Effect of atomic-scale defects on the low-energy electronic structure of graphene: Perturbation theory and local-density-functional calculations

Based on perturbation theory and local-density-functional calculations, we study the effect of atomic-scale defects, whose potentials vary on the scale of interatomic distance, on the electronic structure of graphene in the region of low energies. If defects are identical, for example, vacancies at the same sublattice sites or the Stone-Wales defects with the same orientations, the degeneracy at the Dirac point of graphene is removed with an energy splitting proportional to lambda for low disorder densities (lambda), which is attributed to the breaking of the intrinsic symmetry of the honeycomb lattice by the presence of atomic-scale defects. However, the degeneracy at the Dirac point is nearly restored if physically equivalent disorders, which are generated by the symmetry operations of graphene, such as reflection and rotation, coexist with similar concentrations.