Rozprawy doktorskie na temat „Quantum Hall regime”

This research develops a theoretical model to explain the behaviour of the thermo-power in the quantum Hall regime. It uses the concept that at low temperatures the transport through the system will be caused by thermal activation as well as that caused by the conductance. The model is built up in stages, starting with proving the assumption that Dykhne's theorem will work for an asymmetric distribution of particle transport through the system and deriving the behaviour of the particles in the edge states of the system. It then combines this information with a previously developed simple model for the bulk of the modulation-doped GaAs/AlGaAs heterostructure and compares this with experimental data. This reveals that this simple system is not a viable model to represent the data, and as such the model is made more complex with the inclusion of tunnelling. The different parameters which describe the model are found, the saddle energy gap , the transition value for the edge states c, the current splitting parameter and the tunnelling parameter . This is done either by extracting them from the experimental data, or in the case of considering it as a free parameter. How these values vary with the temperature is investigated before a comparison of the theoretical model including tunnelling is conducted with the experimental data. The result from the comparison show a promising alignment between the model and experiment, and further work is proposed where is no longer considered a constant.

This thesis discusses the development of a novel scanning capacitance microscope (SCM) that enables the investigation of the local capacitance and conductivity of surfaces and near-surface nanostructures at cryogenic temperatures and high magnetic fields. Simultaneous atomic force microscopy (AFM) and SCM measurements can be made at a temperature of 1.5K and a magnetic field of 12T. The AFM/SCM sensor is based on a quartz-tuning fork with an etched metal tip. SCM measurements are made using an RF tuned filter design which allows changes in capacitance to be measured with sub-attofarad resolution and a bandwidth of 200Hz. Test measurements were made over an evaporated gold film. The capacitance distance curve was recovered from the measured quantities using a deconvolution scheme normally used for force-distance curves. Measurements have been made of a two-dimensional electron gas in the quantum Hall effect (QHE) regime. Highly conductive stripes form near the edge of the sample at integer Landau level filling factors in agreement with theoretical predictions. These measurements are the first direct imaging of the compressible stripes at the physical edge of a Hall bar device. Measurements were also made by point spectroscopy in a region that was locally depleted. Around this region a ring-shaped stripe of considerably larger width than at the sample edge is observed. The increased width was explained in terms of a shallower potential gradient compared to the physical edge of the sample. Preliminary measurements have demonstrated that the microscope is capable of imaging edge states whilst passing current through the device.

This is an experimental thesis, intended to discover new features of the Fractional Quantum Hall Effect (FQHE), by the fabrication of lithographic microstructures. It begins with an exposition of the theory required to understand electrical transport in Gallium Arsenide high mobility semiconductor heterostructures at low temperatures, and high magnetic fields. It continues with the various theories which attempt to explain the origin of the FQHE, and some predictions that have been made. The role of fractionally-charged quasiparticles is discussed, particularly with reference to the Aharonov-Bohm effect (AB). Experimental measurements are presented of the Aharonov-Bohm effect in the FQHE, and analysis is presented as to their interpretation. Numerical models are shown to describe the energy dependence of the Aharonov-Bohm oscillations, and fitted to experiment. There is a discussion of how the Fermi liquid model of the edge states in the FQHE produces different predictions than the Luttinger liquid model, and the Fermi liquid model is shown to give a good fit to the experimental data. The edge of the sample in the FQHE is currently a subject of much discussion, so this thesis presents results about the equilibration of electronic edge states in the FQHE. An experimental device was developed which allows a continuous variation in the slope of the electrostatic potential at the edge, thus allowing the equilibration to be altered. Scattering coefficients between edge states, both in the integer and fractional Quantum Hall regimes are derived, and the implications discussed, as is the energy dependence of the scattering, for which a theory is developed. Novel oscillations in the scattering coefficients are also observed, and the system is used to experimentally refute the theoretical prediction that under certain circumstances quasiparticles propagate counter to the standard direction.

L'effet Hall quantique, qui apparaît dans les gaz d'électrons bidimensionnels soumis à un champ magnétique perpendiculaire et à basses températures, a été un sujet de recherche intense pendant les derniers trente ans, en particulier, à cause des manifestations spectaculaires de la mécanique quantique dans les propriétés de transport à l'échelle macroscopique. Dans cette thèse, on étend l'horizon de la recherche au niveau théorique sur ce sujet en considérant les effets du couplage spin-orbite et l'interaction électron-électron de façon analytique dans ce régime.Dans la première partie de ce manuscrit, on considère l'effet simultané du couplage spin-orbite de type Rashba et l'interaction Zeeman dans le régime de l'effet Hall quantique entier. Pour cela, on étend un formalisme de fonctions de Green basé sur des états de vortex cohérents avec l'objectif d'inclure le couplage entre les degrés de liberté orbitaux et de spin dans les états de dérive électroniques. Puis, comme première application, on montre comment obtenir analytiquement, nonperturbativement et de manière contrôlée des fonctionnelles quantiques (spectre et densité d'états locale) pour des potentiels électrostatiques arbitraires et localement plats. Les fonctionnelles sont ensuite analysées dans différents régimes de températures et comparées aux données expérimentales obtenues à partir des sondes de spectroscopie locales. Comme seconde mise en pratique du formalisme, on étudie en profondeur les propriétés de transport de charge et de spin dans un régime hydrodynamique d'équilibre local (ou quasi-équilibre) et dérive des expressions analytiques qui incorporent les caractères non-relativiste et relativiste des gaz d'électrons avec couplage spin-orbite de type Rashba.Dans la deuxième partie de cette thèse, on s'occupe du problème de traiter analytiquement les fortes interactions électron-électron dans le régime de l'effet Hall quantique fractionnaire. A cette fin, on étudie un problème à deux corps généralisé avec du désordre et des corrélations électroniques, en utilisant une nouvelle représentation d'états de vortex cohérents. Des corrélations à longue portée entre les particules sont incorporées de manière topologique à travers la présence d'une métrique non-Euclidienne. Subséquemment, on montre que ces états de vortex forment bien une base d'un espace de Hilbert élargi, puis on dérive l'équation du mouvement pour la fonction de Green. Enfin, on vérifie la consistance de notre théorie pour tout niveau de Landau de paire et on discute la nécessité d'aller au-delà de la limite semiclassique (à champ magnétique infinie) pour obtenir des gaps dans chaque niveau de énergie
The quantum Hall effect, appearing in disordered two-dimensional electron gases under strong perpendicular magnetic fields and low temperatures, has been a subject of intense research during the last thirty years due to its very spectacular macroscopic quantum transport properties. In this thesis, we expand the theoretical horizon by analytically considering the effects of spin-orbit coupling and strong electron-electron interaction in these systems.In the first part of the manuscript, we examine the simultaneous effect of Rashba spin-orbit and Zeeman interaction in the integer quantum Hall regime. Under these conditions, we extend a coherent-state vortex Green's function formalism to take into account the coupling between orbital and spin degrees of freedom within the electronic drift states. As a first application of this framework, we analytically compute controlled microscopic nonperturbative quantum functionals, such as the energy spectrum and the local density of states, in arbitrary locally flat electrostatic potential landscapes, which are then analyzed in detail in different temperature regimes and compared to scanning tunnelling experimental data. As a second application, we thoroughly study local equilibrium charge and spin transport properties and derive analytical useful formulas which incorporate the mixed non-relativistic and relativistic character of Rashba-coupled electron gases.In the second part of this thesis, we deal with the problem of analytically incorporating strong electron-electron interactions in the fractional quantum Hall regime. To this purpose, we consider a generalized two-body problem where both disorder and correlations are combined and introduce a new vortex coherent-state representation of the two-body states that naturally include long-range correlations between the electrons. The novelty of this theory is that correlations are topologically built in through the non-Euclidean metric of the Hilbert space. Next, we show that this kind of vortex states form a basis of an enlarged Hilbert space and derive the equation of motion for the Green's function in this representation. Finally, we check the consistency of our approach for any Landau level of the pair and discuss the necessity of going beyond the semiclassical (infinite magnetic field) approximation to obtain energy gaps within each energy level

L’optique quantique avec électron est un domaine de recherche en expansion depuis ses débuts au cours des années 90 prenant suite aux premières expériences d’interférence avec électrons réalisées dans les années 80. Ce domaine est dédié à la réalisation d’expérience d’optique quantique avec des électrons plutôt que des photons. Leur intérêt est double, d’une part les électrons étant des fermions de nouveaux phénomènes, en comparaison des photons qui sont des bosons, peuvent être observés. L’électron anti-bunching, en comparaison du bunching des photons obtenu dans des expériences de corrélations en est un exemple. Le deuxième avantage des électrons est le fait qu’ils peuvent être contrôlés et manipulés à l’aide de champ électrique, un tel contrôle n’est pas possible avec des photons. Alors que les composants de base pour la réalisation de ces expériences sont déjà existant comme la lame séparatrice, ou encore les sources cohérentes à électrons uniques, la détection immédiate d’un électron unique dans de telles expériences est toujours manquante. La difficulté étant le faible temps d’interaction entre l’électron en déplacement et le détecteur de charge qui est limité typiquement à moins de 1ns principalement à cause de la vitesse élevée de déplacement de l’électron qui est égale à la vitesse de Fermi soit 10-100km/s. Ce temps d’interaction est environ deux ordres de grandeurs plus petits que ce qui est nécessaire pour le meilleur détecteur de charge démontré jusqu’à présent.Dans ce manuscrit est présenté le développement d’un détecteur ultra-sensible pour la détection immédiate d’un électron se déplaçant à la vitesse de Fermi. Notre stratégie est de détecter un électron unique se déplaçant dans les canaux de bords (ECs) de l’effet Hall quantique à partir de la mesure d’une variation de phase d’un bit quantique singlet-triplet, appelé qubit détecteur par la suite. La détection immédiate de cet électron en déplacement n’étant possible que si l’interaction avec ce dernier induit une variation de phase de pi, avec une lecture immédiate de l’état de spin du qubit détecteur.Grâce au développement et à l’utilisation d’un RF-QPC, cette lecture immédiate de l’état de spin est tout d’abord démontrée. Par la suite le développement du qubit détecteur avec la réalisation d’oscillations cohérentes d’échange est décrit. Sa sensibilité en charge est démontrée avec l’observation d’une phase induite par l’interaction avec un courant d’électrons dans les ECs. Ce courant est imposé par l’application d’un biais de tension contrôlant le potentiel chimique de ces ECs.Après optimisation de ce qubit détecteur pour la détection d’un électron unique, il est calibré en utilisant le même procédé de courant imposé par application d’un biais de tension. Cette calibration nous fournie la variation de signal attendue pour l’interaction avec cette charge unique est indique que sa détection immédiate est impossible dans nos conditions expérimentales. Notre détecteur ayant une sensibilité de charge de l’ordre de 8.10-5 pour une bande passante allant de DC à 1THz. Cette sensibilité est environ deux ordres de grandeur trop petite que ce qui est nécessaire pour la détection immédiate de cette charge unique. Finalement, ce qubit détecteur est utilisé pour détecté, dans une expérience moyennée, ce qui est appelé un edge magneto plasmon composé par moins de 5 électrons. Néanmoins, atteindre la détection de la charge unique dans n’a pas été possible, la sensibilité en charge étant légèrement trop petite pour y arriver.Les différentes limites de notre détecteur sont listées et expliquées tout au long du manuscrit, avec une présentation de différents axes de développement qui devraient permettre de réussir cette détection d’un électron unique dans une nouvelle expérience
The electron quantum optics field is a research topic with an interest growing over the years since the 80's and the first interference experiment with electrons. This field is dedicated to the implementation of quantum optics experiments with electrons instead of photon. The advantage is twofold, one is the fermion nature of the electrons which ensure the observation of phenomenon which cannot be observed with photon (boson), the anti-bunching of the electrons in correlation experiments contrary to the bunching for photons illustrates this point. The second advantage is the possibility to interact and control electrons with electric fields since they are charged particles. Such control does not exist with photon. In addition to these fundamental experiments, it has been recently demonstrated that this topic presents a possible candidate for quantum information with so called flying qubit. While the based components to mimic the quantum optics experiments are already demonstrated like the beam splitter, phase shifter or coherent single electron source, the single electron detection in a single shot manner in such system is still lacking. The difficulty being the short interaction time between the travelling charge and the charge detector, being of less than 1ns in such system where the electron propagate at the Fermi velocity 10-100km/s. This interaction is approximately two orders of magnitude shorter than what is required with the actual best on chip charge detector.In this thesis is presented the development of an ultra-sensitive detector for the single shot detection of an electron travelling at the Fermi velocity. Our strategy was to detect a single travelling electron propagating in the edge channels (ECs) of the quantum Hall effect by measuring the induced phase shift of a singlet-triplet qubit, referred as to the qubit detector. The single shot detection being only possible if the interaction with the travelling electron induces a complete π phase shift and the spin readout of the qubit detector being performed in a single shot manner.Thanks to the development and use of a RF-QPC the single shot spin readout of the qubit detector has been first demonstrated. Its development with the implementation of coherent exchange oscillations is then described. The charge sensitivity of the qubit detector is validated in an experiment consisting in recording a phase shift of these oscillations due to the interaction with an imposed flow of electrons in the ECs. This flow of electron was induced by a DC voltage bias applied on the ECs to tune their chemical potential.This qubit detector is then optimised for the single travelling charge detection. Its calibration has been implemented using the same imposed flow of electrons by application of a DC bias. This calibration provides the expected signal variation induced by the interaction with a single travelling electron, and indicates the impossibility to implement this detection in a single shot manner in our experimental conditions. Our detector exhibits a charge sensitivity estimated close to 8.10-5 e/Hz-1/2 for a detection bandwidth from DC to 1 THz. The sensitivity is close to two orders of magnitude smaller than required for a single shot detection. Finally this qubit detector has been employed to detect in average measurements an edge magneto plasmon composed by less than 5 electrons. However, the single electron level could not be reached in statistical measurement neither, the sensitivity of our qubit detector being too limited.The different limitations of our experiment are listed and explained with the presentation of different axes of development which could permit to succeed this detection in another experiment

Using electronic spin rather than charge to replace existing microelectronic systems has been a well studied area of research in the last ten years. More recently, research has focused on using the nuclear spin of GaAs rather than the electron spin. This work has demonstrated that GaAs nuclear spins have many desirable properties and show great potential as quantum information carriers. The challenge in the implementation of nuclear spins lies in the ability to control and retrieve the information that they carry. One proposed method is to dynamically polarize the GaAs nuclear spins using circularly polarized photoexcitation. If successful, this could open new horizons in the field of quantum information processing. This thesis details an investigation into the use of polarized light to manipulate the properties of a GaAs/AlGaAs quantum well sample. The three main topics explored in this thesis are: 1) the design and operation of a polarization controller that is able to shine well-defined and tunable polarized light on to a sample contained in a cryogenic environment at T = 0.27K; 2) the manipulation of the nuclear polarization in GaAs using low power laser light with tunable polarization; and 3) a preliminary investigation into illuminating a quantum Hall sample with unfocused, low power laser light and the transport properties modifications that occur in the quantum Hall regime.
L'utilisation du spin electronique plutot que la charge electronique pour remplacer les systemes microelectroniques a ete un domaine bien etudie de la recherche au cours des dix dernieres annees. Plus recemment, la recherche a porte sur l'utilisation du spin nucleaire du GaAs plutot que le spin electronique. Ce travail a demontre que les spins nucleaires du GaAs ont de nombreuses proprietes desirables et montrent un grand potentiel en tant que transporteurs de l'information quantique. Le defi dans la mise en oeuvre des spins nucleaires reside dans la capacite de controler et de recuperer les informations qu'elles transportent. Une methode proposee consiste a polariser dynamiquement les spins nucleaires du GaAs en utilisant la photoexcitation polarisee circulairement. Ceci pourrait ouvrir de nouveaux horizons dans le domaine du traitement de l'information quantique. Cette these expose en details une enquete sur l'utilisation de la lumiere polarisee pour manipuler les proprietes d'un echantillon puit quantique de GaAs/AlGaAs. Les trois principaux sujets abordes dans cette these sont les suivants: 1) la conception et le fonctionnement d'un controleur de polarisation qui est capable d'emettre une lumiere polarisee bien definie et ajustable sur un echantillon dans un environnement cryogenique a T = 0.27K, 2) la manipulation de la polarisation nucleaire dans le GaAs en utilisant un laser a faible puissance avec une polarisation ajustable, et 3) une enquete preliminaire sur l'illumination d'un echantillon de Hall quantique avec un laser non-focalise a faible puissance et les modifications des proprietes de transport qui se produisent dans le regime de Hall quantique.

Electron systems in the quantum Hall regime change from a liquid state to a Wigner crystal state as the filling factor is lowered below approximately 1/5. This phase transition can be studied with theoretical methods by comparing the ground-state energies of the quantum liquid and the quantum Wigner crystal. Past studies have not included realistic effects such as finite thickness, Landau-level mixing, and disorder on the electron system. We expand upon the classic work by Maki and Zotos to calculate Wigner crystal energies that include a finite thickness of the two-dimensional electron lattice.

O efeito Hall quântico surge em gases de elétrons bidimensionais (2DEG) na presença de altos campos magnéticos B. O campo magnético quantiza o movimento planar dos elétrons em órbitas ciclotrônicas caracterizadas pelos níveis de Landau. Neste regime a resistividade transversal (ou Hall) ρxy em função de B exibe platôs em submúltiplos inteiros de e2/h, i.e., ρxy = ν-1 e2/h, sendo ν o fator de preenchimento dos níveis de Landau. Por sua vez, a resistividade longitudinal ρxx apresenta picos nas transições entre platôs de ρxy. Em primeira instância, ρxx é uma medida indireta da densidade de estados no nível de Fermi g(εF), e os picos dos mesmos indicam cruzamentos do nível de Fermi εF com niveis de Landau. Assim, o diagrama de densidade de elétrons n2D e B dos picos de ρxx ~ g(εF) fornece um mapa topológico da estrutura eletrônica do sistema. Em sistemas de duas subbandas, ρxx(n2D, B) exibe estruturas em forma de anel devido a cruzamentos de níveis de Landau de subbandas distintas [experimentos do grupo do Prof. Jiang (UCLA)]. Estes cruzamentos podem ainda levar a instabilidades ferromagnéticas. Investigamos estas instabilidades usando a teoria do funcional da densidade (DFT) para o cálculo da estrutura eletrônica, e o modelo de Ando (formalismo de Kubo) para o cálculo de ρxx e ρxy. Para temperaturas mais altas (340 mK) obtemos as estruturas em forma de anel em ρxx. Para temperaturas mais baixas (70 mK), observamos uma quebra dos anéis devido a transições de fase ferromagnéticas. Variando-se o ângulo θ de B com relação ao 2DEG observa-se o encolhimento do anel. Nossos resultados mostram que o ângulo de colapso total do anel depende de uma competição entre o termo de troca da interação de Coulomb (princípio de Pauli) e cruzamentos evitados devido ao ângulo θ finito. As transições de fase exibem ainda o fenômeno de histerese. Na região de instabilidade ferromagnética obtemos diferentes soluções variando B de forma crescente ou decrescente. Estas soluções possuem energias total diferentes, de forma que representam estados fundamental e excitado de muitos corpos. Esta observação, juntamente com resultados anteriores do grupo [Freire & Egues (2007)], representam as primeiras realizações teóricas da previsão da possibilidade de estados excitados como mínimos locais do funcional de energia do estado fundamental [Perdew & Levy (1985)]. O modelo aqui proposto fornece excelente acordo com os experimentos considerados. Adicionalmente, a observação sistemática e experimentalmente verificada dos estados excitados valida as previsões de Perdew & Levy. Aplicamos ainda estas mesmas ideias no cálculo da estrutura eletrônica e condutância de fios quânticos na presença de campos magnéticos, mostrando que cruzamentos de modos transversais também exibem instabilidades ferromagnéticas observadas em experimentos recentes [Dissertação de Mestrado de Filipe Sammarco, IFSC/USP], fortalecendo a validade do modelo apresentado nesta tese.
The quantum Hall effect arises in two dimensional electron gases (2DEG) under high magnetic fields B. The magnetic field quantizes the planar motion of the electrons into cyclotron orbits given by the Landau levels. In this regime the transversal (Hall) resistivity ρxy shows plateaus as a function of B at integer sub-multiples of e2/h, i.e., ρxy = ν-1 e2/h, where n is the filling factor of the Landau levels. The longitudinal resistivity ρxx shows peaks at the transition between the plateaus of ρxy. In principle, ρxx is an indirect measure of the density of states at the Fermi level g(εF), so that the peaks indicate when the Fermi level εF crosses a Landau level. Therefore, a density-B-field diagram n2D-B of the ρxx ~ g(εF) peaks shows a topological map of the electronic structure of the system. In two-subband systems, ρxx( n2D, B) shows ringlike structures due to crossings of spin-split Landau levels from distinct subbands [experiments from the group of Prof. Jiang (UCLA)] that could lead to ferromagnetic instabilities. We study these instabilities using the density functional theory (DFT) to calculate the electronic structure, and Ando\'s model (Kubo formalism) for ρxx and ρxy. At higher temperatures (340 mK) we also obtain the ringlike structures in ρxx. At lower temperatures (70 mK) we see broken rings due to quantum Hall ferromagnetic phase transitions. Tilting B by theta with respect to the 2DEG normal we find that the ring structure shrinks. Our results show that the angle of full collapse depends on a competition between the exchange term from the Coulomb interaction (Pauli principle) and the anticrossing of Landau levels due to the finite angle theta. Additionally, at the instabilities we observe hysteresis. Sweeping the B field up or down near these regions we obtain two different solutions with distinct total energies, corresponding to the ground state and an excited state of the many-body system. This result, together with previous results of our group [Freire & Egues (2007)], are the first realizations of the theoretical prediction of the possibility of excited states as local minima of the ground state energy functional [Perdew & Levy (1985)]. The model proposed here shows an excellent agreement with the experiments. Additionally, the systematic and experimentally verified observation of excited states corroborates the predictions of Perdew & Levy. Similar ideas as presented here when applied to the electronic structure and conductance of quantum wires with an in-plane magnetic field show ferromagnetic instabilities at crossings of the wire transverse modes [Master Thesis of Filipe Sammarco, IFSC/USP], also with excellent experimental agreement. This strengthen the range of validity of the model proposed in this Thesis.

Orientadores: Peter A. B. Schulz, John T. Chalker
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin
Made available in DSpace on 2018-08-06T19:00:22Z (GMT). No. of bitstreams: 1 Pereira_AnaLuizaCardoso_D.pdf: 2880300 bytes, checksum: ffd133973b4bc6e23c91694bc47d8794 (MD5) Previous issue date: 2005
Resumo: Esse trabalho é dedicado ao estudo de dois problemas de interesse atual em sistemas quânticos de baixa dimensionalidade. Ambos são relacionados ao processo de localização eletrônica no regime Hall quântico. O primeiro problema diz respeito ao destino dos estados estendidos no limite de baixos campos magnéticos ou forte desordem, onde ocorre a transição de líquido de Hall para o isolante de Hall. O problema é abordado através de simulações numéricas, com um modelo de rede bidimensional tratado por um Hamiltoniano tight-binding, considerando-se tanto desordem tipo ruído branco quanto desordem correlacionada com perfil Gaussiano. Nós observamos que à medida que o campo magnético tende a zero ou a desordem é suficientemente aumentada no sistema, os estados estendidos sofrem um deslocamento em relação ao centro das bandas de Landau, indo em direção às mais altas energias e, eventualmente, ultrapassando a energia de Fermi. Esse mecanismo é chamado na literatura de levitação de estados estendidos. Nossos resultados permitem uma análise quantitativa. Identificamos os seguintes parâmetros como sendo os relevantes para mapear a levitação: (i) a razão entre escalas de energia ¿ entre a energia de separação dos níveis de Landau e o alargamento do nível devido à desordem; e (ii) a razão entre escalas de comprimento ¿ entre o comprimento magnético e o comprimento de correlação da desordem. Analisando uma vasta gama de parâmetros, uma expressão de escala descrevendo a levitação de estados estendidos é estabelecida neste trabalho. O segundo problema abordado nesta tese é relacionado ao processo de blindagem do potencial de desordem e ao mecanismo de formação dos estados localizados em sistemas Hall quânticos. O trabalho analítico apresentado aqui é motivado por recentes resultados experimentais, que mostram imagens de microscopia com medidas locais do potencial eletrostático e da compressibilidade desses sistemas, evidenciando como se dá o processo de carga de estados localizados por cargas inteiras ou fracionárias (quase-partículas). Em um regime onde o comportamento é dominado por interações Coulombianas, estabelecemos um modelo eletrostático que descreve o estado localizado como sendo uma região compressível (quantum dot ou antidot) envolta por um plano incompressível, usando a aproximação de Thomas-Fermi para tratar as interações. O potencial eletrostático nas vizinhanças da região compressível é calculado, fornecendo o tamanho dos saltos que ocorrem no potencial à medida que cada carga é adicionada ou removida do estado localizado. Além de mostrar como estes saltos se tornam menores com o aumento do índice de Landau, nossos resultados mostram a dependência deles com a altura de observação do potencial (ou seja, a altura da ponta de prova em relação ao gás de elétrons). O modelo apresentado pode ser usado para tratar estados localizados observados nos platôs do efeito Hall quântico inteiro ou fracionário
Abstract: This work is devoted to the study of two problems of current interest in low dimensional quantum systems. Both are related to the process of electron localization in the quantum Hall regime. The first problem refers to the fate of extended states in the limit of low magnetic fields or strong disorder, where the transition from quantum Hall liquid to Hall insulator takes place. A numerical approach to the problem is used, with a 2D lattice model treated in a tight-binding framework, considering both white-noise and Gaussian correlated disorder. We observe that as the magnetic field vanishes or the disorder is sufficiently increased in the system, the extended states are shifted from the Landau band centers, going to higher energies and, eventually, rising above the Fermi energy. This mechanism is referred in the literature as levitation of extended states. Our results allow a quantitative analysis. We identify the following parameters as the relevant ones to map the levitation: (i) the energy scales ratio - between the energy separation of consecutive Landau levels and the level broadening due to disorder; and (ii) the length scales ratio - between the magnetic length and the disorder correlation length. Analyzing a wide range of parameters, a scaling expression describing the levitation of extended states is established. The second problem considered in this thesis is related to the screening of the disorder potential and to the mechanism of formation of localized states in quantum Hall systems. The analytical work we present here is motivated by recent imaging experiments, which probe locally the electrostatic potential and the compressibility of these systems, showing the charging of individual localized states by integer or fractional charges (quasiparticles). For a regime where the behavior is dominated by Coulomb interactions, we set out an electrostatic model describing the localized state as a compressible region (quantum dot or antidot) embebed in an incompressible background, using the Thomas-Fermi approximation to treat the interactions. The electrostatic potential in the vicinity of the compressible region is calculated, providing the size of potential steps as each charge is added or removed from the localized state. Besides from showing how the potential steps get smaller for higher Landau levels, our results show the dependence of these steps with the height of observation (i.e., the distance from the scanning probe to the electron gas). The proposed model can be used to treat localized states observed on integer or fractional quantum Hall plateaus
Doutorado
Física da Matéria Condensada
Doutor em Ciências

L'utilizzo di eterostrutture a semiconduttore nella realizzazione di dispositivi per la quantum information ha come principali vantaggi la scalabilità e la possibilità di essere integrati nei circuiti elettrici tradizionali. In particolare, è stata dimostrata teoricamente la possibilità di realizzare un set universale di porte logiche quantistiche attraverso le architetture che sfruttano flying-qubit. Nei nostri studi, ci concentriamo sulla loro implementazione attraverso interferometri elettronici basati su canali edge nel regime di Hall quantistico, e ne studiamo numericamente la realizzabilità e capacità di calcolo. Gli stati di edge sono canali conduttivi e chirali che corrono lungo il bordo di un 2DEG confinato e caratterizzati da una notevole lunghezza di coerenza, il cui percorso può essere controllato tramite gate metallici che modulano localmente il coefficiente di svuotamento del 2DEG. Oltre all'utilizzo di correnti elettroniche, è attualmente possibile iniettare in tali canali pacchetti elettronici a singola carica. In particolare, ne è stata recentemente proposta la iniezione attraverso quantum dot pumps. Contrariamente ad altri tipi di eccitazione, questi pacchetti sono caratterizzati da un'energia molto più elevata del livello di Fermi e la loro forma gaussiana garantisce un maggiore controllo sulla loro dinamica. Dopo aver illustrato i modelli numerici e le realizzazioni sperimentali dei pacchetti a singolo elettrone maggiormente utilizzati, ovvero i levitoni e le quasi-particelle di Landau, presenteremo il nostro approccio dinamico per la simulazione dei pacchetti di stati di edge. Per simularne l'evoluzione in una geometria 2D arbitraria, abbiamo sviluppato un risolutore parallelo della equazione di Schroedinger basato sul metodo Split-Step Fourier. I risultati ottenuti nel nostro schema dinamico sono confrontati con calcoli di supporto basati su geometrie efficaci 1D e sul metodo della matrice di scattering. Infine, abbiamo descritto il costo computazionale del metodo Split-Step Fourier per un sistema a due particelle in una geometria 2D, che richiede la distribuzione della funzione d'onda a 4 dimensioni sui nodi di macchine HPC nel paradigma MPI. Il metodo Split-Step è inizialmente applicato alla simulazione d'interferenza a singolo elettrone in una geometria scalabile dell'interferometro Mach-Zehnder (MZI) a doppio filling factor. Includendo la forma esatta degli stati di edge generati dallo specifico design del dispositivo, abbiamo progettato un beam splitter elettronico che garantisce una più ampia visibilità rispetto a un tradizionale quantum point contact. Successivamente, abbiamo dimostrato la validità del nostro approccio numerico tramite lo studio dell'antibunching a due elettroni in un esperimento Hong-Ou-Mandel nel regime di Hall quantistico intero. Ivi, abbiamo caratterizzato l'origine della probabilità di bunching non nulla, analizzando l'interazione tra la dispersione energetica dei pacchetti d'onda interferenti e la esatta geometria del partitore elettronico. Grazie alla computazione esatta della funzione d'onda 4D, abbiamo incluso l'interazione Coulombiana tra i nostri pacchetti d'onda fortemente localizzati, e introdotto l'effetto dello screening per osservare la transizione da un interferometro dominato dalla interazione di scambio a uno controllato dalla repulsione intraelettronica. Infine, abbiamo presentato un'implementazione a stato solido di uno sfasatore di fase condizionale basato sugli stati di edge, che si realizza concatenando in parallelo due MZI multicanale. Qui, l'esatta simulazione dello scattering di elettroni sotto l'azione del potenziale Coulombiano in 2D ci consente di analizzare l'interazione tra l'azione selettiva della repulsione elettronica, che funge da entangler, e la geometria su vasta scala del dispositivo.
Quantum information processing devices based on semiconductor heterostructures have the potential to be scalable and easier to be integrated in traditional electronic circuitry than systems based on different platforms. In particular, architectures exploiting the flying-qubit paradigm proved theoretically to allow the implementation of a universal set of quantum gates. We focus on coupled electron interferometers based on edge channels in the Integer Quantum Hall regime for the physical implementation of flying qubits and gates, and study numerically the feasibility of such devices and their quantum computing capabilities. Hall edge states are chiral conductive channels running along the border of a confined 2DEG, with a remarkably long coherence length. Their path can be controlled by means of metallic gates that locally deplete the 2DEG in order to engineer electron interferometers. In addition to the use of interfering currents, electron quantum optics is nowadays realized also by injecting single-electron excitations. Specifically, an injection protocol based on non-adiabatic quantum dot pumps has been recently proposed. In contrast to other types of excitations, these wave packets are characterized by an energy much larger than the Fermi level, and their Gaussian shape ensures a robust control of the wave packet dynamics. After reviewing the theoretical modelling and some experimental realizations of the two most common single-electron excitations, i.e. levitons and Landau quasi-particles, we present our numerical approach for the dynamical simulation of wave packets in edge states. To simulate their evolution in a realistic 2D geometry, we developed an in-house a parallel solver of the time-depedent Schroedinger equation based on the Split-Step Fourier algorithm. Our findings in the dynamic framework are compared to simplified analytical models based on effective 1D geometries and the scattering matrix method. We highlight the numerical challenges in the application of the Split-Step Fourier method for a two-particle system in a 2D geometry, which involves data distribution of the ensuing 4D wavefunction on multi-node HPC architectures with the MPI paradigm. Our time-dependent method has been initially applied to simulate self-electron interference in a scalable geometry of the electron Mach-Zehnder interferometer (MZI) at bulk filling factor two. By accounting for the exact shape of the edge states generated by the full-scale design of the interferometer, we engineer an electron beam splitter that ensures a higher visibility with respect to a traditional quantum point contact. Moreover, we prove the validity of our numerical method by studying two-electron antibunching in an Hall-driven Hong-Ou-Mandel interferometer. Here, we characterize the origin of the non-zero bunching probability - observed experimentally - by studying the interplay between the energy broadening of the two interfering wave packets and the exact geometry of the electron beam splitter. Thanks to the computation of the non separable 4D wavefunction, we include exactly Coulomb interaction between our strongly-localized wave packets, and introduce the effect of screening to observe the transition from an exchange-driven to a Coulomb-driven interferometer in the operating regime of interest. Finally, we propose a solid-state implementation of a conditional phase shifter based on edge states, engineered by concatenating in parallel two multichannel MZIs. Here, the exact simulation of Coulomb-driven electron scattering in 2D allows us to analyze the interplay between the selective action of electron repulsion, that acts as the entangler, and the full-scale geometry of the device.

In the first part of this work, we describe a theory of the ground states and charge gaps in the fractional quantum Hall states of graphene. The theory relies on knowledge of these properties for filling fractions smaller than one. Then, by the application of two mapping rules, one is able to obtain these properties for fractional quantum Hall states at arbitrary fillings, by conceiving the quantum Hall ferromagnets as vacua on which correlated electrons or correlated holes are added. The predicted charge gaps and phase transitions between different fractional quantum Hall states are in good agreement with recent experiments. In the second part, we investigate the low energy theory for the neutral Landau level of bilayer graphene. We closely analyze the way different terms in the Hamiltonian transform under the action of particle-hole conjugation symmetries, and identify several terms that are relevant in explaining the lack of such symmetry in experiments. Combining an accurate parametrization of the electronic structure of bilayer graphene with a systematic account of the impact of screening we are able to explain the absence of particle-hole symmetry reported in recent experiments. We also study the energetics of fractional quantum Hall states with coherence between n=0 and n=1 cyclotron quantum numbers, and obtain a general formula to map the two-point correlation function from their well-known counterparts made from only n=0 quantum numbers. Bilayer graphene has the potential for realizing these states which have no analogue in other two-dimensional electron systems such as Gallium Arsenide. We apply this formula to describe the properties of the n=0/n=1 coherent Laughlin state which displays nematic correlations.
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