The exchange instability in a graphene bilayer

We study the problem of the formation of electron-hole pockets in a graphene bilayer(two layers of graphite). By means of a variational calculation in a simple model we show that pockets can be formed due to the combined effects of the Coulomb interaction and a coherent nearest neighbor hopping within the planes and between the layers. The variational calculation shows that the non-interacting ground-state is unstable and that a small net ferromagnetism can be obtained. Upon doping the system away from half filling the instability goes away above a critical value of the doping. We also discuss how modifications to the model such as the inclusion of screening and disorder will affect the instability.