Stationary solutions of two-dimensional heterogeneous energy

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Stationary solutions of two-dimensional heterogeneous energy models with multiple species near equilibrium

Collaborator: A. Glitzky , R. Hünlich

Cooperation with: L. Recke (Humboldt-Universität zu Berlin)

Supported by: DFG: ``Zur Analysis von thermodynamischen Modellen des Stoff-, Ladungs- und Energietransports in heterogenen Halbleitern'' (Analysis of thermodynamic models for the transport of mass, charge and energy in heterogeneous semiconductors)

Description: We consider a stationary energy model for reaction-diffusion processes of electrically charged species with non-local electrostatic interaction. Such problems arise in electrochemistry as well as in semiconductor device and technology modeling (see, e.g., [2]). But, in contrast to [2] we now additionally take thermal effects into account. We consider a finite number of species X_i, $i=1,\dots,n$ (e.g., electrons, holes, dopants, interstitials, vacancies, dopant-defect pairs). Let $\varphi$ and T be the electrostatic potential and the lattice temperature. We denote by $u_i,\, \zeta_i,\, q_i$ the particle density of the i-th species, its electrochemical potential and its charge number. The state equations are assumed to be given by the ansatz (see [1])

$\begin{displaymath} u_i=\bar u_i(x,T)\,\text{e}^{(\zeta_i-q_i\varphi+E_i(x,T))/T},\quad i=1,\dots,n.\end{displaymath}$

We consider a finite number of reversible reactions of the form

$\begin{displaymath} \alpha_1X_1+\dots +\alpha_{n}X_{n}\rightleftharpoons \beta_1X_1+\dots +\beta_{n}X_{n}.\end{displaymath}$

The set of stoichiometric coefficients $(\alpha,\beta)=(\alpha_1,\dots, \alpha_{n}, \beta_1,\dots ,\beta_{n})$ belonging to all reactions is denoted by ${\cal R}$ .According to the mass-action law the reaction rates $R_{\alpha\beta}$ are prescribed by

$\begin{displaymath} R_{\alpha\beta}=r_{\alpha\beta}(x,u,T)\Big( \text{e}^{\sum_{... ...}^{n}{\beta_i\zeta_i}/T} \Big),\quad (\alpha,\beta)\in{\cal R}.\end{displaymath}$

Here u means the vector $(u_1,\dots,u_{n})$ .For the particle flux densities j_i and the total energy flux density j_e we make the ansatz (see [1])

$\begin{displaymath} \begin{split} j_i&=-\sum_{j=1}^n\sigma_{ij}(x,u,T)\big(\nabl... ...,u,T)\nabla T+\sum_{i=1}^{n}(\zeta_i+P_i(x,u,T)T)j_i\end{split}\end{displaymath}$

with conductivities $\sigma_{ij}$ , $\kappa$ fulfilling

$\begin{displaymath} \sigma_{ij}=\sigma_{ji},\quad \sum_{i,j=1}^n\sigma_{ij}(x,u... ...i^2\quad\forall t\in\IR^n,\quad \kappa(x,u,T) \ge \kappa_0(u,T)\end{displaymath}$

where $\sigma_0(u,T),\,\kappa_0(u,T)\gt$ for all non-degenerated states u, T. For the fluxes $(j_1,\dots,j_{n},j_e)$ and the generalized forces $(\nabla[\zeta_1/T],\dots, \nabla[\zeta_{n}/T],-\nabla[1/T])$ , the Onsager relations are fulfilled. The basic equations of the energy model contain n continuity equations for the considered species, the conservation law of the total energy and the Poisson equation,

$\begin{displaymath} \left. \begin{array} {rcl} \nabla\cdot j_i&=&\displaystyle\s... ...u_i\end{array}\quad\right\}\quad\text{ in }\Omega\subset \IR^2.\end{displaymath}$

(1)

Here $\varepsilon$ is the dielectric permittivity and f represents some fixed charge density. System (1) must be completed by suitable mixed boundary conditions. We introduce the new variables $\lambda=(\lambda_1,\dots,\lambda_{n+2})= (\zeta_1/T,\dots, \zeta_{n}/T,-1/T,\varphi)$ .With some functions H_i we can reformulate the state equations in the form

$\begin{displaymath} u_i(x)= H_i(x,\lambda),\quad i=1,\dots,n.\end{displaymath}$

Also the reaction rates $R_{\alpha\beta}$ are expressed in the new variables

$\begin{displaymath} R_{\alpha\beta}(x,\lambda)= r_{\alpha\beta}(x,H_1(\lambda),\... ...}^{n}{\beta_i\lambda_i}} \Big),\quad (\alpha,\beta)\in{\cal R}.\end{displaymath}$

Then the stationary energy model can be written as a strongly coupled nonlinear elliptic system

$\begin{displaymath} -\,\nabla\cdot\left( \begin{array} {ccccl} a_{11}(\lambda)& ... ...\ f+\textstyle\sum_{k=1}^{n} q_kH_k(\lambda)\end{array}\right).\end{displaymath}$

(2)

Here we have omitted the additional argument x of the coefficient functions. We assume that the Dirichlet parts $\Gamma_D$ and the Neumann parts $\Gamma_N$ of the boundary conditions coincide for all quantities. Then we can formulate the boundary conditions in terms of $\lambda$ ,

$\begin{displaymath} \begin{split} \lambda_i=\lambda_i^D,&\quad i=1,\dots,n+2,\te... ...lon\nabla \lambda_{n+2})=g_{n+2}\text{ on }\Gamma_N.\end{split}\end{displaymath}$

(3)

We assume that $\Omega\cup\Gamma_N$ is regular in the sense of [4] and that the boundary values $\lambda_i^{D}$ , $i=1,\dots,n+2$ , are traces of $W^{1,p}(\Omega)$ functions, p>2. We are looking for solutions of (2), (3) in the form $\lambda=\Lambda+\lambda^D$ .Under weak assumptions on the coefficient functions $a_{ij}(x,\lambda)$ and $\varepsilon(x)$ (such that heterostructures are allowed) we found W^1,q formulations $(q\in(2,p])$ for that system of equations,

$\begin{displaymath} F(\Lambda,\lambda^D,f,g)=0,\quad \Lambda\in W^{1,q}_0(\Omega\cup\Gamma_N)^{n+2}.\end{displaymath}$

If the boundary values $\lambda_i^{D*}$ , $i=1,\dots,n+1$ ,are constants, $\lambda^{D*}_{n+1}<0$ and $\lambda_i^{D*}$ , $i=1,\dots,n$ , correspond to a simultaneous equilibrium of all reactions

$\begin{displaymath} \sum_{i=1}^n \alpha_i\lambda_i^{D*}=\sum_{i=1}^n \beta_i\lambda_i^{D*}\quad \forall(\alpha,\beta)\in {\cal R \;,}\end{displaymath}$

and if $\lambda^{D*}_{n+2}$ , $f^*,\, g^*=(0,\dots,0,g_{n+2}^*)$ are arbitrarily given, then there exists a unique solution $\Lambda^*$ of $F(\Lambda^*,\lambda^{D*},f^*,g^*)=0$ . $\lambda^*=\Lambda^*+\lambda^{D*}$ is a thermodynamic equilibrium of (2), (3). The operator F is continuously differentiable and the linearization $\frac{\partial F}{\partial \Lambda} (\Lambda^*,\lambda^{D*},f^*,g^*)$ turns out to be an injective Fredholm operator of index zero. This follows from results in [5] and from a regularity result of Gröger in [4] for systems of elliptic equations with mixed boundary conditions. Therefore we can apply the Implicit Function Theorem and obtain that for $\lambda^D$ near $\lambda^{D*}$ ,f near f^* and g near g^* the equation $F(\Lambda,\lambda^D,f,g)=0$ has a unique solution $\Lambda$ near $\Lambda^*$ . Thus, near $\lambda^*$ there is a locally unique Hölder continuous solution $\lambda=\Lambda+\lambda^D$ of (2), (3). For details and the precise assumptions of our investigations see [3].

Our local existence and uniqueness result for the stationary energy model (2), (3) works in two space dimensions. But let us note that in our model equations cross-terms with respect to all species and temperature are involved.

References:

G. ALBINUS, H. GAJEWSKI, R. HÜNLICH, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002), pp. 367-383.
A. GLITZKY, Elektro-Reaktions-Diffusionssysteme mit nichtglatten Daten, habilitation thesis, Humboldt-Universität zu Berlin, 2001, Logos Verlag, 2002.
A. GLITZKY, R. HÜNLICH, Stationary solutions of two-dimensional heterogeneous energy models with multiple species near equilibrium, in preparation.
K. GRÖGER, A W^1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), pp. 679-687.
L. RECKE, Applications of the implicit function theorem to quasi-linear elliptic boundary value problems with non-smooth data, Comm. Partial Differential Equations, 20 (1995), pp. 1457-1479.