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Obtention de modèles de diffusion à partir d'équations cinétiques. Modélisation, étude mathématique et simulation

Abstract : This thesis deals with the modelling of electron transport in semiconductors. Diffusion models are commonly used (Drift Diffusion models), but they are not accurate enough for small scales (nanometric semiconductors) and for systems out of equilibrium (transient regimes). Though very precise, kinetic models generally prove too numerically costly to allow for the description of physically realistic situations. Spherical Harmonics Expansion (SHE) models are interesting intermediate models between these two classes of models. SHE models are diffusion models in the position-energy space and are theoretically fitted to the modelling of particle systems close to an equilibrium (non necessarily a Maxwellian equilibrium though, unlike Drift Diffusion models), the thermalization of which is due to elastic collisions. In this thesis, coupled SHE models have been proposed to model particles interacting inelastically with the surrounding medium. A coupled SHE model is introduced to describe electron-phonon interactions in semiconductors. Another coupled SHE model is proposed to model situations where diffusion is generated by particle-wall collisions (between electrons and the frontier of the device, for instance). A numerical study shows that SHE and coupled SHE type models give a reliable description of reality (fitting to kinetic descriptions), even when strongly inelastic collisions are considered. Finally, a hierarchy of quantum SHE models is given for the description of very small length scale systems, when electron transport as well as interactions with the device are governed by quantum mechanics. These models dissipate a quantum entropy.
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Contributor : Jean-Pierre Bourgade <>
Submitted on : Thursday, March 17, 2005 - 7:34:57 PM
Last modification on : Friday, July 24, 2020 - 2:36:02 PM
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  • HAL Id : tel-00008808, version 1


Jean-Pierre Bourgade. Obtention de modèles de diffusion à partir d'équations cinétiques. Modélisation, étude mathématique et simulation. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2004. Français. ⟨tel-00008808⟩



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