Localization of electrons in thin films with rough surfaces

Abstract

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 $T=0$ in the $\ensuremath{\omega}\ensuremath{\rightarrow}0$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{0}$ limits. We find that the localization length ${r}_{0}$ depends on the film thickness $d$ as ${r}_{0}\ensuremath{\propto}\mathrm{exp}(\frac{{d}^{3}}{{l}^{3}})$, where ${l}^{3}$ is a constant depending on the Fermi energy and surface roughness.