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Theses

Etude mathématique et numérique de modèles de transport : application à la spintronique

Abstract : This thesis is decomposed into three parts. The main part is devoted to the study of spin polarized currents in semiconductor materials. An hierarchy of microscopic and macroscopic models are derived and analyzed. These models take into account the spin relaxation and precession mechanisms acting on the spin dynamics in semiconductors. We have essentially two mechanisms : the spin-orbit coupling and the spin-flip interactions. We begin by presenting a semiclassical analysis (via the Wigner transformation) of the Schr¨odinger equation with spin-orbit hamiltonian. At kinetic level, the spinor Vlasov (or Boltzmann) equation is an equation of distribution function with 2×2 hermitian positive matrix value. Starting then from the spinor form of the Boltzmann equation with different spin-flip and non spin-flip collision operators and using diffusion asymptotic technics, different continuum models are derived. We derive drift-diffusion, SHE and Energy-Transport models of two-components or spin-vector types with spin rotation and relaxation effects. Two numerical applications are then presented : the simulation of transistor with spin rotational effect and the study of spin accumulation effect in inhomogenous semiconductor interfaces. In the second part, the diffusion limit of the linear Boltzmann equation with a strong magnetic field is performed. The Larmor radius is supposed to be much smaller than the mean free path. The limiting equation is shown to be a diffusion equation in the parallel direction while in the orthogonal direction, the guiding center motion is obtained. The diffusion constant in the parallel direction is obtained through the study of a new collision operator obtained by averages of the original one. Moreover, a correction to the guiding center motion is derived. In the third part of this thesis, we are interested in the description of the confinement potential in two-dimensional electron gases. The stationary one dimensional Schr¨odinger?Poisson system on a bounded interval is considered in the limit of a small Debye length (or small temperature). Electrons are supposed to be in a mixed state with the Boltzmann statistics. Using various reformulations of the system as convex minimization problems, we show that only the first energy level is asymptotically occupied. The electrostatic potential is shown to converge towards a boundary layer potential with a profile computed by means of a half space Schrödinger?Poisson system.
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Submitted on : Wednesday, November 26, 2008 - 7:39:18 PM
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  • HAL Id : tel-00342139, version 1

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Raymond El Hajj. Etude mathématique et numérique de modèles de transport : application à la spintronique. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2008. Français. ⟨tel-00342139⟩

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