<p>In this overview I describe the simplest models for the quantum Hall and quantum spin Hall effects, and give some general indications as to the description of topological insulators. As a background to the theoretical models I will first trace the development leading up to the description of topological insulators . Then I will present Laughlin's original model for the quantum Hall effect and briefly discuss its limitations. After that I will describe the Kane and Mele model for the quantum spin Hall effect in graphene and discuss its relation to a general quantum spin Hall system. I will conclude by giving a conceptual description of topological insulators and mention some potential applications of such states.</p>

In this overview I describe the simplest models for the quantum Hall and quantum spin Hall effects, and give some general indications as to the description of topological insulators. As a background to the theoretical models I will first trace the development leading up to the description of topological insulators . Then I will present Laughlin's original model for the quantum Hall effect and briefly discuss its limitations. After that I will describe the Kane and Mele model for the quantum spin Hall effect in graphene and discuss its relation to a general quantum spin Hall system. I will conclude by giving a conceptual description of topological insulators and mention some potential applications of such states.

La loi de Moore est basée sur l’observation empirique qu’environ chaque deux années, le nombre de transistors dans des circuits denses intégrées double. Cette tendance s'est bien maintenue au cours des dernières décennies (années 1970 et suivantes). Cependant, la miniaturisation continue des transistors entraîne une augmentation significative des pertes d’énergie par le courant de fuite, ce qui augmente la consommation d'énergie de veille. Cette perte d’énergie est devenue un problème majeur dans la microélectronique pendant les dernières années, ce qui rend plus difficile le développement des nouvelles technologies. L’une des solutions est de placer des éléments mémoire non-volatile dans le puce, qui retiennent la configuration du transistor pendant la mise hors tension et permettent de le restaurer à la mise sous tension. Les Magnetic Random Access Memories (MRAM) sont considérées par l'ITRS comme un candidat crédible pour le remplacement potentiel de SRAM et de DRAM au-delà du nœud technologique de 20 nm. Bien que les exigences de base pour la lecture et l'écriture d'un élément de mémoire unique sont remplies, l'approche actuelle basée sur Spin Torque Transfer (STT) souffre d'un manque inné de la flexibilité. Le courant électrique entraine le retournement de l’aimantation de la couche ferromagnétique libre par le transfert du moment angulaire d’une couche ferromagnétique adjacent. Ainsi les éléments de mémoire basées sur STT ont deux terminaux dont les voies de courant pour « écriture » et « lecture » sont définies par la forme de «pillar». L’optimisation indépendant des paramètres d’écriture et de lecture reste, donc, très difficile. Au même temps, la densité de courant trop haute, nécessaire pour écrire, conduit à la vieillissement prémature du jonction tunnel. En conséquence, l’intégration MRAM dans la technologie du semi-conducteur reste, donc, difficile.Démonstrations récentes de reversement d’aimantation entrainées par l’injection d’un courant planaire dans des heterostructures métal lourd/ferromagnet ont attiré l’attention croissante sur les couples de spin basé sur le transfert du moment angulaire par l’effet Hall de spin et les effets d’interface. Contrairement à STT-MRAM, la SOT-MRAM a trois terminaux, dont les voies de courant pour « écriture » et « lecture » sont indépendantes. Cela permet d’améliorer les paramètres « écriture » et « lecture » de manière indépendante. Pour contrôler et optimiser les SOT il est nécessaire de comprendre très bien leur origine. Cela reste l’une des plus importantes questions dont on n’a pas une réponse définitive. Dans ce contexte, plusieurs études ont conclu sur un modèle basé seulement sur l’effet Hall de spin, en même temps que d’autres ont suggéré un modèle basé sur une contribution combiné de l’effet Hall de spin et l’effet d’interface.L’objectif de cette thèse est de réaliser une étude systématique sur les effets d’interface sur les SOT dans des heterostructures métal lourde/ferromagnet a base de Pt, avec aimantation planaire.Dans ce but, cette thèse explore trois voies différentes. Premièrement nous avons modifié le rapport entre les effets d’interface et les effets bulk en changeant l’épaisseur de la couche de Pt et en suivant l’évolution des SOT. En deuxième nous avons exploré des différents empilements métal lourde/ferromagnet afin d’étudier différentes interfaces. Finalement, nous avons changé les propriétés des interfaces soit par changer la structure cristalline soit par oxydation. La technique de mesure, la méthode d’analyse de données associé et les aspects théoriques nécessaires pour l’interprétation des données sont aussi détaillés dans ce manuscrit. / Moore’s law is based on empirical observation and states that every two years approximately, the number of transistors in dense integrated circuits doubles. This trend has held up well in the past several decades (1970s and onwards). However, the continuous miniaturisation of transistors brings about a significant increase in leakage current, which increases the stand-by power consumption. This energy loss has become a major problem in microelectronics during the last several years, making the development of new technologies more difficult. One of the solutions that can address this issue is to place non-volatile memory elements inside the chip, that retain the configuration of the transistor during power-off and allow to restore it at power-on. Magnetic Random Access Memories (MRAM) are considered by the ITRS as a credible candidate for the potential replacement for SRAM and DRAM beyond the 20 nm technological node. Though the basic requirements for reading and writing a single memory element are fulfilled, the present approach based on Spin Transfer Torque (STT) suffers from an innate lack of flexibility. The electric current drives the magnetization switching of a free ferromagnetic layer by transferring angular momentum from an adjacent ferromagnet. Therefore, STT-based memory elements are two terminal devices in which the “pillar” shape defines both the “read” and the “write” current paths. Independent optimisation of the reading and writing parameters is therefore difficult, while the large writing current density injected through the tunnel barrier causes its accelerated ageing, particularly for fast switching. Consequently, the integration of MRAM into semiconductor technology poses significant difficulties.Recent demonstrations of magnetization switching induced by in-plane current injection in heavy metal (HM)/ferromagnet (FM) heterostructures have drawn increasing attention to spin-torques based on orbital-to-spin momentum transfer induced by Spin Hall and interfacial effects (SOTs). Unlike STT-MRAM, the in-plane current injection geometry of SOT-MRAM allows for a three-terminal device which decouples the “read” and “write” mechanisms, allowing the independent tuning of reading and writing parameters. However, an essential first step in order to control and optimise the SOTs for any kind of application, is to better understand their origin. The origin of the SOTs remains one of the most important unanswered questions to date. While some experimental studies suggest a SHE (Spin Hall Effect)-only model for the SOTs, others point towards a combined contribution of the bulk (SHE) and interface (Rashba Effect and Interfacial SHE). At the same time, many studies start with a SHE only hypothesis and do not consider interfacial effects. Furthermore, there are not so many systematic studies on the effects of interfaces. This thesis tries to fill in this gap, by providing a systematic study on the effects of interfaces on the SOTs, in Pt-based NM/FM/HM multilayers with in-plane magnetic anisotropy. For this purpose, this thesis explores three different, but related avenues. First, we changed the interface/bulk effect ratio by modifying the Pt thickness and following the evolution of the SOTs. Second, we explored different HM/FM/NM combinations, in order to study different interfaces. And third, we changed the properties of the interfaces by changing the crystallographic structure of the interface and by oxidation. The measurement technique and associated data analysis method, as well as the theoretical considerations needed for the interpretation of the results are also detailed in this manuscript.

Neste trabalho foram estudadas as fases topológicas não triviais presentes em sistemas formados pela deposição de átomos de grafeno. Encontramos que quando um átomo hibridiza fortemente com o grafeno, apresenta um momento magnético e um forte spin-órbirta é possível a formação de uma rede hexagonal de baricentros que efetivamente gera uma estrutura de bandas característica de um efeito hall quântico anômalo. Especificamente, determinamos que o Ru satisfaz estas características. Quando este metal é depositado em uma configuração triangular no grafeno ocorrem picos na densidade de estados localizados no centro geométrico (baricentro) dos triângulos formados pelos Ru. Estes picos estão distribuídos de forma hexagonal e efetivamente geram uma estrutura de bandas que nas proximidades do nível de Fermi apresenta uma configuração de spin característica do efeito Hall quântico anômalo. Adicionalmente, encontramos que o sistema composto pela absorção de Ba ou Sr no grafeno favorece a formação do efeito Hall quântico de spin. Neste sistema, o acoplamento spin-órbita (SOC) gera um gap mais de 1000 vezes maior ao período no grafeno prístino. Para o estudo destes sistemas, implementamos no código SIESTA a aproximação on-site do acoplamento spin-órbita via o formalismo dos pseudopotenciais relativísticos de norma conservada. Nossa implementação foi testada a partir do estudo de fenômenos já conhecidos: i) o strong spin-splitting gerado no grafeno pela adsorção de Au, ii) o efeito hall quântico de spin no poço quântico de HgTe/CdTe e, iii) a formação de estados topológicos na superfície do Bi2Se3 e as fases magnéticas deste material com átomos de Mn adsorvidos. / In this work, were studied the non-trivial topological phases present in systems formed by deposition of atoms in graphene. We found that when an atom hybridizes strongly with grapheme, has a magnetic moment and a strong spin-orbit it is possible the formation of a hexagonal network of barycentres that effectively generates a structure band characteristic of a quantum anomalous Hall effect. Specifically, we determined that Ru satisfies these characteristics. When this metal is deposited in a triangular configuration in grapheme, peaks occur in the density of localized states in the geometric center (centroid) of the triangles formed by Ru. These peaks are distributed in a hexagonal structure and effectively generates a band structure that near the Fermi level has a spin configuration characteristic of the spin quantum Hall effect anomalous. Additionally, we found that the system composed by the adsorption of Ba or Sr in grapheme, promotes the formation of spin quantum Hall effect. In this system, the spin-orbit coupling (SOC) generates a gap more than 1000 times grater that predicted in pristine praphene. To study these systems, wu implemented in the code SIESTA the on-site approach of the spin-orbit coupling throught the formalism of norm conserved relativistic pseudo potentials. Our implementation was tested from the study of phenomena already known: i) the strong spin-splitting generated in graphene by adsorption of Au, ii) the quantum spin Hall effect in quantum well of HgTe / CdTe and, iii) formation of topological states in the surface of Bi2Se3 and the magnetic of this material with Mn atoms adsorved.

Neste trabalho foram estudadas as fases topológicas não triviais presentes em sistemas formados pela deposição de átomos de grafeno. Encontramos que quando um átomo hibridiza fortemente com o grafeno, apresenta um momento magnético e um forte spin-órbirta é possível a formação de uma rede hexagonal de baricentros que efetivamente gera uma estrutura de bandas característica de um efeito hall quântico anômalo. Especificamente, determinamos que o Ru satisfaz estas características. Quando este metal é depositado em uma configuração triangular no grafeno ocorrem picos na densidade de estados localizados no centro geométrico (baricentro) dos triângulos formados pelos Ru. Estes picos estão distribuídos de forma hexagonal e efetivamente geram uma estrutura de bandas que nas proximidades do nível de Fermi apresenta uma configuração de spin característica do efeito Hall quântico anômalo. Adicionalmente, encontramos que o sistema composto pela absorção de Ba ou Sr no grafeno favorece a formação do efeito Hall quântico de spin. Neste sistema, o acoplamento spin-órbita (SOC) gera um gap mais de 1000 vezes maior ao período no grafeno prístino. Para o estudo destes sistemas, implementamos no código SIESTA a aproximação on-site do acoplamento spin-órbita via o formalismo dos pseudopotenciais relativísticos de norma conservada. Nossa implementação foi testada a partir do estudo de fenômenos já conhecidos: i) o strong spin-splitting gerado no grafeno pela adsorção de Au, ii) o efeito hall quântico de spin no poço quântico de HgTe/CdTe e, iii) a formação de estados topológicos na superfície do Bi2Se3 e as fases magnéticas deste material com átomos de Mn adsorvidos. / In this work, were studied the non-trivial topological phases present in systems formed by deposition of atoms in graphene. We found that when an atom hybridizes strongly with grapheme, has a magnetic moment and a strong spin-orbit it is possible the formation of a hexagonal network of barycentres that effectively generates a structure band characteristic of a quantum anomalous Hall effect. Specifically, we determined that Ru satisfies these characteristics. When this metal is deposited in a triangular configuration in grapheme, peaks occur in the density of localized states in the geometric center (centroid) of the triangles formed by Ru. These peaks are distributed in a hexagonal structure and effectively generates a band structure that near the Fermi level has a spin configuration characteristic of the spin quantum Hall effect anomalous. Additionally, we found that the system composed by the adsorption of Ba or Sr in grapheme, promotes the formation of spin quantum Hall effect. In this system, the spin-orbit coupling (SOC) generates a gap more than 1000 times grater that predicted in pristine praphene. To study these systems, wu implemented in the code SIESTA the on-site approach of the spin-orbit coupling throught the formalism of norm conserved relativistic pseudo potentials. Our implementation was tested from the study of phenomena already known: i) the strong spin-splitting generated in graphene by adsorption of Au, ii) the quantum spin Hall effect in quantum well of HgTe / CdTe and, iii) formation of topological states in the surface of Bi2Se3 and the magnetic of this material with Mn atoms adsorved.

Closed loop Hall-effect current sensors used in power electronic applications require high bandwidth and small transient errors. For this, the behaviour of a closed loop Hall-e ect current sensor is modeled. Analytical expression of the step response of the sensor using this model is used to evaluate the performance of the PI compensator in the current sensor. Based on this expression a procedure is proposed to design parameters of the PI compensator for fast dynamic performance and for small transient error. A prototype closed loop Hall-effect current sensor is built in the laboratory. A PI compensator based on the procedure devised earlier is designed for the sensor. A power electronic converter based current source is designed and fabricated in the labo-ratory for validation of steady state and transient performance of Hall-effect current sensors. A novel hardware topology is proposed, using which the same hardware set-up can produce both step current and sinusoidal current in its designated sections without any modi cation in the hardware con guration. It produces step current of controlled peak value upto 100A and controlled rate of change with both positive and negative dtdi . The step transition time is less than 200ns. The dtdi is adjustable upto a limit of 300A/ s to verify the dtdi following capability of the sensor. The same current source produces continuous sinusoidal current of controlled magnitude upto 75A peak and controlled frequency from 1Hz to 1000Hz. The magnitude and the frequency of the sinusoidal current can be varied on-line like a voltage function generator. The hardware of the current source is designed to consume minimal ac-tive power from mains during continuous sinusoidal current generation. This current source is used in experimental veri cation of the steady state and the transient performance of the designed laboratory current sensor. The transient performance of the laboratory current sensor is observed to be superior to state-of-the-art commercial current sensors available for power electronic applications.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The experimental observation of the quantum Hall effect in a two-dimensional electron gas posed an intriguing question to theorists: Why is the quantization of conductance so precise, given the imperfections of the measured samples? The question was answered a few years later, when a connection was uncovered between the quantum Hall effect and topological quantities associated with the band structure of the material in which it is observed. The Hall conductance was revealed to be an integer topological invariant, implying its robustness to certain perturbations. The topological theory went further than explaining only the usual integer quantum Hall effect in a perpendicular magnetic field. Soon it was realized that it also applies to certain systems in which the total magnetic flux is zero. Thus it is possible to have a quantized Hall effect without Landau levels. We study a carbon nanotube in a magnetic field perpendicular to its axial direction. Recent studies suggest that the application of an electric field parallel to the magnetic field would induce a gap in the electronic spectrum of a previously metallic carbon nanotube. Despite the quasi onedimensional nature of the carbon nanotube, the gapped state supports a quantum Hall effect and is associated with a non zero topological invariant. This result is revealed when an additional magnetic field is applied parallel to the axis of the carbon nanotube. If the flux due to this magnetic field is varied by one flux quantum, exactly one electron is transported between the ends of the carbon nanotube. / AFRIKAANSE OPSOMMING: Die eksperimentele waarneming van die kwantum Hall effek in ’n twee-dimensionele elektron gas laat ’n interessante vraag aan teoretiese fisikuste: Waarom sou die kwantisasie van die geleiding so presies wees al bevat die monsters, waarop die meetings gedoen word, onsuiwerhede? Hierdie vraag word ’n paar jaar later geantwoord toe ’n konneksie tussen die kwantum Hall effek en topologiese waardes, wat verband hou met die bandstruktuur van die monster, gemaak is. Dit is aan die lig gebring dat die Hall geleiding ’n heeltallige topologiese invariante is wat die robuustheid teen sekere steurings impliseer. Die topologiese teorie verduidelik nie net die gewone kwantum Hall effek wat in ’n loodregte magneetveld waargeneem word nie. Dit is ook moontlik om ’n kwantum Hall effek waar te neem in sekere sisteme waar die totale magneetvloed nul is. Dit is dus moontlik om ’n gekwantiseerde Hall effek sonder Landau levels te hˆe. Ons bestudeer ’n koolstofnanobuis in ’n magneetveld loodreg tot die aksiale rigting. Onlangse studies dui daarop dat die toepassing van ’n elektriese veld parallel aan die magneetveld ’n gaping in die elektroniese spektrum van ’n metaliese koolstofnanobuis induseer. Ten spyte van die een-dimensionele aard van die koolstofnanobuis ondersteun die gapings-toestand steeds ’n kwantum Hall effek en hou dit verband met ’n nie-nul topologiese invariante. Hierdie resultaat word openbaar wanneer ’n bykomende magneetveld parallel tot die as van die koolstofnanobuis toegedien word. Indien die vloed as gevolg van hierdie magneetveld met een vloedkwantum verander word, word presies een elektron tussen die twee kante van die koolstofnanobuis vervoer.

published_or_final_version / abstract / Physics / Doctoral / Doctor of Philosophy

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. / text

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