A weak topological insulator state in quasi-one-dimensional bismuth iodide

Robust Weak Topological Insulator in the Bismuth Halide Bi_{4}Br_{2}I_{2}

We apply a topological material design concept for selecting a bulk topology of 3D crystals by different van der Waals stackings of 2D topological insulator layers, and find a bismuth halide Bi_{4}Br_{2}I_{2} to be an ideal weak topological insulator (WTI) with the largest band gap (∼300  meV) among all the WTI candidates, by means of angle-resolved photoemission spectroscopy (ARPES), density functional theory (DFT) calculations, and resistivity measurements. Furthermore, we reveal that the topological surface state of a WTI is not "weak" but rather robust against external perturbations against the initial theoretical prediction by performing potassium deposition experiments. Our results vastly expand future opportunities for fundamental research and device applications with a robust WTI.

Evolution of the flat band and the role of lattice relaxations in twisted bilayer graphene

Magic-angle twisted bilayer graphene exhibits correlated phenomena such as superconductivity and Mott insulating states related to the weakly dispersing flat band near the Fermi energy. Such a flat band is expected to be sensitive to both the moiré period and lattice relaxations. Thus, clarifying the evolution of the electronic structure with the twist angle is critical for understanding the physics of magic-angle twisted bilayer graphene. Here we combine nano-spot angle-resolved photoemission spectroscopy and atomic force microscopy to resolve the fine electronic structure of the flat band and remote bands, as well as their evolution with twist angle from 1.07° to 2.60°. Near the magic angle, the dispersion is characterized by a flat band near the Fermi energy with a strongly reduced band width. Moreover, we observe a spectral weight transfer between remote bands at higher binding energy, which allows to extract the modulated interlayer spacing near the magic angle. Our work provides direct spectroscopic information on flat band physics and highlights the important role of lattice relaxations.

Electronic Devices Based on Heterostructures of 2D Materials and Self-Assembled Monolayers

2D materials (2DMs), known for their atomically ultrathin structure, exhibit remarkable electrical and optical properties. Similarly, molecular self-assembled monolayers (SAMs) with comparable atomic thickness show an abundance of designable structures and properties. The strategy of constructing electronic devices through unique heterostructures formed by van der Waals assembly between 2DMs and molecular SAMs not only enables device miniaturization, but also allows for convenient adjustment of their structures and functions. In this review, the fundamental structures and fabrication methods of three different types of electronic devices dominated by 2DM-SAM heterojunctions with varying architectures are timely elaborated. Based on these heterojunctions, their fundamental functionalities and characteristics, as well as the regulation of their performance by external stimuli, are further discussed.

Lattice-guided assembly of optoelectronically active π-conjugated peptides on 1D van der Waals single crystals

The deployment of organic molecules in high-performance devices strongly relies on the formation of well-ordered domains, which is often complicated by the dynamic and sensitive nature of supramolecular interactions. Here, we engineered the assembly of water-processable, optoelectronic π-conjugated peptides into well-defined organic-inorganic heterointerfaced assemblies by leveraging the long-range anisotropic ordering of 1D van der Waals (vdW) crystals composed of subnanometer-thick transition metal sulfide chains (MS3; M = Nb, Ta) as assembly templates. We found that the monomers can readily form 1D supramolecular assemblies onto the underlying crystal surface, owing to the structural correspondence between the π-π interactions of the quaterthiophene (4T)-based peptide units (DDD-4T) and sulfur atom ordering along the NbS3 (100) surface. The heterointerfaced assemblies exhibited substantially red-shifted photoluminescence and enhanced visible-range photocurrent generation compared to solution-assembled films. Our results underscore the role of lattice matching in forming ordered supramolecular assemblies, offering an emergent approach to assembling organic building blocks endowed with improved physical properties.

Dislocation Majorana bound states in iron-based superconductors

We show that lattice dislocations of topological iron-based superconductors such as FeTe1-xSex will intrinsically trap non-Abelian Majorana quasiparticles, in the absence of any external magnetic field. Our theory is motivated by the recent experimental observations of normal-state weak topology and surface magnetism that coexist with superconductivity in FeTe1-xSex, the combination of which naturally achieves an emergent second-order topological superconductivity in a two-dimensional subsystem spanned by screw or edge dislocations. This exemplifies a new embedded higher-order topological phase in class D, where Majorana zero modes appear around the "corners" of a low-dimensional embedded subsystem, instead of those of the full crystal. A nested domain wall theory is developed to understand the origin of these defect Majorana zero modes. When the surface magnetism is absent, we further find that s± pairing symmetry itself is capable of inducing a different type of class-DIII embedded higher-order topology with defect-bound Majorana Kramers pairs. We also provide detailed discussions on the real-world material candidates for our proposals, including FeTe1-xSex, LiFeAs, β-PdBi2, and heterostructures of bismuth, etc. Our work establishes lattice defects as a new venue to achieve high-temperature topological quantum information processing.

Controlling crystal cleavage in focused ion beam shaped specimens for surface spectroscopy

Our understanding of quantum materials is commonly based on precise determinations of their electronic spectrum by spectroscopic means, most notably angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy. Both require atomically clean and flat crystal surfaces, which are traditionally prepared by in situ mechanical cleaving in ultrahigh vacuum chambers. We present a new approach that addresses three main issues of the current state-of-the-art methods: (1) Cleaving is a highly stochastic and, thus, inefficient process; (2) fracture processes are governed by the bonds in a bulk crystal, and many materials and surfaces simply do not cleave; and (3) the location of the cleave is random, preventing data collection at specified regions of interest. Our new workflow is based on focused ion beam machining of micro-strain lenses, in which shape (rather than crystalline) anisotropy dictates the plane of cleavage, which can be placed at a specific target layer. As proof-of-principle, we show ARPES results from micro-cleaves of Sr2RuO4 along the ac plane and from two surface orientations of SrTiO3, a notoriously difficult to cleave cubic perovskite.

Observation of Anomalous Planar Hall Effect Induced by One-Dimensional Weak Antilocalization

The confinement of electrons in one-dimensional (1D) space highlights the prominence of the role of electron interactions or correlations, leading to a variety of fascinating physical phenomena. The quasi-1D electron states can exhibit a unique spin texture under spin-orbit interaction (SOI) and thus could generate a robust spin current by forbidden electron backscattering. Direct detection of such 1D spin or SOI information, however, is challenging due to complicated techniques. Here, we identify an anomalous planar Hall effect (APHE) in the magnetotransport of quasi-1D van der Waals (vdW) topological materials as exemplified by Bi4Br4, which arises from the quantum interference correction of 1D weak antilocalization (WAL) to the ordinary planar Hall effect and demonstrates a deviation from the usual sine and cosine curves. The occurrence of 1D WAL is correlated to the line-shape Fermi surface and persistent spin texture of (100) topological surface states of Bi4Br4, as revealed by both our angle-resolved photoemission spectroscopy and first-principles calculations. By generalizing the observation of APHE to other non-vdW bulk materials, this work provides a possible characteristic of magnetotransport for identifying the spin/SOI information and quantum interference behavior of 1D states in 3D topological material.

Controlling the spin current around the rectangular cavities in two-dimensional topological insulators

Controlling spin current in topological insulators (TIs) is a crucial requirement for applications in quantum computing and spintronics. Using the non-equilibrium Keldysh Green's function formalism, we demonstrate that such control can be established around rectangular cavities of two-dimensional TIs by breaking their time reversal symmetry via exchange magnetic fields and magnetic defects. In the presence of magnetic defects with xy symmetry or Ising symmetry, the density of states is localized, and the spin current forms a current loop around the rectangular cavity in TIs interfacing with two ferromagnetic stripes. We also observe that the spin direction of the traveling electrons is inverted under the reversal of bias and gate voltages. The change in the spin-polarized current around the cavities is predicted by varying the strength of Rashba spin-orbit coupling. This result allows for the creation and control of nearly fully spin-polarized currents with various spatial patterns around the cavities in TIs, and the design of tunable spin diodes for highly integrated spintronics.

Laser-based angle-resolved photoemission spectroscopy with micrometer spatial resolution and detection of three-dimensional spin vector

We have developed a state-of-the-art apparatus for laser-based spin- and angle-resolved photoemission spectroscopy with micrometer spatial resolution (µ-SARPES). This equipment is realized by the combination of a high-resolution photoelectron spectrometer, a 6 eV laser with high photon flux that is focused down to a few micrometers, a high-precision sample stage control system, and a double very-low-energy-electron-diffraction spin detector. The setup achieves an energy resolution of 1.5 (5.5) meV without (with) the spin detection mode, compatible with a spatial resolution better than 10 µm. This enables us to probe both spatially-resolved electronic structures and vector information of spin polarization in three dimensions. The performance of µ-SARPES apparatus is demonstrated by presenting ARPES and SARPES results from topological insulators and Au photolithography patterns on a Si (001) substrate.

Topological electronic structure and spin texture of quasi-one-dimensional higher-order topological insulator Bi4Br4

The notion of topological insulators (TIs), characterized by an insulating bulk and conducting topological surface states, can be extended to higher-order topological insulators (HOTIs) hosting gapless modes localized at the boundaries of two or more dimensions lower than the insulating bulk. In this work, by performing high-resolution angle-resolved photoemission spectroscopy (ARPES) measurements with submicron spatial and spin resolution, we systematically investigate the electronic structure and spin texture of quasi-one-dimensional (1D) HOTI candidate Bi4Br4. In contrast to the bulk-state-dominant spectra on the (001) surface, we observe gapped surface states on the (100) surface, whose dispersion and spin-polarization agree well with our ab-initio calculations. Moreover, we reveal in-gap states connecting the surface valence and conduction bands, which is a signature of the hinge states inside the (100) surface gap. Our findings provide compelling evidence for the HOTI phase of Bi4Br4. The identification of the higher-order topological phase promises applications based on 1D spin-momentum locked current in electronic and spintronic devices.

Electronic structure in a transition metal dipnictide TaAs2

The family of transition-metal dipnictides has been of theoretical and experimental interest because this family hosts topological states and extremely large magnetoresistance (MR). Recently,TaAs2, a member of this family, has been predicted to support a topological crystalline insulating state. Here, by using high-resolution angle-resolved photoemission spectroscopy (ARPES), we reveal both closed and open pockets in the metallic Fermi surface (FS) and linearly dispersive bands on the (2‾01) surface, along with the presence of extreme MR observed from magneto-transport measurements. A comparison of the ARPES results with first-principles computations shows that the linearly dispersive bands on the measured surface ofTaAs2are trivial bulk bands. The absence of symmetry-protected surface state on the (2‾01) surface indicates its topologically dark nature. The presence of open FS features suggests that the open-orbit fermiology could contribute to the extremely large MR ofTaAs2.

Towards layer-selective quantum spin hall channels in weak topological insulator Bi4Br2I2

Weak topological insulators, constructed by stacking quantum spin Hall insulators with weak interlayer coupling, offer promising quantum electronic applications through topologically non-trivial edge channels. However, the currently available weak topological insulators are stacks of the same quantum spin Hall layer with translational symmetry in the out-of-plane direction-leading to the absence of the channel degree of freedom for edge states. Here, we study a candidate weak topological insulator, Bi4Br2I2, which is alternately stacked by three different quantum spin Hall insulators, each with tunable topologically non-trivial edge states. Our angle-resolved photoemission spectroscopy and first-principles calculations show that an energy gap opens at the crossing points of different Dirac cones correlated with different layers due to the interlayer interaction. This is essential to achieve the tunability of topological edge states as controlled by varying the chemical potential. Our work offers a perspective for the construction of tunable quantized conductance devices for future spintronic applications.

Topology Hierarchy of Transition Metal Dichalcogenides Built from Quantum Spin Hall Layers

The evolution of the physical properties of 2D material from monolayer limit to the bulk reveals unique consequences from dimension confinement and provides a distinct tuning knob for applications. Monolayer 1T'-phase transition metal dichalcogenides (1T'-TMDs) with ubiquitous quantum spin Hall (QSH) states are ideal 2D building blocks of various 3D topological phases. However, the stacking geometry has been previously limited to the bulk 1T'-WTe₂ type. Here, the novel 2M-TMDs consisting of translationally stacked 1T'-monolayers are introduced as promising material platforms with tunable inverted bandgaps and interlayer coupling. By performing advanced polarization-dependent angle-resolved photoemission spectroscopy as well as first-principles calculations on the electronic structure of 2M-TMDs, a topology hierarchy is revealed: 2M-WSe₂ , MoS_2, and MoSe₂ are weak topological insulators (WTIs), whereas 2M-WS₂ is a strong topological insulator (STI). Further demonstration of topological phase transitions by tunning interlayer distance indicates that band inversion amplitude and interlayer coupling jointly determine different topological states in 2M-TMDs. It is proposed that 2M-TMDs are parent compounds of various exotic phases including topological superconductors and promise great application potentials in quantum electronics due to their flexibility in patterning with 2D materials.

Optical bulk-boundary dichotomy in a quantum spin Hall insulator

The bulk-boundary correspondence is a critical concept in topological quantum materials. For instance, a quantum spin Hall insulator features a bulk insulating gap with gapless helical boundary states protected by the underlying Z₂ topology. However, the bulk-boundary dichotomy and distinction are rarely explored in optical experiments, which can provide unique information about topological charge carriers beyond transport and electronic spectroscopy techniques. Here, we utilize mid-infrared absorption micro-spectroscopy and pump-probe micro-spectroscopy to elucidate the bulk-boundary optical responses of Bi₄Br₄, a recently discovered room-temperature quantum spin Hall insulator. Benefiting from the low energy of infrared photons and the high spatial resolution, we unambiguously resolve a strong absorption from the boundary states while the bulk absorption is suppressed by its insulating gap. Moreover, the boundary absorption exhibits strong polarization anisotropy, consistent with the one-dimensional nature of the topological boundary states. Our infrared pump-probe microscopy further measures a substantially increased carrier lifetime for the boundary states, which reaches one nanosecond scale. The nanosecond lifetime is about one to two orders longer than that of most topological materials and can be attributed to the linear dispersion nature of the helical boundary states. Our findings demonstrate the optical bulk-boundary dichotomy in a topological material and provide a proof-of-principal methodology for studying topological optoelectronics.

Exploring strong and weak topological states on isostructural substitutions in TlBiSe[Formula: see text]

Topological Insulators (TIs) are unique materials where insulating bulk hosts linearly dispersing surface states protected by the Time-Reversal Symmetry. These states lead to dissipationless current flow, which makes this class of materials highly promising for spintronic applications. Here, we predict TIs by employing state-of-the-art first-principles based methodologies, viz., density functional theory and many-body perturbation theory (G[Formula: see text]W[Formula: see text]) combined with spin-orbit coupling effects. For this, we take a well-known 3D TI, TlBiSe[Formula: see text] and perform complete substitution with suitable materials at different sites to check if the obtained isostructural materials exhibit topological properties. Subsequently, we scan these materials based on SOC-induced parity inversion at Time-Reversal Invariant Momenta. Later, to confirm the topological nature of selected materials, we plot their surface states along with calculation of Z[Formula: see text] invariants. Our results show that GaBiSe[Formula: see text] is a strong Topological Insulator, besides, we report six weak Topological Insulators, viz., PbBiSe[Formula: see text], SnBiSe[Formula: see text], SbBiSe[Formula: see text], Bi[Formula: see text]Se[Formula: see text], TlSnSe[Formula: see text] and PbSbSe[Formula: see text]. We have further verified that all the reported TIs are dynamically stable, showing all real phonon modes of vibration.

Topology and Symmetry in Quantum Materials

Interest in topological materials continues to grow unabated in view of their conceptual novelties as well as their potential as platforms for transformational new technologies. Electronic states in a topological material are robust against perturbations and support unconventional electromagnetic responses. The first-principles band-theory paradigm has been a key player in the field by providing successful prediction of many new classes of topological materials. This perspective presents a cross section through the recent work on understanding the role of geometry and topology in generating topological states and their responses to external stimuli, and as a basis for connecting theory and experiment within the band theory framework. In this work, effective strategies for topological materials discovery and impactful directions for future topological materials research are also commented.

Coexistence of Strong and Weak Topological Orders in a Quasi-One-Dimensional Material

Topological materials have broad application prospects in quantum computing and spintronic devices. Among them, dual topological materials with low dimensionality provide an excellent platform for manipulating various topological states and generating highly conductive spin currents. However, direct observation of their topological surface states still lacks. Here, we reveal the coexistence of the strong and weak topological phases in a quasi-one-dimensional material, TaNiTe_{5}, by spin- and angle- resolved photoemission spectroscopy. The surface states protected by weak topological order forms Dirac-node arcs in the vicinity of the Fermi energy, providing the opportunity to develop spintronics devices with high carrier density that is tunable by bias voltage.

All topological bands of all nonmagnetic stoichiometric materials

Topological quantum chemistry and symmetry-based indicators have facilitated large-scale searches for materials with topological properties at the Fermi energy (E_F). We report the implementation of a publicly accessible catalog of stable and fragile topology in all of the bands both at and away from E_F in the 96,196 processable entries in the Inorganic Crystal Structure Database. Our calculations, which represent the completion of the symmetry-indicated band topology of known nonmagnetic materials, have enabled the discovery of repeat-topological and supertopological materials, including rhombohedral bismuth and Bi₂Mg₃. We find that 52.65% of all materials are topological at E_F, roughly two-thirds of bands across all materials exhibit symmetry-indicated stable topology, and 87.99% of all materials contain at least one stable or fragile topological band.

Experimental Realization of Two-Dimensional Weak Topological Insulators

We report the experimental realization of a two-dimensional (2D) weak topological insulator (WTI) in spinless Su-Schrieffer-Heeger circuits with parity-time and chiral symmetries. Strong and weak Z2 topological indexes are adopted to explain the experimental findings that a Dirac semimetal (DSM) phase and four WTI phases emerge in turn when we modulate the centrosymmetric circuit deformations. In the DSM phase, it is found that the Dirac cone is highly anisotropic and that it is not pinned to any high-symmetry points but can widely move within the Brillouin zone, which eventually leads to the phase transition between WTIs. In addition, we observe a pair of flat-band domain wall states by designing spatially inhomogeneous node connections. Our work provides the first experimental evidence for 2D WTIs, which significantly advances our understanding of the strong and weak nature of topological insulators, the robustness of flat bands, and the itinerant and anisotropic features of Dirac cones.

Large-Gap Quantum Spin Hall State and Temperature-Induced Lifshitz Transition in Bi4Br4

Searching for quantum spin Hall insulators with large fully opened energy gap to overcome the thermal disturbance at room temperature has attracted tremendous attention because of the robustness of one-dimensional (1D) spin-momentum locked topological edge states in the practical applications of electronic devices and spintronics. Here, we report the investigation of topological nature of monolayer Bi₄Br₄ by the techniques of angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy. The possible topological nontriviality of 1D edge state integrals within the large energy gap (∼0.2 eV) is revealed by the first-principle calculations. The ARPES measurements at different temperatures show a temperature-induced Lifshitz transition, corresponding to the resistivity anomaly evoked by the chemical potential shift. The connection between the emergency of superconductivity and the Lifshitz transition is discussed.

Gate-Tunable Transport in Quasi-One-Dimensional α-Bi4I4 Field Effect Transistors

Bi₄I₄ belongs to a novel family of quasi-one-dimensional (1D) topological insulators (TIs). While its β phase was demonstrated to be a prototypical weak TI, the α phase, long thought to be a trivial insulator, was recently predicted to be a rare higher order TI. Here, we report the first gate tunable transport together with evidence for unconventional band topology in exfoliated α-Bi₄I₄ field effect transistors. We observe a Dirac-like longitudinal resistance peak and a sign change in the Hall resistance; their temperature dependences suggest competing transport mechanisms: a hole-doped insulating bulk and one or more gate-tunable ambipolar boundary channels. Our combined transport, photoemission, and theoretical results indicate that the gate-tunable channels likely arise from novel gapped side surface states, two-dimensional (2D) TI in the bottommost layer, and/or helical hinge states of the upper layers. Markedly, a gate-tunable supercurrent is observed in an α-Bi₄I₄ Josephson junction, underscoring the potential of these boundary channels to mediate topological superconductivity.

Observation of Topological Edge States on α-Bi4Br4 Nanowires Grown on TiSe2 Substrates

A time-reversal invariant two-dimensional (2D) topological insulator (TI) is characterized by the gapless helical edge states propagating along the perimeter of the system. However, the small band gap in the 2D TIs discovered so far hinders their applications. Recently, we predicted that single-layer Bi₄Br₄ is a 2D TI with a remarkable band gap and that α-Bi₄Br₄ crystals can host topological edge states at the step edges. Here we report the growth of α-Bi₄Br₄ nanowires with (102)-oriented top surfaces on the TiSe₂ substrates and the direct observation of the predicted topological edge states at the step edges of the nanowires using scanning tunneling microscopy. The coupling between the edge states leads to the formation of surface states at the (102) top surfaces of the nanowires. Our work demonstrates the existence of topological edge states in α-Bi₄Br₄ and paves the way for developing α-Bi₄Br₄-based devices for a high-temperature quantum spin Hall effect.

Epitaxial Growth of Quasi-One-Dimensional Bismuth-Halide Chains with Atomically Sharp Topological Non-Trivial Edge States

Quantum spin Hall insulators (QSHIs) have one-dimensional (1D) spin-momentum locked topological edge states (ES) inside the bulk band gap, which can serve as dissipationless channels for the practical applications in low consumption electronics and high performance spintronics. However, obtaining the clean and atomically sharp ES which serves as ideal 1D spin-polarized nondissipative conducting channels is demanding and still a challenge. Here, we report the formation of the quasi-1D Bi₄I₄ nanoribbons on the surface of Bi(111) with the support of the graphene-terminated 6H-SiC(0001) and the direct observation of the topological ES at the step edges by the scanning tunneling microscopy (STM) and spectroscopic-imaging results. The ES reside surround the edge of Bi₄I₄ nanoribbons and exhibits noteworthy robustness against nontime reversal symmetry (non-TRS) perturbations. The theoretical simulations verify the topological nontriviality of 1D ES, which is retained after considering the presence of the underlying Bi(111). Our study supports the existence of topological ES in Bi₄I₄ nanoribbons, benefiting to engineer the topological features by using the 1D nanoribbons as building blocks.

Periodic nanostructures: preparation, properties and applications

Periodic nanostructures, a group of nanomaterials consisting of single or multiple nano units/components periodically arranged into ordered patterns (e.g., vertical and lateral superlattices), have attracted tremendous attention in recent years due to their extraordinary physical and chemical properties that offer a huge potential for a multitude of applications in energy conversion, electronic and optoelectronic applications. Recent advances in the preparation strategies of periodic nanostructures, including self-assembly, epitaxy, and exfoliation, have paved the way to rationally modulate their ferroelectricity, superconductivity, band gap and many other physical and chemical properties. For example, the recent discovery of superconductivity observed in "magic-angle" graphene superlattices has sparked intensive studies in new ways, creating superlattices in twisted 2D materials. Recent development in the various state-of-the-art preparations of periodic nanostructures has created many new ideas and findings, warranting a timely review. In this review, we discuss the current advances of periodic nanostructures, including their preparation strategies, property modulations and various applications.

Experimental Observation and Spin Texture of Dirac Node Arcs in Tetradymite Topological Metals

We report the observation of a nontrivial spin texture in Dirac node arcs, i.e., novel topological objects formed when Dirac cones of massless particles extend along an open one-dimensional line in momentum space. We find that such states are present in all the compounds of the tetradymite M_{2}Te_{2}X family (M=Ti, Zr, or Hf and X=P or As) regardless of the weak or strong character of the topological invariant. The Dirac node arcs in tetradymites are thus the simplest possible textbook example of a type-I Dirac system with a single spin-polarized node arc.

Evidence for a higher-order topological insulator in a three-dimensional material built from van der Waals stacking of bismuth-halide chains

Low-dimensional van der Waals materials have been extensively studied as a platform with which to generate quantum effects. Advancing this research, topological quantum materials with van der Waals structures are currently receiving a great deal of attention. Here, we use the concept of designing topological materials by the van der Waals stacking of quantum spin Hall insulators. Most interestingly, we find that a slight shift of inversion centre in the unit cell caused by a modification of stacking induces a transition from a trivial insulator to a higher-order topological insulator. Based on this, we present angle-resolved photoemission spectroscopy results showing that the real three-dimensional material Bi₄Br₄ is a higher-order topological insulator. Our demonstration that various topological states can be selected by stacking chains differently, combined with the advantages of van der Waals materials, offers a playground for engineering topologically non-trivial edge states towards future spintronics applications.

Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2

Time reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi₂ that belongs to the optimal space group P[Formula: see text], which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (E_F). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects.

Topological Quantum Materials from the Viewpoint of Chemistry

Topology, a mathematical concept, has recently become a popular and truly transdisciplinary topic encompassing condensed matter physics, solid state chemistry, and materials science. Since there is a direct connection between real space, namely atoms, valence electrons, bonds, and orbitals, and reciprocal space, namely bands and Fermi surfaces, via symmetry and topology, classifying topological materials within a single-particle picture is possible. Currently, most materials are classified as trivial insulators, semimetals, and metals or as topological insulators, Dirac and Weyl nodal-line semimetals, and topological metals. The key ingredients for topology are certain symmetries, the inert pair effect of the outer electrons leading to inversion of the conduction and valence bands, and spin-orbit coupling. This review presents the topological concepts related to solids from the viewpoint of a solid-state chemist, summarizes techniques for growing single crystals, and describes basic physical property measurement techniques to characterize topological materials beyond their structure and provide examples of such materials. Finally, a brief outlook on the impact of topology in other areas of chemistry is provided at the end of the article.

Observation and control of the weak topological insulator state in ZrTe5

A quantum spin Hall (QSH) insulator hosts topological states at the one-dimensional (1D) edge, along which backscattering by nonmagnetic impurities is strictly prohibited. Its 3D analogue, a weak topological insulator (WTI), possesses similar quasi-1D topological states confined at side surfaces. The enhanced confinement could provide a route for dissipationless current and better advantages for applications relative to strong topological insulators (STIs). However, the topological side surface is usually not cleavable and is thus hard to observe. Here, we visualize the topological states of the WTI candidate ZrTe5 by spin and angle-resolved photoemission spectroscopy (ARPES): a quasi-1D band with spin-momentum locking was revealed on the side surface. We further demonstrate that the bulk band gap is controlled by external strain, realizing a more stable WTI state or an ideal Dirac semimetal (DS) state. The highly directional spin-current and the tunable band gap in ZrTe5 will provide an excellent platform for applications.

Anisotropic Transport and Quantum Oscillations in the Quasi-One-Dimensional TaNiTe5: Evidence for the Nontrivial Band Topology

The past decade has witnessed the burgeoning discovery of a variety of topological states of matter with distinct nontrivial band topologies. Thus far, most materials that have been studied possess two-dimensional or three-dimensional electronic structures, with only a few exceptions that host quasi-one-dimensional (quasi-1D) topological electronic properties. Here we present clear-cut evidence for Dirac Fermions in the quasi-1D telluride TaNiTe₅. We show that its transport behaviors are highly anisotropic, and we observe nontrivial Berry phases via quantum oscillation measurements. The nontrivial band topology is further corroborated by first-principles calculations. Our results may help to guide the future quest for topological states in this new family of quasi-1D ternary chalcogenides.

Ideal Type-II Weyl Phase and Topological Transition in Phononic Crystals

Ideal Weyl points, which are related by symmetry and thus reside at the same frequency, could offer further insight into the Weyl physics. The ideal type-I Weyl points have been observed in photonic crystals, but the ideal type-II Weyl points with tilted conelike band dispersions are still not realized. Here we present the observation of the ideal type-II Weyl points of the minimal number in three-dimensional phononic crystals and, in the meantime, the topological phase transition from the Weyl semimetal to the valley insulators of two distinct types. The Fermi-arc surface states are shown to exist on the surfaces of the Weyl phase, and the Fermi-circle surface states are also observed, but on the interface of the two distinct valley phases. Intriguing wave partition of the Fermi-circle surface states is demonstrated.

Bulk and Surface Electronic Structure of the Dual-Topology Semimetal Pt_{2}HgSe_{3}

We report high-resolution angle-resolved photoemission measurements on single crystals of Pt_{2}HgSe_{3} grown by high-pressure synthesis. Our data reveal a gapped Dirac nodal line whose (001) projection separates the surface Brillouin zone in topological and trivial areas. In the nontrivial k-space range, we find surface states with multiple saddle points in the dispersion, resulting in two van Hove singularities in the surface density of states. Based on density-functional theory calculations, we identify these surface states as signatures of a topological crystalline state, which coexists with a weak topological phase.

Crystal Chemistry and Bonding Patterns of Bismuth-Based Topological Insulators

Bismuth is gaining importance as a key element of functional quantum materials. The effects of spin-orbit coupling (SOC) are at the heart of many exciting proposals for next-generation quantum technologies, including topological materials for efficient information transmission and energy-saving applications. The "heavy" element bismuth and its compounds are predestined for SOC-induced topological properties, but materials design is challenged by a complex link between them and the chemical composition and crystal structure. Nevertheless, a lot can be learned about a certain property by testing its limits with compositional and/or structure modifications. We survey a handful of topological bismuth-based materials that bear structural and chemical semblance to the early topological insulators, antimony-doped elemental bismuth, Bi₂Se₃ and Bi₂Te₃. Chemical bonding via p orbitals and modular structure underlie all considered bismuth chalcogenides, subhalides, and chalcogenide halides and allow us to correlate the evolution of chemical bonding and structure with variability of the topological properties, although materials design should not be regarded as a building blocks set. Over the past decade, material discoveries have unearthed a plethora of topological properties, and bismuth is very fertile as a progenitor of a rich palette of exotic quantum materials, ranging from strong and weak 3D and crystalline topological insulators over topological metals and semimetals to magnetic topological insulators, while preserving the general layered structure motif.

Enhancement of superconductivity in organic-inorganic hybrid topological materials

Inducing or enhancing superconductivity in topological materials is an important route toward topological superconductivity. Reducing the thickness of transition metal dichalcogenides (e.g. WTe₂ and MoTe₂) has provided an important pathway to engineer superconductivity in topological matters. However, such monolayer sample is difficult to obtain, unstable in air, and with extremely low T_c. Here we report an experimentally convenient approach to control the interlayer coupling to achieve tailored topological properties, enhanced superconductivity and good sample stability through organic-cation intercalation of the Weyl semimetals MoTe₂ and WTe₂. The as-formed organic-inorganic hybrid crystals are weak topological insulators with enhanced T_c of 7.0 K for intercalated MoTe₂ (0.25 K for pristine crystal) and 2.3 K for intercalated WTe₂ (2.8 times compared to monolayer WTe₂). Such organic-cation intercalation method can be readily applied to many other layered crystals, providing a new pathway for manipulating their electronic, topological and superconducting properties.

High-Pressure Modification of BiI3

Structural and optical properties as well as chemical bonding of BiI3 at elevated pressures are investigated by means of refinements of X-ray powder diffraction data, measurements of the optical absorption, and calculations of the band structure involving bonding analysis in real space. The data evidence the onset of a phase transition from trigonal (hR24) BiI3 into PuBr3-type (oS16) BiI3 around 4.6 GPa. This high-pressure modification remains stable up to 40 GPa, the highest pressure of this study. The phase exhibits semiconducting properties with constantly decreasing band gap between 5 and 18 GPa. Above this pressure, the absorbance edge broadens significantly. Extrapolation of the determined band gap values implies a semiconductor to metal transition at approximately 35 GPa. The value is in accordance with subtle structural anomalies and the results of band structure calculations. Topological analysis of the computed electron density and the electron-localizability indicator reveal fingerprints for weak covalent Bi-I contributions in addition to dominating ionic interactions for both modifications.

Band-structure engineering of the magnetically Cr-doped topological insulator Sb2Te3 under mechanical strain

Magnetic doping in topological insulator Sb₂Te₃ can produce very novel physical phenomena such as quantum anomalous Hall effect (QAHE). However, experimental observations of QAHE in the magnetic atoms doped Sb₂Te₃ have encountered significant challenges due to the complexity of the electronic structure and the relatively small band gap. Generally, mechanical strain can effectively modulate the band structure, thus we theoretically investigate the electronic structures of Cr-doped Sb₂Te₃ under mechanical strain using first-principles calculations within density functional theory. The band gap of Cr-doped Sb₂Te₃ is 0.031 eV. When the compressive strain η becomes as  -2%, the band gap will be further enlarged to 0.045 eV, which is 45% larger than that of the unstrained material. However, as the compressive strain η exceed  -2%, strong hybridization between Cr and Te atoms will cause the overlap of bands, which leads to the closure of band gap. In addition, when tensile strain is applied to Cr-doped Sb₂Te₃, the decrease in the spacing between quintuple layers can enhance the coupling between Te and Sb atoms, which can also result in the closing of the band gap. Finally, we used HSE06 to calculated the band gaps. The band gaps may be underestimated, but HSE06 and GGA have the same band structures evolution tendency under mechanical strain. Our calculated results provide a guideline for the modulation of band structure by mechanical strain, which pave the way for the observation of QAHE in Cr-doped Sb₂Te₃.

Pressure-induced phase transitions and superconductivity in a quasi-1-dimensional topological crystalline insulator α-Bi4Br4

Great progress has been achieved in the research field of topological states of matter during the past decade. Recently, a quasi-1-dimensional bismuth bromide, Bi₄Br₄, has been predicted to be a rotational symmetry-protected topological crystalline insulator; it would also exhibit more exotic topological properties under pressure. Here, we report a thorough study of phase transitions and superconductivity in a quasihydrostatically pressurized α-Bi₄Br₄ crystal by performing detailed measurements of electrical resistance, alternating current magnetic susceptibility, and in situ high-pressure single-crystal X-ray diffraction together with first principles calculations. We find a pressure-induced insulator-metal transition between ∼3.0 and 3.8 GPa where valence and conduction bands cross the Fermi level to form a set of small pockets of holes and electrons. With further increase of pressure, 2 superconductive transitions emerge. One shows a sharp resistance drop to 0 near 6.8 K at 3.8 GPa; the transition temperature gradually lowers with increasing pressure and completely vanishes above 12.0 GPa. Another transition sets in around 9.0 K at 5.5 GPa and persists up to the highest pressure of 45.0 GPa studied in this work. Intriguingly, we find that the first superconducting phase might coexist with a nontrivial rotational symmetry-protected topology in the pressure range of ∼3.8 to 4.3 GPa; the second one is associated with a structural phase transition from monoclinic C2/m to triclinic P-1 symmetry.