Electron backscattering from surfaces: The invariant-embedding approach

Abstract

Distributions of electrons reflected from a solid surface corresponding to a reflection electron energy loss spectroscopy (REELS) experiment are calculated within the invariant-embedding approach. The technique is based upon a formalism recently developed by Vicanek. In this paper we show that the pertinent balance equations readily allow effective numerical evaluation of various electron distributions relevant for REELS. The solutions are very accurate (with the relative errors readily reduced to $\ensuremath{\lesssim}{10}^{\ensuremath{-}4}$ for given cross sections) and the resulting technique offers significant advantages over the Monte Carlo approach, though it is less generally applicable. The approach delivers, in a single calculation, the probability distributions resolved in the number of elastic as well as inelastic collisions for all angles of incidence and emission. From these results, one readily obtains other related distributions, e.g., path length and energy spectra. Compared to Monte Carlo simulations, the scheme is faster by up to several orders of magnitude while producing a better accuracy. For a wide range of energies and targets, various distributions of backscattered electrons are exemplified and discussed. The elastic-peak results and a procedure for extraction of the inelastic mean free path are tested by comparison to recent Monte Carlo simulations and typically excellent agreement is found; discrepancies observed in a few cases are shown to be due to deficiencies in the Monte Carlo scheme.