Calculation of EELS at a doped semiconductor surface

We calculate electron energy loss spectra (EELS) within a simple model of a doped semiconductor surface. The model is designed to mimic an n-type GaAs surface for which experimental results exist. Coupling is only allowed via the dipole mechanism and the external electron is presumed to follow a specular trajectory. Using a cumulant expansion, we can treat multiple losses and gains as well as Debye-Waller factors. The basic theoretical quantity is an effective surface dielectric function that depends on frequency and wavevector parallel to the surface. Its calculation requires in general the specification of an additional boundary condition (ABC) in order to match fields at the surface. We explore the consequences of two possible ABC choices. At early stages of these calculations there are large differences between results from different ABC's; but by the time all the broadening (e.g. due to Ohmic damping, to multiple losses, and to finite spatial and energy resolution) has been included in order to compare with experiment, there survives much less dependence on ABC. We conclude that it is difficult to probe fine theoretical details, such as the influence of spatial dispersion, with EELS in such systems.