High-energy Landau levels in graphene beyond nearest-neighbor hopping processes: Corrections to the effective Dirac Hamiltonian

We study the Landau level spectrum of bulk graphene monolayers beyond the Dirac Hamiltonian with linear dispersion. We consider an effective Wannier-like tight-binding model obtained from ab initio calculations, that includes long-range electronic hopping integral terms. We employ the Haydock-Heine-Kelly recursive method to numerically compute the Landau level spectrum of bulk graphene in the quantum Hall regime and demonstrate that this method is both accurate and computationally much faster than the standard numerical approaches used for this kind of study. The Landau level energies are also obtained analytically for an effective Hamiltonian that accounts for up to third nearest neighbor hopping processes. We find an excellent agreement between both approaches. We also study the effect of disorder on the electronic spectrum. Our analysis helps to elucidate the discrepancy between theory and experiment for the high-energy Landau levels energies