We study the band structure and the density of states of graphene in the presence of a next-to-nearest-neighbor coupling (N2) and a third-nearest-neighbor coupling (N3). We show that for values of N3 larger or equal to 1/3 of the value of the nearest
-neighbor hopping (NN), extra Dirac points appear in the spectrum. If N3 is exactly equal to 1/3 NN, the new Dirac points are localized at the M points of the Brillouin zone and are hybrid: the electrons have a linear dispersion along the GammaM direction and a quadratic dispersion along the perpendicular direction MK. For larger values of N3 the new points have a linear dispersion, and are situated along the MK line. For a value of N3 equal to 1/2 NN, these points merge with the Dirac cones at the K points, yielding a gapless quadratic dispersion around K, while for larger values each quadratic point at K splits again into four Dirac points. The effects of changing the N2 coupling are not so dramatic. We calculate the density of states and we show that increasing the N3 coupling lowers the energy of the Van Hove singularities, and when N3 is larger than 1/3 NN the Van Hove singularities split in two, giving rise to extra singularities at low energies.
