Localization of electrons in thin films with rough surfaces

A system of noninteracting electrons in a thin metal film which experience arbitrarily weak scattering from a randomly rough surface is shown to be always in localized states. The frequency-and wave-vector-dependent density response function, the frequency-dependent conductivity, and the localization length are calculated at $T=0\mathrm{}$ in the $\omega \to 0$, $\overrightarrow{\mathrm{q}}\to \overrightarrow{0}$ limits. We find that the localization length $r_0$ depends on the film thickness $d$ as $r_0\propto \exp \left(\frac{d^3}{l^3}\right)$, where $l^3$ is a constant depending on the Fermi energy and surface roughness.