{'fraction_non_alphanumeric': 0.039735099337748346, 'fraction_numerical': 0.01587979776401054, 'mean_word_length': 4.4391944229279625, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Enhancement of electron-hole superfluidity in double few-layer graphene

We propose two coupled electron-hole sheets of few-layer graphene as a new nanostructure to observe superfluidity at enhanced densities and enhanced transition temperatures. For ABC stacked few-layer graphene we show that the strongly correlated electron-hole pairing regime is readily accessible experimentally using current technologies. We find for double trilayer and quadlayer graphene sheets spatially separated by a nano-thick hexagonal boron-nitride insulating barrier, that the transition temperature for electron-hole superfluidity can approach temperatures of 40 K.

The prediction of electron-hole superfluidity in spatially separated electron and hole layers has captured the attention of the scientific community [1]. The recent intense interest results from suggestions that some double-layer electron-hole systems offer the possibility of observing a coherent superfluid up to temperatures approaching room temperature [2]. Despite long standing theoretical predictions [1][2][3] and considerable experimental efforts [4] such electron-hole superfluidity in double layered systems has not yet been observed in zero magnetic field.
Soon after the discovery of graphene [5], a two-dimensional lattice of carbon atoms [6,7], renewed efforts were made to look for superfluidity in graphene-based double monolayer devices. Although the early theoretical work on graphene double monolayers predicted room-temperature superfluidity [2], recent Coulomb drag experiments have found no evidence of superfluidity [8]. However, the linear energy dispersion of monolayer graphene makes it very difficult to access the most promising phase space region for superfluidity, the region where the average strength of the Coulomb interactions between carriers is much larger than their average kinetic energy. The reason is the following. The ratio r s = V /E F is a useful measure of the relative strength of the interactions, where E F is the Fermi energy and V = e 2 /(κ r 0 ) the average Coulomb energy for charge carrier density n. r 0 = 1/ √ nπ and κ is the substrate dielectric constant. For monolayer graphene, E F = v F k F , where the graphene Fermi velocity v F 10 6 ms −1 , the Fermi momentum k F = 4πn/g s g v , and the spin (valley) degeneracy for graphene is g s = 2 (g v = 2). This gives for monolayer graphene a value of r s = e 2 /[κ v F ] that is small and constant, independent of the density. The dielectric constant for a hexagonal boronnitride (h-BN) insulating substrate is κ ≈ 3 [9], giving r s the fixed value of only r s = 0.7. Such a small value for r s indicates that monolayer graphene is always a weakly interacting system [10].
Recently, it has been suggested that a pair of bilayer graphene (BLG) sheets is a promising system for observing high temperature superfluidity [11]. In contrast with monolayer graphene, BLG has a quadratic dispersion at low energies so the parameter r s has the inverse density dependence r s ∼ 1/ √ n familiar in metals and semiconductors, making it possible to experimentally access the strongly interacting region at large r s by simply reducing n.
In this paper we investigate the potential of using systems of few-layer graphene to access regions of phase space that are even more strongly interacting. Based on the two-band Dirac-Weyl equation describing the lowest energy band in ABC stacked N -layer graphene, the energy dispersion of the conduction band is given by [12,13] where t ≈ 400 meV is the interlayer hopping term in fewlayer graphene. Figure 1(a) shows E (N ) (k) for N = 1 to 4. We then obtain in N -layer graphene, Table I compares the values of r s for the typical electron densities found in graphene sheets for N -layer graphene, with N ranging from N = 1 (monolayer), to N = 4 (quadlayer). The table shows that few-layer graphene offers dramatic opportunities for producing extremely strongly interacting systems at experimentally accessible densities. The ability to access large r s values in few-layer graphene, and thus to reach the strong electron-hole pairing regime in an experimentally accessible range of densities, motivates us to propose few-layer graphene as a system to observe electronhole superfluidity at greatly enhanced minimum densities and transition temperatures. Experimental realization of few layer graphene is readily within the grasp of today's technology arXiv:1406.2471v2 [cond-mat.supr-con] 11 Jun 2014 since few-layer graphene sheets can be fabricated in large areas by both mechanical exfoliation [14,15] and by chemical techniques [16][17][18] from graphite with controlled stacking order. References [19][20][21] are examples of experimental studies on electronic and transport properties in trilayer graphene. A schematic setup of our proposed system is depicted in Fig. 2. There are two parallel few-layer graphene sheets. The upper sheet of electrons and lower sheet of holes are influenced by the top and back gates, respectively. The two sheets are separated by a thin h-BN insulating barrier to prevent tunneling between the sheets and electron-hole recombination. The separation of the graphene sheets can be as small as d = 1 nm (three h-BN layers) and still provide a potential barrier high enough to suppress tunneling [8,22]. As well as the top and bottom metal gates there are separate electrical contacts to the two sheets, allowing independent control over the carrier density in each sheet.
Our aim is to provide experimental indicators for system design for observing high-T c electron-hole superfluidity for the first time. We first evaluate the superfluid energy gap within an extended mean-field description. Inducing electrons or holes in a few layer graphene sheet using metallic gates imposes a perpendicular electric field. Experiments show that such a perpendicular electric field induces a band gap in the single-particle spectrum in bilayer [23] and trilayer [19,20,24,25] graphene band structures, and recent theoretical studies predict a similar effect in the energy spectrum of all few-layer graphene [26,27]. This induced band gap makes the contributions from the graphene valence band small, and we need consider only contributions from the conduction band.
We fix the electron and hole chemical potentials µ and densities n to be equal. The equations for ∆ k , the wave-vector dependent zero temperature gap generated from pairing of electrons and holes in the conduction band, and chemical potential µ are where The N -layer graphene geometrical form factor F kk = 1 + cos[N (φ k − φ k )] /2 is associated with the square of the overlap between the single-particle states |k and |k [28].
We evaluate V k−k , the static screened Coulomb interaction between electrons and holes in the two N -layer sheets, starting from the random phase approximation (RPA) in the normal state, where v(q) = −2πe 2 /[κq] is the attractive bare Coulomb interaction and Π 0 (q) the normal state particle-hole polarization bubble. The approximate expression in Eq. (5) uses the property that the most favourable conditions for pairing will occur when k F d 1. For small momentum exchange, we can extend Eq. (5) to the broken symmetry phase at T = 0 by writing the full polarization as Π 0 (q) = Π Π (n) 0 (q) and Π (a) 0 (q) are numerically calculated selfconsistently for the few-layer graphene in the superfluid state [11,28]. This procedure follows the approach of Ref. 28. For k F d 1, the full RPA-BCS screened interaction reduces to the present approximation [29]. Results from this approach have been tested successfully against recent Diffusion Quantum Monte Carlo results for electron-hole double layer systems [30,31].
In the superfluid state at low temperature, Π 0 (q) is suppressed at small momenta q because of the opening of the superfluid energy gap ∆ at the Fermi surface [32,33]. With the screening suppressed, the Cooper pairing can then become strong. At finite temperature, the underlying physics leading to the superfluid transition is that for strong electronhole pairing the fluctuations of the order parameter determine a (pseudo)gap at the critical temperature of the same order as the superfluid gap at zero temperature. This large pseudogap should lead to a suppression of the screening similar to the suppression at zero temperature caused by the superfluid gap. A large pseudogap of the same order as the zero-T superfluid gap has been experimentally observed and theoretically investigated in ultracold Fermion gases in both three-dimensional [34] and two-dimensional traps [35]. As a consequence of the different energy dispersions E (N ) (k), the dependence of the density of states (DOS) on energy changes dramatically with the number of layers N , where Ω (N ) (k) = πk 2 is the volume in k-space of the N -layer sheet. Figure 1 We self-consistently solve Eqs. (3) to (6) at zero temperature for the momentum-dependent gap ∆ k at equal electron and hole densities. Figure 3 shows ∆ max , the maximum ∆ k for coupled N-layer graphene with N = 2, 3 and 4 for barrier thickness d = 2 nm. For densities above an onset density, n > n c , if there is any superfluidity at all, the gap would be extremely small, 1 K. However in practice, for such small gaps, pairing is unlikely in the presence of the minimum levels of disorder typical in real graphene systems. At the onset density, n = n c , there is a sudden discontinuous jump in ∆ max to high energies of the order of the chemical potential µ. For N = 3 and 4, the pairing interactions are stronger as compared to N = 2, and the large-gap superfluidity is seen to persist up to significantly higher n c .
Also indicated in Fig. 3 are the BEC and BEC-BCS crossover regimes. We use the criterion that condensate fractions c > 0.8 correspond to the BEC regime of compact electron-hole pairs on the scale of the average carrier spacing, and 0.2 < c ≤ 0.8 to the BEC-BCS crossover regime [31]. c ≤ 0.2 would correspond to the weak-coupled BCS regime, but screening suppresses the superfluidity at all densities n > n c before the BCS regime is reached. In the BEC regime, however, the effect of screening is much less dramatic. In Fig. 3, the peak in ∆ max is located in the BEC regime, and around this peak, we find that screening reduces the calculated gap by a factor of only two, compared with the corresponding gap calculated without screening. The reason is the compact nature of the electron-hole pairs in the BEC regime.
We next determine the variation of the onset density on the number of layers N . Figure 4(a) shows the different n c as functions of d. At a fixed d, the onset density increases with the number of layers N , and the differences become very significant as d decreases. For the double quadlayer system, n c can become as large as 10 12 cm −2 for d = 1 nm.
A central concern for experiments and device applications is the predicted temperature T c for the superfluid phase transition. While in two dimensions T c is not linearly related to the value of the zero temperature gap ∆ [36], nevertheless a large value of ∆ through strong pairing is an essential prerequisite for a high T c . For superfluids in two-dimensions, an upper bound on the transition temperature is the Kosterlitz-Thouless temperature T KT [36]. This is determined from where J(T ) denotes the superfluid stiffness (the average kinetic energy of the Cooper pairs). At zero T , J(0) is proportional to the superfluid density J(0) = ρ s (0)/2, since ρ s (0) controls the phase stiffness of the complex superfluid order parameter. In the mean field approach, ρ s (T ) falls off only slowly with temperature when T is small compared with the zero-temperature gap, that is ρ s (T ) is well approximated by ρ s (0) when k B T ∆(0). In an isotropic system, ρ s (0) at the mean field level is de- termined from [37] where 1/m (k) = ∂ 2 k /∂k 2 is momentum dependent. For bilayer graphene, Eq. (9) gives ρ s (0) = [2( v F ) 2 /t]n. Note it is only for N = 2 that ρ s (0) is proportional to n. Figure 4(b) shows the maximum T KT , which occurs at the onset density n c . This maximum T KT is plotted as a function of d for N = 2, 3, and 4. Results are shown only for cases k B T KT < 0.5∆ max . We see that for the same d, the enhancement of the T KT with increasing N is significantly less than the corresponding enhancement of the onset density n c (Fig.  4(a)). The reason is that in the relevant density range, the effective mass m (k) for N = 3 and N = 4 is larger than the effective mass m for N = 2. This has the effect of reducing the T = 0 superfluid density for N > 2 as compared with the N = 2 case. The larger m (k) arise from the different band curvatures, and they partially compensate both amplification effects in the pairing gap and also the increased suppression of Coulomb screening caused by the Van Hove singularities present in the DOS for N = 3 and N = 4.
Nevertheless, we see in Fig. 4(b) that changing from an N = 2 to an N = 4 sample with the same d ≤ 2 nm, still has the effect of doubling T KT . Therefore in double quadlayer graphene heterostructures at currently experimentally attainable densities, ∼ 10 12 cm −2 , transition temperatures can approach liquid nitrogen temperature. This strongly suggests that electron-hole superfluidity and counterflow superconductivity in these samples should be readily detectable using current technologies.
In summary, close to liquid nitrogen temperature we predict electron-hole superfluidity in double few-layer sheets of graphene at carrier densities as high as 10 12 cm −2 . A key element of the physical mechanism is that increasing the number of graphene layers in each sheet has the effect of greatly enhancing the density of states ( Fig. 1(b)). This enhancement projects the sheets into the strongly interacting regime, leading to strong electron-hole pairing at accessible densities. The experimental parameters of our proposed device have all been attained in related graphene systems [24,25,38].