Ultrarelativistic electron-hole pairing in graphene bilayer

We consider the ground state of an electron-hole graphene bilayer composed of two independently-doped graphene layers when a condensate of spatially separated electron-hole pairs is formed. In the weak coupling regime the pairing affects only the conduction band of the electron-doped layer and the valence band of the hole-doped layer, thus the ground state is similar to an ordinary BCS condensate. At strong coupling, an ultrarelativistic character of the electron dynamics reveals itself and the bands which are remote from Fermi surfaces (valence band of electron-doped layer and conduction band of hole-doped layer) are also affected by the pairing. Analysis of the instability of the unpaired state shows that s-wave pairing with band-diagonal condensate structure, described by two gaps, is preferable. The relative phase of the gaps is fixed, however at weak coupling this fixation diminishes allowing gapped and soliton-like excitations. The coupled self-consistent gap equations for these two gaps are solved at zero temperature in the constant-gap approximation and in the approximation of a separable potential. It is shown that, if the characteristic width of the pairing region is of the order of magnitude of the chemical potential, then the value of the gap in the spectrum is not much different from the BCS estimation. However if the pairing region is wider, then the gap value can be much larger and depends exponentially on its energy width.