Effect of surface roughness on the image potential

Abstract

We have calculated the electrostatic potential of a point charge located near a rough dielectric-vacuum interface. The system consists of a vacuum in the region $x_3>\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)$ and a dielectric characterized by an isotropic dielectric constant $\epsilon$ in the region $x_3<\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)$. Here $\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)$ is a random function of ${\overrightarrow{\mathrm{x}}}_{\parallel }=x_1{\hat{x}}_1+x_2{\hat{x}}_2$ such that $〈\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)〉=0\mathrm{}$, and $〈\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)\zeta \left({{\overrightarrow{\mathrm{x}}}^{\prime }}_{\parallel }\right)〉={\delta }^{2}W({\overrightarrow{\mathrm{x}}}_{\parallel }-{\overrightarrow{\mathrm{x}}}_{\parallel }^{\prime })$, where $\delta$ is the root-mean-square deviation of the interface from flatness, and the angular brackets denote an average over the ensemble of realizations of $\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)$. We obtain the electrostatic potential of a point charge $q$, situated either in the vacuum or in the dielectric, at any point of the system by joining solutions of Poisson's equation above and below the interface at each point of the interface, and then averaging the result over the ensemble of realizations of the function $\zeta \left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)$. Corrections to the results for a planar interface are obtained to $O\left({\delta }^2\right)$. Analytic asymptotic expressions are presented for the averaged potential when the distance from the image charge to the point at which the potential is determined is small and large, respectively, compared with the average distance between consecutive peaks and valleys on the surface, for a Gaussian form of $W\left({\overrightarrow{\mathrm{x}}}_{\parallel }\right)$. These results are used to obtain the probability for electron-surface-plasmon scattering in the case of an electron moving along a specified trajectory above the rough surface. Although surface roughness increases the energy loss suffered by the electron, the effect is qualitatively different from that required to explain the results of a recent experiment by Lecante, Ballu, and Newns.