Trajectory reversal approach for electron backscattering from solid surfaces

Abstract

The backscattering of medium energy electrons from solid surfaces is investigated by analysis of a linearized Boltzmann-type kinetic equation. A closed expression is derived for the Green's function in an infinite medium valid for a spherically symmetric potential describing the interaction with the ionic subsystem. The solution is expressed in terms of fluctuations of the energy loss and scattering angles and the collision statistics associated with them. Since the fluctuation part is independent of the boundary conditions of the considered problem, solution of the backscattering problem requires an appropriate treatment of the collision statistics. In this context, the exact solution for the Oswald--Kasper--Gaukler model is derived and its limitations are analyzed. An exact approach is presented and implemented in an efficient Monte Carlo scheme based on the trajectory reversal technique. The resulting procedure is faster than the conventional Monte Carlo algorithm by several orders of magnitude. Results for the angular distribution are compared with conventional Monte Carlo calculations and perfectly agree with the latter within their statistical uncertainty. A second approximate algorithm is also given. The approximation involved in this second procedure turns out to be very reasonable: deviations from direct Monte Carlo calculations remain below $\ensuremath{\sim}5%$ for energies exceeding $200\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. The integral elastic-backscattering coefficient for normal incidence for a large number of materials in the energy range $50\phantom{\rule{0.3em}{0ex}}\mathrm{eV}--10\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$ is found to approximately exhibit a universal dependence on the ratio of the inelastic and the transport mean free paths, the so-called scattering parameter.