Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains

We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain Ω0 of the domain of definition Ω of the energy balance equation and of the Poisson equation. Here Ω0 corresponds to the region of semiconducting material, Ω \Ω0 represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a W -regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. 1 Stationary energy models for semiconductor devices Semiconductor devices are heterostructures consisting of various materials (different semiconducting materials, passive layers and metals as contacts, for example). A typical situation is shown in Fig. 1. Metals used as contacts are substituted by Dirichlet boundary conditions on a part ΓD of the boundary of the semiconducting material. In the domain Ω involving the passive layer (Ω1) and semiconducting materials (Ω0) we have to formulate a Poisson equation for the electrostatic potential and an energy balance equation with boundary conditions on Γ := ∂Ω = ΓD ∪ ΓN0 ∪ ΓN1, where the subscripts D and N indicate the parts with Dirichlet and Neumann boundary conditions, respectively. Continuity equations for electrons and holes have to be taken into account only in the part Ω0, and here we must formulate boundary conditions on Γ0 := ∂Ω0 = ΓD ∪ ΓN01 ∪ ΓN0. Especially on ΓN01, which corresponds to the interface between semiconducting material and passive layers, no-flux conditions have to be formulated. In this paper we restrict our considerations to the case that the Dirichlet parts of Γ and Γ0 coincide. Let T and φ denote the lattice temperature and the electrostatic potential. Then the state equations for electrons and holes are given by the following expressions n = N(·, T )F (ζn + φ− En(·, T ) T ) , p = P (·, T )F (ζp − φ+ Ep(·, T ) T ) in Ω0, where n and p are the electron and hole densities, N and P are the effective densities of state, ζn and ζp are the electrochemical potentials, En and Ep are the energy band edges, respectively. The function F arises from a distribution function (e.g. F (y) = e y in the case of Boltzmann statistics or F (y) = F1/2(y) in the case of Fermi-Dirac statistics). The electrostatic potential φ fulfils the Poisson equation −∇ · (ε∇φ) = { f − n+ p in Ω0 f in Ω1 . (1.1) 2 A. Glitzky, R. Hünlich