  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 inter-layer hopping which leads to gapless
semiconductor electronic structure, with 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.
