Electronic band structure effects in the stopping of protons in copper

Abstract

We present an ab initio study of the electronic stopping power of protons in copper over a wide range of proton velocities $v=0.02--10\phantom{\rule{0.28em}{0ex}}\mathrm{a}.\mathrm{u}.$ where we take into account nonlinear effects. Time-dependent density functional theory coupled with molecular dynamics is used to study electronic excitations produced by energetic protons. A plane-wave pseudopotential scheme is employed to solve the time-dependent Kohn-Sham equations for a moving ion in a periodic crystal. The electronic excitations and the band structure determine the stopping power of the material and alter the interatomic forces for both channeling and off-channeling trajectories. Our off-channeling results are in quantitative agreement with experiments, and at low velocity they unveil a crossover region of superlinear velocity dependence (with a power of $\ensuremath{\sim}1.5)$ in the velocity range $v=0.07--0.3\phantom{\rule{0.28em}{0ex}}\mathrm{a}.\mathrm{u}.$, which we associate to the copper crystalline electronic band structure. The results are rationalized by simple band models connecting two separate regimes. We find that the limit of electronic stopping $v\ensuremath{\rightarrow}0$ is not as simple as phenomenological models suggest and it is plagued by band-structure effects.