Quantum size effects in the Fermi energy and electronic density of states in a finite square well thin film model

Abstract

Abstract The dependence of the Fermi energy, E F , and the electronic density of states, ρ( E ), of thin metallic films ( L z ≲ 50 A) on film thickness, electron density, and potential well depth, is systematically investigated in a free-electron, finite square well model. Two size-dependent effects are observed: (1) oscillations in E F and ρ (E) due to the size-quantization of the energy levels, and (2) changes in the mean values of these quantities, averaged over several oscillation periods, relative to their bulk values. The mean value of E F is increased relative to its bulk value by as much as 5%–10% for physically reasonable well depths and typical metallic electron densities. For the special case in which the top energy level in the well is occupied, the mean value of E F is equal to its bulk value. The mean value ofρ( E F ) can be either greater than or less than its bulk value, depending on the well depth. In contrast to the small amplitude oscillations in E F , the oscillations in ρ( E F ) may have an amplitude as large as 25% of the mean value for sufficiently thin films. Accurate analytic expressions for the thickness dependence of the Fermi energy and density of states are derived.