Multiple Plasmon Effects in the Energy-Loss Spectra of Electrons in Thin Films

The interaction of a fast electron (energy {} 1 keV) with collective modes in a thin metallic or doped semiconductor film is studied, taking explicitly into account bulk and surface effects. Reducing the problem to an exactly soluble quantum-mechanical model, we obtain results to all orders of the interaction and investigate the multiple-plasmon contribution to the electron energy losses. The theory is applied to transmission as well as to reflection and a complete description of the loss spectra is obtained. It is shown that the zero-energy plasmon mode does not cause singularities in the low-energy excitation spectrum. The strength of the many-body processes is given by the "fine-structure constant" e²v_⊥, where v_⊥ is the electron velocity normal to the surface, and in some cases multiple-plasmon emission becomes important, especially so in the specular reflection at grazing-angle incidence. For a thicker slab (a~500-1000 {}), this results in a series of sharp peaks in the loss spectrum which have maximum strengths corresponding to several excited plasmons. For a thin slab (a~few 100 {}), the dominant contribution to the loss spectrum is a broad structure with a high-energy (ω>ωₚ) tail, characteristic for the recombinational effect of many emitted low-energy plasmons.