Quantum Valley Hall Effect Research Articles

Topological valley, pseudospin, and pseudospin-valley protected edge states in symmetric pillared phononic crystals

We present a symmetric double-sided pillared phononic crystals (PPnCs) that can emulate both quantum spin Hall effect (QSHE) and quantum valley Hall effect (QVHE) by solely imposing different geometric perturbations. Indeed, the Dirac cones can occur in the low (deep subwavelength) and high frequency regime by judiciously turning the parameters of the attached pillars and even a double Dirac cone can be achieved. We realize the valley-protected, the pseudospin-protected or the pseudospin-valley coupled edge states with the proposed platform. Besides, we show a variety of refraction phenomena (positive, negative and evanescent) of the valley-polarized edge state at the zigzag termination when emulating QVHE. Further, we illustrate the valley-dependent feature of the pseudospin-valley coupled edge state and demonstrate the valley based splitting of the pseudospin-protected edge states in a Y-junction wave guide.

Magnetoelectric control of topological phases in graphene

Topological antiferromagnetic (AFM) spintronics is an emerging field of research, which involves the topological electronic states coupled to the AFM order parameter known as the N$\acute{\rm e}$el vector. The control of these states is envisioned through manipulation of the N$\acute{\rm e}$el vector by spin-orbit torques driven by electric currents. Here we propose a different approach favorable for low-power AFM spintronics, where the control of the topological states in a two-dimensional material, such as graphene, is performed via the proximity effect by the voltage induced switching of the N$\acute{\rm e}$el vector in an adjacent magnetoelectric AFM insulator, such as chromia. Mediated by the symmetry protected boundary magnetization and the induced Rashba-type spin-orbit coupling at the interface between graphene and chromia, the emergent topological phases in graphene can be controlled by the N$\acute{\rm e}$el vector. Using density functional theory and tight-binding Hamiltonian approaches, we model a graphene/Cr2O3 (0001) interface and demonstrate non-trivial band gap openings in the graphene Dirac bands asymmetric between the K and K' valleys. This gives rise to an unconventional quantum anomalous Hall effect (QAHE) with a quantized value of $2e^2/h$ and an additional step-like feature at a value close to $e^2/2h$, and the emergence of the spin-polarized valley Hall effect (VHE). Furthermore, depending on the N$\acute{\rm e}$el vector orientation, we predict the appearance and transformation of different topological phases in graphene across the $180^{\circ}$ AFM domain wall, involving the QAHE, the valley-polarized QAHE and the quantum VHE (QVHE), and the emergence of the chiral edge state along the domain wall. These topological properties are controlled by voltage through magnetoelectric switching of the AFM insulator with no need for spin-orbit torques.

Quantum Valley Hall Effect, Orbital Magnetism, and Anomalous Hall Effect in Twisted Multilayer Graphene Systems

We study the electronic structures and topological properties of $(M+N)$-layer twisted graphene systems. We consider the generic situation that $N$-layer graphene is placed on top of the other $M$-layer graphene, and is twisted with respect to each other by an angle $\theta$. In such twisted multilayer graphene (TMG) systems, we find that there exists two low-energy flat bands for each valley emerging from the interface between the $M$ layers and the $N$ layers. These two low-energy bands in the TMG system possess valley Chern numbers that are dependent on both the number of layers and the stacking chiralities. In particular, when the stacking chiralities of the $M$ layers and $N$ layers are opposite, the total Chern number of the two low-energy bands for each valley equals to $\pm(M+N-2)$ (per spin). If the stacking chiralities of the $M$ layers and the $N$ layers are the same, then the total Chern number of the two low-energy bands for each valley is $\pm(M-N)$ (per spin). The valley Chern numbers of the low-energy bands are associated with large, valley-contrasting orbital magnetizations, suggesting the possible existence of orbital ferromagnetism and anomalous Hall effect once the valley degeneracy is lifted either externally by a weak magnetic field or internally by Coulomb interaction through spontaneous symmetry breaking.

Varying topological properties of two-dimensional honeycomb lattices composed of endohedral fullerenes

Honeycomb lattices (HLs) have been widely explored for realization of massless Dirac quasiparticles and intriguing topological properties in both the quantum and classical regimes, with elemental and artificial atoms as the corresponding building blocks. Here we provide a demonstration showing that when the fullerene molecules of ${\mathrm{C}}_{28}$ are used as prototypical building blocks, stable two-dimensional (2D) HLs can also be achieved, with the structural, electronic, and topological properties tuned via proper atom encapsulation. Specifically, whereas a closely packed structure is preferred for the ${\mathrm{C}}_{28}$ lattice, honeycomb structures with different spacial symmetries are energetically favored for both the Bi@${\mathrm{C}}_{28}$ and In@${\mathrm{C}}_{28}$ endohedral fullerenes. In particular, Bi@${\mathrm{C}}_{28}$-HL is revealed to be a quantum spin Hall insulator because of strong spin-orbit coupling effects in the $f$ molecular orbitals, while In@${\mathrm{C}}_{28}$-HL is a quantum valley Hall insulator resulting from mirror symmetry breaking. The present study offers superatom-based platforms for realizing quantum spin Hall and quantum valley Hall effects in 2D systems, with distinctly enhanced tunability and robustness against atomic-scale imperfections.

Reconfigurable topological insulator for elastic waves.

Inspired by the quantum valley Hall effect, a mechanical topological insulator (TI) purposely built for reconfigurability is proposed and experimentally demonstrated. An aluminum plate serves as the host medium with periodically arranged voids and fixed inclusions used to break mirror symmetry. Reconfigurability is derived from the ability to easily alter the imperfection type (void or fixed inclusion) in any unit cell. The corresponding band structure of the proposed hexagonal unit cell is obtained using numerical means, which documents double-folded Dirac cones at the K-points. The breaking of mirror symmetry results in a topologically protected bandgap. Furthermore, topologically protected edge states (TPES) at the interface of two structures with opposite Chern numbers have been demonstrated numerically, and verified experimentally, for different desired trajectories. These TPES are robust against backscattering at defect locations and sharp bends. The proposed reconfigurable TI can be a stepping-stone platform toward building mechanical logic and circuits, which have advantages over electronic equivalents in harsh operating conditions, or to replace wireless systems near dead-zones of metallic and carbon fiber structures.

Quantum valley Hall effects and spin-valley locking in topological Kane-Mele circuit networks

The Kane-Mele model of time-reversal symmetry, which simultaneously contains spin and valley degrees of freedom, is simulated in a topological circuit network. By varying capacitances and inductances, we are able to demonstrate the quantum valley Hall states, in addition to the quantum spin Hall states, as well as the topological phase transition between them. We also study the rich topological interface states between nontrivial phases with different spin and valley Chern numbers via tuning capacitances and inductances so as to tune the couplings and gauge fields. Finally, we demonstrate the spin-valley locking in the Kane-Mele circuit network. The topological circuit network provides us with a platform to simultaneously control spin and valley degrees of freedom so that a spin splitter and a valley splitter can be realized.

Tunable topological bandgaps and frequencies in a pre-stressed soft phononic crystal

Topological insulators (TIs) have recently received significant attention due to the promise of lossless transport of various types of energy. Despite this interest, one outstanding issue is that the topological bandgap and the frequencies that are topologically permitted are typically fixed once the topological structure has been designed and fabricated. Therefore, an open and unresolved question concerns the ability to actively tune both the bandgap magnitude, as well as the frequencies, for which the energy is topologically protected. In this work, we report a mechanically tunable phononic TI using an acoustic analog of the quantum valley Hall effect. We propose a phononic crystal comprised of a soft, hyperelastic material where the phononic band structure is modulated through large deformation of the structure. In doing so, space-inversion symmetry can be broken, which leads to a phase transition between two topologically-contrasted states and the emergence of topologically-protected interface modes according to bulk-edge correspondence. We further demonstrate the robustness of this topological protection of the edge state along the interface, which demonstrates that mechanical deformation can be used to effectively tailor and tune the topological properties of elastic structures.

Experimental observation of valley-Hall edge states in elastic waveguides based on diatomic-graphene-like phononic crystals

Inspired by recent discoveries of topological phases of matter in quantum physics, there has been a rapidly growing research effort in creating their analogs in other classical wave systems, including acoustics. Achieving robust wave transmission even in the presence of disorder and defects could have a profound impact on many practical applications and devices. In this study, we report on the design and experimental validation of a fully continuous and load-bearing phononic structural waveguide capable of one-directional guided modes along the walls of topologically distinct domains. The lattice structure of the waveguide is inspired by diatomic graphene which allows realizing an elastodynamic analog of the quantum valley Hall effect (QVHE). Despite the fact that individual bulk properties are topologically trivial (i.e., associated with a zero Chern number), the dynamic behavior acquires topological significance in the neighborhood of the high symmetry points in momentum space; the so-called valleys. Our theoretical and experimental results confirm the existence of protected edge states traveling along the walls of domains having broken space inversion symmetry.

Topological phases in acoustic and mechanical systems

The study of classical wave physics has been reinvigorated by incorporating the concept of the geometric phase, which has its roots in optics, and topological notions that were previously explored in condensed matter physics. Recently, sound waves and a variety of mechanical systems have emerged as excellent platforms that exemplify the universality and diversity of topological phases. In this Review, we introduce the essential physical concepts that underpin various classes of topological phenomena realized in acoustic and mechanical systems: Dirac points, the quantum Hall, quantum spin Hall and valley Hall effects, Floquet topological phases, 3D gapless states and Weyl crystals. This Review describes topological phenomena that can be realized in acoustic and mechanical systems. Methods of symmetry breaking are described, along with the consequences and rich phenomena that emerge.

Design and experimental validation on acoustic valley Hall edge states in reconfigurable phononic elastic waveguides

We report on the design and experimental validation of a tunable topological elastic lattice capable of supporting guided waves that are robust against back-scattering from disorder and defects. The topological lattice consists in a patterned aluminum thin plate having through-holes arranged in a hexagonal periodic configuration. The occurrence of the topological edge state is attributed to the acoustic analogue of the electron quantum valley Hall effect. By connecting two lattices having broken space inversion symmetry due to the application of opposite tunable strain fields, a topological transition emerges at the domain wall (i.e., the interface between them). Such a domain wall supports the formation and propagation of quasi-unidirectional edge states. The experimental validation of the topological waveguide is conducted on an aluminum plate that is already fabricated in a deformed (i.e., broken symmetry) configuration. The results confirm the existence of quasi-unidirectional edge states traveling along the domain wall and robust against sharp corners back-scattering. The experimental results also confirm the presence of mechanical energy flux vortices within the unit cell that are used to investigate the origin of the edge states robustness against disorder.

Screening Magnetic Two-Dimensional Atomic Crystals with Nontrivial Electronic Topology.

To date, only a few two-dimensional (2D) magnetic crystals have been experimentally confirmed, such as CrI3 and CrGeTe3, all with very low Curie temperatures ( TC). High-throughput first-principles screening over a large set of materials yields 89 magnetic monolayers including 56 ferromagnetic (FM) and 33 antiferromagnetic compounds. Among them, 24 FM monolayers are promising candidates possessing TC higher than that of CrI3. High TC monolayers with fascinating electronic phases are identified: (i) quantum anomalous Halland valley Hall effects coexist in a single material RuCl3 or VCl3, leading to a valley-polarized quantum anomalous Hall state; (ii) TiBr3, Co2NiO6, and V2H3O5 are revealed to be half-metals. More importantly, a new type of fermion dubbed type-II Weyl ring is discovered in ScCl. Our work provides a database of 2D magnetic materials, which could guide experimental realization of high-temperature magnetic monolayers with exotic electronic states for future spintronics and quantum computing applications.

Topology of the valley-Chern effect

The quantum valley Hall effect (QVHE) has been observed in a variety of experimental setups, both quantum and classical. While extremely promising for applications, one should be reminded that QVHE is not an exact topological phenomenon and that, so far, it has been fully understood only qualitatively in certain extreme limits. Here we present a technique to relate QVHE systems with exact quantum spin-Hall insulators that accept real-space representations, without taking any extreme limit. Since the bulk-boundary correspondence is well understood for the latter, we are able to formulate precise quantitative statements about the QVHE regime and its robustness against disorder. We further investigate the effect using a novel experimental platform based on magnetically coupled spinners. Visual renderings, quantitative data and various tests of the domain-wall modes are supplied, hence giving an unprecedented insight into the effect.

Landau Quantization of a Narrow Doubly-Folded Wrinkle in Monolayer Graphene.

Folding can be an effective way to tailor the electronic properties of graphene and has attracted wide study interest in finding its novel properties. Here we present the experimental characterizations of the structural and electronic properties of a narrow graphene wrinkle on a SiO2/Si substrate using scanning tunneling microscopy/spectroscopy. Pronounced and nearly equally separated conductance peaks are observed in the d I/d V spectra of the wrinkle. We attribute these peaks to pseudo-Landau levels (PLLs) that are caused by a gradient-strain-induced pseudomagnetic field up to about 42 T in the narrow wrinkle. The introduction of the gradient strain and thus the pseudomagnetic field can be ascribed to the lattice deformation. A doubly-folded structure of the wrinkle is suggested. Our density functional theory calculations show that the band structure of the doubly folded graphene wrinkle has a parabolic dispersion, which can well explain the equally separated PLLs. The effective mass of carriers is obtained to be about 0.02 me ( me: the rest mass of electron), and interestingly, it is revealed that there exists valley polarization in the wrinkle. Such properties of the strained doubly folded wrinkle may provide a platform to explore some exciting phenomena in graphene, like zero-field quantum valley Hall effect.

Tunable in-plane topologically protected edge waves in continuum Kagome lattices

In this paper, we report the evidence of topologically protected edge waves (TPEWs) in continuum Kagome lattice. According to the bulk edge correspondence principle, such edge states are inherently linked with the topological characteristics of the material band structure and can, therefore, be predicted evaluating the associated topological invariant. Due to the non-trivial band structures shown in the context of quantum valley Hall effect, TPEWs are supported at the interface between two lattices characterized by different valley Chern numbers. The break of lattice symmetry is obtained here, in contrast with other similar works in continuum elastic structures, biasing in the stiffness properties of the unit cell, instead of manipulating mass at sublattice points. This opens new promising possibilities related to waveguide tunability and wave propagation control, exploiting the established techniques for stiffness modulation in elastic structures. A sensitivity analysis of robustness of the supported energy transport is provided, showing the amount of de-localized disorder the waveguide is immune to, and how performances are affected by perturbations in the nominal parameters of the lattice.

Robust quantum valley Hall effect for vortices in an interacting bosonic quantum fluid

Topologically protected pseudospin transport, analogous to the quantum spin Hall effect, cannot be strictly implemented for photons and in general bosons because of the lack of symmetry-protected pseudospins. Here we show that the required protection can be provided by the real-space topological excitation of an interacting quantum fluid: a quantum vortex. We consider a Bose-Einstein condensate at the Γ point of the Brillouin zone of a quantum valley Hall system based on two staggered honeycomb lattices. We demonstrate the existence of a coupling between the vortex winding and the valley of the bulk Bloch band. This leads to chiral vortex propagation on each side of the zigzag interface between two regions of inverted staggering. The topological protection provided by the vortex winding prevents valley pseudospin mixing and resonant backscattering, allowing a truly topologically protected valley pseudospin transport.

Giant spin-valley polarization and multiple Hall effect in functionalized bismuth monolayers

Valleytronic materials, characterized by local extrema (valleys) in their bands, and topological insulators have separately attracted great interest recently. However, the interplay between valleytronic and topological properties in one single system, likely to enable important unexplored phenomena and applications, has been largely overlooked so far. Here, by combining a tight-binding model with first-principles calculations, we find the large-band-gap quantum spin Hall effects (QSHEs) and valley Hall effects appear simultaneously in the bismuth monolayers decorated with hydrogen/halogen elements, denoted as Bi2XY (X, Y = H, F, Cl, Br, or I). A staggered exchange field is introduced into the Bi2XY monolayers by transition-metal atom (Cr, Mo, or W) doping or LaFeO3 magnetic substrates, which together with the strong spin-orbit coupling of bismuth atoms generates a time-reversal-symmetry-broken QSHE and a huge valley splitting (up to 513 meV) in the system. With gate control, QSHE and anomalous charge, spin, valley Hall effects can be observed in the single system. These predicted multiple and exotic Hall effects, associated with various degrees of freedom of electrons, could enable applications of the functionalized bismuth monolayers in electronics, spintronics, and valleytronics.

Band structure of monolayer transition-metal dichalcogenides and topological properties of their nanoribbons: Next-nearest-neighbor hopping

Tight-binding (TB) models exploited in calculating band structure of monolayer transition-metal dichalcogenides (TMDCs), namely ${MX}_{2}$ ($M=$ Mo and W; $X=$ S, Se and Te), can be divided into two groups: one is based on group theory and the other uses Slater-Koster (SK) method. The former in general is lack of flexibility to be extended to confined finite systems with lower symmetry, e.g., nanoribbons (NRs) and quantum dots. Unlike ubiquitous TB models, here we present an improved scheme of the flexible SK TB method in which the second-nearest-neighbor $M\text{\ensuremath{-}}M$ and $X\text{\ensuremath{-}}X$ hopping terms are included. Its improvement, being of comparable accuracy to first-principles calculations, is clearly elucidated through a comprehensive comparison between our results and those produced by widely accepted TB models in literature for monolayer ${\mathrm{MoS}}_{2}$. Besides, its high flexibility allows us to successfully extend our TB model from monolayer TMDCs to the $\mathrm{Mo}{X}_{2}$ ($X=$ S, Se, and Te) and $\mathrm{W}{X}_{2}$ ($X=$ S and Se) NRs of both zigzag and armchair types. We find that the zigzag NR could be either metallic or semiconducting, depending on the spin-orbit strength and band gap of its parent two-dimensional bulk TMDC, which is in contradiction to the usual consensus concerning TMDC NRs which exhibit the metallic behavior only. For a certain Fermi level, remarkably, we discovered that the ${\mathrm{MoS}}_{2}$ NRs demonstrate quantum valley Hall effect, while others only present topological insulator phase.

Transverse electric field-induced quantum valley Hall effects in zigzag-edge graphene nanoribbons

We have investigated gapless edge states in zigzag-edge graphene nanoribbons under a transverse electric field across the opposite edges by using a tight-binding model and the density functional theory calculations. The tight-binding model predicted that a quantum valley Hall effect occurs at the vacuum-nanoribbon interface under a transverse electric field and, in the presence of edge potentials with opposite signs on opposite edges, an additional quantum valley Hall effect occurs under a much lower field. Dangling bonds inevitable at the edges of real nanoribbons, functional groups terminating the edge dangling bonds, and spin polarizations at the edges result in the edge potentials. The density functional theory calculations confirmed that asymmetric edge terminations, such as one having hydrogen at an edge and fluorine at the other edge, lead to the quantum valley Hall effect even in the absence of a transverse electric field. The electric field-induced half-metallicity in the antiferromagnetic phase, which has been intensively investigated in the last decade, was revealed to originate from a half-metallic quantum valley Hall effect.

Design and experimental observation of valley-Hall edge states in diatomic-graphene-like elastic waveguides

We report on the design and experimental validation of a two-dimensional phononic elastic waveguide exhibiting topological Valley-Hall edge states. The lattice structure of the waveguide is inspired by the diatomic graphene and it is imprinted in an initially flat plate by means of geometric indentations. The indentations are distributed according to a hexagonal lattice structure which guarantees the existence of Dirac dispersion at the boundary of the Brillouin zone. Starting from this basic material, domain walls capable of supporting edge states can be obtained by contrasting waveguides having broken space inversion symmetry (SIS) achieved by using local resonant elements. Our theoretical study shows that such material maps into the acoustic analog of the quantum valley Hall effect (QVHE) while numerical and experimental results confirm the existence of protected edge states traveling along the walls of topologically distinct domains.

Topological valley-chiral edge states of Lamb waves in elastic thin plates

We investigate the nontrivial topology of the band structure of Lamb waves in a thin phononic crystal plate. When inversion symmetry is broken, a valley pseudospin degree of freedom is formed around K and K′ valleys for the A0 Lamb mode, which is decoupled from the S0 and SH0 modes in the low-frequency regime. Chiral edge states are explicitly demonstrated, which are immune to defects and exhibit unidirectional transport behaviors when intervalley scattering is weak. The quantum valley Hall effect is thus simulated in a simple way in the context of Lamb waves.