Stationary energy models for semiconductor devices

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Stationary energy models for semiconductor devices with incompletely ionized impurities

Collaborator: A. Glitzky, R. Hünlich

Cooperation with: L. Recke (Humboldt-Universität zu Berlin), G. Wachutka (Technische Universität München)

Supported by: DFG: ``Energiemodelle für heterogene Halbleiterstrukturen'' (Energy models for heterogeneous semiconductor structures)

Description:

The charge transport in semiconductor devices is described by the van Roosbroeck equations. 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 $\varphi$ . 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 [1, 7]). 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 X_j, j = 1,..., k. These species exist in different charge states which are transformed into each other by ionization reactions. Let X_j be an acceptor-like impurity and let X^- be its ion. Then we have to consider the reactions

e^- + X_j $\displaystyle \rightleftharpoons$ X_j^-, h⁺ + X_j^- $\displaystyle \rightleftharpoons$ X_j.

(1)

If X_j is a donor-like impurity and X_j⁺ denotes its ion, then the reactions are

e^- + X_j⁺ $\displaystyle \rightleftharpoons$ X_j, h⁺ + X_j $\displaystyle \rightleftharpoons$ X_j⁺.

(2)

Moreover, a (direct) electron-hole recombination-generation

e^- + h⁺ $\displaystyle \rightleftharpoons$ 0

(3)

takes place. If X_j is a donor (an acceptor), we denote by u_2j-1 the density of X_j (of X_j^-) and by u_2j the density of X_j⁺ (of X_j). Furthermore, we define charge numbers as follows:

q_2j-1 : = $\displaystyle \begin{cases} 0&\text{ if $\text{X}_j$ is a donor}\\ -1&\text{ if $\text{X}_j$ is an acceptor} \end{cases}$ , q_2j : = 1 + q_2j-1, j = 1,..., k.

Then the stationary energy model for devices with incompletely ionized impurities has the form

- $\displaystyle \nabla$ ^.( $\displaystyle \varepsilon$ $\displaystyle \nabla$ $\displaystyle \varphi$ ) = f₀ - n + p + $\displaystyle \sum_{{i=1}}^{{2k}}$ q_iu_i, $\displaystyle \nabla$ ^.j_e = 0,

(4)

$\displaystyle \nabla$ ^.j_n = R₀ + $\displaystyle \sum_{{j=1}}^{k}$ R_j1, $\displaystyle \nabla$ ^.j_p = R₀ + $\displaystyle \sum_{{j=1}}^{k}$ R_j2,

(5)

R_j1 = R_j2, u_2j-1 + u_2j = f_j, j = 1,..., k.

(6)

Here j_n, j_p denote the particle flux densities of electrons and holes, R_j1, R_j2, and R₀ denote the reaction rates of the first and second reaction in (1) or in (2), respectively, and of the reaction (3). Moreover, $\varepsilon$ is the dielectric permittivity and f₀, f_j are a charge density and particle densities which must be prescribed. The system has to be completed by suitable state equations for the species e, h, X₁,..., X_2k and kinetic relations.

Special isothermal models of this kind are presented in [6]. There also results of simulations with WIAS-TeSCA are compared with experimental results.

First, under reliable assumptions concerning the state equations and kinetic relations we eliminate the constraints from the system by evaluating the subsystem (6). We end up with a system of four strongly coupled equations with nonlinear source terms depending on $\varphi$ , T, on the electrochemical potentials $\zeta_{n}^{}$ , $\zeta_{p}^{}$ of electrons and holes, on f₀, and on the local invariants f_j, j = 1,..., k. The resulting stationary energy model can be written in the form

- $\displaystyle \nabla$ ^. $\displaystyle \left(\vphantom{ \begin{array}{llll} \varepsilon&0&0&0\\ 0&\kap... ...}\\ 0&\widehat\omega_2&\sigma_{np}&\sigma_{p}+\sigma_{np} \end{array}}\right.$ $\displaystyle \begin{array}{llll} \varepsilon&0&0&0\\ 0&\kappa+\widehat\omega... ...igma_{np}\\ 0&\widehat\omega_2&\sigma_{np}&\sigma_{p}+\sigma_{np} \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{llll} \varepsilon&0&0&0\\ 0&\kap... ...}\\ 0&\widehat\omega_2&\sigma_{np}&\sigma_{p}+\sigma_{np} \end{array}}\right)$ $\displaystyle \left(\vphantom{ \begin{array}{l} \nabla \varphi\\ \nabla T\\ \nabla \zeta_n\\ \nabla \zeta_p \end{array}}\right.$ $\displaystyle \begin{array}{l} \nabla \varphi\\ \nabla T\\ \nabla \zeta_n\\ \nabla \zeta_p \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{l} \nabla \varphi\\ \nabla T\\ \nabla \zeta_n\\ \nabla \zeta_p \end{array}}\right)$ = $\displaystyle \left(\vphantom{ \begin{array}{l} H\\ 0\\ R\\ R \end{array}}\right.$ $\displaystyle \begin{array}{l} H\\ 0\\ R\\ R \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{l} H\\ 0\\ R\\ R \end{array}}\right)$ in $\displaystyle \Omega$ ,

(7)

where

$\begin{displaymath}\begin{split} &\left( \begin{array}{l} \widehat\omega_1\\ \w... ...(\cdot,\varphi,T,\zeta_n,\zeta_p,f_0,f_1,\dots,f_m) \end{split}\end{displaymath}$

with conductivities $\kappa$ > 0, $\sigma_{n}^{}$ , $\sigma_{p}^{}$ > 0, $\sigma_{{np}}^{}$ $\ge$ 0, and transported entropies P_n, P_p depending in a nonsmooth way on x and smoothly on the state variables. Let $\Gamma$ be the boundary of the domain $\Omega$ , which is occupied by the heterogeneous semiconductor device. $\Gamma_{D}^{}$ and $\Gamma_{N}^{}$ denote disjoint, relatively open parts of $\Gamma$ with mes( $\Gamma$ $\setminus$ ( $\Gamma_{D}^{}$ $\cup$ $\Gamma_{N}^{}$ )) = 0. For (7) we suppose mixed boundary conditions of the form

$\displaystyle \varphi$ = v_D1,	T = v_D2,	$\displaystyle \zeta_{n}^{}$ = v_D3,	$\displaystyle \zeta_{p}^{}$ = v_D4	on $\displaystyle \Gamma_{D}^{}$ ,
$\displaystyle \nu$ ^.( $\displaystyle \varepsilon$ $\displaystyle \nabla$ $\displaystyle \varphi$ ) = g₁,	- $\displaystyle \nu$ ^.j_e = g₂,	- $\displaystyle \nu$ ^.j_n = g₃,	- $\displaystyle \nu$ ^.j_p = g₄	on $\displaystyle \Gamma_{N}^{}$ .

(8)

We use the vectors v = ( $\varphi$ , T, $\zeta_{n}^{}$ , $\zeta_{p}^{}$ ), v_D = (v_D1,..., v_D4), g = (g₁,..., g₄), f = (f₀, f₁,..., f_k), and the triplet of data w = (v_D, g, f ) and look for weak solutions of (7),(8) in the form v = V + v^D, where v^D is a function fulfilling the Dirichlet boundary conditions v_D and V represents the homogeneous part of the solution.

We assume that the boundary values v_Di, i = 1, 2, 3, 4, are traces of W^{1, p} functions v_i^D, p > 2 and find W^{1, q} formulations (q $\in$ (2, p]) for that system of equations,

F(V, w) = 0, V $\displaystyle \in$ W^{1, q}₀( $\displaystyle \Omega$ $\displaystyle \cup$ $\displaystyle \Gamma_{N}^{}$ )⁴,

w $\displaystyle \in$ W^{1-1/p, p}( $\displaystyle \Gamma_{D}^{}$ )⁴ x L( $\displaystyle \Gamma_{N}^{}$ )⁴ x L( $\displaystyle \Omega$ ) x {h $\displaystyle \in$ L( $\displaystyle \Omega$ ) : essinf h > 0}^k.

Using techniques from [5], the operator F turns out to be continuously differentiable. If w^* = (v_D^*, g^*, f^*) is fixed such that v_Di^* = const, i = 2, 3, 4, v_D3^* + v_D4^* = 0, v_D2^* > 0, and g^* = (g₁^*, 0, 0, 0), then there exists a solution V^* of F(V^*, w^*) = 0 which represents a thermodynamic equilibrium. We prove that for suitable q > 2 the linearization ${\frac{{\partial F}}{{\partial V}}}$ (V^*, w^*) is a linear isomorphism from W^{1, q}₀( $\Omega$ $\cup$ $\Gamma_{N}^{}$ )⁴ onto W^{1, q'}( $\Omega$ $\cup$ $\Gamma_{N}^{}$ )^*4. Here we use regularity results for strongly coupled elliptic systems with mixed boundary conditions stated in [4]. We apply the Implicit Function Theorem to obtain that for w = (v_D, g, f ) near w^*, the equation F(V, w) = 0 has a unique solution V near V^*. Details and the precise assumptions of our investigations may be found in [3].

In [2], we investigated a more general energy model with m different species where all species were assumed to be mobile such that the stationary energy model corresponds to an elliptic system with m + 2 equations.

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, Banach Center Publ., 66 (2004), pp. 135-151.
, Stationary energy models for semiconductor devices with incompletely ionized impurities, WIAS Preprint no. 1001, 2005, submitted.
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.
R. SIEMIENIEC, W. SÜDKAMP, J. LUTZ, Determination of parameters of radiation induced traps in silicon, Solid-State Electronics, 46 (2002), pp. 891-901.
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