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.

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.

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.

This Thesis is devoted to the study of topological phases of matter in optical platforms, focusing on non-Hermitian systems with gain and losses involving nonreciprocal elements, and fractional quantum Hall liquids where strong interactions play a central role.In the first part we investigated nonlinear Taiji micro-ring resonators in passive and active silicon photonics setups. Such resonators establish a unidirectional coupling between the two whispering-gallery modes circulating in their perimeter. We started by demonstrating that a single nonlinear Taiji resonator coupled to a bus waveguide breaks Lorentz reciprocity. When a saturable gain is added to a single Taiji resonator, a sufficiently strong unidirectional coupling rules out the possibility of lasing in one of the whispering-gallery modes with independence of the type of optical nonlinearity and gain saturation displayed by the material. This can be regarded as a dynamical time-reversal symmetry breaking. This effect is further enhanced by an optical Kerr nonlinearity. We showed that both ring and Taiji resonators can work as optical isolators over a broad frequency band in realistic operating conditions. Our proposal relies on the presence of a strong pump in a single direction: as a consequence four-wave mixing can only couple the pump with small intensity signals propagating in the same direction. The resulting nonreciprocal devices circumvent the restrictions imposed by dynamic reciprocity. We then studied two-dimensional arrays of ring and Taiji resonators realizing quantum spin-Hall topological insulator lasers. The strong unidirectional coupling present in Taiji resonator lattices promotes lasing with a well-defined chirality while considerably improving the slope efficiency and reducing the lasing threshold. Finally, we demonstrated that lasing in a single helical mode can be obtained in quantum spin-Hall lasers of Taiji resonators by exploiting the optical nonlinearity of the material. In the second part of this Thesis we dived into more speculative waters and explored fractional quantum Hall liquids of cold atoms and photons. We proposed strategies to experimentally access the fractional charge and anyonic statistics of the quasihole excitations arising in the bulk of such systems. Heavy impurities introduced inside a fractional quantum Hall droplet will bind quasiholes, forming composite objects that we label as anyonic molecules. Restricting ourselves to molecules formed by one quasihole and a single impurity, we find that the bound quasihole gives a finite contribution to the impurity mass, that we are able to ascertain by considering the first-order correction to the Born-Oppenheimer approximation. The effective charge and statistical parameter of the molecule are given by the sum of those of the impurity and the quasihole, respectively. While the mass and charge of such objects can be directly assessed by imaging the cyclotron orbit described by a single molecule, the anyonic statistics manifest as a rigid shift of the interference fringes in the differential scattering cross section describing a collision between two molecules.

<p>Topological materials have been under intense investigation for more than 30 years and have experienced astonishing growth  over the last decade. The two-dimensional electron gas has long served as a model system for the exploration of topological physics, supporting a diverse array of strongly correlated emergent phenomena. Indeed, some of the most stunning topological phases in condensed matter systems are the integer and fractional quantum Hall states forming in two-dimensional electron gases.</p> <p>It was realized early on that electron localization in the bulk has an important role in attaining topological phases, where the sample bulk is well described by randomly localized electrons, known as the Anderson insulator. However, a different type of topological phases forms when charge carriers order in the bulk. Such a charge ordering can only occur in the presence of strong electron-electron interactions and low disorder. Localization of this kind is of a collective nature and differs fundamentally from the single particle physics of the Anderson insulator. The nature of charge ordering, however, is more nuanced than first thought. Indeed, in high Landau levels, Hartree-Fock theories predict the proliferation of numerous exotic bulk insulators, where in the limit of no disorder electrons cluster together and form a hexagonal lattice. Initial observations of these highly correlated insulating phases were limited to low disorder two-dimensional electron gases confined to GaAs/AlGaAs heterostructures. However, recent discoveries of charge ordering in two-dimensional electron gases confined to graphene highlight the universality of this phenomena, irrespective of host material. While progress has been made in understanding the collective insulators residing within integer quantum Hall plateaus, many aspects remain unresolved. In this Dissertation, I discuss the transport properties and energetics of collective insulators forming near integer quantum Hall plateaus in the latest generation of very low disorder two-dimensional electron gases.</p> <p>In chapter  1  I briefly introduce recent developments in our current understanding of the integer quantum Hall effect, where the topological phase is described by both a topological invariant as well as a local order parameter related to the Landau symmetry breaking paradigm. Next, I introduce the basic principles of two-dimensional electron gases confined to semiconductor heterostructures and provide a short summary of recent technological breakthroughs in molecular beam epitaxial growth protocols. The chapter concludes with an introduction to the essential physics of both the integer and the fractional quantum Hall effect.</p> <p>Chapter  2  contains a brief review of the existing literature on the collective insulators forming in sufficiently low disorder two-dimensional electron gases. The primary focus of chapter  2  is on the unique magnetotransport patterns seen at various Landau level filling factors, which support the collective insulator  interpretation. Throughout this chapter I tend to lean on theoretical models that describe these collective phases through the lens of the Hartree-Fock theory; however, it is important to note that both density matrix renormalization group theories and direct diagonalization of small electron systems reach similar conclusions.</p> <p>In chapter  3  I present our data displaying the hallmark transport signatures of a collective insulator residing within the flanks of the nu = 1 integer quantum Hall plateau. Our sample belongs to the latest generation of low disorder 2DEGs confined to GaAs/AlGaAs. I provide a detailed analysis of its development in both temperature and filling factor. The distinct transport signatures we observe strongly overlap in filling factor with prior microwave resonance, surface acoustic wave, compressibility, and tunneling measurements, all of which point to the formation of a collective insulator known as the integer quantum Hall Wigner solid. One puzzling aspect, however, is that while the latter measurements exhibit the integer quantum Hall Wigner solid in older generation samples, transport signatures of this phase appear to be present only in the newest and highest mobility samples. By using distinct features in the magnetoresistance, I propose a stability diagram of the integer quantum Hall Wigner solid in nu −T phase space. Analysis of magnetoresistance profiles at fixed filling factors display sharp peaks within the region of integer quantum Hall Wigner solid phase. It is believed that these sharp peaks are a shared property of collective insulators forming in low disorder two-dimensional electron gases and signal the onset of the electron solid formation. Additional analysis of the magnetoresistance profiles suggests activated transport behavior with a gap energy comparable to that of the plateau center. Lastly, I present large signal measurements of the nu = 1 integer quantum Hall Wigner solid. The data displays strong non-linear behavior in the current-voltage characteristics consistent with the depinning and sliding conduction. However, similar threshold conduction is also seen in the current-voltage characteristics near the center of the integer quantum Hall plateau, where the bulk is an Anderson insulator. Much to our surprise, trends in the threshold current are monotonic in filling factor.</p> <p>In chapter  4  I report on the recent emergence of a newly observed collective insulator residing within the nu = 2 integer quantum Hall plateau and centered at filling factor nu = 1.79. Based on the range of filling factors which stabilizes this collective insulator, we find it distinct from the aforementioned integer quantum Hall Wigner solid. Indeed, the transport behavior is eerily reminiscent to the reentrant insulating phase seen at low filling factors between 1/5 < nu < 2/9. Hence, we term this collective insulator the reentrant integer quantum Hall Wigner solid. Evoking concepts of particle-hole symmetry, we find the reentrant integer quantum Hall Wigner solid to be one member of the larger family of Wigner solids, which is intimately linked through this fundamental symmetry of the system.</p> <p>Lastly in chapter  5 , I explore the energetics of the collective insulators which develop in the N = 2 and N = 3 Landau level, specifically the two- and three-electron bubble phase. We extract the onset temperatures of these exotic bulk insulators from the sharp peaks in the magnetoresistance at fixed filling factor. We compare our measured onset energies with the cohesive energies found from numerical calculations. We find the onset temperatures for the both two- and three-electron bubble phase show an approximately linear trend in filling factor within a single Landau level. In addition, we observe that the three-electron bubble phase has a larger onset temperature than the two-electron bubble phase, a result which is inconsistent with some numerical energy calculations. Thus, our measurements of bubble phase energetics call attention to the importance of the short-range Coulomb interaction in the formation of multi-electron bubble phases and is expected to serve as guide towards the refinement of existing theoretical models.</p>