Colloquium: Topological band theory

Recent Advances in Topological Quantum Materials by Angle-Resolved Photoemission Spectroscopy

Condensed matter physics is a vibrant branch of physics, which addresses a very broad spectrum of issues related to electronic, magnetic, thermal, structural and optical proper- ties of condensed phases of matter. The interplay between structural and magnetic phases, interactions between different components of a material, spin-orbit coupling (SOC) and other effects, make condensed matter physics a rich and colorful field. In this thesis I will focus on topological materials, excitonic insulators and atomically thin films, which are being explored intensely both theoretically and experimentally. Specifically, topological materials, including topological (crystalline) insulators and topological Weyl semimetals are covered in Chapters 2 to 4. Chapter 5 is mainly concerned about the excitonic in- sulator (EI) phase in 'slow graphene'. The electronic structure of atomically thin MoS2 films will be discussed in Chapter 6. 3D Topological insulators (TIs), known as quantum spin Hall insulators in 2D, are in- sulating in the bulk while conducting on the surface in sharp contrast with conventional insulators. In Chapter 2, by using first-principles and tight-binding model calculations, we identify a 2-ML (monolayer) Bi(110) thin film as a candidate quantum-spin-Hall in- sulator. In the absence of spin-orbit coupling, 2-ML Bi(110) thin films have two types of Dirac cones in the Brillouin zone (BZ). The Dirac cones, carrying non-zero winding numbers, serve as the starting or ending points of the edge bands in the ribbon spectrum. After the inclusion of the SOC, all Dirac nodes are gapped out. Correspondingly, a Dirac cone formed by the gapless edge states was found at the boundary of the ribbon reflecting the topologically nontrivial nature of the system. In Chapter 3, we present our work on the topological Weyl semimetal phase in the transi- tion metal monopnictide TaAs. A topological Weyl semimetal is a new kind of topological phase, which shows exotic properties in the bulk and on the surface. In the bulk, the valence and conduction bands touch each other at discrete K points, termed as Weyl points or Weyl nodes. On the surface, the Weyl nodes can induce Fermi arcs which are non-closed surface states connecting Weyl nodes with opposite chiralities. TaAs breaks the inversion symmetry and therefore it can be a possible system to host the topologi- cal Weyl semimetal phase. Our first-principles and tight-binding model calculations find that there are 24 Weyl points in the Brillouin zone. The Fermi arcs on (001) surface are identified by surface-state calculations and observed in angle-resolved photoemission experiments (ARPES). Topological crystalline insulators (TCIs) are new kind of topological insulators, which are protected by the crystal symmetries instead of time-reversal symmetry. In Chapter 4, we consider a rotational symmetry protected topological crystalline phase in TaAs2 family of materials. The TaAs2 class crystalizes in a monoclinic structure with space group No. 12. By checking the parity eigenvalues of the occupied bands at time reversal invariant momenta (TRIM), the topological invariants (symmetry indicators) of TaAs2 are found to be (Z2Z2Z2; Z4) = (111; 2), suggesting that TaAs2 can host two Dirac cones on the (010) surface protected by C2 rotational symmetry. Our surface state calculation identified two clear Dirac cones on the (010) surface with the bulk band gap as large as 300 meV. An excitonic insulator instability can arise in narrow gap semiconductors and semimetals when the binding energy of an electron-hole pair exceeds the band gap. Although many experiments have shown some signs of an excitonic insulator state in various systems, conclusive experimental evidence still remains elusive. In Chapter 5 we discuss the ex- citonic instability in 'slow graphene'. We approach the EI transition from two different directions. First, we apply a commonly used mean-field approach that can give us the overall phase diagram of the EI transition. In another approach, we solve the Bethe- Salpeter equation (BSE) and follow the evolution of the lowest excitons. Since graphene is gapless, the presence of an excitonic state at negative energy signals an instability of the assumed ground state. By studying properties of the exciton, we can infer properties of the resulting EI phase, finding overall consistency between the two approaches. Finally, in Chapter 6, we consider the electronic structure of few-layer MoS2. Transition metal dichalcogenides are a family of layered materials, which exhibit metal to semicon- ductor transitions and many other interesting properties as a function of the number of layers. For example, bulk MoS2 is a semiconductor with indirect gap, while monolayer MoS2 has a direct gap at the K-point. I will investigate the evolution of the electronic structure and related properties of MoS2 films as the number of layers is increased within the first-principles density functional theory (DFT) framework. Wannier-function based tight-binding models will be used to gain insight into the first-principles results. I will summarize my thesis in Chapter 7.

1D topological systems for next-generation electronics

<sec> The discovery of topological materials – condensed matter systems that have nontrivial topological invariants – marked the commencement of a new era in condensed matter physics and materials science. Three dimensional topological insulators (3D TIs) are one of the first discovered and the most studied among all topological materials. The bulk material of the TIs have the characteristics of the insulator, having a complete energy gap. Their surface electronic states, on the other hand, have the characteristics of a conductor, with energy band passes continuously through the Fermi surface. The conductivity of this topological surface state (TSS) is protected by the time reversal symmetry of the bulk material. The TSS is highly spin-polarized and form a special spin-helical configuration that allows electrons with specific spin to migrate only in a specific direction on the surface. By this means, surface electrons in TIs can " bypass” the influence of local impurities, achieving a lossless transmission of spin-polarized current. The existence of TIs directly leads to a variety of novel transport, magnetic, electrical, and optical phenomena, such as non-local quantum transport, quantum spin Hall effect, etc., promising wide application prospects. Recently, several research groups have searched all 230 non-magnetic crystal space groups, exhausting all the found or undiscovered strong/weak TIs, topological crystalline insulators (TCI), and topological semimetals. This series of work marks that theoretical understanding of non-magnetic topological materials has gone through a period of one-by-one prediction and verification, and entered the stage of the large-area material screening and optimization.</sec><sec> Parallel to non-magnetic TIs, magnetic topological materials constructed by ferromagnetic or antiferromagnetic long range orders in topological systems have always been an important direction attracting theoretical and experimental efforts. In magnetic TIs, the lack of time reversal symmetry brings about new physical phenomena. For example, when a ferromagnetic order is introduced into a three-dimensional TI, the Dirac TSS that originally intersected at one point will open a magnetic gap. When the Fermi surface is placed just in the gap, the quantum anomalous Hall effect can be implemented. At present, the research on magnetic topology systems is still in the ascendant. It is foreseeable that these systems will be the main focus and breakthrough point of topology material research in the next few years. </sec><sec> Angle-resolved photoemission spectroscopy (ARPES) is one of the most successful experimental methods of solid state physics. Its unique <i>k</i>-space-resolved single-electron detection capability and simple and easy-to-read data format make it a popular choice for both theoretists and experimentalists. In the field of topological materials, ARPES has always been an important experimetnal technique. It is able to directly observe the bulk and surface band structure of crystalline materials, and in a very intuitive way. With ARPES, it is incontrovertible to conclude whether a material is topological, and which type of topological material it belongs to.</sec><sec> This paper reviews the progress of ARPES research on TIs since 2008, focusing on the experimental energy band characteristics of each series of TIs and the general method of using ARPES to study this series of materials. Due to space limitations, this paper only discusses the research progress of ARPES for strong 3D TIs (focusing on the Bi<sub>2</sub>Se<sub>3</sub> series) and magnetic TIs (focusing on the MnBi<sub>2</sub>Te<sub>4</sub> series). Researches involving TCIs, topological Kondo insulators, weak 3D TIs, topological superconductors and heterostructures based on topological insulators will not be discussed. This paper assumes that the reader has the basic knowledge of ARPES, so the basic principles and system components of ARPES are not discussed.</sec>

Angle-resolved photoemission of topological materials

Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern insulators in the $pm$ or $pmm$ groups, $C_{nv}$ topological insulators in the $p4m$, $p31m$ and $p6m$ groups, and topological nonsymmorphic crystalline insulators in the $pg$ and $pmg$ groups. Aside from these existing results, we also obtain the following new results: (1) there are two integer mirror Chern numbers ($\mathbb{Z}^2$) in the $pm$ group but only one ($\mathbb{Z}$) in the $cm$ or $p3m1$ group for both the spinless and spinful cases; (2) for the $pmm$ ($cmm$) groups, there is no topological classification in the spinless case but $\mathbb{Z}^4$ ($\mathbb{Z}^2$) classifications in the spinful case; (3) we show how topological crystalline insulator phase in the $pg$ group is related to that in the $pm$ group; (4) we identify topological classification of the $p4m$, $p31m$, and $p6m$ for the spinful case; (5) we find topological non-symmorphic crystalline insulators also existing in $pgg$ and $p4g$ groups, which exhibit new features compared to those in $pg$ and $pmg$ groups. We emphasize the importance of the irreducible representations for the states at some specific high-symmetry momenta in the classification of topological crystalline phases. Our theory can serve as a guide for the search of topological crystalline insulator phases in realistic materials.

Topological materials discovery has evolved at a rapid pace over the past 15 years following the identification of the first nonmagnetic topological insulators (TIs), topological crystalline insulators (TCIs) and 3D topological semimetals (TSMs). Most recently, through complete analyses of symmetry-allowed band structures — including the theory of topological quantum chemistry (TQC) — researchers have determined crystal-symmetry-enhanced Wilson-loop and complete symmetry-based indicators for nonmagnetic topological phases, leading to the discovery of higher-order TCIs and TSMs. The recent application of TQC and related methods to high-throughput materials discovery has revealed that over half of the known stoichiometric, solid-state, nonmagnetic materials are topological at the Fermi level, over 85 per cent of the known stoichiometric materials host energetically isolated topological bands, and just under two-thirds of the energetically isolated bands in known materials carry the stable topology of a TI or TCI. In this Review, we survey topological electronic materials discovery in nonmagnetic crystalline solids from the prediction of the first 2D and 3D TIs to the recently introduced methods that have facilitated large-scale searches for topological materials. We also discuss future venues for the identification and manipulation of solid-state topological phases, including charge-density-wave compounds, magnetic materials, and 2D few-layer devices. Solid-state materials have emerged as a platform for probing and manipulating topological phases of matter. This Review surveys topological materials discovery in nonmagnetic crystalline solids, focusing on the role of crystal symmetry and geometry in topological material predictions.

Topological phase transition in a single material usually refers to transitions between a trivial band insulator and a topological Dirac phase, and the transition may also occur between different classes of topological Dirac phases. It is a fundamental challenge to realize quantum transition between Z2 nontrivial topological insulator (TI) and topological crystalline insulator (TCI) in one material because Z2 TI and TCI have different requirements on the number of band inversions. The Z2 TIs must have an odd number of band inversions over all the time-reversal invariant momenta, whereas the newly discovered TCIs, as a distinct class of the topological Dirac materials protected by the underlying crystalline symmetry, owns an even number of band inversions. Taking PbSnTe2 alloy as an example, here we demonstrate that the atomic-ordering is an effective way to tune the symmetry of the alloy so that we can electrically switch between TCI phase and Z2 TI phase in a single material. Our results suggest that the atomic-ordering provides a new platform towards the realization of reversibly switching between different topological phases to explore novel applications.

Three-dimensional (3D) Dirac semimetals are 3D analogues of graphene, which display Dirac points with linear dispersion in k-space, stabilized by crystal symmetry. Cd3As2 has been predicted to be 3D Dirac semimetals and was subsequently demonstrated by angle-resolved photoemission spectroscopy. As unveiled by transport measurements, several exotic phases, such as Weyl semimetals, topological insulators, and topological superconductors, can be deduced by breaking time reversal or inversion symmetry. Here, we reported a facile and scalable chemical vapor deposition method to fabricate high-quality Dirac semimetal Cd3As2 microbelts; they have shown ultrahigh mobility up to 1.15 × 10(5) cm(2) V(-1) s(-1) and pronounced Shubnikov-de Haas oscillations. Such extraordinary features are attributed to the suppression of electron backscattering. This research opens a new avenue for the scalable fabrication of Cd3As2 materials toward exciting electronic applications of 3D Dirac semimetals.

We study surface states of topological crystalline insulators and superconductors protected by inversion symmetry. These fall into the category of "higher-order" topological insulators and superconductors which possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. We show that the surface states of higher-order topological insulators and superconductors can be thought of as globally irremovable topological defects and provide a complete classification of these inversion-protected phases in any spatial dimension for the ten symmetry classes by means of a layer construction. Furthermore, we discuss possible physical realizations of such states starting with a time-reversal invariant topological insulator (class AII) in three dimensions or a time-reversal invariant topological superconductor (class DIII) in two or three dimensions. The former can be used to build a three-dimensional second-order topological insulator which exhibits one-dimensional chiral or helical modes propagating along opposite edges, whereas the latter enables the construction of three-dimensional third-order or two-dimensional second-order topological superconductors hosting Majorana zero modes localized to two opposite corners. Being protected by inversion, such states are not pinned to a specific pair of edges or corners thus offering the possibility of controlling their location by applying inversion-symmetric perturbations such as magnetic field.

[EN]In three-dimensional topological Insulators, the topological protected helical surface conducting states exist along with the bulk insulating states. In a class of topological insulators, namely topological crystalline insulators, the protection role of surface states is taken by crystalline symmetry instead of time-reversal symmetry. These exotic characteristics bring about potential applications in logic devices, thermoelectricity or quantum computers. However, due to some critical challenges (like their compatibility with the existing devices, their fabrication processes as well as their compatibility with topological or quantum behavior under external stimuli, working temperature range, cost efficiency, practical structure and ease of use), topological materials are still lagging in device applications. In this regard, this thesis aims at investigating the fundamental properties of a topological crystalline insulator, 𝑃𝑏0.77𝑆𝑛0.23𝑆𝑒, based on Raman characterization at different temperature ranges and low magnetic fields as well as magneto-transport of its Hall bar devices at low temperatures, albeit with meeting challenges in fabrication and characterization. The Raman response of this material at relaxed conditions confirms the presence of topological surface states. Moreover, temperature-dependent Raman characterizations indicate that both surface states and their bulk counterparts contribute to the Raman response considering the interplay of electrons and phonons. Furthermore, our findings based on magnetic-field dependent Raman characterization demonstrate that the surface states are topologically protected by symmetry. These Raman results are also corroborated with magneto-transport characterizations, revealing the prominent role of an inherent attribute of this material based on strong spin-orbit coupling. Our results pave the way for electron studies in field-effect transistors based on topological phase transitions

In this review, we discuss recent progress in the explorations of topological materials beyond topological insulators; specifically, we focus on topological crystalline insulators and bulk topological superconductors. The basic concepts, model Hamiltonians, and novel electronic properties of these new topological materials are explained. The key role of the symmetries that underlie their topological properties is elucidated. Key issues in their materials realizations are also discussed.

Topological materials are of great interest because they can support metallic edge or surface states that are robust against perturbations, with the potential for technological applications. Here, we experimentally explore the light-induced non-equilibrium properties of two distinct topological phases in NaCd4As3: a topological crystalline insulator (TCI) phase and a topological insulator (TI) phase. This material has surface states that are protected by mirror symmetry in the TCI phase at room temperature, while it undergoes a structural phase transition to a TI phase below 200 K. After exciting the TI phase by an ultrafast laser pulse, we observe a leading band edge shift of >150 meV that slowly builds up and reaches a maximum after ∼0.6 ps and that persists for ∼8 ps. The slow rise time of the excited electron population and electron temperature suggests that the electronic and structural orders are strongly coupled in this TI phase. It also suggests that the directly excited electronic states and the probed electronic states are weakly coupled. Both couplings are likely due to a partial relaxation of the lattice distortion, which is known to be associated with the TI phase. In contrast, no distinct excited state is observed in the TCI phase immediately or after photoexcitation, which we attribute to the low density of states and phase space available near the Fermi level. Our results show how ultrafast laser excitation can reveal the distinct excited states and interactions in phase-rich topological materials.

We construct a theory describing phase transitions between the spinless topological crystalline insulator phase with glide symmetry and a normal insulator phase. We show that a spinless Weyl semimetal phase should intervene between these two phases. Here, because all the bands are free from degeneracy in general, a gap closing between a single conduction band and a single valence band at phase transition generally gives rise to a pair creation of Weyl nodes; hence the Weyl semimetal phase naturally appears. We show the relationship between the change of the ${\mathbb{Z}}_{2}$ topological number when the system goes through the Weyl semimetal phase, and the trajectory of the Weyl nodes.

A topological insulator protected by time-reversal symmetry is realized via spin-orbit interaction-driven band inversion. The topological phase in the Bi(1-x)Sb(x) system is due to an odd number of band inversions. A related spin-orbit system, the Pb(1-x)Sn(x)Te, has long been known to contain an even number of inversions based on band theory. Here we experimentally investigate the possibility of a mirror symmetry-protected topological crystalline insulator phase in the Pb(1-x)Sn(x)Te class of materials that has been theoretically predicted to exist in its end compound SnTe. Our experimental results show that at a finite Pb composition above the topological inversion phase transition, the surface exhibits even number of spin-polarized Dirac cone states revealing mirror-protected topological order distinct from that observed in Bi(1-x)Sb(x). Our observation of the spin-polarized Dirac surface states in the inverted Pb(1-x)Sn(x)Te and their absence in the non-inverted compounds related via a topological phase transition provide the experimental groundwork for opening the research on novel topological order in quantum devices.