A Singular Perturbation Analysis of the Fundamental Semiconductor Device Equations

In this paper we present a singular perturbation analysis of the fundamental semiconductor device equations which form a system of three second order elliptic differential equations subject to mixed Neumann–Dirichlet boundary conditions. The system consists of Poisson’s equation and the continuity equations and describes potential and carrier distributions in an arbitrary emiconductor device. The singular perturbation parameter is the minimal normed Debye length of the device under consideration. Using matched asymptotic expansions we demonstrate the occurrence of internal layers at surfaces across which the impurity distribution (appearing as an inhomogeneity of Poisson’s equation) has a jump discontinuity (these surfaces are called “junctions”) and the occurrence of boundary layers at semiconductor-oxide interfaces. We derive the layer-equations and the reduced problem (charge-neutral-approximation) and give existence proofs for these problems. The layer solutions which characterize the solutions of the singularly perturbed problem close to junctions and interfaces respectively are shown to decay exponentially away from the junctions and interfaces respectively. We show that, if the device is in thermal equilibrium, then the solution of the semiconductor problem is close to the sum of the reduced solution and the layer solution assuming that the singular perturbation parameter is small. Numerical results for a two-dimensional diode are presented.