Effect of the non-nearest-neighbor hopping on the electronic structure of armchair graphene nanoribbons

Based on the tight-binding model, the non-nearest-neighbor hopping terms of electrons are taken into account and the energy spectra of the armchair graphene nanoribbons (AGRNs) are given analytically. The changes of the energy band and the gap with the non-nearest-neighbor terms are discussed. The results show that the next-nearest-neighbor term can increase the gap and the third-nearest-neighbor term can narrow the gap. The competition relationship between the edge relaxation and the non-neighbor term is compared. When the width n is odd, the van Hove singularity from graphene sheets leads to the dispersion-less band. When the width of AGRNs goes to infinity, the spectrum of AGRNs tends to that of graphene sheets.