Band dispersion of graphene with structural defects
We study the band dispersion of graphene with randomly distributed structural defects using two complementary methods, exact diagonalization of the tight-binding Hamiltonian and implementing a self-consistent T matrix approximation. We identify three distinct types of impurities resulting in qualitatively different spectra in the vicinity of the Dirac point. First, resonant impurities, such as vacancies or 585 defects, lead to stretching of the spectrum at the Dirac point with a finite density of localized states. This type of spectrum has been observed in epitaxial graphene by photoemission spectroscopy and discussed extensively in the literature. Second, nonresonant (weak) impurities, such as paired vacancies or Stone-Wales defects, do not stretch the spectrum but provide a line broadening that increases with energy. Finally, disorder that breaks sublattice symmetry, such as vacancies placed in only one sublattice, open a gap around the Dirac point and create an impurity band in the middle of this gap. We find good agreement between the results of the two methods and also with the experimentally measured spectra.