Dynamical theory of reflection of very low energy electrons at crystal surfaces

Abstract

Elastic electron reflection at crystal surfaces for electron energies below the threshold of the first diffracted beam (≲ 10eV for most surfaces, near normal incidence) is studied in the framework of the dynamical theory of diffraction. The model considered is an idealized substrate crystal with selvedge (surface region) consisting, e.g., of an ordered layer of adsorbed atoms. The indirect effect of inelastic scattering (absorption) is included phenomenologically by going to complex values of the electron energy or propagation vector. A description of the amplitude of reflection as a function of energy is developed in detail for the quasi two-beam case; the quasi two-beam case is defined by the requirement that the totally-symmetric part of the relevant substrate and vacuum band-structure sections be the same as in the two-beam (one-dimensional) form of the dynamical theory, except for the presence of evanescent waves. It is shown that in the absence of a selvedge, the reflection amplitude function possesses branch points deriving from the substrate band structure but does not possess any finite poles or zeros. In the presence of a selvedge, the reflection amplitude may be expressed as the product of a “background” function representing the substrate band structure and having no poles or zeros, and a factor having poles and zeros symmetrically located relative to the real-energy axis. The poles are identified with surface-state resonances of the crystal, and the conjugate zeros with conditions of destructive interference in surface multiple scattering. It is shown that the surface-state resonances, acting through the associated conditions of destructive interference, can have a dominant effect in very low energy electron reflection.