Numerical solution of the Dirac equation for an armchair graphene nanoribbon in the presence of a transversally variable potential

We compute, using a continuum model based on the Dirac equation, the transverse modes and the longitudinal wave vectors in an armchair ribbon, in the presence of a transversally variable external potential. We show that the application of a standard finite difference method to this problem is not effective, since it can lead to the appearance of spurious solutions and to poor precision. We show how an approach based on Fourier analysis allows to overcome this difficulty and to drastically speed up the calculations, using well-established FFT algorithms.