Band structure of ABC-trilayer graphene superlattice

We investigate the effect of one-dimensional periodic potentials on the low energy band structure of ABC trilayer graphene first by assuming that all the three layers have the same potential. Extra Dirac points having the same electron hole crossing energy as that of the original Dirac point are generated by superlattice potentials with equal well and barrier widths. When the potential height is increased, the numbers of extra Dirac points are increased. The dispersions around the Dirac points are not isotropic. It is noted that the dispersion along the k_y direction for k_x0 oscillates between a non-linear dispersion and a linear dispersion when the potential height is increased. When the well and barrier widths are not identical, the symmetry of the conduction and valence bands is broken. The extra Dirac points are shifted either upward or downward depending on the barrier and well widths from the zero energy, while the position of the central Dirac point oscillates with the superlattice potential height. By considering different potentials for different layers, extra Dirac points are generated not from the original Dirac points but from the valleys formed in the energy spectrum. Two extra Dirac points appear from each pair of touched valleys, so four Dirac points appeared in the spectrum at particular barrier height. By increasing the barrier height of superlattice potential two Dirac points merge into the original Dirac point. This emerging and merging of extra Dirac points is different from the equal potential case.