Stationary solutions of a two-dimensional heterogeneous

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Stationary solutions of a two-dimensional heterogeneous energy model for semiconductor devices near equilibrium

Collaborator: A. Glitzky, R. Hünlich

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

Description: $\minipage{0.45\textwidth}\Projektbild {0.95\textwidth}{fig_gli_1.eps}% {Schematic picture of a modeled semiconductor device} \label{fig1_gli_1} \endminipage$ 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 Figure 1. Equations for the contacts are substituted by Dirichlet boundary conditions on the two parts of the boundary $\Gamma_{{D0}}^{}$ . In the remaining domain $\Omega$ , involving the passive layer ( $\Omega_{1}^{}$ ) and semiconducting materials ( $\Omega_{0}^{}$ ), we have to formulate a Poisson equation and an energy balance equation with boundary conditions on $\Gamma$ = $\Gamma_{{D0}}^{}$ $\cup$ $\Gamma_{{N0}}^{}$ $\cup$ $\Gamma_{{N1}}^{}$ $\cup$ $\Gamma_{{S}}^{}$ , where the subscripts D, N, and S indicate the parts with Dirichlet, inhomogeneous Neumann, and symmetry boundary conditions, respectively. Only in the part $\Omega_{0}^{}$ , continuity equations for electrons and holes have to be taken into account, and here we must formulate boundary conditions on $\Gamma_{0}^{}$ = $\Gamma_{{D0}}^{}$ $\cup$ $\Gamma_{{N01}}^{}$ $\cup$ $\Gamma_{{N0}}^{}$ $\cup$ $\Gamma_{{S}}^{}$ .

Let T and $\varphi$ 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 $\displaystyle \Big($ $\displaystyle {\frac{{\zeta_n+\varphi-E_n(\cdot,T)}}{{T}}}$ $\displaystyle \Big)$ , p = P(^., T)F $\displaystyle \Big($ $\displaystyle {\frac{{\zeta_p-\varphi+E_p(\cdot,T)}}{{T}}}$ $\displaystyle \Big)$ in $\displaystyle \Omega_{0}^{}$ ,

where n and p are the electron and hole densities, N and P are the effective densities of state, $\zeta_{n}^{}$ and $\zeta_{p}^{}$ are the electrochemical potentials, E_n and E_p 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) = $\cal {F}$ _1/2(y) in the case of Fermi-Dirac statistics). The electrostatic potential $\varphi$ fulfils the Poisson equation

- $\displaystyle \nabla$ ^.( $\displaystyle \varepsilon$ $\displaystyle \nabla$ $\displaystyle \varphi$ ) = $\displaystyle \begin{cases} f-n+p&\quad \text{in }\Omega_0\\ f&\quad \text{in }\Omega_1 \end{cases}$ .

(1)

Here, $\varepsilon$ is the dielectric permittivity and f is a given doping profile. Mixed boundary conditions on $\Gamma$ have to be prescribed. For the densities of the particle fluxes j_n, j_p and of the total energy flux j_e, we make the ansatz (see [1])

$\begin{displaymath}\begin{split} j_n&=-(\sigma_n+\sigma_{np}) (\nabla\zeta_n+P_n... ...abla T ,&\quad \text{in } \Omega_1\\ \end{cases}, \end{split}\end{displaymath}$

with conductivities $\sigma_{n}^{}$ = $\sigma_{n}^{}$ (x, n, p, T) > 0, $\sigma_{p}^{}$ = $\sigma_{p}^{}$ (x, n, p, T) > 0, $\sigma_{{np}}^{}$ = $\sigma_{{np}}^{}$ (x, n, p, T) $\ge$ 0, $\kappa$ = $\kappa$ (x, n, p, T) > 0, $\tilde{\kappa}$ = $\tilde{\kappa}$ (x, T) > 0, and transported entropies P_n = P_n(x, n, p, T), P_p = P_p(x, n, p, T). These flux densities fulfil the balance equations

$\displaystyle \nabla$ ^.j_n = - R, $\displaystyle \nabla$ ^.j_p = - Rin $\displaystyle \Omega_{0}^{}$ , $\displaystyle \nabla$ ^.j_e = 0in $\displaystyle \Omega$ ,

(2)

where the net recombination rate R has the form

R = r(^., n, p, T)(e^(+)/T -1)in $\displaystyle \Omega_{0}^{}$ .

Suitable boundary conditions on $\Gamma_{0}^{}$ for the first two continuity equations and on $\Gamma$ for the last energy balance equation have to be added.

We use the variables z = (z₁, z₂, z₃, z₄) = ( $\zeta_{n}^{}$ /T|, $\zeta_{p}^{}$ /T|, -1/T, $\varphi$ ), where z₁, z₂ are defined on $\Omega_{0}^{}$ , while z₃, z₄ live on $\Omega$ . Then the stationary energy model for semiconductor devices can be written in the more compact form

- $\displaystyle \nabla$ ^. $\displaystyle \left(\vphantom{ \begin{array}{llll} a_{11}&a_{12}&a_{13}&0\\ a... ...}&a_{23}&0\\ a_{31}&a_{32}&a_{33}&0\\ 0&0&0&\varepsilon \end{array}}\right.$ $\displaystyle \begin{array}{llll} a_{11}&a_{12}&a_{13}&0\\ a_{21}&a_{22}&a_{23}&0\\ a_{31}&a_{32}&a_{33}&0\\ 0&0&0&\varepsilon \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{llll} a_{11}&a_{12}&a_{13}&0\\ a... ...}&a_{23}&0\\ a_{31}&a_{32}&a_{33}&0\\ 0&0&0&\varepsilon \end{array}}\right)$ $\displaystyle \left(\vphantom{ \begin{array}{l} \nabla z_1\\ \nabla z_2\\ \nabla z_3\\ \nabla z_4 \end{array}}\right.$ $\displaystyle \begin{array}{l} \nabla z_1\\ \nabla z_2\\ \nabla z_3\\ \nabla z_4 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{l} \nabla z_1\\ \nabla z_2\\ \nabla z_3\\ \nabla z_4 \end{array}}\right)$	= $\displaystyle \left(\vphantom{ \begin{array}{l} -R\\ -R\\ 0\\ f-n+p \end{array}}\right.$ $\displaystyle \begin{array}{l} -R\\ -R\\ 0\\ f-n+p \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{l} -R\\ -R\\ 0\\ f-n+p \end{array}}\right)$	in $\displaystyle \Omega_{0}^{}$ ,
- $\displaystyle \nabla$ ^. $\displaystyle \left(\vphantom{ \begin{array}{ll} \tilde a_{33}&0\\ 0&\varepsilon \end{array}}\right.$ $\displaystyle \begin{array}{ll} \tilde a_{33}&0\\ 0&\varepsilon \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ll} \tilde a_{33}&0\\ 0&\varepsilon \end{array}}\right)$ $\displaystyle \left(\vphantom{ \begin{array}{l} \nabla z_3 \\ \nabla z_4 \end{array}}\right.$ $\displaystyle \begin{array}{l} \nabla z_3 \\ \nabla z_4 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{l} \nabla z_3 \\ \nabla z_4 \end{array}}\right)$	= $\displaystyle \left(\vphantom{ \begin{array}{l} 0\\ f \end{array}}\right.$ $\displaystyle \begin{array}{l} 0\\ f \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{l} 0\\ f \end{array}}\right)$	in $\displaystyle \Omega_{1}^{}$ ,

(3)

with coefficient functions a_ik(x, z) = a_ki(x, z), x $\in$ $\Omega_{0}^{}$ , z $\in$ $\IR^{2}_{}$ x (- $\infty$ , 0) x $\IR$ , i, k = 1,..., 3, $\tilde{a}_{{33}}^{}$ (x, z₃), x $\in$ $\Omega_{1}^{}$ , z₃ $\in$ (- $\infty$ , 0), $\varepsilon$ (x), x $\in$ $\Omega$ , and R, n, and p have to be regarded as functions of x $\in$ $\Omega_{0}^{}$ and z $\in$ $\IR^{2}_{}$ x (- $\infty$ , 0) x $\IR$ .

We consider the boundary conditions

$\begin{displaymath}\begin{split} &z_i=z_i^{D0},\quad i=1,2,3,4,\text{ on }\Gamma... ...cdot(\varepsilon\nabla z_4)=0\text{ on }\Gamma_{S}. \end{split}\end{displaymath}$

(4)

We use the vectors z^D = (z^D₁,..., z^D₄), g = (g₁^N0,..., g₄^N0, g₃^N1, g₄^N1), and the triplet of data w = (z^D, g, f ) and look for weak solutions of (3), (4) in the form z = Z + z^D, where z^D corresponds to a function fulfilling the Dirichlet boundary conditions and Z represents the homogeneous part of the solution.

We assume that the boundary values z_i^D, i = 1, 2, 3, 4, are traces of W^{1, p}-functions, p > 2. Under weak assumptions on the coefficient functions a_ij, $\tilde{a}_{{33}}^{}$ , and $\varepsilon$ (for example, $\Omega_{0}^{}$ can be composed of different semiconducting materials), we found W^{1, q}-formulations (q $\in$ (2, p]) for that system of equations,

F(Z, w) = 0, Z $\displaystyle \in$ W^{1, q}₀( $\displaystyle \Omega_{0}^{}$ $\displaystyle \cup$ $\displaystyle \Gamma_{{N0}}^{}$ $\displaystyle \cup$ $\displaystyle \Gamma_{{N01}}^{}$ $\displaystyle \cup$ $\displaystyle \Gamma_{{S}}^{}$ )² x W^{1, q}₀( $\displaystyle \Omega$ $\displaystyle \cup$ $\displaystyle \Gamma_{{N0}}^{}$ $\displaystyle \cup$ $\displaystyle \Gamma_{{N1}}^{}$ $\displaystyle \cup$ $\displaystyle \Gamma_{{S}}^{}$ )².

If w^* = (z^D*, g^*, f^*) is arbitrarily given such that the boundary values z_i^D*, i = 1, 2, 3, are constants, z^D*₁ + z^D*₁ = 0 and z^D*₃ < 0 and g^* = (0, 0, 0, g₄^N0*, 0, g₄^N1*), then there exists a unique solution Z^* of F(Z^*, w^*) = 0. Then z^* = Z^* + z^D* is a thermodynamic equilibrium of (3), (4). Using techniques from [5], the operator F turned out to be continuously differentiable. For suitable q > 2, we proved that its linearization ${\frac{{\partial F}}{{\partial Z}}}$ (Z^*, w^*) is an injective Fredholm operator of index zero. For this purpose we derived new results concerning W^{1, q}-regularity and surjectivity for strongly coupled systems of linear elliptic equations which are defined on different domains. Here we adapted ideas of [4]. We applied the Implicit Function Theorem and obtained that for w = (z^D, g, f ) near w^*, the equation F(Z, w) = 0 has a unique solution Z near Z^*. Thus, near z^* there is a locally unique Hölder continuous solution z = Z + z^D of (3), (4). For details and the precise assumptions of our investigations see [3].

In [2] we investigated an energy model with multiple species, but there the continuity equations, the energy balance equation, and the Poisson equation were defined on the same domain.

References:

G. ALBINUS, H. GAJEWSKI, R. HÜNLICH, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002), pp. 367-383.
A. GLITZKY, R. HÜNLICH, Stationary solutions of two-dimensional heterogeneous energy models with multiple species, WIAS Preprint no. 896, 2003, to appear in: Banach Cent. Publ.
, Stationary solutions of a two-dimensional heterogeneous energy model for semiconductor devices 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, Commun. Partial Differ. Equations, 20 (1995), pp. 1457-1479.