Combined effect of quantum size and disorder in a two-dimensional armchair graphene nanoribbon with s-wave pairing

We investigate the quantum size effect combined with the disorder effect on an armchair graphene nanoribbon with s-wave pairing. In the framework of the Bogoliubov-de Gennes equation, the spatial distribution of the pairing potential is self-consistently determined for both pure and disordered systems. The dependence of the order parameter on the chemical potential is calculated for the zero-disordered case and a gap-like structure with two peaks is found. The relationship of the average pairing potential to the width of ribbon is discussed. Moreover, the disorder effect on the associated behavior is studied.