Valley Hall Insulator Research Articles

Quantum spin-valley Hall effect in AB-stacked bilayer silicene

Our density functional theory calculations show that while AB-stacked bilayer silicene has a non-quantized spin-valley Chern number, there exist backscattering-free gapless edge states within the bulk gap, leading to a quantum spin-valley Hall effect. Using a tight-binding model for a honeycomb bilayer, we found that the interlayer potential difference and the staggered AB-sublattice potential lead to abrupt and gradual change of the valley Chern number from a quantized value to zero, respectively, while maintaining backscattering-free gapless edge states if the valley Chern number is not too close to zero. Under an inversion symmetry-breaking potential in the form of the staggered AB-sublattice potential, such as an antiferromagnetic order and a hexagonal diatomic sheet, a finite but non-quantized (spin-)valley Chern number can correspond to a quantum (spin-)valley Hall insulator.

Optimal quantum valley Hall insulators by rationally engineering Berry curvature and band structure

We propose the concept of ”optimal quantum valley Hall insulators” and formulate it to an optimization formulation. In this new design paradigm, mechanical valley Hall insulators with simple geometries, large relative operating bandwidths and highly localized interface modes can be obtained in a systematic way. Compared to existing classical work based trail-and-error method via breaking symmetry, advantages of the new strategy are: (1) it simultaneously confines the Berry curvature and tailors the band structure to achieve robustness and optimality; (2) it adopts an explicit structural optimization method to describe and optimize the unit cells with a few design variables, which paves a rational way to explore a relatively large design space efficiently; (3) valley Hall insulators with quite different geometries and a better performance can be produced directly through an optimization process without introducing and manually tuning the unit cell with Dirac cone feature. This methodology can not only be easily extended to design optimized topological insulators in other physical systems, but also provides some insight into optimization aided band structure and Berry curvature engineering.

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.

Band structure and topological properties of twisted double bilayer graphene

We study the electronic band structure and the topological property of the twisted double bilayer graphene, or a pair of AB-stacked bilayer graphenes rotationally stacked on top of each other. We consider two different arrangements, AB-AB and AB-BA, which differ in the relative orientation. For each system, we calculate the energy band and the valley Chern number using the continuum Hamiltonian. We show that the AB-AB and the AB-BA have similar band structures, while the Chern numbers associated with the corresponding bands are completely different. In the absence of the perpendicular electric field, in particular, the AB-AB system is a trivial insulator when the Fermi energy is in a gap, while the AB-BA is a valley Hall insulator. Also, the lowest electron and hole bands of the AB-AB are entangled by the symmetry protected band touching points, while they are separated in the AB-BA. In both cases, the perpendicular electric field immediately opens an energy gap at the charge neutral point, where the electron branch becomes much narrower than the hole branch, due to the significant electron-hole asymmetry.

Programmable elastic valley Hall insulator with tunable interface propagation routes

A challenge in the area of topologically protected edge waves in elastic media is how to tune the topological interface propagation route as needed. This paper proposes a tunable elastic valley Hall insulator, whose unit cell consists of two cavities and magnetic fluid with the same volume as the cavity. Interface route with arbitrary shape for propagating topologically protected edge waves can be achieved by controlling the distribution of magnetic fluid in each unit cell through a designed programmable magnet lifting array. As demonstrated through numerical simulations and experimental testing, the flexural wave is confined to propagate along the topological interface routes of different configurations. Finally, it is indicated that the proposed valley Hall insulator can be applied to achieve desired localized elastic energy which is robust to defects, sharp corners and so on.

Valley-projected edge modes observed in underwater sonic crystals

Here, we report an experimental study on the valley-projected edge states for underwater sound, which propagates along the interface separating two topologically distinct acoustic valley Hall insulators. The topological edge states are directly observed in our ultrasound scanning experiments, together with a solid evidence for the valley-selective excitation. Furthermore, we demonstrate an intriguing sound splitter built by topologically different acoustic valley Hall insulators. All experimental data agree well with our numerical results. Prospective applications can be anticipated, such as for underwater sound signal processing and ocean noise control.

Electrically tunable valleytronics in quantum anomalous Hall insulating transition metal trihalides

We propose a different scheme for achieving valley splitting using an electric field in the materials with inversion symmetry but without time-reversal symmetry, and apply this scheme to two-dimensional transition metal trihalides $\text{V}{X}_{3}$ ($X=\text{Cl}$, Br). Based on ab initio calculations and the tight-binding model, we find few layer $\text{V}{X}_{3}$ can be readily tuned from intrinsic quantum anomalous Hall insulators to quantum valley Hall insulators by external electric fields. Especially, the electric-field-induced valley splitting of the bilayer $\text{V}{X}_{3}$ is extremely large, about two orders of magnitude higher than that induced by a magnetic field in the state-of-the-art valleytronic materials (e.g., ${\mathrm{MoS}}_{2}$ and ${\mathrm{WSe}}_{2}$). We further reveal rich topological phases of few layer $\text{V}{X}_{3}$ and valley-polarized states at the phase boundary. These findings may motivate further topology and valleytronics related researches in low-dimensional transition metal compounds.

Topological phase transition in mechanical honeycomb lattice

Topological materials provide a new tool to direct wave energy with unprecedented precision and robustness. Three elastic topological phases, the valley Hall, Chern and spin Hall insulators, are currently studied, and they are achieved separately in rather distinct configurations. Here, we explore analytically various topological phase transitions for in-plane elastic wave in a unified mass-spring honeycomb lattice. It is demonstrated that the three elastic topological phases can be realized in this single lattice by designing mass, stiffness or introducing Coriolis’ effect. In particular, the interface between valley Hall and Chern insulators is found to support topological interface mode for the first time. Perturbation method is used to derive the analytic effective continuum model in the neighbor of band degeneracy, and the physics in topological phase transitions are revealed through evaluation of topological invariants. The topologically protected interface states, their decaying profile as well as the pseudo-spin-indicating polarization specific for elastic wave are systematically analyzed, and these results are further confirmed numerically by Bloch wave analysis of domain wall strip and transient simulation of finite sized sample. This study offers a concise and unified analytical model to explore topology nature of elastic wave, and can provide intuitive guidance to design of continuum mechanical topological materials.

Low-energy band structure very sensitive to the interlayer distance in Bernal-stacked tetralayer graphene

We have investigated Bernal-stacked tetralayer graphene as a function of interlayer distance and perpendicular electric field by using density functional theory calculations. The low-energy band structure was found to be very sensitive to the interlayer distance, undergoing a metal-insulator transition. It can be attributed to the nearest-layer coupling that is more sensitive to the interlayer distance than are the next-nearest-layer couplings. Under a perpendicular electric field above a critical field, six electric-field-induced Dirac cones with mass gaps predicted in tight-binding models were confirmed, however, our density functional theory calculations demonstrate a phase transition to a quantum valley Hall insulator, contrasting to the tight-binding model prediction of an ordinary insulator.

Floquet high Chern insulators in periodically driven chirally stacked multilayer graphene

Chirally stacked N-layer graphene is a semimetal with ±pN band-touching at two nonequivalent corners in its Brillioun zone. We predict that an off-resonant circularly polarized light (CPL) drives chirally stacked N-layer graphene into a Floquet Chern insulators (FCIs), aka quantum anomalous Hall insulators, with tunable high Chern number CF = ±N and large gaps. A topological phase transition between such a FCI and a valley Hall (VH) insulator with high valley Chern number Cv = ±N induced by a voltage gate can be engineered by the parameters of the CPL and voltage gate. We propose a topological domain wall between the FCI and VH phases, along which perfectly valley-polarized N-channel edge states propagate unidirectionally without backscattering.

Edge modes in zigzag and armchair ribbons of monolayer MoS2

We explore the electronic structure, orbital character and topological aspect of a monolayer MoS2 nanoribbon using tight-binding (TB) and low-energy () models. We obtain a mid-gap edge mode in the zigzag ribbon of monolayer MoS2, which can be traced back to the topological properties of the bulk band structure. Monolayer MoS2 can be considered as a valley Hall insulator. The boundary conditions at armchair edges mix the valleys on the edges, and a gap is induced in the edge modes. The spin–orbit coupling in the valence band reduces the hybridization of the bulk states.

Topological nature of in‐gap bound states in disordered large‐gap monolayer transition metal dichalcogenides

We propose a physical model based on disordered (a hole punched inside a material) monolayer transition metal dichalcogenides (TMDs) to demonstrate a large‐gap quantum valley Hall insulator. We find an emergence of bound states lying inside the bulk gap of the TMDs. They are strongly affected by spin–valley coupling, rest‐ and kinetic‐mass terms and the hole size. In addition, in the whole range of the hole size, at least two in‐gap bound states with opposite angular momentum, circulating around the edge of the hole, exist.Their topological insulator (TI) feature is analyzed by the Chern number, characterized by spacial distribution of their probabilities and confirmed by energy dispersion curves (energy vs. angular momentum). It not only sheds light on overcoming low‐temperature operating limitation of existing narrow‐gap TIs, but also opens an opportunity to realize valley‐ and spin‐qubits. (© 2016 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)

Hall and Nernst effects in monolayer MoS2

We study Hall and Nernst transports in monolayer MoS2based on Green’s function formalism. We have derived analytical results for spin and valley Hall conductivities in the zero temperature and spin and valley Nernst conductivities in the low temperature. We found that tuning of the band gap and spin-orbit splitting can drive system transition from spin Hall insulator (SHI) to valley Hall insulator (VHI). When the system is subjected to a temperature gradient, the spin and valley Nernst conductivities are dependent on Berry curvature.

Density functional theory calculations of the electric-field-induced Dirac cones and quantum valley Hall state in ABA-stacked trilayer graphene

We have investigated ABA-stacked trilayer graphene under a perpendicular electric field by using the density functional theory (DFT) calculations, which may contribute to the resolution of the discrepancies between experimental and theoretical results on the electric-field-induced band gap and topological phase transition. We found that the electric field opens a band gap at a low field and closes the gap at a high field, supporting one of the experimental results. While the seven electric-field-induced Dirac cones with mass gaps predicted in recent tight-binding (TB) models are confirmed, our DFT calculations demonstrate a phase transition from a quantum valley Hall insulator to a semimetal, contrasting to the TB model prediction of a topological phase transition between topologically nontrivial insulators at a high electric field.

Quantum valley Hall state in gas molecule-adsorbed bilayer graphene

While a variety of topologically nontrivial insulator phases have been predicted to arise from electron-electron and spin-orbit interactions in bilayer graphene, the trigonal warping of conduction and valence bands leads to a (semi)metallic band structure. An electrostatic potential difference between the two layers due to an external electric field is known to open a bandgap, leading to a topologically nontrivial insulator state. A bandgap may also arise from gas molecules adsorbed on bilayer graphene, implying a topologically nontrivial insulator phase. Here, our density functional theory calculations show that bilayer graphene adsorbing gas molecules is a quantum valley Hall insulator. Thus, adsorption of weak donor (or acceptor) molecules with a large electric dipole moment may be instrumental to realize a topologically nontrivial insulator phase in bilayer graphene even without external electric field.

Perfect valley filter in a topological domain wall

We propose a realization of perfect valley filters based on the chiral domain-wall channels between a quantum anomalous Hall insulator and a quantum valley Hall insulator. Uniquely, all these channels reside in the same valley and propagate unidirectionally, $100\%$ valley-polarizing passing-by carriers without backscattering. The valley index, the chirality, and the number of the channels are protected by topological charges, controllable by external fields, and detectable by circular dichroism.

Topological valley transport at bilayer graphene domain walls.

Electron valley, a degree of freedom that is analogous to spin, can lead to novel topological phases in bilayer graphene. A tunable bandgap can be induced in bilayer graphene by an external electric field, and such gapped bilayer graphene is predicted to be a topological insulating phase protected by no-valley mixing symmetry, featuring quantum valley Hall effects and chiral edge states. Observation of such chiral edge states, however, is challenging because inter-valley scattering is induced by atomic-scale defects at real bilayer graphene edges. Recent theoretical work has shown that domain walls between AB- and BA-stacked bilayer graphene can support protected chiral edge states of quantum valley Hall insulators. Here we report an experimental observation of ballistic (that is, with no scattering of electrons) conducting channels at bilayer graphene domain walls. We employ near-field infrared nanometre-scale microscopy (nanoscopy) to image in situ bilayer graphene layer-stacking domain walls on device substrates, and we fabricate dual-gated field effect transistors based on the domain walls. Unlike single-domain bilayer graphene, which shows gapped insulating behaviour under a vertical electrical field, bilayer graphene domain walls feature one-dimensional valley-polarized conducting channels with a ballistic length of about 400 nanometres at 4 kelvin. Such topologically protected one-dimensional chiral states at bilayer graphene domain walls open up opportunities for exploring unique topological phases and valley physics in graphene.

Quantum spin Hall insulators and quantum valley Hall insulators of BiX/SbX (X=H, F, Cl and Br) monolayers with a record bulk band gap

A large bulk band gap is critical for the application of quantum spin Hall (QSH) insulators or two-dimensional (2D) topological insulators (TIs) in spintronic devices operating at room temperature (RT). On the basis of first-principles calculations, we predicted a group of 2D TI BiX/SbX (X=H, F, Cl and Br) monolayers with extraordinarily large bulk gaps from 0.32 eV to a record value of 1.08 eV. These giant-gaps are entirely due to the result of the strong spin-orbit interaction related to the px and py orbitals of the Bi/Sb atoms around the two valleys K and K′ of the honeycomb lattice, which is significantly different from that consisting of the pz orbital as in graphene/silicene. The topological characteristic of BiX/SbX monolayers is confirmed by the calculated nontrivial Z2 index and an explicit construction of the low-energy effective Hamiltonian in these systems. We demonstrate that the honeycomb structures of BiX monolayers remain stable even at 600 K. Owing to these features, the giant-gap TIs BiX/SbX monolayers are an ideal platform to realize many exotic phenomena and fabricate new quantum devices operating at RT. Furthermore, biased BiX/SbX monolayers become a quantum valley Hall insulator, exhibiting valley-selective circular dichroism.

Topological phases in iridium oxide superlattices: Quantized anomalous charge or valley Hall insulators

We study topological phases in orthorhombic perovskite iridium (Ir) oxide superlattices grown along the $[001]$ crystallographic axis. Bilayer Ir oxide superlattices display topological magnetic insulators exhibiting quantized anomalous Hall effects due to strong spin-orbit coupling of Ir $5d$ orbitals and electronic correlation effects. We also find a valley Hall insulator with counterpropagating edge currents from two different valleys and a topological crystalline insulator with edge states protected by the crystal lattice symmetry based on stacking of two layers. In a single-layer superlattice, a topological insulator can be realized, when a strain field is applied to break the symmetry of a glide plane protecting the Dirac points. It turns into a topological magnetic insulator in the presence of magnetic ordering and/or in-plane magnetic field. We discuss essential ingredients for these topological phases and experimental signatures to test our theoretical proposals.

Topological Edge States at a Tilt Boundary in Gated Multilayer Graphene

The search for new realization of topologically protected edge states is an active area of research. We show that a tilt boundary in gated multi-layer graphene supports topologically protected gapless kink states, associated with quantum valley Hall insulator (QVH). We investigate such kink states from two perspectives: the microscopic perspective of a tight-binding model and an ab-initio calculation on bilayer, and the perspective of symmetry protected topological (SPT) states for general multi-layer. We show that a AB-BA bilayer tilt boundary supports gapless kink states that are undeterred by strain concentrated at the boundary. Further, we establish the kink states as concrete examples of edge states of {\it time-reversal symmetric} ${\mathbb Z}$-type SPT, protected by no valley mixing, electron number conservation, and time reversal $T$ symmetries. This allows us to discuss possible phase transitions upon symmetry changes from the SPT perspective. Recent experimental observations of a network of such tilt boundaries suggest that transport through these novel topological kink states might explain the long standing puzzle of sub-gap conductance. Further, recent observation of gap closing and re-opening in gated bi-layer might be the first example of a transition between two distinct SPT's: QVH and LAF.