Tight-binding description of patterned graphene

The existence of an energy gap of graphene is vital as far as nano-electronic applications such as nano-transistors are concerned. In this paper, we present a method for introducing arbitrary energy gaps through breaking the symmetry point group of graphene. We investigate the tight-binding approximation for the dispersion of π and π* electronic bands in patterned graphene including up to five nearest neighbors. As we show by applying special defects in graphene structure, an energy gap appears at Dirac points and the effective mass of fermions also becomes a function of the number of defects per unit cell.