Theory of Inelastic Scattering of Slow Electrons by Long-Wavelength Surface of Optical Phonons: Multiphonon Processes

Abstract

We describe a quantum-mechanical theory of the inelastic scattering of low-energy electrons by multiphonon processes, from the surface of a semi-infinite crystal. A model introduced in an earlier paper is also employed in this work. The model describes the interaction of an incident low-energy electron with surface optical phonons by means of the macroscopic electric field set up outside the crystal by the ion motion. The model may be used to describe scattering either from ionic crystals, such as ZnO, or from nonionic crystals. In this paper, we find an explicit expression for the wave function of the outgoing electron, and we obtain an expression for the probability that $n$ phonons are created or absorbed in the scattering process. Two cases are considered. First we examine the cross section for scattering off thermal phonons, and second from a coherent surface wave excited by external means. For the first case, our result agrees with the earlier semiclassical theory of Lucas and Sunjic. However, the model here is more general than theirs, since it is fully quantum mechanical. We show explicitly that the energy-loss cross section is proportional to the intensity of the specular beam, for scattering off both ionic and covalent crystals. For the second case (scattering from surface optical phonons generated coherently by an external source), we obtain a closed expression for the cross section. The physical origin of differences between the expressions is discussed.