Thesis advisor: Ziqiang Wang / Graphene has emerged as an important two dimensional electron system with novel physical properties due to its relativistic-like linear energy-momentum dispersion relation at low energy. Alongside two dimensional electron systems in semiconductor heterostructures, it has a rich set of integer and fractional quantum Hall states. Significant progresses have been made recently, but a full understanding of these states is still lacking. The prevailing approach for fractional quantum Hall effects in graphene has been the numerical exact diagonalization. In this work, we develop a fermionic Chern-Simons effective theory for Dirac fermions as a complement to the existing theories, and to bring new insights in our understanding of the phenomena. In particular, we study the possibility for quantum Hall plateaus at even-denominator filling factors. We first construct a unitary Chern-Simons transformation to attach even number of flux quanta to Dirac fermions. To deal with the four-fold spin-valley degeneracy, a set of K-matrices is introduced. At even-denominator filling factors in the zeroth Landau level, the fictitious magnetic field of the Chern-Simons field cancels the external magnetic field on average. It is shown that the Chern-Simons field mediates an effective mutual statistical interaction between composite Dirac fermions. We further show the statistical interaction and Coulomb interaction favor the formation of an exciton condensate. Quasi-particles at finite filling factors can be regarded as excita- tions above the exciton condensate, and can be described as massive Dirac fermions. This means a mass is generated dynamically for Dirac fermions. Different types of K-matrices give rise to different mass gaps. The Chern numbers associated with different massive Dirac band structures can be used to classify the K-matrices. In the last part of the thesis, we study the pairing instability of the composite Dirac fermion liquid. We show the statistical interaction drives a complex p-wave pairing among the quasi-particles. As long as the Coulomb pair breaking effect is weak, the system can develop a superconducting energy gap, thus form a fractional quantum Hall state. / Thesis (PhD) — Boston College, 2012. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Les évolutions rapides du marché des composants électroniques font apparaître de nombreux challenges pour la conception et la fabrication de ces derniers. Lorsque ces éléments deviennent plus petits, leur comportement se complexifie à mesure que de nouveaux phénomènes, liés aux effets d'interférence, entrent en jeu. Comprendre ces derniers nécessite le développement d'outils théoriques avancés. Dans ce contexte cette thèse est consacrée au transport électronique quantique dans des systèmes multi-terminaux. Dans la première partie on développe un formalisme général, utilisant les fonctions de Green de Keldysh, pour le transport électronique quantique dans des systèmes multi-terminaux en présence de perturbations oscillantes. Nous sommes capable d'exprimer toute obervable AC en termes de fonctions de Green à l'équilibre et des self-énergies des contacts. Ceci fait de notre formalisme un outil pratique pour toute une variété de perturbations à fréquence finie. Dans la seconde partie on présente l'idée d'induction d'un fort couplage spin-orbite dans le graphène en déposant à sa surface un certain type d'atomes lourds. Le graphène devient alors un isolant topologique. Nous avons ensuite étudié l'évolution de la phase topologique avec un champ magnétique externe. Une transition entre la phase de Hall quantique et la phase de Hall quantique de spin a été identifiée dans le même système en variant seulement la position du niveau de Fermi. Nous avons montré qu'une hétérojonction entre ces deux phases donnait lieu à un nouveau type d'état chiral à l'interface. / Rapidly changing market of electronic devices sets up a lot of challenges for the manufacturing and design technologies. When electronic circuit elements get smaller, the device behavior becomes increasingly complicated as new physical phenomena due to quantum interference effects come into play. Understanding of the latter necessitates development of advanced theoretical tools. In this thesis we investigate quantum electron transport in multiterminal devices. In the first part making use of the Keldysh Green's functions we develop a general framework for electron quantum transport in multi-terminal systems in the presence of oscillating fields. We are able to express any AC observable in terms of stationary Green's functions and leads self-energies, which makes our formalism a practical numerical tool for a variety of possible finite-frequency perturbations. In the second part we investigate theoretically a proposal to induce strong spin-orbital coupling in graphene by functionalizing its surface with certain type of heavy adatoms. In this case graphene becomes a topological insulator. Then we investigate the evolution of this topological phase in external magnetic field. We were able to see a unique transition between quantum Hall and quantum spin Hall phases in the same system by only varying the position of the Fermi level. A heterojunction between these two phases was shown to give rise to a new type of a chiral state at the interface between the latter.

In the present thesis, we have explored multiple aspects concerning the stochasticity of magnetic domain wall motion during magnetization reversal, all of which originated from our initial study of magnetic field sensing using planar Hall effect. Magnetic field sensors occupy a very important and indispensable position in modern technology. They can be found everywhere, from cellphones to automobiles, electric motors to computer hard disks. At present there are several emerging areas of technology, including biotechnology, which require magnetic field sensors which are at the same time simple to use, highly sensitive, robust under environmental conditions and sufficiently low cost to be deployed on a large scale. Magnetic field sensing using planar Hall effect is one such feasible technology, which we have explored in the course of the thesis. The work was subsequently expanded to cover some fundamental aspects of the stochasticity of domain wall motion, studied with planar Hall effect, which forms the main body of work in the present study. In Chapter 1, we give an introduction to the phenomenology of planar Hall effect, which is the most important measurement technique used for all the subsequent studies. Some early calculations, which had first led to the understanding of anisotropic magnetoresistance and planar Hall effect as being caused by spin-orbit interaction are discussed. In Chapter 2, we discuss briefly the experimental techniques used in the present study for sample growth and fabrication, structural and magnetic characterization, and measurement. We discuss pulsed laser ablation, which is the main technique used for our sample growth. Particular emphasis is given to the instrumentation that was carried out in-house for MOKE and low field magnetotransport (AMR and PHE) measurement. This includes an attempt at domain wall imaging through MOKE microscopy. Some of the standard equipments used for this work, such as the SQUID magnetometer and the acsusceptometer are also discussed in detail. In Chapter 3 we discuss our work on planar Hall sensors that led to the fabrication of a device with a very simple architecture, having transfer characteristics of 650V/A.T in a range of _2Oe. The sensing material was permalloy (Ni81Fe19), and the value had been obtained without using an exchange biased pinning layer. Field trials showed that the devices were capable of geomagnetic field sensing, as well as vehicle detection by sensing the anomaly in Earth's magnetic field caused by their motion. Its estimated detection threshold of 2.5nT made it well suited for several other applications needing high sensitivity in a small area, the most prominent of them being the detection of macromolecules of bio-medical significance. Chapter 4: The work on Barkhausen noise was prompted by reproducibility problems faced during the sensor construction, both between devices as well as within the same device. Study of the stochastic properties led us to the conclusion that the devices could be grouped into two classes: one where the magnetization reversal occurred in a single step, and the other where it took a 0staircase0 like path with multiple steps. This led us to simulations of Barkhausen noise using nucleation models like the RFIM whence it became apparent that the two different groups of samples could be mapped into two regimes of the RFIM distinguished by their magnetization reversal mode. In the RFIM, the nature of the hysteresis loop depends on the degree of disorder, with a crossover happening from single-step switching to multi-step switching at a critical disorder level. Appropriate changes also appear in the Barkhausen noise statistics due to this disorder-induced crossover. By studying the Barkhausen noise statistics for our permalloy samples and comparing them with simulations of the RFIM, we found nearly exact correspondence between the two experimental groups with the two classes resulting from crossing the critical disorder. What remained was to quantify the 0disorder0 level of our samples, which was done through XRD, residual resistivity and a study of electron-electron interaction effects in the resistivity. All these studies led to the conclusion that the samples reversing in multiple steps were more 0defective0 than the other group, at par with the model predictions. This completed the picture with respect to the modeling of the noise. In experiments, it was found that a high rate of film deposition yielded less 0defective0 samples, which severed as an important input for the sensor construction. These results can be viewed from a somewhat broader perspective if we consider the present scenario in the experimental study of Barkhausen noise, or crackling noise in general. Two classes of models exist for such phenomena: front propagation models and nucleation models. Both appear to be very successful when it comes to experiments with bulk materials, while the comparison with experiments on thin films is rather disappointing. It is still not clear whether the models are at fault or the experiments themselves. Through our study, we could demonstrate that there can be considerable variation in the Barkhausen noise character of the same material deposited in the same way, and what was important was the degree of order at the microscopic level. This may be a relevant factor when experimental papers report non-universality of Barkhausen noise in thin films, which can now be interpreted as either insufficient defects or a sample area too small for the study. Chapter 5: Defects in a sample are not the only cause for stochastic behavior during magnetization. In most cases, random thermal 0events0 are also an important factor determining the path to magnetization reversal, which was also true for our permalloy samples. We studied the distribution of the external fields at which magnetization reversal took place in our samples, and tried to explain it in terms of the popular Neel-Brown model of thermal excitation over the anisotropy barrier. The analysis showed that even though the coercivity behaved 0correctly0 in terms of the model predictions, the behavior of the distribution width was anomalous. Such anomalies were common in the literature on switching field distributions, but there seemed to be no unified explanation, with different authors coming up with their own 0exotic0 explanations. We decided to investigate the simplest situations that could result in such a behavior, and through some model-based calculations, came to the conclusion that one of the causes of the anomalies could be the different magnitudes of barrier heights/anisotropy fields experienced by the magnetic domain wall when the reversal occurs along different paths. Though an exact match for the behavior of the distribution width could not be obtained, the extended Neel-Brown model was able to produce qualitative agreement. Chapter 6 contains a study of some interesting 0geometrical0 effects on Barkhausen noise of iron thin films. By rotating the applied magnetic field out-of plane, we could observe the same single-step to multi-step crossover in hysteresis loop nature that was brought about by varying disorder in Chapter 4. We could explain this through simulations of a random anisotropy Ising model, which, apart from exhibiting the usual disorder induced crossover, showed a transition from sub-critical to critical hysteresis loops when the external field direction was rotated away form the average anisotropy direction. Once again, simulation and experiment showed very good agreement in terms of the qualitative behavior. In the second part of this chapter, a study of exchange biased Fe-FeMn system was carried out, where it was observed that the reversal mode has been changed from domain wall motion to coherent rotation. Barkhausen noise was also suppressed. Though many single-domain models existed for this type of reversal, our system was not found to be strictly compatible with them. The disagreement was with regard to the nature of the hysteresis, which, if present, had to be a single step process for a single domain model. The disagreement was naturally attributed to interaction with the nearby magnetic moments, to verify which, simulations were done with a simplified micromagnetic code, which produced excellent agreement with experiment. In Chapter 7, we have studied the temporal properties of Barkhausen avalanches, to compare the duration distributions with simulation. We had used a permalloy sample that was sub-critical according to avalanche size distributions, and our measurement was based on magneto-optic Kerr effect. We measured duration distributions which showed a similar manifestation of finite-size effects as were shown by the size distributions. The power law exponent was calculated, which was deemed 0reasonable0 upon comparison simulations of the sub-critical RFIM. Appendix A contains a study of high-field magnetoresistance of permalloy, which shows that the dominant contribution to magnetoresistance is the suppression of electron-magnon scattering. An interesting correlation is observed between the magnetization of samples and an exchange stiffness parameter d1, that was extracted from magnetoresistance measurements. Here we also re-visit our earlier observation of permalloy thin films possessing a resistance minimum at low temperature. The origin of this minimum is attributed to electron-electron interaction. Appendix B contains the source codes for most of the important programs used for simulation and data analysis. The programs are written in MATLAB and FORTRAN 95. LabView programs used for data acquisition and analysis are not included due to space requirements to display their graphical source codes. Appendix C discusses the studies on a disordered rare-earth oxide LaMnO3. The re-entrant glassy phase is characterized with ac susceptibility and magnetization measurements to extract information about the nature of interactions between the magnetic 0macrospins0 in the system. Appendix D deals with electron scattering experiments performed with spinpolarized electrons (SPLEED) from clean metal surfaces in UHV. A study of the scattering cross sections as a function of energy and scattering angle provides information about spin-orbit and exchange interactions of the electrons with the surface atoms, and can answer important questions pertaining to the electronic and magnetic structure of surfaces. In the course of this study, planar Hall effect is seen to emerge as a powerful tool to study the magnetic state of a thin film, so that it is interesting to apply it to thin films of other materials such as oxides, where magnetization noise studies are next to nonexistent. What also emerged is that there is still a lot of richness present in the details of supposedly well-understood magnetization phenomena, some of which we have explored in this thesis in the context of stochastic magnetization processes.

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.

<div>Strong electronic correlations in many-body systems are cradles of new physics. They give birth to novel collective states hosting emergent quasiparticles as well as intriguing geometrical charge patterns. Two-dimensional electron gas in GaAs/AlGaAs under perpendicular magnetic field is one of the most well-known hosts in condensed matter physics where a plethora of the collective states appear. In the strong magnetic field regime, strong Coulomb interactions among the electrons create emergent quasiparticles, i.e. composite fermions and Cooper-paired composite fermions. In the weak magnetic field regime, modified Coulomb interactions drive electron solid phases having geometrical charge patterns in the shape of stripes and bubbles and lower the spatial symmetry of the states.</div><div><br></div><div>The fascinating charge order in bubble geometry is the electron bubble phase predicted first by the Hartree-Fock theory. In a bubble phase, certain number of electrons cluster as an entity called bubble and the bubbles order into a crystal of triangular lattice. In addition to the Hartree-Fock theory, the density matrix renormalization group and the exact diagonalization methods further support the formation of electronic bubbles.</div><div><br></div><div>Reentrant integer quantum Hall states are commonly accepted as the manifestations of the bubble phases in transport experiment. Soon after the first prediction of the Hartree-Fock theory, the reentrant integer quantum Hall states were observed in the third and higher Landau levels. Since then, the association to the bubble phases has been tested with different experimental techniques for decades.</div><div><br></div><div>Although the experimental results from different methods support the bubble phase picture of the reentrant integer quantum Hall states, the electron confinement under the quantum well structure hindered direct scanning of bubble morphology. Thus none of the experiments could showcase the bubble morphology of the reentrant integer quantum Hall states. Meanwhile, a significant discrepancy still remained in between the bubble theories and the experiments. Even though the bubble theories predict the proliferation of bubble phases with increasing orbital index, none of the experiments could observe multiple reentrant integer quantum Hall states in a high Landau level, which signify the multiple bubble formation. Therefore, the proliferation of bubble phases with increasing Landau level index was pessimistic. </div><div><br></div><div>In this Dissertation, I present my research on solving this discrepancy. In chapter 4, we performed a magnetotransport measurement of reentrant integer quantum Hall states in the third and higher Landau levels at various different temperatures. Then, we scrutinized how each of the reentrant integer quantum Hall states develops with the gradual increase of the temperature. As a result, we observed multiple reentrant integer quantum Hall states in the fourth Landau level which are associated with the two- and three-electron bubble phases. This result strongly supports the bubble phase picture of the reentrant integer quantum Hall states by confirming the possibility of the proliferation of bubble phases in high Landau levels.</div><div><br></div><div>In chapter 5, I analyzed the energetics of newly resolved two- and three-electron bubble phases in the fourth Landau level as well as those of two-electron bubble phases in the third Landau level. Here, I first found, in the fourth Landau level, the three-electron bubbles are more stable than the two-electron bubbles indicating that the multi-electron bubbles with higher electron number are more stable within a Landau level. Secondly, I found distinct energetic features of two- and three-electron bubble phases which are independent of Landau level index throughout the third and the fourth Landau levels. These results highlight the effect of the number of electrons per bubble on the energetics of multi-electron bubble phases and are expected to contribute on improving the existing Hartree-Fock theories.</div>

Une étude expérimentale de gaz d'électrons bidimensionnel confinés dans des puits quantiques de CdTe et de CdMnTe est présentée. L'analyse de données est soutenue par des calculs numériques de la structure de bande des états confinés, utilisant l'approximation de densité locale et de fonction enveloppe. Un calcul de type k.p a été utilisé pour justifier l'approximation parabolique appliquée pour les bandes valence. Les échantillons ont été caractérisés par spectroscopie Raman et par spectroscopie d'absorption de la résonance cyclotron infrarouge. Le magnéto-transport à bas champ est dominé par la contribution semi-classique de Drude et révèle trois contributions plus faibles, qui sont la localisation faible, l'interaction électron-électron et les oscillations Shubnikov-De Haas. La contribution des interactions électron-électron est expliquée dans un modèle semi-classique à trajectoire circulaire. La forme des niveaux de Landau, leurs élargissement, les temps de vie transport et quantique de la diffusion et le mécanisme (long-portée) de la diffusion dominant ont été déterminés. Le magnéto-transport sous champs magnétiques intenses révèle la présence d'états Hall quantique fractionnaires dans les niveaux de Landau N=0 et N=1. Nous avons montré, que les états 5/3 et 4/3 étaient complètement polarisés en spin, en accord avec l'approche des fermions composites pour l'effet Hall quantique fractionnaire. La forme de la photoluminescence à champ magnétique nul et son évolution avec la température sont décrites par un modèle analytique simple. La dépendance en champ magnétique et en température de la photoluminescence indique que le gap de spin est amplifié dans les niveaux de landau entièrement occupés. Ces effets multi-corps de l'amplification du gap du spin ont été décrits avec succès par un modèle numérique simple. L'intensité de la photoluminescence a mise en évidence l'importance des processus non-radiatifs pendant la recombinaison, la dégénérescence des niveaux de Landau, leur taux d'occupation, les règles de sélection et l'influence de l'écrantage. Le mécanisme de la relaxation parallèle de spin d'électron et de trou a été identifié et attribué au mécanisme Bir-Aharonov-Pikus, assistée par les phonons acoustiques. Les spectres de photoluminescence d'excitation reflètent la densité des états caractéristique des systèmes bidimensionnels. Les résonances excitoniques, qui sont observées aux bords des sous-bandes électriques inoccupées, illustrent l'importance de l'écrantage et des champs électriques intrinsèques dans les puits asymétriquement dopés. / Experimental studies of two-dimensional electron gases confined in CdTe and CdMnTe quantum wells are presented. The data analysis is supported by numerical calculations of the band structure of confined states, using the local density and envelope function approximations. Four by four, k.p calculations have been performed to justify the parabolic approximation of valence bands. Samples were characterized by Raman scattering spectroscopy and far infrared cyclotron resonance absorption measurements. Low-field magneto-transport shows the dominant contribution of the semi-classical Drude conductivity and ten times weaker contributions of weak localization, electron-electron interaction and Shubnikov-de Haas oscillations. The contribution of electron-electron interactions is explained within a semi-classical model of circling electrons. The shape of Landau levels, broadening, transport and quantum lifetimes and dominant long-range scattering mechanism have been determined. High-field magneto-transport displays fractional quantum Hall states at Landau levels N=0 and N=1. The ground states 5/3 and 4/3 have been determined to be fully spin polarized, in agreement with the approach of composite fermions for the fractional quantum Hall effect. The form of the photoluminescence at zero magnetic field and its evolution with temperature have been described by simple analytical model. Magnetic field and temperature dependence of the photoluminescence has been found to display the enhanced spin splitting of fully occupied Landau levels. This many body enhanced spin gap has been successfully described by a numerical model. The intensity of the photoluminescence demonstrated the importance of the non-radiative recombination channel, degeneracy of Landau levels, their occupation, selection rules and screening. The mechanism of the simultaneous electron and hole spin-flip was recognized and attributed to the longitudinal acoustical phonon assisted Bir-Aharonov-Pikus spin relaxation mechanism. Photoluminescence excitation spectra embody the characteristic density of states of two-dimensional systems. The excitonic resonances, which are observed at the edges of unoccupied electric subbands, illustrate the importance of screening and internal electric fields in asymmetrically doped quantum wells.