Degenerate perturbation in band-gap opening of graphene superlattice

We study the graphene band-gap engineering by introducing different defects, namely the defects breaking the inversion symmetry and the ones periodically patterning graphene into superlattice such as the regularly arranged antidots, etc. Comparing to the primitive unit cell of graphene, the pseudo-graphene superlattice, referred to the pristine graphene supercell, modulates the boundary condition accordingly. According to the energy band-folding picture, these superlattices can be categorized into two groups on the basis of the Dirac cone position. In some cases, the Dirac points K and K' in primitive cell are folded to the λ point of pseudo-superlattices. The coincidence of Dirac points with λ point results in the fourfold degeneracy. In these systems, a band gap at λ point can be opened by introducing periodically arranged defects such as the antidots, which could be easily utilized in experiment, for example, by making the graphene nanomesh through lithography technique. In the other cases, the twofold degenerate Dirac points remain nonequivalent with λ point in pseudo-superlattice, and the method to break the inversion symmetry could open their band gaps.