103104105106S1·S2(R,t ) t·DρJ=0.200 ρJ=0.225 ρJ=0.250 ρJ=0.275 ρJ=0.300 ρJ=0.325 ρJ=0.350 ρJ=0.375 ρJ=0.400 ρJ=0.425 ρJ=0.450 ρJ=0.475 ρJ=0.500 (b) −0.6−0.4−0.200.20.40.60.811.21.4 10−510−410−310−210−1100101102103104105f1.11(t) t/tcor 1.11ρJ=0.200 ρJ=0.225 ρJ=0.250 ρJ=0.275 ρJ=0.300 ρJ=0.325 FIG. 10. (a) Time-dependent behavior of the correlation function for the distance kFR/π=1.11 and different couplings J.( b )T h e reduced correlation function f1.11(t) plotted against the rescaled time t/tcor 1.11. NRG parameters: λ=6, Ns=2000,Nz=32, and a TD-NRG damping α=0.2. strength Jwhile the long-time value /angbracketleft/vectorS1/vectorS2/angbracketright(1.11,t→∞ )i s reduced. Interestingly, for large couplings ρJ > 0.3 the correlation function ﬁrst increases until its starts to decrease and caneven reach an antiferromagnetic long-time value for couplingsρJ/greaterorequalslant0.475. This behavior is in accordance with equilibrium results at low temperatures such that for the distance k FR/π= 1.11 and couplings ρJ/greaterorequalslant0.475 we also observe small anti- ferromagnetic correlation functions in the equilibrium NRGresults. This effect has already been discussed in Sec. III A . To extract a Jdependent timescale, we again deﬁne a reduced correlation function f 1.11(t)=/angbracketleft/vectorS1/vectorS2/angbracketright(kFR/π=1.11,t)−/angbracketleft/vectorS1/vectorS2/angbracketrightmin /angbracketleft/vectorS1/vectorS2/angbracketrightmax−/angbracketleft/vectorS1/vectorS2/angbracketrightmin, (13) where /angbracketleft/vectorS1/vectorS2/angbracketrightminis the value of the second minimum of /angbracketleft/vectorS1/vectorS2/angbracketright(kFR/π=1.11,t) after the ﬁrst ferromagnetic peak and /angbracketleft/vectorS1/vectorS2/angbracketrightmaxis the value of the maximum of the same function directly after the increase and before the correlation functionstarts to decrease again. Here, we modify the deﬁnition of thecoupling dependent timescale to f 1.11(tcor 1.11)=0.75. 035109-9BENEDIKT LECHTENBERG AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 98, 035109 (2018) 02004006008001000 01234567 t·D kF·R/π−0.35−0.3−0.25−0.2−0.15−0.1−0.0500.050.10.150.2 01020304050 01234567 t·D kF·R/π−0.15−0.1−0.0500.050.1(a) (b) FIG. 11. (a) Time-dependent correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R,t)f o r ρJ=0.2 and a 1D dispersion. (b) The short-time behavior in more detail. The white line indicates the Fermi velocity vF. Note that forkFR/π=nthe correlation function does not evolve towards its equilibrium value and instead remains almost zero (vertical blacklines). NRG parameters: λ=3, Ns=1400, N z=4. The reduced correlation function f1.11(t) for small cou- plings plotted against the rescaled time t/tcor 1.11is depicted in Fig. 10(b) . Due to the rescaling, we ﬁnd a universal behavior for the increase. The coupling dependency of the timescale isonce again given by t cor 1.11∝J−4.1. We can, therefore, conclude that for small couplings Jthe timescale for the long-time behavior is the same and does not depend on whether theRKKY interaction is ferromagnetic or antiferromagnetic. The examination of the timescales for larger couplings, however, turns out to be difﬁcult since, as already mentionedabove, the long-time behavior starts to become more compli-cated than a rather simple increase of the correlation functionand instead starts to decrease for long times. D. Propagation of the correlation In this section, we investigate the propagation of correla- tions through the system. For that purpose, we combine thereal-time dynamics calculations for different but ﬁxed dis-tances of the two impurities into two-dimensional plots wherethe horizontal axis denotes the dimensionless distance betweenthe two impurities and the vertical axis denotes the time. For the coupling ρJ=0.2 and a 1D dispersion Fig. 11(a) depicts the correlation function for times up to tD=1000 and distances up to k FR/π=7. For long times, /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) approaches its equilibrium value, and the steplike oscillations02004006008001000 00 .511 .522 .533 .54 t·D kFR/π−0.25−0.2−0.15−0.1−0.0500.050.10.150.2 FIG. 12. Time-dependent correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R,t)f o ra 2D linear dispersion. The white line indicates the Fermi velocity vF. NRG parameters: λ=3, Ns=1400, Nz=4. as found in equilibrium [see Fig. 2(a)] caused by the RKKY interaction are already clearly visible for times tD > 100. In the center of the ferromagnetic correlations at the magic distances kFR/π=n, the black vertical lines indicate that the correlation function remains almost zero. At these distances theRKKY interaction is maximal ferromagnetic [see Fig. 2(a)]; however, either the even-parity or the odd-parity conductionband decouples from the problem. Therefore, the local impu-rity parity becomes a conserved quantity which leads to a ﬁxedvalue for /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) as already discussed above. Figure 11(b) depicts the same data as in Fig. 11(a) for times up to tD=50 to illustrate the short-time behavior in more detail. At kFR/π=0.5 a ferromagnetic correlation evolves which then propagates with the Fermi velocity, indicated bythe white line, through the conduction band. Directly in frontof the light cone, we observe antiferromagnetic correlationsat distances k FR/π=(n+0.5). Such correlations outside of the light cone were also found for the correlations between theimpurity spin and the spin density of the conduction band atdistance Rand could be traced back to the intrinsic correlations of the Fermi sea [ 39]. These correlations are already present before the impurities are coupled to the conduction band andare a property of the Fermi sea. One can also see that for the distances k FR/π=(n+0.5) the correlation function at ﬁrst evolves towards a ferromagneticvalue for a relatively long time until it later approaches itsexpected antiferromagnetic equilibrium value since the RKKYinteraction is antiferromagnetic for these distances. It becomes apparent that the correlation function remains almost zero for distances k FR/π=nafter the ferromagnetic correlation wave has passed due to the local parity conserva-tion. Note that with increasing distance Rthe frequency of the oscillations in N 1D e/o(/epsilon1,R) increases and, consequently, the width of the gap becomes narrower so that the energy scaleon which the impurities see the gap decreases with 1 /R.T h e decreasing energy scale, on the other hand, leads to a linearlyincreasing timescale ∝Rat which /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t)i sﬁ x e d . In order to demonstrate that the local impurity parity conser- vation is a special feature of certain dispersions, Fig. 12shows the time-dependent correlation function for a linear dispersionin two dimensions. The normalization functions are givenbyN 2D e/o(/epsilon1,R)=/Gamma10{1±J0[kFR(1+/epsilon1 D)]}in this case, with 035109-10EQUILIBRIUM AND REAL-TIME PROPERTIES OF THE … PHYSICAL REVIEW B 98, 035109 (2018) the zeroth Bessel function J0(x)[39,40]. These hybridization functions do not exhibit a gap for any ﬁnite distance R.N o t e that for vanishing distance R=0 the odd conduction band always decouples for all dispersions. In two dimensions we only observe a vanishing correlation function for long times at distances separating the ferromag-netic and antiferromagnetic correlations. Unlike before, theseblack vertical lines are simply caused by a vanishing RKKYinteraction for these distances. This is in contrast to the 1Dcase where /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) remained zero for distances where the RKKY interaction is maximal ferromagnetic. Also note thatthe correlation function decays faster compared to the 1D casefor larger distances at large times, which is directly related tothe faster decaying RKKY interaction ∝1/R 2in comparison to the ∝1/Rdecay for a 1D dispersion. V . SUMMARY AND OUTLOOK The equilibrium properties as well as real-time dynamics of the spin-correlation function between two localized spinsat a distance Rcoupled to one conduction band via a local Heisenberg interaction Jwere investigated using the NRG. Since we did not add a direct exchange between the spins,spin-spin correlations can only be mediated by the indirectRKKY interaction. In order to set the stage for the nonequilibrium dynamics after a local interaction quench, we presented the distancedependent equilibrium spin-spin-correlation function for theTIKM. There is a competition between Kondo physics andRKKY mediated singlet formation [ 20,21,28,29] for an AF coupling J. For a FM coupling, the distance dependent spin- spin-correlation function is only weakly coupling dependentdue to reduction of Jin the RG. For both signs of interactions J, the correlation function oscillates with the distance Ras expected. Although the RKKY interaction varies continuouslywith the well-established cos(2 k FR) oscillations in one dimen- sion, the spin-spin-correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(R,t)s h o w sa steplike behavior that is reminiscent of the zero-temperaturelevel crossing of local singlet-triplet state energies. For distances Rwith generically FM RKKY interactions close to its distance dependent maximum, /angbracketleft/vectorS 1/vectorS2/angbracketright(R,t) clearly reveals the inﬂuence of the Kondo screening. While forR<ξ Kthe correlation function is ferromagnetic as expected, /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) can change its sign once Rexceeds the Kondo correlation length ξK.F o rR→∞ , two independent Kondo singlets are formed and the spin-correlation function vanishes.At ﬁnite distances and R/greatermuchξ K, the sign of /angbracketleft/vectorS1/vectorS2/angbracketright(R,t) depends on the magnitude of the potential scattering terms. Thedifference of these terms in the even and odd channel is relatedto a marginal relevant operator [ 24] that generates a small antiferromagnetic interaction responsible for the sign change. For distances with purely AF RKKY interactions, at dis- tances k FR/π=(n+1/2), we found universality in R/ξ Kfor the amplitude of the correlation function and a 1 /R2decay once the distance exceeds ξK, which is faster than the 1 /R decrease of the 1D RKKY interaction: The Kondo screeningof each impurity spin induces a faster decay of the correlationfunction. In the case of ferromagnetic Kondo couplings J< 0, the amplitude remains constant even for R→∞ since the Kondoeffect is absent. Only ﬁnite temperature evokes a power-law decay of the correlation function, which turns into an expo-nential decay once the length scale of the ﬁnite temperature ξ T is exceeded. The nonequilibrium dynamics of the spin-spin-correlation function after a sudden quench shows distinct behavior forshort and for long times as a function of the distance. The short-time dynamics is governed by the propagation of correlationsvia the conduction band [ 39] with the Fermi velocity: A short ferromagnetic wave is propagating through the systemas a consequence of the total spin conservation since locallyantiferromagnetic correlations between the local spin and thelocal conduction electron spin density are building up. Itsmagnitude is deﬁned by J 3, which can be understood from third-order perturbation theory. We extracted the characteristic long timescale t∗for a ﬁxed short distance reﬂecting the different mechanism in thereal-time dynamics. While for weak coupling Jthe scaling t ∗∝J−4is related to the dominating RKKY interaction, t∗∝ 1/√TKreveals the dominating Kondo effect with increasing local coupling. The most striking feature is, however, the remarkable nonequilibrium dynamics at the distances kFR/π=n.A l - though the RKKY interaction reaches its periodic maxima, thecorrelation function only changes for short times whereas itremains constant for long times. This effect originates fromthe symmetry of the 1D dispersion and is caused by thefact that for these distances conduction electron states witheven-parity ( n=1,3,... ) or odd-parity ( n=0,2,... )s y m - metry decouple from the impurities at low temperatures. Thisdecoupling also enforces a dynamic local parity conservationfor the impurity spins which leads to a conserved value of thecorrelation function for long times. This effect might be very useful for the implementation of spin qubits since the parity symmetry protects the en-tanglement between both spins and prevents the correlationfrom decaying to its equilibrium value. Usually, highly lo-calized electrons in quantum dots are used as qubits sincethe localization reduces the decoherence facilitated by free-electron motion, but simultaneously increases the hyperﬁneinteraction strength between the conﬁned electron spin and thesurrounding nuclear spins [ 58–61]. Making use of symmetries such as the parity to retain the entanglement might, therefore,be a way to employ more delocalized electrons and thusdecrease the hyperﬁne interaction. ACKNOWLEDGMENTS B.L. thanks the Japan Society for the Promotion of Sci- ence and the Alexander von Humboldt Foundation. Partsof the computations were performed at the SupercomputerCenter, Institute for Solid State Physics, University of Tokyoand the John von Neumann Institute for Computing at theForschungszentrum Jülich under Project No. HHB00. APPENDIX: PERTURBATIVE APPROACH FOR THE REAL-TIME CORRELATION FUNCTION In this Appendix we will brieﬂy present a perturbation theory to show that the lowest nonvanishing contribution to 035109-11BENEDIKT LECHTENBERG AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 98, 035109 (2018) the real-time dynamics of the correlation function is given by the third order ∝J3. For this purpose the Hamiltonian is divided into two parts H=H0+HKwithH0=/summationtext σ,/vectork/epsilon1/vectorkc† /vectorkσc/vectorkσ, the free conduction-band dispersion /epsilon1/vectork, andHK=J/summationtext i/vectorSi/vectorsc(ri). The time-dependent spin-correlation function /angbracketleft/vectorS1/vectorS2/angbracketright(t) can be written as /angbracketleft/vectorS1/vectorS2/angbracketright(t)=Tr[ρI(t)/vectorS1/vectorS2], (A1) where the index Iindicates that the operator is transformed into the interaction picture, which is deﬁned for any operatorAas A I(t)=eiH0tAe−iH0t. (A2) /vectorS1and/vectorS1remain time independent since they commute withH0. The von Neumann equation governs the real-time evolution of ρI(t): ∂tρI(t)=i/bracketleftbig ρI(t),HI K(t)/bracketrightbig , (A3) which is integrated to ρI(t)=ρ0+i/integraldisplayt 0/bracketleftbig ρ0,HI K(t1)/bracketrightbig dt1 −/integraldisplayt 0/integraldisplayt1 0/bracketleftbig/bracketleftbig ρI(t2),HI K(t2)/bracketrightbig ,HI K(t1)/bracketrightbig dt2dt1,(A4) where we used the initial condition ρI(0)=ρ0. Replacing ρI(t2)b yρ0in the second integral yields an approximate solution in O(J2). Substituting ( A4)i n t o( A1) and performing a cyclically rotation of the operators under the trace, we obtain /angbracketleft/vectorS1/vectorS2/angbracketright(t)≈Tr/bracketleftbig ρ0/vectorS1/vectorS2/bracketrightbig +i/integraldisplayt 0Tr/bracketleftbig ρ0/bracketleftbig HI K(t1),/vectorS1/vectorS2/bracketrightbig/bracketrightbig dt1 −/integraldisplayt 0/integraldisplayt1 0Tr/bracketleftbig ρ0/bracketleftbig HI K(t2),/bracketleftbig HI K(t1),/vectorS1/vectorS2/bracketrightbig/bracketrightbig/bracketrightbig dt2dt1.(A5)This expression contains only expectation values that in- volve the initial density operator ρ0in which the impu- rity spins and the conduction electrons factorize since inH 0the impurity spins are decoupled from the conduction band. In the absence of magnetic ﬁelds the ﬁrst term van-ishes, Tr[ ρ 0/vectorS1/vectorS2]=/angbracketleft/vectorS1/vectorS2/angbracketright0=0, where the index denotes that the expectation value is taken with respect to theinitial density operator ρ 0. For the integral kernel of the ﬁrst-order correction we obtain /angbracketleftbig/bracketleftbig HI K(t1),/vectorS1/vectorS2/bracketrightbig/angbracketrightbig 0=−J/summationdisplay ijk/epsilon1ijk/parenleftbig/angbracketleftbig Sk 1Sj 2/angbracketrightbig 0/angbracketleftsi(r1,t1)/angbracketright0 +/angbracketleftbig Sj 1Sk 2/angbracketrightbig 0/angbracketleftsi(r2,t1)/angbracketright0/parenrightbig =0, (A6) where the upper index indicates the spin component, /epsilon1ijkis the Levi-Civita symbol, and si(rj,t) is the time-dependent spin component of the conduction-band electrons at positionr jin the interaction picture. Since all occurring expectation values vanish, also the complete ﬁrst-order contribution in J vanishes. 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PhysRevB.90.075407.pdf	PHYSICAL REVIEW B 90, 075407 (2014) Current-conserving and gauge-invariant quantum ac transport theory in the presence of phonon Yunjin Yu,1Hongxin Zhan,1Yadong Wei,1,and Jian Wang2, 1College of Physics and Institute of Computational Condensed Matter Physics, Shenzhen University, Shenzhen 518060, People’s Republic of China 2Department of Physics and the Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, People’s Republic of China (Received 16 April 2014; revised manuscript received 17 June 2014; published 8 August 2014) Using the nonequilibrium Green’s function (NEGF) approach, we develop a microscopic ac transport theory in the presence of electron-phonon interaction. Taking into account the self-consistent Coulomb interaction, thedisplacement current is included. This ensures that our theory satisﬁes the current-conserving and gauge-invariantconditions. Importantly, the inclusion of self-consistent Coulomb interaction naturally connects the NEGFformalism to the density functional theory (DFT). This allows us to calculate the self-consistent Hamiltonianusing DFT within the NEGF framework, which paves the way for ﬁrst principles ac transport calculation ofnanoelectronic devices in the presence of electron-phonon interaction. It is known that the inelastic electrontunneling spectroscopy (IETS) is a powerful tool in studying the inelastic dc quantum transport in moleculardevices. The basic idea of IETS is to obtain the information of vibrational spectrum of molecular devices bymeasuring the second derivative of the dc current with respect to the bias voltage. In the ac transport, we ﬁndthat the phonon spectrum and electron-phonon coupling strength can be obtained from the second derivative ofthe admittance with respect to the frequency which is the working principle of the inelastic electron admittancespectroscopy (IEAS). Hence we propose to use IEAS to probe the effect of the phonon in ac transport. As anexample, dynamic conductance of a quantum dot is discussed in detail and the concept of IEAS is demonstrated. DOI: 10.1103/PhysRevB.90.075407 PACS number(s): 72 .10.Di,72.25.−b,85.65.+h I. INTRODUCTION Quantum transport in nanostructures has been intensively investigated because of promising potential applications innanoelectronics [ 1–3]. Various transport properties includ- ingI-Vcharacteristic, conductivity, current noise, quantum capacitance, etc. have been studied both experimentally[4–9] and theoretically [ 10–16]. Although most theoretical and experimental studies have been concentrated on the dctransport regime, time-dependent transport problems haveattracted more and more attention including steady state actransport [ 17–32] and the transient problems driven by the step function-like bias [ 33,34]. Ac quantum transport properties have also been investigated in the super-conducting hybridsystem [ 35–37]. To examine the phase breaking effect, ac quantum transport with electron-phonon interaction has beenstudied by several groups [ 21,38,39]. For electron transport, current-conserving and gauge-invariant conditions are twofundamental requirements [ 10,11]. In order to satisfy the two conditions in ac transport, one should include self-consistentCoulomb interaction explicitly into the Hamiltonian andcalculate both conduction current and displacement currentdue to the charge pileup in the scattering region [ 10,11,23,24]. Moreover, the inclusion of Coulomb interaction is also the keyfor the ﬁrst principles transport calculation for nanoelectronicswhich bridges the nonequilibrium Green’s function in quantumtransport theory (NEGF) and the density function theory(DFT) characterizing the chemical ingredient of the moleculardevices [ 40]. One of the most important issues in nanoelectronics theory and modeling is the role played by interaction between *Corresponding authors: ywei@szu.edu.cn; jianwang@hku.hkelectrons and the nuclear vibrations (phonons). This is becausethe electron-phonon interaction gives rise to inelastic currentand modiﬁes the elastic current that affects the characteristicsof nanoelectronic devices in an essential way. In addition, theinelastic current due to electron-phonon interaction can beused to measure vibrational spectra of single molecule as hasbeen demonstrated experimentally using inelastic tunnelingspectroscopy. Clearly, predicting quantum transport propertiesof nanoelectronic devices in the presence of electron-phononinteraction must use the ﬁrst principle method including allthe atomic details of nanoelectronic devices. Indeed, various effects on the transport in the molecular devices have been studied such as inelastic electron scatter-ing, atomic position rearrangements [ 3], energy dissipation [41–43] and local heat in device [ 44], phonon-mediated nega- tive differential resistance [ 45] and phonon assisted tunneling [9,46], etc. In the dc situation, the inelastic transport properties of molecular device are studied intensively [ 33,47–54]. The inelastic tunneling spectroscopy (IETS), which is the secondderivative of the current over bias voltage, is usually used asa powerful tool to identify the molecular vibrational modesand the electron-phonon coupling strength [ 55,56]. These ﬁrst principle investigations show that under nonequilibriumcondition vibrational spectra and electron-phonon interactioncan be quite different from the equilibrium ones. For instance,it was found that the dc bias voltage can drastically affectthe electron-phonon coupling strength while the phononfrequencies change only a few percent [ 56]. Since most of the nanoelectronic devices are operated with ac signals and ﬁnitetemperatures, there is a clear need to understand the role playedby electron-phonon interaction in the ac regime. Up to now,most of the theoretical studies of electron-phonon interactionfocus on dc properties [ 57]. There is yet a ﬁrst principle method for calculating vibrational frequencies, electron-phonon 1098-0121/2014/90(7)/075407(12) 075407-1 ©2014 American Physical SocietyYUNJIN YU, HONGXIN ZHAN, YADONG WEI, AND JIAN WANG PHYSICAL REVIEW B 90, 075407 (2014) couplings, and inelastic transport properties under ac condition for nanoelectronic devices. It is the aim of this paper to providesuch a theoretical framework which is suitable for the ﬁrstprinciples calculation in nanoelectronics. In this paper, wedeveloped a microscopic theory for ac transport with electron-phonon coupling using the nonequilibrium Green’s functiontheory. Our formalism emphasizes the current conservationand gauge invariance. The self-consistent Coulomb interactionin the presence of phonon is included explicitly in theHamiltonian making it possible to combine with DFT for theﬁrst principles transport calculation. Similar to the role IETSplayed in the dc transport, we proposed a new tool namedas the ac inelastic electron admittance spectroscopy (IEAS) tocharacterize the electron-phonon interaction including phononspectrum and electron-phonon coupling strength from thedynamic point of view with frequency as an extra handle. The rest of the paper is organized as follows. In Sec. II,t h e formulas of current and conductance are derived for quantumsystems with ac bias and electron-phonon interaction. Usingthe nonequilibrium Green’s function, the dynamic conduc-tance is obtained by expanding ac current and admittance tothe ﬁrst order with respect to the amplitude of ac bias. Inaddition, current conservation and gauge invariance are provedto be satisﬁed when the Coulomb interaction is considered.Furthermore, the idea of IEAS is introduced and analyzed. InSec. III, as an example, the inelastic ac transport of a quantum dot is numerically studied in detail. A summary is given inSec. IV. II. THEORETICAL FORMALISM We consider a scattering region coupled by two or mul- tiple leads. The system can be described by the followingHamiltonian: H=H lead+Hscat+HT. (1) Hleadis the Hamiltonian of the leads, Hlead=/summationdisplay kα/epsilon1kαˆC† kαˆCkα. (2) ˆC† kα(ˆCkα) is the creation (annihilation) operator of the kstate in the lead α(α=L,R for a two-probe system). /epsilon1kα=/epsilon1(0) kα+ qvαcosωtwith/epsilon1(0) kαthe energy level in lead αandvαis the ac bias amplitude on the lead α. As usual, the electron-phonon interaction and Coulomb interaction in the lead are neglected. Hscatis the Hamiltonian of the scattering region which includes three parts as follows: Hscat=He+Hp+Hep. (3) The Hamiltonian of electrons Hein the scattering region can be expressed as He=/summationdisplay n(/epsilon1n+qUn)ˆd† nˆdn, (4) where ˆd† nand ˆdnare the creation and annihilation operators in the scattering region and satisfy the fermion anticom- mutation relation {ˆdα,ˆd† β}=δαβ.Un=/summationtext mVnm/angbracketleftˆd† mˆdm/angbracketrightis the self-consistent internal Coulomb potential inside the scatteringregion with V nmthe matrix element of the Coulomb potential[58]. If we wish to implement this formalism into the ﬁrst principles calculation, we will use the following Hamiltonianthat includes the potentials V exandVcordue to the the exchange and correlation interactions, respectively: He=/summationdisplay n(/epsilon1n+qUn+qVex+qVcor)ˆd† nˆdn. (5) In Eq. ( 3),Hpis the phonon Hamiltonian and can be written as Hp=/summationdisplay ν/planckover2pi1ων/parenleftbigg ˆb† νˆbν+1 2/parenrightbigg , (6) where ωνis the phonon frequency. The phonon creation and annihilation operators, ˆb† νand ˆbν, satisfy the boson commutation relation [ ˆbα,ˆb† β]=δαβ. The Hamiltonian of the electron-phonon interaction is given by Hep=/summationdisplay νnn/primegν nn/prime(ˆb† ν+ˆbν)ˆd† nˆdn/prime, (7) where gν nn/primedescribes the electron-phonon coupling strength. The third term HTin the total Hamiltonian Eq. ( 1) describes the coupling between the scattering region and the leads. Withthe coupling constant t kαn, it can be expressed as HT=/summationdisplay kαn[tkαnˆC† kαˆdn+t∗ kαnˆd† nˆCkα]. (8) Using the Heisenberg equation of motion, one obtains the current in the form of Green’s function Iα(t)=−/summationdisplay kn[tkαnG< n,kα(t,t)]+H.c. (9) After the analytic continuation, we obtain [ 14] Iα(t)=−q/integraldisplay dt1Tr[Gr(t,t1)/Sigma1< eα(t1,t)+G<(t,t1)/Sigma1a eα(t1,t) −/Sigma1< eα(t,t1)Ga(t1,t)−/Sigma1r eα(t,t1)G<(t1,t)], (10) where /Sigma1γ eαmn(t,t/prime)=/summationdisplay kt∗ kαmgγ kα(t,t/prime)tkαn, (11) withγ=<,r,a .H e r e /Sigma1γ eαis the self-energy due to the electron coupling between the scattering region and the lead α. Electron-phonon coupling between the scattering region andthe leads is set to zero since we assume that the phonon existsonly in the scattering region but not in the lead regions. InEq. ( 11), the Green’s functions of isolated leads are g r,a kα(t,t/prime)=∓iθ(±t∓t/prime)exp/bracketleftbigg −i/integraldisplayt t/primedt1/epsilon1kα(t1)/bracketrightbigg (12) and g< kα(t,t/prime)=if/parenleftbig /epsilon1(0) kα/parenrightbig exp/bracketleftbigg −i/integraldisplayt t/primedt1/epsilon1kα(t1)/bracketrightbigg . (13) The effect of phonon is included in the Green’s function Gγin Eq. ( 10) as self-energy /Sigma1γ epwhich will be discussed in detail in Sec. II B. 075407-2CURRENT-CONSERVING AND GAUGE-INV ARIANT . . . PHYSICAL REVIEW B 90, 075407 (2014) A. Nonequilibrium Green’s function at small bias We are interested in the linear response regime where the bias voltage is small. To calculate the ac current and dynamicconductance in this regime, one can expand the Green’sfunction G γ, electronic self-energy /Sigma1γ e, and the self-energy of electron-phonon coupling /Sigma1γ epto the ﬁrst order of the external biasvαas follows: Gγ(t,t1)=Gγ 0(t,t1)+gγ(t,t1), (14) /Sigma1γ eα(t,t1)=/Sigma1γ 0eα(t,t1)+σγ eα(t,t1), (15) and /Sigma1γ ep(t,t1)=/Sigma1γ 0ep(t,t1)+σγ ep(t,t1), (16) where Gγ 0,/Sigma1γ 0eα, and/Sigma1γ 0epare equilibrium Green’s functions and equilibrium self-energies and gγ,σγ eα, andσγ epare the ﬁrst order corrections due to the bias vα. It is straightforward to ﬁnd that the current in the linear regime is written as Iα(t)=−q/integraldisplay dt1Tr/bracketleftbig Gr 0(t,t1)σ< eα(t1,t)+gr(t,t1)/Sigma1< 0eα(t1,t) +G< 0(t,t1)σa eα(t1,t)+g<(t,t1)/Sigma1a 0eα(t1,t) −/Sigma1< 0eα(t,t1)ga(t1,t)−σ< eα(t,t1)Ga 0(t1,t) −/Sigma1r 0eα(t,t1)g<(t1,t)−σr eα(t,t1)G< 0(t1,t)/bracketrightbig . (17) After taking double-time Fourier transform, the expression of current in energy representation is obtained as Iα(/Omega1)=−q/integraldisplaydE 2πTr/bracketleftbig Gr 0(E+)σ< eα(E+,E)+gr(E+,E) ×/Sigma1< 0eα(E)+G< 0(E+)σa eα(E+,E)+g<(E+,E)/Sigma1a 0eα(E) −/Sigma1< 0eα(E+)ga(E+,E)−σ< eα(E+,E)Ga 0(E) −/Sigma1r 0α(E+)g<(E+,E)−σr eα(E+,E)G< 0(E)/bracketrightbig , (18) where E+=E+/Omega1. (We will set /planckover2pi1=1 in the rest of the paper.) To obtain Eq. ( 18), we have used the fact that Fourier transform Gγ 0(E,E/prime)=2πδ(E−E/prime)Gγ 0(E) and similar relation for the equilibrium self-energy. This is becausethe equilibrium Green’s function Gγ 0(t1,t2) and self-energy /Sigma1γ 0e(t1,t2) depend only on the time difference t1−t2. In the energy representation, the expression of the retarded Green’s function is [ 59] Gr=1 E−H0−U−/Sigma1re−/Sigma1rep, (19) where H0=/summationtext n/epsilon1nˆd† nˆdnis the Hamiltonian of isolated scatter- ing region, Uis the self-consistent internal Coulomb potential, and/Sigma1r epis the phonon self-energy to be discussed in detail in the next subsection. Expanding U(t) in terms of the amplitude of the external bias vα(0)=vα,w eh a v e U(t)=Ueq+U1(t)+U2(t)+··· =Ueq+/summationdisplay αuα(t)vα+1 2/summationdisplay αβuαβ(t)vαvβ+··· ,(20) where Ueqis the equilibrium Coulomb potential and uα(t), uαβ(t) are the so-called characteristic potentials [ 11,25]. Here uα(t) corresponds to the ﬁrst order response of the Coulombinteraction due to ac bias and uαβ(t) describes the second order correction, etc. According to the gauge-invariant condition thecurrent should remain unchanged when all the external biasvoltages are shifted to an equal amount at the same time; wehave/summationtext αuα(t)=cosωtand/summationtext αuαβ(t)=/summationtext βuαβ(t)=0i n the presence of phonon [ 11,25]. In the linear response regime, we treat U1,σr e, andσr epas the perturbation to the equilibrium quantity Ueq,/Sigma1r 0e, and/Sigma1r 0ep, respectively. We have from the Dyson equation Gr=Gr 0+Gr 0/parenleftbig U1+σr e+σr ep/parenrightbig Gr 0, (21) where the equilibrium Green’s function Gr 0is Gr 0=1 E−H0−Ueq−/Sigma1r 0e−/Sigma1r 0ep. (22) So from Eqs. ( 14) and ( 21), the ﬁrst order correction for the retarded Green’s function is gr/a=Gr/a 0/parenleftbig U1+σr/a e+σr/a ep/parenrightbig Gr/a 0. (23) Using the Keldysh equation G<=Gr/Sigma1<Ga, and collecting the ﬁrst order terms of the external bias, one ﬁnds g<=Gr 0/Sigma1< 0ga+Gr 0σ<Ga0+gr/Sigma1< 0Ga0, (24) where /Sigma1< 0=/Sigma1< 0e+/Sigma1< 0ep and σ<=σ< e+σ< ep. (25) Note that Eqs. ( 23) and ( 24) are in time space. Taking double-time Fourier transform of these two equations andusing the abbreviation ¯Gγ 0=Gγ 0(E+) and ¯/Sigma1γ 0=/Sigma1γ 0(E+) with E+=E+/Omega1,w eh a v e gr/a(E+,E)=¯Gr/a 0[U1(/Omega1)+σr/a e(E+,E) +σr/a ep(E+,E)]Gr/a 0 (26) and g<(E+,E)=¯Gr 0(¯/Sigma1< 0e+¯/Sigma1< 0ep)ga(E+,E)+¯Gr 0[σ< e(E+,E) +σ< ep(E+,E)]Ga 0+gr(E+,E)(/Sigma1< 0e+/Sigma1< 0ep)Ga 0. (27) The Fourier transform of the ﬁrst order correction of the nonequilibrium self-energy is [ 21] σγ eα=qvα(/Omega1) /Omega1/bracketleftbig /Sigma1γ 0eα−¯/Sigma1γ 0eα/bracketrightbig , (28) where vα(/Omega1)=πvα[δ(/Omega1+ω)+δ(/Omega1−ω)]. Here ωis the driving frequency and /Omega1is the response frequency. In the equations above, U1(/Omega1) is the Fourier transform of U1(t)w i t h U1(/Omega1)=/summationdisplay αuα(/Omega1)vα(/Omega1). (29) From/summationtext αuα(t)=cosωt, we obtain /summationdisplay αuα(/Omega1)=1. (30) Note that here uα(/Omega1) is deﬁned by Eq. ( 29) but not the direct Fourier transform of uα(t). The expressions of /Sigma1γ 0epandσγ ep will be derived in the next subsection. 075407-3YUNJIN YU, HONGXIN ZHAN, YADONG WEI, AND JIAN WANG PHYSICAL REVIEW B 90, 075407 (2014) B. Self-energy due to electron-phonon coupling It can be extremely computationally demanding to include the effect of electron-phonon coupling in the quantum trans-port. This is because the electron distribution function andelectron transport are affected by the presence of phononswhich in turn inﬂuence the phonon distribution functionand the equilibrium positions of the atoms. So to solvethis inelastic transport thoroughly, one should carry out theself-consistent calculation to get the electron density, theCoulomb interaction, as well as the electron-phonon couplingincluding the positions of atoms and the vibration modes. Thismakes the calculation extremely hard if not impossible. Toreduce the complexity of the problem, only the lowest-ordercontributions due to electron-phonon coupling are usually con-sidered [ 39,53,60–62]. At this level, if the phonon calculation is decoupled with the electronic part, we call it the Bornapproximation. Otherwise, it is called the self-consistent Bornapproximation. In order to proceed further, we assume that [ 56,63]t h e phonon is in equilibrium and its lifetime is inﬁnite. Under thisapproximation, we can write the bare phonon Green’s functionas [64] D r ν(/Omega1)=1 /Omega1−ων+i/epsilon1−1 /Omega1+ων+i/epsilon1, D< ν(/Omega1)=−2πi[nνδ(/Omega1−ων)+(nν+1)δ(/Omega1+ων)],(31) D> ν(/Omega1)=−2πi[(nν+1)δ(/Omega1−ων)+nνδ(/Omega1+ων)], where nν=1 eων/kBT−1(32) is the Bose-Einstein distribution function and Da ν(/Omega1)= [Dr ν(/Omega1)]†. Within the self-consistent Born approximation (SCBA), the electron self-energy due to electron-phononcoupling can be written as [ 64] /Sigma1 ep(τ,τ/prime)=i/summationdisplay νgνG(τ,τ/prime)gν†Dν(τ,τ/prime), (33) where Dνis the equilibrium phonon Green’s function and only depends on the time difference τ−τ/prime. After performing the Fourier transform, we have /Sigma1ep(E,E/prime)=i/summationdisplay ν/integraldisplayd/Omega1 2πgνG(E−/Omega1,E/prime−/Omega1)gν†D(/Omega1), (34) where gνis the electron-phonon coupling matrix. Its ﬁrst order correction to the external bias can be written as [ 21] σep(E,E/prime)=i/summationdisplay ν/integraldisplayd/Omega1 2πgνg(E−/Omega1,E/prime−/Omega1)gν†D(/Omega1).(35) From Eq. ( 34), the equilibrium self-energies due to electron- phonon coupling can be solved as [ 63] /Sigma1r 0ep(E)=/summationdisplay νgν/bracketleftbigg (1+nν)Gr 0(E−ων)+nνGr 0(E+ων) +1 2[G< 0(E−ων)−G< 0(E+ων)]/bracketrightbigg gν†,(36)/Sigma1< 0ep(E)=/summationdisplay νgν[(1+nν)G< 0(E+ων) +nνG< 0(E−ων)]gν†, (37) and /Sigma1> 0ep(E)=/summationdisplay νgν[(1+nν)G> 0(E−ων) +nνG< 0(E+ων)]gν†. (38) At zero temperature, nν=0, and the expressions above can be simpliﬁed as /Sigma1r 0ep(E)=/summationdisplay νgν/bracketleftbigg Gr 0(E−ων)+1 2G< 0(E−ων) −1 2G< 0(E+ων)/bracketrightbigg gν†, (39) /Sigma1< 0ep(E)=/summationdisplay νgνG< 0(E+ων)gν†, (40) and /Sigma1> 0ep(E)=/summationdisplay νgνG> 0(E−ων)gν†. (41) Here, the Green’s functions are the equilibrium Green’s functions which also include the electron-phonon interaction. At zero temperature, the ﬁrst order correction of phonon self-energies can be written as σr ep(E+,E)=/summationdisplay νgν/bracketleftbigg gr(E+−ων,E−ων) +1 2g<(E+−ων,E−ων) −1 2g<(E++ων,E+ων)/bracketrightbigg gν†,(42) σ< ep(E+,E)=/summationdisplay νgνg<(E++ων,E+ων)gν†, (43) and σ> ep(E+,E)=/summationdisplay νgνg>(E+−ων,E−ων)gν†.(44) We note that the ﬁrst order correction of phonon self-energies cannot be expressed analytically because Green’s functions gγ themselves contain the phonon self-energy, so they can only be solved numerically. With the phonon self-energies deﬁned,we are able to calculate dynamic conductance in the presenceof phonon. C. Dynamic conductance The dynamic conductance (admittance) Gαβis deﬁned as Iα(/Omega1)=/summationdisplay βGαβ(/Omega1)vβ(/Omega1). (45) Comparing Eq. ( 18) with Eq. ( 45) and deﬁning gγ α(/Omega1) and σγ α(/Omega1) according to gγ(E+,E)=/summationdisplay αgγ α(/Omega1)vα(/Omega1) (46) 075407-4CURRENT-CONSERVING AND GAUGE-INV ARIANT . . . PHYSICAL REVIEW B 90, 075407 (2014) and σγ(E+,E)=/summationdisplay ασγ α(/Omega1)vα(/Omega1), (47) the dynamic conductance is found to be Gαβ(/Omega1)=−q/integraldisplaydE 2πTr/bracketleftbig g< β(/Sigma1a eα−¯/Sigma1r eα)+gr β/Sigma1< eα−¯/Sigma1< eαga β +(¯Grσ< eα−σ< eαGa+¯G<σa eα−σr eαG<)δαβ/bracketrightbig .(48) Note that we have simpliﬁed the notation and use Grto denote the equilibrium Green’s function instead of Gr 0. This applies to other Green’s functions and self-energies. Whenever wehave two energy variables such as G <(E+,E) it refers to a nonequilibrium situation. Moreover, in the above equation,gγ α=gγ α(/Omega1) and σγ eα=σγ eα(/Omega1). To make the expression of dynamic conductance simpler, we will use the wideband limit(WBL), where the linewidth function is independent of energy.It is straightforward to extend it to a non-WBL case. UnderWBL, Eq. ( 28)g i v e s σ r,a eα=0 (49) and σ< eα=iq /Omega1/Gamma1eα(f−¯f). (50) So the ﬁrst order correction of the Green’s function is gr,a α(/Omega1)=¯Gr,a/bracketleftbig uα(/Omega1)+σr,a epα/bracketrightbig Gr,a, (51) where σr,a epα(/Omega1)=∂σr,a ep(E+,E) ∂vα(/Omega1). Under WBL, the expression of the dynamic conductance can be simpliﬁed as Gαβ(/Omega1)=−q/integraldisplaydE 2πTr/bracketleftbig ig< β/Gamma1eα+¯Grσr epβGr/Sigma1< eα +¯GruβGr/Sigma1< eα−¯Gaσa epβGa¯/Sigma1< eα −¯GauβGa¯/Sigma1< eα+σ< eα(¯Gr−Ga)δαβ].(52) Using the relation [¯Gr]−1−[Ga]−1=/Omega1+/Sigma1a−¯/Sigma1r =/Omega1+i/Gamma1e+/parenleftbig /Sigma1a ep−¯/Sigma1r ep/parenrightbig ,(53) where /Gamma1e=/summationtext α/Gamma1eα,w eh a v e Ga−¯Gr=¯Gr/bracketleftbig /Omega1+i/Gamma1e+/parenleftbig /Sigma1a ep−¯/Sigma1r ep/parenrightbig/bracketrightbig Ga =Ga/bracketleftbig /Omega1+i/Gamma1e+/parenleftbig /Sigma1a ep−¯/Sigma1r ep/parenrightbig/bracketrightbig¯Gr.(54) Substituting the above equation into Eq. ( 52), we ﬁnd Gαβ(/Omega1)=−q/integraldisplaydE 2πTr/bracketleftbig i¯Gr(σ< eβ+σ< epβ)Ga/Gamma1eα +¯Gr/parenleftbig σr epβ+uβ/parenrightbig Gr/Sigma1< eα−¯Ga/parenleftbig σa epβ+uβ/parenrightbig Ga¯/Sigma1< eα +i¯Gr/parenleftbig σr epβ+uβ/parenrightbig G</Gamma1eα+i¯G</parenleftbig σa epβ+uβ/parenrightbig Ga/Gamma1eα −¯Grσ< eαGa/parenleftbig /Omega1+i/Gamma1e+/Sigma1a ep−¯/Sigma1r ep/parenrightbig δαβ/bracketrightbig . (55)Now we derive the equation which determines the charac- teristic potential uα(/Omega1). The Fourier transform of the Poisson equation under ac bias voltage is given by ∇2U(x)=−4πρ(/Omega1)(x)=−4πiq/integraldisplaydE 2π[G<(E+,E)]xx. (56) With the Poisson equation at equilibrium ∇2Ueq(x)=−4πρ0(x)=−4πiq/integraldisplaydE 2π[G<(E)]xx,(57) we ﬁnd the relation between the induced charge distribution δρind=ρ(/Omega1)−ρ0and ﬁrst order correction of Coulomb potential U1due to ac voltage ∇2U1=−4πδρ ind(/Omega1)=−4πiq/integraldisplaydE 2π[g<(E+,E)]xx.(58) Taking the derivative with respect to vα(/Omega1) on both sides of the above equation, we ﬁnd (within WBL) ∇2uα=−4πiq/integraldisplaydE 2π/bracketleftbigg∂g<(E+,E) ∂vα/bracketrightbigg xx =−4πiq/integraldisplaydE 2π/bracketleftbig¯Gr/parenleftbig uα+σr epα/parenrightbig G< +¯G</parenleftbig uα+σa epα/parenrightbig Ga+¯Gr(σ< eα+σ< epα)Ga/bracketrightbig xx.(59) Setting uα=0 on the right-hand side of Eq. ( 59), we obtain the generalized injectivity dnα/dE in the presence of phonon, dnα/dE=i/integraldisplaydE 2π/bracketleftbig¯Grσr epα(0)G<+¯G<σa epα(0)Ga +¯Gr(σ< eα+σ< epα(0))Ga/bracketrightbig xx, (60) where σγ epα(0) with γ=r,a,< denotes the phonon self-energy in the absence of Coulomb interaction. Here the generalizedinjectivity dn α/dE describes the density of states inside the scattering region due to the injection of electron in theαlead [ 11]. Since g <(E+,E) depends on external bias vα and Coulomb potential U1[see Eqs. ( 26) and ( 27)], we have ig<(E+,E)=q/summationdisplay αdnα/dEv α+qMU 1 (61) in the Thomas-Fermi approximation [ 11], where dnα/dE= ∂g</∂vαandMis a Lindhard response function to be determined. From Eq. ( 58), we see that if we shift vαby a constant amount v0,U1shifts by the same constant and g<remains the same. Under this voltage shift, we imme- diately ﬁnd from Eq. ( 61)M=−/summationtext αdnα/dE=dn/dE . So the Poisson-like equation can be cast into the familiarform ∇ 2uα=−4πq2dnα dE+4πq2dn dEuα. (62) The physical meaning of the right-hand side of Eq. ( 62)i s clear: the ﬁrst term is the injected charge density, while thesecond term is the induced charge density due to the Coulombinteraction. Once we have all the Green’s functions and self-energies, we can solve for the characteristic potential u αand hence 075407-5YUNJIN YU, HONGXIN ZHAN, YADONG WEI, AND JIAN WANG PHYSICAL REVIEW B 90, 075407 (2014) the dynamic conductance. In the following, we will show that our formalism satisﬁes two fundamental requirements fortransport in the presence of phonon, i.e., current conservationand gauge invariance. Mathematically, they correspond to/summationtext αGαβ=0 and/summationtext βGαβ=0. D. Current conservation and Gauge invariance Before we proceed to show the current conservation and gauge invariance for inelastic ac transport, we want to discussthe basic assumption in the quantum transport of open systems.To deal with quantum transport for an open system suchas a two probe system, we always divide the system intoscattering region and two semi-inﬁnite leads, where thepotential landscape of the lead is assumed to be constant or aperiodic function. With this assumption, we can calculate thewave function analytically in the lead region from which theself-energy of the lead in the NEGF approach can be calculatedand the scattering matrix in the scattering matrix approachcan be constructed. In other words, using this assumption thescattering problem of an open system with inﬁnite degree offreedom can be reduced to a problem of a closed system withﬁnite degree of freedom. In addition, applied ac or dc biasis assumed to shift the potential landscape of the lead by aconstant amount (called adiabatic approximation) [ 15]. The assumption that the potential landscape of the lead is a constantor a periodic function implies that the Coulomb interaction isscreened in the lead so that the electric ﬁeld of the lead is alwayszero [ 11]. From Gauss’s theorem, the total charge Q(t)i n s i d e of the scattering region is always zero, i.e., Q(t)=0, although the electrons might be re-distributed in the scattering regiondue to the existence of the bias voltage. From the continuityequation, we have/summationtext αIα+∂tQ(t)=0; we conclude that/summationtext αIα=0 as a result of Coulomb interaction. In order to make this assumption valid in practice, we assume that thescattering region is large enough so that the boundaries alongthe transport directions are deep inside the leads where theelectrons are assumed to obey equilibrium distribution f L/R= 1 e(E−μL/R−qVL/R)/kBT+1.H e r e μL/Rare the chemical potentials of left/right leads and VL/R are the bias potentials added at left/right leads. In the following, we will ﬁrst derive the continuity equation on the operator level and discuss its implication on currentconservation. We then prove explicitly that the current conser-vation is satisﬁed in the presence of phonon for both dc and accases. 1. Current conservation on the operator level First of all, we give a derivation of the continuity equation on the operator level. Using the Heisenberg equation of motion,one ﬁnds dˆN α dt=−i[ˆNα,ˆH]=−i[ˆNα,ˆHT] =/summationdisplay kn(tkαnˆC† kαˆdn−t∗ kαnˆd† nˆCkα), (63) where ˆNα=/summationtext kˆC† kαˆCkαis the number operator for the electron in the lead αthat commutes with ˆHlead,ˆHe,ˆHp, andˆHep. Deﬁning ˆNscat=/summationtext nˆd† nˆdnthe number operator for the electron in the scattering region, we have dˆNscat dt=−i[ˆNscat,ˆH]=−i[ˆNscat,ˆHT] =/summationdisplay kαn(−tkαnˆC† kαˆdn+t∗ kαnˆd† nˆCkα), (64) from which we obtain the continuity equation on the operator level in the presence of phonon /summationdisplay αdˆNα dt+dˆNscat dt=0. (65) Taking the quantum average we have the usual continuity equation in the presence of phonon /summationdisplay αIα+∂Q ∂t=0, (66) where Qis the total charge in the scattering region. By including the Coulomb interaction, we solve the followingPoisson equation: ∇ 2U(x,t)=−4πρ(x,t), (67) with the requirement that the electric ﬁeld is zero on the bound- ary. From Gauss’s theorem, we have Q(t)=/integraltext dxρ (x,t)=0 or/summationtext αIα=0 from Eq. ( 66). This shows that the ac current in the presence of phonon is conserved. 2. Current conservation in dc case Now we show explicitly that the current is conserved in the dc case within SCBA. The general expression for current inthe dc case is given by I α=−q/integraldisplaydE 2πTr[Gr/Sigma1< eα+G</Sigma1a eα+c.c.] =−q/integraldisplaydE 2πTr[/Sigma1< eαG>−/Sigma1> eαG<]. (68) Note that the total self-energy in a two-probe system with electron-phonon interaction can be written as /Sigma1γ tot=/Sigma1γ eL+/Sigma1γ eR+/Sigma1γ ep(γ=>,<,r,a ). (69) The total electron current can be written as /summationdisplay αIα=−q/integraldisplaydE 2πTr/bracketleftbigg/summationdisplay α/Sigma1< eαG>−/summationdisplay α/Sigma1> eαG</bracketrightbigg =−q/integraldisplaydE 2πTr[/Sigma1< totG>−/Sigma1> totG<] −q/integraldisplaydE 2πTr[/Sigma1< epG>−/Sigma1> epG<]. (70) Using the relationships /Sigma1> tot−/Sigma1< tot=/Sigma1r tot−/Sigma1a tot,Gr/Gamma1totGa=Ga/Gamma1totGr, it is straightforward to show that the ﬁrst term of Eq. ( 70) is zero. Within SCBA, the self-energies due to the 075407-6CURRENT-CONSERVING AND GAUGE-INV ARIANT . . . PHYSICAL REVIEW B 90, 075407 (2014) electron-phonon interaction are [ 61,62] /Sigma1< ep(E)=/summationdisplay ν[nνgνG<(E−/planckover2pi1ων)gν† +(nν+1)gνG<(E+/planckover2pi1ων)gν†], /Sigma1> ep(E)=/summationdisplay ν[(nν+1)gνG>(E−/planckover2pi1ων)gν† +nνgνG>(E+/planckover2pi1ων)gν†]. (71) Plugging these self-energies into Eq. ( 70), it is easy to show that/integraldisplay dETr[/Sigma1< epG>−/Sigma1> epG<]=0. (72) Hence the dc current is conserved within the SCBA. In general, the self-energies due to the leads depend on E−qvα. By including the Coulomb interaction, the electron Green’s functions depend on E−qUas well as self-energies due to the lead and electron-phonon coupling, where U is the Coulomb potential. Thus the self-energies due toelectron-phonon coupling depend on E−qUas well since they contain electron Green’s function [see Eq. ( 71)]. If all the external biases are shifted by a constant amount v 0,U will also be shifted by v0. Hence by changing the variable EtoE+qv0in the energy integration in Eq. ( 68), the current remains unchanged. Therefore, the gauge-invariantcondition is automatically satisﬁed if Coulomb interaction isincluded. 3. Current conservation and gauge invariance in ac case Now we will prove the continuity equation explicitly in the ac case, i.e., we need to prove /summationdisplay αGαβ(/Omega1)=iq/integraldisplaydE 2πTr[g<(E+,E)]. (73) As we have discussed previously, this equation along with the boundary condition of the Poisson equation will ensure thecurrent conservation. Since the phonon self-energy σγ ep(γ= r,a,< ) cannot be solved explicitly, we will use a perturbative approach by expanding the electron-phonon coupling strength\|g ν\|2order by order and show that the current is conserved to all orders of \|gν\|2[forgν,s e eE q .( 34)]. For instance, we have g<=g<(0)+g<(1)+g<(2)+··· , (74) where g<(0)is the lesser Green’s function without phonon and g<(n)is the nth order correction to the lesser Green’s function, i.e., the term containing \|gν\|2n. To simplify the proof, we will use the WBL and start with Eq. ( 55). From the Dyson equation, we can expand the Green’s function to the ﬁrst order of the phonon couplingstrength, G r=Gr e+Gr e/Sigma1r epGre, (75) where Gr e=1 E−H0−Ueq−/Sigma1re(76) is the retarded Green’s function in the absence of phonon. We ﬁrst show that the current is conserved to the ﬁrst orderof the electron-phonon coupling strength. So we will keep only the zeroth and the ﬁrst order terms in /Sigma1γ epandσγ ep(with γ=r,a,< )i nE q .( 55). Similar expansion can be done on Eqs. ( 26) and ( 27); we ﬁnd gr,a α=/parenleftbig¯Gr,a e+¯Gr,a e¯/Sigma1r,a ep¯Gr,a e/parenrightbig/parenleftbig uα+σr,a ep/parenrightbig ×/parenleftbig Gr,a e+Gr,a e/Sigma1r,a epGr,ae/parenrightbig (77) and g< α=/parenleftbig¯Gr e+¯Gr e¯/Sigma1r ep¯Gr e/parenrightbig/parenleftbig¯/Sigma1< e+¯/Sigma1< ep/parenrightbig ga α +/parenleftbig¯Gr e+¯Gr e¯/Sigma1r ep¯Gr e/parenrightbig/parenleftbig σ< eα+σ< epα/parenrightbig/parenleftbig Ga e+Ga/Sigma1a epGae/parenrightbig +gr α/parenleftbig /Sigma1< e+/Sigma1< ep/parenrightbig/parenleftbig Ga e+Ga e/Sigma1a epGae/parenrightbig . (78) The ac conductance is expanded in terms of electron- phonon coupling strength, Gαβ=G(0) αβ+G(1) αβ+G(2) αβ+··· , (79) where G(0) αβis the conductance in the absence of phonon, while G(1) αβcorresponds to the conductance of the ﬁrst order correction due to the phonon. Substituting Eq. ( 75), Eq. ( 77), and Eq. ( 78) into Eq. ( 55), and using/summationtext βuβ(/Omega1)=1, we have /summationdisplay αG(0) αβ=−q/integraldisplaydE 2πTr/bracketleftbig/parenleftbig¯Gr e−Ga e/parenrightbig σ< eβ+¯Gr euβGre/Sigma1< e −¯/Sigma1< e¯Ga euβGae+i¯Gr e/parenleftbig uβGr e/Sigma1< e+σ< eβ +¯/Sigma1< e¯Ga euβ/parenrightbig Ga e/Gamma1e/bracketrightbig (80) and /summationdisplay αG(1) αβ=−q/integraldisplaydE 2πTr[A1+A2+A3+A4+A5].(81) Here, A1=/parenleftbig¯Gr e¯/Sigma1r ep¯Gr e−Ga e/Sigma1a epGae/parenrightbig σ< eβ, (82) A2=¯Gr e/parenleftbig σr epβ+uβGr e/Sigma1r ep+¯/Sigma1r ep¯Gr euβ/parenrightbig Gr e/Sigma1< e −¯/Sigma1< e¯Ga e(σa epβ+uβGa e/Sigma1a ep+¯/Sigma1a ep¯Ga euβ/parenrightbig Ga e, (83) A3=i¯Gr e/parenleftbig¯/Sigma1r ep¯Gr euβGre/Sigma1< e+uβGr e/Sigma1r epGre/Sigma1< e +σr epβGre/Sigma1< e+uβGr e/Sigma1< ep+uβGr e/Sigma1< eGae/Sigma1a ep/parenrightbig Ga e/Gamma1e, (84) A4=i¯Gr e/parenleftbig¯/Sigma1r ep¯Gr e¯/Sigma1< e¯Ga euβ+¯/Sigma1< e¯Ga e¯/Sigma1a ep¯Ga euβ +¯/Sigma1< ep¯Ga euβ+¯/Sigma1< e¯Ga eσa epβ+¯/Sigma1< e¯Ga euβGae/Sigma1a ep/parenrightbig Ga e/Gamma1e, (85) and A5=i¯Gr e/parenleftbig¯/Sigma1r ep¯Gr eσ< eβ+σ< epβ+σ< eβGae/Sigma1a ep/parenrightbig Ga e/Gamma1e.(86) Using the relationship iGa e/Gamma1e¯Gr e=Ga e−¯Gr e−/Omega1Ga e¯Gr e, (87) Ga e−Gr e=iGr e/Gamma1eGae=iGa e/Gamma1eGre, 075407-7YUNJIN YU, HONGXIN ZHAN, YADONG WEI, AND JIAN WANG PHYSICAL REVIEW B 90, 075407 (2014) it is straightforward but tedious to show that /summationdisplay αG(0) αβ=q/Omega1/integraldisplaydE 2πTr/bracketleftbig g<(0) β/bracketrightbig =0 (88) and /summationdisplay αG(1) αβ=q/Omega1/integraldisplaydE 2πTr/bracketleftbig g<(1) β/bracketrightbig =0. (89) One can easily push it to higher order and show that /summationdisplay αG(n) αβ=q/Omega1/integraldisplaydE 2πTr/bracketleftbig g<(n) β/bracketrightbig =0 (90) forn> 1. Finally, we obtain that /summationdisplay αGαβ=0, (91) which is the expected result. Now we show that the gauge-invariant condition is satisﬁed. From Eq. ( 48), we have /summationdisplay βG(0) αβ=−q/integraldisplaydE 2πTr/bracketleftbig/parenleftbig¯Gr e−Ga e/parenrightbig σ< eα+i/parenleftbig f¯Gr eGre −¯f¯Ga eGae+¯Gr eG<e+¯G< eGae+¯Gr eσ< eGae/parenrightbig /Gamma1eα/bracketrightbig . (92) Here, σ< e=σ< e(/Omega1)=/summationdisplay βσ< eβ(/Omega1). (93) Similar to the current conservation, we have the expression of ﬁrst order correction to the conductance in the presence ofphonon /summationdisplay βG(1) αβ=−q/integraldisplaydE 2πTr[B1+B2+B3+B4+B5],(94) where B1=/parenleftbig¯Gr e¯/Sigma1r ep¯Gr e−Ga e/Sigma1a epGae/parenrightbig σ< eα, (95) B2=¯Gr e/parenleftbig σr ep+Gr e/Sigma1r ep+¯/Sigma1r ep¯Gr e/parenrightbig Gr e/Sigma1< eα −¯Ga e/parenleftbig σa ep+Ga e/Sigma1a ep+¯/Sigma1a ep¯Ga e/parenrightbig Ga e¯/Sigma1< eα, (96) B3=i¯Gr e/parenleftbig σr epGre/Sigma1< e+Gr e/Sigma1< eGae/Sigma1a ep+Gr e/Sigma1< ep +Gr e/Sigma1r epGre/Sigma1< e+¯/Sigma1r ep¯Gr eGre/Sigma1< e/parenrightbig Ga e/Gamma1eα,(97) B4=i¯Gr e/parenleftbig¯/Sigma1< e¯Ga eσa ep+¯/Sigma1< e¯Ga e¯/Sigma1a ep¯Ga e+¯/Sigma1< ep¯Ga e +¯/Sigma1< e¯Ga eGae/Sigma1a ep+¯/Sigma1r ep¯Gr e¯/Sigma1< e¯Ga e/parenrightbig Ga e/Gamma1eα, (98) B5=i¯Gr e/parenleftbig σ< ep+σ< eGae/Sigma1a ep+¯/Sigma1r ep¯Gr eσ< e/parenrightbig Ga e/Gamma1eα.(99) Using the relationship of Eq. ( 87) and/summationtext βuβ(/Omega1)=1, it is straightforward to show /summationdisplay βG(0) αβ=0 (100)and/summationdisplay βG(1) αβ=0. (101) So far, we have proved that the zeroth order and ﬁrst order of/summationtext βGαβare zero. In the same way, we can prove that all the higher order terms are zero although it is tedious butstraightforward. Finally, we have /summationdisplay βGαβ=0. (102) That is the condition of gauge invariance. E. First principles calculation We note that a ﬁrst principles formalism of dc quantum transport by doing density function theory (DFT) calculationwithin nonequilibrium Green’s function theory (NEGF-DFT)has been well established and extended to include the electron-phonon interaction [ 40,52]. First principles investigations have also been carried out using the NEGF-DFT approachfor molecular devices in the presence of ac bias [ 65]. In view of the above progress, our formalism presented in thispaper can in principle be implemented within NEGF-DFTframework so that inelastic ac transport calculation can becarried out from ﬁrst principles. To do this, we start fromEq. (5) where the potentials due to the exchange and correlation are included that are functional of charge density which isgiven before the iteration. From Eqs. ( 5), (36), and ( 37)w e ﬁnd the equilibrium Green’s function from which we canconstruct the new charge density. This in turn gives the newpotential due to exchange and correlation. We then solvethe Poisson-like Eq. ( 62) to ﬁnd the characteristic potential which gives a new Hamiltonian. We repeat this iterationuntil it reaches the self-consistency. Finally, we use Eq. ( 55) to calculate the dynamic conductance in the presence ofphonon. So far, we have treated the electron-phonon coupling strength g νas a constant. Actually both electron-phonon coupling and phonon spectrum depend on the external biasin the dc case and the driving frequency in the ac linearregime. In the presence of phonon, we assume that the phononexists only in the scattering region. To calculate the phononspectrum, one has to diagonalize the Hessian matrix (dynamic matrix) which is constructed from the second derivative of the total energy of the scattering region with respect to theposition of the atom (for details, see Chaps. 4 and 5 inRef. [ 66]). Importantly, the total energy is a functional of charge density which depends on external bias in the dc caseand the driving frequency in the ac case. As a result, the phononfrequency and phonon eigenvector depend on the external bias in the dc case. The electron-phonon coupling is also related to the phonon frequency and corresponding eigenvector. Itwas found in Refs. [ 52,56] that due to the external bias many phonon frequencies are renormalized between 10% and 30%for molecular junctions, while the electron-phonon couplingconstant is affected signiﬁcantly by an order of magnitude.Since the Hessian matrix depends on the driving frequency of the external bias in the ac case, the phonon frequency as well as electron-phonon coupling may also be sensitive to the 075407-8CURRENT-CONSERVING AND GAUGE-INV ARIANT . . . PHYSICAL REVIEW B 90, 075407 (2014) driving frequency. To address this issue quantitatively, a ﬁrst principles calculation has to be performed. In this paper wehave laid down the foundation of ac transport theory in thepresence of phonon; the implementation of this formalismin ﬁrst principles calculation will be the subject of futurework. Now we wish to make some comments on the DFT used in NEGF-DFT. In the dc case, the leads are in equilibriumwith well deﬁned Fermi distribution functions. However, thescattering region is out of equilibrium with the charge densityexpressed in terms of nonequilibrium lesser Green’s functionthat depends on external bias. Therefore, in the formalism ofNEGF-DFT, the density matrix or charge density is constructedat nonequilibrium using nonequilibrium Green’s functions.Due to this nonequilibrium nature, there is no minimizationprinciple to converge the charge density in open systems[67]. The above discussion applies to the ac case as well except that one has to use time-dependent DFT (TDDFT)[68] instead of static DFT. To reduce the computational complexity while still capturing the essential physics, peopleusually use the adiabatic local density approximation forthe exchange and correlation functionals in TDDFT. Thisscheme has been used to predict transient dynamics ofmolecular junctions [ 34]. Recently, the applicability of DFT in the open systems has been put on a more rigorousbasis [ 69]. F. Inelastic electron admittance spectroscopy In the dc situation, inelastic electron tunneling spectroscopy (IETS), which is the second derivative of the current withrespect to the bias voltage, is widely used as a powerfultool to identify the molecular vibrational modes and theelectron-phonon coupling strength. Similarly, in the ac situ-ation we will show below that the inelastic electron admit-tance spectroscopy (IEAS), which is the second derivativeof the admittance with respect to ac frequency, can havea similar functionality but with the frequency as an extrahandle. To demonstrate the feasibility of IEAS, we focus on the conductance up to the ﬁrst order in electron-phonon couplingstrength, which gives G LR=−q2/integraldisplaydE 2πf−¯f /Omega1/braceleftbigg Tr[B1]+/summationdisplay νTr[B2]/bracerightbigg −q2/integraldisplaydE 2πf−¯f /Omega1/summationdisplay νf(E+ων)Tr[B3],(103) where f=f(E),¯f=f(E+/Omega1), and Biare expressed in terms of the Green’s function B1=i/Omega1¯Gr euRGae/Gamma1eL−¯Gr 0e/Gamma1eRGa e/Gamma1eL +i/Omega1¯Gr euRGae/parenleftbig /Sigma1a epGae/Gamma1eL+/Gamma1eL¯Gr e¯/Sigma1r ep/parenrightbig −¯Gr e/Gamma1eRGae/parenleftbig /Sigma1a epGae/Gamma1eL+/Gamma1eL¯Gr e¯/Sigma1r ep/parenrightbig ,(104) B2ν=i/Omega1Ga e/Gamma1eL¯Gr egν/parenleftbig¯Gr e−uRGr e−+¯Ga e−uRGa e−/parenrightbig gν†/2, (105)and B3ν=/bracketleftbig i/Omega1Ga e/Gamma1eL¯Gr egν¯Gr e+uR/parenleftbig Ga e+−Gr oe+/parenrightbig gν† −Ga e/Gamma1eL¯Gr egν¯Gr e+/Gamma1eRGa e+gν†/bracketrightbig/slashbig 2 +/bracketleftbig Ga e+/Gamma1eL¯Gr e+gν¯Gr e(i/Omega1uR−/Gamma1R)Ga egν†/bracketrightbig/slashbig 2.(106) In the above equations, Gγ e±=Gγ e(E±ων). From Eqs. ( 42)– (44) we see that the phonon self-energies in general contain Fermi distribution functions through g<andg>. As a result, we see from Eqs. ( 55) and ( 103) that we will have terms involving two Fermi distribution functions f(E) andf(E+ων). As we will see below, these terms are very important for inelasticelectron admittance spectroscopy. At zero temperature, Eq. ( 103) becomes G LR=−q2 /Omega1/integraldisplay0 −/Omega1dE 2π/braceleftbigg Tr[B1]+/summationdisplay νTr[B2ν]/bracerightbigg −q2 /Omega1/integraldisplay0 −/Omega1dE 2π/summationdisplay νf(E+ων)Tr[B3ν],(107) where at zero temperature f(E+ων) is a step function and we have set the Fermi energy EF=0. So the second term in Eq. ( 107) is zero for E>−ων. On the other hand, the lower and upper limits of the integral require −/Omega1<E< 0. Consequently, the second term does not contribute to theintegral when /Omega1<ω ν. As a result, as we vary /Omega1the integrand of the second term in Eq. ( 107) will jump by Tr[ B3ν] whenever /Omega1sweeps through ων. After the integration over energy, the contribution of this term near ωνis proportional to (/Omega1−ων)Tr[B3ν] giving rise to a discontinuity of ∂/Omega1GLR nearων[see Fig. 3(b)]. Furthermore, from Eq. ( 106), Tr[B3ν] is proportional to the electron-phonon coupling strength. Incontrast, the ﬁrst term in Eq. ( 107) is a continuous function of/Omega1and the numerical calculation in Sec. IIIalso shows that this term changes slowly with frequency /Omega1. Hence the second derivative of the second term in G LRwith respect to /Omega1will give peaks at /Omega1=ων, while no peaks are contributed from the ﬁrst term in GLR. Thus we have ∂2GLR ∂/Omega12∼q2 /Omega1/summationdisplay νδ(ων−/Omega1)Tr[B3ν(−/Omega1)]. (108) So this quantity can be used to analyze the inelastic ac quantum transport which we term as ac inelastic electron admittancespectroscopy (IEAS). III. NUMERICAL CALCULATION OF A QUANTUM DOT To study dynamic conductance numerically, we consider a single level quantum dot system connected by two leads.All the quantities we use in the following calculation are inthe Hartree atomic unit. As we did in formulating the theory,we assume that the electron-phonon interaction exists onlyin the quantum dot. Furthermore, we assume that there arefour vibrational modes in the quantum dot and the vibrationalfrequencies ω νare 0.003, 0.005, 0.007, and 0.009, and the corresponding electron-phonon coupling strengths gνare 0.09, 0.07, 0.05, and 0.03, respectively, and we set the electronself-energy /Gamma1 eL=0.2 and/Gamma1eR=0.3. 075407-9YUNJIN YU, HONGXIN ZHAN, YADONG WEI, AND JIAN WANG PHYSICAL REVIEW B 90, 075407 (2014) FIG. 1. (Color online) Real part of admittance vs frequency. The blue dash line is for Re[ GLL(/Omega1)], the green dash dot line for Re[GLR(/Omega1)] and the red dot line for Re[ GRL(/Omega1)] are overlap, and the black solid line is for the real part of Re( GLL+GLR)o r Re(GLL+GRL). Figures 1and 2depict the real and imaginary part of dynamic admittance versus the frequency, respectively. Theblue dash line is for G LL, the green dash dot line is for GLR, and the red dot line is for GRL. When the ac frequency exceeds certain phonon frequency, the effect of this phonon mode isactivated. So the real part and imaginary part of admittanceshow piece-wise behaviors. Furthermore, our results showthatG LRandGRLare the same as expected. The black solid line is for GLL+GLRorGLL+GRLwhich conﬁrm the conservations of inelastic ac current, i.e., GLL+GRL=0 andGLL+GLR=0. In Fig. 3we plot the imaginary part of dynamic admittance Im(GLL), its ﬁrst derivative∂Im(GLL) ∂/Omega1and the second derivative ∂2Im(GLL) ∂/Omega12 versus the frequency in panels (a), (b), and (c), FIG. 2. (Color online) Imaginary part of admittance vs fre- quency. The blue dash line is for Im[ GLL(/Omega1)], the green dash dot line for Im[ GLR(/Omega1)] and the red dot line for Im[ GRL(/Omega1)] are overlap, and the black solid line is for the real part of Im( GLL+GLR)o r Im(GLL+GRL).(a) (b) (c) FIG. 3. (a) Imaginary part of admittance, (b) the ﬁrst derivative of imaginary part of admittance with respective to frequency, and (c) the second derivative of imaginary part of admittance withrespective to frequency versus the frequency. respectively. One can see that∂2Im(GLL) ∂/Omega12gives a peak at each phonon frequency with the peak height determined by theelectron-phonon strength. The larger the strength, the higherthe peak. Similar behavior is found for the real part ofdynamic admittance Re( G LL). This shows that we can use ∂2GLL ∂/Omega12to acquire the information of electron-phonon interaction including its frequency and coupling strength, making theIEAS a useful tool in studying the inelastic ac transport. IV . SUMMARY In this paper, we developed a theoretical formalism for ac transport with electron-phonon interaction based on thenonequilibrium Green’s function method. The Coulomb in-teraction is included self-consistently so that the current-conserving and gauge-invariant conditions are satisﬁed. Ourformalism can be used for ﬁrst principles transport calculationwithin NEGF-DFT formalism. We also proposed that theinelastic electron admittance spectroscopy can be used toprobe the inﬂuence of the electron-phonon interaction on thedynamic conductance in molecular devices. ACKNOWLEDGMENTS We gratefully acknowledge the support by National Natural Science Foundation of China with Grant No. 11074171(Y .D.W.) and No. 11374246 (J.W.) and GRF Grant No. HKU705212P, the UGC grant (Contract No. AoE/P-04/08) from theGovernment of HKSAR. 075407-10CURRENT-CONSERVING AND GAUGE-INV ARIANT . . . PHYSICAL REVIEW B 90, 075407 (2014) [1] M. A. Ratner, Mater. Today 5,20(2002 ). [2] J. R. Heath and M. A. Ratner, Phys. Today 56,43(2003 ). [3] N. A. Zimbovskaya and M. R. Pederson, Phys. Rep. 509,1 (2011 ). [4] T. P. Smith, III, B. B. Goldberg, P. J. Stiles, and M. Heiblum, P h y s .R e v .B 32,2696(R) (1985 ); T. P. Smith, III, W. I. Wang, and P. J. Stiles, ibid.34,2995(R) (1986 ). [5] R. A. Webb, S. Washburn, and C. P. Umbach, Phys. Rev. B 37, 8455 (1988 ). [6] J. B. Pieper and J. C. Price, P h y s .R e v .L e t t . 72,3586 (1994 ). [7] W. Chen, T. P. Smith, III, M. B ¨uttiker, and M. Shayegan, Phys. Rev. Lett. 73,146(1994 ). [8] L. P. Kouwenhoven, A. T. Johnson, N. 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PhysRevB.79.245325.pdf	Fractional quantum Hall state at /H9263=1 4in a wide quantum well Z. Papi ć,1,2G. Möller,3M. V . Milovanovi ć,2N. Regnault,4and M. O. Goerbig1 1Laboratoire de Physique des Solides, Université Paris-Sud, CNRS UMR 8502, F-91405 Orsay Cedex, France 2Institute of Physics, P .O. Box 68, 11 000 Belgrade, Serbia 3Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom 4Laboratoire Pierre Aigrain, Ecole Normale Supérieure, CNRS, 24 rue Lhomond, F-75005 Paris, France /H20849Received 1 April 2009; published 23 June 2009 /H20850 We investigate, with the help of Monte Carlo and exact-diagonalization calculations in the spherical geom- etry, several compressible and incompressible candidate wave functions for the recently observed quantumHall state at the ﬁlling factor /H9263=1 /4 in a wide quantum well. The quantum well is modeled as a two- component system by retaining its two lowest subbands. We make a direct connection with the phenomeno-logical effective-bilayer model, which is commonly used in the description of a wide quantum well and wecompare our ﬁndings with the established results at /H9263=1 /2 in the lowest Landau level. At /H9263=1 /4, the overlap calculations for the Halperin /H208495,5,3 /H20850and /H208497,7,1 /H20850states, the generalized Haldane-Rezayi state and the Moore- Read Pfafﬁan, suggest that the incompressible state is likely to be realized in the interplay between theHalperin /H208495,5,3 /H20850state and the Moore-Read Pfafﬁan. Our numerics show the latter to be very susceptible to changes in the interaction coefﬁcients, thus indicating that the observed state is of multicomponent nature. DOI: 10.1103/PhysRevB.79.245325 PACS number /H20849s/H20850: 73.43.Cd, 73.21.Fg, 71.10.Pm I. INTRODUCTION Advances in fabrication of high-quality GaAs semicon- ductor systems have led to an ever growing collection of theobserved incompressible fractional quantum Hall states in avariety of settings. 1These states occur at particular ratios between the number of electrons Nand the number of mag- netic ﬂux quanta N/H9278that pierce the system in the direction perpendicular to the sample. This commensurability can beexpressed as the ﬁlling factor /H9263=N/N/H9278=p/qin terms of in- tegers p,q, which is the single most important quantity that characterizes the quantum Hall state. In a thin layer, qusually turns out to be an odd integer, the fact which had its pioneering explanation in terms of theLaughlin wave function 2for the case of p=1, q =3,5,7,... and its subsequent generalizations in terms of composite fermions3/H20849CFs/H20850, applicable to general integers p,qas long as qis odd, and hierarchy theory.4However, a state with an even denominator has also been observed5but in the ﬁrst excited Landau level /H20849LL/H20850. One cannot account for it in the usual Laughlin/composite fermion approach andthe idea of pairing has commonly been invoked to explainthe origin of this fraction. 6,7The simplest realization of pair- ing between spin-polarized electrons is the so-called Pfafﬁandeﬁned by the Moore-Read wave function 7and supporting excitations with non-Abelian statistics.8 The possibility of an extra degree of freedom lifts the requirement of Fermi antisymmetry and hence gives anotherroute toward realizing even denominator fractions. The addi-tional degree of freedom can be the ordinary spin or else a“pseudospin” in case of a wide quantum well, where the twolowest electronic subbands correspond to ↑,↓. If the sample is etched in such a way to create a barrier in the middle, thussupressing tunneling between the two “sides,” one can thinkof it as a bilayer with ↑,↓denoting the left and right layers where electrons can be localized. Incompressible quantumHall states for such systems have been theoretically predictedin Ref. 9and experimentally conﬁrmed for cases of bilayer at ﬁlling factor /H9263=1 and /H9263=1 /2.10,11Later on, essentially the same quantum Hall state at /H9263=1 /2 was observed in a sample which had the geometry of a single wide well.12It was ar- gued, on the basis of a self-consistent Hartree-Fock approxi-mation, that in a wide well the electrons /H20849due to their mutual repulsion /H20850reorganize themselves so as to form an effective bilayer distribution of charge. Hence, an equivalence be-tween the two very different samples was claimed and theo-retical works set out to analyze the problem from thispremise. 13,14 On the basis of the quantum mechanical overlap with the ground state obtained in exact diagonalization /H20849ED/H20850, includ- ing a realistic bilayer conﬁnement potential, Ref. 13estab- lished that the ground state is well described by the so-called/H208493,3,1 /H20850Halperin wave function. 15This wave function distin- guishes between two kinds of electrons and the fact that itdescribes the system is what we mean by the system being“multicomponent.” Experimental work gave further insightinto the nature of the multicomponent state at /H9263=1 /2 and strengthened the belief that the /H208493,3,1 /H20850wave function is a correct physical description.12Namely, the behavior of the excitation gap as a function of tunneling amplitude /H9004SAS/H20849i.e., the splitting between the two lowest subbands /H20850was found to have upward cusp at the intermediate value of /H9004SASand the state was quickly destroyed by the application of electro-static bias /H20849charge imbalance /H20850. 12In Ref. 14, a numerical study was able to reproduce the observed upward cusp in theactivation gap by diagonalizing the bilayer Hamiltonian withexplicit interlayer tunneling. A recent experimental paper 16reports the observation of the/H9263=1 /4 quantum Hall state in a wide quantum well. The state is fragile and almost indiscernible when only a perpen-dicular magnetic ﬁeld is applied /H20849although one could expect that with yet higher sample qualities, a small plateau wouldbe developed already at that point /H20850. However, when the mag- netic ﬁeld is tilted, there is a clear dip in the value of longi-PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 1098-0121/2009/79 /H2084924/H20850/245325 /H2084913/H20850 ©2009 The American Physical Society 245325-1tudinal resistance Rxx, signifying the presence of an incom- pressible state. In this paper we analyze the complex interplay between the single- and multicomponent nature of the ground state at /H9263=1 /4 in a wide quantum well, in comparison with the ground state at /H9263=1 /2. Contrary to previous studies,13,14we do not make the ad hoc assumption that the wide quantum well may be described as an effective bilayer. Instead, weconsider the two lowest electronic subbands of the quantumwell, which is modeled by the inﬁnite square well for thesake of convenience but cross-checked with other conﬁne-ment models. The energy splitting between these two sub-bands, the associated wave functions of which are symmetricand antisymmetric, respectively, in the zdirection is given by /H9004 SAS/H20849occasionally referred to as the tunneling amplitude /H20850. Due to the low ﬁlling factor /H20849/H9263=1 /4/H20850, the power of the ED method will be rather limited and other complementary ap-proaches may be needed to fully explain the experimentalﬁndings. The paper is organized as follows. Section IIis devoted to the single-component candidate for /H9263=1 /4 and we study its overlap with the exact Coulomb ground state within variousconﬁnement models. In Sec. III, we deﬁne the multicompo- nent wave functions expected to be relevant at this ﬁllingfactor. The two likely candidates, the Halperin /H208495,5,3 /H20850and /H208497,7,1 /H20850states, are investigated within a simple bilayer model without tunneling. The two-subband model of the quantumwell is introduced and described in Sec. IV. Our main results of ED calculations in the spherical geometry are presented inSec. V. To extend the reach of our numerics, we furthermore deploy Monte Carlo simulations of the trial wave functionsidentiﬁed beforehand to analyze their energetic competition.We summarize with our view on the nature of the state at /H9263=1 /4 in Sec. VI. II. ONE COMPONENT STATE A. Pfafﬁan at /H9263=1 Õ4 There is a natural candidate for the fully polarized quan- tum Hall state at /H9263=1 /4—it is the generalized Moore-Read Pfafﬁan,8 /H9023Pf/H20849z1, ..., zN/H20850=P f/H208731 zi−zj/H20874/H20863 i/H11021j/H20849zi−zj/H208504, /H208491/H20850 expressed in terms of the complex coordinate of the electron in the plane where zj=xj+iyj. The object Pf is deﬁned as PfMij=1 2N/2/H20849N/2/H20850!/H20858 /H9268/H33528SNsgn/H9268/H20863 k=1N/2 M/H9268/H208492k−1/H20850/H9268/H208492k/H20850, acting upon the antisymmetric N/H11003Nmatrix MijandSNis a group of permutations of Nobjects. Pf renders the wave function totally antisymmetric and encodes the same kind ofcorrelations as in the more familiar /H9263=5 /2 case.7In the spherical geometry4,17many-body states are characterized by the number of electrons N, the number of ﬂux quanta N/H9278 generated by a magnetic monopole placed in the center of the sphere and extending radially through its surface, and anadditional topological number which is the shift. For the Pfafﬁan in Eq. /H208491/H20850, the three numbers are related by the for- mula N/H9278=4N−5./H9023Pfis a zero-energy eigenstate of a certain three-body Hamiltonian8but in our calculations it was gen- erated from its root conﬁguration via the squeezingtechnique. 18On the other hand, the Coulomb /H20849two-body /H20850 Hamiltonian commutes with the angular momentum operatorLbecause of rotational invariance and, by Wigner-Eckart theorem, the interaction is parametrized by discrete set ofnumbers V Lknown as the Haldane pseudopotentials.4The motion of electrons is therefore fully described in terms ofthe in-plane /H20849spherical /H20850coordinates /H9258,/H9278and the use of dif- ferent conﬁnement models in the /H20849perpendicular /H20850zdirection /H20849neglecting the in-plane magnetic ﬁeld /H20850will only modify the values of pseudopotentials. B. Finite thickness models Most of the candidate wave functions for quantum Hall fractions have been extensively studied via numerical tech-niques such as ED or Monte Carlo. For the sake of conve-nience but also due to the intrinsic ambiguity which stemsfrom the fact that in a strongly correlated system many inputparameters /H20849e.g., the precise form of the interaction /H20850are un- known, it is natural to start off from the limit of inﬁnitelythin layer of electrons interacting via Coulomb force andhope that the inclusion of, e.g., realistic conﬁnement andsample thickness will have small, perturbative corrections.There have been different proposals to account for the ﬁnitethickness of the sample in the perpendicular direction but theone that is straightforward and most natural from the point ofview of ED is the Zhang–Das Sarma /H20849ZDS /H20850model 19which is simply given by substituting the interaction 1 r→1 /H20881r2+/H20849w/2/H208502/H208492/H20850 /H20849we will always denote by wthe width of the sample and the energy is always expressed in units of e2//H9280lB, where the magnetic length is lB=/H20881/H6036c/eBis given in terms of the per- pendicular magnetic ﬁeld B/H20850. Qualitatively, this substitution softens the interaction19and was studied extensively /H20849to- gether with other conﬁnement models, some of which wewill introduce below /H20850in Ref. 20, where it was advertised to signiﬁcantly stabilize the Moore-Read Pfafﬁan at /H9263=1 /2/H20849the effect being most pronounced in the second LL /H20850but/H20849in most cases /H20850decrease the overlap somewhat for the Laughlin states at/H9263=1 /3 and 1/5. In Ref. 21it was noticed that this kind of interaction can lead to an instability of the composite fer-mion sea, which is believed to describe the compressiblestate at /H9263=1 /2 in the lowest LL, toward the paired state described by the Pfafﬁan. Indeed, the CF Fermi liquid can beregarded as a special member of the general class of pairedCF wave functions, 22of which it represents the limit of van- ishing gap. Although the ZDS model /H208492/H20850has a very simple form, there is no physical wave function that corresponds to thisconﬁnement potential in the zdirection. Other popular choices for the conﬁnement in the zdirection include the inﬁnite square well /H20849ISQW /H20850and Fang-Howard /H20849FH/H20850, whichPAPI Ćet al. PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-2are presumably more realistic than ZDS because they are deﬁned by the actual wave functions of simple model poten-tials for the quantum well, given by /H9278ISQW /H20849z/H20850=/H208812 wsin/H20873/H9266z w/H20874, /H208493/H20850 /H9278FH/H20849z/H20850=/H2088127 2w3ze−3z/2w, /H208494/H20850 respectively. C. Overlaps We have performed ED calculations for various conﬁne- ment models /H20851Eqs. /H208492/H20850–/H208494/H20850/H20852and all system sizes N =6,8,10,12 accessible at present. In Fig. 1we present the overlap /H20841/H20855/H9023Pf/H20841/H9023exact/H20856/H20841between the exact Coulomb ground state at /H9263=1 /4 and /H9023Pf, ﬁnite width being modeled by the ZDS ansatz /H20851Eq. /H208492/H20850/H20852. The size of the Hilbert space at N =12 is noteworthy: the dimension of the Lz=0 sector is 218 635 791. It appears that the overlap of the Pfafﬁan state is rather high for large values of the width /H20849even if it is negligible for small ws/H20850. These values could likely be increased further by considering general pairing wave functions.22However, these overlaps alone cannot be taken as solid evidence for apairing nature of the /H9263=1 /4 for two reasons. First, for N =6 and 12 there is the aliasing problem with composite fer-mion states: Jain states with different physical properties/H20849e.g., Abelian instead of non-Abelian statistics /H20850occur at the same values of NandN /H9278on the sphere /H20849because of ﬁnite system size /H20850. High overlap for the aliased states may there- fore come from other incompressible states different fromthe Pfafﬁan. Second, for the nonaliased states at N=8 and 10, there appears to be a critical value of the width at which theoverlap as a function of wsuffers a sharp jump. By analyzing the entire low energy spectrum on the sphere as a function ofwidth, we have established that the /H20849neutral /H20850gap collapses at the critical point of w/l B. Therefore, in order to get to the Pfafﬁan phase, one must go through a /H20849ﬁrst-order /H20850phase transition. Before the transition, the ground state is obtainedin the L/H110220 sector of the Hilbert space and the overlap with the Pfafﬁan /H20849which resides in L=0 sector /H20850remains zero due to the difference in symmetry. The lack of adiabatic continuity and the aliasing problem cast some doubt on the Pfafﬁan state as a good candidate for /H9263=1 /4 in the lowest LL. We have also checked using other conﬁnement models /H20851Eqs. /H208493/H20850and /H208494/H20850/H20852but in these cases for N=8 and 10 the overlap remains zero for any value of w/lB. Thus our ED results do not yield a deﬁnite answer withrespect to the relevance of /H9023 Pfin the single layer at /H9263=1 /4. We would like to stress the qualitative difference in our results obtained by using ZDS versus other conﬁnementmodels which appears, to the best of our knowledge, to bethe ﬁrst such case in the literature. The smaller overall en-ergy scale /H20849and the smaller gap as well /H20850is very likely to be at the origin of this discrepancy. We note in passing that, con-trary to the ﬁnite-width models which change allpseudopo- tentials at once, one may start from the pure Coulomb inter-action and vary just a few strongest pseudopotentials. 23We have tried varying both V1andV3but this procedure does not stabilize the Pfafﬁan phase in any ﬁnite region of the param-eter space for N=8. III. TWO-COMPONENT STATES Soon after Laughlin’s wave function describing the in- compressible state at /H9263=1 /3 when the electron spins are fully polarized, Halperin15proposed a class of generalized wave functions deﬁned as /H9023mm/H11032n/H20849z1↑, ..., zN↑↑,z1↓, ..., zN↓↓/H20850 =/H20863 i/H11021jN↑ /H20849zi↑−zj↑/H20850m/H20863 k/H11021lN↓ /H20849zk↓−zl↓/H20850m/H11032/H20863 sN↑ /H20863 tN↓ /H20849zs↑−zt↓/H20850n, /H208495/H20850 where the electrons are distributed over two components /H20849la- beled by ↑,↓/H20850. The exponents m,m/H11032denote the “intracom- ponent” correlations originating from the basic Laughlin-Jastrow building blocks within each component, whereas n describes “intercomponent” correlations /H20849we have omitted the ubiquitous Gaussian factors and implicitly assume thatthere is a spinor part to this wave function as well as anoverall antisymmetrization between ↑and↓/H20850. In order for these wave functions to be eligible candidates for the groundstate of the system, one must enforce an additional require-ment that they be eigenstates of the Casimir operator of theSU/H208492/H20850group, i.e., the total spin S 2, as long as the interaction is symmetric with respect to intracomponent and intercom-ponent /H20849e.g., the usual case of electrons with spin /H20850. However, apart from electrons with spin, the wave functions /H20851Eq. /H208495/H20850/H20852 have also been used in bilayer systems where this symmetryis broken as soon as the layer separation is nonzero. In thiscase, the wave functions /H20851Eq. /H208495/H20850/H20852need not be eigenstates of the total spin. There have been generalizations of these wavefunctions in the physics of bilayer systems at total ﬁllingfactor 24–26/H9263=1 and to more than two components,27where further constraints on the possible values of m,m/H11032,nwere derived within the plasma analogy.28In a two-component case, these turn out to be the intuitive requirement that intra-00.20.40.60.81 0 5 10 15 20Overlap w/lBN=6 N=8 N=10 N=12 FIG. 1. /H20849Color online /H20850Overlap /H20841/H20855/H9023Pf/H20841/H9023exact/H20856/H20841between the exact Coulomb state for ﬁnite width /H20849ZDS model /H20850and the Pfafﬁan at /H9263=1 /4.FRACTIONAL QUANTUM HALL STATE AT /H9263=1 4IN … PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-3component interactions are stronger than intercomponent in- teractions: m,m/H11032/H11350n. For the particular case of two compo- nents and m=m/H11032=n+2/H20849which includes /H9023331and/H9023553/H20850, the Halperin wave function /H20851Eq./H208495/H20850/H20852can be analytically cast into a paired form26,29via Cauchy determinant identity /H20849up to the unimportant phase factor /H20850, /H20863i/H11021jN↑/H20849zi↑−zj↑/H20850/H20863k/H11021lN↓/H20849zk↓−zl↓/H20850 /H20863sN↑/H20863tN↓/H20849zs↑−zt↓/H20850= det/H208751 zi↑−zj↓/H20876, where the pairing function is given by det /H208511 zi↑−zj↓/H20852. In the case of the 111 state, this pairing nature was recently exploited to make a connection to paired composite fermion states and toconstruct wave functions interpolating between these tworegimes. 26Halperin wave functions are the exact zero-energy eigenstates of the two-body Hamiltonian H=/H20858 i/H11021j/H20875/H20858 L=0m−1 VL↑↑Pij↑↑/H20849N/H9278−L/H20850+/H20858 L=0m/H11032−1 VL↓↓Pij↓↓/H20849N/H9278−L/H20850/H20876 +/H20858 i,j/H20858 L=0n−1 VL↑↓Pij↑↓/H20849N/H9278−L/H20850, /H208496/H20850 where Pij/H9268/H9268/H11032/H20849L/H20850projects onto the state with angular momen- tum Lof particles iand jwith respective /H20849pseudo /H20850spins/H9268 and/H9268/H11032. Besides offering great convenience for handling Hal- perin wave functions /H20851Eq. /H208495/H20850/H20852in ED, Eq. /H208496/H20850enabled count- ing of the number of excited quasihole states and reafﬁrmingthe idea that the states described by Eq. /H208495/H20850possess Abelian statistics. 8 At the ﬁlling factor /H9263=1 /4, there are three wave functions of form /H208495/H20850that meet the necessary physical requirements, /H9023553/H11013/H208495,5,3 /H20850,/H9023771/H11013/H208497,7,1 /H20850, and/H90235131/H11013/H208495,13,1 /H20850. None of them is an eigenstate of S2, so they are more adapted to the case of a bilayer than that of real spin. In Fig. 2we present the basic overlap characterization of the ﬁrst twowave functions in a simple bilayer model deﬁned by the interaction V ↑↑/H20849r/H20850=V↓↓/H20849r/H20850=1 /r,V↑↓/H20849r/H20850=1 //H20881r2+d2/H20849where d being the distance between the layers /H20850.30/H208495,5,3 /H20850displays a familiar maximum in the overlap for small distance betweenthe layers. /H208497,7,1 /H20850was dismissed in Ref. 16arguing that it would more likely lead to two coupled Wigner crystals thanan incompressible liquid. Our diagonalization scheme is not adapted to address states with broken translation symmetry,so we do not see an a priori reason to reject this state. The results in Fig. 2are for N=8 particles, they are fully consis- tent with those of smaller Nbut direct comparison between /H208495,5,3 /H20850and /H208497,7,1 /H20850is not possible because they are character- ized by different shifts /H20849−5 and −7, respectively /H20850. 28We will address this issue below by extrapolating to the thermody-namic limit the respective trial energies from Monte Carlosimulations for both of these states. The last possibility, /H208495,13,1 /H20850, is a peculiar one because it can only occur in the case of a strong density imbalance.Such an imbalance would lead to an increase in the chargingenergy but if one of the coupled states is a prominent quan-tum Hall state, the gain in correlation energy can outweighthe price of charge imbalance, as it has been experimentallyveriﬁed. 31However, in the present case, our numerical cal- culations conﬁrmed that this candidate can be discarded be-cause it takes unrealistically high values of the sample widthfor this wave function to have any numerical relevance at all. Given the low ﬁlling factor /H9263=1 /4 we are studying, one must also consider the possibility of nearby compressiblestates that can intervene for some values of the external pa-rameters. Apart from the obvious metallic state similar to theFermi-liquidlike state proposed by Rezayi and Read, 32there is in principle also the Haldane-Rezayi /H20849HR/H20850state,6,8which is deﬁned by /H9023HR/H20849/H20853zi↑,zi↓/H20854/H20850= det/H208751 /H20849zi↑−zj↓/H208502/H20876/H20863 i/H11021jN /H20849zi−zj/H208504. The last term is a global Laughlin-Jastrow factor for all par- ticles regardless of their spin. /H9023HRis the zero energy eigen- state of the interaction parametrized by the set of pseudopo-tentials V L=/H208531,1,0,1,0,... /H20854and occurs at the shift of −6. It is also a spin singlet6and compressible on the basis of its nonunitary parent conformal ﬁeld theory.8,33However, its edge theory33is closely related to that of the Abelian /H208495,5,3 /H20850 state, which suggests that the HR state may be in the vicinityof the incompressible state and nonetheless affect the physi-cal properties of the system. Recently there have been pro-posals that compressible states can be molded into incom-pressible ones. 34 IV . QUANTUM-WELL MODEL So far we have discussed the stability of the one- component Pfafﬁan state in different ﬁnite-width models/H20849Sec. II/H20850and two-component states in a bilayer model where each layer is considered as an inﬁnitely narrow quantum well/H20849Sec. III/H20850. In this section, we consider an inﬁnite square well of width win the direction z/H33528/H208510,w/H20852. The electronic motion in the zdirection will then be quantized, yielding an elec- tronic subband structure. A. Two-subband approximation Instead of a full description with all the electronic sub- bands, we only consider the two lowest subbands and iden-00.20.40.60.81 0 2 4 6 8 10Overlap d/lB(5,5,3) (7,7,1) FIG. 2. /H20849Color online /H20850Overlap between the exact bilayer state with the /H208495,5,3 /H20850and /H208497,7,1 /H20850states for N=8 particles.PAPI Ćet al. PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-4tify them with the two pseudospin states, /H9023↑,↓ =/H9278↑,↓/H20849z/H20850YN/H9278/2,N/H9278/2,m/H20849/H9258,/H9278/H20850, where /H9278↑/H20849z/H20850=/H208812 wsin/H20873/H9266z w/H20874, /H208497/H20850 /H9278↓/H20849z/H20850=/H208812 wsin/H208732/H9266z w/H20874, /H208498/H20850 and the Ys represent monopole spherical harmonics with −N/H9278/2/H11349m/H11349N/H9278/2/H20849we assume that the states are entirely within the lowest LL /H20850. We refer to states /H208497/H20850and /H208498/H20850as sym- metric and antisymmetric, respectively, because of their re-ﬂection symmetry with respect to the center of the well. Iftheir energy difference is denoted by /H9004 SAS, the corresponding second quantized Hamiltonian is given by35 H=−/H9004SAS 2/H20858 m/H20849cm↑†cm↑−cm↓†cm↓/H20850 +1 2/H20858 /H20853m/H20854/H20858 /H20853/H9268/H20854Vm1,m2,m3,m4/H92681/H92682/H92683/H92684cm1/H92681†cm2/H92682†cm4/H92684cm3/H92683, /H208499/H20850 where cm/H9268/H20849†/H20850annihilates /H20849creates /H20850an electron in the state m with pseudospin /H9268. The matrix elements Vm1,m2,m3,m4/H92681,/H92682,/H92683,/H92684can be straightforwardly evaluated from the Haldane pseudopotentials for the result- ing in-plane interaction V2D/H92681,/H92682,/H92683,/H92684/H20849r/H60231−r/H60232/H20850 =e2 /H9280lB/H20885dz1/H20885dz2/H9278/H92681/H11569/H20849z1/H20850/H9278/H92682/H11569/H20849z2/H20850/H9278/H92683/H20849z1/H20850/H9278/H92684/H20849z2/H20850 /H20881/H20849r/H60231−r/H60232/H208502+/H20849z1−z2/H208502, /H2084910/H20850 where the position variables are expressed in units of lBsuch that the integral is dimensionless. In this paper we do not make an attempt to quantitatively model the experiment of Ref. 16but we are interested in the possible phases that may occur and the transitions betweenthem. Therefore, we expect the model described by Hamil-tonian /H208499/H20850to be qualitatively correct and in agreement with other conﬁnement models that assume the lowest subband tobe symmetric and the ﬁrst excited one to have a node in thecenter /H20849z=w/2/H20850. Any difference of the conﬁning potential away from the inﬁnite square well will modify the energyeigenvalues and the associated wave functions /H9278/H9268/H20849z/H20850. How- ever, it is expected that the energies are more strongly af-fected than the wave functions. In particular, the nodal struc-ture of the wave functions is robust, such that the two lowesteigenstates of the inﬁnite well faithfully represent the under-lying features. However, we will allow for the general valuesof the level splitting /H9004 SASto account for the variations in the eigenenergies. B. Connection between the quantum-well model and the bilayer Hamiltonian From a more general point of view, the quantum-well model exposed above is a two-component model such as thebilayer model, which has been used in the discussion of thewide quantum well. 12Indeed, the wide quantum well allows the electrons to reduce their mutual Coulomb repulsion byexploring more efﬁciently the zdirection and it has been argued that due to this effect, a spontaneous bilayer may beformed, under appropriate conditions, in a wide quantumwell. 12,13Here, a connection is made between both two- component models, on the basis of Hamiltonian /H208499/H20850. The in- termediate steps in the derivation of the effective model maybe found in the Appendix. Hamiltonian /H208499/H20850may be rewritten in terms of the density and spin-density operators projected to a single Landau level.The Fourier components of the projected density operator ofpseudospin- /H9268electrons read /H9267¯/H9268/H20849q/H20850=/H20858 m,m/H11032/H20855m/H20841e−iq·R/H20841m/H11032/H20856cm/H9268†cm/H11032/H9268, in terms of the two-dimensional /H208492D/H20850wave vector qand the guiding-center operator R, the latter acting on the states la- beled by the quantum numbers m. It is furthermore useful to deﬁne the total /H20849projected /H20850density operator /H9267¯/H20849q/H20850=/H9267¯↑/H20849q/H20850+/H9267¯↓/H20849q/H20850/H20849 11/H20850 and the projected pseudospin density operators, S¯/H9262/H20849q/H20850=/H20858 m,m/H11032/H20855m/H20841e−iq·R/H20841m/H11032/H20856cm/H9268†/H9270/H9268,/H9268/H11032/H9262 2cm/H11032/H9268/H11032, /H2084912/H20850 where /H9270/H9268,/H9268/H11032/H9262are the usual 2 /H110032 Pauli matrices with /H9262 =x,y,z. In terms of the projected /H20849pseudospin /H20850density operators, Hamiltonian /H208499/H20850approximately reads as H/H112291 2/H20858 qVSU/H208492/H20850/H20849q/H20850/H9267¯/H20849−q/H20850/H9267¯/H20849q/H20850+2/H20858 qVsbx/H20849q/H20850S¯x/H20849−q/H20850S¯x/H20849q/H20850 −/H9004˜SASS¯z/H20849q=0/H20850, /H2084913/H20850 where the SU /H208492/H20850-symmetric interaction potential VSU/H208492/H20850/H20849q/H20850 and the symmetry-breaking potential Vsbx/H20849q/H20850are linear com- binations of the Fourier-transformed potentials deﬁned in Eq./H2084910/H20850. Their precise form is given in the Appendix by Eqs. /H20849A2/H20850and /H20849A5/H20850, respectively. Hamiltonian /H2084913/H20850neglects a par- ticular term /H11008S¯z/H20849−q/H20850S¯z/H20849q/H20850, which turns out to constitute the lowest energy scale in the interaction Hamiltonian /H208499/H20850/H20851see Eq. /H20849A9/H20850in the Appendix /H20852. Furthermore, /H9004˜SAS=/H9004SAS−/H9253/H9263e2 /H9280lBw lB/H2084914/H20850 is the effective subband gap. The numerical prefactor /H9253de- pends on the precise nature of the considered conﬁnementpotential and, as shown in the Appendix, expression /H2084914/H20850is derived within a mean-ﬁeld approximation of a particularterm in Hamiltonian /H208499/H20850. Expression /H2084914/H20850is easy to under- stand; whereas the subband gap /H9004 SAStends to polarize the system in the ↑state, namely, in narrow samples, the second term in Eq. /H2084914/H20850indicates that the interactions are weaker in the↓subband. From the interaction point of view, it is there- fore energetically favorable to populate the ﬁrst excited sub-FRACTIONAL QUANTUM HALL STATE AT /H9263=1 4IN … PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-5band. This effect becomes more pronounced in larger quan- tum wells. Notice furthermore that this argument alsodelimits the regime of validity of the two-subband approxi-mation of the wide quantum well; when the term /H9253/H9263/H20849e2//H9280lB/H20850/H11003/H20849w/lB/H20850becomes much larger than the bare sub- band gap /H9004SAS, the electrons may even populate higher sub- bands, which are neglected in the present model and thesystem eventually crosses over into a three-dimensional re-gime. Notice that Hamiltonian /H2084913/H20850has the same form as the Hamiltonian which describes a bilayer quantum Hallsystem, 36up to a rotation from the ztoxaxis. In this rotated reference frame, one may deﬁne the intralayer and interlayerinteractions as V A/H20849q/H20850=VSU/H208492/H20850/H20849q/H20850+Vsbx/H20849q/H20850=1 4/H20851V2D↑↑↑↑/H20849q/H20850+V2D↓↓↓↓/H20849q/H20850 +2V2D↑↓↑↓/H20849q/H20850/H20852+V2D↑↑↓↓/H20849q/H20850/H20849 15/H20850 and VE/H20849q/H20850=VSU/H208492/H20850/H20849q/H20850−Vsbx/H20849q/H20850=1 4/H20851V2D↑↑↑↑/H20849q/H20850+V2D↓↓↓↓/H20849q/H20850 +2V2D↑↓↑↓/H20849q/H20850/H20852−V2D↑↑↓↓/H20849q/H20850. /H2084916/H20850 As for the case of the true bilayer, the thus deﬁned intralayer interaction is stronger than the interlayer interaction, for allvalues of q. Since our ED calculations employ Hamiltonian /H208499/H20850,i no r - der to compare the numerical results with the Halperin states/H20851Eq. /H208495/H20850/H20852which are the native eigenstates of true bilayer Hamiltonian /H208496/H20850, we can apply the mapping between the two models described above in a reverse fashion. As Halperinwave functions are commonly labeled by the single particlestates /H20841↑/H20856,/H20841↓/H20856/H20849which are the eigenstates of S z/H20850and deﬁned by interaction potentials /H20853VA,VE/H20854, we can imagine a linear transformation /H20849rotation from ztox/H20850that transforms them into /H20849unnormalized /H20850symmetric /H20841+/H20856=/H20841↑/H20856+/H20841↓/H20856and antisym- metric /H20841−/H20856=/H20841↑/H20856−/H20841↓/H20856combinations. Then, by inverting the Eqs. /H2084915/H20850and /H2084916/H20850, we obtain the set of interaction potentials that generate Halperin states /H20849m,m/H11032,n/H20850in a quantum-well description. In what follows, Halperin states /H208495/H20850are under- stood to be indexed by /H20841+/H20856,/H20841−/H20856instead of the usual notation /H20841↑/H20856,/H20841↓/H20856, unless explicitly stated otherwise. C. Energetics of trial wave functions To extend the reach of our calculations to system sizes larger than those which can be treated in ED, we set upMonte Carlo simulations of the trial states which haveemerged as good candidates for the ground state. The generalstrategy of this approach is to obtain an estimate of the en-ergy in the thermodynamic limit for the different trial statesbased on a scaling with system size of their energies. As detailed in Sec. IV B above, we expect formation of two-component wave functions where S xis a good quantum number, such that the Halperin wave functions are expressedin terms of the coordinates of electrons in the /H20841+/H20856and /H20841−/H20856 states, and lower well, indexed below by /H9268. We considercases with equal population of electrons in these two bands or full population of the lowest subband in the ISQW for thesingle-component cases. In order to calculate efﬁciently the interaction of electrons in a well of ﬁnite width using Monte Carlo simulations, wereplace the interaction /H20851Eq. /H2084910/H20850/H20852with an effective potential that reproduces all pseudopotential coefﬁcients of the origi-nal potential V 2D. Many such potentials can be constructed. Here, we use an interaction of the form proposed in Ref. 37, built from simple polynomials38 Veff/H9268/H9268/H11032/H20849r/H20850=/H20858 k=−1Nmax/H9268/H9268/H11032 ck/H9268/H9268/H11032rk. /H2084917/H20850 The pseudopotentials of the monomials rncan be evaluated analytically /H20849generalizing Ref. 39/H20850. Choosing ckto match the pseudopotential coefﬁcients of the interaction /H20851Eq. /H2084910/H20850/H20852be- comes a simple linear problem. Crucially, we allow for thecoefﬁcient of the Coulomb term c −1to be varied, also. The number of terms is chosen equal to the minimal numberrequired to match the relevant pseudopotentials /H20849odd pseudo- potentials V 2m+1for intra /H20849pseudo /H20850spin interactions and all N/H9278+1 terms, otherwise /H20850. It is habitual in the literature to introduce a neutralizing background, in order to highlight the correlation energy as-sociated with a wave function. We use a background E bg/H20851/H9278/H20852 that matches the distribution /H20841/H9278/H20849z/H20850/H208412of electrons in their sub- bands, in order to study the correlation energy of the differ- ent states. However, to establish a ﬁnal comparison betweenthe different wave functions, a unique convention for thebackground is required and we adopt the background of thesingle layer conﬁguration as a reference point E ref=Ebg/H20851/H9278↑/H20852. Extrapolation to the thermodynamic limit is undertaken as two separate steps. The correlation energy is obtained bylinear scaling over the inverse system size N −1, using the habitual rescaling of the magnetic length lB/H11032=/H20851N//H20849/H9263N/H9278/H20850/H208521/2lB.40For the two-component states, the dif- ference in background energy Eref−/H20858/H9268Ebg/H9268/H20851/H9278/H9268/H11032/H20852is extrapo- lated separately and added to the correlation energy. V . COMPETING PHASES IN THE QUANTUM-WELL MODEL In order to justify the model of the quantum well, in this section we present the ED study of Hamiltonian /H208499/H20850and ana- lyze the energetics of the relevant trial wave functions inMonte Carlo simulations. We brieﬂy revisit the problem of /H9263=1 /2 extending the results of Refs. 12and14/H20849Sec. VA /H20850 and then present results pertaining to /H9263=1 /4/H20849Sec. VB /H20850. A./H9263=1 Õ2 in a quantum well At this ﬁlling factor the competing phases we consider here are the /H208493,3,1 /H20850Halperin state, the Moore-Read Pfafﬁan, and the HR state. Reference 14demonstrated a competition between the multicomponent /H208493,3,1 /H20850state and the fully po- larized single-component Moore-Read Pfafﬁan. In the regionof small tunneling, the ground state shows high overlap withthe Halperin state; as the tunneling is increased, the HalperinPAPI Ćet al. PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-6state is destroyed and the Pfafﬁan takes over. The point of crossover between the two is related to the upward cusp in the activation gap.14 Figure 3shows our ED results for eight particles in the quantum well at the ﬁlling factor /H9263=1 /2. Figures 3/H20849a/H20850–3/H20849c/H20850 represent the overlap between the exact ground state and the/H208493,3,1 /H20850, the Pfafﬁan, and the HR states, respectively, as a function of the well width w/l Band the bare subband gap /H9004SAS. In general, the latter is a monotonically decreasing function of the well width. Again, we choose wand/H9004SASas independent parameters of the model. Furthermore, we plotthe quantity denoted by /H20855S z/H20856, the expectation value of the Sz=S¯z/H20849q=0/H20850component of the pseudospin which has the meaning of the “order parameter” /H20851Fig. 3/H20849d/H20850/H20852. One notices that /H20855Sz/H20856continuously crosses over from a full polarization in the↑subband at low values of w/lBand a large gap /H9004SASto a polarization in the ↓subband for larger quantum wells and small gaps /H9004SAS. As it is discussed in the previous section, the interactions in a wider quantum well favor a populationof the ﬁrst excited electronic subband ↓because of the node in the wave function in the zdirection and, therefore, de- crease the effective subband gap. Indeed, Eq. /H2084914/H20850indicates that the crossover line from positive to negative /H9004˜SASis char- acterized by a border that is linear in w/lB. This behavior is also apparent in Fig. 3/H20849d/H20850. Notice, however, that for large negative polarizations /H20849large negative /H9004˜SAS/H20850, the two-subband approximation is no longer valid and the occupation of evenhigher electronic subbands must be taken into account, asalready mentioned in Sec. IV B . Note, furthermore, that we have deﬁned our /H208493,3,1 /H20850state to be an eigenstate of the S xoperator in the terminology of the true bilayer and not the usual Szoperator /H20849naively deﬁn- ing the Halperin state to be the eigenstate of Szdoes not give any appreciable overlap with the exact ground state /H20850. Thereis a simple reason why this needs to be done: because the states of the quantum well possess nodal structure /H20851Eq. /H208497/H20850/H20852, the true bilayer states /H20849like the Halperin states /H20850need to be rotated ﬁrst from the ztoxdirection, in order to match this symmetric/antisymmetric property, before direct comparisoncan be made. With this convention, the /H208493,3,1 /H20850state has its largest over- lap/H20849/H113510.95 /H20850with the exact ground state in the vicinity of the crossover line /H20855S z/H20856=0. However, the overlap remains quite large even in regions beyond this line, where the polarizationbecomes nonzero /H20851Fig.3/H20849a/H20850/H20852, in agreement with Ref. 12. This behavior may have two different origins. First, one noticesthat S zis not a good quantum number if the SU /H208492/H20850 symmetry-breaking terms of Hamiltonian /H2084913/H20850in the xdirec- tion are taken into account. Especially in the vicinity of the crossover line /H9004˜SAS/H112290, the symmetry breaking is governed by these terms in the xdirection and S¯x/H20849q=0/H20850, which does not commute with Sz, is expected to be a good quantum number. An alternative origin of the large overlap with the/H208493,3,1 /H20850state even in regions with /H20855S z/H20856/HS110050 may be a possible admixture /H20849/H110115%/H20850of states to the ground state that are or- thogonal the /H208493,3,1 /H20850and possess a ﬁnite polarization in the z direction. The largest values of the overlap between the compress- ible HR state and the exact ground state are also found in the vicinity of the crossover line from positive to negative /H9004˜SAS, though at extremely large values of w/lB. Notice that the overlap /H208490.64 for w/lB=10.0 /H20850is generally much lower than for the /H208493,3,1 /H20850state. At large values of the bare subband gap /H9004SAS/H20849and narrow quantum wells /H20850, the system becomes po- larized in the ↑subband and the ground state crosses over smoothly from the /H208493,3,1 /H20850state to the spin-polarized Pfafﬁan /H20849overlap of /H113510.92 /H20850. However, the increase in /H9004SAS, some- what counterintuitively, does not immediately destroy theHalperin state but at ﬁrst even increases the overlap. Finally, Fig. 4shows the results of our Monte Carlo study of the energies of the /H208493,3,1 /H20850and Pfafﬁan states. The corre- lation energies of both states were obtained from the ﬁnitesize scaling of systems with N=6–18 electrons as described above in Sec. IV C . All data were obtained in Monte Carlo simulations with 10 7samples. The uncertainty in the energy of the two-component states was obtained as the differencebetween linear and quadratic extrapolation of the backgroundenergies /H20849Fig. 4/H20850, as this was larger than the bare numerical errors of the simulation. The energetic competition of thesetwo phases qualitatively recovers the picture gained fromstudying the overlaps with the exact ground state. Again,some ﬁnite amount of tunneling is required for the single-component-paired state to outcompete the Halperin state. As shown in Fig. 4, the critical tunneling value /H9004 SAScabove which the Pfafﬁan state is energetically favored has a similarupturning shape as the boundary of large overlaps for thePfafﬁan state in Fig. 3. However, there are some quantitative differences at small w, where the thermodynamic values in- dicate that polarization occurs at smaller values of the tun-neling. B./H9263=1 Õ4 in a quantum well We proceed with analyzing the quantum well at /H9263=1 /4 /H20849Figs. 5–8/H20850. Because of the rapid increase in size of the Hil-00.20.40.60.81(a)(3,3,1 ) 0 2 4 6 810 w/lB00.020.040.060.08∆SAS 00.20.40.60.81(b)Pfaffian 0 2 4 6 810 w/lB 00.20.40.60.81(c) HR 0 2 4 6 810 w/lB00.020.040.060.08∆SAS -4-3-2-101234(d) <Sz> 0 2 4 6 810 w/lB FIG. 3. /H20849Color online /H20850Overlap between the exact Coulomb state of the quantum well for N=8 particles at /H9263=1 /2 with /H20849a/H20850the Hal- perin /H208493,3,1 /H20850state, /H20849b/H20850the Pfafﬁan, and /H20849c/H20850the HR states. The ex- pectation value of the Szcomponent of the pseudospin is plotted in/H20849d/H20850.FRACTIONAL QUANTUM HALL STATE AT /H9263=1 4IN … PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-7bert space, there are only two system sizes accessible in ED at this ﬁlling factor: N=6 and 8. The dimension of the Lz =0 sector of the Hilbert space of the latter, taking into ac- count discrete Lz→−Lzsymmetry, is on the order of 13 mil- lion, thus making N=6 the only case amenable to study in great detail. However, for N=6 we also must keep in mind the aliasing problem that occurs for /H208495,5,3 /H20850and the Pfafﬁan /H20849there is no such problem for the HR state /H20850. We will present results for both particle numbers because of the importantdifferences between them. In view of the comments in Sec. III, we note that the overlap with the /H208497,7,1 /H20850state is negli- gible in the range of widths w/lB/H1135110.0 and therefore we will exclude it from the present discussion of ED results.Note that, similarly to the /H208493,3,1 /H20850state in Sec. VA, the /H208495,5,3 /H20850state hereinafter is deﬁned as an eigenstate of the S x operator /H20849if deﬁned as an eigenstate of Sz, the overlap with the exact ground state is negligible /H20850. In Fig. 5we plot the overlap between the ground state of the quantum well for N=6 particles at /H9263=1 /4 and the Halp- erin /H208495,5,3 /H20850state /H20849a/H20850, the Pfafﬁan /H20849b/H20850and the HR state /H20849c/H20850, accompanied by the expectation value of the Szcomponent of the pseudospin. These results are reminiscent of /H9263=1 /2 /H20849Fig. 3/H20850; however, due to the smaller energy scale and the01 2345 w[ lB]-0.5-0.45-0.4-0.35-0.3-0.25Ecorr[e2/εlB]Pfaffian 331-state 0 1 2345 w[ lB]0 0.05 0.1∆SASc[e2/εlB]critical tunnelling 0 0.05 0.1 0.15 0.2 N-1-0.06 -0.04 -0.02∆EBG[e2/εlB]d=1 d=2 d=3 d=4 d=5 FIG. 4. /H20849Color online /H20850Energies in the thermodynamic limit for the/H208493,3,1 /H20850and Pfafﬁan states at /H9263=1 /2/H20849left/H20850. Data shown are for the inﬁnite square well as a function of the well width w. The correlation energies are shown with respect to the single-componentbackground. In the absence of tunneling, the /H208493,3,1 /H20850state has lower energy at all w. The critical tunneling strength required to favor the Pfafﬁan state /H20849top right /H20850and a few typical differences in the ex- trapolation of the background energies for different values of thewell width /H20849bottom right /H20850. 00.20.40.60.81(a)(5,5,3 ) 0 2 4 6 810 w/lB00.020.040.060.08∆SAS 00.20.40.60.81(b)Pfaffian 0 2 4 6 810 w/lB 00.20.40.60.81(c) HR 0 2 4 6 810 w/lB00.020.040.060.08∆SAS -3-2-10123(d) <Sz> 0 2 4 6 810 w/lB FIG. 5. /H20849Color online /H20850Overlap between the exact Coulomb state of the quantum well for N=6 particles at /H9263=1 /4 with /H20849a/H20850the Hal- perin /H208495,5,3 /H20850state, /H20849b/H20850the Pfafﬁan, and /H20849c/H20850the HR states. The ex- pectation value of the Szcomponent of the pseudospin is plotted in/H20849d/H20850.00.20.40.60.81(a)(5,5,3 ) 1357911 w/lB0.020.050.080.11∆SAS 00.20.40.60.81(b)Pfaffian 1357911 w/lB 00.20.40.60.81(c) HR 1357911 w/lB0.020.050.080.11∆SAS -4-3-2-101234(d) <Sz> 1357911 w/lB FIG. 6. /H20849Color online /H20850Overlap between the exact Coulomb state of the quantum well for N=8 particles at /H9263=1 /4 with /H20849a/H20850the Hal- perin /H208495,5,3 /H20850state, /H20849b/H20850the Pfafﬁan, and /H20849c/H20850the HR states. The ex- pectation value of the Szcomponent of the pseudospin is plotted in/H20849d/H20850. 00.20.40.60.81 0.01 0.04 0.07 0.1 0.13-4-3-2-101234Overlap ∆SASL>0(5,5,3) Pfaffian HR <Sz> FIG. 7. /H20849Color online /H20850Overlap between the exact Coulomb state of the quantum well for N=8 particles at /H9263=1 /4 and w/lB=10.5 with the Halperin /H208495,5,3 /H20850state, the Pfafﬁan, and the HR states /H20849left axis/H20850. The expectation value of the Szcomponent of the pseudospin is given on the right axis. The shaded region denotes where theground state is no longer rotationally invariant /H20849L/H110220/H20850.PAPI Ćet al. PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-8gap, it is much easier to polarize the system at /H9263=1 /4. For intermediate values of the width and small tunneling, themaximum overlap with the Halperin /H208495,5,3 /H20850state is high /H208490.96 /H20850but the region that would correspond to this phase is quite narrow in comparison to that of /H208493,3,1 /H20850. On the other hand, the Pfafﬁan phase is much more extended. Given theintrinsic tunneling 12of the samples, which is on the order of /H9004SAS //H20849e2//H9280lB/H20850/H113510.1, it seems more likely that the system will be found in this phase than the /H208495,5,3 /H20850. The small island where the overlap abruptly goes to zero for large w/lBis due to the ground state belonging to a sector with L/H110220—this can be due to the admixture of compressible physics at large widths. The HR state appears to be present inthe transition region between one-component and two-component phases, its overlap steadily increasing with wand peaking at 0.7 for w/l B=6.0. Because of the fact that the HR state occurs at a different shift on the sphere, we stress thatthe overlap presented here does not constitute a proof that itis an intermediary phase /H20849moreover, the overlap drops rap- idly when larger systems are considered, see Fig. 7/H20850. In Fig. 6we plot the same quantities for the system of N=8 particles which is expected to display weaker ﬁnite-size effects and does not suffer from the aliasing problem. The/H208495,5,3 /H20850state is found in a sizable parameter range but the maximum overlap is moderate compared to the case previ-ously studied /H208490.74 for w/l B=4.5 /H20850. While the HR state gen- erally has a small overlap /H20849not exceeding 0.2 /H20850and the evo- lution of /H20855Sz/H20856remains smooth, the striking difference in comparison with the N=6 results /H20849Fig. 5/H20850is the Pfafﬁan phase. Although it similarly develops with the increase in/H9004 SAS, once the system reaches full polarization, the phase is destroyed. To shed more light on how this occurs, it is useful to look at the “cross section” of Fig. 6for a ﬁxed value of the widthw/lB=10.5, chosen to represent the region where the Pfafﬁan phase is most clearly pronounced /H20849Fig. 7/H20850. Although the Pfafﬁan overlap peaks in the region where /H208495,5,3 /H20850starts to drop, very abruptly both overlaps fall to zero, and the groundstate is no longer rotationally invariant. The fact that L/H110220i s a hallmark of compressibility. Precisely at the transitionpoint, a small kink is now visible in /H20855S z/H20856. The origin of this kink or the reason why the ground state is obtained in L /H110220 sector is not entirely clear at present. However, the zero overlap with the Pfafﬁan beyond /H9004SAS //H20849e2//H9280lB/H20850=0.1 /H20849where the ground state reduces to a spinless case /H20850agrees with our results of Sec. II. Notice that a compressible ground state with L/H110220 may also indicate a phase with modulated charge density, such as the Wigner crystal. Indeed, an insulatingbehavior, as one would expect for an electron crystal, hasbeen found at ﬁlling factors slightly above /H9263=1 /5.41Such a state is not captured in the present ED calculations on thesphere and the question whether a Wigner crystal is the true ground state at large values of /H9004 SASin a wide quantum well is beyond the scope of the present paper. We refer to Monte Carlo simulations /H20849Fig. 8/H20850to obtain additional information about the candidate incompressiblestates from larger model systems. We include systems withN=6–16 electrons in the ﬁnite size scaling for the ground- state energies, again using 10 7Monte Carlo samples, and taking errors as the difference between linear and quadraticextrapolation of the background energies. The results of thisstudy are summarized in Fig. 8, where we compare the Pfaff- ian to the /H208495,5,3 /H20850and /H208497,7,1 /H20850Halperin wave functions. Again, a two-component state is always preferred in the absence oftunneling. At the layer separations shown, this is /H208495,5,3 /H20850as shown in Fig. 8/H20849a/H20850. These data also conﬁrm that the /H208497,7,1 /H20850 state becomes relevant only at large well width w/H1102210l B.I n Fig. 8/H20849b/H20850, we display the value of tunneling /H9004SAScrequired to polarize the system into the paired Pfafﬁan phase. This fea-ture of the energetic competition of /H208495,5,3 /H20850and the Pfafﬁan is very close to the results obtained in ED for N=6 both quali- tatively and quantitatively: the shape of /H9004 SASc/H20849w/H20850is nearly linear and reproduces the location where the overlaps withthe exact ground state cross over between the two trial states,as was shown in Fig. 5. The splitting /H9004 SASrequired for the Pfafﬁan to be the ground state is signiﬁcant and probablylarger than the splitting in the experiments of Luhman et al., 16which can be estimated to about /H9004SAS/H110150.069 e2//H9280lBat the sample width w/H1101510lBand their baseline electron den- sity. This similarity between the energetics in the thermody- namic limit and the exact spectrum for N=6 particles may be circumstantial. However, there is another indication that thevery different behavior at N=8 might be exceptional. In Fig. 8/H20849c/H20850, we show the correlation energies of the Pfafﬁan state for different system sizes Nand well widths w. This repre- sentation reveals the case of N=8 as having particularly high energy. This may be a ﬁnite-size effect that can be explainedin the composite fermion picture. The Pfafﬁan wave function can be expressed as a paired state of 4CF feeling one quan- tum of negative effective ﬂux.22,42The shell structure of these composite fermions on the sphere yields ﬁlled shellstates for N=6 and 12, whereas for N=8 two CFs remain in the highest, partially ﬁlled shell. In this conﬁguration, CFs02468 1 0 w[ lB]-0.35-0.3-0.25-0.2Ecorr[e2/εlB]Pfaffian (5,5,3)-state (7,7,1)-state 02468 1 0 w[ lB]0 0.04 0.08 0.1 2Energies [ e2/εlB] E771-E553 ∆SASc[Pf] 6 8 10 12 14 N-0.28 -0.27 -0.26 -0.25Ecorr[e2/εl0’]w=3.5 w=4 w=4.5 (a)(b) (c) FIG. 8. /H20849Color online /H20850Results from our Monte Carlo study of states at /H9263=1 /4:/H20849a/H20850correlation energies of the Pfafﬁan, /H208495,5,3 /H20850, and /H208497,7,1 /H20850states with respect to the single-component background, as a function of the well width win the thermodynamic limit; /H20849b/H20850differ- ence in energy between the different Halperin states and, in particu-lar, the tunneling strength /H9004 SAScfor the Pfafﬁan state to be favored over the /H208495,5,3 /H20850state, and /H20849c/H20850correlation energies for the Pfafﬁan state in units of rescaled magnetic length /H20849Ref. 40/H20850 lB/H11032=/H20851N//H20849/H9263N/H9278/H20850/H208521/2lBfor some values of w: note the particularly high value at N=8.FRACTIONAL QUANTUM HALL STATE AT /H9263=1 4IN … PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-9are susceptible to follow Hund’s rule by maximizing their angular momentum and breaking rotational invariance. ForN=8 and 10, Hund’s rule predicts an angular momen- tum of L=4, which is indeed found in ED. This gives us conﬁdence that the system is still described by liquidlikecomposite fermion physics at large /H9004 SAS. We therefore con- sider the competition between a Hund’s rule state and thepaired Pfafﬁan state. For a similar situation with weak pair-ing in a /H9263=1 /2+1 /2 bilayer system at large layer separation, it was argued22that for larger systems, the shell-ﬁlling ef- fects and Hund’s rule should become less important whereasthe pairing effects will remain the same strength, as only /H11011/H20881Nbeneﬁt from Hund’s rule, whereas all /H20849/H11011N/H20850particles within some gap energy of the Fermi surface contribute topairing. Although the above argument speaks in favor of the pos- sibility for a paired Pfafﬁan state to be realized at /H9263=1 /4 for large tunneling gap /H9004SAS, we insist on the variational charac- ter of the Monte Carlo calculations. In these calculations, wehave indeed considered several competing candidate wavefunctions for a liquid ground state at this ﬁlling factor. How-ever, this analysis may not eliminate the possibility that acompressible state, such as that seen in ED, or even otherincompressible phases may indeed be singled out as a trueground state of the system. Finally, we would like to point out that in ED it is pos- sible to calculate the quantity that we refer to as the “chargegap,” /H9004E=E N,N/H9278+1+EN,N/H9278−1−2EN,N/H9278, /H2084918/H20850 where EN,N/H9278is the ground-state energy for a given number of particles Nand number of ﬂux quanta N/H9278. This quantity probes the response of the system to the introduction ofquasiparticles/quasiholes on top of the ground state and itsdependence on /H9004 SAShas been used to delineate between the one-component and two-component phases.14With the ap- propriate ﬁnite-size corrections, Eq. /H2084918/H20850should correspond to the experimentally measurable “activation” gap12that governs the temperature scaling of longitudinal resistanceR xx/H11011exp/H20849−/H9004E/2T/H20850. For states that undergo a typical one- component to two-component transition, such as the one at /H9263=2 /3/H20849for small tunneling, it is the state of two-decoupled Laughlin liquids, /H9263=1 /3+1 /3, which develops into a single- component 2/3 state for large tunneling amplitudes43/H20850, the charge gap /H20851Eq. /H2084918/H20850/H20852displays a minimum as a function of /H9004SASin the center of the transition region.12On the other hand, for /H9263=1 /2 where the tunneling-driven transition con- nects the /H208493,3,1 /H20850state and the Pfafﬁan, the charge gap /H20851Eq. /H2084918/H20850/H20852shows an upward cusp. Our calculations of the charge gap /H20851Eq. /H2084918/H20850/H20852in the case of /H9263=1 /4 indicate that this quan- tity is a less robust way to characterize the nature of theground state than the calculation of the overlaps with trialwave functions. While for N=6 particles at /H9263=1 /4 the charge gap displays a minimum as a function of /H9004SAS, there is a very weak dependence of /H9004Eon/H9004SASwhen a larger system of N=8 particles is considered. Thus ﬁnite-size ef- fects are too strong in order to extract useful informationfrom Eq. /H2084918/H20850in small systems that can be treated by ED.VI. CONCLUSION In this paper we have presented a systematic study of several candidates for the ground-state wave function at the recently observed16fraction /H9263=1 /4. Assuming that the /H20849pseudo /H20850spin plays no role, i.e., in a one-component picture, the generalized Moore-Read Pfafﬁan state /H208491/H20850shows high overlap for the values of the sample width which are on theorder of those in the experiment of Ref. 16but only if the conﬁnement in the perpendicular direction is modeled byZDS model /H208492/H20850. For other conﬁnement models /H20851Eqs. /H208493/H20850and /H208494/H20850/H20852it was not possible to reproduce such high values of the overlap. We believe that this inconsistency means that thehigh overlap must be due to a special softening of thepseudopotentials that occurs as a pathology of ZDS modelbut does not appear in other /H20849more realistic /H20850conﬁnement models. Therefore, the existence of a fractional quantum Hall state at /H9263=1 /4 is necessarily linked to the speciﬁc features of the quantum well used in Ref. 16that enable the multicompo- nent physics to manifest itself. Additional degrees of free-dom in our theoretical study are conveniently taken into ac-count within the quantum-well model, which is the simplestmodel that can naturally interpolate between a single layerand bilayer charge distribution as the parameters wand/H9004 SAS are varied. This two-parameter model is related to the previ- ous studies14of the true bilayer with tunneling at /H9263=1 /2 /H20849which had to assume at least three independent parameters /H20850 by reproducing the same physical picture of the crossoverbetween the /H208493,3,1 /H20850state and the Pfafﬁan. At the ﬁlling factor /H9263=1 /4, we have not been able to produce clear cut evidence for the expected crossover be-tween the /H208495,5,3 /H20850state and the Pfafﬁan in ED due to the strong ﬁnite-size effects in case of the latter. We have shownthat the /H208495,5,3 /H20850state is indeed present for a range of widths and small tunneling gaps /H9004 SASbut its maximum overlap is not as high as that of the /H208493,3,1 /H20850state. ED cannot delimit the range of parameters for the Pfafﬁan phase due to the differ-ence in the results for the two available system sizes, N=6 and 8, and the effect of compressible physics which is difﬁ-cult to treat within the spherical geometry. However, ourMonte Carlo simulations go some way toward clarifying thesituation. The correlation energies of the Pfafﬁan state revealN=8 as a particularly unfavorable system size. We can ex- plain this from the ﬁnite-size effect in terms of ﬁlling shellsof CF orbitals on the sphere. The competing L/HS110050 states at N=8, as well as N=10, seem to be related to Hund’s rule for CFs. However, the competition between Hund’s rule andpairing is likely favorable for the paired state in the thermo-dynamic limit. In addition, projecting from the two-component model onto the fully polarized /H20849spinless /H20850case, on the other hand, can be seen as analogous to the scenario ofLL mixing, 44which may provide another mechanism to sta- bilize the Pfafﬁan state via generating three-body terms inthe effective interaction. Such effects are beyond the scopeof the present paper. By analyzing the competition betweenthe paired single component and the Halperin states fromtheir variational wave functions, we ﬁnd, in the Monte Carlo simulations, that the tunneling gap /H9004 SAScrequired to form a single-component state roughly behaves linearly as 1.0PAPI Ćet al. PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-10/H11003/H20849w/lB/H20850/H1100310−2e2//H9280lB. Although the tunneling splitting indi- cated for the experiment described in Ref. 16is not far below the transition between the Pfafﬁan and /H208495,5,3 /H20850, our numerics still show it safely in the two-component regime of the/H208495,5,3 /H20850wave function. Although we believe that our quantum-well model takes properly into account the effects of ﬁnite thickness, we haveentirely neglected the effect of the in-plane magnetic ﬁeldwhich may nevertheless prove essential in order to stabilizethe incompressible state at /H9263=1 /4. The existing experimental work45on the /H9263=2 /3 state witnessed that the introduction of an in-plane magnetic ﬁeld may lead to a strengthening of theminimum in R xx, thus inducing the same one-component to two-component transition as by varying /H9004SAS. Similar strengthening occurs for /H9263=1 /2 if the tilt is not too large.45 Therefore, the application of the in-plane ﬁeld may be a likely reason to further stabilize the /H208495,5,3 /H20850state at /H9263=1 /4i f the symmetric-antisymmetric gap /H9004SASis sufﬁciently small. However, Ref. 16also pointed out the difference between /H9263=1 /2 state and /H9263=1 /4 state: when the electron density is increased, the former displays a deeper minimum in Rxx while the latter remains largely unaffected. This difference suggests that in the case of /H9263=1 /4 the quantum-well ground state may be effectively fully polarized and in the class of thePfafﬁan rather than the two-component /H208495,5,3 /H20850state. In order to answer without ambiguity which of the two possibilities is actually realized in the quantum well underthe experimental conditions of Ref. 16, it would be useful to know the dependence of the activation gap as a function of/H9004 SASand also as a function of transferred charge from the front to the back of the quantum well using a gate biasing.These results would help to discriminate between the one-component and two-component nature of the ground state. ACKNOWLEDGMENTS This work was funded by the Agence Nationale de la Recherche under Grant No. ANR-JCJC-0003-01. Z.P. wassupported by the European Commission through Marie CurieFoundation Contract No. MEST CT 2004-51-4307 and Cen-ter of Excellence under Grant No. CX-CMCS. M.V .M. wassupported by the Serbian Ministry of Science under GrantNo. 141035. G.M. would like to thank Steven Simon forstimulating discussions. APPENDIX: EFFECTIVE BILAYER DESCRIPTION OF THE WIDE QUANTUM WELL As in Sec. IV, we consider the quantum well to be sym- metric around w/2, i.e., the lowest subband /H20849↑/H20850state is sym- metric and the ﬁrst excited one /H20849↓/H20850is antisymmetric. Fur- thermore, we consider, in this section, the electrons to be inthe 2D plane, for illustration reasons, although the conclu-sions remain valid also in the spherical geometry. In thisAppendix, we yield the derivation of the effective bilayerdescription of the wide quantum well. The interaction part of Hamiltonian /H208499/H20850consists of a density-density interaction and terms beyond, which may bedescribed as a spin-spin interaction. Indeed, the density-density part consists of the effective interactions /H20851Eq. /H2084910/H20850/H20852 V 2D↑↑↑↑,V2D↓↓↓↓, and V2D↑↓↑↓=V2D↓↑↓↑. Notice that the interactions in the ﬁrst excited subband /H20849↓/H20850are generally weaker than in the lowest one /H20849↑/H20850because the wave function /H20851Eq. /H208498/H20850/H20852/H9278↓/H20849z/H20850 possesses a node at w/2, in the center of the well, i.e., V2D↑↑↑↑/H11022V2D↓↓↓↓. With the help of the /H20849spin /H20850density operators /H20851Eqs. /H2084911/H20850and /H2084912/H20850/H20852, the density-density part of the interac- tion Hamiltonian reads as H=1 2/H20858 qVSU/H208492/H20850/H20849q/H20850/H9267¯/H20849−q/H20850/H9267¯/H20849q/H20850+2/H20858 qVsbz/H20849q/H20850S¯z/H20849−q/H20850S¯z/H20849q/H20850 +/H20858 qVBz/H20849q/H20850/H9267¯/H20849−q/H20850S¯z/H20849q/H20850/H20849 A1/H20850 in terms of the SU /H208492/H20850-symmetric interaction VSU/H208492/H20850/H20849q/H20850=1 4/H20851V2D↑↑↑↑/H20849q/H20850+V2D↓↓↓↓/H20849q/H20850+2V2D↑↓↑↓/H20849q/H20850/H20852 /H20849 A2/H20850 and the SU /H208492/H20850-symmetry breaking interaction terms Vsbz/H20849q/H20850=1 4/H20851V2D↑↑↑↑/H20849q/H20850+V2D↓↓↓↓/H20849q/H20850−2V2D↑↓↑↓/H20849q/H20850/H20852 /H20849 A3/H20850 and VBz/H20849q/H20850=1 2/H20851V2D↑↑↑↑/H20849q/H20850−V2D↓↓↓↓/H20849q/H20850/H20852. /H20849A4/H20850 The remaining 12 interaction terms, which may not be treated as density-density interactions, fall into two differentclasses; the eight terms with three equal spin orientations /H9268 and one opposite − /H9268are zero due to the antisymmetry of the integrand in Eq. /H2084910/H20850. The remaining four interaction terms with two ↑spins and two ↓spins are all equal due to the symmetry of the quantum well around w/2, Vsbx/H11013V2D↑↑↓↓=V2D↓↓↑↑=V2D↑↓↓↑=V2D↓↑↑↓. /H20849A5/H20850 They yield the term Hsbz=2/H20858 qVsbx/H20849q/H20850S¯x/H20849−q/H20850S¯x/H20849q/H20850, /H20849A6/H20850 which needs to be added to the interaction Hamiltonian /H20849A1/H20850, as well as the term HSAS=−/H9004SASS¯z/H20849q=0/H20850, /H20849A7/H20850 which accounts for the electronic subband gap between the ↑ and the ↓levels. Collecting all terms, Hamiltonian /H208499/H20850thus becomes H=1 2/H20858 qVSU/H208492/H20850/H20849q/H20850/H9267¯/H20849−q/H20850/H9267¯/H20849q/H20850+2/H20858 qVsbx/H20849q/H20850S¯x/H20849−q/H20850S¯x/H20849q/H20850 +2/H20858 qVsbz/H20849q/H20850S¯z/H20849−q/H20850S¯z/H20849q/H20850+/H20858 qVBz/H20849q/H20850/H9267¯/H20849−q/H20850S¯z/H20849q/H20850 −/H9004SASS¯z/H20849q=0/H20850. /H20849A8/H20850 Several comments are to be made with respect to this result. First, we have checked that for the inﬁnite-square-wellmodel as well as for a model with a parabolic conﬁnementFRACTIONAL QUANTUM HALL STATE AT /H9263=1 4IN … PHYSICAL REVIEW B 79, 245325 /H208492009 /H20850 245325-11potential there is a natural hierarchy of the energy scales in Hamiltonian /H20849A8/H20850, VSU/H208492/H20850/H11022Vsbx/H11407VBz/H11407Vsbz. /H20849A9/H20850 This hierarchy is valid both for the interaction potentials in Fourier space as for the pseudopotentials. Whereas the ﬁrst term of the Hamiltonian describes the SU/H208492/H20850-symmetric interaction, the second and the third one break this SU /H208492/H20850symmetry. Because Vsbx/H20849q/H20850/H11022Vsbz/H20849q/H20850/H110220 for all values of q, states with no polarization in the xand z directions are favored, with /H20855Sx/H20856=0 and /H20855Sz/H20856=0, respectively. Due to the hierarchy /H20851Eq. /H20849A9/H20850/H20852of energy scales, a depolar- ization in the xdirection is more relevant than that in the z direction. These terms are similar to those one encounters inthe case of a bilayer quantum Hall system, where due to theﬁnite layer separation a polarization of the layer isospin inthezdirection costs capacitive energy. 36The fourth term of Hamiltonian /H20849A8/H20850is due to the stron- ger electron-electron repulsion in the lowest electronic sub-band as compared to the ﬁrst excited one, where the wavefunction possesses a node at z=w/2. In order to visualize its effect, one may treat the density, which we consider to behomogeneous in an incompressible state, on the mean-ﬁeld level, /H20855 /H9267¯/H20849q/H20850/H20856=/H9263/H9254q,0, in which case the fourth term of Eq. /H20849A8/H20850becomes /H9263VBz/H20849q=0/H20850S¯z/H20849q=0/H20850and, thus, has the same form as the subband-gap term /H20851Eq. /H20849A7/H20850/H20852. It therefore renor- malizes the energy gap between the lowest and the ﬁrst ex-cited electronic subbands, and it is natural to deﬁne the ef-fective subband gap as /H9004 SAS→/H9004˜SAS=/H9004SAS−/H9263VBz/H20849q=0/H20850=/H9004SAS−/H9253/H9263e2 /H9280lBw lB, /H20849A10 /H20850 where /H9253is a numerical prefactor that depends on the precise nature of the well model. 1W. Pan, J. S. Xia, H. L. Stormer, D. C. Tsui, C. Vicente, E. D. Adams, N. S. Sullivan, L. N. Pfeiffer, K. W. Baldwin, and K. W.West, Phys. Rev. B 77, 075307 /H208492008 /H20850. 2R. B. Laughlin, Phys. Rev. Lett. 50, 1395 /H208491983 /H20850. 3J. Jain, Composite Fermions /H20849Cambridge University Press, Cam- bridge, England, 2007 /H20850. 4F. D. M. Haldane, Phys. Rev. Lett. 51, 605 /H208491983 /H20850. 5R. Willett, J. P. Eisenstein, H. L. Stormer, D. C. Tsui, A. C. Gossard, and J. H. English, Phys. Rev. Lett. 59, 1776 /H208491987 /H20850. 6F. D. M. Haldane and E. H. Rezayi, Phys. Rev. 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PhysRevB.77.155119.pdf	Modeling elastic and photoassisted transport in organic molecular wires: Length dependence and current-voltage characteristics J. K. Viljas,1,2,F. Pauly,1,2and J. C. Cuevas3,1,2 1Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures, Universität Karlsruhe, D-76128 Karlsruhe, Germany 2Forschungszentrum Karlsruhe, Institut für Nanotechnologie, D-76021 Karlsruhe, Germany 3Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain /H20849Received 8 January 2008; revised manuscript received 2 March 2008; published 17 April 2008 /H20850 Using a /H9266-orbital tight-binding model, we study the elastic and photoassisted transport properties of metal- molecule-metal junctions based on oligophenylenes of varying lengths. The effect of monochromatic light ismodeled with an ac voltage over the contact. We ﬁrst show how the low-bias transmission function can beobtained analytically, using methods previously employed for simpler chain models. In particular, the decaycoefﬁcient of the off-resonant transmission is extracted by considering both a ﬁnite-length chain and inﬁnitelyextended polyphenylene. Based on these analytical results, we discuss the length dependence of the linear-response conductance, the thermopower, and the light-induced enhancement of the conductance in the limit ofweak intensity and low frequency. In general, the conductance enhancement is calculated numerically as afunction of the light frequency. Finally, we compute the current-voltage characteristics at ﬁnite dc voltages andshow that in the low-voltage regime, the effect of low-frequency light is to induce current steps with a voltageseparation determined by twice the frequency. These effects are more pronounced for longer molecules. Westudy two different proﬁles for the dc and ac voltages, and it is found that the results are robust with respect tosuch variations. Although we concentrate here on the speciﬁc model of oligophenylenes, the results should bequalitatively similar for many other organic molecules with a large enough electronic gap. DOI: 10.1103/PhysRevB.77.155119 PACS number /H20849s/H20850: 73.50.Pz, 85.65. /H11001h, 73.63.Rt I. INTRODUCTION The use of single-molecule electrical contacts for opto- electronic purposes such as light sources, light sensors, andphotovoltaic devices is an exciting idea. Yet, due to the dif-ﬁculties that light-matter interactions in nanoscale systemspose for theoretical and experimental investigations, the pos-sibilities remain largely unexplored. Concerning experi-ments, it has been shown that light can be used to change theconformation of some molecules even when they are con-tacted to metallic electrodes, thus enabling light-controlledswitching. 1Some evidence of photoassisted processes inﬂu- encing the conductance of laser-irradiated metallic atomiccontacts has also been obtained. 2Theoretical investigations of light-related effects in molecular contacts are morenumerous, 3–19but they are mostly based on highly simpliﬁed models, whose validity remains to be checked by more de-tailed calculations 20,21and experiments. However, for the de- scription of the basic phenomenology, model approaches canbe very fruitful, as they have been in studies of elastic trans-port in the past. Properties of linear single-orbital tight-binding /H20849TB/H20850chains, in particular, have been studied in de- tail, and to a large part analytically. 3,22–32In a step toward a more realistic description of the geometry, symmetries, andthe electronic structure of particular molecules, empirical TBapproaches such as the /H20849extended /H20850Hückel method have proven useful. 4,8,33–35 Based on a combination of density-functional calculations and simple phenomenological considerations, we have re-cently described the photoconductance of metal-oligo-phenylene-metal junctions. 5It was discussed how the linear- response conductance may increase by orders of magnitudein the presence of light. This effect can be seen as the result of a change in the character of the transport from off-resonant to resonant, due to the presence of photoassistedprocesses. 5,7,8Consequently, the decay of the conductance with molecular length is slowed down, possibly even making the conductance length independent.5,8 In this paper, we apply a Hückel-type TB model of oligophenylene-based contacts36combined with Green- function methods4to study the effects of monochromatic light on the dc current in metal-oligophenylene-metal con-tacts. Again, we concentrate on the dependence of these ef-fects on the length of the molecule. We begin with a detailedaccount of the elastic transport properties of the model andshow that the zero-bias transmission function can be ob-tained analytically, similarly to simpler chain models. 23,27We demonstrate how information about the length dependence ofthe transmission function for a ﬁnite wire can be extractedfrom an inﬁnitely extended polymer. Based on these analyti-cal results, we discuss the length dependences of the conduc-tance and the photoconductance for low-intensity and low-frequency light. While the conductance decays exponentiallywith length, its relative enhancement due to light exhibits aquadratic behavior. Here, we also brieﬂy consider the ther-mopower, whose length dependence is linear. Next, we cal-culate numerically the zero-bias photoconductance as a func-tion of the light frequency /H9275and ﬁnd that the conductance enhancement due to light is typically very large.3,5,8In par- ticular, we show that the results of Ref. 5are expected to be robust with respect to variations in the assumed voltage pro-ﬁles. Finally, we describe how the steplike current-voltage/H20849I-V/H20850characteristics are modiﬁed by light. At high /H9275, the most obvious effect is the overall increase in the low-biasPHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 1098-0121/2008/77 /H2084915/H20850/155119 /H2084914/H20850 ©2008 The American Physical Society 155119-1current. At low /H9275, additional current steps similar to those in microwave-irradiated superconducting tunnel junctions37,38 can be seen. Their separation, in our case of symmetric junc- tions, is roughly 2 /H6036/H9275/e. TB models of the type we shall consider neglect various interaction effects /H20849see Sec. V for a discussion /H20850and thus cannot be expected to give quantitative predictions. How-ever, the qualitative features of the results rely only on thetunneling-barrier character of the molecular contacts, whichresults from the fact that the Fermi energy of the metal lies inthe gap between the highest-occupied and lowest-unoccupiedmolecular orbitals /H20849HOMO and LUMO /H20850of the molecule. Thus, these features should remain similar for junctionsbased on many other organic molecules exhibiting largeHOMO-LUMO gaps. The light-induced effects, if veriﬁedexperimentally, could be used for detecting light, or as anoptical gate /H20849or “third terminal” /H20850for purposes of switching. The rest of the paper is organized as follows. In Sec. II, we describe our theoretical approach, discuss the generalproperties of TB wire models, and introduce the Green-function method for the calculation of the elastic transmis-sion function. Then, in Sec. III, we calculate the transmissionfunction of oligophenylene wires analytically. The decay co-efﬁcient for the off-resonant transmission is extracted alsofrom inﬁnitely extended polyphenylene. Following that, inSec. IV, we present our numerical results for the conduc-tance, the thermopower, the photoconductance, and the I-V characteristics. Finally, Sec. V ends with our conclusions andsome discussion. Details on the calculation of the time-averaged current in the presence of light are deferred to theAppendixes. In Appendix A, a simpliﬁed interpretation ofthe current formula is derived, and in Appendix B, a briefaccount of the general method is given. Readers mainly in-terested in the discussion of the results for the physical ob-servables can skip most of Secs. II and III and proceed toSec. IV. II. THEORETICAL FRAMEWORK A. Transport formalism Our treatment of the transport characteristics for the two- terminal molecular wires is based on Green’s functions andthe Landauer-Büttiker formalism, or its generalizations. As-suming the transport to be fully elastic, the dc electrical cur-rent through a molecular wire can be described with I/H20849V/H20850=2e h/H20885dE/H9270/H20849E,V/H20850/H20851fL/H20849E/H20850−fR/H20849E/H20850/H20852. /H208491/H20850 Here, Vis the dc voltage and /H9270/H20849E,V/H20850is the voltage- dependent transmission function, while fX/H20849E/H20850=1 //H20851exp /H20849/H20849E −/H9262X/H20850/kBTX/H20850+1/H20852,/H9262X, and TXare the Fermi function, the elec- trochemical potential, and the temperature of side X=L,R, respectively.39The electrochemical potentials satisfy eV =/H9004/H9262=/H9262L−/H9262R, and we can choose them symmetrically as /H9262L=EF+eV /2 and /H9262R=EF−eV /2, where EFis the Fermi en- ergy. For studies of dc current, we always assume TL=TR =0. Of particular experimental interest is the linear-response conductance Gdc=/H20841/H11509I//H11509V/H20841V=0, given by the Landauer formulaGdc=G0/H9270/H20849EF/H20850, where G0=2e2/hand/H9270/H20849E/H20850=/H9270/H20849E,V=0/H20850.I n most junctions based on organic oligomers, the transport canbe described as off-resonant tunneling. This results in thewell-known exponential decay of G dcwith the number Nof monomeric units in the molecule.40At ﬁnite voltages V, the current increases in a stepwise manner as molecular levelsbegin to enter the bias window between /H9262Land/H9262R/H20849Ref. 24/H20850. We shall consider both of these phenomena below. If a small temperature difference /H9004T=TL−TRat an aver- age temperature T=/H20849TL+TR/H20850/2 is applied, heat currents and thermoelectric effects can arise.36,41,42In an open-circuit situ- ation, where the net current Imust vanish, a thermoelectric voltage /H9004/H9262/eis generated to balance the thermal diffusion of charge carriers. In the linear-response regime, the proportion-ality constant S=−/H20849/H9004 /H9262/e/H9004T/H20850I=0is the Seebeck coefﬁcient. We will brieﬂy consider this quantity below as an example ofan observable with a linear dependence on the molecularlength Nbut will not enter a more detailed discussion of thermoelectricity or heat transport. The quantity we are most interested in is the dc current in the presence of monochromatic electromagnetic radiation,which we refer to as light independently of its source orfrequency /H9275. We model the light as an ac voltage with har- monic time dependence V/H20849t/H20850=Vaccos/H20849/H9275t/H20850over the contact. The current averaged over one period of V/H20849t/H20850can be written in the form3,4,43 I/H20849V;/H9251,/H9275/H20850=2e h/H20858 k=−/H11009/H11009/H20885dE/H20851/H9270RL/H20849k/H20850/H20849E,V;/H9251,/H9275/H20850fL/H20849E/H20850 −/H9270LR/H20849k/H20850/H20849E,V;/H9251,/H9275/H20850fR/H20849E/H20850/H20852. /H208492/H20850 Here, the transmission coefﬁcient /H9270RL/H20849k/H20850/H20849E/H20850, for example, de- scribes photoassisted processes taking an electron from left/H20849L/H20850to right /H20849R/H20850, under the absorption of a total of kphotons with energy /H6036 /H9275. The parameter /H9251=eVac//H6036/H9275describes the strength of the ac drive.44It is determined by the intensity of the incident light and possible ﬁeld-enhancement effects tak-ing place in the metallic nanocontact. 45Again, in addition to the full I-Vcharacteristics, we study in more detail the case of linear response with respect to the dc bias, i.e., the pho-toconductance G dc/H20849/H9251,/H9275/H20850=/H20841/H11509I/H20849V;/H9251,/H9275/H20850//H11509V/H20841V=0. The argu- ments /H9251and/H9275distinguish it from the conductance Gdc, al- though we sometimes omit /H9251for notational simplicity. The calculation of the coefﬁcients /H9270RL /LR/H20849k/H20850/H20849E/H20850is rather complicated in general,4and we defer comments on this procedure to Appendix B. Below, we shall mostly refer to an approximateformula /H20849see Appendix A /H20850that can be expressed in terms of /H9270/H20849E/H20850. This amounts to a treatment of the problem on the level of the Tien-Gordon approach.3,37,46The full Green-function formalism for systems involving ac driving is presented inRef. 4. In noninteracting /H20849non-self-consistent /H20850models, it is, in general, not clear how the voltage drop should be dividedbetween the different regions of the wire and the electrode-wire interfaces. A self-consistent treatment would be in or-der, in particular, for asymmetrically coupled molecules. Weonly concentrate on left-right symmetric junctions, whereboth the dc and ac voltages /H20849VandV ac/H20850are assumed to drop according to one of two different symmetrical proﬁles. TheVILJAS, PAULY, AND CUEVAS PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-2symmetry of the junctions excludes rectiﬁcation effects, such as light-induced dc photocurrents in the absence of a dc biasvoltage. 3,9,45However, light can still have a strong inﬂuence on the transmission properties of the molecular contact, aswill be discussed below. It will be shown that our conclu-sions are essentially independent of the assumed voltage pro-ﬁle. B. Wire models Below, we will specialize to the case of a metal- oligophenylene-metal junction. However, to make some gen-eral remarks, let us ﬁrst consider a larger class of molecularwires that can be described as Nseparate units forming a chain, where only the nearest neighbors are coupled /H20849see Fig. 1/H20850. We only discuss the calculation of the elastic transmission function /H9270/H20849E,V/H20850here, as this will be the focus of our analyti- cal considerations in Sec. III. From this quantity /H20849atV=0/H20850, the various linear-response coefﬁcients such as the conduc-tance and the thermopower can be extracted. Furthermore, asalready mentioned, it sufﬁces for an approximate treatment of the amplitudes /H9270RL/H20849k/H20850/H20849E/H20850as well. We assume a basis /H20841/H9273p/H20849/H9251/H20850/H20856of local /H20849atomic /H20850orbitals, where p=1,..., Nindexes the unit, while /H9251=1,..., Mpdenotes the orbitals in each unit.47For simplicity, the basis is taken to be orthonormal, i.e., /H20855/H9273p/H20849/H9251/H20850/H20841/H9273q/H20849/H9252/H20850/H20856=/H9254/H9251/H9252/H9254pq. The /H20849time-indepen- dent /H20850Hamiltonian Hpq/H20849/H9251,/H9252/H20850=/H20855/H9273p/H20849/H9251/H20850/H20841Hˆ/H20841/H9273q/H20849/H9252/H20850/H20856of the wire is then of the block-tridiagonal form H=/H20898H11H12 H21H22 H23 /GS/GS /GS HN−1,N−2HN−1,N−1HN−1,N HN,N−1HNN/H20899,/H208493/H20850 where Hpqwith p,q=1,..., NareMp/H11003Mqmatrices. /H20849The unindicated matrix elements are all zeros. /H20850 In the nonequilibrium Green-function picture, the effect of coupling the chain to the electrodes is described in terms of “lead self-energies.”48We assume these to be located only on the terminal blocks of the chain, with components /H901811 and/H9018NN. The inverse of the stationary-state retarded propa- gator for the coupled chain will then be of the form F=/H20898F11h12 h21h22 h23 /GS/GS /GS hN−1,N−2hN−1,N−1hN−1,N hN,N−1FNN/H20899. /H208494/H20850 Here, hp,p/H110061=−Hp,p/H110061,hpp=E+1pp−Hpp, and E+=E+i0+, whileF11=h11−/H901811andFNN=hNN−/H9018NN. Charge-transfer ef-fects between the molecule and the metallic electrodes shift the molecular levels with respect to the Fermi energy EF.I n a TB model, these can be represented by shifting the diago-nal elements of H. Once a transport voltage Vis applied, further shifts are induced. In our model, the voltage-inducedshifts will be taken from simple model proﬁles, and the rela-tive position of E Fwill be treated as a free parameter. Effective numerical ways of calculating the propagator G=F−1for block-tridiagonal Hamiltonians exist.49,50In Sec. III, we shall be interested in a special case, where Hp,p−1 =H−1,Hp,p+1=H1, andHpp=H0with the same H1=H−1Tand H0/H20849of dimension Mp=M/H20850for all p, describing an oligomer of identical monomeric units. In such cases also, analyticalprogress in calculating the current in Eq. /H208491/H20850may be pos- sible. Once the Green’s function Gis known, the transmis- sion function is given by 48 /H9270/H20849E,V/H20850=T r /H20851/H900311G1N/H9003NN/H20849G1N/H20850†/H20852, /H208495/H20850 where/H900311=−2 Im /H901811and/H901811/H20849E,V/H20850=/H901811/H20849E−eV /2/H20850, for ex- ample. Typically, EFlies within the HOMO-LUMO gap, result- ing in the exponential decay /H9270/H20849EF/H20850/H11011e−/H9252/H20849EF/H20850Nwith N, charac- teristic of off-resonant transport. The decay coefﬁcient /H9252/H20849EF/H20850 is actually independent of /H901811and/H9018NN. This can be seen by considering the Dyson equation G=G+G/H9018G, where Gand Gare the Green’s function of the coupled and uncoupled wires, respectively, and /H9018is the matrix for the lead self- energies. Assuming that G1Ndecays exponentially with N, then G1N/H11015/H208491−G11/H901811/H20850−1G1N /H208496/H20850 when N→/H11009, and therefore G1Ndecays with the same expo- nent. Thus, one can, in principle, obtain the decay exponentfrom the propagator of an isolated molecule, or even an in-ﬁnitely extended polymer. In the next section, we demon-strate this by extracting the decay exponent of a ﬁnite oli-gophenylene junction from the propagator forpolyphenylene. We note that in doing so, we neglect thepractical difﬁculty of determining the correct relative posi-tion of E F. There are efﬁcient numerical methods for computing the lead self-energies for different types of electrodes and vari-ous bonding situations between them and the wire. Typically,the methods are based on the calculation of surface Green’sfunctions. 51Below, we shall simply treat the self-energies as parameters. III. PHENYL-RING-BASED WIRES In this section, we discuss a special case of the type of wire model introduced above, describing an oligomer of phe-nyl rings coupled to each other via the para /H20849p/H20850position. 36 The bias voltage Vis assumed to be zero. In the special case that we will consider, the inversion of Eq. /H208494/H20850can then be done analytically with the subdeterminant method familiarfrom elementary linear algebra. 23,24,27,32Below, we ﬁrst use this method for calculating the propagator of the ﬁnite-wirejunction and derive the decay exponent /H9252/H20849E/H20850of the transmis- sion function at off-resonant energies. After that, we rederive1 23Σ11 33 Σ H H HH12 23 21 32HHH11 22 33 FIG. 1. /H20849Color online /H20850A ﬁnite block chain of length N=3 con- nected to electrodes at its two ends. This gives rise to self-energies/H9018 11and/H9018NNon the terminating blocks.MODELING ELASTIC AND PHOTOASSISTED TRANSPORT … PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-3the decay exponent by considering an inﬁnitely extended polymer of phenyl rings. A. Oligo- p-phenylene junction Our model for the oligophenylene-based molecular junc- tion is depicted in Fig. 2. Within a simple /H9266-electron picture, the electronic structure of the oligophenylene molecule canbe described with a nearest-neighbor TB model with twodifferent hopping elements − /H9253and −/H9257/H20849Ref. 52/H20850. Here, − /H9253is for hopping within a phenyl ring, between the porbitals oriented perpendicular to the ring plane, while − /H9257describes hopping between adjacent rings. Due to the symmetry of theorbitals, the magnitude of /H9257depends on the angle /H9272between the rings proportionally to cos /H9272/H20849Ref. 53/H20850. We shall assume that/H9257=/H9253cos/H9272, and thus /H20841/H9257/H20841/H33355/H9253. In this way, the natural energy scale of the model is set by /H9253alone. The ring-tilt angle /H9272can be controlled to some extent using side groups. For example, two side groups bonded toadjacent phenyl rings can repel each other sterically, thusincreasing the corresponding tilt angle. 53,54In fact, even the pure oligophenylenes in the uncharged state have /H9272 =30° –40° due to the repulsion of the hydrogen atoms.36,53 However, side groups can introduce also “charging” or “dop- ing” effects, which shift the molecular levels.55 For deﬁniteness, we number the M=6 carbon atoms of a phenyl ring according to the lower part of Fig. 2. The corre- sponding orbitals appear in the basis in this order. Thus, theblocks in Eq. /H208493/H20850are H q,q=/H20898/H9280q/H208491/H20850−/H9253−/H92530 00 −/H9253/H9280q/H208492/H208500−/H92530 0 −/H92530/H9280q/H208493/H208500−/H92530 0−/H92530/H9280q/H208494/H208500−/H9253 0 0−/H92530/H9280q/H208495/H20850−/H9253 0 00 −/H9253−/H9253/H9280q/H208496/H20850/H20899, /H208497/H20850 forq=1,..., N, andHq,q−1=/H2089800000 −/H9257 00000 0 00000 0 00000 0 00000 0 00000 0 /H20899, /H208498/H20850 withHq−1,q=/H20851Hq,q−1/H20852T. Here, the on-site energies /H9280q/H20849/H9251/H20850may be shifted nonuniformly to describe effects of possible sidegroups. 36For simplicity, we shall consider all phenyl rings to have a similar chemical environment, and thus all on-siteenergies are taken to be equal. As a ﬁrst step we note that, assuming /H9280q/H20849/H9251/H20850=/H9280qfor all /H9251, the eigenvalues for the Hamiltonian Hqqof the isolated unit are /H9280q−/H9253,/H9280q+/H9253,/H9280q−/H9253,/H9280q+/H9253,/H9280q−2/H9253, and/H9280q+2/H9253, while the cor- responding orthonormalized eigenvectors are 1 /H208814/H208490,− 1,1,− 1,1,0 /H20850T,1 /H208814/H208490,1,− 1,− 1,1,0 /H20850T, 1 /H2088112/H20849− 2,− 1,− 1,1,1,2 /H20850T,1 /H2088112/H208492,− 1,− 1,− 1,− 1,2 /H20850T, 1 /H208816/H208491,1,1,1,1,1 /H20850T,1 /H208816/H20849− 1,1,1,− 1,− 1,1 /H20850T. /H208499/H20850 The ﬁrst two of the eigenstates have zero weight on the ring- connecting carbon atoms 1 and 6. Therefore, these eigen-states do not hybridize with the levels of the adjacent ringsand consequently cannot take part in the transport. This willbe seen explicitly in the derivation of the propagator. Wenote that these results can also be used to determine a real-istic value for the hopping /H9253from the HOMO-LUMO split- ting of benzene.36 Below, we shall only consider the analytically solvable case, where all on-site energies are set to the same value. We choose this value as our zero of energy: /H9280q/H20849/H9251/H20850=0 for all q =1,..., Nand/H9251=1,..., M. Later on, we shall relax this as- sumption in order to describe externally applied dc and acvoltage proﬁles. In the absence of such voltages, the inversepropagator /H20851Eq. /H208494/H20850/H20852consists of the blocks h p,p=h0,hp,p−1 =h−1, andhp,p+1=h1, where h0=/H20898E+/H9253/H9253 000 /H9253E+0/H925300 /H92530E+0/H92530 0/H92530E+0/H9253 00/H92530E+/H9253 000 /H9253/H9253 E+/H20899,(α)( α)( α) 1 2 31112 3 33 Σεε ε Σ 1 324 56−η −η −γ −γ−γ−γ−γ −γ FIG. 2. /H20849Color online /H20850A ﬁnite chain of length N=3 connected to electrodes at its two ends. This gives rise to self-energies /H901811and /H9018NNon the end sites. The nearest-neighbor hoppings inside the ring /H20849−/H9253/H20850and between the rings /H20849−/H9257/H20850are different. The lower part indi- cates also the numbering of the M=6 carbon atoms within a ring.VILJAS, PAULY, AND CUEVAS PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-4h−1=/H2089800000 /H9257 000000 000000 000000 000000 000000/H20899, /H2084910/H20850 andh1=/H20851h−1/H20852T. The leads are assumed to couple only to the terminal carbon atoms, thus making the self-energy 6 /H110036 matrices of the form /H901811=/H20898/H9018L0¯0 00 ¯0 ]] /GS 0 0000/H20899,/H9018NN=/H208980000 0/GS] ] 0¯00 0¯0/H9018R/H20899. /H2084911/H20850 We also deﬁne the symbol “tilde” /H20849˜/H20850, which means the re- placement of the ﬁrst column of a matrix by /H9257followed by zeros. For example, h˜0=/H20898/H9257/H9253/H9253 000 0E+0/H925300 00 E+0/H92530 0/H92530E+0/H9253 00/H92530E+/H9253 000 /H9253/H9253 E+/H20899. /H2084912/H20850 For the evaluation of Eq. /H208495/H20850, we only need the component G1,MN=/H20851G1N/H208521M. Using the subdeterminants of F=G−1,w e have G1,MN=/H20849−1/H20850MN+1det/H20851F/H20849MN /H208411/H20850/H20852 det/H20851F/H20852. /H2084913/H20850 Here, O/H20849i,..., k/H20841j,..., l/H20850is the submatrix of Oobtained by removing the rows i,..., k, and columns j,..., l. We shall also denote by LandRthe “leftmost” and “rightmost” rows or column of a matrix, respectively. Thus, for example,det/H20851F/H20849MN /H208411/H20850/H20852=det /H20851F/H20849R/H20841L/H20850/H20852. Let us ﬁrst concentrate on the denominator of Eq. /H2084913/H20850.I t is easy to see that det /H20851F/H20852can be written in terms of determi- nants related to the inverse Green’s function F=G −1of the uncoupled wire as follows:23 det/H20851F/H20852= det /H20851F/H20852−/H9018Ldet/H20851F/H20849L/H20841L/H20850/H20852−/H9018Rdet/H20851F/H20849R/H20841R/H20850/H20852 +/H9018L/H9018Rdet/H20851F/H20849L,R/H20841L,R/H20850/H20852. /H2084914/H20850 Furthermore, due to the symmetry of the molecule, det/H20851F/H20849R/H20841R/H20850/H20852=det /H20851F/H20849L/H20841L/H20850/H20852. Thus, we are left with calculat- ing three types of determinants. It can be shown that, for 1/H11021n/H11021N, all of them satisfy a recursion relation of the form/H20873D/H20849n/H20850 D˜/H20849n/H20850/H20874=/H20849E+2−/H92532/H20850Y/H20873D/H20849n−1/H20850 D˜/H20849n−1/H20850/H20874 =/H20849E+2−/H92532/H20850/H20873a−c cb/H20874/H20873D/H20849n−1/H20850 D˜/H20849n−1/H20850/H20874. /H2084915/H20850 For example, in the calculation of det /H20851F/H20852, we have D/H20849n/H20850 =det /H20851F/H20849n/H20850/H20852andD˜/H20849n/H20850=det /H20851F˜/H20849n/H20850/H20852, where the additional super- script /H20849n/H20850on the matrices denotes the number of the M /H11003Mdiagonal blocks. The elements of the matrix Yare given by a=/H20849E+2−/H92532/H20850/H20849E+2−4/H92532/H20850, b=−/H92572/H20849E+2−/H92532/H20850, c=/H9257E+/H20849E+2−3/H92532/H20850. /H2084916/H20850 Only the initial condition /H20849n=1/H20850and the last step of the re- cursion /H20849n=N/H20850will differ for the three determinants. The recursion relations can be solved by calculating Ynexplicitly, which can be done by diagonalizing Y. The eigenvalues of Y are/H92611,2=/H20849a+b/H11007/H20881/H20849a−b/H208502−4c2/H20850/2, while the /H20849unnormalized /H20850 eigenvectors are v1,2=/H20873a−b/H11007/H20881/H20849a−b/H208502−4c2 2c,1/H20874T . /H2084917/H20850 Then, if V=/H20849v1,v2/H20850and/H9011=diag /H20849/H92611,/H92612/H20850, we have Yn =V/H9011nV−1. The result is Yn=/H20873y11/H20849n/H20850y12/H20849n/H20850 y21/H20849n/H20850y22/H20849n/H20850/H20874, /H2084918/H20850 where the components are given by y11/H20849n/H20850=/H20849/H92611n−/H92612n/H20850/H20849b−a/H20850+/H20849/H92611n+/H92612n/H20850/H20881/H20849a−b/H208502−4c2 2/H20881/H20849a−b/H208502−4c2, y22/H20849n/H20850=/H20849/H92611n−/H92612n/H20850/H20849a−b/H20850+/H20849/H92611n+/H92612n/H20850/H20881/H20849a−b/H208502−4c2 2/H20881/H20849a−b/H208502−4c2, y12/H20849n/H20850=−y21/H20849n/H20850=c/H20849/H92611n−/H92612n/H20850 /H20881/H20849a−b/H208502−4c2. /H2084919/H20850 Using these, we can now write explicit expressions for the three required determinants. For det /H20851F/H20852, the recursion can be started at n=1 with the initial conditions D/H208490/H20850=1 and D˜/H208490/H20850 =0 and carried out up to n=N. The result is det/H20851F/H20849N/H20850/H20852=/H20849E+2−/H92532/H20850Ny11/H20849N/H20850. /H2084920/H20850 The other two determinants require special initial and ﬁnal steps, and the results are det/H20851F/H20849N/H20850/H20849L/H20841L/H20850/H20852=/H20849E+2−/H92532/H20850Ny21/H20849N/H20850//H9257, det/H20851F/H20849N/H20850/H20849L,R/H20841L,R/H20850/H20852=/H20849E+2−/H92532/H20850N/H20851y21/H20849N−1/H20850c−y22/H20849N−1/H20850b/H20852//H92572. /H2084921/H20850MODELING ELASTIC AND PHOTOASSISTED TRANSPORT … PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-5Next, we consider the determinant in the numerator of Eq. /H2084913/H20850, det /H20851F/H20849N/H20850/H20849R/H20841L/H20850/H20852=det /H20851F/H20849N/H20850/H20849R/H20841L/H20850/H20852. It can easily be shown that it satisﬁes the recursion relation det/H20851F/H20849N/H20850/H20849R/H20841L/H20850/H20852=2/H9257/H92533/H20849E+2−/H92532/H20850det/H20851F/H20849N−1/H20850/H20849R/H20841L/H20850/H20852 /H2084922/H20850 and so det/H20851F/H20849N/H20850/H20849R/H20841L/H20850/H20852=2N/H20849/H9257/H92533/H20850N/H20849E+2−/H92532/H20850N//H9257. /H2084923/H20850 Now, the Green’s function of Eq. /H2084913/H20850can be written as G1,MN=−/H208492/H9257/H92533/H20850N//H9257 y11/H20849N/H20850+/H9018LRy21/H20849N/H20850//H9257+/H9018L/H9018R/H20849y21/H20849N−1/H20850c−y22/H20849N−1/H20850b/H20850//H92572, /H2084924/H20850 where we used the shorthand /H9018LR=/H9018L+/H9018R. It is notable that the common /H20849E+2−/H92532/H20850Nfactors canceled out from the ﬁnal propagator. These factors apparently cor-respond to the two eigenvectors of h 0/H20851Eq. /H208499/H20850/H20852having zero weight on the ring-connecting atoms 1 and 6. The cancella-tion is a manifestation of the physical fact that such localizedstates cannot contribute to the transport through the mol-ecule. In the inﬁnite polymer to be discussed below, thesestates appear as completely ﬂatbands in the band structure. To conclude this part, we point out that for Einside the HOMO-LUMO gap /H20851more precisely, when /H20849a−b/H20850 2−4c2/H110220/H20852, the eigenvalues /H92611,2are real valued and the decay exponent of the transmission /H9270/H20849E/H20850for large Nis controlled by the one with a larger absolute value. Since inside the gap E/H110150, we ﬁnd that /H92612/H11022/H9261 1/H110220. Then, using Eq. /H208495/H20850and omitting N-independent prefactors, the decay of the transmission for large Nfollows the law /H9270/H20849E/H20850/H11011/H20875/H92612/H20849E/H20850 2/H9257/H92533/H20876−2N =e−2Nln/H20851/H92612/H20849E/H20850//H208492/H9257/H92533/H20850/H20852. /H2084925/H20850 Thus, the decay exponent is given by /H9252/H20849E/H20850=2l n /H20851/H92612/H20849E/H20850//H208492/H9257/H92533/H20850/H20852. /H2084926/H20850 We note that for resonant energies, oscillatory dependence of /H9270/H20849E/H20850onNcan be expected, instead, and for limiting cases also power-law decay is possible.32Next, we shall reproduce the result for the decay exponent by considering an inﬁnitelyextended polymer. B. Poly- p-phenylene For comparison with the “correct” evaluation of the propagator and the decay coefﬁcient for a ﬁnite chain, let usconsider the propagator for an inﬁnitely extended polymer.To describe the polymer, we start from a ﬁnite chain withperiodic boundary conditions. Neglecting curvature effects,the latter actually represents a ring-shaped oligomer, as de-picted in Fig. 3/H20849a/H20850. Let us ﬁrst consider the eigenstates of the periodic chain. The Hamiltonian H pq/H20849/H9251,/H9252/H20850=/H20855/H9273p/H20849/H9251/H20850/H20841Hˆ/H20841/H9273q/H20849/H9252/H20850/H20856is of the general formH=/H20898H0H1 H−1 H−1H0H1 /GS/GS /GS H−1H0H1 H1 H−1H0/H20899, /H2084927/H20850 where H0,/H110061are the M/H11003Mmatrices /H20849M=6/H20850of Eqs. /H208497/H20850and /H208498/H20850, with /H9280q/H20849/H9251/H20850=0. /H20849Again, only nonzero elements are indi- cated. /H20850The normalized eigenvectors /H9274p/H20849n/H20850/H20849k/H20850satisfying /H20858 qHpq/H9274q/H20849n/H20850/H20849k/H20850=E/H20849n/H20850/H20849k/H20850/H9274p/H20849n/H20850/H20849k/H20850/H20849 28/H20850 are of the Bloch form /H9274q/H20849n/H20850/H20849k/H20850=eikqd/H9278/H20849n/H20850/H20849k/H20850//H20881N, where /H9278/H20849n/H20850/H20849k/H20850are the normalized eigenvectors of H/H20849k/H20850=eikdH1+H0+e−ikdH−1 /H2084929/H20850 with the eigenvalue E/H20849n/H20850/H20849k/H20850, and n=1,..., M. Due to the ﬁ- niteness of the wire, the kvalues are restricted to k/H9262 =2/H9266/H9262/Nd, where /H9262is an integer and dis the lattice constant /H20849the length of a single phenyl-ring unit /H20850. The spectral decomposition of the /H20849retarded /H20850propagator g/H20849E/H20850=/H20849E+1−H/H20850−1of the chain is of the form gpq/H20849/H9251,/H9252/H20850/H20849E/H20850=/H20858 /H9262,n/H20855/H9273p/H20849/H9251/H20850/H20841/H9274/H20849n/H20850/H20849k/H9262/H20850/H20856/H20855/H9274/H20849n/H20850/H20849k/H9262/H20850/H20841/H9273q/H20849/H9252/H20850/H20856 E+−E/H20849n/H20850/H20849k/H9262/H20850, /H2084930/H20850 with the Bloch states /H20841/H9274/H20849n/H20850/H20849k/H9262/H20850/H20856=1 /H20881N/H20858 p=− ⌈N/2⌉+1⌊N/2⌋ eik/H9262pd/H20858 /H9251=1M /H9278/H9251/H20849n/H20850/H20849k/H9262/H20850/H20841/H9273p/H20849/H9251/H20850/H20856./H2084931/H20850 In the limit of large N/H20851Fig. 3/H20849b/H20850/H20852, we can use N−1/H20858/H9262 →/H20849d/2/H9266/H20850/H20848−/H9266/d/H9266/ddkto turn the summation into an integral over the ﬁrst Brillouin zone. In this case, there are M=6 bands with energies E/H208491,2/H20850/H20849k/H20850=/H11006/H9253, E/H208493,4/H20850/H20849k/H20850=/H110061 /H208812/H20881/H92572+5/H92532−2B/H20849k/H20850,(a) (b)1234 N −η−γ −γ−γ −γ−γ −γ FIG. 3. /H20849Color online /H20850Phenyl-ring chains: /H20849a/H20850a periodic chain with Nunits and /H20849b/H20850an inﬁnite chain. Case /H20849b/H20850is obtained from /H20849a/H20850 in the limit N→/H11009.VILJAS, PAULY, AND CUEVAS PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-6E/H208495,6/H20850/H20849k/H20850=/H110061 /H208812/H20881/H92572+5/H92532+2B/H20849k/H20850, /H2084932/H20850 where B/H20849k/H20850=1 2/H20881/H20849/H92572+3/H92532/H208502+1 6/H9257/H92533cos/H20849kd/H20850. /H2084933/H20850 Clearly, we have the symmetries E/H208491/H20850/H20849k/H20850=−E/H208492/H20850/H20849k/H20850,E/H208493/H20850/H20849k/H20850 =−E/H208494/H20850/H20849k/H20850, and E/H208495/H20850/H20849k/H20850=−E/H208496/H20850/H20849k/H20850. For n=1,2, the bands are completely ﬂat, and the corresponding eigenvectors /H9278/H208491,2/H20850/H20849k/H20850 are as in Eq. /H208499/H20850, i.e., independent of kand completely local- ized on atoms /H9251=2,3,4,5. Thus, for p/HS11005q, they do not con- tribute to the propagator in Eq. /H2084930/H20850. For n=3,4,5,6, the vectors are very complicated, but they are not needed in thefollowing. To compare with the result of Sec. III A, we should now calculate, for example, the component g pq/H208491,6/H20850. However, ex- pecting the decay exponent to be independent of /H9251and/H9252,w e consider the simpler case Tr /H20851gpq/H20852=/H20858/H9251gpq/H20849/H9251,/H9251/H20850. Due to the ortho- normality /H20858/H9251/H9278/H9251/H20849m/H20850/H9278/H9251/H20849n/H20850=/H9254mn, the dependence on the vector components then drops out. Thus, for p/HS11005q, /H20858 /H9251gpq/H20849/H9251,/H9251/H20850=4EAd 2/H9266/H20885 −/H9266/d/H9266/d dkeikd/H20849p−q/H20850 A2−B2/H20849k/H20850, /H2084934/H20850 where we deﬁned A=E+2−1 2/H20849/H92572+5/H92532/H20850, /H2084935/H20850 such that E+2−/H20851/H9280/H208493,5/H20850/H20849k/H20850/H208522=A/H11006B/H20849k/H20850. Deﬁning now z=eikd, the integral can be turned into a contour integral around the con-tour /H20841z/H20841=1, /H20858 /H9251gpq/H20849/H9251,/H9251/H20850=−2EA 2/H9266i/H9257/H92533/H20886 /H20841z/H20841=1dzzp−q /H20849z−z+/H20850/H20849z−z−/H20850, /H2084936/H20850 where the poles z/H11006are determined from the equation z2 −/H208514A2−/H20849/H92572+3/H92532/H208502/H20852/H208498/H9257/H92533/H20850−1z+1=0. They are given by z/H11006=4A2−/H20849/H92572+3/H92532/H208502 16/H9257/H92533/H11006/H20881/H208754A2−/H20849/H92572+3/H92532/H208502 16/H9257/H92533/H208762 −1 /H2084937/H20850 such that z+=1 /z−, and we choose the signs so that z−is inside the contour /H20841z/H20841=1. In addition to this, assuming that p/H11021q, there is a pole of order q−patz=0. The integral can then be evaluated using residue techniques, with the result /H20858 /H9251gpq/H20849/H9251,/H9251/H20850=2EA /H9257/H92533z+p−q z+−z−. /H2084938/H20850 This leads to an exponential decay of the propagator with growing q−p/H110220 when Eis off-resonant /H20849in which case z/H11006 are real valued /H20850. Using this result, we can give an estimate for the decay of the transmission function /H20851Eq. /H208495/H20850/H20852through a ﬁnite chain of length Nby replacing G1,MNwith Tr /H20851g1N/H20852/M. This yields /H9270/H20849E/H20850/H11011/H20851 z+/H20849E/H20850/H20852−2N=e−2Nln/H20851z+/H20849E/H20850/H20852, /H2084939/H20850 and thus the exponent/H9252/H20849E/H20850=2l n /H20851z+/H20849E/H20850/H20852. /H2084940/H20850 It can be checked that this result is, in fact, equal to the result /H20851Eq. /H2084926/H20850/H20852obtained for the ﬁnite chain. It is thus seen explicitly that the decay coefﬁcient of the off-resonant transmission does not in any way depend on thecoupling of the molecule to the leads. It should be kept inmind, however, that the relative position of E Fwithin the HOMO-LUMO gap depends on the electrode-lead couplingand the charge-transfer effects. This information is stillneeded for predicting the decay exponent /H9252/H20849EF/H20850of the con- ductance. The analytical results presented in this and the previous section can be used for understanding the behavior of thetransmission function upon changes in the parameters. Forexample, it should be noted that when /H9257is made smaller, the band gap around E/H110150 becomes larger, and at the same time the decay exponent /H9252/H20849E/H20850grows. In this way, the conductance of a molecular junction can be controlled, for example, byintroducing side groups to control the tilt angles /H9272between the phenyl rings.36,53 IV . PHYSICAL OBSERV ABLES AND NUMERICAL RESULTS In this section, we present numerical results based on our model. Throughout, we employ the “wide-band” approxima-tion for the lead self-energies, such that /H9018 L/H20849E/H20850=−i/H9003L/2 and /H9018R/H20849E/H20850=−i/H9003R/2, with energy-independent constants /H9003L,R. Furthermore we only consider the symmetric case /H9003L=/H9003R =/H9003. First, we brieﬂy describe how we generalize the theory, as presented above, to take into account static and time-dependent voltage proﬁles. Then, we concentrate on near-equilibrium /H20849or “linear-response” /H20850properties, using as ex- amples the conductance, the thermopower, and theconductance enhancement due to light with low intensity andfrequency. In this case, knowledge of the zero-bias transmis-sion function calculated above is sufﬁcient, and we can dis-cuss the length dependence of the transport properties in asimple way. After that, we consider the dc current in thepresence of an ac driving ﬁeld of more general amplitudeand frequency, ﬁrst concentrating on the case of inﬁnitesimaldc bias and ﬁnally on the I-Vcharacteristics. A. Voltage proﬁles When considering ﬁnite dc or ac biases within a non-self- consistent TB model that cannot account for screening ef-fects, one of the obvious problems is how to choose thevoltage proﬁle. Throughout the discussion, we shall refer totwo possible choices, as depicted in Fig. 4. They are in some sense limiting cases, and the physically most reasonablechoice should lie somewhere in between. Proﬁle A assumesthe external electric ﬁelds to be completely screened insidethe molecule, such that the on-site energies are not modiﬁed,while B corresponds to the complete absence of such screen-ing. In both cases, we can write the time-dependent on-site energies as /H9280p/H20849/H9251/H20850/H20849t/H20850=eV/H20849t/H20850P/H20849zp/H20849/H9251/H20850/H20850, where zp/H20849/H9251/H20850are the distances of the carbon atoms from the left metal surface, and V/H20849t/H20850 =V+Vaccos/H20849/H9275t/H20850. In case A, P/H20849z/H20850=0 inside the junction,MODELING ELASTIC AND PHOTOASSISTED TRANSPORT … PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-7while in case B P/H20849z/H20850=/H20849L−2z/H20850//H208492L/H20850, where L=Nd+d/3 is the distance between the two metal surfaces. The proﬁle B is more complicated, because the voltage ramp breaks the homogeneity of the wire. In this case, thecurrent must be calculated with the method outlined in Ap-pendix B. In the case of proﬁle A, however, the I-Vcharac- teristics can be calculated based on the knowledge of thezero-bias transmission function in the absence of light, /H9270/H20849E/H20850. As discussed in Appendix A, the current is given by3,46,56 I/H20849V;/H9251,/H9275/H20850=2e h/H20858 l=−/H11009/H11009/H20875Jl/H20873/H9251 2/H20874/H208762/H20885dE/H9270/H20849E+l/H6036/H9275/H20850/H20851fL/H20849E/H20850−fR/H20849E/H20850/H20852. /H2084941/H20850 The low-temperature zero-bias conductance then takes the particularly simple form4,5 Gdc/H20849/H9251,/H9275/H20850=G0/H20858 l=−/H11009/H11009/H20875Jl/H20873/H9251 2/H20874/H208762 /H9270/H20849EF+l/H6036/H9275/H20850. /H2084942/H20850 Here, lindexes the number of absorbed or emitted photons, Jl/H20849x/H20850is a Bessel function of the ﬁrst kind /H20849of order l/H20850, and /H9251=eVac//H6036/H9275is the dimensionless parameter describing the strength of the ac drive. Note that Gdc/H20849/H9251,/H9275=0/H20850=Gdc/H20849/H9251 =0,/H9275/H20850=G0/H9270/H20849EF/H20850=Gdc. Equation /H2084941/H20850may equally well be written in the form37,57 I/H20849V;/H9251,/H9275/H20850=/H20858 l=−/H11009/H11009/H20875Jl/H20873/H9251 2/H20874/H208762 I0/H20849V+2l/H6036/H9275/e/H20850, /H2084943/H20850 where I0/H20849V/H20850is the I-Vcharacteristic in the absence of light /H20851Eq. /H208491/H20850/H20852. Below, the results from these formulas are com- pared to the numerical results for proﬁle B. In Fig. 5, we plot the zero-bias transmission functions for wires with Nbetween 1 and 7. Notice that the four energy bands numbered 3–6 in Eq. /H2084932/H20850are all visible, being sepa- rated by the HOMO-LUMO gap at E//H9253/H110150 and the addi- tional gaps at E//H9253/H11015/H110061.7. Here, we use the parameters /H9003//H9253=5.0,/H9272=40° /H20849i.e.,/H9257//H9253/H110150.77 /H20850, and set the Fermi energy toEF//H9253=−0.4. These values are close to those used in Ref. 36, where they were extracted from a ﬁt to results for gold- oligophenylene-gold contacts based on density-functionaltheory /H20849DFT /H20850. We shall continue to use them everywhere below. A DFT calculation for the HOMO-LUMO splitting ofbenzene, together with the results preceding Eq. /H208499/H20850, yields the hopping /H9253/H110153 eV. The length of a phenyl-ring unit is approximately d=0.44 nm, and the largest ac electric ﬁelds Vac/Lconsidered will be on the order of 109V/m. The pho- ton energies /H6036/H9275will mainly be kept below the energy of the HOMO-LUMO gap of the oligophenylene. B. Near-equilibrium properties Let us start by illustrating the usefulness of the analytical results of Sec. III with a few examples. We concentrate onlow temperatures and small deviations from equilibrium. Inaddition to the linear-response conductance G dc=G0/H9270/H20849EF/H20850, /H2084944/H20850 we shall consider the thermopower, or Seebeck coefﬁcient. At low enough temperature T, this is given in terms of the zero-bias transmission function /H9270/H20849E/H20850as28,41,58,59 S=−/H92662kB2T 3e/H9270/H11032/H20849EF/H20850 /H9270/H20849EF/H20850, /H2084945/H20850 where prime denotes a derivative. Thus, it measures the loga- rithmic ﬁrst derivative of the transmission function at E =EF. The sign of this quantity carries information about the location of the Fermi energy within the HOMO-LUMO gapof molecular junction. 41The third quantity we shall consider is the photoconductance. In the limit /H9251/H112701 and/H6036/H9275//H9253/H112701, we can expand /H9270/H20849E/H20850and the Bessel functions in Eq. /H2084942/H20850/H20849see Appendix A /H20850to leading order in these small quantities, yield- ing Gdc/H20849/H9275/H20850=G0/H9270/H20849EF/H20850+G0/H20849/H9251/H6036/H9275/H208502/H9270/H11033/H20849EF/H20850/16. Deﬁning then the light-induced conductance correction /H9004Gdc/H20849/H9275/H20850=Gdc/H20849/H9275/H20850 −Gdc/H20849/H9275=0/H20850, where Gdc/H20849/H9275=0/H20850=Gdc=G0/H9270/H20849EF/H20850, the relative correction becomes /H9004Gdc/H20849/H9251,/H9275/H20850 Gdc=/H20849/H9251/H6036/H9275/H208502 16/H9270/H11033/H20849EF/H20850 /H9270/H20849EF/H20850. /H2084946/H20850 We thus see that this quantity gives experimental access to thesecond derivative of the transmission function at E=EF. Note that in this approximation, which can be seen as and/6 d/3 d/3 d/3(a)d L (b)(z) P B Az FIG. 4. /H20849Color online /H20850/H20849a/H20850The coordinates of the carbon atoms in the direction zalong the molecular wire. The left electrode is at z=0 and the length of a phenyl-ring unit is d./H20849b/H20850Relative variation of the on-site energies for two different voltage proﬁles, A and B.The proﬁle function P/H20849z/H20850describes how the harmonic voltage V/H20849t/H20850=V+V accos/H20849/H9275t/H20850is assumed to drop over the junction, the volt- age at zbeing given by V/H20849z,t/H20850=V/H20849t/H20850P/H20849z/H20850.-2 -1 0 1 2 E/γ10-610-410-2100τ(E)N=1 N=7EF FIG. 5. /H20849Color online /H20850Transmission functions for the oligophe- nylene wires with lengths N=1,3,5,7. The parameters are /H9003//H9253 =0.5,/H9272=40°, and EF//H9253=−0.4, as discussed in the text.VILJAS, PAULY, AND CUEVAS PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-8adiabatic or “classical” limit,57the conductance correction depends only on the driving ﬁeld through the ac amplitudeV ac=/H9251/H6036/H9275/e. As discussed above, it is reasonable to assume that for large enough N, the transmission function /H9270/H20849E/H20850satisﬁes the exponential decay law /H9270/H20849E/H20850/H11011C/H20849E/H20850e−/H9252/H20849E/H20850N/H2084947/H20850 at the off-resonant energies E/H11015EF. Let us furthermore as- sume that C/H20849E/H20850is only weakly Edependent. Then, it is clear that the Seebeck coefﬁcient will have the following simplelinear dependence on N/H20849Refs. 28and36/H20850: S/H11008 /H9270/H11032/H20849EF/H20850//H9270/H20849EF/H20850/H11011−/H9252/H11032/H20849EF/H20850N. /H2084948/H20850 In contrast, the light-induced conductance correction satisﬁes a quadratic law /H9004Gdc/H20849/H9275/H20850/Gdc/H11008/H9270/H11033/H20849EF/H20850//H9270/H20849EF/H20850/H11011−/H9252/H11033/H20849EF/H20850N+/H20851/H9252/H11032/H20849EF/H20850/H208522N2. /H2084949/H20850 Deviations from these laws can follow from the energy de- pendence of C/H20849E/H20850. In Fig. 6, we demonstrate these length dependences within our model for the oligophenylene junctions. Thecircles connected by lines show the results based on thetransmission functions of Fig. 5, using Eqs. /H2084944/H20850–/H2084946/H20850. The separate solid lines are the estimates of Eqs. /H2084947/H20850–/H2084949/H20850, based on the analytic result for /H9252/H20849E/H20850. The result for /H9004Gdc/H20849/H9275/H20850/Gdcis furthermore compared with some example results for ﬁnite /H9251 and/H9275, using /H9251=0.5 and /H6036/H9275//H9253=0.05 /H20849see below /H20850. Although Eq. /H2084946/H20850was derived above by assuming proﬁle A, the result appears to be rather well satisﬁed for proﬁle B as well. C. Zero-bias conductance at ﬁnite drive frequencies and amplitudes Next we consider the zero-bias photoconductance Gdc/H20849/H9275/H20850 for light whose frequencies and intensities are not restrictedto the adiabatic limit. We have discussed this case previously, based on DFT results for gold-oligophenylene-goldcontacts. 5There, however, the analysis was based solely on the simple formula of Eq. /H2084942/H20850. Here, we show that those results are not expected to change in an essential way withina more reﬁned theory, since the results of our TB model arenot very different for the two voltage proﬁles A and B. Thisis seen in Fig. 7, where we show G dc/H20849/H9275/H20850forN=1,...,4 as a function of /H9275for two values of /H9251, and for both proﬁles. The results for proﬁle A again follow from Eq. /H2084942/H20850, but the re- sults for B require a more demanding numerical calculation/H20849see Appendix B /H20850. In both cases, the effect of light is to increase the conductance considerably. The physical reasonis that the photoassisted processes, where electrons emit orabsorb radiation quanta, bring the electrons to energies out-side of the HOMO-LUMO gap, where the transmission prob-ability is higher. This happens when /H6036 /H9275exceeds the energy difference between the Fermi energy and the closest molecu-lar orbital, in this case the HOMO. The main difference be-tween the two proﬁles is that in case B, the sharp resonancesat some frequencies are smeared out, and thus the light- induced conductance enhancement tends to be smaller. Theincrease can still be an order of magnitude or more. The dependence of this effect on the length of the mol- ecule is still illustrated in Fig. 8, where the conductances in the absence of light and in the presence of light with /H6036 /H9275//H9253 =0.5 and /H9251=1.5 are shown as a function of N. While the conductance in the absence of light has a strong exponentialdecay, in the presence of light, this decay is much slower. Forproﬁle A, the conductance actually oscillates periodically,while in the case of proﬁle B, the oscillations are superim-posed on a background of slow exponential decay. In theDFT-based results, 5the oscillations were not present, or at least not visible for the cases N=1, ... ,4 considered there. Indeed, they are likely to be artifacts of the our TB modelthat neglects all other than /H9266-orbital contributions, as well as uses the wide-band approximation. The results of Fig. 8can also be stated in terms of the relative conductance enhancement /H9004Gdc/H20849/H9275/H20850/Gdc. For large /H925110-810-610-410-2100Gdc/G0 048 1 2 N01020304050S/( kB2π2T/3 e γ) 048 1 2 N050100150(∆Gdc/Gdc)/(αh_ω/γ)2 (a) (b)(c) FIG. 6. /H20849Color online /H20850Dependence of observables on the num- ber of units N:/H20849a/H20850conductance, /H20849b/H20850Seebeck coefﬁcient, and /H20849c/H20850the light-induced relative conductance enhancement. The circles corre-spond to values extracted from the /H9270/H20849E/H20850function /H20849Fig.5/H20850using Eqs. /H2084944/H20850–/H2084946/H20850. The red lines correspond to the simple order-of- magnitude estimates of Eqs. /H2084947/H20850–/H2084949/H20850, with the analytically calcu- lated/H9252/H20849E/H20850.I n /H20849c/H20850, the crosses /H20849/H11003for proﬁle A and /H11001for proﬁle B /H20850 show numerical results with the ﬁnite values /H9251=0.5 and /H6036/H9275//H9253 =0.05 /H20849see Sec. IV C /H20850.10-210-1100Gdc(ω)/G0 0 0.2 0.4 0.6 0.8 1 h_ω/γ10-310-210-1100Gdc(ω)/G0 0 0.2 0.4 0.6 0.8 1 h_ω/γN=1N=2 N=3N=4α=2.0 α=0.5(a) (b) (c) (d) FIG. 7. /H20849Color online /H20850Zero-bias conductance for different driv- ing frequencies /H9275and driving strengths /H9251=eVac//H6036/H9275. Panels /H20849a/H20850–/H20849d/H20850 are for N=1, ... ,4. The solid lines correspond to proﬁle A, and the dashed lines to proﬁle B. The lower pair of curves is for /H9251=0.5, and the upper pair for /H9251=2.0.MODELING ELASTIC AND PHOTOASSISTED TRANSPORT … PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-9and/H9275, the increase of this quantity with Nis exponential for both proﬁles A and B. This should be contrasted with thequadratic behavior for small /H9251and/H9275/H20851Eq. /H2084949/H20850/H20852. Thus, the fact that the results indicated by the crosses in Fig. 6exceed the result of Eq. /H2084946/H20850is understandable. D. Current-voltage characteristics Finally, we discuss the effects of light at ﬁnite voltages V. Let us ﬁrst consider the properties of the I-Vcharacteristics in the absence of light. Examples are shown in Fig. 9/H20849a/H20850for the case N=5. They consist of consecutive steps,60which appear every time a new molecular level comes into the biaswindow between /H9262Land/H9262R. These steps are seen as peaks 1 and 2 in the differential conductance dI/dVshown in Fig. 9/H20849b/H20850. The ﬁrst one occurs roughly at the voltage V1=2/H20849EF −EHOMO /H20850/e, where EHOMO is the energy of the HOMO. The factor of 2 arises from the symmetric division of the voltageswith respect to the molecular energy levels. In the case ofproﬁle B, the currents tend to be smaller than for proﬁle A,but the current steps occur at roughly the same voltages. Itshould also be noticed that for proﬁle B, a small negativedifferential conductance is present following some of thesteps. The origin of this is the localization of the moleculareigenstates due to the dc voltage ramp, which suppresses thetransmission resonances. 24This can be seen in the voltage- dependent transmission functions /H9270/H20849E,V/H20850in Fig. 10. In the presence of light, the step structure of the I-V curves is modiﬁed. For proﬁle A, the results follow simplyfrom Eq. /H2084941/H20850or/H2084943/H20850, but for proﬁle B, a fully numerical treatment is again needed. In Fig. 9, the results for /H9251=1.5 and/H6036/H9275//H9253=0.075 are shown as the curves indicated with arrows. In Fig. 9/H20849a/H20850, it is seen that the current for voltages below the steps is increased and decreased above them. Thisremoves the negative differential conductance present in thecase of proﬁle B. These changes are associated with the ap-pearance of additional current steps. Here, we concentrateonly on the additional steps in the low-bias regime at volt-ages V/H11351V 1, as the relative changes are largest there. Figure 9/H20849c/H20850shows the differential conductance on a logarithmicscale in this voltage region. It can be seen that there are multiple extra peaks below the main peak, all of which areseparated by voltages 2 /H6036 /H9275/efrom each other. These peaks are “images” of the main peak at V=V1and are easily un- derstood based on Eq. /H2084943/H20850. For proﬁle B, all the peaks are moved to slightly smaller voltages and their spacing is re-duced, since ﬁnite voltages tend to also suppress the trans-mission gap /H20849see again Fig. 10/H20850. Notice that, in contrast to high dc biases /H20851Figs. 9/H20849a/H20850and9/H20849b/H20850/H20852, in the low-bias regime /H20851Fig.9/H20849c/H20850/H20852, the results depend only weakly on the choice of the voltage proﬁle. Thus, the predictions of the model appearto be robust. To observe the side steps, the radiation fre-quency should be large enough such that the steps are not“lost” under the broadening of the main steps. On the other1234 5 6 7 8 N10-610-510-410-310-210-1100Gdc(ω=0.5γ/h_)/G0A B FIG. 8. Dependence of the conductance on N. Circles represent the conductance in the absence of light, while the squares are forlight with /H6036 /H9275//H9253=0.5 and /H9251=1.5. The solid line is for proﬁle A and the dashed line for proﬁle B.00.020.040.060.08e I/( G0γ) 0 0.2 0.4 0.6 0.8 1 eV /γ00.20.4(dI/dV )/G0 0 0.1 0.2 0.3 0.4 0.5 eV /γ10-310-210-1(dI/dV )/G0(b)(a) N=5 lightlight (c)light 2h_ω/γA B 12 FIG. 9. /H20849Color online /H20850/H20849a/H20850I-Vcharacteristics /H20849N=5/H20850with and without light for proﬁles A /H20849solid lines /H20850and B /H20849dashed lines /H20850. Re- sults in the presence of light with /H9251=1.5 and /H6036/H9275//H9253=0.075 are indicated with an arrow. /H20849b/H20850The corresponding differential conduc- tances. /H20849c/H20850Same as /H20849b/H20850, but concentrating on the low-bias regime and on a logarithmic scale. The vertical dashed lines indicate theapproximate positions of the main peak and the light-induced sidepeaks. They are all separated by 2 /H6036 /H9275/ein voltage. -1 -0.5 0 0.5 1 1.5 E/γ10-510-410-310-210-1100τ(E,V)0.0 0.5 1.0eV / γEF FIG. 10. /H20849Color online /H20850The voltage-dependent transmission function at three voltages for the wire with N=5 and proﬁle B. For proﬁle A, the result is independent of voltage and equal to /H9270/H20849E,V =0/H20850.VILJAS, PAULY, AND CUEVAS PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-10hand, it should be small enough to have at least one step present. Thus, if the voltage broadening of the main step atV=V 1is approximately /H90041/e, then we require /H90041/H11351/H6036/H9275 /H11021EF−EHOMO . Figure 11additionally shows the low-bias differential conductances for N=1, ... ,4, with other parameters chosen as in Fig. 9/H20849c/H20850. It is seen that the effects of light quickly become weaker, as the length of the molecule decreases. Inthe case N=4, small side peaks are still observed. Larger effects could be obtained by increasing the parameter /H9251. Similar-looking additional steps are visible in the I-V characteristics of an extended-Hückel model for xylyl-dithiolin Ref. 8. Despite the differences in magnitudes of param- eters, and slight asymmetries in the geometries, it is likelythat some of those steps have essentially the same origin asexplained above. However, the most striking result in thatreference was the overall order-of-magnitude increase in thecurrent. V . CONCLUSIONS AND DISCUSSION In this paper, we have studied a /H9266-orbital tight-binding model to describe elastic and photoassisted transport throughmetal-molecule-metal contacts based on oligophenylenes. Incontrast with simpler linear chain models that have previ-ously been studied in great detail, our model describes aspeciﬁc molecule, and its parameters can be directly associ-ated with quantities obtainable from DFT simulations, forexample. Models of this type can be of value in analyzingthe results of more detailed ab initio or DFT calculations, 36 and in making at least qualitative predictions in situations where such calculations would be prohibitively costly. We ﬁrst showed that at zero voltage bias the model can be studied analytically in a similar fashion as the simpler linearchain models. In particular, we derived an expression for thedecay exponent of the off-resonant transmission function. Wethen discussed the length dependence of the dc conductance,the thermopower, and the relative light-induced conductanceenhancement in the case of light with a low intensity /H20849 /H9251/H20850and low frequency /H20849/H9275/H20850. The conductance enhancement was foundto scale quadratically with length. For large /H9251and/H9275, the relative enhancement increases exponentially with length. Fi-nally, it was shown, by numerical calculations, that thecurrent-voltage characteristics are modiﬁed in the presenceof light by the appearance of side steps with a voltage spac-ing 2/H6036 /H9275/e. We demonstrated that the predictions of the model are robust with respect to variations in the assumedvoltage proﬁles. This provides further support for our previ-ous results on the photoconductance. 5 In our work, only symmetrical junctions with symmetrical voltage proﬁles were studied. Asymmetries can modify ourresults through the introduction of rectiﬁcation effects 45and can change the positions of the light-induced current steps.The experimental observation of additional steps with aspacing related to the frequency of the light would neverthe-less provide more compelling evidence for the presence ofphotoassisted transport than a conductance enhancementalone. The latter can also have other causes. 2 We note that the light-induced current steps are similar to the steps observed in current-voltage characteristics of mi-crowave-irradiated superconducting tunnel junctions, wherethey result from photoassisted quasiparticle tunneling. 37,61In that case, the main difference is that the energy gap neces-sary for the effect is located in the macroscopic electrodes,while the transmission through the tunnel barrier dependsonly weakly on energy and voltage. As a result, the currentsteps have a voltage spacing of precisely /H6036 /H9275/e. These effects are exploited in the detection of microwaves in radio-astronomy. 57Similarly, one may imagine properly engi- neered molecular contacts as detectors of light in the infrared or visible frequency range. In terms of our model, to increase the chances of observ- ing the light-induced current steps, the aim should be tominimize the broadening /H9004 1/eof the ﬁrst main current step at voltage V1and to maximize /H9251. Also, a wire with a large enough V1should be used. The broadening /H90041is related to the sharpness of the transmission resonances, and thus to thelength of the molecule and its coupling to the electrodes,described by /H9003. A decrease of /H9003, however, increases the im- portance of Coulomb correlations. Their effect on photoas-sisted transport has recently been discussed within simplemodels. 15,62Increase of /H9251through the light intensity, in turn, increases the heating of the electrodes2and the excitation of local molecular vibrations.51These may affect the geometry through thermal expansion45and structural deformations but will also give rise to an incoherent component to thecurrent. 63At high enough photon energies, also the direct excitation of electrons on the molecule may become impor-tant. The relaxation of such excitations due to variousmechanisms /H20849creation of electron-hole pairs in the electrodes, spontaneous light emission /H20850should thus also be considered. 9 Also, conformational changes of the molecule are possible.1 Finally, a proper treatment of screening effects on the mol- ecule and in the electrodes, the excitation of plasmons, andtheir role in the ﬁeld enhancement 45are other issues that should be studied in more detail. Of course, for the investigation of most of these issues, noninteracting models of the type presented above are not0 0.2 0.4 0.6 0.8 1 eV / γ10-210-1100( d I/d V )/G0 0 0.2 0.4 0.6 0.8 eV / γ 0 0.2 0.4 0.6 eV / γ10-310-210-1100( d I/d V )/G0 0 0.2 0.4 0.6 eV / γ(a) (b) (c) (d)N=1 N=2 N=3 N=4 light FIG. 11. /H20849Color online /H20850Same as Fig. 9/H20849c/H20850but for wires with N=1,...,4.MODELING ELASTIC AND PHOTOASSISTED TRANSPORT … PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-11sufﬁcient. Strong time-dependent electric ﬁelds may have ef- fects that can only be captured by self-consistent theoriestaking properly into account the electron correlations due toCoulomb interactions. These interactions may inﬂuence theelectronic structure in a way that would, at least, require theparameters of our model to be readjusted in the presence ofthe light. Even the geometry of the junction can becomeunstable, and so it should, in principle, be optimized with thelight-induced effects included. Time-dependent density-functional theory is showing some promise for the treatmentof such problems. 20,21In addition to DFT, also more ad- vanced computational schemes are being developed tohandle correlation effects. 64,65A systematic investigation of the optical response of metal-molecule-metal contacts, andthus the testing of the predictions of the simple models, 3,6–9 remains an important goal for future research. ACKNOWLEDGMENTS This work was ﬁnancially supported by the Helmholtz Gemeinschaft /H20849Contract No. VH-NG-029 /H20850, by the DFG within the Center for Functional Nanostructures, and by theEU network BIMORE /H20849Grant No. MRTN-CT-2006-035859 /H20850. F.P. acknowledges the funding of a Young InvestigatorGroup at KIT. APPENDIX A: SIMPLIFIED FORMULA FOR THE TIME-A VERAGED CURRENT Consider the expression Eq. /H208492/H20850for the time-averaged /H20849or dc/H20850current. The coefﬁcient /H9270RL/H20849k/H20850/H20849E/H20850, for example, is the sum of the transmission probabilities of all transport channels tak-ing the electron from energy Eon the left to energy E +k/H6036 /H9275on the right. That is, for k/H110220/H20849k/H110210/H20850, it describes electron transmission under the absorption /H20849emission /H20850ofk photons. Assuming the wide-band approximation and thevoltage proﬁle A, Eq. /H208492/H20850can be written in the more trans- parent forms of Eqs. /H2084941/H20850and /H2084943/H20850. This can be demonstrated rigorously using the equations of Appendix B, but it is in-structive to consider the following simpler derivation. Theidea is the same as in the “independent channel approxima-tion” of Ref. 7. For now, we allow the ac voltage drops at the LandR lead-molecule interfaces to be asymmetrical. Thus, we deﬁnethe quantities /H9251Land/H9251R, satisfying /H9251=/H9251L−/H9251R. Since for proﬁle A there is no voltage drop on the molecule, electronictransitions only occur at the lead-molecule interfaces. Thus, the transmission coefﬁcients /H9270RL/H20849k/H20850/H20849E/H20850are given by /H9270RL/H20849k/H20850/H20849E/H20850=/H20858 l=−/H11009/H11009 /H20851Jl−k/H20849/H9251R/H20850/H208522/H9270/H20849E+l/H6036/H9275/H20850/H20851Jl/H20849/H9251L/H20850/H208522, /H20849A1/H20850 where /H20851Jl/H20849/H9251L/H20850/H208522is the probability for absorbing /H20849emitting /H20850l photons on the left interface and /H20851Jl−k/H20849/H9251R/H20850/H208522the probability for emitting /H20849absorbing /H20850l−kphotons on the right interface. The propagation between the interfaces occurs elastically atthe intermediate energy E+l/H6036 /H9275, according to the transmis- sion function /H9270/H20849E/H20850. A similar expression holds for /H9270LR/H20849k/H20850/H20849E/H20850.Using these and the sum formula /H20858k=−/H11009/H11009/H20851Jk/H20849x/H20850/H208522=1, Eq. /H208492/H20850 leads to I/H20849V;/H9251,/H9275/H20850=2e h/H20858 l=−/H11009/H11009/H20885dE/H9270/H20849E+l/H6036/H9275/H20850/H20853/H20851Jl/H20849/H9251L/H20850/H208522fL/H20849E/H20850 −/H20851Jl/H20849/H9251R/H20850/H208522fR/H20849E/H20850/H20854. /H20849A2/H20850 Equation /H2084941/H20850follows by setting /H9251L=/H9251/2 and/H9251R=−/H9251/2, and the equivalent form of Eq. /H2084943/H20850follows by changing summa- tion indices and integration variables. Similarly, other sug-gestive forms may be derived. 3,46,56Forx/H112701 and l/H110220, one may expand J/H11006l/H20849x/H20850/H11015/H20849/H11006x/2/H20850l/l!−/H20849/H11006x/2/H20850l+2//H20849l+1/H20850!. This can be used in the limit /H9251/H112701,/H6036/H9275//H9253/H112701 discussed in the text. APPENDIX B: GREEN-FUNCTION METHOD FOR THE TIME-A VERAGED CURRENT Here, we outline the Green-function method4,7,66used for obtaining the results for voltage proﬁle B. Consider again thedc current of Eq. /H208492/H20850. In the case of a harmonic driving ﬁeld, it is reasonable to assume the existence of time-reversal in-variance, in which case we have the symmetry 3 /H9270LR/H20849k/H20850/H20849E/H20850=/H9270RL/H20849−k/H20850/H20849E+k/H6036/H9275/H20850. /H20849B1/H20850 The current expression of Eq. /H208498/H20850in Ref. 4was derived under this assumption, and that result can be brought into the formof Eq. /H208492/H20850. Using the notation of that reference, 47the coefﬁ- cients can be written as /H9270RL/H20849k/H20850/H20849E/H20850=T r/H9275/H20851Gˆ/H20849E/H20850/H9003ˆ R/H20849k/H20850/H20849E/H20850Gˆ†/H20849E/H20850/H9003ˆ L/H208490/H20850/H20849E/H20850/H20852, /H9270LR/H20849k/H20850/H20849E/H20850=T r/H9275/H20851Gˆ/H20849E/H20850/H9003ˆ L/H20849k/H20850/H20849E/H20850Gˆ†/H20849E/H20850/H9003ˆ R/H208490/H20850/H20849E/H20850/H20852, /H20849B2/H20850 where the hats denote the extended “harmonic” matrices67 and Tr /H9275a trace over them. In particular, Gˆis the matrix for the retarded propagator Gˆ/H20849E/H20850=/H20851/H20849Eˆ−H1ˆ/H20850−Wˆ−/H9018ˆL/H20849E/H20850−/H9018ˆR/H20849E/H20850/H20852−1, /H20849B3/H20850 where His the Hamiltonian of the wire in the absence of voltage proﬁles. The matrix Eˆis deﬁned by /H20851Eˆ/H20852m,n=/H20849E +m/H6036/H9275/H20850/H9254m,n1, where mandnare the harmonic indices. Using the wide-band approximation for the electrodes, the matrices /H9018ˆXand/H9003ˆ X/H20849l/H20850are given by /H20851/H9018ˆX/H20852m,n/H20849E/H20850=/H9254m,n/H9018X, /H20851/H9003ˆ X/H20849l/H20850/H20852m,n/H20849E/H20850=Jm−l/H20849/H9251X/H20850Jn−l/H20849/H9251X/H20850/H9003X, /H20849B4/H20850 with X=L,Rand/H9251L,R=/H11006/H9251/2. Here, /H9018Xis the self-energy matrix of lead X/H20849extended to the size of H/H20850, and /H9003X =−2 Im /H9018X. The matrix Wˆincludes the effect of the proﬁles for the voltage V/H20849t/H20850=V+Vaccos/H20849/H9275t/H20850.I f W/H20849t/H20850=Wdc +Waccos/H20849/H9275t/H20850is a diagonal matrix consisting of the on-site energies /H9280p/H20849/H9251/H20850/H20849t/H20850, thenVILJAS, PAULY, AND CUEVAS PHYSICAL REVIEW B 77, 155119 /H208492008 /H20850 155119-12/H20851Wˆ/H20852m,n=Wdc/H9254m,n+1 2Wac/H20849/H9254m−1,n+/H9254m+1,n/H20850. /H20849B5/H20850 In this formalism, the time-reversal invariance amounts to Gˆ and/H9003ˆ L,R/H20849k/H20850being symmetric, i.e., AˆT=Aˆ. Equation /H20849B1/H20850canthen be proven by using the relations /H20851Gˆ/H20852m+k,n+k/H20849E/H20850 =/H20851Gˆ/H20852m,n/H20849E+k/H6036/H9275/H20850and /H20851/H9003ˆ X/H20849l/H20850/H20852m+k,n+k/H20849E/H20850=/H20851/H9003ˆ X/H20849l−k/H20850/H20852m,n/H20849E+k/H6036/H9275/H20850. We note that /H9003ˆ X/H20849l/H20850is deﬁned with a different sign of lthan in Ref. 4. *janne.viljas@kit.edu 1S. 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PhysRevB.86.174416.pdf	PHYSICAL REVIEW B 86, 174416 (2012) Gate-voltage controlled electronic transport through a ferromagnet/normal/ferromagnet junction on the surface of a topological insulator Kun-Hua Zhang, Zheng-Chuan Wang, Qing-Rong Zheng,and Gang Su† Theoretical Condensed Matter Physics and Computational Materials Physics Laboratory, School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China (Received 14 May 2012; revised manuscript received 6 November 2012; published 16 November 2012) We investigate the electronic transport properties of a ferromagnet/normal/ferromagnet junction on the surface of a topological insulator with a gate voltage exerted on the normal segment. It is found that the conductanceoscillates with the width of normal segment and gate voltage, and the maximum of conductance graduallydecreases while the minimum of conductance approaches zero as the width increases. The conductance can becontrolled by tuning the gate voltage like a spin ﬁeld-effect transistor. It is found that the magnetoresistanceratio can be very large, and can also be negative owing to anomalous transport. In addition, when there existsa magnetization component in the surface plane, it is shown that only the component parallel to the junctioninterface has an inﬂuence on the conductance. DOI: 10.1103/PhysRevB.86.174416 PACS number(s): 72 .25.Dc, 73 .20.−r, 73.23.Ad, 85 .75.−d I. INTRODUCTION Topological insulators are new quantum states discovered recently, which have a bulk band gap and gapless edge statesor metallic surface states due to the time-reversal symmetryand spin-orbit-coupling interaction. 1Two-dimensional (2D) topological insulators were ﬁrst predicted theoretically as aquantum spin Hall state 2,3and then observed experimentally.4 The topological characterization of quantum spin Hall insula- tors can be generalized from the 2D to three-dimensional (3D)case and leads to the discovery of 3D topological insulators(TIs). 5–8TIs in 3D are usually classiﬁed according to the num- ber of Dirac cones on their surfaces. Those strong topologicalinsulators with an odd number of Dirac cones on their surfaceare robust against the time-reversal-invariant disorder, whilethe weak topological insulator is referred to those with aneven number of Dirac cones on their surfaces, which dependson the surface direction and might be broken even withoutbreaking time-reversal symmetry. 5,8When TIs are coated with magnetic or superconducting layers, the surface states couldbe gapped and many interesting properties emerge, such as thehalf-integer quantum Hall effect, 9Majorana fermions,10etc. Topological surface states were observed by several experi- mental groups by angle-resolved photoemission spectroscopy(ARPES) 11–13and scanning tunneling microscopy (STM).14,15 Although the residual bulk carrier density brings much difﬁ- culty to surface-state transport experiments,16,17the signatures of negligible bulk carriers contributing to the transport18and near 100% surface transport in topological insulators19have been found recently in experiments. The low-energy physics of the surface states of strong topo- logical insulators can be described by the 2D massless Diractheory, 7which is different from that in graphene where the spinors are composed of different sublattices.20The topologi- cal surface states show strong spin-orbit coupling, which maybe applied to the spin ﬁeld-effect transistors in spintronics. 21–26 The electronic transport properties of topological insulator sur- faces with magnetization have attracted a lot of attention.27–34 In Refs. 27and 28the results are given in the limit of thin barrier (i.e., the width of barrier L→0 and barrier potential V0→∞ whileV0Lis constant), and the physical origin of thisthin barrier is the mismatch effect and built-in electric ﬁeld of junction interface. References 29and33studied the spin valve on the surface of topological insulators, in which the exchangeﬁelds in the two ferromagnetic leads are assumed to align alongthey-axis direction. References 30–32and 34investigated electron transport through a ferromagnetic barrier on thesurface of a topological insulator. Note that both the electricpotential barrier and the ferromagnetic barrier are the transportchannels in these models. The bulk band gap of topologicalinsulators is usually about 20–300 meV 7,11–13,18in order to keep the transport at the Fermi energy inside the bulk gap, andthe gate voltage on topological insulators should be ﬁnite. In this paper, we study the electronic transport through a 2D ferromagnet/normal/ferromagnet junction on the surfaceof a strong topological insulator where a gate voltage isexerted on the normal segment with a ﬁnite width, and theexchange ﬁelds in the two ferromagnetic leads point mainlyin the z-axis direction. So far such a system has not been well studied. We ﬁnd that the conductance oscillates with thewidth of normal segment and gate voltage, and the maximumof conductance gradually decreases while the minimum ofconductance can approach zero as the width increases. Thisbehavior is more obvious when the gate voltage is lessthan the Fermi energy. This gate-controlled 2D topologicalferromagnet/normal/ferromagnet junction shows the proper-ties of a spin ﬁeld-effect transistor. The magnetoresistance(MR) can be very large and could also be negative owingto the anomalous transport. In addition, when there exists amagnetization component in the 2D plane, it is shown that onlythe magnetization component parallel to the junction interfaceinﬂuences the conductance. This paper is organized as follows: First, we describe the theoretical model for the electronic transport through the topo-logical spin-valve junction. Second, we present our numericalresults and discussions. Finally, a brief summary is given. II. THEORETICAL FORMALISM We consider a 2D ferromagnet/normal/ferromagnet junc- tion on a strong topological insulator surface as shown in Fig. 1. 174416-1 1098-0121/2012/86(17)/174416(7) ©2012 American Physical SocietyZHANG, W ANG, ZHENG, AND SU PHYSICAL REVIEW B 86, 174416 (2012) z y xM1M1 M2M2 Topological InsulatorFI FIV0V0 0 L FIG. 1. (Color online) Schematic layout of a 2D ferromag- net/normal/ferromagnet junction on the surface of a topological insu-lator. An exchange split on the surface underneath the ferromagnetic insulator (FI) is induced by the proximity effect, and the central normal segment is tuned by a gate voltage V 0. The current ﬂows along the xaxis on the surface. The bulk ferromagnetic insulator (FI) interacts with the surface electrons in the TI by the proximity effect, and ferromagnetismis induced in the topological surface states. 27–31,34–37The interfaces between ferromagnet (FM) and normal segment areparallel to the ydirection, and the normal segment is located between x=0 and x=Lwith gate voltage V 0exerted on it.38–40Here we presume, for simplicity, the distance Lbetween two interfaces is shorter than the mean-free path as well as thespin coherence length. With this setup, the Hamiltonian for this system reads 27–31,34 /hatwideH=υF/hatwideσ·/hatwidep+/hatwideσ·/arrowrighttophalfm(r)+V(r), (1) with Pauli matrices /hatwideσ=(/hatwideσx,/hatwideσy,/hatwideσz), the in-plane electron mo- mentum /hatwidep=(/hatwidepx,/hatwidepy,0), and Fermi velocity υF. The piecewise magnetization/arrowrighttophalfm(r) is chosen to be a 3D vector pointing along an arbitrary direction in the left region with/arrowrighttophalfmL= (mLx,mLy,mLz)=mL(sinθcosβ,sinθsinβ,cosθ) and ﬁxed along the zaxis perpendicular to the TI surface in the right region with/arrowrighttophalfmR=(0,0,mRz). We can use a soft magnetic insulator for the left ferromagnet, which is controlled by aweak external magnetic ﬁeld, and a magnetic insulator withvery strong easy-axis anisotropy for the right ferromagnet. Theconﬁguration between the left and right ferromagnets directlydepends on the weak external magnetic ﬁeld, where the in-terlayer [Ruderman-Kittel-Kasuya-Yosida (RKKY)] exchangecoupling between left and right ferromagnets 41is ignored for simplicity. In the middle segment, there is no magnetizationbut, instead, a gate voltage V 0is exerted. Solving Eq. (1), we obtain the wave function in the left region as follows: ψL(x/lessorequalslant0)=A/parenleftbiggυF¯hkx+mLx−i(υF¯hky+mLy) E−mLz 1/parenrightbigg eikxx +B/parenleftbigg−(υF¯hkx+mLx)−i(υF¯hky+mLy) E−mLz 1/parenrightbigg e−i(kx+2mLx υF¯h)x, (2) where the Fermi energy lies in the upper bands of Dirac cone and E> 0. We also deﬁne φas the inci- dent angle. Then kx=[(E2−m2 Lz)1/2cosφ−mLx]/(υF¯h), ky=[(E2−m2 Lz)1/2sinφ−mLy]/(υF¯h) .T h ew a v ef u n c t i o nin normal region ψCdepends on the gate voltage. If V0/negationslash=E, ψC(0/lessorequalslantx/lessorequalslantL)=C/parenleftbiggυF¯h(k/prime x−iky) E−V0 1/parenrightbigg eik/prime xx +D/parenleftbigg−υF¯h(k/prime x+iky) E−V0 1/parenrightbigg e−ik/prime xx, (3) where k/prime x=± { [(E−V0)/υF¯h]2−k2 y}1/2with the ±corre- sponding to the upper and lower bands of the Dirac cone,respectively. If V 0=E,42 ψC(0/lessorequalslantx/lessorequalslantL)=C/parenleftbigg 0 1/parenrightbigg e−kyx+D/parenleftbigg 1 0/parenrightbigg ekyx.(4) The wave function in the right region is ψR(L/lessorequalslantx)=F/parenleftBigg υF¯h(k/prime/prime x−iky) E−mRz 1/parenrightBigg eik/prime/prime xx, (5) withk/prime/prime x=[(E2−m2 Rz)/(υF¯h)2−k2 y]1/2. There exists a trans- lation invariance along the ydirection, so the momentum ky is conserved in the three regions, and we omit the part eikyyin wave functions. These piecewise wave functions are connectedby the boundary conditions ψ L(0)=ψC(0),ψ C(L)=ψR(L), (6) which determine the coefﬁcients A, B, C, D, and F in the wave functions. As a result, according to the Landauer-B ¨uttiker formula,43 it is straightforward to obtain the ballistic conductance Gat zero temperature G=e2wy hπEF υF¯h1 2/integraldisplayπ 2 −π 2dφF∗F A∗A(EF−mLz)υF¯hk/prime/prime x (EF−mRz)EF,(7) where wyis the width of interface along the ydirection, which is much larger than L, and we take EasEFbecause in our case the electron transport happens around the Fermi level. III. NUMERICAL RESULTS AND DISCUSSIONS We focus on the two cases about the electronic transport controlled by a gate voltage through this 2D topological ferro-magnet/normal/ferromagnet junction. One is the conductanceGand the magnetoresistance when the magnetizations in the left and right FM are collinear in the zdirection, and another is the inﬂuence of the magnetization component along the x orydirection on the conductance. A. Conductance and magnetoresistance for collinear magnetization We show the normalized conductance G/G 0as a function ofkFLandV0/EFof parallel [Figs. 2(a) and 2(c)] and antiparallel [Figs. 2(b) and 2(d)] conﬁgurations for two different magnetizations along the zaxis, where G0=e2wy hπEF υF¯h. In Figs. 2(a) and2(b) we choose mLz=mRz=0.95EF, while in Figs. 2(c) and 2(d)mLz=mRz=0.6EF.I nF i g . 2(a) the 174416-2GATE-VOLTAGE CONTROLLED ELECTRONIC TRANSPORT ... PHYSICAL REVIEW B 86, 174416 (2012) FIG. 2. (Color online) Normalized conductance G/G 0as a function of EFL/(υF¯h)a n dV0/EF,mLz=mRz=0.95EFin panels (a) and (b), mLz=mRz=0.6EFin panels (c) and (d). Panels (a) and (c) correspond to the parallel conﬁguration and panels (b) and (d) correspond to the antiparallel conﬁguration. Panels (e) and (f) are the sections of (a) and (c), respectively, for three values of EFL/(¯hυF). gap of surface states in the left and right ferromagnet regions opened by the magnetization along the zaxis is 0 .95EF. The conductance oscillates with gate voltage V0[parameters EFL/(¯hυF) and V0/EFin Fig. 2are dimensionless]. The maximum of conductance gradually decreases as the widthincreases. The minimum of conductance can approach zero.The change of conductance between maximum and minimumby gate voltage is similar to the spin ﬁeld-effect transistor,in which the conductance modulation arises from the spinprecession due to spin-orbit coupling. 21The gate voltage can be used to change k/prime xsuch that the phase factor k/prime xLof quantum interference in the normal segment can be changed.The oscillation period of conductance with respect to V 0 depends on the width Land decreases with the increase of width L. The conductance has a period πwith respect toz=V0L, when V0→∞,L→0, in a 2D topological ferromagnet/ferromagnet junction.27,28 In Fig. 2(b), the conductance changes with the width Land gate voltage V0in the same way as in Fig. 2(a). The difference is that the conductance is maximum in Fig. 2(b) while it is minimum in Fig. 2(a), and vice versa. The conductance in Figs. 2(c) and2(d) shows the same tendency of variation with width Land gate voltage V0as in Figs. 2(a) and2(b), respectively. However, both the maximum and minimum of conductance inFigs. 2(c) and2(d) are larger than those in Figs. 2(a) and2(b), since the gap of surface states in the left and right ferromagnetregions is 0 .6E Fin Figs. 2(c) and 2(d). The conductance changes more obviously with the gate voltage at the side ofV 0/EF<1t h a na tt h es i d eo f V0/EF>1. In Fig. 2, both theFIG. 3. (Color online) MR as a function of width EFL/(¯hυF) with different gate voltage V0.( a )mLz=mRz=0.95EFand (b) mLz= mRz=0.6EF. maximum and minimum of the conductance become smaller when the gate voltage is closer to the Fermi energy, becausethe number of the incident wave functions transported throughthe normal segment by the evanescent waves (imaginary k /prime x) becomes bigger. Figure 2shows that the conductance of this 2D topological ferromagnet/normal/ferromagnet junction couldbe changed in the same way as that in the spin ﬁeld-effecttransistor. As for the reason for the angular spectrum ofelectrons in the surface plane and the linear dispersion relation,how to get a large maximum/minimum ratio of the conductanceis important for a transistor. After obtaining the conductance G Pfor the parallel conﬁg- uration and GAPfor the antiparallel conﬁguration, we can get the MR directly, which is deﬁned as MR=(GP−GAP)/GP. Compared with the conductance in Figs. 2(a) and 2(c),t h e conductance in Figs. 2(b) and2(d) shows a property indicated below. On the one hand, the conductance in the antiparallelconﬁguration can be less than that in the parallel conﬁgurationas in the conventional spin valve 22–24and its counterpart in graphene.44On the other hand, the conductance in the antiparallel conﬁguration can also be larger than that in theparallel conﬁguration, which is an anomalous electronic trans-port property of a topological spin-valve junction. Figure 3 shows the MR as a function of width L. When V 0/EF/negationslash=1, the MR oscillates with the width L. The amplitude and period of oscillation of MR depend on the gate voltage V0. When V0/EF=1, the MR does not oscillate and decreases monotonically with increasing L, because the Fermi surface of the normal segment is at the Dirac point in this case and thecorresponding density of states is zero while the conductanceis not zero, which is a typical property of Dirac fermionsystems. 42The MR could be negative for the anomalous electronic transport.27,45The maximum MR in Fig. 3(a) is 174416-3ZHANG, W ANG, ZHENG, AND SU PHYSICAL REVIEW B 86, 174416 (2012) FIG. 4. (Color online) Transmission probability as a function of incident angle φand width EFL/(¯hυF)w h e r e mLz=mRz=0.95EF. We choose the parallel conﬁguration on the left-hand side and the antiparallel conﬁguration on the right-hand side, and the gate voltagesV 0/EFin (a) and (b), (c) and (d), (e) and (f), (g) and (h) are 0, 0.5, 1 and 1.5, respectively. larger than that in Fig. 3(b) and can approach 100%. The big negative MR (more than −10) in Fig. 3(a) also means a big variation in conductance between the parallel and antiparallelconﬁgurations. Next we discuss the underlying physics quantitatively to more clearly understand the above results. Since electronsfrom all incident angles contribute to the conductance whichis proportional to the electron transmission probability, thephysical origin of conductance oscillating with width Land gate voltage V 0in Fig. 2is a direct result of the summation of electron transmission probability over all incident angles. Figure 4plots the transmission probability as a function of incident angle φand width Lfor different gate voltage V0. We ﬁnd that the transmission probability mainly oscillates with the width L. Its period of oscillation becomes large as the gate voltage increases from V0/EF=0t oV0/EF=1. The reason for such a change can be illustrated in Fig. 5. Because the wave functions in the left and right FMs areconnected through the wave function in the normal segment,the transmission probability depends on the phase factor k /prime xL. Due to the conservation of momentum ky,k/prime xdepends on the gate voltage. When the gate voltage varies from V0/EF=0 to 1, the Fermi surface for the normal region reduces askxky k\ x kxky kykxky k\ x kxky ky(a) (b)FM normalFM FM normalFM FIG. 5. (Color online) Fermi surfaces of the ferromagnet/ normal/ferromagnet junction in momentum space, where the different colored Fermi surfaces in the normal segment stand forthe cases with different gate voltages and the dashed lines have the same length which equals the range of momentum k yof the incident wave function in panels (a) and (b), respectively. (a) mLz=mRz= 0.95EFand (b) mLz=mRz=0.6EF. in Fig. 5, and k/prime xreduces, too, such that the transmission probability has a longer periodicity with width Land changes considerably with incident angle, as shown in Figs. 4(a) or 4(b) and4(c) or4(d). In these cases, the electronic transport through the normal segment occurs in the upper bands ofthe Dirac cone. Although the Fermi surface for the normalsegment in Figs. 4(g) or4(h) is equal to that in Figs. 4(c) or4(d), their transmission probability is different, because in Figs. 4(g) or4(h) the electronic transport through the normal segment occurs in the lower bands of the Dirac cone. Whenthe gate voltage V 0/EF=1, the electronic transport through the normal segment is totally due to the evanescent waves; thetransmission probability is not a periodic function of width L as in Figs. 4(e) or4(f). Now we consider the inﬂuence of magnetization conﬁg- uration on the transmission probability. It is clear that thetransmission probability is an even function of the incidentangle φin the parallel conﬁguration on the left-hand side of Fig. 4, while it is not an even function of the incident angle φin the antiparallel conﬁguration on the right-hand side. This is unusual, because the transmission probabilityis an even function of incident angle φon the antiparallel conﬁguration in its counterpart in graphene. 44This unusual property arises from the unequal spinor parts of the incidentand transmission wave functions. At the normal incidence(φ=0), the period of the transmission probability with width Lin the parallel conﬁguration is the same as that in the antiparallel conﬁguration and the position of maximum of thetransmission probability has a shift of the half period betweentwo conﬁgurations. Now with the help of Figs. 4and 5,t h e properties of conductance in Figs. 2(a) and 2(b) and MR in Fig. 3(a) can be understood explicitly. When the magnetizations in the left and right FMs are taken as 0 .6E Fin Fig. 5(b), one may see that the gaps of the surface states in the left and right ferromagnet regions 174416-4GATE-VOLTAGE CONTROLLED ELECTRONIC TRANSPORT ... PHYSICAL REVIEW B 86, 174416 (2012) FIG. 6. (Color online) Transmission probability as a function of incident angle φand width EFL/(¯hυF), where mLz=mRz=0.6EF, and we choose the parallel conﬁguration on the left-hand side and the antiparallel conﬁguration on the right-hand side, and the gatevoltage V 0/EF=0.5 and 1.5 in panels (a) and (b) and (c) and (d), respectively. decrease, and the Fermi surfaces in the left and right FMs become large. So, the range of kyexpands, and those of k/prime xand the phase factor k/prime xLexpand, too. The transmission probability in Fig. 6changes more dramatically than in Figs. 4(c) and4(d) and in Figs. 4(g) and 4(h). Therefore, as the gap of surface states in the left and right ferromagnet regions decreases, moreincident electronic states will contribute to the conductancesuch that the conductance becomes larger on the whole andmore unsymmetrical about the gate voltage V 0/EF=1.0i n Figs. 2(c) and 2(d). The MR in Fig. 3(b) can be understood similarly. B. Inﬂuence of xand ycomponents of magnetization on conductance Now we consider the inﬂuence of the xandycomponents of magnetization on the conductance. First, we choose the z component of magnetization in the left and right FM to beequal as that in Sec. III A. We ﬁnd that the inﬂuence of the x andycomponents of magnetization on the conductance is quite different. The xcomponent of magnetization has no inﬂuence on the conductance, while the ycomponent of magnetization has a great inﬂuence on the conductance. Because the x component of magnetization just moves the Fermi surfacealong the xaxis, the states contributing to the conductance do not change, while the ycomponent of magnetization shifts the Fermi surface in the left FM along the ydirection and decreases the number of incident electron states that contributeto the conductance. The inﬂuence of m Lyon the conductance is shown in Fig. 7. It is seen that the conductance decreases with increasing \|mLy\|,s oal a r g e \|mLy\|can lead the conductance to be zero. We also discover that the inﬂuence of magnetizationm Lyon the conductance is different from that of −mLy. Second, by keeping the magnetizations in the left and right FMs the same value, the direction of magnetization in theleft FM is changed in the x-zplane ( β=0) or in the y-z plane ( β=π/2), where θandβare indicated as shown in Fig. 1. The conductance as a function of θand the gateFIG. 7. (Color online) Conductance as a function of gate voltage V0for different mLy,w h e r e EFL/(¯hυF)=2a n d mLz=mRz= 0.95EF. voltage V0is plotted in Fig. 8, which is different from that in a ferromagnetic/normal/ferromagnetic graphene junction.45The distinction between Figs. 8(a) and8(b) is more obvious at θ= ±0.5π, where the conductance changes slightly with gate volt- age in Fig. 8(a) while the conductance changes remarkably in Fig. 8(b). These results are from different connections of wave functions between left and right FMs. Since when θ=± 0.5π, the spin in the right FM is parallel to ( υF¯hkR x,υF¯hky,m)t,27 and the spin in the left FM is parallel to ( υF¯hkx1± m,υF¯hky1,0)tin Fig. 8(a) which satisﬁes the relation E= [(υF¯hkx1±m)2+(υF¯hky1)2]1/2, while the spin in the left FM is parallel to ( υF¯hkx2,υF¯hky2±m,0)tin Fig. 8(b) which satisﬁes the relation E=[(υF¯hkx2)2+(υF¯hky2±m)2]1/2.I n this case, the zcomponent of spin in the left FM is 0 in Figs. 8(a) and8(b). Because in Fig. 8(b) the Fermi surface of left FM shifts along the ydirection about ±m, the difference of the xcomponent of spin between the left FM and right FM in Fig. 8(a) is larger than that in Fig. 8(b). Finally, we discuss the realization of our model. The bulk band gap of topological insulator is small and depends on thematerials, which is, for example, about 300 meV in Bi 2Se3, 100 meV in Bi 2Te37,12,13, and 22 meV in HgTe.18Far away from the Dirac point, the surface electronic states exhibit largedeviations from the simple Dirac cone in Bi 2Te3.46The gap of surface states could be induced by putting the magneticinsulator on the surface of a topological insulator (such as FIG. 8. (Color online) Conductance as a function of θand gate voltage V0/EFforEFL/(¯hυF)=2,m=\|(mLx,mLy,mLz)\|= \|(0,0,mRz)\|=0.95EF, the angle θis (a) in the x-zplane ( β=0) and (b) in the y-zplane ( β=π/2). 174416-5ZHANG, W ANG, ZHENG, AND SU PHYSICAL REVIEW B 86, 174416 (2012) EuO, EuS, and MnSe). Depending on the interface match of the topological insulator and ferromagnetic insulator, thegap is several to dozens of meV . 27,35–37The gate electrode could be attached to the topological insulator to control thesurface potential. 38–40The predicted properties of our model may be observed when the Fermi energy of surface statesis about 10 to 100 meV , and the junction width is about 10to 100 nm. The calculated results in this paper are based onthe ballistic transport. In order to observe experimentally ourpredicted properties, a clean 2D topological surface state witha sufﬁciently long mean-free path is needed. It is interesting tonote that the surface of topological insulator with such a longmean-free path can be realized in experiments. 39 IV . SUMMARY In summary, we have studied the electronic transport properties of the ferromagnet/normal/ferromagnet junctionon the surface of a strong topological insulator, where agate voltage is exerted on the normal segment with a ﬁnitewidth. It is found that the conductance oscillates with the width of the normal segment and the gate voltage. Themaximum of conductance gradually decreases as the widthincreases and the minimum of conductance approaches zero.This gate-controlled conductance behaves in the same wayas the spin ﬁeld-effect transistor, but further study is neededto increase the maximum/minimum ratio of the conductance.The magnetoresistance can be very large and could also benegative owing to the anomalous transport. In addition, whenthere exists a magnetization component in the 2D plane, it isshown that only the magnetization component parallel to thejunction interface has an inﬂuence on the conductance. ACKNOWLEDGMENTS One of authors (K. H. 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PhysRevB.100.115132.pdf	PHYSICAL REVIEW B 100, 115132 (2019) Editors’ Suggestion Cooper pairing of incoherent electrons: An electron-phonon version of the Sachdev-Ye-Kitaev model Ilya Esterlis1and Jörg Schmalian2,3 1Department of Physics, Stanford University, Stanford, California 94305, USA 2Institute for Theory of Condensed Matter, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany 3Institute for Solid State Physics, Karlsruhe Institute of Technology, Karlsruhe 76021, Germany (Received 11 June 2019; revised manuscript received 14 August 2019; published 16 September 2019) We introduce and solve a model of interacting electrons and phonons that is a natural generalization of the Sachdev-Ye-Kitaev model and that becomes superconducting at low temperatures. In the normal state, twonon-Fermi-liquid ﬁxed points with distinct universal exponents emerge. At weak coupling, superconductivityprevents the onset of low-temperature quantum criticality, reminiscent of the behavior in several heavy-electron and iron-based materials. At strong coupling, pairing of highly incoherent fermions sets in deep inthe non-Fermi-liquid regime, a behavior qualitatively similar to that in underdoped cuprate superconductors.The pairing of incoherent time-reversal partners is protected by a mechanism similar to Anderson’s theoremfor disordered superconductors. The superconducting ground state is characterized by coherent quasiparticleexcitations and higher-order bound states thereof, revealing that it is no longer an ideal gas of Cooper pairs,but a strongly coupled pair ﬂuid. The normal-state incoherency primarily acts to suppress the weight ofthe superconducting coherence peak and reduce the condensation energy. Based on this, we expect strongsuperconducting ﬂuctuations, in particular at strong coupling. DOI: 10.1103/PhysRevB.100.115132 I. INTRODUCTION Superconductivity is the ultimate fate of a Fermi liquid at low temperatures [ 1–4]. A key assumption that gives rise to this Cooper instability is that the excitations of a Fermiliquid are slowly decaying Landau quasiparticles with thesame quantum numbers as free fermions. The resulting super-conducting ground state can be understood as an ideal gas ofCooper pairs. Since superconductivity occurs in many systemswhere such sharp excitations are absent, the conditions forpairing of incoherent electrons is an important open problem.The emergence of a sharp superconducting coherence peakof small weight from a broad and structureless normal-statespectrum is in fact one of the hallmarks of the cupratesuperconductors [ 5–9], where the weight of the coherence peak was shown to be strongly correlated with the superﬂuidstiffness and the condensation energy [ 9]. Key questions in this context are as follows: Can one form Cooper pairs fromcompletely incoherent fermions? What is the role of quantumcriticality for pairing? Are there sharp quasiparticles in such asuperconductor? Is the Cooper pair ﬂuid that emerges still anideal gas of pairs? To address these questions in a theoretically well- controlled way, it is highly desirable to identify a solvablemodel for nonquasiparticle superconductivity. A crucial issueis the proper interplay of non-Fermi-liquid excitations and thepairing interaction. For example, the spectral function of aFermi liquid right at the Fermi surface, A FL(ω)=ZFLδ(ω), (1) is expected to transform for a quantum-critical system to the power-law form AQC(ω)=A0\|ω\|2/Delta1−1(2)with exponent /Delta1.F o r /Delta1> 0 an evaluation of the pairing susceptibility with instantaneous pairing interaction yieldsno Cooper instability [ 10–12]. Superconductivity would then only occur if the pairing interactions exceeded a thresholdvalue. Then, a superconducting ground state would be theexception rather than the rule. However, for a number ofsystems near a fermionic quantum-critical point, rangingfrom composite-fermion metals, high-density quark matterto metals with magnetic or nematic critical points, the self-consistently determined pairing interaction inherits a singularbehavior V pair(ω)=V0\|ω\|1−4/Delta1(3) with the same exponent /Delta1[13–25]. The singular pairing in- teraction compensates for the weakened ability of non-Fermi-liquid (NFL) electrons to form Cooper pairs. One obtainsa generalized Cooper instability and superconductivity forinﬁnitesimal V 0. These considerations already demonstrate that there is a fundamental distinction between a pairing interaction that is unrelated to or directly linked to the cause of non-Fermi-liquid physics. A particularly dramatic phe-nomenon is the pairing of fully incoherent non-Fermi-liquidstates, e.g., systems with a ﬂat and structureless spectralfunction A IC(ω)=A0+··· . (4) The pairing of such fully incoherent fermions remains an open question. It corresponds to the extreme limit of /Delta1=1 2of the quantum-critical pairing problem. Signiﬁcant progress in our understanding of quantum- critical superconductivity was achieved because of advancesto formulate models that allow for sign-problem free quan-tum Monte Carlo simulations [ 26–34]. The appeal of these 2469-9950/2019/100(11)/115132(20) 115132-1 ©2019 American Physical SocietyILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) computational approaches is that they allow for a detailed analysis of the interplay between quantum criticality, pairing,and other competing states of matter. Advances have also beenmade in clearly specifying how one would sharply distinguishthe pairing state of a non-Fermi liquid from the more con-ventional one. Cooper pairing of quantum-critical fermionsand incoherent pairing should be discernible by analyzingthe frequency and temperature dependence of the dynamicalpair susceptibility [ 18,19,35], a quantity accessible through higher-order Josephson effects. An interesting approach that yields non-Fermi-liquid be- havior is provided by the Sachdev-Ye-Kitaev (SYK) model[36–40] and generalizations thereof [ 41–47]. The SYK model describes Nfermions with a random, inﬁnite-ranged interac- tion and gives rise to a critical phase where fermions have avanishing quasiparticle weight at low energies and tempera-tures. The model is exactly solvable in the limit of inﬁnitelymany fermions, N→∞ , yielding a tractable example of strong-coupling, non-Fermi-liquid behavior. The SYK modelis appropriate for situations where interactions dominate overthe kinetic energy. Thus, it could serve as a toy model forsystems that are characterized by ﬂat bands, such as cupratesuperconductors for momenta near the antinodal points orpossibly twisted bilayer graphene near the magic angle [ 48]. Formulated as a model with inﬁnite-range interactions it canalternatively be understood either as a mean-ﬁeld approachthat ignores strong spatial ﬂuctuations, akin to the dynamicalmean-ﬁeld theory of correlated electron systems [ 49,50], or as a local quantum dot system with an abundance of internaldegrees of freedom and weak interdot coupling [ 46,47]. Either point of view implies that results obtained within the SYKapproach are likely to capture important aspects of strongcorrelations in ﬁnite-dimensional systems at intermediate en-ergies. The randomness of the model, that is crucial for theformal development of the theory, may be understood assimulating real disorder or rather be an effective description ofa clean system with a rich spectrum of low-energy excitations.Such self-generated randomness is a phenomenon known toemerge in strongly frustrated classical and quantum systems[51,52]. Another appeal of this model is that its gravity dual is an asymptotic anti–de Sitter space AdS 2that can be explicitly constructed [ 40,42], an approach that is particularly promising if one wants to include ﬂuctuations that go beyond the leadinglarge- Nlimit [ 53,54]. An exciting question is whether one can construct su- perconducting versions of the SYK model and address thequestion of how pairing occurs in such a non-Fermi-liquidstate of matter. Indeed, in Ref. [ 55] Patel et al. added an additional pairing interaction to the model and demonstratedthat an instantaneous attractive coupling induces a large super-conducting gap in the spectrum. This describes the behaviorof a non-Fermi liquid toward Cooper pairing due to an inter-action that is unrelated to the initial cause of non-Fermi-liquidbehavior. In another setting, of neutral fermions coupled toa single site of an “ordinary” complex spinless fermion,odd-frequency superconductivity was recently discussed inRef. [ 56]. It was also shown recently by Wang in Ref. [ 95] that superconductivity can emerge at O(1/N) in a model similar to that discussed here (but in which superconductivity is absentin the large- Nlimit).A fundamental question is to understand systems where the interaction that causes the breakdown of the quasiparticledescription is equally responsible for pairing. Such quantum-critical pairing is then directly linked to the non-Fermi-liquidstate. As we will see, the SYK strategy allows to constructa solvable model of superconductivity near a quantum-criticalpoint. It can directly address the issue of a generalized Cooperinstability with enhanced pairing interaction balancing theweakened ability of non-Fermi-liquid states to form pairs; seeEq. ( 3). Such a model also has the potential to deepen our understanding of holographic superconductivity [ 57–59]. The SYK model offers an explicit gravity dual that will have todisplay an instability due to the onset of superconductivity. In this paper we present a model of electrons interacting with phonons via a random, inﬁnite-range coupling. It is wellestablished that singlet superconductivity can easily be de-stroyed if one breaks time-reversal symmetry. Thus, we con-sider a distribution function of real-valued electron-phononcoupling constants. This will indeed give rise to supercon-ductivity in the SYK model at leading order in an expansionfor large number of fermions and bosons. The well-knownEliashberg equations of superconductivity [ 60–62], yet with self-consistently determined electron and phonon propagator,turn out to be exact. We ﬁnd a superconducting ground state for all values of the coupling constant. Our calculation reveals that su-perconductivity emerges very differently in the weak- andstrong-coupling regimes of the model. At weak couplingT ccoincides, up to numerical prefactors, with the crossover from Fermi-liquid to non-Fermi-liquid behavior. Such behav-ior, where superconductivity preempts the ultimate quantum-critical state, is reminiscent of that observed in heavy-electron[63–66] and iron-based [ 67–70] superconductors. Thus, the superconducting state masks large parts of the non-Fermi-liquid regime. Similar behavior was recently seen in quantumMonte Carlo simulations of spin-ﬂuctuation-induced super-conductivity [ 34]. At weak coupling we also reproduce a generalized Cooper instability of the type discussed in Eq. ( 3). The nature of the superconductivity changes in the strong-coupling regime, where pairing occurs deep in the non-Fermi-liquid state and T capproaches a universal value times the bare phonon frequency. Pairing at strong coupling is a genuineexample of Cooper pairs made up of completely ill-deﬁnedindividual electrons, a phenomenon that is relevant for theunderdoped cuprate superconductors. A model for incoher-ent fermions in the cuprates due to similarly soft bosons,that also gives rise to magnetic precursors, was discussed inRefs. [ 71,72] and is similar in spirit to the behavior found here in the strong-coupling regime. The resulting phase diagramthat follows from our analysis is given in Fig. 1. The results of this paper are determined from a model of electrons that interact strongly with soft lattice vibrations.In several instances we compare the qualitative features ofour results with observations made in strongly correlatedsuperconductors such as members of the heavy-fermion, iron-based, or cuprate family. Strong evidence exists that thepairing mechanism in these systems is predominantly of elec-tronic origin. The ﬁndings of our analysis can, however, berather straightforwardly extended to models of electrons thatinteract with collective electronic excitations, such as nematic 115132-2COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) T/ω 0 ∼g2 Free fermions 11/10 SC Quantum critical: SYK-NFLImpurity-like NFLQuantum critical: ∼g−2 g FIG. 1. Schematic phase diagram of the SYK model for electron- boson coupling as function of the dimensionless coupling constantg=¯g/ω 3/2 0,w h e r e ω0is the bare phonon frequency. At lowest Tthe normal state would be a non-Fermi-liquid state with anomalous ex- ponents, similar to other SYK models. For g<1 superconductivity sets in at Tc/ω0∝g2, comparable to the temperature where quantum- critical SYK-NFL sets in. Thus, pairing occurs instead of the low- Tquantum critical state. At strong coupling a new intermediate- temperature regime opens up that is characterized by fully incoherent fermions. Coherent pairing of such incoherent fermions is still pos- sible with ﬁnite transition temperature Tc→0.112ω0. or magnetic ﬂuctuations; see also the summary section of this paper. In this more general reasoning we see the justiﬁca-tion of our statements as they pertain to the aforementionedmaterials. II. MODEL We start from the following Hamiltonian: H=−N/summationdisplay i=1/summationdisplay σ=±μc† iσciσ+1 2M/summationdisplay k=1/parenleftbig π2 k+ω2 0φ2 k/parenrightbig +√ 2 NN/summationdisplay ij,σM/summationdisplay kgij,kc† iσcjσφk, (5) with fermionic operators ciσandc† iσthat obey [ ciσ,c† jσ/prime]+= δijδσσ/primeand [ ciσ,cjσ]+=0 with spin σ=±1. In addition, we have phonons, i.e., scalar bosonic degrees of freedom φkwith canonical momentum πk, such that [ φk,πk/prime]−=iδkk/prime. Here, i,j=1...Nrefer to fermionic modes and k=1...Mto the phonon ﬁeld. Below we consider the limit N=M→∞ .W e brieﬂy comment on the behavior for arbitrary M/Nin Ap- pendix C. For simplicity, we assume particle-hole symmetry which yields μ=0 for the chemical potential. Notice, the coupling to phonons usually shifts the particle-hole symmetricpoint to a nonzero value of μ. This is a consequence of the Hartree diagram. However, this contribution vanishes in theN→∞ limit. The electron-phonon coupling constants g ij,kare real, Gaussian-distributed random variables that obey gij,k=gji,k. (6)The distribution function has zero mean and a second moment \|gij,k\|2=¯g2. The unit of ¯ gis energy3/2. Thus, even for μ=0, the model has two energy scales, the bare phonon frequencyω 0and ¯g2/3. For convenience we measure all energies and temperatures in units of ω0and use the dimensionless cou- pling constant g2=¯g2/ω3 0. Whenever it seems useful, we will reintroduce ω0in the ﬁnal results. We perform the disorder average using the replica trick [73]. Since gij,konly occurs in the random part of the inter- action we are interested in the following average: e−Srdm=e−/summationtext ijkgijkOijk, (7) where Oijk=√ 2 N/summationtext σa/integraltextβ 0dτc† iσa(τ)cjσa(τ)φka(τ).Here, a= 1,..., nstands for the replica index and the overbar denotes disorder averages, while τstands for the imaginary time in the Matsubara formalism with β=(kBT)−1the inverse tem- perature. The gij,kare for given kchosen from the Gaussian orthogonal ensemble (GOE) of random matrices [ 74]. We obtain for the disorder average e−/summationtext ijkgijkOijk\|GOE=e¯g2/summationtext ijk(O† ijk+Oijk)2. (8) There is an important distinction between the models with and without time-reversal symmetry for individual disorderconﬁgurations. If we allow for complex coupling constantswith g ij,k=g∗ ji,k, then, for given k,gij,kwould be chosen from the Gaussian unitary ensemble (GUE). Performing thedisorder average for the case of the unitary ensemble yields e−/summationtext ijkgijkOijk\|GUE=e2¯g2/summationtext ijkO† ijkOijk. (9) As can be seen from the distinct behavior of the disorder averages in Eqs. ( 9) and ( 8), the orthogonal ensemble with time-reversal symmetry contains, in addition to terms likeO † ijkOijk, that also occur in the unitary ensemble, the anoma- lous terms O† ijkO† ijkandOijkOijk. The anomalous terms can be analyzed at large Nby introducing anomalous propagators and self-energies. These terms give rise to superconductivity(see Appendix A). The subsequent derivation of the self-consistency equa- tions of the model in the large- Nlimit proceeds along the lines of other SYK models [ 36,39–43,55,56]. Assuming replica diagonal solutions, we obtain a coupled set of equations forthe fermionic and bosonic self-energies and Green’s func-tions. This derivation is summarized in Appendix A.T h e most straightforward formulation can be performed usingthe Nambu spinors c i=(ci↑,c† i↓) in the singlet channel. Then, we obtain the coupled set of equations for the self-energies: ˆ/Sigma1(τ)=¯g 2τ3ˆG(τ)τ3D(τ), (10) /Pi1(τ)=− ¯g2tr(τ3ˆG(τ)τ3ˆG(τ)), (11) with D−1(νn)=ν2 n+ω2 0−/Pi1(νn) and the fermionic Dyson equation in Nambu space ˆG(/epsilon1n)−1=i/epsilon1nτ0+μτ3−ˆ/Sigma1(/epsilon1n), where ταare the 2 ×2 Pauli matrices in Nambu space. Here, /epsilon1n=(2n+1)πTandνn=2nπTare fermionic and bosonic Matsubara frequencies, respectively. These relationscorrespond to the Eliashberg equations of electron-phononsuperconductivity, however, with the inclusion of the fully 115132-3ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) g 1 g 1g=∞ onset of SC (NFL)SYK ﬁxed point Impurity ﬁxed point (NFL)Free fermions (FL) FIG. 2. Renormalization group ﬂow that summarizes the physics of the phase diagram of Fig. 1. The free-fermion ﬁxed point is always unstable and ﬂows at low energies to the quantum-criticalSYK ﬁxed point. At strong coupling, the ﬂow is inﬂuenced for a large energy window by a new strong-coupling ﬁxed point of fully incoherent fermions. At g=∞ this impuritylike ﬁxed point is stable and governs the behavior at all scales. Superconductivity, marked by the red line, at strong coupling occurs in the vicinity of the impuritylike ﬁxed point. At weak coupling it sets in at the crossoverbetween the two ﬁxed points. renormalized boson self-energy. We use the standard parametrization for ˆ/Sigma1in Nambu space [ 60–62]: ˆ/Sigma1(/epsilon1n)=/Sigma1(/epsilon1n)τ0+/Phi1(/epsilon1n)τ1, (12) where we dropped the terms proportional to τ3andτ2due to our assumption of particle-hole symmetry and by ﬁxing thephase of the superconducting wave function, respectively. Wewill also frequently use the parametrization /Sigma1(/epsilon1 n)=i/epsilon1n[1−Z(/epsilon1n)], (13) where Z(/epsilon1n)−1contains information about the quasiparticle weight. III. NON-FERMI-LIQUID FIXED POINTS IN THE NORMAL STATE We ﬁrst solve the coupled equations in the normal state, i.e., assuming that the anomalous self-energy vanishes: /Phi1= 0. As discussed above, this corresponds to the full solutionof a model that breaks time-reversal symmetry for individualconﬁgurations of the g ij,k, chosen from the unitary ensemble. We obtain the following coupled equations for the fermionicand bosonic self-energies: /Sigma1 σ(τ)=¯g2Gσ(τ)D0(τ), (14) /Pi1(τ)=− ¯g2/summationdisplay σGσ(τ)Gσ(−τ), (15) as well as the Dyson equations G−1 σ(/epsilon1n)=i/epsilon1n+μ−/Sigma1σ(/epsilon1n) andD−1(νn)=ν2 n+ω2 0−/Pi1(νn). As sketched in Fig. 2, these coupled equations give rise to two distinct non-Fermi-liquidﬁxed points that govern the low-temperature regime for allcoupling constants and the intermediate temperature regimeat strong coupling, respectively. In what follows we willsummarize the key properties of these ﬁxed points, while adetailed derivation of our results can be found in Appendix B.A. Low-temperature behavior: Quantum-critical SYK ﬁxed point We ﬁrst discuss the solution of this coupled set of equations at low temperatures. The key ﬁnding is the following form ofthe fermionic and bosonic propagators on the Matsubara axis: G(/epsilon1 n)=1 i/epsilon1n/parenleftbig 1+c1/vextendsingle/vextendsingleg2 /epsilon1n/vextendsingle/vextendsingle2/Delta1/parenrightbig, (16) D(νn)=1 ν2n+ω2r+c3/vextendsingle/vextendsingleνn g2/vextendsingle/vextendsingle4/Delta1−1. (17) Here, ω2 r=c2(T/g2)4/Delta1−1(18) is the renormalized phonon frequency. The ciare numerical coefﬁcients of order unity. The value of the exponent /Delta1is generally conﬁned to the interval1 4</Delta1<1 2, and for our problem we ﬁnd /Delta1/revasymptequal0.420374134464041 . (19) In Appendix Bwe derive these results, demonstrate that they agree very well with our numerical solution of Eqs. ( 14) and ( 15), and give analytic and numeric expressions for the coefﬁcients ci(/Delta1). With /Delta1of Eq. ( 19) we ﬁnd c1≈1.154 700, c2≈0.561 228, and c3≈0.709 618. The ﬁndings of Eqs. ( 16)–(18) are summarized in Fig. 3, where these equations have been analytically continued fromthe imaginary to the real frequency axis. Let us discuss themain implications of these ﬁndings. The fermionic propagator(16) is similar to solutions of other SYK models and at low energies is dominated by the self-energy /Sigma1(/epsilon1 n)=−isign(/epsilon1n)c1g4/Delta1\|/epsilon1n\|1−2/Delta1, (20) with anomalous exponent /Delta1. Only the numerical value of /Delta1is different from what can be found in purely fermionic models.Notice, however, that we can vary /Delta1in the intervals ( 1 4,1 2)i f we vary the ratio M/Nof the number of bosonic and fermionic degrees of freedom (see Appendix Cand Ref. [ 45]). The bosonic propagator ( 17) is, at low frequencies, dominated by an anomalous Landau damping term, caused by the couplingto fermions and hence determined by the same anomalousexponent /Delta1. Notice that the system is critical for all values of ω 0 andg. This is a surprising result. The renormalized phonon frequency ω2 r=ω2 0−/Pi1(0) (21) in Eq. ( 18) always vanishes as T→0. One might have expected that /Pi1(0) compensates the bare mass only for one speciﬁc value of the coupling constant g, which would then determine a quantum-critical point. Instead, the system re-mains critical for all values of g, i.e., the ﬁxed point described by Eqs. ( 16) and ( 17)i ss t a b l e( s e eF i g . 2). This stability is a consequence of the diverging charge susceptibility of barefermions with G(i/epsilon1 n)−1≈i/epsilon1n. It is the non-Fermi-liquid state that lifts the degeneracy of the local Fermi liquid and protectsthe system against diverging charge ﬂuctuations. The scaling solution in Eqs. ( 16) and ( 17) is valid in a low-temperature regime T/lessorsimilarT ∗where the self-energies 115132-4COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) FIG. 3. Spectral function A(ω)=−1 πImG(ω) and imaginary part of the bosonic propagator on the real frequency axis for dimensionless coupling constant g=0.5. The phonon spectrum is shown for several temperatures, displaying the softening of the phonon mode ωr. dominate the bare fermion and boson Green’s functions. We can estimate this crossover temperature as T∗=min( Tf,Tb), (22) where Tf∼g2ω0andTb∼g−φω0, where 0 <φ=8/Delta1−2 3−4/Delta1/lessorequalslant2 for the allowed values1 4</Delta1/lessorequalslant1 2. Below we will see that the relevant exponent at large gis/Delta1=1 2, so that φ=2. Thus, the SYK-type quantum-critical regime is conﬁned totemperatures T/lessorsimilarg 2ω0at small gandT/lessorsimilarg−2ω0at large g (see Fig. 1). B. Intermediate-temperature behavior: Impuritylike non-Fermi-liquid ﬁxed point The quantum-critical regime of Eqs. ( 16) and ( 17)i s , however, not the only universal non-Fermi-liquid regime ofthis model. Once g>1 an increasingly wide intermediate- temperature window g −2<T<g2opens up. In this new temperature window we ﬁnd for the electron and phononpropagators the solution G(/epsilon1 n)=−2isign(/epsilon1n)/radicalBig /epsilon12n+/Omega12 0+\|/epsilon1n\|, (23) D(νn)=1 ν2n+ω2r, (24) with a large fermionic energy scale /Omega10=16 3πg2and small phonon energy ω2 r=/parenleftbigg3π 8/parenrightbigg2 T/g2. (25)The ﬁndings of Eqs. ( 23)–(25) are summarized in Fig. 4. Since T/lessmuch/Omega10fermions are “cold” and effectively behave as if they were quantum critical with exponent /Delta1=1 2, i.e., with impuritylike self-energy /Sigma1(/epsilon1n)=−isign(/epsilon1n)8 3πg2. (26) Noninteracting electrons with static impurities give rise to a similar self-energy and can, for a given disorder conﬁguration,be considered a Fermi liquid, essentially by deﬁnition. In ourcase, the situation is different. We have to analyze multiplephonon conﬁgurations, even for a given disorder conﬁgurationof the g ij,k. The resulting state cannot be mapped onto a free-fermion problem. Hence, the term non-Fermi liquid. Thespectral function A(ω) is semicircular with a width 2 /Omega1 0. The low-frequency spectral function is therefore frequencyindependent A(\|ω\|/lessmuch/Omega1 0)=3 8g2, (27) reﬂecting the incoherent nature of the fermion spectrum, as mentioned in Eq. ( 4) in the Introduction. On the other hand, phonons are undamped but “hot,” i.e., thermally excited sinceT/greatermuchω ronce T/greatermuchg−2. Given the large fermionic energy scale/Omega10we can neglect Landau damping terms that we ﬁnd to be∝\|ωn\|//Omega10in the intermediate energy window. While the phonons are sharp excitations with a strongly renormalized,soft frequency, the fermions are highly incoherent. Similarbehavior was discussed in the context of magnetic precur-sors in cuprates [ 71,72]. The impuritylike behavior for the fermionic self-energy is expected given the quasistatic nature FIG. 4. Spectral function and imaginary part of the bosonic propagator on the real frequency axis and for dimensionless coupling constant g=5. The phonon spectrum is shown for several temperatures, displaying the softening of the phonon mode ωr. 115132-5ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) of the phonons. Notice all these results correspond to an anomalous fermionic exponent /Delta1=1 2. This strong-coupling ﬁxed point is unstable and the system eventually crosses overto the low-temperature SYK ﬁxed point. Only for g=∞ does the impurity ﬁxed point describe the ultimate low- Tbehavior (see Fig. 2). The analytic derivation of this strong-coupling criticality is summarized in Appendix Band compared with the full numerical solution of Eqs. ( 14) and ( 15). IV . SUPERCONDUCTIVITY AND PAIRING OF NON-FERMI LIQUIDS In the previous section we analyzed the behavior of the model ( 5) in the normal state. As indicated in Fig. 1the normal state consists of three distinct regions that are separated bycrossover lines. For T>T f≈g2ω0interaction effects are weak and we have essentially free electrons. For T<Tfwe have two distinct interacting regimes. At lowest temperatureswith T<T ∗∼min( g2ω0,g−2ω0), quantum-critical behavior similar to that found in previous SYK-model calculationsoccurs, where phonons are characterized by anomalous Lan-dau damping. For strong coupling, i.e., for g>1, a new universal intermediate-temperature window g −2<T/ω0< g2opens up where strongly incoherent fermions interact with soft phonons. Next, we allow for superconducting solutions and solve the coupled equations for the normal and anomalous self-energies. On the Matsubara axis, these coupled equations are i/epsilon1 n[1−Z(/epsilon1n)]=− ¯g2T/summationdisplay n/primeD(/epsilon1n−/epsilon1n/prime)i/epsilon1n/primeZ(/epsilon1n/prime) (/epsilon1n/primeZ(/epsilon1n/prime))2+/Phi1(/epsilon1n/prime)2, /Phi1(/epsilon1n)=¯g2T/summationdisplay n/primeD(/epsilon1n−/epsilon1n/prime)/Phi1(/epsilon1n/prime) (/epsilon1n/primeZ(/epsilon1n/prime))2+/Phi1(/epsilon1n/prime)2, /Pi1(νn)=−2¯g2T/summationdisplay n/prime[G(/epsilon1n/prime+νn)G(/epsilon1n/prime) −F(/epsilon1n/prime+νn)F(/epsilon1n/prime)]. (28) If we linearize the second equation with respect to the anoma- lous self-energy /Phi1and set /Phi1=0 in the ﬁrst equation, we can determine the superconducting transition temperature. Theresult of this analysis is summarized in Fig. 5. First, our model does indeed give rise to a superconducting ground state forall values of the coupling constant g>0. For small gthe transition temperature behaves as T c(g/lessmuch1)≈0.16g2ω0. (29) Thus, while Tcat weak coupling is numerically smaller than the crossover scale T∗to the quantum-critical regime, both temperature scales have the same parametric dependence. Wewill demonstrate in the next section that indeed superconduc-tivity at g<1 occurs near the onset of the low- Tquantum- critical state. The behavior changes at strong coupling, wherewe ﬁnd that T c(g→∞ )≈0.11188 ω0 (30) approaches a ﬁnite value. In this case we form Cooper pairs deep in the non-Fermi-liquid state. We will analyze the be-havior of this superconducting ground state and demonstrate 0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5 6 7 8Tc /ω0 ~ 0.112Tc /ω0 g 0 0.002 0.004 0.006 0.008 0 0.01 0.02 0.03 0.04T*Tc /ω0 g2 FIG. 5. Superconducting transition temperature as function of the coupling constant from the numerical solution of the coupledequations in the normal state and the analysis of the eigenvalue of the pairing vertex. At weak coupling we obtain T c∝g2ω0, while the transition temperature saturates at strong coupling withT c(g→∞ )≈0.112ω0. that it is characterized by a subtle formation of bound states of Cooper pairs with the dynamical pairing ﬁeld. In Eqs. ( 29) and ( 30) we give our results in terms of the bare phonon frequency ω0and the dimensionless coupling constant g. For any ﬁnite value of gthe phonon frequency takes its bare value at sufﬁciently high temperature T∼g2. Thus, this frequency is in principle observable. However, inthe large- glimit it is unclear whether this bare frequency can be recovered experimentally. The real observable is rather therenormalized frequency for temperatures of the order of T c. Using our results for ωrand the transition temperature, it fol- lows Tc/ωr≈0.284g. In terms of the renormalized frequency, the transition temperature grows without bound [ 75]. For our subsequent discussion it is useful to express the pairing state in terms of the gap function /Delta1(/epsilon1n)=/Phi1(/epsilon1n)/Z(/epsilon1n). (31) This yields the following coupled equations that are formally equivalent to Eq. ( 28): Z(/epsilon1n)=1+¯g2T/summationdisplay n/primeD(/epsilon1n−/epsilon1n/prime)/radicalBig /epsilon12 n/prime+/Delta12(/epsilon1n/prime) ×⎡ ⎣1 Z(/epsilon1n/prime)/radicalBig /epsilon12 n/prime+/Delta12(/epsilon1n/prime)⎤ ⎦/epsilon1n/prime /epsilon1n, /Delta1(/epsilon1n)=¯g2T/summationdisplay n/primeD(/epsilon1n−/epsilon1n/prime)/radicalBig /epsilon12 n/prime+/Delta12(/epsilon1n/prime)⎡ ⎣1 Z(/epsilon1n/prime)/radicalBig /epsilon12 n/prime+/Delta12(/epsilon1n/prime)⎤ ⎦ ×/parenleftbigg /Delta1(/epsilon1n/prime)−/epsilon1n/prime /epsilon1n/Delta1(/epsilon1n)/parenrightbigg , (32) and the same equation for /Pi1(νn). These equations are distinct from the usual Eliashberg theory where the momentum inte-gration over states in a broad band replaces the terms in squarebrackets by πρ 0, where ρ0is the density of states in the normal 115132-6COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) state. In our problem we analyze systems with nondispersing bands, changing the character of the Eliashberg equations. Wewill see below that for very large g, where the interactions give rise to a signiﬁcant broadening of the spectral function, wecan replace the terms in square brackets by a spectral functionA(g→∞,ω)= 3 8g−2times π. In this limit, some known results of the conventional Eliashberg theory [ 62,76–79] can be used to obtain a better understanding of the strong-couplinglimit. The appeal of the reformulation in terms of the gap func- tion in Eq. ( 32) is that it clearly reveals the role of the zeroth bosonic Matsubara frequency for the gap equation. Supposethe bosonic propagator is dominated by the zeroth Matsubarafrequency. This is the case at strong coupling where weobtained with Eqs. ( 24) and ( 25) that D(ν m) is dominated byνm=0, a result that led to the solutions of Eq. ( 23). From Eq. ( 32) it follows that there is no contribution to the pairing problem for /epsilon1n=/epsilon1n/prime. Thus, static phonons do not affect the onset of superconductivity. The same effect is alsoresponsible for the Anderson theorem [ 80–85] where static nonmagnetic impurities will not affect the superconductingtransition temperature. Soft phonons behave somewhat sim-ilar to nonmagnetic impurities [ 86,87]. Superconductivity is then only caused by the remaining quantum ﬂuctuations ofthe phonons. How this happens and what the implications forthe spectral properties of the superconducting state are will bediscussed in the subsequent sections. A. Superconductivity at weak coupling We start our analysis of superconductivity in the weak- coupling regime g<1 and ﬁrst estimate the superconduct- ing transition temperature Tcfrom the linearized version of Eq. ( 28): /Delta1(/epsilon1n)=¯g2Tc/summationdisplay n/primeD(/epsilon1n−/epsilon1n/prime) Z(/epsilon1n/prime)/epsilon12 n/prime/parenleftbigg /Delta1(/epsilon1n/prime)−/epsilon1n/prime /epsilon1n/Delta1(/epsilon1n)/parenrightbigg , (33) where both Z(/epsilon1n) and D(νn) are determined by our normal- state solutions ( 16) and ( 17). Here we use /epsilon1nZ(/epsilon1n)=/epsilon1n+ i/Sigma1(/epsilon1n). For the phonon propagator of Eq. ( 17) we can safely neglect the ν2 nterm in the denominator. When we explicitly write out the temperature dependence in the various terms weobtain the linearized gap equation /Delta1(/epsilon1 n)=a0/summationdisplay n/prime/parenleftbigTf T/parenrightbig2/Delta1sign(/epsilon1n/prime) /parenleftbigT Tf/parenrightbig2/Delta1/vextendsingle/vextendsinglen/prime+1 2/vextendsingle/vextendsingle+/vextendsingle/vextendsinglen/prime+1 2/vextendsingle/vextendsingle1−2/Delta1 ×/Delta1(/epsilon1n/prime) /epsilon1n/prime−/Delta1(/epsilon1n) /epsilon1n m0+\|n−n/prime\|4/Delta1−1, with m0=c2 c3(2π)4/Delta1−1≈0.156 558, a0=1 2πc2 1c2≈0.212 687, andTf=1 2πc1 2/Delta1 1g2≈0.1888 g2. The temperature dependence of the gap equation only occurs in the combination T/Tf. Thus, the scale for the superconducting transition is set byT f.However, this is precisely the temperature scale where the crossover between the univeral low- Tnon-Fermi-liquid ﬁxed point and the high-temperature free-fermion behaviortakes place. This is also the reason why we included theterm ( T Tf)2/Delta1\|n/prime+1 2\|in the denominator, which corresponds to the bare fermionic propagator. Equally, the coefﬁcient m0 occurs as we have to include a ﬁnite phonon frequency at the transition temperature. If we keep all those terms, we obtainT c≈0.0821 g2. Thus, we ﬁnd that the transition temperature is about half of the crossover temperature Tf.Theg2dependence agrees with our numerical ﬁnding shown in Fig. 5.N o t surprisingly, the precise numerical coeffﬁcient in Tccannot be reliably determined as the transition temperature is right in thecrossover regime between free-fermion and quantum-criticalSYK behavior. The reason is that there appear to be correc-tions to the fermionic self-energy that are formally subleadingat low frequencies, yet modify numerical coefﬁcients. Thecorrect behavior was obtained from the full numerical solutionand yields Eq. ( 29); see also Fig. 5. This analysis demonstrates that superconductivity in the weak-coupling regime occurs at the same temperature scalewhere quantum-critical non-Fermi-liquid behavior emerges.Thus, superconductivity occurs instead of the quantum-critical regime. While parametrically the same, the numericalcoefﬁcient of the transition temperature is somewhat smallerthan the crossover scale T fbetween the region of free-fermion and quantum-critical fermion behavior. Thus, in this regime itmight be possible to observe quantum-critical scaling over aregime up to a decade in frequency or temperature. It should,however, not be possible to ﬁnd several decades of universalscaling according to Eqs. ( 16) and ( 17). Superconductivity prevents such a wide quantum-critical regime. Nevertheless, it is very instructive to compare our gap function with results from a previous analysis of the linearizedgap equation in quantum-critical systems; see, in particular,Refs. [ 15,18,20–24]. If we formulate the linearized gap equa- tion merely in terms of the universal contributions to theelectron and phonon self-energies, we obtain /Phi1(/epsilon1 n)=Tc c2 1c3/summationdisplay n/prime/Phi1(/epsilon1n/prime) \|/epsilon1n−/epsilon1n/prime\|4/Delta1−1\|/epsilon1n/prime\|2−4/Delta1, (34) where /epsilon1n=(2n+1)πTc. Here, we can see explicitly what was discussed in the Introduction, namely, that the singularpairing interaction V pair(νn)∝D(νn)∝\|νn\|1−4/Delta1compensates for the less singular fermionic propagator giving rise to a gen-eralized Cooper instability. Self-consistency equations of thistype have been discussed in the context of several scenariosfor quantum-critical pairing in metallic systems [ 13–24]. In this equation, the entire Tdependence disappears given that the two exponents in the denominator add up to unity. Thus,unless this equation is supplemented by appropriate boundaryconditions, it is not possible to determine T c(see Ref. [ 24]). This is achieved by our above solution of the gap equation for/Delta1 n. For a detailed discussion of the gap equation in the form (34), see Refs. [ 20–24]. In Fig. 6we show the spectral function in the weak- coupling regime at low temperatures that was obtained froma numerical solution of the full coupled equations on thereal frequency axis, following the approach of Refs. [ 88,89]. Our main observation is the emergence of a sharp excitation,and of several high-energy structures. We will discuss thesehigh-energy shakeoff peaks in greater detail in the discussionof the strong-coupling limit. Finally, we observe that in this 115132-7ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1g=0.5 Tc ~ 0.03A(ω) ωT=0.021 T=0.022 T=0.024 T=0.026 T=0.028 T=0.029 T=0.030 FIG. 6. Spectral function as function of temperature for g=0.5. The superconducting transition temperature is Tc=0.03ω0.W eﬁ n d higher-order bound states as well as a gap closing as function of temperature. weak-coupling regime the superconducting gap closes as the temperature increases. Overall, the analysis of the pairing problem in this weak-coupling regime closely resembles the behavior thatwas found in a number of metallic quantum-critical points[13–24]. The SYK model proposed here may serve as a starting point to go beyond the mean ﬁeld limit and investigatethe ﬂuctuation corrections by following the advances in the 1/Ncorrections of SYK-type models [ 53,54]. B. Superconductivity at strong coupling The investigation of superconductivity at strong coupling is of particular interest, as it reveals why fully incoherentfermions are able to nevertheless form a coherent supercon-ducting state. We begin again with a determination of thesuperconducting transition temperature from the linearizedgap equation. To this end, we start from Eq. ( 32) to obtain /Delta1(/epsilon1 n)=3π 8Tc/summationdisplay n/prime1 (/epsilon1n−/epsilon1n/prime)2+ω2r ×/parenleftbigg/Delta1(/epsilon1n/prime) /epsilon1n/prime−/Delta1(/epsilon1n) /epsilon1n/parenrightbigg sign(/epsilon1n/prime). (35) Here, we used the normal-state result ( 23) that has the low- frequency behavior \|/epsilon1n\|Z(/epsilon1n)≈8 3πg2. (36) The large normal-state fermionic self-energy is responsible for the fact that the coupling constant ggets canceled in the prefactor of Eq. ( 35). The only dependence on the cou- pling constant in this equation is in the renormalized phononfrequency ω r.A t Tc,ωris determined by the normal-state solution of Eq. ( 25). However, since T/greatermuchωrin the strong- coupling regime and since the zeroth Matsubara frequencydoes not contribute to superconductivity, we can simply set ωrto zero in Eq. ( 35). The linearized gap equation becomes /Delta1n=α/summationdisplay n/prime/negationslash=n/Delta1n/prime 2n/prime+1−/Delta1n 2n+1 (2n−2n/prime)2sign/parenleftbigg n/prime+1 2/parenrightbigg (37) withα=3ω2 0 8π2T2c. One easily ﬁnds that this equation has a solution for αc≈3.034 58, which yields for the transition temperature Tc=/radicalBig 3ω2 0 8π2αc. Inserting the numerical coefﬁcients yields Eq. ( 30). The transition temperature saturates as g→ ∞, in quantitative agreement with the numerical results shown in Fig. 5. This analysis also reveals the reason why pairing of fully incoherent fermions is possible. The lackof fermionic coherence, with large imaginary part of theelectronic self-energy, is caused by the coupling to almoststatic bosonic modes. However, by arguments that in the con-text of disordered superconductors give rise to the Andersontheorem, such static bosons affect the normal and anomalousself-energies /Sigma1and/Phi1, yet they cancel for the actual pairing gap/Delta1=/Phi1/Zwhich is solely affected by the much weaker quantum ﬂuctuations of the bosonic spectrum. Thus, pairingof time-reversal partners occurs even for incoherent fermions,a state that is protected by the same mechanism that makes thesuperconducting transition temperature robust against non-magnetic impurities [ 80–87]. Now that we established that superconductivity sets in at a temperature that is deep in the incoherent strong-couplingregime, we discuss the properties of this superconductingstate. We start with our numerical results for the spectralfunction and the anomalous Green’s function. In Fig. 7we show the fermionic spectral function in the superconductingstate. In contrast to the gap-closing behavior that occurs atweak coupling, we ﬁnd a ﬁlling of the gap, where the positionof the maximum is essentially unchanged with temperature.In addition, higher-order shakeoff peaks occur that becomemost evident in the strong-coupling limit. The value of thesuperconducting gap is, just like the transition temperature,independent of coupling constant and of order of the barephonon frequency ω 0. The lowest excitation of the fermions is/Delta10≈0.640 869 140 625 ω0. This yields 2/Delta10/Tc≈11.456 366 , (38) which is more than three times the BCS value 2 πe−γE≈ 3.527 754. Such large values of 2 /Delta10/Tchave been obtained in the Eliashberg theory at strong coupling and for small phononfrequencies [ 61,62]; for a recent discussion, see [ 90]. Since the spectral weight of the excited state is transferred fromenergies below the gap, we can estimate the weight of the peak as Z coh≈/integraltext/Delta1≈ω0 0Ans(ω)dω∝g−2,w h e r ew eu s e dt h e normal-state spectral function of Eq. ( 27). We will see below that this result can be obtained rigorously at large g. A very intriguing feature of the low- Tspectral function is the occurrence of a large number of shakeoff peaks at discreteenergies /Omega1 lthat are reminiscent of the satellites that emerge as one forms polaronic states due to strong electron-phononcoupling. However, in the conventional polaronic theory theseshakeoff structures exist at energies /epsilon1 0+lωrwhere /epsilon10is the bare fermion energy, ωrthe phonon frequency [ 91], and lan 115132-8COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) 0 0.01 0.02 0.03 0.04 0.05 0.06 0 1 2 3 4 5 6g=4 Tc ~ 0.11A(ω) ωT=0.085 T=0.090 T=0.095 T=0.100 T=0.105 T=0.110 T=0.115 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 2 4 6 8 1T=0A(ω) ωg=4 g=5 g=6 FIG. 7. Left panel: spectral function at strong coupling ( g=4 with Tc≈0.11ω0) for different temperatures. In distinction to the weak- coupling case we ﬁnd gap ﬁlling, rather than gap closing, and a pronounced peak-dip-peak structure. The latter is not due to the coupling to the phonon mode, which has much smaller energy. Right panel: spectral function at T=0 for different coupling constants revealing a large number of shakeoff peaks that reﬂect the bound-state formation in this limit of strongly coupled Cooper pairs. Also, the total weight of theleading coherence peak decreases with increasing coupling strength. integer. In our case, ωris much smaller than the separation of the peaks in the spectral function. In fact, such structures inthe normal and anomalous Green’s function (see Fig. 8)h a v e already been discussed in the context of strong-coupling solu-tions of the Eliashberg theory [ 77–79] and can be considered as self-trapping states of excited quasiparticles in the pairingpotential of the other electrons [ 79]. The excited quasiparticle polarizes the pairing ﬁeld, that deforms and traps it. The posi-tions of the peaks are not equidistant. Following Ref. [ 79]w e ﬁnd at large lthat the energies grow as /Omega1 l≈√ 3π 4√ 2l−1ω0. The ﬁrst 10 peaks are located at /Omega1l=pl/Delta10with pl≈ (1.,2.81,4.05,5.00,5.76,6.47,7.14,7.71,8.29,8.81). The ﬁrst peak corresponds of course to the gap /Omega11=/Delta10. These features are a clear sign of the fact that we have stronglyinteracting Cooper pairs, instead of an ideal gas of such pairs.While most of these shakeoff peaks smear out as the temper-ature increases (see left panel of Fig. 7) the ﬁrst one or two peaks should be visible and serve as potential explanation forthe observed peak-dip-hump structures seen in photoemissionspectroscopy measurements of cuprate superconductors nearthe antinodal momentum [ 5–9]. One way to verify the emergence of these shakeoff peaks due to self-trapping in the pairing ﬁeld is via the ACJosephson effect with current I J(t)=2et2 0[Re/Pi1F(eV)s i n ( 2 eV t)+Im/Pi1F(eV) cos(2 eV t)], (39) where /Pi1F(ω) is the retarded version of the Matsubara func- tion/Pi1F(νn)=T/summationtext mF†(/epsilon1m)F(/epsilon1m−νn). At low applied volt- age\|eV\|<2/Delta10the imaginary part of /Pi1Fvanishes and the Josephson current is proportional to the sinus of the phasedifference [ 92]. As the magnitude of the voltage exceeds 2/Delta1 0, an additional, phase-shifted AC Josepshon current that is proportional to cos(2 eV t)s e t si n[ 93]. The coefﬁcient is proportional to Im /Pi1F(eV) that we show in Fig. 9. Clearly, the sequence of bound states of the spectral function canbe identiﬁed in the cosine AC Josephson response. Mostinterestingly, the sign change of two consecutive bound states,visible in the anomalous propagator in Fig. 8, directly leads to an alternating sign of the phase-shifted Josephson signal.This offers a way to identify the nature of higher-energystructures in the spectral function of superconductors, suchas the bound states discussed here. For example, peaks inthe spectral function due to multiple gaps on different Fermisurface sheets would not display such a sign-changing ACJosephson signal. -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 12 14Re F( ω) ωg=4 g=5 g=6 -0.2-0.15-0.1-0.05 0 0.05 0.1 0 2 4 6 8 10 12 1-Im F( ω)/π ωg=4 g=5 g=6 FIG. 8. Real part (left panel) and imaginary part (right panel) of the anomalous propagator F(ω)a tT=0 and for different coupling strengths. Notice the alternating sign of the peaks in the imaginary part. 115132-9ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) -0.0004-0.0003-0.0002-0.0001 0 0.0001 0.0002 0.0003 0.0004 -10 -5 0 5 10g=5 T=0Im ΠF(ω) ω FIG. 9. Imaginary part of /Pi1F(ω) (deﬁned in the text) for g=5 atT=0. Im/Pi1F(ω) determines the amplitude of the phase-shifted AC Josephson current at higher voltage. The alternating sign of the peaks shown here is a direct consequence of the sign changes of consecutive peaks in the anomalous propagator, shown in Fig. 8. Thus, the AC Josephson response might serve as a tool to identify the internal structure of the Cooper pair states of a strongly coupledsuperconductor. Finally, in Fig. 10we show our results for the softening of the phonon frequency in the superconducting state. In thenormal state the phonon mode is expected to soften, ﬁrstaccording to Eq. ( 25) and below T∼ω 0g−2according to Eq. ( 18). In the normal state, ωralways vanishes for T→0. With the onset of superconductivity, the phonon frequencystill decreases with decreasing T, however, it reaches a ﬁnite value ω sc ratT=0. If we simply determine the phonon renormalization from the high-energy behavior of the spectral 0.01 0.1 0.01 0.1 Tg=4 g=5 g=6 FIG. 10. Softening of the phonon frequency in the superconduct- ing state at strong coupling. The dashed line is the normal-stateresult, continued below T c. While in the normal state the phonon frequency vanishes as T→0, it approaches the ﬁnite T=0v a l u e ωsc r=ω0 2(3π 8)2g−2, indicated by the arrows. Thus, both the electrons and the bosons are gapped in the superconducting state.function in the superconducting state, we ﬁnd ωsc r= ω0 2(3π 8)2g−2which agrees well with our numerical ﬁnding. As expected, the superconducting ground state has gappedfermion and phonon excitations which explains its coherentnature. In the strong-coupling limit one can make contact with re- sults that were obtained in the context of the usual Eliashbergtheory, where conduction electrons with a large bandwidthrequire momentum averaging [ 60–62]. This additional mo- mentum integration is not present in the SYK model, whereone is interested in the behavior of strongly interacting narrowbands. From a purely technical point of view, the effect of themomentum integration in the usual Eliashberg formalism is toreplace the term A(/epsilon1 n)=1 π1 Z(/epsilon1n)/radicalbig /epsilon12n+/Delta12(/epsilon1n), (40) that occurs in square brackets in Eq. ( 32), by the normal-state density of states of the system. We will now show that atstrong coupling the interaction-induced broadening plays arole similar to the momentum integration and we can replaceA(/epsilon1 n) by the broad spectral function of Eq. ( 27), i.e., A(/epsilon1n)≈ 3 8g−2. To demonstrate this we take the T=0 limit for Z(/epsilon1)i n Eq. ( 32): Z(/epsilon1)=1+¯g2/integraldisplayd/epsilon1/prime 2π1 (/epsilon1−/epsilon1/prime)2+/parenleftbig ωscr/parenrightbig2 ×1 Z(/epsilon1/prime)[/epsilon1/prime2+/Delta12(/epsilon1/prime)]/epsilon1/prime /epsilon1. (41) At large gtheT=0 phonon frequency is small and the sharp Lorentzian behaves as a δfunction. Using our above result for ωsc rit follows that Z(/epsilon1)=1+/parenleftbigg8g2 3π/parenrightbigg21 Z(/epsilon1)[/epsilon12+/Delta12(/epsilon1)], (42) which yields at large gthe solution Z(/epsilon1)=8g2 3π1/radicalbig /epsilon12+/Delta12(/epsilon1). (43) Thus, while Z(/epsilon1) and/Delta1(/epsilon1) depend strongly on frequency in the superconducting state, the combination that enters A(/epsilon1)i s a constant. We have veriﬁed that this result for Z(/epsilon1) agrees very well with the full numerical solution for g/greaterorsimilar4. Using Eq. ( 43), the equation for the gap function is given as /Delta1(/epsilon1n)=3π 8T/summationdisplay n/primeD(/epsilon1n−/epsilon1n/prime)/radicalBig /epsilon12 n/prime+/Delta12(/epsilon1n/prime)/parenleftbigg /Delta1(/epsilon1n/prime)−/epsilon1n/prime /epsilon1n/Delta1(/epsilon1n)/parenrightbigg . (44) While the physics we are describing is rather different, for-mally this equation is identical to the usual Eliashberg theory,yet with a dimensionless coupling constant λ= 3 8and a very soft phonon frequency. If we set this phonon frequency tozero, the solution for /Delta1(/epsilon1 n) is fully universal and independent of the coupling constant. Comparing with the numerical solu-tion, we ﬁnd that for g/greaterorsimilar4 this is indeed the case with high accuracy. Our result ( 30) can also be obtained from the well- known strong-coupling solution T c≈0.1827√ λω0by Allen 115132-10COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) -0.03-0.025-0.02-0.015-0.01-0.005 0 0.005 0.01 0.1 1δΩ/N Tg=0.25 g=0.50 g=1.00 g=4.00 g=5.00 g=6.00 -0.04-0.03-0.02-0.01 0 0 1 2 3 4 5 6T=0.005δΩ/N g FIG. 11. Condensation energy δ/Omega1/Nas a function of tempera- tureTfor several values of g. The inset shows δ/Omega1/Nas a function of gatT=0.005ω0. and Dynes [ 76] if one inserts3 8for the coupling constant. This is curious as one is very far from the applicability of thisstrong-coupling Allen-Dynes result for λ=0.375. The reason we can apply this formula is because of the extreme softeningof the phonons in our critical system. In the usual Eliashbergformalism, the frequency that enters the phonon propagatorD(ν n) is the bare, unrenormalized phonon frequency ω0. Then, the Allen-Dynes limit of Tconly becomes relevant for extremely large values of the coupling constant. Using Eq. ( 43) we can also ﬁnd a very efﬁcient way to relate the function /Delta1(ω) on the real frequency axis and the spectral function A(ω)=3 8g2Re/parenleftBigg ω/radicalbig ω2−/Delta1(ω)2/parenrightBigg . (45) Since at large gthe solution for the gap function is inde- pendent of the coupling constant, we immediately see thatthe weight of the superconducting coherence peak must scaleasg −2, a behavior that we estimated earlier based on the conservation of spectral weight. Thus, the key effect of theincoherent nature of the normal state is the reduced weight ofthe coherence peak, not its lifetime. We ﬁnish this discussion with an analysis of the condensa- tion energy as function of temperature and coupling strength.We determine the condensation energy δ/Omega1from the dif- ference of /Omega1/N=−T/summationdisplay ntr log (ˆ1−ˆG0(νn)ˆ/Sigma1(νn)) +T 2/summationdisplay mlog[1−D0(/epsilon1m)/Pi1(/epsilon1m)] −T/summationdisplay ntr(ˆG(νn)ˆ/Sigma1(νn)) (46) in the normal and superconducting state. Here, the trace is performed with respect to the degrees of freedom in Nambuspace. As shown in Fig. 11, the temperature dependence ofthe condensation energy is very different in the weak- and strong-coupling regimes with an almost linear behavior downto very low Tfor large g.I nt h i sr e g i m ew ea l s oﬁ n dac l o s e relation between the condensation energy and the quasipar-ticle weight. At weak coupling g<1 the magnitude of the condensation energy rises precipitously with increasing g.O n the other hand, for g/greaterorsimilar2 the magnitude of the condensation energy drops slowly, consistent with the power-law dropoff ofthe quasiparticle weight. Such a correlation between coherentweight in the superconducting state and condensation energyhas indeed been observed in the cuprate superconductors [ 9]. V . SUMMARY In summary, we introduced and solved a model of inter- acting electrons and phonons with random, inﬁnite-rangedcouplings that is in the class of Sachdev-Ye-Kitaev modelsand allows for an exact solution in the limit of a large numberof fermion and boson ﬂavors. The normal-state phase diagramis summarized in Fig. 1and contains adjacent to a high-energy regime of almost free fermions two distinct non-Fermi-liquidregimes. While the model starts out with a ﬁnite bare bosonmass, soft, critical bosons are generated at low temperatureswithout ﬁne tuning of the coupling strength. In addition to theusual SYK ﬁxed point, characterized by power-law correlatedbosons and fermions, we ﬁnd an inﬁnite-coupling ﬁxed pointthat has no analog in the usual SYK formalism. It describesfully incoherent fermions and extremely soft yet sharp bosons.If the random electron-phonon interaction respects time-reversal symmetry not just on the average, but for each dis-order conﬁguration, the system becomes superconducting forall values of the coupling constant. Superconductivity not onlyemerges instead of non-Fermi-liquid behavior, an observationmade in previous studies of two-dimensional systems [ 13–24] and reproduced in our weak-coupling regime, but also deep inthe strong-coupling non-Fermi-liquid phase. The pairing statethat we ﬁnd is not an ideal gas of Cooper pairs like in theBCS theory or in the theory of preformed pairs undergoingBose-Einstein condensation. Bound states of pairs explain thepeak-dip-hump feature observed in the cuprates. Despite theincoherent nature of normal-state excitations, sharp, coherentexcitations, including higher-order shakeoff peaks, emergebelow T c. The broader the fermionic states above Tc,t h e smaller the weight of the coherence peak below Tc.W ee s - tablished a direct quantitative connection between the degreeof incoherency in the normal state and the reduced weightof a coherent Bogoliubov quasiparticle in the superconduct-ing state, a correlation seen in experiments on the cupratesabout two decades ago [ 9]. The superconducting transition temperature grows monotonically with coupling strength andlevels off at a ﬁnite value that is determined by the barephonon frequency. We remark that a general upper bound onT cin conventional superconductors was recently proposed in Ref. [ 94], with the numerical value Tc/lessorsimilar¯ω/10 comparable to the maximal Tcfound here. The quantity ¯ ωis an appropriately deﬁned maximal renormalized phonon frequency. Given thatfor large gthe bare and renormalized phonon frequencies at T care dramatically different, the comparison between these two bounds is at best possible for intermediate values of thecoupling constant. In addition, the bound obtained in Ref. [ 94] 115132-11ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) is ultimately due to polaron physics at strong coupling, which is absent in the N→∞ limit of the model considered here. In contrast to Tc, which grows with the coupling constant, we ﬁnd the condensation energy is nonmonotonic and largest forintermediate-coupling strength g≈1. Thus, we expect strong ﬂuctuations for large gif one goes beyond the leading large- N limit. Indeed, the appeal of the SYK formalism is that it offersa well-deﬁned avenue to systematically improve the results(see, e.g., Refs. [ 53,54]). The fact that we ﬁnd superconductivity due to the same interactions that cause non-Fermi-liquid behavior leads toa puzzle in the holographic description of SYK. It im-plies that there must be unstable versions of nearly AdS 2 theories that yield superconductivity. Such instabilities areusually identiﬁed through the emergence of complex-valuedscaling exponents [ 57–59]. Our results suggest to analyze whether such instabilities can be related to holographicsuperconductivity. Our analysis can also be used as a starting point for lattice models of coupled strongly interacting superconductors andmay be relevant in the theory of Josephson-junction arraysthat are made up of unconventional superconductors. Finally,our analysis was performed for fermions that interact with aphonon mode, i.e., a scalar boson that couples to the fermionoperator c † iσcjσin the charge channel. It is straightforward to generalize the model and include a spin-1 boson φkthat couples to electrons via gij,kφk·/summationtext σσ/primec† iσσσσ/primedjσ/primewithσthe vector of Pauli matrices in spin space and with two fermionspecies c iσanddjσ. These two fermion species correspond to different bands or different antinodal regions on the sameband, depending on the problem under consideration. Thelarge- Nequations of this model are very similar to Eqs. ( 10) and ( 11), with τ 3→τ0. The superconducting gap function of the two fermion species then has a relative minus sign, justlike the gap function at the two antinodal points of a d-wave superconductor. The formal expression for the gap functionturns out to be the same as the one discussed in this paper.Overall, the approach presented here is a promising startingpoint to understand superconductivity in strongly coupled,incoherent materials. It justiﬁes some of the known results ofthe Eliashberg formalism, in particular, in the strong-couplinglimit, and serves as a starting point to include ﬂuctuations thatgo beyond the Eliashberg theory. Note added. Recently, we learned about an independent study of random imaginary coupling between the fermionsand bosons by Wang [ 95]. Because of the difference in the fermion-boson coupling, pairing occurs at higher order in theexpansion in 1 /N. However, our normal-state results agree with those of Ref. [ 95]. ACKNOWLEDGMENTS We are grateful to D. Bagrets, E. Berg, A. L. Chudnovskiy, J. C. Seamus Davis, S. A. Hartnoll, A. Kamenev, Y . Wang, andin particular A. V . Chubukov, S. A. Kivelson, K. Schalm, andY . Schattner for stimulating discussions. J.S. was funded bythe Gordon and Betty Moore Foundation’s EPiQS Initiativethrough Grant No. GBMF4302 while visiting the GeballeLaboratory for Advanced Materials at Stanford University.I.E. was supported by NSF Grant No. DMR-1608055 atStanford. We are grateful to Y . Wang for sharing his unpub- lished work with us. APPENDIX A: DERIV ATION OF THE SELF-CONSISTENCY EQUATIONS After performing the disorder average with the help of the replica trick, we obtain for the averaged replicated partitionfunction Zn=/integraldisplay Dnc†DcDnφe−S, (A1) where the action is of the form S=S0+Sg. (A2) The bare action is given as S0=/summationdisplay iσa/integraldisplay dτc† iσa(τ)(∂τ−μ)ciσa(τ) +/summationdisplay ia/integraldisplay dτφia(τ)/parenleftbig −∂2 τ+m0/parenrightbig φia(τ),(A3) while the disorder-average induced interaction term is Sg=−g2 4N2/summationdisplay ijk/parenleftBigg/summationdisplay aσ/integraldisplay dτc† iσa(τ)cjσa(τ)φka(τ) +/summationdisplay aσ/integraldisplay dτc† jσa(τ)ciσa(τ)φka(τ)/parenrightBigg2 , (A4) a result that can be rewritten as Sg=g2 2N2/summationdisplay abσσ/prime/integraldisplay dτdτ/primeN/summationdisplay iφia(τ)φib(τ/prime) ×⎡ ⎣N/summationdisplay ic† iσa(τ)ciσ/primeb(τ/prime)N/summationdisplay jc† jσ/primeb(τ/prime)cjσa(τ) −/parenleftBiggN/summationdisplay ic† iσa(τ)c† iσ/primeb(τ/prime)/parenrightBigg⎛ ⎝N/summationdisplay jcjσ/primeb(τ/prime)cjσa(τ)⎞ ⎠⎤ ⎦. (A5) In order to analyze the action, we introduce collective vari- ables G(τ,τ/prime) and Lagrange multiplyer ﬁelds /Sigma1(τ,τ) 1=/integraldisplay DG/productdisplay abττ/primeδ/parenleftBigg NG ba,σ/primeσ(τ/prime,τ)−/summationdisplay ic† iσa(τ)ciσ/primeb(τ/prime)/parenrightBigg =/integraldisplay DGD/Sigma1e/summationtext ab,σσ/prime/integraltext dτdτ/prime[NGba,σ/primeσ(τ/prime,τ)−/summationtext ic† iσa(τ)ciσ/primeb(τ/prime)] ×/Sigma1ab,σσ/prime(τ,τ/prime), (A6) that allow for an efﬁcient decoupling of the interaction terms. Because of the last term in Sgwe also include corresponding 115132-12COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) anomalous propagators and self-energies: 1=/integraldisplay DF/productdisplay abττ/primeδ/parenleftBigg NFba,σ/primeσ(τ/prime,τ)−/summationdisplay iciσa(τ)ciσ/primeb(τ/prime)/parenrightBigg =/integraldisplay DFD/Phi1+e/summationtext ab,σσ/prime/integraltext dτdτ/prime[NFba,σ/primeσ(τ/prime,τ)−/summationtext iciσa(τ)ciσ/primeb(τ/prime)]/Phi1+ ab,σσ/prime(τ,τ/prime), (A7) as well as 1=/integraldisplay DF+/productdisplay abττ/primeδ/parenleftBigg NF+ ba,σ/primeσ(τ/prime,τ)−/summationdisplay ic† iσa(τ)c† iσ/primeb(τ/prime)/parenrightBigg =/integraldisplay DF+D/Phi1e/summationtext ab,σσ/prime/integraltext dτdτ/prime[NF+ ba,σ/primeσ(τ/prime,τ)−/summationtext ic† iσa(τ)c† iσ/primeb(τ/prime)]/Phi1ab,σσ/prime(τ,τ/prime). (A8) Finally, for the bosonic degrees of freedom we use 1=/integraldisplay DD/productdisplay abττ/primeδ/parenleftBigg ND ab(τ,τ/prime)−/summationdisplay iφia(τ)φib(τ/prime)/parenrightBigg =/integraldisplay DDD/Pi1e1 2/summationtext ab/integraltext dτdτ/prime[ND ba(τ/prime,τ)−/summationtext iφia(τ)φib(τ/prime)]/Pi1ab(τ,τ/prime) and obtain an effective action with a sizable amount of integration variables: Zn=/integraldisplay DGD/Sigma1DF+D/Phi1+DFD/Phi1DDD/Pi1Dnc†DncDφe−S, where the collective action is now given as S=/summationdisplay iabσσ/prime/integraldisplay dτdτ/primec† iσa(τ)[(∂τ−μ)δabδσσ/primeδ(τ−τ/prime)+/Sigma1ab,σσ/prime(τ,τ/prime)]ciσ/primeb(τ/prime) +/summationdisplay iabσσ/prime/integraldisplay dτdτ/prime[c† iσa(τ)/Phi1ab,σσ/prime(τ,τ/prime)c† iσ/primeb(τ/prime)+ciσa(τ)/Phi1+ ab,σσ/prime(τ,τ/prime)ciσ/primeb(τ/prime)] (A9) +1 2/summationdisplay iab/integraldisplay dτdτ/primeφia(τ)/bracketleftbig/parenleftbig −∂2 τ+m/parenrightbig δabδ(τ−τ/prime)−/Pi1ab(τ,τ/prime)/bracketrightbig φib(τ/prime) −N/summationdisplay ab,σσ/prime/integraldisplay dτdτ/primeGba,σ/primeσ(τ/prime,τ)/Sigma1abσσ/prime(τ,τ/prime)+N 2/summationdisplay ab/integraldisplay dτdτ/primeDba(τ/prime,τ)/Pi1ab(τ,τ/prime) −N/summationdisplay ab,σσ/prime/integraldisplay dτdτ/primeFba,σ/primeσ(τ/prime,τ)/Phi1abσσ/prime(τ,τ/prime)−N/summationdisplay ab,σσ/prime/integraldisplay dτdτ/primeF+ ba,σ/primeσ(τ/prime,τ)/Phi1+ abσσ/prime(τ,τ/prime) +Ng2 2/summationdisplay abσσ/prime/integraldisplay dτdτ/prime(Gab,σσ/prime(τ,τ/prime)Gba,σ/primeσ(τ/prime,τ)−F+ ab,σσ/prime(τ,τ/prime)Fba,σ/primeσ(τ/prime,τ))Dab(τ,τ/prime). (A10) We use the Nambu spinor ψia(τ)=(ci↑a(τ),ci↓a(τ),c† i↑a(τ),c† i↓a(τ))T and rewrite the ﬁrst two lines of the previous equation as Sferm=−1 2/summationdisplay iab/integraldisplay dτdτ/primeψ† ia(τ)/parenleftBigg G−1 0,ab(τ,τ/prime)−/Sigma1ab(τ,τ/prime) /Phi1ab(τ,τ/prime) /Phi1+ ab(τ,τ/prime) −G−1 0,ba(τ/prime,τ)+/Sigma1ba(τ/prime,τ)/parenrightBigg ψib(τ/prime). Here, we introduced the bare propagator G−1 0,ab(τ,τ/prime)=−(∂τ−μ)δabσ0δ(τ−τ/prime), where σ0is the 2 ×2 identity matrix. Then, we can work with matrices in Nambu space ˆG−1 0,ab(τ,τ/prime)=/parenleftBigg G−1 0,ab(τ,τ/prime)0 0 −G−1 0,ba(τ/prime,τ)/parenrightBigg (A11) 115132-13ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) and ˆ/Sigma1ab(τ,τ/prime)=/parenleftbigg /Sigma1ab(τ,τ/prime)/Phi1ab(τ,τ/prime) /Phi1+ ab(τ,τ/prime)−/Sigma1ba(τ/prime,τ)/parenrightbigg . (A12) Here,/Sigma1ab(τ,τ/prime) and/Phi1ab(τ,τ/prime), etc., are still 2 ×2 matrices in spin space. In addition we use for the bare phonon propagator D−1 0(τ,τ/prime)=/parenleftbig −∂2 τ+m/parenrightbig δ(τ−τ/prime). (A13) We can now integrate out the fermions and bosons: S=−Ntr log/parenleftbigˆG−1 0−ˆ/Sigma1/parenrightbig +N 2tr log/parenleftbig D−1 0(τ,τ/prime)δab−/Pi1ab(τ,τ/prime)/parenrightbig −N/summationdisplay ab,σσ/prime/integraldisplay dτdτ/primeGba,σ/primeσ(τ/prime,τ)/Sigma1abσσ/prime(τ,τ/prime)+N 2/summationdisplay ab/integraldisplay dτdτ/primeDba(τ/prime,τ)/Pi1ab(τ,τ/prime) −N/summationdisplay ab,σσ/prime/integraldisplay dτdτ/primeFba,σ/primeσ(τ/prime,τ)/Phi1abσσ/prime(τ,τ/prime)−N/summationdisplay ab,σσ/prime/integraldisplay dτdτ/primeF+ ba,σ/primeσ(τ/prime,τ)/Phi1+ abσσ/prime(τ,τ/prime) +Ng2 2/summationdisplay abσσ/prime/integraldisplay dτdτ/prime(Gab,σσ/prime(τ,τ/prime)Gba,σ/primeσ(τ/prime,τ)−F+ ab,σσ/prime(τ,τ/prime)Fba,σ/primeσ(τ/prime,τ))Dab(τ,τ/prime). (A14) We assume a replica-diagonal structure such that Zn=Zn. Thus, the average is essentially an annealed one. Now, the replica structure disappears from the action that determines Z: S=−Ntr log/parenleftbigˆG−1 0−ˆ/Sigma1/parenrightbig +N 2tr log/parenleftbig D−1 0−/Pi1/parenrightbig −N/summationdisplay σσ/prime/integraldisplay dτdτ/primeGσ/primeσ(τ/prime,τ)/Sigma1σσ/prime(τ,τ/prime)+N 2/integraldisplay dτdτ/primeD(τ/prime,τ)/Pi1(τ,τ/prime) −N/summationdisplay σσ/prime/integraldisplay dτdτ/primeFσ/primeσ(τ/prime,τ)/Phi1+ σσ/prime(τ,τ/prime)−N/summationdisplay σσ/prime/integraldisplay dτdτ/primeF+ σ/primeσ(τ/prime,τ)/Phi1σσ/prime(τ,τ/prime) +Ng2 2/summationdisplay σσ/prime/integraldisplay dτdτ/prime(Gσσ/prime(τ,τ/prime)Gσ/primeσ(τ/prime,τ)−F+ σσ/prime(τ,τ/prime)Fσ/primeσ(τ/prime,τ))D(τ,τ/prime). (A15) At large Nwe can perform the saddle-point approximation and obtain the stationary equations G(τ,τ/prime)=/parenleftbig G−1 0−/Sigma1/parenrightbig−1 τ,τ/prime,D(τ,τ/prime)=/parenleftbig D−1 0−/Pi1/parenrightbig−1 τ,τ/prime,/Sigma1 σσ/prime(τ,τ/prime)=g2Gσσ/prime(τ,τ/prime)D(τ,τ/prime), /Phi1σσ/prime(τ,τ/prime)=−g2Fσσ/prime(τ/prime,τ)D(τ,τ/prime),/Pi1 (τ,τ/prime)=−g2/summationdisplay σσ/prime(Gσσ/prime(τ/prime,τ)Gσ/primeσ(τ,τ/prime)−F+ σσ/prime(τ/prime,τ)Fσσ/prime(τ,τ/prime)).(A16) If we focus on singlet pairing we have Fσσ/prime(τ)=F(τ)iσy σσ/primeandF+ σσ/prime(τ)=−F+(τ)iσy σσ/prime. Now, we can rewrite these equations in the usual fashion in 2 ×2 Nambu space with ( ci↑,c† i↓) with fermionic Green’s function ˆG(ωn)−1=iωnτ0+μτ3−ˆ/Sigma1(ωn). (A17) For the bosons we use D(νn)=1 ν2n+ω2 0+/Pi1(νn). (A18) Then, the self-energies are given as ˆ/Sigma1(τ)=g2τ3ˆG(τ)τ3D(τ), /Pi1(τ)=−g2tr(τ3ˆG(τ)τ3ˆG(−τ)). (A19) Those are the coupled equations given above. APPENDIX B: DERIV ATION OF THE NORMAL-STATE RESULTS In this Appendix we summarize the derivation of the electron and phonon propagators for the two normal-stateregimes. We start our analysis with the behavior in thelow-temperature quantum-critical SYK regime and continuewith the intermediate-temperature impuritylike behavior at strong coupling. In addition to the analytic derivation, we alsopresent results of the full numerical solution that conﬁrm ouranalytic ﬁndings in detail. 1. Quantum-critical SYK ﬁxed point: Derivation of Eqs. ( 16)–(18) and numerical results We start our analysis at T=0 and make the following ansatz for the fermionic self-energy: /Sigma1(ω)=−iλsign(ω)\|ω\|1−2/Delta1. (B1) To preserve causality, the coefﬁcient λhas to be positive. This is most transparent if one analytically continues thisansatz to the real frequency axis. Here, causality requires thatthe retarded self-energy has a negative imaginary part. WithIm/Sigma1 R(/epsilon1)=− sin (π/Delta1)λ\|/epsilon1\|ηfollows λ> 0f o r0 </Delta1< 1. 115132-14COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) As long as /Delta1> 0, the low-energy fermionic Green’s func- tion is dominated by this singular self-energy G(ω)≈−1 /Sigma1(ω)=−i λsign(ω)\|ω\|−(1−2/Delta1). (B2) On the real axis this corresponds to the spectral func- tionA(/epsilon1)=−1 πImGR(/epsilon1)=sin (π/Delta1)\|/epsilon1\|−(1−2/Delta1) λπ. The bosonic self- energy is /Pi1(ω)=−2¯g2/integraldisplaydω 2πG(ω)G(ω+/Omega1) =2g2 λ2/integraldisplaydω 2πsign(ω)sign(ω+/Omega1) \|ω\|1−2/Delta1\|ω+/Omega1\|1−2/Delta1. (B3) This bosonic self-energy for /Omega1→0 is ultraviolet divergent if/Delta1>1 4, i.e., /Pi1(0)∝/Lambda14/Delta1−1with upper cutoff /Lambda1.T h i s divergence can be avoided if we include the full propagatorand write /Pi1(0)=−2¯g 2/integraldisplaydω 2πG(ω)2=−2g2/integraldisplaydω 2π/parenleftbigg1 iω−/Sigma1(ω)/parenrightbigg2 =2/Delta1−1 2/Delta12sinπ 2/Delta1¯g2λ−1 2/Delta1. (B4) Next, we analyze the dynamic part δ/Pi1(ω)=/Pi1(ω)−/Pi1(0). It is easiest to do this by ﬁrst Fourier transforming the propaga-tor to imaginary time: G(τ)=−/Gamma1(2/Delta1)s i n (π/Delta1) πλsign(τ) \|τ\|2/Delta1, (B5) such that the Fourier transform of the phonon self-energy is given as /Pi1(τ)=2g2(/Gamma1(2/Delta1) sin (π/Delta1) πλ)21 \|τ\|4/Delta1, which yields δ/Pi1(ω)=2/integraldisplay∞ 0/Pi1(τ)[cos(ωτ)−1]dτ =−g2 λ2C/Delta1\|ω\|4/Delta1−1 with coefﬁcient C/Delta1=−8 cos(π/Delta1)s i n3(π/Delta1)/Gamma1(2/Delta1)2/Gamma1(1− 4/Delta1)/π2. Now, we can analyze the bosonic propagator D(ω).We can neglect the bare /Omega12term against the singular bosonic frequency dependence due to the Landau damping. In ad-dition, we can only expect a power-law solution if indeedω 2 0−/Pi1(0)=0. If this is the case, it follows for the bosonic propagator D(ω)≈−1 δ/Pi1(ω)=λ2 ¯g2C/Delta1\|/Omega1\|1−4/Delta1. (B6) The Fourier transform is D(τ)=λ2 g2B/Delta11 \|τ\|2−4/Delta1with B/Delta1= π(1−4/Delta1)c o s( 2 π/Delta1) 8/Gamma1(2/Delta1)2cos (π/Delta1) sin3(π/Delta1)which gives for the self-energy /Sigma1(τ)=−λB/Delta1/Gamma1(2/Delta1)s i n (π/Delta1) πsign(τ) \|τ\|2−2/Delta1. (B7) Fourier transforming this back to the Matsubara frequency axis ﬁnally yields /Sigma1(ω)=−iλA/Delta1sign(ω)\|ω\|1−2/Delta1(B8)with A/Delta1=4/Delta1−1 2(2/Delta1−1)[sec(2π/Delta1)−1]. (B9) Notice, for the Fourier transforms to be well deﬁned, it must hold that1 4</Delta1<1 2. In order to have a self-consistent solution it must of course hold that A/Delta1=1. This determines the exponent /Delta1given in Eq. ( 19). Interestingly, the coefﬁcient λremains undetermined by this procedure. However, our solution still relies on the assumption that the renormalizedphonon frequency vanishes at T=0. We have not yet de- termined when this is the case. We can now always use thefreedom and determine λsuch that ω r(T=0)=0, which yields the condition λ=c1g4/Delta1(B10) in order to generate a critical state for all values of the coupling constant. The numerical coefﬁcient is c1=/parenleftbigg2/Delta1−1 2/Delta12sinπ 2/Delta1/parenrightbigg2/Delta1 . (B11) With/Delta1from Eq. ( 19) follows c1≈0.832 260 211 4. There is one caveat in this argumentation. The relationship be-tween /Pi1(0) and λthat we used to determine the coefﬁcient c 1relied on the simultaneous knowledge of the low- and high-frequency behaviors of the fermionic propagator [seeEq. ( B4)]. To address this, we used an expression that interpo- lates between the two known limits. Such an approach givesthe correct qualitative behavior. Yet, the numerical value for c 1 cannot be reliably determined by such a procedure. To avoid this uncertainty we determine this coefﬁcient from the fullnumerical solution of the problem that conﬁrms our scaling re-sults in detail; see below. This yields c 1≈1.154 700 5 which is somewhat larger than the above estimate. In what followswe will use this result for c 1. Notice, all other coefﬁcients of our analysis, such as C/Delta1orA/Delta1, can be uniquely determined by the universal low-energy behavior and do not have to bedetermined numerically. These results for the phonon frequency allow us to de- termine the coefﬁcient of the dynamic part of the bosonpropagator δ/Pi1(ω)=−c 3/vextendsingle/vextendsingle/vextendsingle/vextendsingleω g2/vextendsingle/vextendsingle/vextendsingle/vextendsingle4/Delta1−1 , (B12) where c3=C/Delta1 c2 1. With /Delta1from Eq. ( 19) and the numerically determined value of c1follows c3≈0.709 618. This analysis further allows us to determine the temper- ature dependence of the phonon frequency, which is deter-mined via ω 2 r(T)=ω2 0−/Pi1(T), (B13) where /Pi1(T)=−2g2T∞/summationdisplay n=−∞G(ωn)2. (B14) 115132-15ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) 1 0.01 0.1 1T=0.002~ \|ωn /g2\|2Δ - 1-g2 Im G( ωn) ωn /g2g=0.25 g=0.50 g=1.00 1 10 100 0.01 0.1~ \|νn /g2\|1 - 4 ΔD(νn) νn /g2 FIG. 12. Numerical solution of the fermionic (left panel) and bosonic (right panel) propagators on the imaginary axis in comparison with the analytic solution given in Eqs. ( 16)a n d( 17). At low but ﬁnite temperatures we use for the propagator our result G(ωn)=1 iωn+iλsign(ωn)\|ωn\|1−2/Delta1. (B15) Using the Poisson summation formula for fermionic Matsub- ara sums gives for the phonon frequency ω2 r(T)=ω2 0−2g2∞/summationdisplay k=−∞(−1)k/integraldisplay∞ 0dω πcos(βωk) (ω+λω1−2/Delta1)2. (B16) The k=0 term corresponds to the T=0 result. Thus, it exactly cancels the bare frequency. The remaining frequencyintegrals are ultraviolet convergent even without the barefermionic propagator included, which ﬁnally gives ω 2 r(T)=4g2 λ2∞/summationdisplay k=1(−1)k+1/integraldisplay∞ 0dω πcos(βωk) ω2−4/Delta1 =c2/parenleftbiggT g2/parenrightbigg4/Delta1−1 , (B17) with numerical coefﬁcient c2=4 πc2 1sin(2π/Delta1)/Gamma1(4/Delta1−1)(1−22−4/Delta1)ζ(4/Delta1−1), (B18) where c1was determined numerically [see text below Eq. ( B11)]. With /Delta1from Eq. ( 19) follows c2≈0.561 228. We ﬁnish this discussion with a comparison of our analyt- ical results with the numerical solutions of the coupled equa-tions in the normal state. In Fig. 12we compare the fermionic and bosonic propagators as function of the imaginary Mat-subara frequency with our analytic solution of Eqs. ( 16) and (17). Finally, in Fig. 13we demonstrate that the phonon frequency agrees with our analytical result ( 18). In particular,this demonstrates that indeed the phonon frequency is soft for all values of g. 2. Impuritylike ﬁxed point: Derivation of Eqs. ( 23)–(25)a n d numerical results Let us assume that the boson propagator behaves as in Eq. ( 24) with renormalized boson frequency ωr, but without additional dynamic renormalizations due to Landau damping.We further assume T/greatermuchω rsomething we need to check below to be consistent. Then it follows that the self-energyis dominated by the lowest bosonic Matsubara frequency, i.e.,bosons behave as classical impurities: /Sigma1(ω n)=g2T/summationdisplay n/primeD(ωn−ωn/prime)G(ωn/prime) =g2T ω2r1 iωn−/Sigma1(ωn). (B19) 0.01 0.1 1 0.001 0.01 0.1 1 10~(T/g2)4 Δ-1ωr2 T/g2g=0.25 g=0.50 g=1.00 FIG. 13. Temperature dependence of the renormalized phonon frequency for several values of the coupling constant gdetermined from the numerical solution of the coupled equations and compared with the analytical expression of Eq. ( 18). 115132-16COOPER PAIRING OF INCOHERENT ELECTRONS: AN … PHYSICAL REVIEW B 100, 115132 (2019) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.001 0.01 0.1 1 10 100T=0.05-g2 Im G( ωn) ωn /g2g=4.0 g=5.0 g=6.0 g=7.0 0 1 2 3 4 5 0 1 2 3 41/D(νn) νn2 FIG. 14. Numerical solution of the fermionic (left panel) and bosonic (right panel) propagators on the imaginary axis in comparison with the analytic solution given in Eqs. ( 23)a n d( 24). This suggests to introduce the energy scale /Omega10=2/radicalBig g2T ω2r which yields /Sigma1(ωn)=−isign(ωn)1 2/parenleftbig/radicalBig ω2n+/Omega12 0−\|ωn\|/parenrightbig (B20) as the solution of the above quadratic equation. For \|ωn\|/lessmuch /Omega10holds/Sigma1(ωn)=−isign(ωn)/Omega10 2while for large frequencies it follows that /Sigma1(ωn)=−isign(ωn)/Omega12 0 4\|ω\|. For the fermionic Green’s function follows then Eq. ( 23). Next, we determine the bosonic self-energy for this problem: /Pi1(ωn)=−2g2T/summationdisplay n/primeG(ωn/prime)G(ωn/prime+ωn). (B21) Let us ﬁrst determine the zero-frequency part /Pi1(0)=−2g2T/summationdisplay n/primeG(ωn/prime)2 =8g2T/summationdisplay n/prime1 /parenleftbig/radicalBig ω2n+/Omega12 0+\|ωn\|/parenrightbig2. (B22) Let us try to determine /Omega10from the condition that the boson frequency goes to zero as Tis extrapolated to T=0. Formally we can just require that /Pi1(0)=ω2 0atT=0.Then, we have /Pi1(0)=8g2/integraldisplay∞ 0dω π1 /parenleftbig/radicalBig ω2+/Omega12 0+ω/parenrightbig2 =16g2 3π/Omega1 0. (B23) This yields /Omega10=16 3πg2. Combining both expressions that we obtained for /Omega10can be used to determine the phonon frequency and gives rise to our result ( 25). The assumption of classical bosons was T/greatermuchωrwhich implies T/greatermuchg−2, consistent in the strong-coupling limit. In addition, as longasT/lessmuchg2we also have T/lessmuch/Omega10and the evaluation of the above fermionic Matsubara sum in the zero-temperature limit is justiﬁed. The frequency dependence of the self-energy forω/lessmuchg 2is then /Sigma1(ωn)=−isign(ωn)8 3πg2. For consistency we have to check that we can indeed ignore the frequency dependence of the bosonic self-energy. Theonly scale that enters the fermionic propagator is /Omega1 0.I nt h e relevant limit T/lessmuch/Omega10the fermions are essentially at zero temperature, where δ/Pi1(ω)=2/integraldisplay∞ 0dτ/Pi1(τ)[cos(ωτ)−1] =−4g2/integraldisplay∞ 0dτG(τ)G(−τ)[cos(ωτ)−1]. The Fourier transform of the fermionic propagator can be determined analytically and expressed in terms of modiﬁedBessel functions and the modiﬁed Struve function. For our 0.001 0.01 0.1 1 0.001 0.01 0.1 1 10(3π/8)2 T/g2ωr2 T/g2g=4.0 g=5.0 g=6.0 g=7.0 FIG. 15. Temperature dependence of the renormalized phonon frequency for several values of the coupling constant gdetermined from the numerical solution of the coupled equations and compared with the analytical expression of Eq. ( 25). 115132-17ILYA ESTERLIS AND JÖRG SCHMALIAN PHYSICAL REVIEW B 100, 115132 (2019) purposes it sufﬁces to analyze the short- and long-time limits: G(τ)=sign(τ)×/braceleftBigg1 /Omega10\|τ\|if\|τ\|/greatermuch/Omega1−1 0, 1 2−2 3π\|τ\|/Omega10if\|τ\|/lessmuch/Omega1−1 0,(B24) which yields δ/Pi1(ω)≈−\|ω\| /Omega10. This Landau damping term is negligible compared to ω2 n forT/greatermuchg−2. Thus, we can indeed approximate the bosonic propagator by Eq. ( 24). We ﬁnish this discussion with a comparison of our analyt- ical results with the numerical solutions of the coupled equa-tions in the normal state. In Fig. 14we compare the fermionic and bosonic propagators as function of the imaginary Matsub-ara frequency with our analytic solution of Eqs. ( 23) and ( 24). Finally, in Fig. 15we demonstrate that the phonon frequency agrees with our analytical result ( 25).APPENDIX C: ON THE ROLE OF DISTINCT FERMION AND BOSON MODES The ratio m=M/Nchanges the relative importance of the fermion and boson self-energies. Changing the ratio mof the number of boson and fermion ﬂavors does not affect the over-all behavior of Eqs. ( 10) and ( 17). The exponent /Delta1changes continuously from /Delta1(m→0)→1/2t o/Delta1(m→∞ )→1/4. The phonon softening still formally follows Eq. ( 18), yet the temperature scale below which this power-law softeningoccurs depends sensitively on the relative importance of thephonon and electron renormalizations. If phonon self-energy effects dominate ( m/lessmuch1) we ﬁnd ω 2 r=m 4π2log 2( T/g2)1−m 2, i.e., phonons are soft below a very large temperature T∗∼ g2/m1−m 2. 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PhysRevB.87.214102.pdf	PHYSICAL REVIEW B 87, 214102 (2013) Lattice constants and cohesive energies of alkali, alkaline-earth, and transition metals: Random phase approximation and density functional theory results Laurids Schimka,Ren´e Gaudoin, Ji ˇr´ı Klime ˇs, Martijn Marsman, and Georg Kresse Faculty of Physics, Universit ¨at Wien, and Center for Computational Materials Science, Sensengasse 8 /12, A-1090 Wien, Austria (Received 29 December 2012; published 13 June 2013) We present lattice constants and cohesive energies of alkali, alkaline earth, and transition metals using the correlation energy evaluated within the adiabatic-connection ﬂuctuation-dissipation (ACFD) framework in therandom phase approximation (RPA) and compare our ﬁndings to results obtained with the meta-GGA functionalrevTPSS and the gradient corrected PBE (Perdew-Burke-Ernzerhof) functional and the PBEsol functional (PBEreparametrized for solids), as well as a van der Waals (vdW) corrected functional optB88-vdW. Generally, theRPA reduces the mean absolute error in the lattice constants by about a factor 2 compared to the other functionals.Atomization energies are also on par with the PBE functional, and about a factor 2 better than with the otherfunctionals. The study conﬁrms that the RPA describes all bonding situations equally well including van derWaals, covalent, and metallic bonding. DOI: 10.1103/PhysRevB.87.214102 PACS number(s): 31 .15.A− I. INTRODUCTION Benchmarking theoretical methods against experimental data is common practice widely adopted within the densityfunctional theory 1,2(DFT) community, in particular, when introducing new functionals. However, remarkably few sys-tematic results are available for the transition-metal series.Although these elements are partly covered in Refs. 3and4, a concise study covering all elements using modern densityfunctionals is so far not available. The tests presented hereclose this gap and include transition metals of the 3 d,4d, and 5dseries, as well as alkali and alkaline earth and coinage metals. We show results for lattice constants and cohesiveenergies obtained with several different approximations tothe exchange correlation energy: the widely applied gener-alized gradient approximation (GGA) of Perdew, Burke, andErnzerhof 5(PBE) and its reparametrized version for solids PBEsol,6the recently published meta-GGA revTPSS (Tao- Perdew-Staroverov-Scuseria),7,8the optB88-vdW (Ref. 9) functional which uses the nonlocal correlation functionalof Dion et al. 10with the exchange functional ﬁtted to reproduce weak interactions in the gas phase, and ﬁnallythe random phase approximation in the adiabatic-connectionﬂuctuation-dissipation (ACFD) framework. 11–13The local density approximation has not been included in this studysince it underestimates the lattice constants of 3 dmetals signiﬁcantly, strongly overestimates the atomization energies,and does not yield accurate results for magnetic transitionmetals. Furthermore, all experimental values are corrected for the effect of zero-point vibrational energies, which were calculatedat the DFT level applying a force constant approach. 14Where necessary, the lattice constants were also extrapolated fromavailable ﬁnite-temperature data to 0 K. Independent of theactual results for the here investigated functionals, these datawill serve as a useful reference for future work. One reason why the 3 d,4d, and 5 dseries have been rarely considered as a benchmark might be that most of the semilocalfunctionals are not particularly good in describing the differentbonding situations encountered in these series. Althoughthe alkali and alkaline earth metals are usually considered to be prototypical metals that can be well described evenusing second-order perturbation theory and a free-electron-gasreference, 15a sizable bonding contribution also stems from van der Waals bonding, in particular, for the soft alkali metals.This contribution originates from the semicore pand to a lesser extent semicore sstates, and can modify the lattice constants by up to 2%–3%. 16As the dﬁlling increases along the series, the bonding changes from s- andp-like bonding in alkali and alkaline earth metals to bonding dominated by thedelectrons. It is commonly assumed that dbonding includes a sizable fraction of covalent bonding with bonding linearcombinations of dstates below the Fermi level and antibonding linear combinations above the Fermi level. 17 Other challenging materials are the metallic 3 delements which exhibit a fairly small band width and are expectedto show strong ﬂuctuations in the ground state. Speciﬁcally,ferromagnetic Fe, Co, and Ni are known to be difﬁcult fordensity functionals, as exempliﬁed by the many attempts to include correlations beyond the mean ﬁeld. 18–21In summary, transition metals include contributions from different kinds ofbonding: van der Waals-type bonding between closed semicoresandpshells, van der Waals bonding from closed semicore d states towards the end of the series (Cu, Ag, and in particularAu), free-electron-like metallic bonding for alkali, alkalineearth metals and the coinage metals, as well as covalent d bonding. As we will also see in this work, the general shortcoming of semilocal functionals in describing bonding between closedshells results in large errors towards the beginning and theend of the series: the “classical” PBE functional is indeedunsatisfactory. With the advent of new functionals that includethe kinetic energy density, the situation has slightly improved, as we will conﬁrm here for the meta-GGA functional revTPSS. However, our main focus is on the random phase approxima-tion, which should capture all important bonding contributionsaccurately. 22–24As a side line, we will also show results for ferromagnetic Fe, Co, and Ni and thereby assess the accuracyof the random phase approximation for magnetic elements. 214102-1 1098-0121/2013/87(21)/214102(8) ©2013 American Physical SocietySCHIMKA, GAUDOIN, KLIME ˇS, MARSMAN, AND KRESSE PHYSICAL REVIEW B 87, 214102 (2013) TABLE I. PAW potentials used in this work. The second column indicates the states treated as valence states. The local potential was generated by replacing the all-electron potential by a soft potential within the cutoff radius rloc(a.u.), which is provided in the “ rloc” column. The number of partial waves and projectors for different angular momentum numbers lis speciﬁed in columns 4–7. The energy cutoff Ecutspeciﬁes the VA S P “default” cutoff in eV for DFT calculations usually guaranteeing convergence of absolute energies to few meV per electron. This cutoff is determined by the largest wave vector of the spherical Bessel functions that are used when theall electron partial wave is replaced by a soft pseudopartial wave. Valence rloc spdfE cut(eV) K3 s3p4s 1.2 3 2 1 249 Ca 3 s3p4s 1.2 3 2 1 281 Sc 3 s3p4s3d 1.2 3 2 1 1 285 Ti 3 s3p4s3d 1.2 3 2 1 1 286 V3 s3p4s3d 1.1 3 2 1 1 323 Fe 3 s3p4s3d 1.0 4 3 1 1 364 Co 3 s3p4s3d 1.1 4 3 1 1 364 Ni 3 s3p4s3d 1.1 4 3 1 1 413 Cu 3 d4s 1.5 2 2 2 1 417 Rb 4 s4p5s 1.8 3 2 2 1 221 Sr 4 s4p5s 1.8 3 2 2 1 225 Y4 s4p5s4d 1.8 3 2 2 1 229 Zr 4 s4p5s4d 1.6 3 2 2 1 282 Nb 4 s4p5s4d 1.6 3 2 2 1 286 Mo 4 s4p5s4d 1.6 3 2 2 1 312 Tc 4 s4p5s4d 1.6 3 2 2 1 318 Ru 4 s4p5s4d 1.6 3 2 2 1 321 Rh 4 s4p5s4d 1.6 3 2 2 1 320 Pd 4 d5s 1.6 2 2 2 2 251 Ag 4 d5s 1.4 2 2 2 2 250 Cs 5 s5p6s 1.8 2 2 2 2 198 Ba 5 s5p6s 1.8 2 2 2 2 237 Hf 5 s5p6s5d 1.6 3 2 2 1 283 Ta 5 s5p6s5d 1.6 3 2 2 1 286 W5 s5p6s5d 1.6 3 2 2 1 317 Re 5 s5p6s5d 1.6 3 2 2 1 317 Os 5 s5p6s5d 1.6 3 2 2 1 320 Ir 5 s5p6s5d 1.6 3 2 2 1 320 Pt 5 s5p6s5d 1.6 3 2 2 1 324 Au 5 d6s 1.6 2 2 2 1 300 II. TECHNICAL DETAILS All calculations were performed using the Vienna ab initio simulation package ( V ASP ),26,27applying the projector- augmented wave (PAW) potentials28,29listed in Table I. The potentials correspond to the GW potentials distributedwith the V ASP package. These potentials are slightly more accurate than the standard V ASP potentials, although the DFT lattice constants agree within 0.15% with the lattice constantsobtained using other PAW potentials with a similar set ofvalence orbitals. 30Furthermore, we note that freezing the semicore states by placing them into the core increases theDFT lattice constants by up to 0.5% for the early transitionm e t a l s( S c ,T i ,V ,Y ,N b ,M o ) . Details for the construction of the pseudopartial waves are discussed in Ref. 31. This speciﬁc construction results in fairlysoft potentials requiring only modest plane wave cutoffs, as listed in Table I. Since the density functional theory calcula- tions are comparatively cheap, the energy cutoff has been setto 800 eV for the revTPSS and optB88-vdW calculations andto 1000 eV for the PBE and PBEsol calculations. For PBE andPBEsol, results at 1000 and 800 eV are identical, guaranteeingthat all reported results are fully converged with respect to theplane wave basis set for semilocal functionals. At 800 eV , themore costly revTPSS and optB88-vdW calculations are alsoessentially exact, as conﬁrmed by repeating some calculationsat a higher plane wave cutoff. For the signiﬁcantly moreexpensive RPA calculations, we have set the energy cutoffto 1.5 times the “default” energy cutoff listed in Table I.A l l RPA calculations were performed using the PBE orbitals andPBE one-electron energies (RPA@PBE), and no attempts toobtain self-consistent results were made. The Brillouin zone (BZ) was sampled by 15 ×15×15k points for the bulk calculations with the density functionals.For the RPA, the BZ sampling was increased from 6 ×6×6 over 8 ×8×8t o1 0 ×10×10kpoints where k-point convergence was observed, except for Fe, where the k-point set had to be increased to 16 ×16×16kpoints. For the hcp structures we used the ideal c/a ratio and a 10 ×10×10 k-point grid. Overall, we found that this setup ensures an accuracy of about 0.25% in the lattice constants (better than1% in the volume). The equilibrium volumes were determinedusing a seven-point ﬁt to a Birch-Murnaghan equation of state,where the volume in the calculations was varied by ±15%. The bulk modulus is not reported here. Because of noisein the RPA data, the changes in the bulk moduli from oneto the next k-point set sometimes exceed 10% (Cu, Ag, Au), although changes of 5% are more common. Furthermore, thebulk moduli show nothing unexpected and follow the usualtrend: if the volume is overestimated, the bulk modulus tendsto be underestimated and vice versa . For the calculations of the atoms, a 14 ×15×16˚A 3cell was used for the density functional theory calculations. Theground states of the atoms were calculated by seeking thelowest-energy conﬁguration allowing for spin polarization andbreaking of the spherical symmetry, but disregarding spin-orbitcoupling. All symmetry-broken ground-state conﬁgurationswere characterized by orbital occupancies of 1 (occupied)or 0 (unoccupied) only. In some cases, we started the DFTcalculations from different starting points, to guarantee thatthe lowest-energy conﬁguration was correctly determined. Inmost (but not all) cases, the DFT ground-state conﬁgurationagrees with the experimental observations (see Sec. III B). For the RPA, three calculations at three different volumes wereperformed (7 ×8×9˚A 3,8×9×10˚A3, and 9 ×10×11˚A3) and the values were extrapolated to the isolated atom limit. Theexact exchange energy (evaluated also using PBE orbitals) wasevaluated for supercells of 10 ×11×12˚A 3,1 1 ×12×13˚A3, and 12 ×13×14˚A3and also extrapolated to the isolated atom limit (for alkali and alkali earth metals even larger unit cellswere used). Depending on the convergence corrections, theexact exchange energy can show spurious ﬁnite-size errors ofthe order 1 /volume before this residual correction, whereas the correlation energy shows residual ﬁnite-size errors of theorder 1 /volume squared before correction. 22Except for Ti, the present RPA calculations for atoms are usually based 214102-2LATTICE CONSTANTS AND COHESIVE ENERGIES OF ... PHYSICAL REVIEW B 87, 214102 (2013) on the PBE ground-state orbitals, disregarding that the true RPA atomic ground state could correspond to a differentatomic conﬁguration. For titanium, the RPA calculationswere initiated from a DFT-PBE calculation with the atomicconﬁguration 3 d 24s2(total spin moment 2 μB) compatible to experiment. This lowered the atomic RPA energy signiﬁcantly. The zero-point vibration corrections to the lattice con- stants and atomization energies were calculated from densityfunctional theory using the same procedure as outlined inRef. 32. The vibrational frequencies were calculated using a2 ×2×2 supercell of the conventional (cubic) unit cell. The BZ sampling was done with 8 ×8×8kpoints. A similar energy cutoff as in the RPA calculations was chosen, yieldingessentially converged results in the phonon frequencies (errorsare below 1% upon further increase of the energy cutoff).For elements with a hexagonal close-packed structure (hcp),the vibrational contributions were estimated using a moreconvenient face centered cubic (fcc) structure. In tests, wefound that applying the fcc instead of the hcp structure yieldsidentical results up to the third digit in the energy (eV). III. RESULTS AND DISCUSSION A. Equilibrium volumes Figure 1shows the relative error of the equilibrium volumes with respect to the experimental values extrapolatedwhere necessary to 0 K. All metals were considered in theirnonmagnetic states, except for Fe, Co, and Ni, which wereconsidered in the ferromagnetic bcc (Fe) and ferromagneticfcc (Co and Ni) structures. We have subtracted the effectof the zero-point vibrational energies from the experimentaldata. In the tables and ﬁgures, the elements are ordered byascending atomic number. Cr and Mn have been excludedfrom this study. Mn exhibits a complicated antiferrimagneticstructure and would require signiﬁcant efforts in the RPA. 36 Cr is antiferromagnetic, with a very strong change of the localmagnetic moment around the equilibrium volume (at leastin density functional theory). In the RPA, this would requireus to scan the energy landscape as a function of the volume -505Error in Volume [%] revTPSS RPA PBEsol K CaSc TiV FeCo NiCu RbSr YZr NbMo TcRu RhPd AgCs BaHf TaW ReOs IrPt Au-505Error in Volume [%] PBE RPA optB88-vdW FIG. 1. (Color online) Relative error in volume compared to experimental data from which the effects of the zero-point energy as well as the thermal effects (where necessary) were subtracted.and magnetic moment, an effort beyond the scope of this study. For the other ferromagnetic metals (bcc Fe, fcc Co,and fcc Ni), we have simply used the PBE density functionaltheory orbitals and one-electron eigenvalues to determine theexact exchange energy, as well as the correlation energy inthe random phase approximation. For the magnetic materials,the magnetic moment is therefore ﬁxed to the values deter-mined in the ground-state DFT-PBE calculations. We will start our discussion with the well-established PBE functional. The PBE functional (blue circles) works fairly wellacross the series, with the errors being noticeably larger for thealkali metals and the coinage metals (Cu, Ag, and Au). Theerrors in the volumes are particularly sizable for Rb (4.9%), Ag(6.3%), Cs (5.9%), and Au (6.9%). It is also well establishedthat PBE yields fairly accurate 3 dlattice constants, but the lattice constants for the 4 dand 5delements are systematically overestimated. What is particularly unsatisfactory is theincrease of the lattice constants along the series with increasingd-band ﬁlling. We found a similar increase also for other pure density functionals, for instance, PBEsol (see Fig. 1) or AM05. Furthermore, a similar behavior is quite generallyfound as the atomic number increases. 3The origin for this is not fully understood. Most likely, the conventional densityfunctionals fail to describe important electronic correlationsbetween neighboring sites. Along this line of arguments, thelarge error for K, Rb, and Cs, as well as the coinage metals Cu,Ag, and Au, is then related to the neglect of correlation effectsbetween closed semicore sandpstates for alkali metals, and between the almost ﬁlled dshells for Cu, Pd, Ag, Pt, and Au. The RPA (red diamonds) yields much improved results. Most notable is the decrease of the lattice constants forthe alkali metals as well as coinage metals. We relate this to thefact that the random phase approximation can account for thecorrelation between closed shells (van der Waals bonding), 22 allowing for an accurate description of the correlation betweenthe semicore sandpstates for K, Rb, and Cs and the ﬁlled dshells for Cu, Ag, and Au. A slight tendency towards too large lattice constants with increasing d-band ﬁlling prevails in the RPA, but this might be also related to some systematicdeﬁciency of the PAW data sets for correlated calculations.Speciﬁcally, we note that the RPA results are sensitive tothe description of the unoccupied states, and although weinclude partial waves for fstates for most elements, we have not made attempts to include gpartial waves as well. Visual inspection of the scattering properties, however, indicates thatthegscattering properties are very accurately described by the local potential. The more likely explanation for the increasein the lattice constant is some residual self-interaction errorwithin the dshell, which will necessarily increase with d-band ﬁlling. The results for the 3 dmetals are also satisfactory for the RPA. For Co and Ni, we ﬁnd a tendency towards too largevolumes, but with volume errors of 3%–4% the errors remainacceptably small. For Fe, the RPA energy-volume curve is verypeculiar, with a double-well structure shown in Fig. 2. We note that this behavior becomes more apparent when 20 ×20× 20kpoints are used, and the corresponding calculations were performed using otherwise less stringent convergence criteriathan for the other calculations. The ﬁrst minimum is deeper,and corresponds very well with the experimentally observed 214102-3SCHIMKA, GAUDOIN, KLIME ˇS, MARSMAN, AND KRESSE PHYSICAL REVIEW B 87, 214102 (2013) 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 Volume [Å³]-42.9-42.8-42.7-42.6-42.5-42.4Energy [eV]fcc structure hcp structure bcc structure FIG. 2. (Color online) Energy-volume curve for nonmagnetic fcc, nonmagnetic hcp, and ferromagnetic bcc iron as obtained for RPA. lattice constant, whereas the second minimum occurs at larger volumes. At this volume, the DFT ground-state calculationsthat we use to determine the orbitals and occupancies showas p i no f2 . 5 μ B, close to a Hund’s rule ferromagnet which we believe to be related to the existence of the secondminimum. In passing, we note that no such minimum wasobserved for the other ferromagnetic transition metals Coand Ni. We also determined the energy difference betweenmagnetic bcc Fe and nonmagnetic hcp and fcc Fe usingthe RPA and found values of /Delta1E bcc-hcp =−130 meV and /Delta1E bcc-fcc =−180 meV . This conﬁrms that the magnetic phase is more stable than competing nonmagnetic phases.Furthermore, the energy differences are slightly larger thanfor the PBE functional ( /Delta1E bcc-hcp =−83 meV , /Delta1E bcc-fcc = −153 meV). We predict a transition pressure of 32 GPa for a pressure-induced transition from ferromagnetic bcc tononmagnetic hcp, but note that the lowest-energy hcp structuremight possess an antiferromagnetic or antiferrimagnetic spinorder possibly lowering its energy. 33Hence, we do not consider the overestimation of the transition pressure compared toexperiment to be an issue. In summary, the RPA yields excellent results with a quite clear tendency towards, on average, 1%–2% too large volumes,as we already observed in our previous studies for s- and p-bonded systems. Compared to PBE, the improvements are also clearly visible in the statistical mean relative error (MRE)and mean absolute relative error (MARE) summarized inTable II. The MRE and MARE drop by almost a factor 2 from PBE to RPA, and the small MARE is particularly noteworthy. The revTPSS results are shown as green squares in the ﬁrst panel of Fig. 1. We will ﬁrst concentrate on the 4 dand 5dmet- als. Disregarding Rb and Cs, it is astounding how closely therevTPSS curve follows the RPA. Furthermore, revTPSS yieldsabout 2% smaller volumes than RPA improving the agreementwith experiment and, most notably, revTPSS exhibits also nosigniﬁcant slope with increasing d-band ﬁlling. Considering the design principles of revTPSS, we can understand thisbehavior. The revTPSS functional uses the kinetic energydensity to distinguish spatial regions where the electron densitystems from a single orbital only from those where the densityis made up by the sum of the density of many (one-electron)orbitals. When the density is made up by many orbitals, the functional behaves very similar to the PBEsol functional,whereas in spatial regions where the density originates fromone orbital only, a functional form is used that largely removesself-interaction errors. This allows the revTPSS functional torecover the exchange and correlation energy of the hydrogenatom almost exactly. As the dband becomes ﬁlled, revTPSS hence gradually switches from a “one-electron” description toa “many-electron” description, becoming gradually identicalto the PBEsol functional at roughly half ﬁlling (compareFig. 1). Below half ﬁlling, the self-interaction free form increases the lattice constants compared to PBEsol, counteract-ing the slope in the PBE and PBEsol functionals. This explainsthe very respectable performance of revTPSS for 4 dand 5d metals. For the alkali metals, however, large errors prevail, andthese are certainly related to the neglect of correlation effectsfrom the ﬁlled semicore states that semilocal functionals cannot handle by construction. 16 The 3dmetals are another issue. Errors for Fe are unfor- tunately fairly large, and the volume almost drops to PBEsolvalues (see also Ref. 8). In this case, the functional is too “PBEsol” like, as the dshell is almost entirely ﬁlled. What was beneﬁcial for the ﬁlled 4 dand 5dshells has clearly a negative impact on the magnetic 3 dmetals. This also signiﬁcantly increases the MARE over that for the RPA, resulting in, overall,an only modest improvement over PBE. Finally, we turn to the optB88-vdW functional 9which uses the vdW-DF correlation functional of Dion et al.10and a modiﬁed B88 exchange functional.37The results for some of the materials have been published before,38namely, the alkali and alkaline earth metals as well as the late dmetals (Cu, Rh, Pd, and Ag). It was observed that this functional gives similarresults as PBE for the late dmetals, while too small equilibrium volumes were obtained for the alkali and alkaline earth metals.This follows the trend already observed here for the otherGGA-based functionals (PBE and PBEsol). However, theslope in the difference to experiment from left to right is evenlarger than for PBE and PBEsol. We checked that the reasonfor the increase in the slope is the vdW correlation functional:replacing the vdW correlation by the PBE correlation recoversthe behavior for other semilocal functionals. We concludethat the vdW functional most likely overestimates dispersioncontributions with particularly sizable errors for the soft alkalimetals (and to a lesser extent alkaline earth metals). B. Atomization energies The accurate prediction of atomization energies is a difﬁcult challenge to density functional theory methods, as well asmany-electron methods. For transition metals, the situation isparticularly severe since transition metals are “strongly” corre-lated with many almost isoenergetic low-energy conﬁgurationsin the Hartree-Fock approximation. Since the true many-electron wave function for the ground state is then a mixtureof many Slater determinants, often multiconﬁguration methodsare needed to make accurate predictions for transition-metalatoms and their compounds. Despite the multiconﬁgurationalmany-electron wave function, density functionals very oftenyield reasonably accurate answers for the atomization energyof transition-metal solids. 4 214102-4LATTICE CONSTANTS AND COHESIVE ENERGIES OF ... PHYSICAL REVIEW B 87, 214102 (2013) TABLE II. Theoretical equilibrium volumes for PBE, revTPSS, and RPA. The columns marked with % report the relative error with respect to experimental data corrected for zero-point vibrational effects. These are shown in the last column, while the uncorrected results are given in parentheses. If not otherwise stated, corrected experimental values are from this work. All elements were considered in the nonmagnetic state, except for Fe, Co, and Ni (ferromagnetic). PBE % PBEsol % revTPSS % optB88-vdW % RPA % Experiment K bcc 73.51 3.8 70.70 −0.1 75.05 6.0 68.67 −3.0 70.02 −1.1 70.79a(71.32)a Ca fcc 42.15 −1.7 40.53 −5.5 41.97 −2.1 40.31 −6.0 42.74 −0.3 42.88a(43.09)a Sc hcp 24.63 −0.4 23.58 −4.6 24.24 −1.9 23.95 −3.1 25.28 2.3 24.72 (25.00)b Ti hcp 17.39 −0.7 16.71 −4.6 16.99 −3.0 17.06 −2.6 18.00 2.8 17.52 (17.66)b V bcc 13.45 −3.2 12.93 −6.2 13.05 −5.0 13.28 −3.7 13.96 1.1 13.78 (13.88)b Fe bcc 11.36 −2.2 10.83 −6.7 10.92 −6.0 11.23 −3.3 11.67 0.5 11.61a(11.71)a Co fcc 10.86 −0.7 10.40 −4.9 10.50 −4.0 10.81 −1.1 11.33 3.6 10.94 (11.08)b Ni fcc 10.78 −0.1 10.34 −4.2 10.39 −3.7 10.83 0.4 11.07 2.6 10.79 (10.94)b Cu fcc 11.97 3.0 11.31 −2.7 11.19 −3.7 11.88 2.3 11.48 −1.2 11.62c(11.69)c Rb bcc 90.99 4.9 86.22 −0.6 93.00 7.2 84.79 −2.2 85.11 −1.9 86.73a(87.10)a Sr fcc 54.53 −1.0 51.71 −6.1 53.95 −2.1 51.79 −6.0 55.11 0.0 55.09a(55.31)a Y hcp 32.84 0.0 31.29 −4.7 32.19 −2.0 31.90 −2.8 32.95 0.4 32.83 (33.18)b Zr hcp 23.37 1.1 22.45 −2.9 22.85 −1.2 23.07 −0.2 23.25 0.5 23.12 (23.27)b Nb bcc 18.14 1.5 17.56 −1.7 17.71 −0.9 18.08 1.2 18.14 1.5 17.87a(17.90)a Mo bcc 15.79 1.9 15.35 −0.9 15.44 −0.4 15.81 2.0 15.70 1.3 15.49a(15.54)a Tc hcp 14.45 2.0 14.02 −1.0 14.06 −0.7 14.48 2.2 14.43 1.8 14.17 (14.30)d Ru hcp 13.77 2.5 13.33 −0.8 13.36 −0.6 13.81 2.7 13.67 1.7 13.44 (13.55)b Rh fcc 14.06 3.0 13.51 −1.0 13.52 −1.0 14.09 3.2 13.83 1.3 13.65c(13.70)c Pd fcc 15.21 4.5 14.43 −0.9 14.46 −0.7 15.18 4.3 14.77 1.5 14.56c(14.61)c Ag fcc 17.81 6.3 16.61 −0.9 16.62 −0.8 17.57 4.8 17.01 1.5 16.76c(16.84)c Cs bcc 116.65 5.9 108.16 −1.8 119.51 8.5 102.69 −6.8 110.96 0.8 110.12e(110.45)e Ba bcc 63.17 1.0 58.01 −7.3 60.99 −2.5 58.95 −5.8 62.59 0.0 62.58a(62.76)a Hf hcp 22.43 1.4 21.50 −2.8 21.62 −2.3 22.00 −0.6 22.22 0.5 22.12 (22.25)d Ta bcc 18.25 1.7 17.61 −1.9 17.66 −1.6 18.09 0.8 18.09 0.7 17.95a(17.98)a W bcc 16.11 2.1 15.68 −0.6 15.67 −0.7 16.10 2.0 15.79 0.1 15.78a(15.81)a Re hcp 14.92 2.1 14.52 −0.7 14.51 −0.7 14.97 2.4 14.69 0.5 14.61 (14.71)b Os hcp 14.29 2.8 13.91 0.1 13.88 −0.1 14.37 3.4 14.01 0.8 13.90 (13.99)b Ir fcc 14.47 2.9 14.02 −0.3 13.98 −0.6 14.59 3.8 14.30 1.7 14.06a(14.15)a Pt fcc 15.63 4.4 15.02 0.3 14.99 0.0 15.74 5.1 15.24 1.7 14.98a(15.01)a Au fcc 17.92 6.9 16.95 1.1 16.95 1.1 17.94 7.1 17.28 3.1 16.76a(16.79)a MRE 1.9 −2.5 −0.8 0.0 1.0 MARE 2.5 2.6 2.4 3.2 1.3 aReference 3. bReference 35. cReference 32. dReference 34. eReference 4. Here, we deﬁne the atomization energy (or cohesive energy) of a material M with Natoms in a unit cell as EAtm(M) =1 N/braceleftBigg/summationdisplay atomsE(X) −E(M)/bracerightBigg . (1) E(M) is the total energy of the solid and E(X) denotes the corresponding energy of the constituent atoms. Withthis deﬁnition, positive errors correspond to an overbinding,whereas negative errors correspond to underbinding. It isclear from Fig. 3that PBE performs quite reasonably for the atomization energies. It is also quite remarkable that,with few exceptions, the atomic electronic conﬁgurationspredicted by PBE agree with experiments (compare Table III). These exceptions are Ti, V , and W. For Ti, the exactexchange energy (EXX) and the RPA atomic energy are considerably lower when the experimental conﬁguration ischosen as starting point for the RPA calculations, which canbe achieved by ﬁxing the magnetic moment in the precedingDFT calculations to 2 μ B(triplet). Therefore, RPA and EXX predict an atomic electronic conﬁguration in agreement withexperiment, whereas PBE fails to predict the correct atomicground state of Ti. For V , PBE, and RPA, as well as EXX,all predict the wrong atomic electronic conﬁguration, andfor W we where unable to stabilize the experimental 5 d 46s2 conﬁguration, as our electronic-structure code always ended up in the 5 d56s1conﬁguration. For PBE, errors are always close to zero and hardly ever exceed 0.5 eV . The RPA inherits this good overall performance 214102-5SCHIMKA, GAUDOIN, KLIME ˇS, MARSMAN, AND KRESSE PHYSICAL REVIEW B 87, 214102 (2013) K CaSc TiV FeCo NiCu RbSr YZr NbMo TcRu RhPd AgCs BaHf TaW ReOs IrPt Au-0.500.511.52Error in Energy [eV]PBE revTPSS RPA PBEsol FIG. 3. (Color online) Error of the theoretic atomization energy in eV compared to experiment. Positive value means that the atomization energy is overestimated by a given functional. from PBE, in particular for the mean absolute error (MAE). The statistical errors compiled in Table IIIindicate that RPA shows the usual underestimation of the binding energies alsoobserved for other elements in the periodic table. 24It has been demonstrated that this error is signiﬁcantly reduced by addingthe second-order screened exchange (SOSEX) contribution, 25 but the corresponding calculations are presently not possiblefor metallic systems. Furthermore, outliners with particularlylarge errors are Ni, Nb, and Pt. These three atoms arecharacterized by PBE one-electron band gaps that are smallerthan 0.15 eV in the atomic ground state. This small band gapcauses a single strong transition in the excitation spectrum,shifting the RPA atomic energies to too negative values. Themagnitude of this small one-electron band gap depends on theDFT functional, and increases by a factor 1.5 for the revTPSSfunctional. When the revTPSS functional is used to generatethe orbitals and one-electron energies for the RPA calculations,the atomization energies of Ni, Nb, and Pt agree slightly betterwith experiment (Ni 4.25, Nb 7.15, Pt 5.14), whereas theatomization energies of other elements hardly change by morethan 50 meV . The improvement is, however, modest, and thesmall changes suggest that the atomization energies are notvery sensitive to the choice of the initial DFT functional. Remarkably, the RPA as well as all density functionals exhibit minima in the binding curve for close to half ﬁlling(Nb and W) and for an entirely ﬁlled dband (Ni, Pd, Ag, Pt, and Au). Note that the dband contains more electrons for equivalent 4 delements than 5 delements (e.g., Mo versus W) since the 6 sshell is pulled down by relativistic effects increasing its occupancy in the 5 dseries. Hence, the minimum for half ﬁlling occurs slightly earlier in the 4 delements (Nb) than in the 5 delements (W). One possible reason for this systematic variation in the atomization energies and the agreement between RPA andPBE is that the interpolation of the correlation energy betweenthe nonmagnetic and fully spin-polarized case (known fromquantum Monte Carlo simulations) is based on the RPAcorrelation energy for a partially spin-polarized electron gas. 39 Possibly, this underestimates the correlation energy of atomswith partially spin-polarized shells, with accurate results only obtained at full spin polarization and zero spin polarization.Finally, we observe that the 3 dmetals behave differently than the 4dand 5dmetals. Speciﬁcally, the PBE overbinds all 3dmetals compared to experiment (recall the too small PBE lattice constants), whereas the RPA yields excellent agreementwith experiment, with a slight tendency towards too smallbinding energies as for the 4 dand 5dseries. The performance of revTPSS and PBEsol for the atom- ization energies is somewhat disappointing. The mean errorincreases from −0.07 eV for PBE to 0.41 for revTPSS. We note that a similar behavior has already been observed for othersolids in our recent work. 8As opposed to semiconductors and insulators where the revTPSS atomization energies arevery good, the revTPSS atomization energies of metals aregenerally close to PBEsol values and signiﬁcantly too large.We can understand this along the same line of argumentsalready discussed above: in metals, and speciﬁcally in tran-sition metals with a largely ﬁlled dshell, the total charge density is the sum of several one-electron orbitals. In this case,the revTPSS functional behaves very similar to the PBEsolfunctional. Although this was clearly beneﬁcial for the latticeconstants, it undesirably increases the atomization energies tothat of the PBEsol functional. We ﬁnally note that PBEsol andrevTPSS seem to be accurate for some elements, for instance,the alkali metals, Au, and Ag, as well as Pd and Pt, elementsthat are often included in benchmark data sets. This highlightsthat too limited test sets might be misleading in judging theoverall quality of a functional. IV . SUMMARY AND CONCLUSIONS The here considered test set of 30 alkali, alkaline earth, transition, and coinage metals turns out to be a signiﬁcantchallenge to present day semilocal density functionals. Thedeﬁciencies of semilocal functionals can be summarized asfollows. (i) Using the PBE functional, the 3 dlattice constants are slightly too small, and the 4 dand 5dlattice constants are too large. (ii) Furthermore, the difference to the experimentalvolumes shows an upwards slope with increasing d-band ﬁlling for 4 dand 5dmetals. Since other semilocal functionals, for instance PBEsol, reduced the volume by roughly the samemagnitude for all metals, none of the semilocal functionalsgives a satisfactory description. The meta-GGA functional revTPSS yields essentially identical results as the PBEsol functional from half ﬁllingon, but improves signiﬁcantly upon the PBEsol functional forless than half ﬁlling. By rectifying issue (ii), the revTPSSfunctional yields the best lattice constants for 4 dand 5dmetals, with sizable errors only prevailing for the alkali metals. Thevolume error for the alkali metals using semilocal functionalsis related to the neglect of dispersion forces related to thesemicore sandpstates, an issue that has already been partly resolved in Ref. 16using pairwise corrections. Unfortunately, issue (i), the underestimation of the lattice constants of 3 d metals, remains unaddressed by the revTPSS functional. As previously observed, the optB88-vdW functional seems to overestimate the dispersion forces in the alkali and alkalineearth metals and gives lattice constants that are too shortat the beginning of the series. Furthermore, towards the 214102-6LATTICE CONSTANTS AND COHESIVE ENERGIES OF ... PHYSICAL REVIEW B 87, 214102 (2013) TABLE III. Theoretical atomization energies in eV for PBE, PBEsol, revTPSS, and RPA. The atomic electron conﬁguration considered as starting point for the RPA calculations is reported in the second column. The lowest atomic electronic conﬁguration of Ti for the DFT functionals is 3 d34s1. The electronic conﬁgurations of V and W also differ from experiment (experiment: V 3 d34s2,W5d46s2). The “Error” columns report the absolute error with respect to experiment. The last column reports the experimental values corrected for phonon zero-pointvibrational effects (uncorrected values are in parentheses). The estimated error bar for the atomization energies of the DFT and RPA calculations (technical convergence with respect to all parameters) is ±20 meV and ±50 meV , respectively. Conﬁguration PBE Error PBEsol Error revTPSS Error RPA Error Experiment K4 s 0.87 −0.07 0.93 −0.01 0.97 0.03 0.86 −0.08 0.94a(0.93)a Ca 4 s21.91 0.05 2.12 0.26 2.06 0.20 1.51 −0.35 1.86 (1.84)b Sc 3 d4s24.11 0.18 4.54 0.61 4.30 0.37 3.75 −0.18 3.93 (3.90)b Ti 3 d24s25.27 0.39 5.83 0.95 5.58 0.70 4.98 0.10 4.88 (4.85)b V3 d44s15.37 0.03 5.97 0.63 5.80 0.46 5.24 −0.10 5.34 (5.31)b Fe 3 d64s24.89 0.59 5.66 1.36 5.24 0.94 4.20 −0.10 4.30 (4.28)b Co 3 d74s24.98 0.56 5.79 1.37 5.38 0.96 4.52 0.10 4.42 (4.39)b Ni 3 d84s24.75 0.27 5.46 0.98 5.24 0.76 4.00 −0.48 4.48 (4.44)b Cu 3 d104s13.50 −0.02 4.06 0.54 4.16 0.64 3.33 −0.19 3.52c(3.49)c Rb 5 s 0.77 −0.09 0.84 −0.02 0.86 0.00 0.83 −0.03 0.86 (0.85)b Sr 5 s21.61 −0.12 1.81 0.08 1.81 0.08 1.50 −0.23 1.73 (1.72)b Y4 d5s24.16 −0.23 4.60 0.21 4.46 0.07 4.04 −0.35 4.39 (4.37)b Zr 4 d25s26.19 −0.08 6.84 0.57 6.53 0.26 6.14 −0.13 6.27 (6.25)b Nb 4 d45s16.96 −0.63 7.67 0.08 7.51 −0.08 6.97 −0.62 7.59 (7.57)b Mo 4 d55s16.28 −0.56 7.09 0.25 6.91 0.07 6.60 −0.24 6.84 (6.82)b Tc 4 d55s26.88 0.00 7.82 0.94 7.46 0.58 6.94 0.06 6.88 (6.85)b Ru 4 d75s16.70 −0.07 7.75 0.98 7.20 0.43 6.61 −0.16 6.77 (6.74)b Rh 4 d85s15.70 −0.08 6.65 0.87 6.28 0.50 5.44 −0.34 5.78c(5.75)c Pd 4 d103.76 −0.18 4.50 0.56 4.46 0.52 3.44 −0.50 3.94c(3.91)c Ag 4 d105s12.52 −0.46 3.09 0.11 3.05 0.07 2.63 −0.35 2.98c(2.96)c Cs 6 s 0.72 −0.09 0.78 −0.03 0.83 0.02 0.81 0.00 0.81 (0.80)b Ba 6 s21.88 −0.03 2.12 0.21 2.09 0.18 1.75 −0.16 1.91 (1.90)b Hf 5 d26s26.42 −0.04 7.08 0.62 6.95 0.49 6.20 −0.26 6.46 (6.44)b Ta 5 d36s28.11 −0.01 8.93 0.81 8.83 0.71 7.88 −0.24 8.12 (8.10)b W5 d56s18.39 −0.53 9.17 0.25 9.17 0.25 8.53 −0.39 8.92 (8.90)b Re 5 d56s27.80 −0.25 8.77 0.72 8.69 0.64 7.76 −0.29 8.05 (8.03)b Os 5 d66s28.34 0.14 9.42 1.22 9.19 0.99 8.19 −0.01 8.20 (8.17)b Ir 5 d76s27.31 0.34 8.35 1.38 8.09 1.12 7.03 0.06 6.97 (6.94)b Pt 5 d96s15.51 −0.35 6.38 0.52 6.27 0.41 5.06 −0.80 5.86 (5.84)b Au 5 d106s13.05 −0.78 3.74 −0.09 3.67 −0.16 3.12 −0.71 3.83 (3.81) ME −0.07 0.56 0.41 −0.23 MAE 0.24 0.57 0.42 0.25 aReference 4. bReference 34. cReference 32. right of the periodic table, the functional essentially recovers the PBE results. Hence, the trend (ii) to overestimate theequilibrium volumes with increasing d-band ﬁlling is even more pronounced for optB88-vdW than for either PBE orPBEsol, a point that needs to be addressed in the future inorder to make vdW functionals fully competitive. The RPA results for lattice constants of 4 dand 5dmetals are remarkably close to the revTPSS results, but since theRPA includes dispersion forces, outliers (errors for the alkalimetals) are not present, supporting our claim that the RPAaccounts equally well for all bonding situations. Furthermore,the RPA results for the 3 dmetals are in good agreement with experiment and do not show the peculiar underestimation ofthe volume observed for standard density functionals.For the atomization energies, we ﬁnd that the RPA and PBE perform roughly equally, although the RPA trend towardstoo weak binding, as for other solids and molecules, prevails.PBEsol and revTPSS atomization energies are very similar andsigniﬁcantly too large compared to experiment. Overall, RPAoffers a well-balanced description with mean absolute errorsbeing smaller than for the density functionals considered here. ACKNOWLEDGMENT Funding by the Austrian Science Fund (FWF) within the special research program ViCoM (grant F41) is gratefullyacknowledged. 214102-7SCHIMKA, GAUDOIN, KLIME ˇS, MARSMAN, AND KRESSE PHYSICAL REVIEW B 87, 214102 (2013) laurids.schimka@univie.ac.at 1P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964). 2W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 3P. Haas, F. Tran, and P. 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PhysRevB.79.205432.pdf	Low-energy theory and RKKY interaction for interacting quantum wires with Rashba spin-orbit coupling Andreas Schulz,1Alessandro De Martino,2Philip Ingenhoven,1,3and Reinhold Egger1 1Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany 2Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, D-50937 Köln, Germany 3Institute of Fundamental Sciences, Massey University, Private Bag 11 222 Palmerston North, New Zealand /H20849Received 25 February 2009; revised manuscript received 5 May 2009; published 29 May 2009 /H20850 We present the effective low-energy theory for interacting one-dimensional /H208491D/H20850quantum wires subject to Rashba spin-orbit coupling. Under a one-loop renormalization-group scheme including all allowed interactionprocesses for not too weak Rashba coupling, we show that electron-electron backscattering is an irrelevantperturbation. Therefore no gap arises and electronic transport is described by a modiﬁed Luttinger liquidtheory. As an application of the theory, we discuss the Ruderman-Kittel-Kasuya-Yosida /H20849RKKY /H20850interaction between two magnetic impurities. Interactions are shown to induce a slower power-law decay of the RKKYrange function than the usual 1D noninteracting cos /H208492k Fx/H20850//H20841x/H20841law. Moreover, in the noninteracting Rashba wire, the spin-orbit coupling causes a twisted /H20849anisotropic /H20850range function with several different spatial oscil- lation periods. In the interacting case we show that one special oscillation period leads to the slowest decay andtherefore dominates the Ruderman-Kittel-Kasuya-Yosida interaction for large separation. DOI: 10.1103/PhysRevB.79.205432 PACS number /H20849s/H20850: 73.63./H11002b, 71.10.Pm, 85.75. /H11002d I. INTRODUCTION Spin transport in one-dimensional /H208491D/H20850quantum wires continues to be a topic of much interest in solid-state andnanoscale physics offering interesting fundamental questionsas well as technological applications. 1Of particular interest to this ﬁeld is the spintronic ﬁeld effect transistor /H20849spin-FET /H20850 proposal by Datta and Das,2where a gate-tunable Rashba spin-orbit interaction /H20849SOI /H20850of strength/H9251allows for a purely electrical manipulation of the spin-dependent current. Whilethe Rashba SOI arises from a structural inversionasymmetry 3–5of the two-dimensional electron gas /H208492DEG /H20850in semiconductor devices hosting the quantum wire, additionalsources for SOI can be present. In particular, for bulk inver- sion asymmetric materials, the Dresselhaus SOI /H20849of strength /H9252/H20850should also be taken into account. By tuning the Rashba SOI /H20849via gate voltages /H20850to the special point /H9251=/H9252the spin- FET was predicted to show a remarkable insensitivity todisorder, 6see also Ref. 7. On top of these two, additional /H20849though generally weaker /H20850contributions may arise from the electric conﬁnement ﬁelds forming the quantum wire. In thispaper, we focus on the case of Rashba SOI and disregard allother SOI terms. This limit can be realized experimentally byapplying sufﬁciently strong backgate voltages, 8–11which cre- ate a large interfacial electric ﬁeld and hence a signiﬁcantand tunable Rashba SOI coupling /H9251. The model studied be- low may also be relevant to 1D electron surface states ofself-assembled gold chains. 12 The noninteracting theory of such a “Rashba quantum wire” has been discussed in the literature13–18and is summa- rized in Sec. IIbelow. We here discuss electron-electron /H20849e-e/H20850 interaction effects in the 1D limit where only the lowest/H20849spinful /H20850band is occupied. The bandstructure at low-energy scales is then characterized by two velocities 19 vA,B=vF/H208491/H11006/H9254/H20850,/H9254/H20849/H9251/H20850/H11008/H92514. /H208491/H20850 These reduce to a single Fermi velocity vFin the absence of Rashba SOI /H20849/H9254=0 for/H9251=0/H20850but they will be different for /H9251/HS110050 reﬂecting the broken spin SU/H208492/H20850invariance in a spin- orbit-coupled system. The small- /H9251dependence /H9254/H11008/H92514fol- lows for the model below and has also been reported in Ref.20. Therefore the velocity splitting /H20851Eq. /H208491/H20850/H20852is typically weak. While a similar velocity splitting also happens in a magnetic Zeeman ﬁeld /H20849without SOI /H20850, 21the underlying phys- ics is different since time-reversal symmetry is not broken bySOI. The bandstructure of a single-channel quantum wire with Rashba SOI should be obtained by taking into account atleast the lowest two /H20849spinful /H20850subbands since a restriction to the lowest subband alone would eliminate spinrelaxation. 15,22,23The problem in this truncated Hilbert space can be readily diagonalized and yields two pairs of energybands. When describing a single-channel quantum wire onethen keeps only the lower pair of these energy bands. Wemention in passing that band-structure effects in the presenceof both Rashba SOI and magnetic ﬁelds have also beenstudied. 24–28In addition, the possibility of a spatial modula- tion of the Rashba coupling was discussed29but such phe- nomena will not be further considered here. Finally disordereffects were addressed in Refs. 30and31. For 1D quantum wires it is well known that the inclusion of e-e interactions leads to a breakdown of Fermi liquidtheory and often implies Luttinger liquid /H20849LL/H20850behavior. This non-Fermi liquid state of matter has a number of interestingfeatures, including the phenomenon of spin-chargeseparation. 32Motivated mainly by the question of how the Rashba spin precession and Datta-Das oscillations in spin-dependent transport are affected by e-e interactions, RashbaSOI effects on electronic transport in interacting quantumwires have been studied in recent papers. 15,20,22,33–37In ef- fect, however, all those works only took e-e forward-scattering processes into account. Because of the Rashba SOIone obtains a modiﬁed LL phase with broken spin-chargeseparation 33,34leading to a drastic inﬂuence on observables such as the spectral function or the tunneling density ofPHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 1098-0121/2009/79 /H2084920/H20850/205432 /H2084910/H20850 ©2009 The American Physical Society 205432-1states. Moroz et al.33,34argued that e-e backscattering pro- cesses are irrelevant in the renormalization-group /H20849RG/H20850sense and hence can be omitted in a low-energy theory. Unfortu-nately their theory relies on an incorrect spin assignment ofthe subbands 15,22which then invalidates several aspects of their treatment of interaction processes. The possibility that e-e backscattering processes become relevant /H20849in the RG sense /H20850in a Rashba quantum wire was raised in Ref. 38where a spin gap was found under a weak- coupling two-loop RG scheme. If valid, this result has im-portant consequences for the physics of such systems andwould drive them into a spin-density-wave type state. Toestablish the spin gap, Ref. 38starts from a strict 1D single- band model and assumes both /H9251and the e-e interaction as weak-coupling constants ﬂowing under the RG. Our ap-proach below is different in that we include the Rashba cou-pling /H9251from the outset in the single-particle sector, i.e., in a nonperturbative manner. We then consider the one-loop RGﬂow of all possible interaction couplings allowed by momen-tum conservation /H20849for not too small /H9251/H20850. This is an important difference to the scheme of Ref. 38since the Rashba SOI eliminates certain interaction processes which become mo-mentum nonconserving. This mechanism is captured by ourapproach. The one-loop RG ﬂow then turns out to be equiva-lent to a Kosterlitz-Thouless ﬂow and for the initial valuesrealized in this problem e-e backscattering processes are al-ways irrelevant. Our conclusion is therefore that no spin gaparises because of SOI and a modiﬁed LL picture is alwayssufﬁcient. We mention in passing that in the presence of amagnetic ﬁeld /H20849which we do not consider /H20850a spin gap can be present because of spin-nonconserving e-e “Cooper” scatter-ing processes; 39,40the effects of e-e forward scattering in Rashba wires with magnetic ﬁeld were studied as well.41–44 Below, we also provide estimates for the renormalized cou- plings entering the modiﬁed LL theory, see Eq. /H2084926/H20850below. When taking bare /H20849instead of renormalized /H20850couplings we recover previous results.22Note that the SOI in carbon nanotubes45or graphene ribbons46leads to a similar yet dif- ferent LL description. In particular, for /H20849achiral /H20850carbon nanotubes, the leading SOI does not break spin-chargeseparation. 45We here only discuss Rashba SOI effects in semiconductor quantum wires in the absence of magneticﬁelds. We apply our formalism to a study of the Ruderman- Kittel-Kasuya-Yosida /H20849RKKY /H20850interaction 47,48between two spin-1/2 magnetic impurities /H90181,2separated by a distance x. The RKKY interaction is mediated by the conduction elec-trons in the quantum wire which are exchange coupled /H20849with coupling J/H20850to the impurity spins. In the absence of both the e-e interaction and the SOI, one ﬁnds an isotropic exchange/H20849Heisenberg /H20850Hamiltonian 48 HRKKY =−J2Fex/H20849x/H20850/H90181·/H90182,Fex/H20849x/H20850/H11008cos/H208492kFx/H20850 /H20841x/H20841,/H208492/H20850 where the 2 kF-oscillatory RKKY range function Fex/H20849x/H20850is speciﬁed for the 1D case. When the spin SU/H208492/H20850symmetry is broken by the SOI, spin precession sets in and the RKKYinteraction is generally of a more complicated /H20849twisted /H20850form. For a noninteracting Rashba quantum wire it has in- deed been established 49–51that the RKKY interaction be- comes anisotropic and thus has a tensorial character. It canalways be decomposed into an exchange /H20849scalar /H20850part, a Dzyaloshinsky-Moriya /H20849DM /H20850-type /H20849vector /H20850interaction, and an Ising-type /H20849traceless symmetric tensor /H20850coupling. On the other hand, in the presence of e-e interactions but withoutSOI, the range function has been shown 52to exhibit a slow power-law decay Fex/H20849x/H20850/H11008cos/H208492kFx/H20850/H20841x/H20841−/H9257with an interaction- dependent exponent /H9257/H110211. The RKKY interaction in inter- acting quantum wires with SOI has not been studied before. For the beneﬁt of the focused reader we brieﬂy summa- rize the main results of our analyis. The effective low-energytheory of an interacting Rashba quantum wire is given in Eq./H2084929/H20850, with the velocities /H2084930/H20850and the dimensionless interac- tion parameters /H2084931/H20850. Previous theories did not fully account for the e-e backscattering, processes and the conspiracy ofthese processes with the broken SU/H208492/H20850invariance due to spin-orbit effects leads to K s/H110211 in Eq. /H2084931/H20850. This in turn implies effects in the RKKY interaction of an interactingRashba wire. In particular, the power-law decay exponent inan interacting Rashba wire, see Eq. /H2084938/H20850, depends explicitly on both the interaction strength and on the Rashba coupling. The structure of the remainder of this paper is as follows. In Sec. II, we discuss the bandstructure. Interaction processes and the one-loop RG scheme are discussed in Sec. IIIwhile the LL description is provided in Sec. IV. The RKKY inter- action mediated by an interacting Rashba quantum wire isthen studied in Sec. V. Finally we offer some conclusions in Sec. VI. Technical details can be found in the Appendix. Throughout the paper we use units where /H6036=1. II. SINGLE-PARTICLE DESCRIPTION We consider a quantum wire electrostatically conﬁned in thezdirection within the 2DEG /H20849xzplane /H20850by a harmonic potential Vc/H20849z/H20850=m/H92752z2/2 where mis the effective mass. The noninteracting problem is then deﬁned by the single-particleHamiltonian 3,13–15,17 Hsp=1 2m/H20849px2+pz2/H20850+Vc/H20849z/H20850+/H9251/H20849/H9268zpx−/H9268xpz/H20850, /H208493/H20850 where/H9251is the Rashba coupling and the Pauli matrices /H9268x,z act in spin space. For /H9251=0 the transverse problem is diagonal in terms of the familiar 1D harmonic-oscillator eigenstates/H20849Hermite functions /H20850H n/H20849z/H20850with n=0,1,2,... labeling the subbands /H20849channels /H20850. Eigenstates of Eq. /H208493/H20850have conserved longitudinal momentum px=kand with the zdirection as spin-quantization axis, /H9268z/H20841/H9268/H20856=/H9268/H20841/H9268/H20856with/H9268=↑,↓=/H11006, the /H9268xpzterm implies mixing of adjacent subbands with associ- ated spin ﬂips. Retaining only the lowest /H20849n=0/H20850subband from the outset thus excludes spin relaxation. We follow Ref.15and keep the two lowest bands n=0 and n=1. The higher subbands n/H113502 yield only tiny corrections which can in prin- ciple be included as in Ref. 17. The resulting 4 /H110034 matrix representing H spin this truncated Hilbert space is readily diagonalized and yields four energy bands. We choose theFermi energy such that only the lower two bands, labeled bys=/H11006, are occupied and arrive at a reduced two-band modelSCHULZ et al. PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-2where the quantum number s=/H11006replaces the spin quantum number. The dispersion relation is Es/H20849k/H20850=/H9275+k2 2m−/H20881/H20873/H9275 2+s/H9251k/H208742 +m/H9275/H92512 2, /H208494/H20850 with eigenfunctions /H11011eikx/H9278k,s/H20849z/H20850. The resulting asymmetric energy bands /H20851Eq. /H208494/H20850/H20852are shown in Fig. 1. The transverse spinors /H20849in spin space /H20850are given by /H9278k,+/H20849z/H20850=/H20873icos/H20851/H9258+/H20849k/H20850/H20852H1/H20849z/H20850 sin/H20851/H9258+/H20849k/H20850/H20852H0/H20849z/H20850/H20874, /H9278k,−/H20849z/H20850=/H20873sin/H20851/H9258−/H20849k/H20850/H20852H0/H20849z/H20850 icos/H20851/H9258−/H20849k/H20850/H20852H1/H20849z/H20850/H20874, /H208495/H20850 with k-dependent spin-rotation angles /H20849we take 0/H11349/H9258s/H20849k/H20850 /H11349/H9266/2/H20850 /H9258s/H20849k/H20850=1 2cot−1/H20873−2sk−/H9275//H9251 /H208812m/H9275/H20874=/H9258−s/H20849−k/H20850. /H208496/H20850 As a result of subband mixing, the two spinor components of /H9278k,s/H20849z/H20850carry a different zdependence. They are therefore not just the result of a SU/H208492/H20850rotation. For /H9251=0 we recover /H9258s =/H9266/2 corresponding to the usual spin-up and -down eigen- states with H0/H20849z/H20850as transverse wave function; the s=+/H20849s= −/H20850component then describes the /H9268=↓/H20849/H9268=↑/H20850spin eigenstate. However, for /H9251/HS110050, a peculiar implication of the Rashba SOI follows. From Eq. /H208496/H20850we have lim k→/H11006/H11009/H9258s/H20849k/H20850=/H208491/H11006s/H20850/H9266/4 such that both s=/H11006states have /H20849approximately /H20850spin/H9268=↓ fork→/H11009but/H9268=↑fork→−/H11009; the product of spin and chirality thus always approaches /H9268sgn/H20849k/H20850=−1. Moreover, under the time-reversal transformation T=i/H9268yCwith the complex conjugation operator C, the two subbands are ex- changede−ikx/H9278−k,−s/H20849z/H20850=sT/H20851eikx/H9278k,s/H20849z/H20850/H20852,E−s/H20849−k/H20850=Es/H20849k/H20850. /H208497/H20850 Time-reversal symmetry, preserved in the truncated descrip- tion, makes this two-band model of a Rashba quantum wirequalitatively different from Zeeman-spin-split models. 21 In the next step, since we are interested in the low-energy physics, we linearize the dispersion relation around the Fermi points /H11006kF/H20849A,B/H20850, see Fig. 1, which results in two veloci- tiesvAandvB, see Eq. /H208491/H20850. The linearization of the dispersion relation of multiband quantum wires around the Fermi levelis known to be an excellent approximation for weak e-einteractions. 32Explicit values for /H9254in Eq. /H208491/H20850can be derived from Eq. /H208494/H20850and we ﬁnd /H9254/H20849/H9251/H20850/H11008/H92514for/H9251→0 in accordance with previous estimates.20We mention that /H9254/H113510.1 has been estimated for typical geometries in Ref. 34. The transverse spinors/H9278ks/H20849z/H20850, see Eq. /H208495/H20850, entering the low-energy descrip- tion can be taken at k=/H11006kF/H20849A,B/H20850where the spin rotation angle /H20851Eq. /H208496/H20850/H20852only assumes one of the two values /H9258A=/H9258+/H20849kF/H20849A/H20850/H20850,/H9258B=/H9258−/H20849kF/H20849B/H20850/H20850. /H208498/H20850 The electron ﬁeld operator /H9023/H20849x,z/H20850for the linearized two- band model with /H9263=A,B=+,− can then be expressed in terms of 1D fermionic-ﬁeld operators /H9274/H9263,r/H20849x/H20850, where r=R,L =+,− labels right and left movers /H9023/H20849x,z/H20850=/H20858 /H9263,r=/H11006eirkF/H20849/H9263/H20850x/H9278rkF/H20849/H9263/H20850,s=/H9263r/H20849z/H20850/H9274/H9263,r/H20849x/H20850, /H208499/H20850 with/H9278k,s/H20849z/H20850speciﬁed in Eq. /H208495/H20850. Note that in the left-moving sector, band indices have been interchanged according to thelabeling in Fig. 1. In this way, the noninteracting second-quantized Hamil- tonian takes the standard form for two inequivalent speciesof 1D massless Dirac fermions with different velocities H 0=−i/H20858 /H9263,r=/H11006rv/H9263/H20885dx/H9274/H9263,r†/H11509x/H9274/H9263,r. /H2084910/H20850 The velocity difference implies the breaking of the spin SU/H208492/H20850symmetry, a direct consequence of SOI. For /H9251=0 the index/H9263coincides with the spin quantum number /H9268for left movers and with − /H9268for right movers and the above formu- lation reduces to the usual Hamiltonian for a spinful single-channel quantum wire. III. INTERACTION EFFECTS Let us now include e-e interactions in such a single- channel disorder-free Rashba quantum wire. With the expan-sion /H208499/H20850andr=/H20849x,z/H20850the second-quantized two-body Hamil- tonian H I=1 2/H20885dr1dr2/H9023†/H20849r1/H20850/H9023†/H20849r2/H20850V/H20849r1−r2/H20850/H9023/H20849r2/H20850/H9023/H20849r1/H20850 /H2084911/H20850 leads to 1D interaction processes. We here assume that the e-e interaction potential V/H20849r1−r2/H20850is externally screened al- lowing to describe the 1D interactions as effectively local.Following standard arguments, for weak e-e interactions, go--kF(A)-kF(B)+kF(B)+kF(A)kE vBvAεF-vB-vA B,L A,L B,R A,R E(+)(k) E(-)(k) FIG. 1. /H20849Color online /H20850Schematic band structure /H20851Eq. /H208494/H20850/H20852of a typical 1D Rashba quantum wire. The red/blue /H20849right/left solid /H20850 curves show the s=/H11006bands and the dotted curves indicate the next subband /H20849the Fermi energy /H9280Fis assumed below that band /H20850. For the low-energy description we linearize the dispersion. It is notationallyconvenient to introduce bands A /H20849solid lines /H20850and B /H20849dashed lines /H20850. Green and black arrows indicate the respective spin amplitudes /H20849ex- aggerated /H20850. The resulting Fermi momenta are /H11006k F/H20849A,B/H20850with Fermi velocities vA,B.LOW-ENERGY THEORY AND RKKY INTERACTION FOR … PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-3ing beyond this approximation at most leads to irrelevant corrections.53We then obtain the local 1D interaction Hamiltonian54 HI=1 2/H20858 /H20853/H9263i,ri/H20854V/H20853/H9263i,ri/H20854/H20885dx/H9274/H92631,r1†/H9274/H92632,r2†/H9274/H92633,r3/H9274/H92634,r4, /H2084912/H20850 where the summation runs over all quantum numbers /H92631,...,/H92634andr1,..., r4subject to momentum conservation r1kF/H20849/H92631/H20850+r2kF/H20849/H92632/H20850=r3kF/H20849/H92633/H20850+r4kF/H20849/H92634/H20850. /H2084913/H20850 With the momentum transfer q=r1kF/H20849/H92631/H20850−r4kF/H20849/H92634/H20850and the par- tial Fourier transform V˜/H20849q;z/H20850=/H20885dxe−iqxV/H20849x,z/H20850/H20849 14/H20850 of the interaction potential, the interaction matrix elements in Eq. /H2084912/H20850are given by V/H20853/H9263i,ri/H20854=/H20885dz1dz2V˜/H20849q;z1−z2/H20850/H11003/H20851/H9278r1kF/H20849/H92631/H20850,/H92631r1†·/H9278r4kF/H20849/H92634/H20850,/H92634r4/H20852/H20849z1/H20850 /H11003/H20851/H9278r2kF/H20849/H92632/H20850,/H92632r2†·/H9278r3kF/H20849/H92633/H20850,/H92633r3/H20852/H20849z2/H20850. /H2084915/H20850 Since the Rashba SOI produces a splitting of the Fermi mo- menta for the two bands, /H20841kF/H20849A/H20850−kF/H20849B/H20850/H20841/H112292/H9251m, the condition /H2084913/H20850eliminates one important interaction process available for/H9251=0, namely, interband backscattering /H20849see below /H20850. This is a distinct SOI effect besides the broken spin SU/H208492/H20850invari- ance. Obtaining the complete “g-ology” classiﬁcation32of all possible interaction processes allowed for /H9251/HS110050 is then a straightforward exercise. The corresponding values of the in-teraction matrix elements are generally difﬁcult to evaluateexplicitly but in the most important case of a thin wire d/H112711 /H20881m/H9275, /H2084916/H20850 where dis the screening length /H20849representing, e.g., the dis- tance to a backgate /H20850, analytical expressions can be obtained.55To simplify the analysis and allow for analytical progress, we therefore employ the thin-wire approximation/H20851Eq. /H2084916/H20850/H20852in what follows. In that case we can neglect the z dependence in Eq. /H2084914/H20850. Going beyond this approximation would only imply slightly modiﬁed values for the e-e inter-action couplings used below. Using the identity /H20885dz/H20851/H9278rkF/H20849/H9263/H20850,/H9263r†·/H9278r/H11032kF/H20849/H9263/H11032/H20850,/H9263/H11032r/H11032/H20852/H20849z/H20850 =/H9254/H9263/H9263/H11032/H9254rr/H11032+ cos /H20849/H9258A−/H9258B/H20850/H9254/H9263,−/H9263/H11032/H9254r,−r/H11032, /H2084917/H20850 where the angles /H9258A,Bwere speciﬁed in Eq. /H208498/H20850, only two different values W0andW1for the matrix elements in Eq. /H2084915/H20850emerge. These nonzero matrix elements are V/H9263r,/H9263/H11032r/H11032,/H9263/H11032r/H11032,/H9263r/H11013W0=V˜/H20849q=0/H20850, V/H9263r,/H9263/H11032r/H11032,−/H9263/H11032−r/H11032,−/H9263−r/H11013W1= cos2/H20849/H9258A−/H9258B/H20850V˜/H20849q=kF/H20849A/H20850+kF/H20849B/H20850/H20850. /H2084918/H20850We then introduce 1D chiral fermion densities /H9267/H9263r/H20849x/H20850¬/H9274/H9263r†/H9274/H9263r:, where the colons indicate normal ordering. The interacting 1D Hamiltonian is H=H0+HIwith Eq. /H2084910/H20850 and HI=1 2/H20858 /H9263/H9263/H11032,rr/H11032/H20885dx/H20849/H20851g2/H20648/H9263/H9254/H9263,/H9263/H11032+g2/H11036/H9254/H9263,−/H9263/H11032/H20852/H9254r,−r/H11032 +/H20851g4/H20648/H9263/H9254/H9263,/H9263/H11032+g4/H11036/H9254/H9263,−/H9263/H11032/H20852/H9254r,r/H11032/H20850/H9267/H9263r/H9267/H9263/H11032r/H11032 +gf 2/H20858 /H9263r/H20885dx/H9274/H9263r†/H9274/H9263,−r†/H9274−/H9263r/H9274−/H9263,−r. /H2084919/H20850 The e-e interaction couplings are denoted in analogy to the standard g-ology, whereby the g4/H20849g2/H20850processes describe for- ward scattering of 1D fermions with equal /H20849opposite /H20850chiral- ityr=R,L=+,− and the labels /H20648,/H11036, and fdenote intraband, interband, and band ﬂip processes, respectively. Since thebands /H9263=A,B=+,− are inequivalent, we keep track of the band index in the intraband couplings. The gfterm corre- sponds to intraband backscattering with band ﬂip. The inter-band backscattering without band ﬂip is strongly suppressedsince it does not conserve total momentum 56and is neglected in the following. For /H9251=0 the g4,/H20648//H11036couplings coincide with the usual ones32for spinful electrons while gfreduces to g1/H11036 andg2,/H20648//H11036→g2,/H11036//H20648due to our exchange of band indices in the left-moving sector. According to Eq. /H2084918/H20850the bare values of these coupling constants are g4/H20648/H9263=g4/H11036=g2/H20648/H9263=W0, g2/H11036=W0−W1,gf=W1. /H2084920/H20850 The equality of the intraband coupling constants for the two bands is a consequence of the thin-wire approximation whichalso eliminates certain exchange matrix elements. The Hamiltonian H 0+HIthen corresponds to a speciﬁc realization of a general asymmetric two-band model wherethe one-loop RG equations are known. 54,57Using RG invari- ants we arrive after some algebra at the two-dimensionalKosterlitz-Thouless RG ﬂow equations dg ¯2 dl=−g¯f2,dg¯f dl=−g¯fg¯2, /H2084921/H20850 for the rescaled couplings g¯2=g2/H20648A 2/H9266vA+g2/H20648B 2/H9266vB−g2/H11036 /H9266vF, g¯f=/H208811+/H9253 2gf /H9266vF, /H2084922/H20850 where we use the dimensionless constant /H9253=vF2 vAvB=1 1−/H92542/H113501. /H2084923/H20850 As usual, the g4couplings do not contribute to the one-loop RG equations. The initial values of the couplings can be readoff from Eq. /H2084920/H20850SCHULZ et al. PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-4g¯2/H20849l=0/H20850=/H20849/H9253−1/H20850W0+W1 /H9266vF, g¯f/H20849l=0/H20850=/H208811+/H9253 2W1 /H9266vF. /H2084924/H20850 The solution of Eq. /H2084921/H20850is textbook material32and g¯fis known to be marginally irrelevant for all initial conditions with /H20841g¯f/H208490/H20850/H20841/H11349g¯2/H208490/H20850. Using Eqs. /H2084918/H20850and /H2084924/H20850, this implies with/H9253/H112291+/H92542the condition V˜/H208490/H20850/H113501 4cos2/H20849/H9258A−/H9258B/H20850V˜/H20849kF/H20849A/H20850+kF/H20849B/H20850/H20850, /H2084925/H20850 which is satisﬁed for all physically relevant repulsive e-e interaction potentials. As a consequence intraband back-scattering processes with band ﬂip, described by the coupling g ¯f, are always marginally irrelevant , i.e., they ﬂow to zero coupling as the energy scale is reduced, g¯f/H11569=g¯f/H20849l→/H11009/H20850=0. Therefore no gap arises and a modiﬁed LL model is the appropriate low-energy theory. We mention in passing thateven if we neglect the velocity difference in Eq. /H208491/H20850, no spin gap is expected in a Rashba wire, i.e., the broken SU/H208492/H20850 invariance in our model is not required to establish the ab-sence of a gap. The above RG procedure also allows us to extract renor- malized couplings entering the low-energy LL description. The ﬁxed-point value g ¯2/H11569=g¯2/H20849l→/H11009/H20850now depends on the Rashba SOI through /H9253in Eq. /H2084923/H20850. With the interaction ma- trix elements W0,1in Eq. /H2084918/H20850, it is given by g¯2/H11569=/H20881/H20851/H20849/H9253−1/H20850W0+W1/H208522−/H20849/H9253+1/H20850W12/2 /H9266vF. /H2084926/H20850 For/H9251=0 we have /H9253=1 and therefore g¯2/H11569=0. The Rashba SOI produces the nonzero ﬁxed-point value /H2084926/H20850reﬂecting the broken SU/H208492/H20850symmetry. IV. LUTTINGER LIQUID DESCRIPTION In this section, we describe the resulting effective low- energy LL theory of an interacting single-channel Rashbawire. Employing Abelian bosonization 32we introduce a bo- son ﬁeld and its conjugate momentum for each band /H9263 =A,B=+,−. It is useful to switch to symmetric /H20849“charge” /H20850, /H9021c/H20849x/H20850and/H9016c/H20849x/H20850=−/H11509x/H9008c/H20849x/H20850, and antisymmetric /H20849“spin” for /H9251=0/H20850,/H9021s/H20849x/H20850and/H9016s/H20849x/H20850=−/H11509x/H9008s, linear combinations of these ﬁelds and their momenta. The dual ﬁelds /H9021and/H9008then allow to express the electron operator from Eq. /H208499/H20850and the “bosonization dictionary,” /H9023/H20849x,z/H20850=/H20858 /H9263,r/H9278rkF/H20849/H9263/H20850,/H9263r/H20849z/H20850/H9257/H9263r /H208812/H9266aeirkF/H20849/H9263/H20850x+i/H20881/H9266/2/H20851r/H9021c+/H9008c+/H9263r/H9021s+/H9263/H9008s/H20852, /H2084927/H20850 where ais a small cutoff length and /H9257/H9263rare the standard Klein factors.32,52,58/H20849To recover the conventional expression for/H9251=0, due to our convention for the band indices in the left-moving sector, one should replace /H9021s,/H9008s→−/H9008s,−/H9021s./H20850Using the identity /H2084917/H20850we can now express the 1D charge and spin densities /H9267/H20849x/H20850=/H20885dz/H9023†/H9023,S/H20849x/H20850=/H20885dz/H9023†/H9268 2/H9023, /H2084928/H20850 in bosonized form. The /H20849somewhat lengthy /H20850result can be found in the Appendix. The low-energy Hamiltonian is then taken with the ﬁxed- point values for the interaction constants, i.e., backscatteringprocesses are disregarded and only appear via the renormal- ized value of g ¯2/H11569in Eq. /H2084926/H20850. Following standard steps, the kinetic term H0and the forward-scattering processes then lead to the exactly solvable Gaussian-ﬁeld theory of a modi-ﬁed /H20849extended /H20850Luttinger liquid H=/H20858 j=c,svj 2/H20885dx/H20873Kj/H9016j2+1 Kj/H20849/H11509x/H9021j/H208502/H20874 +v/H9261/H20885dx/H20873K/H9261/H9016c/H9016s+1 K/H9261/H20849/H11509x/H9021c/H20850/H20849/H11509x/H9021s/H20850/H20874. /H2084929/H20850 Using the notations g¯4=W0//H9266vFand y/H9254=g2/H20648A/H11569−g2/H20648B/H11569 4/H9266vF, y/H11006=g2/H20648A/H11569+g2/H20648B/H11569/H110062g2/H11036/H11569 4/H9266vF, where explicit /H20849but lengthy /H20850expressions for the ﬁxed-point values g2/H20648A/B/H11569andg2/H11036/H11569can be straightforwardly obtained from Eqs. /H2084922/H20850and /H2084926/H20850, the renormalized velocities appearing in Eq. /H2084929/H20850are vc=vF/H20881/H208491+g¯4/H208502−y+2/H11229vF/H20881/H208731+W0 /H9266vF/H208742 −/H208732W0−W1 2/H9266vF/H208742 , vs=vF/H208811−y−2/H11229vF, v/H9261=vF/H20881/H92542−y/H92542/H11229vF/H9254/H208811−/H20873W1 4/H9266vF/H208742 . /H2084930/H20850 In the respective second equalities we have speciﬁed the leading terms in /H20841/H9254/H20841/H112701, since the SOI-induced relative- velocity asymmetry /H9254is small even for rather large /H9251, see Eq. /H208491/H20850. The corrections to the quoted expressions are of O/H20849/H92542/H20850and are negligible in practice. It is noteworthy that the spin velocity vsisnotrenormalized for a Rashba wire, al- though it is well known that vswill be renormalized due to W1for/H9251=0.32This difference can be traced to our thin-wire approximation /H20851Eq. /H2084916/H20850/H20852. When releasing this approximation there will be a renormalization in general. Finally the dimen-sionless LL interaction parameters in Eq. /H2084929/H20850are given by K c=/H208811+g¯4−y+ 1+g¯4+y+/H11229/H208812/H9266vF+W1 2/H9266vF+4W0−W1, Ks=/H208811−y− 1+y−/H112291−/H20881W0W1 /H208812/H9266vF/H20841/H9254/H20841,LOW-ENERGY THEORY AND RKKY INTERACTION FOR … PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-5K/H9261=/H20881/H9254−y/H9254 /H9254+y/H9254/H11229/H208814/H9266vF+W1 4/H9266vF−W1, /H2084931/H20850 where the second equalities again hold up to contributions of O/H20849/H92542/H20850. When the 2 kFcomponent of the interaction potential W1=0, see Eq. /H2084918/H20850, we obtain Ks=K/H9261=1 and thus recover the theory of Ref. 22. The broken spin SU/H208492/H20850symmetry is reﬂected in Ks/H110211 when both /H9254/HS110050 and W1/HS110050. Since we arrived at a Gaussian ﬁeld theory, Eq. /H2084929/H20850, all low-energy correlation functions can now be computed ana-lytically without further approximation. The linear algebraproblem needed for this diagonalization is discussed in theAppendix. V. RKKY INTERACTION Following our discussion in Sec. I, we now investigate the combined effects of the Rashba SOI and the e-e interactionon the RKKY range function. We include the exchange cou-pling H /H11032=J/H20858i=1,2/H9018i·S/H20849xi/H20850of the 1D conduction-electron spin density S/H20849x/H20850to localized spin-1/2 magnetic impurities sepa- rated by x=x1−x2. The RKKY interaction HRKKY , describing spin-spin interactions between the two magnetic impurities,is then obtained by perturbation theory in J. 48In the simplest 1D case /H20849no SOI and no interactions /H20850it is given by Eq. /H208492/H20850. In the general case one can always express it in the form HRKKY =−J2/H20858 a,bFab/H20849x/H20850/H90181a/H90182b, /H2084932/H20850 with the range function now appearing as a tensor /H20849/H9252 =1 /kBTfor temperature T/H20850 Fab/H20849x/H20850=/H20885 0/H9252 d/H9270/H9273ab/H20849x,/H9270/H20850. /H2084933/H20850 Here, the imaginary-time /H20849/H9270/H20850spin-spin correlation function appears /H9273ab/H20849x,/H9270/H20850=/H20855Sa/H20849x,/H9270/H20850Sb/H208490,0/H20850/H20856. /H2084934/H20850 The 1D spin densities Sa/H20849x/H20850/H20849with a=x,y,z/H20850were deﬁned in Eq. /H2084928/H20850and their bosonized expression is given in the Ap- pendix, which then allows to compute the correlation func-tions /H20851Eq. /H2084934/H20850/H20852using the unperturbed /H20849J=0/H20850LL model /H20851Eq. /H2084929/H20850/H20852. The range function thus effectively coincides with the static space-dependent spin-susceptibility tensor. When spinSU/H208492/H20850symmetry is realized, /H9273ab/H20849x/H20850=/H9254abFex/H20849x/H20850, and one re- covers Eq. /H208492/H20850, but in general this tensor is not diagonal. For a LL without Rashba SOI, Fex/H20849x/H20850is as in Eq. /H208492/H20850but with a slow power-law decay.52 If spin SU/H208492/H20850symmetry is broken, general arguments im- ply that Eq. /H2084932/H20850can be decomposed into three terms, namely, /H20849i/H20850an isotropic exchange scalar coupling, /H20849ii/H20850aD M vector term, and /H20849iii/H20850an Ising-type interactionHRKKY /J2=−Fex/H20849x/H20850/H90181·/H90182−FDM/H20849x/H20850·/H20849/H90181/H11003/H90182/H20850 −/H20858 a,bFIsingab/H20849x/H20850/H90181a/H90182b, /H2084935/H20850 where Fex/H20849x/H20850=1 3/H20858aFaa/H20849x/H20850. The DM vector has the compo- nents FDMc/H20849x/H20850=1 2/H20858 a,b/H9280cabFab/H20849x/H20850, and the Ising-type tensor FIsingab/H20849x/H20850=1 2/H20873Fab+Fba−2 3/H20858 cFcc/H9254ab/H20874/H20849x/H20850 is symmetric and traceless. For a 1D noninteracting quantum wire with Rashba SOI, the “twisted” RKKY Hamiltonian/H2084935/H20850has recently been discussed 49–51and all range functions appearing in Eq. /H2084935/H20850were shown to decay /H11008/H20841x/H20841−1,a se x - pected for a noninteracting system. Moreover, it has beenemphasized 50that there are different spatial oscillation peri- ods reﬂecting the presence of different Fermi momenta kF/H20849A,B/H20850 in a Rashba quantum wire. Let us then consider the extended LL model /H20851Eq. /H2084929/H20850/H20852 which includes the effects of both the e-e interaction and theRashba SOI. The correlation functions /H20851Eq. /H2084934/H20850/H20852obey /H9273ba/H20849x,/H9270/H20850=/H9273ab/H20849−x,−/H9270/H20850and since we ﬁnd /H9273xz=/H9273yz=0 the an- isotropy acts only in the xyplane. The four nonzero correla- tors are speciﬁed in the Appendix, where only the long-ranged 2 k Foscillatory terms are kept. These are the relevant correlations determining the RKKY interaction in the inter-acting quantum wire. We note that in the noninteracting case,there is also a “slow” oscillatory component corresponding to a contribution to the RKKY range function /H11008cos/H20851/H20849k F/H20849A/H20850 −kF/H20849B/H20850/H20850x/H20852//H20841x/H20841. Remarkably, we ﬁnd that this 1 /xdecay law is not changed by interactions. However, we will show belowthat interactions cause a slower decay of certain “fast” oscil- latory terms, e.g., the contribution /H11008cos/H208492k F/H20849B/H20850x/H20850. We there- fore do not further discuss the slow oscillatory terms in whatfollows. Collecting everything, we ﬁnd the various range functions in Eq. /H2084935/H20850for the interacting case, F ex/H20849x/H20850=1 6/H20858 /H9263/H20851/H208511 + cos2/H208492/H9258/H9263/H20850/H20852cos/H208512kF/H20849/H9263/H20850x/H20852F/H9263/H208491/H20850/H20849x/H20850 + cos2/H20849/H9258A+/H9258B/H20850cos/H20853/H20851kF/H20849A/H20850+kF/H20849B/H20850/H20852x/H20854F/H9263/H208492/H20850/H20849x/H20850/H20852, FDM/H20849x/H20850=eˆz/H20858 /H9263/H9263 2cos/H208492/H9258/H9263/H20850sin/H208492kF/H20849/H9263/H20850x/H20850F/H9263/H208491/H20850/H20849x/H20850, FIsingab/H20849x/H20850=/H208751 2/H20858 /H9263G/H9263a/H20849x/H20850−Fex/H20849x/H20850/H20876/H9254ab, /H2084936/H20850 with the auxiliary vectorSCHULZ et al. PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-6G/H9263=/H20898cos/H208492kF/H20849/H9263/H20850x/H20850F/H9263/H208491/H20850/H20849x/H20850 cos2/H208492/H9258/H9263/H20850cos/H208492kF/H20849/H9263/H20850x/H20850F/H9263/H208491/H20850/H20849x/H20850 cos2/H20849/H9258A+/H9258B/H20850cos/H20851/H20849kF/H20849A/H20850+kF/H20849B/H20850/H20850x/H20852F/H9263/H208492/H20850/H20849x/H20850/H20899. The functions F/H9263/H208491,2/H20850/H20849x/H20850follow by integration over /H9270from F˜ /H9263/H208491,2/H20850/H20849x,/H9270/H20850, see Eqs. /H20849A1/H20850and /H20849A2/H20850in the Appendix. This implies the respective decay laws for a/H11270/H20841x/H20841/H11270vF/kBT F/H9263/H208491/H20850/H20849x/H20850/H11008/H20841a/x/H20841−1+Kc+Ks+2/H9263/H208491−Kc/K/H92612/H20850/H20849v/H9261K/H9261/vc+vs/H20850, F/H9263/H208492/H20850/H20849x/H20850/H11008/H20841a/x/H20841−1+Kc+1 /Ks. /H2084937/H20850 All those exponents approach unity in the noninteracting limit in accordance with previous results.49,50Moreover, in the absence of SOI /H20849/H9251=/H9254=0/H20850, Eq. /H2084937/H20850reproduces the known /H20841x/H20841−Kcdecay law for the RKKY interaction in a conventional LL.52 Since Ks/H110211 for an interacting Rashba wire with /H9254/HS110050, see Eq. /H2084931/H20850, we conclude that F/H9263/H208491/H20850with/H9263=B, corresponding to the slower velocity vB=vF/H208491−/H9254/H20850, leads to the slowest de- cay of the RKKY interaction. For large distance xthe RKKY interaction is therefore dominated by the 2 kF/H20849B/H20850oscillatory part and all range functions decay /H11008/H20841x/H20841−/H9257Bwith the exponent /H9257B=Kc+Ks−1−2/H208731−Kc K/H92612/H20874v/H9261K/H9261 vc+vs/H110211. /H2084938/H20850 This exponent depends both on the e-e interaction potential and on the Rashba coupling /H9251. The latter dependence also implies that electric ﬁelds are able to change the power-lawdecay of the RKKY interaction in a Rashba wire. The DMvector coupling also illustrates that the SOI is able to effec-tively induce off-diagonal couplings in spin space, reminis-cent of spin-precession effects. Also these RKKY couplings are 2 k F/H20849B/H20850oscillatory and show a power-law decay with the exponent /H2084938/H20850. VI. DISCUSSION In this paper we have presented a careful derivation of the low-energy Hamiltonian of a homogeneous 1D quantumwire with not too weak Rashba spin-orbit interactions. Wehave studied the simplest case /H20849no magnetic ﬁeld, no disor- der, and single-channel limit /H20850and in particular analyzed the possibility for a spin gap to occur because of electron-electron backscattering processes. The initial values for thecoupling constants entering the one-loop RG equations weredetermined and, for rather general conditions, they are suchthat backscattering is marginally irrelevant and no spin gapopens. The resulting low-energy theory is a modiﬁed Lut-tinger liquid, Eq. /H2084929/H20850, which is a Gaussian ﬁeld theory for- mulated in terms of the boson ﬁelds /H9021 c/H20849x/H20850and/H9021s/H20849x/H20850/H20849and their dual ﬁelds /H20850. In this state spin-charge separation is vio- lated due to the Rashba coupling but the theory still admitsexact results for essentially all low-energy correlation func-tions. Based on our bosonized expressions for the 1D charge and spin density, the frequency dependence of various sus-ceptibilities of interest, e.g., charge- or spin-density-wave correlations, can then be computed. As the calculationclosely mirrors the one in Refs. 34and35we do not repeat it here. One can then infer a “phase diagram” from the studyof the dominant susceptibilities. According to our calcula-tions, due to a conspiracy of the Rashba SOI and the e-einteraction, spin-density-wave correlations in the xyplane are always dominant for repulsive interactions. We have studied the RKKY interaction between two mag- netic impurities in such an interacting 1D Rashba quantumwire. On general grounds the RKKY interaction can be de-composed into an exchange term, a DM vector term, and atraceless symmetric tensor interaction. For a noninteractingwire the corresponding three range functions have severalspatial oscillation periods with a common overall decay/H11008/H20841x/H20841 −1. We have shown that interactions modify this picture. The dominant contribution /H20849characterized by the slowest power-law decay /H20850to the RKKY range function is now 2 kF/H20849B/H20850 oscillatory for all three terms with the same exponent /H9257B /H110211, see Eq. /H2084938/H20850. This exponent depends both on the inter- action strength and on the Rashba coupling. This raises theintriguing possibility to tune the power-law exponent /H9257B governing the RKKY interaction by an electric ﬁeld since /H9251 is tunable via a backgate voltage. We stress again that inter-actions imply that a single spatial oscillation period /H20849wave- length /H9266/kF/H20849B/H20850/H20850becomes dominant, in contrast to the nonin- teracting situation where several competing wavelengths areexpected. The above formulation also holds promise for future cal- culations of spin transport in the presence of both interac-tions and Rashba spin-orbit couplings and possibly with dis-order. Under a perturbative treatment of impuritybackscattering, otherwise exact statements are possible evenout of equilibrium. We hope that our work will motivatefurther studies along this line. ACKNOWLEDGMENTS We wish to thank W. Häusler and U. Zülicke for helpful discussions. This work was supported by the SFB TR 12 ofthe DFG and by the ESF network INSTANS. APPENDIX: BOSONIZATION FOR THE EXTENDED LUTTINGER LIQUID In this Appendix, we provide some technical details re- lated to the evaluation of the spin-spin correlation functionunder the extended Luttinger theory /H20851Eq. /H2084929/H20850/H20852. The exact calculation of such correlations is possible within thebosonization framework and requires a diagonalization ofEq. /H2084929/H20850. The one-dimensional /H208491D/H20850charge and spin densities /H20851Eq. /H2084928/H20850/H20852can be written as the sum of slow and fast /H20849oscillatory /H20850 contributions. Using Eq. /H2084917/H20850, the bosonized form for the 1D charge density is /H9267/H20849x/H20850=/H208812 /H9266/H11509x/H9021c−2i /H9266a/H9257AR/H9257ALcos/H20849/H9258A−/H9258B/H20850sin/H20851/H20849kF/H20849A/H20850+kF/H20849B/H20850/H20850x +/H208812/H9266/H9021c/H20852cos/H20849/H208812/H9266/H9008s/H20850. Similarly, using the identityLOW-ENERGY THEORY AND RKKY INTERACTION FOR … PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-7/H20885dz/H20851/H9278rkF/H20849/H9263/H20850,/H9263r†/H9268/H9278r/H11032kF/H20849/H9263/H11032/H20850,/H9263/H11032r/H11032/H20852/H20849z/H20850=/H9254r,r/H11032/H20898cos/H20849/H9258A−/H9258B/H20850/H9254/H9263,−/H9263/H11032 −i/H9263rcos/H20849/H9258A+/H9258B/H20850/H9254/H9263,−/H9263/H11032 /H9263rcos/H208492/H9258/H9263/H20850/H9254/H9263,/H9263/H11032/H20899+/H9254r,−r/H11032/H20898/H9254/H9263,/H9263/H11032 −i/H9263rcos/H208492/H9258/H9263/H20850/H9254/H9263,/H9263/H11032 /H9263rcos/H20849/H9258A+/H9258B/H20850/H9254/H9263,−/H9263/H11032/H20899, the 1D spin-density vector has the components Sx/H20849x/H20850=−i/H9257AR/H9257BR /H9266acos/H20849/H9258A−/H9258B/H20850cos/H20851/H20849kF/H20849A/H20850−kF/H20849B/H20850/H20850x +/H208812/H9266/H9021s/H20852sin/H20849/H208812/H9266/H9008s/H20850−i/H9257AR/H9257AL /H9266acos/H20851/H20849kF/H20849A/H20850+kF/H20849B/H20850/H20850x +/H208812/H9266/H9021c/H20852sin/H20851/H20849kF/H20849A/H20850−kF/H20849B/H20850/H20850x+/H208812/H9266/H9021s/H20852, Sy/H20849x/H20850=i/H9257AR/H9257BR /H9266acos/H20849/H9258A+/H9258B/H20850sin/H20851/H20849kF/H20849A/H20850−kF/H20849B/H20850/H20850x +/H208812/H9266/H9021s/H20852sin/H20849/H208812/H9266/H9008s/H20850−i/H20858 /H9263=A,B=+,−/H9263/H9257/H9263R/H9257/H9263L 2/H9266acos/H208492/H9258/H9263/H20850 /H11003cos/H208512kF/H20849/H9263/H20850x+/H208812/H9266/H20849/H9021c+/H9263/H9021s/H20850/H20852, Sz/H20849x/H20850=1 /H208818/H9266/H20851/H20849cos 2/H9258A+ cos 2/H9258B/H20850/H11509x/H9008s +/H20849cos 2/H9258A− cos 2/H9258B/H20850/H11509x/H9008c/H20852−i/H9257AR/H9257BL /H9266a /H11003cos/H20849/H9258A+/H9258B/H20850cos/H20851/H20849kF/H20849A/H20850+kF/H20849B/H20850/H20850x +/H208812/H9266/H9021c/H20852sin/H20849/H208812/H9266/H9021s/H20850. Note that while /H11509x/H9021cis proportional to the /H20849slow part of the /H20850 charge density, the /H20849slow /H20850spin density is determined by both candssectors. Next we specify the nonzero components of the imaginary-time spin-spin correlation function /H9273ab/H20849x,/H9270/H20850, see Eq. /H2084934/H20850. Using the above bosonized expressions, some alge- bra yields /H9273xx/H20849x,/H9270/H20850=/H20858 /H9263cos/H208492kF/H20849/H9263/H20850x/H20850 2/H208492/H9266a/H208502F˜ /H9263/H208491/H20850/H20849x,/H9270/H20850, /H9273yy/H20849x,/H9270/H20850=/H20858 /H9263cos2/H208492/H9258/H9263/H20850cos/H208492kF/H20849/H9263/H20850x/H20850 2/H208492/H9266a/H208502F˜ /H9263/H208491/H20850/H20849x,/H9270/H20850, /H9273zz/H20849x,/H9270/H20850=/H20858 /H9263rcos2/H20849/H9258A+/H9258B/H20850 2/H208492/H9266a/H208502cos/H20851/H20849kF/H20849A/H20850+kF/H20849B/H20850/H20850x/H20852F˜ /H9263/H208492/H20850/H20849x,/H9270/H20850, and /H9273xy/H20849x,/H9270/H20850=/H20858 /H9263/H9263cos/H208492/H9258/H9263/H20850sin/H208492kF/H20849/H9263/H20850x/H20850 2/H208492/H9266a/H208502F˜ /H9263/H208491/H20850/H20849x,/H9270/H20850. Here the functions F˜ /H9263=A,B=+,−/H208491,2/H20850/H20849x,/H9270/H20850are given byF˜ /H9263/H208491/H20850/H20849x,/H9270/H20850=/H20863 j=1,2/H20879/H9252uj /H9266asin/H20873/H9266/H20849uj/H9270−ix/H20850 /H9252uj/H20874/H20879−/H20849/H9003/H9021c/H9021c/H20849j/H20850+/H9003/H9021s/H9021s/H20849j/H20850+2/H9263/H9003/H9021c/H9021s/H20849j/H20850/H20850 and F˜ /H9263/H208492/H20850/H20849x,/H9270/H20850=/H20863 j=1,2/H20879/H9252uj /H9266asin/H20873/H9266/H20849uj/H9270−ix/H20850 /H9252uj/H20874/H20879−/H20849/H9003/H9021c/H9021c/H20849j/H20850+/H9003/H9008s/H9008s/H20849j/H20850/H20850 /H11003/H20875sin/H20849/H9266/H20849uj/H9270+ix/H20850 /H9252uj/H20850 sin/H20849/H9266/H20849uj/H9270−ix/H20850 /H9252uj/H20850/H20876/H9263/H9003/H9021c/H9008s/H20849j/H20850 . The dimensionless numbers /H9003/H20849j/H20850appearing in the exponents follow from the straightforward /H20849but lengthy /H20850diagonalization of the extended Luttinger liquid /H20849LL/H20850Hamiltonian /H2084929/H20850, where the ujare the velocities of the corresponding normal modes. With the velocities /H2084930/H20850and the dimensionless Lut- tinger parameters /H2084931/H20850, the result of this linear algebra prob- lem can be written as follows. The normal-mode velocitiesu 1andu2are 2uj=1,22=vc2+vs2+2v/H92612−/H20849−1/H20850j/H20875/H20849vc2−vs2/H208502+4v/H92612 /H11003/H20875vcvs/H20873K/H92612 KcKs+KcKs K/H92612/H20874+vc2+vs2/H20876/H208761/2 , and the exponents /H9003/H20849j=1,2 /H20850appearing in F˜ /H9263/H208491,2/H20850/H20849x,/H9270/H20850are given by /H9003/H9021c/H9021c/H20849j/H20850=/H20849−1/H20850jKcvc uj/H20849u12−u22/H20850/H20873vs2−uj2−K/H92612v/H92612vs KcKsvc/H20874, /H9003/H9021s/H9021s/H20849j/H20850=/H20849−1/H20850jKsvs uj/H20849u12−u22/H20850/H20873vc2−uj2−K/H92612v/H92612vc KcKsvs/H20874, /H9003/H9021c/H9021s/H20849j/H20850=/H20849−1/H20850jK/H9261v/H9261 uj/H20849u12−u22/H20850/H20873v/H92612−uj2−KcKsvsvc K/H92612/H20874, /H9003/H9008s/H9008s/H20849j/H20850=/H20849−1/H20850jvs Ksuj/H20849u12−u22/H20850/H20873vc2−uj2−KcKsv/H92612vc K/H92612vs/H20874, /H9003/H9021c/H9008s/H20849j/H20850=/H20849−1/H20850jv/H9261 u12−u22/H20873K/H9261 Ksvs+Kc K/H9261vc/H20874. Since /H20841/H9254/H20841/H112701, we now employ the simpliﬁed expressions for the velocities in Eq. /H2084930/H20850and the Luttinger liquid param- eters in Eq. /H2084931/H20850, which are valid up to O/H20849/H92542/H20850corrections. In the interacting case, this yields for the normal-mode veloci-ties simply u 1=vcandu2=vs./H20851In the noninteracting limit, theSCHULZ et al. PHYSICAL REVIEW B 79, 205432 /H208492009 /H20850 205432-8above equation instead yields u1=vAand u2=vB, see Eq. /H208491/H20850./H20852Moreover, the exponents /H9003/H20849j/H20850simplify to /H9003/H9021c/H9021c/H208491/H20850=Kc,/H9003/H9021c/H9021c/H208492/H20850=/H9003/H9021s/H9021s/H208491/H20850=/H9003/H9008s/H9008s/H208491/H20850=0 , /H9003/H9021s/H9021s/H208492/H20850=Ks,/H9003/H9008s/H9008s/H208492/H20850=1 /Ks, /H9003/H9021c/H9021s/H208491/H20850=v/H9261 vc2−vs2/H20849K/H9261vc+Kcvs/K/H9261/H20850, /H9003/H9021c/H9021s/H208492/H20850=−v/H9261 vc2−vs2/H20849K/H9261vs+Kcvc/K/H9261/H20850, /H9003/H9021c/H9008s/H208491,2/H20850=/H11006/H9003/H9021c/H9021s/H208492/H20850. Collecting everything and taking the zero-temperature limit the functions F˜ /H9263=/H11006/H208491,2/H20850/H20849x,/H9270/H20850take the formF˜ /H9263/H208491/H20850/H20849x,/H9270/H20850=/H20879vc/H9270−ix a/H20879−Kc−2/H9263v/H9261K/H9261vc+Kcvs/K/H9261 vc2−vs2 /H11003/H20879vs/H9270−ix a/H20879−Ks+2/H9263v/H9261K/H9261vs+Kcvc/K/H9261 vc2−vs2 , /H20849A1/H20850 and F˜ /H9263/H208492/H20850/H20849x,/H9270/H20850=/H20879vc/H9270−ix a/H20879−Kc/H20879vs/H9270−ix a/H20879−1 /Ks /H11003/H20875/H20849vs/H9270−ix/H20850/H20849vc/H9270+ix/H20850 /H20849vs/H9270+ix/H20850/H20849vc/H9270−ix/H20850/H20876−/H9263v/H9261/H20849K/H9261vs+Kcvc/K/H9261/H20850 vc2−vs2 ./H20849A2/H20850 The known form of the spin-spin correlations in a LL with /H9251=0 is recovered by putting v/H9261/H11008/H9254=0. 1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 /H208492004 /H20850. 2S. Datta and B. Das, Appl. Phys. Lett. 56, 665 /H208491990 /H20850. 3Y. A. Bychkov and E. I. Rashba, J. Phys. 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In this procedure, it is crucial to understand the products of fermionic operators in the sense of operatorproduct expansions. Corrections to the leading terms are thenirrelevant in the RG sense. 54O. A. Starykh, D. L. Maslov, W. Häusler, and L. I. Glazman, in Proceedings of the International WEH Workshop , Lecture Notes in Physics, Hamburg, July 1999 /H20849Springer, New York, 2000 /H20850. 55Note that Eq. /H2084916/H20850excludes the case of ultralocal contact inter- actions. 56It is in principle possible that such a process becomes important if a collective density readjustment between subbands takesplace in the wire. However, this can only happen for almostequivalent subbands, see Ref. 54for a detailed discussion. Here we assume that the SOI is strong enough to guarantee that sucha readjustment does not occur. 57K. A. Muttalib and V. J. Emery, Phys. Rev. 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PhysRevB.93.245401.pdf	PHYSICAL REVIEW B 93, 245401 (2016) Gap and spin texture engineering of Dirac topological states at the Cr-Bi 2Se3interface H. Aramberri and M. C. Mu ˜noz Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC, Cantoblanco, 28049 Madrid, Spain (Received 18 January 2016; revised manuscript received 9 May 2016; published 1 June 2016) The presence of an exchange ﬁeld in topological insulators reveals novel spin related phenomena derived from the combination of topology and magnetism. In the present work we show the controlled occurrence of eithermetallic or gapped topological Dirac states at the interface between ultrathin Cr ﬁlms and the Bi 2Se3surface. The opening and closing of the gap at the Dirac point is caused by the spin reorientation transitions arising inthe Cr ﬁlms. We ﬁnd that atom-thin layers of Cr adhered to Bi 2Se3surfaces present a magnetic ground state with ferromagnetic planes coupled antiferromagnetically. As the thickness of the Cr ﬁlm increases stepwise from oneto three atomic layers, the direction of the magnetization changes twice from out of plane to in plane and to outof plane again. The out-of-plane magnetization drives the gap opening and the topological surface states acquirea circular meron spin structure. Therefore, the Cr spin reorientation leads to the metal-insulator transition in theBi 2Se3surface and to the correlated modiﬁcation of the surface-state spin texture. Consequently, the thickness of the Cr ﬁlm provides an effective and controllable mechanism to modify the metallic or gapped nature, as wellas the spin texture of the topological Dirac states. DOI: 10.1103/PhysRevB.93.245401 I. INTRODUCTION The recent discovery of three-dimensional (3D) topological insulators (TIs) has led to unique and fascinating physicalphenomena, such as the quantum anomalous Hall (QAH) phase [ 1] and the topological magnetoelectric effect [ 2]. The robustness of the surface metallicity under time-reversalinvariant perturbations and the realization of novel quantizedstates arising from their peculiar coupling to magnetic ﬁeldsare distinct characteristics of this new phase of quantummatter. A central feature of TIs is the existence of helicalsurface states (SSs) with the electron spin locked to the crystal momentum [ 3,4]. The presence in the TI of an exchange ﬁeld, which violates time-reversal symmetry (TRS), liftsthe Kramers degeneracy and discloses novel spin relatedphenomena directly derived from the combination of topologyand magnetism. The QAH effect has already been observedin three-dimensional magnetic TIs [ 5]. Nevertheless, the experimental realization of a magnetoelectric topological insulator, which in fact corresponds to a noninteger quantum Hall effect at the surface [ 1,6], still remains a challenge. To experimentally achieve these topological phases a surface gapped by a TRS breaking perturbation is required.There are three different ways to break TRS in TIs: either byconventional doping with magnetic elements [ 7–9], by proxim- ity to a magnetic ﬁlm at a TI-magnetic interface [ 10,11], or by an external magnetic ﬁeld [ 12]. The effect of magnetic doping with 3 dtransition metals in the Bi 2Se3family of compounds has been extensively investigated both theoretically and exper-imentally [ 7,9,13–19]. It has been shown that the interaction with magnetic impurities modiﬁes the electronic and magneticground state of the 3D TIs. However, the changes in theground state are not universal since they critically depend onthe speciﬁc magnetic atoms, occupation sites of the magneticimpurities [ 14], and experimental conditions. Cr-doped Bi 2Se3 is a prototype magnetic TI, and several works [ 7,9,13–16]h a v e reported magnetically induced effects in this system. First-principles calculations found that substitutional Cr, which isenergetically more favorable than interstitial Cr, preserves theinsulating character in the bulk and that Cr-doped Bi 2Se3 is likely to be ferromagnetic [ 20,21]. However, evidence from the experimental observations is so far inconclusive[7,13,14,17–19]. Both ferro- [ 14] and antiferromagnetism [ 7] have been reported and, recently, the coexistence of both ferro-and antiferromagnetic Cr defects in high quality epitaxialthin ﬁlms has been observed [ 22]. Nevertheless, Cr doping of bulk or thin ﬁlms of Bi 2Se3crystals seems to lead to a gap opening in the Dirac cone, evidencing time-reversalsymmetry breaking [ 7,13,14]. In contrast, surface deposition of Cr atoms on the surface of Bi 2Se3up to≈10% monoatomic layer (ML) coverages preserves the metallic surface [ 23]. The absence of a gap opening at the Dirac point indicates thatfor dilute Cr adatom concentrations there is no long-rangeout-of-plane ferromagnetic order. Despite these works, theinterface between Cr ﬁlms and the Bi 2Se3surface has not been investigated and thus the spin behavior of the topological SSunder the interaction with ultrathin Cr ﬁlms remains unknown. In the present work we explore the spin conﬁguration and topological state at the interface of Bi 2Se3surfaces and Cr ﬁlms in the ultrathin limit, one to three MLs thick. We ﬁnd that thepresence of the Cr magnetic ﬁlm triggers a double transition,from a Dirac-metal to a gapped system, on the topological SSof Bi 2Se3as a function of the Cr thickness. The gap opening at the Dirac point is induced by the proximity of the Cr ﬁlmand thus the observed modulation of the gap is associatedwith the spin reorientation occurring in the magnetic layer.In fact, the magnetization direction in the Cr ﬁlm evolvesfrom out of plane to in plane and once again to out of planeas the Cr thickness increases stepwise from one to two andthree MLs. Correlated with the gap, there is a modulation ofthe spin texture of the topological SSs, which undergoes adouble circular skyrmion to circular meron transition. II. MODEL AND METHODS Density functional theory spin-polarized calculations were carried out with the SIESTA code [ 24] as implemented in the GREEN package [ 25,26], although speciﬁc structures 2469-9950/2016/93(24)/245401(7) 245401-1 ©2016 American Physical SocietyH. ARAMBERRI AND M. C. MU ˜NOZ PHYSICAL REVIEW B 93, 245401 (2016) were also calculated with the Vienna ab initio simulation package ( V ASP )[27]. The generalized gradient approximation with Perdew-Burke-Ernzerhof [ 28] type exchange-correlation functional was used in all cases. In the SIESTA calculations, the spin-orbit coupling is considered via the recently implementedfully relativistic pseudopotential formalism [ 26], while the semiempirical pair-potential approach to van der Waals forcesof Ortmann et al. [29] was employed to correctly account for the weak interquintuple layer (QL) interaction in theBi 2Se3crystal. The numerical atomic orbitals basis set was generated according to the double ζ-polarized scheme with conﬁnement energies of 100 meV . For the computation ofthree-center integrals, a mesh cutoff as large as 1200 Ry was used, equivalent to a real-space grid resolution below 0 .05˚A3. In the V ASP calculations plane wave basis set with a kinetic energy cutoff of 340 eV was used. For the Brillouin zoneintegrations a centered 13 ×13×1ksampling was employed, while the electron temperature was set to k BT=10 meV in both calculation schemes. Bi2Se3has a rhombohedral crystal structure with space group R¯3m(D5 3d). It can be described as a layered compound constituted by QLs along the [0001] direction. A QL containsalternating Se and Bi atomic layers, and within each QL thetwo Bi layers are equivalent, while the Se in the middle isinequivalent to the external Se. The stacking pattern is fcclike, -AbCaB-CaBcA-, where capital and small letters standfor Se and Bi, respectively. The Se-Bi bonds within the QLsare mainly covalent, while at adjacent QLs the Se-Se doublelayer is only weakly bonded through van der Waals forces.The in-plane lattice parameter is a Bi2Se3=4.14˚A, while c= 9.54˚A determines the periodicity along the [0001] direction. Bulk Cr follows a bcc crystal structure with lattice pa- rameter aCr=2.91˚A. Each atom has eight nearest neighbors (NNs). Surfaces perpendicular to the [111] direction exhibitthreefold C 3symmetry and an open structure, since only six out of the eight NNs lie in the adjacent atomic layers,while the remaining two NNs are located three atomic layersabove and below. Along this direction the stacking sequencefollows an ...ABCABC... pattern, analogous to that of theBi 2Se3crystal in the [0001] direction (see Fig. 1). Cr is FIG. 1. (a) bcc unit cell of bulk Cr. Arrows indicate the magnetic moment. The two inequivalent atoms show opposite magnetic moment. (b) Side view of bulk Cr with the [111] direction as indicated in the ﬁgure. Cr exhibits an ABC stacking pattern along this direction,with opposite magnetic moments for alternating atomic planes. Six out of the eight ﬁrst nearest neighbors of each atom lie in the ﬁrst layer above and below, while the remaining two lie three layers aboveand below.unique among the 3 dtransition metals, showing an itinerant antiferromagnetic ground state. It exhibits a spin density wave(SDW) along the [100] direction—or, equivalently, along the[111] direction—with a wave vector almost commensuratewith the lattice, being its N ´eel temperature T N=311 K. Contrary to what happens in bulk crystals, in which all threecrystallographic directions are equivalent, in thin Cr ﬁlms theSDW wave vector is perpendicular to the ﬁlm surface and theSDW is commensurate with the lattice. Since the Cr-Bi 2Se3 systems studied are formed of a maximum of three Cr layers,we can consider Cr as a pure antiferromagnet in the ultrathinﬁlm regime. Thus, the Cr slabs are expected to show atomsin the same atomic layer coupled ferromagnetically, being theinterlayer coupling antiferromagnetic. We model the Cr-Bi 2Se3interfaces by 1 ×1×1 and 2 × 1×1 supercells with the equilibrium in-plane lattice constant of bulk Bi 2Se3. We take the [0001] Bi 2Se3direction as zand the (111) plane as the xyplane. Along the zdirection the supercells contain the Cr ﬁlm on top of either 4 or 6 QLs (20 or 30 atomicplanes) of Bi 2Se3and a vacuum layer larger than 20 ˚A to avoid interaction between opposite surfaces. During the structureoptimization, the Cr overlayers and the QL of Bi 2Se3closest to the interface were fully relaxed until the residual forceswere smaller than 0.02 eV /˚A, while the remaining atoms were ﬁxed to the relaxed geometry of the corresponding Bi 2Se3thin ﬁlm. III. ATOMIC STRUCTURE AND INTERFACE CHARGE TRANSFER We consider commensurate Cr ﬁlms with 1, 2, and 3 ML thicknesses on top of (111) Bi 2Se3surfaces. The atomic structure of the (111) composed slab exhibits threefold C3 symmetry and three reﬂection planes perpendicular to the surface [see Fig. 2(a)]. The in-plane lattice parameters of Bi2Se3(4.14 ˚A) and Cr (4.12 ˚A) show a small lattice mismatch of 0.5%. First, we examine different positions for the Croverlayers, including fcc and hcp (hexagonal-close-packed)hollow sites, and bridge and Se-top sites. As expected, the highsymmetry hollow sites are the energetically most favorable.Figures 2(a)–2(d) show the calculated equilibrium structures. For 1 and 2 ML ﬁlms the interfacial Cr atoms occupy thefcc hollow sites following the Bi 2Se3stacking, ...-BcAbC- A and ...-BcAbC- AB, respectively, where bold letters correspond to Cr atoms. However, for the 3 ML ﬁlm the interface Crmoves into the hcp hollow site on top of the Bi subsurfacelayer and there is a reversal of the stacking sequence [ 30], ...-BcAbC- BAC . This spatial self-organization of the Cr ﬁlm has to be due to the peculiar open structure of the (111) bccsurface in which ﬁrst NNs are in the adjacent layers andin the third layers above and below. In this way, while forthe 1 and 2 ML Cr ﬁlms the interface Cr atoms are almostcoplanar to the surface Se and lie on top of the Se in the centerof the ﬁrst QL, for the 3 ML ﬁlm the Cr-Se interface bonddistance increases notably and the Cr at the interface lies ontop of the outermost Bi. The relaxed bond lengths are givenin Table I. The Cr-Cr distances are close to the bond lengths in bulk Cr, 2.49 ˚A. Note the increase in the Cr1-Se1 bond length for the 3 ML ﬁlm. Additionally, the bond distances forthe nonequilibrium Cr trilayer in the fcc conﬁguration [see 245401-2GAP AND SPIN TEXTURE ENGINEERING OF DIRAC . . . PHYSICAL REVIEW B 93, 245401 (2016) FIG. 2. (a) Top view of the Bi 2Se3(111) surface. Dashed lines depict the mirror planes M1,M2,a n dM3, and the MandKpoints of the Brillouin zone are also indicated. (b)–(d) show the relaxed geometries for 1 to 3 ML Cr coverages on a 4 QL Bi 2Se3slab, along with arrows indicating the magnetization of the Cr layers for the magnetic ground state in each case. (e) Band structure around the center of the Brillouin zone for a pristine 4 QL Bi 2Se3slab. (f)–(h) display the band dispersion diagrams for 1 to 3 Cr MLs on a 4 QL Bi 2Se3slab in the magnetic ground state conﬁguration shown above. (i)–(k) depict the band structure of 1 to 3 Cr MLs on a 4 QL Bi 2Se3slab with the Cr MMs perpendicular to that of the magnetic ground state for each system, i.e., along xfor (i) and (k) and along zfor (j). The projection of the states o nt h eQ Lo fB i 2Se3closest to the interface is shown in red,while the projection on the Cr subsystem is shown in cyan. Fig. 3(a)] are presented at the bottom of the table. In this conﬁguration, similar to the 1 and 2 ML cases, the interface Cr atoms remain almost coplanar to the Se surface at the expense of very large NN Cr-Cr bond distances. The fcc conﬁgurationis about 80 meV more energetic than the equilibrium 3 ML TABLE I. Relaxed bond lengths in ˚A between the Cr layers (columns 2 and 3) and between the interface Cr and the interface Se (column 4). The last row corresponds to the more energetic fccconﬁguration (see Fig. 3) for the three ML Cr system. The adhesion energy E adsis given in eV in the rightmost column. The fcc-like case for the three ML Cr is more than 80 meV less stable. Cr3-Cr2 Cr2-Cr1 Cr1-Se1 Eads 1 ML 2.39 −2.00 2 ML 2.59 2.40 −1.98 3 ML 2.64 2.48 2.84 −1.77 3 ML-fcc 3.11 3.42 2.39 −1.69Cr-Bi 2Se3structure, well above the energy involved in room temperature ﬂuctuations. The calculated binding energies are also given in the table. The binding energy Eadsis obtained as Eads=ECr-Bi 2Se3−EBi2Se3−ECr, (1) where ECr-Bi 2Se3is the total energy for the composed Cr-Bi 2Se3 system, EBi2Se3is the total energy of the isolated 4 QL Bi 2Se3 system, and ECris the total energy of the isolated Cr subsystem in the same ionic and magnetic conﬁguration as it acquires inthe composed Cr-Bi 2Se3system. We found a negative value for the adhesion energy for all the Cr ﬁlms in correspondencewith the exothermic character of dilute Cr adsorbed on Bi 2Se3 surfaces for submonolayer coverages [ 23,31]. The different atomic conﬁguration of the equilibrium struc- tures is clearly reﬂected in the interface charge redistribution.We have calculated the Mulliken charges for the Cr-Bi 2Se3 systems and for the corresponding isolated slabs, a pristine4Q LB i 2Se3slab, and the isolated Cr ﬁlms of 1, 2, and 3 Cr MLs with the same atomic and magnetic conﬁguration as 245401-3H. ARAMBERRI AND M. C. MU ˜NOZ PHYSICAL REVIEW B 93, 245401 (2016) FIG. 3. (a) Side view of the 3 Cr ML on the Bi 2Se3(111) surface with the Cr layers following the stacking pattern of Bi 2Se3, i.e., with the ﬁrst Cr layer occupying the fcc hollow site. The arrows indicate the magnetic ground state for this ionic conﬁguration. Note that the ionic conﬁguration shown in Fig. 2(d) is more stable for the Cr trilayer. The band structure of a 4 QL Bi 2Se3slab with a Cr trilayer in the ionic and magnetic conﬁguration depicted in (a) is shown in (b). The projection of the states on the QL of Bi 2Se3closest to the interface is s h o w ni nr e d ,w h i l et h ep r o j e c t i o no nt h eC rs u b s y s t e mi ss h o w ni n cyan. they present when adhered to Bi 2Se3. The differences between the Mulliken charges of all the Cr-Bi 2Se3systems and those corresponding to the isolated subsystems are displayed inFig.4. In all the cases the charge transfer is small and mostly conﬁned to the Cr ﬁlm and the ﬁrst Bi 2Se3QL. For 1 and 2 Cr ML coverages, the Cr layers acquire charge at the expenseof the Se atoms, both at the interface and in the middle ofthe ﬁrst QL. In the three Cr ML system, on the contrary,the charge transfer is towards the Bi 2Se3. The interfacial Cr donates charge, mainly to the NNs. Se gains electron charge,increasing its ionic radius and consequently increasing theinterface bond length. This different behavior can be attributedto the different adsorption site of the ﬁrst Cr layer (hcp hollowversus fcc hollow for 1 and 2 Cr MLs). Nevertheless, thereis always a chemical interaction at the interface. In addition,the Bi 2Se3free surface presents a small charge gain in all the calculated structures. FIG. 4. Mulliken charge rearrangement of the 1 (cyan), 2 (red), and 3 (orange) Cr ML systems on a 4 QL Bi 2Se3slab. The ﬁgure displays the atomic charge difference between the charge of the composed Cr-Bi 2Se3system and those of the isolated Cr and Bi 2Se3 subsystems. Only the Cr subsystem and the ﬁrst and last QLs are shown since the charge rearrangement in the inner QLs of Bi 2Se3is negligible. The Cr-Se interface is indicated with a dashed line as aguide to the eye. FIG. 5. Spin-resolved DOS for the Cr trilayer adhered to Bi 2Se3. The blue (black) line shows the majority (minority) spin total DOS, while the red (green) line corresponds to the DOS projected on the whole Cr trilayer for the majority (minority) spin bands and thecyan (magenta) indicates the DOS projected on the interfacemost Cr layer for the majority (minority) spin. Magnetism is patent from the difference in the majority and minority curves. IV . MAGNETIC GROUND STATE To model the magnetic ground state of Cr layers we con- sider different conﬁgurations having parallel and antiparallelcollinear Cr magnetizations both between planes and within aplane. We employed an in-plane unit cell with two atoms perplane. We ﬁnd a ferrimagnetic ground state with ferromagneticCr planes coupled antiferromagnetically for all the studied Crﬁlm thicknesses. D u et ot h e C 3symmetry of both Cr and Bi 2Se3layers, the 3 d Cr and 4 pSe orbitals hybridize, as can be clearly appreciated in the spin-resolved total density of states (DOS) for three Crlayers adhered to Bi 2Se3shown in Fig. 5. The hybridization drives the Cr states close to the Fermi level, conﬁned in anenergy region ≈1.5 eV below E F. In addition, a large energy splitting of about 4 eV between the spin-majority and thespin-minority states is obtained, and the majority Cr statesare fully occupied while the minority-spin channel is almostunoccupied. Therefore, the magnetic moments (MMs) of theCr layers are close to the Hund rule value for isolated Cr atoms.The calculated MMs, shown in Table II, are remarkably large at the surface plane ( /greaterorequalslant4μ B/atom) for all the systems, while they decrease for the subsurface Cr layers. For 1 and 2 ML Cr ﬁlms there is an appreciable induced MM on the Se and Bitopmost planes of 0 .2μ B, aligned opposite to the Cr MM at the interface, while the induced MMs in the Bi 2Se3f o rt h e3M LC r TABLE II. Magnetic moments of the Cr layers in Bohr magnetons for 1, 2, and 3 ML coverages. Cr1 (Cr3) corresponds to the Cr layer closest to the interface. μTotis the total magnetization of the whole Cr-Bi 2Se3system for each case. The last row corresponds to the more energetic fcc conﬁguration (see Fig. 3) for the 3 Cr ML system. Cr overlayers grow as a layer-by-layer ferrimagnet with in-plane ferromagnetic coupling. μCr3 μCr2 μCr1 μTot 1 ML 4.3 4.02 ML 4.2 −3.1 1.4 3M L −4.2 3.6 −3.7 −4.3 3 ML-fcc −4.9 4.7 −4.0 −3.8 245401-4GAP AND SPIN TEXTURE ENGINEERING OF DIRAC . . . PHYSICAL REVIEW B 93, 245401 (2016) ﬁlm is almost negligible in correspondence with the different chemical interaction at the interface. Note the larger MMs ofthe 3 ML Cr fcc structure due to larger interlayer distances. Since the spin-orbit coupling is included in the calculations we can determine the direction of the Cr MM relative to thecrystal lattice. The preferential orientation of the Cr magne-tization vector was obtained by comparing the total energiesof in-plane ( M x,My) and out-of-plane ( Mz) orientations of the total magnetization M(thezaxis is deﬁned normal to the surface). It is noteworthy to point out that in the groundstate within the planes the Cr atoms are always coupledferromagnetically, thus the Cr MMs are aligned within eachlayer [see Fig. 2(a)]. The easy magnetization axis for the 1 ML Cr-Bi 2Se3system lies perpendicular to the surface (out of plane), while as thethickness of the Cr ﬁlm increases a double spin reorientationtransition takes place and the magnetization direction changesto in plane for 2 ML and again to out of plane for the 3 ML Crﬁlm. A similar spin reorientation transition has been reportedin ultrathin Co ﬁlms grown on hexagonal Ru (0001) [ 32]. The magnetic anisotropy for the 1 and 2 ML Cr-Bi 2Se3systems is unusually large, of ≈25 and 35 meV , respectively, while for the 3 ML Cr-Bi 2Se3system we obtain a smaller value of 5m e V . V . TOPOLOGICAL SURFACE STATES We additionally analyze the electronic structure of the Cr- Bi2Se3slabs. Figures 2(e)–2(h) show the corresponding band dispersions around the /Gamma1point and that of the pristine Bi 2Se3 4 QL ﬁlm. The band dispersion of the pristine ﬁlm shows the topologically protected metallic surface states with the Fermilevel located at the Dirac point. However, for all the Cr-Bi 2Se3 slabs the position of the Fermi level is shifted up between0.2 and 0.4 eV with respect to the Dirac cone of the freeBi 2Se3surface, which persists in the three cases. As a result the free-surface topological SSs are always electron doped. Next, we focus on the SS when a single Cr overlayer is adhered to the Bi 2Se3surface. A large Dirac gap opens up, and the gap opening only occurs at the interface with themagnetic ﬁlm while the Dirac cone at the free Bi 2Se3surface remains, evidencing the spatially localized character of theeffect. Furthermore, our calculations reveal that the magneticeasy axis is along the out-of-plane direction as shown inFig. 2(b). Therefore, the origin of the gapped Dirac point is the exchange coupling between the TI SS and the out-of-planemagnetization of the Cr ﬁlm, which breaks TRS. As explained above, in the 2 ML system the Cr layers present an in-plane magnetization, and we do not ﬁnd anyappreciable energy difference when the in-plane magnetizationis along or normal to the vertical reﬂection planes of theBi 2Se3thin ﬁlms –[Figure 2(a)]. Thus, we discuss the results for the in-plane magnetization normal to the reﬂection planeM 1. The corresponding band dispersion around the /Gamma1point is represented in Fig. 2(g). The topological surface state survives and there is no shift in momentum space of the Dirac point,which remains at /Gamma1. However, the dispersion is no longer linear and the SS presents a large anisotropic mass. Only along the-K 2-/Gamma1-K2line, perpendicular to the mirror plane, electrons at kand−khave the same energy. The preservation of the Diracpoint can be easily understood considering that although the breaking of TRS occurs for any nonzero magnetization, theslab is invariant under a reﬂection normal to the in-planemagnetization direction, thus the reﬂection symmetry M 1 survives. This result is a clear demonstration that in order to open a gap at the Dirac cone, breaking the TRS and thethree reﬂection symmetries M 1,2,3of the Bi 2Se3lattice is required [ 33]. As in the 1 ML system, the Dirac cone at the free surface of Bi 2Se3remains unmodiﬁed but for an energy shift. For the system consisting of 3 MLs of Cr on top of the Bi2Se3thin ﬁlm, the magnetization points again along the out-of-plane direction. Therefore, its behavior is analogous tothat of the 1 Cr ML slab: a gap opens at the original Diracpoint, although the gap is smaller. Moreover, it is worth notingthat for 3 Cr MLs, the Fermi level lies exactly within thegap of the surface Dirac fermions gapped by the exchangeinteraction. For comparison, we have additionally included the disper- sion relations of the 1, 2, and 3 Cr-Bi 2Se3systems with the magnetization of the Cr layers aligned perpendicular to thatof the corresponding magnetic ground states, i.e., in planealong xfor the 1 and 3 ML Cr and out of plane along z for the 2 ML Cr case [Figs. 2(i)–2(k)]. Now, the behavior of the topological SS is just the opposite, which conﬁrms thecorrelation between the opening of the gap at the Dirac pointand the presence of a perturbation that breaks both TRS and the invariance of the system under the three reﬂection symmetries of the Bi 2Se3lattice. The crossing of the topological SS persists whenever the magnetization is aligned in plane andperpendicular to a reﬂection plane, as in the 1 and 3 Cr MLsystems [Figs. 2(i) and 2(k)]. In both cases the reﬂection symmetry M 1is preserved. On the contrary, a gap opens for the out-of-plane 2 ML Cr ﬁlm, where TRS and the three reﬂectionsymmetries M 1,2,3are broken. The mass enhancement and the induced anisotropy in the topological SS for the 1 and 3Cr MLs are also clearly appreciable. Moreover, the origin ofthe large calculated magnetocrystalline anisotropy energy isevident from the sharp contrast between the band structures ofthese excited states [Figs. 2(i)–2(k)] and their corresponding magnetic ground states [Figs. 2(f)–2(h)]. Finally, the band structure of the nonequilibrium 3 ML Cr ﬁlm with the fccstacking is shown in Fig. 3(b). As expected, there is a gap opening due to the out-of-plane magnetization, analogous tothat developed in the equilibrium 3 ML Cr-Bi 2Se3structure [see Fig. 2(h)]. These results prove that the gap opening of the topological surface states is exclusively due to the interplay of the topologyand the induced magnetization, and independent of thechemical behavior. As noted above the 1 and 2 ML Cr-Bi 2Se3 systems show similar interface chemical interactions—thecharge transfer has the same sign and similar value—andopposite to the interface interaction in the 3 ML slab (seeFig.4). Nevertheless, there is a gap in the 1 and 3 ML Cr-Bi 2Se3 systems, while in the 2 ML Cr-Bi 2Se3structure the degeneracy of the topological SS at the /Gamma1point remains. VI. SPIN TEXTURE OF THE SURFACE STATES As shown above, the magnetization of the Cr layers attached to the surface of the Bi 2Se3ﬁlm provides a local 245401-5H. ARAMBERRI AND M. C. MU ˜NOZ PHYSICAL REVIEW B 93, 245401 (2016) FIG. 6. (a)–(d) Side view of the spin texture of the surface state for Cr overlayers of 0 (pristine Bi 2Se3surface) to 3 MLs. (e)–(h) Top view of the holelike surface state for Cr coverages of 0 to 3 MLs. The expectation value for the spin is shown as an arrow at each kpoint, while the Szcomponent is additionally color coded according to the scale at the left, being the limits ±100 (30)% of the modulus of S=/radicalBig S2 x+S2 y+S2 z for the 1 and 3 (0 and 2) Cr MLs. In the 2 Cr ML system [(c) and (g)] three elliptical black solid lines depict constant energy contours. The circular meron texture is patent in the 1 and 3 ML cases, while the spin texture of the 2 Cr MLs on Bi 2Se3is an anisotropic circular skyrmion. magnetic ﬁeld, which modiﬁes the degeneracy and topology of the SS. Additionally, it induces a spin component alongthe magnetization direction and alters the spin texture of thetopological SS. We examine the spin texture of the SSs in theequilibrium Cr-Bi 2Se3systems close to /Gamma1by calculating the expected value of the spin operator. The results are displayedin Fig. 6, which also includes the spin distribution of the Dirac cone states of the pristine Bi 2Se3surface. For the latter the spin is locked perpendicular to crystal momentum, showing thedistinct helical spin texture protected by TRS, and S zvanishes close to the Dirac point. At large kthere is, however, a ﬁnite smallSzcomponent due to the trigonal warping. Szremains null along the mirror lines /Gamma1-Mand reverses its sign traversing fromKto−K, in correspondence with the trigonal symmetry of the system. The spin texture of the gapped topological SSs (1 and 3 ML Cr systems) is in sharp contrast to that of the free surface.In the vicinity of the gapped Dirac point, the states showan imbalance between S zand−Szat a given energy, and they present a signiﬁcant net out-of-plane spin polarization. Only the in-plane components reverse sign changing from k to−k. Furthermore, the upper and lower Dirac bands have opposite Sz, evidencing that the spin degeneracy is indeed lifted at the /Gamma1point. For larger k, away from /Gamma1, the induced Szcomponent gradually decreases, and the out-of-plane spin distribution results from the competition between the magneticorder that aligns the spin along the out-of-plane direction andthe spin texture imposed by the warping term which forcesadjacent Kpoints to have opposite S z. In the 2 ML Cr slab, the in-plane magnetization exhibited by the Cr layers in theinterfacial plane does not induce observable spin reorientationsof the Dirac state, and its spin texture is analogous to that of the free-surface Dirac cone. However, due to the largeanisotropy of the effective mass, the constant energy lines areno longer circular, but present an elliptical shape. Nevertheless, the SSs exhibit a well-deﬁned spin helicity and the total spin cancels in every constant energy contour. TRS breaking isevident from the spin texture of the three Cr-Bi 2Se3systems analyzed. VII. CONCLUSIONS In summary, we have found that the structural conﬁguration of ultrathin Cr ﬁlms attached to the (111) surface of Bi 2Se3is determinant to establish the topological behavior of Bi 2Se3 SSs. Due to the coupling between Cr 3 dorbitals and the Bi 2Se3electrons, the Cr interface induces simultaneous charge and magnetic doping. However, the properties of thetopological SS critically depend on the Cr ﬁlm thickness andare independent of the speciﬁc chemical interaction at theCr-Bi 2Se3interface. As the thickness of the Cr ﬁlm increases stepwise from one to three MLs, the magnetization of the Crlayers undergoes two reorientation transitions, and changesfrom out of plane (1 ML) to in plane (2 ML) and to out ofplane (3 ML) once again. For the 1 ML and 3 ML Cr-Bi 2Se3 interfaces the magnetic overlayer induces a gap at the Diracpoint, producing massive fermions at the interface. Moreover,the gap already opens for a single Cr ML, and the value of thegap depends on the absolute value of the exchange interaction.In contrast, for the 2 ML Cr system the gapless Dirac cone ispreserved. The complexity of the spin texture of gapped Diracstates signiﬁes a competition between the in-plane helical 245401-6GAP AND SPIN TEXTURE ENGINEERING OF DIRAC . . . PHYSICAL REVIEW B 93, 245401 (2016) component of the spin dictated by the spin-orbit coupling and the out-of-plane TRS breaking component induced bythe proximity to the magnetic Cr. Our results evidencethe importance of the actual structural conﬁguration of themagnetic ﬁlms and show that the thickness of the Cr ﬁlm canbe used to modify in a controlled way the metallic or gappednature of topological Dirac states and their associated spintexture.ACKNOWLEDGMENTS This work has been supported by the Spanish Ministerio de Econom ´ıa y Competitividad through Grants No. MAT2012-38045-C04-04 and No. MAT2015-66888-C3-1-R.We acknowledge the use of computational resources ofCESGA, Red Espa ˜nola de Supercomputaci ´on (RES), and the i2BASQUE academic network. We also acknowledge J. I.Cerd ´a for fruitful discussions. [1] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Science 329,61(2010 ). [2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008 ). [3] H. 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PhysRevB.79.125116.pdf	Evidence for the formation of a Mott state in potassium-intercalated pentacene Monica F. Craciun,1,2Gianluca Giovannetti,3,4Sven Rogge,1Geert Brocks,4 Alberto F. Morpurgo,1,5and Jeroen van den Brink3,6,7 1Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan 3Institute Lorentz for Theoretical Physics, Leiden University, 2300 RA Leiden, The Netherlands 4Faculty of Science and Technology and MESA /H11001Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands 5DPMC and GAP, University of Geneva, quai Ernest-Ansermet 24, CH-1211 Geneva 4, Switzerland 6Institute for Molecules and Materials, Radboud University, 6500 GL Nijmegen, The Netherlands 7Stanford Institute for Materials and Energy Sciences, Stanford University and SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA /H20849Received 5 November 2008; revised manuscript received 9 January 2009; published 24 March 2009 /H20850 We investigate electronic transport through pentacene thin ﬁlms intercalated with potassium. From temperature-dependent conductivity measurements we ﬁnd that potassium-intercalated pentacene shows me-tallic behavior in a broad range of potassium concentrations. Surprisingly, the conductivity exhibits a re-entrance into an insulating state when the potassium concentration is increased past one atom per molecule. Weanalyze our observations theoretically by means of electronic structure calculations, and we conclude that thephenomenon originates from a Mott metal-insulator transition, driven by electron-electron interactions. DOI: 10.1103/PhysRevB.79.125116 PACS number /H20849s/H20850: 73.61.Ph, 71.20.Tx, 71.30. /H11001h I. INTRODUCTION Pentacene /H20849PEN/H20850is a conjugated molecule very well known in the ﬁeld of plastic electronics for its use in high-mobility organic thin-ﬁlm transistors. 1,2Plastic electronic ap- plications rely on the fact that at low density of charge car-riers pentacene ﬁlms effectively behave as weakly dopedsemiconductors. 3–5In this regime, which is studied exten- sively, the interactions between charge carriers can be ne-glected. However, the opposite regime of high carrier densityhas remained virtually unexplored. Since pentacene forms amolecular solid with narrow bandwidth it can be expectedthat at high density the simple assumptions of independentelectron-band theories break down and the electronic corre-lations determine the electronic properties of the material, 6as it happens in other molecular systems. The origin of corre-lated behavior in these systems is the competition betweenthe energy gained by delocalizing the /H9266electrons /H20849given by the electronic bandwidth /H20850and the Coulomb repulsion be- tween two carriers on the same molecule. If the repulsionenergy is larger than the one gained on delocalization, thenthe electrons become localized and a Mott metal-insulatortransition takes place. Among the most studied molecularsystems in the high carrier density regime are the intercalatedC 60crystals7and the organic charge-transfer salts, where the electronic interactions lead to the appearance of highly cor-related magnetic ground states and unconventionalsuperconductivity. 8,9 Our goals are to investigate pentacene compounds at high carrier density—of the order of one carrier per molecule—and to show that electron correlation effects are crucial tounderstand the resulting electronic properties. To this end,we have studied electronic transport through high-qualitypentacene thin ﬁlms similar to those used for the fabricationof ﬁeld-effect transistors. In order to reach high densities of charge carriers we intercalate the pentacene ﬁlms with potas-sium atoms to form K xPEN. Several past experimental stud- ies have addressed the possibility to chemically dope penta-cene thin ﬁlms through the inclusion of alkali atoms andiodine. For all these compounds the structural investigationshave shown that large concentrations of atoms /H20849up to three iodine atoms per pentacene molecule /H20850can intercalate in be- tween the planes of the pentacene molecular ﬁlms. Similar tothe case of intercalated C 60,10the alkali atoms donate their electrons to the lowest unoccupied molecular orbital/H20849LUMO /H20850of the pentacene molecules, whereas the iodine do- nate holes to the highest occupied molecular orbital/H20849HOMO /H20850, enabling the control of the conductivity of the ﬁlms. Earlier studies 11–18indicate that upon iodine or ru- bidium intercalation the conductivity of pentacene ﬁlms canbecome large /H20849in the order of 100 S/cm /H20850and exhibit a me- tallic temperature dependence. The experiments so far, how-ever, have not led to an understanding of the doping depen-dence of the conductivity /H20849e.g., how many electrons can be transferred upon intercalation /H20850or of the microscopic nature of electrical conduction of doped pentacene in the high car-rier density regime. In this paper we present the experimental investigation of the evolution of the temperature-dependent conductivity ofpotassium-intercalated pentacene thin ﬁlms, with increasingtheir doping concentration. We ﬁnd that, upon K intercala-tion, PEN ﬁlms become metallic in a broad range of dopingconcentrations, up to K 1PEN, after which the conductivity re-enters an insulating state. Our experiments also show thatthe structural disorder of PEN ﬁlms plays an important roleon the transport properties of K xPEN, as ﬁlms of poor struc- tural quality do not exhibit metallic behavior. The analysis ofour data shows that at high carrier density the conductivity ofPHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 1098-0121/2009/79 /H2084912/H20850/125116 /H208498/H20850 ©2009 The American Physical Society 125116-1K1PEN cannot be described in terms of independent elec- trons ﬁlling the molecular band originating from the LUMO.Rather, our observations are consistent with the formation ofa Mott insulating state, driven by electron-electron interac-tions, as we show theoretically by calculating the electronicstructure of K 1PEN. II. PREPARATION OF K xPEN FILMS Our choice of working with pentacene thin ﬁlms /H20849as op- posed to single crystals /H20850is motivated by both the relevance for applications—control of doping in organic semiconduc-tors is important for future plastic electronic devices—andby the difﬁculty to grow crystals of alkali-intercalated pen-tacene, which has so far impeded this sort of investigations.All technical steps of our experimental investigations includ-ing the deposition of pentacene ﬁlms, potassium intercala-tion, and temperature-dependent transport measurementshave been carried out in ultrahigh vacuum /H20849UHV /H20850 /H2084910 −11mbar /H20850in a fashion that is similar to our previous stud- ies of intercalated phthalocyanine ﬁlms.19The use of UHV prevents the occurrence of degradation of the doped ﬁlmsover a period of days. As with the phthalocyanines, the PEN ﬁlms /H20849/H1101125 nm thick /H20850were thermally evaporated from a Knudsen cell onto a silicon substrate kept at room temperature. In order to mini-mize the parallel conduction through the silicon, we use ahigh resistive silicon-on-insulator /H20849SOI/H20850wafer as substrate. The SOI wafer consists of 2 /H9262m silicon top layer electrically insulated by 1- /H9262m-thick SiO 2layer from the silicon substrate.19Ti/Au electrodes /H2084910 nm Ti and 50 nm Au /H20850were deposited ex situ on the SOI substrates /H20851see Fig. 1/H20849b/H20850/H20852. After the deposition of the electrodes and prior to loading the sub-strate into the UHV system, a hydrogen-terminated Si sur-face was prepared by dipping the SOI surface in a hydrof-luoric acid solution and rinsing in de-ionized water. The useof such a H-terminated silicon surface proved necessary toachieve sufﬁcient quality in ﬁlm morphology, as we will dis-cuss in Sec. IIIin more detail. Special care was taken to chemically purify pentacene prior to the ﬁlm deposition. As-purchased pentacene powderwas puriﬁed by means of physical vapor deposition in a tem-perature gradient in the presence of a stream of argon gas asdescribed in Ref. 20. After this step, the pentacene powder was loaded in the Knudsen cell in the UHV system and wasfurther puriﬁed by heating it at a temperature just below thesublimation temperature for several days. The ﬁlm thicknesswas determined by calibrating the pentacene deposition rateex situ using an atomic force microscope /H20849AFM /H20850. Potassium doping was achieved by exposing the ﬁlms to a constant ﬂux of K atoms generated by a current-heated gettersource. The source was calibrated and the potassium concen-tration determined by means of an elemental analysis per-formed on PEN ﬁlms doped at several doping levels using ex situRutherford backscattering /H20849RBS/H20850. As shown in the top inset of Fig. 1the ratio of K atoms to PEN molecules, N K/NPEN, increases linearly with increasing the doping time, as expected. Deviations from linearity—approximately10%–20%—are due to inhomogeneity of the potassium con-centration.III. TRANSPORT PROPERTIES OF K xPEN A. Electronic transport through high structural-quality KxPEN ﬁlms The conductance of K xPEN ﬁlms is measured in situ in a two terminal measurement conﬁguration with a contact sepa-ration of approximately 175 /H9262m/H20851see Fig. 1/H20849b/H20850/H20852. The depen- dence of the conductivity on the potassium concentration,hereafter referred to as the “doping curve,” is determined fordifferent PEN ﬁlms as a function of the ratio of K atoms toPEN molecules. The doping curves for different samples arevery similar, as shown in Fig. 1/H20849a/H20850. Upon doping, the con- ductivity initially increases rapidly up to a value of /H9268 /H11011100 S /cm—in the same range as the conductivity of me- tallic K 3C60.21Upon doping further, the conductivity contin- ues to increase more slowly, reaches a maximum at a con-centration of 1 K/PEN, and then drops sharply back to thevalue of the undoped PEN ﬁlm. All of the more than 40 ﬁlmsthat we have investigated exhibit a similar behavior. The observed suppression of the conductivity of penta- cene ﬁlms at high doping /H20849for potassium concentrations higher than 1 K/PEN /H20850allows us to exclude the possibility that the conduction of the intercalated ﬁlms observed in theexperiments is due to an experimental artifact, for instance,the formation of a potassium layer on top of the pentaceneﬁlm. In fact, at doping higher than 1 K/PEN the measuredconductance, and its temperature dependence, is essentiallyidentical to what is measured for pristine ﬁlms. To understand the nature of conduction of pentacene ﬁlms at high carrier density we measured the temperature depen-dence of the conductivity for different values of potassiumconcentration /H20851see Fig. 2/H20849a/H20850/H20852. Pristine PEN ﬁlms have a very low conductivity and the measured conductance of undopedﬁlms is dominated by transport through the substrate’s2- /H9262m-thick Si top layer. The measured conductivity de- creases rapidly with lowering temperature, as expected, con-ﬁrming that undoped /H20849x=0/H20850pentacene ﬁlms are insulating. On the contrary, in the highly conductive state—for xbe- tween 0.1 and 1—the conductance of the ﬁlms remains highdown to the lowest temperature reached in the experiments/H20849/H110115K/H20850, indicating a metallic state. When the potassium concentration is increased beyond approximately 1 K/PEN,the conductivity again decreases rapidly with lowering tem-perature, indicating a re-entrance into an insulating state. Themetallic and insulating nature of pentacene thin ﬁlms at dif-ferent potassium concentrations is conﬁrmed by measure-ments of volt-amperometric characteristics /H20849I-Vcurves /H20850at 5 K. For xbetween 0.1 and 1 the ﬁlms exhibit linear I-Vchar- acteristics, as expected for a metal /H20851Fig.2/H20849b/H20850/H20852. On the con- trary, in the highly doped regime /H20849forx/H110221/H20850, the insulating state manifests itself in strongly nonlinear I-Vcurves and virtually no current ﬂowing at low bias /H20851Fig. 2/H20849c/H20850/H20852. There- fore, the data clearly show that pentacene ﬁlms undergo ametal-insulator transition as the density of potassium is in-creased past one atom per molecule. Since in the overdopedregime the conduction occurs through the Si layer of the SOIsubstrate, it is not possible to gain speciﬁc information aboutthe properties of the insulating KPEN ﬁlms—for instance, todetermine the electronic gap from measurements of the acti-vation energy of the conductivity—by studying dc transporton our samples.CRACIUN et al. PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-2B. Effect of structural disorder on the transport properties of K xPEN The high structural quality of the ﬁlms proves to be the essential ingredient necessary to obtain K xPEN ﬁlms which exhibit metallic conductivity. We ﬁnd that the quality of pen-tacene thin ﬁlms is highly sensitive to the choice of the sub-strate material and sufﬁcient quality can be achieved by us-ing a hydrogen-terminated Si surface. To illustrate thisimportant technical point, we show here that the structuralquality of ﬁlms deposited on a SiO 2surface has a very large impact on their electronic transport, with low quality result-ing in considerably poorer electrical properties. Figure 3/H20849a/H20850shows the doping curve of PEN ﬁlms depos- ited onto 300 nm SiO 2that was thermally grown on a Si substrate. For these ﬁlms, the maximum conductivity that wemeasured experimentally is several orders of magnitudelower than the conductivity measured for ﬁlms deposited ona Si surface. In addition, /H20849on SiO 2/H20850the conductivity was always observed to decrease rapidly with lowering tempera-ture, i.e., the potassium-intercalated ﬁlms are always insulat-ing/H20851see Fig. 3/H20849b/H20850/H20852. Both the magnitude and the temperature dependence of the conductivity that we measured on SiO 2 substrates are comparable to results obtained in earlier workreported in the literature. We attribute the difference in the electrical behavior ob- served for ﬁlms deposited on Si and SiO 2substrates to the difference in ﬁlm morphology, which we have analyzed us-ing an atomic force microscope. Figure 3shows AFM im- ages of two pentacene ﬁlms of similar thickness deposited onthe SiO 2surface /H20851Fig.3/H20849c/H20850/H20852and on the hydrogen-terminated Si/H20851Fig.3/H20849d/H20850/H20852. It is apparent that very different morphologies are observed for the two substrates. PEN ﬁlms deposited onSi surfaces exhibit large crystalline grains with a commonrelative orientation and only relatively small ﬂuctuations inheight. On SiO 2, on the contrary, the grains are much smaller, randomly oriented, and they exhibit much largerheight ﬂuctuations. This conclusion is consistent with past studies 22showing that the growth and morphology of pentacene ﬁlms arestrongly inﬂuenced by the substrate surface. Speciﬁcally, forpentacene ﬁlms grown on SiO 2, a high density of nucleation centers was observed, leading to the growth of small islandsand to a high concentration of grain boundaries. On thehydrogen-terminated silicon surface, on the other hand, themuch smaller density of nucleation centers results in signiﬁ-cantly larger islands and in a reduced density of grain bound-aries. Note that the critical inﬂuence of the ﬁlm morphologyon the electrical characteristics of electron-doped pentaceneﬁlms is also supported by recent experiments studying theconduction of rubidium-intercalated pentacene ﬁlms depos-ited on glass. 17In that work, as-doped ﬁlms exhibited an insulating temperature dependence of the conductivity. How-ever, by performing a high-temperature annealing on thedoped ﬁlms, which results in an improved morphologicalquality, metallic behavior was also observed. The sensitivity of the morphology of pentacene ﬁlms to the substrate, together with the resulting effects on the elec- σ(S/cm) NK/NPEN0 0.50306090120 75150225300 0 G(µS) 1.5 12 NK/NPEN Doping time (min)0 80 16000.81.6 a 012 µm 12bc FIG. 1. /H20849Color /H20850/H20849a/H20850Conductivity /H9268and square conductance G/H17040 of three different K-doped PEN ﬁlms as a function of the ratio NK/NPEN; under the curves a pentacene molecule. Inset: NK/NPEN as a function of doping time. Schematic view /H20849b/H20850of our setup and /H20849c/H20850atomic force microscopy image of a high-quality undoped PEN ﬁlm showing large crystalline grains.0 100 200 300-4-202 T(K)Log10G(µS) 0120150300 NK/NPENG(µS)a -2 0 204 V (V)I (mA)b -2 0 2-202 V (V)I(µA)c FIG. 2. /H20849Color /H20850Temperature dependence of the conductance of potassium-intercalated pentacene ﬁlms. /H20849a/H20850The colored dots in the inset of /H20849a/H20850indicate the doping level at which the temperature- dependent conductivity measurements with corresponding colorwere performed. In black the temperature-dependent conductivityof the Si substrate is shown. The low temperature /H208495K/H20850I-Vchar- acteristics of K xPEN in the /H20849b/H20850conducting and /H20849c/H20850highly doped insulating states.EVIDENCE FOR THE FORMATION OF A MOTT STATE IN … PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-3tronic properties, is common to ﬁlms of many conjugated molecules. In fact, a similar sensitivity was found in ourearlier work on the electronic properties of alkali dopedmetal-phthalocyanine /H20849MPc/H20850ﬁlms. 19Speciﬁcally, for ﬁlms of CuPc, NiPc, ZnPc, FePc, and MnPc, the maximum conduc-tivity which can be achieved upon alkali doping when theﬁlms are deposited on SiO 2substrates is several orders of magnitude lower than the conductivity measured for ﬁlmsdeposited on a Si surface and has always an insulating tem-perature dependence. Also for alkali doped C 60ﬁlms, we observed that the surface termination of the substrate affectsthe morphology and the electronic transport properties of theﬁlms. As illustrated in Fig. 4/H20849a/H20850, the resistivity of K 3C60ﬁlms grown on Si shows a low resistivity at low temperature and atransition to a superconducting state. On the contrary, theK 3C60ﬁlms grown on SiO 2have signiﬁcantly higher resis- tivity, exhibiting thermally activated transport /H20851see Fig. 4/H20849b/H20850/H20852, without a superconducting transition. IV . INTERPRETATION IN TERMS OF A MOTT STATE OF K 1PEN The most striking aspect of our observations, namely, a sharp decrease in the conductivity starting at a carrier con-centration of one electron per molecule concomitant with there-entrance into an insulating state, has not been reported inearlier experiments on intercalated pentacene /H20849in which the density of intercalants could not be determined 14–18/H20850or in studies of pentacene ﬁeld-effect transistors with gate electro-lytes /H20849in which a metallic state has not been observed 23/H20850.I t implies that, contrary to the case of pentacene devices usedin plastic electronics, the electronic properties of pentaceneﬁlms at high carrier density cannot be described in terms ofnoninteracting electrons. In fact, even though it is known thatpentacene molecules can accept only one electron and thatdoubly negatively charged pentacene ions do not exist 24,25 /H20849i.e., in our ﬁlms charge transfer from the potassium atoms saturates at 1 K/PEN /H20850, a carrier concentration of one electron per molecule corresponds to a half-ﬁlled band and, for non-interacting electrons, should result in a metallic state. There-fore, interactions need to be invoked in order to explain ourobservations. An established scenario for the formation of an insulating state at half-ﬁlling is the one of a Mott insulator emergingfrom strong electron-electron interactions. 9,26In a Mott insu- lator a strong Coulomb repulsion prevents two electrons tooccupy the same pentacene molecule. Since at half-ﬁlling themotion of electrons necessarily requires double occupationof molecular sites, electron transport is suppressed and thesystem becomes insulating. This scenario is usually modeledtheoretically using a Mott-Hubbard Hamiltonian, which in-cludes a kinetic-energy term described within a tight-bindingscheme and an on-site repulsion term. The Mott state occurswhen this repulsion /H20849U/H20850is larger than the bandwidth /H20849W/H20850 /H20849determined by the tight-binding hopping amplitudes t/H20850.I n this case the half-ﬁlled band splits into a lower /H20849completely ﬁlled /H20850and an upper /H20849completely empty /H20850Hubbard band, sepa- rated by a /H20849Mott /H20850gap of the order of U, when the interactions are strong. It is realistic that this scenario is realized in amolecular solid such as pentacene, in which the bandwidth isexpected to be small owing to the absence of covalent bondsbetween the molecules. To substantiate the Mott-insulator hypothesis we have analyzed the electronic structure using density-functionaltheory /H20849DFT/H20850calculations to extract the parameters of the Mott-Hubbard model. A main difﬁculty in doing this is thatthe structural knowledge of the intercalated ﬁlms is incom-plete, as our ultrahigh-vacuum setup is not equipped to per-form in situ structural characterization, and ex situ character- ization is impeded by oxidation of potassium when thesample is extracted from the vacuum system where the ﬁlmsDoping time (min)σ(µS/cm) 0 40 80 1200510 a 0 100 200 300 T(K)0369σ(µS/cm)b 11µµmmd 11µµmmc FIG. 3. /H20849Color online /H20850/H20849a/H20850Conductivity /H9268measured at room temperature as a function of doping time for a 25-nm-thick penta-cene ﬁlm deposited on SiO 2./H20849b/H20850Temperature dependence of the conductivity for a pentacene ﬁlm grown on SiO 2and doped into the highest conductivity state. The conductivity is rapidly decreasingwith lowering the temperature as it is typical for an insulator. /H20851/H20849c/H20850 and/H20849d/H20850/H20852AFM images of pentacene ﬁlms grown on SiO 2and on H-terminated Si. /H20849c/H20850Small and randomly oriented grains with large height ﬂuctuations are observed when the PEN ﬁlms are depositedon SiO 2,/H20849d/H20850whereas PEN ﬁlms of similar thickness deposited on Si consist of large crystalline grains with a common relative orienta-tion and only relatively small ﬂuctuations in height. ρ(mΩ.cm) 0.00.20.40.60.8 0 50 100 150 200 T(K)K1K3 K4K6 Doping time (min)KxC60G (mS)/box2 0.00.30.6 04 0 8 0a ρ(mΩ.cm) T(K)0200400 150 200 250 30 0b superconducting transition FIG. 4. /H20849a/H20850Temperature dependence of the resistivity of a high- quality K 3C60ﬁlm grown on Si. As expected, the resistivity exhibits a superconducting transition at 18 K. The inset shows the dopingdependence of the conductivity. The conductance peak is typical ofK 3C60./H20849b/H20850Temperature dependence of the resistivity of a K 3C60 ﬁlm grown on SiO 2showing insulating behavior.CRACIUN et al. PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-4are prepared. Therefore, for the DFT calculations we take advantage of the existing structural information on interca-lated pentacene compounds and we determine the stablecrystal structure of K 1PEN using a computational relaxation procedure which reﬁnes the positions of all the atoms in theunit cell. A. Structural details of K 1PEN It is well known from previous structural studies on pen- tacene ﬁlms that the herringbone arrangement of the mol-ecules is preserved when pentacene is intercalated withiodine 11–13or with different alkali atoms14–18and that inter- calation takes place between the pentacene layers. It is alsoknown that intercalation is accompanied by a considerableexpansion of the unit-cell caxis /H20849by an amount close to the radii of the intercalated ions /H20850while the in-plane lattice pa- rameters aandbare only minorly affected. 15,16 A reliable estimate of the length of the expanded caxis is given by the sum of the radius of the alkali ion and thepristine c-axis parameter: the c-axis lattice constants that are obtained in this way for, e.g., RbPEN and CsPEN are within2% of the experimental values. For K 1PEN we construct the lattice parameters starting from two different polymorphs/H20849one with c=14.33 Å and the other with 14.53 Å /H20850using a K +ionic radius of 1.33 Å. The structures of the two poly- morphs are taken from the experimental results in Ref. 27. They differ slightly in the packing of the pentacene mol-ecules, which enables us to study the inﬂuence of realisticvariations in the packing on the electronic structure. The pre-cise length of the caxis in the K 1PEN is not critical for the resulting electronic structure. Our relaxation and band-structure computations were checked for values up to 8%larger and smaller than the estimated c-axis parameters. We found that even such relatively large variations in cdo not affect our main results /H20849i.e., the values of the calculated bandwidth Wand on-site repulsion U/H20850because the interac- tion between adjacent pentacene layers is weak. After constructing the unit cell of potassium-intercalated pentacene, using the information above to ﬁx a,b, and c,w e reﬁne the positions of the atoms by a computational relax-ation procedure. For all the electronic structure calculationswe used the Vienna ab initio simulation package /H20849 VASP /H20850 /H20849Refs. 28and29/H20850with projector augmented waves /H20849PAWs /H20850 /H20849Ref. 30/H20850and the PW91 density functional.31The self- consistent calculations were carried out with an integrationof the Brillouin zone using the Monckhorst-Pack schemewith a 6 /H110036/H110034k-points grid and a smearing parameter of 0.01 eV and a plane-waves basis set with a cutoff energy of550 eV. To determine the stable structure of K 1PEN all the atom positions in the unit cell are relaxed using a conjugate-gradient method. To avoid possible energy barriers we used anumber of different initial conﬁgurations. In the relaxationprocedure ﬁrst the forces on the K ions are calculated andthen the K positions are relaxed. We observe that the dopantsmove into high-symmetry positions in the plane between thepentacene layers. In the next step the positions of allatoms in the unit cell are relaxed—including the ones of the twoPEN molecules. The ﬁnal stable structure is the same for alldifferent initial conﬁgurations. 32The optimized structure of K 1PEN is shown in Fig. 5.W e checked the reliability of the relaxation procedure on un-doped pentacene and found that the calculated structure in-deed corresponds to the actual known crystal structure of thematerial. In K 1PEN there are two inequivalent PEN mol- ecules per unit cell, just as in the undoped compound. Inter-calation changes the detailed molecular orientations in theunit cell /H20849see Fig. 5/H20850, but we do not observe the formation of superstructures such as, for instance, molecular dimers. Forthe two distinct pentacene polymorphs 27for which we have performed the relaxation procedure, we found that the con-clusions on electronic bandwidths and Coulomb interactionsthat will be presented hereafter hold equally well. B. Electronic structure and electronic correlations in K 1PEN Figure 6/H20849a/H20850shows the DFT band structure of K 1PEN to- gether with the projected density of states on the pentaceneand potassium orbitals for the polymorph associated with therelaxed structure of Fig. 5. The Fermi energy lies in the middle of a half-ﬁlled band that is entirely of pentacene char-acter, originating from its LUMO. The potassium derivedelectronic states are present only at much higher energy,demonstrating that little hybridization takes place and thatthe role of the potassium atoms is limited to transferring itselectrons to the pentacene molecules. The total bandwidth isW=0.7 eV. From a tight-binding ﬁt of the band dispersion /H20851see Fig. 6/H20849b/H20850/H20852we extract the hopping amplitudes t ijthat enter the kinetic-energy part of the Mott-Hubbard Hamil-tonian H=/H20858 ieini+/H20858 /H20855ij/H20856,/H9268tij/H20849ci,/H9268†cj,/H9268+ H.c. /H20850+U/H20858 ini,↑ni,↓, where we have two molecules in the unit cell with on-site energy ei, the electron creation /H20849annihilation /H20850operators on siteiareci,/H9268†/H20849ci,/H9268/H20850, with /H9268as the electron spin, H.c. is the Hermitian conjugate, ni,/H9268=ci,/H9268†ci,/H9268,ni=/H20858/H9268ni,/H9268, and Uis the effective Coulomb interaction between two electrons on thesame molecule. The hopping integrals t ijare different in dif- ferent directions and between nearest- and next–nearest-neighbor molecules /H20849see Table I/H20850. The resulting electronic a b FIG. 5. /H20849Color /H20850Crystal structure of potassium-intercalated pen- tacene K 1PEN obtained from ab initio computational relaxation with /H20849a/H20850showing the herringbone of the PEN molecules and K atoms in the unit cell and /H20849b/H20850is a side view of the stacked layers of PEN and K, illustrating the potassium intercalation in between themolecular planes. The unit-cell parameters are a,b,c =6.239,7.636,15.682 Å and /H9251,/H9252,/H9253=76.98° ,88.14° ,84.42°.EVIDENCE FOR THE FORMATION OF A MOTT STATE IN … PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-5band structure /H20851Fig. 6/H20849a/H20850/H20852displays only very minor differ- ences for the two stable polymorphs and within the presentaccuracy the tight-binding parameters are the same. In order to determine the relative strength of electronic correlations and to compute the magnetic exchange interac-tions, the on-site Coulomb interaction U barefor two electrons on the same pentacene molecule is determined using thetechniques described in Ref. 6. For this the total energy of neutral and charged pentacene molecules is calculated bydensity-functional calculations in the local-density approxi-mation /H20849LDA /H20850using GAMESS with a double zeta plus polar- ization basis set /H20849DZVP /H20850basis set.34The bare value of the Coulomb interaction is found to be Ubare=3.50 eV. In the solid this value is screened, leading to a lower value U.35–37 From the eigenvalues of the charged molecule that is placed inside a cavity of an homogeneous dielectric medium withdielectric constant of 3.3 using the surface and simulation of/H20849volume /H20850polarization for electrostatics /H20851SS/H20849V/H20850PE/H20852model, 6,38one ﬁnds U=1.45 eV. We have also performed an indepen- dent estimate for the value of Uby considering the differ- ence between the band gap of pristine pentacene fromdensity-functional calculations /H208490.7 eV /H20850and its experimental value /H208512.2 eV /H20849Ref. 39/H20850/H20852, which gives U/H110151.5. These two values, determined in two very different ways, are remark-ably close. Very similar values for Uare found also for the second polymorph used in our calculations, indicating thatthese values are not very sensitive to differences in thestructure. 40 From a straightforward self-consistent mean-ﬁeld decou- pling computation on the resulting Hubbard Hamiltonian theground state is found to be a Néel ordered antiferromagnet,with a charge gap of 1.23 eV. The antiferromagnetic ex-change between neighboring molecules in the plane is J =4t /H20849a/H11006b/H20850/22/U/H11229290 K. This value is actually an underestima- tion of the Heisenberg exchange, as certainly a nearest-neighbor Coulomb interaction Vis also present, which has the effect of increasing the value of exchange by a factor ofU//H20849U−V/H20850. 41We ﬁnd that the coupling between molecules in neighboring planes, J/H11036, is 4 orders of magnitude lower than the in-plane J. Consequently K 1PEN is a quasi-two- dimensional antiferromagnet. Finally, an antiferromagneticexchange of /H1122940Kis also present between in-plane next- nearest-neighbor molecules along the aaxis, leading to a weak frustration of the magnetic Néel ordering. V . DISCUSSION AND CONCLUSIONS Using the results of the electronic structure calculations we are now in a position to validate the Mott-state hypoth-esis. With a ratio U/W/H112292.1, electron-electron interactions cause the splitting of the LUMO band and the opening of aMott gap, as shown in Fig. 6/H20849c/H20850. The gap explains the ob- served re-entrance to the insulating state. Note that the Mott-state scenario also explains why the insulating state is onlyobserved for a potassium concentration of 1.1–1.2 atoms permolecule /H20849and not at exactly one /H20851see Fig. 1/H20849a/H20850/H20852/H20850. In fact, at exactly one potassium per pentacene molecule, nonunifor-mity in the potassium concentration—estimated to be ap- proximately 10%–20% in our ﬁlms—effectively dopes theMott insulator causing the conductivity to remain large.However, even in the presence of nonuniformity, a potassiumconcentration slightly larger than 1 K/PEN results in a uni-form electron concentration exactly equal to one electron per molecule since, as we mentioned earlier, only one electron/H20849and not two /H20850can be donated to each pentacene molecule. 24,25It should be noted that imperfections in the material, either due to disorder of the dopants or in the mo-lecular arrangements, will lead to the presence of both disor-der in the bandwidth and a local disorder potential. In gen-eral the physics of disordered Mott-Hubbard systems is veryrich, 42but it is not a priori clear how relevant disorder will be in the present situation as we ﬁnd that the Mott state inK 1PEN is stabilized by a substantial electronic gap. The situ- ation is similar for the detailed dependence of the transportproperties on potassium concentration. The doping curves,Fig.1for instance, show a shoulder/peak in the conductivity at low density of unknown origin. It is clear on the otherXΓ-101 a* bc ΓXZYPEN K EF -101Energy (eV) dEnergy (eV) MY Γ MΓ ZNH O M HUarb. unitx10b a c FIG. 6. /H20849Color /H20850Results of electronic structure calculations for K1PEN. /H20849a/H20850Single-particle band structure, with the Fermi level EF /H20849green line /H20850as the zero of energy /H20849Ref.33/H20850. Valence and conduction bands are indicated by the thick blue lines. /H20849b/H20850Carbon /H20849blue/H20850and potassium /H20849orange /H20850projected density of states. /H20849c/H20850Tight-binding ﬁt to the valence and conduction bands /H20849blue thin lines /H20850and the result- ing lower and upper Hubbard bands from a mean-ﬁeld analysis ofthe corresponding Hubbard Hamiltonian with U=1.45 eV /H20849red thick lines /H20850. The arrows indicate the opening of the Hubbard gap. /H20849d/H20850Reciprocal lattice vectors and the ﬁrst Brillouin zone of K 1PEN. TABLE I. Tight-binding ﬁt parameters to the ab initio band structure of the half-ﬁlled conduction or valence band of KPEN.The on-site energy difference is denoted by eand the hopping pa- rameters along the a,b, and caxes are denoted by t. Parameter meV Parameter meV Parameter meV e 39 t a −33 tb −11 tc 1 t2a −1 ta+b 1 ta−b −9 ta+c −6 tb+c 3 ta+b+c −5 t/H20849a+b/H20850/2 −96 t/H20849a−b/H20850/2 90 t/H208493a+b/H20850/2 −4 t/H208493a−b/H20850/2 9 t3a/2+b/2+c 1 ta/2+3b/2+c −3 t3a/2+3b/2+c −2 t3/H20849a+b/H20850/2 −3 t3/H20849a−b/H20850/2 2CRACIUN et al. PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-6hand that in the presence of strong electron-electron interac- tions and impurity scattering the conductivity needs not belinear in carrier density. 43 Since the coupling between pentacene molecules in dif- ferent layers is very small and the electron-electron interac-tion is sufﬁciently large, the low-energy effective electronicHamiltonian of the K xPEN reduces to the well-known two- dimensional tJmodel9with t/J/H110153–4. Interestingly, the same tJmodel in the same coupling regime describes an- other important class of materials, namely, strongly corre-lated cuprate superconductors such as La 2−xSrxCuO 4. An ap- parent difference between these classes of materials is that indoped organics the formation of lattice polarons is expectedto play a very important role. We conclude that temperature-dependent transport mea- surements and theoretical calculations consistently indicatethat at a doping concentration of one potassium ion per mol-ecule potassium-intercalated pentacene is a strongly corre-lated Mott insulator, whose electronic properties are domi-nated by electron-electron interactions. An immediate consequence is the emergence of magnetism. Our calcula-tions show that the magnetic interactions are dominated by alarge positive magnetic exchange J=4t 2/U/H11229290 K between electrons on nearest-neighbor molecules in the same penta-cene layer. We predict that K 1PEN is therefore an antiferro- magnet. In fact, experimental indications for the presence ofantiferromagnetism in intercalated pentacene have been re-ported in magnetic-susceptibility measurements performed inthe past, 44albeit at very low temperature. ACKNOWLEDGMENTS This work was supported by the Foundation for Funda- mental Research on Matter /H20849FOM /H20850, the Royal Dutch Acad- emy of Sciences, the NWO Vernieuwingsimpuls, theNanoNed, and the Stichting Nationale Computerfaciliteiten.We are grateful to the FOM Institute for Atomic and Molecu-lar Physics /H20849AMOLF /H20850for the RBS analysis of our samples. 1S. F. Nelson, Y.-Y. Lin, D. J. Gundlach, and T. N. Jackson, Appl. Phys. Lett. 72, 1854 /H208491998 /H20850. 2C. D. Dimitrakopoulos and P. R. L. Malenfant, Adv. Mater. /H20849Weinheim, Ger. /H2085014,9 9/H208492002 /H20850. 3H. Sirringhaus, T. Kawase, R. H. Friend, T. Shimoda, M. In- basekaran, W. Wu, and E. P. Woo, Science 290, 2123 /H208492000 /H20850. 4G. H. Gelinck, H. E. A. Huitema, E. van Veenendaal, E. 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Ruiz, B. Nickel, N. Koch, L. C. Feldman, R. F. Haglund, A. Kahn, and G. Scoles, Phys. Rev. B 67, 125406 /H208492003 /H20850. 23M. J. Panzer and C. D. Frisbie, J. Am. Chem. Soc. 127, 6960 /H208492005 /H20850. 24J. Szczepanski, C. Wehlburg, and M. Vala, Chem. Phys. Lett. 232, 221 /H208491995 /H20850. 25T. M. Halasinski, D. M. Hudgins, F. Salama, L. J. Allamandola, and T. Bally, J. Phys. Chem. A 104, 7484 /H208492000 /H20850. 26R. W. Lof, M. A. van Veenendaal, B. Koopmans, H. T. Jonkman, and G. A. Sawatzky, Phys. Rev. Lett. 68, 3924 /H208491992 /H20850. 27C. C. Mattheus, A. B. Dros, J. Baas, A. Meetsma, J. L. de Boer, and T. T. M. Palstra, Acta Crystallogr., Sect. C: Cryst. Struct.Commun. 57, 939 /H208492001 /H20850. 28G. Kresse and J. Hafner, Phys. Rev. B 47, 558 /H20849R/H20850/H208491993 /H20850. 29G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 /H208491996 /H20850. 30G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 /H208491999 /H20850. 31J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 /H208491992 /H20850. 32The data ﬁle with the resulting structure for both polymorphs are available directly from the authors. 33The points in the Brillouin zone are /H9003=/H208490,0,0 /H20850,X=/H208491 2,0,0/H20850, M=/H208491 2,1 2,0/H20850,Y=/H208490,1 2,0/H20850,Z=/H208490,0,1 2/H20850,N=/H208490,1 2,1 2/H20850,H=/H208491 2,1 2,1 2/H20850, andO=/H208491 2,0,1 2/H20850. 34M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen,S. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, Jr., J.EVIDENCE FOR THE FORMATION OF A MOTT STATE IN … PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-7Comput. Chem. 14, 1347 /H208491993 /H20850. 35G. Giovannetti, G. Brocks, and J. van den Brink, Phys. Rev. B 77, 035133 /H208492008 /H20850. 36J. van den Brink, M. B. J. Meinders, J. Lorenzana, R. Eder, and G. A. Sawatzky, Phys. Rev. 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Mori and S. Ikehata, J. Appl. Phys. 82, 5670 /H208491997 /H20850. 45A. Hansson, J. Bohlin, and S. Stafstrom, Phys. Rev. B 73, 184114 /H208492006 /H20850.CRACIUN et al. PHYSICAL REVIEW B 79, 125116 /H208492009 /H20850 125116-8
PhysRevB.76.035112.pdf	Single magnetic impurity in a correlated electron system: Density-matrix renormalization group study S. Nishimoto and P. Fulde Max-Planck-Institut für Physik Komplexer Systeme, D-01187 Dresden, Germany /H20849Received 3 March 2007; revised manuscript received 1 June 2007; published 18 July 2007 /H20850 We study a magnetic impurity embedded in a correlated electron system using the density-matrix renormal- ization group method. The correlated electron system is described by the one-dimensional Hubbard model. Athalf ﬁlling, we conﬁrm that the binding energy of the singlet bound state increases exponentially in theweak-coupling regime and decreases inversely proportional to the correlation in the strong-coupling regime.The spin-spin correlation shows an exponential decay with distance from the impurity site. The correlationlength becomes smaller with increasing the correlation strength. We ﬁnd discontinuous reduction of the bindingenergy and of spin-spin correlations with hole doping. The binding energy is reduced by hole doping; however,it remains of the same order of magnitude as for the half-ﬁlled case. DOI: 10.1103/PhysRevB.76.035112 PACS number /H20849s/H20850: 71.27. /H11001a, 75.20.Hr, 71.10.Fd, 75.30.Hx I. INTRODUCTION Although more than 40 years have passed since the dis- covery of the Kondo effect, it is still one of the most inter-esting topics in condensed matter physics; it lies at the heartof understanding strongly correlated electron systems. 1The Kondo effect, which leads to the quenching of an impurityspin, forms the basis of the physics of a single magneticimpurity embedded in a metal. In order to understand theKondo effect, the Anderson model 2has been applied with great success. In the theoretical studies, one generally as-sumes an impurity level to be embedded in a noninteractingconduction band. In the past, a system of magnetic ions coupled to /H20849strongly /H20850correlated conduction electrons has attracted con- siderable interest in connection with the heavy-fermion be-havior, namely, Nd 2−xCexCuO 4.3This raised the question whether correlations among conduction electrons affect sub-stantially the expected formation of heavy quasiparticles. 4So far, a number of authors have studied models of a singlemagnetic impurity embedded in a host of correlated conduc-tion electrons. Thereby, perturbation theory and other ap-proximation schemes were applied. 5–10For example, it was shown that the Kondo scale can increase exponentially in theweak-coupling regime with increasing interaction of the con-duction electrons. 8,10However, a quantitative theory is still missing. Moreover, the case of strongly correlated conduc-tion electrons with band ﬁlling slightly less than one-half/H20849hole doping /H20850is still an open problem. In this paper, we study a single magnetic impurity coupled to a correlated electron system. The latter is assumed to beone dimensional /H208491D/H20850and described by a Hubbard Hamil- tonian. Using the density-matrix renormalization group/H20849DMRG /H20850method, we calculate the binding energy of the impurity-induced bound state and spin-spin correlation func-tions between the impurity and the correlated electrons in thethermodynamic limit. Special attention is paid to the case ofa nearly half-ﬁlled conduction band with repulsive electron-electron interactions. For a 1D correlated host, there has beena numerical study for similar models 11,12as well as an ana- lytical study for an integrable model.13,14We hope that the present investigation will contribute to better insights.This paper is organized as follows. In Sec. II, we intro- duce our model, i.e., a magnetic impurity coupled to a Hub-bard chain. In Sec. III, we give some numerical details of theDMRG method applied here. In Sec. IV, we ﬁrst presentcalculated results for the binding energy and spin-spin corre-lation functions at half ﬁlling and discuss the effect of thehost-band correlations on the Kondo physics. Then, we con-sider the evolution of the same quantities with hole doping.Section IV contains a summary of the results and the discus-sions. II. MODEL We study a magnetic impurity coupled to a 1D correlated electron system.15The Hamiltonian consists of three terms, H=Hc+Hf+Hcf. /H208491/H20850 The ﬁrst term Hcrepresents 1D correlated electrons. Here, we describe them by a Hubbard Hamiltonian, Hc=t/H20858 i,/H9268/H20849ci+1/H9268†ci/H9268+ H.c. /H20850+U/H20858 ini↑ni↓, /H208492/H20850 where ci/H9268†/H20849ci/H9268/H20850is the creation /H20849annihilation /H20850operator of an electron with spin /H9268/H20849=↑,↓/H20850at site i, and ni/H9268=ci/H9268†ci/H9268is the number operator. Furthermore, tis the hopping integral be- tween neighboring sites and Uis the on-site Coulomb inter- action. The second term Hfis the orbital energy of the mag- netic impurity site. We assume that the impurity contains oneorbital, e.g., 4 f, and the Coulomb repulsion on the orbital U f is inﬁnite. Since double occupancies are excluded, i.e., the f orbital is either empty or singly occupied, the impurity site isgiven by H f=/H9255f/H20858 /H9268fˆ /H9268†fˆ/H9268, /H208493/H20850 with fˆ /H9268†=f/H9268†/H208491−f/H9268¯†f/H9268¯/H20850, where /H9268¯=−/H9268and/H9255f/H110210. For conve- nience, we deﬁne r=−/H9255f/U/H20849/H110220/H20850and label an electron on the impurity site as “ felectron.” The third term Hcfinvolves the interaction between the impurity site and the correlated elec-tron system. The interaction is assumed to be local and de-scribed by a hybridization like in the Anderson model, i.e.,PHYSICAL REVIEW B 76, 035112 /H208492007 /H20850 1098-0121/2007/76 /H208493/H20850/035112 /H208497/H20850 ©2007 The American Physical Society 035112-1the impurity site is hybridized with a single site /H20849denoted as site 0 /H20850of the correlated electron system. Thus, Hcf=V/H20858 /H9268/H20849c0/H9268†fˆ/H9268+fˆ /H9268†c0/H9268/H20850, /H208494/H20850 where Vis the hopping integral between the impurity site and site 0. The lattice structure is shown in Fig. 1. Values of /H20841/H9255f/H20841=2–3 eV and V=0.1–0.2 eV are typical for Ce3+ions in metals. We will work in units where t=1 and take as values /H20841/H9255f/H20841=2–3 and V=0.1–0.2, throughout. III. METHOD We employ the DMRG method, which is one of the most powerful numerical techniques for studying quantum latticemany-body systems including quantum impurity systems. 16 With the DMRG method, we can obtain ground-state andlow-lying excited-state energies as well as expectation valuesof physical quantities quite accurately for very large ﬁnite-size systems. In order to carry out our calculations, we consider N/H20849=N ↑+N↓/H20850electrons /H20849N: even /H20850in a system consisting of a chain of Lsite correlated electron system /H20849L: odd /H20850and a single-impurity site. The electron density is deﬁned as n =N//H20849L+1/H20850. Note that the number of lattice sites must be taken as L+1=4 l−2, with l/H20849/H110221/H20850being an integer to main- tain the total spin of the ground state as S=0. If one chooses it as L+1=4 l, the singlet and triplet states are degenerate. We now apply the open-end boundary conditions to the 1Dcorrelated electron system and assume that the impurity siteis hybridized with the central site of the 1D open chain. Thelatter corresponds to site 0, and sites iand − iare equivalent. In this paper, we restrict ourselves to the half-ﬁlled and hole-doped cases /H20849N/H33355L+1/H20850. Regarding quantum impurity problems, it is generally complicated for ﬁnite-size calculations to obtain accurate re-sults in the thermodynamic limit L→/H11009because of ﬁnite-size effects. In our calculations, the most problematic ﬁnite-sizeeffects are Friedel oscillations due to the open ends of theHubbard chain. Mostly, the energy scale of the Kondo phys-ics is exponentially small; nevertheless, Friedel oscillationscan persist even at the center of the chain as they decay as apower law from the edge sites. Therefore, we study severallong chains with sites L+1=62, 126, 190, 254, 318, 382, 446, and 510, and then perform the ﬁnite-size-scaling analy-sis based on the size-dependent quantities. All DMRG resultsin this paper are extrapolated to the thermodynamic limitL→/H11009. For precise calculations, we keep up to m/H110155000density-matrix eigenstates in the DMRG procedure. In this way, the maximum truncation error, i.e., the discardedweight, is 7 /H1100310 −9, while the maximum error in the ground- state energy is less than 10−8–10−7. IV . RESULTS A. System at half ﬁlling „n=1 … 1. Binding energy We ﬁrst study the binding energy between the felectron and the correlated electrons. It corresponds to an energy gaindue to the formation of a Kondo /H20849or local /H20850singlet bound state. Hence, the binding energy is given by an energy dif-ference between the ﬁrst triplet excited state and the singletground state, /H9004 B= lim L→/H11009/H9004B/H20849L/H20850, /H208495/H20850 with /H9004B/H20849L/H20850=E0/H20849L,N↑+1 ,N↓−1/H20850−E0/H20849L,N↑,N↓/H20850, /H208496/H20850 where E0/H20849L,N↑,N↓/H20850is the ground-state energy in a system of L+1 sites with N↑up-spin and N↓down-spin electrons. Note that, at half ﬁlling, the system is insulating for ﬁnite U. The bound state therefore may be from a local singlet rather thanthe Kondo singlet. Here and in the following, we will speakof a Kondo singlet only if it involves more than the centralsite of the correlated electrons. In Fig. 2/H20849a/H20850, we show the DMRG results of the binding energy /H9004 Bas a function of the Coulomb interaction Ufor various parameter sets. In total, the results for the differentparameter sets are qualitatively the same; as Uincreases, the binding energy rises rapidly for small U, reaches a maximum around U/H110154, and decreases gradually for large U. This be- havior is similar to the dependence of the effective Heisen-berg interaction on the Coulomb interaction in the half-ﬁlledHubbard model. 21Accordingly, the DMRG results show that for large values of Uthe binding energy is approximately proportional to the effective exchange coupling Jcf, between the impurity and site 0.17If we assume that the effective exchange coupling results from second-order perturbation, i.e.,Jcf=2V2 U−/H9255f, we can explain why the results for V=0.2 are about four times larger than those for V=0.1. This estimation of the effective exchange coupling is also consistent with aslight decrease of the binding energy with increasing /H20841/H9255 f/H20841. Let us now consider the behavior in the limiting cases for weak and strong interaction strengths. A magniﬁed view ofthe weak-coupling regime /H20849U/H11021t/H20850for/H9255 f=−3, V=0.2 is given in Fig. 2/H20849b/H20850. When U=0, the system is metallic and essen- tially equivalent to the single-impurity Anderson model/H20849SIAM /H20850in the Kondo limit /H20849U f/V=/H11009/H20850but asymmetric case /H20849/H9255/HS11005−Uf/2/H20850. The orbital energy of the impurity site is lower than the Fermi energy of the conduction band, so that the occupation number of the impurity site is always 1. The ex-change interaction J cfis estimated to be the order of V2//H9255F and, therefore, the binding energy is expected to be very small but ﬁnite. We estimate it to be roughly /H9004B /H1122910−7–10−6. This value is compatible with the Kondo tem- Ꜽ /BY /B4 /BO /BC /B5 /CE /D8 /CD /BP /A0 /BF /A0 /BE /A0 /BD /BC /BD /BE /BF FIG. 1. Lattice structure of the system. Open and solid circles represent correlated electron system and impurity site, respectively.The bottom numbers idenote the site index of correlated electron system and /H20841i/H20841corresponds to a distance between site iand the impurity site.S. NISHIMOTO AND P. FULDE PHYSICAL REVIEW B 76, 035112 /H208492007 /H20850 035112-2perature TKin the asymmetric SIAM.18The introduction of a ﬁnite Coulomb interaction makes the system insulating. Withincreasing U,/H9004 Bincreases gradually when U/t/H113510.2 and rapidly for U/t/H114070.2. There is a crossover from the Kondo singlet to a local singlet around U/t=0.2. Assuming an ex- ponential behavior of /H9004Bwith Uleads to a good ﬁtting of the DMRG data, i.e., /H9004B=/H20881/H9251Uexp/H20849−/H9252/U/H20850with/H9251/H112292.5/H1100310−4 and/H9252/H112290.4. Furthermore, /H9004Bincreases almost linearly in the regime U/t=0.2–2. We thus ﬁnd that the binding energy of the local singlet can be a few orders of magnitude larger thanthat of the Kondo singlet. The DMRG results for the strong-coupling regime /H20849U /H11271t/H20850with/H9255 f=−3 and V=0.2 are plotted in Fig. 2/H20849c/H20850. In this regime, the electrons are strongly localized at each site. Therefore, the system /H20851Eq./H208491/H20850/H20852can be reduced to the Heisen- berg model with Hamiltonian, Heff=J/H20858 isi·si+1+JcfSf·s0, /H208497/H20850 with J=4t2 U. The DMRG data can be ﬁtted quite well by a function /H9004B=/H9261 U−/H9255Fwith/H9261/H112293.1/H1100310−4. Despite the strong lo- calization of the electrons, the binding energy is 2 orders of magnitude smaller than the cfexchange coupling. This is so because for n=1 a spin-density wave /H20849SDW /H20850is forming in the chain for any value of U/H20849/H110220/H20850, which makes the forma- tion of the local singlet state more difﬁcult. This kind of behavior has already been observed before for J/H11022Jcf.19,20 We then note that the behavior of the binding energy for ﬁnite Uis essentially the same as that of the Néel tempera- ture in the half-ﬁlled Hubbard model.212. Spin-spin correlations In the Kondo problem, the spin degrees of freedom around the impurity play an essential role. Therefore, weinvestigate spin-spin correlations between the felectron and the correlated electrons. The correlated system is now de-scribed by the lattice model /H20851Eq./H208492/H20850/H20852, so that we are allowed to study the distance rdependence of correlation functions, like /H20855S f·sr/H20856. Let us ﬁrst derive the spin-spin correlations between the spin on the impurity site and on the central site i=0, i.e., /H20855Sf·s0/H20856. The DMRG results for various parameter sets are shown in Fig. 3/H20849a/H20850as function of the Coulomb interaction U. Since the antiferromagnetic correlation is derived from the cf exchange interaction, /H20855Sf·s0/H20856is negative for all parameter sets and Coulomb interaction strengths. The absolute value of/H20855Sf·s0/H20856increases with increasing Vand with decreasing /H20841/H9255F/H20841, as expected from the behavior of the binding energy. However, the inﬂuence of /H20841/H9255F/H20841is rather smaller. In the limit U→0,/H20855Sf·s0/H20856is antiferromagnetic but the magnitude is very small due to strong charge ﬂuctuations, when the system is metallic /H20851see inset of Fig. 3/H20849a/H20850/H20852. It reﬂects the small binding energy around U=0. The magnitude of /H20855Sf·s0/H20856increases with increasing Uand reaches its maximum value as U→/H11009, which means that one electron is localized on each site inthat limit. We consider next spin-spin correlations between a spin on the impurity site and on the next-nearest-neighbor site i=1, i.e., /H20855S f·s1/H20856. In Fig. 3/H20849b/H20850, the DMRG results for /H20855Sf·s1/H20856are shown as a function of the Coulomb interaction Ufor vari-0 0.2 0.4 0.601234[ /g15210-5]0 2 4 6 8 100123[/g15210-4] 0 100 200012[ /g15210-4]∆B/t U/t∆B/t ∆B/t U/tU /t(a) (b) (c) FIG. 2. /H20849a/H20850Binding energy /H9004Bfor/H9255f=−3, V=0.2 /H20849circles /H20850,/H9255f =−3, V=0.1 /H20849triangles /H20850, and/H9255f=−2, V=0.1 /H20849squares /H20850./H20849b/H20850Magniﬁed view of small Uregion for /H9255f=−3, V=0.2. The data are ﬁtted by a function /H9004B=/H20881/H9251Uexp/H20849−/H9252/U/H20850with/H9251/H112292.5/H1100310−4and/H9252/H112290.4. /H20849c/H20850 /H9004Bfor/H9255f=−3, V=0.2 in the strong-coupling regime /H20849U/H11271t/H20850. The data are ﬁtted by a function /H9004B/H11229/H9261 U−/H9255Fwith/H9261/H112293.1/H1100310−4.-0.05-0.04-0.03-0.02-0.010 0 2 4 6 8 1000.010.020.030.040.050246810-4-3-2-1 0246810-5-4-3-2-1 /CW /CB /CU /A1 /CB /BC /CX /CW /CB /CU /A1 /CB /BD /CX /CD /BP /D8 /B4 /CP /B5 /B4 /CQ /B5 /D0 /D3 /CV /BD /BC /B4 /A0 /CW /CB /CU /A1 /CB /BC /CX /B5 /D0 /D3 /CV /BD /BC /CW /CB /CU /A1 /CB /BD /CX FIG. 3. Spin-spin correlation functions /H20849a/H20850/H20855Sf·s0/H20856and /H20849b/H20850 /H20855Sf·s1/H20856as a function of the Coulomb interaction Ufor/H9255f=−3, V =0.2 /H20849circles /H20850,/H9255f=−3, V=0.1 /H20849triangles /H20850, and /H9255f=−2, V=0.1 /H20849squares /H20850. Inset: semilogarithmic plots of the magnitude of the spin- spin correlation functions.SINGLE MAGNETIC IMPURITY IN A CORRELATED … PHYSICAL REVIEW B 76, 035112 /H208492007 /H20850 035112-3ous parameter sets. One expects ferromagnetic correlations from the effective Hamiltonian /H20851Eq. /H208497/H20850/H20852for ﬁnite values of U, and indeed /H20855Sf·s1/H20856has positive sign for all the parameter sets and Coulomb interaction strengths. Note that the Ruderman-Kittel-Kasuya-Yosida /H20849RKKY /H20850interaction in- duces ferromagnetic correlations, as substitute for the spin-spin interaction /H20851Eq. /H208497/H20850/H20852, in the weak-coupling /H20849U/H110110/H20850and metallic regimes. However, it is difﬁcult to separate the con- tribution from RKKY and the interaction /H20851Eq./H208497/H20850/H20852. The Cou- lomb interaction dependence of /H20855S f·s1/H20856is similar to that of /H20855Sf·s0/H20856. For the same parameter sets, the value of /H20855Sf·s1/H20856is found to be slightly smaller than that of /H20841/H20855Sf·s0/H20856/H20841. This indi- cates a slow decay of the spin-spin correlation /H20855Sf·sr/H20856with distance r. It implies that the spin of the felectron is hardly screened by the spin on site 0. In addition, the inﬂuence of V on the spin-spin correlations is rather small. Note that thebinding energy depends strongly on the hybridization V. Let us now consider the distance dependence of the spin- spin correlation functions. In Fig. 4/H20849a/H20850, we plot the DMRG results for /H20855S f·sr/H20856as a function of distance r/H20849=/H20841i/H20841/H20850.W e choose three Coulomb interactions: /H20849i/H20850U=0.5 in the Kondo- singlet regime, /H20849ii/H20850U=200 in the limit of the local singlet regime, and /H20849iii/H20850U=4 in the intermediate regime where a maximal binding energy is obtained. The results for differentdistances are extrapolated to the thermodynamic limit L →/H11009. We ﬁnd that /H20855S f·sr/H20856decays slowly and the sign changes alternately with r, i.e., /H20855Sf·sr/H20856has a positive /H20849negative /H20850sign for odd /H20849even /H20850r, denoted by solid /H20849empty /H20850symbols in Fig. 4/H20849a/H20850. The interaction /H20851Eq./H208497/H20850/H20852and/or the RKKY interactions cause ferromagnetic /H20849antiferromagnetic /H20850correlations be- tween the spin of the felectron and that of the odd /H20849even /H20850 siter. The absolute value of /H20855Sf·si/H20856increases with increasing Ubecause larger Coulomb interactions stabilize the 2kF-SDW oscillation which accompanies charge localization.Since the system is in a spin-gapped ground state, an ex- ponential decay of the spin-spin correlation with distancemust be expected. In Fig. 4/H20849b/H20850, we present a semilogarithmic plot of /H20855S f·sr/H20856as a function of distance r. For a convenient comparison, we have normalized /H20855Sf·sr/H20856with respect to its value at r=0. The results can be ﬁtted with a function exp/H20849−r /H9264/H20850and thus the exponential decay of the correlation functions is conﬁrmed for all values of U. The correlation lengths are estimated as /H9264=3184, 508, 400 for U=0.5, 4, 200, respectively. They seem to be much longer than those ofother standard spin-gapped systems, e.g., /H9264=3.19 in the two- leg isotropic Heisenberg system. However, it has been foundthat in the zigzag Heisenberg chain, the correlation lengthsincrease rapidly with decreasing binding energy. 22Thus, the very large values of /H9264reﬂect exponentially small binding energies. This also means that spin-polarized electrons arewidely spread around the impurity site, i.e., the Kondoscreening effect is quite weak. Furthermore, we note that thecorrelation functions decay rapidly around r/H112290. The decay rate is dependent on the magnitude of the cfexchange interaction. B. Less than half ﬁlling „n/H110211… We are also interested in doped systems, which are metal- lic even if U/H110220. We thus investigate the properties of the model /H20851Eq. /H208491/H20850/H20852with/H9255f=−3 for various hole concentrations n=1− Nh/L, where Nhis the number of doped holes /H20849Nh /H110220/H20850. For this choice of /H9255f, the occupation number of the impurity site is near unity because the Fermi level lies well above /H9255f. In the strong-coupling limit /H20849U/H11271t/H20850, doubly occu- pied sites are excluded and therefore we can derive an effec- tive model /H20851Eq. /H208491/H20850/H20852by applying degenerate perturbation theory.5The effective Hamiltonian is written as H=Ht+HJ+Hp+HK+H/H11032. /H208498/H20850 Here, Htis the kinetic-energy term of the conduction elec- trons, Ht=/H20858 i/H9268ti/H20849cˆi+1/H9268†cˆi/H9268+cˆi/H9268†cˆi+1/H9268/H20850, ti=−t 2/H208751−V2/H208492+2 r+r2/H20850 2/H9255f2/H208491+r/H208502/H9254i0/H20876, /H208499/H20850 with cˆi/H9268†=ci/H9268†/H208491−ni/H9268/H20850. Furthermore, HJis a spin-coupling term between the conduction electrons, which is of the Heisenberg type, HJ=Ji/H20858 isi·si+1, Ji=2t2 U/H208751−V2 /H9255f/H20849U−/H9255f/H20850/H9254i0/H20876. /H2084910/H20850 The sum of these two terms deﬁnes the 1D correlated elec- tron system. It is essentially equivalent to a t-Jmodel except for small modiﬁcations around site 0 due to the impurity. Theterm H pcorresponds to the one-particle potential around the impurity site, which is given by0 20 40 60 80 100-0.2-0.10-0.04-0.0200.020.04 /CW /CB /CU /A1 /CB /D6 /CX /D0 /D3 /CV /CJ /CW /CB /CU /A1 /CB /D6 /CX /BP /CW /CB /CU /A1 /CB /BC /CX /CL /D6 /B4 /CP /B5 /B4 /CQ /B5 /CD /BP /BG /BE /BC /BC /BC /BM /BH FIG. 4. /H20849a/H20850Spin-spin correlation functions /H20855Sf·sr/H20856as a function of the distance rforU=0.5 /H20849triangles /H20850,4 /H20849squares /H20850, and 200 /H20849circles /H20850./H20849b/H20850Semilogarithmic plot of the magnitude of the spin-spin correlation functions. The data are ﬁtted by a function /H20855Sf·sr/H20856 /H11229exp/H20849−r /H9264/H20850with/H9264=3184, 508, 400 for U=0.5, 10, 200, respectively.S. NISHIMOTO AND P. FULDE PHYSICAL REVIEW B 76, 035112 /H208492007 /H20850 035112-4Hp=−/H9257V2 2/H9255f/H208491+r/H20850/H208491−n0/H20850+V2t2 /H9255f2U/H208491+r/H208502/H20858 i=±1/H208491−ni/H20850, /H9257=2+ r+2t2 /H9255f2/H208491+r/H208502/H208492+7 r+7r2+r3/H20850. /H2084911/H20850 It describes the attraction /H20849repulsion /H20850of a hole at site 0 /H208491/H20850by thefelectron. Furthermore, HKis a spin-spin interaction term in analogy to the cfexchange interaction, HK=2/H9253V2 U−/H9255fSf·s0+tV2/H208492+r/H20850 U/H9255f/H208491+r/H208502Sf·/H20858 i=±1/H20849sˆi0+sˆ0i/H20850,/H2084912/H20850 with sˆii/H11032=/H208491/2/H20850/H20858/H9251/H9252cˆi/H9251†/H9268/H9251/H9252cˆi/H11032/H9252†, where /H9268/H9251/H9252are the Pauli matri- ces. Furthermore, /H9253=1+2 t2//H20849U−/H9255/H208502. The last term H/H11032gives a correction to the effective model, H/H11032=2V2t2 U/H9255f2/H208491+r/H208502/H20858 i=±1Sf·/H20851si/H208491−n0/H20850−s0/H208491−ni/H20850/H20852./H2084913/H20850 The ﬁrst term of Eq. /H2084913/H20850implies an antiferromagnetic inter- action between the impurity site and site ±1 if there is a holeat site 0; on the other hand, the second term gives a correc-tion to the Kondo-type interaction, i.e., the ﬁrst term of Eq./H2084912/H20850, and the antiferromagnetic spin exchange between the impurity site and site 0 may be reduced. 1. Binding energy Of particular interest is the evolution of the binding en- ergy of the impurity-induced bound state upon hole doping.We can easily imagine that the binding energy is suppressedby hole doping due to the enhancement of charge ﬂuctuation.Thus, away from half ﬁlling, the 1D correlated system ismetallic and the bound state changes from a local singlet tothe Kondo singlet. If the bound state survives with hole dop-ing, it has a much larger energy than the standard Kondosinglet. In Fig. 5, we show the binding energy /H9004 Bas a func- tion of band ﬁlling n/H20849/H333551/H20850at/H20849a/H20850V=0.2 and /H20849b/H20850V=0.1 with /H9255f=−3 for various values of U. Filled /H20849empty /H20850symbols refer to the data for n=1/H20849n/H110211/H20850and empty symbols at n=1 rep- resent the values for inﬁnitesimally doped systems /H20849see be- low/H20850. Roughly speaking, /H9004Bis discontinuously reduced at n=1 and decreases with increasing hole doping for all cases except U=0. We ﬁnd, however, that /H9004Bremains of the same order of magnitude as in the half-ﬁlled case even at dopinglevel up to a few percent. Also, the dependence of /H9004 BonUis weaker for higher doping concentrations. More precisely, there are two differences in behavior on the hybridization strength V. One is that in the vicinity of n=1, the binding energy for V=0.2 decreases more rapidly than that for V=0.1 despite larger cfexchange coupling /H20851Eq. /H2084912/H20850/H20852. It must be associated with the attraction between doped holes and the felectron, which is described in detail in the next paragraph. The other is that the binding energy dis-appears at lower doping levels for small values of V;/H9004 Bfor V=0.2 maintains its value at n/H113510.9 and that for V=0.1 goes to zero around n/H112290.8–0.9. It results from the size of the cf exchange coupling Jcf, and thus the critical doping concen- tration is highest at U/H110154, giving a maximal value of Jcf.For the limit n→1, we have extrapolated the ﬁnite-size binding energy /H9004B/H20849L/H20850to the thermodynamic limit L→/H11009for the four-hole-doped system by going up to L+1=510. One notices that the value of the binding energy in the limit n →1 differs from the n=1 undoped value. It reﬂects the fact that the binding energy of the Kondo singlet in the inﬁnitesi-mally doped system is less than that of the local singlet in theundoped system. The reason being that, when the system isdoped by a hole, the carrier tends to move onto site 0 due tothe attraction from the impurity site /H20851Eq. /H2084911/H20850/H20852and thus a spin-singlet formation is prevented. In Fig. 5/H20849c/H20850, we show the hole density n hi=1− niforV=0.2 and U=4 for the 1% hole- doped case. One can see that the doped holes concentratearound the impurity site. The discontinuity is higher for V =0.2 than for V=0.1 because the attractive interaction is en- hanced by the hybridization V. Such a discontinuity of the spin-excitation energy has also been found in studies of lad-der systems. 23,24Note that in the hole-doped case, the Vde- pendence of the binding energy is not simple because Ven- hances two competing effects: /H20849i/H20850the attraction between doped holes and the felectron and /H20849ii/H20850thecfexchange cou- pling between conduction electrons and the felectron. 2. Spin-spin correlations Finally, we study the hole-doping dependence of spin-spin correlations between the fand conduction electrons. The cor- relation is expected to be weakened by hole doping due to anincrease of charge ﬂuctuations. In Fig. 6, we show the spin- spin correlation functions /H20855S f·s0/H20856and /H20855Sf·s1/H20856as a function of band ﬁlling n/H20849/H333551/H20850when /H20849a/H20850V=0.2 and /H20849b/H20850V=0.1 with /H9255f =−3 for various Coulomb interaction strengths. The proper- ties are fundamentally linked to those of the binding energy0.8 0.9 102468[/g15210-5] 0.8 0.9 10123[/g15210-4] /A1 /BU /BP /D8 /D2 /D2 /B4 /CP /B5 /B4 /CQ /B5 -30 -20 -10 0 10 20 3 0 i /D2 /CW /BP /BC /BM /BC /BD /BH /D2 /BP /BC /BM /BL /BL /B4 /CR /B5 FIG. 5. Binding energy /H9004Bfor/H20849a/H20850V=0.2 and /H20849b/H20850V=0.1 with /H9255f=−3 as a function of the band ﬁlling n. The Coulomb interaction strengths are U=0 /H20849crosses /H20850,2 /H20849triangles /H20850,4 /H20849circles /H20850, and 10 /H20849squares /H20850. Filled /H20849empty /H20850symbols correspond to the data for n=1 /H20849n/H110211/H20850, and empty symbols at n=1 represent the values for inﬁni- tesimally doped systems. /H20849c/H20850Calculated hole density nhi=1− nifor V=0.2, U=4, and n=0.99. The size of a dot is proportional to the hole density and is explicitly shown for h=0.015.SINGLE MAGNETIC IMPURITY IN A CORRELATED … PHYSICAL REVIEW B 76, 035112 /H208492007 /H20850 035112-5as follows: /H20849i/H20850correlations are suppressed by hole doping and /H20849ii/H20850there exists a discontinuity at n=1. Let us now investigate the DMRG results for the two V values. When V=0.2, all the correlation functions for ﬁnite U decrease rapidly close to n=1 and decay slowly when n /H113510.9. This behavior is quite similar to that of the binding energy. It is seen that /H20841/H20855Sf·s1/H20856//H20855Sf·s0/H20856/H20841decreases with de- creasing n. The small value corresponds to a rapid decay of /H20855Sf·sr/H20856around r=0, as seen in Fig. 4, e.g., for U=0.5 and n=1. It is accompanied by a transfer from the local singlet to the Kondo singlet. It also suggests a reduction of the RKKYinteraction with doping. In addition, it is surprising that/H20855S f·s0/H20856seems to be enlarged by hole doping for small values ofU/H20849/H113512/H20850. The “exchange hole” around the impurity is as a consequence of the Pauli principle. When V=0.1, all the cor- relation functions decrease monotonously and go to zeroaround n/H112290.9, which is accompanied by a vanishing of the binding energy. For n/H113510.8−0.9, /H20841/H20855S f·s1/H20856/H20841has small negative values for large values of U, which indicates antiferromag- netic correlations between the felectron and the spin at site1. It is derived from the ﬁrst term of Eq. /H2084913/H20850and was pre- viously suggested in Ref. 5. V . CONCLUSION Using the DMRG method, we have studied a magnetic impurity embedded in a correlated electron system, which isassumed to be the 1D Hubbard chain. At half ﬁlling, weconﬁrm that the binding energy increases exponentially inthe weak-coupling regime. There is a crossover from theKondo singlet to the local singlet. The former state involvesa wider spread of spin-polarized electrons around the impu-rity than the latter one. With increasing values of U, the binding energy has a maximum around U/H110154 and afterward decreases inversely proportional to the Coulomb interaction.Due to the formation of a singlet bound state, the spin-spincorrelation function decays exponentially with distance fromthe impurity site for all values of U/H20849/H110220/H20850. The correlation length is quite long when the binding energy is small. It becomes shorter with increasing Coulomb interaction. Forinﬁnitesimally hole doping, we ﬁnd a discontinuous reduc-tion of the binding energy and of the spin-spin correlationsfrom the values at half ﬁlling. For further doping, the bindingenergy is reduced but remains of the same order of magni-tude as in the half-ﬁlled case even for doping concentrationof a few percent. The electron-doped case is not studied here,but we expect qualitatively similar properties as for holedoping. When Ubecomes very large, the effective repulsion of electrons at site 0 is somewhat enlarged and the probabil-ity for double occupancy is correspondingly reduced due tothe presence of the impurity. 5However, there is no disconti- nuity at half ﬁlling. This is so because when an electron isadded to the half-ﬁlled system, it is distributed almost uni-formly over the 1D chain. Possible further extensions of this work include the com- putation of the speciﬁc heat away from half ﬁlling. This is ofinterest because of available experiments on Ce dopedNd 2CuO 4. However, sufﬁciently accurate calculations are not simple and will require considerble efforts. A simple exten-sion is the computation of spectral densities. ACKNOWLEDGMENT We thank T. Takimoto for useful discussions. 1A. C. Hewson, The Kondo Problem to Heavy Fermions /H20849Cam- bridge University Press, Cambridge, 1993 /H20850. 2P. W. Anderson, Phys. Rev. 124,4 1 /H208491961 /H20850. 3T. Brugger, T. Schreiner, G. Roth, P. Adelmann, and G. Czjzek, Phys. Rev. Lett. 71, 2481 /H208491993 /H20850. 4P. Fulde, V. Zevin, and G. Zwicknagl, Z. Phys. B: Condens. Mat- ter92, 133 /H208491993 /H20850. 5T. Schork and P. Fulde, Phys. Rev. B 50, 1345 /H208491994 /H20850. 6D. Poilblanc, D. J. Scalapino, and W. Hanke, Phys. Rev. Lett. 72, 884 /H208491993 /H20850.7J. Igarashi, K. Murayama, and P. Fulde, Phys. Rev. B 52, 15966 /H208491995 /H20850. 8G. Khaliullin and P. Fulde, Phys. Rev. B 52, 9514 /H208491995 /H20850. 9T. Schork, Phys. Rev. B 53, 5626 /H208491996 /H20850. 10W. Hofstetter, R. Bulla, and D. Vollhardt, Phys. Rev. Lett. 84, 4417 /H208492000 /H20850. 11K. A. Hallberg and C. A. Balseiro, Phys. Rev. B 52, 374 /H208491995 /H20850. 12S. Costamagna, C. J. Gazza, M. E. Torio, and J. A. Riera, Phys. Rev. B 74, 195103 /H208492006 /H20850. 13P. Phillips and N. Sandler, Phys. Rev. B 53, R468 /H208491996 /H20850.0.8 0.9 1 0.8 0.9 100.010.020.030.04-0.04-0.03-0.02-0.010 /CW /CB /CU /A1 /CB /BC /CX /CW /CB /CU /A1 /CB /BC /CX /CW /CB /CU /A1 /CB /BD /CX /CW /CB /CU /A1 /CB /BD /CX /D2 /D2 /B4 /CP /B5 /B4 /CQ /B5 FIG. 6. Spin-spin correlation functions /H20855Sf·s0/H20856and /H20855Sf·s1/H20856for /H20849a/H20850V=0.2 and /H20849b/H20850V=0.1 with /H9255f=−3 as a function of the band ﬁlling n. The Coulomb interaction strengths are U=0/H20849crosses /H20850,2 /H20849squares /H20850,4 /H20849triangles /H20850, and 10 /H20849circles /H20850. Filled /H20849empty /H20850symbols correspond to the data for n=1/H20849n/H110211/H20850and empty symbols at n=1 represent the values for inﬁnitesimally doped systems.S. NISHIMOTO AND P. FULDE PHYSICAL REVIEW B 76, 035112 /H208492007 /H20850 035112-614A. A. Zvyagin, Phys. Rev. Lett. 79, 4641 /H208491997 /H20850; P. Schlottmann and A. A. Zvyagin, Phys. Rev. B 56, 13989 /H208491997 /H20850. 15S. R. White, Phys. Rev. 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PhysRevB.71.115303.pdf	Effects of electron interactions at crossings of Zeeman-split subbands in quantum wires Karl-Fredrik Berggren, Peter Jaksch, and Irina Yakimenko Department of Physics and Measurement Technology, Linköping University, S-58183 Linköping, Sweden sReceived 3 November 2004; published 9 March 2005 d Recent experimental studies of Zeeman-split one-dimensional subbands in ballistic quantum wires in an in-plane magnetic ﬁeld show that additional nonquantized conductance structures occur as subbands cross atlow electron densities fA. C. Graham et al., Phys. Rev.Lett. 91, 136404 s2003 dg.These structures are called 0.7 analogs. We analyze the experimental transconductance data within the Kohn-Sham spin-density-functionalmethod, including exchange and correlation effects for an inﬁnite split-gate quantum wire in a parallel, in-planemagnetic ﬁeld B i. Energy levels are found to rearrange abruptly as they cross due to polarization effects driven by exchange and Coulomb interactions. Experimental qualitative features are explained well by this model. DOI: 10.1103/PhysRevB.71.115303 PACS number ssd: 73.21.Hb, 73.23.Ad, 72.25.Dc, 71.70. 2d I. INTRODUCTION The so-called 0.7 anomaly in the conductance Gthrough GaAs/AlGaAs quantum wires has attracted considerable at-tention during the last years. Since the anomaly was exploredin 1996 sRef. 1 dit has been observed in both GaAs/AlGaAs 2–6andp-Si wires.7Two popular theoretical models are based on the effect of spin polarization occurringat low densities, 8–13on Kondo-type conductance,14,15or on combinations of the two.16There are also propositions about electron-phonon interactions17and the formation of aWigner lattice.18Nevertheless, the complete explanation of this anomaly is still missing and it remains to be the “mesoscopicmystery” as discussed in Refs. 19 and 20. However, an im-portant observation is that the 0.7 structure evolves into a 0.5s2e 2/hdplateau when the magnetic ﬁeld is applied.1This fact favors the models based on the spin-polarization effect. As for the two-dimensional electron gas,21one should expect that exchange interactions favor parallel spin ordering at lowdensities beacuse the system gains energy in this way. As indicated there is a number of different scenarios for the 0.7 anomaly, but consensus is yet to be found. Differenttypes of measurements are therefore needed to gain freshinsight into the role of electron interactions. Fortunately,there are recent experimental data of this kind for quantumwires in GaAs/AlGaAs heterostructures in a high in-planemagnetic ﬁeld 22ssee also Refs. 23 and 24 d. A remarkable feature of these measurements is the strong Zeeman splittingbecause of an unusually large g-factor of 1.9. As a conse- quence crossings of higher Zeeman-split subbands arereadily observed with increasing magnetic ﬁeld. Figure 1shows typical experimental data for the transconductancedG/dVas function of applied gate voltage Vand in-plane parallel magnetic ﬁeld B i. As shown by Graham et al.22the gross features of Fig. 1 may be explained quite well from a simple model for nonin-teracting electrons in an inﬁnite quantum wire with a para- bolic conﬁnement U conf=mvy2y2/2+mvz2z2/2, where mis the electron effective mass and yandzrefer to the lateral and perpendicular motions at the interface. This model, whichmay be solved exactly, 22shows that a parallel magnetic ﬁeld introduces a coupling between the two motions. Figure 2shows the splitting of the ﬁve lowest sublevels with "vy =1.85 and "vz=15 meV, g=1.9, and m/me=0.067 for the GaAs effective mass.Assuming that there is a linear relation-ship between the electron density in the wire and the voltageVone may now locate the speciﬁc data points sV,B idat which subbands start to be sdedpopulated. As Graham et al. demonstrated the dark features in Fig. 1 correspond to suchpoints. There are, however, important deviations from this elementary subband model. Thus the intricate, steplike be-havior at the crossings of Zeeman-split levels at points a1, b1, etc. cannot be derived from the simple, one-electron level diagram in Fig. 2. These points are referred to as 0.7 analogstructures indicating that they are part of the same family asthe usual 0.7 conduction anomaly associated with a0at low/ zero magnetic ﬁeld.22 The purpose of this paper is to show that different con- ductance anomalies are related to electron interactions andmay be explained in terms of polarization effects. Because ofthe relative success with the noniteracting inﬁnite wire, 22as outlines above, we will therefore let that elementary modelbe our starting point. We have previously found 24that the very shape of the conﬁnement matters. In this work we have therefore modeledthe electronic structure and the onset subband occupationsfor an extended, realistic GaAs/AlGaAs quantum wire at dif-ferent gate voltages Vand an in-plane, parallel magnetic ﬁeldsB i, using the Kohn-Sham local spin-density-functional theory.25The main objective is to ﬁnd qualitative agreement FIG. 1. Typical gray scale of experimental transconductance datadG/dVat 50 mK as a function of gate voltage Vin volts and magnetic ﬁeld Biin T sadapted from Ref. 24 d. The structures at a1,a2,b1…are referred to as 0.7 conduction analogs. The splitting ata0is the usual low/zero ﬁeld 0.7 conduction anomaly sRef. 1 d.PHYSICAL REVIEW B 71, 115303 s2005 d 1098-0121/2005/71 s11d/115303 s5d/$23.00 ©2005 The American Physical Society 115303-1with experiment rather than ﬁne tuning of a particular de- vice. Our main focus is on polarization effects induced byexchange and correlation effects among the Zeeman-splitsubbands at points a1,a2,b1, etc.26 II. MODELING OFAN EXTENDED QUANTUM WIRE As mentioned, Graham et al.22analyzed the transconduc- tance data assuming that the system may to ﬁrst order beviewed as an inﬁnite parabolic wire. Because of the successof that model in explaining the gross features of the mea-sured data, we will stay with an inﬁnite wire, a choice thatsimpliﬁes the numerical work considerably. To make themodel more realistic we will, however, assume a quantumwire that is made of successive layers of GaAs, n-AlGaAs, andAlGaAs with a metallic gate covering the top of a struc-ture except for a straight slit of width was in Fig. 3. A narrow strip of electrons, whose density may be controlledby an applied gate voltage V, is formed at the GaAs/AlGaAs interface. The device is translationally invariant along the slit. We choose this direction as the xaxis and the direction normal to the heterostructure interface as the zaxis with origin in the middle of the wire as in Fig. 3. Consider the case of an in-plane magnetic parallel magnetic ﬁeld B i=Bixˆ.As we will see, it is convenient to choose the gauge as A=−Bizyˆ. The noninteracting part of the Hamiltonian is therefore H0sx,y,zd=−"2 2m„2+Uconfsy,zd+mvc2 2z2+ieB" mz] ]y +gmBBisˆ/2, s1d whereUconfsy,zdis the conﬁnement potential and vc =ueBiu/mthe cyclotron frequency. The spin operator sˆin the Zeeman term has eigenvalues s=±1 and mBis the usual Bohr magneton. The imaginary term in H0accounts for the coupling between the motions in the yandzdirections as Bi is applied. Because of the large difference between the lon-gitudinal and transverse sublevels at zero ﬁeld, for example, 1.85 and 15 meV for the parabolic wire in Fig. 2, the mag-netic coupling is relatively weak.At this stage we may there-fore ignore the magnetic interaction of the two sets of levels,a step that makes our problem separable. Indeed, the basicphysics behind the analog structures is driven by the crossingof spin-split subbands, not by the magnetic narrowing of thesubband separations. Furthermore, we may restrict ourselvesto the case that only the lowest transverse mode f1szdis occupied. Thus, the general form for the wave function is ck,lssx,yd=eikxwlsydf1szdxssd, s2d where xssdis the spin eigenfunction. The total energy of an electron in this state is therefore Elsskd="2k2 2m+El+" 2˛vc2+vz2+gmBBs/2, s3d whereElare discrete energy levels corresponding to the lat- eral motion. Here we assume that the perpendicular conﬁne-ment is parabolic as above. Because the perpendicular state f1szdis much more con- tracted than wlsydwe ignore its spatial extension. In this way our modeling turns one dimensional for the low-lying lstates of interest here. We now introduce electron interactions in terms of a spin-dependent exchange-correlation potential Uxcs and arrive at the self-consistent one-electron Kohn-Sham equation F−"2 2m]2 ]y2+UeffssydGwlssyd=Elswlssyds 4d for the lateral modes wlssyd. Because of the exchange inter- actions these modes now depend on spin s. The effective potential energy in Eq. s4dis the sum of electrostatic, Hartree, and exchange-correlation terms, FIG. 2. sColor online dCalculated Zeeman-split sublevels in meV as function a parallel magnetic ﬁeld Biin Tesla for noninter- acting electrons in a parabolic well with "vy=1.85 and "vz =15 meV, and a g-factor of 1.9. The GaAs effective mass is m/me=0.067. FIG. 3. Schematic picture of the split gate wire used in the modeling.There is an undoped GaAs cap layer s24 nm dfollowed by a doped AlGaAs layer s36 nm, doping density rD=631017cm−3d, and a spacer layer of AlGaAs s10 nm d. The electron gas resides at the interface between the GaAs substrate and the spacer layer; z0 =70 nm is the distance to the metallic gate. The electron density is controlled by an applied voltage Vbetween the gate and the sub- strate. The width wof the gate opening is 700 nm.BERGGREN, JAKSCH, AND YAKIMENKO PHYSICAL REVIEW B 71, 115303 s2005 d 115303-2Ueffssyd=Uconfsyd+UHsyd+Uxcssyd. s5d For completeness we specify the different contributions for the split-gate wire in Fig. 3. Uconfsydderives from the split gate, the charge density erDin the donor layer, and the occupied surface states at the interface between metallic gateand GaAs ssee, e.g., Ref. 9 for details d, U confsyd=−eVgsyd+s−eVdd+s−eVsd. s6d The electrostatic potental Vg, generated by the split gate, is the solution of Laplace’s equation sincluding mirror charges, see Ref. 30 d Vgsyd=VH1−1 pFarctanSw/2−y z0D+arctanSw/2+y z0DGJ, s7d and the potential created by the donor layer is Vd=erD 2ee0s2c+ddd, s8d wherecanddare the thicknesses of the cap and donor lay- ers, respectively. The effect of the surface states, ﬁnally, isincluded as a simple Schottky barrier with − eV s=0.8 eV. UHsydandUxcsderive from electron interactions. Includ- ing mirror charges to ensure that the Hartree potential van- ishes atz=z0we obtain UHsyd=e2 4pee0Ensy8dSlnsy−y8d2+4z02 sy−y8d2Ddy8 s9d for the extended wire. Here, nsy8dis the electron density, deﬁned as nsy8d=o snssy8d. s10d For a given Fermi energy EFthe density for s-spin electrons is nssy8d=1 po Els,EFS2m "fEF−ElsgD1/2 uwlssy8du2.s11d Here the summation is over all occupied states; uwlssy8du2is normalized to one. For the very important exchange and cor- relation potentialUxcs=d«xcfn",n#g dn=]sn«xcd ]ns. s12d we use a recent parametrization for the exchange-correlation energy «xc.31 The Kohn-Sham local spin-density equation s4dhas been solved for the GaAs/AlGaAs structure in Fig. 3 by iterationin the usual manner.Typically, between 50 and 500 iterationswere needed to achieve self-consistency; 150 mesh points were used for wlssyd. III. EVIDENCE OF ELECTRON INTERACTIONS The onset of subband ﬁllings occurs when the chemical potential coincides with a sublevel as the gate voltage Vand magnetic ﬁeld Biare varied. To single out the effects of exchange and correlation we have computed such data pointsin two steps. First, we have considered the mean-ﬁeld Har-tree approximation only, i.e., exchange/correlation is omittedin the self-consistent procedure. Numerical results are shownin Fig. 4. FIG. 4. sColor online dOnset of subband ﬁllings as function of magnetic ﬁeld Biand gate voltage Vin the Hartree approximation. FIG. 5. sColor online dOnset of subband ﬁllings as function of magnetic ﬁeld Biand gate voltage Vaccording to the spin- dependent Kohn-Sham equations with exchange and correlationincluded. FIG. 6. sColor online dElectron density corresponding to Fig. 5 as function of magnetic ﬁeld Biand gate voltage V. Curves show the occupation onset of the different Zeeman-split subbands.EFFECTS OF ELECTRON INTERACTIONS AT … PHYSICAL REVIEW B 71, 115303 s2005 d 115303-3Evidently the Zeeman-split sublevels cross each other without any dramatic effects at the points a1andb1shown in the graph. Furthermore, there is no splitting at the low/zero ﬁeld points a0andb0. In fact, the results are qualita- tively the same as obtained from the parabolic model fornoninteracting electrons. 22 The situation is radically changed as we include exchange and correlations.As evident from Fig. 5, we have now foundthe desired anomalous features at the crossings of the Zee-man states, and the qualitative agreement with the experi-mental data in Fig. 1 is quite satisfactory. Obviously the anomalous conductance structures derive from electron inter-actions and can be explained in detail within the frameworkof the Kohn-Sham local spin-density approximation. Figure6 shows the total densities as function of voltage and appliedmagnetic ﬁeld.We note that up-spin subbands populate twiceat the anomalies as found in experiments. 22 The discontinuous behavior of dG/dVat the Zeeman level crossings indicate that electron interactions give rise to a gapin the energy spectrum. In general, this is an expected featureat level crossings. The detailed behavior is, however, un-usual. Our modeling shows that the energy gap occurs onlyfor the "-spin states. In this way the system manages to in- crease its content of parallel #-spin electrons at a level cross- ing and thereby lower its total energy. Spontaneous polariza-tion of this kind is evidently related to spin-drivenconductance anomalies at low/zero magnetic ﬁeld s a0,b0d. As mentioned in the introduction one then expects that ex- change interactions favors parallel spin ordering at low elec-trons densities. Related phase transitions from spin-unpolarized state to a spin-polarized state have beendiscussed also for coinciding Landau levels. 32 In summary we have shown that the anomalous 0.7 con- duction analogs observed in GaAs/AlGaAs quantum wires22 are related to spontaneous spin-polarization driven by ex-change and Coulomb interactions. We have found that amodeling based on Kohn-Sham local spin-density formalismis adequate for the present purpose. However, in this workwe have restricted ourselves to the simpliﬁed case of an in-ﬁnite wire. For a detailed agreement with experiments onewould have to model the conductance in a real device withsource and drain as, for example, in Ref. 10. In principle,such a modeling is straightforward but numerically quite de-manding in practice.Although detailed agreement with mea-surements may be achieved principle features are expected toremain the same as found here. ACKNOWLEDGMENT We are grateful to Abi Graham and Michael Pepper for discussions about the experiments and for providing Fig. 1. 1K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace, and D. A. Ritchie, Phys. Rev. Lett. 77, 135 s1996 d. 2A. Kristensen P. E. Lindelof, J. B. Jensen, M. Zaffalon, J. Hollingbery, S. W. Pedersen, J. Nygard, H. Bruus, S. M. Re- imann, C. B. Sorensen, M. Michel, and A. Forchel, Physica B 251, 180 s1998 d. 3A. Kristensen, H. Bruus, A. E. Hansen, J. B. Jensen, P. E. Linde- lof, C. J. Marckmann, J. Nygård, and C. B. Sorensen, F. Beus-ches, A. Forchel et al., Phys. Rev. B 62, 10950 s2000 d. 4D. J. Reilly, T. M. Buehler, J. L. O’Brien, A. R. Hamilton, A. S. Dzurak, R. G. Clark, B. E. Kane, L. N. Pfeiffer, and K. W. West,Phys. Rev. Lett. 89, 246801 s2002 d. 5S. M. Cronenwett, H. J. Lynch, D. Coldhaber-Gordon, L. P. Kou- wenhoven, C. M. Marcus, K. Hirose, N. S. Wingreen, and V.Umansky, Phys. Rev. Lett. 88, 226805 s2002 d. 6P. Roche, J. Sgala, D. C. Glattli, J. T. Nicholls, M. Pepper, A. C. Graham, K. J. Thomas, M. Y. Simmons, and D. A. Ritchie,Phys. Rev. Lett. 93, 116602 s2004 d. 7N. T. Bagarev, Semiconductors 36, 439 s2002 d. 8C.-K. Wang and K.-F. Berggren, Phys. Rev. B 54, R14257 s1996 d;57, 4552 s1998 d. 9K.-F. Berggren and I. I. Yakimenko, Phys. Rev. B 66, 085323 s2002 d. 10A. A. Starikov, I. I. Yakimenko, and K.-F. Berggren, Phys. Rev. B 67, 235319 s2003 d. 11J. P. Bird and Y. Ochiai, Science 303, 1621 s2004 d. 12P. Havu, M. Puska, R. Nieminen, and V. Havu, Phys. Rev. B 70, 233308 s2004 d. 13P. S. Cornaglia, C. A. Balseiro, and M. Avignon, Phys. Rev. B71, 024432 s2005 d. 14Y. Meir, K. Hirose, and N. S. Wingreen, Phys. Rev. Lett. 89, 196802 s2002 d. 15K. Hirose, Y. Meir, and N. S. Wingreen, Phys. Rev. Lett. 90, 026804 s2003 d. 16D. J. Reilly, cond-mat/0403262 sunpublished d. 17G. Seelig and K. A. Matveev, Phys. Rev. Lett. 90, 176804 s2003 d. 18K. A. Matveev, Phys. Rev. Lett. 92, 106801 s2004 d. 19J. Minkel, Phys. Rev. Focus 10,2 4 s2002 d. 20F. Fitzgerald, Phys. Today 55s5d,2 1 s2002 d. 21A. Ghosh, C. J. B. Ford, M. Pepper, H. E. Beere, and D. A. Ritchie, Phys. Rev. Lett. 92, 116601 s2004 d. 22A. C. Graham, K. J. Thomas, M. Pepper, N. R. Cooper, M. Y. Simmons, and D. A. Ritchie, Phys. Rev. Lett. 91, 136404 s2003 d. 23A. C. Graham, K. J. Thomas, M. Pepper, M. Y. Simmons, and D. A. Ritchie, Physica E sAmsterdam d22, 264 s2004 d. 24A. C. Graham, K. J. Thomas, M. Pepper, D. A. Ritchie, M. Y. Simmons, K.-F. Berggren, P. Jaksch, A. Debranova, and I. I.Yakimenko, Solid State Commun. 131, 591 s2004 d. 25R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules sOxford University Press, New York, 1989 d. 26It is sometimes argued that spontaneous spin polarization cannot occur in quantum wires in zero magnetic ﬁeld becuase of theLieb-Mattis theorem sRef. 27 dfor ideal 1D systems. In the present case, however, we generally focus on multisubbandswires in ﬁnite magnetic ﬁeld, and with spin-dependent forcespresent. Therefore the Lieb-Mattis does not apply here. In by-BERGGREN, JAKSCH, AND YAKIMENKO PHYSICAL REVIEW B 71, 115303 s2005 d 115303-4passing, we also ﬁnd polarization effects at zero ﬁeld. Strictly speeking, such polarized solutions should not be accepted from amathematical point of view if the wire is in the ideal 1D limit.However, a real device is never strictly 1D. Guided by experi-mental evidence we therefore take a pragmatic view by accept-ing also these solutions and give them physical signiﬁcance. Infact, Fig. 1 shows a smooth, regular behavior of the observeddata as the magnetic ﬁeld is turned on. From a more formalpoint of view, we suggest that the polarized solutions at zeroﬁeld indicate that local spin order may extend over a large sbut not inﬁnite ddistance. In fact, the correlation length may exceed the dimensions of a real device, and for this reason the questionabout the Lieb-Mattis theorem appears less interesting. Similaraspects on the relation between symmetry and broken and symmetry-adapted solutions for many-electron quantum dots arediscussed in Refs. 28 and 29. 27E. Lieb and D. Mattis, Phys. Rev. 125, 164 s1962 d. 28S. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 s2001 d. 29I. I. Yakimenko, A. M. Bychkov, and K.-F. Berggren, Phys. Rev. B63, 165309 s2001 d. 30J. H. Davies, I. A. Larkin, and E. V. Sukhorukov, J. Appl. Phys. 77, 4504 s1995 d. 31C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Phys. Rev. Lett. 88, 256601 s2002 d. 32S. Koch, R. J. Haug, K. v. Klitzing, and M. Razeghi, Phys. Rev. B 47, 4048 s1993 d.EFFECTS OF ELECTRON INTERACTIONS AT … PHYSICAL REVIEW B 71, 115303 s2005 d 115303-5
PhysRevB.102.014454.pdf	PHYSICAL REVIEW B 102, 014454 (2020) Ultralow Gilbert damping in CrO 2epitaxial ﬁlms Zhenhua Zhang,1Ming Cheng,1Ziyang Yu,1Zhaorui Zou,1Yong Liu,1Jing Shi,1Zhihong Lu ,2,and Rui Xiong1,† 1Key Laboratory of Artiﬁcial Micro- and Nano-structures of Ministry of Education, School of Physics and Technology, Wuhan University, Wuhan 430072, People’s Republic of China 2School of Materials and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, People’s Republic of China (Received 7 April 2020; revised 6 June 2020; accepted 9 July 2020; published 30 July 2020) In this study, we report the observation of ultralow Gilbert damping in epitaxial CrO 2ﬁlms. The dynamic properties of (100)- and (110)-oriented CrO 2epitaxial ﬁlms grown on TiO 2substrates were studied using ferromagnetic resonance measurements based on a resonant cavity and a coplanar waveguide in a large frequencyrange. The Lande gfactor was found to be 1.98, and it was independent of ﬁlm orientation and thickness. The effective damping constant rapidly increased with ﬁlm thickness when the ﬁlm thickness was smaller than 50 nm,which might be attributed to magnon scattering. Extremely low damping was observed in the (110)-orientedCrO 2ﬁlm with a thickness of 364 nm, and the damping constant was obtained as (6 .2±0.4)×10−4. This value is about half an order of magnitude lower than that of ultralow-damping CoFe systems [(1 .3–2.0)×10−3]a n di s comparable with the lowest value observed recently in some Heusler alloy systems. The extremely low dampingbehavior in the CrO 2system is strongly correlated with its half-metallic nature. DOI: 10.1103/PhysRevB.102.014454 I. INTRODUCTION Magnetic half-metallic materials, predicted by de Groot et al. [1], have attracted revived research interest due to the recent remarkable developments in the spintronic ﬁeld. Suchmaterials exhibit high spin polarization near the Fermi energydue to their unique electronic band structure with one spinchannel exhibiting conductor characteristics and the otherpossessing insulator characteristics. Recent reports based onﬁrst-principle calculations have revealed that magnetic halfmetals exhibit low Gilbert damping [ 2–4]. High spin polar- ization and low damping make half-metallic materials partic-ularly promising for applications in spintronic devices such asgiant magnetoresistance (GMR) spin valves, magnetic tunneljunction (MTJ) sensors, and spin-transfer torque magneticrandom access memory (STT-MRAM). Owing to the strongcorrelation between spin-wave propagation length and damp-ing constant, low-damping materials exhibit immense poten-tial for magnonic devices. The propagation length is estimatedto vary from several micrometers to a few millimeters whenthe Gilbert damping constant reduces from 10 −2to 10−5[5]. Chromium dioxide (CrO 2) is an ideal half-metallic mate- rial with excellent conductivity and complete spin polarizationin principle, and its nearly complete spin polarization hasbeen demonstrated experimentally [ 6,7]. High quality epi- taxial CrO 2and doped CrO 2ﬁlms with enhanced thermal stability have been successfully grown on TiO 2substrates [8–10]. Owing to its great application prospects, several stud- ies have focused on CrO 2-based devices over the past few years [ 11–14]. CrO 2-based spintronic devices and heterojunc- tions, such as CrO 2/Cr2O3[11,12], CrO 2/RuO 2/CrO 2[13], zludavid@live.com †xiongrui@whu.edu.cnand CrO 2/MgO/CoFe [ 14], have been realized experimentally. Moreover, superconductor-ferromagnet (SF) hybrids basedon CrO 2have been successfully fabricated, which are ex- tremely promising for superconducting spintronic applica-tions [ 15,16]. For boosting the practical applications of CrO 2ﬁlms in var- ious devices, a clear understanding of their dynamic propertiesis crucial, which has been addressed in several studies. Lubitzet al. [4] investigated the exchange and relaxation effects in CrO 2ﬁlms, and the exchange constant was obtained by analyzing the standing spin-wave spectra. Rameev et al. [17] used the ferromagnetic resonance (FMR) technique to studythe dynamic behavior and analyzed the magnetic anisotropyin thin epitaxial CrO 2ﬁlms at a ﬁxed microwave frequency of 9.8 GHz. The relaxation behavior of CrO 2thin ﬁlm has been experimentally investigated by ultrafast magnetizationdynamics, and a rather long demagnetization time was de-termined, which was found to be related to the half-metallicnature of CrO 2[18–21]. Recently, Durrant et al. [3] explored the magnetiza- tion dynamics of epitaxial CrO 2thin ﬁlms on (100)- oriented TiO 2substrates by time resolved scanning Kerr microscopy (TRSKM) and vector network analyzer FMR(VNA-FMR) techniques, and a low Gilbert damping constant (∼10 −3–10−2) was observed. In their study, the damping constant was extracted by analyzing the low-frequency (1–10 GHz) properties of CrO 2thin ﬁlms. In the earlier ex- perimental studies on the damping properties, the extrinsiclinewidth such as two-magnon scattering (TMS) linewidthand inhomogeneous linewidth were found to signiﬁcantlycontribute to the total linewidth [ 22,23]. Especially, TMS contributes to a frequency-dependent linewidth in the low-frequency region, which may seriously hamper the extractionof intrinsic damping. Further, the strain due to lattice mis-match may enhance the contribution of TMS [ 24]. Therefore, 2469-9950/2020/102(1)/014454(9) 014454-1 ©2020 American Physical SocietyZHENHUA ZHANG et al. PHYSICAL REVIEW B 102, 014454 (2020) FIG. 1. Schematic of FMR measurement with in-plane and out- of-plane scan mode. it is necessary to investigate the damping properties of CrO 2 ﬁlms with different epitaxial crystal orientations in a large frequency range. In this study, we have systematically studied the dynamic response of epitaxial CrO 2ﬁlms with different crystal orien- tations: (100) and (110). Based on linewidth analysis, Gilbertdamping constant as low as (6 .2±0.4)×10 −4was found in (110)-oriented CrO 2ﬁlms, while it was nearly 5 ×10−3for (100)-oriented ﬁlms with large thickness. We determined theLande gfactor as 1.98 for both (100)- and (110)-oriented CrO 2 ﬁlms, which indicated weak correlation between electrons as compared to the results of ﬁrst-principle calculation. By ﬁtting the results under in-plane resonance condition, thestrain ﬁeld was found to be sensitive to the ﬁlms thickness for(100)-oriented ﬁlms, while the strain had a negligible effecton (110)-oriented ﬁlms. Further, the anisotropy ﬁeld obtainedusing FMR method showed good agreement with the resultsbased on vibrating sample magnetometry (VSM). II. THEORETICAL BACKGROUND For convenience, FMR analysis based on the free energy theory of Smit and Beljers [ 25] was used for obtaining the relationship between the resonance ﬁeld or linewidth andthe direction of the external ﬁeld. Since the free energies of(100)- and (110)-oriented CrO 2ﬁlms are similar, we used the former as the research object in this section for cleardemonstration. According to the earlier experimental results,strain may be induced at the ﬁlm-substrate interface due to thelattice mismatch between CrO 2and TiO 2substrate, which is −3.79% along [010] direction and −1.48% along the [001] direction [ 26]. A schematic of our sample and the directions of external ﬁeld and magnetization are displayed in Fig. 1. The free-energy density of (100)-oriented CrO 2ﬁlm with saturation magnetization of Mscan be expressed as F=− HM S[sinθsinθHcos(ϕ−ϕH)+cosθcosθH] +2πM2 Scos2θ−Kσsin2(ϕ−δ)sin2θ −(K1+2K2)sin2ϕsin2θ+K2sin4θsin4ϕ. (1) Here, the ﬁrst term represents the Zeeman energy under an applied magnetic ﬁeld of H, and the second term arises fromthe demagnetizing energy (the demagnetizing factor is taken as 4πbecause the thickness is much smaller than the in- plane scale of CrO 2ﬁlms). The third term represents the strain anisotropy energy, and the last two terms correspondto the contribution of uniaxial magnetocrystalline anisotropyenergy. K 1and K2are the ﬁrst- and second-order uniaxial magnetocrystalline anisotropy energy constants. δis the an- gle between the easy axes of strain anisotropy and uniaxialmagnetocrystalline anisotropy. For convenience, the quadraticterms of strain and magnetocrystalline anisotropy energy canbe expressed as follows [ 27]: K /prime=(K1+2K2)sin2ϕ+Kσsin2(ϕ−δ) =MS 2[H∗sin2(ϕ−ϕ∗)+H∗sin2ϕ∗+Hσsin2δ], (2) where H∗=[(HK1+2HK2)2+H2 σ+2(HK1+2HK2)Hσcos 2δ]1/2 (3) and tan 2ϕ∗=Hσsin 2δ (HK1+2HK2)+Hσcos 2δ. Here, HK1=2K1/MS,HK2=2K2/MS, and Hσ=2Kσ/MS. Substituting the free-energy density ( F) into the general resonance condition for different external ﬁeld directions be-tween resonant angular frequency and free energy, we get w r=γ MSsinθ/parenleftbig FθθFϕϕ−F2 θϕ/parenrightbig1/2, (4) where wr=2πfr, and fris the resonance frequency. γ is the gyromagnetic ratio of the tested sample, which is∼1.7588×10 7rad Hz Oe−1for free electron, and Fθθ,Fϕϕ, and Fθϕare the second-order partial derivatives of free energy with respect to θandϕ. Then, the resonance condition for speciﬁc directions can be obtained. For the in-plane condition ( θ=π/2), wr=γ/braceleftbig/bracketleftbig Hcos(ϕ−ϕH)+Heff d+H∗sin2(ϕ−ϕ∗) −2HK2sin4ϕ/bracketrightbig/bracketleftbig Hcos(ϕ−ϕH) −H∗cos 2(ϕ−ϕ∗)+6HK2sin2ϕcos2ϕ −2HK2sin4ϕ/bracketrightbig/bracerightbig1/2, (5) where Heff d=4πMS+H∗sin2ϕ∗+Hσsin2δ. The magnetization direction can change according to the external ﬁeld, so the equilibrium condition is necessary foranalysis. The equilibrium condition can be obtained using∂F/∂ϕ=0, i.e., Hsin(ϕ−ϕ H)−1 2H∗sin 2(ϕ−ϕ∗) +2HK2sin3ϕcosϕ=0. (6) At the equilibrium position, the direction of magnetization is the same as that of the effective ﬁeld ( Heff=∂F/∂M), and a tiny angular difference is generated between them whenthe magnetization precesses consistently in the resonancecondition, and the magnetization linearly responds to theexternal microwave ﬁeld. A special in-plane magnetizationdirection such as the [010] axis is important due to the precise 014454-2ULTRALOW GILBERT DAMPING IN CRO 2… PHYSICAL REVIEW B 102, 014454 (2020) measurement conditions (chamfer edges are chopped along [001] direction of TiO 2substrate). Speciﬁcally, the resonance condition for the [010] axis is expressed as follows: wr=γ/bracketleftbig/parenleftbig H+Heff d+H∗sin2ϕ∗/parenrightbig ×(H−H∗cos 2ϕ∗)/bracketrightbig1/2,ϕ=0. (7) III. EXPERIMENTAL METHODS The CrO 2ﬁlms were fabricated on 5 ×5 mm (100)-oriented monocrystalline TiO 2substrates by atmospheric pressure chemical vapor deposition (APCVD) technique. Before de-position, hydroﬂuoric acid (HF) was used to treat the surfaceof TiO 2substrates for 2 min in ultrasonic environment. The source, i.e., CrO 3powder, was placed in the low-temperature zone (260 °C), and the substrate was put in a high-temperaturezone (390 °C). After deposition, the CrO 2ﬁlms were pre- served by heat treatment for 60 min. The ﬁlm’s thicknesswas approximately linearly proportional to the growth time.The crystalline phase of the ﬁlms was characterized by x-raydiffraction (XRD; Bede D1). Hysteresis loop measurementswere carried out by using a VSM in a physical propertymeasurement system (PPMS; Quantum Design). The dynamic properties were studied by a FMR measure- ment setup with a resonant cavity and a coplanar waveguide.The scanning ﬁeld mode was used in all the measurements.The resonant cavity is adopted in in-plane angle dependent FMR measurements at a ﬁxed frequency of 10 GHz, where the samples were placed at the center of a copper disk withangle indexes. In addition, the dependence of resonance ﬁeldon the frequency was analyzed by coplanar waveguide basedFMR for stronger signals, and the resonance frequency couldbe adjusted in the range 0–40 GHz by using a frequencydoubler. The external ﬁeld could reach up to several tesla withan accuracy of 0.5 Oe, and before acquiring every data point,the ﬁeld was maintained for several seconds. The appliedmicrowave power was below 10 dbm to reduce the nonlineareffects. IV . RESULTS AND DISCUSSION A. Structural characterization The XRD patterns of (100)- and (110)-oriented CrO 2ﬁlms with different thicknesses are displayed in Fig. 2. Only peaks corresponding to CrO 2ﬁlm and TiO 2substrate are observed in the spectra, and no peak for impurity phase Cr 2O3(often observed at 2 θ∼36.6◦) appears, which is usually generated under inappropriate experimental conditions such as highgrowth temperatures and unsuitable oxygen ﬂow rate, etc.[8,9]. The (200) peaks of (100)-oriented CrO 2ﬁlms are located at slightly larger angles as compared to that of bulk CrO 2 due to the strain induced by lattice mismatch. Strain becomesmore prominent as the ﬁlm thickness decreases, which canbe clearly observed from the increase in the position shiftin Fig. 2(a) (dashed line indicates the peak position of bulk CrO 2). For (110)-oriented CrO 2ﬁlms, the (110) peak position is only slightly deviated from that of bulk CrO 2, even when the thickness is as small as 35 nm, which suggests that the(110)-oriented ﬁlms are almost strain-free. After a 2 θscan, the FIG. 2. XRD patterns of (110)-oriented (a) and (100)-oriented (b) CrO 2ﬁlms epitaxially grown on TiO 2substrates with different thicknesses. The vertical dashed lines represent the peak positions ofbulk CrO 2. rocking curves of (200) peaks for (100)-oriented CrO 2ﬁlms and (220) peaks for (110)-oriented CrO 2ﬁlms were measured, which were ﬁtted by Lorentz function. The full width at halfmaximum (FWHM) for 123-nm-thick (100)-oriented CrO 2 ﬁlm was obtained as 574 arcsec, and the FWHM for (110)-oriented ﬁlm was slightly larger. The small value of FWHMvalidates the excellent ﬁlm quality of our CrO 2ﬁlms. B.gfactor and anisotropy analysis The resonance ﬁeld and peak-to-peak linewidth are ex- tracted from the shape of absorption peak by consideringa mixture of absorption and dispersion phase in the outputsignal due to the possible contribution of eddy current [ 28]. The ﬁtting equation has the following form [ 29]: y=a/parenleftbig Hr−H /Delta1Hr/parenrightbig +9b−3b/parenleftbigHr−H /Delta1Hr/parenrightbig2 /bracketleftbig 3+/parenleftbigHr−H /Delta1Hr/parenrightbig2/bracketrightbig2, (8) where yis the FMR response; Hand Hrare the applied and resonance ﬁelds, respectively; /Delta1Hris the resonance 014454-3ZHENHUA ZHANG et al. PHYSICAL REVIEW B 102, 014454 (2020) FIG. 3. (a) Resonances curve of CrO 2ﬁlms, where the black curves represent the experimental results. Here, the contributions of both absorption (red circles, antisymmetric Lorentzian function) and dispersion (blue circles, symmetric Lorentzian function) are considered. (b) Frequency dependence of resonance ﬁeld and ﬁtting results based on Eq. ( 7), where ftopdenotes the upper limit of frequency used for ﬁtting. (c), (d) Dependence of ftopongfactor for (110)- and (100)-oriented CrO 2ﬁlms, respectively. peak-to-peak linewidth; aand bare the amplitudes of ab- sorption and dispersion signals, respectively. Considering thecontributions of absorption and dispersion signals, we canobtain a good agreement between ﬁtting and experimentalsignal. The linewidth and resonance ﬁeld extracted in thisway have a small deviation from those read directly fromexperimental resonance signal. The ﬁtting result for 113-nm-thick (110)-oriented ﬁlm at 32 GHz is shown in Fig. 3(a).I t is obvious that the inclusion of dispersion alters the linewidthconsiderably. At a certain thickness, dispersion signal of CrO 2 ﬁlm may be signiﬁcant, so the shape of resonance peak canvary with the frequency due to the change in the relativeamplitudes of absorption and dispersion signals. Therefore,it is important to include the contribution of dispersion signalfor accurate ﬁtting. The Lande gfactor ( g=γ¯h/μ B, where ¯ his the Dirac con- stant and μBis the magnitude of Bohr magneton) is crucial for extracting the anisotropy ﬁeld and damping constant from theﬁtting process. Here, we use a precise method of obtaining thegfactor of the CrO 2ﬁlms from data ﬁtting under the resonance condition for a hard axis in Eq. ( 7)[30]. The frequency de- pendence of resonance ﬁeld for 364-nm-thick (110)-orientedCrO 2ﬁlm is shown in Fig. 3(b) as an example, and ftopis the maximum frequency adopted in the ﬁtting process. To obtainthe value of g(f top) for each ftop, the experimental data from the lowest frequency (8 GHz) to ftopare ﬁtted [the points marked in red circles in Fig. 3(b) that are beyond the ﬁtting range for responding ftopare discarded]. The relationship between gfactor and ftopfor (110)- and (100)-oriented CrO 2 ﬁlms is shown in Figs. 3(c) and3(d), respectively. For both the ﬁlms with varying thickness, the gfactors converge to approximately 1.98 with the increase in ftop, which is close to the value of gfactor for free electrons and indicates the weak spin-orbit interaction or large quenching of orbital angularmomentum due to the crystal ﬁeld. Typically, the gfactor is larger than 2 for ferromagnetic materials. However, when thematerial contains cations with half full or less than half fullperipheral electronic shell, the gfactor may be lower than 2[31–33]. According to the simpliﬁed model proposed by Kittel [ 34], the spin-orbit coupling (SOC) term λL·S(Land Sare respectively orbital and spin angular momentum, and λ represents the SOC strength) has a non-negligible effect onthe ground state. Consequently, the contribution of the orbitalpart to the total momentum becomes non-negligible, which isin contrast to the quenching state in crystalline ﬁeld. Therefore, in a SOC system, a new term proportional to −λ//Delta1 should be added due to the change in energy difference between spin-up and spin-down states, i.e., g=2(1+4ε). (9) Here,ε=−λ//Delta1, where /Delta1is the level separation in the crystal ﬁeld. When the peripheral electronic shell of cation is lessthan half full, λis positive, and therefore ε<0. Since the peripheral electronic shell of Cr 4+is 3d2, it is reasonable that thegfactor is smaller than 2. Based on the above analysis, the degree of deviation of the gfactor from 2 indirectly indicates the strength of SOC. For our CrO 2ﬁlms, g=1.98, indicating the existence of weak SOC, which is similar to theobservation for other half-metallic systems [ 35,36]. Actually, x-ray magnetic circular dichroism (XMCD) measurement re-sults have revealed that the orbital magnetic moment of Cris (−0.06±0.02)μ B, and the spin magnetic moment of Cr can be obtained from ﬁrst-principle calculations with the localspin-density approximation (LSDA) or LSDA +Uapproach for different Hubbard-like Uterms, which is taken in the range 0–9 eV based on the XMCD results [ 37,38]. Considering that the absolute value of ratio between orbital magnetic momentand spin magnetic moment ( \|μ L/μS\|) increases with U,t h e g factor [the gfactor can be obtained using g−2=2(μL/μS) [34]] varies from 1.96 to 1.92 as Uincreases from 0 to 9 eV . We obtained the gfactor as ∼1.98, which is close to that in the case of U=0, suggesting that CrO 2may exhibit a weak electron-electron correlation. The resonance ﬁelds as a function of in-plane azimuth angle ( ϕH) for (110)- and (100)-oriented CrO 2ﬁlms with different thicknesses are shown in Figs. 4(a)and4(b), respec- tively. The experiment data were obtained using the rotated-sample method in a resonant cavity at a frequency of 10 GHz.The experiment data are well ﬁtted using Eq. ( 5). The ﬁtting parameters are listed in Table I. A remarkable feature in Figs. 4(a)and4(b) is the quadratic symmetric behavior, which is consistent with our analysis andprevious observation that only uniaxial anisotropy (uniaxialmagnetocrystalline anisotropy and strain anisotropy) exists inplane for (110)- and (100)-oriented CrO 2ﬁlms. According to an earlier study [ 39], the strain anisotropy increases with the decrease in the thickness for (100)-oriented CrO 2ﬁlms. Therefore, it can be inferred that the departure angle ϕ∗ increases due to the increase in strain anisotropy, provided the easy-axis direction of strain is invariable. However, the overalleasy axis for CrO 2ﬁlms is deviated by less than 1° from [001] direction according to the ﬁtting results. This phenomenon isreasonable under a special circumstance in which the strain isalong axis direction, perpendicular to or along the easy axis. 014454-4ULTRALOW GILBERT DAMPING IN CRO 2… PHYSICAL REVIEW B 102, 014454 (2020) FIG. 4. Dependence of resonance ﬁeld and ﬁtting results (solid lines) on the azimuth angle of external ﬁeld for (a) (110)-oriented and (b) (100)-oriented CrO 2ﬁlms with 10-GHz ac ﬁeld. (c),(d) Dependence of resonance on the polar angle for (110)- and (100)- oriented CrO 2ﬁlms, respectively. The theoretical results obtained from in-plane ﬁtting parameters are shown by solid lines with the acﬁeld of 8 GHz for (100)-oriented ﬁlms and 10 GHz for (110)-oriented ones. The tiny deviation may result from errors when placing the samples into the waveguide. Therefore, the in-plane uniaxial anisotropy can be rewritten as K/prime=MS 2[(HK1+2HK2+Hσ[001])sin2ϕ+Hσ⊥cos2ϕ] =MS 2/parenleftbig HK1eff+2HK2/parenrightbig sin2ϕ+MSHσ⊥ 2, (10) where HK1eff=HK1+Hσ[001]−Hσ⊥ =2/parenleftbig K1+Kσ[001]−Kσ⊥/parenrightbig MS. Here, Hσ[001] and Hσ⊥are the components of effective strain ﬁeld along [001] and its perpendicular direction, the[010] direction for the (100)-oriented ﬁlm and the [1–10]direction for the (110)-oriented ﬁlm, respectively. The con-stant term, M SHσ⊥/2, in K/primecannot be discarded due to the θHdependence of the last term of free energy. The effective strain anisotropy ﬁeld along the [010] direction is mainlyconsidered due to large lattice mismatch along this direction. FIG. 5. Hysteresis loops of hard axis [[010] for (100)-oriented ﬁlm and [1–10] for (110)-oriented ﬁlm] and easy axis ([001] for both ﬁlms) for (a) 52-nm-thick (110)-oriented and (b) 24-nm-thick (100)-oriented ﬁlms. Consequently, HK1effdecreases with the thickness of CrO 2 ﬁlms due to the large contribution of strain anisotropy and no easy-axis rotation in this circumstance before easy axisand hard axis exchanging. Therefore, we can set ϕ ∗as zero in the resonance relationship and equilibrium condition in thefollowing analysis. The dependence of H ron the polar angle θHwas measured with coplanar waveguide for strong signals by rotating the ﬁlmfrom the easy-axis direction to the direction normal to the thinﬁlm. The relationship between H randθHwas calculated using the ﬁtting parameters obtained from above in-plane Hr−ϕH analysis. As shown in Figs. 4(c) and4(d), the experimental results are in excellent agreement with the calculated results.The self-consistence of the ﬁtting parameters for differentmeasurement conditions suggest the reliability of our ﬁttingparameters. The magnetic properties of the ﬁlms were alsocharacterized using VSM. The hysteresis loops were mea-sured with an external ﬁeld along the two principle axes direc-tion for (110)- and (100)-oriented ﬁlms. The results are shownin Fig. 5. It is evident that for both the epitaxial ﬁlms, the TABLE I. Comparison of the magnetic parameters of CrO 2ﬁlms obtained using in-plane FMR ﬁtting and VSM. t(nm) gH K1eff(Oe) HK2(Oe) HVSM K(Oe) Meff(emu/cc) MVSM(emu/cc) (110) 364 1.98 1194 −25.68 1084 447 475 104 1.98 1192 −29.38 1127 437 468 52 1.98 1188 −18.96 1093 415 477 (100) 168 1.98 825 −1.53 815 510 452 95 1.98 798 −6.45 795 519 455 24 1.98 526 28.07 563 503 445 014454-5ZHENHUA ZHANG et al. PHYSICAL REVIEW B 102, 014454 (2020) shape of hysteresis loops along the two different measurement directions are nearly square and linear, respectively, whichindicates that the easy axis is along the [001] direction. It is well known that the work done on a unit-volume medium for magnetization from the demagnetized state tosaturation is W=/integraldisplay Ms 0HdM, (11) which is equal to the area ( S) surrounded by the M(H) curve, the Maxis, and the line M=Msparallel to the Haxis. Therefore, the in-plane uniaxial anisotropy constant KVSM (KVSM=K1eff+K2+···+··· Kn) is numerically equal to Sfor the [010] direction. Furthermore, it can be proved that the value of K1eff(ﬁrst-order uniaxial magnetocrystalline anisotropy and strain anisotropy are included) is equal to thearea ( S /prime) surrounded by the tangent line at the original point of the M(H) curve, the Maxis, and the line M=Msparallel to the Haxis. It is clear from Fig. 5that the hysteresis loops along the hard axis exhibit near linear behavior in the processof magnetization to the saturation state. Consequently, thecontributions of higher-order terms in our samples are muchlower than that of the K 1effterm, which can be seen from the ﬁtting results in Table I, where HK2is much smaller than HK1eff. Here, we have compared the effective uniaxial anisotropy ﬁeld ( HK1eff+HK2) obtained by ﬁtting of resonance data with that ( HVSM K=2KVSM/MS) obtained by hysteresis loops measurement. Table Ishows good consistency between two characterization methods for CrO 2 ﬁlms with different thicknesses. For (100)-oriented ﬁlm, HK1eff decreases with the ﬁlm thickness, which can be attributed to the increase in strain, while almost no change of HK1eff is observed for (110)-oriented ﬁlm due to the relaxation of strain. The values of HK1effobtained in our study are consistent with those reported in earlier studies [ 40,41]. However, in our samples, HK2is quite low even at small thickness, i.e., it is nearly two orders of magnitude lower than HK1eff, which is obviously different from the earlier studies. Moreover, bystudying the magnetic anisotropy of (100)-oriented CrO 2ﬁlm u s i n gaB r u k e rE M X X-band electron paramagnetic resonance (EPR) spectrometer at 9.8 GHz, Rameev et al. [17] observed a multipeak absorption behavior in 65-nm-thick and 434-nm-thick ﬁlms. Further, they reported switching of the easy axisfrom the [001] to [010] direction in a 27-nm-thick ﬁlm. Thesefeatures were not observed in our samples. Besides, the effec-tive anisotropy ﬁelds for our ﬁlms with different thicknessesare distinctly larger than those obtained by Rameev et al. The difference in the ﬁlm properties may be attributed to thedifferent fabrication conditions and ﬁlm quality. For the convenience of the comparison between the mag- netic parameters obtained by FMR and those by VSM, H eff d is transformed into effective magnetization MeffasHeff d= 4πMeffand the corresponding effective magnetization values are listed in Table I.Mefffor (110)-oriented ﬁlm is slightly less than the experimental value obtained using VSM measure-ment, which may be due to the weak contribution of surfaceanisotropy [ 42].M effof (100)-oriented ﬁlm is larger than that of the (110)-oriented one due to the larger contribution ofstrain anisotropy ﬁeld according to our previous FMR analysis(H eff d=4πMS+Hσ). FIG. 6. Linewidths extracted from resonance absorption curves as a function of resonance frequency for (a) (110)-oriented and(b) (100)-oriented ﬁlms with different thicknesses. (c), (d) Depen- dence of thickness on the effective damping constant extracted from linewidth-frequency linear ﬁtting. C. Damping analysis The peak-to-peak linewidths were extracted from scan- ﬁeld spectra at different frequencies, and the linewidth asa function of frequency for (110)- and (100)-oriented CrO 2 ﬁlms with different thicknesses is shown in Figs. 6(a) and 6(b), respectively. Under the condition of magnetization in the direction of external ﬁeld, the variation of resonance linewidthand the resonance frequency is obtained from phenomeno-logical analysis based on the Landau-Lifshitz-Gilbert (LLG)equation, i.e., /Delta1H=2αw √ 3γ. (12) However, in an actual situation, the inhomogeneous linewidth /Delta1H0should be included, which stems from the local inhomogeneity of the ﬁlms. Moreover, a simple correctionshould be adopted for the equation when the directions ofmagnetization and the external ﬁeld are not the same [ 43]. The effective damping constant and inhomogeneity linewidth areobtained from the slope and intercept of the ﬁtted line, respec-tively. The effective damping constants for (110)- and (100)-oriented CrO 2ﬁlms with different thicknesses are shown in Figs. 6(c) and6(d), respectively. Considering that the reso- nance absorption signal is proportional to the thickness ofCrO 2ﬁlms grown on the TiO 2substrates of the same size and the signal is inversely proportional to the resonance peaklinewidth for the same ﬁlm [ 44], the absorption signal in the high-frequency range for the 24-nm-thick ﬁlm is so weak thata signiﬁcant error is added in the ﬁtting process, which is notshown in the ﬁgure for clarity. Here, we have used α effto represent the effective damping constant to account for the probable broadening of linewidthdue to the contributions of eddy current and TMS. With theincrease in the ﬁlm thickness, α effdecreases rapidly when the thickness is smaller than 50 nm. Above 50 nm, theeffective damping constant tends to be independent of the 014454-6ULTRALOW GILBERT DAMPING IN CRO 2… PHYSICAL REVIEW B 102, 014454 (2020) thickness. The calculation of Gilbert damping constant based on the tight-binding model including SOC shows that theGilbert damping changes considerably with the thickness offerromagnetic ﬁlm at lower thickness and then stabilizes toa constant value [ 45]. It should be noted that under weak inhomogeneity condition, TMS contributes to the resonancelinewidth, and the linewidth broadening term changes withfrequency and does not affect /Delta1H 0for ultrathin ﬁlms, which is treated by Arias et al. using the approach of the equation of motion for the response functions [ 23]. When the tested sample exhibits an ideal single domain without any defects, itis uniformly magnetized in space. Therefore, magnon scat-tering does not occur due to the conservation of angularmomentum. For practical ﬁlms grown by a variety of methods,defects exist unavoidably, which destroy the conservation ofangular momentum, and energy leaks from the zero-wave-vector magnon (consistent precession) to other magnons inwhich the conservation of energy should be satisﬁed. It isclear in Figs. 6(c) and 6(d) that the effective damping of (100)-oriented CrO 2ﬁlms is distinctly larger than that of (110)-oriented ones. As mentioned above, strain exists in(100)-oriented CrO 2ﬁlms due to the lattice mismatch between CrO 2ﬁlms and TiO 2substrates, which induces TMS in FMR experiment [ 24]. Moreover, a native ultrathin Cr 2O3layer may exist on the surface of pure CrO 2ﬁlm, which can also lead to the occurrence of the magnon scattering [ 8,46]. Ultralow effective damping with a magnitude of 10−4was observed for (110)-oriented CrO 2ﬁlms of thickness larger than∼50 nm. The data points at low frequencies were masked during the ﬁtting process. The low-frequency experimentaldata are complex due to the contributions of TMS and thelocalized spread of anisotropy axis stemming from crystalimperfection in the low ﬁeld, which has a minor effect on thelinewidth in the high ﬁeld on account of the suppression ofspin waves and more consistent magnetization alignment. Thelow-ﬁeld nonlinear behavior is also observed in low-dampingsystems such as CoFe, Co 2FeAl, CoFeSiB, etc. [ 22,47,48]. Theoretical analysis indicates that TMS becomes inactivewhen the magnetization is tipped out of the sample plane inexcess of 45 ◦because the degeneracy between uniform mode and spin-wave modes disappears. Therefore, measurementswith magnetization perpendicular to ﬁlms were conductedin a relatively low-frequency range (a very strong ﬁeld isrequired for higher frequency, which is out of the range of ourequipment). It is clear in Fig. 7that the linewidth is linearly proportional to the frequency, which facilitates a straightfor-ward extraction of damping constant nearly intrinsic. The α ⊥ is slightly smaller than the damping constant obtained from ﬁtted data in high-frequency range under in-plane circum-stance due to the decrease in the contribution of TMS in thisfrequency range. The damping constant for (100)-orientedCrO 2ﬁlms with thickness over 50 nm is slightly smaller than that reported by Durrant et al. [3] using TRSKM and VNA-FMR. In this study, the damping constant for (110)-oriented CrO 2ﬁlms is obtained as low as (6 .2±0.4)×10−4, which is nearly one order of magnitude smaller than that of(100)-oriented ones. This difference may be attributed to thedifferent contribution of TMS resulting from strain and sur-face morphology [ 40]. It is well established that dislocations are always generated to relieve the misﬁt strain, which can FIG. 7. Dependence of linewidth on the resonance frequency of 364-nm-thick ﬁlm for two measurement directions: [1–10] in-planeand [110] perpendicular to the ﬁlm. enhance the contribution of TMS [ 39]. In addition, the change in electronic structure due to the strain from lattice mismatchcan also cause a change in intrinsic damping constant [ 2]. Recently, the low damping phenomenon has been observed in CoFe [ 22,35], CoFeB [ 49], and some Heusler alloy systems [36,50,51]. The damping constant observed in our CrO 2ﬁlms [(6.2±0.4)×10−4] is lower than the ultralow damping value observed in CoFe systems [(1 .3–2.0)×10−3] and is compa- rable to the lowest value obtained in some Heusler alloys[(4.1–9.0)×10 −4][22,35,36]. Presently, the lowest damping is exhibited by yttrium iron garnet (YIG) ﬁlms ( ∼10−5) [52], but their insulation characteristic limits the practical applications in spintronic devices. Since we have obtainedan extremely low value [(6 .2±0.4)×10 −4] of total effective damping constant for (110)-oriented CrO 2ﬁlms with thick- ness above 50 nm, the intrinsic Gilbert damping may be evenlower. By optimizing the fabrication process and enhancingthe ﬁlm quality, the damping constant of CrO 2ﬁlms may be further decreased. Considering the half metallicity andlow Glibert damping, CrO 2is very promising for practical applications in spintronics. Using the framework of Kamberský’s torque correlation model [ 2], the intrinsic Gilbert damping constant can be expressed as α=g2μ2 B γ¯hM Sπ2 2/Omega1at/angbracketleftBigg/summationdisplay m,n\|/Gamma1− mk,nk\|2Wmk,nk/angbracketrightBigg k, (13) where /Omega1atis the atomic volume, nand mare band indices, k is the electron wave vector, /Gamma1− mk,nkis the transition matrix ele- ment, and /angbracketleft/angbracketrightkdenotes the average over the ﬁrst Brillouin zone. Here, spectral overlap function Wmk,nkis strongly related to the spectral density of states (DOS) around the Fermi level. Inthis framework, two kinds of scattering, intraband scattering(breathing of the Fermi surface) and interband scattering(bubbling of the Fermi sea), contribute to damping. Further,the damping due to the intraband term is roughly proportionalto the DOS, and the contribution of the interband term is 014454-7ZHENHUA ZHANG et al. PHYSICAL REVIEW B 102, 014454 (2020) also strongly correlated to the DOS [ 53]. For half-metallic materials, there is a gap around the Fermi level for spin-down electrons, therefore the spin-down channel does notcontribute to the damping in the limit of no SOC, which leadsto the low damping of CrO 2ﬁlms. SOC should be considered for practical materials, which introduces the contribution ofminority spin states. Notably, some studies have attempted toverify the quantitative relationship between damping constantand SOC strength ( λ)[54,55]. By maintaining the DOS at the Fermi surface while varying the SOC by changing thePt/Pd concentration ratio in FePd 1−xPtxsystem, He et al. [55] found that the intrinsic damping constant is proportional tothe square of SOC strength. As explained in the previoussection, the SOC strength is weak in the CrO 2system (the gfactor is close to 2), which is consistent with the low damping observed. The SOC strength is laborious to obtain inexperiment, but fortunately, it can be stated indirectly throughspin polarization, which reﬂects the strength of SOC-inducedspin-ﬂip scattering at the Fermi energy [ 2]. Nearly complete spin polarization has been observed for CrO 2ﬁlms [ 6,7], which is consistent with our result of low damping for CrO 2 ﬁlms. V . CONCLUSIONS We systematically investigated the dynamic properties of CrO 2ﬁlms with varying thickness prepared on TiO 2sub-strates with different epitaxial growth directions. The gfactors for the ﬁlms were obtained by ftop-Hﬁtting, and they con- verged to 1.98. Considering the in-plane magnetocrystallineuniaxial anisotropy and strain anisotropy, we obtained a goodagreement between experimental and theoretical results. For(100)-oriented CrO 2ﬁlms, the strain ﬁeld increased with the decrease in ﬁlm thickness, which weakened the magnetocrys-talline uniaxial anisotropy, while for (110)-oriented CrO 2 ﬁlms, the strain was nearly relaxed. The damping constant for(110)-oriented CrO 2ﬁ l m sw a sa sl o wa s( 6 .2±0.4)×10−4, which is lower than the reported values for metals or commonalloys and is comparable to the recently observed ultralowvalue in some Heusler alloy systems. Moreover, for CrO 2 ﬁlms with thickness less than 50 nm, the effective dampingwas obviously larger than that for thicker ones, which mightbe attributed to the contribution of TMS due to strain ordegeneration of ﬁlm surface. 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PhysRevB.74.155118.pdf	Physical properties of the ferromagnetic heavy-fermion compound UIr 2Zn20 E. D. Bauer,1A. D. Christianson,1,2,3J. S. Gardner,4,5V . A. Sidorov,1,J. D. Thompson,1J. L. Sarrao,1and M. F. Hundley1 1Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3University of California, Irvine, California 92697, USA 4Indiana University, 2401 Milo B. Sampson Lane, Bloomington, Indiana 47408, USA 5NCNR, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA /H20849Received 14 July 2006; published 23 October 2006 /H20850 Measurements of magnetization, speciﬁc heat, neutron diffraction, and electrical resistivity at ambient and applied pressure have been carried out on the cubic compound UIr 2Zn20. A ﬁrst-order-like ferromagnetic transition occurs at TC=2.1 K with a saturation magnetization /H9262sat/H110110.4/H9262B, indicating itinerant ferromag- netism. In this ordered state, the electronic speciﬁc heat coefﬁcient remains large, /H9253/H11011450 mJ/mol K2, classi- fying UIr 2Zn20as one of the very few heavy-fermion ferromagnets. DOI: 10.1103/PhysRevB.74.155118 PACS number /H20849s/H20850: 71.27. /H11001a, 72.15.Qm I. INTRODUCTION Attention has been focused on uranium-based magnets to discern the nature of the strongly correlated electron groundstate in these intermetallic compounds. 1A rich variety of phenomena is found including the coexistence of unconven-tional superconductivity and magnetism, 2pressure-induced superconductivity,3hidden order,4and non-Fermi-liquid behavior.5Of central importance in these uranium-based ma- terials is the degree of localization of the 5 felectrons. Vari- ous measurements on prototypical systems such as UPd 2Al3 indicate a “dual nature” of the felectrons,6with two of three being localized and the remaining one itinerant. Recent workbased upon a dual-nature model 7,8of these strongly corre- lated electron materials show promise for making headwayinto this difﬁcult electronic structure problem. Fortunately,Nature provides an abundance of intermetallic compounds toinvestigate spanning the entire range of localized/itinerantbehavior, from truly localized, as revealed by large-momentmagnetism /H20849e.g., UGa 2/H208509or strong crystalline electric ﬁeld excitations /H20849e.g., UPd 3/H20850,10to itinerant magnets such as UGe 2 or UIr.3,11Yet another class of U-based heavy-fermion anti- ferromagnets possess an extremely large Sommerfeld coefﬁ-cient /H9253/H110111 J/mol K2suggesting a high degree of itineracy at low temperatures, yet have reasonably large orderedmoments /H20849 /H9262ord/H110111/H9262B/H20850indicating localized f-electron behavior.12,13Indeed, neutron scattering experiments on U2Zn17reveal distinct responses characteristic of both itiner- ant and localized felectrons.14It is, therefore, worthwhile to search for other uranium compounds that exhibit “dual-nature” behavior. A new family of lanthanide and actinide intermetallic compounds RX 2Zn20/H20849R=lanthanide, Th, U; X=transition metal /H20850,15–18would appear, at ﬁrst glance, to be ideal candi- dates for investigating magnetism and strong electronic cor-relations. These materials crystallize in the cubic Mg 3Cr2Al18 structure with an R-Rspacing of more than 6 Å. This dis- tance is considerably larger than the Hill limit19for U/H20849dU−UHill=3.5 Å /H20850which roughly delineates two classes of ac- tinide materials, one in which there is signiﬁcant overlap ofthe actinide orbitals resulting in itinerant /H20849paramagnetic /H20850f-electron behavior /H20849d U−UHill/H110213.5 Å /H20850, and the other where there is negligible overlap leading to long-range magneticorder. While strong electronic correlations are manifest in theheavy-fermion behavior observed in the other U X 2Zn20/H20849X =Fe,Ru; Co, Rh /H20850compounds where the Sommerfeld coefﬁ- cient ranges from 50 to 250 mJ/mol K2, only UIr 2Zn20or- ders magnetically. The physical properties of U X2Zn20/H20849X =Fe,Ru; Co, Rh /H20850will be reported elsewhere;17,18here, we focus on the behavior of UIr 2Zn20. We present measurements of neutron diffraction, speciﬁc heat, magnetization, and both magnetic susceptibility andelectrical resistivity at applied pressure on the cubic com-pound UIr 2Zn20. This material undergoes a ﬁrst-order-like transition to a ferromagnetic state at 2.1 K. Within thisstate, a large electronic speciﬁc heat coefﬁcient /H9253 /H11011450 mJ/mol K2is observed. To our knowledge, UIr 2Zn20 is the ﬁrst truly heavy-fermion uranium-based ferromagnet discovered to date. II. EXPERIMENTAL DETAILS Single crystals AIr2Zn20/H20849A=Th,U /H20850were grown in Zn ﬂux.20,21The materials were placed in the ratio A:X:Zn =1:2:100 in a T a crucible and sealed under vacuum in a quartz ampoule. The sample was heated to 600 °C for 12 h,then to 1050 °C for 4 h, followed by a slow cool at 4 °C/h to 700 °C, at which point the excess molten Zn ﬂux wasremoved using a centrifuge. Magnetic measurements were performed in magnetic ﬁelds up to 6.5 T from 1.8 to 300 K using a commercialsuperconducting quantum interference device magnetometer.Speciﬁc heat measurements were carried out in a commercialcryostat from 0.4 to 300 K using an adiabatic heat-pulsetechnique. Four-wire electrical resistivity were also per-formed in a commercial cryostat from 0.4 to 300 K. In somecases, the electrical resistivity measurements were performedin a small magnetic ﬁeld was applied /H20849H=0.2 T /H20850to suppress impurity superconductivity from Zn inclusions. Neutron powder diffraction data were collected using the BT-1 neutron powder diffractometer at the NIST Center forNeutron Research /H20849NCNR /H20850.AC u /H20849311/H20850monochromator pro-PHYSICAL REVIEW B 74, 155118 /H208492006 /H20850 1098-0121/2006/74 /H2084915/H20850/155118 /H208496/H20850 ©2006 The American Physical Society 155118-1duced neutrons with a wavelength /H9261=1.5403 /H208492/H20850Å. Data were collected over the range of 2 /H9258=3–168° with a step size of 0.05°. A 5 g sample of UIr 2Zn20was measured at 0.6 and 4.2 K for 8 h at each temperature in a single shot3He sys- tem. Two high-pressure cells were used for the electrical resis- tivity measurements: a clamped piston cylinder and a toroi-dal anvil cell. The toroidal is a proﬁled anvil system suppliedwith a boron-epoxy gasket and Teﬂon capsule, containingpressure-transmitting liquid, sample and a pressure sensor. 22 The pressure in both cells was determined from the variationof the superconducting transition of lead using the pressurescale of Eiling and Schilling. 23Both ac susceptibility and electrical resistivity measurements were carried out in a4He cryostat between 1 and 300 K using a commercial resistancebridge operating at 15 Hz with excitation currents rangingfrom 500 /H9262A to 1 mA. Two samples were used in this study. Sample No. 1 was placed in the clamped pressure cell afterthe ambient pressure electrical resistivity measurements werecompleted. This sample was judged to have somewhat morefree Zn content from the metallic behavior of the resistivity.Sample No. 2 was used for both the ac susceptibility andelectrical resistivity measurements in the torodial cell. Thebehavior of both was qualitatively similar. III. RESULTS AND DISCUSSION Reﬁnements of the neutron diffraction data yield good agreement with the Mg 3Cr2Al18structure type.15In this or- dered structure, the uranium atoms possess cubic symmetryand are located at the center of 16-fold coordinated Zn poly-hedra. Likewise, the Ir atoms are situated within a Zn icosa-hedra. The lattice constant, atom positions, and isotropicthermal parameters are listed in Table I. All reﬁnements yielded full occupancy of the atomic sites. We were unable toindex several small peaks corresponding to an unknown im-purity phase with a concentration of less than 3%. The magnetic susceptibility /H9273/H20849T/H20850of UIr 2Zn20is displayed in Fig. 1measured in a magnetic ﬁeld H=0.1 T. The data can be ﬁt by a Curie-Weiss law above 100 K as shown in theinset /H20849a/H20850of Fig. 1, yielding an effective moment /H9262eff =3.6/H9262B, close to the value expected for either a 5 f2/H20849/H9262eff=3.58/H9262B/H20850or 5f3/H20849/H9262eff=3.62/H9262B/H20850conﬁguration, and a large /H20849negative /H20850Curie-Weiss temperature /H9258=−123 K. Below 5 K, the magnetic susceptibility increases dramatically and ananomaly is observed at 2.75 K /H20849H=0.1 T /H20850consistent with a ferromagnetic phase transition. /H20849As shown below, the mag- netic transition temperature is quite sensitive to magneticﬁeld and increases from T C=2.1 K in zero ﬁeld to 2.75 K in 0.1 T. /H20850 Isothermal magnetization measurements at 2 and 10 K on UIr 2Zn20shown in Fig. 2conﬁrm the onset of ferromag- netism slightly above 2 K. /H20849An Arrott plot analysis,24i.e., extrapolation of M2→0 from a plot of M2vsH/M, is also consistent with this result. /H20850A full hysteresis loop at 2 K is displayed in the inset of Fig. 2. Both the coercive ﬁeld /H20849Hc /H1101112 Oe /H20850and the remnant magnetization /H20849MR/H110110.05/H9262B/H20850 classify UIr 2Zn20as a soft ferromagnet. A saturation magne- tization of Msat/H110110.4/H9262B/U atom obtained from linear ﬁt to the high ﬁeld data /H20849H/H110224T/H20850at 2 K indicates itinerant mag- netism. An extrapolation of the /H9273/H20849T/H20850data assuming a Bloch law /H20851M=M0/H208491−aT3/2/H20850/H20852 /H20849Ref. 25/H20850below the phase transition at 2.75 K in H=0.1 T yields a spontaneous magnetization M0=0.3/H9262B; it is expected that the zero temperature value will not be too different from these values.TABLE I. Structural reﬁnement of UIr 2Zn20at 0.6 K. Uisois deﬁned as one-third of the trace of the orthogonalized Uijtensor. Uncertainties in the last digit are enclosed in parentheses. Space group Fd3¯ma =14.1783 /H208491/H20850Å, V=2850.20 /H208496/H20850Å3 /H20849No. 227, origin choice 2, Z=8/H20850 /H9267calc=8.996 gm/cm3 Atomic positions Atom Site xy z U iso/H20849102Å2/H20850 U8 a 1/8 1/8 1/8 0.21 /H208498/H20850 Ir 16 d 1/2 1/2 1/2 0.54 /H208494/H20850 Zn/H208491/H20850 16c 0 0 0 1.14 /H2084910/H20850 Zn/H208492/H20850 48f 0.4860 /H208492/H20850 1/8 1/8 0.74 /H208497/H20850 Zn/H208493/H20850 96g 0.0596 /H208491/H20850 0.0596 /H208491/H20850 0.3244 /H208492/H20850 0.70 /H208494/H20850 Reduced /H92732=3.214 Rwp=13.04% Rp=10.42% FIG. 1. Magnetic susceptibility /H9273/H20849T/H20850of UIr 2Zn20atH=0.1 T. Inset: Inverse magnetic susceptibility /H9273−1/H20849T/H20850. The solid line is a linear ﬁt to the data.BAUER et al. PHYSICAL REVIEW B 74, 155118 /H208492006 /H20850 155118-2Neutron diffraction measurements on UIr 2Zn20were em- ployed to determine the nature of the magnetic phase transi-tion at 2.1 K by directly comparing data collected at both 0.6and 4.2 K. The data at 0.6 K do not show any additionalintensity at either antiferromagnetic or ferromagnetic /H20849coin- ciding with nuclear Bragg peaks /H20850positions relative to the data at 4.2 K above the transition. Assuming a simple ferro- magnetic model, the data are consistent with an upper boundof the magnitude of the magnetic moment of less than 1 /H9262B /H20849the neutron absorption of Ir precludes further reﬁnement of this estimate /H20850, in agreement with the magnetization measure- ments /H20849Fig. 2/H20850. However, the UIr 2Zn20sample was shown to depolarize a polarized beam of neutrons at 0.6 K, indicativeof a ferromagnetic component to the low temperature phase.The similarity between the data above and below the phasetransition at 2.1 K, including the goodness of ﬁt, the latticeconstants, and the atomic positions suggests there is no struc-tural distortion associated with this transition. At this point, alarge ferromagnetic component to a more complicated mag-netic structure cannot be ruled out; further measurements arein progress to determine the exact nature of the magnetictransition in this material. For simplicity, we will continue torefer to it a ferromagnetic transition. Figure 3shows the speciﬁc heat, plotted as C/TvsT,o f UIr 2Zn20and the isostructural compound ThIr 2Zn20. A ferro- magnetic transition is observed at TC=2.1 K. Analysis of the heat-pulse decay curves does not reveal features characteris-tic of a strong ﬁrst-order transition in speciﬁc heat; 26how- ever, the symmetry of the peak in C/Tsuggests the transition into the ferromagnetic state is weakly ﬁrst order. After sub-traction of the speciﬁc heat of nonmagnetic ThIr 2Zn20, the 5 f contribution to the speciﬁc heat /H9004C/Tis displayed in the inset of Fig. 3./H9004C/Tincreases monotonically below 10 K reaching a value /H11011450 mJ/mol K2at 2.5 K just before the onset of ferromagnetism. Within the ferromagnetic state, the5fcontribution remains large: a linear extrapolation below 0.4 K yields /H9004C/T/H11011450 mJ/mol K 2. The magnetic entropy S5f/H208492.5 K /H20850/H20848/H20849/H9004C/T/H20850dT/H110111.2 J/mol K, implying itinerant fer- romagnetism in UIr 2Zn20, in agreement with the reduced mo- ment determined from magnetization measurements de-scribed above. At 10 K, the entropy amounts to S 5f=4 J/mol K /H110110.7R ln /H208492/H20850.The 5 fcontribution to speciﬁc heat /H9004C/Tof UIr 2Zn20in magnetic ﬁelds up 9 T is shown in Fig. 4. The ﬁrst-order-like transition at 2.1 K in zero ﬁeld moves higher in temperaturewith increasing ﬁeld for H/H110211.5 T, then increases more slowly above 1.5 T as displayed in Fig. 5/H20849a/H20850. Concomitant with this increase of the transition temperature, the phasetransition evolves from ﬁrst-order-like to more second-order-like for H/H110220.1 T. There is a moderate suppression of the speciﬁc heat coefﬁcient from /H9253/H11011450 mJ/mol K2atH = 0Tt o /H11011250 mJ/mol K2atH=9 T as shown in Fig. 5/H20849b/H20850. The magnetic entropy is shown in the inset of Fig. 4/H20849a linear extrapolation of the /H9004C/Tdata below 0.4 K was used to obtain S5f/H20850. The entropy released below TCremains roughly constant at S5f/H110111.2–2.0 J/mol K in applied ﬁeld despite the change in the shape of the transition above 0.1 T. The electrical resistivity /H9267/H20849T/H20850of UIr 2Zn20is shown in Fig. 6. The room temperature value of /H9267is 175 /H9262/H9024cm and /H92670 =15/H9262/H9024cm, resulting in residual resistivity ratio /H20849RRR /H20850 =12./H9267/H20849T/H20850is weakly temperature dependent at high tempera- tures, passes through a maximum at Tmax/H1101185 K, and de- creases more rapidly below /H1101150K. An obvious change in slope of /H9267/H20849T/H20850denotes the Curie temperature at TC=2.0 K. FIG. 2. Magnetization Mof UIr 2Zn20at 2 K /H20849solid squares /H20850and 10 K /H20849open circles /H20850. Inset: Hysteresis loop M/H20849H/H20850at 2 K. FIG. 3. Speciﬁc heat C/TvsTof UIr 2Zn20/H20849solid squares /H20850and ThIr 2Zn20/H20849line/H20850below 10 K. Inset: /H9004C/TvsT/H20849left axis /H20850andS5f /H20849right axis /H20850. FIG. 4. /H20849Color online /H208505fcontribution to the speciﬁc heat /H9004C/T vsTof UIr 2Zn20in magnetic ﬁelds up to 9 T for H/H20648/H20851111/H20852. Inset: S5f/H20849T/H20850forH/H333559T .PHYSICAL PROPERTIES OF THE FERROMAGNETIC … PHYSICAL REVIEW B 74, 155118 /H208492006 /H20850 155118-3With increasing magnetic ﬁeld, the transition temperature moves to higher temperature, in agreement with the speciﬁcheat measurements discussed above. Below T C, a Fermi- liquid T2temperature dependence of /H9267/H20849H,T/H20850is observed. Fits of the data to /H9267=/H92670+AT2yield a monotonically decreasing Acoefﬁcient with applied ﬁeld /H20849not shown /H20850. The Kadowaki-Woods relation27/H20851A//H92532=1 /H1100310−5/H9262/H9024cm/H20849mol K/mJ /H208502/H20852implies an electronic speciﬁc heat coefﬁcient /H9253=600 mJ/mol K2for H=0 and 220 mJ/mol K2atH=9 T, comparable to the values deter- mined from speciﬁc heat measurements. In a simpliﬁedmodel of the sharp Abrikosov-Suhl resonance at the Fermilevel E Fin the Kondo picture,28the application of a magnetic ﬁeld will broaden the resonance /H20849whose width is proportional to the Kondo temperature TK/H20850and, hence, further populatethe lower spin-up band. This leads to an increase in TKand, hence, a decrease in /H9253/H20849/H110081/TK/H20850, as is observed experimen- tally /H20851Figs. 4and 5/H20849b/H20850/H20852. No superconductivity is observed above 0.4 K. It is known that unconventional superconduc-tivity coexisting with ferromagnetism in such materials asURhGe is extremely sensitive to disorder, 29which may ac- count for the lack of superconductivity in UIr 2Zn20. The electrical resistivity /H9267/H20849T/H20850on sample No. 2 of UIr 2Zn20at various pressures up to P=43 kbar is displayed in Fig. 7. The application of pressure does not signiﬁcantly change the overall shape and magnitude of the /H9267/H20849T/H20850 curves—a result not unexpected given the relative isolation of both the uranium and iridium atoms in this structure. TheCurie temperature increases with applied pressure /H20849inset of Fig. 7/H20850at a rate dT C/dP=0.04 K/kbar up to 25 kbar then increases more slowly for P/H1102225 kbar as shown in Fig. 5/H20849c/H20850. At modest pressures below 13 kbar, the shape of d/H9267/dTis reminiscent of a ﬁrst-order phase transition /H20849not shown /H20850; above 13 kbar, d/H9267/dTacquires the characteristic shape of a second-order transition in speciﬁc heat.30Fits of the data within the magnetic state to a T2temperature dependence reveal decrease of the Acoefﬁcient with applied pressure as shown in Fig. 5/H20849d/H20850./H20851The data can also be reasonably well described by /H9267−/H92670=BTnwith n=2.5 /H20849not shown /H20850./H20852The in- crease of the temperature of the maximum in /H9267/H20849P,T/H20850,Tmax, and the concomitant decrease of Awith applied pressure im- plies that the Kondo temperature increases with P, similar to a number of other Ce- and U-based heavy-fermionmaterials. 31Both the increase of the Curie temperature and the decrease in the Acoefﬁcient with applied pressure, and the large electronic speciﬁc heat coefﬁcient suggests thatUIr 2Zn20is located just to the left of the maximum in the Doniach diagram.32The ac-susceptibility measurements on sample No. 2 up to 53 kbar are displayed in Fig. 8; the Curie temperatures deduced from these curves are in excellentagreement with those determined from electrical resistivity/H20851Fig. 5/H20849d/H20850/H20852. UIr 2Zn20displays all the characteristics of a heavy- fermion ferromagnet. At high temperatures, the f-electron magnetic moments are only weakly hybridized with the con-duction electrons and remain localized, as evidenced by a FIG. 5. Physical properties of UIr 2Zn20./H20849a/H20850Curie temperature TC/H20849H/H20850determined from speciﬁc heat /H20849solid squares /H20850and electrical resistivity /H20849open circles /H20850,/H20849b/H208505fcontribution to the speciﬁc heat /H9004C/TvsH,/H20849c/H20850TC/H20849P/H20850determined from electrical resistivity /H20849solid squares /H20850and ac susceptibility /H20849open circles /H20850at various pressures up to 53 kbar on sample No. 2. /H20849d/H20850T2coefﬁcient of resistivity A/H20849P/H20850of sample No. 1 /H20849open circles /H20850and sample No. 2 /H20849solid circles /H20850. The data at ambient pressure of sample No. 1 has been normalized tothat of sample No. 2 for comparison. FIG. 6. /H20849Color online /H20850Electrical resistivity /H9267/H20849T/H20850of UIr 2Zn20 below 300 K for I/H20648111. Inset: /H9267/H20849T/H20850below 10 K in magnetic ﬁelds up to H=9 T. From left to right the ﬁelds are 0, 0.3, 1.5, 3, 5, 7, and 9T . FIG. 7. /H20849Color online /H20850Electrical resistivity /H9267/H20849T/H20850of UIr 2Zn20 /H20849sample No. 2 /H20850at various pressures up to 43 kbar. Inset: /H9267/H20849P,T/H20850 below 5 K.BAUER et al. PHYSICAL REVIEW B 74, 155118 /H208492006 /H20850 155118-4Curie-Weiss susceptibility /H20849Fig. 1/H20850. As the temperature is lowered, the system evolves continuously to a heavy Fermi-liquid ground state where the f-electrons appear to be itiner- ant; in this case, ferromagnetism intervenes before this zerotemperature Fermi-liquid state is reached. The zero-temperature /H9253in heavy-fermion antiferromagnets such as U2Zn17and UCd 1113is approximately 1/3 of the value at the ordering temperature implying that some of the /H20849itinerant /H20850 heavy quasiparticles are removed from the Fermi surface inthe ordered state; a similar factor of 1/3 appears when com-paring the ratio of the ordered and effective moments /H20851or equivalently, the remaining mean-square ﬂuctuating moment /H20849 /H9262eff2−/H9262ord2/H20850//H9262ord2/H20852.33It is interesting to note this “1/3” rule holds true for many U-based heavy-fermion magnets, despite their different antiferromagnetic structures.13The ferromag- netic transition UIr 2Zn20probably results in a simple mag- netic structure in this cubic material and, hence, does notdrastically alter the Fermi surface upon ordering; this may beone reason for the near equality of the Sommerfeld coefﬁ-cient above and below T C. While this small change in /H9253on either side of the transition is not unexpected if it is ﬁrst-order-like, it is unusual, at least compared to other heavy-fermion antiferromagnets, that such behavior is observedwhen the phase transition appears to be second-order for H /H110220.1 T. In addition, upon entry into the ferromagnetic state, the heavy band /H20849s/H20850of UIr 2Zn20associated with the heavy- fermion state that begin /H20849s/H20850to develop above Tcwill split into spin-up and spin-down bands. However, this splitting will be small /H20849of order TC=2 K /H20850; hence, there will be little effect on the heavy quasiparticle formation within the ferromagneticstate. In contrast, the larger internal magnetic ﬁeld in theantiferromagnets such as UCd 11may have a greater effect on the narrower peak in the density of states /H20849/H9253 /H11011800 mJ/mol/K2/H20850than in UIr 2Zn20. It is difﬁcult to deter- mine the degree of localization in UIr 2Zn20; further measure- ments on UIr 2Zn20are necessary to compare it to other heavy-fermion compounds such as U 2Zn17or UPd 2Al3in which a variety of experiments indicate 2 of the 3 U felec- trons are localized.6,14 To place UIr 2Zn20within the context of other itinerant ferromagnets, it is useful to construct a Rhodes-Wholfarthplot,34,35i.e., the ratio of effective and saturation moments /H9262eff//H9262satvsTC, as shown in Fig. 9. The U-based itinerant ferromagnets /H20851e.g., URhGe /H20849Ref. 29/H20850, UPt /H20849Ref. 36/H20850/H20852have much lower Curie temperatures than the 3 dferromagnets involving dilute magnetic impurities in Pd, consistent with anarrow fband at the Fermi level. 34UIr 2Zn20has a large value of /H9262eff//H9262sat=8.9, comparable to the ferromagnetic, pressure-induced superconductor UIr but with an order ofmagnitude smaller Curie temperature. 11,37Such a large value of/H9262eff//H9262satsuggests predominantly itinerant f-electron char- acter, in marked contrast to the localized ferromagnets /H20849e.g., UGa 2/H20850in which /H9262eff//H9262sat/H110111.34 In summary, the physical properties of UIr 2Zn20have been measured by means of neutron diffraction, magnetiza-tion, speciﬁc heat, and electrical resistivity and ac suscepti-bility under pressure. This material undergoes a phase tran-sition to a ferromagnetic state below T C=2.1 K. Speciﬁc heat measurements indicate the Sommerfeld coefﬁcient is /H9253 /H11011450 mJ/mol K2within the ferromagnetic state, classifying it as a heavy-fermion material. Further neutron diffractionmeasurements are planned to determine the magnetic struc-ture of UIr 2Zn20, while other measurements including photo- emission are in progress to further probe the degree of local-ization itineracy in this interesting material. ACKNOWLEDGMENTS We thank Zach Fisk for valuable discussions. Work at Los Alamos was performed under the auspices of the U.S. DOE.We acknowledge the support of the National Institute ofStandards and Technology, U. S. Department of Commerce,in providing the neutron research facilities used in this work.Work at UC Irvine was supported by the Department of En-ergy /H20849DOE /H20850under Grant No. DE-FG03-03ER46036. Oak Ridge National Laboratory is managed by UT-Battelle, forthe DOE under Contract No. DE-AC05-00OR22725. V .A.S.acknowledges the support of the Russian Foundation for Ba-sic Research /H20849Grant No. 06-02-16590 /H20850and Program “Physics and Mechanics of Strongly Compressed Matter” of thePresidium of Russian Academy of Sciences. FIG. 8. /H20849Color online /H20850Real part of the ac magnetic susceptibility /H9273ac/H20849T/H20850of UIr 2Zn20/H20849sample No. 2 /H20850at various pressures up to P =53 kbar. FIG. 9. /H20849Color online /H20850Rhodes-Wohlfarth plot /H9262eff//H9262satvsTC, for various materials.PHYSICAL PROPERTIES OF THE FERROMAGNETIC … PHYSICAL REVIEW B 74, 155118 /H208492006 /H20850 155118-5Also at: Institute for High Pressure Physics, Russian Academy of Sciences, 142190 Troitsk, Russia. 1V . Sechovský and L. Havela, in Ferromagnetic Materials , edited by E. P. Wohlfarth and K. H. J. Buschow /H20849North-Holland, Am- sterdam, 1988 /H20850, V ol. 4, p. 309. 2C. Geibel, C. Schank, F. Jährling, B. Buschinger, A. Grauel, T. Lühman, P. Gegenwart, R. Helfrich, P. H. P. 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PhysRevB.88.075118.pdf	PHYSICAL REVIEW B 88, 075118 (2013) Tuning thermoelectric power factor by crystal-ﬁeld and spin-orbit couplings in Kondo-lattice materials Seungmin Hong,1Pouyan Ghaemi,1,2,3Joel E. Moore,2,3and Philip W. Phillips1 1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 2Department of Physics, University of California, Berkeley, California 94720, USA 3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 28 January 2013; revised manuscript received 15 July 2013; published 9 August 2013) We study thermoelectric transport at low temperatures in correlated Kondo insulators, motivated by the recent observation of a high thermoelectric ﬁgure of merit ( ZT)i nF e S b 2atT∼10K [A. Bentien et al. ,Eur. Phys. Lett. 80, 17008 (2007) ]. Even at room temperature, correlations have the potential to lead to high ZT,a si n YbAl 3, one of the most widely used thermoelectric metals. At low temperature correlation effects are especially worthy of study because ﬁxed band structures are unlikely to give rise to the very small energy gaps Eg∼5k T necessary for a weakly correlated material to function efﬁciently at low temperature. We explore the possibilityof improving the thermoelectric properties of correlated Kondo insulators through tuning of crystal-ﬁeld andspin-orbit coupling and present a framework to design more efﬁcient low-temperature thermoelectrics based onour results. DOI: 10.1103/PhysRevB.88.075118 PACS number(s): 72 .15.Jf, 75.30.Mb I. INTRODUCTION Thermoelectrics support a voltage drop in response to a modest temperature gradient. Since a temperature gradientaffects the electrons and the lattice degrees of freedom,optimizing thermoelectrics involves not only the thermopoweror Seebeck coefﬁcient ( S), but also the electrical ( σ) and thermal ( κ) conductivities. The Holy Grail of thermoelectrics is to achieve a ﬁgure of merit, ZT=/parenleftbiggS 2σ κ/parenrightbigg T, (1) that exceeds unity at room temperature. Despite great develop- ments in this regard,1this tall order remains a grand challenge problem.2–5Two of promising recent directions focused on either decreasing the thermal conductivity as in the case ofnanocrystalline arrays of Bi xSb2−xTe3in which a ZT of 1.4 was achieved3atT=373 K or on maximizing the power factor S2σthrough strong electron correlations. An example of the latter is the report6that FeSb 2achieves a colossal thermopower of 45 000 μV/K at 10 K, resulting in the largest power factor, S2σ, witnessed to date. In this paper, we follow up on the role strong correlations play in maximizing the power factorby focusing on Kondo insulators. We show explicitly thatmultiorbital physics in Kondo insulators lies at the heart ofthe problem of maximizing the power factor. Because the thermopower is related to the entropy per carrier, particle-hole asymmetry and large density of statesat the chemical potential are central to the optimization ofZ. Of course, other factors, such as the type of impurity, can affect the transport properties. However, the purpose of thepresent paper is to study the effect of engineering the densityof states in a correlated system. In this regard, the Andersonmodel of a single impurity in a metal, 7which is among the few strongly correlated systems solvable exactly, presents a densityof states with demanding features for efﬁcient thermoelectrictransport. For a single SU(2) spin on a localized impurity, the density of states appears as a single inﬁnite symmetric peak atthe chemical potential, leading to a divergent density of states but vanishing Seebeck coefﬁcient by virtue of particle-holesymmetry. Increasing the degeneracy of the localized orbitaland the metallic band to SU(N)(N> 2) softens the peak in the density of states and at the same time moves above thechemical potential leading to an asymmetric density of statesand, as a result, a larger Seebeck coefﬁcient. 8 It makes sense then to consider systems in which such physics is naturally present, for example, Kondo insulators inwhich a regular lattice of Anderson impurities is hybridizedwith multiple bands of itinerant electrons. Electrons in the localorbitals are poorly screened and strong Coloumb repulsionprohibits them from being multiply occupied. Contrary tothe single impurity, the periodic Anderson model is notexactly solvable but multiple mean-ﬁeld-type methods havebeen used 9–11to understand many of their features. Motivated by the single impurity model, we examine the effect ofdegeneracy of the local impurities and the conduction band onthe thermoelectric properties of Kondo insulators. In additionto directly studying the degeneracy of the local and conductionbands, we study the effect of lifting the degeneracy by a crystal-ﬁeld (which mainly affects the local orbitals) and spin-orbitcoupling (which mainly affects the conduction band) have onthermoelectric efﬁciency. In this way, we can continuously liftthe level of degeneracy. Interestingly, we observe that there isan optimum value for the crystal-ﬁeld and spin-orbit coupling.As was shown in a previous study, 12the presence of multiple orbitals close to the chemical potential is a common feature ofKondo insulators. Our results indeed present a possible routefor using strong correlations to enhance the thermoelectricperformance through controlling the orbital degeneracy oflocal and itinerant bands. II. MODEL AND METHODOLOGY Heavy fermion materials typically contain rare-earth or ac- tinide ions forming a lattice of localized magnetic moments.13 The strong Coulomb repulsion of electrons localized in for 075118-1 1098-0121/2013/88(7)/075118(8) ©2013 American Physical SocietyHONG, GHAEMI, MOORE, AND PHILLIPS PHYSICAL REVIEW B 88, 075118 (2013) dorbitals leads to the formation of these local moments,7 which then hybridize with the itinerant electron bands and form the heavy electron bands. If the chemical potential is inthe heavy electron bands, a heavy fermion metal is formed.The volume of the Fermi surface in this correlated statecorresponds to a sum of the number of itinerant and localizedelectrons. If the chemical potential is in the hybridizationgap, the heavy electron band will be fully occupied anda Kondo insulator obtains. 14,15Notice that such an insulat- ing state is fundamentally different from a noninteractinginsulator. For example, in order to reach a ﬁlled valanceband, we need to add the number of localized and itinerantelectrons which are developed solely as a result of stronginteractions. The underlying microscopic model of this correlated system is H=/summationdisplay klσσ/primeεσσ/prime(k)c† klσcklσ/prime+/summationdisplay klσ/epsilon1fld† klσdklσ +/summationdisplay iklσ(Vklσeik·ric† klσdilσ+H.c.) +U 2/parenleftBigg/summationdisplay il,σ/negationslash=σ/primend ilσndilσ/prime+/summationdisplay i,l/negationslash=l/prime,σσ/primend ilσndil/primeσ/prime/parenrightBigg ,(2) where c† klσ(d† klσ) is the creation of a conduction (local) electron with momentum k, orbital l, and spin σ=(↑,↓), and nd ilσ= d† ilσdilσis the number operator of a local dorbital at site ri. The dispersion of the celectron εσσ/prime(k)=/epsilon1kδσσ/prime+/Gamma1k· σσσ/primeincludes spin coupling. The nondispersive energy of local states ( /epsilon1fl) depends on the orbital index l. The pseudovector /Gamma1k represents the amplitude of the spin-orbit (SO) coupling16,17 and its form depends on the crystal symmetry of the underlying lattice (see the Appendix ). Typically, the hybridization matrixelement, V klσ, encodes the complex orbital structures of local states which can have novel effects on the properties of thestrongly correlated heavy fermion phase, 18but as in other studies, we consider Vklσto be independent of ( k,σ)t om a k e the calculation more tractable. Using the model Hamiltonian, Eq. (2), we can capture the effect of the degeneracy of both localized and itinerantbands, as well as the effect of crystal-ﬁeld and SO coupling inbreaking the degeneracy of these bands. As a result of weakscreening of electrons in fanddorbitals, the associated on-site repulsive potential Uis much larger than the hopping energies of the itinerant electrons. To treat the large on-site repulsionterm, we use the U(1) slave-boson mean-ﬁeld theory. 10,19 This method has been widely used to study both heavy fermion metals and Kondo insulators.20–27In this treatment, the creation operator of a local electron d† ilσ=f† ilσbiis partitioned into a neutral fermion f† ilσand a charged boson bithat accounts for annihilation of an empty state. Since the local Hilbert spaceis restricted to be either an empty or a singly occupied state,the additional local constraint, ˜Q i=b† ibi+/summationdisplay lσf† ilσfilσ=1, (3)should be enforced at every site ri. The Hamiltonian in terms of these slave particles then becomes H=/summationdisplay klσσ/primeεσσ/prime(k)c† klσcklσ/prime+/summationdisplay klσ/epsilon1flf† klσfklσ +/summationdisplay iklσ(V∗ le−ik·rif† ilσbicklσ+H.c.)+/summationdisplay iλi(˜Qi−1), (4) where λiis a Lagrange multiplier to maintain the local constraint. In the above Hamiltonian, the effect of the crystalﬁeld is to break the degeneracy of the local orbital states /epsilon1 fl, whereas the SO coupling breaks the spin degeneracy of theconduction band. As a result, by tuning the crystal-ﬁeld andSO coupling, we can change the degeneracy of the local andconduction orbitals in a continuous manner. Consequently,we have a tunable knob to gain the optimum thermoelectricperformance. The mean-ﬁeld approximation to the model Hamiltonian can be obtained by taking the coherent expectation b=/angbracketleftb i/angbracketright= /angbracketleftb† i/angbracketrightandλ=/angbracketleftλi/angbracketright. This corresponds to the condensation of the bosonic ﬁeld bi. However, it has been shown28that in situations sufﬁciently similar to the one we consider here, condensationof the boson does not survive the ﬂuctuations as dictated byElitzur’s theorem. 29Hence, this saddle-point approximation should mirror the actual behavior of the system and shouldnot be taken as a literal statement that the boson condenses. 28 This replacement effectively renormalizes the mixing matrixelement V l→bVland the local energy /epsilon1fl→/epsilon1fl+λand leads to the quadratic Hamiltonian HMF=/summationdisplay klh(/epsilon1k+h\|/Gamma1k\|)c† klhcklh+/summationdisplay klσ(/epsilon1fl+λ)f† klhfklh +/summationdisplay klh(bV∗ lf† klhcklh+H.c.)+λ/summationdisplay i(b2−1).(5) Instead of working in the spin basis, we use a helical basis that diagonalizes the single-electron dispersion εσσ/prime(k)→ [U† kε(k)Uk]hh/prime=(/epsilon1k+h\|/Gamma1k\|)δhh/primewith h,h/prime=± 1. Then cklh(fklh) is accordingly rotated by the unitary matrix Uk from the spin basis, cklσ(fklσ). By performing the Bogoliubov transformation, aklh+=αklhcklh+βklhfklh, (6) aklh−=−βklhcklh+αklhfklh, (7) we obtain the diagonal mean-ﬁeld Hamiltonian, HMF=/summationdisplay klh,±E± klha† klh±aklh±+λ/summationdisplay i(b2−1), (8) where the dispersion is given by E± klh=1 2(/epsilon1k+h\|/Gamma1k\|+/epsilon1fl+λ±Wklh), (9) Wklh=/radicalBig (/epsilon1k+h\|/Gamma1k\|−/epsilon1fl−λ)2+4b2V2 l. (10) The Bogoliubov parameters are /parenleftbiggα2 klh β2 klh/parenrightbigg =1 2/bracketleftbigg 1±(/epsilon1k+h\|/Gamma1k\|)−(/epsilon1fl+λ) Wklh/bracketrightbigg . (11) Minimization of the free energy with respect to the mean-ﬁeld parameters bandλ, and the total chemical potential μleads 075118-2TUNING THERMOELECTRIC POWER FACTOR BY ... PHYSICAL REVIEW B 88, 075118 (2013) to two coupled equations, 1=b2+/summationdisplay klhα2 klhnF(Eklh+)+β2 klhnF(Eklh−),(12) λ=/summationdisplay klhV2 l Wklh[nF(Eklh−)−nF(Eklh+)], (13) ntot=/summationdisplay klh[nF(Eklh−)+nF(Eklh+)], (14) where the total density of electrons is ﬁxed to be ntot=2lmax forl=1,2,..., l max. The transport of this noninteracting mean-ﬁeld Hamiltonian is now tractable. To compute the transport properties, we use the relaxation- time approximation to the Boltzmann equation.30This method has been used before in studying the properties of Kondoinsulators. 19Other more advanced methods have also been used to study transport in Kondo systems.31,32Given that the main purpose of our work is to study the effect of thecorrelation-induced density of states, we use the relaxation-time approximation. A more detailed study of scatteringprocesses is an important next step, which is beyond the scopeof this paper. Under this scheme, the electrical resistivity,ρ=σ −1, and the thermopower tensors, S,a r eg i v e nb y ρ=L−1 0,S=−kB \|e\|L−1 0L1, (15) where the tensors Lmare (Lm)ab=−e2 Volume/summationdisplay klh±∂nF(Eklh±) ∂Eklh± ×τklh±(vklh±)a(vklh±)b/parenleftbiggEklh±−μ kBT/parenrightbiggm , (16) explicitly. Here we set vklh±=1 ¯h∇Eklh±. Considering that the electrons are scattered by Nimpimpurities with an interaction strength of Vimp(i.e.,Hsctt∼Vimpc† k/primelσcklσ), the relaxation time τklh±for each state is given by 1 τklh±=2π ¯hNimp Nsite\|Vimp\|2/bracketleftbigg∂Eklh± ∂(/epsilon1k+h\|/Gamma1k\|)/bracketrightbigg2 ρlh(Eklh±),(17) withρlh(Eklh±) the density of the states of the Bogoliubov quasiparticles. III. RESULTS We now present our results on the dependence of the transport properties on the orbital degeneracies of bothlocalized and itinerant electron bands which form correlatedKondo insulators. In the ﬁrst two sections, we consider doubledegeneracy of conduction and localized bands. This modelis indeed consistent with the models previously proposedfor Kondo insulators. 33The crystal ﬁeld will then split the degeneracy of the two flevels into four and SO coupling breaks the degeneracy of the conduction band states withdiffering helicity. 17In these two sections, we change the size of the degeneracy-breaking gap continuously. Usingthe relaxation-time approximation, we can then calculatethe transport properties of a Kondo insulator. For mostof the materials, the dominant contribution to the thermalconductivity comes from lattice vibrations; as a consequence, the electronic contribution to the thermoelectric performanceis measured through the power factor Z PF=σS2, where σis the electrical conductivity and Sis the Seebeck coefﬁcient. In order to conﬁrm that the enhancement of the thermoelectricefﬁciency can properly be attributed to strong correlations, weconsider two different band structures of itinerant electrons inSecs. III A andIII B and we see that similar features emerge. Finally, in Sec. III C we present the effect of multiple orbital degeneracy. Contrary to the treatment in Secs. III A andIII B , where the double degeneracy is continuously lifted by crystal-ﬁeld and SO couplings, in Sec. III C we discretely change the number of degenerate conduction and localized bands.We show that, indeed, there is also an optimum degeneracyassociated with the maximum power factor. A. Nearly free electron itinerant bands We ﬁrst focus on the effect of crystal-ﬁeld and SO coupling on the power factor within the context of a parabolic band forthe itinerant electrons /epsilon1 k=/epsilon10+W(k/kBZ)2withW=2e V , taken from Ref. 34. In principle, the SO coupling should be expressed as a periodic function under the crystal environment,but we model it to be isotropic as well, /Gamma1 k=γso(k/kBZ) (γso/lessorequalslant0.2 eV). In the following, we carry out the numerical calculation based on this isotropic band dispersion with lmax= 2. Here we choose /epsilon1f1=1.0606 eV , V1=0.2236 eV , and V2= 1.05V1=0.2348 eV . The parameterNimp Nsite\|Vimp\|2=0.045 eV2 (Ref. 34). The other control parameters are temperature (T/lessorequalslant100 K), crystal electric ﬁeld (CEF) splitting ( /Delta1CEF= /epsilon1f2−/epsilon1f1/lessorequalslant15 meV), and the SO coupling ( γso/lessorequalslant0.15 eV). Although we do not include the supporting data here, we found 0200T=5K ΔCEF=5meV γso=20meV 0200T=10 0200 DOST=20 0200 -10 0 10T=50T=10K =0meV =20meV(e) =5 =8 -10 0 10 ω (meV)=11T=10K =1meV =0meV(i) =50 (j) =120 -10 0 10=150 (h)ΔCEF γso γso γso γsoγso CEFΔ CEFΔ CEFΔCEFΔ(a) (f) (b) (c) (g) (k) (d) (l) FIG. 1. (Color online) Density of states (DOS) of the parabolic model for different control parameters. The red dashed lines are for the orbital l=1, the dotted blue lines l=2, and the green lines are for the total DOS. The central gray area displays the thermalwindow ( ∼k BT) for each temperature. Panels (a)–(d) compare the DOS for different temperatures. Panels (e)–(h) correspond to different magnitudes of the CEF ( /Delta1CEF), which breaks the degeneracy of the twoforbitals. Panels (i)–(l) correspond to different magnitudes of the SO interaction ( γSO) that breaks the degeneracy of the conduction bands with different helicity. 075118-3HONG, GHAEMI, MOORE, AND PHILLIPS PHYSICAL REVIEW B 88, 075118 (2013) that our conclusions are insensitive to the strength of V2as long as 0.5/lessorsimilarV2/V1/lessorsimilar2.0. Figure 1 shows the density of states (DOS) for different control parameters. From Fig. 1(a) to Fig. 1(d), we notice that the temperature controls only the number of thermallyactivated charge carriers, while it does not signiﬁcantlychange the DOS compared to the other parameters. Whenthe degeneracy of the two local orbitals is broken by the CEF,one of the hybridized bands moves closer to the chemicalpotential. Consequently, the system is driven from an insulatorto a conductor [Figs. 1(e)–1(h)], at which point the power factor is signiﬁcantly enhanced (see Fig. 2). Likewise, the SO interaction breaks the degeneracy of the two helical modes,which turns an insulator into a metallic state [Figs. 1(i)–1(l)]. We point out that the metallic state is characterized eitherby a local orbital, l=2 [Fig. 1(h)], or by a helicity, h=+ [Fig. 1(l)], since only the bands with corresponding quantum numbers are conducting. In order to further examine the effect of the control parameters, we ﬁrst calculate the transport coefﬁcients asa function of the CEF and SO. In Fig. 2, we show the results of a calculation of the thermopower ( S), the electrical conductivity ( σ), and the power factor ( Z PF). As can be seen from the right column, the power factor is enhanced eitherby ﬁnding the optimal CEF or by adjusting the SO. Sinceboth CEF and SO shift some of the lower energy bandstoward the chemical potential [Figs. 1(e)–1(l)], the number of lower energy bands relevant for thermal transport is controlledby CEF and SO simultaneously. For the temperature rangeT/lessorsimilar20 K, where the thermal windows are sufﬁciently narrow, 0.000.050.10 0.000.050.10 0.000.050.10γSO (eV) 0.000.050.10 0.000.050.10S 050010001500 μV K-1 051015 ΔCEF (meV)σ 012 mΩ-1cm-1 051015ZPF=S2σ 0.000.010.02 μW K-2cm-1 051015T=50K T=30K T=20K T=10K T=5K FIG. 2. (Color online) Transport coefﬁcients for different tem- peratures as a function of crystal-ﬁeld splitting and SO interaction: thermopower S(left column), conductivity σ(middle column), and power factor ZPF(right column). From top to bottom, the temperature varies from 50 K to 5 K, and the solid lines are equally spaced constant contours.CEF and SO compete; hence, there are two distinctive optimal regimes. For a sufﬁciently wide thermal window, attainableat intermediate temperatures, T∼30 K, CEF and SO are working cooperatively to form a single optimal region. ForT> 30 K, the enhancement in Z PFis not as drastic as at low temperature. ZPFis maximized in the vicinity of the insulator- metal transition (see the conductivity σatT=5–20 K), resulting from a competition between Sandσ. For instance, atT=5 K, the thermopower Sdecreases with SO and CEF, while the system acquires a ﬁnite conductivity. Note that themetallic state here has one dominant helical state over theother. In Fig. 3, we repeat the calculation of the transport coefﬁcients for a ﬁxed SO as a function of CEF andtemperature. Consistent with Fig. 2is that the optimal point for the power factor is located in the vicinity where theinsulator-metal transition occurs. For instance, when γ so/lessorsimilar 0.1 eV , there is the optimal CEF and temperature for the power factor, at which point the electric conductivity acquiresa noticeable ﬁnite value. From the left column, one ﬁndsthat the thermopower generally decreases with increasingtemperature as a widened thermal widow implies the reductionof the asymmetry in the DOS within the thermal region[Figs. 1(a)–1(d)]. This obtains because, as the temperature increases more of the bands (lower and upper) are involvedin the thermal transport. In other words, the asymmetry of theDOS within the thermally active region is relieved. Beyonda certain threshold of SO, γ so/greaterorequalslant0.13 eV , there is no phase transition (at mean-ﬁeld level); hence, optimization cannot berealized. 2550 2550 2550T (K) 2550 255005001000 μ ΩV K-1 051015 ΔCEF (meV)0123 m-1cm-1 051015S σ 0.000.010.02 μW K-2cm-1 051015γSO=0.02 eVZPF=S2σ γSO=0.05 eV γSO=0.10 eV γSO=0.13 eV γSO=0.15 eV FIG. 3. (Color online) Transport coefﬁcients for each SO as a function of CEF and temperature: thermopower S(left column), conductivity σ(middle column), and power factor ZPF(right column). The range of SO is 0 .02–0.15 eV from the top to the bottom panels. 075118-4TUNING THERMOELECTRIC POWER FACTOR BY ... PHYSICAL REVIEW B 88, 075118 (2013) B. Tight-binding itinerant electron bands Next, we consider the three-dimensional tight-binding case. We see that, as in the case of a quadratic band, tuning thecrystal-ﬁeld and SO coupling can optimize the thermoelectricperformance. This result indicates that the effect of orbitaldegeneracy in controlling the thermoelectric performance isnot that sensitive to the details of the band structure. Here wechoose the hopping parameter t hop=0.2167 eV (band width W=2.6 eV), and we located the local energy /epsilon1f1=− 0.8thop. The hybridization strength V1=thopandV2=1.01thop.I n the SO, we take the next-nearest-neighbor hopping parameterg 2=0.3. As in the simpliﬁed parabolic model, the roles of CEF and SO are not different; both efﬁciently control the systemto drive it from an insulator to a conductor, as seen fromFig. 4. Compared to the corresponding panels in Figs. 1, however, Figs. 4(e)–4(h) show that the CEF also pushes one of the upper bands toward the chemical potential, hencereducing the gap size signiﬁcantly. In fact, the parabolic modelis rather exceptional since the bottom of the upper bandscorresponds to the point k=0, which is not usual for typical three-dimensional (3D) tight-binding models. Figures 4(i)– 4(l) display the evolution of DOS with the increase of the SO. Even though the degeneracy of the two helical modes arebroken with a ﬁnite SO, it cannot be seen clearly, as was inthe linearized SO case [Figs. 1(i)–4(l)]. The reason is that \|/Gamma1 k\| decreases as kapproaches the boundary of the Brillouin zone due to the periodic form of the SO, while it does not for thelinearized SO. Unlike the CEF, which affected drastically onlyone of the orbitals, the effect of the SO is quite different. Uptoγ so=0.2thop/similarequal40 meV , the changes in the DOS are not signiﬁcant. For γso/greaterorsimilar0.2thop, the system undergoes a phase 0200=0.02 =0.1 0200T=10 0200T=20 0200 -5 0 5T=50=0.0 =0.02 =0.01 =0.02 -5 0 5=0.05 (h)=0.02 =0.0 =0.2 =0.5 -5 0 5=0.7 DOS ω (meV)T=5K ΔCEF γsoT=5KΔ CEF γsoT=5KΔ CEF γso γso γso γsoΔCEF ΔCEF ΔCEF(a) (e) (i) (b) (f) (j) (k) (g) (c) (d) (l) FIG. 4. (Color online) Density of states (DOS) of the 3D tight- binding model for different control parameters. Both /Delta1CEFandγsoare in units of the hoping amplitude, thop=0.216 eV . The red dashed lines are for the orbital l=1, the dotted blue lines are for l=2, and the green lines are for the total DOS. The central gray area indicates thethermally active region for each temperature. Panels (a)–(d) compare the DOS for different temperatures, (e)–(h) for the CEF ( /Delta1 CEF), and (i)–(l) for the SO interaction ( γSO). 0.000.25 0.000.25 0.000.25γso (thop) 0.000.25-400-2000 μV K-1 0.00 0.04 ΔCEF (thop)0.00.51.0 mΩ-1cm-1 0.00 0.04S σ 0.0000.0050.0100.015 μW K-2cm-1 0.00 0.04T=25KZPF=S2σ T=15K T=10K T=5K FIG. 5. (Color online) Transport coefﬁcients for each tempera- ture: thermopower S(left column), conductivity σ(middle column), and power factor ZPF(right column). From top to bottom, the temperature is ﬁxed to 25 K, 15 K, 10 K, and 5 K, respectively. transition to a (helically polarized) metal, beyond which point ZPFis reduced (see Fig. 5). From Fig. 5, we observe consistency with the parabolic model: The power factor can be enhanced by adjusting theCEF, while the SO slightly lowers the optimal value of theCEF. For T> 15 K, the enhancement in Z PFis not as drastic as it was in the low-temperature case. As in the parabolicmodel, this trend occurs because the thermally active regionis too wide to encompass only one band [see Figs. 4(c) and 4(d)]. Comparison with the other columns reveals that Z PFis also maximized near an insulator to a (helical) metal transition (see the conductivity σatT=5K ) ,w h i c hi st h e consequence of the competition between Sandσ. Here one can observe that Sbecomes maximal at /Delta1CEF/similarequal0.01thop, which is a consequence of the choice V2/V1=1.01. With V2/V1=1, Sonly decreases with CEF (not shown). Figure 6similarly conﬁrms the consistency with the parabolic model. The onlydifference is that the effect of SO is not as remarkable, thoughit works to shift the optimal value of the CEF. The reasonmainly lies in the changes of the DOS depending on SO:Linearized SO changes the bandwidth signiﬁcantly, while 3Dtight-binding SO does not, due to its periodic structure (seeFig. 7). C. Effect of multiorbital degeneracy In addition to the continuous control of orbital degeneracy through crystal-ﬁeld and SO coupling, we can speciﬁcallystudy the effect of increasing the number of degenerateorbitals ( l max=1,2,..., 5). To minimize the number of free parameters, we set the orbital degeneracy of the two bands (theallowed values of l) to be equal. Although continuous control is not possible in this case, one can then consider changing thematerial content to achieve a better thermoelectric. Here, thebare conduction electron dispersion is taken to be that of the3D tight binding model. 075118-5HONG, GHAEMI, MOORE, AND PHILLIPS PHYSICAL REVIEW B 88, 075118 (2013) 025 025 025T (K) 025-600-400-2000 μV K-1 0.01.0 mΩ-1cm-1S σ 1.5 0.5 0.00 0.04 0.00 0.04 ΔCEF (thop)0.00 0.040.0000.0050.0100.015 μW K-2cm-1γSO=0.0 (thop) =0.1 =0.2 =0.5 ZPF=S2σ γSO γSO γSO FIG. 6. (Color online) Transport properties are compared for each SO:S(left column), conductivity σ(middle column), and power factor ZPF(right column). The SO couplings are chosen to be 0.0, 0.1, 0.2, and 0.5 in units of the hoping amplitude thop, respectively. First, we compare the DOS depending on the number of orbitals involved (Fig. 8). Aslmaxincreases, the asymmetry between the upper and the lower bands becomes morepronounced. At the same time, the insulating gap increaseswithl maxforlmax/greaterorequalslant2. Note that this feature is quite similar to the single impurity problem with Nﬂ a v o r s .T h ei n s e to fF i g . 8 displays the DOS without adjusting the chemical potential. 0.5 0 0.5 1 1.5 2DOS ω (eV)(a)γso=0.0 eV 0.1 eV 0.2 eV 0.3 eV 0 0.5 1 1.5 2ω (eV)(b) 0.00.51.0 -1 -0.5 0 0.5 1DOS ω (6t )(c)γso=0.0 t hop 0.2 t hop 0.4 t hop 0.6 t hop -1 -0.5 0 0.5 1(d)0.01.0γso=0.0 eV 0.1 eV 0.2 eV 0.3 eV hop ω (6t ) hopγso=0.0 t hop 0.2 t hop 0.4 t hop 0.6 t hop FIG. 7. (Color online) Density of states for bare conduction electrons: quadratic dispersion (top) and 3D tight binding (bottom) and helicity h=− 1 (left, blue curves) and h=1 (right, red lines). For the quadratic dispersion, the SO is taken to be linear in momentum, asγ so\|k\|. The changes in the bandwidth are exactly proportional to γso.I n panels (c) and (d), the SO term, γso/Gamma1k, is taken in accordance with the cubic point group symmetry, and the next-nearest-neighbor hoppingparameter g 2=0.3. Note that the bandwidths are not drastically affected by the SO, while the shapes become more asymmetric with t h ei n c r e a s ei n γso. 0 50 100 150 200 250 -0.01 μ 0.01 0.02DOS/l max ω (eV)lmax=1 2 3 4 5 0510 0 0.5 1 ω (eV) FIG. 8. (Color online) Density of states per orbital for different number of orbitals ( lmax), where the chemical potential at each case is adjusted to the center. The inset displays the DOS without the adjustment. Since the slave-boson method renormalizes the local energy by/epsilon1f→/epsilon1f+λ, the relative location of the Kondo resonance for each case (near each gap) indicates that the amount ofrenormalization λincreases with the number of available orbitals. Since each band below and above the insulating gapshould accommodate one electron, the deformation of thelower bands becomes less signiﬁcant as l maxincreases (see the lower bands for different lmax). In other words, since the area below and above the gap should be equal, the asymmetryof the DOS becomes more signiﬁcant as λ, or equivalently l max, increases. Given the band structures at the mean-ﬁeld level, we proceed to calculate the transport properties as shown in Fig. 9. Aslmaxincreases, the maximum of ZPFalso increases as the temperature is elevated. The thermopower is also enhancedwith the number of available orbitals, which is caused bypronounced asymmetry in the DOS (see Fig. 8). Since the gap size increases, the conductivity generally decreases with the 0 1 2 10 20 30 40 50κ (μW K-1cm-1) T (K)(c) 0 250 500S (μV K-1)(a) lmax=1 2 3 4 5 0 0.5 1 1.5σ (mΩ-1cm-1) (b) 0 0.005 0.01ZPF (μW K-2cm-1)(d) 10-410-310-210-1Z (1/K)(e) 10-310-210-1100 10 20 30 40 50ZT T (K)(f) FIG. 9. (Color online) Transport properties for lmaxorbitals: (a) Seebeck coefﬁcient, (b) electrical conductivity, (c) thermal conduc- tivity, (d) thermopower, (e) ﬁgure of merit, and (f) dimensionless ﬁgure of merit. 075118-6TUNING THERMOELECTRIC POWER FACTOR BY ... PHYSICAL REVIEW B 88, 075118 (2013) number of orbitals for lmax/greaterorequalslant2. Though not shown here, the maximal power factor per orbital, ZPF/lmax, also increases with lmaxuntillmax<7. In Fig. 9(c), the thermal conductivity due to electronic structure is evaluated, excluding any contributionfrom lattice vibrations. Typically, phonons are dominantcontributors to the thermal conductivities, but it may not beso prevalent at the low-temperature range considered here,presumably T/lessorsimilar10 K. The resultant (dimensionless) ﬁgure of merit, assuming κ=κ electron +κphonon/similarequalκ=κelectron ,i s strongly enhanced with the number of orbitals at least by anorder of 10. This is one of the key results of this paper. IV . SUMMARY In this paper we studied one scheme by which tuning the orbital degeneracy can be used to enhance the power factorfor strongly interacting thermoelectrics. Our key ﬁndings arethat the power factor is maximized in strongly correlatedsystems by tuning (1) the gap between nearly degeneratelocalforbitals through the crystal-ﬁeld effect, (2) the gap between nearly degenerate itinerant electron bands throughthe SO coupling, and (3) the the number of degenerate localand itinerant orbitals. The amplitude of the SO coupling andthe degeneracy of orbitals close to the Fermi energy are usuallyknown properties of the materials and might be also tuned bydoping. 35Our results then serve as a guiding tool for designing efﬁcient thermoelectric materials. This approach provides a parameter space for the design of strongly correlated thermoelectric materials. Our resultwas derived using the slave-particle mean-ﬁeld theory, whichis not expected to be quantitatively reliable but shouldcapture general trends. The effect of degeneracy in enhancingthermoelectric performance of strongly correlated systemscould also be investigated using other methods such dynamicalmean-ﬁeld theory 36and ﬁnite frequency methods.37Another direction for future work is to consider more profound effectsof SO coupling combined with correlations, as in the proposed“topological Kondo insulators,” 38whose surface states are currently being sought experimentally; such surface states havethe potential to increase thermoelectric performance at lowtemperatures. 2 Finally, a direction that is difﬁcult theoretically but may be important for actual materials is to ﬁnd ways of interpolatingbetween the effectively itinerant calculation here (i.e., thereare plane-wave states of the slave particles) with the “atomiclimit,” 39where the effects of multiple orbitals have also been considered.40The atomic limit, which is valid when the hopping is the smallest energy scale in the problem,has been argued to be relevant to experiments on sodium cobaltates near room temperature.41,42The results of this paper should motivate continued investigation of correlatedmaterials for thermoelectricity and suggest that a guidedsearch with controlled crystal-ﬁeld splitting may lead to furtherimprovements in thermoelectric ﬁgure of merit, especially inthe low-temperature regime. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Ofﬁce of Basic Energy Sciences, Materials Sciencesand Engineering Division, under Contract No. DE-AC02-05CH11231 (P.G. and J.E.M.). P.G. also acknowledges supportfrom NSF Grant No. DMR-1064319. S.H. and P.W.P. are funded by NSF Grant No. DMR-1104909. APPENDIX: SPIN-ORBIT COUPLING In the presence of SO coupling,16,17the conduction electron dispersion matrix εσσ/prime(k) is given by εσσ/prime(k)=/epsilon1kδσσ/prime+/Gamma1k·σσσ/prime, (A1) where /epsilon1kis the dispersion without the SO interaction and σ are the Pauli matrices. (The SO interactions considered hereoriginate from the absence of an inversion symmetry in thecrystal lattice.) The antisymmetric SO coupling is describedby the real pseudovector /Gamma1 k, which is determined by the point group symmetry of the crystal. For instance, CePt 3Si, CeRhSi 3, and CeIrSi 3belong to tetragonal point group ( G=C4v)i n which /Gamma1k=γso[ˆkxsinkya−ˆkysinkxa +ˆkzg2sinkxasinkyasinkzc(coskxa−coskya)], (A2) in the next-nearest-neighbor approximation for a real γso,t h e lattice spacing a,c, and the next-nearest-neighbor parameter g2. In case of the cubic point group symmetry ( G=O), the pseudovector is given by /Gamma1k=γsoˆkxsinkxa[1−g2(coskya+coskza)] +(positive permutations of x,y,z ). (A3) In real noncentrosymmetric crystals, the typical SO strength ranges up to 200 meV . Instead of working in the spin basis,it is useful to introduce the helical basis that diagonalizes the single-electron dispersion ε σσ/prime(k)→[U† kε(k)Uk]hh/prime=(/epsilon1k+ h\|/Gamma1k\|)δhh/primewithh,h/prime=± 1. 1Y . Ono, T. Matsuura, and Y . Kuroda, J. Am. Ceram. Soc. 96,1 (2013). 2P. Ghaemi, R. S. K. Mong, and J. E. Moore, Phys. Rev. Lett. 105, 166603 (2010). 3B. Poudel et al. ,Science 320, 634 (2008). 4C. Wood, Rep. Prog. 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Vekhter, P h y s .R e v .B 85, 081107(R) (2012). 18P. Ghaemi, T. Senthil, and P. Coleman, Phys. Rev. B 77, 245108 (2008). 19W. Mao and K. S. Bedell, Phys. Rev. B 59, R15590 (1999). 20V . Dorin and P. Schlottmann, Phys. Rev. B 47, 5095 (1993). 21V . Dorin and P. Schlottmann, Phys. Rev. B 46, 10800 (1992). 22P. S. Riseborough, Phys. Rev. B 68, 235213 (2003). 23P. Riseborough, J. Magn. Magn. Mater 127, 226 (2001). 24S. Burdin, A. Georges, and D. R. Grempel, Phys. Rev. Lett. 85, 1048 (2000). 25J. W. Rasul, P h y s .R e v .B 51, 2576 (1995). 26Y . Ono, T. Matsuura, and Y . Kuroda, J. Phys. Soc. Jpn. 63, 1406 (1994).27C. Sanchez-Castro, K. S. Bedell, and B. R. Cooper, P h y s .R e v .B 47, 6879 (1993). 28R. Frsard, H. Ouerdane, and T. Kopp, Nucl. Phys. B 785, 286 (2007). 29S. Elitzur, P h y s .R e v .D 12, 3978 (1975). 30N. M. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976). 31T. J. Scheidemantel, C. Ambrosch-Draxl, T. Thonhauser, J. V .Badding, and J. O. Sofo, P h y s .R e v .B 68, 125210 (2003). 32G. K. Madsena and D. J. Sing, Comput. Phys. Commun. 175,6 7 (2006). 33H. Kontani, J. Phys. Soc. Jpn. 73, 515 (2004). 34C. Sanchez-Castro, Philos. Mag. B 73, 525 (1996). 35C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Phys. Rev. X 1, 021001 (2011). 36A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992). 37B. S. Shatry, Rep. Prog. Phys. 72, 016501 (2009). 38M. Dzero, K. Sun, V . Galitski, and P. Coleman, P h y s .R e v .L e t t . 104, 106408 (2010). 39G. Beni, P h y s .R e v .B 10, 2186 (1974). 40S. Mukerjee, Phys. Rev. B 72, 195109 (2005). 41M. Lee et al. ,Nat. Mater. 5, 537 (2006). 42S. Mukerjee and J. Moore, Appl. Phys. Lett. 90, 112107 (2007). 075118-8
PhysRevB.96.115128.pdf	PHYSICAL REVIEW B 96, 115128 (2017) Defect-induced large spin-orbit splitting in monolayer PtSe 2 Moh. Adhib Ulil Absor,Iman Santoso, Harsojo, and Kamsul Abraha Department of Physics, Universitas Gadjah Mada, BLS 21 Yogyakarta, Indonesia Fumiyuki Ishii and Mineo Saito Faculty of Mathematics and Physics Institute of Science and Engineering Kanazawa University, 920-1192 Kanazawa, Japan (Received 19 July 2017; published 18 September 2017) The effect of spin-orbit coupling on the electronic properties of monolayer (ML) PtSe 2is dictated by the presence of the crystal inversion symmetry to exhibit a spin-polarized band without the characteristic of spinsplitting. Through fully relativistic density-functional theory calculations, we show that large spin-orbit splittingcan be induced by introducing point defects. We calculate the stability of native point defects such as a Sevacancy (V Se), a Se interstitial (Se i), a Pt vacancy (V Pt), and a Pt interstitial (Pt i) and ﬁnd that both the V Seand Seihave the lowest formation energy. We also ﬁnd that, in contrast to the Se icase exhibiting spin degeneracy in the defect states, the large spin-orbit splitting up to 152 meV is observed in the defect states of the V Se.O u r analyses of orbital contributions to the defect states show that the large spin splitting is originated from thestrong hybridization between Pt- d x2+y2+dxyand Se- px+pyorbitals. Our study clariﬁes that the defects play an important role in the spin-splitting properties of the PtSe 2ML, which is important for designing future spintronic devices. DOI: 10.1103/PhysRevB.96.115128 I. INTRODUCTION Much of the recent interest in spintronics has been focused on the manipulation of nonequilibrium materials using spin-orbit coupling (SOC) [ 1,2]. When the SOC occurs in a system with sufﬁciently low crystalline symmetry, an effectivemagnetic ﬁeld B eff∝[∇V(r)×p] is induced [ 3,4], where V(r) denotes the crystal potential and pis the momentum, that leads to spin splitting even in nonmagnetic materials. Current-induced spin polarization [ 5] and the spin Hall effect [ 6] are important examples of spintronics phenomena where theSOC plays an important role. For spintronic device operation[7], semiconductor materials having large spin splitting are highly desirable [ 8]. Besides their electronic manipulability under gate voltages [ 9,10], semiconductors with the large spin splitting enable us to allow operation as a spintronic device atroom temperature [ 11,12]. The two-dimensional (2D) transition-metal dichalco- genides (TMDs) family comprise promising candidates forspintronics due to the strong SOC [ 13–17]. Most of the 2D TMD families have graphenelike hexagonal crystal structureconsisting of transition-metal atoms ( M) sandwiched between layers of chalcogen atoms ( X) with MX 2stoichiometry. However, depending on the chalcogen stacking, there aretwo stable forms of the MX 2in the ground state, namely anHphase having trigonal prismatic hole for metal atoms, and a Tphase that consists of staggered chalcogen layers forming octahedral hole for metal atoms [ 18]. In the H-MX 2 monolayer (ML) systems such as molybdenum and tungsten dichalcogenides (MoS 2, MoSe 2,W S 2, and WSe 2), the absence of inversion symmetry in the crystal structure together withstrong SOC in the 5 dorbitals of transition-metal atoms leads to the fact that a large spin splitting has been established[13–17]. This large spin splitting is believed to be responsible adib@ugm.ac.idfor inducing some of interesting phenomena such as the spin Hall effect [ 19,20], spin-dependent selection rule for optical transitions [ 21], and magnetoelectric effect in TMDs [ 22]. Furthermore, the long-lived spin relaxation and spin coherenceof electrons have also been reported on various H-MX 2TMD MLs, such as MoS 2ML [ 23,24] and WS 2ML [ 24], that could be implemented as energy-saving spintronic devices. Recently, PtSe 2ML, a 2D TMD ML with T-MX 2ML structures, has attracted much attention since it was suc-cessfully synthesized by a single-step fabrication method,a direct selenization at the Pt(111) substrate [ 25], which is in contrast to conventional fabrication methods used inthe various H-MX 2TMD MLs such as exfoliation [ 26]o r chemical vapor deposition (CVD) [ 27,28]. Moreover, the high electron mobility up to 3000 cm2/V/s has been experimentally observed on the PtSe 2ML, which is the largest among the studied TMD MLs [ 29] and thus is of great interest for electronic applications. However, the PtSe 2ML has the crystal inversion symmetry, and, consequently, the SOC leadsto spin-polarized bands without the characteristic of spin splitting. This is supported by the fact that the absence of the spin splitting has been experimentally observed by Yao et al. using spin- and angle-resolved photoemission spectroscopy(spin-ARPES) [ 30]. Because the absence of the spin splitting in the PtSe 2ML provides a natural limit for spintronic applications, it is highly desirable to ﬁnd a method to generatethe spin splitting in the PtSe 2ML, which is expected to enhance its functionality for spintronics. In this paper, by using fully relativistic density-functional theory (DFT) calculations, we show that large spin-orbit splitting in the PtSe 2ML can be induced by introducing point defects. We calculate stability of native point defects such as aSe vacancy (V Se), a Se interstitial (Se i), a Pt vacancy (V Pt), and a Pt interstitial (Pt i) and ﬁnd that both the V Seand Se ihave the lowest formation energy. By taking into account the effect ofthe SOC in our DFT calculations, we ﬁnd that, in contrast to the Se icase having spin degeneracy in the defect states, the large 2469-9950/2017/96(11)/115128(6) 115128-1 ©2017 American Physical SocietyABSOR, SANTOSO, HARSOJO, ABRAHA, ISHII, AND SAITO PHYSICAL REVIEW B 96, 115128 (2017) FIG. 1. The relaxed structures of native point defects induced by vacancy and interstitial in the PtSe 2ML compared with the pristine system: (a) a pristine, (b) a Se vacancy (V Se), (c) a Se interstitial (Se i), (d) a Pt vacancy (V Pt), and (e) a Pt interstitial (Pt i). The Pt-Se, Pt-Pt, Sei-Se, and Pt i-Pt bond lengths are indicated by the red arrows. spin-orbit splitting up to 152 meV is observed on the defect states of the V Se. We clarify the origin of the spin splitting by considering orbital contributions to the defect states and ﬁnd that the large spin splitting is mainly originated from thestrong hybridization between Pt- d x2+y2+dxyand Se- px+py orbitals. Finally, a possible application of the present system for spintronics will be discussed. II. COMPUTATIONAL DETAILS We performed ﬁrst-principles electronic structure calcu- lations based on the DFT within the generalized gradientapproximation (GGA) [ 31]u s i n gt h e OPENMX code [ 32]. We used norm-conserving pseudopotentials [ 33], and the wave functions are expanded by the linear combination ofmultiple pseudoatomic orbitals (LCPAOs) generated using aconﬁnement scheme [ 34,35]. The orbitals are speciﬁed by Pt7.0-s 2p2d2and Se9.0- s2p2d1, which means that the cutoff radii are 7.0 and 9.0 bohr for the Pt and Se atoms, respectively,in the conﬁnement scheme [ 34,35]. For the Pt atom, two primitive orbitals expand the s,p, anddorbitals, while, for the Se atom, two primitive orbitals expand the sandporbitals, and one primitive orbital expands the dorbital. Spin-orbit coupling was included in our DFT calculations. Bulk PtSe 2crystallizes in a centrosymmetric crystal asso- ciated with a Tstructure ( T-MX 2), having space group P3mI for the global structure and polar group C3vandD3dfor the Se and Pt sites, respectively. In the monolayer (ML) phase,one Pt atom is sandwiched between two Se atoms, forming anoctahedral hole for transition-metal atoms and shows trigonalstructure when projected to the (001) plane [Fig. 1(a)]. In ourDFT calculations, we used a periodic slab to model the PtSe 2 ML, where a sufﬁciently large vacuum layer (20 ˚A) is used to avoid interaction between adjacent layers. The geometrieswere fully relaxed until the force acting on each atom was lessthan 1 meV /˚A. We ﬁnd that the calculated lattice constant of the PtSe 2ML is 3.75 ˚A, which is in good agreement with the experiment (3.73 ˚A[25]) and previous theoretical calculations (3.75 ˚A[36–38]). We then introduced native point defects consisting of a Se vacancy (V Se), a Se interstitial (Se i), a Pt vacancy (V Pt), and a Pt interstitial (Pt i) [Figs. 1(b)–1(e)]. To model these point defects, we constructed a 4 ×4×1 supercell of the pristine PtSe 2ML with 48 atoms. The larger supercell (5 ×5×1 and 6×6×1 supercells) was used to test our calculational results, and we conﬁrmed that it does not affect to the main conclusion.We calculated formation energy to conﬁrm stability of thesepoint defects by using the following formula [ 39]: E f=Edefect−Eperfect+/summationdisplay iniμi. (1) In Eq. ( 1),Edefect is the total energy of the defective system, Eperfect is the total energy of the perfect system, niis the number of atom being added or removed from the perfect system, andμ iis the chemical potential of the added or removed atoms corresponding to the chemical environment surrounding thesystem. Here, μ iobtains the following requirements: EPtSe 2−2ESe/lessorequalslantμPt/lessorequalslantEPt, (2) 1 2(EPtSe 2−EPt)/lessorequalslantμSe/lessorequalslantESe. (3) Under Se-rich condition, μSeis the energy of the Se atom in the bulk phase (hexagonal Se, μSe=1 3ESe-hex ), which corresponds to the lower limit on Pt, μPt=EPtSe 2−2ESe. On the other hand, in the case of the Pt-rich condition, μPtis associated with the energy of the Pt atom in the bulk phase (fcc Pt,μ Pt=1 4EPt-fcc) corresponding to the lower limit on Se, μPt= 1 2(EPtSe 2−EPt). III. RESULT AND DISCUSSION First, we examine energetic stability and structural relax- ation in the defective PtSe 2ML systems. Table Ishows the calculated results of the formation energy for the point defects(V Se,VPt,S ei,P ti) corresponding to the Pt-rich and Se-rich TABLE I. Formation energy (in eV) of various point defects in the PtSe 2ML corresponding to the Se-rich and Pt-rich conditions. The theoretical data from the previous report are given for a comparison. Point defects Pt-rich (eV) Se-rich (eV) Reference VSe 1.27 1.84 This work1.24 1.83 Ref. [ 37] 1.82 Ref. [ 40] V Pt 3.06 4.28 This work 3.00 4.19 Ref. [ 37] 3.7 Ref. [ 38] Sei 2.01 1.98 This work Pti 4.68 3.45 This work 115128-2DEFECT-INDUCED LARGE SPIN-ORBIT SPLITTING IN . . . PHYSICAL REVIEW B 96, 115128 (2017) FIG. 2. The electronic band structure of (a) the pristine, (b) the V Se, and (c) the Se i, where the calculations are performed without inclusion the effect of the spin-orbit coupling (SOC). The electronic band structure of (d) the pristine, (e) the V Se, and (f) the Se iwith inclusion of the effect of the SOC. The Fermi level is indicated by the dashed black lines. conditions. Consistent with previous studies [ 37,38,40], we ﬁnd that the V Seand Se ihave the lowest formation energy in both the Pt-rich and the Se-rich conditions, indicating that bothsystems are the most stable point defects formed in the PtSe 2 ML. The found stability of the V Seand Se iis consistent with previous reports that the chalcogen vacancy and interstitialcan be easily formed in the TMD MLs, as found in the MoS 2 [40–42], WS 2[43], and ReS 2[44]. In contrast, the formation of the other point defects (V Ptand Pt i) is highly unfavorable due to the required electron energy. Because the Pt atom iscovalently bonded to the six neighboring Se atoms, addingor removing the Pt atom is stabilized by destroying the Sesublattice, thus increasing the formation energy. Due to the relaxation, the position of the atoms around the point defects marginally changes from the position ofthe pristine atomic positions. In the case of the V Se, one Se atom in a PtSe 2ML is removed in a supercell [Fig. 1(b)], and, consequently, three Pt atoms surrounding the vacancyare found to be relaxed, moving close to each other. Aroundthe V Sesite, the Pt-Se bond length at each hexagonal side has the same value of about 2.509 ˚A. As a result, trigonal symmetry suppresses the V Seto exhibit the C3vpoint group [Fig. 1(b)]. Similar to the V Secase, the V Ptretains threefold rotation symmetry [Fig. 1(c)], yielding the D3hpoint group. We ﬁnd that the Pt-Se bond length in the V Ptis 2.508 ˚A, which is slightly lower than that of the pristine system (2.548 ˚A).The geometry of the Se iundergoes signiﬁcant distortion from the pristine crystal, but the symmetry itself remainsunchanged [Fig. 1(d)]. Here, we investigate various atomic conﬁgurations for the Se iand ﬁnd that the Se adatom structure on top of a host Se atom is the most stable conﬁguration.In this conﬁguration, we ﬁnd that the Se-Se ibond length is 2.313 ˚A. We also ﬁnd two other metastable conﬁgurations of the Se i: (i) bridge position of the Se iwith two host Se atoms on the surface (2.94 eV higher in energy) and (ii) hexagonalinterstitial in the Pt layer bonding with the three host Pt atoms(5.95 eV higher in energy). The Pt iis the most complicated case among the chosen point defects. There are ﬁve atomicconﬁgurations of the Pt i, and we conﬁrmed that the Pt-Pt isplit interstitial along the cdirection [Fig. 1(e)] is the most stable conﬁguration. Four other metastable conﬁgurations are (i) thebridge conﬁguration of Pt iin between two surface Se atoms (0.95 eV higher in energy), (ii) the Pt iat hexagonal hollow center on the surface (1.44 eV higher in energy), (iii) the Pt i hexagonal hollow center in the Pt layer (3.05 eV higher inenergy), and (iv) the Pt ion top of a surface Se atom (3.15 eV higher in energy). In the Pt-Pt isplit interstitial conﬁguration, we ﬁnd that the Pt-Pt ibond length is 1.05 ˚A. Strong modiﬁcation of the electronic properties of the PtSe 2 ML is expected to be achieved by introducing the point defect.Here, we focused on both the V Seand the Se ibecause they have the lowest formation energy among the other point defects. 115128-3ABSOR, SANTOSO, HARSOJO, ABRAHA, ISHII, AND SAITO PHYSICAL REVIEW B 96, 115128 (2017) FIG. 3. Density of states projected to the atomic orbitals for (a) the V Seand (b) Se i. The NN and NNN Se denote the nearest- and next-nearest-neighboring Se atoms, respectively. Figure 2shows the calculated results of the electronic band structure of the defective systems (V Seand Se i) compared with those of the pristine one. In the case of the V Se, without inclusion of the SOC, three defect levels are generated insidethe band gap [Fig. 2(b)]. Due to the absence of an anion in the V Se, two excess electrons occupy the one bonding states near the valence band maximum (VBM), while the twoantibonding states are empty, which are located close to theconduction band minimum (CBM). Our calculational resultsof the density of states (DOS) projected to the atoms nearthe V Sesite conﬁrmed that the two unoccupied antibonding states originated mainly from dx2+y2+dxyorbitals of the Pt atom with a small contribution of px+pyorbitals of the nearest-neighboring (NN) Se atoms [Fig. 3(a)]. On the other hand, admixture of the Pt- dx2+y2+dxyand the small NN Se-pzorbitals characterizes the one occupied bonding state. It is pointed out here that the porbitals of the next-nearest- neighboring (NNN) Se atoms contribute very little to the defectstates, indicating that only the Pt- dand NN Se- porbitals play an important role in the defect states of the V Se. The formation of the Se ialso induces three defect levels inside the band gap, which are two occupied bonding states near the VBM and one unoccupied antibonding state close to the CBM [Fig. 2(c)]. Due to the fact that the Se iis on top conﬁguration [Fig. 1(c)], the Se iatom forms a Se i-Se bond with a host Se atom. In this case, the host Se atom is an anionwhich is in the Se 2−oxidation state in the PtSe 2ML, while the Seiis in the neutral state. Accordingly, four pelectrons occupy the two bonding states ( px,py) near the VBM, while the one antibonding state ( pz) located near the CBM remains empty [Fig. 3(b)]. The fully occupied two bonding levels ( px,py) and one empty antibonding level ( pz) play an important role in the nature of the Se i-Se linear diatomic chemical bonding, which FIG. 4. Relation between the point group symmetry of the V Se, the spin splitting, and the orbital contributions to the defect states in the ﬁrst Brillouin zone. (a) Symmetry operation in the real space of the VSecorresponding to (b) the ﬁrst Brillouin zone. (c) The spin splitting in the defect states calculated along the ﬁrst Brillouin zone. Here, the/Delta11and/Delta12represent the spin splitting for the lower and upper unoccupied antibonding states, respectively, while /Delta13represents the spin splitting of the occupied bonding state. (d) Orbital-resolved electronic band structures calculated in the defect states. The radii ofthe circles reﬂect the magnitudes of spectral weight of the particular orbitals to the band. is similar to those previously reported on the sulfur interstitial of the MoS 2ML [ 41,42]. Turning the SOC, the energy bands are expected to develop a spin splitting, which is dictated by the lack of the inversionsymmetry [ 3,4]. However, in the pristine PtSe 2ML, the presence of the inversion symmetry suppresses the electronicband structures to exhibit spin-polarized bands without thecharacter of the spin splitting [Fig. 2(d)]. This is supported by the fact that the absence of the spin splitting in the pristinePtSe 2ML has been reported by Yao et al. , using spin-ARPES [30]. On the other hand, introducing the Se ileads to the fact that the crystal symmetry of the pristine PtSe 2ML remains unchanged [Fig. 1(d)]; thus, there is no spin splitting induced on the defect states [Fig. 2(f)]. In contrast to the Se icase, large spin splitting is established in the defect states of the V Sebecause the inversion symmetry of the pristine PtSe 2is already broken by the stable formation of the V Se[Fig. 2(f)]. Figures 4(c) and 4(d) show the k dependence of the spin splitting corresponding to the orbital resolved of the defect states along the ﬁrst Brillouin zone[Fig. 4(b)]. In the unoccupied antibonding states, the large spin splitting is observed along the /Gamma1-Kdirection, and becomes maximum at the Kpoint. On the other hand, the substantially small spin splitting is visible along the /Gamma1-M direction [Fig. 4(c)]. Conversely, a complicated trend of the spin splitting is observed in the occupied bonding state: thespin splitting is small at the Kpoint and rises continuously up to maximum at midway between the Kand/Gamma1points, but gradually decreases until the zero spin splitting is achieved at 115128-4DEFECT-INDUCED LARGE SPIN-ORBIT SPLITTING IN . . . PHYSICAL REVIEW B 96, 115128 (2017) the/Gamma1point. The similar trend of the spin splitting is also visible along the /Gamma1-Mdirection. It is noted here that zero spin splitting is observed at the /Gamma1andMpoints due to time reversibility. We identify the spin splitting on the defect states at the K point: /Delta1K(1)=152 meV and /Delta1K(2)=127 meV in the lower and upper unoccupied antibonding states, respectively, and/Delta1 K(3)=5 meV in the occupied bonding states. The large spin splitting found in the unoccupied antibonding states ( /Delta1K(1)and /Delta1K(2)) is comparable with those found in the pristine MoS 2ML (148 meV [ 14]) and defective WS 2ML (194 meV [ 43]), but is much larger than that of conventional semiconductor III-V andII-VI quantum well ( <30 meV [ 10,45]). Indeed, they are fully comparable to the recently reported surface Rashba splitting(of some 100 meV) observed on Au (111) [ 46], Bi (111) [ 47], PbGe (111) [ 11], Bi 2Se3[001] [ 12], and W [110] [ 48] surfaces. To clarify the origin of the observed spin splitting, we consider orbital contribution to the defect states of the V Se projected to the bands structures in the ﬁrst Brillouin zone asshown in Fig. 4(d). We ﬁnd that the unoccupied antibonding states reveal strong hybridization between the Pt- d x2+y2+dxy and Se- px+pyorbitals at the Kpoint, which induces the large spin splitting at the Kpoint. Toward the /Gamma1point, these contributions are gradually replaced by the hybridizationbetween Pt- d z2−r2and Se- pzorbitals, which contributes only minimally to the spin splitting around the /Gamma1point. The same orbital hybridizations are also visible in the Kpoint of the occupied bonding state, resulting in the spin splitting becomingvery small. However, around midway between the Kand/Gamma1 points, the contribution of the Pt- d x2+y2+dxyand Se- px+ pyorbitals to the occupied bonding state increases, which enhances the spin splitting around the /Gamma1point. Remarkably, the in-plane orbital hybridizations (Pt- dx2+y2+dxyand Se- px+pyorbitals) in the defect states induced by the V Seplay an important role for inducing the large spin splitting. To further reveal the nature of the spin splitting in the defect states of the V Se, we consider our system based on the symmetry arguments. As mentioned before, the structuralrelaxation retains the symmetry of the V Seand becomes C3v [Figs. 1(b)and4(a)]. Here, the symmetry itself consists of a C3 rotation and a mirror symmetry operation My−z:x−→ − x, where xis along the /Gamma1-Mdirection [Fig. 4(b)]. Therefore, the spin splitting for general kis determined by time-reversal symmetry and the C3vpoint group symmetry. By using the theory of invariants, the C3vleads to the spin splitting [14,17,49] /Delta1(k,θ)=[α2(k)+β2(k)s i n2(3θ)]1/2. (4) Here, α(k) and β(k) are the coefﬁcient representing the contribution of the in-plane and out-of-plane potential gra- dient asymmetries, respectively, and θ=tan−1(ky/kx)i st h e azimuth angle of momentum kwith respect to the xaxis along the /Gamma1-Mdirection. In Eq. ( 4), due to the \|sin(3θ)\| dependence of the /Delta1, the spin splitting is minimum when θ=nπ/3, where nis an integer number. Therefore, it is expected that the small spin splitting is observed along the/Gamma1-Mdirection. On the other hand, the spin splitting becomes maximum when θ=(2n+1)π/6, which can be visible along the/Gamma1-Kdirection. These predicted spin splittings along the /Gamma1-Mand the /Gamma1-Kdirections are, in fact, consistent with ourcalculational results of the spin splitting in the defect states shown in Fig. 4(c). Thus far, we found that the spin-orbit splitting in the electronic band structures of the PtSe 2ML can be induced by introducing the point defects. Considering the fact thatthe large spin splitting is achieved on the Kpoint of the unoccupied antibonding states, n-type defective PtSe 2ML for spintronics is expected to be realized. This is supported bythe fact that a deep single acceptor induced by the chalcogenvacancy has been predicted on MoS 2ML [ 41]. Moreover, the observed large splittings enable us to allow operation asa spintronics device at room temperature [ 11,12]. As such, our ﬁnding of large spin splitting are useful for realizingspintronics application of the PtSe 2ML system. It is pointed out here that our proposed approach for inducing the large spin splitting by using the point defects isnot only limited on the PtSe 2ML, but also can be extendable to other T-MX 2ML systems such as the other platinum dichalcogenides like PtS 2and PtTe 2[50], vanadium dichalco- genide like VSe 2,V S 2, and VTe 2[18], and rhenium disulﬁdes (ReS 2)[44], where the electronic structure properties are similar. Importantly, controlling the electronic properties ofthese materials by using the point defects has been recentlyreported [ 44,50]. Therefore, this work paves a possible way to engineer the spin-splitting properties of the two-dimensionalnanomaterials, which provide useful information for thepotential applications in spintronics. IV . CONCLUSION We have investigated the spin-orbit-induced spin splitting in the defective of the PtSe 2ML systems by employing the ﬁrst-principles DFT calculations. First, we have obtained theformation energy of the native point defects and found thatboth the Se vacancy (V Se) and the Se interstitial (Se i)a r e the most stable defects formed in the PtSe 2ML. By taking into account the effect of the spin-orbit coupling in our DFTcalculations, we have found that the large spin-orbit splitting(up to 152 meV) is observed in the defect states induced bythe V Se. We have clariﬁed the origin of the spin splitting by considering orbital contributions to the defect states andfound that the large spin splitting is induced by stronghybridization between the Pt- d x2+y2+dxyand Se- px+py orbitals. Recently, the defective of the PtSe 2ML has been extensively studied [ 37,38,40]. Our study clariﬁes that the defects play an important role in the spin-splitting propertiesof the PtSe 2ML, which is important for designing future spintronic devices. ACKNOWLEDGMENTS This work was partly supported by a Fundamental Research Grant (Grant No. 2237/UN1.P.III-DITLIT-LT/2017) funded bythe Ministry of Research and Technology and Higher Educa-tion, Republic of Indonesia. Part of this research was supportedby a BOPTN research grant (2017), founded by Faculty ofMathematics and Natural Sciences, Universitas Gadjah Mada.The computations in this research were performed using thehigh-performance computing facilities (DSDI) at UniversitasGadjah Mada, Indonesia. 115128-5ABSOR, SANTOSO, HARSOJO, ABRAHA, ISHII, AND SAITO PHYSICAL REVIEW B 96, 115128 (2017) [1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. v. Molnár, M. L. Roukes, A. Y . Chtchelkanova, and D. M.Treger, Science 294,1488 (2001 ). [2] I. Žuti ´c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76,323 (2004 ). [ 3 ] E .I .R a s h b a ,S o v .P h y s .S o l i dS t a t e 2, 1109 (1960). [4] G. Dresselhaus, Phys. Rev. 100,580(1955 ). [5] S. Kuhlen, K. Schmalbuch, M. 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PhysRevB.94.144204.pdf	PHYSICAL REVIEW B 94, 144204 (2016) Decay of density waves in coupled one-dimensional many-body-localized systems Peter Prelov ˇsek Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia (Received 15 June 2016; revised manuscript received 20 September 2016; published 10 October 2016) This work analyzes the behavior of coupled disordered one-dimensional systems as modelled by identical fermionic Hubbard chains with the on-site potential disorder and coupling emerging through the interchainhopping t /prime. The study is motivated by the experiment on fermionic cold atoms on a disordered lattice, where a decay rate of the quenched density wave was measured. We present a derivation of the decay rate /Gamma1within perturbation theory and show that, even at large disorder along the chains, the interaction leads to ﬁnite /Gamma1> 0, the mechanism being the interaction-induced coupling of in-chain localized and interchain extended single-fermionstates. Explicit expressions for /Gamma1are presented for a weak interaction U<t , t /prime, but extended also to the regime t>U>t/prime. It is shown that, in both regimes, /Gamma1increases with the interchain hopping t/prime, as well as decreases with increasing disorder. DOI: 10.1103/PhysRevB.94.144204 I. INTRODUCTION The paradigm of the many-body localization (MBL) represents the extension of well-understood single-particleAnderson localization [ 1–3] to fermionic systems with a repulsive interaction. While original proposals for the MBLstate were dealing with systems with a weak disorder [ 4,5], by now numerous theoretical studies conﬁrm the existenceof a MBL-like state in the regime of strong disorder andmoderate interactions. Most studies so far were performedby the numerical investigation of the prototype model, beingthe one-dimensional (1D) model of disordered interactingspinless fermions, equivalent to the anisotropic Heisenbergchain with random local ﬁelds [ 6–17]. Results conﬁrm that, for large disorder, W>W csystems reveal some basic features of the MBL, referring here to those relevant also forexperiments (a) the absence of dc transport at any temperatureT[9,10,16,18–22], (b) generally nonergodic behavior of correlation functions and of quenched initial quantum states[11,13,23–27], (c) the area law instead of volume law for entropy, still with a logarithmic growth in the MBL phase[7,12,25,27,28]. Even within apparently simple 1D models there are essential theoretical and numerical challenges, amongthem also the nature of the MBL transition, e.g., a well-deﬁnedphase transition [ 11,25,29–31] vs a sharp crossover [ 10,21], and the prediction of measurable signatures of the MBLtransition. On the other hand, cold atoms in optical lattices have already provided a direct experimental insight into theMBL phenomenon and have shown the qualitative transitionbetween an ergodic and the nonergodic phase. Studies of 1Ddisordered systems of cold atoms [ 24,32] have been recently extended to coupled 1D systems [ 33] as well to systems with a full two-dimensional (2D) disorder [ 34]. The motivation for this work is the former experiment, which clearly revealsthat, in coupled chains of localized fermions with identicaldisorder, the fermion interaction Uleads to the decay and the thermalization of the initial density-wave (DW) state. Thisimplies also that the 1D nonergodic behavior is destroyed in thepresence of U/negationslash=0 by the interchain coupling, provided that there is no interchain disorder. Such an observation and itsunderstanding may be very important for further explorations of the MBL physics in higher dimensions. Theoretically, thereare few studies discussing MBL physics beyond 1D, e.g., inladders [ 35] and in 2D systems [ 36]. We should also note that, for cold-atom systems, the appropriate model is theHubbard model, which is much less explored with respectto possibility of MBL physics [ 36–38] and might even reveal some qualitative differences (taking into account additionalsymmetries [ 39]) relative to prototype disordered spinless models predominantly studied so far. In this paper we show on the example of coupled identical disordered Hubbard chains that the decay mechanism ofthe initial out-of-equilibrium state is related to the Hubbard interaction U, coupling the in-chain localized and interchain extended single-particle states. In particular, we formulate theanalytical procedure for the calculation of the decay of an initial density-wave (DW) state, as relevant for cold-atom experiments [ 33]. In the latter the measured quantity is time-dependent imbalance I(τ). The breaking of the ergodicity of the latter, i.e., I(τ →∞ )>0 can be considered as a measurable order parameter for the nonergodic state. We do not address here in more detail the possible (or at least slow) decay of initial DW state in uncoupled chains. We show,however, that the interchain coupling introduces even for large disorder a relevant and leading additional decay channel for DW decay. In Sec. IIwe present the model and its representation within the basis of 1D localized states. We introduce also the relevantDW operators studied further on. Section IIIis devoted to the derivation of the DW decay rate within the perturbation theory,leading to an approximation in terms of a Fermi-golden-ruleexpression. Section IVpresents results within the perturbative regime U/lessmucht,t /primefor the DW rate for the case of coupled chains, touching also the relation to the problem of 1D DWdecay and possible generalizations. Conclusions are given inSec. V. II. MODEL To remain close to the experiment [ 33] we consider in the following the (repulsive) fermion Hubbard model on coupled 2469-9950/2016/94(14)/144204(6) 144204-1 ©2016 American Physical SocietyPETER PRELOV ˇSEK PHYSICAL REVIEW B 94, 144204 (2016) chains where the disorder is identical in all chains, H=/summationdisplay jH0j−t/prime/summationdisplay ljs(c† l,j+1,sclj,s+H.c.)+HU, H0j=−t/summationdisplay ls(c† l+1,j,sclj,s+H.c.)+/summationdisplay lhlnlj, (1) HU=U/summationdisplay ljnlj↑nlj↓, with the in-chain (site index l) and interchain (chain index j) nearest-neighbor (n.n.) hopping t,t/prime>0, respectively. nlj=/summationtext snljsand we assume the disorder entering via random and independent local potentials −W<h l<W , the same in all chains. We note that, within the actual experiment [ 33],hl are quasirandom. For further analysis it is relevant that we consider the ﬁlling ¯n< 1 (in the actual experiment ¯n∼1/2), avoiding the scenario of an (Mott) insulating state entirely dueto repulsive U> 0. Further on we also consider only the case of weaker interchain hopping t /prime<t. Let us start by considering a single 1D chain as described byH0jin Eq. ( 1), where we omit for simplicity the index j. One can ﬁnd ﬁrst single-particle eigenfunctions of H0which are localized states for W> 0, \|φms/angbracketright=ϕ† ms\|0/angbracketright=/summationdisplay lφmlc† ls\|0/angbracketright,H 0=/summationdisplay ms/epsilon1m˜nms,(2) where ˜nmsis the occupation of the single-particle localized state. One can then represent HUin terms of such localized states, HU=U/summationdisplay mm/primenn/primeχm/primen/prime mnϕ† m/prime↑ϕ† n/prime↓ϕn↓ϕm↑, (3) χm/primen/prime mn=/summationdisplay lφm/primelφn/primelφnlφml, where coefﬁcients χm/primen/prime mn are by construction invariant on the index permutation, and indices m,m/prime,n,n/primefurther on refer to 1D localized basis, ordered conveniently by the position of themaxima of localized functions. Let us consider many-body (MB) states \|m /angbracketright=/producttext mϕ† ms\|0/angbracketright within such a localized basis. In this representation one termis the diagonal (Hartree–Fock) correction H /prime d=U/summationdisplay mnχmn mn˜nn↑˜nm↓, (4) so that we can separate HU=H/prime d+H/prime/prime, and only H/prime/prime/negationslash=0 can mix different \|m/angbracketright. Our goal is the behavior of the staggered DW operator, deﬁned by A=/summationdisplay l(−1)lnl/√ L. (5) In particular, we wish to follow its time dependence, being directly related to the measured imbalance I(τ)∝/angbracketleftA/angbracketright(τ) emerging from an initial state /angbracketleftA/angbracketright(τ=0)/negationslash=0. Starting in experiment [ 24,33] as well as in numerical studies [ 24,37], with a DW eigenstate A\|/Psi10/angbracketright=A0\|/Psi10/angbracketright, leads to fast initial dynamics (including oscillations) on the timescale τ∼1/t, representing the decomposition of \|/Psi10/angbracketrightinto different localized\|m/angbracketright. We are rather interested in long-time decay, beyond the former short-time transient, which is qualitatively of the formI(τ)=I 0(τ)exp(−/Gamma1τ). In particular, we study decay-rate /Gamma1 emerging from the dominant channel due to the interchaincoupling, as appears also in the experiment [ 33]. For such long-time decay it is more convenient to analyze the modiﬁedDW operator, given already in terms of localized states, B=1 √ L/summationdisplay ms(−1)m˜nms. (6) We can for convenience assume that localized states are ordered by the site mwhere they have maximum amplitude. It is evident that, in the case H/prime/prime=0, the initial state \|m/angbracketrightwould not decay as well as /angbracketleftB(τ)/angbracketrightwould be constant, in contrast to more standard deﬁnition via Eq. ( 5). III. DENSITY-WA VE DECAY RATE: DERIVATION The goal is to evaluate /angbracketleftB(τ)/angbracketrightwhen perturbed from the initial value /angbracketleftB/angbracketright0=0. In actual experiment the deviation can be and actually is large [ 33]. Still we assume that the system under consideration (as well as in experiment [ 33]) is ergodic and approaches the thermal equilibrium. Final DWdecay rate should be therefore determined by the equilibriumand consistent with an analytical approach to the problemwe therefore apply the linear-response theory for the DWdecay to the equilibrium, as characterized by the temperatureT> 0 and the average particle density ¯n. The information is then contained within susceptibility for the modiﬁed DWobservable, i.e., χ B(ω)=−i/integraldisplay∞ 0eiωt/angbracketleft[B(t),B]/angbracketright. (7) To derive the expression for the DW decay rate /Gamma1within per- turbation theory, as used, e.g., for the dynamical conductivity[40], we follow the memory function formalism [ 40–42], since it has the advantage of being easily extended to nonergodiccases (as expected within the MBL phase). Besides χ B(ω)w e deﬁne in the usual way the relaxation function φB(ω)[40–42] and static (thermodynamic) susceptibility χ0 B, φB(ω)=χB(ω)−χ0 B ω,χ0 B=/integraldisplayβ 0dτ/angbracketleftB†B(iτ)/angbracketright, (8) where β=1/T. In an ergodic case, χ0 B=χB(ω→0), while in a nonergodic system one has to consider also the possibilityofχ 0 B>χB(ω→0). Nevertheless, our study deals with the situation where (at least due to interchain coupling) there is adecay towards the equilibrium (thermalization). Due to generalequilibrium properties of φ B(ω), we can represent it in terms of the complex memory function [ 40], φB(ω)=−χ0 B ω+M(ω). (9) Skipping formal representation for the memory function M(ω) [41,42], we turn directly to the simpliﬁed expression valid within the perturbation theory [ 40], M(ω)=χF(z)−χ0 F ωχ0 B, (10) 144204-2DECAY OF DENSITY W A VES IN COUPLED ONE- . . . PHYSICAL REVIEW B 94, 144204 (2016) where χF(z) is deﬁned in analogy to Eq. ( 8), for the operator F=[H,B ]=[H/prime/prime,B]. The latter represents the effective force on the DW operator B, F=2U√ L/summationdisplay mm/primenn/primesχm/primen/prime mnζmm/primeϕ† n/prime,−sϕn,−sϕ† m/primesϕms, (11) where ζmm/prime=0 for even m/prime−mandζmm/prime=(−1)mfor odd m/prime−m. Within perturbation theory and within the eigenbasis of H0 we further get χF(ω)=−1 Z/summationdisplay n,me−βEn−e−βEm ω+i(En−Em)\|/angbracketleftn\|F\|m/angbracketright\|2, (12) where Z=/summationtext me−βEm. For the decay of interest is primarily the low- ωvalue/Gamma1=M(ω→0) [provided that M(ω) depen- dence is modest] and for ω/lessmuchTwe obtain, /Gamma1=/summationdisplay mpm/Gamma1m, (13) /Gamma1m=πβ χ0 B/summationdisplay n\|/angbracketleftn\|F\|m/angbracketright\|2δ(En−Em), where pm=e−βEm/Zis the Boltzmann probability and /Gamma1m are decay rates of particular states. We note that /Gamma1in Eq. ( 13) takes the simple form of generalized Fermi golden rule (FGR)for the considered problem. It should be noted that such aformulation, taking into account the form Eq. ( 11), also yields /Gamma1as well as /Gamma1 mas an intensive quantity, i.e., they do not depend on the system size L. Further simpliﬁcation can be obtained for high T, i.e., where from Eq. ( 8) we get χ0 B=β/angbracketleftB2/angbracketright,/angbracketleftB2/angbracketright= ¯n(1−¯n/2), (14) so that /Gamma1mareTindependent. IV . DECAY RATE: RESULTS A. One-dimensional system Before entering the analysis of the 2D case, we ﬁrst comment the 1D system (uncoupled chains), and speciﬁcallythe stability of the DW perturbation in the presence of theHubbard-type perturbation H U. In contrast to the prototype interacting spinless models (see, e.g., Ref. [ 18]), much less is known on the existence of the nonergodicity within theHubbard model [ 24,36,37], whereby the symmetry arguments may imply also the restriction on the MBL physics [ 39]. Our formulation of the DW decay, Eq. ( 13), allows some additional insight into the problem by considering the condition for /Gamma1> 0 in a macroscopic disordered system. While the density of MBstates entering Eq. ( 13) is continuous and dense (for L→∞ ), matrix elements /angbracketleftn \|F\|m/angbracketrightdo not connect states with En∼Em, since the interaction is local, while degenerate states can appearasymptotically only at large space separation. The interplayand proper treatment of related resonances is in the core of thetheory of single-particle localization [ 1–3] and of the MBL question [ 4–6,30,43]. Let us consider in Eq. ( 11) only thedominant (most local) term, F∼2U √ L/summationdisplay mm/primes˜χmm/primeζmm/prime[ϕ† m/primesϕms−ϕ† msϕm/primes]˜nm,−s.(15) where ˜ χmm/prime=χmm mm/prime∼φmm/prime. Following a simple argument by Mott [ 2] for 1D noninteracting disordered system, single- particle energies on n.n. sites cannot be close, i.e., \|/epsilon1m+1− /epsilon1m\|>2t. In the same way one can get for more distant neighbors [ 2] \|/epsilon1m+r−/epsilon1m\|>2texp(−ξ(r−1)), (16) where ξ∼ln(W0/W) is the effective inverse localization length (averaged over band for large-enough disorder W> W0∼2t). On the other hand, ˜ χm,m+ralso decays as ∝exp(−ξr). So at least for U/lessmuchtwe get the answer quali- tatively consistent with the nonergodicity of DW correlations,/Gamma1 m=0. On the other hand, large U>U c(going beyond simple perturbation approach) are expected to lead to anergodic behavior of DW perturbation with /Gamma1 m>0, although the actual transition is not yet explored in detail within the 1Ddisordered Hubbard model [ 24,37]. B. Coupled identical Hubbard chains The introduction of the interchain hopping t/prime/negationslash=0i nE q .( 1) qualitatively changes the physics in the case of identicaldisorder in all chains. Without interaction, i.e., at U=0, the eigenstates are a product of localized function andperpendicular plane waves. For simplicity we consider a 2Dsystem, so that H 0\|φmqs/angbracketright= (/epsilon1m+˜/epsilon1q)\|φmqs/angbracketright, \|φmqs/angbracketright=1√ N/summationdisplay ljφmleiqjc† ljs\|0/angbracketright=ϕ† lqs\|0/angbracketright,(17) where ˜ /epsilon1q=− 2t/primecosqandNis the number of chains. The interaction mixes such states, HU=U N/summationdisplay mm/primenn/prime qkpχm/primen/prime mnϕ† n/prime,k+q↓ϕnk↓ϕ† m/prime,p−q↑ϕmp↑.(18) The essential difference to possible decay in 1D, Eq. ( 13), is that the interchain dispersion leads to a continuous spectrum ofoverlapping initial and ﬁnal states, so that the matrix elementsin FGR, Eq. ( 13), can have ﬁnite values. Assuming for the moment that we are dealing with a weak perturbation U<t /prime, the evaluation of Eq. ( 13) leads to an effective (Boltzmann) density of decay channels, i.e., the density of states D(ω), where (at β→0) D(ω)=μ˜D(ω),μ=(1−¯n/2)2¯n2/4, (19) ˜D(ω)=1 N3/summationdisplay kpqδ(ω−˜/epsilon1p−q−˜/epsilon1k+q+˜/epsilon1p+˜/epsilon1k), with/integraltext dω˜D(ω)=1. Distribution D(ω) depends linearly on t/prime and has a form as shown in Fig. 1, with a singularity at ω∼0. It is nonzero within the interval −8t/prime<ω< 8t/primewith a width √ ¯ω2∼√ 8t/prime. Taking as the main contribution the reduced F,E q .( 15),/Gamma1 (atβ→0) can be represented as the sum of contributions 144204-3PETER PRELOV ˇSEK PHYSICAL REVIEW B 94, 144204 (2016) 0 0.1 0.2 0.3 0.4 -8 -6 -4 -2 0 2 4 6 8DOS ω D DI FIG. 1. Effective density of states ˜D(ω), emerging from the interchain hopping and entering the evaluation of the decay rate /Gamma1. Plotted is also the incoherent approximant ˜DI(ω). emerging from different distances r,/Gamma1=/Gamma11+/Gamma13+··· , where /Gamma1r=32π˜μU2\|˜χm,m+r\|2˜D(/Delta1/epsilon1r=/epsilon1m−/epsilon1m+r), (20) and ˜μ=μ//angbracketleftB2/angbracketright= ¯n(1−¯n/2)/4. At least nearest neighbors r=1 can be calculated more explicitly, taking into account the actual random distributionofh l. Assuming for simplicity that we are dealing with a two-level noninteracting problem with local potentials hl,hl+1, respectively, we get /Gamma11=32π˜μU2/integraldisplay d˜h\|˜χm,m+1(˜h)\|2P(˜h)D(/Delta1/epsilon1 1(h)),(21) where ˜h=hl−hl+1. In an analogous way one can treat also further neighbors r/greaterorequalslant3, but here with an additional approximation that the effective in-chain hopping is reducedast r∼t(2t/W )r−1. The displayed result /Gamma1vst/prime/t, as shown in Fig. 2,i s calculated by using Eq. ( 21)a tﬁ x e d ¯n=1/2 and for various W/t . In spite of simpliﬁed approximations χm/primen/prime mnas well as for local energies /epsilon1m, several conclusions are straightforward: (a) The decay rate becomes /Gamma1> 0 for any ﬁnite t/prime/negationslash=0 and is proportional to ¯n, consistent with the origin in the interaction U> 0 between fermions. /Gamma1∝U2, at least within the perturbation regime considered analytically. 0 0.05 0.1 0.15 0 0.2 0.4 0.6 0.8Γ t/U2W/t=4 W/t=6 W/t=8 FIG. 2. Decay rate /Gamma1t/U2vs interchain hopping t/prime/tfor different disorders W/t at ﬁxed particle density ¯n=1/2.(b)/Gamma1shows a steady increase with \|t/prime/t\|fort/prime/t < 0.6, consistent with experiments [ 33]. The decay rate /Gamma1vst/prime/t is, at least within the approach used, not a simple function.Namely, for small t /prime/t < 0.2/Gamma1is strongly reduced since the contributions beyond the n.n. term /Gamma11become suppressed. There appears also a saturation of /Gamma1fort/prime/t > 0.6. To some extent such behavior is plausible since excessively wide bandst /prime/t > 1 cannot increase /Gamma1much further. C. Generalizations So far the analysis has been restricted to the regime of weak interaction U/4/lessmucht,t/prime, whereby the factor of four seems to be a fair estimate for the crossover to a nonperturbative case. Sincein the experiment [ 33]t /prime/tis also varied, and of particular interest are results with t/prime/t/lessmuch1, one would wish to have an analytical result for the intermediate regime t/prime<U / 4< t. If we consider in this case just the interchain part of the Hamiltonian, Eq. ( 1) would for U/greatermucht/primetransform into H⊥=/summationdisplay lHl⊥,H l⊥∼−t/prime/summationdisplay js(˜c† l,j+1,s˜cljs+H.c.),(22) where ˜cljs=cljs(1−nlj,−s) are projected fermion operators. Here, we omit possible exchange terms, since we are interestedin systems with ¯n< 1/2, i.e., away from half ﬁlling. As before, the modiﬁed H ⊥commutes with the DW operator, i.e., [H⊥,B]=0, hence it is expected not to inﬂuence signiﬁcantly the form of F,E q .( 11). It is well known [ 44] that eigenstates of the projected model, Eq. ( 22), can be mapped on those of an noninteracting spinless model with the same single-particledispersion /epsilon1 q=− 2t/primecosq. On the other hand, wave functions within the original basis are complicated and selection ruleschanged. We therefore argue that, within the intermediateregime the essential difference appears in the evaluation ofEq. ( 18), whereby the changed coherence factors between q states and eigenstates of Eq. ( 22) lead to a different, rather incoherent D I(ω). For simplicity we assume for the latter the Gaussian form with the same width ¯ ω=√ 8t/prime, i.e., DI(ω)=exp(−ω2/(4t/prime)2)/√ 16πt/prime2. (23) Taking DI(ω) as an input into Eqs. ( 20) and ( 21), results are presented in Fig. 3. Results differ from those in Fig. 2only in some details. In particular, due to continuous DI(ω)t h e 0 0.05 0.1 0.15 0 0.2 0.4 0.6 0.8Γ t/U2W/t=4 W/t=6 W/t=8 FIG. 3. Decay rate /Gamma1t/U2vst/prime/tas calculated within the incoherent approximation for different W/t at ﬁxed ¯n=1/2. 144204-4DECAY OF DENSITY W A VES IN COUPLED ONE- . . . PHYSICAL REVIEW B 94, 144204 (2016) variation of /Gamma1vst/prime/tis more gradual, but still showing a distinctive contributions /Gamma1r>1with strong Wdependence. The question of strong interactions U> 4tis more subtle. One might employ an approximation similar to Eq. ( 22)a l s o for the in-chain terms, i.e., Hj∼−t/summationdisplay ls(˜c† l+1,js˜cljs+H.c.)+/summationdisplay lhinlj. (24) The message of such term is that the decay rate /Gamma1would not increase with U> 4t, but would saturate, being ﬁnally determined by t, as emerging from Eq. ( 24), as well as on t/prime andW. Taking strictly the 1D model, as described by Eq. ( 24), DW perturbation should not decay at all due to the mappingon the spinless fermions and on the noninteracting Andersonmodel. Still, t /prime/negationslash=0 and the emerging 2D problem does not have such a mapping, so that interchain and in-chain fermionstates become coupled again. V . CONCLUSIONS We presented a theory of a DW decay in the case of coupled disordered Hubbard chains, with the identicaldisorder in each chain. It should be pointed out that wedo not address the question of whether the uncoupled 1Dchains already show weak DW decay, but rather discussthe nontrivial additional contribution due to the interchaincoupling. From the perturbation-theory approach the decayemerges due to Hubbard interaction U> 0 mixing the in-chain localized states and interchain extended single-fermion states.The essential ingredient for /Gamma1> 0 (given by transition rates between discrete localized states) are continuous spectra ofoverlapping extended states, i.e., with ﬁnite matrix elements.The latter are the the precondition for an evaluation of /Gamma1within a FGR-type approximation. Taking into account that levelslocalized close in space are (on average) distant in energy,this leads to quite strong dependence of /Gamma1on the ratio t /prime/tas well as on an increase of /Gamma1with decreasing disorder W.T h e nontrivial structure within the dependence on t/prime/temerges from a different regimes which allow for contributions beyondﬁrst n.n. in Eq. ( 20). The saturation of /Gamma1att /prime/t∼1i st o some extent plausible since for t/prime>tthe decay is limited by tand not by t/primebut can be also beyond the feasibility of initial assumptions. An interesting question is also to what extent theDW decay /Gamma1and possible MBL are sensitive to the difference of potentials in each chain [ 45], since even a small difference δ/epsilon1 > t /primecan induce also perpendicular localization and prevent the DW decay discussed above. The theory is motivated by a concrete experiment on cold atoms [ 33]. We ﬁnd that the variation of /Gamma1, as measured via the time-dependent imbalance I(τ) with Uas well as on t/prime/t andWare qualitatively reasonably reproduced. Still, several restrictions on the theoretical description should be taken intoaccount. In actual experiment a quasiperiodic (Aubry–Andr `e) lattice is employed which is different from an Anderson modelwith respect to the character and stability of localized states.Also, most results are available within the strong-interactionregime U/greatermuch4twhere we cannot give an explanation on the same level of validity, although the saturation (or a maximum)of/Gamma1forU> 4tis expected. ACKNOWLEDGMENTS The author acknowledges the explanation of cold-atom experiments by P. Bordia and H. L ¨uschen within the group of I. Bloch, LMU M ¨unchen, and fruitful discussions with F. Heidrich–Meisner and F. Pollmann. The author acknowledgesalso the support of the Alexander von Humboldt Foundation,as well the hospitality of the A. Sommerfeld Center for theTheoretical Physics, LMU M ¨unchen, and the Max–Planck Institute for Complex Systems, Dresden, where this work hasbeen started and major steps have been accomplished. [1] P. W. Anderson, Phys. Rev. 109,1492 (1958 ). 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PhysRevB.72.241311.pdf	Local transport in a disorder-stabilized correlated insulating phase M. Baenninger, A. Ghosh, M. Pepper, H. E. Beere, I. Farrer, P. Atkinson, and D. A. Ritchie Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom /H20849Received 13 October 2005; published 22 December 2005 /H20850 We report the experimental realization of a correlated insulating phase in two-dimensional /H208492D/H20850 GaAs/AlGaAs heterostructures at low electron densities in a limited window of background disorder. This hasbeen achieved at mesoscopic length scales, where the insulating phase is characterized by a universal hoppingtransport mechanism. Transport in this regime is determined only by the average electron separation, indepen-dent of the topology of background disorder. We have discussed this observation in terms of a pinned electronsolid ground state, stabilized by the mutual interplay of disorder and Coulomb interaction. DOI: 10.1103/PhysRevB.72.241311 PACS number /H20849s/H20850: 73.21. /H11002b, 73.20.Qt In the presence of Coulomb interaction, both magnetic ﬁeld and disorder are predicted to stabilize many-body charge-ordered ground states.1,2Strong perpendicular mag- netic ﬁeld B/H11036quenches the vibrational motion of electrons, and has been extensively exploited to realize a charge-density wave /H20849CDW /H20850ground state in systems with weak background disorder. 3,4Despite the effort, however, the na- ture of localization in such systems has been controversial,with both pinned Wigner solid /H20849WS/H20850formation and inhomo- geneity driven percolation transition being suggested. 5On the other hand, disorder stabilizes Coulomb correlation ef-fects by introducing a pinning gap /H9004 pinin the phonon density of states, which provides a long-wavelength cutoff.2This has led to the theoretical prediction of several forms of CDWground states at zero or low B /H11036. Systematic experimental investigations on such possibilities, however, have been rare,and form the subject of this work. Increasing the magnitude of background potential ﬂuctuations increases /H9004 pin, which stabilizes the CDW phases to higher temperatures. In modulation-doped GaAs/AlGaAsheterostructures, where disorder primarily arises from the charged dopant ions, 6/H9004pin/H11011exp/H20849−4/H9266/H9254sp//H208813ree/H20850depends strongly on the setback distance /H9254spthat separates the 2D electron system /H208492DES /H20850and the dopants, where reeis the mean distance between the electrons in the 2DES/H20849Refs. 7 and 8 /H20850. However, disorder affects the ground-state transport in two critical ways. First, the presence of /H9004 pin disintegrates the CDW phase into domains of ﬁnite size /H9261d /H11011sound velocity/ /H9004pin. At strong pinning, /H9261dbecomes micro- scopically small, leading to signiﬁcant averaging in transportmeasurements with conventional macroscopic devices. Sec-ond, strong potential ﬂuctuations can also result in a “freez-ing” of transport below a certain percolation threshold evenwhen electron density /H20849n s/H20850is relatively high, thereby making the regime of strong effective Coulomb interaction inacces- sible. Here, we show that these difﬁculties can be largely over- come by using modulation-doped heterostructures of mesos-copic dimensions. In such devices transport freezes at muchlower n sin comparison to macroscopic devices at the same /H9254spor disorder, thereby allowing transport at a large interac- tion parameter rs=1/aB/H20881/H9266ns/H110117–8 /H20849aBis the effective Bohr radius /H20850, even when /H9254spis relatively small. The typical dimen-sion Lof our devices in the current carrying direction was chosen to be /H110112–4/H9262m, which is also similar in order of magnitude to the /H9261dsuggested by recent microwave absorp- tion studies for pinned WS ground states.4The low- B/H11036mag- netotransport in these devices was found to display a strikinguniversality in that the hopping distance in the localized re- gime was determined by r ee=1//H20881ns, rather than the details of background disorder, indicating an unusual self-localizationof electrons at sufﬁciently low n s. We have used Si modulation-doped GaAs/AlGaAs het- erostructures where /H9254spwas varied from 20 to 80 nm. At a ﬁxed ns, the effect of /H9254spon the strength of potential ﬂuctua- tions is reﬂected in the mobility /H9262, as can be observed from Fig. 1 /H20849b/H20850. Both monolayer /H20849/H9254/H20850- and bulk-doped wafers were used. Relevant properties of the devices are given in Table I. Devices were cooled from room temperature to 4.2 K over24–36 h to allow maximal correlation in the dopant layer/H20851redistribution of charged-donor /H20849DX/H20850centers /H20852. 9This slow cooldown technique also leads to excellent reproducibility over repeated thermal cycles. Electrical measurements werecarried out with a standard low-frequency /H208497.2 Hz /H20850four- probe technique with an excitation current of /H110110.01–0.1 nA to minimize heating and other nonlinear effects. A directmeasurement of n swithin the mesoscopic region was carried out with an edge-state reﬂection-based technique.10 In Fig. 1 /H20849a/H20850we compare the nsscale of localization tran- sition at B/H11036=0 and T=0.3 K in macroscopic and mesoscopic devices from the same wafer. In a standard 100 /H11003900/H9262m2 Hall bar, as illustrated with wafer A2677, the linear conduc- tivity /H9268→0/H20851A77L, inset of Fig. 1 /H20849a/H20850/H20852at/H110113 times the ns compared to the mesoscopic sample /H20849A77/H20850from the same wafer. Further, /H9268in the large sample A77L shows excellent classical percolationlike scaling /H9268/H11011/H20849ns−nc/H20850/H9253/H20849nc=1.72 /H110031010cm−2/H20850, where /H9253/H110152, implying a inhomogeneity driven percolation transition at nonzero T5/H20851solid line in the inset of Fig. 1 /H20849a/H20850/H20852. Similar scaling in the mesoscopic systems, how- ever, was found to be difﬁcult, with unphysically large esti-mates of /H9253/H110113.2–3.7 /H20849not shown /H20850, indicating a different mechanism of localization transition. Asnsis lowered below a sample-dependent characteristic scale ns/H20851denoted by the crosses in Fig. 1 /H20849a/H20850/H20852, the onset of strong localization is identiﬁed by the resistivity /H9267/H20849=1//H9268/H20850 exceeding /H11015h/e2.A t ns/H11270ns, the Tdependence of /H9267at aPHYSICAL REVIEW B 72, 241311 /H20849R/H20850/H208492005 /H20850RAPID COMMUNICATIONS 1098-0121/2005/72 /H2084924/H20850/241311 /H208494/H20850/$23.00 ©2005 The American Physical Society 241311-1ﬁxed nscan be divided into three regimes, as illustrated with A78a: First, transport in the classical regime at T/H11407TFis magniﬁed in Fig. 1 /H20849c/H20850, where TFis the Fermi temperature. In this regime /H9267/H11008T−/H9252, where /H9252/H110111/H20849indicated by the solid line /H20850. AsTis decreased, the onset of the quantum regime /H20849T /H11351TF/H20850results in a stronger increase in /H9267with decreasing T. Note that the clear classical to quantum crossover implies awell-deﬁned TF, and hence a uniform charge-density distri- bution down to the lowest ns/H110116.5/H11003109cm−2/H20849in A77L, in- homogeneity sets in at nsas large as /H110114–5/H110031010cm−2/H20850.I n the quantum regime and for TF/H11022T/H11022T, Fig. 1 /H20849d/H20850shows that the behavior of /H9267is activated with /H9267/H20849T/H20850=/H92673exp/H20849/H92803/kBT/H20850, where /H92803is the activation energy. From the nsandB/H11036depen- dence of the pre-exponential /H92673, we have shown earlier that the transport mechanism in this regime corresponds tonearest-neighbor hopping. 10Below the characteristic scale T/H110111 K, variation of /H9267becomes weak, tending to a ﬁnite magnitude even in the strongly localized regime. This satu-ration in the insulating regime cannot be explained in termsof an elevated electron temperature due to insufﬁcient ther-mal coupling to the lattice since T depends only weakly on electron density up to ns/H11011ns/H20851Fig. 1 /H20849d/H20850/H20852, and the damping of Shubnikov–de Haas oscillations in the metallic regime showsthe base electron temperature to be /H11015300 mK. In order to explore the physical mechanism behind the weak Tdependence of /H9267, we have carried out extensive mag- netoresistivity /H20849MR/H20850measurements at the base T. Figures 2/H20849a/H20850–2/H20849d/H20850show the B/H11036dependence of MR in the insulating regime of four devices with increasing /H9254spfrom 20 to 80 nm. In general, we ﬁnd a strong negative MR in A07, A78, andC67 at low B /H11036, which can be attributed to interference of hopping paths. The negative MR is followed by an exponen-tial rise in /H9267asB/H11036is increased further. We have recently shown that the logarithm of such a positive MR at low B/H11036 varies in a quadratic manner with B/H11036, i.e., /H9267/H20849B/H11036/H20850 =/H9267Bexp/H20849/H9251B/H110362/H20850, where /H9267Band/H9251arens-dependent factors.10 Such a variation, denoted by the solid lines in Fig. 2, is found to be limited to ns/H11351ns, and extends over a B/H11036scale of Bc, where Bcwas found to decrease rapidly as /H9254spis increased. Note that in T46 /H20849lowest disorder /H20850, neither a clear negative MR nor an exponential B/H110362dependence were observed. A physical signiﬁcance of Bcand of the qualitatively different MR behavior of T46 will be discussed later.TABLE I. Geometrical and structural property of the devices. n/H9254 is the density of Si dopants and Wis the width. The background doping concentration is /H113511014cm−3in all devices. Wafer Device/H9254sp /H20849nm/H20850n/H9254 /H208491012cm−2/H20850W/H11003L /H20849/H9262m/H11003/H9262m/H20850Doping A2407 A07a 20 2.5 8 /H110032 /H9254 A07b 20 2.5 8 /H110033 /H9254 A2678 A78a 40 2.5 8 /H110032.5 /H9254 A78b 40 2.5 8 /H110034 /H9254 A2677 A77 40 —a8/H110033 Bulk A77L 40 —a100/H11003900 Bulk C2367 C67 60 0.7 8 /H110033 /H9254 T546 T46 80 1.9 8 /H110033 /H9254 aThe doping concentration of bulk-doped devices is 2 /H110031018cm−3 over a range of 40 nm. FIG. 1. /H20849Color online /H20850/H20849a/H20850Conductivity /H20849/H9268/H20850of mesoscopic samples as a function of electron density nsatT/H110150.3K. The crosses denote nsfor individual samples /H20849see text /H20850. Inset: nsdependence of /H9268for a macroscopic Hall bar A77L. The solid line is the best ﬁt of a classical percolationlike scaling relation /H9268/H11011/H20849ns−nc/H20850/H9253./H20849b/H20850/H9254spde- pendence of mobility at constant nsandn/H9254for heterostructures simi- lar to those used in presented work. /H20849c/H20850Resistivity /H20849/H9267/H20850as a function of temperature measured at B/H11036=1 T. The solid line represents a power law of /H11011T−1; the vertical lines in /H20849c/H20850and /H20849d/H20850indicate the Fermi temperatures TF./H20849d/H20850Activation and saturation of /H9267atB/H11036 =1.5 T. FIG. 2. Typical magnetoresistivity traces in four samples with varying levels of disorder. The vertical lines denote /H9263=1. The num- bers indicate electron density in units of 1010cm−2.Bcdenotes the ﬁeld scale up to which a quadratic B/H11036dependence could be ob- served. The parameters /H9251and/H9267Bwere obtained from the slope and yintercept of linear ﬁts to ln /H20849/H9267/H20850−B/H110362traces, respectively.BAENNINGER et al. PHYSICAL REVIEW B 72, 241311 /H20849R/H20850/H208492005 /H20850RAPID COMMUNICATIONS 241311-2The observed behavior of /H9267can be naturally explained in the framework of tunneling of electrons between two trapsites separated by a distance r ij. In weak B/H11036, such that the magnetic length /H9261=/H20881/H6036/eB/H11036/H11271/H9264, where /H9264is the localization length, the asymptotic form of the hydrogenic wave functionchanges from /H9274/H20849r/H20850/H11011exp/H20849−r//H9264/H20850to/H9274/H20849r/H20850/H11011exp/H20849−r//H9264 −r3/H9264/24/H92614/H20850./H20849Ref. 11 /H20850. This leads to a MR, /H9267/H20849B/H11036/H20850 =/H92670exp/H208492rij//H9264/H20850exp/H20849Ce2rij3/H9264B/H110362/12/H60362/H20850, which implies /H9267B=/H92670exp/H208492rij//H9264/H20850and/H9251=Ce2rij3/H9264/12/H60362. /H208491/H20850 While /H9267Bdepends on the tunneling probability at B/H11036=0,/H9251 denotes the rate of change of this probability when B/H11036is switched on. Importantly, both parameters provide informa-tion on the intersite distance r ij, as well as /H9264independently. The parameter C/H110110.5–1 depends on the number of bonds at percolation threshold in the random resistor network /H20849we shall subsequently assume C/H110151/H20850. Since conventional hop- ping sites are essentially impurity states, both /H9251and/H9267Bare expected to be strongly disorder dependent. Note that, sincewave-function overlap plays a critical role in transport, adirect source-to-drain tunneling is ruled out in our case. 12 From the MR data we have evaluated /H9251and/H9267Bfrom the slope and intercept of the ln /H20849/H9267/H20850−B/H110362traces. Further details of the analysis can be found elsewhere.10In Fig. 3 we have shown /H9251as a function of nsfor ﬁve different samples up to the corresponding ns. Strikingly, the absolute magnitudes of /H9251from different samples are strongly correlated, and can be described by a universal ns-dependent function over nearly two orders of magnitude. At stronger disorder /H20849e.g., A07 /H20850, localization occurs at a higher nsresulting in a lower /H9251, while at lower disorder /H20849e.g., C67 /H20850localization occurs at lower nsyielding a larger magnitude of /H9251. This indicates that magnetotransport in such mesoscopic samples is not deter-mined directly by disorder, but by n sin the localized regime. Qualitatively, the decreasing behavior of /H9251with increasing ns itself is inconsistent with the single-particle localization in an Anderson insulator.10,13 From the strong sample-to-sample correlation in the mag- nitude of /H9251, a disorder-associated origin of rijis clearly un- likely. For example, taking rij/H11011/H9254spwill lead to distinct sets of/H9251for wafers with different /H9254sp. However, in the context ofa pinned CDW ground state, another relevant length scale is ree. Indeed, in a case of tunneling events over a mean elec- tron separation, i.e., rij/H11015ree, we ﬁnd that Eq. /H208491/H20850describes both absolute magnitude, as well as the nsdependence of /H9251 quantitatively. Using rij/H110151//H20881ns, Eq. /H208491/H20850leads to /H9251/H11008ns−3/2,a s indeed observed experimentally /H20849solid line in the inset of Fig. 3 /H20850. Allowing for sample-to-sample variation, we ﬁnd /H9251 =/H208491.7±0.5 /H20850/H110031021/ns3/2T−2from which, using Eq. /H208491/H20850,w eg e t /H9264=9.0±2.6 nm, which is close to aBin GaAs /H20849/H1101510.5 nm /H20850. The analysis can be immediately checked for consistency from the reedependence of /H9267B. From Fig. 4, we ﬁnd that /H9267B increases strongly with increasing reewhen ns/H11270ns,a se x - pected in the simple tunneling framework /H20851Eq./H208491/H20850/H20852. In spite of the scatter, the overall slopes of the ln /H20849/H9267B/H20850−reeplots are similar in different samples /H20849solid lines /H20850with/H9264estimated to be/H1101513±4 nm, agreeing with that obtained from the analysis of/H9251. Note that the /H9267Bdeviates from the exponential depen- dence and tends to saturate as ns→ns. While this is not com- pletely understood at present, we note that the saturation in /H9267Boccurs within the range /H9267B/H110111–2/H11003h/e2, irrespective of sample details. Similar universality in the hopping pre-exponential has been observed in the context of Tdepen- dence of /H9267in variable-range hopping,14and has been sug- gested to indicate an electron-electron interaction mediatedenergy-transfer mechanism. We now discuss the physical scenario which could lead to the electron separation-dependent hopping transport. Weshow that our observations can be explained in the theoreti-cal framework of defect motion in a quantum solid that wasoriginally developed by Andreev and Lifshitz in the contextof solid He 3/H20849Ref. 15 /H20850, and later adapted for a WS ground state.16,17In our case, transport in both the quantum and clas- sical regime can be understood in terms of tunneling of lo-calized defects in an interaction-induced pinned electron solid phase as n sis reduced below the melting point ns. The defects, which act as quasiparticles at low T, can arise from regular interstitials, vacancies, dislocation loops, etc., as wellas from zero-point vibration of individual lattice sites. 15The scale of zero-point ﬂuctuation /H11011h/ree/H20881mUC/H110152/H9266//H20881rs/H114071, is indeed strong in our case over the experimental range ofn s, where UC/H11015e2/4/H9266/H9280reeis the interatomic interaction en- ergy scale. In the quantum regime, the transport at higher T/H20849TF/H11271T /H11271T/H20850is predicted to be thermally activated nearest-neighbor FIG. 3. Absolute magnitude of /H9251obtained from the slope of ln/H20849/H9267/H20850−B/H110362traces for ﬁve different samples. The inset shows the same data in a log-log scale. The slope of the solid line is −3/2. FIG. 4. /H20849Color online /H20850The dependence of /H9267Bon the average electron separation reein ﬁve different samples. The slope of the solid lines gives an estimate of /H9264/H20851Eq./H208491/H20850/H20852.LOCAL TRANSPORT IN A DISORDER-STABILIZED … PHYSICAL REVIEW B 72, 241311 /H20849R/H20850/H208492005 /H20850RAPID COMMUNICATIONS 241311-3hopping of localized defects, while at lower T/H20849/H11270T/H20850tunnel- ing of such defects leads to a T-independent transport.15 While this clearly describes the weak Tdependence of /H9267 atlow temperatures /H20851Fig. 1 /H20849d/H20850/H20852, the strongest support of this picture comes from the fact that the natural length scale oftunneling is indeed the average electron separation r ee. This immediately explains the unusual ns/H20849orree/H20850dependence of both/H9251and/H9267B, as well as the apparent insensitivity of these parameters to local disorder. The negative MR at low B/H11036 caused by destruction of interference is then expected to per- sist up to a B/H11036corresponding to /H9263=nsh/eB/H11036/H110111/H20849one ﬂux quantum /H92780within an area of ree2/H20850, as indeed observed in our experiments /H20849Fig. 2 /H20850. The tunneling of a defect scenario also allows an estimate of the crossover scale kBT* =/H92803/ln/H20849/H9004pin//H9004/H9280/H20850/H20849Ref. 15 /H20850, where /H9004/H9280is the bandwidth. For a pinned WS ground state, using the expression of /H9004pinin Ref. 8, experimentally measured /H92803, and/H9004/H9280/H11011h2/8mree2,w eﬁ n d T/H11011O/H208491K/H20850over the experimental range of nsin A78a, giv- ing good order-of-magnitude agreement to the observed scale of T. Finally, the behavior of /H9267/H11011T−1in the classical regime /H20849T/H11022TF/H20850/H20851Fig. 1 /H20849c/H20850/H20852has also been recently observed,18 and interpreted in terms of transport mediated by defect-typetopological objects /H20849Fermi-liquid droplets /H20850in the WS phase.16 In the presence of pinning, the MR data suggests the asymptotic form of the wave function /H9274/H20849r/H20850/H11011exp/H20849r//H9264/H20850, where /H9264/H11015aB. However, the interplay of conﬁnement arising from the magnetic potential and disorder pinning is expected to becritical in determining /H9274/H20849r/H20850, with disorder pinning dominat- ing at low B/H11036. This is expected to result in the upper cutoff Bcthat decreases with decreasing disorder, as observed ex- perimentally. The intricate interplay between disorder,electron-electron interaction, and magnetic ﬁeld is further il- lustrated by the absence of a clear B /H110362dependence of the MR in T46 /H20849largest /H9254sp/H20850, which could be explained by a prohibi- tively small Bcor the very instability of the solid phase at sufﬁciently low disorder. On the other hand, devices with /H9254sp/H1135110 nm showed inhomogeneity driven Coulomb- blockade oscillations in the localized regime, making the in-vestigation of such a charge-correlated state impossible. Aquantitative understanding of the scale of B c, as well as the speciﬁc spatial structure of the ground state in the interme-diate disorder regime, will require further investigations,which are presently in progress. 1B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 /H208491989 /H20850;A . G. Eguiluz, A. A. Maradudin, and R. J. Elliott, ibid. 27, 4933 /H208491983 /H20850; A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys. Rev. Lett. 76, 499 /H208491996 /H20850. 2J. S. Thakur and D. Neilson, Phys. Rev. B 54, 7674 /H208491996 /H20850;A .A . Slutskin, V. V. Slavin, and H. A. Kovtun, ibid. 61, 14184 /H208492000 /H20850; G. Benenti, X. Waintal, and J.-L. Pichard, Phys. Rev. Lett. 83, 1826 /H208491999 /H20850; R. Jamei, S. Kivelson, and B. Spivak, ibid. 94, 056805 /H208492005 /H20850; S. T. Chui and B. Tanatar, ibid. 74, 458/H208491995 /H20850. 3H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 65, 633 /H208491990 /H20850;V .J . Goldman, M. Santos, M. Shayegan, and J. E. Cunningham, ibid. 65, 2189 /H208491990 /H20850; H. C. Manoharan, Y. W. Suen, M. B. Santos, and M. Shayegan, ibid. 77, 1813 /H208491996 /H20850; J. Yoon, C. C. Li, D. Shahar, D. C. Tsui, and M. Shayegan, ibid. 82, 1744 /H208491999 /H20850. 4P. D. Ye, L. W. Engel, D. C. Tsui, R. M. Lewis, L. N. Pfeiffer, and K. West, Phys. Rev. Lett. 89, 176802 /H208492002 /H20850; Y. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D. Ye, L. N. Pfeiffer, and K.W. West, ibid. 91, 016801 /H208492003 /H20850. 5A. A. Shashkin, V. T. Dolgopolov, G. V. Kravchenko, M. Wendel, R. Schuster, J. P. Kotthaus, R. J. Haug, K. von Klitzing, K. Ploog, H. Nickel, and W. Schlapp, Phys. Rev. Lett. 73, 3141 /H208491994 /H20850; Y. Meir, ibid. 83, 3506 /H208491999 /H20850; S. Das Sarma, M. P. Lilly, E. H. Hwang, L. N. Pfeiffer, K. W. West, and J. L. Reno,ibid. 94, 136401 /H208492005 /H20850. 6A. L. Efros, Solid State Commun. 65, 1281 /H208491988 /H20850; A. L. Efros, F. G. Pikus, and V. G. Burnett, Phys. Rev. B 47, 2233 /H208491993 /H20850.7I. M. Ruzin, S. Marianer, and B. I. Shklovskii, Phys. Rev. B 46, 3999 /H208491992 /H20850. 8S. T. Chui, J. Phys.: Condens. Matter 5, L405 /H208491993 /H20850. 9E. Buks, M. Heiblum, and H. Shtrikman, Phys. Rev. B 49, 14790 /H208491994 /H20850; M. Stopa, ibid. 53, 9595 /H208491996 /H20850. 10A. Ghosh, M. Pepper, H. E. Beere, and D. A. Ritchie, Phys. Rev. B70, 233309 /H208492004 /H20850. 11B. I. Shklovskii, Fiz. Tekh. Poluprovodn. /H20849S.-Peterburg /H2085017, 2055 /H208491983 /H20850/H20851Sov. Phys. Semicond. 17, 1311 /H208491983 /H20850/H20852; B. I. Shklovskii and A. L. Efros, in Electronic Properties of Doped Semiconduc- tors, Springer Series in Solid-State Sciences Vol. 45 /H20849Springer, Berlin, 1984 /H20850. 12A. K. Savchenko, V. V. Kuznetsov, A. Woolfe, D. R. Mace, M. Pepper, D. A. Ritchie, and G. A. C. Jones, Phys. Rev. B 52, R17021 /H208491995 /H20850. 13G. Timp and A. B. Fowler, Phys. Rev. B 33, 4392 /H208491986 /H20850. 14S. I. Khondaker, I. S. Shlimak, J. T. Nicholls, M. Pepper, and D. A. Ritchie, Phys. Rev. B 59, 4580 /H208491999 /H20850; W. Mason, S. V. Kravchenko, G. E. Bowker, and J. E. Furneaux, ibid. 52, 7857 /H208491995 /H20850. 15A. F. Andreev and I. M. Lifshitz, Zh. Eksp. Teor. Fiz. 56, 2057 /H208491969 /H20850/H20851Sov. Phys. JETP 29, 1107 /H208491969 /H20850/H20852. 16B. Spivak, Phys. Rev. B 67, 125205 /H208492003 /H20850. 17G. Katomeris, F. Selva, and J.-L. Pichard, Eur. Phys. J. B 31, 401 /H208492003 /H20850;33,8 7 /H208492003 /H20850. 18H. Noh, M. P. Lilly, D. C. Tsui, J. A. Simmons, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 68, 241308 /H20849R/H20850/H208492003 /H20850.BAENNINGER et al. PHYSICAL REVIEW B 72, 241311 /H20849R/H20850/H208492005 /H20850RAPID COMMUNICATIONS 241311-4
PhysRevB.28.7308.pdf	PHYSICAL REVIEW B VOLUME 28,NUMBER 12 15DECEMBER 1983 Investigation oftheelectronic structure, hyperfine interactions, andradialdensities intheirontetrahedral sulfides withtheuseofthemultiple-scattering Xamethod S.K.LieandC.A.Taft CentroBrasileiro dePesquisas Fisicas,RuaDr.XavierSigaud150,Urea,22290,RiodeJaneiro, RiodeJaneiro, Brazil (Received 23May1983) Spin-polarized multiple-scattering andtheSlaterXalocal-exchange calculations havebeenper- formedonthetetrahedralFeS4,FeS4, andFeS4clusters. Thecalculated chargeandspinden- sitiesattheFenucleus havebeenusedtointerpret theMossbauer hyperfine parameters. Thecalcu- latedenergylevelsandiron3dand4spopulation wereusedtoexplainthe4scontribution, themea- suredmagnetic moment, theobserved crystal-field transition, andthelargereduction ofthefree-ion Fermi-contact term. I.INTRODUCTION Ironisbyfarthemostabundant transition element in theearth'scrustandoccursfrequently indifferent oxida- tionstateswiththechalcogenide andpnictide elementsof whichsulfuristhemostimportant. Iron(II)-, iron(III)-, andiron(IV)-sulfur tetrahedral unitsinvestigated' in thispaperarethebasicpolyhedral unitsinsuchminerals' asspharelite, stannite, semiconductors chalcopyrite, cu- banite,linear-chain one-dimensional single-crystal alkali dithioferrates,''normal' spinelFeCr2S&, insulators, andmixed-valence iron-barium-sulfur systems, battery cellsKLi„FeS2 systems, plantferrodoxins, beefadreno- doxins, protein putidaredoxin, andreduced andoxidized iron-sulfur proteins (respiration andphotosynthesis).' Octahedral FeSindicates metallic conductivity, troilitein- dicatesferroelectricity, andtheironsulfides ingeneral in- dicateparamagnetism, diamagnetism, ferromagnetism, an- tiferromagnetism, andferrimagnetism aswell.'' Abetterunderstanding oftheelectronic andchemical bonding structure oftheseunitsisnecessary toelucidate thewidediversityofoptical,electrical, magnetic, andhy- peHineinteractions, aswellasotherimportant solid-state andbiochemical effectsobserved intheironsulfides.' Themultiple-scattering andtheSlaterXo.'local- exchange (MSXa)method havebeenpreviously' applied toiron-sulfide clusters. Wehavepreviously'" performed MSXacalculations intheiron(III) tetrahedral clusterin anefforttointerpret variousexperimental results. Inthis paperwehaveundertaken amoreextended andgeneral coinparative studyoftheiron(II), iron(III), andiron(IV) tetrahedral sulfides. Weareinterested inthewavefunction fortheground stateandwhatitrevealsaboutthebonding mechanisms in ordertoexplain theobserved experimental results. We havecalculated themagnetic andelectric hyperfine pa- rameters, theenergylevels,andtheatomic populations andinterpreted theMossbauer, optical, andneutron- diffraction experiments. Wehavealsotakenparticular in- terestintheouterradialfunctions andtheimportant in- formation theyprovide regarding spinpolarization andbonding aswellaseffective chargeandspindensities at theFenucleus. II.METHOD OFCALCULATIONS TheMSXnmethod'applied inthispapertothe FeS4,FeS4, andFeS4 clusters isanabinitioone- particle approach inwhichtheorbitals donotdependon atomicorbitals asbasisfunctions asinthelinearcombina- tionofatomic orbitals(I.CAO)approach. Theone- particle Schrodinger equations aresolvednumerically but inordertodosoonefirstmakessomeapproximations. Oneisthatthepotential isapproximated byamuffin-tin potential. Theotherapproximation isthattheexchange- integral operator isapproximated bySlater'slocalaverage exchange whichisdeduced fromatomiccalculations. The Fe-Sdistances usedweretakenfromHoggins andStein- fink.'Theaverage Fe-Sdistance is2.370,2.233,and 2.141Aincompounds whichcontainFe+(FeS4 ),Fe+ (FeS4),andFe+(FeSq),respectively, intetrahedral coordination. Thevaluesofatomicexchange parameters ausedinthesecalculations weretakenfromSchwarz a(Fe)=0.711,a(S)=0.724,anda=0.721intheouterand intersphere region. Themuffin-tin scheme employed as- sumedtheFe-sphere tangenttothesulfurspheres. Wat- sonspheres withchargesof+4,+5,and+6,+7were used,tangenttothesulfurspheres andlimiting theouter regionsoftheclusters, tostabilize theFeS4,FeS4, and FeS4clusters, respectively. Thedependence ofourcal- culations ontheWatson-sphere chargeandFe-Sdistance wasinvestigated. Wealsoinvestigated thedependence of ourcalculations ontheFe-sphere radius. Siqueira etal.' concluded fromtheircalculations ofvarious clusters in different oxidation statesthatitmakessensetocompare theresultsonlyifthesamemuffin-tin radiiareused.Our resultsaregivenandcompared usinginthethreeclusters thesameFeradius(2.21a.u.)usedinourprevious work'" intheFeS4cluster. Wehaveincluded ineach self-consistent-field (SCF)cycleallthe(coreplusvalence) electrons intheclusterandthecalculation wascarriedto self-consistency which wasachieved tobetterthan 7308 Qc1983TheAmerican Physical Society INVESTIGATION OFTHEELECTRONIC STRUCTURE, ... 7309 1)&10Ryintheenergylevelsinallcases. Ineachmuffin-tin sphereaandintheouterregionthe orbitalsaregivenby P(r)=QCI~R&(r) Yrm(~0) WeareusingfortheFeandouterregionl=0fororbi- talsofa~symmetry, I=1,2fort2,I=2foreand3fort&. IntheSspheres weareusinguptol=l.Thewavefunc- tionintheinteratomic regionisexpressed intermsof spherical BesselandHankelfunctions. Weobtainthesec-ularequations fromthecondition thatthewavefunction andtheirderivatives should becontinuous acrossthe sphereboundaries. Theorbitaleigenvalues andeigenvec- torsmaythenbedetermined. Larson's technique isasuitable procedure toderive atomicpopulations forthecalculated MSXawavefunc- tions.Theatomicpopulations aredefinedas 2C;gII—++i (2) sos whereKiistheamplitude ofanatomicorbitalusedas TABLEI.Orbitalenergies andorbitalcharacters forFeS4 OrbitalOrbital energy (Ry) FeCharge' inmuffin-tin sphere Inter Outer Sb atomic sphereOrbital character lacy lalg 2Q)$ 2Q)$ 1t2y 1t2& 3Q1 3ajg 2t2t 2t2l 4a)y 4a)g 3t2t 3t2$ Sa~&,le&,lt~& 4t2$,5t27 Sa~$,leg,lt~$ 4t2&,5t21 6a)y 6a)g 6t2y 6t2g 7Q]T 7t2f 7a)$ 7t24 8t2) 2ef 8a)f 8a)g 9t2) 8t2& 2e& 9t2& 2t~$ 3ef 2t~$ 10t2f' 3e) 10t2J,509.137 509.136 175.987 175.985 175.987 175.985 59.118 58.963 51.116 50.999 15.368 15.367 15.368 15.367 11.464 11.462 6.779 6.454 4.432 4.115 1.312 1.292 1.299 1.280 0.686 0.676 0.675 0.652 0.623 0.612 0.604 0.576 0.552 0.551 0.542 0.480 0.340 0.304100 100 100 100 100 100 99.89 99.87 99.64 99.57 1.64 1.27 1.41 0.93 64.13 59.29 10.14 8.42 1.78 1.37 5.77 16.03 0.09 34.93 0.08 31.22 84.69 73.7125 25 25 25 25 25 25 25 25 25 0.0 0.0 0.0 0.0 19.54 20.28 19.61 20.34 5.27 4.81 12.38 12.63 12.81 12.56 12.81 13.59 15.82 9.08 15.77 10.62 0.97 2.680.11 0.12 0.35 0.41 19.17 15.92 19.08 15.96 12.82 20.06 38.56 39.26 42.15 43.47 37.89 23.62 31.90 23.98 32.00 20.45 9.12 13.070.0 0.0 0.0 0.0 1.05 1.69 1.08 1.74 1.99 1.42 1.77 1.79 4.83 4.92 S.09 5.98 4.73 4.77 4.82 5.87 2.29 2.48Fe1s Fe1s S1s Sls S1s Sls Fe2s Fe2s Fe2p Fe2p S2s S2s S2s S2s S2p Fe3s Fe3s Fe3p Fe3p S3s S3s S3s S3s Fe3d,S3p Fe3d,S3p S3p,Fe4s S3p,Fe4s S3p S3p,Fe3d S3p,Fe3d S3p,Fe3d S3p,Fe3d S3p S3p S3p,Fe3d Fe3d,S3p Fe3d,S3p 'Inpercentofoneelectron charge. PercentofchargeineachSsphere. 'Highest occupied level. 7310 S.K.LIEANDC.A.TAFT 28 reference orbital, C;~aretheamplitudes ofthemolecular orbitals [defined inEq.(I)],andn;istheoccupation ofthe orbitali. III.GRBITALS, ORBITAL ENERGIES, ANDELECTRON POPULATION Theorbitalenergies, orbitalcharacters, andthecharge distributions withinthedifferent muffin-tin regionsfor theFeS4 andFeS4clusters aregiveninTablesIand IIlabeledaccording totheirreducible representation of thesymmetry groupTd.Thecalculated energylevelsfor FeS4aregiveninFig.1.Alltheseorbitalenergies areobtained inaground-state calculation. Fepopulations for thedifferent molecular orbitals forbothclusters are presented inTablesIIIandIV.Todetermine theK~coef- ficients inEq.(2)wehaveusedtheconfiguration 3d4s'and3d4sforatomicFeintheFeS4 and FeSqclusters, respectively. Thischoiceisofcoursear- bitrary; however, itwasmadebyexamining thetotal chargedistributions obtained. SinceFeS4 andFeS4 areopen-shell complexes havingfourunpaired 3delec- trons,theXacalculations havebeencarriedoutinthe spin-unrestricted formalism. Theresulting spinpolariza- tionsplitstheenergy levelsintospin-up andspin-down groupswithanenergydifference whichisinsomecases OrbitalOrbital energy (Ry)TABLEII.Orbitalenergies andorbitalcharacters forFeS4 Charge' inmuffin-tin sphere Inter Outer Fe Sb atomic sphereOrbital character la)y 1QI& 2Q]f 2a)$ 1t2t 1t,~ 3Q1't 3Q&~ 2t2t 2t2$ 4QI& 4a)4 3t2t 3t2$ 5a~g,jje),1t~f 4t2),5t2) 5Q)$,1e$,1t)$ 4t,&,5t,g 6a)g 6a)4 6t27 6t,g 7a~g 7a)$ 7t27 7t24 8t2) 2ef 8a)f 8alg 9t2T 2e], 8t,~ 9t,g 3ef 2t)$ 2t)$ 10tpg' 3e4 10t, &509.193 509.192 176.072 176.068 176.072 176.068 59.183 59.038 51.181 51.071 15.475 15.471 15.475 15.471 11.568 11.563 6.826 6.527 4.479 4.188 1.321 1.299 1.284 1.262 0.728 0.724 0.672 0.643 0.600 0.593 0.592 0.564 0.505 0.487 0.471 0.405 0.367 0.291100 100 100 100 100 100 99.89 99.87 99.62 99.55 3.77 3.39 3.38 2.50 71.92 73.08 12.22 10.86 3.66 22.83 13.89 28.38 21.91 0.24 0.23 23.90 69.83 53.210.0 0.0 0.0 0.0 15.64 15.71 16.71 16.84 3.09 2.16 8.92 8.93 9.56 7.71 7.84 9.33 8.58 12.73 12.67 10.36 2.97 5.940.11 0.12 0.36 0.42 31.79 31.86 26.85 27.11 13.98 17.42 50.70 52.19 52.64 41.91 49.67 29.74 37.25 43.11 43.26 29.41 14.74 20.280.0 0.0 0.0 0.0 1.84 1.91 2.90 3.02 1.73 0.87 1.35 1.23 5.42 4.41 5.07 4.57 6.51 5.73 5.84 5.25 3.53 2.74Fe1s Fe1s Sls Sls Sls Sls Fe2s Fe2s Fe2p Fe2p S2s S2s S2s S2s S2p Fe3s Fe3s Fe3p Fe3p S3s S3s S3s S3s Fe3d,S3p Fe3d,S3p S3p,Fe4s S3P,Fe4s S3p S3p,Fe3d S3p,Fe3d S3p,Fe3d S3p,Fe3d S3p, S3p S3p,Fe3d Fe3d,S3p Fe3d,S3p 'Inpercentofoneelectron charge. PercentofchargeineachSsphere.'Highest occupied level. 28 INVESTIGATION OFTHEELECTRONIC STRUCTURE, ... 7311 6(Ry) 0.0—Spin UpSpi,n down SymmetryAtomic population Spinup SpindownTABLEIV.Electron population onFeformolecular orbitals inthea&,e,andt2symmetries forFes4 -0.8—10t~ 38 2t)9tg 8py 'Pr~& 8t2 -1.0— 7al FIG.1.OrbitalenergylevelsforFeS4cluster.6a) 7a& 8a) 2e 3e 6tp 7t2 8t2 9tp 10t2 Total Charge3$ 4s 4s 3d 3d 3p 3d 3d 3d 3d 3s 3p 3d 4s Netcharge0.99 0.09 0.35 1.42 0.52 2.97 0.04 2.10 0.02 0.53 0.99 2.97 4.63 0.44 onFe(16total)=1.220.99 0.10 0.35 0.37 2.96 0.01 0.32 0.65 0.99 2.96 1.35 0.45 TABLEIII.Electron population onFeformolecular orbitals inthea&,e,andt2symmetries forFeS4 SymmetryAtomic character Spinup Spindown 6ai 7a) 8a) 2e 3e 6t2 7t2 8t2 9t2 10tp Total charge3$ 4s 4s 3d 3d 3p 3d 3d 3d 3d 3$ 3p 3d 4s Netcharge0.99 0.05 0.35 1.16 0.77 2.96 0.01 1.85 0.00 1.04 0.99 2.96 4.83 0.40 onFe(16total)=1.070.99 0.05 0.36 0.09 0.91 2.96 0.01 0.01 0.37 0.99 2.96 1.39 0.41greaterthan2eV. Forbothclusters the6a~and6t2orbitalscorrespond to iron3swhereas the7a~and7t2orbitals correspond to ligand3s.Thet~orbitals areexclusively ligand2pand2t~ orbitals arethemainnonbonding orbitalsofthesystems whicharealmost whollyligandS3pincharacter. Both Sa~orbitals areligand3pwithiron4scomponents. For theFeS4 clusterthehighest-energy orbitalcontaining electrons isthe3elwhichishalf-filled (i.e.,contains one electron) whereasfortheFeS&"thehighest-energy orbital containing electrons isthe10t2twhichispartially filled withtwoelectrons. Forbothclusters themolecular orbitalsatlowerener- giesarecompletely filledwithelectrons andthoseat higherenergies areempty.WethushaveforFeS4unoc- cupiedorbitalsof3e&10t2l symmetry andforFeS4 unoccupied orbitalsof3el,10tql,and10tztsymmetry as well.Theorbitalsofgreatest interestforourpresentstudy aretheSt2,9t2,10t2,2e,and3e.TheSt2gand2e&forFeS4 and3eg,St2&,and2efforFeS4are3d-likeof thecrystal-field typewithsmallS3padmixture. Thecor- responding St&&and2elaredominantly S3p.Theother occupied orbitals areligand3porbitals withdifferent de- greesofFe3dadmixture. SomeFe4pcomponent may alsobepresent inthe9t2and10t2orbitals. Atthetopof whatwouldbetermed thevalence bandistheempty 10t21.Thisorbitalisalmostpure3dandisstrongly anti- bonding. Thecorresponding 10t2)whichisonlypartially fullforFeS4isalsoanantibonding orbitalmuchlower inenergyanddifferent incomposition beingdominantly S3pwithsomeFe3pcharacter. The3elwhichishalf- filledforFeS46andemptyforFeS4,arealmostpure3d antibonding orbitals. Thecorresponding 3e)orbitalis muchlowerinenergyanddominantly S3pincharacter. The9t2 &and9t21orbitalsarealsodominantly S3p.The St2fand2egorbitals areatthebottomofthevalence band.Molecular orbitals belowthevalence bandshowlit- tleornomixingofFeandSatomicorbitals. Themaindifference inmolecular-orbital compositions between theFeS4 andFeS4 clusters occursinthe crystal-field-type orbitals andotheremptyandpartially filledeandtzvalence-band orbitals. Weobservealarger 3dorbitalcharacter andelectron population inthe crystal-field typegt21'and2etorbitalsofFeS44.The highest occupied 10t2ttheempty(10t2t) andpartially filled(3et)orbitals inFeSzindicate alarger3dorbital character andelectron population. Thebonding orbitals inthetwosystems areverysimilarincomposition (Tables I—IV).Uponcomparing theFeS4clusterwithourpre- viousresults' intheFeS4cluster' wedonotfindsub- stantial changes inthemolecular-orbital compositions despitethereduction oftheelectron ingoingfromFe+to Fe+.Thepresenceofatetravalent ironinasulfideisnot verylikely,andindeeditappearsthattheelectron iseffec- tivelybackdonated totheironion,thusreducing its charge. Inotherwords,upongoingfromFe+toFe+, i.e.,fromFeS&toFeS4, theelectron wiHberemoved fromamainlyFeorbital. However, withfurtheroxida- tiontoFeS4theelectron willbebackdonated froman 7312 S.K.LIEANDC.A.TAFT S3psulfurorbital. Thisissubstantiated byourelectron- population analysesofFemolecular orbitalsofeandt2 symmetries (TableII)forFeS4 andFeSz clusters' whichindicated aFe=—1.2effective chargeinbothclusters (TableIV).Thiswouldexplain whydespitethefactthat delocalization ofanelectron withinaFeS4tetrahedron isobserved inBa3FeS5, thecalculated andobserved isomer shiftsindicate''thepresenceofFe+. Thevalence bandinFeS4(6.03eV)andFeS4(5.94 eV)clusters arewiderthanintheFeS4(5.20eV)cluster. Wealsonotethatinthelanguage ofbandtheoryinthe threeclusters theiron3dcrystal-field-type levelsarepart- lyburiedinthesulfurpband.Inthetransition-metal sul- fidestheelectronic structures andhenceelectrical and magnetic properties arecomplicated bythepresenceofd electrons. Generally inthesesystems thevalence-band en- ergylevelsarecomposed ofsulfur3p-and3s-typeorbitals andtheconduction bandofmetalsandporbitals withd orbitals added,whoserelative energy levels(asshownin TablesIandII)mayvarywidely.'Also,mostimportant- ly,thedorbitals mayoverlap withsulfurorbitalstoform bands.Whenthedlevelsarelocalized butbelowthetop ofthevalence bandinenergy(asindicated byourcalcula- tions)thematerial mayexhibit semiconduction' and paramagnetism (sincethedelectrons arenotcompletely paired). TheMSXaresultsthussupport theparamagnet- icsemiconducting behavior observed experimentally for variousiron(II)andiron(III) tetrahedral sulphides.' Thegreatligandcharacter ofthecrystal-field-type molecular orbitalsoftheFeS4,FeS4, andFeS4clus- tersindicates thelargeoverlap between metaldandsulfur porbitals whichtakesplace.Thisoverlap destabilizes the antibonding eandt2crystal-field-type orbitals butstabi- lizesthebonding eandt2orbitals whichisprobably an important factorcontributing tothechalcophilic natureof transition elements assuggested byBurns.''The orbital-energy difference e(10t2i)—e(3et)=b,ecorre- spondstothequantityAt=—,&10Dq whichcanbeob- tained fromabsorption spectra. Thecalculated transition-state calculations donotsignificantly change Ae.Ourcalculated valueofAeis3951cm'whichisin goodagreement withtheexperimental valueof4000 cm—'.IV.HYPERFINE INTERACTIONS, MAGNETIC MOMENTS, ANDRADIAL DENSITIES TheFeS4tetrahedral clusteroccursinFeCr2S4 which hasthenormal spinelstructure withtheFe+ionsoccu- pyingthetetrahedral 2sites.Mossbauer-spectroscopy' andneutron-diffraction measurements' showsthatthe material ordersmagnetically about180Kindicating a magnetic momentof4.2p~.Inatetrahedral sitethefive- fold-degenerate orbitalgroundstate'DofthefreeFe+ ionissplitbythecrystalfieldintoalowerorbitaldoublet Egandanuppertriplet Tzgseparated byanenergyA. Thehyperfine fieldattheFenucleus inFeCr2S4 maybe expressed as jeff~c+~orb+~dip (3) wheregistheelectronic spectroscopic splitting factor,pz istheBohrmagneton, and5isthetotalspin.Thisfield II,maybeinterpreted inaHartree-Fock-type (HF)for- malism bytheexchange-polarization mechanism ofthes shellsbytheunpaired delectrons. Theothertwotermsin (3)aretheorbitalanddipolarcontributions tothehyper- finefield. FromtheopticalandMossbauer spectroscopic experi- mentaldatainFeCrzSq thevalueof ~H, ~=320kOewas determined.'Thehyperfine fieldinthiscompound is unusually smallifoneassumes theirontobeinthe+2 oxidation state.HFXacalculations fortheFe+freeion yields ~H, ~=577kCx(TableV).Thesmallexperimental valuecompares, however, withthevaluesfoundinsome covalent materials, suggesting strongcovalent effectsin theFe—Sbonds. Fromourspin-polarized calculation weobtainthe chargeandspindensities atthenucleusfromthea~orbi- talshavingl=0intheironsphere. TheFermi-contact termwhichisproportional tothedifference inspin-up andspin-down densities atthenucleus couldthenbecal- culated. InTablesVandVIwearegivingthetotalspinThefirsttermin(3)istheFermi-contact termH,whichis givenby H,=',~g„S( ~g(0)~,—ttj(0)~,), TABLE V.SpindensitiesgattheFenucleus andFermi-contact termM,forFe4andtheFe+ freeion.+=4m.g,.[ ~u',(0) ~— ~u'„(0) ~~,whereu'isanoccupied ofa~symmetry oratomicsorbi- tal. FeS4 1al(Fe1s) 3al(Fe2s) 6a~(Fe3s) 7a( 8al M,(kG)Total—0.76—20.30 10.04 0.47 4.07 —6.48—273Fe'+ (Xcz,a=0.711) —0.81—23.31 10.41 —13.71 28 INVESTIGATION OFTHEELECTRONIC STRUCTURE, ... 7313 TABLEVI.SpindensitiesgattheFenucleus andFermi-contact termH,forFeS4andFe+free ion.+=4m.g, I Iu',(0)I— Iu',(0) II,whereu'isanoccupied orbitalofa&symmetry oranatomics orbital. la1(Fe1s) 3a1(Fe2s) 6a1(Fe3s) 7a1 Sa1 Total H,(kG)FeS4 —0.75—18.76 10.32 0.77 3.30—5.12—2161$ 2$ 3$Fe4+ Xu(a=0.711) —1.04—29.13 14.76 —15.41—648 densityattheFenucleus aswellastheindividual contri- butions fromthedifferent orbitals fortheFeSq and FeS4 clusters. Forcomparison thesameinformation fortheFe+andFe+ionswasalsoincluded. Thecalculated hyperfine fieldintheFeS4clusteris IH, I=273kCi(TableV)ingoodagreement withtheex- perimental IH,Ivalueof320kGinFeCrSq. Individual contributions toH,arequitesensitive toco- valentexperimental effects,i.e.,bonding mechanisms whichtendtodonatechargefromtheligand3porbitalsto 4sandtoFe3dorbitals withimportant exchange polariza- tioneffects. TheFe3dand4spopulation inFeS4 are 3dT",3dt',4st,and4st'(TableIII).Themain difference between theactualconfiguration oftheironin FeS4 andtheFe+freeionisanincreaseof0.39elec- tronsinthe3dIlevelsandthepartialoccupation ofthe4s shells.Between theFeS4clusterandFe+freeionwe alsoobservealargeincrease in3dtand3dlelectrons as wellaspartialoccupation ofthe4sshells(TableIV). FromTablesVandVIweseealargepositive 4scontri- butionof191and171kCxinFeS4andFeS4", respec- tively,fromthe7atandSatorbitals. Wealsoobservea modification ofthe2sand3scontributions withrespectto thefree-ion values.ThisisInainly duetotheincreased partialpopulation ofthe3dtand3dTlevelsmentioned above.Viatheexchange polarization mechanism addi- tional3dspin-down electrons willreducethemagnitude of thenegative andpositive 2s-and3s-spincontributions, respectively. Thesmallerincrease inthe3d4electron pop- ulation inFeS4 resultsinasmaller reduction ofthe3s contribution (withrespecttofree-ion values)toH,as compared toFeS&and'FeS&(TablesVIandVII).TheMossbauer isomershiftisdefined as 5=',1TZeS—(z)b,(r)[Iq(0)Ig— Iq(&) Is] I@(O) Isl:rt~p(0) whereb(r)isthechange inthemean-square nuclear- chargeradiiintheMossbauer transition (negative forFe) andtheterminbrackets isthedifference between the squared amplitude oftheelectronic wavefunction atthe nucleusofabsorver andsource.S(z)isacorrection term forrelativistic effects.aistheisomer-shift calibration constant whichcontains allnuclearconstants. Themajordifference intheMossbauer parameters of ironinsulfides compared tomoreionicmaterials isthe smaller valuesoftheisomershiftwhichmaybequalita- tivelyattributed toastronger covalency intheformer compounds. InTablesVIIIandIXwegivethetotal chargedensities andindividual contributions fromthedif- ferentorbitalsforFeS4andFeS4.Thevaluesforthe free-ionFe+andFe+arealsogivenforcomparison. We observe alargedecrease inthe3scontributions with respecttofree-ion valuesduetoincreased shielding ofthe nuclear chargecaused bytheincreased 3dpopulation. Thisdecrease in3scontribution islargerintheFeS4 clusterduetothelargerincrease in3dpopulations inthis configuration. Wealsoobserve thiseffectwhenwecom- parethe3scontributions (TableX)intheFeS4, FeSq andFeS4clusters. Wealsonowhaveinbothclusters an important 4scontribution whichislargerinFeS4 and FeS&clusters duetoscreening effectsofalarger3dpop- TABLEVIII.Electron densities Il((0)attheFenucleus forFeSq andtheFe+freeion.[ Il((0) I=g,Iu'&(0) I +I&'~(o) I'I TABLEVII.NetFechargeand3d-and4s-electron popula- tionsforFeS4,FeS4,andFeS4clusters.FeSFe'+ (Xa) Fe4+S4- Fe+S Fe2+S6— 'FromRef.1.NetFe charge 1.22 1.20 1.073d-electron population 3d)4.633d)1.35 3dt"43d' 3dy3dg'4s-electron population 4~0.444~0.45 4sf0.434$$0.42 4$g0.404$g0.411a1(Fe1s) 3a1(Fe2s) 6a1(Fe3s) 7a1 8a1 Total10752.19 979.41 140.74 0.47 3.17 11875.9810752.46 979.06 141.41 11872.93 S.K.LIEANDC.A.TAFT TABLEIX.Electron densities ~lt(0) ~attheFenucleusfor FeS&4andFe+freeion.[~@(0)(=g,~u'&(0) ~+ ~&q(0) ~'.]0.30 FeS4 la~(Fe1s) 3a)(Fe2s) 6al(Fe3s) 7a& Sa) Total10752.10 979.39 141.19 1.13 3.99 11877.801$ 2s 3$Fe+4 (Xa) 10751.76 978.99 147.73 11878.480.25 0.20 0.15 OI CL 0.10 0.05 ulation intheFeS4cluster. Weinvestigated thedepen- denceofourcalculations onthechargeoftheWatson sphereandtheFe-sphere radius. Thechanges inp(0) whenacharge+7wasusedinsteadof+6forFeS& wasonlyinthesixthsignificant figureandthusunimpor- tantevenforisomershifts.'Wealsoobserved thatp(0) increases astheFe-sphere radiusisincreased andde- creasesastheFe-Sdistance isincreased. The4soccupancy of40%%uointheFeS4clusterisin goodagreement withthedensityof-43%inFeCrS4 de- ducedfromtheMossbauer isomershiftusingDanon's'' calibration constantof—0.2aomm/s. Neutron-diffraction measurements inFeCr2S4 indicate anironmagnetic moment ofpF,——4.2pz,i.e.,3.32un- paired3delectrons (spin-only formula) whichisinvery goodagreement withthe3.44unpaired 3delectrons (Table III)calculated inthiswork.Thisvalueresultsfromanin- creasedpartialoccupation oftheFe3d$orbitals. InFigs.2—7weplottheradialdensities oftheouter orbitals (TablesIandII)ofmostinterest intheFeSq andFeS4clusters. TheHFXnFe+andFe+free-ion radialdensities arealsogivenforcomparison. TheFe3d and4sorbitals, weemphasize, arequitesensitive tobond- ingmechanisms whichtendtodonatechargefromthe1.0 1.5 r(a.u)2.0 2.5 FIG.2.Radialdensities oforbitalsforFeS4cluster. a, Fe+HFXa4s(scale2);b,Fe4s 7(Sa~g);c,Fe4s g(8a~g). ligand3porbitalsto3dand4sorbitals withsignificant ef- fectsontheelectric, magnetic, andhyperfine parameters. Figures2and5showtheradialdensitiesofFe4st (Ba&t) and4st(Sa&t)orbitals whichareligand3pwithaniron 4scomponent. TheFeHFXa(a=0.711)4sradialdensi- tyisgivenforcomparison. Inbothclustersthe4s&densi- tyislargerthanthe4sldensitywhichresultsinasubstan- tialpositive contribution totheFermi-contact term(M,) causingalargecovalent reduction ofthenegative hyper- finefield.Theunpaired spinsinthe3dorbital"attract" the4sspin-up electrons inwards through thePauliex- clusion principle, although thereisagreater inflowof electrontothemoreextended 4sorbitals. Theadditionof a"d"electron resultsinanexpansion oftheoutermost s shellduetotheincrease intheexternal screening ofthe nuclearpotential. Onemustbecareful, however, incon- TABLEX.Contribution to ~ltj(0) ~3sand4sorbitalsatFe nucleus (ina.u.)invariousclusters anddifferent atomicconfigu- rations. Fe-Sdistance is2.141AintheFeS4cluster,2.370A inFeS4', and2.233AinFeS4 Fe configuration 3d4 3d4s 3d' 3d'4s' 3d' 3d64$2147.73 147.00 143.85 143.63 141.41 141.7118.05 13.22 8.920.6 OI lK 0.4 Cluster0.51.01.5 r(a.u) FeS4 FeS4 FeS4(Fe4+) (Fe'+) (Fe'+)141.19 140.98 140.745.12 4.22 3.64FIG.3.Radialdensities oforbitalsforFeS4cluster.a, Fe+3dHFXa;b„Fe3dl (10t24); e,Fe3dg (St&g);d,Fe3dg (10t2&);e,Fe3dg(9t2&). 73/5 STRUCTUR OFTHEEEIPCTRO&IC I~EsTIGATION 1.0—1.0 0.80.8 0.6 04lL CQ 0.40.6 NlL 0.40.2 p.5 r(Q.U)2.00.2 p.5I 1.5 1.0 r(a.u)2.04—lst«. forbitals ),dpe3adjaldensities o Fe3dg(10t2 &FIG.6.Rad' 3d)(8t,t); HF+A F3dg(8tgl)~p (9tg). (lOt~~) e"' Radialdensities 2+3dHP+a;b,Pe ( (3eg)e,Fe3dg—cl forbitalsfo ).dFe3d~ (3el);c, ete'4sorbitalisalso ethediffuse4sori S1g 1morea tiono types 'd)radialfunctio hunoccup atomic1 ep 3egand10t24or1aftionssug- meregionasatom thesulfurlga1ntheSa sferfromt 'screen-maxim™, tchargetrans effective scstingslgnif.hargetrans ' iffectsthehYeFe1 lization, sgldliketono enganndexchang p ters.Wewo 'theFeS4'onaram ' ita»'"~interactlo 3dtypeor hazebeenthestab1»ty 6—clusters 1sst ozalency eff contradictory p.lHazony™ b'taisinthemoin~ol&ing 3 nsionofh supportdorb1tas dtheradialexpa ounds. onbigyp -diffraction measu neutron- i 1.2— 0.30 1.0 0.25 0.8 0.4 0.10 0.2 0.05 1.5 1.0 r(o.u)2.0I 1.5j.o r(a.u)2.02.5 cluster.a, forFe4 forbitalsfo a+HFXu4s(scale2);,e FerFes4cluster. a, ldensities 4e+3dHFXo.';b,Fe (3ef);e,Fe 7316 S.K.LIEANDC.A.TAFT 28 Oneofuse(S.K.L.)isgratefulforaresearch fellowship fromConselho Nacional deDesenvolvimento Cientifico e Tecnologico (Brasil).manganese sulfides, arecontrary tosuchanexpansion ofdifficulties oftenencountered insurveying thephysical the3dorbitals.'Theneedofmorecalculations toclarify properties ofaseriesofsuchcompounds insupposedly different viewpoints havebeenoftenemphasized. 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PhysRevB.96.174207.pdf	PHYSICAL REVIEW B 96, 174207 (2017) Fate of topological states and mobility edges in one-dimensional slowly varying incommensurate potentials Tong Liu,1Hai-Yang Yan,2and Hao Guo1,* 1Department of Physics, Southeast University, Nanjing 211189, China 2Key Laboratory of Neutron Physics, Institute of Nuclear Physics and Chemistry, CAEP , Mianyang, Sichuan 621900,China (Received 1 August 2017; revised manuscript received 16 October 2017; published 15 November 2017) We investigate the interplay between disorder and superconducting pairing for a one-dimensional p-wave superconductor subject to slowly varying incommensurate potentials with mobility edges. With amplitudeincrements of the incommensurate potentials, the system can undergo a transition from a topological phaseto a topologically trivial localized phase. Interestingly, we ﬁnd that there are four mobility edges in the spectrumwhen the strength of the incommensurate potential is below a critical threshold, and a novel topologicallynontrivial localized phase emerges in a certain region. We reveal this energy-dependent metal-insulator transitionby applying several numerical diagnostic techniques, including the inverse participation ratio, the density of states,and the Lyapunov exponent. Since the precise control of the background potential and the p-wave superﬂuid can be realized in the ultracold atomic systems, we believe that these novel mobility edges can be observedexperimentally. DOI: 10.1103/PhysRevB.96.174207 I. INTRODUCTION In recent years, considerable attention has been paid to the topological matters, including topological insulators (TIs)[1,2] and topological superconductors (TSCs) [ 3,4]. Among various models, the one-dimensional (1D) TSC, i.e., thespinless p-wave superconductor model studied originally by Kitaev [ 3], is an important and well known example. A key feature of the 1D TSC is that it hosts the zero-energyMajorana fermion states [ 5–7], which promise a platform for the error-free quantum computation since the information canbe stored in the topologically protected Majorana states and thequbits are immune to the weakly disordered perturbation [ 8]. The search for Majorana fermions in TSCs has been a subjectof intense interest, and many theoretical approaches have beenachieved. The TSCs can be classiﬁed according to the theirsymmetries, such as the time reversal, particle-hole, and chiralsymmetry, and correspondingly there are four classes of TSCs,i.e., BDI, CII, D, and DIII [ 9]. If the time reversal symmetry of the 1D TSC system is broken by the presence of impurities [ 10] or the strength of the disorder is strong enough, the stabilityof the topological phase can be signiﬁcantly affected and atransition driven to the topologically trivial localized phasecan occur. The disorder effects of 1D TSC systems have been studied intensively. So far, most of the theoretical work for theAnderson localization in 1D TSCs focuses on the random dis-order [ 11–14] and the quasiperiodic disorder/incommensurate potential [ 15–21]. Reference [ 15] studies the interplay between the quasiperiodic disorder and superconductivity, and it leadsto the topological phase transition from a topological super-conducting phase to a topologically trivial localized phasewhen the strength of the incommensurate potential increasesabove a critical value. The same model is studied in Ref. [ 16], and a wide critical region in the parameter space is discovered,which is quite different from the Aubry-André (AA) model *guohao.ph@seu.edu.cn[21] where the wave functions are critical only at the phase transition point. However, none of these disorder models, both the random and the quasiperiodic, can host the mobility edge. A studyabout the interplay between the disorder with mobility edgesand the p-wave superconducting pairing is still absent to the best of our knowledge. Here we introduce a class of 1Dpotentials [ 22,23] with analytical expressions for the mobility edges, which enables us to study the interplay between themobility edges and the p-wave superconducting pairing in a more controlled fashion. These deterministic potentialsare neither random nor simply incommensurate, but ratherslowly varying in real space. So we consider the 1D p-wave superconductor in these lattices, which is described by thefollowing Hamiltonian: ˆH= L−1/summationdisplay i=1(−tˆc† iˆci+1+/Delta1ˆciˆci+1+H.c.)+L/summationdisplay i=1Viˆni, (1) where ˆc† i(ˆci) is the fermion creation (annihilation) operator, ˆni=ˆc† iˆciis the particle number operator, and Lis the total number of sites. Here the nearest-neighbor hopping amplitudetand the p-wave pairing amplitude /Delta1are real constants, and V i=Vcos(2πβiv+φ) is the slowly varying incommensurate potential with 0 <v< 1 andV> 0 being the strength of the incommensurate potentials. A typical choice for parameters isβ=(√ 5−1)/2,φ=0, and v=0.4. For computational convenience, t=1 is set as the energy unit. The 1D TSC chain with complex/real superconducting pairing belongs tothe D/BDI class. Since the pairing of our model is real, itbelongs to the BDI class. Models belonging to other classesalso attract growing interest recently. When /Delta1=0 and v=1, this model reduces to the AA model, and the system can undergo a metal-insulator transitionatV=2. When /Delta1=0 and 0 <v< 1, Eq. ( 1) describes a model with slowly varying incommensurate potentials [ 23]. It is well known that this model has two mobility edgeswhen V< 2, i.e., all wave functions with eigenenergy in 2469-9950/2017/96(17)/174207(8) 174207-1 ©2017 American Physical SocietyTONG LIU, HAI-Y ANG Y AN, AND HAO GUO PHYSICAL REVIEW B 96, 174207 (2017) [V−2,2−V] are extended and otherwise localized. When V> 2, all wave functions are localized as in the AA model. When β=0 such that Vibecomes a constant V,E q .( 1) describes Kitaev’s p-wave superconductor model, and the system can undergo a topological phase transition at V=2. When /Delta1/negationslash=0 and v=1, Eq. ( 1) describes the 1D p-wave superconductor in incommensurate potentials. By applyingthis model, Ref. [ 15] determines the phase transition point V /prime=2+2/Delta1both numerically and analytically, and Ref. [ 16] demonstrates that wave functions in the parameter spacebetween V /prime/prime=2−2/Delta1andV/prime=2+2/Delta1are not extended but critical. In this work we study the situation for which /Delta1/negationslash=0 and 0 < v< 1, i.e., the interplay between the disorder with mobility edges and the p-wave superconducting pairing. The main questions that we are interested are: (1) how the slowly varyingincommensurate potentials drive a 1D p-wave superconductor to undergo a transition from a topological phase to a trivialphase, and (2) how localized properties (such as mobilityedges) of this system change besides the topological transition. The rest of the paper is organized as follows. In Sec. II we investigate the phase transition from a topological phase totopologically trivial localized phase. In Sec. IIIwe demonstrate the existence of the four mobility edges by numericallystudying the inverse participation ratio of wave functions, thedensity of states, and the Lyapunov exponent. We conclude and discuss possible experimental observations in Sec. IV. II. PHASE TRANSITION FROM TOPOLOGICAL PHASE TO TOPOLOGICALLY TRIVIAL LOCALIZED PHASE The Hamiltonian ( 1) can be diagonalized by using the Bogoliubov–de Gennes (BdG) transformation [ 24,25]: ˆχ† n=L/summationdisplay i=1[un,iˆc† i+vn,iˆci], (2) where Ldenotes the total number of sites, nis the energy level index, and un,i,vn,iare the two-component wave functions. Hence the Hamiltonian is diagonalized as H=/summationtextL n=1En(ˆχ† nˆχn−1 2) where Enis the eigenenergy of the Hamiltonian. The BdG equations can be expressed as /parenleftbigg ˆm ˆ/Delta1 −ˆ/Delta1−ˆm/parenrightbigg/parenleftbigg un vn/parenrightbigg =En/parenleftbigg un vn/parenrightbigg , (3) where ˆmij=−t(δj,i+1+δj,i−1)+Viδji,ˆ/Delta1ij=−/Delta1(δj,i+1− δj,i−1),uT n=(un,1,..., u n,L), and vT n=(vn,1,..., v n,L). It is widely known that the particle-hole symmetry ˆ χn(En)= ˆχ† n(−En) is conserved in the BdG equtions. By numerically solving Eq. ( 3), we can get the spectrum of the system and the wave functions un,iandvn,i.I n Fig. 1we show the spectrum when /Delta1=0.3 under the open boundary conditions. It can be shown that there is a regimewith nonzero energy gaps in the range V/lessorsimilar2 and there are the zero energy modes. These zero energy modes still existwith the increasing of V. To show the wave functions of the zero energy modes clearly, we introduce γ A i=ˆc† i+ˆci andγB i=(ˆci−ˆc† i)/i, where γAandγBare two species of Majorana fermions, satisfying the relations ( γα i)†=γα iand0 0.5 1 1.5 2 2.5 3 V-1012Eigenenergynth=10000 nth=10001 nth=9999 nth=10002 0123-20210-14 0 2000 4000 6000 8000 10000-101 V=1.9 0 2000 4000 6000 8000 10000-101 V=1.90 2000 4000 6000 8000 10000-101 V=2.5 0 2000 4000 6000 8000 10000-101V=2.5 FIG. 1. The spectrum (only the lowest four eigenenergies close to zero are shown) of the Hamiltonian ( 1) with /Delta1=0.3 as a function ofVunder the open boundary condition. Here the total number o fs i t e si ss e ta s L=10 000. Surprisingly, the lowest excitation, i.e., the 10 000th and 10 001th eigenenergies stay at zero as V increases. The inset shows the blow up of these two eigenenergies. We also carry numerical calculations by choosing other /Delta1’s, and ﬁnd that the zero energy modes are independent of Vtoo. The spatial distributions of φandψfor the lowest excitation with various V’s are shown in the lower ﬁgures. The lower left picture corresponds tothe wave functions of the Majorana zero energy mode, and the lower right picture corresponds to the wave functions of the other zero energy mode. {γα i,γβ i}=2δijδαβwithαandβtaking AorB. Then the Bogoliubove quasiparticle operators can be rewritten as ˆχ† n=1 2L/summationdisplay i=1/bracketleftbig φn,iγA i−iψn,iγB i/bracketrightbig , (4) where φn,i=(un,i+vn,i) andψn,i=(un,i−vn,i). Using this new deﬁnition we plot the spatial distributions of φandψfor the lowest excitation of the spectrum. When V= 1.9,φandψof the zero energy modes are located at the right (left) end and decay very quickly away from the right (left)edge, as shown in Fig. 1. Since there is no overlap between the amplitudes of φandψ, the zero energy modes split into two spatially separated Majorana edge states. However, whenV=2.5 the amplitudes of φandψof the zero energy modes overlap together and are located within a ﬁnite range of thewhole chain. This indicates the corresponding quasiparticle isa localized fermion which cannot be split into two independentMajorana edge states. Therefore, these results demonstrate thatthe system can undergo a transition from a topological phaseto a topologically trivial localized phase when the strengthof the incommensurate potentials Vis increased to a certain level. This is quite unusual because in general if there existsa topological phase transition, the energy gap will close andreopen and the zero energy modes disappear, but in our modelthe system becomes gapless after the energy gap closed andthe zero energy modes still appear. This unique character is due to the proliferation of subgap states, the eigenenergies of which are close to zero. Therehave been a subject of intense study about the disorder-induced 174207-2FATE OF TOPOLOGICAL STATES AND MOBILITY EDGES . . . PHYSICAL REVIEW B 96, 174207 (2017) TABLE I. The 9998th–10 003th eigenenergies with /Delta1=0.3 as a function of Vunder the open boundary condition. Here the total number of sites is set as L=10 000. EE 9998 E9999 E10 000 E10 001 E10 002 E10 003 V=2.0 −0.0181 −0.0178 −3.5×10−15−1.1×10−150.0178 0.0181 V=2.1 −3.3×10−6−2.4×10−6−4.6×10−151.4×10−152.4×10−63.3×10−6 V=2.2 −1.4×10−11−6.9×10−12−8.2×10−151.4×10−156.9×10−121.4×10−11 V=2.5 −1.8×10−15−6.0×10−16−4.2×10−16−4.2×10−16−2.8×10−161.3×10−15 subgap bound states in the superconductor. For adatom chains, a band of subgap Shiba states has been demonstrated tostrongly modify the low-energy properties of the system[26–30] and possibly induce trivial zero-energy features at the chain end [ 31]. We list part of the proliferation of subgap states in Table I. When V=2, the energy gap /Delta1 g=E10 002− E9999=0.0356 is still well deﬁned, and E10 001 andE10 000 denote two Majorana zero energy modes. When V=2.1, the subgap states start to appear. When Vincreases, the energies of subgap states are closer to zero and the system becomesgapless. When V=2.5, the 9998th–10 003th eigenenergies have almost the same numerical accuracy, hence the commonpicture that the energy gap closes and reopens breaks down inour model. We now wonder if there exists a ﬁxed value of Vwhich denotes the gap-closing point. In Fig. 2we plot the variation of energy gap /Delta1 gversus Vfor different /Delta1’s, the energy gap/Delta1gvanishes near V=2. To make the result clear we make a ﬁnite-size analysis in the inset. The scaling behavior 0123456 V01234g=0.2 =0.5 =1 =1.2 =1.5 0.0001 0.0003 0.0005 0.0007 0.0009 1/L00.050.10.150.2 V=1.9 V=2 V=2.1 V=5 FIG. 2. /Delta1gas a function of Vwith various /Delta1’s under the twisted periodic boundary condition. The total number of sites is set to L= 1000. Here /Delta1gis chosen to be twice of the lowest excitation energy. The inset shows the ﬁnite size analysis of /Lambda1=/Delta1g 2nearV=2. The red symbols correspond to /Delta1=0.5 and the blue symbols correspond to/Delta1=1. We can clearly see that when V=1.9,/Lambda1is ﬁnite as L→∞ .W h e n V=2.1,/Lambda1vanishes as L→∞ .W h e n V=5,/Lambda1 also vanishes as L→∞ , which is consistent with the result in Fig. 1 that there exists a widely gapless range. Interestingly, when V=2, /Lambda1→L−awitha> 0, hence /Lambda1→0a sL→∞ , the energy gap vanishes at VT=2.of/Lambda1atV=2 has a power-law decreasing trend, hence we conﬁrm the gap-closing point is VT=2. We numerically ﬁt the curves with ( /Delta1,V )=(0.5,2) and ( /Delta1,V )=(1,2), respectively, and obtain the expressions /Lambda10.5=1.087L−0.4088and/Lambda11= 1.749L−0.4097. However, whether the system still has well- deﬁned topological properties and consequently there exists aprecisely deﬁned quantity to distinguish different topologicalphases is still not clear. Hence we try to calculate a topologicalquantum number/topological invariant. The topological phase transition is characterized by the change of the topological quantum number Q.I na p-wave superconducting wire, the value of Q=(−1) mis determined by the parity of the number mof Majorana bound states at each end of the wire, and Q=− 1 denotes a topological phase. For a periodic translationally invariant p-wave superconductor inkspace, Kitaev [ 3] deﬁned the topological quantum number as QKitaev=sgn{Pf[iH(0)]Pf[ iH(π)]}, (5) where Pf denotes Pfafﬁan operation on a matrix. However, to identify the topologically nontrivial phase of a ﬁnite disorderedchain it is more suitable to work with the scattering matrix[32,33]. The scattering matrix Srelates incoming and outgoing wave amplitudes. The waves can come in from the left/rightend of the chain in two channels, i.e., particle and holechannels, so Sis a 4×4 unitary matrix. The 2 ×2 subblocks R,R /primeandT,T/primeare the reﬂection and transmission matrices at the two ends of the chain, respectively, S=/parenleftbigg RT/prime TR/prime/parenrightbigg , (6) where R=/parenleftbigg reereh rherhh/parenrightbigg . (7) Here reeandrehare the normal and Andreev reﬂection amplitudes, respectively. The BdG Hamiltonian has a particle-hole symmetry PH BdGP−1=−HBdG, (8) where P=τxCwithτxbeing the ﬁrst Pauli matrix and Cbeing the complex conjugation operator. This leads to the followingconstraint on the reﬂection matrix: τ xRτx=R∗, (9) which implies det(R)=det(R)∗. (10) Here we have implicitly applied the condition that Fermi level E=0. 174207-3TONG LIU, HAI-Y ANG Y AN, AND HAO GUO PHYSICAL REVIEW B 96, 174207 (2017) At the Fermi level the transmission Tthrough the nanowire is zero because there are no extended states from one end tothe other. Therefore the reﬂection matrix Ris unitary, i.e., RR †=1, which implies \|det(R)\|=1. (11) Combining with the condition ( 10), we get det( R)=± 1, and consequently the topological quantum number is Q= sgn[det( R)]. The scattering matrix can be obtained by the transfer matrix scheme at the Fermi level /parenleftbiggˆti/Phi1i /Phi1i+1/parenrightbigg =Mi/parenleftBigg ˆt† i−1/Phi1i−1 /Phi1i/parenrightBigg , (12a) Mi=/parenleftBigg 0 ˆt† i −ˆt−1 i−ˆt−1 iˆhi/parenrightBigg , (12b) where /Phi1i=(ui,vi)Tis the two-component wave functions on sitei. Here sites i=0 and i=L+1 represent the electron reservoirs. Waves at the two ends of the chain are related bythe total transfer matrix M=M LML−1···M2M1. (13) We transform to a new basis with right-moving and left-moving waves separated in the upper and lower twocomponents by means of the unitary transformation [ 34,35] ˜M=U †MU, U =/radicalbigg 1 2/parenleftbigg 11 iI−iI/parenrightbigg . (14) Under this basis the transmission and reﬂection matrices are related by /parenleftbigg T 0/parenrightbigg =˜M/parenleftbigg I R/parenrightbigg ,/parenleftbigg R/prime I/parenrightbigg =˜M/parenleftbigg 0 T/prime/parenrightbigg . (15) Finally, the topological quantum number Qis evaluated by calculating the transfer matrix ˜M. Here we adopt the numerical method presented by Ref. [ 36]. Most papers only calculate QL=sgn[det( R)], which de- termines the parity of the number of Majorana bound statesat the left end of the wire. In this paper we also calculateQ R=sgn[det( R/prime)], which determines the parity of the number of Majorana bound states at the right end of the wire.We show our numeric results in Fig. 3. Surprisingly Q L andQRdo not change simultaneously when Vvaries. We also veriﬁed other models including constant, periodic, andquasiperiodic potentials [ 37,38] and found that Q LandQR change simultaneously, which is fully consistent with previous theoretical results. This unsimultaneous change of QLandQR does not depend on the disorder, but the slow varying potential. We also ﬁnd that this phenomenon occurs for the two-periodslow varying potential V i=Vcos(πiv). This phenomenon demonstrates that the scattering method breaks down when calculating the topological quantum num-ber of a p-wave superconductor in slowly varying potentials. The reason for this may be due to the nature of slowlyvarying incommensurate potential V i=Vcos(2πβiv). Since012345 V-1-0.500.51Q=0.3,R =0.3,L =0.5,R =0.5,L =0.8,R =0.8,L =1.2,R =1.2,L FIG. 3. Topological quantum numbers QLandQRversus Vfor systems with various /Delta1’s and L=10 000. its derivative is dVi di=− 2Vπβiv−1sin(2πβiv). (16) Then, in the thermodynamic limit i→∞ we have lim i→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingledV i di/vextendsingle/vextendsingle/vextendsingle/vextendsingle=− lim i→∞2Vπβ\|sin(2πβiv)\| i1−v=0, (17) when 0 <v< 1. Equivalently, lim i→∞(Vi+1−Vi)=0, which implies that the potential Vivaries very slowly and can be safely taken as a constant locally when iis very large. In Fig. 4we plot the potential landscape of Vi, we can see the variation tendencies at the left and right ends are totallydifferent. At the left end where iis small, V i→Vcos(2πβi), so the topological quantum number QLchanges near V/prime= 2+2/Delta1[15]. However, at the right end where iis very large, the asymptotic property of “being constant” of Viis similar to that of the chemical potential of Kitaev’s p-wave model, so the topological quantum number QRchanges near V=2. Hence the topological quantum number QLandQRdo not exactly reﬂect the topological phase transition because the FIG. 4. The potential landscape of Vi=Vcos(2πβiv). The site number iranges from 1 to 10 000. 174207-4FATE OF TOPOLOGICAL STATES AND MOBILITY EDGES . . . PHYSICAL REVIEW B 96, 174207 (2017) -1.8 -1.5 -1.2 -1 0 1 1.2 1.5 1.810-410-2100IPR(,V)=(0.5,0.5) (,V)=(0.5,0.8) (,V)=(0.5,1) -1.8 -1.5 -1.2 0 1.2 1.5 1.8 Eigenenergy10-410-2100IPR(,V)=(0.6,0.5) (,V)=(0.6,1) (,V)=(0.6,1.5)(b)(a) FIG. 5. The distribution of IPR as a function of eigenenergy for various ( /Delta1,V ). “Black dotted lines” correspond to two turning points of IPR located at the mobility edges Ec1=± (2−V)a n d Ec2=± 2/Delta1, respectively. (a) When ( /Delta1,V )=(0.5,0.5) and (0 .5,0.8), Ec1=± 1.2,±1.5a n dEc2=± 2/Delta1=± 1 are located at the spectrum due to V< 2−2/Delta1=1, while when ( /Delta1,V )=(0.5,1), the mobility edges disappear at the spectrum due to V=2−2/Delta1=1. (b) When (/Delta1,V )=(0.6,0.5),Ec1=± 1.5a n d Ec2=± 1.2 are located at the spectrum due to V< 2−2/Delta1=0.8, while when ( /Delta1,V )=(0.6,1) and (0 .6,1.5), there are no mobility edges and all wave functions are localized due to V> 2−2/Delta1=0.8, however, the zero energy modes still exist. Therefore, when the strength of the incommensurate potentials is less than the threshold VL=2−2/Delta1, there exist four mobility edges located at Ec1=± (2−V)a n d Ec2=± 2/Delta1in the spectrum. The number of sites is set as L=5000. Majorana edge states at the left and right ends must appear and disappear by pairs. This may explain why the scatteringmethod fails. Consequently, although we haven’t found atopological invariant, we conjecture that the topological phasetransition point may be the same as the gap-closing pointV T=2. III. MOBILITY EDGES AND TOPOLOGICALLY NONTRIVIAL LOCALIZED PHASE Furthermore, to clarify the localized properties of this model we calculate the inverse participation ratio (IPR)[39–41], which is deﬁned as IPR n=L/summationdisplay j=1/parenleftbig u4 n,j+v4 n,j/parenrightbig , (18) for a normalized wave function. Here nis the energy level index, and un,j,vn,jare the solutions to BdG equations subject to the normalization condition/summationtext i(u2 n,i+v2 n,i)=1. The above deﬁnition can be thought of as an extension of IPR with /Delta1=0. It is well known that the IPR scales as L−1for an extended state. Hence it approaches 0 in the thermodynamic limit, butis ﬁnite for a localized state. Figure 5plots the IPR of the corresponding wave functions as a function of eigenenergy for various ( /Delta1,V ). We ﬁnd that as012345 1/L 10-310-410-310-210-1IPRE=1.1 E=1.3 E=1.4 E=1.6 E=1.1 E=1.3 E=1.4 E=1.6 FIG. 6. The ﬁnite size analysis of IPR corresponding to four typical eigenenergies E=1.1,1.3,1.4,1.6, when ( /Delta1,V )=(0.6,0.5) (blue)/(0 .6,1.5) (red), β=Fm−1/Fm,a n dL=Fm. the eigenenergy varies, the IPR suddenly jumps from the order of magnitude 10−2(a typical value for the localized states) to 10−4(a typical value for the extended states) or inversely at speciﬁc energies. This jumping phenomenon suggests thatthere exist mobility edges in the energy spectrum. We didcalculations for various ( /Delta1,V ) and found that these mobility edges are exactly located at E c1=± (2−V) andEc2=± 2/Delta1, respectively. For the mobility edges to exist there is an implicitcondition that 2 −V> 2/Delta1.I nF i g . 5it is clearly shown that when the strength of the slowly varying incommensuratepotentials is larger than the threshold V L=2−2/Delta1, there are no mobility edges in the spectrum. Remarkably, when V> V Lthe IPR of all wave functions are of the magnitude of 10−2, and none of them appears around 10−4, as shown in Fig. 5. Hence all wave functions are localized in this situation. To make this conclusion solid,we plot IPR as a function of the inverse of the system size(1/L)i nF i g . 6. Here we choose the total number of sites to be L=F mwhere Fmis themth Fibonacci number. The advantage of such choice is that the golden ratio can be approximated by (√ 5−1)/2=limm→∞Fm−1/Fm, which is a conventional practice in the ﬁnite size analysis of the quasiperiodic system.With the increasing of the system size, IPR approaches 0 forE=1.3,1.4 which are in the extended states when ( /Delta1,V )= (0.6,0.5), whereas is ﬁnite for other eigenenergies when (/Delta1,V )=(0.6,0.5) and ( /Delta1,V )=(0.6,1.5). Another intuitive tool to distinguish the extended, localized and critical wavefunctions is the proﬁle of wave functions in the semi-log plot[42,43]. A localized state has linearly decreasing wings in the semi-log plot, while the extended and critical states do not.Shown in Fig. 7, the probability density in Figs. 7(a),7(d), 7(e),7(f),7(g), and 7(h) indeed has linearly decreasing wings, which is consistent with the result of the ﬁnite size analysis.Therefore, the wave functions are indeed localized when V> V L. However, if VL<V <V T, there exists a region [ VL,VT] in which the energy gap does not close and the Majoranazero energy modes still exist as demonstrated in Sec. II.F o r the case ( /Delta1,V )=(0.3,1.9) shown in Fig. 1, although all 174207-5TONG LIU, HAI-Y ANG Y AN, AND HAO GUO PHYSICAL REVIEW B 96, 174207 (2017) 10-4010-20100P(a) 10-1010-5100P(b) 10-1010-5100P(c) 10-4010-20100P(d) 10-4010-20100P(e) 10-4010-20100P(f) 0 2000 4000 6000 site number10-4010-20100P(g) 0 2000 4000 6000 site number10-4010-20100P(h) FIG. 7. The probability density Pi=u2 i+v2 icorresponding to four typical eigenenergies E=1.1,1.3,1.4,1.6, when ( /Delta1,V )= (0.6,0.5) (blue)/(0 .6,1.5) (red), and L=6765. (a) E =1.1; (b) E =1.3; (c) E =1.4; (d) E =1.6; (e) E =1.1; (f) E =1.3; (g) E =1.4; and (h) E =1.6. wave functions are localized due to 1 .9>2−2/Delta1=1.4,φ andψwith the lowest excitation still split into two spatially separated Majorana edge states, therefore a novel topologicallynontrivial localized phase emerges here. We also choosedifferent sets of parameters to ensure that this novel phaseindeed exists. Figures 8and Fig. 9present the eigenstates corresponding to three different eigenenergies with ( /Delta1,V )=(0.5,0.5). In -0.200.2u(a) -0.0500.05v(b) -0.200.2u(c) -0.0200.02v(d) 01000 2000 3000 4000 5000 site number-0.100.1u(e) 01000 2000 3000 4000 5000 site number-0.0500.05v(f) FIG. 8. Eigenstates uandvnear the mobility edge Ec1=1.5, when/Delta1=0.5a n d V=0.5. Here we choose three typical eigenen- ergies (with four signiﬁcant digits): high energy localized state aboveE c1(a) and (b), critical state near Ec1(c) and (d), and low energy ex- tended state below Ec1(e) and (f). (a) E =1.5026 above edge; (b) E = 1.5026 above edge; (c) E =1.5015 near edge; (d) E =1.5015 near edge; (e) E =1.4975 below edge; and (f) E =1.4975 below edge.-0.100.1u(a) -0.100.1v(b) -0.100.1u(c) -0.100.1v(d) 01000 2000 3000 4000 5000 site number-0.100.1u(e) 01000 2000 3000 4000 5000 site number-0.200.2v(f) FIG. 9. Eigenstates uandvnear the mobility edge Ec2= 1.0, when /Delta1=0.5a n d V=0.5. Here we choose three typical eigenenergies (with four signiﬁcant digits): high energy extended state above Ec2(a) and (b), critical state near Ec2(c) and (d), and low energy localized state below Ec2(e) and (f). (a) E =1.0012 above edge; (b) E =1.0012 above edge; (c) E =1.0000 near edge; (d) E =1.0000 near edge; (e) E =0.9991 below edge; and (f) E =0.9991 below edge. Fig. 8the wave function is localized [Figs. 8(a) and 8(b)], critical [Figs. 8(c) and 8(d)], and extended [Figs. 8(e) and 8(f)], when the corresponding eigenenergy is above, near, and below the mobility edge Ec1=2−V=1.5, respectively. In Fig. 9, in contrast, the wave function is extended [Figs. 9(a) and9(b)], critical [Figs. 9(c)and9(d)], and localized [Figs. 9(e) and9(f)], when the corresponding eigenenergy is above, near, and below the mobility edge Ec2=2/Delta1=1, respectively. To strengthen our ﬁndings, we also calculate the density of states (DOS) D(E) and the Lyapunov exponent γ(E)o ft h i s system, which are deﬁned as [ 23] D(E)=L/summationdisplay n=1δ(E−En), γ(En)=1 L−1L/summationdisplay n/negationslash=mln\|En−Em\|. (19) HereEnis thenth eigenenergy. Since the Lyapunov exponent is the inverse of the localization length, then γ=0 for an extended state, whereas γ/negationslash=0 for a localized state. These two quantities are related to each other through the equation γ(E)=/integraldisplay dE/primeD(E/prime)l n\|E−E/prime\|. (20) In Fig. 10we present the behavior of DOS as a function of eigenenergy. Three different sets of parameters ( /Delta1,V )= (0.5,0.4), (0.5,0.6), and (0 .6,0.4) are chosen for not los- ing generality. The energy band consists of two subbandswhich are symmetric around E=0 due to the particle-hole symmetry. Obviously the DOS in our model is singular whilecrossing the mobility edge, and the change of the nature 174207-6FATE OF TOPOLOGICAL STATES AND MOBILITY EDGES . . . PHYSICAL REVIEW B 96, 174207 (2017) -1.8-1.6-1.4-1.2 -1 0 11.21.41.61.8 Eigenenergy00.0050.010.0150.020.0250.03DOS(,V)=(0.5,0.4) (,V)=(0.5,0.6) (,V)=(0.6,0.4) FIG. 10. DOS as a function of eigenenergy with three different sets of parameters ( /Delta1,V )=(0.5,0.4), (0 .5,0.6), and (0 .6,0.4). Obviously a dramatic change occurs when the eigenenergy passesthrough the mobility edges E c1=± (2−V)a n dEc2=± 2/Delta1,w h i c h are in accordance with the IPR predictions. of the eigenstates can be reﬂected by the singularity of the DOS [ 22,23]. Therefore two sharp peaks in both subbands shown in Fig. 10indicate the extended state-localized state transition corresponding to two mobility edges located atE c1=± (2−V) and Ec2=± 2/Delta1.I nF i g . 11we plot the Lyapunov exponent by plugging in the same sets of parametersas in Fig. 10. It also exhibits a singular behavior at the mobility edge. The implications from the numerical resultsare in excellent agreement with those from the IPR and DOS.We also try other sets of parameters and obtain the same resultsas expected. -1.8-1.6-1.4-1.2 -1 0 11.21.41.61.8 Eigenenergy00.050.10.150.20.250.30.35(E)(,V)=(0.5,0.4) (,V)=(0.5,0.6) (,V)=(0.6,0.4) FIG. 11. The Lyapunov exponent γ(E) vs eigenenergy with three different sets of parameters ( /Delta1,V )=(0.5,0.4), (0 .5,0.6), and (0 .6,0.4). When the eigenenergy is located in the intervals [V−2,−2/Delta1]a n d[ 2 /Delta1,2−V],γ(E)→0, indicating that the corresponding state is extended. Otherwise γ(E) is ﬁnite, indicating that the corresponding state is localized.To understand where the mobility edges take place, we provide a possible explanation here. It is a challenge tounderstand how the disorder induces a pronounced transitionfrom a superconducting into an insulating state. One route tothe insulating phase is the direct localization of Cooper pairs,another is that the Cooper pairs are ﬁrst destroyed followed bythe standard localization of single electrons. Since the p-wave pairing amplitude /Delta1is a real constant, the Cooper pairs could be destroyed by the energy >2/Delta1. When the energy <2/Delta1,w e conjecture that the Cooper pairs are directly localized, whichcorresponds to the mobility edge E c2=2/Delta1. When the energy >2/Delta1, the Cooper pairs are ﬁrst destroyed to single electrons, then it is reduced to a single electron localized problem [ 23], which corresponds to the mobility edge Ec1=2−V. Another interesting subject is the speciﬁc form of the critical behavior of the Lyapunov exponent at the mobilityedgeE c1andEc2. In the localized regions of energy spectrum, we have γ(E)∼\|E−Eci\|θ,i=1,2. (21) Similarly, the density of states at the mobility edge behaves like D(E)∼\|E−Eci\|−δ,i=1,2. (22) The critical exponents θandδare related by the equation θ+δ=1. (23) In Fig. 11the singular behaviors of γ(E) are identiﬁed to be linear with Ein the localized region, indicating that θ=1 andδ=0 accordingly. These results are the same as those of the single-particle model [ 23], and we ﬁnd that the parameters V,/Delta1,β, andvare all irrelevant with regard to the critical exponents θandδ. In addition, by varying the parameters, we also ﬁnd that the four mobility edges depends on Vand/Delta1but are irrelevant to βandv. IV . CONCLUSIONS In summary, we study the interplay between the disorder with mobility edges and the p-wave superconducting pairing. With regard to the questions raised in the introduction, we ﬁndfollowing interesting features of this model. (1) Increasing the strength Vof slowly varying incommen- surate potentials can destroy the topological SC phase anddrive the system into a topologically trivial localized phase.The gap-closing point occurs at V T=2. Although we have not found a topological quantum number to denote the topologicalphase transition, we conjecture that it occurs at V T=2 too. (2) There exist four mobility edges located at Ec1=± (2− V) andEc2=± 2/Delta1in the spectrum when the strength of the incommensurate potentials is less than a threshold VL=2− 2/Delta1, otherwise all wave functions are localized. Hence there is a region marking the topologically nontrivial localized phasebetween V LandVT. To the best of our knowledge it has never been proposed in the 1D TSC system yet. We veriﬁed ourpredictions by utilizing several typical numerical techniques, 174207-7TONG LIU, HAI-Y ANG Y AN, AND HAO GUO PHYSICAL REVIEW B 96, 174207 (2017) and all results are consistent with one another. We believe that the interesting features of this model will shed light on a widerange of topological and disordered systems. Finally, we would like to point out that Anderson localiza- tion in disordered systems has been studied extensively in ul-tracold atomic experiments, both for the speckle disorder case[42] and the quasiperiodic disorder case [ 43] in a controlled artiﬁcial method. Experimentally determining the mobilityedge trajectory have been realized in a speckle disorder systemwith sufﬁciently high energy resolution [ 44–46]. It is also possible to induce directly superﬂuid p-wave pairing by using a Raman laser in proximity to a molecular BEC [ 47,48]. These signiﬁcant advances in ultracold atomic systems provide apotential way to experimentally study the interplay between mobility edges and the p-wave superconductor (superﬂuid). Thus we expect that these novel features including mobil-ity edges and the topologically nontrivial localized phasediscovered in this model can be realized experimentally inthe ultracold atomic system. 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PhysRevB.98.054515.pdf	PHYSICAL REVIEW B 98, 054515 (2018) Competition of electron-phonon mediated superconductivity and Stoner magnetism on a ﬂat band Risto Ojajärvi,1Timo Hyart,1,2Mihail A. Silaev,1and Tero T. Heikkilä1 1Department of Physics and Nanoscience Center, University of Jyvaskyla, P .O. Box 35 (YFL), FI-40014 Jyvaskyla, Finland 2Institut für Theoretische Physik, Universität Leipzig, D-04103 Leipzig, Germany (Received 7 March 2018; revised manuscript received 21 June 2018; published 22 August 2018) The effective attractive interaction between electrons, mediated by electron-phonon coupling, is a well- established mechanism of conventional superconductivity. In metals exhibiting a Fermi surface, the criticaltemperature of superconductivity is exponentially smaller than the characteristic phonon energy. Therefore, suchsuperconductors are found only at temperatures below a few kelvin. Systems with ﬂat energy bands have beensuggested to cure the problem and provide a route to room-temperature superconductivity, but previous studiesare limited to only BCS models with an effective attractive interaction. Here we generalize Eliashberg’s theoryof strong-coupling superconductivity to systems with ﬂat bands and relate the mean-ﬁeld critical temperature tothe microscopic parameters describing electron-phonon and electron-electron interaction. We also analyze thestrong-coupling corrections to the BCS results and construct the phase diagram exhibiting superconductivityand magnetic phases on an equal footing. Our results are especially relevant for novel quantum materials whereelectronic dispersion and interaction strength are controllable. DOI: 10.1103/PhysRevB.98.054515 I. INTRODUCTION The overarching idea in quantum materials is to design the electronic (or optical, magnetic, etc.) properties of materialsto perform the desired functionality [ 1]. This goal is aided by generic models and concepts, such as speciﬁc lattice modelsthat lead to certain topological phases. Often the studiedmodels and the resulting topological phases for electronicsystems are noninteracting and do not include the possibilityof spontaneous symmetry breaking. However, such noninter-acting models are platforms for exotic electron dispersionsthat provide a basis for studying symmetry-broken interactingphases. In particular, certain models support approximate ﬂatbands [ 2–10], and here we consider microscopic mechanisms for symmetry-breaking phases in such systems. We analyze the interplay of electron-phonon [ 11] and (screened) electron-electron interaction in providing meansfor a symmetry-broken phase transition, thereby couplingtogether works on ﬂat-band superconductivity [ 2,7,10,12] with those on ﬂat-band (Stoner) magnetism [ 9,13–17]. In both cases the resulting mean-ﬁeld critical temperature is linearlyproportional to the coupling constant [ 18], thus allowing for a very high critical temperature. The two types of inter-action mechanisms work in opposite directions and, in thecase of weak interactions, in a symmetric way. However,upon increasing the coupling strength the retarded natureof the electron-phonon interaction shows up (as opposedto the instantaneous electron-electron interaction), breakingthe symmetry between the two. In particular, we generalizeEliashberg’s strong-coupling theory of superconductivity [ 19], usually formulated for systems with a Fermi surface, forﬂat bands. As a result, we describe the dimensionless BCSattractive interaction [ 20] in terms of the electron-phonon coupling and the characteristic phonon frequency [Eq. ( 8)]. In addition, we provide the generalization of the well-knownMcMillan formula of strong-coupling superconductivity (forFermi surface systems) [ 21] to the case with ﬂat bands in Eq. ( 14). In addition to superconductivity, we consider ﬂat-band Stoner magnetism. Because of the retarded nature of theelectron-phonon interaction, the combined interaction cansimultaneously have attractive and repulsive components, andthus the system can be unstable with respect to both singlet su-perconductivity and magnetism (see a generic strong-couplingphase diagram in Fig. 1). Often one of the phases still dominates and suppresses the other, but we ﬁnd that when the criticaltemperatures of the phases are similar, both phases are localminima of the free energy at low temperatures. We ﬁnd thattheir bulk coexistence and the resulting odd-frequency tripletsuperconducting order [ 22,23] are only realized as an unstable solution. On the other hand, these phases can form metastabledomains inside the sample, and therefore an odd-frequencytriplet order parameter can appear at the domain walls. The structure of this paper is as follows. In Sec. IIwe introduce the model of surface bands with electron-phononand Coulomb interactions. In Sec. IIIwe formulate the Eliashberg model extension for the surface bands, describeall possible ordered states that can appear within this model,and calculate the critical temperatures of the superconductingand antiferromagnetic states. We study the competition andpossible coexistence of these two types of ordering in Sec. IV. Conclusions are given in Sec. V. II. MODEL As a low-energy model for the ﬂat band, we assume two sublattices coupled through an electronic Hamiltonian [ 3] Hel,p=/parenleftbigg0εp εp 0/parenrightbigg ,withεp=ε0/parenleftbiggp pFB/parenrightbiggN , (1) 2469-9950/2018/98(5)/054515(8) 054515-1 ©2018 American Physical SocietyOJAJÄRVI, HYART, SILAEV , AND HEIKKILÄ PHYSICAL REVIEW B 98, 054515 (2018) FIG. 1. Strong-coupling phase diagram for ﬂat-band systems as a function of electron-phonon attraction λfor electron-electron repulsion u=0.5ωE[Eq. ( 8)].TCE Cis the temperature at which the TC’s of magnetic and superconducting order coincide. In the striped region these phases can form metastable domains inside the sample. This diagram is for N→∞ . For ﬁnite Nthe overlap region between the phases is smaller. where an integer Nparametrizes the ﬂatness of the dispersion, andε0is the energy at p=pFB. The model is electron-hole symmetric and the two energy bands have the dispersions ±εp. For large N, the states with low momenta, \|p\|<p FB,a r ea l m o s t at zero energy and the density of states is very high. Thestates with momenta larger than p FBdo not contribute much to the momentum integrals due to their low density of states.Therefore, the results for large Ndo not depend much on the momentum cutoff, as long as it is larger than p FB. In our model we take the cutoff to inﬁnity and consider only the cases N> 2. This is in contrast to models with isolated ﬂat bands extendingthroughout the Brillouin zone. The effects discussed below inthe case of large Nare mostly applicable also to such models (provided they have the type of sublattice degree of freedomdiscussed below), as long as p FBis taken as the size of the Brillouin zone. Equation ( 1) is approximately realized for the surface states of N-layer rhombohedrally stacked graphite. In that system the surface states delocalize into the bulk at theedges of the ﬂat band and this gives a momentum-dependentcorrection in the low-energy Hamiltonian [ 12,24]. In the case ofN→∞ the delocalization of the surface states to the bulk leads to strong amplitude mode ﬂuctuations invalidating themean-ﬁeld theory [ 24]. Therefore, the theory considered in this paper is applicable to rhombohedral graphite only in thecase where Nis not too large. We model the electron-electron interaction as a repul- sive on-site Hubbard interaction [ 25] with energy U.T h e magnitude of Udepends on the microscopic details of the system and its environment. The coupling between electronsand phonons, with strength g, creates an effective attraction between the electrons and makes the system susceptible tosuperconductivity [ 19]. We mostly consider Einstein phonons with constant energy ω q=ωEand discuss generalizations in the Supplemental Material [ 26].The total Hamiltonian incorporating these effects is H=/summationdisplay p,σ/Psi1† pσHel,p/Psi1p,σ+/summationdisplay q,ρωqb† q,ρbq,ρ +U 2N/summationdisplay p,k,q ρ,σ,σ /primeψ† p+q,σρψ† k−q,σ/primeρψk,σ/primeρψp,σρ +g√ N/summationdisplay p,q,σ,ρ(b† −q,ρ+bq,ρ)ψ† p+q,σρψp,σρ, (2) where Nis the number of lattice points in the system and /Psi1† pσ=(ψ† pσA,ψ† pσB) is a pseudospinor in sublattice space. We assume that the low-energy states on the two sublattices ρ= A/B are spatially separated (e.g., localized on the two surfaces in rhombohedral graphite), so that neither the electron-electroninteractions nor the phonons couple them. The only couplingbetween the sublattices comes from the off-diagonal dispersionrelation. In the Supplemental Material we also show thatthe ﬂat-band phenomenology applies to linear, graphenelikedispersion with an electronic Hamiltonian H el,p=vF/parenleftbigg0 px−ipy px+ipy 0/parenrightbigg , (1/prime) and with an energy cutoff εcand Fermi velocity vF, provided the interaction energy scales are large compared to εc. Hence, our results may also apply as an effective model for twistedbilayer graphene close to its “magic” angles [ 30]. In the theory of electron-phonon superconductivity of met- als, the neglect of higher-order diagrams in the perturbationtheory is typically justiﬁed with the help of the Migdal theorem[31]. In that case, the expansion parameter gets an additional factor of ω E/EF, where EFis the Fermi energy. Because of the Migdal theorem, the theory of superconductivity for metalsis not strictly limited to weak coupling with respect to theinteraction parameter. In the ﬂat band, however, the chemical potential is located at the bottom of the band and there is no Fermi energy with whichto compare the Debye energy. Migdal’s theorem cannot be usedin this case. In the intermediate case of narrow electronic bands,corrections in the higher orders of the adiabatic parameterω E/EFhave been studied in Refs. [ 32–35] and the Eliashberg theory has been found also to be in agreement with MonteCarlo results in the weak-coupling regime when ω E/EF=1 in Ref. [ 36]. We ﬁnd that the diagrams beyond the mean-ﬁeld approximation do not inﬂuence the self-energies signiﬁcantlyif the effective pairing constant introduced below in Eq. ( 8)i s small, λ/lessmuch1, and ω E,u/lessmuchε0. Moreover, although the mean- ﬁeld theory is applied beyond its formal limits of validity inthe strong-coupling regime, this theory captures the interestingpossibility that the retarded nature of the electron-phononinteraction can lead to the presence of attractive and repulsivecomponents at the same time. As a result, the system canbe simultaneously unstable with respect to the appearance ofboth singlet superconductivity and magnetism as discussed inSec. IV. 054515-2COMPETITION OF ELECTRON-PHONON MEDIATED … PHYSICAL REVIEW B 98, 054515 (2018) FIG. 2. Quasiparticle dispersions E(p) for different kinds of symmetry breakings with N=5. (a) In the noninteracting case, the spin bands are degenerate with E(p)=±ε(p). (b) For the ferromagnetic (FM) or the superconducting (SC) phase with a θ=π phase shift between the sublattices, one quasiparticle band is shifted up and the other down in energy. In this case, no energy gap is opened.(c) For the antiferromagnetic (AFM) or the SC phase with θ=0a n energy gap is opened and quasiparticle bands are doubly degenerate. III. ORDERED STATES Hamiltonian ( 2) allows for a number of spontaneous symmetry-breaking phases. We restrict our study to spatiallyhomogeneous phases. Therefore, the order parameter canappear in the spin, sublattice (pseudospin), and electron-hole(Nambu) spaces. The general self-energy is /Sigma1(iω n)=3/summationdisplay i,j,k=0/Sigma1ijk(iωn)τiσjρk, (3) where τi,σj, andρkare the Pauli matrices in electron-hole, spin, and sublattice spaces, respectively. We characterize thedifferent components /Sigma1 ijkand determine their values within the self-consistent Hartree-Fock model. This reduces to solv-ing a set of nonlinear integral equations, known as Eliashbergequations in the context of conventional superconductors. To explore the possible phases of the system, we ﬁrst assume that the U(1) gauge symmetry is broken, but the SU(2) spin-rotation symmetry is not. After ﬁxing the overall phaseof the superconducting order parameter, we are left with theself-energy /Sigma1 000(iωn) and three degrees of freedom for the su- perconducting singlet order parameter: the magnitudes of theorder parameter on the sublattices /Delta1 Aand/Delta1Band the relative phase θ. Choosing θ=0 leads to a gapped quasiparticle dispersion [Fig. 2(c)], whereas θ=πwould imply a gapless dispersion [Fig. 2(b)]. Thus, in the case of an instantaneous interaction the total energy is minimized when θ=0 and /Delta1A=/Delta1B. Generalizing the above to the frequency-dependent interactions, we choose the singlet to be proportional to theτ2σ2ρ0component, whose magnitude and the functional form are obtained from the self-consistency equation. The self-energy for the fermionic Matsubara frequency ω nis /Sigma1SC(iωn)=−i/Sigma1ω n1+φnτ2σ2, (4) where /Sigma1ω n=(1−Zn)ωnis the frequency renormalization by the retarded interaction [ 19]. To simplify the equations, we deﬁne renormalized frequencies ˜ ωn=Znωn.W eu s et h e symbol φnfor the “bare” singlet order parameter and /Delta1for the maximum value of the renormalized singlet order parameter/Delta1 n≡φn/Znrelated to the energy gap. When SU(2) spin-rotation symmetry is broken but U(1) gauge symmetry is not, the self-energies describe the frequency renormalization and the magnetization. After ﬁxing the di- rection of the magnetization on one sublattice, the relevantdegrees of freedom are reduced to three similarly as in thesuperconducting case. These can be chosen as the magnitudesof the magnetizations in the two sublattices h AandhBand the relative angle ϕbetween their directions. The quasiparticle dispersion in the magnetic case is the same as in the supercon-ducting case if we identify /Delta1 A,B=hA,Bandθ=π−ϕ(see Fig.2). In this case, the relative angle ϕ=0 leads to a gapless quasiparticle dispersion [Fig. 2(b)], and ϕ=πto a gapped dispersion [Fig. 2(c)]. Thus, the energy minimum is obtained withhA=hBandϕ=π. The stable magnetization is hence antiferromagnetic, with opposite magnetizations on the twosublattices, so that the self-energy is /Sigma1 AFM(iωn)=−i/Sigma1ω n1+hnτ3σ3ρ3, (5) where hnis the frequency-dependent exchange ﬁeld. This result agrees with density functional theory (DFT) studies onrhombohedral graphite [ 37], and similar magnetization struc- ture has been predicted also in the case of ﬂat bands appearingat the zigzag edges of graphene nanoribbons [ 38–40]. We also note that the AFM state is insulating [see Fig. 2(c)]. If the noninteracting dispersion is completely ﬂat at zero energy, thesublattices are uncoupled and the antiferromagnetic state isdegenerate with the ferromagnetic ϕ=0 state. By calculating the Hartree-Fock self-energies, we ﬁnd the self-consistency equations, from which we can determine thevalues of the self-energy terms. For the superconducting (SC)self-energy ( 4), they are φ n=2T∞/summationdisplay m=−∞(λnm−u)/integraldisplay∞ 0dp p p2 FBφm ˜ω2m+ε2p+φ2m, (6) Zn=1+2T∞/summationdisplay m=−∞λnmωm ωn/integraldisplay∞ 0dp p p2 FBZm ˜ω2m+ε2p+φ2m,(7) where the interaction kernel is λnm= λω3 E/[ω2 E+(ωn−ωm)2]. The functional form of the interaction kernel is determined by the phonon propagatorfrom which it is derived. The width in frequency space isdetermined by the characteristic phonon frequency, which inthis case is the Einstein frequency ω E. The effective interaction constants in the ﬂat band are λ=g2 ω2 E/Omega1FB /Omega1BZ,u=U/Omega1FB /Omega1BZ, (8) 054515-3OJAJÄRVI, HYART, SILAEV , AND HEIKKILÄ PHYSICAL REVIEW B 98, 054515 (2018) where/Omega1FBand/Omega1BZare the momentum-space areas of the ﬂat band and of the ﬁrst Brillouin zone, respectively. For an antiferromagnet with self-energy ( 5), the self- consistency equations are hn=2T∞/summationdisplay m=−∞(u−λnm)/integraldisplay∞ 0dp p p2 FBhm ˜ω2m+ε2p+h2m,(9) Zn=1+2T∞/summationdisplay m=−∞λnmωm ωn/integraldisplay∞ 0dp p p2 FBZm ˜ω2m+ε2p+h2m.(10) Superconductivity and magnetism are thus symmetric with each other also on the level of the self-consistency equations,but with the roles of uandλ nmswitched. Tovmasyan et al. have shown that this duality is also broken by taking into accounthigher-order terms in the perturbation theory [ 41]. To solve the self-consistency equations ( 6)–(10), we trun- cate the Matsubara sums with a cutoff ω C∼10ωE. This causes no numerical error if we use the pseudopotential trick andsimultaneously replace uwith an effective value u ∗, which depends on the cutoff [ 42]. For superconductivity (magnetism), cutting off high-energy scatterings is compensated by a reduc-tion (increase) in the low-energy effective interaction. After the pseudopotential trick, the solutions are found by a ﬁxed-point iteration. The iteration is continued until all ofthe components have converged. The ﬁxed-point method onlyﬁnds the stable solutions; to ﬁnd the unstable solutions, weused a solver based on Newton’s method. The number of parameters in Eqs. ( 6)–(10) can be reduced by deﬁning new interaction constants ˜λ≡λ(ω E/ε0)2/Nand ˜u=uω2/N−1 E/ε2/N, so that one parameter is eliminated com- pletely and the results become proportional to ωE. For weak coupling, λ/lessmuch1, the frequency dependence of λnmcan be disregarded and we can approximate Z≈1 and /Delta1≈φ. Assuming λωE>u, the superconducting gap at T=0 and the critical temperature are /Delta10 ωE=1 2/bracketleftBigg (˜λ−˜u)√π/Gamma1/parenleftbig1 2−1 N/parenrightbig Nsin/parenleftbigπ N/parenrightbig /Gamma1/parenleftbig 1−1 N/parenrightbig/bracketrightBiggN N−2 , (11) Tsc C ωE=1 2π/bracketleftBigg (˜λ−˜u)ζ/parenleftbig 2−2 N/parenrightbig/parenleftbig 22−2 N−1/parenrightbig Nsin/parenleftbigπ N/parenrightbig/bracketrightBiggN N−2 . (12) These results are valid for N> 2 as the momentum integrals diverge without a cutoff for N/lessorequalslant2. Note that the T=0 limit can thus be taken before the ﬂat-band limit of large N. Analogous results have been obtained before within the BCSmodel in Ref. [ 12]. For large N,/Delta1 0is linear in the coupling and its magnitude is proportional to the phonon energy scale.Hence the associated critical temperature can be very large.Relabeling /Delta1 0→h0and˜λ↔˜u, we ﬁnd similar equations for magnetism. Here h0is the magnetic order parameter at T=0. At strong coupling, the retardation matters and the results for magnetism and superconductivity diverge from each other.For superconductivity, we can improve on the weak-couplingresult by including some of the corrections from the Eliashbergtheory when N→∞ . We still neglect the full frequency dependence, but we include the electron mass renormalizationas a static factor Z 0=1+λ. The order parameter at zeroFIG. 3. Critical temperatures for superconducting and magnetic phases for N→∞ . (a) Superconductivity is suppressed when λ/lessorsimilar u/ω E. Above the critical point λC(u),Tsc Cis linear in λ. With increasing λ, the electron-phonon renormalization increases and this limits the critical temperature. The dashed line is the approximation in Eq. ( 14). (b) Critical interaction strength for superconductivity as a function of u.W h e n λ<λ C(u), superconductivity is suppressed. The dashed line is the instantaneous approximation. (c) Magnetism is suppressed when u/ω E/lessorsimilarλ. Above the critical point uC(λ),Tm Cis linear in u. (d) Critical interaction strength for magnetism as a function of electron-phonon interaction. When u<u C(λ), magnetism is sup- pressed. The dashed line is the instantaneous approximation. In thisﬁgure, we do not take into account the possible magnetic instability of the superconducting state, or vice versa. temperature becomes /Delta10=λωE−u 2(1+2λ). (13) In metals with a Fermi surface [ 43], the electron-phonon interaction renormalizes the pairing potential with the factor of1+λinstead of 1 +2λas in Eq. ( 13). Thus, for weak coupling, the electron-phonon renormalization is more effective in theﬂat band than in the usual metals. This difference is morepronounced at strong coupling, as we see next. By linearizing Eqs. ( 6) and ( 7) with respect to φ, we can solve for the critical temperature [see Fig. 3(a)]. We ﬁnd that whenN→∞ , the critical temperature scales as T sc C∝λ0.2ωE for large λ. In metals [ 43] the asymptotic scaling goes as Tsc C∝ λ1/2ωE. When u/negationslash=0, there is a critical point λCsuch that for λ<λ C there is no superconducting transition at any temperature. For smallu/ωE,λCis linearly proportional to the Coulomb inter- action. For large u,λCincreases sublinearly [see Fig. 3(b)]. 054515-4COMPETITION OF ELECTRON-PHONON MEDIATED … PHYSICAL REVIEW B 98, 054515 (2018) FIG. 4. Effect of ﬁnite Non critical temperature when u=0. For small ˜λ, the results coincide with the instantaneous approximation of Eq. ( 12) (shown with the dashed lines). For large ˜λ, the strong- coupling corrections limit the increase in Tsc C. An approximate numerical equation for Tsc Cis Tsc C=λωE−u(1−0.3u/ωE) 4(1+2.6λ0.8). (14) This is a ﬂat-band analog of the McMillan equation [ 21], which for the conventional superconductors incorporates theEliashberg and Coulomb corrections to T sc C.T h eu2term in the numerator accounts for the retardation correction to λCas in Fig.3(b). The form of the denominator is chosen to show the λ0.2power-law behavior for large λ. The factor 2.6 is obtained by a ﬁt in the region λ< 1f o ru=0. The ﬁt is shown as the dashed line in Fig. 3(a). The ratio /Delta10/Tsc Cis not constant, but depends on both N andλ.F o rN→∞ , the ratio has the value 2 for weak coupling and increases as λincreases. For λ=1 the ratio is 2.56. For the critical temperature at ﬁnite N, see Fig. 4. The phenomenology of the magnetism can be understood as follows. According to the Stoner criterion, the magnetization isrelated to the competition between the exchange energy gainand the kinetic energy penalty from moving electrons fromone spin band to another. For a ﬂat band with N→∞ , there is no kinetic energy penalty, and at zero temperature withλ=0 even a small exchange interaction leads to a complete magnetization of the ﬂat band. In the presence of the electron-phonon interaction the competition is between the exchangeenergy gain and the electron-phonon energy penalty, whichcoincide at u=u C. If we can neglect the retardation, the total interaction in Eq. ( 9)i su−λωE. The ﬂat band is completely magnetized when u>u C≈λωE. Due to retardation, for large λthe critical point is reduced from the linear estimate [see Fig.3(d)]. Above, we have discussed the superconducting order pa- rameter φ. The other important property of the superconduct- ing state is the existence of a supercurrent. In the ﬂat band theelectronic group velocity vanishes and it is not immediatelyclear that there can be a ﬁnite supercurrent. However, the ﬂat-band surface states of superconducting rhombohedral graphitedo support a ﬁnite supercurrent [ 44] and similarly it is known that quantum Hall pseudospin ferromagnets can support aﬁnite pseudospin supercurrent [ 16]. More generally, Peotta and Törmä [ 7] have shown that for a topological ﬂat bandFIG. 5. Mean-ﬁeld phase diagram for N=∞ obtained by deter- mining the line on which the critical temperatures for superconductiv- ity and antiferromagnetism are equal. The thin dashed line shows thephase boundary λ=u/ω Ein the case of instantaneous interactions. When the energy scales of interactions are small compared to ωEwe recover the BCS results. The phase diagram for ﬁnite Nlooks similar but the retardation effects are weaker, so that the deviation from the BCS approximation is smaller. there is an additional geometric contribution to the superﬂuid weight so that the critical current is ﬁnite. As we have notﬁxed the underlying topology in our model, it can be appliedto topologically nontrivial ﬂat bands. As one can see, the Eliashberg model describes the nucle- ation of both the magnetic and superconducting phases whichcan have rather close critical temperatures as shown in Fig. 3. In the next section we consider the nonlinear problem bycalculating the entire phase diagram of the ordered states tostudy the competition and the possible coexistence betweenthe superconductivity and antiferromagnetism. IV . COMPETITION BETWEEN THE PHASES If the electron-phonon interaction is approximated as in- stantaneous, we can sum the two interactions together andhave either a total interaction, which makes the normal stateunstable to the superconducting transition ( λω E−u> 0) or to the magnetic transition ( λωE−u< 0), but not to both at once. On the other hand, if the electron-phonon interactionis retarded, the situation is different, as the total interactioncan be attractive for low frequencies but repulsive for highfrequencies. There is then a parameter range in which bothphases are local minima of the free energy. This occurs whenλis large enough to overcome the suppressing effect of uin the case of superconductivity [ λ>λ C(u)i nF i g . 3(b)] ,b u ta tt h e same time uis large enough to overcome the suppressing effect ofλand create a magnetic instability [ u>u C(λ)i nF i g . 3(d)]. We study the phase diagram of the system by determining the state with a higher critical temperature as a function of uand λ(Fig. 5). The phase diagram is almost symmetric with respect to SC and AFM phases except that the lack of retardation in 054515-5OJAJÄRVI, HYART, SILAEV , AND HEIKKILÄ PHYSICAL REVIEW B 98, 054515 (2018) electron-electron repulsion favors the AFM phase for strong coupling. Even if there is a parameter region in T,u, andλwhere both SC and AFM self-consistency equations have a ﬁnitesolution, it does not mean that both phases are necessarilysimultaneously present. To determine the stability, we con-struct the coupled self-consistency equations in the case whenboth order parameters are nonzero and interact with each other[26]. By linearizing the coupled self-consistency equation with respect to SC, and solving the AFM part fully, the stabilityof the AFM phase with respect to the SC transition can bedetermined, and vice versa. Figure 1shows the region in λ-T space with ﬁxed u, where the two phases are stable. The ﬁgure shows that in the region where SC is dominant, the AFMphase is unstable near the expected second-order transition (thesolid line between the magnetic and paramagnetic phases) butbecomes a local minimum of free energy at lower temperatures.The same happens for superconductivity when the AFM phasedominates. The transition between SC and AFM phases is ofthe ﬁrst order. When discussing superconductivity in the presence of an exchange ﬁeld (either induced or spontaneous), we have anadditional ingredient in the self-energy, namely, the supercon-ducting triplet order parameter [ 22,45], which has been dis- cussed in the context of the Eliashberg model in Ref. [ 46]. The triplet is spatially isotropic, and in order to satisfy the fermionicantisymmetry, it has to be odd in frequency. It is generated in theself-energy only when there is an odd-frequency componentin the interaction. In the retarded interaction, this is alwayssatisﬁed. When calculating the stability of the AFM phase withrespect to SC, the triplet appears in the linear order. It hencemodiﬁes the boundaries of the region where both AFM andSC phases are stable. We have taken this effect into account inFig.1. Besides the competition between AFM and SC phases, we need to consider the possibility of a coexistence phase inthe dashed region of Fig. 1, where both phases can show up alone. We indeed have numerically found such a coexistencesolution, but tests based on ﬁxed-point iteration revealed it tobe unstable at every temperature that we checked. This ﬁndingis in accordance with a simpliﬁed model where both interactionchannels are instantaneous and independent of each other [ 26]. However, the fact that the two phases are simultaneously local minima of the free energy suggests that this systemcould have domains of antiferromagnetic order coexisting withsuperconducting domains. Such domains would be separatedby a domain wall mixing the two kinds of phases and inducingodd-frequency triplet pairing, as schematically illustrated inFig.6. In addition to providing a mechanism for the appearance of odd-frequency triplet pairing, the domain walls can supportinteresting excitations. In particular, it is known that ﬂat-band ferromagnets can support interesting topological anddomain-wall excitations in the form of different kinds of spintextures [ 16,47], and various combinations of spin textures and superconductivity may lead to the appearance of Majoranazero modes [ 48–52]. Also, alternatively to the intrinsic domain structure generation, the ferromagnetic superconductors cansupport different types of nonuniform magnetic order andspontaneous vortex states [ 53–55]. A detailed analysis of different possibilities goes beyond the scope of this paper.FIG. 6. Sketch of a domain wall between magnetic (red) and su- perconducting (blue) domains. At the domain wall a triplet component(purple) is induced. V . CONCLUSIONS We have proposed a simpliﬁed model of a ﬂat-band system with a retarded electron-phonon interaction and a repulsiveHubbard interaction. For this model, we have determined theself-consistency equations in the Hartree-Fock approximationand all the possible homogeneous phases. Antiferromagnetismand superconductivity are essentially symmetric in this system,with the only difference coming from the retardation ofthe electron-phonon interaction. For large λ, the retardation suppresses the increase in /Delta1more effectively in a ﬂat band than in metals with a Fermi surface. We ﬁnd that the retardationalso creates a situation in which both phases are separatelylocal minima of the free energy, suggesting a possibility ofcoexisting antiferromagnetic and superconducting domainsinside the sample. Our results indicate how ﬂat-band superconductivity can be generated from electron-phonon interaction and providesmeans to estimate the mean-ﬁeld critical temperature whenthe details of the electron-phonon coupling and the screenedinteraction are known. The superﬂuid transition in low-dimensional systems occurs in the form of a Berezinskii-Kosterlitz-Thouless (BKT) transition at a temperature that islower than the mean-ﬁeld transition temperature. That thelatter is nonzero is ensured by the possibility of having anonvanishing supercurrent (see, for example, Refs. [ 7,10,44]) in a ﬂat-band superconductor. Our results are of relevance indesigning novel types of quantum materials for the interplayof superconducting and magnetic order, and the search forsystems exhibiting exotic superconductivity with a very highcritical temperature, up to room temperature. They may alsoshed light on recent evidence of high-temperature supercon-ductivity in graphite interfaces [ 56]. Our results could also explain some of the phenomena associated with the recent experiments on bilayer graphene[30,57]. (For a more microscopic description of that case within the BCS model, see Refs. [ 58,59].) In the experiment, the twist angle between two superimposed graphene layers is chosento a certain magic angle, so that the two Dirac cones in thegraphene layers hybridize, forming a pair of ﬂat bands. Ourmodel can be adjusted to describe this situation with smallchanges (see the Supplemental Material [ 26] for details). When the chemical potential was tuned to the lower of these bands,the system became an insulator. From our point of view, thiscould be the insulating AFM state we describe. When thechemical potential is tuned slightly off from the ﬂat band, asuperconducting dome in the T-μphase diagram was observed on both sides. These domes can be the s-wave SC phases 054515-6COMPETITION OF ELECTRON-PHONON MEDIATED … PHYSICAL REVIEW B 98, 054515 (2018) we describe here. The competition between the particle-hole (AFM) and the particle-particle (SC) channels in the presenceof the chemical potential was considered by Löthman andBlack-Schaffer in Ref. [ 8], and for a range of parameters, they reproduce a similar phase diagram near the ﬂat band, with theAFM state at the level of the ﬂat band and two superconductingdomes with doping away from the ﬂat band (see Fig. 2(b) in Ref. [ 8]). In the experiments, SC domes are only observed on the hole-doped side. The electron-doped side exhibits onlyinsulating behavior near the ﬂat band. One possible explanationis the difference in screening, which changes the relativemagnitude of the repulsive and attractive interactions, so thatthe AFM state covers the SC domes completely. However, we leave the detailed treatment of the effects of doping andscreening (both intrinsic and that provided by the environment)for further work. ACKNOWLEDGMENTS We thank Sebastiano Peotta, Long Liang, and Päivi Törmä for helpful comments. This project was supported by theAcademy of Finland Key Funding (Project No. 305256),Center of Excellence (Project No. 284594), and ResearchFellow (Project No. 297439) programs. 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PhysRevB.103.024104.pdf	PHYSICAL REVIEW B 103, 024104 (2021) Origin of intense blue-green emission in SrTiO 3thin ﬁlms with implanted nitrogen ions: An investigation by synchrotron-based experimental techniques Vishnu Kumar ,1,Anuradha Bhogra,2Manju Bala,1S. C. Haw,3C. L. Chen,3C. L. Dong ,4 K. Asokan ,2,†and S. Annapoorni1,‡ 1Department of Physics and Astrophysics, University of Delhi, Delhi, 110007, India 2Inter-University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi, 110067, India 3National Synchrotron Radiation Research Center, Hsinchu, 30076, Taiwan 4Department of Physics, Tamkang University, Tamsui 25137, Taiwan (Received 2 February 2020; revised 8 November 2020; accepted 15 December 2020; published 12 January 2021) The present study utilizes synchrotron-based x-ray diffraction (XRD), photoluminescence (PL), and x-ray absorption near edge structure (XANES) spectroscopic techniques to comprehend the evolution of optical intenseblue-green emission in 100 keV nitrogen (N) ion implanted SrTiO 3(STO) thin ﬁlms deposited by RF magnetron sputtering technique. The XRD pattern shows a shift in reﬂections at lower N ion ﬂuences and the amorphizationof the ﬁlms at higher ﬂuences. A disordered phase induced by implantation in the STO ﬁlms leads to an intenseblue-green emission due to oxygen (O) vacancies and N (2 p) bound states. A schematic diagram of energy levels has been proposed to explain the origin of PL emission. The XANES spectra at Ti Kedge reﬂect a change in the valency of Ti ions and the local atomic structure of ordered and disordered phases of STO with an increasein N ion ﬂuence. The splitting of peak assigned to e gorbitals, and discrepancy in ratio dz2/dx2−y2observed in the Ti L-a n dO K-edge spectra, conﬁrm a distortion in TiO 6octahedral structure and modiﬁcations in O 2p-Ti 3 dhybridization states. The synchrotron-based techniques reveal that N ion implanted STO can be a good photoluminescent material exhibiting a variety of emissions through bound states of O vacancies and implantedN ions. DOI: 10.1103/PhysRevB.103.024104 I. INTRODUCTION Perovskite oxide materials with ABO 3type structure are employed in a wide range of applications including photocat-alytic activity, semiconductor devices, optoelectronic devices,and also used as an insulating layer in most electronic de-vices [ 1–6]. A well-known prototype perovskite compound, SrTiO 3(STO) shows a cubic structure at room temperature and is widely used as photocatalyst for dye degradation onultraviolet light (UV) irradiation [ 7]. It is well known that pure crystalline STO does not show any photoluminescence(PL) emission at room temperature under UV light, but abroad emission is observed at low temperatures [ 8,9]. Blue light emission at room temperature was ﬁrst reported by Kanet al. due to O vacancies induced by 300 keV argon (Ar +)i o n irradiation in STO thin ﬁlms [ 10]. In another study, Kumar et al. irradiated STO single crystals with 550 eV Ar+ions and showed three distinct PL emissions: ∼430 nm, ∼550 nm, and ∼830 nm [ 11]. These studies show that the optical properties can be modiﬁed or improved by introducing defects in thematerial. Despite extensive study on the electrical propertiesand electronic structures of STO, the origin of photolumi-nescence and its correlation with band structure and induced vkmevphysics@gmail.com †asokaniuac@gmail.com ‡annapoornis.phys@gmail.comdefect states remains unclear. Therefore, the role of defectsin modifying the optical properties like PL is very intriguingand is a very important subject of study [ 5,12]. Defects can be produced during the synthesis of materials or created by sev-eral modiﬁcations induced by external parameters like thermaltreatment [ 13,14] and ion beam irradiation or implantation etc. [ 15]. Controlling the defects to produce desired variation in the properties of semiconductors is a challenging task. Ionbeam implantation technique has widely revolutionized phys-ical properties especially optical and electrical nature of thematerial [ 10,15]. It can induce charge carriers, defects, lattice distortion, vacancies, and trap levels in the material up to thedesired depth with a reproducible precise dose that plays acrucial role in improving the functionality of transition metaloxides [ 10,16]. This technique not only produces the oxygen (O) vacancies and defects but also can introduce self-dopingof the ions in interstitial sites such as O or Ti interstitials [ 17]. These interstitials modify the structural and optical proper-ties of materials [ 18]. Nitrogen (N) ions can occupy O sites resulting in transition metal oxynitrides [ 19]. The hybridized states of N (2 p) are slightly at higher potential energy than that of O (2 p)[20]. Correlation between defects and electronic structures have been studied using x-ray photoelectron spec-troscopy (XPS) [ 18], electron paramagnetic resonance (EPR) [21,22], and x-ray absorption spectroscopy (XAS) [ 5,23,24]. However, there are very few detailed investigations of funda-mental changes in the photoluminescence properties of STOﬁlms induced by N ion implantation. 2469-9950/2021/103(2)/024104(11) 024104-1 ©2021 American Physical SocietyVISHNU KUMAR et al. PHYSICAL REVIEW B 103, 024104 (2021) The present study employs the synchrotron-based char- acterization techniques to investigate the effect of 100 keVN ion implantation on the structural, optical properties, andelectronic structures of STO and also meticulously attemptsto correlate the origin of optical emissions with introduced de-fects and modiﬁed electronic structure. Based on experimentalobservation, N ion implanted STO can be perceived as goodphotoluminescent material. The observed emissions are com-prehended by x-ray absorption near-edge structure (XANES)studies and further corroborated by literature related to theﬁrst principle calculations and explained using schematic di-agram of energy levels of STO comprising localized boundstates of O vacancies and N 2 plevels. II. EXPERIMENTAL DETAILS STO thin ﬁlms were deposited on quartz substrates using a commercial STO target of 2 mm thickness and 5 cm di-ameter by radio frequency (RF) magnetron sputtering (HINDHIV AC, Model 12” MSPT) in the presence of oxygen (20%)balanced with argon gas. The cleaned quartz substrates ofdimensions 10 mm ×10 mm ×2 mm were used to deposit STO thin ﬁlms. The vacuum chamber was maintained at ∼5× 10 −6mbar before deposition. The plasma was processed at a ﬁxed power of 250 W. During the deposition, a mixture ofargon (Ar) and oxygen (O 2) (20%) gas was circulated into the chamber maintaining the pressure of ∼5×10−2mbar. The substrates were kept at room temperature and exposedto the plasma for 40 min. These ﬁlms were annealed at 750 ◦C for 5 h in a horizontal tubular furnace in Ar +O2(20%) gas ﬂow controlled using a needle valve and the gas was efﬂuentthrough the water. Hereafter, these samples are considered as“Pristine” ﬁlms. These pristine ﬁlms were implanted with 100keV N ion beam at room temperature using the low energyion beam facility (LEIBF) housed in Inter-University Accel-erator Centre (IUAC), New Delhi with different ﬂuences viz.∼5×10 14,∼1×1015,∼5×1015, and∼1×1016ions/cm2 and these samples are hereafter referred as N-5E14, N-1E15, N-5E15, and N-1E16, respectively. The time for irradiationwas estimated using the parameters; ion beam current (nA),sample area (cm 2), and the ﬂuence of the ion beam (number of incident ions /cm2) in the equation ( 1). Time (sec) =Fluence ×Area Current ×6.25×109(1) For the structural characterization, synchrotron glancing incidence x-ray diffractometer (GIXRD) operated at 13 keVwas used at MCX Beamline, Elettra Sincrotrone Trieste, Italy[25,26]. The x-ray source was ﬁxed at a glancing angle of 0 .2 ◦ and the detector was rotated with a step size of 0 .01◦to record the intensity of reﬂections in the 2 θrange from 18◦to 45◦. Further, the samples were characterized by synchrotron PLspectroscopy at TLS-03A1, and XAS at TLS-20A1 & TLS-17C1 beamlines of National Synchrotron Radiation ResearchCenter (NSRRC), Hsinchu, Taiwan. The PL measurementswere performed at room temperature in a vacuum chamber(∼10 −7Torr) in reﬂection mode at an incident angle of ∼45◦ of the source beam with the sample. The intensity of PL emissions was measured by using a Photomultiplier Tube(PMT detector; Hamamatsu R943–02) operated at 1700 V 0 100 200 300 400-0.10.10.30.50.70.91.1 (a)Ion Concentration (%) Depth (nm)N-5E14 N-1E15 N-5E15 N-1E16 0 50 100 150 200 250 300 3500.000.030.060.090.120.150.18Number/ Å-ion Depth (nm)O vacancies Ti vacancies Sr vacancies(b) FIG. 1. (a) Calculated N ion concentration in STO thin ﬁlms corresponding to different ion ﬂuences and (b) the possible Ti, Sr, and O vacancies simulated by TRIM in the 225 nm STO thin ﬁlmson quartz substrate. and kept at an angle of 90◦with respect to the beam [ 27,28]. The emission spectra were collected under excitation with240 nm. During these measurements, an optical ﬁlter wasused to remove band emissions from the spectra. Soft XASmeasurements were carried out at 20A1 beamline in a vacuumchamber ( ∼10 −7Torr) and the spectra were recorded in total electron yield (TEY) mode. XAS measurements with hardx rays were performed in air using a sample rotator at 17C1 Wiggler beamline and spectra were recorded in ﬂuores-cence mode. Thickness measurements were performed usingRutherford backscattering (RBS) spectrometry at IUAC NewDelhi. TRIM (Transport of Ions in Matter) and SRIM (TheStopping and Range of Ions in Matter) simulations were usedto determine the ion beam parameters like range, electronicand nuclear stopping power, and straggling. III. RESULTS The plots of ion energy versus energy loss and the depth versus energy loss are shown in Fig. A (see Supplemen-tal Material, SM [ 29]) along with other parameters derived from SRIM-TRIM simulations. The maximum N ion con-centrations were calculated to be 0.05%, 0.1%, 0.5%, and0.9% in N-5E14, N-1E15, N-5E15, and N-1E16, respectively 024104-2ORIGIN OF INTENSE BLUE-GREEN EMISSION IN … PHYSICAL REVIEW B 103, 024104 (2021) FIG. 2. XRD spectra of STO thin ﬁlms: (a) pristine, N-5E14, N-1E15, N-5E15, and N-1E16 and (b) closer view of x-ray peak resulting from the diffraction of the (011) plane of pristine, N-5E14, and N-1E15. [Fig. 1(a)], and the Sr, Ti, and O vacancies were estimated to be 0.070, 0.075, and 0.165 number /Å-ion using TRIM calculations [Fig. 1(b)]. The unit “per Å” is used in order to keep the numerical values as small numbers on the plot.The unit “numbers /ion” stands for the number of vacancies created in the material per incident ion [ 30]( s e eS Mf o r other descriptions [ 29]). The thickness of as-grown ﬁlms was estimated to be ∼220 nm by RBS. The RBS spectra and corresponding depth proﬁle of pristine and N-1E16 are shownin Fig. B (see SM [ 29]). Figure 2(a)shows the GIXRD spectra of pristine, N-5E14, N-1E15, N-5E15, and N-1E16. The observed reﬂections ofpristine match well with the standard data (High Score Plusreference number 98-002-3076) of cubic perovskite poly-crystalline structure of STO. The lattice parameters werecalculated to be ∼3.905 Å for pristine STO ﬁlm. Figure 2(b) compares the XRD peaks resulting from the diffraction ofthe (011) plane for the three samples: Pristine, N-5E14, andN-1E15. The x-ray peaks corresponding to (011) planes in theXRD pattern for N-5E14 and N-1E15 show a shift of 0 .21 ◦ and 0.28◦, respectively, towards lower angle, ascertaining an expansion in the unit cell of STO. The lattice parameters a, b,and c were calculated to be 3.947 Å, 3.938 Å, and 3.951 Åfor N-5E14, and 3.969 Å, 3.948 Å, and 3.973 Å for N-1E15,respectively. A slight expansion along all axes is observedafter N ion implantation. XRD spectra of N-5E15 and N-1E16indicate that higher ion ﬂuences result in amorphization of thesamples (discussed later). The interplanar spacing ( d-spacing) values calculated using the Bragg’s equation for the corre-sponding plane (hkl) are found increasing with N ion ﬂuences(see Table I). The modiﬁcation in crystal structure leads to the changes in optical properties. To identify the possible optical emissionsdue to defects created by N ion implantation, the PL mea-surements were performed for all the samples. No signiﬁcantemission intensity was observed for the pristine sample asseen from Fig. 3and this is consistent with the literature [9,10,31].The PL spectra of N-5E14, N-1E15, N-5E15, and N-1E16 were ﬁtted using Origin 8.6 software with three distinct Gaus-sian peaks. The position, standard error, and FWHM of theGaussian peaks are presented in Table IIand the ﬁgures, illustrating deconvolutions are shown in Fig. C (see SM [ 29]). In the inset of Fig. 3, a typical deconvolution process is shown for N-5E14 and N-1E16. All N ion-implanted samples havetwo common peaks at ∼415 nm (A) and ∼550 nm (D). Apart from these peaks, N-5E14 and N-1E15 have a peak at ∼480 nm (C), and N-5E15 and N-1E16 have at ∼460 nm (B). Peak C appears at lower ﬂuence, which shifts towards lower wave-length at higher ﬂuences and is assigned as peak B. All thesepeaks, at ∼415 nm (blue), ∼460 nm (blue), ∼480 nm (blue), and∼550 nm (green), correspond to the energy of ∼2.98 eV , ∼2.69 eV , ∼2.58 eV , and ∼2.25 eV , respectively. An enhance- ment in the PL intensity is observed for all emissions with anincrease in the ion ﬂuence. This result is consistent with thestudy by Pontes et al. which reported an intense PL emission at room temperature in amorphous STO thin ﬁlms [ 32]. A broad PL emission corresponds to a convolution of mul- tiple transitions involving various energy levels. Moreover,it is affected by the defects, Fermi level, density of states,etc. All these factors contribute to the broadening of the TABLE I. “ dspacing” for corresponding planes (hkl) of pristine and N-5E14, and N-1E15. Pristine N-5E14 N-1E15 hkl 2 θ(◦)d ( Å )2 θ(◦)d ( Å )2 θ(◦)d ( Å ) 011 19.90 2.7606 19.69 2.7897 19.62 2.7996 111 24.43 2.2553 24.17 2.2783 24.06 2.2886002 28.23 1.9559 27.94 1.9758 27.81 1.9849 112 34.80 1.5942 34.45 1.6108 34.29 1.6180 022 40.39 1.3807 39.95 1.3963 39.77 1.4023 013 45.42 1.2348 45.07 1.2446 44.78 1.2522 024104-3VISHNU KUMAR et al. PHYSICAL REVIEW B 103, 024104 (2021) TABLE II. The ﬁtting parameters (in nm) of PL spectra of N ion implanted STO thin ﬁlms. N-5E14 N-1E15 N-5E15 N-1E16 Peak Position FWHM Position FWHM Position FWHM Position FWHM A 415 .29±0.30 44.72 414 .03±0.27 43.74 413 .85±0.14 43.19 413 .91±0.09 42.40 B 463 .18±1.53 79.89 462 .90±1.83 87.71 C 483 .53±1.10 77.67 482 .28±1.22 82.36 D 555 .10±6.30 159.87 553 .24±4.23 167.73 551 .73±8.93 153.03 546 .33±7.87 150.52 experimental PL spectra [ 33]. The transitions at lower wave- lengths are more populated than that of higher wavelengths.Consequently, the blue emission is sharper than the greenishemission. The latter emission is more signiﬁcant at higherﬂuences where an increased number of N ions causes thelocal amorphization leading to the rise of the extended VBand CB along with the tailing localized states [ 34,35]. The FWHM of these emissions increases with an increase in thewavelength (see Table II). The formation of tailing localized states provides a path to the excited electrons to emit thephotons through various defects levels. This results in a broadrange of the emission which is centered at a deﬁned energy,i.e., the center of the Gaussian peak, where the emissions aremaximum for a particular peak. Moreover, the strong electron-phonon coupling may lead to dissipation of some energy ofthe excited electrons and this may also cause variation in theenergy of emitted photons [ 36–38]. The PL emission of N ion implanted STO ﬁlms indeed suggests the presence of defects, interstitials, and vacanciesbut does not give any explicit evidence regarding any spe-ciﬁc type of defect. The synchrotron GIXRD, on one hand,reveals the effect of the N ion implantation on the averagecrystal structure and shift in the reﬂections followed by amor- FIG. 3. Photoluminescence spectra of STO thin ﬁlms: pristine, N-5E14, N-1E15, N-5E15, and N-1E16. The spectra were recorded under excitation with 240 nm and the band emission was eliminatedusing an optical ﬁlter during the measurements. Inset shows a typical deconvolution process for N-5E14 and N-1E16. Note that the blue- green emission evolves on N ion implantation.phization. The evolution of PL is attributed to the possible O vacancies, lower crystallinity, and ion-beam induced de-fects. The localized studies based on the electronic states ofelements may help to verify the aforementioned proposition.Hence, XANES at Ti Kedge was studied to examine the local environment of Ti ions. The Ti K-edge features appear due to the transition of 1 score electrons to the unoccupied states consisting of 3 dand 4 sporbitals strongly hybridized with the O2porbitals above the Fermi level, E f. The XANES spectra at Ti Kedge for the STO thin ﬁlms of pristine, N-5E14, N-1E15, N-5E15, and N-1E16 STO thinﬁlms are shown in Fig. 4(a). The spectra were normalized using Athena software [ 39]. The pre-edge features of all the ﬁlms consist of two peaks appearing in the energy range of∼4966 eV to 4972 eV . The features were deconvoluted using two Gaussian functions. The inset shows a magniﬁed viewof the pre-edge region and a representative deconvolution ofpristine is shown under the curve. The pre-edge peaks areattributed to the transitions to unoccupied 3 dt 2gand 3 deg states. The egpeaks are marked as αin Fig. 4(a). The ﬁtting parameters of the peak are shown in Table III. The peak α departs to the lower energy with an increase in ion ﬂuence.Figure 4(b) shows the change in the position and intensity of the peak αby varying the N ion ﬂuence. To analyze the trend of the variation of pre-edge peak parameters with the changein N ion ﬂuence, one needs to understand the origin of thispeak. The pre-edge features appear just above the Fermi level E f and below the main absorption edge due to the transition of electrons from 1 sto 3dt2gand 3 degorbitals. These transi- tions are forbidden by the dipole selection rules since theseinvolve a change in the orbital angular momentum /triangleL=2 [40]. The experimental observation of pre-edge features is a consequence of the fact that the ﬁnal state of photoelectronspossesses an admixture of p-dcharacters via the hybridization of Ti 3 dand O 2 porbitals [ 40,41]. Therefore, the change in intensity and position of the peaks suggests the modiﬁcationin degree of hybridization after N ion implantation. Based on experimental analysis of Ti Kedge along with multiple scattering calculations, Farges et al. [40] and Frenkel et al. [41,42] related the position and intensity of pre-edge peaks with the coordination number, oxidation state, and off-center displacement of Ti ions [ 40–42]. Farges et al. compared the experimental and theoretical data to provide the physicsinvolved in pre-edge region and to distinguish between theﬁvefold, and mixture of fourfold and sixfold coordinated Tiions (cf. Fig. 2 in Ref. [ 40]). The changes in local environment of Ti ions are expected to be reﬂected in the pre-edge ofXANES spectra in terms of variation in the intensity and posi-tion of peaks [ 43,44]. The larger intensity of peak corresponds 024104-4ORIGIN OF INTENSE BLUE-GREEN EMISSION IN … PHYSICAL REVIEW B 103, 024104 (2021) FIG. 4. (a) Ti K-edge spectra of N ion implanted ﬁlms. Inset shows the pre-edge region, labeled as a, b, c, d, and e corresponding to pristine, N-5E14, N-1E15, N-5E15, and N-1E16, respectively. (b) The variations in the peak position and height of peak αas the function of N ion ﬂuences. to the fourfold coordinated Ti ions whereas the lower intensity to sixfold geometry of the octahedron [ 40]. From Fig. 4(b), the position of the peak αshifts towards lower energy side, and its intensity increases with an increase in N ion ﬂuenceswhich indicates a reduction in the coordination number ofTi ions in N ion implanted STO [ 42]. It has been reported that the amorphous STO has a more intense pre-edge peakand shifts towards lower energy in comparison to crystallineSTO [ 41]. The area under the peak αcan be analyzed to estimate the off-center displacement of Ti ions from the centerof TiO 6octahedron [ 42]. The relation between the off-center displacement dof Ti ion and area under the peak αis related byA=γ 3d2where γis a constant for a particular perovskite [40–42,44]. From Table III, the increase in intensity and area under the pre-edge peak speciﬁes the increase in off-centerdisplacement of Ti ions. The chemical shifts of STO ﬁlms were calculated using the reference compounds of Ti 2O3(Ti+3) and single crystal STO (SSTO, Ti+4) by taking the ﬁrst derivative of Ti K- edge spectra. The average Ti valence states are determinedby extrapolation of the energy of Ti K-edge spectra with the standards and are given in Table IIIalong with estimated com- positions of the ﬁlms. This indicates a mixed-valence state ofTi (+3 and+4) with oxygen vacancy surrounding the Ti ions in N ion implanted ﬁlms. Hence, it is inferred that N-5E14contains ∼62% and ∼38% of Ti ions in +3 and +4 valence states, respectively. Similarly, both N-1E15 and N-5E15 have∼77% and ∼23% of Ti ions in +3 and +4 valence states,respectively. The average valence state of Ti ions in pristine and N-1E16 are considered to match with the standard STOand Ti 2O3, respectively. The analysis of Ti K-edge spectra of these samples gives extensive information about the changes induced by N ionimplantations in the crystal structure and the coordinationnumber of Ti ions. Ti L-edge XANES spectra were recorded in the energy ranges of 440 eV to 490 eV that provide in-formation about the nature of Ti 3 dorbitals in the electronic structure of STO. These spectra are divided into two regionsviz.L 3andL2edge. The L2edge appears broader than the L3 edge due to Coster-Kroning decay [ 45,46]. In the transition metals, the 2 porbitals are split into 2 p3/2(L3) and 2 p1/2(L2) by spin-orbit interaction and the Ti 3 dorbitals into t2gand egorbitals by crystal ﬁeld effects [ 47]. Four absorption peaks are observed corresponding to the following transitions: from2p 3/2to 3dt2gand 3 degand from 2 p1/2to 3dt2gand 3 deg orbitals. The L-edge spectra give explicit information on un- occupied 3 dorbitals which are related to the transition metal ion valency and O vacancies. The normalized Ti L3,2-edge spectra of pristine, N-5E14, N-1E15, N-5E15, and N-1E16are shown in Fig. 5(a). All the peaks were ﬁtted using XP- SPEAK4.1 software to determine the accurate peak position,intensity, and area under each peak. The L 3-edge spectrum of pristine STO ﬁlm is ﬁtted with two peaks corresponding to t2gandegorbitals but one addi- tional peak is needed to ﬁt the spectra of N ion implantedSTO ﬁlms which is assigned to the splitting of 3 de gstates into TABLE III. Peak parameters of pre-edge peak αfor all the samples and estimated valency of Ti ions. Sample Peak Position Height FWHM Area under name (eV) (eV) the peak First derivative Valence state Composition Pristine 4970.01 0.1170 1.555 0.1936 4969.7 4 SrTiO 3 N-5E14 4969.73 0.1648 1.949 0.3419 4968.9 3.38 SrTiO 2.70 N-1E15 4969.46 0.2535 2.209 0.5961 4968.7 3.23 SrTiO 2.62 N-5E15 4969.38 0.3153 2.268 0.7614 4968.7 3.23 SrTiO 2.62 N-1E16 4969.18 0.3292 2.339 0.8199 4968.4 3 SrTiO 2.50 024104-5VISHNU KUMAR et al. PHYSICAL REVIEW B 103, 024104 (2021) FIG. 5. (a) Ti L3,2-edge spectra of pristine and N ion implanted STO ﬁlms. The deconvoluted L3-edge spectra under the pristine and N-5E14 are shown to explain the splitting in egstates. (b) The change in area ratio t2g/eganddz2/dx2−y2as a function of N ion ﬂuence calculated for Ti L3-edge spectra of pristine and N ion implanted STO ﬁlms. dz2anddx2−y2orbitals. The ratios of area under the 3 dt2gand 3degorbitals derived peaks, and ratio dz2/dx2−y2are plotted as a function of N ion ﬂuences as shown in Fig. 5(b) and listed in Table IV. The discrepancy in the ratio dz2/dx2−y2 implies a distorted noncubic TiO 6octahedral structure [ 48]. This can be accredited to a change in the Ti-O bond due TABLE IV . Area ratios t2g/eganddz2/dx2−y2corresponding to Ti L3and O Kedges for pristine and the N-ion implanted STO ﬁlms. TiLedge O Kedge Sr. no. Sample t2g/eg dz2/dx2−y2 t2g/eg dz2/dx2−y2 1 Pristine 0.40 0.60 0.43 2 N-5E14 0.37 1.14 0.57 0.35 3 N-1E15 0.37 0.80 0.36 0.34 4 N-5E15 0.32 0.76 0.29 0.30 5 N-1E16 0.30 0.71 0.24 0.20to variation in the degree of hybridization. The amount of splitting in 3 degstate gradually increases as a function of N ion ﬂuence. Jan et al. observed the off-center displacement of Ti ion from the octahedral site which is reﬂected in the TiLedge of Pb 1−xCaxTiO 3(PCT) as splitting in 3 degorbitals [49]. Mastelaro et al. observed that the degree of distortion in the octahedral structure of Pb 1−xLaxTiO 3(PLT) decreases on increasing the La doping and less pronounced splitting of3de gorbitals in Ti Ledges [ 50]. Therefore, the splitting of 3degorbitals into dz2anddx2−y2is a measure of the degree of deviation from octahedral symmetry [ 50]. The intensity of Ti L3edge is proportional to the density of unoccupied states which is the sum of t2gandegstates [46,48,51]. In principle, the ratio t2g/(t2g+eg) and eg/(t2g+ eg) should be 0.6 and 0.4, respectively, for Ti+4since there are no electrons in 3 dorbitals [ 52]. These ratios can be also written as t2g/egwhich turns out to be 6 /4( o r3 /2). This is a theoretical assumption where all the Ti ions are assumedto be in d 0state and if only one electron is considered in the calculations ignoring the electron-electron correlation. Foroctahedral symmetry, the ratio of unoccupied orbitals is 6 /4 ast 2gandegcan take six and four electrons, respectively [52]. Kuo et al. stated that the intensity of the spectral feature assigned to t2gstates is sensitive to the valence state of Ti ion [48]. The more intense t2gfeature implies an increase of the oxidation state and thus indicates the presence of Ti+4(3d0) with respect to the TiO 2system [ 51]. Janotti et al. performed the HSE (Heyd-Scuseria-Ernzerhof) calculation and reportedthat if one electron is added to the TiO 2system, it goes to Ti ion occupying the nearest neighbor site to the O vacancy [ 53]. According to the density functional theory (DFT) calculationsby Lin and co-workers, the added one electron has a highprobability of occupying the lowest energy state ( t 2g)o ft h e octahedral symmetry [ 54]. Hence, in the case of d1state, the t2gorbital is ﬁlled with one electron being available at the lower energy state. Thus, the ratio t2g/egdecreases from 6 /4 to 5/4. Similarly, for d2and d3state, the ratio will be 4 /4 and 3/4, respectively. Hence, if one considers the ratio t2g/eg,a n y change in t2gstate (or a charge state of Ti) will be evident [ 51]. This is due to change in the density of unoccupied 3 dorbitals. The decrease in the intensity of the t2gpeak with an increase in N ion ﬂuence indicates the presence of Ti+3(3d1) in the N ion implanted samples. The t2g/egratio decreases as the content of N increases. This implies that the oxygen vacancies are alsoincreasing with N ion implantation. The Ti L-edge spectra of these ﬁlms are also compared with an O deﬁcient STO thinﬁlm annealed at 750 ◦Ci nA r +H2(5%) gas ﬂow for 5 h. The t2g/egratio is estimated to be 0.33 for these O deﬁcient ﬁlms. Thus, the electronic conﬁguration of Ti from the Ti-O bondexhibits a combination of 3 d 0in Ti+4and 3 d1in Ti+3in the ground state [ 55,56]. However, the pristine ﬁlm also deviates from the expected value of t2g/egratio. The calculated ratio based on Ti L-edge spectra is 0.4. Wu et al. measured Ti L-edge spectra of TiO 2nanotubes and reported that the t2g/eg ratio is in a range of ∼0.29–0.31 [57]. At ﬁrst observation, as evident from the RBS study (see SM [ 29]), the surface layer is found to be Sr 0.95TiO 2.98. Since the thin-ﬁlm fabrication with perfect stoichiometric composition and uniform thickness isa big challenge, any slight deviation causes change in thevalue of t 2g/egratio [ 58]. There are some detailed reports 024104-6ORIGIN OF INTENSE BLUE-GREEN EMISSION IN … PHYSICAL REVIEW B 103, 024104 (2021) FIG. 6. O K-edge spectra of pristine and N ion implanted STO. The deconvolution of pre-edges are also shown under the corre- sponding spectra. understanding the Ti L-edge spectra using Multiplet calcula- tions [ 59,60]. Kroll et al. observed that the t2g/egratio does not match with the expected value of 3 /2 while considering the spin-orbit coupling and crystal ﬁeld that ﬁt the experimentalspectra [ 60]. In the crystal ﬁeld, 3 dorbitals split into t 2gand eg, while these states are mixed in the ﬁnal states through 2p3dCoulomb interaction and within the same symmetry, the intensity is transferred between states [ 60]. Hence, the ﬁnal states do not show the expected t2g/egratio. Similarly, Laskowski et al. , while calculating the XAS spectra at the L edges of STO by solving the Bethe-Salpeter equation (BSE),also observed that the ratio t 2g/egis not as expected to be 3/2[59]. Wu et al. performed conﬁguration interaction (CI) cluster calculations for STO to reproduce Ti L-edge spectra and observed that the ratio t2g/egincreases with an increase in hybridization strength [ 61]. Hence, the presented Ti L-edge spectra and analysis are in line with the reported multipletscattering calculations of SrTiO 3[59,60,62–65]. The nature of bonding of O ions with constituent ions (Ti, Sr, and implanted N) was investigated by recording theXANES spectra at O Kedge. These spectra are very sensitive to the Ti-O hybridization. Figure 6shows the XANES spectra at O Kedge for pristine, N-5E14, N-1E15, N-5E15, and N-1E16. The prominent spectral features up to 535 eV areassigned to the transitions from O 1 sorbitals to hybridized states between O 2 pand Ti 3 dorbitals and the peaks above 535 eV are the hybridized states between O 2 pand Ti 4 sp orbitals. To determine the accurate positions and area underthe featured peaks, the spectra were ﬁtted using XPSPEAK4.1software. The pre-edges can be deconvoluted using threeFIG. 7. The change in t2g/eganddz2/dx2−y2area ratios as a func- tion of N ion ﬂuence for O K-edge spectra of pristine and N ion implanted STO ﬁlms. peaks which are assigned to 3 dt2gand 3 degorbitals ( dz2and dx2−y2) derived peaks. These peaks are separated due to crystal ﬁeld splitting. In Fig. 6, the area under the 3 dt2gpeak is ﬁlled with red. The width and asymmetry of the 3 degpeak suggests the inclusion of two peaks to ﬁt the pre-edge region which isattributed to the splitting in 3 de gstates into dz2anddx2−y2. These two peaks are shown by ﬁlling the area under the peakwith green and blue, respectively. The t 2gandegpeaks arise due to the transitions from O 2 pto 3dt2gand 3 degorbitals, respectively. The 3 degstates appear at higher energy and have stronger coupling with O 2 pions than the t2gstates because dz2anddx2−y2levels are directed towards O ions. Therefore, 3degstates are more sensitive to the deviations in the sym- metry of octahedral structure [ 66]. Hence, the positions of the 3 degspectral features of O Kedges vary with change in the ion ﬂuence. This can be a consequence of differentlocal environment and ion coordination. The change in thecrystal ﬁeld splitting ( /triangle3d) for all the implanted ﬁlms indi- cates the variation in the Ti-O distance as a result of changein O 2 p-Ti 3 doverlap [ 47]. Also, the O K-edge spectra become sharper after N ion implantation. The sharpness ofthe peaks derived from the O 2 p-Ti 3 dhybridized orbitals implies higher order of covalent bonding between the Ti and Oatoms [ 67]. The ratio t 2g/egdecreases gradually with an increase in the N ion ﬂuence. It implies that the 3 dt2gstates are occupied due to the creation of O vacancies [ 53–56]. The ratio of area under the dz2anddx2−y2peaks also changes with N ion ﬂuence as shown in Fig. 7and Table IV. It suggests the existence of O vacancies and mixed valence states of Ti ion [ 48,51]. Besides, the splitting in the peak, assigned to 3 degstate, can be attributed to the modiﬁcations in O 2 p-Ti 3 dhybridization states and the distortion in the TiO 6octahedral structure [ 48]. IV . DISCUSSION As evident from TRIM simulations, N ions cause dis- placements of the target atoms which may result in vacancies 024104-7VISHNU KUMAR et al. PHYSICAL REVIEW B 103, 024104 (2021) thereof. The knocked-out Ti and O ions possibly occupy the interstitial sites in the octahedra which may be responsible forthe observed expansion in interplanar spacing [ 17]. From the XRD pattern, it is evident that the crystallinity decreases withN ion implantation and amorphization is observed at higherﬂuences. In the previous study, it is revealed that the presenceof O vacancies and N interstitials in the STO thin ﬁlms in-duced by low energy N ion implantation cause a split in theXRD reﬂections and amorphization of the ﬁlms [ 17] which can be comprehended by evoking the ion beam interactionwith materials. When an energetic ion passes through a material, it under- goes a series of collisions with the nuclei and atomic electronsof the material. The incident ion loses its energy via twoprocesses mainly (i) elastic collisions with the target nucleuswhich leads to displacement of the atoms as a whole (nuclearenergy loss, S n) and (ii) inelastic interaction with the electrons that excite or eject the atomic electrons (electronic energyloss, S e)[30]. Low energy ion interactions are dominated by elastic processes, resulting in the ballistic atomic displace-ments of the target atoms. This causes radiation damage inthe target material. At sufﬁciently high doses, it results in thecrystalline to the amorphous transformation of the irradiatedarea because of complete disordering in the crystal lattice[68–70]. This amorphization depends on the type of materials, the mass of ion, and ion irradiation conditions [ 30]. The light mass projectiles create isolated point defects that accumulateto transform crystalline to amorphous structure [ 71] after cer- tain ion ﬂuence. In the case of heavy ions, dense collisioncascades create amorphous pockets which result to transforminto the amorphous structure [ 72]. In the present case, almost complete amorphization is observed with the low energy Nion implantation in N-1E16 STO thin ﬁlm. Hence, this studyfocuses on varying ion ﬂuences from 5 ×10 14ions/cm2to 1×1016ions/cm2.K a n et al. irradiated STO ﬁlms with 300 keV Ar+ions and reported a thin amorphous layer of few nanometers near the surface [ 10]. The XRD results are corrob- orated by XANES studies of Ti Kedge. There is an evolution and a shift in the pre-edge position of Ti Kedge towards lower energies with an increase in the N ion ﬂuences. This indicatesthe amorphization of the ﬁlms, reduction in the coordinationnumber, and a decrease in the valence state of Ti ion (from +4 to+3) [40–42,44]. Both Ti L- and O K-edge spectra depict the modiﬁcations due to N ion implantation and the spectralfeature assigned to e gstate splits into dz2anddx2−y2orbitals. The splitting in egstates is an indication of distortion in the crystal structure [ 50]. This supports the shift in diffraction peak and change in the lattice parameters with N ion ﬂuence.Based on atomic multiplet scattering calculations, Fan et al. simulated the Ti L-edge spectra for ATiO 3(Ca, Sr, Ba) system and reported that there is a drastic change in the electronicstructure of O ion due to different local environment resultingin a strong hybridization between O ion and A cation [ 62]. The intensity of d x2−y2increases with ion ﬂuence. The decrease in the ratio of t2g/egin Ti Land O Kedge implies the decrease in density of unoccupied t2gstates. Also, the discrepancy in ratio dz2/dx2−y2is a direct indication of lattice distortion and change in Ti-O hybridization. In the cubic perovskite structure, Ti ion at the body center is surrounded by six O ions situated at the faces of cubic FIG. 8. (a) Cubic perovskite structure of pristine STO and (b) the possible deformation in local atomic structure of STO after N ion implantation. structure [ 19,73]. If the O ions are knocked out from their sites, the Ti ions are displaced from their center to stabilizethe octahedral structure resulting in off-center displacementand lattice expansion. Kan et al. revealed that the presence of O vacancies is responsible for blue emission in STO [ 10]. The role of N ions is still to be explored to understand the origin ofblue-green PL emission. A few studies based on ﬁrst principlecalculations using the DFT have been reported for N dopedSrTiO 3and TiO 2systems [ 74–76]. In these studies, a few models based on the occupancies of N such as substitutional,interstitial, and oxygen vacancies accompanied by N dopingin STO lattice were considered [ 74–80]. Mi et al. reported that the substitutional occupancy is favored in comparison tointerstitial and stated that the substitutional N 2 plocalized states lie above the top of O 2 pvalence band (VB) [ 74,75]. Miyauchi et al. performed similar ﬁrst principle calculations based on DFT considering the substitutional occupancy of Nand assumed the band structure being narrowed due to N 2 p and Ti +3localized states above the VB and below the conduc- tion band (CB), respectively [ 76]. According to SRIM-TRIM calculation, an increase in ion ﬂuence leads to an increasein vacancies because more number of ions hitting the targetresults in sputtering of oxygen ions due to its high sputteringyield [ 30]. In addition to ion beam induced vacancies, N implantation facilitates the formation of oxygen vacancies tomaintain the charge neutrality [ 78]. Thus, the combined effect of ion implantation and N substitution leads to an increase inthe number of oxygen vacancies [ 81–84]. Based on the above results using synchrotron characteri- zation techniques viz. XRD, PL, and XAS, and the existingliterature based on ﬁrst principle calculations using DFT forN doped SrTiO 3, and TiO 2systems, a schematic crystal struc- ture is proposed as shown in Fig. 8. The pristine sample possesses an ideal cubic crystal structure in which Ti+4ion has coordination number of 6 [Fig. 8(a)]. Implantation of N ions creates oxygen vacancies in the lattice [shown as Ov in Fig. 8(b)], which results in +3 valence state of the body- centered Ti ion. The N and O ions possess almost identicalradii. Hence, N ion can either substitute O atom (shown asN o) or occupy the interstitial sites (shown as Ni). There are six O ions located at the face center of STO cubic structure.The probability of occupancy of any O site by N odepends on the ion implantation conditions like ion species, energy,ﬂuence, current, and the time of exposure to the ion beam 024104-8ORIGIN OF INTENSE BLUE-GREEN EMISSION IN … PHYSICAL REVIEW B 103, 024104 (2021) FIG. 9. Schematic diagram of energy levels illustrating the pos- sible mechanisms for observed PL emission. [11]. Therefore, it is challenging to determine the exact charge state of the O vacancy. To identify the possible transitions, defect levels, and recombination of the electron-hole pair, aschematic diagram of energy levels is proposed as shown inFig. 9and discussed as follows. The band gap of STO is of the order of ∼3.3–3.4 eV which arises from the gap between the Ti 3 dCB and the O 2 pVB [ 11]. Xu et al. studied the effect of crystallization on the band structure of STO ﬁlmsand proposed an energy band structure for amorphous andcrystalline STO [ 34]. As there is no long-range order in the amorphous STO, tailing localized states appear near the ex-tended VB and CB [ 34,35]. The bound states corresponding to the O vacancy donor levels also appear near the E f[10,34,85]. Mitra et al. studied the electronic structure of O vacancies in STO and LaAlO 3using HSE hybrid density functional and suggested that the O vacancies can be neutral ( V0), singly positive ( V+), or doubly positive ( V++) which create bound states, respectively, at 0.7 eV , 0.57 eV , and 0.28 eV belowthe CB in the band gap [ 12,16,85,86]. The implanted N ions form a bound state of hybridized 2 porbitals above the top level of the VB [ 20]. The recombination of excited electrons (in the CB) and the holes (in the VB) through the boundstates of doubly positive O vacancies ( V ++) results in the blue emission at ∼415 nm (A) [ 5,10], whereas the transitions of the excited electrons through the bound states of singly positive(V +) and neutral O vacancies ( V0) give emissions at ∼460 nm (B) and ∼480 nm (C), respectively. If the transitions of excited electrons occur through V0to the N 2 phybridized states or trap levels near the valence band [ 5,10,20], a green emission appears at ∼550 nm (D). At higher ﬂuences (N-5E15 and N-1E16), the number of singly positive O vacancies increasewhich contributes to the emission at ∼460 nm. When the O ions are knocked out by N ions, they leave two unboundelectrons behind with a neutral O vacancy. On increasing theN ion ﬂuence, we observed that the number of O vacancieshaving one unbound electron has increased. In any material,there is a limit of creating O vacancies [ 87]. At higher ﬂuence,the number of implanted N ions increases, and the number of O vacancies approaches the limit. Therefore, to balancethe extra negative charge of implanted N ion, an electron isemitted from the neutral O vacancy. However, it is tricky tocontrol this phenomenon as the creation of defects dependson the ion beam current, energy, and ﬂuence, i.e., exposuretime to beam [ 11]. As evident from Fig. 9,av a r i e t yo fP L emissions are observed in PL spectra through the bound statesof O vacancies and N 2 pstates. Hence, N ion implantation proves to be a very efﬁcient technique to make a tunablephotoluminescent material for optoelectronics applications. V . CONCLUSION STO thin ﬁlms deposited by RF sputtering were subjected to 100 keV N ion beam to investigate the change in struc-tural, optical, and electronic properties. XRD spectra revealdistortion in the lattice structure and amorphization of ﬁlmsat higher ﬂuences. An intense blue-green emission corre-sponding to the bound states of O vacancies and implantedN ions are observed in N ion implanted ﬁlms. This makesthe N ion implanted STO a promising material for the futureoptoelectronics. The XANES at Ti Kedge show the amor- phization of ﬁlms, change in valency states (from +4t o+3), and reduction in the coordination number of Ti ions. Thesplitting of 3 de gstates into dz2anddx2−y2, observed from the TiLand O Kedges, conﬁrms the distortion in lattice along with its expansion. This study using the synchrotron-basedcharacterization techniques explains the origin of blue-greenphotoluminescence emission in N ion implanted STO. ACKNOWLEDGMENTS Authors are thankful to the Department of Science and Technology, Delhi, India, and Elettra Sincrotrone Trieste,Italy for providing the fund to perform the synchrotron x-raydiffraction measurements at MCX beamline Elettra, Italy cor-responding to proposal no. 20175434. Authors thank IUACscientists, Mr. Kedar Mal for his support in low energy Nion implantation, and Mr. Sunil Ojha and Mr. G. R. Uma-pathy for RBS measurements. The authors would like tothank Dr. Bing-Ming Cheng, Dr. Sheng-Lung Chou, and Dr.Jen-Iu lo, National Synchrotron Radiation Research Center(NSRRC), Taiwan for their consistent support during thebeamtime at TLS-03A1 beamline. Authors are also thankfulto NSRRC, Taiwan for XAS measurements at TLS-20A1and 17C1 beamline. V .K. and A.B. gratefully acknowledgethe ﬁnancial support in the form of fellowship given byUGC Delhi, India, and CSIR Delhi, India, respectively. Au-thors thank the Department of Science and Technology, India(SR/NM/Z-07/2015) for the ﬁnancial support and Jawaharlal Nehru Centre for Advanced Scientiﬁc Research (JNCASR)for managing the project. C.L.D., K.A., and V .K. would liketo acknowledge the MoST Project No. MoST 107-2112-M-032-004-MY3 and Taiwan Experience Education Program(TEEP). [1] V . Craciun and R. Singh, Appl. Phys. Lett. 76, 1932 (2000) .[2] H. Inoue, H. Yoon, T. A. Merz, A. G. Swartz, S. S. Hong, Y . Hikita, and H. Y . Hwang, Appl. Phys. Lett. 114, 231605 (2019) . 024104-9VISHNU KUMAR et al. PHYSICAL REVIEW B 103, 024104 (2021) [3] T. Yajima, M. Minohara, C. Bell, H. Hwang, and Y . Hikita, Appl. Phys. Lett. 113, 221603 (2018) . [4] R. N. Schwartz, B. A. Wechsler, and L. 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PhysRevB.88.134519.pdf	PHYSICAL REVIEW B 88, 134519 (2013) Multiorbital and hybridization effects in the quasiparticle interference of the triplet superconductor Sr 2RuO 4 Alireza Akbari1and Peter Thalmeier2 1Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany 2Max Planck Institute for the Chemical Physics of Solids, D-01187 Dresden, Germany (Received 29 August 2013; published 28 October 2013) The tetragonal compound Sr 2RuO 4exhibits a chiral p-wave superconducting (SC) state of its three t2g-type conduction bands. The characteristics of unconventional gap structure are known from experiment, in particularﬁeld-angle-resolved speciﬁc-heat measurements and from microscopic theories. A rotated extremal structureon the main active SC band with respect to the nodal gaps on the passive bands was concluded. We proposethat this gap structure can be further speciﬁed by applying the scanning tunneling microscopy quasiparticleinterference (QPI) method. We calculate the QPI spectrum within a three-band and chiral three-gap model andgive closed analytical expressions. We show that as a function of bias voltage, the chiral three-gap model will leadto characteristic changes in QPI that may be identiﬁed and may be used for a more quantitative gap determinationof the chiral gap structure. DOI: 10.1103/PhysRevB.88.134519 PACS number(s): 74 .20.Rp, 74 .55.+v, 74.70.Pq I. INTRODUCTION The quasi-two-dimensional (quasi-2D) compound Sr2RuO 4is one of the few established cases of triplet superconductivity.1,2This assignment follows from experimental facts such as the absence of a Knight shift,3,4the presence of a spontaneous condensate moment (time-reversalsymmetry breaking), 5,6evidence for a superconducting two-component order parameter,7and the absence of a Hebel-Slichter peak.8The unconventional nature of the order parameter is also witnessed by ﬁeld-angular-resolved speciﬁc-heat9and thermal conductivity10investigations in the vortex phase. These results imply a multiband nodal tripletsuperconducting order parameter. 11 The conduction-band structure is formed by the three t2g orbitals of xz,yz, andxytype, where the former hybridize with each other. It was found2,9,12–14that the gap structure consists of a main gap on the “active” xy-type band which is nodeless but has deep minima along the [100] directions andsmaller (nearly) nodal gaps on the “passive” xz,yz bands with minima or nodes along the [110] directions. Although the basic gap features are clear, it would neverthe- less be desirable to conﬁrm and specify the gap model further.The scanning tunneling microscopy (STM) technique hasrecently proved to be quite powerful in this respect for stronglycorrelated unconventional superconductors. The determina-tion of the tunneling conductance map on a ﬁnite surface arealeads, via Fourier transformation, to the quasiparticle interfer-ence (QPI) pattern that is caused by impurity scattering on thesurface. This pattern contains information on the normal-stateconduction-band Fermi surface as well as on the superconduct-ing gap structure. 15–17In particular, for bias voltage smaller than the gap amplitude, it leads to characteristic QPI featuresat wave vectors that allow conclusions on the kdependence of the gap function. This technique has been used successfully toinvestigate the superconducting gap structure of cuprates, 18Fe pnictides,19and more recently heavy fermion compounds.20–22 We believe it could also be used for further analysis of the chiral p-wave gap structure in Sr 2RuO 4. In fact, the quasiparticle density of states (DOS) (which is the integral overthe QPI function) has already been investigated in Sr 2RuO 4 recently23demonstrating the feasibility of this approach. Be- fore it was also applied to the nonsuperconducting Sr 3Ru2O7 compound.24,25 In this work, we therefore investigate the expected QPI momentum and frequency structure for Sr 2RuO 4in detail. We start from the known parametrization of conduction bands anduse a simple representation of all three gap functions on activeand passive bands that reproduce the microscopic gap structuredetermined by Nomura 12,13quite reasonably, in particular its extremal and nodal structure. Using this well-deﬁned model,we calculate the expected QPI spectrum as a function of biasvoltage. We perform a fully analytical calculation in the Bornapproximation and give a closed solution for the QPI spectrum,including the subtle effects of hybridization in the passivebands. Our approach is complementary to the full t-matrix numerical treatment 26which also uses different gap models. We show that due to the multiband gaps and the rotated gap extrema on active and passive bands, typical changes in the QPIspectrum are to be expected when the bias voltage changes.We identify the characteristic wave vectors that appear in QPIby comparing to the structure of constant quasiparticle energysurfaces. These features, if compared to a future experimentaldetermination of QPI, may be used to further quantify theknown superconducting gap structure of Sr 2RuO 4. II. THREE-ORBITAL MODEL OF QUASI-2D ELECTRONIC BANDS IN Sr 2RuO 4 The quasi-2D bands of Sr 2RuO 4originate from the three t2g3dorbitals dxz,dyz, anddxy, which are denoted by n= a,b,c , respectively. The effective tight-binding (TB) model Hamiltonian for these states may be deﬁned as27–30 H0=/summationdisplay k,n,mhnm 0(k)c† nkcmk, h0(k)=⎡ ⎢⎣/epsilon1xz(k)V(k)0 V(k)/epsilon1yz(k)0 00 /epsilon1xy(k)⎤ ⎥⎦, (1) 134519-1 1098-0121/2013/88(13)/134519(8) ©2013 American Physical SocietyALIREZA AKBARI AND PETER THALMEIER PHYSICAL REVIEW B 88, 134519 (2013) where c† nkcreates the unhybridized conduction electrons. Their dispersion /epsilon1n(k) for each orbital and their hybridization V(k) are parametrized as /epsilon1ak=/epsilon1xz(k)=−/epsilon1/prime 0−2tcoskx−2t⊥cosky, /epsilon1bk=/epsilon1yz(k)=−/epsilon1/prime 0−2tcosky−2t⊥coskx, (2) /epsilon1ck=/epsilon1xy(k)=−/epsilon10−2t/prime(coskx+cosky) +4t/prime/primecoskxcosky, V(k)=Vk=− 2Vmsinkxsinky. Sr2RuO 4is nearly 2D, therefore dispersion along kz is neglected. The in-plane parameters are chosen as in Ref. 28:(/epsilon1/prime 0=0.77,t=1.0,t⊥=0.14), (/epsilon10=1.61,t/prime= 1.39,t/prime/prime=0.45), and Vm=0.1. The absolute energy unit is t. Within the local density approximation (LDA) it is given byt/similarequal0.3e V .27,28The effective bandwidth or effective tis, however, reduced by a factor of 3.5 due to correlations,31i.e., tot=0.085 eV . The spin-orbit (s.o.) coupling32is neglected in our model since we are only interested in the charge QPI withoutresolving the spin channels in the conductance. Its principaleffect on the band structure and superconducting state hasbeen considered in Refs. 33and34. First, due to the presence of inversion symmetry in Sr 2RuO 4, s.o. coupling does not split the Kramers degeneracy of the three bands whichshould now be described in terms of pseudospin degrees offreedom. Therefore, the Fermi surface sheets are qualitativelyunchanged. 34Secondly, the s.o. coupling stabilizes the triplet chiral superconducting state33at ﬁnite temperature; however, forT/lessmuchTcit is already stable without s.o. coupling35due to the feedback effect. For these additional reasons, we do notconsider the effects of s.o. coupling explicitly in our analysis. The TB Hamiltonian may then easily be diagonalized to give the three conduction bands, E 1(k)=E1k=1 2(/epsilon1ak+/epsilon1bk)−1 2/bracketleftbig (/epsilon1ak−/epsilon1bk)2+4V2 k/bracketrightbig1 2, E2(k)=E2k=1 2(/epsilon1ak+/epsilon1bk)+1 2/bracketleftbig (/epsilon1ak−/epsilon1bk)2+4V2 k/bracketrightbig1 2, E3(k)=E3k=/epsilon1ck. (3) HereE1kandE2kare hybridized 2D bands resulting from an anticrossing of quasi-1D a,b bands along the ( ±π,±π) directions. Furthermore, E3kis the unhybridized 2D xyband. The correspondence to conventional band notation is givenby (1,2,3)≡(α,β,γ ). Their dispersions as obtained from the model described above are shown in Fig. 1(a).T h e three conduction bands were determined in angle-resolvedphotoemission spectroscopy (ARPES) experiments 11,36and their associated Fermi surface sheets are shown in Fig. 1(b). The hybridized dispersions fulﬁll the identity E1k+E2k=/epsilon1ak+/epsilon1bk,E 1kE2k=/epsilon1ak/epsilon1bk−V2 k.(4) In the limit of vanishing hybridization, Vk→0, the hybridized bands are given by E1k=/epsilon1ak−(/epsilon1ak−/epsilon1bk)θH(/epsilon1ak−/epsilon1bk) and E2k=/epsilon1bk+(/epsilon1ak−/epsilon1bk)θH(/epsilon1ak−/epsilon1bk), where θH(···)i st h e Heaviside function. Therefore, a small hybridization rear-ranges corrugated quasi-1D Fermi surface (FS) sheets of/epsilon1 ak,/epsilon1bkwhich are parallel to ky,kx, respectively, into the square-shaped 2D FS sheets of the hybridized E1k,E2k(α,β) bands shown in Fig. 1(b). We note that from Fig. 1(b) theΑ Β Γ X M10622Energy ta 0 Π0Π kxkyq1´ q1 1 2q2´q3´ q4´b FIG. 1. (Color online) (a) Hybridized E1,2(k)(α,β) and unhy- bridized E3(k)(γ) band dispersions according to Eq. (3).( b )F e r m i surface sheets of Sr 2RuO 4with one unhybridized band ( γ) resulting fromxy(=c) orbitals and two hybridized ( α,β) bands resulting from xz(=a)a n dyz(=b) orbitals. Parameters are given below Eq. (2). Characteristic QPI wave vectors qiandq/prime iforγandα,β bands are indicated (cf. Fig. 4). curvature of αorβandγsheets (implying a relative rotation byπ/4) is quite similar. Therefore, there is no reason to make a fundamental distinction concerning their quasi-2D character. III. THE CHIRAL p-WA VE SUPERCONDUCTING GAP FUNCTION OF Sr 2RuO 4 There are numerous SC gap models that have been discussed for Sr 2RuO 4.9,29,37,38The multiband nature implies that the gap sizes and phases may be different on differentsheets. From the experiments mentioned in the Introductionand theoretical analysis, 12,13,38it was concluded that the (“active”) unhybridized 2D γband has the largest gap. This gap is nodeless but has deep minima in the [100] and [010]directions. However, this cannot describe the presence ofnearly nodal quasiparticles concluded from transport 10and thermodynamic9,11measurements. Theoretical analysis12,13,38 suggests that the nodes appear on the much smaller gaps of the (“passive”) hybridized 2D α,β bands and are shifted byπ/4 with respect to the minima on the active bands. From these theoretical and experimental investigations, themultiband nodal chiral triplet gap function of Sr 2RuO 4was established2,12,13as (n=α,β,γ ) dn(k)=/Delta1n 0(T)fn(k)ˆz=/Delta1n(k)ˆz. (5) The form factors fn(k) contain high Fourier components because of the sharp minima in /Delta1n(k). Here we restrict to the lowest two Fourier components in the expansion ofform factors which are already sufﬁcient to ﬁx the qualitativeextremal and nodal structure of the three gaps. We use a modelwhere the gap functions on α,β bands are degenerate in the limit of vanishing hybridization V k→0. This means they will also be degenerate in the orbital basis a,b. Explicitly, written separately for the active and passive gaps we have /Delta1c(k)=/Delta10[sinkx(1+Acosky)+isinky(1+Acoskx)], /Delta1a,b(k)≡/Delta1(k)=/Delta1/prime 0[s i n (kx+ky)[1+A/primecos(kx−ky)] +isin(kx−ky)[1+A/primecos(kx+ky)]]. (6) Both unhybridized a,bandcbands then have the same type of modiﬁed nodal chiral p-wave gap function.12,13The different 134519-2MULTIORBITAL AND HYBRIDIZATION EFFECTS IN THE ... PHYSICAL REVIEW B 88, 134519 (2013) 0Π 2Π 202468101214 Θ103taΑorΒ Γ Total 2 1 0 1 20123 Ω 0Dos tVm0.1t b FIG. 2. (Color online) (a) The variation of superconducting gap /Delta1n(k)(n=α,β,γ )o nt h e hybridized Fermi surfaces as a function of azimuthal angle θ=tan−1(ky/kx) counted from the /Gamma1(0,0) point for β,γ and from the M(π,π) point for α. Gap parameters are /Delta10= 0.045t,/Delta1/prime 0=0.01tandA=0.98,A/prime=− 0.7(\|/Delta1αmax\|∼0.004t, \|/Delta1βmax\|∼0.006t,a n d\|/Delta1γmax\|∼0.014t). (b) Quasiparticle DOS in the superconducting state. The asymmetry is due to the underlyingnormal state DOS. Fermi surface radii and the effect of the hybridization will lead to a splitting of gaps of α,β bands on the respective Fermi surface sheets. For /Delta10=/Delta1/prime 0,A=A/prime=0 and going to the continuum representations one obtains the originalchiral p-wave gap /Delta1 n(k)=/Delta10(kx+iky) proposed by Rice and Sigrist37which is the same on all three bands and has no nodes on the Fermi surface of Fig. 1(b) (theπ/4 rotation of coordinates in the a,bcase is implied here). The chiral nature of the gap in Eq. (6)is compatible with the time-reversal symmetry breaking observed in μSR experiments.5 The above model has four parameters, namely gap ampli- tudes/Delta10,/Delta1/prime 0and higher harmonic contents A,A/prime. They may be determined in such a way that we obtain the basic extremal andnodal structure of gap functions /Delta1 n(θ)i nF i g . 2(a). Note that the maxima and minima of active and passive bands are shiftedby an angle θ 0=π 4as a consequence of the correspondingk-space coordinate rotation in /Delta1a,b(k). The parameters of the above model that reproduce the microscopic gap calculation inRefs. 12,13reasonably well are given in the caption of Fig. 2. IV . CALCULATION OF GREEN’S FUNCTIONS For the calculation of the QPI spectrum, we need the Green’s function in the superconducting state. The impurityscattering will be treated in a Born approximation. This issufﬁcient if we are not interested in the resonance phenomenaassociated with strong scattering. 39,40In the calculation, we include all three bands and their active and passive gapfunctions because they may dominate QPI features for differentranges of the bias voltage V(or frequency ω). For the decoupled nonhybridized single band, the expression of the QPI spectrum is known (e.g., Ref. 39) and will be added in the end. Here we treat the more involved hybridized subsystemof passive a,borbitals which will dominate QPI contributions at low frequencies. Their projected mean-ﬁeld BCS Hamiltonian is written in 8×8 matrix form in terms of the eight-component Nambu spinors /Psi1 † k=(ψ† k,ψ−k) with ψ† k=(c† ka↑,c† kb↑,c† ka↓,c† kb↓), where a,b denote the xz,yz orbitals, respectively. We thenhave (n=a,b) HSC=/summationdisplay knσ(εkn−μ)c† knσcknσ +1 2/summationdisplay knσσ/prime/parenleftbig /Delta1σσ/prime knc† −knσc† knσ/prime+H.c./parenrightbig , (7) where the gap function /Delta1nk=dn(k)·σ(iσy) is given by Eqs. (5)and(6)which is of the unitary type with dn×d∗ n=0 anddn(k)=/Delta1nkˆz.H e r e σ=(σx,σy,σz) are the Pauli matrices in spin space; we also deﬁne the unit as σ0=I. The bare 8 ×8 Green’s function (two Nambu, two orbital, two spin degrees of freedom) in the superconducting state isgiven by ˆG−1(k,iωn)=⎡ ⎢⎢⎢⎣(iωn−/epsilon1ak)σ0 −Vkσ0 −/Delta1akσx 0 −Vkσ0 (iωn−/epsilon1bk)σ0 0 −/Delta1bkσx −/Delta1† akσx 0( iωn+/epsilon1ak)σ0 Vkσ0 0 −/Delta1† bkσx Vkσ0 (iωn+/epsilon1bk)σ0⎤ ⎥⎥⎥⎦. (8) This matrix may be inverted and written in terms of 4 ×4 blocks as ˆG(k,iωn)=/bracketleftbiggG(k,iωn) F(k,iωn) F(k,iωn)†−G(−k,−iωn)/bracketrightbigg . (9) Here the block index is the Nambu spin τz(2) and each 4 ×4 block is indexed by orbital κz(2) and spin σz(2) degrees of freedom. The individual blocks may be written as G(k,iωn)=/bracketleftbiggGaa(k,iωn)Gab(k,iωn) Gba(k,iωn)Gbb(k,iωn)/bracketrightbigg ⊗σ0,F (k,iωn)=/bracketleftbiggFaa(k,iωn)Fab(k,iωn) Fba(k,iωn)Fbb(k,iωn)/bracketrightbigg ⊗σx. (10) We restrict here to the relevant case /Delta1ak=/Delta1bk≡/Delta1kof the Sr 2RuO 4gap model in Eqs. (5)and(6). The general solution will be given in Appendix. We obtain for the orbital matrix elements of normal Green’s functions, Gaa(k,iωn)=D(k,iωn)−1[(iωn−/epsilon1bk)(iωn+E1k)(iωn+E2k)−\|/Delta1k\|2(iωn+/epsilon1ak)], Gbb(k,iωn)=D(k,iωn)−1[(iωn−/epsilon1ak)(iωn+E1k)(iωn+E2k)−\|/Delta1k\|2(iωn+/epsilon1bk)], (11) Gab(k,iωn)=Gba(k,iωn)=D(k,iωn)−1Vk[(iωn+E1k)(iωn+E2k)−\|/Delta1k\|2], 134519-3ALIREZA AKBARI AND PETER THALMEIER PHYSICAL REVIEW B 88, 134519 (2013) and for the anomalous part the result is Faa(q,iωn)=D(k,iωn)−1/Delta1k/bracketleftbig (iωn)2−E2 bk−V2 k/bracketrightbig , Fbb(q,iωn)=D(k,iωn)−1/Delta1k/bracketleftbig (iωn)2−E2 ak−V2 k/bracketrightbig , (12) Fab(q,iωn)=Fba(q,iωn)=D(k,iωn)−1/Delta1kVk(/epsilon1ak+/epsilon1bk). Here the determinant D(k,iωn) is given by D(k,iωn)=/bracketleftbig (iωn)2−E2 ak/bracketrightbig/bracketleftbig (iωn)2−E2 bk/bracketrightbig −2V2 k[(iωn)2+(/epsilon1ak/epsilon1bk−\|/Delta1k\|2)]+V4 k,(13) where the unhybridized superconducting quasiparticle ener- giesEak,Ebkare given by ( n=a,b) E2 nk=/epsilon12 nk+\|/Delta1nk\|2. (14) They are distinct from the hybridized normal state quasiparti- cle energies E1k,E2kdeﬁned in Eq. (3). The determinant may also be factorized [Eq. (A5) ] by using the hybridized supercon- ducting quasiparticle energies given by /Omega12 1,2k=E2 1,2k+\|/Delta1k\|2 in the present case of equal gaps. In the normal state ( /Delta1k≡0) the anomalous Green’s function vanishes, i.e., Fαβ(k,iωn)=0, while the normal Green’s function matrix simpliﬁes to Gaa(k,iωn)=(iωn−/epsilon1bk) (iωn−E1k)(iωn−E2k), Gbb(k,iωn)=(iωn−/epsilon1ak) (iωn−E1k)(iωn−E2k), (15) Gab(k,iωn)=Gba(k,iωn)=Vk (iωn−E1k)(iωn−E2k). Finally, when the hybridization vanishes ( Vk=0) then Gnm(k,iωn)=δnm(iωn−/epsilon1nk)−1, where n=a,bis the usual normal state unhybridized Green’s function matrix. This isequivalent to the cband, where G c(k,iωn)=(iωn−/epsilon1ck)−1. V . IMPURITY SCATTERING We describe the effect of normal impurity scattering within the hybridizing a,bsubspace. For the single corbital, results are completely equivalent without involving the trace overorbital subspace. The elastic scattering potential is given by ˆU(q)=[U c(q)τ3σ0+Um(q)τ0σz]κ0=ˆUc+ˆUm, (16) where we assumed that only intraband scattering ( ∼κ0)i s present. Here σ,τ,κ denote Pauli matrices in spin, Nambu, and orbital ( a,b) space, respectively. In the Born approximation, the full Green’s function including the scattering effect is givenby (k /prime=k−q) ˆGs(k,k/primeiωn)=ˆG(k)δkk/prime+ˆG(k,iωn)ˆU(q)ˆG(k/prime,iωn).(17)The single-particle density of states by the scattering is then obtained as (per spin) Ns(q,iωn)=−1 π1 2NIm/summationdisplay k[trστκˆGs(k,k/primeiωn)] =N(iωn)+δN(q,iωn), (18) where N(iωn)=(1/N)/summationtext kκδ(ω−Ekκ) is the background DOS of hybridized bands and δN(q,iωn) is the modiﬁcation of the local DOS due to impurity scattering. It may be writtenin terms of the QPI function ˜/Lambda1(q,iω n)( f o ra,borbitals) as δN(q,iωn)=−1 πIm˜/Lambda10(q,iωn), (19) ˜/Lambda10(q,iωn)=1 2N/summationdisplay ktrστκˆGkˆUˆGk−q. VI. THE QUASIPARTICLE INTERFERENCE SPECTRUM The QPI function in a,borbital subspace for nonmagnetic scattering ( Um=0) in the charge channel in the Born approximation is given by ˜/Lambda10(q,iωn)=Uc/Lambda1/prime 0(q,iωn) with /Lambda1/prime 0(q,iωn)=1 2N/summationdisplay ktrσταˆGkτ3σ0α0ˆGk−q. (20) The calculation of /Lambda1/prime 0(q,iωn) may now proceed numerically as is usually done. However, here we use the fully analyticalclosed solution for the QPI spectrum because it gives consid-erably more insight. In particular, the relation to special casesof the model becomes clearer. For that purpose we performthe traces and use the explicit analytical form of the orbitalmatrix elements of normal and anomalous Green’s functionsin Eqs. (11) and(12). This leads to the QPI function per spin ina,borbital subspace given by ( n,m=a,b) /Lambda1 /prime 0(q,iωn)=1 N/summationdisplay k,nm[Gnm(k)Gnm(k−q) −Fnm(k)Fnm(k−q)∗]. (21) To this the contribution of the unhybridized corbital has to be added, which is explicitly given by /Lambda10(q,iωn)=1 N/summationdisplay k(iωn+/epsilon1ck)(iωn+/epsilon1ck−q)−/Delta1ck/Delta1∗ ck−q (iωn)2−/parenleftbig /epsilon12 ck+\|/Delta1ck\|2/parenrightbig . (22) The total Born QPI spectrum /Lambda1t 0(q,iωn)=/Lambda10(q,iωn)+ /Lambda1/prime 0(q,iωn) of active and passive bands, respectively, is then obtained as a closed solution from Eqs. (21) and (22) and Eqs. (11)–(13) for the individual matrix elements Gnm(k),Fnm(k). Here we made the simplifying assumption that tunneling matrix elements of a,b, andcorbitals are equal. It is useful to consider the result ﬁrst for the normal state(/Delta1 k=/Delta1ck=0). In this case, using Eq. (15) it simpliﬁes to /Lambda1t 0(q,iωn)=1 N/summationdisplay k,n=a,b/bracketleftbigg(iωn−/epsilon1nk)(iωn−/epsilon1nk−q)+VkVk−q (iωn−E1k)(iωn−E2k)(iωn−E1k−q)(iωn−E2k−q)/bracketrightbigg +1 N/summationdisplay k/bracketleftbigg1 (iωn−/epsilon1ck)(iωn−/epsilon1ck−q)/bracketrightbigg .(23) 134519-4MULTIORBITAL AND HYBRIDIZATION EFFECTS IN THE ... PHYSICAL REVIEW B 88, 134519 (2013) It further reduces to /Lambda1t 0(q,iωn)=(1/N)/summationtext nk(iωn− /epsilon1nk)−1(iωn−/epsilon1nk−q)−1with n=a,b,c for unhybridized bands ( Vk=0). The QPI spectrum in Eq. (23) is only determined by the dispersion of the three bands and will mapthe prominent wave vectors of their corresponding surfaces ofconstant energy ω. VII. DISCUSSION OF NUMERICAL RESULTS FOR THE THREE-BAND CHIRAL GAP MODEL The band structure and associated Fermi surface model for Sr 2RuO 4is shown in Fig. 1consisting of the hybridized α,β and one unhybridized γband. Typical wave vectors qi,q/prime i characterizing the FS sheet dimensions are indicated (b). These should appear prominently in the normal-state QPI functions.We note, however, that the full QPI landscape in ( q x,qy) space may not be completely characterized by such characteristicwave vectors and they may not always be unambiguouslyidentiﬁed. The simpliﬁed gap model of Eq. (6)on this Fermi surface is shown in Fig. 2(a). It reproduces the overall extremal and nodal behavior obtained by a fully microscopic model in Refs. 12 and 13, in particular the shifted minima or nodes of the gap functions on active ( γ) and passive ( α,β) bands. Since the model of Eq. (6)includes only two Fourier components for each band, there are, however, quantitative differences to thefull calculation in Refs. 12and 13. This has little inﬂuence on the overall appearance of the QPI spectra. The associatedquasiparticle DOS for this gap model is shown in Fig. 2(b). 0 Π0Π kxkya 0.003 tq5,6´ q7´ 0 Π0Π kxkyb 0.006 t 0 Π0Π kxkyc 0.011 t q25q6 0 Π0Π kxkyd 0.016 t q8,7q3´ FIG. 3. (Color online) Surfaces of constant quasiparticle energy /Omega1nk=ω(n=1–3) for various ωin the superconducting state [cf. Fig. 2(a)]. For small ω, only surfaces connected with α,β bands are present, ﬁrst as arcs around [110]-type directions. For larger energies, the surfaces of the γband also appear, ﬁrst as lenses along [100]-type directions rotated by π/4 with respect to low energy α,β sheets. Characteristic QPI wave vectors qiandq/prime iforγandα,β bands are indicated (cf. Fig. 5). FIG. 4. (Color online) QPI spectrum for the normal state. Charac- teristic QPI wave vectors qiandq/prime iassociated with γandα,β bands are indicated [cf. Fig. 1(b)]. Note that the γ-band DOS and therefore the total DOS are slightly asymmetric. This is due to the behavior of normal-state DOS around the Fermi level. It is determined by theasymmetric behavior of the γband dispersion around the X point [Fig. 1(a)]. The features of QPI functions in the SC state are determined by the shape of constant quasiparticle energy surfaces givenby/Omega1 nk=ω(n=1–3). They are shown in Fig. 3. For small ω/lessmuch/Delta1/prime 0, ﬁrst the double arc-shaped sheets around the [110]- type nodal directions of the α,β bands appear (a). For ω> /Delta1/prime 0/2, one basically obtains the (doubled) constant energy surface sheets of the normal state (b). When the frequency ω increases above the minimum of the γ-band, lens-shaped γ- sheets around the [100]-type extremal directions appear whichare rotated by π/4 with respect to low-energy α-βarc-shaped sheets (c). Finally, when ωis above the maximum gap in Fig. 2, doubled normal constant energy surface sheets split by the gapappear also for the γband. The prominent connecting wave vectors of those sheets, if they appear in the QPI spectrum,should give information on Fermi surface structure, and inthe superconducting state they should give direct evidencefor the nodal structure of the gap function. Several candidatewave vectors are indicated in Fig. 3. For clarity, we denote by q iandq/prime i(i=1,2,3,... ) wave vectors connecting points on equal energy surfaces of γandα,β bands, respectively. We do not distinguish between wave vectors related by fourfoldsymmetry. First we discuss the normal state in Fig. 4where we show the QPI spectrum for four increasing energy values 134519-5ALIREZA AKBARI AND PETER THALMEIER PHYSICAL REVIEW B 88, 134519 (2013) FIG. 5. (Color online) QPI spectrum for the superconducting state. Characteristic QPI wave vectors qiandq/prime iassociated with γ andα,β bands are indicated (cf. Fig. 3). [cf. Figs. 2(a) and3]. At the wave vectors q1,q/prime 1,q/prime 2associated with the main across-Fermi surface scattering processes,clearly line structures are seen in the QPI spectrum at allenergies ω. Note that q 1,q/prime 1are folded back into the ﬁrst BZ. To compare with the vectors in Fig. 1(b), one has to add zone boundary vectors ( π,0) and (0 ,π), respectively. There are also weaker lines emanating from the zone center and forminga split cross, in particular those visible in Fig. 4(c).T h e y can be associated with parallel scattering along ( α,β)-sheets including hybridization-induced interband scattering betweenα,β bands. It will appear according to Eq. (15), although there is no interband scattering potential. The split crossesare obtained by tracking wave vectors of the type q /prime 3(and the one reﬂected at the symmetry plane) in Fig. 1(b) from zero to the zone boundary. Furthermore, the small wave vectoraxis-aligned cross features in Fig. 4are due to scattering parallel to αβsurfaces with wave vector q /prime 4. Therefore, all major features observed in normal state QPI of Fig. 4can be reasonably understood from the hybridized three-band Fermisurface structure. Now we turn to the QPI in the chiral p-wave superconduct- ing state described by Eq. (6). By tuning the bias voltage or frequency, it is clear that the most signiﬁcant information onthe gap function may be obtained in situations like Figs. 3(a) and 3(c), where the small arc-and lens-shaped sheets ﬁrst appear around the nodal or extremal directions, respectively.These small sheets have points of high curvature and may showup as distinguished features in the QPI.In Figs. 5(a) and5(b), the QPI signature of the small gap on theα,β bands at q /prime 5–q/prime 7[Fig. 3(a)] is apparently rather weak. This is due to the smallness of the gap, /Delta1/prime 0/t=10−2. However, clearly the intensities at q/prime 5,6as compared to neighboring wave vectors are enhanced with respect to the normal state. Thesituation here is quite different from the heavy fermion systemCeCoIn 5,20where the gap is only about one order of magnitude less than the effective hopping. Then the QPI in the SC stateshows up more clearly. This situation changes when the energy is raised to the region of the large gap on the active γsheet. The scattering vectors connecting the lens-shaped γsurface sheets in Fig. 3(c) at wave vectors q 2–q5clearly turn up as separate features in the QPI of Fig. 5(c). They partly survive to even higher energy in Fig. 5(d) when the constant energy surfaces of the γband are already reconnected again [Fig. 3(d)]. The observation of this wave-vector quadruplet q2–q5above some threshold energy ω0 would be a clear indication of the active gap having a minimum of the size /Delta1min/similarequalω0in the [100]-type directions. In the low momentum region of Fig. 5(c) it is also possible to identify the intralens scattering vector q6of Fig. 3(c). Furthermore, the vector q/prime 3in Fig. 5(d) is apparently related to the axis parallel scattering in Fig. 3(d) made possible by the doubling of α,β sheets in the superconducting state. VIII. CONCLUSION AND OUTLOOK In this work, we investigated the QPI spectrum of a multi- band chiral p-wave superconductor Sr 2RuO 4.O u rw o r k i n g model is a simpliﬁed version of the microscopic three-bandmodel studied ﬁrst in Refs. 12and 13and experimentally proved in Refs. 2and11. It consists of a nonhybridized xy-type active γband with a Fermi surface that supports the main chiral gap function. The latter has deep minima along [100]-typedirections. A secondary near nodal gap is supported by thehybridizing ( xz,yz )-type α,βbands with a near nodal structure that is rotated by π/4 with respect to the minima of the large γ band gap. In the normal state, the basic across-Fermi surfacescattering appears as clear line features in the QPI spectrumoriginating from all three bands. The QPI changes due the α,β gap opening are quite subtle due to the smallness of the gapand they are caused by low momentum scattering between thearc-shaped α-βsurfaces in Fig. 3(a). A clearer signature in QPI is left by the dominant chiral p-wave gap on the γsurface. The scattering due to lens-type constant energy surfaces [Fig. 3(c)] around the gap minima positions along the [100] direction leads to a quadruplet ofwave vectors that can be identiﬁed in the QPI spectrum. Thisobservation would support the existence of the minimum inthe main gap. Further less prominent wave vectors may alsobe identiﬁed in the QPI structure. In general, the QPI analysis should be focused in those voltage regions where equal energy surfaces have the shapeas shown in Figs. 3(a) and 3(c). Outside these regions the equal energy surfaces are, aside from the doubling, quitesimilar to the normal state [Figs. 3(a) and3(c)] and then little change may be expected. It would be most interesting to seewhether QPI can conﬁrm the relative π/4 rotation of (near) nodal positions on α-βand the extremal positions γband from the characteristic wave vectors in Figs. 3(a) and3(c) and 134519-6MULTIORBITAL AND HYBRIDIZATION EFFECTS IN THE ... PHYSICAL REVIEW B 88, 134519 (2013) Figs. 5(a) and 5(c). Our results suggest that it is worthwhile to investigate Sr 2RuO 4using the QPI method to learn more about its electronic structure, particularly the chiral p-wave gap function. APPENDIX In this appendix, we discuss the most general case of the QPI when the gap functions for hybridizing orbitals/Delta1ak,/Delta1bkmay be unequal. Generally this implies a breaking of fourfold symmetry of QPI spectra in the tetragonal planewhen the gap amplitudes /Delta1 /prime n0forn=a,bare different. There is no evidence of spontaneous fourfold symmetry breakingin the superconducting phase of Sr 2RuO 4from ﬁeld-angle- dependent speciﬁc-heat analysis.9Nevertheless, we include this case here because it may be useful for other multibandsuperconductors. We obtain Gaa(k,iωn)=D(k,iωn)−1[(iωn−/epsilon1bk)(iωn+E1k)(iωn+E2k)−\|/Delta1bk\|2(iωn+/epsilon1ak)], Gbb(k,iωn)=D(k,iωn)−1[(iωn−/epsilon1ak)(iωn+E1k)(iωn+E2k)−\|/Delta1ak\|2(iωn+/epsilon1bk)], (A1) Gab(k,iωn)=D(k,iωn)−1Vk[(iωn+E1k)(iωn+E2k)−/Delta1ak/Delta1∗ bk], Gba(k,iωn)=D(k,iωn)−1Vk[(iωn+E1k)(iωn+E2k)−/Delta1∗ ak/Delta1bk], where the determinant is given by D(k,iωn)=/bracketleftbig (iωn)2−E2 1k/bracketrightbig/bracketleftbig (iωn)2−E2 2k/bracketrightbig −\|/Delta1ak\|2/bracketleftbig (iωn)2−/epsilon12 bk/bracketrightbig −\|/Delta1bk\|2/bracketleftbig (iωn)2−/epsilon12 ak/bracketrightbig +(/Delta1∗ ak/Delta1bk+/Delta1ak/Delta1∗ bk)V2 k+\|/Delta1ak\|2\|/Delta1bk\|2, (A2) or equivalently it can be expressed as D(k,iωn)=/bracketleftbig (iωn)2−E2 ak/bracketrightbig/bracketleftbig (iωn)2−E2 bk/bracketrightbig −2V2 k{(iωn)2+[/epsilon1ak/epsilon1bk−1 2(/Delta1∗ ak/Delta1bk+/Delta1ak/Delta1∗ bk)]}+V4 k. (A3) Likewise the orbital matrix elements of the anomalous Green’s function are obtained as Faa(q,iωn)=D(k,iωn)−1/braceleftbig /Delta1ak/bracketleftbig (iωn)2−E2 bk/bracketrightbig −/Delta1bkV2 k/bracerightbig , Fbb(q,iωn)=D(k,iωn)−1/braceleftbig /Delta1bk/bracketleftbig (iωn)2−E2 ak/bracketrightbig −/Delta1akV2 k/bracerightbig , (A4) Fab(q,iωn)=D(k,iωn)−1Vk[−/Delta1ak(iωn−/epsilon1bk)+/Delta1bk(iωn+/epsilon1ak)], Fba(q,iωn)=D(k,iωn)−1Vk[/Delta1ak(iωn+/epsilon1bk)−/Delta1bk(iωn−/epsilon1ak)]. The determinant may also be written by using the true quasiparticle energies /Omega11,2(k) in the hybridized and superconducting case according to D(k,iωn)=/bracketleftbig (iωn)2−/Omega12 1k/bracketrightbig/bracketleftbig (iωn)2−/Omega12 2k/bracketrightbig , (A5) with /Omega12 1k=1 2/parenleftbig E2 ak+E2 bk/parenrightbig +V2 k−1 2/braceleftbig/parenleftbig E2 ak−E2 bk/parenrightbig2+4V2 k[(/epsilon1ak+/epsilon1bk)2+\|/Delta1ak−/Delta1bk\|2]/bracerightbig1 2, (A6) /Omega12 2k=1 2/parenleftbig E2 ak+E2 bk/parenrightbig +V2 k+1 2/braceleftbig/parenleftbig E2 ak−E2 bk/parenrightbig2+4V2 k[(/epsilon1ak+/epsilon1bk)2+\|/Delta1ak−/Delta1bk\|2]/bracerightbig1 2. For the hybridized case with equal gaps ( /Delta1ak=/Delta1bk=/Delta1k), the above equation simpliﬁes to /Omega12 1,2k=E2 1,2k+\|/Delta1k\|2. 1A. P. Mackenzie and Y . Maeno, Rev. Mod. Phys. 75, 657 (2003). 2Y . Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida, J. Phys. Soc. Jpn. 81, 011009 (2012). 3K. Ishida, H. Mukuda, Y . 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PhysRevB.83.205206.pdf	PHYSICAL REVIEW B 83, 205206 (2011) Response properties of III-V dilute magnetic semiconductors including disorder, dynamical electron-electron interactions, and band structure effects F. V . Kyrychenko and C. A. Ullrich Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA (Received 27 January 2011; revised manuscript received 17 March 2011; published 19 May 2011) A theory of the electronic response in spin and charge disordered media is developed with the particular aim to describe III-V dilute magnetic semiconductors like Ga 1−xMnxAs. The theory combines a detailed k·pdescription of the valence-band, in which the itinerant carriers are assumed to reside, with ﬁrst-principles calculations ofdisorder contributions using an equation-of-motion approach for the current response function. A fully dynamictreatment of electron-electron interaction is achieved by means of time-dependent density-functional theory. Itis found that collective excitations within the valence-band signiﬁcantly increase the carrier relaxation rate byproviding effective channels for momentum relaxation. This modiﬁcation of the relaxation rate, however, hasonly a minor impact on the infrared optical conductivity in Ga 1−xMnxAs, which is mostly determined by the details of the valence-band structure and found to be in agreement with experiment. DOI: 10.1103/PhysRevB.83.205206 PACS number(s): 72 .80.Ey, 75 .50.Pp, 78 .20.Bh I. INTRODUCTION The idea of using both charge and spin of electrons in a new generation of electronic devices constitutes thebasis of spintronics. 1The magnetic properties of the mate- rial combined with its semiconducting nature makes dilutemagnetic semiconductors (DMSs) potentially appealing forvarious spintronics applications. 2In particular, the effect of carrier-mediated ferromagnetism opens up the possibility tocontrol the electron spin and magnetic state of a system ordevice by means of an electric ﬁeld. A lot of attention isdrawn to Ga 1−xMnxAs due to the well-developed technology of the conventional GaAs-based electronics and discovery ofits relatively high ferromagnetic transition temperature, 2with a current record of Tc=185 K.3 Unlike most other III-V DMSs, the nature of the itinerant carriers in Ga 1−xMnxAs is still under debate.4,5It is widely accepted that for low-doped insulating samples the Fermienergy lies in a narrow impurity band. For more heavilydoped, high- T cmetallic samples there are strong indications that the impurity band merges with the host semiconductorvalence band, forming mostly hostlike states at the Fermienergy with some low-energy tail of disorder-related localizedstates. 6First-principles calculations7–9have so far not been fully conclusive regarding the nature of the itinerant carriersin this regime, and further theoretical studies continue tobe necessary. Meanwhile, attention has shifted to modelHamiltonian approaches assuming either the valence-band 10or impurity-band11picture and their ability to adequately describe the experimental results in Ga 1−xMnxAs. The extreme sensitivity of the magnetic and transport prop- erties of Ga 1−xMnxAs to details of the growth conditions12 and postgrowth annealing13–15points to the crucial role played by the defects and their conﬁgurations. This has stimulated in-tense research on the structure of defects and their inﬂuence onthe various properties of the system. 16It is essential, therefore, to develop a theory of electrical conductivity in DMSs withemphasis given to disorder and electron-electron interactions,without neglecting the intricacies of the electronic bandstructure. Several previous theoretical studies of Ga 1−xMnxAs,based on the assumption of the valence-band nature of itinerant holes, treat the band structure in detail, while disorder andmany-body effects are accounted for only by using simplephenomenological relaxation-time approximations and staticscreening models. 17–19Other studies of the magnetic and transport properties of DMSs include microscopic treatmentsof disorder effects, 20–25but use simpliﬁed model descriptions of the band structure. Here we present a comprehensive theory for the electron dynamics in DMSswhich accounts for the complexity of thevalence-band structure of the semiconductor host material andtreats disorder and electron-electron interaction on an equalfooting. In previous work we used a simpliﬁed treatment ofthe semiconductor valence band 26,27or considered only static properties of the system.28In this paper we simultaneously account for the complexity of the valence band, use a ﬁrst-principles approach to describe disorder contributions, andemploy a fully dynamic treatment of electron interactions. To account for the valence-band structure we use the generalized k·papproach 29where a certain number of bands are treated exactly while the contribution from the remotebands is included up to second order in momentum. For ourpurposes (in the optimally annealed regime, with itinerantvalence band holes) the k·papproach is an ideal compromise: It captures the essential features of the band structure whichdominate the infrared response of Ga 1−xMnxAs, while being computationally much less expensive than a fully ab initio treatment. The latter would be more appropriate for acceptorlevels that are spatially localized or deep in the gap. 4 To describe disorder effects we use the equation of motion for the paramagnetic current response function of the fully dis-ordered system. This approach has some similarities to modelsdeveloped earlier using the memory function formalism. 30–32 The advantage of our approach as compared with the memory function formalism is the relative simplicity and transparencyof the derivation and the straightforward possibility to includethe spin degree of freedom. Another advantage is that ourformalism is expressed in terms of a current-current anda set of density and spin-density response functions. Thisenables us to use the powerful apparatus of time-dependent 205206-1 1098-0121/2011/83(20)/205206(13) ©2011 American Physical SocietyF. V . KYRYCHENKO AND C. A. ULLRICH PHYSICAL REVIEW B 83, 205206 (2011) density-functional theory (TDDFT)33to treat many-body effects such as dynamic screening and collective excitationsof the itinerant carriers in principle exactly. The paper is divided into two major sections and con- clusions. For ease of reading, some of the derivations arepresented in appendices. The theory section (Sec. II)i s organized as follows. In Sec. II A we present our general formalism based on the equation of motion of the current-current response function of the disordered system. In Sec. II B we describe the evaluation of the current-current, density andspin-density response functions for the multiband system usinga generalized k·pperturbation approach. Next, in Sec. II C we show the treatment of electron-electron interaction bymeans of TDDFT. In Sec. IIIwe ﬁrst discuss the new features that the valence-band character of itinerant carriers brings intothe system, namely the dominance of the long-wavelengthside of the single-particle excitation spectrum by the interbandspin transitions and the effective suppression of the collectiveplasmon excitations within the valence band for the wholerange of momentum. Next, in Sec. III B we discuss the effect of magnetic doping: spin and charge disorder in the systemand modiﬁcation of the band structure in the magneticallyordered phase. We show that the full dynamic treatment ofelectron-electron interactions allows us to capture the effect ofcollective excitations on the carrier relaxation time. We thencompare our results also with experimental data on infraredconductivity. Finally, in Sec. IVwe draw our conclusions. II. THEORY A. General formalism We discuss a system described by the Hamiltonian ˆH=ˆHe+ˆHm+ˆHd, (1) where ˆHeis the contribution of the itinerant carriers and ˆHm represents the subsystem of localized magnetic spins. These two terms constitute the “clean” part of the total Hamiltonian.The last term in Eq. ( 1) describes disorder in the system: ˆH d=V2/summationdisplay kˆ/vectorU(k)·ˆ/vectorρ(−k), (2) where the four-component charge and spin disorder scattering potential, ˆ/vectorU(k)=1 V/summationdisplay j⎛ ⎜⎜⎜⎜⎜⎝U j(k) −J 2/parenleftbigˆSz j−/angbracketleftS/angbracketright/parenrightbig −J 2ˆS− j −J 2ˆS+ j⎞ ⎟⎟⎟⎟⎟⎠e ik·Rj, (3) is coupled to the four-component vector of charge- and spin- density operators of the itinerant carriers: ˆ/vectorρ=⎛ ⎜⎝ˆρ1 ˆρz ˆρ+ ˆρ−⎞ ⎟⎠=⎛ ⎜⎝ˆn ˆsz ˆs+ ˆs−⎞ ⎟⎠, (4)with the components ˆρμ(k)=1 V/summationdisplay q/summationdisplay nn/prime/angbracketleftun/prime,q−k\|σμ\|un,q/angbracketrightˆa+ n/prime,q−kˆan,q. (5) Here, σμ(μ=1,z,+,−) is deﬁned via the Pauli matrices, where σ1is the 2 ×2 unit matrix, σ±=(σx±iσy)/2, and\|un,q/angbracketrightare the two-component Bloch function spinors with wave vector qand band index n. The summation in Eq. ( 3) is performed over all defects. Note that the mean-ﬁeld part of the p-dexchange interaction between itinerant holes and localized spins is absorbed into the clean system bandstructure Hamiltonian ˆH e; disorder in our model consists of the Coulomb potential of charge defects and ﬂuctuations oflocalized spins around the mean-ﬁeld value /angbracketleftS/angbracketright. The general case of multiple types of defects, including defect correlations, was considered in Ref. 26. For simplicity we here include only the most important defect type, namelyrandomly distributed manganese ions in gallium substitu-tional positions (Mn Ga). Our model treats localized spins as quantum mechanical operators coupled to the band carriersvia a contact Heisenberg interaction featuring a momentum-independent exchange constant J. We use the value of VJ= −55 meV nm 3, which corresponds to the widely used DMS p-dexchange constant N0β=− 1.2e V .10Thezaxis is chosen along the direction of the macroscopic magnetization. Earlier we developed a theory of transport in charge and spin disordered media with emphasis on a treatment of disorderand electron-electron interaction. 27It is based on an equation- of-motion34,35approach for the paramagnetic current-current response of the full, disordered system: χjpαjpβ(r,r/prime,τ)=−i ¯h/Theta1(τ)/angbracketleft[ˆjpα(τ,r),ˆjpβ(r/prime)]/angbracketrightH, (6) where ˆjpα(τ,r)=ei ¯hˆHτˆjpα(r)e−i ¯hˆHτ(7) is the paramagnetic current-density operator in Heisenberg representation and α,β=x,y,z are Cartesian coordinates. During the derivation we assumed our system to be macro- scopically homogeneous, which implies that the coherencelength of the electrons is much shorter than the system size.In this case, summing over all electrons will leave us withan averaged effect of disorder that does not depend on theparticular disorder conﬁguration. For such macroscopicallyhomogeneous systems the response at point rdepends only on the distance \|r−r /prime\|to the perturbation and not on the particular choice of points randr/prime.T h e a posteriori justiﬁcation for this assumption is that we will apply our formalism in theweak-disorder limit on the metallic side of the metal-insulatortransition in Ga 1−xMnxAs. Another major approximation involves the decoupling procedure, where we neglect the inﬂuence of the itinerantcarriers on the localized spins. Therefore, our approachdoes not include magnetic polaron effects and lacks themicroscopic features of carrier mediated ferromagnetism. Thelatter, however, can be reinstated to some extent by introducinga phenomenological Heisenberg-like term in the magneticsubsystem Hamiltonian ˆH m. Details of the derivation are presented in Ref. 27. Thus, instead of calculating the Curie temperature for our DMS system, we take it as an input 205206-2RESPONSE PROPERTIES OF III-V DILUTE MAGNETIC ... PHYSICAL REVIEW B 83, 205206 (2011) parameter to deﬁne the temperature-dependent magnetization of the localized spin subsystem. The coupling to the itinerantcarriers then occurs via the ﬂuctuations of the localized spinsthat come in through the disorder potential, Eq. ( 3). The ﬁnal expression for the total current response reads χ J αβ(q,ω)=χc jpαjpβ(q,ω)+n mδαβ +V2 m2ω2/summationdisplay kkαkβ/summationdisplay μν/angbracketleftˆUμ(k)ˆUν(−k)/angbracketrightHm ×/parenleftbig χρμρν(q−k,ω)−χc ρμρν(−k)/parenrightbig , (8) where χρμρν(k,ω) is the set of charge- and spin-density response functions with respect to operators ( 4) and ( 5) and the superscript “ c” indicates quantities deﬁned in the clean system. By comparing Eq. ( 8) with the Drude formula in the weak-disorder limit ωτ/greatermuch1, χJ D(ω)=n m1 1+i/ωτ≈n m−in mωτ, (9) we identify the tensor of Drude-like frequency- and momentum-dependent relaxation rates of the form τ−1 αβ(q,ω)=iV2 nmω/summationdisplay k μνkαkβ/angbracketleftˆUμ(−k)ˆUν(k)/angbracketrightHm ×/parenleftbig χρμρν(q−k,ω)−χc ρμρν(k,0)/parenrightbig .(10) Note that the right-hand side of Eqs. ( 8) and ( 10) contains the set of spin and charge response functions of the full,disordered system. Therefore, strictly speaking, Eq. ( 8) should be evaluated self-consistently 36with the continuity equations closing the loop. Here we use a simpliﬁed approach based ontwo approximations. First, taking the weak-disorder limit inthe right hand side of Eq. ( 10) we retain terms up to the second order in components of the disorder potential. In other words,the spin and charge response functions of the full system inEq. ( 10) are replaced by their clean system counterparts: χ ρμρν(q−k,ω)→χc ρμρν(q−k,ω). (11) Next we assume that the paramagnetic current response function of the full system may be expressed as the cleansystem response function with a lifetime broadening given byEq. ( 10): χ jpαjpβ(q,ω)≈χc jpαjpβ/parenleftbig q,ω−iτ−1 αβ/parenrightbig . (12) Equations ( 10)–(12) will be used in the following section. B. Multiband k ·p approach To obtain the conductivity through Eqs. ( 10)–(12) we will have to calculate the paramagnetic current response and spin-and charge-density response functions of the clean system. Toproperly describe the complexity of the semiconductor valenceband we are going to implement the multiband k·papproach. First we derive the current and density response functions in the formal basis of the Bloch states \|n,k/angbracketright=1 √ Veik·r\|un,k/angbracketright, (13)which diagonalize the clean system Hamiltonian ˆH=/summationdisplay n,kεn,kˆa+ n,kˆan,k. (14) Within second quantization in basis ( 13), the paramagnetic current in the system with a spin-orbit interaction is given by ˆjp(q)=1 V/summationdisplay n,n/prime,k/bracketleftbigg¯h m0/parenleftbigg k−1 2q/parenrightbigg /angbracketleftun/prime,k−q\|un,k/angbracketright +1 m0/angbracketleftun/prime,k−q\|ˆ/vectorπ\|un,k/angbracketright/bracketrightbigg ˆa+ n/prime,k−qˆan,k, (15) with ˆ/vectorπ=ˆp+¯h 4m0c2[ˆσ×ˆ∇Uc], (16) where Ucis the periodic crystal-ﬁeld potential. Hereafter, when performing the real space integration, we assume thatthe envelope function varies slowly on the scale of a unit cell. Introducing the time dependence of the creation and destruction operators in Eq. ( 15), the paramagnetic current response of the multiband system can be directly evaluated,and one ﬁnds χ c jpαjpβ(q,ω)=1 Vm2 0/summationdisplay n,n/prime,kfn/prime,k−q−fn,k εn/prime,k−q−εn,k+¯hω+iη ×/bracketleftbigg ¯h/parenleftBig kα−qα 2/parenrightBig /angbracketleftun/prime,k−q\|un,k/angbracketright +/angbracketleftun/prime,k−q\|ˆπα\|un,k/angbracketright/bracketrightbigg/bracketleftbigg ¯h/parenleftBig kβ−qβ 2/parenrightBig ×/angbracketleftun,k\|un/prime,k−q/angbracketright+/angbracketleftun,k\|ˆπβ\|un/prime,k−q/angbracketright/bracketrightbigg . (17) A similar procedure for the spin- and charge-density response yields χc ρμρν(q,ω)=1 V/summationdisplay n,n/prime,kfn/prime,k−q−fn,k εn/prime,k−q−εn,k+¯hω+iη /angbracketleftun/prime,k−q\|ˆσμ\|un,k/angbracketright/angbracketleftun,k\|ˆσν\|un/prime,k−q/angbracketright. (18) All we need now for evaluating Eqs. ( 17) and ( 18)i st o determine the form of the periodic Bloch functions \|un,k/angbracketright that diagonalize the clean system Hamiltonian. The commonapproach is to diagonalize the multiband k·pHamiltonian that treats certain bands exactly and treats contributionsfrom remote bands up to second order in momentum. Thederivation of such a Hamiltonian is outlined in Appendix A. By diagonalizing the matrix of this Hamiltonian, however,we obtain the eigenvectors of the modiﬁed Hamiltonian ( A7). Before evaluating the matrix elements between Bloch periodicfunctions \|u n,k/angbracketrightin Eqs. ( 17) and ( 18) we therefore have to perform the unitary transformation Eq. ( A4). Details of these calculations are presented in Appendix B. The ﬁnal expression for the paramagnetic current response function in the long-wave limit q=0 (since we are looking 205206-3F. V . KYRYCHENKO AND C. A. ULLRICH PHYSICAL REVIEW B 83, 205206 (2011) for the optical response) is given by χc jpαjpβ(ω)=1 Vm2 0/summationdisplay n,n/prime,kfn/prime,k−fn,k εn/prime,k−εn,k+¯hω+iη ×/bracketleftBigg/summationdisplay s/primesB∗ s/prime(n/prime,k)Bs(n,k)m0 ¯h∂ ∂kα/angbracketlefts/prime\|¯H\|s/angbracketright/bracketrightBigg /bracketleftBigg/summationdisplay s/primesB∗ s(n,k)Bs/prime(n/prime,k)m0 ¯h∂ ∂kβ/angbracketlefts\|¯H\|s/prime/angbracketright/bracketrightBigg ,(19) where ¯Hdenotes the effective multiband k·pHamiltonian (A7) and B(n,k) is its eigenvector for the state with energy εn,k. The charge- and spin-density response is approximated by χc ρμρν(q,ω)≈1 V/summationdisplay n,n/prime,kfn/prime,k−q−fn,k εn/prime,k−q−εn,k+¯hω+iη ×/summationdisplay s/prime,s,τ,τ/primeB∗ s/prime(n/prime,k−q)Bτ/prime(n/prime,k−q) ×Bs(n,k)B∗ τ(n,k)/angbracketlefts/prime\|ˆσμ\|s/angbracketright/angbracketleftτ\|ˆσν\|τ/prime/angbracketright.(20) If ˆσμ=(ˆσν)+, i.e., for χnn,χszsz, andχs±s∓, the second sum is a real quantity. Then, the imaginary part is /Ifractur/bracketleftbig χc ρμ(ρμ)+(q,ω)/bracketrightbig =−π (2π)3/summationdisplay n,n/prime/integraldisplay d3k(fn/prime,k−q−fn,k) ×δ[¯hω−(εn,k−εn/prime,k−q)] ×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay s,s/primeB∗ s/prime(n/prime,k−q)Bs(n,k)/angbracketlefts/prime\|ˆσμ\|s/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (21) It is seen that in the long-wavelength limit ( q→0) that the imaginary part of the density response ( σμ≡σ1) vanishes as a product of orthogonal states, while the imaginary part ofspin response is, in general, ﬁnite. We conclude from this thatthe long-wavelength spectrum of single-particle excitations isdominated by spin transitions. The calculations were performed within an 8-band k·p model. The basis functions and explicit form of the Hamilto-nian matrix are presented in Appendix C. C. Electron-electron interaction A major advantage of our formalism is that it is expressed in terms of current and density response functions. This allowsus to use the powerful apparatus of TDDFT to account for theeffects of electron-electron interaction. Let us ﬁrst examine the current response of the clean system. In this paper we are considering the optical response, i.e., theresponse to transverse perturbations. Since transverse pertur- bations induce only a transverse response in a homogeneoussystem, there are no density ﬂuctuations directly created byan electromagnetic ﬁeld. The total current response of theinteracting system in this case can be expressed as (χ J(q,ω))−1=(χJ 0(q,ω))−1+4πe ω2−c2q2+q2 ω2vqGT+, (22) where χJ 0is the response of the noninteracting system, vq is the Coulomb interaction, and the local ﬁeld factor GT+ represents corrections from the exchange-correlation (xc) part of the electron interaction. The corrections to the transverse current response function caused by electron-electron interaction are relativisticallysmall in this case and can be neglected. So, for the trans-verse current response of the clean system we will use thenoninteracting form. The set of the density and spin-density response functions of the clean system enters our expression ( 10) for the frequency- and momentum-dependent relaxation rates. TDDFT allowsus to describe all the effects of electron interaction, includingcorrelations and collective modes, in principle, exactly. Withinthe TDDFT formalism the charge- and spin-density responsesof the interacting system can be expressed as: 37 χ−1(q,ω)=χ0−1(q,ω)−v(q)−fxc(q,ω), (23) where all quantities are 4 ×4 matrices and χ0denotes the matrix of response functions of the noninteracting system,v(q) is the Hartree part of the electron-electron interactions, andf xcrepresents xc corrections in the form of local ﬁeld factors. As a simpliﬁcation we use only the exchange part off xcand apply the adiabatic local spin density approximation. Explicit expressions for the local ﬁeld factors of the partiallyspin polarized system are given in Appendix D. In general, f xcis a symmetric 4 ×4 matrix. If, however, the zaxis is directed along the average spin, then the ground-state transversal spin densities vanish, ρ+=ρ−=0, and the matrix fxcbecomes block-diagonal: fxc=⎛ ⎜⎝f11f1z00 f1zfzz00 00 0 f+− 00 f+− 0⎞ ⎟⎠. (24) Performing the matrix inversion in Eq. ( 23) we obtain the tensor of response functions of the interacting system in theform χ≡⎛ ⎜⎜⎜⎜⎜⎜⎜⎝χ nnχnszχns+χns− χsznχszszχszs+χszs− χs+nχs+szχs+s+χs+s− χs−nχs−szχs−s+χs−s−⎞ ⎟⎟⎟⎟⎟⎟⎟⎠=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝χ 0 nn−fzz/Delta1 εLFFχ0 nsz+f1z/Delta1 εLFF00 χ0 szn+f1z/Delta1 εLFFχ0 szsz−(v(q)+f11)/Delta1 εLFF00 00 0χ0 s+s− 1−f+−χ0 s+s− 00χ0 s−s+ 1−f+−χ0 s−s+0⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (25) 205206-4RESPONSE PROPERTIES OF III-V DILUTE MAGNETIC ... PHYSICAL REVIEW B 83, 205206 (2011) EF hh lh sohh- lh FIG. 1. (Color online) Schematic diagram of the possible single- particle excitations in the valence band of a p-type semiconductor. Dashed lines indicate intravalence-band excitations within the heavy-hole band (hh), within the light-hole band (lh), and intervalence-band excitations between heavey-hole and light-hole bands (hh-lh) and between split-off and heavy hole and light hole bands (so). where εLFF=1−/parenleftbig v(q)+f11/parenrightbig χ0 nn(q,ω)−fzzχ0 szsz(q,ω)−f1z/parenleftbig χ0 nsz(q,ω)+χ0 szn(q,ω)/parenrightbig +/parenleftbig fzz(v(q)+f11)−f2 1z/parenrightbig /Delta1, (26) /Delta1=χ0 nnχ0 szsz−χ0 nszχ0 szn=4χ0 ↑χ0 ↓. (27) III. RESULTS AND DISCUSSION We now discuss applications of our formalism for the speciﬁc case of GaMnAs DMSs. The band structure para-meters used in our calculations correspond to those of theGaAs host material: the band gap and spin-orbit splittingareE g=1.519 eV and /Delta1=0.341 eV , Luttinger parameters areγ1=6.97,γ2=2.25, and γ3=2.85, the conduction-band effective mass is me=0.065m0, the Kane momentum matrix element is Ep=27.86 eV , and the static dielectric constant is K=13. The s(p)-dexchange interaction constants within the conduction and valence bands are N0α=0.2 eV and N0β=− 1.2 eV , respectively. A. Clean p-type GaAs Before considering the effects of magnetic impurities and associated charge and spin disorder on the transport properties,we would like to discuss some new features that the valenceband character of the itinerant carriers brings into the system.They stem from the complexity of the semiconductor valenceband: strong spin orbit interaction and the /Gamma1-point degeneracy of the pstates. The multiband nature of the valence-band gives rise to a rich single-particle excitation spectrum. In Fig. 1we show a schematic representation of the valence band structure of ap-type semiconductor. Arrows indicate the possible single- particle excitations. In addition to the intraband excitationswithin the heavy-hole band (analogous to the excitationswithin the conduction band of n-doped semiconductors), here we have intraband excitations within the light-hole bandas well as intervalence-band excitations between light- andheavy-hole bands and between split-off and heavy- and/orlight-hole bands. χ [e V-1 Å-3] -0.006-0.004-0.0020 energy [eV]0 0.2 0.4 0.6q= 0.05 Å-1 Im [χnn] Im [χzz](b)χ [e V-1 Å-3] -0.03-0.02-0.010 energy [eV]0 0.2 0.4q= 0.003 Å-1 Im [χnn] Im [χzz](a) FIG. 2. (Color online) Imaginary part of the noninteracting density and longitudinal spin response functions in p-doped GaAs for different wave vectors (a) q=0.003 ˚A−1and (b) q=0.05˚A−1. The hole concentration is p=3.5×1020cm−3. The variety of the possible single-particle excitations sub- stantially modiﬁes the density and spin response of the system.Some of the modiﬁcations are not very obvious. At the end ofthe Sec. II B we already mentioned the signiﬁcant difference between spin and density responses in the long-wavelengthlimit. Let us consider this in more detail. The spin responseof the noninteracting electron gas coincides with the densityresponse and can be expressed through the Lindhard function.The spin-orbit interaction within the valence band breaks downthis correspondence. In Fig. 2we plot the imaginary part of the noninteracting density and longitudinal spin response functions in p-doped GaAs for different wave vectors. For a small wave vectorq=0.003 ˚A −1the longitudinal spin response exhibits a strong peak around 0.2 eV associated with intervalence-band spinexcitations between heavy- and light-hole subbands. Thecorresponding density excitations are suppressed due to theorthogonality of the initial and ﬁnal states; see Eq. ( 21). As a result, the density response for short wave vectorsis almost nonexistent. If we increase the wave vector toq=0.05˚A −1, the intraband excitations within the heavy-hole band become noticeable in both density and spin responses.The longitudinal spin response, however, still prevails in therange of intervalence-band transitions. This leads us to conclude that the long-wavelength spectrum of the single-particle excitations in p-doped semiconductors is dominated by the intervalence-band spin excitations. Theorigin of this effect is in the spin-orbit interaction, which mixesspin and orbital degrees of freedom. Without the spin-orbitinteraction, vertical spin excitations would be prohibited dueto the orthogonality of the orbital parts of Bloch functions. Another interesting feature of p-doped semiconductors is the effective suppression of the collective modes in thevalence band. In the conventional picture of the conductionband, collective plasmon excitations are well deﬁned onthe long-wavelength side of the excitation spectrum. Withincreasing momentum, the collective mode approaches andthen enters the region of single-particle excitations, where itbecomes rapidly suppressed due to Landau damping. The situation is different for the valence band. In Fig. 3we plot a schematic diagram of the excitation spectrum. 205206-5F. V . KYRYCHENKO AND C. A. ULLRICH PHYSICAL REVIEW B 83, 205206 (2011) FIG. 3. (Color online) Schematic diagram of the excitation spectrum within the semiconductor valence band. Labels indicate the edges of single-particle excitation regions within the heavy-holeband (hh), within the light-hole band (lh), between heavy-hole and light hole bands (hh-lh), and between the split-off band and heavy- and light-hole bands (so); see Fig. 1. As a result, the plasmon mode in the valence band lies entirely within the single-particle excitation spectrum and is effectively suppressed due to Landau damping. The excitation region for single-particle transitions within the heavy-hole band is qualitatively similar to that of the conduc-tion band. In the valence band, however, the single-particleexcitation spectrum is extended due to the intraband transitionswithin the light-hole band and interband transitions betweenheavy- and light-hole bands and between split-off and heavy-/light-hole bands (red and blue arrows in Fig. 1). In Fig. 3 the corresponding regions of single-particle excitations areshaded with different patterns. It can be seen that the collectivemode in the valence band falls entirely within the region ofsingle-particle excitations and, therefore, becomes suppressedeven at the long-wavelength side of the spectrum. Error barsin Fig. 3indicate the plasmon resonance broadening due to Landau damping. To illustrate the effect we have performed numerical calcu- lations of the plasmon dispersion and the lifetime broadeningof the collective excitations in the valence band of p-doped GaAs. The plasmon frequencies were determined as the zerosof the real part of the random phase approximation (RPA)dielectric function and the lifetime broadening is associatedwith the imaginary part of the frequency poles. In Fig. 4 the black and red lines correspond to the dispersion andthe lifetime of the plasmon excitations, respectively. Thedotted lines indicate the regions of the intraband single-particle excitations within the light-hole and heavy-hole bands;compare with Fig. 3. At small wave vectors the plasmon mode falls within the region of intervalence-band single-particleexcitations resulting in a lifetime broadening of the collectiveresonance of about 5 meV . Once the plasmon dispersion entersthe region of single-particle excitations within the light-holeband, the lifetime broadening substantially increases into the30–40-meV range. An additional sharp rise in the damping energy [eV] 00.10.20.30.40.5 lifetime broadenin g [meV] 00.010.020.030.040.05 q [Å-1]0 0.02 0.04 0.06 0.08 0.1 0.12lh hh FIG. 4. (Color online) Dispersion (dashed black) and lifetime broadening (solid red) of the valence band plasmon calculated for thep-doped GaAs with the hole concentration of p=3.5×10 20cm−3. Dotted lines correspond to the onset of the intraband single-particle excitations within the light-hole and heavy-hole bands. takes place when the collective mode enters the region of heavy-hole intraband excitations. We thus conclude that the collective response of valence- band holes in GaAs is substantially different compared withthat of conduction-band electrons. We also mention recentwork by Schliemann, 38,39who pointed out several other interesting features of the structure and response of interactinghole gases in p-doped III-V semiconductors. B. Magnetically doped GaMnAs The introduction of magnetic impurities in GaAs has two consequences. First, charge and spin disorder are broughtinto the system and, second, the mean-ﬁeld part of the p-d exchange interaction between localized spins and itinerantholes causes modiﬁcations of the valence-band structure oncethe system enters the magnetically ordered phase. Let us consider the effect of disorder ﬁrst. In calculating carrier relaxation rates, most theoretical models for GaMnAsuse a static screening approach, where all many-body effectsare reduced to the static screening of the Coulomb disorderpotential. Within our model, however, the momentum- andfrequency-dependent relaxation rates of Eq. ( 10) are expressed through the set of density and spin-density response functionsthat allow us to use the full dynamic treatment of electron-electron interaction, thus accounting for the variety of many-body effects including correlations and collective modes. In Fig. 5we plot the frequency dependence of the total (charge and spin) relaxation rate calculated forGa 0.948Mn 0.052As within the static screening model and using the full dynamic treatment of electron-electron interactionaccording to Eq. ( 10). The difference between the two curves in the static limit is due to the xc part of the electron-electron interaction that affects both charge and spin scattering.The most striking difference, however, is the pronouncedfeature appearing between 0.2 and 0.5 eV associated withthe collective modes. Although we have seen above that thecollective excitations are signiﬁcantly damped in the valence 205206-6RESPONSE PROPERTIES OF III-V DILUTE MAGNETIC ... PHYSICAL REVIEW B 83, 205206 (2011)τxx-1 [ps-1] 050100150200250 [eV] 00.040.080.120.16 ω [eV]0 0.2 0.4 0.6 0.8 1 FIG. 5. (Color online) Total (charge and spin) carrier relax- ation rate for Ga 0.948Mn 0.052As with hole concentration p=3× 1020cm−3. Dashed line: static screening model. Solid line: evalu- ation of Eq. ( 10) with full dynamic TDDFT treatment of electron interaction. See discussion in text. band, they still play an important role in the transport properties of the system, providing an effective channel for momentumrelaxation. Their contributions give up to a 50% increaseto the total carrier relaxation rate. Note that, due to theirlongitudinal character, the plasmon modes do not directlyaffect the optical response and enter only indirectly throughthe tensor of frequency- and momentum-dependent relaxationrates ( 10). In Fig. 6we compare our calculations of the infrared conductivity of ferromagnetic Ga 0.948Mn 0.052As with the experimental data of Singley et al.40The calculations were performed according to Eq. ( 12). The solid line corresponds to a relaxation rate obtained through Eq. ( 10), and the dashed line describes calculations with the ﬁxed τ−1=230 ps−1.T h e theory shows qualitative agreement with the experiment. Theinsensitivity of the calculations to the frequency dependenceof relaxation rate (minor difference between solid and dashedlines in Fig. 6) suggests that effects of the band structure play the dominant role in determining the shape of the infraredconductivity and overshadow the strong frequency dependenceofτobtained within our model and presented in Fig. 5. An alternative possible experimental probe that could reveal the details of the frequency and momentum dependence of thecarrier relaxation rate in more explicit ways is measurement ofthe position and line shape of the plasmon resonance itself. Itwas shown in Ref. 31that these quantities are sensitive to the carrier relaxation time, with both real and imaginary parts ofτand its dynamic nature being essential. Our approach seems to ﬁt well to describe such experiments. As was mentioned before, the magnetic impurities bring localized spins into the system, which interact with the itinerantcarriers through the p-dexchange interaction. The ﬂuctuating part of this interaction constitutes the spin disorder. The mean-ﬁeld part of exchange interaction, which we absorb into theclean system band structure Hamiltonian ˆH e, is responsible for the spin splitting of the valence bands once the systementers the magnetically ordered state. Due to the spin-orbitinteraction within the valence band, this spin splitting strongly σ [cm-1 Ω-1] 050100150200250300 ω [eV]0 0.2 0.4 0.6 0.8 1 FIG. 6. (Color online) Infrared conductivity of ferromagnetic Ga 0.948Mn 0.052As with hole concentration p=3×1020cm−3.C a l - culations are performed according to Eq. ( 12), and using a relaxation rate obtained through Eq. ( 10) (solid line) or a ﬁxed τ−1=230 ps−1 (dashed line). Symbols are the experimental data of Ref. 40. depends on both the magnitude and the direction of the wave vector k. In Fig. 7we plot the band structure of ferromagnetic Ga0.95Mn 0.05As. Strong anisotropy of the valence-band spin splitting is seen between directions along and perpendicularto the magnetization of localized spins ( zdirection). The inset shows a cut of the Fermi surface by the plane k y=0. One can easily see the distortion of the Fermi surface from thespherical shape of the paramagnetic system (for clarity we haveneglected here the valence-band warping, but it is included inour calculations). The modiﬁcation of the Fermi surface andthe suppression of localized spin ﬂuctuations are responsiblefor the signiﬁcant drop in static resistivity of GaMnAs duringthe transition from the paramagnetic to the ferromagnetic state.This effect was considered before. 28,41 Here we point out that the modiﬁcation of the valence- band structure during the transition from the paramagnetic tothe ferromagnetic state also modiﬁes energies and oscillatorstrengths of intervalence-band optical transitions, affectingthus the infrared conductivity as well. To better show theunderlying physics of temperature-induced changes, we plotin Fig. 8the infrared conductivity for the sample parameters of Ref. 40, but with a small lifetime broadening of /Gamma1=5m e V . In the paramagnetic state (solid line) three features can beidentiﬁed: a strong peak around 0.2 eV , corresponding tothe heavy-hole–light-hole transitions; a smaller peak with abroad shoulder around 0.4 eV , associated with the split off tolight-hole transitions; and a wide background of split off toheavy-hole transitions. With the temperature going below T c=70 K, two main phenomena occur. The ﬁrst is the suppression of the highenergy shoulder of the split-off to light-hole transitions. Thesecond is the appearance of the transitions between the spin-split heavy-hole and light-hole bands and the redistributionof the oscillator strength among them. The lowest energypeaks correspond to the transitions between spin-split bands.Calculations were performed for light linearly polarized in theplane perpendicular to the magnetization. Due to the spin-orbit 205206-7F. V . KYRYCHENKO AND C. A. ULLRICH PHYSICAL REVIEW B 83, 205206 (2011)energy [eV] -0.6-0.5-0.4-0.3-0.2-0.100.1 -kx [Å-1] -0.2 0 0.2 0.4 kz [Å-1]EFk[Å] -0.2-0.100.10.2 k[ Å ]-0.2 -0.1 0 0.1 0.2 FIG. 7. (Color online) Band structure of ferromagnetic Ga 0.95Mn 0.05As with hole concentration p=3.5×1020cm−3. Align- ment of localized spins results in strongly anisotropic valence band spin splitting. Inset shows a cut of the Fermi surface by the plane ky=0. interaction within the valence band, the transitions between the spin-split states are optically allowed. The additional peak athigher energy corresponds to heavy-hole–light-hole spin-ﬂiptransitions. As the temperature goes down, the spin splittingincreases and the “spin-ﬂip” transitions gain the intensitiesat the account of “spin-conserving” heavy-hole–light-holetransitions. Real GaMnAs samples are much more disordered. In Fig. 9 we compare experimental data on infrared conductivity ofGa 0.948Mn 0.052As from Ref. 40with calculations using our model of Eqs. ( 12) and ( 10). The large disorder-induced lifetime broadening blankets most of the features discussedabove. The suppression of the high energy shoulder of split-offto light-hole transitions in the ferromagnetic state is seen,however, on both the experimental and theoretical plots.Overall, for energies above the main peak position around0.2 eV , the calculations are in good agreement with theexperimental results. 02004006008001,000 0 0 .2 0.4 0 .6 0.8 70 K 65 K 45 K 5 Kσ [cm-1 Ω-1] ω [eV] FIG. 8. (Color online) Temperature dependence of infrared con- ductivity of Ga 0.948Mn 0.052As with hole concentration p=3× 1020cm−3andTc=70 K calculated with weak-disorder, lifetime broadening of /Gamma1=5m e V . 050100150200250 0 0.2 0.4 0.6 0. 5 K 25 K 45 K 70 K[cσΩm-1 -1] [c ω [ev]σΩm-1 -1]050100150200250 0 0.2 0.4 0.6 0. 5 K 25 K 45 K 70 K FIG. 9. (Color online) Temperature dependence of the infrared conductivity of Ga 0.948Mn 0.052As with hole concentration p=3× 1020cm−3andTc=70 K. Upper panel: experimental data of Ref. 40. Lower panel: results from Eq. ( 12). Note also that, unlike in Ref. 19, our calculations do not require incorporation of an impurity band within the energygap to avoid a drop in conductivity around 0.8–1 eV . Atenergies below the main peak position the agreement with theexperiment is worse. We should mention, however, that this isthe region of ωτ/lessorequalslant1 where our calculations are less reliable due to the approximate nature of expression ( 12). The self- consistent evaluation of Eq. ( 8) should be used there instead. Once the frequency goes to zero, the static conductivityshould more appropriately be calculated using an expressionderived from the semiclassical Boltzmann equation. 18We have investigated this regime before28to describe the drop in static resistivity in the ferromagnetic phase. IV . CONCLUSIONS We have developed a comprehensive theory of transport in spin and charge disordered media. The theory is based onthe equation of motion of the paramagnetic current responsefunction of the disordered system, treats disorder and many-body effects on equal footings, and combines a k·pbased description of the semiconductor valence band structure witha full dynamic treatment of electron-electron interaction bymeans of TDDFT. We have applied our theory to the speciﬁccase of GaMnAs. We have shown that the multiband nature and spin-orbit interaction within the valence band bring new effects forp-doped GaAs as compared with the conventional n-type sys- tems. The density and spin-density responses of noninteractingcarriers within the valence band are not the same anymore.Moreover, the long-wavelength side of the single-particleexcitation spectrum is now completely dominated by theintervalence-band spin excitations. Due to the extended region of single-particle excitations within the valence band, thecollective plasmon mode entirely falls within the region ofthese excitations and, therefore, is effectively damped for allwave vectors. For the magnetically doped system the mean-ﬁeld part of the p-dexchange interaction between itinerant holes and localized spins substantially modiﬁes the semiconductor band 205206-8RESPONSE PROPERTIES OF III-V DILUTE MAGNETIC ... PHYSICAL REVIEW B 83, 205206 (2011) structure once the system enters a magnetically ordered phase. This modiﬁcation signiﬁcantly affects energies and oscillatorstrengths of the intervalence band optical transitions. Ourcalculations are in good agreement with experimental datafor the temperature dependence of the infrared conductivity inGaMnAs. A full dynamical treatment of electron-electron interactions is essential to capture the inﬂuence of the collective excitationson the carrier relaxation rate. Our calculations show that, byproviding an effective channel of momentum relaxation, thecollective excitations within the valence band signiﬁcantly (upto 50%) increase the transport relaxation rate. However, it turns out that the actual infrared absorption spectra are not very sensitive to the details of the frequencydependence of the relaxation rate, but are mostly determinedby the features of the band structure. Direct measurements ofthe position and line shape of the plasmon resonance itself arelikely to be more sensitive to the details of the frequency andmomentum dependences of the carrier relaxation rate. In this paper we considered optical response properties. Since a transversal electric ﬁeld does not directly coupleto longitudinal collective modes, plasmon excitations affectthe carrier dynamics of the system only indirectly throughthe relaxation rates; see Eq. ( 10) and Fig. 5.I tw o u l db e interesting to consider the response to longitudinal ﬁelds,where the collective modes would dominate the carrierdynamics. The disorder-induced damping of such collectivemodes in heterostructures would be of particular interest. Thisrequires a generalization of our formalism for inhomogeneousor lower-dimensional systems. The theory presented here, treating disorder and many-body effects on an equal footing, provides a very general frameworkfor describing electron dynamics in materials. It can, in prin-ciple, be made self-consistent and thus be applied beyond theweak-disorder limit; it can accommodate many different typesof disorder, as well as band structure models. This should makeit well suited for further exploration of the optical and transportproperties of DMSs and other systems of practical interest. ACKNOWLEDGMENTS This work was supported by the DOE under Grant No. DE-FG02-05ER46213. APPENDIX A: GENERALIZED k ·p APPROACH The derivation of the generalized k·pperturbation ap- proach presented here is based on Ref. 29. First, the electronic wave function is expanded in the Luttinger–Kohn basis.42 /Psi1=/summationdisplay n,kAn(k)χn,k=1√ V/summationdisplay n,kAn(k)eikr\|un,0/angbracketright,(A1) where \|un,0/angbracketrightare periodic parts of Bloch functions at k=0 andAn(k) are the expansion coefﬁcients. This results in the following matrix form of the Schr ¨odinger equation: /summationdisplay n,kAn(k)/bracketleftbigg/parenleftbigg εn,0+¯h2k2 2m0−ε/parenrightbigg δn/prime,n+¯h m0k·πn/prime,n/bracketrightbigg =0, (A2)where εn,0are the band edge energies at k=0. The last term in Eq. ( A2) mixes states with different nfor k/negationslash=0. Now we separate the whole set of the bands {n}into those whose contribution we are going to calculate exactly, {s}, and the remote bands {r}that we will treat up to the second order in momentum. Equation ( A2) can be represented as (H0+H1+H2)A=εA, (A3) where Ais the vector of coefﬁcients An(k),H0is the diagonal part of Hamiltonian, and H1andH2correspond to the block- diagonal and off-block-diagonal parts of the k·πmatrix with respect to the included and remote bands. Next, we apply thecanonical transformation A=e SB=eS1+S2B, (A4) withS1andS2being antihermitian operators of ﬁrst and second order in the perturbation, respectively. Matrix equation ( A3) then has the form {e−S1−S2(H0+H1+H2)eS1+S2}B=¯HB=εB.(A5) By choosing H2+[H0,S1]=0,[H0,S2]+[H1,S1]=0,(A6) where [ ...] denotes the commutator, we write up to terms of second order in the perturbations H1andH2; ¯H≈H0+H1+1 2[H2,S1]. (A7) The matrix elements between the Luttinger–Kohn periodic amplitudes \|un,0/angbracketright≡\|n/angbracketrightare /angbracketleftn\|H0\|n/prime/angbracketright=/parenleftbigg εn,0+¯h2k2 2m0/parenrightbigg δn,n/prime, (A8) /angbracketlefts\|H1\|s/prime/angbracketright=/summationdisplay α¯hkαπα s,s/prime m0, (A9) /angbracketlefts\|H2\|r/angbracketright=/summationdisplay α¯hkαπα s,r m0, (A10) /angbracketlefts\|S1\|r/angbracketright=−/angbracketlefts\|H2\|r/angbracketright /angbracketlefts\|H0\|s/angbracketright−/angbracketleftr\|H0\|r/angbracketright =/summationdisplay α¯hkαπα s,r m01 εr,0−εs,0. (A11) For the last term in ( A7) we can then write /angbracketlefts\|[H2,S1]\|s/prime/angbracketright=/summationdisplay r/braceleftBig /angbracketlefts\|H2\|r/angbracketright/angbracketleftr\|S1\|s/prime/angbracketright −/angbracketlefts\|S1\|r/angbracketright/angbracketleftr\|H2\|s/prime/angbracketright/bracerightBig (A12) =/summationdisplay α,β r¯h2kαkβ m2 0/parenleftBigg πα s,rπβ r,s/prime εs/prime,0−εr,0+πβ s,rπα r,s/prime εs,0−εr,0/parenrightBigg . Here we used the fact that the H2andS1operators have only off-block-diagonal matrix elements between the sand rbands. Equations. ( A8)–(A12) deﬁne the matrix of the 205206-9F. V . KYRYCHENKO AND C. A. ULLRICH PHYSICAL REVIEW B 83, 205206 (2011) effective Hamiltonian ( A7). Nonvanishing matrix elements are determined by the symmetry of the crystal. APPENDIX B: EVALUATION OF THE MATRIX ELEMENTS IN EQS. ( 17) AND ( 18) In order to evaluate Eq. ( 17) we need to calculate the following matrix element: ¯h/parenleftBig kα−qα 2/parenrightBig /angbracketleftui/prime,k−q\|ui,k/angbracketright+/angbracketleftui/prime,k−q\|ˆπα\|ui,k/angbracketright =/angbracketleftBig ui/prime,k−q/vextendsingle/vextendsingle/vextendsingle¯h/parenleftBig k α−qα 2/parenrightBig +ˆπα/vextendsingle/vextendsingle/vextendsingleu i,k/angbracketrightBig , (B1) where \|ui,k/angbracketrightis expressed through the amplitudes at the zone center: \|ui,k/angbracketright=/summationdisplay nAn(i,k)\|un,0/angbracketright. (B2) From diagonalization of the effective Hamiltonian ( A7), however, we obtain coefﬁcients Bn(i,k) related to An(i,k) through Eq. ( A4). Expanding eS≈1+S, we express \|ui,k/angbracketright=/summationdisplay sBs(i,k)\|s/angbracketright+/summationdisplay s/summationdisplay r/angbracketleftr\|S(k)\|s/angbracketrightBs(i,k)\|r/angbracketright,(B3) where we have used the fact that the coefﬁcients Bnare nonzero only for exact bands and Shas only off-block-diagonal matrix elements. The bra vector is /angbracketleftui/prime,k/prime\|=/summationdisplay s/primeB∗ s/prime(i/prime,k/prime)/angbracketlefts/prime\|−/summationdisplay s/prime/summationdisplay r/prime/angbracketlefts/prime\|S(k/prime)\|r/prime/angbracketrightB∗ s/prime(i/prime,k/prime)/angbracketleftr/prime\|, (B4) where we have used the antihermiticity of S. Matrix elements of an arbitrary operator ˆFto the lowest order in Scan then be expressed as follows: /angbracketleftui/prime,k/prime\|ˆF\|ui,k/angbracketright=/summationdisplay s/primesB∗ s/prime(i/prime,k/prime)Bs(i,k)/parenleftbigg /angbracketlefts/prime\|ˆF\|s/angbracketright +/summationdisplay r/parenleftbig /angbracketlefts/prime\|ˆF\|r/angbracketright/angbracketleftr\|S(k)\|s/angbracketright−/angbracketlefts/prime\|S(k/prime)\|r/angbracketright/angbracketleftr\|ˆF\|s/angbracketright/parenrightbig/parenrightbigg .(B5) Using Eq. ( A11) for matrix elements of ˆS1,w eh a v e /angbracketleftui/prime,k/prime\|ˆF\|ui,k/angbracketright=/summationdisplay s/primesB∗ s/prime(i/prime,k/prime)Bs(i,k)/parenleftbigg /angbracketlefts/prime\|ˆF\|s/angbracketright−¯h m0 /summationdisplay λ,r/parenleftBigg kλ/angbracketlefts/prime\|ˆF\|r/angbracketright/angbracketleftr\|ˆπλ\|s/angbracketright εr−εs+k/prime λ/angbracketlefts/prime\|ˆπλ\|r/angbracketright/angbracketleftr\|ˆF\|s/angbracketright εr−εs/prime/parenrightBigg/parenrightbigg .(B6) Matrix element ( B1) has thus the following form: /angbracketleftui/prime,k−q\|¯h/parenleftBig kα−qα 2/parenrightBig +ˆπα\|ui,k/angbracketright=/summationdisplay s/primesB∗ s/prime(i/prime,k−q)Bs(i,k) ×/bracketleftbigg ¯h/parenleftBig kα−qα 2/parenrightBig δs/primes+/angbracketlefts/prime\|ˆπα\|s/angbracketright+¯h m0/summationdisplay λ,r/parenleftBigg kλπα s/prime,rπλ r,s εs−εr +(kλ−qλ)πλ s/prime,rπα r,s εs/prime−εr/parenrightBigg/bracketrightbigg .Forq=0 it reduces to /angbracketleftbig ui/prime,k\|¯hkα+ˆπα\|ui,k/angbracketrightbig =/summationdisplay s/primesB∗ s/prime(i/prime,k)Bs(i,k)/bracketleftbigg ¯hkαδs/primes +/angbracketlefts/prime\|ˆπα\|s/angbracketright+¯h m0/summationdisplay λ,rkλ/parenleftBigg πα s/prime,rπλ r,s εs−εr+πλ s/prime,rπα r,s εs/prime−εr/parenrightBigg/bracketrightbigg .(B7) By comparison with the expressions derived in Appendix A, we ﬁnd that this reduces to /angbracketleftbig ui/prime,k\|¯hkα+ˆπα\|ui,k/angbracketrightbig =/summationdisplay s/primesB∗ s/prime(i/prime,k)Bs(i,k)m0 ¯h∂ ∂kα/angbracketlefts/prime\|¯H\|s/angbracketright, (B8) where ¯His the Hamiltonian ( A7). The matrix elements of the spin operator in Eq. ( 18) should also be evaluated through Eq. ( B6): /angbracketleftui/prime,k/prime\|ˆσμ\|ui,k/angbracketright=/summationdisplay s/primesB∗ s/prime(i/prime,k/prime)Bs(i,k)/parenleftbigg /angbracketlefts/prime\|ˆσμ\|s/angbracketright−¯h m0/summationdisplay λ,r/parenleftbiggkλ/angbracketlefts/prime\|ˆσμ\|r/angbracketright/angbracketleftr\|ˆπλ\|s/angbracketright εr−εs+k/prime λ/angbracketlefts/prime\|ˆπλ\|r/angbracketright/angbracketleftr\|ˆσμ\|s/angbracketright εr−εs/prime/parenrightbigg/parenrightbigg . (B9) Let us look now at the sum over remote bands. Since the spin operator acts only on the spin part of the basis functions, onlythose remote bands whose orbital part has the same symmetryas the exact bands will contribute to this sum. If we are considering a 6 ×6 Hamiltonian and neglect inversion asymmetry, the exact states are p-bonding states that transform according to the F + 1representation of the point group Oh(/Gamma1/prime 15small representation). The momentum operator transforms as F− 2, and since the direct product F+ 1×F− 2×F+ 1 does not contain a unit representation, the sum over remote bands vanishes. There may be a small contribution in Td crystals, but it can be considered negligible. If we are working in an 8-band k·pmodel, there are possible contributions to the sum when \|s/angbracketrightand\|r/angbracketrightare/Gamma1/prime 1 states and \|s/prime/angbracketrightis/Gamma1/prime 15and vice versa. Since there is only a small admixture of the conduction-band amplitude to thevalence-band states, these contributions are expected to besmall and therefore can be neglected. Because of this reasoning, we use the following approxi- mation: /angbracketleftu i/prime,k/prime\|ˆσμ\|ui,k/angbracketright≈/summationdisplay s/primesB∗ s/prime(i/prime,k/prime)Bs(i,k)/angbracketlefts/prime\|ˆσμ\|s/angbracketright.(B10) APPENDIX C: 8 ×8 HAMILTONIAN In the basis \|1/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleE,+1 2/angbracketrightbigg =S↑, \|2/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleE,−1 2/angbracketrightbigg =iS↓, \|3/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleHH,+3 2/angbracketrightbigg =1√ 2(X+iY)↑, \|4/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleLH,+1 2/angbracketrightbigg =i√ 6[(X+iY)↓−2Z↑], 205206-10RESPONSE PROPERTIES OF III-V DILUTE MAGNETIC ... PHYSICAL REVIEW B 83, 205206 (2011) \|5/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleLH,−1 2/angbracketrightbigg =1√ 6[(X−iY)↑+2Z↓],(C1) \|6/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleHH,−3 2/angbracketrightbigg =i√ 2(X−iY)↓,\|7/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleSO,+1 2/angbracketrightbigg =1√ 3[(X+iY)↓+Z↑], \|8/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingleSO,−1 2/angbracketrightbigg =i√ 3[−(X−iY)↑+Z↓], the Hamiltonian matrix has the form ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝E g+¯h2k2 2/tildewideme0i√ 2Vk+/radicalbigg 2 3Vkzi√ 6Vk− 0i√ 3Vkz1√ 3Vk− 0 Eg+¯h2k2 2/tildewideme0i√ 6Vk+/radicalbigg 2 3Vkzi√ 2Vk−1√ 3Vk+i√ 3Vkz −i√ 2Vk− 0 P+QL M 0i√ 2L/prime−i√ 2M/prime /radicalbigg 2 3Vkz−i√ 6Vk− L∗P−Q 0 M −i√ 2Q/primei/radicalbigg 3 2L/prime −i√ 6Vk+/radicalbigg 2 3Vkz M∗0 P−Q −L −i/radicalbigg 3 2L/prime∗−i√ 2Q/prime 0 −i√ 2Vk+ 0 M∗−L∗P+Q−i√ 2M/prime∗−i√ 2L/prime∗ −i√ 3Vkz1√ 3Vk−−i√ 2L/prime∗i√ 2Q/primei/radicalbigg 3 2L/primei√ 2M/primeP/prime−/Delta1 0 1√ 3Vk+−i√ 3Vkzi√ 2M/prime∗−i/radicalbigg 3 2L/prime∗i√ 2Q/primei√ 2L/prime0 P/prime−/Delta1⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(C2) with k±=kx±iky, V=−i¯h m0/angbracketleftS\|ˆpx\|X/angbracketright=/radicalBigg Ep¯h2 2m0. Interaction with remote bands results in the intravalence-band terms, P(/prime)=−¯h2 2m0/tildewideγ(/prime) 1k2, Q(/prime)=−¯h2 2m0/tildewideγ(/prime) 2/parenleftbig k2 x+k2 y−2k2 z/parenrightbig , L(/prime)=¯h2 2m0i2√ 3/tildewideγ(/prime) 3kzk−, M(/prime)=−¯h2 2m0√ 3/bracketleftbig /tildewideγ(/prime) 2/parenleftbig k2 x−k2 y/parenrightbig −i/tildewideγ(/prime) 3(kxky+kykx)/bracketrightbig , where renormalization leads to 1 /tildewideme=1 m∗e−1 m0Ep 3/parenleftbigg2 Eg+1 Eg+/Delta1/parenrightbigg ,/tildewideγ1=γ1−Ep 3Eg, /tildewideγ/prime 1=γ1−Ep 3(Eg+/Delta1), /tildewideγ2=γ2−Ep 6Eg, /tildewideγ/prime 2=γ2−Ep 12/parenleftbigg1 Eg+1 Eg+/Delta1/parenrightbigg , /tildewideγ3=γ3−Ep 6Eg, /tildewideγ/prime 3=γ3−Ep 12/parenleftbigg1 Eg+1 Eg+/Delta1/parenrightbigg . This reﬂects the fact that the interaction between conduction and valence bands is taken in our Hamiltonian explicitly. Inwriting matrix ( C2) we have neglected small terms associated with the lack of inversion symmetry in T dcrystals. The matrix of the mean-ﬁeld part of the s(p)-dexchange interaction, which is responsible for the band spin splitting inthe magnetically ordered phase, has the form 205206-11F. V . KYRYCHENKO AND C. A. ULLRICH PHYSICAL REVIEW B 83, 205206 (2011) −1 2/angbracketleftS/angbracketrightxN 0⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝α 00 0 0 0 0 0 0 −α 00 000 0 00 β 00 0 00 000 1 3β 00 i2√ 2√ 3β 0 000 0 −1 3β 00 −i2√ 2√ 3β 000 0 0 −β 00 000 −i2√ 2√ 3β 00 −1 3β 0 000 0 i2√ 2√ 3β 001 3β⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (C3) where the zaxis is chosen in the direction of the magnetization andN0αandN0βare the s-dandp-dexchange constants. The mean-ﬁeld value of localized spins, is determined as the thermodynamic average /angbracketleftS/angbracketright=/angbracketleft ˆSz/angbracketright=1 ZTre−ˆHm kTˆSz, (C4) with the partition function Z=Tre−ˆHm kT. (C5) Within the mean ﬁeld approximation for uncorrelated spins the spin Hamiltonian is ˆHm=−BeffˆSz, (C6) with the effective ﬁeld Beff=/angbracketleftˆSz/angbracketrightJ0, (C7) and J0=3kTc S(S+1). (C8) The Curie temperature Tcis an input parameter of our model; through transcendental equations ( C4) and ( C7) it determines the mean ﬁeld value of /angbracketleftS/angbracketright. APPENDIX D: LOCAL FIELD FACTORS FOR PARTIALLY SPIN-POLARIZED SYSTEMS Expressions for local ﬁeld factors of the partially spin polarized electron gas were derived in Ref. 43, but in a different spin basis. Here we will brieﬂy rederive them in the basis ofEq. ( 4). In the adiabatic approximation (which ignores frequency dependence), the components of the tensor f xcof the local ﬁeld factors in Eq. ( 23) have the form fij=∂2[nexc(n,ξ)] ∂ρi∂ρj, (D1) where excis the xc energy per particle, n≡ρ1is the electron density, and ξis the spin polarization: ξ≡\|/vectorξ\|=1 n/radicalbigg ρ2z+1 2(ρ+ρ−+ρ−ρ+). (D2)We assume here that excdepends on only the absolute value of \|ξ\|. Direct evaluation of Eq. ( D1)g i v e s f11=2∂exc ∂ρ1−2ξ∂2exc ∂ρ1∂ξ+ρ1∂2exc ∂ρ2 1+ξ2 ρ1∂2exc ∂ξ2, f1i=∂ξ ∂ρi/parenleftbigg ρ1∂2exc ∂ρ1∂ξ−ξ∂2exc ∂ξ2/parenrightbigg ,i=(z,+,−), fzz=A+ρ2 zB, fz+=ρzρ− 2B, fz−=ρzρ+ 2B, f++=ρ−ρ− 4B, f−−=ρ+ρ+ 4B, f+−=A 2+ρ−ρ+ 4B, with A=1 ρ1ξ∂exc ∂ξ,B =1 (ρ1ξ)2/parenleftbigg ξ∂2exc ∂ξ2−∂exc ∂ξ/parenrightbigg . Note that fii/prime=fi/primeiand, generally, the tensor of local ﬁeld factors is a symmetric matrix. If, however, the zaxis is directed along the average spin direction, so that the ground-statetransverse spin densities vanish ( ρ +=ρ−=0), then the matrix reduces to the block-diagonal form of Eq. ( 24). We deﬁne the xc energy of the spin-polarized system in the usual manner as43 exc(n,ξ)=exc(n,0)+(exc(n,1)−exc(n,0))f(ξ),(D3) with f(ξ)=(1+ξ)4/3+(1−ξ)4/3−2 2(21/3−1). (D4) This is exact for the exchange part, but only approximately so for the correlation part (which will be neglected anyway in thefollowing). With this, we get ∂e xc ∂ξ=(exc(n,1)−exc(n,0))(1+ξ)1/3−(1−ξ)1/3 3 2(21/3−1), (D5) 205206-12RESPONSE PROPERTIES OF III-V DILUTE MAGNETIC ... PHYSICAL REVIEW B 83, 205206 (2011) ∂2exc ∂ξ2=/parenleftBig exc(n,1)−exc(n,0)/parenrightBig(1+ξ)−2/3+(1−ξ)−2/3 9 2(21/3−1). (D6) This completes the deﬁnition of the local ﬁeld factors for a par- tially spin-polarized system. The only remaining ingredientswe need to perform the actual calculations are the expressionsfor the xc energy for unpolarized and fully spin polarizedsystem, e xc(n,0) and exc(n,1). In this work for simplicity we limit ourselves to the exchange part of exc: ex(n,0)=−3e2 4K/parenleftbigg3n π/parenrightbigg1/3 , (D7)ex(n,1)=21/3ex(n,0), (D8) where Kis the static dielectric constant of the host material. 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PhysRevB.90.155309.pdf	PHYSICAL REVIEW B 90, 155309 (2014) Density matrix model for polarons in a terahertz quantum dot cascade laser Benjamin A. Burnettand Benjamin S. Williams Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, California 90095, USA (Received 25 July 2014; revised manuscript received 26 September 2014; published 16 October 2014) A density matrix based method is introduced for computation of steady-state dynamics in quantum cascade systems of arbitrary size, which incorporates an optical ﬁeld coherently. The method is applied to a model terahertzquantum dot cascade laser system, where a means of treating coherent electron-optical-phonon coupling is alsointroduced. Results predict a strong increase in the upper state lifetime and operating temperature as comparedto traditional well-based terahertz quantum cascade lasers. However, new complications also arise, includingmultiple peaks in the gain spectrum due to strong electron-phonon coupling, and strong parasitic subthresholdcurrent channels that arise due to reduced dephasing. It is anticipated that novel design schemes will be necessaryfor such lasers to become a reality. DOI: 10.1103/PhysRevB.90.155309 PACS number(s): 42 .70.Hj,07.57.Hm,73.63.Nm,73.63.Kv I. INTRODUCTION Quantum cascade (QC) lasers based on intersubband transitions now cover large segments of the mid-infrared andterahertz spectral ranges [ 1–3]. However, because the quantum conﬁnement is in only one dimension, electrons are free tomove in-plane, and hence each subband supports a continuumof states above its minimum energy. This has a major impact onthe electron dynamics, as it allows fast relaxation of electronsbetween subbands via the emission of longitudinal optical(LO) phonons, even when the intersubband energy separationis different from the LO phonon energy ( E LO≈36 meV in GaAs). This is particularly damaging for terahertz (THz) QClasers, which have radiative energies less than E LO(/planckover2pi1ω∼ 5–20 meV), and are still limited to cryogenic temperatures(T max=200 K) [ 4]. At low temperature, electrons reside near the upper subband minimum, and have insufﬁcient energy toemit an LO- phonon. However, as the device warms, electronsgain sufﬁcient in-plane energy to emit an LO phonon, whichleads to an exponential decrease in the upper state lifetime tofar subpicosecond levels at 300 K. This leads to a concomitantdecrease in the population inversion with temperature, andis believed to be the primary inhibitor to room temperatureoperation [ 3,5]. It has been proposed by several authors that room tempera- ture could be reached in THz QC lasers by introducing lateralquantum conﬁnement so that the electronic density of statesbecomes fully discrete, i.e., sublevels instead of subbands[6–8]. In this way, it may be possible to greatly increase the upper radiative state lifetime if LO-phonon scatteringcan be suppressed across the radiative transition by inten-tional misalignment of all associated transitions from E LO, utilizing an effect known as “phonon bottleneck.” Candidateschemes include self-assembled quantum dots [ 9,10], quantum posts [ 11], and nanopillars etched from the top down into planar quantum-cascade material [ 12–14]. The concept of a phonon bottleneck for carrier relaxation has been shownto be of limited validity for interband devices (such asQD diode lasers), unless multiphonon, electron-electron, andelectron-hole scattering processes are carefully prevented [ 15]. bburnett@ucla.eduHowever, dramatic increases in intersublevel relaxation times have been observed experimentally where the relevant condi-tions are met; for unipolar self-assembled quantum dots withintersublevel energy separation less than E LO, relaxation times as long as 1 ns were measured at 10 K, and many tens ofpicoseconds at room temperature [ 16]. A series of theoretical and experimental investigations on the electron-LO-phonon interaction in quantum dots has con-vincingly shown that the usual Fermi’s “golden rule” approachis not appropriate given that there is not a continuum of ﬁnalstates. This suppresses decoherence so that the degenerate LO-phonon modes can form a strongly coupled system with theintersublevel excitation, leading to sustained Rabi oscillationsand the formation of intersublevel polarons [ 17–21]. In isolated quantum dots, this persists until interruption by another inter-action, likely the anharmonic decay of the LO phonon [ 22,23]. This picture dramatically modiﬁes the energy-selectivity ofthe LO-phonon interaction, and introduces a complex seriesof anticrossed energy levels, leading to stark new featuresin transport characteristics and gain spectra which must beproperly modeled if quantum-dot QC lasers are to be realized. Candidate models must incorporate coherent electron- phonon interaction as well as coherent response to the opticalﬁeld. The most detailed approach involves nonequilibriumGreen’s function (NEGF) methods, which provide motivationfor the lateral conﬁnement approach, but are extremelycomputationally intensive and also tedious for the nonexpert[24–29]. Density matrix models are attractive since they allow intuitive use of quantum-cascade wave functions as abasis and have been shown to capture signatures of coherentelectron tunneling as well as coherent response to the opticalﬁeld [ 30–34]. However, such models have not yet been applied to a quantum-dot QC laser where electron-phononcoupling is strong. Moreover, most density matrix methodsuse a fully derived algebraic solution, which quickly becomescumbersome when considering more than three to four states.One exception is Ref. [ 35], which accommodates an arbitrarily large basis, although it stopped short of including coherent op-tical response. The latter point was accounted for in Ref. [ 36], in which the coherent gain and transport characteristics werecalculated in a proposed silicon-based terahertz QCL. In this paper, we present a density matrix approach suitable for steady-state modeling of transport and optical properties 1098-0121/2014/90(15)/155309(11) 155309-1 ©2014 American Physical SocietyBENJAMIN A. BURNETT AND BENJAMIN S. WILLIAMS PHYSICAL REVIEW B 90, 155309 (2014) in QC systems of arbitrary size. While typical models for QC lasers use a Hilbert space of electronic states only, we apply theconcept more generally where the density matrix representsa combined system of electronic (sublevel) and bosonic(LO phonon) degrees of freedom. This allows simultaneousconsideration of coherent electron-LO-phonon interaction,electron tunneling, and the optical ﬁeld, alongside incoherenttransition and dephasing mechanisms. A nanopillar geometryis used as a model for quantum conﬁnement, although ourtreatment is generally applicable to any cascaded quantum dotbased structure. Results predict a complicated multipeakedgain spectrum, and highlight the importance of dephasing onboth transport and gain characteristics. Overall, quantum dotQC lasers will exhibit transport and gain features that arequalitatively different from conventional QC lasers. II. METHOD A. Steady-state solution The unit cell of a QC system is a multiwell module, which contains a ﬁnite number of states. The module is repeateda large number of times with a successive energy differenceimposed by the electrical bias, as conceptually illustrated inFig. 1. These states might represent subbands such as in a conventional QC laser, discrete sublevels as in a quantum-dotsystem, or even product states of a combined system such as theelectron/LO-phonon tensor product Hilbert space used in thiswork. The states are coupled together by a variety of coherentprocesses (such as resonant tunneling, LO-phonon interaction,and optical-ﬁeld dipole coupling) and incoherent processes(such as acoustic phonon scattering and pure dephasing) thatresult in charge transport and optical gain. Reﬂecting this, wewill assume that the system Hamiltonian is known, and allowthat the time evolution of the representative density matrix(ρ) consists of a coherent Liouville-von Neumann component Position Energy modE modL FIG. 1. (Color online) Conceptual schematic of a representative three-level quantum-cascade system, which is periodic in both position and energy. Interactive processes will be incorporated as static couplings (empty double arrows), an optical ﬁeld (green),and irreversible transition processes (solid single arrows). The lower part depicts an example energy structure, where the black lines are the conduction band proﬁle and the probability densities for eachsubband/level are shown.alongside an incoherent component due to transitions and dephasing [ 37,38]: d dtρ=d dtρ/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh +d dtρ/vextendsingle/vextendsingle/vextendsingle/vextendsingleinc =−i /planckover2pi1LHρ+d dtρ/vextendsingle/vextendsingle/vextendsingle/vextendsingleinc .(1) LH≡[H,... ] is known as the Liouville superoperator. Through its tetradic form, it can be understood as relating theelements of ρto the coherent part of its own time evolution, where d dtρab/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh =−i /planckover2pi1/summationdisplay cdLH abcdρcd =−i /planckover2pi1/summationdisplay cd(δbdHac−δacHdb)ρcd. (2) The periodicity of the cascaded modules allows us to express Handρin block matrix form as H=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣...... ... (H 0−/Delta1)(H1)( H2) ··· (H−1)( H0)( H1) ··· (H−2)( H−1)(H0+/Delta1) ... ......⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3) ρ=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣...... ... (ρ 0)(ρ1)(ρ2) ··· (ρ−1)(ρ0)(ρ1)··· (ρ−2)(ρ−1)(ρ0) ... ......⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦. (4) Each submatrix depicted in Eqs. ( 3) and ( 4)i so fs i z e N×N, where Nis the number of levels in each repetitive module. H 0andρ0represent the intramodule Hamiltonian and density matrix, while H±pand and ρ±prepresent the intermodule Hamiltonian and coherences for modules spacedpapart. The matrix /Delta1=E mod1Naccounts for the applied bias, where Emodis the difference in energy per module. Applying block matrix multiplication in ( 1) with these matrices, we obtain the equations for the coherent time evolution of allsubmatrices, which collectively describe the entire system: d dtρp/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoh =−i /planckover2pi1/parenleftbigg/summationdisplay q[Hp−q,ρq]−pE modρp/parenrightbigg . (5) In order to consider interactions with a harmonic electro- magnetic ﬁeld, we expand at steady-state each submatrix of Handρinto harmonics of order αat frequency ω: H p=/summationdisplay αH(α) peiαωt,ρ p=/summationdisplay αρ(α) peiαωt, (6) where the sums could in principle run over all integers (−∞,∞). Substituting ( 6)i n t o( 5), and selecting a particular harmonic m, we obtain with incoherent effects omitted: imωρ(m) p=−i /planckover2pi1/parenleftbigg/summationdisplay qα/bracketleftbig H(α) p−q,ρ(m−α) q/bracketrightbig −p/Delta1ρ(m) p/parenrightbigg . (7) 155309-2DENSITY MATRIX MODEL FOR POLARONS IN A . . . PHYSICAL REVIEW B 90, 155309 (2014) An equation can be obtained at each element in ( 7) by invoking the Liouville superoperator and a change of variables: imωρ(m) p,ab=−i /planckover2pi1⎛ ⎝/summationdisplay qncd/parenleftbig LH(m−n) p−q abcdρ(n) q,cd/parenrightbig −pE modρ(m) p,ab⎞ ⎠.(8) Terms outside the quadruple sum can then be brought inside using Kronecker δfunctions, yielding a system of equations providing the relation/summationtext qncdM(ab)np,(cd)nqρ(n) q,cd=0, where M(ab)mp,(cd)nq=−i /planckover2pi1LH(m−n) p−q abcd+iδpqδmnδacδbd/parenleftbigg pEmod /planckover2pi1−mω/parenrightbigg +S(ab)mp,(cd)nq. (9) The incoherent contribution S(ab)p m,(cd)q nis addressed in Appendix. Once a population sum condition is substituted atas i n g l er o wi n M, the entire steady-state solution is attainable by the matrix equation M×A=B, where Ais a list of the unknowns and Bis a matching vector of zeros with the single exception of a 1 in the sum row. For a module consisting of Nstates, and considering up toPnearest-module couplings and harmonics up to e ±iQωt, the number of unknowns is N2(2P+1)(2Q+1). Although the method as formulated can, in principle, accommodatearbitrarily higher N,P, and Q, in this work, we restrict our analysis to only nearest-module coupling and singleharmonics, such that the number of unknowns is 9 N 2. B. Optical gain Gain is computed through the induced harmonic polariza- tion in response to an optical ﬁeld Eopt=\|E\|eiωt+c.c. If the position operator zis known, the harmonic Hamiltonian is then H(±1)=q\|E\|z, and the polarization is found using P=Ndq/angbracketleftz/angbracketright=NdqTr(ρz). We assume that by some choice of basis, zhas only diagonal submatrices: z=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣...... ... (z 0−/Delta1z)0 0 ... 0 z0 0 ... 00 ( z0+/Delta1z) ... ......⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦, (10) where /Delta1 z=Lmod1Naccounts for the spatial separation between modules. Since the population sum is normalizedto a single module, the trace over one diagonal submatrix iseffectively a trace over the entire problem. Taking only theharmonic component, the optical susceptibility is then χ opt(ω)=Ndq /epsilon10\|E\|Tr/parenleftbig ρ(1) 0z0/parenrightbig , (11) from which the material gain per unit length can be calculated asg=2Im(√/epsilon1s+χoptω/c),/epsilon1sbeing the background dielec- tric constant. In this way, the gain material is treated as aneffective medium. By carrying out the trace in ( 11), it is seen that there are susceptibilities associated with each transition, which adddirectly: χ opt(ω)=/summationdisplay αβχαβ(ω)=Ndq /epsilon10\|E\|/summationdisplay αβρ(1) αβzβα, (12) so that contributions can be examined for different transitions independently. C. Current Current is computed from the expectation value of velocity. However, since the time evolution includes both coherent andincoherent components, the velocity will have contributionsfrom both as well: J=N dqTr(ρvcoh)+Jinc. (13) Using vcoh=i /planckover2pi1[H,z], and assuming that Hhas the form of (3) with only a single harmonic and zhas the form of ( 10), the static component of current due to coherent velocity is foundto be J coh=iNdq /planckover2pi1Tr/summationdisplay pρ(0) p/parenleftbig/bracketleftbig H(0) −p,z0/bracketrightbig −pL modH(0) −p/parenrightbig +ρ(1) p/parenleftbig/bracketleftbig H(−1) −p,z0/bracketrightbig −pL modH(−1) −p/parenrightbig +ρ(−1) p/parenleftbig/bracketleftbig H(1) −p,z0/bracketrightbig −pL modH(1) −p/parenrightbig . (14) The second and third terms are optically induced currents (stimulated emission and absorption). If vanishing opticalintensity is assumed, we are left with the expression forcoherent current density below threshold: J coh=iNdq /planckover2pi1Tr/summationdisplay pρ(0) p/parenleftbig/bracketleftbig H(0) −p,z0/bracketrightbig −pL modH(0) −p/parenrightbig .(15) The incoherent contribution Jincaccounts for the semiclassical “hopping” velocity due to incoherent transitions betweenspatially localized basis states. It is described in more detail inAppendix. III. ELECTRON-LO-PHONON COUPLING Interaction of a discrete electronic density of states with a manifold of nearly degenerate LO-phonon modes is aparticularly distinct problem in that dephasing is weakeneddue to the lack of a continuum of states. In this way, itis similar to an atom strongly coupled to a single opticalcavity mode, where higher-order coherent quantum effectsbecome possible and so the electron-boson interaction cannotbe treated using Fermi’s “golden rule.” The simplest exampleis in a two-state electronic system, where electrons do not relaxirreversibly from the higher energy state to the lower one, butrather undergo a sustained Rabi oscillation which continuesuntil interrupted by another process such as the decay of theemitted phonon or interaction of the electron with the outside[17–20,22,23,39,40]. The excitations of a phonon coupled to an electronic transition are quasiparticles known as polarons. However,rather than use the polaron states as our basis, we choose a basisformed by a tensor product of the electronic sublevel Hilbertspace with LO-phonon number states; these are then coupledtogether by the electron-phonon (Fr ¨ohlich) Hamiltonian. This 155309-3BENJAMIN A. BURNETT AND BENJAMIN S. WILLIAMS PHYSICAL REVIEW B 90, 155309 (2014) choice is important, as it determines that in the limit of weak dephasing the system will form coherent polarons, butwill relax into a separable state as the dephasing becomesstrong in comparison to the polaron splitting. In addition, thisallows for both electron and phonon distributions to reach anonequilibrium steady state, and also simpliﬁes the inclusionof electron tunneling, phonon decay and generation, and theoptical ﬁeld. Our particular application allows considerationof coherent polarons comprising multiple phonon modesand multiple electronic intersublevel transitions, each ofwhich may have (in principle) their own dephasing rates.Furthermore, unlike the case of an isolated quantum dot, wheredecay of the phonon component dominates polaron decay, wecan also include decay of the electronic component, throughtunneling, or some other incoherent scattering mechanism. A comparison can be made between our treatment of polaron effects and that employed in the NEGF simula-tions of Refs. [ 26–29]. In these works, the electron-phonon interaction is accounted for by a phonon Greens functionwhich enters into the electron self-energy; it is thereforerepresented as an average ﬁeld which is assumed to remainat thermal equilibrium. The phonon decay, which broadensthe interaction, is treated by introducing an anharmonicity inthe phonon Greens function. Our work, on the other hand,treats the electron-phonon interaction in a similar manner tothe Jaynes-Cummings model in quantum optics, where the LO phonons themselves become as much a part of the system as the electrons, decaying towards equilibrium by their interactionwith acoustic phonons. In this way, it is the acoustic, ratherthan LO phonons that play the role of the system bath. A. Single transition The electron-LO-phonon interaction is described by the Fr¨ohlich Hamiltonian ˆHf, which includes all modes simulta- neously. Assuming bulk plane-wave LO-phonons with wave vectors /vectork,ˆHf=/summationtext /vectorkˆF/vectork, where ˆF/vectorkis the Fr ¨ohlich Hamiltonian for single mode /vectork, expressed as ˆF/vectork=A k√ V(ei/vectork·/vectorrb/vectork+e−i/vectork·/vectorrb† /vectork), (16) where the constant A=/radicalbig ELOq2 2(1 /epsilon1∞−1 /epsilon1dc).Vis the crystal volume, b/vectorkandb† /vectorkare annihilation and creation operators, and /epsilon1∞and/epsilon1dcare the high- and low-frequency bulk permittivities. For a particular transition which involves electronic states ψ1 andψ2, we form product states with LO-phonon modes /vectorkand deﬁne the matrix elements F/vectork,T≡/angbracketleftψ1;0\|ˆHf\|ψ2;1/vectork/angbracketright=/angbracketleftψ1;0\|ˆF/vectork\|ψ2;1/vectork/angbracketright. (17) We now follow previous works and introduce a particular LO-phonon mode T, which is a superposition of plane-wave modes, deﬁned through any number state \|nT/angbracketright[39,40]: \|nT/angbracketright≡1/radicalBig/summationtext /vectork\|F/vectork,T\|2/summationdisplay /vectorkF∗ /vectork,T\|n/vectork/angbracketright. (18) Under assumption of LO-phonon degeneracy, this mode remains an energy eigenmode. This is valid given that theinteraction strength falls off rapidly for phonon wave vectorsnot much larger than the inverse dot size ( ∼20 nm), so that the relevant phonon modes comprise only a small part of theBrillouin zone close to the /Gamma1point. The coupling strength to modeTis then /angbracketleftψ 1;0\|ˆHf\|ψ2;1T/angbracketright=/radicalBigg/summationdisplay /vectork\|F/vectork,T\|2≡/Omega1pol,T, (19) while it can be shown that the matrix element involving any orthogonal mode is zero. Therefore the problem reduces toone involving only a single mode. In the form of an integral over /vectork, the expression for /Omega1 pol,T becomes /Omega1pol,T=A2 (2π)3/integraldisplay /vectorkd3/vectork\|F(T)(/vectork)\|2/k2, (20) where we have deﬁned the form-factor for the transition F(T)(/vectork)≡/angbracketleftψ1\|ei/vectork·/vectorr\|ψ2/angbracketright. (21) The displacement ﬁeld for such a mode can be constructed using ( 18). To obtain a more physical understanding of this mode, we consider the lowest energy intersublevel transitionin a simple cylindrical quantum dot with a height of 30 nm anda diameter of 20 nm, with an inﬁnite conﬁnement potential onall sides. The wave functions are thus products of the inﬁnitesquare well ground and ﬁrst excited states in the axial directionwith the circular well ground state in the cross-sectional plane.A plot of upward and radial displacements for the associatedphonon mode are depicted in Fig. 2. This helps to justify our use of an unbounded plane wave basis—the results are notvery different than if conﬁned modes were used. 01 12 0n1n2n (a) (b) FIG. 2. (Color online) (a) The particular phonon mode interact- ing with a cylindrical QD lowest-lying transition: (left) upward and (right) radial displacements. Both are in separate arbitrary units. Dashed lines denote the dot boundary ( h=30 nm, d =20 nm). Red and blue areas are maxima opposite in sign, and the radial displacement is shown at a phase π/2 relative to that of the upward. (b) Generation and decay processes represented as transitions betweennumber states of a single phonon mode. 155309-4DENSITY MATRIX MODEL FOR POLARONS IN A . . . PHYSICAL REVIEW B 90, 155309 (2014) B. Extension to two transitions We next consider a system with two intersublevel transi- tions, both of which interact coherently with LO phonons.These will later be identiﬁed as the nonradiative depopula-tion (ψ L→ψI) and radiative lasing ( ψU→ψL) transitions, respectively. In considering more than one transition, it is found that we can deﬁne a particular mode associated with each. However,the problem arises that these modes are not generally the samenor orthogonal to one another. The transitions of our concernwould couple to phonon modes NandR, respectively, but we can choose instead basis modes Nandα, where αis a mode in theNR plane of the mode space, but orthogonal to N.T h i s amounts to an orthonormalization within the basis of modesNandR, which could be performed on a larger number of modes using a Gram-Schmidt process. The matrix element for any electronic transition coupling to any arbitrary phonon mode Qis /angbracketleftψ 1;0\|ˆHf\|ψ2;1Q/angbracketright=/Omega1pol,T/angbracketleftT·Q/angbracketright, (22) where /angbracketleftT·Q/angbracketrightis the normalized inner product of mode Qwith the mode associated with the transition. Therefore the important matrix elements governing the problem whenconsidering phonon modes Nandαare given for the N transition as /angbracketleftψ L;0\|ˆHf\|ψI;1N/angbracketright=/Omega1pol,N, (23) /angbracketleftψL;0\|ˆHf\|ψI;1α/angbracketright=0, and for the Rtransition as /angbracketleftψU;0\|ˆHf\|ψL;1N/angbracketright=/Omega1pol,R/angbracketleftR·N/angbracketright, (24) /angbracketleftψU;0\|ˆHf\|ψL;1α/angbracketright=/Omega1pol,R/angbracketleftR·α/angbracketright. The inner product /angbracketleftR·N/angbracketrightcan be computed by a sum over phonon modes as /angbracketleftR·N/angbracketright=1 /Omega1pol,R/Omega1pol,N/summationdisplay /vectorkF/vectork,RF∗ /vectork,N, (25) or as an integral by /angbracketleftR·N/angbracketright=A2 (2π)31 /Omega1pol,R/Omega1pol,N/integraldisplay /vectorkd3/vectorkF(N)∗(/vectork)F(R)(/vectork)/k2. (26) For simplicity, we choose an overall phase for αsuch that /angbracketleftR·α/angbracketrightis positive real. Then, since αlies in the plane deﬁned byRandN, we have that /angbracketleftR·α/angbracketright=/radicalbig 1−\| /angbracketleftR·N/angbracketright\|2. (27) C. Phonon decay and generation LO phonons have a ﬁnite lifetime due to anharmonic decay, typically into pairs of acoustic phonons [ 41]. A rigorous computation of the relaxation time τrfor the LO-phonon distribution towards equilibrium in spherical quantum dotswas performed by Li and Arakawa [ 42]. It was found to be only weakly size-dependent for GaAs dots of diameters greaterthan 15 nm, and results were nearly identical for the two modesconsidered. In our model, we use the approximate ﬁt to theirresults for all modes: τ r(T)=/bracketleftbigg 8−T 54.5K/bracketrightbigg ps. (28) τrinvolves both the competing decay and generation pro- cesses, where by detailed balance the two are equal and opposite at thermal equilibrium. Speciﬁcally, it is deﬁned as 1 τr≡−/Gamma1+−/Gamma1− δN, (29) where /Gamma1±are the generation and decay rates and δNis the deviation from equilibrium. We are interested in the bare decayand generation rates /Gamma1 ± n, which are the transition rates between number states nas depicted in Fig. 2(b). From the form of the interaction Hamiltonian governing the LO-phonon decay[42–44], it is found that /Gamma1 + n=(n+1)/Gamma1+ 0and that /Gamma1− n=n/Gamma1− 1. Combining this result with ( 29) and enforcing thermodynamic equilibrium, the decay and generation rates can be expressedin terms of the relaxation rate as /Gamma1 − n=1 τrn 1−e−ELO/kBT, (30) /Gamma1+ n=1 τrn+1 eELO/kBT−1=1 τr(n+1)nLO, where nLOis the Bose-Einstein factor evaluated at the LO-phonon energy. At low temperature, the generation isextremely slow such that relaxation is dominated by the decay,but at temperatures approaching 300 K, the generation doesbecome signiﬁcant. IV . APPLICATION We are now in the position to compute the steady-state transport and gain characteristics of a model quantum dotQC laser. We choose perhaps the simplest possible system—atwo dot module containing three electronic states. We treatthe lateral quantum conﬁnement as an inﬁnite cylindricalpotential, which allows separation of variables between theaxial and lateral dimensions. This could approximate forexample the conﬁnement of etched nanopillars [ 13,14], or nanowires grown with a core-shell heterostructure [ 45,46]. In order to keep the problem tractable, we only consider thecase where the lateral quantum conﬁnement is sufﬁcient sothat only the lowest lateral energy state is relevant. In practice,this would require lateral conﬁnement which is strong enoughthat the s-penergy separation is signiﬁcantly above E LO.I n GaAs and approximating the lateral conﬁnement as circular,we obtain a value of 50 meV for the energy separation in aconﬁnement diameter of 20 nm. A. Model system The band structure in the growth direction for our model system in GaAs/Al 0.2Ga0.8As, adapted from Refs. [ 47,48], is shown in Fig. 3. The design features a tunnel injection, followed by a diagonal radiative transition, and resonantphonon depopulation. The diagonality is intended to reduce thestrength of the phonon interaction with the lasing transition.Fully discrete electronic states are formed by products of thepictured axial states with the inﬁnite circular well ground 155309-5BENJAMIN A. BURNETT AND BENJAMIN S. WILLIAMS PHYSICAL REVIEW B 90, 155309 (2014) −300 −200 −100 0 100 200 30050100150200 ψP ψU ψL ψI z (Å)Energy (meV) FIG. 3. (Color online) Band structure in the growth direction for the model system, computed from a two-well tight-binding standpoint. The layer thicknesses in angstroms starting from the injector barrier are 37 /82/38/168. The lasing transition at injection anticrossing is 10.6 meV (2.56 THz), the phonon depopulation transition is 36.7 meV , and the injection anticrossing gap is 3.8 meV . The dipole matrix element for the optical transition is 4.7 nm. (s) state in the lateral directions: ψ(z,ρ,θ )=ψz(z)J0(k/bardblρ), where ψzis the axial wave function, J0is the zero order Bessel function, and k/bardblis an in-plane wave vector, which matches the pillar wall boundary condition at the ﬁrst Besselzero. A diagram of the relevant tunneling and electron-phononinteractions is given in Fig. 4. To begin, tunneling will be considered only through the intended channel betweenthe injector and upper radiative states; a parasitic tunnelingmechanism coupling the injector to the lower radiative state (0,0)(1,0)(0,1)(2,0)(1,1)(0,2) (n ,n )Nα FIG. 4. (Color online) Relevant tunneling and electron-phonon interactions. Arrows represent: red is a coupling with Nphonons, green is a coupling with αphonons, and blue are the tunneling processes. Not shown is a parasitic tunnel coupling between ψ/prime I andψL, which will be neglected until Sec. IV F. The vertical axis represents energy, although states grouped together are degenerate.Dashed arrows represent couplings between phonon numbers 1 and 2, which have a strength of√ 2 times those between 0 and 1. The module boundary is deﬁned at the tunnel coupling, between the injector andupper electronic states.will be accounted for and studied in Sec. IV F. We ﬁnd that the most important of the electron-phonon couplings occuracross the depopulation ( ψ L→ψI) transition and the radiative (ψU→ψL) transition. At design biases, we can safely neglect coupling to the higher energy parasitic state ψP, and so it is not considered as part of our calculation. Basis states for the combined electron-phonon system are constructed as tensor products of the threeelectronic states ψ I,ψL, andψUwith phonon number states in modes Nandα, where these modes are deﬁned in the manner described in the previous section. We allow the total numberof phonons in both modes to reach up to two, and the decayand generation rates pertaining to both are assumed to followthe results from Ref. [ 42] and the previous section. The optical Hamiltonian is constructed from the dipole operator z 0, which is expanded appropriately into the tensor product basis. It must be noted that each module in reality contains its own pair of phonon modes Nandα, and so by constructing our schematic of interactions as shown in Fig. 4, we are implicitly enforcing that the occupations in all modules are perfectlycorrelated. This is of course not the case in a real system;however, this approximation is necessary in order to make theproblem tractable. With faster dephasing, coherences spanningthe entire module are reduced, making the approximationcloser to exact. The values relevant to the electron-phonon interaction were computed as /Omega1 pol,R=2.5m e V , /Omega1pol,N=3.3 meV, and /angbracketleftR· N/angbracketright=0.176. These calculations were greatly simpliﬁed by the assumption of a cylindrical cross-section. B. Results A critical parameter is the pure dephasing time T∗ 2, which encompasses all processes that decohere the various interac-tions without changing level populations. It contributes to thelinewidth broadening for various transitions (for example, fora two level system, the transition linewidth is increased by2/planckover2pi1/T ∗ 2), and also determines the coherence of the various interactions (for example, between states tunnel coupled by/Omega1if/planckover2pi1/T ∗ 2/greatermuch/Omega1, the interaction will be incoherent, whereas if/planckover2pi1/T∗ 2/lessmuch/Omega1, it will be coherent, exhibiting strong coupling where the two states form an anticrossed doublet). Theoretical and experimental works suggest that deco- herence in quantum dots occurs primarily due to both realand virtual acoustic phonon processes [ 49,50]. In Ref. [ 49], T ∗ 2was measured in InAs self-assembled quantum dots via four-wave mixing; values ranged from 90 ps at 10 K to 9 ps at120 K. Dephasing was observed to be strongly temperaturedependent (more so than the sublevel lifetimes) but alsoconnected to the detailed energy structure of the system.As expected however, these times are much longer than inconventional quantum-well QC lasers, where T ∗ 2∼300 fs [31,51]. To simplify this intricate problem, we use a single phenomenological T∗ 2parameter throughout our simulations. Unless otherwise speciﬁed, we assume T∗ 2=5 ps at 300 K, which is a reasonably conservative value and consistent withthe computed values of Ref. [ 50]. Results for the steady-state gain proﬁle and population inversion at 100 and 300 K are shown in Fig. 5for vanishing optical intensity. The electron density was taken to be 155309-6DENSITY MATRIX MODEL FOR POLARONS IN A . . . PHYSICAL REVIEW B 90, 155309 (2014) 0 0.5 1 1.5 2050010001500 ω/ω0Gain (cm−1) 0.475 0.463 0.426100K 0 0.5 1 1.5 20100200 ω/ω0Gain (cm−1) 0.315 0.312 0.301300KT2* = ∞ T2* = 5 ps T2* = 1 ps FIG. 5. (Color online) Computed gain proﬁles at the injection anticrossing bias for pure dephasing times T∗ 2o f1p s ,5p s ,a n d ∞at 100 and 300 K, with vanishing optical intensity. Colored numbers on the left give the inverted population fractions. Nd=1016cm−3, which corresponds to an active medium made up of a nanopillar array spaced on an 80 nm grid anddoped with one electron per well. Pure dephasing times T ∗ 2 of 1 ps, 5 ps, and ∞were applied to all coherences. T∗ 2 is especially crucial for the peak gain and linewidth at low temperature, where the lifetimes of the ( nN,nα)=(0,0) states are extremely long due to the slow generation rate. At bothtemperatures, it is also noted that reduction in peak gain due todephasing is attributed mainly to broadening rather than actualloss of population inversion. Figure 6shows gain proﬁles computed from each phonon occupation state separately at a temperature of 300 K andwithout pure dephasing for clarity. Even at 300 K, the largemajority of the gain comes from the (0 ,0) states, which justiﬁes our truncation at a total of two phonons. This is due to both theirlarger populations and longer lifetimes resulting in narrowerlinewidths. By separating the gain into occupation numberswe also note that while the total gain appears to exhibit ﬁvepeaks, there are in fact more as well as resonance shifts whichoccur in the higher phonon occupation states. C. Tunnel coupling dependence The exact locations of the resonance peaks exhibit a com- plicated dependence on the coupling parameters and energy 0 0.5 1 1.5 2050100150 ω/ω0Gain (cm−1)(0,0) (1,0) (0,1) (2,0) (1,1) (0,2) FIG. 6. (Color online) Gain proﬁle separated into phonon occu- pations for T=300 K and T∗ 2=∞ .00.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 20200400600800100012001400 Ωtun=0meVΩtun=0.4meVΩtun=0.8meVΩtun=1.2meVΩtun=1.6meVΩtun=2meVΩtun=2.4meV ω/ω0Gain (cm−1) FIG. 7. Zero-phonon gain at 300 K and T∗ 2=∞ as the tunnel coupling is turned on. Curves are offset by 200 cm−1for clarity. Thin lines denote the anticipated resonances based on diagonalization of the Hamiltonian for the subspace around the zero-phonon radiative transition not including the phonon coupling across the radiativetransition. These energies are given by E rad±/Omega1tun±√ /Omega12 tun+/Omega12 pol,N andErad±/Omega1tun. structure due to the complex nature of the chain-coupled problem, and also experience other shifts due to the dampingmechanisms. In Fig. 7, we provide some insight by examining the evolution of the zero-phonon gain proﬁle without puredephasing as the injection tunnel coupling (one half of theanticrossing gap) is turned on. In practice, this is equivalent tovarying the thickness of the injection barrier. The peak locations can be partially interpreted by diagonal- izingHonly within the subspaces of states directly coupled to the zero-phonon radiative states, ignoring the phonon couplingacross the radiative transition itself. The upper radiative state issplit into a doublet by the tunnel coupling back to the injector,while the lower radiative state is split into a triplet by thephonon coupling to the next injector followed by the tunnelcoupling to the next radiative state. This model is sufﬁcientat low tunnel coupling, where at /Omega1 tun=0 only the polaronic splitting exists, and the peak at the central frequency is absentdue to its vanishing coupling strength to radiation. As thetunnel coupling strength is increased, the peak near centerfrequency begins to emerge and eventually dominates the gainproﬁle. We attribute the emergence of this peak to the onsetof the phonon coupling across the radiative transition, whichhighly expands the Hilbert space relevant even to only thezero-phonon gain. This polaronic splitting represents a majordifference compared to a conventional quantum-well QC laser,and must be properly accounted for in any design. D. Current versus voltage characteristic Figure 8shows the transport characteristic at 100 and 300 K, alongside the gain proﬁles at 300 K for various bias points.The current is signiﬁcantly higher at 300 K due to the fasterphonon decay which results in overall faster transport. Theproﬁles demonstrate noticeable shifts in peak position andamplitude as the voltage is tuned, varying the alignment ofthe injector states. While this highlights the complexity of the 155309-7BENJAMIN A. BURNETT AND BENJAMIN S. WILLIAMS PHYSICAL REVIEW B 90, 155309 (2014) 30 40 50 601020304050 Bias per module (mV)Current Density (A/cm2)100K 300K 0 1 202004006008001000 35.141.747.452.156.9 ω/ω0Gain (cm−1) FIG. 8. (Left) Transport characteristic for T∗ 2=5 ps. The anti- crossing bias is marked by the dashed vertical line. (Right) Gainproﬁle at 300 K for various bias points, labeled by bias/module in millivolts (offset for clarity). problem, it also suggests that such a device may be a candidate for wide bias-tunability. A noticeable feature absent from Fig. 8is the negative differential resistance (NDR) anticipated for biases above theinjection resonance (design) bias. This is explained by thephonon bottleneck effect itself, which suppresses transportthrough the device but is eased by increasing bias as theradiative transition is tuned closer to E LO.I nF i g . 9,i ti ss h o w n that the NDR does in fact emerge if an additional scatteringmechanism is included across the radiative transition. Thescatterer is considered to be a spontaneous boson emissionrateτ sp, which is accompanied by stimulated emission and absorption rates τst=τabs=τsp/nr, where nris the Bose- Einstein occupation at the radiative energy. In an actual device,this might represent acoustic phonon scattering, for example.While for a rate τ sp=100 ps, a very large increase is seen in the current, the gain remains relatively unaffected for τsp>10 ps. E. Gain saturation A particular advantage to our method is the ability to automatically account for effects of increasing optical intensitydirectly onto the gain proﬁle, allowing us to study gain satu-ration without needing to extract a stimulated emission rate. 30 40 50 60102030405060 τsp=100 ps τsp=1 ns τsp=∞ Bias per module (mV)Current Density (A/cm2) 0 0.5 1 1.5 2050100150 ω/ω0Gain (cm−1)∞ 100ps 10ps 3ps 1ps FIG. 9. (Color online) (Left) Transport characteristic as a scatter- ing mechanism is introduced, for T∗ 2=5 ps. Blue denotes 100 K and red 300 K. Rates τspare∞, 1 ns, and 100 ps in order of increasing current. (Right) Gain at anticrossing bias, 300 K, for various scattering timesτsp.00.2 0.4 0.6 0.8 11.2 1.4 1.6 1.8 2050100150 ω/ω0Gain (cm−1)0W/mm2 100W/mm2 500W/mm2 1000W/mm2 2000W/mm2 FIG. 10. (Color online) Gain saturation as optical intensity is increased, for T=300 K and T∗ 2=5p s . Figure 10shows the change in gain proﬁle as the circulating optical intensity is increased ( I=2/epsilon10nc\|E\|2). Reduction in peak gain is evident due to loss of overall population inversion,redistribution of population among various states, and effectivelifetime broadening. In this way, the steady-state opticalintensity could be estimated in a laser system by clampingthe peak gain to the total cavity losses. F. Parasitic tunneling To this point, we have focused on somewhat of an ideal case, where only tunnel coupling from the injector to upper radiativestate is considered. However, it is well known that a major issuefor THz QC lasers is the existence of a parasitic current channelthat occurs for voltage biases below the injection resonance[5,31,52]. While the details vary between designs, this current channel is associated with tunnel coupling from the injectorto the lower radiative state or the excited state in the widedepopulation well. The presence of this parasitic current setsa ﬂoor on the threshold current density, and if it is too strong,creates a premature NDR, which prevents reaching the designbias. In conventional QC lasers, since this coupling is typically/Omega1 p∼0.2–0.5 meV , the relatively fast dephasing ( T∗ 2∼0.3p s ) helps to suppress this current. Since the dephasing times ina quantum dot QC laser are expected to be 1–2 orders ofmagnitude longer, a concern naturally arises that this parasiticchannel will be too strong. To account for this effect, we now introduce a tunnel coupling from the injector to the lower radiative state,having a value computed from the level anticrossing as/Omega1 p=0.875 meV . Although this channel is well detuned at the injection resonance, it is, however, important at lower bias.Figure 11demonstrates the effect of the parasitic coupling on the transport characteristic for T ∗ 2=5 ps and 1 ps, and the gain at various bias points for T∗ 2=5 ps. Very large current ﬂow is found at biases over a wide range around the parasiticresonance, leading to a considerable NDR. As expected, thegain is signiﬁcantly modiﬁed at lower bias points while athigher bias the parasitic tunneling becomes unimportant as itis further detuned. 155309-8DENSITY MATRIX MODEL FOR POLARONS IN A . . . PHYSICAL REVIEW B 90, 155309 (2014) 30 40 50 6050100150200250300350400450500 Bias per module (mV)Current Density (A/cm2) T2* = 5 psT2* = 1 psΩp=0 Ωp=0.875meV 0 1 202004006008001000 35.141.747.452.156.9 ω/ω0Gain (cm−1) FIG. 11. (Color online) Effects of a parasitic tunneling channel atT=300 K. (Left) Modiﬁed transport characteristic for T∗ 2=5 and 1 ps. (Right) Modiﬁed gain at various bias points for T∗ 2=5p s (labelled in mV/module and offset by 200 cm−1for clarity). A similar current instability was predicted in Ref. [ 28], where the possibility of doubling all barrier thicknesses wasexplored. In our two-well design, where the radiative transitionis diagonal, it is clearly disadvantageous to increase the radia-tive barrier, but, for example, doubling only the injector barrierthickness from 3.7 to 7.4 nm reduces the injection couplingfrom 1.9 to 0.3 meV and the parasitic coupling from 0.875to 0.2 meV . However, this reduction in the injection couplingintroduces other complications, importantly a large splittingin the gain spectrum as shown in Fig. 7. Furthermore, in order to appreciably reduce the parasitic current level, one requiresthe coupling to be /Omega1 p/lessmuch/planckover2pi1/T∗ 2(0.13 meV for T∗ 2=5p s ) , which is difﬁcult to achieve in this simple two-well design. Itis likely that more sophisticated designs will be required thatselectively reduce the parasitic tunnel coupling, although thiswill be at the cost of device and material complexity. V . CONCLUSIONS A density matrix formulation has been derived for com- puting the steady-state gain and current in quantum cascadesystems of arbitrary size driven by a classical light ﬁeld.Gain is calculated coherently from the optical susceptibilitywhich arises from the induced harmonic coherences. Themethod is also useful for other quantum-cascade systems, andcould readily be generalized for the study of nonlinear effectssuch as harmonic, sum frequency, and difference frequencygeneration. The method was applied to a nanopillar-based quantum dot QC laser, where coherent interaction of the discreteelectronic density of states with quantized LO-phonon modeswas accounted for alongside phonon decay processes, electrontunneling, and the light ﬁeld. Results predict a complexdependence on coupling parameters, energy structure, anddamping parameters, and forecast high temperature operation,wide bias tunability, and considerable robustness to addedscattering mechanisms. A simple way to account for gainsaturation was demonstrated, and ﬁnally the effect of parasitictunnel coupling was isolated, leading to predictions of possibleelectrical instability.This work addresses the feasibility of an idealized quantum dot QC laser, where certain practical concerns such as dotinhomogeneity or interface roughness are not accounted for. Afurther limitation is the inclusion of only two phonon-coupledtransitions, restricting our treatment to the regime of smallpillar diameter ( ∼20 nm in GaAs). As the pillar diameter becomes wider, the higher lateral ( p) states become important, thus greatly expanding the necessary Hilbert space and cou-pling parameters. Such a problem is tractable by this method,although it would require an algorithm for automaticallyenumerating the basis states and computing matrix elements. Even in this idealized system, several key conclusions emerge. First, as expected, the formation of intersublevel-LO-phonon polarons is beneﬁcial in the long upper state relaxationtimes, which leads to signiﬁcant population inversion levelseven at room temperature. This leads to peak gain on theorder of 100 cm −1at 300 K, which is sufﬁcient for lasing in a low-loss metal-metal waveguide where the losses are∼15–30 cm −1[53]. The exact peak values depend upon the pure dephasing parameters, which will require further exper-imental and theoretical consideration. Second, the coherentpolaron formation also leads to a series of level splittings onthe order of several meV . This produces a complicated gainspectrum with multiple peaks that depend strongly on bias, theelectron-phonon interaction strength, and the tunnel coupling.Third, in our model system the longer dephasing times of several ps lead to a strong parasitic current channel which may cause electrical instabilities. While the simple two-well designpresented here has few degrees of freedom, it is possible thatnew design strategies could minimize this effect. In summary, future quantum dot QC lasers are predicted to have sufﬁcient gain for room-temperature operation. However,they are likely to encounter fundamentally new transportphysics not present in conventional QC lasers, which mustbe properly accounted for during the design and modelingprocess. Our results suggest that naively scaling existingterahertz QC-laser designs to the quantum dot limit may meetsome difﬁculty. ACKNOWLEDGMENTS This work was partially supported by NSF grants ECCS- 1002387 and ECCS-1202591. The authors thank A. Pan forhelpful discussions. APPENDIX: DERIVATION OF INCOHERENT CONTRIBUTION TO M The incoherent evolution is separated into that due to each transition and pure dephasing: d dtρ/vextendsingle/vextendsingle/vextendsingle/vextendsingleinc =/summationdisplay XLXρ+Dρ. (A1) LXis the Lindblad superoperator for transition X, which is constructed in the form [ 38,54] LXρ=CXρCX†−1 2(CX†CXρ+ρCX†CX), (A2) where CXis the jump operator which induces the transition. For a simple transition ψi→ψfhaving rate /Gamma1i→f,t h e 155309-9BENJAMIN A. BURNETT AND BENJAMIN S. WILLIAMS PHYSICAL REVIEW B 90, 155309 (2014) associated jump operator is C=/radicalbig/Gamma1i→f\|ψf/angbracketright/angbracketleftψi\|. In this case, Cwill have only one nonzero element, but in a combined Hilbert space this may not be true; Citself must be expanded as a tensor product and thus can acquire more than one nonzeroelement, in which case transfers of coherence can occur. For example, we can examine the collapse operator which is due to the transition of phonon mode Nfromn N=1 tonN=0. Given the allowed mode occupations, the col- lapse operator in the space of {\|nN,nα/angbracketright}is then C1N→0N=√ /Gamma1− 1(\|00/angbracketright/angbracketleft10\|+\| 01/angbracketright/angbracketleft11\|).I fNelelectron degrees of freedom are included, C1N→0Nis further expanded to√ /Gamma1− 11Nel⊗ (\|00/angbracketright/angbracketleft10\|+\| 01/angbracketright/angbracketleft11\|). We will neglect correlations in the different transition processes between number states of agiven phonon mode by including separate jump operators foreach.Once the jump operators are obtained, we need to use them to ﬁll out elements of S (ab)mp,(cd)nqin the chain-coupled system. These are deﬁned as [LXρ](m) p,ab≡/summationdisplay qncdSX (ab)mp,(cd)nqρ(n) q,cd, (A3) or in other words the coefﬁcients relating variable ρ(n) q,cd to the evolution of ρ(m) p,ab due to transition X. Importantly, we ﬁrst distinguish between transitions which are correlatedbetween modules and those that are not. In the former, thejump operator itself assumes a chain-coupled form whichforms a single Lindblad superoperator [shown in Eq. ( A4)], whereas in the latter there exists a series of jump operatorswhich form separate Lindblad superoperators, which are thensuperimposed [shown in Eq. ( A5)]: L⎛ ⎜⎜⎜⎜⎝⎡ ⎢⎢⎢⎢⎣... (¯C) (¯C) ...⎤ ⎥⎥⎥⎥⎦⎞ ⎟⎟⎟⎟⎠(A4) ···+L⎛ ⎜⎜⎜⎜⎝⎡ ⎢⎢⎢⎢⎣... (¯C) (0) ...⎤ ⎥⎥⎥⎥⎦⎞ ⎟⎟⎟⎟⎠+L⎛ ⎜⎜⎜⎜⎝⎡ ⎢⎢⎢⎢⎣... (0) (¯C) ...⎤ ⎥⎥⎥⎥⎦⎞ ⎟⎟⎟⎟ ⎠+.... (A5) In the ﬁrst case, we ﬁnd that the elements in SXare SX (ab)mp,(cd)nq=δpqδmn/bracketleftbig¯Cac¯C† db−1 2(δbd[¯C†¯C]ac +δac[¯C†¯C]db)/bracketrightbig , (A6) assuming that ¯Cresides in the diagonal submatrices, and for the second we ﬁnd that SX (ab)mp,(cd)nq=δpqδmn/bracketleftbig δp0¯Cac¯C† db −1 2(δbd[¯C†¯C]ac+δac[¯C†¯C]db)/bracketrightbig , (A7) independent of any displacement of ¯Cfrom the diagonal. Since the collapse operator ¯Cis always positive, we no- tice in the solution for SXthat correlated transitions can transfer intermodule coherence while uncorrelated transitionsdo not. The pure dephasing contribution Dis trivial. For a pure dephasing time T ∗ 2applied to all coherences, it is D(ab)mp,(cd)nq=−1 T∗ 2δpqδmnδacδbd(1−δp0δab), (A8) but can also easily be generalized to incorporate different dephasing times.Finally, we must derive the expression for the incoherent contribution to velocity, and thus Jinc. We are interested in the expectation value of velocity due to incoherent processes, andso we equate /angbracketleftv inc/angbracketright≡d dTr(ρz)/vextendsingle/vextendsingle/vextendsingle/vextendsingleinc =Tr/bracketleftbigg/summationdisplay XLXρz+Dρz/bracketrightbigg . 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PhysRevB.100.075401.pdf	PHYSICAL REVIEW B 100, 075401 (2019) Time-resolved magneto-Raman study of carrier dynamics in low Landau levels of graphene T. Kazimierczuk ,1,A. Bogucki,1T. Smole ´nski,1M. Goryca,1C. Faugeras,2 P. Machnikowski,3M. Potemski,1,2and P. Kossacki1 1Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ulica Pasteura 5, 02-093 Warsaw, Poland 2Laboratoire National des Champs Magnétiques Intenses, CNRS-UGA-UPS-INSA-EMFL, 25 rue des Martyrs, 38042 Grenoble, France 3Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland (Received 4 October 2018; published 1 August 2019) We study the relaxation dynamics of the electron system in graphene ﬂakes under a Landau quantization regime using an approach of time-resolved Raman scattering. The nonresonant character of the experimentallows us to analyze the ﬁeld dependence of the relaxation rate. Our results clearly evidence a sharp increasein the relaxation rate upon the resonance between the energy of the Landau transition and the G band and shedlight on the relaxation mechanism of the Landau-quantized electrons in graphene beyond the previously studiedAuger scattering. DOI: 10.1103/PhysRevB.100.075401 I. INTRODUCTION Despite the whole rapidly expanding ﬁeld of atomically thin semiconductors, graphene is still one of the most im-portant systems with applications already being introduced.Carrier dynamics is one of the relevant issues in the devel-opment of devices. Although they are directly related to thetransport properties, optical tools are often needed to gainbetter insight into the carrier behavior. In particular, the ultra-fast dynamics of the carrier relaxation can be accessed usingoptical pump-probe techniques (for a review, see Ref. [ 1]). Although such an approach has been extensively exploited tostudy the basic problem of relaxation dynamics in graphene atzero magnetic ﬁeld, an independent case of carrier relaxationbetween Landau levels (LLs) emerging upon application ofthe magnetic ﬁeld still remains relatively unexplored. In fact,there were only two time-resolved optical studies dealing withthe carrier dynamics in Landau-quantized graphene [ 2,3]. In both of these studies, the pump and probe pulses were of thesame energy. The pump pulse was utilized to initially populatea certain electronic LL, while the intensity of the probe wasused to measure the dynamics of subsequent depletion of thisLL [see Fig. 1(a)]. Such a depletion was evidenced to be due to efﬁcient Auger scattering regardless of the number of theexcited LLs, being either a high-energetic level ( n∼100) in the case of experiment exploiting a near-infrared Ti:sapphirelaser [ 2] or a low-lying level ( n=0,1) when the graphene is excited with a THz radiation produced by a free-electronlaser [ 3]. Here, we present a study of qualitatively different, slower carrier-relaxation processes in Landau-quantized graphene,which take place after the system reaches its quasiequi- librium state due to fast Auger scattering [see Fig. 1(b)]. Experimentally, it is realized by combining the pump-probetechnique with monitoring the electronic transitions between tomasz.kazimierczuk@fuw.edu.plLLs using Raman scattering spectroscopy. More speciﬁcally, anear-infrared Ti:sapphire laser pulse is exploited to pump thecarriers into some high-energy LLs, from which they Augerscatter occupying lower LLs, the population of which is ﬁnallymeasured based on the intensity of magneto-Raman peakscorresponding to electronic excitation between different LLs.The feasibility of such a technique was recently proven withrespect to phonon transitions at zero magnetic ﬁeld [ 4]. It has an advantage of high spatial resolution, inaccessible in previ-ous experiment exploiting THz sources due to the differencein the wavelength scale. The small diameter of the laser spotsize additionally beneﬁts from superior performance of theoptics in the visible range (high-NA objectives). Altogether,the presented approach allows one to easily achieve submi-crometer resolution in a standard microphotoluminescencesetup, as compared with 0.5 mm spot size in a typical THz ex-periment [ 3]. In such a regime, the intensity of various peaks in the Raman scattering spectrum provides direct access tocarrier dynamics in a given LL, even for very small grapheneﬂakes, such as graphene domains on the surface of the naturalgraphite, which in turn exhibit much better optical propertiesas compared, e.g., to larger epitaxial graphene ﬂakes [ 5,6]. An additional advantage of this system is its inherent neutrality,which makes the results clear from the effects of residualbackground carriers that are possibly present in the graphenesystem [ 7]. II. SAMPLES AND CHARACTERIZATION In our work, we studied graphenelike domains occurring on the surface of the natural graphite. The identiﬁcation ofsuch domains was performed upon application of a strongmagnetic ﬁeld, which reveals qualitative differences in theLL structure between two-dimensional (2D) graphene and3D graphite [ 8,9]. Figure 2(a) presents the magnetic ﬁeld evolution of the Raman scattering spectrum measured forone out of several investigated graphene domains. Apartfrom the well-known phonon-related resonances—G band 2469-9950/2019/100(7)/075401(5) 075401-1 ©2019 American Physical SocietyT. KAZIMIERCZUK et al. PHYSICAL REVIEW B 100, 075401 (2019) FIG. 1. Scheme presenting the spectrum of Landau levels in graphene placed in an external magnetic ﬁeld. Color saturation de- notes the relative population of LLs (a) directly after the pump pulsetuned to a transition between certain low-lying hole and electron LLs, and (b) after a few-hundred fs following a relatively high- energy (near-infrared) laser pulse, during which the system reachesits quasiequilibrium state due to fast Auger scattering. ≈1590 cm−1and 2D band ≈2690 cm−1—the data feature a series of ﬁeld-dependent peaks corresponding to electronicexcitations between different LLs. The optical selection rulesallow transitions between the nth hole Landau level and the mth electron Landau level provided that \|n−m\|/lessorequalslant1[10]. The energy position of such transitions in the spectrum isdescribed by the square-root dependence characteristic for the FIG. 2. Magnetic ﬁeld dependence of the Raman scattering spec- trum measured at T=200 K under excitation with a cw Ti:sapphire laser at λ=781 nm. Each spectrum was corrected by subtracting 80% of the zero-ﬁeld spectrum in order to spotlight the ﬁeld-induced changes. The white dashed lines mark two example Landau-leveltransitions originating from the graphene domain. The signal below 400 cm −1is suppressed due to the long-pass ﬁlter placed in the detection path.Dirac dispersion [ 11], E−n,m=√ 2¯heBvF(/radicalbig \|n\|+/radicalbig \|m\|). (1) The Fermi velocity extracted from the data shown in Fig. 2 yields vF=1.00×106m/s, which is comparable to the results reported in previous studies of such a system [ 6]. In the time-resolved experiments described in the followingsections, we focused mainly on the strongest Landau-leveltransition from the ﬁrst hole level to the ﬁrst electron leveldenoted as L −1,1. III. TIME-RESOLVED RAMAN SCATTERING SPECTROSCOPY The core results presented in this work were obtained using the two-color pump-probe Raman scattering spectroscopytechnique in a Landau-quantization regime. The magneticﬁeld needed for such experiments was applied by placingthe sample inside a cryostat equipped with a superconductivemagnet ( B=0–10 T) oriented in Faraday geometry. An as- pheric lens, mounted on a piezopositioner directly in front ofthe sample, allowed us to obtain spatial resolution of about1μm. Such high resolution was important due to the relatively small dimensions of the graphene domains as well as toachieve high pump laser ﬂuence, which was needed for thetime-resolved experiments. The probe beam used for the Raman scattering spec- troscopy was either a femto- or picosecond Ti:sapphire laseratλ Ti:Sa=775 nm with 76 MHz repetition rate. The resulting Raman scattering signal was dispersed by a 30 cm monochro-mator and recorded with a Si-based CCD camera. The acqui-sition time for a single spectrum was between 20 s (character-ization with the CW laser) and 60 s (measurements with thepulsed laser). In the latter case, the signal was improved byaveraging multiple measurement series. Dichroic ﬁlters in theexcitation and the detection path were employed, respectively,to ﬁlter out the ampliﬁed spontaneous emission (ASE) and toremove the excess of the Rayleigh-scattered laser light. Simul-taneously, the sample was additionally excited with the second(pump) beam produced by an optical parametrical oscillator(OPO) at λ OPO=1200 nm. The Raman signal induced by the OPO laser corresponds to the infrared (IR) range (e.g.,λ=1.48μm for the G band), and thus it was not detected in the experiment. The OPO was pumped with the same Ti:sapphire laser that was used as a probe, which assured the necessary synchro-nization between both laser pulse trains. The delay betweenpump and probe pulses was adjusted using a mechanical delayline. The overall temporal resolution of the experiments waslimited by the duration of laser pulses. For the femtosec-ond conﬁguration, used in most of the experiments, the fullwidth at half maximum (FWHM) of the pump pulses yielded0.21 ps, while the FWHM of the probe pulses was equal to0.44 ps. The main contribution to the latter value was theeffect of the band-pass ﬁlter in the excitation path. Cross-correlation measurements of the pulses from both sourcesrevealed no appreciable jitter, which would lead to reductionof the temporal resolution. In the picosecond conﬁguration,the FWHM of the pump pulses yielded 3.7 ps, whereas theFWHM of the probe pulses was equal to 1.8 ps. 075401-2TIME-RESOLVED MAGNETO-RAMAN STUDY OF CARRIER … PHYSICAL REVIEW B 100, 075401 (2019) FIG. 3. A series of Raman scattering spectra measured using a pulsed laser of different intensity at B=10 T. Each spectrum was normalized using intensity of the G-band peak as the reference. IV . RESULTS The LL population was studied by analysis of the relative intensity of the Raman Ln,mpeaks. Crucially, such a quantity is known to be proportional to the probability of the optical tran-sitions between the involved levels, which become blocked forincreasing occupancy of the LLs. As a result, the intensity ofthe Raman line starts to be quenched for sufﬁciently high car-rier density, which in our case was controlled by changing thepower of the exciting laser. The invoked behavior is illustratedin Fig. 3, which presents a set of Raman spectra measured at B=10 T using different intensities of the pulsed laser. Each of these spectra features two phonon peaks (G band, 2D band)as well as a multitude of electronic peaks ( L −1,1,L−1,2,...). The intensity of the phonon-related Raman peaks scales lin-early with the excitation power. The underlying reason forsuch behavior is the high density of phonon states and thattheir population can be affected signiﬁcantly only by usingmuch stronger pump pulses [ 4]. In contrast, the electronic peaks exhibit the aforementioned saturation behavior due toﬁlling the relevant electron or hole Landau levels, which, inturn, limits the density of states available for further Ramanscattering [ 9,12]. The same phenomenon was exploited in the pump-probe experiment: the strong pump pulse was utilized to initiallypopulate the low Landau levels, while the intensity of theRaman peaks from the probe pulse was used as a measureof this population at later time. Example data measured insuch an experiment are shown in Fig. 4. The power of the pump and probe beams was set to, respectively, 20 and0.6 mW. The presented results clearly show that directly afterthe pump pulse, the electronic Raman signal is weaker dueto the reduced density of states. The full spectrum of thechanges [shown in Fig. 4(a)] evidences that the pump indeed affects only the L n,mpeaks, while the phonon peaks (e.g., the G band) do not exhibit a noticeable variation upon arrivalof the pump pulse. In agreement with previous studies [ 3], the characteristic timescale of the pump-induced perturbationis in the range of a few picoseconds. Based on relativelyfast (subpicosecond) rise time of the signal, we attribute thedecay dynamics directly to the relaxation rate of the quasither-malized electronic system. The value of the relaxation time FIG. 4. (a) Spectrum of the changes induced by the pump beam in the Raman spectrum at T=200 K and B=10 T. The color reﬂects a difference between the measured Raman signal and the base signal at the same energy determined for the negative delay. (b) Thereference Raman spectrum on top of the map marks the position of various peaks in the spectrum. (c) Transient of the integrated intensity of the L −1,1peak. was extracted from the data by ﬁtting the exponential-decay proﬁle to the measured transient. The example data shownin Fig. 4(c) yield a decay time of τ=(3.4±1.0) ps. The employed experimental technique allowed us to follow thedynamics of the population of the Landau level continuouslyupon changes of the magnetic ﬁeld, which was inaccessible inprevious pump-probe experiments [ 3]. Two systematic data series are presented in Fig. 5(a) for T=200 and T=10 K. As seen, the data obtained for theL −1,1andL−2,2transitions at lower temperature overlay each other, which is consistent with our assumption that the FIG. 5. (a) Electron relaxation times as a function of the mag- netic ﬁeld. A set of square-root functions are marked by dashed lines as a guide to the eye. Two arrows indicate the resonant ﬁelds discussed in the text. (b) Pump-induced change in the signal for twomagnetic ﬁelds demonstrating the difference in relaxation time. The straight lines represent the exponential-decay proﬁles ﬁtted to the experimental data. 075401-3T. KAZIMIERCZUK et al. PHYSICAL REVIEW B 100, 075401 (2019) measured decay corresponds to a relaxation of the system remaining in a quasiequilibrium with respect to Auger scat-tering. Importantly, we ﬁnd that the rate of such a relaxationsigniﬁcantly increases around B=5 T and B=7T( m o r e visible for the lower temperature). This is an observation ofthe theoretically predicted [ 13,14] increase in the electron relaxation rate due to the resonance between the energy of theLandau levels and the E 2g(G-band) phonon. In particular, at B=5 T, the resonance occurs for L−2,1andL−1,2transitions, while at B=7T ,i ti st h e L−1,1transition, which coincides with the energy of the invoked phonon. Surprisingly, in ourdata the resonance at 7 T is much broader than the one at 5 T.This ﬁnding remains in contrast with the previous theoreticalpredictions, according to which the resonant increase in therelaxation rate should occur rather for the nonsymmetrictransitions (e.g., −1→2) due to their strong mixing with the optical phonons [ 13,14]. The reason for this disparity between the theory and the experiment is not clear at the moment. The second observation on top of the resonant behavior discussed above is that, in general, the relaxation rate sys-tematically increases with the magnetic ﬁeld and is muchfaster at higher temperature. Such ﬁnding might seem to beexpected as several processes related to the interaction withacoustic phonons exhibit a similar increase of the rate withthe magnetic ﬁeld and temperature. For example, in manysystems, spin relaxation accelerates due to an increase of thenumber of phonons accessible for higher relaxation energy[15]. Similarly, the increase of the temperature results in the increase of the population of acoustic phonons and higherprobability of the relaxation. However, the present case of thegraphene is qualitatively different. The picosecond-timescalerelaxation of the LL occupation is related to cooling of the hot-electron system, which has to be mediated by electron-latticeenergy transfer. The inter-Landau-level transition requires en-ergy much higher than thermal energy, even at moderate mag-netic ﬁeld (e.g., 1 →0 transition at B=5 T corresponds to 670 K). Therefore, the thermal population of active phononsis negligible and no signiﬁcant variation should be observedfor a reasonable temperature range. Thus our experimentalﬁndings unequivocally show that the process cannot be ex-plained by simple phonon-assisted relaxation between LLsand the relaxation is related to more complex processes in-volving low-energy acoustic phonons and other higher-energyexcitations. The simplest mechanism is a two-phonon process,in which most of the energy is carried out by the opticalphonon, while acoustic phonons provide the required contin-uum. Such processes appear in the second order in the carrier-phonon coupling or via optical phonon anharmonicity [ 16]. The former relies on the electron-acoustic-phonon coupling.An electron on the Landau level effectively interacts onlywith phonons of wavelengths not greater than the magneticlength l B, which restricts the available acoustic (in-plane) phonon energy to, at most, ( v/vF)E1, where vis the speed of sound. Since v/vF∼10−2, the energy conservation limits this two-phonon process to a very narrow range of inter-LLseparations around the optical phonon energy. A quantitativeestimate is obtained by treating the two-phonon relaxationas an acoustic-phonon-mediated transition between L −1,1and the electronic ground state with one optical phonon, whichis enabled by an optical-phonon admixture to the LL states.The most resonant optical-phonon admixture to the L −1,1 excitation is the phonon-assisted L0,1orL−1,0state since the most resonant single-phonon state on the electronic groundstate (the G line in the Raman spectrum in Fig. 2) is decoupled from the L −1,1electronic excitation, as witnessed by the lack of resonant anticrossing in the spectrum (which suggests thatthe observed excitation is valley symmetric [ 17]). With only this admixture included and using the description of carrier-phonon interaction in graphene [ 18,19], the maximum values of the relaxation rate at 200 K, in the close vicinity of theresonant magnetic ﬁeld of 7.3 T, are comparable to thosefound in our measurements, while at 10 K, the rates are afew orders of magnitude below the experimental values. Inaddition, the theoretically predicted rate for this relaxationchannel falls off exponentially and decreases by many ordersof magnitude already at 1 T off resonance, in obvious contrastwith the measurements. The anharmonicity-induced relaxation channel yields rates smoothly varying with the magnetic ﬁeld because the an-harmonic decay of the zone-center optical phonon can in-volve acoustic phonons with arbitrary, mutually oppositewave vectors. The resulting rate can be estimated as theproduct of the optical-phonon decay rate and the optical-phonon admixture to the LL. The former is determined, bothexperimentally [ 20] and theoretically [ 21], to be of the order of a few ps. The effective optical-phonon-assisted coupling Vto other LLs, separated by energies of the order of the G-mode energy E G, can be estimated as the typical width of a resonant anticrossing between the G line and the LLexcitations, which is of the order of a few meV . The admixtureis then of the order of the Huang-Rhys factor ( V/E G)2∼ 10−4, yielding the anharmonicity-induced inter-LL relaxation in graphene ineffective. Quantitative calculations indeed yieldrelaxation times of the order of 100 ns at magnetic ﬁeldsaround 7 T. We therefore conclude that the decay of the LL occupation has to be attributed to the overall cooling of the hot-electronsystem via a more complex process, which might explain,in particular, the thermal dependence of the rate, which iscompatible neither with the energies of the effectively cou-pled acoustic phonons nor with the inter-LL separation, norwith the optical-phonon energy. As revealed by the data inFig. 5(a), the decay rate scales roughly as ∝√ Bin the low- temperature regime. Such a magnetic ﬁeld dependence wastheoretically predicted for broadening of LLs due to impurity-induced dephasing [ 22], which, in principle, could be respon- sible for the observed increase of the relaxation rate as longas it is accompanied by phonon scattering since the impuritydephasing alone is not expected to exhibit any temperaturedependence [ 22]. Another feature that is difﬁcult to explain in terms of simple relaxation processes is the pronouncedchange in the character of the magnetic ﬁeld dependence ofthe relaxation rate upon increasing the temperature, which isfound to be almost linear at T=200 K. A detailed discussion of the physical reason for the observed features would requiremore in-depth knowledge about the nature of the involved re-laxation processes, which demands further theoretical studies.Possible mechanisms may involve relaxation between higherLLs or combined phonon-Auger processes involving thoselevels, accompanied by very fast redistribution of occupations 075401-4TIME-RESOLVED MAGNETO-RAMAN STUDY OF CARRIER … PHYSICAL REVIEW B 100, 075401 (2019) between the levels, consistent with the fs-timescale rise of the Raman signal. V . CONCLUSIONS To conclude, our results demonstrate the feasibility of the time-resolved Raman scattering technique in studies ofthe Landau-quantized electrons. The nonresonant characterof the Raman experiment enabled us to vary the magneticﬁeld continuously, which is a distinct advantage over previousapproaches [ 3]. The results of our study qualitatively conﬁrm predictions regarding a resonant increase in the electronicrelaxation rate due to resonances with optical phonons. How-ever, these results also reveal some deﬁciency of the existingtheoretical description of the carrier dynamics in graphene ata strong magnetic ﬁeld. Further theoretical studies are needed to determine whether the detected discrepancy is related tothe nonresonant character of our experiment or perhaps is anindication of the inadequacy of the assumptions made in theexisting models. ACKNOWLEDGMENTS This work was supported by the Polish National Science Centre as research Grants No. DEC-2013 /10/M/ST3/00791 and No. DEC-2015 /17/B/ST3/01219, the EC Graphene Flagship project (Grant No. 785219), and the ATOMOPTOproject carried out within the TEAM programme of the Foun-dation for Polish Science coﬁnanced by the European Unionunder the European Regional Development Fund. [1] E. Malic and A. Knorr, Graphene and Carbon Nanotubes: Ultrafast Optics and Relaxation Dynamics (Wiley-VCH, New York, 2013). [2] P. Plochocka, P. Kossacki, A. Golnik, T. Kazimierczuk, C. Berger, W. A. de Heer, and M. Potemski, P h y s .R e v .B 80, 245415 (2009 ). [3] M. Mittendorff, F. Wendler, E. Malic, A. Knorr, M. Orlita, M. Potemski, C. Berger, W. A. de Heer,H. Schneider, M. Helm et al. ,Nat. Phys. 11,75 (2015 ). [4] J.-A. Yang, S. Parham, D. Dessau, and D. Reznik, Sci. Rep. 7, 40876 (2016 ). [5] P. Neugebauer, M. Orlita, C. Faugeras, A.-L. Barra, and M. Potemski, P h y s .R e v .L e t t . 103,136403 (2009 ). [6] C. Faugeras, M. Amado, P. Kossacki, M. Orlita, M. Kühne, A. A. L. Nicolet, Y . I. Latyshev, and M. Potemski, Phys. Rev. Lett.107,036807 (2011 ). [7] D. Sun, C. Divin, C. Berger, W. A. de Heer, P. N. F i r s t ,a n dT .B .N o r r i s , Phys. Rev. Lett. 104,136802 (2010 ). [8] M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A. de Heer, Phys. Rev. Lett. 97,266405 (2006 ). [9] C. Faugeras, M. Orlita, and M. Potemski, J. Raman Spectrosc. 49,146(2018 ). [10] O. Kashuba and V . I. Fal’ko, Phys. Rev. B 80,241404(R) (2009 ).[11] 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 ). [12] P. Kossacki, C. Faugeras, M. Kuhne, M. Orlita, A. Mahmood, E. Dujardin, R. R. Nair, A. K. Geim, and M. Potemski, Phys. Rev. B 86,205431 (2012 ). [13] Z.-W. Wang, L. Liu, L. Shi, X.-J. Gong, W.-P. Li, and K. Xu, J. Phys. Soc. Jpn. 82,094606 (2013 ). [14] F. Wendler, A. Knorr, and E. Malic, Appl. Phys. Lett. 103, 253117 (2013 ). [15] K. J. Standley, Electron Spin Relaxation Phenomena in Solids , Monographs on Electron Spin Resonance (Springer, New York,1969). [16] L. Jacak, P. Machnikowski, J. Krasnyj, and P. Zoller, Eur. Phys. J. D22,319(2003 ). [17] M. O. Goerbig, J.-N. Fuchs, K. Kechedzhi, and V . I. Fal’ko, Phys. Rev. Lett. 99,087402 (2007 ). [18] H. Suzuura and T. Ando, J. Phys. Soc. Jpn. 77,044703 (2008 ). [19] H. Suzuura and T. Ando, J. Phys. Conf. Ser. 150,022080 (2009 ). [20] M. Kühne, C. Faugeras, P. Kossacki, A. A. L. Nicolet, M. Orlita, Y . I. Latyshev, and M. Potemski, Phys. Rev. B 85,195406 (2012 ). [21] N. Bonini, M. Lazzeri, N. Marzari, and F. Mauri, Phys. Rev. Lett.99,176802 (2007 ). [22] F. Wendler, A. Knorr, and E. Malic, Nanophotonics 4,224 (2015 ). 075401-5
PhysRevB.73.172503.pdf	Chemical potential shift in lightly doped to optimally doped Ca 2−xNaxCuO 2Cl2 H. Yagi, T. Yoshida, and A. Fujimori Department of Physics and Department of Complexity Science and Engineering, University of Tokyo, Kashiwa, Chiba 277-8561, Japan Y . Kohsaka Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan and LASSP , Department of Physics, Cornell University, Ithaca, New York 14853, USA M. Misawa, T. Sasagawa, and H. Takagi Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan M. Azuma and M. Takano Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan /H20849Received 5 October 2005; revised manuscript received 22 March 2006; published 16 May 2006 /H20850 We have deduced the chemical potential shift /H9004/H9262in the high- Tcsuperconductor Ca 2−xNaxCuO 2Cl2 /H20849Na-CCOC /H20850using core-level x-ray photoemission spectroscopy. The derived /H9004/H9262is rigid-band-like, and almost linear in hole concentration x, quantitatively consistent with the shift estimated from a recent angle resolved photoemission spectroscopy study. Also, /H9004/H9262in Na-CCOC is much larger than that in La 2−xSrxCuO 4/H20849LSCO /H20850 and as large as that in Bi 2Sr2CaCu 2O8+y. Qualitatively different behavior of /H9004/H9262between Na-CCOC and LSCO is discussed in relation to the different behaviors of charge ordering. DOI: 10.1103/PhysRevB.73.172503 PACS number /H20849s/H20850: 74.25.Jb, 74.72.Jt, 79.60. /H11002i High- Tcsuperconductivity appears when the parent insu- lating cuprates are doped with holes or electrons. One of themost important but still controversial issues is how the elec-tronic structure evolves with carrier doping from the Mottinsulator to the superconductor. Two scenarios have beenproposed so far, that is, /H20849i/H20850doping creates new states within the charge-transfer gap of the parent insulator, the chemicalpotential /H9262is pinned in these new states and spectral weight is transferred from the upper and lower Hubbard bands tothese states, or /H20849ii/H20850upon carrier doping, /H9262jumps to the valence-band maximum or the conduction-band minimumand spectral weight transfer occurs. Recently, the shift of thechemical potential with carrier doping was measured forvarious kinds of high- T ccuprates using core-level x-ray pho- toemission spectroscopy /H20849XPS /H20850.1–3In La 2−xSrxCuO 4/H20849LSCO /H20850, strong suppression of the chemical potential shift /H9004/H9262was observed in the underdoped region, which indicates the pin-ning of /H9262and suggests the creation of new states inside the charge-transfer gap.1Angle-resolved photoemission spec- troscopy /H20849ARPES /H20850study of underdoped LSCO has also shown that the lower Hubbard band /H20849LHB /H20850does not shift upon hole doping, /H9262is pinned at /H110110.4 eV above the top of the LHB and spectral weight is transferred from the LHB tothe new states created around /H9262.4On the other hand, a ﬁnite shift of /H9262in the underdoped region was also observed in Bi2Sr2CaCu 2O8+y/H20849Bi2212 /H20850.2The electron-doped supercon- ductor Nd 2−xCexCuO 4showed a monotonous shift of /H9262from the underdoped to overdoped regions.3This monotonous shift of /H9262has been attributed to the rather robust long-range antiferromagnetic /H20849AF/H20850ordering and thus the stable AF band structure up to high carrier concentrations. The appearanceof an electron pocket at k/H11011/H20849 /H9266,0/H20850is consistent with the rigid-band-like picture of electron doping. Recently, Shen et al.reported that /H9004/H9262could be accurately estimated from ARPES spectra for Ca 2−xNaxCuO 2Cl2/H20849Na-CCOC /H20850using the nonbonding O 2 pstates as a reference and that /H9262showed alarge, monotonous shift with hole doping.5Remarkably, they showed that the LHB and the nonbonding O 2 pstates were shifted by the same amount. Since it was the ﬁrst reportwhich quantitatively estimated /H9004 /H9262by ARPES, comparing /H9004/H9262derived from core-level XPS with that derived from ARPES is important to check the consistency between ex-perimental methods and ﬁrmly establishing the method todetermine /H9004 /H9262. In this paper, we report on the study of the chemical po- tential shift /H9004/H9262in Na-CCOC with wider doping range by core-level XPS. The parent compound Ca 2CuO 2Cl2is an AF insulator. Hole doping is achieved by chemical substitutionof Na +for Ca2+under high pressures, leading to superconductivity.6The crystal structure of Na-CCOC is a simple K 2NiF 4-type, where single CuO 2plane is sandwiched by the so-called blocking layers of CaCl. Na-CCOC andLSCO are the only materials in which hole concentration canbe changed from zero to optimal doping. Therefore, it wouldprovide valuable information to study the doping dependenceof the electronic structure of the CuO 2plane in Na-CCOC and to compare the result with that of LSCO. Polycrystals of Na-CCOC /H20849x=0, 0.03, 0.06, 0.09, 0.12, 0.18 /H20850were synthesized with a cubic-anvil-type high-pressure apparatus. X-ray photoemission measurements were per-formed using a Mg K /H9251source /H20849h/H9263=1253.6 eV /H20850and a SCI- ENTA SES-100 analyzer. The total energy resolution was about /H110110.8 eV, which was largely due to the width of the photon source. Owing to the highly stabilized power supplyof the analyzer, however, it was possible to determine thebinding energy shifts with the accuracy of /H1101140 meV. Actu- ally, the energy position of the Au 4 fcore levels, which was used for energy calibration, stayed constant within ±10 meVeven for several days. During the measurements, the sampleswere cooled down to /H11011120 K and scraped every 30 min to obtain fresh surfaces. Some measurements were repeated at/H1101150 K, but because no clear temperature dependence wasPHYSICAL REVIEW B 73, 172503 /H208492006 /H20850 1098-0121/2006/73 /H2084917/H20850/172503 /H208494/H20850 ©2006 The American Physical Society 172503-1observed, we present results only for /H11011120 K. The base pressure in the analyzer chamber was 10−10Torr. Figure 1 shows the XPS spectra of the Ca 2 p,C l 2 p, O1s, and Cu 2 pcore levels. The line shapes of the Cu 2 p, Cl 2p, and O 1 score levels slightly changed with composi- tion due to some surface degradation and contamination. Asthe signals from the contamination appeared on the highbinding energy side of the main peak and had little effect onthe low binding energy side, we estimated the core-levelshifts using the low binding energy side of the peak, i.e., themidpoint of each peak. On the other hand, the line shape ofthe Cu 2 pcore level was different between different compo- sitions at both low and high binding energy sides of the peak.That is, a broadening of the Cu 2 pcore levels with carrier doping was observed. This is a common feature of high- T c cuprates and is attributed to the change in the Cu valence.1,2 Accordingly, it was difﬁcult to uniquely determine the shift of the Cu 2 pcore level, and we simply used the peak posi- tion. Figure 2 shows the binding energy shift /H9004E, the shift of the core-level energy measured relative to /H9262, of each core level thus estimated with the x/H110050.03 sample as the reference. /H20849The x/H110050 sample showed an obvious charging effect and could not be used in the analysis. /H20850One can see that the Ca 2 p,C l2 p, and the O 1 score levels show the same shifts and that the Cu 2 pcore level moves in the opposite direction to them. When the band ﬁlling is varied, /H9004Eis given by /H9004E=−/H9004/H9262+K/H9004Q+/H9004VM−/H9004ER.7Here, /H9004/H9262is the change in the chemical potential, /H9004Qis the change in the number of valence electrons on the considered atom and Kis a constant, /H9004VMis the change in the Madelung potential, and /H9004ERis the change in the extra-atomic relaxation energy. The oppositeshift of Cu 2 pis attributed to the change of the Cu valence, that is, to the K/H9004Qterm, which includes both the change inthe electrostatic potential and the change in the intra-atomic relaxation energy. The same shifts of Ca 2 p,C l2 p, and O 2 p suggest that /H9004V Mis screened and becomes negligible be- cause/H9004VMshould shift an anion core level and a cation core level in different ways. Finally, /H9004ERis due to the change in the screening of the core hole potential and, therefore, shouldbe larger for the atoms in the metallic CuO 2plane than for those out of the plane. The same shifts of O 1 s,C a2 p, and Cl 2pindicate that the /H9004ERterm is negligible. From the above considerations, we conclude that the same shifts of Ca 2 p,C l2 p, and O 1 sreﬂect the chemical poten- tial shift /H9004/H9262. We have, therefore, taken the average of these three core-level shifts as /H9004/H9262, and plotted it in Fig. 3. /H9004/H9262 shows a large, monotonous shift of /H20849/H11509/H9262//H11509x/H20850/H11011−2.0 eV/ hole. This is quantitatively consistent with the ARPES results of −1.8±0.5 eV/hole.5shown in the same ﬁgure, which use FIG. 1. Core-level XPS spectra of Ca 2−xNaxCuO 2Cl2./H20849a/H20850Ca 2 p, /H20849b/H20850Cl 2p,/H20849c/H20850O1s, and /H20849d/H20850Cu 2 p. Vertical bars mark the position of the midpoint of the slope on the low binding energy side /H20851/H20849a/H20850–/H20849c/H20850/H20852 and the peak position /H20851/H20849d/H20850/H20852. FIG. 2. /H20849Color online /H20850Energy shift of each core level relative to thex/H110050.03 sample as a function of hole concentration x. FIG. 3. /H20849Color online /H20850Chemical potential shift /H9004/H9262in Na-CCOC as a function of doped hole concentration xdeduced from the core levels compared with that derived from the valenceband, i.e., the nonbonding oxygen 2 p /H9266and 2 pzband /H20849Ref. 5 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 172503 /H208492006 /H20850 172503-2the shift of nonbonding oxygen 2 p/H9266and 2 pzbands to esti- mate/H9004/H9262./H20849The apparent discrepancy between the XPS and ARPES results is largely due to the uncertainty in the datafor the x/H110050 sample. /H20850Therefore, both the XPS study of core levels and the ARPES study of valence bands can be consis-tently used for determining /H9004 /H9262. In Fig. 4, the chemical potential shift in Na-CCOC is compared with those of other high- Tcsuperconductors. −/H11509/H9262//H11509p, where pis the hole concentration per Cu and p=x for Na-CCOC and LSCO, in the underdoped region inNa-CCOC is much larger than that in LSCO but as large asthat in Bi2212, or even larger than it. The large, monotonousshift in Na-CCOC cannot be attributed to a long range AForder unlike Nd 2−xCexCuO 4, because /H9262SR measurements have shown that AF order dissappeared already at x/H110110.02.8 According to the calculation using the t−t/H11032−t/H11033−Jmodel, the large shift of /H9262in Bi2212 compared to LSCO has been at- tributed to a large value of /H20841t/H11032/H20841.9Here, t,t/H11032, and t/H11033are the nearest-neighbor, second next, and third next-nearest neigh-bor hopping between Cu atoms, respectively. The value of/H20841t /H11032/H20841is largely determined by the energy of the Cu sorbital /H9255s.10As the distance between the Cu and apical oxygen in- creases, /H9255sis lowered and /H20841t/H11032/H20841becomes larger. In multilayer cuprates, too, /H9255sis lowered through the formation ofCu 4 s-Cu 4 sbonding states. The observation that /H9004/H9262in Na-CCOC is as large as that in Bi2212 suggests that the /H20841t/H11032/H20841 of Na-CCOC is as large as that of Bi2212. ARPES studieshave shown that the band dispersion width along the “under- lying Fermi surface” in the parent insulator is comparablebetween in Na-CCOC and in Bi2212, which also suggeststhat the magnitude of /H20841t /H11032/H20841is comparable between Na-CCOC and Bi2212 and is larger than LSCO.11 The dramatic difference in /H11509/H9262//H11509pin the underdoped re- gion between Na-CCOC and LSCO needs further remark.Spatial conductance modulations recently observed by scan-ning tunneling microscopy in Na-CCOC showed 4 a 0/H110034a0 checkerboard patterns independently of doping levels,12be- ing reminiscent of a stable charge order. Under such chargeordering which maintains a certain periodicity against carrierdoping, the change of the charge density in real space isexpected to be rather uniform and the chemical potential isexpected to show a rigid-band like shift. 13On the other hand, the stripe-like /H20849dynamical /H20850charge ordering as seen in LSCO changes its periodicity with doping. In such a case, the localcharge density is nearly stable because as holes are doped,the hole rich region expands and the hole poor regionshrinks. Such situation can be viewed as a microscopic“phase separation” and causes the apparent pinning of thechemical potential in analogy with the chemical potentialpinning in the case of a macroscopic phase separation. Thedifferent behaviors of the chemical potential shifts betweenNa-CCOC and LSCO may, therefore, be associated with thedifferent behaviors of charge ordering in addition to the dif-ferent /H20841t /H11032/H20841values. Whether the magnitude of /H20841t/H11032/H20841value di- rectly affects the charge ordering phenomena or not will be asubject of future theoretical studies. In summary, we have determined /H9004 /H9262in Na-CCOC by core-level XPS. The shift is large and monotonous and quan-titatively consistent with the shift deduced from the recentARPES study. The fact that − /H11509/H9262//H11509p is comparable between Na-CCOC and Bi2212 suggests that /H20841t/H11032/H20841is also comparable betweem them. The dramatic difference of /H9004/H9262in Na- CCOC from that in LSCO may also be related to the differ-ent types of charge ordering between these compounds. Useful discussion with K. M. Shen and Z.-X. Shen is gratefully acknowledged. This work was supported by aGrant-in-Aid for Scientiﬁc Research /H20849S17105002 /H20850from JSPS and by a Grant-in-Aid for Scientiﬁc Research in PriorityArea “Invention of Anomalous Quantum Materials”/H2084916076208 /H20850from MEXT, Japan. 1A. Ino, T. Mizokawa, A. Fujimori, K. Tamasaku, H. Eisaki, S. Uchida, T. Kimura, T. Sasagawa, and K. Kishio, Phys. Rev. Lett. 79, 2101 /H208491997 /H20850. 2N. Harima, A. Fujimori, T. Sugaya, and I. Terasaki, Phys. Rev. B 67, 172501 /H208492003 /H20850. 3N. Harima, J. Matsuno, A. Fujimori, Y . Onose, Y . Taguchi, and Y . Tokura, Phys. Rev. B 64, 220507 /H20849R/H20850/H20849 /H208492001 /H20850. 4T. Yoshida, X. J. Zhou, T. Sasagawa, W. L. Yang, P. V . Bogdanov, A. Lanzara, Z. Hussain, T. Mizokawa, A. Fujimori, H. Eisaki,Z.-X. Shen, T. Kakeshita, and S. Uchida, Phys. Rev. Lett. 91, 027001 /H208492002 /H20850.5K. M. Shen, F. Ronning, D. H. Lu, W. S. Lee, N. J. C. Ingle, W. Meevasana, F. Baumberger, A. Damascelli, N. P. Armitage, L. L.Miller, Y . Kohsaka, M. Azuma, M. Takano, H. Takagi, and Z.-X.Shen, Phys. Rev. Lett. 93, 267002 /H208492004 /H20850. 6Z. Hiroi, N. Kobayashi, and M. Takano, Nature /H20849London /H20850371, 139 /H208491994 /H20850. 7S. Hüfner, in Photoelectron Spectroscopy /H20849Springer-Verlag, Ber- lin, 1995 /H20850, Chap. 2, p. 35. 8K. Ohishi, I. Yamada, A. Koda, W. Higemoto, S. R. Saha, R. Kadono, K. M. Kojima, M. Azuma, and M. Takano, cond-mat/0412313 /H20849unpublished /H20850. FIG. 4. /H20849Color online /H20850Chemical potential shift /H9004/H9262in Na-CCOC compared with those in LSCO /H20849Ref. 1 /H20850and Bi2212 /H20849Ref. 2 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 172503 /H208492006 /H20850 172503-39T. Tohyama and S. Maekawa, Phys. Rev. B 67, 092509 /H208492003 /H20850. 10E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen, and O. K. Andersen, Phys. Rev. Lett. 87, 047003 /H208492001 /H20850. 11K. Tanaka, T. Yoshida, A. Fujimori, D. H. Lu, Z.-X. Shen, X.-J. Zhou, H. Eisaki, Z. Hussain, S. Uchida, Y . Aiura, K. Ono, T.Sugaya, T. Mizuno, and I. Terasaki, Phys. Rev. B 70, 092503 /H208492004 /H20850.12T. Hanaguri, C. Lupien, Y . Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi and J. C. Davis, Nature /H20849London /H20850430, 1001 /H208492004 /H20850. 13A. Fujimori, A. Ino, J. Matsuno, T. Yoshida, K. Tanaka, and T. Mizokawa, J. Electron Spectrosc. Relat. Phenom. 124, 127 /H208492002 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 172503 /H208492006 /H20850 172503-4
PhysRevB.79.214417.pdf	Extension of the spin-1 2frustrated square lattice model: The case of layered vanadium phosphates Alexander A. Tsirlin1,2,and Helge Rosner1,† 1Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany 2Department of Chemistry, Moscow State University, 119992 Moscow, Russia /H20849Received 28 January 2009; revised manuscript received 15 May 2009; published 11 June 2009 /H20850 We study the inﬂuence of the spin lattice distortion on the properties of frustrated magnetic systems and consider the applicability of the spin-1/2 frustrated square lattice model to materials lacking tetragonal sym-metry. We focus on the case of layered vanadium phosphates AA /H11032VO/H20849PO4/H208502/H20849AA /H11032=Pb 2, SrZn, BaZn, and BaCd /H20850. To provide a proper microscopic description of these compounds, we use extensive band structure calculations for real materials and model structures and supplement this analysis with simulations of thermo-dynamic properties, thus facilitating a direct comparison with the experimental data. Due to the reducedsymmetry, the realistic spin model of layered vanadium phosphates AA /H11032VO/H20849PO4/H208502includes four inequivalent exchange couplings: J1andJ1/H11032between nearest-neighbors and J2andJ2/H11032between next-nearest-neighbors. The estimates of individual exchange couplings suggest different regimes, from J1/H11032/J1and J2/H11032/J2close to 1 in BaCdVO /H20849PO4/H208502, a nearly regular frustrated square lattice, to J1/H11032/J1/H112290.7 and J2/H11032/J2/H112290.4 in SrZnVO /H20849PO4/H208502,a frustrated square lattice with sizable distortion. The underlying structural differences are analyzed, and the keyfactors causing the distortion of the spin lattice in layered vanadium compounds are discussed. We proposepossible routes for ﬁnding new frustrated square lattice materials among complex vanadium oxides. Fulldiagonalization simulations of thermodynamic properties indicate the similarity of the extended model to theregular one with averaged couplings. In case of moderate frustration and moderate distortion, valid for all theAA /H11032VO/H20849PO4/H208502compounds reported so far, the distorted spin lattice can be considered as a regular square lattice with the couplings /H20849J1+J1/H11032/H20850/2 between nearest-neighbors and /H20849J2+J2/H11032/H20850/2 between next-nearest-neighbors. DOI: 10.1103/PhysRevB.79.214417 PACS number /H20849s/H20850: 75.10.Jm, 75.30.Et, 75.50. /H11002y, 71.20.Ps I. INTRODUCTION Frustrated spin systems represent one of the actively de- veloping topics in solid state physics. The vast interest inmagnetic frustration originates from a number of unusualphenomena /H20849spin-liquid ground state, 1the formation of su- persolid phases in high magnetic ﬁelds,2etc./H20850suggested by theory. The theoretical predictions challenge the experimentthat, however, requires proper frustrated materials. Thesearch for the respective compounds has been a long story ininorganic chemistry, 3,4and the problem turned out to be quite complex. To meet the theoretical predictions, one has to ﬁnda material that reveals frustrated geometry of magnetic atomsand presents spin degrees of freedom only to avoid any for-eign effects /H20849e.g., orbital ordering /H20850tending to lift the frustra- tion. A number of frustrated spin models still lack the properrealizations, especially for the case of spin-1/2, where thestrongest quantum effects and the most interesting phenom-ena are expected. For other models, few appropriate materi-als are known and extensively studied. For example, the min-eral herbertsmithite ZnCu 3/H20849OH/H208506Cl2was recently proposed as a spin-1/2 kagomé material.5However, the actual physics of this compound is still debated due to the Cu/Zn antisitedisorder and the presence of non-magnetic sites within thekagomé layers. 6Other natural kagomé materials–minerals kapellasite and haydeeite–are proposed. Their structures donot suffer from the disorder effects, but the pure kagoméphysics is again modiﬁed due to the non-negligible interac-tions beyond nearest-neighbors. 7 Clearly, the task of ﬁnding an ideal frustrated material is hardly solvable at all. Therefore, it is instructive to examinewhich deviations from the ideal spin model may be allowed and, to a certain extent, do not qualitatively modify the prop-erties of this model. Considering structural distortions and the resulting spin lattice distortions is especially attractive,since lots of known materials have low symmetry in contrastto the high geometrical symmetry of the most theoreticallystudied frustrated spin models. The major part of the frustrated magnetic materials are the so-called geometrically frustrated magnets. In these systems,the frustration arises due to the competition of equivalentexchange couplings: in the most simple case, antiferromag-netic /H20849AFM /H20850couplings on a triangle. Then, any structural distortion should inevitably reduce this competition hencereducing the frustration. In the other group of the frustratedmaterials, the competing interactions are inequivalent buttheir topology and magnitudes can be tuned so that thestrong quantum ﬂuctuations destroy the long-range ordering,similar to the geometrically frustrated magnets. The lattergroup looks more favorable to tolerate the structural distor-tions, since the modiﬁcation of the spin lattice can probablybe balanced by the ratios of the competing interactions hencepreserving the strong frustration. To study this issue in moredetail, we focus on a speciﬁc model–spin-1/2 frustratedsquare lattice /H20849FSL /H20850–and consider a number of recently pro- posed FSL materials. The ideal /H20849regular /H20850FSL model assumes two competing interactions: the nearest-neighbor /H20849NN/H20850interaction J 1run- ning along the side of the square and the next-nearest-neighbor /H20849NNN /H20850interaction J 2running along the diagonal of the square /H20849see the inset of Fig. 1/H20850. The phase diagram /H20849Fig. 1/H20850reveals three ordered phases /H20849ferromagnet, Néel antiferro-PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 1098-0121/2009/79 /H2084921/H20850/214417 /H2084913/H20850 ©2009 The American Physical Society 214417-1magnet, and columnar antiferromagnet /H20850and two critical re- gions around J2/J1/H11229/H110060.5, where a spin-liquid ground state is expected.8,9Recent theoretical studies also considered the extended model with inequivalent NN couplings J1and J1/H11032 /H20849Refs. 10–14/H20850. This spatial anisotropy tends to narrow the critical region15and to destroy it completely at a certain value of J1/H11032/J1. Therefore, the distortion is not favorable for the frustration, but the speciﬁc geometry of the lattice en-ables to preserve the strong frustration and the resulting criti-cal region at moderate distortion. Below /H20849Sec. VI/H20850, we will show that the anisotropy of the NNN couplings on the squarelattice should have an even weaker /H20849and, likely, opposite /H20850 effect on the frustration. Experimental studies of the FSL systems have utilized a number of model compounds. First reports focused onLi 2VOXO 4/H20849X=Si,Ge /H20850materials that lay far away from the critical regions and did not show any speciﬁc propertiescaused by the frustration. 16–21Later on, the vanadyl molyb- date VOMoO 4was found to reveal unusual structural changes upon cooling, and this effect was tentatively as-cribed to the magnetic frustration despite the frustration ratioJ 2/J1was quite small /H20849likely, J2/J1/H110210.2/H20850.22,23The region of ferromagnetic /H20849FM/H20850J1-AFM J2was accessed by studying the layered vanadium phosphate Pb 2VO/H20849PO4/H208502/H20849Refs. 24–26/H20850 and the related AA /H11032VO/H20849PO4/H208502compounds with AA /H11032=SrZn and BaZn.25,27Quite recently, we proposed two more com- pounds, PbVO 3/H20849Refs. 28and29/H20850and BaCdVO /H20849PO4/H208502/H20849Ref. 30/H20850, that lay very close to the critical regions at J2/J1=0.5 and −0.5, respectively. Interestingly, the latter material lacksthe tetragonal symmetry and reveals a distorted FSL. Never-theless, we succeeded to observe two effects predicted forthe regular FSL: the suppression of the speciﬁc heatmaximum 9and the pronounced bending of the magnetization curve.31 In other systems, the problem of the spin lattice distortion may also be crucial. The layered copper oxychloride/H20849CuCl /H20850LaNb 2O7was recently proposed as a promising FSL material, lacking long-range magnetic order.32However, careful studies indicated the structural distortion33that com-pletely changed the underlying spin model and precluded from any interpretations within the FSL framework.34Thus, it is important to understand whether the FSL model can beapplied to BaCdVO /H20849PO 4/H208502and, more generally, to the low- symmetry materials. Below, we study this issue in detail,discuss the whole family of the AA /H11032VO/H20849PO4/H208502compounds /H20849hereinafter, we imply that AA /H11032=Pb 2, SrZn, BaZn, and BaCd /H20850, and provide quantitative estimates for the spin lattice distortion. Our approach combines several computationalmethods /H20849band structure calculations, subsequent analysis of the exchange couplings, and model simulations /H20850in order to analyze magnetic interactions in these systems, derive theproper spin model, and facilitate the comparison with theexperimental data. We show that the structural distortion inthe AA /H11032VO/H20849PO4/H208502compounds is a minor effect as compared to the frustration, and the FSL description holds. The outline of the paper is as follows. We start with an analysis of the crystal structures in Sec. IIand review the computational methods employed in our work /H20849Sec. III/H20850. Then, we address several problems: /H20849i/H20850the realistic spin model and the magnitude of the distortion /H20849Sec. IV/H20850;/H20849ii/H20850 structural factors that inﬂuence the distortion of the spin lat-tice /H20849Sec. V/H20850; and /H20849iii/H20850thermodynamic properties of the ex- tended model /H20849Sec. VI/H20850. In Sec. VII, we discuss the results of our study that suggests an accurate /H20849and, within the Heisen- berg model, exact /H20850way to treat the FSL-like spin systems of the AA /H11032VO/H20849PO4/H208502phosphates. We also present a more gen- eral recipe for ﬁnding strongly frustrated square lattices inthe compounds with similar topology of the magnetic layer,being generic for most of the FSL materials. Finally, weshow how the distortion of the square lattice affects thermo-dynamic properties of the model and the magnitude of thefrustration. II. CRYSTAL STRUCTURES AND EXPERIMENTAL RESULTS Most of the vanadium-based FSL compounds reported so far reveal similar /H20851VOXO 4/H20852magnetic layers. These layers are built of VO 5pyramids and XO 4tetrahedra /H20849see left panel of Fig. 2/H20850with X being a non-magnetic cation /H20849P, Si, Ge, or Mo+6/H20850. The connections via the tetrahedra provide superex- change pathways for both NN and NNN couplings. Vana-dium atoms have the steady oxidation state of +4 /H20849implying the electronic conﬁguration 3 d 1and spin-1/2 /H20850, while the va- lence of the X cation can be changed and controls the ﬁllingof the interlayer space. Thus, the interlayer space is empty inVOMoO 4/H20849Ref. 35/H20850, ﬁlled by lithium atoms in Li 2VOXO 4 /H20849X=Si,Ge /H2085036and ﬁlled by complex interlayer /H20851AA /H11032PO4/H20852 blocks in the AA /H11032VO/H20849PO4/H208502phosphates.37–39The magnetic layer is compatible with the tetragonal symmetry, hence theFSL model can be realized. Yet the overall symmetry of thestructure is sometimes reduced due to the complex conﬁgu-ration of the interlayer block, and this is the case for theAA /H11032VO/H20849PO4/H208502compounds. The crystal structure of the AA /H11032VO/H20849PO4/H208502phosphates /H20849Fig. 2/H20850is fairly ﬂexible with respect to the accommodation of different divalent metal cations A and A /H11032in the interlayer block. Most of the compounds combine a larger /H20849Ba or Sr /H20850J1J2Li VOSiO24 Li VOGeO24 PbVO3 VOMoO4~ 0.4 ~ 0.4/CID2~ 0.7/CID2 JJ21/=/CID2 8~ 0.7Pb VO(PO )24 2BaZnVO(PO )42 SrZnVO(PO )42 BaCdVO(PO )42 NAFCAF FM J1 J2 FIG. 1. /H20849Color online /H20850Phase diagram of the FSL model9and the respective model compounds /H20849see text for references /H20850. The inset shows the regular FSL with the NN /H20849J1/H20850and NNN /H20849J2/H20850couplings denoted by solid and dashed lines, respectively.ALEXANDER A. TSIRLIN AND HELGE ROSNER PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-2and a smaller /H20849Cd or Zn /H20850cation, yielding the orthorhombic symmetry of the structure /H20849space group Pbca /H20850.37,38Y e ti ti s also possible to use the same cation /H20849Pb/H20850for A and A /H11032, and the resulting structure has monoclinic symmetry /H20849space group P21/c/H20850.39Different symmetries produce slightly differ- ent geometries of the magnetic /H20851VOPO 4/H20852layers, hence dif- ferent types of the spin lattice distortion should be expected.In the following, we will focus on the magnetic layer typicalfor the orthorhombic AA /H11032VO/H20849PO4/H208502compounds with differ- ent A and A /H11032/H20849Fig. 2/H20850. To ease the comparison to monoclinic Pb2VO/H20849PO4/H208502, one should convert the crystal structure39to the non-standard setting P21/bwith abeing the monoclinic axis /H20849in this setting, the layer corresponds to one shown in Fig. 2/H20850. Basically, the whole discussion of the spin lattice distortion is applicable to Pb 2VO/H20849PO4/H208502, but due to the lower lattice symmetry this compound has one additional feature:two inequivalent NN couplings along the baxis. However, our estimates show that these couplings nearly match. There-fore, the spin system of Pb 2VO/H20849PO4/H208502can be adequately de- scribed with four parameters similar to the otherAA /H11032VO/H20849PO4/H208502compounds. The lack of the tetragonal symmetry in the AA /H11032VO/H20849PO4/H208502 phosphates gives rise to four different interactions in the magnetic /H20851VOPO 4/H20852layers. According to Fig. 2, we label the NN interactions as J1,J1/H11032and the NNN interactions as J2,J2/H11032. Previous experimental works implicitly assumed the ideal FSL model with J1=J1/H11032andJ2=J2/H11032as a natural, albeit therein unjustiﬁed, approximation.24,25,30In the previous studies, ex- perimental data on the magnetic susceptibility and the spe-ciﬁc heat were ﬁtted with high-temperature series expansions/H20849HTSE /H20850for the regular FSL 19to yield the effective couplings J1expandJ2exp/H20849Table I/H20850. The FM J1-AFM J2regime of the FSL was further supported by analyzing ﬁeld dependence of themagnetization of BaCdVO /H20849PO 4/H208502/H20849Ref. 30/H20850and the ground states of Pb 2VO/H20849PO4/H208502and SrZnVO /H20849PO4/H208502/H20849Refs. 26and 27/H20850. Within the phenomenological approach, the consistent in- terpretation of the experimental results justiﬁes a posteriori the application of the ideal FSL model to the AA /H11032VO/H20849PO4/H208502 compounds. Of course, a microscopic approach requires a more careful consideration of all the four inequivalent ex-change couplings in the magnetic /H20851VOPO 4/H20852layers. However, it is quite difﬁcult /H20849at least, experimentally /H20850to go beyond the regular FSL description due to the lack of theoretical results for the extended J1−J1/H11032−J2−J2/H11032model. In the following, we address the problem using computational methods. Thesemethods are known to provide a reliable microscopic de-scription of complex spin systems, including the FSL com-pounds Li 2VOXO 4/H20849Refs. 18and19/H20850. Yet we also refer to the phenomenological results and show that at sufﬁciently hightemperatures the distorted FSL can be considered as a regularFSL with effective, averaged NN and NNN couplings. III. METHODS AND MODELING Scalar-relativistic band structure calculations were per- formed within the full-potential local-orbital scheme/H20849FPLO7.00–27 /H20850, 40and the exchange-correlation potential by Perdew and Wang41was applied. Our calculations employed experimental crystal structures and a number of modiﬁed andmodel structures, as described below. Different k-meshes were used depending on the size and the geometry the unitcell. In all the calculations, the convergence with respect tothek-mesh was carefully checked. To evaluate the exchange couplings in the AA /H11032VO/H20849PO4/H208502 compounds, we use two different approaches. First, local density approximation /H20849LDA /H20850calculations are performed. These calculations enable to select relevant states and to es-timate hopping parameters /H20849t/H20850for the respective bands by ﬁtting these bands with a tight-binding /H20849TB/H20850model. The TABLE I. Experimental exchange couplings /H20849J1exp,J2exp/H20850evalu- ated within the regular FSL model and the resulting frustration ra-tios in the AA /H11032VO/H20849PO4/H208502compounds AA /H11032 J1exp/H20849K/H20850 J2exp/H20849K/H20850 J2exp/J1expRef. BaCd −3.6 3.2 −0.9 30 Pb2 −5.1 9.4 −1.8 24and25 BaZn −5.0 9.3 −1.9 25 SrZn −8.3 8.9 −1.1 25/CID1 bb J1’ J2’ J1 J1’ J2’J1 J2J2 caa a VOPO4AA PO’4 FIG. 2. /H20849Color online /H20850Crystal structure of the AA /H11032VO/H20849PO4/H208502compounds and the underlying spin model. The left panel shows a single /H20851VOPO 4/H20852layer. The middle panel presents the stacking of the /H20851VOPO 4/H20852layers and the /H20851AA /H11032PO4/H20852blocks as well as the angle /H9272measuring the layer buckling: larger and smaller spheres denote the A and A /H11032cations, respectively. The right panel shows the magnetic interactions in the/H20851VOPO 4/H20852layers /H20849compare with the regular model in the inset of Fig. 1/H20850: solid, dash-dotted, dashed, and dotted lines indicate J1,J1/H11032,J2, andJ2/H11032, respectively.EXTENSION OF THE SPIN-1 2FRUSTRATED … PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-3LDA calculations fail to reproduce the strong correlation ef- fects in the vanadium 3 dshell; therefore, the correlations are included on a model level. The hoppings are introduced to anextended Hubbard model with the effective on-site Coulombrepulsion U eff=4.5 eV /H20849this value is representative for vana- dium oxides, see Refs. 18,19,42, and 43/H20850. The strongly correlated limit t/H11270Ueffand the half-ﬁlling regime justify the reduction to the Heisenberg model for the low-lying excita-tions. Then, AFM contributions to the exchange couplings are estimated as J iAFM=4ti2/Ueff. Within this approach, all the possible AFM couplings are evaluated. Second, we consider the correlation effects within the self-consistent calculations and employ the local spin densityapproximation /H20849LSDA /H20850+Umethod. Total energies for a number of ordered spin conﬁgurations are mapped onto theclassical Heisenberg model to yield the estimates for bothFM and AFM couplings, hence supplementing the TB analy-sis. LSDA+ Utreats correlation effects in a mean-ﬁeld ap- proximation and uses two input parameters, U dand Jd,t o describe the on-site Coulomb repulsion and the intraatomicexchange, respectively. Since the exchange parameter hasminor inﬂuence on the results, we ﬁx J d=1 eV as a repre- sentative value. Yet the choice of the repulsion parameter Ud may be crucial, especially in case the exchange couplings are weak /H20849see Ref. 44for an instructive example /H20850. To reduce the ambiguity related to the choice of Ud, we thoroughly com- pare the computational results with the experimental data, asfurther discussed in Sec. IV B . The LDA calculations employed the full symmetry of the crystal structures, 112-atom orthorhombic unit cells, and thek-mesh of 256 k-points with 75 points in the irreducible part of the ﬁrst Brillouine zone /H20849IBZ/H20850. In case of Pb 2VO/H20849PO4/H208502, the 56-atom monoclinic unit cell and a mesh of 512 k-points /H20849170 in IBZ /H20850were used. To realize different spin orderings within the LSDA+ U calculations, one has to reduce the crystal symmetry and, insome cases, to extend the unit cell. The evaluation of the fourexchange couplings in the AA /H11032VO/H20849PO4/H208502compounds re- quires the doubling of the unit cell in the adirection, hence 224-atom unit cells should be constructed. For such unitcells, full-potential calculations are extremely time-consuming and, likely, not accurate enough for a reliableevaluation of the rather small exchange constants. 45There- fore, in our LSDA+ Ucalculations we simplify the crystal structures of the AA /H11032VO/H20849PO4/H208502compounds and construct a number of modiﬁed structures. The idea resembles our studyof Ag 2VOP 2O7/H20849Ref. 42/H20850: the leading magnetic interactions take place in the V–P–O layers, hence it is essential to usethe correct geometry of the layer, while the stacking of thelayers and the ﬁlling of the interlayer space have minor effecton the leading exchange couplings. To build the simpliﬁed structures of the AA /H11032VO/H20849PO4/H208502 compounds, we keep the exact geometry of the /H20851VOPO 4/H20852 layers, stack these layers one onto another, and ﬁll the inter-layer space with lithium atoms, providing the proper chargebalance. The resulting composition is LiVOPO 4. The inter- layer separation is ﬁxed at 6.5 Å to achieve realistic, sufﬁ-ciently weak interlayer hoppings /H20849below 2 meV /H20850. To justify the structure simpliﬁcation, we perform the TB analysis. Thedifference between the respective hoppings in the experi-mental and modiﬁed structures does not exceed 5%, imply- ing an error below 10% for the exchange couplings. Such anerror is deﬁnitely acceptable for the further LSDA+ Ucalcu- lations. Basically, the structure simpliﬁcation provides apromising computational approach to the magnetic proper-ties of low-dimensional spin systems with complex crystalstructures. In our LSDA+ Ucalculations, we use 64-atom supercells /H208512a/H11003b/H11003c/H20849=6.5 Å /H20850/H20852with triclinic symmetry /H20849space group P1/H20850and a mesh of 108 k-points. Finally, we also performed a number of LDA calculations for model structures in order to study the inﬂuence of indi-vidual structural changes on the spin lattice distortion. Themodel structures were built similar to the simpliﬁed struc-tures described above. The initial geometry was taken fromthe structure of /H9251I-LiVOPO 4that includes regular /H20851VOPO 4/H20852 layers separated by Li cations.46Then, we introduced a num- ber of structural distortions and checked the changes of theexchange couplings /H20849see Sec. Vfor details /H20850. The initial struc- ture reveals a rather low interlayer separation of 4.45 Å andyields sizable interlayer interactions. To reduce the interlayerinteractions and to properly emulate the two-dimensional/H208492D/H20850character of the FSL compounds, we increased the in- terlayer spacing up to 6.5 Å. We also reduced the symmetrydown to the orthorhombic space group Pbma /H20849Pbcm in the standard setting /H20850that allowed the distortions of the /H20851VOPO 4/H20852 layer. A k-mesh of 4096 points /H20849729 in the IBZ /H20850was used. Full diagonalization /H20849FD/H20850simulations were performed for the N=16 /H208494/H110034/H20850cluster using the ALPS simulation package.47Basic thermodynamic quantities /H20849magnetic sus- ceptibility and speciﬁc heat /H20850were evaluated by an internal procedure of the program. In general, the FD simulationssuffer from ﬁnite-size effects, because current computationalfacilities do not allow to perform the calculations for largeclusters. The presently available cluster size /H20849normally, 16 or 20 sites /H20850is sufﬁcient to obtain the quantitatively correct in- formation on thermodynamic properties of one-dimensionalspin systems only. The accuracy of the FD simulations fortwo-dimensional systems is challenged by the experimentaldata and the results of other simulation techniques. 25,30,48Yet, the FD simulations are able to provide qualitatively correcttrends upon the change of the model parameters /H20849e.g., the change of the frustration ratio in the FSL, see Ref. 9/H20850. Keep- ing in mind these considerations, we restrict ourselves to theanalysis of the relative changes in the thermodynamic prop-erties, as the spin lattice is distorted. Investigation of theground state and quantitative simulation of thermodynamicproperties for the extended FSL model are clearly beyond thescope of the present paper that intends to stimulate furtherstudies of the problem. IV. EXCHANGE COUPLINGS IN AA /H11032VO(PO 4)2 A. LDA and tight-binding analysis In this section, we consider the band structures of the AA /H11032VO/H20849PO4/H208502compounds and evaluate individual exchange couplings. The LDA density of states /H20849DOS /H20850for BaCdVO /H20849PO4/H208502is shown in Fig. 3. This plot is representa- tive for the whole family of the AA /H11032VO/H20849PO4/H208502materials. The electronic structure resembles that of other vanadiumALEXANDER A. TSIRLIN AND HELGE ROSNER PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-4phosphates.44The states below −2.5 eV are mainly formed by oxygen orbitals, while the states close to the Fermi levelreveal predominant vanadium contribution. The vanadiumbands are rather narrow and show 3 d-related crystal ﬁeld levels /H20849see the inset of Fig. 3/H20850as expected for the square- pyramidal or distorted octahedral coordination of V +4/H20849Ref. 49/H20850. The lowest-lying vanadium states formed by dxyorbitals lie at the Fermi level. The respective orbitals are located inthe basal planes of the square pyramids hence overlappingwith porbitals of the basal oxygen atoms and facilitating exchange couplings in the abplane. The LDA energy spec- trum is metallic in contradiction to the experimental green oryellow-green color that indicates insulating behavior. Thelack of the energy gap is a typical failure of LDA due to theunderestimate of strong electron correlations in the 3 dshell. Applying LSDA+ U, we ﬁnd insulating spectra with an en- ergy gap of 1.5–2.8 eV depending on the U dvalue. The upper estimate is in reasonable agreement with the experi-mental sample colors.For the TB analysis, we select the eight half-ﬁlled d xy bands lying at the Fermi level /H20849Fig.4/H20850. These bands originate from eight vanadium atoms in the unit cell of AA /H11032VO/H20849PO4/H208502. The unit cell includes two /H20851VOPO 4/H20852layers /H20849see middle panel of Fig. 2/H20850and four vanadium atoms from each of the two layers. The layers are well separated by the /H20851AA /H11032PO4/H20852block; therefore, interlayer interactions are very weak, and thebands are close to double degeneracy in the whole Brillouinzone. The leading interlayer hopping is about 1 meV , imply-ing the AFM interactions of about 0.01 K, well below thein-layer interactions /H20849see Tables IandII/H20850. In Fig. 4, we show the band structures of BaCdVO /H20849PO 4/H208502and SrZnVO /H20849PO4/H208502at the Fermi level. The two plots are rather similar in the over-all behavior and in the energy scale, although notable differ-ences are found near the X,Y, and Tpoints. The TB analysis suggests that the differences are mainly related to the NNN couplings t 2andt2/H11032. In Table II, we list the leading hopping parameters for all the AA /H11032VO/H20849PO4/H208502compounds along with the resulting JAFM values for the two representative and extreme cases, SrZnVO /H20849PO4/H208502and BaCdVO /H20849PO4/H208502. All the hoppings be- yond t2andt2/H11032are negligible /H20849below 2 meV /H20850. We ﬁnd a no- table distortion of the square lattice in all the materials understudy, yet the magnitude of the distortion is rather different.The NNN couplings are AFM /H20849see Table I/H20850and correspond to long V–V separations, hence the FM contributions are ex- pected to be small, and one could consider the J 2/H11032AFM/J2AFM ratio as a good estimate for the distortion ratio, J2/H11032/J2, with respect to the ideal value J2/H11032/J2=1 for the regular FSL. We ﬁnd the least pronounced distortion /H20849J2/H11032AFM/J2AFM=0.83 /H20850in BaCdVO /H20849PO4/H208502and the strongest distortion /H20849J2/H11032AFM/J2AFM =0.37 /H20850in SrZnVO /H20849PO4/H208502. The NN couplings are ferromagnetic /H20849see Table I/H20850, hence the TB model does not yield the direct estimate of J1/H11032/J1. However, one can assume similar FM contributions to J1and J1/H11032due to similar V–V separations. Then, the difference be- tween the NN couplings should originate from the difference between J1AFMandJ1/H11032AFM, and the value of J1/H11032AFM−J1AFMis the most convenient characteristic of the distortion. Both J1AFM and J1/H11032AFMare well belo w1Ki n BaCdVO /H20849PO4/H208502.I n SrZnVO /H20849PO4/H208502,J1/H11032AFMis about 3 K, while J1AFMis still below 1 K. Thus, one can expect the sizable difference between J1 dxy d+ d322 22zr xy-- d+ dxz yz 00050100 2 E(eV)050100 DOS (eV )-1150Total Cd V P O200 3 -3 -6 FIG. 3. /H20849Color online /H20850LDA density of states for BaCdVO /H20849PO4/H208502 /H20849the contribution of barium is not shown, because it is negligible in the whole energy range /H20850. The Fermi level is at zero energy. The inset shows the orbital resolved DOS for vanadium. 0.0 0.0 E(eV) X XBaCdVO(PO )42 SrZnVO(PO )42 M M Y Y Z Z T T /CID3 /CID3 /CID3 /CID30.1 0.1 /CID20.1 /CID20.1 /CID20.2 /CID20.20.2 0.2 FIG. 4. /H20849Color online /H20850LDA band structure /H20849thin light lines /H20850and the ﬁt of the tight-binding model /H20849thick green lines /H20850for BaCdVO /H20849PO4/H208502 /H20849left panel /H20850and SrZnVO /H20849PO4/H208502/H20849right panel /H20850. The Fermi level is at zero energy. The notation of the kpoints is as follows: /H9003/H208490,0,0 /H20850, X/H208490.5,0,0 /H20850,M/H208490.5,0.5,0 /H20850,Y/H208490,0.5,0 /H20850,Z/H208490,0,0.5 /H20850, and T/H208490.5,0,0.5 /H20850/H20849the coordinates are given along kx,ky, and kzin units of the respective reciprocal lattice parameters /H20850.EXTENSION OF THE SPIN-1 2FRUSTRATED … PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-5andJ1/H11032in SrZnVO /H20849PO4/H208502. In summary, we ﬁnd weak distor- tion of the square lattice in BaCdVO /H20849PO4/H208502and a more pro- nounced distortion for AA /H11032=BaZn and Pb 2. The spin lattice of SrZnVO /H20849PO4/H208502is distorted with respect to both the NN and NNN couplings. Below, we will conﬁrm this conclusionwith the LSDA+ Ucalculations /H20849see Table III/H20850. The TB estimates are in reasonable agreement with the experimental data. We ﬁnd the small NN hoppings consistent with the FM nature of J 1exp. On the other hand, the sizable hoppings t2andt2/H11032imply AFM NNN couplings in agreement with the AFM coupling J2exp. We further conﬁrm the FM J1,J1/H11032-AFM J2,J2/H11032scenario with the LSDA+ Ucalculations /H20849see Table III/H20850that perfectly match the results of the TB analysis. The remarkable agreement of the TB and LSDA+Uresults ensures sufﬁcient accuracy of our approach. However, the calculated values of J 2andJ2/H11032are likely over- estimated as compared to J2exp, the averaged NNN coupling /H20849Table I/H20850. We can speculate about several reasons for this discrepancy. First, one should not expect very precise resultsof band structure calculations while analyzing weak ex-change couplings in AA /H11032VO/H20849PO4/H208502. The calculated Jvalues likely include a systematic error that, however, does not in-validate any of the qualitative conclusions: note the studiesof the Li 2VOXO 4/H20849X=Si, Ge /H20850compounds18,19as an instruc- tive example of the correct microscopic scenario, emergingfrom LDA calculations. Second, exchange couplings arehighly sensitive to the geometry of superexchange pathways /H20849see, e.g., Ref. 42/H20850, hence it is essential to use the accurate structural information in order to obtain precise computa-tional estimates. In case of AA /H11032VO/H20849PO4/H208502, the crystal struc- tures are solved from single-crystal diffraction data, and thestructural information should be precise. Yet, there are someunresolved issues /H20849e.g., poorly reproducible superstructure reﬂections for Pb 2VO/H20849PO4/H208502, see Ref. 39/H20850that can be crucial for the accuracy of the computational estimates. Along thispaper, we mainly focus on the qualitative differences be-tween the AA /H11032VO/H20849PO4/H208502compounds, while the quantitative analysis should likely reference to the experimental data /H20849see Sec. VIIfor further discussion /H20850and may require additional structural studies. B. LSDA+ Uresults To get a direct estimate of FM NN couplings in AA /H11032VO/H20849PO4/H208502, we turn to the LSDA+ Uapproach. In this approach, one has to select an appropriate value of Ud, the on-site Coulomb repulsion parameter /H20849see Sec. III/H20850.51A num- ber of previous works have established Ud=3.5–4 eV for several V+4-containing compounds,52–54although the higher value of Ud=6 eV55as well as a lower value of Ud =2.3 eV56were proposed for speciﬁc materials. The reason for the higher Udvalue in Ref. 55was tentatively ascribed toTABLE II. The hopping parameters ti/H20849in meV /H20850and the resulting magnitude of the distortion /H20849J2/H11032AFM/J2AFM/H20850for all the AA /H11032VO/H20849PO4/H208502compounds along with the antiferromagnetic contributions to the exchange integrals JiAFM/H20849in K /H20850for AA /H11032=BaCd and SrZn AA /H11032 t1 t1/H11032 t2 t2/H11032 J2/H11032AFM/J2AFM BaCd 2 −5 45 41 0.83 Pb2 4a15 47 38 0.67 BaZn −11 12 46 36 0.61SrZn 4 17 43 26 0.37 J 1AFMJ1/H11032AFMJ2AFM J2/H11032AFMJ1/H11032AFM−J1AFM BaCd 0.05 0.3 21.0 17.4 0.25 SrZn 0.2 3.0 19.0 7.0 2.8 aThe averaged NN coupling along the baxis, see Sec. IIand Ref. 50 TABLE III. LSDA+ Uestimates of the exchange couplings in BaCdVO /H20849PO4/H208502and SrZnVO /H20849PO4/H208502.Udis the Coulomb repulsion parameter of LSDA+ U. The columns /H20849J1/H11032−J1/H20850andJ2/H11032/J2should be directly compared to the last column of Table II. AA /H11032 Ud/H20849eV/H20850 J1/H20849K/H20850 J1/H11032/H20849K/H20850 J1/H11032−J1/H20849K/H20850 J2/H20849K/H20850 J2/H11032/H20849K/H20850 J2/H11032/J2 BaCd 2.0 −8.6 −6.3 2.3 27 21.6 0.802.5 −2.2 −1.3 0.9 21.9 17.8 0.813.0 1.6 1.6 0.0 18.1 14.6 0.81 SrZn 2.0 −10.0 −5.4 4.6 24.8 11.3 0.462.5 −2.4 1.6 4.0 20.8 8.7 0.423.0 2.7 6.3 3.6 17.5 7 0.40ALEXANDER A. TSIRLIN AND HELGE ROSNER PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-6the reduced p-dhybridization. Still, there is one more uncer- tainty in the selection of this number. The constrained LDAprocedure and the further LSDA+ Ucalculations are usually performed within the LMTO-ASA /H20849linearized mufﬁn-tin or- bitals, atomic spheres approximation /H20850approach that employs mufﬁn-tin orbitals. In our work, we use a different basis set/H20849atomic-like local orbitals of FPLO /H20850, hence the U dvalue may also be different, because the Udpotential is applied to the orbitals from the basis set and depends on the particularchoice of the dfunctions. Indeed, our previous FPLO studies showed that the U dof 5 or even 6 eV was required to repro- duce the magnetic interactions in a number ofV +4-containing phosphates.42,44Keeping in mind the ambi- guity of the Udchoice due to the differences in the compu- tational method /H20849basis set /H20850and in the structural features /H20849dif- ferent p-dhybridization /H20850, we do not use any of the previously established Udvalues. Rather, we perform calcu- lations for a broad range of Udvalues and use well-studied and structurally similar vanadium compounds as a reference. According to Sec. II, the structure of the magnetic layer in the AA /H11032VO/H20849PO4/H208502phosphates is very similar to that of Li2VOXO 4compounds /H20849X=Si,Ge /H20850. Exchange couplings in the latter materials are ﬁrmly established by ﬁtting magneticsusceptibility and speciﬁc heat data with the HTSE. 18–20,25 We calculate the exchange couplings in Li 2VOXO 4for a wide range of Udvalues and compare the results with the experiment /H20849Fig. 5/H20850. In the left and middle panels of Fig. 5, one can see that any reasonable value of Udyields the correct energy scale, but there is no unique Udvalue yielding accu- rate estimates of both J1andJ2. At low Ud,J2is overesti- mated, while high Udvalues tend to overestimate J1. As long as we are interested in the frustration, the essential quantityis the frustration ratio, J 2/J1. This quantity is found with a sizable error bar due to the uncertainty for the low J1values. Computational estimates of J2/J1are also quite uncertain and highly sensitive to the Udvalue /H20849see right panel of Fig. 5/H20850.A tUd/H110223 eV, we ﬁnd very low frustration ratios, contra- dicting the J2/H11271J1regime established experimentally. The narrow range of Ud=2–3 eV is able to reproduce the rea- sonable frustration ratios, while the Udvalues below 2 eV lead to FM J1. Thus, we argue that the Udvalue of 2–3 eV should be optimal for reproducing exchange couplings in thelayered vanadium FSL compounds. Yet one can construct thecorrect picture solely based on the LSDA+ Uresults only in a narrow range of the U dvalues. In the LSDA+ Ustudy, we restrict ourselves to BaCdVO /H20849PO4/H208502and SrZnVO /H20849PO4/H208502as the “edge” membersof the AA /H11032VO/H20849PO4/H208502series: these compounds show the least and the most pronounced distortion, respectively /H20849see Table II/H20850. The LSDA+ Uresults are summarized in Table III. The exchange couplings show a sizable dependence on Ud, even in the narrow range of Ud=2–3 eV. Nevertheless, we ﬁnd the experimentally observed FM NN–AFM NNN regime forU d=2.0 and 2.5 eV . Additionally, the numbers are in excel- lent agreement with the TB results, the difference between J2 andJ2/H11032/H20849quantiﬁed by the J2/H11032/J2ratio /H20850as well as the different AFM contributions to J1andJ1/H11032/H20849quantiﬁed by the difference J1/H11032−J1, similar to Table II/H20850are remarkably reproduced. The FSL-like spin system of the AA /H11032VO/H20849PO4/H208502com- pounds includes four inequivalent exchange couplings. In case of BaCdVO /H20849PO4/H208502, the respective NN /H20849J1,J1/H11032/H20850and NNN /H20849J2,J2/H11032/H20850couplings nearly match and give rise to the almost regular FSL. In BaZnVO /H20849PO4/H208502and Pb 2VO/H20849PO4/H208502, the dis- tortion is more pronounced. In case of SrZnVO /H20849PO4/H208502, the square lattice is strongly distorted: the two NNN couplingsdiffer by a factor of three, and the two NN couplings do not match as well. Using J 1exp/H11229−8.3 K as the averaged value /H20849Table I/H20850and assuming J1/H11032−J1/H112293K /H20849Table II/H20850, we estimate J1/H11032/J1/H112290.69. In the next section, we discuss the structural origin of the spin lattice distortion. V. STRUCTURAL ORIGIN OF THE DISTORTION To study the inﬂuence of individual structural factors on the spin lattice distortion in the AA /H11032VO/H20849PO4/H208502compounds, we construct a number of model structures. The initial struc-ture resembles that of /H9251I-LiVOPO 4/H20849see Sec. III/H20850and in- cludes regular vanadium and phosphorous polyhedra in the/H20851VOPO 4/H20852layers /H20849the left panel of Fig. 2/H20850. The cation–oxygen separations are 1.950 Å for the basal oxygen atoms and1.582 Å for the axial oxygen atom in the VO 5square pyra- mids and 1.543 Å for the PO 4tetrahedra. Then, we intro- duce certain distortions and analyze their inﬂuence on themagnetic interactions. We use the TB approach; therefore,we focus on the AFM NNN couplings and trace the change of the J 2/H11032/J2ratio. In the end of the section, we brieﬂy com- ment on the NN couplings and the J1/H11032vs.J1distortion. The difference between the AA /H11032VO/H20849PO4/H208502compounds originates from different metal cations located between the/H20851VOPO 4/H20852layers. The cation size is reduced along the series from AA /H11032=BaCd to AA /H11032=SrZn, and this trend correlates to the reduction of J2/H11032/J2from /H112290.8 in BaCdVO /H20849PO4/H208502to/H112290.4 in SrZnVO /H20849PO4/H208502. The coincidence of the two trends gives a2 2 2 3 3 3Li VOSiO24 Li VOGeO24 X=G eX=S iLi VOXO24 4 4 4 5 5 5 Ud(eV) Ud(eV) Ud(eV)J1 J1J1J2J2GeSiJ1 J2 J2 6 6 6J(K) J(K)J21/J 00 04 388 6916 12 12 FIG. 5. /H20849Color online /H20850Exchange couplings in Li 2VOSiO 4/H20849left panel /H20850and Li 2VOGeO 4/H20849middle panel /H20850calculated for different values of Coulomb repulsion parameter Udand the resulting frustration ratios J2/J1for both the compounds /H20849right panel /H20850. Shaded stripes show experimental estimates from Refs. 18–20and25.EXTENSION OF THE SPIN-1 2FRUSTRATED … PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-7hint that the cation size should be the origin of the distortion. Still, the inﬂuence of the cation size on the structure of the/H20851VOPO 4/H20852layers is quite complex. Smaller metal cations tend to have shorter metal–oxygen distances and lower coordination numbers. Thus, barium issurrounded by nine oxygen atoms in BaZnVO /H20849PO 4/H208502and BaCdVO /H20849PO4/H208502, while strontium has only eight neighboring oxygens in SrZnVO /H20849PO4/H208502. The same holds for cadmium /H20849coordination number of 6 /H20850vs. zinc /H20849coordination number of 4/H20850. Larger Ba and Cd cations are compatible with the nearly ﬂat/H20851VOPO 4/H20852layers /H20849see the left panel of Fig. 6/H20850. To provide proper oxygen coordination for smaller cations, the layershave to buckle. The Sr and Zn cations occupy the positions atthe points of the downward and upward curvature, respec-tively /H20849see the right bottom panel of Fig. 6/H20850. The layer buck- ling can be quantiﬁed via the angle /H9272, see Table IVand the middle panel of Fig. 2. The structural changes are not conﬁned to the layer buck- ling. In particular, the unit cell parameters change in a pecu-liar manner. As the cation size is decreased, the aand b parameters /H20849in-layer spacings /H20850are increased /H20849see Table IV/H20850, while the cparameter /H20849interlayer spacing /H20850is decreased to provide the overall reduction of the unit cell volume ex-pected for the substitution by a smaller cation. The increaseof the in-layer dimensions can also be understood via the change of the cation coordination numbers. Barium cationscoordinate oxygen atoms from four surrounding PO 4tetrahe- dra in the /H20851AA /H11032PO4/H20852interlayer block /H20849upper left panel of Fig. 6/H20850. Strontium cations coordinate three tetrahedra only /H20849upper left panel of Fig. 6/H20850, while the fourth tetrahedron is “pushed away,” thus expanding the unit cell along the aandbdirec- tions. The picture presented in the last two paragraphs and vi- sualized in Fig. 6does not reﬂect all the structural changes including, e.g., slight shifts of the metal cations and tiltingsof the PO 4tetrahedra. However, this picture grasps the es- sential changes that bear inﬂuence on the magnetic /H20851VOPO 4/H20852 layers and on the exchange couplings. There are two mainstructural changes in the magnetic layers: /H20849i/H20850the buckling and /H20849ii/H20850the stretching in the abplane. The latter is performed via the distortion of the VO 5pyramids, while the PO 4tetra- hedra remain rigid. The speciﬁc distortion of the square pyra-mids is shown in Fig. 7: vanadium atoms are shifted away from the pyramid center and yield two longer and twoshorter V–O bonds. Two types of the V–O bonds are easilydistinguished in Table IV. The V–O /H208492/H20850and V–O /H208498/H20850distances remain nearly constant /H208491.95–2.00 Å /H20850in the whole AA /H11032VO/H20849PO4/H208502series, while the two other distances /H20851V–O /H208496/H20850 TABLE IV . Lattice parameters /H20849a,b,c/H20850and some other geometrical characteristics of the AA /H11032VO/H20849PO4/H208502compounds. The V–O distances are given in units of Å, and the oxygen positions are numbered according to the structural data in Refs. 37and38. The angle /H9272is a measure for the buckling of the /H20851VOPO 4/H20852layers as shown in Fig. 2. AA /H11032 a/H20849Å/H20850 b/H20849Å/H20850 c/H20849Å/H20850 /H9272/H20849°/H20850 V–O /H208492/H20850 V–O /H208498/H20850 V–O /H208496/H20850 V–O /H208499/H20850 Ref. BaCd 8.838 8.915 19.374 172 1.977 1.975 2.011 1.992 37 Pb2a9.016 8.747 9.863b155 1.954 1.975 2.024 2.000 39 BaZn 8.814 9.039 18.538 160 1.956 1.974 2.045 1.993 38 SrZn 9.066 9.012 17.513 150 1.971 1.999 2.110 2.039 37 aNon-standard setting used, see Sec. II bThe unit cell of Pb 2VO/H20849PO4/H208502includes one magnetic layer in contrast to the other AA /H11032VO/H20849PO4/H208502compounds with two /H20851VOPO 4/H20852layers in the unit cell. Yet the cparameter is not the true /H20849shortest /H20850interlayer distance due to the monoclinic symmetry of the structure.b ca a Sr Zn BaCdVO(PO )42 Ba Cd SrZnVO(PO )42FIG. 6. /H20849Color online /H20850Crystal structures of BaCdVO /H20849PO4/H208502/H20849left panels /H20850and SrZnVO /H20849PO4/H208502 /H20849right panels /H20850: the upper panels show the inter- layer /H20851AA /H11032PO4/H20852blocks, while the bottom panels present the buckling of the /H20851VOPO 4/H20852layers. The change of Ba and Cd for Sr and Zn leads to thereduction of the coordination numbers and the re-sulting reorganization of the structure: the layerbuckling and the stretching in the abplane /H20849see text for details /H20850.ALEXANDER A. TSIRLIN AND HELGE ROSNER PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-8and V–O /H208499/H20850/H20852are expanded from about 2.00 Å in BaCdVO /H20849PO4/H208502up to 2.11 Å in SrZnVO /H20849PO4/H208502. Both the buckling and the stretching of the magnetic layers are mostpronounced in SrZnVO /H20849PO 4/H208502. Below, we construct proper model structures and separately analyze the inﬂuence ofthese effects on the exchange couplings. To reproduce the layer buckling /H20849i/H20850, we keep the VO 5and PO4polyhedra rigid and simply change the V–O–P angles to achieve the necessary buckling angle /H9272. To reproduce the layer stretching /H20849ii/H20850, vanadium atoms are shifted away from the centers of the pyramids /H20849see Fig. 7/H20850, and the unit cell is properly expanded in the abplane. Then, two V–O distances remain constant /H20849d=1.95 Å /H20850, while two other distances /H20849d/H11032/H20850 are increased. The expansion in quantiﬁed by the value of/H9004d=d /H11032−d. In Fig. 8, we plot the distortion of the NNN couplings /H20849J2/H11032/J2/H20850as found for the real compounds and for the model structures. We ﬁnd that the buckling of the layers is unable toaccount for the spin lattice distortion observed in most of theAA /H11032VO/H20849PO4/H208502compounds /H20849see open circles in Fig. 8/H20850.O n the other hand, the shifts of the vanadium atoms and theresulting layer stretching perfectly reproduce the distortion,even without considering the layer buckling /H20849see ﬁlled circles in Fig. 8/H20850. Thus, we conclude that the distortion of the NNN couplings in AA /H11032VO/H20849PO4/H208502originates from the distor- tion of the VO 5pyramids. We can also unambiguously assign the weaker interaction J2/H11032to the longer V–O /H208496/H20850and V–O /H208499/H20850 separations consistent with the LSDA+ Uresults for the real compounds. Unfortunately, the trends for the NN couplings are less clear. According to the discussion in Sec. IV, the difference between J1andJ1/H11032originates from different AFM contribu- tions to these couplings. The initial model structure does not show any sizable NN hoppings /H20849both t1andt1/H11032are below 4 meV /H20850, and the shifts of the vanadium atoms within the ﬂat layer do not change these hoppings. The layer buckling en- larges t1and t1/H11032up to 10–15 meV , i.e., the TB results for BaZnVO /H20849PO4/H208502are reproduced /H20849see Table II/H20850. However, the model structures do not show the anisotropy of the NN cou-plings, as found in SrZnVO /H20849PO 4/H208502and Pb 2VO/H20849PO4/H208502. Theanisotropy is likely caused by more subtle changes that are not included in our model structures. Nevertheless, the ﬂat/H20851VOPO 4/H20852layer /H20851as found in BaCdVO /H20849PO4/H208502/H20852does not show any considerable AFM contributions to the NN couplings,hence the anisotropy of the NN couplings in the ﬂat-layercompounds should be small. VI. EXTENDED FSL MODEL In this section, we discuss the properties of the extended FSL model. We address thermodynamic properties, magneticsusceptibility and speciﬁc heat, because these quantities aremeasured experimentally and commonly used for the evalu-ation of the exchange couplings in the FSL compounds. Theextended model includes four independent parameters, butwe are mainly interested in the role of the distortion, i.e., the difference between J 1andJ1/H11032orJ2andJ2/H11032. Therefore, we ﬁx the averaged NN and NNN couplings /H20849J¯1and J¯2, respec- tively /H20850and the effective frustration ratio /H9251=J¯2/J¯1. Then we vary either J1andJ1/H11032orJ2andJ2/H11032. We performed the simu- lations for two representative values, /H9251=−2 and /H9251=−1, to study the frustration regime relevant for Pb 2VO/H20849PO4/H208502and BaZnVO /H20849PO4/H208502or SrZnVO /H20849PO4/H208502and BaCdVO /H20849PO4/H208502, re- spectively /H20849see Table I/H20850. In the following, we present the results obtained at /H9251=−2 /H20849Fig. 9/H20850. The simulations for /H9251= −1 reveal a very similar behavior, thus we do not discussthem in detail. The results for the regular FSL at /H9251=−2 match that of Ref. 9/H20849for the comparison, one should use the frustration angle/H9272f//H9266/H112290.65 with /H9272f=tan−1/H9251/H20850. The distortions of the NN and NNN bonds have different effects on the thermody- namic properties. The distortion of the NNN bonds /H20849J2/H11032vs.J2, left panel of Fig. 9/H20850leads to a slight shift of the susceptibilitybaO(2), O(8) O(6), O(9)V J2’J2 dd’ FIG. 7. /H20849Color online /H20850Distortion of the /H20851VOPO 4/H20852layer as implemented in the model structures. Arrows show displacementsof the vanadium atoms away from their ideal positions in the cen-ters of the VO 5square pyramids. The displacements yield two dif- ferent V–O distances /H20849dand d/H11032/H20850and two inequivalent NNN cou- plings /H20849J2and J2/H11032/H20850. Small spheres with solid and hatched ﬁlling denote different types of oxygen atoms with shorter /H20849white bonds /H20850 and longer /H20849shaded bonds /H20850V–O distances, respectively. Dashed and dotted lines indicate the interactions J2andJ2/H11032.01.0 0.8 0.6 0.4 0.02BaCd BaZn SrZnPb2180 170 160/CID1(deg) 150 0.04layer buckling layer stretching AA VO(PO )’42 Dd(A)o0.06 0.08 0.10JJ22’/ FIG. 8. /H20849Color online /H20850Spin lattice distortion /H20849J2/H11032/J2/H20850for the model structures with the layer buckling /H20849empty circles /H20850and the layer stretching /H20849ﬁlled circles /H20850and for the real AA /H11032VO/H20849PO4/H208502com- pounds /H20849ﬁlled diamonds /H20850. The layer buckling is quantiﬁed by the buckling angle /H9272/H20849Fig. 2/H20850. The layer stretching is imposed by shift- ing vanadium atoms and quantiﬁed by /H9004d=d/H11032−d/H20849see Fig. 7/H20850. For the real compounds, d=/H20849dV-O /H208492/H20850+dV-O /H208498/H20850/H20850/2 and d/H11032=/H20849dV-O /H208496/H20850 +dV-O /H208499/H20850/H20850/2/H20849see Table IV/H20850.EXTENSION OF THE SPIN-1 2FRUSTRATED … PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-9maximum to lower temperatures, while the absolute value at the maximum is increased. The maximum of the speciﬁc heatis also shifted to lower temperatures, but its height is re- duced. The distortion of the NN bonds /H20849J 1/H11032vs.J1, right panel of Fig. 9/H20850leads to opposite and more pronounced changes. The maxima are shifted to higher temperatures, the suscep-tibility maximum is reduced, while the speciﬁc heat maxi-mum is increased. To understand these results, one should recall that the magnetic susceptibility and the speciﬁc heat maxima arecaused by correlated spin excitations. These excitations com-pete with quantum ﬂuctuations caused by the low dimension-ality and the magnetic frustration. Thus, the shift of themaxima to lower/higher temperatures implies theenhancement/reduction of the quantum ﬂuctuations. Then theeffects for the susceptibility and the speciﬁc heat are oppo-site, since spin correlations contribute to the speciﬁc heat andincrease its value, but lead to AFM ordering, hence reducingthe susceptibility. The trends presented in Fig. 9suggest that the distortion of the NNN bonds enhances quantum ﬂuctua-tions /H20849see the explanation below /H20850, while the distortion of the NN bonds reduces the ﬂuctuations, and the latter effect ismore pronounced. The effect of the NN bonds distortion is consistent with the previous theoretical results for the spatially anisotropicFSL model. 11,12,14The narrowing and closing of the critical /H20849spin-liquid /H20850region corresponds to the reduction of the quan- tum ﬂuctuations, as observed in the thermodynamic data.One can get further insight into this effect by consideringenergies of the ordered structures within the classical Heisen-berg model. At /H9251=−2, the competing ground states are the FM and columnar AFM ordering /H20849see Fig. 1/H20850. The columnar AFM state is favored by AFM NNN interactions and by theFM interaction along the directions of columns /H20849say, along thebaxis, i.e., the respective interaction is J 1/H20850. Yet, the FM interaction J1/H11032along the aaxis is unfavorable for the colum- nar ordering. As the absolute value of J1/H11032is reduced and that ofJ1is increased, the columnar AFM state is stabilized, and frustration is released. Applying similar considerations to theNNN bonds distortion, we ﬁnd that the distortion does notchange the energies of the FM and columnar AFM states,hence the magnitude of the frustration should remain un- changed. This conclusion is consistent with the relativelyweak effect of the NNN bonds distortion. Still, the certainenhancement of the frustration is clearly visible in the ther-modynamic data and likely related to quantum effects. We can speculate that the reduction of the J 2/H11032/J2ratio leads to the formation of spin chains within the 2D lattice. The chains areformed by the J 2bonds and run along the baxis /H20849see the right panel of Fig. 2/H20850. Then, this one-dimensional feature of the spin system should enhance quantum ﬂuctuations due tothe effectively reduced dimensionality. Now, we turn to the case of moderate distortion relevant for the AA /H11032VO/H20849PO4/H208502compounds. According to Sec. IV, the strongest distortion is found in SrZnVO /H20849PO4/H208502with J1/H11032/J1 /H112290.7 and J2/H11032/J2/H112290.4. The respective susceptibility curves in Fig.9nearly match that for the regular FSL. Thus, the ﬁtting of the experimental susceptibility data should yield averaged exchange couplings of the distorted FSL, J¯1and J¯2. This conclusion provides a reliable basis for the interpretation ofthe experimental values listed in Table I. The changes in the speciﬁc heat curves are also minor, hence the experimentalspeciﬁc heat will be described by the HTSE. Thus, the ther-modynamic properties of the AA /H11032VO/H20849PO4/H208502compounds should ﬁt the regular FSL model with averaged exchangeparameters as an excellent approximation, and this is thecase. 24,25,30 VII. DISCUSSION AND SUMMARY In this study, we performed a detailed microscopic inves- tigation of the distorted FSL spin systems in theAA /H11032VO/H20849PO4/H208502compounds. We estimated the magnitude of the distortion, found the structural origin of the distortionand analyzed the thermodynamic properties of the extendedFSL model. Below, we consider the consequences of theseﬁndings in several aspects: /H20849i/H20850the physics of the AA /H11032VO/H20849PO4/H208502phosphates; /H20849ii/H20850layered vanadium com- pounds as a playground for the search of new FSL materials;and /H20849iii/H20850lattice distortion in frustrated spin systems. The basic experimental results for the AA /H11032VO/H20849PO4/H208502 compounds can be interpreted within the framework of the0 00 00.04 0.040.08 0.080.12 0.12 /CID3/CID1JN gcA B/22 1 10.5 0.5 1.0 1.0 1.5 1.5 2.0 2.00.40.4 0.2 0.2CR/ CR/ 2 2JJ22’/ JJ11’/ TJ/c TJ/c3 31 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0 4 4 FIG. 9. /H20849Color online /H20850Full diagonalization results for the distorted FSL model: magnetic susceptibility /H20849primary ﬁgures /H20850and speciﬁc heat /H20849insets /H20850. The simulations are performed at ﬁxed averaged couplings J¯1=/H20849J1+J1/H11032/H20850/2 and J¯2=/H20849J2+J2/H11032/H20850/2 and the ﬁxed frustration ratio J¯2/J¯1 =−2. In the left and right panels, J2andJ2/H11032orJ1andJ1/H11032are varied, respectively. The arrows show the changes upon increasing the distortion. The thermodynamic energy scale Jcis deﬁned as Jc=/H20881/H20849J1+J1/H110322+J2+J2/H110322/H20850/2.ALEXANDER A. TSIRLIN AND HELGE ROSNER PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-10regular FSL model.24,25,30According to Sec. VI, thermody- namic measurements lead to relevant, averaged exchange couplings J1expand J2exp. These values place all the AA /H11032VO/H20849PO4/H208502compounds to the columnar AFM region of the FSL phase diagram /H20849Fig. 1/H20850. Indeed, neutron scattering results for Pb 2VO/H20849PO4/H208502and SrZnVO /H20849PO4/H208502/H20849Refs. 26and 27/H20850conﬁrm the columnar ordering. Yet, the experimental situation is not fully clear, since muon spin relaxation /H20849/H9262SR/H20850 studies suggest a broad distribution of local magnetic ﬁeldsin the ordered phase, hence pointing to a possible incommen-surate ground state, at least for some of the AA /H11032VO/H20849PO4/H208502 compounds.57One can suggest that the ground state of these materials is inﬂuenced by the spin lattice distortion. Al-though detailed investigation of the ground state of the ex-tended FSL model lies beyond the scope of the present study,the introduction to the extended model and the evaluation ofthe model parameters is a ﬁrst step toward understanding thelong-range magnetic ordering in AA /H11032VO/H20849PO4/H208502. Further the- oretical and experimental /H20849neutron scattering, NMR /H20850studies on the ground state properties are highly desirable andshould be stimulated by our work. Apart from the thorough studies of the ground state, one can suggest a more simple way for the experimental obser-vation of the spin lattice distortion in AA /H11032VO/H20849PO4/H208502. Ac- cording to Sec. VI, the distortion of the nearest-neighbor bonds /H20849J1vs.J1/H11032/H20850stabilizes the columnar AFM state with respect to the FM state. Within the classical model, the en-ergy difference between the FM and columnar AFM states corresponds to the saturation ﬁeld. Therefore, the J 1/HS11005J1/H11032sce- nario should have an effect on the saturation ﬁeld. In case ofBaCdVO /H20849PO 4/H208502with the nearly regular FSL, the saturation ﬁeld is in excellent agreement with the averaged couplings J1expand J2exp/H20849Ref. 30/H20850. However, the saturation ﬁeld of SrZnVO /H20849PO4/H208502should be different from the ﬁeld estimated using J1expand J2exp. High-ﬁeld magnetization studies of the AA /H11032VO/H20849PO4/H208502compounds will challenge this proposition and enable the quantitative analysis of the spin lattice distor-tion in AA /H11032VO/H20849PO4/H208502. The next important issue is the capability of layered va- nadium compounds to reveal new strongly frustrated FSLmaterials. The basic structural element of these compoundsis the /H20851VOXO 4/H20852layer shown in the left panel of Fig. 2.I n this layer, non-magnetic XO 4tetrahedra mediate either FM or weak AFM NN and AFM NNN interactions. Then thereare two ways to reach the strongly frustrated regime of /H9251 /H11229−0.5: one should either increase the absolute value of J1 /H20851as observed in SrZnVO /H20849PO4/H208502/H20852or decrease J2/H20851as observed in BaCdVO /H20849PO4/H208502/H20852. According to the results of our study /H20849Sec. V/H20850, the NNN interactions /H20849J2/H20850are sensitive to the V–O distances, since the magnitude of any superexchange interac-tion depends on the orbital overlap. Thus, it is possible toreduce the J 2value by increasing the V–O separations in the VO 5square pyramids. The factors inﬂuencing on the value ofJ1are less clear. In the AA /H11032VO/H20849PO4/H208502compounds, the J1 values correlate with the distortion of the NNN bonds. One may suggest the layer buckling or the distortion of the VO 5 pyramids as possible reasons for the increase of J1expfrom BaCdVO /H20849PO4/H208502to SrZnVO /H20849PO4/H208502. However, this conclusion is rather empirical, and other structural factors may be rel-evant as well.The proper FSL material should combine the strong frus- tration with the lack of the distortion or, at least, with arelatively weak distortion of the spin lattice. It is also desir-able to ﬁnd a family of isostructural FSL compounds. Then,the replacement of the metal cations may facilitate the tuningof the system toward the strongly frustrated regime. Our re-sults provide a clear recipe for the search of undistorted FSLmaterials. To get a regular FSL, one should keep the mag-netic layer ﬂat and avoid the distortion of the VO 5square pyramids. Clearly, it is quite difﬁcult to fulﬁll these criteriawithin the AA /H11032VO/H20849PO4/H208502series. Different metal cations re- quire different coordination numbers /H20849see Fig. 6/H20850; therefore, most of the respective compounds reveal a distorted FSL. InBaCdVO /H20849PO 4/H208502, the cation sizes are optimal to yield nearly ﬂat and regular magnetic layers and to result in the weakdistortion of the FSL. Though tuning is possible /H20849see Table I/H20850, most of the metal cations lead to a layer distortion and to a spin lattice distortion as well. To avoid the spin lattice distortion within a compound family, one can try to reduce the number of metal cations andto look for another ﬁller of the interlayer space. For example,one can consider the A /H20849VOPO 4/H208502·4H 2O compounds /H20849A =Ca, Cd, Sr, Pb, Ba, and Mg/Zn /H2085058–65that reveal layered structures with /H20851VOPO 4/H20852layers separated by metal cations and water molecules. The resulting symmetry is orthorhom-bic, monoclinic, or even triclinic, but the respective spin lat-tice distortion should be weak. The layers are nearly ﬂat forall the six compounds reported, and the distortion of the VO 5 pyramids is also negligible. Yet, the change of the metalcation bears inﬂuence on the magnetic interactions, as indi-cated by different Curie-Weiss temperatures. 60,62–65The magnitude of the frustration in the A /H20849VOPO 4/H208502·4H 2O com- pounds remains unknown. All the reports available considerNN interactions only. Clearly, this scenario is oversimpliﬁed,and one has to apply the FSL /H20849rather than a simple square lattice /H20850model while analyzing magnetic properties of A/H20849VOPO 4/H208502·4H 2O. Further studies of these materials should be very promising. The last, but not least, point deals with the inﬂuence of the distortion on the properties of frustrated spin systems. Inthis study, we exempliﬁed this problem by considering thedistortion of the FSL. We found that different types of dis-tortion had different effects on the magnitude of the frustra-tion. The distortion of the NN bonds stabilizes the columnarAFM ordering and releases the frustration. Yet, the distortionof the NNN bonds keeps the strong frustration and even en-hances the quantum ﬂuctuations. These results suggest thatboth regular and distorted frustrated materials should be con-sidered as proper realizations of the frustrated spin models.The case of the AA /H11032VO/H20849PO4/H208502compounds shows how the distorted FSL materials can be successfully treated within theregular FSL model. In these compounds, the thermodynamicproperties at available temperatures nearly match those of theregular model. However, the ground state properties may bedifferent. We believe that the study of frustrated materialswith spin lattice distortion will improve our understanding ofthe frustrated spin systems and facilitate the observation ofinteresting phenomena suggested by theory. In conclusion, we have studied the distorted frustrated square lattice in layered vanadium phosphatesEXTENSION OF THE SPIN-1 2FRUSTRATED … PHYSICAL REVIEW B 79, 214417 /H208492009 /H20850 214417-11AA /H11032VO/H20849PO4/H208502/H20849AA /H11032=Pb 2, SrZn, BaZn, and BaCd /H20850. In these compounds, both nearest-neighbor and next-nearest-neighborbonds of the square lattice are distorted, hence an extendedspin model with four inequivalent exchange couplingsshould be considered. Our estimates of the individual modelparameters suggest the least pronounced distortion inBaCdVO /H20849PO 4/H208502and the most pronounced distortion in SrZnVO /H20849PO4/H208502. The difference between the nearest-neighbor and next-nearest-neighbor interactions in these compoundsoriginates from peculiar structural changes upon substitutinginterlayer metal cations A and A /H11032. The buckling of the mag- netic layers may change the interactions along the side of the square, while the distortion of the vanadium polyhedra andthe resulting stretching of the magnetic layer lead to the dif-ference between the diagonal interactions of the distortedsquare lattice. The distortion of the square lattice inAA /H11032VO/H20849PO4/H208502is moderate from the point of view of the thermodynamic properties. The temperature dependences ofthe magnetic susceptibility and the speciﬁc heat resemble those of the regular square lattice, hence previous experi-mental studies of AA /H11032VO/H20849PO4/H208502reported averaged cou- plings. These couplings can be used to place the compoundson the phase diagram of the regular model. In contrast to themoderate inﬂuence on the thermodynamic properties, thespin lattice distortion may have a larger effect on the groundstate. Further experimental and theoretical studies of thisproblem are highly desirable. ACKNOWLEDGMENTS The authors are grateful to Christoph Geibel for fruitful discussion. Financial support of GIF /H20849I-811-257.14/03 /H20850, RFBR /H2084907-03-00890 /H20850, and the Emmy Noether Program of the DFG is acknowledged. 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PhysRevB.101.134505.pdf	PHYSICAL REVIEW B 101, 134505 (2020) Theory of the orbital moment in a superconductor Joshua Robbins, James F. Annett, and Martin Gradhand H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom (Received 28 February 2018; revised manuscript received 18 March 2020; accepted 20 March 2020; published 10 April 2020) The chiral p-wave superconducting state is comprised of spin-triplet Cooper pairs carrying a ﬁnite orbital angular momentum. For the case of a periodic lattice, calculating the net magnetization arising from thisorbital component presents a challenge as the circulation operator ˆr×ˆpis not well deﬁned in the Bloch representation. This difﬁculty has been overcome in the normal state, for which a modern theory is ﬁrmlyestablished. Here, we derive the extension of this normal-state approach, generating a theory which is valid for ageneral superconducting state, and go on to perform model calculations for a chiral p-wave state in Sr 2RuO 4.T h e results suggest that the magnitude of the elusive edge current in Sr 2RuO 4is ﬁnite, but lies below experimental resolution. This provides a possible solution to the longstanding controversy concerning the gap symmetry ofthe superconducting state in this material. DOI: 10.1103/PhysRevB.101.134505 I. INTRODUCTION An unconventional superconducting state exhibits a lower order of symmetry than the s-wave singlet pairing observed in conventional BCS superconductors. An example of this is thechiral p-wave paired state, which arises in conjunction with a breaking of time-reversal symmetry at the superconductingtransition [ 1]. Such a state consists of spin-triplet Cooper pairs carrying a ﬁnite orbital angular momentum. The symmetrybreaking associated with this pairing theoretically facilitates anumber of new and exotic phenomena, such as the Kerr effect[2,3] and edge currents [ 4,5]. Of major signiﬁcance in the study of this class of materials is the topological nature of superconducting states with chiralsymmetry [ 6,7]. A chiral edge mode in a topological super- conducting state would support a protected Majorana boundstate conﬁned to the edges of the sample [ 8,9]. The existence of these bound states is inextricably linked to the orbitalmoment of the spin-triplet Cooper pairs, as both phenomenaarise from the chiral nature of the superconducting orderparameter. Given this interest, it is surprising that there currently exists no general framework with which to calculate thetotal orbital magnetic moment in a superconducting state.The orbital angular momentum carried by the Cooper pairsshould, in principle, lead directly to an orbital magnetizationin the superconducting lattice. Contributions to the magneticmoment are expected from edge currents [ 4,5], while bulk contributions are also predicted in multiorbital systems [ 10]. The goal of this paper is to present a general approach to thisproblem. A rigorous theory for the orbital magnetization in a normal- state periodic lattice has been deﬁned previously [ 11,12]. Obtaining a formalism of this nature had been an outstandingissue due to the problem of evaluating the circulation oper-ator ( ˆr×ˆp) in a Bloch representation. In an inﬁnite lattice, the position operator ( ˆr) is unbound and the cell-periodicBloch functions [ u k(r)] are not localized. The coexistence of these two factors means that the position expectation valuesof Bloch wave functions cannot be evaluated directly. Thenormal-state theory was developed by reformulating the prob-lem in a localized basis, the Wannier representation [ 12,13]. Here, we extend this formalism to the orbital magnetization inthe superconducting state. The new theory for the orbital moment in an inﬁnite peri- odic lattice has previously been applied to cases of insulatorsand metals, for both single-band and multiband conﬁgurations[12]. The derivation introduced two distinct contributions to the total moment, referred to as the “local” and “itinerant”circulations. The terms correspond to orbital moments gener-ated by the movement of the centers of mass of orbital wavefunctions (itinerant), and the moment due to self-rotationabout their centers of mass (local). Extending this theory to a general superconducting state, we obtain equivalent expressions for the local and itinerantcontributions. We further break down the local contributionby performing a tight-binding expansion, extracting the purelyon-site component deﬁned previously [ 10]. The formalism developed here will then be applied to a multiband tight-binding model of Sr 2RuO 4. II. THEORY We begin our analysis by giving an outline of the derivation of the orbital moment in the superconducting state. In secondquantized form, the operator for the total orbital angularmomentum in an arbitrary state is given by ˆL z=/integraldisplay drˆa†(r)ˆlzˆa(r), (1) where a†,aare Fermi creation and annihilation operators, respectively, and ˆlz=[ˆr×ˆp]z. The total orbital magnetic moment is then given by γ/angbracketleftˆLz/angbracketright, where γ=−e/(2me). 2469-9950/2020/101(13)/134505(6) 134505-1 ©2020 American Physical SocietyROBBINS, ANNETT, AND GRADHAND PHYSICAL REVIEW B 101, 134505 (2020) In order to obtain a second quantized operator valid for a gapped state, we perform the Bogoliubov-Valatin transforma-tion on the creation and annihilation operators [ 14], ˆa=/summationdisplay nkθnk(r)ˆγnk+χ∗ nk(r)ˆγ† nk, (2a) ˆa†=/summationdisplay nkθ∗ nk(r)ˆγ† nk+χnk(r)ˆγnk, (2b) where nis the number of spin-resolved bands, kis the Bloch wave vector, and γ†,γare quasiparticle creation and annihilation operators. The functions θ,χare, respectively, electron and hole components of a Bloch-type wave functionψ. This transformation recasts the equation into an expression for the orbital moment arising from Bogoliubov quasiparticleswhich appear as excitations in a superconductor. To obtain the total orbital moment in an arbitrary super- conducting state, we compute the expectation value of thetransformed operator by applying the following relations: /angbracketleftˆγ † nkˆγn/primek/prime/angbracketright=δnn/primeδkk/primefnk, (3a) /angbracketleftˆγnkˆγ† n/primek/prime/angbracketright=δnn/primeδkk/prime(1−fnk), (3b) /angbracketleftˆγnkˆγn/primek/prime/angbracketright=/angbracketleft ˆγ† nkˆγ† n/primek/prime/angbracketright=0, (3c) where fis the Fermi-Dirac function. The transformed equa- tion and its associated operators then take the following form: /angbracketleftˆLz/angbracketright=/summationdisplay nk/integraldisplay drψ† nk(r)Lzψnk(r), Lz=/parenleftbiggˆlzfnk 0 0−ˆl∗ z(1−fnk)/parenrightbigg ,ψ nk(r)=/parenleftbigg θnk(r) χnk(r)/parenrightbigg .(4) At this point, we can defer to the derivation laid out for the normal state in terms of Wannier orbitals [ 11,12], where we now consider two-component Wannier wave func-tions containing electron and hole amplitudes in correspon-dence with the Bloch-type eigenfunctions. We also introducethe cell-periodic components of the Bloch wave functions,[u nk(r),vnk(r)]=e−ik·r[θnk(r),χnk(r)]. Following the steps of this derivation, we are able to re- move the dependence of Eq. ( 4) on the problematic operators ˆrand ˆv. Performing a Fourier transform on the real-space ex- pressions obtained via this approach, we obtain two reciprocal space expressions which generate the orbital magnetizationvia Brillouin-zone integrals, M LC=−γIm/braceleftBigg/integraldisplay BZdk (2π)3/summationdisplay n[/angbracketleft∂kunk\|× ˆHk\|∂kunk/angbracketrightfnk −/angbracketleft∂kvnk\|× ˆH∗ k\|∂kvnk/angbracketright(1−fnk)]/bracerightbigg , (5) MIC=γIm/braceleftBigg/integraldisplay BZdk (2π)3/summationdisplay nEnk[/angbracketleft∂kunk\|×\|∂kunk/angbracketrightfnk +/angbracketleft∂kvnk\|×\|∂kvnk/angbracketright(1−fnk)]/bracerightbigg , (6) where LC and IC refer to local and itinerant circulations, as deﬁned previously [ 11], and the total magnetization is given by M=MLC+MIC. We have divided by the unit-cellvolume, to convert from the magnetic moment to magneti- zation, and also introduced Dirac notation where, crucially,the expectation values taken in Eqs. ( 5) and ( 6)a r en o w evaluated for the unit cell only. These equations constitute ourcentral result: a comprehensive framework for computing thetotal orbital magnetization in a general bulk superconductingstate. The cell-periodic functions are obtained through self- consistent calculation of the Bogoliubov–de Gennes (BdG)equation, /parenleftbiggˆH k(r)/Delta1(r) /Delta1†(r)−ˆH∗ −k(r)/parenrightbigg/parenleftbigg unk(r) vnk(r)/parenrightbigg =Enk/parenleftbigg unk(r) vnk(r)/parenrightbigg ,(7) where ˆHkis the k-dependent normal-state Hamiltonian [ 15]. The gap function ( /Delta1) enforces the symmetry of the supercon- ducting state in question. In order to perform model calculations, we must recast the Bloch equations into a tight-binding representation. Perform-ing the kderivatives in ( 5) and ( 6) and expanding in terms of the Bloch wave functions, we obtain ∂ kunk(r)=e−ik·r[∂kθnk(r)−irθnk(r)], (8a) ∂kvnk(r)=e−ik·r[∂kχnk(r)−irχnk(r)]. (8b) Substituting Eqs. ( 8)i n t o( 6), we ﬁnd one term containing r×r, which will vanish. For the local component, however, this does not occur and we can split the equation into two partsof the form ∂ kθ∗ nk×ˆH∂kθnkandθ∗ nk[r×ˆHr]θnk, respectively. Using the standard deﬁnition of the velocity operator, r×ˆHr can be rewritten as −iˆlz. Having rewritten Eq. ( 6) in terms of θ,χ, we can subse- quently apply a general tight-binding expansion of the Blochwave function via /parenleftbigg θ nk(r) χnk(r)/parenrightbigg =/summationdisplay L,Reik·R/parenleftbigg unL(k) vnL(k)/parenrightbigg φL(r−R), (9) where Lis the orbital index and φLis the corresponding orbital wave function. Substituting Eq. ( 9)i n t o( 5), we obtain the following terms: M(1) LC=−γIm/braceleftBigg/summationdisplay nLL/prime/integraldisplay BZdk (2π)3[∂ku∗ nL(k)×ˆHLL/prime(k)∂kunL/prime(k)fnk −∂kv∗ nL(k)×ˆH∗ LL/prime(k)∂kvnL/prime(k)(1−fnk)]/bracerightbig ,(10) M(2) LC=γRe/braceleftBigg/summationdisplay nLL/prime/integraldisplay BZdk (2π)3[u∗ nL(k)(ˆlz,LL/prime)unL/prime(k)fnk +v∗ nL(k)(ˆl∗ z,LL/prime)vnL/prime(k)(1−fnk)]/bracerightBigg . (11) The eigenvectors ( unL,vnL) are computed by solving Eq. ( 7) self-consistently in the tight-binding basis. The terms ˆHLL/primerepresent the matrix elements of the tight-binding Hamil- tonian. Similarly, the matrix elements ˆlz,LL/primecorrespond to the orbital angular momentum expectation values of the orbitalscontained in the tight-binding basis. These elements can be 134505-2THEORY OF THE ORBITAL MOMENT IN A … PHYSICAL REVIEW B 101, 134505 (2020) calculated by direct consideration of the spherical harmonics of the basis. The second term, M(2) LC, is identical to the purely on-site orbital moment computed previously [ 10]. We therefore label M(2) LCas the “on-site” component and continue to refer to M(1) LC as the local contribution. III. RESULTS FOR Sr 2RuO 4 Now that the framework for calculating the magnetic moment has been set up, we brieﬂy outline the model forSr 2RuO 4that will be used to perform the calculations. The su- perconducting state of Sr 2RuO 4is widely believed to exhibit chiral p-wave superconductivity below its transition temper- ature of 1.5 K [ 16,17], such that the superconducting order parameter is given by d∼(sinkx±isinky)ˆz. This hypothesis is supported by measurements of spin susceptibility [ 18,19] and indirect observations of time-reversal symmetry breakingatT c[20]. In addition, a ﬁnite Kerr shift has been measured in this material [ 2], providing direct evidence of a macro- scopic orbital magnetization in the bulk superconductingstate. The classiﬁcation of Sr 2RuO 4as a p-wave superconductor remains a point of controversy, however, as phenomenologicaland quasiclassical approaches have predicted that large edgecurrents should accompany the single-band chiral supercon-ducting state [ 4,21,22]. Such currents have remained elusive despite years of intensive experimental work [ 23–25]. A large surface-based current would provide a signiﬁcant contributionto the total orbital magnetization. By generating a full theoret-ical description of the orbital magnetic moment and its varioussources in such a state, we provide a vital avenue throughwhich we can attempt to reconcile these observations withtheory. We have constructed a three-dimensional tight-binding Hamiltonian consisting of three Ru 4 dorbitals ( d xy,dxz, and dyz) contributing to the normal-state Fermi surface, resulting in a two-dimensional (2D) band (denoted γ) and two quasi-1D bands ( αandβ). In many approaches to modeling Sr 2RuO 4, the model is formulated such that superconductivity arisesprimarily on γ, with accompanying gaps on αandβarising only through proximity effects. Here, we treat all three bandson an equal footing, resulting in a fully multiband supercon-ductivity picture. The included 1D bands display horizontalline nodes, leading to the experimentally observed power lawfor the speciﬁc heat. This model has been covered in moredetail previously [ 15,26]. The distinct contributions to the magnetization in this model are plotted in Fig. 1. It should be noted that the contributions M ICandM(1) LCdiverge to plus and minus inﬁnity, respectively, as Tapproaches Tc. This problem arises due to the fact that these components are not separately gaugeinvariant, and thus we must take the sum of the two. Ithas been shown previously that gauge-invariant forms of thenormal-state equations for M LCandMICcan be obtained [27]. However, this requires an absolute distinction between the occupied and unoccupied states in the electron bandstructure. The Bogoliubov transformation enforces mixingof the electron and hole states. This mixing is essential torecover the quasiparticle band structure of the superconduct-050100150200250300 0 0.5 1.0 1.5 2.0Moment (10−6µB) T (K)M(2) LC M(1) LC+MIC FIG. 1. On-site moment M(2) LCalongside the sum of the itinerant and local components MIC+M(1) LCfor the model without SOC. ing state, but prevents any attempts to project excitations onto occupied states, and thus our expressions cannot be separatelyconverted into gauge-invariant forms. The comparison of the itinerant contributions and the on- site component reveal that the latter is almost two orders ofmagnitude smaller than the itinerant orbital moment. Whilethe on-site part corresponds to a magnetic ﬁeld of the order of∼3 mG, it is ∼300 mG for the itinerant part. This places the on-site orbital moment around the the resolution of the mostrecent attempts to experimentally identify an edge current inSr 2RuO 4via magnetometry measurements ( ∼2.5m G[ 25]). On the other hand, the itinerant part, including edge andunit-cell currents, is sizable in comparison to the experimentalresolution and is, in fact, compatible with μSR measurements suggesting ﬁelds of 500 mG [ 20]. However, the latter part is still a bulk property and will be affected by any experimentalsituation where boundaries and ﬁnite size of the sample playany signiﬁcant role. Despite the fact that these results do nottrivially resolve the uncertainty around the magnetic momentin Sr 2RuO 4, it established that the bulk orbital moment is signiﬁcantly smaller than some earlier models have beenpredicting. The reason for this suppression in the orbital moment in comparison to other theoretical approaches likely lies in themultiband, nodal nature of our tight-binding model and gapstructure. Signiﬁcantly, this result agrees with other experi-mental and theoretical observations which support the ideathat multiband superconductivity is prevalent in this material.It has been shown previously that interorbital transitions arenecessary in order for the Kerr effect to arise intrinsically inthe superconducting state [ 15,28]. In order to see the effect in a single band picture, extrinsic mechanisms such as skewscattering must be considered [ 29]. The inclusion of the addi- tional 1D, line nodal bands also leads to the correct speciﬁcheat below T c[28]. The nodeless 2D band would not produce the experimentally observed power laws in heat capacity[30] or nuclear magnetic resonance (NMR) spin-relaxation rate [ 31]. 134505-3ROBBINS, ANNETT, AND GRADHAND PHYSICAL REVIEW B 101, 134505 (2020) -50050100150200250300 0 0.5 1.0 1.5 2.0Moment (10−6µB) T( K )Model Model+SOC FIG. 2. Itinerant magnetic moment MIC+M(1) LCfor the models with and without SOC. A. The effect of spin-orbit coupling We also wish to assess the inﬂuence of spin-orbit cou- pling (SOC) on the magnetic moment in the chiral state.To do this, we compare results using a tight-binding modelwith an additional spin-orbit Hamiltonian derived in an on-site approximation. As was shown previously [ 28], a model including spin-orbit coupling with coupling parameter λ= 12.5meV is able to replicate experimental features such as the Fermi surface, bandwidth, and heat capacity. In the following,we compare the non-SOC case ( λ=0) to the case with SOC (λ=12.5m e V ) . The results for the model including SOC are displayed in Figs. 2and 3. It is clearly visible that SOC leads to a suppression of the orbital magnetic moment. We observe asigniﬁcant quantitative reduction in all contributions, withoutany qualitative differences in the temperature dependence thatis displayed. This suppression is also of similar order to thatseen in the Kerr effect under the inﬂuence of SOC, as reportedpreviously [ 28]. 00.51.01.52.02.53.0 0 0.5 1.0 1.5 2.0Moment (10−6µB) T( K )Model Model+SOC FIG. 3. On-site magnetic moment M(2) LCfor the models with and without spin-orbit coupling.00.20.40.60.81.01.21.4 0 0.5 1.0 1.5 2.0Moment (10−6µB) T (K)Δm(2) LCSpin FIG. 4. Spin moment in the model including SOC alongside the difference in the on-site orbital moment with and without SOC. B. The spin-magnetic moment In order to fully assess the inﬂuence of SOC, it is informa- tive to also compute the spin moment of the chiral state. To dothis, we start with the equation for the spin expectation valuein the orbital basis, /angbracketleftˆS z/angbracketright=/summationdisplay mm/primeσσ/prime/angbracketleftmσ\|¯h 2σz\|m/primeσ/prime/angbracketrightnσσ/prime mm/prime, (12) where m,σare the orbital and spin degrees of freedom, respectively, and nare the single-particle density matrices. The density matrix can be evaluated in terms of solutions to the BdG equation, while the σzmatrix elements are ±1f o r σ=σ/prime=± 1 and m=m/prime. The ﬁnal expression is then /angbracketleftˆSz/angbracketright=/summationdisplay m¯h 2/parenleftbig n↑↑ mm−n↓↓ mm/parenrightbig , (13) nσσ mm=1 N/summationdisplay nk/vextendsingle/vextendsingleuσ nk/vextendsingle/vextendsingle2f(Enk)+/vextendsingle/vextendsinglevσ nk/vextendsingle/vextendsingle2[1−f(Enk)].(14) The spin-magnetic moment is given by γs/angbracketleftˆSz/angbracketright, where γs= −eg/(2me) and gis the spin gyromagnetic ratio. It is interesting to note here that the spin moment in this context becomes nonzero when SOC is included (see Fig. 4), but is zero otherwise. The spin moment in the SOC regime isof similar order to the reduction in the on-site orbital momentinduced by the spin-orbit interaction (which we have denoted/Delta1m (2) LC). This would suggest that the spin-orbit interaction me- diates a transfer of magnetic moment from the orbital degreesof freedom (where it arises from the chiral order parameter)to the spin degrees (which are otherwise disordered). This observation provides an interesting insight into the origin of the Kerr effect, a phenomenon which is driven bythe anomalous Hall conductivity present in systems with aﬁnite orbital moment. The microscopic origin of this effect inunconventional superconductors has been extensively debated[29,32]. The current controversy concerns whether the origin is an extrinsic mechanism, i.e., arising from disorder [ 33–35], or an intrinsic mechanism, i.e., arising from coupling of the 134505-4THEORY OF THE ORBITAL MOMENT IN A … PHYSICAL REVIEW B 101, 134505 (2020) -3 -2 -1 0 1 2 3 kx-3(a) (b) -2 -1 01 2 3ky -12-60612Im[σxy]( 1 0−11e2//planckover2pi1d) -3 -2 -1 0 1 2 3 kx-3 -2 -1 01 2 3ky -2-1012/angbracketleftˆSz/angbracketright(10−6/planckover2pi1) FIG. 5. (a) Berry curvature contributions in the Brillouin zone integrated along kz,T=0, with spin-orbit coupling. (b) kx−ky resolved plot of the spin moment in the Brillouin zone. The kz dependence has been integrated out. pair state to orbital degrees of freedom at the Fermi level [15,28,36]. In the normal-state ferromagnet, the intrinsic mechanism facilitating the Kerr effect is induced by coupling of theordered spins to the orbital component via SOC. Namely, thesymmetry breaking in the spin degree of freedom is trans-ferred to the orbital component via the spin-orbit interaction.This is a clear analog to the results reported here, where orbitalorder arises naturally due to the chiral superconducting orderparameter, and is then reduced via coupling to the disorderedspin component. These results coincide with the observationsreported previously, where the magnitude of the Kerr shift inthe same chiral superconducting model was also shown tobe suppressed by a similar order following the introductionof SOC [ 28]. Our model is thus able to effectively describe an intrinsic origin of the anomalous phenomena observed inSr 2RuO 4. This analysis of the inﬂuence of SOC is further sup- ported by assessing the regions of the Brillouin zone inwhich the spin moment arises (see Fig. 5). Here we see that the spin moment is present in regions of near degeneraciesbetween the orbital degrees of freedom in the band structure.These regions on the Brillouin zone contribute strongly tothe Berry curvature, which gives rise to an anomalous Hallconductivity [ 37]. This implies that these regions contain thehighest density of ordered orbital moments, which in turn suggests that the spin magnetization is arising directly asa result of coupling of the spins to the orbital degree offreedom. IV . CONCLUSION In conclusion, a formalism for computing the orbital mag- netization in a superconductor has been derived and calcula-tions for the model chiral p-wave superconductor Sr 2RuO 4 have been performed. The results suggest that early estima- tions of the itinerant magnetization in this state were toogenerous. With the results that are presented here, the itinerantmoment is comparable to μSR experiments but the on-site moment is probably below the resolution of magnetometry-based investigations. This same model has been shown to alsogive a physically reasonable estimate of the observed Kerreffect [ 15]. An interesting insight into the inﬂuence of SOC on a magnetic superconducting state has also been highlighted.Generally, the SOC reduces the magnetic moment, but for theon-site contribution the quantitative change is compensated bythe generation of an on-site spin-magnetic moment. It should be stressed that the general result here is not restricted to the model used. We note that our theory wouldalso apply to other pairing states which have been proposedfor Sr 2RuO 4, such as the chiral d-wave [ 38],f-wave [ 39], or long-range p-wave [ 40] states. In addition, the equations presented here could be used to investigate the unconventionalpairing symmetries observed in other materials, such as theunderdoped cuprates and heavy-fermion compounds. ACKNOWLEDGMENTS This work was carried out using the computational facili- ties of the Advanced Computing Research Centre, Universityof Bristol [ 41]. J.R. acknowledges support via the CMP- CDT funded by EPSRC and J.F.A. via EPSRC Grant No.EP/P007392/1. M.G. acknowledges ﬁnancial support from theLeverhulme Trust via an Early Career Research Fellowship(ECF-2013-538). [1] C. Kallin and J. Berlinksy, Rep. Prog. Phys. 79,054502 (2016 ). [2] J. Xia, Y . Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. 97,167002 (2006 ). [3] J. Xia, E. Schemm, G. Deutscher, S. A. Kivelson, D. A. Bonn, W. N. 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PhysRevB.84.165315.pdf	PHYSICAL REVIEW B 84, 165315 (2011) Electrical spin injection and accumulation in CoFe/MgO/Ge contacts at room temperature Kun-Rok Jeon,1Byoung-Chul Min,2Young-Hun Jo,3Hun-Sung Lee,1Il-Jae Shin,2Chang-Yup Park,1Seung-Young Park,3and Sung-Chul Shin1,* 1Department of Physics and Center for Nanospinics of Spintronic Materials, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea 2Center for Spintronics Research, Korea Institute of Science and Technology (KIST), Seoul 136-791, Korea 3Nano Materials Research Team, Korea Basic Science Institute (KBSI), Daejeon 305-764, Korea (Received 3 June 2011; revised manuscript received 15 July 2011; published 10 October 2011; corrected 28 October 2011) We report the all-electrical spin injection and detection in CoFe /MgO/moderately doped n-Ge contact at room temperature (RT), employing three-terminal Hanle measurements. A sizable spin signal of ∼170 k/Omega1μm2has been observed at RT, and the analysis using a single-step tunneling model gives a spin lifetime of ∼120 ps and a spin diffusion length of ∼683 nm in Ge. The observed spin signal shows asymmetric bias and temperature dependences which are strongly related to the asymmetry of the tunneling process. DOI: 10.1103/PhysRevB.84.165315 PACS number(s): 72 .25.Dc, 72 .25.Mk, 75 .47.−m, 85.75.−d I. INTRODUCTION The rapid evolution of electronics requires alternative technologies more than scaling down the device size, and spintronics based on the electron spins in semiconductor raises prospects for future electronics.1–6The electrical in- jection of spin-polarized electrons from ferromagnet (FM)into semiconductor (SC) and subsequent detection of theresultant spin accumulation provide a viable route for the realization of SC-based spintronics. 1–6The electrical spin injection into GaAs, InAs, or Si from FM through a spin-dependent tunnel barrier has been demonstrated using opticaldetection in spin lighting emitting diodes 7–10or electrical detection in vertical /lateral (spin valve) structures.11–18With engineering of magnetic tunnel contacts, signiﬁcant spin signals have been observed in Si using Co /NiFe/Al2O3 and Fe /SiO 2tunnel contacts up to room temperature (RT).6,19 Recently, the n-type Ge in conjunction with a crystalline bcc FM /MgO(001)20–24has attracted much attention as a promising candidate for the efﬁcient spin injection in termsof a high tunnel spin polarization (TSP), a small conduc-tivity mismatch, and a negligible interdiffusion /intermixing in FM /oxide/SC contacts. Moreover, considering high elec- tron mobility in Ge (at least twice higher than Si) andits weak dependence on doping concentration, Ge prospec-tively represents an SC channel with a long spin diffu-sion length. 2,25Several important achievements have been recently reported in the ﬁelds of spin transport26,27and spin accumulation28in Ge at low temperature, and in the ﬁeld of spin detection29,30at RT, but all-electrical spin injection and detection in Ge at RT is yet to be investi-gated. Here, we demonstrate the electrical spin injection in spin tunnel contacts consisting of crystalline bccCoFe/MgO (001) /moderately doped n-Ge and the elec- trical detection of the induced spin accumulation at RT.We have analyzed the spin accumulation, spin life time,spin diffusion length in Ge from the measured spinsignal and studied their bias and temperature depen-dences.II. EXPERIMENTAL DETAILS A. Principle of the approach Figure 1(a) illustrates the device geometry and measure- ment scheme used in this paper. We have fabricated a symmet-ric device consisting of ﬁve single crystalline CoFe /MgO/n- Ge tunnel contacts ( a–e) spaced as shown in the inset of Fig.1(a). The contacts a,b, and c(30×100, 20 ×100, and 20 ×100μm 2) are used as spin injectors /extractors and also spin detectors, while the contacts dande(150×100 and 150 × 100μm2) are used as references. The contacts are separated from each other more than 100 μm, which is much longer than the spin diffusion length. The magnetic easy axis of the CoFecontacts are along the [110] direction of Ge in parallel to thelong axes of the contacts. The measurement scheme 6,16,18,19,28 [Fig. 1(a)] using a single contact in the three-terminal ge- ometry provides a simple way to measure the induced spinaccumulation in SC by spin injection or extraction. When the spin-polarized electrons are injected from FM 1 (a/b/c ) to SC, majority spins accumulate in SC (at x1,/Delta1μ+= μ+↑−μ+↓>0); when the electrons (mostly majority-spin electrons) are extracted from SC to FM 1(a/b/c ), minority spins accumulate in SC (at x1,/Delta1μ−=μ−↑−μ−↓<0) as shown in Fig. 1(b). This spin accumulation induced by spin injection or extraction can be detected electrically usingthe same contact by means of the Hanle effect. 16,31,32A transverse magnetic ﬁeld ( B) suppresses the spin accumulation in the SC (at x1) via spin precession and results in a voltage drop between FM 1(a/b/c ) and FM 2(d/e) as a function of the applied ﬁeld ( B) [i.e. negative magnetoresistance (MR)] as depicted in Fig. 1(b). Ignoring recombination effects, the voltage drop ( /Delta1V) can be described approximately by a Lorentzian function, /Delta1V∓(B⊥)=/Delta1V∓(0)/[1+(/Omega1τsf)2],33 with/Delta1V∓(0)=γ/Delta1μ ±(0)/(−2e),/Omega1=gμBB⊥/¯h.Here,γis the TSP of the tunnel contact, gis the Land ´eg-factor, μBis the Bohr magneton, and τsfis the spin lifetime. From the above relation, one can extract the spin lifetime of carriers ( τsf) and spin accumulation ( /Delta1μ)i nS C . Three-terminal Hanle measurement cannot fully uncover whether the measured spin accumulation comes from the bulkSC channel 34or the localized states (LSs) at the interface.18 165315-1 1098-0121/2011/84(16)/165315(10) ©2011 American Physical SocietyJEON, MIN, JO, LEE, SHIN, PARK, PARK, AND SHIN PHYSICAL REVIEW B 84, 165315 (2011) (a) (b)(c) (d) (e) FIG. 1. (Color online) (a) Schematic illustration of device geometry and measurement scheme. Inset: photomicrograph of the symmetric device consisting of ﬁve tunnel contacts ( a–e). (b) Spatial distribution of the induced spin accumulations ( /Delta1μ±) by spin injection ( V−<0) and extraction ( V+>0) without /with an applied transverse magnetic ﬁeld ( B). The arrows between ( x1,y1)a n d( x1,y2) represent the voltage drops by the tunnel contact ( x1,y1), the spin accumulation ( x1,y2), and part of Ge channel ( x1,y2). (c) High-resolution TEM image of the CoFe (5 nm) /MgO (2 nm) /n-Ge tunnel structure. The topmost Cr layer is a capping layer to prevent oxidation of the sample. Left: low-magniﬁcation TEM image of the structure. The zone axis is parallel to the [110] direction of Ge. Middle: in-situ RHEED patterns of the MgO and CoFe layer along the azimuths of Ge [110] and Ge [100], respectively. Right top: SAED covering the whole region of the contact. Right bottom: simulated diffraction pattern of CoFe(001) [100] /bardblMgO(001)[110] /bardblGe(001)[100] along the [110] direction of Ge. (d) J-Vcharacteristics of CoFe (5.0 nm) /MgO ( tMgO=1.5, 2.0, and 2.5 nm) /n-Ge tunnel contacts at 300 K. (e) Associated RA products (at the reverse bias voltages of −0.05,−0.15, and −0.25 V), estimated Schottky barrier heights ( /Phi1B) and depletion regions ( Wd) for the tunnel contacts using the conventional I-V-Tmethod, respectively. It has been argued that the observed Hanle spin signal comes from the LSs in Co /Al2O3/GaAs contact, which have a wide depletion region and large contact resistance.18In contrast, the recent report34studying the NiFe /Al2O3/Cs/n-Si contact, which has a narrow depletion region and small contactresistance, demonstrates that the spin polarization exists in thebulk bands of the SC rather than in LSs. These studies showedthat the measured spin signals are closely associated withthe contact characteristics, such as the width of the depletionregion ( W d) and the resistance area (RA) product. B. Structural and electrical characterization Figure 1(c) shows in-situ reﬂective high-energy electron diffraction patterns of the MgO (2 nm) layer and CoFe(5 nm) layer after annealing at 300◦C, low-magniﬁcation and high-resolution transmission electron microscope (TEM)images, and selected area electron diffraction (SAED) cover-ing the whole region of the CoFe (5 nm) /MgO (2 nm) /n- Ge tunnel structure. These in-situ and ex-situ structural characterizations conﬁrm the single-crystalline nature of thetunnel structure and the in-plane crystallographic relationshipof CoFe(001)[100] /bardblMgO(001)[110] /bardblGe(001)[100], exhibit- ing sharp interfaces in the (001) matching planes. Thiscrystalline tunnel structure with a fourfold in-plane crystallinesymmetry is desirable for efﬁcient spin injection with a highTSP via the symmetry-dependent spin ﬁltering effect of theMgO(001) barrier in conjunction with bcc FM. 7,35 Figure 1(d)shows the typical J-Vcharacteristics of the CoFe (5.0 nm) /MgO ( tMgO=1.5, 2.0, and 2.5 nm) /n-Ge tunnel 165315-2ELECTRICAL SPIN INJECTION AND ACCUMULATION IN ... PHYSICAL REVIEW B 84, 165315 (2011) (a) (b) (c) (d) FIG. 2. (Color online) (a) V oltage changes ( /Delta1V) vs transverse magnetic ﬁeld ( B) over the temperature range 200–300 K at the bias voltages of∓0.15 V (spin injection /extraction condition) for the CoFe /MgO (2 nm) /n-Ge contact. (b) V oltage changes ( /Delta1V)o fC o F e /Cr (tCr=0, 1.5 and 3.0 nm) /MgO/Ge contacts vs transverse magnetic ﬁeld ( B) at 300 K. (c) Electrical Hanle signals ( /Delta1V) and corresponding spin RA products (/Delta1V/J) across the CoFe /MgO/n-Ge tunnel contact as a function of a transverse magnetic ﬁeld ( B) at 300 K. Data are taken with the applied current of −14/+179μA, corresponding to V∓=∓0.15 V at B=0. The solid lines represent the Lorentzian ﬁts with τsf,∓=120/159 ps (V∓=∓0.15 V). (d) Normal ( /Delta1V normal ) and inverted Hanle ( /Delta1V inverted ) effects of the contact for perpendicular ( M⊥B,red) and in-plane ( M//B, blue) measurement, respectively. contacts with the electric resistivity ( ρ) of 7.5–9.5 m /Omega1cm and a moderate doping concentration ( nd)o f2 . 5 ×1018cm−3,w e l l below the metal-insulator transition (1.04 ×1019cm−3),25 at 300 K. As shown in J-Vcurves, a rectifying behavior is gradually reduced with increasing the MgO thickness, indicat-ing that the Schottky characteristics have been considerablysuppressed. For a quantitative analysis, we have estimatedthe RA product ( V/J), the Schottky barrier height (SBH, /Phi1 B) and the depletion width ( Wd) using the conventional I-V-Tmethod. The estimated values are shown in Fig. 1(e). In this ﬁgure, we see that a thicker MgO layer effectivelyreduces the SBH with the cost of increase of tunnel resistance.This result is fairly consistent with the Fermi-level depinning(FLD) mechanism 21–23in metal /insulator /Ge contacts. As a consequence, we have effectively tuned the energy-bandproﬁle of the CoFe /MgO/n-Ge contact by adjusting the MgO thickness (i.e. 2-nm MgO in our system) for the spin injectionand detection approach in moderately doped n-Ge at RT. III. RESULTS & DISCUSSION A. Electrical injection and detection of spin accumulation in Ge at 300 K The spin accumulation in the CoFe /MgO/n-Ge contact is measured by the voltage changes ( /Delta1V)a saf u n c t i o no fa transverse magnetic ﬁeld ( B) at the bias voltages of ∓0.15 V in the temperature range 200–300 K. As shown in the /Delta1V-B⊥plots [Fig. 2(a)], the tunnel contact clearly exhibits the negative MR with a Lorentzian line shape, indicating that the inducedspin accumulation in Ge by spin injection or extraction iseffectively detected. It is noteworthy to mention that the spintunnel contact with a small /Phi1 Bof 0.25 eV and a narrow Wd of 12 nm enables us to observe the spin signals with both forward and reverse bias polarities in the temperature range200–300 K. 16,36Albeit the signiﬁcant suppression of the SBH, the still-remaining Schottky barrier results in a resistive contactat low temperature and makes it difﬁcult to obtain enough /Delta1V signals below 200 K. B. Control experiment The anisotropic MR (AMR) of the FM is negligible in our experiment, since the resistance of the FM contact is at leasttwo orders of magnitude smaller than the tunnel resistance.The Lorentz MR (LMR) of the Ge channel cannot explainthis voltage change, since the resistance of the SC increaseswith the applied magnetic ﬁeld in the LMR effect. In order toexclude any artifacts caused by the stray ﬁeld near the edges ofthe FM, we have conducted the control experiments using theCoFe (5 nm) /Cr (t Cr=1.5 and 3.0 nm) /MgO (2 nm) /Ge tunnel contacts by inserting the nonmagnetic Cr between CoFe andMgO, 6which is effective to reduce the tunnel spin polarization without signiﬁcantly changing the stray ﬁeld (note that nosigniﬁcant changes of the structural and electrical propertieswere observed in the Cr-inserted tunnel contacts compared to 165315-3JEON, MIN, JO, LEE, SHIN, PARK, PARK, AND SHIN PHYSICAL REVIEW B 84, 165315 (2011) the tunnel contact without the Cr layer; see Appendix B). As shown in Fig. 2(b), a strong suppression of the MR signal is observed with increasing the Cr thickness ( tCr), verifying that the observed MR signals in the CoFe /MgO/Ge contact is purely originated from the spin accumulation. C. Estimation of spin accumulation, spin life time, spin diffusion length, and spin polarization in Ge Figure 2(c) shows the electrical Hanle signals ( /Delta1V)a sa function of a transverse magnetic ﬁeld at RT with the appliedcurrents of −14/+179μA, corresponding to V ∓=∓0.15 V atB=0. The most salient feature of Fig. 2(c) is clear and signiﬁcant Hanle signals obtained at RT for both conditions ofspin injection /extraction ( V ∓).A remarkable spin RA product (or spin signal, /Delta1V/J) as large as 170 k /Omega1μm2is obtained across the CoFe /MgO/Ge tunnel contact for the low bias voltage ( V−=− 0.15 V), which is an order of magnitude greater than that of Co /NiFe/AlO/n-Si contact.6 The estimation of the spin accumulation, spin lifetime, spin diffusion length, and spin polarization in Ge from themeasured spin signal strongly depends on a model describingthe tunneling process in the spin tunnel contacts. Taking intoaccount the narrow W d(∼12 nm) and the relatively small RA of the contact ( ∼3×10−5/Omega1m2at−0.15 V), two orders of magnitude smaller than that in Ref. 18, we have analyzed the measured results based on a single-step tunneling processinstead of the two-step tunneling process. 18The two-step tunneling could be possible as long as the interface and the SCbulk channel are sufﬁciently decoupled by a wide Schottkybarrier [see Eq. ( C5) in Appendix C]. A narrow depletion region might facilitate a single-step tunneling from Ge(CoFe)to CoFe(Ge) across the depletion region without loss of spinpolarization. 34Hence, the interface and the Ge bulk channel are directly coupled, which equalizes their spin accumulation[see Eq. ( C4) in Appendix C]. We have calculated the spin accumulation /Delta1μ +≈ (+)2.23 mV at the Ge interface from /Delta1μ+=(−2e)/Delta1V−/γ−, using the measured Hanle signal of /Delta1V−≈(−)0.78 mV . In this calculation, the TSP ( γ−) value of crystalline CoFe /MgO tun- nel contact was assumed to be 0.7,37because the experimental data for the TSP of the CoFe /MgO/Ge contact is not available; this TSP value is likely to be a higher bound. Assuminga parabolic conduction band and a Fermi–Dirac distributionfor each spin and using the calculated spin accumulation,/Delta1μ +≈(+)2.23 mV , we have determined the associated spin polarization in the Ge, n↑−n↓/n↑+n↓≈(+)4.4%, where n↑/n↓≈1.31×1018cm−3/1.20×1018cm−3are the density of spin up /down electrons.25We believe that spin polarization might be larger than ( +)4.4%, since we have used the highest value of γ=0.7. Using a Lorentzian ﬁt and taking an electron gfactor of 1.6 for the n-Ge, we have obtained the spin lifetime of τsf,≈120 ps ( V−=− 0.15 V) in moderately doped n-Ge at RT. Such a timescale is much smaller than the expectedspin lifetime (order of an ns) of conduction electrons inmoderately doped n-Ge from the Elliott–Yafet spin relaxation rate. 2,38,39However, we believe that the true spin lifetime may be longer than τsf,≈120 ps. According to a recent report,40 the local magnetostatic ﬁelds due to the ﬁnite roughness ofthe FM /oxide interface strongly reduce spin accumulation at the SC interface and artiﬁcially broaden the Hanle curve.As proven by the in-plane measurement ( M//B), showing the inverted Hanle effect [Fig. 2(d), blue], the interfacial depolarization effect is considered as a main origin of theunexpectedly broadened Hanle curve in this system. Hence, thetrue spin lifetime is expected to be longer, and its temperaturedependence is masked by the effect of the local magnetic ﬁelds[see Fig. 2(a)]. It should be noticed that the Hanle curve has a slightly broader width for the reverse bias ( V −=− 0.15 V , spin injection) than the forward bias ( V+=+ 0.15 V , spin extraction). The broadening effect of Hanle curves dueto the local magnetic ﬁelds can be quantiﬁed using aparameter /Delta1V inverted//Delta1V normal . As shown in Fig. 2(d),t h e /Delta1V inverted//Delta1V normal is more or less the same for both reverse and forward bias. This implies that the bias dependenceof the spin lifetime could be caused by other mechanisms,for example, unequal momentum scattering rates 38,39for the injected and extracted electrons or differences in the tunnelingprocess (see Section D). In addition, we have calculated the spin diffusion length l sf=/radicalbigDτsfin the Ge, where Dis the diffusion coefﬁcient [D≈38.9 cm2s−1at RT estimated from the Einstein relation using the mobility ( μ) vs doping concentration ( nd) relation].25 Withτsf,≈120 ps, we have obtained the corresponding spin diffusion length lsf,−≈683 nm at 300 K. This value is about three times larger than that of the electron spin diffusion length(230 nm) of the degenerate n-Si (As-doped, ρ=3m/Omega1cm). 6 D. Bias voltage dependence of spin signal The electrical Hanle signal ( /Delta1V) and the spin RA product (/Delta1V/J) of the CoFe /MgO/n-Ge contact show a strong bias dependence [Figs. 3(a) and3(b)]: those data are signiﬁcantly asymmetric with respect to the voltage polarity. The Hanlesignal increases gradually with the reverse bias ( V −<0, spin injection), but varies slightly with the forward bias ( V+>0, spin extraction). The spin RA product shows a similar biasdependence as reported in the Co /NiFe/Al 2O3/n-Si contact.6 In order to understand the asymmetric bias dependence of the spin signal (or spin RA product), we utilize theequation describing the spin signal at the Ge interface: 3,18 /Delta1V/J =γdγi/erch=γdγi/eρ/radicalbigDτsf.Here, γdis the TSP corresponding to the detection of induced spin accumulationat the Ge interface, γ i/eis the TSP of the injected /extracted electrons, and rchis the spin-ﬂip resistance associated with the Ge bulk channel. According to the above equation, the /Delta1V/Jis proportional toγdγi/e√τsfat a given temperature ( T), which depends on V.U s i n gt h e /Delta1V/Jvalues [Fig. 3(b)] and τsfvalues (not shown) extracted from the Lorentzian ﬁt, we have plotted theTSP 2(γdγi/e)v s Vat different temperatures to extract the bias dependence of TSP in Fig. 3(c), where the TSP2data is normalized by the maximum value at each temperature.Interestingly, TSP 2becomes independent of bias voltage for V−<0 [gray line in Fig. 3(c)], but decays exponentially for V+ >0 [black line in Fig. 3(c)]. With the assumption of γd=γi/e, the variation of TSP with Vis then obtained as γ−∝γoandγ+ ∝γoexp(−eV+/0.06). This is qualitatively similar to that of 165315-4ELECTRICAL SPIN INJECTION AND ACCUMULATION IN ... PHYSICAL REVIEW B 84, 165315 (2011) Spin injection (V<0) Spin extraction (V>0) -2.4-1.6-0.80.0 -2.4-1.6-0.80.0 10-1101103105 10-1101103105 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.310-410-2100 10-410-2100 300 K 250 K 225 K 200 KΔV (mV) Spin-RA (k Ω μm2) TSP2 (arb. units) Bias voltage (V)200 225 250 275 30010-1100101102103104105 Spin injectionV=-0.15 V Temperaure (K)ΔV/J (k Ω μm2) 10-210-1100101102103104Applied current (μA)(e) (a) (b) (c) CoFe n-Ge FE + TFE High temperature Jlocal (I/Alocal) >> Jav (I/Ageo) -e -e -e -e -e CoFe n-Ge FE Low temperature Jlocal (I/Alocal) >>> Jav (I/Ageo) -e -e (f) FIG. 3. (Color online) (a) Electrical Hanle signal ( /Delta1V), (b) spin RA product ( /Delta1V/J), and (c) TSP2(γdγi/e) with an applied bias voltage (up to±0.3 V) over the temperature range 200–300 K. (d) Comparison of the measured spin signals ( /Delta1V/J, rectangles) with the expected ones from the single-step ( rsss, circles) and two-step ( rsts, triangles) tunneling process. For this calculation, we have used the representative values of NLS∼5×1013eV−1cm−2for MgO /Ge contact,22NLS∼1×1014eV−1cm−2for Al 2O3/Cs/Si contact,34andNLS∼5×1012eV−1cm−2for SiO 2/Si contact.46The red, magenta, and cyan symbols represent our data, and the data taken from Ref. 34and Ref. 19, respectively. [The closed and open triangles represent calculated spin signals from the two-step tunneling using the measured spin lifetime and optimistic value ( ∼1n s ) , respectively.] (e) Temperature dependence of obtained spin signal ( /Delta1V/J) and applied current ( I) at the bias voltage of −0.15 V . (f) Schematic illustration for lateral inhomogeneity of tunneling current across the tunnel contact and its localization with the temperature decrease. FM/I/NM (nonmagnet) tunnel contacts.41,42The asymmetry of TSP observed in FM /I/NM contacts is mainly due to the intrinsic asymmetry of the tunneling process with respect tobias polarity: 41the electron tunneling out of the FM originates near the Fermi level with relatively large polarization [ V−<0, Fig.4(b)], whereas the electron tunneling into the FM faces hot electron states well above the Fermi level with signiﬁcantlyreduced polarization [ V +>0, Fig. 4(a)]. Therefore, the asymmetric bias dependence of the spin signal in our systemis understood in terms of the asymmetry of TSP caused bythe intrinsic asymmetry in these tunneling processes. 41 E. Comparison of obtained spin signal with existing drift-diffusion model It should be noticed here that the obtained spin signal [/Delta1V/J, red rectangle in Fig. 3(d)] for the reverse bias ( V−<0) is more than three orders of magnitude larger than the expectedvalue from the single-step tunneling [ r sss=γdγi/eρ/radicalbigDτsf, r e dc i r c l ei nF i g . 3(d)]. It is tempting to explain this discrepancy using a different tunneling model. For example, the unexpectedlarge spin signal was also found in Co /AlO/n-GaAs tunnel contact18at low temperature, which was explained by the contribution of the two-step tunneling process through the LSsnearby the SC interface (e.g. interface states at the oxide /SC, ionized impurities in the depletion region), where the LSsact as an intermediate stage for the spin injection ( V −<0) and absorb most of the spin polarization before they reachthe SC. However, the measured spin signal also shows a largediscrepancy with the spin signal estimated from the two-steptunneling [ r sts=γdγi/erLS=γdγi/eτsf/e2NLS, with NLS∼5×1013eV−1cm−2,22red triangle in Fig. 3(d)]T h e calculated spin signal from the two-step tunneling, even withan optimistic spin lifetime ( ∼1 ns), is still about one order of magnitude smaller than that of obtained spin signal [seeopen triangle in Fig. 3(d)]. Moreover, the two-step tunneling process cannot explain the exponential increase of our spinsignal [Fig. 3(e)] with the temperature decrease, as the two-step tunneling predicts only a modest increase of the spinsignal with decreasing the temperature from 300 to 200 K. Because of the limitation of the three-terminal Hanle measurements, the optical or nonlocal measurement of spin 165315-5JEON, MIN, JO, LEE, SHIN, PARK, PARK, AND SHIN PHYSICAL REVIEW B 84, 165315 (2011) (a) (b) (c) (d) FIG. 4. (Color online) (a) and (b) Schematic energy band diagrams for the CoFe /MgO/n-Ge tunnel contact incorporating the variation of depletion region under different bias regimes. Parabolic dispersion E(k) representing majority (red) /minority (blue) spin bands of the ferromagnet is displaced in the energy band diagram. (c) and (d) Associated spin accumulations near the n-Ge interface [localized states ( rLS), Ge bulk channel ( rch)]. (a)/(c) and (b) /(d) represent the forward ( V+>0, spin extraction) and reverse ( V−<0, spin injection) bias region, respectively. signals is required to unambiguously determine whether the observed spin signal in this system originates from the spinaccumulation in the Ge bulk channel or LSs. F. Underestimation of real /local current density A large deviation of the obtained spin signal ( /Delta1V/J) from those estimated from a single-step tunneling modelhas been also reported in the tunnel contacts on moderatelydoped Si [magenta 34and cyan19symbols in Fig. 3(d)].19,34 It has been argued that the unexpected large spin signal (/Delta1V/Jav) is mainly associated with the underestimation of real/local current density ( Jlocal),6not the LSs effect. The lateral distribution of tunneling current across the tunnelcontact is inhomogeneous with the variation of thickness andthe composition of the tunnel barrier 6[note that the contact resistance of CoFe /MgO/Ge is very sensitive to the MgO thickness, see Figs. 1(d) and1(e)]. Hence, the local current density ( Jlocal,I/A local), which induces the spin accumulation at the contact, is expected to be much larger than the averagecurrent density ( J av,I/Ageo) estimated from the geometrical contact area ( Ageo)[ s e eF i g . 3(f)].6Using this picture, we can also explain the exponential dependence of /Delta1V/JavonTin a consistent way. The electron transport in our contacts basically consists of the tunneling (orﬁeld emission, FE) and thermionic ﬁeld emission (TFE) withan SBH of 0.25 eV and a W dof 12 nm. As Tdecreases, the TFE process is strongly suppressed [see I-Tplot in Fig. 3(e)]. Hence, the electron tunneling is conﬁned within narrow paths with arelatively thinner tunnel barrier [Fig. 3(f)], since the tunnel transmission is exponentially dependent on the thicknessof the barrier. This conﬁnement results in the signiﬁcantincrease of the J local(Jlocal>>> J av) by several orders of magnitude. IV . CONCLUSIONS In conclusion, we have experimentally demonstrated the electrical spin accumulation in tunnel contacts consistingof crystalline bcc CoFe /MgO(001) /moderately doped n-Ge at RT, employing three-terminal Hanle measurements. Asizable spin signal of ∼170 k/Omega1μ m 2, spin polarization of ∼(+)4.4%, spin lifetime of ∼120 ps, and spin diffusion length of ∼683 nm are obtained at RT. We ﬁnd that the 165315-6ELECTRICAL SPIN INJECTION AND ACCUMULATION IN ... PHYSICAL REVIEW B 84, 165315 (2011) asymmetric bias dependence of spin signal is strongly related to the asymmetry of tunnel spin polarization. We expect thatour experimental ﬁndings will lead towards the interfaceengineering of FM /MgO/n-Ge systems for efﬁcient spin injection and detection, and eventually pave a way to realizeGe-based spintronics at RT. ACKNOWLEDGMENTS This paper was supported by the National Research Laboratory Program Contract No. R0A-2007-000-20026-0through the National Research Foundation of Korea fundedby the Ministry of Education, Science, and Technology, theKIST institutional program, and the KBSI Grant T31405 forYoung-Hun Jo. APPENDIX A: SAMPLE PREPARATION The single crystalline CoFe (5 nm) /MgO ( tMgO nm)/n-Ge (Sb-doped, ρ≈7.5–9.5 m /Omega1cm) tunnel structures were prepared by molecular beam epitaxy(MBE) system with a base pressure better than 2 × 10 −10torr. To obtain a clean and ﬂat surface, we have conducted the cleaning procedure combining ex-situ chemical cleaning and in-situ ion bombardment and annealing (IBA) process.20All layers were deposited by e-beam evaporation with a working pressure better than 2 ×10−9torr. We used a single crystal MgO source and rod-type CoFe with acomposition of Co 70Fe30.T h etMgO-nm MgO and 5-nm-thick CoFe layers were grown at 125◦C and RT, respectively, and then the samples were subsequently annealed in situ for 30 min at 300◦C below 2 ×10−9torr to improve the surface morphology and crystallinity. Finally, the sampleswere capped by a 2-nm-thick Cr layer at RT to preventoxidation of the sample. The ﬁnal sample structure was aCr (2 nm) /CoFe (5 nm) /MgO ( t MgO nm)/n-Ge(001). The symmetric device consisting of ﬁve tunnel contacts with lateralsizes of 30 ×100/20×100/20×100/150×100/150× 100μm 2was prepared by using microfabrication techniques (e.g. photolithography and Ar-ion beam etching)22for the electrical Hanle measurement. APPENDIX B: STRUCTURAL AND ELECTRICAL CHARACTERIZATION OF CHROMIUM-INSERTED TUNNEL CONTACTS The control experiment to exclude the artifacts caused by the stray ﬁeld should be based on a structurally and electricallyidentical sample, except the Cr insertion layer. In order toconﬁrm this, we have analyzed CoFe (5 nm) /Cr (t Cr= 0, 1.5, and 3.0 nm) /MgO (2 nm) /n-Ge samples by using in-situ reﬂective high-energy electron diffraction (RHEED) and conventional I-V-Tmeasurements for the structural and electrical characterizations, respectively. The Cr layers of CoFe /Cr/MgO/n-Ge samples were grown by e-beam evaporation at RT with a working pressure betterthan 2 ×10 −9torr. Except the insertion of a Cr layer, all layers were prepared under the same growth condition described inAppendix A. It should be noted that the Cr layer on MgO /Ge surface was not grown layer by layer because the Cr does notwet well on the MgO(001) surface due to the substantially large surface energy of Cr(001) (3.98 J /m 2) compared with that to the MgO(001) surface (1.16 J /m2).43,44Thus, RHEED patterns [Fig. 5(a)] of the CoFe(001) layers (with the surface energy of 2.55 J /m2)44grown on three-dimensional Cr /MgO/Ge surface show more distinct spot patterns than the CoFe layerg r o w no nM g O /Ge surface. However, after in-situ annealing at 300 ◦C, the surface morphology and crystallinity of the CoFe layers become comparable to each other, as exhibitedby the streaky patterns in Fig. 5(a). Although chemically inhomogeneous interface might be formed at the CoFe /Cr interface during the post-annealing process, it is known thatthe Fe grown on the Cr system does not show a signiﬁcantinterface alloying because the binding energy of the Cr layeris larger than that of the Fe adatoms. 45It is believed that interdiffusion /intermixing is not signiﬁcant in this system. The J-Vcharacteristics of CoFe (5 nm) /Cr (tCr=0, 1.5, and 3.0 nm) /MgO (2 nm) /n-Ge tunnel contacts [Fig. 5(b)] show quasi-Ohmic behaviors for the entire contacts at RT,except for more symmetric features in the Cr-inserted tunnelcontacts that might be expected due to the lower work functionof Cr (4.5 eV) than CoFe (4.75 eV). Moreover, using theconventional I-V-Tmethod, we have deduced the Schottky barrier height (SBH) of each contact. The SBHs estimated fromthe slope of the Arrhenius plots [In( I R/T2)−1/T) by the linear ﬁt at reverse bias of −0.15 V [Fig. 5(c)] are 0.25, 0.23, and 0.24 eV for the Cr thickness ( tCr) of 0, 1.5, and 3.0 nm, respectively. It indicates that the insertion of Cr layers does not affect majorelectrical features of the CoFe /MgO/n-Ge contact. As a result, we can rule out another possible origin for the strong suppression of the MR signal due to signiﬁcant changesof the structural and electrical properties of the tunnel contactsby the insertion of a Cr layer. APPENDIX C: EXISTING DRIFT-DIFFUSION MODEL To examine the possibility of a two-step tunneling process (or LSs effect) in our system, here, we adopt a model,18taking into account the two-step tunneling process through LSs (e.g.interface states at the oxide /SC, ionized impurities in the depletion region). According to the model, 18the spin accumulations in the Ge [LSs (/Delta1μ LS),n-Ge channel ( /Delta1μ ch)] and the magnetoresistance (/Delta1V/V) are expressed as: /Delta1μ LS≈2eγJrLS(rb+rch) rb+rLS+rch, (C1) /Delta1μ ch≈2eγJrLSrch rb+rLS+rch, /Delta1V V≈γ2 1−γ2/parenleftbiggrLS R∗ b+rb/parenrightbiggrb+rch rb+rLS+rch=γ2 1−γ2/parenleftbiggτsf τn/parenrightbigg (C2) with τsf≈τLS sfNchτLS →+(Nch+NLS)τch sf Nch(τLS→+τLS sf)+NLSτch sf, (C3) τn≈/parenleftBigg 1+Nchτch sf NLSτch sf+NchτLS→/parenrightBigg (τLS ←+τLS →), 165315-7JEON, MIN, JO, LEE, SHIN, PARK, PARK, AND SHIN PHYSICAL REVIEW B 84, 165315 (2011) (a) (b) (c) FIG. 5. (Color online) Structural and electrical characterizations of CoFe /Cr/MgO/Ge tunnel contacts. (a) Evolution of in-situ RHEED patterns during the growth processes of the CoFe (5 nm) /Cr (tCr=0, 1.5, and 3.0 nm) /MgO (2 nm) /Ge samples. The RHEED observations were carried out along the azimuths of Ge[110]. (b) J-Vcharacteristics of CoFe (5 nm) /Cr (tCr)/MgO (2 nm) /n-Ge tunnel contacts with the different Cr thickness of 0, 1.5, and 2.0 nm at 300 K. (c) Arrhenius plots [ln( IR/T2)−1/T] of the tunnel contacts with the different Cr thicknesses. where R∗ b=τLS ←/(e2NLS 3DdLS) is the spin-dependent tunnel resistance of the MgO layer, rb=τLS →/(e2NLS 3DdLS)i st h e bias-dependent leakage resistance between the LSs and the n-Ge bulk channel, and rLS/ch=τLS/ch sf/(e2NLS/ch 3DdLS/ch)a r e the spin-ﬂip resistances associated with these LSs or n-Ge bulk channel. Here, τLS/ch sf,NLS/ch 3D,anddLS/chare the spin lifetime, density of states per unit volume, and thickness ofeach layer, respectively. The τ LS ←/→represent the mean escapetimes of carriers from a LSs into the FM on the left ( ←)o r towards the n-Ge on the right ( →). The τsfis an (average) spin lifetime in the Ge (both LSs and Ge bulk channel) and τnis the (total) mean escape time from the LSs to the FM and the Gebulk channel after creation of spin-polarized carriers at the Ge interface. Here, N LS/ch=NLS/ch 3DdLS/chis the two-dimensional density of states integrated over the thickness of the LSs layeror Ge bulk channel. 165315-8ELECTRICAL SPIN INJECTION AND ACCUMULATION IN ... PHYSICAL REVIEW B 84, 165315 (2011) Forrb/lessmuchrch, when the decoupling between the interface and the SC bulk channel by a Schottky barrier is negligible(i.e. the Schottky barrier is thin enough to facilitate the directtunneling from a FM to SC), Eqs. ( C1), (C2), and ( C3) become as follows. Single-step tunneling ( r b/lessmuchrLS,rch/lessmuchrLS), /Delta1μ LS≈2eγJr ch,/Delta1 μ ch≈2eγJr ch, /Delta1V V≈γ2 1−γ2/parenleftbiggrch R∗ b/parenrightbigg =γ2 1−γ2/parenleftBigg τch sf (Nch/NLS)τLS←/parenrightBigg ,(C4) τsf≈τch sf,τ n≈(Nch/NLS)τLS ←.On the other hand, for rb/greatermuchrch, when the interface is sufﬁciently decoupled from the SC bulk channel by a Schottkybarrier (i.e. the Schottky barrier is too thick to directly tunnelfrom an FM to SC), Eqs. ( C1), (C2), and ( C3) should be considered as follows. Two-step tunneling ( r b/greatermuchrLS,rch/lessmuch rLS), /Delta1μ LS≈2eγJr LS,/Delta1 μ ch≈2eγJrLSrch rb, /Delta1V V≈γ2 1−γ2/parenleftbiggrLS R∗ b+rb/parenrightbigg =γ2 1−γ2/parenleftBigg τLS sf τLS←+τLS→/parenrightBigg ,(C5) τsf≈τLS sf,τ n≈τLS ←+τLS →. *Corresponding author: scshin@kaist.ac.kr 1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ´ar, M. L. Roukes, A. Y . Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). 2I.ˇZuti´c, J. 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PhysRevB.76.174414.pdf	High-energy magnetodielectric effect in kagome staircase materials R. C. Rai,1,J. Cao,1L. I. Vergara,1S. Brown,1J. L. Musfeldt,1D. J. Singh,2G. Lawes,3N. Rogado,4 R. J. Cava,5and X. Wei6 1Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA 2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032, USA 3Department of Physics, Wayne State University, Detroit, Michigan 48201, USA 4DuPont Central Research and Development, Experimental Station, Wilmington, Delaware 19880-0328, USA 5Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544, USA 6National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA /H20849Received 10 July 2007; published 7 November 2007 /H20850 We use a combination of optical spectroscopy, ﬁrst-principles calculations, and energy-dependent magneto- optical measurements to investigate the high-energy magnetodielectric effect in the frustrated kagome staircasecompound Co 3V2O8and develop structure-property relations in this family of materials. The optical spectra show two distinct Co on-site dtodexcitations that can be assigned as deriving from spine and cross-tie sites, respectively. The energy separation between these features is substantially larger in Co 3V2O8than in quasi- isostructural Ni 3V2O8, indicating that the spine and cross-tie crystal ﬁeld environments are more dissimilar in the Co compound compared with those in the Ni analog. Despite the similar appearance of the spectra, orbitalcorrelation effects seem to dominate the optical properties of Co 3V2O8, different from Ni 3V2O8. Through the 6.2 K ferromagnetic transition temperature, Co 3V2O8displays /H110112% dielectric contrast near 1.5 eV, larger than that observed in the static dielectric constant. Co 3V2O8also shows a high-energy magnetodielectric contrast of /H110112% near 1.4 eV at 30 T, smaller than that of Ni 3V2O8/H20849/H1101116% near 1.3 eV at 30 T /H20850. We attribute this result to the lack of strong lattice coupling at the low temperature magnetic phase boundaries. DOI: 10.1103/PhysRevB.76.174414 PACS number /H20849s/H20850: 75.80. /H11001q, 78.20.Ls, 71.20.Be, 75.30.Et I. INTRODUCTION Magnetoelectric effects have been extensively investi- gated in complex materials due to the intriguing physics andpossible applications. 1–8In particular, the coupling of mag- netic ﬁeld and dielectric properties in multiferroics are inter-esting and promising from the device standpoint. Severalrare earth manganites of the family RMnO 3andRMn2O5 /H20849R=Y,Tb,Dy,Ho /H20850, in which spin and lattice degrees of free- dom are intimately coupled, show signiﬁcant static magneto- dielectric effects.3,4,9–13Recent reports of high-energy magnetodielectric contrast in complex oxides such as inho-mogeneously mixed-valent K 2V3O8, frustrated multiferroic HoMnO 3, kagome staircase compound Ni 3V2O8, and several manganites are also important and demonstrate signiﬁcantcoupling between spin, lattice, and charge degrees offreedom. 14–21The large high-energy magnetodielectric effect in Ni 3V2O8/H20849/H1101116% at 30 T near 1.3 eV /H20850/H20849Ref. 15/H20850suggests that frustrated kagome staircase compounds are excellentmodel systems for mechanistic and structure-property inves-tigations. The M 3V2O8/H20849M=Mg,Ni,Co,Cu,Zn /H20850family of materials has several quasi-isostructural members, each with slightly different spin-orbit coupling and magneticanisotropies. 22Here, we use the term quasi-isostructural to indicate that although the space group and atom-atom con-nectivity is identical, there are small differences in the localstructure. Although the title compound, Co 3V2O8, does not display a ferroelectric phase, it provides an important oppor-tunity to explore structure-property relations and potentialtunability of the high-energy magnetodielectric response. Co 3V2O8displays an orthorhombic /H20849Cmca /H20850crystal structure23/H20849Fig. 1/H20850. It consists of layers of edge sharingCo2+O6octahedra separated by nonmagnetic V5+O4tetrahe- dra. Each unit cell contains 4 formula units /H20849f.u./H20850and two kagome layers of Co2+. Unlike in a planar kagome material, the CoO 6octahedra are buckled in the acplane, forming a staircase structure. Local symmetry considerations deﬁnetwo inequivalent Co 2+/H20849S=3/2 /H20850sites, which we refer to as “spine” and “cross-tie” sites. The Co spine centers form chains that run along the adirection. They are connected by Co cross-tie sites in the cdirection, forming nearly equilat- FIG. 1. /H20849Color online /H20850300 K crystal structure of Co 3V2O8. Co- balt occupies inequivalent spine and cross-tie sites. The three poly-hedra /H20849light color octahedra, CoO 6with cross-tie Co sites; dark color polyhedra, CoO 6with spine Co sites; and VO 4tetrahedra /H20850 indicate the packing arrangement /H20849Ref. 23/H20850.N i 3V2O8has a similar structure /H20849Ref. 23/H20850.PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 1098-0121/2007/76 /H2084917/H20850/174414 /H2084912/H20850 ©2007 The American Physical Society 174414-1eral triangles. Based on the Debye-Waller factors, Co 3V2O8 is softer than Ni 3V2O8, especially perpendicular to the chains.23 The magnetic properties of Co 3V2O8are anisotropic, with aandbdirections as the easy and hard axes, respectively.24,25 The zero-ﬁeld transport and neutron scattering data show that Co 3V2O8undergoes a transition from the paramagnetic state to an incommensurate antiferromagnetic state at/H1101111.3 K. A cascade of additional magnetic transitions is ob- served below 11.3 K. Two incommensurate and one com-mensurate antiferromagnetic states have been reported to ex-ist between 11.3 and 6.2 K, and there is a transition from theantiferromagnetic state to a weakly ferromagnetic state at/H110116.2 K. 26–30The complex H-Tphase diagram is due to com- peting magnetic interactions in the system.26–29,31Here, the different phases seem to be distinguished by the commensu-rability of the bcomponent of the spin density vector. While Co 3V2O8displays a small static dielectric anomaly at the 6.2 K transition, ferroelectricity has not been observed inany of the low temperature magnetic phases. 26,29,30Quasi- isostructural Ni 3V2O8also displays a rich H-Tphase dia- gram, different from that of Co 3V2O8.15,32–34The magnetic properties of this S=1 system are less anisotropic. Further, Ni3V2O8has a spontaneous ferroelectric polarization in- duced by the incommensurate magnetic order, which is inti-mately coupled to the magnetic properties. 32,33Muon spin resonance was used to study the local ﬁeld distributions inthe various phases of these compounds. 35Mixed kagome materials with formula of /H20849CoxNi1−x/H208503V2O8exhibit only one phase transition for high enough mixing.36 In order to investigate structure-property relationships in this family of frustrated kagome staircase materials, we mea-sured the optical and magneto-optical properties of Co 3V2O8 and compare the results to those of Ni 3V2O8. We comple- ment these measurements with ﬁrst-principles electronicstructure calculations, ﬁnding that Co 3V2O8has large crystal ﬁeld splitting and important orbital correlation effects. Thelatter is needed to account for both the small gap and thelarge orbital moment. The optical spectra show two distinctCo on-site dtodexcitations that can be assigned as deriving from spine and cross-tie sites, respectively. The energy sepa-ration between these features is substantially larger inCo 3V2O8than in quasi-isostructural Ni 3V2O8, indicating that the spine and cross-tie environments are more dissimilar inthe Co compound compared with those in the Ni analog.This is consistent with the larger distortion in Co 3V2O8com- pared with Ni 3V2O8. High-energy dielectric contrast of /H110112% is observed around the 6.2 K ferromagnetic transition tem-perature. The high-energy magnetodielectric effect is differ-ent. Co 3V2O8displays modest high-energy magnetodielec- tric contrast /H20849/H110112% near 1.4 eV at 30 T /H20850. This is smaller than that of quasi-isostructural Ni 3V2O8/H20849/H1101116% near 1.3 eV at 30 T /H20850, a result that we attribute to the softer lattice and the lack of strong lattice coupling at the low temperature mag-netic phase boundaries in Co 3V2O8. II. METHODS A. Crystal growth Single crystals of Co 3V2O8were prepared by combining K2CO 3,C o 3O4, and V 2O5in a 1.5:1:3 ratio. The mixture wasplaced in dense alumina crucibles and heated in a vertical tube furnace for an hour at 1100 °C. The melt was cooledslowly to 900 °C at 0.1 °C/min and left to cool in the fur- nace to room temperature. The dark colored platelike crystalswere then separated from the ﬂux. Typical crystal dimensionsused for our measurements were 5 /H110035/H110032m m 3. B. Spectroscopic investigations Near-normal reﬂectance of Co 3V2O8was measured over a wide energy range /H208493.7 meV–6.5 eV /H20850using three different spectrometers including a Bruker 113 V Fourier transform infrared spectrometer, a Bruker Equinox 55 Fourier trans-form infrared spectrometer equipped with an infrared micro-scope, and a Perkin Elmer Lambda 900 grating spectrometer.The spectral resolution was 2 cm −1in the far and middle infrared and 2 nm in the near infrared, visible, and near ul-traviolet. Polarizers were employed, as appropriate. For vari-able temperature studies, the sample was mounted on thecold ﬁnger of an open-ﬂow helium cryostat equipped with atemperature controller. Optical constants /H20849 /H92681and/H92801/H20850were calculated by a Kramers-Kronig analysis of the measuredreﬂectance. 37,38We deﬁne the dielectric contrast with respect to temperature as /H9004/H92801//H92801=/H20851/H92801/H20849E,T2/H20850−/H92801/H20849E,T1/H20850/H20852//H92801/H20849E,T1/H20850. The magneto-optical properties of Co 3V2O8were investi- gated between 0.75 and 4.1 eV using a 3/4 m grating spec-trometer equipped with InGaAs and charge-coupled devicedetectors and a 33 T resistive magnet at the National HighMagnetic Field Laboratory in Tallahassee, FL. Experimentswere performed with polarized light /H20849E /H20648aandE/H20648c/H20850in the temperature range between 5 and 18 K for applied magneticﬁelds up to 30 T /H20849H /H20648b/H20850. The ﬁeld-induced changes in the measured reﬂectance were studied by taking the ratio of the reﬂectance at each ﬁeld with the reﬂectance at zero ﬁeld, i.e.,/H20851R/H20849H/H20850/R/H20849H=0 T /H20850/H20852. To obtain the high-ﬁeld optical conduc- tivity /H20849 /H92681/H20850and dielectric response /H20849/H92801/H20850, we renormalized the zero-ﬁeld absolute reﬂectance with the high-ﬁeld reﬂectance ratios and recalculated /H92681and/H92801using Kramers-Kronig techniques.14,37We deﬁne the magneto-dielectric contrast as /H9004/H92801//H92801=/H20851/H92801/H20849E,H/H20850−/H92801/H20849E,0/H20850/H20852//H92801/H20849E,0/H20850. C. Electronic structure calculations First-principles calculations were carried out for Co 3V2O8 using several different techniques as enumerated below. The electronic density of states /H20849DOS /H20850was obtained for a collin- ear ferromagnetic arrangement of the Co spins using the fullpotential linearized augmented plane wave /H20849LAPW /H20850method with local orbitals, 39–41as implemented in the WIEN2K code.42LAPW sphere radii of 1.8 a0and 1.4 a0were used for the metal and O sites, respectively, along with well con-verged basis sets corresponding to RK max=7.5, where Ris the O LAPW sphere radius. The projections of the DOSshown are the projections of a given angular momentumcharacter within these LAPW spheres. III. RESULTS AND DISCUSSION A. Optical properties of Co 3V2O8 Figure 2/H20849a/H20850shows the polarized optical conductivity of Co3V2O8in the paramagnetic phase at 300 and 12 K. TheRAI et al. PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-2spectra show several strong directionally dependent vibra- tional and electronic excitations with an optical energy gapof/H110110.4 eV. Based on our electronic structure calculations of Co 3V2O8/H20849detailed below /H20850, those of quasi-isostructural Ni3V2O8, and comparison with chemically similar Co- containing compounds,15,43–45the excitations centered at /H110110.7 and 1.6 eV in the 12 K spectra are presumed to be Co dtodon-site excitations in the minority spin channel on cross-tie and spine sites, respectively, and will be referred toas such. These dtodexcitations are optically allowed due to the modest hybridization between the Co dand O pstates. The broad feature centered at /H110112.7 eV derives from a com- bination of O 2 pto Co 3 dand O 2 pto V 3 dcharge transfer excitations, and the /H110114.2 eV feature derives from O 2 pto V3dcharge transfer excitations. Figure 2/H20849b/H20850shows a close-up view of the optical conduc- tivity of Co 3V2O8near the Co /H20849spine /H20850dtodon-site excita- tions at 12, 8, and 5 K. This structure is only weakly sensi-tive to changes in the local crystal ﬁeld environment through the cascade of low temperature magnetic transitions, differ-ent from Ni 3V2O8where the Ni dtodon-site excitation associated with the spine site splits into ﬁve different com-ponents at low temperature. 15In particular, the oscillator strength does not change between the 12 K /H20849paramagnetic /H20850 and 8 K /H20849incommensurate antiferromagnetic /H20850phases. It is slightly enhanced at 5 K /H20849ferromagnetic phase /H20850likely due to a small local structural distortion around the Co /H20849spine /H20850cen- ter. The observation of a slightly different CoO 6environment is consistent with the recent report of a lattice distortion andsmall change in the static dielectric constant at the ferromag-netic transition temperature in Co 3V2O8.26,29It is interesting to compare the static dielectric results /H20849/H110110.3% dielectric contrast around the ferromagnetic transition temperature /H2085026,29 with the dielectric properties at higher energy. Figure 3dis- plays the real part of the dielectric constant of Co 3V2O8at 8 and 5 K for E/H20648aandE/H20648c. The inset of Fig. 3shows the dielectric contrast, /H9004/H92801//H92801, across the ferromagnetic phase boundary. The dielectric contrast around the ferromagnetictransition is as large as /H110112% near 1.5 eV, indicative of the spin-charge coupling in Co 3V2O8. The sign of the dielectric contrast is either positive or negative depending on the en-ergy. Figure 4/H20849a/H20850displays a comparison of the c-polarized op- tical conductivity of Co 3V2O8and Ni 3V2O8at 12 K, allow- ing us to explore the chemical structure-optical property re-lationships in this family of kagome staircase materials. Asanticipated for quasi-isostructural compounds, qualitativelysimilar electronic excitations are observed, although the cen-ter positions and splitting patterns of the cross-tie and spineCodtodon-site excitations are different. The energy sepa- ration between the spine and cross-tie excitations is substan-tially larger in Co 3V2O8than in quasi-isostructural Ni 3V2O8, indicating that the spine and cross-tie crystal ﬁeld environ-ments are more dissimilar in the Co compound comparedFIG. 2. /H20849Color online /H20850/H20849a/H20850Polarized optical conductivity of Co3V2O8at 300 and 12 K, extracted from reﬂectance measure- ments by a Kramers-Kronig analysis. The inset shows a close-upview of optical conductivity near the Co /H20849cross tie /H20850dtodon-site excitations. /H20849b/H20850A close-up view of the Co /H20849spine /H20850dtodon-site excitations at 12 K /H20849dotted green line /H20850,8K /H20849dashed purple line /H20850, and 5 K /H20849solid blue line /H20850, respectively.FIG. 3. /H20849Color online /H20850Dielectric constant of Co 3V2O8at 8 and 5 K for light polarized along the aandcdirections. The inset shows the dielectric contrast, /H9004/H92801//H92801=/H20851/H92801/H20849E,8 K /H20850 −/H92801/H20849E,5 K /H20850/H20852//H92801/H20849E,5 K /H20850, across the ferromagnetic phase boundary.HIGH-ENERGY MAGNETODIELECTRIC EFFECT IN … PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-3with those in the Ni analog. Figure 4/H20849b/H20850shows a close-up view of the Co /H20849cross tie /H20850and Ni /H20849cross tie /H20850dtodon-site excitations at 300 and 12 K. Both compounds show splittingof the cross-tie dtodexcitations below /H1101175 K, indicating that a weak structural distortion of the MO 6/H20849M=Co and Ni /H20850 building block in this direction precedes the low temperaturemagnetic transitions. The distortion is much stronger inCo 3V2O8, as evidenced by shoulders at 0.5 and 0.66 eV as well as a ﬁne structure centered at 0.7 eV. It is interesting to compare the aforementioned trends in the low temperature optical properties and the size of thehigh-energy magnetodielectric effect /H20849discussed below /H20850with direct measurements of the lattice. Figure 5displays the 300 K optical conductivity of both Co 3V2O8and Ni 3V2O8, highlighting the vibrational properties of these quasi-isostructural materials. Although a detailed analysis of themode patterns 46is beyond the scope of this work, we can assign many of the structures and connect the observed pat-terns with previously reported x-ray results 23to obtain a bet- ter picture of the magnetoelastic interactions in these mate-rials. For instance, we assign the peaks between 90 and 105 meV as deriving from the well-known triply degenerateVO 4asymmetric stretch.47The additional ﬁne structure is due to the symmetry breaking effects of incorporating theVO 4building block unit into a three-dimensional lattice, and the observed splitting is consistent with an a-cplane orien- tation of the tetrahedron. Focusing on the c-polarized modes of Co 3V2O8, we see that they are redshifted compared with those of the Ni analog, and the splitting is much larger, con-sistent with a softer, more distorted local environment aroundthe VO 4units. The structural environment of the transitionFIG. 4. /H20849Color online /H20850/H20849a/H20850Comparison of the c-polarized optical conductivity of Co 3V2O8and Ni 3V2O8at 12 K /H20849paramagnetic phase /H20850./H20849b/H20850A close-up view of the Co /H20849cross tie /H20850and Ni /H20849cross tie /H20850 dtodon-site excitations, at 300 and 12 K. FIG. 5. /H20849Color online /H20850300 K optical conductivity of Co 3V2O8 and Ni 3V2O8for light polarized along the a/H20849dotted line /H20850andc /H20849solid line /H20850directions, extracted from reﬂectance measurements by a Kramers-Kronig analysis. Panel /H20849a/H20850: stretching modes. Panel /H20849b/H20850: bending modes. The inset to panel /H20849b/H20850shows our use of heat capac- ity to estimate the Debye temperatures of these two kagome latticematerials. The dashed line and red symbols correspond to Co 3V2O8; the solid line and green symbols correspond to Ni 3V2O8. The dashed and solid lines correspond to our ﬁts, as discussed in thetext.RAI et al. PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-4metal ions, as determined by the diffraction measurements of Ref.23, is illustrated in Fig. 6. Normally, high spin Co2+has a modest /H20849t2gdriven /H20850Jahn-Teller distortion, while Ni2+is not Jahn-Teller active. The orthorhombic lattice already allowsthe Jahn-Teller distortion in the average structure and, infact, it may be seen that the distortions of the CoO 6octahe- dra in Co 3V2O8are signiﬁcantly larger than those of the corresponding octahedra in the Ni compound. In addition,further distortions in the local structure beyond the distor-tions in the average diffraction structure cannot be excluded.However, it should be noted in this regard that abnormallylarge O thermal parameters were not found in the reﬁnementof Ref. 23. In any case, the distortions of the CoO 6octahedra are expected to couple to the orbital moments of Co2+via the spin-orbit interaction and, in fact, we ﬁnd evidence for largeorbital moments and anisotropy in our calculations, dis-cussed below. Returning to the structural differences betweenthe Co and Ni compounds, we note that in addition to themore distorted octahedra of the Co compound, the V-O bondlengths are very slightly shorter in Co 3V2O8than in Ni3V2O8, while the Co-O bonds are on average slightly longer than the Ni-O bonds, consistent with the 0.055 Å dif-ference in ionic radii. While, based on their frequency rangeand similarity to modes in other compounds with VO 4tetra-hedra, the modes in the range 90–105 meV are associated with the VO 4tetrahedra, we note that there is a larger split- ting in the Co compound reﬂecting the fact that these arereally collective vibrations with O shared between the differ-ent transition metal sites. The reported larger thermal param-eters in the Co compound and lower speciﬁc heat Debyetemperature are consistent with our results and together in-dicate that the Co 3V2O8has a somewhat softer lattice than Ni3V2O8. Extending the structure analysis to the MO6octahedra, we observe a strongly c-polarized mode at /H1101177 meV in both compounds /H20849Fig.5/H20850. In the absence of baxis data,46there is little to learn from the Co-O-V /H20849or Ni-O-V /H20850motion. Bending modes /H20849discussed below /H20850have more to offer. Comparing the local structures /H20849Fig. 6/H20850, we see that the octahedra on the spine sites are more distorted than those on cross-tie sites;this is true for both compounds. Vibrational structures in the 20–35 and 35–55 meV range /H20851Fig. 5/H20849b/H20850/H20852are assigned to octahedral /H20849NiO 6and CoO 6re- lated /H20850and tetrahedral /H20849VO 4related /H20850bending modes, respec- tively. Despite the overall similarity of the vibrational pat-tern, the features associated with the octahedral bendingmodes are overall much softer in Co 3V2O8than in Ni 3V2O8. Since Co and Ni have nearly the same mass, the redshift ofCoO 6-related bending modes cannot be a mass effect. We conclude that Co 3V2O8has a softer, more ﬂexible lattice and, as a consequence, is likely to distort more strongly. This isconsistent with observations that the Debye temperature ofCo 3V2O8is smaller than that of Ni 3V2O8. The Debye tem- perature of Ni 3V2O8, determined by ﬁtting the heat capacity above the magnetic ordering transitions, but below /H9258D/10 toCp=/H9253T+/H9252T3,i s /H9258D=600 K /H20849/H9252=0.115 mJ/mole K4/H20850, whereas the Deybe temperature of Co 3V2O8is found to be /H9258D=550 K /H20849/H9252=0.152 mJ/mole K4/H20850. These ﬁts are shown in the inset to Fig. 5/H20849b/H20850. B. Electronic structure calculations of Co 3V2O8 The optical spectrum for Co 3V2O8is remarkable and poses a challenge for theory. In particular, it clearly showsthat Co 3V2O8is a small band gap /H20849/H110110.4 eV /H20850insulator with a spectrum very similar to that of Ni 3V2O8, regardless of the fact that Co has one less electron than Ni. In Ni 3V2O8, the band gap is of dtodcharacter and arises from the fact that the Fermi level lies in the crystal ﬁeld gap between minorityspin t 2gand egmanifolds in this narrow band Ni2+/H20849d8/H20850 compound.15This is not feasible in Co 3V2O8because of the different electron count. Experimentally, Co 3V2O8and Ni3V2O8share similar crystal structures,23small band gap insulating character, and complex /H20849but different /H20850ﬁeld- temperature phase diagrams. In the case of Ni 3V2O8, com- parison of optical spectra with local spin density approxima-tion /H20849LSDA /H20850and LDA+ Ucalculations showed an unexpected electronic structure. 15Ni3V2O8like NiO contains Ni2+ions in an octahedral O environment but unlike NiO has a spectrum that shows a small band gap between crystal ﬁeldsplit Ni minority t 2gvalence bands and minority egderived conduction bands. In fact, two peaks are seen in the opticalspectrum at low energy, one centered at /H110110.75 eV and the FIG. 6. /H20849Color online /H20850Comparison of bond lengths and angles for the 300 K crystal structures of Co 3V2O8and Ni 3V2O8,a sd e - termined by x-ray diffraction in Ref. 23. X-ray diffraction measures the average or bulk structure, whereas infrared probes the localstructure, which can be different from the average structure. Thecolor scheme for Co 3V2O8matches that in Fig. 1.HIGH-ENERGY MAGNETODIELECTRIC EFFECT IN … PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-5other at /H110111.35 eV. These two peaks were identiﬁed as the minority t2g-egexcitations. Based on the comparison with LSDA results, the crystal ﬁeld splitting on the Ni1 /H20849cross- link /H20850site is smaller than on the Ni2 /H20849spine /H20850sites, and accord- ingly the lower peak was associated with Ni1 and the higherone with Ni2. Addition of a Coulomb repulsion U, within an LDA+ Uframework, even for low values of U/H110115 eV, changes the electronic structure to a charge transfer insulatorwith a wide gap, similar to the physics in NiO, 48but in contradiction with experimental results for Ni 3V2O8. As mentioned, the optical spectrum of Co 3V2O8is quali- tatively very similar to that of Ni 3V2O8, showing a small band gap insulating behavior, which cannot be understood inthe same way as for the Ni compound. This is evident fromthe electronic structure obtained within the LSDA, as shownin Fig. 7. As may be seen from the DOS, the V occurs as V 5+, as might also be expected from the similar crystal struc- tures of Co 3V2O8and Ni 3V2O8. Therefore, the Co is nomi- nally Co2+and the Fermi energy lies in the minority spin t2g Co manifold, which yields a metallic behavior in contradic-tion with experiment at the LSDA level. The crystal ﬁeld scheme is similar to that of Ni 3V2O8, in which there are clearly deﬁned majority and minority /H20849t2gandeg/H20850manifolds, even for the ferromagnetic ordering. This reﬂects the narrowbands. In the Co 3V2O8case, the minority t2gmanifold con- tains two electrons and one hole. As usual in such cases, a gap can be produced by the LDA+ Umethod. This approach adds an ad hoc correction to the Kohn-Sham Hamiltonian that splits the occupied and un-occupied dstates. This favors integer orbital occupations and was used to successfully describe many properties of bothNiO and CoO. 48We did LDA+ Ucalculations, including spin orbit, with two values of Ueff=U−J, speciﬁcally Ueff =6 eV, which is a value appropriate for describing CoO,48 and a smaller value, Ueff=3 eV. Calculated densities of states are shown in Figs. 8and9, respectively. As expected, the dependence of the LDA+ Uspectra on the magnetization direction is weak, as may be seen from the comparison ofFig. 9with Fig. 10, which shows the density of states for magnetization along c. These LDA+ Ucalculations were done using the so-called self interaction correction /H20849SIC /H20850. 49,50 As may be seen, while both of these yield insulating states, neither of these electronic structures is similar to the LSDAelectronic structure of Ni 3V2O8and neither is compatible with the experimental spectrum, regardless of the magnetiza-tion direction. The calculations with U eff=6 eV yield a large gap, incompatible with the experiment, similar to what wasfound in such calculations for Ni 3V2O8.15Calculations with Ueff=3 eV, which is an unphysically small value, still yield a gap larger than the experiment. Additionally, the character ofthe gap is now different from Ni 3V2O8,a si ti sa t2gtot2g gap. This is because the LDA+ Umethod shifts all unoccu- pied dorbitals up by approximately the same amount. It is also notable that the crystal ﬁeld has been changed so thatthe larger crystal ﬁeld is now on the Co1 site, while in theLSDA it was on the Co2 site, similar to Ni 3V2O8. This result shows that correlation effects beyond the LSDA are needed to understand the electronic structure ofCo 3V2O8and that these correlation effects are not from the static on-site Coulomb repulsion, as described in the LDA-10-50510 -3 -2 -1 0 1 2N(E) / f.u. E(eV)Co1 d Co2 d-20-15-10-505101520 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd OpCod Vd t2g t2g egeg FIG. 7. /H20849Color online /H20850LDA density of states and projections onto the LAPW spheres for Co 2V2O8with ferromagnetic ordering on a per f.u. basis. Majority /H20849minority /H20850spin is shown above /H20849below /H20850 the horizontal axis. Spin orbit is included with the magnetizationdirected along the baxis; the DOS for aorcdirections of the magnetization is very similar. The bottom panel is a blowup aroundthe gap showing the crystal ﬁeld split Co projections.-15-10-5051015 -8 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd FIG. 8. /H20849Color online /H20850Density of states and projections as in Fig.7but using LDA+ U,Ueff=6 eV, applied to Co.RAI et al. PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-6+Umethod. Information about the nature of these correla- tions is provided by the magnetic properties. Like Ni 3V2O8, the phase diagram of Co 3V2O8is complex, but in contrast to Ni 3V2O8, the ground state is ferromagnetic.24,26,27,35Interestingly, however, the ordered moments for the two different sites in the ferromagnetic ground state, as determined by neutron diffraction, are ratherdifferent: 2.73 /H9262Bon the spine site /H20849Co2 /H20850and 1.54 /H9262Bon the cross-tie site /H20849Co1 /H20850.26This difference and the complex higher temperature orderings imply competing interactions andhave been modeled within an Ising picture with competingtemperature dependent exchange interactions. 26Another form of frustration that can be important in ferromagneticsystems is that which can arise due to competing magneto-crystalline anisotropies associated with different sites. 51,52 However, considering the actual noncubic, nontetragonal symmetry of the lattice, this may require large anisotropies toprevent a simple ordering if the exchange interactions are notfrustrated. In any case, the ordered ferromagnetic momentsare much smaller than the effective moments from the tem-perature dependence of the susceptibility: /H110115–6 /H9262Bper Co, with the implication that the Co ions have large orbital mo-ments in this compound. 24,27Large orbital moments, if present, would be consistent with large magneto-crystallineanisotropies arising from the spin-orbit interaction. Starting with the LSDA in a scalar relativistic approxima- tion, as used for Ni 3V2O8, we did calculations for a ferro- magnetic ordering and a ferrimagnetic ordering where theCo1 and Co2 sites are oppositely aligned. This calculation showed a strong ferromagnetic interaction between the spineand cross-tie spins at the LSDA level, with a calculated en-ergy difference of 0.33 eV/f.u. Calculations were also donewith the Perdew-Burke-Ernzerhof generalized gradientapproximation. 53Again, a ferromagnetic alignment was strongly favored, in this case with a lower energy of0.17 eV/f.u., relative to the ferrimagnetic ordering. Since Co 2+has a partially ﬁlled t2gshell, orbital moments and spin-orbit interactions are expected to be important. Assuch, we did calculations including spin orbit, for the ferro-magnetic case with magnetization directions along the threeCartesian axes, a,b, and c. At the LSDA level, small orbital moments are induced. These are parallel to the spin momentin agreement with Hund’s rules and vary according to thespin direction /H20849from 0.16 /H9262Bto 0.20 /H9262Bfor Co1 and from 0.15/H9262Bto 0.19 /H9262Bfor Co2 /H20850. Signiﬁcantly, the direction of magnetization for the maximum orbital moment is differentfor Co1 and Co2 /H20849candb, respectively /H20850, which shows that the different crystal ﬁeld environments of the two sites willlead to competing site anisotropies. This potentially providesa different mechanism for frustration and a complex mag-netic phase diagram from the competing exchange interac-tions discussed in Ref. 26. In this regard, it is interesting that the ordered moments seen in neutron scattering experimentsare very different for the two sites, 1.54 /H9262Bfor Co1 /H20849cross tie/H20850and 2.73 /H9262Bfor Co2 /H20849spine /H20850.26However, the LSDA cal- culations clearly do not describe the electronic ground stateof Co 3V2O8since they yield a metal in disagreement with-15-10-5051015 -8 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd -8-6-4-2024 -3 -2 -1 0 1 2N(E) / f.u. E(eV)Co1 d Co2 d FIG. 9. /H20849Color online /H20850Density of states and projections as in Fig.7but using LDA+ U,Ueff=3 eV, applied to Co. The bottom panel is a blowup around the gap showing the Co projections.-15-10-5051015 -8 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd -8-6-4-2024 -3 -2 -1 0 1 2N(E) / f.u. E(eV)Co1 d Co2 d FIG. 10. /H20849Color online /H20850Density of states and projections as in Fig.9but with the magnetization along the c-axis direction.HIGH-ENERGY MAGNETODIELECTRIC EFFECT IN … PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-7the experiment. This is due to the hole introduced into the minority t2gorbital in going from Ni2+to Co2+, as shown in Fig. 7. Aside from the position of the Fermi energy, this electronic structure is, in fact, quite similar to that ofNi 3V2O8, including the crystal ﬁeld gap between the minor- ityt2gandegstates and smaller splitting for the Co1 /H20849cross- link /H20850. Thus, it is tempting to associate the lower peak in the optical spectrum with the Co1 site. However, such a connec-tion cannot be made without removing the minority t 2ghole. One interesting feature is that the LDA+ Ucalculations lead to a strong enhancement of the orbital moments, which be-come noncollinear. For example, with U eff=3 eV and spin magnetization along c, the Co2 /H20849spine /H20850orbital moment is 0.87/H9262Bpointing close to /H20851101 /H20852. The strong orbital moments suggest an alternative way of obtaining an insulating ground state. This is the orbital po-larization correction of Brooks, 54Eriksson et al. ,55and Norman.56This amounts to a term added to the LSDA Hamiltonian to compensate for the underestimate of the cor-relations that leads to underestimated orbital moments and a weakened third Hund’s rule in the LSDA. This is of the formV OP=cOP/H20855Lz/H20856lz, where cOPis a parameter that can be calcu- lated or adjusted ad hoc andLzandlzare the projections of the total and single orbital momenta along the magnetizationdirection. This term represents a dynamic correlation correc-tion, which arises because electrons orbiting in the samesense can lower their Coulomb repulsion relative to electronsthat are counter-rotating and as such must frequently pass byeach other. While formally such a term is included in theexact density functional, it is difﬁcult to explicitly constructthe interaction from the spin densities since changing theorbital momentum of one of the Kohn-Sham orbitals wouldchange the orbital moment but would not change the spindensity apart from indirect effects, such as breathing of theorbital. The orbital polarization correction was originally derived using the Racah Bparameter appropriate for pstates but was applied with success to a number of dandfsystems. In the case of CoO, however, it was found that a larger correctionthan would be obtained from the ﬁrst-principles Slater inte-grals entering the Racah parameter was needed to obtain aproper insulating ground state. 56This may be justiﬁed, as the more complicated atomic expressions for dstates give a larger average correction,57and furthermore the double counting corrections /H20849i.e., what is included already in the LSDA /H20850, which can in principle be either positive or negative, are unknown. Here, we report calculations both using theﬁrst-principles value of the Racah parameter and also with anenhanced correction, where the parameter is treated as ad-justable in order to see what effect this term can have. In allcases, spin orbit was included, with various magnetizationdirections, and the orbital polarization correction was calcu-lated separately for the two spin channels. Since the majorityspin states of Co 2+are full, Lzis only signiﬁcant for the minority spin; this leads to a spin dependent orbital correc-tion, which is large only in the minority channel. While it can be seen that this approach will also yield a splitting of the t 2gmanifold, it differs from the LDA+ Uap- proach by the dependence on lz. Thus, unlike the LDA+ U approach, for which the unoccupied eglevels remain abovethe unoccupied t2glevels, a strong orbital polarization cor- rection can shift the level of the orbitally polarized t2ghole above the egbands and yield a spectrum like that of Ni3V2O8. As mentioned, we did calculations using the ab initio value of cOPand with larger values. With the ab initio value, we obtain enhanced orbital moments but do not obtain an insulating state. However, with enhanced values ofc OP, we can, as expected, obtain insulating ground states depending on the particular choice of the parameter and onthe magnetization direction. This is illustrated in Fig. 11, which shows the calculated DOS with magnetization along the three crystallographicaxes with an orbital polarization parameter of 0.5 eV /H20849the-20-15-10-505101520 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd -15-10-5051015 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd -15-10-5051015 -6 -4 -2 0 2 4N(E) / f.u. E(eV)Total Co1 d Co2 d Vd FIG. 11. /H20849Color online /H20850Density of states and projections as with an orbital polarization correction using a 0.5 eV parameter. Thethree panels show the DOS with the magnetization direction alonga/H20849top/H20850,b/H20849middle /H20850, and c/H20849bottom /H20850directions.RAI et al. PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-8ab initio value is 0.16 eV /H20850. A sizable gap appears for M/H20648c and a small /H110110.05 eV gap for M/H20648a, while only a pseudogap /H20849DOS minimum /H20850exists for M/H20648b. With an orbital polarization parameter of 0.3 eV, the gap for M/H20648cshrinks to 0.28 eV, while there are only pseudogaps for M/H20648bandM/H20648a. The calculated orbital moments are large and dependent on theorbital polarization parameter, as shown in Table I. Thus, in relation to the experimental data, none of the schemes tested is satisfactory. The LSDA /H20849and also general- ized gradient approximation /H20850produces a ferromagnetic ground state, in accord with experiment, but has muchsmaller orbital moments than those that are inferred fromsusceptibility data, and in addition the electronic structure ismetallic in contrast with the experiment. The LDA+ U method includes a parameter, which when chosen within theusual range for a 3 dtransition metal ion produces a spectrum with a wide gap, in disagreement with the experimental spec-trum. The orbital polarization approach can produce bothgaps and orbital moments in the experimental range but re-lies on the use of an arbitrarily enhanced parameter to do so.Clearly, further work is needed to understand the correlationeffects in Co 3V2O8in relation to the experimental data. However, some features are likely to remain. In particular,the combination of small gaps and large orbital momentssuggest a theory along the lines of the orbital polarizationcorrection. Within such a framework, the large orbital mo-ments would be expected to give strong magnetocrystallinecoupling of the moment directions to the lattice. This couldbe the source of nontemperature dependent but competinginteractions that might lead to a complex phase diagram andalso the large magnetocapacitive effects observed in the vari-ous phases in Co 3V2O8, but not Ni 3V2O8, as was already suggested.58This is also consistent with recent neutron scat-tering measurements, which show noncollinearity of the magnetic moments.29 C. High energy magnetodielectric properties of Co 3V2O8 Figure 12shows the energy-dependent magneto-optical response, R/H20849H/H20850/R/H20849H=0 T /H20850,o fC o 3V2O8at 5 K for H=0 and 30 T /H20849H/H20648b/H20850for light polarized along the aandcdirections. Since this is a normalized response, deviations from unity indicate ﬁeld-induced changes in the measured reﬂectance.TABLE I. Calculated orbital moments in /H9262B, within the LSDA with spin orbit and with the orbital polarization /H20849OP/H20850corrections using different values of the orbital polarization parameter. x,y, and zare Cartesian directions along a,b, and c, respectively. Co1 /H20849x/H20850 Co1 /H20849y/H20850 Co1 /H20849z/H20850 Co2 /H20849x/H20850 Co2 /H20849y/H20850 Co2 /H20849z/H20850 LSDA M/H20648a 0.15 0.00 0.00 0.18 0.00 −0.01 M/H20648b 0.00 0.15 0.00 0.00 0.19 0.00 M/H20648c 0.00 −0.01 0.20 −0.01 0.00 0.15 OP /H208490.16 eV /H20850 M/H20648a 0.40 0.00 0.00 0.53 0.00 −0.02 M/H20648b 0.00 0.33 0.01 0.00 0.62 0.00 M/H20648c 0.00 −0.04 0.86 0.00 0.00 0.47 OP /H208490.3 eV /H20850 M/H20648a 1.54 0.00 0.00 1.49 0.00 0.01 M/H20648b 0.00 1.24 −0.06 0.00 1.45 0.00 M/H20648c 0.00 −0.05 2.06 −0.09 0.00 1.99 OP /H208490.5 eV /H20850 M/H20648a 2.19 0.00 0.00 2.10 0.00 0.11 M/H20648b 0.00 2.10 −0.05 0.00 2.09 0.00 M/H20648c 0.00 −0.04 2.36 −0.10 0.00 2.32 FIG. 12. /H20849Color online /H20850The normalized magneto-optical re- sponse, R/H20849H/H20850/R/H20849H=0 T /H20850,o fC o 3V2O8at 5 K for H=0 and 30 T /H20849H/H20648b/H20850for light polarized along the aandc/H20849inset /H20850directions.HIGH-ENERGY MAGNETODIELECTRIC EFFECT IN … PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-9In a 30 T ﬁeld, the reﬂectance decreases by /H110111–2 % de- pending on the energy. We attribute the changes near/H110111.5 eV and /H110222 eV to ﬁeld-induced modiﬁcations of Co /H20849spine /H20850dtodon-site excitations and O 2 pto Co 3 dcharge transfer excitations, respectively. Note that these ﬁeld-induced changes in the reﬂectance are much smaller thanthose found in quasi-isostructural Ni 3V2O8.15 In order to correlate ﬁeld-induced changes in the reﬂec- tance with the optical constants, we combined the reﬂectanceratio results of Fig. 12with absolute reﬂectance measure- ments and a Kramers-Kronig analysis to extract the opticalconductivity and dielectric response. 38Figure 13displays the polarized optical conductivity of Co 3V2O8atH=0 and 30 T /H20849H/H20648b/H20850. Comparing the 0 and 30 T optical conductivities, we can conﬁrm that the aforementioned ﬁeld-induced changes in reﬂectance correspond to the ﬁeld-induced modiﬁcations ofthe Co /H20849spine /H20850dtodon-site excitations and O 2 pto Co 3 d charge transfer excitations. These changes are slightly largerat 5 K /H20849ferromagnetic phase /H20850compared with 18 K /H20849paramag- netic phase /H20850, an indication that the spin-charge coupling is stronger in the ferromagnetic phase. The reﬂectance ratiochanges discussed above also translate into the ﬁeld-dependent dielectric properties. The insets of Figs. 13/H20849a/H20850and 13/H20849b/H20850show the real part of the dielectric constant under simi- lar conditions. The magnetic-ﬁeld-induced modiﬁcations of /H92801are largest in the dispersive regime. We can calculate the magnetodielectric contrast as /H20851/H92801/H20849E,H/H20850−/H92801/H20849E,0/H20850/H20852//H92801/H20849E,0/H20850 =/H9004/H92801//H92801to see these effects more clearly. Figure 14/H20849a/H20850displays the high-energy magnetodielectric contrast of Co 3V2O8near the Co /H20849spine /H20850dtodon-site exci- tations at 5 K for H=30 T /H20849H/H20648b/H20850. The size and sign of the high-energy dielectric contrast, /H9004/H92801//H92801, depend on the en- ergy./H9004/H92801//H92801is as large as 2% /H20849at 5 K and 30 T /H20850near 1.25FIG. 13. /H20849Color online /H20850/H20849a/H20850Polarized optical conductivity of Co3V2O8at 5 K for H=0/H20849dotted line /H20850and 30 T /H20849solid line /H20850/H20849H/H20648b/H20850. The inset shows the dielectric response under similar conditions. /H20849b/H20850 Polarized optical conductivity of Co 3V2O8at 18 K for H=0/H20849dotted line /H20850and 30 T /H20849solid line /H20850/H20849H/H20648b/H20850. The inset shows the dielectric response under similar conditions.FIG. 14. /H20849Color online /H20850/H20849a/H20850A close-up view of the high-energy dielectric contrast, /H9004/H92801//H92801=/H20851/H92801/H20849E,H/H20850−/H92801/H20849E,0/H20850/H20852//H92801/H20849E,0/H20850,o f Co3V2O8near the Co /H20849spine /H20850dtodon-site excitation at 5 K for H=30 T /H20849H/H20648b/H20850. The inset shows a close-up view of dielectric con- trast near the same electronic excitation at 18 K for H=30 T. /H20849b/H20850 Dielectric contrast of Ni 3V2O8near the Ni /H20849spine /H20850dtodon-site excitation at 5 K for H=30 T /H20849H/H20648b/H20850. Note the substantially ex- panded yaxis. The inset shows a detailed view of the dielectric contrast near the charge transfer excitations under similarconditions.RAI et al. PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-10and 1.62 eV, although with opposite signs.59A similar but smaller magnetodielectric contrast is observed in the para-magnetic phase /H20851inset, Fig. 14/H20849a/H20850/H20852. 60As shown in Fig. 14/H20849b/H20850, the dielectric contrast of the quasi-isostructural Ni 3V2O8is signiﬁcantly larger /H20849/H1101116% near 1.3 eV at 30 T /H20850, a difference that is made manifested by subtle differences in the metalcoordination environment of the two compounds. 61The Co cross-tie center is particularly distorted compared to that inthe Ni analog. Although the magnitude is different, the high-energy magnetodielectric response of Co 3V2O8and Ni 3V2O8 demonstrate an appreciable interplay between the electronicand magnetic properties in this class of materials. Magnetoelastic coupling plays a major role in the magne- toelectric response of frustrated multiferroics. 9,62–68Based on these magnetodielectric studies, magnetoelastic coupling isalso important in the kagome staircase materials. High-energy magnetodielectric effects in Ni 3V2O8derive from ﬁeld-induced changes in the crystal ﬁeld environment aroundNi centers due to a modiﬁcation of the local NiO 6structure. Moreover, Ni 3V2O8is a local moment band insulator with an intermediate gap, and its electronic structure appears to favorstrong magnetodielectric couplings. 15For the case of Co3V2O8, however, we suggest that the local structure of CoO 6is substantially distorted at higher temperature, per- haps preventing the low temperature magnetic transitionsin Co 3V2O8from having a strongly coupled lattice component—a necessary condition to achieve large dielectriccontrasts. The larger Debye-Waller factors in Co 3V2O8com- pared with Ni 3V2O8,23the differences in local structure and vibrational properties, and our estimate of relative Debyetemperatures from speciﬁc heat are consistent with this pic-ture. Comprehensive vibrational studies are in progress totest this hypothesis. IV. CONCLUSION We measured the optical and magneto-optical properties of Co 3V2O8in order to probe structure-property relationships in the M3V2O8/H20849M=Co, Ni /H20850family of frustrated kagomestaircase materials. We assign excitations centered at /H110110.7 and 1.6 eV to Co dtodon-site excitations on cross-tie and spine sites. The energy separation between these features issubstantially larger in Co 3V2O8than in quasi-isostructural Ni3V2O8, indicating that the spine and cross-tie environ- ments are more dissimilar in the Co compound comparedwith those in the Ni analog. The large moment, small gapstate indicates that orbital correlation effects are important.Around the 6.2 K ferromagnetic transition temperature, thedielectric contrast of Co 3V2O8is/H110112% near 1.5 eV, much larger than the /H110110.3% change in the static dielectric con- stant. The broad features centered at /H110112.7 and 4.2 eV are assigned as O pto Co dand O pto V dcharge transfer excitations. Only a very slight change in the dielectric func-tion is observed through the ferromagnetic transition tem-perature in this higher energy range. The high-energy mag-netodielectric contrast of Co 3V2O8is/H110112% near 1.4 eV at 30 T, much smaller than that of Ni 3V2O8/H20849/H1101116% near 1.3 eV at 30 T /H20850. We attribute this difference to the lack of strong lattice coupling at the low temperature magneticphase boundaries in Co 3V2O8. Direct measurements of the lattice indicates that this difference is due to the more dis-torted coordination environment of the Co cross-tie centers. ACKNOWLEDGMENTS Work at the University of Tennessee is supported by the Materials Science Division, Basic Energy Sciences, U.S. De-partment of Energy /H20849DE-FG02-01ER45885 /H20850. Research at ORNL is sponsored by the Division of Materials Sciencesand Engineering, Ofﬁce of Basic Energy Sciences, U.S. De-partment of Energy, under Contract No. DE-AC05-00OR22725 with Oak Ridge National Laboratory, managedand operated by UT-Battelle, LLC. A portion of this researchwas performed at the NHMFL, which is supported by NSFCooperation Agreement No. DMR-0084173 and by the Stateof Florida. Work at Princeton University is supported byNSF through the MRSEC program /H20849NSF MRSEC Grant No. DMR-9809483 /H20850. We are grateful for helpful discussions with O. Eriksson, K. Hoon, and A. Litvinchuk. Present address: Physics Department, Buffalo State College, Buf- falo, New York 14222, USA. 1M. A. Subramanian, T. He, J. Chen, N. S. Rogado, T. G. Calva- rese, and A. W. Sleight, Adv. 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PHYSICAL REVIEW B 76, 174414 /H208492007 /H20850 174414-12
PhysRevB.82.024427.pdf	Magnetic interaction at an interface between manganite and other transition metal oxides Satoshi Okamoto Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6071, USA /H20849Received 15 May 2010; revised manuscript received 13 July 2010; published 27 July 2010 /H20850 A general consideration is presented for the magnetic interaction at an interface between a perovskite manganite and other transition metal oxides. The latter is speciﬁed by the electron number nin the d3z2−r2level as/H20849d3z2−r2/H20850n. Based on the molecular orbitals formed at the interface and the generalized Hund’s rule, the sign of the magnetic interaction is rather uniquely determined. The exception is when the d3z2−r2orbital is stabilized in the interfacial manganite layer neighboring to a /H20849d3z2−r2/H208501or/H20849d3z2−r2/H208502system. In this case, the magnetic interaction is sensitive to the occupancy of the Mn d3z2−r2orbital. It is also shown that the magnetic interaction between the interfacial Mn layer and the bulk region can be changed. Manganite-based heterostructures thusshow a rich magnetic behavior. We also present how to generalize the argument including t 2gorbitals. DOI: 10.1103/PhysRevB.82.024427 PACS number /H20849s/H20850: 73.20. /H11002r, 75.70. /H11002i I. INTRODUCTION Transition metal /H20849TM/H20850oxides have been one of the main subjects of materials science for decades. Experimental andtheoretical efforts are driven by their rich, complex, and po-tentially useful behaviors originating from strong correla-tions between electrons and/or electrons and lattices. 1The recent developments in the crystal-growth techniques, in par-ticular, the /H20849laser /H20850molecular-beam epitaxy, have made us recognize the opportunity to further control their behaviorsand to generate phenomena that are not realized in the bulksystems. 2–10 Here, we focus on the magnetic behavior at an interface between perovskite manganite and other TM oxides. Perov-skite manganites, especially La 1−xSrxMnO 3/H20849LSMO /H20850, are par- ticularly important because of their ferromagnetic /H20849F/H20850metal- lic behavior with relatively high Curie temperature TCand large polarization. Controlling the magnetic interaction at in-terfaces involving manganites would cause a technologicalbreakthrough for electronic devices using, for example, atunneling magnetoresistance /H20849TMR /H20850effect 11,12and an ex- change bias /H20849EB/H20850effect.13This requires the microscopic in- formation on the orbital states, not only on the spin states asdemonstrated for cuprate/manganite interfaces in Refs. 9and 14. However, it remains controversial whether the magnetic moment is induced in the cuprate region 7,14or the dead lay- ers appear in the manganite region.8,15 The difﬁculty dealing with interfaces involving strongly correlated electron systems comes from the small volumefraction which makes the experimental analysis challenging,and strong-correlation effects which hinder someof theoretical treatments. Therefore, if a Goodenough-Kanamori-type 16,17transparent description of the interfacial magnetic interaction becomes available, both experiment andtheory would greatly beneﬁt. In this paper, we present a general consideration for the magnetic interaction at an interface involving manganites.We ﬁrst focus on the interfacial interaction derived by d 3z2−r2 orbitals which have the largest hybridization along the z layer-stacking direction. We see that the sign of the magneticinteraction via the d 3z2−r2orbitals is naturally ﬁxed based on the molecular orbitals formed at the interface and the gener-alized Hund’s rule. The argument uses localized orbitals, and therefore shows only the qualitative trend. The molecularorbitals effectively lift the degeneracy between d 3z2−r2and dx2−y2orbitals by the order of the hopping intensity. In the second part, we perform the model Hartree-Fock calculation and show that the broken degeneracy can lead to the addi-tional change in the magnetic interaction between the inter-facial Mn layer and its neighboring Mn layer. We also dis-cuss how to generalize the molecular-orbital-based argumentfor more complicated situations including t 2gorbitals. II. MOLECULAR-ORBITAL PICTURE In this section, we consider the magnetic interaction be- tween manganite and other TM oxides focusing on the mo-lecular orbitals formed by d 3z2−r2orbitals which have the largest overlap at the interface. The TM region is speciﬁed by the number of electrons occupying a d3z2−r2orbital. Here, t2gelectrons are assumed to be electronically inactive and considered as localized spins when ﬁnite number of electronsoccupy t 2gorbitals. Generalization including these electrons will be discussed later. /H20849d3z2−r2/H208500system. Let us start from the simplest case, an interface between Mn and a /H20849d3z2−r2/H208500system /H20849Fig.1/H20850. In this case, the bonding /H20849B/H20850orbital is occupied by an electron whose spin is parallel to the localized t2gspin in Mn while the antibonding /H20849AB/H20850orbital is unoccupied. When there are other unpaired electrons in the /H20849d3z2−r2/H208500system at dx2−y2 and/or t2gorbitals, their spins align parallel to that of the Mn TM()0 32 2r zd− 2 23 r zd− 2 23 r zd−2 2y xd− gt2 FIG. 1. /H20849Color online /H20850Molecular orbital /H20849middle /H20850formed by d3z2−r2orbitals on Mn /H20849right /H20850and TM with the /H20849d3z2−r2/H208500conﬁgura- tion /H20849left/H20850.PHYSICAL REVIEW B 82, 024427 /H208492010 /H20850 1098-0121/2010/82 /H208492/H20850/024427 /H208496/H20850 ©2010 The American Physical Society 024427-1electron in the B orbital due to the Hund coupling. Thus, the F coupling is generated between Mn and /H20849d3z2−r2/H208500systems. This is equivalent to the double-exchange /H20849DE/H20850interaction originally proposed by Zener.18When dx2−y2is much lower in energy than d3z2−r2and in the interfacial Mn layer /H20849termed dx2−y2order /H20850,M na n d /H20849d3z2−r2/H208500systems are virtually decou- pled. Thus, the magnetic coupling is due to the superex- change /H20849SE/H20850interaction between t2gelectrons. This interac- tion is either F or antiferromagnetic /H20849AF/H20850depending on the orbital state and the occupancy of the TM t2glevel. /H20849d3z2−r2/H208501,2normal. This simple consideration can be eas- ily generalized to /H20849d3z2−r2/H208501and /H20849d3z2−r2/H208502systems. First we consider that the d3z2−r2anddx2−y2are nearly degenerate in the interfacial Mn layer and the unoccupied dx2−y2level in the TM region is much higher than the d3z2−r2level /H20849Fig. 2, top ﬁgures /H20850. We call this conﬁguration “normal” /H20849N/H20850conﬁgura- tion. In the lowest energy conﬁguration, B orbitals and theMnd x2−y2orbital are occupied by electrons. For the /H20849d3z2−r2/H208501 system, the F interaction is favorable as in the /H20849d3z2−r2/H208500sys- tem. On the other hand, for the /H20849d3z2−r2/H208502system, the down electron orbital is hybridized with the minority band in the Mn region. Thus, the “down” B orbital is higher in energyand has larger weight on the TM than the “up” B orbital.Because of the Hund coupling with the down electron in theB orbital, other unpaired electrons, if they exist in d x2−y2 and/or t2gorbitals, tend to be antiparallel to the Mn spin. Since dx2−y2orbitals are predominantly occupied in the interfacial Mn layer due to the B/AB splitting of d3z2−r2-based molecular orbitals, further stabilization of dx2−y2 orbitals in the Mn region, i.e., dx2−y2order, would not affect the interfacial magnetic coupling discussed here. But, this could reverse the magnetic coupling between the interfacialMn layer and the second Mn layer as discussed in the nextsection./H20849d 3z2−r2/H208501,2anomalous. When the dx2−y2level in the TM region becomes lower than the Mn dx2−y2level, the charge transfer occurs. We shall call this conﬁguration “anomalous” /H20849AN/H20850conﬁguration /H20849Fig. 2, middle ﬁgures /H20850. The electron transferred to the TM dx2−y2orbital has the same spin as the higher energy B orbital due to the Hund coupling /H20849indicated by arrows /H20850. Therefore, the sign of the magnetic coupling be- tween the Mn and /H20849d3z2−r2/H208501,2systems is unchanged. Note that this argument is applicable when the hopping probability between dx2−y2orbitals in the Mn and the TM regions is negligibly small. The ﬁnite hopping probability would make the charge transfer continuous. Furthermore,when the hopping probability becomes large, the DE inter-action is generated. Although the DE interaction throughd x2−y2bonds may not be realistic, it cooperatively stabilizes the F spin alignment for the /H20849d3z2−r2/H208501case while it competes with the AF tendency for the /H20849d3z2−r2/H208502case. Magnetic interactions discussed so far are insensitive to the electron density in the interfacial Mn because the inter-actions are mainly derived from the virtual electron excita-tion from the occupied d 3z2−r2orbital in the TM region to the unoccupied counterpart in the Mn region. Next, we consider that the d3z2−r2level is much lower than thedx2−y2in the interfacial Mn layer due to either the local Jahn-Teller distortion or compressive strain originating from the substrate /H20849Fig. 2, lower ﬁgures denoted by “JT” /H20850. The magnetic coupling in this case is sensitive to the electrondensity of the interfacial Mn. /H20849d 3z2−r2/H208501JT.When the Mn d3z2−r2density is close to 1, the SE interaction between the occupied d3z2−r2orbitals becomes AF. On the other hand, when the density is much less than 1, the F interaction between /H20849d3z2−r2/H208500conﬁguration on Mn and /H20849d3z2−r2/H208501becomes dominant. /H20849d3z2−r2/H208502JT.When the Mn d3z2−r2occupancy is close to 1, up electrons are localized on each sites because both B and AB molecular orbitals are occupied while down electronscan be excited or leaked from the /H20849d 3z2−r2/H208502system to the Mn minority level, i.e., down electron density is virtually re- duced in the TM region. As a result, unpaired spins, if theyexist in d x2−y2and/or t2gorbitals, become parallel to the up spin, i.e., F coupling. When the Mn d3z2−r2density becomes much less than 1, the up AB orbital becomes less occupied while keeping the occupancy of B orbitals relatively un-changed. Eventually, the down density in the TM region be-comes larger than the up density, and the magnetic couplingbetween the Mn and /H20849d 3z2−r2/H208502regions becomes AF. III. MODEL HARTREE-FOCK ANALYSIS In the previous section, we discussed the interfacial mag- netic coupling controlled by the molecular orbitals. Theseparation between B and AB molecular levels can becomeas large as the order of t, the hybridization between d 3z2−r2 orbitals along the zdirection. Since the interfacial Mn d3z2−r2 band is represented by the B /H20849AB/H20850d3z2−r2orbital for a /H20849d3z2−r2/H208500/H208491,2/H20850/manganite interface, the egdegeneracy is effec- tively lifted in the interface layer. This degeneracy lifting is expected to affect the magnetic interaction in the Mn region.Normal Normal()2 32 2rzd− 2 2y xd−()1 32 2rzd− Anomalous Anomalous()2 32 2rzd−()1 32 2rzd− JT JT()2 32 2rzd−()1 32 2rzd− FIG. 2. /H20849Color online /H20850Molecular orbitals formed by d3z2−r2or- bitals on Mn and the /H20849d3z2−r2/H208501system /H20849left column /H20850and the /H20849d3z2−r2/H208502system /H20849right column /H20850. In the normal /H20849anomalous /H20850con- ﬁgurations, d3z2−r2anddx2−y2orbitals are nearly degenerate in the interfacial Mn, and the unoccupied dx2−y2orbital in the neighboring TM is higher in energy /H20849lower in energy than the occupied Mndx2−y2/H20850. In the JT case, the d3z2−r2level is signiﬁcantly lower than the dx2−y2level. Black /H20849light /H20850lines indicate the level of major- ity/H20849minority /H20850spins. The up level and down level are exchange split resulting in the level scheme as indicated. The minority levels areneglected in the upper left two because these are irrelevant.SATOSHI OKAMOTO PHYSICAL REVIEW B 82, 024427 /H208492010 /H20850 024427-2In this section, we discuss this effect using the microscopic model calculation. We consider a two-band DE model given by H=/H20858 i/H9004ni/H9251−/H20858 /H20855ij/H20856ab/H20853tijabUijdia†djb+ H.c. /H20854+/H20858 iU˜ni/H9251ni/H9252 +J/H20858 /H20855ij/H20856S/H6023ti·S/H6023tj. /H208491/H20850 Here, an electron annihilation operator at site iand orbital a/H20851=/H9251/H20849d3z2−r2/H20850,/H9252/H20849dx2−y2/H20850/H20852is given by dia,nia=dia†dia, and the level difference between /H9251and/H9252is given by /H9004. We consider the large Hund coupling limit, in which the spin direction ofa conduction electron is always parallel to that of a localizedt 2gspin on the same site, and omit the spin index. Instead, the relative orientation of t2gspins is reﬂected in the hopping matrix; Uijis the unitary transformation representing the ro- tation of the spin direction between sites iand j. For sim- plicity, we only consider nearest-neighboring /H20849NN/H20850hoppings between Mn egorbitals via oxygen 2 pin the middle. Using the Slater-Koster scheme,19the orbital dependence of tijabis written as ti,i+z/H9251/H9251=4ti,i+x/H20849y/H20850/H9251/H9251=t,ti,i+x/H20849y/H20850/H9252/H9252=3t 4,ti,i+x/H20849y/H20850/H9251/H9252=ti,i+x/H20849y/H20850/H9252/H9251 =/H20849−/H20850/H208813t 4, and ti,i+z/H9251/H9252,/H9252/H9251=ti,i+z/H9252/H9252=0. The third term represents inter- orbital Coulomb interaction. Due to the egsymmetry, U˜is related to the intraorbital Coulomb interaction Uand the in- terorbital exchange integral JHasU˜=U−3JH. The last term represents the AF SE interaction between NN t2gspins /H20841S/H6023t/H20841 =3 2. From the optical measurements, the on-site interactions are estimated as U/H110113 eV and JH/H110110.5 eV.20The density- functional theory calculation provides t/H110110.5 eV.21Using the mean-ﬁeld analysis for the Néel temperature /H11011120 K of CaMnO 3, one estimates J/H110111 meV.22A similar value is ob- tained from the magnon excitations in the A-AF phases of50% doped Pr 1−xSrxMnO 3and Nd 1−xSrxMnO 3supposing that the AF interaction is due to the same J.23Thus, in what follows, we take U˜=3t. Considering some ambiguity, the realistic value for JSt2/tis expected to be /H110110.01–0.05. We analyze the model Hamiltonian, Eq. /H208491/H20850, using the Hartree-Fock approximation at T=0 focusing on the doped region /H20849carrier density Nfar away from 1 /H20850. In light of the experimental reports, we compare the energy of the follow-ing eight magnetic orderings: F ordering, planar AF orderingin which spins align ferromagnetically in the xy/H20849xzoryz/H20850 plane /H20851A/H20849A /H11032/H20850/H20852, chain-type AF ordering in which spins align ferromagnetically along the z/H20849xory/H20850direction /H20851C/H20849C/H11032/H20850/H20852, zigzag AF in which spins form ferromagnetic zigzag chainsin the xy/H20849xzoryz/H20850plane /H20851CE /H20849CE /H11032/H20850/H20852, and NaCl-type AF /H20849G/H20850. AtN→1, in addition to the spin symmetry breaking, orbital symmetry can be broken due to the SE mechanism in thepresent model. 24Since we are focusing on the metallic re- gime N/H110211, we do not consider such a symmetry breaking. The numerical results for the bulk phase diagram are pre- sented in Figs. 3/H20849a/H20850–3/H20849c/H20850. Here, all phase boundaries are of ﬁrst order, and those at small Ncan be replaced by canted AF phases or the phase separation between the undoped G-AFphase and doped F or AF phases. The overall feature is con-sistent with the previous theoretical reports. 25–27At/H9004=0,A-AF and A /H11032-AF /H20849C- and C /H11032-, CE- and CE /H11032-/H20850are degenerate but the degeneracy is lifted by the ﬁnite /H9004. We found the CE phase at U˜=/H9004=0 at JSt2/H114070.112 tandN/H114070.5 /H20849not shown /H20850as in the previous reports.28–30AtU˜=3t, the CE phase becomes unstable against A- and C-AF phases and appears only at the positive /H9004with JSt2/H114070.1t.JSt2/H110110.05treproduces the phase diagram of a high TCsystem such as LSMO and Pr1−xSrxMnO 3/H20849Refs. 31and32/H20850fairly well. The main effect of the level separation /H9004is changing the stability of planar-type AF /H20849Ao rA /H11032/H20850with respect to the chain-type AF /H20849Co rC /H11032/H20850and F states. In particular, the A-AF phase is stabilized by the positive /H9004more strongly than the C-AF phase by the negative /H9004. This is because the energy gain by the DE mechanism is favorable for the A-AF thanthe C-AF. The result is semiquantitatively consistent with theprevious report based on the density-functional theory. 27At /H9004=t, the boundary between F and A-AF phases is moved down to JSt2/H110110.02tat 0.3/H11351N/H113510.7. This behavior suggests that, when the d3z2−r2AB level for the /H20849d3z2−r2/H208501,2/manganite interfaces is about thigher than the Mn dx2−y2level, the mag- netic coupling between the interfacial Mn and the second Mn layers is switched to AF while retaining the intraplane Fcoupling. We conﬁrmed this behavior by computing the surface phase diagram considering F phase and two AF phases:A1/H208492/H20850where the surface layer /H20849and the second layer /H20850is an- tiferromagnetically coupled to its neighbor. We introduce0.000.050.100.15 A'CG F 0.000.050.10C GA FA 0.000.050.10 C'CE G A F 0.0 0.2 0.4 0.6 0. 80.000.05 A2A1 NF(b)∆=0 (c)∆=t(a)∆=−tJSt2/t (d)∆=ton surface … … FIG. 3. /H20851/H20849a/H20850–/H20849c/H20850/H20852Mean-ﬁeld phase diagrams of doped mangan- ites as a function of electron density Nand the AF interaction Jfor three choices of the level difference between egorbitals /H9004.A t/H9004 /H11021/H20849/H11022/H208500,d3z2−r2is lower /H20849higher /H20850in energy than dx2−y2. For nota- tions of the magnetic phases, see the main text. /H20849d/H20850Phase diagram for the 20-layer slab with /H9004=tin the surface layers. Nin this case corresponds to the mean electron density. For A1 and A2 phases,schematic spin alignments are also shown. A dashed line is thephase boundary between F and A-AF phases in the bulk calculation.MAGNETIC INTERACTION AT AN INTERFACE BETWEEN … PHYSICAL REVIEW B 82, 024427 /H208492010 /H20850 024427-3positive /H9004only on the surface layers in the 20-layer slab. As shown in Fig. 3/H20849d/H20850, a large part of F phase is replaced by A1 phase compared with the bulk phase diagram /H20849b/H20850./H20849Precise phase boundary requires detailed information of the surfaceor interface. /H20850Although the parameter regime is small, it is also possible that surface three layers are AF coupled whilethe other couplings remain F, A2 phase, before the wholesystem enters A-AF when Jis increased or Nis decreased. When the d 3z2−r2orbital is stabilized, the inter-Mn-layer cou- pling remains F but the intraplane F coupling is reduced. Therefore, in-plane canted AF structure may result for smallN. IV. SUMMARY AND DISCUSSION Summarizing, we presented a general consideration on the magnetic interaction between the doped manganite and othertransition metal oxides when an interface is formed. Usingthe molecular orbital formed at the interface and the gener-alized Hund’s rule, the sign of the magnetic interaction isdetermined /H20849Sec. II/H20850. The bonding/antibonding splitting of the molecular orbitals leads to the degeneracy lifting of e g orbitals on the interface Mn layer. Further, the bulk strain lifts the egdegeneracy. These effects control the magnetic interaction in the interfacial Mn plane and between theinterfacial Mn plane and its neighbor /H20849Sec. III/H20850. Considering these effects, we summarized the magnetic couplings in/H20849d 3z2−r2/H20850n/manganite interfaces in Table I. Although the present argument is rather qualitative, it is physically trans-parent and can be applied to a variety of systems. It is also straightforward to generalize the argument to include other orbitals. Therefore, the present argument will also help amore quantitative analysis with detailed information fromeither the experiment or the ﬁrst principle theory. It is worth discussing the implication of the present results to the real systems. An example of the /H20849d 3z2−r2/H208502system is high- Tccuprate. It has been reported that the magnetic cou- pling between YBa 2Cu3O7/H20849YBCO /H20850and La 1−xCaxMnO 3is AF,7andd3z2−r2anddx2−y2in the interfacial Cu have a similar amount of holes.9This corresponds to the AN situation. F coupling due to the DE remains in the Mn region because ofthe ﬁnite bandwidth of d x2−y2. An example of the /H20849d3z2−r2/H208501 system is BiFeO 3/H20849BFO /H20850. Recently, the EB effect was re- ported at BFO/LSMO interfaces accompanying the “AF”coupling between BFO and LSMO. 13We expect the N situ- ation with dx2−y2ordering at this interface. Although the in- terfacial coupling is F, the AF coupling between the interfa- cial Mn and the second Mn layers results in the AFalignment between BFO and bulk LSMO as observed experi-mentally and is responsible for the exchange bias effect. A question one may ask is what causes the “AN situation” in YBCO and the “N situation” in BFO? A qualitative expla-nation is as follow: in YBCO, the unoccupied Cu d x2−y2state is right above the Mott gap and its position is nearly identical to the occupied band of manganites.33Therefore, the charge transfer from Mn egto Cu dx2−y2can easily occur. On the other hand, the high-spin state is realized in BFO, and the unoccupied dx2−y2state with opposite spin with respect to the majority electrons is located far above the gap. In addition, the very close chemical potentials /H20849i.e. close d3z2−r2levels /H20850of BFO /H20849Ref. 34/H20850and LSMO /H20849Ref. 35/H20850maximize the B and AB splittings. This situation is favorable for the dx2−y2ordering in the interfacial Mn layer and the resulting AF coupling be- tween the ﬁrst and the second Mn layers /H20851see Fig. 3/H20849d/H20850/H20852. Finally, an example of the /H20849d3z2−r2/H208500system may be non- magnetic SrTiO 3, and the coupling with this is expected to affect the magnetic state near the interfacial Mn. For smalldoping xof LSMO, the coupling with SrTiO 3/H20849with the smaller lattice constant of SrTiO 3/H20850increases the d3z2−r2or- bital occupancy suppressing the inplane DE effect. For large doping, SrTiO 3creates the tensile strain stabilizing dx2−y2, and the out-of-plane F coupling is reduced.12Both are ex- pected to cause a more rapid decrease in the ordered momentwith increasing temperature than in the bulk region, 36result- ing in the rapid suppression of the TMR effect inLSMO /SrTiO 3/LSMO junctions.11,12The inplane /H20849out-of- plane /H20850spin canting may also be realized in the former /H20849latter /H20850.12For undoped LaMnO 3, the out-of-plane ferromag- netic coupling may result because the overlap between theoccupied d 3z2−r2in the ﬁrst Mn layer and the unoccupied dx2−z2ordy2−z2orbitals in the second Mn layer is increased, favorable for the F SE interaction between Mn layers. How- ever, since t2gorbitals in titanates are located near /H20849slightly above /H20850the Fermi level of manganite,33one may need to con- sider t2gorbitals more carefully as discussed below. Extension to t 2gsystems. In t2gsystems such as titanates, vanadates, and cromates, coupling between t2gorbitals could become as important as the coupling between egorbitals.TABLE I. Magnetic interaction at an interface between Mn and TM with the /H20849d3z2−r2/H20850nconﬁguration. The interfacial Mn is indicated by Mn /H208491/H20850and Mn in the second layer by Mn /H208492/H20850.M n /H208491/H20850-Mn /H208491/H20850 indicates intraplane interaction while the others interplane interac-tions. The TM-Mn /H208491/H20850interaction is based on the molecular-orbital picture presented in Sec. IIwhile the Mn-Mn interaction is based on the model Hartree-Fock study presented in Sec. III.A td x2−y2order, dx2−y2orbital is stabilized at the /H20849interfacial /H20850Mn layer. At F/H11569, Mn-Mn interaction is weak and the canted AF ordering may result.See the stabilization of the C-AF phase by the JT-type distortion/H9004/H110210 in Fig. 3/H20849a/H20850, the stabilization of the C- and A-AF phases by reducing the carrier density Nin Fig. 3/H20849b/H20850, and the stabilization of the A1 phase by the interfacial d x2−y2order in Fig. 3/H20849d/H20850. n Condition TM-Mn /H208491/H20850Mn/H208491/H20850-Mn /H208491/H20850Mn/H208491/H20850-Mn /H208492/H20850 0 N&J T F F/H11569F Nw / dx2−y2order AF F AF 1N F F F/H11569 Nw / dx2−y2order F F AF AN F F/H11569F/H11569 JT w/ N/H110111A F F/H11569F JT w/small N FF/H11569F 2N A F F F/H11569 Nw / dx2−y2order AF F AF AN AF F/H11569F/H11569 JT w/ N/H110111F F/H11569F JT w/small N AF F/H11569FSATOSHI OKAMOTO PHYSICAL REVIEW B 82, 024427 /H208492010 /H20850 024427-4Here, we discuss how to generalize the molecular-orbital ar- gument presented in Sec. IItot2gsystems. As an example, we consider an interface between titanate with the d0con- ﬁguration and manganites. Extending the argument to othersystems is straightforward. Figure 4/H20849a/H20850shows the level diagram of the titanate/ manganite interface including both e gand t2gorbitals. For simplicity, only bonding orbitals are presented. Because thed 3z2−r2level in titanate is far above the occupied levels in manganite and the unoccupied t2glevels in titanate and man- ganite /H20849down electrons for the latter /H20850are close,33highest oc- cupied molecular orbitals could be either B up d3z2−r2ordx2−y orbital /H20851Fig. 4/H20849b/H20850which is equivalent to Fig. 1/H20852or B down dxz,yzorbitals /H20851Fig. 4/H20849c/H20850/H20852. Note that the interfacial hybridiza- tion between dxyorbitals on Ti and Mn is much smaller than those between dxzand between dyz. In the case of Fig. 4/H20849b/H20850, induced moment in the titanate region is tiny but parallel to the moment in manganite region.On the other hand in the case of Fig. 4/H20849c/H20850, the induced mo- ment in the titanate region could be either parallel or antipar-allel to the manganite moment. This depends on the relativeweight of the down electron density in the B d xz,yzorbitals with respect to the up electron density in the B dxz,yzorbitals. The situation in Fig. 4/H20849c/H20850with antiparallel spin arrangement between titanate and manganite could happen when theoriginal electron density in the manganite egorbital is large andd3z2−r2level both in titanate and manganite is high due, for example, to the in-plane tensile strain. But in general, the difference between two conﬁgurations, Figs. 4/H20849b/H20850and4/H20849c/H20850 with either parallel or antiparallel spin conﬁgurations, wouldbe subtle. Therefore, depending on a variety of conditionsuch as the sample preparation, any situation could be real-ized. When the number of t 2gelectrons is increased, such as doped titanates, vanadates, and cromates, electrons tend toenter the down B d xz,yzorbitals and, then, the down B dxy, resulting in the antiparallel spin conﬁguration. However, theantiparallel conﬁguration becomes unstable against the par-allel conﬁguration when the electron number in t 2gorbitals becomes large and the level separation between t2gandeg orbitals, i.e., 10 Dq, becomes relatively small. In this case, because of the strong on-site Coulomb inter- actions, the energy gain by forming B orbitals becomes smallfort 2gelectrons and comparable to having electrons in both B and AB orbitals with the parallel spin conﬁguration. Theparallel conﬁguration further lowers the energy by formingd 3z2−r2B orbital and by the Hund coupling between the d3z2−r2B orbital and t2gelectrons on the t2gsystem, i.e., the Zener’s double-exchange ferromagnetism discussed in Sec. II. When there are more than three delectrons with relatively large 10 Dq, a low spin state is realized. In this case, three electrons enter B orbitals formed with t2gminority bands of Mn and remaining electrons enter AB orbitals formed witht 2gmajority band of Mn. Thus, the antiparallel conﬁgurations persist. Such a situation may be realized in, for example, aninterface between manganites and SrRuO 3in which Ru4+is in a low spin state with t2g4. This AF conﬁguration can be turned to the F conﬁguration when the double-exchange-typeinteraction becomes dominant due to the formation of d 3z2−r2 B orbital.37 So far, we have considered the ideal lattice structure in which orbitals with different symmetry do not hybridize. Inreality, bond angle formed by two transition metal ions andan oxygen ion in between becomes smaller than 180° allow-ing electrons to hop between orbitals with different symme-try. As a result, additional magnetic channels are generated.A simple argument presented in this paper can be generalizedto deal with such a situation. ACKNOWLEDGMENTS The author thanks P. Yu, R. Ramesh, J. Santamaria, and C. Panagopoulos for stimulating discussions and sharing the ex-perimental data prior to publication, J. Kuneš for discussion,and the Kavli Institute of Theoretical Physics, University ofCalifornia Santa Barbara, which is supported in part by theNational Science Foundation under Grant No. PHY05-51164, for hospitality. This work was supported by the Ma-terials Sciences and Engineering Division, Ofﬁce of BasicEnergy Sciences, U.S. Department of Energy.Mn2 23 rzd−2 2y xd− xyd yzxzd,xyd yzxzd, Ti Ti MnTi Mnor(a) (c) (b) FIG. 4. /H20849Color online /H20850Molecular orbitals formed by 3 dorbitals on Mn and Ti, originally d0. Here, only bonding orbitals are shown. /H20849a/H20850Full level diagram including both egandt2g. Black /H20849light /H20850lines indicate the level of majority /H20849minority /H20850spins. The highest occupied molecular orbitals and the magnetic alignment depend sensitivelyon the detail of the interface as shown in /H20849b/H20850and /H20849c/H20850./H20849b/H20850/H20851 /H20849c/H20850/H20852The up B orbital of d 3z2−r2orbital is lower /H20849higher /H20850in energy than the down B orbitals of dxz,yz.I n /H20849b/H20850, induced magnetic moment in Ti is parallel to Mn, i.e., /H20849d3z2−r2/H208500conﬁguration while in /H20849c/H20850, it depends on the relative occupancy of Ti dxz,yzorbitals in the up and down B orbitals. When the occupancy of the Ti down dxz,yzorbitals is larger than the up dxz,yzorbitals, net moment induced in Ti site becomes antiparallel to the Mn moment.MAGNETIC INTERACTION AT AN INTERFACE BETWEEN … PHYSICAL REVIEW B 82, 024427 /H208492010 /H20850 024427-51M. Imada, A. Fujimori, and Y . Tokura, Rev. Mod. Phys. 70, 1039 /H208491998 /H20850. 2M. Izumi, Y . Ogimoto, Y . Konishi, T. Manako, M. Kawasaki, and Y . Tokura, Mater. Sci. 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PhysRevB.73.045313.pdf	Spin ﬁltering through magnetic-ﬁeld-modulated double quantum dot structures P. Brusheim and H. Q. Xu† Division of Solid State Physics, Lund University, P .O. Box 118, S-221 00 Lund, Sweden /H20849Received 6 June 2005; revised manuscript received 23 November 2005; published 12 January 2006 /H20850 We report on a theoretical study of spin-dependent electron transport in double quantum dot structures, made from a two-dimensional electron gas, with a local magnetic ﬁeld modulation. Spin-dependent conductance andprobability density of electrons in the structures are calculated and the underlying physics of the results isdiscussed. We include in the study not only the magnetic ﬁeld component perpendicular to the two-dimensionalelectron gas plane, but also a consideration of the in-plane component of the magnetic ﬁeld. It is shown thatgiant spin polarization /H20849/H11011100% /H20850of the conductance, with tunable spin polarity, can be achieved with the double-dot structures. It is also shown that the structures can be used as efﬁcient spin ﬁltering devices attemperatures well above that deﬁned by the spin splitting energy. DOI: 10.1103/PhysRevB.73.045313 PACS number /H20849s/H20850: 73.40.Gk, 73.23.Ad, 72.25.Dc, 75.75. /H11001a The search for a source of spin-polarized electrons is an area of current active research.1–21The use of ferromagnetic metal /H20849FM/H20850-semiconductor /H20849SC/H20850junctions as spin polariza- tion devices has been widely considered. However, thesesystems suffer from rather poor spin-injection efﬁciency 4 arising from the conductance mismatch between the FM andSC materials. 5Recent experiments have demonstrated solu- tions of this problem by introducing diluted magnetic semi-conductors as a spin aligner 1,2,7,8and by adding a tunneling barrier.9,10In addition, spin polarization using hybrid FM-SC structures, in which the electron transport takes place onlyinside the semiconductor but can be inﬂuenced by the fringeﬁeld of the ferromagnetic material, has beenproposed. 12,13,15,16In particular, it was recently shown that by employing a FM stripe and a Schottky metal stripe on top ofa two-dimensional electron gas /H208492DEG /H20850formed in a SC het- erostructure, highly spin-polarized electron transport withtunable spin polarity can be achieved. 15This device relies on resonant tunneling through states formed between the doublemetal stripe induced magnetic and electrical barriers in the2DEG. Also, spin ﬁltering based on resonant tunnelingthrough Rashba spin-split levels in a triple barrier structure 17 and on coherent transport through Zeeman-split localizedstates in a double bend structure18has been proposed and studied. In the present work, we propose nanoscale spin ﬁltering devices by incorporating FM materials with SC double quan- tum dot structures. We consider devices made from a planardouble quantum dot structure in a SC heterostructure and astripe of FM material placed on top of the double quantumdot /H20849see the device schematics in Fig. 1 /H20850. Such a double-dot structure can be realized with a SC heterostructure by ad-vanced nanofabrication techniques. 22Here, to demonstrate the device principle, we consider two simple structures,realizable by etching techniques 23,24and/or by employing metal gate electrodes,25,26namely a structure which is formed by introducing a triple barrier in a quasi-one-dimensional /H20849Q1D /H20850channel as shown in Fig. 1 /H20849b/H20850and a structure which is deﬁned by three quantum point contacts/H20849QPCs /H20850in a series as shown in Fig. 1 /H20849d/H20850. Spin-dependent conductance and spin polarization of the double quantum dotstructures under a local magnetic ﬁeld modulation inducedby the FM stripe on top will be calculated. We will show thatgiant spin polarization /H20849/H11011100% /H20850of the conductance, with tunable spin polarity, can be achieved with the double-dot devices. We will further show that the structures can be used FIG. 1. Schematic illustration of /H20849a/H20850aF M stripe positioned on top of a 2DEG. /H20849b/H20850Top view of the FM stripe modulated triple-barrier device./H20849c/H20850Potential barrier and out-of-plane magnetic ﬁeld component proﬁle of the device in /H20849b/H20850./H20849d/H20850 Top view of the FM-stripe-modulated QPC-deﬁned double-dot device.PHYSICAL REVIEW B 73, 045313 /H208492006 /H20850 1098-0121/2006/73 /H208494/H20850/045313 /H208495/H20850/$23.00 ©2006 The American Physical Society 045313-1as efﬁcient spin ﬁltering devices at temperatures well above that deﬁned by the spin splitting energy. In the calculation, we assume that electrons transport along the xdirection and are conﬁned in the ydirection, and the electrical conduction takes place only in the SC hetero-structure. Under the application of a small in-plane magneticﬁeld, the FM stripe can be polarized along the transport di-rection. The fringe ﬁeld of the FM stripe 27leads to a nonho- mogeneous magnetic ﬁeld, B/H20849x/H20850=Bx/H20849x/H20850ex+Bz/H20849x/H20850ez,i nt h e planar double-dot region. The Hamiltonian of an electron in such a planar system under the single-particle effective massapproximation can be written as H=/H208731 2m/H20851p+eA/H20849x,y/H20850/H208522+Uc/H20849y/H20850+UE/H20849x,y/H20850/H20874/H92680 +1 2g/H9262B/H9268·B/H20849x/H20850, /H208491/H20850 where p,m, and gare the momentum, effective mass and effective gfactor of the electron, /H9262B=e/H6036/2meis the Bohr magneton /H20849meis the free electron mass /H20850,/H92680is the 2 /H110032 unit matrix, /H9268is the vector of the Pauli matrices, A/H20849x,y/H20850is the vector potential which, in the Landau gauge in the plane of the 2DEG, is given by A=/H208510,Ay/H20849x/H20850,0/H20852with Ay/H20849x/H20850 =/H20848−/H11009xBz/H20849x/H11032/H20850dx/H11032, and Uc/H20849y/H20850andUE/H20849x,y/H20850are the conﬁning po- tential in the leads and the potential that deﬁnes the double quantum dot, respectively. We take the spin quantization axisto be along the zaxis and assume a vanishing magnetic ﬁeld in the leads. For an electron of energy Einjected from the left lead in mode nand spin /H9268, the wave function, /H9023n/H9268/H20849x,y/H20850, and the transmission amplitude, tmn/H9268/H11032/H9268/H20849E/H20850, associated with the electron to be transmitted to the right in mode mand spin /H9268/H11032, can be calculated using scattering matrix methods.28–30In the linear response regime, the electron probability density of the sys-tem at the Fermi energy E Fis then found from /H9267/H9268/H20849x,y/H20850/H11011/H20858 n/H20841/H9023n/H9268/H20849x,y/H20850/H208412 kn, /H208492/H20850 and the zero-temperature conductance at EFfrom the Land- auer formula, G0/H20849EF/H20850=/H20858 /H9268/H11032/H9268G/H9268/H11032/H9268=e2 h/H20858 mn,/H9268/H11032/H9268/H20841tmn/H9268/H11032/H9268/H20849EF/H20850/H208412. /H208493/H20850 In the above two equations, the sums of mandnare taken over all propagating modes in the leads. The conductance atﬁnite temperature Tis given as a convolution of the zero- temperature conductance and the thermal broadening func- tion, G/H20849E F,T/H20850=/H208480/H11009dEG0/H20849E/H20850/H20849−/H11509f0//H11509E/H20850, where f0/H20849E/H20850=/H208531 +exp /H20851/H20849E−EF/H20850/kBT/H20852/H20854−1is the Fermi-Dirac distribution func- tion. The spin polarization can now be deﬁned as the ratio between the normalized spin conductance and the normal-ized total conductance 19,20at the Fermi energy, Px+iPy=2e2/h G/H20858 mn,/H9268tmn↓/H9268/H20849tmn↑/H9268/H20850,Pz=G↑↑+G↑↓−G↓↓−G↓↑ G. /H208494/H20850 It is seen that the components Px,ycan only be ﬁnite when spin ﬂipping processes are involved, while Pzcan be ﬁnite for spin-conserved transport. In our calculations we take the material parameters to be g=15 and m=0.024 mecorresponding to an InAs quantum well system. For the devices to be studied in this work, theFM stripe is assumed to have a width of d x=80 nm, a thick- ness of dz=40 nm, and a magnetization strength of /H92620M =3 T, which can be achieved with, e.g., Fe 16N2ﬁlms,31,32 and to be located z0=40 nm on top of the quantum well. For simplicity, the lateral conﬁnement potential in the SC hetero-structure is assumed to be the hard-wall type and the wholequantum region is taken to be 100 nm in the transverse di-rection and 200 nm in the transport direction /H20849see Fig. 1 for details /H20850. As a ﬁrst simplifying approximation we will only con- sider the magnetic ﬁeld component perpendicular to theplane, i.e., B x=0. With this approximation, commonly used in the literature,12,13,15,16only spin-conserved conductances remain to be nonzero. We will later show how the results areaffected by the inclusion of the in-plane ﬁeld component.The nature of spin-polarized electron transport for the de-vices under consideration /H20849with B x=0/H20850relies on resonant tun- neling through spin-dependent molecular states formed in thedouble dot. Within each dot, spin-split bound states areformed. These states will interact with bound states of thesame spin on the other dot to form molecular states. It can beshown that when the double-dot structure is symmetric under the operation of space inversion Rˆ xRˆy, where Rˆx/H20849Rˆy/H20850is the reﬂection operator, x→−x/H20849y→−y/H20850, and at the same time the fringe ﬁeld distribution of the FM stripe in the double-dot region is antisymmetric under the operation of Rˆx, the Hamil- tonian of the system is invariant under the operation of TˆRˆxRˆy, where Tˆ=−i/H9268yKˆis the time-reversal operator with Kˆ being the complex conjugation operator and /H9268ythe Pauli spin matrix. Since the operation of TˆRˆxRˆywill map a spin-up /H20849spin-down /H20850state into a spin-down /H20849spin-up /H20850state,15this symmetry implies that the double-dot states are spin degen-erate. Thus no spin polarization can occur in the linear-response conductance of the symmetric double-dot system.However, the symmetry can be broken easily, and spin po-larization can thus be achieved, by making the double-dot structure asymmetric under Rˆ xorRˆy. Here we investigate two simple cases, in which the double-dot structure is no longer symmetric under the operation of Rˆx, in order to dem- onstrate the device principle. We ﬁrst consider the device as shown in Fig. 1 /H20849b/H20850,i n which the double quantum dot was formed by introducing atriple barrier in a Q1D channel. Figure 1 /H20849c/H20850shows the elec- trical potential, U E/H20849x/H20850, assumed for the triple-barrier structure inside the Q1D channel and the zcomponent of the fringe ﬁeld of the FM stripe, Bz/H20849x/H20850, calculated using a formula given in Ref. 27. For the triple-barrier potential as shown by the thin solid line in Fig. 1 /H20849c/H20850, the system is invariant underP. BRUSHEIM AND H. Q. XU PHYSICAL REVIEW B 73, 045313 /H208492006 /H20850 045313-2the operation of TˆRˆxRˆy. Thus no spin-polarized transport can occur. Figure 2 /H20849a/H20850shows the calculated conductance of such a symmetric double-dot system with the triple barrier ofheight 5.79 meV. Here, only the calculation for the spin-upelectron conductance is plotted. The calculated result for thespin-down electron conductance looks exactly the same. Twoenergy-resolved conductance peaks at E=4.02 and 4.64 meV can be seen. These conductance peaks correspond to thetransmission through spin-degenerate lowest bonding andanti-bonding states of the double-dot system. Note that con-ductance peaks arising from the transmission through otherspin-degenerate bonding and anti-bonding states of thedouble-dot system can also be seen at higher energies. Various degrees of breaking of the TˆRˆ xRˆysymmetry and, thus, of spin-polarized electron transport can be achievedwith the double-dot system by, e.g., altering the height of theright-most potential barrier with the use of a Schottky gate.Figure 2 /H20849b/H20850shows the calculated conductances of spin-up /H20849solid line /H20850and spin-down /H20849dashed line /H20850electrons for the double-dot system with the right-most barrier set at a heightof 9.65 meV. It is seen that conduction peaks of spin-up andspin-down electrons split. It is also seen that the conductancepeaks have different heights. In particular, the conductancepeak corresponding to the transmission through the lowestspin-down bonding state is strongly suppressed. This is be-cause this bonding state is largely localized in the left dotregion /H20851see the solid line in Fig. 2 /H20849c/H20850/H20852and therefore has coupled to the left and right leads with very differentstrengths. However, the lowest spin-down antibonding stateis more localized in the right dot region /H20851see the dashed line in Fig. 2 /H20849c/H20850/H20852and therefore its coupling strengths to the left and right leads are of less difference. Thus, the conductancepeak of spin-down electrons corresponding to the transmis-sion through this antibonding state is only slightly reducedfrom the value of e 2/h.33For spin-up electrons, both the lowest bonding and antibonding states are more evenly lo-calized in the two dots, and have therefore a much weakercoupling to the right lead than to the left lead, due to the factthat the right-most potential barrier is now much higher thanthe left-most potential barrier. As a result, the conductancepeaks of spin-up electrons corresponding to the transmissionthrough the lowest spin-up bonding and antibonding statesare reduced. The spin splitting of the conductance peaks asshown in Fig. 2 /H20849b/H20850leads to spin polarized electron transport. The spin polarization, P z, of the conductance of the double- dot system as a function of the Fermi energy is displayed inFig. 2 /H20849d/H20850. Here it is seen that a large spin polarization /H20849/H11011100% /H20850of the conductance can be achieved with the double-dot system. For applications, the performance of the device at ﬁnite temperatures is of interest. At T=0 K the conductance peaks have a ﬁnite width due to the fact that thedot states have a ﬁnite coupling strength to the leads. Thesmearing of the Fermi distribution function at increasingtemperatures will further broaden the conductance peaks,thus diminishing the peaks in the polarization, as can be seenin Fig. 2 /H20849d/H20850. However, because of the relative difference in magnitude of the spin-up and spin-down conductance peaks,a remnant spin polarization is observable at T=5 K. This is a very interesting result, which suggests that the proposed FM-stripe modulated double-dot structure can be used as a spinpolarization device at temperatures much higher than thecritical temperature T cdeﬁned by the spin splitting energy Es in the device /H20849note that Tc=Es/kB/H110151.5 K for the device with parameters considered in Fig. 2 /H20850. We now consider spin-polarized electron transport through a structurally different but conceptually equivalentsystem as schematically illustrated in Fig. 1 /H20849d/H20850. Here, again, the perpendicular, antisymmetrical magnetic ﬁeld compo-nent, B z/H20849x/H20850, created by the FM stripe located 40 nm above the 2DEG is taken into account and the in-plane component, Bx, is neglected, as a ﬁrst simpliﬁed approximation. The double-dot structure is, however, implemented by threequantum point contacts /H20849QPCs /H20850in series. The transverse width of the right dot is made to be variable. This variation,characterized by parameter c, introduces breaking of the TˆRˆ xRˆysymmetry in the system. Figures 3 /H20849a/H20850and 3 /H20849b/H20850show the zero-temperature conductance of spin-up and spin-downelectrons at different values of c. Here, energy separation of the spin-up and spin-down conductance peaks is clearly ob-served. At c=20 nm a complete separation of the spin-up and spin-down conductance peaks is found /H20851Fig. 3 /H20849b/H20850/H20852and thus the spin polarization of the conductance approaching 100%can be realized with the use of the double-dot system at zerotemperature /H20851Figs. 3 /H20849d/H20850/H20852. Also, as in the previous device, the relative difference in the magnitude of the spin-up and spin-down conductance peaks leads to the appearance of spin po-larizations of the conductance at a temperature much higherthan the critical temperature deﬁned by the spin splitting FIG. 2. /H20849a/H20850Zero-temperature spin-up and spin-down conduc- tance of the symmetrical double-dot device as shown in Figs.1/H20849a/H20850–1/H20849c/H20850with a barrier height of 5.79 meV and an antisymmetrical magnetic ﬁeld modulation B z/H20849x/H20850./H20849b/H20850Zero-temperature conductance spectra of spin-up /H20849solid line /H20850and spin-down /H20849dashed line /H20850electrons for the same device as in /H20849a/H20850but with the right-most barrier set at a height of 9.65 meV. /H20849c/H20850Probability density of the bonding /H20849solid line/H20850and antibonding /H20849dashed line /H20850spin-down states at their corre- sponding conductance-peak energies plotted along the center lineover the device. The dash-dotted line in /H20849c/H20850shows the barrier po- tential proﬁle in the device to guide the eye. /H20849d/H20850Calculated spin polarization P zfor the device in /H20849b/H20850atT=0 K /H20849thick solid line /H20850, 0.5 K /H20849dashed line /H20850,1K /H20849dash-dotted line /H20850,2K /H20849dotted line /H20850, and 5K /H20849thin solid line /H20850.SPIN FILTERING THROUGH MAGNETIC-FIELD- … PHYSICAL REVIEW B 73, 045313 /H208492006 /H20850 045313-3energy in the device. An interesting new feature seen in Figs. 3/H20849c/H20850and 3 /H20849d/H20850is that the spin polarizations have opposite signs at the two values of c. Thus in this device both the amplitude and polarity of the spin polarization of the con-ductance can be tuned by variation of the structure parameterc. Now we discuss how the above results are affected by the inclusion of the in-plane component of the fringe ﬁeld of theFM stripe. The in-plane fringe ﬁeld component has a proﬁleof even parity, B x/H20849x/H20850=Bx/H20849−x/H20850, and will introduce an addi- tional Zeemann term to the Hamiltonian. However, it will not affect the electron motion in the 2DEG plane and the vectorpotential is therefore invariant. Since now the Hamiltonianno longer commutes with any of the Pauli matrices, the spinstates will mix and the three polarization vector componentscan be ﬁnite and therefore need to be calculated. In Fig. 4 weshow the calculated conductance and polarization spectra forthe device in Fig. 3 /H20849b/H20850with the inclusion of the in-plane fringe ﬁeld component obtained again using a formula givenin Ref. 27. Note that here only the results of the calculationsfor the device shown in Fig. 1 /H20849d/H20850with the same structure parameters as in Fig. 3 /H20849b/H20850are presented and the results of the calculations for the other type of device are qualitativelysimilar. By looking at the zero-temperature spin-up conduc-tance G ↑↑+G↑↓and spin-down conductance G↓↓+G↓↑, shown in Fig. 4 /H20849a/H20850, it is found that the inclusion of the in-plane ﬁeld will slightly shift the energy positions of the conductancepeaks and produce small spin-up /H20849down /H20850peaks at the posi- tions where dominant spin-down /H20849up/H20850peaks are found. Thus,the effect on the spin polarization is expected to be small. Figure 4 /H20849b/H20850shows the results of the calculation for the three polarization vector components. It is clearly seen that P xand Pyare rather small, Py/H110110.01Px/H110110.1Pz, and the transport is therefore largely spin conserved in this structure. As a con-sequence, the dominant out-of-plane polarization component, P z, in Fig. 4 /H20849c/H20850shows a rather similar behavior as in Fig. 3/H20849d/H20850, namely that the sign of the polarization at T=0 is strongly energy dependent, while at elevated temperatures itis consistently negative due to the dominance of the spin-down peak in the conductance. In conclusion, spin-ﬁltering abilities of double quantum dot structures under a local magnetic ﬁeld modulation in-duced by a FM stripe have been investigated. Both the out-of-plane and in-plane magnetic ﬁeld components have beenconsidered in the investigation. It has been shown that byvarying the potential of one dot, a large spin polarization/H20849/H11011100% /H20850with either polarity can be achieved. It has also been shown that due to the relative difference in magnitude between the spin-split conductance peaks, an appreciablespin polarization can be observed in the devices at tempera-tures well above the temperature T c=Es/kBdeﬁned by the spin splitting energy Es. The authors thank Dr. Feng Zhai for stimulating discus- sions. This work, which was performed in the NanometerStructure Consortium at Lund University, was supported bythe Swedish Research Council /H20849VR/H20850and by the Swedish Foundation for Strategic Research /H20849SSF /H20850. FIG. 3. /H20849a/H20850Zero-temperature spin-up /H20849solid line /H20850and spin-down /H20849dashed line /H20850conductance of the device in Fig. 1 /H20849d/H20850with the struc- ture parameter c=12 nm and the perpendicular magnetic ﬁeld modulation, Bz/H20849x/H20850, given in Fig. 1 /H20849c/H20850./H20849b/H20850The same as /H20849a/H20850but for c=20 nm. The inset in /H20849b/H20850shows the result in a ﬁne energy scale. /H20849c/H20850Spin polarization, Pz, for the device structure in /H20849a/H20850for tempera- tureT=0 K /H20849thick solid line /H20850, 0.5 K /H20849dashed line /H20850,1K /H20849dash-dotted line/H20850,2K /H20849dotted line /H20850, and 5 K /H20849thin solid line /H20850./H20849d/H20850The same as /H20849c/H20850but for the device structure in /H20849b/H20850. FIG. 4. Effect of inclusion of the in-plane component of the fringe ﬁeld of the FM stripe for the device in Fig. 3 /H20849b/H20850./H20849a/H20850Zero temperature spin up conductance G↑↑+G↑↓/H20849solid line /H20850and spin down conductance G↓↓+G↓↑/H20849dashed line /H20850./H20849b/H20850Polarization vector components, Px/H20849dashed line /H20850,Py/H20849thin solid line /H20850, and Pz/H20849thick solid line /H20850./H20849c/H20850Out-of-plane polarization component, Pz,a tT=0 K /H20849thick solid line /H20850,T=0.5 K /H20849dashed line /H20850,T=1 K /H20849dashed-dotted line/H20850,T=2 K /H20849dotted line /H20850, and T=5 K /H20849thin solid line /H20850.P. BRUSHEIM AND H. Q. XU PHYSICAL REVIEW B 73, 045313 /H208492006 /H20850 045313-4*Electronic address: patrik.brusheim@ftf.lth.se †Corresponding author. Electronic address: hongqi.xu@ftf.lth.se 1R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, and L. W. Molenkamp, Nature /H20849London /H20850402, 787 /H208491999 /H20850. 2Y . Ohno, D. K. Young, B. Beschoten, F. Matsukura, M. Ohno, and D. D. Awschalom, Nature /H20849London /H20850402, 790 /H208491999 /H20850. 3S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y . Chtchelkanova, and D. M.Treger, Science 294, 1488 /H208492001 /H20850. 4A. T. Filip, B. H. Hoving, F. J. Jedema, B. J. van Wees, B. Dutta, and S. Borghs, Phys. Rev. B 62, 9996 /H208492000 /H20850. 5G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. B 62, R4790 /H208492000 /H20850. 6T. Van čura, T. Ihn, S. Broderick, K. Ensslin, W. Wegscheider, and M. Bichler, Phys. Rev. B 62, 5074 /H208492000 /H20850. 7B. T. Jonker, Y . D. Park, B. R. Bennett, H. D. Cheong, G. Ki- oseoglou, and A. Petrou, Phys. Rev. B 62, 8180 /H208492000 /H20850. 8A. Slobodskyy, C. Gould, T. Slobodskyy, C. R. Becker, G. Schmidt, and L. W. Molenkamp, Phys. Rev. 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Xu, Phys. Rev. Lett. 94, 246601 /H208492005 /H20850. 20B. K. Nikolic and S. Souma, Phys. Rev. B 71, 195328 /H208492005 /H20850. 21F. Zhai and H. Q. Xu, Phys. Rev. B 72, 085314 /H208492005 /H20850. 22W. G. van der Wiel, S. De Franceschi, J. M. Eizerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev. Mod. Phys. 75,1/H208492003 /H20850, and references therein. 23Q. Wang, N. Carlsson, I. Maximov, P. Omling, L. Samuelson, W. Seifert, W. Sheng, I. Shorubalko, and H. Q. Xu, Appl. Phys.Lett. 76, 2274 /H208492000 /H20850. 24I. Shorubalko, P. Ramvall, H. Q. Xu, I. Maximov, W. Seifert, P. Omling, and L. Samuelson, Semicond. Sci. Technol. 16, 741 /H208492001 /H20850. 25B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Will- iamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon,Phys. Rev. Lett. 60, 848 /H208491988 /H20850. 26D. A. Wharam, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys. 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Datta, Electronic Transport in Mesoscopic Systems /H20849Cambridge University Press, Cambridge, 1995 /H20850.SPIN FILTERING THROUGH MAGNETIC-FIELD- … PHYSICAL REVIEW B 73, 045313 /H208492006 /H20850 045313-5
PhysRevB.87.075401.pdf	PHYSICAL REVIEW B 87, 075401 (2013) Inclusion of screening effects in the van der Waals corrected DFT simulation of adsorption processes on metal surfaces Pier Luigi Silvestrelli and Alberto Ambrosetti* Dipartimento di Fisica e Astronomia, Universit `a di Padova, via Marzolo 8, I–35131, Padova, Italy, and DEMOCRITOS National Simulation Center of the Italian Istituto Ofﬁcina dei Materiali (IOM) of the Italian National Research Council (CNR), Trieste, Italy (Received 9 December 2012; revised manuscript received 21 January 2013; published 4 February 2013) The DFT/vdW-WF2 method, recently developed to include the van der Waals (vdW) interactions in density functional theory (DFT) using the maximally localized Wannier functions, is improved by taking into accountscreening effects and applied to the study of adsorption of rare gases and small molecules, H 2,C H 4,a n dH 2O on the Cu(111) metal surface, and of H 2on Al(111), and Xe on Pb(111), which are all cases where screening effects are expected to be important. Screening is included in DFT/vdW-WF2 by following different recipes,also considering the single-layer approximation adopted to mimic a screened metal substrate. Comparison of thecomputed equilibrium binding energies and distances, and the C 3coefﬁcients characterizing the adparticle-surface van der Waals interactions, with available experimental and theoretical reference data show that the improvementwith respect to the original unscreened approach is remarkable. The results are also compared with those obtainedby other vdW-corrected DFT schemes. DOI: 10.1103/PhysRevB.87.075401 PACS number(s): 68 .43.Bc, 71 .15.Mb, 68 .35.Md I. INTRODUCTION Adsorption processes on solid surfaces represent a very important topic both from a fundamental point of viewand to design and optimize countless material applications.In particular, the adsorption of closed electron-shellparticles, such as rare-gas (RG) atoms, H 2, and methane (CH 4) molecules on metal surfaces is prototypical1for “physisorption” processes, characterized by an equilibriumbetween attractive, long-range van der Waals (vdW)interactions and short-range Pauli repulsion acting betweenthe electronic charge densities of the substrate and the adsorbedatoms and molecules, 2hereafter referred to as “adparticles.” RG adsorption on many close-packed metal surfaces, such as Ag(111), Al(111), Cu(111), Pd(111), Pt(111), etc.,have been extensively studied both experimentally 3–6and theoretically.6–15In spite of this recent substantial progress, the understanding of the interaction of RGs with metal surfacesis not complete yet. 6For instance, due to the nondirectional character of the vdW interactions that should be dominantin physisorption processes, surface sites that maximize thecoordination of the RG adsorbate atom were expected tobe the preferred ones, so that it was usually assumed thatthe adsorbate occupies the maximally coordinated hollow site. However, this picture has been questioned by manyexperimental 3–5and theoretical8–11recent studies, which in- dicate that the actual scenario is more complex: in particular,for Xe and Kr, a general tendency is found 6,8–11for adsorption on metallic surfaces in the low-coordination topsites (this behavior was attributed6,16to the delocalization of charge density that increases the Pauli repulsion effect at the hollow sites relative to the topsite and lifts the potential well upwards both in energy and height). H2represents another interesting case; in fact, particularly for the H 2molecule on low-index Cu surfaces, accurate physisorption data from experiment are available. Actually,H 2is the only molecule for which a detailed mapping of the gas-surface interaction potential has been performed withresonance scattering measurements (see Ref. 17and references therein). We also consider the methane molecule (CH 4) on Cu(111), as representative of the interaction of an organic moleculewith a metal substrate. Metal-organic interfaces are relevantfor many applications, ranging from surface-functionalizationprocesses, chemical sensors, coating, catalysts to organicelectronics, organic ﬁeld effect transistors, and organic spin-based devices (see, for instance, Ref. 18and references therein). Finally, we study the case of H 2O on Cu(111). In fact, water adsorption at well-deﬁned single-crystal metal surfaces represent an important topic19because it is relevant to many areas of science: water is involved in many catalytic surfacereactions and plays a crucial role in understanding wetting and corrosion, while environmental concerns underlie the increasing importance of the fuel cell reaction and interest inphotocatalysis. In the case of the water molecule, differentlyfrom the other cases, the bonding with the metal surface is not only due to vdW interactions: in fact, a weak covalent bond is formed since water tends to act as an electron donor andthe substrate as an electron acceptor 20(typically H 2O donates a charge of about 0 .1eto the metal21); moreover, the water molecule is characterized by a signiﬁcant intrinsic electronic dipole moment, so that electrostatic effects are also importantdue to the interaction between the H 2O permanent dipole and its image beneath the surface.21 Density functional theory (DFT) is a well-established computational approach to study the structural and electronicproperties of condensed matter systems from ﬁrst principlesand, in particular, to elucidate complex surface processessuch as adsorptions, catalytic reactions, and diffusive motions.Although current density functionals are able to describequantitatively condensed matter systems at much lowercomputational cost than other ﬁrst-principles methods, theyfail 22to properly describe dispersion interactions. Dispersion forces originate from correlated charge oscillations in separate 075401-1 1098-0121/2013/87(7)/075401(12) ©2013 American Physical SocietyPIER LUIGI SILVESTRELLI AND ALBERTO AMBROSETTI PHYSICAL REVIEW B 87, 075401 (2013) fragments of matter and the most important component is represented by the R−6vdW interaction,23originating from correlated instantaneous dipole ﬂuctuations, which plays afundamental role in adsorption processes of fragments weaklyinteracting with a substrate (“physisorbed”). This is clearly the case for the present systems, which can be divided into well separated fragments (adparticles and themetal substrate) with negligible electron-density overlap. Thelocal or semilocal character of the most commonly employedexchange-correlation functionals makes DFT methods unableto correctly predict binding energies and equilibrium distanceswithin both the local density (LDA) and the generalized gradi-ent (GGA) approximations. 24Typically, in many physisorbed systems, GGAs give only a shallow and ﬂat adsorption well atlarge adparticle-substrate separations, while the LDA bindingenergy often turns out to be not far from the experimentaladsorption energy; however, since it is well known thatLDA tends to overestimate the binding in systems withinhomogeneous electron density (and to underestimate theequilibrium distances), the reasonable performances of LDAmust be considered as accidental. Therefore a theoreticalapproach beyond the DFT-LDA/GGA framework, that is ableto properly describe vdW effects is required to provide morequantitative results. 9 In the last few years, a variety of practical methods have been proposed to make DFT calculations able to accuratelydescribe vdW effects (for a recent review, see, for instance,Refs. 24–26). We have previously investigated by such an approach, namely the DFT/vdW-WF method 27–29based on the use of maximally localized Wannier functions (MLWFs),30the interaction of the adsorption of RG atoms on the Cu(111) andPb(111) surfaces. 31However, in previous studies, screening effects, which are expected to be of importance in describinginteractions of small molecules with metal surfaces 26,32–34 have been neglected35or taken into account only in a very approximate way.31In particular, for noble-metal surfaces, such as the Cu(111) one, given the high valence-electrondensity, screening effects are certainly relevant. In Ref. 31, we applied the DFT/vdW-WF method and approximated thescreening effect by explicitly considering only the more local-ized MLWFs corresponding to the d-like orbitals, while the s- andp-like electrons were supposed to give a screening-effect contribution which was evaluated by a simple Thomas-Fermimodel. Here, we improve the previous approach to describe adsorption on metal surfaces in two basic ways. First, weuse the new DFT/vdW-WF2 method, 36which is based on the London expression and takes into account the intrafragmentoverlap of the MLWFs, leading to a considerable improvementnot only in the evaluation of the C 6vdW coefﬁcients but also of the C3coefﬁcients, characterizing molecule-surfaces vdW interactions.36Secondly, we describe screening effects more accurately, by adopting three different recipes, as detailed inthe Method section. We apply these new schemes to the case of adsorption of RGs and small molecules, H 2,C H 4, and H 2O on the Cu(111) metal surface, and of H 2on Al(111), and Xe on Pb(111). In particular, the Cu(111) surface has been chosen because of themany experimental and theoretical data available which canbe compared with ours in such a way to validate the presentapproach, whose performances are also compared with those of other vdW-corrected DFT schemes. II. METHOD Basically (more details can be found in Ref. 36), while in the original DFT/vdW-WF method the vdW energy correctionfor two separate fragments was computed using the exchange-correlation functional proposed by Andersson et al. , 37the latest DFT/vdW-WF2 version is instead based on the simpler,well known London’s expression 23where two interacting atoms, AandB, are approximated by coupled harmonic oscillators and the vdW energy is taken to be the change ofthe zero-point energy of the coupled oscillations as the atomsapproach; if only a single excitation frequency is associated toeach atom, ω A,ωB, then ELondon vdW =−3e4 2m2ZAZB ωAωB(ωA+ωB)1 R6 AB, (1) where ZA,Bis the total charge of AandB, andRABis the distance between the two atoms ( eandmare the electronic charge and mass). This approach is clearly applicable to wellseparated fragments only: in the present systems, characterizedby the interaction of adparticles weakly interacting with thesubstrate (“physisorbed”), this condition is always satisﬁed. Now, adopting a simple classical theory of the atomic polarizability, the polarizability of an electronic shell of chargeeZ iand mass mZi, tied to a heavy undeformable ion can be written as αi/similarequalZie2 mω2 i. (2) Then, given the direct relation between polarizability and atomic volume,38we assume that αi∼γS3 i, where γis a proportionality constant, so that the atomic volume isexpressed in terms of the MLWF spread S i. Rewriting Eq. (1) in terms of the quantities deﬁned above, one obtains anexplicit expression (much simpler than the multidimensionalintegrals involved in the Andersson functional 37)f o rt h e C6 vdW coefﬁcient: CAB 6=3 2√ZAZBS3 AS3 Bγ3/2 /parenleftbig√ZBS3/2 A+√ZAS3/2 B/parenrightbig. (3) The constant γcan then be set up by imposing that the exact value for the H atom polarizability ( αH=4.5 a.u.) is obtained (of course, in the H case, one knows the exact analytical spread,S i=SH=√ 3 a.u.). In order to achieve a better accuracy, one must properly deal with intrafragment MLWF overlap (we refer here to charge overlap, not to be confused with wave functionsoverlap): in fact, the DFT/vdW-WF method is strictly validfor nonoverlapping fragments only; now, while the overlapbetween the MLWFs relative to separated fragments is usuallynegligible for all the fragment separation distances of interest,the same is not true for the MLWFs belonging to the samefragment, which are often characterized by a signiﬁcantoverlap. This overlap affects the effective orbital volume,the polarizability, and the excitation frequency [see Eq. (2)], thus leading to a quantitative effect on the value of the C 6 coefﬁcient. We take into account the effective change in 075401-2INCLUSION OF SCREENING EFFECTS IN THE V AN DER ... PHYSICAL REVIEW B 87, 075401 (2013) volume due to intrafragment MLWF overlap by introducing a suitable reduction factor ξobtained by interpolating between the limiting cases of fully overlapping and nonoverlappingMLWFs. In particular, since in the DFT/vdW-WF2 methodtheith MLWF is approximated with a homogeneous charged sphere of radius S i, then the overlap among neighboring MLWFs can be evaluated as the geometrical overlap amongneighboring spheres. 36By extending the approach to partial overlaps, we deﬁne the free volume of a set of MLWFs belonging to a given fragment (in practice, three-dimensionalintegrals are evaluated by numerical sums introducing asuitable mesh in real space) as V free=/integraldisplay drwfree(r)/similarequal/Delta1r/summationdisplay lwfree(rl), (4) where wfree(rl) is equal to 1 if \|rl−ri\|<Sifor at least one of the fragment MLWFs and is 0 otherwise. The corresponding effective volume is instead given by Veff=/integraldisplay drweff(r)/similarequal/Delta1r/summationdisplay lweff(rl), (5) where the new weighting function is deﬁned as weff(rl)= wfree(rl)nw(rl)−1, with nw(rl) that is equal to the number of MLWFs contemporarily satisfying the relation \|rl−ri\|<Si. Therefore the nonoverlapping portions of the spheres (inpractice, the corresponding mesh points) will be associated to aweight factor 1, those belonging to two spheres to a 1 /2 factor, and, in general, those belonging to nspheres to a 1 /nfactor. The average ratio between the effective volume and the freevolume ( V eff/Vfree) is then assigned to the factor ξ, appearing in Eq. (6). We therefore arrive at the following expression for theC6coefﬁcient: CAB 6=3 2√ZAZBξAS3 AξBS3 Bγ3/2 /parenleftbig√ZBξAS3/2 A+√ZAξBS3/2 B/parenrightbig, (6) where ξA,Brepresents the ratio between the effective and the free volume associated to the Ath and Bth MLWF. The need for a proper treatment of overlap effects has been also recentlypointed out by Andrinopoulos et al. , 29who, however, applied a correction only to very closely centered WFCs. Finally, the vdW interaction energy is computed as EvdW=−/summationdisplay i<jf(Rij)Cij 6 R6 ij, (7) where f(Rij) is a short-range damping function, which is introduced not only to avoid the unphysical divergence ofthe vdW correction at small fragment separations, but also to eliminate double countings of correlation effects (in fact, standard DFT approaches are able to describe short-rangecorrelations); it is deﬁned as f(R ij)=1 1+e−a(Rij/Rs−1). (8) The parameter Rsrepresents the sum of the vdW radii Rs= RvdW i+RvdW j, with (by adopting the same criterion chosen above for the γparameter) RvdW i=RvdW HSi√ 3, (9)where RvdW H is the literature39(1.20 ˚A) vdW radius of the H atom and, following Grimme et al. ,40a/similarequal20 (the results are almost independent on the particular value of this parameter).Although this damping function introduces a certain degree ofempiricism in the method, we stress that ais the only ad hoc parameter present in our approach, while all the others are onlydetermined by the basic information given by the MLWFs,namely, from ﬁrst-principles calculations. The evaluation ofthe vdW correction as a post-standard DFT calculation,using the DFT electronic density distribution, represents anapproximation because, in principle, a full self-consistentcalculations should be performed; however, investigations 41 on different systems have shown that the effects due to thelack of self-consistency are negligible, especially in proximityof the equilibrium, lowest-energy conﬁguration, and for wellseparated fragments: in fact, one does not expect that the ratherweak and diffuse vdW interaction substantially changes theelectronic charge distribution. In order to get an appropriate inclusion of screening effects, three different schemes have been adopted, hereafter referredto as DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3, respectively, which are described in the followingsections. A. DFT/vdW-WF2s1 This scheme is similar to that previously applied31to the original DFT/vdW-WF method (see description above),however, now the vdW C 6coefﬁcients are computed by considering not only the more localized d-like MLWFs (as in Ref. 31, of course in the case of the Al(111) substrate thed-like MLWFs are absent) but also the s- and p-like electrons (so that all the MLWFs are taken into account);this being now more justiﬁed because in the DFT/vdW-WF2method (differently from the original DFT/vdW-WF) the effectof relatively delocalized MLWFs is made less relevant bythe proper treatment of intrafragment overlap, as describedabove. Then the screening reduction effect is included by multiplying (as in Ref. 31)t h eC ij 6/R6 ijcontribution in Eq. (7) by a Thomas-Fermi factor: fTF=e−2(zs−zl)/rTFwhere rTFis the Thomas-Fermi screening length relative to the electronicdensity of a uniform electron gas (“jellium model”) equal to theaverage density of the s- andp-like electrons of the substrate, z sis the average vertical position of the topmost metal atoms, andzlis the vertical coordinate of the WFC belonging to the substrate ( l=iif it is the ith WFCs which belongs to the substrate, otherwise l=j); the above fTFfunction is only applied if zl<zs, otherwise it is assumed that fTF=1( n o screening effect). B. DFT/vdW-WF2s2 In this alternative scheme, the screening is taken into account by adopting the following, two-step strategy, aimingat separating the effects of the relatively localized d-like orbitals from those of the more delocalized s- and p-like orbitals. (i) First, we compute the vdW energy correction byonly considering the more localized d-like MLWFs, with the C 6coefﬁcients screened by the same Thomas-Fermi factor adopted for DFT/vdW-WF2s1; then by ﬁtting (as in Ref. 31) 075401-3PIER LUIGI SILVESTRELLI AND ALBERTO AMBROSETTI PHYSICAL REVIEW B 87, 075401 (2013) the calculated binding energies, at different adparticle-surface distances, with the function: Ae−Bz−C3/(z−z0)3,A,B, C3, andz0being adjustable parameters, we get an estimate of the Thomas-Fermi screened C 3dTFcoefﬁcient (and also of the unscreened C3dcoefﬁcient if the Thomas-Fermi reduction factor is omitted). (ii) Then the ﬁnal vdW energy [see Eq. (7)] is evaluated by using “rescaled” C6coefﬁcients, deﬁned as Cij 6r=Cij 6(C3dTF+C3f) C3d, (10) where C3fis theC3coefﬁcient evaluated by assuming the free- electron approximation for the metal surface, that is usuallya reasonable estimate for the more delocalized s- andp-like orbital contribution 42and can be easily computed2as C3f=α0 8¯hω0ωp ω0+ωp, (11) where α0andω0are the static polarizability and the charac- teristic frequency of the adsorbed adparticle, respectively, andω pis the plasma frequency of the metal substrate (appropriate to the electron density relative to sandpelectrons). α0and ωpvalues can be easily found in the literature, and ω0can be expressed43in terms of α0: ω0=/radicalBigg Ze2 mα 0, (12) where Zis the number of valence electrons of the adparticle andeandmare the electronic charge and mass, respectively. In the ﬁtting function Ae−Bz−C3/(z−z0)3, the image-plane position z0can be taken32as half the interlayer distance of the substrate (in fact, half a normal lattice spacing above theoutermost layer of substrate nuclei can be taken as the jellium-edge position 44) andzis the distance of the adsorbed adparticle from the surface. In the case of H 2on Al(111), d-like orbitals are absent and Eq.(10) reduces to Cij 6r=Cij 6C3f C3, (13) where C3is the unscreened C3coefﬁcient, always obtained by ﬁtting the Ae−Bz−C3/(z−z0)3function. This second scheme, DFT/vdW-WF2s2, based on rescaled C6coefﬁcients, follows a strategy similar to that adopted in Ref. 32, where screening effects are included in the TS-vdW method45by using the Lifshitz-Zaremba-Kohn theory46for the vdW interaction between an atom and a solid surface, whichdescribes the many-body collective response of the substrateelectrons. C. DFT/vdW-WF2s3 A simple approach to mimic screening effects in adsorp- tion processes is represented by the so-called “single-layer”approximation in which vdW effects are only restricted to theinteractions of the adparticle with the topmost metal layer; 47 in fact, as a consequence of screening, one expects that thetopmost metal atoms give the dominant contribution. We have implemented this by multiplying the Cij 6/R6 ijfactor in Eq. (7)by a damping function: fSL=1−1 1+e(zl−zr)//Delta1z, (14) where zlis the vertical coordinate of the WFC belonging to substrate ( l=iif it is the ith WFCs, which belongs to the substrate, otherwise l=j), the reference level zris taken as intermediate between the level of the ﬁrst, topmostsurface layer and the second one, and we assume that/Delta1z=(interlayer separation) /4; we found that the estimated equilibrium binding energies and adparticle-surface distancesexhibit only a mild dependence on the /Delta1zparameter. Clearly, this third approach resembles the DFT/vdW-WF2s1 scheme,the basic difference being that the Thomas-Fermi dampingfunction of DFT/vdW-WF2s1 is here replaced by the f SL damping function introduced to just select the WFCs around the topmost surface layer. Although fSLis, in principle, less physically motivated than the Thomas-Fermi function, itspractical effect is expected to be very similar, as conﬁrmedby the applications of the methods (see Results). D. Computational details We here apply the DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods to the case of adsorption ofRGs, H 2,C H 4, and H 2O on the Cu(111) surface and of H 2 on Al(111) and Xe on Pb(111). All calculations have beenperformed with the QUANTUM ESPRESSO ab initio package48 (MLWFs have been generated as a postprocessing calculation using the W ANT package49). Similarly to our previous study,31 we modeled the metal surface using a periodically repeatedhexagonal supercell, with a (√ 3×√ 3)R30◦structure and a surface slab made of 15 Cu, Al, or Pb atoms distributedover ﬁve layers (repeated slabs were separated along thedirection orthogonal to the surface by a vacuum region ofabout 24 ˚A). The Brillouin zone has been sampled using a 6×6×1k-point mesh. In this model system, the coverage is 1/3, i.e., one adsorbed adparticle for each three metal atoms in the topmost surface layer. The (√ 3×√ 3)R30◦structure has been indeed observed4at low temperature by LEED for the case of Xe adsorption on Cu(111) and Pd(111) (actually, thisis the simplest commensurate structure for RG monolayers onclose-packed metal surfaces and the only one for which goodexperimental data exist), and it was adopted in most of theprevious ab initio studies. 7–9,11,12,50The metal surface atoms were kept frozen (of course, after a preliminary relaxationof the outermost layers of the clean metal surfaces) andonly the vertical coordinate (perpendicular to the surface)of the center of mass of the adparticles was optimized, thisprocedure being justiﬁed by the fact that only minor surfaceatom displacements are observed upon physisorption. 8,50–52 Moreover, the adparticles were adsorbed on both sides of the slab: in this way, the surface dipole generated by adsorptionon the upper surface of the slab is canceled by the dipoleappearing on the lower surface, thus greatly reducing thespurious dipole-dipole interactions between the periodicallyrepeated images (previous DFT-based calculations have shownthat these choices are appropriate 9,13,18,31). We have carried out calculations for various separations of the atoms and molecules adsorbed on the tophigh-symmetry 075401-4INCLUSION OF SCREENING EFFECTS IN THE V AN DER ... PHYSICAL REVIEW B 87, 075401 (2013) FIG. 1. (Color online) Plan (top) and side (bottom) views for a single water molecule on the Cu(111) surface, showing the simulationcell and the periodic images. site (on the top of a metal atom), since this is certainly the favored adsorption site for Xe on Cu(111);31in the case of H2,C H 4, and H 2O by adsorption on the top site we mean that the center of mass of these molecules is on top of a Cuatom (see Figs. 1and2), which is assumed to be the preferred adsorption site. 17,19,21,53For the Xe-Cu(111) and Xe-Pb(111) cases, we have also considered adsorption on the hollow site (on the center of the triangle formed by the three surfacemetal atoms contained in the supercell) in order to verifywhether the present schemes are able to correctly predictwhich conﬁguration is energetically favored (see discussionin Ref. 31). In the calculations, the H 2molecule is kept in a ﬂat orientation above the Cu(111) surface (the binding energydepends very little on the orientation 17,54). The same is true for the water monomer since there is a general agreement19 that the water molecule prefers to bind in a top position on theCu(111) substrate, with its molecular plane nearly parallel tothe surface. For a better accuracy, as done in previous applications on adsorption processes, 28,31,35,55,56we have also included the interactions of the MLWFs of the physisorbed fragmentsnot only with the MLWFs of the underlying surface,within the reference supercell, but also with a sufﬁcientnumber of periodically-repeated surface MLWFs (in anycase, given the R −6decay of the vdW interactions, the convergence with the number of repeated images is rapidlyachieved). Electron-ion interactions were described usingnorm-conserving pseudopotentials by explicitly including14, 11, and 3 valence electrons per Pb, Cu, and Al atom,respectively. As in our previous study, 31we chose the PW9157reference DFT functional. The problem of choosing FIG. 2. (Color online) Plan (top) and side (bottom) views for a single hydrogen molecule on the Cu(111) surface, showing thesimulation cell and the periodic images. the optimal DFT functional, particularly in its exchange component, to be combined with long-range vdW interactionsand the related problem of completely eliminating doublecounting of correlation effects [which, in our scheme, isaccomplished by the short-range damping function f(R ij) deﬁned above] still remain open;24however, they are expected to be more crucial for adsorption systems characterized byrelatively strong adparticle-substrate bonds (“chemisorption”)and, for instance, for the determination of the perpendicularvibration frequency 11than for the equilibrium properties of the physisorbed systems, we focus on in our paper. The additional cost of the post-processing vdW correction is basically represented by the cost of generating the maximallylocalized Wannier functions from the Kohn-Sham orbitals,which scales linearly with the size of the system. 30In our speciﬁc applications, the Wannier-function generation is moreexpensive because of the k-point sampling of the Brillouin zone, that is appropriate for metals and make the spread-minimization process less efﬁcient. 30In practice, in our cases, the additional cost of the vdW correction is comparable withthat of the previous standard DFT calculation, however, onemust point out that, for generating the maximally localizedWannier functions, we have just used the public-released scalarversion of the W ANT code49without any attempt to develop a much faster parallelized version or to make the minimizationprocess more efﬁcient. III. RESULTS AND DISCUSSION In Tables I–VIII results are reported for all the systems under consideration; in particular, in Tables I,V, and VII,w e 075401-5PIER LUIGI SILVESTRELLI AND ALBERTO AMBROSETTI PHYSICAL REVIEW B 87, 075401 (2013) TABLE I. Binding energy, Ebin meV , of adparticles in the top conﬁguration on the metal surface computed using the standard PW91 calculation, and including the vdW corrections using our (unscreened) DFT/vdW-WF2, and (screened) DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods. System PW91 DFT/vdW-WF2 DFT/vdW-WF2s1 DFT/vdW-WF2s2 DFT/vdW-WF2s3 H2-Cu(111) −10.5 −49.8 −36.0 −25.6 −33.7 H2-Al(111) −14.7 −35.2 −22.9 −25.9 −26.5 Ne-Cu(111) −17.5 −66.3 −50.8 −34.9 −52.2 Ar-Cu(111) −13.0 −140.1 −91.3 −66.4 −97.8 Kr-Cu(111) −20.3 −196.8 −130.5 −102.2 −131.3 Xe-Cu(111) −23.1 −333.2 −214.5 −242.7 −224.2 Xe-Pb(111) −56.3 −151.9 −100.0 −210.0 −111.9 CH 4-Cu(111) −16.1 −166.1 −111.9 −119.5 −112.7 H2O-Cu(111) −71.0 −425.4 −345.7 −350.1 −345.3 compare quantities evaluated by the standard PW91 approach, and including the vdW corrections using our (unscreened)DFT/vdW-WF2 method, and the screened schemes DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 describedabove; in Tables II–IV,VI, and VIII we instead compare our global, screened DFT/vdW-WF2s estimates (obtained byconsidering the range of values calculated separately by theDFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3methods), to available theoretical and experimental estimatesand to corresponding data obtained using the “seamless”vdW-DF and vdW-DF2 methods of Langreth et al. , 41,58,59 which perform well in a variety of applications, although they are not perfect since they violate some important limits;60 moreover, they do not explicitly take into account screeningeffects of metal surfaces. 17 Thebinding energy Ebis deﬁned as Eb=1/2[Etot−(Es+2Ea)], (15)where Es,arepresent the energies of the isolated fragments (the substrate and the adparticles) and Etotis the energy of the interacting system, including the vdW-correction term (thefactors 2 and 1 /2 are due to the adsorption on both sides of the slab); E sandEaare evaluated using the same supercell adopted for Etot. The experimentally measured adsorption energy Eaoften includes not only the interaction of adparticles with thesubstrate but also lateral vdW interfragment interactions. 13,31 Therefore sometimes it is more appropriate to compare experimental data with the quantity Ea, which can be related toEbby31 Ea=Eb+(El−Ef), (16) where Elis the total energy (per particle) of the 2D lattice formed by the adparticles only (that is as in the adsorptionconﬁgurations but without the substrate and including vdW TABLE II. Binding energy, Ebin meV , of adparticles in the top conﬁguration on the metal surface computed considering our DFT/vdW-WF2s estimates (within the range of values obtained by the DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods), compared to the vdW-DF and the vdW-DF2 methods by Langreth et al.41,58,59and available theoretical and experimental (in parenthesis) reference data. System DFT/vdW-WF2s vdW-DF vdW-DF2 Reference H2-Cu(111) −36↔− 26 −53 −39 −32a(−29b) H2-Al(111) −27↔− 23 −59 −47 −19c−24d(−37e) Ne-Cu(111) −52↔− 35 −56 −37 ··· Ar-Cu(111) −98↔− 66 −106 −91 −85a Kr-Cu(111) −131↔− 102 −136 −116 −119a Xe-Cu(111) −243↔− 214 −168 −156 −280f,−183a,−277g,−270h(−190g,−227i) Xe-Pb(111) −210↔− 100 −186 −136 ··· CH 4-Cu(111) −119↔− 112 −124 −108 ··· H2O-Cu(111) −350↔− 345 −133 −141 ··· aReference 2. bReference 17. cReference 62. dReference 54. eReference 61. fReference 11. gReference 4. hReference 14. iReference 76. 075401-6INCLUSION OF SCREENING EFFECTS IN THE V AN DER ... PHYSICAL REVIEW B 87, 075401 (2013) TABLE III. Adsorption energy, Eain meV (see text for the deﬁnition), of methane and water in the top conﬁguration on the Cu(111) surface computed considering our DFT/vdW-WF2s esti- mates (within the range of values obtained by the DFT/vdW-WF2s1,DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods), compared to the vdW-DF and the vdW-DF2 methods by Langreth et al. 41,58,59and available experimental reference data. System DFT/vdW-WF2s vdW-DF vdW-DF2 Reference CH 4-Cu(111) −185↔− 178 −205 −166 −160a H2O-Cu(111) −446↔− 441 −240 −223 −352b aReference 65. bReference 73. interfragment corrections when vdW-corrected methods are used) and Efis the energy of an isolated (free) adparticle. Ebhas been evaluated for several adsorbate-substrate dis- tances; then the equilibrium distances and the correspondingbinding energies have been obtained (as in Ref. 31, see also the Method section) by ﬁtting the calculated points withthe function: Ae −Bz−C3/(z−z0)3[as illustrated for the H2-Cu(111) case in Fig. 3]. Typical uncertainties in the ﬁt are of the order of 0 .05˚A for the distances and a few meVs for the minimum binding energies. When vdW interactionsdominate, the equilibrium binding energy is expected to beroughly proportional to the adparticle polarizabilities. 44As found in the previous studies31(see Fig. 3and Tables I and V), the effect of the vdW-corrected schemes is a much stronger bonding than with a pure PW91 scheme, with theformation of a clear minimum in the binding energy curve ata shorter equilibrium distance. Moreover, by comparing withunscreened data (we recall that also the vdW-DF and vdW-DF2methods do not take explicitly metallic screening into account),we see that the effect of screening is substantial, leading toreduced binding energies and increased adparticle-substrateequilibrium distances. By ﬁrst considering the adsorption of H 2on Cu(111) for which accurate reference data are available, both theexperimental binding energy ( −29 meV) and the equilibrium H 2-Cu(111) distance ( zeq=3.52 ˚A) are well reproduced (see Tables I,II,Vand VI, and Fig. 3) by our screened methods (with DFT/vdW-WF2s1 and DFT/vdW-WF2s3 thatslightly overestimate the strength of the interaction andDFT/vdW-WF2s2 that slightly underestimates it, the trendbeing reversed for the equilibrium distance). Interestingly, ourresults are much better than those obtained by the vdW-DF 41,58 (Eb=− 53 meV , zeq=3.85˚A), DFT-D340(Eb=− 98 meV , TABLE IV . Difference, /Delta1Eb, in meV , between the binding energy Ebof Xe on metal surfaces in the topandhollow conﬁgurations, computed considering our DFT/vdW-WF2s estimates (within the range of values obtained by the DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods), compared to the vdW-DF and the vdW-DF2 methods by Langreth et al. 41,58,59 System DFT/vdW-WF2s vdW-DF vdW-DF2 Xe-Cu(111) −40↔− 37 −3 −1 Xe-Pb(111) +8↔+ 22 +6 +3zeq=2.86˚A), and TS-vdW45(Eb=− 66 meV , zeq=3.20˚A) methods, and also slightly better than the estimates of vdW-DF2 59(Eb=− 39 meV , zeq=3.64˚A). We therefore conﬁrm the observations of Lee et al. who, by comparison with the ref- erence potential energy curve of H 2on Cu(111), concluded that vdW-DF2 performs relatively well (the remaining discrepancybeing probably due to lack of screening-effect description 17), differently from DFT-D3 and TS-vdW, a behavior attributedto the fact that pair potentials, on which these two methodsare based, center the interactions on the nuclei and do notfully reﬂect that important binding contributions arise in thewave function tails outside the surface. 17Concerning the C3coefﬁcients (see Tables VII and VIII), these represent notorious difﬁcult quantities to evaluate (see, for instance,Refs. 1and 31): in fact, the reliability of reference data is hard to assess, moreover, one should really make estimatesby sampling the asymptotic region, corresponding to largeadparticle-surface distances, where the binding energy isquite small and the relative uncertainty large. Moreover, forcharacterizing the adsorption processes, the focus is mainly onthe equilibrium properties, corresponding to a region not farfrom the minimum of the adparticle-surface binding-energycurve. In any case, for the C 3coefﬁcient of H 2on Cu(111), the agreement with the reference data is less satisfactorythan for E bandzeq, and comparable with that of vdW-DF2, while instead vdW-DF clearly strongly overestimates. Notethat, by using the simple DFT/vdW-WF2s3 approach, for thissystem one gets results comparable (see Fig. 3) with those obtained by DFT/vdW-WF2s1 and DFT/vdW-WF2s2 with theC 3coefﬁcient that is even closer to the reference value. If H 2is instead adsorbed on the Al(111) surface, accurate reference data are more scarce: there is just an indirectexperimental estimate 61for E b(−37 meV), old theoretical calculations based on jellium models62,63or damped dipole- dipole and dipole-quadrupole interactions,54and a study per- formed using a density functional for asymptotic vdW forces.64 By considering the reference binding energies one can seethat, also in this case, the performances of the DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methodsare satisfactory and comparable: all the methods predict(see Table II) values slightly below the experimental estimate, the agreement being better with previous theoretical calcula-tions, while vdW-DF and vdW-DF2 tend to overestimate thestrength of the interaction (again vdW-DF2 performs betterthan vdW-DF). The experimental estimate of the equilibriumdistance appears instead signiﬁcantly smaller than the valuesobtained by all the theoretical methods considered in thepresent study. For the C 3coefﬁcients, the same observations relative to the H 2-Cu(111) case apply. Considering the adsorption of RGs on Cu(111), reference data are available, particularly the “best estimates” reported byVidali et al. , 2that represent averages over different theoretical and experimental evaluations. As can be seen, for Ne, Ar, andKr on Cu(111) the DFT/vdW-WF2s1, DFT/vdW-WF2s2, andDFT/vdW-WF2s3 methods give binding energies compatiblewith those obtained by vdW-DF2, while vdW-DF tends insteadto overbind: this is also conﬁrmed by the fact that vdW-DFpredicts C 3values much larger than the other schemes and comparable to those obtained by our DFT/vdW-WF2 methodwithout any screening correction. Note that, as a general 075401-7PIER LUIGI SILVESTRELLI AND ALBERTO AMBROSETTI PHYSICAL REVIEW B 87, 075401 (2013) TABLE V . Equilibrium adparticle-metal surface distance, in angstroms, of adparticles in the top conﬁguration computed using the standard PW91 calculation, and including the vdW corrections using our (unscreened) DFT/vdW-WF2, and (screened) DFT/vdW-WF2s1, DFT/vdW- WF2s2, and DFT/vdW-WF2s3 methods. System PW91 DFT/vdW-WF2 DFT/vdW-WF2s1 DFT/vdW-WF2s2 DFT/vdW-WF2s3 H2-Cu(111) 4.10 3.24 3.40 3.60 3.49 H2-Al(111) 4.08 3.84 3.93 3.92 3.91 Ne-Cu(111) 3.90 3.38 3.44 3.56 3.43 Ar-Cu(111) 4.50 3.26 3.41 3.54 3.39 Kr-Cu(111) 4.50 3.05 3.36 3.38 3.37Xe-Cu(111) 4.40 2.97 3.12 3.04 3.15 Xe-Pb(111) 4.50 3.98 4.07 3.73 4.06 CH 4-Cu(111) 4.70 3.39 3.49 3.43 3.52 H2O-Cu(111) 2.81 2.40 2.41 2.36 2.43 trend, both vdW-DF and vdW-DF2 give larger equilibrium distances than our DFT/vdW-WF2s1, DFT/vdW-WF2s2, andDFT/vdW-WF2s3 methods. For Xe-Cu(111), the scenarioappears to be more complex: in fact, with respect to the ref-erence values, our screened methods appear to well reproducethe equilibrium binding energy and C 3coefﬁcient, although the equilibrium distances are shorter; instead vdW-DF andvdW-DF2 overestimate the equilibrium Xe-Cu(111) distanceand the C 3coefﬁcient, while they undererestimate the binding energies. This peculiar behavior can be probably explainedby the tendency of Xe to induce a substantial electroniccharge delocalization on the Cu(111) surface, 31thus making screening effects relatively less important than for the otherRGs. Probably in this case the results also depend in amore subtle way on the speciﬁc choice of the underlyingDFT functional. Interestingly, all the considered theoreticalschemes (see Table IV) predict that the top site is favored with respect to the hollow one for Xe on Cu(111) (in agreementwith the experimental evidence 6), while the opposite is true for Xe on Pb(111) (in line with previous calculations31),although vdW-DF and vdW-DF2 clearly tend to minimize the differences. Concerning the case of methane on Cu(111), the experi- mental adsorption energy has been estimated by temperature-programmed-desorption measurements of the activation en-ergy (160 meV) for molecular desorption of methane froma saturated ﬁrst monolayer, 65so that it includes the lateral interactions mentioned above and it is more appropriateto compare this estimate with the E aquantity deﬁned in Eq.(16). As can be seen in Table IIIthe performances of the different schemes exhibit the same trend observed in the previ-ous investigated cases: DFT/vdW-WF2s1, DFT/vdW-WF2s2,DFT/vdW-WF2s3, and vdW-DF2 gives similar adsorptionenergies (with vdW-DF2 that in this case is closer to thereference value), while vdW-DF appears to overbind; the C-Cu(111) equilibrium distance and the C 3coefﬁcient are larger with vdW-DF and vdW-DF2 than with DFT/vdW-WF2s1,DFT/vdW-WF2s2, and DFT/vdW-WF2s3. Coming to our ﬁnal system, namely the water monomer on Cu(111), in this case the experimental characterization is made TABLE VI. Equilibrium adparticle-metal surface distance, in angstroms, of adparticles in the top conﬁguration computed considering our DFT/vdW-WF2s estimates (within the range of values obtained by the DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods), compared to the vdW-DF and the vdW-DF2 methods by Langreth et al.41,58,59and available theoretical and experimental (in parenthesis) reference data. System DFT/vdW-WF2s vdW-DF vdW-DF2 Reference H2-Cu(111) 3.40 ↔3.60 3.85 3.64 2.86a, 3.2a(3.52a) H2-Al(111) 3.91 ↔3.93 3.94 3.75 3.52b Ne-Cu(111) 3.43 ↔3.56 3.68 3.68 ··· Ar-Cu(111) 3.39 ↔3.54 3.86 3.74 3.53c Kr-Cu(111) 3.36 ↔3.38 3.99 3.75 ··· Xe-Cu(111) 3.04 ↔3.15 4.09 3.93 3.2 ↔4.0d(3.6e) Xe-Pb(111) 3.73 ↔4.07 4.30 4.29 ··· CH 4-Cu(111) 3.43 ↔3.52 4.14 3.99 ··· H2O-Cu(111) 2.36 ↔2.43 3.27 3.05 2.25f, 2.36g aReference 17. bReference 54. cReference 77. dReference 31. eReference 4. fReference 21. gReference 78. 075401-8INCLUSION OF SCREENING EFFECTS IN THE V AN DER ... PHYSICAL REVIEW B 87, 075401 (2013) TABLE VII. Estimated C3coefﬁcients, in meV ˚A3, for adparticles in the top conﬁguration on the metal surface computed using our (unscreened) DFT/vdW-WF2, and (screened) DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods. System DFT/vdW-WF2 DFT/vdW-WF2s1 DFT/vdW-WF2s2 DFT/vdW-WF2s3 H2-Cu(111) 1485 1171 984 1216 H2-Al(111) 1442 943 1098 1103 Ne-Cu(111) 1698 1443 1018 1415 Ar-Cu(111) 3078 2277 2030 2235Kr-Cu(111) 5036 3593 2848 3858 Xe-Cu(111) 5601 4016 3480 3995 Xe-Pb(111) 4242 2935 4263 3317 CH 4-Cu(111) 3533 2559 2523 2720 H2O-Cu(111) 2386 1892 1612 1986 difﬁcult by facile water-cluster formation that masks the true H2O-metal interaction.20In any case, previous studies indicate that it is easier to desorb than to dissociate H 2O on the Cu(111) and Cu(110) surfaces (see Ref. 66and references therein). The system has been already studied using pure GGA (mainlybased on PW91 and PBE functionals) or hybrid (B3LYP)approaches, 19,21,52,66–71giving rather spread estimates for the binding energy (between −120 and −660 meV) and the Cu-O equilibrium distance (between 2.2 and 3.9 ˚A), these relatively large differences being mainly attributed to the differentexchange-correlation functionals adopted (besides other tech-nical details, including surface coverage, reference supercell,geometry optimization conditions, number of considered Cuplanes, pseudopotentials, plane-wave energy cutoff, etc.). In allthese studies, a proper description of vdW effects is missing.Higher-level (MP2) ab initio calculations exist, 72that should include vdW interactions, predicting that the energeticallyfavored adsorption conﬁguration is characterized by an H-down conformation (with a binding energy of −166 meV and an equilibrium Cu-O distance of 3.59 ˚A), differentlyfrom the other studies which instead predict an almost planar equilibrium conﬁguration for the water monomer on the Cusurface; however, these results are questionable since theCu(111) surface is modeled by relatively small Cu clusters,which are affected by well-known size-dependent effects. Theenergy values of the H 2O-Cu(111) bond indicate that it lies in the weak chemisorption/physisorption regime;21interestingly, this energy range (about 0.25 eV) also represents the energyof a typical H-bond between water molecules, 20so that adsorbate-adsorbate and adsorbate-substrate interactions arecomparable. Old experimental estimates for water on Cu(111)are available, 20however, these values (in the range from −0.4 to −0.7 eV) are probably overestimated52since they possibly correspond to polycrystalline samples containing alarge number of low-coordinated surface atoms. An estimate 73 for the adsorption energy of water on Cu(111), on the basis ofx-ray photoelectron spectroscopy, gives −352 meV . Although it is believed 74that vdW effects are not crucial for many aspects of structure and bonding of H 2O on Cu(111), nonetheless, due to the high polarizability of the substrate metal atoms, TABLE VIII. Estimated C3coefﬁcients, in meV ˚A3, for adparticles in the top conﬁguration on the metal surface computed using our DFT/vdW-WF2s data (within the range of values obtained by the DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods),compared to the vdW-DF and the vdW-DF2 methods by Langreth et al. 41,58,59and available theoretical reference data. System DFT/vdW-WF2s vdW-DF vdW-DF2 Reference H2-Cu(111) 984 ↔1216 2310 1097 681a, 673b H2-Al(111) 943 ↔1103 2427 1279 605c, 661d, 669e, 706f Ne-Cu(111) 1018 ↔1443 1644 801 488a, 417g Ar-Cu(111) 2030 ↔2277 4690 2641 1621b, 1397g Kr-Cu(111) 2848 ↔3858 6722 3962 2294a, 2110b, 1992g Xe-Cu(111) 3480 ↔4016 9712 6146 3391a, 3390b, 2967g Xe-Pb(111) 2935 ↔4263 8837 5506 ··· CH 4-Cu(111) 2523 ↔2720 6735 3967 ··· H2O-Cu(111) 1612 ↔1986 4167 2297 ··· aReference 79. bReference 2. cReference 64. dReference 62. eReference 54. fReference 63. gReference 80. 075401-9PIER LUIGI SILVESTRELLI AND ALBERTO AMBROSETTI PHYSICAL REVIEW B 87, 075401 (2013) 3 3.5 4 4.5 5 5.5 z (Å)-80-70-60-50-40-30-20-10010Binding energy (meV)PW91 DFT/vdW-WF2 DFT/vdW-WF2s1 DFT/vdW-WF2s2 DFT/vdW-WF2s3 vdW-DF vdW-DF2 FIG. 3. (Color online) Binding energy of H 2on Cu(111), as a function of the distance between the center of mass of H 2and the Cu(111) surface, computed using the standard PW91 calculation andincluding the vdW corrections using our (unscreened) DFT/vdW- WF2, and (screened) DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 methods, and the vdW-DF and the vdW-DF2methods by Langreth et al. ; 41,58,59the triangle indicates the position of the experimental value. they contribute substantially to the water-metal bond, which is an important factor in determining the relative stabilities ofwetting layers and 3D bulk ice. 74 In our study, for the sake of uniformity, we have maintained the (√ 3×√ 3)R30◦supercell also for H 2O on Cu(111), although the 2 ×2 simulation cell would be, in principle, more appropriate in this case (with the smaller (√ 3×√ 3)R30◦ cell the separation between the periodic images of the water molecule is smaller and the coverage is higher than withthe 2×2 supercell, which may lead to stronger adsorbate- adsorbate interactions that affect the adsorption 70). Using this supercell, we have explicitly veriﬁed that the quasi-planarstructure is the favored one for the water monomer on Cu(111). As can be seen in Table III, again DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 give similar results,while vdW-DF and vdW-DF2 predict lower adsorption en-ergies and larger O-Cu(111) equilibrium distances and C 3 coefﬁcients. Note that, concerning the equilibrium distance, whose reference values are restricted within a relatively narrowrange, DFT/vdW-WF2s1, DFT/vdW-WF2s2, and DFT/vdW-WF2s3 perform much better than vdW-DF and vdW-DF2.As expected, in this case a pure (i.e. non vdW-corrected)PW91 calculation gives already a signiﬁcant amount of thebinding energy (about 20% considering DFT/vdW-WF2s1,DFT/vdW-WF2s2 and DFT/vdW-WF2s3, see Table I) and the screening corrections are relatively less relevant than in theprevious systems where the vdW interactions were dominant.In fact, although the water molecule and, for instance, theAr atom have the same number (8) of valence electronsand similar polarizabilities, electrostatic effects are also ofimportance for water due to its intrinsic electronic dipolemoment. In the present study we focus on (111) surfaces only, although our approach is expected to be applicable to other,interesting substrates, as already shown in preliminary applica-tions of the original DFT/vdW-WF scheme on the interaction of Ar, He, and H 2with two different Al surfaces.28Changing the surface face can have different effects on adsorptionprocesses. For instance, in the case of H 2on copper the experimental-based and computed potential-energy curves ofphysisorption of H 2on the Cu(111), Cu(100), and Cu(110) surfaces are very similar;17for H 2on aluminum, the measured physisorption well depth is similar for the (111) and (110) facesof Al but larger than for the intermediate (100) face. 61For water on copper, the interaction is stronger with the open Cu(110) andCu(100) surfaces than with the more closely packed Cu(111)surface. 70 IV . CONCLUSIONS In summary, we have investigated the adsorption of RGs and small molecules, H 2,C H 4, and H 2O on the Cu(111) metal surface, and of H 2on Al(111), and Xe on Pb(111), by considering three different recipes to include screeningeffects in our recently developed DFT/vdW-WF2 method. Byanalyzing the results of our study and comparing them toavailable reference data, we get a substantial improvementwith respect to the original, unscreened approach. Giventhe uncertainties in the reference data, one cannot easilystate which scheme is more appropriate. Considering allthe studied cases and, in particular, H 2-Cu(111) for which more reliable reference data are available, DFT/vdW-WF2s2turns out to be marginally superior which correlates with therelatively higher complexity of this approach. Interestingly,we conﬁrm the conclusion of previous studies (see, Ref. 47 and references therein) which suggest that, particularly forthe close-packed (111) surfaces, the assumption of a one-layer screening depth (single-layer approximation) worksreasonably well. The differences between the values of theequilibrium binding energies and distances predicted by thethree different schemes can be taken as the order of magnitudeof the uncertainty associated to the screened DFT/vdW-WF2method and to estimate its accuracy. Looking at the resultsreported in the tables, it turns out that these differences arerelatively large for the case of Xe on Pb(111), essentiallybecause the DFT/vdW-WF2s2 schemes predict a strongerbonding than DFT/vdW-WF2s1 and DFT/vdW-WF2s3. Thisbehavior is probably due to the fact that the free-electronapproximation for the s- and p-like orbital contribution, on which the DFT/vdW-WF2s2 approach is based [see Eq. 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PhysRevB.85.014401.pdf	PHYSICAL REVIEW B 85, 014401 (2012) Superposition of ferromagnetic and antiferromagnetic spin chains in the quantum magnet BaAg 2Cu[VO 4]2 Alexander A. Tsirlin,1,Angela M ¨oller,2,†Bernd Lorenz,3Yurii Skourski,4and Helge Rosner1 1Max Planck Institute for Chemical Physics of Solids, N ¨othnitzer Str. 40, DE-01187 Dresden, Germany 2Texas Center for Superconductivity, and Department of Chemistry, University of Houston, Houston, Texas 77204-5003, United States 3Texas Center for Superconductivity, and Department of Physics, University of Houston, Houston, Texas 77204-5005, United States 4Dresden High Magnetic Field Laboratory, Helmholtz-Zentrum Dresden-Rossendorf, DE-01314 Dresden, Germany (Received 30 September 2011; revised manuscript received 9 December 2011; published 3 January 2012) Based on density functional theory band-structure calculations, quantum Monte Carlo simulations, and high- ﬁeld magnetization measurements, we address the microscopic magnetic model of BaAg 2Cu[VO 4]2that was recently proposed as a spin-1 2anisotropic triangular lattice system. We show that the actual physics of this compound is determined by a peculiar superposition of ferromagnetic and antiferromagnetic uniform spin chainswith nearest-neighbor exchange couplings of J (1) a/similarequal−19 K and J(2) a/similarequal9.5 K, respectively. The two chains featuring different types of the magnetic exchange perfectly mimic the speciﬁc heat of a triangular spin lattice,while leaving a clear imprint on the magnetization curve that is incompatible with the triangular-lattice model.Both ferromagnetic and antiferromagnetic spin chains run along the crystallographic adirection, and slightly differ in the mutual arrangement of the magnetic CuO 4plaquettes and nonmagnetic VO 4tetrahedra. These subtle structural details are, therefore, crucial for the ferromagnetic or antiferromagnetic nature of the exchangecouplings, and put forward the importance of comprehensive microscopic modeling for a proper understandingof quantum spin systems in transition-metal compounds. DOI: 10.1103/PhysRevB.85.014401 PACS number(s): 75 .30.Et, 75.10.Pq, 71 .20.Ps, 75.50.Gg I. INTRODUCTION Frustration and dimensionality are two crucial parameters underlying the physics of magnetic systems. In insulators, these parameters rarely correlate with the apparent features of the atomic arrangement because superexchange couplingsare highly sensitive to details of the electronic structure andto positions of nonmagnetic atoms linking the magnetic sites.While computational techniques based on electronic-structurecalculations developed into a powerful tool for elucidatingspin lattices of complex materials, simple phenomenological criteria are equally important for the preliminary assessment of the experimental data and the compound under consideration. The best-known phenomenological criterion of the mag- netic frustration is the \|θ\|/T Nratio. It compares the Curie- Weiss temperature θ, which is often considered as an effective energy scale of the magnetic couplings, to the magneticordering temperature T N.1High\|θ\|/TNratios are believed to indicate strong frustration, although this rule will only hold for simple systems with few exchange couplings andwell-established dimensionality. Thus, the \|θ\|/T N/similarequal50–100 ratio is easily obtained even in nonfrustrated quasi-one-dimensional (1D) systems, where strong quantum ﬂuctuationsdue to the weak interchain couplings effectively prevent thesystem from long-range ordering down to low temperatures. 2–5 Another possible scenario is that of magnets with strong dimer correlations, where the long-range-ordered state com-petes with the disordered singlet ground state, and theordering temperature T Nmay be strongly reduced without any frustration involved.6–8The low \|θ\|/TNratio can be equally deceptive because θis in fact a linear combination of different exchange couplings that can be much smaller than the effective energy scale of the system. For example, the coexistence of ferromagnetic (FM) and antiferromagnetic(AFM) couplings renders θand\|θ\|/TNlow even in strongly frustrated magnets.9,10 The phenomenological assessment of the frustration in a magnetic system has to be backed by additional criteria.Magnetic speciﬁc heat is an especially appealing quantitybecause it is expressed in absolute units and does not requirean ambiguous reference to the effective energy scale ofthe system. Further, the magnetic speciﬁc heat distinguishesbetween the effects of dimensionality and frustration, with thelatter leading to a much stronger reduction in the maximum ofthe magnetic speciﬁc heat ( C m). For example, the spin-1 2square lattice (two-dimensional, nonfrustrated) reveals the maximumofC m/R/similarequal0.44, the spin-1 2uniform chain (one-dimensional, nonfrustrated) shows a lower maximum of Cm/R/similarequal0.35, but the speciﬁc-heat maximum for the spin-1 2triangular lattice (two-dimensional, frustrated) is even lower, Cm/R/similarequal 0.22.11The reduced magnetic speciﬁc heat is a seemingly unambiguous measure of the frustration. It can be equallyused to identify strongly frustrated spin systems 9,10,12or to refute premature conclusions on the strong frustration.13 However, this phenomenological criterion is not universal, aswe demonstrate in the following. In our study, the breakdown of the simple relationship between the magnetic speciﬁc heat and the frustration is relatedto a peculiar superexchange scenario in BaAg 2Cu[VO 4]2.T h i s compound has a fairly complex crystal structure with magneticCu 2+ions interspersed between the nonmagnetic [VO 4]3− tetrahedra as well as Ba2+and Ag+cations.14The spatial arrangement of Cu2+(Fig. 1) resembles a weakly anisotropic triangular lattice with the intraplane Cu-Cu distances of 5.45 ˚A (Ja), 5.63 ˚A(Jab1), and 5.69 ˚A(Jab2) and the interplane distance of 7.20 ˚A(Jc). This lattice topology should induce magnetic frustration, as further corroborated by the magneticspeciﬁc heat that reaches the maximum value of C m/R/similarequal0.22 014401-1 1098-0121/2012/85(1)/014401(8) ©2012 American Physical SocietyTSIRLIN, M ¨OLLER, LORENZ, SKOURSKI, AND ROSNER PHYSICAL REVIEW B 85, 014401 (2012) aa Cu1O4[V1O4] [VO4]Cu[V2O4] Cu2O4 Ba Agc JaJab1 Jab2 bb JcJc FIG. 1. (Color online) Top panel: perspective view of the BaAg 2Cu[VO 4]2structure showing alternating layers consisting of VO 4-bridged chains of Cu1 and Cu2, respectively. Different colors (shadings) identify the inequivalent CuO 4plaquettes and their slightly different orientation. Jcrefers to the interlayer coupling. Bottom panel: a single layer in the abplane (left) and the respective spin lattice with the intrachain coupling Jaas well as interchain couplings Jab1andJab2(right). and strongly resembles theoretical predictions for the spin-1 2 triangular lattice.14 In the following, we will show that the reduced Cmhas a different origin and arises from a peculiar superposition ofFM and AFM spin chains. The system is, therefore, quasi-one-dimensional and only weakly frustrated, in contrast tothe straightforward phenomenological assessment. To supportthe one-dimensional scenario, we perform extensive band-structure calculations combined with the ﬁtting of mag-netization and speciﬁc-heat data. We also present originalexperimental results on the high-ﬁeld magnetization thatunequivocally rules out the triangular-lattice spin model forBaAg 2Cu[VO 4]2. II. METHODS Our microscopic magnetic model of BaAg 2Cu[VO 4]2is based on full-potential scalar-relativistic density functionaltheory (DFT) band-structure calculations performed in the FPLO code15implementing the basis set of local orbitals. We used the local density approximation (LDA) with thePerdew-Wang parametrization for the exchange-correlationpotential. 16Thekmeshes of 292 points and 64 points in the symmetry-irreducible part of the ﬁrst Brillouin zonewere chosen for the crystallographic unit cell and supercell,respectively. Correlation effects were treated on a model levelor within the mean-ﬁeld local spin-density approximation (LSDA) +Uapproach, as further described in Sec. III. Thermodynamic properties were calculated with the loop 17 and dirloop sse (directed loop in stochastic series expansion representation)18quantum Monte Carlo (QMC) algorithms implemented in the ALPS simulation package.19Simulations were done for ﬁnite lattices with periodic boundary conditions.We used two independent chains containing L=40 sites each. This chain length is sufﬁcient to eliminate ﬁnite-size effectsfor thermodynamic properties within the temperature rangeunder investigation. Powder samples of BaAg 2Cu[VO 4]2were prepared accord- ing to the method described in Ref. 14. Magnetic susceptibility was measured with MPMS SQUID magnetometer in thetemperature range 2–380 K in the applied ﬁeld of 0.1 T.Magnetization isotherm was collected at 1.5 K using thepulsed magnet installed in Dresden High Magnetic Field Lab-oratory. Details of the experimental procedure are describedelsewhere. 20The low-temperature heat capacity was measured above 0.5 K by a relaxation method using the3He option of the Physical Property Measurement System (PPMS, QuantumDesign). III. MICROSCOPIC MAGNETIC MODEL LDA results for the band structure of BaAg 2Cu[VO 4]2 (Fig. 2) closely follow expectations for a Cu2+-based insulat- ing compound.21–24Oxygen 2 pstates between −6 and−2e V are surmounted by Ag 4 dand Cu 3 dbands. The states above 2 eV originate from unﬁlled V 3 dorbitals. While silver states are mostly found below −0.3e V ,C u3 dstates additionally form narrow bands in the vicinity of the Fermi level. Thecalculated partial densities of states conﬁrm the anticipatedvalences of Ag 1+(4d10), Cu2+(3d9), and V5+(3d0), and identify Cu2+ions as the magnetic sites in the structure. The metallic LDA energy spectrum violates the insulating natureof the compound, as evidenced by the dark-yellow color ofBaAg 2Cu[VO 4]2. This discrepancy is well understood, given the importance of correlation effects for the partially ﬁlled Cu3dshell and the severe underestimation of such correlations in LDA. The missing correlations can be introduced on themodel level, or by a mean-ﬁeld LSDA +Uprocedure. 0 /Minus6/Minus4/Minus2 0 2 4 Energy (eV)2040Total VCu Ag O60 DOS (eV )/Minus1 FIG. 2. (Color online) LDA density of states for BaAg 2Cu[VO 4]2. The Fermi level is at zero energy. 014401-2SUPERPOSITION OF FERROMAGNETIC AND ... PHYSICAL REVIEW B 85, 014401 (2012) 0.10.10.2 0.0 X M Y Z T R AEnergy (eV) FIG. 3. (Color online) LDA bands (thin light lines) and the ﬁt with the tight-binding model (thick dark lines). The kpath is deﬁned as follows: /Gamma1(0,0,0),X(0.5,0,0),M(0.5,0.5,0),Y(0,0.5,0), Z(0,0,0.5),T(0.5,0,0.5),R(0.5,0.5,0.5), and A(0,0.5,0.5), where the coordinates are given in units of the respective reciprocal lattice parameters. Following the ﬁrst approach to the treatment of correlations, we consider in more detail the narrow bands in the vicinity ofthe Fermi level (Fig. 3). The two bands can be assigned to two inequivalent Cu sites in the crystal structure. Both bandshave the d x2−y2orbital character, with xandyaxes directed along shorter Cu–O bonds. In BaAg 2Cu[VO 4]2, the local en- vironment of Cu2+resembles a severely elongated octahedron CuO 4+2, with four short Cu–O bonds (1 .96–1.97˚A) lying in the plane and two long bonds (2.44 ˚A) perpendicular to this plane. Therefore, the dx2-y2orbital is the highest-lying crystal-ﬁeld level in agreement with the LDA results. To ﬁt the dx2-y2bands with the tight-binding model, we construct Wannier functions localized on Cu sites.25The ﬁt perfectly reproduces the calculated band structure (Fig. 3), and yields Cu-Cu hopping parameters ti(Table I). By mapping the tight-binding model onto a one-orbital Hubbard model withthe effective on-site Coulomb repulsion U eff=4.5e V ,21,24we ﬁnd the anticipated strongly correlated regime ( ti/lessmuchUeff), and utilize second-order perturbation theory for analyzingthe lowest-lying (magnetic) excitations. This way, AFMcontributions to the exchange couplings are evaluated asJ AFM i=4t2 i/Ueff. TABLE I. Cu-Cu distances (in ˚A), hoppings ti(in meV), and exchange couplings Ji(in K) in BaAg 2Cu[VO 4]2. The AFM contri- butions JAFM i are calculated as 4 t2 i/UeffwithUeff=4.5 eV; the full exchange couplings Jiare obtained from LSDA +Ucalculations (Ud=6e V ,Jd=1 eV); and JFM i=Ji−JAFM i. The notation of Ji is illustrated in Fig. 1. Distance ti JAFM i JFM i Ji J(1) a 5.45 −11 1 −21 −20 J(2) a 5.45 −43 19 −16 3 J(1) ab1 5.63 −81 −3 −2 J(2) ab1 5.63 0 0 0 0 J(1) ab2 5.69 0 0 −0.3 −0.3 J(2) ab2 5.69 0 0 −0.3 −0.3 Jc 7.20 11 1 −0.30 .7TABLE II. Interatomic distances (in ˚A) and angles (in degrees) in the BaAg 2Cu[VO 4]2structure. The columns refer to the Cu1 and Cu2 layers, as shown in Fig. 1. The notation of individual atoms follows Fig.5(see text for details). Cu1-O1 2 ×1.973 Cu2-O3 2 ×1.969 Cu1-O2 2 ×1.974 Cu2-O4 2 ×1.959 Cu1-O8 2 ×2.436 Cu2-O7 2 ×2.444 V1-O1 1.749 V2-O3 1.757 V1-O2 1.740 V2-O4 1.755V1-O5 1.681 V2-O6 1.674 V1-O8 1.713 V2-O7 1.713 O1-O2 2.884 O3-O4 2.907 Cu1-O1 /prime-O2 113.1 Cu2-O4/prime-O3 113.4 Cu1-O2/prime-O1 145.1 Cu2-O3/prime-O4 144.6 ϕ(1)123.7 ϕ(2)102.2 The results of our model analysis are summarized in Table I.26While AFM couplings in BaAg 2Cu[VO 4]2are mostly weak, we ﬁnd the sizable AFM coupling J(2) aalong theadirection. Remarkably, this AFM coupling along a (denoted Ja) is observed for the Cu2 site and not for the Cu1 site, as emphasized by the superscripts (1) and (2) in thenotation of J i. This observation puts forward one important feature of the BaAg 2Cu[VO 4]2structure. The two Cu sites in BaAg 2Cu[VO 4]2are very similar and look nearly identical with respect to the geometry of individual superexchangepathways (Table II). The Cu1-Cu1 and Cu2-Cu2 distances in the abplane are equal because of the constraints imposed by the lattice translations. However, our microscopic analysisputs forward important differences between the deceptivelysimilar superexchange pathways within the Cu1 and Cu2sublattices (Table I). This difference gives a clue to understand the magnetism of BaAg 2Cu[VO 4]2, and will be discussed in more detail below. The FM part of the superexchange originates from pro- cesses beyond the one-orbital model employed in our tight-binding analysis. In cuprates, FM interactions are generallyascribed to the Hund’s coupling on the ligand site 22and can be evaluated by mapping total energies for different collinearspin conﬁgurations onto the classical Heisenberg model. Thetotal energies are obtained from spin-polarized band-structurecalculations with LSDA +Uas the mean-ﬁeld correction for correlation effects. Following previous studies of Cu 2+- based compounds,21,24we use the around-mean-ﬁeld double- counting correction scheme, the on-site Coulomb repulsionparameter U d=6 eV , and the Hund’s exchange parameter Jd=1 eV . In the case of BaAg 2Cu[VO 4]2, alterations of Udand the double-counting correction scheme have marginal inﬂuence on the results, and do not change the qualitativemicroscopic scenario. The total exchange couplings J ibased on the LSDA +U calculations are listed in the last column of Table I. We ﬁnd comparable FM contributions to the couplings J(1) aandJ(2) a along the adirection. Owing to the larger AFM contribution toJ(2) a, this coupling remains weakly AFM, while J(1) abecomes FM. Other couplings show small FM contributions and hoveraround zero. The LSDA +Ucalculations conﬁrm the leading couplings along aas well as the notable difference between 014401-3TSIRLIN, M ¨OLLER, LORENZ, SKOURSKI, AND ROSNER PHYSICAL REVIEW B 85, 014401 (2012) Cu1O1 O1O2O2 O2O2 V1 V1V1V1 FIG. 4. (Color online) Wannier function based on the Cu dx2-y2 orbital. J(1) aandJ(2) a. Before comparing our magnetic model to the experimental data, we further comment on the microscopicorigin of different exchange couplings in the Cu1 and Cu2sublattices of BaAg 2Cu[(VO 4]2. The sizable FM and AFM contributions are identiﬁed for the exchange couplings J(1) aandJ(2) aonly. This ﬁnding is easily rationalized based on the magnetic dx2-y2orbital of the Cu2+ions. The crystal structure is best viewed in terms of the CuO 4plaquettes entailing the magnetic orbitals. This representation underscores the 1D nature of the structure(Fig. 1), and illustrates the quasi-1D magnetic behavior. However, unlike the well-known spin-chain Cu 2+compounds, such as Sr 2CuO 3(Ref. 3) and CuPzN,4,27BaAg 2Cu[VO 4]2 reveals a combination of two inequivalent spin chains with strikingly different exchange couplings. According to Table I, both J(1) aandJ(2) afeature similar FM contributions, yet very different AFM exchanges arising fromdifferent Cu-Cu hoppings in the effective one-orbital model.To elucidate the origin of these couplings, we consider theWannier functions localized on Cu sites. Apart from the Cud x2-y2orbital forming the core of the Wannier function, we ﬁnd sizable contributions from oxygen 2 pand vanadium 3 d orbitals (Fig. 4). These contributions can also be observed in the LDA energy spectrum (Fig. 2). The Wannier functions of the neighboring Cu atoms overlap on the vanadium sites,where each Wannier function features a different 3 dorbital of vanadium. This leads to the Hund’s exchange on the vanadiumsite and explains the sizable FM contributions to J (1) aandJ(2) a, in contrast to the very low FM contributions to other nearest-neighbor couplings having similar Cu-Cu distances (Table I). Note that a comparable J FM/similarequal−15 K has been found in β- Cu2V2O7, where vanadium 3 dorbitals also contribute to the Cu-based Wannier functions.21 We now consider different AFM contributions to J(1) a andJ(2) a. Geometrical parameters summarized in Table II demonstrate a striking similarity between the respectivesuperexchange pathways for Cu1 and Cu2. The only notabledifference is the orientation of the VO 4tetrahedra with respect to the chains. Naively, the position of the tetrahedra isdescribed by the O8-O2-O1 and O7-O4-O3 angles (Fig. 5). However, these do not account for the different tilting ofthe Cu1O 4and Cu2O 4plaquettes with respect to the aaxis (Fig. 1). Therefore, we use dihedral angles ϕreferring to theO2/CurlyPhi(2)/CurlyPhi(1)O2' O3' O4'O1' O1O3 O4O7 O6 O8O5 V1V2Cu1 Cu2 a bc FIG. 5. (Color online) Comparison of the Cu1 (left) and Cu2 (right) chains in the BaAg 2Cu[VO 4]2structure. Note the different orientations of the VO 4tetrahedra with respect to the CuO 4plaquettes, as quantiﬁed by the respective dihedral angles ϕ(1)andϕ(2). O1/prime-O8-O2 and O1-O2-O1/prime-O2/primeplanes for Cu1 ( ϕ(1)), and to the O3/prime-O7-O4 and O3-O4-O3/prime-O4/primeplanes for Cu2 ( ϕ(2)). According to Table II, the difference between ϕ(1)andϕ(2)is as large as 21.5◦, thus, to be considered as the main feature to account for the different Cu-Cu hoppings t(1) aandt(2) a. To explore the role of the dihedral angles ϕ, we construct ﬁctitious model structures with the VO 4tetrahedra rotated about the O-O edges (O1-O2 and O3-O4 for V1 and V2,respectively). This way, we are able to tune ϕ (1)toward ϕ(2)=102.2◦and enhance t(1) ato 21 meV (compare to −11 meV at the experimental ϕ(1)=123.7◦), or change ϕ(2) toward ϕ(1)=123.7◦, thus reducing t(2) ato 3 meV (compare to−43 meV at the experimental ϕ(2)=102.2◦). Overall, a change of orientation by approximately 22◦is accompanied by a/Delta1(ta) of 32 and 46 meV , respectively. Therefore, the orientation of the nonmagnetic VO 4tetrahedra is of crucial importance for the Cu-Cu hoppings and AFM superexchange.Note, however, that this geometrical parameter is not unique,and the speciﬁc arrangement of the CuO 4plaquettes with respect to the chain direction (Figs. 1and5) is also responsible for the large AFM contribution to J(2) a, compared to the low AFM contribution to J(1) a. IV . EXPERIMENTAL DATA The DFT results summarized in Table Iidentify the spin lattice of BaAg 2Cu[VO 4]2as a system of weakly interacting inequivalent spin chains with the intrachain couplings J(1) aand J(2) a, respectively. While J(1) ais clearly FM, J(2) ais weakly AFM and probably close to zero. This qualitative scenariois veriﬁed by the magnetization isotherm measured at 1.5 K.Previous measurements 14in ﬁelds up to 5 T showed that half of the Cu spins seem to saturate around 1.5 T. Here, we extend ourstudy into the behavior of the magnetization in higher ﬁelds(Fig. 6). Based on these high-ﬁeld measurements, we show that the magnetization of BaAg 2Cu[VO 4]2is further increased between 1.5 and 16 T, where the full saturation with M/similarequal 1.08μB/f.u. is reached. This peculiar behavior apparently contradicts the conjecture on the triangular spin lattice thatwould lead to a smooth increase in the magnetization betweenzero ﬁeld and the saturation ﬁeld. 28 The experimental magnetization curve is readily elucidated by our microscopic model. While half of the spins comprising 014401-4SUPERPOSITION OF FERROMAGNETIC AND ... PHYSICAL REVIEW B 85, 014401 (2012) 10 20 30 0 Field (T)0.40.81.2 0.0Magnetization ( /f.u.)B FM chain, = 19 K J(1) a AFM chain, = 9.5 K J(2) aExperiment ( = 1.5 K) TFit (FM + AFM chains) FIG. 6. (Color online) Magnetization isotherm of BaAg 2Cu[VO 4]2measured at 1.5 K (ﬁlled circles) and the ﬁt with a combination of FM and AFM spin chains (solid line). The contributions of the FM (Cu1) and AFM (Cu2) chains are shown by the dashed and dotted lines, respectively. the FM spin chains (Cu1) align with the ﬁeld already at 1.0–1.5 T once thermal ﬂuctuations are suppressed, the remaining spins (Cu2) are coupled antiferromagnetically andrequire larger ﬁelds to overcome the AFM interactions. Thisbehavior strongly reminds of a two-sublattice ferrimagnet,where half of the maximum magnetization is recovered inlow ﬁelds, while larger ﬁelds are required to ﬂip one of thesublattices. Note, however, that BaAg 2Cu[VO 4]2is not in a magnetically ordered state at 1.5 K, hence, no magnetizationhysteresis is observed. The long-range magnetic order inBaAg 2Cu[VO 4]2is established below TC/similarequal0.7 K and is further discussed in Sec. V. The above qualitative picture can be quantiﬁed by ﬁtting the experimental magnetization data.29In BaAg 2Cu[VO 4]2, ﬁeld dependence of the magnetization (Fig. 6) and temperature dependence of the susceptibility (Fig. 7) are complemen- tary. The magnetization isotherm is sensitive to the AFMexchange J (2) athat determines the saturation ﬁeld, but the alignment of the FM component mostly depends on thermal 0.20.40.6 0.0 10Experiment ( = 0.1 T)0H Fit (FM + AFM chains) FM chain, = 19 K J(1) a AFM chain, = 9.5 K J(2) a 100 Temperature (K)(emu/mol) FIG. 7. (Color online) Magnetic susceptibility of BaAg 2Cu[VO 4]2measured in the applied ﬁeld of 0.1 T (ﬁlled circles) and the ﬁt with a combination of FM and AFM spin chains (solid line). The contributions of the FM (Cu1) and AFM (Cu2) chains are shown by the dashed and dotted lines, respectively.ﬂuctuations so that J(1) acan not be determined precisely. In contrast, the FM chains coupled by J(1) aproduce the dominant contribution to the susceptibility,30which gives an accurate estimate for J(1) a, while leaving certain ambiguity for J(2) a. The two sets of data are successfully ﬁtted with the samemodel parameters: J (1) a/similarequal−19 K, J(2) a/similarequal9.5K ,g/similarequal2.16 (Figs. 6and7). We also included a temperature-independent contribution to the susceptibility χ0/similarequal9×10−4emu/mol, which accounts for the van Vleck paramagnetism and corediamagnetism. Our ﬁtted gvalue is in excellent agreement with the experimental powder-averaged ¯g=2.18. 14While J(1) a closely follows the DFT prediction (Table I), the computational estimate of J(2) ais less accurate, although still acceptable considering the low energy scale of the exchange couplings inBaAg 2Cu[VO 4]2.31 Figures 6and7illustrate the contributions of the FM and AFM components to the magnetization and susceptibility ofBaAg 2Cu[VO 4]2, respectively. The FM chains lead to the sharp increase in the susceptibility at low temperatures, while the 0.10 0.100.20 0.200.30 0.300 0,taehcificepscitenga M/R CmExperiment 0H=3T0H=0T 0H=7TFM+AFMFM chain AFM chain 5 10 15 0 Temperature (K)0.100.20 0 FIG. 8. (Color online) Magnetic part of the speciﬁc heat [ Cm(T)] divided by the gas constant ( R) for BaAg 2Cu[VO 4]2measured in zero ﬁeld (top) and in applied ﬁelds of 3 T (middle) and 7 T (bottom). The simulated curves for the combination of FM and AFM spinchains are shown by solid lines, whereas the dashed and dotted lines denote the contributions of the FM (Cu1) and AFM (Cu2) spin chains, respectively. Experimental data (circles) are taken from Ref. 14.T h e model parameters J (1) a=−19 K and J(2) a=9.5 K are extracted from the ﬁts to the magnetization data (Figs. 6and 7). Therefore, we compare our model to the experiment with no adjustable parameters. 014401-5TSIRLIN, M ¨OLLER, LORENZ, SKOURSKI, AND ROSNER PHYSICAL REVIEW B 85, 014401 (2012) contribution of the AFM chains is barely visible on the same scale. The contribution of the FM chains to the magnetizationisotherm is saturated at low ﬁelds and corresponds to onehalf of the maximum magnetization because half of the Cuatoms belong to the FM chains. The magnetization of theAFM chains is linear at low ﬁelds, bends upward above 7 T,and ﬁnally saturates around 16 T where the full alignment ofspins is achieved. We will now test our quasi-1D model against the experimen- tal speciﬁc-heat data showing the strongly reduced maximumthat might be characteristic of a spin- 1 2triangular lattice. We use the ﬁtted parameters based on the magnetization dataand, therefore, compare our model to the experiment withno adjustable parameters. 32Figure 8presents the magnetic speciﬁc-heat data measured in zero ﬁeld and in two repre-sentative applied ﬁelds along with the simulated curves. Theremarkable agreement between the experiment and the modelprediction conﬁrms our microscopic scenario and suggeststhat the strongly reduced speciﬁc-heat maximum, especiallyin zero ﬁeld, is not an unambiguous footprint of the magneticfrustration. In zero ﬁeld, the speciﬁc-heat maximum closely follows the contribution of the AFM spin chains, while the FM chains withthe stronger coupling J (1) a/similarequal−19 K provide a temperature- independent “background” below 15 K. The applied ﬁeldof 3 T increases the maximum up to C m/R/similarequal0.32. The stronger ﬁeld of 7 T additionally shifts the maximum tohigher temperatures. Both effects are perfectly reproducedby our microscopic model. Magnetic ﬁelds transform thetemperature-independent zero-ﬁeld speciﬁc heat of the FMchains into a small maximum at 3 .5–4.0 K. This maximum of the FM contribution weakly depends on the ﬁeld becausethe FM (Cu1) subsystem is saturated above 2 T (Fig. 6). By contrast, the contribution of the AFM chains shows apronounced ﬁeld dependence that underlies the evolution ofthe experimental magnetic speciﬁc heat in the applied ﬁeld. V . DISCUSSION AND SUMMARY The combination of DFT calculations and QMC ﬁts to the experimental data gives compelling evidence for the quasi-1Dmagnetic behavior of BaAg 2Cu[VO 4]2. The superposition of FM and AFM spin chains with different magnitudes ofthe exchange couplings results in peculiar and perplexingthermodynamic properties. While the zero-ﬁeld speciﬁc heatresembles the typical response of the spin- 1 2triangular lattice, the magnetization isotherm is reminiscent of a system withtwo different magnetic sublattices and underpins the proposedmagnetic model. Based on our microscopic analysis, we establish the spin lattice of BaAg 2Cu[VO 4]2as a peculiar derivative of conventional Heisenberg spin chains with nearest-neighborexchange coupling J. This model was widely studied for both FM and AFM J, 33–35but the combination of FM and AFM spin chains was neither considered theoretically nor encounteredexperimentally. The superposition of inequivalent spin chains is a challenge for “nonlocal” experimental techniques, such as thermody-namic measurements or inelastic neutron scattering, that probethe system as a whole. These methods inevitably blend thesignals of different sublattices, and generally lead to a complex response that can be fully elucidated based on the microscopicapproach only. A more direct experimental information couldbe extracted from “local” methods, which probe different mag-netic sublattices independently. For example, an elegant way tostudy the physics of BaAg 2Cu[VO 4]2further could be nuclear magnetic resonance (NMR) on51V nuclei. The inequivalent vanadium sites V1 and V2 are coupled to Cu1 and Cu2, respec-tively. Owing to the very similar local environment, the signalsfrom these two vanadium sites should perfectly overlap at hightemperatures. At low temperatures, though, the lines will splitbecause of the different Knight shifts resulting from the dis-parate local magnetization in the vicinity of the FM and AFMspin chains. Therefore, the NMR experiment should be a valu-able additional experimental test of our microscopic model. Another interesting problem is the long-range-ordered (LRO) ground state of BaAg 2Cu[VO 4]2. While isolated spin chains do not show the LRO even at zero temper-ature, interchain couplings induce the LRO state at a ﬁ-nite temperature, 36–38irrespective of the weak frustration that could be induced by the triangular arrangement.39In BaAg 2Cu[VO 4]2, speciﬁc-heat measurements reveal the sharp anomaly at TC/similarequal0.7 K in zero ﬁeld. This anomaly is drastically suppressed even in weak magnetic ﬁelds (Fig. 9, see also Ref. 14), as typical for a ferromagnetic transition, or, more generally, for an LRO state with nonzero netmagnetization. Such a ground state can be indeed derivedfrom our microscopic model and explained in terms of atwo-sublayer system with the interlayer exchange couplingJ c. As outlined above, these sublayers are stacked along the c axis in an alternate fashion. Each plane consists either of FM(Cu1) or AFM (Cu2) spin chains, respectively (Fig. 1). The Cu1 spins within the FM spin chains prefer the parallel alignment so that a FM sublattice is formed. The Cu2 spinsare expected to be ordered antiferromagnetically along a and form an AFM sublattice. The nature of the interchaincouplings is more difﬁcult to establish because of their lower FIG. 9. (Color online) Field-dependent magnetic part of the speciﬁc heat Cm(T)d i v i d e db y Rfor BaAg 2Cu[VO 4]2at low temperatures showing the behavior typical for a ferromagnetic or ferrimagnetic transition. 014401-6SUPERPOSITION OF FERROMAGNETIC AND ... PHYSICAL REVIEW B 85, 014401 (2012) energy scale, which might allow for additional, nonisotropic contributions, such as dipolar interactions. However, even theisotropic (Heisenberg) model based on DFT enables us to makea plausible conjecture about the ground state. The couplingsJ ab1andJab2in the abplane (Table I) are compatible with both FM and AFM exchange along a. These couplings should reinforce the formation of the FM sublattice for Cu1 andthe AFM sublattice for Cu2. The AFM coupling J calong cintroduces a weak frustration, but its effect should be small. Altogether, BaAg 2Cu[VO 4]2entails two inequivalent sublattices and presents a peculiar example of a spin-1 2system with nonzero net magnetization. From a phenomenological point of view, a similar ground state with the nonzero net magnetization has been recentlyobserved in the spin- 1 2ferrimagnet Cu 2OSeO 3.40However, unlike conventional ferrimagnets and unlike Cu 2OSeO 3, BaAg 2Cu[VO 4]2does not feature well-deﬁned sublattices with opposite directions of the spin, and rather showsa sequence of FM and AFM layers. Unfortunately, thefrustrated nature of the interlayer coupling J cprevents us from using QMC for simulating the ground-state propertiesand the transition temperature T C. Therefore, we are presently unable to verify the proposed magnetic structure. Furtherexperimental studies, such as neutron diffraction, would berequired to tackle this problem. The microscopic magnetic model of BaAg 2Cu[VO 4]2is furthermore instructive from a structural viewpoint. The Cu1and Cu2 sites look deceptively similar, so that one wouldnot expect any substantial difference between the magneticcouplings within the two sublattices. However, the couplingsare very different, not only in the magnitude but also in thenature, because of the subtle inﬂuence of the VO 4tetrahedra connecting the neighboring CuO 4plaquettes. The effect of the nonmagnetic group is sizable and twofold. Vanadium 3 d orbitals contribute to the Wannier functions and induce a FMsuperexchange, which is weakly dependent on the speciﬁcarrangement of the VO 4tetrahedra. This FM contribution represents a constant term that is superimposed on a variableAFM superexchange. The latter is controlled by the Cu-Cuhoppings, showing dramatic dependence on the mutual orien-tation of the VO 4tetrahedra and CuO 4plaquettes. Depending on the speciﬁc geometry, the AFM contributions may or maynot surpass the FM superexchange, and qualitatively differentexchange couplings emerge. The subtle dependence of AFM superexchange on the orientation of nonmagnetic tetrahedra is reminiscent of thelong-range couplings in BiCu 2PO6, where slight rotations of the bridging PO 4groups modify the interactions by 50–70 K.41More generally, the unusual microscopic scenario of BaAg 2Cu[VO 4]2conﬁrms the crucial importance of non- magnetic bridging groups for the superexchange in magneticinsulators. Other remarkable examples include the effect ofGeO 4tetrahedra on the Cu-based spin chains in CuGeO 3 (Ref. 42), as well as the unusual ferromagnetism of CdVO 3 related to the low-lying 5 sorbitals of Cd atoms.43The effect of the nonmagnetic groups opens broad prospects for tweakingsuperexchange couplings by minor alterations of the crystalstructure. For example, BaAg 2Cu[VO 4]2is likely to sustain cation substitutions in the Ba and Ag positions, thus leadingto further interesting combinations of FM and/or AFM spinchains in a single chemical compound. In summary, we have derived a microscopic magnetic model of BaAg 2Cu[VO 4]2, and presented a consistent inter- pretation of the available experimental data for this compound.The crucial and highly unexpected feature of BaAg 2Cu[VO 4]2 is the dramatic difference between the couplings withinthe Cu1 and Cu2 sublattices. While the Cu1 sublattice isferromagnetic, the Cu2 sublattice is antiferromagnetic. Thisunusual, and so far unreported, combination of weakly coupledFM and AFM spin chains within a single chemical compoundleads to peculiar thermodynamic properties. The speciﬁc heatresembles that of a strongly frustrated two-dimensional spinsystem. The spin lattice of BaAg 2Cu[VO 4]2is, however, only weakly frustrated and quasi-1D, as conﬁrmed by the high-ﬁeldmagnetization measurements, suggesting the ground state withnonzero net magnetization. The different couplings withinsimilar structural units are solely determined by the orientationof the nonmagnetic VO 4tetrahedra with respect to the CuO 4 plaquettes. These results present an instructive example on theimportance of bridging groups for superexchange pathways,and open interesting opportunities for tuning low-dimensionalspin systems within a given structure type. ACKNOWLEDGMENTS We are grateful to O. Janson and D. Kasinathan for stimulat- ing discussions, and to N. E. Amuneke for her help in samplepreparation. The high-ﬁeld magnetization measurements weresupported by EuroMagNET II under the EC Contract No.228043. A.T. acknowledges the funding from Alexander vonHumboldt Foundation. 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PhysRevB.83.115432.pdf	PHYSICAL REVIEW B 83, 115432 (2011) Structures of ﬂuorinated graphene and their signatures H. S¸ahin,1M. Topsakal,1and S. Ciraci1,2,* 1UNAM-Institute of Materials Science and Nanotechnology, Bilkent University, 06800 Ankara, Turkey 2Department of Physics, Bilkent University, 06800 Ankara, Turkey (Received 4 February 2011; published 15 March 2011) Recent synthesis of ﬂuorinated graphene introduced interesting stable derivatives of graphene. In particular, ﬂuorographene (CF), namely, fully ﬂuorinated chair conformation, is found to display crucial features, such ashigh mechanical strength, charged surfaces, local magnetic moments due to vacancy defects, and a wide band gaprapidly reducing with uniform strain. These properties, as well as structural parameters and electronic densitiesof states, are found to scale with ﬂuorine coverage. However, most of the experimental data reported to dateneither for CF nor for other C nF structures complies with the results obtained from ﬁrst-principles calculations. In this study, we attempt to clarify the sources of disagreements. DOI: 10.1103/PhysRevB.83.115432 PACS number(s): 73 .22.Pr, 61.48.Gh, 63 .22.Rc, 71 .20.−b I. INTRODUCTION Active research on graphene1revealed not only numerous exceptional properties2–5but also have prepared the grounds for the discovery of several graphene-based materials. Prepara-tion of freestanding graphene sheets with nonuniform oxygencoverage have been achieved. 6More recently the synthesis of two-dimensional hydrocarbon in a honeycomb structure,so-called graphane 7(CH), showing diverse electronic, mag- netic, and mechanical properties,8–12is reported. According to the Pauling scale, F has an electronegativity of 3.98, which is higher than that of C(2.55), H(2.20), andO(3.44), and hence ﬂuorination of graphene is expected toresult in a material that may be even more interesting than bothgraphene oxide and CH. Before the ﬁrst synthesis of graphene,ﬂuorinated graphite has been treated theoretically. 13,14Owing to promising properties revealed for CH, ﬂuorinated graphenestructures are now attracting considerable interest 15–22despite uncertainties in their chemical compositions and atomicstructures. In an effort to identify the structures of ﬂuorinatedsamples, previous theoretical models attempted to deducethe lowest-energy structures. 13,15In addition, band gaps of different structures calculated within density functional theory(DFT) are compared with the values revealed through speciﬁcmeasurements. 17,18However, the stability of proposed struc- tures has not been questioned, and an underestimation of bandgaps within DFT has not been studied. The Raman spectrumby itself has been limited in specifying C nF structures.18 In this work, we ﬁrst determined stable C nF structures forn/lessorequalslant4. Then we revealed speciﬁc properties (such as internal structural parameters, elastic constants, the formationand binding energies, the energy band gap, and photoelectricthreshold) for those stable structures as signatures to identifythe derivatives probed experimentally. We placed an emphasison fully ﬂuorinated graphene or ﬂuorographene (CF), in whichDand GRaman peaks of bare graphene disappear after a long ﬂuorination period. 17,18The present study reveals that the properties, such as structural parameters, binding energy,band gap, and phonon modes of various ﬂuorinated structures,are strongly dependent on the binding structure of F atomsand their composition. Some of these properties are foundto roughly scale with F coverage. While the stable C 2F chair structure is metallic, CF is a nonmagnetic insulator with a bandgap,Eg, being much larger than 3 eV , i.e., a value attributed experimentally to fully ﬂuorinated graphene. In view of thecalculated diffusion constant, Raman-active modes, and otherproperties, available experimental data suggest that domains(or grains) of various C nF structures with extended and imperfect grain boundaries can coexist after the ﬂuorinationprocess. Hence the measured properties are averaged fromdiverse perfect and imperfect regions. II. COMPUTATIONAL METHODOLOGY Our predictions are obtained from ﬁrst-principles plane- wave calculations23within DFT, which is demonstrated to yield rather accurate results for carbon-based materials. Cal-culations are performed using the spin-polarized local-densityapproximation (LDA) 24and projector augmented wave (PAW) potentials.25The kinetic energy cutoff ¯ h2\|k+G\|2/2mfor a plane-wave basis set is taken as 500 eV . In the self-consistentpotential and total energy calculations of ﬂuorographene a setof (25 ×25×1)k-point samplings is used for Brillouin zone (BZ) integration. The convergence criterion of self-consistentcalculations for ionic relaxations is 10 −5eV between two consecutive steps. By using the conjugate gradient method, allatomic positions and unit cells are optimized until the atomicforces are less than 0.03 eV /˚A. Pressures on the lattice unit cell are decreased to values less than 0.5 kbar. The energy bandgap, which is usually underestimated in DFT, is corrected byfrequency-dependent GW 0calculations.26InGW 0corrections screened Coulomb potential, W, is kept ﬁxed to an initial DFT valueW0and the Green’s function, G, is iterated four times. Various tests are performed regarding vacuum spacing, kineticenergy cutoff energy, number of bands, kpoints, and grid points. Finally, the band gap of CF is found 7.49 eV afterGW 0correction, which is carried out by using (12 ×12×1) kpoints in BZ, a 15- ˚A vacuum spacing, a default cutoff potential for GW 0, 192 bands, and 64 grid points. Phonon frequencies and phonon eigenvectors are calculated using thedensity functional perturbation theory (DFPT). 27 III. STRUCTURES OF FLUORINATED GRAPHENE Each carbon atom of graphene can bind only one F atom, and through coverage (or decoration) of one or two sides of 115432-1 1098-0121/2011/83(11)/115432(6) ©2011 American Physical SocietyH. S¸AHIN, M. TOPSAKAL, AND S. CIRACI PHYSICAL REVIEW B 83, 115432 (2011) graphene, one can achieve diverse C nF structures. Uniform F coverage is speciﬁed by /Theta1=1/n(namely, one F adatom per nC atoms), whereby /Theta1=0.5 corresponds to half ﬂuorination and/Theta1=1 is ﬂuorographene CF. The adsorption of a single F atom to graphene is a precursor for ﬂuorination. When placedat diverse sites of a (4 ×4) supercell of graphene, a simple F atom moves to the top site of a carbon atom and remainsadsorbed there. The resulting structure is nonmagnetic and itsbinding energy is E b=2.71 eV in equilibrium, which is a rather strong binding unlike many other adatoms adsorbed tographene. An energy barrier, Q B=∼ 0.45 eV , occurs along its minimum energy migration path. Our calculations, relatedwith the minimum energy path of a single F atom, followhexagons of the underlying graphene. Namely, the F atommigrates from the highest binding energy site, i.e., the top site(on top of the carbon atom) to the next top site through a bridgesite (the bridge position between two adjacent carbon atoms ofgraphene). The corresponding diffusion constant for a singleFa t o m , D=νae −QB/kBT, is calculated in terms of the lattice constant, a=2.55˚A, and characteristic jump frequency ν≈ 39 THz. Experiments present evidence that energy barrierson the order of 0.5 eV would make the adatoms mobile. 18,28 Moreover, this energy barrier is further lowered even it is collapsed in the presence of a second F atom at close proximity. Γ ΓΜ Κ (b)C F BOAT2 ab XS Yααd d Γ Γ 50010001500 Phonon Frequency (cm )-1 0 C F ab4 α C α C* dCCdCC ab αα(a)C F CHAIR2 (c) Γ ΓΜ Κ Fα = 104dd\|a\| = 2.52 CCd = 1.48d=1.47CF Cα= 116\|b\| = 2.52 Fα = 101 \|a\| = 4.92 CCd = 1.49d=1.43CF Cα= 114\|b\| = 4.92 Cα = 119 CCd = 1.39 Fα = 101\|a\| = 2.54 CCd = 1.51d=1.40CF Cα= 114\|b\| = 4.36 Cα = 118 α = 100 F* CCd = 1.61ο ο ο ο ο ο ο ο ο50010001500 Phonon Frequency (cm )-1 050010001500 Phonon Frequency (cm )-1 0 FIG. 1. (Color online) Atomic structure and calculated phonon bands (i.e., phonon frequencies vs wave vector, k)o fv a r i o u s optimized C nF structures calculated along the symmetry directions of BZ. Carbon and ﬂuorine atoms are indicated by black (dark) andblue (light) balls, respectively. (a) C 2F chair structure. (b) C 2F boat structure. (c) C 4F structure. Units are ˚A for structural parameters and cm−1for frequencies.Consequently, this situation, together with the tendency toward clustering, favors that C nF grains (or domains) of different n on graphene can form during the course of ﬂuorination. Wenote that the energy barrier for the diffusion of a single carbonadatom adsorbed on the bridge sites of graphene was calculatedto be in a similar energy range. Carbon adatoms on graphenewere found to be rather mobile. That energy barrier for a singleC adatom was found to decrease, and even to collapse at a closeproximity to a second adatom. 29 In earlier theoretical studies,13,15,17the total energies and/or binding energies were taken as the criteria for whether a givenC nF structure exists. Even if a C nF structure seems to be in a minimum on the Born-Oppenheimer surface, its stabilityis meticulously examined by calculating frequencies of allphonon modes in BZ. Here we calculated phonon dispersionsof most of the optimized C nF structures. We found that the C4F, the C 2F boat, the C 2F chair (see Fig. 1), and the CF chair (see Fig. 2) structures have positive frequencies throughout the BZ, indicating their stability. Some of phonon branches of C nF structures (for exam- ple, the CF boat) have imaginary frequencies and henceare unstable, in spite of the fact that their structures canbe optimized. The possibility that these unstable structurescan occur at ﬁnite and small sizes is, however, not ex-cluded. For stable structures, the gap between optical andacoustical branches is collapsed, since the optical branches 04001000(b) d = 1.37CF d = 1.55CC δ = 0.49(a) ΜΚ Eg A 1gEg A 1gΩ=245 Ω=681Ω=1264 Ω=1305(c) Graphene CH CF 0 500 1000 1500 2000 2500 3000 Frequency (cm )C F -14(d) 20060080012001400 ab Phonon Frequency (cm )-1 FIG. 2. (Color online) (a) Atomic structure of ﬂuorographene CF.aandbare the lattice vectors ( \|a\|=\|b\|) of a hexagonal structure; dCC(dCF) is the C-C (C-F) bond distance; δis the buckling. (b) Phonon frequencies vs wave vector kof optimized CF calculated along symmetry directions in BZ. (c) Symme-tries, frequencies, and descriptions of Raman-active modes of CF. (d) Calculated Raman-active modes of graphene, CH, CF, and C 4F are indicated on the frequency axis. Those modes indicated by “ +” are observed experimentally. There is no experimental Raman data in the shaded regions. Units are ˚A for structural parameters and cm−1 for frequencies. 115432-2STRUCTURES OF FLUORINATED GRAPHENE AND THEIR ... PHYSICAL REVIEW B 83, 115432 (2011) TABLE I. Comparison of the calculated properties of four stable, ﬂuorinated graphene structures (namely, CF, the C 2F chair, the C 2F boat, and C 4F) with those of graphene and CH. Lattice constant, a=b(a/negationslash=bfor rectangular lattice); C-C bond distance, dCC(second entries with the slash differ from the previous one); C-X bond distance [X indicating H (F) atom for CH (CF)], dCX; the buckling, δ; angle between adjacent C-C bonds, αC; angle between adjacent C-X and C-C bonds, αX; total energy per cell comprising eight carbon atoms ET; formation energy per X atom relative to graphene, Ef; binding energy per X atom relative to graphene, Eb(the value in parentheses, Eb/prime, excludes the X-X coupling); desorption energy, Ed(see the text for formal deﬁnitions); energy band gap calculated by LDA, ELDA g; energy band gap corrected byGW 0,EGW 0g; photoelectric threshold, /Phi1; in-plane stiffness, C; Poisson ratio, ν. All materials are treated in a hexagonal lattice, except for the C 2F boat, which has a rectangular lattice. a(b) dCC dCXδα C αXELDA gEGW 0g ET EfEb(Eb/prime)Ed/Phi1C Material ( ˚A) ( ˚A) ( ˚A) ( ˚A) (deg) (deg) (eV) (eV) (eV) (eV) (eV) (eV) (eV) (J /m2)ν Graphene (Ref. 30) 2.46 1.42 – 0.00 120 – 0.00 0.00 −80.73 – – – 4.77 335 0.16 CH (Ref. 10) 2.51 1.52 1.12 0.45 112 107 3.42 5.97 −110.56 0.39 2.8(2.5) 4.8 4.97 243 0.07 CF 2.55 1.55 1.37 0.49 111 108 2.96 7.49 −113.32 2.04 3.6(2.9) 5.3 7.94 250 0.14 C2F chair 2.52 1.48 1.47 0.29 116 101 Metal Metal −89.22 0.09 1.7(0.9) 1.2 8.6/5.6 280 0.18 C2F boat 2.54(4.36) 1.51/1.61 1.40 0.42 114/118 100/101 1.57 5.68 −92.48 0.91 2.5(1.6) 2.4 7.9/5.1 286(268) 0.05 C4F 4.92 1.49/1.39 1.43 0.34 114/119 104 2.93 5.99 −87.68 1.44 3.0(2.7) 3.5 8.1/5.6 298 0.12 associated with the modes of C-F bonds occur at lower frequencies. This situation is in contrast with the phononspectrum of graphane, 10where optical modes related with C-H bonds appear above the acoustical branches at ∼2900 cm−1. The formation energy of ﬂuorination is deﬁned as Ef= (nF2ET,F2+ET,Gr−ET,CnF)/nFin terms of the total ground- state energies of optimized structures of graphene andﬂuorinated graphenes at different compositions, respec-tively, E T,Gr,ET,CnF, and the total ground-state energy of a single carbon atom, ET,C,o faF 2molecule and a F atom, ET,F2andET,F. Similarly, the binding energy of the F atom relative to graphene including F-F coupling isE b=(ET,Gr+nFET,F−ET,CnF)/nFand without F-F cou- pling Eb/prime=(ET,Gr+ET,n FF−ET,CnF)/nF.H e r e ET,n FFis the total energy of suspended single or double layers of Foccupying the same positions as in C nF. The desorption energy, Edis the energy required to remove one single F atom from the surface of C nF.nF2andnFare numbers of F 2molecules and F atoms, respectively. The total energies are calculatedin periodically repeating supercells comprising eight carbonatoms and keeping all the parameters of calculations describedabove using spin-polarized as well as spin-unpolarized LDA.The lowest (magnetic or nonmagnetic) total energy is used asthe ground-state total energy. Fluorographene (CF), where F atoms are bound to each C atom of graphene alternatingly from top and bottom sides,is energetically the most favorable structure. Upon full ﬂuori-nation, the planar honeycomb structure of C atoms becomesbuckled (puckered) and the C-C bond length increases by∼10%. At the end, while planar sp 2bonding of graphene is dehybridized, the buckled conﬁguration is maintained bysp 3-like rehybridization. In Table I, the calculated lattice constants, internal structural parameters, relevant bindingenergies, and energy band gaps of stable C nF structures are compared with those of bare graphene and CH.10Notably, internal parameters (such as δ, C-C bond length) as well as lattice constants of various C nF structures vary with F coverage, /Theta1. CF has the highest values for Ef,Eb,Eb/prime, and Edgiven in Table I; those of C 4F are second highest among stable C nF structures.Since the Raman spectrum can convey information for a particular structure and hence can set its signature, thecalculated Raman-active modes of stable C 4F and CF struc- tures, together with those of graphene and CH, are alsoindicated in Figs. 2(c) and2(d). It is known that the only characteristic Raman active mode of graphene at 1594 cm −1 is observed so far.31Similarly, for CH the mode at ∼1342 cm−1is observed.7One of two Raman-active modes of C 4F at 1645 cm−1seems to be observed.17In compliance with the theory,32phonon branches of all these observed modes exhibit a kink structure. However, none of the Raman activemodes of CF revealed in Fig. 2has been observed yet. Raman spectroscopy in the low-frequency range may be useful inidentifying experimental structures. IV . ELECTRONIC STRUCTURES Energy bands, which are calculated for the optimized C 4F, the C 2F boat, the C 2F chair, and the CF chair structures are presented in Figs. 3and4, respectively. The orbital projected densities of states (PDOS), together with the total densitiesof states of these optimized structures, are also presented. Ananalysis of the electronic structure can also provide data to re-veal the observed structure of the ﬂuorinated graphene. As seenin Table I, stable C nF structures have LDA band gaps ranging from 0 to 2.96 eV . Surprisingly, the C 2F chair structure is found to be a metal owing to the odd number of valence electrons inthe primitive unit cell. Even if various measurements on theband gap of ﬂuorinated graphene lie in the energy range from68 meV (Ref. 16)t o3e V , 18these calculated band gaps are underestimated by LDA. Incidentally, the band gaps changesigniﬁcantly after they are corrected by various self-energymethods. In fact, the correction using the GW 0self-energy method predicts a rather wide band gap of 7.49 eV for CF.The corrected band gaps for the C 2F boat structure and C 4F are 5.68 and 5.99 eV , respectively. It should be noted that theGW 0self-energy method has been successful in predicting the band gaps of three-dimensional (3D) semiconductors.33 While predicting a much larger band gap for CF, the measured band gap of ∼3 eV reported by Nair et al.18marks the serious discrepancy between theory and experiment. The 115432-3H. S¸AHIN, M. TOPSAKAL, AND S. CIRACI PHYSICAL REVIEW B 83, 115432 (2011) EFEnergy (eV)8 -80 Γ ΓΜ ΚEnergy (eV)8 -80 XSEF Y Γ ΓEnergy (eV)8 -80 Γ ΓΜ ΚEFpz sp +p total Carbon Fluorine Carbon Fluorinepz sp +p total Carbon Fluorinepz sp +p total(b) (c)(a) Electronic DOSElectronic DOSElectronic DOS GW O FIG. 3. (Color online) Energy band structures of various stable CnF structures, together with the orbital PDOS and the total densities of states (DOS). The LDA band gaps are shaded and the zero of energy is set to the Fermi level EF. The total DOS is scaled to 45%. Valence- and conduction-band edges after GW 0correction are indicated by ﬁlled (red) circles. (a) C 2F chair structure. (b) C 2F boat structure. (c) C 4F structure. character of the band structure of CF is revealed from the analysis of PDOS as well as charge densities of speciﬁcbands in Fig. 4(b). The conduction-band edge consists of the antibonding combination of p zorbitals of F and C atoms. Thepzorbitals of C atoms by themselves, are combined to formπbands. The bands at the edge of the valence band are derived from the combination of C-( px+py) and F-( px+py) orbitals. The total contribution of the C orbitals to the valenceband can be viewed as the contribution of four tetrahedrallycoordinated sp 3-like hybrid orbitals of the sandporbitals of the C atoms. However, the deviation from tetrahedralcoordination increases when nincreases or the single side is ﬂuorinated. As a matter of fact, the total DOS presentedin Figs. 3and4marks crucial differences. In this respect, spectroscopy data is expected to yield signiﬁcant informationregarding the observed structures of ﬂuorinated graphenes. The contour plots of the total charge density, ρ T,i nt h e F-C-C-F plane suggests the formation of strong covalent C-Cbonds from the bonding combination of two C- sp 3hybrid orbitals. The difference charge density, /Delta1ρ(which is obtained by subtracting the charges of free C and free F atoms situated Γ-pointC2 C1 V1 V2Κ-point Total Charge ( ρ ) Difference Charge ( Δρ) CarbonFluorine Δρ<0Δρ<0(a) (b)Energy (eV)11 -110 (c) Δρ>0Δρ>0Γ ΓΜ ΚEF Carbon Fluorinepz sp +pxy total ΤC1C2 V1 V2GW o FIG. 4. (Color online) (a) Energy-band structure of CF, together with the orbital PDOS and total DOS. The LDA band gap is shaded and the zero of energy is set to the Fermi level, EF. Valence- and conduction-band edges after GW 0correction are indicated by ﬁlled (red) circles. (b) Isosurfaces of charge DOS corresponds to ﬁrst (V1), second (V2) valence and ﬁrst (C1) and second (C2) conduction bands at the /Gamma1andKpoints. (c) Contour plots of the total charge density ρTand difference charge density /Delta1ρin the plane passing through F-C-C-F atoms. Contour spacings are 0.03 e/˚A3. at their respective positions in CF), indicates charge transfer to the middle of the C-C bond and to F atom, revealingthe bond charge between C atoms and the ionic characterof the C-F bond. However, the value of the charge transferis not unique, but diversiﬁes among different methods ofanalysis. 34Nevertheless, the direction of the calculated charge transfer is in compliance with the Pauling ionicity scale and iscorroborated by calculated Born effective charges, which havein-plane ( /bardbl) and out-of-plane ( ⊥) components on C atoms, Z ∗ C,/bardbl=0.30,Z∗ C,⊥=0.35 and on F atoms Z∗ F,/bardbl=−0.30, Z∗ F,⊥=−0.35. Finally, we note that a perfect CF is a nonmagnetic insulator. However, a single isolated F vacancy attains a net magneticmoment of 1 Bohr magneton ( μ B) and localized defect states in the band gap. Creation of an unpaired πelectron upon F vacancy is the source of a magnetic moment. However, the 115432-4STRUCTURES OF FLUORINATED GRAPHENE AND THEIR ... PHYSICAL REVIEW B 83, 115432 (2011) exchange interaction between two F vacancies calculated in a( 7×7×1) supercell is found to be nonmagnetic for the ﬁrst-nearest-neighbor distances due to spin pairings. Similarto graphane, 10,11it is also possible to attain large magnetic moments on F-vacant domains in CF structures. V . ELASTIC PROPERTIES OF CF Having analyzed the stability of various C nF structures withn=1,2, and 4, we next investigate their mechanical properties. The elastic properties of this structure can beconveniently characterized by its Young’s modulus andPoisson’s ratio. However, the in-plane stiffness Cis known to be a better measure of the strength of single-layer honeycombstructures, since the thickness of the layer hcannot be deﬁned unambiguously. Deﬁning A 0as the equilibrium area of a C nF structure, the in-plane stiffness is obtained as C=(δE2 s/δ/epsilon12)/A0, in terms of strain energy Esand uniaxial strain /epsilon1.12The values of in-plane stiffness C, and Poisson’s ratioν, calculated for stable C nF structures, are given in Table I together with the values calculated for graphene and graphane.For example, the calculated values of CF are C=250 J/m 2 andν=0.14. It is noted that Cincreases with n.F o rC F (i.e.,n=1), the in-plane stiffness is close to that calculated for CH. It appears that the interaction between C-F bonds inCF (or the interaction between C-H bonds in CH) does nothave a signiﬁcant contribution to the in-plane stiffness. Themain effect occurs through dehybridization of sp 2bonds of graphene through the formation C-F bonds (or C-H bonds). A value of the Young’s modulus of ∼0.77 TPa can be calculated by estimating the thickness of CF as h=3.84˚A, namely the sum of the thickness of graphene (3.35 ˚A) and buckling, δ(0.49 ˚A). This value is smaller but comparable with the value proposed for graphene, i.e., ∼1 TPa. Here the contribution of C-F bonds to the thickness of CF is neglected,since the interaction between C-F bonds has only negligibleeffects on the strength of CF. In Fig. 5the variation of strain energy E sand its derivative, δEs/δ/epsilon1, with strain /epsilon1are presented in both elastic and plastic regions. Two critical strain values, /epsilon1c1and/epsilon1c2, are deduced. The ﬁrst one, /epsilon1c1, is the point where the derivative curve attains its maximum value. This means that the structure can ES(eV)C1 C2 dES/dε(eV)(a) Elasticregion 020406080100 0102030 Strain [ Δc/c0]0.0 0.1 0.2 0.3 Band Gaps (eV)(b) 02468 LDA GW0 Strain [ Δc/c0]0.0 0.1 0.2 0.3 FIG. 5. (Color online) (a) Variation of strain energy and its ﬁrst derivative with respect to the uniform strain /epsilon1. Orange (gray) shaded region indicates the plastic range. Two critical strains in the elastic range are labeled as /epsilon1c1and/epsilon1c2. (b) Variation of the band gaps with/epsilon1.L D Aa n d GW 0calculations are carried out using a 5 ×5 supercell having a lattice parameter of c0=5a,a n d /Delta1cis its stretchingbe expanded under a smaller tension for higher values of strain. This point also corresponds to phonon instability12where the longitudinal acoustic modes start to become imaginary for/epsilon1>/epsilon1 c1. The second critical point, /epsilon1c2(/similarequal0.29), corresponds to the yielding point. Until this point the honeycomblike structureis preserved, but beyond it the plastic deformation sets in.We note that for /epsilon1 c1</epsilon1</epsilon1 c2the system is actually in a metastable state, where the plastic deformation is delayed.Under long-wavelength perturbations, vacancy defects andhigh ambient temperatures, /epsilon1 c2approaches to /epsilon1c1. In fact, our further molecular dynamics simulations show that /epsilon1c2→ 0.17 at 300 K and to 0.16 at 600 K. In the presence of aperiodically repeating F vacancy and C +F divacancy, the value of/epsilon1 c2is also lowered to 0.21 and 0.14, respectively. Apart from phonon instability occurring at high /epsilon1, the band gap is strongly affected under uniform expansion. In Fig. 5(b) we show the variation of LDA and GW 0-corrected band gaps under uniform expansion. The LDA gap slightly increasesuntil/epsilon1=0.05 and then decreases steadily with increasing /epsilon1. TheGW 0-corrected band gap essentially decreases with increasing strain. For example, its value decreases by 38%for/epsilon1=0.20. VI. CONCLUSIONS The present analysis of ﬂuorinated graphenes shows that different C nF structures can form at different levels of F coverage. Calculated properties of these structures, such aslattice parameter, d CCdistance, band gap, DOS, work function, in-plane stiffness C, Poisson’s ratio, and surface charge, are shown to depend on nor coverage /Theta1. Relevant data reported in various experiments do not appear to agree with the propertiescalculated for any one of the stable C nF structures. This ﬁnding leads us to conclude that domains of various C nF structures can form in the course of the ﬂuorination of graphene. Therefore,the experimental data may reﬂect a weighted average ofdiverse C nF structures, together with extended defects in grain boundaries. In this respect, imaging of ﬂuorinated graphenesurfaces by scanning tunneling and atomic force microscopy,as well as x-ray photoemission spectroscopy, is expected toshed light on the puzzling inconsistency between theory andexperiment. Finally, our results show a wide range of interesting features of C nF structures. For example, a perfect CF structure, as described in Fig. 2, is a stiff, nonmagnetic, wide-band-gap nanomaterial having a substantial surface charge, but attains asigniﬁcant local magnetic moment through F-vacancy defects.Moreover, unlike graphane, half-ﬂuorinated graphene withonly one side ﬂuorinated is found to be stable, which canbe further functionalized by the adsorption of adatoms to theother side. For example, hydrogen atoms adsorbed to the otherside attain a positive charge and hence a permanent transversalelectric ﬁeld, which can be utilized to engineer electronicproperties. ACKNOWLEDGMENTS This work is supported by TUBITAK through Grant No. 108T234. Part of the computational resources has beenprovided by UYBHM at ITU through Grant No. 2-024-2007. 115432-5H. S¸AHIN, M. TOPSAKAL, AND S. CIRACI PHYSICAL REVIEW B 83, 115432 (2011) We thank the DEISA Consortium (www.deisa.eu), funded through the EU FP7 project RI-222919, for support withinthe DEISA Extreme Computing Initiative. S.C. acknowledgesthe partial support of TUBA, Academy of Science of Turkey. The authors would also like to acknowledge the valuablesuggestions made by D. Alfe. ciraci@fen.bilkent.edu.tr 1K. 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). 2A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007). 3C. Berger, Z. Song, T. Li, X. Li, A. Y . Ogbazghi, R. Feng, Z. Dai, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer,Science 312, 1191 (2006). 4M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat. Phys. 2, 620 (2006). 5H. S¸ahin, R. T. Senger, and S. 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Lett. 93, 185503 (2004). 33HSE. a hybrid functional implemented in VA S P [K. Hummer, J. Harl, and G. Kresse, P h y s .R e v .B 80, 115205 (2009)], is demonstrated to be as successful as the GW 0self-energy correction method in predicting the band gaps of bulk 3D crystals. Consistently, for 2Dhoneycomb structures, HSE is found to yield smaller values thanthose of GW 0. For example, while HSE predicts the band gap of CF as 4.86 eV , GW 0gives 7.49 eV . Similar trends are also found for CH and 2D boron nitride (BN). HSE and GW 0-corrected band gaps of CH (BN) are 4.51 (5.74) and 5.97 (6.86) eV , respectively. 34For example, Bader [G. Henkelman, A. Arnaldsson, and H. Jonsson,Comput. Mater. Sci. 36, 254 (2006)] L¨owdin [P.-O. L ¨owdin, J. Chem. Phys. 18, 365 (1950) ] and Mulliken [R. S. Mulliken, J. Chem. 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PhysRevB.90.115314.pdf	PHYSICAL REVIEW B 90, 115314 (2014) Step energy and step interactions on the reconstructed GaAs(001) surface Rita Magri,Sanjeev K. Gupta,†and Marcello Rosini‡ Dipartimento di Fisica, Informatica e Matematica (FIM) dell’universit ´a degli studi di Modena e Reggio Emilia and S3 research center of CNR-INFM, via Campi 213/A, 41100 Modena, Italy (Received 19 February 2014; revised manuscript received 22 July 2014; published 30 September 2014) Using ab initio total energy calculations we have studied the relation between the step atomic conﬁguration and its properties (step energy, donor/acceptor behavior, and step interaction) on a β2(2×4) reconstructed GaAs (001) surface. The results have been tested against the widely used elastic dipole model for the step energy andstep interaction considered valid for stress-free surfaces. We have found that acceptor-behaving steps have anattractive interaction and donor-behaving steps have a repulsive interaction in contrast with the elastic dipolemodel which predicts always a repulsive interaction between like-oriented steps. To account for the attractiveinteraction we consider the electrostatic dipole interaction having the L −2scaling with the step distance Land therefore compatible with the standard elastic model. Using a model charge distribution with localized pointcharges at the step based on the electron counting model we show that the electrostatic step interaction canindeed be generally attractive and of the same order of magnitude of the negative elastic dipole interaction. Ourresults show however that the usually employed dipole model is unable to account for the repulsive/attractive stepinteraction between donorlike/acceptorlike steps. Therefore, the ab initio results suggest an important electronic contribution to the step interaction, at least at the short step distances accessible to the ﬁrst-principles study. Ourresults explain qualitatively many experimental observations and provide an explanation to the step bunchingphenomenon on GaAs(001) induced by doping or by critical growth conditions as due to the stabilization ofattractively interacting step structures. These ideas would lead to the development of a bottom-up surface stepengineering. DOI: 10.1103/PhysRevB.90.115314 PACS number(s): 68 .35.B−,68.47.Jn,71.15.Mb,71.15.Nc I. INTRODUCTION Extended surface defects such as surface steps play a crucial role in epitaxial growth, deciding the growth mode [ 1], the favorable sites for island nucleation [ 2], the island shape and evolution, roughening and facetting, and structure and stabilityof vicinal surfaces. Despite their importance most theoreticalinvestigations have addressed so far steps on metal or siliconsurfaces [ 3]. However, the progress in semiconductor homo- and heteroepitaxy, leading to spontaneous self-assembly ofsemiconductor and metal nanostructures on semiconductorsurfaces [ 1], requires a better understanding of step stability and dynamics to assess precisely their role in the epitaxialgrowth of interfaces and nanostructures. To study the stabilityand atomic structure of steps and vicinal surfaces, and therelation between their structure and properties is a ﬁrst stepin this direction. Here we address the III-V (001) techno-logically important surfaces. Unfortunately these surfacesshow complex ambient-dependent reconstructions makingthe problem very difﬁcult to tackle. Few semiconductorsurface steps were studied in some detail, among them thestepped (1 ×2)/(2×1) Si(001) surface [ 4–7] and Si(111):H surface [ 8,9] using atomistic (semiempirical or ab initio )o r continuous theory methods. GaAs(001) surface, on the otherhand, forms reconstructions having a much larger periodicityand structural complexity. Because of the inherent large rita.magri@unimore.it †Present address: Department of Physics, St. Xavier’s College, Navrangpura, Ahmedabad 380009, India. ‡Permanent address: Margen, Via Dino Ferrari, Maranello (MO), Italy.dimensions of the surface unit cells required to describe stepped reconstructed semiconductor surfaces, few atomisticcalculations [ 10] and even less ab initio calculations have been attempted [ 11]. Stability of stepped surfaces is usually described in terms of the step energy, which is the energy required to form asingle step line on a ﬂat surface, and the step-step interaction.AtT=0 K steps are free of kinks and for surfaces free of external stresses, such as the lateral strains imposed by growthon mismatched substrates or surface stress anisotropies causedby broken symmetry domains [ 5,7], the step-step interaction is commonly described using the Marchenko and Parshin theory(MP) [ 12] where steps are modeled by periodic lines of point force dipoles on otherwise perfectly ﬂat surfaces. Within thistheory the step-step elastic interaction energy scales as L −2 with the step distance Land is repulsive for like-oriented steps. GaAs(001) step structure and vicinal surfaces to GaAs(001) have been the object of experimental investigations [ 13– 15] that have not found adequate support by theoretical or computational works. In As-rich conditions GaAs(001) showstwo main reconstructions: c(4×4) at lower growth temper- atures and β 2(2×4) at higher temperatures, the transition temperature depending strongly on the As/Ga ﬂux ratio.Experiments have shown that the thinnest steps on GaAs(001)β 2(2×4) reconstructed surfaces are two atomic layers high, since the last layer is always As terminated. Thus GaAs(001)steps correspond to the double layer steps on Si (001) andare essentially of two kinds: steps oriented along [110] [Asteps, corresponding to the DB steps of Si(001)] or along [ 110] [B steps, corresponding to the DA steps of Si(001)]. For theβ 2(2×4) reconstructed surfaces A steps run parallel to the As dimer bond direction, while B steps run orthogonal to them. 1098-0121/2014/90(11)/115314(11) 115314-1 ©2014 American Physical SocietyRITA MAGRI, SANJEEV K. GUPTA, AND MARCELLO ROSINI PHYSICAL REVIEW B 90, 115314 (2014) Heller et al. [13] extracted an estimate of the step energies of A and B steps counting the kinks on β2(2×4) surfaces grown in quasiequilibrium conditions under the assumption that kinksform in a uncorrelated way and found that the step energy of Bsteps is about 6 times [ 13] or 10 times [ 14] higher than that of A steps. As a consequence, B steps are rougher than A steps andgrowing islands are elongated along the [ ¯110] surface direction which is the A step direction. As for the step-step interactionthe constant Kof the MP repulsive KL −2interaction between steps at distance Lwas estimated by analyzing the terrace width distributions and correlation lengths for a sequence ofsteps in thermodynamic equilibrium [ 15]. The authors found for A steps a very large value of K:K=20–30 eV or K=1.5 eV using theirs or Heller’s [ 13] data, respectively, whereas the interaction between B steps was found to be much weaker.The very large Kof A steps was explained speculating that the interaction constant is dominated not by the elastic interactionbut by a strong repulsive electric dipole interaction [ 15]. In these experiments the surface reconstruction was reported to beβ 2(2×4). In this paper we extract the step energy and the step-step interaction from sets of ﬁrst-principles calculations on vicinalsurfaces to the β 2(2×4) reconstructed GaAs(001) at different miscut angles, misoriented towards (111) (A steps) or ( ¯111) (B steps). We have found that A steps of monoloyer height aremore stable than the equally high B steps in agreement withthe experiment. B steps have a weaker step interaction than Asteps also in agreement with the experiment. Interestingly, Asteps have both short-range repulsive or attractive step interac-tions depending on their atomic structure. In particular,As-rich steps behaving as acceptor defects have an attractiveinteraction, while As-poor steps behaving as donor defectshave a repulsive interaction. A “donorlike” step becomesstable at relatively large distances and is likely to be formedat low temperatures. However, the unstable attractivelyinteracting As-rich steps could form when stabilized by donorimpurities such as silicon or As-rich and high temperaturegrowth conditions. Since attractive interactions can be at theorigin of the formation of step bunching our results couldexplain the observed formation of step bunches when thesamples are in those conditions [ 15,16]. The attractive step interaction can be explained accounting for the electrostaticpoint-dipole model. An estimate of the electrostaticpoint-dipole interaction using the electron counting rule(ECR) assuming localized dangling bond charges at thesteps reveals that the interaction is indeed generallyattractive. II. METHOD The vicinal surfaces are modeled through a sequence of monolayer or bilayer-high equally spaced steps oriented alongthe [110] (A steps) or [ ¯110] (B steps) surface directions, respectively. Different step conﬁgurations are considered thatare compatible with STM observations [ 13]. The atomic structural models of A steps are shown in Fig. 1.S e v e n different atomistic models for A and B step geometries,denoted atog, are studied. These atomic conﬁgurations correspond to different ways the β 2reconstruction can be matched at the up and down ledges. For each A stepconﬁguration shown in Fig. 1, further structures are obtained shifting the single or double As dimers up and down of a/2 along the [ ¯110] direction, across the step, where ais the surface lattice constant, a=a0/√ 2, with a0the GaAs lattice constant. We have calculated the surface energy γof the vicinal surfaces and compared it to that of the unstepped β2(2×4) surface. The calculations are performed in the framework of the plane-wave density functional theory (DFT) and the local density approximation (LDA) using the open source package QUANTUM ESPRESSO (http://www.quantum-espresso.org )[17]. The surfaces are modeled through repeated slabs along the [001] direction. The calculations are carried out in a supercell geometry with periodic boundary conditions. The supercelldimension for A steps is 2 aalong the [ 110] direction and ranges from 4 .5ato 17.5aalong the [110] direction. For B steps the surface unit cell is 4 aalong the [110] direction and ranges from to 5 .5ato 11.5aalong the [ 110] direction. The atoms at the bottom layer of the slab are kept ﬁxed atthe theoretical bulk positions, and their dangling bonds arepassivated with pseudohydrogen atoms of fractional charge,in order to mimic the constraint due to the semi-inﬁnite bulk. The remaining atoms have been relaxed until forces were less than 0.005 eV /˚A. The slab is 13 atomic planes thick for A steps and 12 atomic planes thick for B steps excluding thepseudohydrogen plane. The slabs are separated by a vacuumregion of about 15 ˚A in order to minimize the interactions across the boundaries. The As and Ga pseudopotentials for thesandpvalence electrons are norm conserving, separable, and core corrected and the plane wave energy cutoff is 15Ry which yielded structural and elastic parameters for the metal elements and bulk GaAs in good agreement with the experimental values [ 18]. The convergence of the calculated γon the Brillouin zone sampling has been accurately tested ﬁnding that the sampling along the [ 110] direction is crucial. To obtain good converged results (within 0.3 meV /˚A) the grid needs to be dense (at least eight kpoints) along this direction. For this reason we used a grid of 16 kpoints to sample the Brillouin zone. A small metallic smearing (0.26 eV) wasused to account for the metallicity of the step conﬁgurations.The contribution of the hydrogenated backside has been subtracted using similarly calculated energies of stepped slabs of analogous dimensions, hydrogenated on both sides. Theprocedure we employ for the subtraction is schematicallyshown in Fig. 2.I nF i g . 2(a) the supercell of the vicinal surface having steps A aseparated by two β 2terraces is shown (replicated twice along the direction orthogonal to the step line, i.e., the [110] direction, xdirection). The hydrogenated backside (delimited by the dashed rectangle in the ﬁgure)energy contribution is subtracted by calculating also the total energies of the systems (b) and (c) depicted in Figs. 2(b) and 2(c), respectively, and using E vicinal=E(a)−E(b)+1 2E(c), where E(a),E(b), andE(c)are the total energies of the three systems (a), (b), and (c) in Fig. 2. The total energies of the three structures are calculated using equivalent k-point meshes. This procedure is followed for all the structures calculated in this work. Further details about the calculations are givenelsewhere [ 11]. To derive the surface energies γof vicinal surfaces with step termination iand miscut angle αat temperature T=0K 115314-2STEP ENERGY AND STEP INTERACTIONS ON THE . . . PHYSICAL REVIEW B 90, 115314 (2014) FIG. 1. (Color online) Top and side views of A steps with terraces only one β2unit cell long: (a) the β2structure, (b) the step A a, (c) the steps A band A c, (d) the steps A dand A e, (e) the step A f, and (f) the step A g. In the ﬁgure are shown the unit cells of the shortest step structures of each kind. The vertical double arrows relate the top view and the side view indicating the position of the surface As dimers. The step part of the unit cells has been indicated for each structure. Purple balls: Ga atoms; yellow balls: As atoms. we use the expression γi,α(μAs)=/parenleftbig Ei,α−nGaμbulk GaAs+(nGa−nAs)μbulk As/parenrightbig S +(nGa−nAs)/Delta1μ As S, =γi,α(/Delta1ni=0)+/Delta1ni S/parenleftbig μbulk As+/Delta1μ As/parenrightbig ,(1) where Ei,αis the vicinal surface energy. The label irefers to the speciﬁc step atomic conﬁguration, i=a,..., g (see Fig. 1), andαis the miscut angle, tan( α)=h/L,hbeing the step height (1 or 2 ML) and Lthe terrace length. Sis the surface unit cell area and μbulk GaAs the formation energy of one Ga-As pair in bulk GaAs. nGaandnAsare the number of Ga and As atoms in the system. /Delta1ni=nGa−nAsis the surface with step i stoichiometry. /Delta1n=2i st h e β2stoichiometry. Thus /Delta1ni step= /Delta1ni−/Delta1nR S,i, where RS,iis the ratio between the step and β2surface areas, deﬁnes the isolated step stoichiometry. The surface energy depends on the growth conditions via the Gaand As chemical potentials. This dependency is expressed inEq. ( 1) by the deviation of the As chemical potential /Delta1μ As (treated as a variable quantity) from the value it has in the bulk rhombohedral As metal (e.g., /Delta1μ As=0f o rμAs=μbulk As).We have found that for A steps the surface energy change related to the in-plane shifts of (single-single, single-double,and double-double) As dimers parallel to the step idirection (due to the degeneracy of the As dimer position) is an order ofmagnitude smaller ( <0.05 meV /˚A 2for tan α> 0.1) than the energy difference between the step structures i(atog), so we next consider only the actual steps ias shown in Fig. 1.I n Fig. 3the reduced projected surface energies ( γ/prime=γ/cosα) [19] of vicinal surfaces formed by A and B steps, respectively, are plotted as a function of tan αfor two values of the As chemical potential at the end points of the calculated range ofstability of the β 2(2×4) [and the slightly more stable c(2×8)] reconstruction. To extract the step properties the reduced surface energies a r eﬁ tt ot h er e l a t i o n[ 20] γ/prime i(α,μ As)=γ(0,μAs)+/epsilon1i(μAs) htanα+qi(μAs)(tanα)3,(2) where γ(0,μAs)i st h e β2(2×4) (miscut α=0) surface energy, /epsilon1iis the step energy, /epsilon1i=h(dγi/dα)α=0, i.e., the energy per unit step length of a single isolated step of structurei, andq i(μAs) is the contribution of the step-step interaction. 115314-3RITA MAGRI, SANJEEV K. GUPTA, AND MARCELLO ROSINI PHYSICAL REVIEW B 90, 115314 (2014) FIG. 2. (Color online) (a) Ball and stick model of the slab featuring the vicinal surface having steps of kind A a. The terrace (twoβ2unit cell long) and step regions along the [110] direction are shown. The unit cell dimension along the [110] direction is comprised between the two dashed blue lines. (b) The hydrogenated back side ofthe slab. (c) The hydrogenated ﬂat slab used to subtract the energy of the top side of the structure (b). Yellow balls represent arsenic atoms, purple balls represent gallium atoms, and cyan small balls represent pseudohydrogen atoms. III. RESULTS A. Step energy The results of the ﬁts are reported in Fig. 3as solid lines. Equation ( 2) has been used to interpret STM images of stepped surfaces to extract the step parameters in the case of Si and Ge(001) surfaces [ 21]. The step energy /epsilon1 iis generally considered to be positive since the formation of a step goes along with thecreation of additional dangling bonds (DBs). The last termin Eq. ( 2) derives from the assumption that the step-step interaction exhibits a L −2decay. This decay was derived, within isotropic continuous elasticity, for the elastic ﬁeldinteraction between force dipoles localized at δ-like positions on a ﬂat surface [ 12]. The model predicts repulsive interactions between like-oriented steps and attractive interactions between opposite-oriented steps. Equation ( 2) has been used to ﬁt empirically calculated data of single and double step energeticson the Si(001) surface [ 7]. Generally dipolar long-range step-step interactions were shown [ 22] to decay to the lowest order as L −2. The step energies relative to different /Delta1μ Asare given in Table I. We can see that B steps (all one ML high) have a higher formation energy than the one ML high A steps in agreementwith the experiments [ 13,14] by Heller et al. discussed in the Introduction section.FIG. 3. (Color online) Reduced surface energies versus miscut angles tan αfor steps A and B at /Delta1μ As=− 0.32 and /Delta1μ As=− 0.58. We focus now on the more stable A steps. The step energy is roughly related to the number of additional DBs NDBinserted with the step [ 23,24] modiﬁed by the effect of the As chemical potential via the step stoichiometry /Delta1ni step, which changes the degree of each step (un)stability depending on the externalconditions. We ﬁnd that steps A ahave a negative step energy (respective to the β 2“ﬂat” surface) and become stable when sufﬁciently far apart. Negative step energies were calculatedalso for SB and DB steps on the 2 ×1 Si(001) using atomistic interatomic potentials [ 7]. For these steps the destabilizing effect due to the additional dangling bonds introduced bythe step is largely offset by an additional release of thesurface elastic stress. Indeed, while reconstructions lower the surface energy by creating new bonds between the atoms at the surface (formation of dimers, for example), the newbonds introduce also a stress on the subsurface atoms. Thetrade-off between these two effects [electronic (stabilizing)and elastic (destabilizing)] decides the stability of a surfacereconstruction. Looking at Fig. 3we see that different As chemical potentials just shift the calculated γto higher energies and change the relative stability of the vicinal surfaces. Thedeviation of the projected (or reduced) surface energies asa function of the miscut angle from a straight line is due to the 115314-4STEP ENERGY AND STEP INTERACTIONS ON THE . . . PHYSICAL REVIEW B 90, 115314 (2014) TABLE I. Step parameters for the a,b,c,d,e,f,a n dgA and B steps. /epsilon1is the step energy entering Eq. ( 2) evaluated at different /Delta1μ As values. qare the ﬁtted values (from the ab initio calculated values; see text) of the step-step interaction entering Eq. ( 2).Lstepis the step length along the [110] direction, deﬁned as the smallest length between two β2terraces having different structural motifs from the β2structure. /Delta1ni step is the step stoichiometry (number of As versus Ga atoms) relative to that of the β2surface, NDBis the number of additional dangling bonds introduced with the steps, and Qis the excess charge, that is the charge not transferred from the Ga to the As dangling bonds. Q=0 means the ECR is satisﬁed (complete transfer, all Ga DBs empty, and all As DBs completely full). pxandpzare the electrostatic dipole components per step unit length calculated as explained in the text minus those of the β2surface. KesandKelare the estimated electrostatic and elastic K constants, respectively, of the L−2step-step interaction. Steps A abcde f g /epsilon1(/Delta1μ As=0)(meV /˚A) −2.2 21.6 38.2 38.6 16.0 198.4 181.5 /epsilon1(/Delta1μ As=− 0.32)(meV /˚A) −12.3 31.7 48.2 68.9 46.2 168.1 131.0 /epsilon1(/Delta1μ As=− 0.58)(meV /˚A) −20.5 39.9 56.4 93.4 70.8 143.5 90.0 q(meV/˚A2) +864.0 −154.24 −149.52 −59.39 −83.6 +3342.85 +1531.52 Lstep(a) 1.5 2.5 2.5 3.5 3.5 4.5 3.5 /Delta1ni step +0.25 −0.25 −0.25 −0.75 −0.75 +0.75 +1.25 NDB 46688 1 2 1 0 Q(e) −0.5 +0.5 +0.5 +1.5 +1.5 −1.5 −2.5 px(e) −0.20 +0.14 +0.13 +0.52 +0.53 −0.14 −0.45 pz(e) 0.06 0.06 0.06 +0.13 +0.13 +0.22 +0.02 Kes(meV ˚A) −82.58 −38.32 −32.46 −566.93 −589.71 +14.05 −438.89 Kel(meV ˚A) +41.83 +322.37 Steps B abcdef g /epsilon1(/Delta1μ As=0)(meV /˚A) 137.91 162.44 140.47 138.79 134.14 132.35 124.13 step-step interaction qthat does not change with /Delta1μ As. Indeed, by combining Eq. ( 1) with Eq. ( 2) we obtain for /epsilon1i(μAs) a linear dependence on the As chemical potential: /epsilon1i(/Delta1μ As)=/epsilon1i(/Delta1μ As=0)+/Delta1ni step L⊥/Delta1μ As, (3) where L⊥=2a(4a) is the lateral dimension of the A (B) steps surface unit cells. In Table Iwe report the main parameters characterizing the structure of A steps: the step length Lstep, the step stoichiometry /Delta1ni step, and the additional number of dangling bonds NDB. Equation ( 3) stresses that the step relative stability depends on the growth conditions. B. Step-step interaction From Fig. 3we can see that B steps are much less interacting than A steps in agreement with the experiment of Lelarge et al. [15] mentioned in the Introduction. Indeed, we ﬁnd in most cases an almost straight dependence of the surface energy onthe miscut angle. However, the step-step interaction parameter qextracted using Eq. ( 2) is very sensitive to the number of calculated values and to small details of the γversus αcurves. To test the sensitivity of the value of qon the details of the ﬁtted curves we show in Fig. 4other ﬁts where one calculated point was omitted: the one corresponding to the largest α for which the point dipole model should not work well or,alternatively, the γ(0,μ As) ﬁnal point corresponding to an inﬁnite distance between the steps, because of the possibleerror in the alignment of the step surface energies with theβ 2surface energy (estimated within 0.05 meV /˚A2). We can see that the extrapolated β2v a l u ei nt h ec a s eo fs t e pA ais inagreement with the value obtained using all the calculated values, while for step A ewe obtain for the extrapolated γ(0,μAs) a value out of the range of the estimated alignment error. We can see from this test that the numerical value of q is very sensitive to small changes of the γversus αcurvature. The calculations show also that at least four calculated pointsare necessary to obtain a consistent estimate of the step-stepinteraction parameter q. However, the fourth point (calculated for the step A astructure but not for the step A estructure) corresponds to steps separated by four β 2unit cells. This amounts to very large unit cells (500 atoms for the calculatedfourth point of the A avicinal surface having a smaller unit cell size). Unfortunately we are unable to provide an equallyaccurate value for the larger A evicinal surface with a four β 2long terrace using our computational tool and choice of parameters (energy cutoff, k-point grids, norm-conserving pseudopotentials, etc.). Theqparameter appearing in Eq. ( 2) is related to the Kconstant of the L−2step-step interaction by q=K/L3 ⊥. For Aaand Aesteps we obtain qa=864 meV /˚A2andqe= −84 meV /˚A2, that is steps ahave a repulsive interaction and stepseinteract attractively. In the same way the calculations hint to a repulsive interaction for steps A fand A gand to an attractive interaction for steps A b,Ac, and A d, that is the step-step interaction is weakly attractive for the As-richer steps(acceptors Q> 0) and strongly repulsive for the As-poorer steps (donors Q< 0). The important issue here is that some steps seem to interact attractively contrary to the predictions of the elas-tic force-dipole model. Indeed, the elastic line point force dipole components F i=Aid(δ(−→r)) dx,i=x,z (δis the Dirac 115314-5RITA MAGRI, SANJEEV K. GUPTA, AND MARCELLO ROSINI PHYSICAL REVIEW B 90, 115314 (2014) FIG. 4. Above: step A a. Solid line: all ﬁve values, /epsilon1= −2.2m e V /˚A,q=+ 864.0m e V /˚A2; dashed line: four values including γβ2,/epsilon1=− 9.0m e V /˚A,q=+ 1481 meV /˚A2; dotted line: four values without γβ2,/epsilon1=− 0.8m e V /˚A,q=+ 845 meV /˚A2, predicted γβ2+51.27 meV /˚A2, calculated 51.29 meV /˚A2.B e l o w : step A e. Solid line: all four values, /epsilon1=16.0m e V /˚A,q=− 83.6 meV/˚A2; dashed line: three values including γβ2,/epsilon1=17.8m e V /˚A, q=− 291.6m e V /˚A2; dotted line: three values without γβ2,/epsilon1= 9.0m e V /˚A,q=+ 69.2m e V /˚A2, predicted γβ2+51.40 meV /˚A2, calculated 51.29 meV /˚A2. δfunction), located at the step line x(the direction orthogonal to the step direction, in our case x=[110]) can be shown to generate the displacements [ 20]: ux(x)=2(ν2−1) πEAx x,u z(x)=2(ν2−1) πEAz x.(4) E=85.5 GPa is GaAs Young modulus, ν=0.31 is the GaAs Poisson ratio, and Aiare the components of the force dipoles. The elastic energy is given by Wtot el=1 2/summationdisplay n,m/bracketleftbigg/integraldisplay dx/bracketleftbig Fn x(x)um x(x)+Fn z(x)um z(x)/bracketrightbig/bracketrightbigg ,(5) where nandmare the step indexes. From these expressions we can see that the elastic step self-energy (sum of the terms withn=m) is always positive since the force and displacement ﬁelds are equally oriented. The terms with n/negationslash=mconstitute the elastic step interaction. For like-oriented steps the energyis also positive since forces and displacements on the differentsteps distanced by Lare similarly oriented and the elastic stepinteraction energy is given by W el=π(1−ν2) 3EA2 L2=Kel L2. (6) For opposite oriented steps on the contrary the elastic in- teraction energy predicted by the model is attractive. It hasbeen shown in the literature that for like-oriented steps onstress-free surfaces the elastic interaction remains repulsiveeven when orders beyond the dipolar one or better models ofthe force ﬁelds at steps are considered [ 20]. Obviously these extensions of the model, although incapable to change the signof the step interaction, introduce further unknown parameters.Attractive interactions however were inferred experimentallyby the STM analysis of the step surface distributions onCu(001) [ 25]. On the theoretical point of view an elastic attractive behavior between steps was shown to arise onlyin the case of a vicinal surface subjected to an external stresssuch as that induced by the growth on a mismatched substrate[26]. The problem of the possible origin of the step attractive interactions has been largely debated in the literature [ 27] where many different speculations have been proposed butnot much progress has been done since. At T=0Kt h eo n l y other possible contribution to the dipolar step interaction hasan electrostatic nature. This has been inferred in the literature[28] whenever the elastic point dipole model was unable to ﬁt the data points. An explicit account of the electrostaticcontribution was given to model the step energy on II-VI(001) surfaces [ 29], thus explaining the surface island shapes, but the step interaction contribution having the dipolar formK/L 2was never proposed. We derive here the expression of a dipolar electrostatic step interaction following the sameassumptions made for the derivation of the MP elastic pointdipole interaction. C. Electrostatic dipole model and electrostatic step interaction We derive the interaction energy of an inﬁnite sequence of line electrostatic dipoles. We consider a linear densityof point electrostatic dipoles located at the step line (i.e.,atx=0). Differently from the elastic interactions the line dipoles pinteract both through the material and through the vacuum, and the electric ﬁeld components depending on p z are discontinuous at the surface. After integration along the step line ( ydirection) one ﬁnds the expression for the electric ﬁelds at the surface z=0a s Ex(x)=+k4px (1+/epsilon1r)x2, Ez(x)=−k4pz /epsilon1r(1+/epsilon1r)x2material , Ez(x)=−k4pz (1+/epsilon1r)x2vacuum , where kis the vacuum electrostatic constant, pis the dipole linear density, and /epsilon1r=12.9 the GaAs relative dielectric constant of the material. This expression is equivalent to theexpression reported for the displacement ﬁeld under the pointforce dipole, Eq. ( 4), at the step location x=0[20]. 115314-6STEP ENERGY AND STEP INTERACTIONS ON THE . . . PHYSICAL REVIEW B 90, 115314 (2014) The electrostatic interaction energy is then calculated as Wes=−1 2/summationdisplay n,m/negationslash=n/bracketleftbigg/integraldisplay dx/bracketleftbig pn x(x)Em x(x)+pn z(x)Em z(x)/bracketrightbig/bracketrightbigg . After integration we ﬁnd that the electrostatic interaction energy of an inﬁnite sequence of point dipole lines distancedbyLalongxis given by W es=kπ/bracketleftbig −2/epsilon1rp2 x+(1+/epsilon1r)p2 z/bracketrightbig 3/epsilon1r(1+/epsilon1r)1 L2=Kes L2, (7) where we have taken the average value of the electrostatic ﬁeld across the vacuum and the material. From these expressionswe see that, while the dipole elastic interactions between like-oriented steps are always repulsive, see Eq. ( 6), the electrostatic ones can be in principle both repulsive or attractive. To assesswhat the situation is for steps on GaAs(001) we use a pointcharge model based on the electron counting rule (ECR)[30,31]. Following the ECR, III-V surfaces stabilize through a charge transfer from the DBs on Ga atoms, lying at higherenergies, to the DBs on the As atoms, lying at lower energies.For the octet rule each Ga DB has a 0 .75echarge, while each As DB has a 1 .5echarge when bonded to two Ga and one As atom in a dimer. If the number of both kinds of DBs is right all GaDBs become empty while As DBs become fully occupied withtwo electrons. In this case the ECR is said to be satisﬁed: thesurface is semiconductor and the system lowers considerablyits energy. This is the case of the β 2(2×4) surface. Following the charge transfer the undercoordinated surface Ga (As) atomsbecome positively (negatively) charged with q Ga=+ 3/4e (qAs=− 0.5e). The stepped surfaces do not satisfy the ECR:steps, like A a,Af, and A g, have an excess of Ga DBs; thus a charge Qcannot be transferred to As DBs and we assume it remains localized in the original Ga DBs. Since the Ga DBstate energies are closer to or within the conduction band thesesteps behave as donor defects [ 32]. This situation is indicated in Table IwithQ< 0. Steps A b,Ac,Ad, and A ehave instead an excess of As DBs; thus the As DBs remain still partiallyoccupied after all the charge available from the Ga DBs hasbeen transferred to them. The steps are acceptors and Q> 0. This model of charge transfer allows us to ascribe point chargesto the undercoordinated atoms at the surface. On the β 2(2×4) reconstructed terraces the Ga point charges are +0.75eand the As point charges are −0.5e[33]. Using these values and the calculated equilibrium atom positions we can calculate the stepdipoles as−→p=/summationtext iqi−→ri/L⊥. The dipoles depend on how the step charges are distributed. We report here as an example thecase where the untransferred charges remain localized atthe DBs at the step making the corresponding ions less positiveor negative than the β 2’s. A localized charge arrangement of this kind is shown in Fig. 5and the corresponding dipoles (subtracting the β2ones:px=0 and pz=− 0.33) are given in Table Itogether with the electrostatic energy constant Kes of the L−2dipole interaction. We have found that for most charge arrangements, even more delocalized, the dipole-dipoleelectrostatic energy is indeed negative; that is, the electrostaticdipole interaction for monolayer high steps tends to beattractive. We ﬁnd that, interestingly, the calculated−→pdo not depend substantially on the terrace length Lbetween steps, which shows that the atom positions (and displacements) at step i ( w eu s e dt h e ab initio calculated atom positions) are similar and independent of L. FIG. 5. (Color online) Point charge distribution used to calculate step point dipoles. In the step structures the charges assigned to the dangling bonds on the β2terrace are the same as for the β2structure. Yellow balls: arsenic atoms; purple balls: gallium atoms. 115314-7RITA MAGRI, SANJEEV K. GUPTA, AND MARCELLO ROSINI PHYSICAL REVIEW B 90, 115314 (2014) FIG. 6. (Color online) Red line: ﬁt to the calculated atomic dispacements of step A a(dots and solid line). Obtained values for the force dipole components are Ax=− 143.7m e V /˚Aa n d Az=− 54.2m e V /˚A. (a) Uxat atom positions naalong [110]. (b) Uzat atom positions na along [110]. In the middle is a ball and stick side view along [110] of the ﬁrst few atomic planes; yellow dots: As atoms; purple dots: Ga atoms. The distance Lbetween steps if four β2unit cells. D. Elastic step interaction In Table Iwe give an estimate of the analogous elastic energy constant Kelwithin the elastic point dipole model for the steps A aand Ae. The force dipole AxandAzcomponents entering Eq. ( 6) are extracted by the displacements (relative to those of the β2ﬂat surface) of the atoms of the ﬁrst six layers for the steps having smaller miscut angles by ﬁtting these dis-placements to those given by the elastic dipole model Eq. ( 4). We have summed the displacements over the atoms of the ﬁrst six layers located at x: U k(x)=/summationdisplay iui k(x), (8) where ui k(x)a r et h e k=x,zcomponents of atom idisplace- ment at xalong the [110] direction. This procedure allows us to obtain the behavior of a “continuum” surface layercomparable with the dipole model. The atom displacementsatxof the atoms belonging to the layers below the sixth have a negligible effect on the sum. The U xandUzdisplacements so obtained are shown in Figs. 6and7. We can see from the ﬁgures that even for surfaces having complex reconstructionsand a short distance between steps the obtained displacementsU xfollow approximately a dipolar behavior at the step. The U curves go to zero in the region between the steps as requiredby the dipole model. Interestingly, we can see that there areoscillations in the sign of the displacements overimposedon the dipolar behavior with the periodicity of the surfacereconstruction features. In particular, we can see that larger displacements (step A e) are generated by a larger number of As dimers at the step (the displacements relative to stepAbnot shown have values falling between those of steps A a and Ae). This result translates in weaker force dipole components A for step A athan for step A e. We obtain a similar result also in the case of the shortest distance between the steps. The U iare larger for step A ethan for step A a. Within the predictions of the dipole model the repulsive elastic contribution to the stepenergy should be smaller for step A athan for step A e. Since the same force components A ienter also the expression of the elastic contribution to the step energy /epsilon1this implies a smaller elastic step energy for step A awhich would be in agreement with the larger stability we have found with the ab initio calculations. However, our ab initio calculations found also a strong repulsive interaction at short distances for step A a which is contrary to the elastic dipole model predictions. 115314-8STEP ENERGY AND STEP INTERACTIONS ON THE . . . PHYSICAL REVIEW B 90, 115314 (2014) FIG. 7. (Color online) Red line: ﬁt to the calculated atomic displacements of step A e(dots and solid line). Obtained values for the force dipole components are Ax=− 377.3m e V /˚Aa n d Az=− 198.4m e V /˚A. (a) Uxat atom positions naalong [110]. (b) Uzat atom positions na along [110]. In the middle is a ball and stick side view along [110] of the ﬁrst few atomic planes; yellow dots: As atoms; purple dots: Ga atoms. The distance Lbetween steps is three β2unit cells. Summarizing our results we ﬁnd that the (elastic and electrostatic) dipole model predicts weaker dipolar step in-teractions between the Ga-rich steps (like step A a) than for the As-rich steps (like step A e). IV . DISCUSSION AND CONCLUSIONS In this paper we report on direct ab initio calculations to study the structural properties of steps on the GaAs(001)surface reconstructed β 2(2×4). The calculated surface energies of vicinal surfaces featuring different step structuresand orientations have been compared to the standard elasticdipole model of Marchenko and Parshin [ 12]. In this model the action of steps on the ﬂat surface is modeled via lines of point force dipoles which mutually interact with long rangeelastic interactions. The resulting elastic energy was shown toscale with the distance Lbetween steps as KL −2. The ﬁt of the ab initio calculated surface energies to the model allows us to extract the two parameters describing the step properties: thestep energy /epsilon1and the step-step interaction q. The comparison reveals that some step structures interact attractively contraryto the dipole model of the elastic interaction which predictsthat between like-oriented steps the interaction is repulsive.The elastic dipole model was found to be better applicable tosteps whose distances are more than a few lattice parameters apart. For instance in the case of fcc metal surfaces (Ag,Au, Cu, Pd, and Pt) the elastic dipole model was able toﬁt the semiempirically calculated values for step distanceslarger than 3 a 0[28]. The authors of that paper found also that adding an attractive L−3term improved the ﬁt over all distances. The continuum theory is indeed expected tofail at very short step distances where the discreteness ofthe atomic lattice becomes important. In that paper, as ina large part of the following literature, the topic of theorigin of attractive interactions has been debated but not fullyunderstood. We ﬁrst have tackled the problem of a possible attractive interaction within the dipole interaction model recognizingthat relevant charge transfer at the step can create electrostaticdipoles. Likewise for the elastic dipole interaction we haveconsidered the interaction between lines of point electrostaticdipoles obtaining the expression for the electrostatic interac-tion energy scaling as L −2with the step distance. The estimates of this expression using concepts from the electron countingrule have enabled us to show that some step structures indeedinteract attractively. Our estimates show that the elastic andelectrostatic dipole interactions have a similar magnitude. Thesign of the ﬁnal resulting dipole interaction thus comes out 115314-9RITA MAGRI, SANJEEV K. GUPTA, AND MARCELLO ROSINI PHYSICAL REVIEW B 90, 115314 (2014) from the interplay between the elastic contribution (always repulsive) and the electrostatic contribution (attractive orrepulsive) that depend ultimately on the speciﬁc step structure. This analysis however does not explain the ab initio results relative to the fact that “donor” steps have a strong repulsiveinteraction, while “acceptor” steps have a weaker attractiveinteraction. The elastic and electrostatic dipole model doesnot explain our calculated q, since it predicts elastic repulsive interactions weaker for the donor A astep than for the acceptor Aestep, contrary to the ab initio results, and electrostatic attractive dipole interactions stronger for the A estep than for the Aastep, in qualitative agreement with the ab initio results. We notice that in our calculatons the shortest steps are separated by one β 2unit cell, i.e., L=2.83a0. This value ofLfalls within a distance range where the applicability of the dipole interaction model is questionable. Thus we arguethat at such short step distances a different kind of interactionbecomes dominant. The observation that a repulsive behavioris associated to steps with an electronic “donorlike” bandstructure, while an attractive behavior is associated to stepswith an electronic “acceptorlike” band structure suggests anquantum origin for the short distance step-step interactionwhich is accessible precisely to the ab initio calculations that treat electronic and structural degrees of freedom on the samefooting. In the case Q> 0 (acceptorlike step states) the system Fermi energy falls below the top of the valence band; thusthe partially occupied step states have substantially a valencecharacter. The opposite is true for the case Q< 0 where the system Fermi level falls much higher in energy above thebottom of the conduction band. In this case the partiallyoccupied step states lying at higher energies have a moreconduction state character. In the case of short distancesbetween steps it is possible that the step states with energiesnear the gap edges have a substantial overlap between them(and likely with the other surface states). Since usually thisoverlap is larger for the states at the bottom of the conductionband than for those at the top of the valence band (as testiﬁed,for example, by the larger band dispersion and consequentlysmaller effective masses of the states at the bottom of theconduction band than at the top of the valence band in mostIII-V semiconductors and the β 2surface), we speculate that the donorlike step states interact repulsively more stronglythan the acceptorlike states. This repulsive interaction wouldlead to a higher positive contribution to the total energy forthe donor steps and, as a consequence, to a larger repulsivevalue for the step interaction qterm. To explain the repulsive interaction we observe that the total energy of the ground statecan be written as E=/summationdisplay n/angbracketleftψn\|/hatwideT+/hatwideV\|ψn/angbracketright+EH+Exc+Eion-ion,(9) where \|ψn/angbracketrightare the occupied states. /hatwideTis the kinetic energy operator, /hatwideVis the one-body potential energy acting on the electrons, EHis the electron-electron Hartee energy, Excis the electron-electron exchange-correlation energy, and Eion-ion is the ion-ion Ewald interaction energy. Let’s assume that wecan separate the contributions to the Hartree term: e2 2/integraldisplay/integraldisplayn(−→r)n(−→r/prime) \|−→r−−→r/prime\|d−→rd−→r/prime(10) [n(−→r) is the particle density at−→r] due uniquely to the step states. They would read as Estep H=e2 2/integraldisplay/integraldisplay\|ψi(−→r)\|2\|ψj(−→r/prime)\|2 \|−→r−−→r/prime\|d−→rd−→r/prime, (11) where \|i/angbracketrightand\|j/angbracketrightare\|ai/angbracketrightand\|aj/angbracketrightstep acceptor states on neighboring steps iandj(for instance, Wannier functions localized at the steps) or \|di/angbracketrightand\|dj/angbracketrightthe analogous donor states. From this equation we can see that the larger the stepfunctions overlap in space, the more the electronic charge isevenly distributed over all the step and terrace region, leadingto a higher contribution to the Hartree integral. Another wayto look at the same concept is to observe that the sum of thesingle-particle eigenvalues /epsilon1 i(not to be confused with the step energies) is the largest electron contribution to the total energyEifEis expressed in the analogous alternative form: E=/summationdisplay i/epsilon1i−EH+/integraldisplay (/epsilon1xc−Vxc)n(−→r)d−→r+Eion-ion,(12) where /epsilon1xcandVxcare the exchange and correlation energy and potential, respectively. Now we can observe that, in thecase of donor steps, the step-related levels lying within theconduction band are occupied since the Fermi level is withinthe conduction band, while the occupied levels related to theacceptor step states have lower energies since in this casethe Fermi energy falls within the valence band. Thus the ﬁrstterm on the right side of Eq. ( 12) is larger for donor steps than for acceptor steps. This difference is larger when thestep-related occupied states are not a negligible part of all theoccupied states, that is, in the case of close by steps (smallerunit cells). Obviously this hypothesis needs further work to befully understood. Finally, monolayer steps on (2 ×4)/c(2×8) vicinal Si- doped GaAs(001) surfaces ( ntype doping) have been visual- ized using ultrahigh-vacuum scanning tunneling microscopy[34]. The observed step structures correspond to the A b steps of Fig. 1reported to be acceptors. We ﬁnd these steps unstable (i.e., /epsilon1 b>0 and /epsilon1b>/epsilon1aat all/Delta1μ As; see Table I); thus the probability to be formed at a given temperature Twould be much lower than for step A a(which instead has not yet been visualized to our knowledge). The explanationof the experimental observation of step A bcould be given conjecturing that the dopant atoms lower the formation energy of steps A band make them more likely to form. It was shown indeed [ 35] that dopant atoms of nandptypes in semiconductor nanocrystals are preferentially located at thesurface (where the strain they induce in the matrix can be moreeffectively relieved) close to one another because in this way acharge transfer occurs which leads to a considerable loweringof the system energy. The same behavior could be at work alsoin this case with the silicon ndopant states interacting with the acceptorlike states of step A bin such a way as to lower the step Abformation energy. Other surface calculations have shown that a Q< 0 situation (i.e., Fermi level above the bottom of 115314-10STEP ENERGY AND STEP INTERACTIONS ON THE . . . PHYSICAL REVIEW B 90, 115314 (2014) the conduction band) can indeed stabilize acceptor surface defects [ 31]. Since the step energy /epsilon1depends on the As chemical potential it is also greatly inﬂuenced by the epitaxial growthconditions. Clearly, conditions of high temperature (increasingthe probability of formation of the less stable steps) and a highAs ﬂux can increase the probability of formation of the As-richsteps. These considerations lead us to speculate that by doping or through the choice of proper growth conditions one can imposewhat step structures can be formed and, as a consequence,manipulate the step-step interaction. On the other hand, itis known that attractive interactions between steps can leadto step bunching [ 26]; thus we could expect that when the conditions are such as to stabilize the As-rich steps we shouldassist to the formation of step bunchings. To conﬁrm thisexpectation experimental works have indeed found that silicondoping [ 15] or a very high As to Ga ﬂux ratio at high growth temperatures [ 16] lead to step bunching on GaAs(001), whereas step bunching never occurs at a low As to Ga ﬂux inabsence of doping [ 15]. 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PhysRevB.90.121304.pdf	RAPID COMMUNICATIONS PHYSICAL REVIEW B 90, 121304(R) (2014) Unidirectional spin-orbit interaction and spin-helix state in a (110)-oriented GaAs/(Al,Ga)As quantum well Y . S. Chen,1,S. F¨alt,2W. Wegscheider,2and G. Salis1,† 1IBM Research–Zurich, S ¨aumerstrasse 4, 8803 R ¨uschlikon, Switzerland 2Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland (Received 30 April 2014; revised manuscript received 2 September 2014; published 18 September 2014) The Dresselhaus spin-orbit interaction is quantitatively investigated in a (110)-oriented GaAs quantum well by means of time- and spatially resolved Kerr rotation. The experimental results directly demonstrate a unidirectionalout-of-plane spin-orbit interaction that linearly depends on the electron momentum along the [1 10] direction and vanishes for the electron momentum along the [001] direction. Spatially resolved measurements of thediffusion-driven spin precession dynamics provide evidence of the formation of a persistent spin-helix state inthis system. DOI: 10.1103/PhysRevB.90.121304 PACS number(s): 73 .21.Fg,75.70.Tj,75.78.Jp,78.47.db In semiconductor quantum structures, the spin-orbit in- teraction (SOI) represents a practical means to control spinstates, e.g., by gate tuning the SOI strength [ 1–6]o rb y deﬁning the electron momentum [ 7–10]. It also leads to new fundamental phenomena, such as current-induced spinpolarization [ 11–13] and the spin Hall effect [ 12,14–16]. In a system where the spin-orbit (SO) ﬁeld is unidirectional,the predominant Dyakonov-Perel mechanism [ 17]f o rs p i n relaxation in a quantum well (QW) becomes inoperativefor spin polarization along the SO ﬁeld [ 18–20]. Such a situation can be achieved either in a (001)-oriented QW, inwhich the strength of the Rashba contribution is equal tothe Dresselhaus one [ 21,22], or in a (110)-oriented QW, in which the Dresselhaus SOI is intrinsically unidirectional inthe out-of-plane direction [ 18,23]. Theoretical work predicts the emergence of a persistent spin helix (PSH) in both of theseSO systems with an SU(2) symmetry, in which the SO ﬁelddepends linearly on the electron momentum along one speciﬁccrystal direction [ 24,25]. The PSH state has recently been well characterized for (001)-oriented GaAs QWs [ 26–30], and it is found that the cubic Dresselhaus SOI noticeablyperturbs the required symmetry. This limits the spin lifetimeto approximately 1 ns and thus the spin-diffusion lengthtypically to several micrometers. Interestingly, the SO ﬁeldin the (110)-oriented QW is completely unidirectional [ 23], pointing to an ultralong PSH state both in the time domain andin real space. Efﬁcient spin manipulation in combination withan electrically tunable Rashba SOI is also predicted [ 2,31]. The out-of-plane SO ﬁeld in the (110) QW has only beenqualitatively inferred from the large anisotropy of the spinrelaxation rate [ 7,19,32–35]; a quantitative experimental study of the predicted SU(2) symmetry is still missing. In this Rapid Communication, the Dresselhaus SOI of a two-dimensional electron gas (2DEG) symmetrically conﬁnedin a (110)-oriented GaAs/(Al,Ga)As QW is investigated bymeans of time-resolved Kerr rotation (TRKR). We extendthe method presented in Ref. [ 8] to an oblique geometry in which the spin precession axis is monitored as a function ofa dc current in the presence of a magnetic ﬁeld applied at an ych@zurich.ibm.com †gsa@zurich.ibm.comangleβto the QW plane. We determine both the magnitude and the direction of the effective SO magnetic ﬁeld. Weconﬁrm the theoretical prediction that an effective SO ﬁeld isoriented along the out-of-plane direction and depends linearlyon the component of the electron momentum along the [1 10] direction. The strength of the SOI determined corresponds to aDresselhaus coefﬁcient of γ≈−10 eV ˚A 3,t h es a m ev a l u ea s obtained for (001) QWs [ 36]. In addition, we directly observe a change of the precession axis direction that results fromthe vectorial sum of the noncollinear external and SO ﬁelds.Finally, the effect of SOI on diffusing electron spins is spatiallyresolved. In the presence of a ﬁnite in-plane magnetic ﬁeld,we observe spin-diffusion patterns that provide evidence ofthe formation of a spin-helix state in the (110)-oriented QWsystem. The sample studied is a (110)-oriented 20-nm-wide single GaAs QW, which is sandwiched between a 100-nm-thickAl 0.25Ga0.75As layer above and an in total 100-nm-thick GaAs/Al0.25Ga0.75As superlattice below. Owing to remote Si doping inside single 2-nm-thick GaAs/AlAs QWs that are113 nm away from the QW center on both sides, the electrongas investigated has an electronic mobility of up to 1 .1× 10 6cm2/Vs and an electron concentration of 2 .3×1011cm−2 at 4.2 K. Optical lithography is used to pattern semiconductor mesa structures with Ohmic contacts to deﬁne the currentdirection [ 4,8] in the QW plane. TRKR measurements are performed to investigate the SOI in the 2DEG. Circularly polarized pump pulses (power density∼35 W/cm 2) optically generate spin-polarized electrons, and linearly polarized probe pulses ( ∼7W/cm2) monitor the out-of-plane electron spin polarization Szby means of the polar magneto-optical Kerr effect. Both picosecond pulsesare tuned to an energy of 1.535 eV , corresponding to theresonant excitation from the heavy-hole valence band tothe conduction band. This ensures that the optical pulsesexcite and measure spin polarization along the growth axisof the QW [ 37,38]. Each beam is focused onto a spot of ∼12μm diameter. The pump and probe pulses (repetition rate of 79.2 MHz) are delayed with respect to each other by a time t, enabling time-resolved detection of S z. All the measurements are performed with a sample temperature of 20 K, at whichno nuclear spin polarization is observed for the low opticalexcitation density. 1098-0121/2014/90(12)/121304(5) 121304-1 ©2014 American Physical SocietyRAPID COMMUNICATIONS Y .S .C H E N ,S .F ¨ALT, W. WEGSCHEIDER, AND G. SALIS PHYSICAL REVIEW B 90, 121304(R) (2014) (b) (a) ωtotωexty ωextz ωSOαQWBext β (c) (d) Bext = 235 mT, β=45° Bext= 235 mT, β=225°SS// S⊥ ωtotαQW -200 μA +200 μA fits 0 -0.1 -0.2 -0.3 -0.4 -0.5Kerr rotation (Arb. units) 0 500 1000 1500 2000 Time delay (ps)0 500 1000 1500 2000 Time delay (ps)0 -0.1 -0.2 -0.3 -0.4 -0.5 Kerr rotation (Arb. units) 0 Aμ-200 μA +200 μA fits 0 Aμ[110]Electron spin SSO [1 10] [001]ky kxωz FIG. 1. (Color online) (a) Schematic description of the oblique experimental geometry and the SOI vs the electron momentum on the Fermi surface. (b) Schematic description of the electron spin dynamics and the two spin components ( S/bardblandS⊥) detected by the polar magneto-optical Kerr effect. (c), (d) TRKR measurements upon introduction of a positive current (black circles), a negative current (red circles) along the [1 10] direction, and no current (blue circles). The solid and dashed green lines are ﬁtted curves for I=±200μA andI=0μA, respectively. The external magnetic ﬁeld is Bext= 235 mT, with β=45◦for (c) and β=225◦for (d). Rapid momentum scattering averages the SOI on the Fermi circle. By introducing a current Ithrough the 2DEG, the Fermi circle is shifted, and a ﬁnite averaged SO ﬁeldinteracts with the electron spins. This ﬁeld is describedby a precession axis ω SO.A n external magnetic ﬁeld Bext leads to a Larmor precession with an axis ωext. The total precession axis is thus given by ωtot=ωext+ωSO, andωtot≈ ωext+(ωSO·ωext)/ωextdepends on the projection of ωSOonto ωext. This enables us to determine both the magnitude and the direction of ωSO[8,9,36,39]. If we apply ωextalong an in-plane direction of the (110) QW [ β=0i nF i g . 1(a)], no change in spin precession frequency is observed when a current I is passed through the 2DEG (see the Supplemental Material[40]), indicating that the two in-plane components of ω SOare small. To measure the out-of-plane component, Bextis applied at an oblique angle β/negationslash=0. Figure 1(a)illustrates the directions ofBextand the resulting direction of the total spin precession axis,ωtot=(0,ωy ext,ωz ext+ωz SO). Here, /planckover2pi1ωy ext=gyyμBBy extand /planckover2pi1ωz ext=gzzμBBz ext. The in-plane and out-of-plane components of the g-factor are denoted as gyyandgzz;μBis the Bohr magneton, and /planckover2pi1is the reduced Planck constant. Both gyyand gzza r ea s s u m e dt ob en e g a t i v e[ 37,41], as indicated in Fig. 1(a). TRKR measurements are presented in Fig. 1(c)for a current of 200 μA applied along the [1 10] direction of a 100- μm-wide Hall bar channel with Bext=235 mT and β=45◦. The TRKR curves recorded clearly differ for a positive current (blackcircles) and a negative one (red circles): For a positive current,the spins precess faster than for a negative one, and the signaloscillates with a larger offset from the zero level. By referringto the schematics in Fig. 1(b), the TRKR curves measured canbe described by S z(t)=S⊥e−t/T∗ 2cos(ωtott)+S/bardble−t/T 1. (1) The signal consists of an oscillating component with amplitude S⊥=S0sin2αand a nonoscillating component with S/bardbl= S0cos2α. The angle between ωtotand the zaxis is denoted by α.T∗ 2is the spin dephasing time of the electron ensemble, T1is the relaxation time along the precession axis, and S0is the ini- tial spin polarization along zatt=0. From the TRKR curves measured, α=arctan/radicalbigS⊥/S/bardblcan be directly determined. Fitting the data with Eq. ( 1) [lines in Fig. 1(c)], we obtain ωtot=7.81 GHz and α=36.2◦forI=+200μA,ωtot= 7.16 GHz and α=37.8◦forI=0μA, and ωtot=6.66 GHz andα=41.4◦forI=−200μA. A simultaneous increase ofωtotand a decrease of αpoint to an ωz SOwith the same sign as ωz ext[see Fig. 1(a)]. This suggests ωz SO<0f o rI>0, and vice versa ωz SO>0f o rI<0. This scenario is further conﬁrmed by reversing the direction of Bextand thus ωext, i.e., by choosing β=225◦. As seen from the TRKR curves in Fig. 1(d), in this case a positive current now decreases ωtot to 6.37 GHz but increases αto 44.2◦, whereas the negative current increases ωtotto 7.39 GHz and decreases αto 35.1◦ fromωtot=6.79 GHz and α=39.1◦forI=0μA. From Figs. 1(a)–1(d), we conclude that the current-induced out-of- plane SO ﬁeld changes both the magnitude and the directionof the precession axis. To quantify the linear dependence of the SOI on the electron momentum, we extract ω totfrom ﬁts to TRKR data taken at different IandBext. From the value pairs ω+ tot(β=45◦) and ω− tot(β=225◦) for the same Bext, we obtain ωSO≈(ω+ tot− ω− tot)/(2 cos α0). This approximation is valid for ωext/greatermuchωSO, and the factor of cos α0(withα0being the value of αatI=0) corresponds to the projection of ωSOontoωext.I nF i g . 2(a),w e (a) (b)(c) θI yx[110 ] direction [001] direction Current vector 100 μA 200 μA 300 μA156 mT 235 mT 313 mTBextħω (neV) 500 0 -500 500 0 -500300 200 100 0 -100 -200 -300-200 0 200 Current ( μA) -200 0 200 Current ( μA) 0 100 200 300 Angle θ (degree)ħω (neV) ħω (neV) FIG. 2. (Color online) (a), (b) Current-dependent SO energy splitting in the presence of different Bext,Bext=156 mT (circles), Bext=235 mT (triangles), and Bext=313 mT (squares), with currents along the [1 10] direction in (a) and along the [001] direction in (b). The solid line in (a) is a linear ﬁt to all the symbols. (c) Dependence of the SO energy splitting /planckover2pi1ωSOon the direction of an introduced current of 100 μA (cross symbols), 200 μA (plus symbols), and 300 μA (star symbols) with Bext=235 mT. Solid lines are ﬁts. Inset: Schematic description of rotation of the current withrespect to the crystal axis. 121304-2RAPID COMMUNICATIONS UNIDIRECTIONAL SPIN-ORBIT INTERACTION AND . . . PHYSICAL REVIEW B 90, 121304(R) (2014) ﬁnd that /planckover2pi1ωSOdepends linearly on the current Ialong the [1 10] direction, with a same slope of −2.2 neV/μA for all values of Bext. From the slope, the SOI coefﬁcient is determined as γ≈ −10 eV ˚A3(see the Supplemental Material [ 40]), which agrees well with our previously determined value in (001)-orientedGaAs QWs [ 9,36]. As a comparison, TRKR measurements are performed for Ialong the [001] direction. As presented in Fig. 2(b),ω SO is found to be small and to depend little on Ifor all Bext.A residual slope of +0.25 neV /μA is mostly likely related to a small direction misalignment of IandBext. This supports the theoretical prediction that the SOI is absent for an electronmomentum along the [001] direction [ 18,23]. Next, we study the angular dependence of the SO ﬁeld for currents along arbitrary in-plane directions θ(measured from theydirection), as depicted in the inset of Fig. 2(c). For this, a cross-type mesa structure is used with a channel width of150μm. For each current magnitude, the SO energy splitting in the mesa center obtained can be well ﬁtted (lines) by therelation /planckover2pi1ω SO(θ)=/planckover2pi1ωSO,0cos(θ+θ0). The /planckover2pi1ωSO,0obtained depends linearly on I. We ﬁnd 77 neV for I=100μA, 158 neV for 200 μA, and 255 neV for 300 μA. A nonzero value forθ0is due to the nonperfect resistance match of two arms of the cross-type structure, and thus the current ﬂow is slightlytilted in the mesa center (see e.g., Ref. [ 39]). The experimental observations presented in Fig. 2(c) further corroborate that the SOI depends only on the ycomponent of the electron momentum in a (110)-oriented QW [ 23] and thus has the symmetry needed to support a PSH state [ 25]. Now we discuss the effect of the SOI on the direc- tion of the spin precession axis. Figures 3(a) and 3(b) show αas obtained from the S ⊥/S/bardblmeasured for Ialong the [001] direction, where no SO ﬁeld is expected. ForB ext=156 mT, we ﬁnd that αdepends only weakly on I. However, there is a clear current dependence once Bext is increased to 313 mT. This can only be attributed to a current-induced change of the g-factor tensor, which is β=45° β=225 ° (a) (b) (c) (d)156 mT 313 mTAngle α (degree)45 4035 45 40 35 -200 0 200 Current ( μA)-200 0 200 Current ( μA) along [110] along [001] FIG. 3. (Color online) (a), (b) Dependence of the angle αon the current along the [001] direction, as determined experimentally from the Kerr rotation amplitudes S/bardblandS⊥. (c), (d) Dependence of α (symbols) on the current along the [1 10] direction. The lines are calculated from the experimentally determined values of ωSOand the current-dependent g-factor tensor. For (a)–(d), the external magnetic ﬁeld is Bext=156 mT for circles and Bext=313 mT for squares.likely related to cyclotron motion induced by Bz ext[42–44] and the shift of the Fermi circle by a current. As ωSO=0f o r this current direction, we have ωy ext=ωtotsinαandωz ext= ωtotcosα. Therefore, gyy=−/planckover2pi1ωtotsinα/(BextμBsinβ) and gzz=−/planckover2pi1ωtotcosα/(BextμBcosβ) can be determined directly from the ωtotandαmeasured (see the Supplemental Material [40]). Along the [1 10] direction, αdepends differently on I [symbols in Figs. 3(c) and3(d)]. From the linear dependence ofωSOonIshown in Fig. 2(a), a monotonic variation ofαwithIis expected. At larger Bext,αchanges less strongly with I, because ωSObecomes even smaller relative toωext. We can compare the measured αwith the calculated α=arctan[ ωy ext/(ωz ext+ωSO)] [lines in Figs. 3(c) and3(d)], using the ωSOmeasured and including the current-dependent g-factor tensor. The values for αobtained by the two methods agree very well. In the unidirectional SO system of the (110)-oriented QW, a PSH is predicted if the excited spin polarization is perpendicular to the SO ﬁeld [ 25]. In our experiment, the optically generated spin polarization is along the QW growth axis, i.e., parallel to the SO ﬁeld. Therefore, the SO ﬁeld alone is not able to rotate the initial spin polarization and noformation of a PSH is possible. Here we investigate whether an external in-plane magnetic ﬁeld that tilts the spin states into the QW plane can provide the necessary starting condition for the formation of a PSH. Such an external magnetic ﬁeld perturbs the unidirectionality of the effective total magnetic ﬁeld as well as its linear dependence on the ycomponent of the electron momentum [ 27]. As we will show in the following, for a ﬁnite but small enough in-plane B ext, the ﬁngerprint of an emerging PSH can nevertheless be probed. To spatially resolve the diffusing spins, we perform two- color TRKR measurements in the V oigt geometry [ β=0i n Fig. 1(a)] to resolve the spin dynamics in real space. Pump and probe beams are focused to a diameter of ∼2μm. The energy for the pump beam is 1.569 eV with an excitationpower density of ∼1100 W /cm 2, and 1.535 eV for the probe beam with a power density of ∼760 W /cm2. In Fig. 4(a), maps of Szmeasured versus tand the spatial position are plotted for different Bext. In the case of Bext=0, two similar spin evolutions are observed for scans along the[1 10] direction (upper panel) and along the [001] direction (lower panel). From the diffusive expansion of the spinpolarization in space, we determine the electron spin-diffusionconstant D s=0.043 m2/s by using the method detailed in Ref. [ 27]. If an in-plane external ﬁeld of Bext=100 mT is applied along the yaxis, the overall spin polarization precesses about this axis. However, the maps of Szare obviously different for scans along the two different directions. For scans along the[1 10] direction, the spins close to the center precess about Bext, whereas the spins that have diffused away initially maintaina positive S z, leading to an “eye pattern” within the ﬁrst precession period. In contrast, spin diffusion along the [001]direction does not affect the precession phase. Speciﬁcally,at, e.g., t=1000 ps, the spin polarization changes its sign (from negative in the center to positive at about ±20μm) as the position along [1 10] varies, whereas the spins distributed along the [001] direction all have the same precession phase 121304-3RAPID COMMUNICATIONS Y .S .C H E N ,S .F ¨ALT, W. WEGSCHEIDER, AND G. SALIS PHYSICAL REVIEW B 90, 121304(R) (2014) Kerr rotation (Arb. units) (a) 20 0 -20 20 0 -20Position ( μm)[110] [001]Measurements-0.2 0 0.2 0 1000 2000 Time delay (ps)0 1000 2000 Time delay (ps)0 1000 2000 Time delay (ps)0 1000 2000 Time delay (ps)20 0 -20 20 0 -20Position ( μm)[110] [001]Bext=0 m T Bext= 100 mT Bext= 200 mT Bext= 400 mT (b)Simulations FIG. 4. (Color online) (a) Two-dimensional plot of experimentally measured spin dynamics in real space obtained by scanning the pump beam along the [1 10] direction (upper panel) and the [001] direction (lower panel). The external magnetic ﬁeld applied along the [1 10] direction is from left to right sequentially, Bext=0, 100, 200, and 400 mT. (b) Corresponding Monte Carlo numerical simulation results. (individual line scans are shown in the Supplemental Material [40]). On the one hand, Bexttilts the initial spin polarization into the QW plane, providing an increasing spin componentalong the xdirection. On the other hand, ω SOrotates this component into the positive (negative) ydirection for spins that move along the positive (negative) [1 10] direction, leading to a helical spin pattern of the in-plane spin polarization.Although this helix cannot be observed directly in S z,i t explains the anisotropy observed in the maps of Fig. 4(a). Because the SO ﬁeld in the GaAs QW at the Fermi wave number is on the order of 10 T (using the SOI coefﬁcientdetermined—see also Ref. [ 27]), by far exceeding B exthere, the sums of Bextand the SO ﬁeld are completely different for the electron momentum along the two in-plane directions. Alongthe [001] direction, the total effective ﬁeld is in plane, whereasalong [1 10], it mostly points along the zaxis. Therefore spin precession into the QW plane is suppressed for spinsthat move along the [1 10] direction, explaining the observed eye pattern. When Bextis increased to 200 mT, the curved stripe pattern for scans along the [1 10] direction is clearly observed, which indicates a pronounced perturbation by theSOI compared with the straight stripe pattern along the [001] direction. This feature becomes weaker for B ext=400 mT. In the model explained above, the increased magnitude of Bexttilts the total ﬁeld closer to the QW plane and therefore makes the spins precess faster into the plane, so thatthe out-of-plane SO ﬁeld perturbs the spin precession lessefﬁciently. Corresponding Monte Carlo numerical simulations of the spin-diffusion dynamics are presented in Fig. 4(b).F o rt h e simulations, we have taken γ=−10 eV ˚A 3as obtained from the ﬁrst types of measurements and the diffusion constantD s=0.043 m2/s from the measurements of spin-diffusion dynamics at Bext=0 mT. Without any free parameter, the simulation results match perfectly the experimental resultsin Fig. 4(a) of spin diffusion along both the [1 10] and [001] directions for each Bext. Such an excellent numerical reproduction further validates the results obtained in theoblique experimental geometry and, speciﬁcally, conﬁrms theunidirectional SOI with the SU(2) symmetry that leads tothe formation of a PSH state. The work is ﬁnancially supported by the Swiss National Science Foundation within the project of NCCR QSIT. Weacknowledge P. Altmann for experimental assistance andreading the manuscript, and M. Tschudy, R. Grundbacher,and U. Drechsler for help with the sample fabrication. We also thank A. Fuhrer and R. Allenspach for fruitful discussions. [1] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78,1335 (1997 ). [2] O. Z. Karimov, G. H. John, R. T. Harley, W. H. Lau, M. E. Flatt ´e, M. Henini, and R. Airey, P h y s .R e v .L e t t . 91,246601 (2003 ).[3] P. Debray, S. 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PhysRevB.100.195138.pdf	PHYSICAL REVIEW B 100, 195138 (2019) Topological phase transition in the archetypal f-electron correlated system of cerium Junwon Kim,1,Dong-Choon Ryu,1,Chang-Jong Kang,1,†Kyoo Kim,1,2Hongchul Choi,1,‡T.-S. Nam,1and B. I. Min1,§ 1Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 2MPPHC_CPM, Pohang University of Science and Technology, Pohang 37673, Korea (Received 14 June 2019; revised manuscript received 22 September 2019; published 25 November 2019) A typical f-electron Kondo lattice system Ce exhibits a well-known isostructural transition, the so-called γ-αtransition, accompanied by an enormous volume collapse. Most interestingly, we have discovered that a topological phase transition also takes place in elemental Ce, concurrently with the γ-αtransition. Based on the dynamical mean-ﬁeld theory approach combined with density functional theory, we have unraveled that thenontrivial topology in α-Ce is driven by the f-dband inversion, which arises from the formation of a coherent 4fband around the Fermi level. We captured the formation of the 4 fquasiparticle band that is responsible for the Lifshitz transition and the nontrivial Z 2topology establishment across the phase boundary. This discovery provides a concept of a “topology switch” for topological Kondo systems. The “on” and “off” switching knob inCe is versatile in a sense that it is controlled by the available pressure ( /lessorsimilar1 GPa) at room temperature. DOI: 10.1103/PhysRevB.100.195138 The physics of strongly correlated f-electron materials has been a longstanding subject of special interest due to thecomplex interplay between the underlying interactions, suchas strong Coulomb correlations, spin-orbit (SO) coupling, andthe hybridization of the localized fand conduction elec- trons. More intriguing is that the interplay is very sensitiveto small changes in the external parameters. Elemental Ce,which has one occupied felectron in its atomic phase, is a prototypical f-electron Kondo lattice system exhibiting such sensitivity. Indeed, Ce shows a rich phase diagram (seeFig.1) and many interesting physical properties as a function of temperature ( T) and pressure ( P)[1–4]. The ﬁrst-order isostructural volume collapse transition from the γtoαphase of face-centered-cubic (fcc) Ce is the most representativephenomenon that experiences the sensitivity. However, thedriving mechanism of the γ-αtransition is still under debate, between the two well-known models: the Mott transition [ 5] versus Kondo volume collapse [ 6]. The current consensus is that there exists at least a signiﬁcant change in the Kondo hy-bridization between the localized 4 felectrons and conducting electrons across the transition [ 7–10]. This peculiarity in Ce could facilitate the emergence of nontrivial topology in theground-state αphase of Ce. In a recent theoretical work on topological Kondo in- sulators [ 11], it is shown that the Kondo hybridization in f-electron systems can play an important role in the for- mation of nontrivial topology. Since then, many subsequentstudies have been reported to search for nontrivial topolog-ical materials, where the Kondo hybridization gap exists, *These authors contributed equally to this work. †Present address: Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA. ‡Present address: IBS-CCES, Seoul National University, Seoul 08826, Korea. §bimin@postech.ac.kre.g., CeNiSn, CeRu 4Sn6,C e 3Bi4Pt3,S m B 6, SmS, and YbB 12 [11–17]. Despite extensive studies, however, the topological nature of mother elements, Ce, Sm, and Yb, supplying thecorrelated felectrons to the above Kondo insulator com- pounds, has not been explored yet. Here, we report, basedon the dynamical mean-ﬁeld theory (DMFT) approach com-bined with density functional theory (DFT) that has beensuccessful in describing the electronic structures of Ce andCe compounds [ 8,9,18,19], that a narrow f-band metal α-Ce has the nontrivial topology of a topological-insulator (TI)-typeand topological-crystalline-insulator (TCI)-type nature, andthe topological phase transition and the Lifshitz electronictransition occur concomitantly with the γ-αvolume collapse transition in Ce. Figure 2shows the DMFT band structures and the densities of states (DOS) of γ- andα-Ce. In the DMFT calculations, we have used the Coulomb correlation ( U) and the exchange (J) interaction parameters of U=5.5 eV and J=0.68 eV for the Ce felectrons (refer to the Supplemental Material for the computational details [ 21]). The 4 fspectral weights of both phases have three main parts in common: the lowerHubbard band (LHB) at −2.0t o−2.5 eV corresponding to the 4 f 0ﬁnal state, the upper Hubbard band (UHB) at 2–4 eV corresponding to the 4 f2ﬁnal state, and the Kondo resonances near the Fermi level ( EF) corresponding to the 4f1ﬁnal states. The energy positions of LHB and UHB are in good agreement with photoemission spectroscopy (PES)[22–25] and inverse PES experiments [ 26]. One of the most notable features in Fig. 2is that the spectral weight of the Kondo resonance around E Fis much stronger in α-Ce than in γ-Ce, and exhibits the coherent quasiparticle band feature in α-Ce, as is consistent with previous PES [ 22–26] and theoretical reports [ 27–30]. As will be discussed be- low, these contrasting Kondo resonance features betweenthe two phases lead to quite different topological classes:trivial and nontrivial Z 2topologies for γ-Ce and α-Ce, respectively. 2469-9950/2019/100(19)/195138(6) 195138-1 ©2019 American Physical SocietyJUNWON KIM et al. PHYSICAL REVIEW B 100, 195138 (2019) FIG. 1. Left: A phase diagram of Ce [ 20] (see also the Supple- ment Material [ 21]).α-Ce at the red star and γ-Ce at the blue star are selected for a comparison of electronic structures in Fig. 2. The blue dotted line corresponds to the P-Visotherm at 293 K. Right: The bulk BZ of fcc Ce and its (001) and (110) surface BZ. There are twoindependent mirror planes of k y=0 (in blue) and kx=ky(in gray), which, respectively, yield two mirror-symmetry lines along ¯M-¯/Gamma1-¯M and ¯X-¯/Gamma1-¯Xin the (001) surface BZ. Similarly, in the (110) surface BZ, two mirror-symmetry lines are formed along ¯Y-¯/Gamma1-¯Yand ¯X-¯/Gamma1-¯X. The incoherent and coherent 4 fspectral weights for γ-Ce and α-Ce, respectively, are more clearly shown in the ampliﬁed DMFT band structures in Fig. 3. It is seen in Fig. 3(a) that, for γ-Ce, 4 felectrons are incoherent, and so mainly the 5 dband crosses EF, which agrees well with the optical spectroscopy result [ 31]. In contrast, for α-Ce, the coherent 4 fquasiparticle band feature is evident near EFin Fig. 3(b), which is the origin of the effective mass enhancement of charge carriers and the change of the chargecarrier character from 5 dto 4 f. The coherent band fea-ture for α-Ce is corroborated by the fact that the DMFT bands have almost the same dispersion as the renormalizedDFT bands rescaled approximately by 1 /2 [dotted green line in Fig. 3(b)]. The different electronic structures between the two phases are also reﬂected in the Fermi surfaces (FSs). The shapes of FSs in Fig. 3(d) are topologically different, suggesting that the γ-αtransition corresponds to the Lifshitz transition (see the Supplemental Material [ 21]). It is noteworthy in Fig. 3(d) that, while the DMFT FS of γ-Ce is very close to that obtained from the DFT- OPENCORE (“4f-OPENCORE ”) calculation considering the 4 felectrons as core electrons, the DMFT FS of α-Ce is quite similar to the DFT FS. These results indicate that, for γ-Ce, the contribution of 4 felectrons to the FS is negligible, and, for α-Ce, the 4 fquasiparticle band at EFcan be described properly by the DFT band (see Fig. S1 of the Supplemental Material [ 21]). The key ingredient that makes the difference has something to do with the degree of the renormalization factor (quasi-particle weight) Z, arising from the Coulomb correlation interaction of 4 felectrons. The renormalization factor Zis obtained from the self-energy /Sigma1(iω n) at the lowest Matsubara frequency. As shown in Fig. S5 of the Supplemental Material[21], we have obtained qualitatively different behaviors of /Sigma1(ω)’s between the αandγphases, which produce quite distinct electronic structures and resulting physical parame-ters. Indeed, Fig. 3(e) shows that Zincreases discontinuously across the γ-αtransition. As a result, both the hybridization strength /Delta1(ω) [Fig. 3(c)] and the f-fhopping strength, which are to be effectively proportional to Z, are enhanced for α-Ce, which give rise to the enhanced 4 fspectral weight and help to form the coherent 4 fband around E F[29]. The evolution of the electronic structure across the γ-α transition makes the elemental Ce more interesting in a topo-logical sense. The coherent quasiparticle band in α-Ce, which, via the hybridization with the conduction band, brings about FIG. 2. (a) The DMFT electronic structure and DOS for γ-Ce calculated at V=34 Å3(P=ambient pressure) and T=500 K, and (b) those for α-Ce calculated at V=27.76 Å3(P=0.88 GPa) and T=100 K. The 4 fspectral weights of both phases consist of mainly three parts: UHB at 2–4 eV , LHB at −2.0t o−2.5 eV , and the Kondo resonance near EF. In addition to the Kondo resonance near EF(i), the SO side peaks (ii) and (iii) are seen at ∼± 0.3 eV . Note that only α-Ce shows the coherent quasiparticle 4 fband around EF, which is shown more clearly in Fig. 3. 195138-2TOPOLOGICAL PHASE TRANSITION IN THE … PHYSICAL REVIEW B 100, 195138 (2019) FIG. 3. The ampliﬁed DMFT electronic structures near EF:( a )f o r γ-Ce and (b) for α-Ce. For γ-Ce, 4 fstates are hardly seen, because they are incoherent. For α-Ce, the coherent 4 fbands formed around EFproduce, via the hybridization with the conduction band, the separated bands with the gap in-between (colored in gray). There exist clear energy gaps at the TRIM points of /Gamma1,XandL, and also small energy gaps atW, in between L-/Gamma1, and at A along /Gamma1-K. The inset shows the gap formation at A, arising from the same /Gamma15symmetry of the crossing bands [32]. The green dotted lines overlaid with DMFT bands are the DFT bands rescaled by 1 /2. (c) Imaginary part of the DMFT hybridization function /Delta1(ω). (d) DMFT and DFT FSs for both phases (see also Fig. S1 [ 21]). (e) The renormalization factor Zand the energy gaps at /Gamma1and L(/Delta1/Gamma1and/Delta1L) are displayed as a function of pressure (see also Figs. S2–S4 [ 21]). The ﬁrst-order-type phase transition is manifested across theγ-αtransition. (f) The product of the parity eigenvalues of α-Ce at eight TRIM points in the fcc BZ. the hybridization gap in the α-Ce phase, is indicated by the gray-shaded area in Fig. 3(b). The energy gaps are clearly seen at every time-reversal invariant momentum (TRIM) point of/Gamma1,X, and L, while those at Wand in between L-/Gamma1are barely gapped. Then, with respect to the hybridization gap, the 5 d band of even parity and the 4 fband of odd parity are inverted at the TRIM point X. Since the crystal structure is symmetric under the inversion operation, the additional odd parity to theTRIM points yields the nontrivial Z 2topology of α-Ce, as shown in Fig. 3(f). Figure 3(e) shows that the necessary conditions for the nontrivial Z 2topology, the buildup of the coherent 4 fband and the opening of the hybridization gap, are established atthe very starting edge (pressure 0.8 GPa at 293 K) of the αphase in the γ-αtransition. Note that no gaps are present in the γphase, but the gaps at the TRIM points, /Delta1 /Gamma1and /Delta1Lof about 30 meV ( /Delta1X>2 eV), are suddenly developed in the αphase. This implies that the ﬁrst-order topological phase transition would occur concomitantly with the γtoα volume collapse transition. A more detailed evolution of theband structures across the γ-αtransition is given along a P-V isotherm at 293 K in Fig. S2 of the Supplemental Material. Forcomparison, the crossover-type topological phase transition,which is expected to occur above the critical point, is alsodiscussed in the Supplemental Material [ 21]. In order to conﬁrm the nontrivial Z 2topological invariance ofα-Ce, we have performed the surface electronic structure calculations for the slab geometry of α-Ce with a (001) 195138-3JUNWON KIM et al. PHYSICAL REVIEW B 100, 195138 (2019) FIG. 4. (a) The (001) surface electronic structure of α-Ce, calculated by the tight-binding (TB) model with semi-inﬁnite slabs. The TB Hamiltonian is constructed from the DFT band result (rescaled by 1 /2 near EF). D 1: a Dirac point at ¯/Gamma1;D 2: a Dirac point at ¯M; U: projected bulk bands above the indirect gap (cyan colored); L: projected bulk bands below the indirect gap (violet colored). (b), (c) The helical spinstructures of the D 1and D 2Dirac cone energy surfaces, as indicated by (i) and (ii), respectively. (d) The (110) surface electronic structure of α-Ce. (e), (f) Ampliﬁed band structures inside the green square and the red square in (d), respectively. In (e), TSSs of a typical TCI-type nature are revealed with the gapped (red arrow) and protected (black arrow) Dirac points, while in (f), TSSs are mostly buried under the projected bulk bands. surface and explored the existence of topological surface states (TSSs) in Fig. 4(a). Note that, as shown in Fig. 1, one X point is projected onto ¯/Gamma1, while two nonequivalent XandX/prime points are projected onto ¯Mof the (001) surface BZ. Indeed, as shown in Fig. 4(a), the TSSs and corresponding Dirac points emerge in the indirect gap region at ¯/Gamma1(D1) and ¯M(D2). Due to the bulk metallic nature of α-Ce, most parts of the Dirac bands at ¯Mare buried under the projected bulk bands and so the band connectivity is not clear. Nevertheless, it is evident inFig.4(a) that the surface states along ¯/Gamma1-¯Xare the Dirac cone states, because the lower surface band reaches the projectedbulk bands (L) below the indirect gap, while the upper onereaches the projected bulk bands (U) above the indirect gap.The helical spin textures of the corresponding Dirac cone FSsaround ¯/Gamma1and ¯Min Figs. 4(b) and 4(c) also manifest the spin-momentum locking behavior, reﬂecting its topologicalnature. The double Dirac points, which are supposed to be at ¯M due to the projection of two nonequivalent XTRIM points (Fig. 1), are to be separated due to the hybridization between the bands of the double Dirac cones. On the (001) surfaceBrillouin zone (BZ) of α-Ce, there are two mirror-symmetric lines, ¯/Gamma1-¯Xand ¯/Gamma1-¯M, as shown in Fig. 1, which could play a key role in realizing the TCI-type nature. It is thus obvious thatthe band crossing along ¯X-¯Mthat is not a mirror-symmetric line would be gapped, but that along ¯M-¯/Gamma1needs further consideration. However, the surface states along ¯M-¯/Gamma1are completely buried under the projected bulk bands, and so itis not easy to identify the speciﬁc TCI-type band feature inFig.4(a). In view of the surface states inside the dotted black square and those along ¯X-¯Mdesignated by the red arrows in Fig. 4(a), we just conjecture that the band crossing along¯M-¯/Gamma1would be gapped to have Rashba-type surface states, as reported for the golden phase of SmS ( g-SmS) that is expected to have the same topological symmetry as α -Ce [ 33]. In fact, α-Ce is found to have the same mirror Chern numbers as g-SmS [ 34], as shown in Fig. S6 of the Supplemental Material [21]. We have also examined the TSSs for the (110) and (111) surfaces of α-Ce. For the (110) surface, single and double Dirac points are expected to be located at ¯/Gamma1and ¯X, respec- tively, as shown in Fig. 1. For the (111) surface, only the single Dirac point is expected at ¯M, as shown in Fig. S7 of the Supplemental Material [ 21]. As shown in Fig. 4(d) and Fig. S7, however, neither the (110) nor (111) surface statesshow a clear TI-type or TCI-type signature in the hybridiza-tion gap region, because, here too, most of the surface statesnear E Fare buried under the projected bulk bands. In this circumstance, for the (110) surface, one apparent TCI signa-ture is seen at ¯Xnear 240 meV in Fig. 4(e), which demon- strated the gapped and protected Dirac points along ¯X-¯S(red arrow) and ¯X-¯/Gamma1(black arrow), respectively. This suggests that the near- E FTSSs buried under the projected bulk bands in Fig.4(f)would also have the TCI-type band nature. Our ﬁnding highlights that a typical narrow f-band metal α-Ce is a topological Kondo system of TI- and TCI-type na- ture, and the “on” and “off” topology switch can be operativeby using a P-tuning or T-tuning knob, accompanied by a ﬁrst- order volume collapse and Lifshitz transitions. So Ce wouldbe an excellent test bed for investigating the topological phasetransition in f-electron Kondo lattice systems. It is thus highly desirable to explore the topological surface states in α-Ce, preferentially for its (001) surface, by using high-resolutionangle-resolved PES measurements. 195138-4TOPOLOGICAL PHASE TRANSITION IN THE … PHYSICAL REVIEW B 100, 195138 (2019) We would like to thank J. D. Denlinger, J.-S. Kang, and J. H. 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PhysRevB.97.165401.pdf	PHYSICAL REVIEW B 97, 165401 (2018) Tuning Rashba spin-orbit coupling in homogeneous semiconductor nanowires Paweł Wójcik,1,Andrea Bertoni,2,†and Guido Goldoni3,‡ 1AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. A. Mickiewicza 30, 30-059 Krakow, Poland 2CNR-NANO S3,6 Institute for Nanoscience, Via Campi 213/a, 41125 Modena, Italy 3Department of Physics, Informatics and Mathematics, University od Modena and Reggio Emilia, Italy (Received 31 January 2018; published 2 April 2018) We use k·ptheory to estimate the Rashba spin-orbit coupling (SOC) in large semiconductor nanowires. We speciﬁcally investigate GaAs- and InSb-based devices with different gate conﬁgurations to control symmetry andlocalization of the electron charge density. We explore gate-controlled SOC for wires of different size and doping,and we show that in high carrier density SOC has a nonlinear electric ﬁeld susceptibility, due to large reshapingof the quantum states. We analyze recent experiments with InSb nanowires in light of our calculations. Goodagreement is found with the SOC coefﬁcients reported in Phys. Rev. B 91,201413(R) (2015 ), but not with the much larger values reported in Nat. Commun. 8,478(2017 ). We discuss possible origins of this discrepancy. DOI: 10.1103/PhysRevB.97.165401 I. INTRODUCTION Semiconductor nanowires (NWs) are attracting increas- ing interest for (ultrafast) electronic and optoelectronic ap-plications, including single-photon sources [ 1], ﬁeld effect transistors [ 2], photovoltaic cells [ 3], thermoelectric devices [4], lasers [ 5,6], and programmable circuits [ 7]. Recently, special attention is raised for spintronic applications [ 8–10] and topological quantum computing [ 11]. Due to strong spin- orbit coupling (SOC) in InSb- or InAs-based NWs, a helical gap has been observed if a ﬁnite magnetic ﬁeld is appliedorthogonal to the SOC effective ﬁeld, B SOC[12–16]. In this 1D state, carriers with opposite momentum have opposite spin.In combination with the proximity-induced superconductivity[17,18], it imitates a spinless p-wave superconductor (Kitaev chain) [ 19], making the strongly spin-orbit coupled InSb and InAs NWs possible host materials for topologically protectedquantum computing based on Majorana zero modes [ 20–24]. SOC is a relativistic effect where a part of the electric ﬁeld is seen as an effective magnetic ﬁeld in the chargedparticle rest frame. In semiconductor crystals, the electric ﬁeldmay arise from a symmetry breaking that is either intrinsic,i.e., related to the crystallographic structure of the material(Dresselhaus SOC) [ 25], or induced by the overall asymmetry of the conﬁnement potential due to an electrostatic ﬁeld,due to, e.g., compositional proﬁles, strain, or external gates(Rashba SOC) [ 26]. Typically, SOC is the combination of both components [ 27], but zinc-blende NWs grown along [111] posses inversion symmetry, and the Dresselhaus contributionvanishes. This NW direction is the one used in experimentsexploring the existence and nature of Majorana bound states[20,23]. Therefore we shall consider only the Rashba SOC throughout the paper. pawel.wojcik@ﬁs.agh.edu.pl †andrea.bertoni@nano.cnr.it ‡guido.goldoni@unimore.itA critical issue in this context is to engineer devices with strong SOC, as the ratio of the spin-orbit energy relative tothe Zeeman energy determines the magnitude of topologicalenergy gap protecting zero-energy Majorana modes [ 21]. Re- cent studies of 2D InSb wires and planar InSb heterostructuresshow a SOC constant α R=3 meV nm [ 28,29]. Larger values were reported for quantum dots gated in InSb NWs, αR= 16–22 meV nm [ 30,31], which likely includes a contribution from the local electric ﬁelds of the conﬁning gates. The standard method to extract the SOC in semiconductor NWs is by magnetoconductance measurements in low mag-netic ﬁelds, exploiting the negative magnetoresistance due toweak antilocalization [ 32,33]. Recently, this technique was used to extract SO strength in InSb NWs demonstrating verylarge values of the SOC constant, α R=50–100 meV nm [ 34]. Unexpectedly, a much higher value of αRwas reported in Ref. [ 15] where the authors used the conductance measurement technique. The measure of the conductance through the helicalstate and comparison of the data to the theoretical modelgives a spin-orbit energy E SO=6.5 meV , which corresponds toαR=270 meV nm, the highest value reported so far for semiconductor NWs. The determination of SOC strength in semiconductor NWs still remains an open issue, with different measurement tech-niques leading to values of α Rdiffering by almost one order of magnitude. It should be noted that for typical samples,with diameters in the tens of nm range, the symmetry andlocalization of the quantum states is a delicate balance betweendifferent energy scales and it is strongly inﬂuenced by externalﬁelds [ 35–38]. On the other hand, theoretical investigations of SOC, so far [ 39,40], only rely on simple models which do not capture the complexity of the quantum states whose symmetryunderlies the Rashba contribution to SOC nor its tunability byan electric ﬁeld, which is the goal of the present study. In this paper, we evaluate the SOC strength on the basis of ak·ptheory using self-consistent quantum states, which take into account the realistic geometry of large, doped NWs. Our analysis includes external metallic gates and dielectric 2469-9950/2018/97(16)/165401(11) 165401-1 ©2018 American Physical SocietyPAWEŁ WÓJCIK, ANDREA BERTONI, AND GUIDO GOLDONI PHYSICAL REVIEW B 97, 165401 (2018) FIG. 1. Schematic illustration of a NW with a bottom gate. A typical electron gas distribution is shown inside (yellow). A schematic of the semiconductor band structure used within the Kane model isshown. Symbols indicate conduction (c), heavy-hole (hh), light-hole (lh), and split-off (so) bands with corresponding group-theoretical classiﬁcation of zone center states. layers of typical NW-based devices. We evaluated the SOC coefﬁcients as a function of NW size and gate conﬁguration.We ﬁnd that the strong interplay between external ﬁelds andthe localization of quantum states results in a strong nonlinearelectric ﬁeld susceptibility for SOC in the high carrier densityregime. We analyze recent experiments with InSb NWs inlight of our calculations. Good agreement is found with SOCreported in Phys. Rev. B 91,201413(R) (2015 ), but not with the much larger values measured in Nat. Commun. 8,478(2017 ), and we discuss possible origins of this discrepancy. The paper is organized as follows. In Sec. II, we obtain SOC coefﬁcients from the k·ptheory. The effective Hamiltonian that determines the quantum states is devised in Sec. II A, while SOC coefﬁcients in terms of the envelope functions ofthe structure are derived in Sec. II B. In Sec. III, we apply our methodology to GaAs-based (Sec. III A) and InSb-based (Sec. III B) devices, discussing recent experiments in the latter case. A summary of our investigation is drawn in Sec. IV. II. THEORETICAL MODEL Our target systems are NWs with hexagonal cross-section, grown in the [111] direction, see Fig. 1. In these systems, quantum states are determined by several sample parameters,including geometry, Fermi energy, external ﬁelds, etc. Below,we use the 8 ×8 Kane model to derive the (Rashba) SOC con- stants in terms of a realistic description of the quantum states.This allows for quantitative predictions of SOC constants asa function of the gate voltages and geometrical parameters indifferent regimes and gate conﬁgurations. A. Effective Hamiltonian for SOC of conduction electrons Formally, our target system has translational invariance along z. Each component of the envelope function (one for each total angular momentum component) can be developedin a set of subbands ψ n(x,y), the coefﬁcients of the linear combination being determined by the Kane Hamiltonian. Here,nis a subband index and ( x,y) are the space directions in a plane sectioning the NW. Since NWs in our calculations arequite large (with diameters ∼10 2nm), it is usually necessary to include a large number of subbands in calculations. The 8 ×8 Kane Hamiltonian reads [ 8] H8×8=/parenleftbiggHcHcv H† cvHv/parenrightbigg , (1) where Hcis the 2 ×2 diagonal matrix related to the conduction band ( /Gamma16cat the /Gamma1point of Brillouin zone, see Fig. 1), while Hvis the 6 ×6 diagonal matrix corresponding to the valence bands ( /Gamma18v,/Gamma17v) Hc=H/Gamma16(x,y)12×2, (2) Hv=H/Gamma18(x,y)14×4⊕H/Gamma17(x,y)12×2. (3) In the above expressions, H/Gamma16(x,y)=−¯h2 2m0∇2 2D+¯h2k2 z 2m0+Ec+V(x,y),(4) H/Gamma18(x,y)=Ec+V(x,y)−E0, (5) H/Gamma17(x,y)=Ec+V(x,y)−E0−/Delta10, (6) where ∇2D=(∂ ∂x,∂ ∂y),m0is the free electron mass, Ecis the energy of the conduction band edge, E0is the energy gap, /Delta10 is the split-off band gap, and V(x,y) is the potential energy. In doped systems, the potential V(x,y) consists of the sum of the Hartee potential energy generated by the electron gasand ionized dopants, and any electrical potential induced bygates attached to the NW, V(x,y)=V H(x,y)+Vgate(x,y). We adopt the hard wall boundary conditions at the surface of NWs. The off-diagonal matrix Hcvin (1) reads Hcv=⎛ ⎝ˆκ+√ 60ˆκ−√ 2−/radicalBig 2 3κz−κz√ 3κ+√ 3 −/radicalBig 2 3κz−ˆκ+√ 20 −κ−√ 6−κ−√ 3κz√ 3⎞ ⎠, (7) where ˆ κ±=Pˆk±,κz=Pkz,ˆk±=ˆkx±iˆky, and P= −i¯h/angbracketleftS\|ˆpx\|X/angbracketright/m0is the conduction-to-valence band coupling with\|S/angbracketright,\|X/angbracketrightbeing the Bloch functions at the /Gamma1point of the Brillouin zone. Using the folding-down transformation, the 8 ×8 Hamil- tonian ( 1) reduces into the 2 ×2 effective Hamiltonian for the conduction band electrons (details in Appendix) H(E)=Hc+Hcv(Hv−E)−1H† cv. (8) SinceE0and/Delta10are the largest energies in the system, we can expand the on- and off-diagonal elements of the Hamiltonian(8) to second order in the wave vectors. Then H=/bracketleftBigg −¯h 2 2m∗∇2 2D+¯h2k2 z 2m∗+Ec+V(x,y)/bracketrightBigg 12×2, +(αxσx+αyσy)kz, (9) where σx(y)are the Pauli matrices, m∗is the effective mass 1 m∗=1 m0+2P2 3¯h2/parenleftbigg2 Eg+1 Eg+/Delta1g/parenrightbigg , (10) 165401-2TUNING RASHBA SPIN-ORBIT COUPLING IN … PHYSICAL REVIEW B 97, 165401 (2018) andαx,αyare the SOC coefﬁcients given by αx(x,y)≈1 3P2/parenleftbigg1 E2 0−1 (E0+/Delta10)2/parenrightbigg∂V(x,y) ∂y,(11) αy(x,y)≈1 3P2/parenleftbigg1 E2 0−1 (E0+/Delta10)2/parenrightbigg∂V(x,y) ∂x.(12) Without SOC, the conﬁnement in the x-yplane of the NW leads to the formation of quasi-1D spin-degenerate subbands,with the in-plane envelope functions ψ n(x,y)’s determined by the compositional and doping proﬁles, the ﬁeld induced by thefree carriers, and the external gates. The 3D Hamiltonian ( 9) can be represented in the basis set ψ n(x,y)e x p (ikzz). The matrix elements of the spin-orbit term are given by αnm x(y)=/integraldisplay/integraldisplay ψn(x,y)αx(y)(x,y)ψm(x,y)dxdy. (13) These coefﬁcients deﬁne intra- ( αnn x(y)) and intersubband ( αnm x(y)) SOC constants, which are extracted from experiments [ 34] and are estimated in Sec. IIIfor several classes of material and device conﬁgurations. B. Computation of SO coupling constants To obtain the electronic states of a NW ψn(x,y)t ob eu s e d in Eq. ( 13), we employ a standard envelope function approach in a single parabolic band approximation. Electron-electroninteraction is treated at the mean-ﬁeld level by the standardself-consistent Schödinger-Poisson approach. Assuming trans-lational invariance along the growth axis z, we reduce the single-electron Hamiltonian (without SOC) to a 2D problemin the ( x,y) plane: /bracketleftbigg −¯h 2 2m∗∇2 2D+Ec+V(x,y)/bracketrightbigg ψn(x,y)=Enψn(x,y).(14) The above eigenproblem is solved numerically by a box integration method [ 41] on a triangular grid with hexagonal elements [ 42]. While this grid is symmetry compliant if the hexagonal NW is in the isotropic space, avoiding artifactsfrom the commonly used rectangular grid, calculations do notassume any symmetry of the quantum states. Therefore ourcalculations allow to describe less symmetric situations, e.g.,with external gates applied to the NW. After solving Eq. ( 14), we calculate the free electron density n e(x,y)=2/summationdisplay n\|ψn(x,y)\|2/radicalBigg m∗kbT 2π¯h2F−1 2/parenleftbigg−En+μ kbT/parenrightbigg ,(15) where m∗is the effective electron mass along the NW axis, kB is the Boltzmann constant, Tis the temperature, μis the Fermi level, and Fk=1 /Gamma1(k+1)/integraltext∞ 0tkdt et−x+1is the complete Fermi-Dirac integral of order k. Finally, we solve the Poisson equation ∇2 2DV(x,y)=−ne(x,y) /epsilon10/epsilon1, (16) where /epsilon1is the dielectric constant. Equation ( 16)i ss o l v e db ya box integration method on the triangular grid assuming, if notstated otherwise, Dirichlet boundary conditions. The resultingpotential V(x,y) is put into Eq. ( 14), and the cycle is repeateduntil self-consistency is reached. Further details concerning the self-consistent method for hexagonal NWs can be found inRef. [ 43]. The self-consistent potential energy proﬁle V(x,y) and the corresponding envelope functions ψ n(x,y) are ﬁnally used to determine the SOC αnm x(y)from Eq. ( 13). In the present study, we do not include exchange-correlation corrections, since theyresulted to be negligible in the regimes under consideration,both in the local density [ 43–45] and local-spin-density [ 46] approximations. III. RESULTS We used the above methodology to predict SOC coefﬁcients in different classes of materials of direct interest in NW-based spintronics. We put particular emphasis to establish thetunability of the SOC by external gates. Indeed, the latterstrongly shape the quantum states, particularly if NWs areheavily doped, as it turns out. We conclude this section bya qualitative comparison with the latest experiments withInSb-based NWs. A. GaAs GaAs is not a strong SOC material. However, it is the material of choice for transport experiments, due to its highmobility. Recent literature reports high-mobility in dopedGaAs-NWs, comparable to planar structures grown along thesame crystallographic directions [ 47]. Therefore, to establish the potential of GaAs for spintronics, we consider GaAshomogeneous NWs with “ideal” gate conﬁgurations, i.e., withgates directly attached to the NW. Often, in realistic devices,a dielectric spacer layer is used in experiments. Therefore ourcalculations below should be considered as an upper bound forSOC in GaAs NWs. The calculations have been carried out for the follow- ing material parameters [ 48]:E 0=1.43 eV , /Delta10=0.34 eV , m∗=m∗=0.067,EP=2m0P2/¯h2=28.8 eV and dielectric constant /epsilon1=13.18. We consider a temperature T=4.2K .W e assume constant chemical potential μ=0.85 eV . This value ensures that only the lowest electronic state is occupied at Vg= 0. If not stated otherwise, the calculations have been carriedout for the NW width W=87 nm on the grid 100 ×100. In Fig. 2, we show the SOC coefﬁcients α 11 xandα11 y with three different typical gate conﬁgurations, bottom gate, left-bottom gate, and U-shaped gate, as sketched in the top-leftinsets. The gates are held at a voltage V g, which is swept through. We ﬁrst note that at Vg=0 the NW has inversion sym- metry, hence α11 x=α11 y=0. As the gate voltage is switched on,α11 i/negationslash=0, with ibeing the direction of the axis of symmetry broken by the ﬁeld. So, for example, α11 y=0i nF i g s . 2(a)and 2(c), but not in Fig. 2(b). On the other hand, α11 x/negationslash=0i na l l conﬁgurations, since gates remove inversion symmetry aboutxin all cases. The evolution of α 11 x(Vg) is strongly asymmetric, the strongest asymmetry being observed for the conﬁgurationwith a bottom gate. Similarly, α 11 y(Vg) is strongly asymmetric when a left-bottom gate removes inversion symmetry aboutbothxandy. The behavior of the SOC coefﬁcients results from a com- plex interplay between quantum conﬁnement from the NW 165401-3PAWEŁ WÓJCIK, ANDREA BERTONI, AND GUIDO GOLDONI PHYSICAL REVIEW B 97, 165401 (2018) FIG. 2. SOC coefﬁcients α11 xandα11 yas a function of the gate voltage Vgfor three different gate conﬁgurations, as shown in the top-left insets. (a) bottom, (b) left-bottom, and (c) left-bottom-right gates. In each panel, insets show the self-consistent electron density at gate volta ges Vg=−0.4,0,0.4V . interfaces, the gate-induced electric ﬁeld and the self- consistent ﬁeld due to electron-electron interaction. Let usconsider ﬁrst the bottom gate conﬁguration, Fig. 2(a).T h e proﬁles of \|ψ 1(x,y)\|2,ne(x,y), andV(x,y)a r es h o w ni nF i g . 3 at selected gate voltages. To understand the impact of theindividual effects on the SOC, in Fig. 3, the self-consistent potential V(x=0,y) has been divided into two components, the one from the gate, V gate, and the Hartree component, VH. At the negative voltage Vg=−0.4 V , the electron energy is increased by a corresponding quantity near the gate. Therefore FIG. 3. Cross-sections of the self-consistent potential V(red line, left axis), electron density distribution ne(blue line, right axis), and \|ψ1(x=0,y)\|2(green line) along the diameter in the ydirection for the gate voltages (a) Vg=−0.4, (b) 0, (c) 0.4 V . The two components of the self-consistent potential V, namely, the gate voltage Vgateand the electron-electron interaction VH,a r es h o w ni nr e dd a s h e da n dd o t t e d lines, respectively. Arrows attached to the potential curves denote the electric ﬁeld direction. Calculations correspond to the bottom-gateconﬁguration of Fig. 2(a).electrons are pushed away from the bottom facet of the NW, and are localized near the top facets [compare with insets inFig.2(a)]. By the assumption of a constant chemical potential, the NW becomes highly depleted of the charge (compare thescale of the right axes in Fig. 3). Consequently, the electron- electron interaction is negligibly small, and the SOC, in thiscase, is mainly determined by the electric ﬁeld coming fromthe gate. Its low value is related to the localization of the groundstateψ 1near the upper edge, where the gradient of the potential ∂V(x,y)/∂yis very low [Fig. 3(a)]. The opposite situation occurs for the positive gate voltage, Vg=0.4V[ F i g . 3(c)], at which a decrease of the conduction band by the positive voltage results in the accumulation ofcharge in the vicinity of the bottom gate. By the assumption ofa constant chemical potential, the NW becomes highly dopedof the charge. The high value of the SOC in this case is dueto the electron-electron interaction, which, for a high electronconcentration, interplays with the gate electric ﬁeld to increaseSOC. Speciﬁcally, it almost completely compensates the gateelectric ﬁeld in the middle of the NW, simultaneously strength-ening it near the bottom facet, where the envelope function ofthe ground state localizes. Since this effect is stronger for thehigh electron concentration, the SOC coefﬁcients signiﬁcantlyincreases with increasing the gate voltage, in the range V g>0. Figure 4shows the calculated α11 xfor constant chemical po- tential [Fig. 4(a)] and constant electron density [Fig. 4(b)].1α11 x shows similar behavior in both conﬁgurations. Speciﬁcally, forVg<0, it is almost insensitive to the gate voltage, while forVg>0, it increases with Vg, the main difference between the two calculations being that for constant nethe behavior is almost linear, with the slope strongly dependent on the electronconcentration. Note, however, that in contrast to the μ-constant model, for which the asymmetry of α 11 x(Vg) arises from the charging and discharging of the NW by the gate voltage(what determines the Coulomb interaction), for the n e-constant 1As mentioned above, in this gate conﬁguration α11 y=0 by sym- metry. 165401-4TUNING RASHBA SPIN-ORBIT COUPLING IN … PHYSICAL REVIEW B 97, 165401 (2018) FIG. 4. α11 x(Vg) calculated under the assumption of (a) a constant chemical potential μ, and (b) a constant electron density ne.I n s e t in panel (a) shows the comparison between the SOC coefﬁcients of the three lowest subbands, calculated with constant μ=0.85 eV . (c) Same as panel (b) but zooming around symmetry point Vg=0. (d) The SOC electric susceptibility α11 xatVg=0 as a function of the electron concentration ne. model the asymmetry results only from the redistribution of electrons caused by the gate electric ﬁeld. We have checked that, regardless of the electron concen- tration and the calculation model, the behavior of αnn(Vg)f o r a few lowest subbands is almost identical. As an example, inthe inset of panel (a) of Fig. 4, we show the intrasubband SOC coefﬁcients versus V gfor the three lowest subbands. These results, calculated with constant μ=0.858 eV , differ only slightly, mainly in the vicinity of Vg=0, where SOC is small. Therefore we limit ourselves to the analysis of the intrasubbandcoefﬁcient for the ground state α 11 xthroughout. Interestingly, for the high electron concentration (or, anal- ogously, above a certain Fermi energy), the SOC coefﬁcientshoots up around V g=0. In Fig. 4(c), we zoom in α11 x(Vg) around Vg=0. The different behavior at low and high density can be understood in terms of the very different chargeredistribution in the two regimes, as we discuss below. In Fig. 5, we show the electron density maps n e(x,y) and the envelope function ψ1(x,y) at three distinct gate voltages around Vg=0, calculated for a low electron concentration , ne=107cm−1. The right column displays the cross-section of the self-consistent potential energy V(x,y) and the enve- lope function ψ1(x,y) along the facet-facet vertical diameter (upper) and edge-edge diagonal diameter (lower), respectively.In this regime, the electron-electron interaction is negligible,quantum conﬁnement from interfaces dominates, and at V g= 0 the conduction band energy is nearly ﬂat, see Fig. 5(b), right column. As a result, the electron density and the envelopefunction of the ground state are localized in the center of theNW and exhibit a circular symmetry. The charge distribution FIG. 5. Left column: Maps of electron density ne(x,y)( l e f t )a n d envelope function ψ1(x,y) (right) for gate voltage (a) Vg=−0.01, (b) 0, and (c) 0.01 V . Right column: proﬁle of self-consistent potentialV(x,y) (red) and the envelope function ψ 1(x,y) (green) along the facet-facet (upper) and edge-edge (lower) directions, as illustrated by the dashed lines in the top-left hexagon. Calculations performed withn e=107cm−1. is hardly modulated by the potential applied to the gate, and is only slightly shifted upward or downward, depending onthe sign of the gate potential [Figs. 5(a) and5(c)], and the SOC coefﬁcient changes sign accordingly. Moreover, since thegate is located at the bottom of the structure, positive voltagesare slightly more effective in shifting the envelope functiondownward, see Figs. 5(a)and5(c), hence the slight asymmetry between positive and negative voltages shown in Fig. 4(c). At the high concentration regime, the electron-electron interaction dominates and the total energy is minimized byreducing the repulsive Coulomb energy, at the expense oflocalization energy. Accordingly, electrons move outwardsand accumulate near the facets. At sufﬁciently high electronconcentration, charge is localized in quasi-1D channels atthe edges [ 43], a minor part of the charge sits at the facets, while the core of the wire is totally depleted, as shownin Fig. 6(b). The strong localization of the ground state at the six edges of the hexagon explains the shooting of theSOC around V g=0. Indeed, since localization in the core (hence tunneling energy between oppositely localized states)vanishes, symmetric edge localization is easily destroyed byany slight asymmetry introduced by the gate potential. Asimilar, more common situation, occurs in coupled symmetricquantum wells [ 49] when the symmetric and antisymmetric states are nearly degenerate. As presented in Figs. 6(a) and 6(c), any slight positive or negative voltage applied to the gate results in the localization of the ground state in the two loweror upper edges, respectively. Accordingly, the SOC coefﬁcientabruptly changes from zero and almost saturates in a narrowrange around V g=0. In other words, the SOC coefﬁcient is a sensitive probe of the complex localization of the charge density in differentregimes. To make this aspect more quantitative, we deﬁne a 165401-5PAWEŁ WÓJCIK, ANDREA BERTONI, AND GUIDO GOLDONI PHYSICAL REVIEW B 97, 165401 (2018) FIG. 6. Same as Fig. 5for gate voltage (a) Vg=−0.01, (b) 0, and (c) 0.01 V . Calculations performed with ne=3×109cm−1. SOC susceptibility χ=dα11 x/dVg\|Vg=0, i.e., the slope of α11 xat zero gate voltage. Its dependence on the charge density, shownin Fig. 4(d), is clearly nonlinear and correlated to the strength of the Coulomb interaction. Note that, although χgrows with charge density, there is no sign of a critical behavior in ourmean-ﬁeld calculations. Off-diagonal terms in Eq. ( 13) represent spin-ﬂip processes combined to intersubband scattering. Such intersubband SOChas been related to intriguing physical phenomena, such asunusual Zitterbewegung [ 50], intrinsic spin Hall effect in sym- metric quantum wells [ 51] and spin ﬁltering devices [ 52,53]. Below, we analyze intersubband SOC between the ground stateand the two lowest excited states. Figure 7shows \|α 12 x(y)\|and\|α13 x(y)\|as a function of the gate voltage Vgin the low electron concentration regime, ne=107 cm−3. In the whole range of Vg, these coefﬁcients remain almost one order of magnitude smaller than the intrasubbandcoefﬁcient α 11 x. The discontinuity of α13 xin Fig. 7is caused by the crossing of subbands n=3 and 4 at Vg≈0.08 V , as shown FIG. 7. Absolute value of the intersubband SOC couplings α12 x(y) andα13 x(y)as a function of the gate voltage Vg.I n s e ts h o w s En(Vg)f o r the four lowest electronic states. Results for ne=107cm−3. FIG. 8. Envelope functions of the three lowest electronic states together with the self-consistent potential V(x,y)f o rVg=−0.2a n d 0.2 V . in the inset. Note that α12 x=α13 y=0 due to the symmetry of the envelope functions and the self-consistent potential V(x,y), as illustrated in Fig. 8, where the ﬁrst three states and the corresponding potential are reported, for two opposite gatevoltages. Finally, in Fig. 9, we show the behavior of intra- and intersubband SOC couplings with varying NW width, showingmonotonous decrease with increasing width. As expected, inwide NWs, SOC tends to zero. B. InSb Indium antimonide (InSb) is a strong SOC material due to its low-energy gap and small conduction electron mass,which makes this semiconductor the preferred host material forspintronic applications and topological quantum computing. In this section, we investigate SOC in InSb NWs in the context of recent experiments [ 15,34] reporting extremely high value of the SOC coefﬁcients. Calculations shown below havebeen carried out for the following material parameters [ 48]: E 0=0.235 eV , /Delta10=0.81 eV , m∗=m∗=0.01359, EP= FIG. 9. (a) The intrasubband α11 xand (b) intersubband α12 ycou- plings as a function of the NW width W, for different electron concentrations, as indicated, at Vg=0.4V . 165401-6TUNING RASHBA SPIN-ORBIT COUPLING IN … PHYSICAL REVIEW B 97, 165401 (2018) FIG. 10. α11 xvsVgcalculated by the assumption of (a) the constant chemical potential and (b) the constant electron density. Results for W=87 nm and the “ideal” bottom gate conﬁguration. 2m0P2/¯h2=23.3 eV , and dielectric constant /epsilon1=16.8. As in the previous case, we consider a temperature T=4.2K . To compare the SOC in InSb and GaAs NWs, in Fig. 10,w e present α11 xfor the bottom gate conﬁguration and calculation models used in the previous subsection, i.e., W=87 nm and the ideal gate conﬁguration. The behavior is qualitativelysimilar to GaAs NWs [see Figs. 4(a) and 4(b)] but SOC coefﬁcients are two orders of magnitudes larger in InSb NWs.We next investigate two speciﬁc conﬁgurations to compareexplicitly with recently reported experimental setups. 1. Comparison with Ref. [ 34] In Ref. [ 34], the authors used magnetoconductance mea- surements in dual-gated InSb NW devices, with a theoret-ical analysis of weak antilocalization to extract the SOCcoefﬁcients. They obtained SOC coefﬁcients as large as50–100 meV nm. In the measurements, the conductance ofthe NW was controlled by a back gate, separated from the wireby a 285 -nm-thick SiO 2layer, and a /Omega1-shaped gate, separated b yaH f O 2layer 30 nm thick. The schematic illustration of the experimental setup is shown in Fig. 11(a) . As in experiments, we consider a NW with a width W= 100 nm and sweep the gate voltages Vbg=[−10V,10V] and Vtg=[−0.6V,0.6] V . Simulations have been carried out in theμ-constant model, μbeing the only free parameter of the calculations. Its value has been determined on the basis ofthe conductance measurements shown in Fig. 3of Ref. [ 34], which indirectly show the occupation of subsequent electronicstates in the NW while changing both gate voltages V bg,Vtg. Comparing the occupation map from our simulations with theexperimental conductance map, we estimate μ=0.35 eV . For this value, and V bg=Vtg=0,N=15 subbands are occupied, in agreement with the estimated value reported in Ref. [ 34]. Figure 11(b) shows α11 xas a function of the bottom gate voltage, Vbg, with Vtg=0(α11 yis zero by symmetry for this gate conﬁguration). Note that, due to the geometrical asym-metry related to the different position of the gates with respectto the NW, α 11 x/negationslash=0e v e na t Vbg=Vtg=0. The “symmetry point” α11 x=0 is obtained at Vbg≈5.4 V , compensating for the electrostatic asymmetry caused by the experimental geometry.Around this value, α 11 xrapidly changes sign. Moreover, due to the weak coupling of this gate to the NW, α11 xvaries only slightly in the considered voltage range. FIG. 11. (a) Schematic illustration of the simulated device. Two gates are connected to the wire through dielectric layers, as indicated,held at voltages V bgandVtg.( b )α11 xas a function of the bottom gate voltage Vbg, withVtg=0. (c)α11 xas a function of the top gate voltage Vtg, withVbg=0. Insets in (b) and (c) show the electron concentration ne(x,y) (top) and square of the envelope functions of the ground state \|ψ1(x,y)\|2(bottom) at selected gate voltages indicated by arrows. In (c), the range of measured SOC coefﬁcients [ 34]i sm a r k e db yt h e gray area. (d) Map of α11 xas a function of both the gate voltages Vtg andVbg. Results for μ=0.35 eV . The situation is different sweeping the voltage of the strongly coupled top gate. In this case, α11 xgrows by almost two orders of magnitude with increasing Vtg, as shown in Fig. 11(c) . The asymptotic vanishing of α11 xat large, negative Vtgis due to full depletion of the NW. Interestingly, for Vtgin the (positive) range [0 .45−0.6]V, SOC achieves values comparable to experiments, \|α11 x\|≈50–100 meV nm. Since for Vtg=0.6V up to N=45 subbands are occupied, the large value of α11 xis mainly determined by the electron-electron interaction, through the localization mechanism already described forGaAs NWs. For completeness, the full map α 11 x(Vbg,Vtg)i s presented in Fig. 11(d) . For experimental setups characterized by a strong geomet- rical asymmetry, αnn xmay be different for different subbands. In Fig. 12(a) , we present αnn xas a function of Vtgfor the four lowest electronic states. For n=1,2,αxdecreases with increasing Vtg, taking the absolute value 50–100 meV nm in agreement with the experiment. Although near Vtg=0 the curves are different, for large positive voltages, when theelectron-electron interaction is dominant, the curves approacheach other. For states n=3,4, on the other hand, α xquickly saturates at αx≈−10 meV nm and it is almost unaffected by large positive gate voltages. This behavior can be traced to thedifferent localization of the envelope functions for differentsubbbands, shown in the inset of Fig. 12forV tg=0.6V . 165401-7PAWEŁ WÓJCIK, ANDREA BERTONI, AND GUIDO GOLDONI PHYSICAL REVIEW B 97, 165401 (2018) FIG. 12. (a) SOC couplings αnn xas a function of the top gate voltage Vtgcalculated for the four lowest subbands ( Vbg=0). The corresponding \|ψn\|2are shown in the inset for Vtg=0.6 V (vertical dashed line), together with the ycomponent of the electric ﬁeld. (b) Intersubband SOC couplings α12 yandα13 xas a function of the top gate voltage Vtg. Indeed, the envelope functions of states n=3,4 are localized on opposite corners, in regions where the gradient of thepotential changes sign, resulting in a strong reduction of theSOC constant for these subbands [see Eq. ( 13)]. From the experimental point of view, it is interesting to evaluate the intersubband SOC, shown in Fig. 12(b) .F o rt h e present NW diameter, which is quite large, the gap betweenthe lowest two subbands is /Delta1E 12≈4 meV at Vtg=0 and decreases down to ≈2 meV at Vtg=0.6 V , when both states are strongly localized in the two top corners and differ onlyby the parity. As a result, α 12 yreaches values close to that of the intrasubband coefﬁcient α11 x, which means that the spin dynamics of the electron in the ground state is determinedequally by both the intra- and intersubband SOC. 2. Comparison with Ref. [ 15] In Ref. [ 15], the authors reported the conductance mea- surements through the helical gap in InSb NWs. Analysisof the experimental data, taken at different magnetic ﬁeldorientations, using a single-electron model, led to an extremelylarge SO energy E SO=6.5 meV , corresponding to αR≈ 270 meV nm. The conductance of the NW was controlled by abottom gate attached to the wire through a 20-nm-thick Si 3N4 layer, while all other facets were electrostatically free. In thecalculations, the Neumann boundary conditions were applied.The schematic illustration of the sample used in the experimentis reported in Fig. 13(a) . To investigate the device described in Ref. [ 15], we sim- ulated a NW of width W=100 nm and gate voltage in the range V bg=[0 V,0.6 V]. Simulations have been carried out in the μ-constant model, and μwas chosen on the basis of the conductance measurements in Ref. [ 15], reporting the ﬁrst FIG. 13. (a) Schematic illustration of the experimental setup in Ref. [ 15]. The conductance of the NW is controlled by the bottom gateVbgattached to the wire by the 20-nm-thick Si 3N4layer. (b) α11 x as a function of the bottom gate Vbgfor the NW without the sulfur layer (black curve), with the sulfur layer (red curve), and with thedopant concentration included (green curve). conductance step at Vg≈0.1 V . In our simulations, such an occupation is realized with μ=40 meV . Figure 13(b) shows α11 xas a function of the bottom gate Vbg. The calculated value of the SOC coefﬁcient is about nine timeslower than reported in the experiment. In an attempt to explainthis discrepancy we referred to details of the nanofabrication[54]. The precise procedure for the contact deposition includes etching of the native oxide at the InSb NW using sulfur-basedsolution. Inclusion of sulfur at the InSb surface may producea variable donor concentration up to 7 .5×10 18cm−3[55], which results in band bending with electron accumulation nearthe surface. Accordingly, in our calculations we included a5-nm-thick sulfur layer at the InSb NW surface, and consideredtwo cases: without dopants and with a dopant concentrationn d=1017cm−3. As shown in Fig. 1(b), the presence of the sulfur layer decreases the SOC coefﬁcient, due to the lowdielectric constant and the reduction of the electric ﬁeld inthe NW. Even inclusion of the dopants, which bends theconduction band at the interfaces, does not change this behaviorqualitatively, leaving our results well below the experimentallymeasured SOC constant. Therefore, this discrepancy remains unexplained. Note that such a large SOC constant has been reported only in oneexperiment so far [ 15], ﬁtting the helical state conductance measurements to a single-band model which includes neitherthe orbital effects nor the intersubband coupling. Both theseeffects may increase the effective SOC in the ground state andthe use of the simple single-band theory to extract the αvalue can lead to overestimation of this parameter. IV . SUMMARY We have formulated a multiband k·ptheory of SOC in NW- based devices and investigated the behavior of Rashba SOCin GaAs- and InSb-based devices. The strength of the SOCcoefﬁcients is determined by band parameters and externalpotentials. In the absence of any external potentials, the chargedensity shares the symmetry of the structure, hence SOCcoefﬁcients vanish. External gates, breaking the symmetry, can 165401-8TUNING RASHBA SPIN-ORBIT COUPLING IN … PHYSICAL REVIEW B 97, 165401 (2018) tailor SOC. The tunability of the SOC coefﬁcients, however, strongly depends on size and doping. We show, for example,that in the high carrier density regime SOC has a very largesusceptibility. In light of our simulations, we analyzed quantitatively recent experiments with InSb nanowires. Good agreement isfound with SOC reported in Phys. Rev. B 91,201413(R) (2015 ), but not with the much larger values measured in Nat. Commun. 8,478(2017 ). We argue that a possible origin of this discrepancy lies in the model used to extract the parameter,which entails a single-particle, single-band model. Our calcu-lations, on the contrary, show that electron-electron interaction plays a dominant role and intersubband contributions aresubstantial in the investigated samples. ACKNOWLEDGMENTS This work was partially ﬁnanced (supported) by the Faculty of Physics and Applied Computer Science AGH UST deangrant for PhD students and young researchers within subsidyof Ministry of Science and Higher Education and in part byPL-Grid Infrastructure. APPENDIX A: FOLDING-DOWN PROCEDURE We start from the 8 ×8k·pHamiltonian H8×8=/parenleftbiggHcHcv H† cvHv/parenrightbigg , (A1) which in the exact form is given by H8×8=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝¯h2ˆk2 2m0+Ec+V(x,y)01√ 6Pˆk+ 01√ 2Pˆk−−/radicalBig 2 3Pkz−1√ 3Pkz1√ 3Pˆk+ 0¯h2ˆk2 2m0+Ec+V(x,y)−/radicalBig 2 3Pkz−1√ 2Pˆk+ 0 −1√ 6Pˆk−1√ 3Pˆk−1√ 3Pkz 1√ 6Pˆk− −/radicalBig 2 3Pkz Ev(x,y)0 0 0 0 0 0 −1√ 2Pˆk− 0 Ev(x,y)0 0 0 0 1√ 2Pˆk+ 00 0 Ev(x,y)0 0 0 −/radicalBig 2 3Pkz −1√ 6Pˆk+ 00 0 Ev(x,y)0 0 −1√ 3Pkz1√ 3Pˆk+ 00 0 0 E/prime v(x,y)0 1√ 3Pˆk−1√ 3Pkz 00 0 00 E/prime z(x,y)⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (A2) where E v(x,y)=Ec+V(x,y)−E0,E/prime v(x,y)=Ec+V(x,y)−E0−/Delta10,ˆk±=ˆkx±iˆky,ˆk2=ˆk2 x+ˆk2 y+k2 z,Ecis the energy of the conduction band edge, E0is the energy gap, /Delta10is the split-off band gap, V(x,y) is the potential energy, and P= −i¯h/angbracketleftS\|ˆpx\|X/angbracketright/m0is the conduction-to-valence band coupling with \|S/angbracketright,\|X/angbracketrightbeing the Bloch functions at the /Gamma1point of Brillouin zone. The folding-down procedure, H=Hc−Hcv(Hv−E)−1H† cv, (A3) reduces the 8 ×8 Hamiltonian ( A2) to the effective Hamiltonian 2 ×2 for conduction electrons. After some algebraic transformations, H=/parenleftBigg ¯h2ˆk2 2m0+Ec+V(x,y)/parenrightBigg I−(/Lambda10I+/Lambda1xσx+/Lambda1yσy), (A4) where σx(y)are the Pauli matrices and /Lambda10=1 3P2k2 z/parenleftbigg2 Ev(x,y)−E+1 E/primev(x,y)−E/parenrightbigg , /Lambda1x=i 3P2kz/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg ˆky−i 3P2kzˆky/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg , (A5) /Lambda1y=i 3P2kz/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg ˆkx−i 3P2kzˆkx/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg . (A6) 165401-9PAWEŁ WÓJCIK, ANDREA BERTONI, AND GUIDO GOLDONI PHYSICAL REVIEW B 97, 165401 (2018) The Hamiltonian ( A4) can be simpliﬁed to the form H=/bracketleftBigg −¯h2 2m∗∇2 2D+¯h2k2 z 2m∗+Ec+V(x,y)/bracketrightBigg 12×2,+(αxσx+αyσy)kz, (A7) where m∗is the effective mass 1 m∗≈1 m0+2P2 3¯h2/parenleftbigg2 Eg+1 Eg+/Delta1g/parenrightbigg (A8) and the SO coupling constants αx=i 3P2ˆky/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg −i 3P2/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg ˆky, (A9) αy=i 3P2ˆkx/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg −i 3P2/parenleftbigg1 Ev(x,y)−E−1 E/primev(x,y)−E/parenrightbigg ˆkx. (A10) From the fact that E0and/Delta10are the highest energies in the system, we can expand the energy-dependent term in ( A9) and ( A10) in the Taylor series 1 Ev(x,y)−E−1 E/primev(x,y)−E≈/parenleftbigg1 E0+/Delta10−1 E0/parenrightbigg +/parenleftbigg1 E2 0−1 (E0+/Delta10)2/parenrightbigg (Ec+V(x,y)−E). 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PhysRevB.72.094102.pdf	Interlayer contraction in MgB 2upon replacement of Mg by Al: Effect of the covalent bond energy Gabriel Bester and Manfred Fähnle Max-Planck-Institut für Metallforschung, Heisenbergstrasse 3, D-70569 Stuttgart, Germany /H20849Received 19 May 2005; revised manuscript received 5 July 2005; published 6 September 2005 /H20850 The partitioning of the cohesive energy which we derived recently from the total energy expression of density functional theory /H20851J. Phys.: Condens. Matter 13, 11541 /H208492001 /H20850/H20852allows us to quantify the energy Ecov that describes the covalent and the metallic bond energy in a periodic solid. We apply this methodology to analyze various bonds in MgB 2and AlB 2. We ﬁnd that the experimentally observed interlayer contraction when going from MgB 2to AlB 2is consistent with the calculated larger Ecovenergy of the Al-B bond compared to the Mg-B bond. We further explain this result by the ﬁlling of bonding states in the boron- p–Al- p bonds as revealed by the energy resolved covalent bond energy Ecov/H20849E/H20850. DOI: 10.1103/PhysRevB.72.094102 PACS number /H20849s/H20850: 62.20. /H11002x, 74.70.Ad, 71.15.Dx I. INTRODUCTION In the last four decades, calculations based on the elec- tronic density-functional theory1,2/H20849DFT /H20850have become the method of choice to determine and to predict a large varietyof physical properties in many different materials. AlthoughDFT provides the theoretical framework for high accuracycalculations of hundreds of atoms, the interpretation of theresults remains often difﬁcult and necessitates new ideas toattain physical insight. The use of simple models, like thenearly-free-electron model for sp-valent materials or the tight-binding model for transition metals or semiconductorsmakes the interpretation of the results often more tractable.However, the lower accuracy of such methods restricts theirapplication to certain problems, leaving many open questionsto the more accurate DFT methods. Recently, we derived 3a partitioning of the total energy as calculated from DFT that allows the interpretation of theresults using the intuitive chemical language of bonding andantibonding states. This allows us to quantitatively tackle theproblems of the bonding properties of covalent and metallicsystems in general. Here we demonstrate the usefulness ofthe method by addressing the bonding properties of AlB 2and MgB 2. The discovery4of MgB 2with the highest supercon- ducting critical temperature Tc=39 K ever reported for a bi- nary system has triggered tremendous activity to investigatethe electronic 5–12and phononic5,7,11–13properties as well as the electron-phonon coupling5,7,11,13for MgB 2and for related binary and ternary borides /H20849for a review see Ref. 14 /H20850. The structure of MgB 2is graphitelike, i.e., it consists of honey- comb layers of B separated by triangular metal planes. Pre-vious investigations on this compound have shown that /H20849i/H20850 the states at the Fermi level are dominantly derived fromboron porbitals. The B p x,y/H9268-bands are responsible for the strong covalent bonding in the B layers. /H20849ii/H20850The Mg sandp states are admixed to the B states indicating that the intercal-ant Mg does not simply play the role of an electron donor./H20849iii/H20850The partial replacement of the divalent Mg atom by the trivalent Al atom results 15–17in a decrease of the c/aratio, in part in connection with the formation of superstructures dueto Al-layer ordering.It is the objective of the present paper to analyze and compare the covalent bonding properties of MgB 2and AlB 2 quantitatively within our recently developed energy-partitioning scheme. 3A ﬁrst attempt in this direction was made by Ravindran et al.9by means of the crystal-orbital- Hamilton population18/H20849COHP /H20850. It will be shown below, however, that for the use of nonorthogonal basis functionsthe COHP do not have a well-deﬁned physical meaning andcannot be used to compare quantitatively the covalent bond-ing properties between various structures. Our analysis willshow that the interlayer contraction in MgB 2upon replace- ment of Mg by Al is consistent with the larger B-Al covalentbond energy, compared to B-Mg. II. OUR ENERGY-PARTITIONING SCHEME The starting point of the discussion is the expression of the total energy Etotfrom the density functional theory, Etot=/H20858 nfn/H9255n−/H20885n/H20849r/H20850veff/H20849r/H20850d3r+EH+Exc +/H20885n/H20849r/H20850vext/H20849r/H20850d3r+Eii, /H208491/H20850 with the occupation numbers fnand the eigenvalues /H9255nof the Kohn-Sham single-particle wave functions /H9023n, the electron density n/H20849r/H20850, the effective potential veff/H20849r/H20850, the Hartree en- ergy EH, the exchange-correlation energy Exc, the potential vext/H20849r/H20850of the nuclei /H20849or of the ionic cores in the case of a pseudopotential calculation /H20850, and the interaction energy Eii between the nuclei /H20849or between the ionic cores /H20850. The hope is that the trends for the total energies are already well-described by the ﬁrst term /H20849band-structure energy E band /H20850 when comparing various systems. This assumption is madeimplicitly in the common practice to discuss the energeticsvia the electronic density of states. Our energy partitioningscheme is a tool to analyze E bandfurther. The ﬁrst step in the derivation is to expand the single- particle wave functions /H9023nin a set of well-localized nonor- thogonal orbitals /H9272i/H9251, where /H9251describes the angular andPHYSICAL REVIEW B 72, 094102 /H208492005 /H20850 1098-0121/2005/72 /H208499/H20850/094102 /H208496/H20850/$23.00 ©2005 The American Physical Society 094102-1magnetic atom quantum numbers land m, attached to the atoms i/H20849like in a nonorthogonal tight-binding representa- tion /H20850, /H9023n=/H20858 i/H9251ci/H9251n/H9272i/H9251. /H208492/H20850 In this basis we deﬁne overlap and Hamilton matrices as Sj/H9252i/H9251and Hj/H9252i/H9251. It is now possible to rearrange the various terms of the total energy without any approximation, i.e.,conserving the DFT accuracy /H20849see Ref. 3 for the detailed procedure /H20850, to express the cohesive energy as the sum of six terms: E c=Etot−Efree atom=Eprom+Ecf+Epolar+Ecov+Epair+Emb. /H208493/H20850 The last two terms from the right encompass the contri- butions of the ﬁve last terms in Eq. /H208491/H20850. Approximating the charge density by a superposition of the atomic charge den-sities of the respective atoms, they have the meaning of apair-potential term E pairand a many-body potential term Emb which is small for nearly charge neutral atoms. The band- structure contribution encompasses the promotion energyE prom, the covalent bond energy Ecov, the crystal ﬁeld energy Ecf, and the polarization energy Epolar. The key of the analy- sis resides in the simple interpretation of these different con-tributions. The promotion energy, E prom=/H20858 i/H9251/H20849qi/H9251−Ni/H9251free atom/H20850Hi/H9251i/H9251free atom, /H208494/H20850 is a function of the Hamiltonian and of the occupation num- bers of the free atoms before being condensed to the crystal, Hfree atomand Ni/H9251free atom, respectively, and of the Mulliken’s gross charge19 qi/H9251=/H20858 j/H9252/H20858 nfnci/H9251n/H20849cj/H9252n/H20850Sj/H9252i/H9251. /H208495/H20850 Epromdescribes the cost in energy when starting the conden- sation process from free atoms and then redistributing theelectrons among the various orbitals from the occupation numbers N i/H9251free atomto the occupation number qi/H9251found in the crystal and characterized by qi/H9251.45The second term is the crystal-ﬁeld term Ecf=/H20858 i/H9251qi/H9251/H20849Hi/H9251i/H9251−Hi/H9251i/H9251free atom/H20850, /H208496/H20850 which describes the change in energy due to a shift of the on-site energies when the atoms are condensed to form thecrystal so that the potential acting on an electron at atom iis not just the atomic potential of this atom but theenvironment-dependent crystal potential. The polarizationenergy E polar=/H20858 n,i,/H9251,/H9252fnci/H9251n/H20849ci/H9252n/H20850/H20851Hi/H9251i/H9252−/H9254i/H9251i/H9252Hi/H9251i/H9251/H20852/H20849 7/H20850 describes the change in energy due to the hybridization of orbitals localized at one atom when the atom is embedded inthe crystal.Finally, the energy E covis the change in energy arising from the hybridization of orbitals localized at different at-oms, E cov=/H20858 i/H9251,j/H9252 j/HS11005iEcov,i/H9251j/H9252, /H208498/H20850 with Ecov,i/H9251j/H9252=/H20858 nfnci/H9251n/H20849cj/H9252n/H20850/H20851Hj/H9252i/H9251−Sj/H9252i/H9251/H9255j/H9252i/H9251/H20852, /H208499/H20850 /H9255j/H9252i/H9251=1 2/H20849Hi/H9251i/H9251+Hj/H9252j/H9252/H20850. /H2084910/H20850 The energy Ecovis the only term which involves matrix ele- ments between orbitals on different atoms, and thereforeclearly represents the contribution of the interatomic bond-ing. As outlined in the Introduction, there are covalent, me-tallic, and ionic contributions to the cohesive energy ofMgB 2and AlB 2. The covalent bond energy Ecovincludes both covalent and metallic contributions to Ec. In the litera- ture the terms covalent and metallic bonding are often usedto describe systems with directionally structured charge den-sities and homogeneous charge densities, respectively. Forthe latter systems the wave functions /H9023 nare computationally more efﬁciently represented by a set of plane waves ratherthan by atom-localized orbitals. Nevertheless, /H9023 ncan be rep- resented also in the case of delocalized wave functions by aset of atom-localized orbitals, albeit orbitals that are unoccu-pied in the corresponding free atom have to be included tomake the basis set sufﬁciently complete. Thus a covalentbond energy E covarises even for the case of a nearly free electron system. Because our deﬁnition of Ecovis a generali- zation to the case of nonorthogonal basis sets of the covalentbond energy introduced by Sutton et al. 20we keep the his- torically founded nomenclature “covalent bond energy,” al-though this quantity may also contain metallic bonding con-tributions in the above discussed sense. By considering theangular-resolved covalent bond energy E cov,i/H9251j/H9252it is possible to investigate the contributions to the covalent energy of or-bitals which are not occupied in the respective free atoms butgained importance through hybridization in the crystal. E cov,i/H9251j/H9252can be further subdivided into energy-resolved contributions, Ecov,i/H9251j/H9252/H20849E/H20850=/H20858 n/H9254/H20849E−/H9255n/H20850fnci/H9251n/H20849cj/H9252n/H20850/H20851Hj/H9252i/H9251−Sj/H9252i/H9251/H9255j/H9252i/H9251/H20852. /H2084911/H20850 Ecov,i/H9251j/H9252/H20849E/H20850is negative /H20849positive /H20850for bonding /H20849antibonding /H20850 states. In the following we will conﬁne ourselves to the dis- cussion of the covalent bond energy. The energy-partitioning scheme discussed above has the following very attractive property: In a band-structure calcu-lation which deals with an inﬁnitely extended periodic sys-tem the average effective potential does not have a physicalmeaning, and it is therefore set to an arbitrary value which isin most DFT implementations set to the same arbitrary value,regardless of materials and structures. In order to be physi-G. BESTER AND M. FÄHNLE PHYSICAL REVIEW B 72, 094102 /H208492005 /H20850 094102-2cally meaningful the total energy Etotand the considered terms of an energy-partitioning scheme for a band-structurecalculation must therefore be invariant against a constantshift of the effective potential. This is fulﬁlled for E tot/H20849and hence also for Ec/H20850which becomes obvious from Eq. /H208491/H20850: Shifting veffby/H90210yields opposite shifts for the ﬁrst two terms /H20849the remaining terms can be calculated without ambi- guity, as pointed out in Ref. 21 /H20850. This cancellation is a fun- damental prerequisite of total energy DFT calculations. Fur-thermore, the terms E prom,Epolar, and Ecovof Eq. /H208493/H20850as well as their atom- and orbital-resolved contributions /H20849and in ad- dition the respective energy-resolved contributions to Ecov/H20850 are all invariant against such a shift. For instance, if thepotential is shifted by a constant /H9021 0, then the matrix ele- ments Hj/H9252i/H9251are transformed into Hj/H9252i/H9251+/H90210Sj/H9252i/H9251and Hi/H9251i/H9251, /H9255j/H9252i/H9251into Hi/H9251i/H9251+/H90210,/H9255j/H9252i/H9251+/H90210because Si/H9251i/H9251=1, so that /H90210 drops out of the covalent bond energy. Because Ecis also invariant, this must hold for the sum of the terms Epair +Emb+Ecf, too. However, we cannot calculate separately, for instance, the crystal-ﬁeld term Ecfin a band-structure calcu- lation because the matrix element Hi/H9251i/H9251is shifted by the shift of the effective potential of the crystal whereas Hi/H9251i/H9251free atomis not, since for the calculation of the latter quantity the effec-tive potential can always be normalized to zero for distancesfar from the atom. It is therefore physically meaningful onlyto discuss the terms E prom,Epolar,EcovandEpair+Emb+Ecf. In former publications two other quantities have been used which are related to Ecov,i/H9251j/H9252/H20849E/H20850. First, Hughbanks and Hoffmann22have introduced the crystal-orbital-overlap population /H20849COOP /H20850, COOP i/H9251j/H9252/H20849E/H20850=/H20858 n/H9254/H20849E−/H9280n/H20850fnci/H9251n/H20849cj/H9252n/H20850Sj/H9252i/H9251. /H2084912/H20850 Integrating COOP i/H9251j/H9252/H20849E/H20850over Eyields the so-called bond order /H20851for an extensive discussion of these two quantities see the paper by Hoffmann /H20849Ref. 23 /H20850/H20852. Whereas COOP i/H9251j/H9252/H20849E/H20850is able to discuss the bonding character /H20849it is positive for bond- ing states and negative for antibonding states /H20850it cannot ana- lyze quantitatively the contribution of the bonds to the totalenergy and it often exaggerates the antibonding states. There-fore Dronskowski and Blöchl 18have introduced the crystal orbital Hamilton population COHP i/H9251j/H9252/H20849E/H20850, COHP i/H9251j/H9252/H20849E/H20850=/H20858 n/H9254/H20849E−/H9255n/H20850fnci/H9251n/H20849cj/H9252n/H20850Hj/H9252i/H9251, /H2084913/H20850 to characterize the bonding properties. Ecov,i/H9251j/H9252/H20849E/H20850is related toCOOP i/H9251j/H9252/H20849E/H20850andCOHP i/H9251j/H9252/H20849E/H20850via Ecov,i/H9251j/H9252/H20849E/H20850=COHP i/H9251j/H9252/H20849E/H20850−/H9255j/H9252i/H9251COOP i/H9251j/H9252/H20849E/H20850./H2084914/H20850 If orthonormal basis functions are used, COHP i/H9251j/H9252/H20849E/H20850is identical to Ecov,i/H9251j/H9252/H20849E/H20850. However, in the usual chemical analysis nonorthogonal basis sets are used and COHP i/H9251j/H9252/H20849E/H20850 is not invariant against a constant shift of the effective po- tential and therefore does not have a well-deﬁned physicalmeaning in the context of band-structure calculations. Our energy-partitioning scheme has recently been applied successfully for a discussion of the chemical bonding in vari-ous material systems, e.g., TiAl 3and TiSc 3,24perovskitelikeruthenates,25mixed perovskite oxides,26CuTe 2and Cu7Te4,27TlNiO 3,28hydrogen inserted CeNiIn,29VO 2,30 Ca2MnO 4and Ca 2MnO 3.5,31CeCoSi,32YFe 2,33and magnetic oxides.34In the present paper it is applied to MgB 2and AlB 2. III. DETAILS OF THE CALCULATIONS The calculations were performed using the mixed-basis ab initio pseudopotential program35with the new implemen- tation of Ecov3for the bonding analysis. Band structure5–7,9 and charge density difference plots were practically identical to those of previous calculations.5–7,9For the energy- partitioning analysis the /H9023nwere projected3,36–38onto a set of overlapping atom-localized nonorthogonal orbitals. Forthese orbitals we chose /H9272i/H9251/H20849r/H20850=fil/H20849r/H20850ilKlm/H20849rˆ/H20850, /H2084915/H20850 fil/H20849r/H20850=Cil/H9278ilPS/H20849/H9261ilr/H20850/H20877/H208511−e−/H9253il/H20849rilcut−r/H208502/H20852forr/H33355rilcut 0 forr/H33356rilcut,/H20878 /H2084916/H20850 where Cilis a normalization constant, /H9278ilPSis the radial pseudoatomic wave function constructed according to Vanderbilt,39/H9261ildenotes a contraction factor, and rilcutrepre- sents a cutoff length. The parameters /H9261il,/H9253il, and rilcutwere selected in such a way that the spillage3,36–38,40was mini- mized, where the spillage characterizes the loss of the normof the wave functions due to the incompleteness of thepseudoatomic-orbital projection. The optimization of the lo-cal orbitals through minimization of the spillage revealedthat a basis set of s,p, and dorbitals for Mg and B is sufﬁcient to project more than 99% of the wave functions onthe local basis. The band-structure calculated with the pro-jected wave functions is nearly identical to the original bandstructure from the pseudopotential calculation for energiesbelow and not too far above the Fermi level. The implemen-tation of the covalent bond energy is described in detail inRef. 3. IV. RESULTS AND DISCUSSION Table I represents the covalent bond energies for various atom pairs in MgB 2and AlB 2. It should be recalled that the covalent bonding properties of the two materials may becompared only by the measure E covand not by COHP be- cause the latter quantity is not invariant against a constantshift of the potential when nonorthogonal basis functions areTABLE I. Covalent bond energies /H20849in meV /H20850for various atom pairs in the /H208490001 /H20850-planes /H20849intra /H20850and between the /H208510001 /H20852-planes /H20849inter /H20850. B-B intraB-B interM-M intraM-M interB-M Total MgB 2 −2633 18 −512 −37 −1003 −4124 AlB 2 −2453 50 −567 139 −1182 −4325INTERLAYER CONTRACTION IN MgB 2UPON… PHYSICAL REVIEW B 72, 094102 /H208492005 /H20850 094102-3used, which is the usual case for a chemical analysis in terms of atomiclike functions. In Table II the covalent bond ener-gies for the most important atom pairs are further analyzedby considering the dominant angular-resolved contributions.The main results of our calculations are as follows. /H208491/H20850The covalent bonds of the nearest-neighbor intralayer bonds and the nearest-neighbor B-Al and B-Mg bonds arestrongest and all bonds between further distant atoms areconsiderably smaller. Obviously, a nearest-neighbor bondmodel would qualitatively describe MgB 2and AlB 2. This is not at all trivial because in intermetallic compounds likeFeAl, CoAl, and NiAl the bonds between further distant at-oms are essential. 41,42 /H208492/H20850The hierarchy of bond strength is as follows: /H20849a/H20850The B-B intralayer covalent bond energy is largest /H20849in agreement with former investigations5–7,14,43/H20850,/H20849b/H20850the B-Al and B-Mg interlayer bonds are a factor of about 2 smaller but can by nomeans be neglected, /H20849c/H20850the Al-Al and Mg-Mg intralayer bonds are a factor of about 4 smaller than the interlayerbonds, and /H20849d/H20850the interlayer B-B /H20849Al-Al,Mg-Mg /H20850bond is about two orders of magnitude /H20849about one order of magni- tude /H20850smaller than the respective intralayer bonds from /H20849a/H20850 and /H20849b/H20850. We conclude that the structure is mainly maintained by stiff boron planes coupled with the Al or Mg planes by theB-Al and B-Mg bonds. We note that the intralayer B-Bbonds are stronger in MgB 2than in AlB 2despite the fact that the B atoms are further apart in MgB 2compared to AlB 2. This is a signature of the covalent character of the bond. /H208493/H20850The total covalent bond energy is larger in absolute value for AlB 2than for MgB 2. In particular, the interlayercovalent bond energy is larger for the Mg-B bond than for the Al-B bond, and this is consistent with the smaller c/a ratio observed experimentally in AlB 2compared to MgB 2. We therefore suggest that the change in the covalent bondenergy is responsible for the decrease of the c/aratio and not, as suggested previously, 44a change in ionic bonding. To assess qualitatively the magnitude of the ionic contributionto the bonding we calculated the Mulliken 19population on the B and the metal atoms. We ﬁnd for MgB 20.13 electrons missing on B and 0.26 additional electrons on Mg, comparedto their charge neutral conﬁgurations. For AlB 2we ﬁnd vir- tually the same results with 0.13 electrons missing on B and 0.25 additional electrons on Al. These results suggest that theionic contribution of the bonding cannot explain the ob-served trend from MgB 2to AlB 2. /H208494/H20850The angular-momentum-resolved covalent bond ener- gies given in Table II show that for all the considered bondsthep-pcontributions are strongest and the s-scontributions weakest. This is surprisingly also true for the metal-metalbonds where the free-atom electronic conﬁguration is domi-nated by s-electrons. It becomes obvious from Table II that thed-orbitals on the metal atoms make a non-negligible con- tribution to the bonding between the B layers and the metal-lic layers. The occupation of metal d-orbitals is possible in spite of the price which is paid for the promotion energybecause the symmetry of the structure favors the directional-ity of the d-bonds. As discussed in Sec. II, the occupation of d-orbitals which are not occupied in the respective free at- oms can be considered as an indication of the inﬂuence ofmetallic, i.e., delocalized bonding. The importance of themetal d-orbitals has been highlighted previously in connec- tion to the calculation of the de Haas-van Alphen frequenciesin Ref. 43. To analyze the results in more detail we show in Fig. 1 the energy-resolved covalent bond energies for the p-pand s-s contributions of the intralayer B-B, Al-Al, and Mg-Mgbonds and the interlayer B-Mg and B-Al bonds. The inte-grals of these curves give the covalent bond energies fromTable II. The beneﬁt of this representation /H20849Fig. 1 /H20850is that we can understand the electronic origin of the quantitative num-bers given by the covalent bond energy. For two isolatedTABLE II. The dominant orbital-resolved covalent bond ener- gies /H20849in meV /H20850for various atom pairs. B-B intra B-MM-M intra p-p p-pp -ss -sp -pp -dp -ss -s MgB 2−1155 −582 −196 −339 −180 −174 −49 −136 AlB 2−992 −537 −193 −419 −240 −192 −23 −167 FIG. 1. /H20849Color online /H20850Energy resolved cova- lent bond energies for the p-p/H20849full line /H20850and s-s /H20849dashed line /H20850contributions of various atom pairs in MgB 2/H20849upper panels /H20850and AlB 2/H20849lower panels /H20850.G. BESTER AND M. FÄHNLE PHYSICAL REVIEW B 72, 094102 /H208492005 /H20850 094102-4atoms bonding states become progressively ﬁlled before an- tibonding states start to be occupied. In the case of more thantwo interacting atoms the energetic ﬁlling of states canundergo a sequence as bonding-antibonding-bonding-anti-bonding, etc. This is observed in the Mg-Mg intrabond inFig. 1 where we can see that the s-sbond undergoes such a sequence. The main results of the energy-resolved analysisgiven in Fig. 1 can be summarized as follows. /H208491/H20850The s-sbonds are weakened because both bonding and antibonding states are occupied. Without the energy-resolvedanalysis we could erroneously assume that the s-scovalent bond energy is low because the corresponding matrix ele-ments are small. /H208492/H20850The stronger B-B intrabond energy of MgB 2/H20849Table I /H20850 results mainly from a stronger p-pcontribution /H20849Table II /H20850.I n Fig. 1 we see that the reason for the stronger B-B bond inMgB 2is that nearly all bonding states are occupied and none of the antibonding. In AlB 2, the additional electron delivered by the trivalent Al causes a ﬁlling of some antibonding statesand a weakening of the bond. /H208493/H20850The stronger B-Al bond compared to the B-Mg bond /H20849Table I /H20850again results mainly from the stronger p-pcontri- butions. This becomes obvious from the central panels ofFig. 1. The overall shape of the bands is similar in bothmaterials but the additional electron present in AlB 2shifts thep-band down in energy resulting in the occupation of more bonding states in the AlB 2case.V. CONCLUSION We have investigated the bonding properties of MgB 2and AlB 2by our recently developed energy-partitioning scheme3 for the density-functional total energy. This methodology al-lows one to deﬁne a covalent bond energy which is invariantagainst a constant potential shift which is unavoidable indensity functional calculations. This property is a precondi-tion for the comparison of the bonding properties in differentperiodic systems. Our main conclusions are that /H20849i/H20850the bond- ing properties of these materials are strongly dominated bynearest-neighbor interactions with dominant B-B, B-Al, andB-Mg bonds. The material is basically maintained by stiffboron planes coupled to the Al or Mg planes by the B-Aland B-Mg bonds. /H20849ii/H20850The p-pcontributions are strongest for all the bonds, including the bonds between metal atomswhere the d-contribution is surprisingly signiﬁcant. /H20849iii/H20850The interlayer contraction which is observed experimentally forMB 2when going from MgB 2to AlB 2is consistent with the calculated increase of the covalent bond energy when goingfrom the Mg-B bond to the Al-B bond. We show that theoccupation of bonding states in the B- p–Al- pbond is re- sponsible for this effect. /H20849iv/H20850The B-B bonds in MgB 2are optimum since all the bonding states are ﬁlled and all theantibonding states are empty. Replacement of Mg with Altriggers the ﬁlling of antibonding states and weakens thebond. Present address: National Renewable Energy Laboratory, Golden, CO 80401. 1P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 /H208491964 /H20850. 2W. Kohn and L. J. 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PhysRevB.81.081201.pdf	Vacancy defect and defect cluster energetics in ion-implanted ZnO Yufeng Dong,1,F. Tuomisto,2B. G. Svensson,3A. Yu. Kuznetsov,3and Leonard J. Brillson1,4,5 1Department of Electrical and Computer Engineering, The Ohio State University, Columbus, Ohio 43210, USA 2Department of Applied Physics, Helsinki University of Technology, P.O. Box 1100, Helsinki 02015 TKK, Finland 3Department of Physics, University of Oslo, P.O. Box 1048, Blindern, 0316 Oslo, Norway 4Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 5Center for Materials Research, The Ohio State University, Columbus, Ohio 43210, USA /H20849Received 8 January 2010; published 1 February 2010 /H20850 We have used depth-resolved cathodoluminescence, positron annihilation, and surface photovoltage spec- troscopies to determine the energy levels of Zn vacancies and vacancy clusters in bulk ZnO crystals. Dopplerbroadening-measured transformation of Zn vacancies to vacancy clusters with annealing shifts defect energiessigniﬁcantly lower in the ZnO band gap. Zn and corresponding O vacancy-related depth distributions providea consistent explanation of depth-dependent resistivity and carrier-concentration changes induced by ionimplantation. DOI: 10.1103/PhysRevB.81.081201 PACS number /H20849s/H20850: 72.40. /H11001w, 71.55.Gs, 78.60.Hk ZnO is a leading candidate for next generation optoelec- tronic materials because of its large band gap, high excitonbinding energy, thermochemical stability, and environmentalcompatibility. 1,2High quality single-crystal bulk ZnO wafers grown by various methods are commercially available3and ZnO thin-ﬁlm growth has attracted intense interest.4How- ever, despite nearly sixty years of research, several funda-mental issues surrounding ZnO remain unresolved. Chiefamong these have been the difﬁculty of p-type doping and the role of compensating native defects. 5,6Oxygen vacancies /H20849VO/H20850,VOcomplexes, Zn interstitial-related complexes, and residual impurities such as hydrogen and aluminum are allbelieved to be shallow donors in ZnO, while Zn vacancies/H20849V Zn/H20850and their complexes are considered to be acceptors.7,8 Although their impact on carrier compensation is recognized, the physical nature of the donors and acceptors dominatingcarrier densities in ZnO is unresolved. Thus it remains achallenge to correlate the commonly observed 1.9–2.1 eV“red” and 2.3–2.5 eV “green” luminescence emissions with speciﬁc native defects. 9These and other emissions vary widely in ZnO bulk or thin ﬁlms grown by variousmethods. 10–14Previous optical absorption, photolumines- cence, electron paramagnetic resonance, and depth-resolvedcathodoluminescence spectroscopy /H20849DRCLS /H20850/H20849Ref. 15/H20850stud- ies indicate a correlation between the “green” optical transi-tion and O vacancies /H20849V O/H20850.10,16Still controversial, however, is how such visible emissions correlate with the energetics ofZn/O vacancies, interstitials, and their complexes overall.This work clearly identiﬁes the physical nature of the defectsdominating optical features of this widely studied semicon-ductor and, in turn, these defects provide a consistent expla-nation for ZnO’s effective free-carrier densities on a localscale. Contemporary theoretical approaches are also limited in addressing ZnO defect energetics due to major uncertainties,most notably, the “band-gap problem” within density-functional methods. 17Calculations of such basic ZnO defect properties as formation energy and energy-level relative toband edges vary considerably with differentapproximations. 5,18–21Therefore, the determination of energy levels of native point defects and energetics of Zn vacanciesversus their clusters provides a method to evaluate methodsfor calculating deep level energies within ZnO and other semiconductors. Here we augment the depth-resolved luminescence of energy-level transitions involving native defects with recentpositron-annihilation spectroscopy /H20849PAS /H20850results 22,23to deter- mine the energetics of VZnand their complexes in ZnO over both surface and near-surface regions /H208497/H110111500 nm /H20850in ion /H20849Li or N /H20850implanted and annealed bulk ZnO. Both Li and N are among the most important dopants for p-type ZnO dop- ing, yet the roles of the associated defects generated by im-plantation or annealing are not yet clear. Doppler broadeningexperiments with a slow positron beam provide depth distri-butions of neutral or negatively charged vacancy defects, 24,25 in this case, of VZnand vacancy clusters. The correspondence between these PAS native defect distributions and theDRCLS intensity distributions versus depth permits us toidentify the luminescence energy associated with isolatedV Zndefects as well as the energy shift due to vacancy cluster formation. Surface photovoltage spectroscopy /H20849SPS /H20850yields the positions of these levels with respect to the ZnO bandedges. We associate the remaining deep level DRCLS emis-sion with positively charged V O-related defects, which are not detected by PAS, and describe how the balance betweenthese donor and acceptor defects accounts for depth-dependent resistivity in these irradiated crystals. Taking thesedepth-resolved techniques altogether, we clearly identify theoptical transitions and energies of V Znand vacancy clusters, the effects of different annealing methods on their spatialdistributions in ion-implanted ZnO, and the contribution ofV ZnandVOto near-surface resistivity. In order to create well-deﬁned distributions of Zn vacan- cies, we implanted /H208490001 /H20850ZnO wafers with7Li+or14N+and annealed by conventional furnace or ﬂash lamp at varioustemperatures. These crystals were hydrothermally grown, un-intentionally doped with /H110115/H1100310 17Li /cm3,ntype and highly resistive. The wafers were annealed by conventionalfurnace or ﬂash lamp at various temperatures. Details ofthese samples and their preparation appear elsewhere. 22,23 Electron beams with incident energy EB= 1 ,2 ,3 ,4 ,a n d5 keV excited electron-hole pairs for peak DRCLS excitationdepth U 0=7, 18, 32, 50, and 72 nm, respectively for speci- mens at 70 K in UHV. DRCLS with higher EB/H208495–25 keV /H20850atPHYSICAL REVIEW B 81, 081201 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 1098-0121/2010/81 /H208498/H20850/081201 /H208494/H20850 ©2010 The American Physical Society 081201-110 K employed a JEOL 7800F UHV scanning electron mi- croprobe with hemispherical electron analyzer and Oxfordoptical train. For E B=10, 15, 20, and 25 keV, Monte Carlo simulations including backscattering produce U0=215, 530, 950, and 1500 nm, respectively. A two-Gaussian peak ﬁt tothe DRCLS spectra provided characteristic defect intensitiesI Dat nominally /H110112.0 and /H110112.4 eV. Near band-edge /H20849NBE /H20850 intensities at 3.4 eV were corrected for attenuation due toself absorption 26at bulk depths /H20849EB/H110225 keV /H20850. For Li-implanted ZnO, PAS depth proﬁles of the Sparam- eter extracted from Doppler broadening spectra22show: /H20849i/H20850 an increase in the concentration of open volume defects afterimplantation, /H20849ii/H20850formation of vacancy clusters with open volume larger than that of single V Znafter ﬂash /H2084920 ms /H20850 annealing at 1200 °C, and /H20849iii/H20850disappearance of these clus- ters after conventional furnace annealing fo r1ha t8 0 0° C . DRCLS spectra of the as-implanted ZnO /H20849not shown /H20850dis- plays broad deep level emissions extending from/H110211.9–2.5 eV with peak defect intensity I Dnormalized toINBE such that ID/H20849/H110112.0 eV /H20850/INBE /H208495 keV /H20850and ID/H20849/H110112.4 eV /H20850/INBE /H208495 keV /H20850=0.08 and 0.035, respectively. In contrast to conventional furnace anneals, ﬂash anneals of Li–implanted ZnO generates stable and electrically active V Zn clusters. Figures 1/H20849a/H20850and 1/H20849b/H20850show the spectra for Li- implanted samples after fast /H2084920 ms /H20850ﬂash /H20849at 1200 °C /H20850and 1 h furnace /H20849at 800 °C /H20850annealing respectively. Besides /H110112.0, /H110112.4, and 3.4 eV features, phonon replicas appear below the band edge in the 10 K spectra. A 3 eV bulk emis-sion evident in Fig. 1/H20849a/H20850is removed by furnace anneal in Fig. 1/H20849b/H20850. Flash-annealed I D/H20849/H110112.0 eV /H20850/INBE /H208495 keV /H20850=0.57 near the surface, nearly unchanged /H208490.48 /H20850in the bulk /H2084925 keV /H20850, whereas furnace-annealed ID/H20849/H110112.0 eV /H20850/INBE /H208495 keV /H20850=3 near the surface decreasing to 0.16 in the bulk. Furthermore,ﬂash-annealed I D/H20849/H110112.4 eV /H20850/INBEis low /H208490.17 /H20850for both sur- face and bulk, whereas it increases 35 times at the surfaceand decreases 1.7 times in the bulk after furnace annealing.Lower-temperature /H20849500 °C /H20850furnace anneals produce rela- tively few changes. Thus higher-temperature ﬂash and fur-nace annealing produce major changes in the depth distribu-tions of both /H110112.0 and /H110112.4 eV emission intensities.Figure 2shows the correlation of DRCLS I/H20849/H110112.0 eV /H20850 andI/H20849/H110112.4 eV /H20850with PAS V Znand vacancy cluster densities versus depth22on the same Li-implanted ZnO crystals. In order to improve DRCLS depth resolution for higher EB,w e employed a relatively simple subtraction method: we usedMonte Carlo program CASINO /H20849Ref. 27/H20850to renormalize spec- tra from shallower layers for subtraction from deeper layerspectra. This procedure yields I D/INBEproﬁles with reso- lution comparable to the PAS Sparameter depth proﬁles. The 1200 °C ﬂash-annealed ZnO in Fig. 2/H20849a/H20850displays a strong increase in VZnand vacancy cluster defects beginning at /H11011100 nm and peaking at /H110111/H9262m. The latter corresponds to the depth of maximum implantation damage. I/H20849/H110112.0/H20850/INBE also begins to increase at approximately the same depth, in- creases by approximately the same magnitude, and reaches amaximum at the same depth. By contrast, I/H20849/H110112.4/H20850/I NBEis low for depths of 500 nm or more, increasing only graduallyfor deeper excitation. The 800 °C furnace-annealed ZnO inFig. 2/H20849b/H20850again shows a strong correlation between PAS V Zn-related defect densities and I/H20849/H110112.0/H20850/INBE, whereas I/H20849/H110112.4/H20850/INBEexhibits a qualitatively different depth proﬁle. Further evidence for this assignment includes near-surface/H208495 keV /H20850DRCL spectra of 900 °C ﬂash-annealed ZnO /H20849not shown /H20850that display over two orders of magnitude higher I D/H20849/H110112e V /H20850/INBEandID/H20849/H110111.6 eV /H20850/INBE /H20849discussed below /H20850 compared with Fig. 1, in agreement with 2 orders of magni- tude higher isolated VZnmeasured by PAS in this region.22 Note that the depth proﬁle of 3.0 eV emission as shown in Fig.1/H20849a/H20850isnotconsistent with that of the PAS Sparameter. Hence, it may be due to higher order complex defects ratherthan Zn vacancies. From the correlation of depth proﬁles in Fig. 2, the /H110112.0 eV emission can be assigned to Zn vacancies or their complexes. This resolves the many contradictory assign-ments reported previously. 6Vanheusden et al. assigned the 2.45 eV emissions to O vacancies.10Even though PAS is not directly sensitive to O vacancies, our combined PAS-DRCLSresults showing the completely different behavior of 2.4–2.5eV vs the Zn vacancy emissions now demonstrate that opti-cal emissions at energies typically assigned to O vacanciesFIG. 1. /H20849Color online /H2085070/H208491–5/H20850and 10 K /H208495–25 keV /H20850CL spectra for Li-implanted ZnO after /H20849a/H20850ﬂash anneal at 1200 °C and /H20849b/H20850 furnace anneal at 800 °C. Dashed lines represent characteristicemissions at /H110112.0 and /H110112.4 eV as revealed by ﬁtting.FIG. 2. /H20849Color online /H20850PAS and DRCLS defect densities vs. depth for Li-implanted ZnO after /H20849a/H20850ﬂash and /H20849b/H20850furnace anneal- ing. VZndensities and ID/H20849/H110112.0 eV /H20850/INBEcorrelate, in contrast to ID/H20849/H110112.4 eV /H20850/I/NBE.DONG et al. PHYSICAL REVIEW B 81, 081201 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 081201-2are in fact unrelated to Zn vacancies. Indeed the spatial variations of the former that depend on speciﬁc annealingconditions also eliminate any role of bulk impurities. Bothassignments are consistent with calculations showing V Zn and VOvacancies to be the most common native point defects,18and both are the most commonly observed deep level features. Likewise, they agree with assignments basedon metal-oxide and metal-eutectic reactions observed at ZnOSchottky barriers. 16 SPS in a Kelvin probe /H20849i.e., surface potential /H20850force mi- croscope /H20849KPFM /H20850shows that the VZn-related DRCLS emis- sion corresponds to optical transitions from the ZnO conduc-tion band to gap states 2.1 eV below. The SPS measurementconsists of monitoring changes in surface electric potentialwith illumination as photon energy h /H9271sweeps from low to above band-gap values. The corresponding contact potentialdifference /H20849CPD /H20850between the surface and a reference probe changes as h /H9271exceeds threshold values for gap state popu- lation or depopulation.28Forn-type /H20849upward /H20850band bending and gap states of energy EDlocated EC−EDbelow the con- duction band EC, Fig. 3shows that photodepopulation re- moves negative surface charge and reduces the band bend- ing, thereby raising the Fermi level EFtoEF/H11032and lowering the surface potential /H9021by/H9004/H9021. Here, h/H9271slope changes at 2.05, 2.45, and 3.35 eV correspond to thresholds for electronphotodepopulation, population, and free electron-hole pairtransitions, respectively. These SPS features are characteris-tic of ZnO surfaces that exhibit luminescence peaks at theseenergies. Thus the 1.9–2.1 eV peak in Fig. 1corresponds to states 2.05 eV below E Cwhile the 2.45 eV peak corresponds to states 2.45 eV above the valence band. This 2.05 eV SPSfeature is characteristic of ZnO surfaces for which surface-sensitive DRCLS exhibits strong 1.9–2.1 eV emission.Hence the V Zn-related defect luminescence emission at 1.9– 2.1 eV corresponds to an energy level at /H110112.05 eV below the conduction band as revealed by SPS. This 0 /−1VZntran- sition energy is lower than the ﬁrst-principles calculations of/H110112.7 eV using a hybrid functional and ﬁnite-size corrections, 213.2 eV using density-functional theory within the local-density approximation /H20849LDA /H20850and plane-wave pseudopotentials,5and 3.8 eV using the plane-wave pseudo- potential total-energy and force method plus LDA.19 Combined DRCLS and PAS studies of implanted ZnO reveal that the 1.9–2.1 eV emissions in Fig. 1correspond to large vacancy clusters /H20849containing at least 3–4 VZn/H20850and thatthe emission energies for small vacancy clusters /H20849/H113502VZn/H20850 are signiﬁcantly lower.29Previous positron experiments showed that small vacancy clusters are predominant in as-implanted ZnO while 600 °C furnace annealing induces coa-lescence into larger vacancy clusters and substantially Spa- rameter values. 23In Fig. 4/H20849a/H20850, deep level emissions of the same N+-as-implanted crystals are deconvolved into peaks at 1.6 and 1.9 eV with pronounced defect emission at 1.6 eV, inthe near-surface /H208497n m /H20850region, shifting to 1.9 eV at depths above 70 nm. Electron paramagnetic resonance studies con-ﬁrm the Zn vacancy nature of luminescence in this energyrange. 30After the 600 °C anneal, Fig. 4/H20849b/H20850shows that the characteristic emission shifts to higher energy /H208491.9 eV /H20850and ID/INBEincreases. Note the increasing defect energy with increasing depth /H20849overall shift of the defect related emission with the probing depth in Fig. 4/H20850, indicating isolated or small cluster sizes near the free surface. A 1000 °C anneal disso-ciates the larger vacancy clusters, 23and the 1.9–2.1 eV DRCLS feature decreases by nearly an order of magnitudewith a corresponding increase in /H110112.4 eV V O-related emis- sion /H20849not shown /H20850. These results are consistent with vacancy cluster emission at 1.9–2.1 eV versus small vacancy clusteror isolated V Znemission at /H110111.6 eV. They indicate that large vacancy clusters lie /H110220.3 eV lower in the ZnO band gap and are the predominant defect responsible for /H110112 eV “red” photoluminescence. Figure 4also provides a calibration of DRCLS with va- cancy concentrations obtained with positrons. TheN +-as-implanted ZnO contains an estimated concentration of small vacancy clusters /H20851denoted /H20849VZn/H20850n/H20852of 1–2 /H110031018cm−3,23corresponding to I/H20849/H110112.0 eV /H20850/INBE/H110111. Since I/H20849/H110112.0 eV /H20850/INBE/H110111.6 in Fig. 2/H20849a/H20850at the peak implan- tation depth, then /H20849VZn/H20850n/H110111.6/H110031018cm−3, in line with pre- vious estimates.22Calibration at this depth permits estimates of/H20849VZn/H20850nconcentration much closer to the surface than PAS conventionally permits. The relative densities of large and small vacancy clusters or isolated VZn, together with VO, and all as a function of depth account for the ZnO’s local resistance self-consistently.Zn vacancies and vacancy clusters play different roles elec-trically. V Zndefects act as compensating acceptors and in- crease resistance, while vacancy clusters remove isolated VZn and/or deactivate Li dopants, thereby decreasing resistance, as observed previously in experiments with irradiation-induced electrical isolation. 31As Fig. 2showed for Li-FIG. 3. /H20849Color online /H20850/H9004/H9021vsh/H9271with onset of photodepopula- tion of deep levels EDat 2.05 eV below conduction band EC, pho- topopulation at 2.45 eV above valence band EV, and band ﬂattening ath/H9271/H11022band gap EG/H208493.35 eV /H20850.FIG. 4. /H20849Color online /H2085070 K CL spectra /H208491–5 keV /H20850for/H20849a/H20850as- received N-implanted ZnO and after /H20849b/H208501 h 600 °C furnace anneal- ing that induces VZnclustering. Dashed lines represent characteristic emissions at /H110111.6 and /H110111.9 eV as revealed by ﬁtting.VACANCY DEFECT AND DEFECT CLUSTER ENERGETICS … PHYSICAL REVIEW B 81, 081201 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 081201-3implanted samples, 1200 °C ﬂash anneal generates a high concentration /H208491018–1020cm−3/H20850of vacancy clusters in the 500–1500 nm region. These vacancy clusters reduce densi-ties of isolated V Zn, small vacancy clusters, and uncomplexed Li, all of which reduce the compensating acceptor density.Accordingly, the scanning spreading resistance microscopy/H20849SSRM /H20850resistance proﬁle of this crystal in Fig. 5displays a major /H20849three orders of magnitude /H20850decrease within the same region. As Fig. 4showed, N-implantation introduces isolated Zn vacancies in the intimate surface region /H20849/H1102150 nm /H20850. The corresponding surface resistance in Fig. 5increased by over three orders of magnitude. 32 In general, both VZnandVOdensities are needed to ac- count for resistance in ZnO self-consistently. The /H110112.4 eV emission peak attributed to oxygen vacancies acting as deepdonors exhibits a pronounced maximum at /H1101170 nm depth for 800 °C furnace-annealed ZnO in Fig. 2. This 2.4 eV peak maximum can account for the sharp drop in SSRM resistanceat the same depth for this crystal in Fig. 5, whereas the /H110112.0 eV peak intensity is low and varying slowly at these depths. Thus the combination of low concentration V Znand vacancy clusters plus elevated VOin the near-surface /H20849/H11021500 nm /H20850region act to decrease surface resistance bynearly four orders of magnitude relative to the bulk, the low- est near-surface resistance of all Li-implanted ZnO studied.Other recent Sparameter correlations with optical/transport properties include ZnO, 22,23,31GaN,33and InN /H20849Ref. 34/H20850 since the Sparameter reﬂects the vacancy content, which undoubtedly affect the optoelectronic properties of semicon-ductors. In summary, combined PAS, DRCLS, and SPS studies reveal the V Zndefect nature of optical emissions in the range of 1.6–2.1 eV, the energy-level position of vacancy clustersat 1.9–2.1 eV below the conduction band, and the energy-level position of isolated V Zndefects or small clusters 0.3 eV higher above the valence band. DRCLS-measured vacancycluster and V O-related emissions combined with SSRM re- sistance within the same near-surface regions reveal the dif-ferent compensating nature of vacancy clusters on ZnO car-rier concentration and the competing roles of V ZnandVO defects on ZnO resistance. These results resolve the contra- dictory energetic assignments for VZnand add weight to the VO-related defect assignment reported previously. Further- more, these combined results conﬁrm the acceptor-versus do-norlike behavior associated with these two optical emissionsand demonstrate their utility. The physical nature of the de-fects that dominate optical features of this widely studiedsemiconductor and the consistent explanation for ZnO’s ef-fective free-carrier densities on a local scale enable a deeperunderstanding of many ZnO properties and their applica-tions. The authors gratefully acknowledge support from the National Science Foundation Grant No. DMR-0513968/H20849Verne Hess /H20850and the Norwegian Research Council through the NANOMAT and FRINAT programs. F.T. acknowledgesthe support from the Academy of Finland. dong.70@osu.edu 1D. C. Look, Mater. Sci. Eng., B 80, 383 /H208492001 /H20850. 2S. J. Pearton et al. , Prog. Mater. Sci. 50, 293 /H208492005 /H20850. 3D. C. Look, J. Electron. Mater. 35, 1299 /H208492006 /H20850. 4Ü. Özgür et al. , J. Appl. Phys. 98, 041301 /H208492005 /H20850. 5A. Janotti and C. G. Van de Walle, Phys. Rev. B 76, 165202 /H208492007 /H20850. 6M. D. McCluskey and S. J. Jokela, J. Appl. Phys. 106, 071101 /H208492009 /H20850. 7D. C. Look et al. , Phys. Rev. Lett. 95, 225502 /H208492005 /H20850. 8F. A. Selim et al. , Phys. Rev. Lett. 99, 085502 /H208492007 /H20850. 9C. H. Ahn et al. , J. Appl. Phys. 105, 013502 /H208492009 /H20850. 10K. Vanheusden et al. , Appl. Phys. Lett. 68, 403 /H208491996 /H20850. 11Y. W. Heo et al. , J. Appl. Phys. 98, 073502 /H208492005 /H20850. 12M. A. Reshchikov et al. , J. Appl. Phys. 103, 103514 /H208492008 /H20850. 13Q. X. Zhao et al. , Appl. Phys. Lett. 87, 211912 /H208492005 /H20850. 14T. M. Børseth et al. , Appl. Phys. Lett. 89, 262112 /H208492006 /H20850. 15L. J. Brillson, J. Vac. Sci. Technol. B 19, 1762 /H208492001 /H20850. 16L. J. Brillson et al. , Appl. Phys. Lett. 90, 102116 /H208492007 /H20850. 17S. Lany and A. Zunger, Phys. Rev. B 78, 235104 /H208492008 /H20850. 18A. F. Kohan et al. , Phys. Rev. B 61, 15019 /H208492000 /H20850. 19S. B. Zhang et al. , Phys. Rev. B 63, 075205 /H208492001 /H20850. 20P. Erhart et al. , Phys. Rev. B 73, 205203 /H208492006 /H20850.21F. Oba et al. , Phys. Rev. B 77, 245202 /H208492008 /H20850. 22T. Moe Børseth et al. , Phys. Rev. B 74, 161202 /H20849R/H20850/H208492006 /H20850. 23T. M. Børseth et al. , Phys. Rev. B 77, 045204 /H208492008 /H20850. 24Z. Q. Chen et al. , Phys. Rev. B 71, 115213 /H208492005 /H20850. 25F. Tuomisto et al. , Phys. Rev. Lett. 91, 205502 /H208492003 /H20850. 26H. C. Ong et al. , Appl. Phys. Lett. 78, 2667 /H208492001 /H20850. 27D. Drouin et al. , Scanning 29,9 2 /H208492007 /H20850. 28L. Kronik and Y. Shapira, Surf. Sci. Rep. 37,1/H208491999 /H20850. 29Note that these clusters contain also O vacancies as otherwise the positron data would be similar to isolated VZn/H20849for the data to be different as in this case, a larger connected open volume isneeded /H20850. 30L. A. Kappers et al. , Nucl. Instrum. Methods Phys. Res. B 266, 2953 /H208492008 /H20850. 31A. Zubiaga et al. , Phys. Rev. B 78, 035125 /H208492008 /H20850. 32Note: any variations in surface roughness at the outer surface of the cross-sectional proﬁle cuts could perturb outer surfaceSSRM values; however, these do not mask the overall subsur-face and bulk systematics. 33F. Tuomisto et al. , Appl. Phys. Lett. 90, 121915 /H208492007 /H20850. 34F. Tuomisto, A. Pelli, K. M. Yu, W. Walukiewicz, and W. J. Schaff, Phys. Rev. B 75, 193201 /H208492007 /H20850.FIG. 5. /H20849Color online /H20850SSRM resistance depth proﬁles of Li- and N-implanted ZnO in Figs. 2and4, respectively.DONG et al. PHYSICAL REVIEW B 81, 081201 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 081201-4
PhysRevB.83.073403.pdf	PHYSICAL REVIEW B 83, 073403 (2011) Dynamics of the spin Hall effect in topological insulators and graphene Bal´azs D ´ora1,and Roderich Moessner2 1Department of Physics, Budapest University of Technology and Economics, HU-1111 Budapest, Hungary 2Max-Planck-Institut f ¨ur Physik komplexer Systeme, DE-01187 Dresden, Germany (Received 20 December 2010; published 11 February 2011) A single two-dimensional Dirac cone with a mass gap produces a quantized (spin) Hall step in the absence of magnetic ﬁeld. What happens in strong electric ﬁelds? This question is investigated by analyzing time evolutionand dynamics of the spin Hall effect. After switching on a longitudinal electric ﬁeld, a stationary Hall current isreached through damped oscillations. The Hall conductivity remains quantized as long as the electric ﬁeld ( E)i s too weak to induce Landau-Zener transitions, but quantization breaks down for strong ﬁelds and the conductivitydecreases as 1 /√ E. These apply to the spin-Hall conductivity of graphene and the Hall and magnetoelectric response of topological insulators. DOI: 10.1103/PhysRevB.83.073403 PACS number(s): 73 .20.−r, 72.80.Vp, 73 .43.−f The unique electronic properties of graphene can be traced back to the the pseudorelativistic Dirac equation and itslinear energy dispersion with zero bandgap. It exhibits aplethora of interesting and fascinating physical phenomenarelated to electric and heat transport, magnetic ﬁeld effects,valleytronics, and spintronics. 1The “half-integer” quantum Hall effect, in spite of its environmental fragility, has beenobserved at room temperature 2due to the unusual Landau quantization of Dirac electrons. With spin-orbit couplingtaken into account, graphene in principle acts as a spin-Hallinsulator, 3belonging to the class of topological insulators (TI). Moreover, Dirac electrons also occur as surface states of three-dimensional TI.4–6These materials are predicted to display a variety of peculiar phenomena, such as spin-and surface quantum Hall effects and the closely relatedtopological magnetoelectric effect, 7allowing for the control of magnetization by electric ﬁeld. As opposed to the evennumber of Dirac cones in graphene, three-dimensional TIcan have an odd number of Dirac cones on a surface.Due to time-reversal symmetry, these states are robust withrespect to nonmagnetic disorder, similar to the way pairbreaking in s-wave superconductors is suppressed by potential scatterers. The hallmark of (pseudo-) relativistic massive Dirac electrons 8is a single quantum Hall step around half-ﬁlling in the absence of magnetic ﬁeld between σxy=±e2/2h.W e ask how this picture gets modiﬁed in the presence of strongelectric ﬁelds. Generally, a driving electric ﬁeld can producea sizable density of electron and hole excitations around theDirac point in a highly nonthermal, nonstationary momentumdistribution. 9Consequently, the longitudinal transport of Dirac electrons features Klein tunneling10and Schwinger’s pair production9,11–13in a stationary or time-dependent framework, when the electric ﬁeld is represented by a static scalar or a time-dependent vector potential, respectively. The latter approachdirectly yields the nonequilibrium momentum distribution andthe time-dependent current at ﬁnite electric ﬁelds. While it doesnot use any kind of equilibrium or out-of-equilibrium responseformalism (Kubo/Landauer), it still reproduces known resultsand makes predictions for the nonlinear behavior of theelectric current as an example. 9,14On the other hand, a strong electric ﬁeld alters not only the longitudinal transport,9but isexpected to modify the transverse conductivity, involving the nonequilibrium quantum (spin-) Hall breakdown. Commonwisdom has it that while there are no power-law correctionsto the integer Hall conductivity for weak electric ﬁelds,with its quantization “topologically” protected, there can beexponentially small corrections. When these grow with ﬁeld,quantization breaks down. To consider the problem in detail, we elaborate on the time evolution of the Hall current for massive Dirac fermions,after switching on a longitudinal electric ﬁeld. We showthat a stationary transverse current develops for long times,characterized by a quantized Hall conductivity for weak ﬁelds,crossing over to a strongly ﬁeld-dependent Hall response withincreasing ﬁeld. This result applies to the quantum spin-Hallbreakdown of graphene 15as well as for the related4surface Hall and magnetoelectric effects in TI. The low-energy description around the Kpoint in the Brillouin zone of graphene1or on the surface state of a 3D TI4,16(after a π/2 rotation of the spin around ˆz), in the presence of a uniform, constant electric ﬁeld ( E> 0) in the xdirection {switched on at t=0, through a time-dependent vector potential A(t)=[A(t),0,0] with A(t)=Et/Theta1 (t)}is written as i¯h∂t/Psi1p(t)={vF[px−eA(t)],vFpy,/Delta1}·σ/Psi1p(t),(1) where vFis the Fermi velocity, and the Pauli matrices ( σ) encode the two sublattices1of the honeycomb lattice in graphene, or the physical spin in TI. /Delta1> 0 is the mass gap, originating from the intrinsic spin-orbit coupling (SOC) ingraphene, 3or from a thin ferromagnetic ﬁlm covering the surface of TI, lifting the Kramer’s degeneracy of the Diracpoint. To make our analysis more transparent, we perform a two-step unitary transformation, U=U 1U2. Firstly, a time- independent rotation around the σxaxis as U1=C+− iσxC−, with C±=[1±vFpy//radicalbig (vFpy)2+/Delta12]1/2/√ 2, and secondly, a time-dependent one, bringing us to theadiabatic basis as U 2=exp[−iϕ(t)σz/2](σx+σz) with tanϕ(t)=√ p2 y+(/Delta1/v F)2/[px−eA(t)]. The resulting in- stantaneous energy spectrum in the upper Dirac cone 073403-1 1098-0121/2011/83(7)/073403(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 83, 073403 (2011) isεp(t)=/radicalBig /Delta12+v2 F([px−eA(t)]2+p2y). The transformed time-dependent Dirac equation reads i¯h∂t/Phi1p(t)=/bracketleftBigg σzεp(t)−σx¯hvFeE/radicalbig (vFpy)2+/Delta12 2ε2p(t)/bracketrightBigg /Phi1p(t), (2) and/Psi1p(t)=U/Phi1p(t), with initial (ground-state) condition /Phi1T p(t=0)=(0,1), in which the lower (upper) Dirac cone is fully occupied (empty). The electric ﬁeld alters the energyspectrum and induces off-diagonal terms in the Hamiltonian.Two energy scales at the moving Dirac point [ p=(eEt, 0)] in Eq. ( 2) characterize the low-energy physics: the diagonal energy ( /Delta1) and off-diagonal coupling (¯ hv FeE/2/Delta1), which triggers transitions between the two gap edges or levels usingLandau-Zener (LZ) terminology. 17A crossover from weak to strong ﬁeld is thus expected at E∼/Delta12/¯hvFe, irrespective of the explicit value of t, as we conﬁrm in the following by a more detailed analysis. The quantity we focus on is the time-dependent transverse charge current jy=−evFσyin the basis of Eq. ( 1), with spin current and conductivity differing only by a factor ¯ h/ev F.F o r TI,jycoincides with the topological magnetic ﬁeld induced parallel to the applied electric ﬁeld [after the π/2r o t a t i o no f the spin, leading to Eq. ( 1)] and monitors the magnetoelectric effect.7,16 By denoting /Phi1T p(t)=[αp(t),βp(t)], charge conservation implies \|βp(t)\|2=1−n(t), where np(t)=\|αp(t)\|2is the number of electrons which have tunneled into the initiallyempty upper Dirac cone. After multiplying the transformedDirac equation, Eq. ( 2) with /Phi1 + p(t)σxor/Phi1+ p(t) from the left, we get /angbracketleftjy/angbracketrightp(t)=−evF/Delta1¯h εp(t)/braceleftBigg ∂t/bracketleftbig ε2 p(t)∂tnp(t)/bracketrightbig vFeE[(vFpy)2+/Delta12] +vFeE 2ε2p(t)[2np(t)−1]/bracerightBigg , (3) which depends only on np(t) and its time derivatives. We start (Fig. 1) by analyzing its behavior at weak electric ﬁelds ( E/lessmuch/Delta12/¯hvFe) at short times ( t<√¯h/vFeE), determined by the ﬁrst term in Eq. ( 3). The time-dependent Hall current is obtained after determining np(t) perturbatively9 for weak ﬁelds as jy(t)=e2 2h/braceleftbigg/Delta1t ¯h/bracketleftbigg π−2Si/parenleftbigg2/Delta1t ¯h/parenrightbigg/bracketrightbigg +2s i n2/parenleftbigg/Delta1t ¯h/parenrightbigg/bracerightbigg E, (4) where Si( x) is the sine integral, and exhibits damped oscil- lations around the quantized value of the Hall conductivityasσ xy=e2/2h[1+¯hsin(2/Delta1t/¯h)/2/Delta1t] with a frequency of 2/Delta1/¯h, after expanding Eq. ( 4)f o rt/greatermuch¯h//Delta1. Still at short times, but in the opposite small /Delta1and strong Elimit, similar oscillations with a frequency ∼√vFeE/¯hshow up in the response around the nonquantized asymptotic value. Thetransient behavior at very short times (¯ h/W < t < ¯h//Delta1)r i s e s0 2 4 6 8 1000.20.40.60.811.2 0 5 1000.050.10.15 Δt/σxy2h/e2EΔ2/vFe t vFeE/σxy2h/e2 EΔ2/vFe √ FIG. 1. (Color online) The short-time Hall conductivity is shown from Eq. ( 4) (dashed line) together with the numerical solution of Eq. ( 2) (solid line) for weak electric ﬁeld. The inset shows the numerical results for strong ﬁelds with the characteristic oscillations set by the ﬁeld. The transient response in both cases is well described by Eq. ( 5). linearly with /Delta1Et as jy(t)=e2 2hπ/Delta1t ¯hE. (5) For even shorter times ( t<¯h/W ),jy(t)=e2 2hπ/Delta1Wt2 ¯h2E, with Wthe high-energy cut-off. In the long-time limit [ t/greatermuchmin(¯h//Delta1,√¯h/vFeE)], we can use the analogy of Eq. ( 2) to the LZ problem9,17of two-level crossing to determine np(t): np(t)=/Theta1[px(eEt−px)] exp/braceleftbigg −π[(vFpy)2+/Delta12] vF¯heE/bracerightbigg ,(6) which is the pair-production rate by Schwinger11and also the LZ transition probability17between the initial and ﬁnal levels, applicable if ( px,eEt−px)/greatermuch√ p2 y+(/Delta1/v F)2. In this limit, the second term in Eq. ( 3) dominates, and the transverse current reaches a time-independent value jy(t)=σxyE, with σxy=(evF)2/Delta1 4πh/integraldisplay dp1−2np(t) ε3p(t)≈e2 2herf⎛ ⎝/radicalBigg π/Delta12 vF¯heE⎞ ⎠, (7) which is our main result, erf( x) being the error function. The structure of the nonequilibrium Hall conductivity atlong times [Eq. ( 7)] agrees with the conventional equilibrium Kubo expression 18,19after shifting the momentum with the vector potential and replacing the equilibrium Fermi functionswith the nonequilibrium momentum distribution, Eq. ( 6). Alternatively, Eq. ( 7) reﬂects the competition between Berry’s curvature [ /Omega1 p=v2 F/Delta1/2ε3 p(t)], protecting quantization20and the difference of momentum distributions in the upper [ np(t)] and lower [1 −np(t)] Dirac cones in the numerator, spoiling it. When the two distributions are comparable due to tunneling 073403-2BRIEF REPORTS PHYSICAL REVIEW B 83, 073403 (2011) from the lower to the upper Dirac cone, the gap becomes irrelevant, and the conductivity decays. In the limit of small ﬁelds ( E/lessmuchπ/Delta12/vF¯he), we recover the quantized value σxy=e2 h/integraldisplay dp/Omega1p 2π=e2 2h, (8) without higher-order perturbative or power-law (in E) cor- rections. The additional terms contain the nonperturbative,exponential factor exp( −π/Delta1 2/vF¯heE), signaling the robust- ness of Hall quantization21and the half-integer quantized magnetoelectric polarizability.7In the strong-ﬁeld limit ( E/greatermuch π/Delta12/vF¯he), it decays as σxy=e2 2h2/Delta1√vF¯heE. (9) F o rT Iw i t ham a s sg a p( /Delta1/negationslash=0), the magnetization produced by surface currents probes the Hall conductivity throughthe topological magnetoelectric effect, and the magnetizationparallel to the electric ﬁeld follows Eq. ( 7): Its quantization breaks down with increasing ﬁeld similarly to the Hallresponse. When /Delta1=0, the magnetization perpendicular to Ebecomes ﬁnite ∼(e 2π/2h)Ein weak ﬁelds.9 Assuming a small gap of the order of 0.01–1 K (typical for the intrinsic SOC of graphene3,22or TI) the crossover ﬁeld is 0.001–10 V /mf o r vF∼106m/s, easily accessible experimentally. The Hall conductivity together with numericalresults of the Dirac equation is shown in Fig. 2.T h e agreement between the analytically and numerically obtainedconductivities is excellent. We can get acquainted with these results in different ways: First, a similar situation occurs within equilibriumlinear response (small E): the (spin-) Hall conductivity of 10−110010110200.20.40.60.81 EvFe/Δ2σxy2h/e2 FIG. 2. (Color online) The long-time limit of the Hall conductiv- ity is plotted as a function of the applied longitudinal electric ﬁeld. Quantization breaks down when E∼/Delta12/¯hvFe. The circles denote the numerical data from brute-force integration of the Dirac equation, Eq. ( 2), while the dashed line is the approximate expression from Eq. ( 9)a tl a r g eﬁ e l d s .massive Dirac electrons (i.e., graphene with intrinsic SOC and TI23) is quantized to e2/2h, when the chemical potential lies within the gap ( \|μ\|</Delta1). For \|μ\|>/Delta1, the chemical potential cuts into the continuum of band states, and theconductivity decays as e 2/Delta1/2h\|μ\|, surviving even the effect of disorder.24,25Within our time-dependent formalism, the electric ﬁeld can be thought of formally as introducing aneffective chemical potential μ eff∼√¯hvFeE(only\|vFpy\|< μeffcontributes), which upon substituting into the above linear response expressions, parallels our ﬁndings. Second, in the edge-state picture, the Hall conductivity is provided by gapless, one-dimensional ballistic edge states,giving rise to the quantized value, which holds for weak electricﬁelds. For strong ﬁelds, another type of gapless excitationstarts to contribute, due to tunneling between valence andconduction bands (Schwinger’s pair production or Zener’sdielectric breakdown), spoiling the perfect Hall quantization,as demonstrated earlier. Another way to look at this is to consider the complemen- tary stationary problem to Eq. ( 1) of a static electric ﬁeld in the form of a scalar potential ( ∼eEx), and analyze the evolution of the spectrum and edge states as a function ofthe electric ﬁeld. As a tight-binding example, we consider thespectrum of a zigzag graphene ribbon 3with intrinsic SOC, causing a gap with opposite sign between the two valleys,sublattices and spin directions in the continuum limit. In theabsence of an electric ﬁeld, only the edge states, connectingthe two Dirac cones, carry the transverse current, while in astrong electric ﬁeld perpendicular to the edges, the effect ofedge states is supplemented by the appearance of additionallow-energy modes living in two dimensions, because the bandsapproach each other, as seen in Fig. 3. As long as the electric ﬁeld is smaller than a critical value, the band structure remainsqualitatively similar to that in Fig. 3, left panel: Edge states are protected by a ﬁnite gap, above which a continuum ofexcitations exist. The spin-Hall response remains protected in 0 0.5 1−1−0.500.51 0 0.5 1 aky/2π aky/2πEnergy /tcc E=0 E=0 FIG. 3. (Color online) The energy spectrum of a spin-Hall insulator3in graphene; tccis the hopping. Left panel: without electric ﬁeld, showing two gapless, spin-degenerate edge states. Right panel: with ﬁnite critical electric ﬁeld (red/black denoting up/down spinstates), distorting the spectrum, and bringing additional levels into play around zero energy. Consequently, the spin-Hall conductivity is not quantized any more. Similar effects are generated by astrain-induced pseudoelectric ﬁeld having opposite sign in the two valleys, resulting in a valley-Hall effect. For stronger E, band crossing is more signiﬁcant. 073403-3BRIEF REPORTS PHYSICAL REVIEW B 83, 073403 (2011) this range. When the electric ﬁeld exceeds its critical value, Ec, the gap closes (right panel in Fig. 3), and the edge states merge with the continuum and are not protected any more—thespin-Hall breakdown occurs. Note that the time-dependent framework provides a ﬁnite (spin-) Hall conductivity even in the absence of scattering 18,24 because of its intrinsic character, not unlike the analysis of metallic graphene,25where additional disorder-induced corrections were found, which we also expect to occur whenscattering is added to our framework; these will also limit thelongitudinal conductivity. 9 Third, 2D Dirac electrons in crossed stationary in-plane electric ( E) and perpendicular magnetic ( B) ﬁelds exhibit Landau quantization and subsequently quantized Hall con-ductivity. The value ( e 2/2hper spin and valley) and the origin of the lowest quantum Hall plateau agrees with thatof Eq. ( 8); therefore, its breakdown can also share a common origin. Indeed, at E=v FB, all Landau levels collapse26–28 and a different Hall response should arise. Deﬁning the energy gap as the distance between the Landau levels closestto the Dirac point, we have /Delta1 Landau=vF√ 2¯heB, yielding E=/Delta12 Landau/2¯hevFfor the ﬁeld, causing the collapse of Landau levels, which agrees well with the crossover ﬁeldwhere the spin-Hall response changes dramatically. We expectthat some of our results can be transcribed to the quantum Hallbreakdown in graphene, 15as indicated by a Hall conductivity decreasing with the electric ﬁeld, similarly to Eq. ( 9). These results are robust against disorder [if the mean free path lis not shorter than min( vF¯h//Delta1,√vF¯h/eE )], becausethe basic ingredient of the calculation is the nonequilibrium momentum distribution function, Eq. ( 6), which follows also from a semiclassical approach (WKB, expansion in¯h). Inelastic processes in the form of energy relaxation can be taken into account in the LZ model. 29By adding a ﬂuctuating ﬁeld to Eq. ( 2)a sη(t)σz/Phi1p(t), where η(t)i s a Markov Gaussian process with vanishing mean [ η(t)= 0] and η(t)η(t/prime)∼exp(−\|t−t/prime\|/τ), energy ﬂuctuation is modeled, and τis the decay time of ﬂuctuation correlation, the typical timescale of energy relaxation. In the limit of¯h/τ/lessmuch(/Delta1,√ vF¯heE), e.g., when the noise ﬂuctuations are small compared to the transition time,29which corresponds exactly to the above condition for the mean-free path withl=v Fτ, the LZ transition probability, Eq. ( 6), remains valid. Consequently, the dynamics of the spin-Hall effect remainintact. We have studied the breakdown of the spin-Hall effect in graphene and surface Hall and magnetoelectric effects intopological insulators in a ﬁnite electric ﬁeld. The quantizationofσ xyremains intact as long as the Hamiltonian varies smoothly (for weak electric ﬁelds). When non-adiabaticityenters via LZ transitions in strong ﬁelds, quantization is lostand the Hall conductivity as well as the magnetoelectriccoefﬁcient decay as E −1/2. This work was supported by the Hungarian Scientiﬁc Research Fund No. K72613, CNK80991 T ´AMOP-4.2.1/ B-09/1/KMR-2010-0002, and by the Bolyai program of theHungarian Academy of Sciences. Electronic address: dora@kapica.phy.bme.hu 1A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 2K. S. Novoselov, Z. Jiang, Y . Zhang, S. V . Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim,Science 315, 1379 (2007). 3C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 4M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 5B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, 1757 (2006). 6X.-L. Qi and S.-C. Zhang, Phys. Today 63, 33 (2010). 7X.-L. Qi, T. L. Hughes, and S.-C. 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PhysRevB.89.195413.pdf	PHYSICAL REVIEW B 89, 195413 (2014) Valleytronics on the surface of a topological crystalline insulator: Elliptic dichroism and valley-selective optical pumping Motohiko Ezawa Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan (Received 1 January 2014; revised manuscript received 5 March 2014; published 12 May 2014) The low-energy theory of the surface of the topological crystalline insulator (TCI) is characterized by four Dirac cones anisotropic into the xandydirections. Recent experiments have shown that the band gap can be introduced in these Dirac cones by crystal distortion by applying strain to the crystal structure. The TCI surfaceprovides us with a way to valleytronics when gaps are given to Dirac cones. Indeed, the system has the Chernnumber and three valley-Chern numbers. We investigate the optical absorption on the TCI surface. It showsa strong elliptic dichroism though the four Dirac cones have the same chiralities. Namely, it is found that theabsorptions of the right- and left-polarized light are different, depending on the sign of mass and the location ofthe Dirac cones, owing to the anisotropy of the Dirac cone. By measuring this elliptic dichroism it is possible todetermine the anisotropy of a Dirac cone experimentally. DOI: 10.1103/PhysRevB.89.195413 PACS number(s): 78 .67.−n,73.20.At,78.20.Ek I. INTRODUCTION Valleytronics is a promising candidate of the next genera- tion electronics [ 1–7]. It is a technology of manipulating the degree of freedom to which inequivalent degenerate state, anelectron, belongs near the Fermi level. The main target of val-leytronics is the honeycomb lattice system such as graphene.Indeed, the honeycomb structure is an ideal playground ofvalleytronics since it has two inequivalent Dirac cones orvalleys . A key progress in valleytronics is valley-selective optical pumping [ 4,5,8–12]. By applying circular polarized light in a gapped Dirac system, we can selectively exciteelectrons in one valley based on the property that two valleyshave opposite chiralities. It is known as circular dichroism.Valley-selective pumping has been observed [ 13–18]i nt h e transition-metal dichalcogenides such as MoS 2, where there exists a direct gap between the conduction and valence bandsfor Dirac fermions. However, the valleytronics is not restricted to the honey- comb system. Recently, the topological crystalline insulator(TCI) attracted much attention due to its experimental real-izations [ 19–21]i nP b 1−xSnxTe. It is a topological insulator protected by the mirror symmetry [ 22,23]. The remarkable properties of the TCI is that there emerge four topolog-ical protected surface Dirac cones, as has been observed in the angle-resolved photoelectron spectroscopy (ARPES) experiment [ 19–21]. The appearance of several topologically protected Dirac cones enables us to use the TCI as the basicmaterial for the valleytronics. Recent experiments [ 24]s h o w that the band gap can be introduced in the surface Diraccones by crystal distortion by applying strain to the crystalstructure. In this paper we investigate the optical absorption of the TCI surface. The key properties of surface Dirac cones arethat all of them have the same chirality but that each ofthem has a particular anisotropy. Based on the anisotropy,we can selectively excite electrons in different valleys by theelliptically polarized light. This is a type of dichroism differentfrom the circular dichroism. We call it an elliptic dichroism .W e propose an experimental method to determine the anisotropyof the velocities and the band gap of Dirac cones with theuse of elliptic dichroism. Our ﬁnding will open a way of the valleytronics based on the TCI. The present paper is composed as follows. In Sec. IIwe introduce the low-energy Hamiltonians H XandHYvalid near the XandYpoints for the [001] surface, which are related by the C4discrete rotation symmetry. The Hamiltonian contains the pseudospin degree of freedom representing thecation and the anion. The X(Y) point is separated into a pair of the /Lambda1 Xand/Lambda1/prime X(/Lambda1Yand/Lambda1/prime Y) points due to the spin-pseudospin mixing. We then derive the four low-energyHamiltonians describing four Dirac cones at the /Lambda1 X,/Lambda1/prime X,/Lambda1Y, and/Lambda1/prime Ypoints. They have in general Dirac electrons with different masses mX,m/prime X,mY, andm/prime Y. In Sec. IIIwe study the spin and psuedospin structures around the XandYpoints. In Sec. IVwe analyze the Chern number for each Dirac cone. It is simply given by ±1 2depending on the sign of the Dirac mass. Since there are four Dirac cones, there arise the Chernnumber and three valley-Chern numbers. The Chern numberis a genuine topological number, while valley-Chern numbersare symmetry-protected topological numbers. When the massis induced by the strain, the Chern number is zero becauseof the time-reversal symmetry. On the other hand, when themass is induced by the exchange effect, the Chern number is±2 per surface. In Sec. Vwe investigate optical absorption and elliptic dichroism by exciting massive Dirac electrons bythe right or left elliptically polarized light. We show that theoptical absorption is determined by the Chern number of eachDirac cone and that the elliptic dichroism occurs owing to theanisotropy of a Dirac cone. It is interesting that the ellipticdichroism is observable on the surface of the TCI with theDirac mass being induced by the strain. II. HAMILTONIAN Recent ARPES experiments [ 19–21] show that there are four Dirac cones at /Lambda1X,/Lambda1/prime X,/Lambda1Y, and/Lambda1/prime Ypoints in the [001] surface state of the TCI, whose band structure we show inFig.1(a). They may be used as the valley degree of freedom. Two Dirac cones are present at the /Lambda1 Xand/Lambda1/prime Xpoints near theXpoint but slightly away from the Xpoint along the x 1098-0121/2014/89(19)/195413(7) 195413-1 ©2014 American Physical SocietyMOTOHIKO EZAWA PHYSICAL REVIEW B 89, 195413 (2014) FIG. 1. (Color online) (a) Surface Brillouin zone centered at the /Gamma1point and bounded by the XandYpoints. There are low-energy Dirac cones at the /Lambda1X,/Lambda1/prime X,/Lambda1Y,/Lambda1/primeYpoints, and high-energy Dirac cones at the XandYpoints. (b) Detailed band structure in the vicinity of the Xpoint. Two low-energy Dirac cones are formed at the /Lambda1X and/Lambda1/prime Xpoints. (c) The gaps open when the mass term is present. axis in the momentum space. The other two Dirac cones are present at the /Lambda1Yand/Lambda1/prime Ypoints near the Ypoint along the yaxis. It is notable that the Dirac cones reside at the mirror symmetry invariant points along the /Gamma1Xand/Gamma1Ylines rather than at the time-reversal symmetry invariant XandYpoints, implying that the protected symmetry is the mirror symmetryand not the time-reversal symmetry. The Hamiltonian for the [001] surface states of the TCI near theYpoint has been given in literature [ 25–28]a s H Y(k)=v2kxσy−v1kyσx+nτx+n/primeσxτy+mσz.(1) The Hamiltonian near the Xpoint is given by HX(k)=v1kxσy−v2kyσx+nτx+n/primeσyτy+mσz,(2) as we shall soon see. Here σandτare the Pauli matrices for the spin and the pseudospin representing the cation-anion degree of freedom, respectively, nandn /primedescribe the pseudospin mixing. We have set /planckover2pi1=1 for simplicity. Typical values are v1=1.3e V ,v2=2.4e V ,n=70 meV , and n/prime=26 meV [ 23,25]. The term mσzrepresents the exchange magnetization with the exchange ﬁeld m, and acts as the mass term. It may regarded as the Zeeman term without externalmagnetic ﬁeld. It may arise due to proximity coupling to aferromagnet, as it enhances the exchange interaction to alignthe spin direction. We show the band structure without andwith this term in Figs. 1(b) and1(c), respectively.The crystal structure of the Pb 1−xSnxTe is a rocksalt structure. Accordingly, the [001] surface has the inverse C4 discrete rotation such that σx/mapsto→σy,σ y/mapsto→−σx (3) together with kx/mapsto→ky,k y/mapsto→−kx. (4) Using this transformation, we obtain Eq. ( 2) valid near the Xpoint from Eq. ( 1) valid near the Ypoint. Note that the velocities into the xandydirections are different at the Y point from those at the Xpoint, as is a manifestation of the fourfold rotation symmetry. It follows from ( 2) that the energy spectrum is given by E(k)=±/radicalBig f±2√g (5) in the vicinity of the Xpoint with f=n2+n/prime2+v2 1k2 x+v2 2k2 y+m2, (6a) g=(n2+n/prime2)v2 1k2 x+n2v2 2k2 y+n2m2. (6b) The band structure is shown in Fig. 1.T h eg a pc l o s e sa tt h e two points ( kx,ky)=(±/Lambda1,0) with /Lambda1=√ n2+n/prime2/v1without the mass term ( m=0). They are the /Lambda1Xand/Lambda1/prime Xpoints. An intriguing feature of the TCI surface is the mass acquisition [ 23,29] by crystal distortion, as has been observed in recent experiments [ 24]. They are ±/Delta1mXand±/Delta1mYat the /Lambda1X(/Lambda1/prime X) and /Lambda1Y(/Lambda1/prime Y) points, respectively. Combining the massmdue to the exchange effect, the mass reads [ 29] mX=m+/Delta1mX,m/prime X=m−/Delta1mX, mY=m+/Delta1mY,m/prime Y=m−/Delta1mY (7) at each Dirac point. There might be other mechanisms to generate the mass. The mass term is necessary for the valley-selective optical absorption to occur. However, the followinganalysis is independent of detailed origins of the mass term. By linearizing the band structure around the /Lambda1 Xpoint, we obtain the two-component low-energy Hamiltonian formassive Dirac fermions [ 25,27], H /Lambda1X(/tildewidek)=˜v1/tildewidekxσy−˜v2/tildewidekyσx+˜mXσz, (8) which describes physics near the Fermi level, where /tildewidekx= kx−/Lambda1and/tildewideky=ky, with the renormalized velocity, ˜v1=v1/radicaltp/radicalvertex/radicalvertex/radicalbt1−m2 Xn2(n2+n/prime2) /bracketleftbig (n2+n/prime2)2+m2 Xn2/bracketrightbig3/2/similarequalv1,(9a) ˜v2=v2/radicaltp/radicalvertex/radicalvertex/radicalbt1−n2 /radicalBig (n2+n/prime2)2+m2 Xn2 /similarequalv2n/prime//radicalbig n2+n/prime2=0.84 eV, (9b) and the renormalized mass, ˜mX=sgn(mX)/radicalbigg m2 X+2n2+2n/prime2−2/radicalBig (n2+n/prime2)2+m2 Xn2. (10) 195413-2V ALLEYTRONICS ON THE SURFACE OF A TOPOLOGICAL . . . PHYSICAL REVIEW B 89, 195413 (2014) The energy spectrum reads E/Lambda1X=±/radicalBig ˜v2 1/tildewidek2x+˜v2 2/tildewidek2y+˜m2 X. (11) The linearized Hamiltonian around the /Lambda1/prime Xpoint has precisely the same expression as ( 8) except that ˜mXis replaced by ˜m/prime X. In the same way we have the low-energy Hamiltonian aroundthe/Lambda1 Ypoint, H/Lambda1Y(/tildewidek)=˜v2/tildewidekxσy−˜v1/tildewidekyσx+˜mYσz, (12) where/tildewidekx=kxand/tildewideky=ky−/Lambda1, and the similar one around the/Lambda1/prime Ypoint. III. SPIN DIRECTION We illustrate the expectation value of the spin /angbracketlefts/angbracketright= /angbracketleftψ\|s\|ψ/angbracketrightin the vicinity of the Xpoint in Fig. 2(a). There is one up-pointing vortex with anticlockwise vorticity at the X point, and there are two down-pointing vortices with clockwisevorticity at the /Lambda1 Xand/Lambda1/prime Xpoints [ 28,30,31]. They describe the spin directions of electrons in one Dirac cone at the X point, and two Dirac cones at the /Lambda1Xand/Lambda1/prime Xpoints in Fig. 1. FIG. 2. (Color online) (a) Spin direction of the TCI surface in the vicinity of the Xpoint. The red oval indicates the region where the magnitude of spin is quite small. The spin directions are opposite inside and outside the oval. The spin rotation is clockwise (anticlockwise) in the low-energy (high-energy) Dirac cones at the/Lambda1 Xand/Lambda1/prime Xpoints (the Xpoint). (b) Pseudospin direction of the TCI surface in the vicinity of the Xpoint. The red oval indicates the region where the magnitude of pseudospin is quite small. The pseudospin directions are opposite inside and outside the oval. (c) Berry curvature of the highest occupied band. It has a sharp peak(red) at the Xpoint and sharp peaks (blue) at the /Lambda1 Xand/Lambda1/prime Xpoints. The Chern number contribution from the Berry curvature at the X point is exactly canceled out by the one (green) from the Dirac conein the lowest occupied band at the Xpoint.This structure is understood as follows. Let us assume n=0 and n/prime=0i nE q .( 2). Then the two Dirac cones in the conduction and valence bands touch each other at the Fermilevel. The effect of the term nτ xis to shift these Dirac cones to intersect one another, forming an intersection oval. (It isan oval and not a circle since v 1/negationslash=v2.) These two Dirac cones have opposite chiralities, which leads to the oppositespin rotations inside and outside the oval. We now switch onn /prime. Then the level crossing turns into the level anticrossing with the resulting band structure as in Fig. 1(a), where Dirac cones emerge at the /Lambda1Xand/Lambda1/prime Xpoints. The spin rotates around each Dirac cone. The magnitude of spin, s2=s2 x+s2 y+s2 z,i s found to be quite small around the oval [Fig. 2(a)]. We clearly see the directions of the spin rotation are identical in the fourvalleys at /Lambda1 X,/Lambda1/prime X,/Lambda1Y, and/Lambda1/prime Y, which manifests the identical chirality of the four low-energy Dirac cones. On the otherhand, the spin rotation in the two high-energy Dirac cones attheXandYpoints is opposite to the one in the low-energy Dirac cones. The spin direction has been observed by meansof spin-resolved ARPES [ 20,30]. We have also illustrated the expectation value of the pseudospin in the vicinity of the Xpoint in Fig. 2(b). The pseu- dospin vector points the xdirection when n /prime=0i nE q .( 2), since then τxis a good quantum number. The pseudospin direction is inverted at the oval, which is the interceptionof the two Dirac cones. When n /prime/negationslash=0, the magnitude of the pseudospin t2=t2 x+t2 y+t2 zbecomes quite small also around the oval. The fact that the magnitudes of the pure spin and pseudospin are quite small around the oval leads to a strong entanglementof the spin and pseudospin there, as we now argue. TheHamiltonian is described by the 4 ×4 matrix, which results in the SU(4) group structure of the system. The SU(4) groupis decomposed into the pure spin and pseudospin parts andthe spin-pseudospin entangled part. The generators of the purespin (pseudospin) part are given by σ i(τi) with i=x,y,z . On the other hand, those of the spin-pseudospin entangledpart are given by σ iτjwithi,j=x,y,z , which compose the SU(2)⊗SU(2) group. The magnitude of the SU(4) spin is a constant and takes the same value everywhere. Hence, the factthat the pure spin and pseudospin components become quitesmall means that the spin-pseudospin entangled componentssuch as σ zτyandσyτzbecome large. The results implies a rich topological structure in the SU(4) space. IV . CHERN NUMBER AND VALLEY-CHERN NUMBER The Chern number is obtained by the integration over the whole Brillouin zone. We illustrate the Berry curvature F(k)o f the highest unoccupied state in Fig. 2(c). The Berry curvature is found to exhibit sharp peaks at the vortex centers of the spinrotation, which correspond to the tips of the Dirac cones, andbecome zero away from them. Hence, the Chern number isgiven by the sum of the contributions from individual Diraccones. Note that the Berry curvature at the Xpoint is exactly canceled out by the one from the other occupied band, anddoes not contribute to the Chern number. In the vicinity of the /Lambda1 Xpoint we obtain an analytic form for the Berry curvature FX(k) by using the low-energy 195413-3MOTOHIKO EZAWA PHYSICAL REVIEW B 89, 195413 (2014) Hamiltonian ( 8), FX(k)=˜mX˜v1˜v2/parenleftbig ˜v2 1k2x+˜v2 2k2y+˜m2 X/parenrightbig3/2. (13) The Chern number is explicitly calculated as CX=1 2π/integraldisplay F(k)dk=1 2sgn( ˜mX)=1 2sgn(mX),(14) which is associated with the Dirac cone at the /Lambda1Xpoint. The similar formulas are derived for C/prime X,CY, andC/prime Ywith the use of m/prime X,mY, andm/prime Yfor the Dirac cones at the /Lambda1/prime X,/Lambda1Y, and/Lambda1/prime Y points, respectively. At low energy there are four Dirac Hamiltonians such as ( 8) and ( 12), each of which describes a Dirac cone possessing a deﬁnite Chern number depending on the sign of the Diracmass. Hence there are four Chern numbers. The genuine Chernnumber is their sum, C=C X+C/prime X+CY+C/prime Y. (15) This is a genuine topological number. In addition, there are three valley-Chern numbers [ 32], w h i c hw em a yt a k ea s C1=CX+C/prime X−CY−C/prime Y, (16a) C2=CX−C/prime X+CY−C/prime Y, (16b) C3=CX−C/prime X−CY+C/prime Y. (16c) They are symmetry-protected topological numbers. The rele- vant symmetry is the valley symmetry, which is the permu-tation symmetry of Dirac valleys. This is a good symmetrynear the Fermi level, since the system is described by fourDirac Hamiltonians independent of each other. However, athigher energy, the system is described by the tight-bindingHamiltonian, containing intervalley hoppings, where there isno valley symmetry. If we treat the four masses independently there are 16 topological states indexed by ( C,C 1,C2,C3). However, when there are constraints on them, they read as follows: (1) When we apply only the exchange ﬁeld ( /Delta1mX= /Delta1mY=0), we ﬁnd ±(2,0,0,0) with CX=CY=C/prime X=C/prime Y. (2) When we apply only the strain ( m=0), we ﬁnd ±(0,0,2,0) with CX=CY=−C/prime X=−C/prime Yfor/Delta1mX/Delta1mY> 0, and ±(0,0,0,2) with CX=−CY=−C/prime X=C/prime Yfor /Delta1mX/Delta1mY<0. (3) When we apply both the exchange ﬁeld and the strain to the crystal, we ﬁnd ±(1,−1,1,1) for /Delta1mX>m> 0 and m>/Delta1 m Y>0. There are some other cases depending on m,/Delta1mX, and /Delta1mY. We have found that the Chern number may take values 2,1,0,−1,−2. Even if it is zero, the state is topological with respect to the valley-Chern numbers. V . OPTICAL ABSORPTION AND ELLIPTIC DICHROISM An interesting experiment to probe and manipulate the valley degree of freedom is to employ the optical absorp-tion [ 4,5,8–10,12]. It is possible to excite massive Dirac electrons by the right or left circularly polarized light, known ascircular dichroism. Originally, circular dichroism is proposedin honeycomb systems, where the velocities of the Dirac conesare isotropic. On the other hand, they are anisotropic in the TCI surface. This leads to the elliptic dichroism, where the opticalabsorptions are different between the right and left ellipticallypolarized lights. Furthermore, the optical absorptions dependcrucially on the sign of the Dirac mass. A. Kubo formula We explore optical interband transitions from the state \|uv(/tildewidek)/angbracketrightin the valence band to the state \|uc(/tildewidek)/angbracketrightin the conduction band. The fundamental transition is a transitionfrom the highest occupied band to the lowest unoccupied band(Fig. 1). We inject a beam of elliptical polarized light onto the TCI surface. The corresponding electromagnetic potential isgiven by A(t)=(A xsinωt,A ycosωt). The electromagnetic potential is introduced into the Hamiltonian by way of theminimal substitution, that is, by replacing the momentum /tildewidek i with the covariant momentum Pi≡/tildewideki+eAi. The resultant Hamiltonian simply reads H(A)=H+PxAx+PyAy, with Px=∂H ∂/tildewidekx,Py=∂H ∂/tildewideky, (17) in the linear response theory. The optical absorption is governed by the Fermi golden rule. Namely, the imaginary part of the dielectric function arises dueto interband absorption, and is given by the Kubo formula. In the case of elliptical polarized light it reads [ 4] ε θ(ω)=πe2 ε0m2eω2/summationdisplay i/integraldisplay BZd/tildewidek (2π)2f(/tildewidek)\|Pθ(/tildewidek)\|2 ×δ[Ec(/tildewidek)−Ev(/tildewidek)−ω], (18) with the use of the optical matrix element Pθ(/tildewidek), where Ec(/tildewidek) andEv(/tildewidek) are the energies of the conduction and valence bands, while f(/tildewidek) is the Fermi distribution function. The coupling strength with optical ﬁelds is given by the opticalmatrix element between the initial and ﬁnal states in thephotoemission process [ 4,5,8,9], P i(/tildewidek)≡m0/angbracketleftuc(/tildewidek)\|∂H ∂/tildewideki\|uv(/tildewidek)/angbracketright, (19) which is the interband matrix element of the canonical mo- mentum operator. The optical matrix element for ellipticallypolarized light is P θ(/tildewidek)=Px(/tildewidek) cosθ+iPy(/tildewidek)s i nθ, (20) where θis the ellipticity of the injected beam. We call it the right-polarized light for 0 <θ<π and the left one for −π<θ< 0. B. Optical absorption at the Dirac point We ﬁrst investigate optical interband transitions from the valence-band tops to the conduction band bottoms, i.e., at theDirac point. By adjusting the energy of light to the band edge,namely, at /tildewidek=0, ω=E c(0)−Ev(0)=2\|˜m\|, (21) 195413-4V ALLEYTRONICS ON THE SURFACE OF A TOPOLOGICAL . . . PHYSICAL REVIEW B 89, 195413 (2014) we ﬁnd εθ(2\|˜m\|)=πe2 4ε0m2e˜m2\|Pθ(0)\|2(22) at each Dirac point, where ˜mcan be any of ˜mX,˜m/prime X,˜mY,˜m/prime Y. It follows that \|Pθ(0)\|2can be directly observed by optical absorption. The wave functions \|uv(/tildewidek)/angbracketrightand\|uc(/tildewidek)/angbracketrightare obtained explicitly by diagonalizing Eq. ( 2), and we have Px(0)=˜v1,P y(0)=−i˜v2sgn[mX]. (23) It is possible to derive an explicit form of \|P± θ(0)\|2 /Lambda1Xat the/Lambda1X point for arbitrary ellipticity θas \|Pθ(0)\|2 /Lambda1X=m2 0(˜v1cosθ+sgn[mX]˜v2sinθ)2. (24) Similar formulas follow at the other Dirac points. By introducing tanφX=˜v1/˜v2,tanφY=˜v2/˜v1, (25) we rewrite them as \|Pθ(0)\|2 /Lambda1X=m2 0/parenleftbig ˜v2 1+˜v2 2/parenrightbig sin2(φX+sgn[mX]θ),(26a) \|Pθ(0)\|2 /Lambda1/prime X=m2 0/parenleftbig ˜v2 1+˜v2 2/parenrightbig sin2(φX+sgn[m/prime X]θ),(26b) and \|Pθ(0)\|2 /Lambda1Y=m2 0/parenleftbig ˜v2 1+˜v2 2/parenrightbig sin2(φY+sgn[mY]θ),(26c) \|Pθ(0)\|2 /Lambda1/prime Y=m2 0/parenleftbig ˜v2 1+˜v2 2/parenrightbig sin2(φY+sgn[m/prime Y]θ).(26d) We note that φX=0.317π, φ Y=0.183π (27) forv1=1.3e V ,v2=2.4 eV , and that φX+φY=π 2(modπ). (28) There are four functions with the same amplitude in general: See Fig. 3(a). The function (red solid curve) involving \|sin(φX+θ)\|2is the main one. The function (blue solid curve) involving \|sin(φY+θ)\|2is constructed by sifting it so that ( 28) holds. The other two functions (dotted curves) are constructedby changing θ→−θ. For instance, when all masses are positive such as in the case of the exchange effect, it follows that \|P θ(0)\|2 /Lambda1X=\|Pθ(0)\|2 /Lambda1/prime X, as is shown in the red solid lines in Fig. 3(a). It also follows that\|Pθ(0)\|2 /Lambda1Y=\|Pθ(0)\|2 /Lambda1/prime Y, as is shown in blue solid curves in Fig.3(a). For instance, when mXm/prime X<0 and mYm/prime Y<0 such as in the case of the strain effect, it follows that \|Pθ(0)\|2 /Lambda1X= \|P−θ(0)\|2 /Lambda1/prime Xand\|Pθ(0)\|2 /Lambda1Y=\|P−θ(0)\|2 /Lambda1/prime Y. Thus, if mX>0 and mY>0, they are described by the same solid curves at the /Lambda1Xand/Lambda1Ypoints but by the dotted curves at the /Lambda1/prime Xand/Lambda1/prime Y points in Fig. 3(a). A perfect elliptic dichroism is a phenomenon that only one-handed elliptically polarized light is absorbed. It occurs atθ=−φ Xfor the function \|sin(φX+θ)\|2. At the same point the function \|sin(φY+θ)\|2takes the maximum value. More explicitly they occur as θ=θXat the/Lambda1Xpoint and so on, with θX=−sgn[mX]φX,θ/prime X=−sgn[m/prime X]φX, θY=−sgn[mY]φY,θ/prime X=−sgn[m/prime Y]φY. (29)FIG. 3. (Color online) (a) Optical matrix element \|P±θ\|2at the /Lambda1Xand/Lambda1Ypoints with various ellipticity θ[Eq. ( 26)]. Red (blue) solid curves are optical absorption \|Pθ\|2at theX(Y) point, and dotted curves are for \|P−θ\|2. (b) Illustration of optical absorption \|Pθ\|2at (b1)θ=θX, (b2) θ=−θX, (b3) θ=θY, and (b4) θ=−θY.T h e magnitude of arrows indicates the magnitude of optical absorption. We have assumed that all four masses have positive values. We give an example in Fig. 3(a) when all the masses are positive, where θ/prime X=θXandθ/prime X=θX. We have studied analytically the optical matrix element \|Pθ(/tildewidek)\|at the Dirac point. Next we investigate it away from the Dirac point. An analytic solution of the optical matrix elementof right and left elliptically polarized light \|P θ(/tildewidek)\|2is obtained from Eq. ( 2). However, the expression is very complicated. We show the result in Fig. 4atθ=θX, which shows the low-energy Dirac theory captures the essential features. Thereare sharp peaks in optical absorption near the /Lambda1 X(/Lambda1/prime X) points. Figure 4(a) shows the optical matrix element \|PθX(/tildewidek)\|2and \|P−θX(/tildewidek)\|2along the /tildewidekxaxis. We clearly see the difference between the right- and left-polarized lights at the /Lambda1X(/Lambda1/prime X) point. There is large optical absorption in right-polarized light,while no optical absorption in left-polarized light. This is a FIG. 4. (Color online) (a) Optical matrix element \|PθX(/tildewidek)\|2 (red curve) and \|P−θX(/tildewidek)\|2(blue curve) along the /tildewidekxaxis. (b) /tildewidek-resolved optical polarization η(/tildewidek). It has two sharp peaks at the /Lambda1Xand/Lambda1/prime Xpoints. We have taken ˜mX=˜mY=2m e V . 195413-5MOTOHIKO EZAWA PHYSICAL REVIEW B 89, 195413 (2014) dichroism caused by elliptically polarized light, and the key feature of the elliptic dichroism. C. Optical absorption away from the Dirac point We proceed to drive the analytic expression of \|Pθ(/tildewidek)\| away from the /Lambda1Xpoint with the use of the low-energy Hamiltonian ( 8)i n( 19). It is straightforward to ﬁnd that Px(/tildewidek)=˜v1˜v1/tildewidekx˜mX+i˜v2/tildewideky/radicalBig ˜m2 X+˜v2 1/tildewidek2x+˜v2 2/tildewidek2y/radicalBig ˜v2 1/tildewidek2x+˜v2 2/tildewidek2y/radicalBig ˜m2 X+˜v2 1/tildewidek2x+˜v2 2/tildewidek2y,(30a) Py(/tildewidek)=˜v2˜v2/tildewideky˜mX−i˜v1/tildewidekx/radicalBig ˜m2 X+˜v2 1/tildewidek2x+˜v2 2/tildewidek2y/radicalBig ˜v2 1/tildewidek2x+˜v2 2/tildewidek2y/radicalBig ˜m2 X+˜v2 1/tildewidek2x+˜v2 2/tildewidek2y,(30b) sincePx=˜v1σxandPy=˜v2σy.A tθ=θX, it yields a simple form, \|PθX(/tildewidek)\|2=m2 0˜v1˜v2/parenleftbig ±˜mX+/radicalBig ˜m2 X+˜v2 1/tildewidek2x+˜v2 2/tildewidek2y/parenrightbig2 ˜m2 X+˜v2 1/tildewidek2x+˜v2 2/tildewidek2y.(31) We derive the same formula away from the /Lambda1/prime Xpoint just replacing ˜mXwith ˜m/prime X. Similar formulas are derived also with respect to the /Lambda1Yand/Lambda1/prime Ypoints. Representing ( 31) in terms of the energy ( 11), we obtain \|Pθ(/tildewidek)\|2=m2 0˜v1˜v2[±˜m+Ev(/tildewidek)]2 [Ev(/tildewidek)]2, (32) atθ=θX(Y)with the use of ˜m=˜mX(Y), andθ=θ/prime X(Y)with the use of ˜m=˜m/prime X(Y), where we have used the relation εv(/tildewidek)= −εc(/tildewidek) required by the electron-hole symmetry of the energy spectrum.We substitute ( 32)t o( 18) and use the density of state ρ(E)=\|E\| 2π˜v1˜v2/Theta1(E−2\|˜m\|), (33) with the step function /Theta1(x)=1f o rx> 0 and /Theta1(x)=0f o r x< 0, to ﬁnd ε±(ω)=e2m2 0 2ε0m2eω(±˜m+ω/2)2 (/planckover2pi1ω/2)2/Theta1(ω−2\|˜m\|). (34) Hence there is no optical absorption for ω=∓2˜m> 0. (35) We show the optical absorption ( 34)i nF i g . 5. A clear difference is observed between the right- and left-polarizedlights. There is almost no optical absorption for left-polarizedlight for ω> 2\|˜m\|.H e r e ˜mstands for any of ˜m X,˜m/prime X, and ˜m/prime Y. A perfect elliptic dichroism follows that \|PθX(0)\|2=0i f mX>0, while \|P−θX(0)\|2=0i fmX<0. The anisotropy of the Dirac cone is determined by measuring the ellipticityangleθ Xof the injected beam: See Fig. 3. We would expect θX=0.317πas in ( 27). We can also determine the band gap by measuring the energy where the optical absorptionbecomes nonzero ( 34): See Fig. 5. The role of the right- and left-polarized light is inverted when the sign of the mass termis negative. Thus we can determine the sign of the mass termFIG. 5. (Color online) Imaginary part of dielectric function ε±(ω) due to interband absorptions at θ=θX:S e eE q .( 34). A clear difference is observed between the right- and left-polarizedlights. There is almost no optical absorption for left-polarized light forω> 2˜m. We have taken ˜m> 0 for deﬁnateness. by the elliptic dichroism even when the magnitude of the mass term is very small. D. Optical polarization We next investigate the k-resolved optical polarization ηθ(/tildewidek), which is given by [ 4,5,8,9] ηθ(/tildewidek)=\|Pθ(/tildewidek)\|2−\|P−θ(/tildewidek)\|2 \|Pθ(/tildewidek)\|2+\|P−θ(/tildewidek)\|2, (36) w h i c hw es h o wi nF i g . 4(b). This quantity is the difference between the absorption of the left- and right-handed lights(±θ), normalized by the total absorption, around the /Lambda1 Xpoint. Optical polarizations are perfectly polarized at the /Lambda1Xand/Lambda1/prime X points ( /tildewidek=0). Namely, the selection rule holds exactly at the /Lambda1Xand/Lambda1/prime Xpoints. Then, \|ηθ(/tildewidek)\|rapidly decreases to 0 as \|/tildewidek\| increases. E. Valley-selective optical pumping An interesting valleytronics application of the elliptic dichroism would read as follows. Let us adjust the ellipticityof light at θ=θ Xso that the optical absorption near the /Lambda1X point does not occur [Fig. 3(b1)]. Then the optical absorption is not zero at the /Lambda1Ypoint. Namely, we can selectively excite electrons at the /Lambda1Ypoint by left-polarized light. It is a valley-selective optical pumping. In the same way, by adjustingθ=θ Y, we can selectively excite electrons at the /Lambda1Xpoint by left-polarized light [Fig. 3(b3)]. The valley-selective optical pumping is possible since the anisotropy of Dirac cones at /Lambda1X and/Lambda1Ypoints are different. If the Dirac cones were isotropic, we could not differentiate the Dirac cones at /Lambda1Xand/Lambda1Y points since they have the same chirality. This will pave a way to valleytronics in the TCI. VI. CONCLUSIONS We have investigated the optical absorption on the TCI surface when gaps are given to surface Dirac cones. First,the chiralities of all four Dirac cones are identical, whichcan be veriﬁed by studying the spin direction. Nevertheless,it is possible to make a selective excitation between the/Lambda1 X(/Lambda1/prime X) point and the /Lambda1Y(/Lambda1/prime Y) point, because the Dirac cones are anisotropic, where ˜ v2/˜v1=0.65. Furthermore, it 195413-6V ALLEYTRONICS ON THE SURFACE OF A TOPOLOGICAL . . . PHYSICAL REVIEW B 89, 195413 (2014) is also possible to make a selective excitation between the /Lambda1Xand/Lambda1/prime Xpoints when the Dirac masses ˜mXand ˜m/prime X have the opposite signs. Namely, by tuning the ellipticity of the polarized light, we can realize a perfect ellipticdichroism, where only electrons at one valley are excited.Our results will pave a road toward valleytronics based onthe TCI.ACKNOWLEDGMENTS I am very much grateful to N. Nagaosa, Y . Ando, L. Fu, and T. H. Hsieh for many helpful discussions on the subject. Thiswork was supported in part by Grants-in-Aid for ScientiﬁcResearch from the Ministry of Education, Science, Sports andCulture No. 22740196. [1] A. Rycerz, J. 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PhysRevB.92.165104.pdf	PHYSICAL REVIEW B 92, 165104 (2015) Topological superconducting states in monolayer FeSe/SrTiO 3 Ningning Hao and Shun-Qing Shen Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China (Received 2 May 2015; published 5 October 2015) The monolayer FeSe with a thickness of one unit cell grown on a single-crystal SrTiO 3substrate (FeSe/STO) exhibits striking high-temperature superconductivity with transition temperature Tcover 65 K reported by recent experimental measurements. In this work, through analyzing the distinctive electronic structure, and providingsystematic classiﬁcation of the pairing symmetry, we ﬁnd that both s-a n dp-wave pairing with odd parity give rise to topological superconducting states in monolayer FeSe, and the exotic properties of s-wave topological superconducting states have close relations with the unique nonsymmorphic lattice structure which induces theorbital-momentum locking. Our results indicate that the monolayer FeSe could be in the topological nontrivials-wave superconducting states if the relevant effective pairing interactions are dominant in comparison with other candidates. DOI: 10.1103/PhysRevB.92.165104 PACS number(s): 74 .78.−w,74.20.Rp,74.62.Dh,74.70.Xa I. INTRODUCTION Topological superconductors [ 1–4] and iron-based super- conductors [ 5] have been research focuses of condensed matter physics in recent years. Topological superconductorshave a full pairing gap in the bulk and gapless surface oredge Andreev bound states known as Majorana fermions.Recent scanning tunneling microscopy/spectroscopy (STM/S)measurements observed a robust zero-energy bound stateat randomly distributed interstitial excess Fe sites in su-perconducting Fe(Te,Se), and the behavior of zero-energybound state resembles the Majorana fermion [ 6]. Theoret- ically, one possible scenario accounting for this puzzle isthat Fe(Te,Se) could be in a topological superconducting(SC) state. If it is the case, we can expect that nontrivialtopology can integrate into the SC states in iron-basedsuperconductors. Recently, some studies [ 7,8] have revealed that the band structures can be tuned to have nontrivial topological prop-erties in monolayer Fe(Te,Se) and monolayer FeSe/STO.Furthermore, in electron-doped monolayer FeSe/STO, theexperimental measurements have observed high temperaturesuperconductivity with T cover 65 K [ 9–16]. In analogy to the doped topological insulators, which are strongly believedto be topological superconductors [ 4,17–19], a natural ques- tion arises, can the electron-doped monolayer FeSe/STO betopological superconductors? In this paper we propose that the electron-doped monolayer FeSe/STO could be an odd-parity topological superconductorin the spin-triplet orbital-singlet s-wave pairing channel [ 20]. To show this exotic state, we ﬁrst analyze the distinctiveelectronic structure of monolayer FeSe/STO, and present asystematic classiﬁcation of the pairing symmetry in monolayerFeSe/STO from the lattice symmetric group. Second, wediscuss the topological properties of such odd-parity SC states,and extract the minimum effective models to capture theessential physics. Third, we calculate the phase diagram ofSC states according to different scenarios of effective pairinginteraction. Finally, we discuss the experimental signatures ofthe topological SC states.II. PAIRING SYMMETRY CLASSIFICATIONS The lattice structure of monolayer FeSe is shown in Fig. 1(a). The two-Fe unit cell includes two Se and two Fe labeled by A and B. The space group P4/nmm governs the Se-Fe-Se trilayer structure, and belongs to a nonsymmorphicgroup [ 21–24]. Indeed, there exists a n-glide plane described by the operator {m z\|1 21 2}, which involves a fractional transla- tion (1 21 2) combining with the ab-plane mirror. Centered on an Fe atom [see Fig. 1(a)], eight point group operations E,2S4, c2(z),c2(x),c2(y), and 2 σdform a D2dpoint group. Together with an inversion followed by fractional translations (1 21 2), i.e., {i\|1 21 2}, they generate all the elements of P4/nmm . The 16 operations do not form a point group. However, if the fractionaltranslation ( 1 21 2) is stripped off, the 16 operations form a point group, which indeed is D4h. It is convenient to classify the pairing symmetry with the irreducible representation (IR) ofD 4h. For this purpose, one simple way is to recompose the Bloch wave functions in the one-Fe Brillouin zone (BZ). The glide plane symmetry {mz\|1 21 2}divides the ﬁve d orbitals into two groups, ( dxz,dyz) and ( dxy,dx2−y2,dz2), and each group is recomposed to be the eigenstates of the glideplane operation with the deﬁnite orbital parities. The tight-binding Hamiltonian can also be decomposed into two partswith inverse orbital parities, which allow us to transfer thetwo-Fe unit cell picture into a one-Fe unit cell picture [ 21–23]. In momentum space, the tight-binding Hamiltonian in a one-Feunit cell picture can be written as H 0=/summationdisplay k,σψo† σ(k)Ao(k)ψo σ(k)+/summationdisplay k,σψe† σ(k)Ae(k)ψe σ(k).(1) Here the ﬁrst/second term has odd/even orbital parity under the glide plane operation. ψo σ(k)=[dxz,σ(k),dyz,σ(k),dx2−y2,σ(k), dxy,σ(k),dz2,σ(k)]Twithdm,σ(k) denoting the electron anni- hilation operator at the mth orbital with momentum kand spinσ.ψe σ(k)=ψo σ(k+Q) and Ae(k)=Ao(k+Q) with Q=(π,π) (see Appendix Afor details). The energy spectra from Eq. ( 1)a r es h o w ni nF i g . 1, in which Fig. 1(e) is con- sistent with observations of the angle-resolved photoemission 1098-0121/2015/92(16)/165104(10) 165104-1 ©2015 American Physical SocietyNINGNING HAO AND SHUN-QING SHEN PHYSICAL REVIEW B 92, 165104 (2015) FIG. 1. (Color online) (a) The Se-Fe-Se trilayer structure. The black/green balls with deep and light ﬁlling label Fe/Se atoms. Here the deep/light ﬁlling Se atoms are above/below the Fe plane. The red/black dashed squares label the one-Fe/two-Fe unit cells. (b) The Fermi surface of monolayer FeSe/STO is schematically illustrated. The red/blue electron pockets have odd/even orbital parity. The red/blackdashed squares label the one-Fe/two-Fe Brillouin zone. The evolution of the band structure from (c) the free-standing monolayer FeSe to (d) monolayer FeSe/STO with small tensile strain, and to (e) monolayerFeSe/STO with large tensile strain. The red/blue color labels the spectrum with odd/even orbital parity. spectroscopy (ARPES) [ 10,11], and the chemical potential is set to satisfy that 10% electrons is doped per Fe clariﬁedby experiments [ 10–12]. The fundamental difference between Figs. 1(c) and1(f)is referred to the band-renormalization effect induced by the strain from the STO substrate, which stronglymodulates the hopping parameters between the ( d xz,dyz,dxy) orbitals and switches the positions of two doubly degeneratepoints M 1andM3at theMxhigh symmetric point, where the M1point mainly has ( dxz,dyz) orbital weight and the M3point mainly has dxyorbital weight. This picture is the most natural and simplest to account for the distinctive electronic structureof monolayer FeSe/STO compared to other scenarios [ 25–27]. The SC order parameters should follow the IRs of the symmetry group of the system. It is safe to use D 4hto do so in the picture of one-Fe unit cell according to our aforemen-tioned arguments. There exist two kinds of symmetry-allowedCooper pairs, i.e., ( k,−k) and ( k,−k+Q) pairing channels. Previously, the ( k,−k+Q) pairing channels are proposed to coexist with ( k,−k) pairing channels to explain the nodeless and sign-change gap structures in iron-based superconduc-tors [ 21,22]. The price for coexistence of both kinds of pairings is that the orbital parities are mixed and the spatial inversionsymmetry is broken. Here we focus on an SC state with onlyone IR in the ( k,−k) pairing channel and leave to discuss the irrelevant ( k,−k+Q) pairing channel in Appendix B. Moreover, we only need to consider the pairings between thethreet 2gorbitals as the orbital weight for Egorbitals are neglectable on the Fermi surfaces [ 28]. Deﬁne the Nambu basis, /Psi1(k)=[{d↑(k)},{d↓(k)},{d† ↓(−k)},{−d† ↑(−k)}]Twith {dσ(k)}={dxz,σ(k),dyz,σ(k),dxy,σ(k)}. The pairing term in theTABLE I. The IRs of all the possible on-site superconducting pairing in ( k,−k) channels. Here η1/4=∓1 3(λ0+2√ 3λ8)a n dη2/3= 1 3(∓λ0±√ 3λ8∓3λ3/1). (k,−k): /Delta1(k) c2(z) c2(x) σd/braceleftbig i/vextendsingle/vextendsingle1 21 2/bracerightbig IR −iszη1−isxη2−i(sx−sy)η3√ 2s0η4 s0λ0 11 1 1 A(1) 1g s0λ8 11 1 1 A1g s0λ1 1 −11 1 B2g s0(λ4,λ6)( −1,−1) (1 ,−1) s0(λ6,λ4)( −1,−1)Eu iszλ2 11 1 1 A1g sz(λ5,λ7)( −1,−1) (−1,1) −sz(λ7,λ5)(−1,−1)E(1) u i(sx,sy)λ2 (−1,−1) (−1,1) i(sy,sx)λ2 (1,1) Eg i(sxλ5,syλ7) (1,1) (1,1) −i(syλ7,sxλ5)(−1,−1)E(2) u i(syλ5,sxλ7) (1,1) ( −1,−1)−i(sxλ7,syλ5)(−1,−1)E(2/prime) u Bogoliubov–de Gennes (BdG) Hamiltonian can be expressed as Hp=/summationdisplay k/Psi1†(k)/Delta1(k)τx/Psi1(k). (2) Hereτxis one Pauli matrix in Nambu space, and /Delta1(k)i sa 6×6 matrix. Our purpose is to identify the exact form of /Delta1(k). For convenience, we utilize four Pauli matrices ( s0,sx,sy,sz)t o span spin space and nine Gell-Mann matrices ( λ0,..., λ 8)( s e e Appendix Bfor deﬁnitions of Gell-Mann matrices) to span orbital space. In such a way, /Delta1(k) can be decomposed into the product of the Pauli matrices and Gell-Mann matrices, i.e.,/Delta1(k)=f(k)s mλn, in which f(k) is the pairing form factor. We summarize all the possibilities of the ( k,−k) on-site pairing channels according to the IRs of D4hin Table Iand non-on-site pairing channels up to the next-nearest neighbor in Table II. TABLE II. The IRs of all the possible nearest and next- nearest neighbor superconducting pairing in ( k,−k) channels. Here f1/2(k)=coskx±cosky;f4(k)=coskxcosky;[f3(kx),f3(ky)]= [sinkx,sinky];f5(k)=sinkxsinky. (k,−k):/Delta1(k)I R f1/4(k)s0λ0/8,f5(k)s0λ1,f3(kx)s0λ5+f3(ky)s0λ7 A(2) 1g f2(k)s0λ0/8,f3(kx)s0λ5−f3(ky)s0λ7 B(1) 1g f2(k)s0λ1,f3(ky)s0λ5−f3(kx)s0λ7 A2g f5(k)s0λ0/8,f1/4(k)s0λ1,f3(ky)s0λ5+f3(kx)s0λ7 B2g if1/4(k)szλ2,i1/0/0[f3(kx)sz/x/yλ4+if3(ky)sz/y/xλ6] A1g if2(k)szλ2,i1/0/0[f3(kx)sz/x/yλ4−if3(ky)sz/y/xλ6] B1g i1/0/0[f3(ky)sz/x/yλ4−f3(kx)sz/y/xλ6] A2g if5(k)szλ2,i1/0/0[f3(ky)sz/x/yλ4+f3(kx)sz/y/xλ6] B2g if1/2/4/5(k)(sx,sy)λ2 Eg f3(kx)sx/yλ0±f3(ky)sy/xλ0 A(1) 1u [f3(kx),f3(ky)]szλ0 E(3) u 165104-2TOPOLOGICAL SUPERCONDUCTING STATES IN . . . PHYSICAL REVIEW B 92, 165104 (2015) In both Tables Iand IIthe spin-singlet/spin-triplet pairing channels are listed in the ﬁrst/second parts. III. TOPOLOGICAL SUPERCONDUCTING STATES To evaluate the pairing channels that could support the topological SC states, we ﬁrst impose the nodeless gapstructure restrictions to the pairing channels in Tables IandII according to ARPES and STM/S experimental results [ 9–11], i.e.,A(1) 1g,E(1) u,E(2) u, and E(2/prime) uin Table IandA(1) 1gwith f4(k)s0λ0,B(1) 1g,A(1) 1u, andE(3) uin Table II. Second, we focus on the odd-parity pairing channels based on the proposalsthat odd-parity pairings usually support the topological SCstates in doped topological insulators [ 4]. Finally, we consider the SC states with the C 4rotation symmetry veriﬁed by both experimental observations [ 10–13] and our calculations in Sec. IV. This constraint forces the time-reversal (TR) symmetry to be broken spontaneously for some Eustates. With all the above constraints and a turn to the monolayerFeSe/STO, four possible odd-parity pairing states survive: (1)E (1) u, a doubly degenerate TR breaking state with /Delta11(k)= /Delta10sz(λ5±iλ7), (2)E(2) u, a TR invariant state with /Delta12(k)= /Delta10i(sxλ5+syλ7) (note that E(2/prime) uis equivalent to E(2) u), (3) E(3) u, a doubly degenerate TR breaking state with /Delta13(k)= /Delta10[f3(kx)±if3(ky)]szλ0, and (4) A(1) 1u, a TR invariant state with/Delta14(k)=/Delta10[f3(kx)sxλ0+f3(ky)syλ0] [note that all four components in {A(1) 1u:f3(kx)sx/yλ0±f3(ky)sy/xλ0}are equivalent]. Through the bulk-boundary correspondence, wedemonstrate that all these four kinds of odd-parity pair-ing channels support topological SC states in monolayerFeSe/STO. The BdG Hamiltonian describing the SC statescan be obtained by combining the tight-binding HamiltonianH 0in Eq. ( 1) and pairing term Hpin Eq. ( 2), i.e., HBdG=H0+Hp. (3) Note that HBdG in Eq. ( 3) includes both odd-orbital-parity and even-orbital-parity parts. The edge spectra from theodd-orbital-parity parts of H BdG with/Delta11(k)···/Delta14(k)a r e presented in Fig. 2. The even-orbital-parity parts of HBdG give the same spectra if kyis translated to ky+π[see Fig. 1(b) for comparison]. The edge spectra in Fig. 2explicitly support the Andreev bound states which are the identiﬁcationsof topological superconductors. Besides, the bulk propertiesof topological superconductors are usually characterized bysome topological numbers. Here the pairing channels with/Delta1 1(k) and /Delta13(k) break the TR symmetry, and the Chern number [ 29] can be introduced to characterize such two states, i.e., C=i 2π/summationtext En<0/integraltext BZdk/angbracketleft∇kun(k)\|×\| ∇ kun(k)/angbracketright.T h e calculations show that both odd-orbital-parity and even-orbital-parity parts give the Chern numbers C o=Ce=4i n the one-Fe BZ for /Delta11(k) and/Delta13(k) pairing channels. Thus, two such pairing channels are characterized by the total Chernnumber C= 1 2(Co+Ce)=4 in the two-Fe BZ. The Chern number C=4 is equal to the number of edge Andreev bound states shown in Figs. 2(a) and 2(d). For the TR invariant /Delta12(k) and/Delta14(k) pairing channels, the total Chern numbers are zero. However, the spin Chern numbers [ 30,31] can be introduced to characterize the bulk topological propertiesof SC states in /Delta1 2(k)o r/Delta14(k) pairing channels. Namely, FIG. 2. (Color online) The edge spectra of odd-orbital-parity BdG Hamiltonian with /Delta11(k),/Delta12(k),/Delta13(k), and /Delta14(k)i n( a ) ,( b ) , (d), and (e). In the presence of the orbital-parity-broken perturbation, i.e., the staggered potential of Fe sublattices, the edge spectra of BdG Hamiltonian with /Delta12(k)a n d/Delta14(k) are shown in (c) and (f). Here the system has a periodic boundary condition along the ydirection and an open boundary condition along the xdirection with 51 one-Fe unit cell lengths. The red/blue colors label the edge states localizing at theopposite boundaries, and the dashed/solid lines label the edge states with up/down spin directions. Note that the degenerate edge states on the same edge are artiﬁcially split as a guide for the eye. Co/e ↑=1,Co/e ↓=− 1 in the two-Fe BZ. Correspondingly, two Z2topological numbers [ 32] with opposite orbital parities deﬁned by vo/e=1 2(Co/e ↑−Co/e ↓)=1 characterize the bulk topological properties for SC states in /Delta12(k)o r/Delta14(k) pairing channels. Having conﬁrmed that the topological SC states emerge in the nodeless odd-parity pairing channels, we notice that theedge spectra shown in Figs. 2(a) and2(b) and the edge spectra shown in Figs. 2(d) and2(e) are very different. Therefore, it is necessary to extract the minimum effective models to clarifythe essential physics hidden behind. First, we are aware of the/Delta1 3/4(k) pairing channels being in the intraorbital spin-triplet p-wave pairing channels. Thus, the orbital degree of freedom is inessential, and the minimum effective Hamiltonian canbe reduced into the single band space, which is the sameHamiltonian to describe the well-known p±iptopologi- cal superconductors/superﬂuids [ 1,33,34], and the nontrivial topology is referred to the p±ippairing terms. Therefore, we omit our discussions for these “trivial” topological SC states. For/Delta1 1(k) and/Delta12(k), which are the interorbital spin-triplet s-wave pairing channels, the three t2gorbitals are involved and entangled with each other not only in the bands aroundthe Fermi surface shown in Fig. 3(a), but in the pairing terms shown in Fig. 3(d). Note that we should have three bands when we consider three t 2gorbitals. It indicates that the third band mainly with the dxzanddyzweight has to strongly couple with two egorbitals and be gapped and pushed away from the Fermi level. In order to describe the twobands in an exact three orbital basis, we adopt the angularmomentum representation characterized by the azimuthal 165104-3NINGNING HAO AND SHUN-QING SHEN PHYSICAL REVIEW B 92, 165104 (2015) FIG. 3. (Color online) (a) The weight of three t2galong the Fermi surface around Mywith odd-orbital parity. (b) and (c) The effective band dispersions without/with interorbital coupling from the glide plane. (d) Three competitive pairing channels with ϕ=π 2in weak- coupling limit. and magnetic quantum numbers landm. The new electron creation operators are d† (lm=2,±1),σ(k)=∓1√ 2[d† xz,σ(k)± id† yz,σ(k)], then we have ˆ/Delta1† 1(k)∼[d† (2,1),↑(k)d† xy,↓(−k)+ d† (2,1),↓(k)d† xy,↑(−k)] and ˆ/Delta1† 2(k)∼[d† (2,−1),↑(k)d† xy,↑(−k)+ d† (2,1),↓(k)d† xy,↓(−k)]. Now we can only exploit the operators involved in ˆ/Delta11/2(k) to construct the basis to write the minimum effective Hamiltonian, and this approximation is equivalent totreating d xzanddyzorbitals with equal weights. In the effective basis,/Psi11/2(k)=[{ψ1/2↑(k)},{ψ1/2↓(k)}]Twith{ψ1/2,σ(k)}= {d[2,1/−(−1)σ],σ(k),dxy,σ(k),d† xy,¯σ/σ(−k),−d† [2,1/−(−1)σ]¯σ(−k)}, H(1/2)(k)=H(1/2) 1(k)⊕H(1/2) 2(k). (4) Here kis measured from the Mpoint. ¯ σ=−σand (−1)σ= 1/−1f o rs p i n ↓/↑, the orbital parity index is omitted for simplicity. H(1/2) 1(k)=τz[d(1/2) 0(k)+/summationtextz i=xd(1/2) i(k)σi]+ τx/Delta10,H(1) 2(k)=H(1) 1(k) and H(2) 2(k)=H(2)∗ 1(−k). The three Pauli matrices σ1/2/3are introduced to span the effective two-band space. d(1/2) 0(k)=ε1(k)+ε2(k) 2−μ,d(1/2) x(k)=∓Aky, d(1/2) y(k)=−Akx, andd(1/2) z(k)=ε1(k)−ε2(k) 2.H(1)(k) breaks TR symmetry, because only m=1 is involved. H(2)(k) is TR invariant, and characterized by the T−1H(2)(k)T= H(2)∗(−k), where the TR symmetry operator is T=isyτ0σ0K with Kthe complex conjugated operator. The disper- sions ε1/2(k) with deﬁnite orbital parity can be read out from Figs. 1(e) and 3(b). Around Mypoint, we have εe 1/2(k)=e1/2−μ+α1/2k2 x+β1/2k2 yandεo 1/2(k)=e1/2− μ+β1/2k2 x+α1/2k2 y. The signs of α/βare crucial to determine the properties of the topological SC states. In Figs. 3(b) and3(c) we schematically illustrate the evolution of the εo 1/2(k) under the couplings induced by the glide plane around My point, and we can ﬁnd e1<e 2,α1<0,β1>0,α2>0,β2< 0. The effective mass measuring the energy gap EM3−EM1 shown in Fig. 1(e) or 3 (c) is m=e2−e1 2>0. The ﬁnite electron-doped condition μ2+/Delta12 0>m2[35] always supportstopological SC states for H(1/2) 1(k), where the chemical potential μis measured from the middle of the gap. The remarkable feature of the edge spectra in Figs. 2(a) and2(b) is that the edge Andreev bound states have a twist (three timesof crossings) around k y=πand only one crossing around ky=0. This difference can be understood with the “orbital mirror helicity” from the mirror operator in c2(x/y) acting on three t2gorbitals in analogy to the “spin mirror helicity” proposed in Ref. [ 35]. The conservation of mirror helicity force the nontwisted/twisted feature of the edge Andreev edgestates under the nonband/band-inversion conditions betweenε e/o 1(k) andεe/o 2(k) along the xdirection, sgn[( e2−e1)(α2− α1)]>0/sgn[( e2−e1)(β2−β1)]<0 [note that εe 1/2(My+ k)=εo 1/2(Mx+k)]. We are aware of the importance of the nonsymmorphic lattice symmetry which not only inducesthe orbital-momentum locking k×σ·ˆzthrough the glide plane, but protects the exotic behaviors of the edge Andreevbound states. We can verify this point through introducing thestaggered on-site potential, which mixes the orbital parities,breaks the nonsymmorphic lattice symmetry, and destroys thetwist feature of the edge spectra. The results are shown inFigs. 2(c) and 2(f). However, the bulk topological properties are robust against such perturbations. IV . THE EFFECTIVE PAIRING INTERACTIONS Although the high temperature interfacial superconductiv- ity in monolayer FeSe/STO seems to have been establishedbeyond doubt, the mechanism for superconductivity is stillan open question [ 36], and the unique features of mono- layer FeSe/STO further pose a higher barrier to block ourunderstanding of the superconductivity from some standardtheories. For example, the monolayer FeSe/STO is strictlytwo dimensional and has no hole pockets at the BZ center,while its three-dimensional counterpart bulk FeSe resem-bles iron-pnictide with hole pockets. The Fermi surface ofmonolayer FeSe/STO is similar to that of A xFe2−ySe2(A= K, Cs, Rb), except that the small electron pocket around(0,0,π)i nA xFe2−ySe2is absent here. In weak coupling limit, the spin-ﬂuctuation-exchange theory predicts that the {B1g: f2(k)s0λ0}pairing channel is dominant in A xFe2−ySe2and the gap structure has nodes along the kzdirection [ 37,38]. However, the ARPES measurements reported isotropic fullgaps without nodes on all pockets in A xFe2−ySe2[39,40]. In the strong coupling limit, the phenomenological t-J model predicts that the {A1g:f4(k)s0λ0}pairing channel is dominant in A xFe2−ySe2and the gaps have same sign for all the pockets [ 41]. However, the inelastic neutron scattering measurements on A xFe2−ySe2reported a resonance with wave vector Qc=(π,π/ 2) in the superconducting state [ 42], which indicated that there existed a sign change between theFermi surfaces connected by Q c. These contradictions strongly question the standard theories. On the other hand, the studiesof some conﬁrmed systems with interfacial superconductivityincluding bilayer lanthanum cuprate [ 43] and LaAlO 3/SrTiO 3 heterostructure [ 44] could provide us some useful insights to understand the superconductivity in monolayer FeSe/STO.The studies of the aforementioned systems indicate that surfacephonon plays a key role to drive the superconductivity [ 45]. A recent ARPES experiment observed the band replication, 165104-4TOPOLOGICAL SUPERCONDUCTING STATES IN . . . PHYSICAL REVIEW B 92, 165104 (2015) which was attributed to strong coupling between the cross phonon and electrons [ 15], and the cooperation between the cross phonon mode and spin ﬂuctuation is argued to be theorigin to enhance T cin monolayer FeSe/STO. Therefore, it is still possible that the superconductivity in monolayerFeSe/STO is driven by the electron-phonon coupling, andthe surface phonon-mediated SC mechanism in monolayerFeSe/STO has been proposed in Ref. [ 46]. Here, without loss of generality, we consider several possibilities of the effectiveinteractions that can drive superconductivity in differentpairing channels and focus on the parameter regime missedpreviously. We ﬁrst assume the multiorbital Hubbard interactions as a pairing driver, H (1) int=U/summationdisplay i,lnil↑nil↓+V/summationdisplay i,l>l/primenilnil/prime +JH/summationdisplay i,l>l/prime/parenleftbigg 2Sil·Sil/prime+1 2nilnil/prime/parenrightbigg +J/prime/summationdisplay i,l/negationslash=l/primed† i,l↑d† i,l↓di,l/prime↓di,l/prime↑. (5) HereU,V,JH,J/primeare the intraorbital, interorbital, Hund’s coupling, and pairing hopping term. l,l/prime∈(xz,yz,xy ), and Sil=1 2d† ilσsσσ/primedilσ. The spin rotation symmetry requires U= V+2JH, andJH=J/primeat the atomic level. Since the predic- tions from the weak-coupling theory [ 37,38] about H0+H(1) int were not consistent with the experimental reports [ 39,40], the strongly correlative picture with quite large JHis possible and the strongly correlative effects in iron chalcogenides have beenreported by recent ARPES experiments [ 47]. Deﬁne the pairing operators ˆ/Delta1 s,ll/prime=/summationdisplay kˆ/Delta1s,ll/prime(k),ˆ/Delta1α t,ll/prime=/summationdisplay kˆ/Delta1α t,ll/prime(k), ˆ/Delta1s,ll/prime(k)=/summationdisplay σσ/prime[isy]σσ/prime 4[dlσ(k)dl/primeσ/prime(−k)+dl/primeσ(k)dlσ/prime(−k)], ˆ/Delta1α t,ll/prime(k)=/summationdisplay σσ/prime[isysα]σσ/prime 4[dlσ(k)dl/primeσ/prime(−k)−dl/primeσ(k)dlσ/prime(−k)]. (6) The interaction Hamiltonian has the form H(1) int=U/summationdisplay lˆ/Delta1† s,llˆ/Delta1s,ll+JH/summationdisplay l/negationslash=l/primeˆ/Delta1† s,llˆ/Delta1s,l/primel/prime +(V−JH)/summationdisplay ll/primeαˆ/Delta1α† t,ll/primeˆ/Delta1α t,ll/prime +(V+JH)/summationdisplay l/negationslash=l/primeˆ/Delta1† s,ll/primeˆ/Delta1s,ll/prime. (7) When the Hund’s coupling is strong enough, i.e., JH>U / 3, the third term of Eq. ( 7) can give rise to the instability in a spin- triplet channel [ 48,49], which involves the {A1g:iszλ2},E(1) u, andE(2) uIRs in Table I. The detailed discussions about these pairing channels are merged into the third kind of effectiveinteraction in the following. Another standard theory for the superconductivity is the phenomenological Heisenberg model in the strong couplinglimit, we consider the effectively frustrated Heisenberg interaction [ 50] as the pairing force, H(2) int=J1/summationdisplay l,/angbracketlefti,j/angbracketrightSil·Sjl+J2/summationdisplay l,/angbracketleft/angbracketlefti,j/angbracketright/angbracketrightSil·Sjl. (8) HereJ1/2are the nearest and next-nearest neighbor magnetic exchange couplings. A well-know result of H(2) intis that the magnetic ground state is checkerboard antiferromagnetic when2J 2<\|J1\|, and collinear antiferromagnetic when 2 J2>\|J1\|. However, no Fermi surface reconstruction induced by spindensity wave was observed in monolayer FeSe/STO butin mutlilayer FeSe/STO in ARPES experiments [ 12]. The recent ﬁrst-principles calculations proposed that the magneticorder was strongly frustrated in monolayer FeSe/STO with2J 2≈\|J1\|[51]. Another issue is the sign of J1. If both J1 andJ2are antiferromagnetic, the /Delta13/4(k) pairing channels are ruled out, and the SC states fall into {A1g:f4(k)s0λ0}induced byJ2or{B1g:f2(k)s0λ0}induced by J1.I fJ1is ferromagnetic andJ2are antiferromagnetic, the /Delta13/4(k) pairing channels are possible from the symmetry point, but these two odd-paritypairing channels have to compete with the {A 1g:f4(k)s0λ0} induced by J2. The winner is determined by the topology of the Fermi surface [ 52]. For the low electron doped at 0 .1e/Fe, the Fermi pockets locating at Mpoints are quite small. Therefore, the form factor f4(k) has large magnitude, and the SC states favor the {A1g:f4(k)s0λ0}. If the electron-doped level can be tuned in monolayer FeSe/STO without suppressing thesuperconductivity, we can expect that the SC states in overelectron-doped samples would favor /Delta1 3/4(k) pairing channels for ferromagnetic J1, because the Fermi surface locates at theXpoints, where the form factors f3(kx/y) have large magnitudes. We note that such kind of pairing was discussedin underdoped cuprates [ 53]. From the aforementioned arguments about the possibly signiﬁcant role of surface phonon, we consider the third kindof phenomenological interaction to induce the interfacial SCinstability in monolayer FeSe/STO, H (3) int=/summationdisplay l,l/prime,σ,σ/prime,k,k/prime1 2Vσ,σ/prime l,l/prime(k,k/prime)d† k,lσd† −k,l/primeσ/primed−k/primel/primeσ/primedk/prime,lσ.(9) Here we assume Vσ,σ/prime l,l/prime(k,k/prime)=−V0forl=l/prime,σ/prime=¯σand Vσ,σ/prime l,l/prime(k,k/prime)=−V1forl>l/prime. Note that the third term in Eq. ( 7) withJH>U / 3 can also be described by H(3) int. With the pairing operators shown in Eq. ( 6),H(3) inttakes the form H(3) int=−V0/summationdisplay lˆ/Delta1† s,llˆ/Delta1s,ll−V1/summationdisplay l>l/primeˆ/Delta1† s,ll/primeˆ/Delta1s,ll/prime −V1/summationdisplay l>l/primeαˆ/Delta1α† t,ll/primeˆ/Delta1α t,ll/prime. (10) Under the mean-ﬁeld approximation /Delta1s,ll/prime=/angbracketleftˆ/Delta1† s,ll/prime/angbracketright,/Delta1α t,ll/prime= /angbracketleftˆ/Delta1α t,ll/prime/angbracketright,t h eH(3) intcan be decoupled as follows: H(3) int=−V0/summationdisplay l/Delta1s,llˆ/Delta1† s,ll−V1/summationdisplay l>l/prime/Delta1s,ll/primeˆ/Delta1† s,ll/prime −V1/summationdisplay l>l/primeα/Delta1α t,ll/primeˆ/Delta1α† t,ll/prime+H.c.+hcon. (11) 165104-5NINGNING HAO AND SHUN-QING SHEN PHYSICAL REVIEW B 92, 165104 (2015) Herehcon=/summationtext lV0\|/Delta1s,ll\|2+V1/summationtext l>l/prime\|/Delta1s,ll/prime\|2+V1/summationtext l>l/prime,α \|/Delta1α s,ll/prime\|2. Now we consider the odd-orbital-parity parts of the normal-state Hamiltonian. The mean-ﬁeld Hamiltonian takesthe following form: H MF=/summationdisplay k1 2/Psi1†(k)HMF(k)/Psi1(k)+Hcon, (12) where /Psi1(k) has the same form shown in Eq. ( 2) except {dσ(k)}={dxz,σ(k),dyz,σ(k),dxy,σ(k),dx2−y2,σ(k),dz2,σ(k)} now. Then HMF(k)=H0(k)τz+/Delta1(k)τx,H0(k)= Ao(k)⊕Ao(k), and Hcon=/summationtext5 k,m=1Ao,mm(k)+hcon. Assume the HMF(k) can be diagonalized with matrix ˜Uk, i.e., ˜U† kHMF(k)˜Uk=Ek,1⊕Ek,2···Ek,20. Then the mean-ﬁeld self-consistent equations take the forms /Delta1s,ll/prime=20/summationdisplay k,n=1[˜U∗ k,n,l˜Uk,n,l/prime+10+˜U∗ k,n,l+5˜Uk,n,l/prime+15]f(Ek,n) 2, /Delta1x t,ll/prime=20/summationdisplay k,n=1−[˜U∗ k,n,l˜Uk,n,l/prime+15+˜U∗ k,n,l+5˜Uk,n,l/prime+10]f(Ek,n) 2, /Delta1y t,ll/prime=20/summationdisplay k,n=1−i[˜U∗ k,n,l˜Uk,n,l/prime+15−˜U∗ k,n,l+5˜Uk,n,l/prime+10]f(Ek,n) 2, /Delta1z t,ll/prime=20/summationdisplay k,n=1−[˜U∗ k,n,l˜Uk,n,l/prime+10−˜U∗ k,n,l+5˜Uk,n,l/prime+10]f(Ek,n) 2, Ne=20/summationdisplay k,n=110/summationdisplay m=1\|˜U∗ k,n,m\|2f(Ek,n). (13) Here f(x)=1 ex kBT+1is the Fermi distribution func- tion and Neis the electron number. In comparison with Table Iand Eq. ( 11), the relevant IR channels in Table Ican be represented with ( 13). For exam- ple,{A(1) 1g:s0λ0}=s0(/Delta1s,xz,xz⊕/Delta1s,yz,yz⊕/Delta1s,xy,xy ),{E(2) u: i(sxλ5,syλ7)}=i(/Delta1x t,xz,xysxλ5,/Delta1y t,xz,xysyλ7). Likewise, other IR channels can be read out following the same way. It is possible for /Delta1(k) to take the form of linear combina- tions of several different IR channels, but some symmetrieshave to be broken to pay the price for such coexistence. Forexample the inverse symmetry is broken for the SC statesproposed in Refs. [ 21,22]. Likewise, the TR symmetry or lattice symmetry could also be broken when two differentone-dimensional IRs or two components in a two-dimensionalIR coexist. In order to gain some insight before we performthe numerical calculations, we note that all the experimentsreported the isotropic Fermi surface and gap structures withoutany resolvable distortions, and the monolayer FeSe/STO wasconformed to be the cleanest composition with the simpleststructure [ 10–12]. These features rule out the possibilities of some complex orders, such as nematic order found in bulkFeSe. From Table Iwe can ﬁrst eliminate the possibilities of the {B 2g:s0λ1},{A1g:iszλ2},{Eg:i(sx,sy)λ2}, and {A1g:s0λ8}pairing channels, because the leading inter- dxz-dyz hopping term is proportional to sin kxsinky, which is nearly zero around the Fermi surface, and the {A1g:s0λ8}channel has nodes. Second, it is straightforward to check that twocomponents in {Eu:s0(λ4,λ6)}or{E(1) u:sz(λ5,λ7)}give two degenerate strip SC states with nodes. Thus, the TR-broken linear combination of two components is optimal to achievethe isotropic nodeless gap structure and lower the energy. Notethat the coexistence of these two two-dimensional IRs couldraise the energy, because they follow different transformationsunder the lattice symmetric operations and suppress the gapamplitude. Finally, no additionally global symmetries can be broken for {A (1) 1g:s0λ0}and{E(2) u:i(sxλ5,syλ7)}to coexist with each other and with {Eu:s0(λ4,λ6)}or{E(1) u:sz(λ5,λ7)} to avoid breaking the isotropic SC gap structure and achievinglower energy. Therefore, we ﬁnd that these four IRs, i.e., {E u:s0(λ4,λ6)},{E(1) u:sz(λ5,λ7)},{A(1) 1g:s0λ0}, and {E(2) u: i(sxλ5,syλ7)}are independent, and TR symmetry should be spontaneously broken in the ﬁrst two IRs. It is straightforwardto verify these arguments through the following numericalcalculations. Now we perform the numerical calculations to evaluate which pairing channel governs the ground state of thesystem for different V 0andV1. The ground state energy of Eq. ( 12)i sGs(T)=−kBTln Tre−βH MF, and Gs(T∼0)= (Hcon−1 2/summationtext10 k,n=1\|Ek,n\|) at zero temperature. For simplicity we can evaluate the ground state through the minimumof the condensed energy density deﬁned as f g=hcon− 1 8π2/summationtext10 n=1/integraltext d2k\|Ek,n\|−1 4π2/summationtext5 n=1/integraltext d2k\|Eo k,n\|for given elec- tron number, where Eo k,nare the energy spectra of normal state. Solve the self-consistent equations ( 12) and ( 13)f o r parameters ( V0,V1) with respect to the minimum of fg,w e show the evolution of SC order parameters and condensedenergy about ( V 0,V1)i nF i g . 4, and we ﬁnd topologically trivial {A(1) 1g:s0λ0}channel and topologically nontrivial FIG. 4. (Color online) (a) The evolution of three components of SC order parameters in A(1) 1gchannel about V0. (b) The evolution of components of SC order parameters in Eu,E(1) u,a n dE(2) uchannels about V1. (c) The evolution of the condensed energy in different SC states with relevant IRs about V0andV1. (d) The phase diagram is plotted in ( V0,V 1) plane with respect to the lowest energy. We set a 51×51 mesh of k, and the electron number to satisfy electron-doped 0.1e/Fe. The energy scale is measured with eV . 165104-6TOPOLOGICAL SUPERCONDUCTING STATES IN . . . PHYSICAL REVIEW B 92, 165104 (2015) {E(2) u:i(sxλ5,syλ7)}are dominant in relevant regime of ( V0,V1) parameter plane. V . DISCUSSION AND SUMMARY If the superconductivity in monolayer FeSe/STO is driven by the effective interaction H(3) intin Eq. ( 10), the observed isotropic and nodeless s-wave gap structures select both topologically trivial A(1) 1g(s0λ0) and nontrivial E(2) u[/Delta12(k)] as possible candidates. The essential difference lies in that theformer one has even-parity and spin-singlet pairing while thelatter one has odd-parity and spin-triplet pairing. Therefore, itis unambiguous to adopt the experiments which can directlydistinguish the spin states and parities to pin down the possiblecandidate. Particularly, temperature dependence of the nuclearmagnetic relaxation (NMR) rate can be utilized to distinguishthe two different pairings. The well-known result is that theNMR rate has a Hebel-Slichter peak at the SC transitiontemperature for the even-parity and spin-singlet s-wave SC state [ 54]. However, the Hebel-Slichter peak could disappear with the antipeak behavior due to the unique spin, orbital, andmomentum locking effect in topological SC states with oddparity as shown in Ref. [ 55]. The parity of the Cooper pair is characterized by the inverse operator {i\| 1 21 2}. It indicates the odd-parity pairing has a sign change or phase shift of πbetween the top Se and and bottom Se layers along the c axis compared with the even-parity pairing. Thus, the standardmagnetic-ﬂux modulation of dc SC quantum interferencedevices (SQUIDS) measurements [ 4,56,57] provide another scheme to distinguish the odd- and even-parity pairings. Onthe other hand, some transport measurements can also beapplied to detect the topological superconductors, such asthe thermal Hall conductivity [ 58,59]. The challenge for such measurements is that the FeSe is very air sensitive, and theexperimental measurements should be performed under theultrahigh vacuum condition. In the aforementioned discussions about the SC pairings, we assume that the glide plane symmetry is not broken.Actually, there exist some possible effects to break the glideplane symmetry. For example, the atomic spin-orbital couplingcould have non-neglectable effect in iron chalcogenides. It isexplicit that the interorbital spin-orbital coupling can mix thebands with inverse orbital parities, and induce the interorbitalSC pairing in ( k,−k+Q) channels. However, the weight of inter-d xz-dxyspin-orbital coupling is proportional to λso∼ 0.05 eV [ 38], while the inter- dxz-dxyorbital hopping term with deﬁnite orbital parity is proportional to \|2it14 xsinkF\|∼ 0.3e V at the Fermi surface. We can estimate that the ratio betweenthe amplitudes of SC pairing order parameter in ( k,−k+Q) channel and that in ( k,−k) channel should be ∼0.025. It is straightforward to check that the coexistence of the SCpairings in ( k,−k) and ( k,−k+Q) channels does not change the topological natures of the SC states with E (1) uandE(2) uIRs under the condition that the pairings in ( k,−k+Q) channels have the reasonable amplitudes in the physical regime. Thereason lies in that the pairings in the ( k,−k+Q) channel correspond to the interband pairings in the band basis, andcannot drive the gap-closing-reopening process to achieve thequantum phase transition. Another issue should be noticedthat the spin quantum number is adopted to label the SC pairings in the pairing classiﬁcation, and such an approachis not exact when atomic spin-orbital coupling is involved.However, the approximation works well, because the atomicspin-orbital coupling here is quite small. Indeed, it is shownthat the atomic spin-orbital coupling plays a secondary rolein SC states in A xFe2−ySe2[38]. Other issues, such as the coupling between the monolayer FeSe and substrate STO,could also break the glide plane symmetry. Such couplingsare tunable and strongly affected by the fabrication processand the substrate materials [ 13,60]. Here we consider the case that the strength of coupling between the monolayer FeSe andsubstrate is weak in comparison with the relevant hoppingamplitude. Compared with the general topological materials, in which the extended sandporbitals are the bricks to build low-energy electronic structures, and the spin-orbital coupling plays anessential role in inducing the strong linear couplings, the linearcouplings in monolayer FeSe/STO is attributed to effectivecouplings between 3 dorbitals induced by d-phybridizations from the unique nonsymmorphic lattice structures. Suchfeatures provide us an alternative route to search for the newtopological materials in strongly correlated electron systems. In conclusion, we propose that the monolayer FeSe/STO could support the odd-parity topological SC states with thenodeless s-wave gap structures. In contrast with other topolog- ical superconductors [ 2,4] in which the spin-orbital coupling plays a key role, such topological SC states have strongrelations with the unique nonsymmorphic lattice symmetrywhich induces the orbital-momentum locking. Furthermore,we calculate the phase diagram and suggest some experimentalschemes to identify such uniquely nontrivial topological SCstates. ACKNOWLEDGMENTS We thank Professor J. P. Hu for helpful discussions. This work is supported by the Research Grant Council of HongKong under Grant No. HKU703713P. APPENDIX A: THE TIGHT-BINDING HAMILTONIAN FROM SYMMETRY ANALYSES In this Appendix we discuss the properties of the tight- binding Hamiltonian from the symmetric point. The trilayerstructure of the monolayer FeSe is shown in Fig. 1(see main text). We focus on the three space group operations includingglide plane symmetry operator ˆg z={mz\|r0}with r0=(1 21 2) and two reﬂection symmetry operations ˆgx={mx\|r0}and ˆgx/prime={mx/prime\|00}. Besides, the lattice has inverse symmetry denoted by the operator ˆgi={i\|r0}. According to the LDA calculation, we can only focus on Fe atoms, the Bloch wavefunctions for the 3 dorbitals of Fe are deﬁned as \|αη,k /prime/angbracketright=1√ N/summationdisplay neik/prime·r/prime nηφα(r/prime−r/prime nη). (A1) Here r/prime nη=R/prime n+r/prime ηwith lattice vector R/prime nand the position r/prime ηof Fe atom η=A,B , andφαdenotes the dorbital basis function ( α=xz,yz,x2−y2,xy,z2). The symmetry operators acting on the basis function \|αη,k/prime/angbracketrighthave the following 165104-7NINGNING HAO AND SHUN-QING SHEN PHYSICAL REVIEW B 92, 165104 (2015) properties: ˆgx/prime\|αη,k/prime/angbracketright=/summationdisplay βmx/prime,αβ\|βη,m x/primek/prime/angbracketright, ˆgz\|αη,k/prime/angbracketright=/summationdisplay βe−i(ˆmzk/prime)·r0mz,αβ\|β¯η,ˆmzk/prime/angbracketright, (A2) ˆgx\|αη,k/prime/angbracketright=/summationdisplay βe−i(ˆmxk/prime)·r0mx,αβ\|β¯η,ˆmxk/prime/angbracketright. The relevant tight-binding (TB) Hamiltonian can be expressed as H0=/summationdisplay k/prime/Psi1†(k/prime)H(k/prime)/Psi1(k/prime), (A3) with /Psi1†(k/prime)=[ψ† A(k/prime),ψ† B(k/prime)], ψ† η(k/prime)=[d† η,xz(k/prime),d† η,yz(k/prime),d† η,x2−y2(k/prime),d† η,xy(k/prime),d† η,z2(k/prime)]. (A4) In the basis /Psi1(/vectork/prime), the corresponding transformation matrices for the three operations ˆgαhave the following forms: U(ˆgx/prime)=/bracketleftBigg mx/prime 0 0mx/prime/bracketrightBigg , U(ˆgz)=/bracketleftBigg 0 e−i(mzk/prime)·r0mz e−i(mzk/prime)·r0mz 0/bracketrightBigg , (A5) U(ˆgx)=/bracketleftBigg 0 e−i(mxk/prime)·r0mx e−i(mzk/prime)·r0mx 0/bracketrightBigg , where mx/prime=⎡ ⎢⎢⎢⎢⎢⎢⎣01 000 10 00000 −100 00 01000 001⎤ ⎥⎥⎥⎥⎥⎥⎦, m z=⎡ ⎢⎢⎢⎢⎢⎢⎣−1 0000 0−1000 0 0100 0 00100 0001⎤ ⎥⎥⎥⎥⎥⎥⎦, (A6) m x=⎡ ⎢⎢⎢⎢⎢⎢⎣−100 00 010 00001 00 000 −10 000 01⎤ ⎥⎥⎥⎥⎥⎥⎦. The symmetry of the Hamiltonian requires H 0(k/prime)=U(k/prime)H0(Uk/prime)U†(k/prime). (A7) Deﬁne H0(k/prime)=/bracketleftBigg HA(k/prime)HAB(k/prime) HBA(k/prime)HB(k/prime)/bracketrightBigg . (A8)We can get HA/B(kx/prime,ky/prime)=mx/primeHA/B(−kx/prime,ky/prime)mx/prime, (A9) HAB(kx/prime,ky/prime)=mx/primeHAB(−kx/prime,ky/prime)mx/prime, HA(kx/prime,ky/prime)=mzHB(kx/prime,ky/prime)mz, (A10) HAB(kx/prime,ky/prime)=mzHBA(kx/prime,ky/prime)mz, HA(kx/prime,ky/prime)=mxHB(−ky/prime,−kx/prime)mx, (A11) HAB(kx/prime,ky/prime)=mxHBA(−ky/prime,−kx/prime)mx. Moreover, since \|αη,k/prime+G/prime/angbracketright=eiG/prime·r/prime η\|αη,k/prime/angbracketright, HA/B(k/prime+G/prime)=HA/B(k/prime), (A12) HAB(k/prime+G/prime)=eiG/prime·r/prime 0HAB(k/prime), r/prime 0=r/prime B−r/prime A=(1 2,1 2). Considering the operator ˆgz, we can ﬁnd in the entire BZ/bracketleftBigg/bracketleftBigg 0mz mz 0/bracketrightBigg ,/bracketleftBigg HA(k/prime)HAB(k/prime) HBA(k/prime)HB(k/prime)/bracketrightBigg/bracketrightBigg =0. (A13) We have V†/bracketleftBigg 0mz mz 0/bracketrightBigg V=/bracketleftBigg −I5×5 0 0 I5×5/bracketrightBigg , (A14) V=1√ 2/bracketleftBigg AA B−B/bracketrightBigg , (A15) withA=I5×5,B=−mz. It is straightforward to check that H0(k/prime) can also be block diagonalized, i.e., V†H0(k/prime)V=H11(k/prime)⊕H22(k/prime), (A16) withH11(k/prime)=HA(k/prime)−HAB(k/prime)mzandH22(k/prime)=HA(k/prime)+ HAB(k/prime)mz.F r o mE q .( A12), we can get HA/B(kx/prime+2πnx/prime, kx/prime+2πny/prime)=HA/B(kx/prime+2πnx/prime,kx/prime+2πny/prime) and HAB(kx/prime+ 2πnx/prime,kx/prime+2πny/prime)=ei(2πnx/prime1 2+2πny/prime1 2)HAB(kx/prime+2πnx/prime,kx/prime+ 2πny/prime). When ( nx/prime,ny/prime)=(0,1),H11(k/prime)=HA(k/prime)− HAB(k/prime)mzandH22(k/prime)=HA(k/prime+Q/prime)−HAB(k/prime+Q/prime)mz, with Q/prime=(0,2π). Furthermore, the momentum deﬁned in the one-Fe BZ is kx=(kx/prime+ky/prime)/2,ky=(−kx/prime+ky/prime)/2 and Q=(π,π). Under the basis, /Psi1†(k)=[ψ†(k),ψ†(k+Q)], with ψ†(k)=[d† xz(k),d† yz(k),d† x2−y2(k),d† xy(k),d† z2(k)],dl(k)=1√ 2 [dA,l(k/prime)+dB,l(k/prime)], and dl(k+Q)=1√ 2[dA,l(k/prime)−dB,l(k/prime)] forl=xz,yz ,dl(k)=1√ 2[dA,l(k/prime)−dB,l(k/prime)] and dl(k+ Q)=1√ 2[dA,l(k/prime)+dB,l(k/prime)] for l=xy,x2−y2,z2,t h eT B Hamiltonian in the one-Fe BZ takes the following form: H0=/summationdisplay k/Psi1†(k)H0(k)/Psi1(k). (A17) Then, H0(k)=Ho(k)⊕He(k). (A18) HereHe(k)=Ho(k+Q). The TB Hamiltonian in one-Fe BZ Eq. ( A18) have block- diagonal forms, and each block has deﬁnitive orbital parity 165104-8TOPOLOGICAL SUPERCONDUCTING STATES IN . . . PHYSICAL REVIEW B 92, 165104 (2015) with respect to the glide plane symmetry. Besides, the inversion symmetry ˆgi={i\|r0}indicates that the inversion center of monolayer FeSe is at the midpoint of the Fe-Fe link. Thus wecan ﬁnd that d xz/yz (k)/dxy/x2−y2/z2(k) are inversion even/odd, anddxz/yz (k+Q)/dxy/x2−y2/z2(k+Q) are inversion odd/even. In other words, dxz/yz orbitals and dxy/x2−y2/z2orbitals have opposite parities in the subspace with deﬁnitive orbital parity.The TB Hamiltonian in the one-Fe BZ is H o(/vectork)=⎡ ⎢⎢⎢⎢⎢⎢⎣A 11A12A13A14A15 A22A23A24A25 A33A34A35 A44A45 A55⎤ ⎥⎥⎥⎥⎥⎥⎦. (A19) The nonzero terms in A(k) are listed as follows: A 11/22(k)=/epsilon11+2t11 x/ycoskx+2t11 y/xcosky +4t11 xycoskxcosky+2t11 xx/yy cos 2kx +2t11 yy/xx cos 2ky+4t11 xxy/yyx cos 2kxcosky +4t11 xyy/xxy coskxcos 2ky +4t11 xxyycos 2kxcos 2ky, A33(k)=/epsilon13+2t33 x(coskx+cosky)+4t33 xycoskxcosky, A44(k)=/epsilon14+2t44 x(coskx+cosky)+4t44 xycoskxcosky +4t44 xxy(cos 2kxcosky+coskxcos 2ky) +4t44 xxyycos 2kxcos 2ky, A55(k)=/epsilon15, A12(k)=− 4t12 xysinkxsinky, A13/23(k)=± 2it13 xsinky/x±4it13 xysinky/xcoskx/y, A14/24(k)=− 2it14 xsinkx/y+4it14 xysinkx/ycosky/x, A15/25(k)=2it15 xsinky/x+4it15 xysinky/xcoskx/y, A35(k)=2t35 x(coskx−cosky), A45(k)=− 4t45 xysinkxsinky. The on-site orbital energy is /epsilon11=/epsilon12=0.02,/epsilon13= −0.539,/epsilon14=0.014,/epsilon15=− 0.581, and the hopping pa- rameters for the free-standing monolayer FeSe arelisted as follows [ 61]:t 11 x/y=− 0.08/−0.311,t11 xy=0.232, t11 xx/yy=0.009/−0.045,t11 xxy/yyx =− 0.016/0.019,t11 xxyy= 0.02,t33 x=0.412,t33 xy=− 0.066,t44 x=0.063,t44 xy=0.086, t44 xxy=− 0.017,t44 xxyy=− 0.028,t12 xy=0.099,t13 x=0.3,t13 xy= −0.089, t14 x=0.305, t13 xy=− 0.056, t15 x=− 0.18,t15 xy= 0.146,t35 x=0.338,t45 xy=− 0.109. The renormalized parame- ters corresponding to Fig. 1(d) in the main text are t44 xy=0.066, t14 x=0.405,t11 x=− 0.12. The renormalized parameters cor- responding to Fig. 1(e) in the main text are t44 xy=0.076, t44 x=0.183,t14 x=0.405,t11 x=− 0.311,t11 xy=0.19.TABLE III. The IRs of all the possible on-site superconducting pairing in ( k,−k+Q) channels. (k,−k+Q): /Delta1/prime(k) c2(z) c2(x) σd/braceleftbig i/vextendsingle/vextendsingle1 21 2/bracerightbig/primeIR s0λ0 11 1 −1A1u s0λ8 11 1 −1A1u s0λ1 1 −11 −1B2u s0(λ4,λ6)( −1,−1) (1 ,−1) s0(λ6,λ4) (1,1) Eg iszλ2 11 1 −1A1u sz(λ5,λ7)( −1,−1) (−1,1) −sz(λ7,λ5) (1,1) Eg i(sx,sy)λ2 (−1,−1) (−1,1) i(sy,sx)λ2 (−1,−1)Eu i(sxλ5,syλ7) (1,1) (1,1) −i(syλ7,sxλ5) (1,1) Eg i(syλ5,sxλ7) (1,1) ( −1,−1)−i(sxλ7,syλ5) (1,1) Eg APPENDIX B: THE CLASSIFICATIONS FOR THE (k,−k+Q) PAIRING CHANNELS FROM SYMMETRY ANALYSES The nine GellMann matrices λ0–λ8in the main text are listed as follows: λ0=⎡ ⎣100 010 001⎤ ⎦,λ 1=⎡ ⎣010 100 000⎤ ⎦, λ2=⎡ ⎣0−i0 i 00 000⎤ ⎦,λ 3=⎡ ⎣100 0−10 000⎤ ⎦, λ4=⎡ ⎣001 000 100⎤ ⎦,λ 5=⎡ ⎣00 −i 00 0 i00⎤ ⎦, (B1) λ6=⎡ ⎣000 001 010⎤ ⎦,λ 7=⎡ ⎣00 0 00 −i 0i 0⎤ ⎦, λ8=1√ 3⎡ ⎣10 0 01 0 00 −2⎤ ⎦. TABLE IV . The IRs of all the possible non-on-site superconduct- ing pairing in ( k,−k+Q) channels. (k,−k+Q):/Delta1/prime(k)I R f4,ks0λ0/8,f5,ks0λ1,f3,kxs0λ5+f3,kys0λ7 A1u f2,ks0λ0/8,f3,kxs0λ5−f3,kys0λ7 B1u f2,ks0λ1,f3,kys0λ5−f3,kxs0λ7 A2u f5,ks0λ0/8,f1/4,ks0λ1,f3,kys0λ5+f3,kxs0λ7 B2u if1/4,kszλ2,i1/0/0[f3,kxsz/x/yλ4+f3,kysz/y/xλ6] A1u if2,kszλ2,i1/0/0[f3,kxsz/x/yλ4−f3,kysz/y/xλ6] B1u i1/0/0[f3,kysz/x/yλ4−f3,kxsz/y/xλ6] A2u if5,kszλ2,i1/0/0[f3,kysz/x/yλ4+f3,kxsz/y/xλ6] B2u if1/2/4/5,k(sx,sy)λ2 Eu 165104-9NINGNING HAO AND SHUN-QING SHEN PHYSICAL REVIEW B 92, 165104 (2015) The monolayer FeSe has inversion symmetry, thus every IR in Table Ishould have a counterpart with an inverse parity. In other words, ( k,−k+Q) pairing channels should be possible from the symmetry point. For the(k,−k+Q) pairing, we deﬁne the Nambu basis, /Psi1 /prime(k)= [{ψm↑(k)},{ψm↓(k)},{ψ† m↓(−k+Q)},−{ψ† m↑(−k+Q)}]t, with{ψmσ(k)}=[dxzσ(k),dyzσ(k),dxyσ(k)]. 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PhysRevB.71.085115.pdf	Self-consistent random phase approximation: Application to the Hubbard model for ﬁnite number of sites Mohsen Jemaï * Institut de Physique Nucléaire d’Orsay, Université Paris-Sud, CNRS-IN2P3, 15, Rue Georges Clemenceau, 91406 Orsay Cedex, France Peter Schuck† Institut de Physique Nucléaire d’Orsay, Université Paris-Sud, CNRS-IN2P3, 15, Rue Georges Clemenceau, 91406 Orsay Cedex, France and Laboratoire de Physique et Modlisation des Milieux Condenss (LPMMC) (UMR 5493), Maison Jean Perrin, 25 avenue des Martyrs BP 166, 38042 Grenoble cedex 9, France Jorge Dukelsky‡ Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientiﬁcas, Serrano 123, 28006 Madrid, Spain Raouf Bennaceur§ Département de Physique, Faculté des Sciences de Tunis, Université de Tunis El-Manar 2092 El-Manar, Tunis, Tunisie sReceived 8 July 2004; revised manuscript received 16 November 2004; published 22 February 2005 d Within the one-dimensional Hubbard model linear closed chains with various numbers of sites are consid- ered in the self-consistent random phase approximation sSCRPA d. Excellent results with a minimal numerical effort are obtained for s2+4nd-site cases, conﬁrming earlier results with this theory for other models. However, the 4n-site cases need further consideration. The SCRPA solves the two-site problem exactly. It therefore contains the two-electron and high-density Fermi gas limits correctly. DOI: 10.1103/PhysRevB.71.085115 PACS number ssd: 75.10.Jm, 72.15.Nj I. INTRODUCTION The standard random phase approximation ss-RPA dis one of the most popular many-body approaches known. It wasinvented in condensed matter physics ssee, e.g., Ref. 1 dand has subsequently spread to almost all branches of physics,including atomic physics, 2molecular physics,3plasma physics,4relativistic ﬁeld theory,5nuclear physics,6and many more. The deﬁnition of the s-RPA is not uniform, de-pending on whether exchange is included or not. We under-stand it—e.g., as in nuclear physics 6—as the small-amplitude limit of time-dependent Hartree-Fock sTDHF dtheory and therfore with exchange. Its popularity probably stems fromits conceptual simplicity, its numerical tractability sin spite of some serious problems in ﬁnite-size systems d, and most of all its well-behaved properties concerning fulﬁllment of con-servation laws sWard identies d, Goldstone theorem, and res- toration of spontaneously broken symmetries. Though thereexist respectable general theories ssee, e.g., Refs. 7 and 8 d, any practical attempt to go beyond this basic HF-RPAscheme conserving these properties turned out to be techni-cally extremely demanding and no well-accepted general andpractical extension has emerged so far. Nevertheless, thestandard RPA has also quite serious shortcomings and it isdesirable to overcome them. One of the most prominent is itsviolation of the Pauli principle, often paraphrased as the“quasiboson approximation.” It is most critical for only mod-erately collective modes or when the self-interaction of thegas of quantum ﬂuctuations becomes important as in ultras-mall ﬁnite quantum systems. Since a couple of years two ofthe present authors and collaborators have been working on a nonlinear extension of the RPA sRef. 9 dwhich has shown surprisingly accurate results in a number of nontrivialmodels. 10It is called the self-consistent RPA sSCRPA dand can be obtained from minimizing an energy-weighted sumrule. Therefore the s-RPA which is perturbative in the sensethat it sums a certain class of diagrams sthe bubbles dis up- graded in the SCRPA to a nonperturbative variational theorythough it is in general not of the Raleigh-Ritz type. A strongbonus of this extension of the s-RPA is that it generally pre-serves its positive features as conservation laws and restora-tion of symmetries as well as numerical tractability, since itleads to equations of the Schrödinger type. 11In this paper we want to apply this theory to the Hubbard model for stronglycorrelated electrons. Because of its necessarily increased nu-merical complexity over the s-RPA, we ﬁrst want to considerﬁnite clusters in reduced dimensions. Before going into thedetails, let us very brieﬂy repeat the main ideas of theSCRPA. One way of presentation is to outline its strong analogy with the Hartree-Fock-Bgoliubov sHFB dapproach to inter- acting boson ﬁelds b †andb. The HFB canonical transforma- tion reads qn†=o iui,nbi†−vi,nbi. s1d The amplitudes uandvcan be determined12from minimiz- ing the following mean energy senergy-weighted sum rule d:PHYSICAL REVIEW B 71, 085115 s2005 d 1098-0121/2005/71 s8d/085115 s15d/$23.00 ©2005 The American Physical Society 085115-1vn=k0ufqn,fH,qn†ggu0l k0ufqn,qn†gu0l, s2d whereHis the usual many-body Hamiltonian with two-body interactions and the ground state u0lis supposed to be the vacuum to the quasiboson operators qn—i.e., qnu0l=0. s3d With this scheme and the usual orthonormalization condi- tions for the amplitudes uandv, which allows the inversion of Eq. s1d, one derives standard HFB theory6with no need to construct u0lexplicitly. Of course, in this way the fact that the HFB theory is a Raleigh-Ritz variational theory is notmanifest but the scheme has the advantage to be physicallytransparent and to lead to the ﬁnal equations with a minimumof mathematical effort. For the SCRPA we follow exactly the same route. We replace in Eq. s1dthe ideal boson operators by fermion pair operators of the particle-hole sphdtype and form an ansatz for a general transformation of ph-fermion pairs: Q n†=o phsXphnap†ah−Yphnah†apd, s4d with unl=Qn†u0lan excited state of the spectrum. In analogy with Eq. s2dwe minimize a mean excitation energy Vn=k0ufQn,fH,Qn†ggu0l k0ufQn,Qn†gu0l, s5d with u0l, in analogy with Eq. s3d, the vacuum to the operators Qn, i.e., Qnu0l=0, s6d and arrive at equations of the usual RPA type:6 SA B −B−ADSXn YnD=VnSXn YnD, s7d with Aph,p8h8=k0ufah†ap,fH,ap8†ah8ggu0l ˛nh−np˛nh8−np8, Bph,p8h8=−k0ufah†ap,fH,ah8†ap8ggu0l ˛nh−np˛nh8−np8. s8d Here we supposed to work in a single-particle basis which diagonalizes the density matrix snatural orbits d, k0uak†ak8u0l;nkdkk8, s9d and therefore the nk’s are the occupation numbers. For H with a two-body interaction, Eqs. s8donly contain correlation functions of the ka†alandka†aa†altypes and, since Eq. s6d admits the usual RPA orthonormalization relations for the amplitudes XandY,6the relation s4dcan be inverted and with Eq. s6dthe correlation functions in Eq. s8dbe expressed byXandY. However, to be complete, occupation numbers nk =k0uak†aku0land two-body correlation functions with otherindex combinations than two-times particle and two-times hole need extra considerations. That will be done in the maintext. This is, in short, the SCRPA scheme which, as HFBtheory, is obviously non-linear, since the elements AandB in Eq. s7dbecome functionals of the XandYamplitudes.We want to point out that no bosonization of fermion pairs isoperated at any stage of the theory. We want to apply this scheme to the Hubbard model of strongly correlated electrons which is one of the most wide-spread models to investigate strong electron correlations andhigh-T csuperconductivity. Its Hamiltonian is given by H=−to kijlscis†cjs+Uo inˆi"nˆi#, s10d wherecis†andcjsare the electron creation and destruction operators at site iand thenˆis=cis+cisare the number opera- tors for electrons at site iwith spin projection s.As usual tis the nearest-neighbor hopping integral and Uthe on-site Cou- lomb matrix element. In this exploratory work, we will limitourselves to the simplest cases possible; i.e., we will con-sider closed chains in one dimension s1Ddwith an increasing number of sites at half ﬁlling, starting with the two-site prob-lem. It will turn out that the next case of four sites is aconﬁguration with degeneracies which cause problems in theSCRPA, as do all 4 nsn=1,2,3,... dconﬁgurations in 1D.We therefore will postpone the treatment of these cases to future work and directly jump to the case of six-sites and onlyshortly outline at the end why the four-site case is unfavor-able and how the problem can eventually be cured. In thiswork we will stop with the six-site case, considering it assufﬁciently general to be able to extrapolate to the more-electron case. In this way one may hope to approach thethermodynamic limit in increasing the number of sites asmuch as possible. Let us mention that an earlier attempt tosolve the SCRPA in 1D in the thermodynamic limit in astrongly simpliﬁed version of the SCRPA, the so-calledrenormalized RPA sr-RPA d, produced interesting results. 13 In detail our paper is organized as follows: in Sec. II we present the two-site case with its exact solution. In Sec. IIIwe outline the six-site case with a detailed discussion of theresults, and in Sec. IVwe present the difﬁculties encounteredin the four-site case and how, eventually, one can overcomethem. Finally in Sec. V we give our conclusions togetherwith some perspectives of this work. II. TWO-SITE PROBLEM In this section we will apply the general formalism of the SCRPA outlined in the Introduction to the two-site problemat half ﬁlling—i.e., two electrons with periodic boundaryconditions. This case may seem trivial; the fact, however, isthat such popular many-body approximations as the s-RPA,GW, 14Gutzwiller wave function,15the two-particle self- consistent sTPSC dapproach by Vilk, Chen, and Tremblay,16 etc., do not yield very convincing results in this study case, whereas it has recently been shown that the SCRPA solvestwo-body problems exactly. 10,11,17We again will brieﬂy dem- onstrate this here for the two-site problem. First we will transform Eq. s10dinto momentum space. With the usual transformation to plane waves, cj,sJEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-2=s1/˛Ndokak˜,se−ik˜·x˜j, this leads to the standard expression for a zero-range two-body interaction: H=o k˜,ssek−mdnˆk˜,s+U 2No k˜,p˜,q˜,sak˜,s†ak˜+q˜,sap˜,−s†ap˜−q˜,−s, s11d wherenˆk˜,s=ak˜,s†ak˜,sis the occupation number operator of the mode sk˜,sdand the single-particle energies are given by ek˜=−2tod=1Dcosskddwith the lattice spacing set to unity. For our further considerations it is convenient to trans- form Eq. s11dto HF quasiparticle operators via swe switch to 1Dd ah,s=bh,s†,ap,s=bp,s, s12d wherehandpare momenta below and above the Fermi momentum, respectively, so that bk,suHFl=0 for all kwhere uHFlis the Hartree-Fock ground state in the plane-wave ba- sis. For the two-site problem with periodic boundary condi-tions we then write, after normal ordering, the Hamiltonians11din the following way: H=H HF+Hq=0+Hq=p, s13d with HHF=EHF+o sf−e1n˜k1,s+e2n˜k2,sg, e1=−t+U 2,e2=t+U 2, s14d Hq=0=U 2sn˜k2,"−n˜k1,"dsn˜k2,#−n˜k1,#d, s15d Hq=p=−U 2sJ"−+J"+dsJ#−+J#+d, s16d andJs−=b1,sb2,s,Js+=sJs−d+, andn˜ki,s=bi,s†bi,s, where we in- troduced the abbreviation “1” and “2” for the two momenta k1=0 andk2=−pof the system, respectively. The HF ground state is uHFl=b1,"b1,#uvacland the corresponding energy is given by E0HF=kHFuHuHFl=−2t+U 2. s17d The RPA excitation operator corresponding to Eq. s4dcan, because of rotational invariance in spin-space, be separatedaccording to spin-singlet sS=0, charge dand spin-triplet sS=1dexcitations. The latter still can be divided into spin- longitudinal sS=1,m s=0dand spin-transverse sS=1,ms =±1 dexcitations. Let us ﬁrst consider the charge- and spin- longitudinal sectors. For later convenience we will not sepa- rate them and write, for the corresponding RPA operator, Qn†=X"nK"++X#nK#+−Y"nK"−−Y#nK#−, s18d whereKs±=Js±/˛1−kMsl,Ms=n˜1s+n˜2s, and the mean val- ueskﬂlare always taken with respect to the RPA vacuum:QnuRPA l=0. s19d Because of the orthonormality relations o ssXsnXsn8−YsnYsn8d=dnn8, o ssXsnYsn8−YsnXsn8d=0, o nsXsnXs8n−YsnYs8nd=dss8, o nsXsnYs8n−YsnXs8nd=0, s20d one can invert Eq. s18dto obtain Js−=˛1−kMslo nsXsnQn+YsnQn†d, Js+=sJs−d†. s21d The operators Js±and 1−Msform aSUs2dalgebra of spin-1 2 operators and, therefore, using the Casimir relation we obtain Ms=2Js+Js−. s22d In this way we can calculate with Eq. s19dthe following expectation values: kJs8+Js−l=˛k1−Ms8lk1−Mslo nYs8nYsn, kJs8−Js+l=˛k1−Ms8lk1−Mslo nXs8nXsn, kJs8+Js+l=˛k1−Ms8lk1−Mslo nYs8nXsn, kJs8−Js−l=˛k1−Ms8lk1−Mslo nXs8nYsn, s23d with kMsl=2onuYsnu2 1+2onuYsnu2. s24d We will see that in order to close the system of SCRPA equations, expectation values kMsMs8lwill also be needed. It is easy to see that we have MsMs=2Ms s25d and MsMs8=4Js†Js8†Js8JsssÞs8d. s26d With Eq. s21dthe expectation value of Eq. s26dgivesSELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-3kMsMs8l=4s1−kMslds1−kMs8ldo nn8o n1n2YsnYsn8Ys8n1Ys8n2 3kQnQn1Qn2†Qn8†l. s27d For the calculation of the correlation functions which appear on the right-hand side of Eq. s27done commutes the destruc- torsQnto the right and uses Eq. s6d, yielding again correla- tion functions kMsMs8l. One then obtains a closed linear system of equations for the latter. Details are given in Ap- pendix A. The SCRPA matrix elements can be expressed in the fol- lowing way: A","=kfK"−,fH,K"+ggl=2t+B",", A#,#=kfK#−,fH,K#+ggl=2t+B#,#, A",#=kfK"−,fH,K#+ggl=B",#, A#,"=kfK#−,fH,K"+ggl=B#,", s28d B","=−kfK"−,fH,K"−ggl=U˛1−kM#l 1−kM"lo nsX"nY#n+X"nX#nd, B#,#=−kfK#−,fH,K#−ggl=U˛1−kM"l 1−kM#lo nsX"nY#n+Y"nY#nd, B",#=−kfK"−,fH,K#−ggl=−U 2ks1−M"ds1−M#dl ˛s1−kM"lds1−kM#ld, B#,"=−kfK#−,fH,K"−ggl=B",#. s29d With our previous relations s23d,s24d, and s27dwe can en- tirely express the elements of Eqs. s28dands29dby the RPAamplitudes and therefore we have a completely closed sys- tem of equation for the amplitudes X,Y. With the orthonor- mality relations s20dwe furthermore have A","=A#,#=A,A",#=A#,"=A8, B","=B#,#=B,B",#=B#,"=B8, s30d and, therefore, the SCRPA equation can be written in the following form: 1AA 8BB 8 A8AB 8B −B−B8−A−A8 −B8−B−A8−A21X"n X#n Y"n Y#n2=En1X"n X#n Y"n Y#n2.s31d The system s31dhas the two positive roots E1 =˛sA−A8d2−sB−B8d2andE2=˛sA+A8d2−sB+B8d2. The SCRPAequation s31dcan be solved numerically by iteration, leading, as expected, to the exact result. This latter fact canalso be seen analytically in noticing that, by symmetry, X "1=−X#1;Xsp,Y"1=−Y#1;Ysp, X"2=X#2;Xch,Y"2=Y#2;Ych. s32d Therefore the 4 34 equation s31ddecouples into two 2 32 equations corresponding to charge schdand spin sspd. Then we see that the exact ground-state wave function which con- tains only up to 2p-2h excitations u0l~s1+dJ"+J#+duHFls 33d is the exact vacuum to the RPA operators—i.e., QchsspduRPA l=0—under the condition that d=SY XD chsspd;tansfd. s34d We therfore can express the SCRPA equations by the single parameter fand obtain the solution analytically sup to the solution a nonlinear equation for fd. The solution agrees for all quantities with the exact result. For example the ground-state energy is given by E 0SCRPA=−2tcoss2fd+U 2f1−sin s2fdg. s35d This expression can either be derived directly from kHlusing Eq.s33dands34dor one uses a generalization of the standard RPA expression for the ground-state energy:6 E0SCRPA=EHF−1 2o ss1−kMsldfE2uYchu2+E1uYspu2g. s36d It is straightforward to verify that expressions s35dands36d are identical. The standard RPAexpression are recovered from Eq. s31d in replacing in all expectation values the RPA ground stateby the uncorrelated HF determinant. In Fig. 1 we compare FIG. 1. Excitation energies of the standard RPA sdashed lines d, SCRPA scrosses d, and exact solution ssolid lines das a function of U in the channels of charge schdand longitudinal spin sspdfor the two-site case.JEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-4the standard RPA with the SCRPA and exact results for the excitation energies and in Fig. 2 the corresponding ground-state energies together with the HF values are shown. Fromthese ﬁgures one should especially appreciate the long waythe SCRPA has gone from the s-RPA to recover the exactresult. For instance it is clearly seen that the instability of thes-RPA at U=2 is, as expected for such a small system, an artifact and is completely washed out by the self-consistenttreatment of quantum ﬂuctuations contained in the SCRPAapproach. Without explicit demonstration let us also mention that the SCRPA in the spin-transverse channel with Q n†=X1#2"nb2"†b1#†+X1"2#nb2#†b1"†−Y1#2"nb1#b2"−Y1"2#nb1"b2#as well as in the particle-particle channel with Q†=Xb2"†b2#† −Yb1#b1"also gives the exact solution for the two-site prob- lem. How the pp-SCRPAworks can be seen in Ref. 10 where for the pairing problem the two-particle problem is alsosolved exactly. The fact that the SCRPA solves the two-site problem ex- actly is nontrivial, since other well-known many-bodyapproaches, 14–16as already mentioned, so far failed to obtain this limit correctly. III. SIX-SITE PROBLEM After this positive experience with the two-site problem we next will consider the one-dimensional six-sites case, asfor the four-site case problems appear needing particularconsiderations to be outlined in Sec. IV. We again considerthe plane-wave transformation explained in Sec. II with thecorresponding Hamiltonian in momentum space s11d.I nt h e ﬁrst Brillouin zone − płk,pwe have for N=6 the follow- ing wave numbers: k1=0,k2=p 3,k3=−p 3, k4=2p 3,k5=−2p 3,k6=−p. s37d With the HF transformationah,s=bh,s†,ap,s=bp,s, s38d such that bk,suHFl=0 for all k, we can write the Hamiltonian in the following way snormal order with respect to b†,bd: H=HHF+Huqu=0+Huqu=p/3+Huqu=2p/3+Huqu=p,s39d where HHF=E0HF+o sse4n˜4,s+e5n˜5,s+e6n˜6,s−e1n˜1,s −e2n˜2,s−e3n˜3,sd, s40ad Huqu=0=Go i=13 sn˜pi,"−n˜hi,"do j=13 sn˜pj,#−n˜hj,#d,s40bd Huqu=p/3=GhhfsS4",6"−+S6",5"+d−sS2",1"++S1",3"−d +sJ2",4"−+J5",3"+dgfsS6#,4#++S5#,6#−d−sS1#,2#−+S1#,3#−d +sJ4#,2#++J3#,5#−dgj+c.c. j, s40cd Huqu=2p/3=GhhfsS5",4"+−S3",2"+d+sJ1",5"−+J4",1"++J3",6"− +J6",2"+dgfsS4#,5#−−S2#,3#−d+sJ5#,1#++J1#,4#−+J6#,3#+ +J2#,6#−dgj+c.c. j, s40dd Huqu=p=GfsJ1",6"−+J2",5"−+J3",4"−d+c.c. gfsJ1#,6#−+J2#,5#− +J3#,4#−d+c.c. g, s40ed with the deﬁnition of operators n˜k,s=bk,s†bk,s, Jph,s−=bh,sbp,s,Jph,s+=sJph,s−d† Sll8,s+=bl,s†bl8,s, with l.l8Sl8l,s−=sSll8,s+d†,s41d and EHF=−8t+3 4U, e1=−2t+U 2,e2=e3=−t+U 2, e4=e5=t+U 2,e6=2t+U 2, G=U 6. s42d The level scheme is shown in Fig. 3. The hole states are labeledh=h1,2,3 jand the particle states p=h4,5,6 j. The HF ground state is uHFl=a1,"†a1,#†a2,"†a2,#†a3,"†a3,#†u−l. s43d We see that the Hamiltonian for six sites has largely the same structure as the one for two sites. It is only augmented FIG. 2. Ground-state energy in HF sdot-dashed line d, standard RPA sdashed line d, SCRPA scrosses d, and exact solution ssolid line d as a function of Uin the charge and longitudinal spin responses for the two-site case.SELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-5byHuqu=p/3+Huqu=2p/3which contains the Soperators on which we will comment below. There are three different absolute values of momentum transfers as shown in Table I. Since the momentum transferuquis a good quantum number, the RPA equations are block diagonal and can be written down for each uquvalue sepa- rately. For example, for uqu= p/3 we have the following RPA operator for charge and longitudinal spin excitations: Ququ=p/3,n†=X2",4"nK4",2"++X2#,4#nK4#,2#++X3",5"nK5",3"+ +X3#,5#nK5#,3#+−Y2",4"nK2",4"−−Y2#,4#nK2#,4#− −Y3",5"nK5",3"−−Y3#,5#nK3#,5#−, s44d where Kps,hs±=Jps,hs± ˛1−kMps,hsls45d and Mps,hs=n˜p,s+n˜h,s. s46d We write this RPA operator in shorthand notation asQn†=o i=141 ˛1−kMilsXinJi+−YinJi−d, s47d again with the properties unl=Qn†u0l, s48ad Qnu0l=0. s48bd The matrix elements in the SCRPA equation SA B −B−ADSXn YnD=EnSXn YnD are then of the form Ai,i8=kfJi8−fH,Ji+ggl ˛s1−kMi8lds1−kMild, s49ad Bi,i8=kfJi8−fH,Ji−ggl ˛s1−kMi8lds1−kMild. s49bd Since the SCRPA equations have the same mathematical structure as the standard RPA, one also has equivalent ortho- normality relations oisXinXin8−YinYin8d=dnn8, etc., in analogy to Eqs. s20dof the two-site case. This allows us to invert Eq. s47dand to calculate the expectation values which will ap- pear in Eqs. s49adands49bdin complete analogy to Eq. s23d. The missing expectation values kMilcan be expressed by the XandYamplitudes in observing that Ji±andJi0 =1 2sMi−1dform, as in the two-site case, an SUs2dLie alge- bra for spin-1 2particles. Using the Casimir relation one again obtainsMi=2Ji+Ji−and thus kMil=2onuYinu2 1+2onuYinu2. s50d We also will need expectation values of MiMj=4Ji+Jj−Jj+Ji−foriÞj fforMiMi=2Miwe can use Eq. s50dg. Those can again be calculated following the same procedure as outlined in Eq.s27dand Appendix A. In order to solve the SCRPAequations we now practically havepreparedallweneed.Nonetheless,atthispointwehaveto discuss a limitation of our RPA ansatz s44dwhich is not absolutely necessary but which turned out to be convenientfor numerical reasons. The fact is that our RPA ansatz isrestricted to ph and hp conﬁgurations, as this is also the casein standard RPA. In the latter case this is a strict consequence of the use of HF occupation numbers n p0andnh0with values zero or one, respectively. In the SCRPA case with a corre-lated ground state the occupation numbers are different fromzero and one and a priorithere is no formal reason not to include into the RPA operator also pp and hh conﬁgurations of the form a p†ap8;bp†bp8andah†ah8;−bh8†bh. Such terms are usually called scattering or anomalous terms.19With rounded occupation numbers the SCRPA equations satT=0dare for-TABLE I. The various momentum transfers in the six-site case. uqu=2p 3uqu=p uqu=p 3 51!q51=−2p/36 1 !q61=−p42!q42=+p/3 41!q41=+2 p/35 2 !q52=−p53!q53=−p/3 62!q62=+2 p/34 3 !q43=+p 63!q63=−2p/3 FIG. 3. Excitation spectrum of the HF ground state U=0 for the chain with six sites at half ﬁlling and projection of spin ms=0. The occupied states are represented by the solid arrows and those notoccupied are represented by the dashed arrows.JEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-6mally and mathematically equivalent to standard RPA equa- tions at ﬁnite temperature where also pp and hh componentsare to be included, in principle. 18The inclusion of those scat- tering terms18,19ftheSterms in Eq. s39dgusually is of little quantitative consequence,11but entails, however, the impor- tant formal property that, as for the standard RPA, theenergy-weighted sum rule is fulﬁlled exactly. 11,19In spite of this desirable feature, we had to refrain from the inclusion ofthe scattering conﬁgurations in this work because the factors ˛1−kMilby which the SCRPA matrix is divided fsee Eqs. s49adands49bdgcan become very small in these cases and this perturbed the convergence process of the iterative solu-tion of the SCRPAequations. Though we do not exclude thata more adequate numerical procedure could be found to sta-bilize the iteration cycle, we decided to postpone such aninvestigation, because, as already mentioned and as will beshown later, the inﬂuence of the scattering terms is, as foundalready in other studies, 11very small. We will shortly come back to this discussion when presenting the results for theenergy-weighted sum rule below. As a consequence and forconsistency we then also will have to disregard the Sterms of the Hamiltonian sremember that also in standard RPA these terms do not contribute d. Under these conditions we then obtain a completely closed systemof SCRPA equations.For completeness we give some examples of SCRPA matrixelements which correspond to the ansatz s44dforuqu= p/3: A1,1=kfJ2",4"−fH,J4",2"+ggl s1−kM24,"ld =e4−e2−Gh2kJ2",4"−sJ3#,5#−+J4#,2#+dl +ksJ1",4"−+J2",6"−dsJ1#,5#−+J3#,6#−+J4#,1#++J6#,2#+dl +ksJ3",4"−+J2",5"−dfsJ1#,6#−+J2#,5#−+J3#,4#−d +c.c. gljs1−kM24,"ld−1, s51ad A2,1=kfJ2#,4#−fH,J4",2"+ggl ˛s1−kM24,#lds1−kM24,"ld =Ghks1−M24,"ds1−M24,#dl +ksJ4",1"+−J6",2"+dsJ1#,4#−−J2#,6#−dl +ksJ4",3"+−J5",2"+dsJ3#,4#−−J2#,5#−dlj 3hs1−kM24,#lds1−kM24,"ldj−1/2 As 51bd The other matrix elements can be elaborated along the same lines. Of course in the approximation where the expec-tation values in Eqs. s51adands51bdare evaluated with the HF ground state the usual matrix elements of the standardRPA are recovered. We should also mention that in expres-sions s51adand s51bdexpectation values such as, for ex- ample, kJ 1",4"−J4#,1#+lwhich involve momentum transfers other than the one under consideration suq3u=p/3 in the spe- ciﬁc example dmust be discarded. That this implicit channel coupling cannot be taken into account without deterioratingthe quality of the SCRPAsolutions is an empirical law whichwas established quite sometime ago. 20It is part of the decou-pling scheme and it is intuitively understandable that, since each channel is summing speciﬁc correlations, one cannotmix the channels implicitly without perturbing the balance ofthe minimization procedure which is done channel by chan-nel. It can also be noticed that, neglecting the Sterms inH, the channel coupling disappears. We here give for the transfer uqu= p/3 the totality of the elements of the matrix SCRPA, AandB, just as was used in the numerical calculation. For others transfers there will beanalogous expressions. Indeed with the abbreviations i=1;s2",4"d,i=2;s2#,4#d, i=3;s3",5"d,i=4;s3#,5#d, the elements of matrices AandBare given by A 1,1=e4−e2−2GkJ2",4"−sJ3#,5#−+J4#,2#+dl 1−kM24,"l, A2,1=Gks1−M24,"ds1−M24,#dl ˛s1−kM24,#lds1−kM24,"ld, A3,1=A4,1=A3,2=A4,2=0, A2,2=e4−e2−2GksJ3",5"−+J4",2"+dJ2#,4#−l 1−kM24,#l, A3,3=e5−e3−2GkJ3",5"−sJ2#,4#−+J5#,3#+dl 1−kM35,"l, A4,3=Gks1−M35,"ds1−M35,#dl ˛s1−kM35,#lds1−kM35,"ld, A4,4=e5−e3−2GksJ2",4"−+J5",3"+dJ3#,5#−l 1−kM35,#l, s52ad B1,1=−2GkJ2",4"−sJ2#,4#−+J5#,3#+dl 1−kM24,"l, B2,1=B3,1=B4,2=B4,3=0, B4,1=Gks1−M24,"ds1−M35,#dl ˛s1−kM35,#lds1−kM24,"ld, B2,2=−2GksJ2",4"−+J5",3"+dJ2#,4#−l 1−kM24,#l, B3,2=Gks1−M35,"ds1−M24,#dl ˛s1−kM24,#lds1−kM35,"ld, B3,3=−2GkJ3",5"−sJ3#,5#−+J4#,2#+dl 1−kM35,"l,SELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-7B4,4=−2GksJ3",5"−+J4",2"+dJ3#,5#−l 1−kM35,#l. s52bd Let us add that the matrices AandBare symmetric and that the expectation values kﬂlin Eqs. s52adands52bdcan be expressed in an analogous way as the expectation values s23d ands27dby the amplitudes X,Y. The structure of the self-consistent matrix elements s52ad ands52bdis also quite transparent: the bare interaction which survives in the limit of the standard RPA is renormalized—i.e., screened—by two-body correlation functions which arecalculated self-consistently. The general structure of thescheme is in a way similar to the one proposed by Tremblayand co-workers; 16however, the details of the expressions and the spirit of derivation are different. One can also interpretour theory as a mean-ﬁeld theory of quantum ﬂuctuations asthis was done in Ref. 9. Let us now come to the presentation of the results. In Figs. 4, 5, and 6 we display the excitation energies in thethree channels uqu= p,2p/3, and p/3 as a function of U/t. The exact values are given by the solid lines, the SCRPAones by crosses, and the ones corresponding to the standardRPA by the dashed lines. We see that in all three cases theSCRPA results are excellent and a strong improvement overthe standard RPA.As expected, this is particularly importantat the phase transition points where the lowest root of thestandard RPA goes to zero, indicating the onset of a stag-gered magnetization on the mean-ﬁeld level. It is particularlyinteresting that the SCRPA allows one to go beyond themean-ﬁeld instability point. However, contrary to the two-site case where the SCRPA, in the plane-wave basis, solvedthe model for all values of U, here at some values U.U cr the system “feels” the phase transition and the SCRPA stops to converge and also deteriorates in quality. Up to these val-ues ofUthe SCRPA shows very good agreement with the exact solution and in particular it completely smears thesharp phase transition point of the standard RPAwhich is an artifact of the linearization. In Fig. 7 we show the ground-state energy fsee Eq. s36dg E 0SCRPA=EHF−o nEno is1−kMilduYinu2s53d as a function of U. In addition to the exact, SCRPA, and s-RPAvalues we also show the HF energy.Again we see thatthe SCRPAis in excellent agreement with the exact solution.The standard RPA is also good for low values of Ubut strongly deteriorates close to the lowest phase transitionpoint which occurs in the uqu= pchannel at U=12t/5. The HF energies, on the contrary, deviate quite strongly from theexact values. The reader certainly has remarked that our RPA ansatz s44dhas so far not separated charge and spin excitations. In the two-site problem this was automatically and exactly the FIG. 4. Energies of excited states in the standard RPA, SCRPA, and exact cases as a function of Ufor six sites with spin projection ms=0 and for uqu=p. States of the charge response and those of the longitudinal spin response are denoted by chandsp, respectively. FIG. 5. Same as Fig. 4 but for uqu=2p/3. FIG. 6. Same as Fig. 4 but for uqu=p/3.JEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-8case. However, here, since we did not consider the Sopera- tors in the Hamiltonian or the RPA operator, spin symmetryis violated. On the other hand, this permits us to evaluate theimportance of the Soperators. Normally the eigenvectors of the RPA matrix should be such that for charge schdexcita- tions the operators J ph"++Jph#+andJph"−+Jph#−can be factored whereas for spin sspdexcitations the combinations Jph"+ −Jph#+andJph"−−Jph#−hold. Because of our violation of spin symmetry, this factorization is not exact. To have a measureof this violation we plot in Fig. 8 the ratio r=uX ph"nu−uXph#nu uXph"nu+uXph#nu. s54d For exact spin symmetry, rshould be zero. From Fig. 8 we see that the violation is on the level of a fraction of 1%.This,therefore justiﬁes, a posteriori , having neglected the scatter- ing terms sSterms din the Hamiltonian and RPA operator. A further indication that Sterms are not important comes from the energy-weighted sum rule. We know that the sum ruleincluding the Sterms is fulﬁlled in the SCRPA. 13,19However, neglecting them gives a slight violation. Considering the ex-act relation L=R, s55d with L=o nsEn−E0duknuFu0lu2 =o n,uqusEn−E0duk0uQuqu,nFu0lu2 =o n,uqusEn−E0duk0ufQuqu,n,Fgu0lu2 =o n,uqusEn−E0dUo isuqud˛1−MisXin+YindU2,s56ad R=1 2k0ufF,fH,Fggu0l =o isuqud˛1−Mio i8suqud˛1−Mi8sAi,i8−Bi,i8d,s56bd with F=o isuqudsJi++H.c. d, s57d we trace in Fig. 9 the ratio j=sR−Ld/R. Again we see that the violation is on the level of a fraction of 1%, conﬁrming the very small inﬂuence of the scattering terms. A further quantity which crucially tests the ground-state correlations is the occupation numbers. We have no directaccess to them; however, we will use the so-called Cataraapproximation for their evaluation: 21 nps=knˆpsl=o hkJph,s+Jph,s−l=o hs1−kMphsldo nuYphsnu2, s58ad FIG. 7. Energy of the ground state in the HF, standard RPA, SCRPA, and exact cases as a function of Ufor six sites with spin projection ms=0. FIG. 8. The ratio rfEq.s54dgas a function of the interaction U for the ph excitations s2, 4dands3, 5din the channel uqu=p/3. FIG. 9. The ratio j=sR−Ld/Rof the energy-weighted sum rule in the charge response for the six-site case.SELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-9nhs=knˆhsl=o pkJph,s+Jph,s−l=1−o ps1−kMphsldo nuYphsnu2. s58bd We show these quantities in Figs. 10 and 11 in comparison with the exact values and the ones of the standard RPA. Weagain see the excellent performance of the SCRPA. Concluding this section we can say that the expectation we had from the two-site case, with its exact solution, havevery satisfactorily also been fulﬁlled in the six-site case.However, in spite of the very good performance of theSCRPA, there is the limitation that the SCRPA, in the sym-metry conserving basis of plane waves used here, cannot beemployed in the strong- Ulimit. One also may wonder howthe extension to cases with sites number 2+4 nwithn.1 works. For such cases it does not make sense anymore toelaborate the Hamiltonian in its detailed form as given in Eq.s40d. This explicit expression was only given to make clear the detailed internal structure of the approach for a deﬁniteexample. In the general case with many sites one would justtake the form s11dof the Hamiltonian, calculate the double commutators as needed in Eqs. s8d, and then express the resulting correlation functions by the XandYamplitudes. That such a program is feasible in terms of analytic work andnumerical execution was demonstrated in our earlier work onthe multilevel pairing model 10where cases up to 100 levels were treated. However, this number was not considered anupper limit. Though the present model is slightly more com-plicated, we think that a generalization to the case of manysites is perfectly possible. It needs, however, some invest-ment which is planned for the future. This also concerns theD=2 case. Another question to ask is whether the degrada- tion of the SCRPA results going from the N=2 to the N=6 case does not go on considering N=10,14, etc.? One again may cite the experience with the multilevel pairing model 10 where also the N=2 case turned out to be exact in the SCRPA but not the other cases. However, all N.2 cases showed more or less the same degrees of accuracy: excellentresults of SCRPA up to the phase transition point and dete-rioration beyond. Since this behavior has also been found insimpler models, 12we think that this is a generic feature of the SCRPA and that this behavior will also translate to thecase of the present model. Another problem for further work is how to continue the present theory into the strong-coupling regime. Of course,there exists the possibility to perform the SCRPA in thesymmetry-broken basis, but details and how to match withthe symmetry-unbroken phase must still be worked out.Alsothe inclusion of higher-order operators, as will shortly bediscussed in the next section, may be an interesting directionin this respect. IV. FOUR-SITE PROBLEM A. Symmetry-unbroken case The problem of the four-site case is easily located in re- garding the level scheme of Fig. 12 ssee also Ref. 22 dealing with the attractive Hubbard model in 1D d. We see that the FIG. 10. Occupation numbers as function of the interaction U for various values of the momenta kfor states above the Fermi level. For each approximation, s-RPA and SCRPA, the occupationnumbers are represented in increasing order like ks− p, −2p/3,2 p/3d. Let us notice that the modes k=2p/3 andk= −2p/3 are degenerate. FIG. 11. Occupation numbers as a function of the interaction U for various values of the momenta kfor the holes states. For each approximation, s-RPA and SCRPA and exact solution, the occupa-tion numbers are represented like k=0, p/3,− p/3. Let us notice that the modes k=p/3 andk=−p/3 are degenerate. FIG. 12. Level spectrum for U=0 for the half-ﬁlled chain with four sites with spin projection ms=0. The occupied states are rep- resented by the solid arrows and those not occupied are representedby the dashed arrows.JEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-10Fermi energy coincides with the second level which is half ﬁlled. The uncorrelated ground state is therefore degenerateand excitations with momentum transfer uqu= pcost no en- ergy. On the other hand, for excitations with uqu=p/2 there is no problem. The corresponding RPA operator is given by Ququ=p/2,n†=X13,"nK31,"++X24,"nK42,"++X13,#nK31,#++X24,#nK42,#+ −Y13,"nK13,"−−Y24,"nK24,"−−Y13,#nK13,#−−Y24,#nK24,#−. s59d In Fig. 13 we show the results of the s-RPA and SCRPA, together with the exact solution. We see that the lower exci-tation is still very well reproduced by the SCRPA, whereasfor the second excited state the SCRPA only reduces thedifference of the s-RPA to exact by half. The real problemshows up for the transfer uqu= p. The corresponding operator is Ququ=p,n†=X14,"nK41,"++X14,#nK41,#++X23,"nK32,"++X23,#nK32,#+ −Y14,"nK14,"−−Y14,#nK14,#−−Y23,"nK23,"−−Y23,#nK23,#−. s60d The standard RPA produces a doubly degenerate zero mode independent of Uas seen in Fig. 14. As compared with the exact solution, we see that these two zero modes approxi-mate two very low-lying exact solutions. Unfortunately, be-cause of these modes at low energy, the SCRPAcould not bestabilized. The only possibility consisted in excluding the components K 32,"±andK32,#±in the RPA operator. Then self- consistency was achieved without problem and the result isshown in Fig. 14. The result of the SCRPA is halfway be-tween the s-RPA and the exact solution. On the other hand,because of the omission of the two lower states, the ground-state energy cannot correctly be calculated in the SCRPA.Therefore, for the four-site problem in the symmetry-unbroken basis splane waves d, the SCRPA cannot fully ac- count for the situation. B. Symmetry-broken basis An analysis of the HF solution shows that, as soon as U Þ0, the plane-wave state becomes unstable and the system prefers a staggered magnetization. The general HF transfor-mation can be written as 1c1,"† c2,"† c3,"† c4,"†2=1 ˛21v−10u u0−1−v v10u v01−v21a1,"† a2,"† a3,"† a4,"†2,s61ad 1c4,#† c3,#† c2,#† c1,#†2=1 ˛21v−10u u0−1−v v10u v01−v21a1,#† a2,#† a3,#† a4,#†2,s61bd withu=cos sqdandv=sinsqdeiw. The minimization of the ground-state energy, with uHFl=a1,"†a1,#†a2,"†a2,#†u−l, s62d shows that w=0 for any value of Uand the angle qis ob- tained from tan4sqd−U 2ttan3sqd−1=0. s63d The occupation numbers are given by n1,"=n3,"=n2,#=n4,#=1 2f1+sin2sqdg, FIG. 13. Energies of excited states with the standard RPA, SCRPA, and exact solution for four sites with spin projection ms =0 and for uqu=p/2 in the symmetry-unbroken basis. FIG. 14. Energies of excited states with the standard RPA, SCRPA, and exact solution for four sites with spin projection ms =0 and for uqu=pin the symmetry-unbroken basis.SELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-11n1,#=n3,#=n2,"=n4,"=1 2cos2sqd, s64d and shown in Fig. 15 which illustrates the spontaneous sym- metry breaking for any value of U. ForU!‘we have a perfect antiferromagnet. We can now perform a SCRPA calculation in the symmetry-broken basis. The RPA operators are given by Qsn†=X1s,3snK3s,1s++X2−s,4−snK4−s,2−s+−Y1s,3snK1s,3s− −Y2−s,4−snK2−s,4−s−, s65d with s=±1 2. We also have two other excitation operators Q1n†=X1",4"nK4",1"++X1#,4#nK4#,1#+−Y1",4"nK1",4"−−Y1#,4#nK1#,4#− s66d and Q2n†=X2",3"nK3",2"++X2#,3#nK3#,2#+−Y2",3"nK2",3"−−Y2#,3#nK2#,3#−. s67d In Figs. 16 and 17 we give the results. The most striking feature is that the s-RPA and SCRPA are very close and thatthe error with respect to the exact solution does not becomegreater than 25% for any value of U. Though the improve- ment of the SCRPA over the s-RPA is very small in eachchannel, at the end in the ground-state energy this sums to amore substantial correction in the right direction for theground-state energy.This is shown in Fig. 17 as a function ofatansU/td. We see that the HF, s-RPA and SCRPA become exact for U=0 andU!‘. In between the SCRPA deviates, e.g., by 8% from the exact result at U.6fatansU/td .1.4gwhereas this deviation is 20% for the s-RPA. Concluding this section on the four-site case at half ﬁlling we can say that in the symmetry-unbroken basis the SCRPAis unable to account for some low-lying excitations andtherefore fails to reproduce the ground-state energy as well.In the symmetry-broken basis the SCRPA gives very littlecorrection over the s-RPA. However, the maximum error isnot greater than 25% for all values of Ufor the excited statesand the ground-state energy in the SCRPA whereas this is 30% for the standard RPA. This may be an interesting resultin view of the importance of the so-called “plaquettes” ssee, e.g., Ref. 23 din high-T csuperconductivity. Nevertheless, even though one plaquette sfour sites dmay reasonably be described, the present approach cannot account for the situ-ation of many plaquettes in interaction which is the real situ-ation in 2D. For the future it is therefore very interesting todevelop an extension of the present SCRPA which not onlygives an exact solution for the two-site case but equally forthe four-site case. Such a generalization is possible in includ-ing into the RPA operator in addition to the fermion pairoperators also quadruples of fermion operators.This is a gen-eral principle and it has already been demonstrated to holdtrue in the case of the simpler Lipkin model. 24One could call such an extension a second SCRPA in analogy to the well-known standard second RPA which involves in addition tothe ph conﬁgurations also 2p-2h ones. In the case of many FIG. 15. Occupation numbers for site 1, n1,"etn1,#, as a func- tion of interaction Uin the symmetry-broken basis. FIG. 16. Energies of excited states with the standard RPA, SCRPA, and exact solution as a function of Ufor four sites with spin projection ms=0 in the symmetry-broken basis. FIG. 17. Ground-state energies in the HF, standard RPA, SCRPA, and exact solution as a function of atansU/tdfor four sites with spin projection ms=0 in the symmetry-broken basis.JEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-12plaquettes this second SCRPA would then constitute a self- consistent mean-ﬁeld theory for plaquettes. V. DISCUSSION, CONCLUSIONS, AND OUTLOOK In this work a many-body approach which was essentially developed in the nuclear physics context in recent years9has been applied to the Hubbard model for a ﬁnite number ofsites. The theory is an extension of the standard RPA, calledthe self-consistent RPA, which aims to correct its well-known deﬁciencies such as the quasiboson approximationwith its ensuing violation of the Pauli principle and its per-turbation theoretical aspect. Of course the appealing featuresof the RPA, such as, for instance, fulﬁllment of sum rules,restoration of broken symmetries, Goldstone theorem, nu-merical practicability, and physical transparency, should bekept as much as possible. That this is indeed the case withthe SCRPA has in the past been demonstrated with applica-tions to several nontrivial models 10such as, for instance, the many-level pairing sRichardson dmodel10and the three-level Lipkin model.11The SCRPA can be derived by minimizing an energy-weighted sum rule and it is therefore a nonpertur-bative variational approach though it is in general not of theRaleigh-Ritz type. The resulting equations are a nonlinearversion of the RPA type which can be interpreted as themean-ﬁeld equations of interacting quantum ﬂuctuations.Though the SCRPA equations are of the Schrödinger type,their nonlinearity nonetheless makes their numerical solutionquite demanding. We therefore thought it indicated to beginwith applications to the Hubbard model, restricting them tolow-dimensional cases given by a ﬁnite number of siteswhere exact diagonalization can easily be obtained. We thenlogically started out considering the two-site case swith pe- riodic boundary conditions d, increasing the number of sites by steps of 2—i.e., N=2,4,6,... To our satisfaction the SCRPA solves the two-site problem exactly for any value ofU. This, as a matter of fact, did not come entirely as a sur- prise, since the same happened already with the pairing prob-lem for two fermions 10and indeed it can be shown that the SCRPA solves a general two-body problem exactly.17It is nonetheless worth pointing out that other respectable many-body theories fail in the two-particle case, apart from thelow-Ulimit. In the four-site problem at half ﬁlling the SCRPA failed. This, as in all 4 nsn=1,2,3,... dcases, presents the particu- lar problem that the system is unstable with respect to the formation of staggered magnetization for any ﬁnite value ofUand this prevented the SCRPA solution from existing in the plane-wave basis for particular values of the momentumtransfer uqu.At the end of the paper we indicated that extend- ing the present RPAansatz of ph pairs to include quadruplesof fermion operators can solve not only the two-electron butalso the four-electron case exactly. This is particularly inter-esting in view of the fact that the four-site case splaquette d may be very important for the explanation of high- T csuper- conductivity, in considering the many plaquette conﬁgura-tions in 2D. 23In this work we jumped directly to the six-site problem which, as all 2+4 ncases, causes no particular dif- ﬁculties in the SCRPA, even in the symmetry-unbroken basisof plane waves. Of course, in the case of six sites, the SCRPA is not exact anymore. However, it is shown that theresults are still excellent for all quantities considered: excitedstates, ground state, and occupation numbers. Contrary to thetwo-site case, the SCRPA solutions in the plane-wave basiscannot be obtained for all values of U. Somewhere after the point where, as a function of U, the ﬁrst mean-ﬁeld instabil- ity shows up, the SCRPAalso starts to deteriorate and in factdoes not converge any longer. Often the mean-ﬁeld criticalvalue ofUis by passed by 20% up to 50% in the SCRPA, still staying excellent. However, to go into the strong- Ulimit we have to introduce the above mentioned quadruple fer-mion operators or perform a SCRPA calculation in thesymmetry-broken basis. 12Such investigations shall be left for the future. We also gave arguments why we think that,going to the N.6 cases, the precision we found for N=6 will not deteriorate. We therefore think that our formalismwill allow one to ﬁnd precise results for system sizes wherean exact diagonalization becomes prohibitive. Problems in2D with closed-shell conﬁgurations probably also can andshall be considered with the present formalism. Also, asshown in Ref. 10, the extension to ﬁnite temperatures is pos-sible. We also should mention that in this work we neglected the so-called scattering terms of the form a p†ap8orah†ah8—that is, fermion ph operators where either both indices are above or both below the Fermi level. In the standard RPA those con-ﬁgurations automatically decouple from the ph and hpspaces. However, in the SCRPA with its rounded occupationnumbers, there is formally no reason not to include them.Asa matter of fact, as shown in earlier work, 11,19to assure the fulﬁllment of the fsum rule and the restoration of broken symmetries, these scattering terms must be taken into ac-count. In the present case, as well as in earlier studies, thescattering terms seem to be almost linearly dependent withthe ordinary ph and hp conﬁgurations. This fact induced dif-ﬁculties with the iteration procedure, since they correspondto very small eigenvalues of the norm matrix. Though we donot exclude the possibility that this difﬁculty could be mas-tered with a more reﬁned numerical algorithm, we ﬁnallyrefrained from pursuing this effort, since we could show thatthe inﬂuence of the scattering terms on the results is only onthe level of a fraction of percent and also the fsum rule is only violated on this order. In short we showed that the SCRPA, as in previous mod- els, performs excellently in the symmetry-unbroken regimeof the Hubbard model. However, the high- Ulimit and the 4n-site cases need further developements. ACKNOWLEDGMENTS We are very grateful to B. K. Chakraverty and J. Ran- ninger for elucidating discussions. One of us sP.S.dthanksA. M. Tremblay for useful information. One of the authorssJ.D.dacknowledges support from the Spanish DGI under Grant No. BFM2003-05316-C02-02. APPENDIX A: PARTICLE-HOLE CORRELATION FUNCTIONS We give the commutations rules which will be useful in the calculation of the correlations functions in theph channel:SELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-13fQn,Qn8†g=o isXinXin8−YinYin8d1−Mi 1−kMil, fQn,Qn8g=o isYinXin8−XinYin8d1−Mi 1−kMil,fMi,Qng=−2Yino n1sXin1Qn1†+Yin1Qn1d, fMi,Qn†g=2Yino n1sYin1Qn1†+Xin1Qn1d. sA1d Thus, the following average values can be calculated scom- muting the Q’s to the right d: kQn3Qn2†Qn1Qn0†l=o ijsXin3Xin2−Yin3Yin2d s1−kMildsXjn1Xjn0−Yjn1Yjn0d s1−kMjldks1−Mids1−Mjdl, sA2d kQn3fQn1,Qn2†gQn0†l=o ijsXin3Xin0−Yin3Yin0d s1−kMildsXjn1Xjn2−Yjn1Yjn2d s1−kMjldks1−Mids1−Mjdl−2o iXin3Xin2Xin1Xin0−Yin3Yin2Yin1Yin0 s1−kMild. sA3d Finally, one can express the correlation function according to the amplitudes RPA, kMiland of kMiMjlas kQn3Qn1Qn2†Qn0†l=kQn3fQn1,Qn2†gQn0†l+kQn3Qn2†Qn1Qn0†l=2o ijsXin3Xin2−Yin3Yin2d s1−kMildsXjn1Xjn0−Yjn1Yjn0d s1−kMjldks1−Mids1−Mjdl +o ijsXin3Xin0−Yin3Yin0d s1−kMildsXjn1Xjn2−Yjn1Yjn2d s1−kMjldks1−Mids1−Mjdl−2o iXin3Xin2Xin1Xin0−Yin3Yin2Yin1Yin0 s1−kMild. sA4d APPENDIX B: DENSITY-DENSITY CORRELATION FUNCTIONS Given that this RPA formalism preserves the number of particles per spin, ssowing to the fact that the transforma- tion HF does not break the symmetry of spin d, one has Nˆs=Ns+o pn˜ps−o hn˜hs sB1d and the average value kNˆsl=Ns=N/2, which gives us o pkn˜psl=o hkn˜hsl. sB2d On the other hand, one also has NˆsNˆs8=SNs+o pn˜ps−o hn˜hsDSNs8+o p8n˜p8s8−o h8n˜h8s8D, sB3d with the average value kNˆsNˆs8l=Ns+Ns8, which gives usKSo pn˜ps−o hn˜hsDSo p8n˜p8s8−o h8n˜h8s8DL =Ns8KSo pn˜ps−o hn˜hsDL +NsKSo p8n˜p8s8−o h8n˜h8s8DL. sB4d Thus, for our case, there is the relation KSo pn˜p"−o hn˜h"DSo p8n˜p8#−o h8n˜h8#DL =3So pskn˜psl−o hskn˜hslD=0. sB5dJEMAIet al. PHYSICAL REVIEW B 71, 085115 s2005 d 085115-14*Also at the Département de Physique, Faculté des Sciences de Tunis, Université deTunis El-Manar 2092 El-Manar,Tunis,Tunis.Electronic address: jemai@ipno.in2p3.fr †Electronic address: schuck@ipno.in2p3.fr ‡Electronic address: dukelsky@iem.cfmac.csic.es §Electronic address: raouf.bennaceur@inrst.rnrt.tn 1A. L. Fetter and J. D. Walecka, Quantum Theory of Many- Particle Systems sMcGraw-Hill, New York, 1971 d. 2G. D. Mahan, Many-Particle Physics sPlenum Press, New York, 1981 d. 3F. Furche, Phys. Rev. B 64, 195120 s2001 d. 4G. F. Bertsch, C. Guet, and K. Hagino, physics/0306058 sunpub- lished d. 5A. K. Kerman and C. Y. Lin, Ann. Phys. sN.Y.d241, 185 s1995 d; 269,5 5 s1998 d. 6P. Ring and P. Schuck, The Nuclear Many-Body Problem sSpringer, Berlin, 1980 d. 7G. Baym and L. P. Kadanoff, Phys. Rev. 124287 s1961 d;G . Baym,ibid.127, 1391 s1962 d; Phys. Lett. 1, 242 s1962 d. 8J. P. Blaizot and G. Ripka, Quantum Theory of Finite Systems sMIT Press, Cambridge, MA, 1986 d. 9P. Schuck and S. Ethofer, Nucl. Phys. A 212, 269 s1973 d;J . Dukelsky and P. Schuck, ibid.512466s1990 d; J. Dukelsky, G. Röpke, and P. Schuck, ibid.628,1 7 s1998 d. 10P. Krüger and P. Schuck, Europhys. Lett. 27, 395 s1994 d;J .G . Hirsch, A. Mariano, J. Dukelsky, and P. Schuck, Ann. Phys.sN.Y.d296, 187 s2002 d; A. Storozhenko, P. Schuck, J. Dukelsky,G. Röpke, and A. Vdovin, ibid.307, 308 s2003 d. 11D. Delion, P. Schuck, and J. Dukelsky, nucl-th/0405002 sunpub- lishd. 12A. Rabhi, R. Bennaceur, G. Chanfray, and P. Schuck, Phys. Rev. C66, 064315 s2002 d. 13D. S. Schäfer and P. Schuck, Phys. Rev. B 59, 1712 s1999 d. 14F. Aryasetiawan, T. Miyake, and K. Terakura, Phys. Rev. Lett. 88, 166401 s2002 d. 15G. Seibold, F. Becca, and J. Lorenzana, Phys. Rev. B 67, 085108 s2003 d. 16Y. M. Vilk, L. Chen, and A.-M. S. Tremblay, Phys. Rev. B 49, 13 267 s1994 d; Physica C 235–240, 2235 s1994 d;Y .M .V i l ka n d A.-M. S. Tremblay, J. Phys. I 7, 1309 s1997 d; S. Allen and A.-M. S. Tremblay, Phys. Rev. B 64, 075115 s2001 d; B. Kyung, J. S. Landry, and A.-M. S. Tremblay, ibid.68, 174502 s2003 d. 17D. Delion and P. Schuck sunpublished d. 18H. M. Sommermann, Ann. Phys. sN.Y.d151, 163 s1983 d;D .V a u - therin and N. Vinh Mau, Nucl. Phys. B 422140s1984 d;N .V i n h Mau and D. Vautherin, ibid.445, 245 s1985 d. 19M. Grasso and F. Catara, Phys. Rev. C 63, 014317 s2000 d. 20D. Janssen and P. Schuck, Z. Phys. A 339,3 4 s1991 d. 21F. Catara, G. Piccitto, M. Sambataro, and N. Van Giai, Phys. Rev. B54, 17 536 s1996 d. 22K. Tanaka and F. Marsiglio, Phys. Rev. B 60, 3508 s1999 d. 23E. Altman and A. Auerbach, Phys. Rev. B 65, 104508 s2002 d. 24A. Storozhenko and P. Schuck sunpublished d.SELF-CONSISTENT RANDOM PHASE … PHYSICAL REVIEW B 71, 085115 s2005 d 085115-15
PhysRevB.92.235423.pdf	PHYSICAL REVIEW B 92, 235423 (2015) Electron-phonon coupling in metallic carbon nanotubes: Dispersionless electron propagation despite dissipation Roberto Rosati,1Fabrizio Dolcini,1,2and Fausto Rossi1 1Department of Applied Science and Technology, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy 2CNR-SPIN, Monte S. Angelo - via Cinthia, I-80126 Napoli, Italy (Received 26 September 2015; revised manuscript received 20 November 2015; published 14 December 2015) A recent study [Rosati, Dolcini, and Rossi, Appl. Phys. Lett. 106,243101 (2015 )] has predicted that, while in semiconducting single-walled carbon nanotubes (SWNTs) an electronic wave packet experiences the typicalspatial diffusion of conventional materials, in metallic SWNTs, its shape remains essentially unaltered up tomicrometer distances at room temperature, even in the presence of the electron-phonon coupling. Here, byutilizing a Lindblad-based density-matrix approach enabling us to account for both dissipation and decoherenceeffects, we test such a prediction by analyzing various aspects that were so far unexplored. In particular,accounting for initial nonequilibrium excitations, characterized by an excess energy E 0, and including both intra- and interband phonon scattering, we show that for realistically high values of E0the electronic diffusion is extremely small and nearly independent of its energetic distribution, in spite of a signiﬁcant energy-dissipationand decoherence dynamics. Furthermore, we demonstrate that the effect is robust with respect to the variation ofthe chemical potential. Our results thus suggest that metallic SWNTs are a promising platform to realize quantumchannels for the nondispersive transmission of electronic wave packets. DOI: 10.1103/PhysRevB.92.235423 PACS number(s): 72 .10.−d,73.63.−b,85.35.−p I. INTRODUCTION Using wave dynamics as a platform to encode information naturally offers the possibility to exploit the superposition of states and, thereby, to perform an intrinsically paralleltransfer and manipulation of information. To this purpose,a crucial ingredient is to generate sequences of wave packetspropagating coherently without overlapping to each other. Inquantum optics, where sources of single-photon wave packetshave been achieved since long, the control of light propagationand polarization with beam splitters and polarizers is extremely high, and photonic materials are nowadays considered a realistic platform to perform scalable quantum computing [ 1]. The exciting perspective to achieve a similar degree of con- trol using electron waves [ 2,3] has led to the implementation of single electron pumps with various setups [ 4–7]. However, despite a number of proposals [ 8–13], the realization of ﬂying qubits via single-electron wave packets of controllable shapeand phase that propagate ballistically in low-dimensionalconductors still remains a fascinating challenge in physics. A major difference between an electromagnetic and an elec- tronic wave is that, while the velocity of a photon is nearly inde-pendent of its wave vector k, the group velocity of an electron in conventional materials—characterized by a paraboliclikedispersion relation—depends on k, so that its components propagate with different velocities. This leads to an intrinsicspreading of an electron wave packet, even in the absenceof scattering processes. However, in metallic single-walledcarbon nanotubes (SWNTs), in graphene, and in the surfacestates of topological insulators, electrons behave as masslessrelativistic fermions and, just like photons, are characterizedby a linear spectrum, with the Fermi velocity v F∼106m/s playing the role of the speed of light c. This property makes such materials ideal candidates for an electronic alternative tophoton-based quantum information processing. In graphene,for instance, electron supercollimation has been predictedto occur when an external static and long-range disorder issuitably applied [ 14,15]. SWNTs are even more promising, in view of the accuracy reached in their synthesis [ 16,17], their behavior as one-dimensional ballistic conductors [ 18,19], and their versatility in forming perfectly aligned arrays forhigh-performance electronic devices [ 20–22]. Although, in principle, an electron wave packet can prop- agate along a metallic SWNT maintaining its initial shape,in realistic devices, such a property may be affected byscattering processes. Extrinsic scattering due to impuritiescan nowadays be made essentially negligible, by exploitingwell established fabrication techniques yielding ultraclean nanotubes by avoiding exposure to chemicals [ 16,23]. Intrinsic scattering mechanisms involve electron-electron and electron-phonon couplings. The former plays an important role at verylow temperatures, where it has been shown to lead to theCoulomb blockade [ 24] and Luttinger liquid behavior [ 25]. At intermediate and room temperature, however, electron-phononcoupling is the most important scattering mechanism, as experimental results indicate [ 26–28]. For these reasons, in the last few years, various theoretical studies have analyzedthe effects of electron-phonon coupling in SWNTs. Onthe one hand, models based on a classical-like treatmentof the electron-phonon coupling as an external oscillatingpotential [ 29–32] enable one to analyze the time-dependent evolution of single wave packets and to obtain the linear conductance by performing a suitable averaging over the initial state. These approaches, however, fail in capturing theintrinsically dissipative nature of the phonon bath. On the otherhand, treating electron-phonon coupling in SWNTs via theBoltzmann-equation schemes [ 33,34] does not allow one to account for electronic phase coherence. In a recent work [ 35], it has been shown that, while in semiconducting SWNTs an electronic wave packet spreadsalready for a scattering-free propagation, in metallic SWNTs,the shape of the wave packet can remain essentially unaltered,even in the presence of electron-phonon coupling, up to 1098-0121/2015/92(23)/235423(12) 235423-1 ©2015 American Physical SocietyROBERTO ROSATI, FABRIZIO DOLCINI, AND FAUSTO ROSSI PHYSICAL REVIEW B 92, 235423 (2015) micrometer distances at room temperature. Although such a result is quite promising, a number of fundamental questionsremain still open in the problem. In the ﬁrst instance, the caseof nonequilibrium carrier distributions has not been discussedso far. Secondly, the result of Ref. [ 35] is limited to the case of intraband phonon scattering, whereas interband couplingmay be signiﬁcant, especially due to breathing phonon modes.Furthermore, while the spatial dynamics of the wave packethas been discussed, it is still unclear how dissipation anddecoherence affect its energy and momentum distribution.Finally, it is crucial to understand whether and to whatextent the predicted dispersionless propagation is affected bya change of the chemical potential. This paper addresses these relevant problems. To this purpose, we apply a recently developed density-matrix ap-proach [ 36,37] that enables us to account for both energy- dissipation and decoherence effects. Focussing on the caseof a metallic SWNT, we demonstrate that the shape of thewave packet is essentially unaltered, even in the presence ofinterband electron-phonon coupling, provided that the excessenergy of the excitation is realistically high. Thus, despite asigniﬁcant energy-dissipation and decoherence dynamics, theelectronic diffusion in metallic SWNT is extremely small andnearly independent of the wave-packet energetic distribution.Furthermore, we show that this effect is weakly dependenton the chemical potential, at least at room temperature. Our results thus support the conclusion that metallic SWNTs can be considered as an electron-based platform for informationtransfer. The paper is organized as follows. In Sec. II, we describe the SWNT model utilized to account for the electronic andphononic energy spectrum, as well as for the correspondingelectron-phonon coupling. In Sec. III, we brieﬂy summa- rize the main aspects of the Lindblad-based density-matrixformalism developed in Ref. [ 37], providing the explicit expression for the electronic properties needed for the presentinvestigation, namely, the spatial and the energetic carrierdistributions. In Sec. IV, we present simulated experiments that enable us to quantify the impact of intra- as well asinterband carrier-phonon interactions on the propagation ofelectron wave packets for different initial conditions andchemical-potential values. As we shall discuss, the highlynontrivial interplay between energy dissipation and electronicquantum diffusion is crucial for such a purpose. Finally, inSec. V, we summarize our results and draw the conclusions. II. SWNT MODEL In order to describe our SWNT, we adopt the well established model developed by Ando and co-workers (seeRef. [ 38] and references therein), whose main ingredients needed for our analysis are summarized here below. Electronic properties . The low-energy electron dynamics in a SWNT decouples into two valleys around the KandK /prime points, described by the following Hamiltonian matrices in the sublattice basis, HK=/planckover2pi1vFσ·k,H K/prime=−/planckover2pi1vFσ∗·k, (1) where σ=(σx,σy) denote Pauli matrices acting on the twofold sublattice space, and k=(k⊥ n,ν,k) the carrier wavevector [ 38]. Here, kdenotes the continuous component along the SWNT axis ( /bardbl), whereas k⊥ n,ν=(n+vν/3)/Ris the discrete component along the circumference ( ⊥), where n is the electron subband, v=± 1f o rt h e K/K/primevalley, Rthe nanotube radius, and the index ν=0,±1 is deﬁned through the relation exp( iK·C)=exp(−iK/prime·C)=exp(2πiν/ 3), where Cis the vector rolling the graphene lattice into the SWNT. Since typical subband energy separations are of the order of eV, we shall focus on the lowest energy subband ( n=0), whose energy spectrum is independent of the valley v=K/K/prime=± 1 and is given by /epsilon1α=b/planckover2pi1vF/radicalbig k2+(ν/3R)2, (2) where α=(k,b) is the quantum number multilabel, with b=c/v=± 1 denoting the conduction and valence band, respectively. The related eigenvectors are ψαv(r)=/angbracketleftr\|αv/angbracketright=eık·r √ 4πRL/parenleftbigg1 bveıvθk/parenrightbigg , (3) where θkis the polar angle of the two-dimensional wave vector k, andLdenotes the nanotube length, which we assume to be the longest length scale in the problem, L→∞ . While forν/negationslash=0 the energy spectrum is gapped (semiconducting nanotube) and near k=0 is parabolic-like similarly to con- ventional semiconductors, for ν=0, the spectrum is gapless (metallic case), and the typical massless Dirac-cone structure is recovered. All armchair and (3 n,0) zigzag SWNTs are remarkable examples of the metallic case [ 38]. Phonon spectrum . In the long-wavelength phonon limit, the transversal phonon wave vector q⊥vanishes. The SWNT phononic spectrum only depends on the wave vector q along the SWNT axis, and includes zone-center and zone- boundary (ZB) modes [ 38–40]. The former can be grouped into (i) longitudinal (L) stretching modes, characterised by anacoustic (A) branch ω q,LA=vL\|q\|withvL/similarequal1.9×104m/s, and an optical (O) branch with /planckover2pi1ωLO/similarequal0.2 eV; (ii) breathing (Br) modes orthogonal to the nanotube surface, with a roughlyq-independent spectrum /planckover2pi1ω Br/similarequal0.14 eV ˚A/R; (iii) transverse (T) twisting modes, characterised by an acoustic branch withω q,TA=vT\|q\|withvT/similarequal1.5×104m/s and an optical (O) branch with /planckover2pi1ωTO/similarequal0.2 eV . In contrast, ZB modes, primarily corresponding to the Kekul ´e distortions, are characterized by a typical phonon energy /planckover2pi1ωZB/similarequal0.16 eV. Electron-phonon coupling . As far as electron-phonon cou- pling is concerned, a few preliminary remarks are in order.First, while zone-center modes induce intravalley scattering,zone-boundary modes cause intervalley scattering. Secondly,not all the above modes are relevant for our investigation. Inparticular, optical modes and zone-boundary modes typicallybecome important only at very high energies, as observed, e.g.,in transport measurements at high applied voltage bias [ 14,26]. As we shall discuss in detail later, we consider here valuesof nonequilibrium excess energy that are much smaller than/planckover2pi1ω LO,/planckover2pi1ωTO, and /planckover2pi1ωZB. In such a regime, only scattering with acoustic and breathing modes actually matters, whereas thecontribution of optical and zone-boundary modes is deﬁnitelynegligible. In fact, in Sec. IV B , we shall explicitly prove that this is true for TO and LO modes; zone-boundary modes,whose energies are comparable to O modes, are expected to 235423-2ELECTRON-PHONON COUPLING IN METALLIC CARBON . . . PHYSICAL REVIEW B 92, 235423 (2015) have a physically negligible impact too, with the unnecessary computational drawback of coupling the two valleys. Forthese reasons, we shall exclude ZB modes, and considerhenceforth intravalley processes only. The electron dynamicsthus decouples into the two valleys, and in each valley electronsscatter with each vibrational mode ξ=LA,TA,Br,LO,TO. Near the energetically relevant KandK /primepoints, the electron-phonon coupling for each vibrational mode is de-scribed by a 2 ×2 matrix acting on the electronic states of the related valley. In Refs. [ 39,41], explicit expressions for such matrices are given in the sublattice space. Here, in order totreat the electron-phonon coupling with the Lindblad-baseddensity-matrix formalism (see Sec. III), it is more suitable to switch from the sublattice basis to the αbasis of the electron eigenvectors ( 3). Then, the electron-phonon coupling is rewritten as ˆH e−ph=/summationdisplay αα/prime,v,qξ/parenleftbig gqξv− αα/primeˆc† αvˆbqξˆcα/primev+gqξv+ αα/primeˆc† αvˆb† qξˆcα/primev/parenrightbig ,(4) where gqξv± αα/prime=gqξv∓∗ α/primeα describe carrier-phonon matrix entries for the carrier transition α/prime→αoccurring in the valley v= K/K/prime=± 1 and resulting from the absorption ( −) or emission (+) of a phonon with a vibrational mode ξand wave vector q. Furthermore, ˆc† αv(ˆcαv) and ˆb† qξ(ˆbqξ) denote the creation (annihilation) of an electron in the αvsingle-particle states ( 3), and of a qξphonon, respectively. The explicit expression for the coefﬁcients gqξv± α,α/primeis given in Appendix Afor the case of a metallic SWNT. III. LINDBLAD-BASED DENSITY-MATRIX FORMALISM In order to investigate energy dissipation and decoherence as well as quantum-diffusion phenomena induced by thenanotube phonon bath on the otherwise phase-preservingelectron dynamics, we apply the general formalism introducedin Ref. [ 37] to the SWNT model just described. According to such a fully quantum-mechanical treatment, the time evolutionof the single-particle density matrix ρ v α1α2=/angbracketleftˆc† α2vˆcα1v/angbracketrightin the αbasis of the electronic single-particle eigenstates is given by dρv α1α2 dt=/epsilon1α1−/epsilon1α2 ı/planckover2pi1ρv α1α2+dρv α1α2 dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat. (5) In Eq. ( 5), the ﬁrst term on the right-hand side describes the scattering-free propagation, with /epsilon1αdenoting the single- particle electron eigenvalues, whereas the second term is anonlinear scattering superoperator, dρ v α1α2 dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat=1 2/summationdisplay α/primeα/prime 1α/prime 2,ξ/bracketleftbig/parenleftbig δα1α/prime−ρv α1α/prime/parenrightbig Pξv α/primeα2,α/prime 1α/prime 2ρv α/prime 1α/prime 2 −/parenleftbig δα/primeα/prime 1−ρv α/primeα/prime 1/parenrightbig Pξv∗ α/primeα/prime 1,α1α/prime 2ρv α/prime 2α2/bracketrightbig +H.c., (6) expressed via generalized scattering rates Pξv α1α2,α/prime 1α/prime 2, whose explicit form is microscopically derived from the electron- phonon Hamiltonian ( 4). More speciﬁcally, from the general scheme described in Ref. [ 37], one obtains Pξv α1α2,α/prime 1α/prime 2=/summationdisplay q±Aqξv± α1α/prime 1Aqξv±∗ α2α/prime 2(7)with Aqξv± αα/prime=/radicalBigg 2π/parenleftbig N◦ qξ+1 2±1 2/parenrightbig /planckover2pi1gqξv± αα/primeDqξ± αα/prime, (8) where N◦ qξis the Bose occupation number corresponding to the phonon qξ, and Dqξ± αα/prime=lim δ→0exp{−[(/epsilon1α−/epsilon1α/prime±/planckover2pi1ωqξ)/2δ]2} (2πδ2)1/4(9) is the Gaussian regularization of the total energy conservation constraint.1 The fully quantum-mechanical density-matrix equation ( 5) enables us to go beyond the conventional Boltzmann transportequation, whose space-independent version is straightfor-wardly recovered in the diagonal limit ( ρ v α1α2=fv α1δα1α2),2 where the generalized scattering rates reduce to the semiclas- sical rates provided by the standard Fermi’s golden rule: Pξv αα/prime=Pξv αα,α/primeα/prime. (10) The latter provide a qualitative information about the typical time scale of energy dissipation versus decoherence processesinduced by the various phonon modes, and will play a centralrole in understanding the simulated experiments presented inSec. IV. The average value of a generic single-particle opera- torˆa(with matrix entries a v α1α2in valley v) can be ex- pressed in terms of the single-particle density matrix asa=/summationtext v/summationtext α1α2ρv α1α2av α2α1. In particular, in our investigation, two physical quantities play a central role, namely, the spatialcarrier distribution in band b(b=c/v=± 1), n b(r)=/summationdisplay v/summationdisplay α1α2ρv α1α2nvα 2α1(r)δb1,bδb2,b (11) with nv α2α1(r)=/angbracketleftα2v\|r/angbracketright/angbracketleftr\|α1v/angbracketright, (12) and the corresponding (valley-averaged) carrier momentum distribution, fkb=1 2/summationdisplay v/summationdisplay α1α2ρv α1α2fv k,α2α1δb1,bδb2,b (13) with fv k,α2α1=/angbracketleftα2v\|k/angbracketright/angbracketleftk\|α1v/angbracketright. (14) In particular, for the case of a metallic SWNT, which is the focus here, the above expressions reduce to nb(r/bardbl)=1 2πRL/summationdisplay v/summationdisplay k1k2>0ρv k1b,k 2beı(k1−k2)r/bardbl(15) 1As discussed in Ref. [ 36], the choice of this regularization function, which has no speciﬁc impact on the asymptotic system dynamics, allows for a natural time symmetrization, crucial ingredient for the derivation of our Lindblad-like scattering superoperator. 2Notice that the derivation of the space-dependent Boltzmann equation goes beyond the mere diagonal limit mentioned here, and requires to perform a proper spatial coarse-graining procedure, as discussed, e.g., in Ref. [ 42]. 235423-3ROBERTO ROSATI, FABRIZIO DOLCINI, AND FAUSTO ROSSI PHYSICAL REVIEW B 92, 235423 (2015) and fkb=1 2/summationdisplay vρv kb,kb, (16) respectively. An inspection of Eq. ( 15) shows that a nonhomogeneous spatial carrier distribution is intimately related to the presenceof phase coherence between different states, k 1/negationslash=k2.I n particular, the constraint k1k2>0 indicates that the only density-matrix entries contributing to the spatial distributionare those with k 1andk2of equal sign. Such feature plays a crucial role in understanding the strong suppression of carrierdiffusion in metallic SWNTs, as we shall discuss in Sec. IVas well as in Appendix B. IV . SIMULATED EXPERIMENTS In order to show that metallic SWNTs can be utilized as quantum-mechanical channels for the nondispersive trans-mission of electronic wave packets, we have performed anumerical solution of the Lindblad-based nonlinear density-matrix equation (LBE) in Eq. ( 5). We shall henceforth focus on the metallic case [ ν=0i nE q .( 2)], and present results of simulated experiments, where the shape of an initially preparedwave packet is monitored while it evolves under the effect ofthe phonon bath. For any arbitrary electronic state, the density matrix can always be written as ρ v α1α2=ρ◦ α1α2+/Delta1ρv α1α2, (17) where ρ◦ α1α2=f◦ α1δα1α2is the homogeneous equilibrium state, characterized by a Fermi-Dirac distribution f◦ α≡f◦(/epsilon1α)=1 e(/epsilon1α−μ)/kBT+1(18) with chemical potential μand temperature T, and /Delta1ρv α1α2describes a localized excitation. Inserting Eq. ( 17)i n t o Eq. ( 15), the spatial carrier distribution is rewritten as nb(r/bardbl)=n◦ b+/Delta1nb(r/bardbl), (19) where n◦ bis the homogeneous equilibrium charge density and /Delta1nb(r/bardbl)=1 2πRL/summationdisplay v/summationdisplay k1k2>0/Delta1ρv k1b,k 2beı(k1−k2)r/bardbl, (20) is the inhomogeneous density excitation. Similarly, the mo- mentum carrier distribution, obtained by inserting Eqs. ( 17) into ( 16), reads fkb=f◦ kb+/Delta1fkb, (21) where f◦ kbis the equilibrium Fermi-Dirac distribution in Eq. ( 18) and /Delta1fkb=1 2/summationdisplay v/Delta1ρv kb,kb. (22) The spatial and energetic proﬁle (e.g., Gaussian-like) of the excitation /Delta1ρv α1α2can in principle be generated experimentally via a properly tailored optical excitation. While the descriptionof the speciﬁc optical-generation process is beyond the aim ofthe present paper, the localisation of the initial wave packetis a crucial aspect in our analysis. In Ref. [ 35], the excitation /Delta1ρ v α1α2was chosen to arise from the conduction band only and, most importantly, was assumed to have purely equilibriumdiagonal contributions. Here we aim to go beyond such asimpliﬁed scenario, and include nonequilibrium contributions,both in the conduction and the valence band. To this purpose,we take an initial state described by the following intravalleydensity-matrix excitation: /Delta1ρ v α1α2=b1δb1b2Ce−1 2(\|/epsilon1k\|−E0 /Delta1E)2e−/lscript\|k/prime\|, (23) where k=(k1+k2)/2 and k/prime=k1−k2are the usual center- of-mass momentum coordinates, while Ccan be regarded as a sort of excitation amplitude. Notice that the excitation ( 23)i s independent of the valley v=K/K/prime=± 1, and has opposite signs in the conduction ( b1=c=+ 1) and in the valence band ( b1=v=− 1), so that no total net charge excitation is injected into the SWNT. The parameter /lscriptplays the role of a delocalization length: for /lscript→∞ the homogeneous case is recovered, whereas for ﬁnite values of /lscript, an interstate phase coherence (intraband polarization) is present. Moreover, theenergetic distribution of the interband excitation is parame-terized by its average energy E 0, often referred to as excess energy, together with its standard deviation /Delta1E. Indeed, the nonequilibrium density matrix in Eq. ( 23) can be regarded as the after-excitation intraband state generated by an interbandlaser pulse with central photon energy /planckover2pi1ω=2E 0and pulse duration τ=/planckover2pi1/2/Delta1E. We shall focus here on the armchair (10,10) SWNT, a metallic nanotube characterized by a breathing-mode phononenergy /planckover2pi1ω Brof about 20 meV . In all the simulated experiments, we shall adopt as an initial condition the nonequilibriumexcitation in Eq. ( 23), choosing a delocalization length /lscript= 0.2μm (corresponding to a FWHM value of the initial peak of about 0 .4μm) and an energetic broadening /Delta1E=5m e V , corresponding to a laser-pulse duration τ/similarequal70 fs. We shall henceforth focus on the low-excitation regime, and take a valueof the excitation amplitude Cin Eq. ( 23) such as to produce a small deviation in the carrier distribution, i.e., /Delta1f kb/lessmuch1. A. Scattering-free evolution We start our analysis from the scattering-free propagation of the initial state in ( 23) switching off the electron-phonon coupling term in Eq. ( 6). Then, the solution of the density- matrix equation ( 5) is simply given by ρv α1α2(t)=ρv α1α2(0)e−ı(/epsilon1α1−/epsilon1α2)t//planckover2pi1, (24) leading to a density excitation /Delta1nb(r/bardbl,t)=/Delta1nr b(r/bardbl−vFt)+/Delta1nl b(r/bardbl+vFt), (25) where /Delta1nλ b(r/prime /bardbl)=/summationdisplay v/summationdisplay k1,k2∈/Omega1λ b/Delta1ρv k1b,k 2b(0)eı(k1−k2)r/prime /bardbl 2πRL, (26) withλ=r/l=± 1, and b=c/v=± 1. Here, /Omega1λ bdenotes a domain deﬁned as follows: k1/2∈/Omega1λ bifλbk 1/2>0. In Eq. ( 25), the components /Delta1nr/l bof the scattering-free carrier density excitations are straightforwardly identiﬁed as 235423-4ELECTRON-PHONON COUPLING IN METALLIC CARBON . . . PHYSICAL REVIEW B 92, 235423 (2015) right(r)- or left(l)-moving contributions in the bband, as they fulﬁll d/Delta1nr/l b dt=∓vFd/Delta1nr/l b dr/bardbl. (27) The splitting ( 25) of the carrier density evolution into right- or left-moving components is the hallmark of the wellknown symmetry underlying the Hamiltonian ( 1) in the case of metallic SWNTs: the right- and left-moving electronicstates ( 3) are characterized by opposite and k-independent pseudospin eigenvalues. Thus the carrier density, which tracesover the pseudospin degree of freedom [see Eqs. ( 11) and ( 12)], consists of oppositely propagating terms. Explicitly, in theK-valley right-moving carriers have k> 0 in the conduction band ( b=+ 1) and k< 0 in the valence band ( b=− 1) and are all characterized by a pseudospin +1, whereas left-moving components have k< 0 in the conduction band ( b=+ 1) and k> 0 in the valence band ( b=− 1) and are all characterized by pseudospin −1. The opposite pseudospin eigenvalues occur in the K /primevalley. Importantly, for a given propagation direction, all electrons are characterized by the very same velocity vF, so that no wave-packet dispersion occurs. The initial chargepeak thus splits into two components, which travel in oppositedirections with velocity ±v Fand preserve their shape. This is shown in Fig. 1, where the charge excitation ( 20)f o rt h e conduction band ( b=c) is plotted as a function of the position along the SWNT axis, for an excess energy E0of 10 meV (solid curves) and 50 meV (dashed curves) at three different times:t=0 ps (third peak), t=1 ps (second and fourth peaks), and t=2 ps (ﬁrst and ﬁfth peaks). Similarly, an equal and opposite charge excitation arises from the valence band ( b=v) (not plotted here). Importantly, as can be seen from Fig. 1, for a metallic SWNT, the shape and the propagation dynamics of the electron -2 -1 0 1 20.00.51.0charge density (arb. units) position ( µm)E0=1 0m e V E0=5 0m e V FIG. 1. (Color online) Scattering-free dynamics of an electronic wave packet in a metallic SWNT corresponding to the initial condition in Eq. ( 23): conduction-band ( b=c) excitation charge distribution in Eq. ( 20) as a function of the position along the SWNT axis for an excess energy E0of 10 meV (solid curves) and 50 meV (dashed curves) at three different times: t=0 ps (third peak), t=1 ps (second and fourth peaks), and t=2 ps (ﬁrst and ﬁfth peaks). Note that solid and dashed lines almost coincide (see text).wave packet is nearly independent of the initial excess energy E0, which is once again a peculiar feature stemming from the linearity of the band. The above scenario strongly differs from the semiconduct- ing SWNT case in various aspects: in the ﬁrst instance, inthe latter case, right- and left-moving electronic eigenstatesare characterized by a k-dependent pseudospin direction, similarly to a conventional material in the presence of spin-orbit coupling, so that the carrier density is not simply thesum of right- and left-moving terms, but also mixed termsarise. Secondly, because of the nonlinearity of the band,the propagation velocity depends on the wave vector k. As a consequence, the wave packet experiences the typicaldispersion of conventional (i.e., parabolic-band) materials, asobserved in Ref. [ 35]. Finally, a dependence on the initial excess energy E 0arises in semiconducting SWNT. B. Effects of electron-phonon coupling Let us now switch on the electron-phonon coupling and address the crucial question of whether and how-energydissipation and decoherence modify such an ideal dispersion-free scenario. To this purpose, we have performed a set ofsimulated experiments based on the LBE ( 5), including all the relevant phonon modes discussed in Sec. II. 1. Total scattering rates To start our analysis, a useful insight about the typical energy-relaxation time scale is provided by the semiclassical ratesPξv αα/primein Eq. ( 10), via the following total scattering rates: /Gamma1ξ k,b→b/prime=1 2/summationdisplay v/summationdisplay k/prime/negationslash=k(1−f◦ k/primeb/prime)Pξv k/primeb/prime,kb, (28) where the generic (intravalley) transition kb→k/primeb/primeis multi- plied by the Pauli-blocking factor of the ﬁnal state. The totalscattering rates ( 28) are displayed in Fig. 2as a function of the conduction energy ( /epsilon1=/planckover2pi1v F\|k\|) for the (10,10) SWNT. As one can see, for both intraband and interband processes,the dominant (i.e., fastest) dissipation channels are due tooptical (LO and TO) and breathing (Br) phonon modes, whichare expected to induce a signiﬁcant energy dissipation anddecoherence, in view of their strongly inelastic nature. Inparticular, for values of E 0signiﬁcantly smaller than the optical-phonon energy ( /similarequal200 meV), the primary dissipation channel is ascribed to Br phonon modes. Furthermore, dueto the different threshold mechanisms for intraband andinterband scattering (both dictated by the phonon energy/planckover2pi1ω Br/similarequal20 meV), the impact of Br modes is expected to be strongly E0-dependent. In any case, the total scattering rates shown in Fig. 2would suggest that the carrier-phonon scattering induces energy dissipation and decoherence on apicosecond time scale. Note that LA-phonon scattering isabsent for the considered (10,10) SWNT: the only availabletransition is k→k, the so-called self-scattering. The crucial question to address is whether and to what extent such incoherent dynamics modiﬁes the dispersion-free propagation scenario of Fig. 1. Indeed, combining Eqs. ( 5), (17), and ( 20), in the presence of carrier-phonon 235423-5ROBERTO ROSATI, FABRIZIO DOLCINI, AND FAUSTO ROSSI PHYSICAL REVIEW B 92, 235423 (2015) 1E-41E-30.010.1110 intra + interinter 0 50 100 150 200 2501E-41E-30.010.1110intra energy (meV)total scattering rate (1/ps)1E-41E-30.010.1110 TA Br LO TO Tot FIG. 2. (Color online) Room-temperature ( T=300 K and μ=0) total scattering rates in Eq. ( 28) as a function of the conduction energy (/epsilon1=/planckover2pi1vF\|k\|) for intraband (top), interband (middle), and intra plus interband scattering processes (bottom) due to the various phonon modes: ξ=TA (dashed curves), Br (dotted-dotted-dashed curves), LO (short-dotted-dashed curves), TO (dotted-dashed curves), andtheir sum (solid curves). Note that the ξ=LO and ξ=TO curves almost coincide in every panel.scattering, the dispersion-free result in Eq. ( 27) is modiﬁed to d/Delta1nr/l b dt=∓vFd/Delta1nr/l b dr/bardbl+d/Delta1nr/l b dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat(29) with d/Delta1nλ b dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat=/summationdisplay v/summationdisplay k1,k2∈/Omega1λ beı(k1−k2)r/bardbl 2πRLdρv k1b,k 2b dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat, (30) and/Omega1λ bis deﬁned below Eq. ( 26). 2. Intraband scattering Let us start by considering the case of intraband scattering processes only, where all interband dissipation channels areswitched off. Figure 3shows a direct comparison between energetic (left panels) and spatial distributions (right panels)for conduction band excitation carriers at different times,for two values of the excess energy, E 0=10 meV (upper panels) and E0=50 meV (lower panels). The chemical potential is set here at the charge neutrality point, μ=0, so that the valence band excitation distributions are equal in magnitude and opposite in sign to the conduction ones, andare not explicitly shown. As one can see, both the energeticcarrier distributions (left panels) exhibit the typical phonon-replica scenario of ultrafast energy-relaxation experiments. Inparticular, the nature (i.e., number of emitted Br phonons) andtime scale of the dissipation process depend on the value ofthe excess energy E 0, and agree with the intraband scattering 02 0 4 0 6 0 8 00.00.51.0E0=1 0m e V E0=5 0m e VE0=1 0m e V initial intra (1ps) intra (2ps) energy (meV)charge density (arb. units) 012 position ( µm)0.00.51.0 E0=5 0m e V FIG. 3. (Color online) Room-temperature ( T=300 K and μ=0) dynamics of an electronic wave packet in a metallic SWNT corresponding to the initial condition in Eq. ( 23) in the presence of intraband scattering only: conduction-band ( b=c) excitation charge distribution in Eq. ( 22) as a function of the carrier energy /planckover2pi1vF\|k\|(left) and corresponding excitation charge distribution in Eq. ( 20) as a function of the position along the SWNT axis (right) at three different times [ t=0 (solid curves), t=1 ps (dashed curves), and t=2 ps (dash-dotted curves)] for two different excess energies: E0=10 meV (upper panels) and E0=50 meV (lower panels). Here all thick curves show the effects of the intraband carrier-phonon coupling accounted for by the LBE ( 5) while the thin ones in the right panels correspond to their scattering-free counterparts (see text). 235423-6ELECTRON-PHONON COUPLING IN METALLIC CARBON . . . PHYSICAL REVIEW B 92, 235423 (2015) rates reported in the upper panel of Fig. 2. In spite of such picosecond energy-relaxation and decoherence dynamics, thespatial carrier distributions (right panels) clearly show thatthe electron-phonon coupling does not signiﬁcantly alter theshape of the electron wave packet with respect to the idealscattering-free results (thin curves), so that (i) the propagationis essentially dispersionless up to the micrometric scale, evenat room temperature, and (ii) the small diffusion effect is nearlyindependent of E 0. In order to understand the origin of such shape-preserving dynamics, it is worth noting that in the electron-phononcoupling a natural distinction arises between forward andbackward scattering processes, namely processes where theinitial and ﬁnal electronic states have the same and oppositevelocity sign, respectively. In terms of our density-matrixformalism, since a quantum transition involves two pairs of momenta ( k 1,k2)→(k1+q,k 2+q), forward and backward processes can in principle interplay in Eq. ( 6). In semiconduct- ing materials such transitions can lead to scattering nonlocalityand quantum diffusion speed-up phenomena [ 42]; moreover, in Luttinger liquids, the forward component of the electron-phonon coupling can lead to Wentsel-Bardeen instabilitiesof the electron propagator [ 43]. However, in a metallic SWNT, due to the energy and momentum conservation, mixed(forward-backward) processes occupy a vanishing measuresubset of the phase space, and are irrelevant. Furthermore, a detailed investigation summarized in Appendix Bshows that intraband forward processes yield a negligible contributionto the scattering term in Eq. ( 30) and therefore have an extremely small impact on the wave-packet propagation. Thewave-packet dispersion (see right panels in Fig. 3) originates mainly from backward processes. Such conclusion, obtainedfrom a fully quantum-mechanical approach, turns out to beessentially similar to the expectation one would formulate ona semiclassical argument based on the Boltzmann theory. At room temperature, the backward scattering processes may be ascribed to different phonon modes, depending onthe type of SWNT: for armchair SWNT, like the (10,10) one,they are due to TA modes only, whereas for zigzag SWNTsthey are due to Br as well as to LA modes. We stress thatalso LO modes induce backward processes; however, due totheir high phonon energy [ 26], in the simulated experiments of Fig. 3their impact is extremely negligible. In turn, this also conﬁrms that the neglect of the zone-boundary modes—whose energy is comparable to the optical modes—is a goodapproximation. The scenario described so far is conﬁrmed by the forward- versus-backward total scattering rates reported in Fig. 4. As anticipated, the intraband scattering rates in the upperpanel of Fig. 2are dominated by forward processes (see upper panel in Fig. 4) which, in turn, are dominated by Br phonon modes. In contrast, the total scattering rate due tobackward processes (see lower panel in Fig. 4) is due to TA modes only, and is at least one order of magnitude smallercompared to the forward one. Recalling that the diffusionof an electronic wave packet in a metallic SWNT is mainlydetermined by backward processes (see Appendix B) and that the latter are characterized by a much longer time scale, we arethen able to explain the apparent discrepancy in Fig. 3between the energy-relaxation (left panels) and the quantum diffusion1E-41E-30.010.1110 TA Br LO TO Tottotal scattering rate (1/ps)forward 0 50 100 150 200 2501E-41E-30.010.1110 energy (meV)backward FIG. 4. (Color online) Forward-scattering (top) and backward- scattering components (bottom) of the intraband total scattering ratesreported in the upper panel of Fig. 2(see text). time scale (right panels). Moreover, the fact that quantum diffusion is mainly determined by backward processes, andthat the latter involve TA phonons only, explains well how thediffusion dynamics (right panels) is basically independent ofthe excess energy E 0. 3. Effects of interband scattering As a second step, we have included also interband carrier-phonon scattering, and analyzed how the simulatedexperiments of Fig. 3are modiﬁed by the presence of such processes. Figure 5shows again a direct comparison between energy-relaxation (left panels) and spatial-diffusion dynamics(right panels), for the same two values of E 0. While for E0= 10 meV the presence of interband scattering induces strongmodiﬁcations with respect to the intraband results of Fig. 3, forE 0=50 meV , the effect of interband coupling is hardly visible, both in terms of the energetic and the spatial carrierdistributions. Indeed, an inspection of the interband totalscattering rates reported in the central panel of Fig. 2shows that for the considered values of E 0the most efﬁcient (i.e., fastest) interband scattering channel is again ascribed to Br phononmodes; however, such a picosecond scattering mechanism isactive only for carrier energies smaller than /planckover2pi1ω Br. Moreover, in addition to an energetic carrier redistribution, the presence ofinterband transitions leads to a progressive decay of the initialexcitation charge /Delta1n b(r/bardbl)i nE q .( 20) via an interband charge transfer, which can be regarded as a net phonon-mediatedelectron-hole recombination process. The resulting loss ofconduction electrons may affect the nearly dispersion-freescenario of Fig. 3. However, its impact is directly related to the effective time scale of such phonon-induced interband transfer,which, in turn, depends on the fraction of below-threshold(/epsilon1</planckover2pi1ω Br) electrons, and therefore on the value of E0. Such highly nontrivial interplay between the conduction- band energy redistribution and electronic loss due to phonon-induced interband transfer is fully conﬁrmed by the twosimulated experiments of Fig. 5. For a given initial excitation peak with an excess energy E 0(see solid curves in the left panels), the conduction electrons experience a sequence 235423-7ROBERTO ROSATI, FABRIZIO DOLCINI, AND FAUSTO ROSSI PHYSICAL REVIEW B 92, 235423 (2015) 02 0 4 0 6 0 8 00.00.51.0 E0= 10 meV E0= 50 meV E0= 50 meV E0= 10 meV initial intra + inter (1ps) intra + inter (2ps) energy (meV) charge density (arb. units) 012 position ( µm)0.00.51.0 FIG. 5. (Color online) Same quantities as in Fig. 3, but in the presence of both intra- and interband carrier-phonon scattering. While for small excess energy ( E0=10 meV) interband scattering processes affect both the energy and space distributions (compare upper panels with those of Fig. 3), for higher excess energy ( E0=50 meV) interband scattering has a negligible impact (compare lower panels with those of Fig. 3). of Br-phonon emissions and/or absorptions, giving rise to corresponding phonon replica in the excitation charge dis-tribution. The resulting time scale of interband scattering isthen related to the number of emitted phonons needed toenter the below-threshold energy region, and thus increases forincreasing values of E 0. Such a behavior is fully conﬁrmed by the time evolution of the total excess density reported in Fig. 6, which shows that, by increasing E0from 10 to 50 meV , the net interband carrier transfer is reduced by more than one order ofmagnitude. The relevant conclusion is that for excess energiesE 0>/planckover2pi1ωBrthe room-temperature wave-packet propagation is 0120.40.50.60.70.80.91.0total excess density (arb. units) time (ps) E0 = 10 meV E0 = 50 meV FIG. 6. (Color online) Time evolution of the total excess density corresponding to the two simulated experiments of Fig. 5:E0= 10 meV (solid curve) and E0=50 meV (dashed curve) (see text).again essentially dispersionless up to the micrometric scale also in the presence of interband scattering. To conclude this section, we observe that the energy and space carrier distributions shown in Figs. 3and5have been chosen as the most suitable quantities to speciﬁcally addressthe problem of the wave-packet dispersion. The densitymatrix obtained by solving Eq. ( 6)—or equivalently its related Wigner function—encodes further information, however, itsdescription is beyond the purposes of the present paper. Asimilar analysis, carried out on parabolic quantum wires withinthe Wigner function formalism can be found, e.g., in Ref. [ 44]. 4. Effects of the chemical potential So far, all the described simulated experiments (see Figs. 3 and 5) have been performed for a value of the chemical potential μcorresponding to the charge neutrality point: μ=0. We now want to discuss the effects of the chemical potential on the wave-packet propagation. In particular, onewould expect that, as the chemical potential is increasedor decreased, the change in the occupation of initial andﬁnal electronic states available for electron-phonon scatteringalters the relative weight of intra- and interband scatteringcontributions [see Eq. ( 28)]. Furthermore, one expects that, away from the charge neutrality point μ=0, the magnitudes of conduction- and valence-band carriers become different. To analyze these effects, the simulated experiments of Fig. 5 have been repeated with varying the value of the chemicalpotential. Figure 7shows snapshots of the wave-packet spatial distribution taken 2 ps after the initial condition for differentvalues of μ. The upper and lower panels refer to the same two excess energy values of Figs. 1,3, and 5, namely, E 0=10 and 235423-8ELECTRON-PHONON COUPLING IN METALLIC CARBON . . . PHYSICAL REVIEW B 92, 235423 (2015) 0.00.5 E0=1 0m e Vcharge density (arb. units)E0=5 0m e V 1.0 1.5 2.0 2.50.00.5 position ( µm)1.95 2.00 2.050.400.410.420.431.95 2.00 2.050.180.190.20 FIG. 7. (Color online) Snapshot of the wave-packet spatial dis- tribution after 2 ps at room temperature for different values of the chemical potential: μ=0 (thin solid), μ=40 meV (dashed), andμ=− 80 meV (dashed-dotted). Upper and lower panels refer toE0=10 and 50 meV , respectively. A very weak dependence is observed on the chemical potential, which becomes appreciable only when zooming near the peaks, as shown by the two insets (see text). 50 meV , respectively. As one can see, the wave-packet spatial proﬁle exhibits a weak dependence on μ. Similarly, a small modiﬁcation was found on the energy-relaxation process, andhas not been reported here. Surprisingly, such independenceoccurs even for small values of excess energy E 0, where the interband contribution has been shown to modify the intrabandresults, as discussed above (see Figs. 5and6). Furthermore, only a minor difference turns out to arise for μ/negationslash=0 between the magnitudes of the conduction and valence band carrierdistributions: the relative difference of the maximum heightsis less than 2%. In order to explain such seemingly counterintuitive behav- ior, it is useful to describe the effect of the relevant interbandscattering channel, namely Br phonon modes, via a simpletwo-level toy model, which involves just one single conduction(c) and a single valence (v) state. More speciﬁcally, we shalldenote with /epsilon1 c/v=±/planckover2pi1ωBr/2 the corresponding energy levels, withfc/vthe corresponding electron populations, and with Pcv=WN◦ BrandPvc=W(N◦ Br+1) the interlevel absorption and emission rates, respectively ( N◦ Brdenoting the Breathing mode Bose occupation number). Within the conventional semi-classical picture, the time evolution of the electron populationis described by the following Boltzmann-like equation: df c dt=(1−fc)Pcvfv−(1−fv)Pvcfc=−dfv dt. (31) Writing the two electron populations as fc/v=f◦ c/v±/Delta1f (/Delta1f denoting the deviation from the thermal-equilibrium distribution f◦ c/v) and neglecting quadratic terms in /Delta1f, Eq. ( 31) reduces to d/Delta1 f dt=−/Gamma1/Delta1f, (32)where /Gamma1(μ)=(Pcv+Pvc)/parenleftbigg 1−/Delta1f◦ vc(μ) 2N◦ Br+1/parenrightbigg , (33) with/Delta1f◦ vc(μ)=f◦ v(μ)−f◦ c(μ) denoting the difference be- tween valence and conduction Fermi-Dirac functions. Equa-tion ( 32) shows that the initial excess population /Delta1fundergoes an exponential-decay dynamics according to the μ-dependent decay rate in Eq. ( 33), whose relative change with respect to theμ=0 case is given by /Delta1/Gamma1(μ)≡/Gamma1(μ)−/Gamma1(0) /Gamma1(0), (34) i.e., a positive, ﬁnite, and symmetric function of μ. This implies that the decay rate /Gamma1in Eq. ( 33)i sm i n i m a lf o r μ=0 and increases with \|μ\|, reaching a saturation value for \|μ\|→∞ . It is, however, straightforward to verify that for the parametersof the simulated experiments reported in Fig. 7, namely, T=300 K, \|μ\|=40 meV , and /planckover2pi1ω Br/similarequal20 meV , the relative change in Eq. ( 34) is only 3%. Moreover, also for \|μ\|→∞ the latter never exceeds its limiting value of about 7%. Weemphasize that such extremely weak μdependence is ascribed to the room-temperature regime considered here. Indeed,atT=77 K, the 3% value obtained at room temperature increases to about 140%, which implies a strong μdependence in the low-temperature limit, as expected. Regardless of thespeciﬁc μdependence, our analysis shows that the impact of interband carrier-phonon scattering is always minimum forμ=0. V . SUMMARY AND CONCLUSIONS We have investigated in detail the impact of carrier- phonon coupling on the dynamics of an electron wavepacket propagating in metallic SWNTs, utilizing a recentlydeveloped density-matrix approach [ 37] that enables us to account for both energy dissipation and decoherence effects.The recent study in Ref. [ 35] has been extended in various aspects in this paper: (i) we have considered the case ofnonequilibrium carrier distributions; (ii) we have includedinterband carrier-phonon coupling; (iii) we have analyzedthe effects of dissipation and related decoherence phenomenaon the wave-packet energetic distribution; and (iv) we havediscussed the effects of the chemical potential. Based on ouranalysis, we can extend the conclusion that in metallic SWNTsthe shape of the wave packet is essentially unaltered, even inthe presence of intraband as well as interband electron-phononcoupling, up to micrometer distances at room temperature. More speciﬁcally, our investigation has shown that, in spite of a signiﬁcant energy-dissipation and decoherence dynamics,electronic diffusion in metallic systems is extremely smallas well as nearly independent of the wave-packet energeticdistribution, namely, excess energy and chemical potential.Our results thus indicate that metallic SWNTs constitutea promising platform to realize quantum channels for thenondispersive transmission of electronic wave packets. 235423-9ROBERTO ROSATI, FABRIZIO DOLCINI, AND FAUSTO ROSSI PHYSICAL REVIEW B 92, 235423 (2015) ACKNOWLEDGMENTS We are grateful to Massimo Rontani for stimulating and fruitful discussions. We gratefully acknowledge funding bythe Graphene@PoliTo laboratory of the Politecnico di Torino,operating within the European FET-ICT Graphene Flag- ship project ( www.graphene-ﬂagship.eu ). F.D. also acknowl- edges ﬁnancial support from Italian FIRB 2012 projectHybridNanoDev (Grant No. RBFR1236VV). APPENDIX A: ELECTRON-PHONON COUPLING COEFFICIENTS In this Appendix, we provide the explicit expression for the gqξv± αα/primecoefﬁcients appearing in the electron-phonon coupling Hamiltonian ( 4), focusing on the case of a metallic SWNT [ ν=0i nE q .( 2)]. The coefﬁcients can be obtained from the 2 ×2 electron-phonon matrices given for the sublattice basis in Refs. [ 39,41], by changing to the eigenvector basis αdeﬁned in Eq. ( 3). Recalling that in gqξv± αα/primemultilabels for the electronic states are α=(k,b) andα/prime=(k/prime,b/prime), the conservation of total momentum implies that the gqξv± αα/primeexhibit the form gqξv± αα/prime=gξv k,k±q;b,b/primeδk±q,k/prime, (A1) where the gξv k,k±q;b,b/primeacquire the following expressions: gLAv k,k±q;b,b/prime=−/radicalBigg /planckover2pi1\|q\| 2NMv Le−iv(θ−θ/prime)/2/bracketleftbigg g1fs(\|q\|)/parenleftbigg1+bb/prime 2cosθ−θ/prime 2+iv1−bb/prime 2sinθ−θ/prime 2/parenrightbigg +g2vb+b/prime 2cos/parenleftbigg 3η+θ+θ/prime 2/parenrightbigg −g2ib−b/prime 2sin/parenleftbigg 3η+θ+θ/prime 2/parenrightbigg/bracketrightbigg , (A2) gTAv k,k±q;b,b/prime=/radicalBigg /planckover2pi1\|q\| 2NMv Te−iv(θ−θ/prime)/2g2/bracketleftbigg vb+b/prime 2sin/parenleftbigg 3η+θ+θ/prime 2/parenrightbigg +ib−b/prime 2cos/parenleftbigg 3η+θ+θ/prime 2/parenrightbigg/bracketrightbigg , (A3) gLOv k,k±q;b,b/prime=−23/2/planckover2pi1vF a2 0/radicalBigg /planckover2pi1 2NMω LOe−iv(θ−θ/prime)/2sgn(q)/bracketleftbiggb+b/prime 2ivcos/parenleftbiggθ+θ/prime 2/parenrightbigg +b−b/prime 2sin/parenleftbiggθ+θ/prime 2/parenrightbigg/bracketrightbigg , (A4) gTOv k,k±q;b,b/prime=−23/2/planckover2pi1vF a2 0/radicalBigg /planckover2pi1 2NMω TOe−iv(θ−θ/prime)/2sgn(q)/bracketleftbiggb+b/prime 2ivsin/parenleftbiggθ+θ/prime 2/parenrightbigg −b−b/prime 2cos/parenleftbiggθ+θ/prime 2/parenrightbigg/bracketrightbigg , (A5) gBrv k,k±q;b,b/prime=1 R/radicalBigg /planckover2pi1 2NMω Bre−iv(θ−θ/prime)/2/bracketleftbigg g1fs(\|q\|)/parenleftbigg1+bb/prime 2cosθ−θ/prime 2+iv1−bb/prime 2sinθ−θ/prime 2/parenrightbigg −g2vb+b/prime 2·cos/parenleftbigg 3η+θ+θ/prime 2/parenrightbigg +g2ib−b/prime 2sin/parenleftbigg 3η+θ+θ/prime 2/parenrightbigg/bracketrightbigg , (A6) where θ=sgn(k)π/2 andθ/prime=sgn(k±q)π/2 are shorthand notations for the polar angles θkandθk±qof the electron wave vectors kandk±qin the metallic SWNT. In the above equations, Ndenotes the number of unit cells, Rthe nanotube radius, M=19.9×10−27kg the mass of a carbon atom [ 31],a0=1.44˚A is the lattice spacing, and ηis the SWNT chirality angle (e.g., η=0 for zigzag and η=π/6f o r armchair SWNT [ 38]). The values for vL,vT,ωTO,ωLOandωBr are given in Sec. II. Furthermore, g1=30 eV and g2=1.5e V are the coupling constants related to deformation potential andbond-length change [ 39], respectively, while f s(\|q\|) denotes the screening function given in Ref. [ 45]. The hermiticity of the electron-phonon coupling ( 4) ensures gξv k,k±q;b,b/prime=gξv∗ k±q,k;b/prime,b, whereas the additional relation gξv k,k±q;b,b/prime=gξ−v∗ −k,−k∓q;b,b/primestems from time-reversal symmetry. APPENDIX B: ANALYSIS OF INTRABAND FORWARD SCATTERING PROCESSES In this Appendix, we show that in a metallic SWNT the electron diffusion dynamics is not affected by intrabandforward carrier-phonon scattering. To begin with, a comment is in order here. At the level of the electron-phonon Hamiltonian,each term in Eq. ( 4) can be written as a sum of forward and backward processes, where a forward (backward) contribution can be deﬁned as a quantum-mechanical transition wherethe electron group velocity in the ﬁnal state αhas the same (opposite) direction as the one in the initial state α /prime. However, in our approach based on the density matrix, two states are involved in ρv α1,α2, and quantum-mechanical transitions ( α1,α2)→(α/prime 1,α/prime 2) may in general mix backward and forward Hamiltonian contributions. For these reasons, weshall utilize here the term “forward” (“backward”) for those processes where the group velocity is preserved (changed) forboth density-matrix indices. A similar distinction can be made between intra- and interband processes, and weshall refer to intraband transitions as the ones where bothinitial and ﬁnal states are in the same band. In terms of the above deﬁnitions, the case of intraband forward transitions characterizes processes where the sign of both carrier wavevectors k 1andk2is preserved. We shall now argue that they do not contribute to the spatial electronic diffusion. 235423-10ELECTRON-PHONON COUPLING IN METALLIC CARBON . . . PHYSICAL REVIEW B 92, 235423 (2015) To this purpose, we ﬁrst consider the structure of the nonlinear scattering superoperator ( 6) and focus on the low- excitation regime considered in our simulated experiments.We note that, by inserting Eq. ( 17) into Eq. ( 6) and neglecting quadratic terms in /Delta1ρ v α1α2, the original scattering term reduces to the following linear superoperator: dρv α1α2 dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat=1 2/summationdisplay α/prime 1α/prime 2,ξ/parenleftbig Pξv α1α2,α/prime 1α/prime 2/Delta1ρv α/prime 1α/prime 2−Pξv∗ α/prime 1α/prime 1,α1α/prime 2/Delta1ρv α/prime 2α2/parenrightbig +H.c., (B1) with effective (i.e., μ-dependent) scattering rates Pξv α1α2,α/prime 1α/prime 2=/parenleftbig 1−f◦ α1/parenrightbig Pξv α1α2,α/prime 1α/prime 2+f◦ α1Pξv∗ α/prime 1α/prime 2,α1α2. (B2) In the case of intraband forward scattering processes, the effective rates Pin Eq. ( B2) can take a simpler expression. Indeed for intraband processes, the coefﬁcients gξv k,k±q;b,b/prime appearing in Eq. ( A1) further simplify to a form gξv k,k±q;b,b/prime= gξv k,k±q;bδbb/prime. Moreover, when only intraband forward scatter- ing is considered, the above gξv k,k±q;bturn out to acquire an expression that is independent of the magnitude of k, and that we shall denote as gqξv b. This can easily be seen by focusing on an illuminating example. Let us consider, for instance, the conduction band ( b= c) and right-moving electrons ( λ=r), as illustrated by the ﬁgures shown in Sec. IV. In this case, a direct evaluation of Eqs. ( A2)t o( A6)f o rb=b/prime=c and for forward scattering (i.e.,k,k±q∈/Omega1r ccorresponding to θ=θ/prime=π/2), and theuse of Eq. ( A1) reveal that the gqξv± αα/primetake the simple form gqξv± αα/prime=gqξv cδk±q,k/primeδbb/primeδbc. (B3) Furthermore, as a result of the linearity of the band in the metallic SWNT, in this case, the energy-conservation functionin Eq. ( 9) also becomes independent of the magnitude of k: Dqξ± αα/prime=Dqξ c.=lim δ→0exp{−[/planckover2pi1(vFq−ωqξ)/2δ]2} (2πδ2)1/4. (B4) Inserting Eqs. ( B3) and ( B4) into Eq. ( 8), one obtains Aqξv± αα/prime=Aqξv± cδk±q,k/primeδbb/primeδbc, (B5) with Aqξv± c=/radicalBigg 2π/parenleftbig N◦ qξ+1 2±1 2/parenrightbig /planckover2pi1gqξv cDqξ c. (B6) The related generalized scattering rates in Eq. ( 7) are thus given by Pξv α1α2,α/prime 1α/prime 2=δb1b2,b/prime 1b/prime 2δb1b2,cc/summationdisplay q±/vextendsingle/vextendsingleAqξv± c/vextendsingle/vextendsingle2δk1±q,k/prime 1δk2±q,k/prime 2 (B7) and turn out to be real and positive, just like semiclassical rates (here, δi1i2,j1j2is a shorthand notation for δi1j1δi2j2). Inserting Eq. ( B7) into Eq. ( B2), the explicit form of the linear superoperator ( B1) corresponding to forward scattering processes acting on right-moving electrons comes out to be dρv k1,c;k2,c dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat=1 2/summationdisplay qξ±/braceleftbig/vextendsingle/vextendsingleAqξv± c/vextendsingle/vextendsingle2/bracketleftbig/parenleftbig 1−f◦ k1,c/parenrightbig /Delta1ρv k1±q,c;k2±q,c+f◦ k1,c/Delta1ρv k1∓q,c;k2∓q,c/bracketrightbig +H.c./bracerightbig −1 2/summationdisplay qξ±/braceleftbig/vextendsingle/vextendsingleAqξv± c/vextendsingle/vextendsingle2/bracketleftbig /Delta1ρv k1,c;k2,c/parenleftbig 1−f◦ k1∓q,c+f◦ k1±q,c/parenrightbig/bracketrightbig +H.c./bracerightbig . (B8) The spatial diffusion of the excitation charge density can now be determined by inserting Eq. ( B8) into Eq. ( 30). 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PhysRevB.91.075433.pdf	PHYSICAL REVIEW B 91, 075433 (2015) Valley Zeeman effect and spin-valley polarized conductance in monolayer MoS 2 in a perpendicular magnetic ﬁeld Habib Rostamiand Reza Asgari† School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran (Received 7 December 2014; revised manuscript received 2 February 2015; published 26 February 2015) We study the effect of a perpendicular magnetic ﬁeld on the electronic structure and charge transport of a monolayer MoS 2nanoribbon at zero temperature. We particularly explore the induced valley Zeeman effect through the coupling between the magnetic ﬁeld Band the orbital magnetic moment. We show that the effective two-band Hamiltonian provides a mismatch between the valley Zeeman coupling in the conduction and valencebands due to the effective mass asymmetry and it is proportional to B 2similar to the diamagnetic shift of exciton binding energies. However, the dominant term which evolves with Blinearly, originates from the multiorbital and multiband structures of the system. Besides, we investigate the transport properties of the system by calculatingthe spin-valley resolved conductance and show that, in a low-hole doped case, the transport channels at the edgesare chiral for one of the spin components. This leads to a localization of the nonchiral spin component in thepresence of disorder and thus provides a spin-valley polarized transport induced by disorder. DOI: 10.1103/PhysRevB.91.075433 PACS number(s): 75 .70.Ak,72.25.−b,73.43.−f I. INTRODUCTION Monolayer of the molybdenum disulﬁde (ML-MoS 2) has recently attracted great interest because of its potential appli-cations in two-dimensional (2D) nanodevices [ 1–3], owing to the structural stability and lack of dangling bonds [ 4]. The ML-MoS 2is a direct band gap semiconductor with a band gap of 1 .9e V[ 2], and can be easily synthesized by using scotch tape or lithium-based intercalation [ 2–5]. The mobility of the ML-MoS 2can be at least 217 cm2V−1s−1at room temperature using hafnium oxide as a gate dielectric, andthe monolayer transistor shows the room temperature currenton/off ratios of 10 8and ultralow standby power dissipation [ 2]. These properties render Ml-MoS 2as a promising candidate for a wide range of applications, including photoluminescence(PL) at visible wavelengths [ 6], and photodetectors [ 7]. The experimental achievements triggered the theoretical interestsin the physical and chemical properties of the ML-MoS 2 nanostructures to reveal the origins of the observed electrical,optical, mechanical, and magnetic properties, and guide thedesign of MoS 2-based devices. Having deﬁned the valleytronics of graphene, many phys- ical phenomena, originated from the spin of the electron,have been extended to be used for the valley index. Oneis the internal magnetic moments of spin which couplesto an external magnetic ﬁeld through well-known Zeemaninteraction. In a system where the inversion symmetry is broken, the valley degree of freedom can be distinguished. There is a valley dependence orbital magnetic moment whichcan result in a Zeeman-like interaction for the valley index.Gapped graphene is one of the main representatives of mate-rials in which the valley index couples to the perpendicularmagnetic ﬁeld as a real spin [ 8]. However, due to the small value of the gap, this effect has not been yet observedexperimentally. Transition metal dichalcogenides (TMDCs),on the other hand, provide a more applicable paradigm for rostami@ipm.ir †asgari@ipm.irthe valley Zeeman (VZ) effect. The VZ in TMDCs has beenrecently observed [ 9–12] and studied theoretically [ 13]. Those measurements were based on the shift of photoluminescence peak energies as a function of the magnetic ﬁeld interpreted as a Zeeman splitting due to the valley-depended magneticmoments. In order to explore the VZ we do need to perceiveall physical characteristics of the system. Actually, the energyband structure which can be calculated via ab initio methods, contains some information and besides, the Berry curvatureand orbital magnetic moment of the Bloch states, are twomain quantities which provide extra information to the bandstructure [ 14–16]. A peculiar property of the ML-MoS 2is its spin-valley coupled electronic structure which is due to the strong spin-orbit coupling and it induces a spin-orbit splitting in the valenceband [ 17]. Furthermore, many physical properties of TMDCs can be described by using a two-band model which is indeed aprojected model from a higher dimension Hamiltonian. Sincethe projection is an approximation and it is not a perfect unitarytransformation, the two-band Hamiltonian may not provide afull description of the low-energy excitations of the systemespecially when the system is addressed by a perpendicularmagnetic ﬁeld. Basically, some physics related to the multi-band structure such as Berry curvature and orbital magneticmoment properties might be ignored along the projectionprocess. In this work we would like to address these issuesand explore their physical sources in the ML-MoS 2structure. An effective model based on a Dirac-like Hamiltonian has been introduced by Xiao et al. [17] to explore ML-MoS 2 electronic properties. Very recently, it has been shown, based on the tight-binding [ 18,19] and k·pmethod [ 20], that a model going beyond the Dirac-like Hamiltonian (includingeffective mass asymmetry, trigonal warping, and a quadraticmomentum dependent term) is very important. Each termin the Hamiltonian can be as a source of many physicalconsequences. For example, due to the spin-orbit coupling(λ) and the diagonal quadratic term ( α), the two-band model reveals a particle-hole asymmetry and also the diagonal quadratic term of βgives a contribution to the Chern number at each valley [ 21]. A nanoribbon MoS 2in the presence of the 1098-0121/2015/91(7)/075433(11) 075433-1 ©2015 American Physical SocietyHABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 91, 075433 (2015) perpendicular magnetic ﬁeld reveals the Landau level band structure with a VZ term [ 18]. We attempt to clarify the VZ concept based on symmetry arguments, semiclassical (orbitalmagnetic moment) and quantum mechanical (Landau levels)calculations. In other words, we emphasize that a particle-holeasymmetry originating from the orbital magnetic momentoccurs in the presence of the perpendicular magnetic ﬁeldand thus we express the physical reasons of the asymmetry observed in the experiments [ 9–12]. In this paper we further study the electronic structure and two-terminal electronic transport of a zigzag ML-MoS 2in the presence of the perpendicular magnetic ﬁeld. Our calculationsare based on the multiorbital tight-binding approach [ 22] which describes the electronic properties of the monolayerMoS 2based on all dandprelevant orbitals of both the Mo and S atoms, respectively. We calculate the conductance of a cleanand disordered systems in the presence of the perpendicularmagnetic ﬁeld by using a nonequilibrium recursive Green’sfunction method [ 23]. According to the spin-orbit coupling and the valley degen- eracy breaking, a spin-valley polarization (SVP) is expectedin the electronic structure of the bulk system and particularlyin the hole doped case. Most remarkably, in the zigzag ribbon case, there are some metallic edge states which spoil the SVP in a clean system. However, our numerical resultsin the two-terminal conductance show a spin-valley polarizedmode made by the quantum Hall and ﬁnite size edge states inthe presence of on-site disorder. The paper is organized as follows. In Sec. IIwe introduce the formalism that will be used for calculating the electronicstructure, orbital magnetic moment, two terminal conductance,and the valley polarization quantity from the recursive Green’sfunction approach. In Sec. IIIwe present our analytic and numeric results for the dispersion relation in the presence ofthe magnetic ﬁeld. Section IVcontains a brief summary of our main results. II. THEORY AND METHOD A. Tight-binding model The tight-binding Hamiltonian is a common and a powerful technique to explore the transport properties. The modelprovides a reasonable description of the bulk properties ofthe ML-MoS 2including direct band gap [ 22]. We carry out our calculations based on the following real space modelHamiltonian: H=/summationdisplay i,μ/epsilon1a i,μa† i,μai,μ+/epsilon1b i,μ/parenleftbig bt† i,μbt i,μ+bb† i,μbb i,μ/parenrightbig +/summationdisplay i,μ/bracketleftbig t⊥ i,μbt† i,μbb i,μ+H.c./bracketrightbig +/summationdisplay /angbracketleftij/angbracketright,μν/bracketleftbig tab ij,μνa† i,μ/parenleftbig bt j,ν+bb j,ν/parenrightbig +H.c./bracketrightbig +/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketright,μν/bracketleftbig taa ij,μνa† i,μaj,ν+H.c./bracketrightbig +/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketright,μν/bracketleftbig tbb ij,μν/parenleftbig bt† i,μbt j,ν+bb† i,μbb j,ν/parenrightbig +H.c./bracketrightbig ,(1)where /epsilon1aand/epsilon1bindicate on-site energies for Mo and S atoms and tab,taa, and tbbshow the hopping matrixes corresponding to Mo-S, Mo-Mo, and in-plane S-S hoppingprocess, respectively. t ⊥denotes the hoping integral between two sulfur layers, i,jandμ,ν stand for lattice site and atomic orbital indices, respectively. Note that the Hamiltonian isconstructed by dandporbitals of the Mo and S atoms which are listed as follows: dbasis (Mo atoms): d z2,dx2−y2,dxy,dxz,dyz, (2) pbasis (S atoms): px,t,py,t,pz,t,px,b,py,b,pz,b, where the torbsubindex indicates the top or bottom sulfur plane, respectively. A unitary transformation is used to reducethe dimensionality of the Hamiltonian and thus relevantorbitals are only considered. The unitary matrix is given by U=1 √ 2/parenleftbiggIu I −u/parenrightbigg , (3) where Iis a three-dimensional identity matrix and u= diag[1 ,1,−1]. Implementing the unitary matrix on the pbasis of the sulfur atoms, results in two decoupled bases with a symmetric (even) and an antisymmetric (odd) combination of theporbitals of two sulfur layers with respect to the horizontal reﬂection symmetry. These even and odd spaces read as Even :1√ 2(px,t+px,b),1√ 2(py,t+py,b),1√ 2(pz,t−pz,b), Odd :1√ 2(px,t−px,b),1√ 2(py,t−py,b),1√ 2(pz,t+pz,b). (4) The transformation gives rise to an opportunity to suppress direct coupling between two sulfur layers. Based on theHamiltonian in the main orbital space, two sulfur layers aredirectly coupled due to the vertical hopping as H=/parenleftbigght ⊥ t⊥h/parenrightbigg , (5) where h=/epsilon1bwhich indicates the on-site term of the tight- binding Hamiltonian corresponding to the porbitals of the sulfur atoms on both top and bottom layers. Using ut⊥=t⊥u andu/epsilon1b=/epsilon1buone can show that in the new space we have H/prime=UHU†=/parenleftbiggh+ut⊥0 0 h−ut⊥/parenrightbigg , (6) where the ﬁrst (˜ /epsilon1b=/epsilon1b+ut⊥) and second diagonal block belong to the even and odd symmetric subspaces [ 22], respectively. Therefore, the six-band real space Hamiltoniancan be written in the even symmetric subspace which containseven subspace of porbital and even subspace of dorbital (i.e., d z2,dx2−y2,dxy). Besides, in the presence of the perpendicular magnetic ﬁeld, the six-band Hamiltonian reads as H=/summationdisplay i,μ/epsilon1a i,μa† i,μai,μ+˜/epsilon1b i,μb† i,μbi,μ +/summationdisplay /angbracketleftij/angbracketright,μν/bracketleftbig eiφijtab ij,μνa† i,μbj,ν+H.c./bracketrightbig 075433-2V ALLEY ZEEMAN EFFECT AND SPIN-V ALLEY . . . PHYSICAL REVIEW B 91, 075433 (2015) +/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketright,μν/bracketleftbig eiφijtaa ij,μνa† i,μaj,ν+H.c./bracketrightbig +/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketright,μν/bracketleftbig eiφijtbb ij,μνb† i,μbj,ν+H.c./bracketrightbig . (7) Using Eq. ( 6), together with the crystal ﬁelds of the system [22], and also spin-orbit couplings for the valence and conduction bands in atomic limit, i.e., L·S, the on-site energy matrices are given by /epsilon1a i,μ=⎛ ⎜⎝/Delta10 00 0 /Delta12 −iλMˆsz 0 iλMˆsz /Delta12⎞ ⎟⎠, (8) ˜/epsilon1b i,μ=⎛ ⎜⎝/Delta1p+t⊥ xx −iλX 2ˆsz 0 iλX 2ˆsz /Delta1p+t⊥ yy 0 00 /Delta1z−t⊥ zz⎞ ⎟⎠, where λM=0.075 eV and λX=0.052 eV stand for the spin-orbit coupling originating from the Mo (metal) and S(chalcogen) atoms, respectively [ 24]. Notice that s=± indi- cates the zcomponent of the spin degree of freedom. Moreover, we have added an external perpendicular magnetic ﬁeld to the system using Peierls phase factor φ ij=e /planckover2pi1/integraltextj i/vectorA·/vectordrto carry out the orbital effect of the perpendicular magnetic ﬁeld.Interlayer hopping between the sulfur planes is given as t ⊥= diag[Vppπ,Vppπ,Vppσ] based on the Slater-Koster table [ 25]. The numerical values of the tight-binding parameters are /Delta10= −1.096,/Delta12=− 1.512,/Delta1p=− 3.560,/Delta1z=− 6.886,Vddσ= −0.895,Vddπ=0.252,Vddδ=0.228,Vppσ=1.225,Vppπ= −0.467,Vpdσ=3.688, and Vpdπ=− 1.241 in eV units. These parameters will be presented elsewhere [ 26]. We might express that this Hamiltonian provides a very good energy bandstructure in according to the comparison with those resultsobtained within the density functional theory simulations [ 24]. B. Orbital magnetic moments In many semiconductor systems, such as GaAs bulk, the circular polarization of luminescence from circularlypolarized excitation originates from electron or hole spinpolarization [ 27]. However in ML-MoS 2, the optical selection rule originates from the orbital magnetic moments at each K orK/primevalley independent of electron or hole spin [ 28]. In a periodic lattice, the eigenfunctions of the Schr ¨odinger equation are Bloch states un,k, where nandkindicate the band index and crystal momentum, respectively. In semiclassicalmethod, it is common to use a wave packet picture of electrons[14–16]. The wave packet \|W/angbracketrightcan be easily constructed by the linear superposition of the Bloch states. Due to theself-rotation of the wave packet around its own center ofmass, the magnetic moment (or the angular orbital momentumL) deﬁned as M=− e 2m0L=−e 2m/angbracketleftW\|(ˆr−rc)×ˆp\|W/angbracketrightalong thezdirection, where m0is the free electron mass and ˆpis the canonical momentum operator, and moreover the wave packet is also centered at rcin the position space. The orbital magnetic moment of Bloch electrons has a contributionfrom intercellular current circulation governed by symmetryproperties. After straight forward calculations [ 14–16], the orbital magnetic moment is written as M n(k)=ie /planckover2pi1/summationdisplay m/negationslash=n/angbracketleft∇kunk\|×[H(k)−/epsilon1nk]\|∇kunk/angbracketright.(9) This relation can be written in a more practical expression as Mn(k)=−ˆze /planckover2pi1/summationdisplay m/negationslash=nIm/bracketleftbig /angbracketleftunk\|∂kxH(k)\|umk/angbracketright/angbracketleftumk\|∂kyH(k)\|unk/angbracketright/bracketrightbig /epsilon1nk−/epsilon1mk. (10) Up to linear order in the magnetic ﬁeld and in semiclassical limit, the energy dispersion in an external magnetic ﬁeldmodiﬁes as E nk=/epsilon1nk−Mn(k)·B, (11) where /epsilon1nkis the band dispersion of the system without magnetic ﬁeld. It is worth mentioning that the inversion andtime reversal symmetries play vital roles in the nontrivialBerry curvature and the orbital magnetic moment. Accordingto the time reversal symmetry, M(k)=−M(−k) while the presence of the inversion system results M(k)=M(−k). Consequently, the orbital magnetic moment vanishes bygoverning both symmetries. Most importantly, the magneticmoment is nonzero in ML-MoS 2since the inversion symmetry is broken. Similar behavior is expected for the Berry curvatureas well. In order to calculate the orbital magnetic moment,based on the six-band tight-binding model, we carry out aFourier transformation along the xandydirections to ﬁnd the six-band Hamiltonian in the kspace. Moreover, the orbital magnetic moment can be also found through the correspondingtwo-band model around the Kpoint. The two-band model can be extracted by using a L ¨owding partitioning method from the six-band Hamiltonian. The two-band Hamiltonian of themonolayer MoS 2, after ignoring the trigonal warping and the momentum dependence of the spin-orbit coupling, is given by H=/Delta10+λ0τs 2+/Delta1+λτs 2σz +t0a0q·στ+/planckover2pi12\|q\|2 4m0(α+βσz), (12) where s=± andτ=± indicate spin and valley, respec- tively, στ=(τσx,σy) are Pauli matrices, and q=(qx,qy)i s momentum. The numerical values of the two-band modelparameters are given by /Delta1 0=− 0.11 eV, /Delta1=1.82 eV, λ0= 70 meV, λ=− 80 meV, t0=2.33 eV, α=− 0.01, and β= −1.54. The zcomponent of the orbital magnetic moment of the conduction and valence bands in the two-band modelHamiltonian are given by M s c(k)=Ms v(k)=−τe /planckover2pi1t2 0a2 0/parenleftbig /Delta1−2bβa2 0k2+λs/parenrightbig /parenleftbig /Delta1+2bβa2 0k2+λs/parenrightbig2+4t2 0a2 0k2, (13) where b=/planckover2pi12/4m0a2 0≈0.572. Moreover, at two valleys ( k= 0) the contribution from βis eliminated and one can ﬁnd Ms c(k=0)=Ms v(k=0)=−τe /planckover2pi1t2 0a2 0 /Delta1+λs. (14) 075433-3HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 91, 075433 (2015) Note that for the low-energy model parameters we have /planckover2pi1M↑(k=0)/(ea2 0)≈− 3.14τeV and /planckover2pi1M↓(k=0)/(ea2 0)≈ −2.87τeV . It should be noticed that the opposite sign of the orbital magnetic moments at two valleys, which originates from thetime reversal symmetry, leads to the VZ effect when thesystem is imposed by an external perpendicular magneticﬁeld. Moreover, the low-energy Hamiltonian exhibits thesame value of the semiclassical magnetic moment at both thevalence and conduction bands while the recent experimentalstudies showed a different value for the magnetic momentat two bands. In the numerical section we will discuss thisdiscrepancy more carefully. Although the magnitude of the valley splitting in each band has not been measured experimentally, the mismatch was mea-sured in four different experiments. The photoluminescenceintensity of a monolayer transition metal dichalcogenide hasbeen measured in the presence of the external perpendicularmagnetite ﬁeld using circular polarized light as the excitationlight. The shift value of the peak of the luminescence spectrumof MoSe 2[9,12] and WSe 2[10,11] are about 2–5 meV for left- and right-handed polarizations and for both neutral andcharged exciton. The linear dependence of the valley splitting demonstrates a Zeeman-like effect of the valley index. According to thecircular dichroism effect in these materials, the right- (left-)handed light couples just to the K(K /prime) valley. In the magnetic ﬁeld the energy gap between electron and hole states differsin two valleys, whereas E CBM−EVBM=/Delta1+λ+τ(gcon v− gval v)/planckover2pi1ωc/2 and the difference provides an opportunity to the valley Zeeman effect to be measured experimentally.Therefore, due to the circular dichroism effect, the left- andright-handed emitted light have two different frequencies (i.e.,corresponding energy gap) leading to a splitting in the peak ofthe PL spectrum for two polarizations. Being aware of the discrepancy of the two-band model in the magnetic ﬁeld and in order to capture the correct value ofthe orbital magnetic moment of the system, we add a mismatchκ vbetween the semiclassical orbital magnetic moments of the six- and two-band models at the Kpoint to the low-energy two-band Hamiltonian when there is a perpendicular magneticﬁeld. Consequently, in the presence of the magnetic ﬁeld, thelow-energy Hamiltonian, Eq. ( 12), is modiﬁed as H τs=/Delta10+λ0τs 2+/Delta1+λτs 2σz+vπ·στ +\|π\|2 4m0(α+βσz)−1 2τκv/planckover2pi1ωc−1 2sgs/planckover2pi1ωc,(15) where π=p+eAandgs≈2 is the Zeeman coupling for the real spin and the mismatch between the Zeeman coupling ofboth the bands is κ v=1eV /planckover2pi12//parenleftbig 4m0a2 0/parenrightbig/parenleftbiggmc−m2 0 0 mv−m2/parenrightbigg ≈/parenleftbigg−0.62 0 0 −1.50/parenrightbigg , (16) where m2(in units of e2Va2 0//planckover2pi1) is the magnetic moment calculated by the two-band model while mcandmvare theLead Scattering Region Lead 1 2 M . . .1N 2... FIG. 1. (Color online) A top view schematic of a monolayer MoS 2lattice structure in a two-terminal setup. Blue (orange) circles indicate the Mo (S) atoms. The nearest neighbor ( δi)a n dt h en e x t nearest neighbor ( ai) vector are shown in the ﬁgure. Ribbon width and scattering region length are W/a 0=3N/2−1,L / a 0=√ 3M, respectively. magnetic moment obtained within the six-band tight-binding model in the conduction and valence bands, respectively. Thenumerical values of κ v[which is about /planckover2pi1ωc=/planckover2pi1(eB/2m0)] are obtained by using the semiclassical results of the orbitalmagnetic moments presented in Fig. 3at the Kpoint and by averaging over spins. We also deﬁne κ con v=− 0.62 and κval v=− 1.5. C. Conductance and spin-valley polarization Using the Fourier transformation along the ribbon, the energy dispersion can be found as Hk=H00+H01eika+ H† 01e−ika, where H00andH01are the intra- and interprincipal cell Hamiltonian, respectively [ 29]. Note that a=√ 3a0= 0.316 nm stands for the Mo-Mo or in-plane the S-S bond length witha0as the in-plane projection the Mo-S bond length. To calculate the conductance we use the nonequilibrium Green’sfunction method in which the retarded Green’s function isdeﬁned as G r s=(E−Hs−/Sigma1s+i0+)−1by employing the recursive Green’s function method [ 30]. Note that s=↑ or↓for the spin degree of freedom. In the noninteracting Hamiltonian, the self-energy ( /Sigma1s=/Sigma1L s+/Sigma1R s) originates only from the connection of the system to leads (Fig. 1) and it can be calculated by the method that has been developedby Lopez et al. [31]. Using the Landauer formula, the zero temperature conductance for each spin component is given as G ↑(↓)=e2 hT↑(↓), where Ts=Tr/bracketleftbig /Gamma1L sGrs/Gamma1R sGr† s/bracketrightbig (17) and/Gamma1L,R s=− 2/Ifracturm[/Sigma1L,R s] are linewidth functions. Because of the collinear spin structure, the conductance of each spincomponent can be calculated separately. Consequently, inprincipal, a spin polarization quantity can be deﬁned asP=(G ↑−G↓)/(G↑+G↓). 075433-4V ALLEY ZEEMAN EFFECT AND SPIN-V ALLEY . . . PHYSICAL REVIEW B 91, 075433 (2015) III. RESULTS AND DISCUSSION In this section we present our main results in the orbital magnetic moment, Landau levels spectrum, and spin-valleypolarized transport in monolayer MoS 2in the presence of the perpendicular magnetic ﬁeld. We present our extensivenumerical results of the electronic structure by exploring thestructure of the Landau levels in the quantum Hall regimeand the spin-valley resolved transport properties of the zigzagMoS 2nanoribbon. We calculate the conductance in both unipolar electron and hole doped cases and we explore thespin-valley-resolved electronic transport in both clean anddisordered systems. A. Valley Zeeman and Landau levels Before calculating the conductance of the system, we ﬁrst discuss the VZ effect induced by the perpendicular magneticﬁeld in both semiclassical and quantum aspects. First of all,the orbital magnetic moment corresponding to the conductionand valence bands are calculated in the whole Brilloun zone(BZ) using the six-band tight-binding model, specially usingEqs. (7), (8), and (10), and results are shown in the counterplots in Fig. 2. It is obvious that the orbital magnetic moment changes sign in the two valleys owing to the time reversalsymmetry. Indeed, the states near the corners of the BZcontribute mainly to the orbital magnetic moment. Moreover,a comparison between the semiclassical orbital magneticmoment calculated within the two-band, using Eq. ( 13), and the six-band models as a function of the momentum along x axis are shown in Fig. 3for both spin components. As seen in the ﬁgure, a remarkable difference between the value of theorbital magnetic moment in the valence and conduction bandsis obtained by the six-band model Hamiltonian. However, inthe two-band model, the semiclassical magnetic moment isthe same in both the valence and conduction bands [see Eq.(13)] even in the presence of the particle-hole asymmetry terms such as the spin-orbit coupling and effective mass asymmetry.Most remarkably, the mismatch between the orbital magneticmoment of two bands calculated within the six-band modelplays an important role in interpreting the VZ experimentalmeasurements. The difference between the two- and six-band models can be classiﬁed in two intraband and interband categories. Theintraband reason is related to the orbital character of the bands.Using the Slater-Koster table for constructing the tight-bindingmodel provides a platform for taking into account the natureof the relevant atomic orbitals such as panddtypes and also considering the neighboring lattice symmetry. However, theorbital basis of the two-band model is substituted with theband basis and the orbital character can be mainly captured byd-type orbitals. According to Eq. ( 10), similar to the Berry curvature formula and the second order perturbation theory, the orbitalmagnetic moment of each band is affected by virtual transitionsbetween bands corresponding to the interband sector [ 32]. Due to the transition between neighboring energy bands, observinga different value of the orbital magnetic moment of twodifferent bands is awaited, however such virtual transition −3−2−1 0 1 2 3 kxa0−3−2−10123kya0−4−3−2−10 1 2 3 4 −3−2−1 0 1 2 3 kxa0−3−2−10123kya0−4−3−2−10 1 2 3 4 FIG. 2. (Color online) Contour plot of the orbital magnetic mo- ment as function of the momenta along the xaxis at the conduction (top panel) band and the valence (below panel) band. Mis in units of e2Va2 0//planckover2pi1and the spin-orbit coupling is neglected in this ﬁgure. is deﬁnitely eliminated in the two-band case. Consequently, we would like to emphasize that one might be careful inusing the L ¨owdin canonical projection from a multiband to a two-band model, because some information regarding theorbital character and virtual transitions might be ignored. The wave vector point group symmetry of a honeycomb lattice with broken inversion symmetry, like gapped graphene,isC 3hpoint group [ 33,34] near the KandK/primepoints. The irreducible representations of the point group characterizeenergy eigenfunctions at the KandK /primevalleys. According to the character table, the phase winding at each KandK/prime isC3\|c,τ/angbracketright=ωτ\|c,τ/angbracketrightandC3\|v,τ/angbracketright=ω−τ\|v,τ/angbracketright, where ω= ei2π/3due to threefold rotational for the conduction and the 075433-5HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 91, 075433 (2015) −0.4 −0.2 0.0 0.2 0.4 kxa01.01.52.02.53.03.54.0−Mz(k) condution valence 2-band −0.4 −0.2 0.0 0.2 0.4 kxa01.01.52.02.53.03.54.0−Mz(k) condution valence 2-band FIG. 3. (Color online) Orbital magnetic moment as a function of the momentum along the xaxis for both the spin-up and -down components calculated by the six-band and the two-band models. Up (below) panel corresponds to the spin-up (-down) component and M is in units of e2Va2 0//planckover2pi1. valence bands. The relation means that the orbital angular momentum in the conduction band is lc=−τand similarly lv=τfor the valence band. In a semiclassical picture, the angular momentum has been induced from the self-rotation ofthe electron wave packet around its center of mass. This kindof the orbital angular momentum, called Bloch phase shift, iswell studied in the content of gapped graphene which can beexplained by a single p z-orbital tight-binding model. However, in any multiorbital system, another distinct contribution to theorbital angular moment might be expected. At high symmetric points where the Bloch states are in- variant under a g-fold discrete rotation, an azimuthal selection rulel c+gN=lv±1 is expected for interband transitions. According to the ab initio calculations near the K(K/prime) point, the conduction band minimum is mainly formed from the Mod z2orbitals with lz=0 and the valence band is constructed by the Mo dx2−y2+idxy(dx2−y2−idxy) orbital with lz=2 (lz=− 2). Note that there are some contributions from pxand pyorbitals of the S atoms in both band edges. If the mixing from the porbital is ignored, the total angular momentum will belc∼−τandlv∼τ+2τ∼3τincluding the Bloch phase shift and local orbital contribution of the conduction band.Moreover, owing to the selection rule allowed with discretethreefold rotational symmetry, we can add a multiplicand of K/prime K−1.00−0.98−0.96−0.94−0.92−0.90−0.88Ek(eV) 0.840.860.880.900.920.94 0 5 10 15 20 25 B[T]−1.0−0.50.00.51.01.5ΔK−ΔK/prime(meV) 0 5 10 15 200246810 Valley Zeeman splitting (meV)E E E E FIG. 4. (Color online) (Top panel) Landau levels as a function of the momentum in units of eV calculated by a tight-binding approach on a zigzag ribbon where B=100 T. (Bottom panel) Valley Zeeman splitting in units of meV as a function of the magnetic ﬁeld in unitsof tesla for both the conduction and valence bands. In the inset: The mismatch between the valley Zeeman effect of the conduction and valence bands which is the splitting in PL spectrum for right- andleft-handed polarized light as a function of the magnetic ﬁeld in units of tesla. Note that blue (red) lines indicate spin-up (-down) states. We setN=100 as the ribbon width and the real Zeeman effect is not included in this ﬁgure. three to the orbital angular moment of one of the bands in order to satisfy lv−lc=± 1 which is necessary in the dipole absorption limit [ 35]. In this case, we have lv∼0 andlc=−τ. The Landau level spectrum is also calculated within the six-band model (see Fig. 4) of a zigzag ribbon ML-MoS 2 after applying a Peierls substitution in the tight-binding model. Thus, by using the Landau level spectrum resulted fromfull tight-binding calculation, we extract the valley Zeemaneffect of the conduction and valence bands. The mismatchbetween the splitting in two bands, which is the shift of the PLspectrum of right- and left-handed light in the presence of themagnetic ﬁeld, is shown in Fig. 4(bottom panel). This linear dependence of the magnetic ﬁeld magnitude of the energysplitting approves the Zeeman-like coupling and is in goodagreement with those results measured in experiments. Having calculated the orbital magnetic moments in the six- and two-band models, we modiﬁed the two-band modelHamiltonian in the presence of the perpendicular magneticﬁeld given by Eq. ( 15). After a straightforward calculation, the Landau level spectrum of the modiﬁed two-band Hamiltonian, 075433-6V ALLEY ZEEMAN EFFECT AND SPIN-V ALLEY . . . PHYSICAL REVIEW B 91, 075433 (2015) Eq. ( 15) reads as E± n/negationslash=0,τs=±/radicalBigg/bracketleftbigg/Delta1+λτs 2+/planckover2pi1ωc/parenleftBig βn−ατ 2/parenrightBig/bracketrightbigg2 +2/parenleftbiggt0a0 lB/parenrightbigg2 n +/Delta10+λ0τs 2+/planckover2pi1ωc/parenleftbigg αn−βτ 2/parenrightbigg −1 2τκv/planckover2pi1ωc −1 2sgs/planckover2pi1ωc, E− n=0,Ks=/Delta10+λ0s 2−/Delta1+λs 2+/planckover2pi1ωc 2(α−β) (18) −1 2κval v/planckover2pi1ωc−1 2sgs/planckover2pi1ωc, E+ n=0,K/primes=/Delta10−λ0s 2+/Delta1−λs 2+/planckover2pi1ωc 2(α+β) +1 2κcon v/planckover2pi1ωc−1 2sgs/planckover2pi1ωc, in the presence of a constant magnetic ﬁeld B.I tm u s tb e noticed that for the n=0 level, there is no solution of the eigenvalue problem in the conduction band at the Kpoint and similarly in the valence band at the K/primepoint. Having calculated the analytical expression of the Landau level fromthe two-band model, we could deduce a valley splitting theconduction band and adding the contribution from a real Zee-man interaction and multiband correction. The valley splittingcoupling in the conduction and valence bands can be deﬁnedasg con/planckover2pi1ωc=E+ 1,K↑−E+ 0,K/prime↓andgval/planckover2pi1ωc=E− 0,K↑−E− 1,K/prime↓, respectively, with the following explicit expressions: gcon(val)/planckover2pi1ωc=/radicalBigg/bracketleftbigg/Delta1+λ 2+/planckover2pi1ωc/parenleftBig β∓α 2/parenrightBig/bracketrightbigg2 +2/parenleftbiggt0a0 lB/parenrightbigg2 −/Delta1+λ 2−/planckover2pi1ωc/parenleftbigg β∓α 2/parenrightbigg −/parenleftbig κcon(val) v +gs/parenrightbig /planckover2pi1ωc, (19) where −/+stands for the conduction/valence band. This is important that αhas no effect on the semiclassical orbital magnetic moment while it is a source of the mismatch ofthe magnetic moment (i.e., valley splitting) in those bandsfrom a quantum point of view. In other words, in the quantumpicture, the two-band model could produce a mismatchbetween magnetic moments while this is not the case in thesemiclassical picture. It is worth to expand the above relationup to leading order in a weak magnetic ﬁeld as g con,val≈4a2 0m0t2 0 /planckover2pi12(/Delta1+λ)+2a2 0m0t2 0/parenleftBig (±α−2β)(/Delta1+λ) m0−4a2 0t2 0 /planckover2pi12/parenrightBig l2 B(/Delta1+λ)3 −κcon,val v−gs. (20) Here, using the six-band tight-binding model, the relation for the splitting is given by gcon−gval=4a2 0et2 0 /planckover2pi1(/Delta1+λ)2×α×B−/parenleftbig κcon v−κval v/parenrightbig .(21) It is clear that the effective mass asymmetry (i.e., α) yields a quadratic dependence of the mismatch to the magneticﬁeld which can compete with the diamagnetic shift of the exciton binding energies which is also quadratic in B[36–38]. However, that cannot explain those PL experimental datawhile the correction from the multiband and the multiorbitalnature of this material ( κ v) gives rise to a linear shift of the PL spectrum of left- and right-handed light. Therefore, our low-energy model predicts gcon−gval∼− 0.88+7.22a2 0 l2 Bα. Based on the tight-binding model, Fig. 4bottom panel, gcon−gval∼− 0.81 indicating that the proposed Eq. ( 20)i s reasonably good by incorporating the semiclassical approachof the value κ con vandκval v. B. Spin polarization: Two-terminal transport The optical probing, such as the PL approach, can just measure the mismatch between the valley Zeeman effectof electron and hole states since measuring valley Zeemansplitting at each band requires a transition between twovalleys which contains a large momentum difference, whilethe optical method are based on direct transitions. We proposea valley splitting at each band which can be measuredvia a two-terminal unipolar transport setup where a valleypolarization is expected. Although specifying valley index is Spin up Spin down 0.82 0.84 0.86 0.88 0.90 0.920123456 EFGnne2 h Spin up Spin down 0.94 0.92 0.90 0.88 0.8601234 EFGppe2 h FIG. 5. (Color online) Unipolar conductance according to the Landau level spectrum of the low-energy model. The Zeeman interaction corresponding to the real spin is not taken into account inthis ﬁgure. 075433-7HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 91, 075433 (2015) not as easy as spin index, we believe that the valley index can be realized through measuring spin resolved conductance inthe TMDs due to the spin-valley coupling. In the unipolar case,the conductance can be calculated by counting the transportchannel, so that the corresponding conductance for each spincomponent is given as G s nn(pp)=min(νs L,νs R), in units of e2/h between the left and right leads. In this regard, we plotthe conductance based on the Landau level sequence of thetwo-band model in Fig. 5for both electron and hole doped cases and the spin polarization can also be seen. In the valenceband the polarization is more pronounced due to the strongspin-orbit coupling. The sequence of the plateaus for both thecases are different in the low-energy levels. This effect canbe understood based on the strong spin-orbit coupling in thevalence band which decreases the number of the channel ofthe hole doped system to the half of the accessible channel inthe conduction band. Moreover, there are some ﬁnite size metallic edge modes (see Fig. 4) due to the zigzag edges. These edge modes suppress the spin polarization when the system is subjectedto an external magnetic ﬁeld. We calculate the normalizedprojected local density of states (PLDOS) to clarify thateach of those states are mostly localized on which edge andorbital. The PLDOS which can be calculated as ρ(y,n,k,μ )=/summationtext mk/prime\|ψmk/primeμ(y)\|2δ(Enk−Emk/prime) is shown in Fig. 6for spin-up [Figs. 6(a) and 6(d)] and spin-down [Figs. 6(c) and 6(d)] components, respectively. Here ψmk,μ is the wave function in which m(n),k(k/prime), andμstand for the band index, momentum, and orbital index, respectively. The left-going (which is deﬁnedby a negative slope of the dispensation relation) spin-up state,which is connected to the zero Landau level in the valenceband at the Kpoint, lies on the top edge while the right-going one is located on the bottom edge. On the other hand, both 00.20.40.60.81Normalized PLDOSMo:dz Mo:dx−y Mo:dxy S:px S:py S:pz 0 50 100 15000.20.40.60.81 y/a0Normalized PLDOS 0 50 100 150 y/a0(b) (d) (c)(a) Spin up, Left going Spin up, Right going Spin down, Right going Spin down, Left going FIG. 6. (Color online) (a) and (b) Projected local density of states ρ(y,Ek) for spin-up edge modes at EK=− 0.89 eV . The left- and right-going modes are localized on opposite edges. (c) and (d) Thesame as before for spin-down edge modes but the left- and right-going states are localized on the same edges. The edge modes are mostly constructed by d xyanddx2−y2orbitals of the molybdenum atoms.right- and left-going spin-down states are on the bottom edge. This feature tells us that the former pair is chiral, whereas thelater one is not. The nonequilibrium Green’s function method is used in a two-terminal setup to count the number of the transportchannel of a zigzag ribbon geometry. First of all, we calculatethe conductance of a clean system in the presence of theexternal magnetic ﬁeld and the results are illustrated in Fig. 5 which shows the two-terminal conductance plateaus for eachspin component. Obviously there is no the spin polarizationfor the low-hole doped case and it is due to the extra ﬁnite sizeedge modes. Furthermore, in a real material there are also impurities and structural defects which can affect the expected transportproperties of the clean sample. Here we study the effect ofimpurities by adding a simple random on-site energy in therange of [ −δ/2,δ/2] to the Hamiltonian where δstands for the intensity of disorder scattering. In this case, we assume −0.95 −0.93 −0.91 −0.89 −0.87 −0.85 EF(eV)0.00.51.01.52.02.53.03.5G(e2/h)(a)δ= 0,up δ= 0,down δ= 1eV,up δ= 1eV,down δ= 2eV,up δ= 2eV,down δ= 3eV,up δ= 3eV,down −0.95 −0.93 −0.91 −0.89 −0.87 −0.85 EF(eV)0.00.20.40.60.81.0P(b)δ=0 δ=1 e V δ=2 e V δ=3 e V δ=4 e V δ=5 e V FIG. 7. (Color online) (a) Unipolar conductance for a zigzag ribbon as a function of the Fermi energy in the presence of the perpendicular magnetic ﬁeld and random on-site energy. (b) Spinpolarization in the presence of the perpendicular magnetic ﬁeld and random on-site energy. The Zeeman interaction corresponding to the real spin is not taken into account in this ﬁgure. We set N=50, M=10, and B=150 T. 075433-8V ALLEY ZEEMAN EFFECT AND SPIN-V ALLEY . . . PHYSICAL REVIEW B 91, 075433 (2015) that all of the relevant atomic orbitals at each lattice site are affected in a same way from the presence of impurity. Thiskind of impurity which has a uniform distribution only inducesan intravalley scattering rate to relax the momentum. We areonly interested in a simple momentum relaxation to realizewhether ﬁnite size or quantum Hall edge modes are robustwith respect to the randomness. The numerical conductanceresults as a function of the Fermi energy are presented inFig. 7showing that disorder induces a spin-valley polarization. In the clean ribbon with a low-hole doped case, both spincomponents have same contributions to the conductance.The spin-down contribution of the conductance in the lowestplateau is originating from the ﬁnite size edge modes, whilethat corresponding to the spin-up component has a contributionfrom a quantum Hall edge mode which is connected to the zeroLandau level at the Kpoint. After adding random on-site energy, one can clearly see that for a reasonable intensity of the randomness the spin-downedge modes are localized. This is due to the fact they are notchiral and thus they can scatter backward similar to a nonchiralone-dimensional system where a localization always occursin the presence of a randomness. However, in the case of thespin-up states, since they are on the opposite side of the ribbon,they cannot be scattered to each other based on their chiralnature. Hence, the spin-up states are not localized and they cancarry spin-polarized current which is also valley polarized dueto the spin-valley coupling of the hole doped case. Eventually,disorder revives the spin-valley polarized transport in the ﬁnitesize case. Moreover, if we increase the strength of the scatteringfrom impurity, the conductance contribution from both spinwill drop, however the polarization will approximately saturateto a constant value ( P∼0.6).IV . CONCLUSION In this work we have shown that the strength of the valley Zeeman interaction in TMDCs, which mainly originates fromthe broken inversion symmetry, differs in the conduction andvalence bands due to the different orbital character and alsovirtual interband transitions. We have provided a modiﬁed two-band Hamiltonian in the presence of the magnetic ﬁeld whichcan be used to describe recent experimental data. Moreover, wehave shown that the quadratic diagonal momentum dependentterms in the low-energy model contribute in the valley splittingwhich evolves in a quadratic way by varying Bthat might compete with the diamagnetic shift of the exciton bindingenergy. Remarkably, the dominant dependance of the valleysplitting to the magnetic ﬁeld, which evolves linearly with B, originates from the multiorbital and multiband structures ofthe system. Furthermore, we have studied the two-terminal electronic transport of a zigzag ML-MoS 2in the presence of a perpendic- ular magnetic ﬁeld using the nonequilibrium recursive Green’sfunction method. We have found that the conductance is notspin polarized in the clean hole-doped case due to the presenceof the ﬁnite size metallic edge modes in addition to the quantumHall edge modes. Our numerical results in the two-terminalconductance show a spin-valley polarized transport in thepresence of the on-site disorder which is related to the chiralnature of one of the spin components. ACKNOWLEDGMENT We would like to thank F. Guinea for valuable discussions. APPENDIX: HOPPING MATRICES The hopping terms of the system, calculated by the Slater-Koster table [ 39], are listed below for the nearest neighbor hopping, tab 1=√ 2 7√ 7⎛ ⎜⎝−9Vpdπ+√ 3Vpdσ 3√ 3Vpdπ−Vpdσ 12Vpdπ+√ 3Vpdσ 5√ 3Vpdπ+3Vpdσ 9Vpdπ−√ 3Vpdσ −2√ 3Vpdπ+3Vpdσ −Vpdπ−3√ 3Vpdσ 5√ 3Vpdπ+3Vpdσ 6Vpdπ−3√ 3Vpdσ⎞ ⎟⎠, (A1) tab 2=√ 2 7√ 7⎛ ⎜⎝0 −6√ 3Vpdπ+2Vpdσ 12Vpdπ+√ 3Vpdσ 0 −6Vpdπ−4√ 3Vpdσ 4√ 3Vpdπ−6Vpdσ 14Vpdπ 00⎞ ⎟⎠, (A2) tab 3=√ 2 7√ 7⎛ ⎜⎝9Vpdπ−√ 3Vpdσ 3√ 3Vpdπ−Vpdσ 12Vpdπ+√ 3Vpdσ −5√ 3Vpdπ−3Vpdσ 9Vpdπ−√ 3Vpdσ −2√ 3Vpdπ+3Vpdσ −Vpdπ−3√ 3Vpdσ −5√ 3Vpdπ−3Vpdσ −6Vpdπ+3√ 3Vpdσ⎞ ⎟⎠. (A3) The next nearest neighbor hopping process, the hopping along aidirection (see Fig. 1) which corresponds to the hopping among the Mo or the S atoms, reads as taa 1=1 4⎛ ⎜⎜⎝3Vddδ+Vddσ√ 3 2(−Vddδ+Vddσ) −3 2(Vddδ−Vddσ) √ 3 2(−Vddδ+Vddσ)1 4(Vddδ+12Vddπ+3Vddσ)√ 3 4(Vddδ−4Vddπ+3Vddσ) −3 2(Vddδ−Vddσ)√ 3 4(Vddδ−4Vddπ+3Vddσ)1 4(3Vddδ+4Vddπ+9Vddσ)⎞ ⎟⎟⎠, (A4) 075433-9HABIB ROSTAMI AND REZA ASGARI PHYSICAL REVIEW B 91, 075433 (2015) taa 2=1 4⎛ ⎜⎝3Vddδ+Vddσ√ 3(Vddδ−Vddσ)0√ 3(Vddδ−Vddσ) Vddδ+3Vddσ 0 00 4 Vddπ⎞ ⎟⎠, (A5) taa 3=1 4⎛ ⎜⎜⎝3Vddδ+Vddσ√ 3 2(−Vddδ+Vddσ)3 2(Vddδ−Vddσ) √ 3 2(−Vddδ+Vddσ)1 4(Vddδ+12Vddπ+3Vddσ) −√ 3 4(Vddδ−4Vddπ+3Vddσ) 3 2(Vddδ−Vddσ) −√ 3 4(Vddδ−4Vddπ+3Vddσ)1 4(3Vddδ+4Vddπ+9Vddσ)⎞ ⎟⎟⎠, (A6) tbb 1=1 4⎛ ⎜⎝3Vppπ+Vppσ√ 3(Vppπ−Vppσ)0√ 3(Vppπ−Vppσ) Vppπ+3Vppσ 0 00 4 Vppπ⎞ ⎟⎠, (A7) tbb 2=⎛ ⎜⎝Vppσ 00 0 Vppπ 0 00 Vppπ⎞ ⎟⎠, (A8) tbb 3=1 4⎛ ⎜⎝3Vppπ+Vppσ −√ 3(Vppπ−Vppσ)0 −√ 3(Vppπ−Vppσ) Vppπ+3Vppσ 0 00 4 Vppπ⎞ ⎟⎠. 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PhysRevB.72.235303.pdf	Origin of positive magnetoresistance in small-amplitude unidirectional lateral superlattices Akira Endo and Yasuhiro Iye Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan /H20849Received 17 September 2005; published 2 December 2005 /H20850 We report quantitative analysis of positive magnetoresistance /H20849PMR /H20850for unidirectional-lateral-superlattice samples with relatively small periods /H20849a=92−184 nm /H20850and modulation amplitudes /H20849V0=0.015−0.25 meV /H20850.B y comparing observed PMR’s with ones calculated using experimentally obtained mobilities, quantum mobili-ties, and V 0’s, it is shown that contribution from streaming orbits /H20849SOs /H20850accounts for only small fraction of the total PMR. For small V0, the limiting magnetic ﬁeld Beof SO can be identiﬁed as an inﬂection point of the magnetoresistance trace. The major part of PMR is ascribed to drift velocity arising from incompleted cyclo-tron orbits obstructed by scatterings. DOI: 10.1103/PhysRevB.72.235303 PACS number /H20849s/H20850: 73.23.Ad, 75.47.Jn, 73.40. /H11002c I. INTRODUCTION Large mean free path /H20849L/H112711/H9262m/H20850of GaAs/AlGaAs-based two-dimensional electron gas /H208492DEG /H20850and modern nanofab- rication technologies have enabled us to design and fabricate2DEG samples artiﬁcially modulated with length scalesmuch smaller than L. The samples have been extensively utilized for experimental investigations of novel physicalphenomena that take place in the new artiﬁcial environ-ments. 1Unidirectional lateral superlattice /H20849ULSL /H20850represents a prototypical and probably the simplest example of suchsamples; there, a new length scale, the period a, and a new energy scale, the amplitude V 0, of the periodic potential modulation are introduced to 2DEG. These artiﬁcial param-eters give rise to a number of interesting phenomena throughtheir interplay with parameters inherent in 2DEG, especiallywhen subjected to a perpendicular magnetic ﬁeld B. Magne- totransport reveals intriguing characteristics over the wholespan of magnetic ﬁeld, ranging from low ﬁeld regime domi-nated by semiclassical motion of electrons, 2–5through quan- tum Hall regime where several Landau levels are occu-pied, 6–10up to the highest ﬁeld where only the lowest Lan- dau level is partially occupied;11–13in the last regime, semi- classical picture is restored with composite fermions /H20849CFs /H20850 taking the place of electrons. Of these magnetotransport fea-tures, two observed in low ﬁelds, namely, positive mag-netoresistance 4/H20849PMR /H20850around zero magnetic ﬁeld and com- mensurability oscillation2,3/H20849CO/H20850originating from geometric resonance between the period aand the cyclotron radius Rc =/H6036kF/e/H20841B/H20841, where kF=/H208812/H9266nerepresents the Fermi wave number with nethe electron density, have the longest history of being studied and are probably the best known. The PMR has been ascribed to channeled orbit, or stream- ing orbit /H20849SO/H20850, in which electrons travel along the direction parallel to the modulation /H20849ydirection /H20850, being conﬁned in a single valley of the periodic potential.4Electrons that happen to have the momentum perpendicular to the modulation /H20849x direction /H20850insufﬁcient to overcome the potential hill consti- tute SO. In a magnetic ﬁeld B, Lorentz force partially cancels the electric force deriving from the conﬁning potential.Therefore the number of SO’s decreases with increasing B and ﬁnally disappears at the limiting ﬁeld where Lorentzforce balances with the maximum slope of the potential. The extinction ﬁeld B edepends on the amplitude and the shape of the potential modulation, and for sinusoidal modulationV 0cos/H208492/H9266x/a/H20850, Be=2/H9266mV0 ae/H6036kF, /H208491/H20850 where mrepresents the effective mass of electrons. It fol- lows then that V0can be deduced from experimental PMR provided that the line shape of the modulation is known,once B eis determined from the analysis of the experimental trace. An alternative and more familiar way to experimen-tally determine V 0is from the amplitude of CO. In the past, several groups compared V0’s deduced by the two different methods for the same samples.14–17In all cases, V0’s deduced by PMR and by CO considerably disagree, with the formerusually giving larger values. Part of the discrepancy may beattributable to underestimation of V 0by CO, resulting from disregarding the proper treatment of the decay of the COamplitude by scattering. 18–20However, the most serious source of the disagreement appears to lie in the difﬁculty inidentifying the position of B efrom an experimental PMR trace, which was taken, on a rather ad hoc basis, as either the peak,14–16or the position for steepest slope.17It is therefore necessary to ﬁnd out the rule to determine the exact positionofB e. This is one of the purposes of the present paper. We will show below that Becan be identiﬁed, when V0is small enough, as an inﬂection point at which the curvature of PMRchanges from concave down to concave up. Another target ofthe present paper is the magnitude of PMR. The magnitudeshould also depend on V 0as well as on other parameters of ULSL samples. The subject has been treated in theories byboth numerical 21,22and analytical23calculations. However, analyses of experimental PMR is so far restricted to thequalitative level 4that the magnitude increases with V0.T o the knowledge of the present authors, no effort has beenmade to date to quantitatively explain the magnitude ofPMR, using the full knowledge of experimentally obtainedsample parameters V 0,ne, the mobility /H9262, and the quantum, or single-particle mobility /H9262s. Such quantitative analysis has been done in the present paper for ULSL samples with rela-PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 1098-0121/2005/72 /H2084923/H20850/235303 /H2084911/H20850/$23.00 ©2005 The American Physical Society 235303-1tively small periods and modulation amplitudes that allow determining reliable values of V0from the CO amplitude.20 The result demonstrates that magnetoresistance attributable to SO is much smaller than the observed PMR. We proposean alternative mechanism that accounts for the major part ofPMR. After detailing the ULSL samples used in the presentstudy in Sec. II, we delineate in Sec. III a simple analyticformula to be used to estimate the contribution of SO toPMR. Experimentally obtained PMR traces are presentedand compared to the estimated SO-contribution in Sec. IV ,leading to the introduction of another mechanism, the contri-bution from drift velocity of incompleted cyclotron orbits, inSec. V , which we believe dominates the PMR for our presentULSL samples. Some discussion is given in Sec. VI, fol-lowed by concluding remarks in Sec. VII. II. CHARACTERISTICS OF SAMPLES We examined four ULSL samples with differing periods a, as tabulated in Table I. The samples were prepared fromthe same GaAs/AlGaAs single-heterostructure 2DEG wafer with the heterointerface residing at the depth d=90 nm from the surface, and having Al 0.3Ga0.7As spacer layer thickness ofds=40 nm. A grating of negative electron-beam /H20849EB/H20850re- sist placed on the surface introduced potential modulation at the 2DEG plane through strain-induced piezoelectric effect.24 To maximize the effect, the direction of modulation /H20849xdirec- tion /H20850was chosen to be along a /H20855110 /H20856direction. For a ﬁxed crystallographic direction, the amplitude of the strain- induced modulation is mainly determined by the ratio a/d. Figures 1 /H20849b/H20850and 1 /H20849c/H20850display scanning electron micrographs of the gratings. Samples 1, 3, and 4 utilized a simple line-and-space pattern as shown in /H20849b/H20850. For sample 2, we em- ployed a patterned grating depicted in /H20849c/H20850; the “line” of resist was periodically notched in every 575 nm by width 46 nm.The width was intended to be small enough /H20849much smaller than d/H20850so that the notches introduce only negligibly small modulation themselves but act to partially relax the strain.The use of the patterned grating enabled us to attain smallerV 0than sample 1, which has the same period a=184 nm. As shown in Fig. 1 /H20849a/H20850, we used Hall bars with sets of voltage probes that enabled us to measure the section with the grat-ing /H20849ULSL /H20850and that without /H20849reference /H20850at the same time. Resistivity was measured by a standard low-frequency aclock-in technique. Measurements were carried out at T=1.4 and 4.2 K, both bearing essentially the same result. Wepresent the result for 4.2 K in the following. To investigate the behavior of PMR under various values of sample parameters, n ewas varied from about 2.0 to 3.0 /H110031015m−2, employing persistent photoconductivity effectTABLE I. List of samples. No. a/H20849nm/H20850 Hall-bar size /H20849/H9262m2/H20850 back gate 1 184 64 /H1100337 /H11003 2 184 64 /H1100337 /H11003 3 161 44 /H1100316 /H17034 4 138 44 /H1100316 /H17034 FIG. 1. /H20849a/H20850Schematic drawing of the sample with voltage probes for measuring modulated /H20849ULSL /H20850and unmodulated /H20849reference /H20850part. /H20849b/H20850,/H20849c/H20850Scanning electron micrographs of the EB-resist gratings that introduce strain-induced potential modulation. Darker areas correspond to the resist. A standard line-and-space pattern /H20849b/H20850was utilized for samples 1, 3, and 4. Sample 2 employed a patterned grating /H20849c/H20850designed to partially relax the strain.A. ENDO AND Y . IYE PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-2through step-by-step illumination with an infrared light- emitting diode /H20849LED /H20850. Samples 3 and 4 were equipped with a back gate, which was also used to alter neapproximately between 1.7 and 2.0 /H110031015m−2. The electron density newas measured by the period of CO or Shubnikov–de Haas /H20849SdH /H20850 oscillation, and also by Hall resistivity. Concomitant with thechange of n e, parameters associated with the random poten- tial scattering, /H9262and/H9262s, also vary. Plots of /H9262and/H9262s/H20849the latter only for samples 1 and 2 /H20850versus neare presented in Figs. 2 /H20849b/H20850and 2 /H20849c/H20850, respectively. Quantum mobility /H9262sis deduced from the damping of the SdH oscillation25of the unmodulated section of the Hall bar. The amplitude V0of the modulation was evaluated from the amplitude of CO. In a previous publication,20the present authors reported that the oscillatory part of the magnetoresis-tance is given, for V 0much smaller than the Fermi Energy EF,/H9257/H11013V0/EF/H112701, by /H9004/H9267xxosc /H92670=A/H20873/H9266 /H9262WB/H20874A/H20873T Ta/H20874 /H110031 2/H208812/H92661 /H90210/H9262B2/H92622 aV02 ne3/2/H20841B/H20841sin/H208732/H92662Rc a/H20874, /H208492/H20850 where A/H20849x/H20850=x/sinh /H20849x/H20850,kBTa/H11013/H208491/2/H92662/H20850/H20849akF/2/H20850/H6036/H9275cwith/H9275c =e/H20841B/H20841/mthe cyclotron angular frequency, /H90210=h/ethe ﬂux quantum, and /H9262B/H11013e/H6036/2m/H20849/H112290.864 meV/T for GaAs, an analog of the Bohr magneton with the electron mass replacedby the effective mass m /H112290.067 me/H20850. Apart from the factor A/H20849/H9266//H9262WB/H20850, which governs the damping of CO by scattering, Eq. /H208492/H20850is identical to the formula calculated by ﬁrst order perturbation theory.26The parameter /H9262Wwas shown in Ref. 20 to be approximately equal to /H9262s, in accordance with the formula given for low magnetic ﬁeld in the theory by Mirlin and Wölﬂe.27Measured /H9004/H9267xxosc//H92670for the present samples are also described by Eq. /H208492/H20850very well, as exempliﬁed in the inset of Fig. 2 /H20849a/H20850. So far, we have treated the modulation as having a simple sinusoidal proﬁle V0cos/H208492/H9266x/a/H20850, and have tacitly neglected the possible presence of higher harmonics. Although the Fourier transforms of /H9004/H9267xxosc//H92670do reveal small fraction of the second- /H20849and also the third- for samples 1 and 2/H20850harmonics,28their smallness along with the power depen- dence on V0of the relevant resistivities /H20851to be discussed later, see Eqs. /H2084912/H20850and /H2084922/H20850/H20852justiﬁes neglecting them to a good approximation. The parameters V0and/H9262Wobtained by ﬁt- ting Eq. /H208492/H20850to experimental traces are plotted in Figs. 2 /H20849a/H20850 and 2 /H20849c/H20850, respectively. The latter shows /H9262W/H11229/H9262s, conﬁrming our previous result. V0does not depend very much on ne when neis varied by LED illumination, but increases with decreasing newhen the back gate is used, the latter resem- bling a previous report.15The dependence of V0onneis discussed in detail elsewhere.29Since aand dare of compa- rable size, V0rapidly increases with the increase of a/H20849with exception, of course, of sample 2 whose amplitude is close tothat of sample 3 /H20850. Since 6 /H33355E F/H3335511 meV for the range of ne encompassed in the present study, the condition /H9257/H112701 is ful- ﬁlled for all the measurements shown here /H20849/H9257=0.010 −0.034 /H20850.III. CALCULATION OF THE CONTRIBUTION OF STREAMING ORBITS In this section, we describe a simple analytic calculation for estimating the contribution of SO to magnetoresistance.The calculation is a slight modiﬁcation of a theory by Matu-lis and Peeters, 30the theory in which semiclassical conduc- tance was calculated for 2DEG under unidirectional mag-netic ﬁeld modulation with zero average. We modify thetheory to the case for potential modulation V 0cos/H208492/H9266x/a/H20850, FIG. 2. /H20849Color online /H20850Sample parameters as a function of the electron density ne, varied either by LED illumination /H20849open sym- bols /H20850or by back-gate voltage /H20849solid symbols /H20850./H20849a/H20850Modulation am- plitude V0./H20849b/H20850Mobility /H9262./H20849c/H20850Damping parameter /H9262Wof CO. Quantum mobility /H9262sfor samples 1 and 2 are also plotted by /H11003and +, respectively. Inset in /H20849a/H20850shows /H9004/H9267xxosc//H92670experimentally obtained by subtracting a slowly varying background from the magnetoresis-tance trace /H20849for sample 2 at n e=2.20/H110031015m−2, shown by solid trace /H20850and calculated by Eq. /H208492/H20850using V0and/H9262Was ﬁtting param- eters /H20849dotted trace, showing almost perfect overlap with the experi- mental trace /H20850.ORIGIN OF POSITIVE MAGNETORESISTANCE IN … PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-3and extend it to include a uniform magnetic ﬁeld − B/H20849the minus sign is selected just for convenience /H20850. The Hamil- tonian describing the motion of electrons is given by /H9255/H20849x,px,py/H20850=1 2m/H20851px2+/H20849py−eBx /H208502/H20852+V0cos/H208732/H9266x a/H20874,/H208493/H20850 in the Landau gauge A=/H208490,−Bx,0/H20850, where p=/H20849px,py/H20850de- notes canonical momentum. Using electron velocities vx=/H11509/H9255 /H11509px=px m,vy=/H11509/H9255 /H11509py=py−eBx m, /H208494/H20850 the conductivity tensor reads /H20849including the factor 2 for spin degeneracy /H20850 /H9268ij=2e2 /H208492/H9266/H6036/H2085021 Lx/H20885 0Lx dx/H20885 −/H11009/H11009 dpx/H20885 −/H11009/H11009 dpy/H9270svivj/H20873−/H11509f /H11509/H9255/H20874,/H208495/H20850 where Lxrepresents the extent of the sample in the xdirec- tion and /H9270sdenotes an appropriate scattering time to be dis- cussed later.31Since the system is periodic in the xdirection and each SO is conﬁned in a single period, the integration over x,Lx−1/H208480Lxdx, can be reduced to one period, a−1/H208480adx,i n calculating the conductivity from SO. The derivative − /H11509f//H11509/H9255 of the Fermi distribution function f/H20849/H9255/H20850=/H208531+exp /H20851/H20849/H9255 −EF/H20850/kBT/H20852/H20854−1may be approximated by the delta function /H9254/H20849/H9255−EF/H20850at low temperatures, T/H11270EF/kB. Therefore the prob- lem boils down to the integration of /H9270svivjover relevant part of the Fermi surface /H9255/H20849x,px,py/H20850=EFin the /H20849x,px,py/H20850space. Fermi surface is depicted in Fig. 3 for three different values of/H9252/H11013B/Be. Since the Hamiltonian Eq. /H208493/H20850does not explic- itly include y,pyis a constant of motion that specify an orbit; an orbit is given by the cross section of the Fermisurface by a constant- p yplane. The presence of SO is indi- cated by the shaded area in Fig. 3. The ratio of SO to all theorbits is maximum at /H9252=0, decreases with increasing /H9252, and disappears at /H9252=1. Before continuing the calculation, we now discuss an ad- equate scattering time to choose. At variance with Ref. 30,we adopt here unweighted single-particle scattering time /H9270s =/H9262sm/e. The choice is based on the fact that the angle /H9258 =arctan /H20849vx/vy/H20850of the direction of the velocity with respect to they-axis is very small for electrons belonging to SO in our ULSL samples having small /H9257=V0/EF. The maximum of /H20841/H9258/H20841 at a position u/H110132/H9266x/acan be approximately written as /H20851/H9257/H9272/H20849/H9252,u/H20850/H208521/2with/H9272/H20849/H9252,u/H20850/H11013/H208811−/H92522+/H9252arcsin /H9252− cos u−/H9252u, /H208496/H20850 whose maximum over uis given by /H208512/H9257/H9278/H20849/H9252/H20850/H208521/2with/H9278/H20849/H9252/H20850 /H11013/H208811−/H92522+/H9252arcsin /H9252−/H20849/H9266/2/H20850/H9252, where /H20841/H9278/H20849/H9252/H20850/H20841/H333551 for /H20841/H9252/H20841/H333551 /H20849see Fig. 4 /H20850. Since /H20841/H9258/H20841is much smaller than the average scat- tering angle /H9258scat/H11011/H208812/H9262s//H9262/H112290.5 rad estimated for our present 2DEG wafer, electrons are kicked out of SO by vir-tually any scattering event regardless of the scattering angleinvolved, letting /H9270sto be the appropriate scattering time. The integration /H208495/H20850over the shaded area gives the correc- tion to the conductivity owing to SO, to the leading order in /H9257,a s /H9254/H9268xxSO /H92680=−2 2/H92662/H9262s /H9262/H20885 arcsin /H9252u1/H20849/H9252/H208502 3/H20851/H9257/H9272/H20849/H9252,u/H20850/H208523/2du =−32/H208812 9/H92662/H9262s /H9262/H92573/2F/H20849/H9252/H20850, /H208497/H20850 where the minus sign results because electrons trapped in SO cannot carry current over the /H20849macroscopic /H20850sample in x-direction and therefore should be deducted from the con- ductivity, and /H9254/H9268yySO /H92680=2 2/H92662/H9262s /H9262/H20885 arcsin /H9252u1/H20849/H9252/H20850 2/H20851/H9257/H9272/H20849/H9252,u/H20850/H208521/2du=8/H208812 /H92662/H9262s /H9262/H92571/2G/H20849/H9252/H20850, /H208498/H20850 and/H9254/H9268xySO=/H9254/H9268yxSO=0, where /H92680=EFe2/H9270//H9266/H60362represents the Drude conductivity. The factor 2 in the ﬁrst equalities ac-counts for the two equivalent SO areas at the upper and thelower bounds of p y. The functions F/H20849/H9252/H20850and G/H20849/H9252/H20850are deﬁned as F/H20849/H9252/H20850/H110133 16/H208812/H20885 arcsin /H9252u1/H20849/H9252/H20850 /H20851/H9272/H20849/H9252,u/H20850/H208523/2du /H208499/H20850 and FIG. 3. /H20849Color online /H20850Fermi surface in the x-px-pyspace for /H20849a/H20850 /H9252=0, /H20849b/H208500/H11021/H9252/H110211, and /H20849c/H20850/H9252=1. Each electron orbit is speciﬁed by the cross section of the Fermi surface by a constant- pyplane. Streaming orbits are present in the shaded area. FIG. 4. /H20849Color online /H20850Functions F/H20849/H9252/H20850,G/H20849/H9252/H20850,/H9278/H20849/H9252/H20850/H20849thin dotted, solid, and dash-dotted lines, respectively, left axis /H20850and/H92522G/H20849/H9252/H20850 /H20849thick solid line, right axis /H20850.A. ENDO AND Y . IYE PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-4G/H20849/H9252/H20850/H110131 4/H208812/H20885 arcsin /H9252u1/H20849/H9252/H20850 /H20851/H9272/H20849/H9252,u/H20850/H208521/2du, /H2084910/H20850 where the upper limit of integration u1/H20849/H9252/H20850is the solution of /H9272/H20849/H9252,u1/H20850=0 other than arcsin /H9252. Both F/H20849/H9252/H20850and G/H20849/H9252/H20850mono- tonically decrease from 1 to 0 while /H9252varies from 0 to 1, as shown in Fig. 4. Since /H9254/H9268xxSO//H9254/H9268yySO/H11008/H9257,/H9254/H9268xxSO/H11270/H9254/H9268yySOfor/H9257 /H112701. Correction to the resistivity by SO can be obtained by inverting the conductivity tensor /H9254/H9267xxSO /H92670=/H20877/H9254/H9268xxSO /H92680+/H208751+ /H20849B/H9262/H208502/H9254/H9268yySO//H92680 1+/H9254/H9268yySO//H92680/H20876−1/H20878−1 −1 /H11229−/H9254/H9268xxSO /H92680+/H20849B/H9262/H208502/H9254/H9268yySO /H92680 =32/H208812 9/H926621 /H20849/H90210/H9262B/H208503/2/H9262s /H9262V03/2 ne3/2F/H20849/H9252/H20850 +4/H208812 /H92661 /H902101/2/H9262B5/2/H9262s/H9262 a2V05/2 ne3/2/H92522G/H20849/H9252/H20850. /H2084911/H20850 For small /H9257,/H9254/H9268xxSO//H92680can be neglected and consequently /H9254/H9267xxSO /H92670/H112294/H208812 /H92661 /H902101/2/H9262B5/2/H9262s/H9262 a2V05/2 ne3/2/H92522G/H20849/H9252/H20850. /H2084912/H20850 The correction therefore increases in proportion to /H9262,/H9262s, and V05/2, and decreases with aand ne. The function /H92522G/H20849/H9252/H20850is also plotted in Fig. 4, which takes maximum at /H9252/H112290.6 and vanishes at /H9252=1. Our ﬁnal result Eq. /H2084912/H20850is identical to Eq. /H2084941/H20850of Ref. 23, which is deduced for the case /H9257/H11270/H9262s//H9262./H20849For larger /H9257, Ref. 23 gives somewhat different formula that is proportional to V07/2./H20850Note that our /H9278/H20849/H9252/H20850and G/H20849/H9252/H20850are iden- tical to the functions denoted as /H90211/H20849/H9252/H20850and/H9021/H20849/H9252/H20850, respec- tively, in Ref. 23. In the following section, Eq. /H2084912/H20850will be compared with experimental traces. IV. POSITIVE MAGNETORESISTANCE OBTAINED BY EXPERIMENT Figure 5 shows low-ﬁeld magnetoresistance traces for samples 1-4 for various values of ne. Solid curves represent measurements before illumination /H20849nevaried by the back gate /H20850and dotted curves are traces for nevaried by LED illu- mination /H20849back gate voltage=0 V /H20850. The magnitude of PMR shows clear tendency of being large for samples havinglarger V 0. By contrast, the peak positions do not vary much between samples. To facilitate quantitative comparison withEq. /H2084912/H20850, Fig. 5 is replotted in Fig. 6, with both horizontal and vertical axes scaled with appropriate parameters: thehorizontal axis is normalized by B ecalculated by Eq. /H208491/H20850 using experimentally deduced neand V0shown in Fig. 2; the vertical axis is normalized by the prefactor in Eq. /H2084912/H20850with /H9262sreplaced by /H9262W, identifying the two parameters.32Mag- netoresistance owing to SO will then be represented by auniversal function /H92522G/H20849/H9252/H20850, which is also plotted in the ﬁg- ures. It is clear from the ﬁgures that experimentally observed PMR is much larger than that calculated by Eq. /H2084912/H20850. Fur- thermore, the peaks appear at B/H11022Be, i.e., where SO havealready disappeared, for all traces for samples 1-3 and traces with smaller nefor sample 4. The peak position is by no means ﬁxed, but depends on the sample parameters. Thisobservation argues against the interpretation of PMR beingsolely originating from SO. Rather, we interpret that SO ac-counts for only a small fraction of the PMR, as suggested byFig. 6, and that the rest is ascribed to another effect to bediscussed in the next section. In fact, humps that appear tocorrespond to the component /H92522G/H20849/H9252/H20850can readily be recog- nized in traces with larger nefor sample 4, superposed on a slowly increasing component of PMR. The humps terminateat around /H20841 /H9252/H20841=1, where the total PMR changes the sign of the curvature. With the increase of /H9257/H20849/H11008V0/ne/H20850either by de- creasing ne/H20849upper traces for sample 4 /H20850or by increasing V0 /H20849samples 1-3 /H20850,/H92522G/H20849/H9252/H20850makes progressively smaller contri- bution to the total PMR, and becomes difﬁcult to be distin- guished from the background. As has been inferred just above, the interpretation that the contribution /H9254/H9267xxSO//H92670from SO is superimposed on another slowly increasing background component offers an alterna-tive way to determine B e:Becan be identiﬁed with the end of the hump, namely, the inﬂection point Binfwhere the curva- ture of the total PMR changes from concave down, inheritedfrom /H92522G/H20849/H9252/H20850, to concave up. To be more speciﬁc, Binfis determined as a point where the second derivative /H20849d2/dB2/H20850 /H11003/H20849/H9004/H9267xx//H92670/H20850changes sign from negative to positive as illus- trated in Fig. 7 /H20849b/H20850. The inﬂection point Binfis marked by a FIG. 5. /H20849Color online /H20850Magnetoresistance traces for various val- ues of ne. Selected values of neare noted in the ﬁgure /H20849in 1015m−2/H20850. Dotted traces indicate that the neis attained by LED illumination. Note that the vertical scale is expanded by ﬁve times for sample 4.ORIGIN OF POSITIVE MAGNETORESISTANCE IN … PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-5downward open triangle both in /H20849d2/dB2/H20850/H20849/H9004/H9267xx//H92670/H20850/H20849solid /H20850 and/H9004/H9267xx//H92670/H20849dotted /H20850traces. /H20851/H20849d2/dB2/H20850/H20849/H9004/H9267xx//H92670/H20850shows oscil- latory features at low ﬁeld, which are attributed to the geo- metric resonance of Bragg-reﬂected cyclotron orbits.33/H20852Fig- ure 7 /H20849a/H20850illustrates the shift of Binfwith ne. The plot of Binf versus Beshown in Fig. 7 /H20849c/H20850demonstrates that Binfis actu- ally identiﬁable with Be. Thus it is now possible to deduce reliable values of V0from PMR: by replacing Bewith Binfin Eq. /H208491/H20850. Unfortunately this method is applicable only for samples with very small /H9257. For samples 1-3, it is difﬁcult to ﬁnd clear inﬂection points because of the dominance of theslowly increasing component; /H20849d 2/dB2/H20850/H20849/H9004/H9267xx//H92670/H20850only gradually approaches zero from below. In the subsequent section, we discuss the origin of the slowly increasing back-ground component of the PMR. V. DRIFT VELOCITY OF INCOMPLETED CYCLOTRON ORBITS An important point to be noticed is that even at the low magnetic-ﬁeld range /H20841B/H20841/H11021Bewhere SO is present, most of the electrons are in cyclotron-like orbits, namely the cyclo-tron orbits slightly modiﬁed by a weak potential modulation,as evident in Fig. 3; SO accounts for only small fraction,order of /H92571/2, of the whole orbits. Therefore, the contributionof these cyclotronlike orbits to the magnetoresistance should be taken into consideration in interpreting the PMR. We willshow below that the slowly varying component of the PMRis attributable to the E/H11003Bdrift velocity of the electrons in the cyclotronlike orbits that are scattered before completing acycle. It is well established that the E/H11003Bdrift velocity resulting from the gradient of the modulation potential E=−/H11633V//H20849−e/H20850 and the applied magnetic ﬁeld B=/H208490,0, B/H20850is the origin of the CO. 34For unidirectional modulation V/H20849x/H20850=V0cos/H20849qx/H20850 with q=2/H9266/a, the drift velocity vd=/H20849E/H11003B/H20850/B2has only the ycomponent vd,y=qV0 eBsin/H20849qx/H20850. /H2084913/H20850 Electrons acquire vd,yduring the course of a cyclotron revo- lution, whose sign alternates rapidly except for when elec-trons are traveling nearly parallel to the modulation /H20849 /H9258 /H112290,/H9266/H20850, i.e., around either the rightmost /H20849maximum- x/H20850or the leftmost /H20849minimum- x/H20850edges. Therefore, the contribution of the drift velocity to the conductivity comes almost exclu- FIG. 6. /H20849Color online /H20850Replot of Fig. 5 with abscissa normalized by the extinction ﬁeld Beand ordinate by the sample-parameter- dependent prefactor in Eq. /H2084912/H20850,/H9251/H9262W/H9262V05/2a−2ne−3/2, with the coef- ﬁcient /H9251/H110134/H208812/H9266−1/H90210−1/2/H9262B−5/2 /H112294.04/H11003107T2meV−5/2m−1. Vertical scale is expanded twice for sample 4. The function /H92522G/H20849/H9252/H20850is also plotted for comparison. FIG. 7. /H20849Color online /H20850/H20849a/H20850Magnetoresistance traces for sample 4 with the inﬂection point Binfmarked by downward open triangles. Traces are offset proportionally to the change in ne. Selected values ofnein 1015m−2are noted in the ﬁgure. /H20849b/H20850Illustration of the procedure to pick up Binf/H20849an example for ne=2.33/H110031015m−2/H20850. The point at which the second derivative /H20849d2/dB2/H20850/H20849/H9004/H9267xx//H92670/H20850/H20849solid curve, right axis /H20850crosses zero upward /H20849marked by open downward triangle /H20850is identiﬁed as Binf.Binfis marked also on /H9004/H9267xx//H92670/H20849dotted curve, left axis /H20850. Shaded area indicates the contribution from SO. /H20849c/H20850Plot of Binfversus Becalculated by Eq. /H208491/H20850using experimentally obtained V0. The line represents Binf=Be.A. ENDO AND Y . IYE PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-6sively from the two edges as depicted in Fig. 8 /H20849a/H20850, which is actually experimentally veriﬁed in Ref. 35. The CO is theresult of the alternating occurrence by sweeping the mag-netic ﬁeld of the constructive and destructive addition of the effects from the two edges, as illustrated by the top and thebottom cyclotron orbits in Fig. 8 /H20849a/H20850, respectively. With the decrease of the magnetic ﬁeld, cyclotron radius R cincreases and consequently the probability of electrons being scatteredbefore reaching from one to the other edge increases. As aresult, the distinction between the constructive and destruc-tive cases are blurred, letting the CO amplitude diminishmore rapidly than predicted by the theories 26,34neglecting such scattering. The absence of CO at lower magnetic ﬁelds signiﬁes that electrons are mostly scattered before traveling to the otheredge. Although the correlation of the local drift velocities atthe both edges is lost at such magnetic ﬁelds /H20851Fig. 8 /H20849b/H20850/H20852, each edge can independently contribute to the conductivity. It is tothis effect that we ascribe the major part of PMR in ourULSL samples. Note that the onset of CO basically coincidewith the end of the PMR, bolstering this interpretation. It can be shown, by an approximate analytic treatment of the Boltzmann’s equation, that the effect actually gives riseto PMR with right order of magnitude to explain the experi-mentally observed slowly varying component. For this pur-pose, we make use of Chambers’ formula, 36–38representing the relaxation time approximation of Boltzmann’s equation,to obtain, from the drift velocity, the component Dyyof the diffusion tensor Dyy=/H20885 0/H11009 e−t//H9270/H20855vd,y/H20849t/H20850vd,y/H208490/H20850/H20856dt, /H2084914/H20850 where /H20855¯/H20856signiﬁes averaging over all possible initial con- ditions for the motion of electrons along the trajectories. Ein- stein’s relation is then used to obtain corresponding incre-ment in the conductivity, /H9254/H9268yy=e2D/H20849EF/H20850Dyywith D/H20849EF/H20850 =m//H9266/H60362=/H20849/H90210/H9262B/H20850−1the density of states, and ﬁnally it is translated to the resistivity by tensor inversion, /H9254/H9267xx//H92670 =/H20849/H9275c/H9270/H208502/H9254/H9268yy//H92680. We use unperturbed cyclotron trajectory, x =X+Rccos/H9258, for simplicity, neglecting the modiﬁcation of the orbit by the modulation /H20849and accordingly, SO is neglected in this treatment /H20850, which is justiﬁed for small /H9257. Since the initial condition can be speciﬁed by the guiding center posi-tion Xand the initial angle /H92580, we can write /H20855vd,y/H20849t/H20850vd,y/H208490/H20850/H20856=/H20873qV0 eB/H2087421 a/H20885 0a dX1 2/H9266/H20885 −/H9266/H9266 d/H92580sin/H20853q/H20851X +Rccos/H20849/H92580+/H9275ct/H20850/H20852/H20854sin/H20851q/H20849X +Rccos/H92580/H20850/H20852. /H2084915/H20850 Therefore Eq. /H2084914/H20850can be rewritten, performing the integra- tion over tﬁrst, as Dyy=/H20873qV0 eB/H2087421 a/H20885 0a dX1 2/H9266/H20885 −/H9266/H9266 d/H92580sin/H20851q/H20849X+Rccos/H92580/H20850/H20852I/H20849/H92580/H20850, /H2084916/H20850 with I/H20849/H92580/H20850=/H20885 0/H11009 e−t//H9270sin/H20853q/H20851X+Rccos/H20849/H92580+/H9275ct/H20850/H20852/H20854dt. /H2084917/H20850 Evaluation of Eq. /H2084916/H20850for a large enough magnetic ﬁeld reproduces basic features of Eq. /H208492/H20850, as will be shown in the Appendix. Here, we proceed with an approximation forsmall magnetic ﬁelds. The approximation is rather crude butis sufﬁcient for the purpose of getting a rough estimate of theorder of magnitude. Because of the exponential factor, only the time t/H11351 /H9270con- tributes to the integration of Eq. /H2084917/H20850. Due to the rapidly oscillating nature of the sin /H20853/H20854factor and the smallness of /H9275ct/H11351/H9275c/H9270,I/H20849/H92580/H20850takes a signiﬁcant value only when /H92580re- sides in a narrow range slightly below /H110110o r /H11011/H9266, corre- sponding to the situation when electrons travels near theright-most or the left-most edge, respectively, within thescattering time. It turns out, by comparing with the numericalevaluation of Eq. /H2084917/H20850using sample parameters for our present ULSL’s, that the following approximate expressionsroughly reproduce the right order of magnitude and the rightoscillatory characteristics /H20849the period and phase /H20850of Eq. /H2084917/H20850 for low magnetic ﬁeld /H20849/H20841B/H20841/H113510.02 T /H20850: FIG. 8. /H20849Color online /H20850Illustration of E/H11003Bdrift velocity vd affecting the electrons during the cyclotron motion. Orbits are de- picted neglecting the modiﬁcation due to the modulation V/H20849x/H20850 =V0cos/H20849qx/H20850/H20849drifting movement and slight variation of the velocity depending on x/H20850for simplicity. Top diagrams represent slightly larger Bthan bottom ones for both /H20849a/H20850and /H20849b/H20850. On averaging vd,y along an orbit, most contribution comes from minimum- and maximum- xedges, as shown by solid arrows in the ﬁgure. Open arrows indicate the direction of Ex=/H20849qV0/e/H20850sin/H20849qx/H20850at the edges. /H20849a/H20850 For Blarge enough so that electrons can travel cycles before being scattered. Depending on B,vdat both edges are constructive /H20849top diagram, 2 Rc/a=n+1/4 with ninteger /H20850or destructive /H20849bottom dia- gram, 2 Rc/a=n−1/4 /H20850, resulting in maxima and minima in the mag- netoresistance, respectively. /H20849b/H20850For small Bso that electrons are scattered before completing a cycle. The interrelation of vd,yat both edges is not simply determined by B. The edges affect the magne- toresistance independently.ORIGIN OF POSITIVE MAGNETORESISTANCE IN … PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-7I/H20849/H92580/H20850 /H11229/H20877/H9270/H9266/H20851sin/H20849qX/H20850J0/H20849qRc/H20850+ cos /H20849qX/H20850H0/H20849qRc/H20850/H20852 /H20849/H92580/H110110/H20850, /H9270/H9266/H20851sin/H20849qX/H20850J0/H20849qRc/H20850− cos /H20849qX/H20850H0/H20849qRc/H20850/H20852 /H20849/H92580/H11011/H9266/H20850,/H20878 /H2084918/H20850 where J0/H20849x/H20850andH0/H20849x/H20850represent zeroth order Bessel and Struve functions of the ﬁrst kind, respectively. The approxi- mation can be obtained by replacing the exponential factorby a constant /H9275c/H9270and limiting the range of the time integral to include only one edge. Here we noted that the integrationof cos /H20849qR ccos/H9258/H20850and sin /H20849qRccos/H9258/H20850over the range of /H9258in- cluding either of the right-most /H20849/H9258=0/H20850or the left-most /H20849/H9258 =/H9266/H20850edge can be approximated /H20849since only the close vicinity of the edges makes signiﬁcant contribution to the integra- tion /H20850by /H20885 right mostd/H9258/H11229/H20885 −/H9266/2/H9266/2 d/H9258,/H20885 left mostd/H9258/H11229/H20885 /H9266/23/H9266/2 d/H9258, /H2084919/H20850 and used the relations /H20885 −/H9266/2/H9266/2 cos/H20849qRccos/H9258/H20850d/H9258=/H20885 /H9266/23/H9266/2 cos/H20849qRccos/H9258/H20850d/H9258=/H9266J0/H20849qRc/H20850 and /H20885 −/H9266/2/H9266/2 sin/H20849qRccos/H9258/H20850d/H9258=−/H20885 /H9266/23/H9266/2 sin/H20849qRccos/H9258/H20850d/H9258 =/H9266H0/H20849qRc/H20850. /H2084920/H20850 Substituting Eq. /H2084918/H20850to Eq. /H2084916/H20850results in Dyy/H11229/H9266 2/H9270/H20873qV0 eB/H208742 /H20851J02/H20849qRc/H20850+H02/H20849qRc/H20850/H20852, /H2084921/H20850 and with Einstein’s relation one ﬁnally obtains /H9254/H9267xxdrift /H92670=/H20881/H9266 21 /H90210/H9262B2/H92622 aV02 ne3/2/H20841B/H20841. /H2084922/H20850 Here we made use of asymptotic expressions J0/H20849x/H20850 /H11015/H208492//H9266x/H208501/2cos/H20849x−/H9266/4/H20850and H0/H20849x/H20850/H11015/H208492//H9266x/H208501/2sin/H20849x−/H9266/4/H20850 valid for large enough x/H20849corresponding to small enough B/H20850. In order to compare experimentally obtained PMR with Eq. /H2084922/H20850, PMR traces shown in Fig. 5 are replotted in Fig. 9 normalized by the prefactor in Eq. /H2084922/H20850, after subtracting the small contribution from SO represented by Eq. /H2084912/H20850. The scaled traces show reasonable agreement with /H20841B/H20841at low magnetic ﬁelds, as predicted in Eq. /H2084922/H20850, testifying that the mechanism considered here, the drift velocity from incom-pleted cyclotron orbits, generates PMR having the magnitudesufﬁcient to explain the major part of PMR observed in ourpresent ULSL samples. Possible sources of the remnant de-viation, apart from the crudeness of the approximation, are/H20849i/H20850the neglect of higher harmonics and /H20849ii/H20850the neglect of negative magnetoresistance /H20849NMR /H20850component innate to GaAs/AlGaAs 2DEG /H20849Ref. 39 /H20850arising from electron interactions 40,41or from semiclassical effect.42,43The nth har-monic gives rise to additional contribution analogous to Eq. /H2084922/H20850with V0and areplaced by the amplitude Vnof the nth harmonic potential and a/n, respectively, and therefore, in principle, enhances the deviation. In practice, however, theeffect will be small because of the small values of V nand its square dependence. On the other hand, the discrepancy canbe made smaller by correcting for the NMR. We have actu-ally observed NMR, which depends on n eand temperature, in the simultaneously measured “reference” plain 2DEG ad-jacent to the ULSL /H20851see Fig. 1 /H20849a/H20850/H20852. Assuming that the NMR with the same magnitude are also present in the ULSL partand superposed on the PMR /H20849the assumption whose validity remains uncertain at present /H20850, the correction are seen to ap- preciably reduce the discrepancy. The approximation leading to Eq. /H2084922/H20850is valid only for very small magnetic ﬁelds. With the increase of the magneticﬁeld, the cooperation between the left-most and the right-most edges is rekindled, and the magnetoresistance tends tothe expression appropriate for large enough magnetic ﬁeld,outlined by Eq. /H20849A11 /H20850, which includes a nonoscillatory term /H20849the ﬁrst term /H20850as well as the term representing CO /H20849the sec- ond term /H20850. Note that the nonoscillatory term approaches a constant /H9251/H11032/H92622V02a−1ne−3/2/H20849m/2/H9266e/H9270/H20850, at small magnetic ﬁeld, although the exact value of the constant is rather difﬁcult to estimate due to the subtlety in choosing the right scattering FIG. 9. /H20849Color online /H20850Replot of magnetoresistance traces nor- malized by the prefactor in Eq. /H2084922/H20850,/H9251/H11032/H92622V02a−1ne−3/2, with /H9251/H11032 =/H20849/H9266/2/H208501/2/H90210−1/H9262B−2=4.06/H110031014Tm e V−2m−2, after subtracting the contribution from SO, /H9254/H9267xxSO//H92670in Eq. /H2084912/H20850. Contribution attribut- able to drift velocity of incompleted cyclotron orbits is given by /H20841B/H20841, which is also plotted by dash-dotted line.A. ENDO AND Y . IYE PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-8time/H9270, as will be discussed in the Appendix. Therefore the /H20849linear /H20850increase of /H9254/H9267xxdrift//H92670with /H20841B/H20841is expected to ﬂatten out at a certain magnetic ﬁeld. The peak in the PMR roughlymarks the position of this transition, which basically corre-sponds to the onset of the cooperation between the twoedges. Thus the peak position is mainly determined by thescattering parameters and is expected to be insensitive to V 0, in agreement with what has been observed in Fig. 5. Experi-mentally, the peak position B pis found to be well described by an empirical formula Bp/H20849T/H20850=/H208514/H208812/H9262W/H20849m2/V s /H20850/H20852−1, using /H9262Wdetermined from CO. On the other hand, the height of the PMR peak are seen to roughly scale as V02, as inferred from Fig. 9, which reveals that the normalized peak heighttends to fall into roughly the same value /H20849notably the top panel showing two samples having the same aand different V 0/H20850, so long as the period aare the same. This is better shown after correcting for the NMR effect mentioned above.The height of the normalized peak slightly decreases withdecreasing a/H20849roughly proportionally to a/H20850, resulting in an empirical formula for the peak height /H20849/H9004 /H9267xx//H92670/H20850peak/H110113 /H1100310−3/H20851/H9262/H20849m2/V s /H20850/H208522/H20851V0/H20849meV /H20850/H208522/H20851ne/H208491015m−2/H20850/H20852−3/2./H20849Unfortu- nately, sample 4 with larger nesigniﬁcantly deviates from this formula. /H20850 VI. DISCUSSION ON THE RELATIVE IMPORTANCE OF THE STREAMING ORBIT Although PMR was thus far generally interpreted to origi- nate from SO, contribution from mechanisms other than SOwas also implied in theoretical papers. By solving Boltz-mann’s equation numerically, Menne and Gerhardts 21calcu- lated PMR and showed separately the contribution of SOwhich did not account for the entire PMR /H20849see Fig. 4 in Ref. 21/H20850, leaving the rest to alternative mechanisms /H20849although the authors did not discuss the origin futher /H20850. Mirlin et al. 23ac- tually calculated contribution of drifting orbit, which is basi-cally similar to what we have considered in the present pa-per. They predicted cusplike shape for the magnetoresistancearising from this mechanism, which is not observed in ex-perimental traces. In both papers, the major part of PMR isstill ascribed to SO, with other mechanisms playing onlyminor roles. In the present paper, we have shown that therelative importance is the other way around in our ULSLsamples. However, we would like to point out that the domi-nant mechanism may change with the amplitude of modula-tion in ULSL. The reason for the contribution of SO being small in our samples can be traced back to the small amplitude of themodulation, combined with the small-angle nature of thescattering in the GaAs/AlGaAs 2DEG. As mentioned earlier,small /H9257=V0/EFlimits the SO within narrow angle range /H20841/H9258/H20841/H33355/H208812/H9257, letting the electrons being scattered out of the SO even by a small-angle scattering event, hence the use of /H9270sin Eq. /H208495/H20850. This leads to small /H9254/H9268yySO, since /H9270s/H11270/H9270. Within the present framework, relative weight of SO in PMR decreaseswith increasing /H9257, since the ratio of Eq. /H2084912/H20850to Eq. /H2084922/H20850is proportional to /H9257−1/2, in agreement with what was observed in Fig. 6. However, the situation will be considerably alteredwith further increase in /H9257/H20849typically /H9257/H114070.1/H20850. Then, due tothe expansion of the angle range encompassed by SO, elec- trons begin to be allowed to stay within SO after small-anglescattering, requiring /H9270sin Eq. /H208495/H20850to be replaced by larger /H20849possibly B-dependent /H20850values. In the limit that the range of /H20841/H9258/H20841is much larger than the average scattering angle, /H9270sshould be supplanted by ordinary transport lifetime /H20849momentum- relaxation time /H20850/H9270, resulting in much larger /H9254/H9268yySO. This largely enhances the relative importance of SO, possibly toan extent to exceed the contribution from the drift velocity.We presume that the contribution of SO is much larger thanin our case in most of the experiments reported so far whichshowed the shift of PMR peak position with the modulationamplitude. 4,14–17Even in such a situation, however, it will not be easy to obtain simple relation between the peak posi-tion B pand the amplitude V0because of the complication by the remnant contribution from the drift velocity. In most ex-periments, V 0is varied by the gate bias, which concomitantly alters the electron density and scattering parameters, therebyaffecting the both contributions as well. VII. CONCLUSIONS The positive magnetoresistance /H20849PMR /H20850in unidirectional lateral superlattice /H20849ULSL /H20850possesses two different types of mechanisms as its origin: the streaming orbit /H20849SO/H20850and the drift velocity of incompleted cyclotron orbit. Although virtu-ally only the former mechanism has hitherto been taken intoconsideration, we have shown that the latter mechanism ac-count for the main part of PMR observed in our ULSLsamples characterized by their small modulation amplitude.The share undertaken by SO decreases with increasing /H9257 =V0/EF, insofar as /H9257is kept small enough for the electrons in SO to be driven out even by a small-angle scattering char-acteristic of GaAs/AlGaAs 2DEG; /H9257/H333550.034 for our samples fulﬁlls this requirement. In this small /H9257regime, the peak position of PMR is not related to the modulation am-plitude V 0but rather determined by scattering parameters; the peak roughly coincide with the onset of commensurabil-ity oscillation /H20849CO/H20850that notiﬁes the beginning of the coop- eration between the left-most and the right-most edges in acyclotron revolution. The height of the peak, on the other hand, are found to be roughly proportional to V 02. For small enough /H9257, the contribution of SO becomes distinguishable as a hump superposed on slowly increasing component and themagnetic ﬁeld that marks the end of the SO, B e, can be identiﬁed as an inﬂection point of the magnetoresistancetrace where the curvature changes from concave down toconcave up. The extinction ﬁeld B eprovides an alternative method via Eq. /H208491/H20850to accurately determine V0. We have also argued that for samples with /H9257much larger than ours, typi- cally/H9257/H114070.1, the relative importance of the two mechanisms can be reversed and the PMR peak position Bpcan depend onV0, although it will be difﬁcult to deduce a reliable value ofV0from Bp. ACKNOWLEDGMENTS This work was supported by Grant-in-Aid for Scientiﬁc Research in Priority Areas “Anomalous Quantum Materials,”ORIGIN OF POSITIVE MAGNETORESISTANCE IN … PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-9Grant-in-Aid for Scientiﬁc Research /H20849C/H20850/H20849Grant No. 15540305 /H20850and /H20849A/H20850/H20849Grant No. 13304025 /H20850, and Grant-in-Aid for COE Research /H20849Grant No. 12CE2004 /H20850from the Ministry of Education, Culture, Sports, Science and Technology. APPENDIX: APPROXIMATION FOR HIGHER MAGNETIC FIELD In this appendix, we delineate the approximation of Eq. /H2084914/H20850at higher magnetic ﬁeld, which leads to an expression for commensurability oscillation /H20849CO/H20850. When the velocity vd,y/H20849t/H20850is a periodic function of time with period T, Eq. /H2084914/H20850 reduces to38 Dyy=1 1−e−T//H9270/H20885 0T e−t//H9270/H20855vd,y/H20849t/H20850vd,y/H208490/H20850/H20856dt. /H20849A1/H20850 Using here again the unperturbed cyclotron orbit x=X +Rccos/H20849/H9275ct/H20850, one obtains Dyy=/H20873qV0 eB/H2087421 a/H20885 0a dX1 2/H9266/H20885 −/H9266/H9266 d/H92580sin/H20851q/H20849X +Rccos/H92580/H20850/H208521 1−e−T//H9270IT/H20849/H92580/H20850/H20849 A2/H20850 with T=2/H9266//H9275cand IT/H20849/H92580/H20850=/H20885 0T e−t//H9270sin/H20853q/H20851X+Rccos/H20849/H92580+/H9275ct/H20850/H20852/H20854dt./H20849A3/H20850 Again because of the sin /H20853/H20854factor, the main contribution in the integration comes from the narrow band of taround /H92580 +/H9275ct/H110110/H20849or 2/H9266depending on the initial angle /H92580/H20850and/H9266. For large enough /H9275c, the band of tbecomes narrow enough to allow the exponential factor e−t//H9270to be approximated by a constant value at t=−/H20849/H92580−k/H9266/H20850//H9275c/H20849with k=0, 1, and 2 /H20850. Thus, using the relations /H2084920/H20850,IT/H20849/H92580/H20850can be approximately written, depending on the values of /H92580,a s IT/H11021/H20849/H92580/H20850/H11229/H9266 /H9275ce/H92580//H9275c/H9270/H20851sin/H20849qX/H20850J0/H20849qRc/H20850+ cos /H20849qX/H20850H0/H20849qRc/H20850/H20852 +/H9266 /H9275ce/H20849/H92580−/H9266/H20850//H9275c/H9270/H20851sin/H20849qX/H20850J0/H20849qRc/H20850− cos /H20849qX/H20850H0/H20849qRc/H20850/H20852 /H20849A4/H20850 for −/H9266+/H11021/H92580/H110210−and IT/H11022/H20849/H92580/H20850/H11229/H9266 /H9275ce/H20849/H92580−/H9266/H20850//H9275c/H9270/H20851sin/H20849qX/H20850J0/H20849qRc/H20850− cos /H20849qX/H20850H0/H20849qRc/H20850/H20852 +/H9266 /H9275ce/H20849/H92580−2/H9266/H20850//H9275c/H9270/H20851sin/H20849qX/H20850J0/H20849qRc/H20850+ cos /H20849qX/H20850H0/H20849qRc/H20850/H20852 /H20849A5/H20850 for 0+/H11021/H92580/H11021/H9266−, where the superscripts + and − indicate small setbacks to avoid the region where the integration havesigniﬁcant value. When /H92580approaches closer to the bound- ary, IT/H20849/H92580/H20850approaches the average of the values on both sidesIT/H20849/H92580→0/H20850→/H20851IT/H11021/H20849/H92580/H20850+IT/H11022/H20849/H92580/H20850/H20852/2 /H20849A6/H20850 and IT/H20849/H92580→/H9266/H20850→/H20851IT/H11022/H20849/H92580/H20850+IT/H11021/H20849/H92580−2/H9266/H20850/H20852/2, /H20849A7/H20850 which can be shown by using the relations /H20885 −/H9266/20 cos/H20849qRccos/H9258/H20850d/H9258=/H20885 0/H9266/2 cos/H20849qRccos/H9258/H20850d/H9258=/H9266 2J0/H20849qRc/H20850 /H20849A8/H20850 and other related equations corresponding to the halves of Eq. /H2084920/H20850. In the integration by /H92580in Eq. /H20849A2/H20850, only /H92580/H110110 and /H9266contributes to the integral for the same reason as before. The integration, after slightly shifting the limits of the inte- gral from /H20848−/H9266/H9266to/H20848−/H9266+/H9266+ , yields /H9266 2/H9275c/H11003/H20853/H208491+e−2/H9266//H9275c/H9270/H20850/H20851sin2/H20849qX/H20850J02/H20849qRc/H20850+ cos2/H20849qX/H20850H02/H20849qRc/H20850/H20852 +2e−/H9266//H9275c/H9270/H20851sin2/H20849qX/H20850J02/H20849qRc/H20850− cos2/H20849qX/H20850H02/H20849qRc/H20850/H20852/H20854. /H20849A9/H20850 Finally, by averaging over X, Eq. /H20849A2/H20850becomes Dyy=1 2/H20873qV0 eB/H208742/H9266 2/H9275c/H11003/H20877coth/H20873/H9266 /H9275c/H9270/H20874/H20851J02/H20849qRc/H20850+H02/H20849qRc/H20850/H20852 + sinh−1/H20873/H9266 /H9275c/H9270/H20874/H20851J02/H20849qRc/H20850−H02/H20849qRc/H20850/H20852/H20878 /H11229/H9270 2/H20873qV0 eB/H2087421 /H9266qRc/H20875/H9266 /H9275c/H9270coth/H20873/H9266 /H9275c/H9270/H20874 +A/H20873/H9266 /H9275c/H9270/H20874sin/H208492qRc/H20850/H20876, /H20849A10 /H20850 which can be translated to magnetoresistance with the use of Einstein’s relation, resulting in /H9004/H9267xx /H92670=1 2/H208812/H92661 /H90210/H9262B2/H92622 aV02 ne3/2/H20841B/H20841/H20875/H9266 /H9275c/H9270coth/H20873/H9266 /H9275c/H9270/H20874 +A/H20873/H9266 /H9275c/H9270/H20874sin/H208492qRc/H20850/H20876. /H20849A11 /H20850 The formula agree with the asymptotic expression of Eq. /H2084921/H20850in Ref. 27 for large enough magnetic ﬁelds. The second term in Eq. /H20849A11 /H20850represents CO, which reproduces qualita- tive features of Eq. /H208492/H20850. It should be noted, however, that the anisotropic nature of the scattering in GaAs/AlGaAs 2DEGcannot be correctly treated in the present simple relaxation-time-approximation approach employing only one scatteringtime. The anisotropic scattering plays an important role inCO because of its high sensitivity to the small-angle scatter-ing. Therefore Eq. /H20849A11 /H20850is of limited validity to describe CO in our ULSL. However, it is interesting to point out that if weare allowed to replace /H9270only in the scattering damping factor A/H20849/H9266//H9275c/H9270/H20850with the single-particle /H9270s=/H9262sm/e/H11229/H9262Wm/e,w e acquire Eq. /H208492/H20850except for the thermal damping factor. The choice of /H9270sfor the damping is not unreasonable, considering that the factor A/H20849/H9266//H9275c/H9270/H20850stems from the exponential factor in Eq. /H20849A9/H20850, which describe the cooperativeness between theA. ENDO AND Y . IYE PHYSICAL REVIEW B 72, 235303 /H208492005 /H20850 235303-10left-most and right-most edges that is susceptible to a small- angle scattering. /H20851In general, velocity-velocity correlation in Eq. /H2084914/H20850fort/H11011kT/2 with k=1,2,3,…, namely, between the edges separated by more than half revolution of the cyclo-tron orbit, requires precise positioning after the revolution,which is ruined by small angle scattering. Therefore the useof /H9270sis reasonable. However, this does not justify the re- placement only in the damping factor. /H20852The thermal damping factor A/H20849T/Ta/H20850can readily be incorporated by allowing for thermal smearing of the Fermi edge, namely, by taking the average over the energy of the sin /H208492qRc/H20850term weighted by the factor /H20849−/H11509f//H11509/H9255/H20850. Electronic address: akrendo@issp.u-tokyo.ac.jp 1C. W. J. Beenakker and H. van Houten, in Solid State Physics , edited by H. Ehrenreich and D. Turnbull /H20849Academic Press, San Diego, 1991 /H20850, V ol. 44, p. 1. 2D. Weiss, K. v. Klitzing, K. Ploog, and G. Weimann, Europhys. Lett. 8, 179 /H208491989 /H20850. 3R. W. Winkler, J. P. Kotthaus, and K. Ploog, Phys. Rev. Lett. 62, 1177 /H208491989 /H20850. 4P. H. Beton, E. S. Alves, P. C. Main, L. Eaves, M. W. Dellow, M. Henini, O. H. Hughes, S. P. Beaumont, and C. D. W. Wilkinson,Phys. Rev. B 42, 9229 /H208491990 /H20850. 5A. K. Geim, R. Taboryski, A. Kristensen, S. V . Dubonos, and P. E. Lindelof, Phys. Rev. B 46, 4324 /H208491992 /H20850. 6G. Müller, D. Weiss, K. von Klitzing, P. Streda, and G. Weimann, Phys. Rev. B 51, 10236 /H208491995 /H20850. 7M. Tornow, D. Weiss, A. Manolescu, R. Menne, K. v. Klitzing, and G. Weimann, Phys. Rev. B 54, 16397 /H208491996 /H20850. 8B. Milton, C. J. Emeleus, K. Lister, J. H. Davies, and A. R. Long, Physica E /H20849Amsterdam /H208506, 555 /H208492000 /H20850. 9A. Endo and Y . Iye, Phys. Rev. B 66, 075333 /H208492002 /H20850. 10A. Endo and Y . Iye, Physica E /H20849Amsterdam /H2085022, 122 /H208492004 /H20850. 11J. H. Smet, S. Jobst, K. von Klitzing, D. Weiss, W. Wegscheider, and V . Umansky, Phys. Rev. Lett. 83, 2620 /H208491999 /H20850. 12R. L. Willett, K. W. West, and L. N. Pfeiffer, Phys. Rev. Lett. 83, 2624 /H208491999 /H20850. 13A. Endo, M. Kawamura, S. Katsumoto, and Y . Iye, Phys. Rev. B 63, 113310 /H208492001 /H20850. 14M. Kato, A. Endo, and Y . Iye, J. Phys. Soc. Jpn. 66, 3178 /H208491997 /H20850. 15A. Soibel, U. Meirav, D. Mahalu, and H. Shtrikman, Phys. Rev. B 55, 4482 /H208491997 /H20850. 16C. J. Emeleus, B. Milton, A. R. Long, J. H. Davies, D. E. Petti- crew, and M. C. Holland, Appl. Phys. Lett. 73, 1412 /H208491998 /H20850. 17A. R. Long, E. Skuras, S. Vallis, R. Cuscó, I. A. Larkin, J. H. Davies, and M. C. Holland, Phys. Rev. B 60, 1964 /H208491999 /H20850. 18P. Bøggild, A. Boisen, K. Birkelund, C. B. Sørensen, R. Tabo- ryski, and P. E. Lindelof, Phys. Rev. B 51, 7333 /H208491995 /H20850. 19Y . Paltiel, U. Meirav, D. Mahalu, and H. Shtrikman, Phys. Rev. B 56, 6416 /H208491997 /H20850. 20A. Endo, S. Katsumoto, and Y . Iye, Phys. Rev. B 62, 16761/H208492000 /H20850. 21R. Menne and R. R. Gerhardts, Phys. Rev. B 57, 1707 /H208491998 /H20850. 22S. D. M. Zwerschke and R. R. Gerhardts, Physica E /H20849Amsterdam /H20850 256-258 ,2 8 /H208491998 /H20850. 23A. D. Mirlin, E. Tsitsishvili, and P. Wölﬂe, Phys. Rev. B 64, 125319 /H208492001 /H20850. 24E. Skuras, A. R. Long, I. A. Larkin, J. H. Davies, and M. C. Holland, Appl. Phys. Lett. 70, 871 /H208491997 /H20850. 25P. T. Coleridge, Phys. Rev. B 44, 3793 /H208491991 /H20850. 26F. M. Peeters and P. Vasilopoulos, Phys. Rev. B 46, 4667 /H208491992 /H20850. 27A. D. Mirlin and P. Wölﬂe, Phys. Rev. B 58, 12 986 /H208491998 /H20850. 28A. Endo and Y . Iye, J. Phys. Soc. Jpn. 74, 2797 /H208492005 /H20850. 29A. Endo and Y . Iye, J. Phys. Soc. Jpn. 74, 1792 /H208492005 /H20850. 30A. Matulis and F. M. Peeters, Phys. Rev. B 62,9 1 /H208492000 /H20850. 31It can readily be shown that Eq. /H208495/H20850is equivalent to the Cham- bers’ formula, Eq. /H2084914/H20850, in the limit /H9275c/H9270/H112701, after allowing for the energy spread at a ﬁnite temperature. 32From our experience, it was easier to deduce an accurate value of /H9262Wfrom CO than to obtain /H9262Sexactly from SdH oscillation, the latter readily being made inaccurate by small inhomogeneity inn e, see Ref. 25. 33A. Endo and Y . Iye, Phys. Rev. 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PhysRevB.84.205319.pdf	PHYSICAL REVIEW B 84, 205319 (2011) Unraveling of free-carrier absorption for terahertz radiation in heterostructures Andreas Wacker* Mathematical Physics, Lund University, Box 118, S-22100 Lund, Sweden Gerald Bastard, Francesca Carosella, and Robson Ferreira Laboratoire Pierre Aigrain, Ecole Normale Superieure, CNRS (UMR 8551), Universit ´eP .e tM .C u r i e ,U n i v e r s i t ´e D. Diderot, 24 rue Lhomond F-75005 Paris, France Emmanuel Dupont Institute for Microstructural Sciences, National Research Council, Ottawa, Ontario, Canada K1A0R6 (Received 1 November 2011; published 17 November 2011) The relation between free-carrier absorption and intersubband transitions in semiconductor heterostructures is resolved by comparing a sequence of structures. Our numerical and analytical results show how free-carrierabsorption evolves from the intersubband transitions in the limit of inﬁnite number of wells with vanishing barrierwidth. It is explicitly shown that the integral of the absorption over frequency matches the value obtained bythef-sum rule. This shows that a proper treatment of intersubband transitions is fully sufﬁcient to simulate the entire electronic absorption in heterostructure THz devices. DOI: 10.1103/PhysRevB.84.205319 PACS number(s): 78 .67.Pt, 73.40.Kp, 78 .40.Fy, 85 .35.Be I. INTRODUCTION The absorption of electromagnetic radiation due to the inter- action with electrons in bulk crystals is essentially determinedby two distinct effects: (i) the free-carrier absorption (FCA),which is directly related to the electrical conductivity and dropswith frequency on the scale of the inverse scattering time; (ii)interband transitions, which are typically described via thedipole moments induced by the coupling between states indifferent bands. For most crystals these transition energies areof the order of eV and thus this dominates the response aroundthe optical spectrum. In addition to these electronic features,optical phonons provide absorption in the far-infrared region,which is not addressed here. Semiconductor heterostructures provide an additional ef- fective potential for the electron in the conduction bandcausing a further quantization of the electronic states inthe growth direction (denoted by z). Taking into account the degrees of freedom for motion in the x,y plane, this establishes subbands within the conduction band. Commonly,the absorption between these subbands is treated analogouslyto the interband transitions in bulk crystals. The standardtreatment relies on the envelope functions ϕ ν(z)f o rt h e subbands νwith energies Eνand areal electron densities nν using expressions for the absorption coefﬁcient αμ→ν(ω)a s1,2 αμ→ν(ω)=e2\|zμν\|2(Eν−Eμ)(nμ−nν) 2¯hLzc√/epsilon1/epsilon10 ×/Gamma1 (Eν−Eμ−¯hω)2+/Gamma12/4, (1) where Eμ<E νand the counterrotating terms are neglected. Hereeis the elementary charge,√/epsilon1is the refractive index, and /epsilon10is the vacuum permeability (SI units are used). The matrix element zμν=/integraldisplay dzϕμ(z)zϕν(z)( 2 )describes the coupling strength. Throughout this work we assume the polarization of the electric ﬁeld to point in thezdirection and that the wave propagates in a waveguide of effective thickness L zwhich is ﬁlled by the (layered) semiconductor material. This scheme is also routinely appliedfor the calculation of the gain spectrum of quantum cascadelasers (QCLs). 3In this context the broadening /Gamma1can be either added in a phenomenological way4or by detailed calculations; see, e.g., Ref. 5. It can also be seen as a limiting case of a full quantum kinetic calculation.6 While the conventional treatment of intersubband tran- sitions is well accepted for transitions in the infrared, thisapproach is less obvious for THz systems, which have becomeof high interest. 7,8Here, FCA-related features might turn up as a strong competing mechanism to the intersubband gaintransition in analogy to the bulk case where both FCA andinterband transitions occur as separate processes. In order todemonstrate the potential relevance, we consider the standardexpression for FCA in bulk systems 9 αFCA(ω)=nce2τ mcc√/epsilon1/epsilon101 ω2τ2+1, (3) where mcis the effective mass, ncthe volume density of electrons in the conduction band, and τis the scattering time. As an example for GaAs with a doping of 1 ×1016/cm3and τ=0.2 ps (corresponding to a mobility of 6000 cm2/V s at 300 K10) one obtains α=120/cm for a frequency ω/2π= 2 THz. This is larger than typical gain coefﬁcients in THzquantum cascade lasers. 11–13Thus, bulk FCA would provide a strong obstacle in achieving lasing in such structures and itsproper treatment in heterostructures is of crucial importancefor the description of QCLs or other THz heterostructuredevices. (For a typical infrared laser, in contrast, it wasshown that FCA in the cascade structure does not play arole. 14)I nR e f . 4FCA was only considered in the waveguide layers but not the QCL structure itself, where the absorptionwas determined by intersubband transitions. Furthermore, in 205319-1 1098-0121/2011/84(20)/205319(6) ©2011 American Physical SocietyW ACKER, BASTARD, CAROSELLA, FERREIRA, AND DUPONT PHYSICAL REVIEW B 84, 205319 (2011) Ref. 15it was shown that processes as described by Eq. ( 1) dominate the absorption of light (with z-polarized electric ﬁeld) for quantum wells. In this context the question arises as to how such a treatment based on intersubband transitions is related to the FCA in thebulk. Is FCA related to the seemingly dominating intersubbandprocesses or does it stem from further processes not identiﬁedyet? In the latter case, such processes could strongly alterthe THz performance of heterostructure devices. In order toshed light on this important issue we present a detailed studyon the unfolding of FCA starting from different types ofheterostructure. Our main conclusion is that the absorptiondue to intersubband transitions evolves into the bulk FCA forvanishing barrier widths. This shows that a proper treatmentof intersubband transitions provides a complete description ofgain and absorption processes in heterostructure devices. II. FROM SUPERLATTICE TO BULK We consider four GaAs-Al 0.3Ga0.7As superlattices16(SLs) with constant period d=10 nm. The barrier width is set equal to 0.5 nm, 1.5 nm, 2.5 nm, and 3.5 nm, respectively,and a homogeneous doping with n c=6×1016/cm3is used. The sample with the 2.5 nm barrier has been investigated inRef. 17, which motivates our choice. Figure 1(a) shows the calculated minibands assuming effective masses of 0 .067m e and 0.0919mefor GaAs and Al 0.3Ga0.7As, respectively, where meis the free electron mass, as well as a conduction band offset of 276 meV .18Further information on the structures is given in Table I. Here the zero-ﬁeld conductivity σ0is evaluated from the nonequilibrium Green’s function (NEGF) model followingRef. 19, which includes scattering processes from phonons, impurities, interface roughness (with an average height ofone monolayer and a length correlation of 10 nm), and alloydisorder in an approximate way. This program also calculatesthe absorption in linear response to the optical ﬁeld 6as given in Fig. 1(b).U s i n g σ0≈nce2τ/m SLthe conductivity for the 0.5 nm barrier structure provides a scattering time of τ=46 fs. 0 0.05 0.1 0.15 0.2 0.25 photon energy (eV)0510Absorption (100/cm)0.5 nm barrier 1.5 nm barrier 2.5 nm barrier 3.5 nm barrier bulk -1 0 1 kz (π/d)00.10.20.30.40.5band energy (eV)Extension of first band gap(b) (a) 300 K 00 . 0 10100.5 nm1.5 nm2.5 nm3.5 nm77 K FIG. 1. (Color online) (a) Lowest three minibands for the SLs together with the dispersion of bulk GaAs (dotted line) neglecting nonparabolicity. (b) Absorption at 300 K for the SLs, calculated bythe NEGF model together with the Drude expression ( 3)f o rb u l k absorption (dotted line) using m c=mSL=0.071meandτ=46 fs. The inset shows the NEGF calculations (symbols) at 77 K togetherwith the corresponding result of Eq. ( 5)u s i n g σ 0from Table Iand τm=70, 100, 110, 100 fs for the sample with the barrier width of 0.5, 1.5, 2.5, 3.5 nm, respectively (lines).TABLE I. Key parameters obtained for the different SLs. The effective mass mSLis taken for the lowest miniband at k=0i nt h e SL direction. P a r a m e t e r S L1 S L2 S L3 S L4 Barrier width (nm) 0 .51 .52 .53 .5 Miniband width (meV) 42 .72 5 .41 5 .61 0 .1 Effective mass mSL/me 0.071 0 .090 0 .125 0 .178 σ0at 300 K (A /V cm) 11 .26 .42 .40 .8 σ0at 77 K (A /V cm) 17 14 .97 .32 .9 This value agrees roughly with the momentum scattering rate 1 /τm=29/ps (which is the sum of the elastic and inelastic scattering rate20) extracted from several highly doped GaAs/AlAs SLs with narrow barriers at room temperature.21 This value is much smaller than the bulk scattering time of0.2 ps, as scattering is enhanced due to the presence of roughinterfaces in all SLs (which are particular strong scatterersfor small barrier widths, when the wave functions highlypenetrate through the barriers). In addition, the assumptionof a constant scattering time is only expected to be of semi-quantitative nature; the same holds for the approximationsin matrix elements used. (For a more detailed treatment ofroughness scattering in thin barriers, see Ref. 22.) Using τ=46 fs, the Drude expression ( 3) ﬁts the absorption quite well, demonstrating that these small barriers actually providealmost the bulk free-carrier absorption behavior. With increasing barrier thickness the conductivity becomes smaller due to the reduced coupling between the quantumwells. Accordingly, there is a decrease in the low-frequencyabsorption α(ω=0)=σ 0 c√/epsilon1/epsilon10, (4) as follows from electrodynamics.23Here our numerical cal- culations are in full agreement, as we do not employ therotating wave approximation and include broadening in afully consistent way. Furthermore, for thicker barriers, theabsorption between the minibands becomes more prominentand thus the absorption increases close to the photon energyrequired to overcome the gap between the ﬁrst and thesecond miniband, as indicated by the arrows in Fig. 1(b). The shift of the peak positions with respect to the minigapscan be related to scattering-induced level shifts. For the2.5 nm barrier the results are in good agreement with themeasurements reported in Ref. 17. The onset of absorption around 100 meV is slightly sharper in the experiment, whichmay be attributed to less rough interfaces or to the limitedaccuracy of the various approximations used for the scatteringpotentials. For SLs the absorption can be understood within the com- mon miniband picture. For low frequencies intra-minibandprocesses dominate, which are easily treated in semiclassicaltransport models providing for zero electric ﬁeld: 24,25 α(ω)=Re{σ(ω)} c√/epsilon1/epsilon10=σ0 c√/epsilon1/epsilon101 (τmω)2+1. (5) This behavior was experimentally observed in Refs. 26and 27. Here, σ0≈nce2τm/m SLfor large miniband widths. With 205319-2UNRA VELING OF FREE-CARRIER ABSORPTION FOR TERAHERTZ ... PHYSICAL REVIEW B 84, 205319 (2011) decreasing miniband width, the increase of mSLreduces σ0. An even stronger reduction arises if the miniband width dropsbelow either k BTor the Fermi energy; see Ref. 20for details.28 For all superlattice structures studied by our NEGF model, we found good agreement with ( 5) for low frequencies. Some examples are shown in the inset of Fig. 1(b). As a further example, the calculated absorption spectrum at 65 K for thestructure of Ref. 27can be ﬁtted by τ m=0.16 ps (not shown here). This is in good agreement with the experimental valueof 0.18 ps, which demonstrates the quality of the NEGFapproach. For higher frequencies, transitions between the minibands can describe the absorption between 60 and 200 meV verywell. See, e.g., the results of the calculations in Ref. 17, which fully agree with our more sophisticated NEGF approach. We conclude that the absorption of SLs at zero bias can be well described by the Drude-like miniband conductionresult ( 5) for low frequencies and by common inter-miniband transitions for higher frequencies. As shown in Fig. 1(b),t h e combination of both features evolves into the bulk FCA ( 3)i f the barrier width becomes small. III. FROM MULTIPLE WELL TO SUPERLATTICE Now we want to study how the SL absorption arises from the behavior of systems containing few wells, which show distinctabsorption peaks between discrete levels. Figure 2shows the absorption for multi-quantum-well structures, as presentedin Fig. 2(a) for the case of two wells. Here, all parameters correspond to the SL with a 1.5 nm barrier discussed above. Forthe double-well structure, essentially the two lowest subbandsare occupied in thermal equilibrium, and one observes clearabsorption peaks corresponding to the separations between thesubbands; see Fig. 2(b). As the dipole matrix element ( 2) van- ishes for equal parity of the states, not all possible transitionsare visible. The observed peak structure can be directly de-scribed by the standard intersubband expressions ( 1). Further- more, there is zero absorption in the limit of zero frequency asno dc current along the structure is possible; compare Eq. ( 4). With increasing well numbers, the peaks III and IV of the double well split up and form the continuous absorptionbetween 60 and 200 meV due to the transitions between theﬁrst and the second SL miniband; see Fig. 2(c). While this is quite expected, peak I does not show any clear splitting, butshifts to lower frequencies, approaching the intra-minibandabsorption. This behavior can be understood by a detailedstudy of the multi-quantum-well eigenstates. Here, a tight-binding model for Nwells with next-neighbor coupling T 1 shows the following (see Appendix Afor details): (i) There areNeigenstates, labeled by an index νaccording to their energy Eν.H e r e Eν+1−Eνis of the order of 4 \|T1\|/N. (ii) The matrix element zμνfrom Eq. ( 2) is small unless for neighboring states; i.e., μ=ν±1. Thus, the transitions between neighboring states dominate, explaining the strongabsorption around ¯ hω≈4\|T 1\|/Nvisible in Fig. 2(c), where 4\|T1\|essentially corresponds to the miniband width of the inﬁnite structure. Together with a tail at higher frequencies dueto broadening of these transitions this explains the appearanceof the Drude-like miniband absorption for the SL in the limitof large N.F o rω=0 the evolution is not smooth as any0 5 10 15 20 25 z (nm)0100200300conduction band edge (meV)III IIIIV(a) 0 0.05 0.1 0.15 0.2 photon energy (eV)0100200300400500600700absorption (1/cm)superlattice double well(b) I IIIII IV 0 20 40 60 80 100 120 140 160 180 200 Photon energy (meV)0200400600Absorption (1/cm)superlattice 2 well 3 well 4 well(c) FIG. 2. (Color online) (a) A double quantum well (well widths 8.5 nm, barrier width 1.5 nm) with its lowest eigenstates. Dashedand dot-dashed lines refer to symmetric and antisymmetric states, respectively. The arrows depict the transitions associated with the peaks in the absorption spectrum. (b) Absorption spectrumcalculated by the NEGF model for the double quantum well and the corresponding SL. (c) Evolution of the absorption for 2, 3, and 4 wells with the same parameters as the double well from (a). Inorder to obtain absorption in the entire waveguide, it is assumed that the multi-quantum-well structure is periodically repeated with a separation by a 7.5 nm barrier. All calculations are done at T=300 K. ﬁnite sequence of quantum wells has a zero dc conductivity in contrast to an inﬁnite SL and thus the absorption must vanishaccording to Eq. ( 4). IV . THE INTEGRATED ABSORPTION Summing over all possible intersubband transitions ( 1), we obtain the total absorption αIS(ω)=/summationtext μναμ→ν(ω)/Theta1(Eν− Eμ). Here the discrete index νruns over all (inﬁnitely many) eigenstates of the heterostructure of ﬁnite length, includingstates which correspond to unbounded states with energies farabove the barrier potential. Integrating over all frequenciesprovides /integraldisplay ∞ 0dω α IS(ω) =/summationdisplay ν,μπe2\|zμ,ν\|2(Eν−Eμ)(nμ−nν) Lzc/epsilon10√/epsilon1¯h2/Theta1(Eν−Eμ) =/summationdisplay μ,νπe2\|zμ,ν\|2(Eν−Eμ)nμ Lzc/epsilon10√/epsilon1¯h2(6) under the assumption Eν−Eμ/greatermuch/Gamma1; otherwise the counterrotating terms become of relevance, which hadbeen neglected here. In Appendix Bwe show that the same 205319-3W ACKER, BASTARD, CAROSELLA, FERREIRA, AND DUPONT PHYSICAL REVIEW B 84, 205319 (2011) integral relation is more generally obtained for arbitrary level spacings Eν−Eμwithin our NEGF model, which also covers dispersive gain.29,30 Following Ref. 31,E q .( 6) can be simpliﬁed by the Thomas- Reiche-Kuhn sum rule32(also called the f-sum rule33) which reads for a parabolic band with effective mass mc /summationdisplay ν2mc(Eν−Eμ) ¯h2\|zμν\|2=1 and provides the integrated absorption /integraldisplay∞ 0dω α IS(ω)=navπe2 2mcc√/epsilon1/epsilon10, (7) where nav=/summationtext μnμ/Lzis the average three-dimensional carrier density in the waveguide. For a bulk semiconductor, the free-carrier absorption ( 3) provides after integration over energy /integraldisplay∞ 0dω α FCA(ω)=ncπe2 2mcc√/epsilon1/epsilon10, (8) which fully agrees with the intersubband result ( 7)f o r equal total densities nc=nav. Thus the total FCA in a bulk semiconductor equals the total intersubband absorption withinthe conduction band for a ﬁnite heterostructure of ﬁnite length,which shows the direct relation between these. More generally,Eqs. ( 7) and ( 8) establish a general rule for the integrated absorption within the conduction band of a semiconductorunder conditions, where the approximation of a constanteffective mass is justiﬁed. In this context superlattices appearas an intermediate case, where the inter-miniband absorptionand the Drude-like intra-miniband absorption add up to thefull result. 31 Our numerical data in Fig. 1(b) exhibit the inte- grated absorption ¯ h/integraltext dωα (ω)=25.7(±0.3)eV/cm for all curves. The data from Fig. 2provide ¯ h/integraltext dωα (ω)= N N+0.625.6(±0.2)eV/cm, where the additional factor takes into account the undoped region of 6 nm between adjacent multiplequantum wells ( Nis the number of well/barrier combinations with a length of 10 nm each). These values are slightly belowthe value of 27 .3eV/cm given by Eq. ( 8) using the GaAs effective mass. This minor discrepancy of less than 7% canbe easily attributed to some absorption at higher frequenciesand the impact of the barrier material with a larger mass. 34We conclude that the absorption obtained by our NEGF code is inexcellent agreement with the rule ( 7). More generally the effect of the semiconductor heterostruc- ture can be understood as shifting the absorption strengthwithin the frequency space, as explicitly demonstrated by ourcalculations. This perception has actually been used in thedesign of QCL structures, where the unavoidable free-carrierabsorption is deﬂected from the frequency region of operationby a proper choice of heterostructures; 35see, e.g., Ref. 36. V . CONCLUSION We demonstrated how the common bulk free-carrier ab- sorption evolves from standard intersubband absorption inheterostructures for electromagnetic waves with an electricﬁeld pointing in growth direction. Here the well-studied SLabsorption constitutes an intermediate case, which can be entirely understood on the basis of common intersubband ab-sorption processes in the limit of a growing number of quantumwells. For decreasing SL barrier width the combination ofinter- and intra-miniband absorption evolves into the standardFCA of the bulk crystal. This behavior reﬂects a redistributionof absorption strength, while the integrated absorption isconstant. The most relevant consequence is that there is noneed to bother about any additional FCA-related absorptionprocesses, provided all intersubband transitions are properlytaken into account. A consistency check for the calculatedgain/absorption spectrum is whether Eq. ( 4) is satisﬁed in the low-frequency limit and the integrated absorption matchesEqs. ( 7) and ( 8). ACKNOWLEDGMENTS We thank J. Faist for helpful discussions. Financial support from the Swedish Research Council (VR) and the French ANRagency (ROOTS project) is gratefully acknowledged. APPENDIX A: ANALYTICAL CALCULATION FOR COUPLED WELLS We consider a multi-quantum-well structure with Nwells centered at z=nd, where n=1,2,..., N . The ground state of the isolated well nhas the wave function /Psi1g(z−nd) and the energy Eg. Restricting to a nearest-neighbor coupling T1 (which is negative for the lowest subband), the eigenenergies are Eν=Eg+2T1cos/parenleftbiggνπ N+1/parenrightbigg forν=1,2,..., N, (A1) and the eigenstates read ϕν(z)=/summationtext na(ν) n/Psi1g(z−nd) with a(ν) n=/radicalbigg 2 N+1sin/parenleftbiggνπn N+1/parenrightbigg . If the overlap between the states in different wells is negligible, i.e.,/integraltext dz/Psi1 g(z−n/primed)z/Psi1g(z−nd)≈ndδnn/prime, we ﬁnd zμν=/summationtext nnda(μ) na(ν) n, which can be directly evaluated. If ν−μis even we ﬁnd zμν=δμ,ν(N+1)d/2 as both states have the same parity with respect to z=(N+1)d/2. For odd ν−μ, some algebra yields zμν=d 2(N+1)/bracketleftBigg 1 sin2/parenleftbig(μ+ν)π 2(N+1)/parenrightbig−1 sin2/parenleftbig(μ−ν)π 2(N+1)/parenrightbig/bracketrightBigg . Forμ/negationslash=νwe thus have zμν=0 for even ( ν−μ), zμν∼−2(N+1)d π2(μ−ν)2for odd and small ( ν−μ), zμν=O/braceleftbiggd N+1/bracerightbigg for odd and large ( ν−μ). As the square of zμνenters the absorption ( 1), it becomes clear that the transitions with ν=μ±1 highly dominate the absorption spectrum. The energy difference of the correspond-ing states ( A1) for these transitions is less than 2 \|T 1\|π/(N+1) with an average of approximately 4 \|T1\|/N. 205319-4UNRA VELING OF FREE-CARRIER ABSORPTION FOR TERAHERTZ ... PHYSICAL REVIEW B 84, 205319 (2011) APPENDIX B: TOTAL ABSORPTION WITH THE GREEN’S FUNCTION MODEL Here we refer to the formulation of our NEGF model as outlined in Ref. 6. Here gain is evaluated within linear response around the stationary state characterized by the Green’s functions ˜Gμν(k,E). In order to simplify the analysis, nondiagonal ˜Gμν(k,E) are neglected here; they are, however, fully included in our numerical implementation. Then the absorption resulting from thep a i ro fs t a t e s μ,ν can be written as α μν(ω)=e2(Eν−Eμ)\|zμν\|2 cLz¯h/epsilon10√/epsilon12 A/summationdisplay k/integraldisplaydE 2πRe/braceleftbig˜Gret νν(k,E+¯hω)˜G< μμ(k,E)+˜G< νν(k,E+¯hω)˜Gadv μμ(k,E) −˜Gret μμ(k,E+¯hω)˜G< νν(k,E)−˜G< μμ(k,E+¯hω)˜Gadv νν(k,E)/bracerightbig , (B1) which is essentially the last equation of the appendix in Ref. 6with the counterrotating term added. Inserting the spectral function37Aν(k,E)=∓ 2Im{˜Gret/adv ν,ν (k,E)}and its occupied part38Aocc ν(k,E)=−i˜G< νν(k,E) ,w h i c hi sa s s u m e dt ob er e a l ,w e ﬁnd αμν(ω)=e2(Eν−Eμ)\|zμν\|2 2cLz¯h/epsilon10√/epsilon12 A/summationdisplay k/integraldisplaydE 2π/bracketleftbig Aocc μ(k,E)Aν(k,E+¯hω)−Aocc ν(k,E)Aμ(k,E−¯hω) +Aocc μ(k,E)Aν(k,E−¯hω)−Aocc ν(k,E)Aμ(k,E+¯hω)/bracketrightbig . (B2) The terms Aocc μ(k,E)Aν(k,E+¯hω)−Aocc ν(k,E)Aμ(k,E−¯hω) provide the physical origin of dispersive gain as sketched in Refs. 6and30. The signs of the counterrotating terms Aocc μ(k,E)Aν(k,E−¯hω)−Aocc ν(k,E)Aμ(k,E+¯hω) seem to contradict our intuition, as the ﬁrst one appears to relate to emission and the second to absorption. However, in this formulation the signis deﬁned via the difference in energy between the initial and the ﬁnal state, where only one speciﬁc combination is used in theprefactor ( E ν−Eμ). Using the general relations /integraldisplay∞ 0dω[Aμ(k,E+¯hω)+Aμ(k,E−¯hω)]=1 ¯h/integraldisplay∞ −∞dE/primeAμ(k,E/prime)=2π/¯h and2 A/summationdisplay k/integraldisplaydE 2πAocc μ(k,E)=nμ, integration of the terms from Eq. ( B2) over frequency provides /integraldisplay∞ 0dω α μν(ω)=πe2\|zμ,ν\|2(Eν−Eμ)(nμ−nν) Lzc/epsilon10√/epsilon1¯h2, (B3) so that the sum over all different pairs ( μ,ν) equals the second line of Eq. ( 6). Thus the integrated absorption ( 7) also holds for the more involved absorption terms ( B2) of the NEGF model which include the dispersive gain. *andreas.wacker@fysik.lu.se 1M. Helm, in Intersubband Transitions in Quantum wells , Semicon- ductors and Semimetals, V ol. 62, edited by H. Liu and F. Capasso(Elsevier, Amsterdam, 1999), pp. 1–99. 2T. Ando, J. Phys. Soc. Jpn. 44, 765 (1978). 3J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y . Cho, Science 264, 553 (1994). 4L. Ajili, G. Scalari, M. Giovannini, N. Hoyler, and J. Faist, J. Appl. Phys. 100, 043102 (2006). 5T. Unuma, M. Yoshita, T. Noda, H. Sakaki, and H. Akiyama, J. Appl. Phys. 93, 1586 (2003). 6A. Wacker, R. Nelander, and C. Weber, Proc. SPIE 7230 , 72301A (2009). 7B. S. Williams, Nature Photonics 1, 517 (2007). 8M. Lee and M. C. Wanke, Science 316, 64 (2007). 9P. Y . Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 1999).10J. R. Meyer and F. J. Bartoli, P h y s .R e v .B 36, 5989 (1987). 11N. Jukam, S. S. Dhillon, D. Oustinov, J. Mad ´eo, J. Tignon, R. Colombelli, P. Dean, M. Salih, S. P. Khanna, E. H. Linﬁeld, andA. G. Davies, Appl. Phys. Lett. 94, 251108 (2009). 12M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, Opt. Express 19, 733 (2011). 13D. Burghoff, T.-Y . Kao, D. Ban, A. W. M. Lee, Q. Hu, and J. Reno,Appl. Phys. Lett. 98, 061112 (2011). 14M. Giehler, H. Kostial, R. Hey, and H. T. Grahn, J. Appl. Phys. 96, 4755 (2004). 15I. Vurgaftman and J. R. Meyer, Phys. Rev. B 60, 14294 (1999). 16H. T. Grahn (ed.), Semiconductor Superlattices, Growth and Electronic Properties (World Scientiﬁc, Singapore, 1995). 17M. Helm, W. Hilber, T. Fromherz, F. M. Peeters, K. Alavi, and R. N. Pathak, P h y s .R e v .B 48, 1601 (1993). 205319-5W ACKER, BASTARD, CAROSELLA, FERREIRA, AND DUPONT PHYSICAL REVIEW B 84, 205319 (2011) 18I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001). 19S.-C. Lee, F. Banit, M. Woerner, and A. Wacker, Phys. Rev. B 73, 245320 (2006); R. Nelander, Ph.D. thesis, Lund University, 2009. 20A. Wacker, Phys. Rep. 357, 1 (2002). 21E. Schomburg, T. Blomeier, K. Hofbeck, J. Grenzer, S. Brandl, I. Lingott, A. A. Ignatov, K. F. Renk, D. G. Pavel’ev, Y . Koschurinov,B .Y .M e l z e r ,V .M .U s t i n o v ,S .V .I v a n o v ,A .Z h u k o v ,a n dP .S .Kop’ev, P h y s .R e v .B 58, 4035 (1998). 22F. Carosella, R. Ferreira, G. Strasser, K. Unterrainer, and G. Bastard, P h y s .R e v .B 82, 033307 (2010). 23J. D. Jackson, Classical Electrodynamics , 3rd ed. (John Wiley & Sons, New York, 1998). 24S. A. Ktitorov, G. S. Simin, and V . Y . Sindalovskii, Fiz. Tverd. Tela13, 2230 (1971) [Sov. Phys. Solid State 13, 1872 (1972)]. 25A. A. Ignatov, K. F. Renk, and E. P. Dodin, P h y s .R e v .L e t t . 70, 1996 (1993). 26G. Brozak, M. Helm, F. DeRosa, C. H. Perry, M. Koza, R. Bhat,and S. J. Allen, P h y s .R e v .L e t t . 64, 3163 (1990). 27K. Tamura, K. Hirakawa, and Y . Shimada, Physica B 272, 183 (1999).28The sequential tunneling picture provides similar results for σ(ω) (Ref. 20). Thus no major differences are expected for thick barriers. 29R. Terazzi, T. Gresch, M. Giovannini, N. Hoyler, N. Sekine, andJ. Faist, Nature Phys. 3, 329 (2007). 30A. Wacker, Nature Phys. 3, 298 (2007). 31F. M. Peeters, A. Matulis, M. Helm, T. Fromherz, and W. Hilber, Phys. Rev. B 48, 12008 (1993). 32W. Kuhn, Z. Phys. A 33, 408 (1925); F. Reiche and W. Thomas, ibid.34, 510 (1925). 33G. D. Mahan, Many-Particle Physics (Plenum, New York, 1990). 34Indeed we found a consistent change with the barrier thickness: The maximal value of 26 eV /cm was obtained for the 0.5 nm barrier and the minimal value of 25 .4e V/cm for the 3.5 nm barrier in Fig. 1(b). 35J. Faist (private communication). 36C. Walther, G. Scalari, J. Faist, H. Beere, and D. Ritchie, Appl. Phys. Lett. 89, 231121 (2006). 37H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors (Springer, Berlin, 1996). 38In thermal equilibrium we have Aocc ν(k,E)=nF(E)Aν(k,E), where nF(E) is the Fermi-Dirac distribution. 205319-6
PhysRevB.78.155118.pdf	Field effects on the electronic and spin properties of undoped and doped graphene nanodots Huaixiu Zheng and Walter Duley Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 /H20849Received 20 June 2008; revised manuscript received 30 August 2008; published 15 October 2008 /H20850 We report a spin-polarized density-functional theory study of electric-ﬁeld effects on the electronic and spin properties of graphene nanodots. Both undoped and graphene nanodots doped with nitrogen and boron areconsidered. In the presence of nonlocal exchange-correlation interactions, undoped graphene nanodots arefound to be half-semiconductors when a weak electric ﬁeld is applied across the zigzag edge. At high electricﬁelds these graphene nanodots become nonmagnetic semiconductors. When the electric ﬁeld is applied acrossthe armchair edge, these graphene nanodots maintain an antiferromagnetic ground state with the energy gapstrongly dependent on the magnitude of the electric ﬁeld. For graphene nanodots doped with nitrogen or boronwe ﬁnd that energetically the most favorable state among all possible conﬁgurations is the one in which thedopant replaces the carbon atom at the center of the zigzag edge. The substitutional dopant atom at the zigzagedge leads to a spin-polarized half-semiconducting state in which the spin degeneracy is broken. The spin-dependent energy gaps can be tuned within a wide range by applying electric ﬁelds. In addition, we ﬁnd thehalf-semiconducting state under doping occurs even when the electric ﬁeld is very strong. This indicates thatedge doping can signiﬁcantly widen the operating range of applied electric ﬁelds for spintronic applicationsbecause undoped graphene nanodots become spinless semiconductors under certain applied electric ﬁelds. DOI: 10.1103/PhysRevB.78.155118 PACS number /H20849s/H20850: 73.22. /H11002f I. INTRODUCTION Due to their exceptional properties,1–8materials of the graphene family including two-dimensional /H208492D/H20850graphene, one-dimensional /H208491D/H20850graphene nanoribbons /H20849GNRs /H20850, and zero-dimensional /H208490D/H20850graphene nanodots /H20849GNDs /H20850have been the focus of much recent research attention from both experi-mental and theoretical points of view. 6–13The experimental advances of the fabrication of single-layer graphene andquasi-one-dimensional GNRs /H20849Refs. 1–5/H20850also encourage much new research in this ﬁeld. 3,9–11What is more exciting is the very recent report of the successful fabrication ofgraphene quantum dots as small as 20 nm, so that purelygraphene-based single-electron transistors can now be con-structed and studied. 14 Both armchair and zigzag GNRs are semiconducting due to edge deformations and the interactions between the ferro-magnetically ordered edge states. 11At the local-density ap- proximation /H20849LDA /H20850level of theory, an ab initio density- functional theory /H20849DFT /H20850study showed that zigzag GNRs can be driven into a half-metallic state where metallic electronswith one spin orientation coexist with insulating electronshaving the other spin orientation. 12,15–17Edge oxidization has been shown to enhance the half-metallicity of zigzag GNRsby lowering the critical electric ﬁeld needed to induce thehalf-metallic state. 13A theoretical study on a ﬁnite cluster of zigzag GNRs /H20849C472H74, of length 7.1 nm and width 1.6 nm /H20850 at the B3LYP level indicates that the nonlocal exchange in-teraction removes half-metallicity in ﬁnite GNRs. 15The re- sulting ﬁnite GNRs are spin-selective semiconductors and itis believed that the ﬁnite-size effect in ﬁnite GNRs inducesthis difference. 17In addition, chemical decoration18and sub- stitutional doping19,20have also been found to be alternative ways to achieve half-metallicity in zigzag GNRs. In this paper, we present a detailed study of electric-ﬁeld effects in doped and undoped GNDs /H208490D counterpart ofGNRs /H20850. We also examine differences that occur when the electric ﬁeld is applied across zigzag and armchair edges.Electronic structures and optimized geometries have beenobtained using ﬁrst-principles DFT spin-unrestricted calcula-tions, implemented with the GAUSSIAN suite of programs.21 All-electron calculations were carried out with electronicwave functions expanded in a Gaussian-type localizedatomic-centered basis set. We have used the hybridexchange-correlation functionals of Becke, Lee, Yang, andParr /H20849B3LYP /H20850, 22which was shown to give a good represen- tation of the characteristics of electronic structure in nano-scale C-based systems. 15,17,23B3LYP by its construction in- cludes nonlocal exchange interaction, which plays animportant role in spin systems. 15We have adopted the 3-21G basis set,24which we ﬁnd to be adequate when considering both computational efﬁciency and the accuracy of results.25 Following the previous convention,16aM/H11003Nﬁnite graphene nanodot is deﬁned according to the number of dan-gling bonds on the armchair edges /H20849M/H20850and the number of dangling bonds on the zigzag edges /H20849N/H20850, as shown in Fig. 1. Electric-ﬁeld effects were studied by applying electric ﬁeldalong the zigzag edge /H20849xdirection /H20850or along the armchair edge /H20849ydirection /H20850. In Sec. II, we report the inﬂuence of applied electric ﬁelds on electronic and spin properties of undoped GNDs. In Sec.III, the conﬁguration of singly doped GNDs with one carbonatom substituted by one nitrogen or boron atom has beeninvestigated. In what follows, we still use the term “half-metallic” or “half-semiconducting” to refer to the states hav-ing different /H9251/H20849spin-up /H20850and/H9252/H20849spin-down /H20850electron gaps by following the conventional deﬁnition in 1D GNRs.12,13,19But we have to keep in mind that 0D GNDs are not exactly“half-metal” or “half-semiconductor” because they are actu-ally ﬁnite molecules. The resulting GNDs are half-semiconductors with two separate energy gaps between thehighest occupied molecular orbital /H20849HOMO /H20850and the lowest unoccupied molecular orbital /H20849LUMO /H20850for /H9251and/H9252electrons.PHYSICAL REVIEW B 78, 155118 /H208492008 /H20850 1098-0121/2008/78 /H2084915/H20850/155118 /H208497/H20850 ©2008 The American Physical Society 155118-1This half-semiconducting state is maintained when an elec- tric ﬁeld is applied along either the xor the ydirection. Energy gaps are found to be insensitive to the application ofan electric ﬁeld in the ydirection. However, the HOMO- LUMO gap of /H9251electrons and /H9252electrons tends to vary drastically when an electric ﬁeld is applied along the xdirec- tion. Such gap modulation is understood from the point ofview of the localized or delocalized nature of HOMO andLUMO states. An approximate model is used to estimate thelinear-screening factor. A value between 2.12 and 4.24 isobtained, which is in nice agreement with the value of 5estimated from random-phase approximation /H20849RPA /H20850. In addi- tion, we also observe an intriguing symmetry between theﬁeld-modulated energy gaps of the donor /H20849N/H20850-doped and ac- ceptor /H20849B/H20850-doped GNDs. II. FIELD EFFECT IN UNDOPED GRAPHENE NANODOTS In this section, we report the results of a study into the effect of applying an electric ﬁeld in the xorydirection, which is across the zigzag edge and across the armchair edge/H20849Fig.1/H20850. By comparing the energy conﬁguration, we found the antiferromanetic spin singlet /H20849S=0/H20850is always the ground state of undoped GNDs with and without electric ﬁelds. Thisis in agreement with previous work by Hod et al. 16Figure 2 shows the calculated spatial distribution of the spin density/H20851 /H9267/H9251/H20849r/H20850−/H9267/H9252/H20849r/H20850/H20852when the electric ﬁeld is 0.000, 0.082, 0.164, 0.246, and 0.328 V /Å. In the absence of an electric ﬁeld, the antiferromagnetic ground state has the highest spin den-sity on the zigzag edges and decreases rapidly from the zig-zag edge to the middle. This is in agreement with the resultsof other calculations. 15,16Application of a weak electric ﬁeld /H208490.082 V /Å/H20850slightly changes the spin density. But as the ﬁeld increases /H20849to 0.162 and 0.246 V /Å/H20850, the spin density isdramatically reduced. It is found that spin density completely disappears when the ﬁeld increases to 0.328 V /Å, resulting in a diamagnetic ground state. Quantitatively, the decrease inthe local magnetic moment of carbon atoms /H20849M/H20850, indicates how an increase in the electric ﬁeld destroys the spin densityconﬁguration, as shown in the bottom of Fig. 2. Without an electric ﬁeld, the largest magnetic moment of edge atoms M=0.43 /H9262B, which is in nice agreement with the result ob- tained with the Perdew-Burke-Ernzerhof /H20849PBE /H20850exchange- correlation functional.26The disappearance of spin density is attributed to spin transfer between the two zigzag edges in-duced by the applied electric ﬁeld. 12We studied the system with lengths up to 2.5 nm /H20849which is the computational limi- tation of our method /H20850; we do not observe the pattern of spin standing wave reported previously,15where the spin density ﬁrst disappears at the middle of the zigzag edge. We believethis difference occurs because the system that is the subjectof the present work is much shorter along the direction of thezigzag-edge direction than those studied by Rudberg et al. /H208490.74 vs 7.1 nm /H20850. The /H9251and/H9252HOMO-LUMO energy gaps in 12 /H110033 GNDs are shown in Fig. 3/H20849a/H20850plotted vs electric-ﬁeld strength in the xdirection. It can be seen that there are four distinguishable regions of applied electric ﬁeld: 0–0.10, 0.1–0.27, 0.27–0.46,and 0.46–0.82 V /Å. For ﬁelds /H11021about 0.1 V /Å, the elec- tric ﬁeld causes the /H9251spin to experience a rapid increase in energy gap, while the /H9252spin experiences a signiﬁcant de- crease in energy gap. At 0.1 V /Å, the /H9251-spin gap reaches its maximum value and the /H9252-spin gap is minimized. The mini- mum/H9252energy gap is referred as Em. A further increase in ﬁeld strength is seen to cause the /H9251-spin gap to decrease while the /H9252-spin gap remains nearly constant until the ﬁeld FIG. 1. /H20849Color online /H20850The atomic structure of M/H11003Ngraphene nanodots: the carbon atoms /H20849red/H20850are passivated with hydrogen at- oms /H20849green /H20850at both the armchair and zigzag edges. There are Mand Nhydrogen atoms on each armchair edge and zigzag edge, respec- tively. The applied electric ﬁelds /H20849blue arrows /H20850along the armchair and zigzag edges are denoted as ExandEy, respectively. FIG. 2. /H20849Color online /H20850Top: the spin density /H20849difference between /H9251-spin and /H9252-spin density /H20850map of the antiferromagnetic ground state of 12 /H110033 graphene nanodots under cross zigzag edge /H20849along armchair edge /H20850electric ﬁeld with different strengths, 0.000, 0.082, 0.164, 0.246, and 0.328 V /Å, as labeled above the ﬁgures. Red: positive; blue: negative. The isovalue is 0.002. Bottom: the largestlocal magnetic moment /H20849M/H20850of carbon atoms against the applied cross zigzag edge electric ﬁeld /H20849E x/H20850.HUAIXIU ZHENG AND WALTER DULEY PHYSICAL REVIEW B 78, 155118 /H208492008 /H20850 155118-2increases to 0.27 V /Å. At this point, the ﬁeld strength is referred to as Ec. The system is spin-selective half- semiconducting before reaching Ec.15After Ec, the system becomes diamagnetic and the /H9251and/H9252electrons have the same gap. The energy gap then increases until the ﬁeldreaches 0.46 V /Å. After 0.46 V /Å, the gap decreases rap- idly. The minimal gap E mdecreases rapidly as the length of GNDs increases in the xdirection as shown in Fig. 3/H20849a/H20850. Its value determines how close the corresponding GND ap-proaches to becoming a half-metal with a zero gap. Mean-while, E cis the parameter that deﬁnes the operating regime of the spin-selective half-semiconductors since ﬁelds in ex-cess of E cwill destroy the half-semiconducting state. The length dependence of EmandEcis plotted in Fig. 3/H20849b/H20850.A n exponential ﬁtting gives Em=A1e−M/3.06+A2,A1=6.22 eV, A2=0.08 eV. On the other hand, Ecis also a decreasingfunction of length, with an exponential ﬁtting Ec =B1e−M/6.46+B2,B1=0.91 V /Å,B2=0.12 V /Å. It is notice- able that the operating regime of ﬁeld strength for half-semiconducting applications is narrowed for longer GNDs.This limited working range requires special attention in half-semiconducting applications to make sure that the spin stateis not destroyed by application of strong ﬁelds. We have also investigated the impact of applying an elec- tric ﬁeld in the ydirection on the electronic structure of GNDs. The ground state remains antiferromagnetic for 6/H11003N/H20849N=3–5 /H20850GNDs under an applied ﬁeld in the 0–1 V /Å range. The energy gap as a function of ﬁeld strength is plot-ted in Fig. 4. For all three cases /H20849N=3–5 /H20850, it was found that for ﬁelds less than 0.5 V /Å, the HOMO-LUMO energy gap is inversely proportional to ﬁeld strength. The dependence ofthe HOMO-LUMO gap on electric ﬁeld in this range is ap-proximately linear. A similar behavior of ﬁeld-modulated gaphas been found in carbon nanotubes /H20849CNTs /H20850before. 27The decreasing slope /H20849L/H20850is larger for wider GNDs, i.e., LN=3 /H11021LN=4/H11021LN=5and it is obvious that the ﬁeld effect is more pronounced in wider GNDs. This is because the same elec-tric ﬁeld will induce a larger electrostatic potential in widerGNDs. In addition, it should be noted that when N=5 and the ﬁeld is higher than 0.6 V /Å, the energy gap stops de- creasing and begins to increase as the ﬁeld is increased. Thisminimum actually also appears within 0–1 V /Å when N /H110225. The reason we cannot ﬁnd a minimum for N=3,4 is that at these cases a ﬁeld higher than 1 V /Å is necessary to generate a minimum gap. The increasing behavior of energygap after the minimum is probably because the system hasreached the limit of linear response and nonlinear responsemakes the gap an increasing function of ﬁeld. III. FIELD EFFECT IN SINGLE N- OR B-DOPED GRAPHENE NANODOTS Chemical doping is an alternative way to tailor the elec- tronic properties of materials such as CNTs for transport,sensing, and optical applications. 28–30Boron doping in zig-FIG. 3. /H20849Color online /H20850/H20849a/H20850The HOMO-LUMO energy gaps of 12/H110033 GNDs as a function of the strength of x-direction electric ﬁeld. Red squares are for /H9251spin and blue circles are for /H9252spin. The minimal energy gap of /H9252spin /H20849Em/H20850that can be obtained in the whole range of ﬁeld strength is indicated by an olive dashed arrow.The critical ﬁeld strength /H20849E c/H20850that will drive the system to be dia- magnetic is indicated by a pink dashed arrow. /H20849b/H20850The length de- pendence of Em/H20849left axis /H20850andEc/H20849right axis /H20850.Mvaries from 6 to 20 /H20849a range of length from 11.4 nm to 41.2 nm in xdirection /H20850with ﬁxed length in ydirection /H20849N=3/H20850. Note that Mis even.FIG. 4. /H20849Color online /H20850The HOMO-LUMO energy gaps of 6 /H11003N/H20849N=3–5 /H20850GNDs as a function of the strength of y-direction electric ﬁeld. Red squares are for N=3, green circles are for N=4, and blue triangles are for N=5.FIELD EFFECTS ON THE ELECTRONIC AND SPIN … PHYSICAL REVIEW B 78, 155118 /H208492008 /H20850 155118-3zag GNRs has been shown to induce a metal-semiconductor transition in the ferromagnetic state and also breaks thespin-up and spin-down symmetry. 31This work demonstrated that spin-polarized electronic currents can be generated andthe resulting GNRs can be used as spin ﬁlter devices. 31Zig- zag GNRs can also be tuned to be half-metallic with borondoping on the edges. 20In addition, nitrogen doping in both armchair and zigzag GNRs has been studied by several re-search groups. 19,32However, chemical doping has rarely been studied in GNDs, which are the 0D counterparts ofGNRs. To study the effect of doping with a single nitrogen or boron atom, we have compared the total energies /H20849E/H20850for all the possible substitutional sites in a 6 /H110033 GND, with edges passivated by hydrogen atoms. As shown in Fig. 5/H20849a/H20850, there are 42 carbon atoms in a 6 /H110033 GND, but only 12 possible substitutional sites due to the symmetric geometry. A singlesubstitution in a 6 /H110033 GND corresponds to a doping concen- tration of 2.38%. The energy differences between differentconﬁgurations are listed in Table Ifor both nitrogen and boron doping. We ﬁnd that the energetically most favorabledoping site for both nitrogen and boron atoms is at site 1 atthe center atom of the zigzag edge. We refer to this site as thecentral zigzag-edge site. The C-N and C-B bond lengths are1.384 Å and 1.524 Å, respectively. It has been previouslyreported that for zigzag GNRs boron prefers to substitute atthe edge. 31Thus, it is not surprising that a similar doping preference is observed here since a GND is a ﬁnite segmentof a zigzag GNR. To verify this result, we also studied alarge set of GNDs of different widths and lengths, including6/H11003N/H20849N=4–7 /H20850andM/H110033/H20849M=8–12 /H20850structures and found that central zigzag-edge doping is always the favorable con-ﬁguration for a single nitrogen or boron substitution. It isthen generally expected that a single nitrogen or boron atomshould always prefer to replace the central carbon atom ofzigzag edge in the actual doping process. Figure 5/H20849b/H20850shows the DOS for nitrogen- and boron-doped 6/H110033 GNDs as well as for undoped GNDs. Comparison with undoped GNDs indicates that for both nitrogen- and boron-doped GNDs, the spin degeneracy between the /H9251spin and the/H9252spin is broken. The introduction of one donor /H20849or ac- ceptor /H20850atom leaves one /H9251/H20849or/H9252/H20850electron unpaired. The HOMO /H20849LUMO /H20850states of /H9251and/H9252electrons thus do not have the same energy anymore. Speciﬁcally, for nitrogen-doped GNDs, the LUMO level disappears and a new HOMOlevel arises for /H9251-spin electrons. For boron-doped GNDs, the HOMO level disappears and a new LUMO level arises for /H9252-spin electrons. Those energy levels are the donor and ac- ceptor impurity levels induced by nitrogen and boron substi-tutions, respectively. The resulting GNDs are half-semiconductors, in the sense that these are semiconductorshaving different energy gaps for spin-up and spin-down elec-trons. Having explored the geometry and electronic structure of nitrogen- or boron-doped GNDs, we now turn to the effect ofan applied electric ﬁeld on the corresponding half-semiconducting state. We ﬁnd cross armchair-edge /H20849ydirec- tion in Fig. 1/H20850electric ﬁelds have little impact on the elec- tronic structure of doped GNDs. The cross zigzag-edgeelectric-ﬁeld effect on the energy gaps of nitrogen- andboron-doped GNDs is shown in Fig. 6. The half- semiconductivity is retained for ﬁelds from −1 to 1 V /Å. It is interesting that there is an important donor and acceptorsymmetry in the ﬁeld effect. The gap curve of /H9251spin in nitrogen-doped GNDs is the mirror image of that of /H9252spin in boron-doped GNDs with respect to the vertical line of zeroﬁeld. The same symmetry between /H9252spin in nitrogen-doped GNDs and /H9251spin in boron-doped GNDs is also observed in Fig.6. In general, when the electric ﬁeld is applied in the x or −xdirections, the gap of one spin orientation of N-doped GNDs behaves the same as that of the other spin orientation FIG. 5. /H20849Color online /H20850/H20849a/H20850The atomic structure of a 6 /H110033G N D and with possible substitutional sites labeled from 1 to 12. /H20849b/H20850Den- sity of states /H20849DOS /H20850in 6/H110033 GNDs without doping, doped with one nitrogen at site 1, and doped with one boron at site 1. Red linesrepresent the /H9251-spin channel, while blue lines correspond to the /H9252-spin channel. The Fermi level is indicated by the dashed vertical green line.HUAIXIU ZHENG AND WALTER DULEY PHYSICAL REVIEW B 78, 155118 /H208492008 /H20850 155118-4of B-doped GNDs when the electric ﬁeld is reversed. With nitrogen doping, the HOMO-LUMO energy gap for /H9251spin /H208492.686 eV /H20850is larger than the one for /H9252spin /H208492.040 eV /H20850in the absence of electric ﬁeld. A ﬁnite ﬁeld is necessary to elimi-nate the gap difference between /H9251and/H9252electrons. In the presence of a negative ﬁeld of 0.154 V /Å, the energy gaps for/H9251and/H9252electrons become nearly the same as indicated by the intersection between the two red lines in Fig. 6. Start- ing from −0.154 V /Å ﬁeld, if we apply a more positive ﬁeld, the energy gap of /H9251electrons will increase a little and then stays nearly constant after the ﬁeld reaches a certainstrength. However, the gap of /H9252electrons shows a rapid de- crease with increasing positive ﬁeld until about 0.5 V /Å. Conversely, if a negative ﬁeld greater than −0.154 V /Åi s applied, the /H9251and/H9252gaps show opposite behaviors: /H9252gapincreases slightly and then remains approximately constant, while the /H9251gap tends to decrease rapidly. Thus, the same ﬁeld affects the /H9251and/H9252gaps in an opposite way. To explore the mechanism of the gap variation as a func- tion of ﬁeld strength, the HOMO and LUMO states and en-ergy levels of the /H9252electrons in nitrogen-doped 6 /H110033 GNDs at ﬁeld strengths −0.514, −0.257, 0.0, 0.257, and 0.514 V /Å are shown in Fig. 7. In the absence of an applied electric ﬁeld, the /H9252-HOMO state is localized on the right edge, while the/H9252-LUMO state is localized on the left edge. The energy gap between the HOMO and LUMO energy levels is 2.040eV as indicated in Fig. 7/H20849b/H20850. With a negative −0.257 V /Å ﬁeld applied, the HOMO and LUMO states both remain lo-calized. Consequently, the HOMO energy level is shifteddownward and LUMO level is shifted upward by the ﬁeldbecause the electrostatic potential e /H9254Vis negative on the right edge and positive on the left edge. The HOMO andLUMO levels move apart in energy, leading to an enlargedgap. A larger negative ﬁeld /H20849−0.514 V /Å/H20850causes the HOMO and LUMO states to become delocalized. Becausethe electron density is uniformly distributed along the nan-odot, the electrostatic potential within the nanodot has bothpositive and negative components, which compensate eachother and together cause slight change in the HOMO-LUMOenergy levels and thus the energy gap. As a result, for ﬁeldsin excess of −0.514 V /Å, the energy gap of /H9252electrons remains nearly constant as shown in Fig. 6. Conversely, a positive ﬁeld will lift the HOMO level and lower the LUMOlevel, leaving a narrowed gap. When the electric ﬁeld ex-ceeds 0.514 V /Å, the HOMO and LUMO states become delocalized and the gap stops decreasing. Similar analysiscan be made to understand other gap curves as a function ofﬁeld strength, as long as we know the nature of the state/H20849localized or delocalized /H20850. This is because the ﬁeld can dras- tically lift or lower the energy levels having localized states,but can only slightly disturb levels corresponding to delocal-ized states. In addition, we found that the HOMO-LUMO states of /H9252 electrons in N-doped GNDs are always highly localizedTABLE I. Total energy differences for single nitrogen or boron substitution in the 6 /H110033 GND: /H9004EN/B=E/H20851Ni/Bi/H20852−E/H20851N1/B1/H20852fori =1–12. Conﬁguration/H9004EN /H20849eV/H20850 Conﬁguration/H9004EB /H20849eV/H20850 N1 0.000 B1 0.000 N2 1.213 B2 0.533N3 0.828 B3 0.086N4 1.272 B4 0.670N5 1.095 B5 0.483N6 1.381 B6 0.631N7 0.428 B7 0.372N8 1.312 B8 0.543N9 1.051 B9 0.275N10 1.256 B10 1.240N11 0.409 B11 0.299N12 0.952 B12 0.935 FIG. 6. /H20849Color online /H20850Energy gap as a function of electric ﬁeld in the xdirection in N- or B-doped 6 /H110033 GNDs. A negative ﬁeld corresponds to the reverse direction. Red empty /H20849ﬁlled /H20850circle: /H9251/H20849/H9252/H20850 spin of nitrogen-doped GNDs. Blue ﬁlled /H20849empty /H20850square: /H9251/H20849/H9252/H20850spin of boron-doped GNDs. -0.514V/ Ao -0.257V/ Ao 0.000V/ Ao 0.257V/ Ao 0.514V/ Ao -LUMO:β-HOMO:β -LUMO:β -HOMO:β(a) (b) 2.950eV 2.679eV 2.040eV 1.301eV 1.102eV FIG. 7. /H20849Color online /H20850/H20849a/H20850The HOMO and LUMO states of /H9252 electrons in N-doped 6 /H110033 GNDs under the electric ﬁelds: −0.514, −0.257, 0.0, 0.257, and 0.514 V /Å. Red arrows indicate the direc- tion of the applied ﬁeld. Color code: red, positive; green, negative.The isovalue is 0.02. /H20849b/H20850The corresponding energy levels of the states shown in /H20849a/H20850, with the energy gap indicated.FIELD EFFECTS ON THE ELECTRONIC AND SPIN … PHYSICAL REVIEW B 78, 155118 /H208492008 /H20850 155118-5when the ﬁeld is in the range −0.2 to 0.2 V /Å. Since HOMO-LUMO states have these characteristics, we can usean approximate model to estimate the screening factor ac-cording to the linear dependence of the energy gap on ﬁeldstrength. Electron interactions in graphene are believed tolead to the RPA screening of the external ﬁeld E ext.33,34The linear-screening approximation is valid under weak ﬁeldconditions: 33E=Eext/k, where kis the screening factor. When the ﬁeld is between −0.2 and 0.2 V /Å, the HOMO and LUMO states are localized on the right and left edges,respectively. The separation distance Dbetween HOMO and LUMO is approximately between 4 a cand 8 ac, where ac =1.42 Å is the standard C-C bond length. The actual elec- trostatic potential difference between the HOMO and LUMOstates induced by ﬁeld is − eED. As a result, the energy gap variation /H20849 /H9254Eg/H20850due to the applied ﬁeld is approximately equal to − eED since the energy levels are well separated and do not interact strongly under these weak ﬁeld conditions.Using a linear ﬁt to energy gap /H20849E g/H20850as a function of ﬁeld strength /H20849Eext/H20850in the range of −0.2 to 0.2 V /Å, we obtain Eg=aEext+band thus /H9254Eg=aEext/H11015−eED, where /H9254Egis the gap variation caused by Eextcompared to the one without electric ﬁeld. Here, a=−2.681 eÅ,b=2.040 eV. This results in a linear-screening factor k=−eD /a, with 4 ac/H11349D/H113498ac. The value of kis thus between 2.12 and 4.24, which agrees well with the value k=5 estimated from the RPA approxima- tion for an inﬁnite graphene sheet33,34given the approxima- tions used here. IV. CONCLUSION In conclusion, it is found that the application of an electric ﬁeld across the zigzag edges dramatically affects the elec-tronic and spin properties of undoped and N- or B-doped GNDs. For undoped GNDs, the antiferromagnetic groundstate under these conditions becomes half-semiconducting ina weak electric ﬁeld diamagnetic in strong electric ﬁelds. Theminimal energy gap that can be obtained within the wholerange of ﬁeld strength decreases rapidly as the length ofGNDs increases. As a result, we predict that long GNDsbecome half-metallic under certain applied electric ﬁelds.However, the threshold ﬁeld under which a GND can betuned to a diamagnetic semiconductor also decreases withthe length of these GNDs. This limits the possible range ofelectric ﬁelds for creation of half-semiconducting GNDs inpractical applications. We propose that nitrogen or borondoping can solve this problem because there is an unpairedelectron which always gives rise to a half-semiconductingstate. By comparing system energies, we ﬁnd that the mostfavored conﬁguration is one in which the central carbonatom on the zigzag edge is replaced by a dopant atom. Theapplication of the electric ﬁeld across the zigzag edges isshown to change the HOMO-LUMO gap of both /H9251and/H9252 electrons. However, an asymmetry in this gap variation isfound when applying positive and negative electric ﬁelds.This is found to strongly correlate with the localized or de-localized nature of HOMO and LUMO orbitals under ap-plied electric ﬁelds. Finally, based on the linear dependenceof the energy gap on ﬁeld strength, we have estimated alinear-screening factor of between 2.12 and 4.24. 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PhysRevB.90.085148.pdf	PHYSICAL REVIEW B 90, 085148 (2014) Exotic Kondo crossover in a wide temperature region in the topological Kondo insulator SmB 6 revealed by high-resolution ARPES N. Xu,1,C. E. Matt,1,2E. Pomjakushina,3X. Shi,1,4R. S. Dhaka,1,5,6N. C. Plumb,1M. Radovi ´c,1,7P. K. Biswas,8 D. Evtushinsky,9V . Zabolotnyy,9J. H. Dil,1,5K. Conder,3J. Mesot,1,2,5H. Ding,4,10and M. Shi1,† 1Swiss Light Source, Paul Scherrer Insitut, CH-5232 Villigen PSI, Switzerland 2Laboratory for Solid State Physics, ETH Z ¨urich, CH-8093 Z ¨urich, Switzerland 3Laboratory for Developments and Methods, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 4Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 5Institute of Condensed Matter Physics, ´Ecole Polytechnique F ´ed´crale de Lausanne, CH-1015 Lausanne, Switzerland 6Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India 7SwissFEL, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 8Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 9Institute for Solid State Research, IFW Dresden, P . O. Box 270116, D-01171 Dresden, Germany 10Collaborative Innovation Center of Quantum Matter, Beijing, China (Received 1 May 2014; revised manuscript received 14 August 2014; published 28 August 2014) Temperature dependence of the electronic structure of SmB 6is studied by high-resolution angle-resolved photoemission spectroscopy (ARPES) down to 1 K. We demonstrate that there is no essential difference forthe dispersions of the surface states below and above the resistivity saturating anomaly ( ∼3.5 K). Quantitative analyses of the surface states indicate that the quasiparticle scattering rate increases linearly as a function oftemperature and binding energy, which differs from Fermi-liquid behavior. Most intriguingly, we observe that thehybridization between the dandfstates builds gradually over a wide temperature region (30 K <T<110 K). The surface states appear when the hybridization starts to develop. Our detailed temperature-dependence resultsgive a complete interpretation of the exotic resistivity result of SmB 6, as well as the discrepancies among experimental results concerning the temperature regions in which the topological surface states emerge and theKondo gap opens, and give insights into the exotic Kondo crossover and its relationship with the topologicalsurface states in the topological Kondo insulator SmB 6. DOI: 10.1103/PhysRevB.90.085148 PACS number(s): 73 .20.−r,71.20.−b,75.70.Tj,79.60.−i I. INTRODUCTION Recently topological insulators (TIs) [ 1,2] with strong correlation effects have been extensively studied [ 3–8], and particular interest has focused on realizing new types of TIsin exotic materials that contain rare-earth elements. SmB 6,a well-known Kondo insulator (KI), has attracted much attentionbecause it has been proposed to be a promising topologicalKondo insulator (TKI) candidate where both electron correla-tions and nontrivial band topology play important roles [ 3–5]. At high temperature, SmB 6behaves as a correlated bad metal. Upon decreasing temperature, a metal-to-insulator transition(MIT) occurs due to the opening of a hybridization bandgap. However, below ∼3.5 K the resistivity saturates instead of diverging toward absolute zero, indicating the existenceof in-gap states at low temperature [ 9–11]. Strong evidence for surface-dominated transport at low temperature has beenreported [ 12–18], suggesting that the in-gap states have a surface origin. However, in transport measurements it is verychallenging to distinguish the topology of these in-gap states.On the other hand, angle-resolved photoemission spectroscopy(ARPES) experiments have resolved that there is a Fermisurface (FS) formed by an odd number of electron pocketsaround Kramers’ points in the surface Brillouin zone (SBZ)[19–21]. Furthermore, a spin-resolved ARPES experiment has nan.xu@psi.ch †ming.shi@psi.chrevealed that the metallic surface states are spin polarized, and the spin texture fulﬁlls the condition that they are topologicallynontrivial states protected by time-reversal symmetry, thusindicating that SmB 6is the ﬁrst realization of a TKI [ 22]. However, so far, almost all the ARPES measurements havebeen carried out at temperatures near or above the resistivityanomaly ( ∼3.5 K), below which the resistivity saturates. It is highly desirable to explore the detailed electronic structureand low-energy excitations of SmB 6well below the resistivity saturating anomaly in order to understand its low-temperatureelectronic properties and to observe how such a topologicalstate behaves in the presence of strongly correlation effectsin the bulk. More fundamentally, there is controversy aboutthe temperature behavior of the Kondo crossover and itsrelationship with the topological surface states [ 19–21,23–25]. In this paper we present high-resolution ARPES resultsobtained over a large temperature range from the high-temperature metallic phase down to very low temperature(1 K), deep in the Kondo regime of the bulk states. Thecombination of very low sample temperatures, high-energy-resolution ARPES, and high-quality SmB 6single crystals make it possible to trace the detailed dispersions of the surfacestates in the narrow bulk band gap ( ∼20 meV), as well as the spectral function of single-particle excitations as a functionof temperature across the resistivity saturating anomaly. Ourquantitative temperature-dependent ARPES results show thata crossover occurs from the high-temperature metallic phaseto the low-temperature Kondo insulating phase over a widetemperature region. The data furthermore demonstrate how 1098-0121/2014/90(8)/085148(6) 085148-1 ©2014 American Physical SocietyN. XU et al. PHYSICAL REVIEW B 90, 085148 (2014) the topologically nontrivial surface states emerge. Our results give a comprehensive interpretation of the exotic resistivitybehavior of SmB 6in terms of the electronic structure and explain the discrepancies between various experimental resultsin the temperature region in which d-fhybridization and topological surface states emerge. High-quality single crystals of SmB 6were grown by ﬂux method. ARPES measurements were performed with VG Sci-enta R4000 electron analyzers using synchrotron radiation atthe 1 3endstation at BESSY and the SIS beamline at Swiss light source, Paul Scherrer Institut, with circular light polarization.The energy and angular resolutions were ∼5−10 meV and 0.2 ◦, respectively. Samples were cleaved in situ along the (001) crystal plane in an ultrahigh vacuum better than 3 ×10−11 Torr. Shiny mirrorlike surfaces were obtained after cleaving, conﬁrming their high quality. The Fermi level of the sampleswas referenced to that of a gold ﬁlm evaporated onto the sampleholder. II. SURFACE BAND STRUCTURE BELOW THE RESISTIVITY SATURATING ANOMALY To compare the surface electronic structure at temperatures below and above the resistivity saturating anomaly ( ∼3.5K ) , we mapped the FS of the surface states at 1 and 17 K. Figure1(c) shows the FS mapped with hν=26 eV ( k z=4πfor the bulk states) at T=1 K, where the resistivity is fully saturated in transport measurements [ 9–11]. The FS was obtained by integrating the ARPES intensity within a narrowenergy window centered at E F(±2 meV). The deﬁnitions Γ_ X_ M_ ΓXMRky kxkzB B Bky kxS S(b)Sm B1 0 -1 1 0 -1 -1 0 1 kx(π/a)ky(π/a)(c) (d) T= 17 KΓ_ X_M_26 eV 26 eVT= 1 Kαβ LowHigh(a) FIG. 1. (Color online) Real- and momentum-space structure of SmB 6. (a) CsCl-type structure of SmB 6withPm¯3mspace group. (b) First Brillouin zone of SmB 6and the projection on the cleaving surface. High-symmetry points are also indicated. (c) ARPES intensity mapping of the Fermi surface at T=1Kf o rS m B 6plotted as a function of two-dimensional (2D) wave vector. The Fermi surfacewas measured at hν=26 eV ( k z=0 for the bulk electron structure). The intensity is obtained by integrating the spectrum within ±2m e V ofEF. (c) Same as (a), but for T=17 K. The intensity is obtained by integrating the spectrum within ±5m e Vo f EF.of the high-symmetry points and their projections on SBZ of (100) surface are given in Fig. 1(b), which depicts the CsCl-type structure of SmB 6in real space [Fig. 1(a)]. For comparison, in Fig. 1(d) we plot the FS map at T=17 K [ 19], which is well above the resistivity saturating anomaly. One canrecognize that the topology of the FS is essentially the sameon both sides of the resistivity saturating anomaly—namely,t h eF Si sf o r m e db yt h e αpocket centered within the SBZ ( ¯/Gamma1 point) and the βpockets sitting at the midpoints of the SBZ edges ( ¯Xpoints). However the ARPES spectral weight of the FS is signiﬁcantly enhanced at 1 K. This allows us to visualizethe FS of the α /primeband at the ¯Xpoints (folding of the αband resulting from 1 ×2 surface reconstruction) in addition to the β/primeband observed in Refs. [ 19,21]. The identical dispersions of the surface states below and above the resistivity saturatinganomaly revealed in our ARPES experiments conﬁrm that thein-gap states inferred from transport measurements at verylow temperature [ 12–16] and surface states observed in the previous ARPES studies [ 19–21] have essentially the same origin. To trace the ﬁne structure of the surface state inside the narrow Kondo gap, we carried out high-resolution ARPESmeasurement ( <5 meV) at 1 K in order to suppress thermal broadening effects. Figures 2(a)and2(d) display the intensity 0.4 0.0 -0.4 0.4 0.0 -0.4 0.4 0.0 -0.4 0.4 0.0 -0.4(a) (b) (c) (d) (e)0 -40 0 -40 Momentum (Å-1)Momentum (Å-1) E - EF (meV)0 -20-40-60 20Γ X__ M_X_ Γ_X_ αβ'β β α' α' ββα 'E - EF (meV) E - EF (meV) Intensity (a. u.) Intensity (a. u.)26 eV 26 eV1 K 1 Kβα γΔΒ ΔΒ γX_M_(f) E - EF (meV)0 -20-40-60 20LowHigh FIG. 2. (Color online) Surface band dispersions in SmB 6at very low temperature (1 K). (a) Near- EFARPES intensity. (b) corre- sponding plot of the curvature of the EDC intensities. (c) Plot of EDCs measured at hν=26 eV as a function of wave vector and EBalong the cut along the ¯/Gamma1-¯Xdirection. The curve above is the MDC taken at EF. (d)–(f) Analogous to (a)–(c), but for the cut along the¯X-¯Mdirection. The dashed lines in (c) are the band dispersion obtained from the EDCs. The black (red) arrow in (c)/(d) indicates the crossing point of the two surface state branches with differentspin polarizations. 085148-2EXOTIC KONDO CROSSOVER IN A WIDE . . . PHYSICAL REVIEW B 90, 085148 (2014) plots along the ¯/Gamma1-¯Xand ¯X-¯Mdirections, respectively. Similar to the observations at T/greaterorequalslant17 K [ 19], the bulk γband hybridizes with the localized fstates, opens a hybridization band gap /Delta1B∼20 meV , and this feature is enhanced in curvature plots [ 26] [Figs. 2(b) and2(e)], as well as in energy distribution curves (EDCs) [Figs. 2(c) and2(f)]. Inside the hybridization band gap, we clearly see the αandβbands centered at the ¯/Gamma1and ¯Xpoints, which form the FSs shown in Fig. 1(c). With the ultralow temperature and high-energy resolution in the measurements, the spectral weight of thein-gap states is strongly enhanced, especially for the αband, which is clearly observed with a well-deﬁned quasiparticlepeak in the EDCs plot at 1 K [Fig. 2(c)]. The enhanced spectrum weight of the αband makes it possible to observe the folded α /primeband centered at ¯X, as shown in the ARPES intensity plot and the momentum distribution curve (MDC) taken at EF [Fig. 2(d)]. We note that this “missing” folding band has not been observed in previous ARPES experiments. The high-quality data also enable us to trace the dispersion of the surface bands in detail. As shown in Fig. 2(a), our results suggest a conelike dispersion of the αband with a Dirac point (DP) very close to the bulk valence band. At deeper energies,theαband eventually merges with the localized fstates. For theβband, due to photoemission matrix element effects, the left branch in Fig. 2(d) is more enhanced than the right one. Following the dispersion of the βband by ﬁtting MDCs, we ﬁnd that it linearly disperses from E Fdown to EB∼20 meV and then shows a back-bending back behavior. The twobranches cross each other at the ¯Xpoint as indicated by the black arrow, and ﬁnally merge to the bulk fstates as shown in Fig. 2(d). This feature is better visualized in the curvature plot [Fig. 2(e)] and in the EDC plots [Fig. 2(f)]. Thus, the ultralow temperature and high-resolution ARPES results givea clear picture of the dispersions of the surface states, and showhow these in-gap surface states, the αandβbands, connect to the bulk valence bands. The insights about the dispersion oftheβband at the ¯Xpoints naturally explain why its intensity suddenly decreases at a binding energy of ∼20 meV. It should be mentioned that a similar situation, but without a clear DP, isalso observed on the ﬁrst three-dimensional TI Bi 1−xSbx[27]. III. SINGLE-PARTICLE SCATTERING RATE OF THE SURFACE STATES To extract the single-particle scattering rate of the surface states, we ﬁt the MDCs with a single Lorentzian [ 28]. The width of the Lorentzian peak, /Delta1k(ω), is related to the quasipar- ticle scattering rate /Gamma1(ω)=2\|Im/Sigma1(ω)\|=/Delta1k(ω)v0(ω), where v0(ω) is the Fermi velocity and \|Im/Sigma1(ω)\|is the imaginary part of the complex self-energy. Figure 3(b)shows MDCs at various binding energies from the ARPES spectrum shown in Fig. 3(a). The clean spectra at low binding energy near EFenable us to ﬁt the MDCs accurately. However, at EB>15 meV , the spectra are mixed with the bulk states, as well as the bending back partof the βband. As shown in Fig. 3(c), the obtained \|Im/Sigma1(ω)\| has a linear energy dependence that is not expected fromthe three-dimensional ( \|Im/Sigma1(ω)\|∝ω 2) and two-dimensional [\|Im/Sigma1(ω)\|∝(ω2/εF)ln(4εF/ω]) Fermi-liquid theory. On the other hand, this linear dependence of the scattering rate /Gamma1(ω) in SmB 6is similar to that of the Dirac fermions observed in 0.6 0.4 0.2 0.0 Momentum (Å-1) E - EF (meV)-ImΣ (meV) 0510(c)(a) 0 -10X_ M_ 0.6 0.4 0.2 0.0-100E - EF (meV) Intensity (a. u.)β -5 -15(b) EF 15 meV βT= 1 K26 eV 26 eV 100 50 00.4 0.2 Momentum (Å-1)T(d) β EB= 5 meVΓ_ X_ 1 K 3 K 6 K 10 K 16 K 20 K T (K)(e) 26 eV01015 5β sample 1 sample 2X_ M_-ImΣ (meV) FIG. 3. (Color online) Quantitative analysis of the surface state dispersion in SmB 6. (a) Near- EFARPES intensity for SmB 6 measured at hν=26 eV as a function of EBand wave vector along the ¯X-¯Mdirection. The blue curve is the dispersion of the βband traced by MDC ﬁtting. (b) Corresponding MDC plots ﬁtted by Lorentzian peaks (blue curves). (c) The imaginary part of the self-energy (Im /Sigma1) of the Lorentzian-shaped MDC peaks at very low temperature (1 K). Standard deviations from the ﬁtting are within 5% of the obtained value. (d) MDCs taken at EB=5 meV with different temperatures, ﬁtted by Lorentzian peaks (black curves). (e) Temperature dependence of Im /Sigma1of the Lorentzian-shaped MDC peaks at EB=5 meV for two different samples. graphene [ 29], and other 3D topological insulators, such as Bi2Se3[30]. The unusual behavior of the energy dependence of the suppressed scattering rate suggests the topologicallynontrivial nature of the surface state on SmB 6.W eh a v e also ﬁtted the MDCs of the βband at EB=5m e Vf o r different temperatures [Fig. 3(d)] and obtained the temperature dependence of the self-energy as summarized in Fig. 3(d). Im/Sigma1(ω) increases with temperature at a constant rate, which also deviates from the Fermi-liquid theory. It is worthwhileto mention that the scattering rate increases faster than theprevious noninteracting TIs [ 30] with rising temperature. This behavior may be due to the strong correlation effect in theTKI SmB 6because electron-electron correlations can open additional channels to reduce the lifetime of single-particleexcitations. IV. EXOTIC KONDO CROSSOVER IN A WIDE TEMPERATURE REGION So far, the temperature behavior of the surface states and its relation to the Kondo gap arising from the d-fhybridization in SmB 6are still controversial issues. Some ARPES [ 20] and STM [ 23] studies claim that the surface state exists inside the hybridization gap only below 30 K; above 30 K, the surfacestate disappears, accompanied by the complete destruction 085148-3N. XU et al. PHYSICAL REVIEW B 90, 085148 (2014) of the d-fhybridization. On the other hand, other ARPES investigations on the same material [ 19,21,24] indicate that the surface states can exist as high as 110 K. The interplaybetween the surface states and the hybridization between theγband and fstates, as well as their relationship with the exotic resistivity-temperature behavior [ 9–11], are currently under debate. To examine the temperature dependence andevolution of the surface states and the d-fhybridization, we performed detailed high-resolution ARPES measurementsin the temperature range 1–280 K. Figure 4(b) shows the evolution of the EDC located at the k Fof the αband, as indicated by E1 in Fig. 4(a), from 1 to 20 K. The α band has a well-deﬁned quasiparticle peak in the temperatureregion below 3.5 K where the resistivity saturates. Uponincreasing temperature, the coherent spectral weight decreasesmonotonically but does not show any anomaly up to 20 K.Cooling the sample back to 1 K, the spectral weight isrecovered without any degradation caused by aging [1K(R) inFig.2(b)]. A similar temperature dependence is also observed for the βband in Fig. 4(c)[E2 cut in Fig. 4(a)]. Above 20 K, it is difﬁcult to trace the quasiparticle peaks in the EDCs for bothαandβdue to thermal broadening of the strong felectron peak at E B∼20 meV . However, the spectral peak can still beclearly observed in the MDC at EF, as shown in Fig. 4(d) for temperatures from 17 to 150 K. The double peaks of the βband monotonically become weaker with increasing temperatureand disappear between 110 and 150 K. Our results demonstratethat the surface states can exist up to temperatures as high as110 K. However, we notice that at temperatures T/greaterorequalslant45 K, some intensity emerges between the two MDC peaks. Thespectral weight in this region increases as the temperatureis raised and becomes dominant at 150 K, corresponding tothe bulk band ( γ norm), which crosses EFwithout hybridizing with the fstates. The temperature behavior of the γband is conﬁrmed by the EDCs taken at the kFof the γnormband for different temperatures in Fig. 4(g), with the position indicated by E3 in Fig. 5(a). When T/greaterorequalslant45 K, some intensity emerges within the hybridization gap ( EB<20 meV) and becomes dominant at T> 110 K. In order to quantitatively analyze the strength of the d-fhybridization, we plot the MDCs for different temperatures at EB=30 meV in Fig. 4(e), with the position indicated by the black line in Figs. 5(a)–5(f). Besides the double peaks centered around the ¯Xpoint, which are the residual intensity of the β-band surface state, an additional peak on the right shoulder corresponds to the γHband which hybridizes with the fstates [see Figs. 5(a) and 5(b)], as 0.4 0.0 -0.40 Intensity (a. u.) 0 -20 E - EF (meV)E - EF (meV)(a) (b)Γ_ X_ X_ αβα Momentum (Å-1)(c) T 1 K 3 K 6 K 10 K 16 K 20 K 1 K(R) T= 1 K26 eV -40-20E1 E2 E1 -0.5 0.0 0.5 Momentum (Å-1)TβX_ M_ M_ (d)17 K 30 K 45 K 70 K 110 K 150 K-40β T E2 Intensity (a. u.)ΔB 20 1.0 0.5 0.0 200 150 100 50 0 T (K)Intensity γun 1 K 3 K 6 K 10 K 16 K 20 K bad metal insulator R1 R2-20 -10 0 10 E - EF (meV) β γun γH R3 (f) E - EF (meV)-20 20 -40 -60 0Intensity (a. u.)150 K 110 K 70 K 45 K 30 K 17 KE3γ(g) 0.5 0.0 -0.5γH17 K 30 K 45 K 70 K 110 K 150 KX_ M_ M_ TEF EB= 30 meV (e) FIG. 4. (Color online) Temperature dependence of the surface and bulk states in SmB 6. (a) ARPES spectrum as a function of binding energy and wave vector along ¯X-¯/Gamma1-¯X. (b),(c) EDCs taken at the kFpoints of the αandβbands [the EDCs indicated by E1 and E2 in (a)], respectively, at various temperatures. ARPES data taken after thermal cycling are shown by 1K(R) in (b), which demonstrates that the in-gap states are robust and protected against repeated thermal cycling. (d) MDCs taken at EFalong the ¯X-¯Mdirection over the temperature range 17–150 K. (e) MDCs taken at EB=30 meV along the ¯X-¯Mdirection over the same temperature range. (f) Temperature dependence of the intensity of the surface state βband and the bulk conduction band with (without) hybridization with the felectrons γH(γnorm). (g) EDCs taken at the kFpoints for the γband [the positions of the EDCs are indicated by E3 in Fig. 5(a)], at various temperatures. 085148-4EXOTIC KONDO CROSSOVER IN A WIDE . . . PHYSICAL REVIEW B 90, 085148 (2014) -1.0 -0.5 0.0 0.5 1.0 0.5 0.0 Momentum (Å-1)0E - EF (meV)-60-30 0 -60-30 0 -60-30(a) (c) (e)R2 R317 K 70 K 280 Kβ γH γHγnorm γHβ β γnormγnormX_ M__ _ XM R1β γHΔB 4f EFXM γnormE3 (d) (f)_ (b)MM EF FIG. 5. (Color online) The illustrations of surface and bulk band structure at different temperature regions. Panels (a), (c), and (e) show ARPES intensity plots for 17, 70, and 280 K, correspondingto temperature points in R1, R2, and R3 in Fig. 4(e), respectively. Panels (b), (d), and (f) illustrate the band structure in R1, R2, and R3 in Fig. 4(f), respectively. indicated by the red region in Fig. 4(e). The intensity of the γH band shows opposite temperature behavior to the γnormband, decreasing as temperature is raised. We extract the intensityfor the surface band β[peak intensity in the green region in Fig.4(d)] and the bulk band γ norm, which crosses EFwithout hybridizing with the felectrons [intensity in the blue region in Fig. 4(d)], as well as the intensity of the bulk band γH, which hybridizes with the felectrons [intensity in the red region Fig. 4(e)]. These intensities are plotted for different temperatures in Fig. 4(f). Our temperature results suggest that the hybridization between the γband and localized fstates is a crossover process in a large temperature region. At the d-f hybridization crossover region [R2 in Fig. 4(f)], theγnormband crossing EFwithout hybridization with the fstates coexists with the hybridized γHband. The ratio of γnorm/γHdecreases with cooling down the temperature, and reaches the minimumatT< 30 K. This indicates that the Kondo crossover in SmB 6 occurs over a wide temperature range, starting at T=110 K and completing at T=30 K. V. DISCUSSION The aforementioned results, especially the observation of the Kondo crossover in a wide temperature region, provideinsights into the evolution of the electronic structure withtemperature in SmB 6and its connection with the exoticresistivity as a function of temperature [ 9–11]. In the high- temperature region [R3 in Fig. 4(f)], the bulk γnorm band crosses EFwithout hybridizing with the fstates, as illustrated in Fig. 5(f), as well as in the ARPES intensity plot at 280 K in Fig.5(e). Therefore, in this temperature regime, SmB 6shows metallic behavior in transport due to the carriers contributedby the FSs of the γ norm band. When the material is cooled gradually from 110 to 45 K [R2 in Fig. 4(f)], the bulk γnormband starts to hybridize with the fstates, as indicated by the bent back dispersion at EB∼20 meV shown in Figs. 5(c) and5(d). During this crossover region, the partial γnormband still crosses EF, coexisting with the hybridized γH band, with the ratio of relative intensities γnorm/γHdecreasing with falling temperature [Fig. 4(f)]. In the meantime, the surface state emerges when the d-fhybridization starts to develop and band inversion of the fanddelectrons occurs. Due to the partial γnorm band that still crosses EFin the crossover region (R2), SmB 6is still a bad metal in this region. When SmB 6is cooled further down to below 30 K, the hybridization between the γband and the fstates becomes complete, as indicated by a clean gap opening at the kFposition of theγband in Fig. 4(g). The complete bulk hybridization and gap opening [as seen in Figs. 5(a) and5(b)] turns the system into a bulk insulator and corresponds to the MIT transition[9–11]. At very low temperatures (below 3.5 K), the surface states dominate the transport properties, causing the resistivityto saturate instead of diverging as the temperature approacheszero. Our temperature-dependent data unify the seemingly con- ﬂicting observations on SmB 6by different groups. In the crossover region [R2 in Fig. 4(f)], the weak surface state αand βbands can hardly be distinguished from the tail of the broad and strong fstates peak sitting at a shallow binding energy of about 20 meV . As a result, the surface-state bands are onlyobserved in the very near- E FMDCs measured by ARPES [19,21,24]. On the other hand, in the temperature crossover region, due to the existence of the partial γnormband crossing EF, the gap seems closed as observed in density of states for both partially angle-integrated ARPES [ 20] and STM [ 23]. Our temperature-dependent ARPES results on SmB 6give a comprehensive picture of the development of the topologicalsurface states and the Kondo gap due to the d-fhybridization, which could account for its exotic resistivity behavior as afunction of temperature. This constitutes the observation thatthe Kondo crossover in SmB 6takes place over such a wide crossover temperature regime, and the origin of such behaviordeserves further studies. One mechanism candidate is that theKondo temperature near the surface region is different fromthe one in the bulk, following from the fact that the conductionelectrons’ density of states at the surface differ from that in thebulk. ACKNOWLEDGMENTS We acknowledge H. M. Weng, X. Dai, and Z. Fang for helpful discussions. This work was supported by theSino-Swiss Science and Technology Cooperation (ProjectNo. IZLCZ2138954), the Swiss National Science Founda-tion (No. 200021-137783), and MOST (2010CB923000),NSFC. 085148-5N. XU et al. 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PhysRevB.77.235120.pdf	Spectral weight redistribution in strongly correlated bosons in optical lattices C. Menotti1,2and N. Trivedi3 1ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain 2Dipartimento di Fisica and CNR-INFM-BEC, Università di Trento, I-38050 Povo (Trento), Italy 3Department of Physics, Ohio State University, Columbus, Ohio 43210, USA /H20849Received 30 January 2008; published 25 June 2008 /H20850 We calculate the single-particle spectral function for the one-band Bose-Hubbard model within the random- phase approximation /H20849RPA /H20850. In the strongly correlated superﬂuid, in addition to the gapless phonon excitations, we ﬁnd extra gapped modes, which become particularly relevant near the superﬂuid-Mott quantum phasetransition /H20849QPT /H20850. The strength in one of the gapped modes, a precursor of the Mott phase, grows as the QPT is approached and evolves into a hole /H20849particle /H20850excitation in the Mott insulator depending on whether the chemical potential /H9262is above /H20849below /H20850the tip of the lobe. The sound velocity cof the Goldstone modes remains ﬁnite when the transition is approached at constant density; otherwise, it vanishes at the transition. It agreeswell with Bogoliubov theory except close to the transition. We also calculate the spatial correlations for bosonsin an inhomogeneous trapping potential creating alternating shells of Mott insulator and superﬂuid. Finally, wediscuss the capability of the RPA to correctly account for quantum ﬂuctuations in the vicinity of the QPT. DOI: 10.1103/PhysRevB.77.235120 PACS number /H20849s/H20850: 05.30.Jp, 03.75.Lm, 71.45.Gm, 37.10.Jk I. INTRODUCTION Optical lattices have made it possible to explore the prop- erties of ultracold dilute atoms in a new regime of strongcorrelations. 1–3By tuning the strength of the laser ﬁeld, the effective interactions between atoms can be tuned to becomestronger than their kinetic energy. For Bose systems, such acompetition between kinetic energy tand interaction energy Udrives a quantum phase transition 4,5from a kinetic-energy- dominated superﬂuid /H20849SF/H20850phase to an interaction-dominated Mott insulating /H20849MI/H20850phase. The Bose-Hubbard model /H20849BHM /H20850captures the essential physics of this problem4pro- vided the interactions between the bosons are smaller thaninterband energies and the problem can be treated in thesingle-band approximation. While the BHM had been pro-posed long before optical lattice experiments became avail-able, a direct experimental realization was missing. In condensed-matter systems, Josephson junction arrays, 64He in Vycor and aerogels,7vortices in superconductors,8and quantum magnets9,10can be modeled by the BHM. However, the actual systems have additional complications of disorderor longer-range interactions, which make the comparisonsbetween theory and experiment difﬁcult. One of the main advantages of the cold atom systems is that they are clean and much more tunable: The density ofbosons, their effective interaction, the tunneling amplitudebetween the wells, the number of lattice sites, the shape ofthe trapping potential, and aspect ratios can all be variedrather easily, making it possible to study the effects of inho-mogeneity and dimensionality. In addition, it is possible toadd random potentials to study the effects of disorder. Thissets apart the optical lattice systems as a useful testingground for theoretical ideas in the area of strongly interactingbosons and fermions. This model has also provided tremen-dous impetus for the development of measurement tech-niques to address questions about the nature of the excita-tions especially near the transition. 11–22The recent experiments on the dynamics23,24have given a window intothe different time scales operating within the different phases and around the quantum phase transitions. The paper is organized as follows: In Sec. II, we present the Bose-Hubbard model and state our main results. In Sec.III, we discuss the nature of the spectral function calculatedwithin the random-phase approximation /H20849RPA /H20850formalism as it evolves from the SF to the MI phase upon decreasing t/U. The sound velocity in the SF phase and its comparison withBogoliubov theory is in Sec. IV. The momentum distributionand the spatial correlations are calculated in Sec. V. The RPAformalism is generalized to a spatially inhomogeneous trap-ping potential in Sec. VI. We conclude in Sec. VII with someremarks about the comparison between RPA and mean-ﬁeldtheory. There are three appendixes that give the details of thecalculations of Green’s function within RPA /H20849Appendix A /H20850, the Bogoliubov calculation for the Bose-Hubbard model/H20849Appendix B /H20850, and the momentum distribution function within RPA in the Mott regime /H20849Appendix C /H20850. II. MODEL AND MAIN RESULTS The Bose-Hubbard Hamiltonian is deﬁned as H=−t 2z/H20858 /H20855ij/H20856/H20849ai†aj+aiaj†/H20850+U 2/H20858 ini/H20849ni−1/H20850−/H9262/H20858 ini,/H208491/H20850 where aiandai†are bosonic annihilation and creation opera- tors, respectively, and ni=ai†aiis the density operator. The parameter Udescribes the on-site repulsive interaction be- tween bosons, tis the tunneling parameter between nearest neighbors as indicated by the symbol /H20855ij/H20856,/H9262is the chemical potential that ﬁxes the number of particles, and z=2Dis the coordination number in Ddimensions. This Hamiltonian shows a quantum phase transition /H20849QPT /H20850from the SF to the MI phase as a function of t/U/H20849Fig. 1/H20850. The theoretical ap- proaches used to investigate this Hamiltonian include mean-ﬁeld theory, 25perturbation theory,26variational methods,27 and quantum Monte Carlo simulations.28,29PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 1098-0121/2008/77 /H2084923/H20850/235120 /H2084913/H20850 ©2008 The American Physical Society 235120-1In this paper, we use a Green’s-function formalism to in- vestigate the excitations and correlations in the BHM. Westart from the mean-ﬁeld /H20849MF /H20850ground state, which is essen- tially a product of single-site states, and we go beyond it byincluding interwell coupling within RPA. 30,31 Our main results are as follows: /H208491/H20850In the weakly interacting SF /H20849t/U/H11015100 /H20850, the gapless phonons of the SF, or the Goldstone modes arising due to thebroken gauge symmetry, exhaust the sum rule on the totalspectral weight, as expected. For t/U/H1101510 there are already small deviations from Bogoliubov theory and new gappedmodes appear in the SF phase. These gapped modes pick upstrength as t/U/H110151. The sum rule is now satisﬁed only upon including both phonon and gapped modes. /H208492/H20850At the transition, we observe the progression of one of the phonon modes in the SF into a gapped mode in the MI/H20849the one which is gapless at the QPT /H20850. The second gapped mode in the MI instead arises from one of the gapped modesin the SF. Such gapped modes in the SF have been reportedpreviously using several theoretical methods. 13,16–21We ar- gue that these additional gapped modes are a distinctive sig-nature of a strongly correlated SF in proximity to a MI in anoptical lattice. They indicate the redistribution of spectralweight from the coherent phonon modes into incoherent ex-citations, and are a precursor of the MI beyond the QPT. /H208493/H20850We calculate the sound velocity in the RPA formalism and show that it agrees with c=1/ /H20881/H9260m/H11569calculated indepen- dently from the mean-ﬁeld effective mass and compressibil-ity. In a wide range of parameters except very close to theSF-MI phase transition, the above sound velocity compareswell with the predictions of Bogoliubov theory. /H208494/H20850We exploit a special feature of superﬂuids that allows us to extract the condensate fraction n 0from the strength of the phonon modes in the spectral function. /H208495/H20850We calculate the spatial correlations in the case where an inhomogeneous conﬁning potential is superimposed onthe optical lattice. The response to a perturbation is stronglyinﬂuenced by the presence of alternating shells of Mott in-sulator and superﬂuid regions.III. FORMALISM: RPA, SPECTRAL FUNCTION, AND EXCITATIONS We start with the mean-ﬁeld approximation in real space25 obtained by giving the annihilation and creation operators an expectation value deﬁned by /H20855a/H20856=/H20855a†/H20856=/H9272. The order param- eter/H9272identiﬁes the nature of the system: it is nonzero in the SF phase and vanishes in the insulating phase. Substituting a=/H9272+a˜anda†=/H9272+a˜†into Eq. /H208491/H20850, where a˜anda˜†are the ﬂuctuations of the Bose ﬁeld around the mean-ﬁeld value,the Hamiltonian Hcan be rewritten as a sum of on-site Hamiltonians, H iMF=U 2ni/H20849ni−1/H20850−/H9262ni−t/H9272/H20849ai†+ai/H20850+t/H92722, /H208492/H20850 which include the tunneling at the mean-ﬁeld level through the order parameter /H9272. In the MF approximation, we neglect the nonlocal interwell hopping term − /H20849t/2z/H20850/H20858 /H20855ij/H20856/H20849a˜i†a˜j+a˜ia˜j†/H20850, which we will later treat in RPA. TheHiMFcan be diagonalized numerically, leading to a set of on-site eigenstates such that Hi/H20841i/H9251/H20856=/H9280/H9251/H20841i/H9251/H20856. In the Mott limit the eigenstates /H20841i/H9251/H20856are number states, while in the SF regime they are coherent superpositions of several numberstates, allowing the order parameter to be different fromzero. The MF ground-state solution is given by the productstate /H20841/H9021/H20856=/H9016 i/H20841i,0/H20856, equivalent to the one obtained in the Gutzwiller ansatz, where /H20841i,0/H20856is the ground state of HiMF. Within the mean-ﬁeld approximation, the state of the sys- tem is described by a product state over the different wells,neglecting all interwell correlations. However, even in theMI, correlations between neighboring wells do not vanish; infact they get large as t/Uis increased from the Mott side, ultimately diverging at the transition. Experimental evidenceof these correlations is found in the interference picture of anatomic cloud released from a three-dimensional /H208493D/H20850optical lattice, the visibility of which does not suddenly vanish at thephase transition. 32–36These important features are captured by the RPA performed on the nonlocal tunneling terms of theBHM. At the end of this paper, we discuss the limitations ofthe RPA method and how it compares with the mean-ﬁeldapproximation. To go beyond the mean-ﬁeld approximation, we treat the interwell hopping term within RPA, as described in Appen-dix A. 30,31This method allows us to compute Green’s func- tion G/H20849q,/H9275/H20850=/H20855/H20855aq†;aq/H20856/H20856/H9275, deﬁned in Eqs. /H20849A4/H20850and /H20849A6/H20850, and from that, the spectral function A/H20849q,/H9275/H20850=−/H208491//H9266/H20850ImG/H20849q,/H9275/H20850. Due to the commutation relations of the bosonic destruc- tion and creation operators, the spectral function always sat-isﬁes the sum rule /H20885 −/H11009/H11009 A/H20849q,/H9275/H20850d/H9275=1 . /H208493/H20850 From the spectral function, one can extract the excitation spectrum, the strength of the excitation modes, and the re-lated density of states0 0.05 0.1 0.15 0.200.51 t/Uµ/U n=1 FIG. 1. /H20849Color online /H20850Mean-ﬁeld phase diagram in the /H9262/Uvs t/Uplane. The blue line shows the Mott insulating lobe at density n=1. On this diagram, we indicate two points where the QPT hap- pens, which we discuss in this paper: a generic one at /H9262/U=0.5 and t/U/H110150.167 /H20849black /H20850and the tip of the lobe at /H9262/U=/H208812−1 and t/U/H110150.1716 /H20849red/H20850, where the QPT takes place at constant density.C. MENOTTI AND N. TRIVEDI PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-2DOS /H20849/H9275/H20850=/H20885A/H20849q,/H9275/H20850dq. /H208494/H20850 Moreover, the spectral function is an essential ingredient to compute the momentum distribution n/H20849q/H20850=/H20855aq†aq/H20856=/H20885 −/H110090 A/H20849q,/H9275/H20850d/H9275 /H208495/H20850 and the single-particle density matrix given by the Fourier transform of the momentum distribution in real space, /H9267/H20849r,r/H11032/H20850=/H20855ar†ar/H11032/H20856=1 N/H20858 qeiq·/H20849r−r/H11032/H20850n/H20849q/H20850, /H208496/H20850 where Nindicates the number of lattice wells. The long- distance behavior of /H9267/H20849r,r/H11032/H20850as a function of the relative distance approaches the condensate density n0, which is non- zero in a SF and vanishes in the MI. In the following we willcalculate and discuss all these quantities. A fundamental implication of broken symmetry for bosonic systems is that the Goldstone modes are directlyreﬂected in the single-particle spectrum. In other words,phonons which are related to modes of density-density ﬂuc-tuations /H20849or two-particle Green’s function /H20850also show up as the poles in single-particle Green’s function. 37We study the behavior of the poles of Green’s function, their strength, mo-mentum, and frequency dependence, to extract informationabout the excitations of the system. In the two extreme limitsof deep MI and weakly interacting /H20849Bogoliubov /H20850SF, Green’s function can be calculated analytically and from it, the exci-tations frequencies and the momentum distribution n/H20849q/H20850. A. Deep Mott regime In the deep Mott regime /H20849zero tunneling /H20850, for U/H20849n−1/H20850 /H11021/H9262/H11021Un, one ﬁnds GMI/H20849q,/H9275/H20850=1 2/H9266/H20877n+1 /H9275−/H20849Un−/H9262/H20850−n /H9275−/H20851U/H20849n−1/H20850−/H9262/H20852/H20878, /H208497/H20850 /H9275MI/H20849q/H20850=/H20877Un−/H9262/H110220 U/H20849n−1/H20850−/H9262/H110210,/H20878 /H208498/H20850 nMI/H20849q/H20850=n,∀q, /H208499/H20850 where nis the atomic occupation at each lattice site. The spectral function consists of two /H9254functions, one at positive energy Un−/H9262/H20849relative to the chemical potential /H20850, required to add a particle, and one at negative energy U/H20849n−1/H20850−/H9262, re- quired to remove a particle or add a hole to the MI, as seenin Eq. /H208498/H20850. The spectral function A/H20849q, /H9275/H20850obtained using ex- pression /H208497/H20850trivially satisﬁes the sum rule in Eq. /H208493/H20850. The momentum distribution, as deﬁned in Eqs. /H208495/H20850and /H208499/H20850,i s completely ﬂat, corresponding to vanishing site-to-site corre-lations, and normalized to the total number of atoms in thelattice /H20849ntimes the number of sites /H20850. Correspondingly, single- particle density matrix /H208496/H20850, given by the Fourier transform ofthe momentum distribution, shows strictly on-site correla- tions. B. Weakly interacting regime In the weakly interacting SF regime,38we have GBG/H20849q,/H9275/H20850=1 2/H9266/H20875/H20841uq/H208412 /H9275−/H9275q−/H20841v−q/H208412 /H9275+/H9275−q/H20876, /H2084910/H20850 /H9275BG/H20849q/H20850=/H11006/H9275/H11006q, /H2084911/H20850 nBG/H20849q/H20850=n0/H9254q,0+/H20841v−q/H208412, /H2084912/H20850 where uqandvqare the Bogoliubov amplitudes, /H9275qis the Bogoliubov frequency at momentum q/H20849see Appendix B for details /H20850, and n0is the condensate density. In the weakly in- teracting SF regime, the sum rule in Eq. /H208493/H20850is constrained by the Bogoliubov normalization condition /H20841uq/H208412−/H20841v−q/H208412=1. The excitation energies are given by symmetric poles at positiveand negative frequencies corresponding to the energies of theBogoliubov spectrum, highlighting in particular phononicexcitations at low momentum. The momentum distribution, given by integrating the spectral function over negative energies, has a singular con-tribution from the condensate at zero momentum. The inte-gral over all momenta different from zero gives the numberof noncondensed atoms, contributing the depletion from thecondensate. In the regime where Bogoliubov theory is valid,the depletion is negligible compared to the atoms in the con-densate. C. Progression from SF to MI The RPA formalism allows us to calculate the spectral function with special emphasis on the strong correlation re-gion near the QPT. In the deep SF, we ﬁnd phonon collectivemodes reﬂected in the single-particle spectrum. As t/Uis decreased, the spectral weight is redistributed over a multi-mode structure composed of coherent phonon excitations andgapped single-particle excitations. When entering the MIphase at the QPT, the spectral weight reorganizes and isshared by only two gapped modes, describing single-particleexcitations, one at positive energy and one at negative en-ergy. In the following, we will discuss this behavior in moredetails. We use the position of the poles of Green’s function to determine the following results about the excitations of thesystem in the different regimes: /H20849i/H20850For a large number of particles per site /H20849/H11015100 /H20850and weak interactions /H20849t/U/H11015100 /H20850, we exactly recover the Bogo- liubov results. We point out that being able to describe theweakly interacting regime starting from the BHM is not atrivial result because of the large number of basis states re-quired /H20849almost 150 states per site /H20850. /H20849ii/H20850By increasing the interactions and decreasing the number of particles per site, we observe small deviationsbetween the spectrum obtained by RPA and that by Bogoliu-bov theory: additional modes appear at higher frequencies, asshown in Fig. 2/H20849a/H20850fort/U=10, in contrast with BogoliubovSPECTRAL WEIGHT REDISTRIBUTION IN STRONGLY … PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-3theory, which predicts a single excitation mode. While there are also differences in the dispersion of the sound modes atlarge momenta, we ﬁnd in general that the Bogoliubov pre-diction turns out to be quite accurate in describing the low- q part of the spectrum and the sound velocity, even in the caseof strong interactions /H20849see Sec. IV /H20850. /H20849iii/H20850Ast/Ubecomes on order of unity and the effect of strong correlations grows, additional gapped modes in the SFphase are clearly visible and grow in strength as seen in Fig.2/H20849b/H20850. The phonon modes are not sufﬁcient to exhaust the sum rule in Eq. /H208493/H20850. In a strongly interacting SF /H20851e.g., for t/U /H110210.25, as shown in Fig. 2/H20849c/H20850/H20852, many modes /H20849in the cases we have considered, up to four at positive and four at negativeenergy /H20850have to be included in order to exhaust the sum rule in Eq. /H208493/H20850. In the particular case of /H9262/U=0.5 and t/U =0.25, the full dispersion of the modes is shown in Fig. 3/H20849b/H20850. /H20849iv/H20850In the MI, only two excitation modes exist, as shown in Figs. 2/H20849d/H20850and3/H20849a/H20850. The mode at positive energy and the one at negative energy correspond, respectively, to the en-ergy needed to create a particle or a hole in the system. Forany given t/U, the difference between the excitation energies atq=0 exactly coincides with the width of the mean-ﬁeld Mott lobe at the same t/U. D. Strengths of the spectral function The progression of the modes from the strongly correlated SF into the MI is better understood by calculating thestrengths of the excitations S i, deﬁned as follows: A/H20849q,/H9275/H20850=/H20858 iSi/H9254/H20849/H9275−/H9275i/H20850. /H2084913/H20850 Numerically, a small but ﬁnite imaginary part of the energy regularizes the spectral function and provides an accurateﬁtting procedure to determine the position of the poles andtheir strength. We checked that the sum rule in Eq. /H208493/H20850, which when using Eq. /H2084913/H20850implies /H20858 iSi=1, was found to be satis- ﬁed to better than few parts in 10−5for all t/U. We are therefore conﬁdent that we have identiﬁed all the excitationswhich contribute in a non-negligible way to the spectrum. In Figs. 4and5, we plot the position of the resonances and their strengths for a ﬁxed value of q= /H9266/50, varying the parameter t/Uacross the phase transition for ﬁxed chemical potential. As explained above, the multimode spectrum in theSF phase evolves into the two-mode excitation spectrum inthe MI. The transition from the MI to the SF phase occurs when one of the two Mott branches becomes gapless /H20851see Fig. 4/H20849a/H20850/H20852. This is the particle /H20849hole /H20850gapped mode in the MI, depending on whether the chemical potential /H9262is above /H20849be- low /H20850the tip of the lobe. Correspondingly, at the QPT, phononic excitations arise with dominant particle /H20849hole /H20850 character, depending on whether the chemical potential /H9262is−5 0 500.51(a) t/U=10 ω/t→ q/π→ −5 0 500.51(b) t/U=1 ω/t→ q/π→ −5 0 500.51(c) t/U=0.25 ω/t→ q/π→ −5 0 500.51(d) t/U=0.16 ω/t→ q/π→ FIG. 2. /H20849Color online /H20850Spectral function A/H20849q,/H9275/H20850as a function of /H9275for various q. Results obtained by RPA /H20849black /H20850and Bogoliubov theory /H20849green /H20850./H20849a/H20850Weakly interacting SF: t/U=10 with /H1101510 bosons per site; the RPA calculation agrees extremely well withBogoliubov theory; note indications of additional modes at higher /H9275./H20849b/H20850t/U=1 with /H110151.8 bosons per site. /H20849c/H20850Strongly interacting SF:t/U=0.25 with /H110151.1 bosons per site; stronger deviations from Bogoliubov theory are present especially at larger q. Additional modes are clearly visible in the spectrum. /H20849d/H20850Mott insulating phase fort/U=0.16 and 1 boson per site. In all these ﬁgures, /H9262/U=0.5.0 0.5 1−10−50510 q/πωi/t(a) 0 0.5 1−505 q/πSi(c)0 0.5 1−10−50510 q/πωi/t(b) 0 0.5 1−505 q/πSi(d) FIG. 3. /H20849Color online /H20850/H20851 /H20849a/H20850and /H20849b/H20850/H20852Dispersion and /H20851/H20849c/H20850and /H20849d/H20850/H20852 strength of excitation modes at /H9262/U=0.5: /H20849a/H20850and /H20849c/H20850are in the Mott regime t/U=0.1; /H20849b/H20850and /H20849d/H20850are in the SF regime t/U=0.25. For clarity, in /H20849b/H20850the fourth pair of resonances at /H9275/H11015/H1100618.4 is not shown and in /H20849d/H20850only the strength of the four modes at lower energy is shown. Note that modes at positive /H20849negative /H20850energy have positive /H20849negative /H20850strength.C. MENOTTI AND N. TRIVEDI PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-4above /H20849below /H20850the tip of the lobe. The second Mott branch evolves into a gapped superﬂuid mode, and a symmetrical inenergy second superﬂuid gapped mode arises with zerostrength without having a precursor in the Mott phase /H20851see Fig.4/H20849b/H20850/H20852. The behavior at the tip of the lobe is quite interesting. In that case both Mott gapped modes become simultaneouslygapless at the QPT /H20851see Fig. 5/H20849a/H20850/H20852. From them the lowest four modes in the SF arise, two of them becoming phononicmodes and two of them becoming gapped modes when mov-ing away from the transition. However, within our approach,we ﬁnd a similar behavior to the one described above,namely, that one of the Mott modes evolves into a SFphononic mode, while the second one evolves into a SFgapped mode /H20851see Fig. 5/H20849b/H20850/H20852. A similar result was found by Huber et al. 20using an effective three-state approximation and mapping onto aspin-1 Hamiltonian. Gapped modes have also been predictedwithin a quantum phase model. 21In those papers, the mea- surement of the dynamical structure factor using Bragg spec-troscopy and lattice modulations has been suggested as aneffective way to investigate the different modes. The problem was also investigated by Sengupta and Dupuis 16in the strong-coupling regime by deriving an effec- tive action using Hubbard-Stratonovich transformations. Byexpanding the action to quadratic order in the ﬂuctuations,they found gapped excitations in the MI and gapless Gold-stone modes in the SF. They found two additional gappedmodes in the SF, which present a similar behavior to the onediscussed in this paper. E. Density of states A further quantity that one can use to characterize the excitations of the system is the density of states deﬁned inEq. /H208494/H20850. 16,18We calculate it across the QPT for a one- dimensional /H208491D/H20850system,39as shown in Fig. 6. In Fig. 6/H20849a/H20850 one can recognize the multimode structure of a strongly cor-related SF through a clearly enhanced DOS in the energy range of the corresponding excitation branch. In particular,we can see here two phononic branches and two gappedones. When approaching the QPT /H20851t/U=0.17; Fig. 6/H20849b/H20850/H20852,w e encounter a similar structure, where the width of the gappedbranch at negative energy is increased, while the gappedbranch at positive energy is not visible on the scale of thispicture /H20849although existing /H20850, since its strength goes to zero at the QPT. As expected, in the Mott regime, the DOS is dif-ferent from zero only in the energy range of the two gappedexcitation branches, one at negative energy and one at posi-tive energy /H20851Figs. 6/H20849c/H20850and6/H20849d/H20850/H20852. As observed in Fig. 6/H20849c/H20850, the branch at positive energy extends almost to /H9275=0, indi- cating the disappearance of the gap at the QPT. IV . SOUND VELOCITY The presence of phononic modes in the excitation spec- trum is an important signature of superﬂuidity. These modesdisappear in the Mott phase, where sound cannot propagatebecause of a gap in the spectrum. In this section, we discussthe evolution of the sound velocity in the strongly correlatedSF phase as the SF-MI transition is approached. The sound velocity is related to the compressibility /H9260and the effective mass m/H11569/H20849Refs. 40–42/H20850through the relation c=1 /H20881/H9260m/H11569=/H20881/H9267s /H9267/H9260m, /H2084914/H20850 where /H9260−1=/H9267/H20849/H11509/H9262//H11509/H9267/H20850and the SF fraction /H9267s//H9267=m/m/H11569.W e calculate the sound velocity cfrom the slope of the gapless mode in the RPA spectrum by using0 0.1 0.2 0.3−1−0.500.51 t/Uωi/U(a) 0 0.1 0.2 0.3−20020 t/USi(b) FIG. 4. /H20849Color online /H20850/H20849a/H20850Energy and /H20849b/H20850strength of the modes at low q=/H9266/50 as a function of t/Ufor/H9262/U=0.5. Note the pres- ence of both phonons and gapped modes in the strongly interactingSF and their evolution into two gapped modes into the MI. One ofthe gapped modes in the MI evolves from the phonon mode in theSF and the other one from a gapped mode in the SF. The thinvertical line at t/U/H110150.167 indicates the QPT. 0 0.1 0.2 0.3−1−0.500.51 t/Uωi/U(a) 0 0.1 0.2 0.3−20020 t/USi(b) 0.17 0.171 0.172 0.173−20020 FIG. 5. /H20849Color online /H20850/H20849a/H20850Energy and /H20849b/H20850strength of the modes at low q=/H9266/50 as a function of t/Uat/H9262/U=/H208812−1 corresponding to the tip of the lobe. The frequency of the lowest four modes /H20849two phononic and two gapped /H20850in the SF vanish at the QPT. The thin vertical line at t/U/H110150.1716 indicates the QPT. In the inset, the zoom around the QPT in panel /H20849b/H20850is shown.SPECTRAL WEIGHT REDISTRIBUTION IN STRONGLY … PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-5lim /H20841q/H20841→0/H9275/H20849q/H20850=c/H20841q/H20841, /H2084915/H20850 and compare it with the one obtained using the compressibil- ity relation in Eq. /H2084914/H20850. We ﬁnd perfect agreement between the values for the sound velocity extracted by the two differ-ent methods. It is important to note that the method using thecompressibility relation in Eq. /H2084914/H20850requires the knowledge only of the mean-ﬁeld solution, which provides the equationof state /H9262/H20849/H9267/H20850and the SF density /H9267s=/H20841/H9272/H208412. We ﬁnd that when the SF-MI transition, tuned by t/Uand /H9262/U, is approached at a generic point away from the tip of the lobe, the sound velocity vanishes, as shown in Fig. 7/H20849a/H20850. This is due to the fact that at the transition the compressibil-ity remains ﬁnite, but the SF density vanishes. Instead, thetip of the lobe where the phase transition happens at constantdensity and /H11509/H9267//H11509/H9262=0 is a special point: There, a perfect compensation between the divergent inverse compressibilityand the vanishing SF density takes place, which results in aﬁnite sound velocity as seen in Fig. 7/H20849b/H20850. We complete our analysis by comparing the sound veloc- ity calculated above with the results of Bogoliubov theory,which for a tunneling parameter t, on-site interaction U, and coordination number zpredicts the valuec BG=/H208812t zU/H20841/H9272/H208412, /H2084916/H20850 as explained in detail in Appendix B. The Bogoliubov pre- dictions are remarkably good in a wide range of parametersand fail only in the close proximity of the phase transition/H20849see thin lines in Fig. 7/H20850, since Bogoliubov theory does not account correctly for the vanishing of the order parameter atthat point. V . MOMENTUM DISTRIBUTION AND SPATIAL CORRELATIONS From Eq. /H208495/H20850, the momentum distribution n/H20849q/H20850is obtained by integrating the spectral function over negative energies. Itis a quantity of primary importance in cold atom experi-ments, as it is directly accessible by imaging the cloud afterexpansion. 3,35,36We have considered a two-dimensional sys- tem, which allows for the existence of Bose-Einstein conden-sation with long-range order in the SF regime. The two-dimensional /H208492D/H20850momentum distribution is shown in Fig. 8for different values of the parameter t/U.W e have checked that we can reproduce the momentum distribu-tion in the two extreme cases of deep MI /H20851see Eq. /H208499/H20850/H20852and weakly interacting SF /H20851see Eq. /H2084912/H20850/H20852. In the Mott phase with nonvanishing tunneling, we ﬁnd that the momentum distribution presents a modulation show-ing up as interference peaks in the expansion pictures. 33,35 When the QPT is reached, a strong peak develops at q=0 corresponding to the condensate. However, in a SF close tothe phase transition, the background momentum distribution/H20849atq/HS110050/H20850is large, indicating a strong depletion from the condensate due to interactions. The momentum distribution obtained with the RPA method happens to be not correctly normalized to the total−10 0 10−20020 ω/tDOS(ω)(a) t/U=0.25 −5 0 5−20020 ω/tDOS(ω)(b) t/U=0.17 −5 0 5−20020 ω/tDOS(ω)(c) t/U=0.16 −5 0 5−20020 ω/tDOS(ω)(d) t/U=0.1 0 FIG. 6. /H20849Color online /H20850Density of states for several values of t/U./H20849a/H20850t/U=0.25, SF regime; /H20849b/H20850t/U=0.17, SF regime, very close to the QPT; /H20849c/H20850t/U=0.16, MI regime, very close to the QPT; /H20849d/H20850 t/U=0.1, MI regime. All these results are for /H9262/U=0.5.0 0.5 1 1.5 2012345 t/Uc(blue) ,ϕ (red)(a) 0.166 0.168 0.1700.10.2 t/Udρ/dµ 0 0.5 1 1.5 2012345 t/Uc(blue) ,ϕ (red)(b) 0.17 0.171 0.172 0.173 0.17400.05 t/Udρ/dµ FIG. 7. /H20849Color online /H20850Sound velocity c/H20849blue solid line /H20850ex- tracted from the slope of the dispersion of the phonon modes andfrom Eq. /H2084914/H20850; order parameter /H9272/H20849red dashed line /H20850. Also shown for comparison are the sound velocity and the order parameter obtainedfrom Bogoliubov theory /H20849thin lines /H20850. In the inset d /H9267/d/H9262is shown. /H20849a/H20850/H9262/U=0.5; /H20849b/H20850/H9262/U=/H208812−1, corresponding to the tip of the lobe.C. MENOTTI AND N. TRIVEDI PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-6number of atoms. We attribute this feature to the fact that ﬂuctuations are not self-consistently included in the groundstate /H20849see discussion in Appendix C /H20850. Starting from the momentum distribution, one can obtain the single-particle density matrix /H9267/H20849r,r/H11032/H20850, which contains di- rect information about the spatial correlations present in thesystem. In the SF phase, the system is characterized by off-diagonal long-range order; and at long distances, the single- particle density matrix approaches a constant value equal tothe square of the order parameter, or the condensate densityn 0, /H9267/H20849r,r/H11032/H20850/H11013/H20855 a†/H20849r/H20850a/H208490/H20850/H20856→/H92722=n0, /H2084917/H20850 which is nonzero in a SF. This quantity is linked through Fourier transform to the /H9254function at q=0 which appears in the momentum distribution.The single-particle density matrix in Fig. 9shows a marked transition from the MI phase to the SF phase, andcorresponds to a change from an exponential decay of thecorrelations to a /H20849quasi- /H20850long-range order. In the MI, the single-particle density matrix shows that the correlations de-cay over a ﬁnite length scale, which decreases upon decreas-ing the tunneling and moving deeper into the Mott lobe.From those results, we have direct access to the length scaleof the correlations in the insulating regime. The condensate fraction can be in principle extracted from the momentum distribution, by subtracting the depletion/H20858 q/HS110050n/H20849q/H20850from the total density n, or equivalently by looking at the asymptotic value at large distances of the single-particle density matrix /H9267/H20849r,r/H11032/H20850. However, unfortunately we ﬁnd that the present application of RPA gives a violation ofthe total density sum rule and /H20858 q/HS110050n/H20849q/H20850/H11022n/H20849see also discus- sion in Appendix C /H20850, so that neither the momentum distribu- tion nor the single-particle density matrix turns out to be auseful quantity for extracting the condensate density. Thisproblem arises because within the present theoretical de-scription, we have not included the feedback of the collectivemodes and other excitations into the mean-ﬁeld ground state.Possible solutions to this problem will be a topic of furtherresearch. 43 However, our analysis of the excitation modes and their strength allows us to extract the condensate density n0di- rectly, using our knowledge of the sound velocity /H20849from the slope of the phononic mode at small momenta /H20850, combined with the knowledge of the strengths of the spectral functionand the compressibility. Starting from the relation A/H20849q, /H9275/H20850=n0 /H9267smc2/H9254/H20849/H92752−c2q2/H20850, /H2084918/H20850 and assuming that the condensate and SF densities are equal /H20849at least within mean-ﬁeld theory /H20850,w eg e t−101 10−1012(a) qx/π qy/πn(q) −101 10−1012(b) qx/π qy/πn(q) −101 10−1012(c) qx/π qy/πn(q) −101 10−1012(d) qx/π qy/πn(q) FIG. 8. /H20849Color online /H20850Momentum distribution n/H20849q/H20850f o ra2 D system for /H9262/U=0.5 and /H20849a/H20850t/U=0.05 deep in the MI; /H20849b/H20850t/U =0.15 in the MI but closer to the QPT; /H20849c/H20850t/U=0.175 in the SF phase close to the QPT; /H20849d/H20850t/U=0.25 in the SF phase but further from the QPT. −20 0 2010−310−1101 r−r/CID107ρ(r−r/CID107)(a) −20 0 2010−310−1101 r−r/CID107ρ(r−r/CID107)(b) −20 0 2010−310−1101 r−r/CID107ρ(r−r/CID107)(c) −20 0 2010−310−1101 r−r/CID107ρ(r−r/CID107)(d)FIG. 9. /H20849Color online /H20850Density matrix /H9267/H20849r,r/H11032/H20850 as a function of the relative distance r−r/H11032for a 2D system /H20849black line for a cut in the center of the trap and blue line along the diagonal /H20850./H20849a/H20850 t/U=0.05 deep in the MI showing only nearest- neighbor correlations; /H20849b/H20850t/U=0.15 in the MI but closer to the QPT showing an increase in thescale of the short-range correlations; /H20849c/H20850t/U =0.175 in the SF phase close to the QPT showinglong-range order; /H20849d/H20850t/U=0.25 in the SF phase but farther from the QPT. Note that theasymptotic value for large r−r /H11032has been subtracted.SPECTRAL WEIGHT REDISTRIBUTION IN STRONGLY … PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-7n0=/H208492qSph/H208502 m/H11569d/H9262/d/H9267=/H208494/H9266qSph/H208502t d/H9262/d/H9267. /H2084919/H20850 where Sphis the strength of the phononic modes. We have used t=/H60362//H20849m/H11569d2/H20850/H20849with/H6036=1/2/H9266and lattice spacing d=1/H20850to relate the effective mass and the tunneling parameter in theBHM. The limiting behavior of the right-hand side of Eq. /H2084919/H20850 for very small qapproaches the condensate density, which coincides with the predictions of mean-ﬁeld theory /H20849/H20841 /H9272/H208412/H20850,a s shown in Fig. 10. VI. INHOMOGENEOUS SYSTEM: OPTICAL LATTICES IN AN EXTERNAL TRAPPING POTENTIAL We extend the RPA formalism to real space and include a spatially inhomogeneous potential, which is taken into ac-count in the BHM through a site-dependent chemical poten-tial. The self-consistent MF solution produces alternatingshells of insulating and superﬂuid phases moving out from the center to the edge of the trap. 2,44,45 In the inhomogeneous system, the derivation of the equa- tion for Green’s function is the same as that outlined in Ap-pendix A, with the essential caution that the on-site energies /H9280/H9251iand the tunneling coefﬁcients T˜ /H9251/H11032/H9251/H9253/H9253 /H11032ikdepend on position and must be calculated separately for each site and for each pair of neighboring sites. In the presence of an external trapping potential, the den- sity and order parameter become nonuniform as shown inFig.11/H20849a/H20850. With our speciﬁc choice of parameters, one ﬁnds a central Mott core at density n=1, surrounded by a ring of superﬂuid. The sequence of panels /H20849b/H20850–/H20849d/H20850shows G/H20849r,r /H11032,/H9275/H20850 as a function of r/H11032for ﬁxed rand/H9275, which, roughly speak- ing, represent the effect of perturbing the system at differentpoints: perturbations in the SF regions produce a large effectall along the SF ring. As the perturbation moves to regionswith lower-order parameter near the SF-Mott interface, itseffect gets reduced and ﬁnally perturbations in Mott-type re-gions decrease exponentially and produce negligible effects. The results shown in Fig. 11are obtained for a given value of the energy /H9275. At different energies the structure remains the same, but the period of the oscillations along thering changes. By integrating G/H20849r,r /H11032,/H9275/H20850over/H9275, one gets the equal-time correlation function G/H20849r,r/H11032/H20850. This quantity would maintain the ringlike behavior shown Fig. 11and hence be qualitatively similar to the one calculated by Wessel et al.46 Before searching for quantitative results in the nonuni- form system, one should ponder on the consequences of theproblem in the normalization of the momentum distributiondiscussed in Sec. V, which might affect the extension of theRPA method to trapped systems. While this extension istechnically simple, although it might become computation-ally quite expensive, its validity is to be questioned due tothe coexistence in the same system of different phases /H20849MI and SF /H20850, where, as we have explained above, RPA introduces different normalization factors. For this reason, while we be-lieve one can get some insights into the correlations in thesystem, those pieces of information can be trusted only at thequalitative level.0 0.2 0.4 0.6 0.8 101234 q/πn0 FIG. 10. /H20849Color online /H20850Condensate fraction obtained from the strength of the poles of the spectral function, as in Eq. /H2084919/H20850, for the phonon mode at positive /H20849upper green line /H20850and negative /H20849lower red line /H20850frequencies. As q→0 both functions approach the condensate fraction n0obtained with MFT /H20849dashed blue line /H20850. In this ﬁgure /H9262/U=0.5 and t/U=0.5. −10 0 10−10−50510(a) 00.250.5 −10 0 10−10−50510(b) −0.300.3 −10 0 10−10−50510(c) −0.0100.01 −10 0 10−10−50510(d) −0.0100.01FIG. 11. /H20849Color online /H20850Inhomogeneous sys- tem: /H20849a/H20850order parameter in the trap. It is zero in the central MI core and ﬁnite and large in thesurrounding SF ring. The other panels showG/H20849r,r /H11032/H20850for a given value of the energy as a func- tion of r/H11032for a ﬁxed r/H20849black dot /H20850:/H20849b/H20850r =/H20849−6,0 /H20850in the SF ring; /H20849c/H20850r=/H20849−3,0 /H20850at the in- terface between the MI core and the SF ring; /H20849d/H20850 r=/H208490,0 /H20850in the center of the MI core.C. MENOTTI AND N. TRIVEDI PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-8VII. CONCLUDING REMARKS: RPA VERSUS MEAN-FIELD THEORY In this work, we have studied the excitations and the spa- tial correlations of the BHM in RPA. It is interesting to com-pare the information that is obtained by using the mean-ﬁeldapproximation with that by RPA which includes a certainclass of ﬂuctuations: /H208491/H20850The prediction of the SF to MI transition in the /H20851 /H9262/U,t/U/H20852plane calculated from the vanishing of the order parameter /H9272at the mean-ﬁeld level and by the disappearance of the gapless excitation mode within RPA lead to exactly thesame result for the boundaries of the insulating lobes. /H208492/H20850At the mean-ﬁeld level, one can extract information about the sound velocity using the compressibility–effectivemass relation, while at the RPA level the sound velocity isgiven by the slope of the phonon mode. The two methodsagain lead to exactly the same result. /H208493/H20850The condensate fraction is n 0=/H20841/H9272/H208412at the mean-ﬁeld level, whereas in the RPA treatment, it is extracted from thesmall- qbehavior of the spectral function and using the mean- ﬁeld compressibility. Again, the two methods lead to exactlythe same result. If the ﬂuctuations around the mean-ﬁeld state were in- cluded self-consistently, they would renormalize the energyand order parameter. We then expect the SF state to be sus-ceptible to ﬂuctuations and the critical t/Uto be shifted to a higher value than the one obtained by the MF theory. The RPA method further gives information which is not included in the mean-ﬁeld treatment. These include: /H20849i/H20850the excitation spectra, /H20849ii/H20850their strengths, /H20849iii/H20850the existence of new gapped modes in the strongly interacting SF phase, and/H20849iv/H20850the momentum distribution and spatial correlations in the system. As an open question, we are left with the role of quantum ﬂuctuations in the vicinity of the QPT, which as we dis-cussed may have additional effects on physical quantitiessuch as, e.g., momentum distribution or the condensate frac-tion. As explained in Appendix C, it is to be expected that inour approach the momentum distribution is not normalized.In the Mott limit, a clear deviation from the normalization tothe total number of atoms can be calculated analytically.Analogously, in the Bogoliubov regime, exactly recovered inRPA in the dilute limit, where one assumes n=/H20841 /H9272/H208412, the nor- malization is larger than the total number of atoms once thedepletion /H20849given by the integral over all momenta different from zero of /H20841 vq/H208412/H20850is added. However, while in the deep MI and SF regimes, the change in the normalization is just asmall perturbation, close to the phase transition it is a strik-ing effect. We attribute this to the fact that we perform RPAon the mean-ﬁeld ground state, without taking the effect ofRPA self-consistently into account. To this same reason, weattribute the existence of predictions /H208491/H20850–/H208493/H20850, which are equal in the mean-ﬁeld and RPA treatments. ACKNOWLEDGMENTS C.M. acknowledges ﬁnancial support from the EU through an EIF Marie-Curie Action. We thank R. B. Diener,P. Pedri, M. Randeria, and S. Stringari for helpful discus- sions. APPENDIX A: GREEN’S FUNCTION FORMALISMS IN RPA In this appendix we recall the main steps of the derivation of Green’s-function formalism in RPA.30,31RPA includes some ﬂuctuations around the mean-ﬁeld solution, which al-lows us to describe the excitations of the system. However,as explained in Sec. VII, these ﬂuctuations are not includedself-consistently, allowing feedback into the mean-ﬁeldground state. Also ignored are quantum ﬂuctuations of theorder parameter which are especially important close to theQPT /H20849see also discussion in Appendix C /H20850. 1. Mean-ﬁeld decoupling Substituting a=/H9272+a˜anda†=/H9272+a˜†into Eq. /H208491/H20850, we obtain without any approximation H=/H20858 i/H20875U 2ni/H20849ni−1/H20850−/H9262ni−t/H9272/H20849ai†+ai/H20850+t/H92722/H20876 −t 2z/H20858 /H20855ij/H20856/H20849a˜i†a˜j+a˜ia˜j†/H20850. /H20849A1/H20850 The Hamiltonian His thus rewritten as a sum of on-site Hamiltonians HiMF/H20849indicated by the term in the square bracket, which includes hopping at the mean-ﬁeld level /H20850, plus an intersite hopping term, which is assumed to be small. 2. Random-phase approximation In the basis given by the complete and orthonormal set of on-site eigenstates /H20841i/H9251/H20856of the on-site Hamiltonians HiMF, the Hamiltonian in Eq. /H20849A1/H20850takes the form H=/H20858 i/H9251/H9280/H9251iL/H9251/H9251i−t 2z/H20858 /H20855ij/H20856/H9251/H9251/H11032/H9252/H9252/H11032T˜ /H9251/H9251/H11032/H9252/H9252/H11032ijL/H9251/H9251/H11032iL/H9252/H9252/H11032j, /H20849A2/H20850 where we have deﬁned L/H9251/H9251/H11032i=/H20841i/H9251/H20856/H20855i/H9251/H11032/H20841, T˜ /H9251/H9251/H11032/H9252/H9252/H11032ij/H11013/H20855i/H9251/H20841a˜i†/H20841i/H9251/H11032/H20856/H20855j/H9252/H20841a˜j/H20841j/H9252/H11032/H20856+/H20855i/H9251/H20841a˜i/H20841i/H9251/H11032/H20856/H20855j/H9252/H20841a˜j†/H20841j/H9252/H11032/H20856. /H20849A3/H20850 For any pair of single-particle operators Aand B, the re- tarded /H20849/H9257=+1 /H20850or advanced /H20849/H9257=−1 /H20850Green’s function, de- ﬁned as Gr,a/H20849/H9270/H20850=−i/H9257/H9258/H20849/H9257/H9270/H20850/H20855A/H20849/H9270/H20850B/H208490/H20850−B/H208490/H20850A/H20849/H9270/H20850/H20856, /H20849A4/H20850 can be written in the on-site eigenbasis as G/H20849/H9270/H20850=/H20858 /H9251/H9251/H11032/H9252/H9252/H11032/H20855i/H9251/H20841A/H20841i/H9251/H11032/H20856/H20855j/H9252/H20841B/H20841j/H9252/H20856G/H9251/H9251/H11032/H9252/H9252/H11032ij/H20849/H9270/H20850, /H20849A5/H20850 where G/H9251/H9251/H11032/H9252/H9252/H11032ij/H20849/H9270/H20850=/H20855/H20855L/H9251/H9251/H11032i/H20849/H9270/H20850;L/H9252/H9252/H11032j/H20856/H20856. In the energy domain, Green’s function is deﬁned asSPECTRAL WEIGHT REDISTRIBUTION IN STRONGLY … PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-9G/H20849/H9275/H20850/H11013/H20855 /H20855 A;B/H20856/H20856/H9275/H11006=/H20885 −/H11009/H11009dt 2/H9266G/H20849/H9270/H20850r,aei/H9275/H11006/H9270. /H20849A6/H20850 A relation analogous to Eq. /H20849A5/H20850also holds for the energy resolved Green’s functions. In the uniform system at vanishing tunneling /H20849deep Mott regime /H20850,H=/H20858i/H9251/H9280/H9251iL/H9251/H9251iis exactly diagonalized by the on-site eigenstates /H20849Fock basis /H20850. Green’s function G/H20849/H9275/H20850can be cal- culated exactly as GMIt=0/H20849/H9275/H20850=1 2/H9266/H20875n+1 /H9275−/H20849En+1−En/H20850−n /H9275+/H20849En−1−En/H20850/H20876, /H20849A7/H20850 with En=−/H9262n+/H20849U/2/H20850n/H20849n−1/H20850. When the nearest-neighbor tunneling plays a role, the commutator with the transition operators L/H9251/H9251/H11032iwith the tun- neling part of the Hamiltonian produces a coupling to three operators’ Green’s functions. Following the prescriptions ofRPA of replacing the average of a product with the productof averages, one obtains /H20849E− /H9280/H9251/H11032i+/H9280/H9251i/H20850G/H9251/H9251/H11032/H9252/H9252/H11032ij/H20849E/H20850 =1 2/H9266/H20849/H20855L/H9251/H9251i/H20856−/H20855L/H9251/H11032/H9251/H11032i/H20856/H20850/H9254/H9251/H9252/H11032/H9254/H9251/H11032/H9252/H9254ij−t z/H20849/H20855L/H9251/H9251i/H20856 −/H20855L/H9251/H11032/H9251/H11032i/H20856/H20850/H20858 /H20855k/H20856i/H9253/H9253/H11032T˜ /H9251/H11032/H9251/H9253/H9253 /H11032ikG/H9253/H9253/H11032/H9252/H9252/H11032kj/H20849E/H20850. /H20849A8/H20850 At zero temperature results, /H20855L/H9251/H9251i/H20856are equal to 1 for the ground state and vanish otherwise. For nearest-neighbor hopping and for a uniform system, where /H9280/H9251i,/H20855L/H9251/H9251i/H20856, and T˜ /H9251/H11032/H9251/H9253/H9253 /H11032ikare site independent /H20851see Eq. /H20849A3/H20850/H20852, the same equation in momentum space takes the form /H20849E−/H9280/H9251/H11032+/H9280/H9251/H20850G/H9251/H9251/H11032/H9252/H9252/H11032/H20849E,q/H20850 =1 2/H9266/H20849/H20855L/H9251/H9251/H20856−/H20855L/H9251/H11032/H9251/H11032/H20856/H20850/H9254/H9251/H9252/H11032/H9254/H9251/H11032/H9252+/H9280/H20849q/H20850/H20849/H20855L/H9251/H9251/H20856 −/H20855L/H9251/H11032/H9251/H11032/H20856/H20850/H20858 /H9253/H9253/H11032T˜/H9251/H11032/H9251/H9253/H9253 /H11032G/H9253/H9253/H11032/H9252/H9252/H11032/H20849E,q/H20850. /H20849A9/H20850 where /H9280/H20849q/H20850=−/H208492t/z/H20850/H20858icos/H20849qi/H20850, with irunning over the di- mensionality of the system and zbeing the number of nearest neighbors. In practice, for each value of the energy Eand momentum q, the solution of Eq. /H20849A9/H20850amounts to inverting a2/H20849Ns−1/H20850/H110032/H20849Ns−1/H20850matrix, where Nsis the dimension of the number-state basis considered. The solution can also be found analytically to be18 G/H20849q,/H9275/H20850=1 2/H9266/H9016/H20849q,/H9275/H20850 1−/H9280/H20849q/H20850/H9016/H20849q,/H9275/H20850, /H20849A10 /H20850 where /H9016/H20849q,/H9275/H20850=A11+A12A21/H9280/H20849q/H20850 1−/H9280/H20849q/H20850A22, /H20849A11 /H20850A11=/H20858 /H9251y0/H9251y/H92510† /H9275−/H9004/H9251−y0/H9251†y/H92510 /H9275+/H9004/H9251, /H20849A12 /H20850 A22=A11†, /H20849A13 /H20850 A12=/H20858 /H9251y0/H9251y/H92510 /H9275−/H9004/H9251−y0/H9251y/H92510 /H9275+/H9004/H9251, /H20849A14 /H20850 A21=A12†, /H20849A15 /H20850 with /H208410/H20856indicating the ground state, y/H92510†=/H20855/H9251/H20841a†/H208410/H20856/H20849and analo- gously the other terms /H20850, and/H9004/H9251=E/H9251−E0. This equation can be easily evaluated numerically once the mean-ﬁeld ground-state wave function is known. This can be done analyticallyin the MI phase /H20849see Appendix C /H20850, but has to be done nu- merically in the SF phase. In principle, it is possible to also apply this formalism in the nonuniform case. One should then start from Eq. /H20849A8/H20850 considering that both /H9280/H9251iand T˜ /H9251/H11032/H9251/H9253/H9253 /H11032ikgenerally depend on position. Hence, in that case, for each value of the energy, the solution amounts to inverting a 2 /H20849Ns−1/H20850N/H110032/H20849Ns−1/H20850N matrix, where Nis the number of lattice wells considered. APPENDIX B: BOGOLIUBOV THEORY FOR THE BHM In this appendix we will present the details of the Bogo- liubov treatment for the BHM. The results are expected to bevalid in the weakly interacting SF regime, and these werecompared to the results of RPA in Sec. IV. We start from the BHM, H= /H20858 iU 2ai†ai†aiai−/H20858 i/H9262ni−t 2z/H20858 /H20855ij/H20856/H20849ai†aj+aiaj†/H20850,/H20849B1/H20850 and deﬁne, as done before, the ﬂuctuation operators subtract- ing from the operators aanda†their mean value a˜=a−/H9272,a˜†=a†−/H9272/H11569. /H20849B2/H20850 For a uniform system, one gets H=/H20858 i/H20875U 2/H20841/H9272/H208414−/H9262/H20841/H9272/H208412−t/H20841/H9272/H208412/H20876 +/H20858 i/H20851a˜i†/H20849U/H20841/H9272/H208412−/H9262−t/H20850/H9272+ H.c. /H20852 +/H20858 i/H20875U 2/H20849/H92722a˜i†2+4/H20841/H9272/H208412a˜i†a˜i+/H9272/H115692a˜i2/H20850 −/H9262a˜i†a˜i−t 2z/H20858 /H20855j/H20856i/H20849a˜i†a˜j+a˜j†a˜i/H20850/H20876 +/H208493rd + 4th /H20850order in a˜and a˜†. /H20849B3/H20850 To minimize the energy one has to set to zero the ﬁrst order, leading to the discretized version of the Gross-Pitaevskiiequation for the uniform system,C. MENOTTI AND N. TRIVEDI PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-10U/H20841/H9272/H208412−/H9262−t=0⇒/H9272GP=/H20881/H9262+t U. /H20849B4/H20850 The quantity /H20841/H9272/H208412is the density in the uniform system and is related to the chemical potential. Conversely, for a givendensity /H20841 /H9272/H208412, the chemical potential is given by /H9262=U/H20841/H9272/H208412−t, /H20849B5/H20850 namely, the kinetic energy plus interaction energy. Strictly speaking, the kinetic energy of the condensate is zero, and t is just a shift due to the choice of the zero of energy. Thisapproach does not include any phase transition, because theSF fraction /H20841 /H9272/H208412is always equal to the total density. To diagonalize the second-order terms in the Hamiltonian /H20849H2/H20850, we perform a transformation to momentum space, a˜i=1 N/H20858 qe−iq·ria˜q, /H20849B6/H20850 a˜i†=1 N/H20858 qeiq·ria˜q†, /H20849B7/H20850 so that ﬁnally it reads H2=−1 2/H20858 q/H208512U/H20841/H9272/H208412−/H9262+/H9280/H20849q/H20850/H20852+1 2/H20858 q/H20851/H208492U/H20841/H9272/H208412−/H9262+/H9280/H20849q/H20850/H20850 /H11003/H20849a˜q†a˜q+a˜−qa˜−q†/H20850+U/H92722a˜q†a˜−q†+U/H9272/H115692a˜−qa˜q/H20852, /H20849B8/H20850 where, same as before, /H9280/H20849q/H20850=−/H208492t/z/H20850/H20858icos/H20849qi/H20850. Then, we ap- ply the Bogoliubov transformation which diagonalizes H2, /H20873a˜q a˜−q†/H20874=/H20873uqbq+v−q/H11569b−q† u−q/H11569b−q†+vqbq/H20874, /H20849B9/H20850 with the additional condition /H20841uq/H208412−/H20841v−q/H208412=1 to preserve the commutation relations. This is equivalent to the Bogoliubovequations /H20849L q−/H6036/H9275q/H20850uq+U/H92722vq=0 , /H20849Lq+/H6036/H9275q/H20850vq+U/H9272/H115692uq=0 , where Lq=U/H20841/H9272/H208412+/H208494t/z/H20850/H20858isin2/H20849qid/2/H20850. The solution for the Bogoliubov spectrum and the Bogo- liubov amplitudes is given by /H6036/H9275q=/H208814t z/H20858 isin2/H20873qi 2/H20874/H208754t z/H20858 isin2/H20873qi 2/H20874+2U/H20841/H9272/H208412/H20876, /H20849B10 /H20850 uq+vq=/H20881Lq−U/H92722 /H6036/H9275q=/H20881/H208494t/z/H20850/H20858isin2/H20849qi/2/H20850 /H6036/H9275q, /H20849B11 /H20850uq−vq=/H20881/H6036/H9275q Lq−U/H92722=/H20881/H6036/H9275q /H208494t/z/H20850/H20858isin2/H20849qi/2/H20850. /H20849B12 /H20850 For q→0, the spectrum shows a linear behavior in /H20841q/H20841,/H6036/H9275q/H11015/H20841q/H20841/H20881/H208492t/z/H20850U/H20841/H9272/H208412with sound velocity c=/H20881/H20851/H208492t/z/H20850U/H20841/H9272/H208412/H20852. In Bogoliubov theory the momentum distribution is given by n/H20849q/H20850=/H20841/H9272/H208412/H9254q,0+/H20841v−q/H208412. /H20849B13 /H20850 It is evident that starting from the fact that /H20841/H9272/H208412equals the total density in the uniform system, the integral of the mo-mentum distribution /H20848n/H20849q/H20850dqwill exceed this value by the depletion n D=/H20848/H20841v/H20849q/H20850/H208412dq. In the limit of validity of the Bo- goliubov approach, this quantity is very small. APPENDIX C: ANALYTIC CALCULATION OF THE MOMENTUM DISTRIBUTION IN THE MOTT REGIME IN RPA In RPA the momentum distribution in the MI regime can be calculated analytically. In this appendix, we will presentthe results for 1D, 2D, and 3D systems, with the aim ofpointing out that close to the phase transition, quantum ﬂuc-tuations play a major role and are not correctly taken intoaccount by RPA. In the MI regime, the RPA Green’s function in Eq. /H20849A10 /H20850 can be written as G MI/H20849q,/H9275/H20850=1 2/H9266A11/H20849/H9275/H20850 1−/H9280/H20849q/H20850A11/H20849/H9275/H20850, /H20849C1/H20850 where A11has been deﬁned in Eq. /H20849A12 /H20850,A12=A21=0 in the MI, and /H9280/H20849q/H20850=−/H208492t/z/H20850/H20858icos/H20849qi/H20850, same as before. In particular, for the MI at density n/H20849i.e., where the on- site ground state is /H208410/H20856=n/H20850,w eg e t A11/H20849/H9275/H20850=n+1 /H9275−/H20849En+1−En/H20850−n /H9275+/H20849En−1−En/H20850, /H20849C2/H20850 /H20851with En=−n/H9262+/H20849U/2/H20850n/H20849n−1/H20850/H20852, which obviously coincides a part from a factor of 1 /2/H9266with Green’s function for the MI0 0.05 0.1 0.15 0.211.21.41.61.8 t/Unorma lisation FIG. 12. /H20849Color online /H20850Normalization of n/H20849q/H20850for/H9262/U=0.5 as a function of t/Uin one /H20849blue dashed-dotted line /H20850,t w o /H20849green dashed line /H20850, and three /H20849red solid line /H20850dimensions. The thin vertical line indicates the phase transition.SPECTRAL WEIGHT REDISTRIBUTION IN STRONGLY … PHYSICAL REVIEW B 77, 235120 /H208492008 /H20850 235120-11at zero tunneling previously introduced in Eq. /H20849A7/H20850. By using the deﬁnitions in Eqs. /H20849C1/H20850and /H20849C2/H20850and deﬁn- ing/H9004/H11006=En/H110061−En, it is straightforward to see that Green’s function takes the form GMI=1 2/H9266/H20849n+1/H20850/H20849/H9275+/H9004−/H20850−n/H20849/H9275−/H9004+/H20850 /H20849/H9275−/H9004+/H20850/H20849/H9275+/H9004−/H20850−/H9280/H20849q/H20850/H20851n/H9275+/H20849n+2/H20850/H9004−/H20852, /H20849C3/H20850 whose poles /H9275/H11006can be calculated analytically and have a momentum dependence due to the kinetic energy /H9280/H20849q/H20850in Eq. /H20849C1/H20850. Hence, Green’s function close to the poles /H20849/H9275/H11015/H9275/H11006/H20850is GMI/H110151 2/H9266/H20849n+1/H20850/H20849/H9275/H11006+/H9004−/H20850−n/H20849/H9275/H11006−/H9004+/H20850 /H20849/H9275/H11006−/H9275/H11007/H20850/H20849/H9275−/H9275/H11006/H20850. /H20849C4/H20850 Consequently, the momentum distribution reads n/H20849q/H20850=−/H20849n+1/H20850/H20849/H9275−+/H9004−/H20850−n/H20849/H9275−−/H9004+/H20850 /H9275−−/H9275+. /H20849C5/H20850 In the deep MI /H20849t=0/H20850,/H9275/H11006=/H11006/H9004/H11006and the momentum distri- bution is simply given by n/H20849q/H20850=n, which integrated over theallowed momenta /H20849equal to the total number of wells /H20850gives the total number of atoms. At ﬁnite tunneling, the kineticenergy /H9280/H20849q/H20850gives a modulation to the momentum distribu- tion. To ﬁnd the normalization of n/H20849q/H20850, we integrate numeri- cally the analytic expression in Eq. /H20849C5/H20850for different dimen- sions. 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PhysRevB.96.245135.pdf	PHYSICAL REVIEW B 96, 245135 (2017) Spin Hall insulators beyond the helical Luttinger model Vieri Mastropietro University of Milan, Via Saldini 50, 20133 Milan, Italy Marcello Porta* University of Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland (Received 18 September 2017; revised manuscript received 8 December 2017; published 26 December 2017) We consider the interacting, spin-conserving, extended Kane-Mele-Hubbard model, and we rigorously establish the exact quantization of the edge spin conductance and the validity of the helical Luttinger liquid relations forDrude weights and susceptibilities. Our analysis takes fully into account lattice effects, typically neglected inthe helical Luttinger model approximation, which play an essential role for universality. The analysis is basedon exact renormalization-group methods and on a combination of lattice and emergent Ward identities, whichenable the emergent chiral anomaly to be related with the ﬁnite renormalizations due to lattice corrections. DOI: 10.1103/PhysRevB.96.245135 I. INTRODUCTION The remarkable edge transport properties of quantum spin Hall insulators (QSHI), predicted in [ 1–5]( s e e[ 6–8]f o r reviews), have been explained so far via topological argumentsor effective quantum ﬁeld theory (QFT) descriptions. Inthe absence of many-body interactions, and if the spin isconserved, topological arguments ensure the quantization ofthe spin Hall conductance. Many-body interactions, however,break the single-particle picture, and prevent the use of suchmethods. Nevertheless, experiments have shown values ofspin conductances that are approximately quantized [ 9–13]. It is a challenge for theorists to understand a mechanismfor universality, or to predict possible deviations from thequantized value. Due to the reduced dimensionality and to the massless dispersion relation, the edge states form a strongly correlatedsystem. To analytically understand its behavior, the helical Lut- tinger (HL) model [4], a QFT for relativistic one-dimensional fermions with locked spin and chirality, has been proposedas an effective ﬁeld-theoretic description. This model canbe studied via bosonization, see e.g., [ 14,15]; as a result, it exhibits anomalous decay of correlations, and the chiral anomaly . Also, nonuniversal anomalous exponents, velocities, and transport coefﬁcients are related by exact scaling relations . Several generalizations of the HL model have been considered,see [ 16–27]. However, these effective QFT descriptions are insufﬁcient to conclude whether many-body interactions breakor not the quantization of the spin conductance, since theyneglect important lattice effects; it is well known that nonlinearcorrections to the dispersion relation and umklapp terms mightproduce ﬁnite corrections to the transport coefﬁcients, as, forinstance, in graphene [ 28,29]. In this paper, we establish the exact quantization of the edge spin conductance of a truly interacting lattice QSHI,going beyond the effective QFT description. Moreover, weestablish the validity of the HL scaling relations by takingfully into account lattice effects and the nonlinearity of the *Present address: Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.energy bands. We use recently developed nonperturbative RG methods, introduced to prove rigorous universality results fornonsolvable statistical mechanics models [ 30]. II. THE KMH MODEL A. The model A basic model for interacting, time-reversal-invariant topo- logical insulators is provided by the extended, spin-conservingKane-Mele-Hubbard (KMH) model. The KMH model is a time-reversal-symmetric system, describing spinful fermionson the honeycomb lattice. The honeycomb lattice /Lambda1can be represented as the superposition of two triangular sublattices /Lambda1 A,/Lambda1Bof side L,/Lambda1=/Lambda1A+/Lambda1B. We denote by /vector/lscript1,/vector/lscript2the normalized basis vectors of /Lambda1A, and we set /Lambda1B=/Lambda1A+(1,0). We shall denote by x1,x2the coordinates of the point /vectorx∈/Lambda1Ain the/vector/lscript1,/vector/lscript2basis. We introduce fermionic creation/annihilation operators a± /vectorx,σandb± /vectory,σ, with spin labels σ=± , acting on the two triangular sublattices /Lambda1Aand/Lambda1B. In the absence of interactions, the Hamiltonian is H0=−t1/summationdisplay /vectorx,j,σ/bracketleftbig a+ /vectorx,σb− /vectorx+/vectorδj,σ+b+ /vectorx+/vectorδj,σa− /vectorx,σ/bracketrightbig −it2⎡ ⎢⎢⎢⎣/summationdisplay /angbracketleft/angbracketleft/vectorx,/vectory/angbracketright/angbracketright σa+ /vectorx,σ(/vectorσ/vectorν/vectorx,/vectory)a− /vectory,σ+/summationdisplay /angbracketleft/angbracketleft/vectorx,/vectory/angbracketright/angbracketright σb+ /vectorx,σ(/vectorσ/vectorν/vectorx,/vectory)b− /vectory,σ⎤ ⎥⎥⎥⎦ −W/summationdisplay /vectorx,σ/bracketleftbig a+ /vectorx,σa− /vectorx,σ−b+ /vectorx+δ1,σb− /vectorx+δ1/bracketrightbig −μN, (1) where in the ﬁrst sum /vectorx∈/Lambda1A, andj=1,2,3 labels one of its three nearest neighbors in /Lambda1B, connected by the vectors /vectorδj; see Fig. 1. The second and third sums run over next-to-nearest neighbors on the A,B sublattices, connected by the vectors ±/vectorγj,j=1,2,3; we denote by /vectorσ=(σ1,σ2,σ3) the vector of the Pauli matrices, and we set /vectorν/vectorx,/vectory=(/vectord/vectorx,/vectorz×/vectord/vectorz,/vectory)/\|/vectord/vectorx,/vectorz×/vectord/vectorz,/vectory\|, (2) where /vectorzis the intermediate site between /vectorxand/vectory, and /vectord/vectorx,/vectory=/vectorx−/vectory. The third term includes a staggered potential 2469-9950/2017/96(24)/245135(10) 245135-1 ©2017 American Physical SocietyVIERI MASTROPIETRO AND MARCELLO PORTA PHYSICAL REVIEW B 96, 245135 (2017) B A δ1δ2 δ3 x γ1γ2 γ3 FIG. 1. The honeycomb lattice /Lambda1: the empty dots belong to the Asublattice, while the black dots belong to the Bsublattice. ±Won the A,B sublattices, and the last term ﬁxes the chemical potential μ(Nis the number operator). The Hamiltonian of the model is the sum of two copies of the Haldane model [31]:H0=/summationtext σHσ 0, where Hσ 0acts on the σ-spin subsector. The connection between the different spin sectorsisH + 0=CH−C, with Cthe complex conjugation operator, which ensures the invariance under time-reversal symmetry ofthe full Hamiltonian. To reduce the honeycomb lattice to a Bravais lattice, we collect the fermionic operators associated with the sites /vectorx,/vectorx+ /vectorδ 1in a single, two-component fermionic operator (see Fig. 1): φ+ /vectorx,σ=(a+ /vectorx,σ,b+ /vectorx+/vectorδ1,σ)≡(φ+ /vectorx,A,σ,φ+ /vectorx,B,σ). With these notations, we rewrite the noninteracting Hamiltonian as H0=/summationdisplay /vectorx,/vectory/summationdisplay ρ,ρ/prime,σφ+ /vectorx,ρ,σHσ ρρ/prime(/vectorx,/vectory)φ− /vectory,ρ/prime,σ, (3) withHσa one-particle Schrödinger operator, acting on /Lambda1A×C2. Let us now deﬁne the density operator as ρ/vectorx,σ=/summationtext ρ=A,Bρ/vectorx,ρ,σ , with ρ/vectorx,ρ,σ=φ+ /vectorx,ρ,σφ− /vectorx,ρ,σ.T h e interacting Hamiltonian is H=H0+λV, V=/summationdisplay /vectorx,/vectory/summationdisplay ρ,ρ/prime/bracketleftbigg ρ/vectorx,ρ,σ−1 2/bracketrightbigg/bracketleftbigg ρ/vectory,ρ/prime,σ/prime−1 2/bracketrightbigg vρρ/prime(/vectorx,/vectory)( 4 ) forvρρ/prime(/vectorx,/vectory) short-ranged, and where λis the coupling constant. B. Lattice currents and conservation laws LetA(t)=eiHtAe−iHtbe the time evolution of A.T h e density operator satisﬁes the following lattice continuityequation: ∂ tρ/vectorx,σ(t)=i[H,ρ/vectorx,σ(t)]=/summationdisplay /vectory/summationdisplay ρ,ρ/primejρρ/prime;σ /vectorx,/vectory(t), jρρ/prime;σ /vectorx,/vectory=iφ+ /vectory,ρHσ ρρ/prime(/vectory,/vectorx)φ− /vectorx,ρ/prime+H.c. (5) The operator jρρ/prime;σ /vectorx,/vectoryis the bond current operator , corresponding to the pairs of honeycomb lattice sites labeled by ( /vectorx,ρ;/vectory,ρ/prime).Notice that, by the ﬁnite range of the hopping Hamiltonian, the only nonvanishing bond currents are those connecting ( /vectorx,/vectorx± /vector/lscripti), with i=1,2, and ( /vectorx,/vectorx±/vectorγ1), with /vectorγ1=/vector/lscript1−/vector/lscript2. Letjσ /vectorx,/vectory=/summationtext ρ,ρ/primejρρ/prime;σ /vectorx,/vectory. Let us deﬁne the discrete lattice derivative as dif(/vectorx)=f(/vectorx)−f(/vectorx−/vector/lscripti). Then, the continuity equation can be rewritten as ∂tρ/vectorx,σ(t)=−d1j/vectorx,/vectorx+/vector/lscript1−d2j/vectorx,/vectorx+/vector/lscript2−j/vectorx,/vectorx+/vector/lscript1−/vector/lscript2−j−/vector/lscript1+/vector/lscript2+/vectorx,/vectorx ≡−d1j1,/vectorx−d2j2,/vectorx, (6) where we deﬁned jσ 1,/vectorx=jσ /vectorx,/vectorx+/vector/lscript1+jσ /vectorx,/vectorx+/vector/lscript1−/vector/lscript2andjσ 2,/vectorx= jσ /vectorx,/vectorx+/vector/lscript2+jσ /vectorx,/vectorx−/vector/lscript1+/vector/lscript2. We shall collect densities and currents in a single 3-current jσ μ,/vectorx,μ=0,1,2. Also, we deﬁne the charge and spin 3-currents as jc μ,/vectorx=/summationtext σjσ μ,/vectorx,js μ,/vectorx=/summationtext σσjσ μ,/vectorx, which satisfy ∂0j/sharp 0,/vectorx+/summationtext idij/sharp i,/vectorx=0, with /sharp=c,s. We shall study the thermodynamic properties of the model in the grand-canonical ensemble. The Gibbs state ofthe model is /angbracketleft/angbracketright β,L=(1/Zβ,L)Tr e−βH, withZβ,L=Tr e−βH the partition function. We introduce the imaginary-time (or Euclidean) evolution of the fermionic operators as φ± x,ρ,σ:= ex0Hφ± /vectorx,ρ,σe−x0H,x=(x0,/vectorx), with x0∈[0,β), extended an- tiperiodically for all x0∈R. A crucial ingredient in our analysis will be the use of Ward identities , implied by the charge and spin conservation laws. Letd0≡i∂x0. The lattice continuity equation can be rewritten in a compact form as/summationtext μdμjσ μ,x=0. This relation can be used to derive identities among correlations, such as /summationdisplay μdxμ/angbracketleftbig Tjσ μ,x;jσ ν,y/angbracketrightbig β,L=iδ(x0−y0)/angbracketleftbig/bracketleftbig jσ 0,/vectorx,jσ ν,/vectory/bracketrightbig/angbracketrightbig β,L.(7) In Eq. ( 7),Tis the time-ordering operator, and the contact term on the right-hand side is called the Schwinger term . Equation ( 7) is the Ward identity for the current-current correlation functions. In the same way, one can also derive aWard identity relating the vertex functions of the lattice modelto the two-point correlation function: /summationdisplay μdzμ/angbracketleftTj/sharp μ,z;φ− y,σ,ρ/primeφ+ x,σ,ρ/angbracketrightβ,L=iσ/sharp[/angbracketleftTφ− y,σ,ρ/primeφ+ x,σ,ρ/angbracketrightβ,Lδx,z −/angbracketleftTφ− y,σ,ρ/primeφ+ x,ρ/angbracketrightβ,Lδy,z], (8) where δx,z=δ(x0−y0)δ/vectorx,/vectoryandσc=+ ,σs=σ. III. NONINTERACTING TOPOLOGICAL INSULATORS In the absence of interactions, λ=0, the Hamiltonian reduces to the sum of two noninteracting Haldane Hamil-tonians, H 0=/summationtext σ=±Hσ 0. Suppose the model is equipped with periodic boundary conditions. Then [using the factthat the single-particle Hamiltonian is translation-invariant,H σ(z,z/prime)≡Hσ(z−z/prime)], we can introduce the Bloch Hamil- tonian as/hatwideHσ(/vectork)=/summationtext ze−i/vectorz·/vectorkHσ(z)f o r/vectorkin the Brillouin zone B.W eh a v e /hatwideHσ(/vectork)=/parenleftbiggmσ(k)−t1/Omega1∗(k) −t1/Omega1(k)−mσ(k)/parenrightbigg , (9) 245135-2SPIN HALL INSULATORS BEYOND THE HELICAL . . . PHYSICAL REVIEW B 96, 245135 (2017) where mσ(/vectork)=W−2σt2α(/vectork),α(/vectork)=/summationtext3 i=1sin/vectork·/vectorγi, and /Omega1(/vectork)=1+e−i/vectork·/vector/lscript1+e−i/vectork·/vector/lscript2. The corresponding energy bands are Eσ ±(/vectork)=±/radicalBig mσ(/vectork)2+t2\|/Omega1(/vectork)\|2. To make sure that the energy bands do not overlap, we assume thatt2/t1<1/3. The two bands can only touch at the Fermi points /vectork± F=(2π 3,±2π 3√ 3), which are the two zeros of /Omega1(/vectork), around which /Omega1(/vectork± F+/vectork/prime)/similarequal3 2(ik/prime 1±k/prime 2). The condition that the two bands touch at /vectorkω F, with ω=+,−, is that mσ ω=0, with mσ ω≡mσ/parenleftbig/vectorkω F/parenrightbig =W+ωσ3√ 3t2. Therefore, the unperturbed critical points are given by the values of Wsuch that W=± 3√ 3t2. Choosing the chemical potential μ=0, which lies halfway between the two energy bands, the condition W/negationslash=± 3√ 3t2corresponds to the insulat- ing phase for which the correlations decay exponentially fast.In the insulating phase, the system may or may not be in atopologically nontrivial phase, depending on the value of theHall conductivity . This quantity is deﬁned starting from the Kubo formula, which we use directly in its imaginary-timeversion (see [ 32] for a discussion of the Wick rotation): σ σ 12=lim p0→0lim /vectorp→0lim β,L→∞1 p0/integraldisplayβ 0dx0/summationdisplay x(1−e−ip·x)/angbracketleftbig jσ 1,x;jσ 2,0/angbracketrightbig β,L. (10) In the absence of interactions, the Hall conductivity of the Haldane model can be computed explicitly. One ﬁnds σσ 12=νσ 2π,νσ=sgn(mσ −)−sgn(mσ +). (11) Concerning the Kane-Mele model, its net Hall conductivity σc 12=σ+ 12+σ− 12vanishes while the net spin conductivity σs 12= σ+ 12−σ− 12is nonzero: σs 12=σ+ 12−σ− 12=ν+ π. (12) This is the quantum spin Hall effect . In the spin-symmetric case, the quantization of σs 12follows from the quantization of σσ 12, which is ensured for topological reasons. In the absence of spin symmetry, for instance in the presence of Rashbacouplings, one does not expect the spin conductivity to bequantized. Nevertheless, topology survives in the sense thatthe Hamiltonians are classiﬁed by a suitable Z 2invariant [ 1,2]. A remarkable feature of topological insulators is the presence of gapless edge modes . Suppose now the system is equipped with cylindric boundary conditions , say periodic in the /vector/lscript1direction and Dirichlet in the /vector/lscript2direction, on the boundaries at x2=0,x2=L. By translation invari- ance in the /vector/lscript1direction, we can introduce a partial Bloch transformation of the initial Hamiltonian, /hatwideHρρ/prime(k1;x2,y2)=/summationtext z1e−iz1k1Hρρ/prime(z1;x2,y2), with k1∈S1. By construction, the Hamiltonian is symmetric under the action of the time-reversal operator, T ∗/hatwideH(k1)T≡T∗[/hatwideH+(k1)+/hatwideH−(k1)]T= /hatwideH−(−k1)+/hatwideH+(−k1)≡/hatwideH(−k1) since /hatwideHσ(k1)=/hatwideH−σ(−k1). Edge states correspond to solutions of the Schrödingerequation /hatwideH(k1)ξ(k1)=ε(k1)ξ(k1) at the Fermi level μ, which are exponentially localized around one of the two edges: /vextendsingle/vextendsingleξx2(k1)/vextendsingle/vextendsingle/lessorequalslantCe−c\|x2\|,/vextendsingle/vextendsingleξx2(k1)/vextendsingle/vextendsingle/lessorequalslantCe−c\|L−x2\|. (13) These 1D eigenfunctions of /hatwideH(k1) correspond to 2D eigen- functions for the Hamiltonian H,o ft h ef o r m e−ik1x1ξx2(k1); they are responsible for the transport of dissipationless edge currents . In the Haldane model, the edge eigenfunctions can be found explicitly [ 33]: each cylinder edge supports either a zero- or one-edge mode. Consequently, the Kane-Mele HamiltonianH=/summationtext σ=±Hσsupports either zero- or two-edge states per edge. Let ε+,ε−be their dispersion relations. By time-reversal symmetry, ε+(k1)=ε−(−k1): the model displays two Fermi points k± F,k+ F=−k− F, such that ε+(k+ F)=ε−(k− F)=μ. Time-reversal symmetry implies that the edge modes are counterpropagating: v+=∂k1ε+(k+ F)=−v−. The edge transport of the system can be investigated by probing the variation of the density or of the current (ofcharge or of spin) after introducing an external perturbationsupported in a strip of width afrom the x 2=0 edge. We shall study these transport phenomena in the linear-responseregime. To deﬁne the edge transport coefﬁcients, let usintroduce the following notations. Given a local operator O /vectorx, we deﬁne its partial space-time Fourier transform as /hatwideOp,x2=/integraltextβ 0dx0/summationtext x1e−ip·xOx, withp=(p0,p1),p0the Matsubara fre- quency, and x=(x0,x1). Let/angbracketleft/angbracketleft/angbracketleft·/angbracketright/angbracketright/angbracketright∞=limβ,L→∞(βL)−1/angbracketleft·/angbracketrightβ,L. We deﬁne, for /sharp,/sharp/prime=c,s, G/sharp;a ρ,ρ(p)=a/summationdisplay x2=0∞/summationdisplay y2=0/angbracketleftbig/angbracketleftbig/angbracketleftbig Tρ/sharp p,x2;ρ/sharp/prime p,y2/angbracketrightbig/angbracketrightbig/angbracketrightbig ∞, G/sharp;a ρ,j(p)=a/summationdisplay x2=0∞/summationdisplay y2=0/angbracketleftbig/angbracketleftbig/angbracketleftbig Tρ/sharp p,x2;j/sharp/prime 1,−p,y2/angbracketrightbig/angbracketrightbig/angbracketrightbig ∞, (14) G/sharp;a j,j(p)=a/summationdisplay x2=0⎡ ⎣∞/summationdisplay y2=0/angbracketleftbig/angbracketleftbig/angbracketleftbig Tj/sharp 1,p,x2;j/sharp/prime 1,−p,y2/angbracketrightbig/angbracketrightbig/angbracketrightbig ∞−i/Delta1(x2)⎤ ⎦, where /Delta1(x2)=limβ,L→∞/angbracketleft/summationtext σ[tσ /vectorx,/vectorx+/vector/lscript1+tσ /vectorx,/vectorx+/vector/lscript1−/vector/lscript2]/angbracketrightβ,L, with tσ /vectorx,/vectory=/summationtext ρρ/prime−iφ+ /vectory,ρHσ ρρ/prime(/vectory,/vectorx)φ− /vectorx,ρ/prime−H.c. As we shall see later, this function is related to the Schwinger term in Eq. ( 7). The edge spin conductance is σs=lim a→∞lim p0→0+lim p1→0Gc,s;a ρ,j(p). (15) It measures the variation of the spin current after introducing a shift of the chemical potential supported in a region ofwidth afrom the x 2=0 edge. Similarly, the edge charge conductance is σc=lim a→∞lim p0→0+lim p1→0Gc,c;a ρ,j(p). (16) Instead, the edge susceptibilities and Drude weights, of charge or of spin, are κ/sharp=lim a→∞lim p1→0lim p0→0+G/sharp,/sharp;a ρ,ρ(p), D/sharp=− lim a→∞lim p0→0+lim p1→0G/sharp,/sharp;a j,j(p). (17) 245135-3VIERI MASTROPIETRO AND MARCELLO PORTA PHYSICAL REVIEW B 96, 245135 (2017) As we shall see, due to the lack of continuity at p=(0,0) of the expressions in ( 14), the order of the limits in the above deﬁni- tions is crucial. It turns out that, in the absence of interactions,the edge transport coefﬁcients can be computed. One has σ c=0,σs=σs 12, κ/sharp=1 π\|v+\|,D/sharp=\|v+\| π. (18) The equivalence of the edge spin conductance with the bulk spin conductivity is a manifestation of the bulk-edge correspondence : namely, a duality between the presence of edge modes at the Fermi level with the value of thetopologically invariant classifying bulk Hamiltonians (actingon inﬁnite lattices, with no edges). For the IQHE [ 34–37], this duality implies that the sum of the chiralities of theedge states/summationtext eωe, withωe=sgn[∂k1εe(ke F)], equals the Chern number of the Bloch bundle, which ﬁxes the value of theHall conductivity. For time-reversal-invariant systems, instead, 1 2/summationtext e\|ωe\|mod 2 turns out to be equal to the bulk Z2invariant [38–40]; in particular, for the spin-conserving Kane-Mele model, this implies that the edge spin conductance equalsthe bulk spin conductivity. The bulk-edge correspondence hasbeen rigorously established for single-particle Hamiltonians:there is no general argument ensuring its validity for interactingmany-body systems. Finally, notice that in contrast to σ s,t h e edge susceptibility κ/sharpand the Drude weight D/sharpare nonuniver- sal quantities, depending on the velocity of the edge modes. The goal of this paper is to understand the effect of many- body interactions of the edge transport coefﬁcients: the naturalquestion we address here is whether some form of universalitypersists, and in particular if the quantization of σ sholds true. IV. MAIN RESULT Here we shall consider the edge transport properties of the Kane-Mele-Hubbard model, λ/negationslash=0. Our main result is the following theorem. Theorem. Consider the KMH Hamiltonian ( 4) with cylin- dric boundary conditions. Let us choose the chemical potentialμin the gap of the bulk Hamiltonian. Suppose that the single-particle KM Hamiltonian supports a pair of edge modes,ε +(k+ F)=ε−(k− F)=μ, and that v+/negationslash=0. Then, there exists λ0>0 such that, for \|λ\|<λ 0, the following is true. Let ω=sgn(v+). The edge spin conductance is universal: σs=−ω π. (19) Moreover, the Drude weights and the susceptibilities satisfy the helical Luttinger liquid relations: κc=K πv,Dc=vK π,κs=1 πvK,Ds=v πK(20) withK=1+O(λ)/negationslash=1,v=v++O(λ)/negationslash=v+. Finally, the two-point function decays with an anomalous exponent, η= (K+K−1−2)/2. As a corollary, our result combined with the universality of bulk transport, following from the analysis of [ 32,41], provides a rigorous example of bulk-edge correspondencefor an interacting time-reversal-invariant topological insulator(see [ 42] for the analogous result for Hall systems). The lackof many-body corrections to the conductance is in agreement with experimental results [ 9,11]. Notice that, in contrast with the conductance, the susceptibilities and the Drude weightsare interaction-dependent: nevertheless, if combined with thedressed Fermi velocity v, they verify a marginal form of universality, in the sense of the validity of the helical Luttingerliquid relation: κ /sharpv2 D/sharp=1. (21) Moreover, the HL parameter Kallows us to determine the anomalous exponent of the two-point function via the formulaη=(K+K −1−2)/2. The rest of the paper is organized as follows. In Sec. V we introduce a Grassmann integral representation for thetransport coefﬁcients. We then integrate out the “bulk degreesof freedom” corresponding to the energy modes far fromthe Fermi level. As a result, we end up with an effectiveone-dimensional model, which is reminiscent of the helicalLuttinger model up to some crucial differences: the fermionicﬁelds are deﬁned on a lattice, the interaction involves arbitrar-ily high monomials in the ﬁelds, the energy-dispersion relationis nonlinear, and the umklapp scattering process is present.Then, in Sec. VIwe study this lattice QFT via exact RG, which allows us to represent the transport coefﬁcients in termsof renormalized, convergent series. Such expansions can bereorganized by isolating the contributions corresponding to anemergent, effective chiral QFT theory with suitably ﬁne-tunedbare parameters, from a remainder term, that depends on alllattice details. The advantage of this rewriting is that thecurrent-current correlation functions of the emergent QFT canbe computed exactly (see Sec. VII) thanks to the validity of extra chiral Ward identities. This allows us to computethe edge transport coefﬁcients of the KMH model up toﬁnite multiplicative and additive renormalizations, dependingon all the microscopic details of the model. The valuesof these renormalizations are, however, severely constrainedfrom one side by the validity of the Adler-Bardeer anomalynonrenormalization property of the emergent chiral theory,and from the other side by the lattice WIs of the KMH model.As we show in Sec. VIII, these facts imply nonperturbative relations among all ﬁnite renormalizations, from which ourtheorem follows. V. REDUCTION TO AN EFFECTIVE 1 DTHEORY For simplicity, we shall directly consider the case L=∞ , which corresponds to having just one edge. It is useful toswitch to a functional integral representation of the correlationfunctions of the lattice model. We deﬁne the generatingfunctional of the correlations as e W(A)=/integraldisplay P(d/Psi1)e−V(/Psi1)+B(/Psi1;A), (22) where /Psi1± x,σ,ρare Grassmann variables, labeled by x=(x0,/vectorx)∈ [0,β)×/Lambda1A,σ=± ,ρ=A,B ;P(dψ) is a Gaussian Grass- mann integration with a propagator given by the noninteractingEuclidean two-point function, g σ,σ/prime(x,y)=δσσ/prime/integraldisplaydk (2π)2e−ik·(x−y) −ik0+/hatwideHσ(k1)−μ(/vectorx;/vectory),(23) 245135-4SPIN HALL INSULATORS BEYOND THE HELICAL . . . PHYSICAL REVIEW B 96, 245135 (2017) where k=(k0,k1), with k0the fermionic Matsubara frequency andk1the quasimomentum associated with the translation- invariant direction /vector/lscript1. The Grassmann counterpart of the many- body interaction is V(/Psi1)=λ/summationdisplay ρ,ρ/prime σ,σ/prime/integraldisplay dxdynx,ρ,σny,ρ/prime,σ/primevρρ/prime(/vectorx,/vectory)δ(x0−y0), where/integraltext dx=/integraltextβ 0dx0/summationtext /vectorx, andnx,ρ,σis the Grassmann coun- terpart of the density operator. Finally, B(/Psi1;A) is a source term of the form B(/Psi1;A)=/summationdisplay μ,/sharp/integraldisplay dxA/sharp μ,xJ/sharp μ,x (24) withJ/sharp μ,xthe Grassmann counterpart of j/sharp μ,x. We now use the addition principle of the Grassmann variables to write /Psi1=/Psi1(e)+/Psi1(b), with /Psi1(e),/Psi1(b)indepen- dent Grassmann variables, with propagators g(edge)andg(bulk), where g(e)takes into account the energy modes close enough to the Fermi level. That is, g(e) σσ/prime(x,y)=δσσ/prime/summationdisplay e/integraldisplaydk (2π)2e−ik·(x−y) ×χσ(k1) −ik0+εσ(k1)−μPσ k1(x2;y2), (25) withPσ k1=\|ξσ/angbracketright/angbracketleftξσ\|, where ξσis the edge mode of /hatwideHσ(k1), with energy εσ, andχσ(k1)≡χ(\|k1−kσ F\|/lessorequalslantδ) is a compactly supported cutoff function. By construction, the propagatorg (bulk)is gapped; it only depends on the energy modes that are at a distance at least ∼δfrom the Fermi level. Thus, \|g(bulk)(x,y)\|/lessorequalslantCe−c\|x−y\|. Instead, due to the fact that, for k1=k/prime 1+kσ Fandk/prime 1small εσ/parenleftbig k/prime 1+kσ F/parenrightbig −μ=σv+k/prime 1+O/parenleftbig k/prime 12/parenrightbig , (26) the edge propagator in Eq. ( 25) only decays as \|x− y\|−1e−c(\|x2\|+\|y2\|). The ﬁeld /Psi1(b)can be integrated out, expanding the integrand of (22) in the coupling λand using the exponential decay of the bulk propagator together with fermionic cluster-expansiontechniques [ 43]. We then get e W(A)=eW(b)(A)/integraldisplay Pe(d/Psi1(e))e−V(e)(/Psi1(e))+B(e)(/Psi1(e);A), (27) where the new effective interaction V(e)(/Psi1(e)) is a sum over monomials Pin the ﬁelds /Psi1(e)of any order \|P\|=n, with kernels W(e) P(x1,...,xn), exponentially decaying in \|xi−xj\| fori/negationslash=j. Graphically, a given kernel can be represented as a sum of Feynman diagrams with \|P\|external lines, corresponding to the edge ﬁelds, and an arbitrary numberof quartic vertices connected by the bulk propagators. Thisexpansion turns out to be convergent for small λ, thanks to determinant bounds for fermionic ﬁeld theories, combinedwith the good decay properties of the bulk propagators. Thenew effective source term B (e)admits a similar representation, where now external lines corresponding to the Aﬁelds are present as well.Due to the special form of the edge propagator, given by Eq. ( 25), we now notice that the edge ﬁeld can be represented as the convolution of a truly one-dimensional ﬁeld with theedge mode eigenfunctions. That is, /integraldisplay P e(d/Psi1(e))e−V(e)(/Psi1(e))+B(e)(/Psi1(e);A) =/integraldisplay P1D(dψ)e−V(e)(ψ∗ˇξ)+B(e)(ψ∗ξ;A), (28) where P1Dis a Grassmann Gaussian integration for a one- dimensional ﬁeld ψ± /vectorx,σ, with the propagator given, in momen- tum space, by /hatwidegσ,σ/prime(k)=δσσ/primeχσ(k) −ik0+εσ(k1)−μ0, (29) where now χσ(\|k\|)=χ(\|k−kσ F\|/lessorequalslantδ) andμ−μ0=ν0, with ν0=O(λ) a counter term that is chosen so as to ﬁx the value of the interacting chemical potential; and (ψ+∗ˇξ)x,ρ=/summationdisplay y1ψ− (x0,y1),σˇξσx2(x1−y1;ρ), (30) where ˇξσ x2(x1;ρ) is the Fourier transform of χσ(k1)ξσ x2(k1;ρ). This representation of the edge ﬁeld allows us to decouple thex 2variables from the remaining x0,x1variables in the effective interaction. Summing over x2(recalling the exponential decay of the edge modes), one ﬁnally gets eW(A)=eW(b)(A)/integraldisplay P0(dψ)e−V(0)(ψ)+B(0)(ψ;A), (31) where P0≡P1Dand for suitable new effective interaction and source terms, which can be again expressed as sums overmonomials of arbitrary order in the 1D ﬁelds ψ. One has V (0)(ψ)=/integraldisplay dx/bracketleftbigg λ0ψ+ x,+ψ− x,+ψ+ x,−ψ− x,−+/summationdisplay σν0ψ+ x,σψ− x,σ/bracketrightbigg +RV(0)(ψ), (32) where the new coupling constant is λ0=λ/summationdisplay x2,y2 ρ,ρ/prime/hatwidevρρ/prime(0;x2,y2)ξ(1,σ) x2(kF;ρ)ξ(1,σ) x2(kF;ρ) ×ξ(1,σ) y2(kF;ρ/prime)ξ(1,σ) y2(kF;ρ/prime)+O(λ2), (33) andRV(0)collects all the higher-order terms, together with nonlocal terms. All these contributions turn out to be irrelevant in the RG sense. Similarly, B(0)(ψ;A)=/summationdisplay μ,/sharp/integraldisplay dxZ/sharp μ(x2)A/sharp μ,xn/sharp μ,x+RB(0)(ψ;A), (34) where Z/sharp μ(x2) is such that \|Z/sharp μ(x2)\|/lessorequalslantCe−cx2, and it is analytic inλ; and nc 0,x=/summationdisplay σψ+ x,σψ− x,σ,nc1,x=/summationdisplay σσψ+ x,σψ− x,σ, ns 0,x=nc 1,x,ns 1,x=nc 0,x. (35) Let us give a quick proof of Eqs. ( 34) and ( 35). After the integration of /Psi1(b)and the reduction to 1D theory, the effective 245135-5VIERI MASTROPIETRO AND MARCELLO PORTA PHYSICAL REVIEW B 96, 245135 (2017) source term has the following form, in momentum space: B(0)(ψ;A)=/summationdisplay μ,/sharp,x 2/integraldisplaydk (2π)2dp (2π)2 ×/hatwideA/sharp μ,(p,x2)/hatwideψ+ k+p,σ/hatwideψ− k,σ/hatwideW/sharp μ,σ(p,k;x2)+O(A2) (36) for suitable kernels /hatwideW/sharp μ,σ. The higher orders in Aturn out to be irrelevant in the RG sense. Let us localize the kernel by writing /hatwideW/sharp μ,σ(p,k;x2)=/hatwideW/sharp μ,σ(0,kσ F;x2)+R/hatwideW/sharp μ,σ, where the Rerror terms are irrelevant. The effective 1D model is invariant undertime-reversal symmetry [recall that ξσ(k1)=ξ−σ(−k1), and that/hatwideHσ(k1)=/hatwideH−σ(−k1)]: /hatwideA/sharp μ,p,x2→γ/sharpγμ/hatwideA/sharp μ,−p,x2,/hatwideψε k,σ→/hatwideψε −k,−σ,c→¯c,(37) withca generic constant in the action, γc=1=−γs, and γ0=1=−γ1. This symmetry implies that /hatwideW/sharp σ,μ(0,kσ F;x2)= γ/sharpγμ/hatwideW/sharp −σ,μ(0,k−σ F;x2). Also, the model is invariant under complex conjugation: /hatwideA/sharp μ,p,x2→/hatwideA/sharp μ,/tildewidep,x2, /hatwideψ+ k,σ→−/hatwideψ− /tildewidek,σ, /hatwideψ− k,σ→/hatwideψ+ /tildewidek,σ,c →¯c (38) with /tildewidek=(−k0,k1). This last symmetry implies that /hatwideW/sharp σ,μ(0,kσ F;x2) is real. Going back to conﬁguration space, Eq. ( 35) follows. Equation ( 31)i sa n exact (but very involved) representation of the generating functional of the KMH model in terms of aneffective one-dimensional ﬁeld. It differs from the HL modelby the presence of nonlinear corrections in the dispersion andirrelevant terms in the effective interaction. VI. MULTISCALE ANALYSIS OF THE EDGE MODES Due to the absence of a mass gap, the ﬁeld ψcannot be integrated in a single step. Instead, we proceed in a multiscalefashion, exploiting a renormalization procedure at every step.We rewrite the ψﬁeld in terms of single-scale quasiparticle ﬁelds as follows: ψ ± x,σ=e±ikσ Fx10/summationdisplay h=hβψ(h) x,σ, (39) where each ﬁeld varies on a scale 2−h, with h/lessorequalslant0. The last scalehβis ﬁxed by the inverse temperature, hβ∼\|log2β\|. The covariance of the ﬁelds is deﬁned inductively. Weintegrate the ﬁelds in an iterative fashion. From a RG pointof view, the ψ + xψ− xterms are relevant, while the ψ+ x∂μψ− x, ψ+ x,σψ− x,σψ+ x,σ/primeψ− x,σ/primeterms are marginal. After the integration of the scales h+1,..., 0, we obtain the following representation of the generating functional: eW(A)=eW(h)(A)/integraldisplay Ph(dψ(/lessorequalslanth))e−V(h)(√Zhψ)+B(h)(ψ;A),(40) where the new Gaussian Grassmann integration has a propagator: g(/lessorequalslanth) σ,σ/prime(x,y)=δσσ/prime Zh/integraldisplaydk/prime (2π)2e−ik/prime·(x−y)χh(k/prime) −ik0+σvhk/prime 1[1+rh(k/prime)],where χhis a smooth cutoff function supported for \|k/prime\|/lessorequalslant 2h+1;rhis an error term, \|rh(k/prime)\|/lessorequalslantC\|k/prime\|; and Zhandvh are, respectively, the wave-function renormalization and the effective Fermi velocity, whose RG ﬂow, as a function ofh, is marginal. Time-reversal symmetry ( 37) and complex conjugation ( 38) imply that these parameters are real and spin-independent. The new effective interaction is a sum of Grassmann monomials of arbitrary order. We rewrite it as V (h)=LV(h)+ RV(h), where LV(h)takes into account all the relevant and marginal contributions: LV(h)(/radicalbig Zhψ)=/integraldisplay dx/bracketleftbigg λhZ2 hψ+ x,+ψ− x,+ψ+ x,−ψ− x,− +/summationdisplay σ2hZhνhψ+ x,σψ− x,σ/bracketrightbigg , whileRV(h)takes into account all irrelevant terms. By the symmetries ( 37) and ( 38), the parameters λhandνhare again real and spin-independent. In the same spirit, we rewrite B(h)= LB(h)+RB(h), where LB(h)collects all marginal terms (there are no relevant terms in the source term): LB(h)(ψ;A)=/integraldisplay dxZ/sharp h,μ(x2)A/sharp μ,xn/sharp μ,x (41) for suitable (real) running coupling functions Z/sharp h,μ(x2). Let us brieﬂy discuss the ﬂow of the running coupling constants. The (relevant) ﬂow of νhis controlled via a ﬁxed point argument by properly choosing the initial shift ofthe chemical potential ν 0;s e e[ 42] for details in a similar case. Instead, the (marginal) ﬂows of λh,vhare controlled using a highly nontrivial cancellation in the renormalizedexpansions, the vanishing of the beta function [30], giving λ h=λ0+O(λ2) andvh=v0+O(λ)uniformly inh. Instead, the ﬂows of the wave function and vertex renormalizationsdiverge with anomalous exponents, Z h∼2−ηh,Z/sharp h,μ(x2)∼2−ηhZ/sharp 0,μ(x2), (42) withη=λ2 0 8π2v2 0+O(λ4 0). The outcome of this construction is a convergent expansion for the correlation functions in terms of the running couplingconstants, which can be used to prove bounds for the decay ofthe current-current correlations. Convergence follows from theuse fermionic cluster expansion at every step of integration, asin [42], and excludes nonperturbative effects. We have /vextendsingle/vextendsingle/vextendsingle/vextendsinglelim β,L→∞/angbracketleftbig Tj/sharp μ,x;j/sharp/prime ν,y/angbracketrightbig β,L/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantCe−c\|x2−y2\|/(1+\|x−y\|2).(43) This estimate, however, is not for the computation of the edge transport coefﬁcients. In fact, it is not even enough to provethe boundedness of the Fourier transform of the current-currentcorrelation, uniformly in p . To improve on this, we need to exploit cancellations in the renormalized expansion, following from the emergent chiral symmetry of the theory. 245135-6SPIN HALL INSULATORS BEYOND THE HELICAL . . . PHYSICAL REVIEW B 96, 245135 (2017) VII. EMERGENT CHIRAL QFT In this section we introduce an emergent effective chiral QFT theory, deﬁned by the generating functional: eWχ(A)=/integraldisplay PN(dψ)e−λχZχ2/integraltext dxdyv(x−y)nx,+ny,−+B(ψ;A),(44) where PN(dψ) is a Gaussian Grassmann measure with a propagator: gχ σ,σ/prime(x,y)=δσσ/prime Zχ/integraldisplaydk (2π)2e−ik·(x−y)χN(k) −ik0+σvχk1,(45) where χNis an ultraviolet cutoff, supported for \|k\|/lessorequalslant2N+1, forN/greatermuch1 (to be sent to inﬁnity at the end). The source term isB(ψ;A)=/summationtext∞ x2=0/integraltext dxZ/sharp,χ μ(x2)A/sharp μ,xn/sharp μ,x. The interaction potential v(x−y) is nonlocal and short-ranged. The presence of the UV cutoff is crucial to give a nonperturbative meaningto Eq. ( 44). Its ﬁnal removal is done through an ultraviolet multiscale analysis, in which the nonlocal, short-range natureof the interaction plays an essential role [ 30]. The infrared regime of this QFT can be studied as for the lattice model. Let us denote by λ χ h,Zχ h,vχ h, andZ/sharp,χ μ,h(x2) the running coupling constants of the emergent chiral model /angbracketleft/angbracketleft/angbracketleft/angbracketright/angbracketright/angbracketrightχ. The bare parameters Zχ,vχ,λχ, andZ/sharp,χ μwill be chosen in such a way that the running coupling constants of latticeand chiral theory converge to the same limit as h→− ∞ . This fact, together with the convergence of the renormalizedexpansions for both models, implies that the correlations of theKMH model can be written in terms of the correlations of theemergent chiral model, up to ﬁnite multiplicative and additiverenormalizations, depending on all the microscopic details ofthe KMH model: /angbracketleftbig/angbracketleftbig/angbracketleftbig Tj /sharp μ,p,x2j/sharp/prime ν,−p,y2/angbracketrightbig/angbracketrightbig/angbracketrightbig ∞=Z/sharp,χ μ(x2)Z/sharp,χ ν(y2)/angbracketleft/angbracketleft/angbracketleftn/sharp μ,pn/sharp/prime ν,−p/angbracketright/angbracketright/angbracketrightχ +/hatwideH/sharp,/sharp/prime μ,ν(p,x2,y2), (46) where /angbracketleft/angbracketleft/angbracketleft·/angbracketright/angbracketright/angbracketrightχdenotes the correlations of the emergent chiral model, and /hatwideH/sharp,/sharp/prime μ,ν(p;x2,y2) is an error term, continuous inp,i n contrast with the ﬁrst term on the right-hand side of ( 46). The improved regularity of this contribution is due to the fact that itinvolves irrelevant terms in the RG sense, which all come witha dimensional gain: in conﬁguration space, such a term decaysas, for large distances, e −c\|x2−y2\|/(1+\|x−y\|2+ϑ)f o rs o m e ϑ> 0. Thus, even though this term disappears pointwise in the scaling limit of the correlations, it gives a ﬁnite contribution to the Fourier transform of the lattice correlations. Concerningthe multiplicative renormalizations, they verify the bound\|Z /sharp,χ μ(x2)\|/lessorequalslantCe−cx2as a consequence of the exponential decay of the edge states. Similarly, up to subleading terms for small external mo- menta, /angbracketleftbig/angbracketleftbig/angbracketleftbig T/hatwidej/sharp p,z2,μ;/hatwideφ− k+p,x2,ρ,σ/hatwideφ+ k,y2,ρ,σ/angbracketrightbig/angbracketrightbig/angbracketrightbig ∞ =Z/sharp,χ μ(z2)Qσ x2/parenleftbig kω F;ρ/parenrightbig Qσy2/parenleftbig kω F;ρ/parenrightbig /angbracketleft/angbracketleft/angbracketleftn/sharp μ,p;/hatwideψ− k+p,σ/hatwideψ+ k,σ/angbracketright/angbracketright/angbracketrightχ (47) for some functions Qσ, such that Qσ=[1+O(λ)]ξσ, which satisfy the exponential bound \|Qσ x2\|/lessorequalslantCe−c\|x2\|. Moreover, up= Zμ Zν+ Hμ,νχ(a) (b)Zμ χ Dσ(p) = −σσ σσ σ −σ σχ χχ + σ σχ FIG. 2. (a) Graphical representation of Eq. ( 46). “χ” denotes the contributions due to the emergent chiral model. The full dots correspond to the vertex renormalizations, associated with the factors Z/sharp,χ μin Eq. ( 46). (b) Graphical representation of the ﬁrst WI in Eq. ( 49) for a ﬁnite UV cutoff N. The small white circle denotes a correction vertex , corresponding to the insertion of Cσ(p,k)ψ+ k+p,σψ− k,σ.T h e empty bubble is a noninteracting diagram, whose N→∞ value is −1 4π\|v\|Z2D−σ(p). to subleading terms in the external momenta, /angbracketleftbig/angbracketleftbig/angbracketleftbig T/hatwideφ− k,x2,ρ,σ/hatwideφ+ k,y2,ρ/prime,σ/angbracketrightbig/angbracketrightbig/angbracketrightbig ∞=Qσ x2/parenleftbig kω F;ρ/parenrightbig Qσy2/parenleftbig kω F;ρ/prime/parenrightbig /angbracketleft/angbracketleft/angbracketleft/hatwideψ− k,σ/hatwideψ+ k,σ/angbracketright/angbracketright/angbracketrightχ. (48) The advantage of comparing the lattice correlations with those of the emergent model is that the latter can be computed in aclosed form, thanks to chiral Ward identities, following fromU(1) chiral gauge symmetry. Notice that this symmetry is only approximate, due to the presence of the ultraviolet cutoff. Asa result, the UV regularization produces extra terms in theWard identities of the emergent chiral theory, which do notvanish as N→∞ , but rather produce anomalies breaking the conservation of the chiral current; see Fig. 2(b).I nt h e ﬁgure, the white circle corresponds to the insertion of acorrection vertex C σ(p,k)=[χ−1 N(k)−1]Dσ(k)−[χ−1 N(k+ p)−1]Dσ(k+p), with Dσ(p)=−ip0+σvχp1.F o r p= O(1), this vertex insertion ﬁxes the momentum of the incoming and outgoing fermionic lines on the scale of the ultravioletcutoff. In the ﬁgure, we isolated the terms where the fermioniclines incident to the correction vertex meet at the same point;instead, the last term on the right-hand side of Fig. 2(b) collects all contributions corresponding to diagrams where the linesmeet at different points. It turns out that this last term vanishes asN→∞ , thanks to the nonlocality of the interaction, and to the support properties of the correction vertex. See [ 30,44] for a detailed proof of this statement, in a similar case. Setting D σ(p)=−ip0+σvχp1,w eh a v e Dσ(p)/angbracketleft/angbracketleft/angbracketleft/hatwideρp,σ;/hatwideρ−p,σ/angbracketright/angbracketright/angbracketrightχ=−D−σ(p) 4π\|vχ\|Zχ2 +τD−σ(p)/hatwidev(p)/angbracketleft/angbracketleft/angbracketleft/hatwideρp,−σ;/hatwideρ−p,σ/angbracketright/angbracketright/angbracketrightχ, /angbracketleft/angbracketleft/angbracketleft/hatwideρp,σ;/hatwideρ−p,−σ/angbracketright/angbracketright/angbracketrightχ=τD−σ(p) Dσ(p)/hatwidev(p)/angbracketleft/angbracketleft/angbracketleft/hatwideρp,−σ;/hatwideρ−p,−σ/angbracketright/angbracketright/angbracketrightχ, (49) 245135-7VIERI MASTROPIETRO AND MARCELLO PORTA PHYSICAL REVIEW B 96, 245135 (2017) where τ=λχ 4π\|vχ\|is the chiral anomaly . The linearity of the anomaly in the bare coupling constant is a highly nontrivial fact, known as Adler-Bardeen anomaly nonrenormalization . The explicit value of the anomaly can be used to determine thecritical exponents of the emergent chiral model. For instance,the anomalous exponent of the two-point Schwinger functionisη=K+K −1−2 with K=1−τ 1+τ. Thus, supposing that /hatwidev(0)=1, we have, up to subleading terms in p, /angbracketleft/angbracketleft/angbracketleft/hatwideρp,σ/hatwideρ−p,σ/angbracketright/angbracketright/angbracketrightχ=−1 4π\|vχ\|Zχ21 1−τ2D−σ(p) Dσ(p), (50) /angbracketleft/angbracketleft/angbracketleft/hatwideρp,−σ/hatwideρ−p,σ/angbracketright/angbracketright/angbracketrightχ=−1 4π\|vχ\|Zχ2τ 1−τ2. These expressions can be plugged in the representation for the lattice current-current correlation function, ( 46). All we have left to do is to determine the unknown multiplicative andadditive renormalizations. VIII. UNIVERSALITY To ﬁx the values of the ﬁnite multiplicative and additive renormalizations, we use again Ward identities, this time forthe lattice model. These identities introduce nonperturbativerelations between the renormalization coefﬁcients, which,as we shall see, imply a dramatic cancellation in the ﬁnalexpression of the edge transport coefﬁcients. To begin, it isconvenient to rewrite the Schwinger term of the lattice WI ( 7) in the following more explicit way: /angbracketleftbig/bracketleftbig j σ 0,/vectorx,jσ 1,/vectory/bracketrightbig/angbracketrightbig =/parenleftbig δ/vectorx,/vectory−δ/vectorx,/vectory+/vector/lscript1/parenrightbig/angbracketleftbig tσ /vectory,/vectory+/vector/lscript1+tσ /vectory,/vectory+/vector/lscript1−/vector/lscript2/angbracketrightbig +/parenleftbig δ/vectorx,/vectory+/vector/lscript1−δ/vectorx,/vectory+/vector/lscript1−/vector/lscript2/parenrightbig/angbracketleftbig tσ /vectory,/vectory+/vector/lscript1−/vector/lscript2/angbracketrightbig/angbracketleftbig/bracketleftbig jσ 0,/vectorx,jσ 2,/vectory/bracketrightbig/angbracketrightbig =/parenleftbig δ/vectorx,/vectory−δ/vectorx,/vectory+/vector/lscript2/parenrightbig/angbracketleftbig tσ /vectory,/vectory+/vector/lscript2+tσ /vectory,/vectory−/vector/lscript1+/vector/lscript2/angbracketrightbig +/parenleftbig δ/vectorx,/vectory+/vector/lscript2−δ/vectorx,/vectory−/vector/lscript1+/vector/lscript2/parenrightbig/angbracketleftbig tσ /vectory,/vectory−/vector/lscript1+/vector/lscript2/angbracketrightbig (51) withtσ /vectorx,/vectorydeﬁned after ( 14). Summing up ( 7) over y2, one gets dy0/summationdisplay y2/angbracketleftTj/sharp 1,/vectorx;j/sharp 0,/vectory/angbracketright+dy1/summationdisplay y2/angbracketleftTj/sharp 1,/vectorx;j/sharp 1,/vectory/angbracketright =iδ(x0−y0)/parenleftbig δx1,y1−δx1,y1+1/parenrightbig /Delta1(x2), dy0/summationdisplay y2/angbracketleftTj/sharp 0,/vectorx;j/sharp/prime 0,/vectory/angbracketright+dy1/summationdisplay y2/angbracketleftTj/sharp 0,/vectorx;j/sharp/prime 1,/vectory/angbracketright=0.(52) To get these relations, we crucially used that/summationtext y2dy2(···)=0, which is implied by the Dirichlet boundary conditions. Bygoing into Fourier space, we can use the relations ( 52)t o prove identities for the edge transport coefﬁcients: −ip 0G/sharp,/sharp;a j,ρ(p)+p1η(p1)G/sharp,/sharp;a j,j(p)=0, (53) −ip0G/sharp,/sharp/prime;a ρ,ρ(p)+p1η(p1)G/sharp,/sharp/prime;a ρ,j(p)=0, withp1η(p1)=p1+O(p2 1) the Fourier symbol associated with the lattice derivative dy1. Equations ( 53) can be used to determine the p→0limit of the additive renormalization/summationtexta x2=0/summationtext∞ y2=0/hatwideH/sharp,/sharp/prime μ,ν(p;x2,y2) (which exists by continuity in p). For instance, consider the edge charge conductance,Gc,s;a ρ,j(p). We can rewrite the second of Eqs. ( 53)a s Gc,s;a ρ,j(p)=[ip0/p1η(p1)]Gc,s;a ρ,ρ(p); thus, this relation implies that lim p1→0limp0→0Gc,s;a ρ,j(p)=0. This identity, together with the representation ( 46) of the current-current corre- lation function, allows us to compute the p→0 limit of/summationtexta x2=0/summationtext∞ y2=0/hatwideHc,s 0,1(p;x2,y2) in terms of the other unknown renormalized parameters. A similar strategy can be followedfor the other transport coefﬁcients. For simplicity, let us drop the χlabel, and let us set Z /sharp μ≡/summationtext z2Z/sharp,χ μ(z2). The above-mentioned strategy allows us to compute, up to subleading terms in p, lim a→∞Gc,s;a ρ,j(p)=−Zc 0Zs 1 Z2(1−τ2)1 π\|v\|p2 0 p2 0+v2p2 1, lim a→∞G/sharp,/sharp;a j,j(p)=−Z/sharp 1Z/sharp 1 Z2(1−τ2)1 π\|v\|p2 0 p2 0+v2p2 1, (54) lim a→∞G/sharp,/sharp;a ρ,ρ(p)=Z/sharp 0Z/sharp 0 Z2(1−τ2)1 π\|v\|v2p2 1 p2 0+v2p2 1. It remains to determine the multiplicative renormalization in Eqs. ( 54). This is done by comparing the vertex WIs of lattice and emergent models. From Eq. ( 8) we have, setting η0(p1)=−i, 1/summationdisplay μ=0ημ(p1)/summationdisplay z2/angbracketleftbig T/hatwidej/sharp p,z2,μ;/hatwideφ− k+p,x2,σ/hatwideφ+ k,y2,σ/angbracketrightbig β,L =σ/sharp/bracketleftbig/angbracketleftbig T/hatwideφ− k,x2,σ/hatwideφ+ k,y2,σ/angbracketrightbig β,L−/angbracketleftbig T/hatwideφ− k+p,x2,σφ+ k+p,y2,σ/angbracketrightbig β,L/bracketrightbig (55) withσc=1 and σs=σ. On the other hand, the WIs for the emergent chiral model are −ip0/angbracketleft/angbracketleft/angbracketleft/hatwiden/sharp 0,p;/hatwideψ− k+p,σ/hatwideψ+ k,σ/angbracketright/angbracketright/angbracketright+p1v/angbracketleft/angbracketleft/angbracketleftn/sharp 1,p;/hatwideψ− k+p,σ/hatwideψ+ k,σ/angbracketright/angbracketright/angbracketright =σ/sharp Z(1−η/sharpτ)[/angbracketleft/angbracketleft/angbracketleft/hatwideψ− k,σ/hatwideψ+ k,σ/angbracketright/angbracketright/angbracketright−/angbracketleft/angbracketleft/angbracketleft/hatwideψ− k+p,σ/hatwideψ+ k+p,σ/angbracketright/angbracketright/angbracketright] (56) withηc=+,ηs=− . As before, we now express the lattice correlation functions appearing in the lattice WI in termsof those of the emerging chiral model, using Eqs. ( 47) and (48); we therefore get twoidentities for the correlations of the emergent chiral model, one involving the Z /sharp μparameters, the other involving Z,v,τ . Therefore, we can use these identities to prove relations among these coefﬁcients; we get vZ/sharp 0 Z/sharp 1=1,Z/sharp 0 Z(1−η/sharpτ)=1. (57) Remarkably, Eq. ( 57) provides a link between the emergent chiral anomaly and the ﬁnite lattice renormalizations. We cannow use Eq. ( 57) to simplify the expressions in Eqs. ( 54). Setting K c=K,Ks=K−1, we get Z/sharp 0Z/sharp 1 Z2(1−τ2)v=K/sharp,Zc 0Zs 1 Z2(1−τ2)v=1, Z/sharp 1Z/sharp 1 Z2(1−τ2)v=K/sharpv,Z/sharp 0Z/sharp 0 Z2(1−τ2)v=K/sharp v.(58) The second relation implies the quantization of σs[forλ small, sgn( v) is independent of λ]. The last two imply the 245135-8SPIN HALL INSULATORS BEYOND THE HELICAL . . . PHYSICAL REVIEW B 96, 245135 (2017) nonuniversality of D/sharp,κ/sharp, and the helical Luttinger liquid relation D/sharp=v2κ/sharp. IX. CONCLUSIONS We have established the exact quantization of the edge spin conductance for the spin-conserving Kane-Mele-Hubbardmodel. As a corollary, our result provides an example ofbulk-edge correspondence for a nonsolvable, interacting time-reversal-invariant system. In addition, we proved a marginalform of universality for the susceptibilities and the Drudeweights, showing the validity of the helical Luttinger liquidscaling relations for the KMH model. Our strategy is basedon an exact RG construction of the lattice model, and on thecombination of lattice Ward identities, following from latticeconservation laws, with relativistic Ward identities, followingfrom the emergent chiral gauge symmetry of the system. Even though they break the integrability of the interacting system,lattice effects and bulk degrees of freedom play a crucial rolefor universality. As an open problem, it would be interesting to include spin- nonconserving terms in the Hamiltonian, and to quantify thepossible breaking of universality of the edge spin conductance. ACKNOWLEDGMENTS V .M. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation programme (ERC CoG UniCoSM,Grant Agreement No. 724939) and from the Gruppo Nazionaledi Fisica Matematica (GNFM). The work of M.P. has beenpartially supported by the NCCR SwissMAP, and by theSNF grant “Mathematical Aspects of Many-Body QuantumSystems.” [1] C. L. Kane and E. J. Mele, P h y s .R e v .L e t t . 95,226801 (2005 ). [2] C. L. Kane and E. J. Mele, P h y s .R e v .L e t t . 95,146802 (2005 ). [3] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96,106802 (2006 ). [4] C. Wu, B. A. Bernevig, and S.-C. Zhang, Phys. Rev. Lett. 96, 106401 (2006 ). [5] C. Xu and J. E. Moore, P h y s .R e v .B 73,045322 (2006 ). [6] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82,3045 (2010 ). [7] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83,1057 (2011 ). [8] M. Hohenadler and F. F. Assaad, J. Phys.: Condens. Matter 25, 143201 (2013 ). [9] M. König, S. 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PhysRevB.72.165301.pdf	Time-dependent simulations of electron transport through a quantum ring: Effect of the Lorentz force B. Szafran1,2and F. M. Peeters1 1Departement Fysica, Universiteit Antwerpen (Campus Drie Eiken), Universiteitsplein 1, B-2610 Antwerpen, Belgium 2Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, aleja Mickiewicza 30, 30-059 Kraków, Poland /H20849Received 11 March 2005; revised manuscript received 3 May 2005; published 3 October 2005 /H20850 The time-dependent Schrödinger equation for an electron passing through a semiconductor quantum ring of nonzero width is solved in the presence of a perpendicular homogeneous magnetic ﬁeld. We study the effectsof the Lorentz force on the Aharonov-Bohm oscillations. Within the range of incident momentum for which thering is transparent at zero magnetic ﬁeld, the Lorentz force leads to a decrease of the oscillation amplitude, dueto the asymmetry in the electron injection in the two arms of the ring. For structures in which the fast electronsare predominantly backscattered, the Lorentz force assists in the transport, producing an initial increase of thecorresponding oscillation amplitude. Furthermore, we discuss the effect of elastic scattering on a potentialcavity within one of the arms of the ring. For the cavity tuned to shift maximally the phase of the maximumof the wave packet we observe a /H9266shift of the Aharonov-Bohm oscillations. For other cavity depths oscilla- tions with a period of half of the ﬂux quantum are observed. DOI: 10.1103/PhysRevB.72.165301 PACS number /H20849s/H20850: 73.63.Kv I. INTRODUCTION The wave function of an electron passing along a path l acquires a phase shift1from the vector potential Agiven by /H9278=/H208492/H9266//H90210/H20850/H20848lA·dx/H20849/H90210=h/eis the ﬂux quantum /H20850. In a ring conﬁguration the Aharonov-Bohm /H20849AB/H20850effect produces a measurable interference1due to the relative phase shifts of the wave function going through the arms /H9004/H9278=2/H9266/H9021//H90210, where /H9021is the magnetic ﬁeld ﬂux through the area inside the ring. Oscillations of the electric properties with period /H90210 were detected in metal2and semiconductor rings.3–7 The theory8–10of the AB conductance oscillations was developed in a strictly one-dimensional model in which themagnetic ﬁeld is inaccessible for electrons and the only ef-fect of the vector potential is the AB phase shift. 8–11In fact the experiments2–7are performed in homogeneous magnetic ﬁelds and the leads have a ﬁnite width so the magnetic ﬁelddeclines the paths of the current ﬂow. Nevertheless, the scat-tering matrix theories 8,9assumed explicitly transport symme- try with respect to the arms of the ring. Comparable ampli-tudes for wave function in both arms of the ring were alsoassumed in the derivation 10of the multichannel AB conduc- tance formula. The theories of Refs. 8–10 addressed metalsfor which, as we discuss below, these assumptions are justi-ﬁed, since the Lorentz force has a negligible effect on trajec-tories of heavy-effective-mass electrons traveling with highFermi velocities. This is no longer true for semiconductorstructures. The purpose of the present paper is to describe the effect of the Lorentz-force-related deformation of the electron tra-jectories on the AB effect in a semiconductor quantum ring.The effect of the Lorentz force was discussed for biprismdiffraction experiments in vacuum. 12The envelope of the interference pattern for an electron traveling in the magneticﬁeld is shifted by the magnetic force according to the clas-sical laws 12following the Ehrenfest theorem. The electrontrajectories declined by the magnetic ﬁeld were also studied for the injection through a semiconductor quantum pointcontact. 13The effect of the magnetic ﬁeld on the electron trajectories in quantum rings was previously addressed in Ref. 14. However, the boundary conditions applied in thispaper 14are not best suited for the discussion of the Lorentz force effect since the wave function for the outgoing wave isnotassociated with the current ﬂowing out of the ring /H20849see Sec. IV /H20850and the “incident” electron is not necessarily mov- ing toward the ring. This is due to the current ﬂowing in theopposite directions at the edges of the leads in the eigenstatesat high magnetic ﬁelds. The problem of the changing orien-tation of the electron velocity across the leads is in thepresent paper neutralized by the time-dependent approach. Inthis approach the incoming lead is clearly deﬁned by theapplied initial condition for the localization of the wavepacket. The present calculations are performed in a basis ofGaussian functions with embedded gauge invariance. Weconsider the lowest subband transport and neglect inelasticscattering effects. We demonstrate that the Lorentz force pro-duces a preferential injection of the electron wave packet inone of the arms of the ring. The injection imbalance growsmonotonically with the external magnetic ﬁeld and eventu-ally leads to a suppression of the AB oscillations at highmagnetic ﬁelds. We ﬁnd that for high incident momenta theLorentz force can be necessary to guide the transport of elec-trons, which are otherwise, in the absence of the magneticﬁeld, backscattered to the incoming lead. When the transportwindow is opened for the fast electrons, they are directed toboth the arms of the ring and the corresponding AB oscilla-tions amplitude initially increases with the magnetic ﬁeld. The real rings are never ideally clean, and the transport is inﬂuenced by the elastic scattering. We study the scatteringeffects introducing a shallow potential cavity in one of thearms of the ring. The scattering phase shift ﬁlters the reso-nances of the transferred momenta. We ﬁnd that the AB os-PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 1098-0121/2005/72 /H2084916/H20850/165301 /H208498/H20850/$23.00 ©2005 The American Physical Society 165301-1cillation of the packet transfer probability is shifted by /H90210/2 only when the depth of the cavity is tuned such that it pro-duces a /H9266phase shift for the maximum of the wave packet in the momentum space. For other depths we ﬁnd the appear-ance of minima in the transition probability at integer mul-tiples of /H9021 0/2 that leads to a halving of the AB period, which was originally expected15for strongly disordered quantum rings and recently observed in GaAs quantum rings.6 II. THEORY We consider an electron conﬁned in the /H20849x,y/H20850plane with perpendicular magnetic ﬁeld /H208490,0, B/H20850. The Hamiltonian re- lated to the kinetic energy has the form H=/H208491/2m/H20850/H20849−i/H6036/H11633 +eA/H208502with the vector potential taken in the Landau gauge A=/H20849−By,0,0 /H20850andmstands for the electron effective mass /H20849we take the GaAs value m=0.067 m0/H20850. We expand the wave function in a basis of Gaussian functions16centered around chosen points Rn=/H20849Xn,Yn/H20850 /H9023/H20849x,y,t/H20850=/H20858 ncn/H20849t/H20850fn/H20849x,y/H20850, /H208491/H20850 withfn/H20849x,y/H20850= exp /H20851−/H20849r−Rn/H208502/2/H92612 +ieB/H20849x−Xn/H20850/H20849y+Yn/H20850/2/H6036/H20852//H9261/H20881/H9266. /H208492/H20850 The shape of the considered structures is deﬁned by the adopted position of centers /H20849Rn/H20850in functions /H208492/H20850. The centers are chosen along the lines drawn in Fig. 1 with spacings of 20 nm. The parameter /H9261in functions /H208492/H20850is set to 19.8 nm. That choice determines the width of the waveguides /H208492/H9261/H20850and is equivalent to deﬁning a harmonic oscillator conﬁnement potential with the oscillator energy /H6036/H9275=2.9 meV in the di- rection perpendicular to the waveguide. In the studied mag-netic ﬁeld range the increase of the electron localization inthe wires with the magnetic ﬁeld is negligible. 17The imagi- nary part of the exponent is related to the magnetic transla-tion and ensures equivalence of all the points R n, i.e., the gauge invariance. Substituting expansion /H208491/H20850into the time- dependent Schrödinger equation we obtain a system of linearequations for the time derivative of the coefﬁcients c n/H20849t/H20850, Sc˙/H20849t/H20850=Hc/H20849t/H20850/i/H6036, /H208493/H20850 where the elements of the overlap and Hamiltonian matrices are given by Skn=/H20855fk/H20841fn/H20856andHkn=/H20855fk/H20841H/H20841fn/H20856, respectively. However, the scheme based directly on Eq. /H208493/H20850increases the amplitude of the wave function with each time step. A morestable and norm-conserving solution is provided by theAskar and Cakmak 18scheme producing a system of linear equations Sc/H20849t+dt/H20850=Sc/H20849t−dt/H20850−2idtHc/H20849t/H20850//H6036. Equation /H208493/H20850is used only for evaluation of the ﬁrst time step. We consider circular and diamond rings enclosing an area of 1322/H9266nm2/H20849see Fig. 1 /H20850, for which a single ﬂux quantum corresponds to B=75.57 mT. For the incident wave packet we take one of the Gaussians /H208492/H20850, namely, the one localized at the wire at a position yi, 200 nm before the entrance of the ring multiplied by a plane wave, i.e., /H9023/H20849x,y,0/H20850=fi/H20849x,y/H20850exp /H20849iqy/H20850. /H208494/H20850 This product is projected onto the basis /H208491/H20850and the projec- tion is used as the initial condition for the simulations. Thecorresponding probability density in wave vector space, cal-culated as a Fourier transform along the axis of the incoming lead /H20849x=0/H20850, is given by P/H20849k/H20850= /H20881/H9266/H9268exp /H20851−/H20849k−q/H208502//H92682/H20852with /H9268=0.0505/nm. The ﬂux of the ycomponent of the probabil- ity density current FIG. 1. Geometry and dimensions of the circular /H20849a/H20850and dia- mond /H20849b/H20850quantum rings studied in the present paper. In the calcu- lations the positions of the centers of the Gaussians /H208492/H20850are chosen along the drawn lines with a spacing of 20 nm. FIG. 2. /H20849Color online /H20850Charge /H20849contours /H20850and current /H20849vectors /H20850densities for a Gaussian wave packet with kinetic energy /H60362q2/2m =1.42 meV being transferred through a quantum ring of radius 132 nm for zero magnetic ﬁeld. /H20849a/H20850–/H20849d/H20850correspond to t=2.17, 4.35, 6.35, and 8.8 ps. Scale for the charge and current density is the same in all the plots, with the exception of the current density vectors in /H20849a/H20850which were shortened by a factor of 1/2 with respect to the other plots.B. SZAFRAN AND F. M. PEETERS PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-2j=i/H6036 2m/H20849/H9023/H11633/H9023−/H9023/H11633/H9023/H20850+e mA/H9023/H9023/H208495/H20850 integrated over the two-dimensional space equals /H6036q/m, which gives the same initial condition for all B. The central part of the packet in the wires moves in real space accordingto the Ehrenfest theorem as d/H20855r/H20856/dt=/H20855p+eA/H20856/m ,s oi nt h e applied gauge, for the leads oriented along the ydirection, the vector potential has no inﬂuence on the movement of thecenter of the wave packet. The Ehrenfest theorem for thechange of the average momentum in time, d/H20855p+eA/H20856 dt=−e m/H20855p+eA/H20856/H11003B, /H208496/H20850 gives for the ycomponent the expression d/H20855py/H20856 dt=d/H20855−i/H6036/H11509//H11509y/H20856 dt=−eB m/H20883i/H6036/H11509 /H11509x+eyB/H20884. /H208497/H20850 The matrix elements of the operator /H20855fm/H20841i/H6036/H11509//H11509x+eyB /H20841fn/H20856are zero for Xm=Xn/H20851see Eq. /H208492/H20850/H20852, so in the leads /H20849oriented par- allel to the yaxis /H20850the average value of momentum is con- served. In other words, the magnetic ﬁeld cannot deﬂect themomentum of the electron packet moving in the leads de-ﬁned as a sequence of Gaussian basis functions centeredalong the same axis. In that sense the leads in our model areeffectively one-dimensional. Deﬂection is only possible atthe junctions of the leads and the ring. Finally, we have veri-ﬁed using Fourier transform analysis that not only /H20855p y/H20856and /H20855py2/H20856but the entire momentum distribution remains un- changed in time when the wave packet travels through the leads. Summarizing, in our model the magnetic forces arenot active in the leads, the momentum of the packet is con-served, although for B/HS110050 the momentum operator does not commute with the Hamiltonian.The numerical results presented in this paper were ob- tained for an average value of the momentum q=0.05/nm, which corresponds to an average kinetic energy /H6036 2q2/2m* =1.42 meV. The average kinetic energy is equal to the Fermi energy /H20849EF=/H6036/H9266n/m/H20850of the two dimensional electron gas at the carrier concentration n=0.4/H110031011/cm2. At the higher en- ergy end of the packet, i.e., for k=q+/H9268, the kinetic energy equals 5.36 meV. III. RESULTS Figures 2–4 show snapshots of the time evolution of the wave packet in a circular quantum ring /H20851see Fig. 1 /H20849a/H20850/H20852for 0, 0.5, and 4.5 ﬂux quanta. The contour plots show the chargedensities and the arrows display the probability density vec-tors /H208495/H20850. Plots for t=2.17 ps /H20851in parts /H20849a/H20850of Figs. 2–4 /H20852corre- spond to the moment just before the maximum of the chargedensity packet enters the ring. A larger part of the wavepacket is scattered back into the injection lead. The plot for/H9021=0 /H20851Fig. 2 /H20849a/H20850/H20852shows an equal spreading of the wave packet into both arms of the ring. At t=4.4 and 6.5 ps /H20851Figs. 2/H20849b/H20850and 2 /H20849c/H20850/H20852we observe the formation of a maximum at the exit region of the ring where left and right circulating partsof the packet meet. For /H9021=/H9021 0/2/H20849Fig. 3 /H20850the parts of the packet transferred through the left and right arms interferedestructively /H20851Figs. 3 /H20849b/H20850–3/H20849d/H20850/H20852, leading to a zero charge den- sity at the upper exit of the ring. Consequently, /H20849almost /H20850no charge is transferred out of the ring at this exit. The injectionasymmetry due to the Lorentz force directing the wavepacket to the left arm, visible already in Fig. 3 /H20849a/H20850, increases with the magnetic ﬁeld /H20849see Fig. 4 for 4.5 /H9021 0/H20850. We observe also that for 4.5 /H90210the wave packet reaches further into the arm as compared to the effect at lower magnetic ﬁelds. TheLorentz force helps the higher-momentum parts of the packetenter into the ring instead of being reﬂected. In comparisonto the case of 0.5 /H9021 0/H20849Fig. 3 /H20850, we see that due to the injection FIG. 3. /H20849Color online /H20850Same as Fig. 2 but for B=0.0378 T, which corresponds to the ﬂux of themagnetic ﬁeld through the ring/H9021=h/2e=/H9021 0/2. /H20849d/H20850corresponds tot=13.06 ps /H20849the others to t’s as in Fig. 2 /H20850. FIG. 4. /H20849Color online /H20850Same as Fig. 2 but for B=0.34 T, i.e., /H9021 =4.5/H90210.TIME-DEPENDENT SIMULATIONS OF ELECTRON … PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-3imbalance the destructive interference at the upper exit is not complete. The force also guides the packet, which travelsthroughout the left arm and exits the ring /H20851Figs. 4 /H20849c/H20850and 4/H20849d/H20850/H20852. The transmission probability of the wave packet through the circular quantum ring is shown by the solid line in Fig. 5.This quantity was obtained by integrating the probabilitydensity leaving the ring through the upper lead. In contrast tothe strictly one-dimensional model with the assumption ofequal amplitudes of wave functions entering both arms of thering as given by the Büttiker single-channel formula, 9we ﬁnd /H208491/H20850that the amplitude of the oscillations decreases with magnetic ﬁeld, and /H208492/H20850that for half-integer ﬂuxes the value of the transmission probability is no longer zero. The de-creasing amplitude is due to the growing imbalance in theamount of charge transferred through the left and right armsof the ring, which prevents the interference from being com-pletely destructive. The values of the transmission probabil-ity maxima and minima are increasing functions of the mag-netic ﬁeld, which is a consequence of the guiding behavior ofthe Lorentz force that eases the entrance and exit of the wavepacket. The envelope of the maxima is well approximated bythe packet transfer probability through a semicircular wirethat is obtained when the right arm of the circular ring isremoved, plotted with the dashed line in Fig. 5. One couldexpect that the probability of transfer through the semicircu-lar wire /H20851T/H20849B/H20850/H20852will be larger for B/H110220, since then the Lor- entz force tends to deﬂect the trajectories to the left. In fact, the transfer probabilities are independent of the magneticﬁeld orientation /H20851T/H20849B/H20850=T/H20849−B/H20850/H20852. This is a signature of the microreversibility relation for a two-terminal device. 19–21 The time dependence of the charge accumulated in the semi- circular part of the wire, as well as the probabilities of ﬁnd-ing the electron below and above the bend, are plotted in Fig.6 for/H9021=10/H9021 0. The probability density below the ring is at any time exactly the same for both the wires. The transmis-sion probability tends for t→/H11009to the same value for bothwires, but the transport for B/H110210 is delayed with respect to theB/H110220 case. For B/H110220 the Lorentz force directly injects the electron into the bend and then ejects it to the outgoinglead. On the other hand, for B/H110210 both the magnetic-ﬁeld- assisted injection into and the ejection out of the wire bendare realized after the electron velocity changes its sign inreﬂection from the junctions at which the waveguide turns atthe 90° angle, hence the time delay. More information on the nature of the transport is ob- tained from the momentum distribution of the transmittedwave packet, calculated numerically as the square of the ab-solute value of the space Fourier transform of the wave func-tion transmitted through the ring calculated along the axis ofthe output lead /H20849x=0/H20850. The black solid line in Fig. 7 shows the momentum distribution of the transmitted packet for zero magnetic ﬁeld /H20849the momentum distribution of the incident packet is plotted by the dashed curve /H20850. The origin of the pronounced peaks can be understood from the transmissionmechanism illustrated in Fig. 2. The momenta that have thehighest probability to be transferred from the injection to thecollection lead correspond to standing waves with maxima of FIG. 5. The transmission probability of the wave packet through the circular /H20849solid line /H20850, and diamond /H20849dotted line /H20850quantum rings as function of the ﬂux passing through the ring in units of the ﬂuxquantum. The dashed line shows the transmission probabilitythrough a wire of semicircular shape obtained from the circularquantum ring after removal of one of its arms. FIG. 6. Probability of ﬁnding the electron inside the semicircu- lar wire above and below it as function of time for /H9021=10/H90210. The solid /H20849dashed /H20850lines show the results for B/H110210/H20849B/H110220/H20850. The reﬂected probability density for the B/H110210 is marked by the dots. FIG. 7. /H20849Color online /H20850Probability density of the transferred wave packet through the circular ring in wave vector space. Linescorresponding to 0, 1, 2.5, 7, and 7.5 ﬂux quanta passing throughthe ring are labeled. The dashed line shows the shape of the initialwave packet. The arrows show the wave vector values equal to n/R.B. SZAFRAN AND F. M. PEETERS PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-4charge density at the entrance and the exit of the ring. This is realized when the phase shift k/H9266Ralong each of the arms is equal to an integer /H20849n/H20850multiple of /H9266leading to the resonant condition k=n/R/H20849values marked by arrows in the top of the ﬁgure /H20850. Position of the momentum peaks for higher values of kagrees well with these values. For lower momenta the spac- ing between the peaks increases, as if the resonant length forthe slower parts of the packet was shorter. The transferredmomentum spectrum for one ﬂux quantum /H20849see Fig. 7 /H20850is very similar to the one for zero magnetic ﬁeld, and the posi-tion of the peaks is unaltered. For halves of the ﬂux quanta atlower magnetic ﬁelds /H20849see the plot for 2.5 /H9021 0/H20850the nonzero value for the transferred spectra is uniquely due to the injec-tion imbalance. The spectrum possesses characteristicdouble-peak structure in between the maxima for integer ﬂuxquanta. For /H9021=0 we also notice a clear asymmetry in the transferred momentum with respect to its original distribu-tion. The parts of the wave packet that travel faster have lesstime to enter into the arms of the ring before they get re-ﬂected back at the entrance to the ring into the incominglead. The momentum distribution at 7 /H9021 0/H20849blue curve /H20850differs with respect to the /H9021=0 and /H90210distributions in two points: /H208491/H20850the gaps between the peaks are ﬁlled, and /H208492/H20850a visibly larger probability of transfer of the fast parts of the packet.Consequently, the spectrum approaches more closely the ini-tial momentum distribution. The Lorentz-force guided trans-port does not require formation of standing waves withmaxima at the ring-leads junctions, which is the reason whythe resonant relation no longer holds. At high ﬁeld for frac-tional ﬂux quanta /H20849see the plot for 7.5 /H9021 0/H20850the minima in the transferred spectrum are shifted toward distinctly nonzerovalues. Figure 8 shows the transfer probability as a function of the magnetic ﬁeld for ﬁxed values of the wave vector. Fork=0.053/nm and 0.06/nm the magnetic ﬁeld leads to a de- crease of the oscillation amplitude, as in the momentum-averaged packet transfer probability /H20849see Fig. 5 /H20850. The growth of the envelope of transfer probability maxima seen in Fig. 5is due to the Lorentz-force guided transport for high incidentmomenta. For k=0.072/nm the transfer probability growswith decreasing AB oscillation amplitude. On the other hand, already for k=0.091/nm the amplitude increases with B. This can be understood on the basis of the properties of thesemicircular wire discussed above. The electron can be in-jected by the Lorentz force into the left arm of the ring di-rectly from the incoming lead or to the right arm after itsvelocity changes sign at the reﬂection at the junction. Forhigh momenta the Lorentz force ﬁrst allows transportthrough both the arms of the ring. It should be expected thatat higher Bthe injection imbalance will appear and eventu- ally destroy the AB oscillations. Note that the peaks of thetransfer probability for k=0.091/nm are spaced by only around 90% of the nominal AB period, which indicates thatthe effective radius of the ring is larger for fast electrons.This classical feature was already noticed in the enlargedspacings between the resonant peaks at the low- kpart of the spectrum for B=0 in Fig. 7. As compared to the circular ring, in the diamond geom- etry the incoming packet enters the arms of the ring moreeasily and leaves also more easily the ring to the upper lead.Consequently, at B=0 the transmission probability is more than twice larger than for the circular ring /H20849see the dotted line in Fig. 5 /H20850. No pronounced resonance pattern similar to the one obtained for the circular ring /H20849compare Figs. 7 and 9 /H20850is observed. The transmitted momentum spectrum exhibits anasymmetric shift towards higher momenta with respect to theinitial momentum distribution which is opposite to the circu-lar ring case. The electrons with higher momenta are nowmore easily transferred through the diamond ring simply bythe inertia and not by the Lorentz force. That explains whyfor the diamond ring the envelope of the maxima of thepacket transfer probability /H20849Fig. 5 /H20850does not exhibit the growth with Bas in the circular ring case. Since the angle at which the trajectory has to be deﬂected is smaller than forthe discussed circular ring geometry, the minima in the os-cillations increase much faster with the Bﬁeld as compared to the circular ring /H20849see Fig. 5 /H20850. For/H9021/H110225/H9021 0the AB oscil- lations in the diamond ring are no longer observed. Next, we study the effect of elastic scattering on a shallow Gaussian potential cavity placed in the center of the left armof the circular ring. To determine the scattering properties ofthe cavity we solved ﬁrst the strictly one-dimensional prob-lem of transmission through a Gaussian quantum well FIG. 8. /H20849Color online /H20850Transfer probability for ﬁxed values of the incident wave vector as a function of the magnetic ﬁeld ﬂux forthe circular ring. The plots for k=0.072, 0.06, and 0.053 per nm were shifted by 0.5, 1, and 1.5, respectively. FIG. 9. Probability density of the packet transferred through the diamond ring in wave vector space for B=0. The dashed line shows the shape of the initial wave packet.TIME-DEPENDENT SIMULATIONS OF ELECTRON … PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-5/H20851−V0exp /H20849−x2/2/H92612/H20850/H20852. Figure 10 presents the transmission probability and the phase shift as functions of the wave vec- tor. The cavity is transparent for wave vectors larger than0.025/nm, only the phase is changed with respect to the V 0 =0 /H20849no cavity /H20850case. The phase shift for k=0.05/nm /H20849maxi- mum of the probability density used in the time-dependentsimulations /H20850is close to /H9266/4,/H9266/2, and /H9266forV0=1, 2.75, and 5.5 meV, respectively. In the time-dependent simulations for the ring structure we introduce a potential cavity described by the potential V/H20849x,y/H20850=−V0exp /H20853−/H20851/H20849x−Xl/H208502+/H20849y−Yl/H208502/H20852/2/H92612/H20854, /H208498/H20850 where /H20849Xl,Yl/H20850is situated in the middle of the left arm of the ring. The transmission probability of the wave packet is plot- ted as a function of V0in Fig. 11 /H20849a/H20850for different values of the ﬂux. For /H9021=0 the transmission probability has a minimum when the phase shift for the maximum of the wave packet inmomentum space is equal to /H9266, which is achieved at V0 =5.5 meV /H20849cf. Fig. 10 for q=0.05/nm /H20850. Figure 11 /H20849b/H20850shows the comparison of the momentum distribution of the trans-ferred part of the packet for V 0=0 and 5.5 meV at /H9021=0. The phase shift acquired in the dot and the destructive interfer-ence at the exit remove the central part of the wave packet inthe momentum space. For /H9021=/H9021 0/2 the cavity has the oppo- site effect on the transmission probability since it compen- sates for the /H9266shift produced by the AB effect. As a conse- quence the probability increases /H20851see dashed lines in Fig. 11/H20849a/H20850/H20852when V0is increased from 0, and is maximal for V0 =5.5 meV where the compensation of the AB phase shift is obtained. The central part of the transferred momentumspectrum at /H9021=0.5/H9021 0forV0=5.5 meV /H20851see Fig. 11 /H20849c/H20850/H20852is similar to V0=0 in the absence of the magnetic ﬁeld /H20851com- pare black line in Fig. 11 /H20849b/H20850/H20852. Note, that the transmission probability plotted in Fig. 11 /H20849a/H20850is not a smooth function of V0. The changing slope of the curve is due to switching on and off the resonances in the transmitted momentum spec-trum. The magnetic ﬁeld dependence of the packet transfer probability for V 0=5.5 meV is plotted in Fig. 12 by the low- est curve. The transfer probability possess maxima at themagnetic ﬁelds corresponding to halves of the ﬂux quanta, for which the Aharonov-Bohm effect compensates for thescattering /H9266shift. The transfer probability for V0=4 meV is plotted by the second curve from below in Fig. 12. By anal-ogy to the results for V 0=0 and V0=5.5 meV one could ex- pect a similar behavior with minima spaced by /H90210but shifted on the ﬂux scale. However, this would violate the even sym-metry of the two-terminal device properties as a function ofthe external ﬁeld. The transfer probability are subject to thephase-locking 21of the AB oscillations resulting in an extre- mum always present at B=0. For V0=4 meV the transmis- sion probability develops shallow minima at odd multiples of/H9021 0/2. The probability amplitudes for the paths passing through the left and right arms do not meet exactly in phaseat the upper exit from the wire, but on the other hand, therotating left and right parts of the packet meet exactly inphase at the entrance. The depth of the probability minimafor both the odd and even multiples of /H9021 0/2 decrease with increasing ﬂux. At V0=2.75 meV the phase shift for the maximum of the wave packet /H20849q=0.05/nm /H20850is about /H9266/2 /H20849see Fig. 10 /H20850. The minima at the even and odd multiples of FIG. 10. /H20849Color online /H20850Transmission probability /H20849solid lines /H20850 and the phase shift /H20849dotted lines /H20850as functions of the wave vector for a strictly one-dimensional Gaussian potential cavity of width 28 nmand different depths V 0. FIG. 11. /H20849Color online /H20850/H20849a/H20850Transmission probability of the wave packet through a circular ring with a Gaussian quantum well /H20851Eq. /H208498/H20850/H20852in the left arm as a function of the depth of the well V0. Values for ﬂuxes equal to integer ﬂux quanta are plotted with solid linesand for half ﬂux quanta with dashed lines. /H20849b/H20850The transferred mo- mentum distribution for V 0=0 and 5.5 meV at /H9021=0. /H20849c/H20850Same as /H20849b/H20850but now for /H9021=0.5/H90210.B. SZAFRAN AND F. M. PEETERS PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-6/H90210/2 acquire similar depth. As a consequence the transmis- sion probability exhibits oscillations with an effective quasi-period of half of the ﬂux quantum. IV. DISCUSSION The measured conductance of semiconductor rings3,4,6,7,22 deviates from the strict periodicity predicted by the one-dimensional models. 8–10,19–21We indicate that the Lorentz force can be responsible for these deviations. The injectionimbalance leads to a decrease of the AB oscillations ampli-tude with the magnetic ﬁeld as obtained in the magnetocon-ductance measurements 6performed on GaAs/AlGaAs quan- tum rings /H20849see Fig. 1 of Ref. 6 /H20850. The amplitude decreasing with magnetic ﬁeld was also observed in GaAs/AlGaAs na-norings formed by an atomic force microscope tip 4/H20851see the ring current plot in Fig. 2 /H20849c/H20850of Ref. 4 /H20852and in the AB inter- ferometer /H20851see Fig. 3 /H20849a/H20850of Ref. 22 /H20852. A suppression of the periodic magnetoresistance oscillations was also reported inRef. 3. Lorentz force can also be responsible for an increaseof the oscillations’ amplitude /H20849observed for instance in Ref. 7/H20850since in some geometries it opens the transport window for fast electrons. In order to be of signiﬁcant importance the Lorentz force has to deﬂect the electron trajectories at the entrance and atthe exit leads of the ring. The classical formula for the radiusof an electron orbit in a magnetic ﬁeld R=m V/eB /H20849for the average q=0.05/nm taken in our calculations R=32.9 nm T/ B/H20850indicates that the effect will be smaller for fast electrons since the radius will not ﬁt into the width of thejunction. This feature was actually conﬁrmed in the contextof Figs. 7 and 8. In metals the Fermi energies are of order eV ,which compares to meV in a two-dimensional electron gas inGaAs. Consequently, in Au rings 2in which both the electron effective mass /H20849/H11011m0/H20850and the Fermi velocity /H208491.4/H11003106m/s /H20850are about 15 times larger than the effective mass and the velocity /H20849/H6036q/m=0.086 /H11003106m/s for q=0.05/nm /H20850con- sidered in the present paper, the AB oscillations pertain up to8 T covering as much as 10 4ﬂux quanta /H90210. Recently, AB oscillations in a semiconductor quantum ring5pertaining to large ﬂuxes were observed in a device with electrostatic tun-nel barriers in both arms of the ring. In the experiment 5the barriers were tuned to have equal transmission, which cancompensate for the Lorentz force effect described in thepresent paper. The imbalance of the current through the arms of the ring as due to the magnetic ﬁeld was previously 14found in a time-independent simulation. Our results contradict the pro-posed mechanism 14of the direction of the current injection changing from the left to the right arm of the ring periodi-cally with the magnetic ﬁeld. The injection imbalance is dueto the Lorentz force; hence it is monotonic in B. At high ﬁeld for single-channel transport the authors 14obtain the trans- mission probability as a bivalue function equal to zero forodd multiples of /H9021 0/2 and 1 for other ﬂuxes /H20849see Fig. 13 of Ref. 14 for /H9280=20 /H20850. According to our calculations there is no physical reason that would lead to a strict vanishing of thesingle-channel conductance at high magnetic ﬁeld penetrat-ing the arms of the two-dimensional quantum ring. TheHamiltonian eigenstates in the leads /H20849oriented along xaxes /H20850 were used 14as boundary conditions for the incoming and outgoing waves. The eigenstates were calculated as gk/H20849x,y/H20850 =exp /H20849ikx/H20850fk/H20849y/H20850, where fkis the eigenfunction of the one- dimensional Hamiltonian H=−/H20849/H60362/2m/H20850d2/dy2+/H208491/2m/H20850 /H11003/H20849/H6036k−m/H9275cy/H208502with zero boundary conditions on the wall, i.e., for y=±d/2. The probability density current in the x direction for eigenstate gkis given by jx/H20849y/H20850=/H20841f/H20849y/H20850/H208412/H20849/H6036k/m −/H9275cy/H20850. At high magnetic ﬁeld14/H20849when/H9275c/H11271/H6036k/m/H20850the cur- rent jx/H20849y/H20850has the opposite orientation at the top /H20849y/H110220/H20850and bottom /H20849y/H110210/H20850edges of both the injection and the extraction lead.23Moreover, the conductance Landauer formula /H20851Eq. /H2084916/H20850of Ref. 14 /H20852assumes that the current is proportional to the wave vector. This assumption is only correct at B=0. V. CONCLUSION AND SUMMARY We have solved the time-dependent Schrödinger equation for a Gaussian electron wave packet passing through a quan-tum ring in the presence of a homogeneous external mag-netic ﬁeld. In contrast to previous strictly one-dimensionaltheories our results indicate that the Aharonov-Bohm oscil-lations for semiconductor quantum rings disappear in thehigh-magnetic-ﬁeld limit due to the Lorentz force action onthe moving electron. In circular quantum rings the magneticﬁeld changes the mechanism of transport, increasing thestrength of the coupling of the ring to the leads with increas-ing magnetic ﬁeld. The momentum resonances in the tunnel-ing at low magnetic ﬁeld resemble the formation of a quasi-bound temporary state localized in the ring. At high magneticﬁeld the tunneling becomes guided by the Lorentz force. Forrings, in which in the absence of the magnetic ﬁeld the ge-ometry blocks the transport of fast electrons, the Lor- FIG. 12. /H20849Color online /H20850Probability of transmission of the wave packet through a circular ring with a Gaussian quantum well /H20851Eq. /H208498/H20850/H20852in the left arm as a function of the magnetic ﬁeld ﬂux for well depths: V0=5.5, 4, 2.75, 1 and 0 meV. Lines for V0=4, 2.75, 1, and 0 meV have been shifted for clarity by +0.12, +0.24, +0.36, and+0.48, respectively.TIME-DEPENDENT SIMULATIONS OF ELECTRON … PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-7entz force helps them to pass through. This can initially in- crease the amplitude of oscillations before the interference iseventually suppressed. The presence of an elastic scatterer inone of the ring arms leads to a /H9266shift of the oscillations of the wave packet transmission probability or to halving of theperiod of its Aharonov-Bohm oscillations.ACKNOWLEDGMENTS We are grateful to T. Ihn and K. Ensslin for helpful dis- cussions. This research was supported by the Flemish Sci- ence Foundation /H20849FWO-Vl /H20850and the Belgian Science Policy. B.S. was supported by the EC Marie Curie IEF Project No.MEIF-CT-2004-500157. 1Y . Aharonov and D. Bohm, Phys. Rev. 115, 485 /H208491959 /H20850. 2R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, Phys. Rev. Lett. 54, 2696 /H208491985 /H20850. 3G. Timp, A. M. Chang, J. E. Cunningham, T. Y . Chang, P. Man- kiewich, R. Behringer, and R. E. Howard, Phys. Rev. Lett. 58, 2814 /H208491987 /H20850. 4A. Fuhrer, S. Lüscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, Nature /H20849London /H20850413, 822 /H208492001 /H20850. 5W. G. van der Wiel, Yu. V . Nazarov, S. De Franceschi, T. Fujisawa, J. M. Elzerman, E. W. G. M. Huizeling, S. Tarucha,and L. P. Kouwenhoven, Phys. Rev. B 67, 033307 /H208492003 /H20850. 6S. Pedersen, A. E. Hansen, A. Kristensen, C. B. Sorensen, and P. E. Lindelof, Phys. Rev. B 61, 5457 /H208492000 /H20850. 7U. F. Keyser, C. Fühner, S. Borck, R. J. Haug, M. Bichler, G. Abstreiter, and W. Wegscheider, Phys. Rev. Lett. 90, 196601 /H208492003 /H20850. 8Y . Gefen, Y . Imry, and M. Y . Azbel, Phys. Rev. Lett. 52, 129 /H208491984 /H20850. 9M. Büttiker, Y . Imry, and M. Y . Azbel, Phys. Rev. A 30, 1982 /H208491984 /H20850. 10M. Büttiker, Y . Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 /H208491985 /H20850. 11S. Viefers, P. Koskinen, P. Sing‘a Deo, and M. Manninen, Physica E/H20849Amsterdam /H2085021,1/H208492004 /H20850.12S. Olariu and I. I. Popescu, Rev. Mod. Phys. 57, 339 /H208491985 /H20850, and references therein. 13T. Usuki, M. Takatsu, R. A. Kiehl, and N. Yokoyama, Phys. Rev. B50, 7615 /H208491994 /H20850. 14K. N. Pichugin and A. F. Sadreev, Phys. Rev. B 56, 9662 /H208491997 /H20850. 15B. L. Al’tshuler, A. G. Aronov, B. Z. Spivak, D. Yu. Sharvin, and Yu. V . Sharvin, JETP Lett. 35, 588 /H208491982 /H20850. 16B. Szafran, F. M. Peeters, S. Bednarek, and J. Adamowski, Phys. Rev. B 69, 125344 /H208492004 /H20850. 17Actually, the oscillator length /H20849half of the waveguide width /H20850is given by l=/H20881/H6036/m/H9275e, where /H9275e2=/H92752+/H9275c2/4, where /H9275cis the cyclotron frequency. For B=0, 0.5, and 0.75 T, l=19.8, 19.7, and 19.6 nm, respectively. 18A. Askar and A. C. Cakmak, J. Chem. Phys. 68, 2794 /H208491978 /H20850. 19M. Büttiker, Phys. Rev. Lett. 57, 1761 /H208491986 /H20850. 20M. Büttiker, Phys. Rev. B 38, 9375 /H208491988 /H20850. 21A. L. Yeyati and M. Büttiker, Phys. Rev. B 52, R14360 /H208491995 /H20850. 22A. Yacoby, M. Heiblum, D. Mahalu, and H. Shtrikman, Phys. Rev. Lett. 74, 4047 /H208491995 /H20850. 23For the magnetic length shorter than the channel width the lead eigenstates can be identiﬁed with the lowest Landau level. Forthe lack of correspondence between the wave vector quantumnumber kand the current in the lowest Landau level; see, e.g., S. Datta, Electronic Transport in Mesoscopic Systems /H20849Cambridge University Press, Cambridge, England, 1995 /H20850.B. SZAFRAN AND F. M. PEETERS PHYSICAL REVIEW B 72, 165301 /H208492005 /H20850 165301-8
PhysRevB.103.075418.pdf	PHYSICAL REVIEW B 103, 075418 (2021) Effect of magnetic ﬁeld and chemical potential on the RKKY interaction in the α-T3lattice Oleksiy Roslyak,1Godfrey Gumbs ,2Antonios Balassis ,1and Heba Elsayed1 1Department of Physics and Engineering Physics, Fordham University, 441 East Fordham Road, Bronx, New York 10458, USA 2Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA (Received 4 July 2020; revised 16 January 2021; accepted 19 January 2021; published 12 February 2021) The interaction energy of the indirect-exchange or Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic spins localized on lattice sites of the α-T3model is calculated using linear response theory. In this model, the AB-honeycomb lattice structure is supplemented with C atoms at the centers of the hexagonallattice. This introduces a parameter αfor the ratio of the hopping integral from hub to rim and that around the rim of the hexagonal lattice. A valley and α-dependent retarded Green’s function matrix is used to form the susceptibility. Analytic and numerical results are obtained for undoped α-T 3when the chemical potential is ﬁnite and also in the presence of an applied magnetic ﬁeld. We demonstrate the anisotropy of these results when themagnetic impurities are placed on the A, B, and C sublattice sites. Additionally, a comparison of the behavior ofthe susceptibility of α-T 3with graphene shows that there is a phase transition at α=0. DOI: 10.1103/PhysRevB.103.075418 I. INTRODUCTION An effective single-particle model Hamiltonian represent- ing an electronic crystal was recently constructed to representthe low-lying Bloch band of the α-T 3lattice (for a review of artiﬁcial ﬂat-band systems, see Ref. [ 1]). The electronic properties of this material have come under growing scrutinyfor a number of important reasons which are fundamental andtechnological [ 2–22]. The potential tunability of these materi- als, ranging from their optical and transport properties to theirresponse to a uniform magnetic ﬁeld and varying chemical po-tential, presents researchers with the opportunity to investigatenew materials. Regarding their fabrication, it was suggested in[2] that an α-T 3lattice may be constructed with the use of cold fermionic atoms conﬁned to an optical lattice with the help ofthree pairs of laser beams for the optical dice ( α=1) lattice [23]. Jo et al. [9] successfully fabricated a two-dimensional kagome lattice consisting of ultracold atoms by superimpos-ing a triangular optical lattice on another one commensuratewith it and generated by light at speciﬁed wavelengths. Theα-T 3and kagome lattices are related in that they both have ﬂat bands as well as Dirac cones at low energies. In modelingthis structure, an AB-honeycomb lattice like that in grapheneis combined with C atoms at the centers of the hexagonallattice as depicted in Fig. 1. Consequently, a parameter α is introduced to represent the ratio of the hopping integralbetween the hub and the rim ( αt) to that around the rim ( t) of the hexagonal lattice. When one of the three pairs of laserbeams is dephased, it is proposed in [ 23] that this could allow the possible variation of the hopping parameter over the range0<α/lessorequalslant1. Interestingly, it would be informative to explore how the optical and transport properties of α-T 3systems are affected by defects. These include substituting impurities or guestatoms in a hexagonal lattice with fermionic host atoms. Inthis way, one could effectively manipulate the fundamental properties which are inherent in the α-T 3system. The guest atoms could be added to their hosts by chemical vapor de-position or discharge experiments. With doping, the A andB sublattices are no longer equivalent since the πbonding on these lattices may be seriously distorted, which causessigniﬁcant modiﬁcation of the physical properties, includingthe energy band structure with a deviation from the originalDirac cone and ﬂat band. However, at low doping ( <1.5%), the low-energy portion of the band structure is only slightlyaffected. We emphasize that the doping conﬁguration andconcentration in general create unusual band structures withfeature-rich and unique properties. Oriekhov and Gusynin [ 15] took the ﬁrst step of investigat- ing the role played by the sea of background α-T 3fermions on the indirect exchange interaction between a pair of spins local- ized on lattice sites. Local moments like these may occur near extended defects. The doping giving rise to the presence ofthese spins was assumed to have such a low concentration thatthe energy dispersion and the zero band gap remain unaltered.Speciﬁcally, these authors [ 15] were interested in this effect of doping and temperature on the Ruderman-Kittel-Kasuya- Yosida (RKKY) or indirect-exchange coupling as discussedfor different types of two-dimensional materials by others[24–28] between spins via the host conduction electrons of freestanding monolayer graphene [ 29–39] and biased single- layer silicene [ 40]. In this paper, we continue the investigation in [15] by calculating the effect of a uniform magnetic ﬁeld and a variable chemical potential on the RKKY interaction ofα-T 3. It is worth getting a better understanding of the behavior of this topic since one could exploit the RKKY interaction to determine spin ordering as excitations near the Fermi levelare, in part, governed by the indirect exchange interactionbetween local magnetic moments [ 41–43]. 2469-9950/2021/103(7)/075418(10) 075418-1 ©2021 American Physical SocietyROSLYAK, GUMBS, BALASSIS, AND ELSAYED PHYSICAL REVIEW B 103, 075418 (2021) x yrlA B C FIG. 1. Lattice sites of the α-T3model. The “rim” atoms are labeled A and B, whereas C is a “hub” atom. The rest of this paper is organized as follows. In Sec. II,w e present the low-energy α-T3model Hamiltonian and derive the lattice Green’s functions for small magnetic ﬁeld (Zeemaneffect). We calculate the indirect exchange coupling between apair of impurities. We represent the RKKY interaction energyas a Hadamard product of three matrices: a valley matrix,anαmatrix, and a distance matrix. In Sec. III, we present numerical results for the α-dependent exchange interaction in the case of strong magnetic ﬁeld when Landau levels havebeen formed. We demonstrate that the spin susceptibility fortheα-T 3model is different in nature from that for graphene, thereby signaling a magnetic phase transition at α=0. We also analyze the behavior of the spin susceptibility at low andhigh doping. We conclude with a summary in Sec. IV. II. WEAK MAGNETIC FIELD: ZEEMAN EFFECT ON RKKY INTERACTION FOR THE α-T3MODEL The conventional α-T3model describes triplon energy bands. A small magnetic ﬁeld induces nontrivial topologicalcharacter in the triplon energy spectrum. First, we shall intro-duce the lattice-speciﬁc Green’s functions which are essentialfor calculating RKKY interactions. Throughout the paper, weuse two conventions for the notation adopted: bold capitalizedletters stand for 3 ×3 matrices (or 3 ×1 vectors); quantities with tildes are dimensionless. The energy spectrum can bederived from the low-energy Hamiltonian at the KandK /prime points, H=⎛ ⎜⎝/Delta1 fλ,kcosφ 0 f∗ λ,kcosφ 0 fλ,ksinφ 0 f∗ λ,ksinφ −/Delta1⎞ ⎟⎠, (1) where 0 <φ/lessorequalslantπ/4 is the hopping parameter with α=tanφ; fλ,k=λ/epsilon1ke−iλθk, with /epsilon1k=¯hvFk;λ=±1 stands for the val- ley index at the Kand K/primepoints located at ( λ4π 3√ 3a,0);a is the conventional graphene carbon-carbon distance; and vFFIG. 2. Dispersion of (a) the massive /Delta1a/¯hvF=0.1 triplon with φ=π/10, (b) the masless /Delta1a/¯hvF=0 triplon with φ=π/10, and (c) the massive /Delta1a/¯hvF=0.1 spin-1 fermions (dice lattice) with φ=π/4. Changing the magnetic ﬁeld orientation Bz→− Bz,o r ,i n other words, /Delta1→−/Delta1, leads to a ﬂip of the dispersion E→− E. stands for the Fermi velocity. The angle between kand the x axis is given by θk, yielding kx/\|k\|=cosθk,ky/\|k\|=sinθk. The rows and columns of the Hamiltonian are labeled bythe (A, B, C) lattice indices indicated in Fig. 1.T h em a s s term induced by the pseudomagnetic ﬁeld as it follows fromRef. [ 44] is denoted by /Delta1=mv 2 F/2. The energy spectrum corresponding to Eq. ( 1) ﬁrst reported in Ref. [ 45] is shown in Fig. 2. For convenience, we denote by ω=E/E0andδ=/Delta1/E0the normalized energy and normal- ized gap, respectively, where E0=¯hvF/a. In the absence of magnetic ﬁeld, the triplon is built from two Dirac cones aswell as a “ﬂat band.” For the dice lattice φ=π/4, and the effect of the mass term is to open a gap at k=0 such as −δ/lessorequalslant ω/lessorequalslantδ, and we recover the standard spin-1 dispersion. This also breaks time reversal symmetry. Reducing the value inφ, we shall obtain two nonsymmetrical gaps, 0 <ω/lessorequalslantδand −δ/lessorequalslantω/lessorequalslantω δ, where ωδ=−δcos (2φ) (asymptotic value of the middle). The bending of the ﬂat band reveals the nontriv-iality of the energy dispersion topology and may be related toa nonzero Chern number. One of the most striking features oftheα-T 3model is the broken particle-hole symmetry. We deﬁne the Green’s functions by the elements of an inverse matrix involving the energy difference with the Hamil-tonian of Eq. ( 1)a s G(k,E;λ;φ)=[(E+i0 +)I−H]−1 =⎛ ⎜⎝GAA GAB GAC G∗ AB GBB GBC G∗ AC G∗BC GCC⎞ ⎟⎠. (2) In this notation, Iis the unit matrix, and the replacement E→E+i0+guarantees the retarded nature of the Green’s functions. The direct diagonalization of the Green’s tensor 075418-2EFFECT OF MAGNETIC FIELD AND CHEMICAL … PHYSICAL REVIEW B 103, 075418 (2021) yields G(k,E;λ;φ)=D−1⎛ ⎜⎝E(E+/Delta1)−/epsilon12 ksin2φ(E+/Delta1)fλ,kcosφ f2 λ,ksin(2φ)/2 E2−/Delta12(E−/Delta1)fλ,ksinφ E(E−/Delta1)−/epsilon12 kcos2φ⎞ ⎟⎠, (3) with the determinant given by D(k,E)=E(E2−/Delta12)−[E+/Delta1cos(2φ)]/epsilon12 k, whose dispersion is given by its poles, as shown in Fig. 2. Clearly, the Green’s function matrix is Hermitian, and we observe that GBB(k,E;λ;φ)=GAA(k,E;λ;φ=0) is the only element of the Green’s function matrix which does not depend on φ. Consequently, this leads to the RKKY interaction between spins on the B site being unaffected when φis varied. Now, deﬁning the Fourier transform of the total Green’s function at the two valleys, upon shifting to the Dirac points with k→k+λK, we obtain the following expression for the components in real space: Gμν(rll/prime,E;φ)=A (2π)2/summationdisplay λ=±1/integraldisplay B.Z.d2kGμν(k,E;λ;φ)ei(k+λK)·rll/prime, (4) where the integration over the wave vector kis carried out in the Brillouin zone and we have used rll/prime=rl−rl/prime. After some straightforward algebra (see Appendix A) we obtain the Green’s function tensor as a Hadamard product, G(rll/prime,E;φ)=A πa2E0V1/2◦/Phi11/2◦R1/2, (5) where the valley matrix is given by V1/2(rll/prime)=⎛ ⎜⎝cos(K·rll/prime)sin(K·rll/prime−αll/prime)cos(K·rll/prime−2αll/prime) cos(K·rll/prime) sin(K·rll/prime−αll/prime) cos(K·rll/prime)⎞ ⎟⎠ and the α- (or, equivalently, φ-) dependent matrix has the form /Phi11/2=⎛ ⎜⎜⎜⎝ω+δ ω−ωδ/parenleftbig 1−ω−δ ω−ωδsin2φ/parenrightbig/radicalBig ω2−δ2 ω(ω−ωδ)ω+δ ω−ωδcosφω2−δ2 (ω−ωδ)2sin(2φ) 2 ω2−δ2 ω(ω−ωδ)/radicalBig ω2−δ2 ω(ω−ωδ)ω−δ ω−ωδsinφ ω−δ ω−ωδ/parenleftbig 1−ω+δ ω−ωδcos2φ/parenrightbig⎞ ⎟⎟⎟⎠. (6) The position- and energy-dependent distance matrix is given by R1/2=ω⎛ ⎜⎝−K0(−i/Omega1r)−iK1(−i/Omega1r) K2(−i/Omega1r) −K0(−i/Omega1r)−iK1(−i/Omega1r) −K0(−i/Omega1r)⎞ ⎟⎠, where /Omega1=/radicalBig ωω2−δ2 ω−ωδand the dimensionless length ris de- ﬁned by r=rll/primea−1, with adenoting the AB separation on the lattice as shown in Fig. 1. We now consider two magnetic impurities having spins S1 andS2occupying the lattice sites rlandrl/prime, respectively. The effective RKKY exchange interaction energy for this pair ofspins in the sea of Dirac electrons is, by linear response theory,given in the Heisenberg form as [ 23,29,30] E μν(rll/prime;φ)=λ2 0¯h2 4χμν(rll/prime;φ)S1·S2, where λ0is the short-range exchange interaction between the impurity spins and the α-T3electrons and χμν(rll/prime;φ)i s the free-particle charge density sublattice susceptibility whichdepends on the lattice sites μ,ν=A,B,C where the impurityspins are positioned and is given by χ μν(rll/prime;φ,δ,μ )=−2 π/integraldisplay0 −∞dEIm/bracketleftbig G2 μν(E+i0+)/bracketrightbig =/parenleftbigg3√ 3 2πE0/parenrightbigg2 E0Vμν(rll/prime)˜χμν(rll/prime;φ,δ,μ ). (7) Hereμ=EF/E0is a normalized Fermi energy. A new valley matrix is given by the highly oscillatory direct product V= V1/2◦V1/2. We now focus on the dimensionless envelop matrix ele- ments ˜ χμν, given by ˜χ=−2 π/integraldisplayμ −∞dωIm[/Phi1◦R], (8) where /Phi1=/Phi11/2◦/Phi11/2is a smooth function of ωandR= R1/2◦R1/2is the oscillating kernel. It is convenient to 075418-3ROSLYAK, GUMBS, BALASSIS, AND ELSAYED PHYSICAL REVIEW B 103, 075418 (2021) FIG. 3. Equation ( 9) along various directions for small (top panels) and large (bottom panels) distances and the set of numerical parameters φ=π/10,δ=0.1,μ=1.0. separate the above expression by writing ˜χ=˜χ(0)+˜χ(1) =−2 π/integraldisplay−μ −∞dωIm[/Phi1◦R]−2 π/integraldisplayμ −μdωIm[/Phi1◦R].(9) Note that due to the symmetry of the kernel the ˜χ(1)(δ=0) term vanishes; therefore, its contribution is a direct measureof the magnetic ﬁeld inﬂuence. At this point we consider thehigh-doping regime δ/μ/lessmuch1, so that we can neglect the δ effect in ˜χ (0)/similarequal˜χ(0)(δ=0)=−2 π/Phi1◦/integraldisplay−μ −∞dωIm[R]. (10) Its exact expression in terms of the Meijer Gfunctions was ﬁrst obtained in Ref. [ 15] and exhibits Friedel oscillations in the susceptibility. The second contribution to the susceptibility in Eq. ( 9)w a s worked out numerically. Special attention has to be paid tothe gap region −δ<ω<δ since it contains a singularity atω=−δcos(2φ) which is the asymptote of the middle (ﬂat-band) dispersion curve in Fig. 2. The Zeeman kernel Im[/Phi1◦R] becomes highly oscillatory upon approaching the singular point (see Fig. 1in the Supplemental Material [ 46]), and the integral was determined using /integraldisplay δ −δcos(2φ)···=/summationdisplay i/integraldisplayωi+1 ωi···, (11) where ωiare the kernel Rzeros in ascending order. The magnitude of the above summation grows with rll/prime. The kernel with and without the Zeeman effect for small kFris shown in Fig. 2of the Supplemental Material. Note that along the AC direction the kernel singularity occurs even for δ=0. This is a manifestation of the ﬂat-band contribution. The Zeemaneffect deforms the otherwise ﬂat band, and its contribution ispronounced in all magnetic impurities orientations. In Fig. 3, we analyze the ˜χ (1)elements. These are shifted to the right for δ>0 when compared to ˜χ(0)for small valuesofrll/prime. The left shift occurs upon ﬂipping of the magnetic ﬁeld orientation δ→−δfollowing the ﬂip in the dispersion curve E→− E(see Fig. 2). Let us focus on points in the lattice such that ˜χ(0)=0. Switching orientation in the magnetic ﬁeld changes the RKKY interaction from ferromagnetic to antifer-romagnetic. This effect may be useful in spintronics. Anotherinteresting effect occurs at larger k Frwhere the shift may disappear and change its direction in beatlike format. We mayattribute the beats to the broken particle-hole symmetry wheretwo types of Friedel oscillations occur. This is supported bythe fact that the beats disappear upon restoring the symmetryto the lattice as in the dice lattice case of φ=π/4 [see Fig. 4(b)]. It is also worth noting that the dice lattice gives vanishing χ (1) AC. The kernel plays a crucial role in the low-temperature correction kbT/μ/lessmuch1 obtained in the Sommerfeld expansion [15] ˜χ=˜χ(T=0)+π2 6/parenleftbiggkbT E0/parenrightbigg2 ˜χ(2)(ω=μ), ˜χ(2)=d dωIm[/Phi1◦R]. (12) It is clear that the expansion fails around a singular point and the edges of the gap (see Fig. 3in the Supplemental Material). The standard approach to correct the expansion is to deﬁne achemical potential that depends on temperature. That wouldtake us beyond the scope of this investigation. III. STRONG MAGNETIC FIELD EFFECTS ON THE RKKY INTERACTION: FORMATION OF LANDAU LEVELS We performed our calculations using the Landau gauge, for which the vector potential is A=− Bzyˆxand∇×A=Bzˆzis the magnetic ﬁeld. Using the Hamiltonian of Eq. ( 1), one can determine the wave functions and Landau levels for the lattice.Making use of the vector potential and the Peierls substitution ¯hk→p→p+eA, where ¯ hkis the momentum eigenvalue in 075418-4EFFECT OF MAGNETIC FIELD AND CHEMICAL … PHYSICAL REVIEW B 103, 075418 (2021) (a) (b) FIG. 4. Limiting cases of RKKY response along various directions for small (top panel) and large (lower panel) distances and the set of numerical parameters δ=0.1,μ=1.0. the absence of magnetic ﬁeld and pis the momentum operator, we have ˆHK=− ˆH∗ K/prime =EB⎛ ⎜⎝0 cos φˆa 0 cosφˆa+0s i n φˆa 0s i n φˆa+0⎞ ⎟⎠, (13) where EB=√ 2γl−1 Bis the cyclotron energy related to the magnetic length lB=√¯h/(eBz). We also deﬁne the annihilation operator ˆ a=1√2¯heB z(ˆpx−eBzˆy−iˆpy) and the creation operator ˆ a+=1√2¯heB z(ˆpx−eBzˆy+iˆpy)a si nt h e case of the harmonic oscillator. We note that when φ=0, the Hamiltonian submatrix consisting of the ﬁrst two rowsand columns is the one used in [ 41,42] for monolayer graphene. In the most general case, let us denote the eigenstates by{/Psi1 n(r),En}, where the eigenfunctions are orthonormal, i.e.,/integraltext d2r/Psi1T n1(r)/Psi1⋆ n2(r)=δn1,n2. We then write the Green’s function as G(E;rll/prime)=1 EI−H=/summationdisplay n/Psi1⋆ n(rl)/Psi1T n(rl/prime) E−En+i0+. (14) In the presence of magnetic ﬁeld, we have n={λ,s,n,ky}, where λ=±1 denotes the KorK/prime=−Kvalley; s= −1,0,1 stands for the valence, ﬂat, and conduction bands, respectively; n/greaterorequalslant0 is the Landau level index; and kyis the wave vector. The energies can be found by diagonalizing theHamiltonian ( 13)a s E n=EB/epsilon1λ,s,n=EBs√n+χλ, (15) where the auxiliary parameter χλ=[1−λcos (2φ)]/2, with 0/lessorequalslantχλ<1, has been used. The susceptibility components at T=0 K and the Fermi energy EFare given by Eq. ( 7). Using the Green’s function in Eq. ( 14), we obtain χμν=−1 πIm/integraldisplay∞ −∞dEθ(EF−E)G2 μν(E;rll/prime)=−1 πIm/summationdisplay n1,n2/Psi1μν n1;n2(rl,rl/prime)/integraldisplay∞ −∞dEθ(EF−E) (En1−En2) ×/parenleftbigg1 E−En1+i0+−1 E−En2+i0+/parenrightbigg =/summationdisplay n1,n2/Psi1μν n1;n2(rl,rl/prime)/bracketleftbiggθ(EF−En1)−θ(EF−En2) En1−En2/bracketrightbigg . (16) Here, we have used the shorthand notation /Psi1μν n1;n2(rl,rl/prime)= /Psi1⋆μ n1(rl)/Psi1ν n1(rl/prime)/Psi1⋆μ n2(rl/prime)/Psi1ν n2(rl). Mapping the sites of the lattice A, B, C →− 1,0,1 and separating the spatial variables in the wave function, we obtain /Psi1⋆μ n(rl)=ψμ λ,s,nφn+λμ,ky(xl)e−ikyyle−iλKyyl, (17) where the vector components speciﬁc to the given lattice are denoted by ψμ λ,s,n,φn,ky(xl) and are given by the harmonic oscillator wave functions. When s2=1, these components take the following form: ψμ λ,s,n=1√2(n+χλ)⎧ ⎪⎨ ⎪⎩√n(1−χλ),λ μ =−1, sλ√(n+χλ),λ μ =0, √(n+1)χλ,λ μ =1.(18) For the ﬂat band ( s=0), when n>0, the components are ψμ λ,s,n=1√n+χλ⎧ ⎪⎨ ⎪⎩−λ√(n+1)χλ,λ μ =−1, 0,λ μ =0, λ√n(1−χλ),λ μ =1,(19) while for n=0 the components are ψμ λ,s,n=⎧ ⎪⎨ ⎪⎩0,λ μ =−1, 0,λ μ =0, 1,λ μ =1.(20) 075418-5ROSLYAK, GUMBS, BALASSIS, AND ELSAYED PHYSICAL REVIEW B 103, 075418 (2021) By combining Eqs. ( 15), (16), and ( 18) and after some algebra (see Appendix B) we ﬁnally obtain the general form of the susceptibility components: χμν=A EB(2πlB)2˜χμν(rl,rl/prime), ˜χμν(rl,rl/prime)=/summationdisplay λ1,2=±1/summationdisplay s1,2=0,±1/summationdisplay n1,2/greaterorequalslant0ψμν λ1s1n1;λ1s1n1˜/Phi1n1+λ1ν n1+λ1μ(s1;rl,rl/prime)˜/Phi1n2+λ2ν n2+λ2μ(s2;rl/prime,rl)e−iK(λ1−λ2)(yl−yl/prime) ×θ(μF−s1√n1+χλ1)−θ(μF−s2√n2+χλ2) s1√n1+χλ1−s2√n2+χλ2, (21) where we have introduced the normalized Fermi energy μF=EF/EB as well as ψμν λ1s1n1;λ1s1n1= ψμ λ1,s1,n1ψν λ1,s1,n1ψμ λ2,s2,n2ψν λ2,s2,n2. Equation ( 21) is applicable for a wide range of experimental parameters and serves asa basis for the numerical simulations which are presentedbelow. For simplicity, we neglect highly oscillatory intervalleyterms, setting λ 1=λ2=λ=±1. Figures 5present the magnetic ﬁeld dependent suscepti- bility as a function of the spin separation when EF=0a tT=0 K. Three values of φwere chosen in the numerical calculations. They all show regions of ferromagnetic and anti-ferromagnetic behavior with the amplitude of the oscillationsdecreasing with increasing separation between the spins onthe lattice. However, for φ=π/80 in Fig. 5(b),χ CChas the largest amplitude for the oscillations, and χAB+χBA,χAC+ χCA, and χBC+χBAall remain negative and independent of rll/prime. These results are interesting as they demonstrate how one could control the magnetic behavior of the α-T3lattice. Most (a) (b) FIG. 5. Spin susceptibility in units of A/EB(2πlB)2as a function of the interparticle separation for EF=0,T=0K . 075418-6EFFECT OF MAGNETIC FIELD AND CHEMICAL … PHYSICAL REVIEW B 103, 075418 (2021) importantly, the results in Fig. 5(b) signal that the magnetic properties of the α-T3lattice near α=0 need to be com- pared with those for graphene in Fig. 5(a). Remarkably, the susceptibility has one sign for small rll/prime. The component χAA oscillates but remains positive for large spin separation. On the contrary, both χABand the sum χAA+χABremain negative in this limit. This behavior is independent of the position ofthe Fermi level. We point out that in doing the calculationsfor graphene, we ﬁrstsetα=0i nE q .( 13) before calculating the eigenstates, which were in turn employed in the spinsusceptibility. Therefore, the change in behavior discoveredhere is clear when αis ﬁnite and zero. We now turn our attention to two speciﬁc cases where closed-form analytic expressions can be obtained for the spinsusceptibility. A very interesting case occurs when the latticeis undoped, i.e., E F=0, in strong magnetic ﬁeld for which there are well-separated Landau levels at λ=1 and φ→0. The dominant contributions to Eq. ( 21) come from n1,2=0 terms, ˜χμν=/summationdisplay s1,2=0,±1˜/Phi1ν μ(s1;rl,rl/prime)˜/Phi1ν μ(s2;rl/prime,rl) ×ψμ 1,s1,0ψν 1,s1,0ψμ 1,s2,0ψν 1,s2,0θ(−s1sinφ)−θ(−s2sinφ) s1sinφ−s2sinφ. (22) Let us introduce the normalized temperature ˜T=kBT EBand the integral representation of the Fermi function instead of the θ function. For an arbitrarily chosen small temperature, we set ˜T=sin2(φ), and we expand the above equation around small positive φto obtain ˜χμν∼Erf/bracketleftbig1√ 2/bracketrightbig exp/bracketleftbig−r2 ll/prime 2/bracketrightbig 4φ ×⎡ ⎢⎣⎛ ⎜⎝00 0 0−11 01 −1⎞ ⎟⎠+⎛ ⎜⎝00 0 00 0 00 −4⎞ ⎟⎠⎤ ⎥⎦.(23) The ﬁrst matrix is due to transitions between the valence and conduction bands as well as within the conduction band frombelow to above the Fermi level. The second matrix arisesfrom transitions from the ﬂat band to the conduction band.We conclude from these results that the largest change in thespin susceptibility occurs in the limit when φ→0 and there is no smooth transition from ﬁnite φtoφ=0. This in turn indicates that there is a phase transition between graphene(φ=0) and the α-T 3model. This anomaly is short range dueto the exponent and has no counterpart in the K(λ=−1) valley. We also study the case of high doping when the Fermi level nFis deﬁned via /radicalbig nF−1+χλ1/lessorequalslantμF/lessorequalslant/radicalbig nF+χλ2. In this case, there are only intraband s1=s2=1 contributions to the susceptibility. The leading terms (largest contributionsto the sum) come from the states nearest to n F. Speciﬁcally, for large nF, we found numerically that the terms in Eq. ( 21) scale as δ\|n1−n2\|,1. The transitions from the ﬂat band to the conduction band do not follow this rule; they rather scaleas∼1/n F, which allows us to neglect such contributions. A similar approach was adapted by Lozovik [ 47] when he discussed edge magnetoplasmons in graphene (leading con-tributions to the conductivity tensor in the above-mentionedlimit). However, there is an important difference in that themagnetoplasmons are given by the optical conductivity tensorwhere δ \|n1−n2\|,1is the true selection rule which applies for all n. In this limiting case Eq. ( 21) can be written in a compact form as ˜χ=/bracketleftbig I◦/Phi1λ1=λ2+Vλ1=−λ2◦/Phi1λ1=−λ2/bracketrightbig ◦R. (24) Contributions from the same valley λ1=λ2(the ﬁrst term in the square brackets in the above expression) are given by/Phi1=/Phi1 1/2◦/Phi11/2, which is identical to the no-magnetic-ﬁeld caseδ=0i nE q .( 6), /Phi1λ1=λ2=⎛ ⎜⎝cos4φcos2φ1 4sin22φ 1s i n2φ sin4φ⎞ ⎟⎠. (25) However, for mixed valley contributions, λ1=−λ2, we ob- tain highly oscillatory terms Vλ1=−λ2=cos(2 Kyll/prime)Ialong with a peculiar form for the φmatrix, /Phi1λ1=−λ2=⎛ ⎜⎝1 4cot2φ1 2csc2φ −2 csc2(2φ)1 2sec2φ 1 4tan2φ⎞ ⎟⎠. (26) It is informative to look at the top left 2 ×2 submatrix in Eqs. ( 25) and ( 26) corresponding to the graphenelike case of A and B sublattices. While Eq. ( 25) provides a smooth transition to graphene at φ→0, the valley mixing in Eq. ( 26)g i v e s 1/φ2scaling. The absence of the smooth graphene limit can be directly attributed to broken symmetry for the KandK/prime valleys in magnetic ﬁeld. The site-to-site distance and Fermi number dependent ma- trix referred to Eq. ( 24)a r eg i v e nb y R(rll/prime,nF)=1 2πr⎛ ⎜⎜⎜⎝−4 cos 2(2√nFr)e−r2cos(4√nFr)+11 4/bracketleftbig e−r2cos(4√nFr)+1/bracketrightbig −4 cos2(2√nFr) e−r2cos(4√nFr)+1 −4 cos2(2√nFr)⎞ ⎟⎟⎟⎠, (27) 075418-7ROSLYAK, GUMBS, BALASSIS, AND ELSAYED PHYSICAL REVIEW B 103, 075418 (2021) where for convenience we have omitted the subscripts through the replacement rll/prime/√ 2→r. If we formally associate√nF with kF, the oscillations in the above equation correspond to Kohn anomalies in the absence of magnetic ﬁeld, which wasﬁrst reported in Ref. [ 35]. However, they are much larger in range due to the ∼1/rdependence. At larger distances, we can neglect the terms ∼exp(−r 2), and the oscillations for impurities which are placed on different sublattices vanish. IV . CONCLUDING REMARKS AND SUMMARY We have investigated the behavior of the RKKY interaction for undoped and doped α-T3semimetals as well as when they are subjected to a uniform perpendicular magnetic ﬁeld.Speciﬁcally, we have shown the following: (a) For undopedsamples, the RKKY interaction obeys an inverse cubic lawfor the separation between spins located on lattice sites. Thestrength of this interaction is anisotropic and determined bythe adjustable hopping parameter φexcept when both spins are on B sites. Furthermore, the AA, BB, and CC exchangeinteractions are ferromagnetic, but the sign of this interactionis reversed when the spins are located on different sublattices.(b) For the case when the chemical potential is ﬁnite, we wereable to express our closed-form analytic expression for thespin susceptibility in the same algebraic form as in case (a).However, the amplitudes of these interactions are multipliedby an oscillatory factor which could be positive or negative forranges of the spin separations. (c) In the presence of magneticﬁeld, the spin susceptibility oscillates as the spin separationis varied, displaying ranges of ferromagnetism and antifer-romagnetism. When φis small, we found that the behavior of the susceptibility is radically different from when the diceor Lieb phase ( φ=π/4) is approached. These observations conﬁrm that a phase transition occurs as φ→0 and this phase change is signaled through an applied magnetic ﬁeld. A phasechange was also reported in Ref. [ 48] when the softening of a magnetoplasmon mode as the hopping parameter is reducedwas discovered. (d) We were able to obtain analytic expres-sions for the spin susceptibility in the limit of low magneticﬁeld or high doping. Interestingly, the power law behavioras a function of spin separation is ∼1/r. At large distances between the impurities the RKKY interaction exhibits Kohnanomalies only when those are located on the same sublat-tices. These effects are experimentally observable signaturesof the electronic properties of α-T 3semimetals and could serve to motivate others to apply them to future technologies. ACKNOWLEDGMENT G.G. would like to acknowledge the support from the Air Force Research Laboratory (AFRL) through Grant No.FA9453-21-1-0046APPENDIX A: DERIV ATION OF EQUATION ( 5) Here we obtain the analytical form of the following integral in Eq. ( 4): /summationdisplay λ/integraldisplay B.Z.···≈/summationdisplay λ/integraldisplay∞ 0dk/integraldisplay2π 0dθ=/summationdisplay λ/integraldisplay/integraldisplay , (A1) where the upper limit of the kintegral is extended to ∞and we usedθk=θ+αll/prime, with αll/primebeing the angle which rll/primemakes with the positive kxaxis. This leads to GAA=2A (2π)2cos(K·rll/prime)/integraldisplay/integraldisplayE(E+/Delta1)−/epsilon12 ksin2φ Deik·rll/prime, GBB=2A (2π)2cos(K·rll/prime)/integraldisplay/integraldisplayE2−/Delta12 Deik·rll/prime, GCC=2A (2π)2cos(K·rll/prime)/integraldisplay/integraldisplayE(E−/Delta1)−/epsilon12 kcos2φ Deik·rll/prime, GAB=A (2π)2/bracketleftbigg ei(K·rll/prime−αll/prime)/integraldisplay/integraldisplay(E+/Delta1)/epsilon1kcosφ Dei(k·rll/prime−θ) −e−i(K·rll/prime−αll/prime)/integraldisplay/integraldisplay(E+/Delta1)/epsilon1kcosφ Dei(k·rll/prime+θ)/bracketrightbigg , GAC=A (2π)2/bracketleftbigg ei(K·rll/prime−2αll/prime)/integraldisplay/integraldisplay/epsilon12 ksin(2φ) 2E(E2−/epsilon12 k)ei(k·rll/prime−2θ) +e−i(K·rll/prime−2αll/prime)/integraldisplay/integraldisplay/epsilon12 ksin(2φ) 2E(E2−/epsilon12 k)ei(k·rll/prime+2θ)/bracketrightbigg , GBC=A (2π)2/bracketleftbigg ei(K·rll/prime−αll/prime)/integraldisplay/integraldisplay(E−/Delta1)/epsilon1ksinφ Dei(k·rll/prime−θ) −e−i(K·rll/prime−αll/prime)/integraldisplay/integraldisplay(E−/Delta1)/epsilon1ksinφ Dei(k·rll/prime+θ)/bracketrightbigg . The above expressions can also be written in the form GAA=cos(K·rll/prime)FAA(rll/prime,E;φ), GBB=GAA(rll/prime,E;φ=0), GCC=GAA(rll/prime,E;φ+π/2), GAB=sin(K·rll/prime−αll/prime)FAB(rll/prime,E;φ), GAC=cos(K·rll/prime−2αll/prime)FAC(rll/prime,E;φ), GBC=sin(K·rll/prime−αll/prime)FBC(rll/prime,E;φ). (A2) Let us deﬁne the following auxiliary quantities given by the Hankel transforms: FAA=/parenleftbiggA πa2E0/parenrightbigg/integraldisplay∞ 0d qqJ 0(qr)/braceleftbiggω(ω+δ)−q2sin2φ ω(ω2−δ2)−[ω+δcos(2φ)]q2/bracerightbigg =−/parenleftbiggA πa2E0/parenrightbigg ωK0/parenleftBigg −ir/radicalBigg ω(ω2−δ2) ω+δcos(2φ)/parenrightBigg ω+δ ω+δcos(2φ)/bracketleftbigg 1−ω−δ ω+δcos(2φ)sin2φ/bracketrightbigg , 075418-8EFFECT OF MAGNETIC FIELD AND CHEMICAL … PHYSICAL REVIEW B 103, 075418 (2021) FAB=−/parenleftbiggA πa2E0/parenrightbigg/integraldisplay∞ 0dq q J 1(qr)/braceleftbigg(ω+δ)qcosφ ω(ω2−δ2)−[ω+δcos(2φ)]q2/bracerightbigg =− i/parenleftbiggA πa2E0/parenrightbigg ωK1/parenleftBigg −ir/radicalBigg ω(ω2−δ2) ω+δcos(2φ)/parenrightBigg/radicalBigg ω2−δ2 ω[ω+δcos(2φ)]ω+δ ω+δcos(2φ)cosφ, FAC=/parenleftbiggA πa2E0/parenrightbigg/integraldisplay∞ 0d qqJ 2(qr)/braceleftbiggq2 ω(ω2−δ2)−[ω+δcos(2φ)]q2/bracerightbiggsin(2φ) 2 =/parenleftbiggA πa2E0/parenrightbigg ωK2/parenleftBigg −ir/radicalBigg ω(ω2−δ2) ω+δcos(2φ)/parenrightBigg ω2−δ2 [ω+δcos(2φ)]2sin(2φ) 2, FBB=−/parenleftbiggA πa2E0/parenrightbigg ωK0/parenleftBigg −ir/radicalBigg ω(ω2−δ2) ω+δcos(2φ)/parenrightBigg ω2−δ2 ω[ω+δcos(2φ)], FCC=−/parenleftbiggA πa2E0/parenrightbigg ωK0/parenleftBigg −ir/radicalBigg ω(ω2−δ2) ω+δcos(2φ)/parenrightBigg ω−δ ω+δcos(2φ)/bracketleftbigg 1−ω+δ ω+δcos(2φ)cos2φ/bracketrightbigg , FBC=− i/parenleftbiggA πa2E0/parenrightbigg ωK1/parenleftBigg −ir/radicalBigg ω(ω2−δ2) ω+δcos(2φ)/parenrightBigg/radicalBigg ω2−δ2 ω[ω+δcos(2φ)]ω−δ ω+δcos(2φ)sinφ. (A3) Here we employed the well-known integral /integraldisplay∞ 0dxxn+1 x2+C2Jn(xR)=CnKn(−CR), where Kn(x)(n=0,1,2,...) is a modiﬁed Bessel function of the second kind. Together Eqs. ( A2) and ( A3) yield the desired ﬁnal expression. APPENDIX B: DERIV ATION OF EQUATION ( 21) The integration over kyin Eq. ( 16) can be performed ana- lytically using /summationdisplay ky=A 2π/integraldisplay∞ −∞d[Y+(xl/prime+xl)/2+i(yl/prime−yl)/2] 2πl2 B.(B1) Then the expression for the wave function overlap becomes /Phi1n+λν n+λμ(rl,rl/prime) =/summationdisplay kyφn+λμ,ky(xl)φn+λν,ky(xl/prime)e−iky(yl−yl/prime)=A 2πexp/bracketleftbig −r2 ll/prime 4−i(xl+xl/prime)(yl−yl/prime) 2l2 B/bracketrightbig 2π3/2l2 B/radicalbig 2n+λμ(n+λμ)!/radicalbig 2n+λν(n+λν)! ×/integraldisplay∞ −∞dye−y2Hn+λμ(x−y)Hn+λν(z−y), (B2) where Y=kyl2 B,y=Y/lB,x=(xl−xl/prime)+i(yl−yl/prime) 2lB=rll/prime 2exp ( iαll/prime), z=(xl/prime−xl)+i(yl−yl/prime) 2lB=−rll/prime 2exp (−iαll/prime), and rll/prime=2\|x\|= 2\|z\|. Now, let us use the following integral relation: /integraldisplay∞ −∞dy e−y2Hn+λμ(x−y)Hn+λν(z−y) =√π2n⎧ ⎨ ⎩2λν(n+λμ)!zλ(ν−μ)Lλ(ν−μ) n+λμ/parenleftBigr2 ll/prime 2/parenrightBig ,λ μ /lessorequalslantλν, 2λμ(n+λν)!xλ(μ−ν)Lλ(μ−ν) n+λν/parenleftBigr2 ll/prime 2/parenrightBig ,λ μ > λ ν . (B3) Including the ﬂat band in the overlap function, we ﬁnally obtain /Phi1n+λν n+λμ(s;rl,rl/prime)=A (2πlB)2˜/Phi1n+λν n+λμ(s;rl,rl/prime), ˜/Phi1n+λν n+λμ(s,rl,rl/prime)=exp/bracketleftbigg −r2 ll/prime 4−i(xl+xl/prime)(yl−yl/prime) 2l2 B/bracketrightbigg⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩/radicalBig 2λν(n+λμ)! 2λμ(n+λν)!zλ(ν−μ)Lλ(ν−μ) n+λμ/parenleftBigr2 ll/prime 2/parenrightBig ,λ μ /lessorequalslantλν, /radicalBig 2λμ(n+λν)! 2λν(n+λμ)!xλ(μ−ν)Lλ(μ−ν) n+λν/parenleftBigr2 ll/prime 2/parenrightBig ,λ μ > λ ν , 0, n+min(λμ,λν )<0, L0 0/parenleftBigr2 ll/prime 2/parenrightBig , n=0,s=0.(B4) Substituting Eqs. ( 18) and ( B4) into Eq. ( 17) and the resulting equation into Eq. ( 16), we ﬁnally obtain Eq. 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PhysRevB.88.165415.pdf	PHYSICAL REVIEW B 88, 165415 (2013) Andreev magnetointerferometry in topological hybrid junctions Pierre Carmier* CEA-INAC/UJF Grenoble 1, SPSMS UMR-E 9001, Grenoble F-38054, France (Received 10 June 2013; revised manuscript received 20 September 2013; published 16 October 2013) We investigate the fate of topological edge states in the quantum Hall (QH) regime when they experience tunneling processes through a narrow superconducting (S) strip. By computing the charge conductance ﬂowingthrough a graphene-based QH/S/QH junction, the S strip is shown to act as an Andreev interferometer, giving riseto spectacular conductance oscillations as a function of the Landau level ﬁlling factor. We provide a semiclassicalanalysis allowing for a natural interpretation of these oscillations in terms of interferences between electron andhole trajectories propagating along the QH/S interfaces. Our results suggest that nontrivial junctions betweentopologically distinct phases could offer a highly tunable means of partitioning the ﬂow of edge states. DOI: 10.1103/PhysRevB.88.165415 PACS number(s): 72 .80.Vp, 03 .65.Sq, 73 .43.−f, 74.45.+c I. INTRODUCTION Quantum Hall (QH) and superconducting (S) proximity effects are two prominent mesoscopic phenomena, yet theirinterplay has received little attention so far due to thewidespread assumption that their respective ranges of validityare incompatible. However, in modern high mobility two-dimensional electron gases (with mean free paths typicallyexceeding the micron scale), magnetic ﬁeld values Brequired to enter the QH regime have become sufﬁciently small toallow the fabrication of QH/S junctions, using high criticalﬁeld superconductors such as Nb compounds. 1These junctions feature chiral edge states of mixed electron-hole nature dueto the Andreev reﬂection experienced by carriers at theinterface with the S region. 2Evidence for such edge states was successfully demonstrated a few years ago in InAs semicon-ducting heterostructures, 1following seminal experiments3,4 and earlier theoretical proposals.2,5–7The advent of graphene, in which both QH and S proximity effects have been routinelyobserved 8–10owing to the material’s low cost and tunability, has led to a revival of experimental activity in the ﬁeld veryrecently. 11–13 The interface between a QH insulator and a superconductor actually provides a particularly interesting and nontrivialrealization of a topological junction, which is a junctionbetween bulk insulating phases (from a single-particle per-spective) characterized by different topological invariants. 14,15 A deﬁning property of these junctions is the existence of topologically protected edge states propagating along theirinterface. Partition of the information carried by these statesin the available outgoing channels of a mesoscopic systemis an important problem, both from a fundamental point ofview and in the perspective of exploiting the robustness oftopological phases to build novel electronic devices. Thepurpose of this article is to highlight the potentially crucialrole played by quantum interferences in such topologicaljunctions, a spectacular example of which is depicted in Fig. 1 where the charge conductance ﬂowing through a QH/S/QHjunction is plotted as a function of the Landau level ﬁllingfactor ν=(k FlB)2/2, where kFis the Fermi wave vector andlB=√¯h/(eB) the magnetic length. Instead of featuring plateaus at odd values of the spin-degenerate conductancequantum g 0=2e2/h(thick black line), as would be the case in the absence of the S region (or equivalently for supercriticalmagnetic ﬁelds), the conductance is seen to oscillate between extremal values, from a situation where current is essentiallyblocked to one where it is fully transmitted through the Sregion. This conﬁguration therefore effectively behaves asan Andreev interferometer which allows ﬁltering the amountof current ﬂowing through the junction. In the following, Iwill provide a quantitative understanding of this phenomenon,using the classical picture of skipping orbits (see Fig. 2)t o describe QH edge states in the ballistic regime. 7,16–18This trajectory-based approach is justiﬁed in the semiclassical(high-energy) limit, 19which in the present context corresponds to the regime ν/greatermuch1. By expressing the conductance as a sum over the various semiclassical trajectories contributing tothe transmission probability through the QH/S/QH junction,we will see that the magneto-oscillations featured in Fig. 1 can be naturally interpreted in terms of interferences betweenelectron and hole paths propagating along the QH/S interfaces.While suspended graphene should be a well suited candidate totest these predictions, as demonstrated by recent experimentalevidence supporting phase-coherent ballistic transport in thissystem, 20–22the obtained results are essentially independent of graphene’s band structure and should therefore also beobservable in other two-dimensional electron gases. II. MODEL AND ASSUMPTIONS Let us consider the geometry depicted in Fig. 2, where a spin-singlet superconductor (connected to a hidden Sreservoir) is deposited on top of a two-terminal grapheneribbon of width Win the QH regime, thereby opening a proximity-induced superconducting gap /Delta1 S(which will be assumed constant) in a strip of length Linside the ribbon. Assuming phase-coherent ballistic transport inside the system,ﬁnite temperature k BT< k BTc(where Tc≈0.6/Delta1S/kBis the critical temperature of the superconductor) should play no rolebeyond renormalizing the values of the superconducting gapand the critical ﬁeld, and k BTwill thus henceforth be set to zero. In (linear) response to a subgap bias voltage eV < /Delta1 S applied in the left lead, tunneling processes through the S region allow for a charge current to ﬂow in the right lead,characterized by the electrical conductance 23,24 G=g0/summationdisplay n/parenleftbig T(n)−T(n) A/parenrightbig , (1) 165415-1 1098-0121/2013/88(16)/165415(5) ©2013 American Physical SocietyPIERRE CARMIER PHYSICAL REVIEW B 88, 165415 (2013) 5 6 7 89ν048121620G/g0-π-π/2 0 π/2 π kSL [2π]048G/g0 FIG. 1. (Color online) Conductance Gﬂowing through a QH/S/QH junction of length L/ξS=2 and width W/l B=40 as a function of the Landau level ﬁlling factor νforkSL=π/2 [mod 2 π] (green circles). Transmission peaks occur at integer values of ν. Regular QH plateaus (when the S region turns normal) and classicalexpectation are plotted for comparison (thick black and dashed black lines, respectively). Inset: Value of the peak at ν=6 (top blue) and of the dip at ν=5.5 (bottom red) as a function of k SL. withT(n)the probability for mode nto be transmitted as an electron and T(n) Athe probability to be transmitted as a hole (see Fig. 2). Mode ncan also be reﬂected as an electron or as a hole with probabilities R(n)andR(n) A, such that R(n)+R(n) A+ T(n)+T(n) A=1. In order to derive semiclassical approximations for these probabilities, one must describe both the incoming QHedge states and their dynamics at the interface with the Sregion semiclassically. The ﬁrst part is rather straightforward. Provided ν/greatermuch1, the classical skipping orbit picture can be translated into a more rigorous semiclassical description by t LS l QH W RQHT TA A(n) (n)(n) (n) (n)nθ n RrrtA A FIG. 2. (Color online) Cartoon of a QH/S/QH junction with N=3 vertices. Full blue lines are for electron trajectories and dashed red lines for hole trajectories. Semiclassically, states along the QH/S interfaces can be described by skipping orbits propagating betweenequidistant vertices. Given an incoming mode ( n) on the upper left edge, we seek how the outgoing probabilities R (n),R(n) A,T(n),T(n) A scale with W.applying a Bohr-Sommerfeld quantization procedure to the periodic motion of the electrons, which effectively results inturning the continuous family of possible classical trajectoriesinto a set of edge modes, each characterized by a quantizedangle 0 <θ n,±<π. For an armchair edge, this quantization condition can be expressed as 2 θn,±−sin 2θn,±=(2π/ν)(n± 1/4),18,25where ncan be identiﬁed with the Landau level index, and the ±sign refers to the lifting of the twofold valley degeneracy in graphene (which will henceforth be implicit).However the choice of boundary condition at the edgesof the ribbon is qualitatively unimportant for our purposesin the semiclassical regime ν/greatermuch1, where the low-energy selection rules imposed by the valley-polarization constraintscharacterizing the single channel case can be relaxed. 26 Let us now address the dynamics along the interface. Because the Lorentz force acting on a hole is the same as thatacting on an electron, both particles rotate in the same directionwith the magnetic ﬁeld, leading to unidirectional motion alonga given QH/S interface (see Fig. 2). This is due to the fact that even though a hole carries an opposite charge from that of anelectron, this is compensated by the hole’s direction of motionbeing opposite to its momentum. The situation becomes morecomplicated in a QH/S/QH setup, as tunneling processesthrough the S region give rise to states localized along thesecond interface which counterpropagate with respect to thoseon the ﬁrst interface (see Fig. 2). The problem we face therefore boils down to describing the dynamics of coupledcounterpropagating QH states (of mixed electron-hole nature).For simplicity, we shall restrict our analysis to the zero-biaslimit, for which the cyclotron radii of electron and holechannels match. In this case, the semiclassical formalism ismore tractable, since the ensemble of classical trajectoriesreduces to scattering between equidistant vertices along theinterface (see Fig. 2), assuming a sufﬁciently doped S region that the position mismatch between scattering vertices on bothsides of the S region can be neglected. This approximationremains valid when a nonvanishing bias voltage eV/lessorsimilar¯hv F/W (withvFthe Fermi velocity) is taken into account, and the results can in principle be extended to arbitrary eV27using the more technical Green’s function approach based on theFisher-Lee formula. 18,28,29 III. SEMICLASSICAL ANALYSIS To proceed further, it is convenient to introduce the vector (ei,hi)Tcomposed of electron and hole probability amplitudes of leaving vertex ialong the left side of the interface. Its evolution is governed by the 2 ×2m a t r i x Wi, according to the equation ( ei+1,hi+1)T=Wi(ei,hi)T. All of the information regarding the semiclassical dynamics at vertex iis encoded in Wiwhich will be referred to as the local propagator. Noting N=[W/l n] the (integer) number of vertices, where ln= 2lB√ 2νsinθnis the distance separating consecutive scattering events along the interface, it is clear that /parenleftbigg eN hN/parenrightbigg =N/productdisplay i=1Wi/parenleftbigg 1 0/parenrightbigg . (2) Transmission probabilities can then easily be obtained by summing over all possible coordinates for the initial scattering 165415-2ANDREEV MAGNETOINTERFEROMETRY IN TOPOLOGICAL ... PHYSICAL REVIEW B 88, 165415 (2013) vertex; for example, R(n)=1 ln/integraldisplayln 0dl\|eN(l)\|2, (3) withN(l)=1+[(W−l)/ln], such that N/lessorequalslantN(l)/lessorequalslantN+1. The main task we are left with is to compute the local propagator Wi. In order to do so, let us now look in more detail at the scattering processes taking place at the QH/S interfacesbetween consecutive vertices. For an incoming particle withangleθ nat a given vertex, these processes are described by the 2×2 matrices R=/parenleftbiggreiφr/prime Aeiφ rAeiφ/primer/primeeiφ/prime/parenrightbigg ,T=/parenleftbiggteiφt/prime Aeiφ tAeiφ/primet/primeeiφ/prime/parenrightbigg ,(4) where Rcorresponds to reﬂection along a given interface and Tto transmission from one interface to the other. Phases φand φ/primeaccount for the action, Maslov index, and Berry phase17,18 respectively acquired by electron and hole channels during their propagation between scattering events on the interface,such that δφ=φ−φ /prime=2πν. The scattering coefﬁcients in Eq. (4)are the local probability amplitudes of normal reﬂection ( r), Andreev reﬂection ( rA), elastic cotunneling (t), and crossed Andreev reﬂection ( tA). Assuming that the QH/S interfaces are abrupt on the scale of lB, the existence of the magnetic ﬁeld can be locally ignored and the scatteringcoefﬁcients can be determined by matching the quantummechanical wave functions at the interfaces and making use ofmomentum conservation arising from translational invariancein the transverse direction. In the limit L/greaterorsimilarξ S, where ξS= ¯hvF//Delta1Sis the superconducting coherence length, one obtains ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩r=− cosθ n, rA=−isinθn, t=2s i nθn(coskSLsinθn+isinkSL)e−L/ξS, tA=−isin 2θncoskSLe−L/ξS,(5) assuming once more kS/greatermuchkF, with kSthe Fermi wavevector in the S region [primed coefﬁcients in Eq. (4)can be obtained from those of Eq. (5)by reversing the sign of kS]. These coefﬁcients carry the signature of graphene’s unusual bandstructure through their angular dependence. 30In particular, r andtAvanish under normal incidence as a consequence of the absence of backscattering (so-called Klein tunneling). Also,note that tandt Aare, as expected, exponentially suppressed on the scale of ξS. Therefore, in order for tunneling processes to play a role, typical lengths of the S region will be limitedto sizes such that diamagnetic screening currents can beneglected. 31As a consequence, the validity of the plane wave approximation to tunneling coefﬁcients in Eq. (5)will require magnetic lengths lB/greaterorsimilarL. We now have all the necessary ingredients to compute the local propagator. Before presenting the general solution, let usfamiliarize ourselves with it by computing the ﬁrst couple ofterms. The ﬁrst one, W 1=R, is rather obvious: incoming carriers at the ﬁrst vertex must necessarily be reﬂected,or else they will be transmitted through the S region andirrevocably leave the interface (see Fig. 2). The second term can be expressed as an inﬁnite sum, W 2=R/summationtext+∞ m=0T2m, where themth contribution takes into account trajectories where charge carriers have tunneled 2 mtimes through the S region.It can be conveniently rewritten in a self-consistent form, W2=R+W2T2, the meaning of which is the following: unless carriers incoming at the vertex are directly reﬂected(R), they must tunnel twice through the S region ( T 2), at which point one is back to the starting point ( W2). Elaborating on this idea, a general recurrence relation can be derived for i/greaterorequalslant2, Wi=R+i−2/summationdisplay j=0/parenleftBiggj/productdisplay k=0Wi−k/parenrightBigg TRjT. (6) Likewise, an equation similar to Eq. (2)can be written down for the electron and hole probability amplitudes to leave theQH/S/QH junction on the right side, ( e /prime N,h/primeN)T=W/prime N(1,0)T, withW/prime 1=Tand, for N/greaterorequalslant2, W/prime N=T+N−2/summationdisplay j=0/parenleftBiggj/productdisplay k=0WN−k/parenrightBigg TRj+1. (7) In the limit L/ξS/greatermuch1 of a single QH/S interface, Eq. (6) reduces to the uniform solution Wi=Rdescribing the periodic skipping orbit motion along the left interface,and one thus retrieves the solution independently obtainedby Chtchelkatchev in nonrelativistic two-dimensional elec-tron gases 7,16and by the author in graphene QH bipolar junctions.17,18 IV . RESULTS Equations (6)and(7)are the central results of this article and can be solved numerically. There are special cases,however, where they can be exactly solved, an important onefor our purposes being when the condition t+t /primee−iδφ=0i s fulﬁlled, which is equivalent to tanπν=sinθn tankSL. (8) In this case, the local propagator can be shown to take the sim- ple form Wi=αiR, with 0 <αi/lessorequalslant1, and it is then a simple task to show that this implies R(n)+R(n) A/lessorequalslante−N(\|t\|2+\|tA\|2):i n other words, full transmission of current through the S regionis achieved exponentially fast with W(blue curves in Fig. 3). One can also easily prove that W /prime N=α/prime NTif Eq. (8)holds, thereby yielding for the asymptotic value of transmission inthe S region T (n)−T(n) A=\|t\|2−\|tA\|2 \|t\|2+\|tA\|2. (9) In particular, for cos kSL=0, Eqs. (5),(8), and (9)imply that perfect electronic transmission is achieved at integer values ofν, which translates into the conductance peaks shown in Fig. 1. The blue curve in the inset of Fig. 1interestingly suggests that these peaks should survive if cos k SLis not strictly zero. A closer look at Fig. 1shows that the conductance peaks are all the more visible that they are accompanied by dipsat half-integer values of ν. The simpliﬁcation brought by the vanishing of the amplitude of crossed Andreev reﬂectionwhen cos k SL=0 [see Eq. (5)] allows us to support this observation by an analytical statement, as one can then showthat\|h /prime N\|2=0, while \|e/prime N\|2=\|t\|2ifNis odd, and zero 165415-3PIERRE CARMIER PHYSICAL REVIEW B 88, 165415 (2013) 02 0 4 0 60 80 10004812G/g0 02 0 4 0 60 80 100 W/lB00.20.40.60.81RA(2) FIG. 3. (Color online) Upper panel: Conductance Gas a function of the width WforL/ξS=2,kSL=π/2 [mod 2 π], and ν=6 (top blue), ν=5.5 (bottom red). Lower panel: Likewise for the probability of a given mode ( n=2) to be reﬂected as a hole for ν=6 (bottom blue), ν=5.5 (top red). Dashed black lines are the classical expectations. otherwise. This yields T(n)−T(n) A/lessorequalslant\|t\|2; in other words, G this time remarkably does not increase with W(red curves in Fig. 3) and only charge carriers having tunneled through the S region at the ﬁrst vertex may actually contribute toG. This even-odd effect is a signature of the destructive interference of electron and hole paths ( δφ=π) when νis a half-integer, which is already manifest in the limit of a singleQH/S interface. 2,5–7Indeed, taking advantage of the unitarity of the local propagator in this limit, its effect on vector ( ei,hi)T can then be interpreted as rotating the latter on the Bloch sphere with a frequency ωgiven by cos ( ω/2)=cosθncosπν, such that, for half-integer values of ν, the rotation frequency is ω=πand the same even-odd effect is displayed. The red curve in the inset of Fig. 1, however, clearly demonstrates that this interpretation breaks down when cos kSLis no longer zero, as expected from the explicit coupling between νandkSL displayed in Eq. (8). In fact, for cos kSL=1, the positions of the conductance peaks and dips are exchanged with respect tothe previous situation (see inset of Fig. 1), with dips occuring at integer values of νand peaks at half-integer values [in agreement with Eq. (8)]. The value of the peaks will however be of lesser magnitude in this case, since t A/negationslash=0 [see Eq. (9)]. V . DISCUSSION The above predictions are in clear contrast with what would be expected from a purely classical point of view inthis situation (dashed lines in Figs. 1and 3): the classical dynamics along the left interface indeed follows a one-dimensional random walk with backward hopping proba-bility p=\|t\| 2+\|tA\|2, for which Eq. (6)can be solved by recurrence, yielding R(n)+R(n) A/lessorequalslant(1−p)/(1−p+Np). In other words, transmission through the S region is classicallyexpected to increase—albeit only slowly (algebraically)—withW, which can be understood as arising from the fact that charge carriers will have to experience an ever larger numberof scattering events to cross the system (see Fig. 2). That this is not always the case semiclassically (as discussed above)illustrates the crucial role played by quantum interferences inthis setup. As a closing remark, let us brieﬂy comment on the sensitivity to disorder of these interference effects. Whiledisorder away from the S region should bring no meaningfulchange to the results, the presence of strong enough disorder inthe vicinity of the QH/S interfaces will essentially randomizethe phases of the charge carriers, including the phase differenceδφacquired by electron and hole carriers between consecutive vertices, thus likely spoiling the conductance oscillationsdepicted in Fig. 1. However, experimental measurements of the conductance in graphene would still be valuable inthe disordered regime, even in the limit of a single QH/Sinterface, drawing on an analogy between QH/S and bipolarQH junctions which shall be discussed elsewhere: 27they could indeed allow estimating the relevance of charge densityﬂuctuations 32regarding the equipartition of charge carriers observed in bipolar QH junctions.33–35 To summarize, we have seen that spectacular quantum interference effects can arise at the interface between a QHinsulator and a superconductor, and that these effects can bequantitatively understood using an intuitive trajectory-basedsemiclassical approach. We provided a clear experimentalsignature of these quantum interferences by showing thatthe conductance ﬂowing through a QH/S/QH junction shouldfeature characteristic oscillations as a function of the Landaulevel ﬁlling factor ν. We hope our results will motivate further studies of edge state transport at the interface betweentopologically distinct phases. ACKNOWLEDGMENTS I am grateful to D. Badiane, G. Fleury, S. Gu ´eron, M. Houzet, T. L ¨ofwander, J. Meyer, D. Ullmo, and X. Waintal for useful discussions at various stages of this project. I amalso thankful for the hospitality of the SPEC in CEA Saclaywhere part of this work was done, and acknowledge ﬁnancialsupport from STREP ConceptGraphene. *pierre.carmier@centraliens.net 1J. Eroms, D. Weiss, J. DeBoeck, G. Borghs, and U. Z ¨ulicke, Phys. Rev. Lett. 95, 107001 (2005). 2H. Hoppe, U. Z ¨ulicke, and G. Sch ¨on,Phys. Rev. Lett. 84, 1804 (2000).3H. Takayanagi and T. Akazaki, Physica B 249, 462 (1998). 4T. D. Moore and D. A. Williams, P h y s .R e v .B 59, 7308 (1999). 5Y . Takagaki, P h y s .R e v .B 57, 4009 (1998). 6Y . Asano, P h y s .R e v .B 61, 1732 (2000). 165415-4ANDREEV MAGNETOINTERFEROMETRY IN TOPOLOGICAL ... PHYSICAL REVIEW B 88, 165415 (2013) 7N. 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PhysRevB.70.165323.pdf	Surface structure of phosphorus-terminated GaP 001-2ˆ1 N. Kadotani, M. Shimomura, and Y. Fukuda * Research Institute of Electronics, Shizuoka University, Hamamatsu, 432-8011 Japan (Received 26 March 2004; revised manuscript received 28 June 2004; published 29 October 2004 ) The surface structure of phosphorus-terminated GaP s001d-s231dhas been studied by low-energy electron diffraction (LEED ), high-resolution electron energy loss spectroscopy (HREELS ), scanning tunneling micros- copy (STM ), and synchrotron radiation photoemission spectroscopy. HREELS spectra indicate that hydrogen is adsorbed on the surface, leading to formation of a P uH bond. The intensity of the stretching vibration of the PuH remains constant for the s231dsurface annealed at 300–600 K, where s231dLEED patterns also remain. The vibration mode and patterns disappeared simultaneously upon annealing at 700 K. STM ﬁlledstate images show zigzag chain structures. On the other hand, straight rows along the [110]direction are seen in the empty state images. Surface core-level shifts are found: +0.60 eV for Ga3 dand −0.69 and +0.39 eV for P2p. These results can well explain a theoretical model of buckled P-dimers with hydrogen adsorbed in an alternating sequence. DOI: 10.1103/PhysRevB.70.165323 PACS number (s): 68.35.Bs, 81.05.Ea, 68.43.Pq, 68.37.Ef I. INTRODUCTION III-V compound semiconductors containing phosphorus, for example, GaP and InP, are important materials for opto-electronic devices. Since they have been also used as sub-strates for quantum effect devices, surface structures of thesubstrates have been widely studied in an atomic level tofabricate good interfaces between the devices and substrates. Recently, an interesting structure model of an InPs001d-s231d/s232dsurface prepared by metalorganic chemical vapor deposition was proposed based on ﬁrst- principles calculations 1and supported by experiments:2a s231d/s232dreconstruction induced by hydrogen adsorbed in an alternating sequence on buckled P-dimers. It was re- ported that a GaP (001)surface terminated by phosphorus is also reconstructed into a s231d/s232dstructure.3–9The ﬁrst-principles calculations showed that the surface structure of GaP s001d-s231d/s232dis similar to the model for the InPs001d-s231d/s232dsurface.10However, no detailed ex- perimental results to support the model have been reported as yet. Especially, a direct evidence for existence of hydrogenon the surface has not found so far in spite that the hydrogenadsorption is crucially important for surface charge stability(electron counting rules )of the s231d/s232dreconstruc- tion on the phosphorus-terminated GaP (001)surface. In this paper, the GaP s001d-s231dsurface prepared by t-butylphosphine (TBP)is studied by low-energy electron diffraction (LEED ), high-resolution electron energy loss spectroscopy (HREELS ), scanning tunneling microscopy (STM ), and synchrotron radiation photoemission spectros- copy (SRPES )to elucidate the surface structure. Comparison of the present work with the theoretical result is also made. II. EXPERIMENT n-GaP s001d(carrier density: 5.0 31017/cm3)and p-GaP s001d(carrier density: 2.0 31018/cm3)samples are used for present experiments. The samples are degreased by acetone and ethanol, etched by acid solutionsHNO3:HCl=1:1 dfor 1 min, and then rinsed by deionized water. They were immersed in sNH4d2Sxsolution for 30 min to avoid surface oxidation in air, rinsed, and dried in a dry nitrogen stream, followed by being introduced into an ultra-high vacuum chamber. They were cleaned by cycles of sput-tering with Ar ions s0.5 keV dand annealing at 733 K, lead- ing to a s234dreconstruction without sulfur. The s234d surface was exposed to TBP fsCH 3d3CPH22.6310−4Paggas for 10 min at 633 K, resulting in a s231dreconstruction. LEED observations were performed at RT to examine the surface structure after various treatments. Measurements ofHREELS spectra were carried out at an incident energy ofabout 5 eV in the specular direction (60° off normal to the surface )with a total resolution of 15 meV. STM (JEOL: JSTM-4500XT )ﬁlled and empty states images were ob- served using a W tip on the n- andp-GaP s001d-s231dsur- faces, respectively, because the latter images could not be obtained for the n-type surface. Core-level spectra of Ga3 d and P2pwere measured by SRPES (at the 13C beamline in KEK, Tsukuba, Japan )using photon energies of 100 and 170 eV, respectively. The spectra were ﬁtted using a least-squares method. The background of the spectra was sub-tracted by using the Shirley method. Differences between themeasured and ﬁtted spectra were shown in the ﬁgures.A background pressure in the chambers was kept below3310 −8Pa. III. RESULTS AND DISCUSSION The surface structure model presented by Hahn et al.10is shown in Fig. 1 where hydrogen is adsorbed on P-dimer inan alternating sequence. The surface unit cell in the modelshows a s232dreconstruction, but it would lead to the s2 31dif the s32dperiodicity in the f−110 gdirection is lost in the longer length than the coherent one s5–10nm dfor the LEED observation. The latter reconstruction occurs in domi- nant domains on the surface studied here. Figure 2 shows HREELS spectra of the GaP s001d-s2PHYSICAL REVIEW B 70, 165323 (2004 ) 1098-0121/2004/70 (16)/165323 (5)/$22.50 ©2004 The American Physical Society 70165323-134d/cs238dand - s231dsurfaces prepared in situand an- nealed at various temperatures in the ultrahigh vacuum. Only the Fuchs-Kliewer phonon peaks at 395, 782, and 1185 cm−1 are seen on the s234d/cs238dsurface. Two peaks at 2355 and 2984 cm−1are found, except for the Fuchs-Kliewer pho- non peak at 385 cm−1(not shown here )on the s231dsur- face. This implies that the two peaks were caused by prepa- ration of the s231dsurface. The peak at 2355 cm−1can be ascribed to a P uH stretching vibration mode11although a weak PuH2peak might not be separated from the P uH because the wave-number difference Dnis very small (about20 cm−1).12However, since a P uH bond is formed on the GaPs001d-s234dsurface by adsorption ofTBP,13the peak at 2355 cm−1would be due to the P uH bond. The 2984 cm−1 peak can be assigned to the C uH stretching vibration mode,13implying that hydrocarbon species are also adsorbed on the surface. Since the s231dsurface was prepared using TBP gas at 633 K, the hydrocarbon decomposed from TBP would be adsorbed on the surface. The s231dsurface was annealed at 300–700 K. The in- tensity ratio of the P uH vibration mode to the elastic peak remains almost constant and the s231dLEED patterns were not changed upon annealing at 300–600 K. The P uH vi- bration disappeared at 700 K where the s231dpatterns were changed to the s234d. The ratio for the hydrocarbon was gradually decreased upon annealing and it was decreased by 85% at 600 K. This concludes that the hydrocarbon is notresponsible for the s231dreconstruction. If the amount of hydrogen bonded to phosphorus is small and it is contaminant on the surface, the hydrogen would notbe responsible for the reconstruction. The intensity ratio forthe PuH vibration mode was saturated at about 1.2 310 −4 for the GaP (110)surface exposed to hydrogen atoms s600 L d.11On the other hand, it is found to be about 1.5310−4for the present work, showing that both the values are close. The intensity ratio sI/Ieldof a certain vibration mode sIdfor an adsorbate to the elastic peak sIeldis expressed in a dipole scattering at ﬁxed measurement conditions as follows:14 I/Iel~AvNs/s1+SAed2, whereNsis the number of the adsorbate, Anan ionic polar- izability, and Aean electron polarizability. This implies that the ratio depends upon (1)the coverage, (2)the electron polarizability, and (3)the ionic polarizability. Since the hy- drogen would be adsorbed at a considerable part of Psites onthe GaP (110)surface at saturation of the ratio, and (2)and (3)would not be so different for both the GaP (110)and GaPs001d-s231dsurfaces, the value for the latter would be large enough to cover the sufﬁcient area of the surface. An impact scattering might excite the P uH vibration because the P uH bond is tilted from the normal to the GaPs001d-s231dsurface (see Fig. 1 ). On the other hand, the PuH bond on GaP (110)is perpendicular to the surface.The intensity of the dipole-scattered vibration is close to that ofthe impact scattered for a light element, such as hydrogen. 14 The above discussion would allow us the conclusion that thehydrogen is not the contaminant, but is responsible for thes231dreconstruction. STM ﬁlled and empty states images of the GaP s001d-s2 31dsurface are displayed in Figs. 3 (a)and 3 (b), respec- tively. Zigzag chains arranged in the [110]direction are found in Fig. 3 (a). The chains in and out of phase are also seen, resulting in formation of the s232dandcs432dstruc- tures, respectively. This ﬁlled state images are in good agree- ment with the calculation 10and the previous result:9bright protrusions correspond to the phosphorus atom with a ﬁlleddangling bond of P-dimers. The empty state images couldnot be obtained for the n-GaP s001d-s231dsurface. This FIG. 1. A s231d/s232dstructure model of the GaP s001d-s2 31d/s232dsurface.Top (a)and side (b)views are shown. Shaded, ﬁlled, and open circles correspond to hydrogen, phosphorus, andgallium atoms, respectively. FIG. 2. HREELS spectra for the GaP s001d-s231dsurface an- nealed at 300–700 K. The spectrum of the GaP s001d-s234d/cs2 38dsurface is also shown. The intensity ratio of the P uH vibra- tion mode to the elastic peak remains almost constant up to 600 K.KADOTANI, SHIMOMURA, AND FUKUDA PHYSICAL REVIEW B 70, 165323 (2004 ) 165323-2might be due to pinning of the Fermi level near the valence band maximum, analogous to the n-GaP s001d-s234d surface.15The empty state images are obtained on the p-type surface in Fig. 3 (b). The images show straight rows arranged in the [110]direction. The calculation showed that density of empty states (C1, C2, and C3 )spreads over the hydrogen- adsorbed P uP bond.10Therefore, the each protrusion could not be resolved within our resolution.This result is similar tothe calculated images taking into account the resolution. Core-level spectra of Ga3 dare shown in Figs. 4 (a)and 4(b). Fitting parameters of Ga3 dare as follows: the Gauss- ian and Lorentzian widths, 0.55 and 0.20 eV, respectively;the spin-orbit separation (SOS), 0.45 eV; the spin-orbit branching ratio (SOBR ), 1.5 Bulk (B)and S1 components are separated. The ﬁtted spectra, especially in Fig. 4 (b), are in good agreement with the measured, which could rational-ize the ﬁtting. The component S1 is clearly enhanced in in-tensity at the detection angle 80° off normal to the surface,which implies that it can be ascribed to the surface compo-nent. The core-level shift is found to be +0.55 eV that is higher than that s+0.31 eV dof threefold Ga atoms with an empty dangling bond for the cleaved GaP s110d-s131d surface. 16According to the presented model10(Fig. 1 ), the surface Ga atoms exist at the second layer and are bonded totwo bulk-like P, Pwith a ﬁlled dangling bond, and fourfold P.Since the latter two P have excess and deﬁcient electrons,respectively, the Ga atom might be predicted to have a simi-lar binding energy to the bulk Ga. However, the Madelungenergy has to be also taken into account for the surface core-level shift. Comparing the surface core-level shift of Ga3 d for the GaAs s110d-s131dand -cs434dsurfaces, the latter s+0.41 eV d 17is larger than the former s+0.28 eV d.18The sur- face Ga atoms for the latter are located at the third layer and bonded to two bulk-like As atoms, As with a ﬁlled danglingbond, and fourfoldAs with deﬁcient electrons. The chemicalenvironment of this Ga atom is similar to that of the Ga at FIG. 3. STM ﬁlled (a)and empty (b)states images measured at bias voltages of −3.5 and +3.0 V (tunneling currents, 0.20 nA ), respectively. The image sizes are 4 nm 34 nm. The n- and p-GaP s001d-s231dsurfaces are used for the measurements of the ﬁlled and empty states images, respectively. The s232dand cs432dunit cells are shown. FIG. 4. Core-level spectra of Ga3 dfor the GaP s001d-s231d surface. The detection angle is 0° (a)and 80° (b)off normal to the surface. Photon energy of 100 eV was employed. The spectra areseparated into two components: B (bulk)and S1 (surface ). The measured and ﬁtted spectra are shown by small open circles andsolid lines, respectively. Differences between the measured and ﬁt-ted spectra were shown below the spectra.SURFACE STRUCTURE OF PHOSPHORUS-TERMINATED PHYSICAL REVIEW B 70, 165323 (2004 ) 165323-3the second layer of the model. The above discussion suggest us that the surface core-level shift of Ga3 din the present work can explain the model. Core-level spectra of P2 pmeasured at 0° and 80° off normal to the surface are displayed in Figs. 5 (a)and 5 (b), respectively. The spectra are separated into a bulk and threesurface components (S1, S2, and S3 )employing the follow- ing parameters: the Gaussian and Lorentzian widths, 0.63and 0.03 eV, respectively; the SOS, 0.85 eV; the SOBR, 2. The sum of the four spectra is well ﬁtted to the measuredvalue, as seen in Fig. 5. The S2 and S3 components arerelatively enhanced in intensity at 80° and shifted by −0.69and +0.39 eV in the kinetic energy from the bulk energy,respectively. The S1 shifted by −1.10 eV can be ascribed toP clusters at the surface, analogous to the result for theGaAs s001d-cs434dsurface 19and judging from the binding energy (BE)difference between GaP and elemental phosphorus.20Since the intensity of the S1 is not dependent upon the detection angle, it would not be the major surfacecomponent. The S3 can be assigned to phosphorus with theﬁlled dangling bond because the component has the lowerBE(higher kinetic energy )than that of the bulk. This is in good agreement with the shift s+0.40 eV dof the P surface component with the ﬁlled dangling bond on GaP (110). 16The S2 would be ascribed to phosphorus bonded to a more elec-tronegative atom than gallium (electronegativity: 2.1 )be- cause the BE is higher than that of the bulk although theMadelung energy is not taken into account. The componentcorresponds to phosphorus bonded to hydrogen (electronega- tivity: 2.2 ). The trend of the present surface core-level shift for P2pis in agreement with the result for buckling dimers on the Si s001d-s231d 21and InP s001d-s231d22surfaces. IV. SUMMARY The surface structure of phosphorus-terminated GaPs001d-s231dhas been studied by LEED, HREELS, STM, and SRPES. The HREELS spectra indicate that hydro- gen is adsorbed on the surface, leading to formation of thePuH bond. The intensity ratio of the P uH vibration mode to the elastic peak remains almost constant and the s231d LEED patterns were not changed upon annealing at 300–600 K. On the other hand, the bond disappeared uponannealing at 700 K, where the LEED pattern was changed tothes234dstructure. Analysis of the HREELS data con- cludes that the P uH bonding is responsible for the s231d reconstruction. The STM ﬁlled state images show the zigzag chain structures. The straight rows along the [110]direction are found in the empty state images. The surface core-levelshifts are found: +0.60 eV for Ga3 dand −0.69 and +0.39 eV for P2 p. These results can well explain the theo- retical model of buckled P-dimers with hydrogen adsorbed inan alternating sequence. ACKNOWLEDGMENTS We would like to acknowledge partial support by the Min- istry of Education, Science, Sports, and Culture, Japan. *Author to whom correspondence should be addressed. Email: royfuku@rie.shizuoka.ac.jp 1W.G. Schmidt, P.H. Hahn, F. Bechstedt, N. Esser, P. Vogt, A. Wange, and W. Richter, Phys. Rev. Lett. 90, 126101 (2003 ).2G. Chen, S.F. Chen, D.J. Tobin, L. Li, R. Raghavachari, and R.F. Hicks, Phys. Rev. B 68, 121303 (2003 ). 3A.J. Van Bommel and J.E. Crombeen, Surf. Sci. 76, 499 (1973 ). 4I.M. Vitomirov, A. Raisanen, L.J. Brillson, C.L. Lin, D.T. Mcln- FIG. 5. Core-level spectra of P2 pfor the GaP s001d-s231dsur- face. The detection angle is 0° (a)and 80° (b)off normal to the surface. Photon energy of 170 eV was employed. The spectra areseparated into one bulk (B)and three surface (S1, S2, and S3 ) components. The measured and ﬁtted spectra are shown by smallopen circles and solid lines, respectively. Differences between themeasured and ﬁtted spectra are shown below the spectra.KADOTANI, SHIMOMURA, AND FUKUDA PHYSICAL REVIEW B 70, 165323 (2004 ) 165323-4turff, P.D. Kirchner, and J.M. Woodall, J. Vac. Sci. Technol. A 11, 841 (1993 ). 5K. Knorr, M. Pristovsek, U.R. Esser, N. Esser, M. Zorn, and W. Richter, J. Cryst. Growth 170, 230 (1970 ). 6N. Sanada, S. Mochizuki, S. Ichikawa, N. Utsumi, M. Shimo- mura, G. Kaneda, A. Takeuchi, Y. Suzuki, Y. Fukuda, S. Tanaka,and M. Kamata, Surf. Sci. 419, 120 (1999 ). 7M. Zorn, B. Junno, T. Trepk, S. Bose, L. Samuelson, J.-T. Zettler, and W. Richter, Phys. Rev. B 60, 11 557 (1999 ). 8Y. Fukuda, N. Sekizawa, S. Mochizuki, and N. Sanada, J. Cryst. Growth221,2 6 (2000 ). 9L. Töben, T. Hannappel, K. Möller, H.-J. Crawack, C. Petten- kofer, and F. Willig, Surf. Sci. 494, L755 (2001 ). 10P.H. Hahn, W.G. Schmidt, F. Bechstedt, O. Pulci, and R. Del Sole, Phys. Rev. B 68, 033311 (2003 ). 11Y. Chen, D.J. Frankel, J.R. Anderson, and G.J. Lapeyre, J. Vac. Sci. Technol. A 6, 752 (1988 ). 12H. Nienhaus, S.P. Grabowski, and W. Mönch, Surf. Sci. 368, 196 (1996 ). 13G. Kaneda, N. Sanada, and Y. Fukuda, Appl. Surf. Sci. 142,1 (1999 ).14H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic, New York, 1983 ). 15Y. Fukuda, M. Shimomura, N. Sanada, and M. Nagoshi, J. Appl. Phys.76, 3632 (1994 ). 16S. D’Addato, P. Bailey, J.M.C. Thornton, and D.A. Evans, Surf. Sci.377–379, 233 (1997 ). 17M. Larive, G. Jezequel, J.P. Landesman, F. Solal, J. Nagle, B. Lepine, A. Taleb-Ibrahimi, G. Indlehofer, and X. Marcadet,Surf. Sci. 304, 298 (1994 ). 18A.B. McLean, Surf. Sci. 220, L671 (1989 ). 19P.K. Larsen, J.H. Neave, J.F. van der Veen, P.J. Dobson, and B.A. Joyce, Phys. Rev. B 27, 4966 (1983 ). 20J.F. Moulder, W.F. Stickle, P.E. Sobol, K.D. Bomben, and J. Chastain, Handbook of X-ray Photoelectron Spectroscopy (Phys. Electronics, Minnesota, 1992 ). 21E. Landemark, C.J. Karlsson, Y.-C. Chao, and R.I.G. Uhrberg, Phys. Rev. Lett. 69, 1588 (1992 ). 22P. Vogt, A.M. Frisch, Th. Hannappel, S. Visbeck, F. Willig, C. Jung, R. Follath, W. Braun, W. Richter, and N. Esser, Appl.Surf. Sci. 166, 190 (2000 ).SURFACE STRUCTURE OF PHOSPHORUS-TERMINATED PHYSICAL REVIEW B 70, 165323 (2004 ) 165323-5
PhysRevB.72.045105.pdf	Variational study of triangular lattice spin-1/2 model with ring exchanges and spin liquid state in/H9260-„ET …2Cu2„CN …3 Olexei I. Motrunich Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030 /H20849Received 28 January 2005; published 5 July 2005 /H20850 We study triangular lattice spin-1/2 system with antiferromagnetic Heisenberg and ring exchanges using variational approach focusing on possible realization of spin-liquid states. Trial spin liquid wave functions areobtained by Gutzwiller projection of fermionic mean-ﬁeld states and their energetics is compared againstmagnetically ordered trial states. We ﬁnd that in a range of the ring exchange coupling upon destroying theantiferromagnetic order, the best such spin liquid state is essentially a Gutzwiller-projected Fermi sea state. Wepropose this spin liquid with a spinon Fermi surface as a candidate for the nonmagnetic insulating phaseobserved in the organic compound /H9260-/H20849ET/H208502Cu2/H20849CN/H208503, and describe some experimental consequences of this proposal. DOI: 10.1103/PhysRevB.72.045105 PACS number /H20849s/H20850: 75.10.Jm, 75.50.Ee, 75.40. /H11002s, 74.70.Kn I. INTRODUCTION This paper reports a variational study of spin-1/2 Heisen- berg antiferromagnet with ring exchanges on a triangular lat-tice. One motivation for this study is the exact diagonaliza-tion work of LiMing et al. 1and Misguich et al.2on this system proposing that it realizes spin liquid states. We areparticularly interested in spin liquids that may occur near theHeisenberg antiferromagnetic state. Multiple-electron ex-changes are believed to be important near quantum meltingand metal-insulator transitions. The speciﬁc model consid-ered here may also be relevant for the description of a ten-tative spin liquid state observed in the quasi-two-dimensionalorganic compound /H9260-/H20849ET/H208502Cu2/H20849CN/H208503,3which is close to metal-insulator transition. Imada et al.4studied appropriate Hubbard model on the triangular lattice and found an insu-lating regime with no spin order. The ring exchange spinmodel can be viewed as derived from the Hubbard model bya projective transformation, which is appropriate in the pres-ence of the charge gap. The present work attempts to understand possible spin liquid states in the ring exchange model by examining can-didate ground-state wave functions. This is complementaryto the exact diagonalization studies, since knowing the char-acter of a candidate wave function can give signiﬁcant intu-ition. The model Hamiltonian on the triangular lattice is, in the notation borrowed from Ref. 2, /H208491/H20850 The two-spin exchanges are between all nearest neighbors and reduce simply to Heisénberg interactions P12=P12† =2S1·S2+1 2. The four-spin “ring exchanges” are around all rhombi of the triangular lattice. In the following, we consider only antiferromagnetic cou- pling J2/H110220 and positive J4/H333560; for brevity, we set J2=1. When J4=0, the system is the familiar Heisenberg antiferro- magnet on the triangular lattice and has a three-sublatticeantiferromagnetic /H20849AF/H20850order. Exact diagonalization study of Ref. 1 proposes the phase diagram summarized in Fig. 1. TheAF order is preserved for small J 4/H113510.07−0.1, but is de- stroyed for larger J4and a spin gap opens up. However, in the regime 0.1 /H11351J4/H113510.25 reported in Ref. 1, there are appar- ently many singlet excitations below the spin gap. Also, thespin gap starts to decrease for J 4/H114070.175. In the exact diagonalization studies, it is hard to say which physical state is realized in the absence of clear sig-natures of some particular phase. The question of possiblespin liquid states is taken up here by considering variationalspin liquid wave functions on the triangular lattice. Speciﬁ-cally, we consider one family of such states obtained byGutzwiller-projecting singlet fermionic mean-ﬁeld states. 5,6 We determine the result of the competition with the AF or-dered state by comparing against variational wave functionswith long-range magnetic order. 7 The result of the variational study is summarized in Fig. 2. For small J4/H113510.14, the AF state is stable compared with the tried spin liquid states. For larger J4, we ﬁnd spin liquid states that have lower variational energy than the magneti-cally ordered state. For example, we ﬁnd that projected su-perconductor Ansätze perform well in the regime 0.14 /H11351J 4 /H113510.3. More speciﬁcally, Ansätze with anisotropic extended s-wave, dx2−y2, and dx2−y2+idxypairing patterns have very close optimal energies and much lower than the energy of the trial AF state. Unfortunately, we conclude that the presentstudy is not sufﬁcient to address the nature of the spin liquidin this regime, which we indicate with question marks in theﬁgure. Our observation that the improvement in the trial en-ergy is little sensitive to the speciﬁc pairing pattern may bean indication that the present restricted study cannot accessthe correct ground state in this regime. A more robust conclusion from our study of such spin liquids is that the best Ansätze are close to the projected Fermi sea state and become more so for increasing J 4. Thus, forJ4/H114070.3−0.35 the variational /H9004in our Ansätze reduces to essentially zero /H20849below few percent of the hopping ampli- tude/H20850, and the ground state is essentially the projected Fermi sea.PHYSICAL REVIEW B 72, 045105 /H208492005 /H20850 1098-0121/2005/72 /H208494/H20850/045105 /H208497/H20850/$23.00 ©2005 The American Physical Society 045105-1The aptitude of the projected Fermi sea state can be intu- itively understood as follows. We can view the ring exchangeterm with positive J 4as arising from the electron hopping in the underlying Hubbard model /H20849which we assume is in the insulating phase /H20850. Therefore, such ring exchange J4/H110220 wants the fermions to be as “delocalized” as possible, andthis “kinetic energy” is best satisﬁed in the simple hoppingAnsatz . A more formal mean-ﬁeld argument is given in Sec. III. We now discuss the indications of this study for the pos- sible spin liquid state in /H9260-/H20849ET/H208502Cu2/H20849CN/H208503. This material is close to the metal-insulator transition, so the role of the elec- tron kinetic energy is clearly important. Based on the expe-rience with the ring exchange model, we therefore proposethat the projected Fermi sea state is a good candidate groundstate close to the metallic phase. We verify this more explic-itly by considering a model with ring exchanges obtained bya projective transformation of the triangular lattice Hubbardmodel at order t 4/U3. For the /H9260-/H20849ET/H208502Cu2/H20849CN/H208503compound we estimate J4/J2/H112290.3. The results are summarized in Fig. 4. The work of Ref. 4 on the triangular lattice Hubbardmodel can be interpreted as an elaborate numerical studybuilding up on free-fermion states, and hints some support tothe proposed projected Fermi sea phase. The proposed picture has many physical consequences. We have a Fermi surface of spinons, and therefore expect nospin gap and ﬁnite spin susceptibility down to zero tempera-ture consistent with the experimental observations. An accu-rate treatment of the no-double-occupancy constraint andﬂuctuations requires that the spinons are coupled to a ﬂuctu-ating U /H208491/H20850gauge ﬁeld. Such spinon-gauge ﬁeld system has been studied extensively and is expected to exhibit someunusual behavior. 9–16For example, one expects a singular contribution to the speciﬁc heat Csing/H11011T2/3at low tempera- tures in two dimensions; the corresponding enhancement in“spin entropy” has concrete consequences for the phaseboundaries. The rest of the paper is organized as follows. In Sec. II we specify the variational states considered in this work. In Sec.III we seek qualitative understanding of the ring exchangeenergetics by considering a fermionic large- Ntreatment of the ring exchange Hamiltonian. In Sec. IV we consider theconnection with the triangular lattice Hubbard model andpossible application to /H9260-/H20849ET/H208502Cu2/H20849CN/H208503. In particular, we describe experimental signatures of the proposed spinon Fermi surface-gauge system. II. VARIATIONAL STATES AND ENERGETICS In this section, we describe variational states used in the present work. Trial spin liquid states are constructed byGutzwiller projection of fermionic mean-ﬁeld states.5We compare their energetics against AF ordered trial wave func-tions constructed using the approach of Huse and Elser. 7 Spin liquid trial states . The starting point here is the fer- mionic mean ﬁeld treatment of the Heisenberg model. A re-cent and very detailed description can be found in Ref. 5.The setup for constructing trial wave functions is as follows.Each spin operator is written in terms of two fermions c r↑ and cr↓,Sr=cr†/H20849/H9268/2/H20850cr, with precisely one fermion per site. Heisenberg exchange interaction is written as a four-fermion interaction, which is then decoupled in the singlet channel. Aconvenient formulation of the mean ﬁeld is to consider gen-eral spin rotation invariant trial Hamiltonian H trial=−/H20858 rr/H11032/H20851trr/H11032cr/H9268†cr/H11032/H9268+/H20849/H9004rr/H11032cr↑†cr/H11032↓†+H.c. /H20850/H20852,/H208492/H20850 with tr/H11032r=trr/H11032,/H9004r/H11032r=/H9004rr/H11032. For each such trial Hamiltonian we obtain the corresponding ground state. An SU /H208492/H20850invariant formulation of the single occupancy constraint is that the isospin operator Tr/H11013/H9274r†/H20849/H9270/2/H20850/H9274ris zero on each site; here /H9274r↑=cr↑,/H9274r↓=cr↓†, and /H92701–3are Pauli matrices. In the mean ﬁeld, we require that this constraint is satisﬁed on average,which is achieved by tuning appropriate on-site terms. Goingbeyond the mean ﬁeld, the physical spin wave function isobtained by projecting out double occupation of sites. Many such trial states can be constructed, but there is also a gauge redundancy in this construction. Here, one is helpedconsiderably by the recently available classiﬁcation schemeof Wen 5,6that allows one to construct all possible such fer- mionic mean-ﬁeld states that lead to physically distinct spinliquids with speciﬁed lattice symmetries. We numerically evaluate the expectation values of the two-spin and four-spin exchanges in such states using stan-dard determinantal wave function techniques /H20849so-called variational Monte Carlo /H20850. 17We consider Ansätze with differ- ent sets of lattice symmetries, with and without time reversal,but primarily we focus on the nearest-neighbor Ansätze that respect upon projection the lattice translation symmetry. Wethen vary the parameters to optimize the trial energy. AF ordered trial states . We want to compare the ring ex- FIG. 1. Phase diagram of the model /H208491/H20850from the exact diago- nalization study of Refs. 1 and 2. The magnetic order is destroyedforJ 4/H114070.07−0.1; a spin gap is observed in the regime 0.1 /H11351J4 /H113510.25, but also many singlets below the spin gap. The spin gap is decreasing for J4/H114070.175. FIG. 2. Variational phase diagram for the Hamiltonian /H208491/H20850. The AF ordered variational state has the lowest energy for small J4, but becomes unstable for J4/H114070.14 compared with the fermionic spin liquid states. One example of such spin liquid is the projectedd x2−y2+idxysuperconductor Ansatz , with the optimal variational pa- rameter /H20849/H9004/t/H20850var=0.22, 0.13, 0.05, 0.02 for J4=0.15, 0.20, 0.25, 0.30, respectively. Some other Ansätze give very close optimal en- ergy, and the situation is particularly not clear near the AF state. ButforJ 4/H114070.3, our best Ansätze become essentially the projected Fermi sea state. We caution that for signiﬁcantly larger J4states with more complicated magnetic orders—e.g., with four-sublatticeorder—may enter the energetics competition /H20849Ref. 8 /H20850, which is not considered here.OLEXEI I. MOTRUNICH PHYSICAL REVIEW B 72, 045105 /H208492005 /H20850 045105-2change energetics of the spin liquid states with the energetics of the antiferromagnetically ordered states. For this purpose,we use the family of variational states considered by Huseand Elser, 7which capture well the Heisenberg model ener- getics. Our primary goal here is to see how the AF state isdisfavored by the ring exchanges. Since we are comparingwith rather different states and are looking for the energylevel crossing, we do not need to know the ground-state en-ergy very accurately, and the wave functions of Ref. 7 shouldbe sufﬁcient to get rough idea of the ring exchange energet-ics in the AF state. For details on these wave functions andnumerical evaluations, the reader is referred to the originalpaper. Variational results . We compared the trial energies of the AF ordered states and the fermionic spin-liquid states, andthe result is summarized in Fig. 2. For small J 4, the ordered states have lower energy, but for J4/H114070.14 the spin liquid states win. The optimal spin liquid Ansätze have the follow- ing structure. The dominant part is the uniform triangularlattice hopping t rr/H11032, and for J4/H114070.3−0.35 we essentially ﬁnd the projected Fermi sea state. In the intermediate regime 0.14/H11351J4/H113510.3, we ﬁnd that the trial energy is improved upon adding /H9004rr/H11032correlations into the mean-ﬁeld wave func- tion. Somewhat perplexingly, we ﬁnd that the result is not very sensitive to the speciﬁc “pairing” pattern. Thus, opti-mized wave functions with extended anisotropic s-wave, d x2−y2, and dx2−y2+idxypairing patterns have close energies. This may be an indication of an instability towards a state that cannot be captured in the context of the trial fermionicstates. The situation is particularly inconclusive close to theAF phase, where several other trial states have competitiveenergies. In summary, we ﬁnd that ring exchanges disfavor the AF ordered state compared with the fermionic spin liquid states,but our study is not conclusive as to which spin liquid state isrealized when the transition happens. Away from the transi-tion, we suggest that the optimal spin liquid state is the pro-jected Fermi sea state. III. FERMIONIC LARGE NSTUDY OF THE RING EXCHANGE ENERGETICS In this section, we present a fermionic large Nstudy of the ring exchange Hamiltonian. Here, natural “trial” states arepure hopping states, and this approach gives us some insightinto their energetics. In particular, it shows how the ringexchanges favor the uniform hopping state, i.e., the projectedFermi sea state. The treatment below was suggested to thepresent author by Senthil. The mean-ﬁeld analysis is alsorather similar to an early work of Ioffe and Larkin. 18 Consider the following generalization of the ring ex- change Hamiltonian /H208491/H20850to an SU /H20849N/H20850spin model HˆSU/H20849N/H20850=J N/H20858 /H2085512/H20856/H20849c1/H9251†c1/H9252/H20850/H20849c2/H9252†c2/H9251/H20850+K N3/H20858 P/H20851/H20849c1/H9251†c1/H9252/H20850/H20849c2/H9252†c2/H9253/H20850 /H11003/H20849c3/H9253†c3/H9254/H20850/H20849c4/H9254†c4/H9251/H20850+H.c . /H20852. We use conventional fermionic representation with Nfer- mion ﬂavors; spin states on each site are viewed as states ofN/2 fermions, i.e., we have occupancy constraint cr/H9251†cr/H9251=N/2 /H208493/H20850 for each site r. In the above, summation over repeated ﬂavor indices is implied. Our generalization of the exchange opera-tors preserves the character of moving spins around a ring.ForN=2, this Hamiltonian reduces precisely to the spin-1/2 Hamiltonian /H208491/H20850with J=2J 2,K=8J4. /H208494/H20850 A similar large Nformulation was considered in a different context in Ref. 19. We also remark here that the general N formulation allows nontrivial exchanges involving threespins. This is unlike the N=2 case where such three-spin exchange reduces to a combination of two-spin exchanges.The three-spin exchanges can be easily included in the fol-lowing analysis; to stay in line with the rest of the paper, weonly consider the two-spin and four-spin exchanges. We formulate the large Nprocedure in the spirit of the variational approach. Consider a single-particle “trial”Hamiltonian Hˆ trial=−/H20858 /H20855rr/H11032/H20856/H20849trr/H11032cr/H9251†cr/H11032/H9251+H.c . /H20850−/H20858 r/H9262rcr/H9251†cr/H9251. We ﬁnd the ground state and use it as a trial wave function for the Hamiltonian HˆSU/H20849N/H20850. In the mean ﬁeld, the occupancy constraints are implemented on average by tuning the chemi- cal potentials /H9262r. The trial energy to leading order in 1/ Nis given by EMF N=−J/H20858 /H2085512/H20856/H20841/H927312/H208412−K/H20858 P/H20849/H927312/H927323/H927334/H927341+c.c. /H20850, where /H9273rr/H11032/H11013/H20855cr†cr/H11032/H20856is the single-species expectation value. We now have to minimize EMFover the possible trr/H11032in the trial Hamiltonian. This leads to the following self- consistency conditions: /H9011−1trr/H11032=J/H9273rr/H11032+/H20858 P=/H208511234 /H20852=/H20851rr/H1103234/H20852K/H927323/H927334/H927341*, /H208495/H20850 where the last sum is over all ring exchange plackets that contain the bond /H20855rr/H11032/H20856as one of the consecutive bonds. Also, we have explicitly indicated the fact that the trial energy does not depend on the absolute scale in the trial Hamiltonian butonly on the relative pattern of t rr/H11032. We ﬁrst make some general observations about this pro- cedure. First of all, note that the self-consistency conditionsimply that the optimal state can have nonzero t rr/H11032only on the bonds that have nonzero Jrr/H11032or that appear in some ring exchange placket. For the triangular lattice model studied here, we then have to consider only nearest-neighbor trr/H11032. Second, we see quite generally that the ring exchange con- tribution for a given placket has the form − K/H20841/H9273/H208414cos/H20849/H9021P/H20850, where /H20841/H9273/H20841is the geometric mean of the absolute values of /H9273rr/H11032around the placket, while /H9021Pis the “ﬂux” of the correspond- ing phase factors. Thus, the positive ring exchange wants tosmear the fermions over the lattice with no ﬂuxes.VARIATIONAL STUDY OF TRIANGULAR LATTICE … PHYSICAL REVIEW B 72, 045105 /H208492005 /H20850 045105-3To be more precise, let us consider several simple trial states. The uniform ﬂux state has ﬂux /H9278through each tri- angle. The expectation values /H9273rr/H11032=/H20855cr/H11032†cr/H20856have the same pat- tern of ﬂuxes as the input trr/H11032, and the trial energy per site is E/H9278=−3 J/H20841/H9273/H9278/H208412−6K/H20841/H9273/H9278/H208414cos/H208492/H9278/H20850, /H208496/H20850 since the ﬂux through each rhombus is 2 /H9278. Among such ﬂux states, we ﬁnd that for K/H113512.76Jthe best state has /H9266/2 ﬂux through each placket /H20849this state has the largest /H20841/H9273/H9278/H20841/H20850, while forK/H114072.76Jthe best state has zero ﬂux The numerical val- ues of the energy per site in the two states can be obtainedfrom E /H9278=/H9266/2= − 0.120 J+ 0.0096 K, /H208497/H20850 E/H9278=0= − 0.081 J− 0.0044 K. /H208498/H20850 For large enough K/Jthe zero-ﬂux state is stable against adding small ﬂux /H9278because /H20841/H9273/H9278/H208412//H20841/H92730/H208412/lessorapproxeql1+0.2 /H92782. We also considered so-called dimer states such that non- zero trr/H11032form nonoverlapping dimer covering of the lattice. These states break translational invariance, and any dimer covering produces such a state. It is well known that thesestates can have lower Heisenberg exchange energy in thelarge Nlimit. This is because the occupied bonds attain the maximal expectation value /H20841 /H9273rr/H11032/H20841maxand their contribution can be sufﬁcient to produce the lowest total energy. The en- ergy per site in any dimer state is Edimer= − 0.125 J, /H208499/H20850 and is indeed the lowest energy for K=0. However, the dimer states gain no ring exchange energy, and for K/H114079.9J the zero ﬂux state becomes the lowest energy state. Finally,the so called box states have identical two-spin exchangeenergy with the dimer states but also nontrivial ﬂuxes andtherefore do not enter the competition for the ground stateforK/H110220. We performed full optimization over t rr/H11032of the mean-ﬁeld energy considering possible unit cells with up to four sites, and found that the above simple states are indeed sufﬁcientto describe the ground state in the large Nlimit: The optimal state is one of the dimer states for K/H113519.9Jand becomes the zero ﬂux state for larger K. The complete study is summa- rized in Fig. 3. To make better connection with the spin-1/2 system, we remark that we expect the dimer states to be energeticallydisfavored even for small Kin the spin-1/2 case, e.g., com-pared with the ﬂux states discussed above. This is because the Gutzwiller projection enhances local antiferromagneticcorrelations more strongly in the translationally invariantmean-ﬁeld states than in the dimerized states. 20More quan- titatively, the enhancement factor for the Heisenberg energy is roughly gJtransl inv=4 for the translationally invariant states, while it is only gJdimer=2 for the dimer states. Furthermore, we expect even stronger enhancements in the ring exchangeenergy upon the projection. Taking all this into account, weexpect the Fermi sea state to be favored for rather moderateJ 4/J2in the spin-1/2 system. To conclude the mean ﬁeld discussion, our main message is that the positive ring exchange dislikes the ﬂuxes andwants to make the system as uniform as possible. This is bestrealized in the projected Fermi sea state. Going beyond the mean ﬁeld, we obtain a theory of fer- mions coupled to a ﬂuctuating gauge ﬁeld /H20849a 0,a/H20850, where the temporal a0/H20849r,/H9270/H20850enforce the local occupancy constraints while the spatial components represent the relevant ﬂuctua- tions of trr/H11032/H20849/H9270/H20850/H11015/H20841t/H20841eiarr/H11032/H20849/H9270/H20850. The corresponding continuum theory /H20849“relativistic electrodynamics in a metal” /H20850was studied in Refs. 9–16, and we will quote some results in the nextsection. In the Appendix, we study long-distance properties of the Gutzwiller-projected wave function in some detail. As men-tioned in the Appendix, this wave function may be not suf-ﬁcient to capture the long wavelength behavior of the actualphase, since the projection treats only the a 0ﬂuctuations, but does not include the ﬂuctuations of arr/H11032, while the latter are crucial in the effective theory.9–16This is pointing a possible limitation of the projected wave function approach for thespinon-gauge system. We still expect that the variationalstudy of the previous section gets the crude energetics cor-rectly in the ring exchange model. This is also what we ex-pect from the mean-ﬁeld treatment, and leads us to proposethe effective spinon-gauge theory. A ﬁner numerical applica-tion likely requires more advanced techniques, perhaps in thespirit of Ref. 4 for the triangular Hubbard model. It would beinteresting, for example, to look for the 2 k Fsignature13in the more elaborate work of Ref. 4, which may be a more accu-rate realization of the spinon-gauge ground state. IV. APPLICATION TO POSSIBLE SPIN-LIQUID STATE IN/H9260-„ET …2Cu2„CN …3 We now discuss possible spin liquid state in the organic compound /H9260-/H20849ET/H208502Cu2/H20849CN/H208503, which is insulating and shows no magnetic order down to the lowest experimental tempera- tures. It is believed3,4,21that the conducting layer of this ma- terial is well described by a single-band triangular latticeHubbard model at half ﬁlling with t/U/H112291/8 and only small hopping anisotropy of about 6%. Unlike the square lattice case, for the half-ﬁlled triangular lattice we expect a metallic phase for large enough t/U. Ref- erence 4 estimates the metal-insulator transition to occur at/H20849t/U/H20850 MI/H112291/5, so the /H9260-/H20849ET/H208502Cu2/H20849CN/H208503material is on the insulating side. Using an elaborate numerical technique, Ref. 4 ﬁnds a nonmagnetic insulator in this regime. We want todevelop some picture of this state. FIG. 3. Summary of the large Nstudy of the Hamiltonian HˆSU/H20849N/H20850./H20849a/H20850Phase diagram from the mean-ﬁeld energy optimization over translationally invariant states. /H20849b/H20850Full optimization.OLEXEI I. MOTRUNICH PHYSICAL REVIEW B 72, 045105 /H208492005 /H20850 045105-4The ideology we pursue here is that the insulating phase can be described by an effective spin model. Since the sys-tem is close to the metal-insulator transition, it is not enoughto stop at two-spin exchange interactions. Starting with theHubbard model, the effective Hamiltonian to order t 4/U3 was obtained in Ref. 22. Specialized to the triangular lattice, the spin Hamiltonian reads Hˆeff=Hˆring/H20851J2,J4/H20852+/H20858 /H20855/H20855ij/H20856/H20856J/H11033Si·Sj+/H20858 /H20855/H20855/H20855ij/H20856/H20856/H20856J/H11630Si·Sj./H2084910/H20850 Here Hringis the ring exchange Hamiltonian /H208491/H20850with J2=/H208491 −32t2/U2/H208502t2/U,J4=20t4/U3. The effective Hamiltonian has additional Heisenberg exchanges J/H11033=−16 t4/U3between second neighbors /H20849separated by a distance /H208813/H20850and J/H11630 =4t4/U3between third neighbors /H20849separation 2 lattice spac- ings/H20850. Our grouping of the terms in the effective Hamiltonian is intended to make it look as close as possible to the ringexchange model studied in the previous sections. For the /H9260-/H20849ET/H208502Cu2/H20849CN/H208503compound, we estimate J4/J2 /H112290.3, which puts the ring exchange model into the proposed spinon Fermi sea regime. Further neighbor interactions notincluded in the J 2-J4model do not modify this result, even though J/H11033andJ/H11630are roughly of the same magnitude as J4. This stability is because the corresponding further neighborspin correlations are small in the spin liquid regime. To proceed more systematically, we repeat the variational study with the effective Hamiltonian /H2084910/H20850. The resulting phase diagram is shown in Fig. 4 in terms of the Hubbardmodel parameter t/U. From this study, we propose that the insulating ground state is the antiferromagnet for t/U/H113511/9 /H20849this corresponds roughly to the ring exchange parameter J 4/J2/H110150.2-0.25 /H20850. For larger t/U, our best trial state is essen- tially the projected Fermi sea state, and the variational /H9004 /H20849which can be used to improve the trial energy slightly /H20850is small already at the transition from the AF state. In the sameﬁgure, we also indicate the metallic phase expected for t/U /H114071/5. It should be emphasized that we do not treat either Hamil- tonian /H2084910/H20850or/H208491/H20850as more realistic or less realistic, particu- larly since we are dealing with the system near the metal- insulator transition. The above variational study with Hˆ effis presented primarily to illustrate that our results are not de-stabilized by making the Hamiltonian “more realistic.” Weexpect that our main prediction for the spin-liquid state closeto the metal-insulator transition is robust, since the proposed Gutzwiller-projected Fermi sea state is even more favored byincluding further effects of the electron kinetic energy. Also,the results of Ref. 4 give us some indication on the stabilityof the proposed state, since that study is building up on free-fermion states. Physical properties in the spin liquid phase with spinon Fermi surface . The effective description of the proposed phase has spinon Fermi sea coupled to a dynamically gener-ated gauge ﬁeld. It has been argued 9–16that this spinon-gauge system is described by a nontrivial ﬁxed point and showsunusual behavior, which can be tested in experiments. Be-low, we list some thermodynamic properties of this Mottinsulator. This phase is in some sense the closest one can getto the Fermi liquid while remaining a charge insulator, andshares some properties with the metal due to the presence ofthe spinon Fermi surface, but also has some “non-Fermi-liquid” properties. Thus, spin susceptibility is expected to approach a con- stant as temperature Tgoes to zero: /H9273spin/H20849T→0/H20850/H11011/H9262B2/H92630. /H2084911/H20850 This is a consequence of having gapless spinon excitations over the entire Fermi surface and is in fact observed in /H9260-/H20849ET/H208502Cu2/H20849CN/H208503.23Here, /H92630is the density of states at the “Fermi surface” in the spinon band structure determined by the spinon “hopping amplitude” tspinon. The latter is set by the Heisenberg exchange energy tspinon /H11011Jand is different from the bare electron hopping amplitude tel/H20849remember that J /H11011tel2/U/H20850. For the triangular lattice at half-ﬁlling, we have /H92630=0.28/ tspinon per triangular lattice site and including spin. Reference 3 reports /H9273=2.9/H1100310−4emu/mol at low tempera- tures, from which we estimate tspinon /H11015350 K. This compares favorably with the reported magnitude of the Heisenberg ex- change J=250 K. Furthermore, Ref. 24 observes that13C nuclear spin relaxation rate 1/ /H20849T1T/H20850approaches a constant at low temperature, which is what one expects for the spinon Fermi surface. Speciﬁc heat, on the other hand, is expected to show non- Fermi-liquid behavior C/H11011kB/H92630tspinon1/3/H20849kBT/H208502/3. /H2084912/H20850 This is written to contrast with the Fermi liquid /H11011Tbehavior, and means that the spin entropy in this charge insulator is infact larger than in the metallic state at low temperature. Thisis very different from the antiferromagnet or gapped spinliquid insulators which have low spin entropy. In particular,the ﬁnite temperature ﬁrst-order transition line between theproposed spin liquid and the metallic state is expected tobend towards the metallic state with increasingtemperature 25,26 pMI/H20849T/H20850−pMI/H208490/H20850/H11011T5/3. /H2084913/H20850 In the last formula, pis an applied pressure which drives the insulator to metal transition.3,27This tendency is actually ob- served in the /H9260-/H20849ET/H208502Cu2/H20849CN/H208503material.28 FIG. 4. Proposed phase diagram for the triangular lattice Hub- bard model. The present study is based on the effective spin Hamil-tonian /H2084910/H20850and applies only to the insulating regime expected for t/U/H113511/5 from Ref. 4. Close to the metal-insulator transition, we propose the spin liquid state with spinon Fermi surface. For smallert/U/H113511/9, the best state is AF ordered. The /H9260-/H20849ET/H208502Cu2/H20849CN/H208503com- pound has t/U/H112291/8VARIATIONAL STUDY OF TRIANGULAR LATTICE … PHYSICAL REVIEW B 72, 045105 /H208492005 /H20850 045105-5V. CONCLUSIONS In summary, we considered the spin-1/2 ring exchange model on the triangular lattice from the variational perspec-tive and identiﬁed the instability of the antiferromagneticallyordered state towards spin liquid state in the regime of mod-erate ring exchange couplings. Our best trial states becomethe Gutzwiller-projected Fermi sea state for larger J 4. De- spite the limitations of the variational approach, it is hopedthat the present work may give complimentary informationand useful guidance for understanding the exact diagonaliza-tion results. We also studied the effective spin Hamiltonian appropri- ate for describing charge insulator states of the triangularlattice Hubbard model. The effective Hamiltonian includesHeisenberg exchanges as well as ring exchanges, and so isclose to the considered ring exchange model. The study ismotivated by the tentative spin liquid state in the /H9260-/H20849ET/H208502Cu2/H20849CN/H208503compound, which is modeled by the trian- gular lattice Hubbard model in the vicinity of the metal- insulator transition. We ﬁnd that upon including the ring ex-changes but well in the insulating regime, theantiferromagnet gives way to the spin liquid state which isessentially the projected Fermi sea state. In view of this ﬁnd-ing, we propose that the effective description of the nonmag-netic insulator phase has Fermi sea of spinons coupled to thedynamically generated gauge ﬁeld. This spin liquid phasefeatures a number of unusual properties which can be lookedfor in experiments. It would be very exciting if this remark-able state is indeed realized in the /H9260-/H20849ET/H208502Cu2/H20849CN/H208503mate- rial.Note added in proof. A recent preprint by Lee and Lee29 studies half-ﬁlled triangular lattice Hubbard model using slave-rotor representation, and suggests the state with spinonFermi surface as a candidate for the spin liquid observed in /H9260-/H20849ET/H208502Cu2/H20849CN/H208503, similar to the present work. Reference 29 also discusses further experimental consequences for the pro- posed state, and in particular predicts a highly unusual tem-perature dependence of the thermal conductivity. 30 ACKNOWLEDGMENTS The author has beneﬁted from many useful discussions with M. P. A. Fisher, V. Galitski, P. Nikolic, and A. Vish-wanath, and is especially grateful to T. Senthil for motivatingthis problem and sharing many insights throughout thecourse of the study. This work was started at MIT and wassupported by NSF grants Nos. DMR-0213282 and DMR-0201069. The work at KITP was supported through NSFgrant No. PHY-9907949. APPENDIX: PROPERTIES OF THE PROJECTED FERMI SEA WAVE FUNCTION We describe some properties of the projected Fermi sea state. Figure 5 shows spin correlations in the projected wavefunction and also in the free fermion state before the projec-tion. In the free fermion state, the spin correlation behaves as−cos 2/H20849kFr−3/H9266/4/H20850/r3at large distances, which oscillates with the wave vector 2 kFwhile always staying negative. To facili- tate the comparison, Fig. 5 shows the mean-ﬁeld correlationsin the speciﬁc ﬁnite system /H20849the ﬁnite size effects are fairly large because of the gaplessness over the Fermi surface /H20850.W e observe that the effect of the projection is not strong: For therange studied, the mean-ﬁeld result multiplied by theGutzwiller enhancement factor g J=4 gives a reasonable ap- proximation for the actual correlation.20After the projection, the correlation function now swings to positive values aswell, but the overall magnitude is roughly captured by thesimple renormalization factor. We also studied spin chirality correlations /H20849not shown /H20850, and found that these are very small beyond few lattice spac-ings. The effective theory of the proposed phase has Fermisea of spinons coupled to a dynamically generated gaugeﬁeld. 9–15The measured spin correlations in the projected wave function represent some puzzle in this respect: Ref. 13predicts that the spin structure factor is singularly enhancednear 2 k Fin the spinon-gauge system. We ﬁnd that in the projected wave function the structure factor remains ﬁnitethroughout the Brillouin zone and that the overall rate ofdecay of spin correlations is roughly the same as in the freefermion state. One possible source of this difference is thatthe projected wave function has ﬁxed t rr/H11032and therefore does not include the ﬂuctuations of the spatial components of the gauge ﬁeld; only the temporal component is “included” bythe projection. This is a limitation of the projected wavefunction approach for the spinon-gauge system. FIG. 5. Spin correlation in the projected Fermi sea state. Mea- surements are done on a 24 /H1100324 triangular lattice. The mean-ﬁeld wave function is constructed for periodic boundary conditions andexcluding the zero momentum single-particle state in order to avoidFermi surface points while satisfying the lattice rotation symmetryfor the ﬁnite system /H20849this does not affect the long-distance proper- ties of the wave function which is our focus here /H20850. Note the oscil- lating character of the correlation /H20851with the period /H110152 /H9266//H208492kF/H20850,kF /H110152.69/H20852. Also note that the renormalized mean ﬁeld roughly repro- duces the overall magnitude of the correlations.OLEXEI I. MOTRUNICH PHYSICAL REVIEW B 72, 045105 /H208492005 /H20850 045105-61W. LiMing, G. Misguich, P. Sindzingre, and C. Lhuillier, Phys. Rev. B 62, 6372 /H208492000 /H20850. 2G. Misguich, C. Lhuillier, B. Bernu, and C. Waldtmann, Phys. Rev. B 60, 1064 /H208491999 /H20850. 3Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, Phys. Rev. Lett. 91, 107001 /H208492003 /H20850. 4M. Imada, T. Mizusaki, and S. Watanabe, cond-mat/0307022 /H20849un- published /H20850; H. Morita, S. Watanabe, and M. Imada, J. Phys. Soc. Jpn. 71, 2109 /H208492002 /H20850. 5X.-G. Wen, Phys. Rev. B 65, 165113 /H208492002 /H20850; cond-mat/0107071. 6Y. Zhou and X.-G. Wen, cond-mat/0210662 /H20849unpublished /H20850. 7D. A. Huse and V. Elser, Phys. Rev. Lett. 60, 2531 /H208491988 /H20850. 8S. E. Korshunov, Phys. Rev. B 47, 6165 /H208491993 /H20850; T. Momoi, K. Kubo, and K. Niki, Phys. Rev. Lett. 79, 2081 /H208491997 /H20850. K. Kubo, H. Sakamoto, T. Momoi, and K. Niki, J. Low Temp. Phys. 111, 583/H208491998 /H20850. 9M. Y. Reizer, Phys. Rev. B 40, 11 571 /H208491989 /H20850. 10P. A. Lee, Phys. Rev. Lett. 63, 680 /H208491989 /H20850. 11P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 /H208491992 /H20850. 12J. Polchinski, Nucl. Phys. B 422, 617 /H208491994 /H20850. 13B. L. Altshuler, L. B. Ioffe, and A. J. Millis Phys. Rev. B 50, 14 048 /H208491994 /H20850. 14C. Nayak and F. Wilczek, Nucl. Phys. B 417, 359 /H208491994 /H20850;430, 534/H208491994 /H20850. 15Y. B. Kim, A. Furusaki, X. G. Wen, and P. A. Lee, Phys. Rev. B 50, 17917 /H208491994 /H20850; Y. B. Kim, P. A. Lee, and X. G. Wen, ibid. 52, 17 275 /H208491995 /H20850. 16T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 /H208492004 /H20850. 17C. Gros, Ann. Phys. /H20849N.Y./H20850189,5 3/H208491989 /H20850; D. M. Ceperley, G. V.Chester, and M. H. Kalos, Phys. Rev. B 16, 3081 /H208491977 /H20850. 18L. B. Ioffe and A. I. Larkin, Phys. Rev. B 39, 8988 /H208491989 /H20850. 19X.-G. Wen, Phys. Rev. Lett. 88, 011602 /H208492002 /H20850. 20F. C. Zhang, C. Gros, T. M. Rice, and H. Shiba, Supercond. Sci. Technol. 1,3 6/H208491988 /H20850; cond-mat/0311604. 21R. H. McKenzie, Comments Condens. Matter Phys. 18, 309 /H208491998 /H20850; cond-mat/9802198. 22A. H. MacDonald, S. M. Girvin, and D. Yoshioka, Phys. Rev. B 37, 9753 /H208491988 /H20850. 23The author is grateful to A. Vishwanath for emphasizing this experimental observation. 24A. Kawamoto, Y. Honma, and K.-i. Kumagai, Phys. Rev. B 70, 060510 /H20849R/H20850/H208492004 /H20850. 25These results were pointed out to the author by T. Senthil. 26The insulator-metal ﬁrst-order phase boundary can be obtained from the Clapeyron equation dp/dT=/H20849Sins−Smetal/H20850//H20849Vins −Vmetal/H20850. Here the volume difference Vins−Vmetal/H110220 is positive since the metal is more stable at higher pressure. Using Sins /H11011T2/3which dominates over Smetal/H11011Tat low temperatures, we obtain Eq. /H2084913/H20850. 27T. Komatsu, N. Matsukawa, T. Inoue, and G. Saito, J. Phys. Soc. Jpn. 65, 1340 /H208491996 /H20850. 28K. Kanoda, “Mott Criticality and Spin Liquid State Revealed in Quasi-2D Organics,” KITP Program talk /H208492004 /H20850; Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, and G. Saito, cond-mat/0504273 /H20849unpublished /H20850. 29S.-S. Lee and P. A. 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PhysRevB.77.035323.pdf	Reexamination of spin decoherence in semiconductor quantum dots from the equation-of-motion approach J. H. Jiang,1,2Y. Y. Wang,2and M. W. Wu1,2,* 1Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People’ s Republic of China 2Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’ s Republic of China /H20849Received 2 April 2007; revised manuscript received 26 June 2007; published 18 January 2008 /H20850 The longitudinal and transversal spin decoherence times, T1and T2, in semiconductor quantum dots are investigated from the equation-of-motion approach for different magnetic ﬁelds, quantum dot sizes, and tem-peratures. Various mechanisms, such as the hyperﬁne interaction with the surrounding nuclei, the Dresselhausspin-orbit coupling together with the electron–bulk-phonon interaction, the g-factor ﬂuctuations, the direct spin-phonon coupling due to the phonon-induced strain, and the coaction of the electron–bulk- and/or surface-phonon interaction together with the hyperﬁne interaction are included. The relative contributions from thesespin decoherence mechanisms are compared in detail. In our calculation, the spin-orbit coupling is included ineach mechanism and is shown to have marked effect in most cases. The equation-of-motion approach isapplied in studying both the spin relaxation time T 1and the spin dephasing time T2, either in Markovian or in non-Markovian limit. When many levels are involved at ﬁnite temperature, we demonstrate how to obtain thespin relaxation time from the Fermi golden rule in the limit of weak spin-orbit coupling. However, at hightemperature and/or for large spin-orbit coupling, one has to use the equation-of-motion approach when manylevels are involved. Moreover, spin dephasing can be much more efﬁcient than spin relaxation at high tem-perature, though the two only differ by a factor of 2 at low temperature. DOI: 10.1103/PhysRevB.77.035323 PACS number /H20849s/H20850: 72.25.Rb, 73.21.La, 71.70.Ej I. INTRODUCTION One of the most important issues in the growing ﬁeld of spintronics is quantum information processing in quantumdots /H20849QDs /H20850using electron spin. 1–5A main obstacle is that the electron spin is unavoidably coupled to the environment/H20849such as the lattice /H20850which leads to considerable spin deco- herence /H20849including longitudinal and transversal spin decoherences /H20850. 6,7Various mechanisms such as the hyperﬁne interaction with the surrounding nuclei,8,9the Dresselhaus and/or Rashba spin-orbit coupling /H20849SOC /H2085010,11together with the electron-phonon interaction, g-factor ﬂuctuations,12the direct spin-phonon coupling due to the phonon-inducedstrain, 9and the coaction of the hyperﬁne interaction and the electron-phonon interaction can lead to the spin decoherence.There are quite a lot of theoretical works on spin decoher-ence in QD. Speciﬁcally, Khaetskii and Nazarov analyzedthe spin-ﬂip transition rate using a perturbative approach dueto the SOC together with the electron-phonon interaction,g-factor ﬂuctuations, and the direct spin-phonon coupling due to the phonon-induced strain qualitatively. 13–15After that, the longitudinal spin decoherence time T1due to the Dresslhaus and/or the Rashba SOC together with theelectron-phonon interaction was studied quantitatively inRefs. 16–26. Among these works, Cheng et al. 18developed an exact diagonalization method and showed that due to thestrong SOC, the previous perturbation method 14–16is inad- equate in describing T1. Furthermore, they also showed that the perturbation method previously used missed an importantsecond-order energy correction and would yield qualitativelywrong results if the energy correction is correctly includedand only the lowest few states are kept as those in Refs.14–16. These results were later conﬁrmed by Destefani andUlloa. 21The contribution of the coaction of the hyperﬁne interaction and the electron-phonon interaction to longitudi-nal spin decoherence was calculated in Refs. 27and28.I n contrast to the longitudinal spin decoherence time, there arerelatively fewer works on the transversal spin decoherencetime T 2, also referred to as the spin dephasing time /H20849while the longitudinal spin decoherence time is referred to as the spinrelaxation time for short /H20850. The spin dephasing time due to the Dresselhaus and/or the Rashba SOC together with theelectron-phonon interaction was studied by Semenov andKim 29and by Golovach et al.20The contributions of the hyperﬁne interaction and the g-factor ﬂuctuation were stud- ied in Refs. 30–44and in Ref. 45, respectively. However, a quantitative calculation of electron spin decoherence inducedby the direct spin-phonon coupling due to phonon-inducedstrain in QDs is still missing. This is one of the issues we aregoing to present in this paper. In brief, the spin relaxation/H20849dephasing /H20850due to various mechanisms has been studied pre- viously in many theoretical works. However, almost all ofthese works only focus individually on one mechanism. Kha-etskii and Nazarov discussed the effects of different mecha-nisms on the spin relaxation time. Nevertheless, their resultsare only qualitative and there is no comparison of the relativeimportance of the different mechanisms. 13–15Recently, Se- menov and Kim discussed various mechanisms contributedto the spin dephasing, 46where they gave a “phase diagram” to indicate the most important spin dephasing mechanism inSi QD where the SOC is not important. However, the SOC is very important in GaAs QDs. To fully understand the micro-scopic mechanisms of spin relaxation and dephasing and toachieve control over the spin coherence in QDs, 47–49one needs to gain insight into the relative importance of eachmechanism to T 1and T2under various conditions. This is one of the main purposes of this paper.PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 1098-0121/2008/77 /H208493/H20850/035323 /H2084919/H20850 ©2008 The American Physical Society 035323-1Another issue we are going to address relates to different approaches used in the study of the spin relaxation time.The Fermi-golden-rule approach, which is widely used inthe literature, can be used in the calculation of the relaxa-tion time /H9270i→fbetween any initial state /H20841i/H20856and ﬁnal state /H20841f/H20856.12–19,21,23–25,27,28,50–52However, the problem is that when the process of the spin relaxation relates to many states /H20849e.g., when temperature is high, the electron can distribute overmany states /H20850, one should ﬁnd a proper way to average over the relaxation times /H20849 /H9270i→f/H20850of the involved processes to give the total spin relaxation time /H20849T1/H20850. What makes it difﬁcult in GaAs QDs is that all the states are impure spin states with different expectation values of spin. In the existing literature, spin relaxation time is given by the average of the relaxationtimes of processes from the initial state /H20841i/H20856to the ﬁnal state /H20841f/H20856/H20849with opposite majority spin of /H20841i/H20856/H20850weighted by the dis- tribution of the initial states f i,18,51,52i.e., T1−1=/H20858 iffi/H9270i→f−1. /H208491/H20850 This is a good approximation in the limit of small SOC as each state only carries a small amount of minority spin.However, when the SOC is very strong which happens athigh levels, it is difﬁcult to ﬁnd the proper way to performthe average. We will show that Eq. /H208491/H20850is not adequate any- more. Thus, to investigate both T 1andT2at ﬁnite tempera- ture for arbitrary strength of SOC, we develop an equation-of-motion approach for the many-level system via projectionoperator technique 56in the Born approximation. With the rotating wave approximation, we obtain a formal solution tothe equation of motion. By assuming a proper initial distri-bution, we can calculate the evolution of the expectationvalue of spin. We thus obtain the spin relaxation /H20849dephasing /H20850 time by the 1 /edecay of the expectation value of spin op- erator /H20855S z/H20856or/H20841/H20855S+/H20856/H20841 /H20849to its equilibrium value /H20850, with S+/H11013Sx +iSy. With this approach, we are able to study spin relaxation /H20849dephasing /H20850for various temperatures, SOC strengths, and magnetic ﬁelds. For quantum information processing based on electron spin in QDs, the quantum phase coherence is very important.Thus, the spin dephasing time is a more relevant quantity.There are two kinds of spin dephasing times: the ensemble spin dephasing time T 2and the irreversible spin dephasing time T2. For a direct measurement of an ensemble of QDs58 or an average over many measurements at different times where the conﬁgurations of the environment have been changed,59–61it gives the ensemble spin dephasing time T2. The irreversible spin dephasing time T2can be obtained by spin echo measurement.60,61A widely discussed source which leads to both T2and T2is the hyperﬁne interaction between the electron spin and the nuclear spins of the lattice. It has been found that T2is around 10 ns, which is too short and makes a practical quantum information processing difﬁ-cult in electron spin based qubits in QDs. Thus, a spin echotechnique is needed to remove the free induction decay andto elongate the spin dephasing time. Fortunately, this tech-nique has been achieved ﬁrst by Petta et al. for a two elec- tron triplet-singlet system and then by Koppens et al. for a single electron spin system. The achieved spin dephasingtime is /H110111 /H9262s, which is much longer than T2. We therefore discuss only the irreversible spin dephasing time T2through- out the paper, i.e., we do not consider the free inductiondecay in the hyperﬁne-interaction-induced spin dephasing. It is further noticed that Golovach et al. have shown that the spin dephasing time T 2is two times the spin relaxation time T1.20However, as temperature increases, this relation does not hold. Semenov and Kim, on the other hand, re-ported that the spin dephasing time is much smaller than thespin relaxation time. 29In this paper, we calculate the tem- perature dependence of the ratio of the spin relaxation timeto the spin dephasing time and analyze the underlyingphysics. This paper is organized as follows. In Sec. II, we present our model and formalism of the equation-of-motion ap-proach. We also brieﬂy introduce all the spin decoherencemechanisms considered in our calculations. In Sec. III, wepresent our numerical results to indicate the contribution ofeach spin decoherence mechanism to spin relaxation/H20849dephasing /H20850time under various conditions based on the equation-of-motion approach. Then, we study the problem ofhow to obtain the spin relaxation time from the Fermi goldenrule when many levels are involved in Sec. IV. The tempera-ture dependence of T 1andT2is investigated in Sec. V. We conclude in Sec. VI. II. MODEL AND FORMALISM A. Model and Hamiltonian We consider a QD system, where the QD is conﬁned by a parabolic potential Vc/H20849x,y/H20850=1 2m/H927502/H20849x2+y2/H20850in the quantum well plane. The width of the quantum well is a. The external magnetic ﬁeld Bis along the zdirection, except in Sec. IV. The total Hamiltonian of the system of electron together withthe lattice is H T=He+HL+HeL, /H208492/H20850 where He,HL, and HeLare the Hamiltonians of the electron, the lattice, and their interaction, respectively. The electronHamiltonian is given by H e=P2 2m+Vc/H20849r/H20850+HZ+HSO, /H208493/H20850 where P=−i/H6036/H11633+e cAwithA=/H20849B/H11036/2/H20850/H20849−y,x/H20850/H20849B/H11036is the mag- netic ﬁeld along the zdirection /H20850,HZ=1 2g/H9262BB·/H9268is the Zee- man energy with /H9262Bthe Bohr magneton, and HSOis the Hamiltonian of SOC. In GaAs, when the quantum well widthis small or the gate voltage along the growth direction issmall, the Rashba SOC is unimportant. 53Therefore, only the Dresselhaus term10contributes to HSO. When the quantum well width is smaller than the QD radius, the dominant termin the Dresselhaus SOC reads H so=/H92530 /H60363/H20855Pz2/H20856/H92610/H20849−Px/H9268x+Py/H9268y/H20850, /H208494/H20850 with/H92530denoting the Dresselhaus coefﬁcient, /H92610being the quantum well subband index of the lowest one, andJIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-2/H20855Pz2/H20856/H9261/H11013−/H60362/H20848/H9274z/H9261/H20849z/H20850/H115092//H11509z2/H9274z/H9261/H20849z/H20850dz. The Hamiltonian of the lattice consists of two parts, HL=Hph+Hnuclei, where Hph=/H20858q/H9257/H6036/H9275q/H9257aq/H9257†aq/H9257/H20851a†/H20849a/H20850is the phonon creation /H20849annihi- lation /H20850operator /H20852describes the vibration of the lattice, and Hnuclei=/H20858j/H9253IB·Ij/H20849/H9253Iis the gyromagnetic ratios of the nuclei andIjis the spin of the jth nucleus /H20850describes the precession of the nuclear spins of the lattice in the external magneticﬁeld. We focus on the spin dynamics due to hyperﬁne inter-action at a time scale much smaller than the nuclear dipole-dipole correlation time /H2085110 −4s in GaAs /H20849Refs. 33and40/H20850/H20852, where the nuclear dipole-dipole interaction can be ignored.Under this approximation, the equation of motion for thereduced electron system can be obtained which only dependson the initial distribution of the nuclear spin bath. 33The in- teraction between the electron and the lattice also has twoparts H eL=HeI+He-ph, where HeIis the hyperﬁne interaction between the electron and nuclei and He-phrepresents the electron-phonon interaction which is further composed of theelectron–bulk–phonon /H20849BP/H20850interaction H ep, the direct spin- phonon coupling due to the phonon-induced strain Hstrainand phonon-induced g-factor ﬂuctuation Hg. B. Equation-of-motion approach The equations of motion can describe both the coherent and the dissipative dynamics of the electron system. Whenthe quasiparticles of the bath relax much faster than the elec-tron system, the Markovian approximation can be made; oth-erwise, the kinetics is the non-Markovian. For electron-phonon coupling, due to the fast relaxation of the phononbath and the weak electron-phonon scattering, the kinetics ofthe electron is Markovian. Nevertheless, as the nuclear spinbath relaxes much slower than the electron spin, the kineticsdue to the coupling with nuclei is of non-Markoviantype. 30,32,33It is further noted that there is also a contribution from the coaction of the electron-phonon and electron-nucleicouplings, which is a fourth-order coupling to the bath. Forthis contribution, the decoherence of spin is mainly con-trolled by the electron-phonon scattering, while the hyperﬁne/H20849Overhauser /H20850ﬁeld 54acts as a static magnetic ﬁeld. Thus, this fourth-order coupling is also Markovian. Finally, since theelectron orbit relaxation is much faster than the electron spinrelaxation, 55we always assume a thermoequilibrium initial distribution of the orbital degrees of freedom. Generally, the interaction between the electron and the quasiparticle of the bath is weak. Therefore, the ﬁrst Bornapproximation is adequate in the treatment of the interaction.Under this approximation, the equation of motion for theelectron system coupled to the lattice environment can beobtained with the help of the projection operator technique. 56 We then assume a sudden approximation so that the initialdistribution of the whole system is /H9267/H20849t=0/H20850=/H9267e/H208490/H20850/H20002/H9267L/H208490/H20850, where/H9267eand/H9267Lare the density matrix of the system and of the bath, respectively. This approximation corresponds to asudden injection of the electron into the quantum dot, whichis reasonable for the genuine experimental setup. 33As the initial distribution of the the lattice /H9267L/H208490/H20850commutates with the Hamiltonian of the lattice HL, the equation of motion can be written asd/H9267e/H20849t/H20850 dt=−i /H6036/H20851He+T r L/H20851HeL/H9267L/H208490/H20850/H20852,/H9267e/H20849t/H20850/H20852 −1 /H60362/H20885 0t d/H9270TrL„/H20851HeL,U0/H20849/H9270/H20850/H20853Pˆ/H20851HeL,/H9267e/H20849t−/H9270/H20850 /H20002/H9267L/H208490/H20850/H20852/H20854U0†/H20849/H9270/H20850/H20852…, /H208495/H20850 where/H9267e/H20849t/H20850is the density operator of the electron system at time t,T r Lstands for the trace over the lattice degree of freedom, and U0/H20849/H9270/H20850=e−i/H20849HL+He/H20850/H9270is time-evolution operator without HeL.Pˆ=1ˆ−/H9267L/H208490/H20850/H20002TrLis the projection operator. The initial distribution of the phonon system is chosen to be the thermoequilibrium distribution.20It has been shown by previous theoretical studies that the initial state of thenuclear spin bath is crucial to the spin dephasing andrelaxation. 30,32,33Although it may take a long time /H20849e.g., sec- onds /H20850for the nuclear spin system to relax to its thermoequi- librium state, one can still assume that its initial state is thethermoequilibrium one. This assumption corresponds to thegenuine case of long enough waiting time during every indi-vidual measurement. For a typical setup at above 10 mK andwith about 10 T external magnetic ﬁeld, the thermo-equilibrium distribution is a distribution with equal probabil-ity on every state. For these initial distributions of phononsand nuclear spins, the term Tr L/H20851HeL/H9267L/H208490/H20850/H20852is zero. Thus, Pˆ/H20851HeL,/H9267e/H20849t−/H9270/H20850/H20002/H9267L/H208490/H20850/H20852=/H20851HeL,/H9267e/H20849t−/H9270/H20850/H20002/H9267L/H208490/H20850/H20852./H208496/H20850 The equation of motion is then simpliﬁed to d/H9267e/H20849t/H20850 dt=−i /H6036/H20851He,/H9267e/H20849t/H20850/H20852−1 /H60362/H20885 0t d/H9270TrL /H11003/H20853/H20851HeL,/H20851HeLI/H20849−/H9270/H20850,U0e/H20849t/H20850/H9267Ie/H20849t−/H9270/H20850U0e†/H20849t/H20850/H9267L/H208490/H20850/H20852/H20852/H20854,/H208497/H20850 where HeLIand/H9267Ieare the corresponding operators /H20849HeLand /H9267e/H20850in the interaction picture and U0e/H20849t/H20850=e−iHetis the time- evolution operator of He. It should be further noted that the ﬁrst Born approximation cannot fully account for the non-Markovian dynamics due to the hyperﬁne interaction withnuclear spins. 33,57Only when the Zeeman splitting is much larger than the ﬂuctuating Overhauser shift the ﬁrst Bornapproximation is adequate. For GaAs QDs, this requires B /H112713.5 T. 33In this paper, we focus on the study of spin dephasing for the high magnetic ﬁeld regime of B/H110223.5 T under the ﬁrst Born approximation, where the second Bornapproximation only affects the long-time behavior. 33Later, we will argue that this correction of long-time dynamicschanges the spin dephasing time very little. 1. Markovian kinetics The kinetics due to the coupling with phonons can be investigated within the Markovian approximation, where theequation of motion reduces toREEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-3d/H9267e/H20849t/H20850 dt=−i /H6036/H20851He,/H9267e/H20849t/H20850/H20852−1 /H60362/H20885 0t d/H9270 /H11003Trph/H20853/H20851He-ph,/H20851He-phI/H20849−/H9270/H20850,/H9267e/H20849t/H20850/H20002/H9267ph/H208490/H20850/H20852/H20852/H20854./H208498/H20850 Here, Tr phis the trace over phonon degrees of freedom and /H9267ph/H208490/H20850is the initial distribution of the phonon bath. Within the basis of the eigenstates of the electron Hamiltonian, /H20853/H20841/H5129/H20856/H20854, the above equation reads d dt/H9267/H51291/H51292e=−i/H20849/H9255/H51291−/H9255/H51292/H20850 /H6036/H9267/H51291/H51292e −/H208771 /H60362/H20885 0t d/H9270/H20858 /H51293/H51294Trp/H20849H/H51291/H51293e-phH/H51293/H51294Ie-ph/H9267/H51294/H51292e/H20002/H9267eqp −H/H51291/H51293Ie-ph/H9267/H51293/H51294e/H20002/H9267eqpH/H51294/H51292e-ph/H20850+ H.c./H20878. /H208499/H20850 Here, H/H51291/H51293e-ph=/H20855/H51291/H20841He-ph/H20841/H51293/H20856and H/H51291/H51293Ie-ph=/H20855/H51291/H20841He-phI/H20849−/H9270/H20850/H20841/H51293/H20856.A general form of the electron-phonon interaction reads He-ph=/H20858 q/H9257/H9021q/H9257/H20849aq/H9257+a−q/H9257†/H20850Xq/H9257/H20849r,/H9268/H20850. /H2084910/H20850 Here,/H9257represents the phonon branch index, /H9021q/H9257is the ma- trix element of the electron-phonon interaction, aq/H9257is the phonon annihilation operator, and Xq/H9257/H20849r,/H9268/H20850denotes a func- tion of electron position and spin. Substituting this into Eq. /H208499/H20850, we obtain, after integration within the Markovian approximation,49 d dt/H9267/H51291/H51292e=i/H20849/H9255/H51291−/H9255/H51292/H20850 /H6036/H9267/H51291/H51292e −/H20877/H9266 /H60362/H20858 /H51293/H51294/H20858 q/H9257/H20841/H9021q/H9257/H208412/H20851X/H51291/H51293q/H9257X/H51294/H51293q/H9257/H9267/H51294/H51292eCq/H9257/H20849/H9255/H51294−/H9255/H51293/H20850 −X/H51294/H51292q/H9257X/H51293/H51291q/H9257/H9267/H51293/H51294eCq/H9257/H20849/H9255/H51293−/H9255/H51291/H20850/H20852+ H.c./H20878, /H2084911/H20850 in which X/H51291/H51292q/H9257=/H20855/H51291/H20841Xq/H9257/H20849r,/H9268/H20850/H20841/H51292/H20856 and Cq/H9257/H20849/H9004/H9255/H20850 =n¯/H20849/H9275q/H9257/H20850/H9254/H20849/H9004/H9255+/H9275q/H9257/H20850+/H20851n¯/H20849/H9275q/H9257/H20850+1/H20852/H9254/H20849/H9004/H9255−/H9275q/H9257/H20850. Here, n¯/H20849/H9275q/H9257/H20850 represents the Bose distribution function. Equation /H2084911/H20850can be written in a more compact form d dt/H9267/H51291/H51292e=−/H20858 /H51293/H51294/H9011/H51291/H51292/H51293/H51294/H9267/H51293/H51294e, /H2084912/H20850 which is a linear differential equation. This equation can be solved by diagonalizing /H9011. Given an initial distribution /H9267/H51291/H51292e/H208490/H20850, the density matrix /H9267/H51291/H51292e/H20849t/H20850and the expectation value of any physical quantity /H20855O/H20856t=Tr /H20851Oˆ/H9267e/H20849t/H20850/H20852at time tcan be obtained,49/H20855O/H20856t=T r /H20849Oˆ/H9267e/H20850 =/H20858 /H51291¯/H51296/H20855/H51292/H20841Oˆ/H20841/H51291/H20856P/H20849/H51291/H51292/H20850/H20849/H51293/H51294/H20850e−/H9003/H20849/H51293/H51294/H20850tP/H20849/H51293/H51294/H20850/H20849/H51295/H51296/H20850−1/H9267/H51295/H51296e/H208490/H20850, /H2084913/H20850 with/H9003=P−1/H9011Pbeing the diagonal matrix and Prepresenting the transformation matrix. To study spin dynamics, we cal-culate /H20855S z/H20856t/H20849/H20841/H20855S+/H20856t/H20841/H20850and deﬁne the spin relaxation /H20849dephas- ing/H20850time as the time when /H20855Sz/H20856t/H20849/H20841/H20855S+/H20856t/H20841/H20850decays to 1 /eof its initial value /H20849to its equilibrium value /H20850. 2. Non-Markovian kinetics Experiments have already shown that for a large ensemble of quantum dots or for an ensemble of many measurementson the same quantum dot at different times, the spin dephas-ing time due to hyperﬁne interaction is quite short,/H1101110 ns. 58–61This rapid spin dephasing is caused by the en- semble broadening of the precession frequency due to thehyperﬁne ﬁelds. 40When the external magnetic ﬁeld is much larger than the random Overhauser ﬁeld, the rotation due tothe Overhauser ﬁeld perpendicular to the magnetic ﬁeld isblocked. Only the broadening of the Overhauser ﬁeld parallelto the magnetic ﬁeld contributes to the spin dephasing. Todescribe this free induction decay for this high magnetic ﬁeldcase, we write the hyperﬁne interaction into two parts: H eI =h·S=HeI1+HeI2. Here h=/H20849hx,hy,hz/H20850andS=/H20849Sx,Sy,Sz/H20850are the Overhauser ﬁeld and the electron spin, respectively. HeI1=hzSzand HeI2=1 2/H20849h+S−+h−S+/H20850with h/H11006=hx/H11006ihy. The longitudinal part HeI1is responsible for the free induction decay, while the transversal part HeI2is responsible for high order irreversible decay. As the rapid free induction decaycan be removed by spin echo, 60,61elongating the spin dephasing time to /H110111/H9262s which is more favorable for quan- tum computation and quantum information processing, wethen discuss only the irreversible decay. We ﬁrst classify thestates of the nuclear spin system with its polarization. Then,we reconstruct the states within the same class to make itspatially uniform. These uniformly polarized pure states,/H20841n/H20856’s, are eigenstates of h z. They also form a complete- orthogonal basis of the nuclear spin system. A formal expres-sion of /H20841n/H20856is 33 /H20841n/H20856=/H20858 m1¯mN/H9251m1¯mNn/H20002 j=1N /H20841I,mj/H20856. /H2084914/H20850 Here, /H20841I,mj/H20856denotes the eigenstate of the zcomponent of the jth nuclear spin Ijzwith the eigenvalue /H6036mj.Ndenotes the number of the nuclei. The equation of motion for the case with initial nuclear spin state /H92671ns/H208490/H20850=/H20841n/H20856/H20855n/H20841is given by33 d/H9267e/H20849t/H20850 dt=−i /H6036/H20853He+T r ns /H11003/H20851HeI/H92671ns/H208490/H20850/H20852,/H9267e/H20849t/H20850/H20854−1 /H60362/H20885 0t d/H9270Trns/H20853/H20851HeI2,U0eI/H20849/H9270/H20850 /H11003/H20851HeI2,/H9267e/H20849t−/H9270/H20850/H20002/H92671ns/H208490/H20850/H20852U0eI†/H20849/H9270/H20850/H20852/H20854. /H2084915/H20850 As in traditional projection operator technique, the dynamicsJIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-4of the nuclear spin subsystem is incorporated self- consistently in the last term.33,56Here, Tr nsis the trace over nuclear spin degrees of freedom. U0eI/H20849/H9270/H20850=exp /H20851−i/H9270/H20849He+HI +HeI1/H20850/H20852. The Overhauser ﬁeld is given by h=/H20858jAv0Ij/H9254/H20849r −Rj/H20850, where the constants Aandv0are given later. IjandRj are the spin and position of the jth nucleus, respectively. As mentioned above, the initial state of the nuclear spin bath ischosen to be a state with equal probability of each state;therefore, /H9267ns/H208490/H20850=/H20858n1/Nw/H20841n/H20856/H20855n/H20841, with Nw=/H20858n1 being the number of states of the basis /H20853/H20841n/H20856/H20854. To quantify the irrevers- ible decay, we calculate the time evolution of S+/H20849n/H20850for every case with initial nuclear spin state /H20841n/H20856. We then sum over n and obtain /H20648/H20855S+/H20856t/H20648=/H20858 n/H20841/H20855S+/H20849n/H20850/H20856t/H20841. /H2084916/H20850 It is noted that the summation is performed after the absolute value of /H20855S+/H20849n/H20850/H20856t. Therefore, the destructive interference due to the difference in precession frequency /H9275zn, which originates from the longitudinal part of the hyperﬁne interaction /H20849HeI1/H20850, isremoved . We thus use the 1 /edecay of the envelope of /H20648/H20855S+/H20856t/H20648to describe the irreversible spin dephasing time T2. Similar description has been used in the irreversible spin dephasing in semiconductor quantum wells62and the irre- versible interband optical dephasing in semiconductors.63,64 Expanding Eq. /H2084915/H20850in the basis of /H20853/H20841n/H20856/H20854, one obtains d dt/H9267/H51291/H51292e=−i /H6036/H20858 /H51293/H20853/H20849/H9255/H51291/H9254/H51291/H51293+Hn/H51291;n/H51293eI1/H20850/H9267/H51293/H51292e−/H9267/H51291/H51293e/H20849/H9255/H51293/H9254/H51293/H51292 +Hn/H51293;n/H51292eI1/H20850/H20854 −/H208771 /H60362/H20885 0t d/H9270/H20858 n1/H20858 /H51293/H51294/H20851Hn/H51291;n1/H51293eI2Hn1/H51293;n/H51294Ie I2/H9267/H51294/H51292e/H20849t−/H9270/H20850 −Hn/H51291;n1/H51293Ie I2/H9267/H51293/H51294e/H20849t−/H9270/H20850Hn1/H51294;n/H51292eI2/H20852+ H.c./H20878. /H2084917/H20850 Here, Hn/H51291;n1/H51293eI2=/H20855n/H51291/H20841HeI2/H20841n1/H51293/H20856and Hn/H51291;n1/H51293IeI2=/H20855n/H51291/H20841HeI2I/H20849−/H9270/H20850/H20841n1/H51293/H20856. For simplicity, we neglect the terms concerning different or- bital wave functions which are much smaller. For small spinmixing, assuming an equilibrium distribution in the orbitaldegree of freedom, under rotating wave approximation, andtrace over the orbital degree of freedom, we ﬁnally arrive at d dt/H20855S+/H20849n/H20850/H20856t=i/H9275zn/H20855S+/H20849n/H20850/H20856t−1 /H60362/H20885 0t d/H9270/H208771 4/H20858 kn/H11032fk/H20849/H20851h+/H20852knn/H11032/H20851h−/H20852kn/H11032n +/H20851h−/H20852knn/H11032/H20851h+/H20852kn/H11032n/H20850exp/H20851i/H9270/H20849/H9275kn−/H9275kn/H11032/H20850/H20852/H20878/H20855S+/H20849n/H20850/H20856t−/H9270. /H2084918/H20850 Here,/H9275zn=/H20858kfk/H20849Ezk//H6036+/H9275kn/H20850with Ezkrepresenting the elec- tron Zeeman splitting of the kth orbital level. /H20851hi/H20852knn/H11032 =/H20855n/H20841/H20855k/H20841hi/H20841k/H20856/H20841n/H11032/H20856/H20849i=/H11006,z/H20850./H9275kn=/H20851hz/H20852knn+/H9280nzwith/H9280nzdenoting the nuclear Zeeman splitting which is very small and can beneglected. By solving the above equation, we obtain /H20841/H20855S+/H20849n/H20850/H20856t/H20841 for a given /H20841n/H20856. We then sum over nand determine the irre- versible spin dephasing time T2as the 1 /edecay of the en- velop of /H20648/H20855S+/H20856t/H20648. By noting that only the polarization of nuclear spin state /H20841n/H20856determines the evolution of /H20841/H20855S+/H20849n/H20850/H20856t/H20841, the summation over nis then reduced to the summation over polarization which becomes an integration for large N. This integration can be handled numerically. In the limiting case of zero SOC and very low tempera- ture, only the lowest two Zeeman sublevels are concerned. The equation for /H20855S+/H20856twith initial nuclear spin state /H92671ns/H208490/H20850 =/H20841n/H20856/H20855n/H20841reduces to d dt/H20855S+/H20856t=i/H9275zn/H20855S+/H20856t−1 /H60362/H20885 0t d/H9270/H208771 4/H20858 n/H11032/H20849/H20851h+/H20852nn/H11032/H20851h−/H20852n/H11032n +/H20851h−/H20852nn/H11032/H20851h+/H20852n/H11032n/H20850exp/H20851i/H9270/H20849/H9275n−/H9275n/H11032/H20850/H20852/H20878/H20855S+/H20856t−/H9270 =i/H9275z/H20855S+/H20856t−/H20885 0t d/H9270/H9018/H20849/H9270/H20850/H20855S+/H20856t−/H9270. /H2084919/H20850 In this equation, /H9275zn=/H20849g/H9262BB+/H20851hz/H20852nn/H11032/H20850//H6036, /H20851h/H9264/H20852nn/H11032 =/H20855n/H20841/H20855/H92741/H20841h/H9264/H20841/H92741/H20856/H20841n/H11032/H20856/H20849/H9264=/H11006,zand/H92741is the orbital quantum number of the ground state /H20850, and/H9275n=/H20851hz/H20852nn. Similar equation has been obtained by Coish and Loss,33and later by Deng and Hu35at a very low temperature such that only the lowest two Zeeman sublevels are considered. Coish and Loss alsopresented an efﬁcient way to evaluate /H9018/H20849 /H9270/H20850in terms of their Laplace transformations, /H9018/H20849s/H20850=/H208480/H11009d/H9270e−s/H9270/H9018/H20849/H9270/H20850. They gave /H9018/H20849s/H20850=1 4/H60362/H20858 n/H11032/H20849/H20851h+/H20852nn/H11032/H20851h−/H20852n/H11032n+/H20851h−/H20852nn/H11032/H20851h+/H20852n/H11032n/H20850//H20849s−i/H9254/H9275nn/H11032/H20850, /H2084920/H20850 with/H9254/H9275nn/H11032=1 2/H20849/H9275n−/H9275n/H11032/H20850. With the help of this technique, we are able to investigate the spin dephasing due to the hyper- ﬁne interaction. C. Spin decoherence mechanisms In this subsection, we brieﬂy summarize all the spin de- coherence mechanisms. It is noted that the SOC modiﬁes allthe mechanisms. This is because the SOC modiﬁes the Zee-man splitting 18and the spin-resolved eigenstates of the elec- tron Hamiltonian; it hence greatly changes the effect of theelectron-BP scattering. 18These two modiﬁcations, especially the modiﬁcation of the Zeeman splitting, also change theeffect of other mechanisms, such as the direct spin-phononcoupling due to the phonon-induced strain, the g-factor ﬂuc- tuation, the coaction of the electron-phonon interaction, andthe hyperﬁne interaction. In the literature, except for theelectron-BP scattering, the effects from the SOC are ne-glected except for the work by Woods et al. 16in which the spin relaxation time between the two Zeeman sublevels ofthe lowest electronic state due to the phonon-induced strainis investigated. However, the perturbation method they useddoes not include the important second-order energy correc-REEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-5tion. In our investigation, the effects of the SOC are included in all the mechanisms and we will show that they lead tomarked effects in most cases. 1. Spin-orbit coupling together with electron-phonon scattering As the SOC mixes different spins, the electron-BP scat- tering can induce spin relaxation and dephasing. Theelectron-BP coupling is given by H ep=/H20858 q/H9257Mq/H9257/H20849aq/H9257+a−q/H9257†/H20850eiq·r, /H2084921/H20850 where Mq/H9257is the matrix element of the electron-phonon in- teraction. In the general form of the electron phonon interac-tion H e-ph,/H9021q/H9257=Mq/H9257and Xq/H9257/H20849r,/H9268/H20850=eiq·r. /H20841Mqsl/H208412 =/H6036/H90142q/2/H9267vslVfor the electron-BP coupling due to the de- formation potential. For the piezoelectric coupling, /H20841Mqpl/H208412 =/H2084932/H6036/H92662e2e142//H92602/H9267vslV/H20850/H20851/H208493qxqyqz/H208502/q7/H20852for the longitudinal phonon mode and /H20858j=1,2/H20841Mqptj/H208412=/H2085132/H6036/H92662e2e142//H20849/H92602/H9267vstq5V/H20850/H20852 /H11003/H20851qx2qy2+qy2qz2+qz2qx2−/H208493qxqyqz/H208502/q2/H20852for the two transverse modes. Here,/H9014stands for the acoustic deformation poten- tial,/H9267is the GaAs volume density, Vis the volume of the lattice, e14is the piezoelectric constant, and /H9260denotes the static dielectric constant. The acoustic phonon spectra /H9275qql =vslqfor the longitudinal mode and /H9275qpt=vstqfor the trans- verse mode with vslandvstrepresenting the corresponding sound velocities. Besides the electron-BP scattering, electron also couples to vibrations of the conﬁning potential, i.e., the surfacephonons, 28 /H9254V/H20849r/H20850=−/H20858 q/H9257/H20881/H6036 2/H9267/H9275q/H9257V/H20849aq/H9257+a−q/H9257†/H20850/H9280q/H9257·/H11612rVc/H20849r/H20850, /H2084922/H20850 in which/H9280q/H9257is the polarization vector of a phonon mode with wave-vector qin branch/H9257. However, this contribution is much smaller than the electron-BP coupling. Compared tothe coupling due to the deformation potential, for example,the ratio of the two coupling strengths is /H11015/H6036 /H92750//H9014ql0, where l0is the characteristic length of the quantum dot and /H6036/H92750is the orbital level splitting. The phonon wave vector qis de- termined by the energy difference between the ﬁnal and ini-tial states of the transition. Typically, for phonon transitionsbetween Zeeman sublevels and different orbital levels, ql 0 ranges from 0.1 to 10. Bearing in mind that /H6036/H92750is about 1 meV while/H9014=7 eV in GaAs, /H6036/H92750//H9014ql0is about 10−3. The piezoelectric coupling is of the same order as the defor-mation potential. Therefore, spin decoherence due to theelectron–surface-phonon coupling is negligible. 2. Direct spin-phonon coupling due to phonon-induced strain The direct spin-phonon coupling due to the phonon- induced strain is given by65 Hstrain=1 2hs/H20849p/H20850·/H9268, /H2084923/H20850 where hxs=−Dpx/H20849/H9280yy−/H9280zz/H20850, hys=−Dpy/H20849/H9280zz−/H9280xx/H20850, and hzs=−Dpz/H20849/H9280xx−/H9280yy/H20850with p=/H20849px,py,pz/H20850=−i/H6036/H11633and Dbeingthe material strain constant. /H9280ij/H20849i,j=x,y,z/H20850can be expressed by the phonon creation and annihilation operators /H9280ij=/H20858 q/H9257=l,t1,t2i 2/H20881/H6036 2/H9267/H9275q/H9257V/H20849aq,/H9257+a−q,/H9257+/H20850/H20849/H9264i/H9257qj+/H9264j/H9257qi/H20850eiq·r, /H2084924/H20850 in which/H9264il=qi/qfor the longitudinal phonon mode and /H20849/H9264xt1,/H9264yt1,/H9264zt1/H20850=/H20849qxqz,qyqz,−q/H206482/H20850/qq /H20648, /H20849/H9264xt2,/H9264yt2,/H9264zt2/H20850=/H20849qy,−qx,0/H20850/q/H20648for the two transverse phonon modes with q/H20648=/H20881qx2+qy2. Therefore, in the general form of electron-phonon interaction He-ph,/H9021q/H9257=−iD/H20881/H6036//H2084932/H9267/H9275q/H9257V/H20850 andXq/H9257/H20849r,/H9268/H20850=/H20858ijk/H9280ijk/H20849/H9264j/H9257qj−/H9264k/H9257qk/H20850pieiq·r/H9268iwith/H9280ijkdenot- ing the Levi-Civita tensor. 3. g-factor ﬂuctuation The spin-lattice interaction via phonon modulation of the gfactor is given by12 Hg=/H6036 2/H20858 ijkl=x,y,zAijkl/H9262BBi/H9268j/H9280kl, /H2084925/H20850 where/H9280klis given in Eq. /H2084924/H20850andAijklis a tensor determined by the material. Therefore in He-ph,/H9021q/H9257=i/H20881/H6036//H2084932/H9267/H9275q/H9257V/H20850 and Xq/H9257/H20849r,/H9268/H20850=/H20858i,j,k,lAi,j,k,l/H9262BBi/H20849/H9264k/H9257qk−/H9264l/H9257ql/H20850/H9268jeiq·r. Due to the axial symmetry with respect to the zaxis and keeping in mind that the external magnetic ﬁeld is along the zdirection, the only ﬁnite element of HgisHg=/H20851/H20849A33−A31/H20850/H9280zz +A31/H20858i/H9280ii/H20852/H6036/H9262BB/H9268z/2 with A33=Azzzz,A31=Azzxx, and A66 =Axyxy.A33+2A31=0.45 4. Hyperﬁne interaction The hyperﬁne interaction between the electron and nuclear spins is66 HeI/H20849r/H20850=/H20858 jAv0S·Ij/H9254/H20849r−Rj/H20850, /H2084926/H20850 where S=/H6036/H9268/2 and Ijare the electron and nucleus spins, respectively, v0=a03is the volume of the unit cell with a0 representing the crystal lattice parameter, and r/H20849Rj/H20850denotes the position of the electron /H20849the jth nucleus /H20850.A =4/H92620/H9262B/H9262I//H208493Iv0/H20850is the hyperﬁne coupling constant with /H92620, /H9262B, and/H9262Irepresenting the permeability of vacuum, the Bohr magneton, and the nuclear magneton separately. As the Zeeman splitting of the electron is much larger /H208493 orders of magnitude larger /H20850than that of the nucleus spin, to conserve the energy for the spin relaxation processes,there must be phonon-assisted transitions when consideringthe spin-ﬂip processes. Taking into account directly the BPinduced motion of nuclei spin of the lattice leads to a newspin relaxation mechanism, 28 VeI-ph/H208491/H20850/H20849r/H20850=−/H20858 jAv0S·Ij/H20851u/H20849Rj0/H20850·/H11612r/H20852/H9254/H20849r−Rj/H20850, /H2084927/H20850 where u/H20849Rj0/H20850=/H20858q/H9257/H20881/H6036//H208492/H9267/H9275q/H9257v0/H20850/H20849aq/H9257+aq/H9257†/H20850/H9280q/H9257eiq·Rj0is the lattice displacement vector. Therefore, using the notation ofJIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-6Eq. /H2084910/H20850,/H9021=/H20881/H6036//H208492/H9267V/H9275q/H9257/H20850and Xq/H9257=/H20858jAv0S·Ij/H11612r/H9254/H20849r−Rj/H20850. The second-order process of the surface phonon and the BP together with the hyperﬁne interaction also leads to spin re-laxation, V eI-ph/H208492/H20850/H20849r/H20850=/H20841/H51292/H20856/H20877/H20858 m/HS11005/H51291/H20855/H51292/H20841/H9254Vc/H20849r/H20850/H20841m/H20856/H20855m/H20841HeI/H20849r/H20850/H20841/H51291/H20856 /H9255/H51291−/H9255m +/H20858 m/HS11005/H51292/H20855/H51292/H20841HeI/H20849r/H20850/H20841m/H20856/H20855m/H20841/H9254Vc/H20849r/H20850/H20841/H51291/H20856 /H9255/H51292−/H9255m/H20878/H20855/H51291/H20841, /H2084928/H20850 and VeI-ph/H208493/H20850=/H20841/H51292/H20856/H20877/H20858 m/HS11005/H51291/H20855/H51292/H20841Hep/H20841m/H20856/H20855m/H20841HeI/H20849r/H20850/H20841/H51291/H20856 /H9255/H51291−/H9255m +/H20858 m/HS11005/H51292/H20855/H51292/H20841HeI/H20849r/H20850/H20841m/H20856/H20855m/H20841Hep/H20841/H51291/H20856 /H9255/H51292−/H9255m/H20878/H20855/H51291/H20841, /H2084929/H20850 in which /H20841/H51291/H20856and /H20841/H51292/H20856are the eigenstates of He. By using the notations in He-ph,/H9021q/H9257=i /H6036/H20881/H6036//H208492/H9267/H9275q/H9257v0/H20850and Xq/H9257=/H20841/H51292/H20856/H9280q/H9257·/H20877/H20858 m/HS11005/H512911 /H9255/H51291−/H9255m/H20855/H51292/H20841/H20851He,P/H20852/H20841m/H20856 /H11003/H20858 jAv0/H20855m/H20841S·Ij/H9254/H20849r−Rj/H20850/H20841/H51291/H20856+/H20858 m/HS11005/H512921 /H9255/H51292−/H9255m/H20855m/H20841 /H11003/H20851He,P/H20852/H20841/H51291/H20856/H20858 jAv0/H20855/H51292/H20841S·Ij/H9254/H20849r−Rj/H20850/H20841m/H20856/H20878/H20855/H51291/H20841/H2084930/H20850 forVeI-ph/H208492/H20850. Similarly,/H9021q/H9257=Mq/H9257and Xq/H9257=/H20841/H51292/H20856/H20877/H20858 m/HS11005/H51291/H20855/H51292/H20841eiq·r/H20841m/H20856 /H9255/H51291−/H9255m/H20858 jAv0/H20855m/H20841S·Ij/H9254/H20849r−Rj/H20850/H20841/H51291/H20856 +/H20858 m/HS11005/H512921 /H9255/H51292−/H9255m/H20855m/H20841eiq·r/H20841/H51291/H20856/H20858 jAv0/H20855/H51292/H20841S·Ij/H9254/H20849r−Rj/H20850 /H11003/H20841m/H20856/H20878/H20855/H51291/H20841/H20849 31/H20850 forVeI-ph/H208493/H20850. Again, as the contribution from the surface phonon is much smaller than that of the BP, VeI-ph/H208492/H20850can be neglected. It is noted that the direct spin-phonon coupling due to thephonon-induced strain together with the hyperﬁne interactiongives a fourth-order scattering and hence induces a spin re-laxation /H20849dephasing /H20850. The interaction is V eI-ph/H208494/H20850=/H20841/H51292/H20856/H20877/H20858 m/HS11005/H51291/H20855/H51292/H20841Hstrainz/H20841m/H20856/H20855m/H20841HeI/H20849r/H20850/H20841/H51291/H20856 /H9255/H51291−/H9280m +/H20858 m/HS11005/H51292/H20855/H51292/H20841HeI/H20849r/H20850/H20841m/H20856/H20855m/H20841Hstrainz/H20841/H51291/H20856 /H9280/H51292−/H9280m/H20878/H20855/H51291/H20841,/H2084932/H20850 with Hstrainz=hsz/H9268z/2 only changing the electron energy but conserving the spin polarization. It can be written as1 2hsz=−i 2D/H20858 q/H9257/H20881/H6036 2/H9267/H9275q,/H9257V/H20849/H9264y/H9257qy−/H9264z/H9257qz/H20850qzeiq·r./H2084933/H20850 Comparing this to the electron-BP interaction /H20851Eq. /H2084921/H20850/H20852, the ratio is /H11015/H6036Dq //H9014, which is about 10−3. Therefore, the second-order term of the direct spin-phonon coupling due tothe phonon-induced strain together with the hyperﬁne inter-action is very small and can be neglected. Also, the coactionof the g-factor ﬂuctuation and the hyperﬁne interaction is very small compared to that of the electron-BP interactionjointly with the hyperﬁne interaction as /H9262BB//H9014is around 10−5when B=1 T. Therefore, it can also be neglected. In the following, we only retain the ﬁrst and the third order terms VeI-ph/H208491/H20850andVeI-ph/H208493/H20850in calculating the spin relaxation time. The spin dephasing time induced by the hyperﬁne inter- action can be calculated from the non-Markovian kineticequation /H20851Eq. /H2084918/H20850/H20852, for unpolarized initial nuclear spin state /H20841n 0/H20856, resulting in /H20855S+/H20849n0/H20850/H20856t/H11008/H20858 kfkA2v02/H20885dr/H20841/H9274k/H20849r/H20850/H208414cos/H20873Av0 2/H20841/H9274k/H20849r/H20850/H208412t/H20874, /H2084934/H20850 where fkis the thermoequilibrium distribution of the orbital degree of freedom. When only the lowest two Zeeman sub-levels are considered, assuming a simple form of the wave function, /H20841/H9023/H20849r/H20850/H20841 2=1 azd/H206482/H9266exp/H20849−r/H206482/d02/H20850with d/H20648/H20849az/H20850representing the QD diameter /H20849quantum well width /H20850and r/H20648=x2+y2, the integration can be carried out, /H20855S+/H20849n0/H20850/H20856t/H11008cos/H20849t/t0/H20850−1 /H20849t/t0/H208502+sin/H20849t/t0/H20850 t/t0. /H2084935/H20850 Here, t0=/H208492/H9266azd/H206482/H20850//H20849Av0/H20850determines the spin dephasing time. Note that t0is proportional to the factor azd/H206482, where az/H20849d/H206482/H20850is the characteristic length /H20849area /H20850of the QD along the zdirection /H20849in the quantum well plane /H20850. By solving Eq. /H2084918/H20850 for various nand summing over n, we obtain /H20648/H20855S+/H20856t/H20648 =/H20858n/H20841/H20855S+/H20849n/H20850/H20856t/H20841. We then deﬁne the time when the envelop of /H20648/H20855S+/H20856t/H20648decays to 1 /eof its initial value as the spin dephasing time T2. As mentioned above, the hyperﬁne interaction can- not transfer an energy of the order of the Zeeman splitting;thus, the hyperﬁne interaction alone cannot lead to any spinrelaxation. 43 In the above discussion, the nuclear spin dipole-dipole interaction is neglected. Recently, more careful examinationsbased on the quantum cluster expansion method or pair cor-relation method have been performed. 41–43,47In these works, the nuclear spin dipole-dipole interaction is also included.This interaction together with the hyperﬁne mediated nuclearspin-spin interaction is the origin of the ﬂuctuation of thenuclear spin bath. To the lowest order, the ﬂuctuation isdominated by nuclear spin pair ﬂips. 41–43,47This ﬂuctuation provides the source of the electron spin dephasing, as theelectron spin is coupled to the nuclear spin system via hy-perﬁne interaction. Our method used here includes only thehyperﬁne interaction to the second order in scattering. How-ever, it is found that the dipole-dipole-interaction-inducedspin dephasing is much weaker than the hyperﬁne interactionREEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-7for a QD with a=2.8 nm and d0=27 nm until the parallel magnetic ﬁeld is larger than /H1101120 T.42Therefore, for the situ- ation in this paper, the nuclear dipole-dipole-interaction-induced spin dephasing can be ignored. 67 III. SPIN DECOHERENCE DUE TO VARIOUS MECHANISMS Following the equation-of-motion approach developed in Sec. II, we perform a numerical calculation of the spinrelaxation and dephasing times in GaAs QDs. Two magneticﬁeld conﬁgurations are considered, i.e., the magnetic ﬁeldsperpendicular and parallel to the well plane /H20849along the x axis /H20850. The temperature is taken to be T=4 K unless otherwise speciﬁed. For all the cases we considered inthis paper, the orbital level splitting is larger than an energycorresponding to 40 K. Therefore, the lowest Zeeman sub-levels are mainly responsible for the spin decoherence.When calculating T 1, the initial distribution is taken to be in the spin majority down state of the eigenstate of theHamiltonian H ewith a Maxwell-Boltzmann distribution fk=Cexp/H20851−/H9280k//H20849kBT/H20850/H20852for different orbital levels /H20849Cis the normalization constant /H20850. For the calculation of T2, we assign the same distribution between different orbital levels butwith a superposition of the two spin states within the sameorbital level. The parameters used in the calculation are listedin Table I. 8,68,69 A. Spin relaxation time T1 We now study the spin relaxation time and show how it changes with the well width a, the magnetic ﬁeld B, and the effective diameter d0=/H20881/H6036/H9266/m/H92750. We also compare the rela- tive contributions from each relaxation mechanism. 1. Well width dependence In Figs. 1/H20849a/H20850and1/H20849b/H20850, the spin relaxation times induced by different mechanisms are plotted as a function of thewidth of the quantum well in which the QD is conﬁned forperpendicular magnetic ﬁeld B /H11036=0.5 T and parallel mag- netic ﬁeld B/H20648=0.5 T, respectively. We ﬁrst concentrate on the perpendicular-magnetic-ﬁeld case. In Fig. 1/H20849a/H20850, the calcula- tion indicates that the spin relaxation due to each mechanismdecreases with the increase of well width. Particularly, theelectron-BP scattering mechanism decreases much fasterthan the other mechanisms. It is indicated in the ﬁgures thatwhen the well width is small /H20849smaller than 7 nm in thepresent case /H20850, the spin relaxation time is determined by the electron-BP scattering together with the SOC. However, forwider well widths, the direct spin-phonon coupling due tophonon-induced strain and the ﬁrst-order process of hyper-ﬁne interaction combined with the electron-BP scattering be-comes more important. The decrease of spin relaxation dueto each mechanism is mainly caused by the decrease of theSOC which is proportional to a −2. The SOC has two effects which are crucial. First, in the second-order perturbation theSOC contributes a ﬁnite correction to the Zeeman splittingwhich determines the absorbed /H20849emitted /H20850phonon frequency and wave vector. 18Second, it leads to spin mixing. The de- crease of the SOC thus leads to the decrease of Zeemansplitting and spin mixing. The former leads to small phononwave vector and small phonon absorption /H20849emission /H20850 efﬁciency. 18Therefore, the electron-BP mechanism decreases rapidly with increasing a. On the other hand, the other twoTABLE I. Parameters used in the calculation. /H9267 5.3/H11003103kg /m3/H9260 12.9 vst 2.48/H11003103m/s g −0.44 vsl 5.29/H11003103m/s/H9014 7.0 eV e14 1.41/H11003109V/mm0.067 m0 A 90/H9262eV A33 19.6 /H92530 27.5 Å3eV I3 2 D 1.59/H11003104m/s a0 5.6534 Åg-factorstrainV(1) eI−phV(3) eI−phBP B⊥=0 .5T a(nm)T−1 1(s−1) 10 9 8 7 6 5 4 3 21010 105 100 10−5 10−10 B/CID1=0 .5T a(nm)T−1 1(s−1) 10 9 8 7 6 5 4 3 2104 102 100 10−2 10−4 10−6 10−8 10−10 (b)(a) FIG. 1. /H20849Color online /H20850T1−1induced by different mechanisms vs the well width for /H20849a/H20850perpendicular magnetic ﬁeld B/H11036=0.5 T with /H20849solid curves /H20850and without /H20849dashed curves /H20850the SOC and /H20849b/H20850parallel magnetic ﬁeld B/H20648=0.5 T with the SOC. The effective diameter d0 =20 nm and temperature T=4 K. Curves with /H20849/H17039/H20850—T1−1induced by the electron-BP scattering together with the SOC. Curves with/H20849/L50098/H20850—T 1−1induced by the second-order process of the hyperﬁne in- teraction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1in- duced by the ﬁrst-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with/H20849/H12135/H20850—T 1−1induced by the g-factor ﬂuctuation.JIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-8largest mechanisms can ﬂip spin without the help of the SOC. The spin relaxations due to these two mechanisms de-crease in a relatively mild way. It is further conﬁrmed thatwithout SOC, they decrease in a much milder way with in-creasing a/H20849dashed curves in Fig. 1/H20850. It is also noted that the spin relaxation rate due to the g-factor ﬂuctuation is at least 6 orders of magnitude smaller than that due to the leadingspin decoherence mechanisms and can therefore beneglected. It is noted that in the calculation, the SOC is always in- cluded as it has large effect on the eigenenergy and eigen- wave-function of the electron. 18We also show the spin re- laxation times induced by the hyperﬁne interactions /H20849VeI-ph/H208491/H20850 and VeI-ph/H208493/H20850/H20850and the direct spin-phonon coupling due to the phonon-induced strain but without the SOC as in theliterature. 27,28,45It can be seen clearly that the spin relaxation that includes the SOC is much larger than that without the SOC. For example, the spin relaxation induced by thesecond-order process of the hyperﬁne interaction together with the BP /H20849V eI-ph/H208493/H20850/H20850is at least 1 order of magnitude larger when the SOC is included than that when the SOC is ne- glected. This is because when the SOC is neglected,/H20855m/H20841H eI/H20849r/H20850/H20841/H51291/H20856and /H20855/H51292/H20841HeI/H20849r/H20850/H20841m/H20856in Eq. /H2084929/H20850are small as the matrix elements of HeI/H20849r/H20850between different orbital energy levels are very small. However, when the SOC is taken into account, the spin-up and -down levels with different orbitalquantum numbers are mixed and therefore /H20841/H5129/H20856and /H20841m/H20856in- clude the components with the same orbital quantum num-ber. Consequently, the matrix elements of /H20855m/H20841H eI/H20849r/H20850/H20841/H51291/H20856and /H20855/H51292/H20841HeI/H20849r/H20850/H20841m/H20856become much larger. Therefore, spin relaxation induced by this mechanism depends crucially on the SOC. It is emphasized from the above discussion that the SOC should be included in each spin relaxation mechanism. In thefollowing calculations, it is always included unless otherwisespeciﬁed. In particular, in reference to the mechanism ofelectron-BP interaction, we always consider it together withthe SOC. We further discuss the parallel-magnetic-ﬁeld case. In Fig. 1/H20849b/H20850, the spin relaxation times due to different mechanisms are plotted as a function of the quantum well width for sameparameters as Fig. 1/H20849a/H20850but with a parallel magnetic ﬁeld B /H20648=0.5 T. It is noted that the spin relaxation rate due to each mechanism becomes much smaller for small acompared with the perpendicular case. Another feature is that the spinrelaxation due to each mechanism decreases in a muchslower rate with increasing a. The electron-BP mechanism is dominant even at a=10 nm but decreases faster than other mechanisms with a. It is expected to be less effective than theV eI-ph/H208493/H20850mechanism or VeI-ph/H208491/H20850mechanism or the direct spin- phonon coupling due to phonon-induced strain mechanismfor large enough a. The g-factor ﬂuctuation mechanism is negligible again. These features can be explained as follows.For parallel magnetic ﬁeld, the contribution of the SOC toZeeman splitting is much less than in the perpendicular-magnetic-ﬁeld geometry. 21Moreover, this contribution is negative which makes the Zeeman splitting smaller.21There- fore, the phonon absorption /H20849emission /H20850efﬁciency becomes much smaller for small a, i.e., large SOC. When aincreases, the Zeeman splitting increases. However, the spin mixingdecreases. The former effect is weak and only cancels part of the latter; thus, the spin relaxation due to each mechanismdecreases slowly with a. 2. Magnetic ﬁeld dependence We ﬁrst study the perpendicular-magnetic-ﬁeld case. The magnetic ﬁeld dependence of T1for two different well widths is shown in Figs. 2/H20849a/H20850and2/H20849b/H20850. In the calculation, d0=20 nm. It can be seen that the effect of each mechanism increases with the magnetic ﬁeld. Particularly, theelectron-BP mechanism increases much faster than otherones and becomes dominant at high magnetic ﬁelds. Forsmall well width /H208515 nm in Fig. 2/H20849a/H20850/H20852, the spin relaxation in- duced by the electron-BP scattering is dominant except atvery low magnetic ﬁelds /H208490.1 T in the ﬁgure /H20850where contri- butions from the ﬁrst-order process of hyperﬁne interactiontogether with the electron-BP scattering and the direct spin-g-factorstrainV(1) eI−phV(3) eI−phBP a=5 nm B⊥(T)T−1 1(s−1) 5 4 3 2 1 0106 104 102 100 10−2 10−4 10−6 10−8 a=1 0 nm B⊥(T)T−1 1(s−1) 5 4 3 2 1 0101 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 (b)(a) FIG. 2. /H20849Color online /H20850T1−1induced by different mechanisms vs the perpendicular magnetic ﬁeld B/H11036ford0=20 nm and /H20849a/H20850a =5 nm and /H20849b/H2085010 nm. T=4 K. Curves with /H20849/H17039/H20850—T1−1induced by the electron-BP scattering. Curves with /H20849/L50098/H20850—T1−1induced by the second-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1induced by the ﬁrst-order pro- cess of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with /H20849/H12135/H20850—T1−1induced by the g-factor ﬂuctuation.REEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-9phonon coupling due to phonon-induced strain also contrib- ute. It is interesting to see that when ais increased to 10 nm, the electron-BP scattering is the largest spin relaxationmechanism only at high magnetic ﬁelds /H20849/H110221.1 T /H20850. For 0.4 T/H11021B /H11036/H110211.1 T /H20849B/H11036/H110210.4 T /H20850, the direct spin-phonon coupling due to the phonon-induced strain /H20849the ﬁrst-order hyperﬁne interaction together with the BP /H20850becomes the larg- est relaxation mechanism. It is also noted that there is nosingle mechanism which dominates the whole spin relax-ation. Two or three mechanisms are jointly responsible forthe spin relaxation. It is indicated that the spin relaxations induced by different mechanisms all increase with B /H11036. This can be understood from a perturbation theory: when the mag-netic ﬁeld is small, the spin relaxation between two Zeeman split states for each mechanism is proportional to n ¯/H20849/H9004E/H20850 /H11003/H20849/H9004E/H20850m/H20849/H9004Eis the Zeeman splitting /H20850with m=7 for electron-BP scattering due to the deformation potential18,25 and for the second-order process of the hyperﬁne interaction together with the electron-BP scattering due to the deforma- tion potential VeI-ph/H208493/H20850,27m=5 for electron-BP scattering due to the piezoelectric coupling15,18,25and for the second-order process of the hyperﬁne interaction together with theelectron-BP scattering due to the piezoelectric coupling V eI-ph/H208493/H20850,27m=5 for the direct spin-phonon coupling due to phonon-induced strain,15andm=1 for the ﬁrst-order process of the hyperﬁne interaction together with the BP VeI-ph/H208491/H20850. The spin relaxation induced by the g-factor ﬂuctuation is propor- tional to n¯/H20849/H9004E/H20850/H20849/H9004E/H208505B/H110362. For most of the cases studied, /H9004Eis smaller than kBT; hence, n¯/H20849/H9004E/H20850/H11011kBT//H9004Eandn¯/H20849/H9004E/H20850/H20849/H9004E/H20850m /H11011/H20849/H9004E/H20850m−1.m/H110221 hold for all mechanisms except the VeI-ph/H208491/H20850 mechanism; therefore, the spin relaxation due to these mechanisms increases with increasing B/H11036. However, from Eq. /H2084927/H20850, one can see that it has a term with /H11612r, which indi- cates that the effect of this mechanism is proportional to1/d 0. As the vector potential of the magnetic ﬁeld increases the conﬁnement of the QD and gives rise to smaller effectivediameter d 0, this mechanism also increases with the magnetic ﬁeld in the perpendicular-magnetic-ﬁeld geometry. We then study the case with the magnetic ﬁeld parallel to the quantum well plane. In Fig. 3, the spin relaxation induced by different mechanisms is plotted as a function of the par-allel magnetic ﬁeld B /H20648for two different well widths. In the calculation, d0=20 nm. It can be seen that, similar to the case with perpendicular magnetic ﬁeld, the effects of most mecha-nisms increase with the magnetic ﬁeld. Also, the electron-BPmechanism increases much faster than the other ones andbecomes dominant at high magnetic ﬁelds. However, withoutthe orbital effect of the magnetic ﬁeld in the present conﬁgu- ration, the effect of V eI-ph/H208491/H20850changes very little with the mag- netic ﬁeld. For both small /H208515 nm in Fig. 3/H20849a/H20850/H20852and large /H20851 10 nm in Fig. 3/H20849b/H20850/H20852well widths, the electron-BP scattering is dominant except at very low magnetic ﬁeld /H208490.1 T in the ﬁgure /H20850, where the ﬁrst-order process of the hyperﬁne inter- action together with the electron-BP interaction VeI-ph/H208491/H20850also contributes. 3. Diameter dependence We now turn to the investigation of the diameter depen- dence of the spin relaxation. We ﬁrst concentrate on theperpendicular-magnetic-ﬁeld geometry. The spin relaxation rate due to each mechanism is shown in Fig. 4/H20849a/H20850for a small well width /H20849a=5 nm /H20850and Fig. 4/H20849b/H20850for a large well width /H20849a=10 nm /H20850, respectively, with a ﬁxed perpendicular mag- netic ﬁeld B/H11036=0.5 T. In the ﬁgure, the spin relaxation rate due to each mechanism except VeI-ph/H208491/H20850increases with the ef- fective diameter. Speciﬁcally, the effect of the electron-BPmechanism increases very fast, while the effect of the direct spin-phonon coupling due to the phonon-induced strain mechanism increases very mildly. The V eI-ph/H208491/H20850decreases with d0slowly. Other mechanisms are unimportant. The electron-BP mechanism eventually dominates the spin relax-ation when the diameter is large enough. The threshold in-creases from 12 to 26 nm when the well width increases from 5 to 10 nm. For small diameter, the V eI-ph/H208491/H20850and the direct spin-phonon coupling due to the phonon-induced strainmechanism dominate the spin relaxation. The increase /H20849de- crease /H20850of the spin relaxation due to these mechanisms can beg-factorstrainV(1) eI−phV(3) eI−phBP a=5 nm B/CID1(T)T−1 1(s−1) 5 4 3 2 1 0106 104 102 100 10−2 10−4 10−6 10−8 10−10 a=1 0 nm B/CID1(T)T−1 1(s−1) 5 4 3 2 1 0104 102 100 10−2 10−4 (b)(a) FIG. 3. /H20849Color online /H20850T1−1induced by different mechanisms vs the parallel magnetic ﬁeld B/H20648ford0=20 nm and /H20849a/H20850a=5 nm and /H20849b/H20850 10 nm. T=4 K. Curves with /H20849/H17039/H20850—T1−1induced by the electron-BP scattering. Curves with /H20849/L50098/H20850—T1−1induced by the second-order pro- cess of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1induced by the ﬁrst-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with /H20849/H12135/H20850—T1−1induced by the g-factor ﬂuctuation.JIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-10understood from the following. The effect of the SOC on the Zeeman splitting is proportional to d02for small magnetic ﬁeld.18The increase of d0thus leads to an increase of Zee- man splitting; therefore, the efﬁciency of the phonon absorp-tion /H20849emission /H20850increases. Another effect is that the increase ofd 0will increase the phonon absorption /H20849emission /H20850efﬁ- ciency due to the increase of the form factor.18Thus, the spin relaxation increases. Moreover, the spin mixing is also pro-portional to d 0.18This leads to a much faster increase of the effect of the electron-BP mechanism and the VeI-ph/H208493/H20850mecha- nism. However, the spin relaxation due to VeI-ph/H208491/H20850decreases with the diameter. This is because VeI-ph/H208491/H20850contains a term /H11612r /H20851Eq. /H2084927/H20850/H20852which decreases with the increase of d0. Physi- cally speaking, the decrease of the effect of VeI-ph/H208491/H20850is due to the fact that the spin mixing due to the hyperﬁne interactiondecreases with the increase of the number of nuclei withinthe dot Nas the random Overhauser ﬁeld is proportional to 1/ /H20881N. The spin relaxation induced by the gfactor is alsonegligible here for both small and large well widths. We then turn to the parallel-magnetic-ﬁeld case. In the calculation, B/H20648=0.5 T. The results are shown for both small well width /H20851a=5 nm in Fig. 5/H20849a/H20850/H20852and large well width /H20851a =10 nm in Fig. 5/H20849b/H20850/H20852. Similar to the perpendicular-magnetic- ﬁeld case, the effect of every mechanism except the VeI-ph/H208491/H20850 mechanism increases with increasing diameter. The effect of the electron-BP mechanism increases fastest and becomesdominant for d 0/H1102212 nm for both small and large well widths. For d0/H1102112 nm for the two cases, the ﬁrst-order pro- cess of the VeI-ph/H208491/H20850mechanism becomes dominant. The effect of the VeI-ph/H208493/H20850mechanism becomes larger than that of the di- rect spin-phonon coupling due to the phonon-induced strainmechanism. However, these two mechanisms are still unim-portant and become more and more unimportant for largerd 0. Here, the spin relaxation induced by the gfactor is negligible.a=5 nm d0(nm)T−1 1(s−1) 30 25 20 15 10102 100 10−2 10−4 10−6 10−8 g-factorstrainV(1) eI−phV(3) eI−phBP a=1 0 nm d0(nm)T−1 1(s−1) 30 25 20 15 1010−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10 (b)(a) FIG. 4. /H20849Color online /H20850T1−1induced by different mechanisms vs the effective diameter d0forB/H11036=0.5 T and /H20849a/H20850a=5 nm and /H20849b/H20850 10 nm. T=4 K. Curves with /H20849/H17039/H20850—T1−1induced by the electron-BP scattering. Curves with /H20849/L50098/H20850—T1−1induced by the second-order pro- cess of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1induced by the ﬁrst-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with /H20849/H12135/H20850—T1−1induced by the g-factor ﬂuctuation.a=5 nm d0(nm)T−1 1(s−1) 30 25 20 15 10104 102 100 10−2 10−4 10−6 10−8 10−10 g-factorstrainV(1) eI−phV(3) eI−phBP a=1 0 nm d0(nm)T−1 1(s−1) 30 25 20 15 10102 100 10−2 10−4 10−6 10−8 (b)(a) FIG. 5. /H20849Color online /H20850T1−1induced by different mechanisms vs the effect diameter d0with B/H20648=0.5 T and /H20849a/H20850a=5 nm and /H20849b/H20850 10 nm. T=4 K. Curves with /H20849/H17039/H20850—T1−1induced by the electron-BP scattering. Curves with /H20849/L50098/H20850—T1−1induced by the second-order pro- cess of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1induced by the ﬁrst-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with /H20849/H12135/H20850—T1−1induced by the g-factor ﬂuctuation.REEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-114. Comparison with experiment In this subsection, we apply our analysis to the experi- mental data in Ref. 7. The results are plotted in Fig. 6.W e ﬁrst show that our calculation is in good agreement with theexperimental results. Then, we compare contributions fromdifferent mechanisms to spin relaxation as a function of themagnetic ﬁeld. In the calculation, we choose the quantum dotdiameter d 0=56 nm /H20849/H6036/H92750=1.1 meV as in experiment /H20850. The quantum well is taken to be an inﬁnite-depth well with a =13 nm. The Dresselhaus SOC parameter /H92530/H20855kz2/H20856is taken to be 4.5 meV Å and the Rashba SOC parameter is 3.3 meV Å. T=0 K as kBT/H11270g/H9262BBin the experiment. The magnetic ﬁeld is applied parallel to the well plane in the /H20851110 /H20852direction. The Dresselhaus cubic term is also taken into consideration.All these parameters are the same with /H20849or close to /H20850those used in Ref. 24in which a calculation based on the electron-BP scattering mechanism agrees well with the ex-perimental results. For this mechanism, we reproduce theirresults. The spin relaxation time measured by the experi-ments /H20849black dots with error bar in the ﬁgure /H20850almost coin- cides with the calculated spin relaxation time due to theelectron-BP scattering mechanism /H20849curves with /H17039in the ﬁgure /H20850. 71It is noted from the ﬁgure that other mechanisms are unimportant for small magnetic ﬁeld. However, for largemagnetic ﬁeld, the effect of the direct spin-phonon couplingdue to phonon-induced strain becomes comparable to that ofthe electron-BP mechanism. At B /H20648=10 T, the two differs by a factor of /H110115. B. Spin dephasing time T2 In this subsection, we investigate the spin dephasing time for different well widths, magnetic ﬁelds, and QD diameters.As in the previous subsection, the contributions of the differ- ent mechanisms to spin dephasing are compared.70To justify the ﬁrst Born approximation in studying the hyperﬁne-interaction-induced spin dephasing, we focus mainly on thehigh magnetic ﬁeld regime of B/H110223.5 T. A typical magnetic ﬁeld is 4 T. We also demonstrate via extrapolation that in thelow magnetic ﬁeld regime, spin dephasing is dominated bythe hyperﬁne interaction. 1. Well width dependence In Fig. 7, the well width dependence of the spin dephasing induced by different mechanisms is presented under the per-pendicular /H20849a/H20850and parallel /H20849b/H20850magnetic ﬁelds. In the calcu- lations, B /H11036=4 T /H20849B/H20648=4 T /H20850andd0=20 nm. It can be seen in both ﬁgures that the spin dephasing due to each mechanismg-factorstrainV(1) eI−phV(3) eI−phBP B/CID1(T)T−1 1(s−1) 15 12 9 6 3 0104 102 100 10−2 10−4 10−6 10−8 10−10 10−12 FIG. 6. /H20849Color online /H20850T1−1induced by different mechanisms vs the parallel magnetic ﬁeld B/H20648in the /H20851110 /H20852direction for d0=56 nm and a=13 nm with both the Rashba and Dresselhaus SOCs. T =0 K. The black dots with error bar are the experimental results inRef. 7. Curves with /H20849/H17039/H20850—T 1−1induced by the electron-BP scatter- ing. Curves with /H20849/H17039/H20850—T1−1induced by the second-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1induced by the ﬁrst-order process of the hyperﬁne inter- action together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain.Curves with /H20849/H12135/H20850—T 1−1induced by the g-factor ﬂuctuation.B⊥=4 T t(µs)\|\|/CID1S+/CID2t\|\|(a.u.) 65432100.4 0.2 0 a(nm)T−1 2(s−1) 10 9 8 7 6 5 4 3 21020 1015 1010 105 100 10−5 V(1) eI−phV(3) eI−phg-factorstrainhyperﬁneBP B/CID4=4 T a(nm)T−1 2(s−1) 10 9 8 7 6 5 4 3 2105 100 10−5 10−10 10−15 10−20 10−25 (b)(a) FIG. 7. /H20849Color online /H20850T2−1induced by different mechanisms vs the well width for d0=20 nm. T=4 K. /H20849a/H20850B/H11036=4 T with /H20849solid curves /H20850and without /H20849dashed curves /H20850the SOC and /H20849b/H20850B/H20648=4 T only with the SOC. Curve with /H20849/H17039/H20850—T2−1induced by the electron-BP interaction. Curves with /H20849/L50098/H20850—T2−1induced by the hyperﬁne inter- action. Curves with /H20849/H17010/H20850—T2−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with /H20849/H12135/H20850—T2−1in- duced by g-factor ﬂuctuation. /H20849/H17009/H20850—T2−1induced by the second- order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17040/H20850—T2−1induced by the ﬁrst-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. The time evolution of /H20648/H20855S+/H20856t/H20648induced by the hyperﬁne interaction with a =2 nm is shown in the inset of /H20849a/H20850.JIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-12decreases with a. Moreover, the spin dephasing due to the electron-BP scattering decreases much faster than that due tothe hyperﬁne interaction. These features can be understoodas follows. The spin dephasing due to electron-BP scatteringdepends crucially on the SOC. As the SOC is proportional toa −2, the spin dephasing decreases fast with a. For the hyper- ﬁne interaction, from Eq. /H2084935/H20850, one can deduce that the decay rate of /H20648/H20855S+/H20856t/H20648is mainly determined by the factor 1 //H20849azd/H206482/H20850 /H20849here az=a/H20850, which thus decreases with abut in a very mild way. The fast decrease of the electron-BP mechanism makesit eventually unimportant. For the present perpendicular-magnetic-ﬁeld case the, threshold is around 2 nm. For paral-lel magnetic ﬁeld, it is even smaller. A higher temperaturemay enhance the electron-BP mechanism /H20849see discussion in Sec. V /H20850and make it more important than the hyperﬁne mechanism. It is noted that other mechanisms contributevery little to the spin dephasing. Thus, in the following dis-cussion, we do not consider these mechanisms. ComparingFigs. 7/H20849a/H20850and7/H20849b/H20850, one ﬁnds that a main difference is that the electron-BP mechanism is less effective for the parallel-magnetic-ﬁeld case. As has been discussed in the previoussubsection, the spin mixing and the Zeeman splitting in theparallel ﬁeld case is smaller than those in the perpendicularﬁeld case. Therefore, the electron-BP mechanism is weak-ened markedly. Similar to Fig. 1, the SOC is always included in the com- putation as it has large effect on the eigenenergy and eigen-wave-function of the electrons. The spin dephasings calcu-lated without the SOC for the hyperﬁne interaction, thedirect spin-phonon coupling due to phonon-induced strain,and the g-factor ﬂuctuation are also shown in Fig. 7/H20849a/H20850as dashed curves. It can be seen from the ﬁgure that for the spindephasings induced by the direct spin-phonon coupling dueto phonon-induced strain and by the g-factor ﬂuctuation, the contributions with the SOC are much larger than those with-out. This is because when the SOC is included, the ﬂuctua-tion of the effective ﬁeld induced by both mechanisms be-comes much stronger and more scattering channels areopened. However, what should be emphasized is that the spindephasings induced by the hyperﬁne interaction with andwithout the SOC are nearly the same /H20849the solid and the dashed curves nearly coincide /H20850. This is because the change of the wave function /H9023/H20849r/H20850due to the SOC is very small /H20849less than 1% in our condition /H20850and therefore the factor 1 //H20849a zd/H206482/H20850is almost unchanged when the SOC is neglected. Thus, the spin dephasing rate is almost unchanged. In the inset of Fig. 7/H20849a/H20850, the time evolution of /H20648/H20855S+/H20856t/H20648in- duced by the hyperﬁne interaction is shown, with a=2 nm. It can be seen that /H20648/H20855S+/H20856t/H20648decays very fast and decreases to less than 10% of its initial value within the ﬁrst two oscillating periods. Therefore, T2is determined by the ﬁrst two or three periods of /H20648/H20855S+/H20856t/H20648. Thus, the correction of the long-time dy- namics due to higher order scattering33contributes little to the spin dephasing time. For quantum computation and quan-tum information processing, the initial, e.g., 1% decay of/H20648/H20855S +/H20856t/H20648may be more important than the 1 /edecay.42,43In- deed, the spin dephasing time deﬁned by the exponential ﬁtting of 1% decay is shorter than that deﬁned by the 1 /e decay. However, the two differs less than ﬁve times. For arough comparison of contributions from different mecha- nisms to spin dephasing where only the order-of-magnitudedifference is concerned /H20849see Figs. 7–9/H20850, this difference due to the deﬁnition does not jeopardize our conclusions. 2. Magnetic ﬁeld dependence We then investigate the magnetic ﬁeld dependence of the spin dephasing induced by the electron-BP scattering and bythe hyperﬁne interaction for two different well widths /H20849a =3 nm and a=5 nm /H20850with both perpendicular and parallel magnetic ﬁelds. From Figs. 8/H20849a/H20850and8/H20849b/H20850, one can see that the spin dephasing due to the electron-BP scattering in-creases with the magnetic ﬁeld, whereas that due to the hy-perﬁne interaction decreases with the magnetic ﬁeld. Thus,the electron-BP mechanism eventually dominates the spindephasing for high enough magnetic ﬁeld. The threshold is B /H11036c=4 T /H20849B/H20648c=7 T /H20850fora=3 nm with perpendicular /H20849parallel /H20850 magnetic ﬁeld. For larger well width, e.g., a=5 nm with par- allel magnetic ﬁeld or perpendicular magnetic ﬁeld, thethreshold magnetic ﬁelds increase to larger than 8 T. ThehyperﬁneBP B⊥(T)T−1 2(s−1) 8 7.57 6.56 5.55 4.54 3.5108 107 106 105 104 B/CID4(T)T−1 2(s−1) 8 7.57 6.56 5.55 4.54 3.5107 106 105 104 103 (b)(a) FIG. 8. /H20849Color online /H20850T2−1induced by the electron-BP scattering and the hyperﬁne interaction vs /H20849a/H20850the perpendicular magnetic ﬁeld B/H11036and /H20849b/H20850the parallel magnetic ﬁeld B/H20648fora=3 nm /H20849solid curves /H20850 and 5 nm /H20849dashed curves /H20850.T=4 K and d0=20 nm. Curves with /H20849/H17039/H20850—T2−1induced by the electron-BP interaction. Curves with /H20849/L50098/H20850—T2−1induced by the hyperﬁne interaction.REEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-13different magnetic ﬁeld dependences above can be under- stood as follows. Besides spin relaxation, the spin-ﬂip scat-tering also contributes to spin dephasing. 20As has been dem- onstrated in Sec. III A, the electron-BP scattering inducedspin-ﬂip transition rate increases with the magnetic ﬁeld.Therefore, the spin dephasing rate increases with the mag-netic ﬁeld also. In contrast, spin dephasing induced by thehyperﬁne interaction decreases with the magnetic ﬁeld. Thisis because when the magnetic ﬁeld becomes larger, the ﬂuc-tuation of the effective magnetic ﬁeld due to the surroundingnuclei becomes insigniﬁcant. Therefore, the hyperﬁne-interaction-induced spin dephasing is reduced. Similar re-sults have been obtained by Deng and Hu. 44 3. Diameter dependence In Fig. 9, the spin dephasing times induced by the electron-BP scattering and the hyperﬁne interaction are plot-ted as a function of the diameter d 0for small /H20849a=3 nm /H20850and large /H20849a=5 nm /H20850well widths. In the calculation, B/H11036=4 T in Fig.9/H20849a/H20850andB/H20648=4 T in Fig. 9/H20849b/H20850. It is noted that the effect of the electron-BP mechanism increases rapidly with d0,whereas the effect of the hyperﬁne mechanism decreases slowly. Consequently, the electron-BP mechanism eventuallydominates the spin dephasing for large enough d 0. The threshold is d0c=19 /H2084927/H20850nm for the a=3/H208495/H20850nm case with the perpendicular magnetic ﬁeld and d0c=26 /H2084930/H20850nm for the a =3 /H208495/H20850nm case under the parallel magnetic ﬁeld. As has been discussed in Sec. III A, both the effect of the SOC and the efﬁciency of the phonon absorption /H20849emission /H20850increase with d0. Therefore, the spin dephasing due to the electron-BP mechanism increases rapidly with d0.18,21The decrease of the effect of the hyperﬁne interaction is due to the decrease of the factor 1 //H20849azd/H206482/H20850/H20851Eq. /H2084935/H20850/H20852with the diameter d0. IV. SPIN RELAXATION TIMES FROM FERMI GOLDEN RULE AND FROM EQUATION OF MOTION In this section, we will try to ﬁnd a proper method to average over the transition rates from the Fermi golden rule, /H9270i→f−1, to give the spin relaxation time T1. In the limit of small SOC, we rederive Eq. /H208491/H20850from the equation of motion. We further show that Eq. /H208491/H20850fails for large SOC where a full calculation from the equation of motion is needed. We ﬁrst rederive Eq. /H208491/H20850for small SOC from the equation of motion. In QDs, the orbital level splitting is usuallymuch larger than the Zeeman splitting. Each Zeeman sub-level has two states: one with a majority up spin and theother with a majority down spin. We call the former asthe “minus state” /H20849as it corresponds to a lower energy /H20850, while the latter as the “plus state.” For small SOC, the spinmixing is small. Thus, we neglect the much smaller contri-bution from the off-diagonal terms of the density matrix to S z. Therefore, Sz/H20849t/H20850=/H20858i/H11006Szi/H11006fi/H11006/H20849t/H20850where i/H11006denotes the plus/ minus state of the ith orbital state. For small SOC, the spin relaxation is much slower than the orbital relaxation.25,55 This implies that the time to establish equilibrium within the plus/minus states is much smaller than the spin relaxationtime. Thus, we can assume an equilibrium /H20849Maxwell- Boltzmann /H20850distribution between the plus/minus states at any time. The distribution function is therefore given by f i/H11006/H20849t/H20850=N/H11006/H20849t/H20850exp/H20849−/H9255i/H11006/kBT/H20850/Z/H11006. Here, N/H11006/H20849t/H20850=/H20858ifi/H11006/H20849t/H20850is the total probability of the plus/minus states with N+/H20849t/H20850+N−/H20849t/H20850 =1 for a single electron in QD and Z/H11006=/H20858iexp/H20849−/H9255i/H11006/kBT/H20850is the partition function for the plus/minus state. At equilib- rium, N/H11006=N/H11006eq. The equation for Sz/H20849t/H20850is hence d dtSz/H20849t/H20850=d dt/H20851Sz/H20849t/H20850−Szeq/H20852=/H20858 i/H11006Szi/H11006exp/H20849−/H9255i/H11006/kBT/H20850/Z/H11006d dt/H9254N/H11006/H20849t/H20850, /H2084936/H20850 with/H9254N/H11006/H20849t/H20850=N/H11006/H20849t/H20850−N/H11006eq. As the orbital level splitting is usu- ally much larger than the Zeeman splitting, the factor exp/H20849−/H9255i/H11006/kBT/H20850/Z/H11006can be approximated by exp /H20849−/H9255i0/kBT/H20850/Z0 with/H9255i0=1 2/H20849/H9255i++/H9255i−/H20850andZ0=/H20858iexp/H20851−/H9255i0/kBT/H20852. Further using the particle-conservation relation /H20858/H11006/H9254N/H11006/H20849t/H20850=0, one has d dtSz/H20849t/H20850=/H20875/H20858 i/H20849Szi+−Szi−/H20850exp/H20849−/H9255i0/kBT/H20850/Z0/H20876d dt/H9254N+/H20849t/H20850. /H2084937/H20850 As Sz/H20849t/H20850−Szeq=/H20851/H9254N+/H20849t/H20850/Z0/H20852/H20858i/H20849Szi+−Szi−/H20850exp/H20849−/H9255i0/kBT/H20850, one ﬁnds that the spin relaxation time is nothing but the relax-hyperﬁneBPB⊥=4 T d0(nm)T−1 2(s−1) 30 25 20 15 10108 107 106 105 104 103 102 B/CID4=4 T d0(nm )T−1 2(s−1) 30 25 20 15 10107 106 105 104 103 102 101 (b)(a) FIG. 9. /H20849Color online /H20850T2−1induced by the electron-BP scattering and the hyperﬁne interaction vs the effective diameter d0,T=4 K. /H20849a/H20850B/H11036=4 T and /H20849b/H20850B/H20648=4 T for a=3 nm /H20849solid curves /H20850and 5 nm /H20849dashed curves /H20850. Curves with /H20849/H17039/H20850—T2−1induced by the electron-BP interaction. Curves with /H20849/L50098/H20850—T2−1induced by the hyperﬁne interaction.JIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-14ation time of N+. The next step is to derive the equation of d dt/H9254N+/H20849t/H20850, which is given in our previous work,49 d dt/H9254N+/H20849t/H20850=/H20858 id dt/H9254fi+/H20849t/H20850=−/H20858 i,f/H20851/H9270i+→f−−1/H9254fi+/H20849t/H20850−/H9270i−→f+−1/H9254fi−/H20849t/H20850/H20852 =−/H20858 i,f/H20851/H9270i+→f−−1+/H9270i−→f+−1/H20852e−/H9255i0/kBT Z0/H9254N+/H20849t/H20850. /H2084938/H20850 Thus, spin relaxation time is given by 1 T1=/H20858 i,f/H20849/H9270i+→f−−1+/H9270i−→f+−1/H20850e−/H9255i0/kBT Z0. /H2084939/H20850 Furthermore, substituting e−/H9255i0/kBT/Z0by fi/H110060=exp /H20849−/H9255i/H11006/kBT/H20850/Z/H11006, we have 1 T1=/H20858 i,f/H20849/H9270i+→f−−1fi+0+/H9270i−→f+−1fi−0/H20850. /H2084940/H20850 This is exactly Eq. /H208491/H20850. For large SOC or large spin mixing due to the anticross- ing of different spin states,19,25the spin relaxation rate be- comes comparable to the orbital relaxation rate. Furthermore,the decay of the off-diagonal term of the density matrixshould contribute to the decay of S z. Therefore, the above analysis does not hold. In this case, it is difﬁcult to obtainsuch a formula and a full calculation from the equation ofmotion is needed. In Fig. 10/H20849a/H20850, we show /H20849forT=12 K, a=5 nm, B /H11036 =0.5 T, d0=30 nm /H20850the spin relaxation times T1calculated from the equation-of-motion approach and that obtainedfrom Eq. /H2084940/H20850. Here, for simplicity and without loss of gen- erality, we consider only the electron-BP scattering mecha-nism. The discrepancy of T 1obtained from the two ap- proaches increases with /H9253.A t/H9253=10/H92530, the ratio of the two becomes as large as /H110113. In Fig. 10/H20849b/H20850, we plot the spin relaxation times obtained via the two approaches as a func-tion of temperature for /H9253=/H92530with other parameters remain- ing unchanged. It is noted that the discrepancy of T1obtained from the two approaches increases with temperature. Forhigh temperature, the higher levels are involved in the spindynamics where the SOC becomes larger. At 40 K, the dis-crepancy is as large as 60%. The ratio increases very slowlyforT/H1102120 K where only the lowest two Zeeman sublevels are involved in the dynamics. V. TEMPERATURE DEPENDENCE OF SPIN RELAXATION TIME T1AND SPIN DEPHASING TIME T2 We ﬁrst study the relative magnitude of the spin relax- ation time T1and the spin dephasing time T2. We consider a QD with d0=30 nm and a=5 nm at B/H11036=4 T where the larg- est contribution to both spin relaxation and dephasing comesfrom the electron-BP scattering /H20851see Figs. 4/H20849a/H20850and9/H20849a/H20850,w e have checked that the electron-BP scattering mechanism isdominant throughout the temperature range /H20852. From Fig. 11, one ﬁnds that when the temperature is low /H20849T/H110215 K in the ﬁgure /H20850,T 2=2T1, which is in agreement with the discussion inRef. 20. However, T1/T2increases very quickly with Tand forT=20 K, T1/T2/H110112/H11003102. This is understood from the fact that when Tis low, the electron mostly distributes in the lowest two Zeeman sublevels. For small SOC, Golovach etT=1 2 K γ/γ 0 T−1 1(s−1)R106 105 104 103 102 109 8 7 6 5 4 3 2 13 2.5 2 1.5 1 γ=γ0 T(K) T−1 1(s−1)R109 108 107 106 105 104 103 102 101 100 10−1 40 35 30 25 20 15 10 5 01.8 1.6 1.4 1.2 1 0.8 (b)(a) FIG. 10. /H20849Color online /H20850Spin relaxation time T1calculated from the equation-of-motion approach /H20849/H17039/H20850vs that obtained from Eq. /H208491/H20850 /H20849/L50098/H20850as a function of /H20849a/H20850the strength of the SOC for T=12 K and /H20849b/H20850 the temperature for /H9253=/H92530. The well width a=5 nm, perpendicular magnetic ﬁeld B/H11036=0.5 T, and QD diameter d0=30 nm. The ratio of the two Ris also plotted in the ﬁgure. Note that the scale of T1−1is at the right hand side of the frame. T1/T2T2T1 T(K) Spin Decoherence (s)T1/T210−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12 10−13 25 20 15 10 5 0103 102 101 100 10−1 FIG. 11. /H20849Color online /H20850Spin relaxation time T1, spin dephasing time T2, and T1/T2against temperature T.B/H11036=4 T, a=5 nm, and d0=30 nm. Note that the scale of T1andT2is at the right hand side of the frame.REEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-15al.have shown via perturbation theory that the phonon in- duces only the spin-ﬂip noise in the leading order. Conse-quently, T 2=2T1.20When the temperature becomes compa- rable to the orbital level splitting /H6036/H92750, the distribution over the upper orbital levels is not negligible anymore. As men-tioned previously, the SOC contributes a nontrivial part tothe Zeeman splitting. Speciﬁcally, the second-order energycorrection due to the SOC contributes to the Zeeman split-ting. The energy correction for different orbital levels is gen-erally unequal /H20849always larger for higher levels /H20850. When the electron is scattered by phonons randomly from one orbitalstate to another one with the same major spin polarization,the frequency of its precession around the zdirection changes. Continuous scattering leads to a random ﬂuctuationof the precession frequency and thus leads to spindephasing. 29,46Note that this ﬂuctuation only leads to a phase randomization of S+but not ﬂips the zcomponent spin Sz, i.e., not leads to a spin relaxation. Therefore, the spin dephasing becomes stronger than the spin relaxation for hightemperatures. Moreover, this effect increases with tempera-ture rapidly as the distribution over higher levels and thephonon numbers both increase with temperature. We further study the temperature dependence of spin re- laxation for lower magnetic ﬁeld and larger quantum wellwidth where other mechanisms may be more important thanthe electron-BP mechanism. In Fig. 12/H20849a/H20850, the spin relaxation time is plotted as a function of temperature for B /H11036=0.5 T, a=10 nm, and d0=20 nm. It is seen from the ﬁgure that the direct spin-phonon coupling due to the phonon-inducedstrain mechanism dominates the spin relaxation throughoutthe temperature range. It is also noted that for T/H333554 K, the spin relaxation rates induced by different mechanisms all in-crease with temperature according to the phonon number fac- tor 2 n ¯/H20849Ez1/H20850+1 with Ez1being the Zeeman splitting of the lowest Zeeman sublevels. However, for T/H110224 K, the spin relaxation rates induced by the direct spin-phonon couplingdue to phonon-induced strain and the electron-BP interactionincrease rapidly with temperature, while the spin relaxation rates induced by V eI-ph/H208491/H20850andVeI-ph/H208493/H20850increase mildly according to 2n¯/H20849Ez1/H20850+1 throughout the temperature range. These fea- tures can be understood as follows. For T/H333554 K, the distri- bution over the high levels is negligible. Only the lowest twoZeeman sublevels are involved in the spin dynamics. The spin relaxation rates thus increase with 2 n ¯/H20849Ez1/H20850+1 and the relative importance of each mechanism does not change. Therefore, our previous analysis on the comparison of therelative importance of different spin decoherence mecha-nisms at 4 K holds true for the range 0 /H33355T/H333554 K. When the temperature gets higher, the contribution from higher levelsbecomes more important. Although the distribution at thehigher levels is still very small, for the direct spin-phononcoupling mechanism, the transition rates between the higherlevels and those between higher levels and the lowest twosublevels are very large. For the electron-BP mechanism, thetransition rates between the higher levels are very large dueto the large SOC in these levels. Therefore, the contributionfrom the higher levels becomes larger than that from thelowest two sublevels. Consequently, the increase of tempera-ture leads to a rapid increase of the spin relaxation rates.However, for the two hyperﬁne mechanisms, the V eI-ph/H208491/H20850and theVeI-ph/H208493/H20850, the spin relaxation rates do not change much when the higher levels are involved. They thus increase by thephonon number factor. In Fig. 12/H20849b/H20850, we show the temperature dependence of the spin relaxation time for the same condition but with B =0.9 T. It is noted that the spin relaxation rate due to theelectron-BP mechanism catches up with that induced by thedirect spin-phonon coupling due to phonon-induced strain atT=9 K and becomes larger for higher temperature. This in- dicates that the temperature dependence of the two mecha-nisms is quite different. In Fig. 13, we show the spin dephasing induced by electron-BP scattering and the hyperﬁne interaction as afunction of temperature for B /H11036=4 T, a=10 nm, and d0 =20 nm. We choose the conditions so that the spin dephasing is dominated by the hyperﬁne interaction at low temperature.However, the effect of the electron-BP mechanism increasesg-factorstrainV(1) eI−phV(3) eI−phBP B⊥=0 .5T T(K)T−1 1(s−1) 25 20 15 10 5 0105 100 10−5 10−10 10−15 B⊥=0 .9T T(K)T−1 1(s−1) 25 20 15 10 5 0106 104 102 100 10−2 10−4 10−6 10−8 (b)(a) FIG. 12. /H20849Color online /H20850Spin relaxation time T1against tempera- ture Tfor /H20849a/H20850B/H11036=0.5 T and /H20849b/H20850B/H11036=0.9 T. a=10 nm and d0 =20 nm. Curves with /H20849/H17039/H20850—T1−1induced by the electron-BP scatter- ing together with the SOC. Curves with /H20849/L50098/H20850—T1−1induced by the second-order process of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208493/H20850/H20850. Curves with /H20849/H17009/H20850—T1−1induced by the ﬁrst-order pro- cess of the hyperﬁne interaction together with the BP /H20849VeI-ph/H208491/H20850/H20850. Curves with /H20849/H17010/H20850—T1−1induced by the direct spin-phonon coupling due to phonon-induced strain. Curves with /H20849/H12135/H20850—T1−1induced by theg-factor ﬂuctuation.JIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-16with temperature quickly, while that of the hyperﬁne interac- tion remains nearly unchanged. The fast increase of the ef-fect from the electron-BP scattering is due to three factors:/H208491/H20850the increase of the phonon number, /H208492/H20850the increase of scattering channels, and /H208493/H20850the increase of the SOC induced spin mixing in higher levels. On the other hand, from Eq./H2084935/H20850, one can deduce that the spin dephasing rate of the hy- perﬁne interaction depends mainly on the factor 1 //H20849a zd/H206482/H20850 with az/H20849d/H206482/H20850is the characteristic length /H20849area /H20850along the z direction /H20849in the quantum well plane /H20850. For higher levels, the d/H206482is larger but only about a factor smaller than 10. Thus, the effect of the hyperﬁne interaction increases very slowly withtemperature. It should be noted that in the above discussion, we ne- glected the two-phonon scattering mechanism, 15,46,50which may be important at high temperature. The contribution ofthis mechanism should be calculated via the equation-of-motion approach developed in this paper and compared withthe contribution of other mechanisms showed here. VI. CONCLUSION In conclusion, we have investigated the longitudinal and transversal spin decoherence times T1and T2, called spin relaxation time and spin dephasing time, in different condi-tions in GaAs QDs from the equation-of-motion approach.Various mechanisms, including the electron-BP scattering,the hyperﬁne interaction, the direct spin-phonon couplingdue to phonon-induced strain and the g-factor ﬂuctuation, are considered. Their relative importance is compared. There isno doubt that for spin decoherence induced by electron-BPscattering, the SOC must be included. However, for spin de-coherence induced by the hyperﬁne interaction, the directspin-phonon coupling due to phonon-induced strain, g-factor ﬂuctuation, and hyperﬁne interaction combined withelectron-phonon scattering, the SOC is neglected in the ex-isting literature. 27,28,45Our calculations have shown that, as the SOC has marked effect on the eigenenergy and the eigen-wave-function of the electron, the spin decoherence induced by these mechanisms with the SOC is larger than that with-out it. Especially, the decoherence from the second-orderprocess of hyperﬁne interaction combined with theelectron-BP interaction increases at least 1 order of magni-tude when the SOC is included. Our calculations show that,with the SOC, in some conditions some of these mechanisms/H20849except g-factor ﬂuctuation mechanism /H20850can even dominate the spin decoherence. There is no single mechanism which dominates spin re- laxation or spin dephasing in all parameter regimes. The rela-tive importance of each mechanism varies with the wellwidth, magnetic ﬁeld, and QD diameter. In particular, theelectron-BP scattering mechanism has the largest contribu-tion to spin relaxation and spin dephasing for small wellwidth and/or high magnetic ﬁeld and/or large QD diameter.However, for other parameters, the hyperﬁne interaction, theﬁrst-order process of the hyperﬁne interaction combined withelectron-BP scattering, and the direct spin-phonon couplingdue to phonon-induced strain can be more important. It isnoted that the g-factor ﬂuctuation always has very little con- tribution to spin relaxation and spin dephasing which canthus be neglected all the time. For spin dephasing, theelectron-BP scattering mechanism and the hyperﬁne interac-tion mechanism are more important than other mechanismsfor magnetic ﬁeld higher than 3.5 T. For this regime, othermechanisms can thus be neglected. It is also shown that spindephasing induced by the electron-BP mechanism increasesrapidly with temperature. Extrapolated from our calculation,the hyperﬁne interaction mechanism is believed to be domi-nant for small magnetic ﬁeld. We also discussed the problem of ﬁnding a proper method to average over the transition rates /H9270i→f−1obtained from the Fermi golden rule to give the spin relaxation time T1at ﬁnite temperature. For small SOC, we rederived the formula for T1 at ﬁnite temperature used in the existing literature18,51,52from the equation of motion. We further demonstrated that thisformula is inadequate at high temperature and/or for largeSOC. For such cases, a full calculation from the equation-of-motion approach is needed. The equation-of-motion ap-proach provides an easy and powerful way to calculate thespin decoherence at anytemperature and SOC. We also studied the temperature dependence of spin re- laxation T 1and dephasing T2. We show that for very low temperature if the electron only distributes on the lowest twoZeeman sublevels, T 2=2T1. However, for higher tempera- tures, the electron spin dephasing increases with temperaturemuch faster than the spin relaxation. Consequently, T 1/H11271T2. The spin relaxation and dephasing due to different mecha-nisms are also compared. ACKNOWLEDGMENTS This work was supported by the Natural Science Founda- tion of China under Grant Nos. 10574120 and 10725417, theNational Basic Research Program of China under Grant No.2006CB922005 and the Innovation Project of Chinese Acad-emy of Sciences. Y.Y.W. would like to thank J. L. Cheng forvaluable discussions.hyperﬁneBP T(K)T−1 2(s−1) 25 20 15 10 5 01010 108 106 104 102 100 FIG. 13. /H20849Color online /H20850Spin relaxation time T1against tempera- ture T.B/H11036=4 T, a=10 nm, and d0=20 nm. 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JETP 33, 1053 /H208491971 /H20850/H20852.JIANG, WANG, AND WU PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-1866A. Abragam, The Principles of Nuclear Magnetism /H20849Oxford Uni- versity Press, Oxford, 1961 /H20850, Chaps. VI and IX. 67This can be obtained from Eq. /H2084917/H20850in Ref. 42. 68Semiconductors , Landolt-Börnstein, New Series, Vol. 17a, edited by O. Madelung /H20849Springer-Verlag, Berlin, 1987 /H20850. 69W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszewska, D. Bertho, F. Kobbi, J. L. Robert, G. E. Pikus, F. G. Pikus, S. V.Iordanskii, V. Mosser, K. Zekentes, and Yu. B. Lyanda-Geller,Phys. Rev. B 53, 3912 /H208491996 /H20850. 70It should be mentioned that one effect is not included: when the electron is scattered by the phonon from one orbital state toanother, it feels a difference in the spin precession frequency since the strength of longitudinal /H20849along the external magnetic ﬁeld /H20850component of the Overhauser ﬁeld differs with orbital states. This effect randomizes the spin precession phase andleads to a pure spin dephasing. However, this effect is negligiblein our paper. 71The deviation of our calculation from the experimental data at T=14 T is due to the fact that we do not include the cyclotron effect along the z, direction. For B/H1140710 T, the cyclotron orbit length is smaller than the quantum well width, which makes ourmodel unrealistic.REEXAMINATION OF SPIN DECOHERENCE IN … PHYSICAL REVIEW B 77, 035323 /H208492008 /H20850 035323-19
PhysRevB.71.214414.pdf	Effective interaction between the interpenetrating Kagomé lattices in Na xCoO2 Martin Indergand,1Yasufumi Yamashita,2Hiroaki Kusunose,3and Manfred Sigrist1 1Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich, Switzerland 2Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan 3Physics Department, Tohoku University, Sendai 980-8578, Japan sReceived 8 February 2005; published 21 June 2005 d A multiorbital model for a CoO 2layer in Na xCoO2is derived. In this model, the kinetic energy for the degenerate t2gorbitals is given by indirect hopping over oxygen, leading naturally to the concept of four interpenetrating Kagomé lattices. Local Coulomb interaction couples the four lattices and an effective Hamil-tonian for the interaction in the top band can be written in terms of fermionic operators with four differentﬂavors. Focusing on charge- and spin-density instabilities, a big variety of possible metallic states with spon-taneously broken symmetry are found. These states lead to different charge, orbital, spin, and angular momen-tum ordering patterns. The strong superstructure formation at x=0.5 is also discussed within this model. DOI: 10.1103/PhysRevB.71.214414 PACS number ssd: 75.30.Fv, 71.55. 2i, 73.21.Cd I. INTRODUCTION Layered Na xCoO2has been initially studied for its ex- traordinary thermoelectric properties and for its interestingdimensional crossover. 1–4But recently, wider attention has been triggered by the discovery of superconductivity in hy-drated Na 0.35CoO2and the discovery of an insulating phase in Na 0.5CoO2.5–8Since then, various types of charge- ordering phenomena in Na xCoO2have been reported,9–23but also strong spin ﬂuctuations and spin-density-wave transi-tions have been observed. 24–36 The material consists of CoO 2layers where Co ions are enclosed in edge-sharing O octahedra. These layers alternatewith the Na-ion layers with Na entering as Na 1+and donating one electron each to the CoO 2layer. The electronic properties are dominated by the 3 d-t2g electrons of the Co ions which form a two-dimensional tri- angular lattice. However, the spatial arrangement of the Na1+ ions plays a crucial role too for the physics of this material.There are two basic positions for the Na ions, one directlyabove or below a Co site and another in a center position ofa triangle spanned by the Co lattice. The metallic propertiesare unusual and vary with the Na concentration and arrange-ment. A brief overview of the present knowledge of the phase diagram of Na xCoO2leads to the following still rough pic- ture. The most salient and robust feature, at ﬁrst glance, isthe charge-ordered phase for x=0.5 separating the Na-poor from the Na-rich system. The Na ions arrange in a certainpattern inducing an insulating magnetic phase in the CoO 2 layers below 50 K.35On the Na-poor side sx,0.5d, the com- pound behaves like a paramagnetic metal. When it is inter- calated with H 2O, superconductivity appears between x <0.25 and x<0.35. In several respects, more interesting is the Na-rich side where one ﬁnds a so-called Curie-Weissmetal. Here the magnetic susceptibility displays a pro-nounced Curie-Weiss-like behavior after subtracting an un-derlying temperature-independent part: x=C/sT−Qd, where Qranges roughly between −50 and −200 K depending on x, and the Curie constant is consistent with a magnetic momentin the range of 1–1.7 mB. Note that deviations from the Curie-Weiss behavior have been observed at lowtemperatures. 37On the Na-rich side, a transition at high tem- perature ,250–340 K has been observed and interpreted as crystal structure or charge ordering.15,18,28For Na 0.75CoO2,a magnetic transition occurs at 22 K and is most likely a com-mensurate spin-density wave or ferrimagnetic order which israther soft towards magnetic polarization. 27–30Interestingly, this magnetic phase is metallic and has even a higher mobil-ity than the nonmagnetic phase. For Na content xø0.75, several magnetic transitions at a similar critical temperaturehave been observed, but mSR data suggest rather an incom- mensurate spin-density-wave order.29,30,36 The arrangement of the Na ions between the layers de- pends on the Na doping x, and several superstructures have been found.9,10The clearest evidence for the superstructure formation is at x=0.5, where the Na ordering leads to a metal-insulator transition at low temperatures.7,8,12But also away from x=0.5, nuclear magnetic resonance sNMR dex- periments indicate the existence of nonequivalent cobalt sitesand phase separation. 14,16 The complex interplay between Na arrangement and the electronic properties poses an interesting problem. Varioustheoretical studies have mainly focused on single-band mod-els on the frustrated triangular lattice, in particular in con-nection with the superconducting phase ignoring Napotentials. 38–44There has also been work done on multior- bital models45–47and density-functional calculations have been performed.48–55According to local-density approxima- tion sLDA dcalculations, the Fermi surface lies near the top of the 3d-t2gbands.They form a large holelike Fermi surface of predominantly a1gcharacter in agreement with angle- resolved photoemission spectroscopy sARPES dexperi- ments.56–58In addition, the LDA calculations suggest that smaller hole pockets with mixed a1gandeg8character exist in theG-Kdirection on the Na-poor side. At the Gpoint, the states with a1gandeg8symmetry are clearly split, but on average over the entire Brillouin zone the mixing between a1gandeg8is substantial. Koshibae and Maekawa argued that the splitting at the Gpoint originates from the cobalt-oxygen hybridization rather than from aPHYSICAL REVIEW B 71, 214414 s2005 d 1098-0121/2005/71 s21d/214414 s19d/$23.00 ©2005 The American Physical Society 214414-1crystal-ﬁeld effect due to the distortion of the oxygen octa- hedra, because the crystal-ﬁeld effect in a simple ionic pic- ture would lead to the opposite splitting of the a1gandeg8 states.45There is also spectral evidence that the low-energy excitations of Na xCoO2have signiﬁcant O 2 pcharacter.59 Reproducing the LDA Fermi surface with a tight-binding ﬁt for the Co t2gorbitals, it turns out that the direct overlap integral between the cobalt orbitals is much smaller than theindirect hopping integral over the oxygen 2 porbitals. 46 Therefore, it is reasonable to start with a three-band tight- binding model of degenerate t2gorbitals, where the only hop- ping processes are indirect hopping processes over interme-diate oxygen orbitals. This approximation provides aninteresting system of four independent and interpenetratingKagomé lattices, as was already pointed out by Koshibae andMaekawa. Our study will be based on this model band structure which has a high symmetry. Within this model, we examinevarious forms of order that could be possible from on-siteCoulomb interaction. The paper is organized as follows. InSec. II, the tight-binding model and the concepts of Kagoméoperators and pocket operators are introduced. In Sec. III, aneffective Hamiltonian for the local Coulomb interaction isderived, and in Sec. IV this effective interaction is written ina diagonal form, by choosing an appropriate basis of SU s4d generators. In Sec. V, the effects of small deviations from oursimpliﬁed tight-binding model are discussed. In Sec. VI, allpossible charge and spin ordering patterns of our model andthe corresponding phase transitions are brieﬂy described. InSec. VII, the relevance of the above-described collective de-grees of freedom to Na xCoO2is discussed by comparing the different coupling constants and by taking into accountsymmetry-lowering effects. In Sec. VIII, we apply our modelto the Na ordering observed at x=0.5, and we summarize and conclude in Sec. IX. II. TIGHT-BINDING MODEL We base our model on the assumption that the 3 d-t2gor- bitals on the Co ions are degenerate. Their electrons disperseonly via phybridization with the intermediate oxygens oc- cupying the surrounding octahedra sFig. 1 d. As noticed by Koshibae and Maekawa, the resulting electronic structurecorresponds to a system of four decoupled equivalent elec-tron systems of electrons hopping on a Kagomé lattice. 45The different sites, however, are represented by different orbitals.Each of the three orbitals hd yz,dzx,dxyjon a given site par- ticipate in one Kagomé lattice, and the fourth Kagomé lattice has a void on this site. The corresponding tight-bindingmodel has the following form: H tb=o kso mm8ekmm8ckms†ckm8s, s1d whereckms†=s1/˛Ndoreik·rcrm†are the operators in momen- tum space of crms†, which creates a t2gorbital sdyz,dzx,dxyd with index mPh1,2,3 jand spin sPh",#jon the cobalt site r.Nis the number of Co sites in the lattice,eˆk=1−m2tcossk3d2tcossk2d 2tcossk3d−m2tcossk1d 2tcossk2d2tcossk1d−m2, s2d withki=k·ai, cf. Fig. 1. The hopping parameter t=tpd2/D .0, where tpdis the hopping integral between the pyand the dxyordyzorbital shown in Fig. 1. Dis the energy difference between the oxygen pand the Co- t2glevels. The diagonal- ization of the matrix eˆkby a rotation matrix OˆkPSOs3d, o mm8Okimekmm8Okjm8=dijEki, s3d results in the three energy bands Ek1=t+t˛1+8cos sk1dcossk2dcossk3d−m, Ek2=t−t˛1+8cos sk1dcossk2dcossk3d−m, Ek3=−2t−m. s4d These bands have the periodicity Ek+Bjl=Ekl, where the vec- torsBjare deﬁned by ai·Bj=2p ˛3sinsui−ujd,i,jPh1ﬂ3js5d with uj=2pj/3. These three vectors Bjconnect the Gpoint with the three Mpoints in the Brillouin zone sBZd, and the vectors 2 Bjare primitive reciprocal-lattice vectors. The bands of this tight-binding model have therefore a higherperiodicity than the bands of a more general model. Thisleads to the appearance of special symmetry lines sthin lines d and symmetry points sM 8andK8din the Brillouin zone, shown in Fig. 2, where the bands are plotted along the lineG 8-K8-M8-G8. Within a reduced BZ, these bands correspond to the bands of a nearest-neighbor tight-binding model on aKagomé lattice. 45The density of states per spin and per re- duced BZ is also shown in Fig. 2. It has a logarithmic sin- gularity at E=2tand jumps from ˛3/s2ptdt o0a tE=4t. FIG. 1. sColor online dSchematic ﬁgure of a CoO 2plane drawn with cubic unit cells. The edge sharing of the oxygen octahedraaround the Co ions is visualized. The edges of the cubes are ori-ented along the coordinate system sx,y,zd. The triangular lattice of the cobalt is spanned by the vectors a 1,a2,a3sa1+a2=−a3d.a =uaiuis the lattice spacing. An oxygen 2 porbital and the cobalt t2g orbitals hybridizing with it by phybridization are shown.INDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-2The states that are connected by the considered hopping processes form a Kagomé lattice. Since in this way the CoO 2 plane consists of four independent and interpenetratingKagomé lattices, 45it is convenient to label the states belong- ing to the same Kagomé lattice with an index lPh0,1,2,3 j. This can be done with the vectors alof Fig. 1 as aRm†l=cR+al+amm†. s6d In this way, the operators aRm†lwith ﬁxed lcreate all the states off a Kagomé lattice. In the following, these operators willbe called Kagomé operators . Their Fourier transform is given by a Km†l=2 ˛No ReiK·sR+al+amdaRm†l, s7d where the vectors Kbelong to the reduced BZ, labeled 0 in Fig. 2 and Rruns over the lattice spanned by the vectors 2 ai. The BZ consists of four reduced BZs shown in Fig. 2.An alternative labeling of the states is obtained, therefore, bydeﬁning the operators b Km†j=e−iBj·amcK+Bjm†, s8d where the vectors Bjare deﬁned in Eq. s5dand in addition we setB0=0. As shown in Eq. sA1dof Appendix A, the transformation between the Kagomé operators aK†land the pocket operators bK†jcorresponds to a discrete Fourier trans- formation of a 2 32 lattice, and is given by bKm†j=1 2o leiBj·alaKm†l=o lFjlaKm†l, s9d where we have deﬁned the symmetric and orthogonal 4 34 matrixFjl=Flj=Fjl=Fjl−1=1 2eiBj·al. s10d Note that the matrix elements of Fare ±1/2, as the scalar products Bj·alof Eq. s5dequal 0 or ± p. The tight-binding Hamiltonian s1dis diagonal in the pocket indices jfcf. Appendix A, Eq. sA2dg, Htb=o lKso mm8eKmm8bKms†lbKm8sl. s11d From this expression, it is apparent that the tight-binding Hamiltonian is invariant under any U s4dtransformation of the form bKms†j!b˜ Kms†j=o j8Ujj8bKms†j8. s12d Equation s9dis just a special case of Eq. s12d. This shows thatHtbis also diagonal in the Kagomé indices. It is important to notice that the transformations in Eq. s12dinvolve symmetries that are not present in a more gen- eral tight-binding model. For example, a ﬁnite hopping inte-gralt dddue to the shybridization between neighboring t2g orbitals would break this symmetry. We will discuss this as- pect below in more detail and remain for the time being inthis high-symmetry situation. In Na xCoO2, the lower two bands are completely ﬁlled and will be quite inert. For this reason, in the followingsections we will only deal with the operators of the top band E k1whose operators are denoted as aKs†l=o mOK1maKms†landbKs†j=o mOK1mbKmsj,s13d respectively, where OK1mare matrix elements of the rotation matrixOˆKof Eq. s3d. The top band gives rise to four identical Fermi surface pockets in the BZ, one in the Gpoint and three at the M points.Atranslation in the reciprocal space by the vectors Bj maps the pocket around the Gpoint onto a pocket around the Mpoint. However, this fact does not lead to nesting singu- larities in the susceptibility because a hole pocket is mappedonto a hole pocket by the vector B j. The susceptibility of the top band is given by xqo=1 No kfk+q−fk Ek1−Ek+q1=4 No KfK+Q−fK EK1−EK+Q1, s14d wherefk=ffbsEk1−mdgandfis the Fermi function. In the last expression of Eq. s14d, the sum over Kis restricted to the reduced BZ. Qalso lies in the reduced BZ and is given by Q=q+Bj. The susceptibility xq0=xQ0is periodic with respect to the reduced BZ and is just four times the susceptibility ofa single Kagomé lattice. As we have almost circular holepockets with quadratic dispersion around the Gand theM points, the susceptibility is therefore approximately given bythe susceptibility of the free-electron gas in two dimensionswithin each reduced BZ, with circular plateaus of radius K F around the Gand the three Mpoints. FIG. 2. The original Brillouin zone sBZdof the triangular lattice consists of four reduced BZs around the Gpoint s0dand the three M points s1,2,3 d. The symmetry points of the reduced BZs— M8,K8, andG8—are symmetry points for the tight-binding model in Eq. s1d due to the higher periodicity of the bands. It is therefore sufﬁcientto draw the bands along the lines G 8-K8-M8-G8. The Fermi surface sFSdforx=0.5 lies at Ek1<3.16t. The density of states per spin and per reduced Brillouin zone Dis given in units of 1/ t. It has a logarithmic singularity at E=2t.EFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-3III. COULOMB INTERACTION In this section, we introduce the Coulomb interaction be- tween the electrons. As we have spin and orbital degrees offreedom, the on-site Coulomb interaction consists of intraor-bital repulsion U, interorbital repulsion U 8, Hund’s coupling JH, and a pair hopping term J8. These parameters are related byU<U8+2JHandJH=J8, where the ﬁrst relation is exact for spherical symmetry. We can write the on-site Coulombinteraction as H rC=Uo mnrm"nrm#+U8 2o mÞm8o ss8nrmsnrm8s8 +JH 2o mÞm8o ss8crms†crm8s8†crms8crm8s +J8 2o mÞm8o sÞs8crms†crms8†crm8s8crm8s, s15d wherenrms=crms†crms. We obtain an effective Hamiltonian for the Coulomb interaction by rewriting the Hamiltonian in terms of the pocket operators of the top band bKs†ldeﬁned in Eq.s13d. For small k=uKua, we can expand Eq. s13din pow- ers of k2and obtain up to terms of the order k2 bKs†j=1 ˛3o mS1+k2 12cosf2su−umdgDbKmsj, s16d where um=2pm/3. Expanding the energy of the top band around the point G8, we obtain eK1=tS4−k2+k4 12−k6 360coss6ud+Osk8dD. s17d This shows that the pockets around the points G8are almost perfectly circular. The radius kF/aof these pockets depends on the Na doping x. Note that xcorresponds to the density of carriers with x=1 giving a completely ﬁlled top band. We have kF2=ps1−xd/˛3. For the interaction in weak coupling and at low temperatures, the states near the Fermi surface are important. For these states and for not too small Na doping x, we can neglect the second term in the parentheses of Eq. s16d compared to 1. Note that this condition on xis not very restrictive. Even for x=0.35, the second term together with all higher-order terms is on the average one order of magni-tude smaller than 1. Dropping the second term in Eq. s16d spreads the a 1gsymmetry of the states bKs†j, which is exact only for K=0, to all relevant states in the top band. The interaction s15dcan now be rewritten in terms of the a1g symmetric operators bKs†j. Processes involving states of the ﬁlled lower bands are dropped. The dropping of the secondterm in the parentheses of Eq. s16dis a considerable simpli- ﬁcation because it removes all Kdependence of the poten- tial. At this point, it is convenient to introduce density and spin-density operators for the pocket operators of the topband,nˆ Qij=4 No KsbK+Qs†ibKsj,Sˆ Qij=2 No Kss8bK+Qs†isss8bKsj. s18d The resulting effective interaction can be expressed with these operators in the following way: Heff=N 32o QSBijklsSˆ QijSˆ −Qlk+1 4BijklcnˆQijnˆ−QlkD. s19d The symbols Bc/sdepend on the Coulomb integrals and are given by Bijklc/s=±Cs2dijkl−eijkl2d±Ddildjk+Ec/sdijdkl+Fc/sdikdjl, s20d where the dse2dsymbol equals 1 if all the indices are equal sdifferent dand 0 otherwise. The coefﬁcients C,D,Ec/s, and Fc/sare listed in Table I. Note that for small pockets, the momenta Kof the pocket operators bKjin the four fermion terms of Eq. s19dcannot add up to a half a reciprocal-lattice vectorBi. In order to conserve momentum they must there- fore add up to zero. Due to the position of the pockets in theBZ, umklapp processes with low-energy transfer are, how-ever, possible for arbitrary small pockets. In fact, the pro- cesses proportional to eijkl2anddildjks1−dijdare umklapp pro- cesses, as Bi−Bj+Bl−Bkis a nonvanishing reciprocal-lattice vector for eijklÞ0 and for dildjks1−dijdÞ0, and from Eq. s8d the momentum created by the operator bK†jisK+Bj. Some details about the derivation of Eq. s19dare provided in Appendix B. There are different ways of writing this in-teraction in terms of the operators in Eq. s18d. Our formula- tion treats charge and spin degrees of freedom in the sameway. It corresponds to the decomposition of a Hubbard inter- actionn "n#into1 2s1 4n2−S·Sd. In order to express the effective interaction Hamiltonian of Eq. s19din terms of the Kagomé operators aKsl, we deﬁne spin- and charge-density operators from the Kagomé opera- torsaKslas in Eq. s18d, nQij=4 No KsaK+Qs†iaKsj,SQij=2 No Kss8aK+Qs†isss8aKsj. s21d Note that the density operators, which are deﬁned from the pocket operators bKsj, are marked by a hut. The effective Hamiltonian, Heff,o fE q . s19dcan be rewritten as Heff=N 32o QSAijklsSQijS−Qlk+1 4AijklcnQijn−QlkD. s22d From Eqs. s9dands10dit follows thatTABLE I. The coefﬁcients of Eq. s20d. 9C=−3U+2J8+2JH+2U8 9D=+3U+6J8−2JH−2U8 9Ec=+3U−2J8−10JH+14U8 9Es=−3U+2J8−6JH+2U8 9Fc=+3U−2J8+14JH−10U8 9Fs=−3U+2J8+2JH−6U8INDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-4Aijklc/s=FimFjnFkoFlpBmnopc/s. s23d The symbols Ac/sturn out to have a simpler structure, given by Aijklc=8 9F−C 2dijkl+J8dildjk+s2U8−JHddijdkl +s2JH−U8ddikdjlG, Aijkls=8 9F+C 2dijkl−J8dildjk−JHdijdkl−U8dikdjlG.s24d IV. SU(4) GENERATORS The tight-binding Hamiltonian described in Sec. II has a Us4dsymmetry, reﬂecting the fact that it consists of four independent and equivalent Kagomé lattices. The correla-tions introduced by the on-site Coulomb repulsion in Eq.s15dbreak this symmetry and lead to interaction between orbitals belonging to different Kagomé lattices, as the threet 2gorbitals on a given Co site belong to three different Kagomé lattices.The effective Hamiltonian in Eq. s19dis not invariant under general U s4dtransformations, but is still in- variant under a ﬁnite subgroup of U s4d. The symbols Aijklc/s deﬁned in Eq. s24dare invariant under permutation of the indices, i.e., Aijkls/c=APsidPsjdPskdPslds/c,PPS4. s25d From this it follows that the symmetric group S4is a sub- group of G. Multiplying all operators aK,slwith the same Kagomé index lby −1 also leaves the Hamiltonian, Heff, invariant, because the symbols Aijklc/sare nonzero only if the four indices ijklare pairwise equal.These two different sym- metry operations generate a group with 384 elements. ThisgroupGis isomorphic to the symmetry group of the four- dimensional hypercube. In Appendix C, the structure of thegroupGis discussed and a character table is shown. To proceed, let Q r,r=0,...,15, be a basis in the 16- dimensional real vector space, V, of Hermitian 4 34 matri- ces, fulﬁlling the usual orthonormality and completeness re-lations Q ijrQjil=1 2drl,o r=015 QijrQklr=1 2dildjk. s26d This basis can be chosen such that Q0is proportional to the unit matrix, Q1−3are diagonal, Q4−9are real, and Q10−15are imaginary. It is convenient to deﬁne also the dualmatrices Kijr=FimFjnQmnr. s27d In Table II, a choice of a basis Qr, which is particularly suitable for our purposes, is shown together with the dualbasisK r.Arepresentation rof the group GonVis given by rsgdQr=NgTQrNgforgPG, whereNgis the natural four- dimensional representation of Gscf. Appendix C d. The rep- resentation ris reducible and Vis the direct sum of the fourirreducible subspaces V0,V1−3,V4−9, andV10−15spanned by matricesQ0,Q1−3,Q4−9, andQ10−15, respectively. Therefore, the chosen basis is appropriate for the symmetry group G. Deﬁning charge- and spin-density operators nQr=QijrnQij=KijrnˆQij,SQr=QijrSQij=KijrSˆ Qij, s28d the interaction Hamiltonian can be written in a diagonal form as Heff=N 8o r=015 o QSLrsSQrS−Qr+1 4LrcnQrn−QrD. s29d The coupling constants Lrc/sare equal for all Qrbelonging to the same irreducible subspace in V. They are given in Table III. V. REDUCTION OF THE SYMMETRY The tight-binding Hamiltonian in Eq. s11dh a saU s4d symmetry and even after introducing Coulomb interaction,the effective Hamiltonian s29dis invariant under the symme- try group G. In a real CoO 2plane, this symmetry is reduced even in the paramagnetic state.There are terms in the Hamil-tonian of the real system that restrict the symmetry opera-tions ofGto the subgroup, which describes real crystallo- graphic space-group symmetries. A trigonal distortion of the oxygen octahedra by ap- proaching the two O layers to the Co layer is, for example,compatible with the point-group symmetry D 3dof the CoO 2 layer. However, it lifts the degeneracy of the t2gorbitals, leading to a term Htr=Dtro kso mÞm8ckms†ckm8s =Dtro lKso mÞm8bKms†lbKm8sleiBl·sam−am8ds30d in the Hamiltonian, where we used Eq. s8dto obtain the second line. For the top band, we obtain with Eqs. s16dand sB4d Htr=˛2/3Dtr4o lKsfKll4+Osk2dgbKs†lbKsl<˛2/3DtrNn04, s31d where the matrix K4is given in Table II and k=uKuais small for the relevant states near the Fermi pockets, if the pocketsare small enough. Similarly, a ﬁnite direct hopping integralt ddleads to the term Hdd=tddo kms2cos skmdckms†ckms =4˛6tddo lKsfKll4+Osk2dgbKs†lbKsl <˛6tddNn04, s32d where we again dropped the terms involving the lower bands in the second line. In fact, any other additional hopping termor any quadratic perturbation compatible with the spaceEFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-5group is proportional to the ﬁeld n04in the limit of small pockets if the perturbation is diagonal in the spin indices.Asthe trigonal distortion of the octahedra is nonzero and addi-tional hopping terms are present in the CoO 2layer, a term proportional to n04exists in the Hamiltonian acting like a symmetry-breaking ﬁeld. For simplicity, we will refer to a term proportional to n04in the Hamiltonian as the trigonaldistortion , even though this term is rather an effective trigo- nal distortion that also includes the effects of additional hop-ping terms. From the matrix K 4it can be seen that the presence of a ﬁnite ﬁeld, n04, in the Hamiltonian leads to a distinction be- tween the Gand theMpoints in the BZ and the four hole pockets are no longer equivalent. In real space, the fourTABLE II. The matrices Q1−15are a choice of an orthonormal complete basis of the 15-dimensional real vector space of traceless Hermitian matrices, so called generators of SU s4d, that is adequate to the symmetry of the CoO 2layer. The matrices Krare obtained from Qrby Eq. s27d. Note that 1 ¯=−1 and i¯=−i.2˛2Q0=2˛2K0is the 4 34 unit matrix. 1 2˛2Q1sG5ad 11000 0100 001¯0 000 1¯21 2˛6Q4sG1bd 10111 1011 1101 111021 2˛2Q7sG5bd 10100 1000 000 1¯ 001¯021 4Q10sG4d 100i¯i 00ii¯ ii¯00 i¯i0021 4Q13sG5cd 100i¯i¯ 00i¯i¯ ii00 ii002 1 2˛2Q2sG5ad 11000 01¯00 0010 000 1¯21 4Q5sG3d 1011¯0 100 1¯ 1¯001 01¯1021 2˛2Q8sG5bd 10010 000 1¯ 1000 01¯0021 4Q11sG4d 10i0i¯ i¯0i0 0i¯0i i0i¯021 4Q14sG5cd 10i¯0i¯ i0i0 0i¯0i¯ i0i02 1 2˛2Q3sG5ad 11000 01¯00 001¯0 000121 4˛3Q6sG3d 1011 2¯ 102¯1 12¯01 2¯11021 2˛2Q9sG5bd 10001 001¯0 01¯00 100021 4Q12sG4d 10i¯i0 i00i¯ i¯00i 0ii¯021 4Q15sG5cd 10i¯i¯0 i00i i00i 0i¯i¯02 1 2˛2K1sG5ad 10100 1000 0001 001021 2˛6K4sG1bd 13000 01¯00 001¯0 000 1¯21 2˛2K7sG5bd 10100 1000 000 1¯ 001¯021 2K10sG4d 10000 0000 000 i 00i¯021 2K13sG5cd 10i00 i¯000 0000 00002 1 2˛2K2sG5ad 10010 0001 1000 010021 2K5sG3d 10000 0100 001¯0 000021 2˛2K8sG5bd 10010 000 1¯ 1000 01¯0021 2K11sG4d 10000 000 i¯ 0000 0i0021 2K14sG5cd 100i0 0000 i¯000 00002 1 2˛2K3sG5ad 10001 0010 0100 100021 2˛3K6sG3d 10000 0100 0010 000 2¯21 2˛2K9sG5bd 10001 001¯0 01¯00 100021 2K12sG4d 10000 00i0 0i¯00 000021 2K15sG5cd 1000 i 0000 0000 i¯0002INDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-6Kagomé lattices are still equivalent, as they transform under space-group symmetries among themselves. In fact, the ma-trixQ 4is still invariant under permutations of rows and col- umns, i.e., NgTQ4Ng=Q4for allgPS4, butQ4is not invariant under changing the sign of all operators with the sameKagomé index. These sign changes, however, are not space-group symmetries, but gauge symmetries, originating fromthe fact that the charge on the Kagomé lattices is conservedbyH tband also by the Coulomb interaction except for the pair-hopping term proportional to J8in Eq. s15d. This term, however, can only change the number of electrons by two,leading to these gauge symmetries, which are broken as soonas single electron hopping processes between the Kagomélattices are introduced. To classify the states according to the real symmetry group of the CoO 2layer without gauge symmetries it is therefore sufﬁcient to consider the presence of a small ﬁeld n04that restricts the symmetry group Gto a subgroup, con- sisting of space-group symmetries of the CoO 2layer. This subgroup of Gis isomorphic to S4.Td.O. Intuitively, it is understandable that the symmetry of the four-dimensionalcube reduces to the symmetry of a three-dimensional cube ifone of the four hole pockets is not equivalent to the otherthree. From Table II it can be seen that the matrices Q 0,Q1−3, Q4,Q5−6,Q7−9,Q10−12, andQ13−15transform irreducibly un- derS4with the representations G1a,G5a,G1b,G3,G5b,G4, and G5c, respectively, where the superscript letter distinguishes be-tween different subspaces transforming with the same repre-sentation. The appearance of three-dimensional irreducible repre- sentations in the classiﬁcation of the order parameters can beunderstood as follows. The point group Pof a single CoO 2 layer isD3d, and the degree of its irreducible representations isł2. The point group is the factor group S/T, whereSis the space group of the CoO 2layer and Tis the subgroup of all pure translations. For our system, it is convenient to con-sider the factor group P 8=S/2T, where 2Tis the subgroup of Tthat is generated by translations of 2 ai.P8is isomorphic to the cubic group Ohand has irreducible representations of degree 3. The operators nQrandSQrtransform irreducibly un- der the translations in 2 Tfor every r. The symmetry opera- tions ofP8, however, mix operators nQrsorSQrdwith different r, and the irreducible representations as given above or shown inTable II are obtained. Strictly speaking, the basis ofSUs4dgenerators shown in Table II is the correct eigenbasis only for an inﬁnitesimal small trigonal distortion; for a ﬁnite distortion, the representations G 1aandG1bas well as G5aandG5b can hybridize as they transform with the same irreducible representation. Note that G5ctransforms differently under time reversal. The situation here is similar to atomic physics,where a crossover from the Zeeman effect to the Paschen-Back effect with increasing magnetic ﬁeld occurs, because states with the same J zcan hybridize. VI. ORDERING PATTERNS In this section, the different types of symmetry-breaking phase transitions are discussed in a mean-ﬁeld picture. Thesymmetry breaking is due to existence of a ﬁnite order pa-rameter, which is given in our case by the expectation value kn QrlofkSQrl. Note that a ﬁnite expectation value kn00lorkn04l does not break any symmetry of the CoO 2layer. In our tight-binding model, as was discussed in Sec. II, the susceptibility x0is given by four identical plateaux around the Gand theMpoints. In the presence of a trigonal distortion, the susceptibility still keeps a plateau-like struc-ture but the diameter of the plateaux decreases, such that thesusceptibility appears sharply enhanced around the Mand theGpoints. Therefore, we restrict the discussion to the case whereQequals zero and write n randSrinstead of n0randS0r from now on. Note that in our formalism, the states with Q=0 describe periodic states with the enlarged unit cell of the Kagomé lattice. But the internal degrees of freedomwithin this enlarged unit cell still allow for rather compli-cated charge and spin patterns. States with a small but ﬁniteQdescribe modulations of these local states on long wave- lengths. It is therefore important to understand ﬁrst the localstates that are described by Q=0 instabilities. Furthermore, onlyQ=0 states couple to the periodic potential produced by a Na superstructure at x=0.5. TheQ=0 instabilities lead to a chemical potential differ- ence for states belonging to different hole pockets. In gen-eral, the BZ is folded and states of different hole pocketscombine to new quasiparticles. In this case, translationaland/or rotational symmetry is broken. Complex ordering pat-terns can be realized without opening of gaps, i.e., the sys-tem stays metallic. We consider ﬁrst the orderings given by a ﬁnite expecta- tion value of the charge-density operators n r. This expecta- tion value is given by knrl=4 No KsllrkvKs†lvKsll, s33d where llrare the eigenvalues of the matrix QrsUkirQijrU¯ ljr =llrdkldandvKsl=UlnraKsnare the creation operators of the quasiparticles. If only one knrlÞ0, the effective interaction Hamiltonian in the mean-ﬁeld approximation reduces to Lrcknrl 4o KsllrvKs†lvKsl. s34d If the coupling constant Lrcis negative, the interaction energy of the system can be lowered by introducing an imbalanceTABLE III. The coefﬁcients Lrs/c. r 0 1–3 4–9 10–15 Lrc 2 9s3U+12U8−6JHd2 9s3U−4U8+2JHd2 9s−2U8+4JH−2J8d2 9s−2U8+4JH+2J8d Lrs−2 9s3U+6JHd −2 9s3U−2JHd −2 9s2U8−2J8d −2 9s2U8+2J8dEFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-7between the occupation numbers nl=oKskvKs†lvKsll. The op- erators vKs†create Bloch states with momentum Kin the reduced BZ. The amplitudes of the three t2gorbitals on a given Co site with these Bloch states can be obtained from Eqs. s6dand s7dand the relation aK†l<1/˛3omaKm†lwhich follows from Eqs. s9dands16d. For the matrices Q0–4, these Bloch states are given by a singlet2gorbital on each Co site. For the nondiagonal matri- cesQ4–9these Bloch states are on each Co site proportional to a linear combination of t2gorbitals of the form 1 ˛3ssxdx+sydy+szdzdwithsx,sy,szPh±1j.s35d This linear combination is the atomic dorbital w0;Y20par- allel to the body-diagonal fsx,sy,szgof the cubic unit cell around a Co atom. The eigenvectors of the matrices Q10–15are complex. A complex linear combination of t2gorbitals has in general a nonvanishing expectation value of the orbital angular mo-mentum operator L. In Table IV, the angular momentum expectation values, which are relevant for our discussion, areshown. The quasiparticles vKslare expressed in terms of pocket operators by vKsl=Uˆ lmrbKsm, where the unitary matrix Uˆ lmr =UlnrFnmdiagonalizes Kr. From this it follows that if Kris already diagonal, no folding of the BZ occurs and transla-tional symmetry is not broken. Otherwise, the BZ is foldedand states of different pockets recombine to form the newquasiparticles. Now we consider ﬁnite expectation values of the spin- density operators S r. Due to the absence of spin-orbit cou- pling, our model has an SU s2drotational symmetry in spin space.Therefore, the discussion can be restricted to the order parameters kSzrl=kez·Srl, given by kSzrl=2 No KsllrskvKs†lvKsll, s36d where stakes the values 1 and −1 corresponding to spin up and down. If only one kSzrlÞ0, the effective interaction Hamiltonian reduces to LrskSzrl 2o KsllrsvKs†lvKsl. s37d The mean-ﬁeld Hamiltonian s37dis given by the same qua- siparticles and the same eigenvalues llras the Hamiltonian in Eq.s34d. The only difference is that the sign of the splittingof the quasiparticle bands depends on the spin. In the follow- ing, all ordering transitions with order parameters knrland kSzrlforr=0,...,15 are brieﬂy discussed. r=0 Charge: kn0lis the total charge of the system, which is ﬁxed and nonzero, even in the paramagnetic phase. Spin:Aﬁnite kSz0ldescribes a Stoner ferromagnetic insta- bility.The coupling constant L0sgiven inTable III is the most negative coupling constant. In the unperturbed system with-out trigonal distortion, the critical temperature of all continu-ous transitions discussed here only depends on the density ofstates and on the coupling constant in the mean-ﬁeld picture.In this case, ferromagnetism is the leading instability for theunperturbed system. In the real CoO 2plane, this must not necessarily occur, but strong ferromagnetic ﬂuctuations willbe present in any case. r=1–3 Charge:A ﬁnite expectation value kn rlforr=1,2,3 cor- responds to a difference in the charge density on the four Kagomé lattices, because the matrices Q1–3of Table II are diagonal and the quasiparticles vKs†lare just the Kagomé statesaKs†l. From the viewpoint of Fermi surface pockets given by K1–3, which are nondiagonal, this order yields a folding of the BZ, because the quasiparticles vKs†lare linear combinations of states belonging to different hole pockets.This means that the translational symmetry is broken. In thematrixQ 1–3, we ﬁnd two positive and two negative diagonal elements. Consequently, a ﬁnite expectation value kn1–3l leads to a charge enhancement on two Kagomé lattices and to a charge reduction on the other two. As specifying twoKagomé lattices speciﬁes a direction on the triangular lattice,rotational symmetry is broken and crystal symmetry is re-duced from hexagonal to orthorhombic. The phases de-scribed by the matrices Q 1–3have the same coupling constant L1cbecause they transform irreducibly into each other under crystal symmetries with the representation G5a. In order to examine which linear combinations of the three order param-eters kn 1l,kn2l, and kn3lcould be stable below the critical temperature, we consider the Landau expansion of the free energy DF=F−F0, DF=a 2sh12+h22+h32d+bh1h2h3+g1 4sh12+h22+h32d2 +g2 4sh12h22+h22h32+h32h12d, s38d with h1=kn1l,h2=kn2l,h3=kn3l. For g1.maxh0,−g2j, the free energy is globally stable. For g2,0, Eq. s38dhas a minimum of the form h1=h2=h3,i fb2−4as3g1+g2d.0. This phase is described by the symmetric combination Q˜1 =sQ1+Q2+Q3d/˛3 which does not break the rotational sym- metry. In Fig. 3, the folding of BZ and the splitting of the bands sthe dotted line is triply degenerate dand the orbital pattern of the quasiparticles vKs†l=aKs†lare shown. Note that Q˜1has one positive and three negative diagonal elements.TABLE IV. The expectation values of the angular momentum operatorLfor several complex linear combinations of t2gorbitals. v=e2pi/3. sidx+dy+dzd/˛3 kLl="s0,−1,1 d2/3 scyclic d sidx+dy−dzd/˛3 kLl="s0,1,1 d2/3 scyclic d sdx+v2dy+vdzd/˛3 kLl="s1,1,1 d/˛3 sv2dy+vdzd/˛2 kLl="s1,0,0 d˛3/2 scyclic dINDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-8The charge is enhanced or reduced on a single Kagomé lat- tice depending on the sign of the coefﬁcient bin Eq. s38d. The third-order term in the free-energy expansion is allowedby symmetry, because there is no inversionlike symmetrythat would switch the s h1,h2,h3d!s−h1,−h2,−h3d. There- fore, the transition can be ﬁrst order. On the other hand, for g2.0, there is a competition between the terms proportional tog2and bin Eq. s38d. The minimum does not have a simple form. For ubu!g2, however, the transition yields states approximately described by the matrix Q1,Q2,o rQ3. In any case, this phase does break the rotational symmetry. Spin:The spin-density mean ﬁelds kSzil,i=1,2,3trans- form under space-group symmetries like G5aand time- reversal symmetry gives kSzilto −kSzil. Due to the latter, the third-order term in Eq. s38dis forbidden, so that the transi- tion is continuous. For g2,0, Eq. s38dhas again a minimumof the form h1=h2=h3, whereas for g2.0 the minimum is realized for h1Þ0 and h2=h3=0sand permutations difa ,0. The folding of the BZ, the quasiparticles, and the break- ing of space-group symmetries is the same as for the charge-density operators n i. However, the splitting of the bands depends now on the spin and time-reversal symmetry isbroken. These states are spin-density waves, spatial modulations of the spin density with a vanishing total magnetization. Thetwo different types of spin-density modulations for g2.0o r g2,0 are shown in Fig. 4. For g2.0, rotational and trans- lational symmetry is broken, yielding a collinear spin orien-tation along one spatial direction and alternation perpendicu-lar. In contrast, g2.0 yields a rotationally symmetric spin- density wave with a doubled unit cell. This special type ofspin-density wave gives a subset of lattice points, forming atriangular lattice, of large spin density and another subset FIG. 3. sColor online dCharge-ordering instability with ﬁnite expectation value kn˜01lleading to a charge enhancement or reduction on one Kagomé lattice.The folding of the BZ and the splitting of the pockets is shown sthe double dotted line in the BZ indicates a triply degenerate pocket d. On the right, a quasiparticle state that is in this case just a Kagomé lattice state is drawn. FIG. 4. sColor online dThe spin-density-wave pattern corresponding to a ﬁnite expectation value kSz1lis shown on the left. This pattern is stabilized if g2.0 in Eq. s38d. The pattern on the right corresponds to a ﬁnite order parameter kSz1+Sz2+Sz3l, which is stabilized for g2,0.EFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-9with opposite spin density of a third in size, forming a Kagomé lattice. Both states are metallic, because no gaps areopened at the FS. This spin-density wave is not a result ofFermi surface nesting, but due to the complex orbital struc- ture. The coupling constant for this transition, L 1s, is the sec- ond strongest coupling in the model Hamiltonian after theferromagnetic coupling constant, L0s, as is best seen in Fig. 7. r=4 Charge:As discussed in Sec. V, a ﬁnite expectation value ofn4does not break any space-group symmetry. The matrix K4is diagonal with one positive and three negative elements. FIG. 5. sColor online dOrder- ing instabilities, described by realoff-diagonal SU s4dgenerators Q 4–9.sadQ4breaks neither trans- lational nor rotational symmetry.sbdshows the ordering corre- sponding to Q 6. The ordering shown in scdis described in real space and in reciprocal space by the same matrix Q˜7=K˜7and breaks translational symmetry.The corresponding BZ is shown inFig. 3.INDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-10This leads to a change of the band energy of the band at the Gpoint relative to those at the Mpoints fFig. 5 sadg. This results in an orbital order, a pattern as shown in Fig. 5 sad, because the number of holes associated with the hole pocketaround the Gpoint is different from that of the other pockets. The net charge on site vanishes, but the charge distributionhas the quadrupolar form, which results from rsrd~1 4f3ucyz+czx+cxyu2−ucyz−czx+cxyu2 −ucyz+czx−cxyu2−ucyz−czx−cxyu2g =cyzczx+czxcxy+cxycyz+c.c. s39d The corresponding tensor operator belongs to the representa- tionG1of the subgroup D3of the cubic group with the three- fold rotation axis parallel to f111g, i.e., along the caxis per- pendicular to the layer. This quadrupolar ﬁeld would bedriven by the symmetry reduction discussed above, throughtrigonal distortion and direct ddhopping among the t 2gor- bitals. Spin:While the corresponding order parameter kSz4l breaks time-reversal symmetry, space-group symmetry is conserved. This order is spatially uniform analogous to aferromagnet without, however, having a net magnetic mo-ment. This is because the magnetic moment associated withthe Fermi surface pocket at the Gpoint is opposite and three times larger than the moment at the three Mpockets. While the net dipole moment vanishes on every site, this conﬁgu-ration has a ﬁnite quadrupolar spin density corresponding tothe on-site spin-density distribution of the same form as thecharge distribution in Eq. s39d, which also belongs to the G 1 representation of D3. It is also important to note that no third-order terms are allowed due to broken time-reversalsymmetry, such that the transition to this order would becontinuous. r=5,6 Charge: The order parameters kn 5l=h5and kn6l=h6 transform according to the irreducible representation G3of the cubic point group. The Landau expansion of the freeenergy is given by DF=a 2sh52+h62d+b 3h6s3h52−h62d+g 4sh52+h62d2,s40d whose global stability requires g.0. The third-order term, allowed here, induces a ﬁrst-order transition and simulta-neously introduce an anisotropy which is not present in thesecond- and fourth-order terms. We can write s h5,h6d =hscosw,sinwdand obtain DF=a 2h2+b 3h3sin3 w+g 4h4. s41d Depending on the sign of b, the stable angles will be w =sgn sbdp/2+2 pn/3. This yields three degenerate states of uniform orbital order whose charge distribution has the qua- drupolar formrsrd~eiwhsczxcxy+cxyczxd+vscyzczx+czxcyzd +v2scxycyz+cyzcxydj+c.c. s42d with a tensor operator belonging to G3ofD3. Each state is connected with the choice of one Mpocket which has a different ﬁlling compared to the other two fFig. 5 sbdg. The main axis of each state points locally along one of the three cubic body diagonals, f1¯,1,1g,f1,1¯,1g,f1,1,1¯g, and the sign of the local orbital wave function is staggered along the corresponding direction on the triangular lattice, f2¯,1,1g, f1,2¯,1g,f1,1,2¯g. In this way, the rotational symmetry is broken but the translational symmetry is conserved. The ma- tricesQ5andQ6commute with Q4such that the external symmetry reduction has only a small effect on this type oforder. Spin:The spin densities kS z5landkSz6lalso belong to the two-dimensional representation G3of the cubic point group. Here time-reversal symmetry ensures that the Landau expan-sion only allows even orders of the order parameters h5,h6d=hscosw,sinwdfi.e., b=0 in Eq. s40dg. The con- tinuous degeneracy in wis only lifted by the sixth-order term, given by d1 6sh12+h22d3+d2 6h22s3h12−h22d2=d1 6h6+d2 6h2sin23w. s43d Stability requires d1.maxh0,−d2j. The anisotropy is lifted by the d2term, which gives rise to two possible sets of three- fold-degenerate states. Depending on the sign of d2, we have a minimum of the free energy for w=s1−sgn d2dp/4+pn. The corresponding spin densities have no net dipole on every site, but again a quadrupolar form of the same symmetry asfor the charge, given by Eq. s42d. r=7–9 Charge:The order parameters kn ilfori=7,8,9transform irreducibly under space-group symmetries with the represen- tation G5b. The expansion s38dof the free energy holds also for these order parameters. The third-order term makes thetransition ﬁrst order and favors the symmetric rotationallyinvariant combination of the order parameters, described by the matrix Q ˜7=sQ7+Q8+Q9d/˛3=K˜7shown in Fig. 5 scd. The folding of the BZ and the splitting of the bands is the sames in Fig. 3. The orbital pattern of the nondegeneratequasiparticle band is also shown in Fig. 5 scd. It consists of atomic w0orbitals pointing along all four cubic space diago- nals. Translational but not rotational symmetry is broken. Spin:The discussion for the spin-density operators is analogous to the discussion in the section r=1−3. r=10–12 Charge:The order parameters knilfori=10,11,12 trans- form irreducibly under space-group symmetries with the rep- resentation G4. For the G4representation of Td, there is no third-order invariant. All other terms in Eq. s38dare, how- ever, also invariants for G4. The absence of the third-orderEFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-11FIG. 6. sColor online dTransitions to time-reversal symmetry-breaking states, where the expectation value of the orbital angular momen- tum kLWlon the Co sites is ﬁnite. sadStates where the angular momentum does not lie in the plane. K˜10=Q˜10.sbdStates with angular momentum in the plane. K˜13=−Q˜13.scdThe folding of the BZ and the hybridization of the bands for sadare shown. Dotted lines indicate doubly degenerate bands.INDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-12term leads to continuous transition. The stabilized state for a,0 depends on the sign of g2in Eq. s38d. Forg2.0, a nontrivial minimum with kn11l=kn12l=0 ex- ists, which is described by the Hermitian, imaginary matrix Q10.I flis an eigenvalue of Q10, then − lis also an eigen- value of Q10and the corresponding quasiparticles are con- nected by time-reversal symmetry. Therefore, the nonvanish-ing eigenvalues of Q 10belong to quasiparticle states, which are not invariant under time-reversal symmetry. They aregiven by complex linear combinations of t 2gorbitals. For complex linear combinations of t2gorbitals, the expectation value of the orbital angular momentum operator kLldoes not vanish in general, as can be seen from Table IV. In Fig. 6 sad, the pattern of the angular momentum expectation values kLl for a quasiparticle of Q10is shown. It is invariant under translations along a1and staggered under translations along a2anda3. The expectation values are parallel to f011g. The folding of the BZ and the splitting of the bands are shown inFig. 6 scd. Rotational, translational, and time-reversal symme- try is broken and the state has the magnetic point group 2 I/mI. For g2,0, the symmetric combination Q˜10=sQ10+Q11 +Q12d/˛3=K˜10is stabilized. The angular momentum pattern for a quasiparticle with nonvanishing eigenvalue is shown in Fig. 6 sad. Depending on the site, the expectation value points along f100g,f010g,f001g,o rf1¯1¯1¯gand the magnitudes are such that the pattern is rotationally invariant and the expec- tation value of the total angular momentum perpendicular tothe plane vanishes. The folding of the BZ and the splitting ofthe pockets is shown in Fig. 6 scd. This state has the magnetic point group 3 ¯mI. Note that these states can also be considered as a kind of staggered ﬂux states. The matrices Q10−12com- mute with Q4and therefore the transitions are only little affected by a trigonal distortion. Spin:The spin-density order parameters kSzilfori =10,11,12 also transform under space-group symmetries likeG4and, except for the spin-dependent quasiparticle en- ergy, the discussion is the same as for the charge-densityoperators. Note, however, that these spin-density operatorsdo not change sign under time-reversal symmetry, becauseboth the orbital angular momentum and the spin are re-versed. This, however, does not lead to a third-order term inthe Landau expansion, as there is no third-order invariant fortheG 4representation anyway. r=13–15 Charge:The order parameters knilfori=13,14,15 trans- form irreducibly under space-group symmetries with the rep- resentation G5c. The matrices Q13−15are also imaginary and time-reversal symmetry changes the sign of the order param-eters. The Landau expansion of the free energy is given asabove by Eq. s38dwith b=0. Forg2.0 and a,0, a minimum of the free energy is given by the order parameter kn13l. The angular momentum pattern of the quasiparticles is shown in Fig. 6 sbd. The ex- pectation values lie in the CoO 2plane and are parallel to the a1direction. Their sign is staggered along the a2anda3 directions. The quasiparticles consist of states belonging totheGand theMpocket. The folding of the BZ is given in Fig. 6 scd, but with the single dotted line in the center being a doubly degenerate Mpocket. Rotational, translational, and time-reversal symmetry is broken. For g2,0, the symmetric combination Q˜13=sQ13+Q14 +Q15d/˛3=−K˜13is stabilized. The pattern of the quasiparti- cles corresponding to Q˜13is shown in Fig. 6 sbd. It consists of nonmagnetic sites with a w0orbital perpendicular to the plane and of sites with angular momentum expectation val-ues along a i. Rotational symmetry is not broken in this case. The folding of the BZ and the splitting of the bands is shownin Fig. 6 scd. All angular momentum expectation values for these two states lie in the CoO 2plane. Therefore, it is not possible to interpret these states as staggered ﬂux states. Spin: The spin-density order parameters kSzilfori =13,14,15 are invariant under time-reversal symmetry. Therefore, the third-order term in Eq. s38dis allowed and the transition is a ﬁrst-order transition. VII. POSSIBLE INSTABILITIES A. Coupling constants As can be seen from Table III, the coupling constants for the SDW transitions Lsare rather negative, whereas the charge-coupling constants Lctend to be positive. This is not surprising as only local repulsive interaction is consideredhere, which tends to spread out the charge as much as pos-sible. The coupling constants L rc/swithr=0,...,3 depend on the intraorbital Coulomb repulsion U.A sUis the largest Coulomb integral, the absolute value of these coupling con- stants is biggest. The remaining coupling constants Lrc/sdo not depend on U. ForJ8=0, they are also independent of r. For ﬁnite J8, the degeneracy between the real s4d–s9dand imaginary s10d–s15dSUs4dgenerators is lifted. In order to compare the different coupling constants bet- ter, the relations U=U8+2JHandJH=J8, which hold in a spherically symmetric system, can be assumed to hold ap- FIG. 7. sColor online dThe dimensionless coupling constants L˜ rc/s=9Lc/s/s2Udas functions of a=U8/U. The relations U =U8+2JHandJ8=JHare assumed to hold. The solid sdashed dlines denote the charge sspindcoupling constants.EFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-13proximately.The ratio a=U8/Uis positive and usually larger than 1/2 and smaller than 1. These assumptions allow us to order the dimensionless coupling constants L˜c/s =9Lc/s/s2Udaccording to their strength. In Fig. 7, the di- mensionless coupling constants L˜ rc/sare shown as functions ofa. The most negative coupling constant is the ferromag- netic one with L˜ 0s=−6+3 a. For aclose to 1, the coupling constant for spin-density order L˜ 1s=−s2+adis comparable. Smaller but still clearly negative are also the coupling con- stants for the spin-density angular momentum states L˜ 10s =−s1+ad. The coupling constants L˜ 4c=L˜ 4s=1−3 aare also negative. Finally, the coupling constant for time-reversal symmetry-breaking angular momentum states L˜ 10c=3−5 a and for the charge-density order L˜ 1c=4−5 ais rather positive, but can in principle also be negative if ais close enough to 1. In fact, it is quite remarkable that for a.0.8, all coupling constants sexcept L0cdare negative. For a=1, additional de- generacies among the coupling constants appear, as can beseen in Fig. 7. This indicates the existence of a higher sym- metry at this point. In fact, the local Coulomb interaction H rC of Eq. s15ddepends only on the total charge nr=omsnrmson the siterand is given by Unrsnr−1d/2 for a=1. B. Effect of the trigonal distortion In the mean-ﬁeld description, an instability occurs if the Stoner-type criterion is satisﬁed. At zero temperature in thesystem with full symmetry, this criterion reads in our nota-tion as −L rc/s 4DsEFd=1, s44d whereDsEFdis the density of states per spin and per hole pocket. For rather small pockets, DsEFdis given by ˛3/s2ptd<0.28/tin our tight-binding model, but it in- creases with decreasing EFscf. Fig. 2 d. From Eq. s44d,w e can estimate that the critical Umust be larger than 10 tfor having a ferromagnetic instability. With the introduction ofthe trigonal distortion, as it was discussed in Sec. V, theStoner criteria of Eq. s44dare modiﬁed. For the order parameters described by the matrices K 0,K4, K5−6, andK10−12, which commute with the trigonal distortion K4, the change of the Stoner criterion is only due to the changing of the density of states at the Mand the Gpockets by the trigonal distortion, and the Stoner criterion is onlyslightly modiﬁed as long as all four pockets exist. On the other hand, the instabilities toward states where the order parameters with the matrices K 13−15are ﬁnite would be strongly affected by the trigonal distortion, as thepocket states that hybridize in such a transition are no longerdegenerate. Finally, as mentioned above, the order parameters de- scribed by the matrices K 1−3andK7−9transform with the same representation and are mixed by the trigonal distortion.For strong distortions, the mixing tends to odd-even combi-nations and only the odd combinations K 1−K7,K2−K8,K3 −K9commute with the symmetry-breaking ﬁeld, K4, andconnects the still degenerate states of the Mpockets. If the trigonal distortion is so strong that the pockets states at theMpoints lie below the FS, only a spontaneous ferro- magnetic instability can still occur according to the Stonercriterion. First-order transitions, however, are still possible. The ferromagnetism is the leading instability in the sym- metric model and is least affected by the trigonal distortion.Therefore, in real Na xCoO2systems where a rather strong trigonal distortion is unavoidably present, ferromagnetismwould be most robust and is in fact the only type of all thedescribed, exotic symmetry-breaking states that would havea chance to occur spontaneously. However, even if the coupling constants of the more ex- otic states are not negative enough to produce a spontaneousinstability, their corresponding susceptibilities can be largeenough to give rise to an important response of the electronsin the CoO 2plane to external perturbations. In the next sec- tion, we describe how the Na ions can be viewed as an ex-ternal ﬁeld for the charge degrees of freedom. VIII. Na SUPERSTRUCTURES In NaxCoO2, the Na ions separate the CoO 2planes. There are two different Na positions which are both in prismaticcoordination with the nearest O ions. The Na2 position isalso in prismatic coordination with the nearest Co ions, whilethe Na1 position lies along the caxes between two Co ions below and above. This leads to signiﬁcant Na-Co repulsion,suggesting that the Na1 position is higher in energy. In fact,the Na2 position is the preferred site for Na 0.75CoO2, where the ratio of occupied Na1 sites to occupied Na2 sites is about1:2. 12Deintercalation of Na, however, does not lead to a further depletion of the Na1 sites. On the contrary, the occu-pancy ratio goes to 1 for xgoing to 0.5. Further, there is clear experimental evidence that at x=0.5 the Na ions form a com- mensurable orthorhombic superstructure already at roomtemperature. 8For several other values of x, superstructure formation has also been reported, but x=0.5 shows the stron- gest signals and has the simplest superstructure.9,10In addi- tion, forx=0.5 samples a sharp increase of the resistivity at 50 K and 30 K, respectively, was reported.7,11,21 This experimental situation is rather surprising. Naively, one expects commensurability effects to be strongest at x =1/3 or at x=2/3 on a triangular lattice but not at x=1/2. Therefore, it was concluded that structural and electronic de-grees of freedom are coupled in a subtle manner inNa xCoO2.12 In this section, we show how the different ordering pat- terns can couple to the observed Na superstructure at x =1/2.Before going into the details, we note that due to our starting point of interpenetrating Kagomé lattices, commen-surability effects will be strongest for samples where the Naions can form simple periodic superstructures that double orquadruple the area of the unit cell, since specifying a singleKagomé lattice also quadruples the unit cell. For x=1/2, such simple superstructures exist, as shown in Fig. 8. A so-dium superstructure couples to the charge but not to the spindegrees of freedom in the CoO 2layer. In our model, there are 15 collective charge degrees of freedom. From Fig. 7 it canINDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-14be seen that Lrcis most negative for r=4,...,9. Hence, these modes are the “softest” charge modes generating the stron-gest response to a Na pattern.As shown in Fig. 5, the chargeorder corresponding to r=4,5,6does not enlarge the unit cell and therefore does not optimally couple to the Na pat-terns that can be formed with x=0.5. However, the orbital pattern shown in Fig. 5 scdhas lobes of electron density pointing toward selected Na1 and Na2 positions. For x =1/2,itis possible to occupy all these and only these posi- tions. This leads to the left Na superstructure of Fig. 8. Inother words, this Na superstructure couples in an optimalway to this rotationally symmetric charge pattern. Further,the Landau expansion shows that the rotationally symmetriccombination is favored by the third-order term. Therefore, itis clear that the electronic degrees of freedom would favorthis Na superstructure. This pattern, however, does not maxi-mize the Na-ion distances. It is apparent that the averagedistances between the sodium ions can be increased if everysecond of the one-dimensional sodium chains is shifted byone lattice constant, as shown in Fig. 8. In this way, an ortho-rhombic Na superstructure is obtained, which is the one ob-served in experiments. This orthorhombic pattern does notdrive the rotationally symmetric charge pattern shown in Fig. 5scd, which is described by the matrix K ˜7=K7+K8+K9.I t might, however, drive the orthorhombic charge pattern de-scribed by the matrix K7or rather the orthorhombic charge pattern described by K1−K7, as in the presence of trigonal distortion the K1andK7mix and the odd combination will have the most negative coupling constant. This charge pat-tern is shown in Fig. 9. It consists of lines of d xorbitals alternating with lines of the linear combination dy−dzorbit- als. Note that this charge pattern corresponds to the mixedK 1−K7matrix; the charge is not uniformly distributed on the Co atoms. In this charge pattern, the Na1 sites above thesd y−dzdCo sites will be lower in energy than the Na1 sites above the dxCo sites, and similarly the Na2 positions are separated into nonequivalent rows. In reciprocal space, such a charge ordering leads to a fold- ing of the BZ such that the two Mpockets hybridize. The ordering of the Na ions along the chains leads to a furtherfolding of the BZ and to a hybridization of the bands, as isshown in Fig. 9. The schematic FS in Fig. 9 is drawn toillustrate the hybridization occurring due to the translationalsymmetry breaking. Li et al.performed density-functional calculations in order to determine the band structure ofNa 0.5CoO2in the presence of the orthorhombic superstruc- ture from ﬁrst-principles.52Quite generally, one can assume that this superstructure, which speciﬁes a direction on thetriangular lattice, can lead to quasi-one-dimensional bands inthe reduced BZ. For such one-dimensional bands, nestingfeatures are likely to occur and would lead to a SDW-likeinstability, as was observed at 53 K by Huang et al. 8,35Such a transition could open a gap at least on parts of the FS andin this way lead to the drastic increase of the resistivity ob-served at 53 K. 7At higher temperature, the resistivity is comparable in magnitude to the metallic samples and in-creases only slightly with lowering temperature. This weaklyinsulating behavior could be another effect of Na-ion order-ing. Since the rotational symmetry is broken, domains can beformed. The existence of domain walls would be an obstaclefor transport where thermally activated tunneling processesplay a role. It would be interesting to test this idea by remov-ing the domains and see whether metallic temperature depen-dence of the resistivity would result. A bias on the domainscan be given by in-plane uniaxial distortion. To ﬁnish this section, we will discuss a further mechanism that could lead to a nonmagnetic low-temperature instabilityin Na 0.5CoO2. In Sec. VI, we saw that the third-order term in the Landau expansion, Eq. s38d, favors always a rotationally symmetric charge ordering where all three order parameters h1,h2, and h3have the same magnitude. But as argued above, the Na-ion repulsion leads, nevertheless, to an ortho-rhombic charge ordering, where only one order parameter h1 is ﬁnite. From Eq. s38d, we obtain a Landau expansion for the remaining two order parameters h2and h3containing only second- and fourth-order terms. The second-order termis given by a˜ 2sh22+h32d+b˜h2h3, s45d where FIG. 8. sColor online dTwo different Na superstructures in Na1/2CoO2. The left one does not break rotational symmetry and would drive a charge-ordering as shown in Fig. 5 scd. The right one is in fact realized in Na 1/2CoO2; it is obtained from the right one by shifting the Na chains along the arrows. This shift is due to theCoulomb repulsion of the Na ions. FIG. 9. sColor online dThe charge-ordering pattern correspond- ing to the matrix K1−K7consists of alternating rows of dxand dy−dzorbitals. On the left-hand side, the original BZ, the ortho- rhombic BZ due to the charge ordering, and the experimentallyobserved reduced orthorhombic BZ sdarkdare shown.EFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-15a˜=a+Sg1+g2 2Dh12,b˜=bh1. s46d The condition for a second-order phase transition, which leads to ﬁnite values of h2andh3,i sa˜,ub˜u. As we have a.0 and linear growth of ub˜uand quadratic growth of a˜ −awith h1, the condition is fulﬁlled neither for large nor for small values of h1. But for intermediate values of h1,i tc a n be fulﬁlled. This tendency back towards the original hexago-nal symmetry in this or a similar form could be responsiblefor the appearance of additional Bragg peaks at the interme-diate temperature of 80–100 K in Na 0.5CoO2.8Note, how- ever, that it was speculated that these Bragg peaks only existover a narrow range of temperature. IX. CONCLUSIONS In this paper, the properties of a high-symmetry multior- bital model for the CoO 2layer in combination with local Coulomb interaction are discussed. The tight-binding modelis a zeroth-order approximation to the kinetic energy, as itonly includes the most relevant hopping processes usingCo-O phybridization. Nevertheless, it produces the hole pocket with predominantly a1gcharacter around the Gpoint, in agreement with both LDA calculations and ARPES ex-periments. Furthermore, the three further pockets around the Mpoints, although not seen in ARPES experiments, suggest that additional degrees of freedom that cannot be captured ina single-band picture could be relevant. The existence ofidentical hole pockets in the BZ, however, does not producepronounced nesting features. The local Coulomb repulsion of the t 2gorbitals can be taken into account by an effective interaction of fermionswith four different ﬂavors, associated with the four holepockets or the four interpenetrating Kagomé lattices. Thiseffective interaction has a large discrete symmetry group,which allows us to classify the spin- and charge-density op-erators, and to determine for every mode the correspondingcoupling constant. It turns out that with an effective trigonal distortion that splits the degeneracy between the Gand theMpoints, gen- eral corrections to the quadratic part of the Hamiltonian,such as trigonal distortion or additional hopping terms, canbe taken into account, provided they are small. This effectivetrigonal distortion reduces the symmetry of the Hamiltoniandown to the space-group symmetries of the CoO 2plane, by breaking the gauge symmetries of the effective interaction. Most coupling constants are negative for reasonable as- sumptions on the Coulomb integrals U,U8,J8, andJH, but the ferromagnetic coupling constant is most negative andconstitutes the dominant correlation. The charge- and spin-density-wave instabilities without trigonal distortion are eas-ily described in a mean-ﬁeld picture. In reciprocal space thedegenerate bands split, and if bands belonging to differentpockets hybridize, the BZ is folded. In real space, differenttypes of orderings are possible. The occupancy of the differ-entt 2gorbitals on different sites can be nonuniform, resulting in a charge ordering with nonuniform charge distribution onthe Co sites. Further, certain real or complex linear combi-nations of t 2gorbitals can be preferably occupied on certain sites. In this case, the charge is uniformly distributed on thesites, but depending on the linear combinations of the orbit-als, certain space-group symmetries are broken.The complexlinear combinations of t 2gorbitals have in general a nonva- nishing expectation value of the orbital angular momentum. The tendency to these rather exotic states turns out to be smaller than the ferromagnetic tendency, and this dominanceof the ferromagnetic state is even more enhanced by thetrigonal distortion. This is in good agreement with experi-ments, where ferromagnetic in-plane ﬂuctuations have beenobserved by neutron-scattering measurements inNa 0.75CoO2.24,25There are also several reports of a phase transition in Na 0.75CoO2at 22 K to a static magnetic order, which is probably ferromagnetic in-plane but antiferromag-netic along the caxis. 27,29,30 In Na0.5CoO2, a periodic Na superstructure couples di- rectly to a charge pattern in our model and crystallizes al- ready at room temperature, whereas simple ˛33˛3 super- structures, which would correspond to x=1/3 orx=2/3,do not couple. For general values of x, the disordered Na ions provide a random potential that couples to the charge degrees of free-dom. Due to the incommensurability, this does not lead tolong-range order, but the short-range correlations will also beinﬂuenced by the charge degrees of freedom in the CoO 2 layers. This interaction between the Na correlations and thecharge degrees of freedom could be the origin of the charge-ordering phenomena at room temperature and the observa-tion of inequivalent Co sites in NMR experiments. 14,15 The overall agreement of our model with the experimental situation is good. Ferromagnetic ﬂuctuations are dominant inour model and in experiments. Furthermore, our model isbased on a metallic state and allows for charge ordering andspin-density ordering transitions without changing the metal-lic character of the state. Finally, the clear Na superstructuresthat were found at x=0.5 can be understood quite naturally in this model. On the other hand, there are still many open questions for the cobaltates. Mainly, the origin and the symmetry of thesuperconducting state of Na xCoO2·yH2O are still under de- bate. Unfortunately, the Na content x=0.35 is beyond the validity of the approximations made in the derivation of ourmodel. But also the samples with xø0.5 still have many intriguing properties such as the strongly anisotropic mag-netic susceptibility, which shows the unusual Curie-Weisstemperature dependence.Apossible description of the aniso-tropy of the magnetic susceptibility could be achieved byintroducing a spin-orbit term into the kinetic energy. We hope that our model will be useful for a further un- derstanding of the rich experimental situation of the cobal-tates. ACKNOWLEDGMENTS We thank E. Bascones, B. Batlogg, M. Brühwiler, J. Gavilano, H. R. Ott, T. M. Rice, and K. Wakabayashi forfruitful discussions. This work is ﬁnancially supported by agrant from the Swiss National Fund and the NCCR programMaNEP of the Swiss National Fund.INDERGAND et al. PHYSICAL REVIEW B 71, 214414 s2005 d 214414-16APPENDIX A The equivalence of the two deﬁnitions for the “pocket operators” made in Eq. s9dand in Eq. s8dfollows from bKm†j=1 2o leiBj·alaKm†l=1 2o leiBj·al2 ˛No ReiK·sR+al+amdaRm†l =e−iBj·am1 ˛No lReisK+Bjd·sR+al+amdcR+al+amm† =e−iBj·amcK+Bjm†. sA1d The diagonal form of the tight-binding Hamiltonian in Eq. s11dfollows directly from the relation eK+Bjmm8=e−iBj·sam−am8deKmm8. sA2d APPENDIX B In this appendix, we provide some details concerning the derivation of the effective Hamiltonian in Eq. s19d. It is con- venient to treat each term in Eq. s15dseparately. Let us start with Hund’s coupling,JH 2o ro mÞm8crms†crm8s8†crms8crm8s sB1d =JH 2No kqk8q8r o mÞm8ckms†ck8m8s8†cqms8cq8m8s sB2d =JH 2No KK8Qo ijklr o mÞm8eisBi−Bkd·ameisBl−Bjd·am8 3bKms†ib−K+Qm8s8†lb−K8+Qms8kbK8m8sj. sB3d The sum over the momenta in Eq. sB2dis restricted such that k+k8−q−q8equals a reciprocal-lattice vector. Equation sB3dfollows from Eq. sB2dby using the deﬁnition of the pocket operators in Eq. s8d. The sum over the pocket indices is again restricted such that Bi+Bj+Bk+Blequals a reciprocal-lattice vector, whereas the sum over the momentain the reduced BZ is simpliﬁed to an unrestricted sum overthree momenta. Note that this simpliﬁcation is valid forsmall pockets, because all the processes at the Fermi energyare kept. sSmall pockets means here 4 K F,uB1u; this corre-TABLE V. The character table for the symmetry group Gof the effective Hamiltonian Heff. The ﬁrst line labels the classes and gives the number of elements in each class. The letters of the classes indicate classes of the subgroup S4:e=1,f=sabd,g=sabdscdd,h=sabcd, and i=sabcd d. The characters appearing in our effective Hamiltonian are x1forQ0,x7forQ1−3,x15forQ4−9sreal matrices d, and x16forQ10−15 simaginary matrices d.x11is the natural representation of Gdeﬁned in Sec. IV.The last column gives the reduction of the representations into irreducible representations of the subgroup S4, which consists of the classes e1,f1,g1,h1, andi1. No.e1e2e3e4e5f1f2f3f4f5f6g1g2g3h1h2h3h4i1i2Reduction toS4 146411 2 1 2 2 4 2 4 1 2 1 2 1 2 2 4 1 2 3 2 3 2 3 2 3 2 4 8 4 8 x111111 111111111111111 G1 x211111 1¯1¯1¯1¯1¯1¯1111111 1¯1¯ G2 x311¯11¯11 1¯1¯11 1¯11¯11 1¯1¯11 1¯ G1 x411¯11¯11¯11 1¯1¯11 1¯11 1¯1¯11¯1 G2 x522222 000000222 1¯1¯1¯1¯00 G3 x622¯22¯20000002 2¯21¯11 1¯00 G3 x733333 111111 1¯1¯1¯0000 1¯1¯ G5 x833333 1¯1¯1¯1¯1¯1¯1¯1¯1¯000011 G4 x933¯33¯31 1¯1¯11 1¯1¯11¯0000 1¯1 G5 x1033¯33¯31¯11 1¯1¯11¯11¯00001 1¯ G4 x11420 2¯4¯2200 2¯2¯0001 1¯11¯00 G1%G5 x12420 2¯4¯2¯2¯00220001 1¯11¯00 G2%G4 x1342¯02 4¯22¯00 2¯200011 1¯1¯00 G1%G5 x1442¯02 4¯2¯2002 2¯00011 1¯1¯00 G2%G4 x1560 2¯06200 2¯2020 2¯000000 G1%G3%G5 x1660 2¯0602 2¯002 2¯02000000 G4%G5 x1760 2¯060 2¯200 2¯2¯02000000 G4%G5 x1860 2¯06 2¯002 2¯020 2¯000000 G1%G3%G5 x19840 4¯8¯000000000 1¯11¯100 G3%G4%G5 x2084¯04 8¯000000000 1¯1¯1100 G3%G4%G5EFFECTIVE INTERACTION BETWEEN THE … PHYSICAL REVIEW B 71, 214414 s2005 d 214414-17sponds to a doping with x.0.55. dThe next step is to go from orbital operators to the band operators. Restricting our-selves to the top band and taking into account Eq. s16d,w e can simply substitute b Kms†j!s1/˛3dbKs†j. Now we can sum over the orbital indices in Eq. sB3d, and taking into account that the sum over the pocket indices is restricted, we obtainthe sum o mÞm8eisBi−Bkd·sam−am8d=2s4dik−1ds B4d and for the Hund’s coupling term JH 9No KK8Qo ijklr bKs†ib−K+Qs8†lb−K8+Qs8kbK8sjs4dik−1d.sB5d The restriction of the sum can be dropped if we replace s4dik−1dwith s2dijkl−eijkl2−dildjk−dijdkl+3dikdjld. The terms proportional to JHin the interaction of Eq. s19dare now obtained by dividing Eq. sB5dinto two equal parts, rewriting one directly in terms of density-density operators and rewrit-ing the other in terms of density-density and spin-density–spin-density operators using the SU s2drelation 2 daddbg =dagdbd+sag·sbd. Terms which renormalize the chemical potential are dropped. All the other terms in Eq. s15dare treated in the same way. APPENDIX C The symmetry group GofHeffis a ﬁnite subgroup of U s4d that is generated by tthe permutation matrices PPS4and the diagonal orthogonal matrices DPsZ2d4.Gis a semidirectproduct of S4and the normal subgroup sZ2d4. This allows us to ﬁnd the irreducible representations of G, cf. Ref. 60. The elements can be written in a unique way as sP,DdwithP PS4andDPsZ2d4. The product of two elements sP,Dd +sP8,D8dis given by sP+P8,D9d. From this it follows that if sP,Ddis conjugate to sP8,D8d,Pis conjugate to P8, and the class of sP,DdPGcan be labeled by the class of PPS4. The elements of S4can be classiﬁed by writing them as dis- junct cyclic permutations. We label the ﬁve classes as fol-lows:e=1,f=sabd,g=sabdscdd,h=sabcd, andi=sabcd d.I n total, there are 20 classes in G. The character table is shown in Table V. The character corresponding to the natural rep-resentation of Gby orthogonal 4 34 matrices is x11. The representation on the 16-dimensional space Vspanned by Q0−15, which was deﬁned in Sec. IV, acts irreducibly on the subspaces V0,V1−3,V4−9, andV10−15with the characters x0, x7,x15, and x16, respectively. With the help of Schur’s Lemma, it is now easy to show that the interaction Heffin the basis Q0−15is diagonal, i.e., QjirAijklc/sQklr8=drr8Lrc/s, sC1d and that the coupling constant Lrc/sdepends only on the irre- ducible subspace. As discussed in Sec. V, the subgroup sZ2d4describes gauge symmetries that are broken in the real system, whereas the subgroup S4describes the space-group symmetries. The subgroup S4consists of the classes e1,f1,g1,h1, andi1. The irreducible representations of Gare in general reducible for the subgroup S4. For example, we have x7=G5,x15=G1 %G3%G5, and x16=G4%G5. 1T. Tanaka, S. Nakamura, and S. Iida, Jpn. J. Appl. Phys., Part 2 33, L581 s1994 d. 2I. Terasaki, Y. Sasago, and K. Uchinokura, Phys. Rev. B 56, R12 685 s1997 d. 3T. Valla, P. D. Johnson, Z. 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PhysRevB.81.035305.pdf	Four-wave mixing and wavelength conversion in quantum dots David Nielsen and Shun Lien Chuang† Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801, USA /H20849Received 4 August 2009; revised manuscript received 30 October 2009; published 7 January 2010 /H20850 We perform a theoretical analysis based on density-matrix equations to determine the nonlinear suscepti- bilities and gain coefﬁcients for a quantum-dot semiconductor optical ampliﬁer. Our results show that for asingle bound-state quantum-dot, carrier relaxation at large current densities is limited by the carrier capturetime from the continuum to the bound state. We then compare our results with experiment and show that thereis a signiﬁcant contribution from carrier heating in the four-wave mixing efﬁciency. Our results and data ﬁtindicate that efﬁcient four-wave mixing on high-speed signals of greater than 160 Gb/s is possible. DOI: 10.1103/PhysRevB.81.035305 PACS number /H20849s/H20850: 78.67.Hc, 78.67.De I. INTRODUCTION The future of high-speed, wavelength-division multi- plexed networks is dependent on the ability to convert opti-cal signals from one frequency to another to prevent wave-length blocking and reduce the number of frequencychannels needed to operate a network. Importantly, it is de-sirable to achieve this entirely in the optical regime to reducethe number of needed components and thus the cost anddevice footprint. The three main mechanisms employed forthis are cross-gain modulation /H20849XGM /H20850, cross-phase modula- tion /H20849XPM /H20850, and four-wave mixing /H20849FWM /H20850. For high-speed uses, both XGM and XPM are limited by the carrier lifetimeas they are dependent on interband carrier recombination andgeneration. FWM however, has three different physicalmechanisms contributing toward its conversion. The ﬁrstmechanism is the beating between the pump and probe,which causes carrier-density pulsation /H20849CDP /H20850allowing wave mixing by producing a temporal grating in the device. LikeXGM and XPM this mechanism relies on interband pro-cesses and is thus limited by the recombination and genera-tion rates of carriers. However, four-wave mixing also has contributions due to spectral-hole burning /H20849SHB /H20850and carrier heating, which are governed by the much faster carrier-carrier and carrier-phonon-scattering rates allowing for the possibility of con-verting higher speed signals. Spectral hole burning occurs asthe strong pump preferentially depletes resonant carrierswhile leaving carriers in other energy states unaffected, cre-ating a spectral hole. To return to quasiequilibrium, carriersrelax down into the depleted states via carrier-carrier scatter-ing. In quantum wells this can be either intersubband or in-trasubband processes and is usually very fast as a result withrelaxation times of 10–45 fs. 1,2However, the carrier local- ization and discrete density of states in quantum dots /H20849QD /H20850 mean that relaxation must occur through either intersubbandor interdot processes. The large difference in the density ofstates between the dots and the wetting layers means that theinterdot processes are much slower with carrier capture timesfrom the wetting layer to the dots usually in the few to tensof picoseconds. 3,4Excited state to ground-state relaxation is much faster due to electron-hole interactions and is typically100–250 fs. 5–7These slower relaxation mechanisms in quan- tum dots allow for deeper spectral holes to form and thus formore efﬁcient wave mixing. While it is true that these slower time constants reduce the overall bandwidth when comparedto quantum wells, they are still fast enough to allow efﬁcientconversion of signals in the 100 GHz to THz range. The last FWM mechanism is carrier heating, in which the temperature of the carriers is raised above that of the latticeand must cool down through carrier-phonon interactions.Carrier heating occurs because stimulated emission from theground state preferentially removes the lowest energy carri-ers while free carriers absorb photons increasing their energystate. Both of these effects result in raising the mean energyof the carrier distribution and thus its temperature while thelattice temperature remains unchanged. The hot carrier dis-tribution must then cool down through carrier-phonon colli-sions. The large carrier density present in quantum wells andbulk can cause carrier heating to be signiﬁcant due to free-carrier absorption. In quantum dots however, the situation ismore complicated. InAs dots grown on GaAs have a largeconduction-band offset. This, combined with the discrete en-ergy spectrum reduces the carrier density at which gain isachieved. This in turn reduces the free-carrier absorption andcarrier-heating effect. Indeed, previous experimental mea-surements have shown that carrier heating is negligible. 6 However, in InAs dots grown on InGaAsP, which have asmall conduction-band offset, experiments have shown sig-niﬁcant carrier heating. 8Previous theoretical work has fo- cused mostly on spectral-hole burning and is extremely de-tailed in form and difﬁcult to follow 9or has relied on a ladder system of rate equations10making it difﬁcult to deter- mine the underlying physics and key parameters. In the fol-lowing sections we will derive a simpliﬁed model for four-wave mixing in quantum dots based on a simple singlebound state that helps elucidate the physical differences be-tween quantum dots and quantum wells. We will then go onto compare our theory with experiment and discuss the im-plications of our model. II. DENSITY-MATRIX THEORY FOR NONLINEAR SUSCEPTIBILITY OF QUANTUM DOTS WITH WETTING LAYERS Following the method of Uskov et al. ,11we used the density-matrix approach to examine four-wave mixing. Tosimplify our model, we have examined quantum dots withPHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 1098-0121/2010/81 /H208493/H20850/035305 /H2084911/H20850 ©2010 The American Physical Society 035305-1only one bound state, taking into account transitions between the bound state and the continuum of the associated wettinglayer. Furthermore, as carrier heating relies primarily oncarrier-lattice dynamics and not carrier dynamics alone, it isnot expected that the results should differ greatly for quan-tum wells and quantum dots. Thus, we ignore carrier heatingin our QD theory and assume quantum-well /H20849QW /H20850-like be- havior for carrier heating when we perform our ﬁnal calcu-lations. A diagram of the theorized carrier dynamics can beseen in Fig. 1. The set of density-matrix equations that de- scribes this system is /H9267˙cw,k=/H20858 i/H9267cd,i/H208491−/H9267cw,k/H20850 /H9270i,k−/H20858 i/H9267cw,k/H208491−/H9267cd,i/H20850 /H9270k,i−/H9267cw,k /H9270s −/H9267cw,k−fcw,k /H92701+/H9011cw,k, /H208491/H20850 where /H9267is the occupation probability of the state. The sub- script cdindicates dot conduction states and the subscript cw indicates wetting-layer conduction states. kindicates the wave vector in the wetting layer and runs over the quantum-well-like states therein, and iruns over every state in the dot ensemble, including each dot twice to account for the spindegeneracy of the states. The ﬁrst sum is the sum of allcarriers escaping from the idot states into the kwetting-layer state at the rates /H9270i,k; the second term is the reverse, the total number of carriers lost from the kwetting-layer state into all possible dot states at the rates /H9270k,i. The third term represents nonradiative recombination, the fourth is spectral-hole burn-ing inside the wetting layer where the occupation probability relaxes back to the Fermi distribution, f cw,k, at a rate /H92701and the ﬁnal /H9011represents carrier injection. The density-matrix equation for the quantum dots is simi- larly /H9267˙cd,i=−/H20858 k/H9267cd,i/H208491−/H9267cw,k/H20850 /H9270i,k+/H20858 k/H9267cw,k/H208491−/H9267cd,i/H20850 /H9270k,i−/H9267cd,i /H9270s −i /H6036/H20849/H9262vc,i/H9267cdvd,i−/H9262cv,i/H9267vdcd,i/H20850E/H20849t/H20850. /H208492/H20850 Here, the last term is the interaction with light, where /H9262isthe transition dipole moment, /H9267cdvdis the coherence term of the density-matrix equations, and E/H20849t/H20850is the electric ﬁeld of the interacting light. Other, higher-order effects such as spon-taneous emission and Auger recombination have been ig-nored in our model. The governing equation for the coherence terms is simply /H9267˙cdvd,i=− /H20849i/H9275i+1 //H92702/H20850/H9267cdvd,i−i /H6036/H9262cdvd,i/H20849/H9267cd,i+/H9267vd,i−1 /H20850E/H20849t/H20850. /H208493/H20850 Here, decoherence at a rate /H92702has been included phenom- enologically to account for interactions with the outside sys-tem. Since the equations for the valence-band states mirrorthose of the conduction band they need not be typed out andcan be determined simply by interchanging the subscripts c and vin Eqs. /H208491/H20850–/H208493/H20850. From these density-matrix equations, the general rate equations governing the carrier density in both the dots andwetting layer can be determined by summing over all statesand dividing by the volume, V. 1 V/H20858 k/H9267cw,k=Nw, /H208494/H20850 1 V/H20858 i/H9267cd,i=Nd, /H208495/H20850 where Nwrepresents the carrier density in the continuum and Ndis the carrier density trapped inside the dots. To integrate over the summations, we assume the time constants are independent of i/H20849all dots release and capture carriers equally /H20850but dependent on kas continuum states closer to the bound state should relax more easily. This al-lows us to determine normalized expressions for the carrier escape time /H9270eand carrier capture time /H9270cas/H9270e Ck=/H9270e,k NVand/H9270c Ck =/H9270c,k DV, respectively. Dis the total number of states in the quantum dots, twice the number of quantum dots due to spindegeneracy. Here the kdependence on the carrier dynamics has been isolated in C k. The other normalization parameters are D, the density of states in the quantum dots which is equal to twice the dot density due to spin degeneracy, and N=1 V/H20858kCk, the effective number of wetting-layer states per volume. Thereare of course an inﬁnite number of states in the wetting layerif all kstates are considered but we expect C kto fall off with larger kvalues such that Nwill be ﬁnite. However, we expect it to fall off slowly enough that it will be nearly equal to 1 forwetting-layer states that have signiﬁcant occupation levels, allowing us to approximate1 V/H20858kCk/H9267cw,k/H110151 V/H20858k/H9267cw,k=Nw.B y inserting these expressions into the summations we ﬁnd 1 V/H20858 k/H20858 i/H9267cd,i/H208491−/H9267cw,k/H20850 /H9270e,k=1 V/H20858 k,iCk/H9267cd,i/H208491−/H9267cw,k/H20850 NV/H9270e, /H208496/H20850 =/H20858 i/H9267cd,i/H20849N−Nw/H20850 NV/H9270e, /H208497/H20850 FIG. 1. Diagram of quantum-dot band-structure and carrier- relaxation processes.DA VID NIELSEN AND SHUN LIEN CHUANG PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-2=Nd/H208731−Nw N/H20874 /H9270e. /H208498/H20850 Using the same approach the reverse process can be calcu- lated 1 V/H20858 k,i/H9267cw,k/H208491−/H9267cd,i/H20850 /H9270c,k=Nw/H208731−Nd D/H20874 /H9270c. /H208499/H20850 Combining these results with our previous results, we ﬁnd the following rate equations: N˙w=Nd/H208731−Nw N/H20874 /H9270e−Nw/H208731−Nd D/H20874 /H9270c−Nw /H9270s+I qV, /H2084910/H20850 N˙d=−Nd/H208731−Nw N/H20874 /H9270e+Nw/H208731−Nd D/H20874 /H9270c−Nd /H9270s+2a/H20849Nd/H20850E/H20849t/H20850. /H2084911/H20850 Here the sum over the coherence terms has been replaced by a/H20849Nd/H20850=−i /H60361 2V/H20858 i/H20849/H9262vc,i/H9267cdvd,i−/H9262cv,i/H9267vdcd,i/H20850E/H20849t/H20850/H2084912/H20850 the material absorption of the system /H20849Eq. I.37 of Ref. 12/H20850. When normalized and written in terms of the occupationprobabilities f=N d/Dandw=Nw/Nthese equations become the same rate equations which have already been extensivelyused and studied 10,13–15validating our starting equations. w˙=D Nf/H208491−w/H20850 /H9270e−w/H208491−f/H20850 /H9270c−w /H9270s+I qVN, /H2084913/H20850 f˙=−f/H208491−w/H20850 /H9270e+N Dw/H208491−f/H20850 /H9270c−f /H9270s+2an/H20849f/H20850E/H20849t/H20850. /H2084914/H20850 Here anis the absorption renormalized for the occupation probability f. Importantly, in most circumstances the number of states in the continuum is very large compared to thenumber of electrons; thus, we can achieve an excellent ap-proximation by taking the limit that N w/H11270Nand ﬁnd that the rate equations become N˙w=Df /H9270e−Nw/H208491−f/H20850 /H9270c−Nw /H9270s+I qV, /H2084915/H20850 f˙=−f /H9270e+1 DNw/H208491−f/H20850 /H9270c−f /H9270s+2a/H11032/H20849f/H20850E/H20849t/H20850. /H2084916/H20850 To calculate the four-wave mixing efﬁciency, we must determine the susceptibilities. To do this we assume an elec-tric ﬁeld of the form E/H20849t/H20850=E 0e−i/H92750t+E1e−i/H20849/H92750+/H9254/H20850t+E2e−i/H20849/H92750−/H9254/H20850t+ c.c., /H2084917/H20850 which is pictured in Fig. 2. Here /H92750is the pump frequency, /H9254 is the pump-probe detuning, E0is the slowly varying ampli-tude of the pump, E1is that of the probe, and E2is the conjugate formed through nonlinear mixing. Together theseelectric ﬁelds will create a polarization density of the similarform P/H20849t/H20850=P 0e−i/H92750t+P1e−i/H20849/H92750+/H9254/H20850t+P2e−i/H20849/H92750−/H9254/H20850t+ c.c. /H2084918/H20850 inside the material. As the polarization density is directly related to the dipole terms P/H20849t/H20850=1 V/H20858 j=i,k/H9262vc,j/H20849/H9267cv,j+/H9267vc,j/H20850/H20849 19/H20850 we expect the dipole terms to also follow the same form /H9267cv,j=/H9268j,0e−i/H92750t+/H9268j,1e−i/H20849/H92750+/H9254/H20850t+/H9268j,2e−i/H20849/H92750−/H9254/H20850t, /H2084920/H20850 where jincludes both the kcontinuum states and the discrete istates. As the pump light ﬁeld is assumed to be on reso- nance with the quantum dots however and not with the con-tinuum, the contributions from continuum’s kstates can be ignored. Due to beating between the pump and probe, we expect both the state occupation probabilities and carrier density tobeat in time as /H9267c,j=/H9267¯c,j+/H9267˜c,je−i/H9254t+/H9267˜c,j/H11569ei/H9254t, /H2084921/H20850 Nj=Nj+N˜je−i/H9254t+N˜ j/H11569ei/H9254t, /H2084922/H20850 where jcan be replaced by both dandw. As these are inco- herent processes, contributions from the continuum and dotsmust both be considered, unlike the dipole terms where con-tinuum contributions can be ignored. Taking these assump-tions and putting them into the density-matrix equations forthe quantum-dot states, we can determine the polarizations toﬁrst order in E 0 P0=1 V/H20858 j=i,k/H20841/H9262j/H208412 /H6036/H9273ˆj/H20849/H92750/H20850/H20849/H9267¯c,j+/H9267¯v,j−1 /H20850E0, /H2084923/H20850 FIG. 2. Diagram of the assumed electric ﬁeld input with a pump, probe, and conjugate.FOUR-WA VE MIXING AND WA VELENGTH CONVERSION IN … PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-3P1=1 V/H20858 j=i,k/H20841/H9262j/H208412 /H6036/H9273ˆj/H20849/H92751/H20850/H11003/H20851/H20849/H9267¯c,j+/H9267¯v,j−1 /H20850E1+/H20849/H9267˜c,j+/H9267˜v,j/H20850E0/H20852, /H2084924/H20850 P2=1 V/H20858 j=i,k/H20841/H9262j/H208412 /H6036/H9273ˆj/H20849/H92752/H20850/H11003/H20851/H20849/H9267¯c,j+/H9267¯v,j−1 /H20850E2+/H20849/H9267˜c,j/H11569+/H9267˜v,j/H11569/H20850E0/H20852, /H2084925/H20850 where /H9273ˆj/H20849/H9275/H20850=1 /H9275−/H9275j+i//H92702/H2084926/H20850 is the Lorentzian lineshape determined by the decoherence time and is responsible for homogeneous broadening. Tosolve these and ﬁnd the susceptibilities, we must determine /H20849 /H9267¯cd,i+/H9267¯vd,i−1 /H20850and /H20849/H9267˜cd,i+/H9267˜vd,i/H20850which can be done by per- forming a steady-state and small-signal analysis of the den-sity matrix and rate equations. For the steady-state solution, we ﬁnd /H20849 /H9267¯cd,i+/H9267¯vd,i−1 /H20850=/H208732 D/H9270d /H9270cN¯w−1/H20874−2i/H20841/H9262i/H208412/H9270d /H60362/H20849/H9267¯cd,i+/H9267¯vd,i−1 /H20850 /H11003/H20841E0/H208412/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852, /H2084927/H20850 where /H9270d=/H208731 /H9270e+1 DN¯w /H9270c/H20874−1 /H2084928/H20850 andN¯wis the steady-state solution for Nwfrom Eqs. /H2084910/H20850and /H2084911/H20850determined by setting N˙w=0. An examination of these equations will show that the steady-state value will be ulti-mately determined by the injected current and the carrierlifetime including contributions from both nonradiative re- combination and stimulated emission. Thus, N ¯wis an exter- nal parameter that is controlled via the applied current andpump power. It is important to point out that in Eq. /H2084927/H20850we have assumed that the hole dynamics mirror the electron dy-namics in the system. By comparing our result in Eq. /H2084927/H20850with the results of the same calculations done for bulk, 11it is clear that /H9270dis the equivalent of a spectral-hole burning time constant for quan-tum dots. Due to charge localization, electrons trapped inquantum dots have no direct interaction with each other andthus cannot redistribute their energy via carrier-carrier inter-actions to return to thermal equilibrium. Instead, the energyexchange must occur through the continuum with depleteddots capturing new electrons from the continuum and dotswhich are over populated ejecting electrons to the con-tinuum. /H9270drepresents the rate at which the quantum-dot en- semble will relax to thermal equilibrium via these captureand escape dynamics. At low wetting-layer carrier densities,the relaxation is limited by how quickly electrons can escapefrom the overly populated dots; however, as the carrier den-sity in the wetting layer increases, it is the rate of carriercapture that limits the relaxation rate. The above allows us to ﬁnd a steady-state expression for the occupation probabilitiesas /H20849 /H9267¯cd,i+/H9267¯vd,i−1 /H20850=/H208732 D/H9270d /H9270cN¯w−1/H20874 1+2i/H20841/H9262i/H208412/H9270d /H60362/H20841E0/H208412/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852. /H2084929/H20850 When the pump is turned off we expect that the dot occupa- tions probabilities should be the same as the occupation probability under thermal equilibrium, f, such that /H20849/H9267¯cd,i +/H9267¯vd,i−1 /H20850=/H20849fcd+fvd−1 /H20850. By taking E0=0 in Eq. /H2084929/H20850we ﬁnd that /H20849fcd+fvd−1 /H20850=/H208732 D/H9270d /H9270cN¯w−1/H20874 /H2084930/H20850 showing that the occupation probability of the dots is com- pletely dependent on the ratio of /H9270d//H9270cand the wetting-layer ﬁlling factor. All dots have the same occupation probabilityunder thermal equilibrium because we previously assumedthat all dots captured electrons at the same rate. Furthermore,by taking the derivative of Eq. /H2084930/H20850it can be shown that /H20873/H11509fc /H11509N¯w+/H11509fv /H11509N¯w/H20874=2 D/H9270d /H9270c/H208731−/H9270d /H9270cN¯w D/H20874. /H2084931/H20850 Similar to the steady-state analysis, we perform a small- signal analysis as well and ﬁnd that to ﬁrst order in E0 /H20849/H9267˜cd,i+/H9267˜vd,i/H20850=1 1−i/H9254/H9270d/H20877N˜w/H20875/H20849/H9267¯cd,i+/H9267¯vd,i−1 /H20850/H20873−1 D/H9270d /H9270c/H20874 +/H208731 D/H9270d /H9270c/H20874/H20876−2i/H9270d/H20841/H9262i/H208412 /H60362/H20849/H9267¯cd,i+/H9267¯vd,i−1 /H20850 /H11003/H20853/H20851/H9273ˆi/H20849/H92751/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852E0/H11569E1+/H20851/H9273ˆi/H20849/H92750/H20850 −/H9273ˆi/H11569/H20849/H92752/H20850/H20852E0E2/H11569/H20854/H20878. /H2084932/H20850 This result leaves us with the need to determine N˜win order to ﬁnalize our solution. For this we return to the rate equa-tions, Eqs. /H2084913/H20850and /H2084914/H20850, and perform a small signal analysis to ﬁnd that N˜w=−X/H20849L1+L2/H20850 WY−XZ, /H2084933/H20850 where L1=i1 DV/H20858 i/H20841/H9262i/H208412 /H60362/H20849/H9267¯cd,i+/H9267¯vd,i−1 /H20850/H11003/H20853/H20851/H9273ˆi/H20849/H92751/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852E0/H11569E1 +/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92752/H20850/H20852E0E2/H11569/H20854, /H2084934/H20850 L2=i1 DV/H20858 i/H20841/H9262i/H208412 /H60362/H20849/H9267˜cd,i+/H9267˜vd,i/H20850/H20841E0/H208412/H20851/H9273ˆi/H20849/H92751/H20850−/H9273ˆi/H11569/H20849/H92752/H20850/H20852, /H2084935/H20850DA VID NIELSEN AND SHUN LIEN CHUANG PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-4W=1−f¯ /H9270c+1 /H9270s−i/H9254, /H2084936/H20850 X=D /H9270e+N¯w /H9270c, /H2084937/H20850 Y=1 /H9270e+1 DN¯w /H9270c+1 /H9270s−i/H9254, /H2084938/H20850Z=1 D1−f¯ /H9270c. /H2084939/H20850 While this expression may seem complicated, it is funda- mentally an expression which takes into account the beatingof the light ﬁeld in L 1, saturation from the pump in L2and with a bandwidth determined by the carrier lifetime in thequantum dot which can both escape to or be captured fromthe wetting layer or recombine nonradiatively. By taking Eq./H2084932/H20850and substituting it into Eq. /H2084933/H20850we can ﬁnd an expres- sion for the varying wetting-layer carrier density N˜w=−iX1 DV/H20858i/H20841/H9262i/H208412 /H60362/H208732 D/H9270d /H9270cN¯w−1/H20874/H20853/H20851/H9273ˆi/H20849/H92751/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852E0/H11569E1+/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92752/H20850/H20852E0E2/H11569/H20854 WY−XZ+Xi1 DV/H20858i/H20841/H9262i/H208412 /H60362/H20875/H208732 D/H9270d /H9270c/H20874/H208731−/H9270d /H9270cN¯w D/H20874/H20876/H20841E0/H208412/H20851/H9273ˆi/H20849/H92751/H20850−/H9273ˆi/H11569/H20849/H92752/H20850/H20852, /H2084940/H20850 where again we have solved to ﬁrst order by assuming that /H20849/H9267¯cd,i+/H9267¯vd,i−1 /H20850=/H208732 D/H9270d /H9270cN¯w−1/H20874 /H2084941/H20850 and /H20849/H9267˜cd,i+/H9267˜vd,i/H20850=2 D/H9270d /H9270c/H208731−/H9270d /H9270cN¯w D/H20874N˜w. /H2084942/H20850 Taking these expressions and combining them with our earlier expressions for the polarization densities we ﬁnd thepump polarization density and linear susceptibility, /H9273/H20849l/H20850,t ob e P0=1 V/H20858 i/H20841/H9262i/H208412 /H6036/H9273ˆi/H20849/H92750/H20850/H208732 D/H9270d /H9270cN¯w−1/H20874 1+2i/H20841/H9262i/H208412/H9270d /H60362/H20841E0/H208412/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852, /H2084943/H20850 /H9273/H20849l/H20850/H20849/H9275/H20850=1 /H928001 V/H20858 i/H20841/H9262i/H208412 /H6036/H9273ˆi/H20849/H9275/H20850 /H11003/H208732 D/H9270d /H9270cN¯w−1/H20874 1+2i/H20841/H9262i/H208412/H9270d /H60362/H20841E0/H208412/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852. /H2084944/H20850 Similarly, we solve for the probe polarization densityP1=/H92800/H9273/H20849l/H20850/H20849/H92751/H20850E1+1 V/H20858 i/H20841/H9262i/H208412 /H6036/H9273ˆi/H20849/H92751/H20850/H208731 1−i/H9254/H9270d/H20874 /H11003/H208752 D/H9270d /H9270c/H208731−N¯w D/H9270d /H9270c/H20874/H20876N˜wE0+1 V/H20858 i/H20841/H9262i/H208412 /H6036/H9273ˆi/H20849/H92751/H20850 /H11003/H208731 1−i/H9254/H9270d/H20874−2i/H9270d/H20841/H9262i/H208412 /H60362/H208732 D/H9270d /H9270cN¯w−1/H20874 /H11003/H20853/H20851/H9273ˆi/H20849/H92751/H20850−/H9273ˆi/H11569/H20849/H92750/H20850/H20852E0/H11569E1+/H20851/H9273ˆi/H20849/H92750/H20850−/H9273ˆi/H11569/H20849/H92752/H20850/H20852E0E2/H11569/H20854E0. /H2084945/H20850 For P1the induced polarization density is split into three terms. The ﬁrst is the linear polarization density associatedwith gain or absorption in the optical ampliﬁer. The secondterms represent the nonlinear interaction between the pumpand probe due to carrier-density pulsation, and the third termis the nonlinear interaction due to spectral-hole burning. Thepolarization density P 2is identical to that of P1except with the subscripts 1 and 2 interchanged. We seek a way to sim-plify Eq. /H2084945/H20850so that it can be more easily expressed as P 1=/H92800/H9273/H20849l/H20850/H20849/H92751/H20850E1+/H92800/H9273CDP/H20849/H92751;/H92750,/H92751/H20850E1 +/H92800/H9273SHB/H20849/H92751;/H92750,/H92751/H20850E1+/H92800/H9273CDP/H20849/H92751;/H92752,/H92750/H20850E02 /H20841E0/H208412E2/H11569 +/H92800/H9273SHB/H20849/H92751;/H92752,/H92750/H20850E02 /H20841E0/H208412E2/H11569, /H2084946/H20850 where the various contributing factors to the susceptibility are separated from each other. These factors include the lin-ear response and the nonlinear responses due to SHB andCDP. Taking this into account, we can determine generalized susceptibilities due to carrier-density pulsation and spectral-hole burning asFOUR-WA VE MIXING AND WA VELENGTH CONVERSION IN … PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-5/H9273CDP/H20849/H92751;/H92752,/H92753/H20850=2/H92800/H20849c/H9257/H208502dg dNw/H9270s/H20841E0/H208412 /H6036/H92750/H92751g/H20849/H92750/H20850/H20849/H9251+i/H20850 /H208511+i/H20849/H92752−/H92753/H20850/H9270d/H20852/H20900D/H9270s X/H20849WY−XZ /H20850+2/H92800c/H9257dg dNw/H9270s/H20841E0/H208412 /H6036/H92750/H20901, /H2084947/H20850 /H9273SHB/H20849/H92751;/H92752,/H92753/H20850=−2i/H9270d /H60363/H20875/H20841E0/H208412 1+i/H20849/H92752−/H92753/H20850/H9270d/H208761 /H928001 V/H20858 i/H20841/H9262i/H208414/H9273ˆi/H20849/H92751/H20850/H208732 D/H9270d /H9270cN¯w−1/H20874/H20851/H9273ˆi/H20849/H92753/H20850−/H9273ˆi/H11569/H20849/H92752/H20850/H20852, /H2084948/H20850 where we have simpliﬁed the expression for /H9273CDPby apply- ing the identities 1 V/H20858 i/H20841/H9262i/H208412 /H6036/H9273ˆi/H20849/H9275/H208502/H9270d D/H9270c/H208731−/H9270d /H9270cN¯w D/H20874=−/H92800c/H9257 /H9275dg dN/H20849/H9251+i/H20850, /H2084949/H20850 i/H9270s1 V/H20858 i/H20841/H9262i/H208412 /H60362/H208732 D/H9270d /H9270cN¯w−1/H20874/H20851/H9273ˆi/H20849/H9275/H20850−/H9273ˆi/H11569/H20849/H9275/H20850/H20852=2c/H9257/H92800/H9270s /H9275/H6036g/H20849w/H20850, /H2084950/H20850 which have been derived by taking the similar identities from Ref. 11and substituting the equivalent values for /H20849f¯c+f¯v −1 /H20850and /H20849/H11509fc /H11509N+/H11509fc /H11509N/H20850in the quantum-dot system identiﬁed in Eqs. /H2084930/H20850and /H2084931/H20850. These identities also introduce important parameters for comparison to experiment, including the line-width enhancement factor, 16/H9251, the refractive index, /H9257, and the material gain, g/H20849/H9275/H20850, which is calculated from Eq. /H2084944/H20850 g/H20849/H9275/H20850=−/H9275 /H9257cIm/H20851/H9273/H20849l/H20850/H20849/H9275/H20850/H20852. /H2084951/H20850 III. MODEL FOR CONVERSION EFFICIENCY The theoretical results developed in Sec. IIdetermined the nonlinear susceptibilities /H9273CDPand/H9273SHBin addition to the linear susceptibility. For our purpose of examining four-wavemixing, we will use these susceptibilities to calculate theconversion efﬁciency. For wavelength conversion, efﬁciency, /H9257eff, is deﬁned as the power out at the new wavelength di- vided by the power in at the original wavelength, /H9257eff =/H20841E2/H20849L/H20850/H208412 /H20841E1/H208490/H20850/H208412. To calculate this efﬁciency, we use the analytical solution developed by Ref. 17to determine the output power at the conjugate wavelength. The analytical solution for theoutput intensity of the light ﬁelds after propagating through adevice of length Lis E 0/H20849L/H20850=eG¯/2/H208491−i/H9251/H20850/H208751+F−/H20849L,/H9254/H20850/H20841E1/H208490/H20850/H208412 Esat2/H20876E0/H208490/H20850, /H2084952/H20850 E1/H20849L/H20850=eG¯/2/H208491−i/H9251/H20850/H208751+F+/H20849L,/H9254/H20850/H20841E0/H208490/H20850/H208412 Esat2/H20876E1/H208490/H20850, /H2084953/H20850E2/H20849L/H20850=eG¯/2/H208491−i/H9251/H20850F−/H20849L,/H9254/H20850E0/H208490/H208502 Esat2E1/H11569/H208490/H20850, /H2084954/H20850 F/H11006/H20849L,/H9254/H20850=−CeG¯−1 2/H209001−i/H9251 1+/H20841E0/H208490/H20850/H208412 Esat2/H11006i/H9254/H9270+/H20858 x/H9260x/H208491−i/H9251x/H20850 1/H11006i/H9254/H9270/H20901. /H2084955/H20850 In these equations, G¯is the steady-state, integrated device gain deﬁned as the steady-state solution to dG dt=G0−G /H9270−/H20849eG−1 /H20850/H20841E/H208490/H20850/H208412 /H9270, /H2084956/H20850 where /H9270is the gain recovery time. Gis the integrated device gain G=/H20885 0L /H9003g/H20849z,t/H20850dz /H2084957/H20850 andG0is the unsaturated, integrated gain. Cis a phenom- enological parameter used to compensate for the nonplane-wave nature of the waveguide modes 17and has been taken to be 0.8. In Eq. /H2084957/H20850g/H20849z,t/H20850is the material gain and is multi- plied by the conﬁnement factor of the waveguide, /H9003to ac- count for the fact that the entire light ﬁeld does not overlapwith active media. Since in four-wave mixing the dominantlight ﬁeld is the pump, we took the gain at the pump wave- length when determining G ¯. In Eq. /H2084955/H20850, the terms in brackets represent the nonlinear interactions with the ﬁrst being CDP and the sum over x representing all other nonlinear interactions, such as spectral-hole burning and carrier heating whose strengths are deter-mined by the normalized nonlinear gain coefﬁcients /H9260x. Combining Eqs. /H2084952/H20850–/H2084954/H20850the FWM efﬁciency becomes easy to derive as /H9257eff=eG¯/H20841F−/H20849L,/H9254/H20850/H208412/H20875E0/H208490/H208502 Esat2/H208762 . /H2084958/H20850DA VID NIELSEN AND SHUN LIEN CHUANG PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-6While originally derived for a simple quantum-well model, the above, Eqs. /H2084952/H20850–/H2084955/H20850, can be adapted to our rig- orous quantum-dot model. To begin this adaptation we ﬁrstdeﬁne the saturation ﬁeld for the QD system as E sat2=/H6036/H92750 2/H92800c/H9257dg dN/H9270s. /H2084959/H20850 Similarly, the CDP term of F/H11006must be rewritten to account for the more complicated dynamics. This is done by compar-ing the above expression with the solution for the quantum-well susceptibilities calculated in Ref. 11and our derived quantum-dot susceptibilities. From this comparison we ﬁnd F /H11006QD/H20849L/H20850=−CeG¯−1 2/H209001−i/H9251 D/H9270s X/H20849WY−XZ /H20850+/H20841E0/H208490/H20850/H208412 Esat2 +/H20858 x/H9260x/H208491−i/H9251x/H20850 1/H11006i/H9254/H9270/H20901. /H2084960/H20850 An examination of Eq. /H2084955/H20850will show that the CDP non- linear gain coefﬁcient is of the same form as the CDP sus-ceptibility except that the numerator is 1− i /H9251so that /H9260CDP =1. From this observation, we can determine the nonlinear gain coefﬁcient for spectral-hole burning by normalizing theSHB susceptibility to the CDP susceptibility. The result ofthis normalization is /H9260SHB /H208491−i/H9251SHB /H20850 =i2/H9270d/H92750 c/H9257/H92800/H9270sdg dN /H11003/H20900/H20858k/H20841/H9262k/H208414 /H60362/H9273ˆk/H20849/H9275/H20850/H208732 D/H9270d /H9270cN¯w−1/H20874/H20851/H9273ˆk/H20849/H92750/H20850−/H9273ˆk/H11569/H20849/H9275/H20850/H20852 /H20858k/H20841/H9262k/H208412 /H6036/H208732 D/H9270d /H9270cN¯w−1/H20874/H20851/H9273ˆk/H20849/H9275/H20850−/H9273ˆk/H11569/H20849/H9275/H20850/H20852/H20901. /H2084961/H20850 Carrier heating was included in our calculation by relying on the same formulation for the nonlinear susceptibility as isfound in quantum wells and bulk. As shallow quantum dotshave the majority of their free carriers in the wetting andbarrier layer this is considered a good approximation of theactual underlying physics. Keeping with the expression for /H9273CHfound in Ref. 11and normalizing as we did to ﬁnd /H9273SHB we ﬁnd that /H9260CH=/H9270ch /H9270s/H11509g//H11509T /H11509g//H11509N/H9004E hc/H208751+/H9268N g/H20849/H9275/H20850/H6036/H92750 /H9004E/H20876. /H2084962/H20850 Here/H9004Eis the energy difference between the chemical po- tential, the energy needed to add one electron to the con-tinuum, and the energy of an electron in a quantum-dotbound state. /H9270CHis the rate at which the electron gas cools back to the lattice temperature. hcis the heat capacity of thefree electrons assuming a two-dimensional /H208492D /H20850electron-gas model hc=/H9266 3kb2T /H60362m/H11569 l, /H2084963/H20850 where m/H11569is as usual the effective mass for the electrons or holes and lis the effective height of the quantum-dot layer. For our calculations, it was considered to be the distancebetween adjacent quantum-dot layers, which for our samplewas 10 nm. The free-carrier-absorption cross section, /H9268, was calculated from the Drude model to be /H9268=q3/H92612 4/H92662/H92800nm/H115692/H9262/H2084964/H20850 but was found to be too small to have an impact on carrier heating due to the low carrier concentration at which gaincan be achieved in quantum dots. Instead the primary carrier-heating mechanism is not free-carrier absorption but insteadthe gain of the device removing the lowest energy carriesfrom the dots while higher-energy electrons are injected into the sample. The ratio /H11509g /H11509T//H11509g /H11509Ncan be found analytically for the quantum-dot system by observing that g/H11008/H20849fc+fv−1 /H20850and that under large bias the majority of carriers actually residein the barrier and wetting layers. Under these conditions thederivatives can be easily taken giving an analytical solutionof /H11509g /H11509T//H11509g /H11509N=−Nw/H9004E kbT2. /H2084965/H20850 This, when combined with the assumption that carrier heat- ing from free-carrier absorption is insigniﬁcant, results in theexpression /H9260CH=3/H9270chN/H9004E2/H60362L /H9266/H9270s/H20849kbT/H208503m/H11569hc. /H2084966/H20850 This allows for an analytical calculation of the nonlinear gain coefﬁcient due to carrier heating in quantum dots. Changesin temperature also have a line-width enhancement factorassociated with them as the varying occupation probabilitieschange both the real and imaginary parts of the susceptibility.In a quantum dot we expect the line-width enhancement fac-tor due to temperature changes, /H9251CH, to be very close to the line-width enhancement factor due to carrier-densitychanges, /H9251, as the raising and lowering of the carrier tem- perature serves only to change the ratio between the dot andwetting-layer occupation probabilities and thus the numberof carriers in the dots. Therefore, these values were set equalto each other. IV. NUMERICAL RESULTS For theoretical calculations to have merit, it is important that they can be easily compared and matched with experi-ment. For this we have performed a simple four-wave mixingexperiment in a semiconductor optical ampliﬁer composed ofseven layers of InAs QDs grown on InGaAsP which waslattice matched to InP. The total device length was 2 mm.FOUR-WA VE MIXING AND WA VELENGTH CONVERSION IN … PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-7Importantly, gain and photoluminescence measurements showed no excited state in these dots allowing for a directcomparison to our derived model. Figure 3shows the gain spectra of the device at various bias currents. As can be seen in the plot, increasing the biascurrent has two effects. First, the peak gain increases, andsecond, the peak wavelength shifts toward shorter wave-lengths. This blueshifting of the peak shows that not all dotsﬁll at the same rate. Rather, lower energy dots ﬁll ﬁrst. Fur-thermore, this blueshifting will result in a large line-widthenhancement factor. Measurements on a similar quantum-dotsample fabricated into a Fabry-Perot laser measured a line-width enhancement factor of 5. For comparison to experi-ment we thus used /H9251=/H9251CH=5. While this value is large for quantum dots, theoretical results have shown that shallowQDs, such as those used, will have larger line-width en-hancement factors due to increased coupling between thebound state and barrier layer. 18While the shifting gain peak at low bias goes against one of our initial assumptions, thatall dots ﬁll at the same rate, at high bias we can see the shiftis greatly diminished. This is because at large bias current thehigh dot occupation probability causes the energy differencebetween the dots to become a minor factor in the carrierdynamics. This results in all dots ﬁlling at nearly the samerate as assumed in our model. To perform four-wave mixing measurements, we sent both a strong pump and a weaker tunable probe into the QDsample. Though the gain peaks at 1480 nm, the limitations ofour tunable lasers required that the pump laser be placedslightly off of the gain peak at 1490 nm so that we couldscan both positive and negative pump-probe detunings. The tunable probe laser was then swept across the pump and theoutput spectrum measured on an optical spectrum analyzer/H20849OSA /H20850. The ampliﬁed spontaneous emission was then sub- tracted and the efﬁciency calculated by comparing the powerof the output conjugate to the input probe. Due to the reso-lution limitations of our OSA, detunings of less than 150GHz could not be measured as the strong pump would washout the weaker conjugate signals. To ﬁt these experimental conditions to theory, we ﬁrst ﬁt the gain spectra of the device using a simple Gaussian ap-proximation for the distribution of dot sizes. To do this we assumed that /H9003g/H20849 /H9275/H20850=/H9003g0e−/H20849/H6036/H9275−/H6036/H92750/H208502/2/H92682−/H9251i. /H2084967/H20850 Here/H92750is the peak-gain wavelength of 1480 nm. /H9251iis the intrinsic loss assumed to be 5 cm−1./H9268and/H9003g0were ﬁtting parameters representing the width of the dot distribution dueto inhomogeneous broadening and the maximum modal gainof the sample, respectively. The best ﬁt can be seen in Fig. 4, where /H9268was found to be 26 meV and /H9003g0was 36.50 cm−1. While the ﬁt shows excellent agreement near the gain peak,the absorption of long-wavelength light is much higher thanexpected from this simple model. Attempts were made tocorrectly match the entire curve by increasing the intrinsicloss but this resulted in unphysically high values. This extraloss is most likely due to a deviation in the inhomogeneousbroadening from a Gaussian proﬁle. As our data was takennear the peak wavelength and our theory is based on anassumption of operating near the peak wavelength as well,this variation from the theoretical model was not consideredsigniﬁcant for the results presented here. Once the gain was ﬁt, the gain of the QD Device, along with /H9260SHB,/H9270dand the dot occupation probability, f, were calculated using Eqs. /H2084928/H20850,/H2084930/H20850,/H2084944/H20850, and /H2084961/H20850. To perform the summation over all states necessary for calculating /H9260SHB, the material gain, and the quasi-Fermi levels in the wettinglayer, we integrated over the density of states, /H9267/H20849/H9255/H20850. This was assumed to have the form /H9267/H20849/H9280/H20850=/H20902D /H208812/H9266/H92682e−/H20849/H9255−Eb/H208502/2/H92682/H9255/H11021Eb+/H9004E m/H11569 /H9266/H60362l/H9255/H11022Eb+/H9004E/H20903. /H2084968/H20850 This includes a single, inhomogeneously broadened bound state in the quantum dots, and a 2D-like continuum of statesin the barrier layer. m /H11569is the effective mass of the electrons1460 1480 1500 1520 1540 1560 158 0-30-20-100102030QD-SOA Device GainGain(dB) Wavelen gth(nm)50 mA 100 mA 150 mA 200 mA 250 mA 300 mA 350 mA 400 mA 450 mA 500 mA 550 mA 600 mA FIG. 3. /H20849Color online /H20850Gain of the QD-SOA for various bias currents.1460 1480 1500 1520 1540 1560 1580-15-10-5051015202530Device Gain(dB) Wavelen gth(nm) FIG. 4. /H20849Color online /H20850Fit of gain data at 600 mA bias current showing good agreement at the experimental wavelengths of 1490nm. Deviation at low wavelength is most likely due to free-carrierabsorption which was not included in the ﬁtting model.DA VID NIELSEN AND SHUN LIEN CHUANG PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-8andEbrepresents the mean bound-state energy in the dots and is equal to /H6036/H92750. Utilizing this density of states, calculations were per- formed at several current densities by recalculating the quasi-Fermi level for each desired current density and then calcu-lating the desired parameters. Other physical parametersnecessary for the calculations had to be determined as well. The differential gain, dg dn, was determined from Fig. 3to be 6.0/H1100310−16cm2. The carrier-capture time was assumed to be 1 ps in agreement with previous experiments4and the escape time was related through the Boltzman factor such that /H9270e =/H9270ce−/H9004E/kTand/H9004Ewas assumed as 0.075 eV , a typical value for quantum dots. The device temperature corresponded toour experimental condition of 288 K. The total number ofstates in the dots D=2/H1100310 17cm−3was determined from the area dot density of 1011cm−2per dot layer with each layer being 10-nm thick. The factor of 2 is, as stated before, fromspin degeneracy. /H20841 /H9262/H20849/H9275/H20850/H208412was calculated by equating the gain model of Ref. 12with that of Ref. 11to ﬁnd that /H20841/H9262/H20849/H9275/H20850/H208412=e2 m02/H92752/H20841eˆ·pcv/H208412. /H2084969/H20850 For bulk, the momentum matrix element is known /H20841eˆ·pcv/H20841bulk2=m0 6Ep. For quantum dots, we expect the result to be the same as a quantum well as self-assembled quantumdots are much wider than they are tall. For TE-polarized light we thus expect that /H20841eˆ·p cv/H20841dot2=3 2/H20841eˆ·pcv/H20841bulk2for the conduction subband to the top heavy-hole subband transition and ﬁndthat /H20841 /H9262/H20849/H9275/H20850/H208412=e2Ep 4m0/H92752, /H2084970/H20850 where Epis the optical matrix parameter and for InAs dots is 22.2 eV ,12andm0is the free electron mass. The results of these calculations can be seen in Fig. 5 where instead of material gain, the integrated, modal gainG 0/H20849/H92750/H20850=/H9003g0Lhas been plotted. These calculations show two expected trends. First, increasing the carrier density causesthe dot occupation probability to increase from 0 to 1 withthe integrated gain increasing proportionally. Second, /H9260SHBis proportional to /H9270dand decreases with increasing carrier den- sity. This is signiﬁcant for two reasons. First, the proportion-ality between /H9260SHBand/H9270dshows those slower carrier relax- ation times allow for more efﬁcient four-wave mixingproviding a trade off between bandwidth and efﬁciency.Higher efﬁciency results in lower bandwidth while largebandwidth reduces efﬁciency. This is also the fundamentalreason that quantum dots should be more efﬁcient for tele-communications applications than quantum wells at speedsin between 10–160 GHz. These speeds are slow enough thatthe 0.1–1 ps relaxation time of quantum dots can easily con-vert them. The faster, 50–10 fs, 1,2relaxation times present in quantum wells result in less efﬁcient conversion but with amuch larger bandwidth. Furthermore, the decrease in /H9260SHBwith increasing bias is not unexpected. /H9260SHBis a measure of the creation rate of conjugate photons and they are created through the simulta-neous absorption of two pump photons and stimulated emis-sion of a probe and conjugate photon. For this to occur there must be unoccupied dots capable of absorbing pump pho-tons. While this at ﬁrst might cause the belief that the con-version is most efﬁcient at low bias where the dot occupationis low, it is important to remember that the gain and absorp-tion of the sample plays a large role as well. Once a conju-gate beam is started, the gain of the sample will amplify itallowing a small conjugate to quickly grow. As the gainreaches a maximum and plateaus after all dots are ﬁlled, thenonlinear gain-coefﬁcient plateaus as well resulting in an op-timal carrier density. This effect can be seen in Fig. 6where the efﬁciency is plotted vs carrier density showing a clear0.01 0.1 10.00.20.40.60.81.0 Carrier Densit y(x1017cm-3)Dot Occupation-8.0-4.00.04.08.0G0(/CID90/CID19)0.1110/CID87d(ps)0.010.11/CID78SHB FIG. 5. /H20849Color online /H20850/H9260SHB,/H9270d,h0/H20849/H92750/H20850, and the dot occupation probability plotted as a function of carrier density. Solid verticalline is the ﬁtting condition. 0.01 0.1 1-40-20020406080Unnorma lized Efficiency(dBu) Carrier Densit y(x1017cm-3) FIG. 6. /H20849Color online /H20850Unnormalized efﬁciency vs carrier den- sity. Plot does not take into account carrier saturation or pumppower so absolute values should not be considered correct.FOUR-WA VE MIXING AND WA VELENGTH CONVERSION IN … PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-9peaking. It is important to point out that for comparison pur- poses gain saturation and pump power have not been consid-ered in this plot. P/P satwas simply taken to be 1 for the calculation of E2but no saturation effects were applied to the gain. In general saturation can play a large role in the idealpump power. 17This shows that for true optimization both pump power and carrier density must be considered. To compare our four-wave mixing data to theory, we took the previous gain ﬁt and calculated the integrated gain overthe 2-mm-long device and compared it to the calculated in-tegrated gain. With no good measurement of the conﬁnementfactor, it was allowed to drift over typical values for aquantum-dot SOA with the best ﬁt resulting in /H9003=2.7 %for an integrated gain of 7.3. While this conﬁnement factor issmall, this is in the range for a typical quantum-dot device.The vertical line in Fig. 5shows the carrier density, which provides the best ﬁt and is in agreement with our previousgain ﬁt. It shows calculated values for /H9270d=0.5 ps, /H9260SHB =0.11 with /H9251SHB=0.013 being found from the phase of /H9260SHB while /H9251SHBwas included in our calculations, the small mag- nitude resulted in it having no real effect on the outcome.The carrier-heating effect included contributions from bothholes and electrons for a total /H9260CH=0.08. This is larger than for typical quantum well and bulk structures due to theslower thermal relaxation measured by Ref. 8. Other important theoretical parameters were assumed in- cluding /H9270s=200 ps, an assumed value typical of semicon- ductor devices under large bias. /H9270CH=2.5 ps in agreement with experimental measurements in similar quantum dots.8 The input pump value was chosen to match experiment at0.16P sat. A comparison between our theoretical model and our ex- perimental measurements can be seen in Fig. 7. The ﬁt shows generally good agreement between theory and experiment,both in the magnitude of the conversion efﬁciency, and in thesplitting between positive and negative detunings. V. DISCUSSION The efﬁciency plot from our theory shows two plateaus. One with a bandwidth of a few GHz due to carrier densitypulsation and another that extends out to around 200 GHz before falling off. By utilizing the detuning range that lies onthe second plateau it is possible to perform high-efﬁciencywavelength conversion at high-speed frequencies greaterthan 160 Gb/s by utilizing the four-wave mixing effect. Cal-culations on typical quantum wells put the efﬁciency muchlower 17along with previous experimental measurements di- rectly comparing quantum dots and quantum wells.19 Importantly, the second plateau is determined more by carrier heating than by spectral-hole burning. This becomesreadily apparent when the individual contributions to four-wave mixing are plotted in Fig. 8. While at ﬁrst one might expect spectral-hole burning to have a large contribution as /H9260SHB/H11022/H9260CH, the large temperature line-width enhancement factor increases the contribution from carrier heating abovethat of spectral-hole burning. This result demonstrates that inshallow dots with a single bound state the primary four-wavemixing mechanism at large detunings and for high-speed sig-nals is carrier heating. This is in contrast to most other theo-ries which focus mainly on spectral-hole burning 9,10in quan- tum dots. This large contribution from carrier heating ispossible due to the very slow thermal relaxation rate thatoccurs in these dots. This slow relaxation is most likely due to the slow means by which carriers in the wetting layer can relax down into thequantum dots, which have been depleted through stimulatedemission. Indeed the measured thermal relaxation time of 2.5ps is similar to the carrier capture time of 1 ps. As a result weexpect deep quantum dots with large energy offsets betweenthe barrier layer and bound state to perform less efﬁciently asthey have a reservoir of excited states which can quicklyrelax down and buffer the slow carrier capture. The draw-back being that these shallow quantum dots, while beingmore efﬁcient, cannot achieve the same symmetric conver-sion that has been reported in deeper quantum dots 20due to their larger line-width enhancement factor caused by cou-10 100 1000 10000-30-20-1001020FWM Efficiency (dB) Detunin g(GHz)Positive Detuning Negative Detuning FIG. 7. /H20849Color online /H20850Experimental data with ﬁt. Experiment is shown as points while matching theory is solid lines.10 100 1000 10000-20020All (-) All (+) CDP CH SHBConversion Efficiency (dBm ) Detunin g(GHz) FIG. 8. /H20849Color online /H20850Theoretical efﬁciency plots with indi- vidual contributions from carrier-density pulsation, carrier heating,and spectral-hole burning superimposed. All /H20849−/H20850indicates negative detuning while All /H20849+/H20850indicates positive detuning.DA VID NIELSEN AND SHUN LIEN CHUANG PHYSICAL REVIEW B 81, 035305 /H208492010 /H20850 035305-10pling to the continuum states. As both spectral-hole burning and carrier heating are seen to be heavily reliant on a slowcarrier-capture time for high efﬁciency, this factor becomesour limiting value in determining the maximum four-wavemixing efﬁciency and bandwidth in shallow quantum dots. VI. CONCLUSION We have developed a theoretical model for four-wave mixing in quantum dots based on density-matrix theory. Us-ing this theory we have calculated the nonlinear gain coefﬁ-cients due to spectral-hole burning and applied an analyticalsolution to ﬁnd the total conversion efﬁciency. Our modelgives excellent quantitative and qualitative agreement withexperiment, and demonstrates that the unique carrier dynam- ics of quantum dots should allow for efﬁcient wavelengthconversion of high-speed signals near 160 Gb/s using four-wave mixing. ACKNOWLEDGMENTS This work at the University of Illinois was supported by the Defense Advanced Research Project Agency /H20849DARPA /H20850 under the University Photonic Center Program /H20849CONSRT /H20850. The authors would also like to acknowledge Donghan Lee ofChungnam National University in Daejeon, Korea for hiscollaboration in providing the quantum-dot device whosegain measurements were used for ﬁtting parameters /H20849Ref. 19/H20850. Also at the Department of Physics; dcnielse@illinois.edu †s-chuang@illinois.edu 1R. A. Kaindl, S. Lutgen, M. Woerner, T. Elsaesser, B. Nottel- mann, V . M. Axt, T. Kuhn, A. Hase, and H. Künzel, Phys. Rev.Lett. 80, 3575 /H208491998 /H20850. 2W. H. Knox, D. S. Chemla, G. Livescu, J. E. Cunningham, and J. E. Henry, Phys. Rev. Lett. 61, 1290 /H208491988 /H20850. 3D. G. Deppe and H. Huang, IEEE J. Quantum Electron. 42, 324 /H208492006 /H20850. 4J. Urayama, T. B. Norris, H. Jiang, J. Singh, and P. Bhattacharya, Appl. Phys. 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PhysRevB.72.245114.pdf	Quasiparticle bands and optical spectra of highly ionic crystals: AlN and NaCl F. Bechstedt, K. Seino, P. H. Hahn, and W. G. Schmidt * Institut für Festkörpertheorie und -optik, Friedrich-Schiller-Universität, Max-Wien-Platz 1, 07743 Jena, Germany /H20849Received 12 August 2005; revised manuscript received 27 October 2005; published 21 December 2005 /H20850 Based on the ab initio density functional theory we study the inﬂuence of many-body effects on the quasiparticle /H20849QP/H20850band structures and optical absorption spectra of highly ionic crystals. Quasiparticle shifts and electron-hole interaction are studied within the GW approximation. In addition to the electronic screeningthe effect of the lattice polarizability is discussed in detail. Substantial effects are observed for QP bands ofAlN and NaCl that have large polaron constants of 1–2. The effect of electronic and lattice polarization on theoptical spectra is discussed in terms of dynamical screening and vertex corrections. The results are criticallydiscussed in the light of experimental data available. We ﬁnd that measured peak positions can be reproducedwithout lattice polarizability in the screening of the electron-hole interaction and a reduced lattice contributionto the QP shifts. DOI: 10.1103/PhysRevB.72.245114 PACS number /H20849s/H20850: 71.20. /H11002b, 71.15.Qe I. INTRODUCTION Single-particle and two-particle electronic excitations are accompanied by the rearrangement of the remaining elec-trons in a solid. This effect is known as screening of excitedelectrons /H20849above the Fermi level /H20850and excited holes /H20849missing electrons below the Fermi level /H20850. The calculation of such electronic excitations has made substantial progress in the last decades, in particular using the framework of the many-body perturbation theory /H20849MPBT /H20850. 1In the case of clusters and molecular structures also the density-functional responsetheory is applied. 2The most common assumption in the MBPT is the GW approximation /H20849GWA /H20850of Hedin3,4which describes the response of the electrons by a dynamicallyscreened Coulomb potential W. In this approximation the self-energy operator /H9018of an excited particle is given as a product of the potential Wand the Green’s function G. The poles of the Gfunction correspond to the energies of the dressed particles, the quasiparticles. Electron-hole pair exci-tations are described by a special two-particle Green’s func-tion, the so-called /H20849irreducible /H20850polarization function P.I t obeys a Bethe-Salpeter equation /H20849BSE /H20850. 5,6Apart from an electron-hole exchange /H20849local-ﬁeld effect /H20850term proportional to the bare Coulomb potential v, its kernel is dominated by the variational derivative /H9254/H9018//H9254Gand hence by the screened potential Win random-phase approximation /H20849RPA /H20850which is already used in GWA and describes the attractive interactionof quasielectrons and quasiholes. 7 The quasiparticle /H20849QP/H20850band structures of semiconductors and insulators are now well described by means of ab initio methods based on the density-functional theory8/H20849DFT /H20850 within the local-density approximation /H20849LDA /H20850for exchange and correlation /H20849XC/H20850.9For DFT-LDA bands with a correct energetical order the QP effects can be included by meansﬁrst-order perturbation theory with respect to the differenceof the XC self-energy and the XC potential already used inthe Kohn-Sham equation of the DFT. Its numericalimplementation 10,11usually yields single-particle excitation energies in good agreement /H20849with an accuracy of about 0.1 eV /H20850with angle-resolved photoemission/inverse photo- emission experiments.12–14Solutions of the BSE in an abinitio framework also appeared in the literature in the past few years. Optical spectra can now be calculated includingexcitonic effects for semiconductors and insulators, 15–17solid surfaces,18,19and even molecules.17,20,21These effects can also be included in nonlinear optical properties.22All these calculations are based on computations of the dielectric ma-trix within the independent-particle approximation or amodel dielectric function for the electronic system. The samecalculational scheme has been also applied to wide-gap in-sulators, such as LiF and MgO, 23,24and wide gap semicon- ductors, e.g., AlN.25These materials possess a remarkable ionic contribution to the total chemical bonding. The bondionicity on an ab initio scale is given by the charge asymme- try coefﬁcient gwith values g=0.794 /H20849AlN /H20850and g=0.958 /H20849NaCl /H20850. 26 Polar materials are characterized by longitudinal-optical /H20849LO/H20850phonons whose excitation induces large macroscopic electric ﬁelds in the crystal.27These ﬁelds strongly couple to the excited electrons and holes and modify their motion.Therefore, the question arises whether or not the lattice po-larizability contributes to the dressing of the quasiparticlesand the screening of the electron-hole attraction. Ionic crys-tals with big dynamical ion charges should show strong lat-tice polaron effects modifying the electronic states near theband edges. 28Such systems have small static dielectric con- stants/H92550and/H9255/H11009and relatively large longitudinal optical pho- non frequencies /H9275LO. Because the static lattice polarizability /H20849/H92550−/H9255/H11009/H20850is of the same order of magnitude as the static elec- tronic dielectric polarizability /H20849/H9255/H11009−1/H20850at high frequencies /H9275/H11271/H9275LO, large polaron constants /H9251p=/H208491//H9255/H11009−1//H92550/H20850 /H11003/H20849/H6036/2maB2/H9275LO/H208501/2/H20849aB-Bohr radius /H20850result,28for instance /H9251p /H110151.2 or /H110152.0 for binary systems such as AlN and NaCl, respectively. They yield non-negligible polaron shifts/H11007 /H9251p/H6036/H9275LOof about 0.1–0.4 eV if perturbation theory can be applied to electron or hole states. However, it is not clear /H20849i/H20850 how the lattice polarization really inﬂuences the quasiparticlebands and /H20849ii/H20850whether or not the lattice polarization plays a role on the time scale of the formation of a Coulomb-correlated electron-hole pair. There are several open ques-tions concerning the theoretical description of excitations inPHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 1098-0121/2005/72 /H2084924/H20850/245114 /H2084912/H20850/$23.00 ©2005 The American Physical Society 245114-1systems with high lattice polarizability. For instance, the peak positions in the optical absorption of wurtzite AlN withrespect to experimental ﬁndings 25are underestimated, and the position of the bound electron-hole-pair peak in the op-tical absorption and the exciton binding in NaCl 29are not clear. In this work, we study the quasiparticle band structures and optical spectra of NaCl and AlN. The calculations arebased on the screening reaction of the strongly inhomoge-neous electron gases. In addition, we show how the latticepolarizability /H11011/H20849/H9255 0−/H9255/H11009/H20850modiﬁes the results for the single- particle excitation energies and the dielectric function in the framework of the GWA. We proceed in three steps: /H20849i/H20850We use the density functional theory in local density approxima-tion to obtain the structurally relaxed ground state conﬁgu-rations of the ionic crystals, wurtzite /H20849w-/H20850and zinc-blende /H20849zb-/H20850AlN and rocksalt /H20849rs-/H20850NaCl, and the Kohn-Sham /H20849KS/H20850 eigenvalues and eigenfunctions that enter the computation of the single- and two-particle Green’s functions. /H20849ii/H20850The elec- tronic quasiparticle spectrum is obtained within the GW ap-proximation to the exchange-correlation self-energy with adielectric tensor modiﬁed by the lattice polarization, and /H20849iii/H20850 the Bethe-Salpeter equation is solved for coupled electron-hole pair excitations, thereby accounting for the screenedelectron-hole attraction and the unscreened electron-hole ex-change. The paper is organized as follows. In Sec. II, webrieﬂy summarize the basic theory formulation. In Sec. III,we present the quasiparticle band structure results with andwithout lattice polarization. In Sec. IV, our results for theoptical absorption and electron-energy loss spectra are given.We discuss where lattice polarization may play a role. Fi-nally, a short summary is given in Sec. V. II. BASIC THEORETICAL FORMULATION A. Ground state Most of the ground-state properties of the crystals under consideration here have been obtained within density-functional theory 8and local density approximation9as implemented in the VASP code.30The Perdew-Zunger interpolation31has been used for the XC energy in LDA. The interaction of the valence electrons with the nuclei is mod-eled by means of pseudopotentials /H20849PPs /H20850in accordance with the projector-augmented wave /H20849PAW /H20850method 32which are rather similar to the ultrasoft pseudopotentials.33For AlN we have used softer and harder PPs. To achieve convergencecutoff energies of 17 or 26 Ry have been checked. In thecase of NaCl this value has been tested to be 18 Ry. Inaddition, we present results for NaCl that have been obtainedwith a massively parallelized multigrid implementation ofthe DFT-LDA. 34In this case ﬁrst-principles normconserving PPs have been generated within the Hamann scheme.35Non- linear core corrections,36which are particularly important for sodium, have also been taken into account. The structural parameters calculated for w-AlN are listed in Table I. They are in reasonable agreement with experi-mental data 37and results of other calculations /H20849see, e.g., Ref. 38/H20850. The underestimated theoretical a-lattice constant is a consequence of the overbinding effect of the LDA for thegiven exchange-correlation energy. This effect with an almost 1% reduction of the theoretical lattice constantwith respect to the experimental one is also observed forzb-AlN with a 0=4.323 Å /H20849compared to a0=4.38 Å from experiment39/H20850. For rs-NaCl we derived a theoretical cubic lattice constant of a0=5.435 Å from the minimization of the total energy. It is again smaller than the experimental latticeconstant of a 0=5.64 Å40but the deviations are larger than that in the AlN case. Nevertheless we calculated the elec-tronic and optical properties at the theoretical lattice con-stants. For NaCl we have repeated the calculations done withthe real-space code 34by using VASP.30However, we did not found signiﬁcant differences. Especially the Kohn-Sham ei-genvalues agreed well. B. Quasiparticle bands In order to account for the excitations aspect we replace the local XC potential VXC/H20849x/H20850in the Kohn-Sham equation of the ground state by a nonlocal and energy-dependent self- energy operator /H9018/H20849x,x/H11032;/H9255/H20850and obtain the quasiparticle equation.10–14For the XC self-energy we apply the GW approximation,3,4 /H9018/H20849x,x/H11032;/H9255/H20850=i/H6036 2/H9266/H20885d/H9275e−i/H92750+G/H20849x,x/H11032;/H9255−/H6036/H9275/H20850W/H20849x,x/H11032;/H9275/H20850. /H208491/H20850 In practical evaluations, the one-particle Green’s function G is described approximately in terms of the results of theDFT-LDA band structure calculation. The new QP bands /H9255 nQP/H20849k/H20850are obtained from the Kohn-Sham eigenvalues /H9255n/H20849k/H20850 shifted by diagonal matrix elements of the difference be- tween the self-energy and the XC potential calculated withKohn-Sham eigenfunctions /H9274nk/H20849x/H20850by10,41 /H9255nQP/H20849k/H20850=/H9255n/H20849k/H20850+1 1+/H9252nk/H20853/H9018nkCOH+/H9018nkSEX+/H9018nkdyn/H20851/H9255n/H20849k/H20850/H20852−VnkXC/H20854, /H208492/H20850 where the self-energy operator /H9018has been divided into two static contributions, the Coulomb hole /H20849COH /H20850part and the screened exchange /H20849SEX /H20850part,4as well as a dynamic /H20849dyn /H20850 contribution. Thereby, /H9252nkis the linear coefﬁcient in the Tay- lor expansion of /H9018dynaround the KS eigenvalue /H9255n/H20849k/H20850. The major bottleneck in the GW calculations is the com- putation of the screened interaction Wand the inverse dielec-TABLE I. Structural parameters of w-AlN. They are the lateral lattice constant a/H20849in Å /H20850, the ratio c/aof the two lattice constants, and the internal-cell parameter u. The values calculated with soft and hard pseudopotentials are compared with results of a previouscalculation /H20849Ref. 38 /H20850and experimental data /H20849Ref. 37. /H20850 Parameter Soft PP Hard PPPrevious /H20849Ref. 38 /H20850Experiment /H20849Ref. 37 /H20850 a 3.07 3.08 3.08 3.11 c/a 1.607 1.604 1.607 1.601 u 0.3815 0.3817 0.3824 0.3821BECHSTEDT et al. PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-2tric function /H9255−1, respectively. An extreme acceleration can be achieved by using a model dielectric function for the re-sponse of the inhomogeneous electron gas in the presence ofexcited electrons and/or holes. Several functional forms havebeen suggested. 42,43For systems with not too large gaps an accuracy of the band energies with respect to the valence-band maximum /H20849VBM /H20850of the order of 0.1 eV has been achieved. 44,45We use the version suggested by Bechstedt et al.41It allows for analytic solutions for the dynamic contri- bution and the COH term. For instance, the static Coulombhole contribution to the self-energy takes the form of a localpotential. For cubic systems it holds /H9018 COH/H20849x,x/H11032/H20850=−qTF/H20849x/H20850 2/H208811−1 /H9255/H11009/H208751+qTF/H20849x/H20850 kF/H20849x/H20850 /H11003/H208813/H9255/H11009 /H9255/H11009−1/H20876−1/2 /H9254/H20849x−x/H11032/H20850, /H208493/H20850 where the Fermi /H20849kF/H20850and Thomas-Fermi /H20849qTF/H20850wave vectors, respectively, are computed at the local electron density n/H20849x/H20850. Local-ﬁeld effects on the screening in the SEX contribution are approximated by using state-averaged electron densities.All occurring matrix elements are performed with Kohn-Sham eigenfunctions independent of the used multigridrepresentation 45or the PAW representation.46 A more extended description of the details of the applica- tion of a model dielectric function and the approximate treat-ment of the local-ﬁeld and dynamical screening effects hasbeen published in Refs. 41, 43, and 46. The most importantnumerical advantage is that the sum over intermediate statesin/H9018 dyncan be analytically carried out. No explicit depen- dence on the number of conduction bands occurs in the com-putation of this self-energy operator. The results for Si,GaAs, AlAs, and ZnSe show agreement with the full GWcalculations to within 0.2 eV for all the states considered.Successful applications were also made to wide-gap semi-conductors such as GaN, 47SiC,48and BN.49Similar to the standard GW treatment of the quasiparticle band structure,also the scheme based on a model dielectric function ne-glects self-consistency effects and vertex corrections. 13,14 C. Pair excitations and optical spectra Excitation energies obtained within the quasiparticle for- malism describe one-particle excitations, such as those in-volved in direct or inverse photoemission experiments. Forthe description of the optical properties, however, one needsto go beyond the single-particle level. We study the diagonalelements of the macroscopic dielectric function /H9255 jj/H20849/H9275/H20850. They are related to the polarization function Pof the electronic system. Using a representation in Kohn-Sham eigenfunctionsone has in the limit of vanishing photon wave vectors 7,50 /H9255jj/H20849/H9275/H20850=1−8/H9266e2/H60362 V/H20858 c,v,k/H20858 c/H11032,v/H11032,k/H11032/H20853Mcvj/H20849k/H20850Mc/H11032v/H11032j/H20849k/H11032/H20850 /H11003P/H20849cvk,c/H11032v/H11032k/H11032;/H9275/H20850+ c.c. and /H9275↔−/H9275/H20854/H208494/H20850 with matrix elements of the velocity operator vMcvj/H20849k/H20850=/H20855ck/H20841vj/H20841vk/H20856 /H9255c/H20849k/H20850−/H9255v/H20849k/H20850/H208495/H20850 andVthe normalization volume. In /H208494/H20850the sums run over pairs of electrons in empty conduction band states /H20841ck/H20856and holes in occupied valence band states /H20841vk/H20856, which are virtu- ally or physically excited by photons of energy /H6036/H9275. The polarization function Pobeys a BSE. However, one has to introduce additional approximations to derive a closedequation for the polarization function P/H20849c vk,c/H11032v/H11032k/H11032;/H9275/H20850that depends only on one frequency. The contribution to the ker- nel of the screened potential with respect to the single-particle Green’s function has to be neglected. 51Moreover, the screening of the Coulomb attraction of electron and holeis assumed to be static. 7Neglecting the coupling of resonant and antiresonant electron-hole pairs as well as the non-particle-conserving contributions to the electron-holeinteraction, 15the polarization function obeys a BSE of the standard form /H20858 c/H11033,v/H11033,k/H11033/H20853H/H20849cvk,c/H11033v/H11033k/H11033/H20850−/H6036/H20849/H9275+i/H9253/H20850/H9254cc/H11033/H9254vv/H11033/H9254kk/H11033/H20854 /H11003P/H20849c/H11033v/H11033k/H11033,c/H11032v/H11032k/H11032;/H9275/H20850 =−/H9254cc/H11032/H9254vv/H11032/H9254kk/H11032/H208496/H20850 with the effective electron-hole pair Hamiltonian H/H20849cvk,c/H11032v/H11032k/H11032/H20850and a small damping /H9253of the pair excita- tions. The Hamiltonian of pairs of excited electrons and holes, more precisely, of quasielectrons and quasiholes, isgiven by 5,6,15,50 H/H20849cvk,c/H11032v/H11032k/H11032/H20850=/H20851/H9255cQP/H20849k/H20850−/H9255vQP/H20849k/H20850/H20852/H9254cc/H11032/H9254vv/H11032/H9254kk/H11032 +W/H20849cvk,c/H11032v/H11032k/H11032/H20850+v¯/H20849cvk,c/H11032v/H11032k/H11032/H20850/H208497/H20850 with the matrix elements W/H20849cvk,c/H11032v/H11032k/H11032/H20850=−/H20885d3x/H20885d3x/H11032/H9274ck/H20849x/H20850/H9274c/H11032k/H11032/H20849x/H20850 /H11003W/H20849x,x/H11032/H20850/H9274vk/H20849x/H11032/H20850/H9274v/H11032k/H11032/H20849x/H11032/H20850/H20849 8/H20850 and v¯/H20849cvk,c/H11032v/H11032k/H11032/H20850=2/H20885d3x/H20885d3x/H11032/H9274ck/H20849x/H20850/H9274vk/H20849x/H20850v¯/H20849x−x/H11032/H20850 /H11003/H9274c/H11032k/H11032/H20849x/H11032/H20850/H9274v/H11032k/H11032/H20849x/H11032/H20850/H20849 9/H20850 of the /H20849statically /H20850screened Coulomb interaction W/H20849x,x/H11032/H20850and a bare Coulomb interaction v¯/H20849x−x/H11032/H20850. Only the short-range part of the latter is taken into account in agreement with the physical character of expression /H208499/H20850as electron-hole exchange.6The matrix elements /H208495/H20850,/H208498/H20850, and /H208499/H20850are again computed using the real-space representation45,50or within the PAW picture.32,52Usually the static screening in /H208498/H20850is sufﬁcient for reasonable spectral properties on the two-particle level. 53 The eigenvalues and eigenvectors of the two-particle Hamiltonian /H208497/H20850can be used to calculate directly the frequency-dependent dielectric function /H208494/H20850. Thereby we ap-QUASIPARTICLE BANDS AND OPTICAL SPECTRA OF … PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-3ply a typical broadening /H9253=0.15 or 0.20 eV of the electron- hole pairs. The rank of the Hamiltonian matrix /H208497/H20850is gov- erned by the number of valence /H20849v/H20850and conduction /H20849c/H20850bands and the number of kpoints in the Brillouin zone /H20849BZ/H20850.I nt h e case of the cubic crystals with an fcc Bravais lattice, wetypically take four valence and four conduction bands intoaccount. The BZ is sampled by 4000 random kpoints. They are generated by means of a random number generator. Spe-cial points such as of the Monkhorst-Pack type 54may give rise to a faster convergence in the calculations of the inter-band density of states. However, random kpoints distributed over the entire BZ give rise to a faster convergence afterinclusion of the strong electron-hole interaction. This hasbeen recently demonstrated for silicon. 55The resulting num- ber of 48000 pair states is nearly conserved in the wurtzitecase by doubling the number of bands but reducing the num-ber of points in the BZ. Such an approach requires the di-agonalization of large-rank matrices. In order to bypass thediagonalization of the Hamiltonian /H208497/H20850, we have developed a numerically more efﬁcient initial-state method 19,50to calcu- late the optical polarizability, which is essentially the productof the transition matrix elements /H208495/H20850and the polarization function /H208496/H20850. This quantity obeys an evolution equation driven by the Hamiltonian /H208497/H20850. In the case of w-AlN we double the number of bands but restrict the BZ sampling to1000 random kpoints. For NaCl the bands are more ﬂat comparing with AlN. For that reason we slightly reduce theBZ sampling to 300 random kpoints when using a conven- tional unit cell with 8 atoms. D. Lattice polarizability Usually the screened interaction Win Secs. II B and II C only contains the response of the inhomogeneous electrongas. For strongly ionic systems with large lattice polarizabil-ities the question arises how the motion of the nuclei willeffect the energies and strengths of electronic single-particleand pair excitations. An answer may be given by taking theelectron-phonon interaction into account. There are many pa-pers that have been addressed to this problem /H20849see Ref. 28 and references therein /H20850. On the other hand, the GW approxi- mation suggests a simple way to study the inﬂuence of thelattice motion, in particular the motion of charged ions, bymodifying the screening of the coupled electron-lattice sys-tem. The effect of the lattice polarizability may be describedby a modiﬁed frequency-dependent dielectric matrix of thecrystal /H9255/H20849q+G,q+G /H11032;/H9275/H20850=/H9254GG/H11032+4/H9266/H9251el/H20849q+G,q+G/H11032;/H9275/H20850 +4/H9266/H9251lat/H20849q+G,q+G/H11032;/H9275/H20850/H20849 10/H20850 with /H9251el/H20849q+G,q+G;0/H20850 =1 4/H92661 1 /H9255/H11009−1+/H20841q+G/H208412/qTF2+/H20841q+G/H208414//H208734 3kF2qTF2/H20874. The most important electronic contribution /H9251elto the polar-izability of the crystal is taken in a form described elsewhere.41,46,50In the strongly ionic crystals under consid- eration, in addition, there exists a contribution /H9251latof the polarizable lattice. In the long-wave-length limit /H20849G=G/H11032 =0,q→0/H20850it is given as27,56 /H9251lat/H20849q→0,q→0;/H9275/H20850=1 4/H9266/H20858 /H9251=x,y,zqˆ/H92512/H20851/H9255/H9251/H9251/H20849/H9275/H20850−/H9255/H11009/H9251/H20852, /H9255/H9251/H9251/H20849/H9275/H20850=/H9255/H11009/H9251/H208751+/H9275LO2/H20849/H9251/H20850−/H9275TO2/H20849/H9251/H20850 /H9275TO2/H20849/H9251/H20850−/H20849/H9275+i0+/H208502/H20876 /H2084911/H20850 with qˆ=q//H20841q/H20841, the zone-center optical frequencies /H9275LO/H20849/H9251/H20850 and/H9275TO/H20849/H9251/H20850, and /H92550/H9251=/H9255/H9251/H9251/H208490/H20850or/H9255/H11009/H9251=/H9255/H9251/H9251/H20849/H9275/H11271/H9275LO/H20849/H9251/H20850/H20850.I n the case of the uniaxial wurtzite crystals with four atoms in the unit cell, expression /H2084911/H20850is generalized to a direction- dependent quantity because of the two independent tensorcomponents /H9255 xx/H20849/H9275/H20850=/H9255yy/H20849/H9275/H20850and/H9255zz/H20849/H9275/H20850. In this case the pho- non frequencies have to be replaced by those of E1/H20849A1/H20850sym- metry for the xx=yy/H20849zz/H20850component. The quantities /H9255/H11009/H9251in expression /H2084911/H20850represent the static electronic dielectric constants of the semiconductor or insu-lator under consideration. The total static dielectric constants/H9255 0/H9251of the polar crystal are enlarged by the static lattice po- larizability. In a hexagonal or cubic crystal the dielectric con-stants obey the Lyddane-Sachs-Teller relation /H9255 0/H9251//H9255/H11009/H9251 =/H20851/H9275LO/H20849/H9251/H20850//H9275TO/H20849/H9251/H20850/H20852.2,27,56The tensor character of the dielec- tric constants in the wurtzite case has been neglected in the many-body calculations. In the literature there is a body ofvarying dielectric constants. In the many-body calculationswe use reliable values /H9255 /H11009=4.4 and /H92550=9.14 for both w- and zb-AlN.38,57Forrs-NaCl these values are /H9255/H11009=2.35 and /H92550 =5.45.58In the case of AlN the used values are close to such derived from RPA or density-functional perturbation theorycalculations. 38,57 E. Inclusion of lattice polarizability The replacement of the dielectric matrix by expression /H2084910/H20850has a great advantage. The response of both electron gas and ionic lattice can be described simultaneously for anyelectronic excitation, electron, hole, and electron-hole pair.In the limit of small wave vectors and frequencies, /H9251el/H20849q +G,q+G/H11032;/H9275/H20850=/H208491/4/H9266/H20850/H20849/H9255/H11009−1/H20850/H9254GG/H11032, the imaginary part of the inverse matrix reads as /H20849/H9275/H110220/H20850 Im/H9255−1/H20849q+G,q+G/H11032;/H9275/H20850=/H9266 2/H9275LO2−/H9275TO2 /H9275LO/H9255/H11009/H9254/H20849/H9275LO−/H9275/H20850/H9254GG/H11032 =/H9266 2/H208731 /H9255/H11009−1 /H92550/H20874/H9275LO/H9254/H20849/H9275LO−/H9275/H20850/H9254GG/H11032. /H2084912/H20850 The prefactor in /H2084912/H20850,/H11011/H208491//H9255/H11009−1//H92550/H20850, dominates the Fröhlich coupling constant of the interaction between elec- trons and longitudinal optical phonons /H20849Ref. 28 and refer- ences therein /H20850. The expression /H2084912/H20850immediately yields the self-energy of an electron or hole polaron using the spectralrepresentation of the self-energy /H208491/H20850. 28The discussed smallBECHSTEDT et al. PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-4wave-vector and frequency limit is also relevant for weakly bonded electron-hole pairs, the Wannier-Mott excitons.28Be- cause of their characteristic large radii the electron-hole ex-change contributions /H208499/H20850are negligible. The small binding energies allow that the lattice can completely follow the ex-citon formation, and the attractive Coulomb interaction /H208498/H20850 has to be screened by the static dielectric constant /H9255 0which includes the static lattice polarizability besides the electroniceffect. 59–61 For negligible static lattice polarization /H20849/H92550−/H9255/H11009/H20850→0 also the dynamic lattice polarizability /H2084911/H20850vanishes. Then both the quasiparticle effects in /H208492/H20850as well as the electron-hole attraction /H208498/H20850are dominated by the pure electronic screening. Corrections due to the vibrating lattice may be derived in astraightforward manner. For small lattice polarizabilities oneobtains the result of Hedin and Lundqvist 4for the inﬂuence of the vibrating lattice on the screened potential W.I na formal description the polarizability in /H2084910/H20850can be replaced according to 4 /H9266/H9251=vPby the bare Coulomb potential vand the polarization function Pof the system. Taking electronic and lattice polarization into account according to /H2084910/H20850, one ﬁnds formally for the screened interaction W=v/H208511−v/H20849Pel+Plat/H20850/H20852−1. /H2084913/H20850 A Taylor expansion yields in ﬁrst order to W=Wel+WelPlatWel /H2084914/H20850 with Wel=v/H208511−vPel/H20852−1. Expression /H2084914/H20850has been used by Hedin and Lundqvist4to derive analytic formulas for the inﬂuence of phonons on the electron self-energy. The application of the dielectric tensor /H2084910/H20850to describe the screening in the XC self-energy and in the BSE requiresa careful discussion of the contributing characteristic wavevectors and frequencies and, consequently, then allows usappropriate approximations. The limit of complete neglect oflattice polarizability is the original approximation I. The fulllattice polarization /H2084911/H20850acts only substantially in the long- wavelength limit. This is more or less automatically adjustedby formula /H2084910/H20850. For wave vectors /H20841q+G/H20841/H11022q TFthe elec- tronic screening dominates. For that reason, we will use thestatic limit of /H2084910/H20850as one possible approach labeled by ap- proximation II. Practically only the static lattice polarizabil-ity /H20849/H9255 0−/H9255/H11009/H20850is added to the /H20849electronic /H20850dielectric function /H2084910/H20850in the statically screened quantities /H9018COH/H208492/H20850,/H9018SEX/H208492/H20850, andW/H208498/H20850. The inclusion in the quasiparticle calculations is obvious. The dynamics in the self-energy is still dominated by the electronic response, since the QP shifts in /H208492/H20850with respect to the KS eigenvales are large compared to the pho-non frequencies. We will also introduce an approximation IIIwhere the lattice polarizability is only partially taken in thecomputation of the self-energy. In the explicit calculationswe replace the dielectric constant by the average of the twovalues /H9255 /H11009and/H92550. The dynamics of screening inﬂuences very much the at- tractive electron-hole interaction. With the inclusion of thefrequency dependence of the dielectric matrix in W /H208498/H20850no closed BSE /H208496/H20850can be derived for the two-particle polariza- tion function depending only on one frequency. 7,53,59,60For that reason we simulate effects of dynamical screening bystudying the static screening for the two limiting cases. Thestrength of the screening, in particular in the BSE /H208496/H20850, de- pends sensitively on the strength of the Coulomb effects, inparticular the exciton binding energy itself. In the case of theWannier-Mott excitons the binding energies E Bare usually so small that EB/H11021/H6036/H9275LOholds. Dynamical screening does not play a role. The complete static lattice polarizability /H11011/H20849/H92550 −/H9255/H11009/H20850contributes to the screening of the electron-hole attraction.59–61The screening is mainly characterized by the static dielectric constant /H92550. In the opposite limit, EB /H11022/H6036/H9275LO, the lattice cannot follow the formation of bound electron-hole pairs and the Coulomb attraction is onlyscreened by the redistribution of electrons. The electronicbands of NaCl are rather ﬂat and the dielectric constant /H9255 /H11009is small. One expects that the conditions for the second ap-proximation are clearly fulﬁlled. AlN seems to represent anintermediate case. For that reason we investigate both situa-tions, neglect of lattice polarizability in /H208498/H20850, i.e., use of /H9255 /H11009, and inclusion of lattice polarizability, i.e., use of /H92550or an averaged value in the model dielectric function described inRefs. 41 and 43. III. QUASIPARTICLE BAND STRUCTURES The lattice effect on the quasiparticle excitations is illus- trated in Fig. 1. It shows the QP band structures of w-AlN FIG. 1. Quasiparticle bands /H20849solid lines /H20850with- out /H20849a/H20850and with /H20849b/H20850the effect of the lattice polar- ization in comparison with Kohn-Sham bands/H20849dashed lines /H20850for w-AlN. The valence-band maximum is used as energy zero. Soft pseudopo-tentials have been used.QUASIPARTICLE BANDS AND OPTICAL SPECTRA OF … PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-5using averaged dielectric constants /H9255/H11009=4.4 and /H92550=9.14 with and without lattice polarization in comparison to the KSbands. The positive /H20849negative /H20850QP shifts of the conduction /H20849valence /H20850bands are somewhat reduced in the presence of the lattice polarization. This observation is in agreement with thefact that lattice polaron effects shrink the gaps and transitionenergies. 28The effect of the lattice polarizability scales with the QP effects themselves. Therefore, it mainly occurs in thesp-conduction bands and in the deep N2 svalence bands. However, the reduction of the quasiparticle shifts by the lat-tice polarizability is stronger in the case of the empty states. Explicit numbers are given in Table II for w-AlN and in Table III for zb-AlN. They show an opening of the gaps and transition energies by pure electronic QP shifts of2.1–2.5 eV. The full inclusion of the lattice polarizabilityreduces these QP shifts. The lattice-polaron effect shrinks thequasiparticle fundamental gaps by about 0.6–0.9 eV. Thecomparison with direct gaps /H20849w-AlN /H20850and indirect gaps /H20849zb -AlN /H20850derived from measurements remains somewhat inclu- sive. Moreover, the experimental gap values show a consid- erable dependence on temperature. 66In both the wurtzite and zinc-blende cases the QP gaps with and without lattice po-larizability frame the experimental values. However, one hasto take into consideration that gaps have generally been de-rived from optical measurements 63–65/H20849see also body of data in Ref. 67 /H20850. Even in the case that excitonic effects have been separated, the extracted data may be inﬂuenced by vertexcorrections of the gap due to the electron-phonon interaction.According to Mahan 28the polaron shrinkage of the optical pair energies is governed by the difference /H20849ge−gh/H208502of the coupling constants for electrons /H20849ge/H20850and holes /H20849gh/H20850. In our GW quasiparticle calculations we take only the effect on electrons /H20849/H11011ge2/H20850and holes /H20849/H11011gh2/H20850into account. The vertex corrections /H11011−2geghdo not occur. Consequently, the QPgaps calculated with the full inclusion of the lattice polariz- ability should be smaller than the gaps extracted from opticaldata. Because of the partial cancellation of the electron andhole effects due to /H20849g e−gh/H208502, one should expect QP gaps in between the values of Tables II and III computed with and without lattice polarizability. There is another problem in the calculations. Our pure electronic QP openings are larger than the values obtained ina previous calculation 62by 0.2 eV. One reason for this dis- crepancy may be due to the use of a larger dielectric constant/H9255 /H11009=4.84.62More substantial are, however, the discrepancies in the DFT-LDA gaps of w-AlN. We ﬁnd 4.67 eV instead of 3.9 eV.62This cannot only be explained by the use of differ- ent lattice constants, theoretical one /H20849here /H20850and experimental one in Ref. 62. To solve the discrepancy we repeated theDFT-LDA calculations with harder pseudopotentials butfound only a small reduction of the gap value to 4.53 eV /H20851see also Fig. 2 /H20849b/H20850/H20852. The majority of the previously calculated DFT-LDA gaps /H20849Ref. 38; see also collection in Ref. 68 /H20850are close to our value. Within the generalized gradient approxi-mation /H20849GGA /H20850of the XC potential in the KS equation, a gap of 4.74 eV has been computed. Our test calculations withinthe GGA framework 69gave almost the same DFT band struc- tures /H20851see Fig. 2 /H20849a/H20850/H20852. Only the s-like conduction band minima are shifted towards smaller energies by about 0.1 eV. Theindirect Kohn-Sham gap for zb-AlN is 3.33 eV /H20849Table III /H20850. This value is also close to that of other DFT-LDA calcula-tions of 3.1 eV. 70 The effect of the lattice polarizability on other details of the band structure is much weaker. Interestingly the latticepolaron effect tends to narrow also the band widths of thevalence bands /H20849see Table II /H20850. Unfortunately the currently available measurements of the density of states 70/H20849DOS /H20850do not give values for the valence-bands widths with a sufﬁcientprecision. Nevertheless, they indicate two interesting facts:TABLE II. Gaps Egand valence-band widths Ewofw-AlN with and without lattice polarization. Soft pseudopotentials have been used. All values in eV. Present calc. Previous calc. /H20849Ref. 62 /H20850 Expt. Energy KS QP /H20849without /H20850QP /H20849with /H20850 KS QP /H20849without /H20850/H20849 Ref. 63 /H20850/H20849 Ref. 64 /H20850 Eg/H20849/H9003−/H9003/H20850 4.67 6.80 5.95 3.9 5.8 6.11 6.25 Eg/H20849/H9003−K/H20850 5.00 7.28 6.51 4.8 6.7 Ew/H20849upper /H20850 6.18 6.33 6.17 7.4 8.0 Ew/H20849total /H20850 15.15 17.00 16.57 16.3 18.2 TABLE III. Fundamental gaps Eg/H20849in eV /H20850forzb-AlN with and without lattice polarizability. Soft pseudo- potentials have been used. Present calc. Previous calc. /H20849Ref. 62 /H20850 Expt. Gap KS QP /H20849without /H20850 QP /H20849with /H20850 KS QP /H20849without /H20850/H20849 Ref. 65 /H20850 /H9003→/H9003 4.61 6.72 5.86 4.2 6.0 /H9003→X 3.33 5.45 4.74 3.2 4.9 5.34 /H9003→K 5.20 7.67 6.78 /H9003→L 7.66 10.15 9.25 7.3 9.3BECHSTEDT et al. PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-6/H20849i/H20850The DOS of zb-AlN and w-AlN roughly agree in the valence-band region and /H20849ii/H20850the peak maximum in the DOS of the lowest valence bands is shifted to lower energies withrespect to the DOS derived from the DFT-LDA. This value isin rough agreement with the QP shift /H20849at least for the use of /H9255 0/H20850as demonstrated in Fig. 1 and the increase of the valence- band width in Table II. There is also a small inﬂuence of theQP effects on the crystal-ﬁeld splitting. Its absolute value isreduced from −205 meV /H20849KS/H20850to −190 meV /H20849QP, without /H20850 and −182 meV /H20849QP, with lattice polarizability /H20850. The recom- mended experimental value amounts to −169 meV. 67 Quasiparticle energies for rs-NaCl are listed in Table IV and plotted in Fig. 3. The band structures also show thepositions of energy peaks measured for critical points in theBZ by means of angle- and energy-resolved distributions ofphotoelectrons from the /H20849100 /H20850face of NaCl single crystals. The fundamental QP gaps given in Table IV are 8.63 eV /H20849/H9255 /H11009/H20850 and 7.17 eV /H20849/H92550/H20850. Again they frame the experimental value of 8.5 eV derived from ultraviolet photoemission data72or oth- ers of about 8.5 eV73or 8.0 eV74and seem to suggest only a small contribution of the lattice relaxation /H20849however, see dis- cussion in Sec. IV /H20850. For that reason, results for the average 3.9 of the two dielectric constants /H92550and/H9255/H11009and, hence, only 50% of the lattice polarizability are also given in Table IV.The gap opening by pure electronic screening of 3.51 eV isstrongly reduced by 1.48 eV due to the inclusion of the fulllattice polarizability in the static parts of the self-energy /H208492/H20850.The reason is mainly related to the variation of the SEX term. The variation of the COH terms only contributes withabout 25% to the lattice-polaron gap shrinkage. We mentionthat a QP gap opening due to pure electronic polarizationeffects of about 3.7 eV has been already predicted manyyears ago by Carlsson 75and Harrison.76 The comparison with experimental band positions71in Fig. 3 leads to a similar conclusion as the discussion of thefundamental gaps in Table IV. The quasiparticle bands ob-tained for the pure electronic screening /H20851Fig. 3 /H20849a/H20850/H20852are closer to the points measured with respect to the VBM or theconduction-band minimum /H20849CBM /H20850. This holds particularly for the bands X 5c,X4/H11032c,X3c,X1c,X5/H11032v, and X4/H11032vat the Xpoint. The band state /H900325c/H11032is too high in energy whereas the band /H900312cis too low. However, the agreement with the QP bands including partially lattice polarizability /H20851Fig. 3 /H20849b/H20850/H20852is much worse. Altogether, comparing with the PES data of Stein-mann and Himpsel 71it seems that the QP band structure with pure electronic screening better describes the experimentalﬁndings. The reason may be related to the used initial-statetechnique to determine the valence-band states with respectto the VBM and the ﬁnal-state technique to determine theconduction-band states with respect to the CBM. Anotherreason may be related to the fact that different bands areinvolved in the underlying emission processes. According tothe vertex corrections of the polaron effect discussed alreadyfor AlN here also a cancellation should occur. The cancella-tion effects may be supported by the fact that both the high-est valence band and the lowest conduction band are mainlyderived from chlorine states. 77This fact is somewhat in con- trast to the anion character of the lowest conduction band inthe case of the other alkali halides. 76,78 IV. OPTICAL SPECTRA: EXCITONIC EFFECTS As a ﬁrst example the frequency-dependent dielectric function of zb-AlN is shown in Fig. 4 using a broadening parameter /H9253=0.2 eV and 4000 random kpoints in the BZ. Left panels /H20851Fig. 4 /H20849a/H20850/H20852show the real and imaginary parts of the dielectric function within the independent QP approach. That means, the Coulomb effects /H11011Wand /H11011v¯in the pair Hamiltonian /H208497/H20850have been disregared. QP results are pre- sented for pure electronic screening and screening including FIG. 2. Comparison of the Kohn-Sham bands /H20849a/H20850in LDA /H20849solid lines /H20850and GGA /H20849dashed lines /H20850 or/H20849b/H20850using soft /H20849solid lines /H20850and hard /H20849dashed lines /H20850pseudopotentials for zb-AlN. TABLE IV. Quasiparticle shifts of the VBM and the CBM as well as level positions of rs-NaCl with inclusion of the lattice po- larizability in different approximations. All values in eV. The VBMin KS eigenvalues is used as energy zero. Inclusion of lattice polarizability LevelKS eigenvalue QP shiftQP eigenvalue without CBM 5.14 1.95 7.09 VBM 0 −1.55 −1.55 with partial CBM 5.14 0.92 6.06 VBM 0 −1.56 −1.56 with CBM 5.14 0.73 5.87 VBM 0 −1.30 −1.30QUASIPARTICLE BANDS AND OPTICAL SPECTRA OF … PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-7the lattice polarizability. The right panels /H20851Fig. 4 /H20849b/H20850/H20852give the same spectra with the inclusion of excitonic effects. The cal-culated curves are compared with measured spectra. 79The QP spectra demonstrate that the most important effect of theinclusion of the lattice polarizability is an almost rigid red-shift of about 0.8 eV. The lineshape is less inﬂuenced. Onthe other hand, the Coulomb correlations, screened electron-hole attraction and electron-hole exchange in /H208497/H20850, yield a drastic redistribution of the absorption spectrum /H20849more strictly: imaginary part of the dielectric function /H20850. Spectral density is redistributed from the high-energy region closer tothe region following the absorption onset in agreement withprevious observations for other crystals. 50This tendency is combined with an overall redshift of the absorption due tothe Coulomb effects. However, no bound excitons are ob-served below the absorption onset within our numerical ac-curacy. Their reproduction may require denser k-point meshes. The redshift amounts to about 1.2 eV for the pureelectronic screening and is reduced to 0.6 eV after inclusionof the lattice polarization. As a consequence of the differentaction of the lattice polarizability on the QP shifts and theCoulomb attraction, the optical spectra resulting for two dif-ferent screenings, with and without lattice polarization, ex-hibit wide similarities. The spectrum with the larger screen-ing is only less redshifted with respect to that computed for the pure electronic screening effect. The question, which of the two computed spectra better ﬁts to the measured one, is difﬁcult to answer. The low-energy side of the absorption and the peak structure in thereal part ﬁt better to the neglect of the lattice polarizability.The reason may be the partial cancellation of the polaroneffects due to vertex corrections /H20849see discussion in Sec. III /H20850 and the dynamics of the screening in the electron-hole attrac-tion. The spectral redshifts due to the excitonic effects arewith 0.6 or 1.2 eV much larger than the optical phonon en-ergies. As a consequence the spectrum computed with thepure electronic screening may be closer to the measured one.Conclusions within the Wannier-Mott exciton picture con-cerning the correct inclusion of dynamical screening are alsovery difﬁcult. Using the band and dielectric parameters fromRef. 80 one ﬁnds different exciton binding energies of about0.09 eV /H20849/H9255 0/H20850or 0.29 eV /H20849/H9255/H11009/H20850in dependence of the dielectric constant. These values surround the optical phonon energy of 0.10 eV. This fact and the comparison of the theoretical andexperimental spectra in Fig. 4 /H20849b/H20850indicate that further studies are needed with an improved k-point sampling /H20849on the theo- retical side /H20850and improved sample quality /H20849on the experimen- tal side /H20850. FIG. 3. Quasiparticle bands without /H20849a/H20850and with /H20849b/H20850the effect of the lattice polarization for rs-NaCl. The valence-band maximum is used as energy zero. The ﬁlled circles indicate measured band positions /H20849Ref. 71 /H20850. FIG. 4. Frequency-dependent macroscopic di- electric function of zb-AlN within the independent-quasiparticle approximation /H20849a/H20850and for Coulomb-correlated electron-hole pairs /H20849b/H20850. The QP and excitonic effects have been calcu-lated using pure-electronic screening /H20849solid lines /H20850 or under inclusion of the lattice polarizability/H20849dashed lines /H20850. The theoretical spectra are com- pared with experimental ones /H20849dotted lines /H20850/H20849Ref. 79/H20850.BECHSTEDT et al. PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-8The optical absorption spectra for ordinary and extraordi- nary light polarization are presented in Fig. 5 for w-AlN. This ﬁgure again demonstrates the huge excitonic effects inthe AlN case. The spectra calculated within the framework ofindependent quasiparticles are completely redistributed fromhigher to lower photon energies. Thereby, the lineshapechanges remarkably. The imaginary part of the ordinary di-electric function shows not only qualitative but also quanti-tative agreement with the measured spectrum. The positionsof the two main peaks at 7.8 and 9.0 eV /H20849computed spec- trum /H20850are in excellent agreement with the results of spectro- scopic ellipsometry measurement. 79,81Only the steep onset of the absorption is less pronounced in the calculated spec-trum because of the use of random kpoints. They do not give rise to converged contributions from the lowest opticaltransitions from the /H9003-point region to the joint density of states. The peak intensities are somewhat below the experi-mental values. However, in fact in a previous measurement 25 smaller intensities have been obtained. Because of the repro-duction of the peak positions we have only used pure elec-tronic screening in the spectra computation for Fig. 5. Thepartial inclusion of the lattice polarizability, at least in thedetermination of the QP band structure, would give a redshiftof the theoretical absorption spectra in disagreement with theexperimental ﬁndings. The inﬂuence of the many-body effects according to /H208492/H20850 and /H208497/H20850on the optical absorption of rs-NaCl is illustrated in Fig. 6. It presents spectra which account for both excitonicand quasiparticle effects or only for quasiparticle effects. Forcomparison the imaginary part of the dielectric function forindependent Kohn-Sham particles is also shown. Only elec-tronic screening effects have been taken into account. Thespectra have been computed using the real-space approach. 34 A conventional simple cubic /H20849sc/H20850unit cell with 8 atoms has been used. The smaller sc BZ has been sampled with a re-duced number of 300 random kpoints. Comparing the DFT- LDA and the QP spectra an overall characteristic blueshift ofabout 3.7 eV is visible. However, the lineshape remains al-most conserved. The high-energy peaks are seemingly berelated to optical transitions between band states in Fig. 3. However, the inclusion of the screened electron-hole attrac-tion and the electron-hole exchange gives rise to a completeredistribution of the optical spectrum. A strong bound exci-ton peak occurs at the absorption edge while the spectrumfor higher photon energies is remarkably reduced. Such ten-dencies have been also observed for another alkali halide,LiF. 24However, in the case of NaCl it is difﬁcult to derive an exciton binding energy directly from the comparison of theQP spectra with and without excitonic effects. The boundexciton peak whose broadening is dominated by the value /H9253=0.2 eV sits practically at the fundamental QP gap value. The reason is that not only the lowest interband transitionsc vcontribute to the exciton but also higher optical transitions c/H11032v/H11032which are mixed in by the matrix elements of W/H208498/H20850and v¯/H208499/H20850. Consequently, it makes no sense to ask directly for an FIG. 5. Imaginary parts of the macroscopic dielectric function calculated including quasipar-ticle /H20849dashed lines /H20850and excitonic /H20849solid lines /H20850ef- fects for w-AlN. The lattice polarization has been neglected. Ordinary /H20849a/H20850and extraordinary /H20849b/H20850 light polarizations are studied. An experimentalspectrum /H20849Ref. 79 /H20850is shown as dotted line. A Lorentzian broadening of /H9253=0.2 eV and 1000 random kpoints in the BZ have been used. FIG. 6. Imaginary part of the dielectric function of rs-NaCl including quasiparticle and excitonic effects /H20849solid line /H20850, within the independent quasiparticle approximation /H20849dashed line /H20850, and the in- dependent Kohn-Sham-particle approximation /H20849dotted line /H20850. Only pure electronic screening has been taken into account. A broadeningof /H9253=0.2 eV is used.QUASIPARTICLE BANDS AND OPTICAL SPECTRA OF … PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-9exciton binding energy because the question with respect to which QP transition energy cannot be answered. In the experimental work29it was claimed that an addi- tional excitonic feature shifted by about 0.8 eV towardshigher energies has been observed. This feature has beeninterpreted as the n=2 exciton peak of a hydrogenlike series with an exciton binding energy of almost 1.1 eV, though theWannier-Mott-like exciton picture should be inappropriate.The calculated spectrum in Fig. 6 also shows a shoulder atphoton energies of about 1 eV higher in energy. However,we cannot really derive the main character /H20849either due to Coulomb effects or due to interband transitions /H20850of this fea- ture. The pronounced exciton peak, the second peak roughly3.4 eV above the ﬁrst one, and the absolute values of thespectral strength agree well with the corresponding featuresin experimental spectra, at least in that measured at roomtemperature. The low-temperature spectrum exhibits a stron-ger exciton peak which may be simulated by a smallerbroadening parameter. However, there is a discrepancy between the measured 29 and the calculated /H20849Fig. 6 /H20850exciton peak. In the measured spectra this peak is redshifted by about 1 eV. In order tobridge this discrepancy, we include effects of the lattice po-larizability in the many-body calculations, at least on thequasiparticle level. We have also performed test calculations/H20849not presented here /H20850with the inclusion of the lattice polariz- ability in the electron-hole attraction. This leads to a consid-erable reduction of the excitonic effects and, hence, to achange of the lineshape. In particular, the bound excitonicpeak is dramatically reduced. For that reason, we concludethat the dynamics of exciton formation does not allow a sub-stantial contribution of the lattice polarizability to theelectron-hole screening and, hence, omit this contribution.Within the Wannier-Mott picture and a reduced pair mass of0.44 m 29an exciton binding energy of about 1.1 eV would result. This value is indeed large compared with the opticalphonon energy of about /H6036 /H9275LO=0.03 eV. Therefore, the lat- tice polarizability is only taken into account in the QP shifts.In order to account for the vertex corrections we reduce theeffect of the lattice polarizability. Numerically we replace /H9255 0 by the average 3.9 of /H92550and/H9255/H11009. This procedure leads to a QP gap of about 7.7 eV /H20849cf. Table IV /H20850. The resulting imaginary part of the dielectric function is presented in Fig. 7 and com-pared with the experimental room-temperature spectrum. 29It seems that this procedure may roughly explain the measuredlineshape of the absorption and the measured peak positions.There remain differences. After partial inclusion of the latticepolarizability the high-energy peaks are seemingly too muchredshifted and the broad structures around 9.5 or 10.5 eVbetween the two peaks in the computed spectra are too pro-nounced. Probably the vertex polaron corrections are stron-ger for the high-energy transitions compared with the 50%reduction assumed here. A similar tendency has been observed for the energy loss function −Im /H208511//H9255/H20849 /H9275/H20850/H20852ofrs-NaCl in Fig. 8. The comparison of the calculated spectrum with the function constructed from measured optical data29shows qualitative agreement. All the observed peaks occur in the computed spectrum.However, the majority of the high-energy peaks is too muchredshifted and the intensity of the calculated loss spectrum issomewhat to small.V. SUMMARY Using a combination of an ab initio density functional theory for the ground-state properties and the many-bodyperturbation theory to describe electronic properties we havestudied the band structures and optical spectra of the wide-gap semiconductor AlN and the insulator NaCl. Because oftheir high static ionicity of the chemical bonds, these crystalsalso possess large dynamical ion charges /H11011/H208491//H9255 /H11009−1//H92550/H208501/2 and, hence, a large polarizability of the vibrating lattice. Consequently, in addition to the screening reaction of theinhomogeneous electron gas one also expects a response ofthe ion lattice after excitation of electrons, holes or electronhole pairs. In order to simulate the lattice inﬂuence we have added the dynamic lattice polarizability to the electronic effect. In FIG. 7. Imaginary parts of the macroscopic dielectric function calculated including quasiparticle and excitonic effects without/H20849solid line /H20850and with /H20849dotted line /H20850the effect of the lattice polariza- tion for rs-NaCl. The experimental room-temperature spectrum /H20849Ref. 29 /H20850is shown as dashed line. A Lorentzian broadening of 0.2 eV has been used. FIG. 8. The energy-loss function for rs-NaCl. Solid line: derived from a dispersion analysis of the optical-reﬂectance data /H20849Ref. 29 /H20850; dashed line: computed with reduced lattice polaron effects in theQP calculations.BECHSTEDT et al. PHYSICAL REVIEW B 72, 245114 /H208492005 /H20850 245114-10the limit of small frequencies, /H6036/H9275/H11270gap energy, this ap- proach yields the well-known lattice-polaron effect on singlebands in the self-energy of electrons and holes. In the limit ofsmall wave vectors, q/H11021q TF,/H20849which is fulﬁlled for extended effective-mass states /H20850and small frequencies, /H9275/H11021/H9275TO, the Coulomb attraction of electrons and holes can be replaced bythe static dielectric constant of the ionic crystal including thelattice polarizability. The opposite limit of pure frequency-and wave-vector-dependent electronic screening is also in-cluded. In the case of quasiparticle bands and gaps we found hints for a reduced lattice polaron effect for AlN. We have dis- cussed this ﬁnding in terms of vertex corrections. In theNaCl case the situation is less clear. Comparing the results ofthe quasiparticle approach only with band positions derivedfrom initial- and ﬁnal-state photoemission spectroscopy/H20849PES /H20850, it seems that the lattice effect can be widely ne- glected. However, looking for the correct peak positions inoptical absorption spectra, a contribution of the lattice polar-izability to the quasiparticle shifts seems to be necessary. Tobring the bound exciton peak at the absorption onset inagreement with the experimental position a large polaronshift of about 1 eV is needed. For higher optical transitionsthis shift can be smaller because of the more efﬁcient vertexcorrections due to the electron-phonon interaction. The discussion of the lattice contribution to the screened Coulomb attraction is difﬁcult because it requires studies ofthe dynamical screening, which, however, does not lead to aclosed Bethe-Salpeter equation. For that reason we studiedonly limiting cases. 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PhysRevB.51.2098.pdf	PHYSICAL REVIEW B VOLUME 51,NUMBER 4 15JANUARY 1995-II Full-potential bandcalculations onYTi03withadistorted perovskite structure Hideaki Fujitani FujitsuLaboratories Ltd,I.O-IMorinosato Wakamiya, Atsugi2/8-0I,Japan Setsuro Asano Institute ofPhysics, CollegeofArtsandSciences, University ofTokyo,8-8-IKomaba, Meguro ku,-Tokyo158,Japan (Received 21July1994;revisedmanuscript received 19September 1994) Theenergybandstructure ofYTi03 wasexamined usingthefull-potential linearaugmented- plane-wave method withthelocal-density approximation. Ourcalculations showthatexperimentally observed ferromagnetism andlatticedistortion arenecessary conditions forYTi03tohaveaband gapattheFermilevel,although thecubicperovskite structure ofYTi03 showsnosignsofitsbeing aninsulator. I.INTRODUCTION Theoriginofthebandgapintransition-metal ox- idesisamuchdebated topic.IntheMott-Hubbard theory, whenthemagnitude oftheon-siteCoulomb re- pulsion energyUexceedstheone-electron bandwidth W (UjW)1),thesystemisaninsulator.'2Thebandthe- oryforthiswasthoughttopredictthewrong,metal- licgroundstates.Theerstexamples ofMottinsulators weretransition-metal monoxides NiOandMnO,whose electronic structures werestudied byphotoelectron spec- troscopy andagreedwiththeligand6eldtheory and clustercalculations withconfiguration interaction. This wasregarded asevidence thatdelectrons werewelllocal- izedneartransition-metal atoms,andthatthesemateri- alshadsuchstrongelectron correlations thattheband theorydidnotapply. However, usingself-consistent bandcalculations withthelocal-density approximation (LDA), Terakura etal.obtained bandgapsofthesean- tiferromagnetic monoxides, although theirvalueswere smallerthantheexperimental ones.LDAcalculations alsoreproduced theobserved latticeparameters very well.Therefore thebandtheorypartlysucceeded inde- scribing theelectronic structure ofstrongly correlated materials. Recently, BTi03perovskites, whereBisatrivalent rare-earth atom,havebeensingledouttoobserve the doping-induced metal-insulator (MI)transition.~The crystalstructure ofBTi03consistsofanoctahedral net- workofTi06,witharare-earthBionlocated between theoctahedra. Itselectronic structure isdominated by theTi06octahedra; thevalence andconduction bands areformed mainlyof02pandTi3dorbitals. There- forebysubstituting Bforadivalent alkaline-earth ion A,bandfillingofthesematerials canbecontrolled on thevergeoftheMItransition without introducing heavy randomness orsubstantial changes intheelectronic or latticestructure. Therefore, theobserved MItransi- tionsinYqCaTi03andLaqSrTi03compounds arethought tobetheresultofstrongelectron correla- tions,according totheMott-Hubbard theory.Thebandtheoryaccurately describes theelectronic structures ofCaTi03 andSrTi03asinsulators. How- ever,itisthought towrongly predict metallic ground statesforthetypicalMottinsulators YTi03andIaTi03, because LDAcalculations forLaTi03 withacubicper- ovskitestructure showedametallic bandstructure. However, thesematerials havedistorted perovskite structures, andshowmagnetic ordering atlowtem- perature. YTi03isferromagnetic andLaTi03 iscanted antiferromagnetic. Inthispaper, wedescribe howthe energybandofYTi03ischanged byferromagnetism and thesmalllatticedistortion. II.CALCULATIONS Because ofthelatticedistortion, YTi03 hasanor- thorhombic primitive cell(GdFeOs type),whichhasthe spacegroupsymmetry Pnmbandcontains fourYTi03 molecules. Figure1showsitslatticestructure. Toex- aminetheelectronic structure, weassumed thecrystal structure ofatomicpositions andlatticeconstants that wasmeasured byMacLean etaLThelatticelengths were0.5316nmintheaaxis,0.5679nminthe6axis, and0.7611nminthecaxis.Themostimportant feature ofthecrystalstructure istheTi-0-Tibondangles,which are140and144fromexperiment. Theseanglesarere- latedtotheferromagnetism andthebandstructure near theFermilevel(E+).Toclarifytheeffectofdistortion ontheelectronic structure, wealsomadecalculations forthecubicperovskite structure, whoselatticelengthis foundbymatching thecellvolume withonequarterof thevolumeoftheorthorhombic cell. Totakethesmalllatticedistortion intoaccount, weusedthefull-potential linearaugmented-plane-wave (FLAPW) method, whichimposes noshapeapproxi- mations ontheelectron distribution orthepotential insideandoutsidemuon-tin spheres.'Thecalcu- lations usedthescalarrelativistic approximation, ne- glecting spin-orbit interaction. Exchange andcorrela- 0163-1829/95/51(4)/2098(5)/$06. 00 512098 1995TheAmerican Physical Society FULL-POTENTIAL BANDCALCULATIONS ONYTi03WITHA... 2099 ~Tioo ~o~o~~~~XZX ~-~L~+I+g—~~o~~$--—~~55 ~ooo oo—--oooo~=—~o~ ~~ ~~ ~ ,yr Ij~ja~ ~o FIG.1.Distorted perovskite structure ofYTi03.0atoms areatthecornersofoctahedra. tionweredetermined bytheLDA,withparameters from Janak,Moruzzi, andWilliams. Weusedsphereradiiof 2.46a.u.fortheYsphere,2.07a.u.fortheTisphere, and1.56a.u.forthe0sphere. Linearaugmented plane waves(LAPW's) areexpanded byspherical harmonics ineachmuKn-tin spherethroughtol=8.Theelectron distribution andpotential areexpanded withlatticehar- monicsintheYspherethroughtol=4,andintheTi and0spheresthroughtol=3.Weincluded about2200 LAPW's (theplane-wave cutofFwas~K~2=19.2a.u.) andused18nonequivalent kpointsinthefirstBrillouin zonetoobtainself-consistent potentials. Thedensityof states(DOS)wascalculated bytetragonal interpolation with75nonequivalent kpoints. Sincethecubicper- ovskite cellisonequarteroftheorthorhombic cell,we usedabout550LAPW's forit. III.RESULTS ANDDISCUSSIONS Wemadenonmagnetic andferromagnetic calculations forboththecubicperovskite structure andthedistorted orthorhombic structure. Figure2showstheenergyband ofnonmagnetic cubicYTi03. Thetopofthevalence band(E„)isatpointRandthebottomoftheconduc- tionband(E,)isatpointI'.Thebandgapbetween themis2.68eVandthevalence bandwidthis4.98eV. ThezeroenergypointisE~,1.09eVaboveE.Since electron configurations oftheoutershellsintheatoms are4d582forY,3d48forTi,and2pfor0,thereare 19valenceelectrons inthecubicperovskite cell.Theva- lencebandhas18electrons including thespindegeneracy sooneelectron enterstheconduction band.Theconduc- tionbandhassufficiently largeenergydispersion nearE~ toallowelectrons tomoveinthecubiclattice.Therefore, theLDAcalculation indicates thatYTi03withthecubic latticeismetallic. Intheferromagnetic calculations forthecubiclattice, thebandgapandvalence bandwidth hardlydierfrom thenonmagnetic ones.Figure3showstheDOSofcubic ferromagnetic YTi03.Itshowsthatthevalencebandis mainly02porbitals, andthattheconduction bandnear E~ismainlyTi3dorbitals, asexpected. Thetotalener- giesarethesameforthenonmagnetic andferromagnetic stateswithinthecalculable range.Although themag- neticmoment measured byexperiment is0.84p~perTiIo~~j ~j~ooe~y~o~0+~oo~ owo+I~~ =-o----~ ~oooo -10I XI I M WavevectorX FIG.2.Theenergybandstructure ofnonmagnetic YTi03 alongtheselected symmetry linesobtained withthecubic perovskite structure. ThezeroenergypointistheFermilevel. 4J CJ i QJ I A~~ ~~~ ~A~~pter ~~[~4 ~~~—Total Ti3d 0p -10 -5 Energy(eV) FIG.3.Densityofstatesofferromagnetic YTi03withthe cubicperovskite structure. ThezeroenergypointistheFermi level.atom, ourcalculated moment is0.08p~,whichistoo small.Inthecalculations forthecubiclattice,thereare nosignsthatYTi03isaninsulator, although theysug- gestthatsubstituting alkaline-earth atomAfortheY atombringsaboutholedoping. Observed YTi03hasadistorted perovskite structure whoseprimitive cellhasfourtimesthevolumeofthecu- biccell.Nonmagnetic calculations withtheexperimen- tallatticestructure gaveusthebandstructure alongthe symmetry lines(Fig.4).Because ofthebandfolding, E„isatpointI'.ThebandgapbetweenE„andE is3.46eV,andthevalence bandwidth is4.28eV.They are,respectively, 0.78eVwiderand0.7eVnarrower than thoseofthecubiclattice.E~isat0.49eVaboveE„so thewidthbetweenE~andE,isalmosthalfthatofthe 2100 HIDEAKI FUJITANI ANDSETSURO ASANO 51 3 I~~I~~~~~~~~~II,;~~~$g~!~Ilgwu~'~~''!llI...l~~~~I~!~ ~!~~~~I~ ~~~~! ~~~~~!!~~~15 II$ $imj!~ I~ ~I ~~ I)I ~!I ~III~~! ~~~0 ~~~ '~~~~ ~~ ~~~ ~I~!~~I ~~~~I I~~ ge~~ ~~~!~~II~~~~~~~~ ~~ ~~l3! &a!~.. ~~ ~! 0q)-0----Im-o-op-o-o- ~~ys~~~~ ~~!~)O.Q~ O0!~~!0I~ ~~eg-(---R-oBo ~~~I~~~ !!~~~ ~!~ ~1~I~~~II~I~ ~'~~I$I,$jl ~~e~!0 ~~ ~$!I ~~gII sIj iij ~~~ II ~~~!I~~! ~~~~I~~~0I~!Ij! I~~!~~!! ~! ~~~~~ ~I~~~~~I~ ~I ~~~ ~I ~~ ~~~I~~ ~~ ~~ ~~ ~~~ ~~ 1 4! ~s~~~~ ~~ ~~ ~~~~ ~~ ~~ ~~~~ ~~~~ ~~~~~II ~~ ~~!! ~~~II ~'~~~~ ~~ ~I ~~~ ~I~ ~!1!a ~!~~~~$ ~!~~~~II ~!~ ~Is~~ .q.g.g-cI-;-o---,o-os-g- -oo-~-"------ &J)-- gO()OO O()10 5 0&V C5 15I I I I I 1I t Ir,,' 1 ~o~eo~+~~~~~0~""I"- ~J~~~~~ ~oem pe~ ~ ~~~~~ ~i I 1~~ ~~~~~~~~ ~~4'b~ ~e ~)t~ ~~'~~~~~4o,~,i I 0yt~~ ~~~~~t ~~~'~~~~~ ~~0 ~~~\—Total ~~~TI3d 02p -9 II XSI YI R-10 -5 Energy(eV) Wavevector FIG.4.Theenergybandofnonmagnetic YTiO&alongthe selected symmetry linesobtained withtheorthorhombic per- ovskitestructure. ThezeroenergypointistheFermilevel. Largerunfilled circlesarethelowestconduction bandswhich areconnected bycompatibility relationships. cubiclattice. Although anewbandgapappears inthe conduction band,itshouldnotafFectelectrical conduc- tivitybecause itisat1.5eVaboveE~. Thelargerunfilledcirclesatthebottomoftheconduc- tionbandinFig.4areforenergybandsthatarelinked witheachotherbycompatibility relationships. Thislow- estconduction band(LCB)cannot,therefore, bedivided intopieces. Although theLCBoverlaps otherconduc- tionbands,itisnotconnected tothem.TheLCBcon- sistsmainlyofTi3d(bothdeanddp)orbitals. Ifthe wholeLCBwereoccupied byelectrons, itcouldcontain eightelectrons, incl'uding thespindegeneracy. Sincethe orthorhombic cellhasfourtimesasmanymolecules as thecubiccell,fourelectrons havetoentertheconduc- tionband.EveniftheLCBhadalltheconduction band electrons init,E~wouldstilllieathalfoftheLCB. Therefore, nonmagnetic YTi03withadistorted latticeFIG.5.Densityofstatesofferromagnetic YTi03withthe orthorhombic structure. ThezeroenergypointistheFermi level. mustbemetallic Theferromagnetic orthorhombic latticehasatotalen- ergy0.12eVlowerthanthenonmagnetic lattice. The ferromagnetic stateisfavorable tothedistortedYTi03. Figure 5showstheDOSoftheferromagnetic YTi03. Valence bandwidths are4.32eVformajority spins(the upperpanel)and4.36eVforminority spins(thelower panel).Bandgapsare3.18eVand3.59eV.Thedipat themiddleofthevalencebandissmaller inthedistorted latticethaninthecubiclattice(Fig.3).Thisagreeswith thefactthatthedipishardlyvisibleinexperiments. E~is0.62eVaboveEnearasharpdipinthemajority spins,withbothsidesformed mainlyofTi3dorbitals. Figure6showsenergybandsnearE~.TheLCBofthe majority spinsisdenoted byasterisks. Itoverlaps other conduction bandsonlyneartheI'andZpoints. The conduction bandoftheminority spinsisalmostabove E~,exceptnearpointI".Ifweraisetheconduction bandoftheminority spinsaboveE~andlowertheLCB ofthemajority spinsbelowotherconduction bands,a 1.0 ( 0.5—(00~0~) 0 ,P0 ++0 ae$ p5 )( )K AC )A0 000 0 p,' 0 Q) ~&O" 0~@o (6'FIG.6.Theenergy band offerromagnetic YTi03 along theselected symmetry linesob- tainedwiththeexperimental orthorhombic structure. The zeroenergypointistheFermi level.Asterisks arethelow- estconduction bandswhichare connected bycompatibility re- latzonshzps. -1.PI I I I I I I I I I I I I I XSY~ZURTZ&XSY tZURTZ Wavevector 51 FULL-POTENTIAL BANDCALCULATIONS ONYTi03WITHA... 2101 0.59 ( 0 e )K ~~%QL Alp)g1(ooo+0 0o00 )0@o o0 00 gPPIG.7.Theenergybandof ferromagnetic YTi03alongthe selected symmetry lineswith Ti-0-Tibondanglesof136 and140'.Thezeroenergy pointistheFermilevel.Aster- isksarethelowestconduction bandswhichareconnected by compatibility relationships. $0l I I IXSYI I I I I I IZuRTz IXSY zuI I RTZ Wavevector bandgapwillappearandallconduction electrons will entertheLCB.Theenergybandoftheferromagnetic distorted YTi03hasthefeatures neededtoopenaband gapatEF,whilethecubicYTi03 showsnosignsofa bandgapatEF. Ifabandgapopenedtocontain allconduction elec- tronsintheLCBofthemajority spins,themagnetic moment wouldbe1p,~perTiatom.Withtheexperi- mentaldistorted. structure, thecalculated magnetic mo- mentis0.96@~perTiatom,because asmallpartof theconduction bandsoftheminority spinsliesbelow EF.Sinceamagnetic interaction couldbecloselyrelated totheTi-0-Ti bondanglesthrough superexchange, we suspected thatfurtherdistortion raisestheconduction bandsoftheminority spinsaboveEF.Inordertocheck this,theTi-0-Ti bondangleswerefurtherdistorted by 4&omtheexperimental onesbychanging the0atom positions. Figure7showstheenergybandsofthisartifi- cialstructure.Eoftheminority spinsis0.14eVabove EF,whiletheEis0.16eVbelowEFintheexperimen- talstructure. Thecalculated magnetic moment is1@~ perTiatom.TheLCBofthemajority spinsisslightly lowerandoverlaps withotherconduction bandsonlynear pointI".Thebandstructure changed toslightlyreduce remnant overlaps, butabandgapdidnotopen. TheenergybandsnearEFformed byTi3dorbitals areverysensitive tolatticedistortion. Toobtainanac- curatebandstructure forthegroundstate,fulllattice relaxation isneededwithintheLDAcalculation. Butit isdifficulttoobtainaminimum energylatticestructure &omanordinary bandcalculation because thestructural degreeofkeedomoftheorthorhombic cellreachesten,in- cludinglatticeparameters, eveniftheGdFe03 typecell isassumed. Recently, Wentzcovitch, Schulz,andAllen reported LDAcalculations onV02,whichwasconsid- eredtobeastrongly correlated material. Usingpseu- dopotentials andavariable cellshapealgorithm, they obtained afullyrelaxedcrystalstructure oftheground statewhichhadanalmostopenbandgap.Fullyrelaxing thecrystalstructure ofYTi03mightpossibly reducethe overlapoftheLCBwithotherconduction bands,butwe speculate thatabandgapwouldnotopeninferromag- neticYTi03. Thisispartlybecauseofthewell-knownLDAerror,whichcausesthecalculated bandgaptobe smallerthantheexperimental one. Fujimori eta/.measured photoemission spectraof transition-metal oxides, andfoundthattherewasan "incoherent" peakat1.5eVbelowEFinYTi03, whileourlowerd-orbital peakisjustbelowE~(Fig.5). Thereasonforthediscrepancy maybethattheLDA cannotaccurately account forthestrongelectron corre- lationsinYTi03. But,sincethedpeakisbelowEF, theLDAcalculation isprobably notsobadtodescribe theelectronic structure ofYTi03. Tokura etal.mea- suredtheHallcoefficient andtheelectronic specificheat ofYqCaTi03compounds andfoundcarriermassen- hancement inthevicinityoftheMItransition. This mightpossibly berelatedtothecalculated bandwidth betweenEandEF,whichisreduced byorthorhombic distortion. Itwaswidelythoughtthatthebandfillingof YqCaTi03couldbecontrolled. without introducing substantial electronic orlatticestructure changes. How- ever,sincetheenergybandsnearEFaresensitive tolat- ticedistortion, moreinvestigation ofthelatticestructure and.thestoichiometry ofYTi03isneeded. Ourcalculations indicate thattheexperimentally ob- servedferromagnetism andlatticedistortion areneces- sary,butnotsufficient, forYTi03tohaveabandgap atEF.Also,themagnetic moment inthegroundstate mustbe1@~perTiatomifabandgapopenedatEF. Sincetheexperimental magnetic moment is0.84@~per Tiatomat4.2K,itisalittlesmallerthantheex- pected value&omthebandcalculation. Butordinary YTi03ispolycrystalline and,strictlyspeaking, alittle oKstoichiometry becauseofadeficiency of0atoms. Ifaperfectorthorhombic YTi03crystalcanbefound, measuring itsmagnetic moment wouldbeacrucialtest ofthevalidityoftheband.theoryforthegroundstateof theinsulating YTi03. IV.SUMMARY Westudied theelectronic structure ofYTi03 using FLAPW calculations. Intheenergybandofthecubic 2102 HIDEAKI FUJITANI ANDSETSURO ASANO perovskite structure, therewerenosignsthatYTi63 is aninsulator. Withthedistorted perovskite structure, theferromagnetic statewasfavorable toYTi03.Itsen- ergybandhadfeatures necessary forYTi03tohavea bandgapatE~,although abandgapdidnotopeninan experimental latticestructure.ACKNOWLEDGMENTS WewouldliketothankProfessor H.Yoshizawa forhis usefuladvice. Oneoftheauthors (H.F.)isalsograteful toDr.N.Sasaki,S.Hijiya,andT.Itofortheirencour- agement. N.F.Mott,Proc.Phys.Soc.LondonSec.A62,416(1949).J.Hubbard, Proc.R.Soc.LondonSer.A277,237(1961). L.FMa.ttheiss,Phys.Rev.B5,290(1972). D.E.Eastman andJ.L.Freeouf, Phys.Rev.Lett.34,395 (1975). A.Fujimori andF.Minami, Phys.Rev.B30,957(1984). W.KohnandL.J.Sham,Phys.Rev.140,A1133(1965). K.Terakura, T.Oguchi, A.R.Williams, andJ.Kubler, Phys.Rev.B30,4734(1984).J.Yamashita andS.Asano,J.Phys.Soc.Jpn.52,3506 (1983). A.Fujimori,I.Hase,M.Nakamura, H.Namatame, Y.Fu- jishima,Y.Tokura, M.Abbate,F.M.F.deGroot,M.T. Czyzyk,J.C.Fuggle,O.Strebel,F.Lopez,M.Domke, and G.Kaindl,Phys.Rev.B46,9841(1992). Y.Tokura,Y.Taguchi,Y.Okada,Y.Fujishima, T.Arima, K.Kumagai, andY.lye,Phys.Rev.Lett.70,2126(1993). Y.Tokura,Y.Taguchi, Y.Moritomo, K.Kumagai, T. Suzuki, andY.Iye,Phys.Rev.B48,14063(1993).D.A.Crandles, T.Timusk,J.G.Garrett,andJ.E. Greedan, Phys.Rev.B49,16207(1994). L.F.Mattheiss, Phys.Rev.BB,4718(1972). A.Fujimori, I.Hase,H.Namatame, Y.Fujishima, Y. Tokura,E.Eisaki,S.Uchida,K.Takegahara, andF.M. F.deGroot,Phys.Rev.Lett.69,1796(1992). D.A.MacLean, Hok-Nam Ng,andJ.E.Greedan,J.Solid StateChem.30,35(1979).J.P.Goral,J.E.Greedan, andD.A.MacLean,J.Solid StateChem.43,244(1982). H.J.F.JansenandA.J.Freeman, Phys.Rev.B30,561 (1984). L.F.Mattheiss andD.R.Hamann, Phys.Rev.B33,823 (1986).J.F.Janak,V.L.Moruzzi, andA.R.Williams, Phys.Rev. B12,1257(1975). R.M.Wentzcovitch, W.W.Schulz,andP.B.Allen,Phys. Rev.Lett.72,3389(1994).
PhysRevB.92.085417.pdf	PHYSICAL REVIEW B 92, 085417 (2015) Phase-coherent transport in catalyst-free vapor phase deposited Bi 2Se3crystals R. Ockelmann,1,2A. M ¨uller,1J. H. Hwang,1,3,4S. Jafarpisheh,1,2M. Dr ¨ogeler,1B. Beschoten,1and C. Stampfer1,2 1JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany 2Peter Gr ¨unberg Institute (PGI-9), Forschungszentrum J ¨ulich, 52425 J ¨ulich, Germany 3Center for Nanomaterials and Chemical Reactions, Institute for Basic Science, Daejeon 305-701, Republic of Korea 4Graduate School of EEWS, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Republic of Korea (Received 25 June 2015; published 17 August 2015) Freestanding Bi 2Se3single-crystal ﬂakes of variable thicknesses are grown using a catalyst-free vapor-solid synthesis and are subsequently transferred onto a clean Si++/SiO 2substrate where the ﬂakes are contacted in Hall bar geometry. Low-temperature magnetoresistance measurements are presented which show a linearmagnetoresistance for high magnetic ﬁelds and weak antilocalization (WAL) at low ﬁelds. Despite an overallstrong charge-carrier tunability for thinner devices, we ﬁnd that electron transport is dominated by bulkcontributions for all devices. Phase-coherence lengths l φas extracted from WAL measurements increase linearly with increasing electron density exceeding 1 μm at 1.7 K. Although lφis in qualitative agreement with electron-electron interaction-induced dephasing, we ﬁnd that spin-ﬂip scattering processes limit lφat low temperatures. DOI: 10.1103/PhysRevB.92.085417 PACS number(s): 73 .23.−b,73.63.−b,73.50.Jt,73.20.Fz I. INTRODUCTION Topological insulators (TIs) are a new class of materials [1–5], consisting of an insulating bulk and topologically protected conducting surface states. These surface states are spin polarized and robust against scattering from nonmagneticimpurities making them interesting candidates for futurespintronics and quantum computing devices [ 6–8]a sw e l l as potential hosts for Majorana fermions [ 9–12]. Binary Bi chalcogenides (Bi 2Se3,Bi2Te3) belong to the class of three-dimensional (3D) strong topological insulators with asingle Dirac cone at the surface [ 13], which is experimentally observable by angle-resolved photoemission spectroscopy[14]. In particular, Bi 2Se3with a single Dirac cone centered in a bulk band gap of Eg≈350 meV is a promising material for probing surface states by electronic transport. However, the measurement of pure surface states is challenging. So farBi 2Se3crystals are unintentionally n-type doped most likely by Se vacancies [ 15–17] leading to bulk conductivity dominating electronic transport. To increase the surface-to-bulk ratio,ultrathin ﬂakes with thicknesses on the order of 10 nm havebeen investigated [ 18,19]. Thin Bi 2Se3crystals can be produced by mechanical exfoliation of bulk material as is common practice for graphenefabrication [ 20,21]. However, it is much more promising to grow thin Bi 2Se3ﬁlms in situ , which has been successfully achieved with molecular beam epitaxy (MBE) [ 22–24]o r vapor-solid synthesis (VSS) in a tube furnace [ 19,25,26]. In this paper, we show a catalyst-free growth of large free- standing Bi 2Se3ﬂakes with a VSS method. Our freestanding growth approach ensures the synthesis of strain-free few-layersingle-crystal ﬂakes with lateral dimensions up to 25 μm and thicknesses in the range of 6–30 nm, ranking our ﬂakes amongthe largest Bi 2Se3single-crystal ﬂakes. The high structural and surface quality of the Bi 2Se3crystals is veriﬁed by Raman and scanning force microscopies. The freestanding single crystalsare ideal for transport studies. We utilize a wet chemistry-free process which allows transferring these single crystalsonto any desired substrate without introducing additionalcontamination. We studied low-temperature magnetotransport on a series of Bi 2Se3crystals of different thicknesses which were transferred on SiO 2/Si++substrates. We observe linear magnetoresistance (LMR) at high Bﬁelds as well as weak antilocalization (WAL), which both indicate the dominance ofbulk transport contributions. Electron phase-coherence lengthsare in the micrometer range, slightly larger compared to earlierstudies on Bi 2Se3crystals grown directly on SiO 2[27]o r by other growth methods [ 28,29]. We show that electron spin-ﬂip processes limit the phase-coherence length at lowtemperatures. II. CRYSTAL GROWTH AND CHARACTERIZATION With the goal of gaining high quality thin Bi 2Se3crystals we applied a catalyst-free vapor-solid synthesis method. Mostcommonly, MBE [ 30–32] is used to grow thin ﬁlms since it offers the growth of extended ﬁlms of rich chemicalcompositions with excellent thickness control. Yet it suffersfrom a limited number of usable substrates. In contrast toMBE, VSS allows the growth of single-crystalline plateletson a variety of different substrates [ 33–35]. However, strain can still be induced by the growth substrate. Moreover, agrowth catalyst may induce unwanted dopants into the crystal.In this paper we therefore use a catalyst-free VSS methodwhere ﬂakes and ribbons grow freestanding on the substrate.This offers an interesting pathway for the fabrication of highquality devices. Freestanding ﬂakes are neither strained norcontaminated by the substrate material and can be easilytransferred onto on a wide range of different substrates. A standard three-zone tube furnace [Fig. 1(a)] with electric heating coils is used for the VSS growth. As a source materialwe place Bi 2Se3crystals [ 36] in the ﬁrst zone. The Si /SiO2 growth substrates are placed downstream in the second zone. Their exact position was optimized through several growthcycles. Prior to growth, the quartz tube was evacuated to2 mbars with subsequent argon ﬂushing for 5 min with a500 SCCM (where SCCM denotes standard cubic centimeterper minute) ﬂow rate which is regulated by a digital mass ﬂow 1098-0121/2015/92(8)/085417(7) 085417-1 ©2015 American Physical SocietyR. OCKELMANN et al. PHYSICAL REVIEW B 92, 085417 (2015) 5μ m 10 μm 4μ m5μ mAr(a) (d) (e)(b) (c) (f) (g) (h) 40 0[nm] 2μ mBi Se23 xμ m xxxx μμμ mmmmm 2μ m -200 -100 0 100 200 -1wavenumber [cm ]anti-Stokes Stokes 13 nm 16 nm10 nm 0 100 200 300Intensity [arb. units]2A1g2Eg1A1g 1EgT = 700°C T = 325°C substrateT = 25°C FIG. 1. (Color online) (a) Schematic of the three-zone oven used for Bi 2Se3sample growth. (b)–(d) Scanning electron micro- scope images of typical freestanding Bi 2Se3ribbons and ﬂakes. (e)–(g) AFM images of grown ﬂakes transferred onto a SiO 2/Si substrate. (h) Characteristic Raman spectra of grown Bi 2Se3ﬂakes with different thicknesses. controller. After cleaning, the growth zone (second zone) is heated up to 325◦C with a constant argon ﬂow of 100 SCCM to carry away vaporized particles. The second zone is kept at325 ◦C and 25 mbars for 2 h as it is crucial for the temperature and pressure to be stabilized during growth. Finally, the actualgrowth process is executed by heating the ﬁrst zone to 700 ◦C. The source material gradually vaporizes and gets carrieddownstream by a 60 SCCM argon ﬂow. Temperature andpressure were optimized to grow large thin freestanding Bi 2Se3 ﬂakes and ribbons as shown by scanning electron microscopeimages in Figs. 1(b)–1(d) and by atomic force microscope (AFM) images in Figs. 1(e)–1(g). Straight edges with only 60 ◦ and 120◦corners indicate single-crystalline growth. According to AFM measurements the ﬂake thicknesses range between 6and 40 nm, and lateral dimensions can reach up to 25 μm. AFM images also reveal the ﬂake’s surfaces to be steplessconﬁrming a very homogeneous layer-by-layer growth. Raman spectroscopy has emerged as an excellent tool to probe crystal stoichiometry of Bi 2Se3[37–39]. The Raman spectra of our Bi 2Se3ﬂakes is obtained using confocal RamanTABLE I. Geometrical dimensions of the four devices discussed. The dimensions are deﬁned as in Fig. 2(a)and were measured using an AFM. The respective gate lever arms αgare also included. Device No. t(nm) L(μm) W(μm) αg(cm−2V−1) 1 12 3.4 2.4 7 .5×1010 2 16 3.5 2.9 8 .2×1010 3 28 4.7 10.6 7 .5×1010 4 30 3.5 11.8 9 .8×1010 spectroscopy with a laser spot diameter of around 500 nm at a wavelength of 532 nm. The laser spot is precisely positioned onthe ﬂakes using a piezostage. In Fig. 1(h)all four characteristic Raman peaks of Bi 2Se3are clearly seen at 37, 71, 131, and 175 cm−1, which correspond to the E1 g,A1 1g,E2 g, and A2 1g vibrational modes, respectively. The peak positions are very close to previously measured Raman peaks of stoichiometricBi 2Se3crystals [ 37,40,41] indicating the high crystal quality of our ﬂakes. For transport studies, the freestanding Bi 2Se3ﬂakes are dry transferred by gently dabbing a clean room cloth ontothe grown chips and subsequently onto a clean SiO 2/Si++ substrate. This method does not involve solvents or other liquids which could effect the surface quality. These substratesare prepatterned with gold markers to relocate individual ﬂakesand enable consecutive electron-beam lithography (EBL). The ﬂakes to be contacted are ﬁrst chosen by optical microscopy and further characterized by AFM, which is alsoused to determine exact dimensions. The contacts are deﬁnedusing standard EBL techniques and (5 /50)-nm Cr/Au ohmic contacts. Directly before metal evaporation the contact areasare etched for 15 s by oxygen plasma to remove any oxidelayers from the Bi 2Se3surface. This step is crucial for low contact resistances. The contact geometry of ﬂakes with ahigh length/width ratio resembles a fairly good approximationof a Hall bar geometry [cf. Figs. 2(a) and2(b)]. With this method no additional patterning step is needed, allowing us tokeep unetched ﬂake edges as grown in the VSS process. Thedimensions of the four investigated devices are summarized inTable I. III. RESULTS AND DISCUSSION Transport measurements were performed in a4He cryostat at a base temperature of T=1.7 K using low-frequency lock-in techniques. A superconducting solenoid, immersed inliquid helium was used to apply magnetic ﬁelds perpendicularto the sample plane. The backgate characteristics of fourdifferent devices [see Fig. 2(b)] with different Bi 2Se3crystal thickness are shown in Fig. 2(c), which depicts the four- terminal conductivity σas a function of applied backgate voltage Vg. For the 28- and 30-nm-thick Bi 2Se3samples almost no gate tunability is observed, which is in contrast to the twothinner (12- and 16-nm-thick) samples where σcan be tuned by a factor of around 2. In none of our samples do we observean ambipolar transport behavior, which is a ﬁrst indication thatvery high ndoping of Bi 2Se3is present in all our devices. 085417-2PHASE-COHERENT TRANSPORT IN CATALYST-FREE . . . PHYSICAL REVIEW B 92, 085417 (2015) 0123456 −70 −35 0 35024810 V [ V ]g70(b)LSD W 0t 150 [nm](a) (c) (d) 13 -2 [10 cm ]n 30 nm 28 nm 16 nm 12 nmSiOBi Se SiL 13 -2 [10 cm ]n1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60100200300400500600700 thickness [nm]01 0 2 0 3 0012345 (V=0) [10 cm ] nρ [Ω](e)−70 −35 0 35 V [ V ]g706σ-3 -1 [10 Ω ] FIG. 2. (Color online) (a) Schematics of the sample geometry of contacted Bi 2Se3ﬂakes. In the left panel we highlight the width ( W), length ( L), and source-drain distance ( LSD) of a contacted sample. The source-drain distance in our samples is in the range of 14–19 μm. The right panel shows a cross section of our samples highlighting the ﬂake with thickness ( t) resting on SiO 2with a backgate. (b) AFM images of the four investigated devices contacted with elec- trical contacts for Hall effect and magnetoresistance measurements. The scale bars are all 5 μm. For more details on the geometry please see Table I. (c) Conductivity as a function of backgate voltage for all four samples. (d) Two-dimensional carrier density nof the different samples, extracted from the Hall resistance vs backgate voltages.(e) Resistivity as a function of carrier density for all devices. The inset shows the carrier density at zero backgate voltage as a function of Bi 2Se3ﬂake thickness. We performed Hall effect measurements to determine the charge-carrier densities. The extracted two-dimensional (2D)electron density n, which varies linearly with V g,i ss h o w n in Fig. 2(d) for all devices. The slope for the three thinner samples of (7 .5–8.2)×1010cm−2V−1(see Table I)i si n reasonably good agreement with the geometrical gate lever armα g=/epsilon10/epsilon1r/(\|e\|d)≈7.2×1010cm−2V−1, where d=285 nm is the thickness of the SiO 2gate oxide with a dielectric constant of /epsilon1r≈4. From the vertical offsets of nin Fig. 2(d) we can estimate the average bulk charge-carrier density. Theinset in Fig. 2(e) shows the 2D carrier density nat zero gate voltage as a function of the Bi 2Se3crystal thickness. From the slope of this linear dependence (dotted line) weextract a 3D bulk carrier concentration of 1 .5×1019cm−3 in our VSS Bi 2Se3crystals. Finally, we plot the resistivity ρ=1/σof all four devices as a function of n[main panel of Fig. 2(e)]. The observed overall trend highlights a consistent carrier-density dependency on the measured resistivity of allmeasured devices. The increasing gate tunability at lowercarrier densities might be either connected: (i) to the lineardensity of states of the 2D surface states or (ii) to a reductionin the 3D bulk density of states (or diffusion constant) at lowerFermi energy values. The carrier mobility in our VSS Bi 2Se3 crystals can be estimated by μ=σ/(en). For the two thinner samples we therefore obtain Hall mobility values in the rangeofμ≈600–1000 cm 2V−1s−1. As our setup only allows for magnetic ﬁelds of up to 9 T, the requirement μB > 1f o r observing Shubnikov–de Haas oscillations can unfortunatelynot be reached. This is consistent with experiment, and weindeed we do not observe any Shubnikov–de Haas oscillations(see magnetotransport measurements below). Overall, weconclude that it is very likely that transport in our samplesis dominated by bulk transport where the gate tunabilityoriginates from a Fermi-level-dependent 3D bulk density ofstates or diffusion constant. For gaining more insight on the separation of bulk and sur- face transports we performed four-terminal magnetoresistancemeasurements [see Fig. 3(a)]. The two most prominent features in our magnetotransport data are as follows: (i) a LMR at highmagnetic ﬁelds [ 42–45] and (ii) a reasonably strong WAL dip at low magnetic ﬁelds (below 1 T) [ 44,46–49] as shown by the inset of Fig. 3(a). Above B=5 to 6 T all four devices exhibit LMR. Inter- estingly, the strength of the LMR does not solely depend onthe sample thickness nor on the total charge-carrier density.We observe that the thinnest and thickest Bi 2Se3crystals exhibit similar LMR slopes [see the black and orange curvesin Fig. 3(a)], whereas the other two devices [see the red and blue curves in Fig. 3(a)] show a signiﬁcantly larger slope of the LMR. However, within a single device we ﬁnd a systematiccarrier-density dependence of the slope of the LMR at largemagnetic ﬁelds [see Fig. 3(b)]. A more detailed analysis of the carrier-density-dependent LMR slope shows that,interestingly, /Delta1ρ//Delta1B changes linearly as a function of n −2 B ﬁeld [T](a) (b) −1−0.5 0 0.5 101234 0 2 4 6 805101520253035 0ρ-ρ [Ω] 01020304070V 45V 20V 0V −35V −70V ﬁts0ρ-ρ[Ω] 11.5 24567 [10 cm ]1/nΔρ/ΔB [Ω/T] 0.5 B ﬁeld [T]0246828 nm30 nm 16 nm 12 nm FIG. 3. (Color online) (a) Magnetoresistance /Delta1ρ=ρ−ρ0 [where ρ0=ρ(B=0 T)] at a zero backgate voltage for the four devices in Fig. 2. The inset shows WAL dips at small Bﬁelds. (b)/Delta1ρas function of the Bﬁeld for the 16-nm-thick sample for various backgate voltages (see labels). The inset shows the slopes of the linear magnetoresistance at high magnetic ﬁelds (see dashedlines in the main panel) vs 1 /n 2. 085417-3R. OCKELMANN et al. PHYSICAL REVIEW B 92, 085417 (2015) −0.6−0.5−0.4−0.3−0.2−0.10α 11.5 22.5 33.5 4040080012001600 13 -2n [10 cm ]28 nm 16 nm 12 nm(b) (c) −0.2 −0.1 0 0.1 0.2−2.5−2.0−1.5−1.0−0.50 16 nm28 nm28 nm 16 nm 12 nm 20 40 60 80 100 1200500100015002000 g 0(a) (d) 1 1.5 2 2.5 3 3.5 4 13 -2n [10 cm ] B field [T] lφ [nm] lφ [nm]16 nm 12 nm FIG. 4. (Color online) (a) WAL peak in conductivity as a function of the Bﬁeld (solid lines) and ﬁts according to the HLN model (dashed lines) for two different devices. (b) and (c) Extracted ﬁtting values for α(panel b) and lφ(panel c) are shown as a function of the charge-carrier density for three different devices. (d) Phase coherence length as a function of the dimensionless conductivity g/square. Here, the solid line displays the theoretical result by Altshuler- Aronov-Khmelnitsky (AAK) [ 50]. The dashed line highlights the result modiﬁed due to a ﬁnite electron spin-ﬂip scattering time τsf(see text for details). [see the inset in Fig. 3(b)]. This dependence is in agreement with the generic quantum description of galvanomagnetic phe-nomena by Abrikosov [ 51,52] and Hu and Rosenbaum [ 53], leading to ρ∝B/n 2. However, although for the investigated B-ﬁeld range the required condition μB > 1 might be fulﬁlled, we are certainly not in the extreme quantum limit whereonly the lowest Landau level is ﬁlled. Moreover, it shouldbe noted that the ﬁt shown in the inset of Fig. 3(b) does not cross the origin. All these bring us to the conclusion thatthe LMR in our devices is rather dominated by the classicallinear magnetoresistance [ 53] due to bulk inhomogeneities and defects in the Bi 2Se3crystals which may also explain the high bulk carrier densities. These ﬁndings are in contrast to the WAL, which exhibits a clear crystal thickness dependence [see the inset of Fig. 3(a)] and which is therefore—also in agreement with literature[48,49,54]—most likely a better ﬁngerprint for surface-state transport. WAL signatures are indeed inherent to the 2Dstates of TIs [ 3,24,46] As is governed by quantum-mechanical interference, a detailed investigation of resulting correctionsto the conductance allows for learning more about phase-coherent transport properties in these materials. Indeed, WALin TIs has already been studied in great detail [ 55–58], and it has been shown that the so-called Hikami-Larkin-Nagaoka(HLN) model [ 59] can be used to ﬁt the WAL corrections at low Bﬁelds. Within the HLN model the conductivity correction is expressed as /Delta1σ=σ(B)−σ(0) =−αe 2 2π2/planckover2pi1/bracketleftbigg ln/parenleftbigg/planckover2pi1 4Belφ/parenrightbigg −/Psi1/parenleftbigg1 2+/planckover2pi1 4Belφ/parenrightbigg/bracketrightbigg ,(1) where /Psi1is the digamma function and lφis the phase-coherence length. The value of the amplitude αis expected to be −1/2 for perfect WAL in a two-dimensional system. For an ideal 3D TI with two independent and decoupled 2Dsurfaces the expected value for the total WAL amplitude istherefore α=−1[49]. For ﬁtting our data, the symmetric and antisymmetric parts of the overall conductivity were separated.This is necessary considering the imperfect Hall bar geometrydue to the etch-free sample fabrication process. Figure 4(a) shows WAL data ﬁtted with the HLN model given by Eq. ( 1) for the symmetric part of the data with an additional term forquadratic magnetoresistance at low magnetic ﬁelds βB 2.T h e values for αandlφas extracted from the ﬁts are shown in Figs. 4(b) and4(c). For the two thinner samples [orange and red data in Fig. 4(b)]αis a gate tunable around a value of −1/2. This indicates either that the surface states are strongly coupled viathe highly conductive bulk or that transport is purely dominatedby the bulk. For the 28-nm-thick sample [blue data in Figs. 4(b) and 4(c)] no gate dependence is observed indicating the dominance of a bulk conduction channel with a Fermi levelin a regime with a constant 3D bulk density of state whichsuppresses any gate tunability. The thickest sample (30 nm)does not show a distinct WAL peak and is hence excluded fromour WAL analysis. A similar trend is also observed for thephase-coherence length l φwhich increases with larger sample thickness and increasing charge-carrier density. Interestingly, such a gate-tunable phase-coherence length— also observed by other groups [ 29,60]—is in qualitative agree- ment with a scattering mechanism based on electron-electroninteractions as predicted by AAK for a two-dimensionalsystem [ 50,61], l φ=/planckover2pi1g/square(4m∗kBTlng/square)−1/2, (2) where kBis the Boltzmann constant, m∗is the effective mass, andg/square=σh/e2is the dimensionless conductivity, which can be directly extracted from the measured conductivity.Apart from the small logarithmic correction (only becomingimportant for very small conductivities) the phase-coherencelength is a linear function of g /square. By plotting the experimentally extracted lφas a function of g/square[see Fig. 4(d)] we indeed can conﬁrm this nearly linear dependence. Furthermore, byassuming a bulk carrier effective mass of m ∗=0.15me[62], we obtain the solid line in Fig. 4(d) without any further adjustable parameters. These values are a factor of 2 to 3 largerthan the values of l φextracted from our WAL measurements, meaning that there must be some corrections to the effectivemass or (more likely) additional sources for dephasing. This becomes even more apparent when investigating the temperature dependence of the WAL and the extractedl φat different carrier densities as shown in Fig. 5.I n Fig.5(a) we show the WAL peak for the 16-nm-thick sample at different temperatures, highlighting its disappearing at 085417-4PHASE-COHERENT TRANSPORT IN CATALYST-FREE . . . PHYSICAL REVIEW B 92, 085417 (2015) −1−0.50 α(a) −4−3−2−10 1.7K 5K 15K 30K 50K −0.2 −0.1 0 0.1 0.2(b) -70V-35V0V 00 . 51 00 . 51 00 . 510100300400500(d) 200 0100300400500(e) 200 0100300400500lφ [nm](c) 200 0 0.2 0.4 0.6 1/TVg V V07-= g V V53-= g = 0Vlφ [nm] lφ [nm] B field [T] [K-1] 1/√T [K-0.5] 1/√T [K-0.5] 1/√T [K-0.5] FIG. 5. (Color online) (a) Broadening of the WAL peak with increasing temperature for the 16-nm-thick Bi 2Se3sample. (b) Dependence of the parameter αas a function of inverse temperature for different backgate voltages. (c)–(e) Dependence of the phase-coherence length lφ as a function of 1 /√ Tfor three different backgate voltages. The solid and dashed lines resemble the same theoretical models as in Fig. 4(d). elevated temperatures. The peak at small magnetic ﬁelds slowly decreases in amplitude and completely disappears at50 K. By ﬁtting again the HNL model to our data we extractthe temperature-dependent αvalues and phase-coherence lengths [Figs. 5(b)–5(e)]. The prefactor αchanges towards zero for increasing temperature, i.e., decreasing 1 /Tas seen in Fig. 5(b). More insight can be gained when investigating the temperature dependence of the phase-coherence length. Inorder to highlight the expected temperature dependence given by Eq. ( 2)w ep l o t l φas a function of 1 /√ Tin Figs. 5(c)–5(e). Similar to Fig. 4(d), the solid lines illustrate the estimates forlφobtained from the AAK theory ( lφ∝1/√ T) for different backgate voltages, i.e., carrier densities which are color coded in Figs. 5(c)–5(e). Indeed, above T=7 K (below 1 /√ T≈ 0.4K−1/2), the experimentally extracted lφvalues are inversely proportional to the square root of the temperature. However,at lower temperatures, l φshows a carrier-density-dependent saturation behavior, which cannot be explained by electron-electron interaction limiting the phase-coherence length. Toaccount for these discrepancies, we follow Ref. [ 61] and include an additional inelastic electron spin-ﬂip scattering timeτ sf. Thus the phase-coherence time τφ=l2 φ/Dwill be limited byτsfat low temperatures. This leads to an overall scattering rate which is the sum of spin ﬂip and the AAK decoherencerateτ −1 φ=τ−1 sf+kBTlng/square/(/planckover2pi1g/square). We use this expression to estimate lφ=/radicalbigDτφ. By assuming a linear carrier-density dependency of the spin-ﬂip scattering time τsf=βnwith β=1.2×10−24cm2s, we obtain good agreement with all our experimental data [see the dashed lines in Figs. 4(d) and5(c)– 5(e)]. The extracted τsfvalues are on the order of 10 ps. We note that similar density dependence of τsfwas found by some of us in weak localization studies on bilayer graphene [ 61]. Although such density dependence was found for severalspin-relaxation mechanisms in graphene including resonant scattering at magnetic impurities [ 63] and spin-pseudospin dynamics induced by local Rashba spin-orbit interaction [ 64], there is a lack of theory for TIs. We want to point out, however,that the short time scale of only 10 ps most likely results fromthe strong spin-orbit interaction in this material class [ 65]. Furthermore, we emphasize that the τ sfvalue is not attributed to surface but rather to bulk transport properties. IV . CONCLUSION In conclusion, we used a catalyst-free vapor-solid synthesis method for obtaining well-shaped single-crystalline Bi 2Se3 ﬂakes with thicknesses in the range of a few nanometers.We performed low-temperature transport measurements onsuch Bi 2Se3ﬂakes with different layer thicknesses, resulting in different (two-dimensional) doping values. From magneto-transport measurements we extract information on the linearmagnetoresistance as well as on phase-coherent transport prop-erties. In particular from weak antilocalization measurementswe gain detailed insight on the phase-coherence length in bulktransport. We observe that the phase-coherence length linearlydepends on both the conductivity and the electron density. Itsvalues are close to the values imposed by electron-electroninteraction but limited by spin-ﬂip scattering at the lowesttemperatures. 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PhysRevB.70.241303.pdf	Endohedral silicon nanotubes as thinnest silicide wires Traian Dumitric ª, Ming Hua, and Boris I. Yakobson Department of Mechanical Engineering and Materials Science, and Department of Chemistry, Rice University, Houston, Texas 77251, USA (Received 22 September 2004; published 8 December 2004 ) Usingab initio calculations, we describe how the smallest silicon nanotubes of (2,2)and (3,0)chiral symmetries are stabilized by the axially placed metal atoms, to form nearly one-dimensional structures withsubstantial cohesive energy, mechanical stiffness, and metallic density of electronic states. Their further recon-structions lead to thicker and shorter wires, while relative stability can be viewed in a binary ﬁeld diagram of M xSi1−x, and depends on chemical potentials of the components. A comparison with recent epitaxial-growth experiments reveals the equivalence of the (2,2)endohedral nanotubes with the thinnest possible experimental wires. DOI: 10.1103/PhysRevB.70.241303 PACS number (s): 61.46. 1w, 68.65. 2k, 81.07.De The long-standing interest in ﬁne hairlike crystals— whiskers—has shifted over the last decade towards yet thin-ner ﬁlaments, nanotubes (NT)and nanowires (NW). This is largely due to their electronic properties and the advances insynthetic methods, both driven by further device miniaturiza-tion for nano and molecular electronics. 1–3While carbon can form narrow NT cylinders,4another critical element, silicon, so far, could only be produced5as NW, but not in tubular form. Calculations show that even though SiNT might cor-respond to local minima, 6they cannot sustain perturbations and collapse into sp3aggregates. Moreover, sp3-bulk-based NW thinner than 1.2 nm lack stability.7 Compelled to stabilize the thinnest SiNT, we “propped” them up by placing metal (M)inside. Here we present such metal-endohedral silicon nanotubes (M@SiNT ), isomorphic to rescaled carbon tubules of (2,2)and(3,0)symmetry. Com- putations for a series of metals show that M@SiNT arestable, and have substantial cohesive energies sE cdand elas- tic moduli. They appear to be the thinnest one-dimensional (1D)silicon forms, since other reconstructions always lead to thicker wires. Electronic densities of states (DOS )show no band gap, which implies good conductivity. Plotting cohe-sive energies as a function of stoichiometry (molar fraction x)permits formation energy comparison with other known M xSi1−xstructures at different conditions. Finally, there is a remarkable correspondence between these thinnestM-endohedral nanotubes and the synthesized 8–11subnanom- eter disilicide wires. The structures were fully optimized within unrestricted density functional theory (DFT )with the periodic boundary conditions (PBC )algorithm12ofGAUSSIAN 03 .13We used the gradient-corrected functional of Perdew, Burke, andErnzerhof 14(PBE)and the 3–21 G Gaussian basis set to represent both the valence and core orbitals. We veriﬁed thatthis framework, previously used for 1D [carbon NT (CNT ) and boron nitride NT (BNNT )] 15,16and three-dimensional (3D)sUO2d17systems, gives for bulk silicon a bond length of 2.38 Å and Ec(Ref. 18 )of 4.64 eV, in agreement with experiments.19 Pure SiNT of zigzag (3,0)and armchair (2,2)types were unstable in our calculations but both could be underpinnedby M atoms inserted along NT’s axis. In the (3,0)case, M atoms were placed between consecutive zigzag motif rings.For the (2,2)SiNT, M atoms were put at the centers of every other Si rectangle, as shown in Fig. 1. Consequently, we usedin our PBC calculations unit cells with stoichiometry M 2Si12 and MSi 8, for the (3,0)and(2,2)SiNT, respectively. Conver- gence was achieved by employing between 116 and 178 k points. The optimized geometries of Fig. 1 (left), where M=Zr, demonstrate that the reinforcement by internal M-Si bondingstabilizes both zigzag and armchair SiNT topologies. Bothpossess large E cvalues of 4.34 eV for the (3,0)and 4.26 eV for(2,2)Zr@SiNT, and stiffness around 200 GPa (see be- low). We explored other M choices, such as the 3 delements Sc, Ti, Cr, Fe, and Ni, and the alkaline earth Be and Ca. Theperformed optimizations indicated that the (3,0)and (2,2) SiNT cages can be stabilized with M from different groupsof the Periodic Table. The E cvalues (Table I )show a slight increase with the group number. As the M’s elemental radiidecreases with the group number, 19the relaxed M-Si bonds get shorter. E.g., in the (2,2)series there is a 7% bond-length decrease, from lCauSi=2.92 Å to lTiuSi=2.72 Å. This bond- ing effect appears as a limiting factor for the M choices, asl MuSican become too short to stabilize the SiNT cage. Cal- culations identify Cr and Ti as the limiting 3 dmetals for the (2,2)and(3,0)series, respectively. To ensure that stability is retained at ﬁnite length as well, we considered the termination caps shown in Fig. 1 (right ). For the (3,0), an additional Si atom was placed on the axis to reduce the number of dangling bonds. For the (2,2)no addi- tional atoms were needed, as the end atoms reconstruct natu-rally to form a square cap. Computations performed for clus- FIG. 1. (Color online )The thinnest (3,0)and(2,2)Si nanotubes are stabilized by endohedral metals (smaller ball ). The inﬁnite [axial and side views (left)]and end-cap (right )structures were optimized for M=Zr at the PBE /3–21G level.PHYSICAL REVIEW B 70, 241303 (R)(2004 )RAPID COMMUNICATIONS 1098-0121/2004/70 (24)/241303 (4)/$22.50 ©2004 The American Physical Society 241303-1ters Zr 3Si28for armchair and Zr 4Si32for zigzag lead to stable conﬁgurations, with large Ec’s of 4.09 and 4.12 eV, and highest-occupied molecular orbital (HOMO )-lowest unoccu- pied molecular orbital (LUMO )gaps of 0.95 and 0.60 eV, respectively. Thus, both NT types are stable not only as in-ﬁnite tubes, but also can sustain the intrinsic strain (surface tension )associated with the tip ends. To thoroughly investigate the conﬁgurational vicinity of our M@SINTs, other nearly 1D structures were considered.Starting from the (3,0)and(2,2)structures, Fig. 2 (a)sketches the lattice changes that lead, through DFT optimizations, tothe new 1D stable structures shown in Fig. 2 (b). For the (3,0) M@SiNT, the three hexagons were transformed into six sur-face rectangles. Analogous to the hexagon-lattice wrappingindexing,welabelthistubeas [6,0],wherethesquarebrack- ets stand for rectangular surface units. Further, we noticedthat a half-period axial shift of the M chain leads to a previ-ously proposed 1D structure, 20conﬁrmed here as stable and labeled f6,0g8. For the (2,2)shell, the alternative longitudi- nal shifts of the zigzag motifs accompanied by the bonding of Si atoms 2 and 5 in Fig. 2 (a), lead to a [4,4]NT. Next, the hexagonal wall pattern can be regained in the (4,0)zigzag orientation through alternating circumferential shifts. TheEcvalues for all wires, stabilized with different M choices, are plotted in Fig. 3 as a function of optimized unitcell lengths l. Clearly, the (2,2)and(3,0)NTs emerge as thelongest [or thinnest (of smallest diameter d)]NTs within their stoichiometric families. When compared with recon- structed NTs, the energy differences appear notably low, inspite of an obviously larger energy contribution of speciﬁc surface, which scales as ˛lor 1/d, and therefore favors shorter and thicker types (and ultimately of course favors bulk material over any ﬁlaments ). For instance, for M=Zr we obtained that the [6,0]structure is by 12% shorter than (3,0)NT, but its Ecis only 1% larger. By separately comput- ing the energies of the Si and Zr subsystems we could dividethis energy difference into separate contributions over thewhole Zr 2Si12unit cell. While the binding strengthens within both the Si cage (by 2.3 eV )and internal Zr chain (by 1.3 eV, as lZruZrshrinks from 2.94 to 2.59 Å, the embed- ding of the Zr chain into the Si cage (a measure of Zr uSi binding )decreases from 15.6 to 12.8 eV. Next, along this family we found that the internal M-chain shift by half pe-riod into f6,0g 8is unfavorable, and Ecdecreases to 4.16 eV for M=Zr. Turning now to the (2,2)M@SiNT, the transfor- mation into [4,4]is uphill, in spite of 12% length shrinkage; further, (4,0)M@SiNT is the lowest in energy for all con- sidered metals, but it is 19% shorter than the initial (2,2).W e have attempted other transformation possibilities besides theones shown in Fig. 2, which did not, however, lead to stableTABLE I. Cohesive energies Ec(eV/atom )for the (2,2)and (3,0)M@SiNT families. Metal: Be Ca Sc Ti Zr Cr (3,0)3.74 3.11 4.05 4.24 4.34 ﬂ [6,0]3.74 3.18 4.10 4.33 4.39 4.49 f6,0g83.71 2.90 3.85 4.11 4.16 4.36 (2,2)3.65 3.34 4.02 4.15 4.26 4.13 [4,4]ﬂ3.19 3.97 ﬂ4.23 ﬂ (4,0)ﬂ3.59 4.12 ﬂ4.30 ﬂ FIG. 2. (Color online )(a)Pos- sible atomic rearrangements, in2D geodesic projection: s3,0d !f6,0g!f6,0g 8and s2,2d !f4,4g!s4,0d. Arrows indicate collective displacements of same-colored atoms. Dotted lines and“;” denote the incipient and breaking bonds. (b)Axial and side views of actual reconstructed NT:“sharp pencil” [6,0], “blunt pen- cil”f6,0g 8(also discussed in Ref. 20),1 Df c c [4,4], and zigzag (4,0). FIG. 3. (Color online )Cohesive energy Ecvs unit-cell length l for the two MSi 8and M 2Si12NT families. Colors and polygons represent different M-s and structures, respectively.DUMITRIC ˆ, HUA, AND YAKOBSON PHYSICAL REVIEW B 70, 241303 (R)(2004 )RAPID COMMUNICATIONS 241303-21D structures. For example, bonding the Si atoms in the 1 and 4 positions [Fig. 2 (a)]of the (2,2)cage could lead to a [4,0]M@SiNT; our calculations proved it unstable, as also suggested by cluster analysis.21 Considering the length-changing transformations of Fig. 3, one can conjecture if they could be induced by appliedforceF. For example, to evaluate the tension required for the f6,0g!s3,0dtransformation, we calculated the energy ver- suselasticelongation curves.We obtained that both [6,0]and (3,0)Zr@SiNT are quite stiff, with the Young’s moduli of 220 and 160 GPa, respectively (assuming cross-sectional ar- eas as 41 and 38 Å 2, to include Si radii ). Thermodynami- cally, the transformation occurs at critical force Ftwhen the enthalpies H=E−Flof the two NTs are equal. The tension estimateFt=dE/dl(where dE=0.79 eV and dl=0.71 Å are the energy and unit-cell length differences between thestress-free phases 16)yields 1.1 eV/Å, which corresponds to a small 2% strain, suggesting that such transition is viable. While primarily focused on structural stability, our calcu- lations also provided the M@SiNT electronic characteriza-tion. Taking into account 128 kpoints, Fig. 4 presents the band structure and the DOS of (3,0)Zr@SiNT. The DOS shows a series of van Hove peaks and maintains a nonzerovalue at the Fermi level E F=−5.54 eV due the contribution of four electronic bands. Further insight is gained by sepa-rately analyzing the projection of the total DOS on the metal chain and SiNT structure. One notices a dominant contribu-tion atE Ffrom the silicon shell, which explains the metallic character of all other M@SiNT including for M=Cr (an in- direct low band-gap semiconductor in the CrSi 2bulk form ). DOS analysis for (2,2)Zr@SiNT revealed similar metallic behavior at EF=−5.24 eV. The possibility of Peierls distor- tion and a small gap opening is not excluded by this analysis. For a broader perspective, MSi 6and MSi 8structures dis- cussed above might be compared with previously reportedZr@Si 16[Ec=4.16 eV (Ref. 22 )]and Zr@Si12[Ec=3.4 eV(Ref. 23 )]clusters, with 5-Å-narrow (4,0)pure-Si tubules [Ec=3.75 eV (Ref. 6 )], and MSi 5pentagonal wire.24For the latter, our DFT calculations conﬁrmed its stability, with Ec =4.15 eV for M=Zr. However, Ec’s computed per “average” atom bear little signiﬁcance for structures of different com-positions.Their relative stability depends on the constituents’chemical potentials, mMandmSi, which in turn represent en- vironmental conditions. To account for this, we follow theapproach customary in binary phase thermodynamics and de-ﬁne a molar (per atom )Gibbs free energy of formation dG for composition M xSi1−x,a s dGsxd=−Ecsxd−xmM−s1−xdmSi, s1d where the Ecterm neglects thermal contribution for the solid phase. Accordingly, Fig. 5 plots the Ec’s of all nanostruc- tures, along with the values for the bulk Si, Zr, and disilicide FIG. 6. (Color online )Bottom part shows the axial (left)and side (right )views of a two-monolayer-high ScSi 2-nanowire grown in the [110]direction of the Si substrate (Refs. 9 and 10 ). Following the arrows, the upper left shows a magniﬁed detail of the framedportion, which upon detachment and bottom dimerization (thick horizontal arrows )leads to the unsupported (2,2)Sc@SiNT (upper right ). FIG. 4. Electronic band structure (left)and density of states (right )of the (3,0)Zr@SiNT. Squares (circles )mark single (doubly degenerate )Fermi-level crossings. The gray and thin black lines are the projection of the total DOS (thicker black line )on the Zr and Si atoms, respectively. FIG. 5. Cohesive energies Ecfor the (2,2)ZrSi8NT(j)and (3,0)ZrSi6NT(m)plotted as a function of Zr fraction x. For a broader comparison we included the ZrSi 5NT of Ref. 24 and the pure (4,0)SiNT of Ref. 6, along with the clusters ZrSi 16of Ref. 22 and ZrSi 12of Ref. 23. The bulk − Ecvalues for Si, ZrSi 2, and Zr are also shown at x=0,1/3,and1.ENDOHEDRAL SILICON NANOTUBES AS THINNEST PHYSICAL REVIEW B 70, 241303 (R)(2004 )RAPID COMMUNICATIONS 241303-3ZrSi2, as a function of Zr molar fraction 0 ,x,1. This al- lows one to conveniently compare the thermodynamics oftherangeof M-Si binary structures, based on the altitude of each −E csxdpoint from the line connecting the reference mSi andmMvalues at x=0 andx=1. For instance, when choosing the constituent chemical potentials in Fig. 5 at the bulk val- ues mSi=mSibulk=−EcfSigand mZr=mZrbulk=−EcfZrg, crystal ZrSi2appears thermodynamically stable. Indeed, the straight line of slope mZrbulk−mSibulkclears above the ZrSi 2point, with dGs1/3d=−0.56 eV.25On the other hand, all nanostructures appear above this line and are metastable (mainly due to great excessive surface ), while dGstill characterizes their relative stability. The (3,0)Zr@SiNT [ZrSi6with dGs1/7d =0.52 eV ]appears to be slightly better than the (2,2) Zr@SiNT [ZrSi8with dGs1/9d=0.54 eV ], and both of these proposed Zr@SiNT remain more favorable than other clus- ters, than the ZrSi 5NT of Ref. 24 fdGs1/6d=0.74 eV g,o r the pure (4,0)-SiNT of Ref. 6 fdGs0d=0.93 eV g. After establishing the thermodynamic advantage of the proposed M@SiNTs, a connection can be made with experi-ment. At a ﬁrst glance, M@SiNT have stoichiometries (x =1/7 and x=1/9 )very different from disilicide sx=1/3 dor the disilicide nanowires 9,10synthesized recently by con- trolled deposition.8As wires become thinner, their formal composition M xSi1−xchanges towards Si (merely due to Si termination of the exterior ). Structurally, Fig. 6 shows a schematic for such Sc-silicide nanowire with x=1/5grown in the [110]direction on Si substrate. The wire’s top surfaceexhibits usual dimerization. Notably, the framed portion has exactx=1/9 Sc fraction and, if lifted off, would make ex- actly a freestanding (2,2)Sc@SiNT (upper right ).(Similarly, the(3,0)Sc@SiNT can be viewed as a cut from a hexagonal silicide nanowire grown in perpendicular f110gdirection. ) Formally, it shows identity of the introduced here “metal- endohedral nanotubes” with thinnest silicide wires, likelyprecursors of experimentally observed thicker types. Further,although the synthesis process is nonequilibrium, a thermo-dynamic analysis is still instructive: to adjust to the M-depleted conditions, 10themZrvalue in Fig. 5 must be lowered. Under such steeper slope mZrgas−mSibulk, the (2,2) Zr@SiNT sZrSi8dappears as the most favorable nanowire. [E.g., with mZrgas=−6.5 eV for ideal gas at T=1200 °C and p=10−10Torr, its dGs1/9d=0.57 eV. ] In summary, metal-endohedral silicon nanotubes M@SiNT are shown to be stable, yet structurally versatile.Within the stoichiometric families, armchair (2,2)and zigzag (3,0)are the thinnest structures. Unexpected morphological similarity with the thicker disilicide nanowires grown onsubstrate, makes these conducting ﬁlaments of ,0.3 nm ra- dii the realistic miniaturization limit for Si-based electronicjunctions. We acknowledge support of the RobertA. Welch Founda- tion,Air Force Research Laboratory, and the NSF MRI GrantNo. EIA0116289. We thank R.S. Williams for stimulatingdiscussion, and K.N. Kudin and I. 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Lett. 89, 266402 (2002 ). 18For a material with stoichiometry MmSin, the cohesive energy Ec (per atom )is deﬁned as − Ec=sEfMmSing−mEfMg−nEfSigd/sm +nd, whereEfMgandEfSigare the energies of isolated atoms. 19See, for example, C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996 ). 20A. K. Singh, V. Kumar, T. M. Briere, and Y. Kawazoe, Nano Lett. 2, 1243 (2002 ); A. K. Singh, T. M. Briere, V. Kumar, and Y. Kawazoe, Phys. Rev. Lett. 91, 146802 (2003 ). 21T. Miyazaki, H. Hiura, and T. Kanayama, Phys. Rev. B 66, 121403 (R)(2002 ). 22V. Kumar and Y. Kawazoe, Phys. Rev. Lett. 87, 045503 (2001 ). 23J. Lu and S. Nagase, Phys. Rev. Lett. 90, 115506 (2003 ). 24M. Menon, A. N. Andreotis, and G. Froudakis, Nano Lett. 2, 301 (2002 ). 25C. J. Först, P. E. Blöchl, and K. Schwarz, Comput. Mater. Sci. 27, 1(2003 ); C. J. Först (private communication ).DUMITRIC ˆ, HUA, AND YAKOBSON PHYSICAL REVIEW B 70, 241303 (R)(2004 )RAPID COMMUNICATIONS 241303-4
PhysRevB.74.121402.pdf	g-factors and discrete energy level velocities in nanoparticles Eduardo R. Mucciolo,1Caio H. Lewenkopf,2and Leonid I. Glazman3 1Department of Physics, University of Central Florida, P .O. Box 162385, Orlando, Florida 32816-2385, USA 2Instituto de Física, Universidade do Estado do Rio de Janeiro, R. São Francisco Xavier 524, 20550-900 Rio de Janeiro, Brazil 3Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA /H20849Received 21 July 2006; revised manuscript received 16 August 2006; published 15 September 2006 /H20850 We establish relations between the statistics of gfactors and the ﬂuctuations of energy in metallic nanopar- ticles where spin-orbit coupling is present. These relations assume that the electron dynamics in the grain ischaotic. The expressions we provide connect the second moment of the gfactor to the root-mean square “level velocity” /H20849the derivative of the energy with respect to magnetic ﬁeld /H20850calculated at magnetic ﬁelds larger than a characteristic correlation ﬁeld. Our predictions relate readily observable quantities and allow for a parameter-free comparison with experiments. DOI: 10.1103/PhysRevB.74.121402 PACS number /H20849s/H20850: 73.23.Hk, 71.70.Ej It was noted in experiments1,2that the Zeeman splitting of discrete energy levels in nanoparticles is very sensitive to thepresence of spin-orbit interaction. The splitting can be char-acterized by a level-dependent gfactor. Adding to the Al grains only 4% of Au resulted in a change of measured g factors from approximately 1.7 to 0.7. In addition to beingsuppressed, the gfactor in the presence of spin-orbit interac- tion also ﬂuctuates randomly from level to level. These ﬂuctuations of gfactors were described in the framework of random matrix theory 3–5/H20849RMT /H20850and the sup- pression of the gfactor was related to the strength of the spin-orbit interaction and to the elastic mean free path ofelectrons in the grains. 4The limit of strong spin-orbit inter- action corresponds to a short spin-orbit scattering time, /H9270so/H9254//H6036/H112701, where /H9254−1is the mean density of states in the grain at zero ﬁeld. It was predicted that the distribution ofeigenvalues of the g-factor tensor in this limit should have a Gaussian form. Within RMT, this distribution is character-ized by a phenomenological parameter /H20855g 2/H20856. In Ref. 4, this parameter was expressed in terms of the grain size, electron mean free path l, and relaxation time /H9270so. However, a com- parison of the experimental results with theory was not en-tirely satisfactory. On the one hand, there is an indication thatthe distributions of eigenvalues and eigenvector directions oftheg-factor tensor are Gaussian and correspond to a “pure” symplectic ensemble. 6,7On the other hand, it is not clear6 whether the small values of /H20855g2/H20856obtained in the experiments agree with the theoretical values estimated in Ref. 4. The difﬁculty in making the comparison comes from the lack ofinformation about the amount of disorder in the grain. The goal of this work is to provide three relations for the distribution width /H20855g 2/H20856to other quantities that are directly measured in the same set of experiments /H20849and thus do not rely on any additional information about the amount of dis-order in the grains /H20850. These quantities are the variance of the energy level derivative with respect to the magnetic ﬁeld/H20849known as level velocity 8/H20850and the zero-magnetic ﬁeld level curvature /H20849the second derivative of the energy level with re- spect to magnetic ﬁeld /H20850. We begin by stating our main results. Our ﬁrst expression, valid for strong spin-orbit coupling only, /H9270so/H9254//H6036/lessmuch1, is/H20855g2/H20856=12 /H9262B2var/H20875/H20873d/H9255/H9263 dB/H20874 B/greatermuchB/H20876 /H9270so→0, /H208491/H20850 where var /H20851/H20852denotes the variance, /H9262Bis the Bohr magneton, andBis the crossover ﬁeld for breaking time-reversal sym- metry. Equation /H208491/H20850gives a statistical connection between the response at B→0/H20849thegfactor /H20850to that at large magnetic ﬁelds /H20849the level velocity /H20850. It provides a way to check experi- mentally if the grains exhibiting ﬂuctuations of the gfactor indeed belong to a “pure” symplectic ensemble, rather thanto an ensemble describing the crossover between the or-thogonal and symplectic limits. Provided that one collectsdata for the dispersion of energy levels over a sufﬁcientlylarge range of magnetic ﬁelds, the terms on both sides of Eq./H208491/H20850can be found independently using the same data set. This expression is universal and contains no microscopic or ma-terials parameters. The second expression we ﬁnd relates properties of two sets of grains which are equivalent macroscopically exceptfor the value of /H9270so. It reads /H20855g2/H20856=3g02 2/H9266/H6036/H9270so/H9254+3 2/H9262B2var/H20875/H20858 /H9268=↑,↓/H20873d/H9255n/H9268 dB/H20874 B/greatermuchB/H20876 /H9270so→/H11009, /H208492/H20850 where g0denotes the materials bulk value for the gfactor /H20849g0=2 for free electrons /H20850.I nE q . /H208492/H20850,/H20855g2/H20856is evaluated in the strong-spin-orbit coupling limit /H20849/H9270so→0/H20850. However, the sec- ond term on the right-hand side is evaluated for /H9270so→/H11009/H20849ab- sence of spin-orbit coupling /H20850. For example, one may con- sider two sets of Al:Au grains, one with no doping andanother with moderate doping. 1 The two terms on the right-hand side of Eq. /H208492/H20850come from distinct contributions. The ﬁrst term, the “spin part,” isassociated with the Debye mechanism of energy dissipationassociated with spin reorientations; the second one, the “or-bital part,” is due to the eddy currents induced in the grain.We note that the spin-orbit scattering rate 1/ /H9270soapparently may be estimated for a given material or host-dopant pair.6 Equation /H208492/H20850permits us to separate the spin and orbital con- tributions to the ﬂuctuations of the gfactor.PHYSICAL REVIEW B 74, 121402 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 1098-0121/2006/74 /H2084912/H20850/121402 /H208494/H20850 ©2006 The American Physical Society 121402-1The second moment of the gfactor can also be related to the statistics of other spectral quantities, such as the zero-ﬁeld level curvature /H20855g 2/H20856=9/H9254 2/H208812/H9262B2/H20883/H20879/H20873d2/H9255/H9263 dB2/H20874 B=0/H20879/H20884. /H208493/H20850 As Eq. /H208491/H20850, this expression is applicable in the strong-spin- orbit coupling regime only /H20849symplectic ensemble /H20850. It is important to note that for a given metallic grain the g-factor tensor may have three different eigenvalues, even for grains that are statistically isotropic.3,7However, the dis- tribution of the matrix elements of the g-factor tensor is still characterized by a single quantity /H20855g2/H20856. Throughout our manuscript, when establishing relations between /H20855g2/H20856and other quantities, we consider a magnetic ﬁeld applied in some ﬁxed but arbitrary direction. With these relations, onemay use Ref. 3to construct the full statistics of the g-factor tensor. We will now establish Eqs. /H208491/H20850–/H208493/H20850. The main idea behind the derivations is to relate statistical quantities to invariantsof the system, such as the traces of the magnetic momentoperator. For that purpose, let us begin by writing the Hamil-tonian for the disordered /H20849or chaotic /H20850grain in the presence of an applied magnetic ﬁeld as Hˆ/H20849B/H20850=Hˆ 0+BMˆ, where the mag- netic moment operator has both orbital and spin parts: Mˆ =Mˆorb+Mˆspin. To simplify the discussion, we assume that the grain is isotropic. We deﬁne the gfactor of the nth energy level as gn/H110131 2/H9262B/H20879/H20873d/H9255n/H9268 dB/H20874 B=0/H20879, /H208494/H20850 where /H20853/H9255n/H9268/H20854are the eigenvalues of Hˆ0. Note that due to Kramers degeneracy at B=0, the levels are twofold degener- ate. We use the index /H9268to distinguish states that are time- reversal conjugate. The matrix elements of Hˆ0follow either the symplectic /H20849/H9252=4/H20850or orthogonal /H20849/H9252=1/H20850ensemble statis- tics, depending on whether spin-orbit coupling is present or absent, respectively. For both cases, the matrix elements of Mˆ, when expressed in the eigenbasis /H20853/H20841n/H9268;0/H20856/H20854ofHˆ0, ﬂuctu- ate according to a Gaussian distribution with zero mean. For the symplectic ensemble, the variance of the diagonal matrixelements reads 4 /H20855/H20870/H20855n/H9268;0/H20841Mˆ/H20841n/H9268;0/H20856/H208702/H20856/H9252=4=3T r /H20849Mˆ2/H20850 4N2, /H208495/H20850 where the trace runs over the 2 Nstates in the band, with N/greatermuch1 being assumed /H20849the factor of 2 accounts for Kramers degeneracy /H20850. The degeneracy at zero ﬁeld allows us to pick a basis such that /H20855n/H9268;0/H20841Mˆ/H20841n/H9268/H11032;0/H20856is diagonal in the /H9268indices. Using Eq. /H208494/H20850and ﬁrst-order perturbation theory, we ﬁnd that /H20855n/H9268;0/H20841Mˆ/H20841n/H9268/H11032;0/H20856=/H20849−1/H20850/H9268/H9254/H9268/H9268/H11032gn/H9262B/2. Thus, from Eq. /H208495/H20850,w e arrive at9/H20855g2/H20856/H9252=4=3 /H9262B2Tr/H20849Mˆ2/H20850 N2. /H208496/H20850 Notice that quantities on the left-hand side of Eqs. /H208494/H20850,/H208495/H20850, and /H208496/H20850are deﬁned at B=0. Now we have to write the statistical quantities that appear on the right-hand side of Eqs. /H208491/H20850–/H208493/H20850, in terms of Tr /H20849Mˆ2/H20850. Note that the latter is an invariant and therefore takes the same value at zero or large magnetic ﬁelds. Let us ﬁrst consider Eq. /H208491/H20850. The variance of the level velocity can be computed in terms of the variance of thematrix elements of the magnetic moment operator since /H20873d/H9255/H9263 dB/H20874 B=B0=/H20855/H9263;B0/H20841Mˆ/H20841/H9263;B0/H20856. /H208497/H20850 For a sufﬁciently large magnetic ﬁeld B0/greatermuchB, time-reversal symmetry in the grain is broken. In the presence of strong-spin-orbit scattering, orbital and spin degrees of freedom re-main mixed, but the ensemble statistics of the Hamiltonianeigenstates switches from 2 N/H110032Nsymplectic to 2 N/H110032N unitary /H20849 /H9252=2/H20850. Thus, we need to compute the variance of the matrix elements of the magnetic moment operator in the uni- tary regime. For this purpose, we make use of the eigenval-ues and eigenvectors of the magnetic moment operator: Mˆ/H20841k /H9251/H20856=/H20849−1/H20850/H9251Mk/H20841k/H9251/H20856, with k=1,..., Nand/H9251= ±1 due to the time-reversal properties of Mˆ.10This yields /H20855/H9263;B0/H20841Mˆ/H20841/H9263;B0/H20856=/H20858 k,/H9251/H20849−1 /H20850/H9251Mk/H20870/H20855/H9263;B0/H20841k/H9251/H20856/H208702. /H208498/H20850 For the unitary ensemble in the large- Nlimit, the eigenvector amplitudes shown on the right-hand side of Eq. /H208498/H20850ﬂuctuate independently according to the Porter-Thomas distribution.11 One ﬁnds that /H20855/H20870/H20855/H9263;B0/H20841k/H9251/H20856/H208702/H20856=1 2Nand /H20855/H20870/H20855/H9263;B0/H20841k/H9251/H20856/H208704/H20856=1 2N2, independently of state indices. Hence, the average matrix element of Mˆmust vanish and the variance can be written as var /H20851/H20855/H9263;B0/H20841Mˆ/H20841/H9263;B0/H20856/H9270so→0,/H9252=2/H20852=Tr/H20849Mˆ2/H20850 4N2. /H208499/H20850 Putting together Eqs. /H208496/H20850,/H208497/H20850, and /H208499/H20850, we arrive at Eq. /H208491/H20850. To derive Eq. /H208492/H20850, we separate the magnetic moment in terms of spin and orbital parts, Mˆ=Mˆspin+Mˆorb, which are statistically independent from each other. From Eq. /H208496/H20850,w e obtain /H20855g2/H20856/H9252=4=3 /H9262B2N2/H20851Tr/H20849Mˆ spin2/H20850+T r /H20849Mˆ orb2/H20850/H20852. /H2084910/H20850 The spin contribution can be written in terms of the imagi- nary part of the ac spin susceptibility of a free electron gas inthe presence of spin-orbit coupling. 4The susceptibility can then be evaluated using conventional means of rate equationsat frequencies much larger than the mean level spacing in thegrain, yet smaller than the spin-orbit scattering rate. In thelimit of /H9270so/H9254//H6036/lessmuch1 one ﬁnds4MUCCIOLO, LEWENKOPF, AND GLAZMAN PHYSICAL REVIEW B 74, 121402 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 121402-2Tr/H20849Mˆ spin2/H20850 N2=g02/H9262B2 2/H9266/H6036/H9270so/H9254. /H2084911/H20850 This corresponds to the ﬁrst term on the right-hand side of Eq. /H208492/H20850. The orbital contribution in Eq. /H2084910/H20850is ensemble indepen- dent and is the same regardless of the presence or absence ofthe spin-orbit interaction. It is convenient to evaluate it in thelimit of /H9270so→/H11009to decouple spin and orbital degrees of free- dom. To relate the Tr /H20849Mˆ orb2/H20850to the variance of the velocity of spin-resolved levels at large ﬁelds, we can use Eq. /H208497/H20850and note that Mˆspinjust induces a constant slope in the dispersion of the energy levels with magnetic ﬁeld at /H9270so→/H11009, while all ﬂuctuations are caused by Mˆorb. Therefore, we write /H20858 /H9268=↑,↓/H20873d/H9255n/H9268 dB/H20874 B=B0=/H20855n↑;B0/H20841Mˆorb/H20841n↑;B0/H20856/H20849 12/H20850 to isolate the ﬂuctuating part /H20849note that B0/greatermuchB*here as well /H20850. It is important to observe that in the absence of spin-orbitmixing and for large magnetic ﬁelds the statistics of theeigenstate /H20841n↑;B 0/H20856corresponds to a N/H11003Nunitary ensemble /H20849rather than to 2 N/H110032Nwhen spin and orbital parts are strongly coupled /H20850. Similarly to Eq. /H208498/H20850, we can decompose the matrix elements in Eq. /H2084912/H20850using the eigenstates of Mˆorb. In this case, however, due to the change in the ensembledimension, we have /H20855/H20870/H20855n↑;B 0/H20841k/H9261/H20856/H208702/H20856=1/N and /H20855/H20870/H20855n↑;B0/H20841k/H9261/H20856/H208704/H20856=2/N2for large N, independently of the or- bital quantum number n. Finally, combining all these results, we arrive at var/H20875/H20858 /H9268=↑,↓/H20873d/H9255n/H9268 dB/H20874 B=B0/H20876 /H9270so→/H11009=2T r /H20849Mˆorb/H208502 N2. /H2084913/H20850 Inserting this expression into Eq. /H2084910/H20850, we obtain the second term on the right-hand side of /H208492/H20850. We remark that Eq. /H208492/H20850is fully consistent with Eq. /H2084931/H20850of Ref. 12, where a description of the statistical properties of the gfactor including the in- termediate crossover regime was developed in terms of phe-nomenological RMT parameters. In order to obtain Eq. /H208493/H20850we follow an approach similar to that employed in the two previous derivations. We deﬁnethe level curvature at B=0 as K n/H11013/H20873d2/H9255n/H9268 dB2/H20874 B=0=2/H20858 n/H11032/HS11005n/H20858 /H9268/H11032/H20870/H20855n/H9268;0/H20841Mˆ/H20841n/H11032/H9268/H11032;0/H20856/H208702 /H9255n−/H9255n/H11032. /H2084914/H20850 /H20849To simplify the notation, here we set /H9255n/H9268=/H9255n./H20850Since eigen- values and eigenfunctions ﬂuctuate independently in theGaussian ensembles, we ﬁnd that/H20855K n2/H20856/H9252=4=8 /H90042/H20851Š/H20870/H20855n/H9268;0/H20841Mˆ/H20841n/H11032/H9268/H11032;0/H20856/H208704‹ +Š/H20870/H20855n/H9268;0/H20841Mˆ/H20841n/H11032/H9268/H11032;0/H20856/H208702‹2/H20852, /H2084915/H20850 with n/HS11005n/H11032and/H9268,/H9268/H11032taking arbitrary values. The prefactor in Eq. /H2084915/H20850is deﬁned as 1 /H90042/H110132/H9254/H20858 n/H20858 n/H11032/HS11005n/H20883/H9254/H20849/H9255n/H20850 /H20849/H9255n−/H9255n/H11032/H208502/H20884, /H2084916/H20850 where the delta function is used to ﬁx the energy level in the middle of the band. The average of eigenvalues can be per-formed using the appropriate two-level cluster function. 13In the limit N→/H11009,w eﬁ n d /H208791 /H90042/H20879 /H9252=4=/H92662 9/H92542, /H2084917/H20850 with/H9254denoting the inverse of the mean density of states.14 The ensemble average of off-diagonal matrix elements of the magnetization is also easily computed in terms of the trace ofthe magnetization operator in the limit of large N: Š/H20870/H20855n /H9268;0/H20841Mˆ/H20841n/H11032/H9268/H11032;0/H20856/H208702‹n/HS11005n/H11032,/H9252=4=Tr/H20849Mˆ2/H20850 4N2, /H2084918/H20850 and Š/H20870/H20855n/H9268;0/H20841Mˆ/H20841n/H11032/H9268/H11032;0/H20856/H208704‹n/HS11005m,/H9252=4=1 8/H20875Tr/H20849Mˆ2/H20850 N2/H208762 . /H2084919/H20850 Inserting Eqs. /H2084917/H20850–/H2084919/H20850into /H2084915/H20850we arrive at /H20855Kn2/H20856/H9252=4=/H92662 6/H9254/H20875Tr/H20849Mˆ2/H20850 N2/H208762 . /H2084920/H20850 For the symplectic ensemble, one can show15,16that /H20881/H20855Kn2/H20856 =/H20849/H9266/H208813/4 /H20850/H20855/H20841Kn/H20841/H20856. Thus, using this relation and combining Eqs. /H208496/H20850and /H2084920/H20850, we obtain Eq. /H208493/H20850. Equation /H208493/H20850, like Eq. /H208491/H20850, also involves only quantities that are directly measurable. However, in practice, the difﬁ-culty in obtaining large statistics for the second derivative atB=0 from the tunneling conductance data makes it less ap- pealing when applied to experiments. 6,7 Once the variance of the level velocity is obtained from the experimental data, it may also allow for another test ofRMT. Consider the level velocity correlation function 8,17 C/H9262/H20849/H9004B/H20850=1 /H92542/H20875/H20883/H20873d/H9255/H9263 dB/H20874 B=B0+/H9004B/H20873d/H9255/H9263 dB/H20874 B=B0/H20884 −/H20883/H20873d/H9255/H9263 dB/H20874 B=B0/H208842/H20876. /H2084921/H20850 For a pure ensemble, this correlation function can be rescaled to a universal form. Deﬁning the correlation ﬁeld as Bc/H110131//H20881C/H9263/H208490/H20850/H20849 22/H20850 and calling x=/H9004B/Bcand c/H20849x/H20850=Bc2C/H9263/H20849/H9004B/H20850, the dimension- less correlation function in the unitary ensemble has the asymptotes18g-FACTORS AND DISCRETE ENERGY LEVEL ¼ PHYSICAL REVIEW B 74, 121402 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 121402-3c/H20849x/H20850=/H208771−2/H92662x2, x/lessmuch1, −1 / /H20849/H9266x/H208502, x/greatermuch1./H2084923/H20850 The full shape of the correlation function is known from numerical simulations,8,18as well as from analytical calculations.19 Finally, it is interesting to note that the correlation ﬁeld is related to the amount of disorder in the grains when theelectron motion is diffusive. It is straightforward to show thatat /H9270so→/H11009, Bc=/H9260/H90210/L2 /H20881kF2lL, /H2084924/H20850 where /H90210is the ﬂux quantum, Lis the grain linear size, kFis the Fermi wavelength, and /H9260is a dimensionless coefﬁcient that depends on the grain geometry. For a spherical shape, /H9260=3/H9266//H208492/H208815/H20850, in which case Lis the grain radius. By mea- suring the variance of the level velocity at large ﬁelds, one can obtain C/H9263/H208490/H20850and ﬁnd the experimental value of Bcfrom Eq. /H2084922/H20850. Using Eq. /H2084924/H20850, one can then get an independent estimate of the amount of disorder present in the grain. An-other approach is to ﬁt the universal curve 8,18c/H20849x/H20850to theexperimental data and obtain C/H9263/H208490/H20850as a ﬁtting parameter. In summary, we have shown that it is possible to relate the second moment of the gfactor of metallic nanoparticles with strong spin-orbit coupling to other spectral statistics of en-ergy levels without resorting to any microscopic parameter. Our results also show that it is possible to estimate the spinand orbital contributions to the ﬂuctuations of the gfactor by comparing data taken from nanoparticles doped and undoped with a heavy-element metal. We suggest that a ﬁtting of thedata to a universal, dimensionless level velocity correlationfunction may provide an additional test of the applicability ofrandom matrix theory to these systems and allow us to ex-tract information about the intragrain disorder. This work was supported in part by NSF Grants No. DMR 02-37296, No. DMR 04-39026 /H20849L.I.G. /H20850, and No. CCF 0523603 /H20849E.R.M. /H20850. C.H.L. acknowledges partial support in Brazil from CNPq, Instituto do Milênio de Nanotecnologia,and FAPERJ. E.R.M. acknowledges partial support from theInterdisciplinary Information Science and Technology Labo-ratory /H20849I 2Lab /H20850at UCF. We are grateful to Y. Fyodorov, J. Petta, and F. von Oppen for enlightening discussions. L.I.G.and E.R.M. thank the Instituto de Física at UERJ, Brazil, andthe Aspen Center for Physics for the hospitality. 1D. G. Salinas, S. Guéron, D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. B 60, 6137 /H208491990 /H20850. 2D. Davidovi ćand M. Tinkham, Phys. Rev. Lett. 83, 1644 /H208491999 /H20850. 3P. W. Brouwer, X. Waintal, and B. I. Halperin, Phys. Rev. Lett. 85, 369 /H208492000 /H20850. 4K. A. Matveev, L. I. Glazman, and A. I. Larkin, Phys. Rev. Lett. 85, 2789 /H208492000 /H20850. 5R. A. Serota, Solid State Commun. 117, 605 /H208492001 /H20850. 6J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 87, 266801 /H208492001 /H20850. 7J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 89, 156802 /H208492002 /H20850. 8A. Szafer and B. L. Altshuler, Phys. Rev. Lett. 70, 587 /H208491993 /H20850. 9An analogous relation was derived for the variance of the persis- tent current of a mesoscopic ring in the presence of spin-orbitcoupling by V. E. Kravtsov and M. R. Zirnbauer, Phys. Rev. B 46, 4332 /H208491992 /H20850. 10The magnetic moment operator is antisymmetric with respect to time reversal. Therefore, eigenvectors related by time reversalhave eigenvalues equal in amplitude but with opposite signs. 11C. E. Porter, in Statistical Theories of Spectra: Fluctuations , ed- ited by C. E. Porter /H20849Academic Press, New York, 1965 /H20850. 12S. Adam, M. L. Polianski, X. Waintal, and P. W. Brouwer, Phys. Rev. B 66, 195412 /H208492002 /H20850. 13M. L. Mehta, Random Matrices , 3rd ed. /H20849Academic Press, Am- sterdam, 2004 /H20850. 14AtB=0, due to Kramers degeneracy, /H9254is equal to half of the mean level spacing. 15Y. V. Fyodorov and H.-J. Sommers, Z. Phys. B: Condens. Matter 99, 123 /H208491995 /H20850. 16F. von Oppen, Phys. Rev. E 51, 2647 /H208491995 /H20850. 17B. D. Simons and B. L. Altshuler, Phys. Rev. Lett. 70, 4063 /H208491993 /H20850. 18B. D. Simons and B. L. Altshuler, Phys. Rev. B 48, 5422 /H208491993 /H20850. 19I. E. Smolyarenko and B. D. Simons, J. Phys. A 36, 3551 /H208492003 /H20850.MUCCIOLO, LEWENKOPF, AND GLAZMAN PHYSICAL REVIEW B 74, 121402 /H20849R/H20850/H208492006 /H20850RAPID COMMUNICATIONS 121402-4
PhysRevB.76.035402.pdf	Magnetotransport and thermoelectricity in Landau-quantized disordered graphene Balázs Dóra * Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany Peter Thalmeier Max-Planck-Institut für Chemische Physik fester Stoffe, 01187 Dresden, Germany /H20849Received 1 February 2007; revised manuscript received 17 April 2007; published 5 July 2007 /H20850 We have studied the electric and thermal response of two-dimensional Dirac fermions in a quantizing magnetic ﬁeld in the presence of localized disorder. The electric and heat current operators in the presence ofmagnetic ﬁeld are derived. The self-energy due to impurities is calculated self-consistently and dependsstrongly on the frequency and ﬁeld strength, resulting in asymmetric peaks in the density of states at theLandau level energies, and small islands connecting them. The Shubnikov–de Haas oscillations remain peri-odic in 1/ B, in spite of the distinct quantization of quasiparticle orbits compared to normal metals. The Seebeck coefﬁcient depends strongly on the ﬁeld strength and orientation. For ﬁnite ﬁeld and chemical poten-tial, the Wiedemann-Franz law can be violated. DOI: 10.1103/PhysRevB.76.035402 PACS number /H20849s/H20850: 81.05.Uw, 71.10. /H11002w, 73.43.Qt I. INTRODUCTION Recent advances of nanotechnology have made the cre- ation and investigation of two-dimensional carbon, calledgraphene, possible. 1–4It is a monolayer of carbon atoms packed densely in a honeycomb structure. In spite of beingfew atoms thick, these systems were found to be stable andready for exploration. 5One of the most intriguing properties of graphene is that its charge carriers are well described bythe relativistic Dirac’s equation and are two-dimensionalmassless Dirac fermions. 6This opens the possibility of in- vestigating “relativistic” phenomena at a speed of /H11011106m/s /H20849the Fermi velocity of graphene /H20850, 1/300th the speed of light. The linear, Dirac-like spectrum causes the density of states toincrease linearly with energy, which is to be contrasted withthe constant density of states of normal metals. Due to thispeculiar property, the response of graphene to externalprobes is expected to be unusual. This manifests itself in theanomalous integer quantum Hall effect, 7which occurs at half-integer ﬁlling factors, and in the presence of universalminimal value of the conductance. The dependence of thethermal conductivity on applied magnetic ﬁeld has beenmeasured in highly oriented pyrolytic graphite. 8,9 Dirac fermions show up in other systems, at least from the theoretical side. They characterize the low-energy propertiesof orbital antiferromagnets, a density wave system with agap of d-wave symmetry. 10,11A similar model has been pro- posed for the pseudogap phase of high Tccuprate supercon- ductors, known as d-density wave, with peculiar electronic properties.12A similar system was also mentioned in the con- text of heavy fermion material URu 2Si2, which shows a clear phase transition at 17 K without any obvious long-range or-der, detectable by x-ray or NMR experiments. Its low-temperature phase was attributed to another spin-densitywave with a d-wave gap 13,14Experimentally, the aforemen- tioned materials possess unusual electric and thermal re-sponses as a function of temperature and magnetic ﬁeld. 15,16 Therefore, the interest in studying the transport properties of two-dimensional Dirac fermions is not surprising.Sharapov and co-workers have studied exhaustively 17–23theelectric and thermal responses of two-dimensional systems with linear energy spectrum, with special emphasis on theWiedemann-Franz law and magnetic oscillations. However,their self-energy due to scattering from impurities was notdetermined in a self-consistent manner but rather they as-sumed a constant, energy, magnetic ﬁeld, and temperature-independent scattering rate. Moreover, they completely ne-glected the real part of the self-energy, responsible for theshift of energy levels. Nevertheless, they derived beautifulanalytical formulas for the various transport coefﬁcients,which, although suffering from the above limitations, turnedout to be useful in explaining experiments. 7 Impurity scattering can be taken into account in the pres- ence of quantizing magnetic ﬁeld in the usual self-consistentway. 24This program has been carried out, among many others,25by Peres et al.26In their work, the full self- consistent Born approximation was used before taking thestrength of the impurity potential to inﬁnity. They studied thefrequency dependence of the electric conductivity for variousﬁelds but never entered into the realm of thermal transport.Parallel studies have also been performed in the limit ofweak scatterers. 27,28 In this paper, we extend the work of Refs. 17–20, and determine self-consistently the energy and magnetic-ﬁeld-dependent self-energies and study the Seebeck coefﬁcient aswell, and also generalize Ref. 26to include thermoelectricity. We study Dirac fermions in a Landau quantizing magneticﬁeld /H20849B/H20850in the presence of scatterers, allowing for arbitrary ﬁeld orientations. In a way, our study here bridges between the efforts of the previous groups. After the introduction ofthe general formalism, we determine the electric and heatcurrent operators, essential for further steps. By introducingimpurities in the system, we can study the quasiparticle den-sity of states, the electric and heat conductivity, the Seebeckcoefﬁcient, and the Wiedemann-Franz law as a function ofmagnetic-ﬁeld strength and orientation and temperature. Forhigh ﬁelds, the discrete nature of the Landau levels is re-vealed in the density of states in the form of asymmetricpeaks at Landau level energies /H20849far from being Lorentzians /H20850,PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 1098-0121/2007/76 /H208493/H20850/035402 /H208499/H20850 ©2007 The American Physical Society 035402-1which smoothen with decreasing ﬁeld. Shubnikov–de Haas oscillations are visible in all transport coefﬁcients, periodicin 1/ B, similar to normal metals. 29The angular-dependent conductivity oscillations become more pronounced with in-creasing ﬁeld. The chemical-potential dependence of theconductivity resembles closely the experimental ﬁndings. 7 The Seebeck coefﬁcient depends strongly on the appliedﬁeld and temperature. II. LANDAU QUANTIZATION, AND ELECTRIC AND HEAT CURRENTS The Hamiltonian of noninteracting quasiparticles living on a single graphene sheet is given by26,30,31 H0=−vF/H20858 j=x,y/H9268j/H20851−i/H11509j+eAj/H20849r/H20850/H20852, /H208491/H20850 where /H9268j’s are the Pauli matrices and stand for Bloch states residing on the two different sublattices of the bipartite hex-agonal lattice of graphene. 19,26Strictly speaking, the Hamil- tonian above describes quasiparticles around the Kpoints of the Brillouin zone, where the spectrum vanishes. The vectorpotential for a constant, arbitrarily oriented magneticﬁeld reads as A/H20849r/H20850=/H20851−Bycos /H9258,0,B/H20849ysin/H9258cos/H9278 −xsin/H9258sin/H9278/H20850/H20852, where /H9258is the angle the magnetic ﬁeld makes from the zaxis, and /H9278is the in-plane polar angle measured from the xaxis. We have dropped the Zeeman term, its energy would be negligible with respect to energy ofthe Landau levels, Eq. /H208495/H20850, using vF/H11015106m/s, characteristic to graphene. Equation /H208491/H20850applies for both spin directions. In the absence of magnetic ﬁeld, the energy spectrum of the system is given by E/H20849k/H20850=±vF/H20841k/H20841. /H208492/H20850 This describes massless relativistic fermions with spectrum consisting of two cones, touching each other at the endpoints. From this, the density of states per spin follows as/H9267/H20849/H9275/H20850=1 /H9266/H20858 k/H9254/H20851/H9275−E/H20849k/H20850/H20852=1 /H9266Ac 2/H9266/H20885 0kc kdk/H9254/H20849/H9275±vFk/H20850=2/H20841/H9275/H20841 D2, /H208493/H20850 where kcis the cutoff, D=vFkcis the bandwidth, and Ac =4/H9266/kc2is the area of the hexagonal unit cell. In the presence of magnetic ﬁeld, the eigenvalue problem of this Hamiltonian /H20849H0/H9023=E/H9023/H20850can readily be solved.26For the zero energy mode /H20849E=0/H20850, the eigenfunction is obtained as /H9023k/H20849r/H20850=eikx /H20881L/H208750 /H92780/H20849y−klB2/H20850/H20876, /H208494/H20850 and the two components of the spinor describe the two bands. The energy of the other modes reads as E/H20849n,/H9251/H20850=/H9251/H9275c/H20881n+1 , /H208495/H20850 with/H9251=±1 , n=0,1,2,..., /H20849Fig. 1/H20850/H9275c=vF/H208812e/H20841Bcos/H20849/H9258/H20850/H20841is the Landau scale or energy but is different from the cyclotron frequency.32Only the perpendicular component of the ﬁeld enters into these expressions, and by tilting the ﬁeld awayfrom the perpendicular direction corresponds to a smallereffective ﬁeld. The sum over integer n’s is cut off at Ngiven byN+1= /H20849D/ /H9275c/H208502, which means that we consider 2 N+3 Landau levels altogether. For later convenience, we deﬁne a magnetic ﬁeld B0, whose Landau scale is equal to the band- width /H20849/H9275c=D/H20850. The corresponding wave function is /H9023n,k,/H9251/H20849r/H20850=eikx /H208812L/H20875/H9278n/H20849y−klB2/H20850 /H9251/H9278 n+1/H20849y−klB2/H20850/H20876, /H208496/H20850 with cyclotron length lb=1//H20881eB. Here, /H9278n/H20849x/H20850is the nth eigenfunction of the usual one-dimensional harmonic oscil- lator. The electron-ﬁeld operator can be built up from thesefunctions as /H9023/H20849r/H20850=/H20858 k/H20875/H9023k/H20849r/H20850ck+/H20858 n,/H9251/H9023n,k,/H9251ck,n,/H9251/H20876. /H208497/H20850 The Green’s functions of these new operators do not depend onkand read as G0/H20849i/H9275n,k/H20850=1 i/H9275n, /H208498/H20850 G0/H20849i/H9275n,k,n,/H9251/H20850=1 i/H9275n−E/H20849n,/H9251/H20850, /H208499/H20850 forckandck,n,/H9251, respectively, and /H9275nis the fermionic Mat- subara frequency. With the use of these, we can determine the electric and heat current operators of the system. Following Mahan,33we deﬁne the polarization operator asE=0 FIG. 1. /H20849Color online /H20850The structure of the Landau levels is visualized schematically for the ﬁrst few levels.BALÁZS DÓRA AND PETER THALMEIER PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-2P=1 2/H20885dr/H20851r/H9267/H20849r/H20850+/H9267/H20849r/H20850r/H20852, /H2084910/H20850 with/H9267/H20849r/H20850=/H9023+/H20849r/H20850/H9023/H20849r/H20850giving the charge density, and the symmetric combination ensures hermiticity. The total current is its time derivative, which follows as J=/H11509tP=i/H20851H,P/H20852. /H2084911/H20850 By performing the necessary steps, after straightforward cal- culations, this yields26 Jx=vFe/H20858 p,/H9251/H208751 /H208812/H20849cp+cp,0,/H9251+cp,0,/H9251+cp/H20850 +/H20858 n,/H9261/H9261 2/H20849cp,n+1,/H9251+cp,n,/H9261+cp,n,/H9261+cp,n+1,/H9251/H20850/H20876, /H2084912/H20850 Jy=ivFe/H20858 p,/H9251/H208751 /H208812/H20849cp+cp,0,/H9251−cp,0,/H9251+cp/H20850 +/H20858 n,/H9261/H9261 2/H20849cp,n,/H9261+cp,n+1,/H9251−cp,n+1,/H9251+cp,n,/H9261/H20850/H20876, /H2084913/H20850 where /H9261= ±1. The heat current operator for the pure system can be determined similarly. In analogy with polarization,one deﬁnes the energy position operator 34as RE=1 2/H20885dr/H20851rH/H20849r/H20850+H/H20849r/H20850r/H20852, /H2084914/H20850 and the total Hamiltonian is H=/H20848drH/H20849r/H20850. Using this, one deduces the energy current from JE=/H11509tRE. /H2084915/H20850 This leads to JxE=vF 2/H20858 p,/H9251/H20877E/H208490,/H9251/H20850 /H208812/H20849cp+cp,0,/H9251+cp,0,/H9251+cp/H20850+/H20858 n,/H9261/H9261 2/H20851E/H20849n+1 ,/H9251/H20850 +E/H20849n,/H9261/H20850/H20852/H20849cp,n+1,/H9251+cp,n,/H9261+cp,n,/H9261+cp,n+1,/H9251/H20850/H20878, /H2084916/H20850 JyE=ivF 2/H20858 p,/H9251/H20877E/H208490,/H9251/H20850 /H208812/H20849cp+cp,0,/H9251−cp,0,/H9251+cp/H20850+/H20858 n,/H9261/H9261 2/H20851E/H20849n+1 ,/H9251/H20850 +E/H20849n,/H9261/H20850/H20852/H20849cp,n,/H9261+cp,n+1,/H9251−cp,n+1,/H9251+cp,n,/H9261/H20850/H20878. /H2084917/H20850 These follow naturally from the electric current operator, af- ter multiplying each term with the corresponding mode en-ergy. Note that the energy of the state labeled solely by /H20849p/H20850is zero, it belongs to the state situated at the meeting point of the two cones. Finally, the heat current operator is related tothe energy current by the simple formula J Q=JE−/H9262J, where /H9262is the chemical potential. So far, we have considered the particle-hole symmetric case with /H9262=0, but we can easily use a ﬁnite chemical potential to break this symmetry, andintroduce ﬁnite Seebeck coefﬁcient.III. IMPURITY SCATTERING IN THE PRESENCE OF MAGNETIC FIELD In the presence of impurities, an extra term is added to the Hamiltonian: Himp=V/H20858 i=1Ni /H9254/H20849r−ri/H20850, /H2084918/H20850 where Nidenotes the number of impurities. As a result, the explicit form of the previous operators might change. How-ever, using Eq. /H2084918/H20850, the electric current remains unchanged, but the heat current changes due to the noncommutativity ofthe impurity Hamiltonian and the energy position operator. 33 As a result, impurities need to be taken into account not onlyin the calculation of the self-energy but also in the form ofthe operators, and one has to use the same level of approxi-mation for both. However, to avoid this difﬁculty, one can replace the en- ergy terms in J Eby the Matsubara frequency,34since from the poles of the Green’s function, this will pick the appropri-ate energy. This replacement works perfectly in the case ofimpurities as well, when quasiparticle excitations possess ﬁ-nite lifetime. Since graphene is two dimensional, positional long-range order /H20849i.e., lattice formation /H20850is impossible at ﬁnite tempera- tures, since thermal ﬂuctuations will destroy it. 5This is why the introduction of defect is natural in this system. To mimicdisorder, we have chosen to spread vacancies in the honey-comb lattice, which can be modeled by taking the impuritystrength /H20849V/H20850to inﬁnity. To take scattering into account, we have to determine the explicit form of H impin the Landau basis. Then, the standard noncrossing approximation can be used,24which, in the case of graphene, is called the full self-consistent Born approxi-mation due to the neglect of crossing diagrams. 26–28Averag- ing over impurity positions is performed in the standard way.As a result, we arrive to the following set of equations: G/H20849i /H9275n,k,n,/H9251/H20850=1 i/H9275n−E/H20849n,/H9251/H20850−/H90181/H20849i/H9275n/H20850, /H2084919/H20850 G/H20849i/H9275n,k/H20850=1 i/H9275n−/H90182/H20849i/H9275n/H20850, /H2084920/H20850 where /H90181/H20849i/H9275n/H20850=niV 2/H208771 1−Vgc/H20851G/H20849i/H9275n,k/H20850+S/H20849i/H9275n/H20850/2/H20852 +1 1−VgcS/H20849i/H9275n/H20850/2/H20878, /H2084921/H20850 /H90182/H20849i/H9275n/H20850=niV 1−Vgc/H20851G/H20849i/H9275n,k/H20850+S/H20849i/H9275n/H20850/2/H20852, /H2084922/H20850 where gc=1/ /H20849N+1/H20850is the degeneracy of a Landau level per unit cell and niis the impurity concentration per lattice sites. These equations describe impurity effects for arbitrary scat-tering potential V. The summation over Landau levels can be performed to yieldMAGNETOTRANSPORT AND THERMOELECTRICITY IN … PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-3S/H20849i/H9275n/H20850=/H20858 n,/H9251G/H20849i/H9275n,k,n,/H9251/H20850=2z /H9275c/H20851/H9023/H208491−z2/H20850−/H9023/H20849N+2− z2/H20850/H20852, /H2084923/H20850 where z=/H20851i/H9275n−/H90181/H20849i/H9275n/H20850/H20852//H9275c,/H9023/H20849z/H20850is the digamma function. By letting the impurity strength V→/H11009, which would cor- respond to the unitary scattering limit in unconventionalsuperconductors, 35our self-consistency equations simplify to /H90181/H20849i/H9275n/H20850=−ni gc/H208751 2G/H20849i/H9275n,k/H20850+S/H20849i/H9275n/H20850+1 S/H20849i/H9275n/H20850/H20876,/H2084924/H20850 /H90182/H20849i/H9275n/H20850=−2ni gc/H208512G/H20849i/H9275n,k/H20850+S/H20849i/H9275n/H20850/H20852. /H2084925/H20850 Similar equations have been derived in Ref. 26. The self- consistency equations can further be simpliﬁed, and afteranalytic continuation to real frequencies /H20849i /H9275n→/H9275+i0+/H20850,w e can read off /H90181/H20849/H9275/H20850=/H90182/H20849/H9275/H20850 2gc/H90182/H20849/H9275/H20850+2ni/H20851/H9275−/H90182/H20849/H9275/H20850/H20852 gc/H90182/H20849/H9275/H20850+ni/H20851/H9275−/H90182/H20849/H9275/H20850/H20852. /H2084926/H20850 At zero frequency, this simpliﬁes to /H90182/H208490/H20850=/H90181/H208490/H20850/H208752−1 1−ni/H20849N+1/H20850/H20876. /H2084927/H20850 The imaginary part of the self-energy is always negative to ensure causality. This means that the last term in parentheseson the right-hand side must always be positive to assure thesame sign of the imaginary parts of the self-energies. Thistranslates into n i/H333561 N+1. /H2084928/H20850 For each impurity concentration, there is a certain magnetic- ﬁeld strength /H20851when N=/H208491/ni/H20850−1/H20852, above which our approxi- mation breaks down. For higher ﬁeld, the self-energy at zero frequency needs to be zero to fulﬁll Eq. /H2084927/H20850and causality. This means that at a ﬁnite impurity concentration, we stillhave excitations in the system with inﬁnite lifetime. Further,we are going to show that this occurs not only on the zerothLandau level but on all Landau levels for ﬁeld exceeding thecritical one. To improve on this, crossing diagrams need tobe considered, which is beyond the scope of the presentwork. Hence, we restrict our investigation to ﬁelds allowedby Eq. /H2084928/H20850. The larger the impurity concentration, the larger the magnetic ﬁeld we can take into account. We mention thatcausality is also maintained for n i/H110211/2 /H20849N+1/H20850, which trans- lates into a Landau energy /H20849/H9275c/H20850comparable to the bandwidth for realistic concentrations, and is beyond the reach of valid- ity. The quasiparticle density of states can be evaluated from the knowledge of the Green’s function and it reads as/H9267/H20849/H9275/H20850=−gc /H9266/H208751 /H9275−/H90182/H20849/H9275/H20850+S/H20849/H9275/H20850/H20876 =ni /H9266Im/H208751 /H90182/H20849/H9275/H20850−1 /H90182/H20849/H9275/H20850−2/H90181/H20849/H9275/H20850/H20876. /H2084929/H20850 Without impurities, the density of states consists of Dirac- delta peaks located at zero frequency and at E/H20849n,/H9251/H20850.B yi n - troducing impurities in the system, we expect the broadening and shift of these levels, and it can be determined from thesolution of the self-consistency equations. For large magnetic ﬁelds /H20849small N/H20850, we can still solve the self-consistency equations /H20851Eqs. /H2084924/H20850and /H2084925/H20850/H20852, but we dis- cover Dirac-delta peaks at the position of the levels andsmall islands between them /H20849Fig. 2,N=100 /H20850. This signals that the noncrossing approximation is insufﬁcient to providethese peaks with a ﬁnite broadening. As we decrease the ﬁeld/H20849increase N/H20850, the peaks and islands merge, and all excitations possess ﬁnite lifetime, but clean gaps are still observablebetween the levels. By further decreasing the ﬁeld, the gapsdisappear, the density of states becomes ﬁnite for all ener-0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1600.050.10.150.20.250.30.350.40.450.5 ω/DDρ(ω) 0 0.05 0.1 0.15 0.200.10.20.30.40.5 ω/DDρ(ω)(a) (b) FIG. 2. /H20849Color online /H20850The density of states is shown in the upper panel for ni=0.001 for N=100 /H20849red/H20850, 1000 /H20849blue /H20850, 3000 /H20849black /H20850with decreasing /H9275c. The vertical red lines stand for the Dirac-delta peaks for N=100. The lower panel visualizes the ni =0.01 case for N=100 /H20849red/H20850, 200 /H20849blue /H20850, 300 /H20849black /H20850. The clean case without magnetic ﬁeld /H20849N=/H11009/H20850is also plotted for comparison in both panels /H20849blue dashed line /H20850.BALÁZS DÓRA AND PETER THALMEIER PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-4gies, and small successive bumps remain present due to Lan- dau level formation, which tend to be smoothened by furtherdecreasing the ﬁeld. In this limit, the resulting density ofstates is very close to that in a d-wave superconductor, 36 stemming from its linear frequency dependence in the pure case. The broadening of the levels is not symmetric, more spec- tral weight is transferred to the lower-energy part, whicharises from the important energy dependence of the imagi-nary part of the self-energies. It is related to the presence ofresonances near the Landau level energies in the complexplane, similar to the resonance in the Dirac point withoutmagnetic ﬁeld. 37Also, the level position is modiﬁed in the presence of impurities due to the ﬁnite real part of the self-energies, and this shift increases with the impurity concen-tration. This was also found in a similar treatment. 26How- ever, in normal metals with nonrelativistic dispersion, such arenormalization is forbidden due to Kohn’s theorem. 38 The numerical solution of Eqs. /H2084924/H20850and /H2084925/H20850and the re- sulting density of states, is shown in Fig. 2. From this, one can conjecture that a given niandNcan qualitatively well describe different ﬁelds and concentrations, if their product/H20849n iN/H20850is the same. These features, including the non-Lorentzian broadening of the Landau levels and the develop- ment of small islands between the levels, should be observ-able experimentally by scanning tunneling microscopy, forexample. IV. ELECTRIC AND THERMAL CONDUCTIVITIES Using the spectral representation of the Green’s functions, we can evaluate the corresponding conductivities afterstraightforward but lengthy calculations. These are related tothe time-ordered products of the form 33 /H9016i,jAB/H20849i/H9275/H20850=−/H20885 0/H9252 d/H9270ei/H9275/H9270/H20855T/H9270JiA/H20849/H9270/H20850JjB/H208490/H20850/H20856, /H2084930/H20850 where AandBdenote the electric and heat currents, and i and jstand for the spatial component. These can be ex- pressed with the use of the following transport integrals:29 Ln=/H20885 −/H11009/H11009d/H9280 4T/H9268/H20849/H9280/H20850 cosh2/H20851/H20849/H9280−/H9262/H20850/2T/H20852/H20873/H9280−/H9262 T/H20874n , /H2084931/H20850 where /H9268/H20849/H9280/H20850=/H9275c2/H20858 /H9251/H20877Im/H90182/H20849/H9280/H20850 /H20851x−R e/H90182/H20849/H9280/H20850/H208522+/H20851Im/H90182/H20849/H9280/H20850/H208522Im/H90181/H20849/H9280/H20850 /H20851x−E/H208490,/H9251/H20850−R e/H90181/H20849/H9280/H20850/H208522+/H20851Im/H90181/H20849/H9280/H20850/H208522 +1 2/H20858 n,/H9261Im/H90181/H20849/H9280/H20850 /H20851x−E/H20849n,/H9251/H20850−R e/H90181/H20849/H9280/H20850/H208522+/H20851Im/H90181/H20849/H9280/H20850/H208522Im/H90181/H20849/H9280/H20850 /H20851x−E/H20849n+1 ,/H9261/H20850−R e/H90181/H20849/H9280/H20850/H208522+/H20851Im/H90181/H20849/H9280/H20850/H208522/H20878 /H2084932/H20850 is the dimensionless conductivity kernel. With the use of these, we obtain the various transport coefﬁcients as usual: /H9268=2e2 /H9266hL0, /H2084933/H20850 S=1 eL1 L0, /H2084934/H20850 /H9260 T=2 /H9266h/H20873L2−L12 L0/H20874, /H2084935/H20850 L=/H9260 /H9268T=1 e2L2L0−L12 L02. /H2084936/H20850 Here, /H9268is the electric conductivity, Sis the Seebeck coefﬁ- cient, /H9260is the heat conductivity, where the last term ensures that the energy current is evaluated under the condition ofvanishing electric current, and Lis the Lorentz number. Off- diagonal components of the conductivity tensors, such as theNernst coefﬁcient, are also of prime interest, but they cannotbe simply evaluated from Kubo formula. Even in the case ofa normal metal with parabolic dispersion, the Kubo formulaturned out to be invalid, 39,40and additional corrections have been worked out. Their determination for two-dimensionalDirac fermions is beyond the scope of the present investiga-tion. For the particle-hole symmetric case /H20849 /H9262=0/H20850, the Seebeck coefﬁcient is trivially zero. If we consider the zero- temperature, half-ﬁlled case and assume small magneticﬁelds, we obtain the universal conductivity given by /H92680=2e2 /H9266h, /H2084937/H20850 and similarly for the thermal conductivity as /H9260 T=2kB2/H9266 3h, /H2084938/H20850 upon reinserting original units. The Seebeck coefﬁcient is zero. From this, the Lorentz number takes its universal value Lu=/H92662 3/H20873kB e/H208742 , /H2084939/H20850 which means that in this limit, the Wiedemann-Franz law holds.17,24Landau levels always develop around the meetingMAGNETOTRANSPORT AND THERMOELECTRICITY IN … PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-5point of the conical valence and conduction band. If we are at half ﬁlling /H20849/H9262=0/H20850, no levels cross /H9262when varying the magnetic ﬁeld, since they are symmetrically placed below and above. However, when /H9262is ﬁnite, Landau levels can cross its value with changing the ﬁeld, and we expectShubnikov–de Haas oscillations. In general, when the num-ber of levels below /H9262is large /H20849or/H9275c/H11270/H20841/H9262/H20841/H20850, we can conjec- ture the periodicity of these oscillations. Assume that a level /H20849thenth/H20850sits right at the chemical potential /H20849/H9262=/H9275c/H20881n+1/H20850. Then, the distance from the adjacent level determines the period of the oscillations. This is /H20841E/H20849n+1 ,/H9251/H20850−E/H20849n,/H9251/H20850/H20841 /H11015/H9275c 2/H20881n+1=/H9275c2 2/H9262=vF2e/H20841Bcos/H20849/H9258/H20850/H20841 /H9262/H11011B /H2084940/H20850 provided that n/H112711. This means that albeit the Landau levels show an unusual /H11008/H20881ndependence of the level index com- pared to that in a normal metal /H11008n, the Shubnikov–de Haas oscillations turn out to be still periodic as a function of 1/ B. The comparison of the coefﬁcient of the magnetic ﬁeld inEq. /H2084940/H20850to that in a parabolic band 29suggests that the cyclo- tron mass can be deﬁned as mc=/H9262/vF2. Even though the spec- trum is linear, the ﬁnite chemical potential provides us withan energy scale for m c.7This can readily be checked in Fig. 3, where not only the ﬁeld but the angle dependence of the conductivity is shown for different ﬁeld strengths. The largerthe magnetic ﬁeld, the more visible the oscillations are, al-though these can be smeared by increasing the concentra-tions. When the Landau energy exceeds the value of thechemical potential /H20849 /H9275c/H11022/H9262/H20850, oscillations disappear for higher magnetic ﬁelds, because no Landau levels remain to cross /H9262. The explicit value of the chemical potential, which is ﬁxed by the particle number at a given temperature and ﬁeld,should also be determined self-consistently. However, no se-rious deviations from its initial values have been detectedduring the evaluation process, and these did not affect thedependence of physical quantities on TandBin the investi- gated range of parameters. Presumably, taking a large valueof the chemical potential would require its self-consistentdetermination as well. In Fig. 4, we show the magnetic-ﬁeld dependence of the heat conductivity. It resembles closely to the electric one atlow temperatures. However, at higher temperatures, eachpeak in the oscillations splits into two. This occurs becausein the electric conductivity, the kernel is sampled by the1/cosh 2/H20851/H20849/H9280−/H9262/H20850/2T/H20852function, which gathers information about excitations at the chemical potential. However, an ex- tra /H20849/H9280−/H9262/H208502factor appears in the heat response, which mea- sures the immediate vicinity of /H9262above and below, within a window 2 T, which gives the splitting. The oscillations be- come smoothened with decreasing ﬁeld, in contrast to Ref.19, where large oscillations were found even at small ﬁelds. The difference can be traced back to our ﬁeld-dependentscattering rate /H20851Eqs. /H2084924/H20850and /H2084925/H20850/H20852, as opposed to the ﬁeld independent one used in Ref. 19. Similar features have been observed in highly oriented pyrolytic graphite. 8,9By decreas- ing the ﬁeld, Nincreases, and the density of states becomes similar to that of a d-wave superconductor,36without signiﬁ-cant deviations from linearity. Both /H9268and/H9260decrease with ﬁeld, a feature already present at /H9262=0. As we increase the ﬁeld,/H9275cincreases, and so does the distance between Landau levels. Then, at a given temperature, a smaller number ofstates will be present for excitations around /H9262; hence, the corresponding conductivity decreases. The Seebeck coefﬁ-cient shows sharp oscillations which die out with tempera-ture. Its background value, after subtracting the oscillations,is found to be almost magnetic ﬁeld independent butsmoothly increases with temperature. The Lorentz numberremains close to 1, if we subtract the oscillations. However,due to the double /H20849single /H20850peak structures in the heat /H20849elec- tric/H20850response, their ratio shows wild but sharp deviations from unity at speciﬁc ﬁelds, where the Wiedemann-Franz0 10 20 30 40 50 60 70 80 90051015202530354045 θ(deg )σπh/e2 0 0.5 1 1.5 2 x1 0−305101520253035404550 \|Bcos(θ)\|/B0σπh/e2 1000 2000 3000 4000 5000 6000051015202530354045 B0/\|Bcos(θ)\|σπh/e2(a) (b) FIG. 3. /H20849Color online /H20850The angular-dependent magnetoconduc- tivity oscillations are visualized for /H9262=0.05 D,ni=0.001, and T =0.0001 D, for magnetic ﬁelds N=600 /H20849red/H20850, 1000 /H20849blue /H20850,/H208492000 /H20850 /H20849black /H20850,/H208493000 /H20850/H20849green /H20850, and 5000 /H20849magenta /H20850in the upper panel from bottom to top. With increasing ﬁeld /H20849decreasing N/H20850, the oscillations become more pronounced, signaling the discrete Landau levelstructure. The lower panel shows the electric conductivity for /H9262 =0.05 D,ni=0.001, and T/D=0.0001 /H20849red/H20850, 0.001 /H20849black /H20850, and 0.01 /H20849blue /H20850with decreasing oscillations. For higher ﬁeld, we arrive to the region, where crossing diagrams need to be taken into account. Theinset shows the electric conductivity as a function of 1/ /H20841Bcos/H20849 /H9258/H20850/H20841to emphasize its periodicity.BALÁZS DÓRA AND PETER THALMEIER PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-6law is violated. In contrast to this, one would have encoun- tered large and wide oscillations in the Lorentz number as afunction of ﬁeld in the presence of phenomenological, con-stant scattering rate. In Fig. 5, we show the evolution of the electric and heat conductivities and the Seebeck coefﬁcient as a function ofchemical potential. In accordance with experiment in Ref. 7, we also ﬁnd oscillations, corresponding to Landau levels,which also smoothen with temperature. Interestingly, thesplitting of the peaks in the heat conductivity is nicely ob-servable as a function of /H9262. These occur in such a way that they produce antiphase oscillations with respect to the elec-tric one and lead to the violation of the Wiedemann-Franzlaw. The Seebeck coefﬁcient shows peculiar behavior. At theparticle-hole symmetric case, it is zero and remains mainlyso apart from large oscillations. The temperature dependence of the electric and heat re- sponses is shown in Fig. 6. Both increase steadily with tem- perature, since more available states are accessible with T. However, at small temperatures, a small decrease is observ-able in low ﬁelds, in accordance with other studies. 17,26The Seebeck coefﬁcient ﬁrst increases, and after a broad bump,decreases with T. For higher temperatures, the bandwidth D makes its presence felt. The Wiedemann-Franz law remainsintact at low temperatures and ﬁelds but becomes violatedfor higher TorB. Our results for the electric and heat conductivities and the Lorentz number agree in general with those found in Refs.17–23 for a constant scattering rate. However, the Shubnikov–de Haas oscillations become asymmetric in boththe electric and heat conductivities due to the energy-dependent scattering rate, determined self-consistently in ourwork. These oscillations are suppressed as one lowers themagnetic ﬁeld as is seen in Fig. 4, as opposed to Ref. 19. The periodic structures are also suppressed with temperature,which feature was not directly observable in previous works.The periodic violation of the Wiedemann-Franz law /H20849Fig. 4/H20850 becomes stronger and sharper with increasing ﬁeld comparedto Ref. 19, since the broadening of the Landau levels de- creases /H20849Fig. 2/H20850, as is borne out from our self-consistent cal- culation of the self-energies. In addition, we considered thetemperature, ﬁeld, and chemical-potential dependence of theSeebeck coefﬁcient in detail. By considering a magnetic ﬁeldwith a component parallel to the plane, we were able to studythe angular-dependent magnetoconductivity as well. V. CONCLUSION We have studied the effect of localized impurities in two- dimensional Dirac fermions in the presence of quantizing,arbitrarily oriented magnetic ﬁeld. The energy spectrum de- pends on the level index as /H11008/H20881n, as opposed to the n+1/2 linear dependence in normal metals.24Expressions for both the electric and heat currents in the presence of magneticﬁeld were worked out. The self-energy in the full Born ap-proximation obeys self-consistency conditions, resulting inimportant magnetic ﬁeld and frequency dependence of scat-tering rate and level shift. In the density of states, only a500 1000 1500 2000 2500 3000 3500 4000 4500 500000.511.522.5 B0/\|Bcos(θ)\|L/Lu 0 0.5 1 1.5 2 x1 0−3050100150 \|Bcos(θ)\|/B0κπh/Tk2 B 0 0.5 1 1.5 x1 0−3−2−10123 \|Bcos(θ)\|/B0Se/k B(a) (b) FIG. 4. /H20849Color online /H20850The upper panel shows the Lorentz num- ber as a function of the inverse magnetic ﬁeld to stress the periodicviolation of the Wiedemann-Franz law for /H9262=0.05 D,ni=0.001, and T/D=0.0001 /H20849red/H20850, 0.001 /H20849black dashed line /H20850, and 0.01 /H20849blue /H20850. The lower panel shows the heat conductivity and the Seebeck coefﬁcient/H20849inset /H20850for the set of same parameters.0 0.02 0.04 0.06 0.08 0.1051015202530 µ/Dσπh/e2,κ3h/Tπk2 B 0 0.02 0.04 0.06 0.08 0.1−2−1012 µ/DSe/kB FIG. 5. /H20849Color online /H20850The electric /H20849blue solid line /H20850and heat /H20849red dashed line /H20850conductivities and the Seebeck coefﬁcient /H20849inset /H20850are shown as a function of the chemical potential for T=0.001 D,N =1000, and ni=0.001. Due to the antiphase oscillations, the Wiedemann-Franz law is violated.MAGNETOTRANSPORT AND THERMOELECTRICITY IN … PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-7small island shows up close to zero frequency for small ﬁelds, similar to d-wave superconductors.35By increasing the ﬁeld, oscillations become visible, corresponding to Lan-dau levels. By further increasing the ﬁeld, these becomeseparated from each other, and clean gaps appear betweenthe levels, in which intragap states, small islands show up athigh ﬁeld. The non-Lorentzian broadening of Landau levelsand the intragap features differ from previous studies assum-ing a constant scattering rate and should be detected experi-mentally in graphene. Both the electric and thermal conductances show Shubnikov–de Haas oscillation in magnetic ﬁeld, which dis-appear for small ﬁelds and higher temperatures. These areperiodic in 1/ B, similar to normal metals, in spite of the different Landau quantization. The Seebeck coefﬁcientshares these features, but its oscillations are really large asopposed to /H9268and/H9260. The Wiedemann-Franz law stays close to unity, except at certain ﬁelds, where large deviations areencountered, which vanish with decreasing ﬁeld. Besides os-cillations, both /H9268and/H9260decrease with ﬁeld, since the larger the Landau energy, the smaller the probability of ﬁndingstates around /H9262. These are in agreement with experiments on the thermal conductivity of highly oriented pyrolyticgraphite. 8,9Oscillations are also present as a function of chemical potential, similar to experimental ﬁndings.3 The temperature dependence of the conductivities is rather conventional, both /H9268and/H9260increase with temperature steadily, regardless of the value of the chemical potential.The Seebeck coefﬁcient exhibits a broad bump around T /H11011 /H9262and decreases afterward. The Wiedemann-Franz law is obeyed for small temperatures and ﬁeld but violated forhigher values. ACKNOWLEDGMENTS We are thankful to A. H. Castro Neto for useful discus- sions. This work was supported by the Hungarian ScientiﬁcResearch Fund under Grant No. OTKA TS049881. *Electronic address: dora@kapica.phy.bme.hu 1C. Berger, Z. M. Song, T. B. Li, X. B. Li, A. Y . Ogbazghi, R. Feng, Z. T. Dai, A. N. Marchenkov, E. H. Conrad, and P. N.First, J. Phys. Chem. B 108, 19912 /H208492004 /H20850. 2K. 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 /H208492004 /H20850. 3K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V . V . Khot- kevich, S. V . Morozov, and A. K. Geim, Proc. Natl. Acad. Sci.U.S.A. 102, 10451 /H208492005 /H20850. 4A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, Nat. Phys. 3,3 6 /H208492007 /H20850. 5A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 /H208492007 /H20850.6S. Y . Zhou, G.-H. Gweon, J. Graf, A. V . Fedorov, C. D. Spataru, R. D. Diehl, Y . Kopelevich, D.-H. Lee, S. G. Louie, and A.Lanzara, Nat. Phys. 2, 595 /H208492006 /H20850. 7K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, M. I. Katsnelson, I. V . Grigorieva, S. V . Dubonos, and A. A. Firsov,Nature /H20849London /H20850438, 197 /H208492005 /H20850. 8R. Ocana, P. Esquinazi, H. Kempa, J. H. S. Torres, and Y . Kopelevich, Phys. Rev. B 68, 165408 /H208492003 /H20850. 9K. Ulrich and P. Esquinazi, J. Low Temp. Phys. 137, 217 /H208492004 /H20850. 10A. A. Nersesyan and G. E. Vachnadze, J. Low Temp. Phys. 77, 293 /H208491989 /H20850. 11A. A. Nersesyan, G. I. Japaridze, and I. G. Kimeridze, J. Phys.: Condens. Matter 3, 3353 /H208491991 /H20850.0 0.02 0.04 0.06 0.08 0.1 0.12050100150200250300350 T/Dσπh/e2,κh/Tπk2 B 0 0.02 0.04 0.06 0.08 0.1 0.12−1−0.500.511.52 T/DSe/kB 0 0.02 0.04 0.06 0.08 0.10.511.522.533.54 T/DL/L u(a) (b) FIG. 6. /H20849Color online /H20850The temperature dependence of the elec- tric /H20849blue solid line /H20850and heat /H20849red dashed line /H20850conductivities is shown in the upper panel for ni=0.001, /H9262=0.05 D, and N=600, 1000, 3000, and 10 000 from bottom to top. Note the 1/ /H92662reduc- tion of the heat conductivity. The upper panel shows the Seebeckcoefﬁcient and the Lorentz number /H20849inset /H20850for the same parameters from top to bottom, with dashed line for N=600. Note the violation of the Wiedemann-Franz law at low temperatures at high ﬁelds/H20849smaller N/H20850.BALÁZS DÓRA AND PETER THALMEIER PHYSICAL REVIEW B 76, 035402 /H208492007 /H20850 035402-812S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B 63, 094503 /H208492001 /H20850. 13H. Ikeda and Y . Ohashi, Phys. Rev. Lett. 81, 3723 /H208491998 /H20850. 14A. Virosztek, K. Maki, and B. Dóra, Int. J. Mod. Phys. B 16, 1667 /H208492002 /H20850. 15K. Behnia, R. Bel, Y . Kasahara, Y . Nakajima, H. Jin, H. Aubin, K. Izawa, Y . Matsuda, J. Flouquet, Y . Haga, Y . Ōnuki, and P. Lejay, Phys. Rev. Lett. 94, 156405 /H208492005 /H20850. 16R. Bel, H. Jin, K. Behnia, J. Flouquet, and P. Lejay, Phys. Rev. B 70, 220501 /H20849R/H20850/H208492004 /H20850. 17S. G. Sharapov, V . P. Gusynin, and H. Beck, Phys. Rev. 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PhysRevB.103.144508.pdf	PHYSICAL REVIEW B 103, 144508 (2021) Metamagnetic phase transition in the ferromagnetic superconductor URhGe V . P. Mineev* Universite Grenoble Alpes, CEA, IRIG, PHELIQS, F-38000 Grenoble, France and Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia (Received 16 February 2021; accepted 30 March 2021; published 12 April 2021) Ferromagnetic superconductor URhGe has orthorhombic structure and possesses spontaneous magnetization along the caxis. Magnetic ﬁeld directed along the baxis suppresses ferromagnetism in the cdirection and leads to a metamagnetic transition into polarized paramagnetic state in the bdirection. The theory of these phenomena based on the speciﬁc magnetic anisotropy of this material in the ( b,c) plane is given. Line of the ﬁrst order metamagnetic transition ends at a critical point. The Van der Waals-type description of behavior of physicalproperties near this point is developed. The triplet superconducting state destroyed by orbital effect is recreatedin the vicinity of the transition. It is shown that the reentrance of superconductivity is caused by the sharp increaseof magnetic susceptibility in the bdirection near the metamagnetic transition. The speciﬁc behavior of the upper critical ﬁeld in the direction of spontaneous magnetization in UCoGe and in UGe 2related to the ﬁeld dependence of magnetic susceptibility is discussed. DOI: 10.1103/PhysRevB.103.144508 I. INTRODUCTION Investigations of uranium superconducting ferromagnets UGe 2, URhGe, and UCoGe continue to attract attention mostly due to the quite unusual nature of its superconductingstates created by the magnetic ﬂuctuations (see the recentexperimental [ 1] and theoretical [ 2] reviews and references therein). They have orthorhombic crystal structure and theanisotropic magnetic properties. The spontaneous magneti-zation is directed along the aaxis in UGe 2and along the c axis in URhGe and UCoGe. The ferromagnetic state in thetwo last materials is suppressed by the external magnetic ﬁeldH ydirected along bcrystallographic direction. In URhGe at ﬁeld Hy=Hcr≈12 T the second order phase transition to ferromagnetic state is transformed to the transition of the ﬁrstorder [ 3]. The superconducting state suppressed [ 4] in much smaller ﬁelds H y≈2 T reappears in the vicinity of the ﬁrst order transition in ﬁeld interval (9 ,13) T. The phenomenolog- ical theory of this phenomenon has been developed in Ref. [ 5] (see also Ref. [ 2]). According to this theory the state arising in ﬁelds above the suppression of spontaneous magnetizationin the cdirection is the paramagnetic state. There was established, however [ 3,6,7], that in ﬁelds above H crthe magnetization along the bdirection looks like it has ﬁeld independent “spontaneous” component My=My0+χyHy. (1) This state is called polarized paramagnetic state. The forma- tion of this state is related to so-called metamagnetic transitionobserved in several heavy-fermion compounds (see the pa-per [ 8] and the more recent publication [ 9] and references therein). To take into account the formation of polarized *vladimir.mineev@cea.frparamagnetic state one must introduce deﬁnite modiﬁcationsin the treatment performed in Ref. [ 5]. Here I present the corresponding derivation. The paper is organized as follows. In Sec. IIafter the brief reminder of results of the paper [ 5] the description of the metamagnetic transition is presented. It is based on thespeciﬁc phenomenon of magnetic anisotropy in URhGe ob-tained with local spin-density approximation calculations byAlexander Shick [ 10]. After the general consideration of the metamagnetic transition the modiﬁcations introduced by theuniaxial stress are considered. Then the Van der Waals-typetheory of phenomena near the metamagnetic critical point isdeveloped and some physical properties are discussed. The phenomenon of the reentrant superconducting state is explained in Sec. III. It is shown that the recreation of super- conductivity is caused by the sharp increase in the magneticsusceptibility [ 7]i nt h e bdirection near the metamagnetic transition. This section also contains the qualitative descrip-tion of the speciﬁc behavior of the upper critical ﬁeld indirection of spontaneous magnetization in UCoGe and inUGe 2related to the ﬁeld dependence of magnetic suscepti- bility. The Conclusion contains the summary of the results. II. METAMAGNETIC TRANSITION IN URhGe As in the previous publications (Refs. [ 2,5] )Is h a l lu s e x,y,zas the coordinates pinned to the corresponding crys- tallographic directions a,b,c. The Landau free energy of an orthorhombic ferromagnet in magnetic ﬁeld H(r)=Hyˆyis F=αzM2 z+βzM4 z+δzM6 z+αyM2 y+βyM4 y+δyM6 y +βyzM2 zM2 y−HyMy. (2) Here αz=αz0/parenleftbig T−Tc c0/parenrightbig ,α y>0, (3) 2469-9950/2021/103(14)/144508(8) 144508-1 ©2021 American Physical SocietyV . P. MINEEV PHYSICAL REVIEW B 103, 144508 (2021) and I bear in mind the terms of the sixth order in powers of Mz,Myand also the fact that in the absence of a ﬁeld in the x direction the magnetization along the hard xdirection Mx=0. A. Transition ferro-para Let us remind ﬁrst the treatment developed in Ref. [ 5] undertaken in the assumption βy>0. Then in the constant magnetic ﬁeld H=Hyˆythe equilibrium magnetization pro- jection along the ydirection My≈Hy 2/parenleftbig αy+βyzM2z/parenrightbig (4) is obtained by minimization of free energy ( 2) in respect to My neglecting the higher order terms. Substituting this expression back to ( 2) we obtain F=αzM2 z+βzM4 z+δzM6 z−1 4H2 y αy+βyzM2z, (5) that gives after expansion of the denominator in the last term, F=−H2 y 4αy+˜αzM2 z+˜βzM4 z+˜δzM6 z+..., (6) where ˜αz=αz0(T−Tc0)+βyzH2 y 4α2y, (7) ˜βz=βz−βyz αyβyzH2 y 4α2y, (8) ˜δz=δz+β2 yz α2yβyzH2 y 4α2y. (9) Thus, in a magnetic ﬁeld perpendicular to the direc- tion of spontaneous magnetization the Curie temperaturedecreases as T c=Tc(Hy)=Tc0−βyzH2 y 4α2yαz0. (10) The coefﬁcient ˜βzalso decreases with Hyand reaches zero at Hy=H⋆=2α3/2 yβ1/2 z βyz. (11) At this ﬁeld under fulﬁllment the condition, αz0βyzTc0 αyβz>1 (12) the Curie temperature ( 10) is still positive and the phase transition from the ferromagnetic to the paramagnetic statebecomes the transition of the ﬁrst order [Fig. 1(a)]. The point (H ⋆,Tc(H⋆)) on the line paramagnet-ferromagnet phase tran- sition is a tricritical point. The qualitative ﬁeld dependences of the normalized Curie temperature tc(Hy)=Tc(Hy) Tc0andb(Hy)= ˜βz βzare plotted in Fig. 1(a). On the line of the ﬁrst order phase transition from the ferromagnet to the paramagnet state the Mzcomponent of magnetization drops from M⋆ zto zero [ 2]. The Mycom- ponent jumps from My≈H⋆ 2(αy+βyzM⋆2z)toMy≈H⋆ 2αy. Then at FIG. 1. (a) Schematic behavior of the normalized Curie tempera- turetc(Hy)=Tc(Hy) Tc0and coefﬁcient b(Hy)=˜βz βz. FM and PM stand for ferromagnetic and paramagnetic phases. (b) Schematic dependenceM y(Hy)a tT<TcrandHcr<H⋆. ﬁelds Hy>H⋆ My≈Hy 2αy(13) proportional to the external ﬁeld. This contradicts experimen- tal observations [ 3,6,7] which demonstrate the presence of a “spontaneous” part of magnetization in the ﬁeld above thetransition in accordance with Eq. ( 1). B. Transition ferro - polarized para The part of free energy depending on My, Fy=αyM2 y+βyM4 y+δyM6 y+βyzM2 zM2 y−HyMy,(14) can be used also far from the transition to the ferromagnetic state in the temperature region where Mzis not small. The important fact obtained with the local spin-density approxi-mation calculations [ 10] is that the coefﬁcient β y<0. In the frame of the isotropic Fermi liquid model the negativeness ofthe fourth order term in the expansion of the free energy inpower of magnetic moment is usually ascribed to the peculiarbehavior of the electron density of states (see the review [ 11] and references therein). In the orthorhombic URhGe this spe-ciﬁc magnetocrystalline anisotropy reveals itself in the systemof magnetic moments localized on the uranium atoms [ 12]. TheM ycomponent of magnetization is determined by the equation 2˜αyMy+4βyM3 y+6δyM5 y=Hy, (15) where ˜αy=αy+βyzM2 z. (16) 144508-2METAMAGNETIC PHASE TRANSITION IN THE … PHYSICAL REVIEW B 103, 144508 (2021) Taking into account the third order term we obtain My≈Hy 2˜αy−βyH3 y 2˜α4y. (17) The coefﬁcient βy<0 and we see that the increase of magne- tization occurs faster than it was according to Eq. ( 4). The shape of My(Hy) depends on the temperature and pres- sure dependence of coefﬁcients αy,βy,δy. In particular, the coefﬁcient ˜ αy(T) is decreasing function of temperature and at temperature decrease the ﬁeld dependence of Mytransfers from the monotonous growth taking place at β2 y<5 3˜αyδyto the S-shape dependence realizing at β2 y>5 3˜αyδy. This trans- formation occurs at some temperature Tcrsuch that in the dependence Hy(My) appears an inﬂection point. It is deter- mined by the equations ∂Hy ∂My=0,∂2Hy ∂M2y=0 (18) having common solution M2 cr=−βy 5δy, (19) atβ2 y=5 3˜αyδy. The corresponding critical ﬁeld is Hcr=Hy(Mcr)=16 5√ 3˜α3/2 y \|βy\|1/2. (20) AtT<Tcrthe inequality β2 y>5 3˜αyδy (21) is realized and the equation∂Hy ∂My=0 acquires two real solu- tions, hence, the ﬁeld dependence of Myacquires the S shape plotted at Fig. 1(b). Equilibrium transition from the lower to the upper part of the curve My(Hy) corresponds to a vertical line connecting the points M1andM2deﬁned by the Maxwell rule/integraltext2 1M(H)dH=0. The integration is performed along the curve My(Hy). The Mycomponent of magnetization jumps from M1toM2[see Fig. 1(b)]. At temperatures above Tcrthe jump transforms into the crossover which is the temperature-ﬁeld region character-ized by the fast growth M y. The lower boundary of this region roughly coincides with the Curie temperature (seeFig. 2). The Curie temperature decreasing with growth of magnetization M y Tc(Hy)=Tc0−βyzM2 y αz0(22) falls down to zero or even to negative value at sharp increase ofMyin the vicinity of the critical ﬁeld Hcrand the ferromag- netic order along the zdirection disappears. Thus, at T<Tcr andHy=Hcrwe have the phase transition of the ﬁrst order from the ferromagnetic state with spontaneous magnetizationalong the zdirection to the polarized paramagnetic state with induced magnetization along the ydirection (Fig. 2). The described jumplike transition is realized in the cylin- drical specimen in the magnetic ﬁeld parallel to the cylinderaxis. In specimens of arbitrary shape with demagnetizationfactor nthe transition occurs in some ﬁeld interval where the FIG. 2. Phase diagram UCoGe in magnetic ﬁeld parallel to the b-crystallographic direction. PM, FM, and PPM denote paramag- netic, ferromagnetic, and polarized paramagnetic phases. CEP is thecritical end point. SC and RSC are the superconducting and reentrant superconducting states. specimen is ﬁlled by the domains with different magnetiza- tion. When the critical ﬁeld Hcris smaller than the critical ﬁeld of transition ferro-para H⋆, the ferro-para transition discussed in the previous section does not occur. At T<Tcrin ﬁelds Hy exceeding Hcr, the ﬁeld dependence of Mycomponent of mag- netization behaves in accordance with Eq. ( 1) corresponding to the experimental observations. C. Uniaxial stress effects It is known that a hydrostatic pressure applied to URhGe crystals stimulates ferromagnetism and at the same timesuppresses the superconducting state [ 13] and the reentrant superconducting state [ 14] as well. The latter is also shifted to a bit higher ﬁeld interval. On the contrary, the uniaxialstress along the bdirection suppresses the ferromagnetism decreasing the Curie temperature and stimulates the supercon-ducting state so strongly that it leads to the coalescence of thesuperconducting and reentrant superconducting regions in the (H y,T) phase diagram [ 15]. The phenomenological descrip- tion of these phenomena was undertaken in the paper [ 16]. There it was shown that both coefﬁcients αzandαyin the Landau free energy Eq. ( 2) acquire the linear uniaxial pressure dependence αz(Py)=αz0(T−Tc0)+AzPy, (23) αy(Py)=αy−\|Ay\|Py (24) corresponding to the moderate uniaxial pressure suppression of the Curie temperature Tc(Py)=Tc0−AzPy αz0, (25) reported in Ref. [ 15] in the absence of an external ﬁeld. How- ever, under the external ﬁeld along the ydirection the drop of the Curie temperature Eq. ( 10) is accelerated Tc(Hy,Py))≈Tc0−AzPy αz0−βyzH2 y 4(αy(Py))2αz0(26) 144508-3V . P. MINEEV PHYSICAL REVIEW B 103, 144508 (2021) in correspondence with the observed behavior. Moreover, the uniaxial stress causes strong decrease of the critical ﬁeldEq. ( 20) H cr=Hy(Mcr)=16 5√ 3(˜αy(Py))3/2 \|βy\|1/2. (27) D. Van der Waals-type theory near the critical point The critical end point temperature for the ﬁrst order transi- tion in URhGe is Tcr=4 K and the critical ﬁeld is Hcr=12T. Let us expand the function Hy(My) at temperature slightly deviating from critical temperature T=Tcr+tand the mag- netization near its critical value My=Mcr+m.W eh a v e h=Hy−Hcr=bt+/bracketleftbigg∂Hy ∂My/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=0+2at/bracketrightbigg m +1 2∂2Hy ∂M2y/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=0m2+1 6∂3Hy ∂M3y/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=0m3. (28) Here, we neglected by the temperature dependence of the second and the third order terms. Taking into account that ∂Hy ∂My\|t=0=∂2Hy ∂M2y\|t=0=0 we obtain h=bt+2atm+4Bm3, (29) which obviously corresponds to the expansion of pressure p=P−Pcrin powers of density η=n−ncrnear the Van der Waals critical point [ 17]. Att<0 according to the Maxwell rule the magnetization densities of two phases in equilibrium with each other are: m2=−m1=/radicalbigg −at 2B. (30) The line of phase equilibrium between the two phases below and above the transition is given by the equation h=bt,t<0. (31) 1. Speciﬁc heat The speciﬁc heat at ﬁxed external ﬁeld (see Ref. [ 17]) is Ch∝T/parenleftbig∂h ∂t/parenrightbig2 m/parenleftbig∂h ∂m/parenrightbig t. (32) Then, using Eq. ( 29) we obtain Ch∝b2T 2at+12Bm2. (33) Thus, the contribution to heat capacity according to the equa- tion of state ( 29) near the critical point grows so long m2 decreases until to m2 1and then begins to fall when m2increases starting from m2 2(see Fig. 3). This is the contribution to the speciﬁc heat of the whole system and cannot be directly at-tributed to the speciﬁc heat of itinerant electrons proportionalto the electron effective mass. The low temperature behavior of the URhGe speciﬁc heat in magnetic ﬁeld has not been established by a direct measure-ment but was derived [ 6] by the application of the Maxwell relation ( ∂S ∂Hy) T=(∂My ∂T)Hyfrom the temperature dependence of the magnetization My(T,Hy) in the ﬁxed ﬁeld. The changes FIG. 3. Schematic behavior Ch/T(see the main text). of the ratio C(T)/Thave been ascribed to the electron ef- fective mass dependence from magnetic ﬁeld [ 6,18]. This was done in the assumption that URhGe is a weak itinerantferromagnet, in other words, all the low temperature degreesof freedom in this material belong to the itinerant electron sub-system. As we already mentioned above, the strong magneticanisotropy of this material [ 10] points on the importance of the magnetic degrees of freedom localized on the uranium ionsand related with crystal ﬁeld levels [ 2,12]. 2. Resistivity The magnetic ﬁeld dependence of effective mass was also found [ 18,19] by the application of the Kadowaki-Woods relation A(Hy)∝(m⋆)2where coefﬁcient Ais a prefactor in the low-temperature dependence of resistivity ρ=ρ0+AT2. TheA(Hy) behavior is determined by the processes of inelastic electron-electron scattering which in the multiband metalsinterfere with scattering on impurities (see Refs. [ 20–24]) and on magnetic excitations with ﬁeld dependent spectrum. Thenonspherical shape of the Fermi surface sheets and the screen-ing of el-el Coulomb interaction can introduce deviationsfrom T 2resistivity dependence. So, the physical meaning of the coefﬁcient A(Hy) behavior is not so transparent and its relationship with the electron effective mass is questionable. One can also note that the temperature ﬁt of the experimen- tal data was done in a very narrow temperature interval andtheT 2temperature dependence claimed in Ref. [ 19] seems somewhat unreliable. Compare with the results reported inRefs. [ 13,25]. 3. Correlation function The correlation function of ﬂuctuations of the magnetiza- tion density mnear the critical point at t<0 behaves similar to the speciﬁc heat [ 17] ϕ(k)=T 2(at+6Bm2+γijkikj). (34) This is in correspondence with a marked increase of the NMR relaxation rate 1 /T2with ﬁeld Hyincreasing toward 12 T reported in Refs. [ 26,27]. III. PHASE TRANSITION TO SUPERCONDUCTING STATE The superconducting state in URhGe is completely sup- pressed by the magnetic ﬁeld Hc2(T=0)≈2 T in the y direction due to the orbital depairing effect. Then supercon-ductivity recovers in the ﬁeld interval 9–13 T around the 144508-4METAMAGNETIC PHASE TRANSITION IN THE … PHYSICAL REVIEW B 103, 144508 (2021) critical ﬁeld Hcr≈12 T of the transition of the ﬁrst order from the ferromagnetic state with spontaneous magnetization alongthezdirection to the state with induced magnetization along theydirection. Evidently such type behavior is possible if the magnetic ﬁeld somehow stimulates the pairing interactionsurmounting the orbital depairing effect. In numerous publications starting from the paper by A. Miyake et al. [18] the treatment of this phenomenon was related with the assumption of an enhancement of electroneffective mass m ⋆=m(1+λ) leading to the enhancement of pairing interaction and consequently of the temperatureof transition to superconducting state according to the Mc-Millan-like formula [ 28] T sc≈/epsilon1exp/parenleftbigg −1+λ λ/parenrightbigg (35) derived in the paper [ 29] for the superconducting state with p pairing in an itinerant isotropic ferromagnetic metal. Similarto the liquid He-3 in this model there are two independentphase transition to the superconducting state in the subsys-tems with spin-up and spin-down electrons. The constant λ determined by the Hubbard four-fermion interaction [ 29,30] increases as we approach but not get too close to ferromag-netic instability. In the frame of this model the question ofwhy the growth of the magnetic ﬁeld H yapproaches the ferro- magnetic transition remains unanswered. The following development of this type approach has been undertaken by Yu. Sherkunov and co-authors [ 31]. The reen- trant superconductivity and mass enhancement have beenassociated with the Lifshitz transition [ 32] which occurs in one of the bands in a ﬁnite magnetic ﬁeld stimulating the split-ting of spin-up and spin-down bands. There was establishedmodest enhancement of the transition critical temperature inthe ﬁeld about 10 T. Thus, the model can claim to the qual-itative explanation of the superconducting state reentrance.However, it should be noted that the measured [ 32] quasipar- ticle mass in the corresponding band does not increase butdecreases and remains ﬁnite, implying that the Fermi velocityvanishes due to the collapse of the Fermi wave vector. Thecross section of the Fermi surface of this band correspondsto 7% of the Brillouin zone area. Thus, the reentrance of su-perconductivity hardly could be associated with the observedLifshitz transition. The models [ 29,31] describe the physics of pure itinerant electron subsystem. Such a treatment is approved in appli-cation to the 3He Fermi liquid. The measurements by x-ray magnetic circular dichroism [ 12] point to the local nature of the URhGe ferromagnetism. Namely, the comparison of thetotal uranium moment μ U totto the total magnetization Mtotat different magnitude and direction of magnetic ﬁeld indicatesthat the uranium ions dominate the magnetism of URhGe. Thesame is true also in the parent compound UCoGe [ 33]. So, the magnetic susceptibility χ ij(q,ω) is mostly determined by the localized moments subsystem. Hence, an approach basedon the exchange interaction between conduction electronsand magnetic moments localized on uranium atoms seemsmore appropriate. This type theory has been developed inthe paper by Hattori and Tsunetsugu [ 34]. Here, there will be undertaken another approach allowing explicitly takinginto account the enhancement of magnetic susceptibility nearthe metamagnetic transition from the ferromagnet state with spontaneous magnetization along the caxis to the magnetic state polarized along the baxis. Using the standard functional-integral representation of the partition function of the system (see Ref. [ 35]), we obtain the following term in the fermionic action describing an effectivetwo-particle interaction between electrons: S int=−1 2I2/integraldisplay dxdx/primeSi(x)Dij(x−x/prime)Sj(x/prime), (36) where S(r)=ψ† α(r)σαβψβ(r) is the operator of the electron spin density, x=(r,τ) is a shorthand notation for the co- ordinates in real space and the Matsubara time,/integraltext dx(...)=/integraltext dr/integraltextβ 0dτ(...),Iis the exchange constant of interaction of itinerant electrons with localized magnetic moments, Dij(x− x/prime) is the spin-ﬂuctuation propagator expressed in terms of the dynamical spin susceptibility χij(q,ω). Making use of the interaction ( 36) one can calculate the electron self energy and ﬁnd the dependence of the electroneffective mass from magnetic ﬁeld as well the temperatureof transition to the superconducting state with triplet pairing.The energy of electronic excitations in the temperature regionwhere the superconducting state is realized is much smallerthan typical energy of magnetic excitations. Hence, in calcu-lation of the superconducting properties one can neglect thefrequency dependence of susceptibility. A. Upper critical ﬁeld parallel to the caxis in UCoGe In application to UCoGe in magnetic ﬁeld parallel to di- rection of spontaneous magnetization this program has beenaccomplished in the paper [ 36]. There has been considered transition into the equal-spin pairing superconducting state intwo-band (spin-up, spin-down) orthorhombic ferromagneticmetal. According to this paper in the simpliﬁed case of asingle-band (say spin-up) equal-spin pairing superconductingstate the critical temperature without including the orbitaleffect of the ﬁeld is T sc=/epsilon1exp/parenleftbigg −1+λ/angbracketleftbig N0(k)χuzz/angbracketrightbig I2/parenrightbigg , (37) where, as in the McMillan formula, 1 +λcorresponds to the effective mass renormalization, whereas the pairing amplitudeexpressed through the odd in momentum part of static suscep-tibility χ u zz=1 2[χzz(k−k/prime)−χzz(k+k/prime)], which is the main source of the critical temperature depen- dence from magnetic ﬁeld. Here, χzz(k)=1 χ−1z+2γijkikj, (38) andχz=χz(Hz)i st h e zcomponent of susceptibility in the ﬁ- nite ﬁeld Hz. Its magnitude at Hz→0, and we will denote χz0. The angular brackets denote averaging over the Fermi surfaceandN 0(k) is the angular dependent density of electronic states on the Fermi surface, /angbracketleftbig N0(k)χu zz(Hz)/angbracketrightbig ≈2/angbracketleftbig N0(k)ˆk2 z/angbracketrightbig k2 Fχz (2χz)−1+4γzzk2 F. (39) 144508-5V . P. MINEEV PHYSICAL REVIEW B 103, 144508 (2021) The denominator in the exponent of Eq. ( 37) can be expressed through its value at Hz→0 /angbracketleftbig N0(k)χu zz(Hz)/angbracketrightbig /angbracketleftbig N0(k)χuzz(Hz→0)/angbracketrightbig=χz χz01+4(ξmkF)2 χz0 χz+4(ξmkF)2. (40) Here the product 2 γzzk2 Fχz0=(ξmkF)2is expressed through the magnetic coherence length ξmwhich near the zero tem- perature is of the order of several interatomic distances. In assumption ( ξmkF)2/greatermuch1 one can rewrite Eq. ( 40)a s /angbracketleftbig N0(k)χu zz(Hz)/angbracketrightbig ≈χz(Hz) χz0/angbracketleftbig N0(k)χu zz(Hz→0)/angbracketrightbig . (41) This very rough estimation presents the qualitative depen- dence of exponent in equation ( 37) from magnetic ﬁeld. The longitudinal susceptibility drops with the augmentation ofmagnetic ﬁeld parallel to the spontaneous magnetization (seeFig. 3 in the paper [ 37]) leading to the suppression of the temperature of transition to the superconducting state withoutincluding the orbital effect according to Eq. ( 37). Taking into account the orbital effect one can write the ﬁeld dependence of critical temperature of transition to thesuperconducting state in the Ginzburg-Landau region T orb sc(H)=Tsc(H)−H ATsc(H), (42) where Ais a constant. Thus, the decreasing of Tsc(H) with magnetic ﬁeld causes not only faster drop but also the pe-culiar upward curvature in the critical temperature T orb sc(H) dependence from magnetic ﬁeld in correspondence with theexperimental data reported in Ref. [ 38]. B. Reentrant superconductivity in URhGe In the ﬁeld perpendicular to the spontaneous magnetization the similar approach applied to the simpliﬁed single bandmodel in weak coupling approximation yields (see Eq. (169)in the review [ 2]) the critical temperature T sc≈/epsilon1exp/parenleftBigg −1/bracketleftbig/angbracketleftbig N0(k)χuzz/angbracketrightbig cos2ϕ+/angbracketleftbig N0(k)χuyy/angbracketrightbig sin2ϕ/bracketrightbig I2/parenrightBigg , (43) where tan ϕ=Hy/handhis the exchange ﬁeld acting on the electron spins. This is the critical temperature of transition tothe superconducting state without including the orbital effect. The orbital effect suppresses the superconducting state and near the upper critical ﬁeld at zero temperature H c2y(T=0)=H0=cT2 sc (44) the actual critical temperature is Torb sc=a/radicalbig H0−Hy, (45) where a√cis the numerical constant of the order of unity. This is the usual square root BCS dependence of the criticaltemperature from magnetic ﬁeld in low temperature-high ﬁeldregion such that T orb sc(Hy=H0)=0. However, in the present case the magnitude H0itself is a function of the external ﬁeld Hy. Let us look on its behavior.Similar to Eq. ( 41) we get /angbracketleftbig N0(k)χu zz(Hy)/angbracketrightbig cos2ϕ+/angbracketleftbig N0(k)χu yy(Hy)/angbracketrightbig sin2ϕ ≈χz(Hy) χz0/angbracketleftbig N0(k)χu zz(Hy→0)/angbracketrightbig cos2ϕ +χy(Hy) χy0/angbracketleftbig N0(k)χu yy(Hy→0)/angbracketrightbig sin2ϕ. (46) Here, χz(Hy) and χy(Hy)a r et h e zand ycomponents of susceptibility in ﬁnite ﬁeld Hyandχz0andχy0are the cor- responding susceptibilities at Hy→0. Unlike Eq. ( 41)t h e ﬁeld dependence of Eq. ( 46) is not so visible. One can note, however, the different ﬁeld dependence of two summands inEq. ( 46). (i) The susceptibility along the zdirection χ z(Hy) increases with magnetic ﬁeld Hyfollowing to the decreasing of the Curie temperature according to Eq. ( 22). The growth of sus- ceptibility along the zdirection at the approaching ﬁeld Hyto Hcris conﬁrmed by the ﬁeld dependence of the NMR scatter- i n gr a t e1 /T1reported in Refs. [ 26,27]. At the same time, the increase of χz(Hy) is limited by the decrease of cos2ϕ.W ed o not know how fast it is because the magnitude of the exchangeﬁeld is not known. (ii) As the ﬁeld approaches to H crthe low temperature susceptibility χy(Hy) has a high delta-function-like peak [ 7] with magnitude more than 10 times greater than it is at Hy→ 0. The factor sin2ϕis also increased. This indicates that in URhGe, more important is the second term connected withthe metamagnetic transition. Thus, in the vicinity of metamagnetic transition one can ex- pect the increase of the critical temperature estimated withoutincluding the orbital effect according to Eq. ( 43). The radicand in equation ( 45) after being negative in some ﬁeld interval acquires the positive value as the ﬁeld approaches to H cr.T h e critical temperature Eq. ( 45) reaches maximum in the vicinity of metamagnetic transition, see Fig. 2. Similar arguments in favor of stimulation superconductiv- ity near the metamagnetic transition in the ﬁeld parallel tothebaxis can be applied to the recently discovered other superconducting compound UTe 2[39–41] isostructural with URhGe. However, in view of many particular properties ofthis material we leave this subject for future studies. In the parent compound UCoGe the metamagnetic tran- sition is absent (at least at H y<40 T) [ 42]. Hence, in this material the unusual temperature dependence of the uppercritical ﬁeld parallel to the baxis is probably mostly deter- mined by the ﬁrst term in Eq. ( 46). Near H y=Hcrat temperatures T<Tcrthe NMR spectrum is composed of two components indicating that the transitionis of the ﬁrst order accompanied by the phase separation [ 26]. Thus, in almost whole interval near H crthe superconductivity is developed in a mixture of ferromagnetic state with polar-ization along the zdirection and the ﬁeld polarized state with polarization along the ydirection. C. Upper critical ﬁeld near metamagnetic transition in UGe 2 A peculiar example of superconductivity stimulation in the vicinity of metamagnetic transition is realized in theother ferromagnetic compound UGe 2. This material has 144508-6METAMAGNETIC PHASE TRANSITION IN THE … PHYSICAL REVIEW B 103, 144508 (2021) FIG. 4. The schematic P,Tphase diagram of UGe 2[45,46]. Thick lines represent ﬁrst-order transitions and thin lines denotesecond-order transitions. The dashed line indicates a crossover while the dots mark the positions of critical points. The superconducting region is represented in the red area at the bottom. orthorhombic structure with spontaneous magnetization di- rected along the acrystallographic direction. The magnetism in UGe 2has an even more localized nature [ 2,43,44] than in related compounds URhGe and UCoGe. The superconduc-tivity exists inside of the ferromagnetic state in the pressureinterval shown in Fig. 4. Inside of this interval at P=P xthere is a metamagnetic transition from ferromagnetic state FM1 toferromagnetic state FM2 characterized by the jump of sponta-neous magnetization from smaller to larger value [ 45]. At a bit higher pressure P=P x+δPthe transition from FM1 to FM2 occurs in a ﬁnite magnetic ﬁeld applied along the direction ofspontaneous magnetization [ 47]. Near this transition in a ﬁnite ﬁeld the magnetic susceptibility along the a-axisχ astrongly increases. Hence, the critical temperature without includingthe orbital effect T sc=/epsilon1exp/parenleftbigg −1+λ/angbracketleftbig N0(k)χua/angbracketrightbig I2/parenrightbigg (47) growths up. As a result the upper critical ﬁeld in acrystal- lographic direction measured at P=Px+δPacquires non- monotonic temperature dependence shown in Fig. 5[48,49]. It is worth noting that at pressures far from metam- agnetic transition the upper critical ﬁeld parallel to the a direction does not reveal an upward curvature [ 48,49]. This important distinction from the upper critical ﬁeld behaviorin UCoGe considered in Sec. III A is related to the differ- ence of susceptibility dependence from magnetic ﬁeld alongspontaneous magnetization in these two materials. Whereas inUCoGe the susceptibility χ calong the caxis is strongly ﬁeld FIG. 5. Temperature dependence of Hc2for a ﬁeld parallel to the aaxis in UGe 2at 1.35 GPa, which is just above Px. The metamagnetic transition is detected at Hxbetween FM1 and FM2 [ 48,49]. dependent [ 37], in UGe 2the susceptibility χaalong the aaxis is practically ﬁeld independent [ 45,50]. IV . CONCLUSION We have demonstrated that in the orthorhombic ferromag- net URhGe the ferromagnetic ordering along the caxis is suppressed in the process of increase of magnetization in theperpendicular bdirection induced by the external magnetic ﬁeld. This process is accelerated by the tendency to the meta-magnetic transition which occurs at H y=Hcr=12 T. The transition of the ﬁrst order is accompanied by the suppressionof the ferromagnetic state with polarization along the caxis and the arising of magnetic state polarized along the baxis. The line of ﬁrst order phase transition is ﬁnished at the criticalend point with temperature T=T cr=4K . The uniaxial stress along the baxis causing moderate sup- pression of the Curie temperature in the absence of magneticﬁeld accelerates the Curie temperature drop in ﬁnite mag-netic ﬁeld H yand quite effectively decreases the critical ﬁeld of metamagnetic transition. As a result, the superconductingstate recovers itself in a much smaller ﬁeld and can evenbe merged with the superconducting state in the small ﬁeldsregion. The superconducting pairing is determined by the ex-change interaction between the conduction electrons and themagnetic moments localized on uranium atoms. In UCoGe the upward curvature of the upper critical ﬁeld along the caxis is mostly determined by the longitudinal magnetic susceptibility decrease along with the magnetizationsaturation. In URhGe the superconducting state suppressed inﬁeld H y≈2 T is recovered in ﬁelds interval (9–13) T near the critical ﬁeld. 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PhysRevB.66.085110.pdf	Electronic structure and isomer shifts of neptunium compounds A. Svane,1L. Petit,1W. M. Temmerman,2and Z. Szotek2 1Institute of Physic and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark 2Daresbury Laboratory, Daresbury, Warrington WA4 4AD, United Kingdom ~Received 15 March 2002; published 15 August 2002 ! The electronic structures of aNp metal and 28 Np compounds are calculated with the generalized gradient approximation to density-functional theory, implemented with the full-potential linear-mufﬁn-tin-orbitalmethod. The calculations are compared to experimental isomer shifts providing a calibration of the 237Np isomeric transition with a value of D^r2&5(240.161.3)31023fm2for the difference in nuclear radius between the excited isomeric level and the ground state. The isomer shift is primarily determined by thechemical environment. Decreasing the volume, either by external or chemical pressure, causes an f!s1d charge transfer on Np, which leads to a higher electron contact density. The possible f-electron localization in Np compounds is discussed using self-interaction corrections, and it is concluded that f-electron localization has only a minor inﬂuence on the isomer shift. DOI: 10.1103/PhysRevB.66.085110 PACS number ~s!: 71.20.Gj, 76.80. 1y I. INTRODUCTION Mo¨ssbauer spectroscopy exploiting the isomeric transition of the237Np isotope is an extremely valuable tool for inves- tigations into the electronic and magnetic properties of ac-tinide compounds. 1–6The data extracted from such measure- ments include the isomer shift, the hyperﬁne ﬁeld, and thequadrupole splitting parameter. The isomer shift is given asthe centroid position of the measured spectra with respect toa reference substance, and reﬂects the electronic density at the 237Np nucleus. The hyperﬁne ﬁeld and the quadrupole splitting are both determined from splittings of the Mo ¨ss- bauer absorption lines.3The hyperﬁne ﬁeld is related to the ordered magnetic moments and vanishes at the magnetictransition temperature. The quadrupole splitting originatesfrom an electric-ﬁeld gradient at the nuclear position, whichcan arise from either a low symmetry ~i.e., less than cubic !of the surroundings, or from a ﬁnite orbital moment in conjunc-tion with magnetic ordering. The central question in the in-terpretation of measured isomer shifts is to what extent thelocal chemical environment of the Np atom is reﬂected bythis quantity. This is the main issue addressed in the presentpaper.Ab initio calculations of the electronic structure of aNp metal and 28 Np compounds are presented, and it will be demonstrated that the isomer shift can be evaluated withan accuracy matching experimental accuracy. Furthermore,the variations in electronic structure due to alterations inchemical bonding will be discussed and related to the isomershifts. The quantum-mechanical understanding of the physics of actinide compounds presents a challenge due to the intricate nature of the partially ﬁlled 5 fshell. Compared to the rare earths, for which the 4 fstates are most often completely localized, e.g., exhibiting an atomiclike multiplet structure, 5fstates in the actinides are less inert and can play a signiﬁ- cant role in bonding, depending on the speciﬁc actinide ele-ment and the chemical environment. This is most convinc-ingly demonstrated in the elemental metals, for which alocalization transition occurs when going from Pu to Am. Inthe early actinides Th, Pa, U, Np, and Pu, the relatively de-localized 5 felectrons actively contribute to bonding, and their atomic volumes decrease in a parabolic fashion, simi-larly to the behavior seen across the transition metal series. 7 In Am, the f-electron localization is accompanied by an abrupt 16% increase in the atomic volume, and for the heavier elements Cm, Bk, and Cf this volume either remainsconstant or decreases only slightly. Pu lies at the borderline,and its very complex phase diagram suggests that thef-electron properties are of a particularly intricate nature. De- pending on the chemical properties of the ligands, actinidecompounds may exhibit different degrees of f-electron local- ization for the same actinide element. In particular, in theactinide monopnictides and monochalcogenides, all of whichcrystallize in the NaCl structure at ambient conditions, theactinide-actinide separations are larger than in the elementalmetals, and the tendency toward f-electron localization can already be observed from Np compounds onwards. 5,8–13The issue of the degree of f-electron localization in Np com- pounds, and its inﬂuence on the isomer shift, is also dis-cussed in the present paper. Over the past 30 years, the local-spin-density ~LSD!and semilocal @generalized gradient ~GGA !#approximations to density-functional theory 14,15have proven very useful and accurate in describing bonding properties of solids withweakly correlated electrons, demonstrating that the cohesiveenergy data for the homogeneous electron gas, that underliethese approximations, are representative of the conduction states in real materials. However, when 4 f-electrons are in- volved, the atomic picture with localized partially ﬁlled f shells is usually a better starting point for calculations, while for 5felectrons the most appropriate picture depends on the speciﬁc system. The most well-known extensions of LSD,capable of describing electron localization, include the self-interaction corrected ~SIC!-LSD, 16,17local density approxi- mation ~LDA!1U,18and orbital polarization methods.19 Electronic structure calculations treating felectrons as band states quite succesfully describe the equilibrium volumes ofthe early actinide metals. 20In an early study of Am, Skriver et al.21found the f-electron localization, signaled by the on- set of spin-polarization, giving rise to an almost full, andPHYSICAL REVIEW B 66, 085110 ~2002! 0163-1829/2002/66 ~8!/085110 ~8!/$20.00 ©2002 The American Physical Society 66085110-1hence nonbonding, spin-polarized f7band. Also, the high- pressure phases of Am have been succesfully described bythe standard LSD theory. 22,23The SIC-LSD method was re- cently applied to a number of actinide metals24from Np to Fm, correctly describing the itinerant nature of Np, the triva-lency of Am, Cm, Bk, and Cf, and the shift to divalency inEs and Fm. SIC-LSD calculations of Am ~Ref. 25 !and Pu ~Ref. 26 !compounds successfully described the reduction of bonding of the strongly correlated felectrons. When applied to Np compounds, 27a mixed localized/delocalized f-electron manifold was revealed. The formation of spin and orbitalmagnetic moments in Np compounds was described theoreti-cally with the orbital polarization extension of LSD by Eriks-sonet al. 28 The remainder of the paper is organized as follows. In Sec. II, a brief description of the calculational details is pre-sented. Section IIIA is concerned with our results for thecalculated properties of Np compounds, including the cali- bration of the 237Np isomeric transition, and a discussion of the electronic structure. In Sec. IIIB the inﬂuence of hydro-static pressure on Np isomer shifts is discussed, while in Sec.IIIC a possible f-electron localization is considered. Section IV concludes the paper. II. THEORETICAL METHODS Two different density-functional, based methods are used to investigate the electronic structure of Np compounds. Foran accurate determination of the charge density and compari-son to experimental isomer shifts, the full-potential ~FP! linear-mufﬁn-tin-orbital ~LMTO !method 29is used, while the energetics of f-electron localization is investigated with the SIC-LSD method.17The latter is also implemented using LMTO basis functions, but with the atomic spheres approxi-mation ~ASA!, whereby the crystal volume is approximated by slightly overlapping atom centered spheres, inside whichthe potential is taken to be spherically symmetric. The isomer shift Sis directly proportional to the elec- tronic charge density at the nucleus, the electron contactdensity; 2 S5a@rs~0!2ra~0!#, ~1! where rs(0) and ra(0) are the electron contact densities of the source and absorber materials, respectively. ais the cali- bration constant, which is proportional to the change innuclear radius upon deexcitation of the isomeric nuclearlevel, a5bD^r2&, ~2! withb59.5a03mm/(sfm2) for237Np~Ref. 2 !, wherea0is the Bohr radius. In this work, the electron contact density is calculated from ﬁrst principles and compared to experimental isomershifts with the aim to conﬁrm the above linear relationship.For this purpose, the GGA ~Ref. 15 !to density functional theory is employed in the FP-LMTO formalism. The Npnucleus is modeled as a homogeneously charged sphere of radius 1.2 A 1/3fm withA5237, the appropriate value for NpMo¨ssbauer spectroscopy. The radial mesh for the electron wave function is chosen such that 360 points fall inside theNp nucleus to guarantee an accurate representation of theelectron charge density in this region. The electron contact density is calculated as the electron density averaged overthe nuclear volume. The electron contact density is directlyinﬂuenced by the number of selectrons ~and a small contri- bution from relativistic p 1/2-electrons !occupied in the solid, since only these radial waves have a ﬁnite overlap with the nuclear region. Indirectly, the occupancy of non- selectrons has an effect on the isomer shift also, through the shieldingof thesorbitals. 30In the FP-LMTO method29the basis func- tions are smooth Hankel functions which are augmented in-side~nearly touching !mufﬁn-tin spheres with numerically determined radial wave functions, which are solutions of theradial Dirac equation in the self-consistenly determined crys-tal potential. In this way the electronic wave functions aretailored to the crystalline environment.All nonspherical con-tributions to the potential are included, and no shape ap-proximations of the crystal geometry are invoked. Three dif-ferent decay constants for the Hankel functions have beenused, and fully converged calculations have been obtained byhaving LMTO’s of s,p,d, andfcharacters of all three decay constants for MTO’s centered on Np, while for the ligands,generally one ffunction and three s,p, anddfunctions have been used. The kspace integration has been done with the tetrahedron method 31using ;200–400 kpoints in the irre- ducible wedge of the Brillouin zone, which ensures the con-vergence of the electron contact density within an accuracy of;0.1a 023. The semicore states have been included as part of the self-consistent band states using local orbitals.32These are most notably the 6 sand 6pstates of Np, but also those of the ligands which fall in the same energy range, 2–4 Ry below the Fermi level, e.g., the 3 dstates of Ga, Ge,As, and Se, the 4 dstates of In, Sn, Sb, and Te, etc. The deep core states have all been treated fully relativistically and calcu-lated self-consistently in the crystal potential, while the semi-core and valence states were treated in the scalar relativisticapproximation, i.e., including all relativistic effects exceptthe spin-orbit coupling. While spin-orbit coupling is an im-portant energy contribution in a heavy element like Np, itsinﬂuence on the charge density is minute. The present method has recently been applied to 119Sn isomer shifts33 and57Fe isomer shifts,34and reproduced results found ~for 57Fe) by the FP-LAPW method.35 The GGA as implemented above inherently relies on an itinerant view of the electrons in the solid. The considerablecorrelations among the electrons are taken into account in asubtle way, through the exchange-correlation functional.Thisscheme is quite reliable in most cases, 15and the charge den- sities even of free atoms are generally well reproduced.However, when on-site Coulomb correlations are so strongas to force felectrons to localize on speciﬁc atoms the scheme fails. The SIC-LSD method, on the other hand, al-lows for both itinerant and localized electron behavior, andthe energetics of various localization scenarios may becompared. 17,36In the SIC-LSD approach the competing en- ergies are the band formation energy and localization energy.The former is the energy gained when felectrons are allowedA. SVANE, L. PETIT, W. M. TEMMERMAN, AND Z. SZOTEK PHYSICAL REVIEW B 66, 085110 ~2002! 085110-2to hybridize with the other available electron states, while the latter quantity is assumed to be given by the self-interaction of a localized forbital. 17 The SIC-LSD scheme was implemented17within the tight-binding linear-mufﬁn-tin orbitals method.37The Np semicore 6 sand 6pstates were described with a separate energy panel. Spin-orbit coupling was fully included in theself-consistency cycles. For simplicity, we have assumed aferromagnetic arrangement of the magnetic moments. Due tothe increased complexity compared to normal band-structurecalculations, the SIC-LSD method is implemented in theASA. Unfortunately, this approximation does not allow theelectron contact density to be calculated with the requiredaccuracy. The geometrical approximations involved in theASA render the electron contact density too sensitive to thespeciﬁc choice of atomic radii. Since the experimental iso-mer shifts generally are quoted with an uncertainty of ;1 mm/s, the calculational uncertainty on the electron con- tact density should be <;10a 023, a limit which is unfortu- nately exceeded in the ASA. III. RESULTS A. Isomer shifts The electron contact density of aNp metal and 28 Np compounds was calculated with the FP-LMTO method usingthe GGA, as outlined in Sec. II. The compounds studied comprise eight Np M 3intermetallics ( M5Sn, In, Pd, Rh, Si, Ge, Ga, and Al !in the Cu 3Au structure, eight Np M2inter- metallics ~M5Fe, Co, Ni, Mn, Os, Ir, Al, and Ru !in the cubic Laves structure ~C15!, and nine Np X(X5C, N, P,As, Sb, Bi, S, Se, and Te !binary compounds in the NaCl struc- ture. Finally, the ionic compound NpO 2and the two ternary compounds NpCo 2Si2and NpCu 2Si2have also been studied. These compounds were selected on the basis of their widespread in bonding characteristics as well as their Mo ¨ssbauer data availability. 5All calculations have been done in the ex- perimentally observed structures.38 The electron contact density of237Np in the sequence of Np compounds studied is depicted against experimental iso-mer shifts in Fig. 1. All isomer shifts are given relative to a standard NpAl 2source, and taken from Refs. 1–3, 5 and 39. The linear relationship expected from Eq. ~1!is well repre- sented by the data. The best ﬁt with a straight line leads to acalibration constant a520.381 60.013a03mm/s. ~3! Using Eq. ~2!, this corresponds to the value D^r2& 520.0401 60.0013 fm2. An earlier estimate2of this con- stant was D^r2&520.027 fm2, obtained on the basis of free-ion Hartree-Fock calculations, assuming that Np31and Np41ions are representative of the Np conﬁguration in NpF 3 and NpF 4, respectively. Another estimate was given in Ref. 40, which found D^r2&520.009 fm2, obtained in a similar fashion, although no details were given in this case. The variations in electron contact density reﬂect the chemical conditions of the Np atom in the varying solid-statesurroundings. To investigate this in more detail the occupan-cies of the Np waves of speciﬁc angular characters have been monitored. The occupancies are given as the projections of the electronic eigenstates onto the Np radial waves, fl: nl5( kocc. u^ckufl&Ru2~4! forl56s,7s,6p,7p,6d, and 5f. These projections are calculated within the Np mufﬁn-tin sphere29of radius R, which depends on the compound studied. Thus some volumedependence goes into the occupancies and renders their com-parison from compound to compound somewhat uncertain.This reﬂects the fact that the concept of occupancies is illdeﬁned, and that the number of electrons of a given angularmomentum character is not an observable quantity. Never-theless, these quantities allow us to discuss trends in bondingproperties in a qualitative manner. To correct partly for thevolume dependence, we compute a similar quantity for thefree atom, n l0~R!5^flatomuflatom&R, where the integration is performed only over a sphere of radiusR, equal to the radius of the mufﬁn-tin sphere associ- ated with Np in a particular solid studied. Taking the differ- encenl2nl0(R) provides a volume-speciﬁc measure of the change in the electronic structure of the Np atom in the solid with respect to the free atom. The free atom has been calcu-lated relativistically, by solving the Dirac equation, in the 7s 26d15f4ground-state conﬁguration. As an example, let us consider NpAl 2, for which a mufﬁn-tin radius of R53.16a0is appropriate. Inside this sphere we compute nl52.33, 6.06, 1.33, and 4.00, respec- tively, for l5s,p,d, andf~including both 6 sand 7sand 6p FIG. 1. Comparison of measured isomer shifts ~in mm/s relative to a NpAl2source !toab initio calculated electron contact density ~ina023) of 28 Np compounds and the two non equivalent Np atoms inaNp. A large constant (7892000 a023) has been subtracted from the contact densities. The dashed line represents the best linear ﬁt@Eq.~3!#. Filled circles are Np M 3, withM5Sn, In, Ge, Ga, Al, Si, Pd, and Rh, in order of increasing contact density. Open circles areNpM 2Laves compounds with M5Al, Ir, Os, Ru, Ni, Mn, Co, and Fe.Triangles are Np Xchalcogenides with X5Te, Se, and S, as well as NpC.Asterisks are Np Xpnictides with X5Bi, Sb,As, P, and N, and diamonds are NpCo2Si2, NpCu2Si2, NpO2, and a-NpIIand a-NpI.ELECTRONIC STRUCTURE AND ISOMER SHIFTS O F... P H Y SICAL REVIEW B 66, 085110 ~2002! 085110-3and 7p). For the free atom a sphere of radius R53.16a0 containsnl0(R)52.56, 5.83, 0.59, and 3.89 electrons of char- acters l5s,p,d, andf, respectively. Comparing these occu- pancies it may be concluded that Np in NpAl 2has experi- enced a change in electronic structure given by Dnl 520.23, 10.23, 10.74, and 10.11 for l5s,p,d, andf, respectively. Hence a signiﬁcant charge increase occurs, as anatural consequence of the compression experienced uponformation of the solid. Most prominent is the increase in thedchannel, while fewer selectrons are present, which of course has a dramatic effect on the electron contact density.Since the sphere for the free atom contains 1.99 electrons of 6scharacter, only 0.57 of the two 7 selectrons present are actually inside the sphere, reﬂecting the rather large extent of the 7sorbital. The atom does not contain any 7 pelectrons, and only 97% of the 6 pelectrons are contained inside the sphere, demonstrating that the 6 p’s are not completely inert but start to form bands and must be treated self-consistently,as is indeed done in the present work. Similarly, only 59% of the single atomic 6 delectron and 3.89 ~97%!of the four atomicfelectrons are contained inside the sphere, reﬂecting the spatial delocalization of the former and localization ofthe latter. Analysis similar to the above for NpAl 2was carried out for all Np compounds studied, and the occupancy differences Dnlare plotted against the calculated electron contact den- sity in Figs. 2 ~a!–2~d!. Several trends may be seen in these ﬁgures. Most markedly, the Dnsquantity increases and Dnf clearly decreases with the increase of the electron contact density, while the variations of DnpandDndare less clear. Dnsis in all cases negative, which is due to the comparisonwith the free atom, whose sshells are fully occupied.Aposi- tiveDnswould indicate either a larger soccupancy in the solid than in the atom, which is not possible, or a signiﬁcant compression of the 7 swave, which does not occur. The Dnp is roughly constant ;0.2 electrons for all compounds, with the exceptions of NpCo 2Si2and NpCu 2Si2which, however, also have distinctly larger Np mufﬁn-tin spheres than any of the other systems studied. Dndshows an irregularly increas- ing tendency with increasing electron contact density, againwith the exception of the above two compounds. The corre- lation between Dn sand the electron contact density is of course to be expected, but on the other hand it is clear fromFig. 2 that the two quantities are only qualitatively related. Through the series of Np Xcompounds with the NaCl structure ~triangles and asterisks in Fig. 2 !, the electron con- tact density increases in the sequence X5Te, Se, Bi, S, Sb, As, P, N, and C. The f-electron count Dn fis monotonically decreasing in this sequence from 10.3 in NpTe to 20.4 in NpC. The accompanying increase in Dnsis only of the order 0.2 electrons, while the DnpandDndare roughly constant. Hence, the calculations reveal that Np experiences a net lossof electrons, primarily of fcharacter, with only partially com- pensating sgain, throughout the series. This charge transfer does not correlate with the standard electronegativity of theligand, but is rather related to the speciﬁc volume and thevalency of the ligand. The difference in absolute foccupancy is 0.96 between the extreme cases of NpTe and NpC among the NaCl com- pounds (n f54.18 and 3.22, respectively !, i.e., Np in NpC is in fact close to the tetravalent f3conﬁguration, while NpTe is close to a trivalent f4conﬁguration. The corresponding FIG. 2. Variations in Np occupancies Dnl, for l5s,p,d, andfcharacters in Np compounds.The occupancies are given relative to those of a free Np atom, as explained in the text. The isostructural series are marked similarly to Fig. 1.A. SVANE, L. PETIT, W. M. TEMMERMAN, AND Z. SZOTEK PHYSICAL REVIEW B 66, 085110 ~2002! 085110-4experimental isomer shift difference ~46 mm/s !is similar to that separating NpF 4and NpF 3~Ref. 5 !. The difference in electron contact density between NpC and NpTe is calculated to be 121 a023, which is only two-thirds of the difference calculated for the free Np41and Np31ions of 181 a023~Ref. 5!. This is the main reason for the larger avalue derived here @Eq.~3!#, compared to that of Kalvius et al.5The mufﬁn-tin radii used for Np in NpTe and NpC are R 53.03a0andR52.37a0, respectively. The free atom in its ground state conﬁguration contains nf(R)53.87 and nf(R) 53.64 electrons inside these spheres, i.e., of the above 0.96 electron difference in foccupancy, only 0.23 electron can be associated with the volume dependence of the occupancy,while the remaining 0.73 electron difference reﬂects a truechange in electronic structure. In fact, the changes in elec-tronic structure with respect to the free atomic charges within spheres of the respective sizes are Dn l520.08, 10.21, 10.34, and 20.42, for Np in NpC, and Dnl520.30, 10.17, 10.31, and 10.31, for Np in NpTe, respectively, for l5s,p,d, andf. Evidently, a charge loss occurs in NpC compared to NpTe, primarily in the fchannel, which is only partly compensated for by a gain in the schannel. In NpM3compounds with a Cu 3Au structure the electron contact density increases in the sequence M5Sn, In, Ge, Ga, Al, Si, Pd, and Rh. For group-III and -IV ligands, the elec- tronic structure of Np follows the same trends as in the Np X compounds, i.e., Dnfdecreases and Dnsincreases as the electron contact density increases. In this case, a slight in- crease of Dndalso contributes to the compensation of the decreasing foccupancy. For the two compounds with transition-metal ligands, M5Pd and Rh, the doccupancies are similar to the other compounds, while the Dnfvalues are less negative for the Np M3compounds than for the Np X compounds of similar high electron contact density. Among Np M2compounds with cubic Laves structure, the electron contact density increases in the sequence M5Al, Ir, Os, Ru, Ni, Mn, Co, and Fe. One observes a signiﬁcantly higher Dndfor these compounds than for the Np Xand NpM3compounds.Thevariationsthroughtheseriesof Dnp, Dnd, and Dnfare only minute, and the electron contact density increases mainly because the Dnsdoes so. In accord with experiment, the calculations ﬁnd almost no difference in electronic structure of the Np atom in the NpMn 2, NpFe 2, and NpCo 2compounds. For Np metal in the aNp structure,41both the isomer shift and the electric-ﬁeld gradient have been calculated. Thepresent results and experimental values are quoted inTable I. In the aNp structure there are two inequivalent Np sites, which according to experiment42have a modest 2.7 mm/s isomer shift difference, while the respective electric-ﬁeld gradients differ by a factor of ;3. The calculated isomer shifts reproduce the experimental ﬁndings quite well, with the isomer shift of the Np Isite lowest ~nomenclature of Ref. 41!. Experimentally, it is not possible to determine which set of Mo¨ssbauer parameters belongs to which crystallographic position. The calculations also ﬁnd signiﬁcantly differentvalues of the electric-ﬁeld gradient, although the ratio is only2.3. The signs of the electric-ﬁeld gradients have not beendetermined experimentally, but in the calculations the electric-ﬁeld gradient is found to be negative for the Np IIsite and positive for the Np Isite. As seen experimentally, the electric-ﬁeld gradient is largest on the atom with the mostnegative isomer shift, thus corroborating the assignment ofthe signals proposed above ~i.e., the I and II positions are opposite in Ref. 42, as compared to Ref. 41 and the presentwork!. The anisotropy parameter is given as h5UVxx2Vyy VzzU, whereVxx,Vyy, andVzzare the principal values of the electric ﬁeld tensor, with Vzz~the electric-ﬁeld gradient !nu- merically largest. The calculated anisotropy parameter is larger on the Np IIsite than on the Np Isite, again in accor- dance with the above assignment of the experimental data.Some uncertainty is associated with the experimental extrac-tion of the electric-ﬁeld gradient due to the uncertainty in the nuclear quadrupole moments ( Q54.0bis used in Ref. 42 !. In the calculations, the main uncertainty is in the treatment ofthefelectrons, where many-body effects may be important. With this in mind, the theory and experiment must be con- cluded to be in good agreement for aNp. For NpO 2the electronic structure in terms of Np occupan- cies resembles that of NpP and does not resemble that of an ideal Np41ion~nor that of Np in NpC !. Thus a considerable covalency persists in this compound. The NpO 2isomer shift is roughly halfway between NpF 3and NpF 4, and the electron contact density considerably lower than that of NpC. In conclusion, the Np isomer shift in intermetallics and covalent compounds cannot be directly correlated with thenumber of felectrons in the solid. The electron contact den- sity is a more complex quantity, inﬂuenced by several as-pects of the chemical bonding. B. Pressure coefﬁcients The variation of Np isomer shifts with pressure has been investigated for several cases.6,44–46Here we study the ef- fects of hydrostatic compression for the representative cases of NpSn 3, NpAs and NpAl 2. Results for the variation of the electron contact density, i. e., dr(0)/dlnV, are quoted in Table II. Experimentally, the pressure coefﬁcient of the Np isomer shift in NpSn 3is found45to be dS/dPTABLE I. The aNp metal isomer shift S@in mm/s relative to NpAl2, using the calibration of Eq. ~3!#, electric ﬁeld gradient Vzz ~in 1022V/m2), and anisotropy parameter h. Experimental values are from Ref. 42, using the isomer shift of NpO2relative to NpAl2 of26.10~4!mm/s ~Ref. 43 !. The two inequivalent Np atoms NpI and NpIIrefer to the nomenclature of Ref. 41, while the experimen- tal Mo¨ssbauer parameters tentatively are assigned by the present authors. SVzz h Site expt. theory expt. theory expt. theory NpI 26.8~3!211.0 4.57 ~5!2.99 0.24 ~2!0.15 NpII 24.1~3! 28.0 1.45 ~3!21.30 0.62 ~4!0.85ELECTRONIC STRUCTURE AND ISOMER SHIFTS O F... P H Y SICAL REVIEW B 66, 085110 ~2002! 085110-5520.35 mm/sGPa. Using the bulk modulus B574.3 611.6 GPa ~Ref. 44 !for NpSn 3, together with the calibra- tion of the present work @Eq.~3!#, this corresponds to an experimental value of dr(0)/dlnV5270a023, which is quoted in Table II. For NpAs and NpAl 2the volume varia- tion of the electron contact density can be deduced from Ref.44, which gives the measured relative size of volume varia- tions between NpAs, NpAl 2and NpSn 3as 2.1:2.5:1.0, which together with the above value for NpSn 3leads to the absolute experimental values quoted in Table II. The calculated pressure coefﬁcients deviate from the ex- perimental values by 164%, 215%, and 114%, for NpSn3, NpAl 2and NpAs, respectively, which is satisfactory for the latter two cases, but disappointingly far off for NpSn3. We have no explanation for this fact. In all cases the pressure coefﬁcient is negative implying that the electroncontact density increases with compression. The calculatedvariations in occupancies, again measured relative to the freeatom, are also quoted inTable II. Most signiﬁcantly, the rela-tivesoccupancy is seen to increase with compression, ex- plaining the increasing electron contact density. At the sametime a signifcant increase in doccupancy takes place, while withfcharacter is decreasing. Thus, in total, compression results in an f!s1dtransfer, which may also be interpreted as a transition of electrons from the spatially localized f states into the broad conduction bands, i.e., the effective va- lency of Np ~given as the number of non- felectrons !in- creases with pressure. C.f-electron localization The calculations of the Np isomer shifts as presented in Sec. IIIB assume that the Np f-electrons can be treated as normal band electrons, as described within the GGA. Thisseems to be an appropriate picture for Np metal, 47,48but in Np compounds, where the direct Np-Np distances are larger,the picture may be different. 8The degree of f-electron local- ization in the Np monopnictides and monochalcogenides wasrecently investigated on the basis of the SIC-LSD method. 27 There it was concluded that the f-electron manifold is best described as mixed localized and delocalized. In NpP, NpAs,NpS, and NpSe the most stable Np conﬁguration consists of a localized f 3shell with four additional itinerant valence electrons, of which ;1.2–1.5 attain fcharacter. In NpSb, NpBi, and NpTe there is a near degeneracy between local- izedf3andf4shells, while for NpN the most favorableconﬁguration has only two localized ( f2)felectrons and hence more delocalized felectrons. The electron contact density in principle does not register whether the electron states building the total charge densityof the crystal are localized or delocalized. Neither does itregister whether particular orbitals are heavily hybridized ornot. Hence it would be hard to determine the degree off-electron localization experimentally solely from the isomer shift. From the discussion in Sec. IIIB it is clear that thisonly holds provided the switch from localized to delocalizedbehavior is not accompanied by a change in angular charac- tersn l. The hyperﬁne ﬁeld is better suited for distinguishing between localized and delocalized f-electron behaviors, since the spin and angular momenta of a localized fnshell couple according to Hund’s rules, which leads to a signiﬁcant mag-netic moment. Narrow bands may, however, also have non-vanishing spin and orbital moments, and the fact that mea-sured moments in Np compounds 49usually differ signiﬁcantly from ideal values of Russell-Saunders coupled fnions, reﬂects the activation of the felectrons in the bond- ing. In the present work, the inﬂuence of f-electron localiza- tion on the isomer shift was studied for some selected cases, namely, the NpAl 2, NpAl 3, NpSn 3intermetallics, as well as the NpP, NpAs, NpSb, NpBi, NpS, NpSe, and NpTe binaries.The SIC-LSD method, as outlined in Sec. II, has been usedhere despite the less accurate determination of the electroncontact density in this approach. It is assumed that by vary-ing only the number of localized felectrons on Np, while keeping all other parameters ﬁxed ~in particular the atomic sphere radii !, one may be able to discuss trends in the contact density due to f-electron localization. The effect of f-electron localization is qualitatively the same in all cases. When more felectrons are treated as lo- calized in the calculations, the electron contact density de-creases. The total foccupancy increases at the expense of primarily dcharacter, while the sandpoccupancies remain unchanged. The d!ftransfer leads to a more effective shielding of the Np 6 sand 7spartial waves, and hence to decrease of the contact density. The effect is quite smallwhen going from the case when all electrons are treated asband states ~LSD!to the scenarios, described within the SIC- LSD theory, when only one or two of the felectrons are localized on each Np atom. The reason is that in the SIC-LSD approach the electron localization is accomplished es-sentially by making a Wannier transformation of occupied f bands, meaning that the same degrees of freedom are de-scribed in two different ways within LSD and SIC-LSD ap- proaches, respectively. For instance, for NpAl 3the electron contact density is lower by 1.2 a023and 2.2a023in the local- izedf1andf2conﬁgurations, respectively, as compared with the itinerant LSD ( f0) conﬁguration. Similarly, for NpAs the corresponding Np contact density differences are 1.3 a023and 1.9a023. When three felectrons are localized on each Np atom, a more pronounced drop in contact density is seen ~by 8.8a023, 5.4a023, 5.6a023, 4.7a023, 6.9a023, and 6.0 a023in NpP, NpAs, NpS, NpSe, NpAl 3, and NpAl 2, respectively, compared to the f2conﬁguration !. Finally, when four felec-TABLE II. The change in electron contact density and occupan- cies upon compression, in NpAs, NpAl2, and NpSn3. The unit of the electron contact density is a023. A negative sign means that the corresponding quantity increases upon compression. dr(0) dlnVdDnl dlnVCompound theory expt. spdf NpAs 2167 2147 20.42 20.10 20.89 10.74 NpAl2 2148 2175 20.38 10.18 20.94 10.30 NpSn3 2115 270 20.44 10.01 20.94 10.39A. SVANE, L. PETIT, W. M. TEMMERMAN, AND Z. SZOTEK PHYSICAL REVIEW B 66, 085110 ~2002! 085110-6trons are localized on each Np atom even more drastic changes occur, both with respect to d!ftransfer and contact density decrease. The latter quantity drops by 24.4 a023, 16.7a023, 11.2a023, 9.1a023, 14.8a023, and 13.2 a023in NpP, NpAs, NpSb, NpBi, NpSe, and NpTe, respectively, relative to thef3conﬁguration. The changes are most signiﬁcant for those compounds for which the f4conﬁguration is energeti- cally most unfavorable,27i. e., NpP, NpSe, and NpAs.The f4 conﬁguration has been found to be stable only for Np in NpBi, which is also the compound for which the electron contact density differs least between the f3andf4Np con- ﬁgurations. In all other Np Xcompounds, the f3localized conﬁguration was energetically favored, while for NpAl 2and NpAl3the most stable conﬁguration was found to be that of localized f2shells. The scatter of points on the calibration line in Fig. 1 corresponds to an average calculational uncer- tainty of ;8a023on the electron contact density, and we conclude that the effect of f-electron localization is only of the same order of magnitude. However, the trend is uni-formly toward lower contact density for the more localizedsystems, and since generally the f-electron localization is more pronounced for those compounds having large positiveisomer shifts in Fig. 1, this means that the localization mighttilt the calibration line slightly toward a lower value of the calibration constant a. IV. CONCLUSIONS Electronic structure calculations of aNp metal and 28 Np compounds have been presented, with an emphasis on theisomer shifts. The expected linear relationship between ex-perimental isomer shifts and calculated electron contact den-sities has been well reproduced leading to the value for thechange in nuclear radius ofD ^r2&520.0401 60.0013 fm2 between the excited isomeric level of237Np and the ground state. Earlier interpretations of Np isomer shifts relied uponfree-ionic Hartree-Fock calculations, while in the presentwork the electronic structure of the Np atom has been calcu-lated self-consistently in the proper crystalline environment,using the generalized gradient approximation to account forexchange and correlation effects. The trends in the electroncontact density have been discussed in terms of the variationin the local chemical environment as manifested through theoccupations of the Np partial waves. The isomer shift isdominated by charge transfer from the Np fstates to the conduction bands, which takes place as an effect of decreas-ing volume, caused either by external pressure or by chemi-cal pressure. Locally, within the Np mufﬁn-tin sphere this transfer appears as an f!sorf!s1dtransfer. Further methodological developments would be required to describe on an ab initiobasis the magnetic properties, and hence the Mo ¨ssbauer hyperﬁne ﬁeld, of Np compounds. 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PhysRevB.85.125127.pdf	PHYSICAL REVIEW B 85, 125127 (2012) Finite-size effects in transport data from quantum Monte Carlo simulations Rubem Mondaini,1K. Bouadim,2Thereza Paiva,1and Raimundo R. dos Santos1 1Instituto de Fisica, Universidade Federal do Rio de Janeiro Cx.P . 68.528, 21941-972 Rio de Janeiro RJ, Brazil 2Department of Physics, Ohio State University, 191 West Woodruff Ave., Columbus Ohio 43210-1117, USA (Received 1 July 2011; revised manuscript received 13 March 2012; published 26 March 2012) We have examined the behavior of the compressibility, the dc conductivity, the single-particle gap, and the Drude weight as probes of the density-driven metal-insulator transition in the Hubbard model on a squarelattice. These quantities have been obtained through determinantal quantum Monte Carlo simulations at ﬁnitetemperatures on lattices up to 16 ×16 sites. While the compressibility, the dc conductivity, and the gap are known to suffer from “closed-shell” effects due to the presence of artiﬁcial gaps in the spectrum (caused by theﬁniteness of the lattices), we have established that the former tracks the average sign of the fermionic determinant(/angbracketleftsign/angbracketright), and that a shortcut often used to calculate the conductivity may neglect important corrections. Our systematic analyses also show that, by contrast, the Drude weight is not too sensitive to ﬁnite-size effects, beingmuch more reliable as a probe to the insulating state. We have also investigated the inﬂuence of the discreteimaginary-time interval ( /Delta1τ)o n/angbracketleftsign/angbracketright, on the average density ( ρ), and on the double occupancy ( d): we have found that /angbracketleftsign/angbracketrightandρare more strongly dependent on /Delta1τaway from closed-shell conﬁgurations, but dfollows the/Delta1τ 2dependence in both closed- and open-shell cases. DOI: 10.1103/PhysRevB.85.125127 PACS number(s): 73 .63.−b, 71.10.Fd I. INTRODUCTION Metal-insulator transitions (MIT) are still a topic of intense activity.1In clean systems, an otherwise metallic system can become an insulator through the opening of a gap inthe spectrum due to electronic repulsion; they become whatare known as Mott insulators. 2Alternatively, band insulators correspond to systems in which the valence band is completelyﬁlled, even in the absence of repulsive interactions. Whenthe onsite energies are different (but regularly distributed),due to, say different atomic species, electrons may becometrapped: in this case, the system is a charge-transfer insulator.In addition, in the presence of disorder, the system maybecome an insulator as a result of electrons being unableto diffuse throughout the lattice; i.e., they may undergoan Anderson localization transition. One clear experimentalsignature of the insulating state is a vanishing conductivity asthe temperature is decreased. However, from the theoreticalpoint of view, and in the context of quantum Monte Carlo(QMC) simulations 3–7in particular, detecting an insulating state is not always straightforward. First, one necessarilydeals with systems of ﬁnite size, hence with gaps in thespectrum which may be of the same magnitude as the onesresponsible for the insulating behavior. These gaps occurat ﬁlling factors corresponding to “closed shells,” and giverise to atypical behavior in several quantities of interest;further, these closed-shell effects, which are readily seen in thenoninteracting case, can persist in the presence of interactions(see below). Second, QMC simulations are plagued by the“minus-sign problem,” 6,7which precludes the study of several low-temperature properties of the system as the electronicdensity is varied continuously. And, ﬁnally, in spite of the widevariety of quantities at our disposal to probe a MIT, such asthe compressibility, the dc conductivity, the single-particle gap,and the Drude weight, 8to name a few, they yield conﬂicting information in some cases, the origin of which is still notfully understood. For instance, under somewhat restrictiveconditions, 9–13the dc conductivity can be calculated in aconvenient way, without resorting to analytic continuation of imaginary-time QMC data to real frequencies, which may bea delicate matter; 14,15however, in the case of the Hubbard model, for some particular combinations of lattice size andelectronic densities (away from half-ﬁlling), the conductivitybehaves as if the system were insulating, which casts doubts onwhether the conditions are really met, or if it is a manifestationof closed-shell effects, or both. In the case of homogeneousversions of well-studied models, one may be able to generatedata for many different lattice sizes for a given electronicdensity (minus-sign problem permitting); in this way, a trendwith system size can be established, and any deviation fromit should be readily identiﬁed. However, this may not bethe case of systems with an overlying structure, such as asuperlattice, 17,18a checkerboard lattice, or even in the presence of staggered onsite energies (the ionic Hubbard model).19–22 Our purpose here is to shed light into these discrepancies, and to compare different approaches to detect a MIT fromQMC data; as a by-product, we will also establish a connectionbetween the behavior of the compressibility and the infamoussign problem of the fermionic determinant. The layout of thepaper is as follows. In Sec. II, we introduce the Hubbard model, and outline the computational approach used. In Sec. III,w e discuss the predictions from the electronic compressibility, when the effects of closed shells manifest themselves as a major ﬁnite-size effect. Section IVis devoted to ﬁnite-size effects on the dc conductivity and the density of states,as obtained through an inverse Laplace transform of thecurrent-current correlation function and the single-particleGreen’s function, respectively; in this section, we also providenumerical estimates for the errors involved when the dcconductivity is calculated setting the imaginary time τ=β/2, where, as usual, β≡1/T, in units such that the Boltzmann constant is unity. In Sec. V, we discuss the Drude weight in detail, and show that it does not suffer from closed-shell effects. The single-particle excitation gap is considered inSec. VI, and we ﬁnd that it suffers from the same closed-shell 125127-1 1098-0121/2012/85(12)/125127(10) ©2012 American Physical SocietyMONDAINI, BOUADIM, PAIV A, AND DOS SANTOS PHYSICAL REVIEW B 85, 125127 (2012) effects as the other probes of the insulating state, apart from the Drude weight. In Sec. VII, a systematic study leads to a connection between the sign of the fermionic determinantand the compressibility; we also discuss the inﬂuence of theimaginary-time interval on some of the data. And, ﬁnally,Sec. VIII summarizes our ﬁndings. II. MODEL AND CALCULATIONAL DETAILS The simplest model to capture the physics of Mott insulators is the repulsive Hubbard model, which is characterized by theHamiltonian H=−t/summationdisplay /angbracketlefti,j/angbracketright,σ(c† iσcjσ+c† jσciσ) +U/summationdisplay i/parenleftbigg ni↑−1 2/parenrightbigg/parenleftbigg ni↓−1 2/parenrightbigg −μ/summationdisplay ini,(1) where, in standard notation, ciσis the fermion destruction operator at site iwith spin σ=↑,↓,niσ=c† iσciσ, andni= ni↑+ni↓. We only consider nearest-neighbor hopping (indi- cated by /angbracketlefti,j/angbracketright) on a two-dimensional L×Lsquare lattice, and work in the grand-canonical ensemble; the chemical potentialμis tuned to yield the desired density ρ=/summationtext i/angbracketleftni/angbracketright/N, where N=L2is the number of lattice sites. The hopping parameter tsets the energy scale, so we take t=1; throughout this paper, we have considered the weak- to intermediate-coupling regimeU/lessorequalslant4, for which size effects are more severe. We use determinant quantum Monte Carlo (DQMC) simulations 3–5,7,23to investigate the properties of the Hubbard model. In this method, the partition function is expressed asa path integral by using the Suzuki-Trotter decompositionof exp( −βH), introducing the imaginary-time interval /Delta1τ. The interaction term is decoupled through a discrete Hubbard-Stratonovich transformation, 23which introduces an auxiliary Ising ﬁeld. This allows one to eliminate the fermionic degreesof freedom, and the summation over the auxiliary ﬁeld (whichdepends on both the site and the imaginary time) is carriedout stochastically. Initially, this ﬁeld is generated randomly,and a local ﬂip is attempted, with the acceptance rate givenby the Metropolis algorithm. The process of traversing theentire space-time lattice trying to change the auxiliary ﬁeldvariable constitutes one DQMC sweep. For most of the datapresented here, we have used typically 1000 warmup sweepsfor equilibration, followed by 4000 measuring sweeps, whenthe error bars are estimated by the statistical ﬂuctuations;when necessary, the data were estimated over an averageof simulations with different random seeds. Typically, wehave set /Delta1τ=0.125, but often data were also collected for /Delta1τ=0.0625, just to conﬁrm that systematic errors are indeed small; further, for some quantities, we have also performedextrapolations toward /Delta1τ→0 from up to eight distinct values of/Delta1τ. One should also keep in mind that since we do not use a checkerboard breakup of the lattice, our equal imaginary-timedata for U=0 are exact, so that they do not depend on the imaginary-time discretization; the τ-dependent quantities result from sampling even for U=0, but the statistical errors are negligible in this case. With the updating being carried outon the Green’s functions, 3,4,7at the end of each sweep we haveat our disposal both equal-“time” and τ-dependent quantities, which we discuss in turn. III. ELECTRONIC COMPRESSIBILITY Let us ﬁrst consider the electronic compressibility κ= ρ−2∂ρ/∂μ . Being a direct measure of the charge gap, it may be used to detect insulating phases; a major computationaladvantage is that it is a local quantity, thus ﬂuctuating verylittle within the DQMC approach. In Fig. 1(a), the density ρis plotted as a function of the chemical potential, for different lattice sizes, for the free case U=0, and at a ﬁxed temperature. If taken at face value, plateaus in the ρ×μcurves would be identiﬁed with incompressible phases, and hencewith insulating regions. However, a closer look reveals thatboth the width of the plateaus, as well as their positions, arestrongly dependent on the ﬁnite system sizes used. Given thatforU=0 the system is certainly metallic for all densities, the presence of these plateaus can be traced back to gapsin the energy spectrum of the noninteracting Hubbard modelon a ﬁnite square lattice, which is given in the usual waybyE=/summationtext q/lessorequalslantqF(ρ);σε(q), with ε(q)=− 2t(cosqx+cosqy), where qF(ρ) is the Fermi wave vector for the density ρ.I n Fig. 2, the total energy is shown as a function of the electronic density for a 10 ×10 lattice: the energy gaps do not have the same magnitude, and one should notice, in particular, the gapatρ=0.42, which is quite large in comparison with the those between levels with E<−2. This gap appears as a plateau in the data for the 10 ×10 lattice in Fig. 1(a), indicated by the horizontal dashed line. The existence of this “gap” is a 0.00.30.60.91.21.5 -4 -3 -2 -1 0 10.00.30.60.91.21.5ρL=6 L=1 0 L=1 4 ρ=0 . 4 2U=0 ; β=1 6(a) ρ=0 . 4 2L = 10; β=1 6ρ μU=0 U=2 U=4(b) FIG. 1. (Color online) Electronic density vs chemical potential: (a) for the free case, at β=16, and for different linear lattice sizes L; (b) for the L=10 lattice and different interactions U. The horizontal dashed lines highlight the speciﬁc density ( ρ=0.42) at which one plateau appears for the L=10 lattice. 125127-2FINITE-SIZE EFFECTS IN TRANSPORT DATA FROM ... PHYSICAL REVIEW B 85, 125127 (2012) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-4-3-2-101234E ρgap at ρ=0 . 4 2L = 10; U = 0; T = 0 FIG. 2. (Color online) Total energy Eas a function of electronic density ρfor the L=10 lattice at T=0 in the noninteracting limit (U=0). manifestation of what is referred to as the closed-shell problem and is characteristic of the ﬁniteness of the lattice. It should bestressed that such effects are still present when the interactionis switched on, at least up to intermediate values of U:f r o m Fig. 1(b), we see that the gap moves toward smaller values of μ asUis increased, although without any noticeable decrease in magnitude; in what follows, we illustrate further consequencesof these closed-shell effects. As the lattice size is increased,the gaps become smaller, and the plateaus in the electronicdensity become narrower, until they completely vanish in thebulk limit L→∞ . For this reason, from now on we will refer to these plateaus as pseudoinsulating states. The use of thecompressibility to locate insulating regions must therefore besupplemented with thorough analyses of the robustness andthe width of the plateaus with system size and temperature. IV . CONDUCTIVITY AND DENSITY OF STATES The optical conductivity and the density of states (DOS) are other probes of the insulating state, which are worth discussingin depth; this is especially in order, given that the use of theshortcut to calculate the dc conductivity (see below) has beenincreasingly widespread, 24,25even beyond QMC.26 First, we recall that the simulations yield imaginary-time quantities, such as the real-space single-particle Green’sfunction G(r≡i−j,τ)=/angbracketleftc iσ(τ)c† jσ(0)/angbracketright,0/lessorequalslantτ/lessorequalslantβ (2) and the current-current correlation functions /Lambda1(q,τ)≡/angbracketleftjx(q,τ)jx(−q,0)/angbracketright, (3) where jx(q,τ) is the Fourier transform of the time-dependent current-density operator jx(i,τ)≡eHτjx(i)e−Hτ, with jx(i)=it/summationdisplay σ(c† i+ˆx,σci,σ−c† i,σci+ˆx,σ). (4)Now, the ﬂuctuation-dissipation theorem yields27 /Lambda1(q=0,τ)=/integraldisplay∞ −∞dω πe−ωτ 1−e−βωIm/Lambda1(q=0,ω),(5) and linear response theory implies28 Im/Lambda1(q=0,ω)=ωReσ(ω); (6) similarly, we have27,28 G(r=0,τ)=/integraldisplay∞ −∞dωe−ωτ 1+e−βωN(ω). (7) The calculation of σ(ω) andN(ω) is then reduced to numer- ically invert these Laplace transforms at a given temperature.Here, we employ an analytical continuation method, 15through which the conductivity and the DOS can be obtained for thewhole spectrum ω. 16While there has been some debate over which type of analytic continuation method is best suited toperform these Laplace transforms, 14,29our purpose here is not to perform a systematic study of the outstanding issues;instead, we adopt one of the procedures 15to extract estimates forσ(ω), which, in turn, will be used to test the trends in the calculation of σdc, as discussed below. In Fig. 3, we compare the DOS at density ρ=0.42 for the free and interacting cases, obtained through the methoddescribed in Ref. 15. It is clear that irrespective of the value of U, the DOS vanishes at the Fermi energy for L=10, while being nonzero for L=6 and 12. Figure 4shows the optical -1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.00.30.60.9L=6 L=1 0 L=1 2N(ω) ωρ=0 . 4 2 ; β=1 6 0.00.30.60.9 U=2N(ω)U=0(a) (b) FIG. 3. (Color online) DOS spectrum for different lattice sizes atρ=0.42, and for U=0 (a) and U=2 (b). The Fermi energy (ω=0) is shown as a dashed line. The error bars represent statistical errors from different realizations. 125127-3MONDAINI, BOUADIM, PAIV A, AND DOS SANTOS PHYSICAL REVIEW B 85, 125127 (2012) 0102030 0.0 0.2 0.4 0.6 0.8 1.0 1.20102030ρ=1 . 0 U=2 β=1 6 ρ=0 . 4 2σ(ω) ωΔτ=0.0625 Δτ=0.125 L=6 L=6 L=1 0 L=1 0 L=1 2 L=1 2(a) (b) FIG. 4. (Color online) Optical conductivity from inverted Laplace transform (see text) at (a) half-ﬁlling ρ=1 and (b) for ρ=0.42, at givenUand inverse temperature, for different lattice sizes and /Delta1τ. The error bars represent statistical errors from different realizations. conductivity for the interacting case ( U=2), calculated with the same inversion method,15both at half-ﬁlling and for ρ= 0.42. We see that, while at half-ﬁlling the insulating behavior is apparent for all system sizes [ σdc(T)=limω→0σ(ω,T )→ 0], for ρ=0.42 one would be led to identify an insulating behavior if only data for a 10 ×10 lattice were available. One should also note that data for both /Delta1τ=0.125 and 0.0625 are the same, within error bars. The origin of this “false insulating”behavior can therefore be traced back to the closed-shell effectsdiscussed above, although here σ(ω) is particularly affected by the large gap required to add an electron to the closed shell of42 electrons; an analogous problem occurs at the closed-shelldensity of ρ=86/144 for the 12 ×12 lattice (not shown). In addition to suffering from the closed-shell problem, the inversion procedure adopted 15can be very costly in computer time, due to the need of very small error bars in the datafor/Lambda1. An alternative method 9–12to obtain σdc(T) consists of setting τ=β/2≡1/2Tin Eq. (5), and assuming σ(ω) admits a Taylor expansion near ω=0, the integral can, in principle, be carried out term by term in the surviving even powers of ω, and we get σdc(T)≈σ(0) dc(T)+σ(2) dc(T), (8) plus higher-order terms, with σ(0) dc(T)=1 πT2/Lambda1(q=0,τ=1/2T)( 9 ) and σ(2) dc(T)=−T2π2/parenleftbigg∂2σ ∂ω2/parenrightbigg ω=0. (10) Note that if one wants to take σ(0) dc(T) as an approximation forσdc(T),σ(2) dc(T) must be small; this should occur if the temperature is low enough, and the frequency dependence ofthe conductivity is smooth, i.e., if T/lessmuch/Omega1, where /Omega1sets a small energy scale of the problem. 10While it is hard to assess ap r i o r i0.00 0.04 0.08 0.12 0.16 0.200.111010010000.111010010000.11101001000 Tρ=0 . 6 6ρ= 0.42σ(0) dc σ(0) dc+σ(2)dc D/e2ρ= 1.0 (a) (b) (c) FIG. 5. (Color online) Temperature dependence of the dc con- ductivity [circles, zeroth order in ω, and triangles, up to second order; see Eqs. (8)–(10)], and of the Drude weight (squares; see Sec. V), forU=2, on a 10 ×10 lattice, and for different electronic densities. The error bars for σdcare due to the averaging process, while those forD/e2are due to extrapolations toward ωm→0. if this condition is satisﬁed, in the present case we have data for σ(ω) at our disposal for several temperatures; this allows us to calculate σ(2) dc(T), and check the errors involved in neglecting it in Eq. (8).F o rρ=1, we see from Fig. 5(a) thatσ(0) dc(T)r i s e s as the temperature is lowered [(red) circles], but eventually bends down at some temperature, consistently with σ(0) dc→0 asT→0; in a generic situation, in which QMC data for these lowest temperatures were not available, one could be misled to state that the system is metallic. However, when σ(2) dc(T) is included [(blue) triangles in Fig. 5(a)], the conductivity acquires the correct steady decrease with decreasing T/lessorsimilar0.15; this shows that higher-order terms may indeed be crucial attemperatures not so low. Figure 5(b) shows that for ρ=0.42, σ(0) dcsteadily decreases as Tdecreases, which is suggestive of insulating behavior; the inclusion of data for σ(2) dc(T) does not revert this trend. Since for other densities the metallic behavioris unequivocal [see, e.g., Fig. 5(c)], one concludes that the spurious effect for ρ=0.42 is yet another manifestation of the closed-shell density. We have found that these overall featuresare also present for U=4; in particular, the contribution ofσ (2) dc(T) when ρ=0.42, although signiﬁcant, is again not sufﬁcient to yield a metallic behavior, thus conﬁrm-ing that the false insulating state is indeed a closed-shelleffect. 125127-4FINITE-SIZE EFFECTS IN TRANSPORT DATA FROM ... PHYSICAL REVIEW B 85, 125127 (2012) From this analysis we conclude that extreme care must be taken when examining lim T→0σ(0) dc(T) to indicate whether the ground state is metallic or insulating; in addition, whilethe overall trend may be captured (away from closed-shell densities), attempts to ﬁt experimental data with σ(0) dc(T) should lead to error, if the temperatures involved are not too low. V . DRUDE WEIGHT We now discuss the Drude weight D, deﬁned through lim T→0Reσ(ω,T )=Dδ(ω)+σreg(ω), (11) where σreg(ω) is the regular (or incoherent) response. Approx- imants to Dare readily available from QMC simulations as8,12 ˜Dm(T) πe2≡[/angbracketleft−kx/angbracketright−/Lambda1(q=0,iωm)], (12) where ωm=2mπT is the Matsubara frequency, and /angbracketleftkx/angbracketrightis the average kinetic energy of the electrons per lattice dimension.The Drude weight is then given by lim T,m→0˜Dm≡lim T→0D(T)=D. (13) In actual calculations, both limits should be taken through extrapolations of sequences of low-temperature frequency-dependent data ˜D m(T);30ﬁnite-size effects and ﬁnite- /Delta1τ effects must also be taken into consideration when analyzingthe data. Figure 6(a) illustrates how the uniform current-current correlation function at half-ﬁlling depends on the Matsubarafrequency, with both βandUﬁxed, for different system sizes. While /angbracketleft−k x/angbracketrightis hardly dependent on the system size (see solid symbols in Fig. 6), the same does not hold for /Lambda1(0,ωm). 0123450.00.51.0U=0 U=2 U=4Λ(q=0,ωm) ωm/2πTL = 12; β=1 6 ; ρ=1 . 00.00.40.81.21.6 L=6 L=8 L=1 0 L=1 2U=2 ; β=1 6 ; ρ=1 . 0 (a) (b) FIG. 6. (Color online) Current-current correlation function /Lambda1(q=0,ωm) at half-ﬁlling ρ=1.0, as a function of ωm/2πT,w h e r e ωmis the Matsubara frequency, at a ﬁxed inverse temperature β=16. The solid symbols denote /angbracketleft−kx/angbracketright. In (a), the onsite repulsion is kept ﬁxed and the data correspond to different linear lattice sizes; in (b), data are for a 12 ×12 lattice, but for different values of U. The error bars represent statistical errors from different realizations.0123450.00.30.6U=0 U=2 U=4Λ(q=0,ωm) ωm/2πTL = 10; β= 16; ρ=0 . 4 20.00.30.6U=2 ; β= 16; ρ=0 . 4 2 L=6 L=8 L=1 0 L=1 2(a) (b) FIG. 7. (Color online) Same as Fig. 6,b u tf o r ρ=0.42; in (b), data are for a 10 ×10 lattice. Nonetheless, approximants to the Drude weight, as given by Eq.(12), do indeed approach zero with growing linear lattice sizeL, as it should for an insulating state. Figure 6(b) displays the same quantity, now for a ﬁxed system size, but for differentvalues of U; we see that as m→0,˜D m→0f o rU/negationslash=0, while ˜Dmapproaches a nonzero value for U=0. Data for ρ=0.42 and U=2 are shown in Fig. 7.W es e e that ˜Dm/πe2[Eq. (12)]f o rt h e1 0 ×10 lattice does not show any false insulating behavior, as it did for other quantities: inFig. 7(a) the difference between /angbracketleft−k x/angbracketrightand lim ωm→0/Lambda1(0,ωm) does not display a signiﬁcant change with lattice size, while inFig. 7(b) the data show that the closed-shell problem does not manifest itself over a wide range of values of U. In order to extract more quantitative data, we adopt the following procedure: For ﬁxed L,U, and β,w ep l o t ˜D m as a function of m≡ωm/2πT, and extrapolate to m→0 with the aid of a parabolic ﬁt to the data for the smallest m’s (ﬁgure not shown); we then obtain the temperature-dependentDrude weight D(T) appearing in Eq. (13). By varying the temperature, system size, and U, we can generate plots of D(T), examples of which are shown in Figs. 8and 9.A s shown in Fig. 8,f o ra1 2 ×12 lattice at half-ﬁlling, in the noninteracting case the Drude weight clearly extrapolates to anonzero value as T→0. For U> 0,D(T) vanishes at some temperature T 0(L,U ), which increases with Ufor a given L. Data for half-ﬁlling in Fig. 9show that at a ﬁxed temperature, the Drude weight vanishes as the lattice size increases; that is,the points below T 0in Fig. 8should approach the D=0 line for sufﬁciently large L. Away from half-ﬁlling, the minus-sign problem prevents us from analyzing the size dependence at very low temperatures,and we are restricted to data for β=16 for the densities ρ=0.42, and 0.66, while keeping U/lessorequalslant2. Nonetheless, some important conclusions can be drawn from our analyses ofthe data for D(T) on ﬁnite-sized lattices: (1) we have found no evidence of a vanishing Drude weight at ﬁxed, ﬁnitetemperatures in the limit L→∞ , as previously suggested for the one-dimensional case; 31(2) the dependence of Dwith 1/L, for ﬁxed both temperature and onsite repulsion, is rather weak, 125127-5MONDAINI, BOUADIM, PAIV A, AND DOS SANTOS PHYSICAL REVIEW B 85, 125127 (2012) 0.1 1-0.6-0.4-0.20.00.20.40.60.81.01.2 U=0 U=2 U=4D(T) / πe2 TL=1 2 ρ=1 . 0 Δτ= 0.125 FIG. 8. (Color online) Drude weight approximants as a function of temperature for ρ=1.0f o ra L=12 lattice for different values ofU. The error bars result from uncertainties in the extrapolations ωm/2πT→0 (see text). without suffering from closed-shell effects, thus rendering extrapolations toward L→∞ trustworthy. Once again, data for/Delta1τ=0.125 are the same as those for /Delta1τ=0.0625, within error bars. The small dependence of the Drude weight on /Delta1τ is shown in Fig. 10. Our results therefore show that the Drude weight has been hitherto unjustiﬁably overlooked as a reliable probe of themetal-insulator transition; its use should be more widespread,given that it is free from closed-shell effects, and its clear-cuttemperature dependence allows for an unambiguous charac-terization of insulating states. 0.00 0.03 0.06 0.09 0.12 0.15 0.18-1.5-1.2-0.9-0.6-0.30.00.30.60.9 ρ= 0.42 U=0 U=2 U=2 Δτ=0.0625 ρ=1 . 0 U=0 ρ=0 . 6 6 U=2 U=0 U=2 Δτ=0.0625 U=2 U=4 U=2 Δτ=0.0625D/πe2 1/L FIG. 9. (Color online) Size dependence of the (normalized) Drude weight at a ﬁxed, ﬁnite temperature β=16 for different den- sities: ρ=0.42 (circles), ρ=0.66 (squares), and ρ=1 (triangles). Empty, half-ﬁlled, and ﬁlled symbols, respectively, correspond toU=0, 2, and 4; data are for /Delta1τ=0.125, except those with crossed symbols. Error bars result from uncertainties in the extrapolations ω m/2πT→0 (see text) and are only appreciable for ρ=1.0.00 0.02 0.04 0.06-0.20.00.20.40.60.81.0 L=1 0 β=1 0 U=2 ρ=1 . 0 U=4 ρ=1 . 0 U=2 ρ=0 . 4 2 U=4 ρ=0 . 4 2D/πe2 Δτ2 FIG. 10. (Color online) Dependence of the (normalized) Drude weight with the square of the “time” interval, at ﬁxed temperature and lattice size at half-ﬁlling (triangles), and at ρ=0.42 (circles). Error bars result from uncertainties in the extrapolations ωm/2πT→0 (see text). VI. SINGLE-PARTICLE EXCITATION GAP Another quantity used to infer the transport properties of the system is the single-particle excitation gap /Delta1sp(q), which is the minimum energy necessary to extract onefermion from the system and is, essentially, related to thegap measurable in photoemission experiments. It can beobtained from the imaginary-time–dependent Green’s functionin reciprocal space, which for large τdecays exponentially, i.e., G(q F,τ)∼e−/Delta1sp(qF)τ(see, e.g., Ref. 32). We can therefore obtain /Delta1spthrough ﬁts of QMC data for the Green’s function, calculated at the Fermi wave vector for the electronic densitiesof interest. Figure 11shows the imaginary-time dependence of the Green’s function for the half-ﬁlled case. In the upperpanel, the absence of a decay in the noninteracting case isa signature of a metallic state, while the exponential decayin the lower panel results from a ﬁnite gap. The inset inFig. 11(b) compares data obtained for two values of /Delta1τ:t h e time-dependent Green’s functions lie on the same exponentialcurve, which illustrates that this quantity is also negligiblydependent on the /Delta1τused. The size dependence of the gap is shown in Fig. 12for different values of U;f o rU=2, one also sees that data for a smaller /Delta1τlie on the same curve. The limiting (i.e., L→∞ ) value of /Delta1 spincreases from zero with increasing U, as expected; it is again clear that the value of /Delta1τdoes not inﬂuence this extrapolation procedure. Figure 13shows data for the Green’s function for the density ρ=0.42. In the noninteracting case, and discarding the data forL=10, we see that the slope decreases as Lincreases, leading to a vanishing gap as L→∞ , as one would expect for a metallic system; the data for L=10 are completely off the mark, again as a result from the closed-shell densityfor this L. For the interacting case [Fig. 13(b) ], the Green’s function for L/negationslash=10 behaves in a way similar to that for the free case; again, the L=10 case behaves completely differently from the others, bearing a negative gap as the signature of 125127-6FINITE-SIZE EFFECTS IN TRANSPORT DATA FROM ... PHYSICAL REVIEW B 85, 125127 (2012) 0.20.30.40.50.60.70.8 0123450.10.20.30.40.50.60.70.80.9 U=2 ρ=1.0L=6 L=8 L=1 0 L=1 2 L=1 4U=0 ρ=1.0G(qF,τ) τ2.10 2.25 2.400.240.280.32(a) (b) FIG. 11. (Color online) Log-linear plot of the imaginary-time dependence of the Green’s function G(q,τ)a tt h eF e r m iw a v e vector qFfor different lattice sizes at half-ﬁlling ρ=1.0 for the noninteracting (a) and interacting (b) cases. The error bars in (b) are due to statistical errors from averaging over different realizationsand equivalent q Fpoints; here, /Delta1τ=0.125. The inset includes data obtained with /Delta1τ=0.0625, denoted by the corresponding crossed symbols from the main panel. 0.00 0.03 0.06 0.09 0.12 0.15 0.180.00.20.40.60.81.01.21.4 Δτ=0.125 Δτ=0.0625 U=0 U=2 U=2 U=4Δsp(q=qF) 1/Lρ=1.0 FIG. 12. (Color online) Finite-size dependence of the single- particle excitation gap /Delta1sp(qF) at half-ﬁlling for different values of the onsite repulsion. The error bars are due to the exponential ﬁts to the data for G(qF,τ) (see text). The crossed symbols denote the corresponding data for /Delta1τ=0.0625.10-410-310-210-1100101102 012345610-310-210-1100101102103104L=6 L=8 L=1 0 L=1 2 L=1 4G(qF,τ) U=2 ρ=0.42U=0 ρ=0.42 τ10.0 10.1 10.2 10.3 10.40.1(a) (b) FIG. 13. (Color online) Same as Fig. 11, but now for the electronic density ρ=0.42. the closed-shell problem. In this respect, it is interesting to have in mind that the single-particle excitation gap provides avery clear indication that a closed-shell incident is at play for agiven combination of ρandL. For completeness, we note that, similarly to half-ﬁlling (Fig. 11), the dependence with /Delta1τis negligible. VII. MINUS-SIGN PROBLEM In the present formulation of the QMC method, once the fermionic degrees of freedom are traced out, the role ofBoltzmann factor in the partition function is played by theproduct of two determinants (see, e.g., Refs. 4,6, and 7). Since one can not guarantee that this product is positive deﬁnitefor each conﬁguration of the auxiliary ﬁelds, the averages arecarried out in the ensemble of positive Boltzmann weights, atthe expense of having to divide these averages by the averagesign of the product of determinants /angbracketleftsign/angbracketright. Therefore, when /angbracketleftsign/angbracketrightbecomes signiﬁcantly smaller than 1, the average values of most quantities of interest become meaningless: this is theinfamous “minus-sign problem.” It should be noted that otherimplementations of the QMC method also run into similarproblems (see, e.g., Ref. 33). This problem has eluded a variety of attempts of solution proposed over the years (see, e.g., Ref. 7for a partial list of references). For instance, once realized that simply ignoringthe negative sign leads to serious discrepancies, 6attempts to use different Hubbard-Stratonovich transformations turned out 125127-7MONDAINI, BOUADIM, PAIV A, AND DOS SANTOS PHYSICAL REVIEW B 85, 125127 (2012) 0.00.51.0L=4 0.00.51.0 L=8 0.00.51.0 L=1 0 0.00.51.0 L=1 2 0.0 0.2 0.4 0.6 0.8 1.00.00.51.0 ρL=1 6 FIG. 14. (Color online) Average sign (circles) of the fermionic determinant and ˜ κ[(green) thick line for U=0, and (green) triangles forU=2; see text for the deﬁnition of ˜ κ] as functions of electronic density for different system sizes. Filled, half-ﬁlled, and emptycircles, respectively, denote U=2, 3, and 4; data for U=0a r e withβ=30, while for U/negationslash=0 data are with β=16. For the sake of clarity, error bars were omitted since they are smaller than datapoints. to be fruitless;34,35the minus-sign problem has been alleviated with implementations of QMC constraining the samplingprocess, 36–39from which a ground-state wave function is ob- tained. Other frameworks have been proposed to improve thesign problem, 40–42but systematic implementations comparing results for, e.g., correlation functions in the Hubbard modelare, as far as we know, still unavailable. More recently, arguments have been given 43suggesting that there is no generic solution to the sign problem; instead, in themost favorable scenario, one may ﬁnd special solutions forspeciﬁc models. 43In view of this, it is imperative to gather as much information as possible about /angbracketleftsign/angbracketright. With this in mind, we deﬁne a quantity ˜ κ≡1−ρ2κ, directly related to the compressibility κdeﬁned in Sec. III. Figure 14shows that ˜κreaches the value 1 at the densities corresponding to closed shell, as already discussed. In the same ﬁgure, wealso show /angbracketleftsign/angbracketrightas a function of ρ: interestingly, we see that it tracks ˜ κ, in the sense that, at least for U/lessorequalslant2, it is harmless at densities such that ˜ κ≈1(κ≈0), but it can be seriously deleterious to the QMC averaging process when thesystem is more compressible, especially at larger values ofU. Since a larger compressibility, in turn, corresponds to stronger density ﬂuctuations, one may conclude that theseare inherently linked with the minus-sign problem. It isworth noticing that improvements on convergence have beenachieved within both projector 44,45and ﬁxed-node39QMC simulations if closed-shell conﬁgurations are used as initialstates; in addition, in Ref. 44, it was also pointed out that the choice of closed-shell initial states led to larger /angbracketleftsign/angbracketrightthan when open-shell initial states were taken. On the other hand,shell effects have also disrupted the density dependence neededin the search for phase separation in the t-Jmodel. 46,47Thus, while indications of an interplay between closed-shell and the0.00.20.40.60.81.0 0.00 0.02 0.04 0.06 0.08 0.10 0.120.00.20.40.60.81.0U=4 ρ=0 . 4 2 μ=-3.2041 U=4 ρ≈0.5μ=-2.66 U=4 ρ≈0.7μ=-2.00(a)β=10<sign>U=2 ρ=0.42 μ=-2.4355 U=2 ρ=0.50 μ=-1.9 U=2 ρ=0.88 μ=-0.35 U=2 ρ=0.90 μ=-0.30 (b)β=16 Δτ2 FIG. 15. (Color online) Average sign of the fermionic determi- nant as a function of the square of Suzuki-Trotter time interval fora1 0×10 lattice, with (a) β=10 and (b) β=16. Black squares and (green) up triangles, respectively, correspond to the closed-shell density ρ=0.42 and to ρ≈0.5; half-ﬁlled and ﬁlled symbols, respectively, correspond to U=2 and 4. minus-sign problem have been suggested in the past, Fig. 14 presents the ﬁrst systematic evidence of this connection. It is also instructive to examine the behavior of /angbracketleftsign/angbracketrightwith /Delta1τ. Figure 15compares data for one lattice size L=10, but for different values of U,β, and the chemical potential μ. Forβ=10, we see that for the closed-shell conﬁguration ρ=0.42,/angbracketleftsign/angbracketright≈1 for all /Delta1τ in the range considered, for both U=2 and 4; this feature is maintained when β is increased to 16, illustrating the harmlessness of /angbracketleftsign/angbracketrightat the closed-shell density. Doping slightly away, e.g., for anopen-shell conﬁguration with ρ≈0.5,/angbracketleftsign/angbracketrightremains almost independent of /Delta1τ forU=2, but acquires a signiﬁcant dependence for U=4, leading to low values for small /Delta1τ; forβ=16,/angbracketleftsign/angbracketright≈0 for all /Delta1τin the relevant range. Worse still, for U=4 andρ≈0.7,/angbracketleftsign/angbracketrightis very close to zero for all values of /Delta1τconsidered, for both β=10 and 16; this should not come as a surprise, since Fig. 14shows that ˜ κvanishes near this density for the 10 ×10 lattice. In Fig. 16, we display the dependence of two average local quantities with /Delta1τ2for two ﬁxed values of the chemical potential, and for U=4 andβ=10. Notwithstanding the fact that systematic errors of order /Delta1τ2are expected as a result of the Suzuki-Trotter decomposition, Fig. 16(a) shows that for μ≈− 3.2, the proportionality constant is quite small, so that ρ=0.42 over the whole range of /Delta1τ; in the open-shell case, for which /angbracketleftsign/angbracketrightdeteriorates with decreasing /Delta1τ(see Fig. 15), the dependence of ρwith/Delta1τ2is noticeable. By contrast, Fig. 16(b) shows that the double occupancy d≡/angbracketleftni↑ni↓/angbracketright, (14) follows the expected linear dependence with /Delta1τ2in both cases. This indicates that whenever /angbracketleftsign/angbracketrightis strongly dependent on /Delta1τ, one can still obtain meaningful averages by using solely the data for the largest values of /Delta1τ to extrapolate toward /Delta1τ=0; although with less conﬁdence, the same procedure could be adopted for β=16. 125127-8FINITE-SIZE EFFECTS IN TRANSPORT DATA FROM ... PHYSICAL REVIEW B 85, 125127 (2012) 0.400.450.500.55 0.00 0.02 0.04 0.06 0.08 0.10 0.120.010.020.03μ=-3.2041 μ=-2.66ρU=4; L=10; β=10d Δτ2(a) (b) FIG. 16. (Color online) Dependence of average values of (a) electronic density and (b) double occupancy (see text) with the square of the Suzuki-Trotter time interval for two values of the chemical potential; U,β, and lattice size are ﬁxed. The extrapolated values, obtained from the ﬁtting of a straight line through all points, are shown in red at /Delta1τ=0. VIII. CONCLUSION In conclusion, we have thoroughly examined the behavior of several quantities obtained through QMC simulations atﬁnite temperatures for the homogeneous Hubbard model onthe square lattice and commonly used to locate insulatingbehavior. Our results show that “closed-shell” effects, whichintroduce important (though artiﬁcial) gaps in the spectrum,may lead to false insulating behavior of the compressibility, of the conductivity, and of the charge gap at certain combinationsof occupation and linear lattice size L; in situations in which a long series of lattice sizes can not be obtained,this may jeopardize extrapolations toward L→∞ .W eh a v e also assessed corrections to the dc conductivity, which areneglected when a Laplace transform is avoided through asimplifying prescription, and found that the latter is not gener- ically valid due to the absence of a sufﬁciently small energyscale in the problem; although quite appealing, ﬁttings toexperimental data with the conductivity thus obtained shouldbe avoided. The Drude weight, on the other hand, suffers frommore controllable ﬁnite-size and ﬁnite-temperature effects.At half-ﬁlling, and at a ﬁxed low temperature, it vanishesw i t hap o w e rl a wi n1 /L, the exponent of which depends onU; away from half-ﬁlling, the Drude weight is only weakly dependent on either temperature and system size, beingfree from the spurious behavior found in other quantities.Therefore, amongst all quantities discussed here, the Drudeweight is certainly the most reliable one to use in situationsfor which the data are limited to a restricted set of systemsizes. In addition, we have also presented numerical evidence showing that the sign of the fermionic determinant tracksthe compressibility: for densities at which the system is“incompressible,” as a result of a gap due to the ﬁniteness of thelattice, /angbracketleftsign/angbracketright≈1, at least for U/lessorequalslant2. However, in-between two successive incompressible densities, /angbracketleftsign/angbracketrightdeteriorates steadily as Uincreases. This behavior is suggestive that strong density ﬂuctuations may be linked to the minus-sign problem.We have also investigated the inﬂuence of the imaginary-timeinterval /Delta1τ on the behavior of /angbracketleftsign/angbracketrightand of some (local) average quantities. All analyzed quantities can be ﬁtted to a linear dependence with/Delta1τ 2, as expected from the Suzuki-Trotter discretization, although at the closed-shell density, the slopes for both /angbracketleftsign/angbracketright and the density ρare very small. The /Delta1τ2dependence is indicative that for some densities, one can conﬁdently use datafor “large” /Delta1τ(i.e., those leading to /angbracketleftsign/angbracketright/greaterorsimilar0.5) to perform extrapolations (toward /Delta1τ→0) of average values. ACKNOWLEDGMENTS We are grateful to R. T. Scalettar for useful discussions. 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PhysRevB.99.075306.pdf	PHYSICAL REVIEW B 99, 075306 (2019) Fe/GeTe(111) heterostructures as an avenue towards spintronics based on ferroelectric Rashba semiconductors Jagoda Sławi ´nska,1,2Domenico Di Sante,3Sara Varotto,4Christian Rinaldi,4Riccardo Bertacco,4and Silvia Picozzi1 1Consiglio Nazionale delle Ricerche, Istituto SPIN, UOS L’Aquila, Sede di lavoro CNR-SPIN c/o Universitá “G. D’Annunzio, ” 66100 Chieti, Italy 2Department of Physics, University of North Texas, Denton, Texas 76203, USA 3Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland Campus Süd, Würzburg 97074, Germany 4Department of Physics, Politecnico di Milano, 20133 Milano, Italy (Received 8 August 2018; revised manuscript received 24 January 2019; published 19 February 2019) By performing density functional theory and Green’s functions calculations, complemented by x-ray photoemission spectroscopy, we investigate the electronic structure of Fe/GeTe(111), a prototypicalferromagnetic/Rashba-ferroelectric interface. We reveal that such a system exhibits several intriguing propertiesresulting from the complex interplay of exchange interaction, electric polarization, and spin-orbit coupling.Despite a rather strong interfacial hybridization between Fe and GeTe bands, resulting in a complete suppressionof the surface states of the latter, the bulk Rashba bands are hardly altered by the ferromagnetic overlayer.This could have a deep impact on spin-dependent phenomena observed at this interface, such as spin-to-chargeinterconversion, which are likely to involve bulk rather than surface Rashba states. DOI: 10.1103/PhysRevB.99.075306 I. INTRODUCTION Ferroelectric Rashba semiconductors (FERSC) are a novel class of relativistic materials whose bulk spin texture is inti-mately linked to the direction of the ferroelectric polarization,thus allowing direct electrical control over the spin degreesof freedom in a nonvolatile way [ 1–7]. Such property holds a large potential for spintronics, or more speciﬁcally for spin or-bitronics [ 8], aiming at injection, control, and detection of spin currents in nonmagnetic materials. Whereas the Rashba effecthas been mostly studied at surfaces where inversion symmetryis intrinsically broken, in FERSC the so-called Rashba bulkbands originate from inversion symmetry breaking due to thepresence of a polar axis existing by deﬁnition in ferroelectrics.Moreover, it has been predicted that the spin texture switchesby changing the sign of polarization, thus it can be reversedby electric ﬁeld. This fundamental prediction of spin texture switchability via changing the sign of electric polarization ( /vectorP) has been recently conﬁrmed experimentally in the prototype material GeTe [ 9], representing a ﬁrst milestone towards the exploita- tion of the GeTe in spintronic devices, such as, for example,the Datta-Das spin transistor [ 10,11]. However, the design process of future applications requires a more detailed charac- terization due to the need of spin injection in any spintronics devices. Theoretical and experimental studies of GeTe-based interfaces containing ferromagnets are particularly important.For this purpose, Fe thin ﬁlms seem to be a natural tar- get material [ 12]. Importantly, Fe/GeTe heterostructures have been recently realized experimentally and have been shown to yield a spin-to-charge conversion (SCC) in spin pumping experiments [ 13], thus opening a realistic perspective for the FERSC-based spintronics and making a need of further theoretical input even more urgent.In this paper, we employ density functional theory (DFT) to investigate realistic Fe/GeTe interfaces, modeled by Te-terminated α-GeTe(111) surfaces capped by multilayer ﬁlms of bcc Fe. As mentioned above, the Fe layers on GeTesurfaces are interesting for spin injection, but they can bealso considered as a two-phase multiferroic [ 14–16]. Such composites have been the subject of intensive studies in the past years given the perspective of controlling ferroelectricity(ferromagnetism) by the magnetic (electric) ﬁeld due to thecoupling between the magnetic and the ferroelectric proper-ties in these materials. Whereas Fe is a standard ferromagneticcomponent considered in two-phase multiferroics, Fe /BaTiO 3 being the prototype material [ 17–23], GeTe has never been considered as a ferroelectric counterpart. Therefore, in orderto clearly understand the coupling mechanisms occurring atthe interface, we will ﬁrst analyze the structural, electronicand magnetic properties of the interfacial atoms, assumingdifferent thicknesses of Fe ﬁlms ranging from 1 monolayer(ML) to 6 MLs. Such a strategy, apart from providing essentialinformation about the magnetoelectric coupling, will alsoallow us to identify when the interface properties become robust, an aspect relevant for the design of novel GeTe-based devices. As a next step, we will focus on the Fe/GeTe spinstructure. The peculiar spin texture of bulk and surface GeTebands was studied in detail in our previous works [ 1,3,9]; here, we will focus on the inﬂuence of Fe on GeTe bulk Rashbabands and their hybridization. We will analyze not only thedependence of the spin texture on the thickness of the ferro-magnetic ﬁlm, but also on the electric polarization /vectorPwhich can be parallel or antiparallel to the surface’s normal, and,ﬁnally, on the magnetic anisotropy. Our theoretical analysisis complemented by x-ray photoemission spectroscopy (XPS)measurements on Fe overlayer deposited on (111)-orientedGeTe thin ﬁlms. 2469-9950/2019/99(7)/075306(8) 075306-1 ©2019 American Physical SocietyJAGODA SŁAWI ´NSKA et al. PHYSICAL REVIEW B 99, 075306 (2019) II. METHODS Te-terminated α-GeTe(111) surface has been modeled using a hexagonal supercell consisting of a sequence ofﬁve ferroelectric bulk GeTe unit cells stacked along the z axis [ 24]. The slabs contain one additional Te layer at the top surface which allowed us to simultaneously study twodifferent conﬁgurations with the dipole pointing outwards(P out) and inwards ( Pin), represented by the bottom and top sides of the slab, respectively [see Figs. 1(a) and1(a/prime)]. As demonstrated in our previous works [ 3,9], for bare GeTe surfaces only the Poutsurface is stable, which can be rational- ized recalling that ferroelectric GeTe consists of alternatinglong and short Ge-Te bonds, and the preferred terminationcorresponds to the breaking of (weaker) long bonds; as aconsequence, the Te-terminated surface always relaxes to theP outconﬁguration. Below, we present a detailed characteri- zation of the two conﬁgurations as our results indicate thatthe capping with Fe layers can stabilize both P outand Pin phases. The Fe/GeTe interfaces have been modeled assuming the pseudomorphic matching between GeTe and bcc Fe(111)surfaces; this seems a reasonable strategy given a relativelysmall mismatch of 4% between the in-plane lattice parameterof the GeTe surface (4.22 Å) and the lattice constant ofthe bcc Fe (2.86 Å). Moreover, recent low-energy electrondiffraction results clearly indicate the hexagonal symmetry ofthe interface which further supports suitability of our model[13]. Next, we consider different stacking orders of Fe layers with respect to the substrate. The GeTe(111) hexagonal cellcontains three different high-symmetry sites to place the Featom: above the topmost Te atom (top), above the topmost Geatom (hcp), or above the second Te atom (fcc). Stacking of twoor more Fe layers can arrange in six different conﬁgurations.We have considered all possible stacking orders of 1–3-MLand 6-ML Fe and further analyzed properties of the moststable ones. Our spin-polarized DFT calculations were performed using the Vienna Ab initio Simulation Package (V ASP) [ 25,26] equipped with the projector augmented-wave method forelectron-ion interactions [ 27,28]. The exchange-correlation (XC) interaction was treated in the generalized gradient ap- proximation in the parametrization of Perdew, Burke, andErnzerhof (PBE) [ 29]. In all simulations, the electronic wave functions were expanded in a plane-wave basis set of 400 eV ,whereas the total energy self-consistency criterion was set to,at least, 10 −7eV. The integrations over the Brillouin zone (BZ) were performed with (10 ×10×1) Monkhorst-Pack /Gamma1- centered k-point mesh, which was increased to (18 ×18×1) for magnetization anisotropy energy (MAE) calculations. Partialoccupancies of wave functions were set according to the ﬁrst-order Methfessel-Paxton method with a smearing of 0.1 eV .As for the considered slabs, in all relaxations, we kept ﬁxedthe central most bulklike block and allowed all other atomsto move until the forces were smaller than 0.01 eV /Å. The surfaces energies were evaluated from additional calculations performed in symmetrized supercells, composed of two equiv-alant surfaces on both sides of the slab and a paraelectric central bulk where Ge and Te atoms remain equidistant; thesame symmetric supercells were employed in the accurate ( ) ( )( ) ( ) ( )( ) ( ) FIG. 1. Schematic side view of optimized (a) α-GeTe(111) (b) 3-ML-Fe/GeTe(111), and (c) 6-ML-Fe/GeTe(111) Poutsurfaces. (a/prime)–(c/prime) The same as (a)–(c) for Pinsurfaces; in (a/prime), the geometry is unrelaxed because the Pinsurface turns out to be unstable. Te, Ge, and Fe atoms are represented by green, red, and yellow balls, respectively. Only the topmost surface layers are shown in eachcase. The primitive hexagonal bulk unit cells (marked by black rectangles) contain six atoms; in the surface calculations, we use, at least, ﬁve such bulk blocks stacked along the zdirection. Gray arrows denote the direction of /vectorP. The interlayer distances are given in angstroms. (d) The same interlayer distances plotted vs number of atomic layers. Our slabs by construction contain both P outand Pinsurfaces, therefore, the left-hand (right-hand) side of the plot represent the interlayer distances of the former (latter), whereas the central part corresponds to constant values in the bulk GeTe. The interlayer distances in GeTe(111), 3-ML Fe/GeTe(111) and 6-ML Fe/GeTe(111) are plotted in black (diamonds), red (circles), and blue(square), respectively; note that, due to the fact that the relaxations never lead to the P instate within a bare GeTe(111) surface, the corresponding line ends in the bulk region. The Pinsurface is omitted, and only the Poutsurface is included. calculations of total energies and MAEs. Dipole corrections were used for the modeling of bare GeTe(111) surfaces. The electronic structures and spin textures shown in the form of projected density of states (PDOS) ( /vectork,E) maps and corresponding maps of spin-polarization /vectors(/vectork,E) were calcu- lated employing the GREEN package [ 30] interfaced with the 075306-2Fe/GeTe(111) HETEROSTRUCTURES AS AN A VENUE … PHYSICAL REVIEW B 99, 075306 (2019) ab initio SIESTA code [ 31]. For these reasons, our most stable conﬁgurations were recalculated self-consistently with SIESTA using similar calculation parameter values (XC functional,ksamplings, etc.). The atomic orbital basis set consisted of Double-Zeta Polarized (DZP) numerical orbitals strictlylocalized by setting the conﬁnement energy to 100 meV .Real-space three-center integrals were computed over three-dimensional grids with a resolution equivalent to 1000 Ryd-bergs mesh cutoff. The fully-relativistic pseudopotential for-malism was included self-consistently to account for the SOC[32]. The electronic and spin structures for the semi-inﬁnite surfaces have been computed following Green’s functionsmatching techniques following the procedure described inRefs. [ 33–36]. To experimentally support the calculations, the chemical interaction between Te and Fe has been monitored by XPSas reported in the Supplemental Material [ 37]. Photoelectrons were excited using an Al Kαx-ray source ( hν=1486.67 eV) and analyzed through a 150-mm hemispherical energy ana-lyzer Phoibos 150 (SPECS TM), yielding an acceptance angle of 6◦and a ﬁeld of view of 1 .4m m2. III. RESULTS AND DISCUSSION A. Structural, electronic, and magnetic properties Figures 1(a)–1(c) and 1(a/prime)–1(c/prime)show the most stable geometries for PoutandPinsurfaces, respectively. Since bare GeTe(111) surfaces have been already studied in our previousworks, their structures are shown here only for comparisonwith Fe/GeTe(111) interfaces. We have omitted the geome-tries of the simplest cases of 1 ML and 2 ML (both areincluded in the Supplemental Material [ 37]) because they are clearly unlikely to be used in real devices where the metalliccontacts for spin injection require stable ferromagnetic ﬁlmsof several layers which ensure preservation of the magneticmoments. We brieﬂy note that the case of 1-ML Fe revealsa strong preference of the atoms to interdiffuse into thesubsurface; in fact, we found such behavior for two moststable among three studied stacking conﬁgurations and forboth P outandPinsurfaces. Such a tendency can be attributed to the fact that the lattice constant of GeTe is large enoughto allow Fe atoms to ﬁt easily below the surface especiallywhen adsorbed at the fcc or hcp sites of the GeTe(111)surface. Certainly, the geometry of GeTe containing buried Featoms induces a strong reorganization of the electric dipolesclose to the surface, leading to changes in the electronicstructures including a partial suppression of the bulk Rashbabands. Our calculations revealed a similar interdiffusion alsofor two out of the six studied 2-ML-Fe/GeTe conﬁgurations(see the Supplemental Material [ 37]). Similar to the case of 1-ML Fe/GeTe, the initial conﬁgurations with Fe atoms athcp and fcc sites clearly preferred to interdiffuse, whereasthose containing Fe atoms in the topconﬁgurations seem to be protected from such structural reorganization, most likelybecause it would require also an in-plane shift of the adatom.This tendency explains the lack of interdiffusion in the 3-ML-Fe/GeTe slabs as in bcc stacking in our high-symmetrymodels at least one of three Fe atoms must occupy the topsite. Remarkably, we have found a very similar trend ofinterdiffusion in analogous 1-ML Co/GeTe and 2-ML Co/GeTe indicating that the ﬁnal GeTe(111) reconstructioncritically depends on the exact positions of the adatoms. As a matter of fact, the XPS investigation of chemical properties at the Fe/GeTe interface indicates a clear tendency to interdiffusion. This is seen already in thin ﬁlms of Fe grown on GeTe at room temperature (RT) by molecular beam epitaxy, and the phenomenon is enhanced by annealing at 200 ◦C. Even though the experiments have been performed on 3-nm-thick Fe layers, as at the ultralow coverages considered in this paper an island growth has been observed, the XPS results qualitatively conﬁrm the theoretical trend. Of course, real ﬁlms studied at RT are far more complex than the ideal systems used for the simulations with defects and vacan-cies largely affecting the interdiffusion. Simulations of such systems would require signiﬁcantly larger supercells; such a detailed structural analysis is beyond the scope of this paper. As can be noted from Figs. 1(b)–1(b /prime), in 3-ML Fe/GeTe the iron atoms are not found anymore to diffuse in the subsur-face, although the geometries of the interface still reveal somepeculiarities which emerge due to the ultrathin character of thecapping layers. For example, although the relaxations of theP outside of the slab performed for different stacking orders of Fe lead to several metastable ﬁnal geometries, the Pinsurfaces always end up in the conﬁguration presented in panel (b/prime), mainly because the 3-ML stacking order is removed in thiscase. Such behavior can be clearly excluded in case of thickerﬁlms as will be shown below. Noteworthy, the presence of Fe not only allows for the existence of stable P intermination but even makes this con- ﬁguration more favorable ( +1.19×10−2eV/˚A2) compared to the Poutsurface. We attribute its stability to a formation of a strong bond between Fe and the topmost Te layer whichcompensates an unfavorable breaking of the short bond attheP insurface. Finally, Figs. 1(c)–1(c/prime)show the structural properties of the most stable 6-ML-Fe/GeTe(111) conﬁgura-tions. Although the GeTe surfaces remain roughly the sameas in case of capping with 3-ML Fe, the ferromagnetic lay-ers adapt different geometries; the preferred stacking orderis different than in the 3-ML Fe/GeTe(111) and identicalfor the P outandPinmodels. Interestingly, both PoutandPin surfaces reveal shorter adsorption distances, which is better captured in panel (d) where all the interlayer distances ofconsidered interfaces are summarized. Finally, we note thatin 6-ML Fe/GeTe(111) all initial Fe conﬁgurations for bothpolarization phases preserved their stacking after relaxation;this can be intuitively explained by the fact that the structure of6-ML Fe already approaches a crystalline one, thus preventingany severe reordering of the outer layers. Again, the P in conﬁguration was found to be signiﬁcantly more stable than Pout(+1.14×10−2eV/˚A2). Further insight on Fe/GeTe(111) interfaces have been gained by performing the calculations of MAEs due to inter-facial magnetocrystalline (single-ion) anisotropy, neglectingdipolar contributions; the corresponding values are listed inTable Ifor 6-ML Fe coverage. For both directions of /vectorP, magnetocrystalline anisotropy favors a perpendicular-to-planeconﬁguration of the Fe magnetic moment (MM). On theother hand, the P inconﬁguration reveals a notably larger magnetocrystalline MAE (by as much as 0.3 meV) which 075306-3JAGODA SŁAWI ´NSKA et al. PHYSICAL REVIEW B 99, 075306 (2019) TABLE I. MAE, spin magnetic moments, and orbital moments of the topmost surface atoms calculated for the PoutandPinphases. MAEs are evaluated as ( E[001]– E[100]) per surface unit cell, Te 1 and Ge 1refer to the interfacial surface atoms, whereas Fe nwith n=1–6 denote iron atoms stacked as shown in Fig. 1with Fe 1 denoting the one closest to the GeTe surface. The magnetic moments are expressed in μB. Pout Pin MAE =−0.43 meV MAE =−0.73 meV Atom MS L[001] MS L[001] Ge1 −0.01 0.00 0.00 0.00 Te1 −0.02 0.00 −0.03 0.00 Fe1 2.06 0.06 2.18 0.08 Fe2 2.62 0.06 2.60 0.06 Fe3 2.32 0.06 2.40 0.07 Fe4 2.70 0.06 2.68 0.06 Fe5 2.56 0.07 2.63 0.07 Fe6 2.82 0.08 2.82 0.08 conﬁrms the existence of a magnetoelectric coupling in the interfaces with thin Fe layers. The MAE dependence on theFe thickness is a delicate issue. It is reported in the literaturethat the single-ion anisotropy in pure iron thin ﬁlms stronglyoscillates with the number of Fe layers up to quite largethicknesses [ 38,39]. In our Fe/GeTe case, for Fe thicknesses larger than 6 ML, the simulations become too expensive fromthe computational point of view, and results with the requiredaccuracy cannot be reported. However, the simulations of8-ML Fe/GeTe(111) and 10-ML Fe/GeTe(111) conﬁrmed thatmagnetocrystalline anisotropy favors the perpendicular-to-plane conﬁguration. The impact of interfacial magnetocrystalline MAE on the real arrangement of Fe magnetization can be understood bycomparing it with the magnetostatic energy term responsiblefor shape anisotropy. As previously reported by Bornemannet al. [40], for small Fe thickness the dipolar energy can be estimated by using the classical concept of magnetostaticenergy [ 39], which quantitatively reproduces the quantum- mechanical results. For a thin ﬁlm, the volume magnetostaticenergy density can be written as E M=1/2μ0M2 Scos2θ, (1) where μ0is the vacuum permittivity, MSis the saturation magnetization, and θis the angle between the sample magne- tization and the out-of-plane direction. The shape anisotropydensity per surface unit cell, to be compared with the MAEvalues reported in Table I, can be calculated multiplying E M by the volume of the unit cell. This is given by the product of the area of the surface unit cell ( A=15.45˚A2as the hexagonal cell of GeTe has a lattice parameter of 4 .22˚A and three Fe atoms per cell) by the average layer spacing in bcc-like Fe/GeTe along the pseudocubic [111] direction (about 0.7˚A) multiplied by the number of layers ( n). For the case of 6 ML considered in Table I, assuming a Fe bulk saturation magnetization of M S=1.74×106A/m, we obtain a shape anisotropy energy density per unit cell of 0.77 meV . Thisvalue is very close to that of the single-ion MAE for the P in FIG. 2. (a) Density of states projected on interfacial (a) Te and (b) Fe atoms calculated in a 6-ML-Fe/GeTe(111) slab without in- cluding spin-orbit coupling. Spin majority (minority) is shown inthe upper (lower) panel. The solid (dashed) lines correspond to the P out(Pin) surface, whereas the shaded area denotes the PDOS of the bulklike atoms; we report in (a) the Te atom in the middle of the slab (bulk α-GeTe phase) and in (b) the atom in the middle of the Fe multilayer. polarization and larger than that for Pout, thus indicating that for 6 ML the large change in MAE induced by polarizationreversal can inﬂuence the overall anisotropy displayed by theFe ﬁlm. Ultrathin Fe ﬁlms can have an out-of-plane easyaxis, whereas at larger Fe thickness, the volume magnetostaticcontribution largely exceeds the single-ion MAE, which isconﬁned at the interface, and the magnetization reorients inthe ﬁlm plane. From the estimation above, the spin reorienta-tion transition should take place at a critical thickness on theorder of 6 ML, corresponding to about 0.42 nm. This is fullyconsistent with our previous result showing that 5 nm of Fe onGeTe(111) display a clear in-plane hysteresis loop [ 13]. In addition, Table Ireports the values of MMs calculated for the surface atoms, including the orbital moments obtainedfor the [100] magnetic orientation. Any differences betweenP outandPinconﬁgurations can be noted mainly at the Fe atoms located close to the semiconductor; the interaction between Feand Te seems to be responsible for the appreciable reductionof Fe MMs which experience a sizable decrease (on the orderof 0.1μ B) when changing from the Pinto the Poutsurface. We emphasize that in both cases the interfacial Te atom reveals asmall magnetic moment (0 .02–0.03μ B) antiferromagnetically coupled to that of Fe. When inspecting the DOS projectedon interfacial Te and Fe presented in Fig. 2we can, indeed, observe a strong hybridization of Fe and Te states within thewhole considered energy window, including the region closeto the Fermi energy where the Fe 3 dstates induce both spin 075306-4Fe/GeTe(111) HETEROSTRUCTURES AS AN A VENUE … PHYSICAL REVIEW B 99, 075306 (2019) majority and minority in the DOS of Te. X-ray photoemission data reported in the Supplemental Material [ 37] support the existence of a preferential interaction between Fe and Te.Whereas Ge peaks do not move in energy upon formation ofthe Fe/GeTe interface, Te display a core-level shift towardshigher binding energy, compatible with that reported in caseof Fe ﬁlms deposited on Bi 2Te3[41]. In closer detail, although the presence of Fe induces new states, the GeTe gap decreasesand makes both interfaces conducting; for P outthere is a sort ofpseudogap very close to the Fermi energy, whereas for Pinthe metallic behavior becomes robust. The value of DOS projected on the interface Te atom increases by ∼2.5 times at EFwhen changing from PouttoPin, a result which might have important consequences for any spin-injection-related processor exploitation for ME junctions. B. Electronic structures and spin texture Figures 3(a)–3(a/prime)show the GeTe(111) band structures cal- culated in the form of projected density of states PDOS( /vectork,E) for each polarization conﬁguration; the surface and bulk pro-jections are distinguished by using white and red shades, re-spectively. The folded bulk Rashba bands are indicated by thearrows. Next, we present in panels 3(b)–3(b /prime), side by side, the analogous electronic structure maps calculated for the 3-MLFe/GeTe(111). Although the geometry of the surface is hardlyaffected compared with bare GeTe [see the interlayer dis-tances in Fig. 1(d)], the inﬂuence of Fe on the band structure is indeed huge. In particular, the surface states are completelyremoved at the P outside and strongly suppressed at the Pin surface due to the several Fe states residing inside the bulk gap (highlighted in blue in the maps). Similar intense states coverpractically the whole displayed energy region but withoutaffecting the most relevant bulk Rashba bands. In order togain further insight on the screening properties of GeTe withrespect to interface electronic states, we additionally present,in panels 3(c) and3(d) [correspondingly 3c /primeand 3d/primefor the Pinsurface), the density of states projected only on topmost surface atoms, i.e., ﬁrst and second Te-Ge bilayers. Certainly,the projection on the two topmost atomic layers (Ge 1+Te1) reveals the presence of Fe-induced states, which points fora strong hybridization at the interface. However, these bandsfade out quite rapidly with the depth; at the third and fourthatomic layers (Ge 2+Te2) we can observe only weak traces of few of them. Instead, the bulk Rashba bands are alreadyclearly visible, showing that interface states are efﬁcientlyscreened by GeTe, consistent with its semiconducting behav-ior. It is worthwhile to note that different results were found byKrempaský et al. in an apparently similar multiferroic system Ge (1−x)Mn xTe where the structure of bulk bands depends on Mn concentration [ 42,43]. In particular, it was found that the bulk Rashba bands possess a Zeeman gap between the Diracpoints, whose presence is attributed to rather strong exchangeinteraction and its interplay with SOC. Our results do notreveal such effect in Fe/GeTe due to the fact that Fe induceschanges mainly at the surface of GeTe(111), in contrast toGe (1−x)Mn xTe where magnetic impurities are homogeneously distributed in the sample. In fact, even in case of strongerinteraction (such as interdiffusion in 1-ML Fe/GeTe), we havenot found any traces of the Zeeman gap.Panels 3(e)–3(e /prime)display the corresponding spin-resolved density of states, /vectors(/vectork,E) calculated for the quantization axis (QA) normal to the surface, which was found to be the moststable one (see Table I). We visualized the spin textures separately for three components s x,sy, and sz. In the case of in-plane projections, we omitted the directions of the BZalong which the spin texture was negligible. These directionsare consistent with the expected Rashba-like spin-momentumlocking, i.e., the spin components are found to be nonzeroonly when perpendicular to the momentum. Expectedly, thestrongly spin-polarized iron bands manifests mainly in thes zcomponent parallel to the QA, overlaying the still visible bulk states, whereas the in-plane projections sxandsyreveal mainly the spin texture of bulk bands, hardly modiﬁed bythe interaction with Fe. Setting the QA along x(y)3(f)–3(f /prime), yields a similar scenario, with spin textures of Fe clearlydominating the s x(sy) components, and purely bulk Rashba bands manifesting in the complementary projections of the /vectors. This shows that the hybridization does not strongly dependon the QA. Finally, the electronic/spin properties of 6-MLFe/GeTe(111) reported in Fig. 4resemble those calculated for 3-ML Fe/GeTe(111); the only differences are several new Festates well visible in PDOS but hardly interacting with thebulk continuum. This conﬁrms the robustness of the interfaceelectronic structure, both with respect to the Fe thickness andwith respect to the stacking order. Overall, our electronic and spin structure calculations show that the Fe/GeTe(111) interface, in general, produces stronginterface hybridization but leaves the bulk Rashba bandshardly altered already at the subsurface level, which seemspromising for their further exploration and exploitation. Re-cent spin-pumping experiments have indeed revealed the SCCin this system, which could originate from interface or bulkRashba states, according to the inverse Edelstein or inversespin Hall effects, respectively [ 44–46]. Our results shed light on this subject as we have seen that the creation of theFe/GeTe interface tends to suppress surface Rashba states.Thus, we suggest that SCC phenomena in this system couldbe mainly related to bulk Rashba states, whose dispersionand spin character are almost unaffected by the presence ofthe Fe/GeTe interface. On the other hand, DFT calculationsindicate that the creation of the Fe/GeTe interface has a deepimpact on the GeTe band structure at the interface. Startingfrom the typical band lineup of a p-doped material, consistent with the large concentration of Ge vacancies in real ﬁlms,the bulk Rashba states in the valence band shift downwardsby about 0.5 eV , and the Fermi level moves towards thecenter of the gap. In these conditions, spin transport at theinterface is expected to involve also states from the conductionband, having different Rashba parameters and, thus, possiblyleading to a different behavior with respect to that expected incase of p-doped GeTe. A detailed explanation of the mech- anism, including determining the exact role of bulk or/andinterface states would require additional out-of-equilibriumspin-transport calculations, which are, however, beyond thescope of the present paper. On the other hand, in analogy withprevious works [ 13], our results point to the crucial role of the interface between a ferromagnet and a Rashba materialin determining the spin-transport properties. The engineering 075306-5JAGODA SŁAWI ´NSKA et al. PHYSICAL REVIEW B 99, 075306 (2019) FIG. 3. (a) Momentum and energy-resolved density of states projected on the surface and bulk principal layers of the bare GeTe(111) Pout surface calculated within the semi-inﬁnite model via the Green’s functions method. The red shades represent the bulk continuum of states, whereas the white lines correspond to purely surface bands. The yellow arrows indicate the folded bulk Rashba bands. The inset shows the Brillouin zone and high-symmetry points of the hexagonal surface unit cell. (b) The same as (a) for 3-ML Fe/GeTe(111). The main color scheme same as in (a); the projections on iron atoms are additionally highlighted in blue. (c) Density of states analogous to (b) but projected only at ﬁrst topmost Te and Ge atoms at the surface. (d) The same as (c), but for second layers of Te and Ge atoms. (e) Spin texture corresponding tothe density of states displayed in (b) assuming QA perpendicular to the surface. The left-hand, middle, and right-hand panels represent its three components s x,sy,a n d sz, respectively. Since sx(sy) achieve non-negligible values only along /Gamma1-M(/Gamma1-K), the perpendicular /Gamma1-K(/Gamma1-M) lines are omitted. The orange (green) shades correspond to positive (negative) values of spin-polarization density. (f) The same as (e) for the QA setin-plane along the xaxis. (a /prime)–(f/prime) The same as (a)–(f) calculated for the Pinsurfaces. 075306-6Fe/GeTe(111) HETEROSTRUCTURES AS AN A VENUE … PHYSICAL REVIEW B 99, 075306 (2019) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) FIG. 4. (a) The electronic structure of 6-ML Fe/GeTe(111) calculated within semi-inﬁnite surface model for the Poutsurface. (b)–(d) Spin texture projected on the x,y,a n d zaxes, respectively. In (b), the /Gamma1-Kdirection is omitted because the spin texture was found to be zero. The QA was set perpendicular to the surface. The color scheme is the same as in Fig. 3.( a/prime)–(d/prime) The same as (a)–(d) but calculated for the Pin surface. of the interface, by properly choosing the ferromagnet and/or by inserting an intermediate layer, provides an additionaldegree of freedom to optimize spin-dependent effects, suchas SCC. This calls for further theoretical and experimentalinvestigations of this system. IV . SUMMARY To summarize, we have performed a detailed analysis of multilayer Fe ﬁlms deposited on α-GeTe(111) surfaces. First, we have revealed that the Fe capping layers stabilize the GeTesurfaces with the two different polar conﬁgurations close tothe surface with the electric dipole pointing outwards andinwards, in contrast to bare GeTe surfaces where the latteris unstable. The ultrathin Fe thicknesses (1 ML and 2 ML)modify the structure of GeTe(111), consistent with the exper-imental results pointing to a large interdiffusion of Fe ionswithin the GeTe substrate. However, starting from 3 ML,the topmost surface atoms in GeTe remain hardly affected,indicating that for any practical purposes rather thick ferro-magnetic ﬁlms should be employed. Finally, we unveiled theelectronic structures and spin textures, including the effectsof both directions of /vectorPand different thicknesses of the Fe overlayer. In all cases, the Fe states strongly hybridize withthe GeTe surface, leading to a suppression of the Rashbasurface states. Importantly, the bulk Rashba bands remainalmost electronically unaffected and are only altered at theinterfacial GeTe layer, consistently with the expected good screening properties of GeTe. In conclusion, our theoretical and experimental work paves the way for the understanding of the microscopic mechanismsat the heart of potentially useful new generations of interfaces.The key idea of combining ferromagnetic overlayers withactive ferroelectric Rashba semiconductors may grasp theavenue to engineer ground-breaking spintronics devices bymaking use, for example, of the already proven efﬁciency ofFe/GeTe heterostructures [ 13]. ACKNOWLEDGMENTS We are grateful to Dr. J. I. Cerdá for helpful comments on the calculation strategy in SIESTA. The work at CNR-SPINwas performed within the framework of the NanoscienceFoundries and Fine Analysis (NFFA-MIUR Italy) Project.This work has been supported by Fondazione Cariplo andRegione Lombardia, Grant No 2017-1622 (Project ECOS).The experimental work reported in the Supplemental Ma-terial [ 37] has been partially carried out at Polifab, the micro- and nanofabrication facility of Politecnico di Mi-lano. 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PhysRevB.91.235307.pdf	PHYSICAL REVIEW B 91, 235307 (2015) Theory of magnetothermoelectric phenomena in high-mobility two-dimensional electron systems under microwave irradiation O. E. Raichev Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospekt Nauki 41, 03028 Kyiv, Ukraine (Received 10 March 2015; revised manuscript received 17 May 2015; published 8 June 2015) The response of two-dimensional electron gas to a temperature gradient in perpendicular magnetic ﬁeld under steady-state microwave irradiation is studied theoretically. The electric currents induced by the temperaturegradient and the thermopower coefﬁcients are calculated taking into account both diffusive and phonon-dragmechanisms. The modiﬁcation of thermopower by microwaves takes place because of Landau quantizationof the electron energy spectrum and is governed by the microscopic mechanisms which are similar to thoseresponsible for microwave-induced oscillations of electrical resistivity. The magnetic-ﬁeld dependence ofmicrowave-induced corrections to phonon-drag thermopower is determined by mixing of phonon resonancefrequencies with radiation frequency, which leads to interference oscillations. The transverse thermopower ismodiﬁed by microwave irradiation much stronger than the longitudinal one. Apart from showing prominentmicrowave-induced oscillations as a function of magnetic ﬁeld, the transverse thermopower appears to be highlysensitive to the direction of linear polarization of microwave radiation. DOI: 10.1103/PhysRevB.91.235307 PACS number(s): 73 .43.Qt,73.50.Lw,73.50.Pz,73.63.Hs I. INTRODUCTION Electron transport in two-dimensional (2D) electron sys- tems placed in a perpendicular magnetic ﬁeld remains oneof the most important subjects in condensed matter physics.Recently, it was established that a variety of interestingtransport phenomena takes place [ 1] in the region of mod- erately strong magnetic ﬁeld, where the Shubnikov–de Haasoscillations of the electrical resistivity are suppressed becauseof thermal smearing of the Fermi level. In particular, there areseveral kinds of magnetoresistance oscillations [ 1] observed in high-mobility 2D systems such as GaAs quantum wells,strained Ge quantum wells, and electrons on liquid heliumsurface. Among these phenomena, the microwave-inducedresistance oscillations (MIRO) appearing under microwave(MW) irradiation of 2D electron gas [ 2–5] are studied most extensively. Their origin is brieﬂy described as follows. Inthe presence of the MW excitation, when absorption andemission of radiation quanta by the electron system take place,both the distribution function and scattering probabilities ofelectrons are modiﬁed. These modiﬁcations correlate withthe oscillating density of electron states owing to Landauquantization in the magnetic ﬁeld B, thus leading to corre- sponding oscillating contributions to resistivity determined bythe ratio of the radiation frequency ωto the cyclotron frequency ω c=\|e\|B/mc . The period and phase of MIRO, as well as the temperature and power dependence of their amplitudes,are in agreement with this physical picture supported by adetailed consideration of microscopic mechanisms of MIROin the past years [ 6–11]. According to both experiment and theory, the MW irradiation strongly affects the longitudinal(dissipative) resistivity and has a much weaker effect on thetransverse (Hall) resistivity. More recent experiments uncoverthe existence of small corrections, sensitive to the directionof the electric ﬁeld of microwaves (MW polarization), toboth longitudinal and Hall resistivities [ 12,13], also in general agreement with theory. Apart from its inﬂuence on electrical resistivity, the MW excitation is expected to modify other transport coefﬁcientsof 2D electrons, for the same reasons as explained above. The thermoelectric coefﬁcients are of special interest in this connection. The study of thermoelectric phenomena inmagnetic ﬁelds has a long history, and the fundamentals ofthis topic, with applications to bulk conductors, are reviewedin Ref. [ 14]. The electrical response to temperature gradient ∇Tis described by the longitudinal (Seebeck) and transverse (Nernst-Ettingshausen) components of thermoelectric power (brieﬂy, thermopower). These coefﬁcients are determined by two mechanisms: the diffusive one, when electrons aredirectly driven by the diffusion force due to temperaturegradient in electron gas, and the phonon-drag one, whenelectrons are driven by a frictional force between them andphonons propagating along the temperature gradient. The con-tribution of both these mechanisms in magnetothermopowerof 2D electron systems has been studied in a number of theoretical and experimental works [ 15–21]( s e ea l s or e v i e w paper Ref. [ 22] and references therein). The quantum effects are commonly observed in strong magnetic ﬁelds, whereShubnikov–de Haas oscillations of thermopower coefﬁcientstake place [ 22]. In high-mobility GaAs quantum wells, the phonon-drag thermopower shows another kind of quantumoscillations, related to resonant phonon-assisted scattering of electrons between Landau levels [ 20]. This occurs in the region of moderately strong magnetic ﬁelds, below the onsetof the Shubnikov–de Haas oscillations, which is favorable forobservation of MW-induced quantum effects. There are two main ways in which the MW irradiation can inﬂuence the thermopower. First, this irradiation leadsto nonequilibrium electron distribution that has nontrivialdependence not only on electron energy but also on the tem-perature of electron gas. Both the diffusive and phonon-dragcontributions to thermoelectric coefﬁcients should be sensitiveto such changes. Next, the MW irradiation in the presenceof magnetic ﬁeld considerably inﬂuences electron-phononscattering. This causes an effect on electrical resistivity [ 23] under conditions when the probability of electron-phononscattering is comparable to that of elastic scattering byimpurities. At temperatures below 4.2 K these conditions 1098-0121/2015/91(23)/235307(16) 235307-1 ©2015 American Physical SocietyO. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) are realized only in very pure 2D systems. In contrast, the effect of microwaves on electron-phonon scattering is alwaysimportant for thermoelectric properties, since the phonon-drag mechanism determined by this scattering gives a verysigniﬁcant [ 15], if not a major, contribution to thermopower of 2D electrons. The above consideration also suggests that in spite of the same microscopic mechanisms involved in both cases, theeffect of microwaves on magnetothermoelectric coefﬁcientsof 2D electrons should be different from their effect onmagnetoresistance. The classical Mott relation between thediffusive current responses to temperature gradient and toelectric ﬁeld is not expected to be valid under MW excitation,even for the moderately strong magnetic ﬁelds. Moreover, onemay presume that both longitudinal and transverse componentsof thermopower oscillate with magnetic ﬁeld in a way differentfrom MIRO, and their dependence on MW polarization is alsodifferent. Therefore, there is enough motivation for a theoret-ical study of the inﬂuence of MW irradiation on thermopowerof 2D electron systems in perpendicular magnetic ﬁeld. Thepresent paper is devoted to this previously unexplored problem. In the linear response regime considered in the following, the current density jis given by the general expression j=ˆσE−ˆβ∇T, (1) where Eis the electric ﬁeld in the plane ( x,y) of the 2D electron system. It is assumed that the 2D system is macroscopicallyhomogeneous so that the chemical potential μdoes not depend on 2D coordinate r. Under conditions when no conduction cur- rents ﬂow in the system, one gets E=ˆα∇T. The thermopower tensor ˆ αdescribes the voltage drop as a result of temperature gradient. It is given by ˆ α=ˆρˆβ, where the resistivity tensor ˆρis the matrix inverse of the conductivity tensor ˆ σ.T h e theoretical approach presented below is based on calculationof thermoelectric tensor ˆβin the presence of the ac ﬁeld of microwaves by using the method of the quantum Boltzmannequation [ 1,8,10,23] established in the previous calculations of the conductivity tensor ˆ σ. As both ˆβand ˆσare known, the thermopower is found straightforwardly. The results arepresented for the case of moderately strong magnetic ﬁeld,when the Shubnikov–de Haas oscillations are still suppressed,but quantum oscillations due to Landau quantization exist inhigh-mobility 2D systems. Such oscillations are caused byinelastic scattering of electrons between Landau levels asa result of electron interaction with acoustic phonons of aresonant frequency ω ph(magnetophonon effect [ 20,24–29]) and with microwaves of frequency ω. These two kinds of inelastic processes actually interfere, leading to combined fre-quencies ω ph±ωwhose ratio to ωcdetermines the periodicity of the quantum magneto-oscillations [ 23]. As shown below, such oscillations exist in both longitudinal and transversethermopower caused by the phonon drag, while the diffu-sion part of the thermopower follows the MIRO periodicitydetermined by the single frequency ω. The phonon-drag part of the MW-induced contribution to transverse thermopower isfound to be comparable with that of longitudinal thermopower.Since the transverse thermopower is much smaller than thelongitudinal one in classically strong magnetic ﬁelds, it isdramatically affected by MW irradiation, demonstrating giantmicrowave-induced oscillations and a high sensitivity to the direction of MW polarization. The paper is organized as follows. Section IIdescribes the main formalism including description of ac electric ﬁeld gen-erated by incident electromagnetic radiation, electric currentin the presence of temperature gradient, and kinetic equationfor 2D electrons with collision integrals for electron-impurityand electron-phonon interactions. In Sec. IIIthe kinetic equation is solved and the tensor ˆβis presented and discussed both for the equilibrium case and under MW irradiation.Section IVcontains expressions for longitudinal and transverse components of thermopower tensor ˆ α, their discussion, and presentation of the results of numerical calculations of thesecomponents as functions of magnetic ﬁeld and polarizationangle. More discussion and concluding remarks are given inthe last section. II. GENERAL FORMALISM Throughout the paper, one uses the system of units where Planck’s constant /planckover2pi1and Boltzmann constant kBare both set to unity. The electron spectrum is assumed to be isotropicand parabolic, with effective mass m. The Zeeman splitting of electron states is neglected. Consider a monochromatic electromagnetic wave normally incident on the surface containing a 2D layer (the direction ofincidence coincides with the direction of the magnetic ﬁeld,along the zaxis). The electric ﬁeld of this wave near the layer is written, in the general form, as E (i) t=E(i) ωRe[ee−iωt] =E(i) ω√ 2Re/braceleftbigg/bracketleftbigg e−/parenleftbigg 1 i/parenrightbigg +e+/parenleftbigg 1 −i/parenrightbigg/bracketrightbigg e−iωt/bracerightbigg , (2) where eis the polarization vector. The second part of this equation represents the wave as a sum of two circularly polarized waves, e ±=(ex±iey)/√ 2=κ±e±iχ;χis the angle between the main axis of polarization of E(i) tand the x axis, and κ±are real numbers characterizing ellipticity of the incident wave (they are normalized according to κ2 ++κ2 −= 1). A circular polarization means that either κ+orκ−is equal to zero. In the case of linear polarization, κ+=κ−=1/√ 2s o that e ±=e±iχ/√ 2. The screening of electromagnetic waves due to the presence of free carriers in the 2D layer changes thepolarization angle and ellipticity [ 30,31], so the electric ﬁeld in the layer, E t, differs from E(i) tand has the following form: Et=Eω√ 2Re/braceleftbigg/bracketleftbigg (ω−ωc)s−/parenleftbigg 1 i/parenrightbigg +(ω+ωc)s+/parenleftbigg 1 −i/parenrightbigg/bracketrightbigg e−iωt/bracerightbigg , (3) where s±=e± ω±ωc+iωp. (4) Here ωpis the radiative decay rate that determines the cyclotron line broadening because of the electrodynamicscreening effect. It is given by ω p=2πe2ns/mc√ /epsilon1∗, where nsis the electron density,√ /epsilon1∗=(1+√/epsilon1)/2, and /epsilon1is the dielectric permittivity of the sample material. Next, 235307-2THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) Eω=E(i) ω/√ /epsilon1∗.I nE q s .( 3) and ( 4), it is assumed that transport relaxation rate, νtr, which determines electron mobility, is much smaller than either \|ω±ωc\|orωp. The relation νtr/lessmuchωp is a very good approximation for high-mobility samples with typical electron density ns>1011cm−2. Apart from the ac ﬁeld Et, the electron system is driven by a weak static (dc) ﬁeld E. To take into account the inﬂuence of both these ﬁelds on 2D electrons, it is very convenientto use a transition to the moving coordinate frame (seeRef. [ 10] and references therein). Then, the quantum kinetic equation for electrons derived by using Keldysh formalism fornonequilibrium electron systems (see details in Refs. [ 10,23]) contains the effect of external ﬁelds only in the collisionintegral. The radiation power is assumed to be weak enoughto neglect the inﬂuence of microwaves on the energy spectrumof electrons: the spectrum remains isotropic and the densityof states is not affected by the radiation. Further, the magneticﬁeld is assumed to be weak enough so there is a large numberof Landau levels under the Fermi energy. The kinetic equationwritten for the electron distribution function f εϕaveraged over the period 2 π/ω takes the form pεϕ m·∇fεϕ+ωc∂fεϕ ∂ϕ=Jεϕ,Jεϕ=Jim εϕ+Jph εϕ,(5) where εis the electron energy, pεϕ=pε(cosϕ,sinϕ) with pε=√ 2mεis the electron momentum in the 2D layer plane, andϕis the angle of this momentum. Since the dependence of all quantities on the spatial coordinate ris considered as a parametric one, the coordinate index at the distributionfunction and collision integrals is omitted. The density ofelectric current is given by the expression j=e π/integraldisplay dεD ε/integraldisplay2π 0dϕ 2πpεϕfεϕ−σ⊥ˆ/epsilon1E−cˆ/epsilon1∇Mz,(6) where σ⊥=e2ns/mω c=\|e\|nsc/B is the classical Hall con- ductivity and Dεis the density of electron states expressed in the units m/π .N e x t ,ˆ /epsilon1=(01 −10) is the antisymmetric unit matrix in the space of 2D Cartesian indices. The last term in the expression (6) is given by the spatial gradientof magnetic moment Mof electrons per unit square. This moment arises because of diamagnetic currents circulating inthe electron system. Actually, the last term in Eq. ( 6) does not contribute to the total current across any ﬁnite sample. However, the necessity of taking into account this term (its bulkanalog is −c[∇×M]) in the expression for the local current density was emphasized in studies of magnetothermoelectricphenomena a long time ago [ 14,32]. Being expressed through the distribution function, the magnetic moment comprises twoterms: M z=−m πB/integraldisplay dε[Dεε−/Pi1ε]fε, (7) where fεis the isotropic (averaged over ϕ) part of elec- tron distribution function, and /Pi1ε=/integraltextε −∞dε/primeDε/primeis the an- tiderivative of Dε. In the ideal 2D electron system, the ﬁrst and the second terms in Mzcorrespond to magnetization due to bulk and edge currents, respectively [ 33]. In the case of the equilibrium Fermi distribution function fε= {exp[(ε−μ)/T]+1}−1, it is easy to transform Eq. ( 7)t oa well-known thermodynamic expression Mz=−∂/Omega1/∂B , where /Omega1=− (Tm / π )/integraltext dεD εln{1+exp[(μ−ε)/T]}is the thermodynamic potential per unit area. In the absence of any collisions, Jεϕ=0, the local current is nondissipative, j=j(n), where j(n)=−cm πB/integraldisplay dε/Pi1 εˆ/epsilon1∇fε−σ⊥ˆ/epsilon1E. (8) If coordinate dependence of fεexists solely due to tem- perature gradient, one has ∇fε=(∂fε/∂T )∇T. The integral term in Eq. ( 8) is reduced to nondissipative thermoelectric current −ˆβ∇Tﬂowing perpendicular to ∇T, with ˆβ= (cm/πB )/integraltext dε/Pi1 ε(∂fε/∂T )ˆ/epsilon1. If chemical potential μentering fεalso depends on coordinate, the integral in Eq. ( 8) produces an additional term proportional to ∇μ. This term, with the aid of the identity ∂fε/∂μ=−∂fε/∂ε, can be combined with the last term of Eq. ( 8), leading to the form σ⊥ˆ/epsilon1∇ζ, where ζ=/Phi1+μ/e is the electrochemical potential and /Phi1 is the electrostatic potential determining the electric ﬁeldE=−∇/Phi1. The electric ﬁeld or, in general, the gradient of the electrochemical potential induced as a result of a temperaturegradient is derived from the expression j (n)=0. This leads to diagonal thermopower tensor ˆ α=ˆ1α, where ˆ1 is the unit 2×2 matrix and α=m πens/integraldisplay dε/Pi1 ε∂fε ∂T. (9) Substituting the equilibrium distribution function into Eq. ( 9), one gets the well-known result α=−S \|e\|ns,S=−∂/Omega1 ∂T, (10) where Sis the entropy of 2D electron gas per unit area. For strongly degenerate electron gas, μ=εF, one has S= (π2/3)nsT/εF. The collision integrals Jim εϕandJph εϕstanding in Eq. ( 5) describe, respectively, electron-impurity and electron-phononscattering [ 23]: J im εϕ=/integraldisplay2π 0dϕ/prime 2π∞/summationdisplay n=−∞ν(\|qεn\|)[Jn(\|Rω·qεn\|)]2 ×Dε+nω+γn[fε+nω+γnϕ/prime−fεϕ], (11) Jph εϕ=/integraldisplay2π 0dϕ/prime 2π/summationdisplay λ/integraldisplay∞ −∞dqz 2πm ×∞/summationdisplay n=−∞/braceleftbig MλQ−[Jn(\|Rω·q− εn\|)]2[(NλQ−+fεϕ) ×fε−ωλQ−+nω+γ−nϕ/prime−(NλQ−+1)fεϕ] ×Dε−ωλQ−+nω+γ−n+MλQ+[Jn(\|Rω·q+ εn\|)]2 ×[(Nλ−Q++1−fεϕ)fε+ωλQ++nω+γ+nϕ/prime −Nλ−Q+fεϕ]Dε+ωλQ++nω+γ+n/bracerightbig , (12) 235307-3O. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) where Jnis the Bessel function, ν(q)=mw(q) is the isotropic elastic scattering rate expressed through the Fourier transformw(q) of the correlation function of random potential of impurities, q εn=pεϕ−pε+nωϕ/primeis the momentum transferred in scattering in the presence of ac ﬁeld, and Rωis a complex vector describing the coupling of the electron system to thisﬁeld: R ω=eEω√ 2mω(s++s−,(s+−s−)/i). (13) The interaction with phonons is considered under approx- imation of bulk phonon modes. The phonons are charac-terized by the mode index λand three-dimensional phonon momentum Qwith out-of-plane component q z. The squared matrix element of the electron-phonon interaction potential isrepresented as M λQ=CλQIqz. The squared overlap integral Iqz=\| /angbracketleft0\|eiqzz\|0/angbracketright\|2is determined by the conﬁnement potential which deﬁnes the ground state of 2D electrons, \|0/angbracketright.T h e function CλQcharacterizes electron-phonon scattering in the bulk. The in-plane momenta transferred in electron-phononcollisions, q ± εn, are found from the equation q± εn=pεϕ− pε±ωλQ±+nωϕ/prime, where Q±=(q± εn,qz) and ωλQis the phonon frequency. The effect of the static electric ﬁeld on thecollision integrals is given by the energies γ n=VD·qεn andγ± n=VD·q± εn, where VD=c[E×B]/B2=(c/B)ˆ/epsilon1E is the drift velocity in the crossed electric and magneticﬁelds. In the case of electrons interacting with long-wavelength acoustic phonons in cubic lattice, the expressions for C λQand dynamical equations needed for determination of ωλQare the following: CλQ=1 2ρMωλQ⎡ ⎣D2/summationdisplay ijeλQieλQjqiqj +(eh14)2 Q4/summationdisplay ijk,i/primej/primek/primeκijkκi/primej/primek/primeeλQkeλQk/primeqiqjqi/primeqj/prime⎤ ⎦,(14) /summationdisplay j/bracketleftbig Kij(Q)−δijρMω2 λQ/bracketrightbig eλQj=0, (15) Kij(Q)=[(c11−c44)q2 i+c44Q2]δij+(c12+c44) ×qiqj(1−δij). (16) HereDis the deformation potential constant, h14is the piezoelectric coupling constant, and ρMis the material density. The sums are taken over Cartesian coordinate indices. Thecoefﬁcient κ ijkis equal to unity if all the indices i,j,k are different and equal to zero otherwise. Next, e λQiare the components of the unit vector of the mode polarization, andK ij(Q) is the dynamical matrix expressed through the elastic constants c11,c12, andc44. Finally, NλQin Eq. ( 12) is the distribution function of phonons. In the presence of thermal gradients this functiondepends not only on the frequency ω λQbut also on the direction ofQ. In the general case, NλQcan be represented as a sum of symmetric ( s) and antisymmetric ( a) parts satisfying the relations Ns λ−Q=Ns λQandNa λ−Q=−Na λQ, respectively. The drag of electrons by phonons is caused by the antisymmetricpart. In the linear regime, Nais proportional to ∇TwhileNsis reduced to the equilibrium distribution function. In particular,one often uses the following form [ 34]: N λQ=NωλQ+∂NωλQ ∂ωλQωλQ TτλuλQ·∇T, (17) obtained from a linearized kinetic equation for phonons in the relaxation time approximation. Here NωλQ=[exp(ωλQ/T)− 1]−1is the equilibrium (Planck) distribution function, τλis the relaxation time of phonons, and uλQ=∂ωλQ/∂Qis the phonon group velocity. Notice that a simple expression uλQ= sλQ/Qrelating the group velocity to the sound velocity sλis valid only in the isotropic approximation. For elastic wavesin real cubic crystals the direction of u λQdoes not generally coincide with the direction of Q, though the symmetry relation uλ−Q=−uλQis always valid. Substituting Eq. ( 17) into the expression for the collision integral Jph εϕ, it is convenient to write the latter as a sum of two parts, Jph εϕ=Jph(0) εϕ+Jph(1) εϕ, (18) where Jph(0) εϕ contains the equilibrium phonon distribution NωλQonly, while Jph(1) εϕ is determined by the anisotropic nonequilibrium correction to phonon distribution [second termin Eq. ( 17)] and is proportional to ∇T. The second term in Eq. ( 18) is responsible for the phonon-drag contribution to electric current. By using Eqs. ( 11) and ( 12), one can directly check the identity/integraltext dεD ε/integraltext dϕJεϕ=0 expressing the electron conser- vation requirement. It is worth emphasizing that the collisionintegrals Eqs. ( 11) and ( 12) are written in the general form valid for an arbitrary relation between radiation frequency ω, phonon frequency ω λQ, and electron energy ε.I nR e f .[ 23] the collision integrals are written in a simpler form validunder the assumptions ω/lessmuchεandω λQ/lessmuchε. For degenerate electron gas, the electrons contributing to electric current haveenergies εclose to the Fermi energy, and these assumptions usually work very well for microwave frequencies and acousticphonon scattering. However, in the problem of diffusivethermocurrent an extra accuracy is required, so the terms ofthe ﬁrst order in ω/ε are to be retained at least in the isotropic part of the electron-impurity collision integral [see Eq. ( 25) below]. III. SOLUTION OF KINETIC EQUATION When searching for the response to temperature gradients only, the effect of the dc ﬁeld in the collision integrals isomitted, γ n=γ± n=0. It is also assumed that the main cause of momentum relaxation of electrons comes from electron-impurity scattering rather than from electron-phonon scatter-ing. In GaAs quantum wells with electron mobility of about10 6cm2/V s this approximation holds al low temperatures T< 10 K (for GaAs quantum well of typical width 20 nm the phonon-limited mobility is estimated as 1 .3×107cm2/V sa tT=4.2 K and 3 .8×106cm2/Vsa t T=10 K). Thus, one may neglect the contribution Jph(0) εϕ in comparison toJim εϕ, but the contribution Jph(1) εϕ leading to phonon drag must be retained. It is convenient to expand the distribution 235307-4THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) function in the angular harmonics, fεϕ=/summationtext kfεkeikϕ.T h e electric current density given by Eq. ( 6) is determined by the components with k=± 1. Only the effects linear in MW power are considered below. The distribution function is represented as a sum of two terms, f(0) εk+f(MW) εk , where f(MW) εk is proportional to MW power. For k/negationslash=0 the ﬁrst term is given by the expression comprising the diffusive and phonon-dragparts: f(0) εk=−1 ikωc+ν(k)Dε/braceleftbiggpε 2m/parenleftbigg∂fεk+1 ∂T∇+T+∂fεk−1 ∂T∇−T/parenrightbigg +/integraldisplay2π 0dϕ 2π/integraldisplay2π 0dϕ/prime 2π/summationdisplay k/primeei(k/prime−k)ϕ ×ˆM/braceleftBigg/summationdisplay l=±1lDε−lωλQ(fε−lωλQk/primee−ik/primeθ−fεk/prime)/bracerightBigg/bracerightBigg ,(19) where ∇±=∇x±i∇y,ν(k)=νθ[1−cos(kθ)] (the line over the expression denotes angular averaging), and νθ= ν[2pεsin(θ/2)]. The integral operator ˆMis proportional totemperature gradient and deﬁned as ˆM{A}=/summationdisplay λ/integraldisplay∞ −∞dqz 2πmMλQτλF/parenleftBigωλQ 2T/parenrightBig ×/bracketleftbigg1 Q2q·∇T+1 q2ωλQ∂ωλQ ∂ϕqq·ˆ/epsilon1∇T/bracketrightbigg A, (20) withF(x)=[x/sinh(x)]2. It is taken into account that ωλQ/lessmuchε, which allows one to use the quasielastic approxima- tion, when the transferred 2D momentum q± εnis replaced by q, with absolute value q=2pεsin(θ/2) depending on electron energy and scattering angle θ=ϕ−ϕ/prime. The angle of the vector qisϕq=π/2+φ, where φ=(ϕ+ϕ/prime)/2. The phonon fre- quency can be written as ωλQ=sλQQ, where Q=/radicalbig q2+q2z andsλQis the sound velocity that depends on the mode index and direction of vector Q. If the quantum well is grown in the [001] crystallographic direction, as assumed in the following,bothω λQandMλQare periodic in ϕqwith the period π/2. To ﬁnd f(MW) εk with the accuracy up to the linear terms in MW power, only the contributions with low-order, \|n\|/lessorequalslant1, Bessel functions Jnare to be taken in Eqs. ( 11) and ( 12). Physically, this corresponds to a neglect of multiphotonabsorption processes. If k/negationslash=0, then f(MW) εk=Pω(ε)/4 ikωc+ν(k)Dε/integraldisplay2π 0dϕ 2π/integraldisplay2π 0dϕ/prime 2π(1−cosθ)[1−be2iφ−b∗e−2iφ]/summationdisplay k/primeei(k/prime−k)ϕ ×/summationdisplay n=±1/braceleftBigg νθ[Dε+nω(fε+nωk/primee−ik/primeθ−fεk/prime)−Dεfεk/prime(e−ik/primeθ−1)] −ˆM/braceleftBigg/summationdisplay l=±1l[Dε−lωλQ+nω(fε−lωλQ+nωk/primee−ik/primeθ−fεk/prime)−Dε−lωλQ(fε−lωλQk/primee−ik/primeθ−fεk/prime)]/bracerightBigg/bracerightBigg , (21) where Pω(ε)=2e2E2 ωε mω2(\|s+\|2+\|s−\|2) (22) is the dimensionless function proportional to MW power [see Eq. ( 4) for deﬁnition of s±] and b=s−s∗ + (\|s+\|2+\|s−\|2)(23) is a complex dimensionless coefﬁcient which depends on the direction of MW polarization and determines the sensitivity oftransport properties of electrons to this direction. The neglectof multiphoton processes implies P ω(ε)/lessmuch1. The expression (21) comprises both electron-impurity and electron-phononparts, though only the electron-phonon part is essential below. To ﬁnd the isotropic ( k=0) part of the distribution func- tion, it is necessary to include electron-electron scattering intoconsideration. Though the corresponding collision integralJ ee εis not written in Eq. ( 5) explicitly, it can be found in Refs. [ 9,23]. The kinetic equation is written as Jim ε+Jph(0) ε+Jee ε=0, (24)where only the isotropic part of the distribution function is retained under the collision integrals. It is essential that Jee εis not affected by MW irradiation, while Jim εis nonzero only in the presence of MW irradiation. The distribution fεis represented as a sum of smooth part f(0) εand rapidly oscillating partf(MW) ε . The smooth part is controlled by electron-electron scattering and, therefore, can be approximated by a heatedFermi distribution with effective electron temperature T e; the latter is to be found from the energy balance equation/integraltext dεD εε[Jim ε+Jph(0) ε]=0. For the oscillating part, one gets the following expression: f(MW) ε=Pω(ε) 4τinνtr/summationdisplay n=±1/parenleftBig 1+nω 2ε(1−Ztr)/parenrightBig ×δDε+nω(fε+nω−fε), (25) where δDε=Dε−1,νtr=τ−1 tr=ν(±1)is the transport relax- ation rate, and Ztr=∂lnτtr ∂lnε(26) is the logarithmic derivative of the transport time over energy. The inelastic scattering time τinentering Eq. ( 25) describes relaxation of the isotropic oscillating part of electron 235307-5O. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) distribution [ 9]. This relaxation is caused mostly by electron- electron scattering and scales with temperature as τin∝T−2 e. The electric current is given by Eq. ( 6), where the distribution function, found from Eqs. ( 19), (21), and ( 25) with the accuracy up to the terms linear in both ∇TandPω, is substituted. The thermoelectric tensor ˆβdetermining the thermocurrent is represented below as a sum of four parts: ˆβ=ˆβ(0) d+ˆβ(0) p+ˆβ(MW) d+ˆβ(MW) p, (27) where diffusive ( d) and phonon drag ( p) parts are written separately. Two ﬁrst terms correspond to dark thermocurrent,in the absence of MW irradiation, while the next two terms areMW-induced corrections. While ˆβ (0) dand ˆβ(0) pare determined only by f(0) εkfrom Eq. ( 19), the MW-induced parts are found in a more elaborate way, by combining together the results givenby Eqs. ( 19), (21), and ( 25), as described in Sec. III B . A. Dark thermocurrent In the absence of MWs, the linear response to temperature gradient is found from Eq. ( 19)f o rk=± 1 with isotropic (k/prime=0) distribution functions substituted in the right-hand side. The thermoelectric coefﬁcients are given by the followingexpressions: ˆβ(0) d=\|e\| π/integraldisplay dε∂f(0) ε ∂Teωc/Pi1εˆ/epsilon1−νtrεD2 εˆ1 ω2c+ν2 trD2ε(28) and ˆβ(0) p=\|e\| 2π/integraldisplay dεωcεDεˆ/epsilon1−νtrεD2 εˆ1 ω2c+ν2 trD2ε ×ˆP1/braceleftBigg/summationdisplay l=±1lDε−lωλQ/parenleftbig f(0) ε−lωλQ−f(0) ε/parenrightbig ω−1 λQ/bracerightBigg ,(29) where ˆPnis the integral operator deﬁned as ˆPn{A}=/integraldisplay2π 0dθ 2π/integraldisplay2π 0dϕq 2π/summationdisplay λ/integraldisplay∞ 0dqz π ×(1−cosθ)nm2MλQτλF/parenleftbiggωλQ 2T/parenrightbigg2ωλQ Q2A.(30) The matrices given by Eqs. ( 28) and ( 29) contain diagonal symmetric ( ∝ˆ1) and nondiagonal antisymmetric ( ∝ˆ/epsilon1) parts, so their symmetry is the same as the symmetry of the electricalconductivity. The expressions ( 28) and ( 29) describe the thermoelectric tensor in a wide region of temperatures and magnetic ﬁelds. Quantum oscillations of ˆβ(0) dand ˆβ(0) poccur because of the oscillating dependence of the density of states, Dε.I nt h e following, the approximation of overlapping Landau levels isused:D ε=1−2dcos(2πε/ω c), where d=exp(−π/\|ωc\|τ) is the Dingle factor ( d/lessmuch1) and τis the quantum lifetime of electrons, given at low temperatures by τ=1/νθ. Apart from the condition d/lessmuch1, the validity of the expression for Dεimplies ετ/greatermuch1. Under the same requirements, /Pi1ε=ε− (ωc/π)dsin(2πε/ω c). The integrals over energy in Eqs. ( 28) and ( 29) are calculated below under the assumption of strongly degenerate electron gas, and the quantum effects up to thesecond order in the Dingle factors are retained. To take intoaccount energy dependence of the Dingle factor due to energy dependence of τ, the logarithmic derivative Z=∂lnτ/∂lnε is introduced. The diffusive part is given by the followingexpression: ˆβ (0) d=π\|e\|Te 3/parenleftbig ω2c+ν2 tr/parenrightbig/braceleftbigg ωc/bracketleftbigg 1+2Ztrν2 tr ω2c+ν2 tr+6dcos2πεF ωcB X/bracketrightbigg ˆ/epsilon1 −νtr/bracketleftbigg 1−Ztrω2 c−ν2 tr ω2c+ν2 tr−12dεF πTeBsin2πεF ωc +2d2/parenleftbigg 1−Ztr+2πZ \|ωc\|τ/parenrightbigg/bracketrightbigg ˆ1/bracerightbigg (31) with X=2π2Te/ωcandB=∂(X/sinhX)/∂X=(1− XcothX)/sinhX. All energy-dependent quantities, namely νtr,τ,Ztr, andZ,i nE q .( 31)a r et a k e na t ε=εF. The classical terms and the quantum term proportional to din the diagonal part of ˆβ(0) dhave been reported previously [ 22]. Calculating the phonon-drag part from Eq. ( 29) under the same approximations, one gets the result ˆβ(0) p=\|e\|ns m/parenleftbig ω2c+ν2 tr/parenrightbig/braceleftBigg ωc[/Gamma11+2d2/Gamma1c1]ˆ/epsilon1 −νtr[/Gamma11(1+2d2)+4d2/Gamma1c1]ˆ1 −4d/Gamma1s1X sinhXcos2πεF ωc[ωcˆ/epsilon1−(3/2)νtrˆ1]/bracerightbigg .(32) S i m i l a r l yt oE q .( 31), this expression contains both classical terms and quantum terms proportional to dandd2.T h e dimensionless functions /Gamma1iused here and below are deﬁned as ⎛ ⎝/Gamma1n /Gamma1cn /Gamma1sn⎞ ⎠=ˆPF n⎧ ⎪⎪⎨ ⎪⎪⎩1 cos 2πωλQ ωc ωc 2πωλQsin2πωλQ ωc⎫ ⎪⎪⎬ ⎪⎪⎭, (33) where ˆPF ndenotes ˆPnatε=εF. The function /Gamma11determines theclassical contribution [ 34] to phonon-drag thermoelectric response and does not depend on the magnetic ﬁeld. Thiscontribution has been considered previously in the isotropicapproximation for the phonon spectrum, when there are onelongitudinal phonon branch with velocity s land two transverse branches with velocity st. For high temperatures, when Tex- ceeds both sλpFandπsλ/a(pFis the Fermi momentum and a is the quantum well width), /Gamma11is temperature-independent. At low temperatures, T/lessmuchsλpF, the Bloch-Gruneisen transport regime is realized, when electron scattering by phonons occursat small angles, θ/lessmuch1. In this regime [ 35],/Gamma1 1scales with temperature as T2(or as T4if only the deformation-potential mechanism of electron-phonon interaction is present). Thefunctions /Gamma1 cnand/Gamma1snstanding in the quantum contributions depend on the magnetic ﬁeld and can be analytically calculatedonly in certain limits (see Appendix). The terms ∝din Eqs. ( 31) and ( 32) describe the Shubnikov–de Haas oscillations of the thermocurrent. Theforthcoming consideration, however, is focused at the caseof\|ω c\|/lessmuch 2π2Te, which means that X/sinhXis expo- nentially small so the Shubnikov–de Haas oscillations aresuppressed and the quantum corrections are given by theterms ∝d 2only. Using ns=mεF/π, one may check that 235307-6THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) the tensor (31) under these conditions satisﬁes the Mott re- lation ˆβ(0) d=− (π2Te/3\|e\|)(∂ˆσ/∂ε F), where ˆ σ=ˆ1σd−ˆ/epsilon1σ⊥ is the conductivity tensor whose components are σd= e2nsνtr(1+2d2)/[m(ω2 c+ν2 tr)] and σ⊥=e2nsωc/[m(ω2 c+ ν2 tr)]. The quantum corrections both in these expressions and in Eq. ( 31) are essential only in the classically strong magnetic ﬁelds, so the terms ∝(νtr/ωc)d2are neglected in comparison to the terms ∝d2. The diffusive thermoelectric coefﬁcients do not oscil- late before the onset of Shubnikov–de Haas oscillations.In contrast, quantum magneto-oscillations of phonon-dragthermoelectric coefﬁcients persist under the assumed condition\|ω c\|/lessmuch 2π2Te, because of the presence of /Gamma1c1inˆβ(0) p. Indeed, the oscillating nature of the function cos(2 πωλQ/ωc) is not completely washed out after the integration under ˆP.T h e major contribution to such integrals comes from the regionof variables around q z=0 and θ=π, which physically corresponds to backscattering of electrons as a result ofemission or absorption of phonons moving in the quantum wellplane; the wave number of these phonons is close to 2 p F. Thus, there exist resonant phonon frequencies, roughly estimated as2p Fsλ, which lead to magneto-oscillations of phonon-drag thermopower observed [ 20] in high-mobility samples. With decreasing temperature, the oscillations are exponentially sup-pressed in the Bloch-Gruneisen regime (see Appendix). Thesame kinds of oscillations are observed in electrical resistivity;they are known as acoustic magnetophonon oscillations orphonon-induced resistance oscillations [ 24–29]. B. Microwave-induced thermocurrent The distribution functions fε1andfε−1determining the electric current under MW irradiation are to be found upto the terms linear in P ω(ε). There are two types of such MW-induced contributions. The direct ones are obtained intwo ways: (i) by calculating f(MW) ε±1from Eq. ( 21), where the isotropic distribution function f(0) εis retained under the integral (only the phonon part is essential), and (ii) by calculating f(0) ε±1from Eq. ( 19), where the isotropic MW- induced distribution function f(MW) ε is placed in the right-hand side. The indirect contributions assume calculation of f(0) ε±1 andf(MW) ε±1by substituting anisotropic parts of f(MW) εk and f(0) εk, respectively, in the right-hand sides of Eq. ( 19) and Eq. ( 21). A similar technique has been used for calculation of the MW-induced conductivity. Following the notations ofRef. [ 10], one may denote the direct contributions (i) and (ii) as the “displacement” and “inelastic” ones, respectively, andthe indirect contributions as the “quadrupole” ones. Strictlyspeaking, there exists one more indirect contribution calledthe “photovoltaic” one [ 10], which is determined by the MW- generated time-dependent part of the distribution function andcannot be obtained from the kinetic equation Eq. ( 5) because the latter is written for time-independent f εϕ. The indirect contributions to ˆβbegin with the terms of the order νtr/ωc compared to direct contributions. Since all the MW-induced contributions are of quantum nature and important only inthe region of classically strong magnetic ﬁeld, ω c/greatermuchνtr,t h e indirect contributions are less signiﬁcant than the direct onesand can be safely neglected in the thermopower coefﬁcientspresented in the next section. Therefore, the attention below is focused at the direct contributions only. The current is calculated in the regime when electron gas is degenerate. Within the required accuracy, the solution ofEq. ( 25) is given by the following expression: f (MW) ε=d 2Pω(ε)τinνtr/braceleftbigg sin2πε ωcsin2πω ωc ×/parenleftbig f(0) ε+ω−f(0) ε−ω/parenrightbig −cos2πε ωccos2πω ωc ×/bracketleftBig f(0) ε+ω+f(0) ε−ω−2f(0) ε +ω 2ε/parenleftbigg 1−Ztr+2πZ \|ωc\|τ/parenrightbigg/parenleftbig f(0) ε+ω−f(0) ε−ω/parenrightbig/bracketrightbigg/bracerightbigg . (34) The ﬁrst term of this expression gives the main contribution sufﬁcient for calculation of the MW-induced resistance [ 9]. The second term represents a correction of the order ω/ε, which is necessary for calculation of the diffusive ther-mopower. The term proportional to the factor sin(2 πω/ω c)i n Eq. ( 34) also enters Eqs. ( 35), (37), (43), (45), and ( 48) below, where the contribution of inelastic mechanism is present. Thisfactor reﬂects the property [ 9] that the strongest modiﬁcation of the electron distribution function under MW irradiationin the presence of weak Landau quantization occurs whenω/ω c=n±1/4(nis an integer). The correction proportional to the factor cos(2 πω/ω c) appears because the resonance absorption of MW radiation at ω/ω c=nalso has an effect on the distribution function. The consideration below assumes the approximation \|ωc\|/lessmuch 2π2Te, when Shubnikov–de Haas oscillations are ther- mally averaged out. This dramatically simpliﬁes calculation ofthe integrals over energy because one can average the productsof rapidly oscillating functions such as D εandf(MW) ε over the period ωcbefore integration over the energy. After substituting Eq. ( 34) into the ﬁrst part of the right-hand side of Eq. ( 19) and calculating the current according to Eq. ( 6) [one may equally use Eq. ( 28) with f(0) εreplaced by f(MW) ε ], the diffusive part ofˆβtakes the form ˆβ(MW) d=\|e\|d2τinνtrω2PωTin πTe/parenleftbig ω2c+ν2 tr/parenrightbig ×/bracketleftbiggω2 c 2πωsin2πω ωcˆ/epsilon1−νtr(1−Ztr) cos2πω ωcˆ1/bracketrightbigg , (35) where all energy-dependent quantities are taken at ε=εF,i n particular, Pω≡Pω(εF). The main contribution to the deriva- tive over temperature in Eq. ( 19) comes from temperature dependence of the inelastic relaxation time, expressed throughthe logarithmic derivative T in=∂lnτin ∂lnTe/similarequal− 2. (36) The factor sin(2 πω/ω c) typical for MW-induced conductiv- ity [9] does not appear in the diagonal part of thermoelectric tensor Eq. ( 35), because of different dependence of the 235307-7O. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) diffusive thermoelectric current on electron energy distribution as compared to the drift current. The ﬁrst term of f(MW) ε is averaged out in the diagonal components of ˆβ(MW) d , while the second term of f(MW) ε , proportional to cos(2 πω/ω c), survives this averaging. For the phonon-drag part of ˆβthe result is the following: ˆβ(MW) p=2\|e\|nsd2Pω m/parenleftbig ω2c+ν2 tr/parenrightbig/braceleftbigg νtrτin/Gamma1s12πω ωcsin2πω ωc ×[−ωcˆ/epsilon1+(3/2)νtrˆ1] +/parenleftbigg /Gamma1c2sin2πω ωc+/Gamma1s2πω ωcsin2πω ωc/parenrightbigg ×[ωc(−ˆ/epsilon1+ˆg0)+2νtr(ˆ1+ˆh0)] +/parenleftbigg ˜/Gamma1c2sin2πω ωc+˜/Gamma1s2πω ωcsin2πω ωc/parenrightbigg ×[ωcˆg1+2νtrˆh1]/bracerightbigg , (37) where ˆg0=b/primeˆσx+b/prime/primeˆσz,ˆg1=b/primeˆσx−b/prime/primeˆσz, (38) ˆh0=b/primeˆσz−b/prime/primeˆσx,ˆh1=b/primeˆσz+b/prime/primeˆσx; ˆσz=(10 0−1) and ˆσx=(01 10) are the Pauli matrices, while b/prime andb/prime/primedenote real and imaginary parts of b[see Eq. ( 23)], respectively. The quantities ˜/Gamma1idiffer from /Gamma1iby placing the factor cos(4ϕq)−sin(4ϕq)/parenleftbigg 1+q2 z 4p2εsin2θ/2/parenrightbigg1 ωλQ∂ωλQ ∂ϕq(39) under the integral operator ˆPin Eq. ( 33). In the general case, both terms in Eq. ( 39) are essential for calculation of˜/Gamma1i. In the isotropic approximation for phonon spec- trum the second term in Eq. ( 39) vanishes, but ˜/Gamma1iis still nonzero, because the piezoelectric-potential part of MλQ remains angular-dependent. If the anisotropy of the phonon spectrum is weak, the second term in Eq. ( 39) can be neglected in the calculation of the piezoelectric-potentialcontribution. The expression ( 37) includes contributions from both inelastic (ﬁrst term) and displacement (second and third terms)mechanisms. The inelastic-mechanism contribution can beobtained from Eq. ( 29) after replacing f (0) εwithf(MW) ε .T h e displacement-mechanism contribution has a form similar tothat of the MW-induced contribution to conductivity [ 8], as it contains the factors sin(2 πω/ω c) and sin2(πω/ω c). The ﬁrst of these factors has extrema at ω/ω c=n±1/4, corresponding to the conditions of maximal displacement of electrons along theeffective drag force or against this force under photon-assistedscattering, similar to the case of a response to dc ﬁeld [ 7]. The second factor describes the enhancement of photon-assisted scattering probabilities in the resonance, ω/ω c=n, and their suppression in the antiresonance, ω/ω c=n+1/2. The displacement-mechanism contribution depends on MWpolarization direction through the terms with the matrices ofEq. ( 38).The fundamental difference between ˆβ (MW) p and the MW- induced contribution to conductivity is given by the factors/Gamma1 iand ˜/Gamma1i, which are not merely constants but functions of the magnetic ﬁeld describing the magnetophonon oscillations.The products of these magnetophonon oscillating factorsby the MW-induced oscillating factors sin(2 πω/ω c) and sin2(πω/ω c) physically correspond to the interference of these two kinds of oscillations and can be viewed as a result of photonand phonon frequency mixing in the scattering probabilities. The phonon-drag part of ˆβdepends on electron tem- perature T ethrough the inelastic scattering time τin. The quantities /Gamma1iand ˜/Gamma1iare determined by the lattice temperature T. There is an important question of whether the thermo- electric tensor ˆβsatisﬁes the symmetry with respect to time inversion (Onsager symmetry). In the absence of microwaves,this symmetry, of course, is satisﬁed. Under MW irradiation,when electrons are out of equilibrium, the Onsager symmetrycan be broken [ 10]. In application to the problem of electrons in the presence of electromagnetic waves, the time inversionimplies, apart from the magnetic ﬁeld reversal ω c→−ωc, the transformations e→e∗andk→− k, where kis the wave vector of the electromagnetic wave. e→e∗means that e±→e∗ ∓, which is equivalent to κ±→κ∓(see the beginning of Sec. II), while k→− kmeans that the sign at ωpin Eq. ( 4)f o rs±is inverted, as follows from reversibility of the wave transmission problem [ 30] employed for derivation of Eq. ( 3). Therefore, the denominator in Eq. ( 4) transforms asω±ωc+iωp→ω∓ωc−iωp, which results in s±→s∗ ∓ under the time inversion. The Onsager symmetry relation takes the form βij(ωc,s−,s+)=βji(−ωc,s∗ +,s∗ −), (40) and similar relations can be written for the other transport coefﬁcients including the conductivity. From s±→s∗ ∓one can see that both \|s+\|2+\|s−\|2ands−s∗ +are invariants with respect to time inversion; thus the function bdeﬁned by Eq. ( 23) is also an invariant. The MW-induced part of ˆβ given by Eq. ( 37) does contain the terms violating the Onsager symmetry Eq. ( 40); these are the terms at the matrices ˆg0and ˆg1. These terms are invariant under permutation of Cartesian indices. IV . THERMOPOWER COEFFICIENTS Having found ˆβ, one may calculate the thermopower tensor ˆα, which is presented below as a sum of dark and MW-induced parts, ˆ α=ˆα(0)+ˆα(MW). Because of the presence of terms which depend on MW polarization, this tensor is a generalmatrix. In the absence of MW irradiation, ˆ αhas the same symmetries as the resistivity tensor, α (0) xx=α(0) yyandα(0) xy= −α(0) yx. By using the expressions ρ(0) xy=mωc/e2nsandρ(0) xx= mν tr(1+2d2)/e2nstogether with Eqs. ( 31) and ( 32), where the Shubnikov–de Haas terms are neglected, one obtains, withinthe accuracy up to d 2, the following results: α(0) xx=−π2 3\|e\|Te εF/parenleftbigg 1+Ztrν2 tr ω2c+ν2 tr/parenrightbigg −1 \|e\|(/Gamma11+2d2/Gamma1c1),(41) 235307-8THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) FIG. 1. (Color online) Longitudinal (a) and transverse (b) ther- mopower in the absence of microwave excitation plotted for three temperatures. The calculations are done for a GaAs quantum well ofwidth 14 nm with electron density n s=5×1011cm−2and mobility 2×106cm2/V s. The quantum lifetime of electrons is τ=7p s . α(0) xy=ωcνtr ω2c+ν2 tr/braceleftbiggπ2 3\|e\|Te εF/bracketleftbigg Ztr+2d2/parenleftbigg Ztr−2πZ \|ωc\|τ/parenrightbigg/bracketrightbigg −1 \|e\|2d2/Gamma1c1/bracerightbigg . (42) In the classical case, the thermopower coefﬁcients have the usual forms found in literature [ 22]. The Landau quantization leads to additional terms proportional to d2. In the phonon-drag part of thermopower, these terms are determined by thefunction /Gamma1 c1oscillating with the magnetic ﬁeld. Because of these quantum corrections, the transverse phonon-drag ther-mopower is nonzero. In Fig. 1the longitudinal and transverse thermopower are plotted as functions of magnetic ﬁeld for arectangular GaAs quantum well of width 14 nm, with electrondensity n s=5×1011cm−2and mobility 2 ×106cm2/Vs . The quantum lifetime τ=7 ps is assumed, which corresponds to the ratio τtr/τ/similarequal11. The phonon scattering time τλis chosen as 0.2μs for each mode, which approximately corresponds to a 1 mm mean-free path for phonons [ 36]. The elastic coefﬁcients for GaAs in units 1011dyn/cm2arec11=12.17,c12= 5.46, and c44=6.16. The deformation potential, piezoelectric coefﬁcient, and density are D=7.17 eV , h14=1.2V/nm, andρ=5.317 g/cm3, respectively. The energy dependence of the transport time and quantum lifetime is assumed to be∝ε 3/2and∝ε1/2, respectively, which corresponds to ν(q)∝ exp(−lcq) under the condition of small-angle scattering, when lcpF/greatermuch1. The oscillations of the thermopower coefﬁcients are caused by magnetophonon resonances. At low temperature,the oscillations are barely visible because the system falls intothe Bloch-Gruneisen regime, but they are essential at highertemperatures. The last peak of α (0) xxis due to the scattering of electrons by high-energy (longitudinal) phonons; this peakdisappears ﬁrst with lowering temperature. The nonoscillating,proportional to 1 /B, part of α (0) xyis determined by the diffusive contribution. Let us consider now the thermopower coefﬁcients in the presence of MW excitation. While Eqs. ( 41) and ( 42)a r e valid for both classically strong and classically weak magneticﬁelds, the MW-induced contributions are important only in the limit of classically strong magnetic ﬁelds. For this reason, onlya part of the terms presented in Eqs. ( 35) and ( 37) are essential for calculation of thermopower in this limit. In particular,the longitudinal thermopower in classically strong magneticﬁelds is written simply as α xx=ρxyβyx. The inﬂuence of microwaves on Hall resistivity ρxyis weak [ 12], soαxxis directly determined by βyx. Neglecting the contributions of higher order in νtr/ωcin Eqs. ( 35) and ( 37), one obtains α(MW) xx=2d2Pω \|e\|/braceleftbigg νtrτin2πω ωcsin2πω ωc/parenleftbigg /Gamma1s1−Tinω2 c 8π2εFTe/parenrightbigg +[(1+b/prime)/Gamma1c2+b/prime˜/Gamma1c2]s i n2πω ωc +[(1+b/prime)/Gamma1s2+b/prime˜/Gamma1s2]πω ωcsin2πω ωc/bracerightbigg , (43) whileα(MW) yy differs from this expression by changing the sign atb/prime. The ﬁrst term in Eq. ( 43) is caused by modiﬁcation of the isotropic distribution function of electrons by microwaves(inelastic mechanism) and includes both the phonon-drag andthe diffusive contributions. Since the diffusive term increaseswith decreasing temperature, it may become comparable tothe phonon-drag one. However, inevitable heating of electrongas by microwaves tends to hinder the contribution of thediffusive term. The remaining terms in Eq. ( 43) describe the phonon-drag thermopower caused by the displacementmechanism. They contain contributions proportional to b /prime, which change the symmetry of the thermopower coefﬁcients.The dependence of these contributions on the polarizationangleχcan be illustrated for the case of linear polarization of the incident wave, when bis represented in the form b=1 2e−2iχω2−ω2 c+ω2 p−2iωcωp ω2+ω2c+ω2p. (44) Since b/prime≡Re(b) contains the terms both even and odd in magnetic ﬁeld, αxx, in general, is not symmetric in B(the reversal of magnetic ﬁeld means alteration of the sign of ωcin all equations). The “inelastic” contribution in Eq. ( 43) should dominate at low enough temperatures, when νtrτin>1. The “displacement” terms become more important with increasingtemperature. It is worth emphasizing that the oscillations inthese terms due to the factor sin 2(πω/ω c) are comparable by amplitude with the oscillations due to the factor sin(2 πω/ω c). This behavior is in contrast with that for MW-induced resis-tance. In the resistance, the contribution at sin(2 πω/ω c) domi- nates because it overcomes the oscillating part of sin2(πω/ω c) by the factor 2 πω/ω cwhich is numerically large in the region ω>ω cwhere MIRO are observed. As a consequence, the MW-induced resistance magneto-oscillations due to thedisplacement mechanism are very similar to the magneto-oscillations due to the inelastic mechanism [ 9], so these two mechanisms are difﬁcult to separate experimentally. In thephonon-drag thermopower, the contributions at sin(2 πω/ω c) and sin2(πω/ω c) are proportional to the functions /Gamma1s2and/Gamma1c2, respectively, and /Gamma1c2is larger than /Gamma1s2. Moreover, /Gamma1c2/greatermuch/Gamma1s2 in the region of low magnetic ﬁelds, \|ωc\|/lessmuch 4πpFsλ;s e et h e Appendix. The ratio of the amplitudes of sin(2 πω/ω c) and sin2(πω/ω c) oscillations in the “displacement” part of the 235307-9O. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) thermopower is estimated as ω/2pFsλ, which is of the order of unity for typical electron densities and MW frequencies. Thesame is true for the “displacement” contribution to transversethermopower described below by Eq. ( 47). A more careful analysis is required for evaluation of the transverse (Nernst-Ettingshausen) thermopower, because thelatter is determined by both diagonal and nondiagonal partsofˆβand is sensitive to MW-induced modiﬁcations of the longitudinal resistivity. Indeed, α xy=ρxyβyy+ρxxβxy.T h e inﬂuence of microwaves on ρxyis weak and not essential for determination of αxy, while their inﬂuence on ρxxis strong. Under the assumed condition that the electron-impurityscattering is more important than electron-phonon scattering,the longitudinal resistivity correction due to MW irradiation iswritten as [ 9] ρ (MW) xx=−2d2mν2 trτin e2nsPω2πω ωcsin2πω ωc, (45) andρ(MW) yy=ρ(MW) xx . Equation ( 45) implies that ρ(MW) xx is governed by the inelastic mechanism. The displacement mech-anism for electron-impurity scattering is less important at lowtemperatures, especially in the case of small-angle scatteringprocesses relevant for high-mobility 2D systems [ 9]. In contrast, for electron-phonon scattering determining phonon-drag thermopower, the displacement mechanism is signiﬁcantunder the condition ω λQ/lessorequalslant2Twhen the main contribution to oscillating functions /Gamma1c2and/Gamma1s2comes from large-angle scattering processes (backscattering). Among the “displace-ment” terms contributing into the transverse thermopowerα (MW) xy there is a strong polarization-dependent term coming from the diagonal part of the matrices ˆg0and ˆg1in Eq. ( 37). The other contributions to α(MW) xy contain a small factor νtr/ωc. Out of them, only the “inelastic” ones can compete withthe mentioned polarization-dependent contribution. Therefore,with the assumed accuracy up to d 2, the result is written as a sum of two terms: α(MW) xy/similarequal/Delta1αxysin(2χ+ηB)+αin xy, (46) where /Delta1αxy=2d2Pω \|e\|\|b\|/bracketleftbigg (/Gamma1c2−˜/Gamma1c2)s i n2πω ωc +(/Gamma1s2−˜/Gamma1s2)πω ωcsin2πω ωc/bracketrightbigg , (47) and αin xy=−2d2Pων2 trτin \|e\|ωc/bracketleftbigg2πω ωcsin2πω ωc/parenleftbigg /Gamma11+π2Te 3εF−/Gamma1s1 2/parenrightbigg −ω2Tin 2TeεF/parenleftbiggωc 2πωsin2πω ωc−(1−Ztr) cos2πω ωc/parenrightbigg/bracketrightbigg . (48) To obtain α(MW) yx , one should change the sign at the second term in Eq. ( 46). Since the effects under consideration are linear in MW intensity, the polarization-dependent term is a harmonicfunction of the doubled polarization angle; a similar angulardependence is expected for electrical resistivity [ 37]. This term is characterized by the amplitude /Delta1α xyand the phase angle ηBwhich are, respectively, a symmetric and an antisymmetricfunction of the magnetic ﬁeld. For linear polarization, when Eq. ( 44) is valid, the phase angle is deﬁned as tan ηB= 2ωcωp/(ω2−ω2 c+ω2 p). One may introduce the effective polarization angle χB=χ+ηB/2 describing the direction of the ac electric ﬁeld in the 2D plane, which is different from thepolarization of the incident wave. The polarization-dependentterm, in general, is not antisymmetric under reversal of B, though for special orientation of the incident ac ﬁeld along x oryaxes the symmetry property α (MW) xy (B)=−α(MW) xy (−B)i s preserved. If the angle χBis equal to π/2 or 0, which means that the electric ﬁeld in the 2D plane is polarized along yorxaxes (i.e., along or perpendicular to the temperature gradient), thepolarization-dependent term is equal to zero. The contributionof this term can be experimentally distinguished from the othercontributions by its dependence on the polarization. The polarization-independent term given by Eq. ( 48) contains several contributions of different origin, though all ofthem are caused by the inelastic mechanism. The ﬁrst part [theﬁrst line of Eq. ( 48)] comprises three different contributions. The ﬁrst one, at /Gamma1 1, comes from the MW-induced correction to resistance if the thermoelectric current is due to thephonon-drag mechanism. The second contribution comesfrom the MW-induced correction to resistance if the ther-moelectric current is due to the diffusive mechanism. Thesetwo contributions can be distinguished from each other bytheir temperature dependence. At low temperatures (roughly estimated as T e<0.5 K), the second contribution can exceed the ﬁrst one, as it decreases with Teslower [see Eq. ( A11)f o r low-temperature behavior of /Gamma11]. However, the MW heating of electron gas renders this regime practically unrealizable.The third contribution, at /Gamma1 s1, is caused by the MW-induced correction to the phonon-drag part of thermoelectric tensor. Incontrast to the ﬁrst and second contributions, this one containsmagnetophonon oscillations. However, in the region of ﬁeldswhere these oscillations exist, \|ω c\|<2pFsλ,t h et e r m /Gamma1s1/2i s much smaller than /Gamma11. The second part [the last line of Eq. ( 48)] contains the contributions due to MW-induced correction todiffusive part of thermoelectric tensor. This part does notexceed the contribution proportional to π 2Te/3εFin the second line of Eq. ( 48) under the assumed condition \|ωc\|/lessmuch 2π2Te. Therefore, the contribution proportional to /Gamma11dominates over the others in Eq. ( 48) in the relevant region of parameters. This means that magneto-oscillations of αin xyare determined only by the ratio ω/ω cand are similar to MIRO. The magneto-oscillations of the polarization-dependent term aremore complicated, because they also have the magnetophononconstituent due to the factors /Gamma1 c2−˜/Gamma1c2and/Gamma1s2−˜/Gamma1s2[see Eq. ( 47), Fig. 6, and its discussion below]. Therefore, the two terms in Eq. ( 46) can be distinguished from each other not only by polarization dependence and B-inversion symmetry but also by the behavior of magneto-oscillations. It is important to emphasize that the components of the thermopower tensor given by Eqs. ( 43) and ( 46) do not violate the Onsager symmetry. This fact requires an explanation inview of the observation (see the end of Sec. III) that some terms in ˆβviolate this symmetry. Indeed, ˆ αis formed as a result of matrix multiplication of ˆ ρand ˆβand its full form does contain terms violating the Onsager symmetry. However,such terms are small in comparison to the terms included inEqs. ( 43) and ( 46), so they are neglected. 235307-10THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) FIG. 2. (Color online) Longitudinal (left) and transverse (right) diffusive thermopower at T=1.5Ka n d T=4.2 K under the linearly polarized MW excitation of frequency 130 GHz and electric ﬁeld Eω=2V/cm. The parameters of the system are the same as in Fig. 1. The dashed lines show the dark thermopower (no MW excitation).The narrow solid line in the right-hand part shows the result of approximation α (MW) xy/similarequalρ(MW) xxβ(0) xyforT=1.5K .T h ei n s e tp r e s e n t s the calculated behavior of the longitudinal resistance. Coming to presentation of numerical results, let us consider ﬁrst the diffusive contribution to thermopower coefﬁcients. This contribution is given by Eqs. ( 41), (42), (43), and ( 46), where all /Gamma1iand ˜/Gamma1iare set to zero. The inelastic scattering time here and below is estimated according to [ 9]τin=εF/T2.T h e diffusive thermopower is not sensitive to MW polarization.The longitudinal diffusive thermopower α xxis modiﬁed by the microwaves in two ways: through the heating of 2Delectrons and through the quantum correction in Eq. ( 43). The calculations (see Fig. 2) demonstrate that the heating mechanism is more essential. In particular, it leads to a peakat cyclotron absorption frequency and to oscillations at smallBcaused by the oscillations of absorbed MW power due to Landau quantization. The transverse diffusive thermopowerα xy, in contrast, is considerably affected by the MW-induced quantum corrections from Eq. ( 48). Among these corrections there is a term ρ(MW) xxβ(0) xy, whose oscillations directly reproduce the MIRO pattern shown in the inset of Fig. 2. The calculations demonstrate that the other terms, those in the last line ofEq. ( 48), are equally important, although their contribution becomes weaker with increasing temperature. Consider now the inﬂuence of microwaves on the ther- mopower coefﬁcients in the presence of both diffusiveand phonon drag mechanisms. Theoretical and experimentalstudies of GaAs quantum wells show that for temperaturesabove 0.5 K the phonon-drag contribution dominates overthe diffusive one. Consequently, the behavior of thermopoweris governed mostly by the inﬂuence of MW excitation onthe phonon-drag contribution. For the typical parameters ofMW excitation, the oscillating quantum corrections givenby Eq. ( 43) are of the order of several μV/K. The partial contributions due to inelastic mechanism [the ﬁrst term inEq. ( 43)] and displacement mechanism (the remaining terms) are shown in Fig. 3. The role of the displacement mechanism increases with increasing temperature. At low temperaturesFIG. 3. (Color online) Microwave-induced corrections to longi- tudinal thermopower at T=1.5Ka n d T=4.2 K due to inelastic (a) and displacement (b) mechanisms, for linearly polarized MW excitation of frequency 130 GHz and electric ﬁeld Eω=2V/cm. The parameters of the system are the same as in Fig. 1. Two plots for T=4.2 K in (b) correspond to two angles of MW polarization. (Bloch-Gruneisen regime), the period of the oscillations is determined by the ratio ω/ω c. With increasing temperature, the magnetophonon resonances become important and the picture of oscillations becomes more rich. The sensitivity of the displacement mechanism to MW polarization is illustratedby plotting its contribution for two angles of electric ﬁeld ofthe incident wave, χ=0 andχ=π/4. However, the relative change of the longitudinal component α xxunder MW irradiation is not strong. The terms due to phonon drag in Eq. ( 43) are proportional to the functions /Gamma1s1,/Gamma1c2, and /Gamma1s2, which are small in comparison to /Gamma11 in the important region of parameters \|ωc\|<2pFsλQand \|ωc\|/lessmuch 2π2Te, where magnetophonon oscillations take place but Shubnikov–de Haas oscillations are suppressed (see amore detailed comparison in the Appendix). The ratio ofthe relative change of α xxdue to MW irradiation to the relative change of the resistivity ρxxis estimated by a small factor /Gamma1s1//Gamma1 1. This means that even in the case when MW- induced resistance oscillations are strong, the MW-inducedoscillations of the longitudinal thermopower still may be weak.The magnetic-ﬁeld dependence of α xxat low temperature is presented in Fig. 4(a).F o rT=1.5 K one can see changes in the oscillation picture, in particular, inversion of the minimumaround 0.18 T and a considerable enhancement of the lastpeak. The vertical shift of α xxas a whole with respect to α(0) xxis caused mostly by the diffusive mechanism contribution, due toheating of electrons by microwaves; see Fig. 2. With increasing temperature, the relative effect of microwaves on α xxbecomes weaker because α(0) xxincreases faster than α(MW) xx . The transverse thermopower αxy, in contrast, is strongly changed by microwaves, because the dark thermopower α(0) xyis small itself. At low temperature [see Fig. 4(b)] the modiﬁcation is almost entirely governed by the oscillations of resistivity,which means that the approximation α (MW) xy/similarequalρ(MW) xxβ(0) xyworks well. This approximation is no longer valid when temperatureincreases and the polarization-dependent contribution, the ﬁrstterm in the expression Eq. ( 46), becomes signiﬁcant. This is 235307-11O. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) FIG. 4. (Color online) Longitudinal (a) and transverse (b) ther- mopower at T=1.5 K under the linearly polarized MW excitation of frequency 130 GHz and electric ﬁeld Eω=2V/cm. The parameters of the system are the same as in Fig. 1. The dashed lines show the dark thermopower. The narrow solid line shows the result of approximation α(MW) xy/similarequalρ(MW) xxβ(0) xyfor transverse thermopower. demonstrated in Fig. 5, where αxyis plotted for two directions of ac electric ﬁeld: along the xaxis (χ=0) and at the angle ofπ/4 to this axis. With increasing B, when the ratio νtr/ωc becomes smaller, αxydeviates from the simple dependence ∝ρ(MW) xx and becomes strongly sensitive to polarization. The polarization dependence of αxyfor different magnetic ﬁelds is characterized by the amplitude /Delta1αxyg i v e nb yE q .( 47). This function is plotted in Fig. 6for different temperatures. The complicated oscillating behavior of /Delta1αxyis caused by the interference of magnetophonon oscillations with microwave-induced oscillations. At small T, when the system is in the Bloch-Gruneisen regime, /Delta1α xyis small. With increasing T, /Delta1αxyincreases and saturates around 10–15 K. The inset in Fig. 6shows how the rotation of the MW polarization angle changes the total transverse thermopower. FIG. 5. (Color online) Transverse thermopower at T=4.2K under the MW excitation of frequency 130 GHz and electric ﬁeldE ω=2V/cm, for two different directions of linear polarization of incident wave. The parameters of the system are the same as in Fig. 1. The dashed line shows the dark thermopower. The narrow solid lineshows the result of approximation α (MW) xy/similarequalρ(MW) xxβ(0) xy.FIG. 6. (Color online) Magnetic-ﬁeld dependence of polarization-sensitive part of transverse thermopower at different temperatures, for the MW excitation of frequency 130 GHz and electric ﬁeld Eω=2V/cm. The parameters of the system are the same as in Fig. 1. The inset shows dependence of thermopower on the polarization angle at B=0.5T . The relative contribution of the polarization-dependent part can be further enhanced at higher MW intensity and at highermobility, because the second term in Eq. ( 46) is proportional to the factor ν 2 trτinwhich goes down when inelastic scattering timeτin∝T−2 edecreases because of microwave heating of electron gas and when the transport scattering rate νtr(inversely proportional to the mobility) decreases. In the case of circular polarization or nonpolarized radi- ation (chaotic polarization) the polarization-dependent termvanishes and α (MW) xy is determined by the second term in Eq. ( 46). Since the most important part of this term is given by ρ(MW) xxβ(0) xy, the oscillations of transverse thermopower under these conditions follow the MW-induced resistanceoscillations. The longitudinal and transverse thermopower components α xxandαxyare directly measured in the Hall bars. The longitudinal thermopower can also be measured in the Corbinodisk geometry [ 38]. In this case, polarization-dependent terms do not appear and the voltage between inner and outer contactsis determined by the thermopower α d=βd/σd, where βd andσdare the diagonal parts of the tensors ˆβand ˆσin the absence of MW polarization. Since σdis modiﬁed by microwaves stronger than βd, the behavior of thermopower in MW-irradiated Corbino disks is determined mostly byMW-induced oscillations of σ d. The theory developed in this paper does not take into account temperature dependence of the density of states.Such a dependence appears mostly due to contribution ofelectron-electron scattering into the inverse quantum lifetime1/τ(see Ref. [ 11] and references therein). This effect leads to an exponential suppression of all quantum contributions in thetransport coefﬁcients, including those considered above, withincreasing T e. Formally, this occurs because the Dingle factor d acquires a multiplier exp( −π/τee(Te)\|ωc\|), where 1 /τee(Te)∼ T2 e/εF. This effect tends to decrease the quantum part of dark thermopower and MW-induced corrections to thermopowerwith increasing temperature. Since the main (phonon-drag) 235307-12THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) contribution to thermopower, in contrast, increases with in- creasing temperature at T< p Fsλ, it is important to investigate possible competition of these opposite trends in the quantum(proportional to d 2) terms in thermopower. Assuming that Te/similarequal T, the exponential dependence of these terms on temperature in the Bloch-Gruneisen regime ( T/lessmuchpFsλ) is written as e−/Phi1T, where /Phi1T/similarequal2πT2/εF\|ωc\|+2pFsλ/T is a nonmonotonic function of temperature. This function decreases at T< T 0and increases at T> T 0, where T0/similarequalpFsλ(\|ωc\|/4πms2 λ)1/3. Since the estimate for GaAs gives T0>pFsλeven for magnetic ﬁelds as small as 0.05 T, one may conclude that the temperaturedependence of the density of states does not alter the thermalincrease of the quantum contributions to thermopower atT< p Fsλ. However, at T> p Fsλall these contributions, both in the dark thermopower and MW-induced corrections,decrease with temperature instead of going to saturation. V . DISCUSSION AND CONCLUSIONS The inﬂuence of MW irradiation on the energy distribution of electrons and on electron scattering by phonons andimpurities has a profound effect on transport properties of2D electron systems in perpendicular magnetic ﬁeld. Whilethe effect of microwaves on the electrical resistance is widelystudied, the related behavior of the other kinetic coefﬁcientshas not received proper attention. This paper reports a theoretical study of possible MW-induced quantum effects in thermopower. Such effects can exist in the samples with highelectron mobility in the moderately strong magnetic ﬁelds, thatis, under the same conditions when the MW-induced quantumoscillations of the electrical resistance are observed. In contrast to electrical resistance, which at low tem- peratures is determined by electron-impurity scattering, thethermopower is determined mostly by electron-phonon scat-tering, through the phonon drag mechanism. The theory ofphonon-drag thermoelectric response in quantizing magneticﬁelds remains an issue of interest even under quasiequilibriumconditions, in the absence of MW irradiation. A furtherdevelopment of such theory is presented in this paper. Inparticular, an anisotropy of the acoustic phonon spectrum hasbeen taken into account and analytical expressions valid inthe regime of overlapping Landau levels with the accuracyup to the square of the Dingle factor have been derived; seeEqs. ( 32), (33), (41), and ( 42). The theory gives a clear picture of the origin of magnetophonon oscillations observed [ 20]i n the longitudinal thermopower of high-mobility GaAs quantumwells and predicts similar oscillations in the transversethermopower (Fig. 1). For typical parameters of GaAs wells, the oscillations are clearly visible for temperatures above2 K, while at lower temperatures they become exponentiallysuppressed because the Bloch-Gruneisen regime is reached. Inthe experiment [ 20], however, the oscillations were resolved between 0.5 K and 1 K. This discrepancy can be explainedby taking into account that the phonon distribution functionin the experiments on thermopower is not reduced to theform of Eq. ( 17) commonly applied by theorists. Even at low temperatures of the sample, there can exist high-energyphonons able to cause backscattering of electrons. Indeed,since the phonon mean-free path at low temperatures isvery large (of 1 mm scale), it is quite possible that suchhigh-energy phonons may arrive at the 2D system directly from the heater, via ballistic propagation. Another possiblereason, which is especially relevant at low temperatures, is thatthe modiﬁcation of phonon distribution function is strong andcannot be represented in the form of a small correction linearin temperature gradient. In any case, a quantitative agreementwith experiment can be reached only if the phonon distributionis known. The theory presented in this paper can be generalizedto the case of arbitrary phonon distribution by substitutingthe antisymmetric part of actual phonon distribution functioninstead of the second term in Eq. ( 17). The inﬂuence of MW irradiation on the longitudinal α xxand transverse αxycomponents of the thermopower has been studied above by using the approved methodsapplied earlier to calculation of the resistivity. It is foundthat the MW irradiation has a considerable effect on boththese components. In contrast, for electrical resistance themicrowaves strongly modify only the longitudinal componentρ xx. Both the diffusive and phonon-drag contributions to thermopower are shown to be affected by MW irradiation. TheMW-induced quantum corrections to diffusive thermopowerincrease with decreasing electron temperature, in contrastto classical diffusive thermopower, which is proportional tothis temperature. However, since the phonon-drag contributiondominates, the MW-induced quantum corrections to phonon-drag thermopower appear to be more important. These effects are of the order of several μV/K for typical parameters of the 2D system and MW excitation, and can be detectedexperimentally. The oscillating behavior of MW-inducedcorrections as functions of the magnetic ﬁeld reﬂects theproperties of electron scattering by phonons under condi-tions when the electron distribution function acquires anMW-induced oscillating component (inelastic mechanism)and when MW-assisted scattering takes place (displacementmechanism). Both these mechanisms are important, and bothprovide a mixing of resonant phonon frequencies with MWfrequency ω, thereby leading to interference oscillations of the thermopower. In terms of relative values, the MW-induced changes in the longitudinal thermopower are much smaller than thecorresponding effect in the resistivity. In contrast, the relativeMW-induced changes in the transverse thermopower are large,because in the classically strong magnetic ﬁelds the transversethermopower itself is much smaller than the longitudinalone. At lower temperatures and weaker magnetic ﬁelds, theoscillations of transverse thermopower α xyfollow the picture of MW-induced resistance oscillations (MIRO) [Fig. 4(b)]. As the temperature and magnetic ﬁeld increase, the oscillations ofα xyno longer follow the MIRO picture and become strongly sensitive to polarization of the incident wave. The polarizationdependence of α xyis much stronger than the corresponding dependence of the electrical resistivity under MW irradia-tion. These ﬁndings may stimulate experimental studies ofthe transverse thermopower of MW-irradiated 2D electrongas. The appearance of a large polarization-dependent term in the MW-induced transverse thermopower is one of themain results of the present study. The nature of this effectcan be easily understood by considering the collisionlessapproximation (no electron-impurity scattering, ν tr=0), when 235307-13O. E. RAICHEV PHYSICAL REVIEW B 91, 235307 (2015) the transverse thermopower does not appear without MW irradiation. The drag of electrons by the phonons drifting alongthe temperature gradient ∇Tcan be described [ 22] in terms of a dragging force due to effective electric ﬁeld E ph∝∇T.T h e electrons in the magnetic ﬁeld are drifting perpendicular toE ph. To compensate this drift, a real electric ﬁeld E=−Eph develops. Thus, the longitudinal thermopower is equal to \|E\|/\|∇T\|while the transverse thermopower is zero. When a polarized ac ﬁeld is applied to the system, the effective electricﬁeldE ph, in general, is not directed along ∇Tand becomes sensitive to polarization. This occurs because Ephis formed as a result of electron-phonon interaction assisted by emissionand absorption of radiation quanta, and this interaction isstronger when the in-plane components of phonon momentaare parallel to the polarization-dependent vector R ω;s e e Eqs. ( 12) and ( 13). Consequently, the real electric ﬁeld E=−Ephis not parallel to ∇T, which means that there exists a transverse component of thermopower. This component isgiven by the ﬁrst term in Eq. ( 46). Beyond the collisionless approximation, the other, polarization-independent terms inα xyare also important. A larger relative contribution of the polarization-dependent term is expected in 2D electronsystems with higher mobility (smaller ν tr). An important issue left beyond the above consideration is the behavior of thermopower at zero longitudinal resistance. Inhigh-mobility 2D systems, intensive MW irradiation leads to a remarkable phenomenon of zero-resistance states [ 3–5], which means that the longitudinal resistance vanishes in certainintervals of magnetic ﬁelds corresponding to MIRO minimaat lower MW intensity. This effect is often explained (seeRef. [ 1] and references therein) as a result of the instability of homogeneous current ﬂow under condition of negative localresistance, which leads to spontaneous formation of domainswith different directions of the currents and Hall ﬁelds. Sincethe longitudinal resistivity formally enters the expression forthermopower and, as shown above, considerably affects thetransverse thermopower in the presence of MW irradiation,the magnetic-ﬁeld dependence should demonstrate the regionsof nearly constant α xyin the intervals of ρxx=0, while αxx is not expected to be sensitive to zero-resistance states. Of course, this conclusion looks somewhat naive, because thepresence of domains may affect the behavior of measuredthermopower. It is not clear, however, which kind of domainpicture is realized under zero-resistance state conditions inthermoelectric experiments, when there is no electric currentsthrough the contacts. Future studies should shed light on thisparticularly interesting problem. ACKNOWLEDGMENT The author is grateful to G. Gusev for helpful discussions. APPENDIX: ASYMPTOTIC BEHA VIOR OF THE FUNCTIONS /Gamma1iAND ˜/Gamma1i In the approximation of the isotropic phonon spectrum, the integral over the polar angle ϕqin the operator ˆPncan becarried out analytically, and Eq. ( 33) is reduced to the form ⎛ ⎝/Gamma1n /Gamma1cn /Gamma1sn⎞ ⎠=m2 ρM/integraldisplayπ 0dθ π(1−cosθ)n/integraldisplay∞ 0dqz πIqz ×/summationdisplay λ=l,tτλGλF/parenleftbiggsλQ 2T/parenrightbigg⎛ ⎜⎝1 cos2πsλQ ωc ωc 2πsλQsin2πsλQ ωc⎞ ⎟⎠, (A1) where Q=/radicalbig q2+q2z,q=2pFsin(θ/2),Gl=D2+ (eh14)29q4q2 z/2Q8, andGt=(eh14)2(8q2q4 z+q6)/2Q8.F o r ˜/Gamma1ione should replace GlandGtby˜Gl=− (eh14)29q4q2 z/4Q8 and ˜Gt=(eh14)2(8q4q2 z−q6)/4Q8, respectively. The functions Gtand ˜Gtdescribe interaction of electrons with transverse phonon modes due to piezoelectric-potentialmechanism, while G land ˜Gldescribe interaction with longitudinal phonon modes due to both deformationpotential and piezoelectric-potential mechanisms. Analyticalexpressions for the functions /Gamma1 cn,/Gamma1sn,˜/Gamma1cn, and ˜/Gamma1sncalculated from Eq. ( A1) are given below in some limiting cases. In the limit ωc/lessmuch4πsλpF, when cos(2 πsλQ/ω c) and sin(2πsλQ/ω c) are rapidly oscillating functions of θand qz/pF, the main contribution to the integrals in Eq. ( A1) comes from the region of small qz, when Iqz/similarequal1, and from two regions of θaround θ=0 (corresponding to forward scattering of electrons) and θ=π(backscattering), because these are the regions of slowest variation of Qas a function of θandqz. Under the requirement \|ωc\|/lessmuch 2π2T, which is already stated as the condition when the Shubnikov–de Haas oscillations aresuppressed, one obtains /Gamma1 c1=γt 2/epsilon1tF/parenleftBigstpF T/parenrightBig cos/epsilon1t−59γt 28/epsilon12 t−45γl 28/epsilon12 l +4γl /epsilon1l/bracketleftbigg F/parenleftBigslpF T/parenrightBig cos/epsilon1l+3 /epsilon13 l/bracketrightbigg/parenleftbiggDpF eh14/parenrightbigg2 ,(A2) /Gamma1c2=γt /epsilon1tF/parenleftBigstpF T/parenrightBig cos/epsilon1t+261γt 27/epsilon14 t+189γl 27/epsilon14 l +8γl /epsilon1l/bracketleftbigg F/parenleftBigslpF T/parenrightBig cos/epsilon1l−45 /epsilon15 l/bracketrightbigg/parenleftbiggDpF eh14/parenrightbigg2 ,(A3) /Gamma1s1=γt 2/epsilon12 tF/parenleftBigstpF T/parenrightBig sin/epsilon1t+59γt 28/epsilon12 t+45γl 28/epsilon12 l +4γl /epsilon12 l/bracketleftbigg F/parenleftBigslpF T/parenrightBig sin/epsilon1l−1 /epsilon12 l/bracketrightbigg/parenleftbiggDpF eh14/parenrightbigg2 ,(A4) /Gamma1s2=γt /epsilon12 tF/parenleftBigstpF T/parenrightBig sin/epsilon1t−87γt 27/epsilon14 t−63γl 27/epsilon14 l +8γl /epsilon12 l/bracketleftbigg F/parenleftBigslpF T/parenrightBig sin/epsilon1l+9 /epsilon14 l/bracketrightbigg/parenleftbiggDpF eh14/parenrightbigg2 ,(A5) ˜/Gamma1c1=−γt 4/epsilon1tF/parenleftBigstpF T/parenrightBig cos/epsilon1t−5 29/parenleftbiggγt /epsilon12 t−9γl /epsilon12 l/parenrightbigg ,(A6) ˜/Gamma1c2=−γt 2/epsilon1tF/parenleftBigstpF T/parenrightBig cos/epsilon1t−21 28/parenleftbiggγt /epsilon14 t+9γl /epsilon14 l/parenrightbigg ,(A7) 235307-14THEORY OF MAGNETOTHERMOELECTRIC PHENOMENA IN . . . PHYSICAL REVIEW B 91, 235307 (2015) ˜/Gamma1s1=−γt 4/epsilon12 tF/parenleftBigstpF T/parenrightBig sin/epsilon1t+5 29/parenleftbiggγt /epsilon12 t−9γl /epsilon12 l/parenrightbigg ,(A8) ˜/Gamma1s2=−γt 2/epsilon12 tF/parenleftBigstpF T/parenrightBig sin/epsilon1t+7 28/parenleftbiggγt /epsilon14 t+9γl /epsilon14 l/parenrightbigg ,(A9) where γλ=τλm2(eh14)2 πρMpF,/epsilon1λ=4πsλpF \|ωc\|. (A10) For comparison, it is useful to present also the expression for/Gamma11: /Gamma11=177ζ(3) 29γt/parenleftbiggT stpF/parenrightbigg2 +135ζ(3) 29γl/parenleftbiggT slpF/parenrightbigg2 +/parenleftbiggDpF eh14/parenrightbigg2 15ζ(5)γl/parenleftbiggT slpF/parenrightbigg4 , (A11) where ζ(k) is the Riemann zeta function. This expression is valid in the limit of T/lessmuchsλpFand can be used for order-of- value estimates at T/similarequalsλpF. From the deﬁnition (A10), the applicability region for Eqs. ( A2)–(A9) can be written as /epsilon1λ/greatermuch1. The magneto-oscillations of the functions described by Eqs. ( A2)– (A9) occur because of the terms with cos /epsilon1λand sin /epsilon1λ.T h e amplitudes of these oscillating terms are always much smallerthan/Gamma1 1of Eq. ( A11) in the case /epsilon1λ/greatermuch1. IfT/similarequalsλpF, this smallness is given by the factors /epsilon1−1 λfor/Gamma1c1,/Gamma1c2,˜/Gamma1c1, and ˜/Gamma1c2and/epsilon1−2 λfor/Gamma1s1,/Gamma1s2,˜/Gamma1s1, and ˜/Gamma1s2. With lowering T, the oscillations are exponentially suppressed because of F(sλpF/T)/similarequal(2sλpF/T)2exp(−2sλpF/T)a tT/lessmuchsλpF.I n the case of strong exponential suppression, the absolute valuesof the functions given by Eqs. ( A2)–(A9) are determined by their nonoscillating parts which are proportional to powersofω c. The nonoscillating parts of n=1 functions ( /Gamma1c1,/Gamma1s1, ˜/Gamma1c1, and ˜/Gamma1s1) are much smaller than /Gamma11due to parameters (ωc/2π2T)2for the piezoelectric-potential contribution and (ωc/2π2T)4for the deformation-potential contribution. The nonoscillating parts of n=2 functions ( /Gamma1c2,/Gamma1s2,˜/Gamma1c2, and ˜/Gamma1s2) contain extra small factors /epsilon1−2 λ, because these functions are much smaller than n=1 functions at small-angle scattering, θ/lessmuch1. In stronger magnetic ﬁelds, when ωcis comparable to 4πsλpF, analytical expressions can be obtained at T> s λpF and under a wide-well approximation, the latter means that the quantum well width ais much larger than π/pFso that the convergence of the integral over qztakes place at qz/lessmuchpFand is governed by the function Iqz. Introducing q0=π−1/integraltext∞ 0dqzIqz(for inﬁnitely deep rectangular wellq0=3/2a), one obtains /Gamma1cn=2nm2q0 ρM/bracketleftbigg (−1)nτlD2I2n(/epsilon1l) +(−1)n−1τt(eh14)2 8p2 FI2n−2(/epsilon1t)/bracketrightbigg , (A12) /Gamma1sn=2nm2q0 ρM/bracketleftbigg (−1)nτlD2 /epsilon1lI2n−1(/epsilon1l) +(−1)n−1τt(eh14)2 8p2 F/epsilon1tI2n−3(/epsilon1t)/bracketrightbigg , (A13) ˜/Gamma1cn=−2nm2q0(eh14)2τt 16p2 FρM(−1)n−1I2n−2(/epsilon1t), (A14) ˜/Gamma1sn=−2nm2q0(eh14)2τt 16p2 FρM/epsilon1t(−1)n−1I2n−3(/epsilon1t), (A15) where Ik(x)=dkJ0(x) dxk(A16) is thekth-order derivative of the Bessel function J0(x). Such derivatives can be expressed through the other Bessel functionsJ i(x). In the special case of /Gamma1s1, there is a term with the function I−1(/epsilon1t), which should be treated as the antiderivative of J0(/epsilon1t). This term is expressed through the Bessel functions and Struvefunctions H i: I−1(x)≡/integraldisplayx 0dx/primeJ0(x/prime)=xJ0(x) +π 2x[J1(x)H0(x)−J0(x)H1(x)].(A17) In the regime of validity of Eqs. ( A12)–(A15) the function /Gamma11 is given by /Gamma11=m2q0 ρM/bracketleftbigg τlD2+τt(eh14)2 4p2 F/bracketrightbigg . (A18) For large arguments /epsilon1λ, the functions ( A12)–(A15)a r e reduced to combinations of oscillating factors sin /epsilon1λand cos /epsilon1λ, similarly to the case described by Eqs. ( A2)–(A9), and are small in comparison to /Gamma11.I f/epsilon1λ/similarequal1, these functions become comparable to /Gamma11. Actually, the wide-well limit a/greatermuchπ/pF is hardly attainable for single-subband occupation in the quantum well. The expressions ( A12)–(A15) are nevertheless useful for estimates of the maximal possible values of thequantities /Gamma1 iand ˜/Gamma1i. [1] I. A. Dmitriev, A. D. Mirlin, D. G. Polyakov, and M. A. Zudov, Rev. Mod. Phys. 84,1709 (2012 ). 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PhysRevB.96.165442.pdf	PHYSICAL REVIEW B 96, 165442 (2017) Identiﬁcation of Ni 2C electronic states in graphene-Ni(111) growth through resonant and dichroic angle-resolved photoemission at the C K-edge G. Drera,1,C. Cepek,2L. L. Patera,2,3F. Bondino,2E. Magnano,2S. Nappini,2C. Africh,2A. Lodi-Rizzini,2N. Joshi,4 P. Ghosh,4A. Barla,5S. K. Mahatha,5S. Pagliara,1A. Giampietri,1C. Pintossi,1and L. Sangaletti1 1I-LAMP and Universitá Cattolica del Sacro Cuore, via dei Musei 41, I-25121 Brescia, Italy 2CNR-IOM, Laboratorio Nazionale TASC, S.S. 14, km 163.5, I-34012 Trieste, Italy 3Department of Physics, Universitá degli Studi di Trieste, via Alfonso Valerio 2, 34127 Trieste, Italy 4Department of Chemistry and Physics, Indian Institute of Science Education and Research, Pune-411021, India 5Istituto di Struttura della Materia (ISM), Consiglio Nazionale delle Ricerche (CNR), S.S. 14 Km 163.5, I-34149 Trieste, Italy (Received 6 March 2017; revised manuscript received 19 September 2017; published 30 October 2017) The graphene-Ni(111) (GrNi) growth via chemical vapor deposition has been explored by resonant, angle- resolved, and dichroic photoemission spectroscopy (soft x-ray Res-ARPES) in order to identify the possiblecontributions to the electronic structure deriving from different phases that can coexist in this complex system.We provide evidences of electronic states so far unexplored at the ¯/Gamma1point of GrNi, appearing at the C K-edge resonance. These states show both circular dichroism (CD) and kdependence, suggesting a long-range orbital ordering, as well as a coherent matching with the underlying lattice. Through a comparison of core-levelphotoemission, valence band resonances, and constant initial-state spectroscopy, we demonstrate that these statesare actually induced by a low residual component of nickel carbide (Ni 2C). These results also show that caution must be exercised while interpreting x-ray magnetic circular dichroism collected on C K-edge with Auger partial yield method, due to the presence of CD in photoelectron spectra unrelated to magnetic effects. DOI: 10.1103/PhysRevB.96.165442 I. INTRODUCTION Nickel(111) is a reference substrate for the growth of epitaxial graphene (hereafter referred as GrNi), due to thenearly perfect lattice parameter match [ 1]. The superposition of nickel 3 dand carbon 2 p zorbitals leads to a strong hybridization, which partially disrupts the usual free-standinggraphene electronic band structure. In particular, πbands are shifted towards larger binding energies (BE) of several eV , andthe Dirac cone at the ¯Kpoint of the reciprocal space cannot be directly observed [ 2].Ab initio calculations also suggest that graphene should exhibit a Ni-induced ferromagnetism, whichhas been investigated by x-ray magnetic circular dichroism(XMCD) measurements [ 3,4] carried out at room temperature. The dichroic signal is reported for the C 1 sabsorption edge, involving either the entire π ∗edge [ 3] or only a pre-edge feature [ 4]. In both cases, the p-dhybridization is invoked as the source of magnetism. The GrNi growth by chemical vapor deposition (CVD) is a very complex process, strongly dependent on the highcarbon solubility in nickel [ 5] and on the initial carbon doping level of the substrate [ 5–8]. In addition to epitaxial graphene, several graphene rotated phases have been observed, whoserelative coverage depends on the CVD growth conditionsand on the substrate pretreatments [ 6,7]. An important fact is the formation of surface nickel carbide, which has beenobserved as a graphene precursor for growth at temperaturelower than 450 ◦C and as an interlayer in the high-temperature processes during the cooling at room temperature, throughcarbon bulk diffusion [ 6,7]. In the latter case, carbide is formed exclusively below the rotated graphene phases, andelectronically decouples these rotated domains from the giovanni.drera@unicatt.itunderlying Ni substrate, as evidenced by laterally resolvedx-ray-photoemission spectroscopy (XPS) and ARPES, whichshow the typical features of noninteracting graphene [ 7]. For all these reasons, a complete high-quality epitaxial layer maybe difﬁcult to obtain, as well as a perfectly carbon-free Nicrystal before graphene growth. In general, the combinationof a single C 1 score-level peak, a clear 1 ×1 LEED pattern, and a standard band dispersion in ARPES is recognized as thecommon signature of a monolayer epitaxial GrNi growth. While several studies have been focused on the epitaxial GrNi properties, a proper identiﬁcation of elemental selectiveelectronic states of graphene precursors (i.e., surface nickelcarbides) is still lacking in literature. Carbides contribution toGrNi magnetism is also unexplored, while several polycrys-talline Ni xC are known to be ferromagnetic metals [ 9]. Here, we exploited resonant photoelectron spectroscopy technique to investigate the carbon-related electronic struc-tures in CVD growth carried out on Ni(111) surfaces usingethylene. Graphene studies have already beneﬁted from syn-chrotron investigations, especially for x-ray absorption andphotoemission [ 3,4,6,10] or band dispersion [ 11–16] mea- surements. The large accessibility of experimental facilitiesand the improvement of the detection techniques now allowresearchers to simultaneously combine several degrees offreedom in a photoelectron spectroscopy experiment [ 17]. Progress in surface and interface physics has already beenfavored by new concepts in resonant photoemission [ 18,19] (ResPES); the key feature in these experiments is the pos-sibility to work in resonance conditions, which boosted theexploration of quite elusive spectral features otherwise missedby conventional photoemission probes. Moreover, while ARPES was initially limited to the low (<100 eV) photon energy range, improvements in the angle- resolved electron analyzers technology now open the ﬁeld tosoft x-ray ARPES. In this regime, the lower photon energy 2469-9950/2017/96(16)/165442(9) 165442-1 ©2017 American Physical SocietyG. DRERA et al. PHYSICAL REVIEW B 96, 165442 (2017) resolution can be traded off for chemical selectivity, exploited through x-ray core-level resonances. Resonant ARPES hasbeen carried out, for instance, on LaAlO 3-SrTiO 3perovskite heterostructures [ 20] with the aim to map the electronic structure of interface states at the basis of the two-dimensional(2D) electron gas observed in these systems. The combinationof ARPES and resonant photoemission could also allowto discriminate the presence of secondary phases, with theselectivity given by the choice of the appropriate photon energyand the strong enhancement of cross section given by theresonant process [ 21,22]. In this work, we extend synchrotron-based photoemission studies on graphene to a combination of techniques thatallowed us to disclose unexpected role of nickel carbidesin the electronic structure. We report the analysis of severalsamples grown with CVD process on Ni(111), carried out withcircular dichroic, angular-resolved, resonant photoemission atthe carbon K-edge, namely, valence band (VB) photoemission, performed by scanning the photon energy across an absorp-tion edge (ResPES), collected with left and right circularlypolarized x rays (circular dichroism) by an angle-resolvedanalyzer (ARPES). The schematic of the technique is givenin Fig. 1(a). We report the discovery of previously unexplored resonant features at the π ∗edge, which can be detected in different GrNi samples. The origin of this resonant electronicstructure is identiﬁed through a careful comparison of several growth stages, taking into account each available experimental degrees of freedom (photon energy, kinetic/binding energy,crystal momentum, and photon polarization). Additionally, wealso show how the circular dichroic properties of the resonantelectronic structures can signiﬁcantly affect the reliabilityof XMCD measurements collected in Auger partial yieldmethods. The paper is organized as follows: full description ofsample growth and characterization (Sec. II A); description of the resonant ARPES experimental technique (Sec. II B); DFT calculation results for various GrNi reconstructions (Sec. II C); experimental ResPES and Res-ARPES results analysis of GrNiwith the comparison of VB resonances measured at variousgrowth stages (Sec. III); and conclusions (Sec. IV). II. EXPERIMENTAL AND COMPUTATIONAL DETAILS A. Samples growth In this study, several C-Ni samples have been investigated. For the sake of simplicity, a speciﬁc label has been assignedto the three cases shown in this work, summarized in Table I. All samples have been grown ex situ in an ultrahigh vacuum chamber (with a base pressure of 10 −10mbar), then transferred and annealed in vacuum in the ARPES experimental chamber;this procedure was required in order to reconstruct the GrNiphase and to remove atmospherical contaminations. Sample A has been grown by CVD of ethylene on a thin (5-monolayer) Ni ﬁlm deposited over a Mo(110) single crystal;the gas exposure has been carried out at a pressure of p= 5×10 −6mbar and at a temperature of T≈480◦C, followed by a 2-min post-annealing, at T≈580◦C. Sample B has been grown by exposing a Ni(111) single crystal to ethylenepressure of 10 −7mbar at T≈650◦C. This sample has been cooled as fast as possible to minimize the formation of carbide FIG. 1. (a) Schematics of the resonant ARPES experiment. (b) Detail of the ARPES geometry for the graphene experiment,showing the incoming and reﬂected x rays (red arrows, lying on the red vertical plane), the analyzer angle dispersion plane (yellow plane), the 1×1 LEED geometry, and the aligned Brillouin zone. (c) Sketch of resonant photoemission intermediate (XAS) and autoionization processes, compared to direct core and VB photoemission. underneath rotated phases [ 6,7]. Additional measurements have been carried out on a single-phase nickel carbide surface(sample C), grown by exposing the Ni(111) single crystalsurface to ethylene at a lower temperature (300 ◦C). Samples A and B have been carefully characterized by LEED, XPS, and UPS in order to verify the orientation andthe presence of impurities, before and after the experiments. 165442-2IDENTIFICATION OF Ni 2C ELECTRONIC STATES . . . PHYSICAL REVIEW B 96, 165442 (2017) TABLE I. Summary of the samples analyzed in this work. The last three columns show the relative area of each peak ﬁtting component shown in Fig. 2. Label Main phase Substrate Growth temperature C int Cnonint Csurf-carb A Interacting C-Ni Ni thin ﬁlm on Mo(110) 580◦C 80.2% 17.2% 2.6 % B Interacting C-Ni Ni (111) single crystal 650◦C 55.6% 44.0% 0.4% CN i 2C Ni (111) single crystal 300◦C 100% For instance, the electronic band dispersion collected at low photon energy (47.5 eV) on sample A is shown in Fig. 2(a); theπ,σ, and Ni 3 dband dispersions are clearly detectable and the πband position at ¯/Gamma1point is located at the expected binding energy ( ≈10 eV). All the C 1 sspectra have been analyzed by following the ﬁt procedure and parameters already used in Refs. [ 6,7], supposing that each spectrum is composed by three Doniach-Sunjic components superimposed to a Shirley background.The results are shown in Fig. 2(b).T h eC1 score-level photoemission spectra [Fig. 2(b)] show the presence of three components, corresponding to interacting graphene (BE ≈ 284.8 eV), noninteracting graphene (BE =284.4 eV), and nickel carbide (BE =283.4 eV), indicating the coexistence of several phases. To calculate the relative amount of each surface component (listed in Table I), we supposed that the noninteracting graphene is only due to the presence of carbide underneath therotated graphene, possibly due to the annealing procedure inthe ARPES chamber. The formation of second-layer graphenehas not been considered. The stoichiometry has been evaluatedby supposing that the graphene layer has a thickness of 3.1 ˚A and that the escape depth in graphene of the photoelectronat the measured kinetic energy is ≈4.3˚A[23]. The relative amount of each surface component is shown in Table I, where the carbide component is due to unreacted carbide withoutgraphene on top. Intensity (arb. units) 290 288 286 284 282 Binding Energy (eV) DATA Interacting Gr Non-Interacting Gr Carbide Background FitA BC 1s C -0.2 0.0 0.2 k // (Å-1)14121086420Binding Energy (eV)πσNi 3dΓ (a) (b) FIG. 2. (a) ARPES band dispersion around ¯/Gamma1point collected on sample A at a photon energy of 47.7 eV , on the ¯M-¯/Gamma1-¯Mdirection. (b) C 1 score-level photoemission on the samples reported in Table I, collected at a photon energy of 370 eV , together with peak ﬁtting results. Experimental data are normalized to the nickel 3 ppeak intensity (a 10 ×multiplicative factor is included for sample C).From the C 1 sanalysis it comes out that the surfaces of both samples A and B are completely covered by single-layergraphene phases, only a few percent of the surface couldbe composed by a nonreacted carbide phase (2.6% and0.4%, respectively). Both samples present rotated phases,as highlighted by the presence of the noninteracting C 1 s component. In the case of sample A, the noninteractinggraphene is ≈17%, while in the case of sample B it is 44%. Both samples show only a sharp 1 ×1 GrNi LEED pattern, indicating that other phases should be present insmall, randomly oriented domains, not detectable by LEED;to further conﬁrm this point, no evidence of rotated and/ornoninteracting graphene phase has been detected in ARPESmaps, both at ¯/Gamma1and ¯Kpoints. Finally, in sample C, only the carbide peak (C 1) is detected, conﬁrming that no graphene nucleated at the surface. Thissample has been annealed (300 ◦C) and checked every few ResPES single spectra acquisition (6–8 h long) in order toremove surface CO contamination. B. Experimental resonant ARPES details In a ResPES experiment, VB photoemission is measured with a photon energy tuned to be resonant with a speciﬁccore level, i.e., across an x-ray absorption edge. In suchprocess, electrons can be promoted to empty states leading toadditional decay channels [see Fig. 1(c)]. In fact, together with direct photoemission, an autoionization process can also takeplace, which may involve (participator decay) or not (spectatordecay) the promoted electrons. Because of the identical ﬁnalstates, interference of photoemission and autoionization mayalso take place [direct photoemission and participator decay,Fig. 1(c)], leading to a sudden increase of selected spectral weight in the VB. While the dependence on photoelectron momentum in standard ARPES is well known, the Auger-type processes(including autoionization) are usually interpreted in terms oflocalized core-hole transitions, independent from the crystalstructure. However, a strong kdependence in resonance spectra has already been observed in several systems [ 20,24]. According to Molodtsov et al. [25], a momentum conservation is expected in resonant photoemission only when the interme-diate state decays into the ﬁnal state (emitted electron) beforeits coherence is lost (and thus phonon assisted decay doesnot take place). Some technical precaution should be appliedwhile measuring resonant ARPES; in fact, a change in photonenergy results in a different electron kinetic energy, whichin turn leads to a variable momentum probing range (with aﬁxed analyzer angular acceptance). Although these effects areminimal for a relatively small photon energy variation, theirinﬂuence is often not negligible. 165442-3G. DRERA et al. PHYSICAL REVIEW B 96, 165442 (2017) The combination of ARPES and circular polarization is usually known as CDAD (circular dichroism in the angulardistribution). The CDAD results in a relative shift of the bandspectral weight in the reciprocal space, while preserving theband dispersion shape [ 26]. This effect, due to the dipole transition matrix element, has already been observed in weaklyinteracting graphene [ 12], but it is expected to appear in any crystal structure when the incidence plane and the mirrorplane of the crystal are matched [ 27,28], as well as in ordered molecular layers [ 29]. In the graphene case, a CDAD node has been observed around the ¯Kpoint of the band structure [ 13]; however, in our experimental geometry an asymmetric CDAD is also expected in the ARPES dispersionof the nickel substrate bands around ¯/Gamma1point. It should be noted that magnetism is not a prerequisite for the detection ofdichroic band dispersion; in general, ferromagnetism wouldgive an additional contribution, superimposed to the substratenatural CDAD. Resonant photoemission has been carried out in a wide (≈280–310 eV) photon energy range across the C K-edge, with a ±10.5 ◦angular dispersion around the sample normal (i.e., around the ¯/Gamma1point in the band dispersion). The angular dispersion plane was set to probe the ARPES in the ¯/Gamma1-¯M direction; this orientation was identiﬁed by the LEED analysisand is corresponding to the vertical yellow plane in Fig. 1(b). The sample has not been physically rotated for ARPES measurements since at the C K-edge the analyzer dispersion range combined to the high electron kinetic energy allowedus to directly measure most of the ¯M-¯/Gamma1-¯Mcrystal momentum dispersion. The x-ray direction formed an angle φ=60 ◦with the surface normal [see Fig. 1(b)]. Data have been properly normalized to the VB intensity and each photoemission spectrum has been aligned with theFermi edge. The data collection has been carried out at theBACH beamline at the Elettra synchrotron, which providesboth tunable x-ray circular polarization and an angle-resolvedelectron analyzer (VG-Scienta R3000 [ 30]). All measure- ments have been carried out on nonmagnetized samples atroom temperature. The absence of magnetism, conﬁrmed byXMCD at the Ni L 3,2-edges, was needed in order to avoid additional effects on the ARPES spectral weight redistributionupon polarization switch. The combined energy resolution(photon energy linewidth and analyzer response) was about0.3 eV . It should be noted that graphene itself poses several exper- imental constraints to VB x-ray spectroscopy measurements.For instance, due to its 2D nature, ARPES must be performedat photon energy lower than 50 eV , in order to maximizethe surface sensitivity. At higher photon energies, VB pho-toemission is not usually carried out, as in off-resonancecondition the spectral weight of C states is overwhelmed bythe Ni 3 demission; in fact, at the typical photon energies for ResPES at C K-edge ( hν≈285 eV), the predicted ratio of C 2 pphotoemission cross section with respect to the Ni 3done is approximately [ 31] 1%. The overall low carbon amount also makes more favorable the collection of x-rayabsorption spectrum (XAS) and magnetic circular dichro-ism (XMCD) through partial yield electron-based detectiontechnique (PEY), rather than of total yield techniques (TEY)which are applied for the substrate Ni L-edge XAS/XMCD;such kinds of detection modes have been used to measure the magnetism by XMCD [ 3,4]. C. DFT calculations In order to predict the GrNi electronic structure, a pre- liminary discussion of the possible bonding geometry isrequired. Several graphene bonding sites can be considered,deﬁned by the relative C position with respect to surfaceNi, shown in Fig. 3. Initially, according to EELS (electron energy-loss spectroscopy) measurements, Rosei et al. [32] proposed a model where C atoms sit on top of the Nisecond-row atoms, hereafter labeled as hcp sites, ﬂoating at adistance of 2.80 ˚A from the surface [Fig. 3(a)]. However, DFT calculations [ 33] suggest that at this distance the graphene electronic structure should not strongly interact with Ni,leading to an unperturbed graphene band structure, at oddswith ARPES results. Alternatively, the R3msymmetry of the Ni(111) surface is conserved when C atoms are bound on top-and third-row Ni atoms [top-fcc reconstruction, FCC in short,Fig. 3(b)] or on top and second Ni planes [top-hcp, TOP in short, Fig. 3(c)]. In these reconstructions, calculations [ 34] predict a graphene-surface distance of nearly 2.0 ˚A, similar to the interplanar distance of bulk nickel and compatible with astrong interaction picture. In addition, a bridgelike structurehas also been proposed [Fig. 3(d)], where the closest bond site lies in-between two Ni surface atoms [ 35]. A survey of the computed density of states (DOS) on the aforementionedreconstructions is given in Fig. 3. In general, TOP and FCC reconstructions are expected to be extremely close in energy,so that their combination can create extended defects inGrNi [ 2]; in fact, several reconstructions have been observed simultaneously on the same sample [ 36,37]. For this work, we refer to spin-resolved GGA-PBE DFT calculations, as reported in the literature [ 38], in order to obtain the band structure of sp 2-Ni 3dhybridized states from the atomic projected DOS (Fig. 3), to be compared with angle-resolved ResPES results; in fact in top, fcc, and hcpC adsorption sites, new hybridized C 2 pstates appear in the Ni 3dVB region, with a noticeable energy shift [arrows in Fig. 3(b)]. Such localized electronic states could be easily resolved by photoemission spectroscopy. In FCC and TOPreconstructions, the top site electronic structures are verysimilar, with a sharp contribution 2 eV below Fermi level.In both cases, a clear difference of spin-up and spin-downcarbon states can be seen, as a consequence of a net magneticmoment on graphene. The HCP conﬁguration, which shows anearly free-standing graphene band dispersion, has not beenconsidered in this work since its presence has already beenexcluded by experimental studies [ 1]. III. EXPERIMENTAL RESULTS First, we show the experimental results on the strongly interacting GrNi samples A and B. Due to their complexity,resonant ARPES data can be processed and represented inseveral ways. The integral ResPES maps [Figs. 4(a)and4(b)] can help to underline the main features of the photoemissionspectra. Auger C KLL peaks, linearly dispersing with photon energy, can be easily distinguished if the photoemission spectra 165442-4IDENTIFICATION OF Ni 2C ELECTRONIC STATES . . . PHYSICAL REVIEW B 96, 165442 (2017) FIG. 3. Calculated spin-resolved density of states for epitaxial graphene bonding site over Ni(111), for the main R3mreconstructions. In each graph, the C-projected DOS spectra color is related to the speciﬁc bonding site: red for top sites (upper Ni layer), blue for hcp sites (second Ni layer), and green for fcc (third Ni layer). Spin up and down are shown on positive and negative left axes, respectively. are referred to the binding energy scale; the Auger shape is consistent with other analyses [ 39]. By performing a pre-edge subtraction, i.e., by subtracting the pre-edge spectrum from the resonant ones, the VB resonantspectral weight (RSW) clearly stands out [Fig. 4(b)]. Two different peaks, at BE =1.8 and 0.5 eV , can be detected at photon energies corresponding to the π ∗andσ∗resonances,respectively. The possibility to observe these resonances (interpreted as participator autoionization decay) is itselfremarkable since in the ResPES of most organic compounds,the dominant KLL Auger is usually superimposed to the resonant spectral weight [ 40]. The integral ResPES maps do not display evident asymme- tries by polarization reversal. However, thanks to the angular 305 300 295 290 285Photon Energy (eV) 20 15 10 5 0 Binding Energy (eV)σ∗ π∗ VB (Ni3d)Ni satellite Auger CIS 20 15 10 5 0 Binding Energy (eV)π∗ resonanceσ∗ resonance C KLL Auger(a) (b) -1.0 -0.5 0.0 0.5 1.0 K// (Å-1)1086420Binding Energy (eV)kA kB kC(c) FIG. 4. (a) Resonant photoemission map collected for circular left polarization (sample A) with a reference CIS spectra (red line, left axis), collected at BE =13 eV . (b) Pre-edge subtracted ResPES map (black dotted lines are a guide to the eye to show the Auger C KLL peaks linearly dispersing with photon energy). (c) Unpolarized ARPES map at pre-edge photon energy with labeled integrated kregions. 165442-5G. DRERA et al. PHYSICAL REVIEW B 96, 165442 (2017) 300 290 Photon Energy (eV)Auger CIS LCP CIS 0.5 eV CIS 1.8 eV CISIntensity (arb. units) 300 290 Photon Energy (eV)Auger CIS 0.5 eV CIS 1.8 eV CISIntensity (arb. units) 10 8 6 4 2 0 Binding Energy (eV)Zone kA {-1,-1/3}Zone kB {-1/3,1/3}Zone kC {1/3,1} Full k-rangePre-edge π∗ σ∗ π* diff.(x2) σ* diff.(x2)Left circular pol. Right circular pol. 10 8 6 4 2 0 Binding Energy (eV)Zone kA {-1,-1/3}Zone kB {-1/3,1/3}Zone kC {1/3,1} Full k-rangePre-edge π* diff.(x2) π∗ σ* diff.(x2) σ∗(a) (b) (c) (d) FIG. 5. k-resolved photoemission [(a) and (b)] and constant initial-state spectra [(c) and (d)] for left and right circular polarization,collected on strongly interacting GrNi (sample A). Vertical dashed line on CIS graph corresponds to C 1 score-level binding energy. resolution, it is possible to analyze the ResPES with crystal momentum resolution (Fig. 5) and point out dichroic effects. In order to better show the data, we divided the {−1,1}˚A−1 fullkrange in three sections of equal width (2 /3˚A−1), labeled askA,kB, andkC[kBbeing centered at the ¯/Gamma1point, as shown in Fig. 4(c)]. Selected on- and off-resonance photoemission spectra are shown in Figs. 5(a) and5(b) for each polarization (sample A). A large asymmetry in the krange is then observed, as well as mutual correspondences upon polarization switch; asan example, the spectra of zone k Ain left circular polarization (LCP) display the same shape of zone kCin right circular polarization (RCP). Most of the spectral weight at the π∗edge is due to RSW at the ¯/Gamma1point (in zone kB), while the spectral weight at the σ∗resonance, lying at Fermi edge, is observed inkAandkCzones. From ResPES images it is also possible [Figs. 5(c)and5(d)] to extract the constant initial-state spectra (CIS), i.e., tomeasure the variation of photoemission intensity at a ﬁxed BEwhile changing the photon energy. Remarkably, CIS spectracollected at BE =13 eV (in black, labeled as Auger CIS in Fig. 5) closely follow the expected XAS at C K-edge [ 3]i n our experimental geometry (40 ◦grazing photon incidence), except for the different background due to the Auger shiftat higher BE. In fact, these CIS curves have been used inthis work to better identify π ∗andσ∗resonances. ResPES thus allows to recover C K-edge XAS-like spectra, even in ultrathin carbon layers and without the need of an additional300 295 290 285Photon Energy (eV)Sample B300 295 290 285Photon Energy (eV)Sample A CIS intensity (a.u.) CIS 1.8 eV CIS 0.5 eV CIS Auger CIS intensity (a.u.) 300 295 290 285Photon energy (eV) 10 8 6 4 2 0 Binding energy (eV)Sample C CIS intensity (a.u.) 300 295 290 285 280 Photon Energy (eV) FIG. 6. Left column: ResPES map summary for each sample, pre- edge subtracted; right column: corresponding CIS spectra collected on a 1-eV interval around 0.5 eV ( σ∗resonance, red line), 1.8 eV (π∗resonance, blue), and on the main Auger peak (5–10 eV range, black). reference, such as the photoinduced drain current on a gold grid across the beam or beamline mirrors (which may alsobe contaminated by carbon). In our case, the wide BE rangeallowed us to use valence band intensity measurement asan additional normalization reference. A close comparison(Fig. 5, bottom right panel) of Auger CIS for the different circular polarization does not reveal any dichroism, as expectedfor these nonmagnetized samples. The VB CIS at 1.8 BE eV shows a peak enhancement at hν=284.8 eV , i.e., the dominant C 1 score-level binding energy [ 6]. Its intensity is completely quenched well before the end of the π ∗absorption edge; moreover, its maximum is clearly shifted with respect to the Auger CIS (proportionalto XAS), indicating that the resonances are mostly locatedat the XAS pre-edge. Such shift in the photon energy isusually related [ 40] to a relatively large delocalization time of the electron promoted in the ResPES intermediate state[x-ray absorption in Fig. 1(c)]. Remarkably, the position and shape of this CIS is also rather similar to one of the XMCDmeasurements reported in literature [ 4]. The comparison of the ResPES maps of different samples, shown in Fig. 6(left column), can clarify the origin of the observed resonance. In fact, the VB resonating features,underlined with orange ( σ ∗edge) and blue ( π∗edge) circles, are detectable in every sample and in particular in sample C,where only nickel carbide is present on the surface. The data ofFig.6also reveal some major differences among the samples. First, the Auger spectral features show a stronger intensityin GrNi (samples A and B) as compared to sample C; thisdifference can be clearly seen by the relative intensity of theAuger CIS spectra (black lines in Fig. 6, right column) with respect to VB CIS (blue and red lines). 165442-6IDENTIFICATION OF Ni 2C ELECTRONIC STATES . . . PHYSICAL REVIEW B 96, 165442 (2017) FIG. 7. ARPES map summary in resonant and nonresonant conditions, for LCP, RCP, LCP+RCP, and LCP-RCP cases, collected on sample A. Resonant ARPES data are already subtracted by the corresponding pre-edge (nonresonant) data. Moreover, the relative photon energy shift between π∗VB resonance and the Auger peak maximum, already shown inFig. 5, is observed only in the GrNi CIS spectra (A and B). Anyway, in each case the Auger CIS clearly shows thepresence of well-resolved π ∗andσ∗edges, even in the carbide C sample. Several phenomena should be considered to explain these results. The Auger intensity quenching is for sure the mostpeculiar effect; in fact, although the C:Ni ratio is not con-stant among the different samples, each carbon atom shouldcontribute in the same way to the Auger intensity. An overalldecrease of carbon content should then result in the quenchingof both VB and Auger signal. However, for low carboncontent sample C the relative amount of carbon atoms directlycontributing to VB resonance seems to be higher with respectto the GrNi ones, at least in the probed momentum range (i.e.,the electron detector angular acceptance). We then conclude that the resonant maps of interacting graphene (Fig. 6) should be interpreted as the superposition of carbide ResPES (which accounts for the VB resonances) andthe normal Auger dispersion, mostly due to the largest absolutequantity of carbon atoms in the GrNi case found by XPS.The presence of both π ∗andσ∗edges is not unexpected; in particular, similar results can be found in the literature for thicknickel carbide ﬁlms, for relatively high C:Ni stoichiometry ratios [ 9]. The asymmetric resonant distribution of spectral weight in the crystal momentum space, already shown for the GrNicase in Fig. 5, can be observed in each sample. CDAD effects can be easily detected through ARPES maps (Fig. 7), both in resonant and nonresonant conditions. In order toimprove the statistical quality of the data, resonant ARPESmaps have been obtained by averaging over the π ∗and σ∗edges, whose range has been deﬁned by CIS spectra. Pre-edge band maps follow the expected band dispersion forthe¯/Gamma1-¯Mdirection [ 2]. As expected from the experimental setup, upon polarization switch the spectral weight is shiftedfrom one side of the ¯/Gamma1point to the other, preserving the same angular dispersion shape. Such result can be rationalized interms of the experimental geometry; in fact, in this workthe analyzer angular dispersion plane was perpendicular tothe x-ray reﬂection plane [as shown in Fig. 1(b)], so that the only ¯/Gamma1point (i.e., the sample normal direction) belongs to the crystal mirror plane in which geometrical CDAD effectsare absent [ 28]. The pre-edge subtracted maps reveal the band dispersion of resonant electronic states, where the opposite sign of CDADatπ ∗andσ∗edges (LCP-RCP data in Fig. 7) becomes 165442-7G. DRERA et al. PHYSICAL REVIEW B 96, 165442 (2017) FIG. 8. (a) High-quality RCP+LCP pre-edge subtracted ARPES map collected at the π∗resonance. (b) C pzk-resolved DOS for top sites (see Fig. 3). (c) Calculated total C pzDOS around ¯/Gamma1point. (d) Cpzk-resolved DOS, calculated for the TOP geometry, for hcp (d) sites. Please note that (a) and (c) are centered on ¯/Gamma1point while (b) and (d) are calculated around ¯K; calculations have been performed for the TOP geometry. now evident. The CDAD sign swap with respect to the nickel substrate could be also interpreted in terms of theexperimental geometry. In fact, a CDAD sign swap fromp zandpx/pyorbitals combination (as in sp2hybridization) has been predicted by a theoretical study [ 41] by Dubs et al. Although the data are shown for sample A, identical band dispersions have been obtained in each sample. Thepresence of isolated, randomly distributed impurities cannot give a satisfactory explanation of these results since inthis case a ﬂat band dispersion would be expected. For thisreason, the measured carbon electronic states are due to along-range-ordered structure, such as a uniform Ni 2C surface layer. The position of resonating bands seems to be pinned by the underneath Ni band structure, while resonances resultin a different intensity modulation as a function of crystalmomentum. The position of the C p zprojected DOS in the case of GrNi (Fig. 3) can match the π∗resonance BE. Given the preponderance of graphene-related XPS signal in samplesA and B, such RSW should be assigned to Ni 2C, at odds with the previously shown CIS results. However, the experimentalπ ∗band dispersion is centered at the ¯/Gamma1point [high-resolution data are given in Fig. 8(a)] while DFT calculations predict the hybridization of πand Ni 3 dstates to be located at the ¯Kpoint [see yellow circles in Figs. 8(b) and8(d)]. In both TOP and FCC calculations, the calculated carbon projected DOS at the ¯/Gamma1point is nearly absent, as shown in Fig. 8(c). Alternatively, the detection of resonant band dispersion at the¯/Gamma1point in the GrNi band structure could be explained byinvoking a complex, phonon-mediated, momentum transfer mechanism from ¯Kand ¯Msimilar to the one reported by low-energy ARPES experiments [ 14–16]; the XAS process itself is strongly dependent on the crystal momentum and thusmay lead to a strong ResPES intermediate state localizationaround ¯K. However, without stronger evidences, the most simple and plausible explanation seems to be the presence of orderedNi 2C, precursor for graphene formation. This effect in samples A and B is related then to the carbide layer both at the surfaceand below the rotated graphene phase, according to XPS data. Although the observed resonant bands should be assigned to aN i 2C reconstruction and not to graphene, their properties are still intriguing. The Ni 2C band dispersion is still unreported in the literature; this work demonstrates the possibility to directlymeasure the carbon contribution hidden beneath the Ni 3 d electronic states by exploiting the resonant enhancement at CK-edge. Our results acquire further signiﬁcance in the light of the extremely weak intensity of XPS C 1 scarbide peak, which could be easily missed in low-resolution measurements, evenwhen LEED and ARPES analysis suggests a fully epitaxialreconstruction. It should be also pointed out that the graphene magnetism has been observed through XMCD exactly at the photonenergy of the observed maximum of VB π ∗CIS [ 3,4]. By considering the measured CDAD signal, our investigation suggests that a small change of XMCD electron detector angle with respect to the sample normal while using partial yield(PY) techniques may deeply inﬂuence the measurement ofthe total Auger intensity. In fact, resonating structures in theVB also contribute to electron inelastic background belowthe Auger features; such a contribution may lead to a netXMCD signal not due to magnetism but to the opposite CDADsigns at the σ ∗andπ∗thresholds, when PY is performed on total Auger intensity. For example, a net XMCD signalcan be obtained from our data, simply by integrating the CISspectra on half of the angular probing range (correspondingto either −10 ◦to 0◦or 0◦to 10◦electron takeoff angular ranges). IV . CONCLUSIONS In conclusion, with the combination of ARPES, ResPES, and circular polarized x rays, we have been able to revealthe hidden hybridization bands in the complex carbon-nickelsystem, where several phases can be easily obtained. Theresonance mechanism allows the simultaneous collection ofXAS-like CIS spectra and the elemental-speciﬁc band struc-ture, in a case where the interesting electronic states are mixedto the substrate valence band. In GrNi, unexpected electronicstates are observed at the ¯/Gamma1point of the band structure, while graphene hybridization with Ni is usually thought tobe conﬁned to the ¯Kpoint; we ascribe these electronic states to ordered nickel carbide at the surface (Ni 2C), whose presence has been detected by high-resolution C 1 sXPS and discriminated by a comparative res-ARPES study on a purelycarbide sample. The peculiar shape of the resonant bandsindicates that Ni 2C itself shows a characteristic electronic structure which is worth further investigations. Finally, thepresence of CDAD asymmetric effects in C:Ni(111) systems 165442-8IDENTIFICATION OF Ni 2C ELECTRONIC STATES . . . PHYSICAL REVIEW B 96, 165442 (2017) also suggests caution while evaluating XMCD results at the C K-edge within the Auger partial yield detection method, due to the opposite sign of the dichroic effect on π∗andσ∗edges. 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PhysRevB.75.054516.pdf	Bound roton pairs in HeII under pressure: Analysis of Raman spectra M. Shay, O. Pelleg, E. Polturak, and S. G. Lipson Department of Physics, Technion–Israel Institute of Technology, Haifa 32000, Israel /H20849Received 29 August 2006; revised manuscript received 1 January 2007; published 21 February 2007 /H20850 We have investigated the properties of bound roton pairs in superﬂuid4He as a function of pressure using Raman scattering. Raman spectra at small energy shifts of 10–30 K were measured at several pressures up tothe melting curve. The spectra reveal an asymmetric peak at about 2 times the energy of a roton, consistentwith previous works. Our data, as well as previous measurements by Ohbayashi et al. , were analyzed according to a model for the two roton density of states. The value of the l=2 component of the interaction energy of a roton pair as a function of pressure was extracted from the analysis and compared with theoretical predictions.We ﬁnd that this interaction changes sign from attractive to repulsive at pressure around 10 bar. The lifetimeof a single roton as a function of temperature was also determined. DOI: 10.1103/PhysRevB.75.054516 PACS number /H20849s/H20850: 67.40.Db, 78.30. /H11002j Following Halley’s suggestion,1the Raman spectrum of HeII was ﬁrst measured by Greytak and Yan in 1969 /H20849Ref.2/H20850 and subsequently by others.3–5Soon after Greytak’s experi- ment a theory to explain the spectrum was developed byseveral authors. 6–9The observed asymmetric peak was inter- preted as a result of a second order Raman scattering processin which the scattered photon loses energy to create tworotons. The spectrum therefore reﬂects the joint density ofstates of two rotons, and is inﬂuenced by roton-roton inter-action. A model for the joint density of states at saturatedvapor pressure /H20849SVP /H20850was constructed, in which interaction between rotons is attractive, so that roton pairs can form abound state, and this bound state dominates the joint densityof states. 8–11According to the model, the Raman spectrum from superﬂuid helium contains information about the en-ergy of the bound state E b, the lifetime of a single roton /H9253−1, and about /H9004, the minimum energy of a roton. Raman spectra from superﬂuid helium up to the melting pressure were mea-sured by Ohbayashi et al. 12,13and also in the present work. Ohbayashi also analyzed the polarization of the Raman scat-tered light and found that scattering is due to pairs of rotonswith l=2 total angular momentum. In this work we analyze new high pressure spectra and those from Ref. 13according to the above model in order to extract the dependence of E b and/H9253on pressure and temperature. We compare our results to a theoretical prediction by Bedell, Pines, andZawadowski 14/H20849BPZ /H20850that the two-roton bound state with an- gular momentum l=2 should exist only below 5 bar. In Raman scattering from superﬂuid helium the Stokes spectrum is the result of a second order transition in whichthe incoming photon loses energy to create two elementary excitations with total momentum equal zero. In such a pro-cess, the line shape of the scattered light reﬂects the densityof states of two elementary excitations with zero totalmomentum. 15This can be seen if we compare the roton den- sity of states to the Raman transition probability calculatedusing Fermi’s golden rule. The joint density of states, /H92672,i s deﬁned as /H92672/H20849E/H20850=1 /H208492/H9266/H208503/H20885d3k/H9254/H20851E−2E/H20849k/H20850/H20852. /H208491/H20850 The Raman transition probability density is given by Fermi’s golden rule,dP /H20849E/H20850 dk1=2/H9266 /H6036/H20841/H20855M /H20856/H208412/H9254/H20851E−E/H20849k1/H20850−E/H20849k2/H20850/H20852, /H208492/H20850 where E=/H6036/H9275i−/H6036/H9275sand/H6036/H9275i,/H6036/H9275sare the energies of the incident and scattered photon, k1,k2are the wave vectors of elementary excitations, E/H20849k/H20850is the dispersion relation, and Mis the transition matrix. An integration over the set of ﬁnal states yields the total transition probability. Fixing thedirection of the scattered photon and its energy to a deﬁnitevalue /H6036 /H9275s, the set of ﬁnal states can be counted by integrat- ing over k1.k2is ﬁxed by the condition of zero total momen- tum. Neglecting the momentum of the photons we getk 1=−k2,E/H20849k1/H20850=E/H20849k2/H20850. Assuming the matrix element to be independent of k, the scattering probability can be written as P/H20849E/H20850=2/H9266 /H6036/H20841/H20855M /H20856/H208412/H20885/H9254/H20851E−2E/H20849k/H20850/H20852d3k=/H208492/H9266/H208504 4/H6036/H20841/H20855M /H20856/H208412/H92672/H20849E/H20850. /H208493/H20850 It is seen from Eq. /H208493/H20850that the Raman scattering probability is proportional to the joint density of states times the squareof the matrix element. The scattered photon can eitherchange its polarization or not change its polarization. Thisrestricts the angular momentum transfer to l=0 or l=2, while for all other cases the matrix element is zero. Polarizationanalysis of the Raman spectra of superﬂuid helium 11,12shows that the scattering state is of angular momentum l=2 at all pressures. Therefore, the Raman scattering from superﬂuidhelium is sensitive to the joint density of states of excitationswith total momentum K=0 and angular momentum l=2. The density of states around the roton minimum in the dispersionrelation is illustrated in Fig. 1. Figure 1/H20849a/H20850shows the density of states for the simplest model, that of noninteracting ro-tons. In this model a parabolic approximation is used for thedispersion relation around the roton minimum: E/H20849k/H20850=/H9004 + /H60362 2/H9262/H20849k−k0/H208502, where /H9004is the roton minimum energy, k0is the wave vector at the minimum and /H9262is the roton effective mass. Substituting this expression into Eq. /H208491/H20850, the joint den- sity of states around the roton minimum at T=0 becomes6,16PHYSICAL REVIEW B 75, 054516 /H208492007 /H20850 1098-0121/2007/75 /H208495/H20850/054516 /H208495/H20850 ©2007 The American Physical Society 054516-1/H92672/H20849E/H20850/H11008/H20849E−2/H9004/H20850−1/2/H9008/H20849E−2/H9004/H20850. /H208494/H20850 It is interesting to note that Eq. /H208494/H20850is similar to the density of states for a free particle in one dimension. Of course, thesystem is three dimensional and the 1D-like behavior can betraced to the ﬁnite value of k 0. The presence of roton-roton interaction changes /H92672/H20849E/H20850in a way which depends on the sign of the interaction. The density of states for the case of attractive interaction is shown in Fig. 1/H20849b/H20850. An attractive in- teraction leads to a formation of a bound state with a bindingenergy E b, adding a /H9254function at 2 /H9004−Ebto Eq. /H208494/H20850.I na d - dition, Ebis added to the denominator of Eq. /H208494/H20850, removing the divergence at 2 /H9004. If the interaction is repulsive, the den- sity of states is shown in Fig. 1/H20849c/H20850. In this case, there is no bound state and the /H9254function is not present. However, Ebin the denominator of Eq. /H208495/H20850remains, representing the energy of the interaction. The coupling constant of the roton-rotoninteraction, gis related to E bthrough g2/H11008Eb. This model was used by Greytak et al.10,11to analyze their data at low /H20849SVP /H20850pressure. The density of states for the case of attrac- tive interaction is given in Eq. /H208495/H20850, /H92672/H20849E/H20850/H110082Eb1/2/H9254/H20849E−2/H9004+Eb/H20850+/H20849E−2/H9004/H208501/2 /H20849E−2/H9004+Eb/H20850/H9008/H20849E−2/H9004/H20850. /H208495/H20850 Note that this model coincides with the free roton model when the binding energy is zero. At ﬁnite temperature, thedensity of states becomes /H92672/H20849E/H20850/H11008k02/H92621/2/H208752/H20881Eb/H9253 /H20849E−2/H9004+Eb/H208502+/H92532 +/H20873/H20881/H20849E−2/H9004/H208502+4/H92532+/H20849E−2/H9004/H20850 /H20849E−2/H9004+Eb/H208502+4/H92532/H208741/2/H20876. /H208496/H20850 The temperature dependence of /H92672is through /H9253, the energy linewidth of the roton. Equation /H208496/H20850coincides with that given by Zawadowski et al. ,9based on the analysis of the two rotonGreen’s function /H20849this was checked by plotting both formulas on the same graph /H20850. The comparison of Eq. /H208496/H20850to the expres- sion given by Zawadowski et al. /H20851Eq. /H208493.24 /H20850from Ref. 9/H20852 shows that /H9253indeed represents the energy linewidth of a single roton. Because of the parabolic approximation to thedispersion relation in Eq. /H208496/H20850, the energy range where this equation is valid is limited. This range is between zero toabout 2 times the energy of the maxon. The measured intensity, I d, is a convolution of the instru- mental resolution function and the scattered intensity Is, Id/H20849E/H20850=C/H20885R/H20849E−/H9255/H20850Is/H20849/H9255/H20850d/H9255=C/H20885R/H20849E−/H9255/H20850/H92672/H20849/H9255/H20850d/H9255, /H208497/H20850 where R/H20849E/H20850is the instrumental resolution function. For a double grating monochromator this resolution function can be measured by scanning the elastically scattered light, sincethe energy linewidths of the Rayleigh and Brillouin scatter-ing are much narrower than the resolution of the spectrom-eter. The ﬁne structure of Eq. /H208496/H20850is below the resolution of the double grating monochromator. The lower panel of Fig. 1 shows the effect of ﬁnite resolution on the line shape of the FIG. 1. /H20849Color online /H20850Upper panel, two-roton joint density of states with K=0 from Eq. /H208496/H20850plotted for three situations: /H20849a/H20850Non- interacting rotons /H20849g=0 /H20850,/H20849b/H20850with a bound state /H20849g/H110210/H20850,/H20849c/H20850with a repulsive roton-roton interaction /H20849g/H110220/H20850. Lower panel /H20849e/H20850–/H20849g/H20850, cor- responding line shapes of the detected light intensity taking intoconsideration the instrumental resolution. The vertical line /H20849red on- line /H20850marks the position of 2 /H9004. FIG. 2. /H20849Color online /H20850Raman spectra of HeII at different pres- sures. Solid lines /H20849blue online /H20850are ﬁts to Eq. /H208496/H20850taking into account instrumental broadening. The vertical dashed line marks the valueof 2/H9004taken from Ref. 17, and the horizontal dashed line marks the dark count level. The increasing background at low energy is thetail of the elastic peak. The dashed lines /H20849red online /H20850in panels /H20849a/H20850 and /H20849b/H20850are ﬁts to a model of noninteracting rotons. In panel /H20849c/H20850the dashed line is a ﬁt with a small positive coupling constant. It showsthat at 10 bar the interaction is still attractive.SHAY et al. PHYSICAL REVIEW B 75, 054516 /H208492007 /H20850 054516-2measured spectrum. We remark that the plots in Fig. 1rep- resent a high resolution system. Although the ﬁne details ofthe density of states cannot be resolved due to ﬁnite resolu-tion, there are clear differences in both the peak position andin the line shape of the spectra. The position at which theRaman peak is observed depends on resolution. This shift ofthe Raman peak towards higher energy at a lower spectrom-eter resolution is well known. 5To be more speciﬁc, a Raman peak at an energy higher than 2 /H9004does not mean that there is no bound state. It is necessary to use the model convolutedwith the resolution function in order to compare with experi-mental data. In our experiment, a BeCu sample cell with three indium sealed windows is mounted on a 3He refrigerator with optical access. The pressure in the cell is measured outside the cry-ostat using a high accuracy pressure gauge connected to theﬁlling line. The cell temperature is measured using a cali-brated germanium resistor. An argon ion laser beam is fo-cused inside the sample cell. The intensity that enters the cellis 65 mW at 5145 Å. The beam exits the cryostat and isreﬂected back by a mirror to double the incident intensity. Achopper is used to reduce the heating of the cell caused bythe laser beam. Using a duty cycle of 1:5, a steady tempera-ture of 0.6 K could be achieved. The 90° scattered light iscollected using a f# /8.5 lens system that images the scatter- ing volume onto the entrance slit of a computer controlleddouble grating spectrometer /H20849Spex 1403 /H20850equipped with ho- lographic gratings having 1800 grooves per mm. The imageis aligned to the entrance slit using a Dove prism. A cooledphoton counting photomultiplier with dark count of about3 cps /H20849Hamamatsu R6358P /H20850converts the scattered light into electrical pulses. Each spectrum was scanned 10 times inintervals of 0.1 cm −1between points. At every point the ex- posure time was 10 seconds so that the total exposure time ateach point is 100 seconds. To determine the resolution func-tion, the elastic peak was measured before each inelasticscan. In order to eliminate drifts, the wave number of eachpoint is measured relative to the position of the elastic peakmeasured in the same scan.Raman spectra from superﬂuid helium were obtained at several pressures from SVP to 20 bar. These spectra areshown in Fig. 2. We use Eqs. /H208496/H20850and /H208497/H20850to ﬁt our experi- mental data with C,/H9004,E b, and/H9253as ﬁtting parameters. The measured line shape at SVP /H20851Fig.2/H20849a/H20850/H20852is reminiscent of Fig. 1/H20849f/H20850. A prominent, rather symmetrical peak is observed. The width of the peak is limited by the resolution of the spec-trometer. This line shape is characteristic of the existence ofa bound state. On the other hand, the high pressure spectra/H20851Figs. 2/H20849d/H20850and2/H20849e/H20850/H20852, have line shapes that do not exhibit the bound state characteristics and are reminiscent of Fig. 1/H20849g/H20850. The sensitivity of the line shape to the presence of the boundstate can be clearly seen in Fig. 2/H20849c/H20850. Here, the solid line is a ﬁt with a bound state with a very small E b, while the dashed line is a ﬁt with the same Ebbut a repulsive interaction. These observations immediately suggest that at elevatedpressures the interaction changes from an attractive to repul-FIG. 3. /H20849Color online /H20850Raman spectra of HeII at different pres- sures from Ref. 13. The solid lines /H20849blue online /H20850are ﬁts of Eq. /H208496/H20850to the data. The dashed lines /H20849red online /H20850are a ﬁt to a model of non- interacting rotons. FIG. 4. /H20849Color online /H20850Pressure dependence of the binding en- ergy from this experiment /H20849black squares /H20850and from the analysis of the experimental data from Ref. 13 /H20851open triangles /H20849blue online /H20850/H20852. Error bars are the uncertainty of the ﬁt. FIG. 5. Pressure dependence of the l=2 coupling constant. Black squares, extracted from Raman scattering data. The solid lineis the prediction of BPZ. The dashed line represents calculationsfrom Ref. 19, based on Raman scattering data.BOUND ROTON PAIRS IN HeII UNDER PRESSURE: … PHYSICAL REVIEW B 75, 054516 /H208492007 /H20850 054516-3sive. In addition to our results, we also ﬁt the high resolution data from Ref. 13. The ﬁts are shown in Fig. 3. In this ﬁgure the solid line is a ﬁt to the interacting roton model and thedashed line is the ﬁt to the noninteracting model. It is evidentthat the interacting model ﬁts the data well while the nonin-teracting model does not. The results of the ﬁts of our data and that from Ref. 13are consistent. The values of E bas a function of pressure are shown in Fig. 4. We ﬁnd that Eb decreases with pressure. At SVP Eb=0.25±0.1 K in agree- ment with previous works.10,11At a pressure of 10 bar Ebis at its lowest value of less than 0.1 K, and above 10 bar thedata can only be ﬁtted without a bound state. At these pres-sures E bstill has a small positive value which represents the interaction energy rather than the binding energy. Using therelation between E band the coupling constant gwe plot in Fig.5the pressure dependence of g, and compare it to the theoretical prediction of BPZ. In addition, Nakajima andNamaizawa 19proposed a different type of an interaction pseudopotential. In their work, the pseudopotential can eitherdiverge or not, depending on the choice of the parameters.The values of gcalculated for the nondivergent case are shown in Fig. 5as the dashed line. Regarding the ﬁtted val- ues of /H9004, these are in agreement with neutron scattering val- ues /H20849taken from Ref. 17/H20850for all pressures, as shown in Fig. 6. We remark that the ﬁt value of E bis sensitive to both the peak position and the line shape of the Raman spectrumwhile the ﬁt value of /H9004is only inﬂuenced by the position of the Raman peak. Therefore, the analysis gives a tighterbound on E bthan on /H9004. We now turn to the analysis of the temperature depen- dence of the high resolution Raman spectra obtained by Oh-bayashi et al. 13The spectra which we analyze are at 4.9 bar and at four different temperatures between 0.75 K to 2.45 K.The original analysis 13shows that the high temperature spec- tra are consistent with a convolution of the low temperaturespectrum and a Lorenzian, and that the temperature depen-dence of the width of that Lorenzian is well described by theBPZ model. We have ﬁtted Eq. /H208496/H20850to the same data. Theresult is shown in Fig. 7. As mentioned, at this pressure the low temperature line shape is too narrow to be explainedwithout a bound state. At 0.75 K the values of the ﬁttingparameters are E b=0.2±0.05 K, /H9253/H110210.01 K. Most of the temperature dependence of the spectra is due to the lifetimeof a single roton. There is also a weak temperature depen-dence of k 0and/H9262. With the values for k0and/H9262from Ref. 17, The constant Cin Eq. /H208496/H20850is indeed found to be temperature independent to within 5%. The use of the physical modelexpressed in Eq. /H208496/H20850for the analysis enables us to extract the temperature dependence of /H9253, at the pressure of 4.9 bar. The values of /H9253, extracted from both the data of Ohbayashi et al. and from our data at 1.2 K, are in good agreement with thevalues obtained by Ohbayashi’s analysis. In Fig. 8we com- pare the measured temperature dependence of /H9253at 5 bar to FIG. 6. /H20849Color online /H20850Pressure dependence of /H9004from this ex- periment /H20849black squares /H20850, from the analysis of the experimental data from Ref. 13 /H20851open triangles /H20849blue online /H20850/H20852, and from neutron scat- tering data /H20849Ref. 17/H20850/H20851open circles /H20849red online /H20850/H20852. FIG. 7. /H20849Color online /H20850Fits of Eq. /H208496/H20850to the Raman spectra at 4.9 bar. From Ref. 13. At this pressure a bound state still exists. The change of the spectrum induced by temperature is dominated by /H9253−1, the lifetime of a single roton. FIG. 8. /H20849Color online /H20850Temperature dependence of /H9253/H20849black squares /H20850at 5 bar extracted from Raman scattering data. The solid line is an approximation to the theoretical value given by BPZ at5 bar. Values of /H9253measured by neutron scattering /H20849Ref. 18/H20850are shown as open circles /H20849red online /H20850. The dashed line is the BPZ theory at SVP.SHAY et al. PHYSICAL REVIEW B 75, 054516 /H208492007 /H20850 054516-4an approximation to the theoretical prediction of the BPZ model that is /H9253/H20849T/H20850=42.3 /H208491+0.0588 T1/2/H20850T1/2exp /H20849/H9004/H20849T/H20850 T/H20850. The approximation that we use is /H9004/H20849T/H20850=/H9004. This approximation also worked well at SVP for neutron scattering data.18The neutron data and the BPZ prediction at SVP is also shown inthe ﬁgure. The general trend predicted by the BPZ model, that the width increases with pressure is evident. In conclusion, the Raman spectra of superﬂuid 4He at several pressures were measured. The results are in agree-ment with previous experiments. The spectra are very welldescribed by a model of interacting rotons presented by Ru-valds and Zawadowski 7,9and Iwamoto8and developed by Bedell, Pines, and Zawadowski.14At SVP an l=2 bound state of two rotons exists with a binding energy of0.3±0.05 K. This bound state also exists at a pressure of5 bar. Above 10 bar, the l=2 bound state seems to disappear yet the line shape is incompatible with the free rotons model.This observation suggests that the coupling constant of thel=2 component of the roton-roton interaction changes signaround 10 bar. Fitted values of the coupling constant are in a broad agreement with the pseudopotential theory presentedby BPZ, 14in the sense that the interaction changes sign at some pressure. The temperature dependence of the lifetimeof a single roton is also in good agreement with the theory ofBPZ. According to the models, 8,9if the roton-roton interac- tion is repulsive a peak should appear in the spectrum at 2times the maxon energy. However, existing experimentaldata shows no trace of such peak. The absence of such apeak when gis positive may suggest that the overall interac- tion between rotons remains attractive at high pressures,however it involves scattering via channels with l/H110222, which are not Raman active. The authors thank E. Akkermans, A. Kanigel, and E. Farhi for useful discussions. The authors are grateful to S. Hoida,L. Iomin, and A. Post for technical support. The authors ac-knowledge the ﬁnancial support of the Israel Science Foun-dation and of the Technion Fund for Research. 1J. W. Halley, Phys. Rev. 181, 338 /H208491969 /H20850. 2T. J. Greytak and J. Yan, Phys. Rev. Lett. 22, 987 /H208491969 /H20850. 3E. R. Pike and J. M. Vaughan, J. Phys. C 4, L362 /H208491971 /H20850. 4C. M. Surko and R. E. Slusher, Phys. Rev. Lett. 30, 1111 /H208491973 /H20850. 5K. Ohbayashi and M. Udagawa, Phys. Rev. B 31, 1324 /H208491985 /H20850. 6M. J. Stephen, Phys. Rev. 187, 279 /H208491969 /H20850. 7J. Ruvalds and A. Zawadowski, Phys. Rev. Lett. 25, 333 /H208491970 /H20850. 8F. Iwamoto, Prog. Theor. Phys. 44, 1135 /H208491970 /H20850. 9A. Zawadowski, J. Ruvalds, and J. Solana, Phys. Rev. A 5, 399 /H208491972 /H20850. 10T. J. Greytak, R. Woerner, J. Yan, and R. Benjamin, Phys. Rev. Lett. 25, 1547 /H208491970 /H20850. 11C. A. Murray, R. L. Woerner, and T. J. Greytak, J. Phys. C 8, L90 /H208491975 /H20850. 12M. Udagawa, H. Nakamura, M. Murakami, and K. Ohbayashi,Phys. Rev. B 34, 1563 /H208491986 /H20850. 13K. Ohbayashi, M. Udagawa, and N. Ogita, Phys. Rev. B 58, 3351 /H208491998 /H20850. 14K. Bedell, D. Pines, and A. Zawadowski, Phys. Rev. B 29, 102 /H208491984 /H20850. 15R. Loudon, Adv. Phys. 13, 423 /H208491964 /H20850. 16M. J. Stephen, in The Physics of Liquid and Solid Helium , edited by K. H. Bennemann and J. B. Ketterson /H20849Wiley, New York, 1976 /H20850, p. 307. 17M. R. Gibbs, K. H. Andersen, W. G. Stirling, and H. Schober, J. Phys.: Condens. Matter 11, 603 /H208491999 /H20850. 18K. H. Andersen, J. Bossy, J. C. Cook, O. G. Randl, and J. L. Ragazzoni, Phys. Rev. Lett. 77, 4043 /H208491996 /H20850. 19M. Nakajima and H. Namaizawa, J. Low Temp. Phys. 95, 441 /H208491994 /H20850.BOUND ROTON PAIRS IN HeII UNDER PRESSURE: … PHYSICAL REVIEW B 75, 054516 /H208492007 /H20850 054516-5
PhysRevB.96.134506.pdf	PHYSICAL REVIEW B 96, 134506 (2017) Magnetic and superconducting properties of an S-type single-crystal CeCu 2Si2probed by63Cu nuclear magnetic resonance and nuclear quadrupole resonance Shunsaku Kitagawa,1,Takumi Higuchi,1Masahiro Manago,1Takayoshi Yamanaka,1 Kenji Ishida,1,†H. S. Jeevan,2and C. Geibel2 1Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2Max-Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany (Received 3 August 2017; revised manuscript received 25 September 2017; published 9 October 2017) We have performed63Cu nuclear-magnetic-resonance/nuclear-quadrupole-resonance measurements to investi- gate the magnetic and superconducting (SC) properties on a “superconductivity dominant” ( S-type) single crystal of CeCu 2Si2. Although the development of antiferromagnetic (AFM) ﬂuctuations down to 1 K indicated that the AFM criticality was close, Korringa behavior was observed below 0.8 K, and no magnetic anomaly was observedabove T c∼0.6 K. These behaviors were expected in S-type CeCu 2Si2. The temperature dependence of the nuclear spin-lattice relaxation rate 1 /T1at zero ﬁeld was almost identical to that in the previous polycrystalline samples down to 130 mK, but the temperature dependence deviated downward below 120 mK. In fact, 1 /T1in the SC state could be ﬁtted with the two-gap s±-wave model rather than the two-gap s++-wave model down to 90 mK. Under magnetic ﬁelds, the spin susceptibility in both directions clearly decreased below Tc,w h i c hi s indicative of the formation of spin-singlet pairing. The residual part of the spin susceptibility was understoodby the ﬁeld-induced residual density of states evaluated from 1 /T 1T, which was ascribed to the effect of the vortex cores. No magnetic anomaly was observed above the upper critical ﬁeld Hc2, but the development of AFM ﬂuctuations was observed, indicating that superconductivity was realized in strong AFM ﬂuctuations. DOI: 10.1103/PhysRevB.96.134506 I. INTRODUCTION Since the discoveries of unconventional superconductivity in heavy-fermion (HF) [ 1–4], organic [ 5,6], and cuprate compounds [ 7–9], many studies have attempted to elucidate the pairing mechanism of these superconductors. Identifyingthe superconducting (SC) gap structure is one of the mostimportant issues since the SC gap structure is closely related tothe SC pairing mechanism. In particular, k-dependent pairing interactions lead to non- s-wave symmetry in unconventional superconductors. Among the HF superconductors, the pairingsymmetry of CeCoIn 5has been identiﬁed to be dx2−y2-wave from ﬁeld-angle-resolved experiments [ 10,11] and scanning tunneling microscopy measurements [ 12]; thus the supercon- ductivity is considered to be mediated by antiferromagnetic(AFM) ﬂuctuations, as in the case of the cuprate superconduc-tivity. The ﬁrst HF superconductor discovered in 1979 [ 1], CeCu 2Si2, was also considered to be a nodal unconventional superconductor since the SC phase was located on the verge ofthe AFM phase. Moreover, the T 3dependence of the nuclear spin-lattice relaxation rate 1 /T1, together with the absence of a coherence peak [ 13–15] and the T2-like temperature dependence of the speciﬁc heat [ 16] in the SC state, indicated a line nodal SC gap in CeCu 2Si2. Finally, a clear spin excitation gap was observed in the SC state with inelastic neutronscattering, suggesting that AFM ﬂuctuations were the mainorigin of superconductivity in CeCu 2Si2[17,18]. The clear decrease of the nuclear magnetic resonance (NMR) Knightshift below T c[19] and the strong limit of the upper critical ﬁeld Hc2[20], plausibly originating from the Pauli-paramagnetic kitagawa.shunsaku.8u@kyoto-u.ac.jp †kishida@scphys.kyoto-u.ac.jpeffect, indicated that the SC pairs were singlets. These results were considered to be evidence of a d-wave gap symmetry with line nodes in CeCu 2Si2, such as a dx2−y2-o rdxy-wave. One difﬁculty in studying CeCu 2Si2is that a stoichiometric CeCu 2Si2is located very close to a magnetic quantum critical point, resulting in a ground state that is quite sensitiveto the actual stoichiometry [ 21,22]. After careful sample- dependence experiments as well as experiments with chemical(Ge substitution) and hydrostatic pressures, the ground state ofthe stoichiometric CeCu 2Si2was found to be the SC state coexisting with an unusual magnetic state called an “ A” phase [14,23–25]. In this coexisting “ A/S” sample, superconduc- tivity expels the magnetic Aphase below Tcand becomes dominant at T→0[23]. The ground state of the Aphase was unclear for a long time. The ground state was revealedby elastic neutron scattering with the A-type single-crystal CeCu 2Si2[26], and the nature of the Aphase was shown to be a spin-density-wave (SDW) instability from the observation of long-range incommensurate AFM order. Thus, an SC samplethat does not show A-phase behavior is located at the Cu-rich side, e.g., CeCu 2.2Si2, which is called an “ S”-type sample. Another difﬁculty in studying CeCu 2Si2is that large single- crystal samples showing superconductivity were not availablebefore 2000, and thus most measurements were performedon well-characterized polycrystalline samples. Consequently,axial-dependent and angle-resolved measurements have notbeen performed. However, large single crystals with well-deﬁned properties have been synthesized and have recentlybeen used for various experiments. In particular, recentspeciﬁc-heat measurements on an S-type CeCu 2Si2single crystal down to 40 mK strongly suggested that CeCu 2Si2 possesses a full gap with a multiband character [ 27]. In addition, the small H-linear coefﬁcient of the speciﬁc heat at low temperatures and its isotropic H-angle dependence under 2469-9950/2017/96(13)/134506(9) 134506-1 ©2017 American Physical SocietySHUNSAKU KITAGAWA et al. PHYSICAL REVIEW B 96, 134506 (2017) a rotating magnetic ﬁeld within the abplane sharply contrast the expected behaviors in nodal d-wave superconductivity. In this study, we have performed63Cu-NMR/nuclear quadrupole resonance (NQR) measurements to investigate theSC and magnetic properties of an S-type single crystal of CeCu 2Si2. As far as we know, this is the ﬁrst NMR/NQR measurement on a single-crystal CeCu 2Si2down to 90 mK. Comparison between the NMR results of previouspolycrystalline and single-crystal samples is very importantto understand the nature of superconductivity in CeCu 2Si2. We found that the temperature dependence of 1 /T1at zero ﬁeld was almost the same as that in previous polycrystallineS- and A/S-type samples down to 130 mK, but it deviated downward below 120 mK. The Tdependence of 1 /T 1down to 90 mK could be reproduced by the two-gap s±-wave and the two-band d-wave model. Taking into account the recent results of the ﬁeld-angle dependence of the speciﬁc heat, thetwo-gap s ±-wave model is plausible. The Knight shift parallel and perpendicular to the c-axis decreased in the SC state, in good agreement with previous results. The magnitude of theresidual Knight shift was analyzed with the 1 /T 1result in magnetic ﬁelds and was ascribed to the ﬁeld-induced densityof states originating from the vortex effect. In addition, we alsoinvestigated whether magnetic ordering was observed abovethe upper critical magnetic ﬁeld H c2since this anomaly was reported above Hc2with magnetoresistance and de Haas– van Alphen measurements [ 28–30]. No magnetic ordering was observed in the present S-type single crystal, but the development of AFM ﬂuctuations was observed. II. EXPERIMENT Single crystals of CeCu 2Si2were grown by the ﬂux method [ 22]. In the present NMR/NQR measurements, we used high-quality S-type single crystals from the same batch as those used in the speciﬁc-heat and magnetizationmeasurements [ 27,31]. A single-crystal sample was used for NQR measurements without being powdered, and the NQRresults of the single crystal were compared with the previousresults measured in polycrystalline samples. Low-temperatureNMR/NQR measurements were carried out with a 3He-4He dilution refrigerator, in which the sample was immersed intothe 3He-4He mixture to avoid rf heating during measurements. The external ﬁelds were controlled by a single-axis rotator withan accuracy better than 0 .5 ◦.T h e63Cu-NMR/NQR spectra (nu- clear spin I=3/2, and nuclear gyromagnetic ratio63γ/2π= 11.285 MHz /T) were obtained as a function of frequency in a ﬁxed magnetic ﬁeld. The NMR measurements were doneatμ 0H∼1.4T(<μ 0Hc2∼2 T) and ∼3.5T(>μ 0Hc2). The63Cu Knight shift of the sample was calibrated by the 63Cu signals from the NMR coil. The63Cu nuclear spin-lattice relaxation rate 1 /T1was determined by ﬁtting the time vari- ation of the spin-echo intensity after saturation of the nuclearmagnetization to a theoretical function for I=3/2[32,33]. III. EXPERIMENTAL RESULTS The inset of Fig. 1(a) shows the63Cu-NQR spectrum as a function of frequency. When I/greaterorequalslant1, the nucleus has an electric quadrupole moment Qas well as a magnetic dipole moment;FIG. 1. (a) Temperature dependence of63Cu-NQR frequency. The dotted line is an empirical relation of νQ(T)=νQ(0)(1−αT3/2). Inset: Frequency dependence of the63Cu-NQR spectrum at 1.8 K. (b) Field-swept NMR spectrum at 4.2 K and f=19.8M H zf o r H/bardblc. thus, the degeneracy of the nuclear-energy levels is lifted even at zero magnetic ﬁeld due to the interaction between Qand the electric ﬁeld gradient (EFG) Vzz=eqat the nuclear site. The electric quadrupole Hamiltonian HQcan be described as HQ=νzz 6/braceleftbigg/parenleftbig 3I2 z−I2/parenrightbig +1 2η(I2 ++I2 −)/bracerightbigg , (1) where νzzis the quadrupole frequency along the principal axis (caxis) of the EFG, deﬁned as νzz≡3e2qQ/ 2I(2I−1) witheq=Vzz, andηis the asymmetry parameter of the EFG expressed as ( Vxx−Vyy)/VzzwithVαα, which is the second derivative of the electric potential Valong the αdirection (α=x,y,z ). The parameter ηshould be zero at the Cu site in CeCu 2Si2because of the fourfold symmetry. The obtained NQR frequency νNQR=3.441 MHz at 1.8 K was almost the same as that in the polycrystalline samples. The full width athalf-maximum (FWHM) in the 63Cu-NQR spectrum, which depended on crystalline homogeneity, was 41 kHz and wasalmost temperature-independent. The obtained FWHM wasbroader than that in high-quality polycrystalline CeCu 2.05Si2 (FWHM ∼13 kHz) characterized as an A/S-type sample and that in Ce 1.025Cu2Si2(FWHM ∼26 kHz) characterized as an S-type sample. The FWHM result indicated that the crystal homogeneity in the present single-crystal sample was not asgood as that in the polycrystalline A/S-type CeCu 2.05Si2.T h i s is consistent with previous results that an S-type sample is located at the Cu-rich region in the qualitative Ce-Cu-Si phasediagram of CeCu 2Si2[21]. As shown in Fig. 1(a),νNQR increases with decreasing temperature. The temperature variation of νNQR followed the empirical relation of νQ(T)=νQ(0)(1−αT3/2)d o w n to 50 K due to a thermal lattice expansion and/or latticevibrations [ 34–36] and deviated downward from the relation. A 134506-2MAGNETIC AND SUPERCONDUCTING PROPERTIES OF AN . . . PHYSICAL REVIEW B 96, 134506 (2017) FIG. 2. Temperature dependence of the Cu-NQR intensity ( I) multiplied by T,I(T)T, normalized by ITat 1.5 K for the present single-crystal CeCu 2Si2, and compared with the various polycrystalline samples [ 14]. The dotted line indicates Tc, and the broken lines provide a guide to the eye. similar temperature dependence has been observed in various Ce-based ﬁlled skutterudites [ 37,38]. No clear change of νQ was observed around 15 K, where the 4 felectron character changed from a localized to an itinerant nature, as we discusslater. This suggested that the Ce valence in CeCu 2Si2did not change when the HF state was formed at ambient pressure. Figure 2shows the temperature dependence of the 63Cu-NQR intensity ( I) multiplied by T,I(T)T, which is nor- malized by ITat 1.5 K for the present single-crystal CeCu 2Si2, compared to various polycrystalline samples [ 14]. The value ofITdecreases rapidly below Tcdue to the SC shielding effect of the rf ﬁeld. As we reported in previous papers [ 14], ITin the AandA/S-type samples decreased signiﬁcantly below about 1.0 K due to the appearance of the magneticfraction related to the Aphase. On the other hand, the loss of the NQR intensity in the S-type polycrystalline Ce 1.025Cu2Si2 was small down to Tc. Since the temperature dependence of ITin the present single-crystal CeCu 2Si2was similar to that of the S-type polycrystalline Ce 1.025Cu2Si2, the present single crystal was also characterized as an S-type sample. Figure 3shows the temperature dependence of 1 /T1of the single-crystal CeCu 2Si2, along with those of the poly- crystalline S-type Ce 1.025Cu2Si2andA/S-type CeCu 2.05Si2, measured by63Cu-NQR. In the present single crystal, 1 /T1 was quite similar to 1 /T1in the polycrystalline samples. In all samples, 1 /T1was almost constant at high temperatures and started to decrease below T∗∼15 K. Here, T∗is deﬁned as the characteristic temperature of the Ce 4 felectrons. With further cooling, 1 /T1Tin the single-crystal sample showed almost constant behavior below 0.8 K. The formation ofthe Fermi-liquid state above T cis one of the characteristic features of S-type samples. On the other hand, the A/S-type sample showed that 1 /T1Tcontinued to increase down to Tcaccompanied by a gradual decrease of the NQR signal intensity. These are the anomalies related to the Aphase.FIG. 3. Temperature dependence of 1 /T1measured with NQR on the present S-type single-crystal CeCu 2Si2. The NQR-1 /T1results on the polycrystalline S-type Ce 1.025Cu2Si2andA/S-type CeCu 2.05Si2 are also plotted [ 14]. The linear scale plot of 1 /T1Taround Tcis shown in the inset. In the SC state, 1 /T1in all samples showed no clear coherence (Hebel-Slichter) peak just below Tc, and 1 /T1was proportional to T3at low temperatures down to 130 mK. TheT3dependence of 1 /T1was consistent with the T-linear dependence of C/T in the intermediate temperature range between Tcand 200 mK. Below 120 mK, 1 /T1in the single- crystal sample deviated downward from the T3dependence, which was consistent with the exponential behavior of C/T in the temperature region between 50 and 200 mK [ 27]. Low-temperature 1 /T1below 90 mK could not be measured due to the limits of the refrigerator in our laboratory. A possiblegap structure will be discussed based on the temperature de-pendence of 1 /T 1in the single-crystal sample later in Sec. IV. For the NMR measurement, we applied magnetic ﬁelds to lift the degeneracy of the spin degrees of freedom, even thoughthe nuclear-energy levels were already split by the electricquadrupole interaction. The total effective Hamiltonian couldbe expressed as H=H Z+HQ=−γ¯h(1+K)IH+HQ, (2) where Kis the Knight shift and His an external ﬁeld. Four nuclear spin levels were well separated, and we observed threeresonance lines for each isotope ( 63Cu and65C u )a ss h o w ni n Fig.1(b). Since the position of the resonance line depended on the angle between the applied magnetic ﬁeld and the principalaxis of the EFG ( /bardblcaxis in CeCu 2Si2), we could determine the ﬁeld direction with respect to the caxis from the NMR peak locus. The misalignment of the caxis with respect to 134506-3SHUNSAKU KITAGAWA et al. PHYSICAL REVIEW B 96, 134506 (2017) FIG. 4. Temperature dependence of 1 /T1Ton the present single crystal at 0 T (NQR), 1.4 T, and 3.5 T for H/bardblcandH⊥c.T h e dotted line is a Curie-Weiss dependence estimated from the ﬁtting below 2 K [ C/(T+θ) with C=75 s−1andθ=3.5 K]. The small θindicates that the system is close to a quantum critical point. the ﬁeld-rotation plane was estimated to be less than 2◦from the NMR spectrum analyses, and Kwas determined from the central line of the63Cu-NMR spectrum. Figure 4shows the temperature dependence of 1 /T1Tat zero ﬁeld, 1.4 T ( <μ 0Hc2) and 3.5 T ( >μ 0Hc2) parallel and perpendicular to the caxis, respectively. In the normal state, (1 /T1T)H⊥cwas larger than (1 /T1T)H/bardblcby a factor of 1.32 [(1 /T1T)H⊥c=1.32(1/T1T)H/bardblc], while the temperature dependence was almost identical between the two directions.The anisotropy of 1 /T 1Twas considered to originate from the anisotropy of the hyperﬁne coupling constant and spinsusceptibility. As mentioned above, 1 /T 1Tmeasured at zero ﬁeld became constant below 0.8 K, but 1 /T1Tcontinued to increase as the temperature decreased to 150 mK whensuperconductivity was suppressed by the ﬁeld above μ 0Hc2. In the ﬁeld lower than μ0Hc2, the constant 1 /T1Twas observed at low temperatures in the SC state, which was indicative of thepresence of the ﬁeld-induced residual density of states ascribedto vortex cores. Figure 5(a)shows the temperature dependence of K i(i=⊥ andc) measured at 1.4 and 3.5 T for both directions. The Knight shift Kiis described as Ki=Ahf,iχspin,i+Korb,i, (3) where Ahf,i,χspin,i, and Korb,iare the hyperﬁne coupling constant, spin susceptibility, and orbital part of theKnight shift in each direction, and K orb,iis usually temperature-independent. In the normal state, K⊥increased upon cooling and became constant below 4 K. The temperaturedependence of K cwas similar to that of K⊥, with opposing sign due to the anisotropic Ahf, which is understood by c-f hybridization [ 39]. In contrast to the constant behavior below 1Ki n3 . 5T( >μ 0Hc2), the absolute value of Kidecreased below Tcat 1.4 T, which is indicative of the decrease of the spin susceptibility in the SC state. This decrease will bediscussed quantitatively later.FIG. 5. (a) Temperature dependence of the Knight shift at 1.4 T and 3.5 T for H/bardblcandH⊥c. In contrast with constant behavior below 1 K at 3.5 T ( >μ 0Hc2), the absolute value of Kidecreases below Tcat 1.4 T, reﬂecting the decrease of the spin susceptibility in the SC state. (b)Temperature dependence of spin susceptibilitynormalized at T c. IV . DISCUSSION A. Spin dynamics in the normal state In general, 1 /T1provides microscopic details about the low-energy spin dynamics, and thus we analyze 1 /T1to quantitatively discuss the character of low-energy spin dy-namics of Ce moments. In temperatures higher than thecoherent temperature T ∗, the Ce moments are in a well- localized regime; thus, the observed 1 /T1value in CeCu 2Si2 is approximately decomposed into conduction electrons and localized Ce felectrons as (1/T1)obs=(1/T1)c+(1/T1)f, (4) where the former contribution can be approximately known from 1 /T1of the LaCu 2Si2[40]. The latter contribution is dominated by ﬂuctuations of the Ce spins and can be given bythe Fourier component of /angbracketleftS(t)S(0)/angbracketrightat the Larmor frequency, where the time dependence arises from the ﬂuctuations of theCe spins. In general, 1 /T 1is expressed as [ 41] 1 T1=γ2 nkBT 2μ2 Blim ω→0/summationdisplay q[A(q)]2χ/prime/prime(q,ω) ω, (5) where A(q)i st h e q-dependent hyperﬁne coupling constant, χ/prime/prime(q,ω) is the imaginary part of the dynamical susceptibility, and the sum is over the Brillouin zone. At higher temperatures, 134506-4MAGNETIC AND SUPERCONDUCTING PROPERTIES OF AN . . . PHYSICAL REVIEW B 96, 134506 (2017) FIG. 6. The temperature dependence of the characteristic energy of the spin ﬂuctuations /Gamma1(T) evaluated with the NMR quantities is shown, along with the temperature dependence of the half-width of the quasielastic neutron-scattering line. The dotted curve is theT 1/2dependence, which is a high-temperature approximation of the theoretical calculation of /Gamma1based on the impurity Kondo model by Cox et al. [42]. The ﬁtting is fairly good above 20 K. Inset: temperature dependence of the characteristic energy of the spin ﬂuctuations /Gamma1(T) as a function of the square root of T. the spin dynamics are determined by independent Ce moments, and the local-moment susceptibility is given by [ 42] χL(ω)=χ0(T) 1−iω//Gamma1 (T), (6) where χ0is the bulk susceptibility and /Gamma1is the characteristic energy of spin ﬂuctuations of Ce moments. We assume that the qdependence of A(q) can be negligibly small, and the dynamical susceptibility is isotropic. Then,Eq. ( 5) can be described as [ 43,44] /parenleftbigg1 T1/parenrightbigg f∼Nγ2 nkBTA2 μ2 Bπ¯hχ0(T) /Gamma1(T), where (1 /T1)fis estimated by subtracting 1 /T1of LaCu 2Si2 from 1 /T1of CeCu 2Si2measured with the63Cu-NQR, and Nis the number of nearest-neighbor Ce sites. Using this equation, /Gamma1(T)/kBis expressed with the NMR quantities as /Gamma1(T) kB=Nγ2 nπ¯h/parenleftbiggA⊥ μB/parenrightbigg TK⊥(T1)f, (7) where K⊥is the Cu Knight shift perpendicular to the caxis. Here,A⊥is the hyperﬁne coupling constant perpendicular to thecaxis, which is evaluated from the K-χplot in the Trange from 8 and 80 K [ 39], since the bulk susceptibility is easily affected by an extrinsic impurity contribution. Figure 6shows the temperature dependence of /Gamma1(T)/kB estimated by Eq. ( 7), as well as /Gamma1(T)/kBdirectly measured with neutron quasielastic scattering (NQS) [ 45]. A similar comparison has been performed with29Si-NMR results on a polycrystalline CeCu 2Si2[46], but the agreement was notas good as that from the current study, probably due to the impurity-phase contribution in the bulk susceptibility. In thepresent analyses based on the 63Cu-NMR results, the agree- ment is rather good, and both /Gamma1(T)/kBshow a very similar Tdependence, although the NQS result is somewhat larger than the NMR estimation. In particular, /Gamma1(T)/kBfollows a T1/2dependence above 20 K. In HF compounds containing Ce and Yb ions, /Gamma1(T) was calculated for independently screened local moments based on an impurity-Kondo modelfor Ce 3+(4f1) and Yb3+(4f13)b yC o x et al. [42]. The T1/2 dependence is the high-temperature approximation of the theoretical calculation of /Gamma1/k B, and it has been observed in various HF compounds. As shown in Fig. 6,/Gamma1/k Bdeviated from the T1/2dependence and remained at a constant value below around 15 K due to the formation of the low-temperaturecoherence ground state. In fact, the resistivity showed a broadmaximum at around 15 K, and thus the resistivity and 1 /T 1 results showed the occurrence of local-moment screening below 15 K by the “ Kondo effect .” A ss h o w ni nF i g . 5(a), the static susceptibility became constant below 4 K, whereas 1 /T1Tprobing q-summed dynamical susceptibility continued to increase as temperaturedecreased to 0.8 K at zero ﬁeld. Thus, AFM ﬂuctuationsbecome dominant at low temperatures. The nature of theAFM ﬂuctuations was investigated by neutron-scatteringmeasurements and is revealed to be of the incommensurate SDW-type with a propagation vector Q AF=(0.22,0.22,0.53), which is the same propagation vector of the A-phase ordered state [ 17,18]. Finally, we discuss the possibility of the ﬁeld-induced AFM state in the present S-type CeCu 2Si2. The ﬁeld-induced mag- netic anomaly was reported from magnetoresistance and deHaas–van Alphen measurements in a previous single-crystalsample [ 29,30]. In general, when magnetic ordering occurs, 1/T 1Tshows a peak at magnetic ordering temperature TM, and the NMR spectra show broadening and/or splitting belowT M. However, in this study, 1 /T1Tdoes not show such a peak but continues to increase as the temperature decreasesto 150 mK, following the Curie-Weiss dependence shown bythe dotted curve in Fig. 4when 3.5 T ( >μ 0Hc2)i sa p p l i e d perpendicularly to the caxis. A similar continuous increase of 1/T1Twas observed in the ﬁeld parallel to the caxis, indicating the development of AFM ﬂuctuations. The smallbut ﬁnite Weiss temperature estimated from the ﬁtting below2K( θ∼3.5 K) suggests that the present S-type sample is still in the paramagnetic state, although it is close to aquantum critical point. These results are consistent with recentneutron-scattering results [ 16]. In addition, no clear reduction of NMR intensity related to the A-phase anomaly was observed [28]. Our NMR results indicate the absence of the ﬁeld-induced magnetic anomaly in the present S-type single crystal. B. Superconducting gap symmetry Here, we discuss a plausible SC gap model for explaining the temperature variation of 1 /T1at zero ﬁeld. The 1 /T1results showing a T3dependence were considered to be evidence of the presence of a line node in CeCu 2Si2, and these results can be reproduced by the two-dimensional d-wave model, as shown in Fig. 7. However, recent speciﬁc-heat measure- 134506-5SHUNSAKU KITAGAWA et al. PHYSICAL REVIEW B 96, 134506 (2017) FIG. 7. Log-log plot of the calculations of normalized 1 /T1with each SC model, and the experimental result of the normalized 1 /T1 results at zero ﬁeld. The inset shows the linear scale plot of normalized 1/T1Tand the calculations. ments indicate the absence of nodal quasiparticle excitations and the presence of a ﬁnite gap with a small magnitudeof/Delta1 0∼0.30 K ( ∼0.43Tc) at low temperatures, although C/T increases linearly with temperature for T> 0.2K ,a s shown in Fig. 8. These results, as well as the absence of C/T oscillation in the ﬁeld-angle dependence measurements, suggest that CeCu 2Si2is a multiband full-gap superconductor. In addition, a multiband full-gap superconductor without signchange ( s ++-wave) and a fully gapped two band d-wave superconductor (two-band d-wave) were recently proposed by electron irradiation experiments [ 47] and penetration depth measurements [ 48], respectively. A multigap SC model with more than two full gaps of different gap sizes was not generallyknown before the discovery of Sr 2RuO 4[49,50], MgB2 [51,52], and Fe-based superconductors [ 53–55], and thus such a multigap model was not applied to reproduce experimentalresults in unconventional superconductors before the year2000. Furthermore, due to the complex Fermi surfaces in HFsuperconductors, the single-band analysis was conventionallyadopted for simplicity. However, after the discovery of theFe-based superconductors, it was clear that the T 3dependence of 1/T1could be reproduced not only by the line nodal SC gap but also by the multiband full gap. In fact, the low-temperatureT 3behavior of 1 /T1observed in LaFeAs(O 0.89F0.11) is not consistent with the d-wave model with line nodes since deviation of the T3dependence, which is expected in a d-wave superconductor, was not observed even in inhomogeneousFIG. 8. Log-log plot of the speciﬁc heat Cdivided by temperature [27] and the square root of 1 /T1TofS-type CeCu 2Si2. The broken and dotted lines are plotted to guide the eye. samples, as shown with75As-NQR measurements [ 56,57]. Furthermore, the multiband full-gap structure was actuallydetected from angle-resolved photoemission spectroscopy[58], and thus the multiband SC model has been accepted as a realistic model for interpreting experimental results.Therefore, as already discussed by Kittaka et al. [27], we must identify whether the present NQR results can be consistentlyunderstood by the two-band SC model. The temperature dependence of 1 /T 1Tin two-gap super- conductors is calculated using the following equations: 1 T1T∝/integraldisplay∞ 0⎧ ⎨ ⎩/bracketleftBigg/summationdisplay iNi s(E)/bracketrightBigg2 +/bracketleftBigg/summationdisplay iMi s(E)/bracketrightBigg2⎫ ⎬ ⎭ ×f(E)[1−f(E)]dE, Ni s(E)=ni/integraldisplay∞ 0E/prime /radicalBig E/prime2−/Delta12 i1/radicalBig 2πδ2 iexp/bracketleftbigg −(E−E/prime)2 2δ2 i/bracketrightbigg dE/prime, Mi s(E)=ni/integraldisplay∞ 0/Delta1i/radicalBig E/prime2−/Delta12 i1/radicalBig 2πδ2 iexp/bracketleftbigg −(E−E/prime)2 2δ2 i/bracketrightbigg dE/prime. Here, Ni s(E),Mi s(E),/Delta1i,δi, andf(E) are the quasiparticle density of states (DOS), the anomalous DOS arising fromthe coherence effect of Cooper pairs, the amplitude of theSC gap, the smearing factor to remove divergence of N i s(E) atE=/Delta1i, and the Fermi distribution function, respectively. The parameter nirepresents the fraction of the DOS of the ith SC gap, and two SC gaps are assumed for simplicity, thus n1+n2=1. We multiply Ni s(E) andMi s(E) by a Gaussian distribution function to suppress the coherence peak. We alsocalculate 1 /T 1Tusing a single-gap two-dimensional d-wave 134506-6MAGNETIC AND SUPERCONDUCTING PROPERTIES OF AN . . . PHYSICAL REVIEW B 96, 134506 (2017) TABLE I. Superconducting gaps /Delta1i, smearing factor δi,a n d weight of the primary band used for the calculation of T1. Model /Delta11 /Delta12 δ1//Delta1 1 δ2//Delta1 2 n1 2-gap s++ 2.1 0.8 0.2 0.2 0.65 2-gap s± 2.1 −0.8 0.2 0.2 0.65 1-gap d 2.1 1.0 two-band d 2.1 0.4 1.0 model and a two-band d-wave model discussed in Ref. [ 46]a s follows: 1 T1T∝/integraldisplay∞ 0Nd s(E)2f(E)[1−f(E)]dE, Nd s(E)=/integraldisplay2π 0dφ 4π/integraldisplayπ 0dθsinθE/radicalbig E2−/Delta1(θ,φ)2, /Delta1(θ,φ)=/Delta10cos(2φ) (single-gap d-wave) , /Delta1(θ,φ)=/radicalbig [/Delta11cos(2φ)]2+[/Delta12sin(2φ)]2 (two-band d-wave) , where Nd s(E) is the quasiparticle DOS in a d-wave supercon- ductor, and /Delta10is the maximum of the SC gap. Figure 7shows the calculated 1 /T1in each model together with experimental data as a function of the normalizedtemperature. All parameters used for the calculations arelisted in Table I.T h e1 /T 1Tbehavior in the two-gap s++-wave shows a clear coherence peak, which seems to be inconsistent with the experimental results. As discussedby Kittaka et al. [31], large and/or temperature-dependent smearing factors originating from quasiparticle damping byAFM ﬂuctuations might suppress the coherence peak. How-ever, such a large smearing factor generally suppresses theSC transition temperature. In addition, the coherence peakwas not observed even in pressure-applied CeCu 2Si2, where the AFM ﬂuctuations were signiﬁcantly suppressed [ 15]. Thus, the suppression of the coherence peak by the damping effectof AFM ﬂuctuations seems to be unlikely. Rather, the two-gaps ±-wave, two-dimensional d-wave, and two-band d-wave can closely reproduce the experimental results near Tc.T h e experimental 1 /T1value deviated from T3behavior below 0.2Tc, which agreed with the two-gap s±-wave and two-band d-wave behavior. However, the d-waves seem inconsistent with the absence of the oscillation of C/T in the ﬁeld-angle dependence [ 27]. We can safely say that 1 /T1Tresults down to 90 mK can be reproduced by the two-gap s±-wave, which was suggested by recent speciﬁc-heat measurements [ 27]. In fact, the square root of 1 /T1Tshows almost the same temperature dependence as Ce/Tdown to 90 mK, as shown in Fig. 8. In the plausible s±state of CeCu 2Si2, the sign of the SC gap would change at the electron Fermi surface that is locatedaround the Xpoint with a loop-shaped node. However, as suggested by Ikeda et al. , because this nodal feature is not symmetry-protected, the loop node can be easily lifted by theslight mixture of on-site pairing due to an intrinsic attractiveon-site interaction, and the corrugated heavy-electron sheetbecomes fully gapped with a small magnitude of the SC gap[59]. The small full gap observed by various experiments in CeCu 2S2can be understood by this scenario. Recently, Yamashita et al. [47] reported that the supercon- ductivity of S-type CeCu 2Si2is robust against the impurity scattering induced by electron-irradiation-creating point de-fects, which strongly suggested that the superconductivity is ofthes ++-wave type without sign reversal. As mentioned above, thes++-wave seems to be inconsistent with the temperature dependence of 1 /T1just below Tc. The absence of the coherence peak immediately below Tcand the robustness of superconductivity against the impurity scattering shouldbe interpreted on the same footing. The same discrepancyhas been also identiﬁed in an iron-based superconductorwith “1111” structure [ 60]. To settle this discrepancy, the Fermi-surface properties of CeCu 2Si2should be clariﬁed with experiments such as de Haas–van Alphen, angle-resolved pho-toemission spectroscopy, and scanning tunneling microscopemeasurements. Finally, we illustrate the differences between 1 /T 1of CeCu 2Si2and 1/T1of CeCoIn 5in the SC state. Various experiments have suggested the presence of a line node inCeCoIn 5not only from the temperature dependence but also from the ﬁeld-angle dependence, and CeCoIn 5is considered to be ofd-wave symmetry [ 10,11,61]. Although both compounds show a similar temperature dependence of 1 /T1(1/T1∝T3) and the absence of a coherence peak immediately below Tc,a clear difference was observed at low temperatures. As shown in Fig.3,1/T1shows a T3dependence down to 130 mK, but 1 /T1 of CeCoIn 5deviated upward from the T3dependence below 300 mK and showed T-linear behavior below 100 mK [ 61,62]. The deviation seems to depend on the quality of the samples:larger deviations are observed in lower quality samples.Because this deviation, which originates from the residualDOS at the Fermi energy, has been commonly observed inunconventional superconductors with symmetry-protected linenodes such as cuprate superconductors [ 63,64], the absence of an appreciable deviation from the T 3dependence even in nonstoichiometric CeCu 2Si2cannot be understood by such a line node. Instead, this result does suggest that the SC state isnot ad-wave. C. Spin susceptibility below Tc Next, we discuss the spin susceptibility in the SC state. The Knight shift measurement in the SC state is known to beone of few measurements to give information about the spinstate of superconductors. Since the Knight shift consists ofspin and orbital components, as shown in Eq. ( 3), we need to estimate the orbital part to determine the spin susceptibility.Ohama et al. measured the Knight shift and 1 /T 1Tof29Si and 63Cu in a magnetically aligned powder sample of CeCu 2Si2, and they reported that the Knight shift and 1 /T1Tof the Cu site were determined by a conduction-electron effect athigher-temperature regions. The present 1 /T 1Tvalue and Knight shift at high temperatures in CeCu 2Si2were similar values to those of YCu 2Si2[39]. Thus, we assume Korb∼0 at both directions, as in the case of YCu 2Si2. Figure 5(b) shows the temperature dependence of the spin component ofthe Knight shift ( K s) normalized by the value at Tc(Kn). Here, (Ks/Kn)H/bardblc=(Ks/Kn)H⊥c=0.6 at the lowest temperature 134506-7SHUNSAKU KITAGAWA et al. PHYSICAL REVIEW B 96, 134506 (2017) underμ0H∼1.4 T. This residual Knight shift originated from the ﬁeld-induced normal state due to vortex cores becauseK s/Knat the lowest temperature became smaller in lower ﬁelds and thus the spin susceptibility would become zeroat 0 K near zero ﬁelds, which provides strong evidence ofa spin-singlet superconductor [ 19]. However, the residual normalized DOS estimated from 1 /T 1Twas 0.4 for H/bardblc and 0.7 for H⊥c, which was slightly different from the estimation from Ks/Kn. We propose this discrepancy to be due to the SC diamagnetic ﬁeld. Assuming the residual Ks/Kn to be equal to the residual DOS (estimated from 1 /T1T) implies a diamagnetic Knight shift Kdiaof about 0.03%. In fact,Kdiais estimated as 0.03% from the formula of Hdia= Hc1{ln[βd/√(e)]/ln(κ)}. Here, the lower critical ﬁeld Hc1= 30 Oe, β=0.38 in the triangular vortex lattice, the distance between vortices d=412˚A at 1.4 T, and the Ginzburg-Landau parameter κ=141 are used for the estimation [ 31,65]. These results suggest that the spin susceptibility in both directionsbecomes zero near zero ﬁeld in CeCu 2Si2because 1 /T1T at the lowest temperatures becomes zero at low ﬁelds. Notethat the normal-state K s, which was enhanced with decreasing temperature, disappeared completely below Tcin CeCu 2Si2, which is indicative of singlet pairing by the pseudospin J.O n the other hand, the decrease of Ksin the SC state is usually very small in U-based heavy-fermion superconductors. Inaddition, even in Ce compounds, the decrease of K sis small in noncentrosymmetric superconductors [ 66,67]. The difference of the decrease of Kspinin the SC state is considered to be related with the strength of spin-orbit coupling interaction, andthus a systematic Knight-shift study in HF superconductivityis required.V . CONCLUSION In conclusion, we have performed63Cu-NMR/NQR mea- surements using S-type single-crystal CeCu 2Si2in order to investigate its SC and magnetic properties. The temperaturedependence of 1 /T 1at zero ﬁeld was almost identical to that in polycrystalline samples down to 130 mK but deviateddownward below 120 mK. The 1 /T 1dependence in the SC state could be reproduced by the two-gap s±-wave and the two-band d-wave. Taking into account the recent results of the ﬁeld-angle dependence of the speciﬁc heat, the two-gaps ±-wave model is plausible. In magnetic ﬁelds, the spin susceptibility in both directions clearly decreased below Tc. The residual part of the spin susceptibility was well understoodby the residual density of state arising from the vortex coresunder a magnetic ﬁeld. Above H c2, no obvious magnetic anomaly was observed in S-type CeCu 2Si2down to 150 mK, although the AFM ﬂuctuations were enhanced upon cooling.Thus, the present S-type single-crystal sample was in the paramagnetic state close to a quantum critical point, andsuperconductivity emerges out of the strong AFM ﬂuctuations. ACKNOWLEDGMENTS The authors acknowledge F. Steglich, S. Yonezawa, Y . Maeno, Y . Tokiwa, Y . Yanase, S. Shibauchi, H. Ikeda, Y . Matsuda, Y . Kitaoka, and S. Kittaka for fruitful discussions. This work was partially supported by Kyoto University LTMcenter, and Grant-in-Aids for Scientiﬁc Research (KAKENHI)(Grants No. JP15H05882, No. JP15H05884, No. JP15K21732,No. JP25220710, No. JP15H05745, and No. JP17K14339). [1] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. 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PhysRevB.99.115131.pdf	PHYSICAL REVIEW B 99, 115131 (2019) Spin separation in the half-ﬁlled fractional topological insulator Sutirtha Mukherjee1and Kwon Park1,2,* 1Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea 2School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea (Received 7 December 2018; published 21 March 2019) All topological insulators observed so far are the lattice analogs of the integer quantum Hall states with time-reversal symmetry, composed of two decoupled copies of the Chern insulator with opposite chiralitiesfor different spins. The fractional topological insulator (FTI) has been similarly envisioned as being composedof two decoupled copies of the fractional Chern insulator (FCI), which is in turn the lattice analog of thefractional quantum Hall state (FQHS). An important question is if such a vision can be realized for the Coulombinteraction, whose strength is irrespective of spin. To address this question, we investigate the effects of theinterspin correlation in the spin-holomorphic Landau levels, where electrons with one spin reside in the usualholomorphic lowest Landau level, while those with the other in the antiholomorphic counterpart. By performingexact diagonalization of the Coulomb interaction Hamiltonian in the spin-holomorphic Landau levels, here,we show that no fractionally ﬁlled states in the spin-holomorphic Landau levels can occur as two decoupledcopies of the FQHS, suggesting that no FTIs can occur as those of the FCI in the lattice either. Fractionallyﬁlled states in this system are generally compressible except at half ﬁlling, where a transport gap developswith spontaneous breaking of the space rotational symmetry in the thermodynamic limit, leading to the spatialseparation of different spins, i.e., spin separation. It is predicted that there is a novel bulk-edge correspondenceat half ﬁlling, representing the hallmark of the half-ﬁlled spin-separated FTI. DOI: 10.1103/PhysRevB.99.115131 I. INTRODUCTION Expected to occur in the (nearly) ﬂat Chern band [ 1,2], the fractional Chern insulator (FCI) [ 3–11] is the lattice analog of the fractional quantum Hall state (FQHS) [ 12]. The analogy between the FQHS and FCI can be made concrete by usingthe basis function mapping between the lowest Landau levelwave functions and the hybrid Wannier functions, which arelocalized in one direction, but extended in another [ 7,8]. The fractional topological insulator (FTI) [ 13–24] is distinguished from the FQHS and FCI in the sense that the former pre-serve the time-reversal symmetry, while the latter do not.Conceptually, a FTI can be constructed by combining twodecoupled copies of the FQHS or FCI with opposite chiralitiesfor different spins [ 14,16,18,19,22,23]. An important question is if the correlation between elec- trons with different spins can be really ignored for theCoulomb interaction, whose strength is irrespective of spin.We seek to ﬁnd an answer to this question by investigatingwhat happens to the fractionally ﬁlled states in the lowestLandau levels with spin-dependent chirality, or the spin-holomorphic Landau levels, where electrons with one spinreside in the usual holomorphic lowest Landau level, whilethose with the other in the antiholomorphic counterpart.The spin-dependent holomorphicity corresponds to the spin-dependent Chern number in the lattice. Speciﬁcally, we perform exact diagonalization of the Coulomb interaction Hamiltonian in the spin-holomorphic *kpark@kias.re.krLandau levels. As a result, here, we show that no fractionallyﬁlled states in the spin-holomorphic Landau levels can occuras two decoupled copies of the FQHS, suggesting that no FTIscan occur as those of the FCI in the lattice either. In thissystem, fractionally ﬁlled states are generally compressibleexcept at half ﬁlling, where a transport gap develops withspontaneous breaking of the space rotational symmetry inthe thermodynamic limit, leading to the spatial separation ofdifferent spins, i.e., spin separation. As an application, the spinseparation can be potentially useful in spintronics [ 25], pro- viding a robust interaction-driven spin ﬁlter , sorting electrons with different spins into two spatially separated regions. It ispredicted that there is a novel bulk-edge correspondence athalf ﬁlling, representing the hallmark of the half-ﬁlled spin-separated FTI. Finally, we discuss the spin polarization ofthe half-ﬁlled spin-separated FTI by relaxing the time-reversalsymmetry. II. SPIN-HOLOMORPHIC LANDAU LEVELS We begin by constructing an appropriate model Hamil- tonian generating the spin-holomorphic Landau levels. Inci-dentally, Bernevig and Zhang [ 13] proposed essentially the same model Hamiltonian as ours to describe the dynamicsof an electron conﬁned in the two-dimensional parabolicquantum well with effective spin-orbit coupling induced viaan appropriate strain gradient. In this work, we consider asomewhat different physical mechanism generating the samemodel Hamiltonian. Fundamentally, any spin-orbit coupling owes its origin to the relativistic nature of the Dirac Hamiltonian. Speciﬁcally, 2469-9950/2019/99(11)/115131(13) 115131-1 ©2019 American Physical SocietySUTIRTHA MUKHERJEE AND KWON PARK PHYSICAL REVIEW B 99, 115131 (2019) (a) (b) (c) FIG. 1. Formation of the spin-holomorphic Landau levels. (a) The red and blue (mixed-color) circles denote the (coincidental) energy levels of spin up and down electrons, respectively, in the two-dimensional harmonic oscillator as a function of the z-component angular momentum eigenvalue lz. Note that ¯ hω0is subtracted from the original energy eigenvalues for convenience. For a guide to eye, the circles are threaded by the respectively colored lines. The energy levels are independent of spin in the absence of spin-orbit coupling. (b) With addition of spin-orbitcoupling, the energy levels of spin up and down electrons evolve differently. Speciﬁcally, the energy levels of spin up and down electrons, threaded by the red and blue lines, rotate around each circle at l z=0 to the opposite directions, as depicted by the respective arrows. In the ﬁgure, the spin-orbit coupling constant αis set to be ω0/2. Note that the sign of the spin-orbit coupling constant is not important since different signs just interchange the role of spin up and down electrons. (c) At an appropriate value of the spin-orbit coupling constant, i.e., α=ω0,t h e energy levels form the effective Landau levels with opposite magnetic ﬁelds for different spins. The lowest energy levels among these effective Landau levels are called the spin-holomorphic Landau levels. the usual L·sterm can be obtained from the ( p×E)·sterm, which is generated by expanding the Dirac Hamiltonian inthe nonrelativistic limit. If the electric ﬁeld Eis induced by a radial electrostatic potential φ(r), i.e., E=− 1 rdφ drr,t h eD i r a c Hamiltonian can be expanded in the nonrelativistic limit asfollows: H=p 2 2me+eφ(r)+C1 rdφ(r) drL·σ, (1) where Cis equal to μB/2mec2in vacuum, but assumed to be varied in material. If eφ(r) is a parabolic potential energy under the strong conﬁnement to the xyplane, Eq. ( 1) can be further simpliﬁed as follows: H=p2 2me+1 2meω2 0r2−αLzσz, (2) where the spin-orbit coupling constant α(=−Cmeω2 0/e)i s independent of r. Note that L·σreduces to Lzσzdue to the two-dimensional conﬁnement. Similarly, p2=p2 x+p2 yand r2=x2+y2. There is a close similarity between Eq. ( 2) and the Hamilto- nian for the Landau levels in the circular gauge, A=B 2ˆz×r: H=p2 2me+1 2me/parenleftbiggωc 2/parenrightbigg2 r2−ωc 2Lz, (3) where the cyclotron frequency ωc=eB/mec. When α=ω0, the Hamiltonian in Eq. ( 2) generates essentially the same effective Landau levels as that in Eq. ( 3) except for the salient distinction that electrons with different spins now feel oppo-site effective magnetic ﬁelds. Consequently, the holomorphic-ity of the lowest effective Landau level now depends on spin: φ m↑(r)=/angbracketleftr\|c† m↑\|0/angbracketright∝ zme−zz∗/4l2 0andφm↓(r)=/angbracketleftr\|¯c† m↓\|0/angbracketright∝ (z∗)me−zz∗/4l2 0, where c† m↑and ¯ c† m↓are the respective creation operators for spin up and down electrons in the holomorphicand antiholomorphic orbitals with quantum number m. Here, the natural length scale is set by l 0=√¯h/2meω0, replacing the usual magnetic length. Also, it is importantto note that the actual z-component angular momentumeigenvalue l zis±mforφm↑(r) and φm↓(r), respectively. For clarity, the antiholomorphic creation operators are dis-tinguished from the holomorphic counterparts by puttingthe bar on top. Let us call these lowest effective Landaulevels the spin-holomorphic Landau levels. At general α, the Hamiltonian in Eq. ( 2) can be regarded as two copies of the Fock-Darwin Hamiltonian with opposite magneticﬁelds for different spins, whose energy eigenvalues are givenbyE=¯hω 0(2n+1)±¯h(ω0−α)lzwith n=0,1,2,... and lz=∓ n,∓(n−1),∓(n−2),... for spin up and down elec- trons, respectively. Note that the energy levels with ﬁxed ncan be regarded as the nth “tilted” Landau level with chiral edge modes. See Fig. 1for illustration. Spin-holomorphic geometries The main goal of this work is to analyze the effects of the Coulomb interaction in the spin-holomorphic Landau levels.To this end, we have to choose an appropriate geometry forthe system. One of the most convenient geometries is the spherical geometry, where the system is placed on the surface of asphere with a Dirac monopole located at the center [ 26–28]. Mathematically, all the basis functions in the planar geometrycan be one-to-one mapped to those in the spherical geometryvia stereographic mapping. That is, the L zeigenstates in the planar geometry with lz={0,..., M}are mapped to those in the spherical geometry with lz={M/2,...,−M/2}, where M determines the system size and thus the ﬁlling factor. Note thatthe sign of Mis reversed for the opposite magnetic ﬁeld. In the spin-holomorphic situation, we consider a spin- dependent Dirac monopole. Speciﬁcally, the spin-dependentmonopole strength Q ↑/↓is related to the spin up /down elec- tron number N↑/↓via 2 Q↑/↓=±(ν−1 ↑/↓N↑/↓+S), where ν↑/↓ is the spin up /down ﬁlling factor and Sis the so-called ﬂux shift. Let us call this geometry the spin-holomorphic sphericalgeometry. In this work, unless stated otherwise, we focus onthe time-reversal symmetric situation with N ↑=N↓andν↑= ν↓. The total number of electrons is given by N=N↑+N↓. 115131-2SPIN SEPARATION IN THE HALF-FILLED FRACTIONAL … PHYSICAL REVIEW B 99, 115131 (2019) 0.00.51.01.52.0(a) 0.00.51.01.52.0(d) 0.00.51.01.52.0(c)0.00.51.01.52.0(b) l0l0 l0 l0 FIG. 2. Electron density as a function of the residual conﬁning potential strength γat half ﬁlling in the spin-holomorphic disk geometry. γ=ω0−αwithω0being the natural frequency of the parabolic potential well and αbeing the spin-orbit coupling constant. (a) ¯hγ=0.25, (b) 0.125, (c) 0.105, and (d) 0.0 in units of e2//epsilon1l0, where l0=√¯h/2meω0is the natural length scale of the system. Scale bars in the ﬁgure denote l0, providing a measure for the overall size of the electron droplet. Also, N=N↑+N↓=16 and mmax=15. We deﬁne νtot=ν↑+ν↓and Q=\|Q↑/↓\|. Due to the spin degree of freedom, half ﬁlling is deﬁned as ν↑=ν↓=1/2 and thus νtot=1, being similar to the deﬁnition of half ﬁlling in the Hubbard model. Another convenient geometry is the planar geometry just as described in Eq. ( 2), but in the presence of an appropriate boundary condition, without which any droplet of interactingelectrons would spread out unboundedly. A natural way toprevent the spreading of the electron droplet is to introduce anadditional conﬁning potential. Fortunately, there is a naturalconﬁning potential present in the system. That is, when thespin-orbit coupling constant αis not exactly equal to ω 0, there is the residual conﬁning potential γLzσzwithγ=ω0−α.I f γgets too large, electrons are all squeezed tightly into the center. On the other hand, if γgets too small, the electron droplet falls apart completely, and electrons are all pushed toan artiﬁcial system boundary at r/similarequal√ 2mmaxl0, where mmaxis a preset maximum value of \|lz\|. In this situation, the electron density would form a ring, whose radius increases as a func-tion of m max.O n l yi f γlies within the right range, the electron droplet can have a roughly uniform electron density in thenatural disk area with its radius being independent of m max. Let us call this geometry the spin-holomorphic disk geometry.We ﬁnd that, at half ﬁlling with N/similarequal10–16 ,¯hγ/similarequal0.1e 2//epsilon1l0 gives rise to a healthy competition between the residual conﬁning potential and the Coulomb interaction energies. Inthis situation, the electron density is maintained to be unitymore or less uniformly throughout the entire electron droplet,which does not change much for any m maxlarger than N−1. Figure 2shows the evolution of the electron density as a function of γ.(a) (b) (c) (d) FIG. 3. Exact energy spectra as a function of the total angu- lar momentum quantum number Ltotat half ﬁlling in the spin- holomorphic spherical geometry. The particle number Nis varied with (a) N=10, (b) 12, (c) 14, and (d) 16. Here, the z-component total angular momentum quantum number, Ltot,z, is set equal to zero. Similar to the usual spherical geometry, there is a 2 Ltot+1 degeneracy for each Ltotwith Ltot,zranging from −LtottoLtot. In what follows, we perform exact diagonalization of the Coulomb interaction Hamiltonian in both spin-holomorphicspherical and disk geometries. Speciﬁcally, we diagonalizethe following Hamiltonian in the spin-holomorphic sphericalgeometry: H=P SHLL⎛ ⎝e2 /epsilon1l0/summationdisplay i<j1 rij⎞ ⎠PSHLL, (4) where PSHLL is the projection operator onto the spin- holomorphic Landau levels. In the spin-holomorphic diskgeometry, there is an additional term due to the residualconﬁning potential: H /prime=γ/summationtext iLz,iσz,i. See Appendix Afor the details of the Coulomb matrix elements in both spin-holomorphic geometries. III. SPIN SEPARATION VIA SPONTANEOUS SYMMETRY BREAKING In this section, we present the results obtained via exact diagonalization of the Coulomb interaction Hamiltonian in thespin-holomorphic Landau levels. Speciﬁcally, Fig. 3shows the exact energy spectra as a function of the total angularmomentum quantum number L totat half ﬁlling in the spin- holomorphic spherical geometry. Before discussing the physical meaning of the results in detail, however, it is important to note that the total angular momentum operator, Ltot=/summationtextN i=1Li, should be appropriately generalized so that the eigenvalue of L2 totremains as a good quantum number even in the spin-holomorphic situation, 115131-3SUTIRTHA MUKHERJEE AND KWON PARK PHYSICAL REVIEW B 99, 115131 (2019) which is also Ltot(Ltot+1) as usual. See Appendix Bfor the details of the angular momentum operators in the spin-holomorphic spherical geometry. Also, note that, similar to the usual spherical geometry, a precise determination of the ﬁlling factor requires an appropri-ate choice of the ﬂux shift in the spin-holomorphic sphericalgeometry. For half ﬁlling, we choose 2 Q=N+Swith the ﬂux shift S=−1 since, in this way, the total number of electrons is exactly half the number of all available (spatialand spin) orbitals in the spin-holomorphic Landau levels.Actually, it can be shown that our results do not depend onthe speciﬁc choice of Sso long as Sis within a reasonable range. See Sec. III C for details. A. Macroscopic degeneracy of the ground state Figure 3reveals that the behavior of the exact energy spectra in the half-ﬁlled spin-holomorphic Landau levels isin stark contrast to that of the usual fractional quantum Hallground states. Most remarkably, the ground states in the spin-holomorphic spherical geometry occur at the maximum valueof the total angular momentum quantum number, L max tot= N2/4, which is allowed by the half ﬁlling condition. Usually, the fractional quantum Hall ground states occur at Ltot=0 with well-developed energy gaps, indicating that they areuniform, incompressible states. Even when the system be-comes compressible, the ground state is supposed to occur ata random value of L tot, not its maximum value. In the current situation, the ground states occur consistently at Ltot=Lmax tot, which diverges even in the proper scaling, i.e., Lmax tot/√Q∼ N3/2. This does not only indicate that the ground states are nonuniform, but also that there would be a macroscopicallylarge number of the degenerate ground states in the thermo-dynamic limit. What would this mean physically? To understand the physical meaning of this result, it is im- portant to realize that there are two speciﬁc states among thevast degenerate multiplets at L tot=Lmax tot, whose exact wave functions are uniquely determined by the symmetry alone.One is the state at L tot,z=Lmax tot, whose exact wave function is uniquely obtained by ﬁlling all the orbitals in the upperand lower hemispheres with spin up and down electrons,respectively. The other is the state at L tot,z=− Lmax tot, whose exact wave function is similarly obtained with the roles of spinup and down electrons interchanged. This means that differentspins are spatially separated in these states. Let us elaborateon this conclusion in the following section by writing theexplicit form of the wave function for the maximum angularmomentum states. B. Explicit form of the wave function The spin separation in the maximum angular momentum states at Ltot,z=± Lmax totcan be shown clearly by examining the explicit form of the wave function: /Psi1±Lmax tot=Q/productdisplay m=1/2c† ±m↑¯c† ∓m↓\|0/angbracketright, (5) where c† m↑and ¯ c† m↓are the respective creation operators for spin up and down electrons in the spin-holomorphicspherical geometry. It is very important to note that the index of multiplication is varied within the range of m= {1/2,3/2,..., Q−1,Q}, being only half the entire range of m. Intuitively, /Psi1±Lmax totcan be denoted as \|↑,...,↑,↓...,↓/angbracketright and\|↓,...,↓,↑,...,↑/angbracketright, respectively, showing the spatial separation of different spins. To be more explicit, /Psi1Lmax totcan be also written in terms of the real-space coordinates [ 28],u=cos (θ/2)e−iφ/2andv= sin (θ/2)eiφ/2, as follows: /Psi1Lmax tot=/productdisplay j∈↑/productdisplay k∈↓(ujv∗ k)N/2/Psi1(1¯10), (6) where /Psi1(1¯10)is the wave function for two decoupled integer quantum Hall states (IQHSs) at νtot=2 with ν↑=ν↓=1. In other words, /Psi1(1¯10)=/productdisplay i<j,↑(uivj−viuj)/productdisplay k<l,↓(u∗ kv∗ l−v∗ ku∗ l). (7) It is important to note in Eq. ( 6) that the factor/producttext j∈↑/producttext k∈↓(ujv∗ k)N/2builds two, very large correlation holes with the N/2-th power for the spin up and down electrons in the south and north poles, respectively. The role of these cor-relation holes is that they push electrons of each spin speciesto their own spatially conﬁned region. Note that /Psi1 −Lmax totcan be obtained by interchanging uandvin the above wave function so that the conﬁned regions of spin up and down electrons arealso interchanged. The above wave function is not for the usual decoupled bipartite state. The usual decoupled bipartite state at halfﬁlling is the product state of two decoupled composite fermion(CF) seas with complex conjugation applied to the spin-downpart: /Psi1 dec.bip.=/Psi12CFS↑⊗/Psi1∗ 2CFS↓, (8) where2CFS stands for the Fermi sea of CFs carrying two vortices [ 28–31]. Actually, it is quite interesting to investigate how the spin-separated state at half ﬁlling undergoes a tran-sition to this product state of two decoupled CF seas. To doso, in Sec. III D, we vary the electron-electron interaction as a function of the tuning parameter, which varies the relativestrength of the interspin Coulomb interaction to the intraspincounterpart. As a result, it is shown that the spin-separatedstate at half ﬁlling is very robust in the vicinity of the realisticCoulomb point, where the interspin and intraspin interactionshave the same strength. This means that the correlation be-tween electrons with different spins cannot be ignored for therealistic Coulomb interaction. See Sec. III D for details. Meanwhile, there is some relationship between the current spin-separated state and the spin-holomorphic version of theHalperin (111) state [ 32–34]: /Psi1 (1¯11)=Q/productdisplay m=−Q(c† m↑+¯c† m↓)\|0/angbracketright, (9) which can be written in terms of the real-space coordinates in the planar geometry as /Psi1(1¯11)=/productdisplay i<j(zi−zj)/productdisplay m<n(ω∗ m−ω∗ n)/productdisplay k,l(zk−ω∗ l),(10) 115131-4SPIN SEPARATION IN THE HALF-FILLED FRACTIONAL … PHYSICAL REVIEW B 99, 115131 (2019) where zandωdenote the coordinates of the spin up and down electrons, respectively. For convenience, let us call theabove state the spin-holomorphic (111) state. Incidentally,it is interesting to mention that Bernevig and Zhang [ 13] have previously proposed the spin-holomorphic version ofthe Halperin ( mmn ) state as the potential FTI state at ν tot= 2/(m+n) with ν↑=ν↓=1/(m+n). Unfortunately, /Psi1(1¯11) is energetically a very poor state since the interspin Coulomb correlation is not properly taken care of. Simply put, /Psi1(1¯11) does not vanish when zk=ωl. The poorness of /Psi1(1¯11)is demonstrated by its overlaps with the exact Coulomb groundstates, which are very low, <6%, for all values of L tot,z.N o t e that/Psi1(1¯11)is not even the angular momentum eigenstate. While /Psi1(1¯11)is a very poor state by itself, it is interesting to note that /Psi1(1¯11)contains /Psi1±Lmax totas two constituent states among many others. As an analogy, /Psi1(1¯11)is a paramagnetic state containing various ferromagnetic constituent states inaddition to many other prevalent ﬂuctuations. /Psi1 ±Lmax totrepresent two of such ferromagnetic constituent states. Below, we makeuse of this analogy further to elucidate the role of spontaneoussymmetry breaking for the emergence of spin separation. The other degenerate multiplets with L tot,z/negationslash=± Lmax totgener- ally have very complicated wave functions, whose amplitudesare spread over various many-body basis states in a seeminglyuncoordinated fashion. It is, however, important to realizethat all these states can be uniquely obtained by applyingthe angular momentum lowering and raising operator to thespin-separated states at L tot,z=± Lmax tot. That is, they are all related with each other via rigid rotation, which, combinedwith the very existence of the macroscopic degeneracy, meansthat these states can be linearly resuperposed to produce otherspin-separated states just like those at L tot,z=± Lmax totwith their spin-separation lines being different from the equator.In other words, the spin separation line can be freely rotatedto become any of the great circles. Eventually, such a freedomwould manifest itself as spontaneous breaking of the spacerotational symmetry in the thermodynamic limit. C. Pair correlation function To provide concrete evidence for the spin separation, we compute the pair correlation function, which measures theprobability of ﬁnding an electron at position rwhen a ref- erence electron is placed at the origin. Among the variousdegenerate multiplets at L tot=Lmax tot, we focus on the state atLtot,z=0, which is supposed to be linearly superposed with various spin-separated constituent states with their spinseparation lines all being the great circles connecting the northand south poles. As shown later, this state is particularlyconvenient since it can render a direct comparison between thepair correlation functions obtained in both spherical and diskgeometries, which are naturally connected via stereographicmapping. Placing the origin on the equator would select a certain spin-separated constituent state, pinning the spin-separationline. After pinning, the interspin pair correlation functionshould be large at long distance, while small at short distance.The intraspin pair correlation function should show exactlythe opposite behavior except for an obvious drop at the origindue to the formation of an exchange hole, as required by the (a) (b) (c) (d) FIG. 4. Pair correlation functions in the half-ﬁlled spin- holomorphic Landau levels. (a) Intraspin pair correlation function, g↑↑(r), and (b) interspin pair correlation function, g↑↓(r), in the spin-holomorphic spherical geometry for the state at Ltot=64 and Ltot,z=0 with N=16. Note that the reference electron is chosen to be spin up and placed at the location indicated by the red arrow on theequator. Similar pair correlation functions, (c) g ↑↑(r)a n d( d ) g↑↓(r), in the spin-holomorphic disk geometry with N=16. Again, the spin-up reference electron is placed at the location indicated by thered arrow. Here, the residual conﬁning potential strength is set to be ¯hγ=0.105e 2//epsilon1l0to ensure the uniform electron density. Pauli exclusion principle. Figures 4(a)and4(b)show that this is indeed the case, clearly conﬁrming the spin separation. Next, we check if such a spin separation also occurs in the spin-holomorphic disk geometry. Figures 4(c) and4(d) show the intraspin and interspin pair correlation functions,respectively, in the spin-holomorphic disk geometry. As onecan see, the pair correlation functions behave exactly theway that the spin-separated state should, consistent with thespin-holomorphic spherical geometry. It is important to note that the occurrence of the spin- separated state does not depend on the speciﬁc choice ofthe ﬂux shift Sso long as Sis within a reasonable range. In Fig. 5, we show the pair correlation functions for various values of Sranging from the so-called Pfafﬁan value S=−3 to the anti-Pfafﬁan value S=1. As one can see, electrons with different spins are spatially separated within this entirerange of S, accompanied by the fact that the ground states always occur at the maximum angular momentum sector. Insummary, the spin separation is robust regardless of the choiceofS. The spin separation is a unique property of the half-ﬁlled spin-holomorphic Landau levels. To show this, we investi-gate the evolution of the ground state as a function of theconﬁning potential in the spin-holomorphic disk geometry,which can tune effectively the total ﬁlling factor of thesystem by squeezing the electron droplet. Physically, theground state is expected to undergo a transition from the spin-separated state at ν tot=1 to two decoupled IQHSs at νtot=2 115131-5SUTIRTHA MUKHERJEE AND KWON PARK PHYSICAL REVIEW B 99, 115131 (2019) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) FIG. 5. Intraspin pair correlation functions g↑↑(r) [(a)–(e)] and interspin pair correlation functions g↑↓(r) [(f)–(j),] in the spin-holomorphic spherical geometry for various values of the ﬂux shift Swith the ﬂux-particle relation 2 Q=N+S. Note that the particle number is N=14 in this ﬁgure. (that is, /Psi1(1¯10)) as the conﬁning potential increases. Figure 6 shows that, indeed, there is no spin separation at a sufﬁcientlystrong conﬁning potential. Speciﬁcally, the conﬁning potentialin Fig. 6is chosen to be ¯ hγ=0.25e 2//epsilon1l0, corresponding to the situation in Fig. 2(a), where electrons are all squeezed tightly into the center. Fundamentally, the spin separation is due to a pecu- liar structure of the Coulomb matrix elements in the spin-holomorphic Landau levels. In the usual quantum Hall system,electrons can scatter away from each other regardless ofspin. In the spin-holomorphic system, however, the angularmomentum conservation law dictates that, after scattering,electrons with different spins move together to the same radialdirection, making it difﬁcult to reduce the Coulomb energyunless electrons with different spins are spatially separatedfrom the outset and thus have no chance to encounter eachother. See Appendix Afor the details of the Coulomb matrix elements in the spin-holomorphic Landau levels. Finally, it is interesting to investigate what happens if the Coulomb interaction is screened. Note that the Coulombinteraction can be screened via the ﬁnite thickness of the 2Delectron system present in real experiments. We have consid-ered the ﬁnite thickness effect of the 2D electron system byusing the well-known Zhang-Das Sarma interaction, V ZDS= 1//radicalbig r2+(d/2)2, where dis the ﬁnite thickness parameter (a) (b) FIG. 6. Pair correlation functions in the spin-holomorphic disk geometry under a strong conﬁning potential. (a) Intraspin pair cor-relation function g ↑↑(r) and (b) interspin pair correlation function g↑↓(r), in the spin-holomorphic disk geometry with N=16 and ¯hγ=0.25e2//epsilon1l0, which corresponds to the situation in Fig. 2(a).roughly representing the width of the system in the third direc- tion [ 35]. As a result, we have found that the spin separation persists all the way up to where the ﬁnite thickness parameterreaches roughly four times the natural length scale of thespin-holomorphic Landau levels, l 0. For a larger value of the ﬁnite thickness parameter, the system undergoes the phasetransition to a gapless state. We believe that the increasedthickness of the system provides opposite spins with someextra space to be separated in the third direction, destroyingthe spin separation in the lateral directions. D. Phase transition to two decoupled CF seas To investigate the phase transition from the spin-separated state to two decoupled CF seas, one can deviate from therealistic Coulomb interaction by constructing a model inter-action, whose strength depends on spin. For this purpose, it isconvenient to vary the Hamiltonian in the spin-holomorphicspherical geometry as a function of the tuning parameter, λ, which varies the relative strength of the interspin Coulombinteraction to the intraspin counterpart as follows: H(λ)=P SHLL(Vintraspin +λVinterspin )PSHLL, (11) which reduces to the realistic Coulomb interaction in Eq. ( 4) atλ=1. We compute two quantities, overlap and pair cor- relation function, as a function of λto determine the phase transition from the spin-separated state to two decoupled CFseas. Figure 7shows the square of overlap between the ex- act ground state of H(λ),/Psi1 H(λ), and those obtained at two limits: (i) /Psi1H(0)atλ=0, i.e., two decoupled CF seas, and (ii)/Psi1H(1)atλ=1, i.e., the spin-separated state at half ﬁlling. The squares of overlap, O2=\| /angbracketleft/Psi1H(λ)\|/Psi1H(0)/angbracketright\|2and \|/angbracketleft/Psi1H(λ)\|/Psi1H(1)/angbracketright\|2, indicate how close two decoupled CF seas and the spin-separated state are to the exact ground state as afunction of λ, respectively. Before explaining the physical meaning of the results, we would like to mention that there is a tricky technicalissue in the spherical geometry when one tries to make adirect comparison between the spin-separated state and twodecoupled CF seas in a single ﬁnite-size system. That is to 115131-6SPIN SEPARATION IN THE HALF-FILLED FRACTIONAL … PHYSICAL REVIEW B 99, 115131 (2019) FIG. 7. Squares of overlap, O2=\| /angbracketleft/Psi1H(λ)\|/Psi1H(0)/angbracketright\|2and \|/angbracketleft/Psi1H(λ)\|/Psi1H(1)/angbracketright\|2, as a function of the tuning parameter λ. Here, /Psi1H(λ)represents the exact ground state of the model Hamiltonian H(λ) at half ﬁlling in the spin-holomorphic spherical geometry. Exact diagonalization is performed for (a) N=12 and (b) 14 in the Ltot,z=0 sector. say, the two states do not necessarily occur in the same particle number and ﬂux sector. Speciﬁcally, the spin-separated stateoccurs at 2 Q=N−1. Meanwhile, the CF sea can be deﬁned as the state with a completely ﬁlled CF Landau level structure.After some algebra by using the CF theory [ 28], it can be shown that this means that 2 Q=(1+ 1 2n)N−(n+2) with n indicating the number of completely ﬁlled CF Landau levels.By equating the above two conditions, one can show that theparticle number for the two states to coexist in a single ﬁnite- size system should be N=2n(n+1)=4,12,24,40,.... The N=12 system is a perfect system to study the phase transition between the two states. This is probably the reasonwhy the collapse of two decoupled CF seas seems to coin-cide smoothly with the emergence of the spin-separated statearound λ=0.78. It is important to note that the spin-separated state is very robust at λ/greaterorsimilar0.78, where the exact ground state occurs in the L tot=Lmax totsector, uniquely determined by the symmetry alone. That is, the overlap \|/angbracketleft/Psi1H(λ)\|/Psi1H(1)/angbracketright\|2is exactly unity at λ/greaterorsimilar0.78. Essentially the same overlap behaviors are obtained for the N=14 system except that, here, there is a region near the phase transition, 0 .6/lessorsimilarλ/lessorsimilar0.78, where neither of the two states is a good state. We think that this is a ﬁnite-size effectsince the N=14 system does not support the completely ﬁlled CF Landau level structure. In this system, the exactground state at λ=0 can be regarded as two decoupled almost-CF-sea-like states with one additional quasiparticle ontop of each CF Fermi sea. The phase transition from the spin-separated state to two decoupled CF seas can be also shown through the behavior ofthe pair correlation function. Figure 8shows the ratio between the interspin pair correlation functions at the origin and itsantipode, g ↑↓(r=2R)/g↑↓(r=0), as a function of the tuning parameter λ. As one can see, the phase transition from the spin-separated state to two decoupled CF seas manifests itselfas an abrupt change in this ratio, occurring around the samecritical value of λdetermined by the overlap. IV . NOVEL BULK-EDGE CORRESPONDENCE AT HALF FILLING At half ﬁlling, the total number of electrons is exactly half the number of all available (spatial and spin) orbitals in FIG. 8. Ratio between the interspin pair correlation functions at the origin and its antipode, g↑↓(r=2R)/g↑↓(r=0) as a function of the tuning parameter λ. The insets show g↑↓(r) directly plotted on the sphere at three different values of λ=0.01,0.78, and 0.8025, whose locations are indicated by the arrows. This result is obtained from exact diagonalization results obtained in the N=16 system. the spin-holomorphic Landau levels. Meanwhile, it is shown above that different spins are spatially separated in the half-ﬁlled spin-holomorphic Landau levels. We argue that thesetwo facts lead to the conclusion that the spin-separated stateat half ﬁlling is incompressible. Combined together, the above two facts lead to the com- plete occupation of all available orbitals by either spin upor down electrons without any vacant space between the twoincompressible droplets of each spin species. In this situation,any additional electrons are to be pushed to the region withthe opposite spin since they cannot enter the region with thesame spin due to the Pauli exclusion principle. However, therewould be a large Coulomb energy cost for this to happen dueto the spin separation. Consequently, the spin-separated stateat half ﬁlling should become incompressible. Actually, the same logic suggests that fractionally ﬁlled states should be generally compressible at less than halfﬁlling, where there are vacant spaces between two poten-tially spin-separated regions. In this situation, any additionalelectron can nestle nicely into these vacant spaces withoutcosting too much Coulomb energy, eventually destroying thespin separation itself. Fractionally ﬁlled states at greater thanhalf ﬁlling are also expected to be compressible owing to theparticle-hole symmetry. A. Transport gap and helical edge states at half ﬁlling To be concrete, we compute the transport gap of the half- ﬁlled state in the spin-holomorphic spherical geometry byusing the following formula: /Delta1=E N+1,Q+EN−1,Q−2EN,Q, (12) where EN,Qis the ground-state energy of Nparticles at ﬂux Q. Above, we increase and decrease the particle number by one from Nsatisfying the half ﬁlling condition, N=2Q+1. It is important to note that the transport gap is given as 115131-7SUTIRTHA MUKHERJEE AND KWON PARK PHYSICAL REVIEW B 99, 115131 (2019) (a) (b) (c) FIG. 9. Incompressibility of the half-ﬁlled fractional topological insulator in the spin-holomorphic Landau levels. (a) Transport gap as a function of inverse particle number 1 /Nin the spin-holomorphic spherical geometry. (b) Schematic diagram showing the helical edgestates in the spin-holomorphic disk geometry. (c) Electron density difference between before and after adding two extra electrons with both spins in the system with N=14. Here, the residual conﬁning potential strength ¯ hγis set to be 0 .105e 2//epsilon1l0. the sum of EN+1,Q−EN,Qand EN−1,Q−EN,Qto take into account the chemical potential shift associated with theparticle number change at a ﬁxed Q. It is interesting to mention that the ground states always occur at the maximumangular momentum values allowed by the particle numbereven in the N+1 and N−1 systems. Figure 9(a) shows that, plotted as a function of 1 /N, the transport gap can be nicely linearly extrapolated to a ﬁnite value ∼0.3e 2//epsilon1l0in the thermodynamic limit, conﬁrming the incompressibility of thehalf-ﬁlled spin-separated state. The ﬁnite transport gap meansthat the half-ﬁlled spin-separated state can be regarded as alegitimate fractional topological insulator . In the disk geometry, incompressibility manifests itself as an absence of the low-energy excitations in the bulk. Ifso, any additional electrons would be pushed to the edge,creating the low-energy edge excitations. Moreover, such edgeexcitations should occur in the form of the mutually counter-rotating orbitals for different spins due to the spin-dependentchirality, resulting in the helical edge states. See Fig. 9(b)for a schematic diagram depicting the situation. To conﬁrm this scenario, we compute the electron density difference between before and after adding two extra elec-trons with both spins in the half-ﬁlled spin-separated state.Figure 9(c) shows that, indeed, almost all of the probability weights of two extra electrons are pushed to the edge, consis-tent with the above mentioned scenario. Now, we argue that the helical edge states formed at the edge of the half-ﬁlled spin-holomorphic Landau levels exhibita novel bulk-edge correspondence. Usually, the bulk-edgecorrespondence is obtained in such a way that the bulk ﬁllingis directly related with the Hall conductivity of the systemvia the Landauer-Büttiker theory [ 36]. Speciﬁcally, when thebulk ﬁlling is νfor each spin species, the Landauer-Büttiker theory [ 36] predicts that the spin-dependent Hall conductivity should be quantized as ±νe 2/h, respectively, in the presence of time-reversal symmetry. In the current situation, the bulk ﬁlling is 1 /2 for each spin species. There would be no transport gap at this ﬁlling inthe usual lowest Landau level. As explained above, however,the half-ﬁlled spin-separated state in the spin-holomorphicLandau levels is predicted to be incompressible in the bulkdue to spontaneous breaking of the space rotation symmetry inthe thermodynamic limit. Meanwhile, the edge is an (1 +1)- dimensional quantum system including both space and timedegrees of freedom. As well known, the Mermin-Wagner-Hohenberg theorem dictates that no continuous symmetriescan be spontaneously broken in two or less dimensions,preventing the spin separation at the edge and therefore main-taining the helical edge states as freely ﬂowing as before. This dichotomy between the bulk and edge gives rise to the novel bulk-edge correspondence that the spin-dependentHall conductivities are to be quantized as ±e 2/hjust like in the usual 2D topological insulators [ 37–39], while the bulk ﬁlling is 1 /2 for each spin species. This novel bulk-edge correspondence can be regarded as the hallmark of the half-ﬁlled spin-separated FTI. Finally, it is important to note that the spin-separated state is gapped against charged excitations, while gaplessagainst neutral excitations. The gapless neutral excitationsare associated with a continuous deformation of the spinseparation line in the absence of any signiﬁcant magneticﬁeld inhomogeneity locally pinning the spin via the Zeemaneffect. Meanwhile, the usual fractional quantum Hall states ofsingle-component fermions are gapped against both chargedand neutral excitations. However, the current situation of thespin-separated state is not unusual if multicomponent degreesof freedom are considered. Speciﬁcally, the incompressible nature of the spin- separated state is closely analogous to that of the multicom-ponent fractional quantum Hall states with either the spin orlayer degree of freedom. For example, the ν=1 quantum Hall state with the spin degree of freedom is gapped againstcharged excitations due to the Landau level gap, while gaplessagainst neutral excitations associated with the spin wave in theabsence of the Zeeman effect. Similarly, the bilayer quantumHall state at total ﬁlling factor ν tot=1 at small interlayer distance is well described by the Halperin (111) state, whichis gapped against charged excitations, while gapless againstneutral excitations in the absence of interlayer tunneling.Here, the gapless neutral excitations are associated with a con-tinuous deformation of the interlayer phase difference, or theGoldstone mode of the bilayer exciton condensate [ 40–42]. In summary, the gapless neutral excitations will not destroythe quantization of the Hall conductivity in the spin-separatedstate, which is as incompressible as the ν=1 quantum Hall state with the spin degree of freedom and the bilayer quantumHall state at total ﬁlling factor ν tot=1. B. Collapse of the transport gap away from half ﬁlling In this section, we show that a fractionally-ﬁlled state in the spin-holomorphic Landau levels at less than half ﬁlling, say, 115131-8SPIN SEPARATION IN THE HALF-FILLED FRACTIONAL … PHYSICAL REVIEW B 99, 115131 (2019) FIG. 10. Transport gap at 1 /3 ﬁlling in the spin-holomorphic spherical geometry as a function of inverse particle number 1 /N. Similar to half ﬁlling, 1 /3 ﬁlling is deﬁned as ν↑=ν↓=1/3a n d thusνtot=2/3. The ﬂux shift is set to be exactly the same as that of the Laughlin state, i.e., 2 Q=3N/2−3. The insets show the exact energy spectra for N=8, 10, and 12, whose corresponding transport gaps are indicated by the respective arrows. The transport gap is computed via /Delta1=EN+1,Q+EN−1,Q−2EN,Q. Here, only the spin-unpolarized states are considered, i.e., N↑=N↓=N/2. ν↑=ν↓=1/3, has a vanishing transport gap in the thermo- dynamic limit. This conclusion is consistent with the previousresult obtained by Chen and Yang [ 19] that a sufﬁciently strong interspin interaction generates a compressible state atν ↑=ν↓=1/3. Figure 10shows the transport gap at 1 /3 ﬁlling in the spin-holomorphic spherical geometry as a function of inverseparticle number 1 /N. Similar to half ﬁlling, 1 /3 ﬁlling is deﬁned as ν ↑=ν↓=1/3 and thus νtot=2/3. As before, the transport gap is computed via Eq. ( 12). It is important to note that the ﬂux shift is set to be exactly the same as that of theLaughlin state, i.e., 2 Q=3N/2−3. As one can see, the transport gap at 1 /3 ﬁlling shows a very different behavior in comparison with that at half ﬁlling,which follows a straight line as a function of 1 /N, nicely extrapolated to a ﬁnite value in the thermodynamic limit. Bycontrast, the transport gap at 1 /3 ﬁlling is ﬁtted to a curve, which seems to collapse after 1 /N/lessorsimilar0.05, i.e., N/greaterorsimilar20. Based on this behavior, we conclude that it is highly likely thatthe 1/3-ﬁlled state in the spin-holomorphic Landau levels is compressible. Due to the exponential increase in the Hilbertspace dimension as a function of N, however, it has not been possible to perform exact diagonalization in sufﬁcientlylarge systems to directly conﬁrm the collapse of the transportgap. Fortunately, the exact energy spectra provide additionalevidence strongly supporting the above conclusion. The exact energy spectra are shown in the insets of Fig. 10 forN=8, 10, and 12, whose corresponding transport gaps are indicated by the respective arrows. Initially at small particlenumbers, say, N=8 and 10, the ground states occur at the maximum angular momentum value, L max tot=N(N−2)/2,(a) (b) FIG. 11. Ground-state energy as a function of the spin po- larization, P=(N↑−N↓)/(N↑+N↓), at half ﬁlling in the spin- holomorphic spherical geometry. Note that P=0a n d ±1 indicate the unpolarized and fully polarized situations, respectively. The particle number Nis set to be (a) 14 and (b) 16. allowed by the 1 /3 ﬁlling. This means that there is a tendency towards the spin separation even at 1 /3 ﬁlling. However, after the particle number becomes sufﬁciently large, say, N= 12, the energy spectrum shows a characteristic sign for thedisordered state that the ground state occurs at a randomangular momentum value, being neither L max totnor 0. In fact, one can even observe a slight softening of the ground-stateenergy curve in the vicinity of L tot=Lmax totasNchanges from 8t o1 0 . In summary, we conclude that the compressible state at 1/3 ﬁlling is likely to be a disordered state, which can be susceptible to random disorders. We think that the situationis similar at general ﬁllings away from half ﬁlling. V . SPIN SEPARATION WITHOUT TIME-REVERSAL SYMMETRY Finally, we would like to discuss what happens to the half- ﬁlled spin-separated state without time-reversal symmetry.Speciﬁcally, we compute the ground-state energy as a functionof the spin polarization, P=(N ↑−N↓)/(N↑+N↓), via exact diagonalization. Figure 11shows that the ground-state energy decreases as the spin polarization changes from being unpolarized ( P=0) to being fully polarized ( P=±1). Therefore, eventually, the ground state would become fully polarized in the presenceof time-reversal symmetry breaking sources such as magneticimpurities. Once fully polarized, the ground state reduces tothe usual ν=1 IQHS. We would like to stress, however, that the ground states occur always at the maximum total angular momentum valueallowed by the given spin polarization, i.e., L max tot=N↑N↓= N2 4(1−P2). This means that different spins are always maxi- mally separated regardless of the value of the spin polariza-tion. Based on this result, we predict that, in the presenceof time-reversal symmetry breaking sources, the half-ﬁlledstate in the spin-holomorphic Landau levels undergoes a slowevolution from the unpolarized spin-separated state to thefully polarized IQHS, while maintaining the maximal spinseparation during the entire process. 115131-9SUTIRTHA MUKHERJEE AND KWON PARK PHYSICAL REVIEW B 99, 115131 (2019) VI. DISCUSSION In this work, we have provided evidence that no fraction- ally ﬁlled states in the correlated topological band can occuras two decoupled copies of the FQHS or FCI with oppositechiralities for different spins. The Coulomb interaction, whichcould generate the FQHS or FCI for each spin species, in-evitably creates a destabilization of the simple product statebetween the two decoupled copies. Speciﬁcally, we perform exact diagonalization of the Coulomb interaction Hamiltonian in the spin-holomorphicLandau levels, where electrons with different spins experienceopposite effective magnetic ﬁelds. It is shown that the FTIoccurring at half ﬁlling of the spin-holomorphic Landau levelsis susceptible to an inherent spontaneous breaking of the spacerotation symmetry in the thermodynamic limit, leading to thespatial separation of different spins. As an application, the half-ﬁlled spin-separated FTI can be potentially useful in spintronics since it can serve as arobust interaction-driven spin ﬁlter , sorting electrons with different spins into two spatially separated regions. Onceembedded in their respective spin-separated regions, spinsare to be protected against various decoherence mechanismsby the Coulomb interaction. A spin-ﬁltered current can ﬂowby attaching a lead deep inside the desired spin-separatedregion, where the edge current surrounding the attached leadis entirely composed of the single spin species correspondingto the region. Now, let us discuss brieﬂy how the spin-separated state can be realized in experiments. Our model Hamiltonian generat-ing the spin-holomorphic Landau levels is based on a two-dimensional electron gas conﬁned in the parabolic conﬁningpotential with strong spin-orbit coupling, which can be inprinciple realized by constructing a semiconductor quantumwell or heterostructure on the substrate made of a strong spin-orbit-coupled material. Another way of generating essentiallythe same model Hamiltonian is to apply an appropriate straingradient in the two-dimensional parabolic quantum well, asproposed by Bernevig and Zhang [ 13]. Perhaps, a more exciting possibility can be obtained in the half-ﬁlled (nearly) ﬂat Chern band in the lattice. In principle,the half-ﬁlled spin-separated state in the spin-holomorphicLandau levels can be mapped onto its lattice version in the(nearly) ﬂat Chern band via the basis function mapping be-tween the lowest Landau level wave functions and the hybridWannier functions [ 7,8]. If so, our study predicts that a similar spin separation can occur in the half-ﬁlled (nearly) ﬂat Chernband in the lattice. Finally, the spin separation can be directly conﬁrmed via the Kerr rotation measurement [ 43] showing the accumulation of opposite spins in the respective halves of the sample. Thequantization of the spin Hall conductivity can be veriﬁedeither in a spin-ﬁltered experiment or in a charge transportexperiment by measuring the four-terminal resistance, asshown by König et al. [39]. Also, the incompressibility of the half-ﬁlled spin-separated FTI can be observed via thethermally activated behavior in the longitudinal resistance.This experimental evidence taken altogether can establish thenovel bulk-edge correspondence at half ﬁlling, which is thehallmark of the half-ﬁlled spin-separated FTI.ACKNOWLEDGMENTS The authors are grateful to Changsuk Noh and Hyun Woong Kwon for insightful discussions. The authors thank theKIAS Center for Advanced Computation (CAC) for providingcomputing resources. APPENDIX A: COULOMB MATRIX ELEMENTS IN THE SPIN-HOLOMORPHIC LANDAU LEVELS In the spin-holomorphic Landau levels, the interaction Hamiltonian can be written in second quantization as follows: H=/summationdisplay m1,m2,m3,m4c† m1↑c† m2↑cm4↑cm3↑/angbracketleftm1,m2\|V\|m3,m4/angbracketright +/summationdisplay m1,m2,m3,m4¯c† m1↓¯c† m2↓¯cm4↓¯cm3↓/angbracketleftm1,m2\|V\|m3,m4/angbracketright +/summationdisplay m1,m2,m3,m4c† m1↑¯c† m2↓¯cm4↓cm3↑/angbracketleftm1,m2\|V\|m3,m4/angbracketright,(A1) where c† m↑and ¯ c† m↓are the respective creation operators for spin up and down electrons in the holomorphic and anti-holomorphic orbitals, respectively, with quantum number m. Note that mis the negative of the actual z-component angu- lar momentum eigenvalue l zin the antiholomorphic orbitals, while being the same as lzin the holomorphic orbitals. The ﬁrst two terms in Eq ( A1) are exactly the same as those in the usual quantum Hall systems. What is different in thespin-holomorphic Landau levels is the last term describing theinterspin interaction. Concretely, in the spin-holomorphic disk geometry, the Coulomb matrix elements between spin-up electrons are writ-ten as follows: /angbracketleftm 1,m2\|V(\|r1−r2\|)\|m3,m4/angbracketright =/integraldisplay d2k˜Vk/angbracketleftm1,m2\|eik·(r1−r2)\|m3,m4/angbracketright =/integraldisplay d2k˜VkAm1m3(k)Am2m4(−k), (A2) where ˜Vkis the Fourier component of V(r), and Amm/prime(k)= /angbracketleftm\|eik·r\|m/prime/angbracketright=/integraltext d2rφ∗ m(r)eik·rφm/prime(r) with φm(r) being the lowest Landau level eigenstate with the quantum number m. Now, by using some analytical properties of φm(r), one can show [ 44] that Amm/prime(k)=(iκ)m−m/primeLmm/prime(k)e−k2/2, (A3) where Lmm/prime(k)=/radicalBig 2m/primem/prime! 2mm!Lm−m/prime m/prime(k2/2) with κ=kx+iky= keiθand Lr n(x) being the generalized Laguerre polynomial. Then, due to the separation of variables between kandθ, Equation ( A2) can be rewritten as follows: /angbracketleftm1,m2\|V(\|r1−r2\|)\|m3,m4/angbracketright =im1−m3(−i)m2−m4 ×/integraldisplay kdk˜VkLm1m3(k)Lm2m4(k)e−k2km1+m2−m3−m4 ×/integraldisplay dθeiθ(m1+m2−m3−m4), (A4) 115131-10SPIN SEPARATION IN THE HALF-FILLED FRACTIONAL … PHYSICAL REVIEW B 99, 115131 (2019) where the last factor imposes the selection rule for the min- dices, m1+m2=m3+m4, indicating the usual angular mo- mentum conservation. The Coulomb matrix elements betweenspin-down electrons are exactly the same as those betweenspin-up electrons. Meanwhile, the Coulomb matrix elements between differ- ent spins are given as follows: /angbracketleftm 1,m2\|V(\|r1−r2\|)\|m3,m4/angbracketright =/integraldisplay d2k˜Vk/angbracketleftm1,m2\|eik·(r1−r2)\|m3,m4/angbracketright =/integraldisplay d2k˜VkAm1m3(k)Am4m2(−k) =/integraldisplay d2k˜VkAm1m3(k)A∗ m2m4(k), (A5) w h e r ew eh a v eu s e d Amm/prime(k)=A∗ m/primem(−k). Again, due to the separation of variables between kandθ, the above equation can be rewritten as follows: /angbracketleftm1,m2\|V(\|r1−r2\|)\|m3,m4/angbracketright =im1−m3(−i)m2−m4 ×/integraldisplay kdk˜VkLm1m3(k)Lm2m4(k)e−k2km1+m2−m3−m4 ×/integraldisplay dθeiθ(m1−m2−m3+m4), (A6) where it is important to note that the selection rule for the mindices is now changed to m1−m2=m3−m4. At ﬁrst sight, this selection rule may seem as if it breaks the angularmomentum conservation law. However, this is not true sincethe actual angular momenta for spin-down electrons are l z= −m2and−m4, and therefore the above selection rule is in fact exactly the angular momentum conservation law for theinterspin interaction. The comparison between Eqs. ( A4) and (A6) tells us that the Coulomb matrix elements between the same and different spins would be exactly the same if it werenot for this change in the selection rule. In fact, this change in the selection rule is the fundamental origin of spin separation at half ﬁlling of the spin-holomorphicLandau levels. To understand this, it is important to notethat the mindices denote the radial locations of the lowest Landau level eigenstates. The peculiar selection rule for theinterspin interaction in the spin-holomorphic Landau levelsmake electrons with different spins move together to the sameradial direction after scattering in order to keep the m-index difference the same. This means that electrons with differentspins cannot avoid each other effectively unless there is aspontaneous breaking of the space rotation symmetry so thatthey are spatially separated from the outset and thus have nochance to encounter each other. The same logic applies to the spin-holomorphic spherical geometry. Speciﬁcally, in the spin-holomorphic spherical ge-ometry, the Coulomb matrix elements between spin-up elec-trons (and between spin-down electrons) are given as follows: /angbracketleftm 1,m2\|V(\|r1−r2\|)\|m3,m4/angbracketright =/integraldisplay d/Omega11/integraldisplay d/Omega12Y∗ QQm 1(r1)Y∗ QQm 2(r2)1 \|r1−r2\| ×YQQm 3(r1)YQQm 4(r2)=1√Q2Q/summationdisplay l=0l/summationdisplay m=−l/angbracketleftQ,m1;l,m\|Q,m3/angbracketright/angbracketleftQ,m4;l,m\|Q,m2/angbracketright ×/angbracketleftQ,Q;l,0\|Q,Q/angbracketright2, (A7) where YQlmrepresents the monopole harmonics with the monopole strength Q, the angular momentum quantum num- berl, and the z-component angular momentum quantum num- berm./angbracketleftj1,m1;j2,m2\|J,M/angbracketrightis the Clebsch-Gordan coefﬁcient. Above, we have used the expansion of the Coulomb potentialon the surface of a sphere in terms of the spherical harmonics[28]: 1 \|r1−r2\|=4π R∞/summationdisplay l=0l/summationdisplay m=−l1 2l+1Y∗ 0lm(/Omega11)Y0lm(/Omega12),(A8) where Ris the radius of a sphere, which is set equal to√Qas usual. Meanwhile, the Coulomb matrix elements between differ- ent spins are given as follows: /angbracketleftm1,m2\|V(\|r1−r2\|)\|m3,m4/angbracketright =/integraldisplay d/Omega11/integraldisplay d/Omega12Y∗ QQm 1(r1)YQQm 2(r2)1 \|r1−r2\| ×YQQm 3(r1)Y∗ QQm 4(r2) =1√Q2Q/summationdisplay l=0l/summationdisplay m=−l/angbracketleftQ,m1;l,m\|Q,m3/angbracketright/angbracketleftQ,m2;l,m\|Q,m4/angbracketright ×/angbracketleftQ,Q;l,0\|Q,Q/angbracketright2, (A9) where it is important to note that the only difference between Eqs. ( A7) and ( A9)i st h a t m2and m4are swapped. This gives rise to the change in the selection rule for the mindices from m1+m2=m3+m4tom1−m2=m3−m4, similar to the spin-holomorphic disk geometry. APPENDIX B: ANGULAR MOMENTUM OPERATORS IN THE SPIN-HOLOMORPHIC SPHERICAL GEOMETRY The total angular momentum operator is deﬁned as Ltot=/summationtext iLi. Speciﬁcally, L2 tot=/summationdisplay i,j/bracketleftbigg1 2(L+,iL−,j+L−,iL+,j)+Lz,iLz,j/bracketrightbigg , (B1) where L±,iare the angular momentum raising and lowering operators of the ith electron, respectively, while Lz,iis the z- component angular momentum operator of the same electron.Concretely, L ±,i=e±iφi/parenleftbigg ±∂ ∂θi+icotθi∂ ∂φi+ˆQ sinθi/parenrightbigg , (B2) Lz,i=− i∂ ∂φi, (B3) where θiandφiare the polar and azimuthal angles of the ith electron. In the usual spherical geometry with a single magnetic monopole, ˆQis just a number representing the monopole 115131-11SUTIRTHA MUKHERJEE AND KWON PARK PHYSICAL REVIEW B 99, 115131 (2019) strength. In the spin-holomorphic spherical geometry, how- ever, electrons with different spins experience opposite effec-tive magnetic ﬁelds, which are generated by the respectivemagnetic monopoles with opposite strengths. In this situation,one has to treat ˆQas an operator. Speciﬁcally, ˆQY Qlm(θ,φ)= QY Qlm(θ,φ) and ˆQY∗ Qlm(θ,φ)=− QY∗ Qlm(θ,φ). As a consequence, the operation rules for the angular mo- mentum raising /lowering operators in the spin-holomorphic Landau levels are generalized as follows: L±\|Q,Q,m/angbracketright=/radicalbig Q(Q+1)−m(m±1)\|Q,Q,m±1/angbracketright,(B4)L±\|Q,Q,m/angbracketright=−/radicalbig Q(Q+1)−m(m∓1)\|Q,Q,m∓1/angbracketright, (B5) Lz\|Q,Q,m/angbracketright= m\|Q,Q,m/angbracketright, (B6) Lz\|Q,Q,m/angbracketright=− m\|Q,Q,m/angbracketright, (B7) where /angbracketleftθ,φ\|Q,Q,m/angbracketright=YQQm(θ,φ) and /angbracketleftθ,φ\|Q,Q,m/angbracketright= Y∗ QQm(θ,φ). Above, the particle index iis dropped for sim- plicity. [1] E. Tang, J.-W. Mei, and X.-G. 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PhysRevB.80.220509.pdf	Thermal conductivity measurements of the energy-gap anisotropy of superconducting LaFePO at low temperatures M. Yamashita,1N. Nakata,1Y. Senshu,1S. Tonegawa,1K. Ikada,1K. Hashimoto,1H. Sugawara,2,T. Shibauchi,1and Y. Matsuda1 1Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan 2Faculty of Integrated Arts and Sciences, The University of Tokushima, Tokushima 770-8502, Japan /H20849Received 14 July 2009; revised manuscript received 20 October 2009; published 28 December 2009 /H20850 The superconducting gap structure of LaFePO /H20849Tc=7.4 K /H20850is studied by thermal conductivity /H20849/H9260/H20850at low temperatures in ﬁelds Hparallel and perpendicular to the caxis. A clear two-step ﬁeld dependence of /H9260/H20849H/H20850 with a characteristic ﬁeld Hs/H20849/H11011350 Oe /H20850much lower than the upper critical ﬁeld Hc2is observed. In spite of the large anisotropy of Hc2,/H9260/H20849H/H20850in both Hdirections is nearly identical below Hs. Above Hs,/H9260/H20849H/H20850grows gradually with Hwith a convex curvature, followed by a steep increase with strong upward curvature near Hc2. These results indicate multigap superconductivity with active two-dimensional /H208492D/H20850and passive three- dimensional bands having contrasting gap values. Together with the recent penetration depth results, wesuggest that the 2D bands consist of nodal and nodeless ones, consistent with extended s-wave symmetry. DOI: 10.1103/PhysRevB.80.220509 PACS number /H20849s/H20850: 74.25.Fy, 74.20.Rp, 74.25.Op, 74.70. /H11002b Recent discovery of a new class of Fe-based superconductors1has attracted much attention. Among them, FeAs-based compounds have aroused great interest becauseof the high transition temperature T c. Undoped arsenide LaFeAsO is nonsuperconducting and has a spin-density- wave /H20849SDW /H20850ground state but becomes superconducting /H20849Tc=25 K /H20850when electron doped.2By changing the rare- earth ion, Tcreaches as high as 55 K in SmFeAs /H20849O,F/H20850.3A key question is the origin of the pairing interaction. Since thesymmetry of the superconducting order parameter is inti-mately related to the pairing interaction at the microscopiclevel, its identiﬁcation is of primary importance. Fully gapped superconducting states in FeAs-based super- conductors have been reported by the penetration depth mea-surements of PrFeAsO 1−y,4SmFeAsO 1−xFy,5and Ba1−xKxFe2As2,6angle-resolved photoemission,7thermal conductivity,8and NMR /H20849Ref. 9/H20850measurements of Ba1−xKxFe2As2. Some of them give evidence of multiband superconductivity with two distinct gaps. On the other hand,the NMR of LaFeAsO 1−xFx/H20849Ref. 10/H20850and PrFeAsO 0.89F0.11 /H20849Ref. 11/H20850and the penetration depth measurements of Ba/H20849Fe1−xCox/H208502As2/H20849Ref. 12/H20850suggest the presence of low- lying excitations, which could be indicative of nodes. Theo-retically, it is proposed that a good nesting between hole andelectron pockets prefers the “ s /H11006” symmetry where the gap is ﬁnite at all Fermi surfaces but changes its sign on differentbands. 13–15Recent neutron resonant scattering16and the im- purity effects on the penetration depth6,17are consistent with this symmetry. The phosphide LaFePO /H20849Ref. 18/H20850has quite different mag- netic and superconducting properties from LaFeAsO, e.g.,LaFePO is nonmagnetic in the normal state, 19while they are isomorphic and share a similar electronic structure.20,21Re- cently, a superconducting state of LaFePO has been sug-gested to possess line nodes in the gap function by a lineartemperature dependence of the penetration depth at lowtemperatures. 22,23However, on which Fermi surfaces the nodes locate in the multiband electronic structure is not yetclariﬁed, and while several candidates have been theoreti-cally proposed, 24,25the superconducting symmetry in LaFePO remains elusive. Thus the clariﬁcation of the de-tailed gap structure of LaFePO is expected to provide impor-tant clues to the origins of magnetism and superconductivityof Fe-based compounds. Here, to shed further light on the gap symmetry of LaFePO, we present the thermal conductivity measurementsat low temperatures. The thermal conductivity probes delo-calized low-energy quasiparticle excitations and is an ex-tremely sensitive probe of the anisotropy of the gap ampli-tude. We provide strong evidence of the multigapsuperconductivity in a more dramatic fashion than FeAs-based superconductors, with two very different gap values.We show that there are passive three-dimensional /H208493D/H20850bands and two kinds of active two-dimensional /H208492D/H20850bands; one is fully gapped and the nodes inferred from the penetrationdepth measurements 22,23are most likely on the other 2D bands. This is compatible with the extended s-wave /H20849nodal s/H11006/H20850symmetry for the gap structure of LaFePO. Single crystals with dimensions of /H110110.8/H110030.4 /H110030.05 mm3were grown by a Sn-ﬂux method.26We care- fully removed the Sn ﬂux at the surface of the crystals byrinsing in diluted hydrochloric acid. The resistivity and sus-ceptibility measurements show the sharp superconductingtransition at T c=7.4 K /H20849determined by the midpoint of the resistive transition /H20850, which is slightly higher than the values reported by other groups.22,27The thermal conductivity /H9260 was measured by a standard four-wire steady method for a heat current qwithin the abplane. The temperature dependence of the in-plane resistivity /H9267 in the zero ﬁeld /H20849inset of Fig. 1/H20850depends on Tas/H9267=/H92670 +AT2, with /H92670=4.9/H9262/H9024cm and A=3.3/H1100310−3/H9262/H9024cm /K2 below 50 K down to Tc. The residual resistivity ratio /H20849RRR /H20850 is 28. Clear de Haas–van Alphen /H20849dHvA /H20850oscillations were observed in samples from the same batch with nearly thesame RRR value. 26The upper critical ﬁelds at T→0 K es- timated by the resistivity measurements are /H92620Hc2c=1.0 T forH/H20648caxis and /H92620Hc2ab=8.6 T for H/H20648abplane. This large anisotropy of the upper critical ﬁeld Hc2indicates that thePHYSICAL REVIEW B 80, 220509 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 1098-0121/2009/80 /H2084922/H20850/220509 /H208494/H20850 ©2009 The American Physical Society 220509-1bands active for the superconductivity have a very aniso- tropic 2D electronic structure. Figure 1depicts the temperature dependence of /H9260/Tin zero ﬁeld and in the normal state above Hc2ab. As the tempera- ture is lowered, /H9260/Tdecreases below Tc. It can be clearly seen that the electron contribution dominates well the pho-non heat contribution because the electronic contribution in /H9260at 1 K estimated by the Wiedemann-Franz law, /H9260=L0T//H9267 /H20849L0=2.44 /H1100310−8/H9024W /K is the Sommerfeld value /H20850,i s 0.5 W /K2m which is close to the observed value 0.65 W /K2m. Further, the phonon contribution measured in a related compound BaFe 2As2/H20849Ref. 28/H20850is one order of mag- nitude smaller in low temperatures. First we discuss the thermal conductivity in zero ﬁeld. A residual term /H926000/TatT→0K i n /H9260/Tis clearly resolved. In the nodal superconductors, such a residual term appears as aresult of the impurity scattering which induces quasiparticleseven at T=0 K. In the presence of line nodes in a single band superconductor, /H926000/Tis roughly estimated as /H110112/H20849/H9264ab//H5129/H20850·/H20849/H9260n/T/H20850, where /H9264abis the in-plane coherence length, /H5129is the mean free path and /H9260nis the thermal conduc- tivity in the normal state. Using /H5129=94 nm from the dHvA measurements26and/H9264ab=/H20881/H90210//H208492/H9266Hc2c/H20850=18 nm, /H926000/Tis estimated to be /H110110.19 W /K2m. This value is comparable to the observed /H926000/T/H110110.30 W /K2m, but we note that this estimate includes large ambiguities due to the multiband ef-fect which could alter effective /H9264aband/H9260n. So this compari- son alone cannot be taken as the evidence for line nodes inthe superconducting gap. It should be also noted that theresidual term may arise from an extrinsic origin, such asnonsuperconducting metallic region 27with high thermal con- ductivity although the sharp superconducting transition andthe observation of the dHvA oscillation indicate a good qual-ity of the crystal. More vital information on the gap structure can be pro- vided by the ﬁeld dependence of /H9260at low temperatures. The ﬁeld-dependent part of /H9260/H20849H/H20850in a mixed state mainly stems from the superconducting part of the crystals even if a non-superconducting region was present in the crystal. Moreover,the phonon scattering at the low temperatures is governed bystatic defects and is therefore ﬁeld independent. Further, it iswell known that fully gapped and nodal superconductors show a contrasting ﬁeld dependence, 29as illustrated in the lower inset of Fig. 2/H20849a/H20850. In fully gapped superconductors, quasiparticles excited by vortices are localized and unable totransport heat until these vortices are overlapped each other.Consequently, /H9260/H20849H/H20850shows a slow growth with Hin low ﬁelds and a rapid increase near Hc2/H20849the dotted line /H20850,a sr e - ported in Nb.30In sharp contrast, the heat transport in super- conductors with nodes or with a large anisotropy in the gapis dominated by contributions from delocalized quasiparti-cles outside vortex cores. 31In the presence of line nodes where the density of states has a linear energy dependence, /H9260/H20849H/H20850increases in proportion to /H20881H/H20849Ref. 32/H20850/H20849the solid line /H20850. If nodeless and nodal gaps are mixed in a multiband systemwithout interband scatterings, an inﬂection point emerges in1.0 0.8 0.6 0.4 0.2 0.0κ/T (W/K2m) 8 7 6 5 4 3 2 1 0 T (K)0T 9T( H/ /a b ) 12 8 4 0ρ(µΩ ·cm) 2000 1000 0 T2(K2) FIG. 1. /H20849Color online /H20850Temperature dependence of /H9260/Tin zero ﬁeld and at /H92620H=9 T /H20851above Hc2ab/H20849H/H20648ab,H/H11036q/H20850/H20852. Inset: /H9267plotted as a function of T2.1.0 0.8 0.6 0.4 0.2 0.0[κ(H) - κ( 0 )]/[ κ(Hc2)-κ(0)] 8 6 4 2 0 µ0H (T)0.46 K, H // abHc21.0 0.8 0.6 0.4 0.2 0.0[κ(H) - κ( 0 )]/[ κ(Hc2)-κ(0)] 1.0 0.8 0.6 0.4 0.2 0.0 µ0H (T)0.46 K, H // c Hc2 Hs1.0 0.8 0.6 0.4 0.2 0.0 1.00.80.60.40.20.0 H/Hc2line node full gap 0.10 0.08 0.06 0.04 0.02 0.00 0.06 0.04 0.02 0.00 µ0H (T)HsH/ /c ,F C H/ /c H/ /a b0.30 0.20 0.10 0.00 0.25 0.00 µ0H (T)1.3 K 1.7 K0.46 K(a) (b) FIG. 2. /H20849Color online /H20850/H20849a/H20850Field dependence of /H9260/H20849H/H20850−/H9260/H208490/H20850nor- malized by /H9260/H20849Hc2c/H20850−/H9260/H208490/H20850forH/H20648catT=0.46 K. Upper inset: the same plot in low ﬁelds for H/H20648cat 0.46 K /H20849circles /H20850, 1.3 K /H20849triangles /H20850, and 1.7 K /H20849squares /H20850. Lower inset: schematic ﬁeld dependence of /H9260/H20849H/H20850−/H9260/H208490/H20850in superconductors with a full gap /H20849dotted line /H20850, with line nodes /H20849solid line /H20850, and with two kinds of gaps with and without nodes /H20849dash-dotted line /H20850./H20849b/H20850Field dependence of /H20851/H9260/H20849H/H20850 −/H9260/H208490/H20850/H20852//H20851/H9260/H20849Hc2ab/H20850−/H9260/H208490/H20850/H20852forH/H20648abplane at 0.46 K. Inset: a compari- son of the low-ﬁeld data for H/H20648abwith those for H/H20648cmeasured in the zero-ﬁeld /H20849ﬁlled circles /H20850and ﬁeld cooling /H20849FC/H20850conditions /H20849open circles /H20850.YAMASHITA et al. PHYSICAL REVIEW B 80, 220509 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 220509-2the intermediate ﬁeld regime /H20849the dash-dotted line /H20850.33 Figures 2/H20849a/H20850and2/H20849b/H20850depict /H9260/H20849H/H20850−/H9260/H208490/H20850normalized by /H9260/H20849Hc2/H20850−/H9260/H208490/H20850forH/H20648candH/H20648ab, respectively, measured at T=0.46 K /H208490.062 Tc/H20850by sweeping Hafter zero-ﬁeld cooling. We note that little difference was observed between the datameasured in zero-ﬁeld and ﬁeld cooling conditions, indicat-ing that the ﬁeld trapping effect is very small. For both ﬁelddirections, the overall Hdependence of /H9260is quite similar. At very low ﬁelds, /H9260exhibits a pronounced increase with in- creasing H. Remarkably, in spite of the large anisotropy of Hc2,/H9260/H20849H/H20850is nearly identical for both Hdirections at low ﬁelds and almost saturates at around Hs/H11011350 Oe, as shown in the inset of Fig. 2/H20849b/H20850. Above Hs,/H9260/H20849H/H20850becomes anisotropic with respect to the ﬁeld direction and is governed by theanisotropy of H c2. For both H/H20648candH/H20648ab,/H9260/H20849H/H20850grows gradually with Hwith a convex curvature followed by a rapid increase with a concave curvature up to Hc2; there is an inﬂection point at /H11011Hc2/4/H20849/H11011Hc2/8/H20850forH/H20648c/H20849H/H20648ab/H20850. The steep increase and subsequent gradual increase in /H9260/H20849H/H20850above Hsfor both H/H20648candH/H20648abindicate that a sub- stantial portion of the quasiparticles is already restored at Hs, much below Hc2. Such a two-step ﬁeld dependence has been reported in MgB 2,34PrOs 4Sb12,35and URu 2Si2,36providing direct evidence for the multiband superconductivity. Here Hs is interpreted as a “virtual upper critical ﬁeld” that controls the ﬁeld dependence of the smaller gap of the “passive”band. Its superconductivity is most likely induced by theproximity effect of the “active” bands with primary gap. Theratio of the large and small gaps is roughly estimated to be /H9004 L//H9004S/H11011/H20881Hc2c/Hs/H110116. If we take /H9004L/H110111.7kBTc/H1101113 K, we obtain /H9004S/H110112K . We note that the steep increase in /H9260/H20849H/H20850below Hsis not due to the inﬂuence of the ﬁrst vortex penetration ﬁeld,which is expected to be anisotropic and much smaller thanH c1/H11011100 Oe if the demagnetization is taken into account.37 We can also rule out a possibility that the steep increase is caused by the remanent Sn ﬂux because /H9260/H20849T/H20850, magnetization, and microwave surface impedance measurements38show no anomaly at Tcof Sn /H20849=3.72 K /H20850. Moreover, the low-ﬁeld steep increase disappears at T/H110111.5 K well below Tcof Sn /H20851the upper inset of Fig. 2/H20849a/H20850/H20852, which is rather in good agree- ment with the gap size estimation. The observed multiband superconductivity is compatible with the band structure of LaFePO. The band structure cal-culations show that Fermi surface consists of two electroniccylinders centered at the Mpoint and two hole cylinders centered at the /H9003point, together with a single hole pocket with 3D-like dispersion at the Zpoint in the Brillouin zone /H20849see the sketch in Fig. 3/H20850. 20,21,39The 3D band is suggested to appear in LaFePO, not in LaFeAsO, and has a character ofthe 3 d 3z2−r2orbital which is expected to have a weak cou- pling to other 2D bands.21Note that only the 2D cylindrical electron and hole bands have been reported byphotoemission 40and dHvA measurements.26,39 Nearly isotropic /H9260/H20849H/H20850with respect to the ﬁeld direction below Hsshown in the inset of Fig. 2/H20849b/H20850indicates that the smaller gap is present most likely in the 3D hole pocket. Thisis consistent with the expected weak coupling between the3D and 2D bands. This passive 3D band is inferred to befully gapped because of the following reasons. The ﬁeld de-pendence of /H9260below Hsdoes not show strong /H20881Hdepen- dence expected for line nodes. Moreover, since the coher- ence length of the smaller gap, /H9264s=/H20881/H92780//H208492/H9266Hs/H20850/H11229100 nm, is comparable to the mean free path, it is unlikely that a nodalsuperconductivity can survive against such a “dirty” condi-tion /H20849 /H9264s/H11229/H5129/H20850. Next we discuss the gap structure of the 2D bands from /H9260/H20849H/H20850above Hs, where essentially all quasiparticles of the 3D band with smaller gap have already contributed to the heattransport. As shown in Fig. 3, /H9260/H20849H/H20850forH/H20648cincreases as /H11011/H20881Hjust above Hsto/H110110.4Hc2/H20849the/H20881Hdependence is not clear below Hsin our resolution /H20850. This Hdependence and the appearance of the inﬂection point from convex to concave H dependence are in sharp contrast to the Hdependence of the simple fully gapped superconductors, in which Hdepen- dence is always concave well below Tc.3Such a convex /H20849sublinear /H20850Hdependence at low ﬁelds appears when the gap is highly anisotropic with a large amplitude modulation. Onthe other hand, the concave Hdependence just below H c2has never been reported in superconductors with large aniso-tropic gap, such as Tl 2Ba2CuO 6+/H9254,32CePt 3Si,41and LuNi 2B2C.42Therefore, it is likely that at least one of the active 2D bands is fully gapped without nodes. In fact, the H dependence of /H9260/H20849H/H20850with two kinds of gaps with and without nodes33shows an inﬂection behavior /H20851see the dash-dotted line in the lower inset of Fig. 2/H20849a/H20850/H20852, which qualitatively re- produces the data. This result, along with the ﬁnite /H926000/T observed in our dHvA-available clean crystal, supports thenodal superconductivity suggested by the linear temperaturedependence of the superﬂuid density. 22,23Thus the whole H dependence of /H9260above Hsimplies that the 2D bands consist of two kinds; one has nodes and the other is fully gapped. We note that this multi-gap feature we deduce is compat- ible with the penetration depth measurements;22,23the nodal gap dominates the linear temperature dependence in lowtemperatures where the effects of large nodeless gap and theΓ /CID48 2 /cH H1.0 0.8 0.6 0.40.20.0[κ(H)-κ(0)]/ [κ(Hc2)-κ(0)] 1.0 0.8 0.6 0.4 0.2 0.0H // cHs FIG. 3. /H20849Color online /H20850The same data in the main panel of Fig. 2/H20849a/H20850plotted against /H20881H/Hc2. The solid line is a guide for the eyes. The inset illustrates the extended s-wave /H20849nodal s/H11006/H20850gap structure /H20849Ref. 25/H20850in the unfolded Brillouin zone. The sign of the gap changes between the solid and dotted lines. The small band at thezone corner represents the 3D band around the Zpoint.THERMAL CONDUCTIVITY MEASUREMENTS OF THE … PHYSICAL REVIEW B 80, 220509 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 220509-3small gap are negligible because the former is already satu- rates far below Tcand the latter has a tiny density of states /H20849less than 5% of total density of states /H20850. Finally we discuss the position of the nodes. As candi- dates of the gap structure with line nodes, the “nodals /H11006-wave” and “ d-wave” symmetries have been proposed for LaFePO.25We infer that the d-wave can be excluded because it possesses line nodes in the 3D band /H20849as well as the 2D hole bands /H20850, which is unlikely for the reasons discussed above. The nodal s/H11006-wave structure has a nodal gap on the 2D electron band around Mpoint and the 2D and 3D hole bands are fully gapped /H20849see the sketch in Fig. 3/H20850. The gap size of the 2D electron and hole bands can be comparable to eachother. 25Therefore, we suggest that the nodal s/H11006-wave struc- ture can be the best candidate for the gap symmetry ofLaFePO.In summary, from the measurements of the thermal con- ductivity, LaFePO is found to be a multigap superconductorwith 2D active and 3D passive bands. The peculiar ﬁelddependence of /H9260provides a stringent constraint on the super- conducting gap structure in this system: there exist fullygapped 2D and 3D bands and the nodes locate most likely onthe other 2D bands. These results are consistent with thenodal s /H11006-wave symmetry proposed for the superconducting state of LaFePO. We thank R. Arita, A. Carrrington, H. Ikeda, K. Kontani, K. Kuroki, and I. Vekhter for valuable discussion. This workwas supported by KAKENHI from JSPS and the Grant-in-Aid for the Global COE Program “The Next Generation ofPhysics, Spun from Universality and Emergence” fromMEXT. 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PhysRevB.84.094435.pdf	PHYSICAL REVIEW B 84, 094435 (2011) Crossover from antiferromagnetic to ferromagnetic ordering in the semi-Heusler alloys Cu1−xNixMnSb with increasing Ni concentration Madhumita Halder, S. M. Yusuf,and Amit Kumar Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India A. K. Nigam Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India L. Keller Laboratory for Neutron Scattering, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland (Received 3 May 2011; revised manuscript received 30 July 2011; published 22 September 2011) The magnetic properties and transition from an antiferromagnetic (AFM) to a ferromagnetic (FM) state in semi-Heusler alloys Cu 1−xNixMnSb, with x<0.3 have been investigated in detail by dc magnetization, neutron diffraction, and neutron depolarization. We observe that for x<0.05, the system Cu 1−xNixMnSb is mainly in the AFM state. In the region 0.05 /lessorequalslantx/lessorequalslant0.2, with decrease in temperature, there is a transition from a paramagnetic to a FM state, and below ∼50 K, both AFM and FM phases coexist. With an increase in Ni substitution, the FM phase grows at the expense of the AFM phase, and for x> 0.2, the system fully transforms to the FM phase. Based on the results obtained, we have performed a quantitative analysis of both magnetic phases and proposea magnetic phase diagram for the Cu 1−xNixMnSb series in the region x<0.3. Our study gives a microscopic understanding of the observed crossover from the AFM to FM ordering in the studied semi-Heusler alloys Cu1−xNixMnSb. DOI: 10.1103/PhysRevB.84.094435 PACS number(s): 75 .50.Ee, 61 .05.fg, 75.30.Et I. INTRODUCTION Heusler and semi-Heusler alloys have become a subject of investigation, both theoretically and experimentally, in recent years because of their interesting physical properties.1–4 This class of materials has become a potential candidate for spintronics application because of their half-metallic character, structural similarity with semiconductors, and Curietemperature above room temperature. Semi-Heusler alloy NiMnSb is one of the best-known examples of half-metallic ferrromagnets. De Groot et al. , 5based on electronic structure calculation, predicted that NiMnSb should exhibit 100% spin polarization at the Fermi level. Another interesting physical property of this class of materials is the martensitic transformation at low temperatures,6which gives rise to some interesting properties, like magnetic shape memory effect,inverse magnetocaloric effect, etc. 1,4These properties are promising for future technological applications. From a basic understanding point of view, these systems show a rich varietyof magnetic behaviors ranging from itinerant to localized magnetism with a wide diversity in the magnetic properties like ferromagnetism, ferrimagnetism, antiferromagnetism, andother types of noncollinear ordering. 7–12 The semi-Heusler alloys XMnSb ( X=3delements) belong to a class of materials with high local magnetic momentson the Mn atoms. The Mn-Mn distance in these alloys isfairly large ( d Mn−Mn>4˚A) for a direct-exchange interaction to propagate. The magnetic-exchange interaction in the Mn-based semi-Heusler alloys varies from ferromagnetic (FM)Ruderman–Kittel–Kasuya–Yosida (RKKY) type exchange toantiferromagnetic (AFM) superexchange interactions withSb (sp) and X(3d) atoms playing a role in mediating the exchange interactions between Mn atoms. 12The semi-Heusleralloy NiMnSb is a ferromagnet with Curie temperature of TC=750 K.13It crystallizes in the C1 bstructure with four interpenetrating fcc sublattices.13The band structure calculations show that the magnetic properties of NiMnSb aredue to the magnetic moments localized only on the Mn atomsinteracting via itinerant electrons in the conduction band, i.e.the exchange mechanism is of RKKY type. 14CuMnSb alloy also has the same crystal structure but antiferromagnetic withN´eel temperature T N=55 K.15The magnetic moment is only on the Mn atom and is aligned perpendicular to theferromagnetic (111) planes with neighboring planes orientedin antiparallel. 16In case of CuMnSb, the exchange interaction is of superexchange type. The AFM to FM phase transitionin Cu 1−xNixMnSb is a consequence of the dominance of ferromagnetic RKKY-type exchange interaction over theantiferromagnetic superexchange interaction which occurs bytuning of the X(nonmagnetic 3 datoms Cu/Ni). Change in the electron concentration (i.e. difference in Cu and Ni valencies),modiﬁes the density of states at the Fermi surface, whichaffects the exchange interaction between Mn-Mn spins in theMn sublattice, resulting in the AFM to FM transition. Thereare reports on electronic, magnetic, and transport propertiesand on the magnetic phase transition in Cu 1−xNixMnSb, both theoretically17,18and experimentally,15,19,20which show that there is a decrease in magnetization and electrical conductivitywith decreasing x,f o r x<0.3. For x> 0.3, compounds of the Cu 1−xNixMnSb series are ferromagnetic in nature with a nearly constant value of Mn moment ( ∼4μB/atom). The magnetic ordering temperature increases continuously withxfor the entire series. The theoretical studies, based on the density functional theory, have shown that for x<0.3, antiferromagnetic superexchange coupling dominates, 17and the FM phase decays into a complex magnetic phase which 094435-1 1098-0121/2011/84(9)/094435(9) ©2011 American Physical SocietyHALDER, YUSUF, KUMAR, NIGAM, AND KELLER PHYSICAL REVIEW B 84, 094435 (2011) can be viewed as the onset of disorder in the orientation of the Mn spins.18However, there is no detailed experimental study reported in the x<0 . 3r e g i o no fC u 1−xNixMnSb series, where the transition from the AFM to the FM state occurs. Moreover,the reported experimental studies 15,19,20are based on bulk techniques, such as magnetization and resistivity. There is nomicroscopic understanding of the nature of AFM-to-FM phasetransition. This motivated us to investigate the Cu 1−xNixMnSb series in detail, in the region x<0.3, by dc magnetization, neutron diffraction, and neutron depolarization techniques inorder to have a detailed understanding of the nature of thisAFM-to-FM transition. Our results suggest electronic phaseseparation in the 0.05 /lessorequalslantx/lessorequalslant0.2 region, i.e. both AFM and FM phases coexist. Our study also gives a quantitative analysis forboth magnetic phases and a magnetic phase diagram for theCu 1−xNixMnSb series in the x<0.3 region. This paper will be useful for understanding the nature of magnetic orderingas well as for tuning magnetic and electronic properties ofdifferent Heusler and semi-Heusler alloys. II. EXPERIMENTAL DETAILS The polycrystalline Cu 1−xNixMnSb samples ( x=0.03, 0.05, 0.07, 0.15, and 0.2) with constituent elements of 99.99%purity were prepared by arc melting under argon atmosphere.An excess of Mn and Sb (2 wt.%) was added to the startingcompositions to compensate the evaporation losses. For betterchemical homogeneity, the samples were remelted manytimes. After melting, they were annealed in a vacuum-sealedquartz tube at 650 ◦C for seven days. The powder x-ray diffraction (XRD) using the Cu-K αradiation in the 2 θrange of 10–90◦with a step of 0.02◦was carried out on all samples at room temperature. The dc magnetization measurements werecarried out on the samples using a superconducting quantuminterference device (SQUID) magnetometer (QuantumDesign, MPMS model) as a function of temperature andmagnetic ﬁeld. The zero-ﬁeld-cooled (ZFC) and ﬁeld-cooled(FC) magnetization measurements were carried out over thetemperature range of 5–300 K under 200 Oe ﬁeld. Magnetiza-tion as a function of magnetic ﬁeld was measured for x=0.05, 0.07, 0.15, and 0.2 samples at 5 K over a ﬁeld variation of ±= 50 kOe. Neutron diffraction patterns were recorded atvarious temperatures over 5–300 K for x=0.03, 0.05, 0.07, 0.15, and 0.2 samples using the powder diffractometer II(λ=1.2443 ˚A) at the Dhruva reactor, Trombay, Mumbai, India. For the x=0.15 sample, the temperature-dependent neutron diffraction experiments were also performed downto 1.5 K on the neutron powder diffractometer DMC(λ=2.4585 ˚A) at the Paul Scherrer Institute (PSI), Villigen, Switzerland. The one-dimensional neutron-depolarizationmeasurements were carried out for x=0.03, 0.05, 0.07, and 0.15 samples down to 2 K using the polarized neutronspectrometer (PNS) at the Dhruva reactor ( λ=1.205 ˚A). FC neutron-depolarization measurements were carried outby ﬁrst cooling the sample from room temperature down to2 K in the presence of a 50-Oe ﬁeld (required to maintainthe neutron beam polarization at sample position) and thencarrying out the measurements in warming cycle under thesame ﬁeld. The incident neutron beam was polarized alongthe−zdirection (vertically down) with a beam polarization01000200030004000 20 40 60 8002000400006001200 04008001200 060012001800 (220)(200)(111) x = 0.03(311) (222) (400) (331) (420) (422) (333) (511)(a) x = 0.2 x-ray Counts (arb. units)x = 0.05 x = 0.07 2θ (degree)x = 0.15 0.00 0.05 0.10 0.15 0.206.066.076.086.09Lattice Constant ( Å) x (Ni Concentration )(b) FIG. 1. (a) X-ray diffraction patterns for x=0.03, 0.05, 0.07, 0.15, and 0.2 samples at room temperature. The ( hkl)v a l u e s corresponding to Bragg peaks are marked. (b) Variation of latticeconstant with Ni concentration. of 98.60(1)%. The transmitted neutron beam polarization was measured along the +zdirection, as described in an earlier paper.21 III. RESULTS AND DISCUSSION Figure 1(a) shows the XRD patterns for all the samples at room temperature. The Rietveld reﬁnement (using the FULLPROF program22) of the XRD patterns at room temperature conﬁrms that all samples are in single phase with C1 b-type cubic structure and space group F¯43m.From the Rietveld reﬁnement, we ﬁnd that Cu/Ni atoms occupy the sublattice(000), while Mn and Sb atoms occupy the other two sublattices 094435-2CROSSOVER FROM ANTIFERROMAGNETIC TO ... PHYSICAL REVIEW B 84, 094435 (2011) 0 50 100 150 200 250 3000.00.2(e)M (emu g-1) Temperature (K)x = 0.030.00.51.0(d)x = 0.050123(c) x = 0.07071421 (b)x = 0.157142128 ZFC FCx = 0.2 (a) FIG. 2. (Color online) Temperature dependence of FC and ZFC magnetization Mforx=0.03, 0.05, 0.07, 0.15, and 0.2 samples at 200 Oe applied ﬁeld. (1 41 41 4) and (3 43 43 4), respectively, as known for the C1 b-type cubic structure.16The fourth sublattice (1 21 21 2) is unoccupied. From the Rietveld reﬁnement, we conﬁrm that the (1 21 21 2) sublattice is unoccupied for all samples. Since Mn, Ni, and Cuare nearby elements on the periodic table, the XRD technique isnot sensitive enough to conﬁrm any interchange of the Cu/Niand the Mn site atoms. The absence of the interchange ofthe atoms between (000) and ( 1 41 41 4) sites (generally present in these types of structures) has been conﬁrmed by neutrondiffraction study (discussed later). The present XRD studyshows that the lattice parameter decreases with increasing Nisubstitution in CuMnSb [Fig. 1(b)]. A. The dc magnetization study Figure 2shows the ZFC and FC magnetization ( M)v s temperature ( T) curves under an applied ﬁeld of 200 Oe for the x=0.03, 0.05, 0.07, 0.15, and 0.2 samples. An antiferromagnetic peak is observed at around 54, 51, and50 K for the x=0.03, 0.05, and 0.07 samples, respectively, in both FC and ZFC M(T) curves, which can be estimated as the antiferromagnetic transition temperature. However, forthex=0.05 and 0.07 samples, a bifurcation (in the FC and ZFC curves) is observed below ∼45 K. For the FC case, the magnetization attains a constant value at lower temperaturesindicating the presence of some ferromagnetic contribution for both x=0.05 and 0.07 samples. For higher Ni substitution (the x=0.15 sample), the antiferromagnetic peak is still present; however, it becomes broad. The FC and ZFC curves showa bifurcation only below ∼45 K. A bifurcation in FC and ZFC curves and a constant value of magnetization in the FCM(T) curve are expected in these compounds when competing AFM and FM interactions are present. The constant valueof magnetization in FC curves of the x=0.05, 0.07, and 0.15 samples below 45 K indicates that, on substituting Niin the AFM CuMnSb, some FM-like clusters appear in theAFM matrix. During the ZFC process, the FM clusters freezein random directions, resulting in the random orientation ofmagnetization of individual clusters. While in the FC process,the FM clusters align along the direction of the applied ﬁeldand contribute to a higher and constant value of magnetizationbelow 45 K. On further increase in the Ni concentration, i.e.for the x=0.2 sample, the antiferromagnetic peak disappears, and a negligible bifurcation between FC and ZFC curvesoccurs, indicating that the nature of the M(T) curve is that of a typical ferromagnetic system. Also the value of magnetizationincreases with increasing Ni concentration. The transitiontemperature also increases with Ni substitution as reportedin literature. 19 Figure 3(a) shows the Mvs applied ﬁeld ( H) curves at 5 K over a ﬁeld range of ±50 kOe (all four quadrants) for x= 0.05, 0.07, 0.15, and 0.2 samples. The enlarged view of the lowﬁeld region of MvsHcurves for x=0.05 and 0.07 samples is shown in the top left inset of Fig. 3(a), and for x=0.15 and 0.2 samples, it is shown in the bottom right inset of Fig. 3(a).T h e observed hysteresis for all four samples indicates the presenceof ferromagnetism. The value of magnetization ( M Max), at the maximum applied ﬁeld (50 kOe) in our study, increases withincrease in Ni concentration [shown in Fig. 3(b)], indicating that FM phase increases with increase in Ni concentration.The coercive ﬁeld [shown in Fig. 3(b)] ﬁrst increases as we increase Ni substitution from x=0.05 to 0.07, which could be due to large anisotropy of the isolated FM-like clusters inthese samples, and then decreases with further increase in Niconcentration ( x=0.07 to 0.2), indicating a soft ferromagnetic nature for high-Ni concentration samples. The Arrott plots forthex=0.05 and 0.07 samples at 5 K are shown in Fig. 3(c).T h e linear extrapolation of the Arrott plots at high ﬁelds, interceptsthe negative M 2axis, indicating that there is no spontaneous magnetization for these samples, whereas the presence ofspontaneous magnetization is evident from the Arrott plots forthex=0.15 and 0.2 samples [shown in Fig. 3(d)], indicating a dominant FM nature of these samples. The volume fractionsof the AFM and FM states have been estimated for thex=0.05, 0.07, 0.15, and 0.2 samples from their maximum magnetization, ( M Max) values at 50 kOe. From band structure calculation, it was concluded that, in FM NiMnSb, the momentwas localized on Mn, and the net moment was found tobe 3.96 μ B/f.u.2,4μB/f.u,23and 4 μB/f.u.24This is due to the large exchange splitting of the Mn 3 delectrons in comparison to the Ni 3 delectrons, which are weakly spin polarized. The experimentally found moment per Mn atom forNiMnSb is 3.85 μ B/atom.25In the case of AFM CuMnSb as well, only Mn carries the moment and is ∼3.9μB/atom.16In the present study, MMaxfor the x=0.2 sample is 3.5 μB/f.u, 094435-3HALDER, YUSUF, KUMAR, NIGAM, AND KELLER PHYSICAL REVIEW B 84, 094435 (2011) -60 -40 -20 0 20 40 60-150-100-50050100150 -2 -1 0 1 2-404 -0.01 0.00 0.01-20-1001020M (emu g-1) Magnetic Field (kOe) x = 0.2 x = 0.15 x = 0.07 x = 0.05(a) 5 KM (emu g-1) H (kOe)(i) x = 0.07 x = 0.05 M (emu g-1) H (kOe) (ii) x = 0.15 x = 0.2 0.00.40.81.2 0.05 0.10 0.15 0.2001234 Hc (kOe)MMax (μB/f.u.) x (Ni concentration)5 K(b) 03 0 6 0 9 0030006000 0 200 400 6000100200 x = 0.2 x = 0.15M2 (emu g-1)2 H/M (Oe g emu-1)(d) 5 K x = 0.07 x = 0.05 H/M (Oe g emu-1)(c) 5 K FIG. 3. (Color online) (a) The MvsHcurves over all the four quadrants for x=0.05, 0.07, 0.15, and 0.2 samples at 5 K. Inset (i) shows the enlarged view of the MvsHcurves for x=0.05 and 0.07 samples, where a clear hysteresis is observed. Inset (ii) shows theenlarged view of the MvsHcurves for x=0.15 and 0.2 samples. (b) The variation of maximum magnetization (at 50 kOe) and coercive ﬁeld with Ni concentration. Arrott plots for (c) x=0.05 and 0.07 and (d) 0.15 and 0.2 samples at 5 K. The solid lines are the linear extrapolation of the Arrott plots at high ﬁelds. whereas the expected moment is ∼3.85μB/f.u. for 100% FM Cu 1−xNixMnSb samples, considering that only Mn atoms carry the moment. So the volume fraction of the FM phase forthex=0.2 sample is about 91%. The remaining fraction can be considered as the AFM phase. Similarly, the FM and AFMphase fractions have been calculated for the x=0.05, 0.07, and 0.15 samples, and the FM phase fractions are around 10,17, and 60%, respectively. B. Neutron diffraction study To study the transition from AFM to FM state in Cu1−xNixMnSb semi-Heusler alloys, in detail, we performed the neutron diffraction study on the x=0.03, 0.05, 0.07, 0.15, and 0.2 samples at various temperatures in magneticallyordered as well as paramagnetic (PM) states. The measureddiffraction patterns were analyzed by the Rietveld reﬁnementtechnique using the FULLPROF program.22The reported values of the atomic positions and lattice constants for CuMnSb wereused as the starting values for the present Rietveld reﬁnementfor all samples. 16The analysis reveals that the crystal structure has four interpenetrating fcc sublattices, i.e. C1 b-type structure as observed in the XRD. Here, neutron diffraction easilydistinguishes between Ni and Cu due to difference in theirscattering lengths (1.03 ×10 −12and 0.77 ×10−12cm for Ni and Cu, respectively). We conﬁrm that the entire Ni issubstituted at the Cu site. The low temperature (at 5 K) neutrondiffraction patterns for the x=0.03, 0.05, and 0.07 samples (Fig. 4) show a number of additional Bragg peaks when compared with diffraction patterns recorded in the PM statefor these samples. These peaks appear below 50 K and can beindexed in terms of an antiferromagnetic unit cell having latticeparameters twice that of the chemical unit cell, similar to that ofCuMnSb, with magnetic moments aligned perpendicular to theferromagnetic (111) planes and neighboring planes oriented inantiparallel. 16Thex=0.03 sample shows a pure AFM phase with a moment of 3.14(3) μBper Mn (Table I). However, forx=0.05 and 0.07 samples, a small ferromagnetic phase contribution ( ∼10 and ∼17%, respectively) was obtained from dc magnetization data, which could not be detected in theneutron diffraction data, possibly due to the low neutron ﬂuxat our instrument. Therefore, for the magnetic reﬁnement,only the AFM phase has been considered with 100% phasefraction (for x=0.05 and 0.07), and the derived values of the Mn moment per atom are given in Table I. The diffraction pattern for the x=0.2 sample [Fig. 4(g)] shows no extra peaks at low temperature (5 K), but an extra Bragg intensity tothe lower angle fundamental (nuclear) Bragg peaks has beenobserved, apparently indicating a pure ferromagnetic nature ofthe sample. However, from our dc magnetization study for thex=0.2 sample, it was concluded that ∼9% volume fraction of the sample is AFM. This small AFM phase fraction could notbe detected in our neutron diffraction study. Therefore, only theFM phase has been considered (100%) to derive Mn momentper atom (Table I). For the x=0.15 sample, neutron diffraction measurements carried out at Paul Scherrer Institute (PSI) inSwitzerland at 1.5 K [Fig. 5(a)], show a number of additional Bragg peaks (as compared to the diffraction patterns at 50K and above) as well as observable extra Bragg intensity tothe lower angle fundamental (nuclear) peaks. The extra Braggpeaks can be indexed to an antiferromagnetic structure similarto that found for other samples ( x=0.03, 0.05, and 0.07). These extra peaks disappear above 50 K. The difference patternobtained by subtracting 250 K data (PM state) from 1.5 K datais shown in Fig. 5(d). Both AFM and FM contributions to the intensity are observed. The difference pattern obtained bysubtracting 250 K data from 50 K data [Fig. 5(e)] shows only the FM contribution to the intensity at 50 K. In this case, forthe magnetic reﬁnement at 1.5 K, we have considered both FMand AFM phases, and the corresponding Mn moment per atomfor each phase is given in Table I. The magnetic phase fraction has also been derived. The ferromagnetic moment for x=0.15 and 0.2 samples is found to align along the crystallographicaxes. The corrected FM Mn moment for the x=0.2 sample, derived by considering the appropriate phase fraction,is 3.66(5) μ B. The corrected values of the AFM Mn moment, obtained by considering the appropriate AFM phase fraction 094435-4CROSSOVER FROM ANTIFERROMAGNETIC TO ... PHYSICAL REVIEW B 84, 094435 (2011) 0700014000 123456707000140000200040006000 02000400060000600012000 06000120000600012000 0600012000 (g)Neutron Counts (arb. units) 5 K x = 0.2 (h) Q (Å-1)300 K x = 0.2*5 K x = 0.07(c) (f)(e) 240 K x = 0.07(d)5 K x = 0.05(a) 100 K x = 0.05(b)5 K x = 0.03 150 K x = 0.03 FIG. 4. (Color online) Neutron diffraction patterns for x=0.03, 0.05, 0.07, and 0.2 samples at below and above the magnetic ordering temperatures. The open circles show the observed patterns. The solid lines represent the Rietveld reﬁned patterns. The difference betweenobserved and calculated patterns is also shown at the bottom of each panel by solid lines. The vertical bars indicate the allowed Bragg peaks position for chemical (top row) and magnetic (bottom row)phase. Asterisks mark the additional AFM Bragg peaks. estimated from magnetization data, are 3.12(4) and 3.05(7) μBat 5 K for x=0.05 and 0.07 samples, respectively. Here, for estimation of the Mn moment, we have considered the factthat, for a given intensity in neutron diffraction, the moment isinversely proportional to the square root of the scale factor(volume phase fraction) in the Rietveld reﬁnement. 22The lesser value of the AFM Mn moment at 5 K could be dueto the large value of T/T Nratio. The neutron diffraction study,therefore, indicates that there is a crossover from an AFM to an FM state on substituting Ni in CuMnSb. It is alsoevident that the derived values of both AFM and FM Mnmoments (after correcting for appropriate phase fractions asobtained from dc magnetization data) remain almost constantacross the studied series. The appearance of the FM Mnmoment in AFM CuMnSb on substituting Ni may be viewedas some of the AFM Mn spins change their direction andalign parallel to each other. These uncompensated spins align(parallel) along the crystallographic axes and can be treatedas FM-like clusters in the AFM matrix. This is in agreementwith the theoretical model of uncompensated disordered localmoment proposed by Kudrnovsk ´yet al. 18As more and more Mn atoms align (parallel) with increasing Ni concentration,the disordered AFM moment appears in equal amount as FMmoment. There could be two possible models for the transitionfrom the AFM to FM state. First is the inhomogeneous model,where in the intermediate (0.05 /lessorequalslantx/lessorequalslant0.2) concentration region, both AFM and FM phases coexist. 26As we change the Ni concentration from x=0.05 to 0.2, the volume fraction of the two phases changes. The second model is thehomogeneous model, where the AFM and FM contributionsto the Mn moment result from a canted magnetic structure. 26 Szytula proposed a canted spin structure for the AFM-to-FMphase transition in Cu l−xNixMnSb.27It was suggested that the magnetic moments are canted at an angle of 45◦to the cube edge, which when resolved into components would giveboth AFM and FM contributions. We have tried to analyzethe neutron diffraction data with both models and ﬁnd thatthe inhomogeneous model with phase coexistence only ﬁts thedata. A canted behavior was not observed from the analysisof the neutron diffraction data for any of the present samplesas evident from nearly constant and almost full values of thederived FM and AFM Mn moments. If a canted structure exists,as suggested by Szytula, then the vertical component of themoment (along the crystallographic axes), as found in ourneutron diffraction study, would give the FM contribution,and the horizontal component (in the basal plane) should givethe AFM contribution. Therefore, the AFM and FM momentsshould be perpendicular to each other. But in the presentcase, the intensity of the antiferromagnetic peaks ﬁts onlyif we consider the moment direction to be perpendicular to the(111) plane that makes an angle of 45 ◦with the FM moment. Furthermore, for a canted structure, the disappearance of theantiferromagnetic order at higher temperature (above 50 K)would indicate that the magnetic moment changes its direction,and the system becomes a collinear ferromagnet. In that case,there would be a sudden increase in the intensity of the FMBragg peaks. We have plotted the integrated intensities of the(111), (200), and (220) nuclear Bragg peaks for the x=0.15 sample as a function of temperature [shown in Fig. 5(g)]. These nuclear Bragg peaks have ﬁnite contribution to the intensityarising from the FM ordering of Mn moments. We ﬁnd thatthe intensity of these Bragg peaks gradually decreases with in-creasing temperature. So our neutron diffraction data infer thecoexistence of both AFM and FM phases (for 0.05 /lessorequalslantx/lessorequalslant0.2) under the inhomogeneous model. Figure 6shows the variation of the AFM and FM phases with the Ni concentration atlow temperature, as obtained from dc magnetization andneutron diffraction experiments. The phase fraction obtained 094435-5HALDER, YUSUF, KUMAR, NIGAM, AND KELLER PHYSICAL REVIEW B 84, 094435 (2011) TABLE I. Derived FM and AFM moments per Mn atom from neutron diffraction data for various compositions. Cu1−xNixMnSb Temperature (K) FM Moment ( μB) AFM Moment ( μB) x=0.03 5 3.14(3) x=0.05 5 –a2.74(4) x=0.07 5 –a2.42(2) x=0.15 1 .5 3.75(4) 3.38(3) x=0.2 5 3.48(4) -a aMagnetic moment per Mn atom for this phase is below the detection limit of our neutron diffraction experiment. from the analysis of neutron diffraction data for x=0.15 sample is close to that of the results obtained from themagnetization study. A coexistence of two magnetic phases,i.e. AFM and FM, was reported in Cu 1−xPdxMnSb28and Pd2MnSn xln1−x26Heusler alloys series, but no quantitative analysis of magnetic phases and their evolutions as a functionof temperature or increasing atomic substitution were studied.Our study gives a microscopic understanding of the AFM-to-FM phase transition and the variation of the two phases asa function of Ni concentration in the Cu 1−xNixMnSb series. Moreover, a magnetic phase diagram in the temperature andNi-concentration plane is proposed here. C. Neutron depolarization study The dc magnetization and neutron diffraction experiments indicate the appearance of FM clusters/domain in the AFMmatrix with increasing Ni content in CuMnSb. To studysuch type of magnetic inhomogenities (FM clusters/domainsin the AFM matrix) on a mesoscopic length scale, neutrondepolarization is a powerful technique. We have carried outthe one-dimensional neutron depolarization study for thex=0.03, 0.05, 0.07, and 0.15 samples. Typically, the neutron depolarization results for various kinds of magnetic materialsare as follows. 29–33In the case of an unsaturated ferromagnetic or ferrimagnetic material, the magnetic domains exert a dipolarﬁeld on the neutron polarization and depolarize the neutronsdue to Larmor pression of the neutron spins in the magneticﬁeld of domains. In pure antiferromagnetic materials, thereis no net magnetization, hence no depolarization occurs. Ina paramagnetic material, the neutron polarization is unableto follow the variation in the magnetic ﬁeld as the temporalspin ﬂuctuation is too fast (10 −12s or faster). Hence, no de- polarization is observed. However, depolarization is expectedin the case of clusters of spins with net magnetization. 21,31–33 Thus, neutron depolarization technique is a mesoscopic probe which detects the magnetic inhomogenities in the length scaleof 100 ˚A to several microns. Figure 7shows the temperature dependence of the transmitted neutron beam polarization P for an applied ﬁeld of 50 Oe applied parallel to the incidentneutron beam polarization. For x=0.03 sample, there is no change in value of P(shown in Fig. 7as well as in the inset), which indicates that the sample is antiferromagneticin nature. For the x=0.05 sample, shown in Fig. 7as well as in the inset, there is a slight decrease in the value of Pat T<70 K, indicating the presence of small ferromagnetic-like clusters in the antiferromagnetic matrix below ∼70 K. For further increase in Ni concentration, i.e. for x=0.07 and 0.15 samples, Pshows a continuous decrease from ∼76 and∼173 K, respectively. Here, Pattains constant values below ∼50 K, indicating that there is no further growth of the domains at lower temperatures. The temperature at which the value ofPstarts decreasing can be considered as the ferromagnetic transition temperature T C. The neutron beam polarization in a depolarization experiment can be represented by the followingexpression 29,34: P=Piexp/bracketleftbigg −α/parenleftbiggd δ/parenrightbigg /angbracketleft/Phi1δ/angbracketright2/bracketrightbigg , (1) where PiandPare the initial and transmitted neutron beam polarization, αis a dimensionless parameter ≈1/3, dis effective sample thickness, δis the average domain size, and/Phi1δ=(4.63 ×10−10G−1˚A−2)λδBthe precession angle. Here, λis the wavelength of the neutron beam and B(=4πMSρ,MSbeing spontaneous magnetization and ρ density of the material) is the average magnetic inductionof a domain/cluster. The above equation is valid only whenthe precession angle /Phi1 δis a small fraction of 2 πover a typical domain/cluster length. The increasing observedneutron beam depolarization at temperatures below T Cwith increasing the xindicates the presence of larger ferromagnetic domains/clusters consistent with neutron diffraction and dcmagnetization experiments. It may be noted here that theArrott plots for the x=0.05 and 0.07 compounds [Fig. 3(c)] do not yield any spontaneous magnetization. However, wehave observed neutron depolarization for these samples. Thisinteresting observation indicates the dynamics of the FMclusters in these compositions. Generally, neutron polarizationvector senses ﬂuctuating magnetic ﬁelds averaged over a timescale of the order of the Larmor precession time, which isof the order of 10 −8seconds for 1 kG magnetic induction (B) of the domain. So if the ﬂuctuation time of these FM clusters/domains is larger than the Larmor precession time, onewould get neutron depolarization from such clusters/domains.The dc magnetization measurements, on the other hand, aretime averaged (over several seconds) measurements resultingin a zero spontaneous magnetization over the experimentaltime scale. Neutron depolarization study indicates that withincrease in the Ni concentration in CuMnSb, i.e. fromx=0.05 to 0.07, the ferromagnetic-like clusters with net magnetic moment (in the antiferromagnetic matrix) increasein size. Further increase in the Ni concentration drives thesystem towards an FM state. This is consistent with theresults obtained from dc magnetization and neutron diffractionexperiments. The magnetic behavior in Cu 1−xNixMnSb semi-Heusler alloys can be interpreted in terms of a delicate balance between 094435-6CROSSOVER FROM ANTIFERROMAGNETIC TO ... PHYSICAL REVIEW B 84, 094435 (2011) 0120000240000 01200002400000120000240000 04000080000120000 0.5 1.0 1.5 2.0 2.5 3.004000080000120000(b) 50 K (c) 250 K* Neutron Counts (arb. units)1.5 K x = 0.15(a) (d) 1.5 K - 250 K (111)FM (200)FM (220)FM(311)AFM (331)AFM (531)AFM(220)FM(200)FM(111)FM(e) Q (Å-1)50 K - 250 K 1234567030006000Neutron Counts (arb. units) Q (Å-1)300 Kx = 0.15(f) 0 50 100 150 200 2500.60.70.80.91.01.1 (111) (200) (220)Integrated Intensity (arb. units)Temperature (K)x = 0.15 (g) FIG. 5. (Color online) Neutron diffraction patterns for x=0.15 sample at various temperatures. The open circles show the observedpatterns. The solid lines represent the Rietveld reﬁned patterns. The difference between observed and calculated patterns is also shown at the bottom of each panel by solid lines. The vertical bars indicatethe position of allowed Bragg peaks (top for chemical and bottom for magnetic phases). Asterisks mark the additional AFM Bragg peaks. (d) Difference pattern obtained by subtraction of the neutrondiffraction pattern at 250 K from the neutron diffraction pattern at 1.5 K, which shows both AFM and FM contributions. (e) Difference pattern obtained by subtraction of the neutron diffraction pattern at250 K from the neutron diffraction pattern at 50 K, which shows the FM contribution. (f) Neutron diffraction pattern for x=0.15 sample at 300 K. (g) Variation of intensities with temperature for (111), (200),and (220) Bragg peaks.0.00 0.05 0.10 0.15 0.20 0.25020406080100Volume Phase Fraction (%) x (Ni concentration) FM(magnetization) AFM(magetization) FM(neutron) AFM(neutron) AFM FM FIG. 6. (Color online) Variation of the AFM and the FM phases with change in the Ni concentration at ∼5 K. Open squares and open triangles represent the FM and AFM phase fractions derivedfrom magnetization data, respectively. ( +)a n d( ×) represent the FM and AFM phase fractions derived from neutron diffraction data, respectively. the two competitive exchange interactions, i.e. ferromagnetic RKKY-type exchange and antiferromagnetic superexchangeinteraction. Here, we observe that the FM state appears inAFM CuMnSb with small substitution of Ni at the Cu site (i.e.x=0.05). Our results suggest an electronic phase separation in the 0.05 /lessorequalslantx/lessorequalslant0.2 region. The quenched disorder in the Cu/Ni sublattice causes a disorder in the orientation of the spins at theMn sublattice. 18As we increase the Ni concentration, more and more spins align parallel, i.e. the FM clusters grow in size, andﬁnally the system becomes completely ferromagnetic. This isevident from the dc magnetization, neutron diffraction, andneutron depolarization studies. Based on the results of ourexperimental studies, we propose a magnetic phase diagramof the present Cu 1−xNixMnSb ( x=0 to 1) semi-Heusler alloys series (shown in Fig. 8). The values of the Curie temperature for some of the samples of the series are taken from Ref. 19. The N ´eel temperature for the x =0.05, 0.07, and 0.15 samples and the Curie temperature for the x =0.2 sample have been obtained from our dc magnetization data, while the Curietemperature for the x =0.07 and 0.15 samples have been obtained from our neutron depolarization data. We observe 0 50 100 150 200 2500.30.40.50.60.70.80.91.0 03 0 6 0 9 0 1 2 00.9820.9840.986x = 0.03 x = 0.07 x = 0.03 x = 0.05 x = 0.07 x = 0.15 P Temperature (K)H = 50 Oe x = 0.15x = 0.05 P Temperature (K)x = 0.05x = 0.03 FIG. 7. (Color online) Temperature dependence of the transmit- ted neutron beam polarization Pat an applied ﬁeld of 50 Oe for x=0.03, 0.05, 0.07, and 0.15 samples. Inset enlarges the temperature dependence of polarization Pforx=0.03 and 0.05 samples. 094435-7HALDER, YUSUF, KUMAR, NIGAM, AND KELLER PHYSICAL REVIEW B 84, 094435 (2011) 0.1 1110100FMTransition Temperature (K) x (Ni concentration )PM FMAFMAFM + FM FIG. 8. Magnetic phase diagram of the Cu 1−xNixMnSb Heusler alloys series from x=0.03 to 1. Solid circles denote the Curie temperature for the series taken from Ref. 19. Hollow squares and hollow triangles denote the Curie temperature and N ´eel temperature, respectively, of the samples from the present work. that only a narrow region of x(/lessorequalslant0.05) has the pure AFM phase, and in the region 0.05 /lessorequalslantx/lessorequalslant0.2, with decrease in temperature, there is a transition from the PM to FM state, and below ∼50 K, both AFM and FM phases coexist. With increase in xfurther, most of the phase diagram is dominated by the FM phase.Similar results were observed in Pd 2MnSn xIn1−xby Khoi et al. ,26which is an example of bond randomness transition in a Heisenberg system, where nuclear magnetic resonance(NMR) data showed the coexistence of both AFM and FMdomains. Their results showed that the intermediate regionhad an inhomogeneous structure with two different typesof coexisting order (AFM and FM) separated in space. Thetheoretical phase diagram of such quenched random alloys,where one end is FM and other AFM, shows that the twophases are either separated by a mixed phase or by a ﬁrst-orderphase line. 35Our results suggest that crossover from AFM to FM transition in Cu 1−xNixMnSb series is separated by a mixed phase. A possible reason could be due to the long-rangenature of the exchange interaction (RKKY-type) present in thesystem. The transition from the AFM state to FM state occurscontinuously with increase in the Ni content, and no abrupt change is observed. The electronic structure calculation showsthat, with increase in the Ni substitution, the ferromagneticRKKY-type exchange increases due to an increase in spinpolarization of the conduction electrons at the Fermi level, andthe superexchange interaction decreases as the Fermi energymoves away from the unoccupied Mn 3d density of states. 18 As a result, there is a crossover from the AFM to FM state inthe Cu 1−xNixMnSb with increase in Ni concentration. IV . SUMMARY AND CONCLUSIONS Here, we have investigated the Cu 1−xNixMnSb series in the region x<0.3 to bring out the microscopic nature of the AFM- to-FM transition by dc magnetization, neutron diffraction, and neutron depolarization techniques. We observe that the FMstate appears in AFM CuMnSb with small substitution of Niat the Cu site (i.e. 5%). We ﬁnd that below x=0.05 the system is mainly in the AFM state. In the region 0.05 /lessorequalslantx/lessorequalslant0.2, with decrease in temperature, there is a transition from the PM toFM state, and below ∼50 K, both AFM and FM phases coexist. Above x=0.2, the system is mainly in the FM state. The FM state can be viewed as some of the antiferromagneticallyaligned Mn spins change their orientation and align parallelto each other, forming an FM-like cluster in the AFM matrix.These clusters grow in size with increase in Ni content as moreand more spins align parallel and ﬁnally drives the systemto the FM state. The results are consistent with the reportedelectronic band structure calculation. The results of the presentinvestigation show a path in tuning magnetic and electronicproperties of different Heusler and semi-Heusler alloys forvarious practical applications and can be used to fabricatematerials with desired physical properties. 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PhysRevB.103.144425.pdf	PHYSICAL REVIEW B 103, 144425 (2021) Large anomalous Hall angle accompanying the sign change of anomalous Hall conductance in the topological half-Heusler compound HoPtBi Jie Chen ,1,2Xing Xu,1Hang Li ,2Tengyu Guo,1Bei Ding,2Peng Chen,1Hongwei Zhang,2 Xuekui Xi,1,2and Wenhong Wang1,2,* 1Songshan lake Materials Laboratory, Dongguan, Guangdong 523808, China 2State Key Laboratory for Magnetism, Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Received 7 February 2021; revised 1 April 2021; accepted 2 April 2021; published 19 April 2021) Controlling the anomalous Hall effect (AHE) in magnetic topological materials is an important property. Because of the close relationship between anomalous Hall conductance (AHC) and topological band (strongBerry curvature), AHC can be effectively tuned by magnetic ﬁeld combined with strong spin-orbit interactionand special band structure. In this work, we observed a magnetic ﬁeld driving the nonmonotonic magneticﬁeld dependence of anomalous Hall resistivity and the sign change in magnetic-ﬁeld-induced Weyl semimetalHoPtBi. The tunable ranges of the AHC and the anomalous Hall angle are −75∼73/Omega1 –1cm–1and−12.3∼ 9.1%, respectively. Anisotropic measurements identiﬁed the magnetic ﬁeld is the key factor in controlling the additional Hall term sign. Further analysis indicated that it originated from the ﬁeld-induced shift of the Weylpoints via exchange splitting of bands near the Fermi level. The large tunable effect of the magnetic ﬁeld on theelectronic band structure provides a path to tune the topological properties in this system. These ﬁndings suggestthat HoPtBi is a good platform for tuning the Berry phase and AHC with the magnetic ﬁeld. DOI: 10.1103/PhysRevB.103.144425 I. INTRODUCTION Ternary half-Heusler compounds crystallize in the cubic MgAgAs-type crystal structure. They constitute a large familyof materials characterized by various physical and chemicalproperties [ 1,2]. Half-Heusler compounds have been inten- sively studied in the last decades regarding, superconductivity[3–5], giant magnetoresistance [ 6,7], heavy fermions [ 8,9], half-metals [ 10,11], magnetocaloric effect [ 12], and Seebeck effect [ 13], because they possess unique multifunctionality that can be easily tuned by small modiﬁcations in their com-position, morphology or some external factors. Some pioneering theoretical works [ 14–16] have recently suggested that rare-earth ( R)-based half-Heusler compounds (RTX), where Tis ad-electron transition metal (Ni,Pd,Pt, …), and Xis a pnictogen (Sb, Bi), are promising candidates for topological materials, exhibiting various unconventionalphysical phenomena, e.g., topological insulators/semimetals[17,18], topological superconductivity [ 3,19], and spin-wave excitations [ 20], topologically coupled with the electromag- netic ﬁeld. All those rare features may emerge in RTX phases because of strong Rashba-type spin-orbit interactions, re-sulting from the lack of inversion symmetry in crystallinesystems. The nontrivial topological nature of selected RTX compounds has been theoretically predicted using electronicband structure calculations and, after that, experimentally con-ﬁrmed [ 21,22]. *wenhong.wang@iphy.ac.cnAmong these topological candidates, the magnetic Weyl semimetals attracted growing attention since their large Berrycurvature resulted in a strong intrinsic anomalous Hall ef-fect (AHE) when the Weyl nodes were close to the Fermilevel [ 23]. This scenario was well-studied in magnetic topo- logical materials, like Co 3Sn2S2[24,25], Co 2MnGa [ 26], and Fe 3GeTe 2[27], in which the band crossing points and nodal lines with a topological order host strong Berry curva-ture [ 23,28–30]. Therefore, the position of topological bands strongly affects the value and the sign of the anomalous Hallconductance (AHC). In our previous works, we observedthe negative magnetoresistance (MR) and extracted the π Berry phase in the rare-earth-based half-Heusler compounds,HoPtBi, which identiﬁed the chiral-anomaly-induced negativelongitude magnetoresistance and conﬁrmed the presence ofWeyl fermions [ 31]. This paper presents a systematic analysis of the experimental results of the temperature- and crystalorientation-dependent AHE in HoPtBi single crystals. Ourresults reveal a large anisotropic AHE in HoPtBi with thetunable anomalous Hall angles (AHAs) of −12.3 to 9.1%, depending on the geometric conﬁgurations of the appliedmagnetic ﬁeld ( B) and electronic current ( I). This feature is ascribed to the tunable position of Weyl points relative to theFermi level, which may induce the inversion of Berry cur-vature. However, more studies in this direction are certainlynecessary. II. EXPERIMENTAL SECTION HoPtBi single crystals, with a typical size of 1 ×1×1 mm, were grown by a solution growth method using a Bi ﬂux [ 7]. 2469-9950/2021/103(14)/144425(8) 144425-1 ©2021 American Physical SocietyJIE CHEN et al. PHYSICAL REVIEW B 103, 144425 (2021) FIG. 1. (a) The crystal structure of HoPtBi with the space group F-43m (216). (b) Brillouin zone and (c) band structure of HoPtBi along the X- /Gamma1-X-W- /Gamma1-W-K- /Gamma1-K-L-/Gamma1-L high symmetry line. (d) Single crystal XRD pattern of the (111) plane. The inset is the typical shape of the HoPtBi single crystal. (e) The thermal magnetization of HoPtBi from 2 to 300 K at an external magnetic ﬁeld B=0.05 T. The inset represents the ﬁtting of the whole paramagnetic region of the thermal magnetization curve by Curie-Weiss law: the extension of the ﬁtting curve intersects the negative xaxis, AFM exchange interaction is dominant. (f) Temperature dependence of resistivity curves for sample #1 with current I// [1–10]. The crystal structure was identiﬁed using the x-ray diffraction (XRD) method with Cu- Kαradiation. The transport and mag- netization properties were measured in the physical propertiesmeasurement system (PPMS, 9T). The samples were polishedinto a rectangle shape with a thickness lower than 0.1 mmfor transport measurement. A six-probe method was appliedto simultaneously measure the magnetoresistance and Hallsignals. The misalignment of the electrode was removed bysymmetrizing the data between negative and positive mag-netic ﬁelds. The electronic band structures were calculatedusing the WIEN 2Kcode, based on the framework of density functional theory [ 32]. The Perdew-Burke-Ernzerhof gener- alized gradient approximation [ 33] was used to calculate exchange correlation potentials. We set the cutoff energy of−6.0 Ry, deﬁning the separation of the valence and core states. Due to heavy elements, we included spin-orbit cou-pling (SOC) in the calculation. A large exchange parameter, U eff=0.6 Ry, was applied to Ho. III. RESULTS Figure S1 shows a uniform distribution of Ho, Pt, Bi, indicating high-quality HoPtBi single crystals grown by theBi-ﬂux method [ 34]. HoPtBi exhibits a cubic MgAgAs-type crystal structure, Fig. 1(a), with a crystal lattice parameter a=6.6344 Å [ 7]. The corresponding Brillouin zone is shown in Fig. 1(b). The high-symmetry lines /Gamma1-Land/Gamma1-Xrepresent [111] and [100] crystal planes, respectively. Since the bandscross the Fermi level around the /Gamma1point, Fig. 1(c)shows the band structure along different high-symmetry lines crossingthe/Gamma1point with ferromagnetic states and spin-orbit coupling. The semimetallic band structure is similar to that of other half-Heusler compounds, like GdPtBi [ 17,35], TbPtBi [ 22], and YbPtBi [ 18]. The theoretical calculation of GdPtBi suggested that the direction of magnetic moment signiﬁcantly inﬂuencedthe number and position of Weyl points [ 35]. The magnetic ﬁeld can easily change the symmetry of the magnetic structurein a paramagnetic material. In theory, besides the strong spin-orbit coupling, the magnetic ﬁeld can effectively tune the bandstructure and topological states. Therefore, the tunable effectcould be observed in the HoPtBi compound. The orientation of a single crystal is determined based on the corresponding XRD pattern. As shown in Fig. 1(c), two peaks indicate the (111) crystal plane of sample #1. The inset represents a photograph of the HoPtBi single crystalwith the (111) plane making triangle (red dashed line). Thethermal magnetization curves are measured from 300 to 2 Kunder 0.05 T, Fig. 1(d), indicating paramagnetic states in the whole Trange. HoPtBi host an antiferromagnetic state inT<T N=1.2K [ 36]. The solid line in the inset is the ﬁtting curve of Curie-Weiss law. The effective magnetic mo-mentμ eff=10.2μBis closed to the theoretical magnetic moment of μHo3+=10.6μB. Figure 1(f)shows the temper- ature dependence of ρxxfor sample #1 with 0 T and I//[110]. The transition of resistivity ρxxfrom semiconductor behav- ior to metallic behavior and small activation energy 18 meV 144425-2LARGE ANOMALOUS HALL ANGLE ACCOMPANYING … PHYSICAL REVIEW B 103, 144425 (2021) FIG. 2. AHE for sample #1. Magnetoresisitance (a) and Hall resistivity (b) for sample #1. The inset of (a) is the conﬁguration of the magnetic ﬁeld B//[111] and current I//[1–10]. (c) Magnetization as a function of Bat different Twith B//[111]. (d) The extracting progress of the anomalous Hall resistivity at 2 K. The lower ﬁgure shows Hall resistivity at 2 and 20 K. The upper one shows the additional term /Delta1ρA xyat 2 K. The total curve can be divided into three regions I, II, and III. The additional term /Delta1ρA xyat 2 K was obtained by subtracting the linear Hall resistivity at 20 K. (e) The additional term /Delta1ρA xyfor sample #1 at T<20 K. (Fig. S4) indicates HoPtBi hosts a small gap, which is coin- cide with band structure. One of our main ﬁnding related to HoPtBi is a large non- monotonic Hall effect. The anomalous behavior in Fig. 2is explained in sample #1. MR and Hall resistivity of sample #1are shown in Figs. 2(a)and2(b), respectively. Compared to B-linear dependence of the normal Hall effect, ρ xyof sample #1 shows an unconventional behavior. The curves deviatefrom a linear behavior, forming a swell at a high magneticﬁeld, similar to the AHE of TbPtBi [ 37]. This swell gradually diminishes and shifts to a high magnetic ﬁeld with the tem-perature increase, disappearing entirely at T=20 K and B= 9 T. The ﬁtting of the normal Hall resistivity indicates thatsample #1 shows very high mobility of 5059 cm 2V–1s–1,e x - ceeding the maximum value of GdPtBi (1 ×103cm2V–1s–1) [38] and TbPtBi (2 .2×103cm2V–1s–1)[39] single crystals. The high quality of the synthesized single crystal also re-ﬂects on the magnetoresistance effect. The MR value reaches1031% at 2 K and 9 T. We also note the complex MR behaviorat a low magnetic ﬁeld, contributing to quantum coherence(weak antilocalization effect and weak localization effect) [ 7]. Figure 2(c)shows the magnetic ﬁeld dependence of magne-tization at low TandB//[111]. The Brillouin-function-like behavior induces the magnetization curves saturate at a highmagnetic ﬁeld reaching 60 emu/g at 9 T. In general, Hallresistivity, ρ xy, in magnetic materials can be expressed as follows: ρxy=RHB+ρA xy, (1) ρA xy=RSM, (2) where RHand Rsare ordinary and anomalous Hall coef- ﬁcients. ρA xyis anomalous Hall resistivity proportional to magnetization in conventional ferromagnetic or ferrimagneticmaterials. Apparently, ρ xyof sample #1 is not proportional to magnetization. An additional term, /Delta1ρA xy, with a non- monotonic dependence on M, occurs. Therefore, ρxycan be expressed as follows: ρxy=RHB+ρA xy+/Delta1ρA xy. (3) Since the curves at B<0.6 T and T<20 K almost overlap and the additional term disappears completely at 20 K.Therefore, the additional Hall resistivity /Delta1ρ A xycan be obtained 144425-3JIE CHEN et al. PHYSICAL REVIEW B 103, 144425 (2021) FIG. 3. Hall resistivity ρxyand magnetoresistance ρxxat different temperatures for three samples with different conﬁgurations of BandI. (a) and (d) are the magnetoresistance ρxxand Hall resistivity ρxy, respectively, with B//[110] and I//[1–10]. The inset of (b) shows the Hall resistivity at a low magnetic ﬁeld. (c) and (d) are magnetoresistance ρxxand Hall resistivity ρxyfor sample #3, respectively, with B//[001] and I//[1–10]. (e) and (f) are the results for sample #4 with B//[001] and I//[100]. by subtracting ρxy(20 K). Figure 2(d) shows the extracted progress of /Delta1ρA xy. The lower ﬁgure is Hall resistivity at 2 and 20 K, and the upper ﬁgure is the additional Hall resistivity/Delta1ρ A xyat 2 K. According to the /Delta1ρA xyvalue, the whole range can be divided into three parts I, II, and III. In part I ( B<0.6 T), ρxy(2 K) and ρxy(20 K) show B-linear dependence, indicating that the normal Hall effect dominates the Hall signal. Hence,the extracted /Delta1ρ A xyis almost zero for the overlapping ρxy(2 K) andρxy(20 K) curves. In part II (0.6 T <B<2.1 T), the ρxy(2 K) curve deviates from the linear behavior and forms a small positive swell. The critical magnetic ﬁeld of AHEisB c=0.6 T, which is signiﬁcantly lower than Bc=7T f o r TbPtBi [ 37]. Large magnetic moment and smaller lattice pa- rameter maybe the main factor that HoPtBi have a smallercritical magnetic ﬁeld. Because the formation of Weyl pointsin magnetic RPtBi compounds was attributed to an externalmagnetic-ﬁeld-induced Zeeman splitting [ 35]. The magnetic moment of Ho 3+(10.2μB) is larger than Tb3+(9.57μB) and smaller lattice constant may lead to RPtBi need smallerZeeman energy to form the band crossing (Weyl points). AtB=2.1T , t h e /Delta1ρ A xydecreases to almost zero. At this point, the additional signal /Delta1ρA xyis missing, which is a phenomenon for AHE. In part III (2. 1 T <B), another large positive swell appears and extends to ﬁelds larger than B=9 T. The peak of /Delta1ρA xy(2 K) reaches 0.7037 m /Omega1cm, which is larger than that of GdPtBi ( /Delta1ρA xy=0.18 m /Omega1cm) [ 38] and TbPtBi ( /Delta1ρA xy= 0.6798 m /Omega1cm) [ 37]. The extracted /Delta1ρA xyfor sample #1 is shown in Fig. 2(e). With the temperature increase, /Delta1ρA xygrad- ually reduces and completely disappears at 20 K, and both ofswells are shifting to high magnetic ﬁeld. Next, we measured in detail the anisotropic transport prop- erties of this compound. Magnetic ﬁeld ( B) and current ( I),as two main external tunable factors, have a great inﬂuence on magnetotransport properties, especially for the Hall effect[40–44]. To clarify the tuning effect of BandI, we designed a series of experiments with different BandIconﬁgurations. Besides sample #1, samples #2 and #3 keep the directionof current ( I//[1–10]) unchanged and rotate Bfrom [111] to [110] and [001]. When Bturns to the [110] direction (sample #2),ρ xx[Fig. 3(a)] and ρxy[Fig. 3(a)] show curves similar to sample #1, having lower MR values and smaller swells.However, there is a completely opposite additional Hall sig-nal when Bturns to [001]. Figure 3(d) shows the negative nonmonotonic magnetic ﬁeld dependence of Hall signals at alow magnetic ﬁeld, and the peaks shift to a high magnetic ﬁeldwith temperature increase. The anomalous Hall signal persists up to 50 K and then reverts to a normal Hall effect. Because the current is ﬁxed ( I//[1–10]), only the magnetic ﬁeld is changing. We can reasonably deduce that the external mag-netic ﬁeld can effectively tune this nonmonotonic magneticﬁeld dependence of AHE. To identify this fact undoubtedly,we pick another sample with B//[001] and I//[100] (sample #4). Comparing to sample #3, Bis ﬁxed, but Ichanges from [1–10] to [100]. The negative nonmonotonic magnetic ﬁeld dependence of the Hall signal is also observed. It furtheridentiﬁes that the magnetic ﬁeld but not the current is the keyfactor of the sign change. The sign change of AHE is anothermain ﬁnding in HoPtBi. To explore the origin of the anomalous Hall resistivity and the sign change, we discuss the possible origins of /Delta1ρ A xyfur- ther. First, /Delta1ρA xy, a nonmonotonic magnetic ﬁeld dependence, reminds us that a spin texture, such as frustrate magnets andmagnetic skyrmion lattices, can provide a ﬁctitious magneticﬁeld and induce a nonproportional Hall effect (topological 144425-4LARGE ANOMALOUS HALL ANGLE ACCOMPANYING … PHYSICAL REVIEW B 103, 144425 (2021) FIG. 4. The extracting progress of the anomalous Hall resistivity for sample #4. (a) Hall resistivity for sample #4 in the 2 −50 K range. The dashed lines are a guide for the eye. (b) The anomalous Hall resistivity for sample #4 at T=2 K. (c) The anomalous Hall resistivity contains two terms, ρA xy, and additional term, /Delta1ρA xy,T<20 K. (d) The additional term /Delta1ρA xyfor sample #4. Hall effect) during the magnetization progress [ 45–49]. In generally, the topological Hall effect usually observed in cer- tain special magnetic materials, such as magnetic skyrmions.However, as shown in Figs. 1(e) and2, HoPtBi is param- agnetic state in 2 −300 K, and topological Hall effect will vanish when magnetization tends toward saturation. It is not coincident with the result in sample #1. Another evidence of excluding topological Hall effect as the origin is that the signof the topological Hall effect would be determined by the signof the normal Hall effect [ 50]. Thus, the sign change of the additional Hall signal in HoPtBi could hardly be reconciled with the constant carrier sign. As shown in Figs. 2and3,t h e slope of the normal Hall resistivities retains a positive signas the magnetic ﬁeld changes from [111] to [110] and [001].Therefore, the sign change of the additional Hall signal depen- dent on the Bdirection cannot be explained by the mechanism of the topological Hall effect. The AHE in other rare-earth-based half-Heusler compounds was studied and attributed tothe Berry phase of electronic band [ 18,21,22,35,38,39]. As previously explained, the magnetic ﬁeld can easily change the symmetry of magnetic structure in paramagnet. The magnetic ordering can tune electronic structure through the strong SOC.Combined with the experimental results that the sign dependson the Bdirection, the /Delta1ρ A xysign change should contribute to the change of topological band. We need to point out that the AHE sign change is observed in this family of materials. Italso indicates that HoPtBi possesses a more special structurecomparing to GdPtBi, TbPtBi, and YbPtBi. The large changeof band structure of HoPtBi is enough to reverse the total Berry curvature. The Hall resistivities of samples #3 and #4 are different from that of samples #1 and #2. In samples #3 and #4, thetermρ A xyin Eq. ( 3), which is proportional to the magnetization, cannot be neglected. Here, we take sample #4 as an example toseparate the unconventional Hall effect. In Fig. 4(a),t h eH a l l resistivity forms a negative swell at a medium magnetic ﬁeld.With temperature increase, the swell peaks shift toward highmagnetic ﬁelds. The additional Hall term persists to 50 K at 9T, which is much higher than that of samples #1 and #2. TheFig.4(b) shows the Hall contribution of ρ A xyand/Delta1ρA xyafter subtracting the normal Hall ρN xyat 2 K for sample #4. The dark line represents the contribution of ρA xy+/Delta1ρA xyand both are negative. The blue line is magnetization Mﬁtting to ρA xy. The red line is the additional Hall resistivity, /Delta1ρA xy, mainly distributed in the 0.6 T <B<3 T region. The critical mag- netic ﬁeld Bcis the same as that of sample #1, identifying that HoPtBi needs a small external magnetic ﬁeld to form the Weylstates. The /Delta1ρ A xyoscillation in high magnetic ﬁeld originates from the quantum oscillation which has been reported in ourprevious result [ 31]. Figure 3(d) shows the anomalous Hall resistivity at T<20 K for sample #4. IV . DISCUSSIONS In Fig. 5, we show the AHC, /Delta1σA xyand the corresponding AHA as a function of T.T h e /Delta1σA xyand AHA are deﬁned 144425-5JIE CHEN et al. PHYSICAL REVIEW B 103, 144425 (2021) FIG. 5. (a) The additional Hall conductance /Delta1σA xyand (b) the corresponding AHA for samples #1 – #4 at 2 K. All data are obtained from the peak of /Delta1ρA xy.II represents the data gotten from the II region (low magnetic ﬁeld), and III represents the data from the III region (high magnetic ﬁeld). (c) The sketches of the mechanism of the AHC and the sign change for HoPtBi. The upper panel: in the absence of an external magnetic ﬁeld, AHC tends to zero. The lower panel: under various magnetic ﬁeld directions, the tformed Weyl points shift around EFand induced the AHC sign change. as follows: /Delta1σA xy=−/Delta1ρA xy/(ρ2 xy+ρ2 xx) and AHA =100× /Delta1σA xy/σxx, respectively. II and III represent that peaks come from the low magnetic ﬁeld and high magnetic ﬁeld regions,respectively. /Delta1σ A xyfor samples #1 and #2 shows negative values and decreases with the temperature. However, for sam-ples #3 and #4, /Delta1σ A xyshows positive values and remains almost constant with the temperature. /Delta1σA xyfor samples #1- II, #2-II, and samples #3, #4 at 2 K are in the range of 60–80 /Omega1–1cm–1, which is comparable to other half-Heusler compounds [ 35,38]. In fact, Pavlosiuk et al. [36] have ob- served the anomalous Hall effect in single crystals of HoPtBi.However, the sign change of anomalous Hall resistivity in HoPtBi observed in the current work is rare. On the one hand, we should note that the total Berry curvature change becauseof the Fermi level ( E F) shifting when the temperature changed from room temperature to low temperatures induced the AHE sign change in SrRuO 3[29]. On the other hand, La doping of a magnetic oxides EuTiO 3ﬁlm also showed an AHE sign change since the electron doping shifted the Fermi level EF [42]. All these indicate the sign change is closely related to the position of the topological band around EF. The giant magneto-band-structure effect whereby a change of magnetization direction signiﬁcantly modiﬁes the elec-tronic band structure need several criteria, such as magneti-cally coupled electrons, strong SOC, and a reduced symmetryto maximize SOC anisotropy [ 51]. Therefore, it is challenging to achieve the AHE sign change in conventional materials us-ing the magnetic ﬁeld, B, as another factor that may inﬂuence the band structure. RPtBi, a rare-earth-based half-Heusler compound with heavy atoms and absence of inversion centers,was proved to be a magnetic-ﬁeld-induced Weyl semimetal [17,22,35,52]. Besides, a simple band structure and the cross- ing point close to E Fmake the RPtBi family an ideal platform to control of the Berry phase and AHE by tuning bands witha magnetic ﬁeld. In Fig. S5, we show the band structures ofHoPtBi with magnetic moment parallel to [001], [110], and[111] directions, respectively [ 34]. The shift crossing point between M// [001] and M// [110], [111] identiﬁed the large tunable effect. The chiral anomaly effect and nontrivial Berryphase extracted by Shubnikov–de Haas oscillations in our pre-vious study identiﬁed the HoPtBi topological properties [ 31]. To describe tunable effect and the relationship between the change of the topological band and the sign change of theAHC clearly, we present a simpliﬁed schematic diagram inFig.5(c). In the absence of magnetic ﬁeld [antiferromagnetic (AFM) states], the AHC is ﬁxed to zero. Because 4 felectrons host strong SOC and high electronic tunability, the magneticﬁeld can tune the topological response via rare-earth atoms[35,38]. When the magnetic ﬁeld is stronger than B c,t h e shifting bands induced by the exchange splitting formthe Weyl points around E F. The crossing points shift around theEFbecause of the anisotropic tunable effect of magnetic ﬁeld on the bands, which further leads to the AHC signchange. V . CONCLUSIONS In summary, we observed that a magnetic ﬁeld induced a large anomalous Hall angle and the sign change inmagnetic-ﬁeld-induced Weyl semimetal HoPtBi. The Hallmeasurements of four samples with different conﬁgurations 144425-6LARGE ANOMALOUS HALL ANGLE ACCOMPANYING … PHYSICAL REVIEW B 103, 144425 (2021) demonstrated that the magnetic ﬁeld was the key factor to control the sign of the additional anomalous Hall term. Thetunable AHC and AHA ranges reach −75–73 /Omega1 –1cm–1and −12.3–9.1%, respectively. Based on the understanding of RPtBi half-Heusler compounds, the Berry phase mechanismwas considered as the origin of the nonmonotonic magneticﬁeld dependence of AHE. The results suggested that the signchange of AHE could be ascribed to special band structureand a large tunable effect of external magnetic ﬁeld on Weylnodes relative to E F. Accordingly, the total Berry curvature ﬂipped, and corresponding AHC changed the sign during thisprogress. The large tunable effect of the magnetic ﬁeld on theelectronic band structures provides a feasible pathway to tunethe topological properties in this system. These results suggest that HoPtBi is a good platform for tuning the Berry phase andAHE with a magnetic ﬁeld. ACKNOWLEDGMENTS This work was supported by the National Science Foun- dation of China (Grants No. 11974406 and No. 12074415),the Strategic Priority Research Program (B) of the Chi-nese Academy of Sciences (CAS) (XDB33000000), andChina Postdoctoral Science Foundation (Grant No. 3662020M680734). [1] K. Manna, Y. Sun, L. Muechler, J. Kübler, and C. Felser, Nat. Rev. Mater. 3, 244 (2018) . [2] F. Casper, T. Graf, S. Chadov, B. Balke, and C. Felser, Semicond. Sci. Technol. 27, 063001 (2012) . [3] H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak, P. Syers, L. Wang, H. Hodovanets, J. D. Denlinger, P. M. R. Brydon,D. F. Agterberg, M. A. Tanatar, R. Prozorov, and J. Paglione,Sci. Adv. 4, eaao4513 (2018) . [4] Y. Nakajima, R. Hu, K. Kirshenbaum, A. Hughes, P. Syers, X. Wang, K. Wang, R. Wang, S. R. Saha, D. Pratt, J. W. Lynn, andJ. Paglione, Sci. Adv. 1, e1500242 (2015) . [5] G. Xu, W. Wang, X. Zhang, Y. Du, E. Liu, S. Wang, G. Wu, Z. Liu, and X. X. Zhang, Sci. Rep. 4, 5709 (2014) . [6] Z. 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PhysRevB.91.085312.pdf	PHYSICAL REVIEW B 91, 085312 (2015) Two-electron n-pdouble quantum dots in carbon nanotubes E. N. Osika and B. Szafran AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Krak ´ow, Poland (Received 28 November 2014; revised manuscript received 12 February 2015; published 27 February 2015) We consider electron states in n-pdouble quantum dots deﬁned in a semiconducting carbon nanotube (CNT) by an external potential. We describe formation of extended single-electron orbitals originating from the conductionand valence bands conﬁned in a minimum and a maximum of the external potential, respectively. We solve theproblem of a conﬁned electron pair using an exact diagonalization method within the tight-binding approach,which allows for a straightforward treatment of the conduction- and valence-band states, keeping an exactaccount for the intervalley scattering mediated by the atomic defects and the electron-electron interaction.The exchange interaction, which in the unipolar double dots is nearly independent of the axial magnetic ﬁeld(B) and forms singletlike and tripletlike states, in the n-psystem appears only for selected states and narrow intervals of B. In particular, the ground-state energy level of a n-pdouble dot is not split by the exchange interaction and remains fourfold degenerate at zero magnetic ﬁeld also for a strong tunnel coupling between thedots. DOI: 10.1103/PhysRevB.91.085312 PACS number(s): 73 .21.La,73.63.Fg I. INTRODUCTION Due to the absence of the hyperﬁne interaction the graphene-based [ 1] materials are an attractive medium for spin control and manipulation. In semiconductor carbonnanotubes [ 2,3] formation of the energy gap prevents the Klein tunneling [ 4] and allows for conﬁnement of charge carriers in quantum dots formed by external voltages. The transportspectroscopy [ 5–7] experiments resolve the signatures of the spin-orbit coupling that appears [ 8–15] with folding of the graphene plane into a nanotube. The spin-orbit interaction in-duces formation of spin-valley states [ 8–15] through coupling of the orbital magnetic moments with spin. The effects ofspin and valley dynamics are monitored in the electric dipolespin-valley resonance experiments [ 16–18] by lifting the valley and/or spin blockade [ 19] of the current ﬂow through a pair of quantum dots connected in series. The pair of quantum dots conﬁning localized electron spins [ 20] is the basic element of the quantum information processing circuitry. The effective spin exchange interaction that splits the singlet and triplet energy levels is a necessary prerequisite for construction of a universal quantum gate [ 20]. In single [ 6,21–23] and double CNT quantum dots [ 16,24–26] the coupling of the spin and valley degrees of freedom results in formation of singletlike and tripletlike states of the electron pair. These states are no longer spin eigenstates but they stillpossess a deﬁnite symmetry of the spatial wave function withrespect to the electron interchange. The graphene is an ambipolar material and the external potentials easily sweep the conduction- and valence-band extrema above or below the Fermi energy [ 16,27]. In the spin-valley resonance experiments [ 16,17] the double quantum dot is set in a n-pconﬁguration for which the Pauli blockade is most pronounced. In the present paper we describe formationof electron orbitals extended over the n-pdouble quantum dot. Next, we study the spin-valley structure of the two-electron states with a single electron in the fourfold degenerateconﬁned state per dot (see Fig. 1), which in experimental papers [ 16,17] is addressed as (3h,1e): a charge conﬁguration with three holes in one quantum dot and a single electronin the other. The two-electron system in the n-pdouble dot is usually considered similar [ 16] to the electron-pair in the n-ndouble dots [ 24–26]. Here we demonstrate that the electronic structure of the double-dot n-psystem differs in a few elementary aspects: (i) the energy level splitting bythe spin-exchange interaction is missing in the two-electronground-state which is fourfold degenerate also when the tunnelcoupling between the dots is strong, (ii) the splitting resultingfrom the exchange energy is found only in the excited partof the spectrum and for a limited range of magnetic ﬁelds,and (iii) formation of singletlike and tripletlike spatial orbitalsappears only within avoided crossings induced by the externalelectric or magnetic ﬁelds. We indicate that these featuresresult from an opposite electron circulation in the conductanceand valence bands for a given valley. Formation of extended orbitals in the n-pdouble dots in presence of the spin-orbit coupling introduces a dependence of the electron distributionon the spin and valley, which produces a ﬁne structure ofthe two-electron spectrum at low B. For completeness we include a brief tight-binding analysis of the n-nsystem, which has been considered in the continuum approximation inRefs. [ 24,25]. The present study is based on the exact diagonalization approach using the tight-binding method that allows for aconsistent description of conduction- and valence-band states, intervalley mixing due to the atomic disorder, and the short- range component of the Coulomb interaction [ 28] and does not require an additional parametrization. The intervalleyscattering due to the electron-electron interaction is usuallyneglected by effective mass theories [ 29]. The tight binding ap- proach at the conﬁguration-interaction level [ 30,31] accounts for all the intervalley scattering processes which result from the electron-electron interaction, including the backward and umklapp scattering [ 28]. Inclusion of the intervalley scattering effects to the low-energy theories is possible [ 32,33]b u t far from straightforward. Finally, the tight-binding approachaccounts for even large modulation of the external potentialdeﬁning the ambipolar quantum dots, which is not necessarilythe case for the low-energy continuum approximations. 1098-0121/2015/91(8)/085312(13) 085312-1 ©2015 American Physical SocietyE. N. OSIKA AND B. SZAFRAN PHYSICAL REVIEW B 91, 085312 (2015) (a) (b) (c) FIG. 1. (Color online) (a) Schematics of the considered system. The external magnetic ﬁeld is oriented along the axis zof the zigzag CNT of radius Rand length L. The inset explains the angles used for the deﬁnition of the interatom hopping elements of the tight-binding Hamiltonian in presence of the spin-orbit interaction. We study the system of a double n-ndot (b) or n-pdot (c) induced by external voltages. The discussed states correspond to a single electron per dot: occupying one of the fourfold degenerate conﬁned energy levels. The green line indicates the Fermi energy. In the considered low-energystates the electrons occupy mostly separated quantum dots. II. SINGLE-ELECTRON STATES: THEORY We consider a semiconducting nanotube of length Land radius R[see Fig. 1(a)]. Most of the results are obtained for a zigzag CNT of length L=53.1 nm with 20 atoms along the circumference (diameter 2 R=1.56 nm). The properties of the low-energy two-electron states in the double dots asdetermined for the zigzag CNT [ C h=(20,0)] are reproduced for any semiconducting CNT. For demonstration we providebelow (Sec. VF) also the results for C h=(20,6) CNT chirality (see Fig. 2). We use the tight-binding Hamiltonian of the form H=/summationdisplay {i,j,σ,σ/prime}/parenleftbig c† iσtσσ/prime ijcjσ/prime+H.c./parenrightbig +/summationdisplay i,σ,σ/primec† iσ/bracketleftBig WQD(ri)+gμb 2σσσ/prime·B/bracketrightBig ciσ/prime, (1) where the ﬁrst summation runs over pzspin orbitals of nearest neighbor pair of atoms, c† iσ(ciσ) is the particle creation (annihilation) operator at ion iwith spin σinzdirection, and tσσ/prime ijis the hopping parameter. The second summation in Eq. ( 1) accounts for the external potential and the Zeeman interaction.In Eq. ( 1)g=2 is the Land `e factor and σdenotes the vector of Pauli matrices. The external magnetic ﬁeld B=(0,0,B)i s applied along the axis of the CNT. The energy gap of the considered CNTs allows for electrostatic conﬁnement of the carriers. The quantum dotFIG. 2. (Color online) Schematics of the CNT folding for the chiral vector Ch=(20,0) (a zigzag CNT) and Ch=(20,6) which are considered in this work. conﬁnement is induced by external potentials modeled by a sum of Gaussian functions: WQD(r)=Vlexp(−(z+zs)2/d2)+Vrexp(−(z−zs)2/d2), (2) where zsis the shift of the dots from the center of the CNT (z=0) and VlandVrare potentials of the left and the right dot, respectively. The paper is focused on the states with a single electron per quantum dot [cf. Figs. 1(b) and 1(c)]. For separated electrons the details of the single-dot potential are of secondaryimportance for the qualitative properties of the system as longas the tunnel coupling between the dots is present. Mostof the discussion is carried for small quantum dots with2d=4.4 nm, with the shift between their centers 2 z s=10 nm. For these small quantum dots the single-electron energy levelspacing is large ( /similarequal100 meV), which is useful for analysis of the properties of the exchange interaction, since a limitednumber of multiplets contribute to the two-electron wavefunctions. Nevertheless, the single-particle level spacings inCNT quantum dots is of the order of a few meV , up to 10 meV atmost [ 7,8]. In order to demonstrate that the identiﬁed properties of the n-psystem are qualitatively independent of the size of the dots we provide in Sec. VEalso the results for larger QDs. The hopping parameters t σσ/prime ijbetween the nearest-neighbor spin orbitals—including the curvature induced spin-orbit cou-pling [ 9,12,13]—are introduced in the following form [ 9,13]: t↑↑ ij=Vπ ppcos(θi−θj) −/parenleftbig Vσ pp−Vπ pp/parenrightbigR2 a2 C[cos(θi−θj)−1]2 +2iδ/braceleftbigg Vπ ppsin(θi−θj) +/parenleftbig Vσ pp−Vπ pp/parenrightbigR2 a2 Csin(θi−θj)[1−cos(θi−θj)]/bracerightbigg =t↓↓ ij∗, (3) t↑↓ ij=−δ(e−iθj+e−iθi)/parenleftbig Vσ pp−Vπ pp/parenrightbigRZji a2 C[cos(θi−θj)−1] =−t↓↑ ij∗, (4) where Vπ pp=− 2.66 eV , Vσ pp=6.38 eV [ 34]aC=0.142 nm is the nearest-neighbor distance, θiindicates the localization angle of atom iin the ( x,y) plane [see the inset to Fig. 1(a)], 085312-2TWO-ELECTRON n-pDOUBLE QUANTUM DOTS IN . . . PHYSICAL REVIEW B 91, 085312 (2015) andZji=Zj−Ziis the distance between atoms iandj along the CNT axis. The SO coupling parameter is taken δ= 0.003 [ 9,13] unless explicitly stated otherwise. Orbital effects of the external magnetic ﬁeld are introduced by Peierls phase shifts tσσ/prime ij→tσσ/prime ijei2π(e/h)/integraltextrj riA·dl.We apply the Landau gauge A=(0,Bx, 0). In the following we refer to electron currents circulating along the circumference of the nanotube. In the tight-bindingmodel the operator of the probability current [ 35]ﬂ o w i n g along the πbonds between k-th and l-th neighbor ion spin orbitals is given by the formula J σσ/prime kl=i /planckover2pi1/parenleftbig c† kσtσσ/prime klclσ/prime−H.c./parenrightbig , (5) which accounts for the spin-precession due to the spin-orbit interaction. In the following discussion we refer to the domi-nating, i.e., the spin-conserving components of the current. III. SINGLE-ELECTRON STATES: RESULTS A. Separate nand pquantum dots Figure 3shows the energy spectrum for a single external Gaussian potential introduced as a minimum [ n-type quantum dot, Fig. 3(a) forVl<0,Vr=0] or a maximum [ p-type quantum dot, Fig. 3(b) forVl=0,Vr>0] inside the carbon nanotube. The energy levels plotted in grey (red) correspond tostates localized inside the n- [Fig. 3(a)]o rp-type quantum dot [Fig. 3(b)]. With the external potential that is introduced to the CNT, the energy spectrum is no longer symmetric with respectto the zero energy. The spectrum for the n-type dot [Fig. 3(a)] with the localized states evolving from the conduction bandis opposite to the spectrum for the pdot [Fig. 3(b)] with the localized states that evolve from the valence band. All thelocalized energy levels are nearly fourfold degenerate withrespect to the valley and spin—the SO coupling energy /Delta1 SO is below the resolution of this plot. Figure 4shows the calculated energy spectrum as a function of the external magnetic ﬁeld for the single-electronstates localized inside the n-type [Fig. 4(a)] andp-type dots -400-300-200-1000100200300400 0 0.2 0.4 0.6 0.8 1E [meV] V [eV]-400-300-200-1000100200300400 0 0.2 0.4 0.6 0.8 1E [meV] V [eV](a) (b) FIG. 3. (Color online) Energy spectrum for a CNT with a local potential minimum (a) and maximum (b) introduced by an externalpotential as functions of the depth (a) and height (b) of the Gaussian potential well (a) and barrier (b). With grey lines (red) we plotted the energy levels that correspond to electron localization inside theGaussian (within the central segment of length 2 d) by at least 50%.z[ n m ]-20 0 20 -20 0 200 0 0 0 z[ n m ](a) (b)(c) (d) FIG. 4. (Color online) Energy levels for the single-electron states localized inside a single separate n-quantum dot (a) or p-quantum dot (b) as function of the external magnetic ﬁeld for V=± 0.42 eV . The energy levels are labeled by valley K/K/prime,s p i n↑↓,a n dlandrdenote the left and right dots. In (c) [(d)] we plotted the circumferential component of the electron current calculated at y=0 for the lowest (highest) energy states of the conduction (valence) bands for V=0 in the absence of the spin-orbit coupling. [Fig. 4(b)]f o rV=± 0.42 eV . In the n-type dot for B=0 one ﬁnds a Kramers doublet ( K/prime↑,K↓) ground-state split by the spin-orbit interaction from higher-energy doublet ( K/prime↓, K↑)[36]. The spin-orbit splitting of the energy levels of Fig. 4is/Delta1SO=1.55 meV. The degenerate KandK/primestates have an opposite orientation of the current circulation aroundthe axis of the nanotube [ 38]. For illustration we plotted the circumferential component of the current calculated [ 35]f o r y=0 and V l=Vr=0 in the lowest state of the conduction band. The conduction-band low-energy K/primestates that we deal with produce orbital magnetic moment which is oriented inthezdirection, i.e., parallel to the external magnetic ﬁeld. The electron circulation in the Kstates of conduction band is opposite [Fig. 4(c)]. Formation of the degenerate pairs of spin- valley energy levels ( K /prime↑,K↓) and ( K/prime↓,K↑) results from the curvature-induced spin-orbit coupling [ 9–11,13,14]. For the electrons localized inside the p-type dot [Fig. 4(b)]—ﬁlling the states of the valence band—the spin-orbit coupling pro-duces a lower-energy doublet ( K↑,K /prime↓) and a higher-energy one (K↓,K/prime↑). The orbital moments for a given valley are opposite in the states of conduction and valence bands [ 38]— cf. the calculated electron current orientation in Figs. 4(c) and 4(d)—thus in the lower-energy Kramers doublets of the p- andn-type dots the valleys are interchanged. As we discuss below, this fact has a pronounced inﬂuence on the propertiesof the two-electron states for the n-pdouble quantum dots. B. Double quantum dots Figure 5(a) shows the energy spectrum for a double unipolar n-nquantum dot in the (1e,1e) charge conﬁguration as a function of the depth of the Gaussian quantum dots. For 085312-3E. N. OSIKA AND B. SZAFRAN PHYSICAL REVIEW B 91, 085312 (2015) -400-300-200-1000100200300400 0 0.2 0.4 0.6 0.8 1E [meV] V [eV]-400-300-200-1000100200300400 0 0.2 0.4 0.6 0.8 1E [meV] V [eV](a) (b) FIG. 5. (Color online) Energy levels for a system of n-n(a) and p-pdouble dots (b) as a function of the depth (height) of the Gaussian quantum dots (antidots). comparison in Fig. 5(b) the energy spectrum for the p-p dot in the (3h,3h) charge states is shown. For the doublen-n[Fig. 5(a)] andp-pdots [Fig. 5(b)] we observe that the energy levels move in pairs with V. The pairs correspond to bonding and antibonding orbitals extended over both thequantum dots [ 25]. Each energy level within the pair is nearly fourfold degenerate with respect to the valley and the spin. Theenergy splitting between bonding and antibonding orbital /Delta1 ba is a few times larger than the spin-orbit splitting /Delta1SObetween Kramers doublets within each of the orbitals (i.e., for V=0.55 eV for the lowest localized n-nstates /Delta1SO≈1.4 meV and /Delta1ba≈7.5m e V ) . For the n-pdouble quantum dot [Fig. 6(b)] the energy levels originating from the conduction and valence bandsmove symmetrically with respect to the neutrality point. Theextended orbitals are only formed when the energies of thestates localized in the n- andp-type dots are close to each other. Figure 6(c) shows the charge density near the anticrossing of the localized energy levels from the n-type and p-type dots. The anticrossing indicates a presence of a tunnel couplingbetween the two quantum dots and a lack of any hiddensymmetry difference between the states of conduction andvalence bands. The orbitals in the n-psystem change their character from ionic to extended as functions of the potential depth-heightwith a 50%-50% distribution at the center of the avoidedcrossing. Note that the avoided crossings for each of theKramers doublets is shifted one with respect to the other alongtheVscale. C. Single-electron wave functions For the discussion of the two-electron interaction matrix elements, it is useful to look at the form of the single-electronwave functions. For illustration (Fig. 7) we consider a single n-type quantum dot and the K /prime↑state, i.e., the lowest-energy quantum-dot-conﬁned state for B> 0 (the center of the n-type quantum dot is set at z=0). Figure 7(a) shows the real part of the spin-up component with a rapid variation of the wavefunction from ion to ion (blue and red colors correspond toopposite signs). The spatial variation of the wave functions in-400-300-200-1000100200300400 0 0.2 0.4 0.6 0.8 1E [meV] V [eV] 00.010.020.03 00.010.020.03 -20 -10 0 10 20 z [nm]-10010 0.43 0.44 0.45 E [meV] V[ e V ](a) (c)(b) FIG. 6. (Color online) (a) Energy spectrum for a n-pdouble dot as a function of the depth or height of the Gaussian potential −Vl= Vr=V. (b) Zoom at the avoided crossing of valence- and conduction- band states near the neutrality point. (c) Charge densities integratedalong the circumference of the CNT for V=0.42 eV . The rapid oscillation results from contributions of A and B sublattices which are both smooth but shifted one with respect to the other. the nanotube can be put in an approximate form, /Psi1K(/prime)↑=exp(iK(/prime)·r+iκ(/prime)Rθ)u(r), (6) 01.534.5 -1 0 1R[nm] z [nm]-1 0 1 z [nm]-1 0 1 z [nm](a) (b) (c) FIG. 7. (Color online) The lowest-energy conﬁned K/prime↑state in a single n-type quantum dot. A short fragment of the nanotube within the dot is considered. (a) Real part of the majority spin componentof/Psi1 K/prime↑wave functions. The values that are plotted in red and blue correspond to positive and negative values, respectively. [(b) and (c)] Real part of the envelope u(r)[ s e eE q .( 6)] on sublattices A and B, respectively. 085312-4TWO-ELECTRON n-pDOUBLE QUANTUM DOTS IN . . . PHYSICAL REVIEW B 91, 085312 (2015) where uis an envelope function [ 14] and for the zigzgag nanotube with 20 atoms along the circumference we have κ/prime= (m−1/3)/RforK/primevalley ( κ=(m+1/3)/RforKvalley), where K/prime=(2π/a)(−1/3,1/√ 3),K=(2π/a)(1/3,1/√ 3), mis an integer, and a=0.246 nm. The nonzero value of κ(/prime)accounts for the amount that the wave vector satisfying the periodic boundary conditions misses the exact valleyposition [ 38]. The lowest-energy conﬁned states correspond to m=0 and Figs. 7(b) and 7(c) show the real part of the envelope function u(r), i.e., the wave function /Psi1 K/prime↑(r) upon extraction of the rapidly varying valley factor exp( iK/prime·r+iκ/primeRθ). The envelope u(r) is a smooth function separately on each of the nanotube sublattices A [Fig. 7(b)] and B [Fig. 7(c)]. We ﬁnd that in the weak magnetic ﬁeld and in the absence ofthe spin-orbit coupling, the envelope uis valley independent. In presence of the spin-orbit coupling the envelope functionfor the majority spin component is nearly the same for allthe four lowest-energy states independent of the spin-valleyquantum numbers. Some subtle differences can only beresolved for the avoided crossings of the valence- and theconduction-band states [see Fig. 6(b) and the discussion below in Sec. VC]. Generally, for the majority spin components of the four low-energy states, we have an approximate relation/Psi1 K=/Psi1K/primefKK/prime, with the fKK/primefactor rapidly varying in space that transforms the wave functions of K/primeintoKvalley, fKK/prime=exp(i(K−K/prime)·r+i(κ−κ/prime)Rθ). IV . TWO-ELECTRON STATES: THE METHOD For the two-electron system we work with the energy operator including the electron-electron interaction, H2e=/summationdisplay a/epsilon1ag† aga+1 2/summationdisplay abcdVab;cdg† ag† bgcgd, (7) where g† ais the electron creation operator in the eigenstate a of the single-electron Hamiltonian, /epsilon1ais the single-electron energy level, and Vab;cdare the Coulomb matrix elements. The Coulomb matrix elements are integrated in the real and spinspace, as V ab;cd=/angbracketleftψa(r1,σ1)ψb(r2,σ2)\|Hc\|ψc(r1,σ1)ψd(r2,σ2)/angbracketright, (8) according to the formula Vab;cd=/summationdisplay i,σi;j,σj;k,σk;l,σlαa∗ i,σiαb∗ j,σjαc k,σkαd l,σlδσi;σkδσj;σl ×/angbracketleftbig pi z(r1)pj z(r2)\|HC\|pk z(r1)pl z(r2)/angbracketrightbig , (9) where αa i,σiis the contribution of pi zorbital of spin σito the single-electron eigenstate aandHcis the Coulomb electron- electron interaction potential HC=e2 4π/epsilon1/epsilon1 0r12(10) withr12=\|r1−r2\|. We adopt the silicon dioxide dielectric constant /epsilon1=4 as for the gated CNT coated in glass [ 39]. For calculation of the interaction matrix elements over theatomic orbitals we use the two-center approximation [ 40]:/angbracketleftp i zpj z\|1 r12\|pk zpl z/angbracketright=1 rijδikδjlfori/negationslash=j. For the on-site inte- gral (i=j)w et a k e /angbracketleftpi zpj z\|HC\|pi zpj z/angbracketright=16.522 eV (after Ref. [ 30]). In the following for the n-n(p-p) system we set Vl=Vr= −0.55 eV ( +0.55 eV) and for the n-psystem Vr=−Vl= 0.42 eV , unless stated otherwise. We consider charging the energy levels which are the closest to the neutrality point. Forthen-ndouble dot in the summation over ain Hamiltonian ( 7) we include 8 energy levels of the bonding-antibonding pair,which correspond to the energy of /similarequal− 100 meV at the vertical green line in Fig. 4(a) and additionally a number of higher- energy levels (the number necessary for convergence dependson the size of the dot). For the n-pdouble dot we consider the pair of energy levels of the avoided crossing marked bythe green rectangle of Fig. 6(a) at the avoided crossing of the conduction and valence bands and a number of higher-energylevels. We assume that all the energy levels below are ﬁlled by electrons. The higher-energy single-electron states introduce additional Slater determinants to the conﬁguration-interactionbasis. Their contribution for the short quantum dots (2 d= 4.4 nm) is small, and reliable results are obtained already for bases including eight single-electron lowest-energy levelsonly. However, a signiﬁcant—also qualitative—contributionof higher multiplets is present for larger quantum dots (2 d= 30 nm) that are considered in Sec. VE. Section VE includes also the discussion of the convergence of the results. V . TWO ELECTRON STATES: RESULTS In Fig. 8we plotted the energy levels for the n-pdouble dot forVr=0.42 eV as a function of Vl=−V. The ground state of the system in a wide range of Vlcorresponds to (1,1) electron distribution over the dots (or (1e,3h) according to notation ofRef. [ 16,17]). The system goes to the (0,2) ([0e,2h]) charge conﬁguration at V l=− 0.15 eV and to (2,0) ([2e,0h]) at Vl= −0.7 eV . The (1,1) energy level is nearly 16-fold degenerate, while the (2,0) and (0,2) levels are 6-fold degenerate. The 16 lowest-energy two-electron states in the n-p,n-n, andp-pdouble dots are displayed in Figs. 9(a),9(b), and 9(c), respectively. The electrons in the 16 lowest-energy statesoccupy different dots [see the inset for (1,1) state in Fig. 8]; for the clarity of the discussion it is useful to consider thebasis of single-electron states conﬁned mostly in the left orright quantum dot. The n-pdouble quantum dot is essentially asymmetric and the single-electron wave functions exhibit adominant localization in one of the dots [see Fig. 6(c)]. We denote the states localized in the left and right dots as landr, respectively. The adopted external potential of the n-ndouble quantum dot is symmetric and the electron occupation of boththe dots is 50%-50% in both the bonding and antibondingstates. In this case the landrwave functions can be constructed by a sum and a difference of the bonding and antibonding wavefunctions. The 16 lowest-energy two-electron levels at B=0 can be divided into three groups (see Figs. 9and10). The contributing basis elements for each of the groups are listed in Table I.T h e four lowest-energy conﬁgurations that form the lowest energylevels at B=0o fF i g s . 9(a)–9(c), is addressed as group “1” in Fig. 9,F i g . 10, and Table I. In this group the electron in each of 085312-5E. N. OSIKA AND B. SZAFRAN PHYSICAL REVIEW B 91, 085312 (2015) -10 0 10 z[nm]-10010z[nm] 0-10 0 10 z[nm]-10 0 10-10010z[nm] z[nm]0 -2000200400600 0.1 0.2 0.3 0.4 0.5 0.6 0.7E [meV] V[ e V ] FIG. 8. (Color online) Energy spectrum for the electron pair in n-psystem as a function of the depth V=−Vlof the n-dot. The p-dot potential is constant and set to Vr=0.42 eV . In the insets: probability densities as functions of coordinates z1andz2(integrated over the CNT circumference) for both electrons. There are 16 states for the electron distribution (1,1) and 6 states for conﬁgurations (2,0) and (0,2).(a) (b) FIG. 10. (Color online) Schematics of the two-electron systems considered in this paper in a double dot (a) and in the n-pdouble dot (b). The ﬁlled (empty) circles correspond to occupied (unoccupied)single-electron orbitals. Valleys and spins of the single-electron energy levels split by the spin-orbit interaction are displayed. The arrows with labels 1, 2, 3 correspond to the dominant contributionsto the two-electron energy levels that are discussed below. In (b) the localized states in the p-dot originate from the valence band, and only one of four accessible energy levels is occupied—the conﬁgurationcorresponds to (1e,3h) charge state of the n-pdouble quantum dot. -125-124-123-122-121-120 0 1 2 3 4 5 6E [meV] B[ T ]} }} 198199200201202 0 1 2 3 4 5 6E [meV] B[ T ]} }}(b) (c)36373839404142 0 1 2 3 4 5 6E [meV] B[ T ]} }}(a) FIG. 9. (Color online) Energy spectrum for the electron pair in the n-psystem (a), a double n-dot (b), and double p-dot (c) as a function of the magnetic ﬁeld B. With the red (blue) color we plotted the energy levels of spin polarized up (down). The green levels correspond to states ofSz=0. The integers 1, 2, and 3 number the group of energy levels. The single-electron energy levels which contribute to these groups are explained in Fig. 10. At the left of the plot we list the dominant conﬁgurations that are found for a nonzero magnetic ﬁeld (see the gray vertical belts). In (b) and (c) we added labels S and T for singletlike and tripletlike states of spatial wave functions: symmetric and antisymmetric with respect to the electron interchange, respectively (see text). In the avoided crossings opened by the exchange interaction in (a) we denote the approximate form of the wave function as expressed with the Slater determinants fjthat are listed in Table I. Parameters of the system: distance 2zs=10 nm, (a) Vl=−Vr=− 0.42 eV , (b) Vl=Vr=− 0.55 eV , (c) Vl=Vr=0.55 eV . 085312-6TWO-ELECTRON n-pDOUBLE QUANTUM DOTS IN . . . PHYSICAL REVIEW B 91, 085312 (2015) TABLE I. 16 lowest-energy Slater determinants basis elements for the n-ndouble dot ( ei)a n dn-pdouble dot ( fi) with electrons occupying separate quantum dots. Ais the antisymmetrization operator with normalization factor 1 /√ 2,l(r) denote the state localized in the left (right) quantum dot, and (1), (2) denote the coordinates of the ﬁrst and second electron, respectively. The numbers in the second column indicate the group of energy levels thedeterminant contribute to—see Fig. 9and10. i Group n-ndotei n-pdotfi 11 A(l↑ K/prime(1)r↑ K/prime(2)) A(l↑ K/prime(1)r↑ K(2)) 21 A(l↓ K(1)r↓ K(2)) A(l↓ K(1)r↓ K/prime(2)) 31 A(l↓ K(1)r↑ K/prime(2)) A(l↓ K(1)r↑ K(2)) 41 A(l↑ K/prime(1)r↓ K(2)) A(l↑ K/prime(1)r↓ K/prime(2)) 52 A(l↑ K/prime(1)r↑ K(2)) A(l↑ K/prime(1)r↑ K/prime(2)) 62 A(l↑ K/prime(1)r↓ K/prime(2)) A(l↑ K/prime(1)r↓ K(2)) 72 A(l↓ K(1)r↑ K(2)) A(l↓ K(1)r↑ K/prime(2)) 82 A(l↓ K(1)r↓ K/prime(2)) A(l↓ K(1)r↓ K(2)) 92 A(l↑ K(1)r↑ K/prime(2)) A(l↑ K(1)r↑ K(2)) 10 2 A(l↑ K(1)r↓ K(2)) A(l↑ K(1)r↓ K/prime(2)) 11 2 A(l↓ K/prime(1)r↓ K(2)) A(l↓ K/prime(1)r↓ K/prime(2)) 12 2 A(l↓ K/prime(1)r↑ K/prime(2)) A(l↓ K/prime(1)r↑ K(2)) 13 3 A(l↑ K(1)r↑ K(2)) A(l↑ K(1)r↑ K/prime(2)) 14 3 A(l↓ K/prime(1)r↓ K/prime(2)) A(l↓ K/prime(1)r↓ K(2)) 15 3 A(l↓ K/prime(1)r↑ K(2)) A(l↓ K/prime(1)r↑ K/prime(2)) 16 3 A(l↑ K(1)r↓ K/prime(2)) A(l↑ K(1)r↓ K(2)) the dots occupies one of the twofold degenerate single-particle ground states (see Fig. 10). In Fig. 9at the left-hand side of the plots we specify the dominant Slater determinant in the energy order that corre-sponds to the gray belt marked in the Figs. 9(a)–9(c). We use the notation of Table Ionly with skipped antisymmetrization symbol. The dominant Slater determinants for the two-electronstates in the n-nandp-psystems differ by the inversion of valley indices ( K↔K /prime). All the systems, including the n-p dot, have an overall similar spin structure (see Szvalue as marked by colors in Fig. 9). The plots contain the lowest 16 energy levels for the (1,1) electron conﬁguration. In then-nandp-pspectra there are 6 pairs of energy level of the same component of the spin along the zdirection which move parallel in B. The corresponding states differ in the symmetry of the two-electron spatial envelope which is either symmetricor antisymmetric with respect to the electron interchange,forming the singletlike and tripletlike states [ 16,17,24,25]. The energy difference between energy levels of each couple isdetermined by the exchange energy, which remains essentiallyunchanged by B. The corresponding pairs of energy levels for the n-psystem are nearly degenerate [Fig. 9(a)]. The n-nandp-psystems [Figs. 9(b) and 9(c)]a tB=0h a v e a nondegenerate ground state and a threefold degenerateexcited state—as in the single-triplet structure of III-V doubledots [ 20,24,25]. On the other hand, for the n-pdouble dot [Fig. 9(a)] we ﬁnd a fourfold degenerate ground state which indicates a vanishing exchange energy.A. Exchange energy in the n-nsystem The spin-orbit coupling in CNTs changes the energies of the states depending on the relative orientation of the spin andangular momentum and introduces only a small contribution ofthe minority spin to the eigenstates. Therefore, in the followinganalysis we refer to the majority spin component only. Let usconsider e 1,e2,e3, ande4basis elements of Table Iforming the lowest energy group of energy levels denoted by (1) inFig. 9(b) and Fig. 10. For the spin polarized e 1ande2basis elements the spin-valley degree of freedom is separable fromthe spatial envelope, which is tripletlike, i.e., antisymmetricwith respect to the electron interchange e 1=1√ 2(l(1)r(2)−r(1)l(2))K/prime↑(1)K/prime↑(2) (11) and /angbracketlefte1\|HC\|e1/angbracketright=/angbracketlefte2\|HC\|e2/angbracketright=C+X, (12) where Cis the Coulomb integral, C=/angbracketleftl(1)r(2)\|HC\|l(1)r(2)/angbracketright, (13) andX> 0 is the exchange integral, X=− /angbracketleftl(1)r(2)\|HC\|r(1)l(2)/angbracketright. (14) The singletlike energy levels are shifted down on the energy scale with respect to the tripletlike energy levels by theexchange energy (2 X) which is nearly independent of the magnetic ﬁeld [see Fig. 9(b)]. The interaction integrals for the parameters of Fig. 9areC=38.75 meV for the Coulomb and 2X=0.22 meV for the exchange energy. In the two-electron basis e 3ande4with zero spin component in thezdirection ( Sz=0) one cannot separate the spin-valley from the spatial coordinates in a similar manner. The Coulombinteraction mixes the e 3ande4conﬁgurations. The diagonal interaction element for the third and fourth basis elements are /angbracketlefte3\|HC\|e3/angbracketright=/angbracketleftl↓ K(1)r↑ K/prime(2)\|HC\|l↓ K(1)r↑ K/prime(2)/angbracketright =C=/angbracketlefte4\|HC\|e4/angbracketright, (15) and the nondiagonal /angbracketlefte3\|HC\|e4/angbracketright=− /angbracketleft l↓ K(1)r↑ K/prime(2)\|HC\|r↓ K(1)l↑ K/prime(2)/angbracketright =− /angbracketleftl(1)r(2)\|HC\|r(1)l(2)/angbracketright=X. (16) As a result, we have a 2 ×2 Hamiltonian matrix, HXC=/parenleftbigg CX XC/parenrightbigg , (17) with the energy eigenvalue C−Xfor the singletlike ground states34=e3−e4andC+Xfor the excited tripletlike eigenstate t34=e3+e4. The latter is degenerate with e1and e2. The singletlike ground-state wave function is of the form s34=1 2(lK↓(1)rK/prime↑(2)−rK/prime↑(1)lK↓(2) −lK/prime↑(1)rK↓(2)+rK↓(1)lK/prime↑). (18) Upon replacement K=K/primefKK/prime, one obtains s34=K/prime(1)K/prime(2)[l(1)r(2)+r(1)l(2)][fKK/prime(1)↓(1)↑(2) −↑(1)↓(2)fKK/prime(2)], (19) 085312-7E. N. OSIKA AND B. SZAFRAN PHYSICAL REVIEW B 91, 085312 (2015) and, similarly, t34=K/prime(1)K/prime(2)[l(1)r(2)−r(1)l(2)][fKK/prime(1)↓(1)↑(2) +↑(1)↓(2)fKK/prime(2)]. (20) Ins34andt34states the spin and valley are nonseparable due to the presence of the intervalley scattering term fKK/primein the spin part of the formulas. Nevertheless, the spatial wave functionseparates from the spin-valley and has a deﬁnite symmetrywith respect to the electron interchange: symmetric for s 34 (singletlike state) and antisymmetric for t34(tripletlike state); see the ﬁrst bracket in Eqs. ( 19) and ( 20). For the two-electron states of the other two groups of energy levels (“2” and “3” in Table I) the mixing of basis elements by the electron-electron interaction occurs in a similar manner. Inthe spectrum one ﬁnds six pairs of two-electron energy levelsthat preserve their energy spacing by 2 XwhenBis varied. B. The n-psystem The lowest-energy group of the two-electron energy levels “1” (f1,f2,f3, and f4in Table I) corresponds to each of electrons occupying the single-electron ground state in one ofthe dots (cf. Fig. 10). The spin-polarized elements f 1andf2 separate from the rest of the group as in the n-ndouble dot. For the n-pdouble dot the lowest-energy states of the left and right dot of the same spin correspond to opposite valleys [seeFig. 10(b) ]. Using the f KK/primeintervalley scattering function, f1 can be written as f1=K/prime↑(1)K/prime↑(2)(l(1)r(2)fKK/prime(2)−r(1)fKK/prime(1)l(2)). (21) The interaction energy for this state is approximately equal to the Coulomb integral /angbracketleftf1\|HC\|f2/angbracketright=C, since the exchange in- tegral /angbracketleft(l(1)r(2)fKK/prime(2)\|HC\|r(1)fKK/prime(1)l(2)/angbracketrightinvolves valley scattering for each of the electrons and thus it is negligiblysmall [ 22]. For the same reason the off-diagonal matrix ele- ment/angbracketleftf 3\|HC\|f4/angbracketrightvanishes, with the diagonal matrix elements equal to C. We are thus left with the fourfold degeneracy of the ground state as in Fig. 9(a). In none of the four lowest-energy eigenstates one can separate the spatial part of the spin-valleypart and, in consequence, no singletlike or tripletlike states interms of the spatial envelope are formed. In Fig. 9(a) one ﬁnds two avoided crossings: one at B=0 for the S z=0 states (green curves) and another below 4T for the spin-up polarized states (blue curves). The avoidedcrossing near 3.5T involves the f 2state (group “1”) and f14(group “3”—see Table I). Both these basis elements have the same ( K↓,K/prime↓) spin-valley conﬁguration. The energy level corresponding to f14(f2) decreases (increases) with increasing B, which is consistent with the behavior of the lowest single-electron energy levels of the nand pdots (Fig. 4). The interaction matrix element is then /angbracketleftf2\|HC\|f14/angbracketright=− /angbracketleft l↓ K(1)r↓ K/prime(2)\|Hc\|r↓ K(1)l↓ K/prime(2)/angbracketright=X. Thus the avoided crossings between these energy levels appear as dueto the exchange interaction, which is, for the n-psystem, activated only when the single-electron energies are set equalby the external magnetic ﬁeld. In this sense, the externalmagnetic ﬁeld induces formation of singletlike and tripletlikestates within the avoided crossing of energy levels.For the n-psystem the two-electron energy levels of the central group (2) [near 39 meV at B=0– see Fig. 9(a)] move in pairs with Bas for the n-nsystem, but now the pairs are nearly degenerate and not split by the exchange energy.The eight energy levels of group (2) correspond to an electronin the ground state of one of the dots and an electron inthe excited state of the other dot [see Fig. 10(b) ]. The pair of spin-down basis elements f 8andf11correspond to both electrons in KandK/primevalleys, respectively. For this reason the interaction matrix elements is negligibly small and no avoidedcrossing between the energy levels is observed near B=0. Inf 8andf11the valley and the spin are the same for both electrons and the wave function has a separable form, f11=K/prime(1)K/prime(2)↓(1)↓(2)[l(1)r(2)−l(2)r(1)], (22) and both the spin-down basis elements f8,f11produce tripletlike states. The diagonal interaction matrix element isC+Xfor both these states. The same applies for the spin-up polarized states f 5andf9. The remaining four Sz=0 states of group (2) can be divided into pairs in which the electrons occupy the same combina-tions of spin-valleys: K /prime↑,K↓for (f6,f7) and K/prime↓,K↑for (f10,f12). For each of the pairs the diagonal matrix elements is Cand off-diagonal interaction matrix element is X. We obtain two singletlike states, s67=f6−f7,s10,12=f10−f12,o f interaction energy C−Xwith s67=K/prime(1)K/prime(2)[l(1)r(2)+r(2)l(1)] ×[↑(1)↓(2)fKK/prime(2)−↓(1)fKK/prime(1)↑(2)],(23) and two tripletlike states t67=f6+f7,t10,12=f10+f12with energy C+X.F o rt h e f6basis element one electron occupies the conduction band K/prime↑energy level and the other electron the valence band K↓energy level which both decrease in B; see Fig. 4. The energy for its partner f7—with interchanged bands for a given spin-valley—increases with B.F o rB> 0.5 T the difference of the single-electron energies lifts the effectsof the exchange interaction and the energy levels become linearfunctions of B. For the n-psystem the interaction energies are very similar to the n-ndots with C=38.76 meV and 2 X=0.25 meV— in spite of the difference in \|V l/r\|values. This similarity is characteristic to coupling of small quantum dots only (seeSec. VE). C. Fine structure of the central level group atB=0f o rt h e n-psystem According to the above discussion, in energy level group (2) at B=0 we should have a twofold-degenerate lower energy level of singletlike states and a fourfold-degeneratetripletlike energy level. In fact, we ﬁnd (see a zoom inFig. 11) that the energy levels are additionally split by an energy of /Delta1/similarequal0.06 meV . This splitting is nota result of the single-electron effects—a difference in SO energy splitting inthe valence and conduction bands, for instance. In the presentmodel the SO splitting energy is exactly the same in boththe dots. The ﬁne structure is an interaction-mediated effectof the varied distribution of electrons within the n-psystem. Let us look back at the avoided crossing of conduction- andvalence-band energy levels of Fig. 6(c). The pair of nearly 085312-8TWO-ELECTRON n-pDOUBLE QUANTUM DOTS IN . . . PHYSICAL REVIEW B 91, 085312 (2015) 38.638.83939.2 0 0.2 0.4 0.6E [meV] B [T] FIG. 11. (Color online) A fragment of Fig. 9(a) for the central group of energy levels (number 2). degenerate energy levels of the conduction and valence bands have inverted valley indices. The avoided crossing between theconduction- and valence-band states for K↓,K /prime↑spin-valley conﬁguration appears for a lower value of Vthan for K/prime↓,K↑ states. Exactly at the center of each avoided crossings theelectron distribution within the n-pdot pair is 50%-50%. At V=0.42 eV for K↓andK /prime↑we are closer to the avoided crossing, and we ﬁnd that each of the states of the n-pdot exhibits a slightly increased presence of the probability densitydistribution in the other dot. The difference is small, and sois the value of /Delta1. The energy increase results from a larger electron-electron interaction for K↓,K /prime↑spin valleys because of a less complete electron separation. The avoided crossingbetween the conduction and valence bands is the only casewe encountered when the spatial localization depends on thespin-valley state. The spin-polarized states in Fig. 11correspond to singlet- like and tripletlike spatial symmetry for any B. On the other hand, the S z=0 states acquire a determined spatial symmetry with respect to the electrons interchange only at the center ofthe avoided crossing ( B=0) that is opened by the exchange interaction. D. Atomic disorder and valley mixing effects for the n-pspectrum The results presented so far were obtained for a clean CNT. In order to estimate the effect of the valley mixinginduced by the lattice disorder we removed one carbon atomat a distance of 8.5 nm to the left from the center of thesystem. The results for the two-electron spectrum in then-pdot are displayed in Fig. 12. The valley mixing opens an avoided crossing near 3.5 T for the energy levels thatcrossed near 3.2 T for a clean CNT [Fig. 9(a)]. The crossing energy levels corresponding to states A[lK /prime↓(1)rK↑(2)] and A[lK↓(1)rK↑(2)] differ by the valley index for one of the two electrons. The lattice disorder induces valley mixingand opens an avoided crossing between the correspondingenergy levels of Fig. 12. Outside these avoided crossings the spectrum resembles the one for a clean CNT [Fig. 9(a)]. In particular, the near twofold degeneracy of these energylevels—in which both the electrons occupy different valleys [(f 2,f3),(f5,f6),(f7,f8),(f9,f10),(f14,f15), see Table I]—36373839404142 0 1 2 3 4 5 6E [meV] B [T]} }} FIG. 12. (Color online) Two-electron energy levels for the n-p dot in the presence of the atomic disorder. An atom at a distance of8.5 nm to the left from the center of the system is removed. is preserved also for B/negationslash=0. The fourfold ground-state degeneracy at B=0 is not affected by the atomic disorder. E. Larger quantum dots In the experimental setups the quantum dots deﬁned electro- statically in CNTs are longer, and, in consequence, the single-electron energy level spacings are smaller than in the resultspresented above. For longer quantum dots the contribution ofhigher single-electron spin-orbitals to the two-electron statesare more signiﬁcant and the tunnel coupling for a ﬁxed barrierwidth is reduced along with the conﬁnement energy. In order to verify the conclusions reached for the model of small quantum dots we performed calculations for the lengthof the dots increased from 2 d=4.4n mt o2 d=30 nm, which required dilatation of the nanotube from L=53.1n mt o L= 106.3 nm. The centers of the dots were placed at a distance of 2z s=24 nm. The results for the single-electron spectra are displayed in Figs. 13(a) and 13(b) , with a pronounced reduction of the level spacing as compared to Fig. 5(a) and Fig. 6(a). The results for two-electrons in the n-ndot calculated for V=0.31 eV are displayed in Fig. 13(c) for the basis of 8 (gray dotted lines), 16 (light blue curves), and 24 single-electron functions spanning the conﬁguration-interaction basisof the Slater determinants. For each choice of the basis wedisplay 16 lowest-energy two-electron levels. For 8 basiselements the 6 highest-energy levels (with energy above−137 meV) correspond to the singletlike states which climb up on the energy scale with respect to the tripletlike states.The variational overestimate for the singletlike states is muchlarger than for the 10 tripletlike states. The slower convergenceof the conﬁguration-interaction method for spin-singlets isfound also for III-V quantum dots [ 41] and results from the fact that for the spin triplets the antisymmetry of the spatialwave functions (Pauli exclusion) keeps the electrons away,with the electron-electron correlation at least partly includedin the symmetry of the wave functions. The results for 16 and24 single-electron basis elements are nearly identical, and thespectrum, once the convergence is reached, is qualitativelythe same as the one found for smaller quantum dots [cf. 085312-9E. N. OSIKA AND B. SZAFRAN PHYSICAL REVIEW B 91, 085312 (2015) -300-200-1000100200300 0 0.1 0.2 0.3 0.4E [meV] V [eV]-300-200-1000100200300 0 0.1 0.2 0.3 0.4E [meV] V [eV] -98-96-94-92-90 0 1 2 3 4 5 6E [meV] B[ T ] 1012141618 0 1 2 3 4 5 6E [meV] B[ T ]-150-145-140-135-130 0 1 2 3 4 5 6E [meV] B [T](a) (b) (c) (d) FIG. 13. (Color online) Spectra for the system of the parameters L=106.36 nm, 2 d=30 nm, zs=12 nm [ zs=15 nm for (d)]. (a)/(b) Energy levels for a system of n-n/n-pdouble dots as a function of the depth V=−Vl=−Vr/depth and height −Vl=Vr=Vof the Gaussian potential traps. (c)/(e) Energy spectrum for the electron pairin the n-n/n-psystem as a function of magnetic ﬁeld B. In (c) the convergence of the results is shown with 8 (gray dots), 16 (light blue), and 24 single-electron states spanning the basis of the Slaterdeterminants. The results for 24 basis elements are given in the color palette for the spin-valleys as used in precedent ﬁgures. (d) The same as (c) but for z s=15 nm. Fig. 9(b) for 8 single-electron basis elements]. For the n-n dots with a larger interdot barrier [Fig. 13(d) for 2zs=30 nm] the exchange energy becomes negligible. The spectrum forthen-pdot displayed in Fig. 13(d) exhibits no effects of the exchange interaction already at 2 z s=24 nm. The exchange energy vanishes along with the overlap of the single-electron wave functions localized in both the dots[cf. Eq. ( 14)]. As the size of the quantum dots increases, the tunnel coupling between the dots disappears faster for then-psystem as compared to the unipolar n-norp-pquantum dots. For the n-pdot, the electron of the type- ndot needs to climb the potential hill deﬁning the type- pdot to form an extended state. Note that the experimental results of Fig. 1(c) of Ref. [ 16] for the current as a function of V landVrvoltages indeed demonstrate that lifting of the Coulomb blockade forthe unipolar dots appears for a wider range of gate voltages2830323436 0 1 2 3 4 5 6E[ m e V ] B[ T ]-300-200-1000100200300 00.1 0.2 0.3 0.4 0.5 0.6E [meV] V [eV] 30.430.630.831 0 0.2 0.4 0.6E[ m e V ] B[ T ](a) (b) FIG. 14. (Color online) (a) Single-electron spectrum for the n-p quantum dot with 2 d=14 nm and 2 zs=10 nm. (b) The two-electron spectrum. The inset shows the zoom on the central part of the spectrum. The colors denote the spin conﬁguration with the paletteof Fig. 9. than for the n-pdot, suggesting a reduced tunnel coupling between the ambipolar dots. For a Gaussian proﬁle of the conﬁnement potential the reduction of the exchange energy for the n-pdots appears already for smaller dots; see Fig. 14for 2d=14 nm and 2zs=10 nm, for which the exchange energy is 2 X=0.1m e V . For larger dots the exchange energy in the n-psystem appears when the n-pjunction is shorter. In Fig. 15we present a calculation for the conﬁnement potential of the form VQD(r)=⎧ ⎨ ⎩−Vexp(−(z+zs)2/d2)f o r z<−zs Vsin (πz/(2zs)) for −zs/lessorequalslantz/lessorequalslantzs Vexp(−(z−zs)2/d2)f o r z>z s (24) forV=0.23 eV , zs=3n m , d=20 nm. This potential proﬁle is plotted in Fig. 15(b) with the black line. An overlap of the wave functions of both dots appear [see Fig. 15(b) ] near the center of the system, and the exchange energy isagain signiﬁcant (2 X=0.22 meV). Then the two-electron energy spectrum takes the form [Fig. 15(c) ] known from the discussion of small quantum dots [Fig. 9(a)]. 085312-10TWO-ELECTRON n-pDOUBLE QUANTUM DOTS IN . . . PHYSICAL REVIEW B 91, 085312 (2015) -300-200-1000100200300 0 0.1 0.2 0.3 0.4E [meV] V [eV] 00.0040.0080.012 -40 -20 0 20 40 z [nm] 242526272829303132 0 1 2 3 4 5 6E[ m e V ] B[ T ]\|\|2(a) (b) (c) FIG. 15. (Color online) Single- (a) and two- (c) electron energy spectra for the n-pquantum dot with conﬁnement potential given by Eq. ( 24) and plotted with the black line in (b). In (b) the blue and red lines show the wave functions for the single-dot eigenstates. F. CNT chirality and the spin-orbit coupling parameter δ The presented results are qualitatively independent of the chirality of the CNT, as long as it is semiconducting. Forpresentation we return to the parameters of the small Gaussianquantum dots and consider a C h=(20,6) CNT (Fig. 2). For 2zs=10 nm—the distance between the centers of the dots for the zigzag CNT considered in Sec. III B —a wide avoided crossing is found in the single-electron states fromthe conduction and the valence bands [Fig. 16(a) ] and the exchange energy is as large as 2 X=0.83 meV [see Fig. 16(b) ]. For 2z s=11.26 nm the width of the avoided crossing of the single-electron energy levels is reduced to 7.7 meV [exactlyas for the zigzag dot of Fig. 6(b)], and the exchange energy is-15-10-5051015 0.38 0.4 0.41 0.42 0.43E [meV] V[ e V ] 323436384042 0 1 2 3 4 5 6E [meV] B[ T ](a) (b) FIG. 16. (Color online) (a) Avoided crossing of valence- and conduction-band single-electron states for a n-pquantum dot within (20,6) CNT. [The results for the zigzag CNT were displayed in Fig. 6(b)]. (b) Two-electron energy spectrum. Two values of interdot separation are considered 2 zs=10 nm and 2 zs=11.26 nm. 2X=0.25 meV . The qualitative character of the n-pspectrum, including the pattern of the avoided crossings is the same asfor the zigzag CNT [cf. Fig. 16(b) and Fig. 9(a)]. The spin-orbit interaction in the applied model is deter- mined by the parameter δ[Eqs. ( 3) and ( 4)]. The sign of δ determines the sign of the spin-orbit splitting /Delta1 SObetween (K/prime↑,K↓) and ( K/prime↓,K↑) energy levels. Depending on the sign of δthe energy levels of the multiplet cross as a function of the magnetic ﬁeld in the lower [ 14] or higher [ 7] pair of energy levels. Both types of crossings are observed in experiments forvarious samples [ 8], depending on the chirality of the nanotube in particular [ 10,12]. The direct dependence of the SO energy on the chirality requires inclusion of the second-neighbor hop-ping elements to the tight-binding Hamiltonian [ 42]. In order to demonstrate that the conclusions of the present study are inde-pendent of the sign of /Delta1 SO, we performed calculations for the small dots within the zigzag CNT adopting δ=− 0.003. The single-particle energy spectra changes are displayed in Fig. 17 for a single carrier and in Fig. 18for the n-pquantum dot. For the negative value of δthe spin-orbit splitting favors parallel alignment of the orbital and the spin magneticmoments, in contrast to the results presented above for thepositive δ. This leads to the switched order of the two Kramers 085312-11E. N. OSIKA AND B. SZAFRAN PHYSICAL REVIEW B 91, 085312 (2015) (a) (b) FIG. 17. (Color online) The same as Fig. 4(a) and 4(b) but for δ=− 0.003. doublets on the energy scale for B=0 (Fig. 17). Furthermore, the crossing of the single electron states in positive magneticﬁeld appear now in higher pair of states— K(K /prime)f o rn-type (p-type) dot—instead of lower [cf. Fig. 4(a) and 4(b)]. Changes in the single electron spectra are projected directlyto the electron-pair spectrum (Fig. 18). Comparing Fig. 18 to Fig. 9(a), we conclude that the avoided crossing in the central part of the spectrum for B/similarequal3.5Tis observed either for the spin-down states [Fig. 9(a)] or spin-up states (Fig. 18) depending on the sign of δ. VI. SUMMARY AND CONCLUSION We have described formation of extended single-electron orbitals in n-pquantum dots deﬁned in a carbon nanotube and the two electron states corresponding to the (1e,3h) charge stateof the double dot. The electronic structure was determined bythe conﬁguration interaction approach within the tight-bindingmethod with a complete account for the intervalley scatteringdue to the atomic disorder and electron-electron interactionwithout any additional parameters describing the coupling ofthe conduction- and valence-band states.} }} 36373839404142 0 1 2 3 4 5 6E [meV] B[ T ] FIG. 18. (Color online) The same as Fig. 9(a) but for δ=− 0.003. The present study indicates that the exchange energy for then-pdots appears only for ﬁnite intervals of the magnetic ﬁeld and only in some parts of the spectrum. In particular,the spin exchange interaction is missing in the ground state,which is fourfold degenerate at B=0. The reason for this unusual behavior of the exchange interaction—as comparedton-nquantum dots—is the fact that for a given valley the orbital momenta are opposite in the conduction and valencebands. Formation of singletlike and tripletlike orbitals appearsonly brieﬂy on the Bscale and the ground state is fourfold degenerate. For a general value of Bthe exchange integral vanishes by the valley orthogonality. The basic structure of thetwo-electron spectrum turns out to be robust against the atomicdisorder, chirality, the sign of /Delta1 SO, and the size of the dots, provided that a tunnel coupling between the quantum dots ispresent. 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PhysRevB.100.184511.pdf	PHYSICAL REVIEW B 100, 184511 (2019) Observation of topological surface states in the high-temperature superconductor MgB2 Xiaoqing Zhou,1,Kyle N. Gordon,1Kyung-Hwan Jin,2Haoxiang Li,1Dushyant Narayan,1Hengdi Zhao,1Hao Zheng,1 Huaqing Huang,2Gang Cao,1Nikolai D. Zhigadlo ,3,4Feng Liu,2,5and Daniel S. Dessau1,6,† 1Department of Physics, University of Colorado at Boulder, Boulder, Colorado 80309, USA 2Department of Physics, University of Utah, Salt Lake City, Utah 84112, USA 3Department of Chemistry and Biochemistry, University of Bern, CH-3012 Bern, Switzerland 4CrystMat Company, CH-8046 Zurich, Switzerland 5Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China 6Center for Experiments on Quantum Materials, University of Colorado at Boulder, Boulder, Colorado 80309, USA (Received 20 November 2018; revised manuscript received 22 October 2019; published 21 November 2019) Most topological superconductors known to date suffer from low transition temperatures ( Tc) and/or high fragility to disorder and dopant levels, which is hampering the progress in this promising ﬁeld. Here, utilizinga combination of angle-resolved photoemission spectroscopy measurements and density-functional theorycalculations, we show the presence of a type of topological Dirac nodal line surface state on the [010] facesof the T c=39 K BCS superconductor MgB2. This surface state should be highly tolerant against disorder and inadvertent doping variations and is expected to go superconducting via the proximity effect to the bulksuperconductor that this state is intimately connected to. This would represent a form of high-temperaturetopological superconductivity. DOI: 10.1103/PhysRevB.100.184511 I. INTRODUCTION As its name suggests, a topological superconductor needs two essential ingredients: nontrivial topology and thesuperconducting order. Initially, the exploration of topologicalsuperconductivity was limited to the 5 /2 quantum Hall state in electron gas systems [ 1–3] and p-wave superconductors such as Sr 2RuO 4[4–6], in which the chiral superconducting order parameter is topologically nontrivial by itself. However,these systems are extremely sensitive to disorder, very scarcein nature, and have transition temperatures well below liquidhelium temperature—each of which imposes great difﬁcultiesin the exploration of topological superconductivity. Thediscovery of topological band structures [ 7,8] introduces an alternative and arguably more favorable recipe—that the“topological” part of a topological superconductor can besubstituted by a topological surface state (TSS). In systemswith both topologically nontrivial band inversions and con-ventional s-wave superconducting gaps [ 9], the topological surface state can be gapped by the superconducting gap[10–12] through the proximity effect and enables topological superconductivity. To avoid the complications of interfacephysics, it is preferable to use a single system, such as the onediscussed here. To make a singular topological superconductor, researchers have devoted a great deal of effort into doping known bulktopological material, which has seen some success inmaterials such as Cu-doped Bi 2Se3[13–15]. However, given Xiaoqing.Zhou@Colorado.edu †Dessau@Colorado.eduthat the discoveries of high-temperature superconductivity have been largely accidental, it is unclear how far thisapproach can go. Here we adopt an alternative approachby looking for topological surface states in knownhigh-temperature superconductors—and MgB 2ranks high on this list. At ambient pressure, its superconducting transitiontemperature of 39 K [ 16] is the highest among conventional s-wave superconductors, and second only to certain members of the cuprate and pnictide family among all knownsuperconductors. Very recently, a particularly promisingcandidate FeTe 1−xSex[17] was found in the family of pnictide high-temperature superconductors, with a transition tempera-ture of 14.5 K setting the current record. While this discoveryis exciting, the required proximity of a very small (20-meVscale) spin-orbit-coupled gap to the Fermi energy means thatthe system should be highly sensitive to inadvertent dopingvariations. Although MgB 2as a high-temperature superconductor has been extensively studied by many techniques including angle-resolved-photoemission spectroscopy (ARPES) [ 18–22], its ability to harbor topological surface states has never beenappreciated. In this work, we use a combination of ﬁrst-principle density-functional theory (DFT) calculations andARPES to look for topological surface states in MgB 2.O u r DFT calculations [ 23] have predicted the existence of pairs of topological Dirac nodal lines [ 24] at the Brillouin-zone boundaries, as well as topological surface bands that connectthese nodal lines. The calculations further predict that thetopological surface states should be robust against realisticdoping variations, and readily gapped by the superconduct-ing order, as expected by the intimate and inherent contactbetween the bulk superconducting states and the topologicalsurface states. 2469-9950/2019/100(18)/184511(6) 184511-1 ©2019 American Physical SocietyXIAOQING ZHOU et al. PHYSICAL REVIEW B 100, 184511 (2019) FIG. 1. (a) Crystal structure of MgB2with hexagonal lattice in the abplane, determined by x-ray diffraction. The face of the edge cleave is shown in blue, and the cleaved surface can be eitherMg- or B terminated. (b) The 3D Brillouin zone and the projected “zigzag” [010] surface Brillouin zone (shown as the blue sheet). High-symmetry points K/H andK /prime/H/primecome in mirror-symmetric pairs, with Berry phase of πand−π, respectively. (c) Calculated DFT bulk band structure along the high-symmetry cut showing Dirac points (red arrows) along K-H. (d) Illustration of a Dirac nodal line along K-H, with Dirac point exactly meeting EFmidway along the cut. The kxdirection is normal to the [010] surface and so is covered by varying photon energy, in our case from 30 to 140 eV . II. RESULTS The key to the topological surfaces states are the topo- logical Dirac nodal lines, with associated Dirac points onhigh-symmetry cuts [highlighted by the arrows in the band-structure plot of Fig. 1(c)]. As illustrated by Fig. 1(d),t h e Dirac nodal line disperses across E Fin the kzdirection(normal to the honeycomb layers) over a few eV range, so that Dirac band crossings as well as the corresponding topologicalsurface states will always be present at E Ffor essentially any conceivable amount of doping or band-bending effects. Asshown in Fig. 1(b), the Dirac nodal lines predicted by the DFT calculation are located at the zone boundary of the 3DBrillouin zone along the K-HandK /prime-H/primehigh-symmetry lines that run along the zaxis. Our calculations (see Supplemental Material [ 25] and references [ 1,2] therein) show that each of these nodal lines in MgB2is wrapped by a Berry phase of π, i.e., they support a Z2topology. Similar to the case in graphene, the K/prime-H/primeline can be regarded as the mirror image of the K-H, so the associated Berry phase for the K/prime-H/primenodal line is −πinstead of πfor the K-H nodal line. Because of their kzdispersion, the Dirac nodal lines can be best accessed from the side so that the dispersion is inthe experimental plane a geometry different from all previousARPES experiments that studied the sample from the caxis, which is also the natural cleavage face. For our experiment wecleaved the samples from the “side” [blue plane of Figs. 1(a) and1(b)] and performed ARPES on that thin [010] face—a challenging but achievable task. The cleaved surface viewedwith a scanning electron microscope [see Fig. 2(b)]a sw e l la s an atomic force microscope (see Supplemental Material [ 25] and references [ 3–8] therein) shows an atomically ﬂat region, upon which high-quality ARPES spectra were observed at10 K. We measured the band dispersion along the momentumperpendicular to the cleavage face ( k x) by scanning the photon energy hνfrom 30 to 138 eV along the /Gamma1-Khigh-symmetry line with linear spolarization. Since at these photon energies the photon momentum is negligible, we have [ 26] kx≈/radicalbigg 2me ¯h2(hν−∅− EB+V0)−/parenleftbig k2y+k2z/parenrightbig , (1) FIG. 2. Edge-on ARPES gives in-plane bulk electronic structure. (a), (b) In-plane and edge-cleaved views of our crystals, respectively. Crystal orientation is obtained through x-ray diffraction. (c) In-plane Brillouin-zone points. (d)–(g) In-plane isoenergy ARPES plots at energie s from EFto−3e V .T h e kydirection is parallel to the cleaved surface (panel b) and so is covered by varying the emission angle. The kxdirection is normal to the edge-cleaved surface and is covered by varying the incident photon energy from 30 eV (low kx) to 138 eV (higher kx). The solid line representing 86 eV cuts through the K/H Brillouin zone point. 184511-2OBSERV ATION OF TOPOLOGICAL SURFACE STATES IN … PHYSICAL REVIEW B 100, 184511 (2019) FIG. 3. (a) An illustration of how the 2D topological water-slide surface state (light magenta sheet) connects the 1D nodal lines. (b) Illustration of the K/H andK/prime/H/primeDirac nodal lines with the Z2Berry-phase monopoles ( +and−) projected to the 2D cleaved surface. The plot is made for the photon energy 86 eV [ kx≈4 in the units of Fig. 2(c)] that due to the proper kxvalue can access the bulk Fermi surface in the center of the plot. The dashed bulk Fermi surface shown at the left and right are at incorrect kxvalues to be observed. A topological surface state connecting the +1a n d−1 monopoles on the projected surface Brillouin zone is drawn in by hand. (c)–(f) DFT spectral function convoluted by a Gaussian function (to simulate band broadening) in the ky−kzplane at 86 eV at a variety of binding energies. (g)–(j) Experimentally measured isoenergy contours using 86-eV photons. In addition to the excellent agreement between the predicted and measured bulk states, we identiﬁed an additional set of surface states (open circles) through Lorentzian ﬁtting of momentum distribution curves along kz(see Supplemental Material [ 25], Fig. S3). This surface state connects the pairs of Dirac points as predicted; therefore it is labeled as TSS (Topological Surface State or surface Fermi arc). The weak spectral intensity of the surface states is expected for a surface with cleavage imperfections or disorder. where ∅= 4.3 eV is the work function, EBthe binding energy, andV0=17 eV the inner potential. This describes the ARPES spectra measured on a “spherical sheet,” the radius of whichis proportional to√ hν−∅− EB+V0. By stitching together many spectra taken over a wide range of photon energies from30 to 138 eV , we reconstruct the isoenergy plots in the [001]plane at k z=0, as shown in Figs. 2(d) to2(g). The data show a spectra consistent with previous ARPES studies onthe [001] cleavage plane [ 18–22] (see Supplemental Material [25]). This conﬁrms that we are able to observe the proper bulk band structure of MgB 2from the cleaved [010] plane. As illustrated in Fig. 2(c), in our experimental geometry the K-H Dirac nodal line acts as a monopole of Berry phase and is neighbored by three of its “mirror-image” K/prime-H/primelines with opposite charge, each of which has bulk bands accessible onlyunder different experimental conditions (i.e., photon energiesand experimental angles). We choose the photon energy hν=86 eV (black line in the panels), which allows us to directly access one of the K-H high-symmetry lines by varying the k z momentum axis. To better investigate the Dirac nodal line, Fig. 3focuses on the region near the high-symmetry line K-H atkx≈ 8π/√ 3a,ky=4π/3a,andkz∼π/cto 2π/c.A ss h o w n in Fig. 3(a), on the projected surface Brillouin zone we should expect a 2D topological surface state (TSS) connecting a pairof 1D Dirac nodal lines with opposite charges [ 27], as any loop encircles the nodal lines will pick up a Berry phase ofπor−π. This “water-slide” TSS is analogous to the ﬂat “drumhead” surface state in Dirac nodal loop systems [ 28], but follows the dispersions of the nodal lines over a range of∼4 eV . Viewed from the sample projection [Fig. 3(b)], the topological surface states could connect the +πnodal line (k y=4π/3a)t ot h e −πnodal line on the left ( ky=2π/3a), or the one on the right ( ky=8π/3a), but not both. As shown 184511-3XIAOQING ZHOU et al. PHYSICAL REVIEW B 100, 184511 (2019) FIG. 4. TSS’s emanating from Dirac points. (a)–(e) ARPES spectra from the ﬁve cuts of Fig. 3(c), overlapped by DFT calculations (white dashed lines). The Dirac band crossings evolve from below EFto above EFconsistent with the diagram of Fig. 1(d). Through second derivative analysis (see Supplemental Material [ 25], Fig. S4), we identiﬁed nearly ﬂat topological surface states (red open circles) in cuts (b)–(d), which connect to the Dirac points and similarly move up in energy. In contrast, regular (nontopological) surface states (indicated by the blue arrows) remain relatively “static” in energy in all ﬁve cuts. (f)–(j) DFT simulation of spectral intensity, convolved by a Gaussian function to simulate band broadening. The calculated DFT points have been shifted up by 0.5 eV relative to the measured spectra. in Figs. 3(b) to3(i), we have excellent agreement between the DFT bulk band calculations and the ARPES isoenergy plots,in which the shifting touching point of an electron pocketand a hole pocket indicates nontrivial band crossings dispers-ing along K-H. Importantly, through momentum distribution curves analysis (see Supplemental Material [ 25]) we found an additional feature originating from the touching point that isabsent from the DFT calculations of the bulk bands. It lookssimilar to a “Fermi arc” in Weyl semimetal [ 29], but persists for most binding energies ( E-E F) and connects towards the −πbulk band crossing point at ky=2π/3a, even though the corresponding bulk bands are not experimentally accessi-ble at 86 eV . These look like the TSS’s drawn in Fig. 3(b) and we label them as such, though conﬁrmation from an energycut (Fig. 4) is still required. To further conﬁrm the topological nature of the surface state, in Fig. 4we plot the dispersion along ﬁve cuts across the touching points [dotted lines in Fig. 3(c)]. As the pre- dicted Dirac nodal line should disperse along the K-H high- symmetry cut, the band crossings should evolve continuouslyfrom below E Fto above EFas shown in Fig. 1(c).T h i s is exactly what we observed, conﬁrming the Dirac nodalline nature of these states. We directly overlay the DFTcalculations of the bulk band dispersions (white dashed lines) with the ARPES spectra with no alteration of the massesor velocities, and a −0.5 eV offset in the DFT chemical potential (possibly due to the depletion of Mg or B atomson the surface). The agreement is overall excellent, with thechemical potential shift indicative of extra electron chargeleaving the cleaved surface. In particular, we stress that over ahuge range of possible chemical shifts (such as those inducedby unintentional doping), here the Dirac nodal lines dispersingalong K-H guarantees Dirac points and topological surface states right at E F, in sharp contrast to the case of FeTe 1−xSex (see Supplemental Material [ 25] for more discussions). In addition to the agreement with the bulk Dirac nodal line,the experiment shows some additional weak features whichis captured by second derivative analysis (see SupplementalMaterial [ 25]). As all bulk bands are accounted for, these should be of nonbulk origin, i.e., surface states. Most interest-ing of these are the ones highlighted in red that are observedto connect to the Dirac points in both energy space (Fig. 4) and momentum space (Fig. 3), with this effect observed over a wide range of energies and momenta. Away from the Diracpoint its dispersion is much ﬂatter than the bulk band in thes-pmetal MgB 2. This is fully consistent with the existence of 184511-4OBSERV ATION OF TOPOLOGICAL SURFACE STATES IN … PHYSICAL REVIEW B 100, 184511 (2019) a topological surface state that we theoretically predicted to connect the Dirac points on this particular surface [ 23], and we label it as such. This state contrasts with a topologicallytrivial surface state (blue arrow) that is largely insensitiveto the energy of the Dirac point as it disperses from cutto cut. III. DISCUSSION These topological surface states are expected to go super- conducting via the proximity effect, which would make thismaterial by far the highest transition temperature and mostrobust topological superconductor, and an excellent platformfor a multitude of future studies. Even higher-energy res-olution ARPES than what we have carried out here couldbe utilized to directly detect a superconducting gap in theTSS’s, although this is nontrivial since the highest-resolutionARPES facilities are mostly laser ARPES [ 30,31], which lack the ability to reach the relevant k-space locations. Scanning tunneling microscopy (STM) could also potentially be utilizedto detect such a gap. Regardless, since the contact betweenthe topological surface states with the bulk superconductivityis almost guaranteed to be near ideal since the surface statesare an intrinsic part of the electronic structure, this next stepis highly likely to be realized. On the other hand, furtherexplorations of MgB 2still face quite a few of their own technical difﬁculties—that the (010) faces of single crystalsare very small and difﬁcult to work with, calling for atomicallyﬂat thin ﬁlms [ 32] on the [010] face. A different probe, such as STM [ 33], might provide vital insights into this exploration, too. The observation of the topological surface state conﬁrms the theoretical prediction of MgB 2as a promising topological superconductor candidate. Given that there are no topologicalsurface states near the Fermi energy in cuprate superconduc-tors, and that the transition temperature in the best pnictidessuperconductors are not much higher, the potential topological superconductivity in MgB 2would not only set the current record for Tcamong topological superconductors but also approach the realistic limit. More important than the high Tc, however, is the fact that the topological surface state that isnow expected in this material should be much less sensitiveto disorder or dopant variations than in other topological su-perconductors, including the discovered state in FeSe 0.45Te0.55 [17]. MgB2thus has the best of these two ingredients, i.e., a high superconducting transition temperature and a robusttopological surface state. Last, the success in the conventional superconductor MgB 2 helps guide the hunt of topological superconductors to a dif-ferent direction: since the topological surface state associatedwith Dirac nodal lines and loops seems to be more abun-dant [ 34] than that of high-temperature superconductivity, we should optimize the weak link, and search for superconduc-tors that have topological band structures or can be madetopological. ACKNOWLEDGMENTS We thank Drs. D. H. Lu, Dr. M. Hashimoto, and Dr. T. Kim for technical assistance on the ARPES measurements.We thank Dr. R. Nandkishore and Dr. Qihang Liu for usefuldiscussions. The photoemission experiments were performedat beamline 5-2 of the Stanford Synchrotron Radiation Light-source and the Diamond Light Source beamline I05 (ProposalNo. SI17595). 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PhysRevB.96.174511.pdf	PHYSICAL REVIEW B 96, 174511 (2017) Intrinsic ac anomalous Hall effect of nonsymmorphic chiral superconductors with an application to UPt 3 Zhiqiang Wang,1John Berlinsky,1Gertrud Zwicknagl,2and Catherine Kallin1,3 1Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 2Institut für Mathematische Physik, Technische Universität Braunschweig, 38106 Braunschweig, Germany 3Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 (Received 4 September 2017; published 16 November 2017) We identify an intrinsic mechanism of the anomalous Hall effect for nonsymmorphic chiral superconductors. This mechanism relies on both a nontrivial multiband chiral superconducting order parameter, which is a mixtureof pairings of even and odd angular momentum channels, and a complex normal-state intersublattice hopping,both of which are consequences of the nonsymmorphic group symmetry of the underlying lattice. We apply thismechanism to the putative chiral superconducting phase of the heavy-fermion superconductor UPt 3and calculate the anomalous ac Hall conductivity in a simpliﬁed two-band model. From the ac Hall conductivity and opticaldata we estimate the polar Kerr rotation angle and compare it to the measured results for UPt 3[Schemm et al. , Science 345,190(2014 )]. DOI: 10.1103/PhysRevB.96.174511 I. INTRODUCTION Understanding unconventional superconductors has been one of the central goals in condensed matter research. Amongthe various unconventional superconductors, chiral supercon-ductors have attracted a great deal of attention in recent years,in part because they provide a platform to study the interplaybetween spontaneous symmetry breaking and topology [ 1]. In a chiral superconductor, a Cooper pair carries a nonzerorelative orbital angular momentum whose projection alonga certain direction is also nonzero. Choosing this directionas the angular momentum quantization axis z, different chiral superconductors that are eigenstates of angular momentum canbe characterized by the Cooper-pair orbital angular momentumquantum numbers L=1,2,3,... andL z=± 1,±2,... .A general chiral superconducting order, however, need not bean angular momentum eigenstate. For example, chiral fwave may mix with chiral pwave, etc. One of the deﬁning properties of a chiral superconductor is its spontaneous breaking of parity and time-reversal symmetry.As a consequence, there can be a nonzero anomalous Halleffect (i.e., a Hall effect in the absence of an external magneticﬁeld), which can be detected by polar Kerr effect measure-ments [ 2]. Experimentally, a frequency-dependent rotation angle between the polarization of incident and reﬂected light ismeasured. This Kerr angle θ K(ω) is related to the ac anomalous Hall conductivity σH(ω)b y[ 3] θK(ω)=4π ωIm/bracketleftbiggσH(ω) n(n2−1)/bracketrightbigg , (1) where nis the frequency-dependent index of refraction. A nonzero Kerr signal has been observed in the superconductingphase of several unconventional superconductors includingSr 2RuO 4[4], UPt 3[5], URu 2Si2[6], PrOs 4Sb12[7], and Bi /Ni bilayers [ 8]. Sr 2RuO 4is widely thought to be a chiral p-wave superconductor [ 9,10], while the heavy-fermion superconduc- tor UPt 3is expected to be a chiral f-wave superconductor withE2usymmetry, corresponding to L=3,Lz=± 2i nt h e continuum limit [ 11,12].However, parity and time-reversal symmetry breaking are necessary but not sufﬁcient conditions for a nonzeroanomalous Hall effect. Breaking of additional symmetries,translation and particle hole, are needed for a nonzero σ H(ω). Consequently, the size of the effect depends crucially on themechanism by which these symmetries are broken. As pointedout previously [ 13–15],σ H(ω) vanishes at all frequencies for a Galliean invariant chiral superconductor. One way to breaktranslation symmetry is by extrinsic impurity scattering, whichhas been studied by several groups in the context of Sr 2RuO 4 [15–17]. This impurity effect does not contribute to σHin the lowest-order Born approximation and therefore requireshigher-order scattering [ 16]. However, both Sr 2RuO 4and UPt 3 are very clean, and it is not clear if the observed effect is due to disorder. Even without impurities, translation symmetry canbe broken by certain intrinsic mechanisms, which turn out tobe rather subtle. There have been two intrinsic mechanismsproposed previously. One is based on a collective mode [ 18], combined with the small but ﬁnite momentum of the incidentphoton and the breaking of inversion symmetry along theincident external electromagnetic wave propagation direction.However, the estimated angle for this mechanism is too smallto account for experiments [ 4]. The other intrinsic mechanism invokes a multiband effect [ 19–23], arising from structure within the crystal unit cell, which also involves interbandpairing. Here, we will study a generalization of this multibandmechanism. All of these theories (impurity effects, collective modes, and the multiband effect) have so far only been studied for the case of chiral p-wave superconductors. This has led to a better understanding of the Kerr effect in Sr 2RuO 4. However, UPt 3is thought to be a chiral f-wave superconductor in its lower superconducting transition temperature phase. One might think that the conclusions obtained for the Kerreffect in a chiral p-wave superconductor can be directly generalized to higher-chirality superconductors with \|L z\|/greaterorequalslant2 without much difﬁculty. However, such a naive generalization is problematic. As recent studies on nontopologically protected quantities, such as the integrated edge current [ 24] and the total orbital angular momentum [ 25,26], have demonstrated 2469-9950/2017/96(17)/174511(15) 174511-1 ©2017 American Physical SocietyW ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) e1e2e3r1r2 r3z xy FIG. 1. Crystal structure of UPt 3. Blue disks denote the positions of U atoms. There is a Pt atom (not shown) between each nearest- neighbor intralayer pair of U atoms. The vectors eiandriconnect two nearest-neighbor intralayer and interlayer U atoms, respectively. The coordinate system is chosen such that ˆx/bardble1. explicitly, chiral superconductors with \|Lz\|/greaterorequalslant2 can behave very differently from the chiral p-wave case. Given that the anomalous Hall conductivity σH(ω) is also a nontopologically protected quantity [ 13,15], unlike its thermal Hall counterpart, we expect that σH(ω) of chiral superconductors with \|Lz\|/greaterorequalslant2 can be quite different from that of \|Lz\|=1. In fact, as has already been pointed out by Goryo in Ref. [ 16], in the continuum limit, the skew impurity scattering diagram for the lowest-order impurity contribution to σH(ω) is nonzero only for chiral superconductors with \|Lz\|=1 and vanishes for\|Lz\|/greaterorequalslant2. More generally, to have a nonzero σHin the continuum limit, the azimuthal angular integral of kxky/Delta11/Delta1∗ 2, where /Delta11,2are the two components of the chiral order parameter, must be nonzero. While the details differ somewhat for the different mechanisms, the kxkyin the angular integral effectively arises from the current (or velocity) operators in σxyand/Delta11/Delta1∗ 2is the lowest-order contribution that directly brings in the chirality to which σHis proportional. It follows thatσH/negationslash=0 only for \|Lz\|=1. The vanishing of σHfor higher-chirality superconductors in the continuum limit is a concern for UPt 3because the observed Kerr signal in UPt 3 [5] is actually larger than in Sr 2RuO 4[4]. To get a nonzero anomalous Hall conductivity for UPt 3from chiral f-wave order, one needs to include lattice or band-structure effects. UPt 3exhibits multiple superconducting phases in its temperature magnetic ﬁeld phase diagram [ 27–29]. At zero ﬁeld it undergoes two separate superconducting transitionsatT + c≈0.55 K and T− c≈0.5K[ 30–34]. A nonzero Kerr rotation [ 5] has been observed only in the superconducting phase below T− c. To study whether this UPt 3Kerr effect can arise from the multiband mechanism, one needs a model with atleast two bands. The simplest case is two bands arising from theABAB stacking of the hexagonal planes of the U atoms alongthe crystal caxis. (See Fig. 1.) Due to this stacking, the crystal has a close-packed hexagonal lattice structure correspondingto the nonsymmorphic space group P6 3/mmc . One can ask if the two bands resulting from this stacking can give rise to anonzero Kerr effect. In fact, as will be discussed later, one canshow that a simple chiral d-o rf-wave pairing on a triangular lattice with ABAB stacking gives zero, even including latticeeffects beyond the continuum limit. Recently, Yanase [ 35] argued that, due to the nonsym- morphic space group, the spin triplet superconducting orderparameter is not a simple chiral fwave or a combination of onlyfandpwaves. Chiral d-wave pairing also mixes with the symmetry of the E 2urepresentation of the crystal lattice point group D6h. In this model, chiral fandpwaves are even in the sublattice index, which can be thought of as anextra pseudospin index, while chiral dwave is odd in that index and, consequently, chiral fpairing is a triplet in the AB-sublattice subspace while the chiral d-wave pairing is a singlet. Both fanddcomponents involve nearest-neighbor interlayer pairing and are of the same magnitude, while thechiral p-wave component involves pairing within the basal plane and is expected to be smaller. The smaller p-wave pairing amplitude is presumably conjectured because of the relativelylarger in-plane U-U atom distance [ 36] and perhaps also because the chiral pcomponent is energetically unfavorable since it pairs only one spin component. The mixing of chiralfanddwaves leads to a more complex chiral f+dpairing order parameter that is nonunitary [ 35]. As a simple model, following Yanase, we study the two bands, resulting from the ABAB stacking, that model the“starﬁshlike” Fermi surfaces [ 30,37], centered on the Apoint at the top and bottom of the Brillouin zone (BZ). There arealso four other Fermi surface sheets resolved experimentally[30,37], which, however, will not be considered in this paper. The four other Fermi sheets are not simply related by stackingsince, in general, the two bands due to the stacking (the bonding and antibonding bands) are well separated in energy and only one of them crosses the Fermi energy. However, in the caseof the “starﬁsh” Fermi surfaces on the BZ boundary, withoutspin-orbit coupling (SOC) the two bands are degenerate bysymmetry on the top and bottom BZ faces. With SOC, banddegeneracies remain along six directions on the top and bottomsurfaces. These bands give a particularly simple two-bandmodel for studying the intrinsic multiband mechanism of theKerr effect. In this paper, we show that this two-band model with a mixed ( f+d)-wave superconducting order parameter can give rise to a nonzero Kerr effect with or without the smallchiral p-wave pairing component. We ﬁnd that mixing of the chiraldcomponent with the chiral f-wave pairing is essential for a nonzero σ H. We also ﬁnd that the nature of the terms that contribute to σHare distinct from the terms that give a nonzero contribution for the Sr 2RuO 4case [ 19]. From σHwe estimate the Kerr angle and ﬁnd it to be about 10% of the experimentalvalue in UPt 3[5]. Factors that might increase (or decrease) this estimate are discussed. Although our work is not a complete theory of the Kerr effect for UPt 3, it captures a key possible contribution and more generally illustrates the necessary ingredients for a nonzeroKerr effect for a higher chirality superconductor, a case whichis noticeably more subtle than that of chiral pwave. The paper is organized as follows. In Sec. IIwe describe the Bogoliubov–de Gennes (BdG) Hamiltonian that we use for thestarﬁshlike Fermi surface. In Sec. IIIwe derive an approximate expression for σ H(ω) for this BdG Hamiltonian, evaluate it numerically, and identify the key ingredients of the result.The estimation of the Kerr angle from σ Hand comparison to experiment are given in Sec. IV. Section Vcontains our conclusions and further discussions. Some technicalcomputational details are relegated to the Appendices. 174511-2INTRINSIC AC ANOMALOUS HALL EFFECT OF . . . PHYSICAL REVIEW B 96, 174511 (2017) II. MODEL We focus on a two-band model proposed by Yanase [ 35]t o describe the starﬁsh Fermi surface (FS) of UPt 3. With the two sublattices and two spin components, the BdG Hamiltoniancan be written in terms of an eight-component spinor /Psi1(k) whose transpose is deﬁned as /Psi1 T k≡(ck1↑,ck2↑,ck1↓,ck2↓,c† −k1↑,c† −k2↑,c† −k1↓,c† −k2↓),(2) where ckisis the annihilation operator for an electron with momentum k, sublattice index i, and spin quantum number s. In this basis, the BdG Hamiltonian can be written as HBdG=1 2/summationdisplay k∈BZ/Psi1† kˆHBdG(k)/Psi1k, (3) with ˆHBdG(k)=/parenleftBiggˆE(k) ˆ/Delta1(k) ˆ/Delta1†(k)−ˆET(−k)/parenrightBigg , (4) where ˆE(k) is the normal-state Hamiltonian and ˆ/Delta1kis the superconducting order parameter, both 4 ×4 matrices. A. Normal-state Hamiltonian and Fermi surfaces Using σαandsαto denote the four Pauli matrices for the two sublattices and spin, respectively, we can write the normal-stateHamiltonian ˆE(k)a s ˆE(k)=ξ kσ0s0+/epsilon1k√ 2σ+s0+/epsilon1∗ k√ 2σ−s0+gk·sσ3,(5) where σ±=(σ1±iσ2)/√ 2 andξk,/epsilon1k, and gkare given by ξk=2t3/summationdisplay i=1cosk/bardbl·ei+2tzcoskz−μ, (6a) /epsilon1k=2t/primecoskz 23/summationdisplay i=1eik/bardbl·ri, (6b) gk=ˆzα3/summationdisplay i=1sink/bardbl·ei. (6c) Here, ξkcontains all nearest-neighbor (NN) hoppings within the same sublattice, both in-plane hopping with parameter t and intrasublattice NN hopping along the caxis with parameter tz,μis the chemical potential and k/bardbl=(kx,ky,0). The three unit vectors ei=(cosφi,sinφi,0) with φi=(i−1)2π 3and i={1,2,3}, are deﬁned within the plane as shown in Fig. 1. (All lattice spacings are set to unity.) /epsilon1kdescribes intersub- lattice NN hopping with parameter t/prime. The prefactor coskz 2in /epsilon1kcomes from the fact that these hoppings are deﬁned on the intersublattice bonds which are described by three nonprim-itive lattice vectors: r i=(1√ 3cosφ/prime i,1√ 3sinφ/prime i,1 2), with φ/prime i= π 6+(i−1)2π 3.gk·sis a Kane-Mele–type spin-orbit coupling (SOC) [ 38,39] that is allowed since the local symmetry of each U atom is D3h, which does not have inversion. Note this SOC term cannot exist between two different sublatticesbecause the center of the intersublattice U-U bond is inversionsymmetric. Also, the SOCs for the two sublattices must haveopposite signs in order for the U lattice to restore its global D 6hsymmetry which preserves inversion [ 35]. This explains the presence of the Pauli matrix σ3in the SOC term in the expression of ˆE(k). The parameter αingkcharacterizes the SOC strength. Diagonalizing the Hamiltonian ˆE(k) gives the two normal- state band dispersions E(n) ±(k)=ξk±√ g2 k+\|/epsilon1k\|2, each of which is twofold degenerate. The Fermi surfaces are shown inFig. 2 for the parameters ( t,t z,t/prime,α,μ )=(1,−4,1,2,12) from Ref. [ 35]. Figure 2(a) shows that the FS is centered around the Apoint of the BZ, while Fig. 2(b) presents a cut of the FS on the zone boundary kz=πplane. Note, from Fig. 2(b),t h e two Fermi surfaces intersect at six points on that plane since/epsilon1 k=0f o rkz=πandgkvanishes along the sixfold-symmetric directions: ky/kx=tanθiwithθi=π 6+(i−1)π 3. B. Superconducting order parameter ˆ/Delta1(k) The superconducting order parameter ˆ/Delta1(k) proposed in Ref. [ 35]i sa n E2ustate that can be written as ˆ/Delta1(k)=η1ˆ/Gamma11(k)+η2ˆ/Gamma12(k). Here, ˆ/Gamma11(k) and ˆ/Gamma12(k)a r et w o basis functions of the E2urepresentation, and ( η1,η2)= /Delta10(1,iη)//radicalbig 1+η2, with overall pairing magnitude /Delta10andηa real number that controls the anisotropy of the order parameter.Due to the relative phase between η 1andη2,ˆ/Delta1(k)i sc h i r a l , with the chirality determined by the sign of η. ˆ/Gamma11(k) and ˆ/Gamma12(k) are both triplets in spin as suggested by experiments [ 11,30,40]. The spatial parts of ˆ/Gamma11(k) and ˆ/Gamma12(k) contain not only f- andp-wave components but also ad-wave component as discussed above. Spatial inversion operation not only transforms k→− kbut also interchanges the two sublattices. The f- andp-wave components are odd functions of kand triplets in the sublattice index, while the d component is an even function of kbut a sublattice singlet. As mentioned above, the pairing amplitudes of the f- and d-wave components connect different sublattices while the p-wave component pairs sites on the same sublattice. The fanddcomponents are of similar magnitude while the p wave is smaller. In the following, we will ignore this small p component. Then, the two basis functions ˆ/Gamma11and ˆ/Gamma12can be written as [ 35] ˆ/Gamma11(k)={f(x2−y2)z(k)σ1−dyz(k)σ2}s1, (7a) ˆ/Gamma12(k)={fxyz(k)σ1−dxz(k)σ2}s1, (7b) where, for nearest-neighbor intersublattice pairing, f(x2−y2)z(k)=− sinkz 2/bracketleftbigg coskx 2cosky 2√ 3−cosky√ 3/bracketrightbigg ,(8a) fxyz(k)=√ 3s i nkx 2sinky 2√ 3sinkz 2, (8b) dyz(k)=− sinkz 2/bracketleftbigg coskx 2sinky 2√ 3+sinky√ 3/bracketrightbigg ,(8c) dxz(k)=−√ 3s i nkx 2cosky 2√ 3sinkz 2. (8d) In the expressions for ˆ/Gamma11(k) and ˆ/Gamma12(k), the spin Pauli matrix s1=s3is2indicates that the spin triplet pairing dvector is 174511-3W ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) −2 −1 0 (a) (b)1 2−2−1012 kxky FIG. 2. Starﬁsh Fermi surface (FS). (a) FS in the three-dimensional Brillouin zone of UPt 3; (b) FS contours in the plane of kz=π.T h e red (blue) line is the E(n) +(k)=0[E(n) −(k)=0] constant energy contour. Parameters used are ( t,tz,t/prime,α,μ )=(1,−4,1,2,12). along the ˆzdirection (or the crystal caxis). The presence of sublattice Pauli matrices σ1andσ2comes from the fact that the f- and d-wave components are derived from the real and imaginary parts, respectively, of a pairing amplitudefor electrons from NN intersublattice U ions. Because of themixing between the f- andd-wave components, ˆ/Delta1(k)ˆ/Delta1 †(k)={ \|fk\|2+\|dk\|2}σ0s0−i{fkd∗ k−f∗ kdk}σ3s0 (9) has a term which is not proportional to the identity matrix σ0s0, which makes ˆ/Delta1(k) nonunitary [ 41]. In Eq. ( 9), fk≡η1f(x2−y2)z(k)+η2fxyz(k), (10a) dk≡η1dyz(k)+η2dxz(k). (10b) C. Reduction of the BdG Hamiltonian The expressions for ˆE(k) and ˆ/Delta1(k) deﬁned above can now be substituted into the BdG Hamiltonian given by Eq. ( 4). One ﬁndsHBdG(k) reduces to two decoupled 4 ×4 blocks: HBdG=H(a)+H(b) =1 2/summationdisplay i=a,b/summationdisplay k∈BZ/bracketleftbig /Psi1(i) k/bracketrightbig†ˆH(i)(k)/Psi1(i) k, (11) with ˆH(a)=⎛ ⎜⎜⎜⎝ξk+gk /epsilon1k 0 /Delta112(k) /epsilon1∗ k ξk−gk/Delta121(k)0 0 /Delta1∗ 21(k)−ξk−gk −/epsilon1k /Delta1∗ 12(k)0 −/epsilon1∗ k −ξk+gk⎞ ⎟⎟⎟⎠, (12a) ˆH(b)=⎛ ⎜⎜⎜⎝ξ k−gk /epsilon1k 0 /Delta112(k) /epsilon1∗ k ξk+gk/Delta121(k)0 0 /Delta1∗ 21(k)−ξk+gk −/epsilon1k /Delta1∗ 12(k)0 −/epsilon1∗ k −ξk−gk⎞ ⎟⎟⎟⎠. (12b)The two bases are /Psi1(a) k=(ck1↑ck2↑c† −k1↓c† −k2↓), (13a) /Psi1(b) k=(ck1↓ck2↓c† −k1↑c† −k2↑). (13b) In the above equations, gk≡ˆz·g(k),/Delta112(k)≡fk+idk, and /Delta121(k)≡fk−idk, where 1,2 are sublattice labels. The two blocks are connected to each other by spin inversion ↑↔↓ , which leaves all matrix elements of ˆH(a)(k) and ˆH(b)(k) unchanged except for a change in the sign of the SOC term gk. However, as will be shown later, the Hall conductivity σH(ω) is an even function of gk. Therefore, we only need to focus on one block, say ˆH(a)(k), and multiply the σHcomputed for that block by a factor of 2. An additional factor of1 2, arising from the double counting of degrees of freedom in BdG theory, willcancel this factor of 2. Hereafter, we drop the superscript ( a) inˆH (a)(k) and simply denote it as ˆH(k) for brevity. Note that this decomposition into two 4 ×4 blocks is only possible in the absence of the intralayer p-wave pairing. From ˆH(k) one can obtain the Bogoliubov quasiparticle energies, which have line nodes on the kz=±πplane that form six rings, as shown in Fig. 3. These nodal rings are coun- terexamples to Blount’s theorem [ 42–47] and are topologically protected as a joint consequence of both the nonsymmorphicgroup symmetries and the nonzero spin-orbital coupling, asdiscussed in Refs. [ 45–47]. III. COMPUTATION OF THE ANOMALOUS HALL CONDUCTIVITY σH(ω) The Hall conductivity σH(ω) can be computed from the Kubo formula [ 19,48] σH(ω)=i 2ωlim q→0{πxy(q,ω)−πyx(q,ω)}, (14) where πxy(q,ω) is the electric current density ˆJx-ˆJycorrelator. At the one-loop level πxyis given by (setting e=¯h=c=1) πxy(q=0,iνm)=/summationdisplay kT/summationdisplay nTr{ˆvx(k)ˆG(k,iωn+iνm) ×ˆvy(k)ˆG(k,iωn)}, (15) 174511-4INTRINSIC AC ANOMALOUS HALL EFFECT OF . . . PHYSICAL REVIEW B 96, 174511 (2017) −2 −1 0 1 2−2−1012 kxky FIG. 3. Bogoliubov quasiparticle energy line nodes of the BdG Hamiltonian ˆH(k)a tkz=π. The parameter /Delta10=0.1t.O t h e r parameters used are the same as in Fig. 2. where Tis the temperature (set to T=0 at the end of the calcu- lation) and ωn=(2n+1)πTandνm=2mπT are fermionic and bosonic Matsubara frequencies, respectively. ˆG(k,iωn)i s the Green’s function of the 4 ×4 block Hamiltonian ˆH(k) with inverse deﬁned by ˆG−1(k,iωn)=iωn−ˆH(k). (16) From det ˆG−1(k,iωn)=0 one obtains the Bogoliubov quasi- particle energies of the Hamiltonian ˆH(k). However, the equation to be solved is not a quadratic equation for ω2 n but a quartic equation in ωn[see Eq. ( A2) in Appendix A]. Consequently, the analytic expressions for the quasiparticleenergies as well as the ﬁnal expression for σ H(ω) are quite lengthy, and these results are summarized in Appendix A in Eqs. ( A4)–(A8b). From these expressions it is difﬁcult to identify which ingredients are essential to obtain a nonzeroσ H(ω), and so we also compute σHperturbatively to obtain a much simpler expression that is valid at intermediate to highfrequencies. We treat the d-wave component of the superconducting order parameter as a perturbation and write ˆH(k)=ˆH 0(k)+ ˆH/prime(k) where ˆH0=⎛ ⎜⎜⎜⎝ξk+gk /epsilon1k 0 fk /epsilon1∗ k ξk−gk fk 0 0 f∗ k −ξk−gk −/epsilon1k f∗ k 0 −/epsilon1∗ k −ξk+gk⎞ ⎟⎟⎟⎠(17) and ˆH/prime=⎛ ⎜⎜⎜⎝000 id k 00 −idk 0 0 id∗ k 0 −id∗ k 000⎞ ⎟⎟⎟⎠. (18)ˆH 0and ˆH/primewill be taken as the “unperturbed” and “perturbed” Hamiltonian, respectively. We choose this particular partitionbecause it is precisely the d-component superconducting order-parameter part that makes the Bogoliubov quasiparticleenergy expression complicated [see Eq. ( A2) in Appendix A] and also because, as we will see later, the leading-ordercontribution to σ His linear in dk. Since we are including the effect of ˆH/primeonly perturbatively, the results are only reliable for sufﬁciently large ω. Actually, the perturbative expansion is in βk∝i(fkd∗ k−f∗ kdk)gk, not justdk[see Eq. ( A2) in Appendix Afor details]. So, the perturbative results are reliable for ω/greatermuchβk∼(/Delta12 0α)1/3, where αis the SOC strength. The full Green’s function results and the perturbative results for σH(ω) are compared in Appendix Ain Figs. 7and8, showing the two are essential identical beyond ω/greaterorsimilar4t. Since the laser frequency at which the Kerr effect has been measured is ω≈0.8e V[ 5], which is>20tin our model, the perturbative results can be used to compare to experiment. A. Perturbative calculation Here, we discuss the perturbative calculation of σH, with further details given in Appendix B. Quantities of different order in ˆH/primeare represented by superscripts (0) ,(1),... . First, consider zeroth-order described by the Hamiltonian ˆH0(k). The Bogoliubov quasiparticle energies E±are E±=/radicalBig a±/radicalbig a2−b, (19) with a=ξ2 k+g2 k+\|/epsilon1k\|2+\|fk\|2, (20a) b=/parenleftbig ξ2 k−g2 k+\|fk\|2−\|/epsilon1k\|2/parenrightbig2+\|fk\|2(/epsilon1k+/epsilon1∗ k)2,(20b) which are slightly different from those of the full Hamiltonian ˆH(k). However, E−still has nodal rings on the kz=±π plane that are almost identical to those obtained from the fullHamiltonian ˆH(k), plotted in Fig. 3. These nodal rings are protected by the nonsymmorphic space-group symmetry andspin-orbit coupling [ 35,45]. The velocity operators, which appear in Eq. ( 15), are deﬁned by the normal-state Hamiltonian ˆH N(k), which can be written in terms of the sublattice Pauli matrices σα: ˆHN(k)=ξkσ0+h·σ, (21) with h=(/epsilon1k√ 2,/epsilon1∗ k√ 2,gk) and σ=(σ+,σ−,σ3). Then, ˆ vx= ∂kxˆHN(k)τ0[15,19], where τ0is the identity matrix for the Nambu space or, written out explicitly, ˆvx=⎛ ⎜⎜⎜⎝∂ kxEa(k)∂kx/epsilon1k 00 ∂kx/epsilon1∗ k∂kxEb(k)0 0 00 ∂kxEa(k)∂kx/epsilon1k 00 ∂kx/epsilon1∗ k∂kxEb(k)⎞ ⎟⎟⎟⎠, (22) withE a(k)≡ξk+gkandEb(k)≡ξk−gk.ˆvycan be ob- tained from ˆ vxby the substitution ∂kx→∂ky. With ˆ vx,ˆvyand ˆG(0)≡{iωn−ˆH0(k)}−1, one can compute the zeroth-order 174511-5W ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) current-current correlator π(0) xy(iνm) from Eq. ( 15). However, a direct computation shows that π(0) xy(iνm)−π(0) yx(iνm)=0, so thatσ(0) H(ω)≡0. In other words, a chiral f-wave supercon- ducting order parameter alone does not give rise to a nonzeroanomalous Hall conductivity from the multiband mechanism ifthe two bands arise from ABAB stacking. The mixing betweenf- andd-wave components is crucial for a nonzero σ Hand one needs to go to ﬁrst order to calculate a nonzero σH(ω). From the full Green’s function ˆG=ˆG(0)+ˆG(0)ˆH/primeˆG(0)+ ..., one can deﬁne the ﬁrst-order Green’s function as ˆG(1)= ˆG(0)ˆH/primeˆG(0)and, from Eq. ( 15), the ﬁrst-order current-current correlator is π(1) xy(iνm)=/summationdisplay kT/summationdisplay n{Tr[ ˆvxˆG(0)(k,iωn+iνm)ˆvy ×ˆG(1)(k,iωn)]+{(0)↔(1)}}. (23) This [or, more precisely, π(1) xy(iνm)−π(1) yx(iνm)] is evaluated in Appendix Bby ﬁrst writing the velocity operators and Green’s functions as linear combinations of Pauli matrices to simplifycomputing the trace and then doing the Matsubara sum. Afterperforming a Wick rotation iν m→ω+iδ, one obtains the ﬁnal expression for the Hall conductivity σ(1) H(ω)=/summationdisplay k4i[fkd∗ k−f∗ kdk]ξk/braceleftbigg 8igkh·∂kxh×∂kyh ×Sk(ω) ω+/Omega1xyTk(ω) ω/bracerightbigg , (24) where for brevity we have suppressed the inﬁnitesimal imag- inary part iδinω+iδ./Omega1xyis an antisymmetrized velocity factor given by /Omega1xy≡−i/bracketleftbig ∂kx/epsilon1k∂ky/epsilon1∗ k−∂kx/epsilon1∗ k∂ky/epsilon1k/bracketrightbig . (25) We have also introduced two frequency-dependent functions in Eq. ( 24), which are deﬁned as (for details see Appendix B) Sk(ω) ω≡F1(k,ω)−ξ2 k−g2 k−\|/epsilon1k\|2 E+E−F2(k,ω) (26) and Tk(ω) ω≡F3(k,ω), (27) withF1(k,ω),F 2(k,ω), and F3(k,ω) given by F1(k,ω)≈C++ ω2−4E2 ++C−− ω2−4E2 −+C+− ω2−(E++E−)2, (28a) F2(k,ω)≈D++ ω2−4E2 ++D−− ω2−4E2 −+D+− ω2−(E++E−)2, (28b) F3(k,ω)≈B+− ω2−(E++E−)2. (28c) The≈sign means only the leading-order terms in fkand dkhave been kept. There are seven frequency-independent coefﬁcients in the numerators of F1,F 2, and F3. Theirexpressions are C++=−D−−=E+ (E2 −−E2 +)3, (29a) D++=−C−−=E− (E2 −−E2 +)3, (29b) C+−=E2 ++E2 − 2E+E−(E++E−)3(E+−E−)2, (29c) D+−=−1 (E++E−)3(E+−E−)2, (29d) B+−=1 2E+E−(E++E−). (29e) The subscripts {+ +,−−,+− } in these coefﬁcients directly reﬂect the corresponding physical processes that they areassociated with, which can be inferred from the denominator ofeach term in the expressions of F 1(k,ω),F2(k,ω), andF3(k,ω). For example, the ﬁrst term in F1(k,ω) with coefﬁcient C++ corresponds to a process where a Cooper pair, with momentum (k,−k), is broken and a Bogoliubov quasiparticle pair with energies E+(k) andE+(−k) are excited by the incident photon with a frequency ω. The two Bogoliubov quasiparticles have the same momentum ( k,−k) as the broken Cooper pair because the incident photon momentum q≈0 relative to k. Energy conservation of this process requires ω=E+(k)+E+(−k)= 2E+, which explains the denominator ω2−(2E+)2in the ﬁrst term in F1(k,ω). Other terms in F1(k,ω),F2(k,ω), and F3(k,ω) can be interpreted in a similar way. Notice that in the expressions for F1(k,ω),F2(k,ω), and F3(k,ω) there is no term with a denominator ω2−(E+−E−)2, which would correspond to a T> 0 process where a preexisting Bogoliubov quasiparticle with an energy E−gets excited to a higher-energy level of E+by the incident photon. Finally, as noted below Eq. ( 13b), we can see from Eqs. ( 26) and ( 27) thatσ(1) H(ω)i sa ne v e nf u n c t i o no f gksince the two functions Sk(ω) and Tk(ω) depend on konly through E±, which are even in gk[see Eq. ( 19)];/Omega1xydoes not depend on gk[see Eq. ( 25)], and the factor gkh·∂kxh×∂kyhis also even ingkbecause the mixed product contributes one and only one gksince h=(/epsilon1k/√ 2,/epsilon1∗ k/√ 2,gk). Next, we evaluate the expression for σ(1) H(ω)i nE q .( 24) numerically. Replacing ωwithω+iδin Eq. ( 24), the imaginary part can be written as Imσ(1) H=−π 2ω/summationdisplay k4i[fkd∗ k−f∗ kdk]ξk/braceleftbig 8igkh·∂kxh ×∂kyhA1(k,ω)+/Omega1xyA2(k,ω)/bracerightbig , (30) where A1(k,ω) andA2(k,ω)a r e A1(k,ω)≡/bracketleftbigg C++−ξ2 k−g2 k−\|/epsilon1k\|2 E+E−D++/bracketrightbigg ×{δ(ω−2E+)+δ(ω+2E+)} +/bracketleftbigg C−−−ξ2 k−g2 k−\|/epsilon1k\|2 E+E−D−−/bracketrightbigg ×{δ(ω−2E−)+δ(ω+2E−)} 174511-6INTRINSIC AC ANOMALOUS HALL EFFECT OF . . . PHYSICAL REVIEW B 96, 174511 (2017) 0 2 4 6 8 10 12 14−1. × 10-6−5. × 10-705. × 10-71. × 10-6 /tIm[ H]/[e2/d] 10 20 30 40 50 60−6. × 10-8−4. × 10-8−2. × 10-802. × 10-84. × 10-86. × 10-8 /tIm[ H]/[e2/d] FIG. 4. Numerical results for Im σ(1) H(ω). Left panel: small frequency regime ω/t/lessorequalslant14; right panel: large frequency regime ω/t/greaterorequalslant10. Note that the vertical axis scales of the two ﬁgures are different. The unit of σHise2/¯hd,w i t h dtheˆc-axis lattice spacing of UPt 3. Parameters used are ( t,tz,t/prime,α,μ,/Delta1 0,η)=(1,−4,1,2,12,0.1,1.0). +/bracketleftbigg C+−−ξ2 k−g2 k−\|/epsilon1k\|2 E+E−D+−/bracketrightbigg ×{δ[ω−(E++E−)]+δ[ω+(E++E−)]}, (31a) A2(k,ω)≡B+−{δ[ω−(E++E−)]+δ[ω+(E++E−)]}. (31b) Theksummation in Eq. ( 30) is calculated numerically for eachωand the results are plotted in Fig. 4over two different ranges of ω/t so that the details at larger ω/t, where \|Imσ(1) H\| is smaller, can be clearly seen. Im σ(1) H(ω) has several sign changes as a function of ωbecause the different factors in Eq. ( 30) change sign at different kpositions with different quasiparticle energies. Also, note that Im σ(1) H(ω) is nonzero for arbitrarily small ωsince the external ﬁeld can excite quasiparticle pairs at arbitrarily small energy near the line nodes in the superconducting gap. Although σ(1) H(ω) vanishes asω→0, this feature is not visible in Fig. 4(left panel) because the crossover to small- ωbehavior occurs at very small frequency ω< 0.01t(see Fig. 6 of Ref. [ 35]). The real part Re σ(1) H(ω) can be computed from the data for Imσ(1) H(ω) by the Kramers-Kronig transformation Reσ(1) H(ω)=2 πP/integraldisplay∞ 0νImσ(1) H(ν) ν2−ω2dν, (32) where Pstands for Cauchy principal value integral. The results for Re σ(1) H(ω) are plotted in Fig. 5. In the right panel of Fig. 5, the red dashed line is an exact high-frequency asymptoticresult, whose expression is given by [ 49] σH(ω→∞ )=i ω2/angbracketleft[ˆJx,ˆJy]/angbracketright+O/parenleftbigg1 ω4/parenrightbigg , (33) where [ ˆJx,ˆJy] is an equal-time commutator and the expectation value /angbracketleft.../angbracketrightis with respect to the ground state of the BdG Hamiltonian. In Appendix C, we compute /angbracketleft[ˆJx,ˆJy]/angbracketrightto ﬁrst order in ˆH/primeand ﬁnd /angbracketleft[ˆJx,ˆJy]/angbracketright(1)≈−i2.2×10−5t2e2/(¯hd). Similar to Im σ(1) H(ω), Reσ(1) H(ω) has further structure at very low frequency, ω< 0.01t. It saturates to a constant with a zero slope as ω→0. Again, due to the large frequency range in Fig. 5(left panel), this feature is not visible. B. Discussions of σ(1) H From Eq. ( 24), we can identify the necessary ingredients forσ(1) Hto be nonzero. As emphasized previously, both the chiral f-wave and the chiral d-wave components need to be present. In particular, the dependence of σ(1) Hon these two parameters is through the combination i[fkd∗ k−f∗ kdk], which is proportional to the chirality. Under time reversal, this combination, and consequently σ(1) H, changes sign. This can be seen explicitly from the fact that under time reversal,/Delta1 12(k)→−/Delta1∗ 12(−k),/Delta121(k)→−/Delta1∗ 21(−k), and 2 i[fkd∗ k− f∗ kdk]=/Delta121(k)/Delta1∗ 21(k)−/Delta112(k)/Delta1∗ 12(k). This is the only com- bination quadratic in /Delta112and/or /Delta121that is odd under time reversal. It is also this term that makes the order parameter ˆ/Delta1(k) nonunitary. The second important ingredient for σHis the complex intersublattice hopping /epsilon1ksince both velocity terms appearing in Eq. ( 24),h·∂kxh×∂kyhand/Omega1xy, vanish if /epsilon1kis real. These 0 2 4 6 8 10 12 14−5. × 10-705. × 10-71. × 10-6 /tRe( H)/[e2/d] 20 40 60 80−4. × 10-8−2. × 10-802. × 10-84. × 10-8 /tRe( H)/[e2/d] FIG. 5. Numerical results for Re σ(1) H(ω). Left panel: small frequency regime; right panel: large frequency regime. Note that the scales of the vertical axes in the two ﬁgures are different. In the right ﬁgure, the red dashed line is a high-frequency asymptotic result. Parameters usedare the same as in Fig. 4. 174511-7W ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) velocity terms are consistent with another general requirement forσHto be nonzero in the multiband mechanism. Namely, some antisymmetrized products of the velocity operatorsv x abvy cd−vy abvx cd(where a,b label orbitals or, in our case, sublattices) need to be nonzero. Note that SOC, gk, is not necessary for a nonzero σH. Of the two terms in Eq. ( 24), only the ﬁrst term vanishes if gk=0. The second term, with /Omega1xy, only depends on gkthrough the Bogoliubov quasiparticle energies E±and remains nonzero if the SOC is absent. The two key ingredients identiﬁed above, the mixing of the chiral f- andd-wave order parameters and the complex intersublattice hopping /epsilon1k, are both direct consequences of the nonsymmorphic symmetry of UPt 3. They would both be absent if the lattice were symmorphic. In this sense, the termsthat we have identiﬁed for σ Hare unique to nonsymmorphic chiral superconductors. The two terms in Eq. ( 24) can be represented by Feynman diagrams, which are shown in Fig. 6. For each diagram in Fig. 6, the time-reversed diagram needs to be subtracted. There are two types of diagrams. In Fig. 6(a), only one of the two vertices involves two different orbitals, while in Fig. 6(b) both the vertices involve transitions between different orbitals.Of the two terms in Eq. ( 24), the term ∝/Omega1 xyonly contributes to Fig. 6(b), while the other term, ∝h·∂kxh×∂kyh, is a mixture of Figs. 6(a) and6(b). This is because h·∂kxh×∂kyhcan be written as a sum of /epsilon1k∂kx/epsilon1∗ k∂kygk+/epsilon1∗ k∂kxgk∂ky/epsilon1k−{x↔y} andgk∂kx/epsilon1k∂ky/epsilon1∗ k−{x↔y}, of which the former and latter correspond to Figs. 6(a) and 6(b), respectively. In the band basis, the /Omega1xyterm in Eq. ( 24) corresponds to Fig. 6(a) (with i,jnow labeling bands), rather than Fig. 6(b) as in the orbital basis, while the whole h·∂kxh×∂kyhterm corresponds to Fig. 6(b). It is clear in the band basis that both Figs. 6(a) and 6(b) vanish if the interband pairing is zero, similar to what was found in Ref. [ 19]. Note that Fig. 6(b) type of diagram is absent in Ref. [ 19] because the model studied there has a real interorbital hopping/epsilon1 k, which makes the contribution from Fig. 6(b) with the photon polarization ( i,j)=(x,y) exactly cancel the same diagram with ( i,j)=(y,x). On the other hand, Fig. 6(a) vanishes in the current model unless /epsilon1kis complex, while ω+ν ωi ja,s a,sb,¯s a,¯sω+ν ωi jb, s a,sa,¯s b,¯s (a) (b) FIG. 6. Diagrammatic representation of the nonvanishing con- tributions to σH, where wiggly lines are photons and double solid lines with arrows are Green’s functions given by Eq. ( 16). The photon polarization is labeled by i,j=x,y.a,bare sublattice labels, andsis the spin label. If s={ ↑,↓},t h e n ¯s={ ↓,↑}. Note that, in each diagram, the spin labels on a right vertex are opposite to that on the corresponding left vertex. This is because, in eachdiagram, each Green’s function contributes one superconducting order parameter that pairs electrons of opposite spin, while all normal-state Hamiltonian matrix elements, including SOC, onlyconnect electrons of the same spin.it survives in Ref. [ 19] for real interorbital hopping, due to the different way the interorbital pairing arises in the twomodels. σ H(ω) also needs to obey the following two sum rules [50,51]: /integraldisplay∞ 0dωReσH(ω)=0, (34) /integraldisplay∞ −∞dωωImσH(ω) π=−i/angbracketleft[ˆJx,ˆJy]/angbracketright, (35) where Eq. ( 35) is analogous to the well-known optical conductivity f-sum rule. In Appendix C, we show these sum rules are satisﬁed, both analytically and numerically, by σ(1) H(ω). Lastly, we mention that the Hall conductivity, quite gen- erally, needs to satisfy several symmetry constraints. Undertime reversal, all vertical mirror reﬂections, and particle- hole interchange, σ Hmust reverse its sign. Both σ(1) Hgiven in Eq. ( 24), and the full Green’s function result of σH given in Appendix Aare consistent with these symmetry constraints. IV . ESTIMATION OF THE KERR ROTATION ANGLE θK From the numerical results of σH(ω), the Kerr rotation angle θKcan be estimated using Eq. ( 1), which also involves the complex index of refraction n(ω). Here, we use our results to estimate the Kerr angle for UPt 3, where θKwas measured [ 5] at a laser frequency ω≈0.8e V . We ﬁrst estimate n(ω=0.8 eV) from experimental data. By deﬁnition n(ω)=√/epsilon1(ω), where /epsilon1(ω) is related to the conductivity σ(ω)b y/epsilon1(ω)=/epsilon1∞+i4πσ(ω)/ωand/epsilon1∞is the high-frequency limit dielectric constant. We extract σ(ω= 0.8e V ) ≈(1.7+i0.4)×1015s−1from the experimental data of Ref. [ 52]. Taking /epsilon1∞=1, we obtain /epsilon1(ω=0.8e V ) ≈ −3.1+i17.5, which gives an index of refraction n(ω=0.8e V ) ≈2.7+i3.2. (36) To obtain a value for σH(ω≈0.8 eV), we need to estimate the in-plane hopping parameter tin eV since we have scaled all energies by t. This can be obtained by comparing the normal-state band dispersions of our two-band model alongthe symmetry directions A −L−H−Ai nt h e k z=πplane to the corresponding ﬁrst-principles calculation results fromRef. [ 53]. The comparison gives t≈36 meV (for details, see Appendix D). This value of tcorresponds to ω/t≈22.2a t ω=0.8 eV. From our numerical results for σ H(ω)i nF i g s . 4 and5we obtain, at ω/t≈22.2, σH(ω≈0.8e V ) ≈− (2.3+i5.1)×10−8e2 ¯hd, (37) where d=4.9˚Ai st h e c-axis lattice spacing of UPt 3.F r o m Eqs. ( 36), (37), and ( 1), the Kerr angle is then θK≈34×10−9rad. (38) Our estimated θKis about an order of magnitude smaller than the experimental value of about 350 nanoradians mea-sured at the lowest temperatures [ 5]. However, it may still 174511-8INTRINSIC AC ANOMALOUS HALL EFFECT OF . . . PHYSICAL REVIEW B 96, 174511 (2017) be a signiﬁcant contribution to the explanation for the Kerr measurement on UPt 3[5] given that there are uncertainties in the optical constants, the band parameters, and the magnitudeof/Delta1 0used for this estimate. We brieﬂy comment on these uncertainties. Ideally, one would like measurements of n(ω)o nt h es a m e crystal used for the Kerr measurements. Other optical data onUPt 3would give somewhat different results [ 54–56], although we estimate that the uncertainty in the optical data is unlikelyto change the estimated Kerr angle by more than a factor of 3or so. As to the band parameters, uncertainty comes both from the value of tand from the fact that a very simpliﬁed nearest- neighbor hopping model has been used to approximate thetwo bands which give rise to the starﬁsh Fermi surface. Thislikely introduces a larger uncertainty than that from errors inthe estimate of n(ω). The other parameter that can greatly affect the size of θ K is/Delta10, the amplitude of the gap function written in the orbital basis. Note that /Delta10is not the gap that one would observe in tunneling measurements. Deﬁning /Delta1gas the position of the coherence peak in the Bogoliubov quasiparticle densityof states spectrum, one ﬁnds /Delta1 g≈0.16/Delta10(see Fig. 6 of Ref. [ 35]). Experiments have found values for /Delta1gof 0.04 meV [ 57], 0.1 meV [ 58], and more recently 0.5 meV [ 31]. The parameters we used, taken from Yanase [ 35], with t= 36 meV , correspond to /Delta1g=0.58 meV , roughly consistent with the most recent experimental value. Since the Kerr anglescales quadratically with the gap magnitude, smaller valuesof/Delta1 gwould give much smaller values of θK. For example, setting Tc=0.53 K, we ﬁnd /Delta1g≈0.11 meV for our model in the weak-coupling limit, which would reduce θKby a factor of 26. Lastly, there are several other Fermi surface sheets that we did not take into account, which might contribute to θK. These additional contributions could either increase or decrease thetotalθ K, depending on their relative magnitude and sign. With these uncertainties in mind, we conclude that the θK that we have identiﬁed here can be signiﬁcant for explaining the Kerr measurement on UPt 3, even if it is not large enough to account for the whole experimentally observed signal. Furtherexperiments and theoretical studies are needed to resolve theabove uncertainties. V . CONCLUSION AND DISCUSSIONS To summarize, by considering a simpliﬁed two-band model that results from ABAB stacking for the starﬁshlike Fermisurface of UPt 3, we have identiﬁed a contribution to the ac anomalous Hall conductivity for UPt 3within the intrinsic multiband chiral superconductivity mechanism. The Kerrangle estimated from the computed Hall conductivity canbe signiﬁcant for understanding the Kerr measurement onUPt 3. This mechanism requires nonzero interband pairing. Since intraband and interband pairing are indistinguish-able at the six points on the k z=±πplane where the starﬁshlike Fermi surfaces of UPt 3intersect, this is a useful model for studying the multiband chiral superconductivitymechanism.We have identiﬁed two crucial ingredients for the nonzero σ H: a complex intersublattice hopping between U sites and a superconducting order parameter that involves mixing betweenchiral f-wave and chiral d-wave pairing. Both of these are consequences of the nonsymmorphic group symmetry of theUPt 3crystal lattice. If the intersublattice hopping is real or if one of the chiral f- andd-wave pairing components is absent, thenσHandθKvanish. This is a generalization of, albeit some- what distinct from, the multiband chiral superconductivity mechanism for the anomalous ac Hall effect in a chiral p-wave superconductor [ 19]. The σHandθKcontribution that we have discussed here can also be applied to other nonsymmorphicchiral superconductors. In our analysis we have identiﬁed two types of terms that contribute to σ H(ω) at each kpoint, as can be seen from Eq. ( 24). One term does not require SOC, while the other does. The two make comparable contributions to σH. However, these two contributions in general can have different signs atdifferent kpoints, which results in multiple sign changes of σ H(ω)a saf u n c t i o no f ω. Because of these sign changes, the estimated Kerr angle can be sensitive to the band parametersas well as to the laser frequency used in the Kerr measurement. Therefore, future Kerr measurements at different frequencies would be very helpful in determining how relevant the Kerrangle contribution identiﬁed here is to UPt 3. We should mention that in our calculation we have ne- glected a small chiral p-wave component pairing in the original proposed superconducting order parameter of Ref. [ 35]. This component is also symmetry allowed but is expected to beenergetically less favorable compared with the dominant chiralfanddcomponents. In the two-band model, we consider, thisp-wave component alone can also give rise to a nonzero σ H(ω). This contribution relies on the nonunitary nature of the p-wave pairing (it pairs only one spin component if η=1), and requires nonzero SOC and complex intersublattice hopping.Presumably, the admixture of this neglected small p-wave component will not signiﬁcantly alter the estimated Kerr anglesimply because its pairing amplitude is thought to be verysmall. Recently, the authors of Ref. [ 59] suggested that the Kerr rotation in UPt 3can not be understood without invoking pairing in completely ﬁlled or empty bands because thelaser frequency used in the Kerr angle measurement [ 5], ω≈0.8 eV, is bigger than the normal-state bandwidth of the partially ﬁlled bands of UPt 3. However, this does not need to be the case for two reasons. First, since the incidentphoton breaks a Cooper pair and generates two Bogoliubovquasiparticles, the maximum energy cost is not the bandwidth,but twice the energy difference between the Fermi level andthe bottom or top of the band (whichever is greater). FromRef. [ 60], this maximum energy along the symmetry direction A−L−H−Ai nt h e k z=πplane is about 0 .68 eV, while from Ref. [ 53], this is about 0 .84 eV. The latter (which we used to determine the hopping tin our model) allows energy- conserving transitions within the band at 0.8 eV . Second, bothReσ H(ω) and Im σH(ω) can make signiﬁcant contributions to θK. Even if the laser frequency is larger than the excitation energy of two quasiparticles within the band, Re σH(ω) will still be nonzero at ω=0.8 eV. Consequently, the observation 174511-9W ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) of nonzero θKin UPt 3at 0.8 eV may still be understood within a model of partially ﬁlled bands. ACKNOWLEDGMENTS We would like to thank T. Timusk and S. Kivelson for helpful discussions. This work is supported in part byNSERC (C.K. and Z.W.), the Canada Research Chair program(C.K.), the National Science Foundation under Grant No.NSF PHY11-25915 (A.J.B., C.K., G.Z.), the Gordon andBetty Moore Foundation’s EPiQS Initiative through GrantNo. GBMF4302 (A.J.B. and C.K.), the ANR-DFG grantFermi-NESt (G.Z.), and a grant from the Simons Foundation(Grant No. 395604 to C.K.). A.J.B., C.K., and G.Z. greatlyappreciate the hospitality provided by the Kavli Institute forTheoretical Physics at UCSB and (for A.J.B., C.K., and Z.W.)the hospitality of the Stanford Institute for Theoretical Physics,where part of the work was completed. APPENDIX A: FULL GREEN’S FUNCTION CALCULATION OF σH(ω) As mentioned in the main text, the full Green’s function calculation is much more involved than the perturbativecalculation. Here, we present some main steps for the fullcalculation of σ H(ω), omitting detail of derivations. We ﬁrst establish some notation. We denote the four Bogoliubov quasiparticle energies of the BdG Hamiltonian ˆH(a)(k), from Eq. ( 12a) of the main text, as Ei, withi={1,2,3,4}.T h eEiare solutions to det{ω−ˆH(a)(k)}=0, (A1) which can be expanded as ω4+αkω2+βkω+γk=0, (A2) where the three coefﬁcients are given by αk=− 2/parenleftbig ξ2 k+g2 k+\|fk\|2+\|dk\|2+\|/epsilon1k\|2/parenrightbig , (A3a) βk=4i(fkd∗ k−f∗ kdk)gk, (A3b) γk=/parenleftbig ξ2 k−g2 k−\|/epsilon1k\|2+\|fk\|2+\|dk\|2/parenrightbig2+4\|dk\|2\|/epsilon1k\|2 +(\|fk\|2−\|dk\|2)(/epsilon1k+/epsilon1∗ k)2+(f∗ kdk−fkd∗ k)2 −i(fkd∗ k+f∗ kdk)/parenleftbig /epsilon12 k−(/epsilon1∗ k)2/parenrightbig . (A3c) Equation ( A2) is a quartic equation for ωrather than a quadratic equation in ω2due to the βkωterm. Because of this, the solutions Eido not occur as {+E,−E}particle-hole pairs. However, this does not contradict the particle-hole symmetryof the full superconducting BdG Hamiltonian which is restoredwhen ˆH (a)is combined with the other 4 ×4 block ˆH(b)(k), given in Eq. ( 12b), to form the full ˆHBdG. Also, because of the βkωterm in Eq. ( A2), the expressions for the Ei, in terms of the three coefﬁcients {αk,βk,γk}, are much more complicated than in the case of βk=0. For brevity we will not present them here. With the coefﬁcients {αk,βk,γk}andEideﬁned above, we can now write the ﬁnal result for σH(ω) as follows: σH(ω)=/summationdisplay k16iξkh·∂kxh×∂kyh/braceleftbig/tildewideF1(k,ω)+/parenleftbig ξ2 k−g2 k−\|/epsilon1k\|2/parenrightbig/tildewideF2(k,ω)/bracerightbig +4iξk(fkd∗ k−f∗ kdk)/Omega1xy/tildewideF3(k,ω) −8ξkOh(k)/tildewideF2(k,ω), (A4) where /Omega1xywas deﬁned in Eq. ( 25). In Eq. ( A4), the three frequency-dependent functions are deﬁned as /tildewideF1(k,ω)=−1 24/summationdisplay i=1\|Ei\|ω4−ω2/parenleftbig 4E2 i−αk/parenrightbig +/parenleftbig 3E4 i−αkE2 i+3γk/parenrightbig /producttext4 j=1,j/negationslash=i(Ej−Ei){(Ej−Ei)2−ω2}, (A5a) /tildewideF2(k,ω)=−1 24/summationdisplay i=1\|Ei\|−2ω2+/parenleftbig 9E2 i+αk+γk/E2 i/parenrightbig /producttext4 j=1,j/negationslash=i(Ej−Ei){(Ej−Ei)2−ω2}, (A5b) /tildewideF3(k,ω)=−1 24/summationdisplay i=1sgn(Ei)ω4−ω2/parenleftbig 6E2 i−αk/parenrightbig +/parenleftbig 12E4 i+4γk/parenrightbig /producttext4 j=1,j/negationslash=i(Ej−Ei){(Ej−Ei)2−ω2}. (A5c) /tildewideF1,/tildewideF2, and/tildewideF3are connected to the three functions Fi(k,ω), that we introduced in our perturbative calculations, by /tildewideF1(k,ω)=βk 2F1(k,ω)+O/parenleftbig β3 k/parenrightbig ,/tildewideF2(k,ω)=−βk 2E+E−F2(k,ω)+O/parenleftbig β3 k/parenrightbig ,/tildewideF3(k,ω)=F3(k,ω)+O/parenleftbig β2 k/parenrightbig , (A6) where E±are the two Bogoliubov quasiparticle energies of the zeroth-order Hamiltonian [see Eq. ( 19)]. From these relations we see that the parameter that controls our perturbativecalculation is β krather than simply dk. TheOh(k)/tildewideF2(k,ω) term in Eq. ( A4) contains terms of higher powers, fourth order in fkanddk, compared with the other terms that are second order in fkanddk(ignoring thefkdependence through the quasiparticle energies E±). This is clear from Eq. ( A3b), the expression for βk, and from Oh(k)=(\|fk\|2+\|dk\|2)gk/Omega1xy −(\|fk\|2−\|dk\|2)/braceleftbig Re[/epsilon1k]/Omega1(1) xy+Im[/epsilon1k]/Omega1(2) xy/bracerightbig +(fkd∗ k+f∗ kdk)/braceleftbig Re[/epsilon1k]/Omega1(2) xy−Im[/epsilon1k]/Omega1(1) xy/bracerightbig ,(A7) 174511-10INTRINSIC AC ANOMALOUS HALL EFFECT OF . . . PHYSICAL REVIEW B 96, 174511 (2017) 1st order Full G 0 2 4 6 8 10 12 14−1.×10-6−5.×10-705.×10-71.×10-6 /tIm[ H]/[e2/d] 1st order Full G 10 20 30 40 50 60−6.×10-8−4.×10-8−2.×10-802.×10-84.×10-86.×10-8 /tIm[ H]/[e/d]2 FIG. 7. Comparison between the numerical results for Im σ(1) H(thick black line) and that for Im σH(dashed red line). Left panel: small ω/t/lessorequalslant14; right panel: large ω/t/greaterorequalslant10. Notice that the vertical axis scales of the two panels are different. Parameters used are the same as in Fig. 4. where we have introduced two additional antisymmetrized velocity products /Omega1(1) xyand/Omega1(2) xy, deﬁned as follows: /Omega1(1) xy=2/braceleftbig ∂kxgk∂kyIm[/epsilon1k]−∂kxIm[/epsilon1k]∂kygk/bracerightbig ,(A8a) /Omega1(2) xy=2/braceleftbig ∂kxgk∂kyRe[/epsilon1k]−∂kxRe[/epsilon1k]∂kygk/bracerightbig .(A8b) FromσH(ω+iδ)i nE q .( A4) we can derive its imaginary part Im σH(ω). Then, we can numerically evaluate Im σH(ω) and compare the results with our perturbation results for Imσ(1) H(ω) in the main text. The comparison is given in Fig. 7. We see that the two are quite different for ω/lessorsimilar2t, but they are essentially indistinguishable for ω/greaterorsimilar4t. We can also compute Re σH(ω) by the Kramers-Kronig transformation and compare the results with Re σ(1) H, presented in the main text. This comparison is shown in Fig. 8.A g a i na t ω/greaterorsimilar4t, the two agree well. APPENDIX B: DERIVATION OF σ(1) H In order to compute σ(1) H, using Eqs. ( 14) and ( 23), we introduce the function F(1) xy(k;iωn,iνm) such that π(1) xy(iνm)−π(1) yx(iνm)=T/summationdisplay k,ωnF(1) xy(k;iωn,iνm). (B1) From the expression for π(1) xy(iνm)i nE q .( 23), we can write F(1) xyas follows: F(1) xy≡{Tr[ ˆvxˆG(0)(k,iωn+iνm)ˆvyˆG(1)(k,iωn)] +{(0)↔(1)}} − {x↔y}. (B2)This expression contains traces of products of 4 ×4 matrices ˆvx,ˆG(0),ˆvy, and ˆG(1). To complete these traces we decompose the 4×4 matrices into linear combinations of σατβ, where σα andταare Pauli matrices for the sublattice and particle-hole Nambu subspaces, respectively. Then, ˆvx=vx ασατ0,ˆvy=vy ασατ0, (B3) ˆG(0)=G(0) αβσατβ,ˆG(1)=G(1) αβσατβ. (B4) We choose the following basis for the above decomposition: σα≡(σ0,σ+,σ−,σ3), (B5) τα≡(τ0,τ+,τ−,τ3), (B6) where σ±=(σ1+iσ2)/√ 2 and τ±=(τ1+iτ2)/√ 2. In Eq. ( B4), and elsewhere, summations over repeated indices are assumed. In order to extract the coefﬁcients vx α,vy α,G(0) αβ, andG(1) αβit will be convenient to introduce both the conjugate ofσα, denoted as ¯ σα, and also the conjugate of α, denoted as ¯α. Their deﬁnitions are ¯σα≡[σα]†=(σ0,σ−,σ+,σ3)≡σ¯α. (B7) Different components of the 4-vectors σαand ¯σαsatisfy an orthonormal relation: Tr {σα¯σβ}=2δα,β. Using this relation we can obtain the coefﬁcients in Eq. ( B4) as follows: vx α=1 4Tr[ ˆvx¯σατ0],vy α=1 4Tr[ ˆvy¯σατ0], (B8) G(0) αβ=1 4Tr[ˆG(0)¯σα¯τβ],G(1) αβ=1 4Tr[ˆG(1)¯σα¯τβ]. (B9) 1st order Full G 0 2 4 6 8 10 12 14−5.×10-705.×10-71.×10-6 /tRe( H)/[e2/d] 1st order Full G 10 20 30 40 50 60 70 80−4.×10-8−2.×10-802.×10-84.×10-8 /tRe( H)/[e2/d] FIG. 8. Comparison between the numerical results for Re σ(1) H(thick black line) and that for Re σH(dashed red line). Left panel: small ω/t/lessorequalslant14; right panel: large ω/t/greaterorequalslant10. Note that the vertical axis scales of the two panels are different. 174511-11W ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) Substituting Eq. ( B4) into the expression for F(1) xyin Eq. ( B2) gives F(1) xy=/braceleftbig vx αG(0) βγvy α/primeG(1) β/primeγ/primeTr [σασβσα/primeσβ/prime]T r[τ0τγτ0τγ/prime] +{(0)↔(1)}/bracerightbig −{x↔y}, (B10) where we have suppressed the arguments of the Green’s functions. However, it should be kept in mind that in each ofthe two Green’s function products, the ﬁrst Green’s functionshould be evaluated at ( k,iω n+iνm), while the second should be evaluated at ( k,iωn). The trace over ταPauli matrix products in Eq. ( B10) is trivial: Tr [ τ0τγτ0τγ/prime]=2δγ,γ/prime. The other trace, Tr [σασβσα/primeσβ/prime], is nonzero only for two cases: (1) all four indices {α,β,α/prime,β/prime}are different from each other; (2) the four indices consist of two identical pairs. However, the lattercontribution is even with respect to the interchange x↔yand therefore contributes zero to F(1) xyafter the antisymmetrization −{x↔y}. Therefore, the only nonzero contribution comes from the case with all four indices different. Because each ofthe indices {α,β,α /prime,β/prime}can take four possible values {0,+,−, 3}there are 4! =24 different terms in total. However, half of them are zero because of the following three identities: G(0) +γG(1) −¯γ−G(0) −γG(1) +¯γ+{(0)↔(1)}=0,(B11a) G(0) −γG(1) 3¯γ−G(0) 3γG(1) −¯γ+{(0)↔(1)}=0,(B11b) G(0) 3γG(1) +¯γ−G(0) +γG(1) 3¯γ+{(0)↔(1)}=0.(B11c) Then, we are left with F(1) xy=8/braceleftbig/braceleftbig vx −vy 3−vx 3vy −/bracerightbig/braceleftbig G(0) 0γG(1) +¯γ−G(0) +γG(1) 0¯γ+{(0)↔(1)}/bracerightbig +/braceleftbig vx 3vy +−vx +vy 3/bracerightbig/braceleftbig G(0) 0γG(1) −¯γ−G(0) −γG(1) 0¯γ+{(0)↔(1)}/bracerightbig +/braceleftbig vx +vy −−vx −vy +/bracerightbig/braceleftbig G(0) 0γG(1) 3¯γ−G(0) 3γG(1) 0¯γ+{(0)↔(1)}/bracerightbig/bracerightbig . (B12) In obtaining this equation we have used the trace identity Tr[ σ0σ+σ−σ3]=2 as well as its permutations. Next we need to complete the the Matsubara summation T/summationtext ωnin Eq. ( B1). This can be done for each of the three terms in Eq. ( B12). The derivations are quite lengthy, and we do not present them here. The ﬁnal results are T/summationdisplay nG(0) 0γG(1) +¯γ−G(0) +γG(1) 0¯γ+{(0)↔(1)}=4i{fkd∗ k−f∗ kdk}ξkgk√ 2/epsilon1kSk(iνm), (B13a) T/summationdisplay nG(0) 0γG(1) −¯γ−G(0) −γG(1) 0¯γ+{(0)↔(1)}=4i{fkd∗ k−f∗ kdk}ξkgk√ 2/epsilon1∗ kSk(iνm), (B13b) T/summationdisplay nG(0) 0γG(1) 3¯γ−G(0) 3γG(1) 0¯γ+{(0)↔(1)}=4i{fkd∗ k−f∗ kdk}ξkgk2gkSk(iνm)−2i{fkd∗ k−f∗ kdk}ξkTk(iνm).(B13c) For brevity, we have introduced two frequency-dependent functions Sk(iνm) andTk(iνm), which are deﬁned as Sk(iνm)≈M1−/parenleftbig ξ2 k−g2 k−\|/epsilon1k\|2/parenrightbig M2, (B14) Tk(iνm)≈−iνm 2E+E−(E++E−){(E++E−)2+ν2m}, (B15) where the ≈sign means only terms of leading order in fkanddkhave been kept. M1andM2are given by M1=−iνm/braceleftBigg C++ 4E2 ++ν2m+C−− 4E2 −+ν2m+C+− (E++E−)2+ν2m+C/prime +−/braceleftbig (E++E−)2+ν2m/bracerightbig2/bracerightBigg , (B16a) M2=−iνm E+E−/braceleftBigg D++ 4E2 ++ν2m+D−− 4E2 −+ν2m+D+− (E++E−)2+ν2m+D/prime +−/braceleftbig (E++E−)2+ν2m/bracerightbig2/bracerightBigg , (B16b) where C++,C−−,C+−,C/prime +−,D++,D−−,D+−, andD/prime +−are eight νmindependent coefﬁcients. The expressions for C++,C−−, C+−,D++,D−−, andD+−were given in Eqs. ( 29a)–(29e). The other two coefﬁcients are as follows: C/prime +−=D/prime +−=−2 (E++E−)(E+−E−)2. (B17) Notice that both the C/prime +−term in Eq. ( B16a ) and the D/prime +−term in Eq. ( B16b ) have a second-order pole at νm=±i(E++E−) on the complex νmplane, while all other terms have ﬁrst-order poles. The second-order poles appear only in the perturbativecalculation but not in the full ˆGcalculation. Numerically, wefound that the second-order pole contributions to σ(1) Hfrom Eqs. ( B16a ) and ( B16b ) are negligible at ω/greatermuchα, where α is the SOC coupling strength. Hence, we will ignore themhereafter. Performing a Wick rotation iν m→ω+iδ,w es e e thatSk(ω)/ωandTk(ω)/ωare given by Eqs. ( 26) and ( 27). 174511-12INTRINSIC AC ANOMALOUS HALL EFFECT OF . . . PHYSICAL REVIEW B 96, 174511 (2017) Now, inserting the results from Eqs. ( B13a )–(B13c )i n t o the expression for F(1) xyin Eq. ( B12) we obtain T/summationdisplay nF(1) xy=64i{fkd∗ k−f∗ kdk}ξkgkSk(iνm)h·∂kxh ×∂kyh+8{fkd∗ k−f∗ kdk}ξkTk(iνm)/Omega1xy, (B18) w h e r ew eh a v eu s e d h·∂kxh×∂kyh =/bracketleftbig vx −vy 3−vx 3vy −/bracketrightbig /epsilon1k/√ 2+/bracketleftbig vx 3vy +−vx +vy 3/bracketrightbig /epsilon1∗ k/√ 2 +/bracketleftbig vx +vy −−vx −vy +/bracketrightbig gk, (B19) and also introduced a notation /Omega1xyfor the following antisym- metrized velocity factor: /Omega1xy≡− 2i[vx +vy −−vx −vy +]=−i[∂kx/epsilon1k∂ky/epsilon1∗ k−∂kx/epsilon1∗ k∂ky/epsilon1k]. (B20) With these compact notations one can substitute T/summationtext nF(1) xy from Eq. ( B18) back into Eq. ( B1) and obtain the ﬁnal expression for the Hall conductivity as a function of frequencygiven in Eq. ( 24). APPENDIX C: ASYMPTOTIC RESULT FOR LARGE ωAND SUM RULES In this Appendix, we compute /angbracketleft[ˆJx,ˆJy]/angbracketrighton the right- hand side of Eq. ( 33) for the BdG Hamiltonian ˆH(a)(k) in Eq. ( 12a) up to ﬁrst order in ˆH/prime. Denote the ba- sis of the Hamiltonian ˆH(a)(k)f r o mE q .( 12a)a s/Psi1≡ (/Psi11,/Psi12,/Psi13,/Psi14)T. Then, the current operator can be written as ˆJi=/summationtext k/summationtext αβ/Psi1† α(k)vi αβ/Psi1β(k), with i={x,y}. The velocity operator matrix vi αβis given in Eq. ( 22). Using the fact that the equal-time expectation value /angbracketleft/Psi1† α/Psi1β/angbracketright=T/summationtext nˆGβα(k,iωn), we obtain /angbracketleft[ˆJx,ˆJy]/angbracketright=/summationdisplay kT/summationdisplay n{A{G11−G22+G33−G44} +B{G21+G43}−B∗{G12+G34}},(C1) withAandBgiven by A=∂kx/epsilon1k∂ky/epsilon1∗ k−∂kx/epsilon1∗ k∂ky/epsilon1k, (C2a) B=2(∂kxgk∂ky/epsilon1k−∂kx/epsilon1k∂kygk). (C2b) On the right-hand side of Eq. ( C1), all Green’s function matix elements are evaluated at ( k,iωn). In Eq. ( C1), if we use the zeroth-order result G(0) αβfor all the Green’s function matrix elements, then we obtain/angbracketleft[ˆJx,ˆJy]/angbracketright(0)=0. This is consistent with Eq. ( 33) and the fact thatσ(0) H(ω)≡0. The nonzero /angbracketleft[ˆJx,ˆJy]/angbracketrightcomes from the next-order contri- bution: /angbracketleft[ˆJx,ˆJy]/angbracketright(1). Substituting the matrix elements of the ﬁrst-order Green’s function ˆG(1)≡ˆG(0)ˆH/primeˆG(0)into Eq. ( C1) and completing the Matsubara summation /angbracketleft[Jx,Jy]/angbracketright(1)=i/summationdisplay k−2iξk(fkd∗ k−f∗ kdk) E+E−(E++E−) ×/braceleftbigg /Omega1xy+8igkh·∂kxh×∂kyh (E++E−)2/bracerightbigg ,(C3) where /Omega1xyis deﬁned in Eq. ( 25). The remaining ksummation in Eq. ( C3) can be evaluated numerically and the ﬁnal result is/angbracketleft[Jx,Jy]/angbracketright(1)≈−i2.2×10−5t2e2/(¯hd). Then, Eq. ( 33) becomes σ(1) H(ω→∞ ) e2/¯hd=2.2×10−5 (ω/t)2+O/parenleftbigg1 (ω/t)4/parenrightbigg . (C4) It is also possible to perform the integral in Eq. ( 35) analytically using Im σ(1) H(ω)f r o mE q .( 30). The result is identical to −itimes Eq. ( C3). Similarly, the integral of Eq. ( 34) can be performed analytically using Eqs. ( 24)–(28c). The zero result follows from the analytic structure of theF i(k,ω)i nE q s .( 28a)–(28c). We also numerically evaluate the two sides of Eqs. ( 34) and ( 35) using the data from Figs. 4 and5and conﬁrm that Eqs. ( 34) and ( 35) are well satisﬁed. APPENDIX D: ESTIMATION OF THE NN HOPPING t We plot the two normal-state energy band dispersions along high-symmetry directions in Fig. 9. From the dispersions along A −L−H−A, the corresponding bandwidth in the kz=0 plane is W≈14t. We can ﬁt this to the ﬁrst-principles calculation results from Ref. [ 53]. From the Supplemental Material Fig. S1(b) of Ref. [ 53], we estimate that the bandwidth of the dispersions along A −L−H−Ai s W≈0.5e V . Therefore, as an estimation, 14 t≈0.5e V⇒t≈36 meV. MK A LH A−30−25−20−15−10−505E/t FIG. 9. Normal-state energy dispersions along high-symmetry directions of the hexagonal Brillouin zone at kz=π.T h et w oe n e r g y band dispersions are E(n) ±(k)=ξk±/radicalbig g2 k+\|/epsilon1k\|2, with E(n) +plotted in full blue line and E(n) −in the dashed red line. The two bands are degenerate along the symmetry axis A −L because /epsilon1k=0a tkz=π and the SOC vanishes along these directions as well. 174511-13W ANG, BERLINSKY , ZWICKNAGL, AND KALLIN PHYSICAL REVIEW B 96, 174511 (2017) We note that the bands along /Gamma1−M−K−/Gamma1in Fig. 9 are far below the Fermi energy, which is inconsistent with therealistic ﬁrst-principles calculation result in Ref. [ 53]. This is due to the oversimpliﬁcation of our model which consists ofonly two bands resulting from the ABAB stacking. Due tothis oversimpliﬁcation, the dispersions along /Gamma1−M−K−/Gamma1 are not realistic. In order to estimate how these unrealisticdispersions affect our calculations of θ K, we have recomputedθKby excluding all kpoints that satisfy E(n) ±(k)/lessorequalslantE(n) −(k= H), where E(n) −(k=H) is the band bottom of the dispersions along A −L−H−Ai nF i g . 9. The result is similar to the value obtained in the main text without this truncation. 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PhysRevB.94.235134.pdf	PHYSICAL REVIEW B 94, 235134 (2016) Double quantum dot Cooper-pair splitter at ﬁnite couplings Robert Hussein,1Lina Jaurigue,2Michele Governale,2and Alessandro Braggio1,3,4 1SPIN-CNR, Via Dodecaneso 33, 16146 Genova, Italy 2School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, P .O. Box 600, Wellington 6140, New Zealand 3NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, Pisa I-56127, Italy 4INFN, Sez. Genova, Via Dodecaneso 33, 16146 Genova, Italy (Received 1 August 2016; revised manuscript received 8 November 2016; published 14 December 2016) We consider the subgap physics of a hybrid double-quantum dot Cooper-pair splitter with large single-level spacings, in the presence of tunneling between the dots and ﬁnite Coulomb intra- and interdot Coulomb repulsion.In the limit of a large superconducting gap, we treat the coupling of the dots to the superconductor exactly. Weemploy a generalized master-equation method, which easily yields currents, noise, and cross-correlators. Inparticular, for ﬁnite inter- and intradot Coulomb interaction, we investigate how the transport properties aredetermined by the interplay between local and nonlocal tunneling processes between the superconductor andthe dots. We examine the effect of interdot tunneling on the particle-hole symmetry of the currents with andwithout spin-orbit interaction. We show that spin-orbit interaction in combination with ﬁnite Coulomb energyopens the possibility to control the nonlocal entanglement and its symmetry (singlet/triplet). We demonstratethat the generation of nonlocal entanglement can be achieved even without any direct nonlocal coupling to thesuperconducting lead. DOI: 10.1103/PhysRevB.94.235134 I. INTRODUCTION Recent developments in quantum technologies [ 1–3]h a v e shown an enormous potential for applications. Quantum keydistributions in quantum cryptography [ 4] have became almost a standard technology. This progress was mainly realizedin optical systems. In order to enable the full potential ofquantum technologies, spintronics [ 5], and topotronics [ 6], in solid state systems, it is crucial to be able to generateentangled states. A promising route to entanglement gener-ation is offered by hybrid superconducting nanostructures.The enormous advancement in the production and control of nanotubes and nanowires [ 7,8] opened up the possibility to couple nanosystems, in a very controlled way, to super-conductors [ 9–11] taking advantage of their properties such as the spin-orbit (SO) interaction. This type of system hasvery rich physics. For example, the possibility to emulatetopological superconductors in low dimensions with, possibly,the creation of Majorana bound states has clearly shown arevolutionary potential [ 12–15]. Quantum phase transitions and anomalous current-phase relations have been studied inhybrid semiconductor-superconductor devices [ 16–21]. SO interaction in the presence of superconducting correlationsmay lead to the generation of triplet ordering in nanowires[22] or quantum wells [ 23]. Superconductors are a natural source of electron-singlets (Cooper pairs) which may provide nonlocal entangled elec-trons when split [ 5,24–27]. Semiconductor-superconductor- hybrid devices have been the object of experimental studiesinvestigating signatures of nonlocal transport in charge cur-rents and cross-correlations [ 28–30]. Cooper-pair splitting and charge transfer has been inves- tigated in Josephson junctions [ 31–35] and QED cavities [36–39]. Spin entanglement [ 40–45] and electron transport [46–54] in hybrid systems have been theoretically investigated also using full counting statistics (FCS) [ 55–59]. Otherstudies have investigated the effects of external magnetic ﬁelds [ 60] and thermal gradients [ 61,62] on Cooper-pair splitting. Quantum dots increase the efﬁciency of Cooper-pairsplitting since sufﬁciently large intradot Coulomb interactionsuppresses local Cooper-pair tunneling [ 26,29,40,60,63,64]. Finally, the efﬁciency can be improved using spin-ﬁltering[65, 66] which may be also crucial for the nonlocal entangle- ment detection using different witness measures and the testof Bell’s inequalities [ 67]. It has been also discussed how nonlocal entanglement can be detected using transport [ 68], current noises, high-order- and cross-correlations [ 58,69], electrically driven spin resonance [ 70] or with light emission [71]. Recently, the spin-orbit interaction has been proposed as novel ingredient for entanglement detection [ 44,72]. Typically, Cooper-pair splitters based on quantum dots are investigated assuming a very strong on-site Coulombinteraction. In the present paper, we consider the possibility of a weaker Coulomb interaction which complicates the analysis as it introduces additional transport channels. We ﬁnd thatthis is not necessarily a limitation in the creation of nonlocalentanglement, instead it offers a different route to achievenonlocal entanglement in the presence of interdot tunnelingwith or without spin-orbit coupling. The model studied in thispaper is a Cooper-pair splitter based on a double quantum dot (DQD) circuit that is tunnel coupled to one superconductor and to two normal leads, see Fig. 1. This is an extension of the model studied by Eldridge et al. [73], to ﬁnite interdot tunneling and SO interaction. In this work, we investigate theeffect of both local and nonlocal Cooper-pair tunneling on thecurrent and conductance in the presence of ﬁnite Coulombenergies. Finally, we will discuss how interdot tunneling with or without SO interaction affects the generation of nonlocal entanglement. This work is organized as follows. In Sec. II, we introduce the model and the formalism employed for our calculations. 2469-9950/2016/94(23)/235134(13) 235134-1 ©2016 American Physical SocietyHUSSEIN, JAURIGUE, GOVERNALE, AND BRAGGIO PHYSICAL REVIEW B 94, 235134 (2016) l UteikSOl ΓNL ΓNRΓSL ΓSR N NS FIG. 1. Double quantum dot circuit coupled to an s-wave superconductor acting as a Cooper-pair splitter. Electron singlets nonlocally injected by the superconductor ( S) into the double quantum dot can leave the system through opposite normal leads ( N). Hence this system can be operated as a source of nonlocal entangled electron pairs. In Sec. III, we provide an overview of the transport properties in the absence of interdot tunneling. The effect of interdottunneling and SO interaction is discussed in Sec. IV. Finally, Sec. Vis devoted to conclusions. II. MODEL AND MASTER EQUATION A. Model of the hybrid double quantum-dot system The system under consideration, depicted schematically in Fig. 1, consists of two quantum dots tunnel coupled to a common s-wave superconductor and each individually to a separate normal lead [ 26,29]. The double-quantum-dot (DQD) system is modeled by the Hamiltonian HDQD=/summationdisplay α,σ/epsilon1αnασ+/summationdisplay αUαnα↑nα↓+U/summationdisplay σ,σ/primenLσnRσ/prime +/parenleftbiggt 2/summationdisplay σesgn(σ)iφd† LσdRσ+H.c./parenrightbigg , (1) where α=L,R labels the left and right dots, respectively, andσ=↑,↓denotes the spin. The orbital levels /epsilon1αare spin degenerate, and UαandUdenote the intra- and interdot Coulomb interaction, respectively. We deﬁne the numberoperator n ασ=d† ασdασ, where dασis the annihilation operator for an electron with spin σin dot α. The last term in Eq. ( 1) describes interdot tunneling through a barrier with SO coupling[51]. The phase φis the phase acquired by a spin-up electron when tunneling from the right dot to the left one and itcan be expressed as φ≡k SOl/negationslash=0, where the SO strength is measured in terms of the wave number kSO, andlis the interdot distance. Here, we used the convention sgn( ↑)=+1 and sgn( ↓)=−1.1The SO coupling may become relevant for InAs [ 74,75] and InSb [ 76,77] nanowire devices where one ﬁnds values of 1 /kSOof typically 50–300 nm, which are comparable to the typical distance between the two dots in these nanodevices. The model Hamiltonian, Eq. ( 1), gives an accurate description of the system when the single-particlelevel spacings in the dots is large compared to the other energy 1Due to the absence of an applied magnetic ﬁeld, it is possible to choose the spin quantization axis such that the interdot tunneling in the presence of the SO coupling is diagonal in the spin space.scales. In this limit, for kBT/lessmuchUα, at most four electrons can occupy the double-quantum-dot system. The normal leads ( η=L,R) are modeled as fermionic baths while the superconducting lead ( η=S) is described by the mean-ﬁeld s-wave BCS Hamiltonian, Hη=/summationdisplay kσ/epsilon1ηkc† ηkσcηkσ−δη,S/Delta1/summationdisplay k(cη−k↓cηk↑+H.c.).(2) Here, cηkσ(c† ηkσ) are the fermionic annihilation (creation) operators of the leads and /epsilon1ηkare corresponding single-particle energies. Without loss of generality, the pair potential inthe superconductor, /Delta1, is chosen to be real and positive. For convenience, we choose the chemical potential of thesuperconductor to be zero and use it as reference for thechemical potentials of the normal leads. The quantum dots are coupled to the normal leads and the superconductor via the Hamiltonian H DQD-leads =/summationtext ηαHtunnel ηα , where the coupling of dot αwith lead η=L,R,S is described by the standard tunneling Hamiltonian Htunnel ηα=/summationdisplay kσ(Vηαc† ηkσdασ+H.c.). (3) Here,VLR=VRL=0, since the left (right) dot is not directly coupled to the right (left) lead. The effective tunneling rates are/Gamma1 ηα=(2π//planckover2pi1)\|Vηα\|2ρη, where the density of states ρηin lead ηis assumed to be energy independent in the energy window relevant for the transport. For a better readability, we introduce/Gamma1 Nα≡/Gamma1ααto emphasize the coupling to the normal leads with a subscript N. As we are interested in Cooper-pair splitting and in general subgap transport, we assume the superconducting gap to bethe largest energy scale in the system, /Delta1→∞ . In this limit, the quasiparticles in the superconductor are inaccessible andthe superconducting lead can be traced out exactly [ 78–80]. Thus the system dynamics reduces to the effective Hamiltonian[45,56,73] H S=HDQD−/summationdisplay α=L,R/Gamma1Sα 2(d† α,↑d† α,↓+H.c.) −/Gamma1S 2(d† R,↑d† L,↓−d† R,↓d† L,↑+H.c.), (4) where /Gamma1Sdescribes the nonlocal proximity effect. This nonlocal coupling decays with the interdot distance l,a s /Gamma1S∼√/Gamma1SL/Gamma1SRe−l/ξ, with ξbeing the coherence length of the Cooper pairs [ 40]. So, only values 0 /lessorequalslant/Gamma1S/lessorequalslant√/Gamma1SL/Gamma1SR are physically admissible. The second term describes the local Andreev reﬂection (LAR) processes where Cooper pairstunnel locally from the superconductor to dot α. The last term describes cross-Andreev reﬂection (CAR), that is, a nonlocalCooper-pair tunneling process where Cooper pairs split intothe two dots. Due to CAR, electrons leaving the systemthrough opposite normal leads are potentially entangled. Onthe contrary, the LAR process does not contribute to thenonlocal entanglement production. The LAR process is usuallyattenuated by large intradot couplings, U α. Albeit the effective Hamiltonian ( 4) no longer preserves the total particle number for the double-dot system, it stillpreserves the parity of the total occupation,/summationtext ασnασ. A de- composition, HS=Heven S⊕Hodd S, of the system Hamiltonian 235134-2DOUBLE QUANTUM DOT COOPER-PAIR SPLITTER AT . . . PHYSICAL REVIEW B 94, 235134 (2016) into an even and an odd parity sector is provided in Ap- pendix A. In conclusion, the Hilbert space for the proximized double-dot system has the dimension 16. A generalization toinclude more charge states, to treat, for instance, smaller levelspacings or higher temperatures, is straightforward and can betreated within the master-equation approach presented below.Lowest-order corrections in 1 //Delta1can be also included in the system Hamiltonian according to Ref. [ 81]. In the following, we consider the case of the quantum dots weakly coupled to the normal leads in comparison tothe superconducting one, /Gamma1 Sα/greatermuch/Gamma1Nα/prime. In this limit, quantum transport is mainly characterized by the transitions between theeigenstates of H S, the Andreev bound states [ 56,73]. Those tunneling events with the normal leads either adda single charge to the DQD or remove one from it and, thus,change the parity of the DQD. B. Master equation and transport coefﬁcients We calculate the stationary transport properties, such as the current and the conductance, by means of the master-equationformalism using standard FCS techniques. All the relevanttransport properties can be related to the Taylor coefﬁcients ofthe cumulant generating function [ 82–86] and obtained in an iterative scheme [ 87,88]. In this work, we limit our analysis to the current and the differential conductance, however, alsohigher cumulants, such as noise and cross-correlations, can beeasily obtained. The master-equation formalism is derived by using real- time diagrammatics [ 47,89], which is a perturbative approach in the coupling to the normal leads. It is in principle able tohandle also high-order corrections, but since we consider theregime /Gamma1 Nα/lessmuchkBT, we can limit to the ﬁrst order, i.e., Fermi’s golden rule. The tunnel couplings to the superconductor,the charging energies, and the interdot tunneling are treatedexactly within the model under consideration. This leads tothe master equation ˙P a=/summationtext a/prime(wa←a/primePa/prime−wa/prime←aPa)f o rt h e occupation probabilities Paof the eigenstates \|a/angbracketrightof the system Hamiltonian, where wa←a/primeare Fermi golden rule rates. The tunneling rates for the tunneling-in contribution read wασ,in a←a/prime(χ)=e−iχα/Gamma1Nαfα(Ea−Ea/prime)\|/angbracketlefta\|d† ασ\|a/prime/angbracketright\|2.(5) Here,Eaand\|a/angbracketrightrefer to the eigenenergies and the eigenstates ofHS, andfα(/epsilon1)={1+exp[(/epsilon1−μα)/kBT]}−1denotes the Fermi function of normal lead αwith chemical potential μα and temperature T. We only attach [ 82,84] counting variables to the normal leads, χ=(χL,χR). An easy generalization of this methodology to calculate energy and heat ﬂuxes isalso possible, however, for simplicity, we do not explore thisline here [ 86,90,91]. The stationary current through the super- conductor I Scan be easily expressed in terms of the cur- rents through the left and right leads, IS=−IL−IR.T h e tunneling-out contribution can be obtained from the substi-tution {d † ασ,fα(/epsilon1),χα}→{dασ,¯fα(−/epsilon1),−χα}, where ¯fα(/epsilon1)= 1−fα(/epsilon1). Summation over the spin and lead indices yields the full rates wa←a/prime=/summationtext ασ(wασ,in a←a/prime+wασ,out a←a/prime). Single-electron tunneling changes the parity of the system. So, the only transitions that occur are between the eigenstates\|e i/angbracketrightofHSwith even occupation number and those with odd occupation numbers, \|oj/angbracketright. Here, the indices i,j=1,..., 8TABLE I. Choice of the system basis, subdivided into states with even parity (top cell) and states with odd parity (bottom cell). Note that the indices αandσin\|ασ/angbracketrightand\|tασ/angbracketrightrefer to the singly occupied unpaired electron. \|0/angbracketright empty state \|S/angbracketright=1√ 2(d† R↑d† L↓−d† R↓d† L↑)\|0/angbracketright singlet state \|dα/angbracketright=d† α↑d† α↓\|0/angbracketright doubly occupied states \|dd/angbracketright=d† R↑d† R↓d† L↑d† L↓\|0/angbracketright quadruply occupied state \|T0/angbracketright=1√ 2(d† R↑d† L↓+d† R↓d† L↑)\|0/angbracketright unpolarized triplet state \|Tσ/angbracketright=d† Rσd† Lσ\|0/angbracketright polarized triplet states \|ασ/angbracketright=d† ασ\|0/angbracketright singly occupied states \|tασ/angbracketright=d† ασd† ¯α↑d† ¯α↓\|0/angbracketright triply occupied states label the eigenstates of the even and odd parity sectors, respectively. We can write the eigenstates of the even sector inthe basis of Table I, \|e i/angbracketright=ei,0\|0/angbracketright+ei,S\|S/angbracketright+/summationdisplay αei,dα\|dα/angbracketright +ei,dd\|dd/angbracketright+ei,T0\|T0/angbracketright+/summationdisplay σei,T σ\|Tσ/angbracketright.(6) Similarly, the eigenstates of the odd sector can be expressed as \|oj/angbracketright=/summationdisplay ασ(oj,ασ\|ασ/angbracketright+oj,tασ\|tασ/angbracketright). (7) Finally, we can evaluate the matrix elements of the fermionic operators, /angbracketleftoj\|d(†) ασ\|ei/angbracketright, and therewith express the transitions from the state \|ei/angbracketrightto the state \|oj/angbracketrightas woj←ei=/summationdisplay ασ/Gamma1Nα¯fα(Eei−Eoj)eiχα/vextendsingle/vextendsingle/vextendsingle/vextendsingleo∗ j,α¯σei,dα+o∗ j,tα ¯σei,dd +1√ 2o∗ j,¯α¯σ(ei,S−ασei,T0)−ασo∗ j,¯ασei,T σ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/summationdisplay ασ/Gamma1Nαfα(Eoj−Eei)e−iχα/vextendsingle/vextendsingle/vextendsingle/vextendsingleo∗ j,ασei,0+o∗ j,tασei,d¯α −1√ 2o∗ j,t¯ασ(ei,S+ασei,T0)−ασo∗ j,t¯α¯σei,T¯σ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ,(8) with the coefﬁcients ei,a=/angbracketlefta\|ei/angbracketrightandoj,a=/angbracketlefta\|oj/angbracketrightandEei (Eoj) the eigenenergy corresponding to \|ei/angbracketright(\|oj/angbracketright). The bar on the indices indicates their complement, i.e., ¯L=R,¯↑=↓ and so forth. The rate for the inverse transition wei←ojfollows straightforwardly. Stationary transport properties can be obtained in the standard FCS scheme [ 82–84,87,88] from the cumulant gener- ating function S(χ,μ)=limt→∞∂ ∂tln/summationtext aPa(χ,μ) where the terms Pa(χ,μ) are the counting ﬁeld dependent probabilities obtained from the n-generalized master-equation approach. The interested reader can ﬁnd the full details of the formalismin the FCS literature; here, we just sketch some of theresults. In particular, the currents through the normal leadareI α=e0∂S/∂iχ α\|χ=0withα=L,R and the corresponding differential conductance is Gα,β=−e0∂Iα/∂μβ\|χ=0, where 235134-3HUSSEIN, JAURIGUE, GOVERNALE, AND BRAGGIO PHYSICAL REVIEW B 94, 235134 (2016) e0is the electron charge. Both the current [ 87,88] and the conductance [ 86] can be calculated in the usual iterative scheme. One can easily show that the stationary current maybe written as [ 83] I α=−ie0 /planckover2pi1/summationdisplay a,a/prime∂wa←a/prime(χ) ∂χα/vextendsingle/vextendsingle/vextendsingle/vextendsingle χ=0Pstat a/prime (9) withPstat a/primestationary populations obtained from the master equation. This formula can also be obtained in the real-timediagrammatic approach [ 89] in lowest order in the coupling with the normal lead /Gamma1 N. Finally, the FCS formalism can be also generalized to include higher-order corrections [ 84] (cotunneling) in the coupling with the normal leads, but thisis beyond the regime we will consider in the following, i.e.,/Gamma1 Nα/lessmuchkBT. III. TRANSPORT IN ABSENCE OF INTERDOT TUNNELING In this section, we give an overview of how the local and nonlocal proximization affects quantum transport in absenceof interdot tunneling, t=0. In particular, we start discussing the two limits /Gamma1 Sα/greatermuchUandU/greaterorsimilar/Gamma1Sα. Both limits feature a resonant current originating from the CAR process. The formercase of weak interdot Coulomb energy is typically realized in experiments [ 26]. The limit of strong interdot Coulomb energy additionally permits to study resonant currents whichare entirely characterized by LAR. Throughout this work,we consider identical quantum dots, i.e., U L=UR≡UC, /Gamma1SL=/Gamma1SR≡/Gamma1Sα, and/Gamma1N≡/Gamma1NL=/Gamma1NR. In the main text, we consider the case of equal orbital levels in the two quantumdots,/epsilon1≡/epsilon1 L=/epsilon1R, and equal chemical potentials μ≡μL=μR in which the currents ILandIRcoincide. Experimentally [26,27,60], it may be not so easy to obtain a symmetric conﬁguration, however, the results we report hereafter arenot substantially affected by an asymmetry. We will brieﬂydiscuss some possible consequences of an asymmetric setupin Appendix B. Before going into detail, we would like to recall that the proximization of the double-quantum dot system affectsthe transport properties mainly by means of the Andreevbound-state spectrum. For instance, this can be seen in thedensity plot in Fig. 2(a), where the current through the superconductor I Sis shown as a function of the dots’ level /epsilon1, which can be tuned by gate voltages, and the chemical potential μof the normal leads. Transport channels open/close above/below the Andreev-bound state addition energies (blacklines). Those lines include all the possible differences betweenthe energies of the even and of the odd sector. The couplingof the superconductor with the quantum-dot system stronglyinﬂuences the energy behavior of these resonances openinggaps between states which are coupled by the superconductor[see Eq. ( 4)]. The ﬁgure clearly shows how the current is activated by the entering of the Andreev addition energiesin the bias window, as it is typical in quantum-dot transport.In the white regions the current is blockaded. We now comparethe current I S=−IL−IR, shown in panel (a) of Fig. 2 on a logarithmic scale (recall that IL=IR), with IRin Fig. 4(a), which corresponds to the same conditions but uses −1.5−1.0 −1.0−0.5 −0.50.0 0.00.51.0 −1−10−2−10−41 10−2 10−4)b( )a( 0ασS, T0,Tσ\|GS/Gmax S\| IS/e0ΓNαμ/U C C FIG. 2. Current ISthrough the superconductor (a) and intensity of the differential conductance GS≡dIS/dV (b) as a function of the gate voltage /epsilon1=/epsilon1L=/epsilon1Rand the chemical potential μ=μR=μL. The differential conductance is normalized by its maximum Gmax Sfor /epsilon10=−(UC/2+U). Parameters are /Gamma1S=/Gamma1Sα/3=2.5×10−2UC, U=0.25UC,kBT=2.5×10−3UC,/Gamma1Nα=2.5×10−4UC,a n d t=0. The solid lines in panel (a) indicate the Andreev bound state addition energies. The current (a) can be directly connected with the “mirror” differential conductance, panel (b), since for this regimethe transport is symmetric around the particle-hole symmetric point /epsilon1 0=−(UC/2+U). a linear scale. This clearly illustrates that the current is mostly exponentially suppressed except in a few resonant regions. For sufﬁciently low temperatures, the Andreev bound state addition energies can be resolved in the differentialconductance as it is shown in Fig. 2(b). However, not every transition is observable since for a transition to be observablethe corresponding initial state must be occupied. This is thereason why some of the resonances are not seen, especiallyinside the Coulomb blockade diamonds where the ground stateinvolves, depending on the level spacing, the states \|0/angbracketright,\|ασ/angbracketright (single occupation), ( \|S/angbracketright,\|T0/angbracketright,\|Tσ/angbracketright) (singlet and triplets). A. Weak interdot Coulomb energy, U≈0 In Fig. 3(a), we show a density plot (linear scale) of the current in the right lead IR(/epsilon1,μ). We notice that the current in the normal leads obeys the symmetry Iα(/epsilon1,μ)=−Iα(2/epsilon10− /epsilon1,−μ) with /epsilon10=−(UC/2+U). This symmetry is due the particle-hole (PH) symmetry of the Hamiltonian ( 4)i nt h e absence of interdot tunneling. For simplicity, in Fig. 3,w e have chosen U=0. We focus on the situation μ< 0, which corresponds to the transport of Cooper pairs from the superconductor to thedouble-dot system. Two resonances can be seen in Fig. 3(a): one at /epsilon1=/epsilon1 CAR=−U/2, which is caused only by CAR and another at /epsilon1=/epsilon10which originates from both CAR and LAR. The current is asymmetrical with respect to the chemicalpotentials μ. The bias asymmetry of the CAR peak can be related to the triplet blockade: for μ> 0, tunneling of electrons 235134-4DOUBLE QUANTUM DOT COOPER-PAIR SPLITTER AT . . . PHYSICAL REVIEW B 94, 235134 (2016) 0.5 0.25 0.0 −0.25 −0.5 −0.75 −1.0−1.5−1.0 −1.0−0.8 −0.5−0.5−0.5 −0.5−0.40.0 0.0 0.00.0 0.00.0 0.00.4 0.50.50.5 0.50.8 1.01.01.0 ΓS=0ΓS=√ΓSLΓSR/3ΓS=√ΓSLΓSR/3 ΓS=√ΓSLΓSRΓS=√ΓSLΓSR P0PS αPdαασPασ\|tασ 0CAR CAR(a) (b) (c) (d)GS/Gmax S IR[e0ΓNα/]IR[e0ΓNα/] μ/U Cμ/U C CC C FIG. 3. (a) Current IRthrough the right lead as a function of the dots’ level positions /epsilon1≡/epsilon1L=/epsilon1R, and the chemical potential μ≡μL=μR, where the solid lines indicate the condition under which the chemical potential is equal to the Andreev addition energies. Parameters are /Gamma1S=/Gamma1Sα=7.5×10−2UC,t=U=0, and kBT=2.5×10−3UC,a n d/Gamma1Nα=2.5×10−4UC. (b) Current IRas a function of the dots’ level positions /epsilon1≡/epsilon1L=/epsilon1Rat constant μ=−UCfor intermediate nonlocal coupling /Gamma1S=/Gamma1Sα/3 (dashed line), and maximal nonlocal coupling /Gamma1S=/Gamma1Sα(solid line). The value μ=−UCis indicated in panel (a) by a dotted line. The arrow indicates the transition \|tασ/angbracketright→\| 0/angbracketright. (c) Differential conductance GS≡dIS/dV at constant /epsilon1=−0.625UCfor/Gamma1Sα=10kBT=0.125UC, normalized by its maximum Gmax Sat constant /epsilon1=−UC/2. (d) Occupation probabilities as a function of the level position for μ=−UCand maximal nonlocal coupling /Gamma1S=/Gamma1Sα. from the leads can bring the double-dot in a triplet state whose spin symmetry is incompatible with the BCS superconductor,hence blocking the CAR [ 73]. The bias asymmetry in some regimes of the ( /epsilon1,μ) plane can also be explained by energetic consideration as has been previously reported and discussedin the literature for the simple case of a single quantum dothybrid device [ 56,68]. Along the level position axis, the CAR resonance is centered at/epsilon1 CARand its broadening is√ 2/Gamma1S. This can be seen in panel ( b )o fF i g . 3, which shows the current at constant μ=−UC for two different values of the nonlocal coupling: /Gamma1S=√/Gamma1SL/Gamma1SR/3 (dashed line) and /Gamma1S=√/Gamma1SL/Gamma1SR(solid line). The CAR broadening is proportional to the nonlocal coupling/Gamma1 Sbut its height does not depend on it. The CAR resonance instead follows the singlet population, i.e., /planckover2pi1ICAR R/e0/Gamma1NR≈ 2PS, as can be seen in panel (d). The states involved in the CAR process are \|0/angbracketright,\|ασ/angbracketright,\|S/angbracketrightand in fact one only observes the corresponding populations, P0+/summationtext ασPασ+PS≈1. On the contrary, the resonance at /epsilon1=−UC/2 is mainly due to the LAR, but involves also CAR as indicated by the nonvanishingsinglet population at the LAR resonance [see panel (d)]. We will discuss now that strong superconducting coupling may also generate negative differential conductance (NDC)when single electron tunneling events with the normal leads are accompanied by a simultaneous exchange of a Cooper pair.For instance, if one of the dots is doubly occupied, while theother is singly occupied, it can occur that an electron leavesthe system through a normal-metal lead and the two remainingelectrons tunnel (locally or nonlocally) to the superconductor.If the process is energetically admissible, the total current is reduced instead of increased by the opening of the new resonance and NDC is observed. This is indeed observed in panel (c) of Fig. 3having deﬁned the conductance as G S≡dIS/dV.W es h o w GSas a function of the chemical potential, for a ﬁxed level posi-tion/epsilon1=/epsilon1 0−0.125UC. The differential conductance becomes negative around μ≈3/epsilon1+UC(leftmost peak). This extra resonance corresponds energetically to the transition from thetriply occupied states to the empty state, \|tασ/angbracketright→\| 0/angbracketright, where two electrons tunnel in the superconductor and the remainingelectron tunnels in one of the normal leads. This involvesonly the exchange of a nonlocal Cooper pair and is no longer present in the absence of nonlocal coupling [dot-dashed linein panel (c) of Fig. 3]. In order to increase the visibility of the NDC, we have chosen a stronger nonlocal coupling /Gamma1 S (by increasing /Gamma1Sα) to obtain a higher peak value and slightly 235134-5HUSSEIN, JAURIGUE, GOVERNALE, AND BRAGGIO PHYSICAL REVIEW B 94, 235134 (2016) 0.0 0.00.0 −0.25 −0.25−0.25 −0.5 −0.5−0.5 −0.75 −0.75−0.75 −1.0 −1.0−1.0 −1.5−1.0 −1.0−0.8−0.5 −0.5−0.40.0 0.0 0.00.00.0 0.00.4 0.50.50.50.50.8 1.01.01.0 1.0 μ=−UC μ=−(UC+U)/2 ΓS=√ΓSLΓSR/3ΓS=0.9√ΓSLΓSRΓS=0.99√ΓSLΓSR P0PS αPdαασPασCAR LAR(a) (b) (c) (d)IR[e0ΓNα/] IR[e0ΓNα/]IR[e0ΓNα/]μ/U C C CC C FIG. 4. (a) Current IRthrough the right lead as a function of the gate voltage /epsilon1≡/epsilon1L=/epsilon1R, and the chemical potential μ≡μL=μRfor ﬁniteU=0.25UCand/Gamma1S=/Gamma1Sα/3. Other parameters are as in Fig. 3. (b) Corresponding slices at constant μ=−(UC+U)/2 (dashed line), andμ=−UC(solid line). The slice at μ=−(UC+U)/2 is indicated in panel (a) by a dotted line. (c) Current IRat constant μ=−(UC+U)/2 for various values of /Gamma1S. (d) Corresponding population probabilities as a function of the gate voltage /epsilon1at ﬁxed μ=−UC. higher temperatures to increase the linewidth of this resonance in comparison to other ﬁgures. B. Finite interdot Coulomb energy, U/greaterorsimilar/Gamma1Sα For ﬁnite interdot Coulomb energy, the LAR dominated res- onance in Fig. 3(a)splits into two resonances at gate voltages /epsilon1LAR=−UC/2 and /epsilon10=/epsilon1LAR−Uas can be seen in panel ( a )o fF i g . 4. The former current resonance is purely affected by LAR involving the states \|0/angbracketright,\|ασ/angbracketright, and\|dα/angbracketright. In fact, in Fig.4(d), one observes that only the corresponding populations are non vanishing, i.e., P0+/summationtext ασPασ+/summationtext αPdα≈1 and that the current is proportional to the population of the doublyoccupied state, /planckover2pi1I LAR R/e0/Gamma1NR≈4PdR. We still observe the asymmetry of the nonlocal-current resonances in the chemicalpotential. Again, this asymmetry of the LAR resonances canbe explained partly by the triplet blockade mechanism andpartly by energy considerations. The central resonance at /epsilon1 0=/epsilon1LAR−Uis affected both by the LAR and the CAR processes. For an intermediate valueof the chemical potentials [see dashed line in panel (b) ofFig.4] its width is roughly proportional to√ /Gamma1SL/Gamma1SR−/Gamma1Sand vanishes if /Gamma1Sbecomes maximal. In panel (c), we demonstrate the effect of the nonlocal Cooper-pair tunneling on thecurrent resonances, for an intermediate value of the chemicalpotentials and for different values of the nonlocal coupling/Gamma1 S. The red-dashed line corresponds to the red-dashed linein panel (b). When /Gamma1Sapproaches its maximum, the width of the central resonance (left peak) tends to zero while itsheight remains unaffected. We suspect that the behavior of thewidth of this central resonance is due to the mutual exclusionof the local Cooper-pair tunneling process and the nonlocalone, and originates from a destructive interference of the twochannels (see also later). On the contrary, if both processeswere independent, the linewidth would be the sum of bothcontributions. In conclusion, this regime of ﬁnite interdotCoulomb energy can be helpful to assess the strength of thenonlocal coupling /Gamma1 Sin comparison to the local terms. IV . INFLUENCE OF INTERDOT TUNNELING AND SPIN-ORBIT INTERACTION In this section, we consider the effect of ﬁnite interdot tunneling and SO interaction on the current IR. For the sake of simplicity, we consider in the following only the case φ=0 (no SO coupling) and φ=±π/2 (ﬁnite SO coupling with kSOl=π/2). Let us ﬁrst focus on the general behavior of the current as a function of the level position and chemicalpotential as shown in the density plots of Fig. 5. For simplicity, we consider the case without interdot Coulomb energy, U=0, which describes well the situation of /Gamma1 Sα/greatermuchU. Finally, in order to see stronger signatures of the interdot tunneling termwe generally consider U C/greatermucht/greatermuch/Gamma1S,/Gamma1Sα. 235134-6DOUBLE QUANTUM DOT COOPER-PAIR SPLITTER AT . . . PHYSICAL REVIEW B 94, 235134 (2016) −1.5−1.0 −1.0−1.0 −0.8−0.8 −0.5 −0.5−0.5 −0.4−0.4 0.00.0 0.0 0.00.0 0.40.4 0.5 0.50.5 0.80.8 1.01.0 (a) (b)IR[e0ΓNα/]μ/U C μ/U C C FIG. 5. (a) Current IRthrough the right lead as a function of the gate voltage /epsilon1=/epsilon1L=/epsilon1Rand chemical potential μ=μR=μL for ﬁnite interdot tunneling with t=0.4UCand the SO angle φ= 0. Other parameters are U=0,/Gamma1S=/Gamma1Sα=7.5×10−2UC,/Gamma1Nα= 2.5×10−4UC,a n dkBT=2.5×10−3UC. (b) Current IRfor the same parameters as in panel (a) but for ﬁnite SO interaction with an SOangle of φ=±π/2. In the top panel, we show the case of ﬁnite interdot tunneling in the absence of SO coupling, i.e., φ=0, which can be directly compared with the density plot of Fig. 3(a) where the interdot tunneling was absent. One immediately sees thatthe Andreev resonant lines (black solid lines) are generallysplit in comparison to the case without interdot tunneling,giving rise to an even richer Andreev-bound-state spectrum.The most general observation is that the PH symmetry ofthe transport properties, as discussed in Sec. III, is broken, i.e.,I α(/epsilon1,μ)/negationslash=−Iα(2/epsilon10−/epsilon1,−μ) with /epsilon10=−(UC/2+U). The breaking of the PH symmetry in transport is observedif both the quantities /Gamma1 S,/Gamma1Sα/negationslash=0. On the other hand if one of these quantities vanishes the PH symmetry is re-stored. We discuss PH-symmetry breaking in more detail inSec. IV A . In the bottom panel of Fig. 5, instead, we show how the current is affected by tunneling in the presence of theSO coupling, for the case φ=±π/2. We see that in this case the PH symmetry is again restored for any value ofof/Gamma1 Sand/Gamma1Sα. Note that the Andreev addition energiesspectrum becomes also quite intricate and it is not so useful to enter in the details of the behavior of any resonant line.In general, one can see that in comparison to the top panelcrossings and avoided crossings occur between different pairsof Andreev levels. This is a natural consequence of the differentsymmetry of the tunnel coupling between the two dots in thetwo cases. Finally, for /Gamma1 S,/Gamma1Sα/negationslash=0, the CAR peaks are split along the level-position axis and an extra resonance appears.We will discuss in detail the nature of this extra resonancein Sec. IV B . A. Interdot tunneling and breaking of PH To investigate the PH symmetry breaking, we apply the PH transformation dασ→d† α−σto Eq. ( 4). It is easy to check that indeed this transformation leaves obviously unaffectedthe local and nonlocal pairing terms but is equivalent to achange of sign of the interdot tunneling term, i.e., t→−t. Therefore the symmetry obeyed by the current is I α(/epsilon1,μ,t )→ −Iα(2/epsilon10−/epsilon1,−μ,−t), which we have numerically veriﬁed. Notice that the sign of tin the tunneling Hamiltonian cannot be gauged away only if both the local and nonlocal pairing terms are present in Eq. ( 4). Finally, we notice that for \|φ\|=π/2 the sign of tis unessential due to Kramer’s degeneracy and therefore the PH symmetry is restored in this specialcase. The question remains, why the sign and more generally a phase of tis detectable in the transport properties of the system. This is essentially due to the interference between twopaths connecting the empty state with the singlet state. Onepath is the nonlocal Andreev tunneling with rate /Gamma1 S,w h i l e the other is the process where a Cooper pair virtually tunnelsinto one of the dots bringing it in the doubly occupied stateand subsequently this state is converted into a singlet state byinterdot tunneling. The interference between the two paths isclearly affected by the phase (not only the sign) of t. In order to observe this interference effect, the doubly occupied state ofa single dot needs to be accessible. We have veriﬁed that, forU C→∞ , an overall phase of tdoes not affect the transport properties of the system. B. Weak interdot Coulomb energy, /Gamma1Sα/greatermuchU We focus on the effect of the interdot tunneling on the CAR resonance. In Figs. 6(a)–6(c), we show the evolution of the CAR current peak for different values of \|t\|for φ=0. For increasing strength of the interdot tunneling, the position of the CAR resonance shifts to the right and at thesame time the resonance linewidth changes. The peak shiftisδ/epsilon1 CAR/UC≈(1/2)(t/UC)2fort/lessmuchUC.T h i si ss h o w ni n Fig.6(d) where the position of the CAR peak maximum /epsilon1max is plotted as a function of τ=t/UC. For different values of the nonlocal coupling /Gamma1S[different point styles in panel (d)], the peak position follows the same universal function of τ (solid line). Instead, the linewidths, shown in Figs. 6(e) and 6(f), exhibit quite different behaviors depending on the value of/Gamma1S, the strength of tand also its sign. These observations can be explained by making use of a reduced Hilbert space, which describes well the system inthe vicinity of the CAR resonance. This simpliﬁed model is 235134-7HUSSEIN, JAURIGUE, GOVERNALE, AND BRAGGIO PHYSICAL REVIEW B 94, 235134 (2016) 0.500.500.50 0.250.250.25 0.000.000.00 01 /81 /410−310−2 10−110−110−1 100100100 10−1100(a) (b) (c)(d) (e) (f) ¯τ=¯τ=¯τ= 00 222 333 444 555 666 ∝τ2 τ>0 τ<0ΓS=0ΓS=0.2ΓSLΓS=ΓSLIR[e0ΓNα/] IR[e0ΓNα/] IR[e0ΓNα/] wCAR/ΓSα wCAR/ΓSα max/UC C \|τ\|=\|t\|/UC FIG. 6. (a)–(c) Current IRin the CAR resonance as a function of the gate voltage for different values of the tunneling amplitude, ¯ τ=8\|t\|/UC, for the nonlocal term /Gamma1S/√/Gamma1SR/Gamma1SL=1( a ) ,0 .2 (b), 0 (c), and other parameters as in Fig. 3. The solid lines correspond to t/UC>0 while the dashed lines in panel (b) correspond to t/UC<0. (d) Gate position at the maximum of the CAR resonance, /epsilon1CAR, as a function of the scaling variable τ=t/UCfor the different values of the nonlocal term considered in (a)–(c). (e)–(f) CAR resonance linewidth wCARas a function of the scaling variable \|τ\|[τ> 0 for (e) and τ< 0 for (f)] for the different values of the nonlocal term considered in (a)–(c); the solid and dotted lines in (d)–(f) are the theoretical predictions in the simpliﬁed model discussed in the main text. sketched in Fig. 7. The relevant states for the CAR resonance are the empty state \|0/angbracketright, the singlet state \|S/angbracketright, and the singly occupied states \|ασ/angbracketright. The states in the even sector \|0/angbracketrightand\|S/angbracketright are connected via the nonlocal term /Gamma1S, and they are connected to the singly occupied states \|ασ/angbracketrightvia the tunneling rate to the normal lead, /Gamma1N. In the absence of interdot tunneling, the CAR resonance linewidth is only determined by the nonlocal term/Gamma1 S, see Sec. III. However, for ﬁnite intradot Coulomb energy in the presence of local terms /Gamma1Sαand strong interdot tunneling t(withφ=0), we need to consider also another possibility: when the quantum dots are in the empty state, a Cooper pair canbe virtually transferred by means of the local term /Gamma1 Sαin the doubly occupied state \|dα/angbracketright, which is converted to the singlet state via the interdot tunneling. One can see in Eq. ( A1) that the tunneling amplitude ( t/√ 2) cos( φ) couples the \|dα/angbracketrightstates with the singlet state \|S/angbracketright. We will quantitatively show that the interference of this alternative channel with the standardnonlocal process fully determines the observed behavior of theCAR peak. When /Gamma1 S/lessmucht, the peak shift can be understood in terms of the level repulsion of the singlet state with the doubly occupiedstate. We ﬁrst note that the interdot coupling removes thedegeneracy of the double occupancies and yields the states\|d±/angbracketright = (\|dR/angbracketright±\|dL/angbracketright)/√ 2. Only the symmetric state \|d+/angbracketrightis affected by the level repulsion with \|S/angbracketright. In this model, the hybridized states \|±/angbracketright ≈ α\|S/angbracketright±β\|d+/angbracketrightwithα,βc-numbers have the energies2 /epsilon1± UC=1+4/epsilon1/UC±√ 1+4τ2 2, (10) where τ=t/UC. The position of the CAR resonance is the solution of equation /epsilon1−(/epsilon1CAR)=0, the resonance condition between \|−/angbracketright, and the empty state \|0/angbracketright. The peak position is /epsilon1CAR=(√ 1+4τ2−1)UC/4, which ﬁts well the shifting of the peak position [see solid line in Fig. 6(d)]. In the limit UC→ ∞(τ→0), the doubly occupied states are unaccessible, even virtually, and the transport becomes independent of the interdottunneling. Finite interdot tunneling also modiﬁes the linewidth of the CAR peak as can be seen in Figs. 6(a)–6(c) and more clearly in Figs. 6(e) and6(f)where we show the linewidth w CARof 2Only in the limit /Gamma1S→0 the hybridized states \|±/angbracketrightcan be written as linear combination of \|S/angbracketrightand\|d+/angbracketright. 235134-8DOUBLE QUANTUM DOT COOPER-PAIR SPLITTER AT . . . PHYSICAL REVIEW B 94, 235134 (2016) \|0 \|S \|T0\|d\|d+ \|ασΓSt, φ=0t, φ=π 2ΓSα ΓN ΓNΓNU2 C+4t2 FIG. 7. Effective level structure at the CAR resonance. Finite interdot tunneling ( φ=0) leads to a level repulsion between the symmetric state \|d+/angbracketright = (\|dR/angbracketright+\|dL/angbracketright)/√ 2 and the singlet state \|S/angbracketright,s e eE q .( 10). SO interaction ( φ=π/2) leads instead to a level repulsion between the unpolarized triplet state \|T0/angbracketrightand the symmetric state \|d+/angbracketright. The symmetric state, virtually occupied by local Cooper-pair tunneling, plays the role of a dark state. the CAR resonance. For /Gamma1S=√/Gamma1SL/Gamma1SR(black circles), the width is roughly proportional to the nonlocal coupling, whilefor/Gamma1 S=0 (red triangles), it increases with τ. Intriguingly, for an intermediate value of /Gamma1S(blue small circles) and τ> 0, the linewidth almost vanishes for a speciﬁc value of τ[see Fig.6(e)]. This behavior is not seen for τ< 0[ s e eF i g . 6(f)]. One may ask if for certain values of τthe CAR peak can really vanish. Albeit within the sequential tunneling approximationits linewidth can become arbitrary small, in such case onewould need to include high-order corrections in the couplingto the normal leads. So, we expect that the minimal linewidthis of the order of /Gamma1 Nbeing this the natural linewidth of the resonance. We can explain these results by making use again of the simpliﬁed model shown in Fig. 7. In the absence of the interdot tunneling the linewidth of the CAR peak is onlydetermined by the strength of the coupling between the emptystate\|0/angbracketrightand the singlet state \|S/angbracketright. Essentially, it is given by the off-diagonal matrix element w CAR≈2\|/angbracketleft0\|HS\|S/angbracketright\| =√ 2/Gamma1S. Any additional process that contributes to that coupling between \|0/angbracketrightand\|S/angbracketright, also through a virtual high energy state, will affect the linewidth. This correction may beobtained considering the effective Hamiltonian H S=H0+V, which represents the model shown in Fig. 7, with H0=/summationtext iEi\|i/angbracketright/angbracketlefti\|−(/Gamma1S/√ 2)(\|0/angbracketright/angbracketleftS\|+\|S/angbracketright/angbracketleft0\|)f o ri=0,S,d+,d− and the perturbation V=[t\|d+/angbracketright/angbracketleftS\|−(/Gamma1Sα/√ 2)\|d+/angbracketright/angbracketleft0\|+ H.c.]. Calculating the off-diagonal matrix element up to second order in the perturbation, O(V3), yields [ 92] /angbracketleft0\|HS\|S/angbracketright=/angbracketleft 0\|H0\|S/angbracketright+/angbracketleft0\|V\|d+/angbracketright/angbracketleftd+\|V\|S/angbracketright E(0) S−E(0) d+(11) withE(0) S−E(0) d+=−UC. We ﬁnd for the linewidth wCAR≈√ 2\|/Gamma1S−/Gamma1Sατ\|. This estimation of the linewidth is indicated by dotted lines in Figs. 6(e) and 6(f). It turns out to be a quite good approximation for τ≪1 but it worsens for increasing /Gamma1S(see, for example, the black case /Gamma1S=√/Gamma1SL/Gamma1SR). A better approximation, however, is obtained0.50 0.25 0.00 01 /81 /4¯τ=02 3 4 5 6IR[e0ΓNα/] C FIG. 8. Current IRin the CAR resonance as a function of the gate voltage for different values of the tunneling amplitude, ¯ τ=8\|t\|/UC, φ=±π/2f o r /Gamma1S=/Gamma1Sα/3 (solid line) and /Gamma1S=0 (dashed line) k e e p i n gﬁ x e dt h e UC. Other parameters as in Fig. 6. by the substitution E(0) S−E(0) d+→/epsilon1−−/epsilon1+=−UC√ 1+4τ2, which includes the energy renormalization effects induced bythe level repulsion discussed before. Therefore the linewidthcan be approximated by w CAR≈√ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/Gamma1 S−/Gamma1Sατ√ 1+4τ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (12) which ﬁts (solid lines) well the numerical results based on the full Hamiltonian, see Figs. 6(e) and6(f). Interestingly, Eq. ( 12) explains also why for positive (negative) sign of tthe linewidth can decrease (increase) due to destructive (constructive) interference. This is seen comparing the resultsfor/Gamma1 S=0.2√/Gamma1SL/Gamma1SRin panels (e) and (f) of Fig. 6. Finally, one intriguing consequence of the virtual-state processinvolving the state \|d+/angbracketrightis that it generates nonlocal entangled electrons even in the absence of a direct nonlocal coupling. 3 We now turn our attention to the case with the SO coupling andφ=±π/2. We see in panel (b) of Fig. 5that the CAR resonance splits into two lines, one at /epsilon1≈0 and the other shifted towards higher values of /epsilon1.I nF i g . 8, we show the behavior of the CAR peak with increasing values of tfor constant values of UCand/Gamma1S. First, we notice that for /Gamma1S=0 (dashed lines) the CAR peak does not split but it shifts tothe right for increasing values of τand no resonance is present at /epsilon1≈0. Instead, for /Gamma1 S/negationslash=0, the resonance splits into two resonances, one ﬁxed at /epsilon1≈0 and the other right- shifted with δ/epsilon1rs/UC≈(1/2)(t/UC)2. This demonstrates the connection with the nonlocal term /Gamma1Sof the CAR peak at/epsilon1≈0. We numerically observed that the current of the right- shifted peak follows the population of the unpolarized tripletstate, /planckover2pi1I rs R/e0/Gamma1NR≈2PT0. These observations suggest that 3The reported results are done in a completely symmetric con- ﬁguration between right and left dots; in Appendix B, we include some observations on how an asymmetry may potentially modify the reported physics. 235134-9HUSSEIN, JAURIGUE, GOVERNALE, AND BRAGGIO PHYSICAL REVIEW B 94, 235134 (2016) a resonant mechanism involving the virtual occupation of the\|d+/angbracketrightis established with the unpolarized triplet state \|T0/angbracketright, as depicted schematically in Fig. 7. This mechanism is analogous to the one induced by the nonlocal singletproximity in the case of interdot coupling where φ=0. We refer to this resonance as triplet CAR resonance, sinceit generates nonlocal entanglement with triplet symmetry.The position and linewidth of this right-shifted resonanceare described by Eqs. ( 10) and ( 12) setting /Gamma1 S=0, re- spectively. This is a consequence of the fact that a s-wave superconductor cannot induce directly triplet correlations. Thisshows how the presence of SO coupling can neverthelessinduce nonlocal triplet superconducting correlations evenwhen the only superconducting lead has s-wave pairing symmetry [ 22,23]. V . CONCLUSIONS We have presented a comprehensive study of a Cooper- pair splitter based on a double-quantum dot. Employing amaster-equation description, in the framework of FCS, wehave calculated the current injected into the normal leads. Wehave considered a ﬁnite intradot interaction which allows thelocal transfer of Cooper pairs from the superconductor to anindividual quantum dot. We have studied the signatures of localand nonlocal Andreev reﬂection in the current injected in thenormal leads. The interdot Coulomb interaction separates thelocal and nonlocal resonances. The effect of interdot tunnelingboth with and without SO coupling has been considered, too.In particular, we ﬁnd that the interdot tunneling can inducenonlocal entanglement starting from local Andreev reﬂection.Furthermore, a process including the virtual doubly occupiedstates of the individual dots leads to modiﬁcations of theposition and linewidth of the current resonances. For the case with SO coupling, we ﬁnd that a nonlocal triplet pair amplitudecan be generated in the system. This mechanism involving thevirtual occupation of the doubly occupied states is active onlyfor ﬁnite intradot Coulomb interaction. The effects reportedin this article can be useful, on one hand, for entanglementgeneration in Cooper-pair splitter devices and, on the otherhand, to indirectly extract information about the spin-orbitinteraction in nanowire systems or to detect the symmetry ofentangled states. ACKNOWLEDGMENTS We thank F. Giazotto, S. Roddaro, and S. Kohler for valuable discussions. This work has been supported byItalian’s MIUR-FIRB 2012 via the HybridNanoDev projectunder Grant no. RBFR1236VV and the EU FP7/2007-2013under the REA Grant Agreement No. 630925-COHEAT. A.B.acknowledges support from STM 2015, CNR, the VictoriaUniversity of Wellington and the Nano-CNR in Pisa where thework was partially done. APPENDIX A: MATRIX REPRESENTATION OF THE SYSTEM HAMILTONIAN In this section, we provide the decomposition of the system Hamiltonian HS=Heven S⊕Hodd Sinto sectors with even and odd parity. We assume the single-particle levelspacing in the quantum dot to be large compared to U, U αand the interdot tunneling t, so the total dimension of the system Hilbert space reduces to 16 states (8 even + 8 odd). Here, we express Eq. ( 4) in the even sector basis {\|0/angbracketright,\|S/angbracketright,\|dL/angbracketright,\|dR/angbracketright,\|dd/angbracketright,\|T0/angbracketright,\|T↑/angbracketright,\|T↓/angbracketright}stated in Table I. The Hamiltonian for the even charge sector reads Heven S=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0 − 1√ 2/Gamma1S −1 2/Gamma1SL −1 2/Gamma1SR 00 0 0 −1√ 2/Gamma1S/epsilon1L+/epsilon1R+Ut√ 2cos(φ)t√ 2cos(φ) +1√ 2/Gamma1S 000 −1 2/Gamma1SLt√ 2cos(φ)2 /epsilon1L+UL 0 −1 2/Gamma1SR it√ 2sin(φ)0 0 −1 2/Gamma1SRt√ 2cos(φ)0 2 /epsilon1R+UR −1 2/Gamma1SL it√ 2sin(φ)0 0 0 +1√ 2/Gamma1S −1 2/Gamma1SR −1 2/Gamma1SL 2(/epsilon1R+/epsilon1L)+UR+UL+4U 000 00 −it√ 2sin(φ)−it√ 2sin(φ)0 /epsilon1L+/epsilon1R+U 00 00 0 0 0 0 /epsilon1L+/epsilon1R+U 0 00 0 0 0 0 0 /epsilon1L+/epsilon1R+U⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (A1) Note that the interdot tunneling preserves the spin of the tunneling electrons (being time reversal invariant) and the total parity of the DQD. In absence of the SO interaction, φ=0, all triplet states \|Ti/angbracketrightare completely decoupled from the other even parity states. When φ/negationslash=πk, where kinteger, the unpolarized triplet state \|T0/angbracketrightcouples with the doubly occupied states \|dα/angbracketright.T h e Hamiltonian for the odd charge sector, in the basis {\|R↑/angbracketright,\|R↓/angbracketright,\|L↑/angbracketright,\|L↓/angbracketright,\|tR↑/angbracketright,\|tR↓/angbracketright,\|tL↑/angbracketright,\|tL↓/angbracketright}, is given by Hodd S=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝/epsilon1 R 0t 2e−iφ0 −1 2/Gamma1SL 0 +1 2/Gamma1S 0 0 /epsilon1R 0t 2e+iφ0 −1 2/Gamma1SL 0 +1 2/Gamma1S t 2e+iφ0 /epsilon1L 0 +1 2/Gamma1S 0 −1 2/Gamma1SR 0 0t 2e−iφ0 /epsilon1L 0 +1 2/Gamma1S 0 −1 2/Gamma1SR −1 2/Gamma1SL 0 +1 2/Gamma1S 0 EtR↑ 0 −t 2e−iφ0 0 −1 2/Gamma1SL 0 +1 2/Gamma1S 0 EtR↓ 0 −t 2e+iφ +1 2/Gamma1S 0 −1 2/Gamma1SR 0 −t 2e+iφ0 EtL↑ 0 0 +1 2/Gamma1S 0 −1 2/Gamma1SR 0 −t 2e−iφ0 EtL↓⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (A2) 235134-10DOUBLE QUANTUM DOT COOPER-PAIR SPLITTER AT . . . PHYSICAL REVIEW B 94, 235134 (2016) 0.0 0.0 0.0 −0.25 −0.25 −0.25 −0.5 −0.5 −0.5 −0.75 −0.75 −0.75 −1.0 −1.0 −1.00.0 0.0 0.00.50.5 0.5 1.01.0 1.0 1.52.0 η=1 η=1/5 η=1 5)c( )b( )a(IL[e0ΓNα/] IR[e0ΓNα/] −IS[e0ΓNα/] C C C FIG. 9. Currents IL(a),IR(b), and IS=−IL−IR(c) as a function of the level position /epsilon1≡/epsilon1L=/epsilon1Rfor asymmetric local couplings to the superconductor: /Gamma1SL=/Gamma1SC/(1+η),/Gamma1SR=η/Gamma1SC/(1+η). Parameters are /Gamma1S=√/Gamma1SL/Gamma1SR/3,/Gamma1SC=1.5×10−1UC,/Gamma1Nα=2.5×10−4UC, U=0.25UC,kBT=2.5×10−3UC,μ=−UC,a n dt=0. where Etασ=2/epsilon1¯α+U¯α+/epsilon1α+2Uwith α=R,L (¯α= L,R) andσ=↑,↓. APPENDIX B: ASYMMETRIC QUANTUM DOT SETUP In this section, we provide some basic comments concern- ing the case of an asymmetric quantum dot conﬁguration.The asymmetry can be introduced essentially in two ways:either externally (tuning of the biases and/or the gate voltagesasymmetrically) or structurally (asymmetric coupling betweenthe leads and Coulomb energies). The former case may leadto an asymmetric conﬁguration, even in the case of a fullsymmetry at the structural level. Since bias and gate voltagescan be externally tuned, such an asymmetry is experimentallycontrollable. Hereafter, we comment on the effects of the struc-tural asymmetry introduced by asymmetric couplings betweenthe dots and the leads. We will also comment on the possibilityto have different Coulomb energies of the right and left dot. Theconclusion will be that in order to get the maximal efﬁciency ofthe Cooper-pair splitting, the structural symmetry is, by far, themost convenient regime. When the asymmetry is introduced atthe level of the couplings to the normal leads /Gamma1 Nα, it basically rescales the currents through lead α=L,R sinceIα∝/Gamma1Nα— in lowest order in the tunneling regime discussed in this work. More intriguing is to consider a possible asymmetry introduced by the couplings with the superconductors /Gamma1Sα. To illustrate this, let us consider the parametrization /Gamma1SL=1 1+η/Gamma1SC,/Gamma1 SR=η 1+η/Gamma1SC (B1) of the local couplings with η=/Gamma1SR//Gamma1SLthe asymmetry parameter; for η=1 one recovers the symmetric case. This parametrization has the advantage to keep the arithmeticmean ( /Gamma1 SL+/Gamma1SR)/2 constant. Changing the local couplings will affect the Andreev spectrum since for /Gamma1SL/negationslash=/Gamma1SRand thecoupling induced by the tunneling of local Cooper pairs are different depending on the dot involved. These nonuniversalspectral effects do not substantially modify the physicsdescribed in the symmetric case and we do not discuss themin detail. The main effect of the asymmetry is to change the height of the LAR contribution to the current through the normalleads asymmetrically leaving unaffected the height of the CARpeak. Figures 9(a) and9(b) show the effect of the asymmetry in comparison to the symmetric case (black solid line) forthe left I L[panel (a)] and the right IR[panel (b)] current. Increasing (decreasing) the asymmetry parameter ηdirectly increases /Gamma1SR(/Gamma1SL) and, thus, increases the corresponding LAR peak in IR(IL). At the same time the total current in the superconductor IS=−IL−IRcorresponding to the LAR peak is unchanged since the arithmetic mean is constant.One can also see some small effects in the linewidths of theresonances. For the CAR peak, this can be explained by thechange of the geometric mean√ /Gamma1SR/Gamma1SL, which essentially determines its linewidth. This implies that any asymmetryin the superconducting coupling will reduce the visibility(linewidth) of the CAR peak since the geometric mean isalways smaller or equal than the arithmetic mean. Finally, we wish to comment on the possibility of different intradot Coulomb energies, U L/negationslash=UR. Still, we will assume these Coulomb energies to be the largest energy scale inthe problem besides the superconducting gap. A ﬁrst ob-servable effect is a further breaking of the degeneracies inthe Andreev bound states leading to a quite intricate levelstructure. For \|U L−UR\|/lessmuch/Gamma1Sα, this modiﬁcation does not change the physics of the Cooper-pair splitter substantially.However, an asymmetry in the Coulomb energies may modifythe effects related to the interdot tunneling discussed inSec. IV B . 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PhysRevB.87.155304.pdf	PHYSICAL REVIEW B 87, 155304 (2013) Quasiparticle band structures and optical properties of strained monolayer MoS 2and WS 2 Hongliang Shi,1Hui Pan,1Yong-Wei Zhang,1,and Boris I. Yakobson2,† 1Institute of High Performance Computing, ASTAR, Singapore 138632 2Department of Mechanical Engineering and Materials Science, Department of Chemistry, and the Smalley Institute for Nanoscale Science and Technology, Rice University, Houston, Texas 77005, USA (Received 24 November 2012; revised manuscript received 2 March 2013; published 9 April 2013) The quasiparticle (QP) band structures of both strainless and strained monolayer MoS 2are investigated using more accurate many-body perturbation GW theory and maximally localized Wannier functions (MLWFs) approach. By solving the Bethe-Salpeter equation (BSE) including excitonic effects on top of the partiallyself-consistent GW 0(scGW 0) calculation, the predicted optical gap magnitude is in good agreement with available experimental data. With increasing strain, the exciton binding energy is nearly unchanged, while optical gapis reduced signiﬁcantly. The sc GW 0and BSE calculations are also performed on monolayer WS 2, similar characteristics are predicted and WS 2possesses the lightest effective mass at the same strain among monolayers Mo(S,Se) and W(S,Se). Our results also show that the electron effective mass decreases as the tensile strainincreases, resulting in an enhanced carrier mobility. The present calculation results suggest a viable route totune the electronic properties of monolayer transition-metal dichalcogenides (TMDs) using strain engineeringfor potential applications in high performance electronic devices. DOI: 10.1103/PhysRevB.87.155304 PACS number(s): 73 .22.−f, 71.20.Nr, 71 .35.−y I. INTRODUCTION Bulk TMDs consisting of two-dimensional (2D) sheets bonded to each other through weak van der Waals forceshave been studied extensively owing to their potential ap-plications in photocatalysis 1and catalysis.2,3MoS 2,W S 2, MoSe 2, and WSe 2are examples of such TMDs. Recently, their 2D monolayer counterparts were successfully fabricatedusing a micromechanical cleavage method. 4Since then, these monolayer materials have attracted signiﬁcant attention.5–12 For monolayer MoS 2, a strong photoluminescence (PL) peak at about 1.90 eV , together with peaks at about 1.90and 2.05 eV of the adsorption spectrum, indicated that MoS 2 undergoes an indirect to direct band gap transition whenits bulk or multilayers form is replaced by a monolayer. 6–8 Shifts of PL peak for the monolayer MoS 2were also observed experimentally, which was attributed to the strainintroduced by covered oxides. 13Theoretical studies which employed density functional theory (DFT) method also pre-dicted monolayer MoS 2to have a direct gap of 1.78 eV .5It is known however that DFT does not describe excited stateof solids reliably. Furthermore, an important character in low-dimensional systems is their strong exciton binding due to theweak screening compared to bulk cases. Therefore, the goodband gap agreement between theoretical and experimentalresults for monolayer MoS 2may be a mere coincidence. As a channel material for transistor application, theoreticalsimulations show that monolayer WS 2performs better than monolayer MoS 2.14In order to address the above questions, it is important and necessary to employ a more accuratecalculation method beyond DFT to investigate the electronicstructures of strained monolayer MoS 2and WS 2. The most common method to circumvent drawback of DFT is the GW approximation,15in which self-energy operator /Sigma1 contains all the electron-electron exchange and correlationeffects. The sc GW 0approach, in which only the orbitals and eigenvalues in Gare iterated, while Wis ﬁxed to the initial DFT W0, was shown to be more accurate in many cases to predict band gaps of solids.16The off-diagonal components ofthe self-energy /Sigma1should be included in sc GW 0calculations since this inclusion has been proved particularly useful formaterials such as NiO and MnO. 17It is noted that /Sigma1within the GW approximation is deﬁned only on a uniform kmesh in the Brillouin zone, due to its nonlocality. Therefore, unlike DFTband structure plot, the QP eigenvalues at arbitrary kpoints along high symmetry lines cannot be performed directly. 18 Started from the sc GW 0calculation, the QP band structure can be interpolated using the MLWFs approach. This combinationwas demonstrated to be accurate and efﬁcient for the sc GW band structure. 18TheGW results were shown to agree well with the photoemission data,19while in order to reproduce the experimental adsorption spectra, the consideration of attraction between quasielectron and quasihole (on top of GW approximation) by solving BSE is indispensable,19particularly for the low-dimensional systems with strong excitonic effect.The main goal of this study is to accurately predict the QPband structures and optical spectra of monolayer MoS 2as a function of strain by adopting the DFT-sc GW 0-BSE approach. Strain in monolayer MoS 2can be produced either by epi- taxy on a substrate or by mechanical loading. It is well knownthat strain can be used to tune the electronic properties ofmaterials. This is particularly important for two-dimensionalmaterials, which can sustain a large tensile strain. In fact, shiftsof PL peak observed experimentally in monolayer MoS 2was attributed to strain,13and the magnetic properties of MoS 2 nanoribbons could be tuned by applying strain.20 By adopting the aforementioned approach, we systemat- ically investigate how the electronic structures and optical properties of monolayer MoS 2evolve as a function of strain. Our results show that exciton binding energy is insensitiveto the strain, while optical band gap becomes smaller asstrain increases. Based on the more accurate band structures interpolated by MLWFs methods based on sc GW 0results, the effective masses of carriers are calculated. In addition, thiscalculation approach is also employed to investigate othermonolayer TMDs, that is, WS 2, MoSe 2, and WSe 2. Our results demonstrate that the effective mass is decreased as the strain increases, and monolayer WS 2possesses the lightest carrier 155304-1 1098-0121/2013/87(15)/155304(8) ©2013 American Physical SocietySHI, PAN, ZHANG, AND Y AKOBSON PHYSICAL REVIEW B 87, 155304 (2013) among the TMDs, suggesting that using monolayer WS 2as a channel material can enhance the carrier mobility and improve the performance of the transistor. II. DETAILS OF CALCULATION Our DFT calculations were performed by adopting the gen- eralized gradient approximation (GGA) of PBE functional21 for the exchange correlation potential and the projector aug-mented wave (PAW) 22method as implemented in the Vienna ab initio simulation package.23Twelve valence electrons are included for both Mo and W pseudopotentials. The electronwave function was expanded in a plane wave basis set withan energy cutoff of 600 eV . A vacuum slab more than 15 ˚A (periodical length of cis 19 ˚A) is added in the direction normal to the nanosheet plane. For the Brillouin zone integration, a12×12×1/Gamma1centered Monkhorst-Pack k-point mesh is used. In the following GW QP calculations, both single-shot G 0W0and more accurate sc GW 0calculations are performed. 180 empty conduction bands are included. The energy cutofffor the response function is set to be 300 eV , the obtainedband gap value is almost identical to the case of 400 eV . Theconvergence of our calculations has been checked carefully.For the Wannier band structure interpolation, dorbitals of Mo (W) and porbitals of S (Se) are chosen for initial projections. Our BSE spectrum calculations are carried outon top of sc GW 0. The six highest valence bands and the eight lowest conduction bands were included as basis forthe excitonic state. BSE was solved using the Tamm-Dancoffapproximation. Notice that the applied strain in the presentstudy is all equibiaxial, unless stated otherwise.III. RESULTS AND DISCUSSIONS We ﬁrst analyze the density of states (DOS) for monolayer MoS 2.T h e dorbitals of Mo and porbitals of S contribute most to the states around the band gap, similar to previousstudies. 9–11Figure 1shows the projected dorbitals of Mo and porbitals of S as well as the decomposed dorbitals for mono- layer MoS 2at the lattice of 3.160 ˚A (the experimental lattice constant aof bulk MoS 29) and under 3% tensile strain. Based on the DOS, the dorbitals of Mo and porbitals of S are chosen as the initial projections in the Wannier interpolated method.Figure 2shows the identical DFT band structures of monolayer MoS 2obtained by the non-self-consistent calculation at ﬁxed potential and Wannier interpolation method, respectively,conﬁrming that our choice of the initial projections and innerwindow energy is appropriate. Based on the good results formonolayer MoS 2, the same procedure is also employed for remaining monolayer TMDs. A. QP band structures of strained monolayer MoS 2 The QP band structures of monolayer MoS 2at four lattice constants of 3.160, 3.190 (the optimized value from thepresent work), 3.255, and 3.350 ˚A are plotted in Fig. 3, corresponding to 0%, 1%, 3%, and 6% tensile strains (withreference to 3.160 ˚A), respectively. As shown in Fig. 3(a), the band structure obtained by DFT for strainless MoS 2is a direct band gap semiconductor with a band gap energy of1.78 eV , while the indirect band gap of 2.49 eV is predictedbyG 0W0. Obviously this G0W0indirect band gap is contrary to the PL observations.6–8The QP band structures predicted by our sc GW 0calculation show that MoS 2is aKtoKdirect FIG. 1. (Color online) Projected density of states of dorbitals of Mo and porbitals of S [(a) and (c)] and decomposed dorbital of Mo [(b) and (d)] for monolayer MoS 2at lattice constants of 3.160 [(a) and (b)] and 3.255 ˚A [(c) and (d)], respectively. The latter corresponds to 3% tensile strain. 155304-2QUASIPARTICLE BAND STRUCTURES AND OPTICAL ... PHYSICAL REVIEW B 87, 155304 (2013) FIG. 2. (Color online) DFT band structures of monolayer MoS 2 at lattice constant of 3.160 ˚A. Red solid line: Original band structure obtained from a conventional ﬁrst-principles calculation. Black dashdot: Wannier-interpolated band structure. The Fermi level is set to zero. band gap semiconductor with a band gap energy of 2.80 eV . This prediction is in excellent agreement with the recentcalculation for MoS 2at the experimental lattice using full- potential linearized mufﬁn-tin-orbital method (FP-LMTO)24, which predicted a KtoKdirect band gap of 2.76 eV . It should be noted that in the 2D materials, the excitonic effect is strong due to the weak screening. Thus it is importantto consider the attraction between the quasielectron andquasihole by solving the BSE discussed below in order to makethe predicted optical gap consistent with the optical spectra.Figure 3(b) shows the band structure of monolayer MoS 2at 3.190 ˚A corresponding to 1% strain. The DFT result predicts the monolayer MoS 2to be an indirect band gap with Kto/Gamma1of 1.67 eV . Previous DFT studies also found that monolayer MoS 2 already becomes an indirect semiconductor under a tensilestrain of 1%. 12After GW correction, both of the G0W0and scGW 0QP band structures show that MoS 2is still a direct semiconductor with KtoKband gaps of 2.50 and 2.66 eV , respectively. As the strain increases, shown in Figs. 3(c) and 3(d),t h eD F T , G0W0, and sc GW 0all predict monolayer MoS 2 to be indirect. The calculated indirect band gaps from DFT, G0W0, and sc GW 0are 1.20 (0.63), 2.19 (1.56), and 2.23 (1.59) for monolayer MoS 2under strain of 3% (6%), respectively. As shown in Fig. 3, the value of band gap decreases as the tensile strain increases, accompanying a shift of valence bandmaximum (VBM) from Kto/Gamma1point and resulting in a direct to indirect band gap transition, which was consistent withprevious results. 9,12 The KtoKdirect and /Gamma1toKindirect band gaps of monolayer MoS 2obtained by DFT and sc GW 0as a function of tensile strain are plotted in Fig. 4. Clearly our DFT and sc GW 0 results have the same trends, and accord well with reported DFT12(cyan triangle) and sc GW24(green solid square) results, respectively. Due to the more accurate description of many-body electron-electron interaction, the sc GW 0band gaps are enlarged about 1 eV compared to DFT results. The optical gapshown in Fig. 4will be discussed in the next subsection. B. Excitonic effect in monolayer MoS 2 In this subsection the optical properties of monolayer MoS 2are discussed in detail. From the technical view, optical transition simulation needs the integration over the irreducibleBrillouin zone using sufﬁciently dense k-point mesh. Naturally the convergence of k-point sampling is important. First, for monolayer MoS 2at strainless case (3.16 ˚A), the optical FIG. 3. (Color online) DFT, G0W0,a n ds c GW 0QP band structures for monolayer MoS 2at lattice constants of (a) 3.160, (b) 3.190 (the optimized lattice constant from this work), (c) 3.255, and (d) 3.350 ˚A corresponding to 0%, 1%, 3%, and 6% tensile strain (with reference to 3.160 ˚A), respectively. The Fermi level is set to be zero. 155304-3SHI, PAN, ZHANG, AND Y AKOBSON PHYSICAL REVIEW B 87, 155304 (2013) FIG. 4. (Color online) Band gaps for monolayer MoS 2obtained by DFT, sc GW 0, and BSE. Reported experimental (Expt.),7DFT,12 and sc GW24results are also shown. adsorption spectra ε2(εxx=εyy) obtained by different k-point meshes are illustrated in Fig. 5(a), in which the independent- particle (IP) picture is adopted within DFT (DFT-IP) and no FIG. 5. (Color online) DFT-IP and sc GW 0+BSE adsorption spectra for monolayer MoS 2at an experimental lattice of 3.160 ˚A (strainless case) obtained by different k-point meshes.local ﬁled effect is included at the Hartree or DFT level. The ﬁrst peak at about 1.78 eV is observed clearly in all thecases, corresponding to the K-K direct transition. The second signiﬁcant peak located at about 2.75 eV is converged for12×12×1 and 15 ×15×1k-point meshes. Other peaks in adsorption spectra between the two aforementioned dominatedpeaks mainly originate from different irreducible kpoints with unequal weights in different k-point meshes. According to our analysis of projected density of states, the two signiﬁcantpeaks located at 1.78 and 2.75 eV correspond to d-d and p-dtransitions, respectively. Considering the dipolar selection rule only transitions with the difference /Delta1l=± 1 between the angular momentum quantum numbers lare allowed, i.e., the atomic d-dtransition is forbidden. However, in the monolayer MoS 2, due to the orbital hybridization, the VBM and conduction band minimum (CBM) still have porbital contributions, especially the former; thus the VBM to CBMtransition dominated by d-d transition is still allowed. As expected, the strength of this d-dtransition is weaker than thep-dtransition as shown in Fig. 5(a). As for the BSE calculations, in order to reduce the computational cost, we adopt 400 and 200 eV for the planewave energy cutoff and response function energy cutoff (shortfor 400 and 200 eV for energy cutoffs), respectively, whilethe accuracy still can be guaranteed. Taking the strainlessmonolayer MoS 2for example, the sc GW 0band gap is 2.78 eV , resulting in only 0.02 eV difference compared to 2.80 eVaforementioned using 600 and 300 eV for energy cutoffs. Thecalculated BSE spectra for strainless monolayer MoS 2are plotted in Fig. 5(b). It is clear that as the k-point mesh reﬁnes, the ﬁrst peaks have a blueshift. For k-point meshes 6 ×6×1, 9×9×1, 12×12×1, and 15 ×15×1, the sc GW 0band gaps are 2.99, 2.84, 2.78, and 2.76 eV , respectively; the ﬁrstadsorption peaks (optical band gaps) are 1.96, 2.08, 2.16, and2.22 eV . Correspondingly, the exciton binding energies are1.03, 0.76, 0.62, and 0.54 eV , inferred from the differencebetween the QP (sc GW 0) and optical (sc GW 0-BSE) gaps. These calculated QP band gaps, optical gaps, and excitonbinding energies are also listed in Table I. The convergence trend is obvious, particularly for the electronic band gap.However, due to the limitation of computation resource,scGW 0calculations with more dense k-point mesh are not performed here. Note that previous theoretical results showeda large value of exciton binding energy for monolayer MoS 2. For example, a value of 0.9 eV for monolayer MoS 2(3.16 ˚A) was obtained using empirical Mott-Wannier theory;24and a value of 1.03 eV was obtained by G0W0-BSE calculations for monolayer MoS 2(3.18 ˚A) using 6 ×6×1k-point mesh and including spin-orbital coupling,25which is the same as our above results using the same k-point mesh without spin-orbital coupling. Binding energy of 0.54 eV reported here is alsoconsistent with 0.5 eV adopting GW and BSE calculations. 26 Experimentally, two close peaks observed in adsorption spectrum of monolayer MoS 2around 1.9 eV are due to the valence band splitting caused by spin-orbital coupling.In our calculations, the spin-orbital coupling is omittedunless otherwise stated and this will not alter our mainconclusions presented in the current study. In order to make acomparison, we also performed the sc GW 0-BSE calculations with spin-orbital coupling using 6 ×6×1k-point mesh 155304-4QUASIPARTICLE BAND STRUCTURES AND OPTICAL ... PHYSICAL REVIEW B 87, 155304 (2013) TABLE I. QP band gap, optical band gap, and exciton binding energy for monolayer MoS 2and WS 2are obtained by QP sc GW 0and BSE with and without spin-orbital coupling (SOC) adopting different energy cutoffs and k-point mesh. All energies are in the unit of eV . Energy cutoffs kpoint Eg Eg(optical) Binding energy Monolayer MoS 2(3.160 ˚A) 400 and 200 6 ×6×1(SOC) 2.89 1.87 1.02 6×6×1 2.99 1.96 1.03 9×9×1 2.84 2.08 0.76 12×12×1 2.78 2.16 0.62 15×15×1 2.76 2.22 0.54 600 and 300 12 ×12×1 2.80 2.17 0.63 Monolayer MoS 2(3.190 ˚A) 600 and 300 12 ×12×1 2.66 2.04 0.62 Monolayer WS 2(3.155 ˚A) 400 and 200 6 ×6×1(SOC) 3.02 1.97 1.05 6×6×1 3.28 2.21 1.07 9×9×1 3.12 2.34 0.78 12×12×1 3.06 2.43 0.63 15×15×1 3.05 2.51 0.54 600 and 300 12 ×12×1 3.11 2.46 0.65 Monolayer WS 2(3.190 ˚A) 600 and 300 12 ×12×1 2.92 2.28 0.64 and 400 and 200 eV for energy cutoffs. The two peaks in BSE adsorption spectrum located at 1.87 and 2.05 eVand the corresponding exciton binding energy is 1.02 eV ,consistent with the aforementioned G 0W0-BSE calculations using the same k-point mesh and energy cutoffs with different pseudopotentials.25Notice that the exciton binding energy obtained with and without spin-orbital coupling for monolayerMoS 2as shown in Table Iis nearly the same, while the optical gap in the former case shifts about 0.1 eV towards lower energydue to the top valence band splitting of 0.17 eV according toour sc GW 0calculation. For the evolution of exciton binding energy as a function of strain, our results demonstrate that it is almost unchanged,i.e., 0.63 eV (strainless), 0.62 eV (1% strain), 0.62 eV (3%strain), and 0.59 eV (6% strain) (using 600 and 300 eV forenergy cutoffs and 12 ×12×1k-point mesh). The direct optical gaps are 2.17, 2.04, 1.81, and 1.52 eV for the fourcases shown in Fig. 3, respectively, and also shown in Fig. 4 using the orange left triangles. The experimental optical gapfor monolayer MoS 2was shown to be about 1.90 eV .7Since there was no mention of speciﬁc lattice parameter, here itis assumed to be the strainless case as shown in Fig. 4. Notice that the consistency is good between our theoreticaland experimental results. If spin-orbital coupling is taken intoaccount, the consistency will be improved further since the ﬁrstpeak in the adsorption spectrum moves towards lower energydue to the top valence band splitting. Most importantly, ourresults demonstrate that the optical gap of monolayer MoS 2 is very sensitive to tensile strain, which can be tuned bydepositing monolayer MoS 2on different substrates,13whereas the exciton binding energy is insensitive to it according toour current results. This insensitivity is mainly because thehole and electron are derived from the topmost valence andlowest conduction edge states close to VBM and CBM thatare signiﬁcantly localized on Mo sites (contributed by Modorbitals) irrespective of the magnitude of strain according to our DOS analysis. We also notice that layer-layer distance or the length of vacuum zone implemented in the periodical supercell methodshas an important inﬂuence on the magnitude of the GWband gap and the exciton binding energy. 27–29In order to obtain an accurate exciton binding energy, the convergenceofk-point mesh, the truncation of Coulomb interaction, 28 and the resulting accurate QP band structure ( G0W0or scGW) are necessary. Compared to exciton binding energy of 1.1 eV obtained by interpolation of G0W0band gap,29our exciton binding energy obtained using a denser kpoint is underestimated,30due to the ﬁnite thickness of vacuum layer adopted in our periodical supercell calculations. However, themagnitude of the optical gap is not affected by the vacuum layerheight according to our test (not shown here). An interestingobservation is that the optical gap of monolayer MoS 2is sensitive to the strain while the exciton binding energy is not.Our results also show that the spin-orbital coupling does notchange the magnitude of exciton binding energy, while theoptical gap reduces towards the experimental result due to theband splitting at Kpoints and better consistency is achieved. C. Chemical bonding properties of monolayer MoS 2 In order to gain further insight into the electronic structures, we revisit the DOS shown in Fig. 1. For the strainless case, the VBM states at Kmainly originate from Mo ( dxy+dx2−y2), and S(px+py) (decomposed porbitals not shown in Fig. 1). The CBM at Kis mainly contributed by Mo dz2and S ( px+py). The Mo dand S porbitals hybridize signiﬁcantly, therefore Mo and S form a covalent bond. Bader charge analysisfurther shows that ionic contribution exists in Mo-S bonds. 31 Notice that MLWFs can also illustrate the chemical bondingproperties of solids. 32T h eM L W F ss h o w ni nF i g . 6were constructed in two groups. The ﬁrst group was generated fromdguiding functions on Mo. The energy window contains the topmost valence band. Isosurface plots of the Mo d xyMLWFs shown in Fig. 6(a) showdxyorbitals form covalent bonding withpx(py) orbitals and also with a certain ionic component. The second group that MLWFs for the lowest-lying conductionband were also generated from was Mo dguiding functions. Isosurface plots of the Mo d z2MLWFs shown in Fig. 6(b) showdz2orbitals form antibonding with px(py) orbitals. The 155304-5SHI, PAN, ZHANG, AND Y AKOBSON PHYSICAL REVIEW B 87, 155304 (2013) FIG. 6. (Color online) Isosurface plots of (a) valence-band and (b) conduction band MLWFs for MoS 2(at constant lattice of 3.16 ˚A), at isosurface values ±0.9a n d±1.6/√ V, respectively, where Vis the unit cell volume, positive value red, and negative value blue. (a) is a Modxy-like function showing bonding with the S px(py)o r b i t a l ,a n d (b) is a Mo dz2-like function showing antibonding with the S px(py) orbital. chemical bonding characters demonstrated by MLWFs are consistent with our DOS analysis shown in Fig. 1. D. QP band structures and optical properties of strained monolayer WS 2 The QP band structures of monolayer WS 2under tensile strain are also investigated, motivated by its better performancethan monolayer MoS 2used as a channel in transistor devices.14 The calculation results are illustrated in Fig. 7. Similar to monolayer MoS 2,t h es c GW 0QP band structures of monolayer WS 2also undergo a direct to indirect band gap transition as tensile strain increases. The direct band gaps for thestrainless (at the experimental lattice of 3.155 ˚A 9) and under 1% tensile strain cases are 3.11 and 2.92 eV , respectively,and the latter corresponds to the optimized lattice constant formonolayer WS 2from this work. The corresponding indirect band gaps under 3% and 6% tensile strains are 2.49 and1.78 eV , respectively. Note that for the strainless case, ourDFT result predicts monolayer WS 2to be an indirect band gap semiconductor with CBM only about 16 meV lower than thelowest conduction band at Kpoints, which is contrary to recent full potential methods. 9The difference may be originated from the technical aspect of these calculations, such as the employedpseudopotential method. 33However, after the GW correction, a correct direct band gap is achieved. For optical properties of monolayer WS 2, our calculated QP band gaps, optical gaps, and exciton binding energies are alsolisted in Table I. It is obvious that the monolayer WS 2presents many similar properties compared to monolayer MoS 2,f o r example, the gaps and exciton binding energy also demonstratea convergence trend as k-point mesh increases; the spin-orbital coupling has little inﬂuence on the magnitude of the exciton binding energy. Notice that our sc GW 0calculation predicts the top valence band splitting of monolayer WS 2to be 0.44 eV , larger than that of monolayer MoS 2of 0.17 eV , because W is much heavier than Mo. The resulting ﬁrst peak in BSEadsorption spectrum shifts 0.26 eV towards lower energy, alsolarger than that of monolayer MoS 2of 0.1 eV correspondingly. As for the strain effect, the BSE optical gap at our optimized lattice constant of 3.190 ˚A is 2.28 eV , while at 3.16 ˚Ai t i s 2.46 eV , as shown in Table I. The former corresponding to 1% tensile strain, results in 0.18 eV reduction of band gaps.This demonstrates that the band gaps and optical gaps are alsovery sensitive to tensile strain, whereas the exciton bindingenergy is not. Based on above analysis, we predict the exciton binding energy of monolayer WS 2is similar to that of MoS 2. Experimentally, the PL maximum of monolayer WS 2locates between 1.94 and 1.99 eV .34Considering the large shift of the peak in the BSE adsorption spectrum caused by spin-orbital coupling, our results at optimized lattice of 3.190 ˚Aa r e consistent with experimental results. 34,35 FIG. 7. (Color online) DFT, G0W0,a n ds c GW 0QP band structures for WS 2at lattice constants of (a) 3.155, (b) 3.190 (optimized lattice constant this work), (c) 3.250, and (d) 3.344 ˚A , corresponding to 0%, 1%, 3%, and 6% tensile strain (with reference to 3.155 ˚A), respectively. The Fermi level is set to be zero. 155304-6QUASIPARTICLE BAND STRUCTURES AND OPTICAL ... PHYSICAL REVIEW B 87, 155304 (2013) According to our above sc GW 0and BSE calculations for monolayer MoS 2and WS 2, it is clear that the self energy within the sc GW 0calculations enlarges the band gap by accounting for the many-body electron-electron interactionsmore accurately, while the strong excitonic effect results ina signiﬁcant reduction of the band gap. Combining the twoopposite effects on band gaps, the ﬁnal resulting optical gap isconsistent with DFT band gaps. Therefore, the good band gapagreement between DFT and experiment is only a coincidencedue to the fact that QP band gap correction is almost offset byexciton binding energy. This phenomenon was also observedin hexagonal boron nitride systems, which also have strongexcitonic effect. 27,36 We also perform the sc GW 0QP band structures for monolayer MoS 2and WS 2under 1% compressive strains. Our results show that the compressed MoS 2has a direct band gap of 2.97 eV , while the compressed WS 2has an indirect band gap of 3.13 eV and KtoKdirect gap of 3.30 eV . Our sc GW 0results show that both MoSe 2and WSe 2are also a direct semiconductor at the strainless state. The experimentallattice constants 9for MoSe 2and WSe 2are 3.299 and 3.286 ˚A and the optimized lattice constants are 3.327 and 3.326 ˚A, respectively; their direct K-K band gaps are 2.40 and 2.68 eV at experimental lattices and 2.30 and 2.50 eV at the optimizedlattices. Compared to the experimental lattice, the optimizedlattice corresponds to 0.86% (1.22%) tensile strain for MoSe 2 (WSe 2), and the band gap also decreases with increasing tensile strain. E. Effective mass Based on the more accurate sc GW 0QP band structures, the effective mass of carriers for TMDs are calculated by ﬁtting the bands to a parabola according to E=¯h2k2 2mem∗, where me is the electron static mass. A k-point spacing smaller than 0.03 ˚A−1is used to keep parabolic effects. Electron and hole effective masses ( m∗) at different strains are collected in Table II.F o rM o S 2under different strains, the CBM always locates in Kpoint, and the electron effective mass Ke increases with increasing compressive strain while decreases with increasing tensile strain. As for the hole, initially theeffective mass also decreases as the tensile strain increases.After the direct to indirect gap transition, VBM shifts to /Gamma1 with a heavier hole, which also decreases as the tensile strainincreases. Compared to the effective masses of 0.64 and 0.48for the hole and electron at Kpoint based on DFT calculation performed at the experimental lattice 9for MoS 2, the effective masses are reduced due to the GW correction in our study. It is noted that the carrier effective masses obtained by our scGW 0calculations do not include the spin-orbital coupling effect. Compared with those including spin-orbital effect formonolayer MoS 2,24it is found that the electron effective masses are in good agreement while the present hole effectivemass is slightly smaller. This is mainly because the spin-orbitalcoupling alters the curvature of the topmost valence band closeto VBM, while the lowest conduction band close to CBM isnot affected. The large difference between the sc GW (scGW 0) andG0W0result25may be due to the poor k-points sampling and non-self-consistent (one-shot) GW calculations of thelatter. For WS 2, MoSe 2, and WSe 2, their masses also show similar behaviors. It is noted that at the same strain level,the electron effective mass of WS 2is the lightest; and electron effective mass decreases as strain increases, making WS 2more attractive for high performance electronic device applicationssince a lighter electron effective mass can lead to a highermobility. Theoretical device simulations also demonstratedthat as a channel material, the performance of WS 2is superior to that of other TMDs.14 IV . SUMMARY In summary, the QP band structures of monolayer MoS 2 and WS 2at both strainless and strained states have been studied systematically. The sc GW 0calculations are found to be reliable for such calculations. Using this approach, we ﬁndthey share many similar behaviors. For the optical propertiesof monolayer MoS 2, exciton binding energy is found to be insensitive to the strain. Our calculated optical band gap isalso consistent with experimental results. In addition, we ﬁndthat the electron effective masses of monolayer MoS 2,W S 2, MoSe 2, and WSe 2decrease as the tensile strain increases, and WS 2possesses the lightest mass among the four monolayer materials at the same strain. Importantly, the present work TABLE II. Electron and hole effective masses ( m∗) derived from partially sc GW 0QP band structures for monolayer MoS 2,W S 2,M o S e 2, and WSe 2at different strains. The effective masses at Kand/Gamma1points are along K/Gamma1andM/Gamma1directions, respectively. Compressive (1%) Experimental lattices Optimized lattices Tensile (3%) Tensile (6%) MoS 2 Ke 0.40 0.36 (0.35,a0.60b) 0.32 0.29 0.27 Kh 0.40 0.39 (0.44,a0.54b) 0.37 /Gamma1h 1.36 0.90 WS 2 Ke 0.27 0.24 0.22 0.20 Kh 0.32 0.31 /Gamma1h 1.24 0.79 MoSe 2 Ke 0.38 0.36 Kh 0.44 0.42 WSe 2 Ke 0.29 0.26 Kh 0.34 0.33 aEffective masses listed here are averages of the longitudinal and transverse values in Ref. 24. bEffective masses listed here are averages of the curvatures along the /Gamma1K andKM directions in Ref. 25. 155304-7SHI, PAN, ZHANG, AND Y AKOBSON PHYSICAL REVIEW B 87, 155304 (2013) highlights a possible avenue to tune the electronic properties of monolayer TMDs using strain engineering for potentialapplications in high performance electronic devices.ACKNOWLEDGMENTS Work at Rice was supported by the U.S. Army Research Ofﬁce MURI grant W911NF-11-1-0362, and by the RobertWelch Foundation (C-1590). *zhangyw@ihpc.a-star.edu.sg †biy@rice.edu 1E. Fortin and W. Sears, J. Phys. Chem. 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PhysRevB.83.155450.pdf	PHYSICAL REVIEW B 83, 155450 (2011) Transport properties of graphene quantum dots J. W. Gonz ´alez and M. Pacheco Departamento de F ´ısica, Universidad T ´ecnica Federico Santa Mar ´ıa, Casilla 110 V , Valpara ´ıso, Chile L. Rosales* Departamento de F ´ısica, Universidad T ´ecnica Federico Santa Mar ´ıa, Casilla 110 V , Valpara ´ıso, Chile and Instituto de F ´ısica, Pontiﬁcia Universidad Cat ´olica de Valpara ´ıso, Casilla 4059, Valpara ´ıso, Chile P. A. Orellana Departamento de F ´ısica, Universidad Cat ´olica del Norte, Casilla 1280, Antofagasta, Chile (Received 24 September 2010; revised manuscript received 10 March 2011; published 27 April 2011) In this work we present a theoretical study of transport properties of a double crossbar junction composed of segments of graphene ribbons with different widths forming a graphene quantum dot structure. The systemsare described by a single-band tight binding Hamiltonian and the Green’s function formalism using real spacerenormalization techniques. We show calculations of the local density of states, linear conductance, and I-V characteristics. Our results depict a resonant behavior of the conductance in the quantum dot structures, whichcan be controlled by changing geometrical parameters such as the nanoribbon segment widths and the distancebetween them. By application of a gate voltage on determined regions of the structure, it is possible to modulatethe transport response of the systems. We show that negative differential resistance can be obtained for low valuesof applied gate and bias voltages. DOI: 10.1103/PhysRevB.83.155450 PACS number(s): 61 .46.−w, 73.22.−f, 73.63.−b I. INTRODUCTION In the last few years, graphene-based systems have attracted a lot of scientiﬁc attention. Graphene is a single layer of carbonatoms arranged in a two-dimensional hexagonal lattice. In theliterature, several experimental techniques have been reportedto obtain this crystal, such as mechanical peeling or epitaxialgrowth. 1–3On the other hand, graphene nanoribbons (GNRs) are stripes of graphene which can be obtained by differentmethods like high-resolution lithography, 4controlled cutting processes,5or unzipping multiwalled carbon nanotubes.6 Different graphene heterostructures based on patterned GNRshave been proposed and constructed, such as graphenejunctions, 7graphene ﬂakes,8graphene antidot superlattices,9 and graphene nanoconstrictions.10The electronic and transport properties of these nanostructures are strongly dependenton their geometric conﬁnement, allowing the possibilityto observe quantum phenomena like quantum interferenceeffects, resonant tunneling, and localization. In this sense, thecontrolled modiﬁcation of these quantum effects by means ofexternal potentials which change the electronic conﬁnementcould be used to develop new technological applications suchas graphene-based composite materials, 11molecular sensor devices,12,13and nanotransistors.14 In this work we study the transport properties of quantum- dot-like structures, formed by segments of graphene ribbonswith different widths connected to each other, forming adouble-crossbar junction. 17,18These graphene quantum dots (GQDs) could be versatile experimental systems which allowa range of operational regimes from conventional single-electron detectors to ballistic transport. The systems we haveconsidered are conductors formed by two symmetric crossbarjunctions of width N Band length LB, and a central region that separates the junctions, of width NCand length LC. Two semi-inﬁnite leads of width NL=NCare connectedto the ends of the central conductor. A schematic view of the considered system is presented in Fig. 1. We studied the different electronic states appearing in the system as functionsof the geometrical parameters of the GQD structure. Wefound that the GQD local density of states (LDOS) as afunction of the energy shows the presence of a variety of sharppeaks corresponding to localized states and also states thatcontribute to the electronic transmission which are manifestedas resonances in the linear conductance. By changing thegeometrical parameters of the structure, it is possible to controlthe number and position of these resonances as functions ofthe Fermi energy. On the other hand a gate voltage appliedat selected regions of the conductors allows the modulationof their transport properties, exhibiting a negative differentialconductance (NDC) at low values of the bias voltage. II. MODEL All considered systems have been described by using a single- π-band tight binding Hamiltonian, taking into account nearest-neighbor interactions with a hopping parameter γ0= 2.75 eV . In addition, we have considered hydrogen passivation by setting a different hopping parameter for the carbon dimersat the ribbon edges, 19γedge =1.12γ0. The electronic properties of the systems have been calculated using the surface Green’s function matchingformalism. 13,21In this scheme, we divide the heterostructure into three parts, two leads composed of semi-inﬁnite pristineGNRs, and the conductor region composed of double GNRcrossbar junctions, as shown in Fig. 1. In the linear response approach, the electronic conductance is calculated by the Landauer formula. In terms of the 155450-1 1098-0121/2011/83(15)/155450(8) ©2011 American Physical SocietyGONZ ´ALEZ, PACHECO, ROSALES, AND ORELLANA PHYSICAL REVIEW B 83, 155450 (2011) FIG. 1. Schematic view of a GQD structure based on leads of widthNL=9, and a conductor region composed of two symmetrical junctions of width NB=21 and length LB=3 separated by a central structure of length LC=4 and width NC=9. conductor Green’s functions, it can be written as22 G=2e2 h¯T(E)=2e2 hTr/bracketleftbig /Gamma1LGR C/Gamma1RGAC/bracketrightbig , (1) where ¯T(E) is the transmission function of an electron crossing the conductor region, and /Gamma1L/R =i[/Sigma1L/R −/Sigma1† L/R] is the coupling between the conductor and the respectiveleads, given in terms of the self-energy of each lead: /Sigma1 L/R = VC,L/RgL/RVL/R,C . Here, VC,L/R are the coupling matrix elements and gL/Ris the surface Green’s function of the corresponding lead.13The retarded (advanced) conductor Green’s functions are determined by22 GR,A C =/bracketleftbig E−HC−/Sigma1R,A L −/Sigma1R,A R/bracketrightbig−1, (2) where HCis the Hamiltonian of the conductor. In order to calculate the differential conductance of the system, wedetermine the I-Vcharacteristics by using the Landauer formalism. 22At zero temperature, it reads I(V)=2e h/integraldisplayμ0+V/2 μ0−V/2¯T(E,V )dE, (3) where μ0is the chemical potential of the system in equilibrium and ¯T(E,V ) is deﬁned by Eq. ( 1). The Green’s functions and the coupling terms depend on the energy and the bias voltage.We consider a linear voltage drop along the longitudinaldirection of the conductor, and the gate voltage is includedin the on-site energy at the regions in which this potential isapplied. In what follows the Fermi energy is taken as the zeroenergy level, all energies are written in terms of the hoppingparameter γ 0, and the conductance is written in units of the quantum of conductance G0=2e2/h. III. RESULTS AND DISCUSSION In Fig. 2, we display results of the linear conductance for a graphene quantum dot structure formed by two armchair rib-bon leads of width N L=5 and a conductor region composed of two symmetric crossbar junctions of width NB=17 and variable lengths LB(from 1 up to 7). Two distances between the junctions[Fig. 2(a)LC=5 and Fig. 2(b)LC=10] are considered and the conductance of a pristine NL=5 armchair nanoribbon is included as a comparison (light green dottedline).FIG. 2. (Color online) Conductance as a function of the Fermi energy for a graphene quantum dot structure composed of twoarmchair ribbon leads of width N L=5 and a double symmetric crossbar junction of width NB=17 and variable length, from LB=1 up to LB=7. The central region has a width NC=5 and two separations (a) LC=5a n d( b ) LC=10. Light green dotted line corresponds to the conductance of a pristine NL=5 ribbon. All curves have been shifted by 2 G0for a better visualization. In both panels it is possible to observe a series of peaks at deﬁned energies in the conductance curves. This resonantbehavior of the electronic conductance arises from the inter-ference of the electronic wave functions inside the structure,which travel forth and back forming stationary states in theconductor region (well-like states). In order to understandthese results, it is convenient to deﬁne two energy regions,the low-energy range from 0 up to 0 .7γ 0(corresponding to the ﬁrst quantum of conductance for the pristine N=5 armchair ribbon) and the high-energy range, from 0 .7γ0to 1.2γ0(corresponding to the second step of conductance of the N=5 pristine system). In the low-energy range, it is clear that the conductance peaks correspond to resonant states belonging to the centralregion of the conductor. By increasing the relative distanceL Cof the central part of the system, the number of allowed well-like states also increases and, as a consequence, the con-ductance curves exhibit more resonances. 13,17,23The well-like states remain almost invariant under geometrical modiﬁcationsof the crossbar junctions. However, for certain energy rangesand for particular junction lengths, the electronic transmissionof the system exhibits an almost constant value. For instance,in both panels of Fig. 2, for the cases of L B=1 and 4 at the energy range 0.4 γ0to 0.65 γ0. This effect corresponds to a constructive interference between well-like states from thecentral region with states belonging to the crossbar junction 155450-2TRANSPORT PROPERTIES OF GRAPHENE QUANTUM DOTS PHYSICAL REVIEW B 83, 155450 (2011) regions. The different interference effects will be clariﬁed by analyzing the LDOS of these systems, which is done further inthis paper. In the high-energy region, the conductance curvesexhibit a complex behavior as a function of the geometricalparameters of the GQD structures. There is not a predictablebehavior of the conductance as the width and length of thecrossbar junction are increased. It is important to point out, from the analysis of Fig. 2, that it is possible to identify some interesting effects associated withwell-known quantum phenomena. For instance, in Figs. 2(a) and2(b), for the cases L B=5,6, and 7 at energies around E=0.5γ0, it is possible to observe a nonsymmetric line shape, which corresponds to a convolution of a Fano-like24and a Breit-Wigner25resonance. This kind of line shape, has been observed before in other mesoscopic systems by Orellana andco-workers. 26In that reference, a simple model is used of two localized states with the same energy ωcdirectly coupled to each other by a coupling τand indirectly coupled throughout a common continuum. The corresponding resonances have beenadjusted by using the following expression: T(ω)=4η 2[(ω−ωc)x−τ]2 [(1 −x2)η2−(ω−ωc)2+τ2]2+4(ω−ωc−τx)2, (4) where ηis the width of a localized state coupled to the continuum and xdeﬁnes the degree of asymmetry of the system. We realize that a possible interference mechanism occur- ring in our considered system can be explained with theabove model, which helps to get an intuitive understandingof the origin of some conductance line shapes. In Fig. 3 we have plotted a particular conductance resonance and thecorresponding ﬁtting given by the model represented byEq. ( 4), where good agreement is observed between the curves. In what follows we focus our analysis on the resonant behavior exhibited by the conductance curves, analyzingthe different electronic states in the conductor. We haveperformed calculations of the spatial distribution of LDOS forcertain energies corresponding to different states present in theconductor. In the bottom panel of Fig. 4we show results for the FIG. 3. (Color online) Numerical adjustment of a convolution of a Fano and a Breit-Wigner line shape (red dashed line) and that of the conductance resonance in the system (black solid line) with η=0.01γ0,ωc=0.59γ0,x=0.7, and τ=0.02γ0.FIG. 4. (Color online) LDOS for a GQD formed by a double crossbar junction of width NB=17 and length LB=3 separated by a central region of width NC=5 and length LC=5. (a), (b), and (c) correspond to the contour plots of some sharp LDOS resonances marked in the bottom plot. As a reference, the LDOS of a pristineN=5 armchair ribbon is plotted as a dotted green line. LDOS as a function of the Fermi energy, for a GQD structure formed by a double crossbar junction of width NB=17 and length LB=3, separated by a central region of width NC=5 and length LC=5. We start our analysis by focusing on some sharp states present in the curve of LDOS vs energy of this ﬁgure. Wehave marked the ﬁrst three sharp states in this LDOS plotwith the letters (a), (b), and (c) and have calculated thespatial distribution of these states, representing them by thecorresponding contour plots exhibited in the ﬁgure. Thesestates are completely localized at the crossbar junctions, andthey correspond to bound states in the continuum (BICs) 27–29 as we established in a previous work.30It is not expected that this kind of state plays a role in the electronic transport ofthese GQD structures, which is shown in the correspondingconductance curves of Fig. 2. In what follows we will focus our analysis on those states that contribute to the conductance of the systems. Figure 5 shows the spatial distribution of LDOS for a GQD formed 155450-3GONZ ´ALEZ, PACHECO, ROSALES, AND ORELLANA PHYSICAL REVIEW B 83, 155450 (2011) FIG. 5. (Color online) LDOSs for a GQD structure formed by a double crossbar junction of width NB=17 and length LB=3 separated by a central region of width NC=5 and length LC=5. (a), (b), (c), and (d) correspond to the contour plots of those resonant states marked in the LDOS plot displayed at the bottom. by a double crossbar junction of width NB=17 and length LB=3, separated by a central region of width NC=5 and length LC=5. As a reference, in the bottom panel we have included a plot with the LDOS versus Fermi energy of theG Q Ds y s t e m ,w h e r ew eh a v em a r k e dw i t ht h el e t t e r s( a ) ,( b ) ,(c), and (d) four particular energy states. The correspondingcontour plots are displayed in the upper parts of the ﬁgure. In these plots, it is possible to observe the resonant behavior of these states, which are completely distributed along theGQD structure, presenting a maximum of the probabilitydensity at the center of the system. This condition favorsthe alignment of the electronic states of the leads with the resonant states in the conductor and consequently a unitarytransmission at those energy values is expected. This behavioris reﬂected as a series of resonant peaks in the conductance ofthe system that could be controlled by means of the geometricalparameters of the GQD, as shown in Fig. 2. At higher energies there is an interplay between localized states in the crossbarjunctions and resonant states in the central region of theconductor. As shown in Fig. 5(d), some states are strongly dependent on the geometry of the junctions; therefore for someparticular conﬁguration it is possible to observe a non-nulltransmission at these energies, while for other cases, there aredestructive interference mechanisms that suppress completelythe transmission in that energy region. We have studied different conﬁgurations of GQD structures, systematically varying some geometric parameters. We haveobserved a quite general behavior of such resonant conductorswith the presence of sharp and resonant states. Dependingon each particular system, changes can be observed in thenumber and position in energy of these states, as well asin their spatial distribution. The different intensity of thepeaks in the LDOS curves depends on the spatial distributionof the states. There are states completely extended alongthe conductor [as in Fig. 5(b)] which generate wider and less intense peaks. On the other hand, there are resonantstates which are more concentrated in certain regions of theconductor [as in Figs. 5(c) and5(d)] which generate sharper and more intense peaks in the LDOS. Now we focus our analysis on the effects of an applied gate voltage on the transport properties of GQD structures. Resultsfor the conductance as a function of the Fermi energy, fordifferent values of a gate voltage applied in selected regions ofa GQD, are shown in Fig. 6. The systems are composed of leads of width N L=5, two crossbar junctions of width NB=17, and a central part of width NC=5 and length LC=5. Figures 6(a)and6(c)correspond to junctions of length LB=2, while Figs. 6(b) and6(d) correspond to junctions of length LB=3. Finally, the upper panels correspond to a gate voltage appliedat the crossbar junction regions, whereas the lower panelscorrespond to a gate voltage applied at the central part of thestructure. In these contour plots of conductance, it is possible to observe the behavior of the resonant states of the system witha gate voltage applied at different regions of the structure.The lower panels of Fig. 6show the case of a gate voltage applied at the central region of the considered GQDs. In theseplots the linear dependence of the conductance resonances as afunction of the gate voltage is manifested. It can be shown thatthe electronic states of a pristine armchair graphene ribbon areregularly spaced in the whole energy range; 15,16,20therefore, as the gate voltage is applied, there will be a high probability thatthe lead states become aligned with the resonant states in thecentral region of the structure. This behavior in completelygeneral and independent of the width L Bof the crossbar junctions. The linear behavior of the conductance peaks couldbe useful in nanoelectronic devices, due to the possibilityof controlling the current ﬂow through these systems. Thisargument will become more clear with the analysis ofthe differential conductance, which is shown below in thispaper. 155450-4TRANSPORT PROPERTIES OF GRAPHENE QUANTUM DOTS PHYSICAL REVIEW B 83, 155450 (2011) FIG. 6. (Color online) Conductance as a function of Fermi energy and gate voltage for GQDs composed of leads of width NL=5, two crossbar junctions of width NB=17, and a central part of width NC=5 and length LC=5. (a) and (c) correspond to junctions of length LB=2, while (b) and (d) correspond to junctions of length LB=3. In the upper panels the gate voltage is applied at the crossbar regions, and in the lower panels the gate voltage is applied at the central structure. The black segmented lines highlight different slopes discussed in thetext. The case of a gate voltage applied at the crossbar junction regions is shown in the upper panels of Fig. 6. The conductance behavior is more complicated to analyze; nevertheless, it isstill possible to observe a linear dependence of the resonantstates on the gate voltage. However, two different slopes canbe noticed, for states belonging to the crossbar junctions(lower slope) and for states belonging to the central regionof the conductor (higher slope). In addition, the panels exhibitan important reduction in the conductance gap, for differentvalues of gate voltage. This effect is mainly produced by anenergy shift of the localized states at the junctions, whichinduces a less destructive interference with the resonant states.It is important to point out that this effect can be observed onlybecause the gate potential is applied simultaneously to bothcrossbar junctions, otherwise, the conductance gap would notbe noticeably affected by the gate potential. Finally, the dark(blue online) regions present in Fig. 6occur at energy ranges around the LDOS singularities of the pristine N=5 armchair GNR. At these energies, the second and the third allowed statesappear, which interrupt the linear behavior of the conductanceresonances as a function of the gate voltage. In order to understand the presence of two different slopes in the upper panels of Fig. 6, we present a simple model which keeps the underlying physics of the considered system andallows us to explain our results qualitatively. The scheme showed in Fig. 7is a simple representation of our conductor. The system is composed of a linear chain ofthree sites, which are connected to two semi-inﬁnite leads. Wehave considered four quantum dots connected to the extremesof the chain forming a double crossbar junction, at which wehave applied symmetrically a gate voltage V g. This potential will modify the on-site energy of the dots by a linear shift ofenergy proportional to the gate voltage amplitude. By use of the Dyson equation, it is possible to calculate the Green’s function of the central site of the chain labeled by 0,which takes the form G 00=1 ω−ε0−/Sigma1, (5)where ωis the energy of the incident electrons, ε0is the central on-site energy, and the self-energy /Sigma1is given by the following expression (see the Appendix for a detailed derivation): /Sigma1=v2 /parenleftbig ω−v2 ω−Vg/parenrightbig2+/tildewide/Gamma12/bracketleftbigg/parenleftbigg ω−v2 ω−Vg/parenrightbigg +i/tildewide/Gamma1/bracketrightbigg .(6) In this model, the self-energy of the Green’s functions of the central region acquires a real part that depends on thegate voltage applied to the crossbar junction region. As aconsequence, two different slopes appear in the behavior ofthe conductance peaks as a function of the gate voltage. Oneof these slopes corresponds to the direct evolution of the statesbelonging to the crossbar junctions as a function of the gatevoltage (lower slope), and the higher slope corresponds to theindirect states belonging to the central region. Now we focus our analysis on the I-Vcharacteristics and the differential conductance of these resonant GQDs. Figure 8 shows results of these transport properties for a conductorformed by two crossbar junctions of width N B=17, length LB=2 and a central region of width NC=5. In these plots the length of the central structure is varied from LC=1u p toLC=20. In Fig. 8(a) it is possible to observe that for a very small separation between the two junctions, the I-V characteristics exhibit abrupt slope changes and oscillations forcertain ranges of the bias voltage. This behavior is producedby the increasing number of resonant states as a result of FIG. 7. (Color online) A simple model of two crossbar junctions formed by two quantum dots, coupled to a linear chain of sites. 155450-5GONZ ´ALEZ, PACHECO, ROSALES, AND ORELLANA PHYSICAL REVIEW B 83, 155450 (2011) (a) (b) FIG. 8. (Color online) (a) Current versus bias voltage and (b) differential conductance as a function of bias voltage for a GQD composed of two crossbar junctions of length LB=2 and width NB=17 separated by a central region of width NC=5a n dv a r i a b l e length from LC=1u pt o LC=20. All curves have been shifted for better visualization. the enlargement of the conductor central region. The applied bias voltage allows the continuum alignment of the resonantstates of the system with the electronic states of the leads,leading to variations of the current intensity. On the otherhand, every I-Vcurve shows a wide gap of zero current until a certain bias voltage. This threshold value exhibits a lineardependence of the length of the central region of the conductor.As the distance between the junctions is increased, there aremore resonant states available at lower energies because theelectronic conﬁnement in this region is weaker; therefore,electronic transmission under a bias voltage is possible at lowervoltage values. The abrupt changes of the current as a function of the bias voltage are clearly reﬂected in the differential conductanceof these systems. In Fig. 8(b) it is possible to observe the behavior of the dI/dV -Vcurves as a function of the length of the central region of the conductor. As this region is enlarged,the oscillations in the differential conductance become moreevident. Each time the bias voltage aligns the resonant statesof the conductor with the lead states, the current will increaseand a positive change in the differential conductance occurs.However, if the bias voltage is not enough to align the states,the current drops and the differential conductance becomesnegative in a range of voltage. This can be seen in the casesL C=10, 15, and 20 in Fig. 8(b). The bias voltage value at which negative differential conductance31–33occurs depends directly on the distance between the crossbar junctions. Now we focus our analysis in the effects of a gate voltage on the differential conductance of the systems. In Fig. 9we present results for the differential conductance of a GQD composed oftwo crossbar junctions of length L B=2 and width NB=17 separated by a central region of width NC=5 and length LC=5. The gate potential has been applied at selected regions of the structure: (a) at crossbar junction regions and (b) at thecentral region. In Fig. 9(b) of this ﬁgure a periodic modulation of the differential conductance can be observed as a functionof the gate voltage applied at the central region of the GQD.This behavior is directly related to the linear evolution ofthe resonant states of the conductor, and consequently to theFIG. 9. (Color online) Differential conductance of a GQD structure composed of two crossbar junctions of length LB=2a n d width NB=17 separated by a central region of width NC=5a n d length LC=5. The gate voltage is applied at selected regions of the conductor: (a) on the crossbar junction region and (b) on the central region. peaks of conductance of the system [Figs. 6(c) and6(d)]. For instance, in the conﬁguration considered in this ﬁgure, abruptchanges of the differential conductance as a function of thegate voltage can be noted for bias voltages values around0.8γ 0, which indicates an abrupt increase of the current ﬂowing through the conductor. This behavior is very general for othersystems studied, which suggests possible applications in thedevelopment of GQD-based electronic devices. In the case in which the gate voltage is applied simul- taneously at the crossbar junction regions [Fig. 9(a)]i ti s still possible to note a certain regularity in the dependenceof the differential conductance as a function of the externalpotentials, although a periodic modulation is not clearlyobserved. However, in this conﬁguration of applied potentialsour results show that the GQD structure exhibits NDC at verylow values of bias voltage and gate voltages. This behaviorcan be understood by observing Figs. 6(a)and6(b), where the contour plots show areas where the conductance is completelysuppressed at low energies, for different values of gate voltage.We have observed this kind of behavior in every conﬁgurationconsidered in this work. In relation to the practical limitations of our calculations, we would like to mention that although the sizes of the structuresused in this work are below the limit of those experimentallyrealizable, our calculations can be scaled to structures of biggersizes. On the other hand our model does not include disorder orelectron-electron interaction; nevertheless, we are convincedthat our results will be robust under these kinds of effect as 155450-6TRANSPORT PROPERTIES OF GRAPHENE QUANTUM DOTS PHYSICAL REVIEW B 83, 155450 (2011) they are in mesoscopic systems. For instance, it is known that in quantum dots the resonant tunneling and the Fano effectsurvive the effect of electron-electron interaction. 34,35 IV. SUMMARY In this work we have analyzed the transport properties of a GQD structure formed of a double crossbar junctionmade of segments of GNRs of different widths. We havefocused our analysis on the dependence of the electronicand transport properties on the geometrical parameters of thesystem, looking for the modulation of these properties throughexternal potentials applied to the structure. Our results depicta resonant behavior of the conductance in the quantum dotstructures which can be controlled by changing geometricalparameters such as the nanoribbon widths and the distancebetween them. We have explained our results in terms of ananalysis of the different electronic states of the system. Thepossibility of modulating the transport response by applyinga gate voltage on determined regions of the structure hasbeen explored, and it has been found that negative differentialconductance can be obtained for low values of the gateand bias applied voltages. Our results suggest that possibleapplications with GQDs can be developed for new electronicdevices. ACKNOWLEDGMENTS The authors acknowledge the ﬁnancial support of USM Internal Grant No. 110971 and FONDECYT program GrantsNo. 11090212, No. 1100560, and No. 1100672. L.R. alsoacknowledges PUCV-DII Grant No. 123.707/2010. APPENDIX: GREEN’S FUNCTION OF THE SIMPLE MODEL In Sec. III, we have introduced a simple model in order to explain the different slopes of Fig. 6. This model is composed of a linear chain of three sites, of the same energy, at whichwe have coupled four quantum dots (QDs) of energies ε n(n =u,d), forming a crossbar junction conﬁguration exhibited in Fig. 7. This simple scheme is very useful in explaining qualitatively the electronic behavior of the graphene quantumdot that we have studied. Let us start with the Hamiltonian of the system described by Fig. 7: H T=Hleads +Hc+Hc,leads, (A1) where the Hamiltonian of the leads, Hleads, is given by Hleads =/summationdisplay k,α(L,R)εk,αc† k,αck,α, (A2)the conductor Hamiltonian Hcis given by Hc=1/summationdisplay i=−1εif† ifi+t1/summationdisplay i=0(f† i−1fi+H.c.) +/summationdisplay m(−1,1)/summationdisplay n(u,d)[εm,nd† m,ndm,n +v(d† m,nfm+H.c.)], (A3) and ﬁnally the leads-conductor Hamiltonian is given by Hc,leads =/summationdisplay k,α(L,R)/summationdisplay m(−1,1)Vα(f† mck,α+H.c.).(A4) By using the Dyson equation, it is possible to calculate the Green’s function of site 0. Following a standard procedure wehave obtained G 00=g0+g0vG 10+g0vG −10, (A5) G10=g1vG 00+g1/summationdisplay kVRGkR,0, (A6) G−10=g0vG −10+g−1/summationdisplay kVLGkL,0, (A7) where GkR,0=gkVRG10andGkL,0=gkVLG−10. Replacing these expression in the previous set of equations, we obtain G10=g1vG 00 1−g1/summationtext kRV† RgkRVR, (A8) G−10=g−1vG 00 1−g−1/summationtext kLV† LgkLVL. (A9) Considering g0=1/(ω−ε0), and replacing the above expressions for G10andG−10,G00reads G00=1 ω−ε0−/Sigma1, (A10) where the self-energy is deﬁned by the following expression: /Sigma1=g1v2 1−g1i/Gamma1R+g−1v2 1−g−1i/Gamma1L(A11) with/Gamma1R=/summationtext kRV† RgkRVRand/Gamma1L=/summationtext kLV† LgkLVL. Using the expressions for the on-site Green’s functions for the sites −1 and 1 given by g−1=1/(ω−ε−1) and g1= 1/(ω−ε1), and considering a gate voltage Vgapplied to the QDs, which redeﬁnes their on-site energies by ˜ εn=εn+Vg, it is possible to write an expression for the self-energy of thesystems as /Sigma1=v 2 /parenleftbig ω−v2 ω−Vg/parenrightbig2+/tildewide/Gamma12/bracketleftbigg/parenleftbigg ω−v2 ω−Vg/parenrightbigg +i/tildewide/Gamma1/bracketrightbigg ,(A12) where we have considered a symmetric system ( /Gamma1L=/Gamma1R). In this approach, it is possible to write a compact form for the self-energy given in Eq. ( A12), which contains a real (gate-voltage dependent) and an imaginary part. 155450-7GONZ ´ALEZ, PACHECO, ROSALES, AND ORELLANA PHYSICAL REVIEW B 83, 155450 (2011) *luis.rosalesa@usm.cl 1K. 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). 2C. Berger, Z. Song, T. Li, X. Li, A. Y . 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PhysRevB.82.144528.pdf	Effect of an Ohmic environment on an optimally controlled ﬂux-biased phase qubit Amrit Poudel and Maxim G. Vavilov Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA /H20849Received 6 August 2010; revised manuscript received 4 October 2010; published 28 October 2010 /H20850 We analyze the effect of environment on the gate operation of ﬂux-biased phase qubits. We employ the master equation for a reduced density matrix of the qubit system coupled to an Ohmic environment, describedby the Caldeira-Leggett model. Numerically solving this equation, we evaluate the gate error as a function ofgate time, temperature, and environmental coupling strength for experimentally determined qubit parameters.The analysis is presented for single-quadrature microwave /H20849control /H20850pulses as well as for two-quadrature pulses, which lower the gate error signiﬁcantly for idealized systems in the absence of environment. Our resultsindicate that two-quadrature pulses with ﬁxed and variable driving frequency have similar performance, whichoutweighs the performance of single-quadrature pulses, in the presence of environment. DOI: 10.1103/PhysRevB.82.144528 PACS number /H20849s/H20850: 85.25.Cp, 03.65.Yz, 03.67.Lx I. INTRODUCTION Superconducting circuits containing Josephson junctions are promising candidates for a scalable quantum-informationprocessing. 1–3However, small separations between succes- sive quantum energy states in these circuits4,5do not permit selective manipulation of the qubit in a two-dimensional sub-space and results in a dynamical leakage of a quantum stateto a broader Hilbert space of the circuit. 6To reduce this leakage, Motzoi et al.7proposed a derivative removal by adiabatic gate /H20849DRAG /H20850method, which reduces the gate error to 10−5for an experimentally optimal gate time of 6 ns. This error is well below the required error threshold of 10−3for fault tolerant quantum computation.8 In addition to the dynamic leakage, any realistic model of a qubit must also address coupling of the qubit to environ-ment, which leads to further destruction of qubit states. Sev-eral efforts have already been made toward the study of ac-curate control of a qubit system. 6,9,10However, the effect of an environment on optimally controlled qubit has only beenstudied in a phenomenological model, 7which leads to the evolution of density matrix of the qubit in Lindblad form.11 In this paper, we resort to a microscopic approach to the modeling of the environment. We employ the Caldeira-Leggett model of the system-environment coupling 12,13to describe time evolution of a ﬂux-biased phase qubit, drivenby the DRAG pulses. 7Numerically solving equation of mo- tion for the qubit density matrix, we study the dependence ofthe gate error on temperature, gate time, and environmentalcoupling strength. Although numerous potential sources of decoherence in phase qubits have been studied experimentally, 14–18in this paper, we focus on decoherence due to the Gaussian noise,which is introduced to the qubit system within the Caldeira-Leggett model. The study of the effect of ubiquitous low-frequency 1 /fnoise on the gate error is out of the scope of this paper. We speciﬁcally study the role of dissipation in the gate error of NOT gate operation. We ﬁnd that for phase qubits with relaxation time T1/H11015700 ns,19two-quadrature DRAG pulses proposed in Ref. 7result in the gate error exceeding 7/H1100310−3, which is too high for fault tolerant quantum com-putation, for a gate time of 6 ns. We then address the limita- tion posed by the environmental coupling on two-quadraturepulses. Here we ﬁnd that for optimal DRAG pulses, 7the coupling to environment must be reduced nearly by a factorof 6 to suppress the gate error below the required threshold.We also investigate the gate error for simple pulse shapingwhere the pulse amplitude of the ﬁrst quadrature variessmoothly according to a Gaussian-shaped function while theamplitude of the second quadrature is proportional to thederivative of the ﬁrst. In this case, however, the microwavedrive frequency is ﬁxed. For this pulse shape, we concludethat the gate error reduces to 10 −3for a gate time /H110157n s when the coupling to environment is reduced by an approxi-mate factor of 10 from the coupling in currently used phasequbits. II. MODEL A ﬂux-biased phase qubit consists of a Josephson junction /H20849JJ/H20850embedded in a superconducting loop.2Finite resistance of the JJ results in dissipation processes in the qubit and canbe accounted for by the Caldeira-Leggett model. 12,13The full Hamiltonian of the qubit and the environment is Hˆ=Hˆq+Pˆ/H20849t/H20850+HˆR+Vˆ. /H208491/H20850 The Hamiltonian of the qubit Hˆqis written in terms of op- erators Qˆand/H9254ˆ, the charge and phase difference of the JJ, respectively, Hˆq=Qˆ2 2C+/H92780 2/H9266/H20875/H92780 4/H9266L/H20873/H9254ˆ−2/H9266/H9278ext /H92780/H208742 −I0cos/H9254ˆ/H20876, /H208492/H20850 where L/H20849C/H20850is the loop inductance /H20849junction capacitance /H20850, /H9278extis the external magnetic ﬂux applied to the phase qubit, I0is the critical current of the JJ, and /H92780=h/2eis the ﬂux quantum. The qubit is capacitively coupled to microwavecurrent source, used to induce coherent transitions betweenthe qubit states. 2This coupling introduces time-dependent part in the HamiltonianPHYSICAL REVIEW B 82, 144528 /H208492010 /H20850 1098-0121/2010/82 /H2084914/H20850/144528 /H208496/H20850 ©2010 The American Physical Society 144528-1Pˆ/H20849t/H20850=/H92780I/H20849t/H20850 2/H9266/H9254ˆ. /H208493/H20850 Here I/H20849t/H20850=Ix/H20849t/H20850cos/H9275dt+Iy/H20849t/H20850sin/H9275dtis microwave current with frequency /H9275d. The environment is introduced as a set of harmonic oscil- lators /H20849reservoir /H20850with the Hamiltonian HˆR=/H20858/H9251=1N /H20849m/H9251/2/H20850/H20849pˆ/H92512/m/H92512+/H9275/H92512xˆ/H92512/H20850. The coupling between the qubit sys- tem and the reservoir is bilinear in the JJ phase /H9254ˆand oscil- lator displacements xˆ/H9251 Vˆ=/H20858 /H9251=1N /H9253/H9251xˆ/H9251qˆ,qˆ/H11013/H9254ˆ−2/H9266/H9278ext /H92780, /H208494/H20850 where parameters /H9253/H9251determine the coupling strength be- tween the qubit and reservoir mode /H9251. Our goal is to describe the time evolution of the qubit density matrix /H9267ˆ/H20849t/H20850. The qubit is initially prepared in a pure state, corresponding to the density matrix /H9267ˆ/H208490/H20850. Assuming that the environment is in a thermal equilibrium at tempera- ture T, the master equation for /H9267ˆ/H20849t/H20850takes the following form:20 d/H9267ˆ/H20849t/H20850 dt=1 i/H6036/H20851Hˆq/H20849t/H20850,/H9267ˆ/H20849t/H20850/H20852−Lˆ/H20853/H9267ˆ/H20849t/H20850/H20854 /H20849 5/H20850 and the dissipative term is Lˆ/H20851/H9267ˆ/H20849t/H20850/H20852 /H110131 /H60362/H20885 0t dt/H11032/H92571/H20849t/H11032/H20850/H20851qˆ,/H20851q˜ˆ/H20849−t/H11032/H20850,/H9267ˆ/H20849t/H20850/H20852/H20852 −1 /H60362/H20885 0t dt/H11032/H92572/H20849t/H11032/H20850/H20851qˆ,/H20853q˜ˆ/H20849−t/H11032/H20850,/H9267ˆ/H20849t/H20850/H20854/H20852, /H208496/H20850 where q˜ˆ/H20849t/H20850is a Heisenberg operator. In Eq. /H208495/H20850,/H92572/H20849t/H20850is the damping part and /H92571/H20849t/H20850represents the quantum noise of the environment21 /H92571/H20849t/H20850=/H6036/H20885 0/H11009 J/H20849/H9275/H20850/H208511+2 N/H20849/H9275/H20850/H20852cos/H9275td/H9275, /H208497/H20850 /H92572/H20849t/H20850=i/H6036/H20885 0/H11009 J/H20849/H9275/H20850sin/H9275td/H9275. /H208498/H20850 The master Eq. /H208495/H20850is time local, however, it contains time- dependent coefﬁcients, which capture memory effects of thenoise due to the heat bath. The spectral density J/H20849 /H9275/H20850 =/H20858/H9251=1N/H9253/H92512//H208492m/H9251/H9275/H9251/H20850/H9254/H20849/H9275−/H9275/H9251/H20850for an Ohmic environment is J/H20849/H9275/H20850=/H9264C 4e2/H60362/H92750/H9275e−/H9275//H9275s, /H208499/H20850 where /H9264is a dimensionless coupling parameter, /H6036/H92750is the energy difference between the qubit states, and /H9275sis a cut-off frequency that exceeds all other frequency scales of the qubitsystem. The Planck’s function N/H20849 /H9275/H20850=1 //H20851exp/H20849/H6036/H9275/T/H20850−1/H20852de- ﬁnes an average excitation number of environment modeswith frequency /H9275.In typical experiments with phase qubits,17,19,22,23the “po- tential” part of Hˆqin Eq. /H208492/H20850has one deep minimum and another very shallow minimum that disappears at the criticalﬂux /H9278c. External ﬂux /H9278extis chosen in such a way that only a few levels are localized in the shallow well but these levelsare still separated from levels localized in the deep well byimpenetrable barrier. 24As a result, we truncate the qubit Hamiltonian, Eqs. /H208492/H20850and /H208493/H20850, to three localized levels and obtain the following Hamiltonian: Hq/H20849t/H20850=/H6036/H20858 j=12 /H20851/H9275j−1/H9016ˆj+a/H9261j/H9268ˆj++a/H11569/H9261j/H9268ˆj−/H20852+Hˆnr, /H2084910/H20850 where /H9016ˆj=/H20841j/H20856/H20855j/H20841is the projector for the jth level, /H9268ˆj+=/H20841j/H20856/H20855j −1/H20841is the raising operator, a=/H20849Ix−iIy/H20850ei/H9275dt/2 is the amplitude of microwave drive, /H9261j=/H92780/H20855j/H20841/H9254ˆ/H20841j−1/H20856/2/H9266/H6036is the matrix ele- ment of the phase operator, /H9275j=/H20849/H9255j+1−/H92550/H20850//H6036,/H9255jis an energy eigenvalue of time-independent Hamiltonian Hqand Hˆnr contains nonresonant terms. In this three-level model, the lower two energy levels comprise qubit states while the thirdlevel accounts for a leakage level. III. GATE ERROR AND DRAG METHOD In order to quantify the error during gate operation we use gate ﬁdelity averaged over two initial input states in a two-dimensional Hilbert space, similar to one deﬁned in Ref. 25 F g=1 2/H20858 j=12 Tr/H20851Uˆideal/H9267ˆj/H208490/H20850Uˆ ideal†/H9267ˆj/H20849tg/H20850/H20852. /H2084911/H20850 Here Uˆidealrepresents an ideal evolution, /H9267ˆj/H20849t/H20850is an actual density matrix of the qubit system with /H9267ˆj/H208490/H20850=/H9267ˆj/H208490/H20850, and /H9267ˆj/H208490/H20850 represents two initial axial states in a Bloch sphere. The gate error Eis deﬁned as E=1− Fg. A simple approach to minimize leakage of quantum infor- mation from qubit subspace is to use a single-quadratureGaussian envelope pulse given by I x/H20849t/H20850=I/H9266/H20849t/H20850=Ae−/H20849t−tg/2/H208502/2/H92682−B,Iy/H20849t/H20850=0 , /H2084912a /H20850 where tgis a gate time and /H9268=tg/2. For a NOTgate operation, which we choose to focus on without any loss of generality,constant Bis chosen so that the Gaussian pulse starts and ﬁnishes off at zero, and Ais deﬁned by /H20885 0tg I/H9266/H20849t/H20850dt=/H9266. /H2084912b /H20850 This pulse shape results in a large gate error for reasonably short pulses. The DRAG method reduces the gate error to order of 10−5 for a gate time of 6 ns /H20849Ref. 7/H20850by using two quadratures and time-dependent detuning d1/H20849t/H20850=/H92750−/H9275d=/H20849/H92612−4/H20850I/H92662/H20849t/H20850/4/H9004, where the anharmonicity parameter /H9004/H11013/H92751−2/H92750, and/H9261mea- sures relative strength of 0 →1 and 1 →2 transitions, that is, /H9261/H11013/H92612//H92611. We note that the laboratory frame is more suitable for the solution of the reduced density matrix of the qubitcoupled to the environment. Therefore, we preserve the formof the quadrature amplitudes as in Ref. 7AMRIT POUDEL AND MAXIM G. VAVILOV PHYSICAL REVIEW B 82, 144528 /H208492010 /H20850 144528-2Ix=I/H9266+/H20849/H92612−4/H20850I/H92663 8/H90042,Iy=−I˙/H9266 /H9004/H2084913a /H20850 and obtain the following equation for the microwave driving frequency for the Hamiltonian Eq. /H2084910/H20850in the laboratory frame: t/H9275˙d/H20849t/H20850+/H9275d/H20849t/H20850=/H92750−d1/H20849t/H20850,/H9275d/H208490/H20850=/H92750. /H2084913b /H20850 Although the DRAG correction is successful in reducing the gate error below the required threshold, a practical imple-mentation may not be feasible due to stringent requirementto vary microwave frequency. For this reason, we also con-sider two-quadrature pulses with ﬁxed driving frequency /H9275d=/H92750.26We transform Hamiltonian /H2084910/H20850to a frame rotating with frequency /H9275dwith respect to the laboratory frame and obtain HˆR=/H6036/H20858 j=12/H20875dj/H9016ˆj+Ix/H20849t/H20850 2/H9261j/H9268ˆj−1,jx+Iy/H20849t/H20850 2/H9261j/H9268ˆj−1,jy/H20876,/H2084914/H20850 where d2=/H9004+2d1, and for /H9275d=/H92750, the detuning d1=0. W e introduce operators /H9268ˆj,kx=/H20841k/H20856/H20855j/H20841+/H20841j/H20856/H20855k/H20841and /H9268ˆj,ky=i/H20841k/H20856/H20855j/H20841 −i/H20841j/H20856/H20855k/H20841. To analyze the dynamics of rotating frame Hamiltonian HR, it is convenient to perform an adiabatic transformation7 Dˆ/H20849t/H20850=exp /H20851−iIx/H20849t/H20850/H20849/H9251/H9268ˆ0,1y+/H9261/H9268ˆ1,2y/H20850/2/H9004/H20852, which preserves the form of the gate, if Ix/H20849t/H20850starts and ﬁnishes off at zero. This condition is satisﬁed by our choice of Ix/H20849t/H20850/H20851see Eq. /H2084912a/H20850/H20852. The parameter /H9251appearing in Dˆis a dimensionless scaling parameter. After performing the transformation, the Hamil-tonian, to ﬁrst order in I x//H9004, takes the following form: HD /H6036/H11015Ix 2/H9268ˆ0,1x+/H208494/H9251−/H92612/H20850Ix2 4/H9004/H9016ˆ1+/H20875/H9251I˙x 2/H9004+Iy 2/H20876/H9268ˆ0,1y +/H20877/H9004+/H20849/H92612+2/H9251/H20850Ix2 4/H9004/H20878/H9016ˆ2+/H20875I˙x 2/H9004+Iy 2/H20876/H9261/H9268ˆ1,2y +/H208492−/H9251/H20850/H9261Ix2 8/H9004/H9268ˆ0,2x+/H9261/H20849/H9251−1/H20850IxIy 4/H9004/H9268ˆ0,2y. /H2084915/H20850 We then require resonant condition for the microwave /H9266 pulse in the qubit subspace and also eliminate the imaginary inertial term from the subspace, that is, require /H9016ˆ1and/H9268ˆ0,1y terms in Eq. /H2084915/H20850to vanish, to obtain /H9251=/H92612 4,Ix/H20849t/H20850=I/H9266/H20849t/H20850,Iy/H20849t/H20850=−/H9251I˙/H9266/H20849t/H20850 /H9004, /H2084916/H20850 where I/H9266/H20849t/H20850is deﬁned by Eq. /H2084912a/H20850. The contributions to the gate error due to transitions to the third level come from the second and third lines of Eq. /H2084915/H20850except for /H9016ˆ2term, which is not directly responsible for the gate error. Using the aboveexpression for I y/H20849t/H20850and Eq. /H2084912b /H20850, we estimate the magnitude of these terms as/H9268ˆ1,2y:/H20875Ix 2/H9004+Iy 2/H20876/H110111 /H9004tg2, /H9268ˆ0,2x:/H208492−/H9251/H20850/H9261Ix2 8/H9004/H110111 /H9004tg2, /H9268ˆ0,2y:/H9261/H20849/H9251−1/H20850IxIy 4/H9004/H110111 /H90042tg3. These estimates show that the error due to /H9268ˆ1,2yand/H9268ˆ0,2x terms are comparable and results in the leading contribution to the gate error. In the case of time-dependent detuning, thechoice of pulses is such that it eliminates the error associated with /H9268ˆ1,2yterm and rescaling of the pulse intensity /H20851I/H92663term in Eq. /H2084913a/H20850/H20852removes the contribution to the gate error due to /H9268ˆ0,2xterm. This elimination of /H110111//H20849/H9004tg2/H20850explains high effec- tiveness of variable driving frequency DRAG pulses. Forﬁxed frequency pulses, the pulse rescaling only marginallyreduces the gate error. IV . RESULTS Qubit parameters used below in our simulation are typical of phase qubits:17,19,22,23C=1 pF, I0=1.5 /H9262A,/H9252L =2/H9266I0L//H92780=3.2, and /H9278ext=0.955 /H9278c, where /H9278cis a critical ﬂux. Numerical simulation indicates that small variations ofqubit parameters do not incur any noticeable change in thegate error as long as there are at least three energy levels inthe shallow well of the potential. For these experimental pa-rameters, we numerically solve the time-independentSchrödinger’s equation with the Hamiltonian given by Eq./H208492/H20850. From this simulation, we obtain the following numerical values /H20849rounded up to two decimal places /H20850: /H92750=39.43 GHz, /H9261=1.42, and /H9004=−2.43 GHz. In Fig. 1, we plot the gate error for the DRAG pulses with and without time-dependent detuning for an ideal phase qu-150 200 250 300 35010−610−510−410−310−210−1 Gate Time (tgω0)Gate Error (E) FIG. 1. /H20849Color online /H20850Gate error vs gate time in log-normal scale with /H20849thick lines /H20850and without /H20849thin lines /H20850dissipation for a single quadrature Gaussian /H20849/H9268=0.5 tg/H20850pulse /H20849solid black /H20850, the Gaussian /H20849/H9268=0.5 tg/H20850pulse with ﬁrst-order DRAG correction and dynamical detuning /H20849dashed-dotted blue /H20850, and the Gaussian /H20849/H9268 =0.5 tg/H20850pulse with ﬁxed driving frequency /H9275d=/H92750and/H9251=0.5 /H20849dashed red /H20850, all in the laboratory frame. For the dissipative case, temperature T=0.1/H6036/H92750, the cut-off frequency /H9275s=10/H92750, and the coupling parameter /H9264=2.EFFECT OF AN OHMIC ENVIRONMENT ON AN … PHYSICAL REVIEW B 82, 144528 /H208492010 /H20850 144528-3bit without environment. We ﬁnd that pulses with two quadratures and ﬁxed driving frequency /H20849thin dashed red /H20850 perform much better than single quadrature Gaussian pulses/H20849thin solid black /H20850but are not as effective as pulses with double quadratures and time-dependent driving frequency/H20849thin dashed-dotted blue /H20850. We verify numerically that the ﬁxed frequency DRAG pulses give the minimal gate error for the choice of param-eter /H9251according to Eq. /H2084916/H20850. As shown in Fig. 2/H20849a/H20850, mini- mum value of the error occurs at around /H9251=0.5 for different gate times, cf. dashed blue curve for tg/H92750=250 and solid black curve for tg/H92750=350. This result is consistent with Eq. /H2084916/H20850, since for the phase qubit /H9261=1.42, which implies /H9251 =0.5. For transmon qubits, discussed in Ref. 27,/H9251=0.4 ow- ing to different value of /H9261. In order to study the effect of dissipation on the DRAG pulses, we integrate the master Eq. /H208495/H20850numerically using the fourth- and ﬁfth-order Runge-Kutta method. First, we con-sider relaxation of the qubit from the ﬁrst excited state to theground state in the absence of microwave drive, shown inFig. 2/H20849b/H20850. For this simulation, we choose the cut-off fre- quency /H9275s=10/H92750/H20849throughout this paper /H20850, temperature T =0.1/H6036/H92750, and the coupling parameter /H9264=2 so that the relax- ation time corresponds to experimentally observed decaytime T 1/H11015700 ns for phase qubits.19We note that spontane- ous relaxation rate of the ﬁrst excited state can also be evalu-ated from the master Eq. /H208495/H20850as /H9003=1 T1=2/H9266/H6036/H927502/H9264C 4e2/H20841q01/H208412,q01=/H208550/H20841qˆ/H208411/H20856. /H2084917/H20850 For the above choice of the dimensionless coupling param- eter/H9264, temperature T, and the cut-off frequency /H9275s, we study the effect of dissipation on two-quadrature pulses. In Fig. 1, we observe a nonmonotonic behavior of the gate error with gate time for pulses with the DRAG corrections.We ﬁnd that for shorter gate times, two-quadrature pulseswith time-dependent driving frequency are less affected bydissipation /H20849thick dashed-dotted blue /H20850. However, for longer gate times, dissipation has a substantial effect on two-quadrature pulses. For instance, for a gate time of /H92750tg =250 /H20849tg/H110156n s /H20850, the gate error increases from 10−5to higherorder of 10−3for two-quadrature pulses with dynamical de- tuning, when dissipation is taken into account. This increasein the gate error is due to relaxation of the qubit from excitedstate to the ground state, which becomes prominent forlonger gate times. For comparison, we plot the gate error forthree different types of pulses: single-quadrature Gaussianpulse /H20849thick solid black /H20850, the Gaussian pulse with ﬁrst-order DRAG correction and time-dependent driving frequency/H20849thick dashed-dotted blue /H20850and the Gaussian pulse with two quadratures and ﬁxed driving frequency /H20849thick dashed red /H20850. One can conclude from these plots that the performance oftwo-quadrature pulses without detuning is comparable to theDRAG pulses with dynamical detuning when dissipation isincluded. Next, we study the effect of the environmental coupling strength on the gate error. In Fig. 3, we plot the gate error for different coupling parameters /H9264for the phase qubit driven by two-quadrature pulses with dynamical detuning. In thissimulation, we consider temperature T=0.1/H6036 /H92750, and cou- pling parameters: /H9264=0 /H20849thin dashed-dotted blue /H20850,/H9264=0.1 /H20849thick solid green with circles /H20850,/H9264=0.3 /H20849thick solid red /H20850, and /H9264=2 /H20849thick dashed-dotted blue /H20850.A t/H9264=0 the gate error origi- nates entirely due to microwave-induced leakage of the qubitstate from the lowest two level subspace. The environment-induced transition rates increase with increase in the environ-mental coupling strength /H9264/H20851see Eq. /H2084917/H20850/H20852. As a result, the gate error also increases, which is corroborated by Fig. 3. One can infer from the plot that two-quadrature pulses withtime-dependent driving frequency suppress the gate error to10 −3for/H9264=0.3 and a gate time /H92750tg/H11015200 /H20849tg/H110155.5 ns /H20850. This indicates that an increase in the relaxation time nearly by afactor of 6 from the currently observed value is necessary tosuppress the error below the threshold. We further analyze the effect of the environmental cou- pling on ﬁxed frequency two-quadrature pulses for a range ofgate times. For this case, gate errors for different values of /H9264 are plotted in Fig. 4, where temperature is the same as above and coupling parameters are: /H9264=0 /H20849thin dashed red /H20850,/H9264=0.1 /H20849thick solid green with circles /H20850,/H9264=0.2 /H20849thick solid blue /H20850,/H92640 0.2 0.4 0.6 0.8 100.0050.010.0150.02(a) Alpha ( α)Gate Error (E)tgω0= 350 tgω0= 250 0 2 4 6 8 10 x1 0400.20.40.60.81 Time (t ω0)Probability(b) \|1〉 \|0〉 FIG. 2. /H20849Color online /H20850/H20849a/H20850Gate error vs alpha for gate times /H92750tg=250 /H20849dashed blue /H20850and/H92750tg=350 /H20849solid black /H20850. The phase qubit is driven by two-quadrature pulses with driving frequency /H9275d=/H92750in the absence of environment. /H20849b/H20850Probability vs time for temperature T=0.1/H6036/H92750, coupling parameter /H9264=2 and cut-off fre- quency /H9275s=10/H92750. The microwave pulse is turned off and the qubit is initially prepared in /H208411/H20856state /H20849solid black /H20850, which relaxes to /H208410/H20856 state /H20849dashed blue /H20850due to dissipation.150 200 250 300 35010−610−510−410−310−210−1 Gate Time ( ω0tg)Gate Error (E)ξ=0 ξ= 0.1 ξ= 0.3 ξ=2 FIG. 3. /H20849Color online /H20850Gate error vs gate time in log-normal scale for the Gaussian /H20849/H9268=0.5 tg/H20850pulse with variable frequency DRAG correction for temperature T=0.1/H6036/H92750and cut-off frequency /H9275s=10/H92750. The environmental coupling parameters: /H9264=0 /H20849thin dashed-dotted blue /H20850,/H9264=0.1 /H20849thick solid green with circles /H20850,/H9264=0.3 /H20849thick solid red /H20850, and /H9264=2 /H20849thick dashed-dotted blue /H20850.AMRIT POUDEL AND MAXIM G. VAVILOV PHYSICAL REVIEW B 82, 144528 /H208492010 /H20850 144528-4=0.5 /H20849thick solid black with triangles /H20850, and /H9264=2 /H20849thick dashed red /H20850. These plots indicate that the DRAG pulses with ﬁxed driving frequency can effectively suppress the gate er-ror if the environmental coupling strength is weakened andgate times are slightly longer than 6 ns. More speciﬁcally, for /H9264=0.2 and a gate time of /H92750tg/H11015300 /H20849tg/H110157n s /H20850, the gate error is close to 10−3. Therefore, we conclude that the relax- ation time must be nearly a factor of 10 longer than thecurrently observed value to attain the threshold of the gateerror for ﬁxed frequency DRAG pulses. This is a much betterimprovement compared to single-quadrature pulses forwhich the gate error never reduces to the threshold for areasonable choice of gate times even in an ideal case, that is, /H9264=0, as shown in Fig. 1. Finally, we investigate the effect of temperature on the gate error. In Fig. 5, we plot the gate error normalized to the error at zero temperature for two different gate times: /H92750tg =150 /H20849dashed-dotted blue /H20850and/H92750tg=350 /H20849dashed black /H20850. The plot shows a monotonic growth of the gate error astemperature increases due to enhancement in the relaxationrate. We compare results of numerical solution of the master Eq. /H208495/H20850and the simple picture of the error due to coupling to the environment in terms of the “Fermi-Golden rule” transi-tion rates. Considering the environment at zero temperatureand assuming that the contribution to the error Efrom the environment is small, E/H112701, we can evaluate the error as the probability of an excitation of a reservoir mode during thequbit operation, which happens with rate /H9003:E/H20849T=0/H20850 =/H9003t g/H926711/H20849t/H20850, where /H926711/H20849t/H20850=/H208480tg/H926711/H20849t/H20850dt/tgis the time average of probability of qubit being in the ﬁrst excited state. At ﬁnitetemperature, processes with excitation of environment hap-pen with rate /H9003/H20849T/H20850=/H9003/H208511+N/H20849 /H9275/H20850/H20852. In addition, the qubit can absorb an excitation from the environment with rate /H9003N/H20849/H9275/H20850. We combine the qubit excitations from the ground to ﬁrststate and the ﬁrst to second state with the relaxation from theﬁrst to ground state, and obtain the following estimate for thegate error due to coupling to the environment:E/H20849T/H20850 /H9003tg=/H20851/H208531+N/H20849/H92750/H20850/H20854+/H92612N/H20849/H92751−/H92750/H20850/H20852/H926711/H20849t/H20850+N/H20849/H92750/H20850/H926700/H20849t/H20850. /H2084918/H20850 For an average occupation of the ground and ﬁrst excited states being /H110151/2, and for a weak anharmonicity of the qu- bit system /H20841/H9004/H20841/H11270/H92750, the gate error reduces to E/H20849T/H20850 E/H208490/H20850/H110151+4 N/H20849/H92750/H20850. /H2084919/H20850 The estimated normalized gate error /H20849solid red /H20850is plotted in Fig. 5together with the gate error obtained from the nu- merical simulation. The rate equation estimation of the erroris fairly close to the error obtained from direct numericalsimulation for a longer gate time /H20849dashed black /H20850. However, for a shorter gate time /H20849dashed-dotted blue /H20850, the estimated error deviates from the exact numerical simulation consider-ably suggesting that the rate equation description may not bevalid for shorter gate times and higher temperatures. V . DISCUSSION AND CONCLUSIONS In this paper we compared possible choices of microwave pulses for NOT gate operation in ﬂuxed-biased qubits. Par- ticularly, we considered three options: single-quadraturepulses and two-quadrature microwave /H20849control /H20850pulses with both variable and ﬁxed frequency. Two-quadrature pulses ledto signiﬁcant suppression of the gate error compared tosingle-quadrature pulses. However, the presence of dissipa-tive environment increased the gate error even for two-quadrature pulses signiﬁcantly above the required thresholdfor fault tolerant quantum computation in currently availablephase qubits. We further investigated how the environmentalcoupling strength affects the gate error and found that animprovement of the qubit relaxation time is crucial for effec-tiveness of the DRAG pulses. We determined that two-quadrature pulses with ﬁxed driving frequency suppress thegate error below the required threshold for a reasonable gate0 0.2 0.4 0.6 0.8 10.511.522.533.5Gate Error [E (T)/E(0)] Temperature (T) [h ω0/2π]tgω0=350 Eq. (19) tgω0=150 FIG. 5. /H20849Color online /H20850Normalized gate error vs temperature for the Gaussian /H20849/H9268=0.5 tg/H20850pulse with varying frequency DRAG cor- rection. In numerical simulation, environmental coupling parameter /H9264=2, and cut-off frequency /H9275s=10/H92750are used for gate times /H92750tg =150 /H20849dashed-dotted blue /H20850and/H92750tg=350 /H20849dashed black /H20850. Analytical rate equation estimation of the normalized gate error /H20849solid red /H20850.150 200 250 300 35010−410−310−210−1Gate Error (E) Gate Time ( ω0tg)ξ=0 ξ=0.1 ξ= 0.2 ξ= 0.5 ξ=2 FIG. 4. /H20849Color online /H20850Gate error vs gate time in log-normal scale for the Gaussian /H20849/H9268=0.5 tg/H20850pulse with ﬁxed frequency DRAG correction for temperature T=0.1/H6036/H92750and cut-off frequency /H9275s =10/H92750. The environmental coupling parameters: /H9264=0 /H20849thin dashed red/H20850,/H9264=0.1 /H20849thick solid green with circles /H20850,/H9264=0.2 /H20849thick solid blue /H20850,/H9264=0.5 /H20849thick solid black with triangles /H20850, and /H9264=2 /H20849thick dashed red /H20850.EFFECT OF AN OHMIC ENVIRONMENT ON AN … PHYSICAL REVIEW B 82, 144528 /H208492010 /H20850 144528-5time of 7 ns but for qubits with the relaxation time ten times longer than the currently observed relaxation times. Simi-larly, our analysis indicated that two-quadrature pulses withdynamical detuning can also effectively reduce the gate errorbelow the required threshold if the relaxation time is longerby at least a factor of 6. In comparison to ﬁxed frequencyDRAG pulses, this is a moderate improvement over thelonger relaxation-time requirement, yet not impressiveenough to outshine the difﬁculty associated with implement-ing control pulses with variable driving frequency. We expectthat in a trade-off between complicated driving frequencyand longer relaxation times, the DRAG pulses with ﬁxed frequency are viable alternatives for reducing the gate error.We emphasize that for single-quadrature pulses, reduction inthe gate error below the error threshold of 10 −3is not pos- sible for reasonable gate times, even in an ideal case withoutany dissipation.In addition, we observed a monotonic increase in the gate error with temperature, which is due to increase in the relax-ation rate with temperature. We found that temperature de-pendence of the gate error for longer pulses can be capturedby a simple error estimation based on the rate equations.Nonetheless, the simple estimation of the error for shorterpulses differs from the gate error obtained from direct nu-merical solution of the reduced density matrix. Therefore,full density-matrix solution is necessary to calculate the errorfor shorter gate times. ACKNOWLEDGMENTS We are grateful to Robert Joynt and Robert McDermott for fruitful discussions. The work was supported by NSFGrant No. DMR-0955500. 1D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Science 296, 886 /H208492002 /H20850. 2M. H. Devoret and J. M. Martinis, Quantum Inf. Process. 3, 163 /H208492004 /H20850. 3J. Clarke and F. K. Wilhelm, Nature /H20849London /H20850453, 1031 /H208492008 /H20850. 4J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, Phys. Rev. Lett. 89, 117901 /H208492002 /H20850. 5I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Science 299, 1869 /H208492003 /H20850. 6M. Steffen, J. M. Martinis, and I. L. Chuang, Phys. Rev. B 68, 224518 /H208492003 /H20850. 7F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Phys. Rev. Lett. 103, 110501 /H208492009 /H20850. 8E. Knill, Nature /H20849London /H20850434,3 9 /H208492005 /H20850. 9P. Rebentrost, I. Serban, T. Schulte-Herbrüggen, and F. K. Wil- helm, Phys. Rev. Lett. 102, 090401 /H208492009 /H20850. 10P. Rebentrost and F. K. Wilhelm, Phys. Rev. B 79, 060507 /H208492009 /H20850. 11G. Lindblad, Commun. Math. Phys. 48,1 1 9 /H208491976 /H20850. 12A. O. Caldeira and A. J. Leggett, Ann. Phys. /H20849N.Y. /H20850149, 374 /H208491983 /H20850. 13A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59,1/H208491987 /H20850. 14D. J. Van Harlingen, T. L. Robertson, B. L. T. Plourde, P. A. Reichardt, T. A. Crane, and J. Clarke, Phys. Rev. B 70, 064517 /H208492004 /H20850.15R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Phys. Rev. Lett. 93, 077003 /H208492004 /H20850. 16J. Claudon, A. Fay, L. P. Lévy, and O. Buisson, Phys. Rev. B 73, 180502 /H208492006 /H20850. 17R. C. Bialczak et al. ,Phys. Rev. Lett. 99, 187006 /H208492007 /H20850. 18J. M. Martinis et al. ,Phys. Rev. Lett. 95, 210503 /H208492005 /H20850. 19J. M. Martinis, Quantum Inf. Process. 8,8 1 /H208492009 /H20850. 20J. P. Paz and W. H. Zurek, Lecture Notes in Physics 587,7 7 /H208492002 /H20850. 21U. Weiss, Quantum Dissipative Systems , 2nd ed. /H20849World Scien- tiﬁc, Singapore, 1999 /H20850. 22R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis,Science 307, 1299 /H208492005 /H20850. 23J. Lisenfeld, A. Lukashenko, M. Ansmann, J. M. Martinis, and A. V. Ustinov, Phys. Rev. Lett. 99, 170504 /H208492007 /H20850. 24The levels in the deep well can also be accounted in the present model, however, our numerical results indicate that the gate er-ror does not change signiﬁcantly if those levels are also includedin the calculation for chosen values of parameters. 25M. D. Bowdrey, D. K. L. Oi, A. J. Short, K. Banaszek, and J. A. Jones, Phys. Lett. A 294, 258 /H208492002 /H20850. 26We also made constant detuning of the driving frequency from /H92750but did not see any improvement compared to /H9275d=/H92750case. 27J. Chow, L. DiCarlo, J. Gambetta, F. Motzoi, L. Frunzio, S. Girvin, and R. Schoelkopf, arXiv:1005.1279 /H20849unpublished /H20850.AMRIT POUDEL AND MAXIM G. VAVILOV PHYSICAL REVIEW B 82, 144528 /H208492010 /H20850 144528-6
PhysRevB.75.224506.pdf	Competition between proximity-induced superconductivity and pair breaking: Ag sandwiched between Nb and Fe H. Stalzer,1A. Cosceev,1C. Sürgers,1and H. v. Löhneysen1,2 1Physikalisches Institut and DFG Center for Functional Nanostructures, Universität Karlsruhe, D-76128 Karlsruhe, Germany 2Forschungszentrum Karlsruhe, Institut für Festkörperphysik, D-76021 Karlsruhe, Germany /H20849Received 12 April 2007; published 13 June 2007 /H20850 The magnetization of superconductor and/or normal-metal /H20849Nb/Ag /H20850double layers is investigated in depen- dence on temperature Tand magnetic ﬁeld B. Screening currents in the normal-metal induced by the proximity effect give rise to a diamagnetic transition in a weak magnetic ﬁeld at a temperature Tb. The phase transition is suppressed when Fe is in direct contact with Ag. Surprisingly, the diamagnetic signal of Ag is recovered forsmall Ag ﬁlm thickness d Ag. These ﬁndings are qualitatively explained by the competition in the Ag layer between proximity-induced superconductivity by the Nb layer and pair breaking by the ferromagnetic Fe layer. DOI: 10.1103/PhysRevB.75.224506 PACS number /H20849s/H20850: 74.45. /H11001c, 74.25.Ha, 74.78. /H11002w Heterostructures with superconducting elements, exploit- ing the macroscopic quantum coherence of the superconduc-tive wave function, have become of increasing interest inrecent years. The integration of superconducting materials/H20849S/H20850in electronics leads to interface contacts with normal metals /H20849N/H20850and also with ferromagnets /H20849F/H20850playing an impor- tant role in spintronics. The “proximity effect,” i.e., the pen- etration of the superconductive pair amplitude of Sinto the adjacent NorFmetal, is mediated by the microscopic pro- cess of Andreev reﬂection. At the interface, incident elec-trons and retroreﬂected holes form correlated but mutuallyindependent Andreev pairs over a coherence length /H9264N/H20849T/H20850. The relevant length scales for a S/Nsystem, where Sis considered as inﬁnitely thick, are the electron mean free pathl N, the thickness dN, and/H9264N/H20849T/H20850. Early investigations of the proximity effect in thin S/N double layers have focused on measurements of their transi- tion temperatures. These experiments could be well de-scribed by the quasiclassical approach 1in the “dirty limit,” i.e., for lN/H11270/H20849dN,/H9264N/H20850. The magnetic properties, e.g., the dia- magnetic response, of S/Nsystems in an external magnetic ﬁeld Bhave been investigated theoretically and experimentally.2–7For commercial sheets and wires, the data clearly deviate from the dirty-limit behavior.4,7,8In particu- lar, data obtained on well-annealed coaxial Nb/Ag wires9are better described by taking into account the large lN, i.e., con- sidering the “ballistic regime.” Besides, a paramagnetic re-entrance behavior was observed at low temperatures. 10,11The dirty or clean limits are characterized by different tempera- ture dependences of /H9264N, i.e., /H9264Nd/H20849T/H20850=/H20849/H6036vFlN/6/H9266kBT/H208501/2and /H9264Nc/H20849T/H20850=/H6036vF/2/H9266kBT, respectively.3Measurements of the mag- netization M/H20849T,B/H20850allow the determination of the critical ﬁeld Bb/H20849T/H20850and the characteristic Andreev temperature TA=/H6036vF/2/H9266kBdNof the Nlayer, deﬁned by /H9264N/H20849TA/H20850=dN.Bb andTAcan be considered as parameters reﬂecting the stabil- ity of the proximity-induced superconductivity /H20849PIS /H20850inN. While previous experiments focused on commercial S/N sheets and wires, little is known about the magnetic responseof clean S/Ndouble layers with thickness d Nin the sub- micrometer range in contact with a ferromagnetic layer. InS/Fcontacts, the pair amplitude in Fdecays exponentially with distance from the interface, superposed with a periodicmodulation due to the ferromagnetic exchange interaction I ex.12–14In particular, for strong ferromagnets with Iex/H9270/H11022h /H20849/H9270, elastic-scattering time /H20850, the pair-condensate amplitude de- cays on a length scale of the order of the electron mean freepath l FinF.15Cladding of micrometer-thick coaxial Nb/Cu wires with Fe results in a strong depression of the proximityeffect. 5InS/Nsystems the diamagnetic transition of the N layer shifts to higher temperatures with decreasing dNindi- cating an enhanced stability of PIS, whereas the cladding ofthe outer Nsurface by a ferromagnetic metal gives rise to additional pair breaking. Hence, in S/N/Fsystems with ap- propriate Nlayer thickness, a concurrent inﬂuence of PIS and pair breaking by FonNshould be observed, as will be demonstrated in this paper. We report on magnetization mea-surement on high-quality Nb/Ag double layers with d Ag /H110211/H9262m and with a ﬁnite but long lAgof the order of dAg.W e observe a diamagnetic phase transition of the Ag layer at atemperature T bdepending on BanddAgfrom which we de- termine Bband TA. In particular, we focus on Nb/Ag/Fe triple layers and ﬁnd that for thick Ag layers, the diamagneticscreening in Ag is suppressed by the proximity of Fe. On theother hand, for small d Ag, the diamagnetic signal in Ag reap- pears due to a delicate balance between PIS in Ag and pairbreaking in contact with the ferromagnetic Fe. The samples were epitaxially grown by electron-beam evaporation in ultrahigh vacuum /H20849base pressure 1/H1100310 −10mbar /H20850. Nb was deposited with a ﬁxed thickness of 200 nm on /H20849112¯0/H20850-oriented sapphire substrates /H20849width w =0.7–5 mm, length L=4–8 mm /H20850at a substrate temperature TS=920 K and covered by 35–550 nm Ag at TS=470 K. The crystalline quality was checked by in situ reﬂection high- energy electron diffraction /H20849RHEED /H20850and ex situ by x-ray diffraction, indicating growth directions of Ag /H20851111 /H20852/H20648Nb/H20851110 /H20852 with mosaic spreads of 0.65° and 0.78° for Nb and Ag, re-spectively. 40 nm Fe were deposited onto the Ag layer atroom temperature without further annealing. For some ﬁlms,a SiO 2barrier was introduced between the Ag and Fe layers. The observed RHEED streaks indicate a smooth growth ofFe along /H20851110 /H20852on Ag /H20849111 /H20850. The in-plane magnetization of the Fe layer was checked by vibrating-sample magnetometryyielding a Fe moment of m Fe=/H208492.1±0.1 /H20850/H9262Bin agreement with that of bulk Fe, see Fig. 1/H20849inset /H20850. A coercivity ofPHYSICAL REVIEW B 75, 224506 /H208492007 /H20850 1098-0121/2007/75 /H2084922/H20850/224506 /H208494/H20850 ©2007 The American Physical Society 224506-1/H1101515 mT was obtained for all Nb/Ag/Fe triple layers. The samples were, ﬁnally, covered by 5 nm SiO 2or Si to protect them from oxidation in ambient air. Four-point measurements of the residual resistivity /H9267Nb and of the upper critical ﬁeld on a single 200 nm Nb ﬁlmyield an electron mean free path l Nb=27 nm using16/H9267NblNb =3.75/H1100310−16/H9024cm2and an upper critical ﬁeld Bc2/H208492K /H20850 =0.65 T. The following superconducting parameters were obtained from standard BCS relations and material param-eters of bulk Nb: 16,17coherence length /H9264Nb=19 nm; penetra- tion depth /H9261Nb/H1101542 nm /H11021dNb; lower critical ﬁeld Bc1/H208492K /H20850 =64 mT. Since dNb/H11271/H20849/H9261Nb,/H9264Nb/H20850, the Slayer can be regarded as inﬁnitely thick. From resistance measurements on several Nb/Ag double layers with the current in plane, a lower limit lAgmin/H11015dAgwas estimated in comparison with the single Nb ﬁlm. The magnetization Mat constant Bwas measured as a function of Tin a coaxial dBz/dzgradiometer coupled to a superconducting quantum interference device /H20849SQUID /H20850. Af- ter cooling down in zero magnetic ﬁeld to temperatures ofabout 60 mK, magnetization signals M/H20849T/H20850were recorded during warm-up. A homogeneous and stable magnetic ﬁeld was applied nearly parallel to the sample surface bymagnetic-ﬂux enclosure of an external magnetic ﬁeld, usinga superconducting Pb or NbTi/Nb/Cu cylinder 18surround- ing the sample holder. Although care was taken to measureall samples at similar positions in the gradiometer, an abso-lute measurement of the magnetization was not possible dueto the strong dependence of the Nb signal from a possible tiltangle between ﬁlm plane and ﬁeld, which could be con-trolled only to ±0.5°. 19Therefore, the M/Bdata are given inarbitrary units. After each M/H20849T/H20850cycle at constant ﬁeld, the superconducting cylinder was heated to well above its tran- sition temperature for complete expulsion of trapped mag-netic ﬂux. Figure 1/H20849a/H20850shows M/H20849T/H20850/Bof a Nb/Ag double layer with d Ag=550 nm. The sharp diamagnetic signal at Tc/H110159.1 K is due to the superconducting transition of the Nb layer. At alower temperature T b, a further diamagnetic transition of height /H9004MAg/Boccurs which is attributed to the proximity- induced diamagnetic screening currents in the Ag layer. /H20851The paramagnetic signal at T/H11021300 mK is due to the oxidized surface of the copper sample holder. This signal is absent insubsequent measurements made with a silver sample holder,see, e.g., Fig. 1/H20849a/H20850/H20849middle curve /H20850or Fig. 2/H20852. We mention that with increasing magnetic ﬁeld, the transition at T bbecomes sharper and shifts toward lower temperatures. This sharpen-ing is observed for all samples studied. For the presentsamples with small d Ag, the transition occurs at a few Kelvin and is, therefore, broadened by the temperature-dependentpenetration depth /H9261 Ag/H20849T/H20850.20The sharpening of the transition in increasing ﬁelds is also inferred from the calculated non- linear susceptibility of S/Ndouble layers.3 The discrete energy levels of Andreev bound states in N are given by2 En=/H6036vx 4dN/H20875/H208492n+1/H20850/H9266−2/H9266 /H92780/H20886A/H6023/H20849r/H6023/H20850dr/H6023/H20876 /H208491/H20850 /H20849vx, component of Fermi velocity perpendicular to the inter- face; A/H6023, vector potential; /H92780, superconducting ﬂux quantum /H20850. ForT/H11270TAandB/H11270Bb, each Andreev pair contributes coher- ently to the macroscopic screening current because only thelowest Andreev level is occupied. At temperatures above T A, the coherence is continuously destroyed by inelastic-scattering events and thermal excitations, changing the popu-lation of the Andreev states. In addition, in a magnetic ﬁeld,the Andreev pairs acquire an additional phase shift by thevector potential according to Eq. /H208491/H20850leading to a randomiza- tion of the Andreev currents and the destruction of the co-herence at the critical ﬁeld B b/H20849T/H20850. The Bb/H20849T/H20850phase diagram in Fig. 1/H20849b/H20850is in qualitative FIG. 1. /H20849a/H20850M/H20849T/H20850/Bfor samples with dAg=550 nm and dFe =40 nm in a ﬁeld of B=8 mT. TcandTbindicate the diamagnetic transition of the Nb and Ag layers, respectively. Inset shows themagnetization curve of the Nb/Ag/Fe sample taken at T=10 K. /H20849b/H20850 Semilogarithmic plot of B b/H20849T/H20850. Solid lines indicate a behavior lnBb/H11008−T/TA. Dashed lines serve as guides to the eye toward Bb/H208490/H20850. FIG. 2. M/H20849T/H20850/Bof Nb/Ag/Fe samples with dFe=40 nm for dif- ferent dAgand applied ﬁelds B. See text for details.STALZER et al. PHYSICAL REVIEW B 75, 224506 /H208492007 /H20850 224506-2agreement with thermodynamic calculations for the clean limit, as previously reported for coaxial Nb/Ag wires.20,21The Bb/H20849T/H20850dependence for temperatures T/H33356TA, i.e., when /H9264Ag/H11021dAg, nicely obeys a behavior ln Bb /H11008−T/TA, in contrast to the dirty limit where ln Bb/H11008−/H20881T. For dAg=550 nm, the Bb/H20849T/H20850behavior can be described only partly by the dirty limit by using a very long lAg=105 nm violating the condition lAg/H11270/H9264Ag. The characteristic tempera- ture TAobtained from the slope of Bb/H20849T/H20850/H20849Ref. 20/H20850isTA =3.02 K, in very good agreement with the theoretical value of 3.06 K estimated from TA=1680 K nm/ dAg. While our data show semiquantitative agreement with the thermody-namic phase diagram, we should point out that we actuallymeasure the superheated ﬁeld B sh/H20849T/H20850, which may be some- what larger than the thermodynamic critical ﬁeld for PIS in Ag. In summary, the Nb/Ag data are quite well described bytheory for the clean limit, although the smooth transitions arenot expected for ﬁrst-order transitions. Hence, the ﬁlmsshould be classiﬁed to fall in the ballistic regime. In what follows, we focus on the pair-breaking effect by a ferromagnetic Fe layer on the PIS. Deposition of a 40 nmthick Fe layer directly onto Ag with d Ag=550 nm suppresses the diamagnetic signal down to below the lowest temperatureof/H1101560 mK, see Fig. 1/H20849a/H20850, even in a weak external ﬁeld of 0.5 mT /H20849not shown /H20850. The small signal variations are caused by thermal instabilities of the SQUID system during thatmeasurement. As already mentioned, in S/Fcontacts, the pair amplitude in Fdecays exponentially with distance from the interface. This is also expected for the Andreev pairspenetrating into the ferromagnetic layer from the “normalconducting” Nlayer in a S/N/Fstructure. The Andreev pairs experience an additional phase shift in Fe, which destroysthe phase coherence in Ag. This has also been reported ear-lier for coaxial /H20849165 /H9262mN b / 2 7 /H9262m Cu/0.09 /H9262mF e /H20850wires.5 An alternative explanation might be the presence of a mag- netic stray ﬁeld from domain walls, which is minimized onlyfor ﬁelds above the coercive ﬁeld. For another sample, a 5 nm thick insulating SiO 2layer was ﬁrst deposited on top of the Ag layer before depositionof 40 nm Fe. As expected, the inﬂuence of the Fe layer isconsiderably weakened and a diamagnetic signal of Ag reap-pears. However, as Fig. 1/H20849a/H20850shows, the diamagnetic transi- tion of Ag appears at a much lower T bwhen compared to the Nb/Ag double layer. The destructive effect of Fe on theproximity effect in Ag also gives rise to a lowerT A=1.86 K and Bb/H208490/H20850=11 mT /H20851Fig. 1/H20849b/H20850/H20852compared to TA=3.02 K and Bb/H208490/H20850=18 mT obtained for the Nb/Ag double layer. The Andreev pairs have a ﬁnite probability to tunnel into Fe via SiO 2and back again, so that their phase coherence is compromised by Iexof Fe. In addition, pinholes in the oxide barrier may play a role. Both effects lead to areduced stability of PIS against magnetic and thermal pertur-bations. In other words, for a ﬁxed temperature, smaller ex-ternal ﬁelds are sufﬁcient for the destruction of coherence 22 in comparison with Nb/Ag double layers. Surprisingly, diamagnetic screening by Ag without a SiO 2 barrier reappears in Nb/Ag/Fe samples with much smaller dAg=35 and 43 nm in a certain range of magnetic ﬁeld. Fig- ure2clearly shows transitions around 3 K, which shift onlyslightly to lower temperatures with increasing ﬁeld Bto- gether with an increase of the jump /H9004MAg/B. Moreover, the ﬁelds where the transitions are observed are much larger thanthe upper limit for the Nb/Ag samples discussed above. Atthese large values, T cof the Nb layer is already reduced. Furthermore, a broadening of the transition is observed dueto the ﬂux penetration for ﬁelds exceeding the lower criticalﬁeld B c1. Figure 3displays the height of the diamagnetic jump/H9004MAg/Bvs magnetic ﬁeld Bfor the different samples in order to illustrate the reappearance of the diamagnetic sig-nal of Ag for certain magnetic ﬁelds. One could argue that the effect is due to the presence of a magnetic stray ﬁeld from the Fe layer. Although a thin mag-netic layer does not exhibit a stray ﬁeld close to the surfaceplane, a stray ﬁeld arising from surface roughness 23or gen- erated by the domain structure of the ferromagnetic layer24 can play a role. At this point, we cannot conclusively dismissthe possibility that the reappearance of the PIS in a higherapplied magnetic ﬁeld is due to the disappearance of thedomains leading to suppression of the stray ﬁeld. We alsocannot exclude effects due to the ﬂux trapped by the Nblayer cooled in the presence of such stray ﬁelds. However,the effect of a magnetic stray ﬁeld from the Fe layer on thediamagnetic screening in Ag is considered to be negligible,as conﬁrmed by zero-ﬁeld measurements with a magnetizedFe layer; i.e., after complete demagnetization of the ﬁeldcoil, no diamagnetic signal of Nb or Ag was observed. Wealso mention that the Ag layer shows a complete Meissnereffect /H20849not shown /H20850and, therefore, cannot contribute to any ﬂux pinning. The observed diamagnetic screening of Nb/Ag/Fe triple layers with thin d Agcannot be explained by present theories for semi-inﬁnite S/Nbilayers, since some initial assumptions are not valid. For instance, the variation of the superconduct-ing energy gap /H9004ofSacross the S/Ninterface cannot be approximated anymore by a step function becaused Ag/H11015/H9264Nb. Moreover, the effect of pair breaking due to the contact wit h a F layer is not considered. However, the result can be at least qualitatively explained in the following way.Ford Ag=35 and 43 nm, the Andreev energy corresponds to TA=48 and 39 K, respectively, so that the coherence in the Ag layer will not be destroyed in contact with the ferromag-netic Fe layer. Theoretically, the lowest-energy state /H20849n=0/H20850at perpendicular incidence /H20849maximum vx/H20850with respect to the interface would lie above the Fermi energies EFof 21 and FIG. 3. /H9004MAg/BvsBof Nb/Ag/Fe samples with dFe=40 nm and different dAg. Lines are guides to the eye.COMPETITION BETWEEN PROXIMITY-INDUCED … PHYSICAL REVIEW B 75, 224506 /H208492007 /H20850 224506-317 meV, respectively /H20851Eq. /H208491/H20850/H20852, and well above /H9004=1.2 meV of Nb. No bound Andreev levels exist for tra- jectories with small incident angles with respect to the inter-face normal. The ﬁnite spin polarization in Fe and the exter-nal ﬁeld Bcause a shift of the Andreev levels to lower energies. The occupation of these levels gives rise to diamag-netic screening currents by correlated phase-coherent An-dreev pairs. The magnitude of the diamagnetic signal is alsodetermined by the density of Andreev pairs. Higher densitiescan be reached by increasing the total number of states be-tween E Fand EF+/H9004with increasing B. The height of the jump/H9004MAg/Balso increases with increasing magnetic ﬁeld until it vanishes above ﬁelds, which completely destroy thephase coherence. Roughly speaking, the PIS in Ag is stabi-lized for thinner d Ag/H20849higher TA/H20850but at the same time, it is weakened by the pair breaking due to the contact with Fe.The balance between these effects can lead to the observa-tion of a diamagnetic transition in Ag for certain d AgandB. This behavior is visualized in Fig. 3. The external magnetic ﬁeld Bacts on both Nb and Ag layers, whereas the pair breaking of Fe only acts on the Ag layer. In this picture, it isclear that for a constant thickness d Ag, the transition tempera- ture Tbbecomes almost independent of Bdue to the strong internal magnetic ﬁeld in Fe which is some orders of mag-nitude larger than the external ﬁeld B. In strong ferromagnets with I ex/H9270/H11022h, such as Fe, the con- densation amplitude decays on length scales of the order ofl Fe.15This suggests that for lFe/H33356dFe, the coherence can bemaintained over the whole layer thickness. Indeed, compari- son of the resistivities of different Nb/Ag/Fe ﬁlms yields lower limit for the mean free path lFemin/H11015dFe. Therefore, the coherence of the Andreev pairs is not destroyed despite thelarge thickness d Fe. Pair breaking by spin-ﬂip scattering can be neglected, since at low temperatures single Fe spins canhardly be ﬂipped against the exchange ﬁeld of theirsurrounding. 25 In conclusion, we have observed a diamagnetic screening of the normal metal in high-quality Nb/Ag double layerswith large electron mean free paths l Ag. For thick Ag layers with dAg/H11271/H20849/H9264Nb,/H9261Nb/H20850, an additional Fe layer on top of Ag destroys the coherence of Andreev pairs. However, the dia- magnetic signal of Ag is recovered if the Ag layer thicknessis strongly reduced. This is due to the competition of inducedsuperconductivity by Nb and pair breaking by Fe. This quali-tative interpretation requires further theoretical investiga-tions. Finally, we wish to point out that the proximity effect inS/N/F/Scontacts may be used in order to realize a tun- able /H9266contact by application of an external magnetic ﬁeld. The authors gratefully acknowledge valuable discussions with W. Belzig, M. Eschrig, P. Wölﬂe, and A. C. Mota. Thiswork was partly supported by the Graduiertenkolleg“Angewandte Supraleitung.” They thank the Nippon SteelCorporation for providing the NbTi/Nb/Cu cylinder used inthe SQUID system. 1For a review of early work, see G. Deutscher and P. de Gennes, in Superconductivity , edited by R. D. Parks /H20849Dekker, New York, 1969 /H20850. 2A. D. Zaikin, Solid State Commun. 41, 533 /H208491982 /H20850. 3W. Belzig, C. Bruder, and G. Schön, Phys. Rev. B 53, 5727 /H208491996 /H20850. 4A. C. Mota, P. Visani, and A. Pollini, J. Low Temp. Phys. 76, 465 /H208491989 /H20850and references therein. 5Th. Bergmann, K. H. Kuhl, B. Schröder, M. Jutzler, and F. Pobell, J. Low Temp. Phys. 66, 209 /H208491987 /H20850. 6Yu. N. Ovchinnikov, B. I. Ivlev, R. J. Soulen, J. H. Claasen, W. E. Fogle, and J. H. Colwell, Phys. Rev. B 56, 9038 /H208491997 /H20850. 7Y . Oda and H. Nagano, Solid State Commun. 35, 631 /H208491980 /H20850. 8A. Sumiyama, T. Endo, Y . Nakagawa, and Y . Oda, J. Phys. Soc. Jpn. 70, 228 /H208492001 /H20850. 9F. B. Müller-Allinger, A. C. Mota, and W. Belzig, Phys. Rev. B 59, 8887 /H208491999 /H20850. 10P. Visani, A. C. Mota, and A. Pollini, Phys. Rev. Lett. 65, 1514 /H208491990 /H20850. 11F. B. Müller-Allinger and A. C. Mota, Phys. Rev. Lett. 84, 3161 /H208492000 /H20850. 12A. I. Buzdin and M. V . Kupriyanov, JETP Lett. 52, 487 /H208491990 /H20850. 13Z. Radovi ć, M. Ledvij, L. Dobrosavljevi ć-Gruji ć, A. I. Buzdin, and J. R. Clem, Phys. Rev. B 44, 759 /H208491991 /H20850. 14T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Phys. Rev. Lett.86, 304 /H208492001 /H20850. 15F. S. Bergeret, A. F. V olkov, and K. B. Efetov, Phys. Rev. B 65, 134505 /H208492002 /H20850. 16H. W. Weber, E. Seidl, C. Laa, E. Schachinger, M. Prohammer, A. Junod, and D. Eckert, Phys. Rev. B 44, 7585 /H208491991 /H20850. 17M. Tinkham, Introduction to Superconductivity /H20849McGraw-Hill, New York, 1996 /H20850. 18I. Itoh and T. Sasaki, IEEE Trans. Appl. Supercond. 3, 177 /H208491993 /H20850. 19H. Stalzer, A. Cosceev, C. Sürgers, H. v. Löhneysen, J.-M. Brosi, G.-A. Chakam, and W. Freude, Appl. Phys. Lett. 84, 1522 /H208492004 /H20850. 20A. L. Fauchère and G. Blatter, Phys. Rev. B 56, 14102 /H208491997 /H20850. 21Bb/H208490/H20850was determined by the lowest ﬁeld for which no diamag- netic screening signal was observed down to the lowest tempera-ture of 60 mK. 22M. Zareyan, W. Belzig, and Yu. V . Nazarov, Phys. Rev. Lett. 86, 308 /H208492001 /H20850. 23S. Demokritov, E. Tsymbal, P. Grünberg, W. Zinn, and I. K. Schuller, Phys. Rev. B 49, 720 /H208491994 /H20850. 24A. Yu. Aladyshkin, A. I. Buzdin, A. A. Fraerman, A. S. Mel’nikov, D. A. Ryzhov, and A. V . Sokolov, Phys. Rev. B 68, 184508 /H208492003 /H20850. 25C. Strunk, C. Sürgers, U. Paschen, and H. v. Löhneysen, Phys. Rev. B 49, 4053 /H208491994 /H20850.STALZER et al. PHYSICAL REVIEW B 75, 224506 /H208492007 /H20850 224506-4

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