# Analyse mathématique d'équations de semi-conducteurs avec mobilités non constantes et identification des frontières libres dans les jonctions PN

Abstract : The description of the mechanisms of conduction in the semiconductor devices by drift-diffusion model (DD) leads to a system of three nonlinear partial differential equations strongly coupled. This thesis is composed of three parts. The first is devoted to the setting of the equations and the description of physical parameters as well as the simplification of the model in the case of a junction PN. The second part consists in identifying the depletion region in a junction $pn$. By using the variational inequalities, we show that the problem admits a solution. The numerical originality of this part is the use of the nodes on the free boundaries as unknowns. We propose two algorithms of resolution which we test by using the finite and boundary element methods. In the third part, we are interested in the mathematical analysis of the steady state DD model written with Slotboom variables. We show the existence of a solution, when the laws of mobilities depend on the electric field, by applying the convex analysis techniques. Then, we consider that the term of avalanche is non-zero, we give a priori estimates and we prove an existence result. In order to study the uniqueness, we first expose a condition to be satisfied in order to ensure the uniqueness. This condition is then verified in the case of sufficiently small domain or large enough permittivity. We give a theorem of local uniqueness in the presence of avalanche term and sufficient regularity of solution.
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Cited literature [125 references]

https://tel.archives-ouvertes.fr/tel-00007195
Contributor : Laurent Guillopé <>
Submitted on : Sunday, October 24, 2004 - 11:06:11 AM
Last modification on : Monday, March 25, 2019 - 4:52:05 PM
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• HAL Id : tel-00007195, version 1

### Citation

Abdellatif Ellabib. Analyse mathématique d'équations de semi-conducteurs avec mobilités non constantes et identification des frontières libres dans les jonctions PN. Mathématiques [math]. Université de Nantes, 2000. Français. ⟨tel-00007195⟩

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