Roughness of equipotential lines due to a self-affine boundary

Abstract

In this work, the characterization of the roughness of a set of equipotential lines {\ell } , due to a rough surface held at a nonzero voltage bias, is investigated. The roughness of the equipotential lines reflects the roughness of the profile, and causes a rapid variation in the electric field close to the surface. An ideal situation was considered, where a well known self-affine profile mimics the surface, while the equipotential lines are numerically evaluated using Liebmann's method. The use of an exact scale invariant profile helps to understand the dependency of the line roughness exponent \alpha ({\ell }) on both the value of the potential (or on the average distance to the profile) and the profile's length. Results clearly support previous indications that: (a) for a system of fixed size, higher values of α characterize less corrugated lines far away from the profile; (b) for a fixed value of the potential, α decreases with the length of the profile towards the value of the boundary. This suggests that, for a system of infinite size, all equipotential lines share the same value of α.