Difference equation solutions for non-steady-state diffusion of charged particles in solids

Abstract

The hopping transport of charged particles through a solid is described by means of difference equations based on the concept of classical thermal motion over discrete energy barriers. Homogeneous electric fields and concentration gradients are considered to be the driving forces for transport. Transient and steady-state currents are derived, and the concentration profiles are obtained for the mobile charged defect species. For the case of a slab geometry the discreteness of the potential barriers leads to a nonlinear dependence of current on voltage in the high electric field limit, with a more rapid increase of current with voltage than would be expected from an extrapolation of the low field linear dependence. The field-dependent relaxation of a non-steady-state defect concentration profile to the corresponding steady-state profile can be nearly exponential in the limit of large fields. Tracer distributions for the cases of semi-infinite and unbounded diffusion mediums are likewise affected appreciably in the high field limit. The velocity of the peak is increased over that obtained by linear extrapolation from the low field limit. It is concluded that the combination of high electric fields with the natural microscopic discreteness of a solid-state diffusion medium can result in readily observable nonlinear electric field effects which increase approximately exponentially with the atomic separation distances of the discrete barriers in hopping transport. Some of this nonlinear behavior can be retained in differential equations derived from the difference equations by means of Taylor series expansions of the carrier concentration with respect to position.