An entropic Quantum Drift-Diffusion model for electron transport in resonant tunneling diodes

International audienceWe present an entropic Quantum Drift Diffusion model (eQDD) and show how it can be derived on a bounded domain as the diffusive approximation of the Quantum Liouville equation with a quantum BGK operator. Some links between this model and other existing models are exhibited, especially with the Density Gradient (DG) model and the Schrödinger-Poisson Drift Diffusion model (SPDD). Then a finite difference scheme is proposed to discretize the eQDD model coupled to the Poisson equation and we show how this scheme can be slightly modified to discretize the other models. Numerical results show that the properties listed for the eQDD model are checked, as well as the model captures important features concerning the modeling of a resonant tunneling diode. To finish, some comparisons between the models stated above are realized

Wave propagation: laser propagation and quantum transport

2021 Spring.Includes bibliographical references.This dissertation consists of two independent projects, where wave propagation is the common theme. The first project considers modeling the propagation of laser light through the atmosphere using an approximation procedure we call the variational scaling law (VSL). We begin by introducing the Helmholtz equation and the paraxial approximation to the Helmholtz equation, which is the starting point of the VSL. The approximation method is derived by pairing the variational formulation of the paraxial Helmholtz equation with a generalized Gaussian ansatz which depends on the laser beam parameters. The VSL is a system of stochastic ODEs that describe the evolution of the Gaussian beam parameters. We will conclude with a numerical comparison between the variational scaling law and the paraxial Helmholtz equation. Through exploring numerical examples for increasing strengths of atmospheric turbulence, we show the VSL provides, at least, an order-one approximation to the paraxial Helmholtz equation. The second project focuses on quantum transport by numerically studying the quantum Liouville equation (QLE) equipped with the BGK-collision operator. The collision operator is a relaxation-type operator which locally relaxes the solution towards a local quantum equilibrium. This equilibrium operator is nonlinear and is obtained by solving a moment problem under a local density constraint using the quantum entropy minimization principle introduced by Degond and Ringhofer in \cite{degondringhofer}. A Strang splitting scheme is defined for the QLE in which the collision and transport of particles is treated separately. It is proved that the numerical scheme is well-defined and convergent in-time. The splitting scheme for the QLE is applied in a numerical study of electron transport in different collision regimes by comparing the QLE with the ballistic Liouville equation and the quantum drift-diffusion model. The quantum drift-diffusion model is an example of a quantum diffusion model which is derived from the QLE through a diffusive limit. Finally, it is numerically verified that the QLE converges to the solution to the quantum drift-diffusion equation in the long-time limit

Mathematical modeling and numerical simulation of innovative electronic nanostructures

Dans cette thèse, nous nous intéressons à la modélisation et la simulation de dispositifs nanoélectroniques innovants. Premièrement, nous dérivons formellement un modèle avec masse effective pour décrire le transport quantique des électrons dans des nanostructures très fortement confinées. Des simulations numériques illustrent l'intérêt du modèle obtenu pour un dispositif simplifié mais déjà significatif. La deuxième partie est consacrée à l'étude du transport non ballistique dans ces mêmes structures confinées. Nous analysons rigoureusement un modèle de drift-diffusion et puis nous décrivons et implémentons une approche de couplage spatial classique-quantique. Enfin, nous modélisons et simulons un nanodispositif de spintronique. Plus précisement, nous étudions le renversement d'aimantation dans un matériau ferromagnétique multi-couches sous l'effet d'un courant de spin.In this PhD thesis, we are interested in the modeling and the simulation of innovative electronic nanodevices. First, we formally derive an effective mass model describing the quantum motion of electrons in ultra-scaled confined nanostructures. Numerical simulations aim at testing the relevance of the obtained model for a simplified (but already significant) device. The second part is devoted to non-ballistic transport in these confined nanostructures. We rigorously analyse a drift-diffusion model and afterwards we describe and implement a classical-quantum spatial coupling approach. In the last part, we model and simulate a spintronic nanodevice. More precisely, we study the magnetization switching of a ferromagnetic material driven by a spin-current