A qualitative analysis of the fundamental semiconductor device equations

We present a qualitative analysis of the fundamental static semiconductor device equations which is based on singular perturbation theory. By appropriate scaling the semiconductor device equations are reformulated as singularly perturbed elliptic system (the Laplacian in Poisson’s equation is multiplied by a small parameter λ², the so-called singular perturbation parameter). Physically the singular perturbation parameter is identified with the square of the normed minimal Debye length of the device under consideration. Using matched asymptotic expansions for small A we characterize the behaviour of the solutions locally at pn junctions, Schottky contacts and oxide-semiconductor interfaces and demonstrate the occurrence of exponential internal/boundary layers at these surfaces. The derivatives of the solutions blow up within these layer regions (as λ² decreases) and they remain bounded away from the layers. We demonstrate that the solutions of the ‘zero-space charge approximation’ are close to the solutions of the “full’ semiconductor problem (when A is small) away from layer regions and derive a second-order ordinary differential equation which (when subjected to appropriate boundary/interface conditions) ‘describes’ the solutions within layer regions.