Theory of the Range of Hot Electrons in Real Metals

Abstract

The equations of Quinn and Ferrell and of Quinn for the rate of energy loss of a hot electron in a free-electron gas are generalized to take solid-state effects into account. A general equation is derived, which in addition to a principal term which reduces to Quinn's result in the free-electron gas limit, contains terms which result from umklapp processes and local-field corrections. The additional terms are evaluated for aluminum on a one-OPW model and are found to result in a 16-30% decrease in the rate of energy loss. The effect of Fermi surface shape on the principal term is discussed in detail, with the aid of an exact recasting of the term into a form which explicitly shows its dependence on the equations of the energy surfaces. It is shown that nonspherical Fermi surfaces lead to an anisotropic hot-electron energy-loss rate, and that for certain shapes of Fermi surface the rate of energy loss is more singular than ${(E_{\mathrm{p}}-E_0)}^3$ near the Fermi surface. It is found that the "flatter" the Fermi surface is, the greater is the rate of hot-electron energy loss. This is suggested as a possible explanation for the anomalously small hot-electron range observed in copper by Crowell et al.