One-Dimensional Potential Model for Image States on Free-Standing Graphene

In the framework of the non-local dielectric theory the static non-local self-energy of an electron near an ultra-thin polarizable layer has been calculated and applied to study image-states near free-standing graphene. The corresponding series of eigenvalues and eigenfunctions have been obtained by solving numerically the one-dimensional Schr{\"o}dinger equation. We compare with the Rydgberg's series for a perfect metal and with experimental values measured on graphene layers grown on Ir and Ru surfaces. For free standing films, the appearance of states with binding energies in between the classical series is discussed.