Dissertationen zum Thema „Topological semimetals“

The importance and promise that topological materials hold has been recently underscored by the award of the Nobel Prize in Physics in 2016 ``for theoretical discoveries of topological phase transitions and topological phases of matter." This dissertation explores the novel qualities and useful topologically protected surface states of topological insulators and semimetals. Topological materials have protected qualities which are not removed by weak perturbations. The manifestations of these qualities in topological insulators are spin-momentum-locked surface states, and in Weyl and Dirac semimetals they are unconventional open surface states (Fermi arcs) with anomalous electrical transport properties. There is great promise in utilizing the topologically protected surface states in electronics of the future, including spintronics, quantum computers, and highly sensitive devices. Physicists and chemists are also interested in the fundamental physics and exotic fermions exhibited in topological materials and in heterostructures including them. Chapter 1 provides an introduction to the concepts and methods of topological band theory. Chapter 2 investigates the spin and spin-orbital texture and electronic structures of the surface states at side surfaces of a topological insulator, Bi2Se3, by using slab models within density functional theory. Two representative, experimentally achieved surfaces are examined, and it is shown that careful consideration of the crystal symmetry is necessary to understand the physics of the surface state Dirac cones at these surfaces. This advances the existing literature by properly taking into account surface relaxation and symmetry beyond what is contained in effective bulk model Hamiltonians. Chapter 3 examines the Fermi arcs of a topological Dirac semimetal (DSM) in the presence of asymmetric charge transfer, of the kind which would be present in heterostructures. Asymmetric charge transfer allows one to accurately identify the projections of Dirac nodes despite the existence of a band gap and to engineer the properties of the Fermi arcs, including spin texture. Chapter 4 investigates the effect of an external magnetic field applied to a DSM. The breaking of time reversal symmetry splits the Dirac nodes into topologically charged Weyl nodes which exhibit Fermi arcs as well as conventionally-closed surface states as one varies the chemical potential.
Ph. D.

Thesis advisor: Kenneth S. Burch
Since their first experimental realizations in the 2000s, bulk electronic topological materials have been one of the most actively studied areas of condensed matter physics. Among the more recently discovered classes of topological materials are the Weyl semimetals whose low energy excitations behave like massless, relativistic particles with well-defined chirality. These material systems display exotic behavior such as surface Fermi arc states, and the chiral anomaly in which parallel magnetic and electric fields lead to an imbalance of left- and right-handed particles. Much of the research into these materials has focused on the electronic properties, but relatively little has been directed towards understanding the vibrational properties of these systems, or of the interplay between the electronic and vibrational degrees of freedom. Further, the technological potential of these materials is still underdeveloped, with the search for physical properties enhanced by the topological nature of these materials being sought after. In this dissertation we address both of these issues. In Chapters III and IV we present temperature dependent Raman investigations of the the Weyl semimetals WP2, NbAs, and TaAs. Measurements of the optical phonon linewidths are used to identify the available phonon decay paths, with ab-initio calculations and group theory used to aid the interpretation of these results. We find that some phonons display linewidths indicative of dominant decay into electron-hole pairs near the Fermi surface, rather than decay into acoustic phonons. In light of these results we discuss the role of phonon-electron coupling in the transport properties of these Weyl semimetals. In Chapter V, we discuss the construction of our "PVIC" setup for the measurement of nonlinear photocurrents. We discuss the experimental capabilities that the system was designed to possess, the operating principles behind key components of the system, and give examples of the operating procedures for using the setup. The penultimate chapter, Chapter VI, presents the results of photocurrent measurements using this setup on the Weyl semimetal TaAs. Through careful analysis of the photocurrent polarization dependence, we identify a colossal bulk photovoltaic effect in this material which exceeds the response displayed by previously studied materials by an order of magnitude. Calculations of the second-order optical conductivity tensor show that this result is consistent with the divergent Berry connection of the Weyl nodes in TaAs. In addition to these topics, Chapter II addresses the results of Raman measurements on thin film heterostructures of the topological insulator Bi2Se3 and the magnetic semiconductor EuS. By investigating the paramagnetic Raman signal in films with different compositions of EuS and Bi2Se3 we provide indirect evidence of charge transfer between the two layers. We also track the evolution of phonon energies with varying film thicknesses on multiple substrates which provides insight into the interfacial strain between layers. We conclude the dissertation in Chapter VII with a summary of the main results from each preceding chapter, and give suggestions for future experiments that further investigate these topics
Thesis (PhD) — Boston College, 2020
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Physics

The topological consequences of time reversal symmetry breaking in two dimensional electronic systems have been a focus of interest since the discovery of the quantum Hall effects. Similarly interesting phenomena arise from breaking inversion symmetry in three dimensional systems. For example, in Dirac and Weyl semimetals the inversion symmetry breaking allows for non-trivial topological states that contain symmetry-protected pairs of chiral gapless fermions. This thesis presents our work on the linear and nonlinear electromagnetic responses in topological semimetals using both a semiclassical Boltzmann equation approach and a full quantum mechanical approach. In the linear response, we find a ``gyrotropic magnetic effect" (GME) where the current density $j

B$ in a clean metal is induced by a slowly-varying magnetic field. It is shown that the experimental implications and microscopic origin of GME are both very different from the chiral magnetic effect (CME). We develop a systematic way to study general nonlinear electromagnetic responses in the low-frequency limit using a Floquet approach and we use it to study the circular photogalvanic effect (CPGE) and second-harmonic generation (SHG). Moreover, we derive a semiclassical formula for magnetoresistance in the weak field regime, which includes both the Berry curvature and the orbital magnetic moment. Our semiclassical result may explain the recent experimental observations on topological semimetals. In the end, we present our work on the Hall conductivity of insulators in a static inhomogeneous electric field and we discuss its relation to Hall viscosity.

Topological insulators(TIs) constitute a novel state of quantum matter which possesses non-trivial topological properties. Although discovered only in the recent few years, TIs have attracted intensive interest among the community of condensed matter physics and material science. TIs are insulating in the bulk but have conductive gapless edge or surface states on the boundaries, which have their origin in the nontrivial bulk band topology that is induced by the strong spin-orbital interactions in the materials. Existing in all dimensions, TIs exhibit a variety of exotic physics such as quantum spin Hall effect, momentum-spin locked surface states, Dirac fermion transport, quantized anomalous Hall effect, Majorana fermions, etc. In this thesis, I study the transport properties of 2D and 3D TIs by numerical approaches. As an introduction, a brief review of TIs is given. A detailed description of the numerical methods is also presented. The results can be summarized in four aspects. First, disorder is found be able to induce a non-trivial TI from an originally trivial band insulator, where the conductance of a two terminal device drops to nearly zero and then rises to form an anomalous plateau as disorder strength is increased, and finally all the states become localized. The real space Chern number calculation as well as the effective medium theory suggests that disorder is fundamentally responsible for the emerging of the extended helical edge states in this system. We also present a levitation and pair annihilation picture of the extended states for this model. Second, by making the 2D TIs into singly connected quantum point contacts(QPCs), I show a coherent and fast Aharonov-Bohm oscillation of conductance caused by the quantum interference of the helical edge states. This oscillation not only happens against weak magnetic field but also against the gate voltage in the zero-field condition. This results in a giant edge magnetoresistance of the device in weak magnetic fields. The amplitude of the magnetoresistance is controllable by adjusting either the QPCs' slit width or the interference loop size in the device. The oscillation is found robust against disorder. Third, by applying a uniform spin-splitting Zeeman field in the bulk of the 3D TI whose surface states can be viewed as massless Dirac fermions, I find chiral edge states on the gapped surfaces of the 3D TI, which can be considered as interface states between domains of massive and massless Dirac fermions. Effectively these states are result of splitting of a perfect interface conducting channel. This picture is confirmed by the Landauer-B?ttiker calculations in four-terminal Hall bars. Finally, I propose the concept of topological semi-metals. By calculating the local density of states on the surfaces, I demonstrate that surface states and the gapless Dirac cone already exist in the system although the bulk is not gapped. We show how the uni-axial strain induces an insulating band gap and turn the semi-metal into true TI. We predict existence of quantum spin Hall effect in the thin films made of these materials, which can be significantly enhanced by disorders.
published_or_final_version
Physics
Doctoral
Doctor of Philosophy

Topological insulators and superconductors, which are featured by not only the topological characteristics of their gapped bulk band structure but also the special edge or surface states, have attracted great attention in the past few years. A complete classification of topological insulators and superconductors in terms of symmetry and spatial dimension has been established, while the application of their surface states remains a challenge. The gapless phases which have topologically stable Fermi surfaces could also exhibit peculiar surface states and topological transport phenomena in the bulk. In this thesis, the topological DIII-classs superconducting chains and the application of its Majorana edge states are studied. On the other hand, Weyl semimetals, as the representative example of topological gapless phases, and its exotic transport phenomena are also investigated. Majorana edge states have been a focus of condensed matter research for their potential applications in topological quantum computation, which appear in the topological DIII-class superconducting chains protected by both the particle-hole and time reversal symmetries. We utilize two charge-qubit arrays to explicitly simulate one type of DIII-class superconducting chains and the universal quantum operations performed on the Majorana edge states. It is shown that combined with one braiding operation, universal single-qubit operations on a Majorana-based qubit can be implemented by a controllable inductive coupling between two charge qubits at the ends of the arrays. It is further shown that in a similar way, a controlled-NOT gate for two topological qubits can be simulated in four charge-qubit arrays. Although the current scheme may not truly realize topological quantum operations, we elaborate that the operations in charge-qubit arrays are indeed robust against certain local perturbations. Weyl semimetals possess nontrivial Fermi surface topology in that the pair of Weyl points with opposite topological charges is separated from each other in momentum space. The physical manifestations of this Fermi surface topology are protected surface states and exotic transport phenomena including the anomalous Hall effect as well as the chiral magnetic effect. By studying the path integral measure under the chiral transformation, it is shown that these transport phenomena can be described by the chiral anomaly which appears when the chiral Weyl fermion couples to the topologically nontrivial gauge field. The case of the gauge anomaly for the Weyl fermion coupled to a non-Abelian gauge field is also discussed.
published_or_final_version
Physics
Doctoral
Doctor of Philosophy

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.

Nodal semimetals host either degenerate points (Dirac/Weyl points) or lines whose band topology in Brillouin zone can be classified either as trivial (normal nodal semimetals) or non trivial (topological nodal semimetals). This thesis investigates the electronic structure of two different categories of topological nodal semimetals probed by angleresolved photoemission spectroscopy (ARPES): The first material is Indium Bismuth (InBi). InBi is a semimetal with simple tetragonal structure with P4/nmm space group. This space group is predicted to host protected nodal lines along the perpendicular momentum direction at the high symmetry lines of the Brillouin zone boundary even under strong spin-orbit coupling (SOC) situation. As a semimetal with two heavy elements, InBi is a suitable candidate to test the prediction. The investigation by ARPES demonstrates not only that InBi hosts the nodal line in the presence of strong SOC, it also shows the signature of type-II Dirac crossing along the perpendicular momentum direction from the center of Brillouin zone. However, as the nodal line observed is trivial in nature, there is no exotic drumhead surface states observed in this material. This finding demonstrates that Dirac crossings can be protected in a non-symmorphic space group. The second material is NbIrTe₄ which is a semimetal that breaks inversion symmetry predicted to host only four Weyl points. This simplest configuration is confirmed by the measurement from the top and bottom surface of NbIrTe₄ showing only a pair of Fermi arcs each. Furthermore, it is found that the Fermi arc connectivity on the bottom surface experiences re-wiring as it evolves from Weyl points energy to the ARPES Fermi energy level. This change is attributed to the hybridisation between the surface and the bulk states as their projection lie within the vicinity of each other. The finding in this work demonstrates that although Fermi arcs are guaranteed in Weyl semimetals, their shape and connectivity are not protected and may be altered accordingly.

In this work we exploit the unique characteristics of a Dirac semimetal material to be symmetry-protected, to investigate dierent topological phase transitions provided by chemical dopings, focusing in particular on the electronic, magnetic and topological properties of the doped systems, studied by the mean of rst-principles methods based on density functional theory (DFT) approach. In particular these doped systems, besides being of interest for investigating the role of topology in solid state physics, could have a great potential for practical application since the dierent topological phases that come along with the chemical dopings allow one to exploit the unique properties of topological materials. The starting point for our study will be the material called cadmium-arsenide (Cd3As2), an example of a topological Dirac semimetal, which is chemically stable at ambient conditions. Chapter I presents a general introduction to topology, especially in condensed matter physics, and to the main physical properties of the topological materials we mentioned. Then, in chapter II, we briey present the methods and the computational tools that we used for our study. In chapter III a more detailed introduction to our work is given, along with a schemetic view of the path we followed, together with the results that we obtained for pristine Cd3As2, which we use as bench mark for our computational methods. Finally, in chapter IV and V, the results for the doped systems are presented and discussed, respectevely for the non-magnetic (IV) and magnetic (V) dopings. Our study has enabled us to discern how doping can give rise to see dierent topological phase transitions. Specically our work shows that dierent realizations of non-magnetic doping gives rise to dierent topological phases: the topological Weyl semimetal phase, which is of great interest since it can support a robust quantum spin Hall eect, and a very special mixed Dirac + Weyl phase, where surprisingly both a Dirac and a Weyl phase can coexist in the same system. Furthermore, magnetically doped systems show the emergence of a magnetic Weyl phase, which can support a quantum anomalous Hall eect. Our work can be the starting point for future studies, both theoretical and experimental, in which the unique physical properties we found in the doped Cd3As2 systems can be further investigated, in order to exploit them for practical applications.

Thesis advisor: Fazel Tafti
Since the idea of topology was realized in real materials, the hunt is on for new candidates of topological semimetals with novel electromagnetic responses. For example, topological states can be highly conductive due to a topological protection, which can be destroyed in a magnetic field and lead to an extremely high magnetoresistance. In Weyl semimetals, a transverse current that would usually require a magnetic field to emerge, can be generated by intrinsic Berry curvature without a magnetic field -- the celebrated anomalous Hall effect. In this dissertation, both phenomena mentioned above are studied in rare-earth monopnictides and RAlX material family (R=rare-earths, X=Ge/Si), respectively. The monopnictides are ideal for the study of extreme magnetoresistance because of their topological transitions and abundant magnetic phases. In LaAs, we untied the connection between topological states and the extreme magnetoresistance, the origin of which is clarified. In HoBi, we found an unusual onset of extreme magnetoresistance controlled by a magnetic phase dome. On the other hand, RAlX material family is a new class of Weyl semimetals breaking both inversion and time-reversal symmetries. In particular, in PrAlGeₓSi₁₋ₓ (x=0-1), we unveiled the first transition from intrinsic to extrinsic anomalous Hall effect in ferromagnetic Weyl semimetals, and the role of topology is discussed. In CeAlSi, we found that the Fermi level can be tuned as close as 1 meV away from the Weyl nodes; moreover, a novel anomalous Hall response appears only when the Fermi level is tuned to be near the Weyl nodes. Thus, we established a new transport response solely induced by Weyl nodes
Thesis (PhD) — Boston College, 2021
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Physics

Propriétés topologiques de systèmes photoniques non gappés modulés périodiquement. La photonique est une plate-forme où les ondes électromagnétiques (ou photons) se propagent à l'intérieur d'un cristal (comme les ondes de Bloch) formé par les degrés de liberté discrets sous-jacents, par exemple des réseaux de guides d'ondes. Ces ondes ne peuvent pas se propager si la fréquence incidente se situe dans la bande interdite photonique, alors ces ondes sont connues sous le nom d'ondes évanescentes. Ainsi, le cristal se comporte comme un réflecteur de ces ondes. Cependant, s'il existe des modes pour lesquels il existe des ondes limites qui relient la bande interdite, alors ces ondes peuvent exister à la limite sans s'infiltrer dans la masse. Ceci est analogue au mouvement chiral des électrons aux bords du Hall quantique, avec un ingrédient supplémentaire de symétrie d'inversion du temps qui se brise dans les cristaux photoniques via certaines propriétés gyromagnétiques de l'échantillon, ou la dépendance inhérente au temps du système. Dans ce dernier cas, lorsque le système, en particulier, est commandé périodiquement, on peut également observer les phases de non équilibre plus exotiques dans ces réseaux.Dans ce travail, nous explorons les propriétés topologiques de ces réseaux photoniques à commande périodique. Par exemple, comment les symétries fondamentales, par exemple la symétrie particule-trou, peuvent être mises en oeuvre pour concevoir la topologie en 1D. Nous trouvons un lien entre les symétries cristallines et les symétries fondamentales, qui facilitent une telle mise en oeuvre. De plus, une dimension synthétique peut être introduite dans ces treillis qui simulent la physique des dimensions supérieures. La différence entre la dimension synthétique et la dimension spatiale devient apparente lorsqu'une symétrie cristalline spécifique, comme l'inversion, est rompue dans ces systèmes. Cette rupture transforme une bande interdite directe en une bande interdite indirecte qui se manifeste par l'enroulement de bandes dans le spectre de la bande quasi-énergétique. Si elle est rompue dans la dimension synthétique, il en résulte une interaction de deux propriétés topologiques : l'une est l'enroulement des bandes de quasi-énergie, et l'autre est la présence d'états de bord chiraux dans la géométrie finie. Cette ancienne propriété de l'enroulement se manifeste par des oscillations de Bloch des paquets d'ondes, où nous montrons que les points stationnaires de ces oscillations sont liés au nombre d'enroulements des bandes. Cette propriété topologique peut donc être sondée directement dans une expérience par la technologie de pointe. Cependant, si cette symétrie est rompue dans la dimension spatiale, l'enroulement des bandes se manifeste comme une dérive quantifiée de la position moyenne, qui est toujours caractérisée par un nombre d'enroulement des bandes.En outre, nous montrons qu'un régime sans lacune différent peut également être conçu tout en préservant la symétrie d'inversion. Dans ce régime, la topologie peut être saisie en enfermant les dégénérescences dans l'espace des paramètres et en calculant le flux de Berry qui traverse la surface enfermée. Dans ce cas, certaines des dégénérescences peuvent héberger des états chiraux de bord avec d'autres protégés à la même quasi-énergie
Photonics has emerged a platform where electromagnetic waves (or photons) propagate inside a crystal (likeBloch waves) formed by the underlying discrete degrees of freedom, e.g., waveguide arrays. These waves cannotpropagate if the incident frequency lies within the so-called photonic bandgap, then these waves are known asevanescent waves. Thus, the crystal behaves as a reflector to these waves. However, if there are modes for whichthere exist boundary waves that connect the bandgap, then these waves can exist at the boundary without leakinginto the bulk. This is analogous to the chiral motion of electrons at the quantum Hall edges, with an extraingredient of time-reversal symmetry breaking in photonic crystals via some gyromagnetic properties of thesample, or inherent time dependence of the system. In the latter case, when the system, specifically, drivenperiodically then the more exotic non-equilibrium phases can also be observed in these lattices.In this work, we explore the topological properties in these periodically driven photonic lattices. For instance,how fundamental symmetries, e.g., particle-hole symmetry, can be implemented to engineer topology in 1D. Wefind a connection between crystalline symmetries and the fundamental symmetries, which facilitate suchimplementation. Moreover, a synthetic dimension can be introduced in these lattices that simulate higherdimensional physics. The difference between synthetic and spatial dimension becomes apparent when a specificcrystalline symmetry, like inversion, is broken in these systems. This breaking changes a direct bandgap to anindirect one which manifests in the winding of bands in the quasienergy band spectrum. If it is broken in thesynthetic dimension, it results in an interplay of two topological properties: one is the winding of the quasienergybands, and the other one is the presence of chiral edge states in the finite geometry. This former property ofwinding manifests as Bloch oscillations of wavepackets, where we show that the stationary points in theseoscillations are related to the winding number of the bands. This topological property can thus be probed directlyin an experiment by the state-of-art technology. However, if this symmetry is broken in the spatial dimension, thewinding of bands manifest as a quantized drift of mean position, which is still characterized by a winding numberof the bands.Furthermore, we show that a different gapless regime can also be engineered while preserving the inversionsymmetry. In this regime, the topology can be captured by enclosing the degeneracies in parameter space andcalculating the Berry flux piercing through the enclosed surface. In this case, some of the degeneracies can hostchiral edge states along with other protected ones at the same quasienergy

Orientador: Iakov Veniaminovitch Kopelevitch
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin
Made available in DSpace on 2018-08-22T01:33:07Z (GMT). No. of bitstreams: 1 Baring_LuisAugustoGomes_D.pdf: 21081310 bytes, checksum: 275f0ba5ff80d6f9f19f53cf8316e1a6 (MD5) Previous issue date: 2012
Resumo: No presente trabalho estudamos os semimetais bismuto Bi, antimônio Sb e Bi1-xSbx, materiais com propriedades topologicamente não triviais. Observamos a ocorrência de supercondutividade intrínseca em bismuto com TC »= 8:5K. Construímos, a partir dos dados de magnetização e resistência, o diagrama de fase do campo crítico H versus a temperatura T. Esse diagrama de fase, pode ser ajustado segundo modelos da literatura válidos para supercondutividade granular. Detectamos, no bismuto, o aumento da corrente Josephson e acoplamento intergranular no limite quântico devidos à quantização de Landau. Isso se manifesta como uma supercondutividade reentrante. Foi também encontrada transição tipo metal-isolante induzida por campo magnético em todos os materiais estudados. O diagrama de fase H versus T mostra uma extraordinária semelhança entre os três materiais. A amostra Bi1-xSbx, com x = 0:052, revelou a ocorrência de transição semimetal-isolante topológico já em campo magnético zero. Fizemos uma comparação com resultados anteriores da literatura, analisando a dependência da temperatura em que ocorre essa transição em relação à concentração de antimônio x e ao campo magnético B e demonstramos a similaridade entre eles. Observamos, também, supercondutividade nos semimetais bismuto, antimônio e no Bi1-xSbx, induzida por dopagem com os metais ouro e índio, e mostramos que a supercondutividade está associada à interface entre os metais e os semimetais. Finalmente, encontramos a indução de supercondutividade mediante a aplicação de campo magnético em bismuto, consistente com a ocorrência de férmions de Majorana na interface entre esse material e a tinta prata usada para os contatos. Tal observação pode ser devida, também, à ocorrência de um estado supercondutor fora do equilíbrio.
Abstract: In this work we studied the semimetals bismuth Bi, antimony Sb and Bi1-xSbx, all of them with non-trivial topologic properties. We observed an intrinsic superconductivity in bismuth, with TC »= 8:5 K. The phasediagram of the critical field H versus the temperature T, based upon the magnetization and resistance data, may be well fitted according to theoretical models valid for granular superconductivity. We also detected, in bismuth, the increase of the Josephson current and interganular coupling in the quantum limit due to Landau quantization. This manifests itself as a reentrant superconducting state. Our results revealed a metal-insulator transition triggered by magnetic field, for all the studied materials. The phase diagram H ¡T shows a striking similarity between them. The sample Bi1-xSbx with x = 0:052 demonstrated a semimetal-insulator transition even at zero field. We compared our results with previous results of other groups and analyzed the temperature dependence of the transition as a function of the antimony amount x and the magnetic field B and demonstrated their similarity. We also observed supeerconductivity in the semimetals bismuth, antimony and Bi1-xSbx, triggered by doping with the metals gold and indium, and showed that the superconductivity is associated to the interface between the metals and the semimetals. Finally, we found the superconductivity induced by the aplication of magnetic field in bismuth, consistent with the Majorana fermions present in the interface between this material and the silver paste contacts. This may also be related to a non-equilibrium superconduting state.
Doutorado
Física
Doutor em Ciências

The signature of the long-sought Majorana fermion from a heterostructure of a superconductor and a topological material is the zero bias conductance peak (ZBCP). Topological semimetal Antimony is a good material in making such heterostructure. Since it is of a bilayer crystal structure, it is expected to be cleaved between bilayers. However, we found that on its cleaved surface there can be steps with step heights corresponding to the intrabilayer distance, indicating that there is a broken layer underneath. The dI/dV spectrum observed using scanning tunneling microscope on these abnormal steps are quite different from the usual Sb spectrum and there is a pronounced ZBCP. Using quasiparticle interference imaging, Landau level spectroscopy and density functional theory (DFT), we found that the ZBCP is originated from the changed band structure through van Hove singularity. This shows that when we try to probe the signature of Majorana fermion in the heterostructure, we need to make sure the ZBCP is not from this trivial origin due to the imperfectness of the topological material.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

Dans cette thèse je me suis intéressé au caractère relativiste de matériaux topologiques tridimensionnels : les semi-métaux de Weyl et les isolants topologiques. Après une introduction aux états de surfaces et aux matériaux topologiques, je discute leurs propriétés de covariance sous les rotations trigonométriques et hyperboliques. Ces transformations me permettent de traiter les équations du mouvement d'un électron dans un champ magnétique ou à la surface, sous l'influence d'un champ électrique ou d'une inclinaison de la relation de dispersion. En première partie, je l'illustre dans le cas de la réponse magnéto-optique des semi-métaux de Weyl, en présence d'une inclinaison. Ces calculs sont en lien avec ma collaboration avec les expérimentateurs du LNCMI à Grenoble pour la caractérisation de la structure de bande de Cd₃As₂ où l'on montre que ce matériau est un semi-métal de Kane et non un semi-métal de Dirac dans la gamme de potentiels chimiques expérimentalement accessible. L'autre partie de cette thèse porte sur les états de surface des isolants topologiques où l'on montre qu'il existe des états de surface massifs au-delà de l'état de surface chiral. Ces états semblent avoir été observés par des études en ARPES d'échantillons de Bi₂Se₃ et Bi₂Te₃ oxydés et par des mesures de transport sur HgTe déformé. J'ai ainsi eu l'occasion de travailler avec les expérimentateurs du LPA à Paris sur le comportement des états de surface de HgTe sous forts effets de champ. Je termine par une discussion des états à l'interface entre un semi-métal de Weyl et un isolant dans le cas où le gap de ce dernier est suffisamment petit pour observer l'effet d'un champ magnétique et d'une inclinaison de la relation de dispersion sur les états de surface
During my PhD studies I focused on the relativistic properties of threedimensional topological materials, namely Weyl semimetals and topological insulators. After introducing surface states and topological materials I discuss their covariance in trigonometric and hyperbolic rotations. These transformations help to solve the equations of motion of an electron in a magnetic field or at the surface with an applied electric field or with a tilt in the band dispersion. In a first place, I illustrate these transformations for the magneto-optical response of tilted Weyl semimetals. This work is related to my collaboration with experimentalists at LNCMI, Grenoble for characterizing the band structure of Cd₃As₂ where we show that this material is a Kane semi-metal instead of a Dirac semi-metal in the experimentally accessible range of chemical doping. The other part of this thesis is concerned with the surface states of topological insulators. I show that massive surface states can also exist in addition to the chiral surface state due to band inversion. Such states may have already been observed in ARPES measurement of oxidized Bi₂Se₃ and Bi₂Te₃ and in transport measurement of strained bulk HgTe. I show the work we performed with experimentalists at LPA, Paris on the behavior of HgTe surface states for strong field effects. Finally, I discuss the states at the interface of a Weyl semimetal and a small gap insulator. In this situation, an applied magnetic field or the tilt of the band dispersion can strongly affect the observed surface states

La réalisation de nouveaux matériaux bidimensionnels est un domaine en plein essor de la matière condensée, à la fois pour les aspects fondamentaux, avec les propriétés exotiques émergeant de la dimensionnalité réduite, et pour les applications technologiques potentielles, avec des promesses telles que des courants sans dissipation et des hétérostructures 2D plus performantes que la technologie actuelle à base de silicium à une fraction de la taille. Dans ce travail, nous avons adopté une approche expérimentale pour la réalisation et la caractérisation de matériaux prédits pour accueillir des lignes nodales de Dirac (DNL), qui malgré de nombreuses prédictions théoriques ont vu peu de réalisations expérimentales rapportées jusqu'à présent. Ces matériaux appartiennent à la classe récemment mise en évidence des semi-métaux topologiques, dont la spécificité est un croisement de bandes protégé par symétrie entre les bandes de valence et de conduction le long d'une ligne dans l'espace réciproque, avec une dispersion linéaire. Dans un premier temps, nous nous sommes concentrés sur Cu2Si, le premier matériau 2D dans lequel des DNL ont été mis en évidence lorsqu'il est préparé sur un substrat Cu(111). Après avoir reproduit avec succès les résultats existants, nous avons montré à l'aide de l'ARPES et du XPS que, contrairement aux attentes, la structure électronique et les DNL étaient préservées après le dépôt de Pb sur la surface. Nous avons ensuite étudié Cu2Si/Si(111), et constaté que malgré une structure atomique fortement liée, le substrat Si(111) interagit assez fortement avec les orbitales hors plan de la couche Cu2Si pour empêcher l'existence des lignes nodales. Nous nous sommes ensuite penchés sur le système 2D Cu2Ge, prédit pour accueillir la DNL, et avons tenté de le synthétiser en déposant du Ge sur Cu(111). En combinant nos résultats LEED, XPS et ARPES, nous avons constaté que toutes les mesures correspondaient étroitement à ce que l'on attendait d'une monocouche de Cu2Ge libre, ce qui montre l'absence presque totale d'interactions entre le substrat Cu(111) et la couche de Cu2Ge superficielle formée sur celui-ci. Il s'agit de la première réalisation expérimentale rapportée de Cu2Ge. Dans une étude miroir, nous avons déposé Cu sur Ge(111) et observé une structure de bande dissemblable. À l'aide du STM, nous avons expliqué ces différences par une structure atomique différente, résultant d'un substrat à forte interaction. Nous soulignons par ce travail l'influence du substrat, qu'il soit métallique ou semi-conducteur, sur les propriétés électroniques des systèmes 2D à DNL
The realization of new two-dimensional materials is a booming field of condensed matter, at once for the fundamental aspects, with the exotic properties emerging from the reduced dimensionality, and for the potential technological applications, with promises such as dissipationless currents and 2D heterostructures outperforming the current silicon-based technology at a fraction of the size. In this work, we took an experimental approach to the realization and characterization of materials predicted to host Dirac nodal lines (DNLs), which despite many theoretical predictions have seen few experimental realizations reported so far. These materials belong to the recently evidenced class of topological semimetals, whose specificity is a symmetry-protected band crossing of the valence and conduction bands along a line in momentum space, with linear dispersion. As a first step, we focused on Cu2Si, the first 2D material in which DNLs have been evidenced when prepared on a Cu(111) substrate. After successfully reproducing existing results, we showed using ARPES and XPS that contrary to expectations, the DNLs were preserved after deposition of Pb on the surface without any gap, and that a band splitting occurred. We followed by the investigation of Cu2Si/Si(111), and found that despite a strongly related atomic structure, the Si(111) substrate interacts strongly enough with the out-of-plane orbitals of the Cu2Si layer to prevent the existence of the nodal lines. We then looked at the 2D Cu2Ge system, predicted to host DNL, and attempted to synthesize it by depositing Ge on Cu(111). By combining our LEED, XPS and ARPES results we found that all measurements matched closely what was expected from a free-standing Cu2Ge monolayer, showing the almost complete absence of interactions between the Cu(111) substrate and the surface Cu2Ge layer grown on it. This is the first reported experimental realization of the two-dimensional Dirac nodal line semimetal Cu2Ge. In a mirroring study, we deposited Cu on Ge(111) and observed a dissimilar band structure. Helped by STM, we explained those differences by a different atomic structure, and by a strongly interacting substrate. We highlight through this work the influence of the substrate, whether metallic or semiconductor, on the electronic properties of 2D DNL systems

This thesis describes two topological phases of matter, the Weyl semimetal and the line node semimetal, that are related to but distinct from topological insulator phases. These new topological phases are semimetallic, having electronic energy bands that touch at discrete points or along a continuous curve in momentum space. These states are achieved by breaking time-reversal symmetry near a transition between an ordinary insulator and a topological insulator, using a model based on alternating layers of topological and ordinary insulators, which can be tuned close to the transition by choosing the thicknesses of the layers. The semimetallic phases are topologically protected, with corresponding topological surface states, but the protection is due to separation of the band-touching points in momentum space and discrete symmetries, rather than being protected by an energy gap as in topological insulators. The chiral surface states of the Weyl semimetal give it a non-zero Hall conductivity, while the surface states of the line node semimetal have a flat energy dispersion in the region bounded by the line node. Some transport properties are derived, with a particular emphasis on the behaviour of the conductivity as a function of the impurity concentrations and the temperature.

Topology, an important topic in physics since several years, is handled as possible solution to many current-state problems in electronics and energy. It could allow to dramatically shrink computational devices or increase their speed without the current problem of heat dissipation, or topological principles can be used to introduce room temperature high-conduction paths within materials. Unfortunately, while many promising materials have been presented yet, the one breakthrough material is still missing. Current style materials are either consisting of toxical elements, obstructing possible use cases, or their electronic structure is too complex to investigate the interplay of all the facets of the electronic structure present in the mateirals. In this thesis, two very promising materials will be thoroughly introduced, namely TaIrTe4 and GaGeTe. Both materials have the potential, to lift one of the shortcomings mentioned. First, TaIrTe4 will be presented. TaIrTe4 is a simplistic Weyl semimetal in terms of its electronic and topological structure - the simplest yet known material. It hosts four Weyl points, the minimum amount of Weyl nodes possible in a non-centrosymmetric material. Predictions state, that these nodes are well separated throughout the Brillouin zone, and are connected by nearly parallel Fermi arcs. The existance of the topological states is proved in this thesis through angle-resolved photoemission spectroscopy (ARPES) and confirmed by spin polarization measurements on these states. GaGeTe is predicted to be a Bi2Se3-style topological insulator, but ARPES data presented shows, that no direct band gap could be observed. Yet, a topological state is still believed to be present. This makes this material interesting in many ways: its elemental composition is less toxic than bismuth and selenium, as well as it is the first realization of such a specific electronic structure. A full discussion of the electronic states close to the Fermi level including the possible existance of topological states is shown in this thesis.

Weyl-Semimetalle haben bemerkenswerte Eigenschaften. Ihr elektrischer Widerstand steigt linear und unsaturiert mit einem angelegten Magnetfeld, diverse Ergebnisse deuten darauf hin, dass sie einen unordnungsinduzierten Metall-Isolator-Phasenübergang aufweisen und ihre Ladungsträger zeigen die chirale Anomalie, d.h., die Nichtkonservierung der chiralen Ladung. Diese Eigenschaften haben ihren Ursprung in der Niedrigenergiephysik der Weyl- Semimetalle, die von Weyl-Punkten, Berührungspunkten zwischen Leitungs- und Valenzband an der Fermi-Energie mit einer linearen Dispersionsrelation, dominiert wird. Diese Berührungspunkte sind topologisch geschützt, d.h., kleine Störung können ihnen nichts anhaben. Weyl-Semimetalle sind daher Beispiele für topologische Semimetalle, Materialien mit geschützten niedrigdimensionalen Berührungenspunkten, -linien, oder -oberflächen an der Fermi-Energie. In dieser Arbeit zeigen wir, wie die Eigenschaften von Weyl-Semimetallen durch Unordnung, Magnetfelder und Deformationen beeinflusst werden. Wir zeigen außerdem eine Querverbindung zwischen Weyl-Semimetallen und nodal line-Semimetallen, topologisch geschützten Semimetallen mit einer eindimensionalen Fermi-Fläche. Durch die Nutzung von Gitter- und Niedrigenergiekontinuumsmodellen können wir Wege aufzeigen, wie man unsere Ergebnisse sowohl aus einer Festkörperphysik- als auch aus einer Hochenergiephysikperspektive verstehen kann. Insbesondere identifizieren wir eine experimentelle Signatur der chiralen Anomalie: die blaue Note, ein charakteristisches Muster in Form einer Note, das mit Hilfe von winkelaufgelöster Photoelektronenspektroskopie gemessen werden kann. Ein weiteres wichtiges Charakteristikum ist der Magnetwiderstand, der in Weyl-Semimetallen vom Winkel zwischen einem angelegten Magnetfeld und der Transportrichtung abhängt. Durch den Einfluss der chiralen Anomalie ist der longitudinale Magnetwiderstand negativ, der transversale Widerstand hingegen wächst linear und grenzenlos mit dem angelegten Magnetfeld. In dieser Dissertation untersuchen wir beide Charakteristiken analytisch und numerisch. Inspiriert durch Experimente, in denen ein scharfes Leitfähigkeitsmaximum für parallele elektrische und Magnetfelder observiert wurde, zeigen wir, dass die Leitfähigkeit vom Winkel zwischen den angelegten Feldern und dem Abstandsvektor der Weyl-Punkte abhängt und dass sie insbesondere für Felder parallel zum Abstandsvektor ein scharfes Maximum aufweist. Dieser Effekt ist besonders ausgeprägt, wenn nur das niedrigste Landau-Niveau zur Leitfähigkeit beiträgt, er bleibt aber auch bei höheren Energien beobachtbar. Für parallelen Magnettransport untersuchen wir starke Unordnung, die außerhalb des von der Störungstheorie abgedeckten Bereichs liegt, numerisch und beobachten einen positiven Magnetwiderstand, qualitativ ähnlich zu experimentellen Daten. Aus Deformationen in Weyl-Semimetallen entstehen sogenannte chirale oder auch axiale Felder, die ähnliche Konsequenzen wie externe elektromagnetische Felder haben, wobei noch viele Details im Verborgenen liegen. Wir untersuchen Deformationen aus zwei verschiedenen Perspektiven: zunächst zeigen wir, wie zwei widersprüchliche Vorhersagen aus der Quantenfeldtheorie, die konsistenten und kovarianten Anomalien, in einem Gittermodell beobachtbar sind. Dann untersuchen wir elektrischen Transport unter Einfluss von axialen Magnetfeldern und zeigen, dass Moden, die sich in unterschiedliche Richtungen bewegen, räumlich getrennt sind. Diese räumliche Trennung hat eine unübliches Wachstums des elektrischen Leitwerts mit der transversalen Systembreite zur Folge. Des weiteren zeigen wir, wie ein nodal line-Semimetall aus einem Weyl-Semimetall entstehen kann, das einer Supergitterstruktur ausgesetzt ist. Wir interpretieren die Oberflächenzustände mit Hilfe der interzellulären Zak-Phase und zeigen zwei verschiedene Mechanismen, die die Bandstruktur vor der Öffnung einer Bandlücke schützen, auf. Um unsere Diskussion abzuschließen, untersuchen wir Transport in nodal line-Semimetallen in Kürze und stellen ihre Quantenfeldtheorie vor. Schließlich wenden wir uns wechselwirkenden Phasen zu und zeigen, welche Konsequenzen die Symmetrieklassifizierung des Sachdev- Ye-Kitaev-Modells hat – ein Modell von Teilchen mit zufälligen Wechselwirkungsstärken, dessen Topologie von der Anzahl der enthaltenen Teilchen bestimmt wird.:1 Introduction 2 Topological Band Theory 2.1 Geometric Phase and Berry Phase 2.1.1 The Adiabatic Theorem 2.1.2 The Zak Phase 2.2 Tenfold Classification of Topological Insulators and Superconductors 2.3 Topological Semimetals 2.3.1 Weyl Semimetals 2.3.2 Nodal Line Semimetals 2.4 Bulk-boundary Correspondence from the Intercellular Zak Phase 2.4.1 Intra- and Intercellular Zak Phase 2.4.2 Bulk-boundary Correspondence 2.4.3 Conclusion 3 Field Theory Perspective on Topological Phases 3.1 Topological Insulators 3.2 Weyl Fermions and the Chiral Anomaly 3.3 Visualizing the Chiral Anomaly with Photoemission Spectroscopy 3.3.1 The Chiral Anomaly in Condensed Matter Systems 3.3.2 Model and Methods 3.3.3 ARPES Spectra for Weyl and Dirac Semimetals 3.3.4 Experimental Details 3.3.5 Summary and Conclusion 3.4 The Consistent and Covariant Anomalies 3.5 Consistent and Covariant Anomalies on a Lattice 3.5.1 Model and Methods 3.5.2 Lattice Results for Consistent and Covariant Anomalies 3.5.3 Influence of the Mass Term 3.5.4 The Quest for One Third 3.6 The Action of Nodal Line Semimetals 4 Transport in Topological Semimetals 4.1 Longitudinal Magnetoresistance in Weyl Semimetals 4.2 Transversal Magnetoresistance in Weyl Semimetals 4.2.1 Model 4.2.2 Mesoscopic Transport in Clean Samples 4.2.3 Numerical Magnetotransport in the Presence of Disorder 4.2.4 Born-Kubo Analytical Bulk Conductivity 4.2.5 Numerical Results in Disordered Samples 4.2.6 Conclusion 4.3 Transport in the Presence of Axial Magnetic Fields 4.3.1 Model and Methods 4.3.2 Longitudinal Magnetotransport for Axial Fields 4.3.3 Conclusion 4.4 Transport in Nodal Line Semimetals 5 Nodal Line Semimetals from Weyl Superlattices 5.1 Weyl Semimetal on a Superlattice 5.2 Emergent Nodal Phases 5.3 Symmetry Classification of the Nodal Line 5.4 Surface States 5.5 Stability against Wave Vector Mismatch 5.6 Time-reversal Symmetric Weyl Semimetal 5.7 Conclusion 6 Symmetry Classification of the SYK Model 6.1 Model and Topological Classification 6.2 Overlap of Time-reversed Partners 6.2.1 Even Number of Majoranas 6.2.2 Odd Number of Majoranas 6.3 Spectral Function 6.3.1 Zero Temperature 6.3.2 Infinite Temperature 6.4 Symmetry-breaking Terms 6.5 Lattice Model 6.6 Conclusion 7 Conclusion and Outlook Appendix A Zak Phase and Extra Charge Accumulation Appendix B Material-specific Details for ARPES B.1 Relaxation Rates B.2 ARPES in Finite Magnetic Fields B.3 Estimates of the Chiral Chemical Potential Difference Appendix C Weyl Nodes in a Magnetic Field C.1 Scattering between Different Landau Levels C.2 Analytical Born-Kubo Calculation of Transversal Magnetoconductivity C.2.1 Disorder Scattering in Born Approximation C.2.2 Transversal Magnetoconductivity from Kubo Formula Appendix D Transfer Matrix Method D.1 Longitudinal Magnetic Field D.2 Transversal Magnetic Field Bibliography Acknowledgments List of Publications Versicherung
Weyl semimetals have remarkable properties. Their resistance grows linearly and unsaturated with an applied transversal magnetic field, and they are expected to show a disorder-induced metal-insulator transition. Their charge carriers exhibit the chiral anomaly, i.e., the nonconservation of chiral charge. These properties emerge from their low-energy physics, which are dominated by Weyl nodes: zero-dimensional band crossings at the Fermi energy with a linear dispersion. The band crossings are topologically protected, i.e., they cannot be lifted by small perturbations. Thus, Weyl semimetals are examples of topological semimetals, materials with protected lower-dimensional band crossing close to the Fermi surface. In this work, we show how the properties of Weyl semimetals are affected by disorder, magnetic fields, and strain. We further provide a link between Weyl semimetals and nodal line semimetals, topological semimetals with a one-dimensional Fermi surface. By using both lattice and low-energy continuum models, we present ways to understand the results from a condensed-matter and a quantum-field-theory perspective. In particular, we identify an experimental signature of the chiral anomaly: the blue note, a characteristic note-shaped pattern that can be measured in photoemission spectroscopy. Another important signature is the magnetoresistance. In Weyl semimetals, its behavior depends on the angle between the magnetic field and the transport direction. For parallel transport, a negative longitudinal magnetoresistance as a manifestation of the chiral anomaly is observed; for orthogonal transport, the transversal magnetoresistance shows a linear and unsaturated growth. In this thesis, we investigate both regimes analytically and numerically. Inspired by experiments that show a sharply peaked magnetoresistance for parallel fields, we show that the longitudinal magnetoresistance depends on the angle between applied fields and the Weyl node separation, and that it is sharply peaked for fields parallel to the node separation. This effect is especially strong in the limit where only the lowest Landau level contributes to the magnetoresistance, but it survives at higher chemical potentials. For transversal magnetotransport, we numerically investigate the strong-disorder regime that is beyond the reach of perturbation theory and observe a positive magnetoresistance, qualitatively similar to recent experiments. Strain in Weyl semimetals creates so-called axial fields that result in phenomena similar to the ones driven by electric and magnetic fields, but with some yet unknown consequences. We investigate strain from two perspectives: first, we show how two different predictions from quantum field theory, the consistent and covariant anomalies, manifest on a lattice. Second, we investigate transport in the presence of axial magnetic fields and show that counterpropagating modes are spatially separated, resulting in an unusual scaling of the conductance with the system’s width. We further show how a nodal line semimetal can emerge from a Weyl semimetal on a superlattice. We interpret the presence of surface states in terms of the intercellular Zak phase and show two distinct mechanisms that protect the spectrum from opening a gap. To complete our discussion, transport in nodal line semimetals is briefly discussed, as well as the quantum field theory that describes the low-energy features of these materials. Finally, we conclude this work by showing manifestations of the different symmetry classes that can be realized in the Sachdev-Ye-Kitaev model—a model of randomly interacting particles whose topology is deeply connected to the number of particles.:1 Introduction 2 Topological Band Theory 2.1 Geometric Phase and Berry Phase 2.1.1 The Adiabatic Theorem 2.1.2 The Zak Phase 2.2 Tenfold Classification of Topological Insulators and Superconductors 2.3 Topological Semimetals 2.3.1 Weyl Semimetals 2.3.2 Nodal Line Semimetals 2.4 Bulk-boundary Correspondence from the Intercellular Zak Phase 2.4.1 Intra- and Intercellular Zak Phase 2.4.2 Bulk-boundary Correspondence 2.4.3 Conclusion 3 Field Theory Perspective on Topological Phases 3.1 Topological Insulators 3.2 Weyl Fermions and the Chiral Anomaly 3.3 Visualizing the Chiral Anomaly with Photoemission Spectroscopy 3.3.1 The Chiral Anomaly in Condensed Matter Systems 3.3.2 Model and Methods 3.3.3 ARPES Spectra for Weyl and Dirac Semimetals 3.3.4 Experimental Details 3.3.5 Summary and Conclusion 3.4 The Consistent and Covariant Anomalies 3.5 Consistent and Covariant Anomalies on a Lattice 3.5.1 Model and Methods 3.5.2 Lattice Results for Consistent and Covariant Anomalies 3.5.3 Influence of the Mass Term 3.5.4 The Quest for One Third 3.6 The Action of Nodal Line Semimetals 4 Transport in Topological Semimetals 4.1 Longitudinal Magnetoresistance in Weyl Semimetals 4.2 Transversal Magnetoresistance in Weyl Semimetals 4.2.1 Model 4.2.2 Mesoscopic Transport in Clean Samples 4.2.3 Numerical Magnetotransport in the Presence of Disorder 4.2.4 Born-Kubo Analytical Bulk Conductivity 4.2.5 Numerical Results in Disordered Samples 4.2.6 Conclusion 4.3 Transport in the Presence of Axial Magnetic Fields 4.3.1 Model and Methods 4.3.2 Longitudinal Magnetotransport for Axial Fields 4.3.3 Conclusion 4.4 Transport in Nodal Line Semimetals 5 Nodal Line Semimetals from Weyl Superlattices 5.1 Weyl Semimetal on a Superlattice 5.2 Emergent Nodal Phases 5.3 Symmetry Classification of the Nodal Line 5.4 Surface States 5.5 Stability against Wave Vector Mismatch 5.6 Time-reversal Symmetric Weyl Semimetal 5.7 Conclusion 6 Symmetry Classification of the SYK Model 6.1 Model and Topological Classification 6.2 Overlap of Time-reversed Partners 6.2.1 Even Number of Majoranas 6.2.2 Odd Number of Majoranas 6.3 Spectral Function 6.3.1 Zero Temperature 6.3.2 Infinite Temperature 6.4 Symmetry-breaking Terms 6.5 Lattice Model 6.6 Conclusion 7 Conclusion and Outlook Appendix A Zak Phase and Extra Charge Accumulation Appendix B Material-specific Details for ARPES B.1 Relaxation Rates B.2 ARPES in Finite Magnetic Fields B.3 Estimates of the Chiral Chemical Potential Difference Appendix C Weyl Nodes in a Magnetic Field C.1 Scattering between Different Landau Levels C.2 Analytical Born-Kubo Calculation of Transversal Magnetoconductivity C.2.1 Disorder Scattering in Born Approximation C.2.2 Transversal Magnetoconductivity from Kubo Formula Appendix D Transfer Matrix Method D.1 Longitudinal Magnetic Field D.2 Transversal Magnetic Field Bibliography Acknowledgments List of Publications Versicherung

archives@tulane.edu
The discovery of topological semimetals has attracted enormous interest since they not only possess many unusual exotic properties, but also offer a fertile ground for searching for new fermions in the low energy spectrum. The first established example of a topological state of matter is the quantum Hall effect, which supports a gapless edge state protected by topological invariance. Later the concept of topology has been extended to describe electronic band structure of solid state materials and this effort leads to discoveries of many new topological quantum states, such as Dirac cone state in graphene, quantum spin Hall insulator states in semiconductor quantum wells, 3D topological insulators, etc. The recently discovered Dirac/Weyl semimetals can be viewed as a 3D analog of graphene. This thesis work aims to discover new Dirac/Weyl semimetals through single crystal synthesis and characterization. This thesis is organized as follows: In chapter 1, I will first briefly review several basic concepts of topological properties and introduce a few prototype topological semimetals related to my thesis work. Since one important part of my thesis work involves single crystal growth of topological semimetals, I will introduce the crystal growth methods used in my research in chapter 2. In chapters 3, 4 and 5, I will present my experimental discoveries of new topological semimetals, including YSn2, CaSn3 and TbPtBi. I will not only show property characterization of these material, but also discuss their underlying physics. For YSn2, my work reveals that its slightly distorted square lattice of Sn generates multiple topologically non-trivial bands, one of which likely hosts nodal line and tunable Weyl semimetal state induced by the Rashba spin-orbit coupling (SOC) and proper external magnetic field. The quasiparticles described as relativistic fermions from these bands are manifested by nearly zero mass, and non-trivial Berry phases probed in de Haas–van Alphen (dHvA) oscillations. The dHvA study also reveals YSn2 has a complex Fermi surface (FS), consisting of several 3D and one 2D pocket. Our first principle calculations show the point-like 3D pocket at Y point on the Brillouin zone boundary hosts the possible Weyl state. Our findings establish YSn2 as a new interesting platform for observing novel topological phases and studying their underlying physics. In the study of CaSn3, we not only found it possesses non-trivial band topology, but also discovered its intrinsic superconductivity at 1.178 K. Its topological fermion properties, including the nearly zero quasi-particle mass and the non-trivial Berry phase accumulated in cyclotron motions, were revealed from the dHvA quantum oscillation studies of this material. Our findings make CaSn3 a promising candidate for exploring new exotic states arising from the interplay between non-trivial band topology and superconductivity, e.g., topological superconductivity. For the Half-Heusler compound TbPtBi, we have studied its field-induced Weyl semimetal state. We have observed remarkable transport signatures of its Weyl state, including the chiral anomaly, intrinsic anomalous Hall effect (AHE), and in-plane Hall effect. Moreover, we found TbPtBi exhibits a much larger AHE than the previously reported field-induced Weyl semimetal state in GdPtBi. The distinct aspect of TbPtBi is that Tb ions carry greater magnetic moments than Gd ions in GdPtBi (9.0B/Tb vs.7.0B/Gd). We find that such a moment increase in TbPtBi drastically enhances its AHE, with its anomalous Hall angle reaching as large as 0.50-0.76 in its antiferromagnetic (AFM) state. This finding not only strongly supports that the Zeeman effect due to the large exchange field from 4f electrons plays a critical role in creating the field-included Weyl state, but also provides clear evidence for the theoretical prediction that the intrinsic anomalous Hall conductivity is proportional to the separation of the Weyl points with opposite chirality.
1
Yanglin Zhu

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.

Topological insulators have garnered enormous amount of attention owing to their unique topological properties. The surface states protected by the topology of the system hold limitless potentials including topological quantum computation. WTIs are one branch of this exotic field which is new to the family of the topological insulators. Initially, weak topological insulators (WTIs) are assumed to be a mere version of ordinary insulators recent reports claim the opposite. The WTIs are in fact much more robust than its counterpart strong topological insulators (STIs). Unfortunately, there are a few materials known to be as WTIs which implies the difficulty in finding this phase in real materials. Since WTIs are thought to be an adiabatic stack of 2D quantum spin Hall layers, many proposals came up with the idea of super lattice structures where a stack of a trivial insulator with a STI could possibly give rise to this most sought after phase. However many of them are still waiting to be experimentally verifi ed. In this regard, our work is a much relief that was needed. We have discovered a new WTI, which is easily cleavable. In this thesis, we are going to discuss about the newly discovered WTI, BiSe, more extensively

In den letzten Jahren führte die Verbindung zwischen Topologie und kondensierter Materie zur Entdeckung vieler interessanter und exotischer elektronischer Effekte. Während sich die Forschung anfangs auf elektronische Systeme mit einer Bandlücke wie den topologischen Isolator konzentrierte, erhalten in letzter Zeit topologische Halbmetalle viel Aufmerksamkeit. Das bekannteste Beispiel sind Weyl-Halbmetalle, die an beliebigen Punkten in der Brillouin-Zone lineare Kreuzungen von nicht entarteten Bändern aufweist. An diese Punkte ist eine spezielle Quantenzahl namens Chiralität gebunden, die die Existenz von Weyl-Punktpaaren erzwingt. Diese Paare sind topologisch geschützt und wirken als Quellen und Senken der Berry-Krümmung, einem topologischen Feld im reziproken Raum. Diese Berry-Krümmung steht in direktem Zusammenhang mit dem anomalen Hall-Effekt, der die Entstehung einer Querspannung aus einem Längsstrom in einem magnetischen Material beschreibt. Analog existiert auch der anomale Nernst-Effekt, bei dem der longitudinale Strom durch einen thermischen Gradienten ersetzt wird. Dieser Effekt ermöglicht die Umwandlung von Wärme in elektrische Energie und ist zudem stark an die Berry-Krümmung gebunden. In dieser Arbeit werden die anomalen Transporteffekte zunächst in fundamentalen Modellsystemen untersucht. Hier wird eine Kombination aus analytischen und numerischen Methoden verwendet, um Quantisierungen sowohl des Hall- und Nernst- als auch des thermischen Hall-Effekts in zweidimensionalen Systemen mit und ohne externen Magnetfeldern zu zeigen. Eine Erweiterung in drei Dimensionen zeigt eine Quasi-Quantisierung, bei der die Leitfähigkeiten Werte der jeweiligen zweidimensionalen Quanten skaliert durch charakteristische Wellenvektoren annehmen. Im nächsten Schritt werden verschiedene Mechanismen zur Erzeugung starker Berry-Krümmung und damit großer anomaler Hall- und Nernst-Effekte sowohl in Modellsystemen als auch in realen Materialien untersucht. Dies ermöglicht die Identifizierung und Isolierung vielversprechender Effekte in den einfachen Modellen, in denen wichtige Merkmale untersucht werden können. Die Ergebnisse können dann auf die realen Materialien übertragen werden, wo die jeweiligen Effekte erkennbar sind. Hier werden sowohl Weyl-Punkte als auch Knotenlinien in Kombination mit Magnetismus als vielversprechende Eigenschaften identifiziert und Materialrealisierungen in der Klasse der Heusler-Verbindungen vorgeschlagen. Diese Verbindungen sind eine sehr vielseitige Materialklasse, in der unter anderem auch magnetische topologische Metalle zu finden sind. Um ein tieferes Verständnis der anomalen Transporteffekte zu erhalten sowie Faustregeln für Hochleistungsverbindungen abzuleiten, wurde eine High-Throughput-Rechnung von magnetisch-kubischen Voll-Heusler-Verbindungen durchgeführt. Diese Berechnung zeigt die Bedeutung von Spiegelebenen in magnetischen Materialien für große anomale Hall- und Nernst-Effekte und zeigt, dass einige der Heusler-Verbindungen die höchsten bisher berichteten Literaturwerte bei diesen Effekten übertreffen. Auch andere interessante Effekte im Zusammenhang mit Weyl-Punkten werden untersucht. Beim bekannten Weyl-Halbmetall NbP weisen die Weyl-Punkte aufgrund der hohen Symmetrie des Kristalls eine hohe Entartung auf. Die Anwendung von einachsigem Zug reduziert jedoch die Symmetrien und hebt damit die Entartungen auf. Eine theoretische Untersuchung zeigt, dass die Weyl-Punkte bei einachsigem Zug energetisch verschoben werden und, was noch wichtiger ist, dass sie bei realistischen Werten das Fermi-Niveau durchschreiten. Dies macht NbP zu einer vielversprechenden Plattform, um die Weyl-Physik weiter zu untersuchen. Die theoretischen Ergebnisse werden mit experimentellen Messungen von Shubnikov-de-Haas-Oszillationen unter einachsigem Zug kombiniert und es wird eine gute Übereinstimmung mit den theoretischen Ergebnissen gefunden. Als erster Schritt in Richtung neuer Berechnungsmethoden wird die Idee eines Weyl-Halbmetall-basierten Chiralitätsfilters für Elektronen untersucht. An der Grenzfläche zweier Weyl-Halbmetalle kann in Abhängigkeit von den genauen Weyl-Punktparametern nur eine Chiralität übertragen werden. Hier wird ein effektives geometrisches Modell erstellt und zur Untersuchung realer Materialgrenzflächen eingesetzt. Während im Allgemeinen eine Filterwirkung möglich erscheint, zeigten die untersuchten Materialien keine geeignete Kombination. Hier können weitere Studien mit Fokus auf magnetische Weyl-Halbmetalle oder Multifold-Fermion-Materialien durchgeführt werden.:List of publications Preface 1. Theoretical background 1.1. Berry curvature and Weyl semimetals 1.1.1. From the adiabatic evolution to the Berry phase 1.1.2. From the Berry phase to the Berry curvature 1.1.3. Topological phases of condensed matter 1.1.4. Weyl semimetals 1.1.5. Dirac semimetals 1.1.6. Nodal line semimetals 1.2. Density-functional theory 1.2.1. Born-Oppenheimer approximation 1.2.2. Hohenberg-Kohn theorems 1.2.3. Kohn-Sham formalism 1.2.4. Exchange-correlation functional 1.2.5. Pseudopotentials 1.2.6. Basis functions 1.2.7. VASP 1.3. Tight-binding Hamiltonian from Wannier functions 1.3.1. Wannier functions 1.3.2. Constructing Wannier functions from DFT 1.3.3. Generating a Wannier tight-binding Hamiltonian 1.3.4. Necessity of the tight-binding Hamiltonian 1.4. Linear response theory 1.4.1. General introduction to linear response 1.4.2. Anomalous Hall effect 1.4.3. Anomalous Nernst effect 1.4.4. Anomalous thermal Hall effect 1.4.5. Common features of anomalous transport effects 1.4.6. Symmetry considerations for Berry curvature related transport effects 1.4.7. Magneto-optic Kerr effect 1.4.8. About the efficiency of the calculations 2. (Quasi-)Quantization in the Hall, thermal Hall, and Nernst effects 2.1. Quantization with an external magnetic field 2.1.1. Two-dimensional case 2.1.2. Three-dimensional case 2.2. Quantization without an external field 2.2.1. Two-dimensional case 2.2.2. Three-dimensional case . 2.3. A remark on the spin Hall effect 2.4. A remark on the quasi-quantization of the three-dimensional conductivities 2.5. Conclusions 3. Understanding anomalous transport 3.1. Anomalous transport without a net magnetic moment 3.1.1. Toy model 3.1.2. Ti2MnAl and related compounds 3.2. Large Berry curvature enhancement from nodal line gapping 3.2.1. Toy model 3.2.2. Fe2MnP and related compounds 3.2.3. Co2MnGa 3.3. Topological features away from the Fermi level and the anomalous Nernst effect 3.3.1. Toy model . 3.3.2. Co2FeGe and Co2FeSn 3.4. Conclusions 4. Heusler database calculation 4.1. Workflow 4.2. Importance of mirror planes 4.3. The right valence electron count 4.4. Correlation between anomalous Hall and Nernst effects 4.5. Selected special compounds 4.6. Conclusions 5. NbP under uniaxial strain 5.1. NbP and its symmetries 5.2. The influence of strain on the electronic structure 5.2.1. Shifting of the Weyl points 5.2.2. Splitting of the Fermi surfaces 5.3. Comparison with experimental results 5.4. Conclusions 6. A tunable chirality filter 6.1. Concept 6.2. Geometrical simplification and expansion for more Weyl points 6.3. Material selection 6.3.1. Workflow 6.3.2. Results for NbP and TaAs 6.3.3. Results for Ag2Se and Ag2S 6.4. Conclusions and perspective . Summary and outlook A. Numerical tricks A.1. Hamiltonian setup at several k points at once A.2. Precalculating prefactors B. Derivation of the conductivity (quasi-)quanta B.1. Two dimensions B.1.1. General formula and necessary approximations B.1.2. Useful integrals B.1.4. Quantized thermal Hall effect B.1.5. Quantized Nernst effect B.1.6. Flat bands and the Nernst effect B.2. Three dimensions B.2.1. General formula B.2.2. Three-dimensional electron gas B.2.3. Three-dimensional Weyl semimetal C. Heusler database tables D. Details on the NbP strain calculations E. Details on the geometrical matching procedure References List of abbreviations List of Figures List of Tables Acknowledgements Eigenständigkeitserklärung
In recent years, the connection between topology and condensed matter resulted in the discovery of many interesting and exotic electronic effects. While in the beginning, the research was focused on gapped electronic systems like the topological insulator, more recently, topological semimetals are getting a lot of attention. The most well-known example is the Weyl semimetal, which hosts linear crossings of non-degenerate bands at arbitrary points in the Brillouin zone. Tied to these points there is a special quantum number called chirality, which enforces the existence of Weyl point pairs. These pairs are topologically protected and act as sources and sinks of the Berry curvature, a topological field in reciprocal space. This Berry curvature is directly connected to the anomalous Hall effect, which describes the emergence of a transverse voltage from a longitudinal current in a magnetic material. Analogously, there also exists the anomalous Nernst effect, where the longitudinal current is replaced by a thermal gradient. This effect allows for the conversion of heat into electrical energy and is also strongly tied to the Berry curvature. In this work, the anomalous transport effects are at first studied in fundamental model systems. Here, a combination of analytical and numerical methods is used to reveal quantizations in both the Hall, the Nernst, and the thermal Hall effects in two-dimensional systems with and without external magnetic fields. An expansion into three dimensions shows a quasi-quantization, where the conductivities take values of the respective two-dimensional quanta scaled by characteristic wavevectors. In the next step, several mechanisms for the generation of strong Berry curvature and thus large anomalous Hall and Nernst effects are studied in both model systems and real materials. This allows for the identification and isolation of promising effects in the simple models, where important features can be studied. The results can then be applied to the real materials, where the respective effects can be recognized. Here, both Weyl points and nodal lines in combination with magnetism are identified as promising features and material realizations are proposed in the class of Heusler compounds. These compounds are a very versatile class of materials, where among others also magnetic topological metals can be found. To get a deeper understanding of the anomalous transport effects as well as to derive guidelines for high-performance compounds, a high-throughput calculation of magnetic cubic full Heusler compounds was carried out. This calculation reveals the importance of mirror planes in magnetic materials for large anomalous Hall and Nernst effects and shows that some of the Heusler compounds outperform the highest so-far reported literature values in these effects. Also other interesting effects related to Weyl points are investigated. In the well-known Weyl semimetal NbP, the Weyl points have a high degeneracy due to the high symmetry of the crystal. However, the application of uniaxial strain reduces the symmetries and therefore lifts the degeneracies. A theoretical investigation shows, that the Weyl points are moved in energy under uniaxial strain and, more importantly, that at reasonable strain values they cross the Fermi level. This renders NbP a promising platform to further study Weyl physics. The theoretical results are combined with experimental measurements of Shubnikov-de Haas oscillations under uniaxial strain and a good agreement with the theoretical results is found. As a first step in the direction of new ways of computation, an idea of a Weyl semimetal based chirality filter for electrons is investigated. At the interface of two Weyl semimetals, depending on the exact Weyl point parameters, it is possible to transmit only one chirality. Here, an effective geometrical model is established and employed for the investigation of real material interfaces. While in general, a filtering effect seems possible, the investigated materials did not show any suitable combination. Here, further studies can be made with the focus on either magnetic Weyl semimetals of multifold-fermion materials.:List of publications Preface 1. Theoretical background 1.1. Berry curvature and Weyl semimetals 1.1.1. From the adiabatic evolution to the Berry phase 1.1.2. From the Berry phase to the Berry curvature 1.1.3. Topological phases of condensed matter 1.1.4. Weyl semimetals 1.1.5. Dirac semimetals 1.1.6. Nodal line semimetals 1.2. Density-functional theory 1.2.1. Born-Oppenheimer approximation 1.2.2. Hohenberg-Kohn theorems 1.2.3. Kohn-Sham formalism 1.2.4. Exchange-correlation functional 1.2.5. Pseudopotentials 1.2.6. Basis functions 1.2.7. VASP 1.3. Tight-binding Hamiltonian from Wannier functions 1.3.1. Wannier functions 1.3.2. Constructing Wannier functions from DFT 1.3.3. Generating a Wannier tight-binding Hamiltonian 1.3.4. Necessity of the tight-binding Hamiltonian 1.4. Linear response theory 1.4.1. General introduction to linear response 1.4.2. Anomalous Hall effect 1.4.3. Anomalous Nernst effect 1.4.4. Anomalous thermal Hall effect 1.4.5. Common features of anomalous transport effects 1.4.6. Symmetry considerations for Berry curvature related transport effects 1.4.7. Magneto-optic Kerr effect 1.4.8. About the efficiency of the calculations 2. (Quasi-)Quantization in the Hall, thermal Hall, and Nernst effects 2.1. Quantization with an external magnetic field 2.1.1. Two-dimensional case 2.1.2. Three-dimensional case 2.2. Quantization without an external field 2.2.1. Two-dimensional case 2.2.2. Three-dimensional case . 2.3. A remark on the spin Hall effect 2.4. A remark on the quasi-quantization of the three-dimensional conductivities 2.5. Conclusions 3. Understanding anomalous transport 3.1. Anomalous transport without a net magnetic moment 3.1.1. Toy model 3.1.2. Ti2MnAl and related compounds 3.2. Large Berry curvature enhancement from nodal line gapping 3.2.1. Toy model 3.2.2. Fe2MnP and related compounds 3.2.3. Co2MnGa 3.3. Topological features away from the Fermi level and the anomalous Nernst effect 3.3.1. Toy model . 3.3.2. Co2FeGe and Co2FeSn 3.4. Conclusions 4. Heusler database calculation 4.1. Workflow 4.2. Importance of mirror planes 4.3. The right valence electron count 4.4. Correlation between anomalous Hall and Nernst effects 4.5. Selected special compounds 4.6. Conclusions 5. NbP under uniaxial strain 5.1. NbP and its symmetries 5.2. The influence of strain on the electronic structure 5.2.1. Shifting of the Weyl points 5.2.2. Splitting of the Fermi surfaces 5.3. Comparison with experimental results 5.4. Conclusions 6. A tunable chirality filter 6.1. Concept 6.2. Geometrical simplification and expansion for more Weyl points 6.3. Material selection 6.3.1. Workflow 6.3.2. Results for NbP and TaAs 6.3.3. Results for Ag2Se and Ag2S 6.4. Conclusions and perspective . Summary and outlook A. Numerical tricks A.1. Hamiltonian setup at several k points at once A.2. Precalculating prefactors B. Derivation of the conductivity (quasi-)quanta B.1. Two dimensions B.1.1. General formula and necessary approximations B.1.2. Useful integrals B.1.4. Quantized thermal Hall effect B.1.5. Quantized Nernst effect B.1.6. Flat bands and the Nernst effect B.2. Three dimensions B.2.1. General formula B.2.2. Three-dimensional electron gas B.2.3. Three-dimensional Weyl semimetal C. Heusler database tables D. Details on the NbP strain calculations E. Details on the geometrical matching procedure References List of abbreviations List of Figures List of Tables Acknowledgements Eigenständigkeitserklärung

Topological phases of matter represent a new phase which cannot be understood in terms of Landau’s theory of symmetry breaking and are characterized by non-local topological properties emerging from purely local (microscopic) degrees of freedom. It is the non-trivial topology of the bulk band structure that gives rise to topological phases in condensed matter systems. Quantum Hall systems are prominent examples of such topological phases. Different quantum Hall states cannot be distinguished by a local order parameter. Instead, non-local measurements are required, such as the Hall conductance, to differentiate between various quantum Hall states. A signature of a topological phase is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations which respect appropriate symmetries. Examples of such properties are the presence of protected gapless edge states at the boundary of the system for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall states. The robustness of these properties can be under-stood through the existence of a topological invariant, such as the Chern number for quantum Hall states which is quantized to integer values and can only be changed by closing the bulk gap. Two other examples of topological phases of matter are topological superconductors and Weyl semimetals. The study of transport in various kinds of junctions of these topological materials is highly interesting for their applications in modern electronics and quantum computing. Another intriguing area to study is how to generate new kind of gapless edge modes in topological systems. In this thesis I have studied various aspects of topological phases of matter, such as electronic transport in junctions of topological insulators and topological superconductors, the generation of new kinds of boundary modes in the presence of granularity, and the effects of periodic driving in topological systems. We have studied the following topics. 1. transport across a line junction of two three-dimensional topological insulators, 2. transport across a junction of topological insulators and a superconductor, 3. surface and edge states of a topological insulator starting from a lattice model, 4. effects of granularity in topological insulators, 5. Majorana modes and conductance in systems with junctions of topological superconducting wires and normal metals, and 6. generation of new surface states in a Weyl semimetal in the presence of periodic driving by the application of electromagnetic radiation. A detailed description of each chapter is given below. • In the first chapter we introduce a number of concepts which are used in the rest of the thesis. We will discuss the ideas of topological phases of matter (for example, topological insulators, topological superconductors and Majorana modes, and Weyl semimetals), the renormalization group theory for weak interactions, and Floquet theory for periodically driven systems. • In the second chapter we study transport across a line junction which separates the surfaces of two three-dimensional topological insulators. The velocities of the Dirac electrons on the two surfaces may be unequal and may even have opposite signs. For a time-reversal invariant system, we show that the line junction is characterized by an arbitrary real parameter α; this determines the scattering amplitudes (reflection and transmission) from the junction. The physical origin of α is a potential barrier that may be present at the junction. If the surface velocities have the same sign, edge states exist that propagate along the line junction with a velocity and orientation of the spin which depend on α and the ratio of the velocities. Next, we study what happens if the two surfaces are at an angle φ with respect to each other. We study the scattering and differential conductance across the line junction as functions of φ and α. We also show that there are edge states which propagate along the line junction with a velocity and spin orientation which depend on φ. Finally, if the surface velocities have opposite signs, we find that the electrons must necessarily transmit into the two-dimensional interface separating the two topological insulators. • In the third chapter we discuss transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. This junction is more complicated than the line junction discussed in the previous chapter because of the presence of the superconductor. In a topological insulator spin-up and spin-down electrons get coupled while in a superconductor electrons and holes get coupled. Hence we have to use a four-component spinor formalism to describe both spin and particle-hole degrees of freedom. The junction can have three time-reversal invariant barriers on the three sides. We compute the subgap charge conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters which characterize the barriers. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in the charge conductances. We discuss the effects of bound states on the superconducting side on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. • In the fourth chapter we derive the surface Hamiltonians of a three-dimensional topological insulator starting from a microscopic model. (This description was not discussed in the previous chapters where we directly started from the surface Hamiltonians without deriving them form a bulk Hamiltonian). Here we begin from the bulk Hamiltonian of a three-dimensional topological insulator Bi2Se3. Using this we derive the surface Hamiltonians on various surfaces of the topological insulator, and we find the states which appear on the different surfaces and along the edge between pairs of surfaces. The surface Hamiltonians depend on the orientation of the surfaces and are therefore quite different from the previous chapters. We use both analytical methods based on the surface Hamiltonians (which are derived from the bulk Hamiltonian) and numerical methods based directly on a lattice discretization of the bulk Hamiltonian in order to find surface and edge states. We find that the application of a potential barrier along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering and conductance across the edge are studied as a function of the edge potential. We show that a magnetic field applied in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states. • In the fifth chapter we study a system made of topological insulator (TI) nanocrystals which are coupled to each other. Our theoretical studies are motivated by the following experimental observations. Electrical transport measurements were carried out on thin films of nanocrystals of Bi2Se3 which is a TI. The measurements reveal that the entire system behaves like a single TI with two topological surface states at the two ends of the system. The two surface states are found to be coupled if the film thickness is small and decoupled above a certain film thickness. The surface state penetration depth is found to be unusually large and it decreases with increasing temperature. To explain all these experimental results we propose a theoretical model for this granular system. This consists of multiple grains of Bi2Se3 stacked next to each other in a regular array along the z-direction (the c-axis of Bi2Se3 nanocrystals). We assume translational invariance along the x and y directions. Each grain has top and bottom surfaces on which the electrons are described by Hamiltonians of the Dirac form which can be derived from the bulk Hamiltonian known for this material. We introduce intra-grain tunneling couplings t1 between the opposite surfaces of a single grain and inter-grain couplings t2 between nearby surfaces of two neighboring grains. We show that when t1 < t2 the entire system behaves like a single topological insulator whose outermost surfaces have gapless spectra described by Dirac Hamiltonians. We find a relation between t1, t2 and the surface state penetration depth λ which explains the properties of λ that are seen experimentally. We also present an expression for the surface state Berry phase as a function of the hybridization between the surface states and a Zeeman magnetic field that may be present in the system. At the end we theoretically studied the surface states on one of the side surfaces of the granular system and showed that many pairs of surface states can exist on the side surfaces depending on the length of the unit cell of the granular system. • In the sixth chapter we present our work on junctions of p-wave superconductors (SC) and normal metals (NM) in one dimension. We first study transport in a system where a SC wire is sandwiched between two NM wires. For such a system it is known that there is a Majorana mode at the junction between the SC and each NM lead. If the p-wave pairing changes sign at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the subgap conductance between the leads and the SC. We derive an analytical expression as a function of and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the energies oscillate and decay exponentially as L is increased. The energies exactly match the locations of the peaks in the conductance. We find that the subgap conductances do not change noticeably with the sign of . So there is no effect of the extra Majorana modes which appear inside the SC (due to changes in the signs of Δ) on the subgap conductance. Next we study junctions of three p-wave SC wires which are connected to the NM leads. Such a junction is of interest as it is the simplest system where braiding of Majorana modes is possible. Another motivation for studying this system is to see if the subgap transport is affected by changes in the signs of . For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads. In addition, depending on the signs of the Δ’s in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various subgap conductances have peaks occurring at the energies of all these modes; we therefore get a rich pattern of conductance peaks. Next we study the effects of interactions between electrons (in the NM leads) on the transport. We use a renormalization group approach to study the effect of interactions on the conductance at energies far from the SC gap. Hence the earlier part of this chapter where we studied the transport at an energy E inside the SC gap (so that − < E < Δ) differs from this part where we discuss conductance at an energy E where |E| ≫ . For the latter part we assume the region of three SC wires to be a single region whose only role is to give rise to a scattering matrix for the NM wires; this scattering matrix has both normal and Andreev elements (namely, an electron can be reflected or transmitted as either an electron or a hole). We derive a renormalization group equation for the elements of the scattering matrix by assuming the interaction to be sufficiently weak. The fixed points of the renormalization group flow and their stabilities are studied; we find that the scattering matrix at the stable fixed point is highly symmetric even when the microscopic scattering matrix and the interaction strengths are not symmetric. Using the stability analysis we discuss the dependence of the conductances on the various length scales of the problem. Finally we propose an experimental realization of this system which can produce different signs of the p-wave pairings in the different SCs. • In the seventh chapter we show that the application of circularly polarized electro-magnetic radiation on the surface of a Weyl semimetal can generate states at that surface. The surface states can be characterized by their momenta due to translation invariance. The Floquet eigenvalues of these states come in complex conjugate pairs rather than being equal to ±1. If the amplitude of the radiation is small, we find some unusual bulk-boundary relations: the Floquet eigenvalues of the surface states lie at the extrema of the Floquet eigenvalues of the bulk system when the latter are plotted as a function of the momentum perpendicular to the surface, and the peaks of the Fourier transforms of the surface state wave functions lie at the momenta where the bulk Floquet eigenvalues have extrema. For the case of zero surface momentum, we can analytically derive interesting scaling relations between the decay lengths of the surface states and the amplitude and penetration depth of the radiation. For topological insulators, we again find that circularly polarized radiation can generate states on the surfaces; these states have much larger decay lengths (which can be tuned by the radiation amplitude) than the topological surface states which are present even in the absence of radiation. Finally, we show that radiation can generate surface states even for trivial insulators.

Topological phases of matter represent a new phase which cannot be understood in terms of Landau’s theory of symmetry breaking and are characterized by non-local topological properties emerging from purely local (microscopic) degrees of freedom. It is the non-trivial topology of the bulk band structure that gives rise to topological phases in condensed matter systems. Quantum Hall systems are prominent examples of such topological phases. Different quantum Hall states cannot be distinguished by a local order parameter. Instead, non-local measurements are required, such as the Hall conductance, to differentiate between various quantum Hall states. A signature of a topological phase is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations which respect appropriate symmetries. Examples of such properties are the presence of protected gapless edge states at the boundary of the system for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall states. The robustness of these properties can be under-stood through the existence of a topological invariant, such as the Chern number for quantum Hall states which is quantized to integer values and can only be changed by closing the bulk gap. Two other examples of topological phases of matter are topological superconductors and Weyl semimetals. The study of transport in various kinds of junctions of these topological materials is highly interesting for their applications in modern electronics and quantum computing. Another intriguing area to study is how to generate new kind of gapless edge modes in topological systems. In this thesis I have studied various aspects of topological phases of matter, such as electronic transport in junctions of topological insulators and topological superconductors, the generation of new kinds of boundary modes in the presence of granularity, and the effects of periodic driving in topological systems. We have studied the following topics. 1. transport across a line junction of two three-dimensional topological insulators, 2. transport across a junction of topological insulators and a superconductor, 3. surface and edge states of a topological insulator starting from a lattice model, 4. effects of granularity in topological insulators, 5. Majorana modes and conductance in systems with junctions of topological superconducting wires and normal metals, and 6. generation of new surface states in a Weyl semimetal in the presence of periodic driving by the application of electromagnetic radiation. A detailed description of each chapter is given below. • In the first chapter we introduce a number of concepts which are used in the rest of the thesis. We will discuss the ideas of topological phases of matter (for example, topological insulators, topological superconductors and Majorana modes, and Weyl semimetals), the renormalization group theory for weak interactions, and Floquet theory for periodically driven systems. • In the second chapter we study transport across a line junction which separates the surfaces of two three-dimensional topological insulators. The velocities of the Dirac electrons on the two surfaces may be unequal and may even have opposite signs. For a time-reversal invariant system, we show that the line junction is characterized by an arbitrary real parameter α; this determines the scattering amplitudes (reflection and transmission) from the junction. The physical origin of α is a potential barrier that may be present at the junction. If the surface velocities have the same sign, edge states exist that propagate along the line junction with a velocity and orientation of the spin which depend on α and the ratio of the velocities. Next, we study what happens if the two surfaces are at an angle φ with respect to each other. We study the scattering and differential conductance across the line junction as functions of φ and α. We also show that there are edge states which propagate along the line junction with a velocity and spin orientation which depend on φ. Finally, if the surface velocities have opposite signs, we find that the electrons must necessarily transmit into the two-dimensional interface separating the two topological insulators. • In the third chapter we discuss transport across a line junction lying between two orthogonal topological insulator surfaces and a superconductor which can have either s-wave (spin-singlet) or p-wave (spin-triplet) pairing symmetry. This junction is more complicated than the line junction discussed in the previous chapter because of the presence of the superconductor. In a topological insulator spin-up and spin-down electrons get coupled while in a superconductor electrons and holes get coupled. Hence we have to use a four-component spinor formalism to describe both spin and particle-hole degrees of freedom. The junction can have three time-reversal invariant barriers on the three sides. We compute the subgap charge conductance across such a junction and study their behaviors as a function of the bias voltage applied across the junction and the three parameters which characterize the barriers. We find that the presence of topological insulators and a superconductor leads to both Dirac and Schrodinger-like features in the charge conductances. We discuss the effects of bound states on the superconducting side on the conductance; in particular, we show that for triplet p-wave superconductors such a junction may be used to determine the spin state of its Cooper pairs. • In the fourth chapter we derive the surface Hamiltonians of a three-dimensional topological insulator starting from a microscopic model. (This description was not discussed in the previous chapters where we directly started from the surface Hamiltonians without deriving them form a bulk Hamiltonian). Here we begin from the bulk Hamiltonian of a three-dimensional topological insulator Bi2Se3. Using this we derive the surface Hamiltonians on various surfaces of the topological insulator, and we find the states which appear on the different surfaces and along the edge between pairs of surfaces. The surface Hamiltonians depend on the orientation of the surfaces and are therefore quite different from the previous chapters. We use both analytical methods based on the surface Hamiltonians (which are derived from the bulk Hamiltonian) and numerical methods based directly on a lattice discretization of the bulk Hamiltonian in order to find surface and edge states. We find that the application of a potential barrier along an edge can give rise to states localized at that edge. These states have an unusual energy-momentum dispersion which can be controlled by applying a potential along the edge; in particular, the velocity of these states can be tuned to zero. The scattering and conductance across the edge are studied as a function of the edge potential. We show that a magnetic field applied in a particular direction can also give rise to zero energy states on certain edges. We point out possible experimental ways of looking for the various edge states. • In the fifth chapter we study a system made of topological insulator (TI) nanocrystals which are coupled to each other. Our theoretical studies are motivated by the following experimental observations. Electrical transport measurements were carried out on thin films of nanocrystals of Bi2Se3 which is a TI. The measurements reveal that the entire system behaves like a single TI with two topological surface states at the two ends of the system. The two surface states are found to be coupled if the film thickness is small and decoupled above a certain film thickness. The surface state penetration depth is found to be unusually large and it decreases with increasing temperature. To explain all these experimental results we propose a theoretical model for this granular system. This consists of multiple grains of Bi2Se3 stacked next to each other in a regular array along the z-direction (the c-axis of Bi2Se3 nanocrystals). We assume translational invariance along the x and y directions. Each grain has top and bottom surfaces on which the electrons are described by Hamiltonians of the Dirac form which can be derived from the bulk Hamiltonian known for this material. We introduce intra-grain tunneling couplings t1 between the opposite surfaces of a single grain and inter-grain couplings t2 between nearby surfaces of two neighboring grains. We show that when t1 < t2 the entire system behaves like a single topological insulator whose outermost surfaces have gapless spectra described by Dirac Hamiltonians. We find a relation between t1, t2 and the surface state penetration depth λ which explains the properties of λ that are seen experimentally. We also present an expression for the surface state Berry phase as a function of the hybridization between the surface states and a Zeeman magnetic field that may be present in the system. At the end we theoretically studied the surface states on one of the side surfaces of the granular system and showed that many pairs of surface states can exist on the side surfaces depending on the length of the unit cell of the granular system. • In the sixth chapter we present our work on junctions of p-wave superconductors (SC) and normal metals (NM) in one dimension. We first study transport in a system where a SC wire is sandwiched between two NM wires. For such a system it is known that there is a Majorana mode at the junction between the SC and each NM lead. If the p-wave pairing changes sign at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the subgap conductance between the leads and the SC. We derive an analytical expression as a function of and the length L of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the energies oscillate and decay exponentially as L is increased. The energies exactly match the locations of the peaks in the conductance. We find that the subgap conductances do not change noticeably with the sign of . So there is no effect of the extra Majorana modes which appear inside the SC (due to changes in the signs of Δ) on the subgap conductance. Next we study junctions of three p-wave SC wires which are connected to the NM leads. Such a junction is of interest as it is the simplest system where braiding of Majorana modes is possible. Another motivation for studying this system is to see if the subgap transport is affected by changes in the signs of . For sufficiently long SCs, there are zero energy Majorana modes at the junctions between the SCs and the leads. In addition, depending on the signs of the Δ’s in the three SCs, there can also be one or three Majorana modes at the junction of the three SCs. We show that the various subgap conductances have peaks occurring at the energies of all these modes; we therefore get a rich pattern of conductance peaks. Next we study the effects of interactions between electrons (in the NM leads) on the transport. We use a renormalization group approach to study the effect of interactions on the conductance at energies far from the SC gap. Hence the earlier part of this chapter where we studied the transport at an energy E inside the SC gap (so that − < E < Δ) differs from this part where we discuss conductance at an energy E where |E| ≫ . For the latter part we assume the region of three SC wires to be a single region whose only role is to give rise to a scattering matrix for the NM wires; this scattering matrix has both normal and Andreev elements (namely, an electron can be reflected or transmitted as either an electron or a hole). We derive a renormalization group equation for the elements of the scattering matrix by assuming the interaction to be sufficiently weak. The fixed points of the renormalization group flow and their stabilities are studied; we find that the scattering matrix at the stable fixed point is highly symmetric even when the microscopic scattering matrix and the interaction strengths are not symmetric. Using the stability analysis we discuss the dependence of the conductances on the various length scales of the problem. Finally we propose an experimental realization of this system which can produce different signs of the p-wave pairings in the different SCs. • In the seventh chapter we show that the application of circularly polarized electro-magnetic radiation on the surface of a Weyl semimetal can generate states at that surface. The surface states can be characterized by their momenta due to translation invariance. The Floquet eigenvalues of these states come in complex conjugate pairs rather than being equal to ±1. If the amplitude of the radiation is small, we find some unusual bulk-boundary relations: the Floquet eigenvalues of the surface states lie at the extrema of the Floquet eigenvalues of the bulk system when the latter are plotted as a function of the momentum perpendicular to the surface, and the peaks of the Fourier transforms of the surface state wave functions lie at the momenta where the bulk Floquet eigenvalues have extrema. For the case of zero surface momentum, we can analytically derive interesting scaling relations between the decay lengths of the surface states and the amplitude and penetration depth of the radiation. For topological insulators, we again find that circularly polarized radiation can generate states on the surfaces; these states have much larger decay lengths (which can be tuned by the radiation amplitude) than the topological surface states which are present even in the absence of radiation. Finally, we show that radiation can generate surface states even for trivial insulators.