equation in the semiclassical limit.
Le but de ce chapitre est d’écrire un schéma numérique asymptotiquement stable pour la limite semi-classique sur la formulation fluide de l’équation de Schrödinger, à savoir le système de Madelung (2.88)–(2.90). Ce système consiste en un modèle d’Euler sans pression où un terme quantique additionnel est ajouté: le potentiel de Bohm. Ces équations sont non linéaires contrairement à l’équation de Schrödinger mais leur avan- tage est que les inconnues macroscopiques ne développent pas d’oscillations d’ordre ε (dans ce chapitre, ε est la constante de Planck adimensionnée), ce qui est le cas de la fonction d’onde dans la formulation de Schrödinger. Ceci est un avantage sérieux dans la limite semi-classique où ε tend vers 0. Différentes stratégies de maillage ont été adoptées dans plusieurs articles sur des schémas pour l’équation de Schrödinger [29,30,3] mais même la meilleure méthode nécessite de prendre pour pas d’espace et de temps ∆x = o(ε) et ∆t = O(ε). Plus proche de cette note et en chimie quantique, des méthodes particulaires dans une formulation lagrangienne ont été employées pour résoudre le système de Madelung [6,43].
Dans ce chapitre, nous proposons d’utiliser un schéma semi-implicite qui a le même coût que le schéma explicite pour résoudre le système de Madelung en formulation eulerienne et lagrangienne. Pour un potentiel extérieur nul, l’analyse du schéma pour le système linéarisé autour d’une densité constante et d’un courant nul montre qu’un critère de stabilité est donné par ∆t < ∆xεπ22. Pour palier l’hypothèse de faible courant
et au vu de la même analyse linéaire, le système de Madelung en coordonnées lagrang- iennes est discrétisé de manière analogue.
Les résultats numériques pour le schéma en formulation eulerienne et lagrangienne avec des potentiels extérieurs constants nous confirment que les schémas sont asymp- totiquement stables et que pour un pas d’espace ∆x fixé, on peut augmenter le pas de temps comme 1/ε. L’inconvénient de ces schémas réside dans leur instabilité lorsque la densité est très proche de zéro. Par ailleurs le schéma en formulation eulerienne est instable lorsque le courant est trop fort alors que le schéma en formulation lagrangi- enne dont les coordonnées bougent avec le fluide reste stable. L’étude de problèmes plus complexes faisant intervenir l’apparition de caustiques et/ou faisant intervenir des potentiels extérieurs non constants est en cours.
General introduction (english
version)
In this thesis, we are interested in some very special particle transport models: the quantum fluid models based on the entropy minimization principle. In this introduc- tion, we are going to give first some motivations for studying such models, then we will remind how the models have been derived and for this purpose, we will sum- marize articles [14, 11, 12] which are the foundations of this thesis subject. Finally, we will introduce some notions on semiconductors which constitute a possible field of application of these models.
1
Motivations
Physics being not unified yet, there does not exist a universal theory which would allow to describe any particle system. Instead there exists some different theories which apply in certain particular domains. For instance, classical mechanics devel- oped during the 17th century by Newton is applicable only for particles with small velocities v compared to the speed of light c. Moreover, classical mechanics is only useful at relatively large space scales. In order to describe correctly particles with very high velocities, one should use relativistic mechanics developed by Einstein at the beginning of the 20th century, and for very small space scales, one should use quantum mechanics developed by Bohr, Dirac, de Broglie, Heisenberg, Jordan, Pauli, Planck and Schrödinger in the first quarter of the 20th century. Figure1is describing some theories and their links according to three constants which appear or not in the theories: the speed of light c, the gravitational constant G and the Planck constant ~. In this thesis, we are interested in the quantum theory and its links with the classical one according to what we call the semiclassical limit.
In addition, inside each theory, there exists different description scales. The most precise scale is the microscopic1 (or particle) scale, the intermediate scale is the meso- scopic (or kinetic) scale, and the scale we are interested in in this thesis is the macro- scopic (or fluid) scale. Of course, passing from one scale to another leads to a precision loss, but the modeling becomes less expensive from a numerical point of view.
1
Be careful, words “microscopic”, “mesoscopic” and “macroscopic” do not qualify necessarily the space scale, but the level of description that we choose. It is indeed possible to use a microscopic model in order to describe the planet movements inside the solar system, and we can use a macroscopic model in order to describe electron transport inside an integrated circuit!
Classical mechanics Quantum mechanics
Quantum Gravity Newtonian Gravity
General Relativity
Special Relativity Quantum Field Theory
G
c
Theory of everything?
h
Figure 1: Different theories and their constants, c being the speed of light, G being the gravitational constant and ~ the Planck constant.
Fluid models in classical mechanics have been employed since a long time ago to describe huge particle systems, the most famous being probably the Navier-Stokes model which allows to model air movements in the atmosphere, ocean currents, water flows in a pipe, etc... In the semiconductor industry, the Drift-Diffusion model [10,
19,31,33] is widely used to model electron transport. However, the size of the devices being smaller and smaller (it can reach about 100 nanometers), this model is attaining its limits and quantum effects appear. Some devices (such the resonant tunneling diodes) behaviors are even based on effects that only quantum mechanics can explain. In order to model such devices, very few quantum fluid models exist and it is often compulsory to use microscopic models which are very expansive from a numerical point of view, and which do not take into account collisions. The only existing fluid models are often classical fluid models with additional quantum correction terms.
Models studied in this thesis have been derived in 2003 [14] by Degond and Ringhofer and in 2005 [11] by Degond, Méhats and Ringhofer and are fully quan- tum. Table 1 gives different classical and quantum models for particle transport at the three different scales described above. The four models studied in this thesis are quantum and macroscopic:
1. the Quantum Drift-Diffusion (QDD) model, 2. the Isothermal Quantum Euler model,
3. the Quantum Energy Transport(QET) model, 4. the Quantum Hydrodynamic (QHD) model.
The goal of this thesis is to study more in detail these models and to implement them numerically. Before giving the results obtained in this thesis, we are going to
Classical Quantum
Microscopic Newton Schrödinger
Mesoscopic Vlasov/Boltzmann Liouville/Wigner Drift-Diffusion Quantum Drift-Diffusion Macroscopic Isothermal Euler Isothermal Quantum Euler
Energy Transport Quantum Energy Transport Hydrodynamic Quantum Hydrodynamic
Table 1: Some classical and quantum models describing particle systems at different scales.
remind how these models have been derived. To this aim, we are going to derive the macroscopic models from the microscopic ones in the classical setting first. This will introduce the hydrodynamic and diffusive limits and we will give then the derivation in the quantum setting which is based on the same method.