Consistency of Semiconductor Modeling: An Existence/Stability Analysis for the Stationary Van Roosbroeck System

Abstract

This paper examines the solution map of the stationary version of the Van Roosbroeck model for the flow of electrons and holes in a crystalline semiconductor. Thermodynamic equilibrium and the Einstein relations linking mobility and diffusion are assumed to hold for this model. Incorporated into the model is the quantum-statistical assumption that the carrier densities satisfy Fermi-Dirac statistical laws with localization in the “Boltzmann tail.” This latter approximation, which is seen to be inessential, characterizes the conduction and valence bands as nondegenerate. The boundary conditions are of two types, and correspond to applied potential differences and doping on the contact portions of the device, and insulation on the remainder. The complete system, then, involves a coupled set of three nonlinear partial differential equations, with their boundary conditions, for the electrostatic potential, and for the quasi-Fermi levels corresponding to the electron and hole concentrations. In the absence of the classical nondegeneracy approximation, the model remains valid, but without a natural physical interpretation for two of the three dependent variables. We demonstrate the existence of solutions for this system by associating a solution mapping T, such that the fixed points of T define solutions of the system. “A priori” estimates, or weak maximum principles, permit T to act invariantly on a naturally chosen subset of a Hilbert product space. The definition of T roughly follows the Gummel procedure for uncoupling systems; the Schauder fixed point theorem is used to prove the existence of a fixed point for T. Uniqueness is not demonstrated, and very possibly does not hold for these systems, without special restrictions on the boundary conditions. We also introduce a discrete version of the mapping T which is defined in terms of a family of finite difference approximations. This mapping enjoys the stability property that it acts invariantly on a rectangular region in Euclidean space; the dimensions are the same as those specified in the pointwise bounds which define the domain of T. A second existence theory framework, with quasi-constructive aspects is also studied. This is based upon pseudomonotone operators. The presentation is technology independent except for a section on MOS-FET devices. These typically induce a breakdown of uniform ellipticity, and may be handled by modifications of the major arguments. They also exemplify a class of devices for which the electrostatic field is defined on a strictly larger domain than the carriers themselves.