Band structure of ABC-stacked graphene trilayers

The ABC-stacked N-layer-graphene family of two-dimensional electron systems is described at low energies by two remarkably flat bands with Bloch states that have strongly momentum-dependent phase differences between carbon pi-orbital amplitudes on different layers and large associated momentum-space Berry phases. These properties are most easily understood using a simplified model with only nearest-neighbor interlayer hopping which leads to gapless semiconductor electronic structure and p(N) dispersion in both conduction and valence bands. We report on a study of the electronic band structures of trilayers which uses ab initio density-functional theory and k.p theory to fit the parameters of a pi-band tight-binding model. We find that when remote interlayer hopping is retained, the triple Dirac point of the simplified model is split into three single Dirac points located along the three KM directions. External potential differences between top and bottom layers are strongly screened by charge transfer within the trilayer but still open an energy gap at overall neutrality.