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Topological matter plays a central role in today's condensed matter research. Zirconium pentatelluride (ZrTe5) has attracted attention as a Dirac semimetal at the boundary of weak and strong topological insulators (TI). Few-layer ZrTe5 is anticipated to exhibit the quantum spin Hall effect due to topological states inside the band gap, but sample degradation inflicted by ambient conditions and processing has so far hampered the fabrication of high quality devices. The quantum Hall effect (QHE), serving as the litmus test for 2D systems to be considered of high quality, has not been observed so far. Only a 3D variant on bulk was reported. Here, we succeeded in preserving the intrinsic properties of thin films lifting the carrier mobility to ∼3500 cm2 V-1 s-1, sufficient to observe the integer QHE and a bulk band gap related zero-energy state. The magneto-transport results offer evidence for the gapless topological states within this gap.

The complex nature of filling factor ν = 0 of monolayer graphene is studied in magnetotransport experiments. As a function of perpendicular magnetic field a metal-insulator transition is observed, which is attributed to disorder-induced Landau level broadening in the canted antiferromagnetic phase. In the metallic regime a separation of the zeroth Landau level appears and signs of the quantum spin Hall effect are seen near ν = 0. In addition to local transport, nonlocal transport experiments show results being consistent with helical edge transport.

Janus Sb and Bi monolayers as a new class of 2D topological insulator materials, which could be fulfilled by asymmetrical functionalizations with methyl or hydroxyl, are demonstrated by first-principles spin–orbit coupling (SOC) electronic structure calculations to conflate nontrivial topology, Rashba splitting and valley-contrast circular dichroism. Cohesive energies and phonon frequency dispersion spectra indicate that all Janus Sb and Bi monolayers possess a structural stability in energetic statics but represent virtual acoustic phonon vibrations of the hydrogen atoms passivating on monolayer surfaces. Band structures of Janus Sb and Bi monolayers and their nanoribbons demonstrate they are nontrivial topological insulators. Rashba spin splitting at G point in Brillouin zone of Janus Bi monolayers arises from the strong SOC px and py orbitals of Bi bonding atoms together with the internal out-of-plane electric field caused by asymmetrical functionalization. Janus Sb and Bi monolayers render direct and indirect giant bandgaps, respectively, which are derived from the strong SOC px and py orbitals at band-valley Brillouin points K and K′ where valley-selective circular dichroism of spin valley Hall insulators is also exhibited.

Topological insulation is widely predicted in two-dimensional (2D) materials realized by epitaxial growth or van der Waals (vdW) exfoliation. Such 2D topological insulators (TI's) host many interesting physical properties such as the quantum spin Hall effect and superconductivity. Here, we extend the search of 2D TI's into the exfoliatable non-vdW 2D crystals. We find that three-dimensional Dirac semimetals A3Bi (A = Na, K, Rb) (Pc1) can be exfoliated into 2D materials with exfoliation energies of 0.479–0.990 J m−2. Our careful examination of the topological invariants of exfoliated A3Bi monolayers/multilayers by using two well-established approaches reveals that bilayer and tetralayer Na3Bi are 2D TI's. It is found that the band gap of 2D TI's can be significantly increased by external strain. We further find that the predicted 2D TI's possess interesting hidden Rashba-like spin textures. Our results suggest a new arena to search for two-dimensional topological insulators and spintronic materials.

Two-dimensional topological materials attracted intense interest in condensed matter physics due to their topologically protected edge states and potential applications in electronic devices. Here, based on first-principles calculations, we found that a two-dimensional CX3 (X = Sb, Bi) monolayer is a quantum spin Hall insulator with a large band gap. With the strong spin-orbit coupling effect, CX3 exhibits noticeable bulk band gaps up to 470 meV, sufficiently large for realizing the quantum spin Hall effect at room temperature. The topological characteristic is confirmed by the Z2 invariant since the system preserves time-reversal symmetry. Particularly, the CSb3 monolayer displays unique topologically entangled Rashba-splitting edge states, resembling nearly free-electron quadratic dispersion. Such topologically entangled Rashba-like edge states derive from the spin-orbit coupling effect and inversion symmetry breaking on the edges. Moreover, we demonstrate that the topological properties are perfectly preserved in the CX3 monolayer even with a h-BN substrate. The nontrivial quantum spin Hall state in the CX3 monolayer will provide possibilities for studying a novel phenomenon of edge states and potential applications in low-dissipation electronic devices.

A new model with analysis for the propagation of flexural waves in a phononic plate at nanoscale is developed. The Gurtin-Murdoch theory for surface elasticity is adopted to model the surface heterogeneity. The Mindlin (or first-order) plate theory is further generalized to establish the governing equations for flexural waves in a phononic plate with surface effect, for which the plane wave expansion method is applied to derive the dispersion relation. A numerical model is developed using the finite element method and very good consistency between theory and numerical solution is observed. It is found that the surface density and the surface residual stress play the main role that affects the band structures. The surface effect can be approximately regarded as the competition between frequency decrease due to surface density and frequency increase caused by surface residual stress, which effectively increases the low-frequency bands but decreases the high-frequency bands. The quantum spin Hall effect is observed in the phononic plate at nanoscale, and the surface effect is studied numerically. By applying the k.p perturbation method, a theoretical framework is established to calculate the spin Chern number, which is an important topological invariant that determines the quantum spin Hall effect. Based on the topological analysis, an efficient waveguide with a zig-zag path is designed, in which a topologically protected wave in the interface state can robustly propagate along the path against disorders. The theory and numerical study developed in this paper will help better understand the size-dependent quantum spin Hall effect in nanostructures and it may also provide guidance for the design of topological wave devices at nanoscale.

As one of the most intriguing elemental 2D materials beyond graphene, stanene is a unique material possessing strong quantum spin Hall effect and is promising for spintronics applications. Since most of these exotic phenomena are associated with the edge states of stanene and their responses under external stimuli, here, we first investigate the electronic and transport behavior of the edge states of stanene. Through examining the acoustic phonon-limited scattering of transporting carriers, we reveal a staggering behavior in the effective mass and mobility varying with the width of stanene nanoribbons. Remarkably, an opposite oscillating trend of the quantum confinement effect with respect to the electrons and holes is found and this trend is in sharp contrast to graphene. Moreover, through group-theory analysis, we further analyze the symmetry-permitted light absorptions and predict a much smaller band gap at Γ compared with other IV-group 2D materials like graphene and silicene, allowing for a red-shift of optical π-π* absorption in stanene. The presence of the narrow flat bands along the M-K path in stanene suggests appreciable density of states of low-energy carriers and a strong light-matter interaction for low-energy photons, which are beneficial for its applications in low-frequency optoelectronics.

Two-dimensional (2D) transition metal dichalcogenides (TMDCs) have been emerged as a new class of van der Waals materials since the successful isolation of graphene. The strong spin-orbit coupling (SOC) and two-dimensionality give rise to plenty of novel physics including metal-insulator transition, charge density wave (CDW), valleytronics, quantum spin Hall effect, and unconventional superconductivity, which make TMDCs an ideal platform to study the fundamental physics and potential applications. In this review, we firstly introduce the crystal structure of 2D TMDCs materials. Then, we summarize the recent progress in the synthesis, novel physical properties, and applications of 2D TMDCs materials. Finally, a summary and an outlook on the topological superconductivity in this field are presented. 2D TMDCs have a chemical formula of MX2 (M=W, Mo and X=Te, Se, S) with a layered crystal structure. Depending on the coordination environments of M, TMDCs can crystallize in a variety of polytypic structures such as 2 H , 1 T , 1 T ′, and T d phases. In the monolayer 2 H -TMDCs, the breaking of an in-plane mirror symmetry and the presence of the out-of-plane mirror symmetry lead to an Ising spin–orbit coupling (SOC), which serves as an effective out-of-plane field acting on the copper pair and pins the electron spins to out-of-plane directions. This is called Ising superconductivity, which has been observed in gated MoS2, monolayer NbSe2 and TaS2. However, due to the inversion symmetry in monolayer 1 T ′-TMDCs, the introduction of SOC makes them a class of large-gap quantum spin Hall insulators such as monolayer 1 T ′-WTe2. Therefore, if we could consecutively tailor the TMDCs’ structure from 2 H to T d phase, the long-sought topological superconductivity may be realized in one substance by incorporating superconductivity and quantum spin Hall effect together. To explore the extraordinary physics and nanodevice applications, we develop a universal molten-salt-assisted chemical vapor deposition (CVD) method to prepare atomically thin TMDCs, including high-quality 2D superconductors such as monolayer MoTe2 and NbSe2. With the powerful sample growth technique , we demonstrate for the first time that a consecutive structural phase transition from T d to 1 T ′ to 2 H polytype can be realized by increasing the Se concentration in Se-substituted MoTe2. More importantly, the Se-substitution is found to dramatically enhance the superconductivity of the MoTe2 thin film, which is interpreted as the introduction of two-band superconductivity. Furthermore, in bilayer 1 T d-MoTe2, we find that the in-plane upper critical field goes beyond the Pauli paramagnetic limit and shows an emergent two-fold symmetry, which is different from the isotropic in-plane upper critical field in 2 H -TMDCs. We show that this is a result of a new type of asymmetric SOC in 1 T d-TMDCs, which has expanded the well-known Ising SOC. The polytypic structures and strong SOC in 2D TMDCs have led to a variety of novel physics and applications. Recent theoretical works have already shown that 2D TMDCs can be a platform to search for topological superconductivity. With the further development of sample preparation, 2D TMDCs will play an important role in realization of topological quantum computation.

The atomic buckling in 2D “Xenes” (such as silicene) fosters a plethora of exotic electronic properties such as a quantum spin Hall effect and could be engineered by external strain. Quantifying the buckling magnitude with subangstrom precision is, however, challenging, since epitaxially grown 2D layers exhibit complex restructurings coexisting on the surface. Here, we characterize using low-temperature (5 K) atomic force microscopy (AFM) with CO-terminated tips assisted by density functional theory (DFT) the structure and local symmetry of each prototypical silicene phase on Ag(111) as well as extended defects. Using force spectroscopy, we directly quantify the atomic buckling of these phases within 0.1-Å precision, obtaining corrugations in the 0.8- to 1.1-Å range. The derived band structures further confirm the absence of Dirac cones in any of the silicene phases due to the strong Ag-Si hybridization. Our method paves the way for future atomic-scale analysis of the interplay between structural and electronic properties in other emerging 2D Xenes.

Topological corner state has attracted much research interests since it does not obey the conventional bulk-edge correspondence and enables tightly confined light within small volumes. In this work, we demonstrate an enhanced second harmonic generation (SHG) from a topological corner state and its directional emission. To this end, we design an all-dielectric topological photonic crystal based on optical quantum spin Hall effect. In this framework, pseudospin states of photons, topological phase, and topological corner state are subsequently constructed by engineering the structures. It is shown that a high Q-factor of 3.66×1011 can be obtained at the corner state, showing strong confinement of light at the corner. Consequently, SHG is significantly boosted and manifests directional out-of-plane emission. More importantly, the enhanced SHG has robustness against a broad class of defects. These demonstrated properties offer practical advantages for integrated optical circuits.

Originating from the study of topological photonic crystals (TPCs), analogues of the quantum spin Hall effect have been used as a potential way to control the propagation of electromagnetic waves. Due to the topological robustness of the spin TPCs, the edge states along the interface between the trivial and topological areas are topologically protected and not reflected from structural defects and disorders. Here, on the basis of the time-spatial reversal symmetry and topological defect theory, we demonstrate broadening of the edge state bandwidth in spin TPCs made of regular metallic cross structures by simultaneously deforming the hexagonal honeycomb lattice and adjusting the rotation angle. Due to the simultaneous tuning of the two parameters, the designed spin TPCs possess more flexibility. Topologically protected one-way propagating edge states are observed in the terahertz regime, where electromagnetic waves propagate along sharp corners without backscattering. Our findings offer the potential application for topological devices in terahertz technology and are beneficial for the development of 6G mobile communications.

Higher-order topological insulators, providing a horizon beyond the conventional bulk-interface correspondence, have zero-dimensional topological corner states at different corners, such as a right-angled corner. Here, we numerically demonstrate a hierarchy of band topology in a two-dimensional square-lattice sonic crystal. In our system, the acoustic bulk bands mimic the quantum spin Hall effect, while gapped interface states emerge in the bulk gap due to symmetry reduction and carry quantized Zak phases, which result in zero-dimensional topological corner states in the interface gap. By direct acoustic measurement, we observe that zero-dimensional topological corner states are confined at not only the right-angled corner, but also the straight-angled corner. Our results present this type of topological shape-dependent corner state, which possibly possesses potential application in controlling and localizing acoustic waves.

Two-dimensional (2D) honeycomb lattices beyond graphene, such as germanene, promise new physical properties such as quantum spin Hall effect. While there have been many claims of growth of germanene, the lack of precise structural characterization of the epitaxial layers synthesized hinders further research. The striped layer formed by Ge deposition on Ag(111) has been recently ascribed as a stretched germanene layer. Using surface X-ray diffraction and density functional theory calculations, we demonstrate that it corresponds in fact to a Ag2Ge surface alloy with an atomic density 6.45% higher than the Ag(111) atomic density. The overall structure is formed by stripes associated with a face-centered cubic top-layer alignment, alternating with stripes associated with an hexagonal-close-packed top-layer alignment, in great analogy with the (22×√3) Au(111) reconstruction.

The quantum Hall effect, a quantum phenomenon that appears in macroscopic length scale, is one of the most important topics in condensed matter physics, is. It has long been expected that the quantum Hall effect may occur without Landau levels so that no external magnetic field nor high sample mobility is required for its studies and applications. Such a quantum Hall effect free of Landau levels can be realized in a topological insulator with its time-reversal symmetry broken by ferromagnetism as the quantized version of the anomalous Hall effect, i.e. the quantum anomalous Hall effect (see Figure 1 for the schematics of the quantum Hall effect and quantum anomalous Hall effect). Combing molecular beam epitaxy, scanning tunneling microscopy and angle-resolved photoemission spectroscopy, we prepared high quality thin films of magnetically doped (Bi, Sb)2Te3 topological insulator with well-controlled composition, thickness and chemical potential and systematically studied their transport properties. In 5 quintuple layer thick Cr-doped (Bi, Sb)2Te3 films we experimentally observed the quantization of the Hall resistance at h/e2 at zero field, accompanied by a considerable reduction in the dissipation of electron transport. Under a strong magnetic field, the longitudinal resistance vanishes, whereas the Hall resistance remains at the quantized value (see Figure 2). The observations unambiguously demonstrate the occurrence of the quantum anomalous Hall effect. The temperature, thickness and magnetic-doping-concentration dependences of the quantum anomalous Hall effect were systematically studied, which clarifies the roles of the band structure, electron localization and magnetic order in the effect and provides clues for obtaining the effect at a higher temperature. The experimental progresses in the quantum anomalous Hall effect pave the ways for applications of dissipationless quantum Hall edge states in low-energy-consuming devices and for realizations of other novel quantum phenomena such as chiral topological superconductivity and axion electrodynamics.

In a closed system, it is well known that the time-reversal symmetry can lead to Kramers degeneracy and protect nontrivial topological states such as quantum spin Hall insulator. In this letter we address the issue whether these effects are stable against coupling to environment, provided that both environment and the coupling to environment also respect the time-reversal symmetry. By employing a non-Hermitian Hamiltonian with the Langevin noise term and ultilizing the non-Hermitian linear response theory, we show that the spectral functions for Kramers degenerate states can be split by dissipation, and the backscattering between counter-propagating edge states can be induced by dissipation. The latter leads to the absence of accurate quantization of conductance in the case of quantum spin Hall effect. As an example, we demonstrate this concretely with the Kane-Mele model. Our study could also be extended to interacting topological phases protected by the time-reversal symmetry., 5+7 pages, 1+2 figures

We have constructed photonic crystals (PCs) of a standard honeycomb lattice composed of solid dielectric columns, hollow dielectric columns, and composite dielectric columns and studied the influence of their dielectric constant and filling ratio on the topological boundary states. The optical quantum spin Hall effect is realized by compressing and stretching the standard honeycomb lattice. In the waveguide structure with sharp bends, cavity, and disorder, the electromagnetic (EM) wave is transmitted unidirectionally and robustly, which verifies the transmission characteristics of the topological edge states. In addition, by replacing several solid dielectric columns near the junction of PCs with different topological states with composite dielectric columns, disturbance is introduced to the edge states, and the influence of the relative effective permittivity of the composite dielectric columns on the edge states is studied. It is found that the EM wave in the gap between the upper boundary state and the lower boundary state cannot be transmitted, and with the increase of the relative effective permittivity of the composite dielectric columns, the gap becomes larger and larger, and the position of the gap gradually moves down. Based on this, a photonic crystal (PC) waveguide structure that has higher transmission efficiency, robust unidirectionality, and higher optical localization performance, is designed. It has more potential development in filters, optical switches, optical communication systems, and semiconductor technology.

Although Pb harbors a strong spin-orbit coupling effect, pristine plumbene (the last group-IV cousin of graphene) hosts topologically trivial states. Based on first-principles calculations, we demonstrate that epitaxial growth of plumbene on the BaTe(111) surface converts the trivial Pb lattice into a quantum spin Hall (QSH) phase with a large gap of ∼0.3 eV via a selective substrate-orbital-filtering effect. Tight-binding model analyses show the pz orbital in half of the Pb overlayer is selectively removed by the BaTe substrate, leaving behind a pz-px,y band inversion. Based on the same working principle, the gap can be further increased to ∼0.5-0.6 eV by surface adsorption of H or halogen atoms that filters out the other half of the Pb pz orbitals. The mechanism of selective substrate-orbital-filtering is general, opening an avenue to explore large-gap QSH insulators in heavy-metal-based materials. It is worth noting that plumbene has already been widely grown on various substrates experimentally.

Topological insulator lasers (TILs) are a recently introduced family of lasing arrays in which phase locking is achieved through synthetic gauge fields. These single frequency light source arrays operate in the spatially extended edge modes of topologically non-trivial optical lattices. Because of the inherent robustness of topological modes against perturbations and defects, such topological insulator lasers tend to demonstrate higher slope efficiencies as compared to their topologically trivial counterparts. So far, magnetic and non-magnetic optically pumped topological laser arrays as well as electrically pumped TILs that are operating at cryogenic temperatures have been demonstrated. Here we present the first room temperature and electrically pumped topological insulator laser. This laser array, using a structure that mimics the quantum spin Hall effect for photons, generates light at telecom wavelengths and exhibits single frequency emission. Our work is expected to lead to further developments in laser science and technology, while opening up new possibilities in topological photonics., Topological insulator lasers offer robustness and efficiency due to their unique properties but usually require cryogenic temperatures or optical pumping. Here the authors demonstrate an electrically pumped topological insulator laser operating at room temperature.

Over the past few years, the rapid development in the fields of condensed matter physics, electronic, and photonic systems have inspired the design and experimental demonstration of various acoustic topological insulators (TIs). Among these, the topologically protected one-way propagation is a phenomenon that is gaining increased attention. Pseudospin states, which is the analogue of the quantum spin Hall effect from electronic systems, has been proven to enable topological edge states in acoustics. Similarly, the valley Hall (VH) effect is also observed in acoustic systems and provides a pair of valley vortex states with opposite chirality. These valley vortex states can similarly form topologically protected edge states and, in turn, realize robust one-way propagation. However, the differences in the physics behind these acoustic systems give rise to distinct features such as different angle selections and immunization levels to various types of defects. This article conducts a comparison study between topological states in valley Hall phononic crystals and TIs that reveals the differences and similarities in several aspects. Both of them have topologically protected edge states and thus the robust one-way propagation. For the maximum transmission incident angle and defect immunization, however, VH topological waveguides and TI waveguides show different characteristics.

The completion of quantum Hall trio, namely the quantum Hall effect (QHE), the quantum spin Hall effect (QSHE), and the quantum anomalous Hall effect (QAHE), has greatly broadened our understandings about the electron movement in solids. Likewise, the discovered scale-invariant dissipationless conduction in these three quantum transport states has inspired us to the development of practical low-power-consumption electronics/spintronics applications. In this review article, we outline the fundamental physics and relations between different Hall effects. In general, QSHE is a result of band inversion due to the large spin-orbit coupling; since the time-reversal symmetry (TRS) is well-protected in such QSH insulators, helical edge states with both spin-up and spin-down electrons are allowed and the conduction is immune to non-magnetic impurities. On the other hand, when TRS is broken by either perpendicular magnetic field or magnetic order, helical edge states are reduced to the single chiral channel, which in turn gives rise to QHE and QAHE, and backscattering from any impurity is strictly forbidden owing to the nature of chirality. Moreover, the interplay between this quantum Hall trio in certain conditions unveils new ways to explore emerging chiral materials and related quantum phenomena.

The topological states of matters arising from the nontrivial magnetic configuration provide a better understanding of physical properties and functionalities of solid materials. Such studies benefit from the active control of spin orientation in any solid, which is yet known to rarely take place in the two-dimensional (2D) limit. Here we demonstrate by the first-principles calculations that spin-orientation dependent topological states can appear in the geometrically frustrated monolayer antiferromagnet. Different topological states including quantum anomalous Hall (QAH) effect and time-reversal-symmetry (TRS) broken quantum spin Hall (QSH) effect can be obtained by changing spin orientation in the NiTl2S4 monolayer. Remarkably, the dilated nc-AFM NiTl2S4 monolayer gives birth to the QAH effect with hitherto reported largest number of quantized conducting channels (Chern number C = -4) in 2D materials. Interestingly, under tunable chemical potential, the nc-AFM NiTl2S4 monolayer hosts a novel state supporting the coexistence of QAH and TRS broken QSH effects with a Chern number C = 3 and spin Chern number C_s = 1. This work manifests a promising concept and material realization toward topological spintronics in 2D antiferromagnets by manipulating its spin degree of freedom.

General nonequilibrium quantum transport equations are derived for a coupled system of charge carriers, Dirac spin, isospin (or valley spin), and pseudospin, such as either one of the band, layer, impurity, and boundary pseudospins. Limiting cases are obtained for one, two or three different kinds of spin occurring in a system. We show that a characteristic integer number Ns determines the formal form of spin quantum transport equations, irrespective of the type of spins or pseudospins, as well as the maximal entanglement entropy. The results may shed a new perspective on the mechanism leading to zero modes and chiral/helical edge states in topological insulators, integer quantum Hall effect topological insulator (QHE-TI), quantum spin Hall effect topological insulator (QSHE-TI) and Kondo topological insulator (Kondo-TI). It also shed new light in the observed competing weak localization and antilocalization in spin-dependent quantum transport measurements. In particular, a novel mechanism of localization and delocalization, as well as the new mechanism leading to the onset of superconductivity in bilayer systems seems to emerge naturally from torque entanglements in nonequilibrium quantum transport equations of spin and pseudospins. Moreover, the general results may serve as a foundation for engineering approximations of the quantum transport simulations of spintronic devices based on graphene and other 2-D materials such as the transition metal dichalcogenides (TMDs), silicene, as well as based on topological materials exhibiting quantum spin Hall effects. The extension of the formalism to spincaloritronics and pseudo-spincaloritronics is straightforward.

The Dirac oscillator was initially introduced as a Dirac operator which is linear in momentum and coordinate variables. In contrast to the usual 2D Dirac oscillator, the 2D Kramers–Dirac oscillator admits the time-reversal symmetry, which is a reason for the present nomenclature. It is shown that there exists a family of eigenstates associated with an eigenvalue linear in the control parameter, and the eigenvalue in question goes down from positive values to negative values as the parameter varies in the positive direction. The other eigenvalues are broken up into two bands, positive and negative. The 2D Dirac and the 2D Kramers–Dirac oscillators are compared in their physical grounds and in their spectral structure from the viewpoint of the time-reversal symmetry.

We propose a computational methodology to perform inverse design of quantum spin hall effect (QSHE)-based phononic topological insulators. We first obtain two-fold degeneracy, or a Dirac cone, in the band structure using a level set-based topology optimization approach. Subsequently, four-fold degeneracy, or a double Dirac cone, is obtained by using zone folding, after which breaking of translational symmetry, which mimics the effect of strong spin-orbit coupling and which breaks the four-fold degeneracy resulting in a bandgap, is applied. We use the approach to perform inverse design of hexagonal unit cells of C6 and C3 symmetry. The numerical examples show that a topological domain wall with two variations of the designed metamaterials exhibit topologically protected interfacial wave propagation, and also demonstrate that larger topologically-protected bandgaps may be obtained with unit cells based on C3 symmetry.

Stanene, the tin analogue of graphene, has been predicted to be a two-dimensional topological insulator, providing an ideal platform for the realization of the quantum spin Hall effect even at room temperature. Here, continuous stanene has been successfully formed on the Au(111) substrate, and its crystalline structure, phonon properties, and electronic structures are investigated by scanning tunneling microscopy and in situ Raman spectroscopy combined with first-principles calculations. The surface Sn-Au alloy with a coverage-dependent structural evolution is first identified. At coverage above a critical value, the Au-Sn alloy is gradually converted into epitaxial stanene with a √3 × √7 superstructure. Distinctive vibrational phonon modes are discovered in √3 × √7 stanene through in situ Raman spectroscopy, which are correlated with the tensile strain evoked by its singular buckled structure. Our results present clear evidence for the existence of epitaxial stanene and provide a platform for exploration of the exotic properties of this strained two-dimensional material.

Topological insulators with unique edge states have revolutionized the understanding of solid-state materials. Recently, higher-order topological insulators (HOTIs), which host both gapped edge states and in-gap corner/hinge states, protected concurrently by band topology, were predicted and observed in experiments, unveiling a new horizon beyond the conventional bulk-edge correspondence. However, the control and manifestation of band topology in a hierarchy of dimensions, which is at the heart of HOTIs, have not yet been witnessed. Here, we propose theoretically and observe experimentally that tunable two-dimensional sonic crystals can be versatile systems to visualize and harness higher-order topology. In our systems, the two-dimensional acoustic bands mimic the quantum spin Hall effect, while the resultant one-dimensional helical edge states are gapped due to broken space-symmetry and carry quantized Zak phases, which then lead to zero-dimensional topological corner states. We demonstrate that topological transitions in the bulk and edges can be triggered independently by tuning the geometry of the sonic crystals. With complementary experiments and theories, our study reveals rich physics in HOTIs, opening a new route towards tunable topological metamaterials where novel applications, such as the topological transfer of acoustic energy among two-, one- and zero-dimensional modes, can be achieved. By tuning the geometry of a two-dimensional sonic crystal, its one-dimensional helical edge states become gapped and zero-dimensional topological corner states emerge. The band topology is thus manifested in a hierarchy of dimensions.

Ferroelasticity and band topology are two intriguing yet distinct quantum states of condensed matter materials. Their coexistence in a single two-dimensional (2D) lattice, however, has never been observed. Here, we found that the 2D tetragonal HfC monolayer allowed simultaneous presence of ferroelastic and topological orders. By using first-principles calculations, we found that it could allow a low switching barrier with reversible strain of 17.4%, indicating that the anisotropic properties are achievable experimentally for a 2D tetragonal lattice. More interestingly, the tuning of topological behaviors with strain led to spin-separated and gapless edge states, that is, the quantum spin Hall effect. These findings from the coupling of two quantum orders offer insights into ferroelastic control over topological edge states for achieving multifunctional properties in next-generation 2D nanodevices.

The quantum spin Hall effect, the valley Hall effect and ferroelectricity are all extensively studied yet distinct properties of insulators and it would be very interesting to see if they could coexist in certain two-dimensional (2D) materials. We present a family of fluorinated methyl-functionalized bismuthene (Bi2C2H6−xFx) films to achieve the 2D ferroelectric valley topological insulator (FEVTI) state, due to strong spin–orbit coupling (SOC) and non-centrosymmetric layer structures. Bi2C2H6−xFx films have topological non-trivial band gaps up to 1.08 eV, which are robust against the coverage and distribution of fluorine atoms, and the strongest out-of-plane ferroelectric polarization is up to 0.25 × 10−10 C m−1. Moreover, some Bi2C2H6−xFx configurations exhibit an extra valley degree of freedom by the valley splitting at the K and K′ points in the Brillouin zone due to the symmetry breaking, resulting in the coexistence of quantum spin and valley Hall states. These phenomena could also be found in other similarly decorated 2D materials, therefore offering unique quantum material platforms for discovering novel physics and exploring innovative applications.

Quantum spin Hall (QSH) effect with great promise for the potential application in spintronics and quantum computing has attracted extensive research interest from both theoretical and experimental researchers. Here, we predict monolayer ${\mathrm{Ta}}_{2}{\mathrm{Pd}}_{3}{\mathrm{Te}}_{5}$ can be a QSH insulator based on first-principles calculations. The interlayer binding energy in the layered van der Waals compound ${\mathrm{Ta}}_{2}{\mathrm{Pd}}_{3}{\mathrm{Te}}_{5}$ is $19.6\phantom{\rule{0.16em}{0ex}}\mathrm{meV}/{\AA{}}^{2}$; thus, its monolayer/thin-film structures could be readily obtained by exfoliation. The band inversion near the Fermi level (${E}_{F}$) is an intrinsic characteristic, which happens between $\mathrm{Ta}\text{\ensuremath{-}}5d$ and $\mathrm{Pd}\text{\ensuremath{-}}4d$ orbitals without spin-orbit coupling (SOC). The SOC effect opens a global gap and makes the system a QSH insulator. With the $d\text{\ensuremath{-}}d$ band-inverted feature, the nontrivial topology in monolayer ${\mathrm{Ta}}_{2}{\mathrm{Pd}}_{3}{\mathrm{Te}}_{5}$ is characterized by the time-reversal topological invariant ${\mathbb{Z}}_{2}=1$, which is computed by the one-dimensional (1D) Wilson loop method as implemented in our first-principles calculations. The helical edge modes are also obtained using surface Green's function method. Our calculations show that the QSH state in ${\mathrm{Ta}}_{2}{M}_{3}{\mathrm{Te}}_{5}$ ($M=\mathrm{Pd}$, Ni) can be tuned by external strain. These monolayers and thin films provide feasible platforms for realizing QSH effect as well as related devices.

Because phononic topological insulators have primarily been studied in discrete, graphenelike structures with ${C}_{6(v)}$ or ${C}_{3(v)}$ hexagonal symmetry, an open question is how to systematically achieve double Dirac cones and topologically nontrivial structures using continuous, nonhexagonal unit cells. Here, we address this challenge by presenting a computational methodology for the inverse design of continuous two-dimensional square phononic metamaterials exhibiting ${C}_{4(v)}$ and ${C}_{2(v)}$ symmetry. This leads to the systematic design of square unit cell topologies exhibiting a double Dirac degeneracy, which enables topologically protected interface propagation based on the quantum spin Hall effect (QSHE). Numerical simulations prove that helical edge states emerge at the interface between two topologically distinct square phononic metamaterials, which opens the possibility of QSHE-based pseudospin-dependent transport beyond hexagonal lattices.

The quantum spin Hall (QSH) insulator, ensured by the time-reversal ($\mathcal{T}$) symmetry, has been a conceptual milestone of complex topological phases and well known to realize the quantum anomalous Hall effect when $\mathcal{T}$ symmetry is broken by introducing ferromagnetism. Here, based on first-principles calculations and a tight-binding model, we show that, unlike previous $\mathcal{T}$-symmetry breaking, the QSH phase can survive under the antiferromagnetic long-range order in a RbCuSe/CsMnP heterobilayer consisting of a nonmagnetic QSH insulator RbCuSe and an antiferromagnetic insulator CsMnP. The calculated spin Chern number, ${\mathcal{Z}}_{2}$ invariant, and gapless edge state confirm the topological nontrivial phase clearly. Moreover, the role of effective and Rashba spin-orbit coupling and the magnitude of antiferromagnetism are discussed to reveal the underlying physical mechanism. Our results may lead to further scientific and technological advances in topological magnetism and antiferromagnetic spintronics.

The quantum spin Hall effect arises due to band inversion in topological insulators, and has the defining characteristic that it hosts helical edge channels at zero magnetic field, leading to a finite spin Hall conductivity. The spin Hall conductivity is understood as the difference of the contributions of two spin states. In the effective four-band BHZ model, these two spin states appear as two uncoupled blocks in the Hamiltonian matrix. However, this idea breaks down if additional degrees of freedom are considered. The two blocks cannot be identified by proper spin $S_z$ or total angular momentum $J_z$, both not conserved quantum numbers. In this work, we discuss a notion of block structure for the more general k.p model, defined by a conserved quantum number that we call isoparity, a combination of parity $z\to-z$ and spin. Isoparity remains a conserved quantity under a wide range of conditions, in particular in presence of a perpendicular external magnetic field. From point-group considerations, isoparity is fundamentally defined as the action of $z\to-z$ on the spatial and spinorial degrees of freedom. Since time reversal anticommutes with isoparity, the two blocks act as Kramers partners. The combination of conductivity and isoparity defines spin conductivity. This generalized notion of spin Hall conductivity uncovers the meaning of 'spin': It is not the proper spin $S_z$, but a crystal symmetry that is realized by a spinorial representation., Comment: 12 pages, 4 figures; Secs. IV-VI shortened, title changed

We report on the formation of critical states in disordered graphene, at the origin of variable and unconventional transport properties in the quantum Hall regime, such as a zero-energy Hall conductance plateau in the absence of an energy band gap and Landau-level degeneracy breaking. By using efficient real-space transport methodologies, we compute both the dissipative and Hall conductivities of large-size graphene sheets with random distribution of model single and double vacancies. By analyzing the scaling of transport coefficients with defect density, system size, and magnetic length, we elucidate the origin of anomalous quantum Hall features as magnetic-field-dependent impurity states, which percolate at some critical energies. These findings shed light on unidentified states and quantum-transport anomalies reported experimentally., This research is partly funded by the European Union Seventh Framework Programme under Grant No. 604391 Graphene Flagship. ICN2 acknowledges support from the Severo Ochoa Program (MINECO, Grant No. SEV-2013-0295). F.O. would like to thank the Deutsche Forschungsgemeinschaft for financial support (Grant No. OR 349/1-1).

First the significance of topology in physics and topological sound waves is introduced. Then acoustical quantum Hall effect is described with theories and applications. These are followed by the introduction of the acoustical quantum spin Hall effect and acoustical valley Hall effect. The chapter is concluded with future directions.

Under the terms of the Creative Commons Attribution License 3.0 (CC-BY)., We report an intriguing transition from the quantum spin Hall phase to the spin Hall effect upon segregation of thallium adatoms adsorbed onto a graphene surface. Landauer-Büttiker and Kubo-Greenwood simulations are used to access both edge and bulk transport physics in disordered thallium-functionalized graphene systems of realistic sizes. Our findings not only quantify the detrimental effects of adatom clustering in the formation of the topological state, but also provide evidence for the emergence of spin accumulation at opposite sample edges driven by spin-dependent scattering induced by thallium islands, which eventually results in a minimum bulk conductivity ~4e2/h, insensitive to localization effects., S. R. acknowledges the Spanish Ministry of Economy and Competitiveness for funding (MAT2012-33911), the Severo Ochoa Program (MINECO SEV-2013-0295), and the Secretaria de Universidades e Investigación del Departamento de Economía y Conocimiento de la Generalidad de Cataluña. The research leading to these results has received funding from the European Union Seventh Framework Programme under Grant Agreement No. 604391 Graphene Flagship.

We discuss the phase-dependent nonlocal thermoelectric effect in a topological Josephson junction in contact with a normal-metal probe. We show that, due to the helical nature of topological edge states, nonlocal thermoelectricity is generated by a purely Andreev interferometric mechanism. This response can be tuned by imposing a Josephson phase difference, through the application of a dissipationless current between the two superconductors, even without the need of applying an external magnetic field. We discuss in detail the origin of this effect and we also provide a realistic estimation of the nonlocal Seebeck coefficient which turns out to be of the order of a few $\ensuremath{\mu}\text{V}/\text{K}$ at temperatures of a few kelvin.

We present the characteristics of a simple waveguiding structure constructed by anisotropic birefringent crystal-metal-chiral medium, anisotropic metal chiral in short, and reveal the chiral-dependent dispersion and propagation properties of the surface plasmon polaritons (SPPs). We demonstrate its unprecedented discrimination ability to the magnitude and sign of both the real and imaginary part of the chirality parameter. The anisotropy plays a key role in such performance and shows tunable ability in enantiomeric discrimination even when the chirality parameter is complex valued. Most importantly, the physical origin of chiral discrimination stems from the extrinsic chirality of the system, which arises from the mutual orientation of the SPPs and the optical axis. Moreover, we also clarify the fundamental physics behind the chiral sensing behavior by associating the intrinsic quantum spin Hall effect of the SPPs with the electromagnetic field analysis. This structure does not rely on complicated fabrication but provides the opportunity of on-chip surface-sensitive biosensing. We anticipate that our work will stimulate intensive research to investigate the anisotropy-induced chiral sensing techniques in plasmonic platforms.

We present a method to design autonomous active metamaterials that can create arbitrary physical interactions on a single, reprogrammable platform, including non-local, non-linear, time-dependent or non-Newtonian couplings. The underlying principle includes application of external forces to a regular mechanical lattice, and processing them in real-time through closed-loop controllers that are pre-programmed to create desired couplings between the metamaterial sites. As an example, we demonstrate that a lattice constrained to out-of-plane vibration can be programmed to exhibit an analogy either of the quantum spin Hall effect, or the quantum Hall effect, thereby enabling a versatile unconventional guiding of mechanical waves.

We investigate thermoelectric transport properties of ferromagnetic graphene with CT-invariant quantum spin Hall (CT-QSH) effect. Considering a strong magnetic field, we calculate the charge Seebeck coefficient ${S}_{c}$, spin Seebeck coefficient ${S}_{s}$, charge Nernst coefficient ${N}_{c}$, and spin Nernst coefficient ${N}_{s}$ based on the nonequilibrium Green's function and Landauer-B\"uttiker formula. Due to the coexistence of the CT-QSH and quantum Hall (QH) effects in ferromagnetic graphene, thermoelectric coefficients are divided into the QSH and QH types appearing at the zeroth and nonzero Landau levels, respectively. We find both the charge thermoelectric coefficients are determined by the filling factor ${\ensuremath{\nu}}_{\ensuremath{\sigma}}$. The $n\mathrm{th}$ peak heights of the QH-type ${N}_{c}$ and ${S}_{c}$ satisfy $|{S}_{c,n}|={N}_{c,n}=ln2/(|n|+\frac{1}{2})$, exhibiting the half-integer QH effect. However, the $m\mathrm{th}$ peak height of the QSH-type ${N}_{c}$ satisfies ${N}_{c,m}=ln2/|m|$, similar to the integer QH effect. The peak height of ${N}_{s}$ remains ${N}_{s}=\mathrm{sgn}(s)2ln2$, and its sign depends on the spin of Landau level, either the QH or QSH type. In addition, the peak height of the QH-type ${S}_{s}$ remains $2ln2$. In the clean system, the QSH-type ${S}_{c}$ and ${S}_{s}$ are zero, while the QSH-type ${N}_{c}$ and ${N}_{s}$ appear at the zeroth Landau levels, which is different from the zero ${N}_{c}$ and ${N}_{s}$ in the conventional QSH system. In the presence of disorders, the QH-type thermoelectric coefficients are more robust than the QSH. For the QH-type thermoelectric coefficients, ${S}_{c}$ and ${S}_{s}$ are more robust than ${N}_{c}$ and ${N}_{s}$. Notably, the QSH-type ${S}_{c}$ and ${S}_{s}$ are no longer zero in dirty systems.

It has been shown recently that the coexistence of Van Hove singularity and geometric phase in ${\mathrm{Pb}}_{3}\mathrm{Bi}\text{/}\mathrm{Ge}(111)$ as a prototypical Rashba system can result in chiral topological superconductivity [W. Qin, L. Li, and Z. Y. Zhang, Nat. Phys. 15, 796 (2019)]. Here, we use first-principles calculations to systematically investigate the structural, electronic, and topological properties of the ${\mathrm{Pb}}_{3}\mathrm{Bi}\text{/}\mathrm{Ge}(111)$ alloyed systems by considering not only the configuration that harbors topological superconductivity considered previously (labeled as the ${\mathrm{T}}_{1}$ phase) but also two other energetically nearly degenerate configurations with high symmetries (labeled as ${\mathrm{H}}_{3}$ and ${\mathrm{T}}_{4}$). We first show that all the three structural configurations have giant Rashba-type spin splittings, resulting from the strong spin-orbit coupling effect of the heavy elements (Pb, Bi) together with inversion symmetry breaking. Next, we demonstrate that each of the saddle- or paraboliclike band dispersions can lead to the emergence of Van Hove singularities, whose positions can be substantially tuned by the giant Rashba effect to the vicinity of the Fermi level. Furthermore, we reveal the existence of topological states in both the ${\mathrm{H}}_{3}$ and ${\mathrm{T}}_{4}$ phases, and the corresponding Rashba-induced band gaps are given by 25 and 50 meV, respectively. Given the energetic affinity of the three phases, these findings suggest the feasibility of realizing quantum phase transitions between quantum spin Hall effect and topological superconductivity within the same materials platform. The present study is expected to give new insights in searching for topological quantum states including 2D topological superconductivity based on Rashba systems.

We study theoretically the band structures and angle-resolved photoemission spectroscopy of both monolayer and bilayer kagome lattices irradiated by variously polarized monochromatic light based on the Floquet-Bloch theory. The polarization of light plays distinctive roles in the different energy regimes of kagome bands. The linearly polarized light predominantly affects the flat bands at the bottom of the kagome band, while the circularly polarized light engenders spin-resolved massive Dirac cones whose spin species can be switched by the polarization direction. The edge states of kagome ribbons exhibit a spatially anisotropic quantum spin Hall effect conforming to the nontrivial topology of kagome bands. In kagome bilayers, we take into account the electric field component of monochromatic light which results in a band gap in flat bands in the case of oblique incidence. This shows that the polarized light provides us a versatile tool to manipulate not only the two-dimensional and quasi-one-dimensional band structures, but also the anomalous Hall conductivity by either the polarization or the incident plane of monochromatic light.

The boundary modes of topological insulators are protected by the symmetries of the non-trivial bulk electronic states. Unless these symmetries are broken, they can give rise to novel phenomena, such as the quantum spin Hall effect in one-dimensional (1D) topological edge states, where quasiparticle backscattering is suppressed by time-reversal symmetry (TRS). Here, we investigate the properties of the 1D topological edge state of bismuth in the absence of TRS, where backscattering is predicted to occur. Using spectroscopic imaging and spin-polarized measurements with a scanning tunneling microscope, we compared quasiparticle interference (QPI) occurring in the edge state of a pristine bismuth bilayer with that occurring in the edge state of a bilayer, which is terminated by ferromagnetic iron clusters that break TRS. Our experiments on the decorated bilayer edge reveal an additional QPI branch, which can be associated with spin-flip scattering across the Brioullin zone center between time-reversal band partners. The observed QPI characteristics exactly match with theoretical expectations for a topological edge state, having one Kramer's pair of bands. Together, our results provide further evidence for the non-trivial nature of bismuth, and, in particular, demonstrate backscattering inside a helical topological edge state induced by broken TRS through local magnetism.

The quantum spin Hall effect lays the foundation for the topologically protected manipulation of waves, but is restricted to one-dimensional-lower boundaries of systems and hence limits the diversity and integration of topological photonic devices. Recently, the conventional bulk-boundary correspondence of band topology has been extended to higher-order cases that enable explorations of topological states with codimensions larger than one such as hinge and corner states. Here, we demonstrate a higher-order quantum spin Hall effect in a two-dimensional photonic crystal. Owing to the non-trivial higher-order topology and the pseudospin-pseudospin coupling, we observe a directional localization of photons at corners with opposite pseudospin polarizations through pseudospin-momentum-locked edge waves, resembling the quantum spin Hall effect in a higher-order manner. Our work inspires an unprecedented route to transport and trap spinful waves, supporting potential applications in topological photonic devices such as spinful topological lasers and chiral quantum emitters., The quantum spin Hall effect is limited to one-dimensional lower boundary states which limits the possibilities for its exploitation in photonic devices. Here, the authors demonstrate a higher-order quantum spin Hall effect in a photonic crystal and observe opposite pseudospin corner states.

Topological insulator materials have wide applications in electronic and spintronic devices by utilizing the protected edge states. In this paper, based on these topological edge states and energy gaps, we propose some types of spin/charge current router in a three-terminal system consisting of silicene-like nanoribbons (SiNRs). The current is well controlled by the helical edge states of zigzag SiNRs (ZSiNRs) and external fields. Using the tight-binding model and non-equilibrium Green's function theory, we investigate three types of such router. The first type is a spin current shunter which separates the spin-up and spin-down current into different leads. The second type is a spin filter which separates the spin-polarized electrons into one of those leads. The last type is a charge current switcher which switches the charge current from one lead to the other lead. The local current distribution is calculated for the specific electron path. We find that the small Rashba does not destroy the filtering properties of the system. Besides, as an example, we employ the Landauer-Buttiker formula to obtain the current-voltage curves of the first type router and investigate the transmittance reciprocity relations in such a three-terminal system. We believe these proposed spin/charge current routers, which can separate the specific current into the expected lead, have potential applications in the future spintronics designs.

Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-can extend Euclidean Bravais lattices and the consequent band theory to non-Euclidean geometry. Here we demonstrate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analogue of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, showing the unique spectral sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory., 26 pages, 8 figures

Owing to the small energy differences between its polymorphs, MoTe2 can access a full spectrum of electronic states, from the 2H semiconducting state to the 1T semimetallic state, and from the Td Weyl semimetallic state to the superconducting state in the 1T and Td phase at low temperature. Thus, it is a model system for phase transformation studies as well as quantum phenomena such as the quantum spin Hall effect and topological superconductivity. Careful studies of MoTe2 and its potential applications require large area MoTe2 thin films with high crystallinity and thickness control. Here, we present cm2 scale synthesis of 2H MoTe2 thin films with layer control and large grains that span several microns. Layer control is achieved by controlling the initial thickness of the precursor MoOx thin films, which are deposited on sapphire substrates by atomic layer deposition and subsequently tellurized. Despite the van der Waals epitaxy, the precursor-substrate interface is found to critically determine the uniformity in thickness and grain size of the resulting MoTe2 films: MoTe2 grown on sapphire show uniform films while MoTe2 grown on amorphous SiO2 substrates form islands. This synthesis strategy decouples the layer control from the variabilities of growth conditions for robust growth results, and is applicable to grow other transition metal dichalcogenides with layer control., Comment: 6 figure

We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator $H_0$ does not commute with the spin operator in view of Rashba interactions, as in the typical models for the Quantum Spin Hall effect. A gapped periodic one-particle Hamiltonian $H_0$ is perturbed by adding a constant electric field of intensity $\varepsilon \ll 1$ in the $j$-th direction, and the linear response in terms of a $S$-current in the $i$-th direction is computed, where $S$ is a generalized spin operator. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (Unit Cell Consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper $S$-conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory) and allows moreover to compute the response of the adiabatic $S$-current as the trace per unit volume of the $S$-current operator times the NEASS. This technique can be applied in a general framework, which includes both discrete and continuum models., Comment: 49 pages, no figures. Keywords: quantum transport theory, spin conductivity, spin conductance, topological insulators, Spin Chern number

Orientador: Prof. Dr. Adalberto Fazzio Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, Santo André, 2020. Neste trabalho focamos no estudo do transporte eletronic de materiais topológicos em duas dimensões, onde o acoplo de spin-órbita(SOC) consegue criar uma interação entre estados das bandas de condução e valência tal que, um novo estado da matéria apareça, o qual tem propriedades muito particulares como por exemplo, a existência de estados de borda contendo dois estados de spin indo em direções opostas. Este novo estado da matéria é protegido por algumas simetrias como, a simetria de reversão temporal, onde é necessário que em alguns pontos da zona de Brillouin os estados da borda tenham um cruzamento e formem um par de Kramer, a degenerância então é protegida fazendo com que o sistema fique sempre sem gap devido a existência da simetria temporal que não é quebrada pela perturbação, ou pelo menos, em média a simetria é respeitada. Para tentar entender o que significa ¿em média¿, vamos usar duas teorias bem estabelecidas,para descrever a estrutura eletrônica usamos a teoria funcional da densidad(DFT) que ao longo dos anos demonstrou ser uma excelente ferramenta para o estudo de materiais. Quando tem-se a perda da simetria translacional por conta da desordem usamos a teoria das funções de Green fora do equilíbrio(NEGF), misturando-se essas duas metodologias conseguimos estudar o transporte eletrônico que pode dar interessantes resultados em termos de comprimentos de coerência de spin e assim entender o comportamento destes materiais mostrando as vezes a possibilidade de ter um filtro de spin ou mais interessante ainda.Seria possível ter um efeito Hall quântico anômalo? Em outra parte da tese estudamos o transporte eletrônico em plataformas onde as propriedades do grafeno podem ser modificadas considerando efeitos de proximidade levando este material mais perto de propostas realistas. Também falamos sobre a possibilidade de usar a metodologia Kernel polinomial aplicada a sistemas topológicos usando modelos brinquedos e também um modelo mais elaborado tal que o estudo de sistemas com desordem possa ser conseguido com um custo computacional menor devido a forma como escalam linearmente as contas usando este método. In this work, we aim to study the electronic transport in two-dimensional topological insulators, where the Spin-Orbit Coupling(SOC) makes possible an interaction between states coming from the valence and conduction bands in such a way that a new state previously unknown arises with peculiar characteristics, amongst them, the existence of edge states with counter-propagating modes carrying opposite spins. This new state of matter is protected by symmetries, like Time-Reversal Symmetry (TRS) for instance, where it's necessary that at some point in the Brillouin zone, the edge states mentioned above crossed and form a Kramer¿s pair, this degeneracy protects the opening of a gap as long as the perturbation under the system does not break the underlying symmetry very badly, in colloquial terms, as long as the symmetry remains unbroken in average, but, what does this ¿in average¿ exactly mean?. To obtain quantitative results we use together two wellestablished theories, first of all, for the description and electronic structure of these systems we use Density Functional Theory(DFT), which has been proved to be an excellent tool to explore in detail the electronic nature that takes places in these materials. When we talk about disorder, where the system loses part of the translational symmetry as the inclusion of structural defects or other atoms, or when the TRS could be broken by the inclusion of magnetic atoms, we refer to the theory of Non-Equilibrium Green¿s Functions(NEGF¿s), in this manner we explore the electronic properties in these systems performing calculations using the data from DFT simulations, this allows to find quantities, for instance, the spin coherence lengths in terms of the Transmission which maybe characterizes some Spin Filter behaviour in our samples, or even better, maybe shed light on the quest of the so-called Quantum Anomalous Hall effect (QAHE). On the other hand, we also report new studies on graphene, which initially was proposed to display a Quantum Spin Hall Effect(QSHE) phase, but the lack of strong SOC made it difficult to realize experimentally. In this sense, our results based on proximity effects give new insights into the electronic behavior of this material. We also address a proposal to study several spin-related phenomena, using the Kernel Polynomial Method(KPM) we talk about the possibility of using a toy model, therefore providing a playground to explore different systems probing the topological features coming from it, along with some tests realized in a realistic tight-binding model.

Two-dimensional photonic crystals made of six air holes on a core-shell dielectric material has been proposed to study the newly emerged photonic quantum spin Hall insulator. Specifically, radii modification of the air holes and core-shell without breaking time-reversal (TR) symmetry are supported by the C 6 point group symmetry upon a proposed scheme. It is shown that multiple topological transitions from an ordinary insulator with zero spin Chern number (Cs) to a topological insulator with a non-zero Cs can be achieved by modifying the geometry of the photonic structure. Studying the two counter-propagating helical edge modes which have the opposite group velocities are of individual importance for various optical purposes like scattering-free waveguides protected to various defects, disorders and strong light-matter interactions. We show that topological edge states demonstrate slow light characteristics. The findings emphasize the fact that exploring topological phase transition can be applied as a unique approach for realizing light transport, robust energy transportation and slow light in integrated photonic circuits and devices.

Layered topological semimetals (TSMs), as new states of quantum matter, have been regarded as one of the most intriguing materials with strong spin–orbit coupling (SOC), nontrivial band topology and unusual features of monolayer. TSMs have attracted broad interests for both the fundamental research and the realization of potential spintronic applications. In this chapter, we will review the progresses of two-dimensional TSMs from the spintronic point of view. The chapter will cover topics related to strong SOC, Fermi arcs, and quantum spin Hall effect in TSMs. An outlook will also be provided on the possible realization of the Majorana fermions in TSMs for quantum computing.