# Analyse mathématique et numérique de modèles quantiques pour les semiconducteurs

Abstract : This thesis is concerned with the mathematical study and the numerical resolution of quantum models of electronic transport in the nanostructures. The model that we use is that of Schrödinger. We took in consideration two approaches. The first is mono-band where two stationary models are studied. The first that we approach takes into account the variation of the effective mass according to semiconductor material. It is the Schrödinger with variable mass model. The second is a model where the non-parabolic effects in the dispersion relation are taken into account. It is the Kohn-Luttinger model. The second approach is a bibande one obtained starting from the Kane model which also rises from the $k.P$ method. The model is the two bands Schrödinger. The theoretical part contains the results of the existence of the solutions (using the fixed point Leray Schauder theorem) and of asymptotic behavior. In the various cases, we derived the transparent boundary conditions. We established a result concerning the semi-classical limit when $\hbar$ tends toward zero of the one dimensional Schrödinger with variable mass model. We showed the existence and uniqueness of solutions only for the energies different of the energies corresponding to eigenvalues of the discrete spectrum of the Kohn-Luttinger operator. We show the existence of solutions of the two bands Schrödinger model in the non-linear case (the electrostatic field is calculated self-consistent). Finally, in the numerical part, we used a Hermitian finite elements for the Kohn-Luttinger model and a finite difference method for the two bands Schrdinger model. For the coupled system, we used an iterative diagram based on Gummel method. We could carry out numerical simulations of devices type intraband (resp. interband) resonant tunneling diode RTD (resp. RITD) to describe the mono-band (resp. multiband) approach. We obtain the characteristics current-voltage, the coefficients of transmission and the profile of the charge density.
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Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00005116
Contributor : Jihene Kefi <>
Submitted on : Wednesday, March 10, 2004 - 4:51:25 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Monday, September 20, 2010 - 11:58:20 AM

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• HAL Id : tel-00005116, version 2

### Citation

Jihene Kefi. Analyse mathématique et numérique de modèles quantiques pour les semiconducteurs. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2003. Français. ⟨tel-00005116v2⟩

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