Band-gap scaling of graphene nanohole superlattices

Based on the tight-binding model, we investigate band structures of graphene nanohole (GNH) superlattices as a function of NH size and density. One common origin of band gaps for GNH superlattices with NHs of either armchair or zigzag edges is the quantum-confinement effect due to the periodic potential introduced by the NHs, which turns the semimetallic sheet into a direct-gap semiconductor. Additional band gaps also open for GNH superlattices with NHs of zigzag edges in a ferromagnetic ground state, arising from the staggered sublattice potential on the zigzag edges due to edge magnetization. Our calculations reveal a generic scaling relation that both types of band gaps increase linearly with the product of NH size and density.