jintanakan/wittr
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Many-body electron correlations in graphene | 0 |
David Neilson\textsuperscript{1}, Andrea Perali\textsuperscript{1} and Mohammad Zarenia\textsuperscript{2} | 1 |
\textsuperscript{1} Dipartimenti di Fisica e di Farmacia, Università di Camerino, 62032 Camerino (MC), Italy | 1 |
\textsuperscript{2} Department of Physics, University of Antwerp, B-2020 Antwerpen, Belgium | 1 |
E-mail: david.neilson@unicam.it | 1 |
Abstract. The conduction electrons in graphene promise new opportunities to access the region of strong many-body electron-electron correlations. Extremely high quality, atomically flat two-dimensional electron sheets and quasi-one-dimensional electron nanoribbons with tuneable band gaps that can be switched on by gate... | 0 |
1. Introduction | 0 |
The Coulomb repulsion between conduction electrons in conventional metals and semiconductors affects their properties only through the relatively weak effects of linear screening and corrections to the values of the Landau Fermi liquid parameters. This is associated with the high densities of conduction electrons found... | 0 |
1.1. New phenomena for strongly correlated conduction electrons | 0 |
If the density of conduction electrons could be lowered sufficiently to make electron interactions dominate over Fermi energies, a wealth of interesting new quantum phenomena driven by strong many-body electron correlations are predicted to appear. These phenomena include: | 0 |
- Wigner crystal of electrons | 0 |
- Charge density waves and other striped ground states | 0 |
- Metal-insulator transition | 0 |
- Quantum glass in the presence of defects | 0 |
- Coherent superconductor and electron-hole superfluid quantum states | 0 |
- Crossover from BCS superconductivity to Bose-Einstein condensation | 0 |
1.2. When are many-body correlations expected to be important? | 0 |
A two-dimensional (2D) electron layer is expected to have correlations that are stronger than the correlations in a corresponding three-dimensional system at the same density, because of the smaller kinetic energy contributions. The dimensionless parameter $r_s$ provides a measure of when electron correlations will be ... | 0 |
2. Strongly correlated conduction electron systems | 0 |
2.1. 2D Electron Liquid in Si MOSFETs and GaAs heterostructures | 0 |
In 2D systems, the new phenomena driven by strong correlations are predicted to appear at very low densities, generally only for $r_s \gtrsim 10$ [1]. For many years quasi-2D electron layers at the interfaces of Si MOSFET devices and in narrow quantum wells in GaAs heterostructures (Fig. 1) have shown great promise as ... | 0 |
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2.2. Graphene | 0 |
In contrast to the quasi-2D electron layers in semiconductor systems, a graphene sheet is atomically flat and hence strictly 2D. There are no finite-width effects to weaken the Coulomb interactions. Levels of defects in graphene are extremely low so that electron freeze-out should be postponed to much lower electron de... | 0 |
2.2.1. Monolayer graphene | 0 |
However nature has not been kind. Because the dispersion of the energy bands in monolayer graphene... | 0 |
is linear at low energies, $E_\pm(k) = \pm \hbar v_F k$ (Fig. 2), the Fermi energy $E_F = \hbar v_F k_F = \hbar v_F \sqrt{\pi n}$ and $\langle KE \rangle$ depend only linearly on $r_s^{-1}$. A result of this is that the $r_s = e^2/(\kappa \hbar v_F)$ in monolayer graphene does not depend on the electron density. The Fe... | 0 |
 | 0 |
2.2.2. Bilayer graphene | 0 |
It should, nevertheless, be possible to access the strongly correlated region in graphene. One workaround is to substitute a graphene bilayer sheet in place of the graphene monolayer. A symmetrically biased graphene bilayer with AB stacking is a semiconductor with parabolic dispersion of the energy bands [5] (Fig. 3). ... | 0 |
 | 0 |
The low density regime in bilayer graphene is dominated by disorder in current samples, making a lower limit for the electron density of $n \sim 10^{10}$ cm$^{-2}$, corresponding to a large maximum value for | 0 |
\( r_s = 23 \). At very low densities a trigonal warping of the bands could transform the parabolic bands into sets of Dirac-like linear bands [7], but residual disorder will mask this effect, and it could be further reduced if necessary by applying an electric field to open up an energy band gap. | 0 |
In summary, extreme high quality, atomically flat bilayer graphene sheets have tuneable electron densities and band gaps that should permit the bilayers to readily access new quantum phenomena predicted for strong electron correlations. | 0 |
### 2.2.3. Few-layer graphene | 0 |
Increasing the number of graphene layers in the sheet beyond bilayers greatly enhances the density of states (DOS), and this projects the sheets even more dramatically into the region of strong correlations at accessible densities [8]. Electron graphene multilayers should be able to access regions of phase space with v... | 0 |
\[ | 0 |
E^{(N)}(k) = \left\{ (\hbar v_F)^N / t^{N-1} \right\} k^N , | 0 |
\] | 0 |
where \( t \approx 400 \text{ meV} \) is the interlayer hopping term in few-layer graphene. Figure 4(a) shows \( E^{(N)}(k) \) for \( N = 1 \) to 4. Using Eq. (1) to determine the Fermi energy \( E_F \) at density \( n \), we obtain for \( N \)-layer graphene, | 0 |
\[ | 0 |
r_s = \langle PE \rangle / \langle KE \rangle = \left\{ e^{2tN-1} \over \kappa (\hbar v_F)^N \sqrt{\pi^{N-1}} \right\} {1 \over n^{(N-1)/2}} . | 0 |
\] | 0 |
 | 0 |
**Figure 4.** (a) Lowest positive energy band in monolayer (\( N = 1 \)), bilayer (\( N = 2 \)), trilayer (\( N = 3 \)), and quadlayer (\( N = 4 \)) graphene. (b) Comparison for \( N = 1 \) to \( N = 4 \) of the density of states at the Fermi energy \( DOS^{(N)}(E_F) \) as function of electron density [8]. The van Hove... | 0 |
A consequence of the different energy dispersions \( E^{(N)}(k) \) is that the dependence of the density of states on energy changes dramatically with the number of layers \( N \), | 0 |
\[ | 0 |
DOS^{(N)}(E) = \frac{d\Omega^{(N)}}{dE} = \frac{2\pi}{N} \frac{t^{2(N-1)/N}}{(\hbar v_F)^2} E^{(2/N)-1} , | 0 |
\] | 0 |
where \( \Omega^{(N)}(k) \) is the volume in \( k \)-space of the \( N \)-layer sheet. Figure 4(b) shows the dependence of \( DOS^{(N)}(E_F) \) at the Fermi energy on electron density \( n \). For monolayer graphene \( DOS^{(1)}(E_F) \) depends linearly on \( n \), for bilayer graphene \( DOS^{(2)}(E_F) \) is a constan... | 0 |
and $DOS^{(2)}(E_F)$. Eventually, at very high densities lying well outside our range of interest, the $DOS^{(3)}(E_F)$ and $DOS^{(4)}(E_F)$ are smaller than $DOS^{(1)}(E_F)$ and $DOS^{(2)}(E_F)$. | 0 |
Table I compares the values of $r_s$ for the typical electron densities found in graphene sheets for $N$-layer graphene, for $N = 1$ (monolayer) to $N = 4$ (quadlayer). The table shows that few-layer graphene offers dramatic opportunities for producing extremely strongly interacting electron systems at experimentally a... | 0 |
Experimental realisation of few layer graphene is within the grasp of current technology since few-layer graphene sheets can be fabricated in large areas by both mechanical exfoliation [11,12] and by chemical techniques [13,14,15] from graphite with controlled stacking order. References [16,17,18] are examples of exper... | 0 |
### 2.2.4. Graphene nanoribbons | 0 |
Electrons can be confined in nanoribbons that are only a few nanometres in width, etched on monolayers of graphene. The nanoribbons can have multiple energy subbands that are occupied. Their electronic properties depend on the type of edge termination [19]. We discuss here only armchair-edge terminated graphene nanorib... | 0 |
Electron correlations in armchair-edge terminated nanoribbons will be further boosted by the quantum confinement of the electrons along the nanoribbons, and also by quantum size effects and van Hove singularities in the quasi-one-dimensional density of states that are accessed if the Fermi energy is increased so it ent... | 0 |
Figure 5 shows the single-particle energy subbands $j$ in the continuum model, | 0 |
$$\epsilon_j(k_y) = (\sqrt{3}ta_0/2)\sqrt{k_y^2 + k_j^2}, \quad j = 1, 2, \ldots$$ | 0 |
for an armchair graphene nanoribbon of width $W = 2$ nm, where $t = 2.7$ eV is the intralayer hopping energy [21] and $a_0 = 0.24$ nm is the graphene lattice constant. We take the $y$-direction parallel to the nanoribbon, with the electrons confined in the transverse $x$-direction. The quantised wave-number for the $j$... | 0 |
$$k_j = [j\pi/W] - [4\pi/(3\sqrt{3}a_0)].$$ | 0 |
The lower panels in Figure 5 show the corresponding density of states $DOS(E)$. The van Hove singularities are clearly visible at the bottom of each subband. | 0 |
### 2.3. Graphene in a periodic magnetic field | 0 |
A periodic magnetic field applied perpendicular to the graphene monolayers can preserve the isotropic Dirac cones of the monolayer energy bands while reducing the slope of the Dirac cones [23,24,25,26]. We represent the magnetic field perpendicular to the monolayer as a one-dimensional array with period $2d$ of success... | 0 |
Figure 5. Top: armchair-edge terminated graphene nanoribbon of width $W$. Lower panels: lowest single-particle energy subbands $\epsilon_j(k_y)$, $j = 1, 2, \ldots$ for an armchair nanoribbon of width $W = 2$ nm, and corresponding density of states DOS($E$) in the nanoribbon [22]. Van Hove singularities are visible at ... | 0 |
Electronic band structure of the graphene monolayers. In this field, the single-particle energy dispersion of the monolayer graphene remains linear, but the velocity $\alpha_d v_F$ is less than the original Fermi velocity $v_F$, so | 0 |
$$\epsilon(k) = \pm \hbar (\alpha_d v_F)|k| (1 + \delta(k)) .$$ | 0 |
(4) | 0 |
The non-linear correction term $\delta(k)$ is small, with $|\delta(k)| \lesssim d_B^2 k_x^2/6$. The constant $\alpha_d \leq 1$, representing the decrease in the Fermi velocity, depends on $d$ [23, 26], | 0 |
$$\alpha_d \simeq 1 - d^4/60 \quad d \ll 1$$ | 0 |
$$\alpha_d \simeq \frac{2d}{\sqrt{\pi}} e^{-d^2/4} \quad d \gg 1 .$$ | 0 |
(5) | 0 |
Figure 6. Variation of the interaction parameter $r_s$ for monolayer graphene in a perpendicular magnetic field of periodicity $2d = 2d_B/\ell_B$ (see text). Without the magnetic field, $r_s$ is a constant less than unity, and is independent of electron density. | 0 |
Recalling for graphene monolayers that the density-independent interaction strength parameter is \( r_s = \frac{e^2}{(\kappa h v_F)} \), we see that an effect of the reduction of the Fermi velocity in Eq. 4 is to increase the value of the \( r_s \) parameter by a factor \( \alpha_d^{-1} \). Figure 6 shows that we can s... | 0 |
3. Electron-hole superfluidity in graphene | 0 |
Two graphene monolayers of electrons and holes separated by a very thin insulating barrier has been proposed to observe an electron-hole superfluid [27, 28, 29]. A hexagonal boron nitride (hBN) separation barrier as thin as 1 nm can efficiently insulate the two monolayers from each other [30]. However, theory suggests ... | 0 |
This poses a challenge as to whether new structures can be designed and fabricated using atomically thin crystals, structures in which a superfluid transition can be observed. The graphene sheets discussed in Sections 2.2.2 to 2.2.4 can all access the region \( r_s \gg 2.3 \), and hence they constitute promising candid... | 0 |
In superfluid systems in graphene, it would be straightforward to access the BCS-BEC crossover and BEC regimes using the electric potential on metal gates and by tuning the sample parameters. This possibility opens up interesting new connections with the physics of ultracold fermions and high-\( T_c \) superconductors. | 0 |
We have predicted the existence of electron-hole superfluidity in the graphene structures described in Sections 2.2.2 to 2.2.4. We find mean field zero temperature superfluid gaps that are consistently large, of the order of several hundred Kelvin. Unlike in three-dimensions, however, the superfluid phase transition te... | 0 |
We find the resulting \( T_{KT} \) is typically of order 15 K for the electron-hole coupled graphene bilayers and the electron-hole coupled graphene monolayers in a periodic magnetic field, for sheet separations \( \sim 2 \) nm and typical experimental carrier densities, \( n \sim 10^{12} \text{ cm}^{-2} \). Switching ... | 0 |
4. Conclusions | 0 |
With graphene and related atomically thin crystals, we are at an exciting many-body threshold to realise and exploit novel quantum phases with tuneable properties. High quality, atomically flat two-dimensional electron graphene sheets and quasi-one-dimensional electron graphene nanoribbons with tuneable electron densit... | 0 |
Acknowledgments | 1 |
We thank Lucian Covaci and François Peeters (University of Antwerp), Alexander Hamilton (University of New South Wales), and Luca Dell’Anna (University of Padua) for useful discussions. We acknowledge support by the University of Camerino FAR project CESEMN (DN and AP), the Flemish Science Foundation (FWO-Vl) (MZ) and ... | 1 |
References | 1 |
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