This thesis is a study of three categories of problems in fermionic systems for which topology plays an important role: (i) The properties of zero modes arising in systems of fermions interacting with a bosonic background, with a special focus on Majorana modes arising in the superconductor state. We propose a method for counting Majorana modes and we study a mechanism for controlling their number parity in lattice systems, two questions that are of relevance to the protection of quantum bits. (ii) The study of dispersionless bands in two dimensions as a platform for correlated physics, where it is shown the possibility of stabilizing the fractional quantum Hall effect in a flat band with Chern number. (iii) The extension of the hierarchy of quantum Hall fluids to the case of time-reversal symmetric incompressible ground states describing a phase of strongly interacting topological insulators in two dimensions. / Physics

This thesis describes electronic transport experiments in graphene from the hopping to the ballistic regime. The first experiment studies dual-gated bilayer graphene devices. By applying an electric field with these dual gates, we can open a band gap in bilayer graphene and observe an increase in resistance of over six orders of magnitude as well as a strongly non-linear behavior in the transport characteristics. A temperature-dependence study of resistance at large electric field at the charge neutrality point shows the change in the transport mechanism from a hopping dominated regime at low temperature to a diffusive regime at high temperature. / Physics

Graphene is widely considered to be a promising candidate for a new generation of electronics, but there are many outstanding fundamental issues that need to be addressed before this promise can be realized. This thesis focuses on the production and properties of graphene grown epitaxially on the carbon terminated face (C-face) of hexagonal silicon carbide leading to the construction of a novel graphene transistor structure. C-face epitaxial graphene multilayers are unique due to their rotational stacking that causes the individual layers to be electronically decoupled from each other. Well-formed C-face epitaxial graphene single layers have exceptionally high mobilities (exceeding 10,000 cm^2/Vs), which are significantly greater than those of Si-face graphene monolayers. This thesis investigates the growth and properties of C-face single layer graphene. A field effect transistor based on single layer graphene was fabricated and characterized for the first time. Aluminum oxide or boron nitride was used for the gate dielectric. Additionally, an all graphene/SiC Schottky barrier transistor on the C-face of SiC composed of 2DEG in SiC/Si2O3 interface and multilayer graphene contacts was demonstrated. A multiple growth scheme was adopted to achieve this unique structure.

The primary subject of this thesis is graphene and how the rudimentary attributes of its charge carriers, and local moments on its surface, can be directly manipulated and controlled with electrostatic potentials. We first consider bilayer graphene subject to a spatially varying electrostatic potential that forms two neighbouring regions with opposite interlayer bias. Along the boundary, 1D chiral `kink' states emerge. We find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid whose properties can be tuned via an external gate. Next, we consider superlattices in bilayer graphene. Superlattices are seen to have a more dramatic effect on bilayer graphene than monolayer graphene because the quasiparticles are changed in a fundamental way; the dispersion goes from a quadratic band touching point to linearly dispersing Dirac cones. We illustrate that a 1D superlattice of either the chemical potential or an interlayer bias generates multiple anisotropic Dirac cones. General arguments delineate how certain symmetries protect the Dirac points. We then map the Hamiltonian of an interlayer bias superlattice onto a coupled chain model comprised of `topological' edge modes. We then discuss the relevance of spatially varying potentials to recent transport measurements. This is followed by another study that considers the effect of a magnetic field on graphene superlattices. We show that magnetotransport measurements in a weak perpendicular (orbital) magnetic field probe the number of emergent Dirac points and reveal further details about the dispersion. In the case of bilayer graphene, we also discuss the properties of kink states in an applied magnetic field. We then consider the implications of these results with regards to scanning tunnelling spectroscopy, valley filtering, and impurity induced breakdown of the quantum Hall effect. Finally, we investigate local moment formation of adatoms on bilayer graphene using an Anderson impurity model. We construct various phase diagrams and discuss their many unusual features. We identify regions where the local moments can be turned on or off by applying a external electric fields. Finally, we compute the RKKY interaction between local moments and show how it too can be controlled with electric fields.

The primary subject of this thesis is graphene and how the rudimentary attributes of its charge carriers, and local moments on its surface, can be directly manipulated and controlled with electrostatic potentials. We first consider bilayer graphene subject to a spatially varying electrostatic potential that forms two neighbouring regions with opposite interlayer bias. Along the boundary, 1D chiral `kink' states emerge. We find that these 1D modes behave as a strongly interacting Tomonaga-Luttinger liquid whose properties can be tuned via an external gate. Next, we consider superlattices in bilayer graphene. Superlattices are seen to have a more dramatic effect on bilayer graphene than monolayer graphene because the quasiparticles are changed in a fundamental way; the dispersion goes from a quadratic band touching point to linearly dispersing Dirac cones. We illustrate that a 1D superlattice of either the chemical potential or an interlayer bias generates multiple anisotropic Dirac cones. General arguments delineate how certain symmetries protect the Dirac points. We then map the Hamiltonian of an interlayer bias superlattice onto a coupled chain model comprised of `topological' edge modes. We then discuss the relevance of spatially varying potentials to recent transport measurements. This is followed by another study that considers the effect of a magnetic field on graphene superlattices. We show that magnetotransport measurements in a weak perpendicular (orbital) magnetic field probe the number of emergent Dirac points and reveal further details about the dispersion. In the case of bilayer graphene, we also discuss the properties of kink states in an applied magnetic field. We then consider the implications of these results with regards to scanning tunnelling spectroscopy, valley filtering, and impurity induced breakdown of the quantum Hall effect. Finally, we investigate local moment formation of adatoms on bilayer graphene using an Anderson impurity model. We construct various phase diagrams and discuss their many unusual features. We identify regions where the local moments can be turned on or off by applying a external electric fields. Finally, we compute the RKKY interaction between local moments and show how it too can be controlled with electric fields.

Nesta tese, apresentamos estudos de magnetotransporte em poços quânticos duplos (DQWs) a campos magnéticos de baixo e sob a aplicação de um campo elétrico externo (potencial de porta). Medidas de magnetorresistência foram realizadas tanto no regime linear quanto no regime não linear. Relatamos a observação de oscilações magnéticas de inter-sub-banda (MIS) pela primeira vez. Estas oscilações MIS já foram estudadas em poços quânticos simples (QWs) com duas sub-bandas ocupadas; um DQW´e o sistema mais apropriado para o estudo das oscilações MIS. As oscilações MIS são atribuídas ao espalhamento inter-sub-banda, e a intensidade delas depende da largura da barreira (relacionada ao gap de energia entre as duas sub-bandas ocupadas, SAS). O estudo das oscilações MIS é uma ferramenta importante para poder acessar ao tempo de vida quântico dos elétrons a temperaturas onde as oscilações Shubnikov-de Haas (SdH) já não são observadas. Em nossas amostras, as oscilações MIS persistem até 25 K. Explicamos estes resultados num modelo teórico considerando um potencial de espalhamento de curto alcance com uma contribuição significativa do tempo de espalhamento elástico dos elétrons e uma contribuição do espalhamento elétron-elétron (e-e) com o aumento da temperatura. A aplicação de um campo elétrico externo (correntes dc) modifica fortemente as oscilações MIS. Descrevemos este efeito não linear causado pelo campo elétrico dc com uma função de distribuição oscilatória. Considerando o aquecimento dos elétrons pelo campo elétrico, é extraído o tempo de espalhamento inelástico. Para correntes dc grandes são encontradas discrepâncias entre o experimento e a teoria. Finalmente, consideramos medidas de magnetotransporte como função de potenciais de porta (porta na superficie) levando ao desbalance do DQW. Encontramos que as contribuições clássica e quântica são necessárias para a descrição teórica da magnetorresistência. Descrevemos as contribuições da magnetorresistência em termos das taxas de espalhamento inter e intra sub-banda utilizando uma função gaussiana como função da correlação do potencial. / In this thesis we present studies of magnetotransport in double quantum wells (DQWs) in low magnetic fields and under application of an external electric field (gate potential). Measurements of magnetoresistance have been carried out in both linear and non-linear regime. We report on the observation of magneto-intersubband (MIS) oscillations for the first time. These MIS oscillations have been studied already in quantum wells (QWs) with two occupied subbands, DQW is the most convenient system for studies of MIS oscillations. They are attributed to intersubband scattering and the strength of MIS oscillations depends on the barrier width (´delta´SAS). Analysis of MIS oscillations is an important tool to access quantum lifetime of electrons at high temperatures where Shubnikov-de Haas (SdH) oscillations are already absent. For our samples, MIS oscillations still exist up to 25 K. We explain these results in a theoretical model considering short-range scattering potential with a significant contribution of el´astic scattering time of electrons and a contribution of electron-electron (e-e) scattering if one increases temperature. Application of an external electric field (here a dc currents) strongly modifies the MIS oscillations. We describe this non-linear effect caused by a dc electric field with nonequilibrium part of the electron distribution function. Including the heating of electrons by the electric field, we are able to extract inelastic scattering time. For a strong dc current, a discrepancy between experiment and theory is found. Finally, we consider gate-dependent (top gate) magnetotransport measurements and drive de DQWs out of balance. We find that both cl´assical and quantum contributions are necessary for theoretical description of the magnetoresistance. We express both contributions in terms of inter and intrasubband scattering rates using a gaussian function as correlation function of the potential.

Les progrès des techniques de nanofabrication des dix dernières années ont permis la mise en place de protocoles visant à manipuler les charges uniques dans les nanostructures. Ces nouvelles techniques permettent d'envisager la réalisation d'expériences d'optique quantique avec des électrons. Cette thèse s'inscrit dans ce contexte. Le but de ce travail a été la construction d'un formalisme adapté à la description de telles expériences. Ce formalisme, construit en analogie avec la théorie de la cohérence quantique du champ électromagnétique de Glauber, souligne les similitudes et différences entre les photons se propageant dans le vide, et le transport électronique dans des conducteurs balistiques unidimensionnels. En particulier, il rend compte de la décohérence et de la relaxation en énergie des excitations électroniques en présence d'interactions. Un autre aspect de cette thèse a été la proposition de protocoles permettant de mesurer des quantités directement reliées aux propriétés de cohérence décrites par le formalisme de l'optique quantique électronique. En particulier, un protocole de tomographie quantique reposant sur l'effet Hanbury Brown et Twiss a été proposé pour reconstruire la cohérence à un corps d'une source monoélectronique. Ce protocole peut aussi être envisagé pour obtenir des informations sur les mécanismes de décohérence. / The last ten years saw tremendous progress in nanofabrication techniques. These progresses allowed the realization of experimental protocols aiming at the manipulation of single electrons in nanostructures. Thus, the advent of these technologies permit to envision the realization of electronic analogues of quantum optics experiments. This thesis is devoted to the theoretical study of quantum optics with electrons propagating in quantum Hall edge channels, in analogy with Glauber's theory for the quantum coherence of the electromagnetic field. The proposed formalism underlines the analogies and differences between photons propagating in the vacuum and electrons in ballistic conductors. In particular, it takes into account the decoherence and relaxation of electronic excitations under the influence of a linear electromagnetic environment. All along this thesis, efforts have been made to propose protocols aiming at accessing experimental quantities related to the coherence properties described by the electron quantum optics formalism. A particular example is a single electron quantum tomography protocol which reconstructs the single particle coherence from current noise measurements. This protocol can also be envisioned to probe decoherence mechanisms.

Quantum Hall states are prototypical topological states of matter whose Hall conductance is topologically quantized to an integer or rational fraction multiple of the fundamental conductance quantum. A significant consequence of this quantization is that the Hall conductance value can be made independent of variations from device to device, within acceptable limits. Such topologically quantized properties are thus highly desirable for metrology or industrial purposes. Formulating a microscopic picture of fractional quantum Hall states and the characterization of all topological responses of quantum Hall states are frontier areas of condensed matter research, with far reaching technological consequences such as realizing anyonic topological quantum computation. In this dissertation, I will present my research on these topics.<br>

Die vorliegende Arbeit untersucht Stromdichterverteilungen in Quanten-Hall-Systemen, sowie Transporteigenschaften von translationsinvarianten Systemen und Quantenpunk- tkontakten (QPCs) im Quanten-Hall-Regime. Zunächst soll die beste Näherung zur Beschreibung von Coulomb-Wechselwirkungen identifiziert werden. Hierzu werden die Hartree Näherung, die Lokale Spindichte Näherung (LSDA) und die Hartree-Fock Näherung anhand translationsinvarianter Systeme untersucht. Es stellt sich heraus das in der Hartree Näherung der meiste Strom vom Bulk getragen wird und das die Hinzu- nahme von Austauschwechselwirkung, zunächst in LSDA den Strombeitrag am Rand erhöht, bis schließlich in Hartree-Fock der Strom hauptsächlich von Randzuständen ge- tragen wird und nur noch ein kleiner Anteil vom Bulk. Damit deckt sich Hartree-Fock am ehesten mit dem Bild nicht-wechselwirkender Elektronen und dem Randstrombild, das in der theoretischen Physik immer wieder sehr erfolgreich angewendet wurde und somit wird Hartree-Fock als die beste Näherung identifiziert. Um dies weiter zu untermauern werden Randgeschwindigkeiten in allen Nährungen berechnet, wobei Hartree-Fock sich sehr gut mit experimentellen Werten deckt und Hartree die Randgeschwindigkeiten stark unterschätzt. Weiterhin werden drei bildgebende Verfahren untersucht: Scanning Force Microscopy, Scanning Capacitance Imaging und Scanning Gate Microscopy. In den er- sten beiden Fällen kann mit den erfolgten Rechnungen der Zusammenhang zwischen Messsignal und lokaler Kompressibilität des Systems gezeigt und die Messungen qual- itativ bestätigt werden. Weiterhin wurden zwei numerische Studien zu Scanning Gate Experimenten an QPCs durchgeführt. Da numerisch nur kleinere Systeme als im Ex- periment betrachtet werde können, wird analog zum Experiment mit zwei Referenz- modellen verglichen, beziehungsweise können Verhältnisse zwischen Plateau und kom- pressiblen Bereichen verglichen werden. Bei höheren Füllfaktoren wird gute Überein- stimmung gefunden, während bei niedrigen Füllfaktoren im Experiment Korrelationsef- fekte eine wichtige Rolle zu spielen scheinen und die hier gefundenen Ergebnisse in der Hartree Näherung entfernen sich von den experimentellen Werten. Es wird gezeigt das die breiten Randzustände die im Experiment gemessen werden eine Folge des Zusam- menspiels von Temperatureffekten und Wechselwirkung sind und das keiner der beiden Effekte alleine zu den Beobachtungen im Experiment führt. Außerdem zeigen sich bei sehr niedrigen Temperaturen Oszillationen im Leitwert, hervorgerufen durch Quanten- interferenz, welche generisch für QPCs mit Wechselwirkung zu sein scheinen. Wech- selwirkung führen zu ausgedehnten kompressiblen Bereichen im QPC, wenn sich dieser schließt. Diese können Oszillationen hervorrufen, ähnlich wie im Modell einer Rechteck- Streubarriere, welches schon in der Vergangenheit diskutiert wurde. Abschließend ließ sich zeigen, dass Signaturen für Quanteninterferenz auch bei höheren Temperaturen noch sichtbar sind.

Two-dimensional electron systems in GaAs/AlGaAs heterostructures have provided a platform for investigating numerous phenomena in condensed matter physics. The quantum Hall effect is a particularly remarkable phenomenon due to its topological properties, including chiral edge states with quantized conductance. This report describes progress made in interference measurements of these edge states in electronic Fabry-Perot interferometers. Previous interference experiments in the quantum Hall regime have been stymied by Coulomb charging effects and poor quantum coherence. These Coulomb charging effects have been dramatically suppressed by the implementation of a novel GaAs/AlGaAs heterostructure which utilizes auxiliary screening wells in addition to the primary GaAs quantum well. Using this heterostructure, Aharonov-Bohm interference is measured in very small devices which have greatly improved coherence. Robust Aharonov-Bohm interference is reported at fractional quantum Hall states nu = 1/3 and nu = 2/3. Discrete jumps in phase at nu = 1/3 consistent with anyonic braiding statistics are observed. The report concludes with proposed future experiments, including extending these results to possible non-Abelian quantum Hall states.