Electronic properties of graphene nanoribbons with AA-stacking order

We study analytically, by making use of the tight-binding model, the electronic properties of armchair and zigzag AA-stacked bilayer graphene nanoribbons (BLGNRs). We apply hard-wall boundary conditions to determine the discretion dominating on the Bloch wavefunctions in the confined direction. First we investigate armchair AA-stacked BLGNRs, examining effects of edge deformation and an electric field applied perpendicular to layers on their electronic band structure and their Bloch wavefunction. We show that their electronic properties are strongly size-dependent and also can be tuned by the perpendicular electric field. Then we calculate the electronic band structure and Bloch wavefunction of zigzag AA-stacked BLGNRs and also consider effects of a perpendicular-to-layers electric field on them. We find that the electronic band structure of zigzag AA-stacked BLGNRs shows the main features of zigzag SLGNRs, having the edge states and being a metallic band structure. Moreover we investigate the effect of increasing the number of nanoribbons layers on the electronic band structure of both armchair and zigzag nanoribbons with AA-stacking order.