On energy estimates for electro-diffusion equations arising in semiconductor technology

The design of modern semiconductor devices requires the numerical simulation of basic fabrication steps. We investigate some electro-reaction-diffusion equations which describe the redistribution of charged dopants and point defects in semiconductor structures and which the simulations should be based on. Especially, we are interested in pair diffusion models. We present new results concerned with the existence of steady states and with the asymptotic behaviour of solutions which are obtained by estimates of the corresponding free energy and dissipation functionals

Global existence result for pair diffusion models

In this paper we prove a global existence result for pair diffusion models in two dimensions. Such models describe the transport of charged particles in semiconductor heterostructures. The underlying model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs involving drift, diffusion and reaction terms, the corresponding equations for the immobile species are ODEs containing reaction terms only. Forced by applications to semiconductor technology these equations have to be considered with non-smooth data and kinetic coefficients additionally depending on the state variables. Our proof is based on regularizations, on a priori estimates which are obtained by energy estimates and Moser iteration as well as on existence results for the regularized problems. These are obtained by applying the Banach Fixed Point Theorem for the equations of the immobile species, and the Schauder Fixed Point Theorem for the equations of the mobile species

Electro-reaction-diffusion systems in heterostructures

The paper is devoted to the mathematical investigation of a general class of electro-reaction-diffusion systems with nonsmooth data which arises in applications to semiconductor technology. Besides of a basic problem, a reduced problem is considered which is obtained if the kinetics of the free carriers is fast. For two dimensional domains we prove a global existence and uniqueness result. In addition, asymptotic properties of solutions are studied. Basic ideas are energy estimates, Moser iteration, regularization techniques and an existence result for electro-diffusion systems with weakly nonlinear volume and boundary source terms which is proved in the paper, too. The relationship between the property that the energy functional decays exponentially in time to its equilibrium value and the existence of global positive lower bounds for the densities of the species is investigated. We illustrate relations between the model and its reduced version in general and for concrete examples. Finally, we discuss the special features of heterostructures for simplified model problems

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

Electro-reaction-diffusion systems including cluster reactions of higher order

In this paper we consider electro-reaction-diffusion systems modelling the transport of charged species in two-dimensional heterostructures. Our aim is to investigate the case that besides of reactions with source terms of at most second order so called cluster reactions of higher order are involved. We prove the unique solvability of the model equations and show the global boundedness and asymptotic properties of the solution. In order to get necessary a priori estimates we apply an anisotropic iteration scheme followed by usual Moser iterations. Then existence is obtained by cutting off the reaction terms

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 $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ 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 $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem

Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures

We treat a wide class of electro-reaction-diffusion systems with nonsmooth data in two dimensional domains. Forced by applications in semiconductor technology a nonlinear and nonlocal Poisson equation is involved. We state global existence, uniqueness and asymptotic properties of solutions to the evolution problem. Essential tools in our investigations are energetic estimates, Moser iteration, regularization techniques and results for electro-diffusion systems with weakly nonlinear volume and boundary source terms. Especially, we discuss the connection between the existence of global lower bounds for the chemical potentials and the property that the energy functional decays exponentially to its equilibrium value as time tends to infinity

On general boundary value problems and duality in linear elasticity. II

summary:The present part of the paper completes the discussion in Part I in two directions. Firstly, in Section 5 a number of existence theorems for a solution to Problem III (principle of minimum potential energy) is established. Secondly, Section 6 and 7 are devoted to a discussion of both the classical and the abstract approach to the duality theory as well as the relationship between the solvability of Problem III and its dual one

Global properties of pair diffusion models

The paper deals with global properties of pair diffusion models with non-smooth data arising in semiconductor technology. The corresponding model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs containing drift, diffusion and reaction terms. The corresponding equations for the immobile species are ODEs involving reaction terms only. Starting with energy estimates obtained by methods of convex analysis we establish global upper and lower bounds for solutions of the initial boundary value problem. We use Moser iteration for the diffusing species, the non-diffusing species are treated separately. Finally, we study the asymptotic behaviour of solutions