Stationary energy models for semiconductor devices with incompletely ionized impurities

The paper deals with two-dimensional stationary energy models for semiconductor devices, which contain incompletely ionized impurities. We reduce the problem to a strongly coupled nonlinear system of four equations, which is elliptic in nondegenerated states. Heterostructures as well as mixed boundary conditions have to be taken into account. For boundary data which are compatible with thermodynamic equilibrium there exists a thermodynamic equilibrium. Using regularity results for systems of strongly coupled linear elliptic differential equations with mixed boundary conditions and nonsmooth data and applying the Implicit Function Theorem we prove that in a suitable neighbourhood of such a thermodynamic equilibrium there exists a unique stationary solution, too. 1. Model equations. The charge transport in semiconductor devices is described by the van Roosbroeck equations (see [17] and e.g. [4, 11, 14]). They consist of two continuity equations for the densities n and p of electrons e and holes h, respectively, and a Poisson equation for the electrostatic potential φ. Physical parameters occurring in these equations depend on the device temperature T . Therefore, under nonisothermal conditions a balance equation for the density of total energy must be added, and a so called energy model arises (see [2, 18]) . Finally, if incompletely ionized impurities (for example radiation induced traps or other deep recombination centers) are taken into account, we have to consider further continuity equations for the densities of (in general immobile) species Xj , j = 1, . . . , k. These species exist in different charge states which are transformed into each other by ionization reactions. For the sake of simplicity we assume that each reaction is a binary one. Let Xj be an acceptor-like impurity which can accept an electron e or deliver a hole h and let X−j be its ion. Then we have to consider the reactions (1) e + Xj X − j , h + + X−j Xj . If Xj is a donor-like impurity which can deliver an electron e or accept a hole h and X + j denotes its ion, then the reactions are (2) e + X+j Xj , h + + Xj X + j . If Xj is a donor (an acceptor) we denote by u2j−1 the density of Xj (of X − j ) and by u2j the density of X+j (of Xj). Furthermore, we define charge numbers as follows: q2j−1 := { 0 if Xj is a donor −1 if Xj is an acceptor , q2j := 1 + q2j−1, j = 1, . . . , k. Then the continuity equations have the form (3) ∂n ∂t +∇ · jn = R0 + k ∑ j=1 Rj1, ∂p ∂t +∇ · jp = R0 + k ∑