One-dimensional potential for image-potential states on graphene

In the framework of dielectric theory the static non-local self-energy of an electron near an ultra-thin polarizable layer has been calculated and applied to study binding energies of 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. Image-potential-state wave functions accumulate most of their probability outside the slab. We find that a Random Phase Approximation (RPA) for the non-local dielectric function yields a superior description for the potential inside the slab, but a simple Fermi-Thomas theory can be used to get a reasonable quasi-analytical approximation to the full RPA result that can be computed very economically. Binding energies of the image-potential states follow a pattern close to the Rydberg series for a perfect metal with the addition of intermediate states due to the added symmetry of the potential. The formalism only requires a minimal set of free parameters; the slab width and the electronic density. The theoretical calculations are compared to experimental results for work function and image-potential states obtained by two-photon photoemission.