Electronic compressibility of a graphene bilayer

We calculate the electronic compressibility arising from electron-electron interactions for a graphene bilayer within the Hartree-Fock approximation. We show that, due to the chiral nature of the particles in this system, the compressibility is rather different from those of either the two-dimensional electron gas or ordinary semiconductors. We find that an inherent competition between the contributions coming from intraband exchange interactions (dominant at low densities) and interband interactions (dominant at moderate densities) leads to a nonmonotonic behavior of the compressibility as a function of carrier density.