No description available.

Topological insulators (TIs) are recently predicted, and much studied, new quantum materials. These materials are characterised by their unique surface electronic properties; namely, behaving as band insulators within their bulk, but with spin-momentum locked surface or edge states at their interface. These surface/edge crossing states are protected by the underlying time-reversal symmetry (TRS) of the bulk band structure, leading to a robust topological surface state (TSS) that is resistant to scattering from impurities which do not break TRS. Their surface band dispersion has a characteristic crossing at time reversal invariant momenta (TRIM) called a Dirac cone. It has been predicted that the introduction of a TRS breaking effect, through ferromagnetic order for instance, will open a band-gap in this Dirac cone. It can be seen that magnetic fields are not time reversal invariant by considering a solenoid. If time is reversed, the current will also reverse in the solenoid and so the magnetic field will also be reversed. So it can be seen that magnetic fields transform as odd under time reversal, the same will be true of internal magnetisation. By manipulating this gapped surface state a wide range of new physical phenomena are predicted, or in some cases, already experimentally observed. Of particular interest is the recently observed quantum anomalous Hall effect (QAHE) as well as, e.g., topological magneto-electric effect, surface Majorana Fermions and image magnetic monopoles. Building on these novel physical effects, it is hoped to open new pathways and device applications within the emerging fields of spintronics and quantum computation. This thesis presents an investigation of the nature of magnetic doping of the chalcogenide TIs Bi₂Se₃ and Bi₂Te₃ using 3d transition-metal dopants (Mn and Cr). Samples were grown by molecular beam epitaxy (MBE), an ideal growth method for the creation of high-quality thin film TI samples with very low defect densities. The grown films were investigated using a range of complementary lab-based and synchrotron-based techniques to fully resolve their physical structure, as well as their magnetic and electronic properties. The ultimate aim being to form a ferromagnetic ground state in the insulating material, which may be expanded into device applications. Samples of bulk Mn-doped Bi₂Te₃ are presented and it is shown that a ferromagnetic ground state is formed below a measured T_C of 9-13 K as determined by a range of experimental methodologies. These samples are found to have significant inhomogeneities within the crystal, a problem that is reduced in MBE-grown crystals. Mn-doped Bi₂Se₃ thin films were grown by MBE and their magnetic properties investigated by superconducting quantum interference device (SQUID) magnetometry and x-ray magnetic circular dichroism (XMCD). These reveal a saturation magnetisation of 5.1 μ_B/Mn and show the formation of short-range magnetic order at 2.5 K (from XMCD) with indication of a ferromagnetic ground state forming below 1.5 K. Thin films of Cr-doped Bi₂Se₃ were grown by MBE, driven by the recent observation of the QAHE in Cr-doped (Bi_1−xSb_x)₂Te₃. Investigation by SQUID shows a ferromagnetic ground state below 8.5 K with a saturation magnetisation of 2.1 μ_B/Cr. Polarised neutron reflectometry shows a uniform magnetisation profile with no indication of surface enhancement or of a magnetic dead layer. Further studies by extended x-ray absorption fine structure (EXAFS) and XMCD elucidate the electronic nature of the magnetic ground state of these materials. It is found that hybridisation between the Cr d and Se p orbitals leads to the Cr being divalent when doping on the Bi^3+ site. This covalent character to the electronic structure runs counter to the previously held belief that divalent Cr would originate from Cr clusters within the van der Waals gap of this material. The work overall demonstrates the formation of a ferromagnetic ground state for both Cr and Mn doped material. The transition temperature, below which ferromagnetic order is achieved, is currently too low for usable device applications. However, these materials provide a promising test bed for new physics and prototype devices.

Cette thèse poursuit deux directions dans le domaine des isolants et supraconducteurs topologiques.Dans la première partie de la thèse nous étudions des isolants en deux dimensions sur réseau, présentant un effet Hall quantique anormal (c'est-à-dire en l'absence d'un champ magnétique externe), induit par la présence d'un flux magnétique inhomogène dans la maille. Le système possède des phase isolantes caractérisés par un invariant topologique, le nombre de Chern, qui est lié à la conductance portée par le bord états. Nous montrons que les modèles à deux bandes admettent des phase à nombre de Chern arbitraire, ou, de façon équivalente, un nombre arbitraire d'états de bord, quand on augmente la portée des couplages sur réseau. Cette compréhension est rendue possible grâce à la démonstration d'une formule montrant que le nombre de Chern d'une bande dépend de certains propriétés d'un ensemble discret de points dans la zone de Brillouin, les points de Dirac en l'absence du gap. Ces idées sont rendues plus concrètes dans l'étude du modèle de Haldane et dans la création d'un modèle artificiel avec cinq phases de Chern dont les états de bord sont déterminés en détail. La deuxième partie de la thèse porte sur les supraconducteurs topologiques unidimensionnels qui exhibent des états exotiques d'énergie zéro: les états liés de Majorana. Nous étudions ici la présence de fermions de Majorana dans des fils de semiconducteurs à fort couplage spin-orbite sous l’effet de proximité d'un supraconducteur d'onde s. Nous montrons que la polarisation de spin des degrés de liberté électroniques dans la fonction d'onde Majorana dépend du poids relatif du couplage spin-orbite Dresselhaus et Rashba. Nous étudions également les fermions de Majorana dans des jonctions linéaires longues supraconducteur-normal et supraconducteur-normal-supraconducteur (SNS) où ils apparaissent comme des états étendus dans la jonction normale. En outre, la géométrie d'anneaux peut être mise en correspondance avec une jonction SNS, et, sous l'action de gradients dans la phase supraconductrice, des fermions Majorana étendus se forment encore à l'intérieur du fil normal. Enfin, un modèle à deux bandes avec des fermions de Majorana multiples est traité. Nous démontrons que les jonctions Josephson construites à partir de ce modèle maintiennent l'une des signatures remarquables des fermions de Majorana, à savoir la périodicité 4π de l'effet Josephson fractionnaire. / This thesis follows two threads in the field of topological insulators and superconductors. The first part of the thesis is devoted to the study of two-dimensional quantum anomalous Hall insulators on a lattice, in the absence of an external magnetic flux, but induced by an inhomogeneous flux in the unit cell. The system possesses several gapped phases characterized by a topological invariant, the Chern number, that is related to the conductance carried by the edge states. Here we show that two-band models admit an arbitrary large number of Chern phases or, equivalently, an arbitrary number of edge states, by adding hopping between distant neighbor sites. This result is based on a formula proving that the Chern number of a band depends on certain properties of a finite set of points in the Brillouin zone, i.e. the Dirac points for the gapless system. These ideas are made more concrete in the study of a modified Haldane model, and also by creating an artificial model with five Chern phases, whose edge states are determined in detail. The second part of the thesis focuses on one-dimensional topological superconductors with exotic zero-energy edge states: the Majorana bound states. Here we investigate the presence of Majorana fermions in spin-orbit coupled semiconducting wire in proximity to an s-wave superconductor. We show that the spin-polarization of the electronic degrees of freedom in the Majorana wave function depends on the relative weight of Dresselhaus and Rashba spin-orbit couplings. We also investigate Majorana fermions in linear superconductor-normal and long superconductor-normal-superconductor (SNS) junctions where they appear as extended states in the normal junction. Furthermore, ring geometries can be mapped to an SNS junction, and, we have shown that under the action of superconducting phases gradients, extended Majorana fermions can form again inside the normal wire. Finally a two-band model with multiple Majorana fermions is treated and we show that Josephson junctions built from this model maintain the 4π periodicity for the fractional Josephson effect, one of Majorana fermions signatures.

Nessa tese de doutorado apresentamos um estudo da estrutura eletronica de materiais isolantes topologicos. A teoria fundamental dos isolantes topologicos foi abordada atraves de invariantes topologicos Z2, assim como os seus metodos para o calculo desses invariantes topologicos e as consequencias da topologia de bandas nao-trivial. Assim como as propriedades atomisticas e energeticas, as propriedades eletronicas de alguns isolantes topologicos foram calculadas atraves de metodos de primeiros principios baseados na Teoria do Funcional da Densidade. Apresentamos nessa tese o estudo de quatro sistemas de interesse fisico: (1) Em isolantes topologicos do tipo Bi2Se3 e Bi2Te3 com falhas de empilhamentos, encontramos que o Bi2Te3 com falhas de empilhamentos apresentam estados metalicos na regiao do defeito; (2) Na interface Bi2Se3/GaAs com tratamento de Se na regiao do GaAs, encontramos que a interacao entre o cone de Dirac do Bi2Se3 com a banda de valencia do GaAs abre um gap de energia no ponto ; (3) Em nanoestradas de germaneno imersas em germanano com interfaces zigzag, encontramos que a partir de uma largura critica podemos observar o efeito Hall quantico de spin; e (4) nas ligas desordenadas hexagonais de SixGe1-x em duas dimensoes, o sistema desordenado compartilha a mesma topologia de bandas do siliceno e do germaneno, enquanto que a liga ordenada Si0.5Ge0.5 e um isolante trivial. As estruturas eletronicas desses sistemas foram investigadas no intuito de entender as consequencias fisicas da topologia de bandas nao-trivial nos estados de Bloch de bulk e de superficies/interfaces. / In this doctoral thesis we present a study of the electronic structure of topological insulators materials. The fundamental theory of topological insulators was addressed through the Z2 topological invariants, as well as their methods to calculate these topological invariants and the consequences of non-trivial band topology. Just as atomistic and energetic properties, the electronic properties of some topological insulators were calculated using first-principles methods based upon Density Functional Theory. We present in this thesis the study of four systems of physical interest: (1) In topological insulators like Bi2Se3 and Bi2Te3 with stacking faults, we found that the Bi2Te3 with stacking faults presents metallic states in the region of the defect; (2) For Bi2Se3/GaAs interface with Se-treatment in the GaAs region, we found that the interaction between the Dirac cone of the Bi2Se3 and the valence band of the GaAs opens a bandgap at the -point; (3) In germanene nanoroads embedded on germanane with zigzag interfaces/edge, we found that from a critical width we can observe the quantum spin Hall effect; and (4) For SixGe1x two-dimensional hexagonal disordered alloy, the system shares the same non-trivial band topology of the silicene and germanene, while the ordered alloy Si0.5Ge0.5 is a trivial insulator. The electronic structures of these systems were investigated in order to understand the physical consequences of non-trivial band topology in the bulk and surfaces/interfaces Bloch states.

Neste trabalho estudamos uma rede hexagonal com uma cadeia de impurezas nas bordas e com supercondutividade induzida, de forma a mostrar a existência de fases com férmions de Majorana. Para tal, começamos introduzindo invariantes topológicos, número de Chern e Z2 e mostramos dois modelos para rede hexagonal. O primeiro, modelo de Haldane, fazemos como motivação histórica. O segundo, modelo de Kane-Mele, é usado como base para todo o trabalho. Seguimos introduzindo supercondutividade e como ela ocorre quando aplicada junto do Modelo de Kane-Mele, o método auto-consistente e quais as condições necessárias para termos supercondutividade apenas nas bordas. Continuamos com efeitos de impurezas magnéticas nas bordas e introduzimos férmions de Majorana que são os alvos principais dos resultados. Mostramos então, que existe fases topológicas em cadeias de impureza magnética, com momentos em espiral, contudo o diagrama de fase depende de várias condições. Por fim, mostramos que a variação da fase topológica se deve a oscilações nos níveis de energia em que o invariante topológico também varia, contrariando resultados obtidos para a rede quadrada. Concluímos esse trabalho com implicações experimentais desse resultado e possíveis caminhos que podem ser seguidos. / In this work, we study a honeycomb lattice with induced superconductivity and edge impurity in order to show the existence of a phase that host Majorana bound state. To do so, we start introducing topological invariants, Chern number and Z2, and we show two models for honeycomb lattice. The first, Haldane\'s Model, due its historical importance. The second, Kane-Mele model, because it will be used during all this work. Then we review superconductivity, showing the self-consistent method, and we apply it to Kane-Mele model, in which we find some necessary conditions to induce superconductivity only at the edges. From this point, we study the effect of magnetic impurities at the edges, and we introduce Majorana bound state, that will be the main objective of our results. In our results, we show the existence of topological non-trivial phases for spiral magnetic chain in the zigzag edge. With this we make a phase diagram. We also find oscillation in the energy spectrum and the topological phase changes with the oscillation, this is different from square lattice in which we should not have a change in the topological phase. We conclude this work with experimental implications of our result and possible developments.

The following dissertation presents a comprehensive study of the interaction between Dirac fermion quasiparticles (DFQs) and surface phonons on the surfaces of the topological insulators Bi2Se3 and Bi2Te3. Inelastic helium atom surface scattering (HASS) spectroscopy and time of flight (TOF) techniques were used to measure the surface phonon dispersion of these materials along the two high-symmetry directions of the surface Brillouin zone (SBZ). Two anomalies common to both materials are exhibited in the experimental data. First, there is an absence of Rayleigh acoustic waves on the surface of these materials, pointing to weak coupling between the surface charge density and the surface acoustic phonon modes and potential applications for soundproofing technologies. Secondly, both materials exhibit an out-of-plane polarized optical phonon mode beginning at the SBZ center and dispersing to lower energy with increasing wave vector along both high-symmetry directions of the SBZ. This trend terminates in a V-shaped minimum at a wave vector corresponding to 2kF for each material, after which the dispersion resumes its upward trend. This phenomenon constitutes a strong Kohn anomaly and can be attributed to the interaction between the surface phonons and DFQs. To quantify the coupling between the optical phonons experiencing strong renormalization and the DFQs at the surface, a phenomenological model was constructed based within the random phase approximation. Fitting the theoretical model to the experimental data allowed for the extraction of the matrix elements of the coupling Hamiltonian and the modifications to the surface phonon propagator encoded in the phonon self energy. This allowed, for the first time, calculation of phonon mode-specific quasiparticle-phonon coupling λⱱ(q) from experimental data. Additionally, an averaged coupling parameter was determined for both materials yielding ¯λ^Te ≈ 2 and ¯λ^Se ≈ 0.7. These values are significantly higher than those of typical metals, underscoring the strong coupling between optical surface phonons and DFQs in topological insulators. In an effort to connect experimental results obtained from phonon and photoemission spectroscopies, a computational process for taking coupling information from the phonon perspective and translating it to the DFQ perspective was derived. The procedure involves using information obtained from HASS measurements (namely the coupling matrix elements and optical phonon dispersion) as input to a Matsubara Green function formalism, from which one can obtain the real and imaginary parts of the DFQ self energy. With these at hand it is possible to calculate the DFQ spectral function and density of states, allowing for comparison with photoemission and scanning tunneling spectroscopies. The results set the necessary energy resolution and extraction methodology for calculating ¯λ from the DFQ perspective. Additionally, determining ¯λ from the calculated spectral functions yields results identical to those obtained from HASS, proving the self-consistency of the approach.

Orientadores: Pascoal José Giglio Pagliuso, Ricardo Rodrigues Urbano / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-24T14:58:04Z (GMT). No. of bitstreams: 1 Garitezi_ThalesMacedo_D.pdf: 27111125 bytes, checksum: d3a94f446fa3f67a500d9adc5f733dc0 (MD5) Previous issue date: 2014 / Resumo: Nesta tese, são abordadas duas principais classes de materiais: os supercondutores à base de ferro e arsênico (FeAs) e os isolantes topológicos tridimensionais. Os supercondutores à base de FeAs foram descobertos em 2008 e desde então despertaram grande interesse na comunidade científica como candidatos à supercondutividade não-convencional de alta temperatura. Dentre as várias estruturas à base de FeAs descobertas, em particular, a família BaFe2As2 (122) é uma das mais estudadas até o momento por ser um composto intermetálico possível de ser crescido com relativa facilidade e alta qualidade comparada às outras famílias. Este composto puro possui uma transição estrutural de tetragonal para ortorrômbica à uma temperatura Ts ? 139 K, e, diminuindo a temperatura, há uma transição magnética de paramagnético (PM) para uma ordem anti-ferromagnética tipo onda de densidade de spin (SDW, no inglês) em TSDW ? 134 K. Estas transições em Ts e TSDW são gradualmente suprimidas tanto por substituição química quanto por pressão aplicada, e, antes que sejam completamente suprimidas, supercondutividade é induzida. Nesta região onde a supercondutividade (SC) ainda está emergindo, há coexistência e/ou competição de SC e SDW, com a presença de momentos quase localizados do Fe. Estes momentos podem agir como centros eficientes de aprisionamento de vórtices (centros de pinning) que são suprimidos gradativamente por pressão aplicada, formando um efeito de pinning magnético que se manisfestaria na corrente crítica dessas amostras. Para estudar os possíveis efeitos de pinning magnético neste material, foram crescidas amostras com diferentes substituições químicas e realizados estudos de densidades de corrente crítica sob pressão e campos magnéticos aplicados. Enquanto observamos que a densidade de corrente crítica nas amostras estudadas é dominada pela evolução da temperatura crítica Tc, encontramos evidências indiretas para a presença de pinning magnético em nossas amostras, através da comparação da dependência com a pressão observada para a temperatura crítica supercondutora Tc e para a densidade de corrente crítica Jc. Quanto às transições estrutural e magnética, discute-se na literatura se nesses sistemas há dopagem eletrônica efetiva quando se realiza substituição química, ou se ocorre uma sintonização das propriedades pelo ajuste da geometria dos tetraedros de FeAs nos planos do material. Além disso, na faixa de temperatura entre Ts e TSDW, há certo debate se o material torna-se inteiramente ortorrômbico e PM, ou se há coexistência entre as fases tetragonal/PM e ortorrômbica/SDW. Neste trabalho, foram crescidos monocristais de Ba(Fe,M)2As2 (M = Co, Cu) e neles estudadas as transições estrutural e magnética e a fase entre elas utilizando técnicas de ressonância magnética nuclear (RMN), difração de raios-X e calor específico. Nossos resultados sugerem que, independente do substituinte, não há dopagem eletrônica e que a geometria nos planos de FeAs regula as propriedades do material. Além disso, mostramos que, de fato, existe coexistência entre as fases tetragonal/PM e ortorrômbica/SDW nas amostras. Estes resultados corroboram nossas próprias observações em amostras de (Ba,K)Fe2As2, o que também indica a independência dos resultados com o tipo, concentração e localização dos átomos substituintes na matriz 122. Os isolantes topológicos (ITs) são uma nova classificação da matéria proposta teoricamente e observada experimentalmente em 2006, e consistem de maneira simplificada em materiais que são isolantes de banda em seu volume, porém possuem estados de superfície metálicos robustos. As propriedades exóticas destes estados de superfície tornam estes materiais candidatos a aplicações em computação quântica e spintrônica. Muitas famílias foram propostas como ITs e estudadas recentemente, em particular a família 23 ((Bi,Sb)2(Se,Te)3) e a família dos half-Heuslers, que são o foco desta Tese. Porém, ainda há muitos aspectos destes materiais que não estão bem determinados, como por exemplo a penetração dos estados de superfície, e sua resposta a campos magnéticos e a radiações incidentes. Neste trabalho, sintetizamos cristais dos ITs da família 23 e o half-Heusler YBiPt, e os estudamos utilizando a técnica de ressonância de spin eletrônico (RSE). Para o YBiPt dopado com Nd3+, observamos um comportamento da forma de linha nãoconvencional do espectro de RSE do Nd3+, onde a forma da linha depende fortemente da potência da micro-ondas, das dimensões da amostra e da concentração de Nd3+. A este comportamento não convencional atribuímos a presença de um efeito de fônon bottleneck e à presença dos estados de superfície. Para o Bi2Se3, nossos resultados preliminares sugerem que os estados de superfície têm uma interação maior com as sondas de Gd3+ abaixo de 40 K, e que estes estados de superfície possuem caráter de orbitais p / Abstract: In this thesis, two main material classes are studied: the iron arsenide (FeAs) superconductors and the three-dimensional topological insulators. The FeAs based superconductors were discovered in 2008 and since then have aroused great interest in the scientific community as possible unconventional high-temperature superconductors. Amongst the many FeAs bases structures that were discovered until now, the BaFe2As2 (122) family is one of the most studied for being an intermetallic compound which can be grown with reasonable ease and high quality when compared to other families. This pure compound presents a structural transition at Ts ? 139 K from tetragonal to orthorhombic, and a magnetic transition at a lower TSDW ? 134 K from paramagnetic (PM) to a spin density wave (SDW) anti-ferromagnetic order. Both transitions at Ts and TSDW are gradually suppressed by either chemical substitution or applied pressure, and, before both are completely suppressed, superconductivity emerges. In this region where superconductivity (SC) is still emerging, there is coexistence and/or competition between SC and SDW order, with the presence of Fe quasi-localized magnetic moments. Those moments may act as efficient pinning centers that are gradually suppressed by applied pressure, producing a magnetic pinning effect that would manifest itself in the critical currents of such samples. To study magnetic pinning effects in the 122 family, samples with different chemical substitutions were grown, and critical current density studies were performed under applied pressure and magnetic fields. While the behavior of critical current densities is dominated by the evolution of critical temperature Tc with pressure, we have found indirect evidences for the presence of magnetic pinning in our samples. Regarding the structural and magnetic transitions, there are evidences that, in the 122 system, there is no effective electronic doping when a chemical substitution is performed but a tuning of the system properties by the geometric configuration of FeAs tetrahedra in the material. Also, there is some debate if in the temperature range between Ts and TSDW the material becomes completely orthorhombic and PM, or if there is coexistence of tetragonal/PM and orthorhombic/SDW phases. In this work, Ba(Fe,M)2As2 (M = Co, Cu) single crystals were grown and their structural and magnetic transitions were investigated by nuclear magnetic resonance (NMR), X-ray diffraction and specific heat. Our results show that there is no effective electronic doping induced by any of the substitutions, and that it is the geometric configuration of FeAs tetrahedra that tunes the system properties. Also, we have shown that there is coexistence between tetragonal/PM and orthorhombic/SDW phases in these samples. The results confirm our previous observations made in (Ba,K)Fe2As2, which also indicates that our results do not depend on the type, concentration and placement of substituted atoms in the 122 matrix. Topological insulators (TIs) are a new class of condensed matter which was proposed theoretically and experimentally observed in 2006. They are, in a simplified view, materials which are band insulators in the bulk, but possess roubst intrinsic metallic surface states. The exotic properties of those surface states make these materials possible candidates in quantum computing and spintronic applications. Many different materials were proposed as TIs and recently studied, in particular the 23 family ((Bi,Sb)2(Se,Te)3) and the half-Heusler family, which are the focus of this thesis. Yet, there are several aspects of these materials which are not yet fully understood, such as the penetration of the surface states, and their response to magnetic fields and incident radiation. In this work, single crystals of the 23 and half-Heusler families were grown, and were studied by electronic spin resonance (ESR). For YBiPt, we observed an anomalous lineshape behavior, to which we attribute a phonon bottleneck effect and the presence of the surface states. For Bi2Se3, our preliminary results indicate that the surface states have an enhanced exchange interaction with the Gd3+ magnetic probes below 40 K, and that those surface states have a p-orbital character / Doutorado / Física / Doutor em Ciências

This thesis studies the physics of multilayer heterostructures grown from topological insulators (TIs), primarily bismuth selenide and antimony telluride, and other topologically trivial materials. This is done by extending a standard low-energy 3D TI Hamiltonian and varying its associated material parameters across the simulation domain. New results arising from the position-dependent TI interface model are found. For the first time, this method is incorporated into a density-functional theory (DFT) solver in order to study the self-consistent charge density in multilayer TI heterostructures due to the interface states. The thesis is structured as follows. The introduction (Ch. 1) presents a pedagogical review of the theory of 3D TIs and low-energy Hamiltonians used to study them, as well as typical methods in solid state physics that are made use of throughout the thesis. Chapter 2 presents the position-dependent Hamiltonian, showing new evidence for topological features of bulk states including varying degrees of band mixing and inversion; also, interface state tunnelling is shown to be affected by atomic layer orbital overlap, and incomplete localisation of surface states is demonstrated for antimony telluride. Chapter 3 presents a new DFT model of TI heterostructure interfaces and shows how conduction through TI interface states can be controlled with an electric field. Chapter 4 covers the extension of the model in Ch. 1 to 2D cross-sections of TI wires and heterostructures, showing for the first time evidence of localisation of conduction almost entirely within the inner interfaces of a 2D heterostructure wire. Chapter 5 presents our work with magnetic fields, demonstrating evolution of interface and bulk states with changing magnetic field and Landau level, as well as presenting new evidence for more complex spin structures in bismuth selenide arising from Landé factor signs. Our conclusions are presented in Chapter 6.

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

Topological insulators (TI) are a new phase of matter with very exotic electronic properties on their surface. As a direct consequence of the topological order, the surface electrons of TI form bands that cross the Fermi surface odd number of times and are guaranteed to be metallic. They also have a linear energy-momentum dispersion relationship that satisfies the Dirac equation and are therefore called Dirac fermions. The surface Dirac fermions of TI are spin-polarized with the direction of the spin locked to momentum and are immune from certain scatterings. These unique properties of surface electrons provide a platform for utilizing TI in future spin-based electronics and quantum computation. The surface bands of 3D TI can be directly mapped by angle-resolved photoemission spectroscopy (ARPES) and the spin polarization can be determined by spin-resolved ARPES. These types of experiments are the first to establish the 3D topological order, which demonstrates the power of ARPES in probing the surface of strongly spin-orbit coupled materials. Extensive investigation of TI has ranged from understanding the fundamental electronic and lattice structure of various TI compounds to building TI-based devices in search of more exotic particles such as Majorana fermions and magnetic monopoles. Surface-sensitive techniques that can efficiently disentangle the charge and spin degrees of freedom have been crucially important in tackling the multi-faceted problems of TI. In this thesis, I show that laser-based ARPES in combination with a time-of-flight spectrometer is a powerful tool to study the spin structure and charge dynamics of the Dirac fermions on the surface of TI. Chapter 1 gives a brief introduction of TI. Chapter 2 describes the basic principles behind ARPES and time-resolved ARPES (TrARPES). Chapter 3 provides a detailed account of the experimental setup to perform laser-based ARPES and TrARPES. In Chapters 4 and 5, how these two techniques are effectively applied to investigate two unique electronic properties of TI is elaborated. Through these studies, I have obtained a complete mapping of the spin texture of several prototypical topological insulators and have uncovered the cooling mechanism governing the hot surface Dirac fermions. / Physics