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.

我們主要研究拓樸絕緣體與鐵磁物質之間的自旋幫浦效應（spin pumping effect），我們選用的鐵磁材料是具有鐵磁性的鎳鐵合金（Py），厚度固定為40nm，而拓樸絕緣體則是選用碲化鉍（Bi2Te3），厚度範圍是0～100nm，碲化鉍已被確定為一個三維拓撲絕緣體，拓撲絕緣體其表面電子態呈線性色散關係，本身中心是絕緣體，但其表面容許有導電態。此導電態一個最有用的特性是其電子的動量與自旋維持一定方向關係（spin-momentum locking），這使得以自旋來傳遞訊息成為可能。但是實驗上要達到中心是絕緣體相當困難。 過去的實驗已驗證鐵磁共振（Ferromagnetic resonance，FMR）現象在鐵磁/一般金屬雙層膜以及鐵磁/半導體雙層膜，可以使其鐵磁層產生一純自旋流流向非磁性層，這被稱為自旋幫浦效應（spin pumping effect）。當此自旋流跨越膜面介面時，不同自旋的電子由於自旋軌道耦合作用（Spin–orbit interaction），將發生逆自旋霍爾效應（ISHE）並產生一橫向電荷流。在我們的研究中，鐵磁共振（FMR）現象透過網路分析儀在設定的外加磁場下掃描頻率。測得的共振頻率與磁場作圖並以Kittel equation擬合（fitting）出有效場（effective field）。我們發現於絕對溫度5K，隨著碲化鉍（Bi2Te3）膜厚從0nm到15nm增加時，其有效場也增加，但當薄膜厚度大於15nm時，有效磁場將下降。我們分析碲化鉍（Bi2Te3）的表面態（surface state）與塊材（bulk）對有效場變化之貢獻。

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

Cette thèse présente une étude expérimentale des propriétés électroniques de deux matériaux topologiques, notamment l’isolant topologique tridimensionnel irradié Bi₂Te₃ et le super-réseau topologique naturel Sb₂Te. Les deux systèmes ont été étudiés par des techniques basées sur la spectroscopie de photoémission. Les composés Bi₂Te₃ ont été irradiés par des faisceaux d’électrons de haute énergie. L’irradiation avec des faisceaux d’électrons est une approche très prometteuse pour réaliser des matériaux qui sont vraiment isolants dans le volume, afin de mettre en évidence le transport quantique dans les états de surface protégés. En étudiant une série d’échantillons de Bi₂Te₃ par la technique de spectroscopie de photoémission résolue en temps et en angle (trARPES), nous montrons que les propriétés topologiques des états de surface de Dirac sont conservées après irradiation électronique, mais leurs dynamiques ultra-rapides de relaxation sont très sensibles aux modifications reliées aux propriétés du volume. De plus, nous avons étudié la structure électronique des bandes occupées et inoccupées du Sb₂Te. En utilisant la microscopie de photoémission d’électrons à balayage (SPEM), nous avons constamment trouvé diverses régions non équivalentes sur la même surface après avoir clivé plusieurs monocristaux de Sb₂Te. Nous avons pu identifier trois terminaisons distinctes caractérisées par différents rapports stœchiométriques de surface Sb/Te et possédant des différences claires dans leurs structures de bandes. Pour la terminaison dominante riche en tellure, nous avons également fourni une observation directe des états électroniques excités et de leurs dynamiques de relaxation en ayant recours à la technique trARPES. Nos résultats indiquent clairement que la structure électronique de surface est fortement affectée par les propriétés du volume du super-réseau. Par conséquent, pour les deux systèmes, nous montrons que la structure électronique de surface est absolument connectée aux propriétés du volume. / This thesis presents an experimental study of the electronic properties of two topological materials, namely, the irradiated three-dimensional topological insulator Bi₂Te₃ and the natural topological superlattice phase Sb₂Te. Both systems were investigated by techniques based on photoemission spectroscopy. The Bi₂Te₃ compounds have been irradiated by high-energy electron beams. Irradiation with electron beams is a very promising approach to realize materials that are really insulating in the bulk, in order to emphasize the quantum transport in the protected surface states. By studying a series of samples of Bi₂Te₃ using time- and angle-resolved photoemission spectroscopy (trARPES) we show that, while the topological properties of the Dirac surface states are preserved after electron irradiation, their ultrafast relaxation dynamics are very sensitive to the related modifications of the bulk properties. Furthermore, we have studied the occupied and unoccupied electronic band structure of Sb₂Te. Using scanning photoemission microscopy (SPEM), we have consistently found various nonequivalent regions on the same surface after cleaving several Sb₂Te single crystals. We were able to identify three distinct terminations characterized by different Sb/Te surface stoichiometric ratios and with clear differences in their band structure. For the dominating Te-rich termination, we also provided a direct observation of the excited electronic states and of their relaxation dynamics by means of trARPES. Our results clearly indicate that the surface electronic structure is strongly affected by the bulk properties of the superlattice. Therefore, for both systems, we show that the surface electronic structure is absolutely connected to the bulk properties.

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

The discovery of topological quantum states of matter has required physicists to look beyond Landau’s theory of symmetry-breaking, previously the main paradigm for<br>studying states of matter. This has led also to the development of new topological theories for describing the novel properties. In this dissertation an investigation in this<br>frontier research area is presented, which looks at the interplay between the quantum geometry of these states, defects and disorder. After a brief introduction to the topological quantum states of matter considered herein, some aspects of my work in this area are described. First, the disorder-induced band structure engineering of topological insulator surface states is considered, which is possible due to their resilience from Anderson localization, and believed to be a consequence of their topological origin.<br>Next, the idiosyncratic behavior of these same surface states is considered, as observed in experiments on thin film topological insulators, in response to competition between<br>hybridization effects and an in-plane magnetic field. Then moving in a very different direction, the uncovering of topological ‘gravitational’ response is explained: the<br>topologically-protected charge response of two dimensional gapped electronic topological states to a special kind of 0-dimensional boundary – a disclination – that encodes spatial curvature. Finally, an intriguing relation between the gravitational response of quantum Hall states, and their response to an apparently unrelated perturbation – nonuniform electric fields is reported. <br>

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