Alternative Approach to the Solution of Added Carrier Transport Problems in Semiconductors

A novel method of solving added carrier transport problems in semiconductors is presented. The usual procedure in treating problems of this type is to derive a continuity equation for charge carriers on the basis of carrier conservation, allowing for generation and recombination, and to solve this equation under appropriate boundary conditions. The resulting fluxes or currents are obtained from diffusion and drift current equations, which involve the concentrations and concentration gradients. In the formulation presented here, equations embodying conservation of flux (again with due allowance for generation and recombination) which incorporate the proper boundary conditions from the outset are solved in the steady-state one-dimensional case to yield a Green's function for the desired carrier fluxes directly. The method is more general than the commonly used continuity equation formulation in that the physical dimensions of the system and the diffusion lengths are not restricted to be large compared to the mean free path; in particular it is unnecessary to assume Fick's law for diffusion processes. Otherwise the method is equivalent to the continuity equation analysis. An example involving carrier generation in a plane region bounded on one side by a surface of arbitrary reflection coefficient (or recombination velocity) and on the other by a collecting $p-n$ junction is worked out. The results are shown to reduce to those obtained via the continuity equation in the appropriate limiting case.