  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.
