Edge states in rationally terminated honeycomb structures.

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of "zigzag type" and those of "armchair type," generalizing the classical zigzag and armchair edges. We prove that zero-energy/flat-band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l.

Characterization of the Edge States in Colloidal Bi2Se3 Platelets.

The remarkable development of colloidal nanocrystals with controlled dimensions and surface chemistry has resulted in vast optoelectronic applications. But can they also form a platform for quantum materials, in which electronic coherence is key? Here, we use colloidal, two-dimensional Bi2Se3 crystals, with precise and uniform thickness and finite lateral dimensions in the 100 nm range, to study the evolution of a topological insulator from three to two dimensions. For a thickness of 4-6 quintuple layers, scanning tunneling spectroscopy shows an 8 nm wide, nonscattering state encircling the platelet. We discuss the nature of this edge state with a low-energy continuum model and ab initio GW-Tight Binding theory. Our results also provide an indication of the maximum density of such states on a device.

Double Quantum Spin Hall Phase in Moiré WSe2.

Quantum spin Hall (QSH) insulators are topologically protected phases of matter in two dimensions that can support a pair of helical edge states surrounding an insulating bulk. A higher (even) number of helical edge state pairs is usually not possible in real materials because spin mixing would gap out the edge states. Here, we report experimental evidence for a QSH phase with one and two pairs of helical edge states in twisted bilayer WSe2 at Moiré hole filling factor ν = 2 and 4, respectively. We observe nearly quantized (within 10%) resistance plateaus of hνe2 and large nonlocal transport at ν = 2 and 4 while the bulk is insulating. The resistance is independent of the out-of-plane magnetic field and increases under an in-plane magnetic field. The results agree with quantum transport of helical edge states in a material with high spin Chern bands protected by Ising spin conservation symmetry.

Formation of Supernarrow Borophene Nanoribbons.

Borophenes have sparked considerable interest owing to their fascinating physical characteristics and diverse polymorphism. However, borophene nanoribbons (BNRs) with widths less than 2 nm have not been achieved. Herein, we report the experimental realization of supernarrow BNRs. Combining scanning tunneling microscopy imaging with density functional theory modeling and ab initio molecular dynamics simulations, we demonstrate that, under the applied growth conditions, boron atoms can penetrate the outermost layer of Au(111) and form BNRs composed of a pair of zigzag (2,2) boron rows. The BNRs have a width self-contained to ∼1 nm and dipoles at the edges to keep them separated. They are embedded in the outermost Au layer and shielded on top by the evacuated Au atoms, free of the need for post-passivation. Scanning tunneling spectroscopy reveals distinct edge states, primarily attributed to the localized spin at the BNRs' zigzag edges. This work adds a new member to the boron material family and introduces a new physical feature to borophenes.

Evidence of Topological Edge States in Buckled Antimonene Monolayers.

Two-dimensional topological materials have attracted intense research efforts owing to their promise in applications for low-energy, high-efficiency quantum computations. Group-VA elemental thin films with strong spin-orbit coupling have been predicted to host topologically nontrivial states as excellent two-dimensional topological materials. Herein, we experimentally demonstrated for the first time that the epitaxially grown high-quality antimonene monolayer islands with buckled configurations exhibit significantly robust one-dimensional topological edge states above the Fermi level. We further demonstrated that these topologically nontrivial edge states arise from a single p-orbital manifold as a general consequence of atomic spin-orbit coupling. Thus, our findings establish monolayer antimonene as a new class of topological monolayer materials hosting the topological edge states for future low-power electronic nanodevices and quantum computations.

Edge States at Nematic Domain Walls in FeSe Films.

Quantum spin Hall (QSH) effect is an intriguing phenomenon arising from the helical edge states in two-dimensional topological insulators. We use molecular beam epitaxy (MBE) to prepare FeSe films with atomically sharp nematic domain boundaries, where tensile strains, nematicity suppression, and topological band inversion are simultaneously achieved. Using scanning tunneling microscopy (STM), we observe edge states at the Fermi level that spatially distribute as two distinct strips in the vicinity of the domain boundaries. At the end point of the boundaries, a bound state at the Fermi level is further observed. The topological origin of the edge states is supported by density functional theory calculations. Our findings not only demonstrate a candidate for QSH states but also provide a new pathway to realize topological superconductivity in a single-component film.

Edge State Engineering of Graphene Nanoribbons.

Zigzag edges of graphene nanoribbons, which are predicted to host spin-polarized electronic states, hold great promise for future spintronic device applications. The ability to precisely engineer the zigzag edge state is of crucial importance for realizing its full potential functionalities in nanotechnology. By combining scanning tunneling microscopy and atomic force microscopy, we demonstrate the zigzag edge states have energy splitting upon fusing manganese the phthalocyanine molecule with the short armchair graphene nanoribbon termini. Moreover, the edge state splitting can be reversibly switched by adsorption and desorption of a hydrogen atom on the magnetic core of manganese phthalocyanine. These observations can be explained by tuning the zigzag edge local doping through the charge transfer process, which provides a new route to functionalize graphene-based molecular devices.

Controlled Growth of 1D MoSe2 Nanoribbons with Spatially Modulated Edge States.

Two-dimensional (2D) transition metal dichalcogenides (TMDCs) possess interesting one-dimensional (1D) properties at its edges and inversion domain boundaries, where properties markedly different from the 2D basal plane, such as 1D metallicity and charge density waves, can be observed. Although 2D TMDCs crystals are widely grown by chemical vapor deposition (CVD), the fabrication of 1D TMDCs ribbons is challenging due to the difficulty to confine growth in only one dimension. Here we report the controlled growth of MoSe2 nanoribbons with an aspect ratio >100 by using prepatterned Se reconstructions on Au(100). Using scanning tunneling microscope and spectroscopy (STM/STS), the atomic and electronic structure of MoSe2 nanoribbons are studied. The ultranarrow ribbons show metallic behavior, while wider ribbons show a crossover from metallic to semiconducting behavior going from the edge to the center of the ribbon. The observed conductance modulations of the ultranarrow ribbons are attributed to 1D Moiré pattern. Remarkably, it shows a different periodicity compared with the 2D Moiré pattern in wider ribbons indicating that the 1D system is softened due to the high ratio of edge to basal plane bonds. Further, we demonstrated that the nanoribbons are stable against ambient conditions, which suggests that 1D TMDCs can be exploited for further applications.

Coexistence of Topological Edge State and Superconductivity in Bismuth Ultrathin Film.

Ultrathin freestanding bismuth film is theoretically predicted to be one kind of two-dimensional topological insulators. Experimentally, the topological nature of bismuth strongly depends on the situations of the Bi films. Film thickness and interaction with the substrate often change the topological properties of Bi films. Using angle-resolved photoemission spectroscopy, scanning tunneling microscopy or spectroscopy and first-principle calculation, the properties of Bi(111) ultrathin film grown on the NbSe2 superconducting substrate have been studied. We find the band structures of the ultrathin film is quasi-freestanding, and one-dimensional edge state exists on Bi(111) film as thin as three bilayers. Superconductivity is also detected on different layers of the film and the pairing potential exhibits an exponential decay with the layer thicknesses. Thus, the topological edge state can coexist with superconductivity, which makes the system a promising platform for exploring Majorana Fermions.

Spectral and Dynamical Properties of Single Excitons, Biexcitons, and Trions in Cesium-Lead-Halide Perovskite Quantum Dots.

Organic-inorganic lead-halide perovskites have been the subject of recent intense interest due to their unusually strong photovoltaic performance. A new addition to the perovskite family is all-inorganic Cs-Pb-halide perovskite nanocrystals, or quantum dots, fabricated via a moderate-temperature colloidal synthesis. While being only recently introduced to the research community, these nanomaterials have already shown promise for a range of applications from color-converting phosphors and light-emitting diodes to lasers, and even room-temperature single-photon sources. Knowledge of the optical properties of perovskite quantum dots still remains vastly incomplete. Here we apply various time-resolved spectroscopic techniques to conduct a comprehensive study of spectral and dynamical characteristics of single- and multiexciton states in CsPbX3 nanocrystals with X being either Br, I, or their mixture. Specifically, we measure exciton radiative lifetimes, absorption cross-sections, and derive the degeneracies of the band-edge electron and hole states. We also characterize the rates of intraband cooling and nonradiative Auger recombination and evaluate the strength of exciton-exciton coupling. The overall conclusion of this work is that spectroscopic properties of Cs-Pb-halide quantum dots are largely similar to those of quantum dots of more traditional semiconductors such as CdSe and PbSe. At the same time, we observe some distinctions including, for example, an appreciable effect of the halide identity on radiative lifetimes, considerably shorter biexciton Auger lifetimes, and apparent deviation of their size dependence from the "universal volume scaling" previously observed for many traditional nanocrystal systems. The high efficiency of Auger decay in perovskite quantum dots is detrimental to their prospective applications in light-emitting devices and lasers. This points toward the need for the development of approaches for effective suppression of Auger recombination in these nanomaterials, using perhaps insights gained from previous studies of II-VI nanocrystals.

Circularly polarized near-field optical mapping of spin-resolved quantum Hall chiral edge states.

We have successfully developed a circularly polarized near-field scanning optical microscope (NSOM) that enables us to irradiate circularly polarized light with spatial resolution below the diffraction limit. As a demonstration, we perform real-space mapping of the quantum Hall chiral edge states near the edge of a Hall-bar structure by injecting spin polarized electrons optically at low temperature. The obtained real-space mappings show that spin-polarized electrons are injected optically to the two-dimensional electron layer. Our general method to locally inject spins using a circularly polarized NSOM should be broadly applicable to characterize a variety of nanomaterials and nanostructures.

Dual-functional pentamode metamaterial with water-like and topological transmission properties.

In this paper, a water-like pentamode metamaterial (PM) with a single metallic material is designed and the topological edge-state transmission properties of elastic waves in the PM are thoroughly investigated. Numerical results indicate that by introducing structural perturbation into PM, the Dirac point degeneracy atK-point can be opened and topological band inversion can be generated. Topological edge states are also obtained by organizing PM structural units, which are robust to defects such as bending and cavities. In addition, it also has the mimics water in acoustic properties over a wide frequency range, i.e. it exhibits transparency when surrounded by water. Therefore, it will have both good transmission efficiency and acoustic stealth performance when used as an underwater waveguide. The dual-functional PM proposed in this study provides theoretical guidance for designing underwater stealth acoustic waveguides.

Designing edge states from fractional polarization insulators.

We theoretically investigated disconnected dispersive edge states in an anisotropic honeycomb lattice without chiral symmetry. When both mirror and chiral symmetries are present, this system is defined by a topological quantity known as fractional polarization (FP) term and exhibits a bulk band gap, classifying it as an FP insulator. While the FP insulator accommodates robust, flat topological edge states (TES), it also offers the potential to engineer these edge states by deliberately disrupting a critical symmetry that safeguards the underlying topology. These symmetry-breaking terms allow the edge states to become dispersive and generate differing configurations along the open boundaries. Furthermore, disconnected helical-like and chiral-like edge states analogous to TES seen in quantum spin and anomalous hall effect are achieved by the finite size effect, not possible from the symmetry-breaking terms alone. The demonstration of manipulating these edge states from a FP insulator can open up new avenues in constructing devices that utilize topological domain walls.

Design of higher Chern number two-band structures from topological defect perspective.

In this article, we propose two methods for designing higher Chern number models from the topological defect perspective. Based on the fact that the Chern number is equal to a summation of the charges of meron defects, we show that the higher Chern number structures can be realized by either moving the positions of merons or increasing the amount of them. The combination of the two methods is also verified to be a viable approach. We shall construct several models and investigate their energy spectrum. More than one gapless state can be observed on the edges of these models. Expectedly, our theory promises to provide not only a simple approach to obtain the Chern number without computing any integrals, but also a practical technique for new material design.

Terminal-Matched Topological Photonic Substrate-Integrated Waveguides and Antennas for Microwave Systems.

In engineered photonic lattices, topological photonic (TP) modes present a promising avenue for designing waveguides with suppressed backscattering. However, the integration of the TP modes in electromagnetic systems has faced longstanding challenges. The primary obstacle is the insufficient development of high-efficiency coupling technologies between the TP modes and the conventional transmission modes. This dilemma leads to significant scattering at waveguide terminals when attempting to connect the TP waveguides with other waveguides. In this study, a topological photonic substrate-integrated waveguide (TPSIW) is proposed that can seamlessly integrate into traditional microstrip line systems. It successfully addresses the matching problem and demonstrates efficient coupling of both even and odd TP modes with the quasi-transverse electromagnetic modes of microstrip lines, resulting in minimal energy losses. In addition, topological leaky states are introduced through designed slots on the TPSIW top surface. These slots enable the creation of TP leaky-wave antennas with beam steering capabilities. A wireless link based on TPSIWs are further established that enables the transmission of distinct signals toward different directions. This work is an important step toward the integration of TP modes in microwave systems, unlocking the possibilities for the development of high-performance wireless devices.

Improved performance of temperature sensors based on the one-dimensional topological photonic crystals comprising hyperbolic metamaterials.

This paper seeks to progress the field of topological photonic crystals (TPC) as a promising tool in face of construction flaws. In particular, the structure can be used as a novel temperature sensor. In this regard, the considered TPC structure comprising two different PC designs named PC1 and PC2. PC1 is designed from a stack of multilayers containing Silicon (Si) and Silicon dioxide (SiO2), while layers of SiO2 and composite layer named hyperbolic metamaterial (HMM) are considered in designing PC2. The HMM layer is engineered using subwavelength layers of Si and Bismuth Germinate, or BGO ( Bi 4 Ge 3 O 12 ). The mainstay of our suggested temperature sensor is mainly based on the emergence of some resonant modes inside the transmittance spectrum that provide the stability in the presence of the geometrical changes. Meanwhile, our theoretical framework has been introduced in the vicinity of transfer matrix method (TMM), effective medium theory (EMT) and the thermo-optic characteristics of the considered materials. The numerical findings have extensively introduced the role of some topological parameters such as layers' thicknesses, filling ratio through HMM layers and the periodicity of HMM on the stability or the topological features of the introduced sensor. Meanwhile, the numerical results reveal that the considered design provides some topological edge states (TESs) of a promising robustness and stability against certain disturbances or geometrical changes in the constituent materials. In addition, our sensing tool offers a relatively high sensitivity of 0.27 nm/°C.

Exchange couplings and edge states in two-dimensional topological insulators.

We consider two-dimensional honeycomb-lattice topological insulators of electrons which are exposed to exchange couplings at the edges, paying particular attention to the energy dispersions of edge states. The energy band structure of edge states is found to depend strongly on the direction of exchange couplings. Energy band splitting occurs for the exchange couplings perpendicular to the plane of topological insulators. In contrast, the parallel exchange couplings turn out to generate a finite energy gap in the edge-state bands for zigzag edges while armchair nanoribbons does not show such energy gaps. We also discuss the dependence of energy band structures of edge states on the magnetic structure of exchange couplings as well as the disorder effects on the edge-state density of states.

Giant nonlocal edge conduction in the axion insulator state of MnBi2Te4.

The recently discovered antiferromagnetic (AFM) topological insulator (TI) MnBi2Te4 represents a versatile material platform for exploring exotic topological quantum phenomena in nanoscale devices. It has been proposed that even-septuple-layer (even-SL) MnBi2Te4 can host helical hinge currents with unique nonlocal behavior, but experimental confirmation is still lacking. In this work, we report transport studies of exfoliated MnBi2Te4 flakes with varied thicknesses down to the few-nanometer regime. We observe giant nonlocal transport signals in even-SL devices when the system is in the axion insulator state but vanishingly small nonlocal signal in the odd-SL devices at the same magnetic field range. In conjunction with theoretical calculations, we demonstrate that the nonlocal transport is via the helical edge currents mainly distributed at the hinges between the side and top/bottom surfaces. The helical edge currents in the axion insulator state may find unique applications in topological quantum devices.

Quasi-periodic scattering of topological edge states induced by the vacancies in chloridized gallium bismuthide nanoribbons.

The chloridized gallium bismuthide was predicted to be a two-dimensional topological insulator with large topological band gap. It may be beneficial for achieving the quantum spin Hall effect and its related applications at high temperatures. To better understand the quantum transport in topological nanoribbons, we investigated the effect of vacancy on the quantum transport of topological edge states in the armchair chloridized gallium bismuthide nanoribbons by combining density functional theory and nonequilibrium Green's function. The results suggest the vacancies at center are more likely to cause the scattering of topological edge states. The average scattering is insensitive to the enlargement of vacancy along the transport direction. More interestingly, the obvious scattering of topological edge states can only be found at some special energies, and these special energies are distributed quasi-periodically. The quasi-periodic scattering may be used as a kind of fingerprint of vacancies. Our studies may be helpful for the application of topological nanoribbons.

Weyl Fermions and broken symmetry phases of laterally confined3He films.

Broken symmetries in topological condensed matter systems have implications for the spectrum of Fermionic excitations confined on surfaces or topological defects. The Fermionic spectrum of confined (quasi-2D)3He-A consists of branches of chiral edge states. The negative energy states are related to the ground-state angular momentum,Lz=(N/2)ℏ, forN/2Cooper pairs. The power law suppression of the angular momentum,Lz(T)≃(N/2)ℏ[1-23(πT/Δ)2]for0⩽T≪Tc, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the chiral edge Fermions. We discuss the effects of wave function overlap, and hybridization between edge states confined near opposing edge boundaries on the edge currents, ground-state angular momentum and ground-state order parameter of superfluid3He thin films. Under strong lateral confinement, the chiral A phase undergoes a sequence of phase transitions, first to a pair density wave (PDW) phase with broken translational symmetry atDc2∼16ξ0. The PDW phase is described by a periodic array of chiral domains with alternating chirality, separated by domain walls. The period of PDW phase diverges as the confinement lengthD→Dc2. The PDW phase breaks time-reversal symmetry, translation invariance, but is invariant under the combination of time-reversal and translation by a one-half period of the PDW. The mass current distribution of the PDW phase reflects this combined symmetry, and originates from the spectra of edge Fermions and the chiral branches bound to the domain walls. Under sufficiently strong confinement a second-order transition occurs to the non-chiral 'polar phase' atDc1∼9ξ0, in which a single p-wave orbital state of Cooper pairs is aligned along the channel.