Flux Methods for Transport Problems in Solids with Nonconstant Electric Fields

Publisher Website
Google Scholar
Abstract
Certain recently developed methods for the solution of one-dimensional steady-state transport problems, based on flux conservation in diffusion-recombination systems involving charged particles, have been extended to cases where nonconstant electric fields act upon the particles of the system. In situations where the electric field is small enough so that the drift velocity is much less than the mean thermal speed, solutions for a system of coupled flux-conservation equations for the diffusion-recombination system can be obtained by use of the WKB method of approximation. From these solutions, expressions which involve integrals containing the field can be derived for the reflection and transmission coefficients of a material layer of arbitrary thickness. For the case where recombination may be neglected, the solutions are exact. Iterative methods are presented for obtaining the reflection and transmission coefficients of a material layer of arbitrary thickness under conditions where the requirements for the validity of the WKB method are not fulfilled. These methods are especially suitable for computer calculations. Expressions for the low-field reflection and transmission coefficients are obtained for a semiconductor surface space-charge layer in the case where bulk recombination is negligible. In the high field "hot-carrier" region, the fraction of flux which is reversed within a differential scattering layer is deduced empirically from symmetry properties and behavior in limiting cases. These expressions, though lacking any rigorous mathematical justification, are obtained independently of the carrier velocity distribution.