Excitonic condensation of massless fermions in graphene bilayers
Graphene, a single sheet of graphite with honeycomb lattice structure, has massless carriers with tunable density and polarity. We investigate the ground state phase diagram of two graphene sheets (embedded in a dielectric) separated by distance where the top layer has electrons and the bottom layer has holes, using mean-field theory. We find that a uniform excitonic condensate occurs over a large range of carrier densities and is weakly dependent on the relative orientation of the two sheets. We obtain the excitonic gap, quasiparticle energy and the density of states. We show that both, the condensate phase stiffness and the mass of the excitons, with massless particles as constituents, vary as the square-root of the carrier density, and predict that the condensate will not undergo Wigner crystallization.
Introduction: Over the past three years, graphene has emerged as the unique candidate that provides a realization of two-dimensional massless fermions whose carrier density and polarity are tunable by an external gate voltage Nov (). Subsequent experimental and theoretical investigations have led to a thorough re-examination of some of the properties of linearly dispersing massless fermions ris (); kya (a). The truly two dimensional (2D) nature of graphene permits control and observation of local carrier density and properties mar (); lev (); yac (). In graphene bilayers, the ability to change the carrier polarity of an individual layer implies that the interlayer Coulomb interaction can be tuned from repulsive to attractive. This raises the possibility of formation of electron-hole bound states or indirect excitons, albeit with massless fermions as its constituents. Properties of such bound states of massless particles are an open question; the only other example, to our knowledge, is the proposed color superconductivity in dense quark matter alf (). Graphene bilayers provide an ideal and unique candidate for straightforward experimental investigations of such phenomena.
A uniform Bose-Einstein condensate of excitons in electron-hole bilayers occurs when the interlayer distance is comparable to the distance between the particles within each layer sno (); Lit (). These excitons have mass where () is the band mass of the electron (hole). At high densities, dipolar repulsion between the excitons can lead to a condensate ground state with broken translational symmetry: a supersolid Joglekar et al. (2006). Biased bilayer quantum Hall systems near total filling factor one have shown uniform excitonic condensation ja (). In this case, the exciton mass is determined solely by interlayer Coulomb interaction and is independent of the bias voltage kya (b); bia (a). These observations raise the questions: What is the mass of an exciton with massless constituents? Will such an excitonic condensate lead to a supersolid if the dipolar repulsion between such excitons (with a nonzero mass) is increased?
In this paper, we investigate the excitonic condensation in two graphene sheets embedded in a dielectric and separated by a distance (=1.4Å is the honeycomb lattice size) so that the tunneling between the layers is negligible, but interlayer Coulomb interaction is not. The layers have opposite polarity and equal density of carriers . We remind the Reader that in graphene, in the continuum limit, the length-scale and the energy-scale are both set by the density of carriers ( is the Fermi momentum, is the Fermi energy, and is the speed of massless carriers). Therefore, the ground state phase diagram depends only on one dimensionless parameter . This is markedly different from conventional bilayer systems parameterized by () where is the band Bohr radius and Lit (), as well as biased bilayer quantum Hall systems, parameterized by () where is the magnetic length and is the filling factor imbalance han (); bia (a); tut (); bia (b).
We use the mean-field theory to obtain the ground-state phase diagram as a function of . We find that a) excitonic condensation occurs at all densities as long as . b) the condensate properties are weakly sensitive to the relative orientation of the two sheets (stacking). c) the superfluid phase stiffness and the exciton mass have a dependence. d) the excitonic condensate does not undergo Wigner crystallization in spite of dipolar repulsion between excitons with a nonzero mass.
The plan of the paper is as follows. In the next section, we present the mean-field Hamiltonian neg () and briefly sketch the outline of our calculations. In the subsequent section, we show the results for the excitonic gap , the quasiparticle energy , and the quasiparticle density of states . We discuss the density dependence of the superfluid stiffness and the mass of the excitons. In the last section, we show that these results are equivalent to absence of Wigner crystallization, and mention the implications of our results to experiments.
Mean-field Model: We consider two graphene sheets embedded in a dielectric separated by distance with chemical potentials in the two layers adjusted so that the top layer (denoted by pseudospin ) has electrons and the bottom layer (denoted by pseudospin ) has holes with the same density. We consider two stackings: the Bernal stacking that occurs naturally in graphite, and the hexagonal stacking in which each sublattice ( and ) in one layer is on top of the corresponding sublattice in the other layer. Since the Hamiltonian in the continuum description is SU(4) symmetric in the spin and valley indices, we ignore those indices for simplicity. In the continuum limit, the single-particle Hamiltonian for carriers in layer is Semenoff (1984)
where is the momentum measured from the -point and denote the conduction and valance bands that result from diagonalizing the Hamiltonian in the sublattice-basis. () is creation (annihilation) operator for an electron in band in layer with momentum . We point out that for the hexagonal stacking, is independent of the layer index . For the Bernal stacking, the creation operators in the two layers are related by complex conjugation, where . The interaction Hamiltonian consists of intralayer Coulomb repulsion and interlayer Coulomb attraction . ( is the dielectric constant). Using standard mean-field techniques neg (), we obtain the following mean-field Hamiltonian
Here creates an electron in the conduction band () in the top layer () and creates a hole in the valance band () in the bottom layer (). The term contains single-particle energy, capacitive Hartree self-energy and the intralayer exchange self-energy. The off-diagonal term is proportional to the excitonic condensate order parameter . The eigenvalues of the mean-field Hamiltonian are given by . We consider mean-field states with a real , and spatially uniform density. It is straightforward to diagonalize the Hamiltonian and obtain the mean-field equations Lit ()
where , is the capacitance per unit area, and . The form factor for the two stackings are
We point out that the self-energy in Eq.(3) takes into account both intrinsic and extrinsic contributions that cancel the -dependent terms in the form factor and make the results independent of the ultra-violet cutoff int (); all (a). Therefore the intra-layer self-energy in Eq.(3) is the same as that for a conventional system int (); all (a). The chemical potential is determined by the carrier density that takes into account the four-fold spin and valley degeneracy
It is straightforward to derive similar equations for a conventional electron-hole system Lit (). They are obtained by changing the single-particle dispersion to a quadratic and replacing the form factor by a constant . We solve Eqs. (3), (4) and (6) iteratively to obtain self-consistent results.
Results: Figure 1 shows the excitonic gap and the quasiparticle energy for the Bernal (green dashed) and the hexagonal (red solid) stacking. The excitonic gap is maximum at the Fermi momentum where the quasiparticle energy is minimum. Since the electron-hole Coulomb interaction is always attractive, the excitonic condensate order parameter is nonzero down to the bottom of the Fermi sea, . Our results predict that the hexagonal-stacked system will have a larger excitonic gap than the Bernal-stacked system.The quasiparticle energy becomes linear at large , since the constituent particles of the exciton have a linear dispersion. The speed of these quasiparticles is increased due to intralayer exchange self-energy all (a); ds (); rol () although the increase is modest, . Corresponding results for a conventional electron-hole system (blue dotted) are also shown in Fig. 1.
Figure 2 shows the dependence of the maximum excitonic gap on . We find that is weakly dependent on the stacking and decays rapidly when . This result implies that the excitonic condensation is a robust phenomenon that will not require precise alignment of the two graphene sheets when they are being embedded in a dielectric. With typical graphene carrier densities /cm and Å or , the excitonic gap is appreciable, meV.
A direct probe of the excitonic gap is the quasiparticle density of states. For graphene with no interactions, the density of states is linear, . In the excitonic condensate phase, for intermediate energies there are two rings in the phase-space consistent with that energy: one with and the other with . Therefore the quasiparticle density of states is given by where () denotes the density of states from respective rings. Figure 3 shows (green dashed) and (red solid); they are both zero for and diverge at as is expected. Note that for , since there are no states for with energies higher than . The asymmetry in and for is due to the linear dispersion of carriers and the nonzero electron-hole pairing that extends to the bottom of the Fermi sea, foo (). The inset shows corresponding results a conventional system, where the density of states without interactions is constant, .
Superfluidity of a uniform Bose-Einstein condensate is characterized by a non-zero phase stiffness that quantifies the energy of a condensate with a linearly winding phase, where is the area of the sample and the phase of the condensate varies as . For graphene, since is the sole energy scale (at zero temperature), it follows from dimensional analysis that phase stiffness must scale linearly with the Fermi energy, where is a dimensionless function that satisfies cav (a) when and for . Hence, the phase stiffness is given by . The condensate energy can also be expressed, in the particle-picture, as the kinetic energy of excitons that have condensed in a state with center-of-mass momentum . Thus, where () is the mass (number) of condensed excitons cav (b). Equating the two expressions for energy implies . Thus we predict that the phase stiffness and the exciton mass will both vary as the square root of the carrier density. We emphasize that these results are unique to graphene and, as we will show in the next section, are equivalent to the absence of excitonic Wigner crystallization in graphene bilayers Chen and Quinn (1991); Joglekar et al. (2006).
Discussion: In this paper, we have investigated the properties of excitonic condensates in graphene bilayers. Our calculations predict that excitonic condensation will occur at all carrier densities as long as , and that the strength of the condensate, as measured by the excitonic gap is relatively insensitive to the stacking.
The mean-field results presented in this paper are obtained at zero temperature =0. (Finite temperature analysis gives a critical temperature or meV. This is an artifact of the mean-field approximation.) In two dimensions, the critical temperature for Bose-Einstein condensation is zero, but the superfluid properties survive for where is the Kosterlitz-Thouless transition temperature. Therefore, our results will be valid at nonzero temperature all (b). A weak disorder will suppress the excitonic condensate order parameter and reduce the excitonic gap, an effect equivalent to increasing the value of . Therefore, we have ignored the effects of a weak disorder potential.
In our analysis, we have only considered excitonic condensation with uniform density. In conventional (quantum Hall electron-hole) bilayers, varying and () leads to excitonic condensates with lattice structure Chen and Quinn (1991); qhs (); Joglekar et al. (2006). The origin of the lattice structure is Wigner crystallization of carriers in an isolated layer at large (small ). Graphene does not undergo Wigner crystallization as its carrier density is changed Dahal et al. (2006). Therefore, we expect that the excitonic condensate in graphene bilayers remains uniform. Now we show that this result is equivalent to our predictions for density dependence of and . The quantum kinetic energy of an exciton, associated with localizing it within a distance , is . The potential energy due to the dipolar repulsion between them is . Hence their ratio is given by . Wigner crystallization occurs when the ratio . This ratio will solely be a function of - no matter what the value of is - if and only if . Therefore, results in the last section show that the excitonic condensate in graphene will not undergo Wigner crystallization in spite of the dipolar repulsion between excitons with a quadratic dispersion. This result, too, is unique to graphene and is markedly different from the behavior of dipolar excitonic condensates in conventional bilayers. It is interesting that the mass of these effective bosons has the same density dependence and order of magnitude as the cyclotron mass of fermionic carriers in graphene Nov ().
The onset of excitonic condensation can be detected by a divergent interlayer drag ga (). A uniform in-plane magnetic field between the two graphene sheets is expected to induce a (counterflow) supercurrent in such a condensate ayp (), . The phase stiffness and its density dependence can be directly obtained from experimental measurements of the counterflow supercurrent. The verification (or falsification) of our predictions, including the density dependence of and , will deepen our understanding of properties and condensation of excitons with massless fermions as constituent particles.
Acknowledgments: This work was supported by the Research Support Funds Grant at IUPUI. After this work was completed, we became aware of a recent related work all (b).
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