Helicity-dependent photoconductance of the edge states in the topological insulator Bi_(2)Te_(3)

The helicity-dependent photoconductance of the edge states in three-dimensional topological insulator Bi_(2)Te_(3)films is investigated.It is revealed that the helicity-dependent photoconductivity current on the left edge of the Bi_(2)Te_(3)film shows an opposite sign with that on the right edge.In addition,the helicity-dependent photoconductivity current increases linearly with the applied longitudinal electric field,and it reverses the sign with the reversal of the electric field.As the thickness of the Bi_(2)Te_(3)film increases,the helicity-dependent photoconductivity current also increases.Theoretical analysis suggests that the helicity-dependent photo-conductivity current may come from the intrinsic spin orbit coupling(SOC)or the SOC introduced by the chiral impurities or defects.

High-performance chiral all-optical OR logic gate based on topological edge states of valley photonic crystal

For all-optical communication and information processing,it is necessary to develop all-optical logic gates based on photonic structures that can directly perform logic operations.All-optical logic gates have been demonstrated based on conventional waveguides and interferometry,as well as photonic crystal structures.Nonetheless,any defects in those structures will introduce high scattering loss,which compromises the fidelity and contrast ratio of the information process.Based on the spin-valley locking effect that can achieve defect-immune unidirectional transmission of topological edge states in valley photonic crystals(VPCs),we propose a high-performance all-optical logic OR gate based on a VPC structure.By tuning the working bandwidth of the two input channels,we prevent interference between the two channels to achieve a stable and high-fidelity output.The transmittance of both channels is higher than 0.8,and a high contrast ratio of 28.8 dB is achieved.Moreover,the chirality of the logic gate originated from the spin-valley locking effect allows using different circularly polarized light as inputs,representing“1”or“0”,which is highly desired in quantum computing.The device’s footprint is 18μm×12μm,allowing high-density on-chip integration.In addition,this design can be experimentally fabricated using current nanofabrication techniques and will have potential applications in optical communication,information processing,and quantum computing.

Topological edge states and electronic structures of a 2D topological insulator: Single-bilayer Bi (111)

Providing the strong spin-orbital interaction, Bismuth is the key element in the family of three-dimensional topological insulators. At the same time, Bismuth itself also has very unusual behavior, existing from the thinnest unit to bulk crystals. Ultrathin Bi （111） bilayers have been theoretically proposed as a two-dimensional topological insulator. The related experimental realization achieved only recently, by growing Bi （111） ultrathin bilayers on topological insulator Bi2Te3 or Bi2Se3 substrates. In this review, we started from the growth mode of Bi （111） bilayers and reviewed our recent progress in the studies of the electronic structures and the one-dimensional topological edge states using scanning tunneling microscopy/spectroscopy （STM/STS）, angle-resolved photoemission spectroscopy （ARPES）, and first principles calculations.

Floquet bands and photon-induced topological edge states of graphene nanoribbons

Floquet theorem is widely used in the light-driven systems. But many 2 D-materials models under the radiation are investigated with the high-frequency approximation, which may not be suitable for the practical experiment. In this work,we employ the non-perturbative Floquet method to strictly investigate the photo-induced topological phase transitions and edge states properties of graphene nanoribbons under the light irradiation of different frequencies(including both low and high frequencies). By analyzing the Floquet energy bands of ribbon and bulk graphene, we find the cause of the phase transitions and its relation with edge states. Besides, we also find the size effect of the graphene nanoribbon on the band gap and edge states in the presence of the light.

Erratum to“Floquet bands and photon-induced topological edge states of graphene nanoribbons”

Figures 2(a)and 2(b)in our original paper[1] should be corrected by the following ones.In Figs.2(a)and 2(b),in the region of hω0>5.96γ0(phase A),the Chern number should be corrected to 1,as indicated in the paper and Table 1.

Random-Gate-Voltage Induced Al'tshuler–Aronov–Spivak Effect in Topological Edge States

Helical edge states are the hallmark of the quantum spin Hall insulator. Recently, several experiments have observed transport signatures contributed by trivial edge states, making it difficult to distinguish between the topologically trivial and nontrivial phases. Here, we show that helical edge states can be identified by the randomgate-voltage induced Φ_(0)/2-period oscillation of the averaged electron return probability in the interferometer constructed by the edge states. The random gate voltage can highlight the Φ_(0)/2-period Al'tshuler–Aronov–Spivak oscillation proportional to sin^(2)(2πΦ/Φ_(0)) by quenching the Φ_(0)-period Aharonov–Bohm oscillation. It is found that the helical spin texture induced π Berry phase is key to such weak antilocalization behavior with zero return probability at Φ = 0. In contrast, the oscillation for the trivial edge states may exhibit either weak localization or antilocalization depending on the strength of the spin-orbit coupling, which has finite return probability at Φ = 0. Our results provide an effective way for the identification of the helical edge states. The predicted signature is stabilized by the time-reversal symmetry so that it is robust against disorder and does not require any fine adjustment of system.

Evidence for topological superconductivity: Topological edge states in Bi2Te3/FeTe heterostructure

Majorana fermions have been predicted to exist at the edge states of a two-dimensional topological superconductor.We fabricated single quintuple layer(QL)Bi2Te3/FeTe heterostructure with the step-flow epitaxy method and studied the topological properties of this system by using angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy.We observed the coexistence of robust superconductivity and edge states on the single QL Bi2Te3 islands which can be potential evidence for topological superconductor.

Persistent Currents and Edge States in a Quasi－One－Dimensional Mesoscopic Ring with a Screened Interaction

The effect ofthe edge state on the persistent current in quasi-1D mesoscopic rings with a screened interactionwhich exists only between nearest-neighboring particles is studied with the Hartree-Fock approximation. The theoreticalvalue of the current magnitude is greatly enhanced by both the edge state and the Coulomb interaction, and pinningthe electrons into a lattice is good for the enhancement if screening happens. In high dimensional systems the screeningeffect can make the interacting range show anisotropy, and create a tendency of gathering for particles with a repulsivepotential.

Edge states enhanced by long-range hopping:An analytical study

We analyze the behavior of edge states in long-range(LR)interacting systems.In terms of lattice model Hamiltonian with the LR coupling,we determine analytically the condition of existence of edge states within the transfer matrix method(TMM).The expressions we obtain are general and hold for any choice of the LR hopping.The reason why edge states can appear is the transfer matrix in the bulk different from that in the boundary layers.Our predictions are in good agreement with numerical results by exact diagonalization.Our result is helpful in solving novel edge states in oneand two-dimensional(2D)superconductors with LR hopping and pairing.

Quantum transport signatures of non-trivial topological edge states in a ring-shaped Su–Schrieffer–Heeger double-chain system

In the ring-shaped Su–Schrieffer–Heeger(SSH)double-chain,the quantum interference between the two different electron tunneling paths of the upper and lower chains has an important influence on the electron transport properties of non-trivial topological edge states.Here,we have studied the electron transport signatures of non-trivial topological edge states in a ring-shaped SSH double-chain system based on the wave-guide theory and transfer-matrix method.In the ringshaped SSH double-chain with the upper chain being different from the lower one,it is demonstrated that the electron transmission probability displays the four and two resonance peaks associated with the non-trivial topological edge states in the weak and strong coupling regimes,respectively.Whereas in the case of the upper chain being the same as the lower one,the two transmission resonance peaks associated with the non-trivial topological edge states in the weak coupling regime are only found,and that in the strong coupling regime disappear that originated from the destructive interference between the two different electron tunneling paths of the upper and lower chains.Consequently,the variation of the number of transmission resonance peaks associated with the non-trivial topological edge states in the weak and strong coupling regimes suggests that an alternative scheme for detecting non-trivial topological edge states in the ring-shaped SSH doublechain system.

Symmetry-controlled edge states in graphene-like topological sonic crystal

Unique topological states emerged in various topological insulators (TI) have been proved with great application value for robust wave regulation. In this work, we demonstrate the parity inversion related to the definition of the primitive cell in one common lattice, and realize a type of symmetry-controlled edge states confined on the zigzag interfaces of the graphene-like sonic topological crystal. By simply sliding the selected 'layer' near the interface, the coupling of the pseudospin states induced by the multiple scattering for the C6v lattice results in the adjustment of the edge states. Based on the physics of the states, we experimentally propose a prototype of acoustic topological filter hosting multiple channels with independent adjustable edge states and realize the selective high transmission. Our work diversifies the prospects for the applications of the gapped edge states in the robust wave regulation, and proposes a frame to design new topological devices.

An Anderson Impurity Interacting with the Helical Edge States in a Quantum Spin Hall Insulator

Using the natural orbitals renormalization group（NORG）method,we investigate the screening of the local spin of an Anderson impurity interacting with the helical edge states in a quantum spin Hall insulator.It is found that there is a local spin formed at the impurity site and the local spin is completel.y screened by electrons in the quantum spin Hall insulator.Meanwhile,the local spin is screened dominantly by a single active natural orbital.We then show that the Kondo screening mechanism becomes transparent and simple in the framework of the natural orbitals formalism.We project the active natural orbital respectively into real space and momentum space to characterize its structure.We conilrm the spin-momentum locking property of the edge states based on the occupancy of a Bloch state on the edge to which the impurity couples.Furthermore,we study the dynamical property of the active natural orbital represented by the local density of states,from which we observe the Kondo resonance peak.

Distinct edge states and optical conductivities in the zigzag and armchair silicene nanoribbons under exchange and electric fields

Based on the tight binding model, we investigate the low energy bandstructures, edge states, and optical absorptions for the silicene nanoribbons （SiNRs） with different terminations under an in-plane exchange field and/or a perpendicular electric field. We find that the zigzag SiNRs are gapped by the exchange field, but they could reenter the metallic state after the application of the electric field. Contrarily, a certain kind of armchair SiNRs remain gapless even if a weak exchange field is present. Furthermore, the combination of the exchange and electric fields could effectively modulate the penetration length and the components of the edge states in the SiNRs. The corresponding optical conductivities for the SiNRs are also calculated, which show remarkable dependence on the edge types of the SiNRs and the two external fields.

Band structure and edge states of star-like zigzag graphene nanoribbons

Connecting three zigzag graphene nanoribbons（ZGNRs） together through the sp^3 hybrid bonds forms a star-like ZGNR（S-ZGNR）. Its band structure shows that there are four edge states at k = 0.5, in which the three electrons distribute at three outside edge sites, and the last electron is shared equally（50%） by two sites near the central site. The lowest conductance step in the valley is 2, two times higher than that of monolayer ZGNR（M-ZGNR）. Furthermore, in one quasithree-dimensional hexagonal lattice built, both of the Dirac points and the zero-energy states appear in the band structure along the z-axis for the fixed zero k-point in the x-y plane. In addition, it is an insulator in the x-y plane due to band gap 4 eV, however, for any k-point in the x-y plane the zero-energy states always exist at kz = 0.5.

Topological edge and corner states of valley photonic crystals with zipper-like boundary conditions

We present a stable valley photonic crystal(VPC)unit cell with C_(3v)symmetric quasi-ring-shaped dielectric columns and realize its topological phase transition by breaking mirror symmetry.Based on this unit cell structure,topological edge states(TESs)and topological corner states(TCSs)are realized.We obtain a new type of wave transmission mode based on photonic crystal zipper-like boundaries and apply it to a beam splitter assembled from rectangular photonic crystals(PCs).The constructed beam splitter structure is compact and possesses frequency separation functions.In addition,we construct a box-shaped triangular PC structures with zipper-like boundaries and discover phenomena of TCSs in the corners,comparing its corner states with those formed by other boundaries.Based on this,we explore the regularities of the electric field patterns of TESs and TCSs,explain the connection between the characteristic frequencies and locality of TCSs,which helps better control photons and ensures low power consumption of the system.

Highly efficient conversion from classical guided waves to topological chiral edge states

Electromagnetic topological chiral edge states mimicking the quantum Hall effect have attracted a great deal of attention due to their unique features of free backscattering and immunity against sharp bends and defects.However,the matching techniques between classical waveguides and the topological one-way waveguide deserve more attention for real-world applications.In this paper,a highly efficient conversion structure between a classical rectangular waveguide and a topological one-way waveguide is proposed and demonstrated at the microwave frequency,which efficiently converts classical guided waves to topological one-way edge states.A tapered transition is designed to match both the momentum and impedance of the classical guided waves and the topological one-way edge states.With the conversion structure,the waves generated by a point excitation source can be coupled to the topological one-way waveguide with very high coupling efficiency,which can ensure high transmission of the whole system(i.e.,from the source and the receiver).Simulation and measurement results demonstrate the proposed method.This investigation is beneficial to the applications of topological one-way waveguides and opens up a new avenue for advanced topological and classical integrated functional devices and systems.

On-surface synthesis and edge states of NBN-doped zigzag graphene nanoribbons

Zigzag graphene nanoribbons(ZGNRs)with spin-polarized edge states have potential applications in carbon-based spintronics.The electronic structure of ZGNRs can be effectively tuned by different widths or dopants,which requires delicately designed monomers.Here,we report the successful synthesis of ZGNR with a width of eight carbon zigzag lines and nitrogen-boronnitrogen(NBN)motifs decorated along the zigzag edges(NBN-8-ZGNR)on Au(111)surface,which starts from a specially designed U-shaped monomer with preinstalled NBN units at the zigzag edge.Chemical-bond-resolved non-contact atomic force microscopy(nc-AFM)imaging confirms the zigzag-terminated edges and the existence of NBN dopants.The electronic states distributed along the zigzag edges have been revealed after a silicon-layer intercalation at the interface of NBN-8-ZGNR and Au(111).Our work enriches the ZGNR family with a new dopant and larger width,which provides more candidates for future carbonbased nanoelectronic and spintronic applications.

Tunable topological edge state in plasma photonic crystals

In this study, we found a kind of edge state located at the interface between plasma photonic crystals(PPCs) and traditional photonic crystals, which depends on the property of the photonic band gap rather than the surface defect. Simulation and theoretical analysis show that by adjusting the plasma density, we can change the topological characteristics of the photonic band gap of PPCs. This makes it different from the photonic band gap of traditional PCs, and thus excites or closes the topological edge states. We further discussed the influence of plasma parameters on edge state characteristics, and the results showed that as the plasma density increased, the first photonic band gap(PBG) of the PPCs closed and then reopened, resulting in band inversion and a change in the PBG properties of the PPCs. We can control the generation of edge states through plasma and adjust the frequency and strength of the edge states. After the appearance of edge states, as the plasma density further increases, the first PBG of the PPCs will shift towards high frequencies and deepen. The frequency of edge states will shift towards higher frequencies, and their strength will also increase. We increased the first PBG depth of the PPCs by increasing the number of arrays and found that when the number of the PPCs arrays increased, only the intensity of the edge states would increase while the frequency remained unchanged. Therefore, flexible adjustment of edge state frequency and intensity can be achieved through plasma density and array quantity parameters. Our study demonstrates the properties of topological edge states in plasma photonic crystals, which we believe can provide some guidance for applications based on edge states.

Exploration of channel width scaling and edge states in transition metal dichalcogenides

We explore the impact of edge states in three types of transition metal dichalcogenides （TMDs）, namely metallic Td-phase WTe2 and semiconducting 2H-phase MoTe2 and MoS2, by patterning thin flakes into ribbons with varying channel widths. No obvious charge depletion at the edges is observed for any of these three materials, in contrast to observations made for graphene nanoribbon devices. The semiconducting ribbons are characterized in a three-terminal field-effect transistor （FET） geometry. In addition, two ribbon array designs have been carefully investigated and found to exhibit current levels higher than those observed for conventional one-channel devices. Our results suggest that device structures incorporating a high number of edges can improve the performance of TMD FETs. This improvement is attributed to a higher local electric field, resulting from the edges, increasing the effective number of charge carriers, and the absence of any detrimental edge-related scattering.

Energy-resolved spin filtering effect and thermoelectric effect in topological-insulator junctions with anisotropic chiral edge states

Topological edge states have crucial applications in the future nano spintronics devices.In this work,circularly polarized light is applied on the zigzag silicene-like nanoribbons resulting in the anisotropic chiral edge modes.An energy-dependent spin filter is designed based on the topological-insulator(TI)junctions with anisotropic chiral edge states.The resonance transmission has been observed in the TI junctions by calculating the local current distributions.And some strong Fabry–Perot resonances are found leading to the sharp transmission peaks.Whereas,the weak and asymmetric resonance corresponds to the broad transmission peaks.In addition,a qualitative relation between the resonant energy separation TR and group velocity vf is derived:TR=πhvf n/L,that indicated TR is proportional to vf and inversely proportional to the length L of the conductor.The different TR between the spin-up and spin-down cases results in the energyresolved spin filtering effect.Moreover,the intensity of the circularly polarized light can modulate the group velocity vf.Thus,the intensity of circularly polarized light,as well as the conductor-length,play very vital roles in designing the energy-dependent spin filter.Since the transmission gap root in the Fabry–Perot resonances,the thermoelectric(TE)property can be enhanced by adjusting the gap.A schedule to enhance the TE performance in the TI-junction is proposed by modulating the electric field(Ez).The TE dependence on Ez in the nanojunction is investigated,where the appropriate Ez leads to a very high spin thermopower and spin figure of merit.These TI junctions have potential usages in the nano spintronics and thermoelectric devices.