Influence of the boundary conditions on the fermi energy and density of states in a free-electron solid of sub-micron dimensions

Asymptotic expressions for the distribution of the eigenvalues of the Helmholtz-Schrödinger equation are used to anlyze the dependence of the Fermi energy, E_F, and the density of states, ρ(E), on sample size, shape, and electron density, in a free-electron model with Dirichlet boundary conditions. It is found that for very small samples E_F is increased relative to its asymptotic (i.e., bulk) value and ρ(E) is decreased relative to its bulk value. These effects are more pronounced for samples with low electron density and with a large surface-to-volume ratio. In general E_F and ρ(E_F) deviate significantly from their bulk values only for systems with fewer than 50,000 electrons and/or with linear dimensions of 100 Å or less. The use of smoothing functions to represent the density of states obtained from the exact eigenvalue distribution is also discussed. It is shown that an oscillating density of states leads to small cusps in the plot of E_F as a function of sample size. This is in qualitative agreement with the results of experiments on size-dependent oscillations in field emission from thin metallic films. Comparison is also made between photoemission experiments from thin films and other results obtained in this study.