Inapplicability of the Sugiyama phase sum rule to very thin films

The Sugiyama phase sum rule, derived for a free-electron model, implies that the Fermi energy of a thin film cannot be size dependent. This rule is shown to be invalid for very thin films ($L_z\lesssim 50$ Å) in a square-well model, since it depends on the incorrect assumption of local charge neutrality near the center of the film. The displacement of the potential barriers outward from the surfaces of very thin films in order to produce local charge neutrality in the interior is discussed. The size dependence found for the free-electron model is expected to persist in the presence of electron-electron interactions, though possibly with reduced magnitude.