An overlayer system composed of a thin film on the top of a semi-infinite substrate was studied in this work for electron inelastic interactions. Analytical expressions for the depth-dependent inelastic differential and integral inverse mean free paths were derived for both incident and escaping electrons. The interface ͑film-substrate͒ effect and the surface ͑vacuum-film͒ effect were analyzed by comparing the results of an overlayer system and a semi-infinite system. It was found that the interface effect extended to several angstroms on both sides of the interface for a 500 eV electron incident into or escaping from the vacuum-SiO 2 -Si and the vacuum-Au-Ni systems. An application of the spatial-varying inelastic differential inverse mean free paths was made by Monte Carlo simulations of the electron elastic backscattering from an overlayer system. Good agreement was found between results calculated presently and data measured experimentally on the elastic reflection coefficient.for zϽϪD/2, B͑,Q,,z ͒ϭ 21ͭ 2 cos ͫ ͑zϩD/2͒ z ͬ Ϫexp͓ϪQ͑zϩD/2͔͒ ͮ Ϫ 21 23 exp͑ϪQD͒exp͓Q͑zϪD/2͔͒ ⑀ 2 ͑ Q,,zϩD/2͓͒1Ϫ 21 23 exp͑Ϫ2QD͔͒ ϩ 23 exp͓Q͑zϪD/2͔͒Ϫ 23 21 exp͑ϪQD͒ ͭ 2 cos ͫ ͑zϩD/2͒ z ͬ Ϫexp͓ϪQ͑zϩD/2͔͒ ͮ ⑀ 2 ͑ Q,,zϪD/2͓͒1Ϫ 21 23 exp͑Ϫ2QD͔͒ ϩ ͫ 1 ⑀ 2 ͑q ,͒ ͬ , ͑10͒ for ϪD/2ϽzϽD/2 and 8249 21ͭ 2 cos ͫ ͑zϩD/2͒ z ͬ Ϫexp͓ϪQ͑zϩD/2͔͒ ͮ exp͓ϪQ͑zϩD/2͔͒Ϫ 21 23 exp͑Ϫ2QD͒ ⑀ 2 ͑Q,͕͒1Ϫ 21 23 exp͑Ϫ2QD͖͒ ϩ 23 exp͓Q͑2zϪD ͔͒Ϫ 23 21 exp͑ϪQD͒ ͭ 2 cos ͫ ͑zϩD/2͒ z ͬ Ϫexp͓ϪQ͑zϩD/2͔͒ ͮ exp͓Q͑zϪD/2͔͒ ⑀ 2 ͑ Q, ͓͒1Ϫ 21 23 exp͑Ϫ2QD͔͒ ϩ ͫ 1 ⑀ 2 ͑Q,͒ ͬ , ͑16͒ for ϪD/2ϽzϽD/2 and B͑,Q,,z ͒ϭ 23 ͓1ϩ 21 exp͑Ϫ2QD͔͒ ͭ 2 cos ͫ ͑zϪD/2͒ z ͬ Ϫexp͓ϪQ͑zϪD/2͔͒ ͮ exp͓ϪQ͑zϪD/2͔͒ ⑀ 3 ͑ Q, ͕͒1Ϫ 21 23 exp͑Ϫ2QD͖͒ Ϫ 21 ͓1ϩ 23 ͔ ͭ 2 cos ͫ ͑zϩD/2͒ z ͬ Ϫexp͓ϪQ͑zϩD/2͔͒ ͮ exp͓ϪQ͑zϩD/2͔͒ ⑀ 3 ͑ Q, ͓͒1Ϫ 21 23 exp͑Ϫ2QD͔͒ ϩ ͫ 1 ⑀ 3 ͑Q,͒ ͬ , ͑17͒ for zϾD/2.8250

An energetic charged particle moving parallel to the surface of an overlayer system was studied. This system was composed of a thin film on the top of a semi-infinite substrate. Based on the dielectric response theory, the induced potential was formulated by solving the Poisson equation and matching the boundary conditions. The stopping force was built-up using the energy-momentum conservation relations and the extended Drude dielectric functions with spatial dispersion. Surface ͑vacuum-film͒ and interface ͑film-substrate͒ excitations were included in the formulations of the interaction between charged particles and the overlayer system. Results of the wake potential were presented for protons moving parallel to a vacuum-copper-silicon system. Dependences of the induced potential and the stopping force on film thickness, distance of the proton from surface, and proton velocity were investigated.