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.

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.

In a crystalline ordered magnet, coherent excitations called spin waves, or magnons, propagate in the material forming band structures in an analogous way to electrons. Spin waves can possess non trivial topology associated with novel response functions of fundamental and potential technological interest. In particular, topologically protected surface states of magnons offer a new path towards coherent spin transport for spintronics applications. One of the central issues in this area is to establish the conditions under which band topology can arise in magnons and explore its variety. In this work we harness the full power of symmetry as applied to magnetism, to facilitate the discovery of new topological magnon models and materials. We show how to efficiently identify such systems by adapting the electronic topological quantum chemistry scheme to magnons, using constraints imposed by time reversal and crystalline symmetries to determine possible gapped and nodal topology in magnon models. Further, we explore enhanced symmetries beyond this paradigm, which are nevertheless natural for magnons: the spin-space groups. Exploring spin-space symmetry, which has wholly or partially decoupled magnetic and lattice degrees of freedom, reveals a proliferation of nodal points, lines, and planes beyond the standard crystalline symmetries. Linear spin wave theory is one of the most valuable techniques to study magnons, however, it can fail in different scenarios. Because of its importance to the community, we explore cases where it contains spurious symmetries which can potentially hide important physics in the spectra, like topology. We provide therefore a simple way to identify and resolve such cases within the linear theory. Finally, a pressing issue in magnons is the experimental detection and manipulation of topological surface states. Even more, the characterisation of generic 2D magnetism is an open problem. We contribute to this by devising an experimental setup based on quasi-particle interference which potentially could solve this long-lasting challenge. / Kohärente Anregungen, wie Spinwellen, auch Magnonen genannt, formen Bandstrukturen in kristallin geordneten Materialien. Diese Magnonen können eine nicht triviale Topologie aufwei- sen, welche neuartige Antwortfunktionen erzeugen können. Sie sind daher von technologischem Interesse. Insbesondere die topologisch geschützten Oberflächenzustände der Magnonen ermöglichen eine Realisierung von kohärentem Spin Transport und erlauben eine potentielle Anwendung in der Spintronik. Zentraler Punkt der aktuellen Forschung sind Bedingungen, unter denen eine nicht triviale Magnon-Bandtopologie entstehen kann und welche Möglichkeiten diese eröffnen. In dieser Arbeit untersuchen wir diese neuartigen topologischen Phasen für verschiedene Mo- delle unter Nutzung verschiedener Symmetrien. Die Erweiterung des elektronischen “topological quantum chemistry” Ansatzes für Magnonen erlaubt eine effiziente Identifikation dieser topologischen Eigenschaften. Der Ansatz basiert auf verschiedenen Einschränkungen, welche von der Zeitumkehr und kristallinen Symmetrien induziert werden. Darüber hinaus untersuchen wir die Anwendung von weiteren Symmetrien, welche relevant für Magnonen sind: die Spin-Raumgruppen. Die Erforschung der Spin-Raum-Symmetrie, welche magnetische Freiheitsgrade und Gittersyme- trien ganz oder teilweise entkoppelt, führt zur Ausbreitung von Knotenpunkten, Linien und Ebenen jenseits der standardmäßigen Kristallsymmetrien. Die lineare Spinwellentheorie ist eine der erfolgreichsten Methoden zur Untersuchung von Magnonen, kann jedoch unter verschiedenen Umständen versagen, da künstliche Symmetrien essenzielle Physik, wie beispielsweise topologische Eigenschaften, verbergen. Ansätze, die im Rahmen dieser Dissertation erarbeitet worden sind, helfen dabei, solche Fälle zu identifizieren und zu verstehen. Aktuelle Experimente zur Manipulation topologischer Oberflächenzustände von Magnonen, sowie die allgemeine Untersuchung von Magnetismus in zwei Dimensionen, fehlen. Daher präsentieren wir einen möglichen experimentellen Aufbau, basierend auf Quasi-Teilchen-Interferenz, welcher einen möglichen Ausweg aufzeigt.