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

Abstract

The purpose of this paper is to present a quantum-mechanical theory of the inelastic scattering of slow electrons by long-wavelength surface optical phonons for simple models of an ionic crystal and for a nonionic crystal such as silicon. It is argued that a quantum-mechanical approach is necessary for this problem. However, the expression we obtain for the one-phonon cross section is found to be identical to the one that follows from the earlier classical theory of Lucas and co-workers, provided one replaces their parameter $P_0$ by the quantum-mechanical reflection coefficient for specular reflection. The angular distribution of the scattered electrons and the energy dependence of the one-phonon cross section are discussed for the case of ZnO and silicon, where the surface optical modes have a very different character. For the surface mode in silicon, we define a dipole-moment effective charge, which is nonzero by virtue of the absence of inversion symmetry in the surface region. A quantitative estimate of the magnitude of this parameter is extracted from the data of Ibach.