Die vorliegende Arbeit befasst sich mit der Anwendung des kürzlich entwickelten 'Functional Approach' zur Elektrodynamik in Medien auf das Modell des freien homogenen Elektronengases. Basierend auf einer ausschließlich mikroskopischen Feldtheorie wird gezeigt, dass mittels universell gültiger Relationen zwischen Antwortfunktionen sowohl alle relevanten optischen als auch magnetischen (linearen) Materialeigenschaften allein aus der Strom-Strom-Korrelation gewonnen werden können. Dabei ist es essentiell, alle Berechnungen auf dem vollen Stromdichteoperator aufzubauen, also auf der Summe aus diamagnetischem, orbitalem und spinoriellem Anteil. Weiterhin wird anhand der magnetischen Suszeptibilität demonstriert, dass im Allgemeinen die Unterscheidung zwischen eigenen und direkten Antwortfunktionen nicht zu vernachlässigen ist. Schließlich wird mit dem „Lindhard-Integral-Theorem“ bewiesen, dass nicht nur der longitudinale, sondern auch der transversale Anteil des vollen frequenz- und wellenvektorabhängigen fundamentalen Antworttensors des freien Elektronengases komplett durch das charakteristische Lindhard-Integral bestimmt ist.:Introduction I Microscopic electrodynamics in media 1 Classical electrodynamics 1.1 Covariant formulation 1.2 Temporal gauge 1.3 Free Green function 1.4 Total functional derivatives 2 Electrodynamics in media 2.1 Field identifications 2.2 Fundamental response tensor 2.3 Universal response relations 2.4 Direct and proper response 2.5 Isotropic and combined limits 2.6 Full Green function 2.7 Wave equations in media and dispersion relations II Application to the free electron gas 3 Fundamental response tensor 3.1 Electromagnetic current density 3.2 Kubo-Greenwood formulae 3.3 Diamagnetic, orbital and spinorial contribution 3.4 Spin susceptibility vs. spinorial current response 4 London model and diamagnetic response 4.1 Interpretation as response function 4.2 Application of universal response relations 4.3 Spin correction 5 Full current response 5.1 Dimensionless formulae 5.2 Lindhard integral theorem 5.3 Laurent expansions 5.4 Optical properties 5.5 Magnetic properties Conclusion Appendix A - Notation Appendix B - Formulary B.1 Basic analysis and vector calculus B.2 Special relativity theory B.3 Fourier transformation B.4 Functional derivatives B.5 Projectors and Helmholtz' theorem B.6 Complex analysis Appendix C - Yang-Mills gauge theory C.1 Field strength tensor C.2 Minimal coupling principle C.3 Gauge invariant quantities and equations Appendix D - Periodic solids D.1 Partitioning of reciprocal space D.2 Homogeneous limit Appendix E - Electromagnetic spectrum Bibliography Acknowledgements Errata / This thesis is concerned with the application of the recently developed 'Functional Approach' to electrodynamics of media to the model of the free homogeneous electron gas. Based on an exclusively microscopic field theory it is shown that with the help of universally valid relations between response functions, all relevant optical and magnetic (linear) materials properties can be extracted from the mere current-current response. For this purpose, it is essential to base all calculations on the full current density operator, i.e. the sum of diamagnetic, orbital and spinorial contributions. Furthermore, we use the example of the magnetic susceptibility to demonstrate that the distinction between proper and direct response functions is in general crucial. Lastly, with the “Lindhard integral theorem” we prove that not only the longitudinal but also the transverse part of the full frequency- and wavevector-dependent fundamental response tensor of the free electron gas is completely determined by the characteristic Lindhard integral.:Introduction I Microscopic electrodynamics in media 1 Classical electrodynamics 1.1 Covariant formulation 1.2 Temporal gauge 1.3 Free Green function 1.4 Total functional derivatives 2 Electrodynamics in media 2.1 Field identifications 2.2 Fundamental response tensor 2.3 Universal response relations 2.4 Direct and proper response 2.5 Isotropic and combined limits 2.6 Full Green function 2.7 Wave equations in media and dispersion relations II Application to the free electron gas 3 Fundamental response tensor 3.1 Electromagnetic current density 3.2 Kubo-Greenwood formulae 3.3 Diamagnetic, orbital and spinorial contribution 3.4 Spin susceptibility vs. spinorial current response 4 London model and diamagnetic response 4.1 Interpretation as response function 4.2 Application of universal response relations 4.3 Spin correction 5 Full current response 5.1 Dimensionless formulae 5.2 Lindhard integral theorem 5.3 Laurent expansions 5.4 Optical properties 5.5 Magnetic properties Conclusion Appendix A - Notation Appendix B - Formulary B.1 Basic analysis and vector calculus B.2 Special relativity theory B.3 Fourier transformation B.4 Functional derivatives B.5 Projectors and Helmholtz' theorem B.6 Complex analysis Appendix C - Yang-Mills gauge theory C.1 Field strength tensor C.2 Minimal coupling principle C.3 Gauge invariant quantities and equations Appendix D - Periodic solids D.1 Partitioning of reciprocal space D.2 Homogeneous limit Appendix E - Electromagnetic spectrum Bibliography Acknowledgements Errata

No description available.

The exact exchange-correlation kernel is a functional derivative of the exact time-dependent exchange-correlation (XC) potential with respect to the time-dependent density, evaluated at the ground-state density. As the XC potential is not known, the exact kernel is also unavailable. Therefore, it must be modeled either using many-body perturbation theory or by satisfying the exact constraints for various prototype systems such as the paradigm uniform electron gas (UEG). The random phase approximation (RPA) neglects the kernel, therefore, fails to provide the accurate ground- and excited-state properties for various systems from a simple uniform electron gas to more complex periodic ones. There are numerous corrections to RPA available, including kernel-corrected RPA, often called the beyond-RPA (bRPA) methods. In this work, we employed various bRPA methods for a diverse set of systems together with RPA. At first, we applied RPA based methods to study the phase stability of the cesium halides. Cesium halides phase stability is one of the stringent tests for a density functional approximation to assess its accuracy for dispersion interaction. Experimentally, CsF prefers the rocksalt (B1) phase, while the other halides CsCl, CsBr, and CsI prefer the cesium chloride (B2) phase. Without dispersion interaction, PBE and PBE0 predict all halides to prefer the B1 phase. However, all RPA based methods predict the experimental observations. The bRPA methods usually improve the quantitative prediction over RPA for the ground-state equilibrium properties of cesium halides. Next, we explored binary intermetallic alloys, where we showed that RPA successfully predicts the accurate formation energies of weakly bonded alloys. However, a kernel corrected RPA is needed when dealing with strongly bonded alloys with partially filled d-band metals. We utilized the renormalized ALDA (rALDA) and rAPBE kernel as bRPA methods. Exact constraints and appropriate norms such as the uniform electron gas are very useful to construct various approximations for the exchange-correlation potentials in the ground-state, and the exchange-correlation kernel in the linear-response theory within the TDDFT. These mathematical formulations not only guide us to formulate more robust nonempirical methods, but they also have more predictive power. We showed the importance of these constraints by calculating plasmon dispersion of the uniform electron gas using the non-local, energy-optimized (NEO) kernel using only a few constraints. More predictive power comes with more constraint satisfaction. As a result, we developed a new wavevector- and frequency-dependent exchange-correlation kernel that satisfies all the constraints that it should satisfy with a real frequency. It gives accurate ground-state correlation energy and describes the charge density wave in low-density UEG. It also predicts an accurate plasmon dispersion with a finite lifetime at wavevectors less than the critical one, where the plasmon dispersion meets the electron-hole continuum. / Physics

No description available.

Nous avons étudiés des fils quantiques dopés de semi-conducteurs gravés par spectroscopie de diffusion Raman. Nous avons observés les excitations du gaz d'électrons. Celles-ci présentent des règles de sélection différentes de celles établies pour les systèmes bi-dimensionnels. Nous avons montré théoriquement qu'elles proviennent de la modification de la structure du champ électromagnétique local provoquée par la géométrie particulière des fils gravés. Pour cela nous avons dû calculer le champ local et l'introduire dans la section efficace de diffusion Raman pour en déduire les règles de sélection de toutes les excitations. Cela a permis de déterminer sans équivoque la nature des excitations qui sont des plasmons. Aucune excitations à une particule ni fluctuations de densité de spin n'a été observées. Nous avons étudié l'évolution continue des dispersions de ces plasmons lorsque la largeur du fil est réduite de 1 micromètre à 30 nm. Jusqu'à 60 nm, celles-ci sont en très bon accord avec les résultats d'un modèle hydrodynamique. Au dessous de 60 nm, la comparaison avec un modèle RPA s'impose. Le plasmon intra-bande dispersif est observé jusqu'à 45 nm, largeur en dessous de laquelle les spectres Raman sont dominés par des excitations localisées qui nécessitent une analyse ultérieure pour en déterminer clairement leur nature. Nous montrons à l'aide du modèle RPA que nous avons atteint la limite quantique pour un fil de largeur 55 nm. Une gamme étroite de fils dont les largeurs sont comprises entre 55 nm à 45 nm permet donc l'étude de gaz strictement unidimensionnel.<br />Nous avons cherché à déterminer la contribution de la forte illumination dans les conclusions précédentes. Nous avons utilisé pour cela la spectroscopie de magnéto-transmission infra-rouge qui ne modifie pas les conditions d'équilibre du gaz d'électrons. Une largeur critique de 130 nm a été extraite, en dessous de laquelle nous n'avons plus aucun signe de la présence d'électrons libres. La comparaison des mesures Raman et infra-rouge a permis l'établissement et la validation d'un modèle microscopique du potentiel de confinement présent dans les fils. Enfin nous avons fabriqués des échantillons de géométries plus complexes. L'observation et l'analyse par diffusion Raman des plasmons dans ces fils a montré que nous pouvions contrôler la géométrie du potentiel confinant les électrons et a mis en évidence des effets nouveaux tels que le repliement et le confinement de plasmons unidimensionnels.

Les propriétés d'un gaz électronique bidimensionnel soumis à champs magnétiques intenses et à bas champs magnétiques sont étudiés par la spectroscopie optique: l'effet Hall quantique fractionnaire (FQHE) par photoluminescence et diffusion inélastique de la lumière, puis un nouvel effet oscillatoire de la résistance induit par micro-ondes (MIROs) par transport et absorption des micro-ondes. Les effets des interactions entre électrons du 2DEG sont à l'origine de FQHE. Jusqu'à maintenant, les expériences d'optiques n'ont pas permis les études des interactions entre électrons sur toute la gamme de fractions. Les fractions 1/3, 2/5, 3/7, 3/5, 2/3 et 1 sont clairement observées dans les spectres non traités et montrent une symétrie autour du facteur de remplissage 1/2. La symétrie des fermions composites ets observée dans les spectres. A bas champ magnétique, sous irradiation micro-onde, les propriétés de transport s'écartent nettement des oscillations bien connues de Shubnikov - de Haas pour évoluer vers une série d'états de résistance zéro. Les résultats des mesures d'absorption des micro-ondes sont présentés pour deux échantillons. L'échantillon de basse mobilité montre seulement une absorption autour de la résonance cyclotron (CR). L' échantillon de haute mobilité montre aussi des signaux d'absorption aux harmoniques de la CR. Les mesures ont permis d'inférer l'existence de deux processus d'absorption différents et séparés. L'absorption non résonante est mieux visible en transport et observée comme MIROs, et l'absorption résonante, mieux observée dans les mesures d'absorption, suit probablement les règles de polarisation de résonance cyclotron.

This work is concerned with the theoretical description of the Scanning Gate Microscopy (SGM) in general and with solving particular models of the quantum point contact (QPC) nanostructure, analytically and numerically. SGM is an experimental technique, which measures the conductance of a nanostructure, while a charged AFM tip is scanned above its surface. It gives many interesting results, such as lobed and branched images, interference fringes and a chequerboard pattern. A generally applicable theory, allowing for unambiguous interpretation of the results, is still missing. Using the Lippman-Schwinger scattering theory, we have developed a perturbative description of non-invasive SGM signal. First and second order expressions are given, pertaining to the ramp- and plateau-regions of the conductance curve. The maps of time-reversal invariant (TRI) systems, tuned to the lowest conductance plateau, are related to the Fermi-energy charge density. In a TRI system with a four-fold spatial symmetry and very wide leads, the map is also related to the current density, on any plateau. We present and discuss the maps calculated for two analytically solvable models of the QPC and maps obtained numerically, with Recursive Green Function method, pointing to the experimental features they reproduce and to the fundamental difficulties in obtaining good plateau tuning which they reveal.

The charge and spin density correlations in a one dimensional electron gas (1DEG) confined in a semiconductor quantum wire structure at zero temperature are studied. The dielectric formulation of the many--body problem is employed and the longitudinal dielectric function, local-field correction, static structure factor, pair correlation function, ground state energy, compressibility, spin-dependent effective interaction potentials, paramagnon dispersion and static spin response function of the 1DEG are computed within the self-consistent field approximations of Singwi et al., known as the STLS and SSTL. The results are compared with those of other groups, and those obtained for two-dimensional electron gas systems whenever it is possible. It is observed that the SSTL satisfies the compressibility sum rule better than the STLS. Calculating the ground state energy of the 1DEG in unpolarized and fully polarized states, it is shown that both STLS and SSTL predict a Bloch transition for 1DEG systems at low electron densities. Finally, the coupled plasmon-phonon modes in semiconductor quantum wires are calculated within the Fermi and Luttinger liquid theories. The coupling of electrons to bulk longitudinal optical phonons without dispersion and to acoustic phonons via deformation potential with a linear dispersion are considered. Using the dielectric formalism, a unified picture of the collective coupled plasmon-phonon modes is presented. Considerable differences between the predictions of the Fermi and Luttinger liquid approaches at large wave vector values, which may be observed experimentally, are found.

Ce travail porte sur la description théorique de la microscopie à grille locale (SGM) et sur la résolution de modèles particuliers de contacts quantiques (QPC), analytiquement et numériquement. SGM est une technique expérimentale, qui mesure la conductance d'une nanostructure, lorsqu'une pointe de microscope a force atomique chargée balaye la surface, sans contacter cette dernière. Les images de SGM révèlent de nombreuses traits intéressants, tels que des lobes, des branches, des franges d'interférence et des motifs de damier. Aucune théorie généralement applicable, donnant une interprétation univoque, n’est disponible à ce jour. En utilisant la théorie de la diffusion de Lippman–Schwinger, nous avons développé une description perturbative de signal de SGM non invasive. Les expressions du premier et du second ordre ont été données, se rapportant aux régions de marche et de plateau de la courbe de conductance. Dans les systèmes invariants par renversement du temps (TRI), adaptés au premier plateau de conductance, les images SGM sont liées à la densité de charge à l`énergie de Fermi. Dans un système TRI, avec une symétrie spatiale centrale et de très larges contacts, les images sont aussi liées à la densité de courant, quelque soit le plateau. Nous présentons et discutons les images calculées pour deux modèles analytiques de QPC et les images obtenues numériquement avec la méthode des fonctions de Green récursives, reproduisant certains motifs observés expérimentalement, et pointant les difficultés fondamentales a se bien positionner sur le plateau de conductance. / This work is concerned with the theoretical description of the Scanning Gate Microscopy (SGM) in general and with solving particular models of the quantum point contact (QPC) nanostructure, analytically and numerically. SGM is an experimental technique, which measures the conductance of a nanostructure, while a charged AFM tip is scanned above its surface. It gives many interesting results, such as lobed and branched images, interference fringes and a chequerboard pattern. A generally applicable theory, allowing for unambiguous interpretation of the results, is still missing. Using the Lippman-Schwinger scattering theory, we have developed a perturbative description of non-invasive SGM signal. First and second order expressions are given, pertaining to the ramp- and plateau-regions of the conductance curve. The maps of time-reversal invariant (TRI) systems, tuned to the lowest conductance plateau, are related to the Fermi-energy charge density. In a TRI system with a four-fold spatial symmetry and very wide leads, the map is also related to the current density, on any plateau. We present and discuss the maps calculated for two analytically solvable models of the QPC and maps obtained numerically, with Recursive Green Function method, pointing to the experimental features they reproduce and to the fundamental difficulties in obtaining good plateau tuning which they reveal.

Two dimensional (2D) systems with low carrier density is an outstanding platform for studying a wide spectrum of physics. These include both classical and quantum effects, arising from disorder, Coulomb interactions and even non-trivial topological properties of band-structure. In this thesis, we have explored the physics at low carrier number density in GaAs/AlGaAs heterostructure and bilayer graphene, by investigating in a larger phase space using a variety of electrical measurement tools. A two-dimensional electron system (2DES) formed in a GaAs/AlGaAs heterostructure offers an avenue to build a variety of mesoscopic devices, primarily because its surface gates can very effectively control its carrier density profile. In the first half of the thesis, we study the relevance of disorder in two kinds of devices made in a 2DES. A very strong negative gate voltage not only reduces the carrier density of the 2DES, but also drives it to a disordered state. In this state, we explore a new direction in parameter space by increasing in-plane electric field and investigating its magneto-resistance (MR). At sufficiently strong gate voltage and source-drain bias, we discover a remarkably linear MR. Its enormous magnitude and weak temperature dependence indicate that this is a classical effect of disorder. In another study, we examine a specially designed dual-gated device that can induce low number density in a periodic pattern. By applying appropriate gate voltages, we demonstrate the formation of an electrostatically tunable quantum dot lattice and study the impact of disorder on it. This work is important in paving way for solid state based platform for experimental simulations of artificial solids. The most striking property of bilayer graphene is the ability to open its band gap by a perpendicular electric field, giving the prospects of enabling a large set of de-vice applications. However, despite a band gap, a number of transport mechanisms are still active at very low densities that range from hopping transport through bulk to topologically protected 1D transport at the edges or along 1D crystal dislocations. In the second half of the thesis, we have used higher order statistical moment of resistance/conductance fluctuations, namely the variance of the fluctuations, to complement averaged resistance/conductance, and study and infer the dominant transport mechanism at low densities in a gapped bilayer graphene. Our results show possible evidence of percolative transport and topologically protected edge transport at different ranges of low number densities. We also explore the same phase space by studying its mesoscopic conductance fluctuations at very low temperatures. This is the first of its kind systematic experiment in a dual-gated bilayer graphene device. Its conductance fluctuations have several anomalous features suggesting non-universal behaviour which is at odds with conventional disordered systems.