  The formation of image-potential states at the interface between a graphene
layer and a metal surface is studied by means of model calculations. An
analytical one-dimensional model-potential for the combined system is
constructed and used to calculate energies and wave functions of the
image-potential states at the Gamma-point as a function of the graphene-metal
distance. It is demonstrated how the double series of image-potential states of
free-standing graphene evolves into interfacial states that interact with both
surfaces at intermediate distances and finally into a single series of states
resembling those of a clean metal surface covered by a monoatomic spacer layer.
The model quantitatively reproduces experimental data available for
graphene/Ir(111) and graphene/Ru(0001), systems which strongly differ in
interaction strength and therefore adsorption distance. Moreover, it provides a
clear physical explanation for the different binding energy and lifetime of the
first (n=1) image-potential state in the valley and hill areas of the strongly
corrugated moire superlattice of graphene/Ru(0001).
