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 $d$ 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.Comment: 4 pages, 3 figures; substantial text revisio

Graphene, a single free-standing sheet of graphite with honeycomb lattice structure, is a semimetal with carriers that have linear dispersion. A consequence of this dispersion is the absence of Wigner crystallization in graphene, since the kinetic and potential energies both scale identically with density of carriers. We study the ground state of graphene in the presence of strong magnetic field focusing on states with broken translational symmetry. Our mean-field calculations show that at integer fillings a uniform state is preferred whereas at non-integer fillings, Wigner crystal states (with broken translational symmetry) have lower energy. We obtain the phase diagram of the system. We find that it is qualitatively similar to that of quantum Hall systems in semiconductor heterostructures. Our analysis predicts that non-uniform states, including Wigner crystal state, will occur in graphene in the presence of a magnetic field and will lead to anisotropic transport in high Landau levels.

Graphene, with its massless linearly-dispersing carriers, in the quantum Hall regime provides an instructive comparison with conventional two-dimensional (2D) systems in which carriers have a nonzero band mass and quadratic dispersion. We investigate the influence of Landau level mixing in graphene on Wigner crystal states in the $n^\mathrm{th}$ Landau level obtained using single Landau level approximation. We show that the Landau level mixing does not qualitatively change the phase diagram as a function of partial filling factor $\nu$ in the $n^\mathrm{th}$ level. We find that the inter-Landau level mixing, quantified by relative occupations of the two Landau levels, $\rho_{n+1}/\rho_{n}$, oscillates around 2% and, in general, remains small ($< 4%$) irrespective of the Landau level index $n$. Our results show that the single Landau level approximation is applicable in high Landau levels, even though the energy gap between the adjacent Landau levels vanishes.Comment: 6 pages, 3 figure